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abstract: 'In this paper, we aim to design sparse D-optimal (determinant-optimal) pose-graph SLAM problems through the synthesis of sparse graphs with the maximum weighted number of spanning trees. Characterizing graphs with the maximum number of spanning trees is an open problem in general. To tackle this problem, several new theoretical results are established in this paper, including the monotone log-submodularity of the weighted number of spanning trees. By exploiting these structures, we design a complementary pair of near-optimal efficient approximation algorithms with provable guarantees. Our theoretical results are validated using random graphs and a publicly available pose-graph SLAM dataset.'
author:
- Kasra Khosoussi
- |
Gaurav S. Sukhatme\
Shoudong Huang
- Gamini Dissanayake
bibliography:
- 'graph.bib'
- 'slam.bib'
title: |
Designing Sparse Reliable Pose-Graph SLAM:\
A Graph-Theoretic Approach
---
Introduction
============
Graphs arise in modelling numerous phenomena across science and engineering. In particular, estimation-on-graph (EoG) is a class of (maximum likelihood) estimation problems with a natural graphical representation that arise especially in robotics and sensor networks. In such problems, each vertex corresponds to an unknown state, and each edge corresponds to a relative noisy measurement between the corresponding states. Simultaneous localization and mapping (SLAM) and sensor network localization (SNL) are two well-studied EoGs.
Designing sparse, yet “well-connected” graphs is a subtle task that frequently arises in various domains. First, note that graph sparsity—in EoGs and many other contexts—lead to computational efficiency. Hence, maintaining sparsity is often crucial. It is useful to see graph connectivity as a spectrum, as we often need to compare the connectivity of connected graphs. In engineering, well-connected graphs often exhibit desirable qualities such as *reliability*, *robustness*, and *resilience* to noise, outliers, and link failures. More specifically, a well-connected EoG is more resilient to a fixed noise level and results in a more reliable estimate (i.e., smaller estimation-error covariance in the Loewner ordering sense). Consequently, maintaining a sufficient connectivity is also critical. Needless to say, sparsity is, by its very essence, at odds with well-connectivity. This is the case in SLAM, where there is a trade-off between the cost of inference and the reliability of the resulting estimate. This problem is not new. Measurement selection and pose-graph pruning have been extensively studied in the SLAM literature (see, e.g., [@6094414; @huang2013consistent]). However, in this paper we take a novel graph-theoretic approach by reducing the problem of designing sparse reliable SLAM problems to the purely combinatorial problem of synthesizing sparse, yet well-connected graphs. In what follows, we briefly justify this approximate reduction.
First, note that by estimation reliability we refer to the standard D-optimality criterion, defined as the determinant of the (asymptotic) maximum likelihood estimator covariance matrix. D-optimality is a standard and popular design criterion; see, e.g., [@Joshi2009; @Khosoussi2014] and the references therein. Next, we have to specify how we measure graph connectivity. Among the existing combinatorial and spectral graph connectivity criteria, the number of spanning trees (sometimes referred to as *graph complexity* or *tree-connectivity*) stands out: despite its combinatorial origin, it can also be characterized solely by the spectrum of the graph Laplacian [@Godsil2001]. In [@Khosoussi2014; @Khosoussi2015graph; @kasra16icra], we shed light on the connection between the D-criterion in SLAM—and some other EoGs—and the tree-connectivity of the underlying graph. Our theoretical and empirical results demonstrate that, under some standard conditions, D-optimality in SLAM is *significantly* influenced by the tree-connectivity of the graph underneath. Therefore, one can accurately estimate the D-criterion without using any information about the robot’s trajectory or realized measurements (see Section \[sec:robot\]). Intuitively speaking, our approach can be seen as a dimensionality reduction scheme for designing D-optimal SLAM problems from the joint space of trajectories and graph topologies to only the space of graph topologies [@kasra16icra].
Although this work is specifically motivated by the SLAM problem, designing sparse graphs with the maximum tree-connectivity has several other important applications. For example, it has been shown that tree-connectivity is associated with the D-optimal incomplete block designs [@gaffke1982d; @cheng1981maximizing; @bailey2009combinatorics]. Moreover, tree-connectivity is a major factor in maximizing the connectivity of certain random graphs that model unreliable networks under random link failure (*all-terminal network reliability*) [@kelmans1983multiplicative; @weichenberg2004high]. In particular, a classic result in network reliability theory states that if the *uniformly-most reliable* network exits, it must have the maximum tree-connectivity among all graphs with the same size [@bauer1987validity; @myrvold1996reliable; @boesch2009survey].
### Known Results. {#known-results. .unnumbered}
Graphs with the maximum weighted number of spanning trees among a family of graphs with the same vertex set are called *$t$-optimal*. The problem of characterizing unweighted $t$-optimal graphs among the set of graphs with $n$ vertices and $m$ edges remains open and has been solved *only* for specific pairs of $n$ and $m$; see, e.g., [@shier1974maximizing; @cheng1981maximizing; @kelmans1996graphs; @petingi2002new]. The span of these special cases is too narrow for the types of graphs that typically arise in SLAM and sensor networks. Furthermore, in many cases the $(n,m)$ constraint alone is insufficient for describing the true set of “feasible” graphs and cannot capture implicit practical constraints that exist in SLAM. Finally, it is not clear how these results can be extended to the case of (edge) weighted graphs, which are essential for representing SLAM problems, where the weight of each edge represents the precision of the corresponding pairwise measurement [@kasra16icra].
### Contributions. {#contributions. .unnumbered}
This paper addresses the problem of designing sparse $t$-optimal graphs with the ultimate goal of designing D-optimal pose-graph SLAM problems. First and foremost, we formulate a combinatorial optimization problem that captures the measurement selection and measurement pruning scenarios in SLAM. Next, we prove that the weighted number of spanning trees, under certain conditions, is a monotone log-submodular function of the edge set. To the best of our knowledge, this is a new result in graph theory. Using this result, we prove that the greedy algorithm is near-optimal. In our second approximation algorithm, we formulate this problem as an integer program that admits a straightforward convex relaxation. Our analysis sheds light on the performance of a simple deterministic rounding procedure that have also been used in more general contexts. The proposed approximation algorithms provide near-optimality certificates. The proposed graph synthesis framework can be readily applied to any application where maximizing tree-connectivity is desired.
### Notation. {#notation. .unnumbered}
Throughout this paper, bold lower-case and upper-case letters are reserved for vectors and matrices, respectively. The standard basis for $\mathbb{R}^{n}$ is denoted by $\{\ee_{i}^n\}_{i=1}^{n}$. Sets are shown by upper-case letters. $|\cdot|$ denotes the set cardinality. For any finite set $\mathcal{W}$, $\binom{\mathcal{W}}{k}$ is the set of all $k$-subsets of $\mathcal{W}$. We use $[n]$ to denote the set $\{1,2,\dots,n\}$. The eigenvalues of symmetric matrix $\MM$ are denoted by . $\mathbf{1}$, $\II$ and $\mathbf{0}$ denote the vector of all ones, the identity and the zero matrices with appropriate sizes, respectively. $\ES_1\succ\ES_2$ (resp. $\ES_1 \succeq \ES_2$) means $\ES_1 -
\ES_2$ is positive definite (resp. positive semidefinite). Finally, $\diag(\WW_1,\dots,\WW_k)$ is the block-diagonal matrix whose main diagonal blocks are $\WW_1,\dots,\WW_k$.
Preliminaries {#sec:pre}
=============
### Graph Matrices. {#graph-matrices. .unnumbered}
Throughout this paper, we usually refer to undirected graphs $\GG = (\VV,\EE)$ with $n$ vertices (labeled with $[n]$) and $m$ edges. With a little abuse of notation, we call $\widetilde{\AAA} \in \{-1,0,1\}^{n \times m}$ the incidence matrix of $\GG$ after choosing an arbitrary orientation for its edges. The Laplacian matrix of $\GG$ is defined as $\widetilde{\LL} \triangleq
\widetilde{\AAA}\widetilde{\AAA}^\top$. $\widetilde{\LL}$ can be written as $\sum_{i=1}^{m} \widetilde{\LL}_{e_i}$ in which $\widetilde{\LL}_{e_i}$ is the *elementary Laplacian* associated with edge $e_i = \{u_i,v_i\}$, where the $(u_i,u_i)$ and $(v_i,v_i)$ entries are $1$, and the $(u_i,v_i)$ and $(v_i,u_i)$ entries are $-1$. [Anchoring]{} $v_0 \in \VV$ is equivalent to removing the row associated with $v_0$ from $\widetilde{\AAA}$. Anchoring $v_0$ results in the *reduced* incidence matrix $\AAA$ and the *reduced* Laplacian matrix $\LL \triangleq
\AAA \AAAT$. $\LL$ is also known as the *Dirichlet*. We may assign positive weights to the edges of $\GG$ via $w : \EE \to
\mathbb{R}_{>0}$. Let $\WW \in \mathbb{R}^{m \times m}$ be the diagonal matrix whose $(i,i)$ entry is equal to the weight of the $i$th edge. The *weighted* Laplacian (resp. reduced *weighted* Laplacian) is then defined as $\widetilde{\LL}_w \triangleq \widetilde{\AAA}\WW\widetilde{\AAA}^\top$ (resp. $\LL_w \triangleq \AAA\WW\AAA^{\hspace{-0.1cm}\top}$). Note that the (reduced) unweighted Laplacian is a special case of the (reduced) weighted Laplacian with $\WW =
\II_m$ (i.e., when all edges have unit weight).
### Spanning Trees. {#spanning-trees. .unnumbered}
A spanning tree of ${\GG}$ is a spanning subgraph of ${\GG}$ that is also a tree. Let $\mathbb{T}_{\GG}$ denote the set of all spanning trees of $\GG$. $t(\GG)\triangleq
|\mathbb{T}_{\GG}|$ denotes the number of spanning trees in $\GG$. As a generalization, for graphs whose edges are weighted by $w : \EE \to \mathbb{R}_{>0}$, we define the *weighted number of spanning trees*, $$t_w(\GG) \triangleq \sum_{\TT \in \mathbb{T}_{\GG}}
\mathbb{V}_{\hspace{-0.05cm}w}(\TT).$$ We call $\mathbb{V}_{\hspace{-0.05cm}w} : \mathbb{T}_\GG \to \mathbb{R}_{>0}$ the *tree value function* and define it as the product of the edge weights along a spanning tree. Notice that for unit edge weights, $t_w(\GG)$ coincides with $t(\GG)$. Thus, unless explicitly stated otherwise, we generally assume the graph is weighted. To prevent overflow and underflow, it is more convenient to work with $\log t_w(\GG)$. We formally define *tree-connectivity* as, $$\tau_{w}(\GG) \triangleq
\begin{cases}
\log t_w(\GG) & \text{if $\GG$ is connected,} \\
0 & \text{otherwise.}
\end{cases}
\label{}$$ For the purpose of this work, without loss of generality we can assume $w(e) \geq 1$ for all $e \in
\EE$, and thus $\tau_w(\GG) \geq 0$.[^1] The equality occurs only when either $\GG$ is not connected, or when $\GG$ is a tree whose all edges have unit weight. Kirchhoff’s seminal matrix-tree theorem is a classic result in spectral graph theory. This theorem relates the spectrum of the Laplacian matrix of graph to its number of spanning trees. The original matrix-tree theorem states that, $$\begin{aligned}
t(\GG) & = \det \LL \\ & =
\frac{1}{n}
\prod_{i=2}^{n} \lambda_i(\widetilde{\LL}).\end{aligned}$$ Here $\LL$ is the reduced Laplacian after anchoring an arbitrary vertex. Kirchhoff’s matrix-tree theorem has been generalized to the case of edge-weighted graphs. According to the generalized theorem, $t_w(\GG) = \det \LL_w =
\frac{1}{n}
\prod_{i=2}^{n} \lambda_i(\widetilde{\LL}_w)$.
### Submodularity. {#submodularity. .unnumbered}
Suppose $\mathcal{W}$ is a finite set. Consider a set function $\xi : 2^{\mathcal{W}} \to \mathbb{R}$. $\xi$ is called:
1. *normalized* iff $\xi(\varnothing) = 0$.
2. *monotone* iff $\xi(\Bcal) \geq \xi(\Acal)$ for every $\Acal$ and $\Bcal$ s.t. $\Acal \subseteq \Bcal \subseteq \mathcal{W}$.
3. *submodular* iff for every $\Acal$ and $\Bcal$ s.t. $\Acal \subseteq \Bcal \subseteq \mathcal{W}$ and we have, $$\xi(\Acal \cup \{s\}) - \xi(\Acal) \geq \xi(\Bcal \cup \{s\}) -
\xi(\Bcal).
\label{}$$
D-Optimality via Graph Synthesis {#sec:robot}
================================
In this section, we discuss the connection between D-optimality and $t$-optimality in SLAM by briefly reviewing the results in [@Khosoussi2014; @Khosoussi2015graph; @kasra16icra]. Consider the 2-D pose-graph SLAM problem where each measurement consists of the rotation (angle) and translation between a pair of robot poses over time, corrupted by an independently-drawn additive zero-mean Gaussian noise. According to our model, the covariance matrix of the noise vector corrupting the $i$th measurement can be written as $\diag(\sigma_{p_i}^{2}\II_2,\sigma_{\theta_i}^{2})$, where $\sigma_{p_i}^2$ and $\sigma_{\theta_i}^{2}$ denote the translational and rotational noise variances, respectively. As mentioned earlier, SLAM, as an EoG problem, admits a natural graphical representation $\GG = (\VV,\EE)$ in which poses correspond to graph vertices and edges correspond to the relative measurements between the corresponding poses. Furthermore, measurement precisions are incorporated into our model by assigning positive weights to the edges of $\GG$. Note that for each edge we have two separate weight functions $w_p$ and $w_\theta$, defined as $w_p : e_i \mapsto \sigma_{p_i}^{-2}$ and $w_{\theta} : e_i \mapsto \sigma_{\theta_i}^{-2}$.
Let $\mathbb{V}\mathrm{ar}[\hat{\xx}_{\textsf{mle}}]$ be the asymptotic covariance matrix of the maximum likelihood estimator (Cramér-Rao lower bound) for estimating the trajectory $\xx$. In [@Khosoussi2014; @Khosoussi2015graph; @kasra16icra], we investigated the impact of graph topology on the criterion ($\det
\mathbb{V}\mathrm{ar}[\hat{\xx}_{\textsf{mle}}]$) in SLAM. The results presented in [@kasra16icra] are threefold. First, in [@kasra16icra Proposition 2] it is proved that $$-2\,\tau_{w_p}(\GG) - \log\det(\LL_{w_\theta}+\gamma\II) \leq
\log\det\mathbb{V}\mathrm{ar}[\hat{\xx}_{\textsf{mle}}] \leq
-2\,\tau_{w_p}(\GG) - \tau_{w_\theta}(\GG)
\vspace{-3pt}$$ in which $\gamma$ is a parameter whose value depends on the maximum distance between the neighbouring robot poses normalized by $\sigma_{p_i}^2$’s; e.g., this parameter shrinks by reducing the distance between the neighbouring poses, or by reducing the precision of the translational measurements (see [@kasra16icra Remark 2]). Next, based on this result, it is easy to see that [@kasra16icra Theorem 5], $$\lim_{\gamma \to 0^+} \log\det\mathbb{V}\mathrm{ar}[\hat{\xx}_{\textsf{mle}}]
= - 2\,\tau_{w_p}(\GG) - \tau_{w_\theta}(\GG).
\label{eq:SLAMth}
\vspace{-3pt}$$ Note that the expression above depends only on the graphical representation of the problem. Finally, the empirical observations and Monte Carlo simulations based on a number of synthetic and real datasets indicate that the RHS of provides a reasonable estimate for $\log\det\mathbb{V}\mathrm{ar}[\hat{\xx}_{\textsf{mle}}] $ even in the non-asymptotic regime where $\gamma$ is not negligible. In what follows, we demonstrate how these results can be used in a graph-theoretic approach to the D-optimal measurement selection and pruning problems.
### Measurement Selection. {#measurement-selection. .unnumbered}
Maintaining sparsity is essential for computational efficiency in SLAM, especially in long-term autonomy. Sparsity can be preserved by implementing a measurement selection policy to asses the significance of new or existing measurements. Such a vetting process can be realized by (i) assessing the significance of any new measurement before adding it to the graph, and/or (ii) pruning a subset of the acquired measurements if their contribution is deemed to be insufficient. These ideas have been investigated in the literature; for the former approach see, e.g., [@Joshi2009; @shamaiah2010greedy], and see, e.g., [@6094414; @huang2013consistent] for the latter.
Now consider the D-optimal measurement selection problem whose goal is to select the optimal $k$-subset of measurements such that the resulting $\log\det\mathbb{V}\mathrm{ar}[\hat{\xx}_{\textsf{mle}}]$ is minimized. This problem is closely related to the D-optimal sensor selection problem for which two successful approximation algorithms have been proposed in [@Joshi2009] and [@shamaiah2010greedy] under the assumption of linear sensor models. The measurement models in SLAM are nonlinear. Nevertheless, we can still use [@Joshi2009; @shamaiah2010greedy] after linearizing the measurement model. Note that the Fisher information matrix and $\log\det\mathbb{V}\mathrm{ar}[\hat{\xx}_{\textsf{mle}}]$ in SLAM depend on the true $\xx$. Since the true value of $\xx$ is not available, in practice these terms are approximated by evaluating the Jacobian matrix at the estimate obtained by maximizing the log-likelihood function using an iterative solver.
An alternative approach would be to replace $\log\det\mathbb{V}\mathrm{ar}[\hat{\xx}_{\textsf{mle}}]$ with a graph-theoretic objective function based on . Note that this is equivalent to reducing the original problem into a graph synthesis problem. The graphical approach has the following advantages:
1. *Robustness*: Maximum likelihood estimation in SLAM boils down to solving a non-convex optimization problem via iterative solvers. These solvers are subject to local minima. Hence, the approximated $\log\det\mathbb{V}\mathrm{ar}[\hat{\xx}_{\textsf{mle}}]$ can be highly inaccurate and lead to misleading designs if the Jacobian is evaluated at a local minimum (see [@kasra16icra Section VI] for an example). The graph-theoretic objective function based on , however, is independent of the trajectory $\xx$ and, therefore, is robust to such convergence errors.
2. *Flexibility*: To directly compute $\log\det\mathbb{V}\mathrm{ar}[\hat{\xx}_{\textsf{mle}}]$, we first need a nominal or estimated trajectory $\xx$. Furthermore, for the latter we also need to know the realization of relative measurements. Therefore, any design or decisions made in this way will be confined to a particular trajectory. On the contrary, the graphical approach requires only the knowledge of the topology of the graph, and thus is more flexible. Note that the $t$-optimal topology corresponds to a range of trajectories. Therefore, the graphical approach enables us to assess the D-optimality of a particular design with minimum information and without relying on any particular—planned, nominal or estimated—trajectory.
We will investigate the problem of designing $t$-optimal graphs in Section \[sec:Synthesis\].
Synthesis of Near-$t$-Optimal Graphs {#sec:Synthesis}
====================================
### Problem Formulation. {#problem-formulation. .unnumbered}
In this section, we formulate and tackle the combinatorial optimization problem of designing sparse graphs with the maximum weighted tree-connectivity. Since the decision variables are the edges of the graph, it is more convenient to treat the weighted tree-connectivity as a function of the edge set of the graph for a given set of vertices ($\VV = [n]$) and a positive weight function $w : \binom{[n]}{2} \to \Rset_{\geq 1}$. $\logtree_{n,w}:2^{\binom{[n]}2} \to
\Rset_{\geq 0} : \EE \mapsto \tau_w([n],\EE)$ takes as input a set of edges $\EE$ and returns the weighted tree-connectivity of graph $([n],\EE)$. To simplify our notation, hereafter we drop $n$ and/or $w$ from $\logtree_{n,w}$ (and similar terms) whenever $n$ and/or $w$ are clear from the context.
Suppose the following are given:
- a *base graph* $\GGi = ([n],\EEi)$
- a weight function $w : \binom{[n]}{2} \to
\Rset_{\geq 1}$
- a set of $c$ *candidate* edges (either $\Cplus$ or $\Cminus$)
- an integer $k \leq c$
Consider the following edge selection problems (ESP):
- $k$-$\espPlus{}$ $$\begin{aligned}
& \underset{\EE \subseteq \Cplus \subseteq \binom{[n]}{2} \setminus \EEi}{\text{maximize}}
& & \logtree(\EEi \cup \EE) \quad \text{\normalsize subject to} \quad |\EE| = k.
\end{aligned}
\label{eq:addEdge}
%\tag{$k$-$\espPlus{}$}$$
- $k$-$\espMinus{}$ $$\begin{aligned}
& \underset{\EE \subseteq \Cminus \subseteq \EEi}{\text{maximize}}
& & \logtree(\EEi \setminus \EE) \quad \text{\normalsize subject to}
\quad |\EE| = k.
\end{aligned}
\label{eq:delEdge}
%\tag{$k$-$\espMinus{}$}$$
It is easy to see that any instance of can be expressed as an instance of and vice versa. Therefore, without loss of generality, in this work we only consider $k$-$\espPlus$.
### $1$-$\espPlus$. {#espplus. .unnumbered}
Consider the simple case of $k = 1$. $\Delta_{uv} \triangleq \aaa_{uv}
\LL^{-1} \aaa_{uv}$ is known as the *effective resistance* between vertices $u$ and $v$. Here $\aaa_{uv} \in
\{-1,0,1\}^{n-1}$ is the vector $\ee_u^n - \ee_v^n$ after crossing out the entry that corresponds to the anchored vertex. Effective resistance has emerged from several other contexts as a key factor; see, e.g., [@ghosh2008minimizing]. In [@kasraArxiv16 Lemma 3.1] it is shown that the optimal choice in $1$-$\espPlus$ is the candidate edge with the maximum $w(e)\Delta_{e}$. The effective resistance can be efficiently computed by performing a Cholesky decomposition on the reduced weighted Laplacian matrix of the base graph $\LLi$ and solving a triangular linear system (see [@kasraArxiv16]). In the worst case and for a dense base graph $1$-$\espPlus$ can be solved in $O(n^3 + c \,n^2)$ time.
Approximation Algorithms for $k$-$\espPlus$
-------------------------------------------
Solving the general case of $k$-$\espPlus$ by exhaustive search requires examining $\binom{c}{k}$ graphs. This is not practical even when $c$ is bounded (e.g., for $c=30$ and $k=10$ we need to perform more than $3 \times 10^7$ Cholesky factorizations). Here we propose a complementary pair of approximation algorithms.
### I: Greedy.
The greedy algorithm finds an approximate solution to $k$-$\espPlus$ by decomposing it into a sequence of $k$ $1$-$\espPlus$ problems, each of which can be solved using the procedure outlined above. After solving each subproblem, the optimal edge is moved from the candidate set to the base graph. The next $1$-$\espPlus$ subproblem is defined using the updated candidate set and the updated base graph. If the graph is dense, a naive implementation of the greedy algorithm requires less than $O(k c n^3)$ operations. An efficient implementation of this approach that requires $O(n^3 + kcn^2)$ time is described in [@kasraArxiv16 Algorithm 1].
### Analysis. {#analysis. .unnumbered}
Let $\GGi = ([n],\EEi)$ be a *connected* base graph and $w : \binom{[n]}{2} \to
\Rset_{\geq 1}$. Consider the following function. $$\begin{aligned}
\mathcal{X}_{w}: \EE \mapsto \logtree({\EE \cup \EEi}) - \logtree({\EEi}).
\label{}\end{aligned}$$ In $k$-$\espPlus$, we restrict the domain of $\mathcal{X}_{w}$ to $2^{\Cplus}$. Note that $\logtree({\EEi})$ is a constant and, therefore, we can express the objective function in $k$-$\espPlus$ using $\mathcal{X}_w$, $$\begin{aligned}
& \underset{\EE \subseteq \Cplus}{\text{maximize}}
& & \mathcal{X}_{w}(\EE) \quad \text{\normalsize subject to} \quad |\EE| = k.
\end{aligned}
\label{eq:kPlus}$$
$\mathcal{X}_{w}$ is normalized, monotone and submodular. \[th:logTG-sub\]
Omitted due to space limitation—see the technical report [@kasraArxiv16].
Maximizing an arbitrary monotone submodular function subject to a cardinality constraint [can be]{} NP-hard in general (see, e.g., the Maximum Coverage problem [@hochbaum1996approximation]). Nemhauser et al. [@nemhauser1978analysis] in their seminal work have shown that the greedy algorithm is a constant-factor approximation algorithm with a factor of $\eta \triangleq (1-1/e) \approx 0.63$ for any (normalized) monotone submodular function subject to a cardinality constraint. Let $\OPT$ be the optimum value of , $\EE_\text{greedy}$ be the edges selected by the greedy algorithm, $\tau_{\text{greedy}} \triangleq
\logtree(\EE_{\text{greedy}}\cup\EEi)$ and $\tau_{\text{init}} \triangleq
\logtree(\EEi)$.
$\normalfont
\tau_{\text{greedy}} \geq \eta\,\OPT + (1-\eta)\,\tau_{\text{init}}$.
### II: Convex Relaxation.
In this section, we design an approximation algorithm for $k$-$\espPlus$ through convex relaxation. We begin by assigning an auxiliary variable $0 \leq \pi_i \leq 1$ to each candidate edge $e_i \in\Cplus$. The idea is to reformulate the problem such that finding the optimal set of candidate edges is equivalent to finding the optimal value for $\pi_i$’s. Let $\ppp \triangleq [\pi_1 \,\, \pi_2 \,\, \cdots \,\, \pi_c]^\top$ be the stacked vector of auxiliary variables. Define, $$\LL_w(\ppp) \triangleq \LLi +
\sum_{\mathclap{e_i \in \Cplus}} \pi_{i} w(e_i) \LL_{e_i} = \AAA
\WW^\pi\hspace{-0.09cm} \AAAT,
\label{eq:Lpi}$$ where $\LL_{e_i}$ is the reduced elementary Laplacian, $\AAA$ is the reduced incidence matrix of $\GG_{\bullet} \triangleq
([n],\EEi \cup \Cplus)$, and $\WW^\pi$ is the diagonal matrix of edge weights assigned by the following weight function, $$w^\pi(e_i) =
\begin{cases}
\pi_i w(e_i) & e_i \in \Cplus, \\
w(e_i) & e_i \notin \Cplus.
\end{cases}
\label{}$$
If $\GGi$ is connected, $\LL_w(\ppp)$ is positive definite for any $\ppp \in
[0,1]^{c}$.
As before, for convenience we assume $\GGi$ is connected. Consider the following optimization problems over $\ppp$.
$$\begin{aligned}
& \underset{\ppp}{\text{maximize}}
& & \log\det {\LL_w(\ppp)}\\
& \text{subject to}
&& \|\ppp\|_0 = k,\\
&&& 0\leq \pi_i \leq {1}, \, \forall i \in [c].
\end{aligned}
\label{eq:conv0}
\tag{P$_1$}$$
$$\begin{aligned}
& \underset{\ppp}{\text{maximize}}
& & \log\det {\LL_w(\ppp)}\\
& \text{subject to}
&& \|\ppp\|_1 = k,\\
&&& \pi_i \in \{0,1\}, \, \forall i \in [c].
\end{aligned}
\label{eq:conv0bin}
\tag{P$^\prime_1$}$$
\[eq:conv0\] is equivalent to our original definition of $k$-$\espPlus$. First, note that from the generalized matrix-tree theorem we know that the objective function is equal to the weighted tree-connectivity of graph $\GG_\bullet = ([n],\EEi \cup \Cplus)$ whose edges are weighted by $w^\pi$. The auxiliary variables act as selectors: the $i$th candidate edge is selected iff $\pi_i = 1$. The combinatorial difficulty of $k$-$\espPlus$ here is embodied in the non-convex $\ell_0$-norm constraint. It is easy to see that in \[eq:conv0\], at the optimal solution, auxiliary variables take binary values. This is why the integer program \[eq:conv0bin\] is equivalent to \[eq:conv0\]. A natural choice for relaxing \[eq:conv0bin\] is to replace $\pi_i \in \{0,1\}$ with $0 \leq \pi_i \leq 1$; i.e., $$\begin{aligned}
& \underset{\ppp}{\text{maximize}}
& & \log\det {\LL_w(\ppp)}\\
& \text{subject to}
&& \|\ppp\|_1 = k,\\
&&& 0\leq \pi_i \leq {1}, \, \forall i \in [c].
\end{aligned}
\label{eq:conv1}
\tag{P$_2$}$$ The feasible set of \[eq:conv1\] contains that of \[eq:conv0bin\]. Hence, the optimum value of \[eq:conv1\] is an upper bound for the optimum of \[eq:conv0\] (or, equivalently, \[eq:conv0bin\]). Note that the $\ell_1$-norm constraint here is identical to $\sum_{i=1}^{c} \pi_i = k$. \[eq:conv1\] is a convex optimization problem since the objective function (tree-connectivity) is concave and the constraints are linear and affine in $\ppp$. In fact, \[eq:conv1\] is an instance of the $\mathrm{MAXDET}$ problem [@vandenberghe1998determinant] subject to additional affine constraints on $\ppp$. It is worth noting that \[eq:conv1\] can be reached also by relaxing the non-convex $\ell_0$-norm constraint in \[eq:conv0\] into the convex $\ell_1$-norm constraint $\|\ppp\|_1
= k$. Furthermore, \[eq:conv1\] is also closely related to a $\ell_1$-regularised variant of $\mathrm{MAXDET}$, $$\begin{aligned}
& \underset{\ppp}{\text{maximize}}
& & \log\det {\LL_w(\ppp)} - \lambda \, \|\ppp\|_1\\
& \text{subject to}
&& 0\leq \pi_i \leq {1}, \, \forall i \in [c].
\end{aligned}
\label{eq:conv1b}
\tag{P$_3$}$$ This problem is a penalized form of \[eq:conv1\]; these two problems are equivalent for some positive value of $\lambda$. Problem \[eq:conv1b\] is also a convex optimization problem for any non-negative $\lambda$. The $\ell_1$-norm in \[eq:conv1b\] penalizes the loss of sparsity, while the log-determinant rewards stronger tree-connectivity. $\lambda$ is a parameter that controls the sparsity of the resulting graph; i.e., a larger $\lambda$ yields a sparser vector of selectors $\ppp$. \[eq:conv1b\] is closely related to graphical lasso [@friedman2008sparse]. \[eq:conv1\] (and \[eq:conv1b\]) can be solved globally in polynomial time using interior-point methods [@Boyd2004; @Joshi2009]. After finding a globally optimal solution $\ppp^\star$ for the relaxed problem \[eq:conv1\], we ultimately need to map it into a feasible $\ppp$ for \[eq:conv0bin\]; i.e., choosing $k$ edges from the candidate set $\Cplus$.
$\ppp^\star$ is an optimal solution for $k$-$\normalfont \espPlus$ iff $\ppp^\star \in \{0,1\}^c$.
### Rounding. {#rounding. .unnumbered}
In general, $\ppp^\star$ may contain fractional values that need to be mapped into feasible integral values for \[eq:conv0bin\] by a *rounding procedure* that sets $k$ auxiliary variables to one and others to zero. The most intuitive deterministic rounding policy is to pick the $k$ edges with the largest $\pi^\star_i$’s.
The idea behind the convex relaxation technique described so far can be seen as a graph-theoretic special case of the algorithm proposed in [@Joshi2009]. However, it is not clear yet how the solution of the relaxed convex problem \[eq:conv1\] is related to that of the original non-convex $k$-$\espPlus$ in the integer program \[eq:conv0bin\]. To answer this question, consider the following randomized strategy. We may attempt to find a suboptimal solution for $k$-$\espPlus$ by randomly sampling candidates. In this case, for the $i$th candidate edge, we flip a coin whose probability of heads is $\pi_i$ (independent of other candidates). We then select that candidate edge if the coin lands on head.
Let the random variables $k^{\ast}$ and $t_w^\ast$ denote, respectively, the number of chosen candidate edges and the corresponding weighted number of spanning trees achieved by the above randomized algorithm. Then, $$\begin{aligned}
\mathbb{E}\,[k^\ast] & = \sum_{i=1}^{c} \pi_i, \\
\mathbb{E}\,[t_w^\ast] & = \det\LL_w(\ppp).
\end{aligned}$$ \[th:random\]
See [@kasraArxiv16] for the proof.[^2]
According to Theorem \[th:random\], the randomized algorithm described above on average selects $\sum_{i=1}^c \pi_i$ candidate edges and achieves $\det\LL_w(\ppp)$ weighted number of spanning trees in expectation. Note that these two terms appear in the constraints and objective of the relaxed problem \[eq:conv1\], respectively. Therefore, the relaxed problem can be interpreted as the problem of finding the optimal sampling probabilities $\ppp$ for the randomized algorithm described above. This offers a new narrative:
The objective in \[eq:conv1\] is to find the optimal probabilities $\ppp^\star$ for sampling edges from $\normalfont\Cplus$ such that the weighted number of spanning trees is maximized in *expectation*, while the *expected* number of newly selected edges is equal to $k$.
In other words, \[eq:conv1\] can be seen as a convex relaxation of \[eq:conv0\] at the expense of maximizing the objective and satisfying the constraint, both *in expectation*. This new interpretation can be used as a basis for designing randomized rounding procedures based on the randomized technique described above. If one uses $\ppp^\star$ (the fractional solution of the relaxed problem \[eq:conv1\]) in the aforementioned randomized rounding scheme, Theorem \[th:random\] ensures that, on average, such a method attains $\det\LL(\ppp^\star)$ by picking $k$ new edges in expectation. Finally, we note that this new interpretation sheds light on why the deterministic rounding policy described earlier performs well in practice. Note that randomly sampling candidate edges with the probabilities in $\ppp^\star$ does not necessarily result in a feasible solution for \[eq:conv0bin\]. That being said, consider every feasible outcome in which exactly $k$ candidate edges are selected by the randomized algorithm with probabilities in $\ppp^\star$. It is easy to show that the deterministic procedure described earlier (picking $k$ candidates with the largest $\pi^\star_i$’s) is in fact selecting the most probable feasible outcome (given that exactly $k$ candidates have been selected).
### Near-Optimality Certificates.
It is impractical to compute $\OPT$ via exhaustive search in large problems. Nevertheless, the approximation algorithms described above yield lower and upper bounds for $\OPT$ that can be quite tight in practice. Let $\tau^\star_\text{cvx}$ be the optimum value of \[eq:conv1\]. Moreover, let $\tau_\text{cvx}$ be the suboptimal value obtained after rounding the solution of \[eq:conv1\] (e.g., picking the $k$ largest $\pi_i^\star$’s). The following corollary readily follows from the analysis of the greedy and convex approximation algorithms.
\[cor:bound\] $$\normalfont
\max\,\Big\{ \tau_\text{greedy},\tau_\text{cvx} \Big\}
\leq
\OPT
\leq
\min\,\Big\{ \mathcal{U}_\text{greedy},\tau^\star_\text{cvx} \Big\}
\label{eq:optbound}$$ where $\mathcal{U}_\text{\normalfont greedy} \triangleq
\zeta\tau_\text{\normalfont greedy} +
(1-\zeta)\tau_\text{\normalfont init}$ in which $\zeta \triangleq \eta^{-1} \approx 1.58$.
The bounds in Corollary \[cor:bound\] can be computed by running the greedy and convex relaxation algorithms. Whenever $\OPT$ is beyond reach, the upper bound can be used to asses the quality of any feasible design. Let $\mathcal{S}$ be an arbitrary $k$-subset of $\Cplus$ and $\tau_\mathcal{S}
\triangleq \logtree(\mathcal{S} \cup \EEi)$. $\mathcal{S}$ can be, e.g., the solution of greedy algorithm, the solution of \[eq:conv1\] after rounding, an existing design (e.g., an existing pose-graph problem) or a suboptimal solution proposed by a third party. Let $\mathcal{L}$ and $\mathcal{U}$ denote the lower and upper bounds in , respectively. From Corollary \[cor:bound\] we have, $$\max \, \Big\{0,\mathcal{L}-\tau_{\mathcal{S}}\Big\} \leq \underbrace{\OPT -
\tau_\mathcal{S}}_{\text{optimality gap}} \leq \mathcal{U} -
\tau_\mathcal{S}.
\label{}$$ Therefore, $\mathcal{U} - \tau_\mathcal{S}$ (or similarly, $\mathcal{U}/\tau_\mathcal{S} \geq \OPT/\tau_\mathcal{S}$) can be used as a near-optimality certificate for an arbitrary design $\mathcal{S}$.
### Two Weight Functions.
In the synthesis problem studied so far, it was implicitly assumed that each edge is weighted by a single weight function. This is not necessarily the case in SLAM, where each measurement has two components, each of which has its own precision, i.e., $w_p$ and $w_\theta$ in . Hence, we need to revisit the synthesis problem in a more general setting, where multiple weight functions assign weights, simultaneously, to a single edge. It turns out that the proposed approximation algorithms and their analyses can be easily generalized to handle this case.
1. *Greedy Algorithm*: For the greedy algorithm, we just need to replace $\mathcal{X}_{w}$ with $\mathcal{Y}_{w} : \EE \mapsto 2 \, \mathcal{X}_{w_p}(\EE) +
\mathcal{X}_{w_\theta}(\EE)$; see . Note that $\mathcal{Y}_{w}$ is a linear combination of normalized monotone submodular functions with positive weights, and therefore is also normalized, monotone and submodular.
2. *Convex Relaxation*: The convex relaxation technique can be generalized to the case of multi-weighted edges by replacing the concave objective function $\log\det\LL_w(\ppp)$ with $2\,\log\det\LL_{w_p}(\ppp) +
\log\det\LL_{w_\theta}(\ppp)$, which is also concave.
Recall that our goal was to design sparse, yet reliable SLAM problems. So far we considered the problem of designing D-optimal SLAM problems with a given number of edges. The dual approach would be to find the sparsest SLAM problem such that the determinant of the estimation-error covariance is less than a desired threshold. Take for example the following scenario: find the sparsest SLAM problem by selecting loop-closure measurements from a given set of candidates such that the resulting D-criterion is $50\%$ smaller than that of dead reckoning. The dual problem can be written as, $$\begin{aligned}
& \underset{\EE \subseteq \Cplus}{\text{minimize}}
& & |\EE| \quad \text{\normalsize subject to} \quad \mathcal{X}_{w}(\EE)
\geq \tau_{\textsf{min}}.
\end{aligned}
\label{eq:dual1}$$ in which $\tau_{\textsf{min}}$ is given. In [@kasraArxiv16] we have shown that our proposed approximation algorithms and their analyses can be easily modified to address the dual problem. Due to space limitation, we have to refrain from discussing the dual problem in this paper.
Experimental Results
--------------------
[0.48]{} ![$k$-$\espPlus$ on randomly generated graphs with $\Cplus =
\binom{[n]}{2} \setminus \EEi$.[]{data-label="fig:esps"}](varying_num_edges-04-Apr-2016__16-08.eps "fig:"){width="\textwidth"}
[0.48]{} ![$k$-$\espPlus$ on randomly generated graphs with $\Cplus =
\binom{[n]}{2} \setminus \EEi$.[]{data-label="fig:esps"}](varying_num_edges-04-Apr-2016__16-12.eps "fig:"){width="\textwidth"}
\
[0.5]{} ![$k$-$\espPlus$ on randomly generated graphs with $\Cplus =
\binom{[n]}{2} \setminus \EEi$.[]{data-label="fig:esps"}](varying_horizon-04-Apr-2016__16-15.eps "fig:"){width="\textwidth"}
The proposed algorithms were implemented in MATLAB. \[eq:conv1\] is modelled using YALMIP [@YALMIP] and solved using SDPT$3$ [@tutuncu2003solving].
### Random Graphs.
Figure \[fig:esps\] illustrates the performance of our approximate algorithms in randomly generated graphs. The set of candidates in these experiments is $\Cplus = \binom{[n]}{2} \setminus \EEi$. Figures \[fig:varEdgesOpt\] and \[fig:varEdges\] show the resulting tree-connectivity as a function of the number of randomly generated edges for a fixed $k=5$ and, respectively, $n = 20$ and $n = 50$. Our results indicate that both algorithms exhibit remarkable performances for $k = 5$. Note that computing $\mathsf{OPT}$ by exhaustive search is only feasible in small instances such as Figure \[fig:varEdgesOpt\]. Nevertheless, computing the exact $\mathsf{OPT}$ is not crucial for evaluating our approximate algorithms, as Corollary \[cor:bound\] guarantees that $ \tau^{\star}_{\text{greedy}}\leq \OPT \leq
{\tau}^\star_{\text{cvx}}$; i.e., the space between each black $\mathrm{\cdot}$ and the corresponding $\times$. Figure \[fig:varK\] shows the results obtained for varying $k$. The optimality gap for $\tau_\text{cvx}$ gradually grows as the planning horizon $k$ increases. Our greedy algorithm, however, still yields a near-optimal approximation.
### Real Pose-Graph Dataset.
We also evaluated the proposed algorithms on the Intel Research Lab dataset as a popular pose-graph SLAM benchmark.[^3] In this scenario, $\EEi$ is chosen to be the set of odometry edges, and $\Cplus$ is the set of loop closures. The parameters in this graph are $n = 943$, $|\EEi| = 942$ and $|\Cplus| = 895$. Note that computing the true $\OPT$ via exhaustive search is clearly impractical; e.g., for $k=100$, there are more than $10^{134}$ possible graphs. For the edge weights, we are using the original information (precisions) reported in the dataset. Since the translational and rotational measurements have different precisions, two weight functions—$w_p$ and $w_\theta$—assign weights to each edge of the graph, and the objective is to maximize $2\,\tau_{w_p}(\GG) +
\tau_{w_\theta}(\GG)$. Figure \[fig:intel\] shows the resulting objective value for the greedy and convex relaxation approximation algorithms, as well as the upper bounds ($\mathcal{U}$) in Corollary \[cor:bound\].[^4] According to Figure \[fig:intel\], both algorithms have successfully found near-$t$-optimal (near-D-optimal) designs. The greedy algorithm has outperformed the convex relaxation with the simple deterministic (sorting) rounding procedure. For small values of $k$, the upper bound $\mathcal{U}$ on $\OPT$ is given by $\mathcal{U}_\text{greedy}$ ( curve). However, for $k \geq 60 $, the convex relaxation provides a significantly tighter upper bound on $\OPT$ ( curve). In this dataset, YALMIP+SDPT3 on an Intel Core i5-2400 operating at 3.1 GHz can solve the convex program in about $20$-$50$ seconds, while a naive implementation of the greedy algorithm (without using rank-one updates) can solve the case with $k=400$ in about $25$ seconds.
[0.48]{} ![$k$-$\espPlus$ for pose-graph SLAM on the Intel Research Lab dataset.[]{data-label="fig:intel"}](intel_esp-10-Jul-2016__13-51.eps "fig:"){width="\textwidth"}
[0.42]{} ![$k$-$\espPlus$ for pose-graph SLAM on the Intel Research Lab dataset.[]{data-label="fig:intel"}](intel0322.eps "fig:"){width="\textwidth"}
Conclusion {#sec:conclusion}
==========
We presented a graph-theoretic approach to the problem of designing sparse reliable (i.e., near-D-optimal) pose-graph SLAM. This paper demonstrated that this problem boils down to a combinatorial optimization problem whose goal is to find a sparse graph with the maximum weighted number of spanning trees. The problem of characterizing $t$-optimal graphs is an open problem with—to the best of our knowledge—no known efficient algorithm. We designed two efficient approximation algorithms with provable guarantees and near-optimality certificates. First and foremost, we introduced a new submodular graph invariant, i.e., weighted tree-connectivity. This was used to guarantee that the greedy algorithm is a constant-factor approximation algorithm for this problem with a factor of $(1-1/e)$ (up to a constant normalizer). In another approach, we formulated the original combinatorial optimization problem as an integer program that admits a natural convex relaxation. We discussed deterministic and randomized rounding schemes. Our analysis sheds light on the connection between the original and the relaxed problems. Finally, we evaluated the performance of the proposed approximation algorithms using random graphs and a real pose-graph SLAM dataset. Although this paper specifically targeted SLAM, we note that the proposed algorithms can be readily used to synthesize near-$t$-optimal graphs in any domain where maximizing tree-connectivity is useful. See, e.g., [@chemistryGraph; @kim2013network; @boesch2009survey; @kasraArxiv16] for applications in Chemistry, RNA modelling, network reliability under random link failure and estimation over sensor networks, respectively.
[^1]: Replacing any $w : \EE \to
\mathbb{R}_{\geq 0}$ with $w^\prime : \EE \to
\mathbb{R}_{\geq 1} : e \mapsto
\alpha_w w(e)$ for a sufficiently large constant $\alpha_w$ does not affect the set of $t$-optimal graphs.
[^2]: A generalized version of this theorem that covers the more general case of [@Joshi2009] is proved in [@kasraArxiv16].
[^3]: <https://svn.openslam.org/data/svn/g2o/trunk/data/2d/intel/intel.g2o>
[^4]: See also <https://youtu.be/5JZF2QiRbDE> for a visualization.
|
---
abstract: |
The spectra of low-lying pair excitations for an imbalanced two-component superfluid Fermi gas are analytically derived within the path-integral formalism taking into account Gaussian fluctuations about the saddle point. The spectra are obtained for nonzero temperatures, both with and without imbalance, and for arbitrary interaction strength. On the basis of the pair excitation spectrum, we have calculated the thermodynamic parameters of state of cold fermions and the first and second sound velocities. The parameters of pair excitations show a remarkable agreement with the Monte Carlo data and with experiment.
PACS numbers: 03.75.Ss, 05.30.Fk, 03.75.Lm
author:
- 'S. N. Klimin$^{1}$, J. Tempere$^{1,2}$, and Jeroen P. A. Devreese$^{1}$'
title: Pair excitations and parameters of state of imbalanced Fermi gases at finite temperatures
---
Introduction \[sec:intro\]
==========================
Recent experimental breakthroughs in the manipulation of ultracold Bose and Fermi gases have opened new prospects for advancing many-body physics [@Bloch2008]. The dimensionality of these gases can be controlled with optical lattices, and the interaction strength can be tuned using Feshbach resonances. The experimental control over geometry and interactions in ultracold atomic gases has turned these systems into powerful quantum simulators that can test and generalize many-body theories originally developed for solid state systems. Recently, much attention has been paid to ultracold atomic gases with strong interactions because of their possible relation to some striking natural phenomena including high-temperature superconductors and neutron stars [@Petick].
In particular, great experimental and theoretical effort has been devoted to the study of superfluidity arising from pairing in ultracold Fermi gases [@Radzihovsky]. Specifically, the effect of population imbalance (between the pairing partners) on the superfluid pairing mechanism is a topic of current investigation. Of no less interest in the context of high-temperature superconductivity is the study of the crossover between the Bardeen-Cooper-Schrieffer (BCS) and the molecular Bose-Einstein condensation (BEC) regimes.
In order to describe the superfluid phase transition as well as the broken-symmetry phase below a critical temperature $T_{c}$, several methods based on the $T$-matrix approach have been developed [@NSR; @deMelo1993; @Pieri; @Chen; @Combescot; @Haussmann; @Diener2008]. Amongst those, the Nozières–Schmitt-Rink (NSR) theory [@NSR] and its path-integral reformulation [@deMelo1993] have been very successful and remain widely used. Here, we use an improved version of the NSR scheme, that we will denote as the Gaussian pair fluctuation theory (GPF) [@Drum; @Drum2], and that effectively works both at low temperatures and above $T_{c}$.
As shown in Ref. [@Salasnich2010], the thermodynamic properties of the superfluid Fermi gas at sufficiently low temperatures can be derived within a simple model using the spectrum of low-lying elementary excitations. In that model, the thermodynamics of the superfluid Fermi gas [@Salasnich2010] is based on the fermion-boson model with phenomenological parameters whose values are determined from the Monte Carlo calculations [@MC1; @MC2; @MC3; @Bulgac] (in the limit of zero temperature). The model exploited in Ref. [@Salasnich2010] describes well the thermodynamic properties of a balanced unitary Fermi gas at low temperatures. Moreover it was demonstrated [@Taylor2007] that in the BEC limit, the imbalanced Fermi superfluid indeed reduces to a simple Bose-Fermi mixture of Bose-condensed molecules and unpaired fermions. The goal of the present work is to extend these results to non-zero temperatures, and to arbitrary scattering lengths.
For this purpose, we calculate the parameters needed for the fermion-boson model from the NSR and/or GPF theories. Our results are compared with the Monte Carlo data at unitarity. We analytically derive the spectra of low-lying elementary excitations for an imbalanced Fermi gas in 3D at finite temperatures in the whole range of the BCS-BEC crossover. These spectra are obtained using the path-integral representation [@deMelo1993] of the NSR theory extended to imbalanced Fermi gases [@PRA2008; @PRB2008] as well as the GPF approach [@Drum; @Drum2]. Using the obtained spectra of the elementary excitations, thermodynamic parameters such as the internal energy, the chemical potential, the first and second sound velocities, are calculated.
Formalism \[sec:theory\]
========================
Path-integral GPF approach for imbalanced Fermi gases
-----------------------------------------------------
We consider a two-component Fermi gas within the path-integral approach. The path-integral formulation [@deMelo1993] of the NSR scheme has been extended in Refs. [@PRA2008; @PRB2008] to the case of unequal ‘spin up’ and ‘spin down’ populations of fermions. In the present work, the treatment of the imbalanced Fermi gas is performed using the NSR scheme [@PRA2008; @PRB2008] and its improved version, the GPF theory [@Drum] extended to the imbalanced case.
The thermodynamic parameters of the imbalanced Fermi gas are completely determined by the thermodynamic potential $\Omega$ of the grand-canonical ensemble. The thermodynamic potential $\Omega$, the same as in Refs. [@PRA2008; @PRB2008], is the sum of the saddle-point thermodynamic potential $\Omega_{sp}$ and the fluctuation contribution $\Omega_{fl}$. These thermodynamic potentials are provided, respectively, by the zeroth-order and quadratic terms of the expansion of the Hubbard-Stratonovich pair-field action around the saddle point.
The saddle-point thermodynamic potential for the imbalanced Fermi gas with $s$-wave pairing is [@PRA2008] $$\Omega_{sp}=-V\int\frac{d\mathbf{k}}{\left( 2\pi\right) ^{3}}\left[
\frac{1}{\beta}\ln\left( 2\cosh\beta\zeta+2\cosh\beta E_{\mathbf{k}}\right)
-\xi_{\mathbf{k}}-\frac{\Delta^{2}}{2k^{2}}\right] -V\frac{\Delta^{2}}{8\pi
a_{s}} \label{Wsp}$$ where $V$ is the system volume, $\beta$ is the inverse to the thermal energy $k_{B}T$, $\Delta$ is the amplitude of the gap parameter, $a_{s}$ is the scattering length, $\xi_{\mathbf{k}}=k^{2}-\mu$ is the fermion energy, and $E_{\mathbf{k}}=\sqrt{\xi_{\mathbf{k}}^{2}+\Delta^{2}}$ is the Bogoliubov excitation energy. The chemical potentials of imbalanced fermions are expressed through the averaged chemical potential $\mu=\left( \mu_{\uparrow
}+\mu_{\downarrow}\right) /2$ and the chemical potential imbalance $\zeta=\left( \mu_{\uparrow}-\mu_{\downarrow}\right) /2$. We choose the units with $\hbar=1$, the fermion mass $m=1/2$, and the Fermi energy $E_{F}\equiv\hbar^{2}\left( 3\pi^{2}n\right) ^{2/3}/\left( 2m\right) =1$ ($n$ is the total fermion density). The fluctuation contribution to the thermodynamic potential $\Omega_{fl}$ is the same as in Refs. [@PRA2008; @PRB2008].
The gap parameter is found from the gap equation minimizing the saddle-point thermodynamic potential,$$\left. \dfrac{\partial\Omega_{sp}(T,\mu,\zeta;\Delta)}{\partial\Delta
}\right\vert _{T,\mu,\zeta}=0$$ from which we can extract $\Delta(T,\mu,\zeta)$. For an imbalanced gas, the saddle-point thermodynamic potential can have two minima: one at $\Delta=0$ and one at $\Delta\neq0$. This can result in a first-order superfluid phase transition [@Bedaque]. With our notation, we emphasize that the thermodynamic potential is a function of $T,\mu,\zeta$ (and actually $V$, but this dependency drops out). However, $\Delta$ is treated as an additional parameter on which the thermodynamic potential depends. It is this treatment of $\Delta$ as an additional parameter (in the broken-symmetry phase with $\Delta\neq0$) that leads to a distinction between the NSR approach and the GPF approach. When calculating the gap equation one should use$$\begin{aligned}
n & =-\left. \dfrac{\partial\Omega\left( T,\mu,\zeta;\Delta\right)
}{\partial\mu}\right\vert _{T,\zeta,\Delta}-\left. \dfrac{\partial\Omega
_{fl}\left( T,\mu,\zeta;\Delta\right) }{\partial\Delta}\right\vert
_{T,\zeta,\mu}\left. \dfrac{\partial\Delta(T,\mu,\zeta)}{\partial\mu
}\right\vert _{T,\zeta},\\
\delta n & =-\left. \dfrac{\partial\Omega\left( T,\mu,\zeta;\Delta\right)
}{\partial\zeta}\right\vert _{T,\mu,\Delta}-\left. \dfrac{\partial\Omega
_{fl}\left( T,\mu,\zeta;\Delta\right) }{\partial\Delta}\right\vert
_{T,\zeta,\mu}\left. \dfrac{\partial\Delta(T,\mu,\zeta)}{\partial\zeta
}\right\vert _{T,\mu}.\end{aligned}$$ In the standard NSR approach the last terms (involving the derivatives of $\Delta$) are omitted. The GPF approach suggested in Refs. [@Drum; @Drum2] takes into account the additional derivatives for the balanced case, and corrects the NSR densities for changes in $\Delta$ as $\mu$ and $\zeta$ are varied. The GPF method presented here is the path-integral formulation of the GPF theory of Refs. [@Drum; @Drum2] extended to the imbalanced case. As shown in Ref. [@Drum2], the GPF theory provides the best overall agreement of its analytic results with experiment and with Monte Carlo data, except in close vicinity to $T_{c}$.
The GPF corrections are not present when one uses the saddle-point approximation and calculates$$\begin{aligned}
n_{sp} & =-\left. \dfrac{\partial\Omega_{sp}\left( T,\mu,\zeta
;\Delta\right) }{\partial\mu}\right\vert _{T,\zeta,\Delta},\\
\delta n_{sp} & =-\left. \dfrac{\partial\Omega_{sp}\left( T,\mu
,\zeta;\Delta\right) }{\partial\zeta}\right\vert _{T,\mu,\Delta}.\end{aligned}$$ However, the correction terms will be relevant for the calculation of the fluctuation contributions, $n_{fl}=n-n_{sp}$ and $\delta n_{fl}=\delta
n-\delta n_{sp}$. These fluctuation contributions to the density $n_{fl}$ and $\delta n_{fl}$ are given by the same expressions as in Ref. [@PRA2008] $$\begin{aligned}
n_{fl} & =-\int\frac{d\mathbf{q}}{\left( 2\pi\right) ^{3}}\left[ \frac
{1}{\pi}\int_{-\infty}^{\infty}\operatorname{Im}\frac{J\left( \mathbf{q},\omega+i\gamma\right) }{e^{\beta\left( \omega+i\gamma\right) }-1}d\omega+\frac{1}{\beta}\sum_{n=-n_{0}}^{n_{0}}J\left( \mathbf{q},i\nu
_{n}\right) \right] ,\label{nfl}\\
\delta n_{fl} & =-\int\frac{d\mathbf{q}}{\left( 2\pi\right) ^{3}}\left[
\frac{1}{\pi}\int_{-\infty}^{\infty}\operatorname{Im}\frac{K\left(
\mathbf{q},\omega+i\gamma\right) }{e^{\beta\left( \omega+i\gamma\right)
}-1}d\omega+\frac{1}{\beta}\sum_{n=-n_{0}}^{n_{0}}K\left( \mathbf{q},i\nu
_{n}\right) \right] .\label{dnfl}$$ Here, $n_{0}$ is an arbitrary positive integer, and the parameter $\gamma$ lies between two bosonic Matsubara frequencies $\nu_{n_{0}}<\gamma<\nu
_{n_{0}+1}$, $\nu_{n}\equiv2\pi n/\beta$. The spectral functions $J\left(
\mathbf{q},z\right) $ and $K\left( \mathbf{q},z\right) $ of complex frequency $z$ are$$\begin{aligned}
J\left( \mathbf{q},z\right) & =\frac{M_{1,1}\left( \mathbf{q},-z\right)
\frac{\partial M_{1,1}\left( \mathbf{q},z\right) }{\partial\mu}-M_{1,2}\left( \mathbf{q},-z\right) \frac{\partial M_{1,2}\left(
\mathbf{q},z\right) }{\partial\mu}}{M_{1,1}\left( \mathbf{q},z\right)
M_{1,1}\left( \mathbf{q},-z\right) -M_{1,2}^{2}\left( \mathbf{q},z\right)
},\label{FJ}\\
K\left( \mathbf{q},z\right) & =\frac{M_{1,1}\left( \mathbf{q},-z\right)
\frac{\partial M_{1,1}\left( \mathbf{q},z\right) }{\partial\zeta}-M_{1,2}\left( \mathbf{q},-z\right) \frac{\partial M_{1,2}\left(
\mathbf{q},z\right) }{\partial\zeta}}{M_{1,1}\left( \mathbf{q},z\right)
M_{1,1}\left( \mathbf{q},-z\right) -M_{1,2}^{2}\left( \mathbf{q},z\right)
},\label{FK}$$ where $M_{j,k}\left( \mathbf{q},z\right) $ are the matrix elements of the pair field propagator. The matrix elements $M_{j,k}\left( \mathbf{q},z\right) $ are given by the expressions [@PRA2008] $$\begin{aligned}
M_{1,1}\left( \mathbf{q},z\right) & =\int\frac{d\mathbf{k}}{\left(
2\pi\right) ^{3}}\left\{ \frac{1}{2k^{2}}+\frac{X\left( E_{\mathbf{k}}\right) }{2E_{\mathbf{k}}}\left[ \frac{\left( z-E_{\mathbf{k}}+\varepsilon_{\mathbf{k}+\mathbf{q}}\right) \left( E_{\mathbf{k}}+\varepsilon_{\mathbf{k}}\right) }{\left( z-E_{\mathbf{k}}+E_{\mathbf{k}+\mathbf{q}}\right) \left( z-E_{\mathbf{k}}-E_{\mathbf{k}+\mathbf{q}}\right) }\right. \right. \nonumber\\
& \left. \left. -\frac{\left( z+E_{\mathbf{k}}+\varepsilon_{\mathbf{k}+\mathbf{q}}\right) \left( E_{\mathbf{k}}-\varepsilon_{\mathbf{k}}\right)
}{\left( z+E_{\mathbf{k}}-E_{\mathbf{k}+\mathbf{q}}\right) \left(
z+E_{\mathbf{k}+\mathbf{q}}+E_{\mathbf{k}}\right) }\right] \right\}
-\frac{1}{8\pi a_{s}},\label{M11}$$$$\begin{aligned}
M_{1,2}\left( \mathbf{q},z\right) & =-\Delta^{2}\int\frac{d\mathbf{k}}{\left( 2\pi\right) ^{3}}\frac{X\left( E_{\mathbf{k}}\right)
}{2E_{\mathbf{k}}}\left[ \frac{1}{\left( z-E_{\mathbf{k}}+E_{\mathbf{k}+\mathbf{q}}\right) \left( z-E_{\mathbf{k}}-E_{\mathbf{k}+\mathbf{q}}\right) }\right. \nonumber\\
& \left. +\frac{1}{\left( z+E_{\mathbf{k}}-E_{\mathbf{k}+\mathbf{q}}\right) \left( z+E_{\mathbf{k}}+E_{\mathbf{k}+\mathbf{q}}\right) }\right]
.\label{M22}$$ Here, the distribution function $X\left( E_{\mathbf{k}}\right) $ is$$X\left( E_{\mathbf{k}}\right) =\frac{\sinh\beta E_{\mathbf{k}}}{\cosh\beta
E_{\mathbf{k}}+\cosh\beta\zeta}.\label{X}$$ The derivatives in (\[FJ\]) and (\[FK\]) within the GPF scheme are determined as mentioned above – taking into account a variation of the gap parameter. Within the NSR scheme, these derivatives are calculated assuming $\Delta$ to be an independent variational parameter.
The equation of state of the imbalanced Fermi gas is thus determined as a joint solution of the saddle-point gap equation and the number equations accounting for Gaussian fluctuations. Within both the NSR and GPF schemes, the Gaussian fluctuations do not feed back into the saddle-point gap equation.
Low-lying pair excitations
--------------------------
In order to obtain the spectrum of low-lying pair excitations for the imbalanced Fermi gas, we perform the long-wavelength and low-energy expansion of the matrix elements $M_{j,k}\left( \mathbf{q},z\right) $ as proposed in Ref. [@Diener2008]. We take into account the terms up to quadratic order in powers of $q$ and $z$, and find:$$\begin{aligned}
M_{1,1}\left( \mathbf{q},z\right) & \approx A+Bq^{2}+Cz+Qz^{2},\nonumber\\
M_{1,2}\left( \mathbf{q},z\right) & \approx D+Eq^{2}+Hz^{2}.
\label{series1}$$
The coefficients of the expansion (\[series1\]) are derived straightforwardly. After some algebra, we arrive at their expression through the integrals:$$\begin{aligned}
A & =\frac{1}{2\pi^{2}}\int k^{2}dk\left( \frac{1}{2k^{2}}-\frac{E_{k}^{2}+\xi_{k}^{2}}{4E_{k}^{3}}X\left( E_{\mathbf{k}}\right) -\frac{\Delta
^{2}}{4}\frac{X^{\prime}\left( E_{k}\right) }{E_{k}^{2}}\right) -\frac
{1}{8\pi a_{s}},\nonumber\\
B & =\frac{1}{48\pi^{2}}\int k^{2}dk\frac{2E_{k}^{4}k^{2}-3E_{k}^{4}\xi
_{k}+9\xi_{k}^{3}E_{k}^{2}+14E_{k}^{2}\xi_{k}^{2}k^{2}-20\xi_{k}^{4}k^{2}}{E_{k}^{7}}X\left( E_{\mathbf{k}}\right) \nonumber\\
& +\frac{\Delta^{2}}{24\pi^{2}}\int k^{2}dk\frac{1}{E_{k}^{4}}\left(
\frac{3\xi_{k}\left( E_{k}^{2}-2\xi_{k}k^{2}\right) }{E_{k}^{2}}X^{\prime
}\left( E_{k}\right) \right. \nonumber\\
& \left. +\frac{6\xi_{k}^{2}k^{2}-E_{k}^{2}\left( 3\xi_{k}+2k^{2}\right)
}{2E_{k}}X^{\prime\prime}\left( E_{k}\right) -\frac{2k^{2}\xi_{k}^{2}}{3}X^{(3)}\left( E_{k}\right) \right) ,\nonumber\\
C & =-\frac{1}{8\pi^{2}}\int k^{2}dk\frac{\xi_{k}}{E_{k}^{3}}X\left(
E_{\mathbf{k}}\right) -\frac{\Delta^{2}}{8\pi^{2}}\int k^{2}dk\frac
{X^{\prime}\left( E_{k}\right) }{\xi_{k}E_{k}^{2}},\nonumber\\
D & =\frac{\Delta^{2}}{8\pi^{2}}\int k^{2}dk\frac{X\left( E_{\mathbf{k}}\right) }{E_{\mathbf{k}}^{3}}-\frac{\Delta^{2}}{8\pi^{2}}\int k^{2}dk\frac{X^{\prime}\left( E_{k}\right) }{E_{k}^{2}},\nonumber\end{aligned}$$$$\begin{aligned}
E & =\frac{\Delta^{2}}{48\pi^{2}}\int k^{2}dk\frac{20\xi_{k}^{2}k^{2}-E_{k}^{2}\left( 9\xi_{k}+6k^{2}\right) }{E_{k}^{7}}X\left( E_{\mathbf{k}}\right) \nonumber\\
& +\frac{\Delta^{2}}{24\pi^{2}}\int k^{2}dk\frac{1}{E_{k}^{4}}\left(
\frac{E_{k}^{2}\left( 3\xi_{k}+2k^{2}\right) -6\xi_{k}^{2}k^{2}}{2E_{k}^{2}}\left[ 2X^{\prime}\left( E_{k}\right) -E_{k}X^{\prime\prime}\left(
E_{k}\right) \right] \right. \nonumber\\
& \left. -\frac{2k^{2}\xi_{k}^{2}}{3}X^{(3)}\left( E_{k}\right) \right)
,\nonumber\\
Q & =-\frac{1}{32\pi^{2}}\int k^{2}dk\frac{E_{k}^{2}+\xi_{k}^{2}}{E_{k}^{5}}X\left( E_{\mathbf{k}}\right) ,\nonumber\\
H & =\frac{\Delta^{2}}{32\pi^{2}}\int_{0}^{\infty}k^{2}dk\frac{X\left(
E_{\mathbf{k}}\right) }{E_{k}^{5}}. \label{coefs}$$
The low-lying pair excitations correspond to the poles of the spectral functions (\[FJ\]) and (\[FK\]). Therefore the dispersion equation for the energies of the pair excitations $\omega=\omega_{q}$ is $$M_{1,1}\left( \mathbf{q},\omega\right) M_{1,1}\left( \mathbf{q},-\omega\right) -M_{1,2}^{2}\left( \mathbf{q},\omega\right) =0.
\label{disp}$$ This equation is solved expanding $\omega_{q}^{2}$ up to the terms of the order of $q^{4}$. We then obtain the energies $\omega_{q}$ in a form structurally similar to the collective excitations in Ref. [@Salasnich2010]:$$\omega_{q}=\sqrt{c^{2}q^{2}+\lambda q^{4}}, \label{spectrum}$$ where the parameters $c$ and $\lambda$ are related to the coefficient of the expansion (\[series1\]) as follows,$$\begin{aligned}
c & =\left( \frac{2A\left( B-E\right) }{C^{2}+2A\left( H-Q\right)
}\right) ^{1/2},\label{c1a}\\
\lambda & =\frac{C^{2}\left( B-E\right) \left( 4A\left( BH-EQ\right)
+C^{2}\left( B+E\right) \right) }{\left( C^{2}+2A\left( H-Q\right)
\right) ^{3}}. \label{lambda}$$
For small pair momentum $q$, the energy of the pair excitation becomes linear in the momentum. Thus $\omega_{q}$ at small $q$ represents a Bogoliubov–Anderson mode, which is gapless in accordance with the Nambu–Goldstone theorem. The parameter $c$ has the dimensionality of velocity. In the zero-temperature limit for a balanced gas, all fermions are in the superfluid state, and the velocity parameter for the pair excitations tends to the first sound velocity for the whole fermion system. The so-called gradient parameter $\lambda$ provides a growth of kinetic energy due to a spatial variation of the density [@Salasnich2008; @S2].
The pair excitation spectra obtained in the present work generalize the long-wavelength expansion of Ref. [@Diener2008] to the case of non-zero temperatures and unequal ‘spin-up’ and ‘spin-down’ fermion populations.
\[ptb\]
[SI-Fig1.EPS]{}
In Fig. 1, the parameters $c$ and $\lambda$ characterizing pair excitations are plotted as a function of temperature for the balanced Fermi gas in the unitarity regime ($1/a_{s}=0$). The parameters of the equation of state for a given temperature are determined from the joint solution of the gap and number equations taking into account fluctuations in the number equation. The fluctuation contributions to the fermion density are calculated within the path-integral GPF and NSR schemes.
As found in Ref. [@Drum], the NSR approach becomes inaccurate in the vicinity of the critical temperature $T_{c}$. It was also shown that the NSR scheme reveals a re-entrant behavior of the parameters in the state above $T_{c}$, leading to an artificial first-order superfluid phase transition [@Fukushima2007]. The re-entrant behavior of the parameters $c$ and $\lambda$ obtained in the NSR approach is clearly seen in Fig. 1. The critical temperature $T_{c}$ for the balanced gas$,$ indicated by a dash-dotted line in Fig. 1, is the same within the NSR and GPF approaches. However, the GPF method leads to better results with respect to NSR for the broken-symmetry phase. This can be seen, for example, in the inset of Fig. 1 where we plot the chemical potential as a function of temperature below $T_{c}$ calculated within the NSR and path-integral GPF approaches and compared with the Monte Carlo results of Ref. [@Bulgac].
In the zero-temperature limit, the sound velocity parameter $c$ for the pair excitations obtained within both the path-integral GPF and NSR approaches exhibits an excellent agreement with the numerical results obtained using different Monte Carlo algorithms [@MC1; @MC1a; @MC2; @MC2a; @MC3]. Also the gradient parameter $\lambda$ at zero temperature lies within the range of the values of $\lambda$ obtained in Refs. [@Salasnich2010; @S2] as the best fitting parameters for the ground state energy of fermions compared with Monte Carlo data. This agreement demonstrates the accuracy of the present approach for the broken-symmetry phase of cold fermions.
\[h\]
[SI-Fig2.EPS]{}
In Fig. 2, we plot the parameters $c$ and $\lambda$ as a function of the inverse scattering length $1/a_{s}$ for the balanced gas (the left-hand panels) and at the chemical potential imbalance $\zeta=0.2$ (the right-hand panels). At $k_{B}T=0.01E_{F}$ the sound velocity parameter monotonously decreases with increasing $1/a_{s}$. For non-zero temperatures, $c$ exhibits a maximum, which shifts to higher coupling strengths for higher temperatures. The parameter $\lambda$ at finite temperatures has a minimum, which almost vanishes in the zero-temperature limit. In the weak-coupling regime, both $c$ and $\lambda$ are sensitive to temperature. When moving towards the strong-coupling regime, $c$ and $\lambda$ gradually become almost independent on $T$. The imbalance leads to the appearance of a critical inverse scattering length such that for smaller $1/a_{s}$, there is no superfluid state (see, e. g., Ref. [@PRB2008]).
Parameters of state \[sec:thermpars\]
=====================================
Thermodynamic functions
-----------------------
Using the spectra of the elementary fermionic and pair excitations derived in Sec. \[sec:theory\], we can obtain the thermodynamic functions of the superfluid Fermi gas at finite temperature. In Ref. [@Salasnich2010], a similar description of the thermodynamic properties was performed for a unitary balanced Fermi gas using the zero-temperature spectra of elementary excitations.
In Ref. [@Salasnich2010], a pair excitation spectrum of the form of expression (\[spectrum\]) is used, where the zero-temperature sound velocity $c$ is taken from the Monte Carlo data [@MC1; @MC1a; @MC2; @MC2a; @MC3] and the gradient parameter $\lambda$ is determined from a fit of the thermodynamic properties to the Monte Carlo results. In the present calculation, the pair excitation spectra are obtained using the analytic path-integral GPF approach without any fit.
In this section we consider the thermodynamic functions of the cold Fermi gas within the model of fermionic and pair excitations. The grand-canonical thermodynamic potential is the sum of the saddle-point and pair excitation contributions$$\Omega=\Omega_{sp}+\Omega_{p}. \label{W}$$ The saddle-point thermodynamic potential $\Omega_{sp}$ is given by Eq. (\[Wsp\]). The contribution of pair excitations is [@Salasnich2010]$$\Omega_{p}=\frac{V}{\beta}\int\frac{d\mathbf{q}}{\left( 2\pi\right) ^{3}}\ln\left( 1-e^{-\beta\omega_{q}}\right) . \label{Wp}$$
The entropy $S$ is found using its relation to the grand-canonical thermodynamic potential,$$S=-\left. \frac{\partial\Omega}{\partial T}\right\vert _{V,\mu,\zeta
}.\label{S}$$ Using the thermodynamic potential $\Omega$ with (\[Wsp\]) and (\[Wp\]) we find that the entropy is expressed as$$\begin{aligned}
S & =\frac{V\beta^{2}}{3\pi^{2}}\int_{0}^{\infty}\frac{\xi_{k}}{E_{k}}\frac{E_{k}\left( \cosh\beta E_{k}\cosh\beta\zeta+1\right) -\zeta\sinh\beta
E_{k}\sinh\beta\zeta}{\left( \cosh\beta\zeta+\cosh\beta E_{k}\right) ^{2}}k^{4}dk\nonumber\\
& +\frac{V}{2\pi^{2}}\int_{0}^{\infty}\left( \beta\omega_{q}\frac
{e^{-\beta\omega_{q}}}{1-e^{-\beta\omega_{q}}}-\ln\left( 1-e^{-\beta
\omega_{q}}\right) \right) q^{2}dq.\label{S1}$$ Finally, the internal energy $E$ of cold fermions is calculated using the relation between $E$ and the grand-canonical thermodynamic potential, $E=\Omega+TS+\mu N,$where $N$ is the total number of fermions.
In Fig. 3, the internal energy per particle for a unitary Fermi gas calculated in different approaches is plotted as a function of temperature for the broken-symmetry phase at $T\leqslant T_{c}$. The critical temperature determined within the path-integral GPF model is the same as within NSR, $T_{c}\approx0.225E_{F}/k_{B}$. The results of the present calculation within the model of fermionic and low-lying pair excitations with parameters determined using the path-integral GPF and NSR methods are shown with short-dashed and dot-dashed curves, respectively. The other results represented in Fig. 3 are (after Ref. [@Salasnich2010]): the internal energy calculated within the low-temperature fermion-boson (FB) model [@Salasnich2010], and the result of the analytic model proposed by Bulgac, Drut, and Magierski (BDM) [@Bulgac]. The analytic results are compared with those of the Monte Carlo calculations from Ref. [@Bulgac] and with the experimental data of Ref. [@Horikoshi] for a gas of $^{6}$Li atoms at unitarity.
\[h\]
[SI-Fig3.EPS]{}
As reported in Ref. [@Salasnich2010], the low-temperature fermion-boson model works well in the broken-symmetry phase where the internal energy resulting from this model is close to the Monte Carlo data of Ref. [@Bulgac]. The present study is performed with the parameters of the elementary excitations obtained using the analytic approaches rather than a fit to the Monte Carlo simulations. The internal energy calculated in the present work with the parameters determined using the path-integral GPF method is very close to the Monte Carlo results at low temperatures $T\lessapprox
0.55T_{c}$. Furthermore, our result is in good agreement with the experiment [@Horikoshi] in the whole temperature range below $T_{c}$.
Neither the model of fermionic and pair excitations used in the present work, nor the low-temperature fermion-boson model of Ref. [@Salasnich2010] predicts the superfluid phase transition: the critical temperature $T_{c}$ is determined within the path-integral GPF method before the long-wavelength expansion is performed in Sec. \[sec:theory\]. However, the latter model can describe well the broken-symmetry phase of cold fermionic atoms.
Superfluid density
------------------
The total density of cold fermions within the model of fermionic and pair excitations is given by the sum of fermion and boson contributions$$n=\frac{1}{2\pi^{2}}\int_{0}^{\infty}k^{2}dk\left( 1-\frac{\xi_{k}}{E_{k}}X_{k}\right) +\frac{1}{\pi^{2}}\int_{0}^{\infty}q^{2}dq\frac{1}{e^{\beta\omega_{q}}-1}.\label{n}$$ The total density is a sum of the normal and superfluid densities $n=n_{n}+n_{s}$. The superfluid density $n_{s}$, as well as the total density, is constituted by the saddle-point result for an imbalanced Fermi gas and the contribution due to the pair excitations, $$\begin{aligned}
n_{s} & =\frac{1}{2\pi^{2}}\int_{0}^{\infty}\left( 1-\frac{\xi_{k}}{E_{k}}X_{k}-k^{2}Y_{k}\right) k^{2}dk\nonumber\\
& +\frac{1}{\pi^{2}}\int_{0}^{\infty}\left( \frac{1}{e^{\beta\omega_{q}}-1}-\frac{\beta}{3}q^{2}\frac{e^{-\beta\omega_{q}}}{\left( e^{-\beta
\omega_{q}}-1\right) ^{2}}\right) q^{2}dq,\label{nsp}$$ where the function $Y_{k}$ is given by$$Y_{k}\equiv\frac{\partial X_{k}}{\partial E_{k}}=\beta\frac{\cosh\beta
E_{k}\cosh\beta\zeta+1}{\left( \cosh\beta E_{k}+\cosh\beta\zeta\right) ^{2}}$$ (see the analogous expression for the Fermi gas in 2D in Ref. [@TKD2009]). In the balanced case, the superfluid density becomes equivalent to the corresponding expression of Ref. [@Salasnich2010], but with other values of the parameters of the pair excitations, as discussed above and shown in Fig.1.
\[h\]
[SI-Fig4.EPS]{}
In Fig. 4, the superfluid density divided by the total density calculated within the model of fermionic and pair excitations is plotted as a function of temperature. As seen from the figure, the superfluid density calculated using the parameters of the pair excitations obtained within the NSR model exhibits a re-entrant behavior above $T_{c}$ similarly to the sound velocity parameter in Fig. 1. The analogous bend-over of the superfluid density above $T_{c}$ was reported in the full NSR approach in Ref. [@Fukushima2007].
Sound velocities
----------------
We consider the sound propagation in a superfluid Fermi gas using the approach of the two-fluid hydrodynamics [@Landau; @Khalatnikov] in the same way as in Ref. [@Salasnich2010]. The first sound velocity $u_{1}$ in the two-fluid hydrodynamics is determined by the formula$$u_{1}=\left( 2\left. \frac{\partial P}{\partial n}\right\vert _{\bar{S},V}\right) ^{1/2}, \label{u1}$$ with the entropy per particle $\bar{S}=S/N$. The pressure is proportional to the grand-canonical thermodynamic potential: $P=-\Omega/V$. We adopt the expression [@Taylor2009] $$\left. \frac{\partial P}{\partial n}\right\vert _{\bar{S},V}=\frac{5}{3}\frac{P}{n} \label{dP}$$ and use the grand-canonical thermodynamic potential given by Eq. (\[W\]) with (\[Wsp\]) and (\[Wp\]).
The second sound velocity $u_{2}$ characterizes the temperature waves in which the motion of the normal and superfluid fractions is out-of-phase. It is determined by the formula$$u_{2}=\left( \frac{2\bar{S}^{2}}{\left. \frac{\partial\bar{S}}{\partial
T}\right\vert _{N,V}}\frac{n_{s}}{n_{n}}\right) ^{1/2}. \label{u2}$$
The formulae (\[u1\]) and (\[u2\]) are valid as far as the first-sound and second-sound modes are decoupled. Following Ref. [@Salasnich2010], we assume that the above condition is fulfilled for a cold Fermi gas.
\[h\]
[SI-Fig5.EPS]{}
In Fig. 5, the first and second sound velocities (divided by the zero-temperature Fermi velocity $v_{F}$) are plotted as a function of temperature. They are calculated within the model of fermionic and pair excitations using the parameters of pair excitations determined by the path-integral GPF and NSR methods.
The first and second sound velocities obtained using the model of fermionic and pair excitations are in a reasonable agreement with the results of the analysis based on the full NSR thermodynamics [@Taylor2009; @Arahato]. In the zero-temperature limit, the first sound velocity tends to the same limit as the sound velocity parameter $c$ for pair excitations, which is extremely close to the Monte Carlo data [@MC1; @MC1a; @MC2; @MC2a; @MC3].
Conclusions \[sec:comclusions\]
===============================
In the present work, the GPF modification [@Drum; @Drum2] of the NSR scheme has been formulated in the path-integral representation and extended to the case of imbalanced fermions. Within this path-integral GPF approach, we have analytically derived the spectra of low-lying pair excitations of the imbalanced Fermi gas with $s$-wave pairing at finite temperatures and extracted the parameters $c$ and $\lambda$ for the pair excitations from these results. Using these spectra, the finite-temperature thermodynamics of the Fermi gas in the superfluid state has been analyzed. The obtained internal energy demonstrates a good agreement with the Monte Carlo results and is remarkably close to the experimental data for the Fermi gas at unitarity. The zero-temperature value of the first sound velocity is in a good agreement with the results of the Monte Carlo simulations. The present method allows us to obtain the spectra of the elementary excitations and, consequently, the thermodynamic parameters of the state for an arbitrary scattering length, at non-zero temperatures, and for non-zero imbalance.
The authors gratefully acknowledge discussions with L. Salasnich. This work was funded by the Fonds voor Wetenschappelijk Onderzoek-Vlaanderen (FWO-V) projects G.0356.06, G.0370.09N, G.0180.09N, and G.0365.08. JPAD acknowledges financial support in the form of a Ph.D. Fonds voor Wetenschappelijk Onderzoek-Vlaanderen (FWO-V).
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|
---
author:
- |
J. Klusoň$^a$, Bum-Hoon Lee $^b$, Kamal L. Panigrahi $^c$ and Chanyong Park $^b$\
$^a$ Department of Theoretical Physics and Astrophysics\
Faculty of Science, Masaryk University\
Kotlářská 2, 611 37, Brno, Czech Republic\
$^b$ Center for Quantum Spacetime (CQUeST), Sogang University,\
Seoul 121-742, Korea\
$^c$ Department of Physics, Indian Institute of Technology Guwahati,\
Guwahati-781 039, India\
E-mail:
title: 'Magnon like solutions for strings in I-brane background'
---
Introduction and Summary {#first}
========================
AdS/CFT correspondence relates the spectrum of free strings on AdS$_5\times$ S$^5$ with that of the spectrum of operator dimensions in the ${\cal N}=4$ super Yang-Mills (SYM) in four dimensions. This mapping is highly nontrivial and challenging. A better understanding of this mapping will be to look at both the gauge and gravity theories at certain limits such as large angular momentum limit and then compare the spectrum. In the understanding the gauge/gravity duality, an interesting observation is that the ${\cal N}=4$ SYM theory can be described by the integrable spin chain model where the anomalous dimension of the gauge invariant operators were found . It was further noticed that the string theory also has as integrable structure in the semiclassical limit and the anomalous dimension in the ${\cal N}=4$ SYM can be derived from the relation between conserved charges of the worldsheet solitonic string solution on the dual string theory on AdS$_5 \times$ S$^5$. In this connection, Hofman and Maldacena (HM) considered a special limit where the problem of determining the spectrum of both sides becomes rather simple. The spectrum consists of an elementary excitation known as magnon which propagate with a conserved momentum $p$ along the long spin chain. In the dual formulation, the most important ingredient is semiclassical string solutions, which can be mapped to long trace operator with large energy and large angular momenta. Once this connection was established, there was a lot of work devoted to understanding of this correspondence (see for example[@Dorey:2006dq]-[@David:2008yk]).
To understand the AdS/CFT like correspondence in a more general background, it is interesting to find out dispersion relation among various conserved charges in case of classical rotating strings in the gravity side and then to look for the corresponding operators in the dual theory. For example, the magnon like dispersion relation in NS5-brane background was considered in [@Kluson:2007qu].
An interesting background configuration is the I-brane background [@Itzhaki:2005tu], which is the intersection of two stacks of NS5-branes in type II string theory on $R^{1,1}$. When all the five branes are coincident, the near horizon geometry of this configuration is given by [@Itzhaki:2005tu] [^1] R\^[2,1]{} R\_ SU(2)\_[k\_1]{} SU(2)\_[k\_2]{}, \[i-brane\]where $R_{\ph}$ is one combination of two radial directions away from two sets of NS5-branes, $k_1$ and $k_2$ are the number of NS5-branes in each stack. The coordinates of $R^{2,1}$ are $x^0,
x^1$ and one more combination of two radial directions. The two $SU(2)$s corresponding to two angular three spheres corresponding to $(R^4)_{2345}$ and $(R^4)_{6789}$. The very fact that (\[i-brane\]) is an exact solution to string equations of motion allows us to obtain information about the intersecting brane system more, which is not accessible via gauge theory. As it is clear from (\[i-brane\]) the exhibits a higher symmetry than the full intersecting brane configuration. In particular, the combination of radial directions away from the intersection that enters $R^{2,1}$ appears symmetrically with the other spatial directions, and the background has a higher Poincare symmetry, $ISO(2, 1)$, than the expected $ISO(1, 1)$. The holographic mapping between field theory living on the I-brane and the corresponding bulk theory was studied in [@Itzhaki:2005tu] [^2]. On the other hand it is also well known from the study of AdS/CFT correspondence that it is possible to derive more information about the boundary CFT theory from the study of the semiclassical string and D1-brane configurations in the bulk of $AdS_5$ [^3].
Motivated by the recent developments in the AdS/CFT correspondence, we will investigate the classical string dynamics in the I-brane background to understand the rotating string solutions. These string solutions correspond to multispin string solutions on the I-brane background and solitonic string solution in the worldsheet point of view. Due to the lack of sufficient knowledge about the theory on the worldvolume of the I-brane we can’t compare them with the dual theory. However the knowledge about the semiclassical rotating string on the I-brane certainly gives information about the possible nature of operators in some dual theory.
In this background, the solitonic string solution which are static and uniformly wrapping the two transverse spheres, and the dispersion relation among various conserved charges were studied in . Here, we will generalize this solution to the rotating one in the transverse spheres and find the dispersion relation among various charges. To do so, we will first solve the equations of motion for a string rotating simultaneously on both the spheres and then by using the Virasoro constraints the dispersion relation will be expressed in terms of various conserved charges that the background obeys in general. Usually, in case of the giant magnon on $R
\times S^3$, the energy and one of the angular momenta for one giant magnon are infinite but their difference is finite. In the I-brane background, on the other hand, which has also $R
\times S^3$ or $R \times S^3 \times
S^3$, the string soliton is composed of multiple magnon-like solutions and one of them, we call this magnon-like solution, has a similar shape to the giant magnon on $R \times S^3$. However, the dispersion relation for magnon-like one is different in that it contains additional linear momenta in two radial directions and is described by the finite conserved charges. We would like to note that on the I-brane background all conserved charges are regularized, so have finite values. If we choose the range of the world sheet string coordinate as $-\infty < \s <
\infty$ like HM case or $-\pi/2
< \s < \pi/2$ , the string soliton becomes a combination of infinite magnon-like solutions or finite numbers. For the closed string case, since $-\pi < \s < \pi$ the solitonic string is given by the finite numbers of the magnon-like solutions.
We would like to mention that in general getting magnon/spike solutions in I-brane background is rather cumbersome. However, we will present in this paper, a parameter space of solutions where there exists a magnon like shape when we restrict the motion of the string along both the spheres. It would be certainly interesting to find out more general rotating string solutions in this backgrounds. A greater challenge will be to find out the dual operators in the worldvolume theory which correspond to the semiclassical string solutions presented in this paper.
The rest of the paper is organized as follows. In section 2, we present the background solution corresponding to the intersection of two five-branes on $R^{1,1}$ in type IIB string theory and study the equations of motion and Virasoro constraints of the fundamental string rotating simultaneously along both the spheres. Section-3 devoted to the study the rotating string solution interpreted as a single magnon like solution while the motion is restricted to only one sphere. We present the corresponding dispersion relation along various charges and interpret that solution as a single magnon solution. Further we present a more general solution when the string moves simultaneously along both the spheres. We analyze the results in a particular parameter space of solutions. Finally in section-4, we present our conclusions.
F-string in the background of I-brane {#third}
=====================================
In this section we will study the dynamics of fundamental string in the background studied in [@Itzhaki:2005tu]. Namely, we consider the intersection of two stack of NS5-branes on $R^{1,1}$. We have $k_1$ number of NS5-branes extended in $(0,1,2,3,4,5)$ directions and another set of $k_2$ number of NS5-branes extended in $(0,1,6,7,8,9)$ directions. Let us define $$\begin{aligned}
\by=(x^2,x^3,x^4,x^5) \ , \nonumber \\
\bz=(x^6,x^7,x^8,x^9) \ .\end{aligned}$$ We have $k_1$ NS5-branes localized at the points $\bz_n \
n=1,\dots,k_1 , $ and $k_2$ NS5-branes localized at the points $\by_a \ , a=1\dots,k_2$. Every pairs of fivebranes from different sets intersect at different point $(\by_a,\bz_n)$. The supergravity background corresponding to this configuration takes the form $$\begin{aligned}
\label{bg}
\Phi(\bz,\by) &=& \Phi_1(\bz)+
\Phi_2(\by) \ , \nonumber \\
g_{\mu\nu}&=&\eta_{\mu\nu} \ ,
\ (\mu,\nu=0,1) \ , \nonumber \\
g_{\alpha\beta}&=&e^{2(\Phi_2-
\Phi_2(\infty))}\delta_{\alpha\beta} , \nn
\mathcal{H}_{\alpha\beta\gamma}&=&
-\epsilon_{\alpha\beta\gamma\delta}
\partial^\delta \Phi_2 \ , \ (\alpha,\beta,\gamma,\delta=
2,3,4,5 ) \ , \nonumber \\
g_{pq}&=&e^{2(\Phi_1-\Phi_1(\infty))}
\delta_{pq} \ , \nn
\mathcal{H}_{pqr}&=&-\epsilon_{pqrs}
\partial^s\Phi_1 \ , \ \ ( p,q,r,s=6,7,8,9 )\ ,\end{aligned}$$ where $\Phi$ on the first line means the dilaton and where $$\begin{aligned}
e^{2(\Phi_1-
\Phi_1(\infty))}=1+
\sum_{n=1}^{k_1}
\frac{l_s^2}{|\bz-\bz_n|^2} \ ,
\nonumber \\
e^{2(\Phi_2-
\Phi_2(\infty))}=1+\sum_{a=1}^{k_2}
\frac{l_s^2}{|\by-\by_a|^2} \ .\end{aligned}$$ Our goal is to find solutions for rotating string in this background when $\bz_n=\by_a=0$. To simplify our notation let us denote $$e^{2(\Phi_1-\Phi_1(\infty))}=H_1(\bz)\ , \ \ \
e^{2(\Phi_2-\Phi_2(\infty))}=H_2(\by) \ ,$$ where for coincident branes we have $$H_1=1+\frac{k_1l_s^2} {|\bz|^2} \ , \ \ \
H_2=1+\frac{k_2l_s^2} {|\by|^2} \ .$$ Let us now consider the probe brane in the near horizon limit where $$\frac{k_1l_s^2}
{|\bz|^2}\gg 1 \ , \ \ \
\frac{k_2l_s^2}
{|\by|^2}\gg 1 $$ so that we can write $$H_1=\frac{\lambda_1} {r^2_1} \ , \quad
\lambda_1=k_1l_s^2 \ , \quad
H_2=\frac{\lambda_2} {r^2_2} \ , \quad
\lambda_2=k_2l_s^2 \ .$$ Then the metric takes the form $$\begin{aligned}
\label{NS5bac}
ds^2=-dt^2+\frac{\lambda_1}{r_1^2}dr_1^2+
\frac{\lambda_2}{r_2^2}dr_2^2+
\lambda_1d\Omega^{(3)}_1+
\lambda_2d\Omega^{(3)}_2 \ ,
$$ where $d\Omega_1^{(3)}$ and $d\Omega_2^{(3)}$ correspond to the line elements on the unit sphere. To describe them better we introduce the following coordinates $$\begin{aligned}
x^2+ix^3&=&r_1 \cos
\theta_1e^{i\phi_1}\ ,
\quad
x^4+ix^5=r_1\cos \theta_1
e^{i\psi_1}
\ ,
\nonumber \\
x^6+ix^7&=&r_2 \cos
\theta_2e^{i\phi_2}
\ , \quad
x^8+ix^9=r_2\cos \theta_2
e^{i\psi_2} \nonumber \\
$$ so that $$\begin{aligned}
\la{NS5bacd}
d\Omega^{(3)}_1&=& d\theta^2_1+
\sin^2\theta_1 d \phi_1^2+
\cos^2\theta_1 d\psi_1^2 , \nonumber \\
b_{\phi_1\psi_1}&=&
\lambda_1\cos^2\theta_1 \ ,
\quad 0 < \theta_1 < \frac{\pi}{2} \
, \quad 0 = \phi_1,\psi_1 <
2\pi \ ,
\nonumber \\
d\Omega^{(3)}_2&=& d\theta^2_2+
\sin^2\theta_2 d \phi_2^2+
\cos^2\theta_2 d\psi_2^2 , \nonumber \\
b_{\phi_2\psi_2}&=&
\lambda_2\cos^2\theta_2 \ ,
\quad 0 < \theta_2 < \frac{\pi}{2} \ ,
\quad
0 = \phi_2,\psi_2 < 2\pi \ .\end{aligned}$$ As usual our starting point is the Polyakov form of the string action in the background (\[NS5bac\]) $$\begin{aligned}
\label{actPol}
S&=&-\frac{1}{4\pi \alpha'}
\int_{-\pi/2}^{\pi/2} d\sigma d\tau
[\sqrt{-\gamma}\gamma^{\alpha\beta}
g_{MN}\partial_\alpha x^M\partial_\beta x^N
-e^{\alpha\beta}
\partial_\alpha x^M\partial_\beta x^N b_{MN}]+
\nonumber \\
&+&\frac{1}{4\pi}\int_{-\pi/2}^{\pi/2}
d\sigma d\tau \sqrt{-\gamma}
R\Phi \ ,\end{aligned}$$ where $\gamma^{\alpha\beta}$ is a world-sheet metric and $R$ is its Ricci scalar. Further, $e^{\alpha\beta}$ is defined as $e^{01}=-e^{10}=1$. Finally, the modes $x^M, M=0,\dots,9$ parameterize the embedding of the string in the background (\[NS5bac\]). The variation of the action (\[actPol\]) with respect to $x^M$ implies following equations of motion $$\begin{aligned}
-\frac{1}
{4\pi\alpha'}\sqrt{-\gamma}\gamma^{\alpha\beta}
\partial_K
g_{MN}\partial_\alpha x^M\partial_\beta x^N
+\frac{1}{2\pi\alpha'}\partial_\alpha[
\sqrt{-\gamma}\gamma^{\alpha\beta}
g_{KM}\partial_\beta x^M]-
\nonumber \\
-\frac{1}{2\pi\alpha'}
\partial_\alpha[
\epsilon^{\alpha\beta}
\partial_\beta x^M b_{KM}]
+\frac{1}{4\pi\alpha'}
\epsilon^{\alpha\beta}
\partial_\alpha x^M\partial_\beta x^N
\partial_K b_{MN}
+\frac{1}{4\pi}\partial_K \Phi
\sqrt{-\gamma}R=0 \ . \nonumber \\\end{aligned}$$ Finally the variation of the action with respect to the metric implies the constraints $$\begin{aligned}
\label{gravcons}
-\frac{4\pi}{\sqrt{-\gamma}}
\frac{\delta S}{\delta \gamma^{\alpha
\beta}}&=&
\frac{1}{\alpha'}
g_{MN}\partial_\alpha x^M\partial_\beta x^N-R_{\alpha
\beta}+
\nonumber \\
&+&(\nabla_\alpha \nabla_\beta x^M)
\partial_M \Phi+(\partial_\alpha x^M\partial_\beta
x^N)\partial_M\partial_N\Phi
-\nonumber \\
&-&\frac{1}{2}
\gamma_{\alpha\beta}
\left(\frac{1}{\alpha'}
\gamma^{\gamma\delta}
\partial_\gamma x^M\partial_\delta
x^N g_{MN}-R\Phi+2 \nabla^\alpha
\nabla_\alpha \Phi\right) \ . \nonumber
\\\end{aligned}$$ As the first step let us introduce two modes $\rho_1$ and $\rho_2$ defined through the relations $$\label{RSp}
r_1=e^{\frac{\rho_1}{\sqrt{\lambda_1}}
}\ , \quad
r_2=e^{\frac{\rho_2}{\sqrt{\lambda_2}}}
\ .$$ Then, following [@Itzhaki:2005tu] we introduce two modes $ r,
y$ through the relation $$\label{phix2}
Qr=\frac{1}{\sqrt{\lambda_1}}\rho_1+
\frac{1}{\sqrt{\lambda_2}}\rho_2 \ ,
\quad
Qy=\frac{1}{\sqrt{\lambda_2}}\rho_1-
\frac{1}{\sqrt{\lambda_1}}\rho_2 \ ,$$ where $$Q=\frac{1}{\sqrt{\lambda}} \ ,
\frac{1}{\lambda}=\frac{1}{\lambda_1}+
\frac{1}{\lambda_2} \ .$$ Note that the inverse transformations of (\[phix2\]) take the forms $$\begin{aligned}
\label{phili}
\rho_1=\frac{1}{\sqrt{\lambda_1+\lambda_2}}
\left(\sqrt{\lambda_1}y+\sqrt{\lambda_2}r
\right) \ , \nonumber \\
\rho_2=\frac{1}{\sqrt{\lambda_1+\lambda_2}}
\left(\sqrt{\lambda_1}r-
\sqrt{\lambda_2}y\right) \ .\end{aligned}$$ Note that this result implies that the dilaton is a function of $r$ only $$\begin{aligned}
\Phi&=&\Phi_1+\Phi_2=
\frac{1}{2}(H_1+H_2)+\Phi_1(\infty)+
\Phi_2(\infty)=\nonumber \\
&=& -\frac{1}{\sqrt{\lambda_1}}\rho_1
-\frac{1}{\sqrt{\lambda_2}}\rho_2+\Phi_0=-Qr+\Phi_0
\ .\end{aligned}$$ With the help of the variables $r,y$ the action for string in $I$-brane background takes the form $$\begin{aligned}
\label{actPol2}
S&=&-\frac{1}{4\pi \alpha'}
\int_{-\pi/2}^{\pi/2} d\sigma d\tau
[\sqrt{-\gamma}\gamma^{\alpha\beta}
(-\partial_\alpha t\partial_\beta t
+\partial_\alpha r\partial_\beta r+
\partial_\alpha y\partial_\beta y+\nonumber \\
&+&g_{mn}\partial_\alpha
x^m\partial_\beta x^n -e^{\alpha\beta}
\partial_\alpha x^m\partial_\beta x^n b_{mn}]-
\frac{1}{4\pi}\int_{-\pi/2}^{\pi/2} d\sigma
d\tau \sqrt{-\gamma} RQr \ ,\end{aligned}$$ where $x^{m,n}$ label angular coordinates corresponding to $S^3_1,S^3_2$ respectively.
Looking at the form of the background (\[NS5bac\]) and (\[NS5bacd\]) we observe that the action (\[actPol2\]) is invariant under following transformations of fields $$\begin{aligned}
t'(\tau,\sigma)&=&t(\sigma,\tau)+\epsilon_t \ ,
\nonumber \\
y'(\tau,\sigma)&=&y(\tau,\sigma)+\epsilon_y \ ,
\nonumber \\
\psi'_{1}(\tau,\sigma)&=&
\psi_{1}(\tau,\sigma)+\epsilon_{\psi_1} \ ,
\nonumber \\
\psi'_{2}(\tau,\sigma)&=&
\psi_{2}(\tau,\sigma)+\epsilon_{\psi_2} \ ,
\nonumber \\
\phi'_{1}(\tau,\sigma)&=&
\phi_{1}(\tau,\sigma)+\epsilon_{\phi_1} \ ,
\nonumber \\
\phi'_{2}(\tau,\sigma)&=&
\phi_{2}(\tau,\sigma)+\epsilon_{\phi_2} \ ,\end{aligned}$$ where $\epsilon_t,\epsilon_y,\epsilon_{\phi_1},\epsilon_{\phi_2},
\epsilon_{\psi_1},\epsilon_{\psi_2}$ are constants. Then it is straightforward to determine corresponding conserved charges $$\begin{aligned}
\label{CGcon}
P_t&=&-\frac{1}{2\pi\alpha'}
\int_{-\pi/2}^{\pi/2} d\sigma
\sqrt{-\gamma}\gamma^{\tau\alpha}
\partial_\alpha t \ , \nonumber \\
P_{\psi_1}&=&
\frac{1}{2\pi\alpha'}
\int_{-\pi/2}^{\pi/2} d\sigma
[\sqrt{-\gamma}\gamma^{\tau\alpha}
g_{\psi_1\psi_1}\partial_\alpha \psi_1
-\partial_\sigma \phi_1 b_{\phi_1\psi_1}] \ ,
\nonumber \\
P_{\psi_2}&=&
\frac{1}{2\pi\alpha'}
\int_{-\pi/2}^{\pi/2} d\sigma
[\sqrt{-\gamma}\gamma^{\tau\alpha}
g_{\psi_2\psi_2}\partial_\alpha \psi_2
-\partial_\sigma \phi_2 b_{\phi_2\psi_2}] \ ,
\nonumber \\
P_{\phi_1}&=&
\frac{1}{2\pi\alpha'}
\int_{-\pi/2}^{\pi/2} d\sigma
[\sqrt{-\gamma}\gamma^{\tau\alpha}
g_{\psi_1\psi_1}\partial_\alpha \psi_1
+\partial_\sigma \psi_1 b_{\phi_1\psi_1}] \ ,
\nonumber \\
P_{\phi_2}&=&
\frac{1}{2\pi\alpha'}
\int_{-\pi/2}^{\pi/2} d\sigma
[\sqrt{-\gamma}\gamma^{\tau\alpha}
g_{\psi_2\psi_2}\partial_\alpha \psi_2
+\partial_\sigma \psi_2 b_{\phi_2\psi_2}] \ ,
\nonumber \\
P_{y}&=&
\frac{1}{2\pi\alpha'}
\int_{-\pi/2}^{\pi/2} d\sigma
[\sqrt{-\gamma}\gamma^{\tau\alpha}
g_{yy}\partial_\alpha y] \ .\end{aligned}$$ Note that $P_t$ is related to the energy as $P_t=-E$. In [@Kluson:2007st] the homogeneous string and D1-brane solutions in I-brane background have been studied [^4]. It was argued that it is necessary to find the configuration when string moves on both two spheres simultaneously. For that reason we have to consider an ansatz where string moves simultaneously on both the spheres $S^{3}_1$ and $S^{3}_2$ as follows $$\begin{aligned}
t&=&t(\tau), \quad r= r(\tau), \quad
y=y(\tau), \quad
\nonumber \\
\theta_1&=&\theta_1(m) \ , \quad
\psi_1=\omega_1\tau+g_1(m) \ , \quad
\phi_1=\nu_1\tau + h_1(m) \ ,
\nonumber \\
\theta_2&=&\theta_2(m) \ , \quad
\psi_2=\omega_2\tau+g_2(m) \ , \quad
\phi_2=\nu_2\tau + h_2(m) \ , \nonumber
\\\end{aligned}$$ where $$\label{defm}
m=\alpha \sigma+\beta \tau$$ and we also consider solution in the conformal gauge $\gamma^{\alpha\beta}=
\eta^{\alpha\beta}$. In this gauge the constraints (\[gravcons\]) that now follow from the variation of the action (\[actPol2\]) take simpler forms $$\begin{aligned}
\label{Tcon}
T_{\sigma\sigma}&=&-4\pi \frac{\delta
S}{\delta \gamma^{\sigma \sigma}}=
\frac{1}{2\alpha'}
(g_{MN}\partial_\sigma
x^M\partial_\sigma x^N
+g_{MN}\partial_\tau x^M\partial_\tau
x^N)- Q\partial_\tau^2 \rho \ ,
\nonumber \\
T_{\tau\tau}&=&-4\pi \frac{\delta
S}{\delta \gamma^{\tau \tau}}=
\frac{1}{2\alpha'}
(g_{MN}\partial_\sigma
x^M\partial_\sigma x^N
+g_{MN}\partial_\tau x^M\partial_\tau
x^N)- Q\partial_\sigma^2 \rho \ ,
\nonumber \\
T_{\tau\sigma}&=&-4\pi \frac{\delta
S}{\delta \gamma^{\sigma\tau }}=
\frac{1}{\alpha'} g_{MN}\partial_\sigma
x^M\partial_\tau x^N
-Q\partial_\sigma\partial_\tau \rho \ .\end{aligned}$$ Further, in conformal gauge the equations of motion for $t,y,r$ take the form $$\begin{aligned}
\partial_\alpha[\eta^{\alpha\beta}\partial_\beta
t]=0 \ , \quad
\partial_\alpha[\eta^{\alpha\beta}\partial_\beta
r]=0 \ , \quad
\partial_\alpha[\eta^{\alpha\beta}\partial_\beta
y]=0 \\end{aligned}$$ that have solutions $$t=\kappa \tau \ , \quad r=v_r \tau+r_0
\ , \quad y=v_y\tau+y_0 $$ for constants $\kappa,v_y,v_r,r_0,y_0$.
Now we are going to study the dynamics of fundamental strings on both three $S^{3}_1,S^{3}_2$. We start then with the equations of motion for $\phi_1$ and $\psi_1$. It can be easily shown that these equations imply two differential equations $$\begin{aligned}
\label{hg1}
h_1'&=&\frac{1}
{\lambda_1\sin^2\theta_1(\alpha^2-\beta^2)}
(C_1-\lambda_1\alpha\omega_1\cos^2\theta_1+\lambda_1\beta
\nu_1 \sin^2\theta_1) \ , \nonumber \\
g'_1&=&\frac{1}{
(\alpha^2-\beta^2)\lambda_1\cos^2\theta_1}
(D_1+\lambda_1\beta\omega_1\cos^2\theta_1+
\lambda_1\nu_1\alpha \cos^2\theta_1) \
.
\nonumber \\\end{aligned}$$ where $h'(m)=\frac{dh}{dm}$ and where $C_1,D_1$ are integration constants. In the same way we consider the equation of motion for $\psi_2,\phi_2$ with the result $$\begin{aligned}
\label{hg2}
h'_2&=&\frac{1}{(\alpha^2-\beta^2)
\lambda_2\sin^2\theta_1} (C_2+\lambda_2
\sin^2\theta_2 \beta \nu_2
-\omega_2\alpha
\lambda_2\cos^2\theta_2) \ , \nonumber \\
g'_2&=&\frac{1}{
(\alpha^2-\beta^2)\lambda_2\cos^2\theta_2}
(D_2+\lambda_2\beta\omega_2\cos^2\theta_2+
\lambda_2\nu_2\alpha \cos^2\theta_2) \
, \nonumber \\\end{aligned}$$ where again $C_2,D_2$ are integration constants. Let us now start to solve the Virasoro constraints for the given model. The constraint $T_{\tau\sigma}=0$ implies $$\begin{aligned}
& & g_{\phi_1\phi_1}\alpha
h'_1(\nu_1+\beta h'_1)+
g_{\psi_1\psi_1}\alpha
g'_1(\omega_1+\beta g'_1)
+g_{\theta_1\theta_1}\alpha \beta
\theta'^2_1 +\nonumber \\
&& + g_{\phi_2\phi_2}\alpha
h'_2(\nu_2+\beta h'_2)+
g_{\psi_2\psi_2}\alpha
g'_2(\omega_2+\beta g'_2) +g_{\theta_2
\theta_2}\alpha\beta\theta'^2_2=0 \ .
$$ On the other hand the Virasoro constraints $T_{\tau\tau}=T_{\sigma\sigma}=0$ take the form $$\begin{aligned}
0 &=& g_{\phi_1\phi_1}[(\nu_1+\beta
h'_1)^2+\alpha^2h'^2_1]+
g_{\psi_1\psi_1}[(\omega_1+\beta
g'_1)^2+\alpha^2 g'^2_1]+
g_{\theta_1\theta_1}(\alpha^2+\beta^2)\theta'^2_1+
\nonumber \\
&& g_{\phi_2\phi_2}[(\nu_2+\beta h'_2)^2+
\alpha^2 h'^2_2]+ g_{\psi_2\psi_2}
[(\omega_2+\beta g'_2)^2+ \alpha^2
g'^2_2]+g_{\phi_2\phi_2}
(\alpha^2+\beta^2)\theta'^2_2
-\nonumber \\
&& -\kappa^2+v_r^2+v_y^2 \ . \nonumber \\\end{aligned}$$ Now, if we combine these constraints as $-\frac{(\alpha^2+\beta^2)}{\alpha\beta}T_{\sigma\tau}
+T_{\tau\tau}$ we obtain $$\label{const}
0= -\nu_1 C_1-\omega_1 D_1
-\nu_2C_2-\omega_2 D_2
+\beta(-\kappa^2+v_r^2+v_y^2) \ .$$ Then if we use (\[hg1\]), (\[hg2\]) together with (\[const\]) in the constraint $T_{\tau\tau}=0$ we obtain $$\begin{aligned}
\label{mas1}
&& (\beta^2+\alpha^2)(\lambda_1\theta'^2_1+
\lambda_2\theta'^2_2) \nn
&& =
\frac{(\alpha^2+\beta^2)^2}
{(\alpha^2-\beta^2)^2}(
\kappa^2-v_r^2-v_y^2)-\frac{(\alpha^2+\beta^2)\alpha^2}{
(\alpha^2-\beta^2)^2}(g_{\phi_1\phi_1}\nu_1^2+
g_{\psi_1\psi_1}\omega_1^2)
(\frac{g_{\phi_1\phi_1}g_{\psi_1\psi_1}+
b_{\phi_1\psi_1}^2}{g_{\phi_1\phi_1}
g_{\psi_1\psi_1}})\nonumber \\
&&-\frac{\alpha^2+\beta^2}
{(\alpha^2-\beta^2)^2}[\frac{C_1^2}{g_{\phi_1\phi_1}}+
\frac{D_1^2}{g_{\psi_1\psi_1}}+
+2\alpha
\frac{b_{\phi_1\psi_1}}
{g_{\psi_1\psi_1}g_{\phi_1\phi_1}}
(D_1\nu_1 g_{\phi_1\phi_1}- C_1
\omega_1 g_{\psi_1\psi_1})]
\nonumber \\
&&- \frac{(\alpha^2+\beta^2)\alpha^2}{
(\alpha^2-\beta^2)^2}(g_{\phi_2\phi_2}\nu_2^2+
g_{\psi_2\psi_2}\omega_2^2)
(\frac{g_{\phi_2\phi_2}g_{\psi_2\psi_2}+
b_{\phi_2\psi_2}^2}{g_{\phi_2\phi_2}
g_{\psi_2\psi_2}})-\nonumber \\
&&-\frac{\alpha^2+\beta^2}
{(\alpha^2-\beta^2)^2}[\frac{C_2^2}{g_{\phi_2\phi_2}}+
\frac{D_2^2}{g_{\psi_2\psi_2}}
+2\alpha
\frac{b_{\phi_2\psi_2}}
{g_{\psi_2\psi_2}g_{\phi_2\phi_2}}
(D_2\nu_2 g_{\phi_2\phi_2}- C_2
\omega_2 g_{\psi_2\psi_2})]\end{aligned}$$ This differential equation determines the most general evolutions of $\theta$’s. As it is clear from the above, solving for general $\theta$ is quite hard. Hence, in what follows we will analyze it in some special situations.
Magnon solutions in $R \times S^{3}_1$
--------------------------------------
Let us start with the situation when $\theta'_2=h'_2=g'_2=0 \ ,
\omega_2=\nu_2=0$. For our convenience, we set $D_1 = \lambda_1 d$, $C_1 = \lambda_1 c$ and $\kappa^2-v_r^2-v_y^2 = \lambda_1 \gamma$. Then, the above equation takes the form $$\begin{aligned}
\theta_1 '^2 &=& \frac{1}{(\alpha^2-\beta^2)^2}
\left[(\alpha^2+\beta^2) \gamma -\alpha^2 (\nu_1^2 - \omega_1^2) - 2 \alpha (d \nu_1 + c\omega_1)
\right. \nonumber \\
&& \qquad \left. - \frac{(\alpha \omega_1-c)^2}{\sin^2 \theta_1}
- \frac{d^2}{\cos^2\theta_1}\right] \ .\end{aligned}$$ Let us write the above differential equation in the following form $$\theta_1'^2= A^2-\frac{B^2}{\sin^2\theta_1}
-\frac{{d'}^2}{\cos^2\theta_1} \ ,$$ where A\^2 &=& , B\^2 &=& , [d’]{}\^2 = . Then \_1’ = A , where \^2\_[max]{} &=& ,\
\^2\_[min]{} &=& . Let us try to find the solution when $\sin^2\theta_{max}=1$. This occurs when $$\begin{aligned}
D_1=0 \ , \quad {\rm with} \
\sin\theta_{min}=\frac{B}{A} \ .\end{aligned}$$ The final form of $\th_1 '$ is $$\theta_1 '=A \frac{\sqrt{\sin^2\theta_1-
\sin^2\theta_{min}}}{\sin\theta_1} \ .$$ Note that the range of $\th_{1,min} \le
\th_1 \le \frac{\pi}{2}$ corresponds to the half of a magnon-like solution. For the convenience, we call one element of the solitonic string solution having the magnon shape as a magnon-like solution. Actually, the solution obtained here is a combination of these magnon-like solution. So the number of the magnon-like solution contained in the solitonic string solution is given by the followings. If the range of the string world sheet is given by $-\frac{\pi}{2} \le \s \le
\frac{\pi}{2}$ for an open string, the number of magnon-like solution is determined by = \_[-/2]{}\^[/2]{} d = \_[\_[1,min]{}]{}\^[/2]{} , where $n$ means the number of the magnon-like solutions. After calculating the last equation, the number of the magnon-like solutions is given by $n= \a A$.
From now on, we will concentrate on the conserved quantities for magnon-like solutions which will give a dispersion relation for one magnon-like solution. Using the definitions of the conserved charges (\[CGcon\]) they can be rewritten as the integral form over $\theta_1 $ P\_t &=& I ,\
P\_y &=& I ,\
P\_r &=& I ,\
P\_[\_1]{} &=& - ( c + \_1 ) I ,\
P\_[PS. \_1]{} &=& - ( I - I’ ) , where I &=& \_[\_[min]{}]{}\^[/2]{} d \_1 = , I’ &=& \_[\_[min]{}]{}\^[/2]{} d \_1 = . The angle difference in the $\ph_1$-direction is given by \_1 = . To obtain the dispersion relation, we first consider the following quantity P\_t\^2 - P\_r\^2 -P\_y\^2 &=& (\^2 -v\_r\^2 -v\_y\^2) I\^2 &=& - I\^2 where the Virasoro constraints are used in the last equation. From the definitions of charges, $\n_1 c \ I^2$ in the above equation can be determined in terms of other charges and the angle difference - I\^2 = . Finally, we obtain the dispersion relation P\_t\^2 - P\_r\^2 -P\_y\^2 &=& &=& , where $T_1 =
\frac{\l_1}{2 \pi \a'}$ and we have identified the angle difference $\D
\ph_1$ with the world sheet momentum $p$.
Magnon solutions on $R\times S^{3}_1 \times S^{3}_2$
----------------------------------------------------
The equations of motion for $\ph_i$ and $\ps_i$ ($i = 1,2$) are summarized as h\_i’ &=& (C\_i- \_i \_i\^2\_i+ \_i \_i \^2\_i) , g\_i ’&=& (D\_i + \_i \_i \^2\_i + \_i \_i \^2\_i ) . ($\ref{mas1}$) becomes \_[i=1]{}\^[2]{} ł\_i \_i ’\^[2]{} = (\^2-v\_r\^2-v\_y\^2) - \_[i=1]{}\^[2]{} K\_i (\_i), where K\_i (\_i) &=& . Without loss of generality, we can set ł\_i \_i ’\^[2]{} + K\_i (\_i) ł\_i \_i , where $\G_i$ are some constants satisfying $\l_1 \G_1 + \l_2 \G_2 =
\frac{\alpha^2+\beta^2}{(\alpha^2-\beta^2)^2}
(\kappa^2-v_r^2-v_y^2)$.
When we set $D_i = 0$ and $C_i = \l_i c_i$, $\th_i'$ are given by \_i’ = A\_i , where A\_i\^2 &=& \_i - (\_i\^2-\_i\^2)+ , \^2 \_[i,min]{} &=& . In this background, the number of the magnon-like solution in each sphere is given by the similar relation in = \_[\_[i,min]{}]{}\^[/2]{} = , where $n_i$ means the number of the magnon-like solution in $i$-th sphere. From the above equation, the number of the magnon becomes $n_i = \a A_i$. If we set the ratio between magnon-like numbers as $r \equiv \frac{n_2}{n_1}$, then this ratio is given by r = . From now on, we set a magnon-like solution as one in the first sphere, which corresponds to $r$ magnon-like solutions in the second sphere.
The conserved charges for a magnon-like solution are P\_t &=& I\_1 ,\
P\_y &=& I\_1 ,\
P\_r &=& I\_1 ,\
P\_[\_i]{} &=& - ( c\_i + \_i ) I\_i ,\
P\_[PS. \_i]{} &=& - ( I\_i - I\_i’ ) ,\
\_i &=& , where I\_1 &=& \_[\_[1,min]{}]{}\^[/2]{} d \_1 = , I\_1’ &=& \_[\_[1,min]{}]{}\^[/2]{} d \_1 = , I\_2 &=& r \_[\_[2,min]{}]{}\^[/2]{} d \_2 = , I\_2’ &=& r \_[\_[2,min]{}]{}\^[/2]{} d \_2 = , with $B_i^2 = (\alpha \omega_i-c_i)^2$. Then the relation (\[const\]) $\kappa^2 -
v_r^2 - v_y^2 =\frac{1}{\b} (-\nu_1 C_1
-\nu_2 C_2)$ can be rewritten in terms of charges P\_t\^2 - P\_r\^2 -P\_y\^2 &=& \_[i=1]{}\^2 &=& \_[i=1]{}\^2 , where $T_i =
\frac{\lambda_i}{2\pi\alpha'}$ with $i=1,2$. This corresponds to the dispersion relation for the string soliton on the I-brane background when the it moves simultaneously on both the spheres, and this does not depend on the previous parameterization, $\G_i$.
Discussion
==========
In this paper we have studied the solutions for rotating strings in the background generated by a 1+1 dimensional intersection of two stacks of five branes in type IIB string theory. We have solved the motion of rotating string in this background and have analyzed the dispersion relation among various conserved charges. We have taken advantage of the fact there exists a parameter space where the motion in the two spheres effectively decoupled, and one could study the single magnon like solution in this background. Knowing the results of the present paper, it would be tempting to study the corresponding states in the dual theory exactly in case of the AdS$_5\times$ S$^5$ background. It would certainly be interesting to check whether these magnon solutions are BPS from the bulk theory view point, which will give us clue about the boundary operators. We wish to come back to this issue in future.
.2in [**Acknowledgements:**]{} KLP would like to thank the hospitality at Institute of Physics, Bhubaneswar, India where a part of this work was done. This work was partially supported by the Science Research Center Program of the Korea Science and Engineering Foundation through the Center for Quantum Spacetime (CQUeST) of Sogang University with grant number R11 - 2005 - 021. The work of JK was supported by the Czech Ministry of Education under Contract No. MSM 0021622409.
[20]{}
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[^1]: See [@Lin:2005nh] for related background and for the gauge theory results.
[^2]: For some relevant works, see [@Hung:2006nn; @Berg:2006ng; @Hung:2006jh; @Grisa:2006tm; @Antonyan:2006pg; @Antonyan:2006qy; @Antonyan:2006vw; @Kluson:2006wa; @Kluson:2005qq; @Kluson:2005eb]
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|
---
author:
- Amr Ahmadain
title: 'A (1+1)-dimensional Lifshitz Weyl Anomaly From a Schr$\mathrm{\ddot{o}}$dinger-invariant Non-relativistic Chern-Simons Action'
---
Introduction
============
Anomalies are symmetries of the classical action that are broken at the quantum level. Gravitational anomalies of one-loop quantum effective actions arise after coupling classical field theories to curved background geometry and integrating out all dynamical fields in the partition function. Since the quantum energy-momentum tensor, by definition, encodes the response of effective actions to infinitesimal variations in the underlying background metric, they are the central objects in studying gravitational anomalies. In particular, a gravitational conformal (or Weyl anomaly) is the statement that the quantum effective action is not invariant under local rescaling of the background metric. The trace of the expectation value of the energy-momentum tensor is the canonical test of whether the theory is Weyl anomalous or not. If the trace is non-zero, the quantum theory suffers a conformal anomaly.
Recently, there has been a considerable level of activity in studying Weyl anomalies in non-relativistic field theories [@Gomes2012; @Boer2012; @Oz2014; @Nardelli2016; @Oz2016; @Filip2016; @Moshe2017; @Mitra2017; @Grinstein2017; @Nardelli2017]. Non-relativistic field theories do not place space and time on an equal footing and thus introduce a degree of anisotropy between them. Lifshitz field theories are important examples of non-relativistic theories which are locally symmetric under foliation-preserving diffeomorphism (FPD), as opposed to full diffeomorphism invariance in relativistic theories of gravity, and *anisotropic* Weyl scaling transformations characterized by a dynamical scaling exponent $z$. In FPD-invariant theories, the spacetime is naturally foliated into *equal-time slices* with a smooth timelike 1-form $n_{\mu}$ *normal* to the foliation leaves which is normalized by the spacetime metric $g_{\mu\nu}$. Along with Schrodinger field theories, they are used to study and characterize several condensed matter systems near or at the quantum critical points [@Henkel94; @Fradkin04; @Troyer11].
Studying quantum anomalies of non-relativistic field theories typically requires coupling to non-relativistic geometries. Newton-Cartan (NC) spacetime with a *torsion* tensor, or torsional NC geometry (TNC) has recently been the focus of intense study. TNC geometry has appeared in different physical setups, for example, in boundary effective actions of non-relativistic holographic theories [@Rollier14; @Rollier14-2; @Obers2015] and in effective field theories of quantum Hall states [@Son13; @Wu2015]. Weyl-invariant field theories coupled to flat NC spacetime were constructed in [@Obers14; @Obers15].
Recently, Weyl anomalies of Lifshitz field theories coupled to NC geometry with *temporal torsion*, where the 1-form $n_{\mu}$ satisfies the Frobenius condition, $n\wedge dn = 0$, have been calculated in several spacetime dimensions and for multiple values of the scaling exponent $z$ by solving the Wess-Zumino consistency condition [@Oz2014]. It was found in [@Oz2014] that while the conformal anomalies of (1+1)-dimensional relativistic conformal field theories are *type-A*, those of Lifshitz field theories belong to *type-B*. Anomalies in the latter class satisfy *trivial descent* equations, or equivalently, consist of Weyl-invariant scalar densities with effective actions that are scale-dependent.
In 1+1 dimensions, which is the focus of this paper, and for *any* value of $z$, only one *trivial descent* anomaly i.e. a trivial descent cocyle modulo a coboundary term, was found in the parity-odd, mixed-derivative sector of the 1+1 Lifshitz cohomology of the relative Weyl operator [@Oz2014]. The rest of the cocycles were shown to be trivial descent *coboundaries* and thus, can be removed by local counterterms. Also, the (1+1)-dimensional Weyl anomaly breaks time-reversal invariance. Using the ADM coordinates, we will illustrate that the 1+1 Lifshitz Weyl anomaly can be directly interpreted as the curvature of the torsion 1-form $ a_\mu$ in the TNC geometry. In terms of the lapse function $N(x,t)$ of the ADM parametrization, we will see that the anomalous degree of freedom is a direct consequence of the *time-dependence* of the lapse function. Hence, the presence of this 1+1 Lifshitz Weyl anomaly necessarily implies that energy is not conserved in the system. More concretely, we will show that the $x$-component of the torsion (or acceleration) vector of the NC geometry defined by $a_x(x,t) = \frac{{\partial}_xN(x,t)}{N(x,t)}$, is not conserved as a result of the Weyl anomaly, and hence it physically represents *jerk* in the system.
In a separate yet related track, Horava-Lifshitz (HL) theories of gravity have been introduced as a power-counting renormalizable non-relativistic quantum theory of gravity with anisotropic scaling symmetry [@Horava1-09; @Horava2-09]. The key idea behind HL gravity theories is that by introducing terms with higher spatial derivatives, the ultraviolet (UV) behavior of the graviton propagator is improved and the theory eventually becomes power-counting renormalizable. When the number of spatial dimensions equals the dynamical scaling exponent $z$, Weyl-invariant actions can be found. HL actions break the principle of general covariance by foliating spacetime with space-like surfaces and introducing extra geometric data that affect the number and dynamics of degrees of freedom in the theory. As a result, not only do they describe the dynamics of the helicity-2 modes of the spatial metric but also an extra *helicity-0* scalar mode. Since this foliation mode is an excitation of the global time, it is usually called a *scalar khronon* [@Blas2011]. Therefore, it is natural to expect that gravitational Weyl anomalies of Lifshitz quantum effective actions coupled to background NC geometry with temporal torsion will somehow encode this extra foliation structure. In fact, this is precisely what 1+1 Lifshitz Weyl anomaly encodes: the time derivative of the lapse function in the ADM (or unitary gauge) or two time derivatives of the khronon field in a general coordinate system [@Blas2011]. It is well known that the breaking of general covariance in HL gravity theories leads to infrared instablities which has cosmological implications as shown in [@Sunny16].
The connection between dynamical NC geometry, with and without torsion, to HL gravity theories was demonstrated in [@ObersDynmicTNC2015]. More specifically, it was shown that dynamical NC geometries without torsion gives rise to projectable HL gravity while those with twistless torsion (TTNC) i.e. those that obey the Frobenius condition and do not allow torsion on the spatial slices, give rise to the non-projectable version of HL gravity. Projectable HL gravity theories are those where the lapse function in the ADM decomposition of spacetime is only dependent on time, i.e. $ N(t) $ whereas the non-projectable version emerges when it is a function of both space and time, i.e. $ N(x,t) $ and hence contain the acceleration vector as a dynamical quantity. Weyl-invariant theories of HL gravity can only be non-projectable [@Thompson2012]. Gauging a symmetry algebra is tightly related to spacetime geometry. Just as gauging the Poincare algebra gives rise to Riemannian geometry that couples to relativistic field theories, it was shown in [@Roo2011] and [@Rosseel2015] that gauging the Bargmann and Schrodinger algebras leads to NC geometries without and with torsion respectively. More specifically, as noted in [@ObersDynmicTNC2015], adding torsion to the NC geometry amounts to making it locally scale-invariant by gauging the Schrodinger algebra. Therefore, it stands to reason that the 1+1 Lifshitz anomaly is directly linked to the torsion vector of the NC geometry, which as shown in [@ObersDynmicTNC2015], maps directly to the acceleration vector in HL gravity theories.
By gauging the non-relativistic Bargmann and centrally-extended Schrodinger algebras, the authors in [@ObersNRSCS] constructed a (2+1)-dimensional non-relativistic Bargmann-invariant and Schr$\mathrm{\ddot{o}}$dinger-invariant Chern-Simons (NRSCS) actions, respectively. While the former gives projectable HL theory of gravity, the latter, which is the focus of this paper, is equivalent to $z=2$ conformal i.e Weyl-invariant non-projectable HL gravity. CS actions are known to be gauge-invariant up to total derivative terms. On manifolds with boundaries, these total derivative terms can generate anomalies of boundary quantum effective actions. For example, in the context of AdS/CFT, under a diffeomorphism or Lorentz transformation, the boundary term of the gravitational CS (gCS) action added to a three-dimensional on-shell gravitational action generates a diffeomorphism or Lorentz anomaly, respectively, of a two-dimensional *boundary* CFT effective action [@Larsen2006]. Analogously, by placing the NRSCS action on a manifold with a boundary, it will be shown in this paper that under a Weyl transformation, the NRSCS action changes by a total derivative term that precisely matches the boundary Weyl anomaly of a $z=2$ Lifshitz effective action coupled to background TTNC geometry. This is the main result of this paper. More concretely, we will show that the 1+1 Lifshitz Weyl anomaly can be derived holographically from a specific term in the three-dimensional NRSCS action constructed from the gauge fields of the Weyl and special conformal symmetry generators of the Schrodinger algebra. Throughout this paper, we call this term the torsional CS (tCS) term. We will show that the tCS term added to a three-dimensional Weyl-invariant HL gravity action plays a role similar to what the gCS term plays when added to a three-dimensional diffeomorphism-invariant action.
We also focus in this paper on the $z=1$ Lifshitz Weyl anomaly, where space and time scale *relativistically*. This anomaly possesses some interesting properties. In addition to being universal i.e. a $z$-independent anomaly in 1+1 dimensions, it was shown that it is the *Weyl partner* of the Lorentz anomaly of 1+1 CFT effective actions [@Oz2014]. To explicitly demonstrate the latter property, the authors shifted the Lorentz anomaly of a 1+1 CFT effective action $W_{\text{conf}}[e^a_{\mu}]$ where $e^a_{\mu}$ are the local orthonormal frame fields, into foliation dependence, i.e. to *dependence* on $n_\mu$ of the background foliation geometry. By doing so, the original CFT effective action $W_{\text{conf}}[e^a_{\mu}]$ turns into a $z=1$ FPD-invariant Lifshitz theory and thus the physical content of the anomalous Ward identity, the expectation value of the trace of the energy-momentum tensor with respect to $e^a_{\mu}$, $\left\langle T^\mu_{(e)\mu}\right\rangle$, now reflects a Lifshitz Weyl anomaly rather than a Lorentz anomaly of the original CFT effective action. In other words, the anomalous local frame rotations of $W_{\text{conf}}[e^a_{\mu}]$ is exchanged for anomalous Weyl transformations of the foliation geometry.
One of the goals of this paper is to further elicit the unique relationship between the $z=1$ Lifshitz Weyl anomaly and Lorentz anomaly in 1+1 dimensions. First, we will illustrate that the $\left\langle T^\mu_{(e)\mu}\right\rangle$ can be interpreted as the scalar curvature of the *dual* Lorentz connection 1-form in a local two-dimensional gravity theory with conformal and Lorentz anomalies constructed in [@Solod90]. In addition, we will show that restoring the local Weyl symmetry of the Weyl-anomalous effective action $W[e^a_{\mu}]$, where $e^a_{\mu}$ are the frame fields, eliminates the gauge redundancy associated with the time dependence of the lapse function and thus makes $a_x$ a conserved quantity. Insisting on the Weyl invariance of $W[e^a_{\mu}]$ amounts to solving an equation of motion for a chiral boson at rest after imposing appropriate appropriate boundary conditions at the spatial boundaries. We show that solving this equation yields the Rindler metric for hyperbolically accelerated observers.
This paper is organized as follows. In Section \[NC\_Geom\], we give the basic definitions of NC geometry with temporal torsion required for the discussion in the paper. In Section \[ADMParam\], after introducing the ADM coordinates, we define the torsion and curvature tensors and give the expression of the anomalous Ward identity for the Lifshitz Weyl symmetry. In Section \[LifWeylAnomGeom\], we use the ADM gauge to elicit the geometric properties of the 1+1 Lifshitz Weyl anomaly. In Section \[AnomalyDerivation\], we proceed to demonstrate the main result of this paper where we will show how the 1+1-dimensional Lifshitz Weyl anomaly can be derived from the torsional Chern-Simons (tCS) term in (2+1)-dimensional NRSCS action and how the tCS term added to the Weyl-invariant HL gravity action plays a role similar to what the gCS term plays when added to a three-dimensional diffeomorphism-invariant action. In Section \[Weyl-Lorentz-Anom-Relation\], we shift our focus on the $z=1$ Lifshitz Weyl anomaly where we will first review its connection to the Lorentz anomaly in 1+1 CFT effective action as pointed out in [@Oz2014] and then show how it is related to the scalar curvature of the dual Lorentz connection in a local two-dimensional gravity theory with conformal and Lorentz anomalies. In Section \[AnomalyCancellation\], we discuss how canceling the Weyl anomaly leads to an equation of motion for a stationary chiral boson which by solving, we obtain the Rindler metric. We also use Darboux’s coordinates to explain how as a result of canceling the Weyl anomaly, we get a symplectic manifold with a *Hamiltonian* function and Hamiltonian vector field. In Section \[Discussion\], we discuss several directions for future work and open questions.
The (1+1)-dimensional Lifshitz Weyl Anomaly {#Section2}
===========================================
In this section, after very briefly reviewing the basic structure of the NC spacetime geometry with temporal torsion in Section \[NC\_Geom\], we introduce the ADM parametrization in Section \[ADMParam\] and then define the anomalous Lifshitz Ward identity before we get to discussing the geometric nature of the Weyl anomaly, true for $z\geq 1$, in Section \[LifWeylAnomGeom\]. In Sections (\[Weyl-Lorentz-Anom-Relation\]) and (\[AnomalyCancellation\]), we focus on the $z=1$ case. Throughout this section, the indices $\mu, \nu$ refer to spacetime coordinates and $i,j$ to spatial ones. For the local tangent frame coordinates i.e. vilbeins, $a, b, \dots = 0,1, \dots d$ and $A, B = 1, \dots d$ are used to denote spacetime and spatial indices, respectively. We closely follow the notations and conventions used in [@Oz2014].
The Newton-Cartan (NC) Geometry {#NC_Geom}
-------------------------------
NC geometry, as opposed to Riemannian geometry in relativstic theories, is what couples naturally to non-relativistic field theories. Anomalies in non-relativistic quantum field theories coupled to NC geometry are prime examples of how the geometrical objects of the NC geometry become manifest. In non-relativistic field theories, the *time* direction plays a major role and spacetime is naturally foliated into *equal-time slices* or surfaces of simultaneity. Assuming the existence of a scalar field globally defined on the foliated spacetime manifold, a smooth local vector field $t_{\mu}$ *normal* to the foliation leaves and normalized by the spacetime metric $g_{\mu\nu}$ given by $ n_\mu = t_\mu /\sqrt{|g^{\beta\gamma} t_\beta t_\gamma|} $ is the most basic and intrinsic geometrical object defined on this manifold. Tangent vector fields to the foliation leaves are then defined as the kernel of $ n_\mu $ i.e. they satisfy $n_\mu V^\mu = 0 $.
More formally, the basic geometrical structure on a $D=$ (*d*+1)-dimensional NC manifold $M$ consists of an everywhere smooth *temporal* metric $t_\mu t_\nu$, a *degenerate* symmetric spatial component $h^{\mu\nu}$ with signature $(0,+, \dots, +)$ i.e. corank-1 tensor [@Geracie15] and a notion of a covariant derivative $\nabla$ all satisfying the following constraints $$h^{\mu\nu}t_{\mu} = 0 , \quad t_{\mu}t^{\nu} = -1, \quad \nabla_{\mu}t_{\nu} = \nabla_{\mu}\gamma^{\nu \lambda} = 0 \,.$$ While the 1-form $n_{\mu}$ provides a notion of a *clock*, its inverse $n^{\mu}$ denotes the direction of time often called the velocity field. Using the NC geometrical objects defined above, one can construct a non-degenerate symmetric rank-2 tensor $g_{\mu \nu}$ with a Lorentzian signature (-1,1,…1) that has a temporal component $n_\mu n_\nu$ as well as a spatial component $\gamma_{\mu \nu}$, i.e. $g_{\mu \nu} = \gamma_{\mu \nu} - n_\mu n_\nu$. Note that $g_{\mu \nu}$ is not a Lorentzian metric as it would normally be in a relativistic theory. For a more formal and thorough definition of the NC spacetime, see [@Karch14; @Geracie15].
Following [@ObersDynmicTNC2015], there are three different constraints on the foliation 1-form $ n_{\mu} $ that each give a different type of NC geometry:
1. *Torsionless* NC geometry: *$ dn = 0 $* where the connection $\Gamma^{\lambda}_{\mu \nu} $ is torsionless
2. *Twistless Torsion* Newton-Cartan (TTNC) or *temporal torsion* geometry $$\label{TTNCCondition}
\partial_{\mu}n_{\nu} - \partial_{\nu}n_{\mu} = a_{\mu}n_{\nu} - a_{\nu}n_{\mu}\, ,$$ where the *acceleration or torsion vector* $a_{\mu}$ is defined as the Lie derivative of the foliation 1-form along the $ n^{\mu} $ velocity vector field $ n_{\mu} $ $$a_{\mu} = \mathcal{L}_{n}n_{\mu} \, ,$$
The TTNC constraint in (\[TTNCCondition\]) is an expression of the solution of the *Frobenius condition*, an integrability condition that states a *local* 1-form defines a codimension-1 foliation if and only if it satisfies the following constraint: $$\label{FrobCon}
n \wedge dn = 0\,.$$ Imposing the Frobenius condition means that it is always possible to find a coordinate system in which the spacetime manifold is foliated into equal-time hypersurfaces or foliation leavs $\Sigma_t$ to which the unit time-like vector field $n_{\mu}$ is orthogonal. The Frobenius condition makes the TTNC spacetime *causal* in the sense that if it does not hold, then each point $p \in M$, has a neighborhood within which all points are spacelike separated. It is also important to mention that TTNC spacetimes, while being causal, still lack the notion of an *absolute* time measured by all observes along their worldlines. The difference between the total coordinate time measured by two observers starting at different points on $\Sigma_{t_1}$ and traveling to another time slice $\Sigma_{t_2}$ along their respective wordlines is exactly the torsion in *time* $dn = a\wedge n$ [@Geracie15]. This point is key to understanding the physical as well the geometrical meaning of the Weyl anomaly.
3. *Torsional* NC or TNC geometry where $ n_{\mu} $ is not constrained and has therefore arbitrary torsion.
We will later see how the (1+1)-dimensional Lifshitz Weyl anomaly is directly related to the TTNC geometry.
The ADM Parametrization {#ADMParam}
-----------------------
In the ADM decomposition, one chooses coordinates $(t,x^i)$ such that the leaves of the foliation are given by constant-time slices $t=const$ and $ x^i $ for the coordinates in each leaf. The ADM metric assumes a frame, the *unitary or synchronous* gauge where the time of the *spatial* foliation hypersurfaces coincides with coordinate time $t$ such that $n_t = N(x,t)$ and $ n_i = 0 $ and the result is a spacetime metric with a well-defined notion of global time. In this gauge, the ADM metric describes the TTNC geometry where the Frobenius condition given in (\[FrobCon\]) is *automatically* satisfied. In these *preferred* coordinates, the metric $g_{\mu \nu} = \gamma_{\mu \nu} - n_\mu n_\nu$ takes the form [^1] $$\label{MetricADM}
g_{\mu\nu}=
\begin{pmatrix}
g_{tt} & g_{ti} \\
g_{jt} & g_{ii}
\end{pmatrix}
=
\begin{pmatrix}
-N^2 + N^i N_i & N_i \\
N_j & h_{ij}
\end{pmatrix} \,,$$ while the components of the inverse metric are given by $$\label{MetricADM}
g^{\mu\nu}=
\begin{pmatrix}
g^{tt} & g^{ti} \\
g^{jt} & g^{ii}
\end{pmatrix}
=
\begin{pmatrix}
-\frac{1}{N^2} & \frac{N^i}{N^2} \\
\frac{N^j}{N^2} & h^{ij} - \frac{N^iN^j}{N^2}
\end{pmatrix} \, ,$$ where $h_{ij} $ is the induced metric on the foliation leaves, $ N^i $ is the shift vector and $ N(x,t) $ is the lapse function. The covariant volume element in these coordinates is given by: $$\sqrt{-g}\,d^{(d+1)}x = N\sqrt{h}\,dt\,d^d x .$$ The timelike normal to the foliation is given by $$\label{ADMTimelikeFol}
\begin{split}
& n_\mu = N(-1, 0) \, ,\\
& n^\mu = \frac{1}{N} (1,-N^i)\, .
\end{split}$$ In $d$ spatial dimensions, the Lie derivative of a foliation tangent tensor $V_{ijk\dots} $ is given by $$\label{LieADM}
\mathcal{L}_n V_{ijk\dots} = \frac{1}{N} \partial_t V_{ijk\dots} -\frac{1}{N} \mathcal{L}^{(d)}_{\vec N} V_{ijk\dots} \,,$$ where $ \mathcal{L}^{(d)}_{\vec N} $ is the Lie derivative inside the foliation leaf taken along the direction of the shift vector $ N^i $.\
\
In 1+1 dimensions, the extrinsic curvature tensor is simply given by $$\label{ExtrCurvADM}
K_{xx} = \frac{1}{2N} (\partial_t h_{xx}) \,.$$ whereas the $x$-component of the acceleration vector in 1+1 dimensions is given by $$\label{TorsionADM}
a_x = \frac{{\partial}_x N}{N}\,.$$ and the temporal component is $a_t= N^x a_x$. In 1+1 dimensions, with a non-zero shift vector $N^x$, the ADM metric is given by [^2] $$\label{ADMMetric}
ds^2 = -N^2\,dt^2 + N_x\, dxdt + h_{xx} \, dx^2 \,.$$ We will later discuss the consequences of having a non-zero $a_t$ with a constant $N^x$. In 1+1 spacetime dimensions with a zero shift vector, the spacetime metric has only two degrees of freedom: the lapse function $ N(x,t) $ and the spatial metric $h_{ij}$. The spatial metric $ h_{ij} $ is a rank-0 tensor, i.e. a function $ h_{xx}(x,t) $. The volume element $dV$ is then $ \sqrt{-g}\,dt\,dx = N\sqrt{h}\,dt\,dx $ .
The 1+1 Weyl Anomaly And Anomalous Ward Identity {#LifWeylAnomGeom}
------------------------------------------------
In this section, we attempt to illustrate the geometric nature of the 1+1-dimensional Lifshitz Weyl anomaly and how it is closely related to the NC geometry with temporal torsion. To that effect, we use the ADM coordinates to define some basic TTNC objects required to understand the geometric nature of the 1+1 Weyl anomaly. As mentioned in the introduction, dynamical TTNC gives rises to non-projectable Horava-Lifshitz theory of gravity. Since this approach is useful for our purposes in this section, we use some of the definitions in [@Obers2015] and [@ObersDynmicTNC2015].
The antisymmetric part of the torsion tensor $\Gamma^{\lambda}_{\mu \nu}$, is expressed as $$\label{TorsionSpinConn}
\Gamma^{\lambda}_{[\mu \nu]} = n^{\lambda}\partial_{[\mu}n_{\nu]} = n^{\lambda}a_{[\mu}n_{\nu]} = n^{\lambda}R_{\mu\nu}(H)\, ,$$ where $R_{\mu\nu}(H)$ is the curvature 2-form of $n_{\nu}$ defined as the *gauge field* of the generator of time translation symmetry, i.e. the Hamiltonian $(H)$ $$\label{CurvatureFoliation}
R(H) = (\partial_{\mu}n_{\nu} - \partial_{\nu}n_{\mu}) \, dx^{\mu}\wedge dx^{\nu} \,.$$ In 1+1 spacetime dimensions, imposing the Frobenius condition and using the ADM gauge, the only non-vanishing component of the torsion 2-form $\Gamma$ as defined by equation (2.27) in [@ObersDynmicTNC2015] is given by $$\label{TraceTorsionTensor}
\Gamma = \frac{1}{2}\Gamma^{t}_{[x t]} \,dx\wedge dt = n^{t}(a_{x}n_{t} - a_{t}n_{x}) \,dx\wedge dt = a_{x} \,dx\wedge dt = \frac{\partial_x N(x,t)}{N(x,t)} \,dx\wedge dt \,,$$ Equivalently, $$\label{CurvForm}
R(H) = \frac{1}{2} n_t\Gamma^{t}_{[x t]} \,dx\wedge dt = (a_xn_t - a_tn_x)\,dt\wedge dx = (a_xn_t)\,dx\wedge dt = R_{xt}(H)\, \,dx\wedge dt \,.$$ Now we can see that in *torsionless* NC geometry, $dn$ is a closed 1-form, i.e. $ dn=0 $ that corresponds to $ R_{\mu\nu}(H) = 0 $. This, in turn, translates to zero curvature in the gauge field i.e. a flat connection, corresponding to the time translation symmetry generated by the Hamiltonian. On the other hand, the TTNC case corresponds to a non-zero $R_{\mu\nu}(H)$ or $ dn\neq0 $. The Frobenius condition then tells us that this curvature is given by the torsion tensor $a_{\mu}$: $ \partial_{[\mu}n_{\nu]} = a_{[\mu}n_{\nu]}$. Lifshitz field theories with classical Weyl invariance couple to TTNC geometry and the Weyl anomaly will be directly related to this torsion or acceleration vector field.
To derive the anomalous Ward identity, we start with a classical action $S[{\phi}, N_i, h_{ij}]$ with matter fields ${\phi}$ coupled to background TTNC geometry and which is invariant under infinitesimal anisotropic local Weyl transformation with scaling exponent $z$ $$\delta N = z\sigma N, \quad h_{ij} = 2\sigma h_{ij} \,,$$ where $\sigma(x,t)$ is the infinitesimal Weyl transformation parameter. Quantum mechanically, however the regularization of UV infinities of the partition function $Z = e^{-W[N,N_{i},h_{ij}]} = e^{-W[N,N_i,h_{ij}]}$ breaks the local Weyl invariance of the quantum effective action $W[N,h_{ij}]$ resulting in a Weyl anomaly. More concretely, the presence of a Weyl anomaly in the effective action necessarily means that $$\delta W = \int N \sqrt{h}\, dt\ dx \, \sigma\, \mathcal{A} \,,$$ where quantum mechanically $ \mathcal{A} $ is given by the expectation values of the trace of the energy-momentum tensor $$\mathcal{A} = z\mathcal{\langle E \rangle } + \langle \mathcal{P}^{i}_{i}\rangle = z\left\langle T^t_t \right\rangle +\left\langle T^x_x \right\rangle \neq 0 \,.$$ It is important to note that although [@Oz2014] in their cohomological classification does not explicitly say that the background geometry to which they couple the Lifshitz theory is a TTNC spacetime, it actually implicitly is. For the cohomological classification of Weyl anomalies in FPD-invariant Lifshitz field theories in all spacetime dimensions, the foliation 1-form $n_{\mu}$ satisfies the Frobenius condition which is the key defining property of TTNC geometry. Section 2.4 of [@Oz2016] contains more information on the relationship between the notations and conventions used in [@Oz2014] and standard NC geometry.
We now move to demonstrate the geometric and physical nature of the Lifshitz Weyl anomaly after rewriting it in terms of the ADM coordinates defined above. We emphasize that the discussion in this section is valid for *all* values of $z$. We will present two different yet related pictures. While the first stresses that $n_{\mu}$ and $n^{\mu}$ are the fundamental objects in the TTNC geometry, the second one stresses the key role of the torsion vector itself $ a_{\mu}$ in generating the Lifshitz Weyl anomaly. The latter picture will turn out to be useful in Section \[AnomalyDerivation\] and Section \[Discussion\] when the anomaly is derived from the (2+1)-dimensional NRSCS action.
### The 1-form Picture
In terms of the ADM gauge in (\[ADMTimelikeFol\]) and (\[LieADM\]), the Weyl anomaly is given by the variation of the one-loop effective action of the (1+1)-dimensional Lifshitz effective action $W[g]$ $$\begin{aligned}
\label{Weylanomaly}
\delta W &=& \int \sqrt{-g}\ \sigma \, \tilde \epsilon^\mu \ \mathcal{L}_n a_\mu \\ \nonumber
&=& \int \sqrt{-g}\ \sigma \, n_x\epsilon^{xt} \, \mathcal{L}_n a_t + n_t\epsilon^{tx} \, \mathcal{L}_n a_x \\ \nonumber
&=& \int \sqrt{-g}\ \sigma \, n_t\, \mathcal{L}_n a_x \,,
\end{aligned}$$ where $ \tilde{\epsilon^\mu} = n_\alpha \epsilon^{\alpha \mu} $ is the foliation-projected Levi-Civita tensor, $\epsilon^{tx}=1$, $ a_t = N^xa_x = 0 $, and $ \sigma(x,t) $ is the Weyl transformation parameter. Using the definition of the Lie derivative in (\[TorsionADM\]), the Weyl anomaly is given by $$\begin{aligned}
\label{AnomnalyTorsion}
\delta W\, &=& -\int dt dx\ N^2 \sqrt{h}\, \sigma \, \mathcal{L}_na_x \\ \nonumber
&=& -\int dt dx\ N \sqrt{h}\, \sigma \, \left(\dfrac{{\partial}a_x}{{\partial}t} \right) \,.
\end{aligned}$$ In terms of the lapse function $ N(x,t)$, using (\[CurvForm\]) and (\[TraceTorsionTensor\]), it takes the following form $$\begin{aligned}
\label{AnomalyLapse}
\delta W &=& - \int dt dx\ N\sqrt{h}\, \sigma \, \left( \frac{1}{N}\partial_t\partial_x N - \frac{1}{N^2} {\partial}_t N{\partial}_x N \right) \nonumber \\
&=& - \int dt dx\ N\sqrt{h}\, \sigma \, \left( \frac{1}{N}\partial_t R_{xt}(H) - \frac{1}{N} \partial_t N a_x\right) \nonumber \\
&=& - \int dt dx \sqrt{h}\, \sigma \, \left(\partial_t R_{xt}(H) - \partial_t N a_x\right) \,.
\end{aligned}$$ Expressing $\left\langle T^{\mu}_{\mu}(x)\right\rangle = \tilde \epsilon^\mu \ \mathcal{L}_n a_\mu $ in terms of local tangent frame coordinates and using differential forms will better reveal its geometric nature. Using that the vielbeins for the temporal and spatial components of the NC metric $g_{\mu \nu} = \gamma_{\mu \nu} - n_\mu n_\nu$ can be expressed as $$\label {ADMMetricVielbein}
n_{\mu \nu} = n_{\mu} n_{\nu} \,, \quad h_{\mu \nu} = e_{\mu}{}^{A}\delta_{AB}e_{\nu}{}^{B} \,,$$ the vielbeins for the ADM metric in 1+1 dimensions in (\[ADMMetric\]) are then given by $$n=N dt, \qquad e^1 = (N^x dt + e^1{}_x dx )\,,$$ and the torsion coefficients are given by $$\begin{aligned}
\label{TorCur}
dn&=&\left({\partial}_x N \right) dx\wedge dt = \frac{{\partial}_1N}{N} e^1\wedge n \\ \nonumber
de^1&=&\left(\frac{{\partial}_xN^x}{2N} -\frac{e_1^x{\partial}_t e_x^1}{2N}
\right) e^1\wedge n = -\frac{K}{2}\ e^1\wedge n \,,
\end{aligned}$$ where $ K $ is the trace of the extrinsic curvature tensor defined in (\[ExtrCurvADM\]). We now use *Cartan’s formula* $ \mathcal{L}_X d\omega = d\mathcal{L}_X \omega $, which relates the Lie derivative along a vector field $ X $ of a $ k $-form $ d\omega $ to the exterior derivative of the $ (k-1)$-form ${\mathcal{L}_{X}\,} \omega$. Acting with the Lie derivative on $ dn$ along $n^{\mu}$, we get $$\begin{aligned}
\label{CartanFormula}
{\mathcal{L}_{n}\,} (dn) = d{\mathcal{L}_{n}\,}n &=& d{\mathcal{L}_{n}\,}\,(N\, dt) \\ \nonumber
&=& d( \frac{1}{N}{\partial}_t N - N^x {\partial}_x a_x) dt \\ \nonumber
&=& ({\partial}_t a_x) dt\wedge dx \,,
\end{aligned}$$ where we used the definition of the Lie derivative in (\[LieADM\]) and (\[TorsionADM\]) and chose $N^x$ to be zero. The expectation value of the energy-momentum tensor is therefore given by $$\label{AnomTraceEM}
z\left\langle T^t_t \right\rangle +\left\langle T^x_x \right\rangle = {\partial}_t a_x = {\partial}_t \left(\frac{R_{xt}(H)}{N(x,t)}\right)\,.$$ From equations (\[CartanFormula\]) and (\[AnomTraceEM\]), we can see that the 1+1 Lifshitz Weyl anomaly, in the 1-form picture, is naturally given by the time derivative of the curvature of the timelike foliation 1-form $ n_\mu $ or equivalently as the time derivative of $a_x$, the solution of the Frobenius condition (\[CurvForm\]). This makes explicit the relationship between TTNC geometry, the Frobenius condition and the role they both play in the generating the 1+1 Lifshitz Weyl anomaly.
### The 2-form Picture {#2-form-picture}
In the 2-form picture, the torsion vector is expressed as a 1-form $$\label{Tor-1form}
a = a_\mu dx^{\mu} = a_t \, dt + a_x \, dx \,.$$ The curvature 2-form $D_{\mu \nu}$ of the torsion 1-form $a_{\mu}$ is then given by $$\label{Tor-curv}
D = da = \left(\dfrac{{\partial}a_t}{{\partial}x} -\dfrac{{\partial}a_x}{{\partial}t} \right)\,dx\wedge dt \,.$$ Using $ a_{t} = N^{x}a_{x} $, $ da $ can be expressed as: $$\label{Lie}
da = \left(\dfrac{{\partial}a_x}{{\partial}t} - N^{x}\dfrac{{\partial}a_x}{{\partial}x}\right)\,dt\wedge dx \,.$$ Setting $a_t = 0$ or equivalently, $ N^x =0 $, we get $$\label{Curvature-2-form}
da = \left(\dfrac{{\partial}a_x}{{\partial}t}\right)\,dt\wedge dx \,.$$ which illustrates the 1+1 Lifshitz Weyl anomaly can be directly interpreted as the curvature of the torsion 1-form $ a_\mu$. We can clearly see that the anomalous *gravitational* degree of freedom is a direct consequence of the *time-dependence* of the lapse function $N(x,t)$. Hence, the presence of this 1+1 Lifshitz Weyl anomaly necessarily implies that energy is not conserved in the system. Equivalently, since the acceleration vector is time-dependent, the system experiences *jerk*. To get an intuitive interpretation of this Weyl anomaly and what it physically implies, we use the picture of Lie dragging in Fig. \[fig:lie-drag2\]. Expressed in this way, $a_\mu$ is the principal $G $-connection 1-form on a principal $G$-bundle $P$ over a smooth manifold $M$ with values in the Lie algebra $ \mathfrak{g} $ of $G$ while $D_{\mu \nu}$ is its curvature form. The associated vector bundle in our case, is the *dual* or cotangent frame bundle over $M$. In Section 3.3, we will see that $ G = SO^+(1,1)$, the identity component of the indefinite orthogonal group $SO(1,1)$ of area-preserving *squeeze* transformations which also happens to be the restricted group of Lorentz boosts in 1+1 dimensions.
![**Figure 1:** Geometrical depiction of the Weyl anomaly. Lie dragging the tangent vector $a_x(x,t) {\partial}_x$ with end points $x_1$ and $x_2$ defined on a spatial slice $\Sigma_t$ at coordinate time $t$ to another slice $\Sigma_{t+dt}$ using the diffeomorphism $f_{dt}$ associated with the vector $n_{\mu} = Ndt$. The points $x_1$ and $x_2$ are Lie dragged to $f_{dt}(x_1)$ and $f_{dt}(x_2)$ respectively, while the vector $a_x(x,t) {\partial}_x$ is dragged to $a_x(x,t+dt) {\partial}_x$. The Lie derivative of the vector $a_x(x,t) {\partial}_x$ along $n_{\mu}$ is then the difference between actual value of the vector at $f_{dt}$ and its value as a result of the Lie drag at $f_{dt}(x_2)$. The difference is the Weyl anomaly $dt\,{\partial}_t a_x$. Notice that as a result of the Weyl anomaly, the vector $a_x$ on $\Sigma_t$ has been *scaled* on $\Sigma_{t+dt}$ and therefore the spacetime area $(x_2 - x_1)dt$ has been *squeezed*.[]{data-label="fig:lie-drag2"}](Lie-Drag3){width="0.7\linewidth"}
As we will also point out in Section \[AnomalyDerivation\], in the process of gauging the Schr$\mathrm{\ddot{o}}$dinger algebra in 2+1 dimensions, the torsion 1-form $a_{\mu}$ appears as the gauge connection of the Weyl symmetry and thus it is natural to expect that the Weyl anomaly would involve the curvature $da$.
Derivation of the Weyl Anomaly From the NRSCS Action {#AnomalyDerivation}
====================================================
In this section, we derive the $z=2$ Lifshitz Weyl anomaly from a 2+1-dimensional (3D) non-relativistic Schr$\mathrm{\ddot{o}}$dinger-invariant Chern-Simons action on a manifold with a boundary. The boundary theory is a $z=2$ Lifshitz theory coupled to TTNC geometry. This 3D NRSCS action was recently constructed by gauging the centrally-extended Schr$\mathrm{\ddot{o}}$dinger algebra which made dynamical the TTNC geometry. In the metric formalism, it was then shown that the NRSCS action is indeed equivalent to a three-dimensional non-projectable $z=2$ *conformal* or Weyl-invariant HL theory of gravity which is the counterpart of relativistic conformal gravity. As we will discuss below, this NRSCS action contains two terms which do not contribute to the solution of the bulk equation of motion. In this section, we place the 3D NRSCS action on a manifold with *boundary* and show that one of these two terms, the tCS term, does in fact generate the $z=2$ (1+1)-dimensional Lifshitz Weyl anomaly in (\[Weylanomaly\]). In fact, the authors in [@ObersNRSCS] wondered if one of these two terms would correspond to a boundary Weyl anomaly. Let us emphasize again that the (1+1)-dimensional Lifshitz Weyl anomaly we are discussing in this paper is *universal* i.e. true for *all* values of $z$. Therefore, throughout the discussion in this section, the relevant dual boundary theory is a $z=2$ Lifshitz theory with a background TTNC geometry.
Key Proprties of the NRSCS Action
---------------------------------
By gauging the *extended* Schordinger algebra, i.e. by letting the gauge field $ A $ take its value in the centrally-extended Schr$\mathrm{\ddot{o}}$dinger algebra, $A$ can be expanded as a linear combination of the generators of the Schr$\mathrm{\ddot{o}}$dinger group [@ObersNRSCS] $$\begin{aligned}
A & = & H \tau +P_a e^a +G_a \omega^a + J \omega + N m +W a +K f \nonumber\\
&&+S\zeta+Y \alpha + Z\beta \,, \label{eq:ASch}\end{aligned}$$ \[NRSCS\]where $H$, $P_a$, $G_a$, $J$, $N$, $W$, $K$ are the generators of time translations (the Hamiltonian), momentum translation, Galilean boosts, rotations, central element, Weyl transformations and special conformal transformations, respectively.[^3] The three central extensions of the Schr$\mathrm{\ddot{o}}$dinger group are $S$, $Y$ and $Z$ respectively. Using the metric on this non semi-simple Lie algebra, the NRSCS action as given in [@ObersNRSCS] is $$\begin{aligned}
S_{NRSCS} [A] &=& \frac{k}{4\pi} \int_M Tr \left[A \wedge dA + \frac{2}{3} A \wedge A \wedge A \right] \label{action3} \nonumber \\
&=& \frac{k}{2\pi} \int_M Tr\ c_1 \mathcal{L}_{NRSCS} {+c_2\left[a\wedge da-\tau\wedge df+2a\wedge\tau\wedge f\right]}+2c_3\omega\wedge d\omega \\
&=& \frac{k}{2\pi} \int_M Tr\ c_1 \mathcal{L}_{NRSCS} +c_2\,\mathcal{L}_{tCS} +2c_3\omega\wedge d\omega\,,\end{aligned}$$ which for $ c_2 = c_3 = 0 $ is equivalent to a bulk 3D action of non-projectable conformal HL gravity. The arbitrary constants $ c_i $ are defined in terms of the symmetric bilinear form invariant under the Schr$\mathrm{\ddot{o}}$dinger algebra, i.e. $ B(V_i, V_j) = c_i $ and $ V_i $ is a generator of the algebra. For example, $ B(D,D) = 2c_2 $. A key observation is that $a_{\mu}$ is the gauge field of the dilatation symmetry. The curvature of the torsion vector is given by $R_{\mu \nu}(D) = da - 2df$ where $f_{\mu}$ is the gauge field associated with the generator of special conformal transformations $K$. (In this section $K$ is not the trace of the extrinsic curvature). Therefore, one should expect that a boundary Weyl anomaly would be generated by the tCS term in the action.
With $ c_{2} = c_{3} = 0 $, the NRSCS action (\[NRSCS\]), satisfies a bulk equation of motion $F=dA+A\wedge A=0$ that gives a $ z=2 $ Lifshitz metric $ ds^2=-\frac{dt^2}{r^4}+\frac{dr^2}{r^2}+\frac{dx^2}{r^2}\,$. Therefore, the bulk theory represented by the NRSCS action is *Weyl-invariant*. However, the tCS term whose coefficient is $c_2$, transforms under the $SL(2,\mathbb{R}) $ subgroup of the Schr$\mathrm{\ddot{o}}$dinger algebra and as discussed in [@ObersNRSCS] cannot be removed by a field redefinition and therefore, as noted in [@ObersNRSCS], it may lead to a Weyl anomaly at the boundary. The tCS term is an Abelian CS term composed entirely of $a_{\mu}$, the gauge connection of the local Weyl symmetry and therefore, under a Weyl transformation on a manifold with a boundary, it is natural to expect that the total derivative term leads to a Weyl anomaly of the boundary effective action. In other words, the sole contribution of the tCS term when added to an on-shell bulk HL theory of gravity is to generate the Weyl anomaly of the *dual* boundary theory since, as discussed above, it does *not* contribute to the solution of the bulk $z=2$ HL gravity theory. It may be worth pointing out that the $SL(2,\mathbb{R})$ is isomorphic to $SO^+(2,1)$ which is the group of Lorentz transformation in three dimensions. At the boundary, where there is a Weyl anomaly, the torsion 1-form $a_{\mu}$ transforms under $SO^+(1,1)$ as we explained in the previous section. Recently, it was shown that the NRSCS action can be reformulated into a manifestly 3D relativistic form due to the presence of the $SL(2,R)$ subalgebra in the extended Schroedinger algebra [@Sorokin19] and therefore can be interpreted as a relativistic CS theory.
The Lifshitz Weyl Anomaly from the tCS Term {#AnomalyFromtCS}
-------------------------------------------
Denote the (2+1) on-shell HL gravity action by $S_{HL}$. Using the developed machinery of non-relativistic holography [@Thompson2012; @KarchNRHolo], which started when Lifshitz and Schr$\mathrm{\ddot{o}}$dinger spacetime solutions to relativistic actions of gravity were found [@Son08; @McGreevy08; @Kachru08; @Taylor08], the variation of the on-shell HL action at low energies and to leading order in the metric can be expressed in terms of the TTNC geometry on the boundary $$\label {NRHolog}
\delta S_{HL} = \int d^3x \sqrt{g^{(0)}} T^{ij}g_{ij}^{(0)} \,,$$ where $d^3x \equiv dt dx dr$ and $\sqrt{g^{(0)}} = N^{(0)}\sqrt{h^{(0)}}$ is the metric in terms of the boundary lapse and shift vectors and $T^{ij}g_{ij}$ is identified with the trace of the expectation value of the boundary theory effective action $W[N^{(0)},h_{ij}^{(0)}]$ coupled to a metric that is *anisotropically* conformal to $N^{(0)}h_{ij}^{(0)}$.[^4] If the action $S_{HL}$ is Weyl-invariant as for example the one constructed in [@ObersNRSCS], then $\delta S_{HL} = 0$ under the variations $\delta N = z\sigma N^{(0)} \, \, \mathrm{and} \, \, h_{ij}^{(0)} = 2\sigma h_{ij}^{(0)}$. However, if we were to trust the machinery of non-relativistic holography especially for HL gravity and asymptotically Lifshitz spacetimes, we have to be able to deal with a Weyl-anomalous boundary theory and assume a non Weyl-invariant bulk theory of gravity with a non-vanishing $T^{ij}g_{ij}^{(0)} = 2\left\langle T^t_t \right\rangle +\left\langle T^x_x \right\rangle $. Adding the tCS term to the on-shell gravity action $S_{HL}$ is our way out. As we show below, under a Weyl transformation, the tCS term is invariant up to a boundary term. If we assume the coefficient of the tCS term matches that of the boundary Weyl anomaly, then it cancels with the variation of the bulk on-shell action. More concretely, the variation of the on-shell bulk gravity action $S= S_{HL} + S_{tCS}$ under a Weyl transformation with parameter $\sigma(x,t)$ should be given by $$\label{Var_On-shell_Gravity_Action}
\delta_{\sigma} S =\delta_{\sigma} S_{tCS} = \frac{k}{2\pi} c_2 \int_{{\partial}M} Tr [\delta a\, da] \,.$$ Let us see how we do that. We set $c_3 = 0$ in the NRSCS action and start by integrating out the connection $ \beta$ in NRSCS action. The corresponding equation of motion is $ df = -2b\wedge f$. Substituting this solution into the tCS term, we see that $- \tau \wedge df $ cancels with $ 2b\wedge \tau\wedge f $ such that the tCS term can be written as $$\label{tCSAction}
\mathcal{L}_{tCS}[a] = 2c_{2} \left(a\wedge da \right) \,.$$ In terms of differential forms, a variation of the torsion field $ a $ in $ S_{tCS}[a] $ gives $$\label{TorsionalActionVar}
\delta S_{tCS}[a] = \frac{k}{2\pi}c_2 \int_M \, Tr[\delta a\,\wedge D] + \int_{{\partial}M} Tr [\delta a\, da]\,,$$ where $k$ is the level of the NRSCS action. In components with coordinates $(t,x,r)$, the $S_{tCS}[a]$ action reads $$S_{tCS}[a] = \frac{k}{2\pi} c_2 \int d^3x\ \epsilon^{\mu\nu\rho}a_\mu \partial_{\nu}a_{\rho}\label{tCS} \,.$$ The variation of the tCS action is then given by $$\begin{aligned}
\delta S_{tCS}[a] &=& \frac{k}{2\pi} c_2\int d^3x\ {\epsilon^{\mu\nu\rho}}\left[ \delta a_\mu\partial_\nu a_\rho + a_\mu \partial_\nu \delta a_\rho\right] \nonumber\\
&=& \frac{k}{2\pi} c_2\int d^3x\ {\epsilon^{\mu\nu\rho}}\left[ \delta a_\mu D_{\nu\rho} + \partial_\mu (a_\nu \delta a_\rho)\right]
\end{aligned}$$ $D_{\mu\nu}=0$ is the equation of motion that minimizes the tCS action. The last term must be set to zero on the boundary at $r=0$. One choice is $$\label{tCSBoundaryGaugeChoice1}
(a_t - N^x a_x) \Big|_{r=0} = 0, \quad N^r =0 \,,$$ since by definition, $ a_t = N^x a_x + N^r a_r $. The other sets only $a_t$ to zero $$\label{tCSBoundaryGaugeChoice2}
a_t = N^x a_x = 0 \,.$$ The choice in (\[tCSBoundaryGaugeChoice1\]) is however more general. Under infinitesimal local Weyl transformation with parameter $\sigma(x,t)$, the gauge connection $a_{\mu}$ transforms as $$\delta_{\sigma}a = d\sigma \,,$$ and the tCS action varies by a the total derivative term $$\delta_{\sigma} S_{tCS}[a] = \frac{k}{2\pi} c_2\int_{{\partial}M} Tr (\sigma da)$$ In components, this becomes $$a_\mu \rightarrow a_\mu + \partial_\mu \sigma \nonumber$$ and $$S_{tCS}\rightarrow S_{tCS} + \frac{k}{2\pi} c_2\int_{r=0} dxdt\ \sigma (\partial_t a_x - \partial_x a_t)\nonumber \,,$$ which has precisely the same *form* of the (1+1)-dimensional Weyl anomaly in (\[AnomnalyTorsion\]) and (\[Tor-curv\]) of the Lifshitz boundary theory. Thus, the bulk remains Weyl-invariant while the boundary theory does not. We can then conclude that the tCS term added to a 3D Weyl-invariant HL gravity action plays a role similar to what the gCS term plays when added to a 3D diffeomorphism-invariant action. This is the main result of this paper. However, it is important to observe that without knowing the exact value of the coefficient $c_2$ and matching it with that of the anomaly computed in an example Lifshitz field theory, for example, using heat kernel methods, it would be difficult to claim the derivation is exact.
It stands to reason that we should be able to find the $ a\wedge da $ term in the parity-odd sector of the cohomology of the relative Weyl operator in 2+1 dimensions. Indeed, we found such a term in [@Oz2014]. The term is given by $$\begin{aligned}
\label{OztCS}
\tilde{\epsilon}^{\alpha \beta} a_{\alpha}{\mathcal{L}_{n}\,}a_\beta &=& \tilde{\epsilon}^{x r} a_{x}{\mathcal{L}_{n}\,}a_r + \tilde{\epsilon}^{r x} a_{r}{\mathcal{L}_{n}\,}a_x \nonumber \\
&=& a_{x} {\partial}_{t} a_r - a_{r} {\partial}_{t} a_x \,,
\end{aligned}$$ where we have used that $ {\mathcal{L}_{n}\,} a_{r} = \frac{1}{N} \left({\partial}_{t} a_r - N^r {\mathcal{L}_{N^r}\,} a_{r} \right) $, $\tilde{\epsilon}^{\alpha \beta} = n_{\gamma}\epsilon^{\gamma \alpha \beta} $ and $a_t = 0 $. Now let us show that $ a\wedge da $ can be expressed as (\[OztCS\]). Let us start by expanding the $ a\wedge da $ in coordinate bases as $ a = a_\mu\,dx^\mu = a_x\, dx + a_r\, dr + a_t\, dt $. The $ a\wedge da $ term can be expanded as follows $$\begin{split}
a\wedge da &= \epsilon^{ijk} a_i{\partial}_j a_k \nonumber
\\
&
= a_t\left( {\partial}_x a_r - {\partial}_r a_x \right) dt dx dr - a_x\left( {\partial}_t a_r - {\partial}_r a_t \right) dx dr dt
\\
&
+ a_r\left( {\partial}_t a_x - {\partial}_x a_t \right) dr dx dt \,,
\end{split}$$ where $di\, dj\, dk\, \equiv di \wedge dj \wedge dk$. The exterior derivative $ da $ is given by $$\begin{aligned}
a\wedge da &=& \epsilon^{xr}a_x{\partial}_ta_r \nonumber \\
&=& a_x{\partial}_ta_y - a_r{\partial}_ta_x
\end{aligned}$$ which matches the one given in (\[OztCS\]).
The $z=1$ Lifshitz Weyl Anomaly
===============================
In this section, we focus on the interesting case of the $z=1$ Lifshitz Weyl anomaly. We will illustrate that $\left\langle T^\mu_{(e)\mu}\right\rangle$ can be interpreted as the scalar curvature of the *dual* Lorentz connection 1-form in a two dimensional local gravity theory constructed in [@Solod90] with conformal and Lorentz anomalies.
The $z=1$ Lifshitz Weyl Anomaly as the Scalar Curvature of the Dual Lorentz Connection {#Weyl-Lorentz-Anom-Relation}
--------------------------------------------------------------------------------------
The authors in [@Oz2014] revealed that the $z=1$ (1+1)-dimensional Lifshitz Weyl anomaly is the *Weyl partner* of the Lorentz anomaly in 1+1 CFT. The idea was to shift the Lorentz anomaly of a $1+1$-dimensional CFT to a foliation dependence, i.e. to a *dependence* on $ n_\mu $ and rewrite the *anomalous* CFT quantum effective action in terms of $ n_\mu$ as follows $$\label{compartoconf:lifactiondef}
W_{\text{Lif}}[e^a{}_\mu,t^a] \equiv W_{\text{conf}} [-e^a{}_\mu n_a, e^a{}_\mu \widetilde{n}_a] = W_{\text{conf}}[-n_\mu, \widetilde{n}_\mu],$$ where $n_{\nu}$ and $\widetilde{n}^a \equiv \epsilon^{ab} n_b$ are arbitrary foliation vectors aligned with the frame fields $e^a{}_\mu$ which are defined for a relativistic spacetime as $$\begin{aligned}
g^{\mu \nu} e^{a}_{\mu} e^{b}_{\nu} &= \eta^{ab} \\
\eta_{ab} e^{a}_{\mu} e^{b}_{\nu} &= g_{\mu \nu} \,,
\end{aligned}$$ where $\eta^{ab}$ is the flat metric in the two-dimensional tangent frame basis. The Lorentz spin connection, in terms of $ e^{a}_{\mu} $, is defined as $$\label{SpinConnection}
\omega_\mu{}^a{}_b = - e_b{}^\nu \nabla_\mu e^a{}_\nu \,.
$$ and the connection 1-form $ \omega_\mu $ is given by $$\omega_{\mu} = \epsilon^{ab}\omega_{ab,\mu} = \epsilon^{ab}\partial_{\mu}\omega_{ab}$$ It is well known [@BertlandBook] that in a 1+1 CFT with local Lorentz anomaly [^5], for example, in a chiral CFT, the Weyl anomaly has an extra term in addition to the Ricci scalar $ R $. This extra term is essentially how the Lorentz anomaly *manifests* itself in $\left\langle T^\mu_{(e)\mu}\right\rangle$ taken as the variation of $ W_{\text{Lif}}[e^a{}_\mu]$ with respect to the viebein 1-forms $e^a{}_\mu $. This additional term is a total divergence of the Lorentz (spin) connection 1-forms $\omega^\mu{}_{ab}$ defined in (\[SpinConnection\]) $$\label{LorentzWeylPartner}
\left\langle T^\mu_{(e)\mu}\right\rangle = -2\epsilon^{ab} \nabla_\mu \omega^\mu{}_{ab} \,.$$ After identifying local tangent frame i.e. the vielbeins with the foliation 1-forms $$\label{vielbein-Iden}
e^{0}_{\mu} \equiv n_{\mu}, \quad e^{1}_{\mu} \equiv \widetilde{n}_\mu \,,$$ the authors in [@Oz2014], were able to demonstrate that $\left\langle T^\mu_{(e)\mu}\right\rangle$ is indeed the Weyl partner of the Lorentz anomaly up to the coboundary terms $(a_\rho K + {\widetilde\nabla}_\rho K)$
[align]{}
T\^\_[(e)]{} = & -2 a \^[ab]{} \_\^\_[ab]{}\
&= -2a \_(\^[ab]{} e\_[a]{} \[TraceEMLorentz\] \^e\_[b]{}\^ )\
& = 4a \^([\_[n]{}]{} a\_+ a\_K + \_K) ,
where ${\widetilde\nabla}_{\mu}$ is the foliation-projected covariant derivative of a foliation-tangent tensor $V_{\alpha \beta}$.[^6] This identification essentially maps the Poicare-invariant tangent vector space of the relativistic spacetime manifold to Schr$\mathrm{\ddot{o}}$dinger-invariant (or Bargmann-invariant) tangent vector space of the TTNC spacetime manifold to which Lifshitz field theories naturally couple. The original CFT effective action $W_{\text{conf}}[e^a_{\mu}]$ is technically no longer Lorentz-anomalous but rather foliation-dependent and hence the physical content of $\left\langle T^\mu_{(e)\mu}\right\rangle$ now reflects a Weyl anomaly rather than a Lorentz anomaly. In other owrds, the anomalous local frame rotations of $W_{\text{conf}}[e^a_{\mu}]$ is exchanged for anomalous Weyl transformations of the foliation in $W_{\text{conf}}[-n_\mu, \widetilde{n}_\mu]$.
We now illustrate that the $\left\langle T^\mu_{(e)\mu}\right\rangle$ is related to the scalar curvature of the dual Lorentz connection 1-form of the local gravity action in [@Solod90]. In [@Solod90], a *local* action of two-dimensional gravity was constructed out of the frame fields $e^{a}_{\mu}$ $$\begin{aligned}
\label{SolodukhinAction}
W[e^{a}_{\mu}] &=& \frac{1}{4} \int d^2 x \,e\, T^{a}{}_{\mu\nu}T^{\mu\nu}{}_{a} \\ \nonumber
&=& \frac{1}{4} \int d^2 x \,e \, ({\partial}_{\mu}e^{a}{}_{\nu} - {\partial}_{\nu}e^{a}{}_{\mu})\, ({\partial}^{\mu}e_{a}{}^{\nu} - {\partial}^{\nu}e_{a}{}^{\mu}) \,,
\end{aligned}$$ where $e$ = det $e^{a}{}_{\mu}$. Under a local conformal transformation $\delta e^{a}{}_{\mu} = \sigma e^{a}{}_{\mu}$ with infinitesimal Weyl parameter $\sigma$, the action suffers a conformal anomaly $R$ while under a local Lorentz transformation $\delta e^{a}{}_{\mu} = \chi \epsilon^{a}{}_{b} e^{b}{}_{\mu}$ with infinitesimal Lorentz parameter $\chi$, it has a Lorentz anomaly $U$. Using (\[SpinConnection\]) and the fact that the two-dimensional Levi-Civita tensor obeys $\epsilon_{ab} + \epsilon_{ba} = 0$, the Ricci scalar $R$ can be expressed in terms of the curvature 2-form of the Lorentz connection $\omega$ $$\mathcal{R} = d\omega \,,$$ while the scalar $U$ of the Lorentz anomaly can be expressed in terms of the curvature of the dual Lorentz connection $\star\omega$ $$\mathcal{U} = d\star\omega \,,$$ where $\star$ is the Hodge dual operator. If the Lorentz connection 1-form is expressed in terms of $a_x$ and $K$ as $$\omega = a_x dt + \frac{K}{N} dx$$ then $\int d\omega$ gives the foliation-projected decomposition of $R$ (not to be confused with the $R(H)$ of the Weyl anomaly) $$\label{RicciFoliation}
R = \int \mathcal{R} = \int dt dx\, N\sqrt{h} \left( K^2 + \frac{1}{N}{\partial}_tK + \left( \frac{{\partial}_xN}{N}\right)^2 - \frac{{\partial}_x^2N}{N}\right) \,.$$ Similarly, if we define the dual Lorentz connection (which interestingly enough, only in two dimensions is also a 1-form) as $$\star \omega = a_x dx - \frac{K}{N} dt \,,$$ then $$\label{DualLorentzConn}
d\star \omega = \left( {\partial}_t a_x + \frac{1}{N} \left({\partial}_x K - a_xK \right)\right) dt\wedge dx \,,$$ and hence $\left\langle T^\mu_{(e)\mu}\right\rangle$ can be expressed as the scalar curvature 2-form $d\star \omega$ $$\label{LorentzScalar}
U\equiv \left\langle T^\mu_{(e)\mu}\right\rangle = 2 \nabla_{\mu}{}(\omega^{\mu}) = 2\nabla_{\mu}{}\left(\epsilon_{ab}e^{a}{}_{\nu}T^{b\mu\nu} \right) \,.$$ By comparing equations (\[LorentzScalar\]) with (\[TraceEMLorentz\]), we see that they have precisely the same form when decomposed in terms of foliation geometry. Therefore, we see that the 1+1 Lifshitz Weyl anomaly $\tilde \epsilon^\rho {\mathcal{L}_{n}\,} a_\rho$ is directly related to the curvature scalar $d\star \omega$ of the dual Lorentz connection (modulo the coboundary terms in (\[TraceEMLorentz\]) and (\[DualLorentzConn\])) when expressed in terms of the foliation 1-forms in (\[vielbein-Iden\]). We note that the same curvature scalar $U$ of $\star \omega$ also appears in the process of quantizing a *non-local* chiral gravity action which is known to have local Weyl as well as Lorentz anomalies [@Myers92]. It was also pointed out in [@Myers92] that the gauge group defined on the dual frame bundle related to $\star \omega$ connection 1-form is multiplication by positive real numbers which is consistent with the Lie drag picture in Fig. \[fig:lie-drag2\].
More importantly, it was pointed out in [@Solod90] and [@Myers92] that $U$ like $R$ is also topological invariant i.e. $$\label{IntegratedLifAnomaly}
\lambda = \int_{M} \, e\, d^2x \,U + \int_{{\partial}M} \,\sqrt{h\,} d\tau \,\mathrm{(boundary \, \, term)} = \mathrm{topological \, \, invariant} \,,$$ where $e$ is det $e^a{}_{\mu}$. In fact, $U$ was obtained from the index of a generalized Dirac operator (see equation 28 in [@Solod90] and by using the conformal-Lorentz gauge, $\lambda$ was expressed as the boundary integral of the divergence of a unit tangent vector in the conformal-Lorentz gauge (here tangent to the *spatial* foliation leaf) $$\label{AnomalyTopo}
\lambda = \int_{{\partial}M}\, d\tau\, \nabla_{\mu} \hat{V}^{\mu} \,.$$ The physical meaning of $\lambda$ as a conserved boundary charge is still not clear but we will discuss this observation further in Section \[Discussion\] in light of recent progress in constructing boundary conformal invariants of type-B anomalies.
Weyl Anomaly Cancellation in $ z=1 $ Lifshitz Effective Action {#AnomalyCancellation}
--------------------------------------------------------------
In this section, we discuss how canceling the Weyl anomaly leads to an equations of motion for a stationary chiral boson, which by solving, we obtain the Rindler metric. We then use Darboux’s coordinates to explain how as a result of canceling the Weyl anomaly, the cotangent bundle of the spacetime manifold is a symplectic manifold with a *Hamiltonian* function.
To restore the local Weyl symmetry of the induced effective action, the Weyl anomaly must be canceled such that $a_x$ becomes conserved. Insisting on the Weyl invariance of the quantum effective action $W[e^a_{\mu}]$, amounts to satisfying the equation of motion in (\[AnomnalyTorsion\]) or (\[AnomalyLapse\]) by putting appropriate boundary conditions on ${\partial}_tN$ at the spatial boundaries. The Weyl anomaly in (\[Weylanomaly\]) and (\[AnomnalyTorsion\]) assumes a zero shift vector $N^x = 0$. If $N^x = 0$, then the equation of motion is simply given by $$\label{EOMZeroShift}
\partial_t\partial_x N = 0 \,.$$ Since, physically, the Weyl anomaly represents a *time-dependent* acceleration, or a non-uniform gravitational field where energy is not conserved, restoring local Weyl invariance in the effective action is tantamount to having observers with constant, i.e. uniform proper acceleration in *flat* spacetime or having a uniform gravitational field where energy is conserved. This necessarily means getting rid of the time dependence of the lapse function $N(x,t)$. Mathematically speaking, restoring local Weyl symmetry requires making $a_{\mu}$ a closed 1-form, i.e. a flat connection $da=0$ with zero curvature $D_{\mu \nu} = 0$.
A natural question to ask is what the implications are of having a time-independent lapse function, one that only depends on the spatial coordinates $N(x)$. After all, in a projectable HL gravity theory, the lapse function is either only time-dependent $N(t)$ or time and spatially-dependent $N(x,t)$ in the non-projectable version. So, what does it mean to have time-independent lapse function as a result of canceling of the $z=1$ Lifshitz Weyl anomaly? We will comment on this peculiarity at the end of this section.
The equation of motion in (\[EOMZeroShift\]) is that of a *stationary* 1+1 gravitational *chiral* boson whose general solution of (\[EOMZeroShift\]) is given by $$N(x,t) = N_1(x) + N_2(t) \,.$$ Imposing the boundary condition ${\partial}_t N = 0$ at the spatial boundaries i.e. the spatial leaves of the foliation, necessarily means eliminating the $N_2(t)$ gauge degree of freedom and therefore (\[EOMZeroShift\]) is automatically satisfied. By putting a boundary condition that sets ${\partial}_t N$ to zero at the spatial boundaries, the Weyl anomaly is canceled and as a result, $a_x$ becomes the conserved charge of the local Weyl gauge symmetry. A spatially-dependent lapse function $N(x)$ then gives a family of arbitrary time-independent solutions each of which on a hypersurface of constant time $t$. Choosing $ N(x) $ to be linear is a particularly important choice of coordinates, since with this choice and $h_{xx}=1$, the background spacetime metric in ADM coordinates becomes $$\label{RindlerMetric}
ds^2 = -(\alpha x)^2\,dt^2 + \, dx^2 \,.$$ which is the *Rindler* metric of a hyperbolically accelerated reference frame with coordinates $(x,t)$ with rapidity $\eta=\alpha t$. If we label the flat Minkowski spacetime coordinates by $(X,T)$ and choose Rindler observer with constant proper acceleration $\alpha =1$ and proper time $\tau$ equal to coordinate time $t$, then $(X,T)$ are related to Rindler coordinates by the following transformations $$\label{RindlerWorldline}
T=x \, \mathrm{sinh(t)}, \quad X = x\,\mathrm{ cosh(t)} \,.$$ These linear transformations preserve the hyperbolae $X^2 - T^2 = N^2(x) = x^2$ which describe the worldlines of a family of Rindler observers at rest for *fixed* $x$. These transformations can be represented by elements of the one-parameter group of Lorentz boosts $SO^+(1,1)$ with boost parameter $\eta=\alpha t$. An element in $SO^+(1,1)$ is represented by a $2\times2$ real matrix $$\label{SOMatrixElement}
M(\eta)=
\begin{bmatrix}
\mathrm{cosh}(\eta) & \mathrm{sinh}(\eta) \\
\mathrm{sinh}(\eta) & \mathrm{cosh}(\eta)
\end{bmatrix} \,.$$ In light-cone coordinates, $U=X+T, \ V=X-T$, $ M(\eta) $ is diagonalized $$\label{SODiagonalMatrixElement}
M(\eta)=
\begin{bmatrix}
e^{\eta} & 0 \\
0 & e^{-\eta}
\end{bmatrix} \,,$$ such that area $U*V = X^2 - T^2$ of the hyperbola is preserved. Therefore, the group $SO^+(1,1)$, in addition to being the group of Lorentz boosts in 1+1 dimensions is also the group of scale (actually squeeze) transformations that preserve the area $U*V$ of the hyperbolic worldline of a Rindler observer at a fixed $x=x_0$.\
\
If we define a frame fields $e^0$ and $e^1$ as $$\label{Lapse-redshift}
e^0 = x\,dt, \quad e^1 = dx \,,$$ which in terms of the dual basis vector field, is given by $$\label{KillingVector}
n^t = \frac{1}{x}\,{\partial}_t, \quad n^x = {\partial}_x \,,$$ then the unit timelike vector $n^{\mu}$ defines integral curves consisting of the world lines of a family of Rindler observers each at fixed $x=x_0$. For each such observer, $n^t$ is a *Killing* vector of the Rindler metric which, when expressed in Minkowski coordinates, becomes the generator of Lorentz boosts in the $X$-direction $$\label{BoostKilling}
{\partial}_t = ({\partial}_tT){\partial}_T + ({\partial}_tX){\partial}_X = X{\partial}_T + T{\partial}_X \,.$$ Since the Lie derivative of the torsion vector $a_{\mu}$ along $n^{\mu}$ after canceling the Weyl anomaly is now ${\mathcal{L}_{n}\,}a_{\mu} = 0$, $a_{\mu}$ is conserved and $n_{\mu}$ satisfies the Frobenius condition $n\wedge dn = 0$. It is interesting to note that the *vorticity-free* condition of the worldlines of Rindler observers i.e. the vanishing of the rotation tensor in the Raychaudhuri equation, is the twistless torsion condition in equation (6.8) of [@ObersDynmicTNC2015].
Some comments are in place. First, we observe that by making the lapse function time-independent, we have eliminated the extra foliation degree of freedom in HL gravity theories which as a result become Weyl-invariant [@Blas2011]. To understand the consequence of canceling the Weyl anomaly a bit further, we use *Darboux’s theorem* to show that the cotangent bundle of the flat spacetime is a symplectic manifold with a *Hamiltonian* function $N(x) = x$ of a Hamiltonian vector field $n^{\mu}$ which is we what should expect anyway. Starting from the Frobenius condition, one can define local Darboux coordinates on a two-dimensional manifold $M$ with a 1-form $n_{\mu}$ $$n=x\, dt \,.$$ Taking the exterior derivative of $n$ gives the *symplectic* 2-from $R(H)$ on $M$ $$R(H)=dn= dx\wedge dt \,.$$ since $R_{xt}(H)={\partial}_x N = 1$ by definition. Furthermore, using Cartan’s formula, one can formally show that the Lie derivative of $R(H)$ vanishes $${\mathcal{L}_{n}\,}R(H) = 0 \Leftrightarrow d(\iota_n R(H)) + \iota_n dR(H) = d(d\,N)+ dR(H) = 0 \,,$$ where $\iota_n$ is the interior product, $dR(H)=0$ is automatically satisfied on a 2-dimensional manifold, $\iota_n R(H) = dN = dx$, $d(dN(n^{\mu}))$ and $d(dx(\frac{1}{x}{\partial}_t)) =0$ essentially means that $R(n^{\mu},n^{\mu}) = 0$. Hence, by choosing $N(x) = x$, we have made the Hamiltonian constant along flow lines as we explained before. Thus, as a result of canceling the local Weyl anomaly in a $z=1$ (1+1)-dimensional Lifshitz field theory, the *cotangent* bundle of the spacetime manifold is a symplectic manifold with a Hamiltonian function $N(x) = x$ and a Hamiltonian vector field $n^{\mu}$.
Discussion and Outlook {#Discussion}
======================
Edge Physics of the NRSCS Action
--------------------------------
It is well known that the Floreannini-Jackiw (FJ) action [@FJ87] describes massless chiral self-dual edge bosons for the Abelian Laughlin fractional quantum Hall (FQH) state [@WenQFT; @FradkinQFT; @TongNotes]. In fact, it is the Wess-Zumino-Witten (WZW) low-energy *boundary* CS action for the Laughlin state. The *local* FJ action is given by $$\label{FJ_Action}
S_{FJ} = \int dt\,dx\ \partial_t\phi\partial_x\phi - v_x(\partial_x\phi)^2 \,,$$ with equation of motion $$\partial_t\partial_x\phi - v_x\partial_x^2N= {\partial}_t \rho (x,t) - v_x{\partial}_x\rho(x,t) = 0 \,, \label{FJEOM}$$ where $\rho(x,t) \equiv {\partial}_x \phi(x,t)$ is the chiral boson *excitation* expressed as spatial derivative of the gauge degree of freedom $\phi(x)$. This equation has solutions of the form $\phi(x+vt)$ which describes a chiral wave propagating with constant velocity $v_x$. Replacing $\phi(x,t)$ with $N(x,t)$, $\rho(x,t)$ with $R_{xt}(x,t)$ and $v_x$ with a constant $N^x$, the FJ action becomes $$\label {}
S_{FJ} = \int dt\,dx\ \partial_tN\partial_xN - N^x(\partial_xN)^2 \,,$$ with an equation of motion $$\label{chiralEOM}
\partial_t R_{xt}(H) - N^x\partial_x R_{xt}(H)=0 \,.$$ We observe that while the first term of (\[chiralEOM\]) is the 1+1 Lifshitz Weyl anomaly, a trivial descent cocycle in the parity-odd, mixed-derivative sector of the cohomology of the Lifshitz Weyl operator, the second term $\partial_x R_{xt}(H)$ is a *coboundary* term that belongs to the parity-even two-spatial derivatives sector. It is interesting to note, as pointed out in [@Maio2000], that in the FJ action, it is as if the chiral boson is propagating in *curved* spacetime with background metric $N^x$.
Note that in deriving the boundary CS action in (\[FJ\_Action\]) from the tCS action (\[tCSAction\]), one usually works in Galilean-boosted coordinates where the temporal component of the gauge field $a_t$ is set to zero (see equations 6.7-6.9 in [@TongNotes]. By doing so, one also sets the velocity of the chiral boson $N^x$ to zero and hence the chiral boson $\rho(x,t)$ is stationary, i.e. with equation of motion ${\partial}_t \rho(x,t)=0$. Analogously, in the process of making the TTNC geometry dynamical, there is complete freedom in deciding the value of $a_t = N^x a_x + N^r a_r$ which fixes the special conformal transformation in the $SL(2,R)$ subgroup of the Schr$\mathrm{\ddot{o}}$dinger algebra [@ObersDynmicTNC2015]. Choosing $N^x = N^r = 0$ directly produces the Lifshitz Weyl anomaly in (\[AnomnalyTorsion\]). On the other hand, setting only $N^r = 0$ with a constant $N^x$ amounts to a boundary condition where $a_t = N^x a_x$ which then adds the *coboundary* term $({\partial}_xN)^2$ to (\[chiralEOM\]) and gives the FJ action in (\[FJ\_Action\]). We would like to further understand the relationship, if any, between the Weyl anomaly of the the $z=1$ Lifshitz theory and the FJ action in the context of the FQHE.
Anomaly Cancellation by Anomaly Inflow
--------------------------------------
In light of the previous discussion and deriving the 1+1 Lifshitz Weyl anomaly from a 3D non-relativistic Abelian tCS action in Section (\[AnomalyDerivation\]) leads one naturally to wonder if the Weyl anomaly is actually somehow related to chiral edge states of a FQH theory. We discuss this possibility here. According to the classification in [@Fradkin15; @AbanovBoundary], four distinct CS terms can appear in the low-energy effective action of the QH state for a microscopic theory with the following symmetries: (1) $ U(1) $ gauge transformations, (2) general covariance, and (3) local $ SO(2) $ rotations. Written in terms of differential forms, the four CS terms are $$\label{CSFQH}
S_{CS} = \frac{\nu}{4\pi}\int_{\mathcal{M}} A \wedge dA
+ 2 \bar{s} A \wedge d\omega + \overline{s^2} \omega \wedge d\omega
+ \frac{c}{96\pi}\int_{\mathcal{M}} \Gamma \wedge \Gamma \wedge \Gamma \,,$$ where $\Gamma^{\mu}{}_{\nu} \equiv \Gamma^{\mu}{}_{\nu\rho}dx^{\rho}$. The first term is the $ U(1) $ electromagnetic Hall conductance term, while the second and third are known as the *Wen-Zee* terms, and the last is the gravitational Chern-Simons (gCS). On a manifold $ \mathcal{M} $ with a boundary, the four CS terms appearing in $ S_{CS} $ defined above are no longer invariant gauge-invariant because boundary terms spoil gauge-invariance. According to [@AbanovBoundary], there are then two possibilities for each CS term: (1) it represents a relevant anomaly of the low-energy effective action that cannot be canceled by adding local boundary terms, or (2) it is a *trivial* anomaly which can be canceled by adding local boundary terms. The electromagnetic Hall conductance and relativistic $gCS$ terms belong to the first. The electromagnetic pure CS term can thus be made invariant as follows $$\label{Anomalyinflow}
\delta_{\sigma} S_{edge} = - \delta_{\sigma} S_{bulk}
= - k\int_{\partial\mathcal{M}}d^2x\, \sigma\,\epsilon^{\alpha\beta}\partial_{\alpha} A_{\beta} \,,$$ where $\sigma$ is the gauge transformation parameter. This is an example of *anomaly inflow* [@Harvey85] where there is an influx of charge into the boundary where they are absorbed by the anomalous gapless edge modes and as a result, the anomaly of the boundary theory gets *canceled* by the total derivative term of a CS action. The Lifshitz Weyl is precisely of that nature $$\delta_{\sigma} W_{edge} = - \delta_{\sigma} S_{tCS}
= -k \int_{{\partial}\mathcal{M}} \sqrt{h} \ N\ \sigma \ dt dx\ \left( \epsilon^{ij}\ {\partial}_i a_j
\right) \,.
\nonumber$$ Since the (1+1) Lifshitz Weyl anomaly, as we discussed in Section \[LifWeylAnomGeom\] is non-trivial, in the sense that it cannot be canceled by a local boundary term, then according the classification by [@AbanovBoundary], it belongs to the first class. Thus, if the microscopic theory of the Abelian QH state, in addition to having the three symmetries in (\[CSFQH\]), is also symmetric under *anisotropic local Weyl* transformations such that $W_{edge} + S_{tCS}$ is Weyl-invariant, could the boundary tCS term $ a\wedge da $ represent a new kind of *torsional anomaly inflow* where torsional (or gravitational) degrees of freedom flow into the Weyl-anomalous boundary Lifshitz effective action? If so, what universal quantity, if any, does the coefficient of the tCS action represent? More importantly, is there is a physical scale-invariant FQH system? Does the NRSCS action contain the two WZ terms in (\[CSFQH\])? We leave these questions for future work.
Another related topic where anomaly cancellation by anomaly inflow is relevant is the thermal Hall effect. In [@Stone12], it was shown that the thermal Hall current does not vanish in equilibrium and hence, *Luttinger’s* idea of coupling the system to a uniform gravitational field that such that the gravitational potential gradient exactly balances out the energy flux induced by a thermal gradient cannot be used and thus the thermal Hall conductance can not be determined by its gravitational counterpart as it was argued before in [@Ryu12]. In other words, it was argued in [@Stone12] that a *uniform* gravitational field can not induce a bulk thermal current and thermal energy must therefore be carried entirely by the (1 + 1)-dimensional edge modes. The relationship the we did in to what we did in Section (\[AnomalyCancellation\]), is to observe that as a result of canceling the Weyl anomaly and restoring Weyl invariance, the system is in *equilibrium*. In fact, if we choose the lapse function $N = e^\psi = x$ such that $a_x = {\partial}_x \psi = \frac{1}{x}$ and the Luttinger potential $\Phi(x) = \psi(x) = $log($x$), then $$N(x) {\partial}_x a_x = - {\partial}_x \Phi(x) \,,$$ $$\frac{1}{T}\frac{{\partial}T}{{\partial}x} = - \frac{{\partial}\Phi}{{\partial}x}\,.$$ if we identify the lapse function with inverse temperature and the torsion with the temperature gradient $$N(x) = \beta(x) = \frac{1}{T(x)} = x, \quad a_x = T(x) = \frac{1}{x} \,.$$ More relevant to the work in this paper is the work in [@AbanovThermal] where a non-relativistic analogue of part of the work in [@Stone12] was introduced. The authors in [@AbanovThermal] coupled a (2+1)-dimensional non-relativistic field theory to a NC geometry with torsion.[^7] However, since TTNC geometry only couples to the energy density, they turned on the spatial components of the timelike vector field $n_{\mu}$ and $n^{\mu}$ to couple to the energy current.[^8] They proceeded then to construct the most general partition function with time-independent, local space and time translations and gauge symmetries. Using the Euclidean path integral to calculate the partition function, the authors in [@AbanovThermal] derived an expression for the thermal current. However, they did not discuss the possibility of Weyl-anomalous effective actions in the context of their work as was done in [@Stone12] where it was shown how the gravitational anomaly of the boundary-induced effective action can be canceled by the inflow of the spatial and temporal components of the bulk energy-momentum tensor computed from the three-dimensional gCS term (\[CSFQH\]). Similarly, understanding the nature of this inflow requires calculating the operator conjugate to $a_{\mu}$ in the bulk which we leave for future work.
Anomaly inflow has also been used to cancel the gravitational anomaly of a chiral field theory and obtain the Hawking radiation as was first discussed in [@Wilczek05] in order who further explored the relationship between Weyl anomalies and the thermal flux of the Hawking radiation as pointed out in [@Fulling77]. Using anomaly inflow, the authors in [@Wilczek05] found that the influx required to cancel the gravitational anomaly at the horizon is proportional to $T^2$ with $T=\frac{\kappa}{2pi}$ which is blackbody radiation with the Hawking temperature. This is interesting since, if we assume the field theory near the black hole horizon is the action in (\[SolodukhinAction\]), then according to the discussion in Sections \[Weyl-Lorentz-Anom-Relation\] and \[AnomalyCancellation\], canceling the Lorentz anomaly of this theory (which we recall can always be traded for a diffeomorphism anomaly by a local counterterm), is equivalent to canceling the Weyl anomaly in a $z=1$ (1+1) Lifshitz theory.
Boundary Conformal Charges
--------------------------
Recently, there has been some activity in constructing boundary conformal charges of type-B anomalies which are known to be Weyl-invariant densities i.e. constructed from the Weyl tensor and its covariant derivatives [@Solod15]. The boundary terms corresponding to the integrated anomalies are of *Gibbons-Hawking-York* (GHY) type, the boundary term that must be added to the Einstein-Hilbert action to compensate for the normal variation of the metric on the boundary and thus have valid variational principle. All the Weyl anomalies of Lifshitz field theories found in [@Oz2014] are type B including the one we discussed in this paper. What is interesting about the 1+1 Weyl anomaly we study here, is that although it is type-B in a Lifshitz field theory, it is related to the dual of a type-A anomaly, the Ricci scalar, whose integral gives the Euler characteristic of the 2-manifold. In addition, the integral of the Weyl anomaly over the boundary gives a topological invariant (\[AnomalyTopo\]). We would like to understand this relationship further as well as the nature of the boundary terms in (\[AnomalyTopo\]) within the context of the discussion in [@Solod15; @Padman14; @Chakra17] especially Section 3 in [@Chakra17] where a comparison between the boundary term of the Einstein-Hilbert action and the standard GHY term has been shown to involve time derivatives of the lapse and shift functions. However, one can perhaps use the duality between the Ricci scalar $R$ and its dual $U$ to make a speculation as to what this topological invariant intuitively means. The foliation decomposition of $R$ in 1+1 dimensions involve terms with two temporal and two spatial derivatives. The GHY boundary term corresponding to $R$ is the trace of the extrinsic curvature (in the conformal gauge) $$\chi = \int _{M} \,d^2x \, e\, R + \int_{{\partial}M} \, d\tau \, \sqrt{h} \, \mathrm{(boundary \, \, term)} = \int _{{\partial}M} \, d\tau \,\sqrt{h} \,\, {\partial}_{\mu} \tilde{n}^{\mu} \,,$$ where $\chi$ is the Euler characteristic of the 2-manifold and $\tilde{n}^{\mu}$ is a unit normal vector. Analogously, when $U$ is projected onto the foliation, it involves terms with mixed temporal and spatial derivatives while its corresponding boundary term is the divergence of a unit tangent vector $\hat{V}^{\mu}$ (\[AnomalyTopo\]). A foliation-tangent vector we have at our disposal in 1+1 dimensions is the $x $-component of the acceleration $a_x$. Therefore, one can speculate that the boundary term corresponding to this new topological invariant may be related to $a_x$.
For future work, constructing a concrete example of $z\geq1$ FPD-invariant (1+1)-dimensional Lifshitz field theory from which the Weyl anomaly discussed in this paper can be explicitly derived using heat kernel methods is important for a deeper understanding of the physics associated with this anomaly as well as the bulk theory. We only hope that the anomaly coefficient does not vanish after all.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank Israel Klich, Shinsei Ryu, Aron Wall, Oleg Lunin and Philip Argyres for valuable and insightful discussions. Special thanks go to Shira Chapman, Niels Obers, Jelle Hartong, and Lei Wang for reviewing large parts of the manuscript and giving constructive comments, suggestions and feedback. I would like to also thank Andrey Gromov for reviewing subsections 5.1 and 5.2 of this paper and providing very interesting ideas for future work.
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[^1]: For details on how the ADM metric is obtained from the fundamental NC objects in the process of gauging the Bargmann algebra, see section 8 in [@ObersDynmicTNC2015].
[^2]: By working in the ADM preferred coordinates, the shift vector $N^x$ can always be removed by an FPD transformation.
[^3]: The generator of scale or dilatation transformation in [@ObersNRSCS] is denoted by $D$ which we reserve here for the curvature of the $a_\mu$ connection.
[^4]: To properly define an asymptotically Lifshitz spacetime, we assume the notion of anisotropic conformal infinity of the $D+1$-dimensional Lifshitz geometry at $r\rightarrow 0$ where there is an asymptotic codimension-one foliation [@Thompson10].
[^5]: A diffeomorphism anomaly in 1+1 CFT can be shifted to a local frame anomaly by a local counterterm.
[^6]: See equation 2.35 in [@Oz2014] for a definition of the foliation-projected covariant derivative.
[^7]: Note that in [@AbanovThermal], the torsion tensor is the curvature in the Hamiltonian $R_{xt}(H) = {\partial}_x\psi e^{\psi}$
[^8]: As noted in [@Son13], turning on the spatial components of the 1-form $n_{\nu}$ does not violate the Frobenius condition.
|
---
abstract: |
Spectral and transport properties of small molecule single-crystal organic semiconductors have been theoretically analyzed focusing on oligoacenes, in particular on the series from naphthalene to rubrene and pentacene aiming to show that the inclusion of different electron-phonon couplings is of paramount importance to interpret accurately the properties of prototype organic semiconductors. While, in the case of rubrene, the coupling between charge carriers and low frequency inter-molecular modes is sufficient for a satisfactory description of spectral and transport properties, the inclusion of electron coupling to both low frequency inter-molecular and high frequency intra-molecular vibrational modes is needed to account for the temperature dependence of transport properties in smaller oligoacenes.\
For rubrene, a very accurate analysis in the relevant experimental configuration has allowed to clarify the origin of the temperature dependent mobility observed in these organic semiconductors. With increasing temperature, the chemical potential moves into the tail of the density of states corresponding to localized states, but this is not enough to drive the system into an insulating state. The mobility along different crystallographic directions has been calculated, including vertex corrections that give rise to a transport lifetime one order of magnitude smaller than the spectral lifetime of the states involved in the transport mechanism. The mobility always exhibits a power-law behavior as a function of temperature in agreement with experiments in rubrene.\
In systems gated with polarizable dielectrics, the electron coupling to interface vibrational modes of the gate has to be included in addition to the intrinsic electron-phonon interaction. While the intrinsic bulk electron-phonon interaction affects the behavior of mobility in the coherent regime below room temperature, the coupling with interface modes is dominant for the activated high temperature contribution of localized polarons.\
Finally, the effects of a weak disorder largely increase the activation energies of mobility and induce the small polaron formation at lower values of electron-phonon couplings in the experimentally relevant temperature window.
author:
- 'C. A. Perroni $^{1,}$\*, F. Gargiulo $^{2}$, A. Nocera $^{3}$, V. Marigliano Ramaglia $^{1}$ and V. Cataudella $^{1}$'
title: 'Effects of different electron-phonon couplings on spectral and transport properties of small molecule single-crystal organic semiconductors'
---
Introduction
============
In recent years, the interest in plastic electronics has grown considerably. The realization of devices such as organic field-effect transistors (OFET) represents a key step in this field. Single-crystal OFET made of ultrapure small molecule semiconductors are characterized by mobilities up to one order of magnitude larger then those typical of thin film transistors [@takeya]. The most promising are those based on oligoacenes, such as pentacene and rubrene, which exhibit a strong anysotropy and the largest mobility measured in organic semiconductors [@morpurgo].
In spite of many applications based on such devices, the intrinsic transport mechanism acting in high mobility organic semiconductors is not fully understood. Transport measurements from $100$ K to room temperature in single crystal semiconductors, such as rubrene, show a behaviour of the charge carrier mobility $\mu$ which can be defined band-like ($\mu\propto T^{-\gamma}$, with the exponent $\gamma$ close to two) similar to that observed in crystalline inorganic semiconductors [@morpurgo]. However, the order of magnitude of mobility is much smaller than that of pure inorganic semiconductors, and the mean free path for the carriers has been theoretically estimated to be comparable with the molecular separation at room temperature [@Cheng]. Therefore, the Ioffe-Regel limit is reached with increasing temperature. Moreover, in some systems, starting from room temperature, a crossover from band-like to activated hopping behavior can take place [@corop; @cheng1; @hanne3]. The crossover has been interpreted as due to the formation of the polaron, that is the quasi-particle formed by the electron (or hole) and the surrounding phonon cloud [@alex]. For example, in naphthalene and anthracene, while the mobility along the $a$ and $b$ axis shows only a slightly change with the temperature, that along the $c$ axis is characterized by a temperature activated behavior at higher temperature with energy barrier of the order of $15$ meV [@warta]. The experimental data in these compounds suggest that the coherent band transport is gradually destroyed and the transport due to polaron hopping evolves as a parallel channel dominating at sufficiently high temperature (which can be larger than room temperature) [@warta].
![\[naphta\] Sketch about the interplay between low frequency inter-molecular and high frequency intra-molecular vibrational modes in a crystal of naphtalene. The charge carrier deforms the benzene ring when it is on the molecule. Moreover, the charge carrier displaces two neighbor molecules when it jumps.](Fig1.eps){width="75.00000%"}
In systems with polarizable gates, scaling laws of the mobility as a function of the dielectric constant of solid [@stassen; @nature] and liquid [@ono] gates have been discovered pointing out that the nearby dielectric has a strong influence. Actually, if the difference between the dielectric constant of the organic semiconductor and of the gate is small, at temperatures close or higher than $100$ K, the mobility $\mu$ of these systems exhibits the power-law band-like behavior. On the other hand, if the dielectric constant mismatch is high, an activated insulating behavior is found with much smaller values of mobility at room temperature [@nature]. A possible explanation of this behavior is that the injected charge carriers undergo a polaronic localization due to the interaction with modes at the interface with the polarizable dielectric gate [@nature; @fratini].
Extended vs. localized features of charge carriers appear also in spectroscopic observations. Angle resolved photoemission spectroscopy (ARPES) supports the extended character of states [@Ostrogota; @Ostrogota1; @arpes2] showing that the quasi-particle energy dispersion does exhibit a weak mass renormalization even if the width of the peaks of the spectral function increases significantly with temperature. For pentacene, the bandwidth is reduced only by about $15 \%$ going from $75 K$ to $300 K$ indicating moderate values of electron-phonon coupling. On the other hand, some spectroscopic probes, such as electron spin resonance (ESR) [@Marumoto1; @Marumoto2], THz [@Laarhoven], and modulated spectroscopy [@Sakanoue] are in favor of states localized within few molecules. Actually, in rubrene and in pentacene, to ascribe the presence of localized features to small polarons is not likely since the electron-phonon coupling is not large enough to justify the polaron formation [@corop]. Therefore, one of the main theoretical problems is to conciliate band-like with localized features of charge carriers [@Troisi; @Orlandi].
First-principle calculations have pointed out that charge carriers are affected by the coupling to inter-molecular modes with low frequency in comparison with typical electron hoppings [@corop; @bredas]. A model that is to some extent close to the Su-Schrieffer-Heeger (SSH) [@SSH] hamiltonian has been recently introduced to take into account this interaction [@Troisi; @Orlandi]. It is a minimal one-dimensional (1D) system, valid for the most conductive crystal axis of high mobility systems, where the effect of the electron-phonon coupling is reduced to a modulation of the transfer integral [@Troisi; @Rubrene]. A dynamic approach where vibrational modes are treated as classical variables has been used in 1D and in a recent generalization to two dimensions (2D) [@Troisi2D]. Within this method, the temperature dependence of computed mobility is in agreement with experimental results. However, the role of dimensionality of the system is not clear: in fact, in the 1D case, one has $\mu\propto T^{-2}$, while, in the 2D case, the decrease of the mobility with temperature is intermediate between $\mu\propto T^{-2}$ and $\mu\propto T^{-1}$. In any case, the computed mobility is larger than that measured (at least a factor of two). Moreover, the dynamics of only one charge particle is studied neglecting completely the role of the chemical potential. Finally, the effects on charge carrier dynamics due to the coupling with vibrational modes are included in an approximate way [@licorop] and the corresponding coupled dynamics do not recover the right thermal equilibrium on long times.
Recently, the transport properties of the 1D SSH model have been analyzed within a different adiabatic approach [@Ciuchi; @Fratini] mapping the problem onto that of a single quantum particle in a random potential (generalized Anderson problem [@Anderson]). Very recently, some of us have made a systematic study of this 1D model including the vertex corrections into the calculation of the mobility [@vittoriocheck]. While finite frequency quantities are properly calculated in this 1D model, the inclusion of vertex corrections leads to a vanishing mobility unless an ad-hoc broadening of the energy eigenvalues is assumed.
It is clear that 1D adiabatic models suffer of severe limitations, of which the main is that electronic states are always localized [@Anderson]. Moreover, features such as band anisotropy, small but finite carrier density are necessary for a correct description of the systems. Therefore, in this review, we first analyze a generic three-dimensional (3D) model such as the anisotropic Holstein model [@hannewald] in order to discuss the relevant issue of the band anisotropy at finite carrier density. This model is studied within the adiabatic approach focusing on the weak to intermediate el-ph coupling regime which is relevant for high mobility organic semiconductors [@meholstein]. Next, we analyze a realistic model for rubrene which represents an extension of the 1D SSH model to the quasi 2D case since this is the relevant geometry for OFET [@fernando].
Spectral and transport properties calculated within these two models are discussed in this review. The spectral functions show peaks which are weakly renormalized in comparison with those of the bare bands. However, with increasing temperature, the width of the spectral functions gets larger and larger making the quasi-particles less defined. The marked width of the spectral functions gives rise to densities of states with a low energy exponential tail increasing with temperature. At low temperatures, this tail corresponds to localized states and gives rough indications for the energy position of the mobility edge. With increasing temperature, in the regime of low carrier doping appropriate to most OFET, the chemical potential always enters the energy region of the tail. The features of the spectral function and the behavior of the chemical potential allow to reconcile the band-like description (ARPES data) with the finding that charge carriers appear more localized at high temperature (ESR and modulated spectroscopy data). The study of spectral properties also clarifies that the states that mainly contribute to the conduction process have low momentum and are not at the chemical potential. The mobility $\mu$ is studied as a function of the electron-phonon coupling, the temperature and particle density. Not only the order of magnitude and the anisotropy ratio between different directions are in agreement with experimental observations, but also the temperature dependence of $\mu$ is correctly reproduced in the model for rubrene since it scales as a power law $T^{-\gamma}$, with $\gamma$ close or larger than two. The inclusion of vertex corrections in the calculation of the mobility is relevant, in particular, to get a transport lifetime one order smaller than the spectral lifetime of the states involved in the transport mechanism. Moreover, with increasing temperature, the Ioffe-Regel limit is reached since the contribution of itinerant states to the conduction becomes less and less relevant.
[*Ab-initio*]{} calculations have clarified that charge carriers in organic semiconductors are not only coupled to low frequency inter-molecular modes, but also to intra-molecular modes with high frequency in comparison with typical electron hoppings [@corop; @hanne2] (see Fig. \[naphta\] for a sketch in naphtalene crystal). An important point is that the reorganization energy (related to the polaron binding energy) decreases with increasing the number of benzene rings in oligoacenes (for example, going from naphthalene to pentacene). In order to fully explore the effects of the different modes on prototype single crystal organic semiconductors, such as oligoacenes, a model with intermediate coupling to both intra- and inter-molecular modes is analyzed in this review [@meholpssh]. We will show that the interplay between local and non local electron-phonon interactions is able to provide a very accurate description of the mobility and to shed light on the intricate mechanism of band narrowing with increasing temperature [@arpes2].
When the organic semiconductor is grown on a polarizable gate, it is important to analyze the effects of electron coupling to surface vibrational modes of the gate at the interface with the semiconductor mediated by a long-range electron-phonon interaction [@substrato; @mebipo]. In this review, we analyze a model which combines the effects of interface and intrinsic bulk electron-phonon couplings on the transport properties at finite temperature. We show that the coupling to the organic semiconductor bulk phonon modes affects the behavior of mobility below room temperature enhancing the coherent contribution, but it is ineffective on the incoherent small polaron contribution dominated by the interface coupling at high temperatures.
In order to improve the modeling of organic semiconductors, the effect of a weak disorder due to bulk and interface traps is included [@substrato]. In particular, the interplay between long-range electron-phonon interactions and disorder effects is investigated within a model. The disorder effects are able to enhance the hopping barriers of the activated mobility and to drive the small polaron formation to lower values of electron-phonon interactions. We point out that disorder is a key factor to get agreement with experimental data in rubrene OFET grown on polarizable gate dielectrics, such as the $Ta_2 O_5$ oxide [@nature].
The paper is organized in the following way. In section II, the effects of electron coupling to low frequency vibrational modes on the spectral and transport properties are discussed in high-dimensional Holstein-like and SSH-like models. In section III, the effects of electron coupling to both low frequency inter-molecular and high frequency intra-molecular modes on the spectral and transport properties are investigated. In section IV, the influence of gates made of polarizable dielectrics and the interplay between electron-phonon couplings and disorder strength on the transport properties are emphasized. In section V, conclusions and final discussions.
Effects of low frequency vibrational modes
==========================================
The anisotropy of the electronic properties is a key ingredient in the description of organic semiconductors [@hannewald]. Therefore, in subsection $2.1$, we will introduce a simple anisotropic tight-binding model including a Holstein-like electron coupling to low frequency modes in order to focus on the effects of electronic structure. However, the results will be discussed later in subsection $2.4$ in comparison with those obtained by a more detailed model based on inter-molecular low frequency modes that will be introduced in subsection $2.2$.
Band anisotropy
---------------
We assume the following model hamiltonian: $$H= - \sum_{\vec{R}_i, \vec{\delta}} t_{\vec{\delta}} c_{\vec{R}_i}^{\dagger}c_{\vec{R}_i+\vec{\delta}}+
\sum_{\vec{R}_i} \frac{{p}^2_{\vec{R}_i}}{2 m}+\sum_{\vec{R}_i} \frac{ k x_{\vec{R}_i}^{2}}{2}+H_{el-ph},
\label{h}$$ where $t_{\vec{\delta}}$ is the bare electron hopping toward the nearest neighbors $\vec{\delta}$, $c_{\vec{R}_i}^{\dagger}$ and $c_{\vec{R}_i}$ are the charge carrier creation and annihilation operators, respectively, relative to the site $\vec{R}_i$ (of a cubic lattice with parameter $a$ for the Holstein model and of an orthorombic lattice with constants $a$, $b$, $c$ for the rubrene model of the next section), $x_{\vec{R}_i}$ and ${p}_{\vec{R}_i}$ are the oscillator displacement and momentum, respectively, $m$ the oscillator mass, and $k$ the elastic constant. In eq.(\[h\]), $H_{el-ph}$ represents the electron-phonon coupling term.
A simple model able to capture the anisotropy of the electronic properties of these materials is the anisotropic Holstein hamiltonian which is generic for high mobility organic semiconductors such as oligoacenes [@meholstein]. This model assumes a cubic lattice with the following anisotropic hopping integrals in Eq. (\[h\]): $t_z \simeq$ 100 meV, $t_x \simeq 50$ meV, $t_y \simeq 20$ meV. Moreover, typical values of phonon frequency $\omega_0 =\sqrt{k/m}$ are of the order of $10 meV$ leading to a very low adiabatic ratio $\gamma=\omega_0/t_z = 0.1$ (adiabatic regime) [@hannewald]. Finally, we assume a very general electron-phonon interaction inspired by Holstein model [@hannewald; @holstein1; @holstein2]. Therefore, in Eq.(\[h\]), the electron-phonon hamiltonian is $$H_{el-ph}=\alpha \sum_{\vec{R}_i} x_{\vec{R}_i} c_{\vec{R}_i}^{\dagger} c_{\vec{R}_i},
\label{hel}$$ where $\alpha$ is the coupling constant controlling the link between the local electron density and lattice displacement. The following dimensionless quantity $\lambda_{Hol}$ $$\lambda_{Hol}=\frac{\alpha^2}{4 k t_z} \label{lamb}$$ correctly describes the strength of the electron-phonon coupling.
Inter-molecular vibrational modes
---------------------------------
In this section, we analyze the influence of the electron coupling to low-frequency inter-molecular modes on the properties of small molecule organic semiconductors.
We consider a realistic quasi-2D model which simulates the properties of rubrene [@fernando]. Therefore, it not only includes the anisotropy of rubrene crystals (shared with many other small molecule organic semiconductors), but also a more appropriate electron-phonon coupling. First, we derive the effective low-energy electronic model, then the lattice parameters and the appropriate electron-phonon interaction.
![\[figrub\] (a) Molecular structure of rubrene. (b) Crystal structure along the (a,b) crystallographic plane. The red lines denote the long axis of the molecule.](Fig2.eps){width="75.00000%"}
Since we are mostly interested in $dc$ conductivity and low frequency spectral properties, we need to determine an effective Hamiltonian for the electron degrees of freedom valid for low energy and particle density. We start from the orthorhombic lattice of rubrene with two molecules per unit cell and $a$, $b$, $c$ lattice parameter lengths along the three crystallographic vectors of the conventional cell [@corop]. We follow the ARPES experiments by Ding [et al.]{} [@Ostrogota] in order to extract the transfer integrals of the highest occupied molecular orbital (HOMO) bands. The dispersion law of the lowest HOMO is accurately fitted with nearest neighbors tight binding parameters for small values of momentum providing the following estimates: $t_{a}=118.6meV$ and $t_{b}=68.6meV$ [@fernando]. For $t_{c}$ there is no experimental measure but theoretical estimates seem to agree that it has to be small compared with other directions owing to the large interplanar separation of rubrene. In the following, we assume $t_{c}$ much smaller than $t_{a}$ and $t_{b}$ (in the following we assume $t_c=0.18$ $t_{a}$ ). The total volume is $V=L_{a}*L_{b}*L_{c}$, and $L_i$ size along the axis $i=a,b,c$. We consider two crystalline layers along c because in OFET the effective channel of conduction covers only few planes [@Shehu].
In Eq.(\[h\]), the whole lattice dynamics is ascribed to an effective phononic mode whose frequency $\omega_{0}=\sqrt{K/M}$ is assumed of the order of $5-6meV$ [@Troisi; @Rubrene]. This assumption provides the low adiabatic ratio $\hbar\omega_0/t_{a}\simeq0.05$ (adiabatic regime).
The electron-phonon interacton $H_{el-ph}$ in Eq.(\[h\]) is $$H_{el-ph}=\sum_{\vec{R}_i, \vec{\delta}} \alpha_{|\vec{\delta}|} \left( x_{\vec{R}_i}- x_{\vec{R}_i+\vec{\delta}} \right)c_{\vec{R}_i+\vec{\delta}}^{\dagger}c_{\vec{R}_i}, \quad\vec{\delta}=\vec{a},\vec{b},\vec{c},\label{eq:Tight binding SSH}$$ where $\alpha_{\vec{\delta}}$ is electron-phonon parameter controlling the effect of the ion displacements in the direction $\vec{\delta}$ on the transfer integral. Once fixed $\alpha_{a}$, we impose $\alpha_{b}/\alpha_{a}=t_{b}/t_{a}$ and, in the same way, $\alpha_{c}/\alpha_{a}=t_{c}/t_{a}$. The dimensionless quantity $\lambda$ $$\lambda=\frac{\alpha_{a}^{2}}{4kt_{a}}\label{eq:coupling}$$ is the relevant parameter to quantify the electron-phonon coupling strength. By comparing ab-initio calculations and average properties of a simple 1D model, Troisi et al. provided an estimate of $\lambda \simeq 0.087$[@Troisi; @Rubrene; @Bologna; @ab-initio; @Bologna; @Raman]. However, in our 3D model the 1D estimate is not correct. Indeed, the hopping $t_b$ is about half $t_a$ and the average kinetic energy is, then, larger with respect to the one-dimensional case. In order to reproduce the same effective one dimensional $\lambda$, we have chosen the larger value $\lambda=0.12$.
In the following part of the section, we use units such that lattice parameter $a=1$, Planck constant $\hbar=1$, Boltzmann constant $k_B=1$, and electron charge $e=1$.
Calculation method
------------------
For the organic semiconductors studied in this paper, the adiabatic ratio is very low implying that the adiabatic limit is appropriate for studying low-frequency intermolecular modes. Consequently, it is possible to adopt a semiclassical approach: the electron dynamics is fully quantum, while the ion dynamics is assumed classical. This assumption will limit the temperature range where the results can be considered valid, indeed all the results are valid for temperatures $T \geq \omega_0 \simeq 100 K $. Finally, we will focus on the weak to intermediate electron-phonon regime that seems to be appropriate for high mobility organic semiconductors [@corop].
Within the adiabatic regime, the calculation is equivalent to the classical problem of quantum particles in the presence of an external disordered potential given by the ion displacements $\{ x_{\vec{R}_i} \}$ and controlled by electron-phonon coupling. Each configuration of ion displacements is generated according to the probability function of the $P \left( \{ x_{\vec{R}_i} \} \right)$, that has to be self-consistently calculated as a function of electron-phonon coupling, temperature and particle density $n=N_p/V$, with $N_p$ number of particles. The adiabatic approach has been also used for the study of molecular junctions and carbon nanotubes at thermodynamical equilibrium and in non-equilibrium conditions [@alberto1; @alberto2; @alberto3; @mepumping1; @mepumping2].
In most OFET, the induced doping is not high, therefore, in the following, we will focus on the regime of low doping (up to $n=0.01$). For this regime of parameters, the probability function of the lattice displacements $P \left( \{ x_{\vec{R}_i} \} \right)$ shows very tiny deviations from the distribution of free oscillators [@vittoriocheck], which is therefore used in the following subsection. Quantities, like spectral function, density of states, and conductivity are calculated through exact diagonalization of the resulting electronic problem at fixed displacements $\{ x_{\vec{R}_i} \}$ and through Monte-Carlo approach for the integration over the distribution $P \left( \{ x_{\vec{R}_i} \} \right)$. In the case of conductivity, for each configuration of the lattice displacements, we calculate the exact Kubo formula [@mahan] $$Re\left[\sigma_{\rho,\rho}\left(\omega\right)\left(\left\{ u_{\mathbf{R}_{i}}\right\} \right)\right]= \frac{\pi\left(1-e^{-\beta\omega}\right)}{V\omega}\sum_{r\neq s}p_{r}\left(1-p_{s}\right)\nonumber \\
\left|\left\langle r\left|J_{\rho}\right|s\right\rangle \right|^{2}\delta\left(E_{s}-E_{r}+\omega\right),\label{eq:Kubo formula}$$ where $\rho=a,b,c$, $\beta=1/T$, and $p_{r}$ is the Fermi distribution $$p_{r}=\frac{1}{1+\exp\left(\beta\left(E_{r}-\mu_{p}\right)\right)}\label{eq:Fermi factor}$$ corresponding to the exact eigenvalue $E_{r}$ at any chemical potential $\mu_{p}$. Finally $\left\langle r\left|J_{\rho}\right|s\right\rangle $ is the matrix element of the current operator $J_{\rho}$ along the direction $\hat{e}_{\rho}$, defined as $$J_{\rho}=i\sum_{\vec{R}_{i},\vec{\delta}}\bar{t}_{\vec{\delta}}\left(\mathbf{R}_{i}\right)
\left(\vec{\delta}\cdot\hat{e}_{\rho}\right)c_{\vec{R}_{i}}^{\dagger}c_{\vec{R}_{i}+\vec{\delta}},
\label{current}$$ with $$\bar{t}_{\vec{\delta}}\left(\mathbf{R}_{i}\right)=t_{\vec{\delta}}-\alpha_{\vec{\delta}}
\left(u_{\mathbf{R}_{i}}-u_{\mathbf{R}_{i}+\vec{\delta}}\right), \quad\vec{\delta}=\vec{a},\vec{b},\vec{c}.$$ We notice that, in contrast with spectral properties, the temperature enters the calculation not only through the displacement distribution, but also directly for each configuration through the Fermi distributions $p_{r}$. We point out that the current-current correlator is not evaluated at the lowest order as convolution of two single-particle Green functions, but the linear response conductivity is exactly calculated within the Kubo formulation presented in Eq. (\[current\]) [@mahan]. Therefore, the numerical calculation of the conductivity is able to include the vertex corrections (terms in the correlator beyond the convolution of two Green functions) discarded by previous approaches [@mahan].
Finally, the mobility $\mu$ is calculated as the ratio between zero-frequency conductivity and carrier density: $$\mu_{\rho}=\lim_{\omega\rightarrow0^{+}}\frac{Re\left[\sigma_{\rho,\rho}\left(\omega\right)\right]}{n}.\label{eq:Mobility}$$
Summarizing, the numerical method provides approximation-free results in the adiabatic regime. The only limitation is due to the computational time being controlled by matrix diagonalizations.
Results about spectral and transport properties
-----------------------------------------------
![\[fig1\] The spectral function (in units of $1/t_a$) for rubrene model at momentum $\mathbf{k}=0$ as a function of the frequency (in units of $t_a/\hbar$) at $\lambda=0.12$ and $n=0.002$ for different temperatures T. We assume the hopping parameter $t_c=0.18 t_a$, with $t_{a}=118.6meV$.](Fig3.eps){width="75.00000%"}
Spectral and transport properties, studied within the two models based on Eq. (\[h\]), will be discussed in this subsection. The study of spectral properties is important to individuate the states that mainly contribute to the conduction process. The spectral properties within the anisotropic Holstein model bear strong resemblance with the rubrene model, so that we will discuss only this last model.
In Fig. \[fig1\], we report the spectral function at momentum $\mathbf{k}=0$ for the model parameters of rubrene. We point out that states close to $\mathbf{k}=0$ are weakly damped. Moreover, with increasing temperature, the peak position of the spectral function is only poorly renormalized in comparison with the bare one, in agreement with results of 1D SSH model [@vittoriocheck]. Therefore, these states keep the itinerant character of the bare ones, and they will be involved into the diffusive conduction process (see discussion in the next paragraph). We notice that the spectral function at $\mathbf{k}=0$ is extremely small at the chemical potential $\mu_{p}$ for all the temperatures. For example, at $T=275$ K, the spectral weight is concentrated in an energy region higher than that in which the chemical potential is located ($\mu_{p}=-3.74 t_a$ for $n=0.002$). Actually, the spectral weight is mainly in the region between $\mu_{p}+2T$ and $\mu_{p}+3T$.
![\[fig2\] The DOS (in units of $1/t_a$) for rubrene model as a function of the frequency (in units of $t_a/\hbar$) at $\lambda=0.12$ and $n=0.002$ for different temperatures T. The squares indicate the chemical potential $\mu_{p}$ (in units of $t_a$) at fixed temperature. $E_{c}$ (dot line, in units of $t_a$) is the free electron band edge close to the mobility edge. We assume the hopping parameter $t_c=0.18 t_a$, with $t_{a}=118.6meV$. ](Fig4.eps){width="75.00000%"}
We have checked that the spectral functions with low momentum are more peaked, while, with increasing $\mathbf{k}$, they tend to broaden. The density of states (DOS) can be calculated as the sum of the spectral functions $A_{\mathbf{k}}$ over all the momenta $\mathbf{k}$. The tail in the DOS is due to the marked width of the high momentum spectral functions. In Fig. \[fig2\], the DOS is shown for different temperatures at $\lambda=0.12$ and $n=0.002$. As shown in logarithmic scale, the DOS has a tail with a low energy exponential behavior. This region corresponds to localized states [@economou]. We have, indeed, checked analyzing the wave functions extracted from exact diagonalizations that, actually, states with energies deep in the tail are strongly localized (one or two lattice parameters along the different directions as localization length). On the other hand, close to the shoulder ($\omega \simeq -3.4 t_a$, see Fig. \[fig2\]), the itinerant nature of states is clearly obtained. This analysis allows to give an estimation for the mobility edge energy (the energy that divides localized and itinerant states) which can be located very close to the band edge $E_{c}$ for free electrons (in our case, $E_{c}=-3.52 t_a$).
The number of localized states available in the tail increases with temperature. It is important to analyze the role played by the chemical potential $\mu_{p}$ with varying the temperature. Actually, $\mu_{p}$ enters the energy tail and will penetrate into it with increasing temperature. At fixed particle density $n=0.002$, for $T=165$ K, one has $\mu_{p}=-3.49 t_a$, while, for $T=330$ K, $\mu_{p}=-3.88 t_a$ (see squares of Fig. \[fig2\] for the values of the chemical potential). One important point is that the quantity $E_{c}$ and the close mobility edge are significantly larger than $\mu_{p}$. Therefore, in the regime of low density relevant for OFET, the itinerant states are not at $\mu_p$, but at higher energies. We point out that those are the states relevant for the conduction process. Therefore, the analysis of the properties of a high dimensional model points out that both localized and itinerant states are present in the system. This is a clear advantage of our work over previous studies in low dimensionality [@Troisi; @Orlandi; @Troisi2D; @Ciuchi; @Fratini] in which all states are localized: more localized at very low energy and less localized close to the free electron edge. Summarizing, in our system, with increasing temperature, all the states up to $\mu_{p}$ become localized and the itinerant states become statistically less effective due to the behavior of the chemical potential. Eventually, the effect of penetration of $\mu_{p}$ in the tail, due to the Fermi statistics, will overcome the effects of available itinerant states around $E_c$. We have checked that the penetration of the chemical potential towards the energy region of the tail is enhanced with increasing the electron-phonon coupling.
![\[fig3\] Upper Panel: Mobility along the z direction within the anisotropic Holstein model as a function of temperature T for different particle densities at $\lambda_{Hol}=0.8$. Lower panel: Mobility along the $a$ direction within the rubrene model as a function of temperature T for different particle densities at $\lambda=0.12$. We assume the hopping parameter $t_c=0.18 t_a$, $t_{a}=118.6meV$.](Fig5.eps){width="56.00000%"}
We devote the last part of the subsection to analyze the transport properties in both the anisotropic Holstein model and the rubrene model. As shown in the upper panel of Fig. \[fig3\], we first discuss the mobility of the anisotropic Holstein model along $z$ direction as a function of temperature with changing the particle density. Within this model, the carrier density strongly affects the behavior of mobility showing a cross-over from metallic to insulating behavior at low temperatures. For densities around one per cent, the chemical potential at low temperature is above the mobility edge and, then, the mobility is metallic-like. On the other hand the situation is different for densities below one per cent. At those densities, the chemical potential is below the mobility edge, so that mobility shows an insulating character. Translated in a polaronic framework, this would mean that the single particle sees a potential well due to electron-phonon coupling, hence, at very low densities, a polaron-like activated mechanism could set in. Actually, the mobility results are consistent with those obtained within the picture of polaron formation [@hannewald].
The strong charge density dependence on the mobility is, instead, lost at higher temperatures where the chemical potential is always in the tail regions for all the charge densities studied, but more states at higher energies (of itinerant nature) get involved providing the main contribution to mobility. As a consequence of the different behavior with charge densities at low and high temperatures, the mobility changes its character at intermediate temperatures for low charge carrier densities.
At high temperatures, the mobilities look very similar for all the densities considered in the upper panel of Fig. \[fig3\]. At room temperature, the mobility is about $20 cm^2 /(V \cdot s)$, a value that recovers the right order of magnitude of experimental data in oligoacenes [@morpurgo]. However, the mobility decreases with the temperature as $1/T$, not in agreement with measurements in oligoacenes, such as rubrene. Moreover, in this model, polaronic localization seems to take place in the low temperature range for enough low charge densities even if experimental data do not seem to support this scenario. Therefore, a more accurate model for the electron-phonon coupling is needed. As already discussed in subsection $2.2$, the SSH-like coupling with intermolecular vibrational modes is what we need. In the following part of this subsection, we will analyze the transport properties of that model.
In the lower panel of Fig. \[fig3\], the mobility along the $a$ direction is reported as a function of the temperature at fixed coupling $\lambda=0.12$ and different concentrations $n$. The plot shows that the absolute magnitude of the mobility substantially agrees with the experimental estimates being $\mu\simeq10$ $cm^{2}/(V s)$ at room temperature. Furthermore, the mobility exhibits a band-like power-law $T^{-\gamma}$ behaviour for all the concentrations. The exponent $\gamma$ is evaluated from fits of the mobility providing values in the range $2-2.4$, where the highest value is related to the lowest concentration. This trend is in agreement with experimental measures that for rubrene establish $\gamma\simeq2$ for temperatures $T>170-180K$ [@morpurgo]. A feature, in contrast with the Holstein model, is that the mobility increases with decreasing the concentration of carriers. This trend, already found in 1D SSH model [@vittoriocheck], points out that the there is no room for a polaronic (bond) localization [@capone] within the regime of rubrene parameters explored in this review.
Another important property is the anisotropy of the transport properties along different in-plane directions [@hanne1]. In our model for rubrene, the anisotropy of the mobility along different cristallographic directions is essentially the same as the anisotropy of the effective mass. From the estimates of in-plane hoppings $t_a$ and $t_b$, the anisotropy of the mobility is evaluated to be of the order of $(t_b/t_a)^2= 0.335$. From experiments [@morpurgo], the anisotropy ratio is about $0.375$ at room temperature, therefore in agreement with our estimate. However, the experimental data shows that this ratio increases with decreasing temperature. In a recent paper [@ishii], where the electron-phonon coupling is slightly more complex than the SSH-like interaction considered in our model, the anisotropy of the mobility is less marked than that of the mass. Within this paper, the anisotropy ratio between the two in-plane mobilities is calculated to be of the order of 0.44 at room temperature (higher than the experimental value), and 0.5 at T=150 K, therefore, it increases with decreasing temperature in agreement with experimental trend. Even if the electron-phonon models are slightly different, the conclusions of our work and these recent calculations are qualitatively consistent. Finally, it has been suggested [@hanne6] that the change in mobility anisotropy upon temperature variation can be explained by a change in transport characteristics (from band transport to hopping).
Starting from the mobility, we can determine the scattering time $\tau_{tr}$ from the relation $\mu=e\tau_{tr}/m$. Since the mass is weakly renormalized from the electron-phonon interaction, one can assume $m$ as the bare mass at $\mathbf{k}=0$. We point out that $\tau_{tr}$ is on the scale of the fs, so that it is one order of magnitude lower than the damping time of the states important for the spectral properties (on the scale of ten fs). Therefore, the transport processes amplify the effects of the electron-phonon interaction and the vertex corrections introduced within our approach are fundamental to take into account this effect.
From the scattering time, one can deduce the mean free path as $l_{tr}=v_{av}\tau_{tr}$, where $v_{av}$ is the average velocity of the charge carriers. The quantity $l_{tr}$ is always on the scale of a few lattice parameters. The most important feature is its temperature behavior. As a consequence of the electron-phonon effects, close to room temperature, it becomes of the order of half lattice parameter $a$. This means that the Ioffe-Regel limit is reached [@gunnarsson]. The decrease of the mobility in the Ioffe-Regel limit is not due to a mass renormalization (dynamic and/or static) but it is due to a reduction of the available itinerant states (the only ones able to transport current) with the temperature. We remark that this result is due to the fundamental role played by vertex corrections (introduced in the previous subsection about the computational methods) in the calculation of the mobility.
Effects of combined low frequency inter-molecular and high frequency intra-molecular vibrational modes
======================================================================================================
For rubrene and pentacene, the carrier mobility is dominated by inter-molecular phonons since the interaction with intra-molecular modes is almost negligible [@Troisi; @Rubrene] and, then, the model discussed in the previous section is considered adequate. On the other hand, in other oligoacenes with a smaller number of benzene rings, the coupling with local modes cannot be neglected [@corop].
Indeed, the decrease of the number of benzene rings affects the reorganization energy, which can be related to the binding energy of the polaron, that is the quasi-particle formed by the electron (or hole) and the surrounding phonon cloud. Actually, going from pentacene to naphthalene, the reorganization energy increases nearly twice suggesting a much stronger coupling with local modes [@corop]. Therefore, for systems with reduced number of benzene rings like naphthalene, we expect a larger interplay between intra- and inter-molecular modes (see Fig. \[naphta\]) within the intermediate electron-phonon coupling regime for both modes. The next step of the review is to combine the effects of high frequency local vibrational (antiadiabatic) modes with non local low frequency (adiabatic) ones [@meholpssh]. For computational ease, we will restrict our analysis to the 1D case assuming the inter-molecular phonons classical, but considering the intra-molecular modes fully quantum.
Model hamiltonian
-----------------
We consider a one-dimensional model with coupling to intra- and inter-molecular modes [@meholpssh; @piegari] similar to one recently introduced, where the treatment only concerns the study of spectral properties [@ciuchi]. The coupling to intra-molecular modes is Holstein-like, that to inter-molecular modes is SSH-like (see Fig \[naphta\] for a sketch of the two couplings). It can be summarized in the following hamiltonian: $$H= H_{el}^{(0)}+H_{Intra}^{(0)}+H_{Inter}^{(0)}+H_{el-Intra}+H_{el-Inter}.
\label{hgen}$$
In Eq. (\[hgen\]), the free electronic part $H_{el}^{(0)}$ is $$H_{el}^{(0)}=-t \sum_{i} \left( c_{i}^{\dagger}c_{i+1}+ c_{i+1}^{\dagger}c_{i} \right),
\label{helgen}$$ where $t$ is the bare electron hopping between the nearest neighbors on the chain, $c_{i}^{\dagger}$ and $c_{i}$ are the charge carrier creation and annihilation operators, respectively, relative to the site $i$ of a chain with lattice parameter $a$. For the transfer hopping the [*ab-initio*]{} estimate is: $t \simeq 50-100 meV$ [@corop]. We consider a single-band one-dimensional electronic structure since it represents the simplest effective model in anisotropic organic semiconductors to analyze the low energy features responsible for the mobility properties.
In Eq. (\[hgen\]), $H_{\alpha}^{(0)}$, with $\alpha= Intra, Inter$, is the Hamiltonian of the free optical molecular modes $$H_{\alpha}^{(0)}= \sum_{i} \frac{{p}^2_{\alpha,i}}{2 m_{\alpha}}+ \sum_{i} \frac{ k_{\alpha} x_{\alpha,i}^{2}}{2},
\label{hintragen}$$ where $p_{\alpha,i}$ and ${x}_{\alpha,i}$ are the oscillator momentum and position of the mode $\alpha$, respectively, $m_{\alpha}$ the oscillator mass and $k_{\alpha}$ the elastic constant of the mode $\alpha$. The inter-molecular modes are characterized by small frequencies ($\hbar \omega_{Inter} \simeq 5-10 meV$) in comparison with the transfer hopping [@corop; @Troisi; @Orlandi]. On the contrary, the most coupled intra-molecular modes have large frequencies ($\hbar \omega_{Intra} \simeq 130-180 meV$) [@corop].
In Eq. (\[hgen\]), $H_{el-Intra}$ is the Holstein-like Hamiltonian describing the electron coupling to intra-molecular modes $$H_{el-Intra}= \alpha_{Intra} \sum_{i} x_i n_i,
\label{hcoupling}$$ with $\alpha_{Intra}$ coupling constant to local modes and $n_i=c_{i}^{\dagger}c_{i}$ local density operator. The dimensionless constant $$g_{Intra}= \alpha_{Intra} / \sqrt{2 \hbar m_{Intra} \omega_{Intra}^3}$$ is used to describe this electron-phonon coupling [@holstein1]. In single crystal organic semiconductors, $g_{Intra}$ is in the weak to intermediate regime (of the order of unity) [@corop].
Finally, in Eq. (\[hgen\]), $H_{el-Inter}$ represents the SSH-like term with electron coupling to inter-molecular modes $$H_{el-Inter}= \alpha_{Inter} \sum_{i} (y_{i+1}-y_i) \left( c_{i}^{\dagger}c_{i+1}+ c_{i+1}^{\dagger}c_{i} \right),
\label{hcouplinggen1}$$ with $\alpha_{Inter}$ coupling constant to non local modes. In the adiabatic regime for non local modes ($\hbar \omega_{Inter} \ll t$), the dimensionless quantity $$\lambda_{Inter}=\alpha_{Inter}^2/4 k_{Inter} t$$ fully provides the strength of the electron coupling to inter-molecular modes. The typical values of $\lambda$ are in the intermediate (of the order of $0.1$) coupling regime [@fernando].
In the following part of this section, we will use units such that lattice parameter $a=1$, Planck constant $\hbar=1$, Boltzmann constant $k_B=1$, and electron charge $e=1$. We will analyze systems in the thermodynamic limit and we will measure energies in units of $t \simeq 80 meV$. We fix $\omega_{Intra}=2.0 t$ as model parameter with the highest energy [@corop].
Calculation method
------------------
Since a very low carrier density is injected into the organic semiconductor, we will study the case of non interacting particles. The temperature range where intrinsic effects are relevant is $\omega_{Inter} \leq T \ll t < \omega_{Intra} $. Therefore, the dynamics of intermolecular modes can be assumed classical. On the other hand, it is important to retain the quantum nature of high frequency local vibrational modes.
Actually, the electron motion is strongly influenced by the statistical “off diagonal” disorder, that, in the limit of low carrier density, is described by the probability function $P \left( \{ y_j \} \right) $ of free classical harmonic oscillators. At a fixed configuration of non local displacements $\{ y_j \}$, Eq. (\[hgen\]) is equivalent to a Holstein model with displacements $\{ x_i \}$, where the electron hopping depends on the specific nearest neighbor sites throughout the assigned $\{ y_j \}$. The resulting inhomogeneous Holstein model can be accurately studied within the modified variational Lang-Firsov approach via a unitary transformation $U \left( \{ y_j \} \right)$, depending on the non local displacements $\{ y_j \}$ and appropriate in the anti-adiabatic regime ($\omega_{Intra} > t $) [@lang; @memanganiti]. The electron mass is renormalized by the coupling with local modes (polaronic effect), and the Holstein-coupled oscillators $\{ x_i \}$ are displaced from their equilibrium position to a distance proportional to the electron-phonon interaction. For each fixed configuration $ \{ y_j \}$, one has to calculate quantities, such as spectral function, density of states, and mobility (calculated as the ratio between conductivity and carrier density), within the Lang-Firsov approach. Then, the effect of non-local adiabatic inter-molecular modes can be taken into account making the integral over the distribution $P \left( \{ y_i \} \right)$ by means of a Monte-Carlo procedure.
The method exposed above is very accurate in the regime $\omega_{Inter} \ll t$ and $\omega_{Intra} > t $ appropriate to high-mobility organic semiconductors. It properly takes into account the quantum effects of high frequency local vibrational modes. Moreover, the approach is able to include spatial correlations relevant in quasi one-dimensional systems, in particular vertex corrections in the calculation of mobility.
Results on spectral and transport properties
--------------------------------------------
![The DOS (in units of $1/t$) as a function of the frequency (in units of $t/\hbar$) for different temperatures T at $\lambda_{Inter}= 0.09$ and $g_{Intra}=1.30$. We consider $\omega_{Intra}=2t$ (in units of $t$, with $t= 80 meV$).[]{data-label="fig4"}](Fig6.eps){width="75.00000%"}
In Fig. \[fig4\], we report the density of states (DOS) for $g_{Intra}=1.3$ and $\lambda_{Inter}=0.09$ at different temperatures. For all the temperatures, there is a strong renormalization of the bare band whose width becomes twice smaller (from $4t$ to roughly $2t$) and move to lower energies (from $-2t$ to roughly $-4t$). Furthermore high energy satellite bands appears at multiples of the vibrational frequency $\omega_{Intra}=2t$ [@mahan] providing a DOS extending from $-4t$ to $4t$. These effects can be easily ascribed to the local modes since they survives also at $T=0$, where the effect of non local modes is weak. The intrisic reduction of the bare band due to local modes provides a simple and direct explanation of the difference in the bandwidth evidenced in the series of oligoacenes from naphthalene (effective band of the order of $40 meV$) to pentacene (effective band of the order of $80 meV$). Indeed, it can be ascribed to the decrease of the reorganization energy with increasing the benzene rings of the single molecules that in turn reduces the renormalization effects [@corop]. Within the polaron theory [@mahan], the narrowing of the main band is related to the spectral weight Z of the quasi-particle, which is estimated to be about $0.5$ from the calculations. Therefore, our estimate of Z compares favorably with recent ab-initio results for which Z relative to the electron channel is of the order of 0.7 [@vukmi].
At finite temperature, the shape of the spectra is changed due to the non local coupling. Actually, any band shows a new small maximum due to the coupling to inter-molecular modes. It is well known that, in the polaron theory, the band narrowing increases strongly with temperature. In order to be more quantitative, we notice that, for $\lambda_{Inter}=0$, the principal band at $T=325$ K is reduced of about $42 \%$ of the band at $T=0$. On the other hand, for $\lambda_{Inter}=0.09$, the principal band at $T=325$ K is reduced of only $7 \%$ of the band at $T=0$. Unexpectedly, in our model, the band narrowing is strongly reduced due to the non local coupling. The narrowing of the principal band results from a subtle equilibrium between the two opposite tendencies. Actually, the coupling to non local modes has the main effect to induce scattering into the single bands of the density of states, preventing the narrowing induced by the coupling to local modes. The interplay between local and non local modes is able to produce a modest narrowing as function of the temperature even if the coupling to local modes is not weak. Our prediction is that this effect should be present not only in pentacene [@arpes2], but also in naphthalene and anthracene.
![Mobility and its different contributions as a function of the temperature T at $\lambda_{Inter}= 0.09$ and $g_{Intra}=1.3$. We consider $\omega_{Intra}=2$ (in units of $t$, with $t= 80 meV$).[]{data-label="fig5"}](Fig7.eps){width="75.00000%"}
Next, we analyze the mobility in the intermediate regime for both intra- and inter-molecular modes (see Fig. \[fig5\]). The mobility can be divided into two contributions: the coherent one, where the scattering of the renormalized electron (the only effect due to local electron-phonon coupling is here the reduction of the bandwidth) with non local modes is included, and the incoherent one, where, in addition to non local modes, scattering with multiple real local phonons is considered.
The coherent term of mobility, relevant at low temperatures, bears a strong resemblance with the mobility of the system at $g_{Intra}=0$, even if, as expected, it is smaller. The local coupling is able to affect but to not destroy the low temperature behavior dominated by the non local coupling. Actually, for $g_{Intra}=0$, the mobility scales as $1/T^{1.89}$, while, with increasing $g_{Intra}$, the power-law becomes slightly less pronounced. In the case $g_{Intra}=1.3$, the mobility goes as $1/T^{1.60}$, still compatible with experiments in naphthalene and it has the correct order of magnitude [@warta].
The incoherent term of mobility starts at a temperature of about $T=230$ K and becomes predominant only at temperatures much higher than room temperature. The role of the local coupling here is to promote an activated behavior in the incoherent regime which is effective only at high temperature. Consequently, the local coupling provides a negligible contribution to the mobility up to room temperature. Actually, the combined effect between intra- and inter-molecular modes is able to provide an activation energy $\Delta$ of only about $20 meV$, therefore less than one half of that for $\lambda_{Inter}=0$ and close to that extracted by experimental data in naphthalene (about $15$ meV for mobility along c-axis) [@warta]. We stress that the small activation energy is found even if the reorganization energy related to intra-molecular modes derived from ab-initio calculations [@corop] is not small. Actually, the polaronic binding energy is given by $g_{Intra}^2 \omega_{Intra}$, which is of the order of $1.7*2*t \simeq 270$ meV. Therefore, the non-local SSH interaction is able to strongly quench the tendency towards localization of the local Holstein coupling. As a result, the activation energy of the transport properties is estimated to be much smaller than the value of the polaronic local energy if other electron-phonon non local interactions are playing a relevant role.
Summarizing, the proposed model is able to capture many features of the mobility in oligoacenes.
Effects of gates made of polarizable dielectrics and disorder
=============================================================
In the last part of this review, we investigate the effect of a polarizable gate on the transport properties of organic semiconductors [@substrato; @mebipo] (see Fig. \[figdie\] for a sketch about the coupling between charge carrier and polarization in the dielectric). This analysis is important to interpret experimental data in rubrene OFET grown on polarizable dielectric gates, such as the $Ta_2 O_5$ oxide [@nature].
Model Hamiltonian
-----------------
![Sketch of the effects induced by the charge carrier in the conducting channel on the gate close to the interface. The electron in the organic semiconductor induces a polarization within the dielectric that, in turn, affects the electron dynamics.[]{data-label="figdie"}](Fig8.eps){width="60.00000%"}
We study a one-dimensional Hamiltonian model with coupling to bulk and interface vibrational modes [@substrato]. This model is similar to that of the previous section. Actually, the free electronic hamiltonian is the same, and the bulk modes of this model correspond to the inter-molecular modes.
The model is described by the following Hamiltonian $$H= H_{el}+H_{Bulk}^{(0)}+H_{el-Bulk}+H_{Int}^{(0)}+H_{el-Int}.
\label{hgen1}$$
In Eq. (\[hgen1\]), the electronic part $H_{el}$ is given by Eq. (\[helgen\]) of the previous section, with $t$ bare electron hopping (estimated to be among $80$ meV and $120$ meV) between the nearest neighbors sites.
In Eq. (\[hgen1\]), $H_{Bulk}^{(0)}$ corresponds to Eq. (\[hintragen\]) for free intermolecular modes with elastic constant $k$, mass $m$, and $\hbar \omega_{Bulk} \simeq 5-10$ meV much smaller than transfer hopping t [@corop; @Troisi; @Orlandi].
In Eq. (\[hgen1\]), $H_{el-Bulk}$ represents the term similar to the SSH [@SSH] interaction for the coupling to intermolecular modes given in Eq. (\[hcouplinggen1\]). As in the previous section, one can define $\lambda_{Bulk}$ whose typical values are in the intermediate coupling regime (in this section, we take the value $\lambda_{Bulk}=0.1$ suitable for rubrene) [@vittoriocheck].
In Eq. (\[hgen1\]), $H_{Int}^{(0)}$ is the Hamiltonian of free interface phonons $$H_{Int}^{(0)}= \hbar \omega_{Int} \sum_{q} a_q^{\dagger} a_q,
\label{hintra}$$ where $\omega_{Int}$ is the frequency of optical modes, $a_{q}^{\dagger}$ and $a_{q}$ are creation and annihilation operators, respectively, relative to phonons with momentum $q$.
In Eq. (\[hgen1\]), $H_{el-Int}$ is the Hamiltonian describing the electron coupling to interface vibrational modes $$H_{el-Int}= \sum_{i,q} M_q n_i e^{i q R_i} \left( a_q + a_{-q}^{\dagger} \right),
\label{hcouplingel}$$ where $ n_i$ is the density operator, $M_q$ is the interaction electron-phonon term $$M_q= \frac{g \hbar \omega_{Int} }{\sqrt{L}} \sum_{i} e^{i q R_i} \frac{R_0^2}{R_0^2+R_i^2},
\label{hcouplingel1}$$ with $g$ dimensionless coupling constant, $L$ number of lattice sites, $R_i$ position of the site $i$, and $R_0$ cut-off length of the order of the lattice spacing $a$. This electron-phonon coupling describes the long-range interaction induced on the electron at the interface with the dielectric gate. In order to quantify this coupling, we use the dimensionless quantity $$\lambda_{Int}=\sum_q \frac{M_q^2}{2 \hbar \omega_{Int} t}.$$ In this work, we take $R_0=0.5 a$ and $\hbar \omega_{Int}=0.5 t$ [@bussac].
In the following part of this section, we will use units such that $a=1$, $\hbar=1$, $e=1$, and Boltzmann constant $k_B=1$. We will analyze systems in the thermodynamic limit measuring energies in units of $t \simeq 100$ meV. The calculation method is similar to that in the previous section since the role of intra-molecular modes is here played by interface modes.
Results about transport properties
----------------------------------
![Mobility $\mu$ as a function of the temperature T at $\lambda_{Bulk}=0.1$ for different values of $\lambda_{Int}$. []{data-label="fig6"}](Fig9.eps){height="50.00000%"}
In Fig. \[fig6\], we report the mobility as a function of the temperature for different values of $\lambda_{Int}$ at bulk coupling $\lambda_{Bulk}=0.1$ (appropriate to rubrene). The quantity $\mu$ shows a coherent band-like behavior at low temperatures, but, with increasing $T$, it goes towards the activated behavior where the bulk coupling is not effective. Actually, the mobility interpolates between the behaviors with only bulk and interface phonons. At low temperature, the diffusive contribution is ascribed to the modulation of the electron kinetic energy due to the bulk modes with SSH interactions. This coherent contribution is weakened with increasing $\lambda_{Int}$, but it does not disappear. This element is in contrast with experimental data which show a more or less marked insulating behavior from $150$ K to $300$ K [@nature]. Therefore, the theoretical prediction of mobility is not accurate even if bulk and interface electron-phonon couplings are active.
Interplay between electron-phonon coupling and disorder strength
----------------------------------------------------------------
In order to explain the experimental data, it is necessary to include also disorder effects. Indeed, there is evidence of traps in the bulk and at the interface with gates [@morpurgo]. Therefore, it is of paramount importance to investigate the role of disorder on the transport properties.
The model bears a strong resemblance with that of the previous subsections. Actually, the only modification is related to the electronic hamiltonian which includes here a disorder term. Therefore, in Eq. (\[hgen1\]), there is a new term given by
$$H_{dis} = \sum_{i} \epsilon_i n_{i} ,
\label{heldis}$$
where $\epsilon_i$ is a local energy whose fluctuations in the range $[-W,W]$ simulate disorder effects in the bulk and at the interface with gate, $ n_i= c_{i}^{\dagger} c_{i}$ is the density operator, with $c_{i}^{\dagger}$ and $c_{i}$ electron creation and annihilation operators, respectively, relative to the site $R_i$. Due to the presence of shallow traps [@morpurgo], disorder is not overwhelming and it is distributed according to a flat probability function. The calculation method is analogous to that of the previous section.
![Mobility $\mu$ as a function of the temperature for different disorder strengths $W$ at $\lambda_{Bulk}=0.1$ and $\lambda_{Int}=1.3$. The quantity $\Delta$ is the polaron activation energy.[]{data-label="fig7"}](Fig10.eps){height="50.00000%"}
In Fig. \[fig7\], we show the mobility as a function of the temperature with increasing the strength of disorder for $\lambda_{Int}=1.3$ and $\lambda_{Bulk}=0.1$. There are two main results. The first one is related to the suppression of the coherent metallic behavior with increasing $W$. The second one is the strong enhancement of the activation energy $\Delta$ up to $67$ meV even for the small amount of disorder $W=70$ meV. Furthermore, the decrease of the magnitude of the mobility is not so marked. Therefore, weak disorder effects are able to provide a very accurate description of the mobility resulting as key quantities for the interpretation of experimental data. Finally, another important effect of disorder is to drive the small polaron formation at lower electron-phonon couplings.
Conclusions
===========
In this review, we have theoretically analyzed the effects of different electron-phonon couplings on spectral and transport properties of small molecule single-crystal organic semiconductors. Focus has been on oligoacenes, in particular on the series from naphthalene to rubrene and pentacene.
First, we have discussed the effects of the electron coupling to low frequency inter-molecular vibrational modes on the spectral and transport properties. The resulting adiabatic models have been studied through numerical approaches with varying electron-phonon coupling and temperature. For rubrene, the model has considered the role of the electron-phonon coupling leading to a modulation of the particle hopping integral. With increasing temperature, the density of states is characterized by a larger exponential tail corresponding to localized states. Consequently, the chemical potential moves into the tail of the density of states, but this is not enough to drive the system into an insulating state. Not only the order of magnitude and the anisotropy ratio between different directions are accurate, but also the temperature dependence of the mobility is correctly reproduced in the model for rubrene. With increasing temperature, the Ioffe-Regel limit is reached since the contribution of itinerant states to the conduction becomes less and less relevant.
Then, we have analyzed the effects of electron coupling to both low frequency inter-molecular and high frequency intra-molecular modes on the spectral and transport properties. The interplay between local and non local electron-phonon interactions has been able to provide a very accurate description of the mobility of oligoacenes and to shed light on the intricate mechanism of band narrowing with increasing temperature. The band narrowing is a complicated phenomenon which could also be affected by the thermal expansion of the crystal structure [@libredas] (an effect which has not been analyzed in this review).
In the last part of the review, we have considered the influence of gates made of polarizable dielectrics on the transport properties. This effect has been studied in a model which has combined bulk and long-range interface electron-phonon couplings. We have pointed out that the bulk coupling affects the behavior of mobility below room temperature enhancing the coherent contribution, but it is ineffective on the incoherent small polaron contribution dominated by the interface coupling at high temperatures.
Finally, we have emphasized the interplay between electron-phonon couplings and disorder strength on the transport properties. The presence of disorder is important to improve the modeling of the materials studied in this review. In particular, for systems gated with polarizable dielectrics, we have shown that disorder effects are able to enhance the hopping barriers of the activated mobility and to drive the small polaron formation at lower values of electron-phonon interactions. Therefore, disorder represents a key factor to get agreement with experimental data.
Some issues have not been covered in this review. Indeed, the transport properties could be affected by the nonlocal electron coupling not only to optical but also acoustic vibrations [@licorop]. The coupling to acoustic vibrations should be effective at low temperatures where it would be interesting also to investigate the role of quantum lattice fluctuations. These quantum effects are small in the adiabatic limit, however, they could be important in the regime where the presence of traps also influences the transport properties. Finally, we believe that concepts and methods discussed in this review can be a starting point for the study of related (such as durene crystals [@hanne4]) and more complex systems [@hanne5].
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|
---
abstract: 'This paper proposes an agent-based model which reproduces different structures of animal groups. The shape and structure of the group is the effect of simple interaction rules among individuals: each animal deploys itself depending on the position of a limited number of close group mates. The proposed model is shown to produce clustered formations, as well as lines and V-like formations. The key factors which trigger the onset of different patterns are argued to be the relative strength of attraction and repulsion forces and, most important, the anisotropy in their application.'
author:
- 'Emiliano Cristiani[^1]'
- 'Paolo Frasca[^2]'
- 'Benedetto Piccoli[^3]'
title: Effects of Anisotropic Interactions on the Structure of Animal Groups
---
0.5cm **Keywords.** Anisotropic interactions, animal groups, coordinated behavior, self-organization, agent-based models. 0.5cm **M.S.C.** 92D50, 92B05. 0.5cm
Introduction {#sec:intro}
============
Group behavior in animals has greatly interested scientists and researchers in the past, and has received further attention in the last decades as a test case of self-organization. Recently it has attracted attention not only from biologists and ethologists, but also from physicists, mathematicians, and engineers. This interest has produced a huge amount of literature, which is well documented and reviewed. See, for instance, [@JK-GDR:02; @DS:06; @IG:08] and [@FB-JC-SM:09]. Loosely speaking, the basic idea behind these works is that complex collective behavior arises from simple interactions among close animals. Following this idea, the aim of this paper is to investigate a simple model of interactions within groups which is able to reproduce rather different patterns and structures.
Let us briefly introduce the main features of our model and its relationships with literature. A detailed description is postponed to Section \[sec:model\]. First, our model is [*agent-based*]{}, in the sense that each animal is singularly considered. Furthermore it is [*leaderless*]{}, meaning that all animals act following the same set of rules and their behavior is not imposed by others. These assumptions are widely considered in literature, and accepted as biologically suitable for a variety of species. See, among others, [@KW-JL:91; @AH-CW:92; @JKP-SVV-DG:02; @IC-JK-RJ-GR-NF:02; @IDC-JK-NRF-SAL:05].
Second, our model is purely based on [*attraction-repulsion*]{} interactions between group mates. Attraction allows the group to be formed and stay tight, while repulsion allows to avoid collisions between group mates and keeps them well spaced. In the majority of papers, attraction and repulsion are combined with velocity alignment. Here we keep aside the issue of alignment which has received a considerable attention in itself [@TV-AC-EBJ-IC-OS:95; @FC-SS:07], and focus on (the superposition of) attraction and repulsion. Our approach is close to [@KW-JL:91; @AM-LE-LB-AS:03] in this respect. Third, each animal interacts with a [*limited number*]{} of group mates. The idea of having a limited number of interacting neighbors is not new. The work [@WDH:71] already considers attraction towards [*the*]{} nearest neighbor, while later experimental investigations found interaction with the closest two-four individuals [@IA:80]. This fact has been included in several models, among others [@KW-JL:91; @AH-CW:92; @YI-KK:02; @JKP-SVV-DG:02; @RL-YXL-LE:09], but it has not always been included in recent models [@SG-SAL-DIR:96; @IC-JK-RJ-GR-NF:02; @HK-CKH:03; @IDC-JK-NRF-SAL:05], in favor of a purely metric notion of neighborhood: interactions occur among group mates which are less than a threshold apart. Very recently, the former idea has again been brought to attention by [@AC-IG:08a], where the authors present experimental results regarding fairly large flocks of starlings. Results show that interaction occurs with up to six-seven neighbors, no matter how far they are. The same paper also gives some simulation results, which suggest that a better cohesion of the flock can be guaranteed in that way. We also want to point out a third approach to neighborhood definition, based on Voronoi partitions. Its application in biology dates back to [@WDH:71], and it is well documented in the physicists [@GG-HC-YT:03] and engineers [@FB-JC-SM:09] literatures.
Fourth, each animal only interacts with group mates which are in a suitably defined [*sensitivity zone*]{}. Restricting the interactions to a sensitivity zone raises the issue of defining its size and shape. On this matter, there is a significant amount of works considering limited visual (or sensing) fields. A limitation in the animal’s angle of vision has been incorporated in most models, assuming a blind rear zone [@AH-CW:92; @YI-KK:02; @IC-JK-RJ-GR-NF:02; @CKH-HH:08; @RL-YXL-LE:09], or distinguishing between front and rear sensitivity [@SG-SAL-DIR:96]. In [@HK-CKH:03] authors assume repulsion and alignment regions to be elliptical (taking into account body shapes), and include blind areas. In the recent paper [@AN-VCB:08], the authors clearly distinguish between visual field and sensitivity zones, stating that the behavioral rules they use in the model apply only in the front zone. However, up to our knowledge, the anisotropy of sensitivity zones has always been taken just as a given constraint, and not as a potential resource able to shape the group geometry. Here is the main contribution of our paper: showing that a restricted sensitivity angle can be a key element in determining the structure and shape of an animal group. Indeed, we show that changing two parameters, namely two sensitivity angles for attraction and repulsion, we obtain [*cluster*]{} formations, [*line*]{} formations and [*V-like*]{} formations. Note that similar patterns have already been obtained by means of other mathematical models (see for example [@GG-HC-YT:03; @SG-SAL-DIR:96; @AN-VCB:08]), the novelty here is that our model is able to reproduce all of them, depending on few parameters. We believe that this can offer new biological insights.
Besides investigating the main issue of the role of anisotropy, we also present some results on the effects of limiting the number of considered neighbors, and on the influence of the relative strength of attraction and repulsion on the inter-animal distance. The latter problem relates to the results in [@AM-LE-LB-AS:03].
The observations drawn from simulations are accompanied by a formal mathematical investigation of the model. One can find that a popular approach to the analysis of agent-based flocking models consists in defining a suitable potential function, called “virtual” or “artificial” potential, whose gradient gives the dynamics. This variational approach has been followed by both engineers [@HGT-AJ-GJP:07] and biologists [@AM-LE-LB-AS:03; @YL-RL-LEK:08] leading to important results. However, it is not possible to apply it to our model. With respect to other models, ours has two main features: [*state-dependent switching dynamics*]{}, and [*asymmetry of interactions*]{}. The former is due to the state-dependent definition of the set of group mates which interact with a given animal. This discontinuous state dependence has been included in previous literature: see for instance the treatment in [@HGT-AJ-GJP:07], based on non-smooth potentials. The latter feature is due to the fact that the limited number of interacting neighbors and the restricted sensitivity angles imply that interactions need not to be symmetrical (reciprocal). This fact prevents the application of the virtual potential approach. The paper [@HS-LW-TC:06] is a partial attempt in this direction, because it considers a second-order system in which velocity alignment is achieved by asymmetric interactions. Nevertheless agents’ positions are controlled using symmetrical information exchange. As a consequence, its results are not useful to us, as we are interested in spatial configurations. Overall, we conclude that the mathematical analysis of our model requires a novel approach, which has to include discontinuous and asymmetric interactions among animals.
The rest of the paper is organized as follows. The details about the model are given in Section \[sec:model\]. Then, extensive simulation results are presented in Section \[sec:results\], whereas Section \[sec:analysis\] contains the mathematical analysis. Later, Section \[sec:discussion\] discusses the implications of our findings and their biological soundness. We conclude presenting some lines of future research.
Model definition {#sec:model}
================
The animals in the model are represented by point particles, which have simple continuous-time dynamics. Given $N\in {{\mathbb{N}}}$, for all $i\in {\{1,\dots, N\}}$ and $t\in {\ensuremath{\mathbb{R}}_{\geq0}}$, let $x_i(t)\in {\ensuremath{\mathbb{R}}}^2$ represent the position of the $i$-th animal, whose evolution is described by the differential equation $$\label{ode_fondamentale}
\dot x_i(t)=v_i(x(t)),$$ where $x(t)$ is the vector $(x_1(t),\ldots,x_N(t))$. As in [@AM-LE-LB-AS:03], we use a coordinate system moving with the group centroid. This means that we are modelling relative movements of the individuals and group structure, rather than its global motion. The velocity $v_i(x)$ is the sum of two contributions, expressing the effects of attraction and repulsion, $$v_i(x)={{v^{\textup{a}}}}_i(x)-{{v^{\textup{r}}}}_i(x).$$ In more detail, each of these contributions depends on the relative position of the other animals, $$\begin{aligned}
{{v^{\textup{a}}}}_i(x)=&\sum_{j\in {\mathcal{A}_{i}^{n}}}{{f_{\textup{a}}}}(\|x_j-x_i\|)\frac{x_j-x_i}{\|x_j-x_i\|}\\
{{v^{\textup{r}}}}_i(x)=&\sum_{j\in {\mathcal{R}_{i}^{n}}}{{f_{\textup{r}}}}(\|x_j-x_i\|)\frac{x_j-x_i}{\|x_j-x_i\|}.\end{aligned}$$ The following definitions have been used.
- The function ${{{f_{\textup{a}}}}: {\ensuremath{\mathbb{R}}_{\geq0}}\rightarrow {\ensuremath{\mathbb{R}}_{\geq0}}}$ (resp., ${{{f_{\textup{r}}}}: {\ensuremath{\mathbb{R}}_{\geq0}}\rightarrow {\ensuremath{\mathbb{R}}_{\geq0}}}$) describes how each animal is attracted (resp., repelled) by a neighbor at a given distance, assuming $\|\cdot\|$ denotes the Euclidean norm in ${\ensuremath{\mathbb{R}}}^2$.
- The [*attraction neighborhood*]{} ${\mathcal{A}_{i}^{n}}$ (resp., the [*repulsion neighborhood*]{} ${\mathcal{R}_{i}^{n}}$) is the set of the $n$ animals closest to the $i$-th one, which are inside the attraction (resp., repulsion) sensitivity zone.
The above model is very general, and we need to specialize it by choosing the interaction functions and the shape of the sensitivity zones. We make the following assumptions.
1. The functions ${{f_{\textup{a}}}}$ and ${{f_{\textup{r}}}}$ are assumed to be $${{f_{\textup{a}}}}(\|x_j-x_i\|)= {{F_{\textup{a}}}}\|x_j-x_i\|\,, \qquad {{f_{\textup{r}}}}(\|x_j-x_i\|)=\frac{{{F_{\textup{r}}}}}{\|x_j-x_i\|},$$ where ${{F_{\textup{a}}}}$ and ${{F_{\textup{r}}}}$ are two positive constants.
2. The sensitivity zones are depicted in Figure \[fig:regions\] and illustrated as follows. Let the center point be the animal’s position, and let the horizontal axis (arrow-headed) represent the direction of motion. Attraction is active in a frontal cone whose width is given by the angle ${{\alpha_{\textup{a}}}}\in(0,360{^\circ\!}]$ (dashed line). Repulsion is active both inside a disk of radius ${{R_{\textup{sr}}}}>0$ (*short-range* repulsion) and in a frontal cone of width ${{\alpha_{\textup{r}}}}\in(0{^\circ\!},360{^\circ\!}]$ (solid line). We stress that the sensitivity zones do not necessarily coincide with the visual field of the animal. They rather represent the zones which attraction and repulsion are focused on.
3. The speed of each animal $\|v_i\|$ is bounded from above by a constant ${{v_{\textup{max}}}}.$
![The shape of the sensitivity zones.[]{data-label="fig:regions"}](zones.eps){width=".6\columnwidth"}
The above assumptions result in the system $$\label{eq:system-simulated}
\dot x_i(t)={{F_{\textup{a}}}}\sum_{j\in {\mathcal{A}_{i}^{n}}} (x_j-x_i) - {{F_{\textup{r}}}}\sum_{j\in {\mathcal{R}_{i}^{n}}} \frac{(x_j-x_i)}{\|x_j-x_i\|^2} .$$ Some remarks are in order.
1. The definition of the interaction neighborhoods ${\mathcal{A}_{i}^{n}}$ and ${\mathcal{R}_{i}^{n}}$ allows to have a priori bound on the number of effective neighbors, and therefore on the sensing and “computational” effort which is required for each animal. This fact, which copes with animals’ intrinsic limitations, has been experimentally observed in biology, for fish [@IA:80] and birds [@AC-IG:08a]. The latter paper calls this neighborhood definition [*topological*]{}, as opposed to [*metric*]{} definitions, based on distance only.
2. Our assumption of unbounded sensitivity regions does not intend to imply that animals sensing capabilities extend on an unlimited range, but rather that group dynamics happen in a relatively small area.
3. If ${{\alpha_{\textup{a}}}}={{\alpha_{\textup{r}}}}=360{^\circ\!}$, the parameter ${{R_{\textup{sr}}}}$ has no effect, and the interaction is completely isotropic as in the simulations presented in [@AC-IG:08a; @DC-SAL:99]. Note that, even if there is no preference for any specific direction, the limitation of the number of considered neighbors makes the interactions not reciprocal, i.e. the fact that the $i$-th animal interacts with the $j$-th does not imply that the $j$-th interacts with the $i$-th.
4. By specializing the functions ${{f_{\textup{a}}}}$ and ${{f_{\textup{r}}}}$, one can obtain various interaction models. Indeed many proposals can be found in the literature, as reviewed in [@KW-JL:91] and [@AM-LE-LB-AS:03]. However, the shape of these functions is not the main point in our paper, and thus we have decided to focus on a simple choice, in order to highlight the innovative part of our approach, i.e. the angle-dependent interactions. Similar considerations are valid also for some features introduced in other models, such as a *neutral zone* around animals [@JHT-SAL-DIR:04; @DC-SAL:99] or a hierarchical decision tree which allows the repulsion force to have the priority over the attraction force [@SG-SAL-DIR:96; @DC-SAL:99].
Simulations results {#sec:results}
===================
In this section we make use of the agent-based model in order to show the effects of the anisotropic interactions on the shape of the group. Namely, depending on the angles ${{\alpha_{\textup{a}}}}$, ${{\alpha_{\textup{r}}}}$, and the ratio between repulsive and cohesive forces, we shall obtain either [*clusters*]{}, or [*lines*]{} or [*V-like*]{} formations. These patterns are described in the sequel.
To perform the simulations, it is instrumental the introduction of the constant $\xi=\sqrt{\frac{{{F_{\textup{r}}}}}{{{F_{\textup{a}}}}}}$, which allows to rewrite as $$\label{eq:system-adim}
\dot x_i(t)=\sum_{j\in {\mathcal{A}_{i}^{n}}} (x_j-x_i) - \xi^2 \sum_{j\in {\mathcal{R}_{i}^{n}}} \frac{(x_j-x_i)}{\|x_j-x_i\|^2}.$$ In equation , the unit of length is chosen to be the body length () of the animal, and the time unit () is the inverse of ${{F_{\textup{a}}}}$. Simulations are obtained solving the system of equations via an explicit forward adaptive Euler scheme. Each run starts from a randomly generated initial configuration (contained in square of edge ${L}$), and ends when the system reaches a steady state. To take into account the uncertainties in sensing and motion of the animals, we include small additive random disturbances on the direction of the velocity, uniformly distributed in $[-\alpha_{\textrm{noise}},\alpha_{\textrm{noise}}]$. All the steady-state configurations described in the sequel are robust to such noise. A summary of the parameters and their values is given in Table \[table:parameters\]. In the sequel we discuss the role of $N$, $n$, $\xi$, ${{\alpha_{\textup{a}}}}$ and ${{\alpha_{\textup{r}}}}$, which are most significant to us, whereas ${{R_{\textup{sr}}}}$, ${{v_{\textup{max}}}}$, $\alpha_{\textrm{noise}}$ and $L$ are kept fixed.
Name Symbol Unit Values explored
-------------------------------- --------------------------- ------------------------ -----------------
Forces ratio $\xi$ 0.1
Attraction angle ${{\alpha_{\textup{a}}}}$ degrees $({^\circ\!})$ 0360
Repulsion angle ${{\alpha_{\textup{r}}}}$ degrees $({^\circ\!})$ 0360
Number of animals $N$ adimensional 2200
Number of considered neighbors $n$ adimensional 1$(N-1)$
Short-range repulsion radius ${{R_{\textup{sr}}}}$ 1
Maximum speed ${{v_{\textup{max}}}}$ / 230
Size of initial domain ${L}$ 15
Noise magnitude $\alpha_{\textrm{noise}}$ degrees $({^\circ\!})$ 010
: Model parameters.[]{data-label="table:parameters"}
Clusters
--------
In this paragraph, we describe simulation results when interactions are assumed to be isotropic. These results are not dissimilar from others in the literature (e.g., [@AM-LE-LB-AS:03; @YL-RL-LEK:08]): we include them for two reasons. First, for comparison with the less usual patterns described in the following paragraphs. Second, because they allow some interesting remarks about the role of the model’s parameters $\xi$ and $n$.
Thus, let us assume that ${{\alpha_{\textup{a}}}}={{\alpha_{\textup{r}}}}=360{^\circ\!}$. As a consequence, the outcome of the simulations is a cohesive and well spaced cluster. For a better understanding, we make use of an indicator which is largely used in the literature (see for instance [@AH-CW:92; @HK-CKH:03]): the [*mean distance to the nearest neighbor*]{} NND, defined as $$\textrm{NND}=\frac{1}{N}\sum_{i=1}^N \min_{j\neq i} \|x_i-x_j\|.$$ We investigate the dependence of NND on the parameter $\xi$: simulation results are shown in Figure \[fig:2crystal\].
![NND as a function of the ratio $\xi$, for different values of $n$. Error bars denote variance across individuals. Plots assume $N=30$ (left) and $N=100$ (right).[]{data-label="fig:2crystal"}](dmin30.eps "fig:"){width=".49\columnwidth"} ![NND as a function of the ratio $\xi$, for different values of $n$. Error bars denote variance across individuals. Plots assume $N=30$ (left) and $N=100$ (right).[]{data-label="fig:2crystal"}](dmin100.eps "fig:"){width=".49\columnwidth"}
We observe that NND is an increasing function of $\xi$, in particular it increases roughly linearly in $\xi$. This linear dependence is observed for any choice of $n$. Two other features are noticeable: first, if $n=1$, animals asymptotically converge to a [*comfortable distance*]{} which is equal to $\xi$. Second, for any fixed $\xi$, NND decreases as $n$ increases. Moreover, all these remarks do not depend on $N$.
These results can be compared with those in [@AM-LE-LB-AS:03]: in that paper, the authors assume that all animals in the group interact among each other, and they conclude that, the larger the group, the closer packed it is. Our simulations, instead, suggest that the significant parameter is $n$, the number of neighbors which is taken into account, rather than $N$, the global number of animals. However, from the biological point of view, the number of neighbors $n$ is not truly a free parameter: $n$ can not be too large because of the limited sensing and analysis capabilities of the animals, and can not be too small either. For instance, a low value of $n$ sounds unsafe from the point of view of collision avoidance. Figure \[fig:2crystal\] also shows that the variance of the distance to the nearest neighbor among the animals is quite small. This means that groups are internally uniform, in terms of spacing. This uniformity is also apparent if one looks at the steady-state configurations: three examples are shown in Figure \[fig:cluster-views\].
Nevertheless, as $n$ varies, significant modifications are apparent in terms of relative positions. If $n=1$, the animals are deployed to form an hexagonal lattice, reminiscent of a crystal. For intermediate $n$’s the internal structure is rather disordered, and if $n=N-1$ it is made of concentric circles. Note that crystal-like patterns have also been found in [@GG-HC-YT:03] and in [@YL-RL-LEK:08], where they are compared with the less regular structures obtained in [@YC:07]. Such patterns might be of interest for engineering application to environmental deployment of robots [@FB-JC-SM:09] or sensors [@PF-PM-BP:09a].
Lines {#subsec:lines}
-----
In this paragraph we show how restricting the sensitivity field (i.e. reducing ${{\alpha_{\textup{r}}}}$ and ${{\alpha_{\textup{a}}}}$) induces the formation of an elongated group. The key element for the formation of these patterns is a restricted frontal sensitivity field. Let us denote by $e$ the oriented [*elongation*]{} of the group (see for instance [@SG-SAL-DIR:96; @HK-CKH:03]), defined as the ratio of the vertical to the horizontal side of the smallest rectangle containing the group, oriented parallel to the direction of the movement. Here we study how $e$ depends on the angles (${{\alpha_{\textup{r}}}}$, ${{\alpha_{\textup{a}}}}$), and on $n$. Results are summarized in Figure \[fig:elongation\], which shows the average and the extreme values of $e$ over 100 runs as a function of the angles (${{\alpha_{\textup{r}}}}$, ${{\alpha_{\textup{a}}}}$), for $n=1,7,N-1$. It is clear that reducing angles from $(360{^\circ\!},360{^\circ\!})$ to $(40{^\circ\!},180{^\circ\!})$ affects the elongation of the group.
![Mean elongation of the group as a function of the sensitivity angles, for $N=30$ and different values of $n$. The angles are $({{\alpha_{\textup{r}}}}$, ${{\alpha_{\textup{a}}}})=(360{^\circ\!}-k 16{^\circ\!},360{^\circ\!}-k 9 {^\circ\!})$, $k=0,\dots,20$. Data come from 100 runs. Error bars are the ranges of the outcomes.[]{data-label="fig:elongation"}](elongations1new.eps "fig:"){width=".49\columnwidth"} ![Mean elongation of the group as a function of the sensitivity angles, for $N=30$ and different values of $n$. The angles are $({{\alpha_{\textup{r}}}}$, ${{\alpha_{\textup{a}}}})=(360{^\circ\!}-k 16{^\circ\!},360{^\circ\!}-k 9 {^\circ\!})$, $k=0,\dots,20$. Data come from 100 runs. Error bars are the ranges of the outcomes.[]{data-label="fig:elongation"}](elongationsN-1new.eps "fig:"){width=".49\columnwidth"}\
![Mean elongation of the group as a function of the sensitivity angles, for $N=30$ and different values of $n$. The angles are $({{\alpha_{\textup{r}}}}$, ${{\alpha_{\textup{a}}}})=(360{^\circ\!}-k 16{^\circ\!},360{^\circ\!}-k 9 {^\circ\!})$, $k=0,\dots,20$. Data come from 100 runs. Error bars are the ranges of the outcomes.[]{data-label="fig:elongation"}](elongations7new.eps "fig:"){width=".7\columnwidth"}
![A line formation, obtained with ${{\alpha_{\textup{r}}}}=40{^\circ\!}$, ${{\alpha_{\textup{a}}}}=180{^\circ\!}$, $n=7$, $\xi=10$, $N=30$. The group is moving horizontally from left to right. See text for the explanation of the irregularities in the head of the line.[]{data-label="fig:line"}](line.eps){width=".7\columnwidth"}
Wide angles (roughly, $360{^\circ\!}>{{\alpha_{\textup{r}}}}>200{^\circ\!}$ and $360{^\circ\!}>{{\alpha_{\textup{a}}}}>270{^\circ\!}$) induce an average elongation greater than 1, i.e. the group stretches along the transverse direction. Moreover, the range of the outcomes is large, meaning that different initial conditions affect significantly the evolution of the system. In the majority of runs the system does not reach an equilibrium in a reasonable time, and the simulation is stopped after a maximum number of iterations. Conversely, small angles ($200{^\circ\!}>{{\alpha_{\textup{r}}}}>40{^\circ\!}$ and $270{^\circ\!}>{{\alpha_{\textup{a}}}}>180{^\circ\!}$) lead to small values of $e$, with small differences among the runs. The steady-state configurations are strongly elongated in the direction of motion: in the limit case, we obtain a line as in Figure \[fig:line\]. One can also observe a dependence on $n$: larger values of $n$ show a sharper transition to lines than with $n=1$. The obtained lines are not in general stable: this fact relates to the phenomenon of [*string instability*]{} described in the engineering literature [@PS-AP-JKH:04]: in a line of vehicles which are tracking their forerunners positions, perturbations propagate down the line in cascade, leading to instabilities. Control-theoretic results presented in [@DS-JKH:96] state that weakening the interaction forces should reduce these negative effects. Consistently, we have observed that damping down to zero small repulsion forces improves the stability of the lines.
Finally, we note that in Figure \[fig:line\] some “border effects” are visible in the head of the line (right side): since in this case we chose $n=7$, the animals in the front can not interact with a sufficient number of group mates and then they do not form a single-file line.
Vees
----
In this paragraph we show that a restricted frontal [*repulsion*]{} range (${{\alpha_{\textup{r}}}}<180{^\circ\!}$) induces the formation of V-like patterns. V-like formations have been recently obtained in the literature [@AN-VCB:08], using an [*ad hoc*]{} model motivated by aerodynamics considerations. In our model, instead, V-like formations arise as one of the anisotropy’s effects. Following [@FHH:74], we adopt a broad definition of V-like formations, which includes asymmetric formations (J-like, and echelons) as well. To understand the role of anisotropy in the emergence of such patterns, we study how the configurations depend on the repulsion angle ${{\alpha_{\textup{r}}}}$, while we keep fixed ${{\alpha_{\textup{a}}}}=360{^\circ\!}$.[^4]
In Figure \[fig:roses\] we plot the distribution of the angles between the nearest neighbor and the direction of motion, for four values of ${{\alpha_{\textup{r}}}}$. A few remarks are in order. If ${{\alpha_{\textup{r}}}}=360{^\circ\!}$, animals do not show any angle preference. If ${{\alpha_{\textup{r}}}}=270{^\circ\!}$, animals show a preference of the front/back positions versus side positions. Finally, if ${{\alpha_{\textup{r}}}}<180{^\circ\!}$, animals clearly prefer to keep a specific angle, namely $\frac{{{\alpha_{\textup{r}}}}}{2}$, with respect to their nearest neighbor.
![Distribution of the nearestneighbor angle for ${{\alpha_{\textup{r}}}}=360{^\circ\!}$, $270{^\circ\!}$, $120{^\circ\!}$, $60{^\circ\!}$. Data from 100 runs.[]{data-label="fig:roses"}](rose1.eps "fig:"){width=".49\columnwidth"} ![Distribution of the nearestneighbor angle for ${{\alpha_{\textup{r}}}}=360{^\circ\!}$, $270{^\circ\!}$, $120{^\circ\!}$, $60{^\circ\!}$. Data from 100 runs.[]{data-label="fig:roses"}](rose2.eps "fig:"){width=".49\columnwidth"} ![Distribution of the nearestneighbor angle for ${{\alpha_{\textup{r}}}}=360{^\circ\!}$, $270{^\circ\!}$, $120{^\circ\!}$, $60{^\circ\!}$. Data from 100 runs.[]{data-label="fig:roses"}](rose3.eps "fig:"){width=".49\columnwidth"} ![Distribution of the nearestneighbor angle for ${{\alpha_{\textup{r}}}}=360{^\circ\!}$, $270{^\circ\!}$, $120{^\circ\!}$, $60{^\circ\!}$. Data from 100 runs.[]{data-label="fig:roses"}](rose4.eps "fig:"){width=".49\columnwidth"}
Second, to obtain more quantitative results about the transition from clusters to Vees, we introduce an Alignment Index (${\textup{AI}}$), defined as follows. ${\textup{AI}}(\theta)$ is the percentage of individuals whose nearest neighbor is positioned (up to a small tolerance ${{{\varepsilon}_{\textup{angle}}}}$) at a given angle $\theta$ with respect to them. The dependence on ${{\alpha_{\textup{r}}}}$ of this novel index is shown in Figure \[fig:angles\]. The figure plots both ${\textup{AI}}({{\alpha_{\textup{r}}}}/2)$ and ${\textup{AI}}(30{^\circ\!})$, computed as the average over 100 runs, with ${{{\varepsilon}_{\textup{angle}}}}=3{^\circ\!}$. If ${{\alpha_{\textup{r}}}}>180{^\circ\!}$ the alignment index is as low as in a random configuration; if instead ${{\alpha_{\textup{r}}}}<180{^\circ\!}$, the high index confirms the preference for an $\frac{{{\alpha_{\textup{r}}}}}{2}$-alignment.
![ ${\textup{AI}}(30{^\circ\!})$ and ${\textup{AI}}({{\alpha_{\textup{r}}}}/2)$ as functions of ${{\alpha_{\textup{r}}}}$. Average of 100 runs. The reported reference value $6\%$ is the expected value of ${\textup{AI}}$ from a random uniform distribution.[]{data-label="fig:angles"}](ai.eps){width=".7\columnwidth"}
Qualitative analysis of the obtained configurations confirm these results: one observes that for a wide range of ${{\alpha_{\textup{r}}}}$, a scenario sets up, in which the animals form (several) V-like formations. Examples are given in Figure \[fig:Vee\]. Finally, it can be noted from Figure \[fig:shots\] that the number of considered neighbors $n$ affects the ability to form V-like configurations: if $n$ is too small ($n=1$) or too large ($n=N-1$), the interesting patterns do not show up.
![V-like formations obtained with ${{\alpha_{\textup{r}}}}=60{^\circ\!},$ $N=30$. The plot on the right is a close-up on one of the Vees. The group is moving horizontally from left to right.[]{data-label="fig:Vee"}](vee-scaled.eps "fig:"){width=".45\columnwidth"} ![V-like formations obtained with ${{\alpha_{\textup{r}}}}=60{^\circ\!},$ $N=30$. The plot on the right is a close-up on one of the Vees. The group is moving horizontally from left to right.[]{data-label="fig:Vee"}](vee-scaled-zoom.eps "fig:"){width=".40\columnwidth"}
![Configurations obtained with $n=1,3,7,N-1$, respectively, and ${{\alpha_{\textup{r}}}}=60{^\circ\!}$, $N=60$. The group is moving horizontally from left to right.[]{data-label="fig:shots"}](vee_n=1.eps "fig:"){width=".49\columnwidth"} ![Configurations obtained with $n=1,3,7,N-1$, respectively, and ${{\alpha_{\textup{r}}}}=60{^\circ\!}$, $N=60$. The group is moving horizontally from left to right.[]{data-label="fig:shots"}](vee_n=3.eps "fig:"){width=".49\columnwidth"} ![Configurations obtained with $n=1,3,7,N-1$, respectively, and ${{\alpha_{\textup{r}}}}=60{^\circ\!}$, $N=60$. The group is moving horizontally from left to right.[]{data-label="fig:shots"}](vee_n=7.eps "fig:"){width=".49\columnwidth"} ![Configurations obtained with $n=1,3,7,N-1$, respectively, and ${{\alpha_{\textup{r}}}}=60{^\circ\!}$, $N=60$. The group is moving horizontally from left to right.[]{data-label="fig:shots"}](vee_n=N-1.eps "fig:"){width=".49\columnwidth"}
Analytical results {#sec:analysis}
==================
In this section we develop a framework for the analysis of the presented model which helps the interpretation of the numerical results. Several analytical tools have been developed and used in literature for the analysis of flocking algorithms. The typical method for their analysis consists in defining a suitable potential function, called “virtual” or “artificial” potential, whose gradient gives the dynamics. Hence a well developed theory on potential systems can be used to make a full mathematical analysis, see e.g. [@YL-RL-LEK:08; @AM-LE-LB-AS:03]. We have anticipated in the introduction that our model has two main features: [*state-dependent switching*]{}, and [*asymmetry of interactions*]{}. Let us illustrate them. It is clear from their definition that the attraction and repulsion neighborhoods ${\mathcal{A}_{}^{n}}$ and ${\mathcal{R}_{}^{n}}$ depend in a discontinuous way on the configuration (i.e. on the positions of the animals). As a consequence, the system’s evolution switches among a finite collection of equations. From a system-theoretical point of view, equation is a switching system [@DL:03] with state-dependent switches. These issues have already been taken into account in flocking studies, as many papers consider the case of animals which interact when they are closer than a certain threshold. See, among others, [@HGT-AJ-GJP:07]. The virtual potentials approach can be extended to these problems, provided the potential is allowed to be non-differentiable at the switching points. However, our model has the distinctive feature that, because of the limitation in the number of neighbors and in the shape of the sensitivity zones, interactions need not be symmetric. Example giving, for two animals $i,j$, the inclusion $j\in{\mathcal{A}_{i}^{n}}$ does not imply that $i\in{\mathcal{A}_{j}^{n}}$. This fact prevents us from using a virtual potentials approach, as the operation of differentiating the potential does not keep any directionality information. Actually, in literature there is no general approach available for systems with directed interactions. A partial asymmetry has been taken into account in other works [@HS-LW-TC:06], but the given treatment is far for being sufficient for our purposes. In what follows we lay down the basics of a theory that we believe is able to catch the specific features of our model.
Let us start by defining solutions of a switching system in an useful sense. Indeed, a differential equation of the form ${\dot x=f(x)}$ with a discontinuous right-hand-side can not have a solution in the classical sense, i.e. a solution which is differentiable. Let us consider the right-hand side of equation . That expression is not defined on the set of [*degenerate configurations*]{}, that is $${S}={\{x\in ({\ensuremath{\mathbb{R}}}^2)^N \, | \; \exists \,i,j \in {\{1,\dots, N\}} \text{ s.t. } x_i=x_j\}}.$$ Moreover, it is not always well defined on the set of [*switching configurations*]{}, in which two or more agents are equidistant from another, $${D}={\{x\in ({\ensuremath{\mathbb{R}}}^2)^N \, | \; \exists \,i,j,k \in {\{1,\dots, N\}} \text{ s.t. } \|x_i-x_j\|=\|x_i-x_k\|\}},$$ because of the ambiguity in the definition of the “$n$ closest neighbors”.
To have a proper definition, we shall consider the following differential equation $$\label{eq:system-abstract}
\dot x(t)=f(x(t)),$$ where the flow ${f: ({\ensuremath{\mathbb{R}}}^2)^N\setminus ({D}{\operatorname{\cup}}{S}) \rightarrow ({\ensuremath{\mathbb{R}}}^2)^N}$ is defined componentwise as in equation , $$f_i(x)=\sum_{j\in {\mathcal{A}_{i}^{n}}} (x_j-x_i) - \xi^2 \sum_{j\in {\mathcal{R}_{i}^{n}}} \frac{(x_j-x_i)}{\|x_j-x_i\|^2}\qquad\forall i\in{\{1,\dots, N\}}.$$ Note that $f$ can not be extended with continuity to the set ${D}{\operatorname{\cup}}{S}$. Hence, a solution involving, for instance, two animals equidistant from a third one, can not be defined in the classical sense. In what follows, we shall extend the definition of the solutions of equation to include the set ${D}$. For such extension, we shall follow the approach in [@AFF:88], which requires defining a suitable [*differential inclusion*]{}, derived from .
To this goal, let $B(y,\delta)$ denote the Euclidean ball of radius $\delta$, centered in $y$, and set $${\mathcal{K}}f(y)= {\bigcap}_{\delta>0}{\bigcap}_{\mu(\Lambda)=0}{\operatorname{\overline{co}}}\left\{f(B(y,\delta)\setminus \Lambda)\right\},$$ where the operator ${\operatorname{\overline{co}}}$ denotes closed convex hull, and $\mu$ denotes Lebesgue measure. The map ${x: {\ensuremath{\mathbb{R}}_{\geq0}}\rightarrow ({\ensuremath{\mathbb{R}}}^2)^N}$ is said to be a [*Filippov solution*]{} of the system if it is absolutely continuous and it satisfies the differential inclusion $$\dot x(t)\in {\mathcal{K}}f(x(t))$$ for almost any $t>0.$
From now on, we restrict ourselves to the case in which ${{\alpha_{\textup{a}}}}=360{^\circ\!}$, ${{\alpha_{\textup{r}}}}=360{^\circ\!}$, and $n=1$. Hence, each animal interacts just with its closest mate, and ${\mathcal{A}_{i}^{1}}={\mathcal{R}_{i}^{1}}$ for every $i$. We make this assumption because, while the analysis for this case is simpler than for the general one, still the significant features of switching and asymmetry are apparent. Indeed, provided $N>2$, the relation of “being the closest to” needs not to be symmetrical. Moreover, simulations show for this case the formation of regular structures, as in Figure \[fig:cluster-views\], which can be of intrinsic interest.
In the case we are considering, it is useful to define the set of the closest neighbors of a given animal $i\in{\{1,\dots, N\}}$ as $${{\textup{closest}_{i}}}(x)=\arg\min\limits_{j\neq i}\{\|x_i-x_j\|\}.$$ Notice that ${{\textup{closest}_{i}}}(x)$ may be multivalued when $x\in{D}$, and let ${{|{{\textup{closest}_{i}}}(x)|}}$ denote its cardinality, that is the number of closest neighbors of animal $i$.
Given the above definitions, we are able to prove the existence of Filippov solutions.
\[th:existence\] Let $n=1$. Then, for any initial condition $x^0\in({\ensuremath{\mathbb{R}}}^2)^N\setminus{S}$, equation has at least one Filippov solution $x$, such that $x(0)=x^0$ and $x(t)\in ({\ensuremath{\mathbb{R}}}^2)^N\setminus{S}$ for every $t>0$.
We remark that $f$ is piecewise continuous in the following sense. Let $$\label{eq:LabelSet}
{\mathcal{V}}={\{v\in{\{1,\dots, N\}}^N \, | \; \forall i\in{\{1,\dots, N\}},\, v_i\neq i\}}.$$ For any $v\in{\mathcal{V}}$, let us also define the open, possibly empty, set $$E_v={\{x\in({\ensuremath{\mathbb{R}}}^2)^N\setminus{S}\, | \; \forall i\in{\{1,\dots, N\}},\, \|x_{v_i}-x_i\|<\|x_{k}-x_i\|, k\notin\{i,v_i\} \}}$$ which is the set of the configurations such that $v_i$ is $i$’s closest neighbor. Note that $E_v{{\ensuremath{\operatorname{\cap}}}}E_u=\emptyset$ if $u\neq v$ and the measure of the boundary of each $E_v$ is zero. Moreover, denoting by $\bar E_v$ the closure of $E_v$ in the induced topology of $({\ensuremath{\mathbb{R}}}^2)^N\setminus{S}$, it holds that $\bigcup_{v\in {\mathcal{V}}} \bar E_v=({\ensuremath{\mathbb{R}}}^2)^N\setminus{S}$. In the interior of each “piece” $E_v$, that is if $x\in E_v$, we have that $f(x)=f^v(x)$, where the function $f^v$ is defined componentwise as $$f^v_i(x)=\left(x_{v_i}-x_i\right)-\xi^2 \frac{x_{v_i}-x_i}{\|x_{v_i}-x_i\|^2}.$$ Moreover, each function $f^v$ is continuous in $\bar E_v$. These facts imply that for every $x\in ({\ensuremath{\mathbb{R}}}^2)^N\setminus{S}$, the set ${\mathcal{K}}f(x)$ is bounded, nonempty, closed and convex, and the map $x \mapsto {\mathcal{K}}f(x)$ is upper semicontinuous. Before we conclude, we need to show that, provided $x(0)\in ({\ensuremath{\mathbb{R}}}^2)^N\setminus{S}$, the solution can not reach ${S}$ either in finite time or asymptotically. By contradiction, let $$\label{eq:min}\lim_{t\to L} \min_{h,k\in{\{1,\dots, N\}}}{||x_h(t)-x_k(t)||}=0,$$ for $L\in (0,\infty].$ Note that for each $t\in[0,L)$, there exist a pair $(h^*,k^*)\in {\{1,\dots, N\}}^2$, possibly depending on time, such that it attains the minimum in , and thus ${h^*}={{\textup{closest}_{{k^*}}}}(x(t))$ and ${k^*}={{\textup{closest}_{{h^*}}}}(x(t))$. By continuity of the solution, it exists $t_0\in(0,L)$ and ${\varepsilon}\in (0,\xi)$ such that for $t\in [t_0,L)$, we have $||x_{h^*}(t)-x_{k^*}(t)||\le {\varepsilon}<\xi$. This implies that repulsion is larger than attraction, and then the animals are moving away from each other, that is $||x_{h^*}(t)-x_{k^*}(t)||$ is increasing in $t$. This contradicts equation .
At this point, we can apply [@AFF:88 §7 Theorem 1], and obtain that the differential inclusion $\dot x(t)\in {\mathcal{K}}f(x(t))$ has at least one solution $x(t)$, for all $t>0$ and for any initial condition $x^0\in({\ensuremath{\mathbb{R}}}^2)^N\setminus{S}.$
The above proof can be extended to the case $n>1$ and ${{{\alpha_{\textup{a}}}}<360{^\circ\!}}$, ${{{\alpha_{\textup{r}}}}<360{^\circ\!}}$, modulo a suitable redefinition of the “pieces” $E_v$, in order to account for the more complex neighborhood relationships: the notational setup would be cumbersome, and we do not detail it.
Loosely speaking, we would expect that a configuration having the closest neighbor(s) at distance $\xi$ for all animals, as in the lattice configuration of Figure \[fig:cluster-1\] would be an equilibrium configuration. Indeed, each animal $i$ is driven by the attraction-repulsion force component $f_i$ towards keeping a distance $\xi$ from its neighbors. The following result technically clarifies this intuition. Before the statement, we need to define a configuration $x^*$ to be a [*Filippov equilibrium*]{} of $f$ when $0\in K\,f(x^*)$.
\[prop:6-equilibria\] Let $x^*\in({\ensuremath{\mathbb{R}}}^2)^N\setminus{S}.$ If for all $i\in{\{1,\dots, N\}}$ and for all $k\in {{\textup{closest}_{i}}}(x^*)$, it holds $\|x^*_i-x^*_k\|=\xi$, then $x^*$ is a Filippov equilibrium for the system , and moreover $1\le{{|{{\textup{closest}_{i}}}(x^*)|}}\le 6$, for all $i\in {\{1,\dots, N\}}.$
We consider ${{\textup{closest}_{i}}}(x^*)$ for every $i\in {\{1,\dots, N\}}$, and distinguish the cases in which their cardinalities are equal to, or larger than 1. If ${{|{{\textup{closest}_{i}}}(x^*)|}}=1$ for all $i\in{\{1,\dots, N\}}$, then $f$ is smooth at $x^*$, and hence ${\mathcal{K}}f(x^*)=\{f(x^*)\}=\{0\}$. If instead there exists $h\in{\{1,\dots, N\}}$ such that ${{|{{\textup{closest}_{h}}}(x^*)|}}>1$, then let us consider the set ${\mathcal{V}}$ defined in , and take $v\in {\mathcal{V}}$ such that $v_i\in{{\textup{closest}_{i}}}(x^*),$ for every $i\in{\{1,\dots, N\}}$. Since $x^*$ is an accumulation point of $E_v$, let us consider a sequence $\{x^l\}_{l\in {{\mathbb{N}}}}\subset E_v$, such that $x^l\to x^*$ as $l\to+\infty$. Hence, $f(x^l)\to 0$ as $l\to\infty$, and this implies, by the definition of the differential inclusion, that $0\in {\mathcal{K}}f(x^*).$
By definition, ${{|{{\textup{closest}_{i}}}(x^*)|}}\ge 1$. The fact that ${{|{{\textup{closest}_{i}}}(x^*)|}}\le 6$ can be shown by contradiction. Let ${{|{{\textup{closest}_{i}}}(x^*)|}}=m$, with $m>6$, for some $i$. Then there are $m$ points of ${\ensuremath{\mathbb{R}}}^2$, representing positions, which belong to a circle of radius $\xi$ centered in $x^*_i$. But then the distance between two of them has to be less than $\xi$, which is a contradiction.
Hexagonal lattice configurations, in which animals are at distance $\xi$ from their closest neighbor(s), are observed in simulations in the case ${{\alpha_{\textup{a}}}}={{\alpha_{\textup{r}}}}=360{^\circ\!}$ and $n=1$, as reported in Figure \[fig:cluster-1\]. Indeed, Proposition \[prop:6-equilibria\] shows that such a lattice is an equilibrium: loosely speaking, we can say that it is the most closely packed configuration among the equilibria pointed out by this result. Notice moreover that such lattice equilibria are actually switching configurations, belonging to the set ${D}$: this gives an a posteriori justification of the effort that we have done for a careful extension of the solutions to this set.
Discussion {#sec:discussion}
==========
Our agent-based model has been conceived from assumptions which are widely accepted from the biological point of view, and it shows the onset of group structures and patterns which are observed in nature. In some sense, our model can be seen as a generalization of other similar models, since in the special case $({{\alpha_{\textup{r}}}}, {{\alpha_{\textup{a}}}})=(360{^\circ\!},360{^\circ\!})$ we recover results which are by now consolidated in literature. The novelty resides in that by modifying a small set of key parameters, we obtain other patterns, which are experimentally observed in animal groups. Our results suggest that apparently large differences in group patterns may arise just from differences in the attraction and repulsion sensitivity zones. In this perspective, we would like to point out that the group structure is not only a function of the species, but also of the external conditions. For instance, surf scoters [@RL-LEK:09] and other animals can form either clusters or lines, depending on the environmental conditions. This suggests that ${{\alpha_{\textup{r}}}}$ and ${{\alpha_{\textup{a}}}}$ are not only animal-dependent, but they can also vary depending on the type of motion, environmental conditions, presence of predators, and aim of the displacement. In this paper, the choices of the angles were made to explore the model’s properties and they do not come from experimental data. Nevertheless, the correspondence of the simulated patters with those experimentally observed suggests that future research should distinguish between the (variable) sensitivity zones and the (fixed) visual field, as in [@SG-SAL-DIR:96; @AN-VCB:08].
Now we review the results obtained in the previous sections. The goal is to propose some biological insights, especially about the role of anisotropy in the interactions. As we did presenting the simulations, we distinguish three cases: isotropic attraction and repulsion; anisotropic attraction and repulsion; isotropic attraction and anisotropic repulsion.
In the first case, with completely isotropic interactions, our model produces clusters of individuals. Clusters are common for small birds (e.g., starlings [@AC-IG:08e]) and for fish, whose sensing abilities allow an almost complete perception around them. Moreover, these animals often change the direction of motion following complex trajectories, and this makes useful to keep under control all the space around them. By means of isotropic interactions, they are able to change rapidly the direction of motion of the whole group, with no need to significantly rearrange the shape of the group. This would not be possible in less symmetric formations like lines or Vees.
Instead, when interactions are not isotropic, also the obtained formations are not isotropic, and we observe that their onset depends on spontaneous leader-following mechanisms. When both repulsion and attraction are focused in front, the leader-following mechanism induces the formation of lines. Lines are commonly observed in slow-moving animals as lobsters, elephants and penguins. The small repulsion angle ${{\alpha_{\textup{r}}}}$ appears to be crucial to obtain such a pattern (see Figure \[fig:elongation\]). This could be related to the fact that such animals keep a steady direction of motion, and, once the line is established, repulsion needs to be active only against the forerunners. The choice ${{\alpha_{\textup{a}}}}=180{^\circ\!}$ can also be significant from the biological point of view. Indeed, this would mean that individuals pay more attention to the group mates in front, while they do not respond to what happens behind them. For example, they would not perceive a disconnection of the group. Hence, a model assuming ${{\alpha_{\textup{a}}}}=180{^\circ\!}$ seems suitable either for animals with a restricted frontal visual field or for animals which are not particularly interested in the cohesion of the whole group. A different leader-following mechanism induces V-like formations when attraction is isotropic and repulsion is restricted to the group mates in front. Our simulations reproduce several significant features of natural V-like formations, described in the experimental literature [@LLG-FHH:74; @FHH:74; @FRH:87]. First, echelons and J-like formations are as common as perfect Vees. In our model all these formations appear, and actually the behavior of single individuals does not depend on the global shape of the formation. Second, V-like formations are not stable, but rather they often disband and quickly reform. Third, in our model each Vee is made of a limited number of individuals (see Figure \[fig:Vee\]), independently of the total number of animals $N$. In other words, increasing $N$ leads to the formation of a larger number of V-like groups, but not to larger groups. This fact has been experimentally observed [@FRH:87] and explained [@PS-AP-JKH:03] resorting to the argument of string instability, that we introduced in Section \[subsec:lines\]. Our results should be compared to those in [@AN-VCB:08], where authors propose an ad-hoc formation algorithm based on aerodynamic arguments in which the number of V-like groups is constant for increasing $N$.
The function of V-like formation has been greatly investigated (see for instance [@FRH:87; @CC-JS:94; @JRS-DB:98; @HW-JM-YC-PA-SJ:01; @PS-AP-JKH:03] and references therein). Two hypothesis are the most considered: aerodynamic advantage, and visual communication advantage. The former is based on the fact that each flying individual creates an upwash region behind it, just off the tips of its wings, so that another individual can benefit placing itself in that region. The latter is instead based on the fact that flying in a skewed position with respect to the bird in front is useful to avoid collisions, and allows an unhindered visual communication with all the group mates [@CC-JS:94]. Our results appear to support the hypothesis of visual communication advantage, since V-like formations are not obtained imposing individuals to stay in upwash regions. Instead, they are obtained just from frontal repulsion and, for the sake of group cohesion, isotropic attraction.
Motivated by some literature [@IA:80; @AH-CW:92; @AC-IG:08a], we have also investigated the dependence of the configurations on $n$, the number of interacting neighbors. We have seen that cluster configurations can be obtained with any $n$, but the value of $n$ influences the internal structure of the cluster. Moreover, it appears that an intermediate number of neighbors is preferable to obtain lines or V-like formations. These results seem to confirm the reasonableness of topological interactions with a limited number of neighbors. However, this point deserves further investigations.
Future research {#future-research .unnumbered}
---------------
We are keen on developing our research in three respects. First, it should be noted that although the presented model is bidimensional, it can readily be extended to a three-dimensional one. We are planning to study a 3D version of our model since the novel set of simulations might show interesting phenomena. We are especially interested in studying the effect of the anisotropy in 3D V-like formations, because the skewed formations described in [@FHH:74] are inherently three-dimensional.
Second, we want to further develop the theoretical analysis of the model, in order to include a thorough description of the equilibria of , and their stability analysis in the suitable switching systems framework. Moreover, we would be interested in a variational interpretation of the model. Indeed, while it is apparent that each animal is trying to minimize a “private” potential depending on its neighbors, it is unclear whether and when this would result in a configuration minimizing some global objective function.
Third, we believe that the simplicity of the interaction rules we have proposed, and the limited number of group mates to be kept into account, can be interesting features for engineering applications. In particular, we will investigate the design of robust algorithms for environmental deployment of robots or sensors and the formation cruising of unmanned vehicles.
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[^1]: CEMSAC, Università di Salerno, Italy and IAC-CNR, Rome, Italy. E-mail: `[email protected]`
[^2]: DIIMA, Università di Salerno, Italy and IAC-CNR, Rome, Italy. E-mail: `[email protected]`
[^3]: IAC-CNR, Rome, Italy. E-mail: `[email protected]` Mail: IAC-CNR c/o Dipartimento di Matematica dell’Università di Roma Tor Vergata, Via della Ricerca Scientifica, 00133, Rome, Italy
[^4]: To obtain sharper evidence from simulations, it is useful to strengthen repulsion with respect to attraction, taking a value of $\xi$ larger than for lines: in the following simulations we set $\xi=13$, and we also set $n=7$ and ${{v_{\textup{max}}}}=10$.
|
---
abstract: 'The IDE and verification system facilitates the use of a wide range of Satisfiability Modulo Theories (SMT) solvers through a driver-based architecture. We present : a portfolio-based approach to discharge proof obligations. We use data analysis and machine learning techniques on static metrics derived from program source code. Our approach benefits software engineers by providing a single utility to delegate proof obligations to the solvers most likely to return a useful result. It does this in a time-efficient way using existing and solver installations — without requiring low-level knowledge about SMT solver operation from the user.'
author:
- 'Andrew HealyRosemary MonahanJames F. Power'
bibliography:
- 'predicting.bib'
title: Predicting SMT Solver Performance for Software Verification
---
Introduction {#sec:intro}
============
The formal verification of software generally requires a software engineer to use a system of tightly integrated components. Such systems typically consist of an IDE that can accommodate both the implementation of a program and the specification of its formal properties. These two aspects of the program are then typically translated into the logical constructs of an intermediate language, forming a series of goals which must be proved in order for the program to be fully verified. These goals (or “proof obligations”) must be formatted for the system’s general-purpose back-end solver. Examples of systems which follow this model are Spec\# [@spec] and Dafny [@Dafny] which use the Boogie [@Boogie] intermediate language and the Z3 [@Z3] SMT solver.
[@why:whereprograms] was developed as an attempt to make use of the wide spectrum of interactive and automated theorem proving tools and overcome the limitations of systems which rely on a single SMT solver. It provides a driver-based, extensible architecture to perform the necessary translations into the input formats of two dozen provers. With a wide choice of theorem-proving tools now available to the software engineer, the question of choosing the most appropriate tool for the task at hand becomes important. It is this question that answers.
As motivation for our approach, Table \[table:avgtimes\] presents the results from running the tool over the example programs included in the distribution (version 0.87.1), using eight SMT solvers at the back-end. Each file contains a number of theories requiring proof, and these in turn are broken down into a number of goals for the SMT solver; for the data in Table \[table:avgtimes\] we had 128 example programs, generating 289 theories, in turn generating 1048 goals. In Table \[table:avgtimes\] each row presents the data for a single SMT solver, and the three main data columns give data totalled on a per-file, per-theory and per-goal basis. Each of these three columns is further broken down to show the number of programs/theories/goals that were successfully solved, their percentage of the total, and the average time taken in seconds for each solver to return such a result. Program verification by modularisation construct is particularly relevant to the use of on the command line as opposed to through the IDE.
Table \[table:avgtimes\] also has a row for an imaginary “theoretical” solver, , which corresponds to choosing the best (fastest) solver for each individual program, theory or goal. This solver performs significantly better than any individual solver, and gives an indication of the maximum improvement that could be achieved *if it was possible to predict in advance which solver was the best for a given program, theory or goal*. In general, the method of choosing from a range of solvers on an individual goal basis is called *portfolio-solving*. This technique has been successfully implemented in the SAT solver domain by SATzilla [@Satzilla] and for model-checkers [@DPVZ15:CAV][@MUX]. presents a unique opportunity to use a common input language to develop a portfolio SMT solver specifically designed for software verification.
The main contributions of this paper are:
1. The design and implementation of our portfolio solver, , which uses supervised machine learning to predict the best solver to use based on metrics collected from goals.
2. The integration of into the user’s existing work-flow by imitating the behaviour of an orthodox SMT solver.
3. A set of metrics to characterise goal formulae.
4. Statistics on the performance of eight SMT solvers using a dataset of 1048 goals.
[@l|YYY|YYY|YYY@]{} & & &\
& \# proved & % proved & Avg time & \# proved & % proved & Avg time & \# proved & % proved & Avg time\
& **48** & **37.5%** & **1.90** & **190** & **63.8%** & **1.03** & **837** & **79.9%** & **0.42**\
**Alt-Ergo-0.95.2** & 25 & 19.5% & 1.45 & 118 & 39.6%& 0.77 & 568 & 54.2% & 0.54\
**Alt-Ergo-1.01** & 34 & 26.6% & 1.70 & 142 & 47.7% & 0.79 & 632 & 60.3% & 0.48\
**CVC3** & 19 & 14.8% & 1.06 & 128 & 43.0% & 0.65 & 597 & 57.0% & 0.49\
**CVC4** & 19 & 14.8% & 1.09 & 117 & 39.3% & 0.51 & 612 & 58.4% & 0.37\
**veriT** & 5 & 4.0% & 0.12 & 79 & 26.5% & 0.20 & 333 & 31.8% & 0.26\
**Yices** & 14 & 10.9% & 0.53 & 102 & 34.2% & 0.22 & 368 & 35.1% & 0.22\
**Z3-4.3.2** & 25 & 19.5% & 0.56 & 128 & 43.0% & 0.36 & 488 & 46.6% & 0.38\
**Z3-4.4.1** & 26 & 20.3% & 0.58 & 130 & 43.6% & 0.40 & 581 & 55.4% & 0.35\
\[table:avgtimes\]
Section \[sec:overview\] describes how the data was gathered and discusses issues around the accurate measurement of results and timings. A comparison of prediction models forms the basis of Section \[sec:predselection\] where a number of evaluation metrics are introduced. The tool is compared to a range of SMT tools and strategies in Section \[sec:eval\]. The remaining sections present a review of additional related work and a summary of our conclusions.
System Overview and Data Preparation {#sec:overview}
====================================
Due to the diverse range of input languages used by software verification systems, a standardised benchmark repository of verification programs does not yet exist [@Dagstuhl]. For our study we chose the 128 example programs included in the distribution (version 0.87.1) as our corpus for training and testing purposes. The programs in this repository are written in WhyML, a dialect of ML with added specification syntax and verified libraries. Many of the programs are solutions to problems posed at software verification competitions such as VerifyThis [@verifythis], VSTTE [@Klebanov2011] and COST [@bormer:hal-00789525]. Other programs are implementations of benchmarks proposed by the VACID-0 [@Leino10vacid-0:verification] initiative. It is our assumption that these programs are a representative software verification workload. Alternatives to this dataset are discussed in Section \[sec:related\].
We used six current, general-purpose SMT solvers supported by : Alt-Ergo [@AltErgo] versions 0.95.2 and 1.01, CVC3 [@CVC3] ver. 2.4.1, CVC4 [@CVC4] ver. 1.4, veriT [@veriT], ver. 201506[^1], Yices [@Yices] ver. 1.0.38[^2], and Z3 [@Z3] ver. 4.3.2 and 4.4.1. We expanded the range of solvers to eight by recording the results for two of the most recent major versions of two popular solvers - Alt-Ergo and Z3.
![The relative amount of *Valid/Unknown/Timeout/Failure* answers from the eight SMT solvers (with a timeout of 60 seconds). Note that no tool returned an answer of *Invalid* for any of the 1048 proof obligations.[]{data-label="fig:barcharts"}](barcharts.pdf){width="0.7\linewidth"}
When a solver is sent a goal by it returns one of the five possible answers *Valid*, *Invalid*, *Unknown*, *Timeout* or *Failure*. As can be seen from Table \[table:avgtimes\] and Fig. \[fig:barcharts\], not all goals can be proved Valid or Invalid. Such goals usually require the use of an interactive theorem prover to discharge goals that require reasoning by induction. Sometimes a splitting transformation needs to be applied to simplify the goals before they are sent to the solver. Our tool does not perform any transformations to goals other than those defined by the solver’s driver file. In other cases, more time or memory resources need to be allocated in order to return a conclusive result. We address the issue of resource allocation in Section \[sec:independant\].
Problem Quantification: predictor and response variables
--------------------------------------------------------
Two sets of data need to be gathered in supervised machine learning [@Mitchell]: the independent/predictor variables which are used as input for both training and testing phases, and the dependent/response variables which correspond to ground truths during training. Of the 128 programs in our dataset, 25% were held back for system evaluation (Section \[sec:eval\]). The remaining 75% (corresponding to 96 WhyML programs, 768 goals) were used for training and 4-Fold cross-validation.
### Independent/Predictor Variables {#sec:independant}
Fig. \[fig:types\] lists the predictor variables that were used in our study. All of these are (integer-valued) metrics that can be calculated by analysing a proof obligation, and are similar to the *Syntax* metadata category for proof obligations written in the TPTP format [@TPTP]. To construct a feature vector from each task sent to the solvers, we traverse the abstract syntax tree (AST) for each goal and lemma, counting the number of each syntactic feature we find on the way. We focus on goals and lemmas as they produce proof obligations, with axioms and predicates providing a logical context.
Our feature extraction algorithm has similarities in this respect to the method used by for computing goal “shapes” [@why:preserving]. These shape strings are used internally by as an identifying fingerprint. Across proof sessions, their use can limit the amount of goals in a file which need to be re-proved.
child [node\[level 2\] (c0) [ops]{} child [node\[level 2, yshift=10pt\] (c1) [divisor]{}]{} child [node\[level 2, yshift=10pt\] (c2) [conds]{}]{} ]{} child [node\[level 2, yshift=-32pt, xshift=55pt\] (c3) [leaves]{}]{} child [node\[level 2, yshift=-32pt, xshift=55pt\] (c4) [quants]{}]{} ;
(c11) [and]{}; (c12) [or]{}; (c13) [not]{}; (c14) [let]{}; (c15) [as]{}; (c16) [eps]{}; (c17) [func]{};
(c21) [if]{}; (c22) [iff]{}; (c23) [imp]{}; (c24) [case]{};
(c31) [var]{}; (c32) [true]{}; (c33) [false]{}; (c34) [wild]{}; (c35) [zero-ar]{}; (c36) [int]{}; (c37) [float]{};
(c41) [forall]{}; (c42) [exists]{};
(c43) [depth]{}; (c44) [avg-arity]{};
in [1,...,7]{} (c1.195) |- (c1.west);
in [1,...,4]{} (c2.195) |- (c2.west);
in [1,...,7]{} (c3.195) |- (c3.west);
in [1,...,2]{} (c4.195) |- (c4.west);
### Dependent/Response Variables {#sec:dependant}
Our evaluation of the performance of a solver depends on two factors: the time taken to calculate that result, and whether or not the solver had actually proven the goal.
In order to accurately measure the time each solver takes to return an answer, we used a measurement framework specifically designed for use in competitive environments. The BenchExec [@benchexec] framework was developed by the organisers of the SVCOMP [@SVCOMP] software verification competition to reliably measure CPU time, wall-clock time and memory usage of software verification tools. We recorded the time spent on CPU by each SMT solver for each proof obligation. To account for random errors in measurement introduced at each execution, we used the methodology described by Lilja [@LiljaJ] to obtain an approximation of the true mean time. A 90% confidence interval was used with an allowed error of $\pm$3.5%.
By inspecting our data, we saw that most *Valid* and *Invalid* answers returned very quickly, with *Unknown* answers taking slightly longer, and *Failure/Timeout* responses taking longest. We took the relative utility of responses to be $\lbrace Valid, Invalid\rbrace>Unknown>\lbrace Timeout,Failure\rbrace$ which can be read as “it is better for a solver to return a *Valid* response than *Timeout*”, etc. A simple function allocates a cost to each solver $S$’s response to each goal $G$: $$\small
cost(S,G) =
\begin{cases}
time_{S,G}, \text{ if answer}_{S,G} \in \lbrace Valid, Invalid \rbrace \\
time_{S,G} + \text{timeout}, \text{ if answer}_{S,G} = Unknown \\
time_{S,G} + (\text{timeout} \times 2), \text{ if answer}_{S,G} \in \lbrace Timeout, Failure \rbrace
\end{cases}$$
Thus, to penalise the solvers that return an *Unknown* result, the timeout limit is added to the time taken, while solvers returning *Timeout* or *Failure* are further penalised by adding double the timeout limit to the time taken. A response of *Failure* refers to an error with the backend solver and usually means a required logical theory is not supported. This function ensures the best-performing solvers always have the lowest costs. A ranking of solvers for each goal in order of decreasing relevance is obtained by sorting the solvers by ascending cost.
Since our cost model depends on the time limit value chosen, we need to choose a value that does not favour any one solver. To establish a realistic time limit value, we find each solver’s “Peter Principle Point” [@Sutcliffe200139]. In resource allocation for theorem proving terms, this point can be defined as the time limit at which more resources will not lead to a significant increase in the number of goals the solver can prove.
Fig. \[fig:line-graph\] shows the number of *Valid/Invalid/Unknown* results for each prover when given a time limit of 60 seconds. This value was chosen as an upper limit, since a time limit value of 60 seconds is not realistic for most software verification scenarios. , for example, has a default time limit value of 5 seconds. From Fig. \[fig:line-graph\] we can see that the vast majority of useful responses are returned very quickly.
By satisfying ourselves with being able to record 99% of the useful responses which would be returned after 60 seconds, a more reasonable threshold is obtained for each solver. This threshold ranges from 7.35 secs (veriT) to 9.69 secs (Z3-4.3.2). Thus we chose a value of 10 seconds as a representative, realistic time limit that gives each solver a fair opportunity to return decent results.
![The cumulative number of *Valid/Invalid/Unknown* responses for each solver. The plot uses a logarithmic scale on the time axis for increased clarity at the low end of the scale. The chosen timeout limit of 10 secs (*dotted vertical line*) includes 99% of each solver’s useful responses[]{data-label="fig:line-graph"}](line-graph.pdf){width="0.7\linewidth"}
Choosing a prediction model {#sec:predselection}
===========================
Given a goal, a ranking of solvers can be obtained by sorting the cost for each solver. For unseen instances, two approaches to prediction can be used: (1) classification — predicting the final ranking directly — and (2) regression — predicting each solver’s score individually and deriving a ranking from these predictions. With eight solvers, there are $8!$ possible rankings. Many of these rankings were observed very rarely or did not appear at all in the training data. Such an unbalanced dataset is not appropriate for accurate classification, leading us to pursue the regression approach.
Seven regression models were evaluated[^3]: Linear Regression, Ridge Regression, K-Nearest Neighbours, Decision Trees, Random Forests (with and without discretisation) and the regression variant of Support Vector Machines. Table \[table:truncated\] shows the results for some of the best-performing models. Most models were evaluated with and without a weighting function applied to the training samples. Weighting is standard practice in supervised machine learning: each sample’s weight was defined as the standard deviation of solver costs. This function was designed to give more importance to instances where there was a large difference in performance among the solvers.
Table \[table:truncated\] also shows three theoretical strategies in order to provide bounds for the prediction models. *Best* always chooses the best ranking of solvers and *Worst* always chooses the worst ranking (which is the reverse ordering to *Best*). *Random* is the average result of choosing every permutation of the eight solvers for each instance in the training set. We use this strategy to represent the user selecting SMT solvers at random without any consideration for goal characterisation or solver capabilities. A comparison to a *fixed* ordering of solvers for each goal is not made as any such ordering would be arbitrarily determined.
We note that the *Best* theoretical strategy of Table \[table:truncated\] is not directly comparable with the theoretical solver from Table \[table:avgtimes\]. The two tables’ average time columns are measuring different results: in contrast to , *Best* will call each solver in turn, as will all the other models in Table \[table:truncated\], until a result *Valid/Invalid* is recorded (which it may never be). Thus Table \[table:truncated\]’s *Time* column shows the average *cumulative* time of each such sequence of calls, rather than the average time taken by the single *best* solver called by .
Evaluating the prediction models
--------------------------------
Table \[table:truncated\]’s *Time* column provides an overall estimate of the effectiveness of each prediction model. We can see that the discretised Random Forest method provides the best overall results for the solvers, yielding an average time of 14.92 seconds.
The second numeric column of Table \[table:truncated\] shows the Normalised Discounted Cumulative Gain (***nDCG***), which is commonly used to evaluate the accuracy of rankings in the search engine and e-commerce recommender system domains [@NDCG]. Here, emphasis is placed on correctly predicting items higher in the ranking. For a general ranking of length $p$, it is formulated as: $$\small
nDCG_p = \frac{DCG_p}{IDCG_p}
\quad\text{ where }\quad
DCG_p = \sum_{i=1}^{p} \frac{2^{rel_i} - 1}{log_2(i+1)}$$ Here $rel_i$ refers to the relevance of element $i$ with regard to a ground truth ranking, and we take each solver’s relevance to be inversely proportional to its rank index. In our case, $p = 8$ (the number of SMT solvers). The $DCG_p$ is normalised by dividing it by the maximum (or *idealised*) value for ranks of length $p$, denoted $IDCG_p$. As our solver rankings are permutations of the ground truth (making $nDCG$ values of 0 impossible), the values in Table \[table:truncated\] are further normalised to the range \[0..1\] using the lower $nDCG$ bound for ranks of length 8 — found empirically to be 0.4394.
The third numeric column of Table \[table:truncated\] shows the $R^2$ score (or coefficient of determination), which is an established metric for evaluating how well regression models can predict the variance of dependent/response variables. The maximum $R^2$ score is 1 but the minimum can be negative. Note that the theoretical strategies return rankings rather than individual solver costs. For this reason, $R^2$ scores are not applicable. Table \[table:truncated\]’s fourth numeric column shows the $MAE$ (Mean Average Error) — a ranking metric which can also be used to measure string similarity. It measures the average distance from each predicted rank position to the solver’s index in the ground truth. Finally, the fifth numeric column of Table \[table:truncated\] shows the mean regression error (*Reg. error*) which measures the mean absolute difference in predicted solver costs to actual values.
[@lYYYYY@]{} & **Time (secs)** & **nDCG** & $ R^2 $ & **MAE** & **Reg. error**\
***Best*** & 12.63 & 1.00 & - & 0.00 & 0.00\
***Random*** & 19.06 & 0.36 & - & 2.63 & 50.77\
***Worst*** & 30.26 & 0.00 & - & 4.00 & 94.65\
Random Forest & 15.02 & 0.48 & **0.28** & 2.08 & **38.91**\
Random Forest (discretised) & **14.92** & 0.48 & -0.18 & 2.13 & 39.19\
Decision Tree & 15.80 & 0.50 & 0.11 & 2.06 & 43.12\
K-Nearest Neighbours & 15.93 & **0.53** & 0.16 & **2.00** & 43.41\
Support Vector Regressor & 15.57 & 0.47 & 0.14 & 2.26 & 47.45\
Linear Regression & 15.17 & 0.42 & -0.16 & 2.45 & 49.25\
Ridge & 15.11 & 0.42 & -0.15 & 2.45 & 49.09\
\[table:truncated\]
Discussion: choosing a prediction model
---------------------------------------
An interesting feature of all the best-performing models in Table \[table:truncated\] is their ability to predict *multi-output* variables [@multisurvey]. In contrast to the Support Vector model, for example, which must predict the cost for each solver individually, a multi-output model predicts each solver’s cost simultaneously. Not only is this method more efficient (by reducing the number of estimators required), but it has the ability to account for the correlation of the response variables. This is a useful property in the software verification domain where certain goals are not provable and others are trivial for SMT solvers. Multiple versions of the same solver can also be expected to have highly correlated costs.
After inspecting the results for all learning algorithms (summarised in Table \[table:truncated\]), we can see that random forests [@RandomForests] perform well, relative to other methods. They score highest for three of the five metrics (shown in bold) and have generally good scores in the others. Random forests are an ensemble extension of decision trees: random subsets of the training data are used to train each tree. For regression tasks, the set of predictions for each tree is averaged to obtain the forest’s prediction. This method is designed to prevent over-fitting.
Based on the data in Table \[table:truncated\] we selected random forests as the choice of predictor to use in .
Implementing in OCaml
=====================
’s interaction with is inspired by the use of machine learning in the Sledgehammer tool [@Sledgehammer] which allows the use of SMT solvers in the interactive theorem prover Isabelle/HOL. We aspired to Sledgehammer’s ‘zero click, zero maintenance, zero overhead’ philosophy in this regard: it should not interfere with a user’s normal work-flow nor should it penalise those who do not use it.
We implement a “pre-solving” heuristic commonly used by portfolio solvers [@sunny-cp][@Satzilla]: a single solver is called with a short time limit before feature extraction and solver rank prediction takes place. By using a good “pre-solver” at this initial stage, easily-proved instances are filtered with a minimum time overhead. We used a ranking of solvers based on the number of goals each could prove, using the data from Table \[table:avgtimes\]. The highest-ranking solver installed locally is chosen as a pre-solver. For the purposes of this paper which assumes all 8 solvers are installed, the pre-solver corresponds to Alt-Ergo version 1.01. The effect pre-solving has on the method uses to return responses is illustrated in Alg. \[algo:where4\].
The random forest is fitted on the entire training set and encoded as a JSON file for legibility and modularity. This approach allows new trees and forests devised by the user (possibly using new SMT solvers or data) to replace our model. When the user installs locally, this JSON file is read and printed as an OCaml array. For efficiency, other important configuration information is compiled into OCaml data structures at this stage: e.g. the user’s `why3.conf` file is read to determine the supported SMT solvers. All files are compiled and a native binary is produced. This only needs to be done once (unless the locally installed provers have changed).
The command-line tool has the following functionality:
1. Read in the WhyML/Why file and extract feature vectors from its goals.
2. Find the predicted costs for each of the 8 provers by traversing the random forest, using each goal’s feature vector.
3. Sort the costs to produce a ranking of the SMT solvers.
4. Return a predicted ranking for each goal in the file, without calling any solver .
5. Alternatively, use the API to call each solver (if it is installed) in rank order until a *Valid/Invalid* answer is returned (using Alg. \[algo:where4\]).
\[algo:where4\]
If the user has selected that be available for use through , the file which lets know about supported provers installed locally is modified to contain a new entry for the binary. A simple driver file (which just tells to use the Why logical language for encoding) is added to the drivers’ directory. At this point, can be detected by , and then used at the command line, through the IDE or by the OCaml API just like any other supported solver.
Evaluating ’s performance on test data {#sec:eval}
======================================
The evaluation of was carried out on a test set of 32 WhyML files, 77 theories, 263 goals (representing 25% of the entire dataset). This section is guided by the following three Evaluation Criteria:
[@l|YYY|YYY|YYY@]{} & & &\
& \# proved & % proved & Avg time & \# proved & % proved & Avg time & \# proved & % proved & Avg time\
**** & 11 & 34.4% & 1.39 & 44 & 57.1% & 0.99 & 203 & 77.2% & 1.98\
*Best* & [(-.5em,.8em)–(-.5em,-1em);]{}& [(-.5em,.8em)–(-.5em,-1em);]{}& 0.25 & [(-.5em,.8em)–(-.5em,-1em);]{}& [(-.5em,.8em)–(-.5em,-1em);]{}& 0.28 & [(-.5em,.8em)–(-.5em,-1em);]{}& [(-.5em,.8em)–(-.5em,-1em);]{}& 0.37\
*Random* & [(-.5em,.8em)–(-.5em,-1em);]{}& [(-.5em,.8em)–(-.5em,-1em);]{}& 4.19 & [(-.5em,.8em)–(-.5em,-1em);]{}& [(-.5em,.8em)–(-.5em,-1em);]{}& 4.02 & [(-.5em,.8em)–(-.5em,-1em);]{}& [(-.5em,.8em)–(-.5em,-1em);]{}& 5.70\
*Worst* & [(-.5em,1em)–(-.5em,0em);]{}& [(-.5em,1em)–(-.5em,0em);]{}& 14.71 & [(-.5em,1em)–(-.5em,0em);]{}& [(-.5em,1em)–(-.5em,0em);]{}& 13.58 & [(-.5em,1em)–(-.5em,0em);]{}& [(-.5em,1em)–(-.5em,0em);]{}& 18.35\
**Alt-Ergo-0.95.2** & 8 & 25.0% & 0.78 & 37 & 48.1%& 0.26 & 164 & 62.4% & 0.34\
**Alt-Ergo-1.01** & 10 & 31.3% & 1.07 & 39 & 50.6% & 0.26 & 177 & 67.3% & 0.33\
**CVC3** & 5 & 15.6% & 0.39 & 36 & 46.8% & 0.21 & 167 & 63.5% & 0.38\
**CVC4** & 4 & 12.5% & 0.56 & 32 & 41.6% & 0.21 & 147 & 55.9% & 0.35\
**veriT** & 2 & 6.3% & 0.12 & 24 & 31.2% & 0.12 & 100 & 38.0% & 0.27\
**Yices** & 4 & 12.5% & 0.32 & 32 & 41.6% & 0.15 & 113 & 43.0% & 0.18\
**Z3-4.3.2** & 6 & 18.8% & 0.46 & 31 & 40.3% & 0.20 & 145 & 55.1% & 0.37\
**Z3-4.4.1** & 6 & 18.8% & 0.56 & 31 & 40.3% & 0.23 & 145 & 55.1% & 0.38\
\[table:avgtimes2\]
EC1: How does perform in comparison to the 8 SMT solvers under consideration?
-----------------------------------------------------------------------------
![The relative amount of Valid/Unknown/Timeout/Failure answers from the eight SMT solvers. Shown on the right are results obtainable by using the top solver (only) with the 3 ranking strategies and the predicted ranking (with an Alt-Ergo-1.01 pre-solver).[]{data-label="fig:barchart2"}](barcharts2.pdf){width="0.9\linewidth"}
![The effect of using a cost threshold. (*top*) The average time taken for to return an answer compared to 8 SMT solvers. (*bottom*) The number of Valid/Invalid answers returned by compared to 8 SMT solvers. For the 7 solvers other than Alt-Ergo-1.01, the number of provable goals is indicated by a mark on the y-axis rather than an intersection with ’s results.[]{data-label="fig:thresholds"}](thresholds.pdf){width="0.8\linewidth"}
![The cumulative time each theoretical strategy, and to return all *Valid/Invalid* answers in the test dataset of 263 goals[]{data-label="fig:line-graph-eval-provers"}](line-graph-eval-provers.pdf){width="0.8\linewidth"}
When each solver in ’s ranking sequence is run on each goal, the maximum amount of files, theories and goals are provable. As Table \[table:avgtimes2\] shows, the difference between and our set of reference theoretical strategies (*Best, Random*, and *Worst*) is the amount of time taken to return the *Valid/Invalid* result. Compared to the 8 SMT provers, the biggest increase is on individual goals: can prove 203 goals, which is 26 (9.9%) more goals than the next best single SMT solver, Alt-Ergo-1.01.
Unfortunately, the average time taken to solve each of these goals is high when compared to the 8 SMT provers. This tells us that can perform badly with goals which are not provable by many SMT solvers: expensive *Timeout* results are chosen before the *Valid* result is eventually returned. In the worst case, may try and time-out for all 8 solvers in sequence, whereas each individual solver does this just once. Thus, while having access to more solvers allows more goals to be proved, there is also a time penalty to portfolio-based solvers in these circumstances.
At the other extreme, we could limit the portfolio solver to just using the best predicted individual solver (after “pre-solving”), eliminating the multiple time-out overhead. Fig. \[fig:barchart2\] shows that the effect of this is to reduce the number of goals provable by , though this is still more than the best-performing individual SMT solver, Alt-Ergo-1.01.
To calibrate this cost of against the individual SMT solvers, we introduce the notion of a *cost threshold*: using this strategy, after pre-solving, solvers with a predicted cost above this threshold are not called. If no solver’s cost is predicted below the threshold, the pre-solver’s result is returned.
Fig. \[fig:thresholds\] shows the effect of varying this threshold, expressed in terms of the average execution time (top graph) and the number of goals solved (bottom graph). As we can see from both graphs in Fig. \[fig:thresholds\], for the goals in the test set a threshold of 7 for the cost function allows to prove more goals than any single solver, in a time approximately equal to the four slower solvers (CVC4, veriT and both versions of Z3).
EC2: How does perform in comparison to the 3 theoretical ranking strategies?
----------------------------------------------------------------------------
Fig. \[fig:line-graph-eval-provers\] compares the cumulative time taken for and the 3 ranking strategies to return the 203 valid answers in the test set. Although both and *Random* finish at approximately the same time, is significantly faster for returning *Valid/Invalid* answers. ’s solid line is more closely correlated to *Best*’s rate of success than the erratic rate of the *Random* strategy. *Best*’s time result shows the capability of a perfect-scoring learning strategy. It is motivation to further improve in the future.
EC3: What is the time overhead of using to prove goals?
-------------------------------------------------------
The timings for in all plots and tables are based solely on the performance of the constituent solvers (the measurement of which is discussed in Sec. \[sec:dependant\]). They do not measure the time it takes for the OCaml binary to extract the static metrics, traverse the decision trees and predict the ranking. We have found that this adds (on average) 0.46 seconds to the time takes to return a result for each file. On a per goal basis, this is equivalent to an increase in 0.056 seconds.
The imitation of an orthodox solver to interact with is more costly: this is due to printing each goal as a temporary file to be read in by the solver individually. Future work will look at bypassing this step for WhyML files while still allowing files to be proved on an individual theory and goal basis.
Threats to Validity
-------------------
We categorise threats as either *internal* or *external*. Internal threats refer to influences that can affect the response variable without the researcher’s knowledge and threaten the conclusions reached about the *cause* of the experimental results [@experimentation]. Threats to external validity are conditions that limit the generalisability and reproducibility of an experiment.
### Internal {#sec:internal}
The main threat to our work’s internal validity is selection bias. All of our training and test samples are taken from the same source. We took care to split the data for training and testing purposes on a *per file* basis. This ensured that was not trained on a goal belonging to the same theory or file as any goal used for testing. The results of running the solvers on our dataset are imbalanced. There were far more *Valid* responses than any other response. No goal in our dataset returned an answer of *Invalid* on any of the 8 solvers. This is a serious problem as would not be able to recognize such a goal in real-world use. In future work we hope to use the TPTP benchmark library to remedy these issues. The benchmarks in this library come from a diverse range of contributors working in numerous problem domains [@Sutcliffe200139] and are not as specific to software verification as the suite of examples.
Use of an independent dataset is likely to influence the performance of the solvers. Alt-Ergo was designed for use with the platform — its input language is a previous version of the Why logic language. It is natural that the developers of the examples would write programs which Alt-Ergo in particular would be able to prove. Due to the syntactic similarities in input format and logical similarities such as support for type polymorphism, it is likely that Alt-Ergo would perform well with any dataset. We would hope, however, that the gulf between it and other solvers would narrow.
There may be confounding effects in a solver’s results that are not related to the independent variables we used (Sec. \[sec:independant\]). We were limited in the tools available to extract features from the domain-specific Why logic language (in contrast to related work on model checkers which use the general-purpose C language [@DPVZ15:CAV][@MUX]). We made the decision to keep the choice of independent variables simple in order to increase generalisability to other formalisms such as Microsoft’s Boogie [@Boogie] intermediate language.
### External
The generalisability of our results is limited by the fact that all dependent variables were measured on a single machine.[^4] We believe that the number of each response for each solver would not vary dramatically on a different machine of similar specifications. By inspecting the results when each solver was given a timeout of 60 seconds (Fig. \[fig:line-graph\]), the rate of increase for *Valid/Invalid* results was much lower than that of *Unknown/Failure* results. The former set of results are more important when computing the cost value for each solver-goal pair.
Timings of individual goals are likely to vary widely (even across independent executions on the same machine). It is our assumption that although the actual timed values would be quite different on any other machine, the *ranking* of their timings would stay relatively stable.
A “typical” software development scenario might involve a user verifying a single file with a small number of resultant goals: certainly much smaller than the size of our test set (263 goals). In such a setting, the productivity gains associated with using would be minor. is more suited therefore to large-scale software verification.
Discussion {#sec:eval-discuss}
----------
By considering the answers to our three Evaluation Criteria, we can make assertions about the success of . The answer to EC1, ’s performance in comparison to individual SMT solvers, is positive. A small improvement in *Valid/Invalid* responses results from using only the top-ranked solver, while a much bigger increase can be seen by making the full ranking of solvers available for use. The time penalty associated with calling a number of solvers on an un-provable proof obligation is mitigated by the use of a *cost threshold*. Judicious use of this threshold value can balance the time-taken-versus-goals-proved trade-off: in our test set of 263 POs, using a threshold value of 7 results in 192 *Valid* responses – an increase of 15 over the single best solver – in a reasonable average time per PO (both *Valid* and otherwise) of 4.59 seconds.
There is also cause for optimism in ’s performance as compared to the three theoretical ranking strategies — the subject of Evaluation Criterion 2. All but the most stubborn of *Valid* answers are returned in a time far better than *Random* theoretical strategy. We take this random strategy as representing the behaviour of the non-expert user who does not have a preference amongst the variety of supported SMT solvers. For this user, could be a valuable tool in the efficient initial verification of proof obligations through the system.
In terms of time overhead — the concern of EC3 — our results are less favourable, particularly when is used as an integrated part of the toolchain. The costly printing and parsing of goals slows beyond the time overhead associated with feature extraction and prediction. At present, due to the diversity of languages and input formats used by software verification tools, this is an unavoidable pre-processing step enforced by (and is indeed one of the system’s major advantages).
Overall, we believe that the results for two out of three Evaluation Criteria are encouraging and suggest a number of directions for future work to improve .
Comparison with Related Work {#sec:related}
============================
*Comparing verification systems:* The need for a standard set of benchmarks for the diverse range of software systems is a recurring issue in the literature [@Dagstuhl]. The benefits of such a benchmark suite are identified by the SMTLIB [@SMTLIB] project. The performance of SMT solvers has significantly improved in recent years due in part to the standardisation of an input language and the use of standard benchmark programs in competitions [@SMTEVAL2013][@SVCOMP]. The TPTP (Thousands of Problems for Theorem Provers) project [@TPTP] has similar aims but a wider scope: targeting theorem provers which specialise in numerical problems as well as general-purpose SAT and SMT solvers. The TPTP library is specifically designed for the rigorous experimental comparison of solvers [@Sutcliffe200139].
*Portfolio solvers:* Portfolio-solving approaches have been implemented successfully in the SAT domain by SATzilla [@Satzilla] and the constraint satisfaction / optimisation community by tools such as CPHydra [@CPHydra] and sunny-cp [@sunny-cp]. Numerous studies have used the SVCOMP [@SVCOMP] benchmark suite of C programs for model checkers to train portfolio solvers [@MUX][@DPVZ15:CAV]. These particular studies have been predicated on the use of Support Vector Machines (SVM) with only a cursory use of linear regression [@MUX]. In this respect, our project represents a more wide-ranging treatment of the various prediction models available for portfolio solving. The need for a strategy to delegate goals to appropriate SMT solvers is stated in recent work looking at verification systems on cloud infrastructures [@rodinplugin].
*Machine Learning in Formal Methods:* The FlySpec [@Flyspec] corpus of proofs has been the basis for a growing number of tools integrating interactive theorem provers with machine-learning based fact-selection. The MaSh engine in Sledgehammer [@Sledgehammer] is a related example. It uses a Naive Bayes algorithm and clustering to select facts based on syntactic similarity. Unlike , MaSh uses a number of metrics to measure the *shape* of goal formulæas features. The weighting of features uses an inverse document frequency (IDF) algorithm. ML4PG (Machine Learning for Proof General) [@ML4PG] also uses clustering techniques to guide the user for interactive theorem proving.
Our work adds to the literature by applying a portfolio-solving approach to SMT solvers. We conduct a wider comparison of learning algorithms than other studies which mostly use either SVMs or clustering. Unlike the interactive theorem proving tools mentioned above, is specifically suited to software verification through its integration with the system.
Conclusion and Future Work
==========================
We have presented a strategy to choose appropriate SMT solvers based on syntactic features. Users without any knowledge of SMT solvers can prove a greater number of goals in a shorter amount of time by delegating to than by choosing solvers at random. Although some of ’s results are disappointing, we believe that the platform has great potential for machine-learning based portfolio-solving. We are encouraged by the performance of a theoretical *Best* strategy and the convenience that such a tool would give users.
The number of potential directions for this work is large: parallel solving, minimal datasets for practical local training, larger and more generic datasets for increased generalisability, etc. The TPTP repository represents a large source of proof obligations which can be translated into the Why logic language. The number of goals provable by could be increased by identifying which goals need to be simplified in order to be tractable for an SMT solver. Splitting transforms would also increase the number of goals for training data: from 1048 to 7489 through the use of the `split_goal_wp` transform, for example. An interesting direction for this work could be the identification of the appropriate transformations. Also, we will continue to improve the efficiency of when used as a solver and investigate the use of a minimal benchmark suite which can be used to train the model using new SMT solvers and theorem provers installed locally.
Data related to this paper is hosted at `github.com/ahealy19/F-IDE-2016`. is hosted at `github.com/ahealy19/where4`.
### Acknowledgments. {#acknowledgments. .unnumbered}
This project is being carried out with funding provided by Science Foundation Ireland under grant number 11/RFP.1/CMS/3068
[^1]: The most recent version - 201506 - is not officially supported by but is the only version available
[^2]: We did not use Yices2 as its lack of support for quantifiers makes it unsuitable for software verification
[^3]: We used the Python Sci-kit Learn [@sklearn] implementations of these models
[^4]: All data collection was conducted on a 64-bit machine running Ubuntu 14.04 with a dual-core Intel i5-4250U CPU and 16GB of RAM.
|
---
author:
- 'R.G. Gratton, S. Lucatello, A. Bragaglia, E. Carretta, Y. Momany, E. Pancino, E. Valenti'
date: 'Received: ; accepted: '
title: 'Na-O Anticorrelation And HB. III. The abundances of from FLAMES-UVES spectra [^1]'
---
Introduction
============
is a very luminous ($M_V=-9.18$: Harris 1996) and massive Globular Cluster located in the inner regions of our Galaxy: according to Harris (1996) it is located about 2.1 kpc from the Galactic center on the other side of the Galaxy with respect to the Sun. has raised a considerable interest because, in spite of being a metal-rich cluster (\[Fe/H\]=$-0.53\pm 0.11$, Armandroff & Zinn 1988), it has a very extended horizontal branch, with several stars on the blue side of the instability strip (Rich et al. 1997). Layden et al. (1999) estimated that about 17% of the horizontal branch stars are found in a feature bluer and brighter than the red clump, a peculiarity shared with the other massive, metal-rich bulge cluster (Rich et al. 1997). has also a quite rich and peculiar population of RR Lyrae (Layden et al. 1999; Pritzl et al. 2001), characterized by long periods in spite of the high metallicity of the Cluster. The color-magnitude diagram of presents other peculiarities: the red giant sequence is broad, and the clump is clearly tilted. These features may at least in part be justified by differential reddening (see e.g. Layden et al. 1999). However, given that is bright and massive, some star-to-star spread in the metal abundances cannot be excluded. Such a possibility was raised in order to explain the odd properties of the horizontal branch and of the RR Lyrae population (Piotto et al. 1997; Sweigart 2001; Pritzl et al. 2001); however, very recently Clementini et al. (2005) have shown that most of the RR Lyrae of are metal-rich, likely in accordance to the other cluster stars. The peculiarities of must then still find a comprehensive explanation.
All the previous discussion seems to imply that is further evidence of the second parameter effect in Globular Clusters; here, like in , the second parameter effect is manifest within the same cluster. A peculiarity of (and of ) is that the blue horizontal branch is brighter than the red one. This might be explained by a higher content in Helium (Sweigart & Catelan 1998). The existence of He-rich populations have been recently proposed to justify various features of some Globular Clusters (D’Antona & Caloi 2003; Bedin et al. 2004; Piotto et al. 2005; D’Antona et al. 2005). In particular, a He-rich population might indicate a second generation of stars born from the ejecta of the most massive of the intermediate-mass stars of the earlier generation, required to explain the observations of O-Na and Mg-Al abundance anticorrelations, ubiquitous in Globular Clusters but not observed among field stars (see the review by Gratton et al. 2004). If this scenario is correct, there should be a correlation between the distribution of the O/Na abundance ratios, observed e.g. on the red giant branch of a cluster, and the color distribution of stars along the horizontal branch of the same cluster.
In order to explore whether such a correlation indeed exists, we have undertaken an extensive spectroscopic survey of Globular Cluster stars with the FLAMES multi-fiber facility at ESO (Pasquini et al. 2002). With its high multiplexing capabilities, FLAMES can get spectra of hundreds of stars in Globular Clusters within a few hours of VLT observing time. FLAMES may simultaneously feed two spectrographs: GIRAFFE, which is most suited for observation of extensive samples of stars in restricted wavelength regions, and UVES, which provides extensive spectral coverage for a few stars. The latter higher resolution (R$\simeq$40,000) spectra may be used to carefully derive the chemical composition of the cluster, and possibly discover the existence of peculiarities due to its chemical evolution history, by comparing the abundance pattern with that obtained for field stars of similar characteristics.
was an obvious candidate to be included in our sample. In this paper, we present an analysis of the FLAMES+UVES data we gathered for stars in this cluster, as well as for a few stars in the same field that turned out not to be members of the cluster. A separate paper will be devoted to the discussion of the GIRAFFE data.
Observations
============
--------------- ------------ ---------- ----------- ---------- -------------
Fiber Date Time Exp. Time Seeing Airmass
Configuration (UT) (sec) (arcsec)
\#1 2004-07-06 04:21:42 5300 1.59 1.040-1.174
2004-07-11 02:48:19 5300 1.35 1.029-1.052
2004-07-11 04:19:18 5300 0.91 1.055-1.221
\#2 2004-07-17 05:18:20 5300 0.69 1.202-1.605
2004-07-26 03:39:54 5300 1.04 1.077-1.282
--------------- ------------ ---------- ----------- ---------- -------------
\[t:journal\]
Data used in this paper are based on exposures obtained in July 2004 with FLAMES at Kueyen (=VLT2) used in service mode. Details of the observations are given in Table \[t:journal\].
is a quite far and concentrated cluster ($c=1.85$: Harris 1996) lying in a very crowded region, close to the direction of the Galactic center ($l=353.53^{\circ}$, $b=-5.01^{\circ}$). Unluckily, when we started our project there was no suitable membership study available. These facts made identification of stars appropriate for abundance analysis difficult, given the safety constraints on fiber-to-fiber separation of the Oz-Poz FLAMES fiber positioner, and the need to avoid stars whose images are blended with neighbors. Our targets were carefully selected from high quality photometry.
The photometric data have been obtained within the observing program 69.D-0582(A) on June 2002 at the 2.2 m ESO/MPI telescope at La Silla Observatory (Chile), using the optical camera WFI. The exceptional capabilities of this imager provide a mosaic of eight CCD chips, each with a field of view of ${\sim}8'{\times}16'$, for a total field of view of ${\sim}34'{\times}33'$. The cluster center was roughly centered on Chip \#7, and the images were taken through the $V$ and $I_c$ broad band filters with typical exposure times of 80 sec and 50 sec, respectively. During the night the average seeing was quite good (full width half maximum FWHM${\approx}0\farcs9$).
Stellar photometry was performed using the standard and tested DAOPHOT II and ALLFRAME programs (Stetson 1994). In crowded fields such as that surrounding NGC 6441, these programs provide high quality stellar photometry by point spread function (PSF) fitting (see Momany et al. 2002, 2004). The single PSFs were constructed for each image after a carefull selection of isolated and well-distributed stars across each chip. The instrumental PSF magnitudes of NGC 6441 were normalized to 1 s exposure and zero airmass, and corrected for the aperture correction. The aperture corrections are necessary to convert the instrumental profile-fitting magnitudes of NGC 6441 on the scale of the standard star measurements (based on aperture photometry, see below). The instrumental NGC 6441 magnitudes are therefore:
$$m^{'}=m_{\rm apert.}+2.5\log(t_{exp})-K_{\lambda}X$$
where $m_{\rm apert.}=m_{\rm PSF}-{\rm apert.~correction}$, $X$ is the airmass of the reference image for each filter, and the adopted mean extinction coefficients for La Silla are: $K_V=0.16$ and $K_I=0.07$. The aperture corrections were estimated upon isolated bright stars, uniformally distributed across the chip. For these we obtained aperture photometry after subtracting eventual nearby companions within 5 FWHM. The aperture magnitudes measured in circular apertures of 6 arcseconds in radius (close to the photoelectric aperture employed by Landolt 1992) were then compared to the PSF magnitudes, and the difference is assumed to be the aperture correction to apply to the PSF magnitudes.
Separately, aperture magnitudes of standard stars from 3 Landolt (1992) fields were measured in circular apertures of 6 arcseconds in radius. These aperture magnitudes were normalized to their corresponding airmass and exposure times, and compared with those tabled in Landolt (1992). A least square fitting procedure provided the calibration relations. The $r.m.s.$ scatter of the residuals of the fit ($0.009$ and $0.011$) are assumed to represent our calibration uncertainties in $V$ and $I$ respectively. The total zero-point uncertainties, including the aperture corrections and calibration errors, are $0.013$, and $0.016$ mag in $V$ and $I$ respectively. The PSF photometry of NGC 6441 (normalized and corrected for aperture correction) was then calibrated using the relations:
$$V=v^{'}-0.077(V-I)+24.199$$
$$I=i^{'}+0.097(V-I)+23.142$$
The [*Guide Star Catalog (GSCII)*]{} was used to search for astrometric standards in the entire WFI image field of view. Several hundred astrometric [*GSCII*]{} reference stars were found in each chip, allowing us an accurate absolute positioning of the sources. An astrometric solution has been obtained for each of the eight WFI chips independently, by using suitable catalog matching and cross-correlation tools developed at the Bologna Observatory. At the end of the entire procedure, the [*rms*]{} residuals are ${\leq}$$0\farcs$2 both in right ascension and declination.
Only uncrowded stars were considered as possible targets, that is stars not showing any companion brighter than $V_{\rm target}+2$ mag within a 2.5 arcsec radius, or brighter than $V_{\rm target}-2$ mag within 10 arcsec. The targets were selected to lie close to the cluster mean locus in the color-magnitude diagram.
The fibers feeding the UVES spectrograph were centered on stars close to the tip of the RGB ($15.8<V<16.3$; see Figure \[f:6441cmd\]). We used two different fiber configurations: in the first set, we observed six stars, with two fibers dedicated to the sky; in the second one, seven fibers were used for the stars, and one for the sky. The spectra cover the wavelength range 4700-6900 Å, and have $35<S/N<85$ for the combined exposures.
As part of the service mode observations, the spectra were reduced by ESO personnel using the dedicated UVES-FLAMES pipeline (uves/2.1.1 version). We found that this UVES pipeline does not accurately subtract the background between orders in the green-yellow part of the spectra. Luckily, the redder part of the spectra included enough lines for our purposes. We hence relied on these pipelines reductions; the bluer portions of the spectra were then not considered in the present analysis.
Cluster membership
==================
--------- ------- ---------- ------------ ---------- -------- ------- ------- --------------- ----- -------------- ---------- -----------
Star Fiber RA Dec. Dist. V V-I V-K RV S/N $\sigma$(EW) \[Fe/H\] Notes
Conf. (degree) (degree) (arcsec) (mag) (mag) (mag) (km s$^{-1})$ (mÅ)
6003734 1 267.422 $-$36.9533 518 15.992 2.083 4.574 +83.5 39 8.7 $-$0.25
6004360 2 267.475 $-$37.1471 414 16.127 1.979 4.403 +89.7 37 8.1 0.27
6005308 1 267.463 $-$37.0816 284 16.290 1.973 $-$144.1 36 8.5 $-$0.17
6005341 2 267.447 $-$36.9682 429 16.294 2.095 4.543 $-$41.2 50 9.4 0.06
7003717 2 267.593 $-$36.9987 219 15.989 2.113 4.522 $-$134.0 50 fast rot.
7004050 1 267.543 $-$37.0793 106 16.062 2.168 4.775 +24.9 52 8.0 $-$0.45 member
7004329 1 267.581 $-$37.1614 404 16.120 1.997 4.258 $-$23.1 51 10.0 0.12
7004434 2 267.541 $-$37.0849 127 16.142 2.167 4.719 +13.6 53 10.0 $-$0.34 member
7004453 1 267.525 $-$37.0109 167 16.144 2.029 4.382 $-$49.3 49 6.7 $-$0.04
7004463 2 267.506 $-$37.0572 140 16.146 2.160 +11.6 77 9.1 $-$0.38 member
7004487 2 267.520 $-$37.0535 99 16.150 2.088 4.552 +32.3 82 6.1 $-$0.50 member
8002961 2 267.670 $-$37.0725 342 15.798 2.185 4.803 +54.1 86 5.8 $-$0.28 member?
8003092 1 267.753 $-$37.0622 573 15.832 2.203 4.923 $-$53.0 37 7.8 $-$0.25
--------- ------- ---------- ------------ ---------- -------- ------- ------- --------------- ----- -------------- ---------- -----------
\[t:uvesphot\]
Table \[t:uvesphot\] gives details on the main parameters for the observed stars, providing for each one of them the distance in arcsec from the cluster center, the magnitudes, mean radial velocities, signal-to-noise ratios of the combined spectra and the $\sigma$ of the EWs residuals, as well as the derived metallicity. Star designations are according to Valenti et al. (2006), from which photometric data were also taken. Coordinates (at J2000 equinox) are from our astrometry (Valenti et al. 2006); distances from the cluster center were obtained considering the nominal position given by Harris (1996). Radial velocities were measured from our spectra, using typically about 100 atomic lines. Errors in these velocities should typically be of a few hundreds of m/s. The signal-to-noise ratio S/N was estimated from the pixel-to-pixel scatter in spectral regions free from absorption lines at about 6200 Å.
Four of the thirteen observed stars (\#7004050, 7004434, 7004463, and 7004487) are very likely members of the cluster, on the basis of the following criteria:
- they are projected close to the nominal cluster center, as given by Harris (1996). These four stars are the closest in our sample to the nominal cluster center, and are all within 150 arcsec from this position. Note that according to Harris (1996) the core radius of is about 7 arcsec, and the tidal radius is about 467 arcsec.
- their radial velocities (+24.9, +13.6, +11.6, and +32.3 km s$^{-1}$ for stars 7004050, 7004434, 7004463, and 7004487, respectively) are close to the average radial velocity listed by Harris ($+16.4\pm 1.2$ km s$^{-1}$). The average velocity given by these four stars is $+20.6\pm 4.9$ km s$^{-1}$ ($r.m.s.$ scatter of 9.8 km s$^{-1}$), in good agreement with the value listed by Harris.
- additionally, the four stars are confined in a very small range of colors, with $2.088<V-I<2.168$, with respect to the rather broad color distribution along the apparent RGB of (see Figure \[f:6441cmd\]). This by itself is not a strong membership criterion, since the interstellar reddening is expected to be quite variable over the field (see Layden et al. 1999 for an extensive discussion). However, the close agreement between the colors of the confirmed member stars might indicate that differential reddening is less of a problem than expected, at least for these particular stars (it should be noted that all these stars are in the south-east quadrant of the cluster, quite close to each other).
- a posteriori, our analysis indicates that these stars have quite similar chemical composition. Again, this is not by itself a strong membership criterion, since there might be some spread in metallicity amongst the stars of and because their metallicity is not unusual for bulge or disk field stars.
![$(V,V-I)$ color magnitude diagram for selected stars in the field of (from Valenti et al. 2006). Large symbols are stars observed with UVES FLAMES. Filled symbols mark stars member of the cluster, on the basis of radial velocities and location in the field close to the cluster center; open symbols are non-member stars.[]{data-label="f:6441cmd"}](gratton_6441_0.eps){width="8.8cm"}
Among the remaining stars, \#8002961 is also a possible cluster member, essentially on the basis of the not too large difference between its radial velocity (+54.1 km s$^{-1}$) and that of the cluster[^2], and its location in the color magnitude diagram, since it lies close to the position occupied by the member stars. This star is projected at a quite large distance from the cluster center (341 arcsec), however still well within the tidal radius of . A posteriori, also the chemical composition supports cluster membership. It is noteworthy that the inclusion or exclusion of this particular stars from the group ob objects considered cluster members does not affect our conclusions.
All remaining stars are unlikely to be cluster members, because of their discrepant radial velocities, large projected distances from the cluster center, much bluer colors, and often quite discrepant chemical composition. The rather large fraction of non-members did not come as a surprise, given the difficulties in identifying good candidates in this cluster and the strong contamination by field bulge stars. We notice that only observations by means of a multi-object facility like FLAMES could ensure the success of the program.
There are various circumstantial arguments that favor the membership to the bulge of all remaining stars. Three of them (\#6005341, 7004329, and 7004453) have small negative radial velocities ($-$41.2, $-$23.1, and $-$49.3 km s$^{-1}$, respectively) and nearly solar Fe abundance. However, given the location of at $b=-5.01$, they should be within 2-3 kpc from the Sun to be members of the thin disk (i.e. at no more than 200-300 pc from the galactic plane: see e.g. Bilir et al. 2005). Were this the correct distance of the stars, they would be sub-giants, with a surface gravity of about $\log g\sim 3$[^3]. This is clearly excluded by their spectra, that indicate a much lower surface gravity of $\log g<2$. These stars are then at more than 6 kpc from the Sun, and at more than 500 pc from the galactic plane. The same surface gravity argument can be applied to the remaining stars. We also note that stars \#6003734 and \#8003092 have a chemical composition similar to that of , but have very different radial velocities (+83.5 and $-53.0$ km s$^{-1}$) and are projected far from the cluster center (518 and 573 arcsec respectively, more than the tidal radius). Also star \#6003734 is distinctly bluer than the cluster members considered above, while star \#8003092 has magnitude and color very similar to those of star \#8002961, that we considered a possible member. Star \#6005308 has a chemical composition similar to that of : however its radial velocity is very different ($-$144.1 km s$^{-1}$), clearly incompatible with membership. Finally, star \#6004360 has a large positive velocity of 89.7 km s$^{-1}$ and a large metal content. All these stars are distinctly bluer than the cluster members, although more metal-rich. This might be due to one of these three causes: smaller interstellar reddening (however there is no clear indication of this from our analysis); shorter distances (putting the stars close to the galactic center), or younger ages. As we will see, a shorter distance by about 20% is also suggested by the equilibrium of ionization. We conclude that all these stars most likely belong to the bulge population.
Finally, one of the target stars (\#7003717) turned out to have very broad lines, likely widened by rotation. We measured a large negative radial velocity ($-$134.0 km s$^{-1}$), with clear indications of quite large variations over the small temporal range covered by our observations. Due to difficulties in the analysis, we did not further consider this star.
---------- --------- ------ ---------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
El. Wavel. E.P. log gf 4050 4434 4463 4487 2961
EW log A EW log A EW log A EW log A EW log A
(Å) (eV) (mÅ) (mÅ) (mÅ) (mÅ) (mÅ)
O[i]{} 6300.31 0.00 $-$9.75 56.2 8.23 83.9 8.61 63.4 8.45 68.7 8.39
O[i]{} 6363.79 0.02 $-$10.25 29.7 8.44 50.9 8.76 15.8 8.14 38.5 8.65 33.3 8.44
Na[i]{} 6154.23 2.10 $-$1.57 99.4 6.36 105.4 6.14 140.8 6.62 100.4 6.20 110.5 6.34
Na[i]{} 6160.75 2.10 $-$1.26 111.1 6.25 137.5 6.24 154.4 6.49 123.3 6.19 136.2 6.40
Mg[i]{} 6318.71 5.11 $-$1.94 59.1 7.30 73.9 7.38 61.7 7.20 75.1 7.41 77.8 7.51
Mg[i]{} 6319.24 5.11 $-$2.16 48.2 7.31 64.6 7.46 48.5 7.22 53.0 7.29 70.8 7.61
Al[i]{} 6696.03 3.14 $-$1.32 99.9 6.30 128.3 6.40 164.2 6.95 106.6 6.22 127.4 6.54
Al[i]{} 6698.67 3.14 $-$1.62 77.7 6.18 92.4 6.18 119.9 6.62 71.1 6.00 89.4 6.21
Si[i]{} 5948.55 5.08 $-$1.23 67.7 7.41 85.4 7.47 103.0 7.75 84.4 7.44 84.4 7.57
Si[i]{} 6125.03 5.62 $-$1.57 27.9 7.35
Si[i]{} 6145.02 5.62 $-$1.49 27.8 7.33 34.4 7.48 36.2 7.47 45.4 7.78
Ca[i]{} 5857.46 2.93 0.24 164.0 5.75 178.9 5.55 207.6 5.88 169.2 5.63 175.7 5.70
Ca[i]{} 5867.57 2.93 $-$1.49 56.5 5.59 77.5 5.74 92.8 6.01 69.9 5.76 77.4 5.82
Ca[i]{} 6161.30 2.52 $-$1.27 109.0 5.88 141.1 5.89 135.4 6.03 144.7 6.16
Ca[i]{} 6166.44 2.52 $-$1.14 103.8 5.64 136.5 5.92 136.0 5.89
Ca[i]{} 6169.04 2.52 $-$0.80 145.2 6.02 158.1 5.69 169.9 5.90 160.5 5.95 156.9 5.89
Ca[i]{} 6169.56 2.52 $-$0.48 136.7 5.58 176.2 5.62 187.9 5.80 146.2 5.42 169.4 5.74
Ca[i]{} 6439.08 2.52 0.39 192.1 5.48 252.0 5.63 241.6 5.58 202.6 5.36 239.7 5.68
Ca[i]{} 6455.60 2.52 $-$1.29 93.5 5.56 145.4 5.98 137.2 5.89 114.3 5.68 125.6 5.85
Ca[i]{} 6462.57 2.52 0.26 246.0 5.99 309.9 6.12 296.6 6.08 261.3 6.00 280.9 6.07
Ca[i]{} 6471.67 2.52 $-$0.69 140.9 5.93 194.2 6.11 162.7 5.72 158.4 5.85 173.9 6.08
Ca[i]{} 6493.79 2.52 $-$0.11 166.6 5.72 209.8 5.73 209.7 5.77 180.5 5.59 198.8 5.82
Ca[i]{} 6499.65 2.52 $-$0.82 113.4 5.56 176.3 5.99 163.7 5.86 138.5 5.64 167.6 6.13
Sc[ii]{} 6245.62 1.51 $-$1.05 71.6 2.72 92.7 2.72 104.7 2.95 93.4 2.83 88.9 2.75
Sc[ii]{} 6279.74 1.50 $-$1.16 85.8 3.18 97.6 2.93 115.5 3.26 81.3 2.75 78.1 2.69
Sc[ii]{} 6604.60 1.36 $-$1.15 91.6 3.03 115.3 2.94 108.5 2.86 92.1 2.68 103.0 2.88
Ti[i]{} 5866.46 1.07 $-$0.84 150.8 4.78 210.1 4.94 221.4 5.14 163.7 4.54 196.5 5.01
Ti[i]{} 5922.12 1.05 $-$1.47 128.8 4.88 170.0 4.81 185.9 5.16 142.0 4.66 165.9 5.02
Ti[i]{} 5978.55 1.87 $-$0.50 111.7 4.77 137.1 4.54 148.6 4.81 122.7 4.58 133.7 4.74
Ti[i]{} 6091.18 2.27 $-$0.42 103.3 4.49 114.8 4.74 103.1 4.72 109.9 4.78
Ti[i]{} 6126.22 1.07 $-$1.42 128.6 4.81 181.9 4.92 169.9 4.78 149.5 4.71 149.2 4.62
Ti[i]{} 6258.11 1.44 $-$0.36 176.6 4.99 220.7 4.95 223.2 5.04 188.3 4.86 187.4 4.80
Ti[i]{} 6554.24 1.44 $-$1.22 151.1 4.91
Ti[ii]{} 6606.98 2.06 $-$2.90 41.6 5.14 45.4 5.07 32.8 4.84 29.8 4.77 51.4 5.19
V[i]{} 6002.31 1.22 $-$1.77 59.2 3.76 63.8 3.64 79.3 3.87 70.9 3.92 58.2 3.56
V[i]{} 6039.73 1.06 $-$0.65 144.5 3.43 159.8 3.76 138.3 3.60
V[i]{} 6081.45 1.05 $-$0.58 168.9 3.92 174.6 4.10 146.8 3.85 158.3 3.99
V[i]{} 6090.22 1.08 $-$0.06 133.0 3.62 193.8 3.83 183.2 3.76 158.5 3.59 172.8 3.79
V[i]{} 6111.65 1.04 $-$0.71 180.7 3.97 178.9 4.02 163.0 4.04
V[i]{} 6119.53 1.06 $-$0.32 130.3 3.77 164.9 3.57 174.2 3.80 146.6 3.58 166.2 3.87
V[i]{} 6135.38 1.05 $-$0.75 118.6 3.80 149.4 3.54 162.2 3.81 140.0 3.71 147.3 3.76
V[i]{} 6199.19 0.29 $-$1.28 175.0 4.04 217.7 4.01 224.9 4.19 187.9 3.90
V[i]{} 6216.36 0.28 $-$0.81 215.6 3.46 224.9 3.68 197.6 3.55 202.6 3.54
V[i]{} 6251.82 0.29 $-$1.34 188.5 3.63 210.5 4.09 180.9 3.89
V[i]{} 6256.90 0.28 $-$2.01 111.1 3.65 167.3 3.88 176.6 4.10 143.4 3.74
V[i]{} 6274.65 0.27 $-$1.67 121.4 3.58 168.6 3.54 174.5 3.72 158.1 3.65
V[i]{} 6285.17 0.28 $-$1.51 135.2 3.76 188.6 3.71 194.5 3.89 167.6 3.76 163.8 3.62
V[i]{} 6292.83 0.29 $-$1.47 152.5 4.03 194.6 3.79 205.5 4.03 174.3 3.86 180.9 3.93
V[i]{} 6531.43 1.22 $-$0.84 152.5 3.83 128.4 3.77
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\[t:ewidths\]
--------- --------- ------ --------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
El. Wavel. E.P. log gf 4050 4434 4463 4487 2961
EW log A EW log A EW log A EW log A EW log A
(Å) (eV) (mÅ) (mÅ) (mÅ) (mÅ) (mÅ)
Cr[i]{} 6330.10 0.94 $-$2.87 139.6 5.52 189.0 5.51 173.1 5.34 144.6 5.11 157.5 5.28
Mn[i]{} 6013.50 3.07 $-$0.25 139.8 5.18 169.3 5.05 187.1 5.30 158.6 5.07 172.2 5.29
Mn[i]{} 6016.65 3.07 $-$0.09 125.3 4.88 156.5 4.78 162.3 4.92 145.9 4.79 162.2 5.09
Mn[i]{} 6021.80 3.08 0.03 122.2 4.83 165.6 4.91 174.9 5.07 141.9 4.73 163.9 5.11
Fe[i]{} 5855.09 4.61 $-$1.48 37.3 7.09 58.7 7.34 44.0 7.08 45.2 7.14
Fe[i]{} 5856.10 4.29 $-$1.57 46.3 6.98 72.7 7.23 87.5 7.52 66.3 7.17 77.0 7.46
Fe[i]{} 5858.78 4.22 $-$2.19 23.3 6.89 37.9 7.13 38.0 7.15 36.2 7.12
Fe[i]{} 5859.60 4.55 $-$0.70 68.6 7.02 88.0 6.98 111.3 7.40 90.8 7.10 95.1 7.30
Fe[i]{} 5862.37 4.55 $-$0.50 79.5 7.07 99.8 6.99 122.2 7.39 97.3 7.03 104.2 7.28
Fe[i]{} 5905.68 4.65 $-$0.76 55.1 6.89 83.3 7.10 89.4 7.23 72.2 6.95 75.4 7.10
Fe[i]{} 5927.80 4.65 $-$1.06 40.4 6.80 72.7 7.21 72.5 7.22 62.4 7.06 64.6 7.17
Fe[i]{} 5929.68 4.55 $-$1.24 49.8 7.10 65.0 7.12 71.8 7.25 59.7 7.06 55.8 7.03
Fe[i]{} 5930.19 4.65 $-$0.29 74.9 6.92 100.5 6.94 116.3 7.26 86.4 6.76 110.0 7.34
Fe[i]{} 5934.67 3.93 $-$1.15 92.0 7.17 107.3 6.90 136.6 7.44 110.0 7.08 117.4 7.32
Fe[i]{} 5956.71 0.86 $-$4.60 130.0 7.09 187.6 7.24 186.5 7.29 160.3 7.06 176.5 7.36
Fe[i]{} 5976.79 3.94 $-$1.33 90.3 6.79 101.5 7.02 97.4 7.03 101.0 7.20
Fe[i]{} 5984.83 4.73 $-$0.39 125.9 7.52
Fe[i]{} 6003.02 3.88 $-$1.08 90.5 7.00 120.7 7.00 124.3 7.10 110.4 6.95 121.3 7.26
Fe[i]{} 6007.97 4.65 $-$0.82 56.3 6.96 74.4 6.99 82.9 7.16 81.8 7.18 77.4 7.18
Fe[i]{} 6008.57 3.88 $-$0.96 96.7 7.03 119.0 6.86 135.8 7.18 108.9 6.80 129.6 7.29
Fe[i]{} 6027.06 4.08 $-$1.23 72.9 6.98 108.0 7.21 111.0 7.29 100.3 7.18 93.9 7.14
Fe[i]{} 6056.01 4.73 $-$0.42 65.1 6.88 91.9 7.03 86.6 6.95 90.7 7.08 104.1 7.47
Fe[i]{} 6079.02 4.65 $-$0.95 68.8 7.15
Fe[i]{} 6082.72 2.22 $-$3.57 86.3 7.02 146.0 7.57 141.8 7.53 131.3 7.57
Fe[i]{} 6089.57 4.58 $-$1.28 85.3 7.56
Fe[i]{} 6093.65 4.61 $-$1.32 42.6 7.07 60.3 7.20 58.1 7.19 59.1 7.26
Fe[i]{} 6094.38 4.65 $-$1.56 34.3 7.15 33.9 7.00 44.1 7.22
Fe[i]{} 6096.67 3.98 $-$1.77 58.8 7.06 79.1 7.09 72.3 6.99 75.3 7.09 75.2 7.14
Fe[i]{} 6098.25 4.56 $-$1.81 38.5 7.38
Fe[i]{} 6137.00 2.20 $-$2.95 125.6 7.28 183.9 7.47 171.7 7.34 158.9 7.32 163.8 7.46
Fe[i]{} 6151.62 2.18 $-$3.30 99.1 6.99 156.6 7.38 144.2 7.20 127.8 7.06 138.1 7.32
Fe[i]{} 6173.34 2.22 $-$2.88 112.4 6.98 159.7 7.07 156.2 7.06 144.2 7.00 153.9 7.26
Fe[i]{} 6188.00 3.94 $-$1.63 85.4 7.00 84.6 7.00 77.9 6.94 78.8 7.01
Fe[i]{} 6200.32 2.61 $-$2.44 106.0 6.94 146.4 7.02 132.1 6.90 136.7 7.07
Fe[i]{} 6213.44 2.22 $-$2.54 149.4 7.26 169.1 6.88 173.9 7.00 156.5 6.89 165.4 7.09
Fe[i]{} 6219.29 2.20 $-$2.43 134.5 6.92 188.3 7.00 188.9 7.07 173.4 7.02 177.4 7.13
Fe[i]{} 6226.74 3.88 $-$2.08 46.1 6.89 80.0 7.27 57.6 6.90 70.2 7.16 61.3 7.02
Fe[i]{} 6232.65 3.65 $-$1.22 97.7 6.96 133.3 7.00 136.3 7.09 130.3 7.09 121.0 7.03
Fe[i]{} 6240.65 2.22 $-$3.23 95.8 6.89 131.9 6.93 144.8 7.20 121.3 6.93 130.8 7.17
Fe[i]{} 6265.14 2.18 $-$2.55 147.4 7.19 208.5 7.34 209.2 7.39 181.6 7.22 187.4 7.34
Fe[i]{} 6270.23 2.86 $-$2.46 121.3 6.92 121.5 6.96 114.7 7.03
Fe[i]{} 6297.80 2.22 $-$2.74 146.7 7.42 191.9 7.39 185.0 7.36 159.4 7.14 180.7 7.50
Fe[i]{} 6301.51 3.65 $-$0.72 116.7 6.80 161.0 6.89 158.2 6.88 164.1 7.15
Fe[i]{} 6311.50 2.83 $-$3.16 76.0 7.19 115.6 7.44 107.7 7.33 101.3 7.38
Fe[i]{} 6315.81 4.08 $-$1.68 46.0 6.76 78.4 7.12 72.2 7.02 61.0 6.86 72.2 7.12
Fe[i]{} 6322.69 2.59 $-$2.43 119.2 7.16 154.8 7.04 144.8 6.90 137.3 6.92 145.7 7.16
Fe[i]{} 6330.85 4.73 $-$1.22 52.0 7.37 59.0 7.24 49.8 7.08 70.7 7.56
Fe[i]{} 6335.34 2.20 $-$2.27 220.2 7.17 193.5 7.10
Fe[i]{} 6380.75 4.19 $-$1.37 96.3 7.27 92.7 7.23 80.9 7.07 97.7 7.49
Fe[i]{} 6392.54 2.28 $-$3.97 78.3 7.25 107.6 7.30 83.8 6.92 90.3 7.14
Fe[i]{} 6400.32 3.60 $-$0.23 263.3 7.28 225.1 7.17
Fe[i]{} 6411.66 3.65 $-$0.60 129.7 6.87 172.7 6.92 164.8 7.04
Fe[i]{} 6421.36 2.28 $-$2.03 235.4 7.16
Fe[i]{} 6481.88 2.28 $-$2.98 119.3 7.26 168.3 7.34 152.2 7.13 140.4 7.08 158.7 7.47
--------- --------- ------ --------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
---------- --------- ------ --------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
El. Wavel. E.P. log gf 4050 4434 4463 4487 2961
EW log A EW log A EW log A EW log A EW log A
(Å) (eV) (mÅ) (mÅ) (mÅ) (mÅ) (mÅ)
Fe[i]{} 6498.94 0.96 $-$4.70 216.4 7.73 185.8 7.37 162.0 7.22 184.3 7.59
Fe[i]{} 6518.37 2.83 $-$2.46 114.0 7.40 151.5 7.34 125.1 6.92 120.9 6.97
Fe[i]{} 6533.94 4.56 $-$1.29 89.4 7.58 58.1 7.05 49.0 6.90 68.6 7.33
Fe[i]{} 6574.25 0.99 $-$5.00 125.7 7.46 183.6 7.60 150.3 7.12 133.9 7.01 150.7 7.34
Fe[i]{} 6581.22 1.49 $-$4.68 157.5 7.63 145.5 7.48 119.8 7.19 134.4 7.48
Fe[i]{} 6593.88 2.43 $-$2.42 137.6 7.17 186.3 7.21 166.6 6.98 153.2 6.92 175.1 7.30
Fe[i]{} 6608.04 2.28 $-$3.96 73.0 7.07 115.8 7.39 99.8 7.16 87.4 7.05 105.7 7.41
Fe[i]{} 6609.12 2.56 $-$2.69 165.3 7.38 153.5 7.24 142.3 7.19 150.5 7.42
Fe[i]{} 6627.56 4.55 $-$1.50 45.4 7.02 38.1 6.89 61.9 7.39
Fe[i]{} 6633.76 4.56 $-$0.82 74.6 7.28 101.6 7.33 80.1 6.97 82.3 7.05 96.5 7.43
Fe[i]{} 6703.58 2.76 $-$3.01 92.7 7.30 128.1 7.34 108.6 7.04 91.0 6.84 109.6 7.23
Fe[i]{} 6713.75 4.80 $-$1.41 41.3 7.37
Fe[i]{} 6725.36 4.10 $-$2.21 45.4 7.29 35.5 6.92 49.8 7.22
Fe[i]{} 6726.67 4.61 $-$1.10 43.9 6.86 70.7 7.12 60.4 6.96 54.3 6.87 71.0 7.24
Fe[i]{} 6733.15 4.64 $-$1.44 42.7 7.21 42.4 7.02 35.1 6.88 50.5 7.23
Fe[i]{} 6750.16 2.42 $-$2.62 124.7 7.09 181.9 7.28 161.3 7.03 146.3 6.94 157.9 7.20
Fe[i]{} 6786.86 4.19 $-$1.90 52.7 7.28 68.4 7.30 49.8 6.99 61.1 7.26
Fe[ii]{} 6247.56 3.89 $-$2.33 24.0 6.91 19.2 6.50 31.7 6.86 28.7 6.92
Fe[ii]{} 6432.68 2.89 $-$3.58 22.1 6.80 48.2 7.35 25.3 6.74 26.3 6.70 42.6 7.29
Fe[ii]{} 6456.39 3.90 $-$2.10 33.3 7.07 61.5 7.52 38.2 6.93 34.8 6.75 42.5 7.16
Fe[ii]{} 6516.08 2.89 $-$3.38 39.2 7.21 67.2 7.56 42.7 6.95 47.8 7.23
Ni[i]{} 5847.01 1.68 $-$3.44 79.3 5.99 108.3 5.99 131.8 6.49 104.5 6.09 110.4 6.28
Ni[i]{} 5996.74 4.24 $-$1.06 22.7 5.70 34.9 5.90 31.1 5.82 31.4 5.82 41.8 6.09
Ni[i]{} 6053.69 4.24 $-$1.07 31.5 5.97 53.6 6.28 40.5 6.03 46.0 6.14
Ni[i]{} 6086.29 4.27 $-$0.47 46.2 5.82 61.6 5.87 47.5 5.61 47.6 5.61 68.0 6.10
Ni[i]{} 6108.12 1.68 $-$2.49 117.7 5.92 174.9 6.12 157.1 5.90 147.9 5.90 156.2 6.11
Ni[i]{} 6111.08 4.09 $-$0.83 55.4 6.17 72.4 6.17 53.6 5.84 61.5 6.00 63.5 6.10
Ni[i]{} 6128.98 1.68 $-$3.39 76.7 5.80 116.0 6.02 110.6 5.96 101.8 5.93 111.2 6.15
Ni[i]{} 6130.14 4.27 $-$0.98 30.4 5.90 49.1 6.15 38.0 5.93 39.4 5.96 50.1 6.23
Ni[i]{} 6176.82 4.09 $-$0.26 77.0 6.14 81.3 5.76 87.3 5.88 88.3 5.95 83.8 5.96
Ni[i]{} 6177.25 1.83 $-$3.60 63.4 5.90 88.4 5.98 89.7 6.02 90.6 6.14 82.7 5.99
Ni[i]{} 6186.72 4.11 $-$0.90 50.4 6.14 74.2 6.30 46.7 5.80 65.6 6.17 60.0 6.13
Ni[i]{} 6204.61 4.09 $-$1.15 45.2 6.22 66.3 6.38 54.8 6.18 50.8 6.11 61.5 6.38
Ni[i]{} 6223.99 4.11 $-$0.97 46.0 6.09 49.6 5.93 39.8 5.74 53.1 6.00 37.9 5.73
Ni[i]{} 6230.10 4.11 $-$1.20 43.5 6.26 57.9 6.31 53.3 6.23 56.0 6.29 51.0 6.24
Ni[i]{} 6322.17 4.15 $-$1.21 45.4 6.14 20.6 5.59 30.5 5.83 43.3 6.14
Ni[i]{} 6327.60 1.68 $-$3.09 128.0 6.09 138.8 6.35
Ni[i]{} 6378.26 4.15 $-$0.82 49.4 6.08 73.9 6.27 51.4 5.86 50.3 5.85 70.3 6.32
Ni[i]{} 6384.67 4.15 $-$1.00 46.1 6.17 69.4 6.36 44.8 5.92 36.2 5.75 61.1 6.30
Ni[i]{} 6482.81 1.93 $-$2.85 93.0 5.86 148.4 6.24 136.9 6.11 122.4 6.00 126.1 6.10
Ni[i]{} 6532.88 1.93 $-$3.42 95.0 6.14 94.2 6.14
Ni[i]{} 6586.32 1.95 $-$2.79 99.6 6.12 142.1 6.21 125.1 5.96 108.3 5.78 126.0 6.17
Ni[i]{} 6598.61 4.24 $-$0.93 59.0 6.23 36.3 5.80 42.8 5.93 55.1 6.23
Ni[i]{} 6635.14 4.42 $-$0.75 46.3 6.30 57.9 6.28 34.4 5.82 27.1 5.64 52.2 6.24
Ni[i]{} 6767.78 1.83 $-$2.11 131.8 5.87 189.5 6.01 165.8 5.74 148.9 5.61 173.1 6.05
Ni[i]{} 6772.32 3.66 $-$1.01 71.3 6.09 94.8 6.09 74.6 5.76 64.6 5.62 92.7 6.22
Y[i]{} 6435.02 0.07 $-$0.82 54.0 1.05 98.9 1.50 112.1 1.77 62.6 1.23 90.2 1.47
Zr[i]{} 6127.46 0.15 $-$1.06 86.5 2.15 121.9 2.08 116.3 2.05 96.8 1.97 84.9 1.58
Zr[i]{} 6134.57 0.00 $-$1.28 67.2 1.59 99.4 1.70 104.3 1.84 91.8 1.87 87.6 1.62
Zr[i]{} 6140.46 0.52 $-$1.41 36.8 1.57 45.1 1.85 41.5 1.83 26.8 1.73 29.7 1.54
Zr[i]{} 6143.18 0.07 $-$1.10 81.6 1.94 121.0 1.98 106.6 1.80 93.5 1.83 81.8 1.44
Zr[i]{} 6445.72 1.00 $-$0.83 21.9 1.56 18.3 1.49 12.1 1.43 15.4 1.31
Ba[ii]{} 5853.69 0.60 $-$1.00 111.5 2.09 148.4 1.84 162.9 2.18 125.2 1.69 140.5 2.09
Ba[ii]{} 6141.75 0.70 0.00 159.2 1.89 220.6 1.93 205.8 1.82 180.3 1.68 182.9 1.74
Ba[ii]{} 6496.91 0.60 $-$0.38 182.2 2.32 221.6 2.14 205.3 1.99 177.4 1.83 196.3 2.10
Eu[ii]{} 6437.64 1.32 $-$0.28 34.0 0.65 22.5 0.40 25.1 0.50 24.6 0.42
Eu[ii]{} 6645.11 1.38 0.20 39.3 0.55 54.6 0.61 48.6 0.52 31.0 0.23 41.0 0.38
---------- --------- ------ --------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
---------- --------- ------ -------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
El. Wavel. E.P. log gf 3734 4360 5308 5341 4329 4453 3092
EW log A EW log A EW log A EW log A EW log A EW log A EW log A
(Å) (eV) (mÅ) (mÅ) (mÅ) (mÅ) (mÅ) (mÅ) (mÅ)
O[i]{} 6300.31 0.00 -9.75 86.2 8.83 63.7 8.82 83.6 8.98 77.0 8.89 69.2 8.78 95.7 8.89
O[i]{} 6363.79 0.02 -10.25 44.0 8.82 39.2 9.03 29.7 8.77 39.3 8.92 50.0 8.78
Na[i]{} 6154.23 2.10 -1.57 91.1 6.19 166.3 7.22 100.9 6.42 125.3 6.70 108.9 6.62 113.2 6.50 91.9 6.26
Na[i]{} 6160.75 2.10 -1.26 98.9 6.00 169.6 6.96 95.5 6.03 136.0 6.55 129.6 6.63 120.4 6.30 129.6 6.60
Mg[i]{} 6318.71 5.11 -1.94 81.4 7.64 132.4 8.40 80.8 7.63 94.4 7.88 89.7 7.82 74.2 7.48 72.1 7.63
Mg[i]{} 6319.24 5.11 -2.16 57.6 7.44 102.6 8.16 39.2 7.14 70.4 7.71 80.9 7.88 61.1 7.49 64.2 7.69
Al[i]{} 6696.03 3.14 -1.32 88.7 6.05 144.7 6.88 118.1 6.66 115.0 6.46 103.6 6.46 118.1 6.52 98.2 6.28
Al[i]{} 6698.67 3.14 -1.62 59.2 5.87 109.9 6.64 99.3 6.64 77.1 6.14 64.1 6.09 79.3 6.21 69.8 6.02
Si[i]{} 5948.55 5.08 -1.23 81.3 7.55 121.7 8.18 62.6 7.13 85.1 7.68 104.5 7.85 93.4 7.67 58.2 7.33
Si[i]{} 6125.03 5.62 -1.57 84.3 8.55 48.0 7.76
Si[i]{} 6145.02 5.62 -1.49 45.2 7.77 87.6 8.51 37.5 7.53 25.7 7.38 50.7 7.74 36.0 7.51 28.7 7.55
Ca[i]{} 5857.46 2.93 0.24 168.3 5.77 213.9 6.25 165.6 5.86 195.1 6.02 201.9 6.32 204.5 6.14 128.0 5.34
Ca[i]{} 5867.57 2.93 -1.49 58.8 5.65 97.1 6.30 70.3 6.01 105.2 6.49 83.3 6.31 79.1 6.04 80.0 6.12
Ca[i]{} 6161.30 2.52 -1.27 129.5 6.16 160.8 6.56 120.8 6.16 147.7 6.40 142.2 6.32 110.3 5.93
Ca[i]{} 6166.44 2.52 -1.14 121.0 5.88 163.0 6.47 124.7 6.10
Ca[i]{} 6169.04 2.52 -0.80 159.5 6.14 192.2 6.52 166.6 6.20 162.4 6.44 158.7 6.12 121.5 5.68
Ca[i]{} 6169.56 2.52 -0.48 159.4 5.82 196.9 6.25 195.4 6.20 179.2 6.32 167.5 5.92 149.2 5.74
Ca[i]{} 6439.08 2.52 0.39 211.3 5.64 243.6 5.93 211.9 5.78 223.3 5.72 222.4 6.00 229.5 5.82 185.4 5.41
Ca[i]{} 6449.82 2.52 -0.50 171.2 6.10 193.2 6.29 175.1 6.30 196.3 6.35 177.3 6.44 190.8 6.30
Ca[i]{} 6455.60 2.52 -1.29 106.4 5.75 141.0 6.24 122.2 6.22 123.4 5.99 118.6 6.21 128.8 6.10 101.4 5.77
Ca[i]{} 6462.57 2.52 0.26 271.5 6.16 315.6 6.50 272.3 6.31 274.2 6.20 262.9 6.39 281.3 6.30 226.1 5.84
Ca[i]{} 6471.67 2.52 -0.69 148.0 5.94 185.6 6.38 156.0 6.22 155.6 6.30 157.2 6.02 125.8 5.71
Ca[i]{} 6493.79 2.52 -0.11 185.7 5.90 217.3 6.19 188.0 6.06 186.6 5.86 177.3 6.05 191.6 5.93 154.3 5.57
Ca[i]{} 6499.65 2.52 -0.82 145.2 6.02 177.6 6.39 141.7 6.11 151.9 6.06 147.0 6.28
Sc[ii]{} 6245.62 1.51 -1.05 99.5 3.24 91.3 3.15 105.6 3.44 89.9 3.21 87.9 2.96 56.1 2.45
Sc[ii]{} 6279.74 1.50 -1.16 72.6 2.80 82.0 3.06 97.2 3.41 109.9 3.65 71.5 2.98 85.1 3.04 59.5 2.66
Sc[ii]{} 6604.60 1.36 -1.15 94.5 2.98 133.5 3.70 76.6 2.74 71.8 2.73 116.6 3.36 84.0 2.96
Ti[i]{} 5866.46 1.07 -0.84 162.8 4.86 200.9 5.40 164.7 5.14 183.9 5.16 165.9 5.31 186.7 5.25
Ti[i]{} 5922.12 1.05 -1.47 143.5 5.03 141.3 5.25 162.8 5.30 137.1 5.30 148.1 5.06 110.1 4.42
Ti[i]{} 5978.55 1.87 -0.50 114.8 4.72 144.0 5.14 127.1 5.20 147.9 5.29 124.3 5.24 130.4 4.97 95.3 4.36
Ti[i]{} 6091.18 2.27 -0.42 111.4 5.16 132.3 5.42 94.4 4.96 119.1 5.24 90.2 4.95 104.9 4.96 96.4 4.93
Ti[i]{} 6126.22 1.07 -1.42 147.0 5.02 159.8 5.11 145.5 5.24 160.0 5.17 122.2 4.90 157.8 5.15 129.0 4.84
Ti[i]{} 6258.11 1.44 -0.36 159.9 4.71 190.3 5.12 159.8 4.94 198.4 5.24 172.6 4.89 128.5 4.32
Ti[i]{} 6554.24 1.44 -1.22 114.6 4.62 155.8 5.26 112.5 4.82 127.5 5.21 138.2 5.02
Ti[ii]{} 6606.98 2.06 -2.90 24.5 4.75 66.0 5.72 28.6 4.92 36.5 5.19 38.1 5.21 63.7 5.60 30.4 4.95
V[i]{} 6002.31 1.22 -1.77 38.7 3.51 89.0 4.39 52.1 3.98 72.3 4.08 45.6 3.96 56.6 3.94 71.4 4.03
V[i]{} 6039.73 1.06 -0.65 110.9 3.46 159.9 4.26 109.1 3.67 140.2 3.99 117.0 3.96 132.6 3.85 117.3 3.79
V[i]{} 6081.45 1.05 -0.58 126.1 3.80 163.2 4.38 159.7 4.36 130.6 4.26 157.4 4.32 128.9 4.03
V[i]{} 6090.22 1.08 -0.06 148.7 3.76 183.2 4.27 138.8 3.83 177.6 4.19 138.8 3.97 167.2 4.04 147.9 3.87
V[i]{} 6111.65 1.04 -0.71 135.9 4.20 174.0 4.54 136.8 4.35 147.7 4.08
V[i]{} 6119.53 1.06 -0.32 134.1 3.71 185.3 4.50 128.1 3.83 162.8 4.16 125.4 3.90 147.0 3.88 131.3 3.82
V[i]{} 6135.38 1.05 -0.75 119.8 3.70 166.4 4.39 118.3 3.91 145.9 4.09 116.9 3.98 135.2 3.92 108.0 3.56
V[i]{} 6199.19 0.29 -1.28 170.3 3.94 202.1 4.42 172.6 4.26 211.8 4.56 171.7 4.39 202.6 4.48 159.5 3.85
V[i]{} 6216.36 0.28 -0.81 179.7 3.58 211.2 4.04 172.3 3.74 215.9 4.09 171.6 3.88 198.1 3.88
V[i]{} 6242.81 0.26 -1.55 174.1 4.15 138.7 3.94
V[i]{} 6251.82 0.29 -1.34 158.7 4.14 190.8 4.38 147.0 4.07 188.2 4.37 129.1 3.40
V[i]{} 6256.90 0.28 -2.01 125.6 3.90 166.3 4.50 112.5 3.91 145.6 4.18 114.8 4.06 137.9 4.08 120.0 3.93
V[i]{} 6274.65 0.27 -1.67 132.2 3.97 163.0 4.14 125.1 3.94 153.5 3.99 118.5 3.54
V[i]{} 6285.17 0.28 -1.51 134.4 3.86 175.8 4.25 146.9 4.22 166.0 4.08 136.8 3.83
V[i]{} 6292.83 0.29 -1.47 155.4 4.21 207.2 4.66 137.8 4.03
V[i]{} 6531.43 1.22 -0.84 97.3 3.48 169.1 4.62 107.0 3.91 127.9 4.02 122.9 3.94 105.5 3.77
---------- --------- ------ -------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
\[t:ewidths2\]
--------- --------- ------ -------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
El. Wavel. E.P. log gf 3734 4360 5308 5341 4329 4453 3092
EW log A EW log A EW log A EW log A EW log A EW log A EW log A
(Å) (eV) (mÅ) (mÅ) (mÅ) (mÅ) (mÅ) (mÅ) (mÅ)
Cr[i]{} 6330.10 0.94 -2.87 121.9 5.04 180.7 5.98 131.6 5.46 155.2 5.60 149.4 5.48 150.2 5.72
Mn[i]{} 6013.50 3.07 -0.25 144.8 5.14 203.3 5.84 139.8 5.16 182.6 5.61 183.0 5.81 178.5 5.53 121.1 4.98
Mn[i]{} 6016.65 3.07 -0.09 155.2 5.18 192.8 5.66 141.4 5.10 172.4 5.43 168.0 5.58 160.1 5.23 113.6 4.72
Mn[i]{} 6021.80 3.08 0.03 138.3 4.96 215.1 5.85 136.4 5.02 174.3 5.44 166.6 5.55 171.0 5.36 121.9 4.89
Fe[i]{} 5835.11 4.26 -2.18 35.5 7.23 76.3 8.30 63.2 7.80
Fe[i]{} 5855.09 4.61 -1.48 43.7 7.19 49.8 7.35 49.0 7.36 60.8 7.64 54.9 7.40 37.4 7.17
Fe[i]{} 5856.10 4.29 -1.57 69.1 7.42 95.5 7.90 60.3 7.25 88.6 7.89 81.1 7.80 81.7 7.60
Fe[i]{} 5858.78 4.22 -2.19 29.7 7.05 33.5 7.50 42.5 7.42 46.3 7.52 42.7 7.36
Fe[i]{} 5859.60 4.55 -0.70 84.0 7.20 81.6 7.16 114.2 7.80 115.7 7.94 111.4 7.63 73.6 7.25
Fe[i]{} 5862.37 4.55 -0.50 99.1 7.31 116.3 7.54 81.3 6.95 110.4 7.54 124.4 7.88 104.0 7.30 84.3 7.28
Fe[i]{} 5881.28 4.61 -1.76 39.3 7.36 47.8 7.58 68.4 8.04 59.6 7.76
Fe[i]{} 5902.48 4.59 -1.86 49.9 7.71 41.1 7.55 53.7 7.84 43.3 7.52
Fe[i]{} 5905.68 4.65 -0.76 81.7 7.38 124.1 8.10 81.3 7.37
Fe[i]{} 5927.80 4.65 -1.06 63.4 7.26 87.3 7.69 58.0 7.15 81.0 7.67 78.4 7.66 65.1 7.22 49.1 7.14
Fe[i]{} 5929.68 4.55 -1.24 73.5 7.53 100.5 8.00 51.0 7.05 86.4 7.83 79.6 7.74 73.8 7.44 49.0 7.17
Fe[i]{} 5930.19 4.65 -0.29 150.8 8.06 91.0 7.11 125.9 7.77 125.9 7.88 118.9 7.52 92.2 7.45
Fe[i]{} 5934.67 3.93 -1.15 98.1 7.12 131.1 7.65 123.5 7.65 145.4 7.96 134.5 7.94 129.3 7.59 101.9 7.50
Fe[i]{} 5956.71 0.86 -4.60 159.0 7.41 147.5 7.41 163.0 7.52 143.9 7.51 129.3 7.58
Fe[i]{} 5976.79 3.94 -1.33 88.1 7.10 84.9 7.07 119.7 7.76 114.7 7.82 111.6 7.48 68.5 6.92
Fe[i]{} 5984.83 4.73 -0.39 103.6 7.48 89.5 7.22 135.1 7.98 99.2 7.65
Fe[i]{} 6003.02 3.88 -1.08 110.3 7.24 142.9 7.72 107.3 7.22 135.2 7.68 134.9 7.82 127.7 7.44 107.9 7.47
Fe[i]{} 6007.97 4.65 -0.82 67.8 7.09 101.9 7.70 73.9 7.24 81.0 7.40 97.6 7.82 84.4 7.35
Fe[i]{} 6008.57 3.88 -0.96 109.9 7.12 146.9 7.68 102.8 7.02 129.6 7.48 134.2 7.70 130.5 7.38 88.6 6.97
Fe[i]{} 6027.06 4.08 -1.23 104.6 7.55 118.0 7.69 97.7 7.44 114.9 7.76 101.9 7.62 102.7 7.39 88.2 7.50
Fe[i]{} 6056.01 4.73 -0.42 86.8 7.25 111.2 7.62 88.0 7.28 102.6 7.58 104.2 7.71 92.7 7.25
Fe[i]{} 6078.50 4.80 -0.48 127.8 7.99 100.5 7.60 110.9 7.80 108.6 7.85 99.9 7.49 71.0 7.28
Fe[i]{} 6079.02 4.65 -0.95 76.9 7.45 104.6 7.92 66.5 7.23 89.0 7.73 79.1 7.40 68.5 7.55
Fe[i]{} 6082.72 2.22 -3.57 112.7 7.50 119.4 7.77 135.0 7.96 116.1 7.82 123.3 7.60
Fe[i]{} 6089.57 4.58 -1.28 94.0 8.03 85.6 7.93 93.8 7.90
Fe[i]{} 6093.65 4.61 -1.32 64.5 7.49 60.6 7.42 59.0 7.40 61.4 7.49 61.5 7.36 58.2 7.59
Fe[i]{} 6094.38 4.65 -1.56 43.4 7.31 40.2 7.27 41.1 7.32 48.0 7.48 45.2 7.34
Fe[i]{} 6096.67 3.98 -1.77 73.2 7.26 109.2 7.93 78.1 7.41 86.0 7.56 85.1 7.64 79.0 7.30 54.1 7.02
Fe[i]{} 6098.25 4.56 -1.81 47.1 7.52 46.4 7.54 58.5 7.82 55.3 7.78 55.9 7.68
Fe[i]{} 6137.00 2.20 -2.95 135.3 7.26 157.4 7.56 146.0 7.56 164.4 7.77 147.8 7.73 170.5 7.77
Fe[i]{} 6151.62 2.18 -3.30 135.2 7.59 148.8 7.72 118.7 7.38 123.3 7.34 121.7 7.56 137.1 7.50 103.0 7.21
Fe[i]{} 6165.36 4.14 -1.50 70.1 7.19 96.8 7.60 63.1 7.02 88.7 7.56 84.5 7.55 77.0 7.20 50.9 6.89
Fe[i]{} 6173.34 2.22 -2.88 147.7 7.44 184.9 7.94 165.7 7.74 144.0 7.62 154.8 7.47 126.0 7.35
Fe[i]{} 6188.00 3.94 -1.63 81.0 7.22 76.9 7.18 97.6 7.59 97.8 7.73 81.2 7.15 59.2 6.96
Fe[i]{} 6200.32 2.61 -2.44 151.2 7.59 160.4 7.66 121.6 7.18 159.4 7.74 138.9 7.62 149.2 7.47
Fe[i]{} 6226.74 3.88 -2.08 73.2 7.42 79.1 7.50 59.1 7.17 66.0 7.30 72.0 7.51 61.0 7.13 57.0 7.26
Fe[i]{} 6232.65 3.65 -1.22 140.0 7.52 179.2 8.03 123.7 7.34 141.8 7.59 131.6 7.58 135.4 7.38
Fe[i]{} 6240.65 2.22 -3.23 128.9 7.45 148.5 7.69 108.1 7.14 121.5 7.29 133.3 7.42 101.3 7.15
Fe[i]{} 6246.33 3.60 -0.73 140.4 6.98 136.2 6.97
Fe[i]{} 6265.14 2.18 -2.55 169.2 7.35 168.6 7.47 147.5 7.27
Fe[i]{} 6270.23 2.86 -2.46 137.1 7.60 104.8 7.19 129.8 7.60 125.4 7.74 122.7 7.34 90.9 7.07
Fe[i]{} 6297.80 2.22 -2.74 155.0 7.52 168.2 7.64 148.6 7.55 170.8 7.59 128.7 7.26
Fe[i]{} 6301.51 3.65 -0.72 142.6 7.04 162.7 7.32 152.3 7.20 160.1 7.44
Fe[i]{} 6311.50 2.83 -3.16 101.5 7.65 121.2 7.92 96.7 7.62 116.2 7.94 95.3 7.68 108.4 7.67 93.2 7.76
Fe[i]{} 6315.81 4.08 -1.68 78.0 7.39 91.8 7.59 77.5 7.41 90.9 7.68 70.4 7.32 82.6 7.40 78.1 7.67
Fe[i]{} 6322.69 2.59 -2.43 140.6 7.34 175.0 7.80 148.4 7.56 151.2 7.53 147.3 7.68 156.3 7.50 120.5 7.28
Fe[i]{} 6330.85 4.73 -1.22 49.2 7.20 61.4 7.47 71.1 7.70 70.6 7.73 48.3 7.15
Fe[i]{} 6380.75 4.19 -1.37 80.0 7.28 96.6 7.65 103.2 7.77 101.8 7.85 102.1 7.62 76.8 7.48
Fe[i]{} 6392.54 2.28 -3.97 79.6 7.16 111.8 7.76 83.4 7.36 84.4 7.30 82.7 7.44 95.3 7.44 77.7 7.34
Fe[i]{} 6411.66 3.65 -0.60 143.2 6.92 155.4 7.12 167.8 7.18
Fe[i]{} 6481.88 2.28 -2.98 136.0 7.36 179.4 7.99 150.3 7.71 152.5 7.67 137.5 7.64 160.1 7.68 125.3 7.47
Fe[i]{} 6498.94 0.96 -4.70 147.3 7.35 198.5 8.10 160.0 7.74 174.3 7.83 184.5 7.89 139.2 7.47
--------- --------- ------ -------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
---------- --------- ------ -------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
El. Wavel. E.P. log gf 3734 4360 5308 5341 4329 4453 3092
EW log A EW log A EW log A EW log A EW log A EW log A EW log A
(Å) (eV) (mÅ) (mÅ) (mÅ) (mÅ) (mÅ) (mÅ) (mÅ)
Fe[i]{} 6518.37 2.83 -2.46 108.5 7.06 140.6 7.54 112.7 7.33 127.0 7.30 93.5 7.04
Fe[i]{} 6533.94 4.56 -1.29 59.2 7.24 92.1 7.84 84.9 7.82 78.5 7.73
Fe[i]{} 6574.25 0.99 -5.00 125.2 7.23 165.7 7.89 122.7 7.35 136.4 7.46 121.8 7.46 147.8 7.58 109.9 7.20
Fe[i]{} 6581.22 1.49 -4.68 123.0 7.52 129.4 7.87 125.5 7.67 116.3 7.74 139.9 7.83 111.0 7.66
Fe[i]{} 6593.88 2.43 -2.42 144.2 7.11 160.5 7.47 155.8 7.33 158.4 7.58 175.3 7.50 133.0 7.20
Fe[i]{} 6608.04 2.28 -3.96 82.0 7.18 131.2 8.08 94.6 7.56 97.3 7.53 96.1 7.69 92.2 7.35 83.6 7.45
Fe[i]{} 6609.12 2.56 -2.69 130.9 7.34 174.5 7.95 134.6 7.50 149.3 7.68 141.6 7.75 164.7 7.80 108.0 7.21
Fe[i]{} 6627.56 4.55 -1.50 40.2 7.03 55.4 7.38 55.7 7.42 59.5 7.52 82.7 7.85 35.4 7.04
Fe[i]{} 6633.76 4.56 -0.82 83.2 7.30 119.5 7.88 100.0 7.65 88.8 7.43 109.4 7.92 107.6 7.66 77.3 7.48
Fe[i]{} 6703.58 2.76 -3.01 87.7 7.02 129.3 7.75 101.6 7.42 102.1 7.34 100.8 7.49 119.4 7.57 83.6 7.20
Fe[i]{} 6713.75 4.80 -1.41 42.5 7.33 45.9 7.41 30.8 7.13 38.1 7.28 56.2 7.57
Fe[i]{} 6725.36 4.10 -2.21 70.1 7.72 50.0 7.37 40.4 7.20 60.7 7.64 64.0 7.58 34.4 7.08
Fe[i]{} 6726.67 4.61 -1.10 57.2 7.07 82.4 7.52 69.7 7.34 77.2 7.55 75.7 7.37 37.8 6.78
Fe[i]{} 6733.15 4.64 -1.44 32.4 6.91 79.3 7.85 54.1 7.40 61.2 7.59 54.6 7.46 80.2 7.86 53.6 7.60
Fe[i]{} 6750.16 2.42 -2.62 136.3 7.11 167.2 7.53 160.8 7.61 157.2 7.49 150.9 7.59 169.0 7.56 123.2 7.18
Fe[i]{} 6786.86 4.19 -1.90 41.9 6.97 85.2 7.80 61.8 7.42 49.3 7.19 61.9 7.47 78.4 7.66 48.7 7.28
Fe[ii]{} 5991.38 3.15 -3.56 31.0 7.31 46.0 7.67 37.0 7.42
Fe[ii]{} 6084.10 3.20 -3.80 30.2 7.75 35.0 7.70 35.2 7.67
Fe[ii]{} 6113.33 3.22 -4.13 27.1 7.78 23.8 7.70
Fe[ii]{} 6149.25 3.89 -2.73 49.1 7.86 21.1 7.22 27.1 7.21 29.2 7.27
Fe[ii]{} 6247.56 3.89 -2.33 52.5 7.64 54.3 7.60 54.3 7.87 43.2 7.33 41.5 7.24 27.6 7.27
Fe[ii]{} 6369.46 2.89 -4.21 25.5 7.47 28.0 7.64 32.1 7.62 28.7 7.55 27.1 7.48
Fe[ii]{} 6416.93 3.89 -2.70 33.4 7.49 29.1 7.50
Fe[ii]{} 6432.68 2.89 -3.58 35.3 7.15 65.4 7.93 56.4 7.64 30.6 7.18 38.9 7.22 45.0 7.34 24.1 7.07
Fe[ii]{} 6456.39 3.90 -2.10 39.5 7.10 75.6 7.97 58.3 7.50 47.6 7.48 44.7 7.16 55.6 7.40 21.8 6.82
Fe[ii]{} 6516.08 2.89 -3.38 53.5 7.45 63.9 7.62
Ni[i]{} 5847.01 1.68 -3.44 102.3 6.39 132.0 6.92 104.1 6.55 113.0 6.67 90.3 6.39
Ni[i]{} 5996.74 4.24 -1.06 38.9 6.10 67.6 6.72 39.1 6.12 42.6 6.27 54.0 6.51 47.8 6.28 45.0 6.44
Ni[i]{} 6053.69 4.24 -1.07 37.5 6.08 84.3 7.06 28.8 5.88 64.7 6.76 55.7 6.56 48.9 6.31
Ni[i]{} 6086.29 4.27 -0.47 52.5 5.87 98.2 6.78 65.2 6.16 88.0 6.70 77.5 6.50 73.0 6.24 55.2 6.18
Ni[i]{} 6108.12 1.68 -2.49 139.7 6.13 178.3 6.70 134.8 6.17 154.1 6.41 136.9 6.34 162.0 6.43 124.7 6.17
Ni[i]{} 6111.08 4.09 -0.83 65.4 6.28 95.4 6.82 66.2 6.30 65.1 6.31 73.1 6.52 71.1 6.32 49.1 6.11
Ni[i]{} 6128.98 1.68 -3.39 107.8 6.41 132.7 6.80 89.7 6.10 108.0 6.42 107.6 6.63 110.8 6.37 81.0 6.02
Ni[i]{} 6130.14 4.27 -0.98 32.7 5.92 77.0 6.86 34.3 5.97 46.6 6.31 47.0 6.31 52.4 6.33 37.8 6.20
Ni[i]{} 6176.82 4.09 -0.26 91.9 6.29 110.0 6.53 87.8 6.22 93.6 6.34 93.2 6.41 91.9 6.17 51.1 5.60
Ni[i]{} 6177.25 1.83 -3.60 80.0 6.19 102.3 6.63 78.9 6.28 104.3 6.76 77.8 6.35 77.1 6.12
Ni[i]{} 6186.72 4.11 -0.90 43.0 5.87 89.7 6.81 59.4 6.25 78.0 6.69 70.6 6.55 57.7 6.14
Ni[i]{} 6204.61 4.09 -1.15 61.4 6.51 99.9 7.23 58.7 6.46 37.4 6.04 64.5 6.64 65.7 6.53
Ni[i]{} 6223.99 4.11 -0.97 49.9 6.09 75.0 6.60 52.2 6.16 67.6 6.53 58.8 6.36 47.0 6.00 28.0 5.68
Ni[i]{} 6230.10 4.11 -1.20 47.1 6.26 83.9 7.00 53.6 6.42 69.0 6.79 52.6 6.45 52.3 6.34 31.7 6.02
Ni[i]{} 6322.17 4.15 -1.21 34.8 6.03 75.6 6.88 33.8 6.03 25.5 5.89 48.3 6.40 30.9 5.95
Ni[i]{} 6327.60 1.68 -3.09 117.8 6.28 121.4 6.47 132.8 6.57 124.1 6.65 142.5 6.64 101.0 6.20
Ni[i]{} 6378.26 4.15 -0.82 51.5 6.02 107.7 7.11 77.0 6.60 57.6 6.22 66.7 6.43 65.8 6.27 54.1 6.32
Ni[i]{} 6384.67 4.15 -1.00 38.6 5.91 85.5 6.86 66.5 6.54 56.7 6.38 58.4 6.42 70.4 6.54 39.5 6.10
Ni[i]{} 6482.81 1.93 -2.85 110.0 6.08 149.3 6.72 128.4 6.47 129.1 6.45 106.9 6.27 132.4 6.41 103.4 6.14
Ni[i]{} 6532.88 1.93 -3.42 81.0 6.12 117.8 6.81 93.5 6.50 106.5 6.71 93.1 6.60 94.2 6.36 66.4 5.98
Ni[i]{} 6586.32 1.95 -2.79 108.3 6.12 158.6 6.96 122.7 6.42 114.9 6.48 133.3 6.50 92.3 6.05
Ni[i]{} 6598.61 4.24 -0.93 31.1 5.78 81.8 6.83 59.7 6.43 48.6 6.26 50.5 6.28 67.1 6.52 52.1 6.50
Ni[i]{} 6635.14 4.42 -0.75 32.6 5.88 80.0 6.86 51.6 6.32 34.2 6.00 51.5 6.36 46.5 6.17 29.1 5.92
Ni[i]{} 6767.78 1.83 -2.11 145.3 5.92 182.6 6.41 157.4 6.22 152.4 6.07 146.0 6.17 168.9 6.20 124.4 5.86
Ni[i]{} 6772.32 3.66 -1.01 68.9 5.89 103.1 6.51 79.4 6.15 78.0 6.14 84.5 6.32 87.3 6.19 63.9 6.04
Y[i]{} 6435.02 0.07 -0.82 61.7 1.30 126.3 2.54 50.1 1.41 75.7 1.57 56.4 1.62 84.0 1.82 64.2 1.27
Zr[i]{} 6127.46 0.15 -1.06 85.2 2.00 118.9 2.64 78.8 2.18 89.6 2.07 72.4 2.16 86.7 2.10 79.6 2.02
Zr[i]{} 6134.57 0.00 -1.28 87.0 2.03 96.2 2.21 73.7 2.07 78.0 1.84 60.3 1.91 78.0 1.94 85.7 2.16
Zr[i]{} 6140.46 0.52 -1.41 26.7 1.75 25.1 1.79 39.0 2.40 42.7 2.26 38.2 1.75
Zr[i]{} 6143.18 0.07 -1.10 74.7 1.69 116.3 2.51 73.0 1.98 98.7 2.17 72.7 2.09 87.9 2.04 98.8 2.40
Zr[i]{} 6445.72 1.00 -0.83 14.0 1.50 26.9 2.07
Ba[ii]{} 5853.69 0.60 -1.00 128.3 2.28 121.3 1.94 124.3 2.35 122.9 2.16 132.4 2.69 147.2 2.47 102.3 2.10
Ba[ii]{} 6141.75 0.70 0.00 185.0 2.08 212.4 2.44 170.5 2.07 197.6 2.30 172.1 2.26 190.1 2.14 150.7 1.86
Ba[ii]{} 6496.91 0.60 -0.38 194.3 2.44 166.5 2.24 183.0 2.36 167.1 2.41 173.3 2.10 168.5 2.25
Eu[ii]{} 6437.64 1.32 -0.28 35.7 1.04 33.4 0.94 18.6 0.58 29.9 0.94 28.9 0.80 15.2 0.27
Eu[ii]{} 6645.11 1.38 0.20 28.9 0.30 50.6 0.92 46.0 0.81 39.6 0.69 34.3 0.64 44.5 0.72 31.1 0.40
---------- --------- ------ -------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- -------
Metallicities
=============
Equivalent Widths
-----------------
The equivalent widths (EWs) were measured on the spectra using the ROSA code (Gratton 1988; see Table \[t:ewidths\] for stars member of , and Table \[t:ewidths2\] for the field stars) with Gaussian fittings to the measured profiles: these exploit a linear relation between EWs and FWHM of the lines, derived from a subset of lines characterized by cleaner profiles (see Bragaglia et al. 2001 for further details on this procedure). Since the observed stars span a very limited atmospheric parameter range, errors in these EWs may be estimated by comparing EWs measures for individual stars with the EW values for each single line averaged over the whole sample. The values listed in Column 9 of Table \[t:uvesphot\] as errors on the EWs measurements are the $r.m.s.$ of residuals around the best fit line for EWs vs. $<$EWs$>$ (after eliminating a few outliers). These errors may be slightly overestimated, due to real star-to-star differences. They are roughly reproduced by the formula $\sigma$(EW)$\sim
380/(S/N)$ mÅ. Considering the resolution and sampling of the spectra, the errors in the EWs are about 4 times larger than expectations based on photon noise statistics (Cayrel 1988), showing that errors are dominated by uncertainties in the correct positioning of the continuum level; the observed errors could be justified by errors of about 1% for the best cases and about 2% for the worst ones. Due to the problems in background subtraction in the green-yellow part of the spectra, only lines with wavelength $>5800$ Å were used.
[lcccc]{} Star &T$_{\rm eff}$ &$\log g$ &$[$A/H$]$ &$v_t$\
& (K) & & &(km s$^{-1}$)\
\
7004050 & 3956 & 1.35 & $-$0.40 &0.90\
7004434 & 3984 & 1.40 & $-$0.40 &1.75\
7004463 & 4000 & 1.41 & $-$0.40 &1.70\
7004487 & 4074 & 1.48 & $-$0.40 &1.55\
8002961 & 3942 & 1.23 & $-$0.40 &1.40\
\
6003734 & 4061 & 1.53 & $-$0.28 &1.20\
6004360 & 4163 & 1.68 & 0.26 &1.55\
6005308 & 4214 & 1.77 & $-$0.17 &1.20\
6005341 & 4079 & 1.67 & 0.05 &1.35\
7004329 & 4257 & 1.73 & 0.11 &1.15\
7004453 & 4176 & 1.69 & $-$0.05 &1.45\
8003092 & 3887 & 1.32 & $-$0.32 &0.80\
\[t:uvesatmo\]
Atmospheric Parameters
----------------------
We performed a standard line analysis on the equivalent widths measured on our spectra, using model atmospheres extracted by interpolation from the grid by Kurucz (1992; models with the overshooting option switched off). Atmospheric parameters defining these model atmospheres were obtained as follows.
Whenever possible, effective temperatures were derived from de-reddened $V-K$ colors, obtained by combining our $V$ magnitudes with $K$ones drawn from the 2MASS catalog (Cutri et al. 2003), using the calibration by Alonso et al. (1999). The reddening we adopted is $E(B-V)=0.49$, which is the average value between various literature determinations. Zinn (1980) and Reed et al. (1988) obtained $E(B-V)=0.47$ and $E(B-V)=0.49$ respectively from integrated photometry; Layden et al. (1999) and Pritzl et al. (2001) found $E(B-V)=0.45\pm 0.05$ and $E(B-V)=0.51\pm 0.02$ mag respectively from the blue edge of the RR Lyrae instability strip. For comparison, the maps by Schlegel et al. (1998) give $E(B-V)=0.616$, but these are known to overestimate reddening for objects close to the galactic plane, like . Adopted $E(B-V)$ values were transformed into $E(V-K)$ ones using the formula $E(V-K)=2.75~E(B-V)$ (Cardelli et al. 1989). Errors in these temperatures are not easy to evaluate. Internal photometric errors in the 2MASS $V-K$ colors are generally small ($<0.04$ mag), leading to random errors of $<25$ K. However, interstellar reddening is probably variable in the field of (Layden et al. 1999; Pritzl et al. 2001); four of the observed stars lie in the south-east sector of the cluster, where Layden et al. found a reddening value in the range $0.41<E(B-V)<0.54$. Assuming an $r.m.s.$ scatter of 0.05 mag in $E(B-V)$ (0.14 mag in $E(V-K)$), the random (star-to-star) uncertainties in the effective temperatures may be as large as $\pm
80$ K. Systematic uncertainties are likely larger, including errors in the Alonso et al. color-temperature calibration, in the photometric calibration, and on the average reddenings we assumed. Hereinafter we will assume possible systematic errors of $\pm 100$ K.
For stars lacking the 2MASS photometry, we used our $V-I$colors. Since these colors lack a proper photometric calibration, we may use those stars having known $V-K$ colors to construct a correlation between the two colors, that turned out to be very narrow (this is not unexpected, since both colors have a weak dependence on metal abundance). The mean relation is $V-K=0.479+1.952~(V-I)$, derived from more than 250 stars spanning over 4 magnitudes in $V-K$color; the $r.m.s.$ scatter around this mean relation is 0.098 mag in ($V-K$). We could derive consistent temperatures also for stars with only $V-I$ colors, the errors due to uncertainties in the colors being smaller than those related to the assumptions about reddening.
We may compare these temperatures derived from colors with those that we could deduce from excitation equilibrium for Fe I lines. We found that on average temperatures derived from Fe I excitation equilibrium are lower by $70\pm 27$ K, with an $r.m.s.$ scatter for individual stars of 95 K. This small difference can be attributed to several causes (errors in the adopted temperature scale, inadequacies of the adopted model atmospheres, etc.). On the whole we do not consider this difference as important. The star-to-star scatter in the difference between temperatures from colors and excitation equilibrium is not much larger than the internal errors in the excitation temperatures alone (68 K). This leaves space for errors in temperatures from colors of about 66 K, to be attributed to the effect of differential reddening. This corresponds to an r.m.s scatter of about 0.043 mag in the reddening, compatible with results from photometry alone.
Surface gravities were obtained from the location of the stars in the color-magnitude diagram. This procedure requires assumptions about the distance modulus; (we adopted $(m-M)_V=16.33$ from Harris (1996) for the cluster members, while for the field bulge stars we assumed that the stars are at the same distance of the galactic center: $7.9\pm
0.3$ kpc from Eisenhauer et al. (2003), the bolometric corrections (from Alonso et al. 1999), and the masses (we assumed a mass of 0.9 M$_\odot$, close to the value given by isochrone fittings). Uncertainties in these gravities are small for cluster stars (we estimate internal star-to-star errors of about 0.06 dex, due to the effects of possible variations in the interstellar absorption; and systematic errors of about 0.15 dex, dominated by systematic effects in the temperature scale).
We may compare these values for the surface gravities with those deduced from the equilibrium of ionization of Fe. On average, abundances from Fe II lines are $0.01\pm 0.05$ dex smaller than that derived from Fe I lines. The agreement is obviously very good. The star-to-star scatter in the residuals is 0.17 dex, but it is as small as 0.12 dex if only stars member of are considered; this last value is close to the value expected considering that only three to four Fe II lines were typically used in the analysis. The larger spread obtained for field stars is not unexpected, since these stars may be at different distances from the Sun.
Micro-turbulence velocities $v_t$ were determined by eliminating trends in the relation between expected line strength and abundances (see Magain 1984). The error on the micro-turbulent velocities is determined by the change required on the derived $v_t$ to vary the expected line strength vs abundances slope by 1$\sigma$. This implies an expected random error in the micro-turbulence velocities of $\pm 0.12$ km s$^{-1}$.
Finally,the model atmospheres were iteratively chosen with model metal abundances in agreement with derived Fe abundance. The adopted model atmosphere parameters are listed in Table \[t:uvesatmo\].
[lcccccc]{} Star & & Fe I & & & Fe II &\
& n & \[Fe/H\] & $r.m.s.$ & n & \[Fe/H\] & $r.m.s.$\
\
7004050 & 52 & $-$0.45 & 0.18 & 4 & $-$0.49 & 0.18\
7004434 & 52 & $-$0.34 & 0.23 & 2 & $-$0.05 & 0.12\
7004463 & 57 & $-$0.38 & 0.18 & 4 & $-$0.56 & 0.46\
7004487 & 51 & $-$0.50 & 0.13 & 4 & $-$0.68 & 0.11\
8002961 & 59 & $-$0.28 & 0.16 & 4 & $-$0.34 & 0.16\
\
\
\
6003734 & 63 & $-$0.25 & 0.20 & 6 & $-$0.16 & 0.17\
6004360 & 44 & +0.27 & 0.24 & 7 & +0.49 & 0.27\
6005308 & 63 & $-$0.17 & 0.21 & 8 & +0.17 & 0.11\
6005341 & 60 & +0.06 & 0.23 & 6 & +0.01 & 0.28\
7004329 & 57 & +0.12 & 0.16 & 7 & $-$0.08 & 0.24\
7004453 & 62 & $-$0.04 & 0.20 & 7 & $-$0.09 & 0.14\
8003092 & 46 & $-$0.25 & 0.23 & 3 & $-$0.44 & 0.23\
\[t:uvesfe\]
------------------- ----------- --------------- ----------- ----------- ---------- ------- ---------- ------------
Element Average $T_{\rm eff}$ $\log{g}$ $[$A/H$]$ $v_t$ EWs Total Total
n\. lines Internal Systematic
Variation 100 0.30 0.20 0.20
Internal 50 0.06 0.07 0.12 0.194
Systematic 100 0.15 0.08 0.12 0.194
$[$Fe/H$]$[i]{} 55.0 $-$0.012 +0.063 +0.028 $-$0.114 0.026 0.075 0.053
$[$Fe/H$]$[ii]{} 2.7 $-$0.210 +0.199 +0.046 $-$0.046 0.119 0.167 0.262
$[$O/Fe$]$[i]{} 1.8 +0.041 +0.057 +0.009 +0.101 0.145 0.161 0.155
$[$Na/Fe$]$[i]{} 2.0 +0.106 $-$0.069 $-$0.025 +0.039 0.137 0.152 0.177
$[$Mg/Fe$]$[i]{} 2.0 $-$0.024 $-$0.010 $-$0.013 +0.084 0.137 0.149 0.141
$[$Al/Fe$]$[i]{} 2.0 +0.087 $-$0.060 $-$0.028 +0.062 0.137 0.152 0.166
$[$Si/Fe$]$[i]{} 2.2 $-$0.102 +0.020 $-$0.004 +0.079 0.131 0.151 0.168
$[$Ca/Fe$]$[i]{} 11.4 +0.131 $-$0.102 $-$0.016 $-$0.006 0.057 0.093 0.152
$[$Sc/Fe$]$[ii]{} 2.8 $-$0.017 +0.065 +0.008 +0.027 0.115 0.120 0.121
$[$Ti/Fe$]$[i]{} 6.3 +0.167 $-$0.050 $-$0.014 $-$0.061 0.078 0.123 0.187
$[$Ti/Fe$]$[ii]{} 1.0 $-$0.048 +0.071 +0.009 +0.095 0.194 0.206 0.205
$[$V/Fe$]$[i]{} 12.8 +0.178 $-$0.041 $-$0.011 $-$0.077 0.054 0.117 0.188
$[$Cr/Fe$]$[i]{} 1.0 +0.150 $-$0.025 $-$0.015 $-$0.055 0.194 0.212 0.246
$[$Mn/Fe$]$[i]{} 3.0 +0.053 $-$0.063 +0.007 $-$0.026 0.112 0.120 0.128
$[$Ni/Fe$]$[i]{} 23.5 $-$0.028 +0.029 $-$0.001 +0.032 0.040 0.054 0.052
$[$Y/Fe$]$[i]{} 1.0 +0.231 $-$0.044 $-$0.018 +0.074 0.194 0.230 0.252
$[$Zr/Fe$]$[i]{} 4.8 +0.207 $-$0.025 $-$0.014 +0.043 0.091 0.141 0.213
$[$Ba/Fe$]$[ii]{} 3.0 +0.029 +0.029 +0.014 $-$0.111 0.112 0.134 0.120
$[$Eu/Fe$]$[ii]{} 1.8 $-$0.004 +0.058 +0.009 +0.093 0.145 0.156 0.091
------------------- ----------- --------------- ----------- ----------- ---------- ------- ---------- ------------
\[t:errorparam\]
Fe Abundances
-------------
Individual \[Fe/H\] values are listed in Table \[t:uvesfe\], as well as averages over the whole sample. Reference solar abundances are as in Gratton et al. (2003). Throughout our analysis, we use the same line parameters discussed in Gratton et al. (2003); in particular, collisional damping was considered using updated constants from Barklem et al. (2000). We note that, for internal consistency, we used four Fe[II]{} lines for all member stars. In the case of star \# 7004463 the lines yield considerably different Fe[II]{} abundances, therefore the derived [*r.m.s.*]{} is larger than in other cases.
Table \[t:errorparam\] lists the impact of various uncertainties on the derived abundances for the elements considered in our analysis. Variations in parameters of the model atmospheres were obtained by changing each of the parameters at a time for the star \#7004487, assumed to be representative of all the stars considered in this paper. The first three rows of the Table give the variation of the parameter used to estimate sensitivities, the internal (star-to-star) errors (for member stars of ), and the systematic errors (common to all stars) in each parameter. The first column gives the average number of lines $n$ used for each element. Columns 3-6 give the sensitivities of the abundance ratios to variations of each parameter. Column 7 gives the contribution to the error given by uncertainties in EWs for individual lines: that is $0.194/\sqrt{n}$, where 0.194 is the error in the abundance derived from an individual line, as obtained by the median error over all the stars (note that the errors in the EWs have much smaller impact for the stars having better spectra, like \#7004487 itself). The two final Columns give the total resulting internal and systematic errors, obtained by summing quadratically the contribution of the individual sources of errors, weighted according to the errors appropriate for each parameter. For the systematic errors, the contribution due to equivalent widths and to micro-turbulence velocities (quantities derived from our own analysis), were divided by the square root of the number of cluster members observed. Note that this error analysis does not include the effects of covariances in the various error sources, which are however expected to be quite small for the program stars.
Errors in Fe abundances from neutral lines are dominated by uncertainties in the micro-turbulent velocity. We note that internal errors in Fe [I]{} abundances are dominated by errors in $v_t$. We estimate total random errors of 0.075 dex, and systematic errors of 0.053 dex. From Table \[t:uvesfe\], the average Fe abundance from all stars of is \[Fe/H\]=$-0.39\pm 0.04$ (error of the mean), with an $r.m.s.$ scatter of 0.09dex from 5 stars. The first result of our analysis is that the metallicity of is \[Fe/H\]=$-0.39\pm
0.04\pm 0.05$, where the first error bar includes the uncertainties related to star-to-star random errors, and the second one the systematic effects related to the various assumptions made in the analysis The offset between abundances given by neutral and singly ionized Fe lines is only 0.03dex, supporting our analysis. The star-to-star scatter (0.09dex) is slightly larger than expected on the basis of our error analysis (0.075 dex); the scatter for Fe II is larger, but in good agreement with expectations based on our error analysis. While this difference is not significant, given the small number of stars, it is possible that stars \#7004050 and \#7004487 are indeed slightly more metal-poor (by about 0.15dex) than the average of the remaining three cluster members. It is also worth mentioning that star \#7004050 is also slightly bluer than the other member stars of , by about 0.08 mag in the $V-I$ color: this is roughly the color difference expected for the metallicity difference found in our analysis. More data on a wider sample of stars of are clearly required to definitely set the issue about an intrinsic spread in metal abundances in this cluster.
To our knowledge, this is the first determination of the metal abundance of from high dispersion spectra of individual stars. We found in the literature only three previous metallicity determinations for . The first two are based on integrated light. Zinn & West (1984) used the Q39 index to obtain a metallicity of \[Fe/H\]=$-0.59\pm 0.15$. A somewhat higher metallicity of \[Fe/H\]=$-0.47\pm 0.12$ dex was obtained by Armandroff & Zinn (1988) from spectroscopy of the Ca II IR triplet, calibrated on the Zinn & West scale. Armandroff and Zinn averaged these two values to produce the value of \[Fe/H\]=$-0.53\pm 0.11$. That is listed by Harris (1996) and usually adopted in the discussion of this cluster. Very recently, Clementini et al. (2005) have measured metal abundances for a few RR Lyrae in using a variant of the $\Delta S$ method, obtaining \[Fe/H\]=$-0.69\pm 0.06$ on the Zinn & West scale, and \[Fe/H\]=$-0.41\pm 0.06$ on the Carretta & Gratton (1997) scale, with a quite considerable star-to-star scatter of more than 0.3 dex ($r.m.s.$). The present metal abundance should be compared with that on the Carretta & Gratton scale, which is based on an analysis method quite consistent with that here considered. Indeed, this last comparison is quite good. Finally, we note that if we add the point of to the correlation between abundances from high dispersion spectra by our group (see Carretta et al. 2001) and those by Zinn & West (1984), a linear relationship between the two scales would be favored, rather than a cubic one as considered by Carretta et al. (2001).
![Spectral region including the \[OI\] line at 6300.3 Å for the member stars of . The spectra have been corrected for the individual radial velocities and offset vertically for clarity[]{data-label="f:ona1"}](gratton_6441_1.eps){width="8.8cm"}
![Spectral region including the \[OI\] line at 6363.8 Å for the member stars of . The spectra have been corrected for the individual radial velocities and offset vertically for clarity[]{data-label="f:ona2"}](gratton_6441_2.eps){width="8.8cm"}
![Spectral region including the Na I doublet at 6154-60 Å for the member stars of . The spectra have been corrected for the individual radial velocities and offset vertically for clarity[]{data-label="f:ona3"}](gratton_6441_3.eps){width="8.8cm"}
![Upper Panel: \[Na/Fe\] ratio as a function of \[O/Fe\], for stars member of , the curve indicates the mean \[O/Fe\] vs \[Na/Fe\] locus for a collection of $\sim$20 Globular clusters (Carretta et al. 2006). Lower panel: \[Al/Fe\] ratio as a function of \[Mg/Fe\], for the same stars.[]{data-label="f:ona4"}](gratton_6441_4.eps){width="8.8cm"}
{width="13cm"}
{width="13cm"}
------------------- ---- --------- ---------- ---- --------- ---------- ---------- --------- ---------- ---- --------- ---------- ---- --------- ---------- --------- ------
Element Cluster $r.m.s.$
N mean $r.m.s.$ N mean $r.m.s.$ N mean $r.m.s.$ N mean $r.m.s.$ N mean $r.m.s.$ Average
$[$Fe/H$]$ 53 $-$0.46 0.19 54 $-$0.38 0.23 59 $-$0.45 0.18 53 $-$0.50 0.13 59 $-$0.36 0.15 $-$0.43 0.06
$[$O/Fe$]$[i]{} 2 +0.10 0.08 2 +0.32 0.11 1 $-$0.15 2 +0.30 0.14 2 +0.03 0.04 +0.12 0.20
$[$Na/Fe$]$[i]{} 2 +0.58 0.08 2 +0.38 0.10 2 +0.79 0.10 2 +0.53 0.00 2 +0.49 0.08 +0.55 0.15
$[$Mg/Fe$]$[i]{} 2 +0.31 0.01 2 +0.34 0.06 2 +0.20 0.02 2 +0.41 0.09 2 +0.44 0.08 +0.34 0.09
$[$Al/Fe$]$[i]{} 2 +0.27 0.08 2 +0.23 0.15 2 +0.56 0.23 2 +0.19 0.16 2 +0.25 0.21 +0.30 0.15
$[$Si/Fe$]$[i]{} 1 +0.27 2 +0.17 0.09 2 +0.45 0.17 3 +0.33 0.06 2 +0.41 0.16 +0.33 0.11
$[$Ca/Fe$]$[i]{} 12 0.00 0.19 11 $-$0.01 0.20 10 +0.07 0.15 12 +0.06 0.22 12 +0.01 0.16 +0.03 0.04
$[$Sc/Fe$]$[ii]{} 3 +0.30 0.24 3 +0.09 0.12 3 +0.29 0.20 3 +0.13 0.07 3 $-$0.05 0.09 +0.15 0.15
$[$Ti/Fe$]$[i]{} 5 +0.39 0.09 6 +0.21 0.21 6 +0.41 0.18 6 +0.27 0.11 7 +0.19 0.15 +0.29 0.10
$[$Ti/Fe$]$[ii]{} 1 +0.50 1 +0.34 1 +0.18 1 +0.18 1 +0.43 +0.33 0.14
$[$V/Fe$]$[i]{} 9 +0.35 0.17 15 +0.18 0.18 14 +0.40 0.17 12 +0.42 0.15 11 +0.11 0.14 +0.29 0.14
$[$Cr/Fe$]$[i]{} 1 +0.39 1 +0.27 1 +0.12 1 +0.03 1 $-$0.06 +0.15 0.18
$[$Mn/Fe$]$[i]{} 3 +0.09 0.19 3 $-$0.07 0.13 3 +0.15 0.20 3 +0.05 0.18 3 +0.14 0.12 +0.07 0.09
$[$Ni/Fe$]$[i]{} 21 +0.18 0.16 23 +0.18 0.17 23 +0.03 0.20 25 +0.09 0.20 24 +0.16 0.14 +0.13 0.07
$[$Y/Fe$]$[i]{} 1 $-$0.34 1 +0.01 1 +0.31 1 $-$0.11 1 $-$0.10 $-$0.05 0.24
$[$Zr/Fe$]$[i]{} 4 $-$0.28 0.28 5 $-$0.35 0.20 5 $-$0.33 0.18 5 $-$0.29 0.22 5 $-$0.74 0.11 $-$0.40 0.19
$[$Ba/Fe$]$[ii]{} 3 +0.39 0.21 3 +0.14 0.15 3 +0.19 0.18 3 +0.07 0.08 3 +0.06 0.18 +0.17 0.13
$[$Eu/Fe$]$[ii]{} 1 +0.49 2 +0.48 0.03 2 +0.37 0.07 2 +0.34 0.19 2 +0.22 0.04 +0.32 0.11
------------------- ---- --------- ---------- ---- --------- ---------- ---------- --------- ---------- ---- --------- ---------- ---- --------- ---------- --------- ------
\[t:abundgc6441\]
------------------- ---- --------- ---------- ---- --------- ---------- ---- --------- ---------- ---- --------- ----------
Element
N mean $r.m.s.$ N mean $r.m.s.$ N mean $r.m.s.$ N mean $r.m.s.$
$[$Fe/H$]$ 62 $-$0.33 0.18 44 +0.16 0.17 64 $-$0.22 0.21 62 $-$0.03 0.23
$[$O/Fe$]$[i]{} 2 +0.36 0.01 2 $-$0.04 0.15 2 +0.22 0.15 1 +0.10
$[$Na/Fe$]$[i]{} 2 +0.24 0.13 2 +0.74 0.19 2 +0.26 0.28 2 +0.44 0.10
$[$Mg/Fe$]$[i]{} 2 +0.39 0.13 2 +0.63 0.17 2 +0.14 0.34 2 +0.33 0.12
$[$Al/Fe$]$[i]{} 2 $-$0.15 0.12 2 +0.13 0.17 2 +0.43 0.01 2 $-$0.13 0.21
$[$Si/Fe$]$[i]{} 2 +0.35 0.16 3 +0.61 0.20 2 $-$0.07 0.29 2 $-$0.08 0.20
$[$Ca/Fe$]$[i]{} 13 $-$0.06 0.19 13 $-$0.05 0.17 11 +0.13 0.17 11 $-$0.04 0.23
$[$Sc/Fe$]$[ii]{} 2 +0.01 0.12 3 $-$0.05 0.32 2 +0.28 0.19 3 +0.07 0.46
$[$Ti/Fe$]$[i]{} 7 +0.24 0.20 6 +0.09 0.14 7 +0.34 0.17 6 +0.28 0.06
$[$Ti/Fe$]$[ii]{} 1 $-$0.08 1 +0.39 1 $-$0.02 1 +0.04
$[$V/Fe$]$[i]{} 10 +0.20 0.16 11 +0.34 0.16 15 +0.36 0.17 15 +0.43 0.22
$[$Cr/Fe$]$[i]{} 1 $-$0.29 1 +0.14 1 +0.03 1 $-$0.05
$[$Mn/Fe$]$[i]{} 3 +0.08 0.13 3 +0.27 0.11 3 $-$0.03 0.07 3 +0.17 0.11
$[$Co/Fe$]$[i]{} 1 +0.05 1 +0.01 1 0.00
$[$Ni/Fe$]$[i]{} 25 +0.07 0.19 24 +0.27 0.19 24 +0.14 0.19 25 +0.05 0.25
$[$Y/Fe$]$[i]{} 1 $-$0.23 1 +0.42 1 $-$0.23 1 $-$0.31
$[$Zr/Fe$]$[i]{} 5 $-$0.46 0.20 4 $-$0.42 0.24 3 $-$0.29 0.09 4 $-$0.61 0.17
$[$Ba/Fe$]$[ii]{} 2 +0.25 0.13 3 $-$0.17 0.30 3 +0.17 0.13 3 +0.03 0.13
$[$Eu/Fe$]$[ii]{} 1 +0.04 2 +0.22 0.10 2 +0.49 0.09 2 +0.07 0.07
Element
N mean $r.m.s.$ N mean $r.m.s.$ N mean $r.m.s.$
$[$Fe/H$]$ 58 +0.06 0.16 64 $-$0.10 0.20 48 $-$0.38 0.23
$[$O/Fe$]$[i]{} 2 +0.15 0.10 2 +0.39 0.07
$[$Na/Fe$]$[i]{} 2 +0.37 0.00 2 +0.30 0.15 2 +0.57 0.23
$[$Mg/Fe$]$[i]{} 2 +0.32 0.05 2 +0.11 0.02 2 +0.52 0.05
$[$Al/Fe$]$[i]{} 2 $-$0.23 0.25 2 $-$0.01 0.20 2 +0.06 0.17
$[$Si/Fe$]$[i]{} 2 +0.19 0.06 2 +0.06 0.10 2 +0.15 0.17
$[$Ca/Fe$]$[i]{} 11 $-$0.03 0.15 11 $-$0.01 0.17 10 $-$0.14 0.23
$[$Sc/Fe$]$[ii]{} 3 $-$0.30 0.23 3 +0.01 0.20 3 $-$0.19 0.24
$[$Ti/Fe$]$[i]{} 6 $-$0.13 0.18 7 $-$0.18 0.13 5 $-$0.08 0.28
$[$Ti/Fe$]$[ii]{} 1 $-$0.02 1 +0.33 1 +0.14
$[$V/Fe$]$[i]{} 14 +0.19 0.17 14 +0.33 0.20 12 +0.26 0.20
$[$Cr/Fe$]$[i]{} 1 $-$0.07 1 +0.39
$[$Mn/Fe$]$[i]{} 3 +0.26 0.15 3 +0.12 0.16 3 $-$0.16 0.14
$[$Co/Fe$]$[i]{} 1 +0.19 1 +0.15
$[$Ni/Fe$]$[i]{} 24 +0.02 0.12 24 +0.04 0.17 20 +0.06 0.23
$[$Y/Fe$]$[i]{} 1 $-$0.31 1 +0.03 1 $-$0.31
$[$Zr/Fe$]$[i]{} 4 $-$0.50 0.20 4 $-$0.41 0.14 4 $-$0.24 0.22
$[$Ba/Fe$]$[ii]{} 3 +0.13 0.22 3 +0.06 0.18 3 +0.14 0.21
$[$Eu/Fe$]$[ii]{} 2 +0.14 0.22 2 +0.27 0.06 2 +0.10 0.07
------------------- ---- --------- ---------- ---- --------- ---------- ---- --------- ---------- ---- --------- ----------
\[t:abundfield\]
------------------- --------- ---------- --------- ---------- --------- ---------- --------- ---------- --------- ---------- --------- ---------- --------- ----------
Element
mean $r.m.s.$ mean $r.m.s.$ mean $r.m.s.$ mean $r.m.s.$ mean $r.m.s.$ mean $r.m.s.$ mean $r.m.s.$
$[$Fe/H$]$ $-$0.43 0.08 $-$0.31 0.08 $-$0.02 0.08 +0.07 0.02 $-$0.16 0.08 $-$0.33 $-$0.59 0.07
$[$O/Fe$]$[i]{} +0.14 0.20 +0.32 0.09 +0.13 0.04 +0.07 0.11 +0.50 0.13 +0.03 0.18
$[$Na/Fe$]$[i]{} +0.46 0.18 +0.22 0.17 +0.27 0.07 +0.40 0.12 +0.21 0.37
$[$Mg/Fe$]$[i]{} +0.34 0.09 +0.35 0.19 +0.25 0.12 +0.14 0.09 +0.41 0.10 +0.35 0.14 $-$0.11 0.07
$[$Al/Fe$]$[i]{} +0.30 0.15 +0.11 0.29 $-$0.12 0.12
$[$Si/Fe$]$[i]{} +0.33 0.11 +0.14 0.21 +0.06 0.14 +0.36 0.07 +0.14 0.18 +0.18 0.24 +0.07 0.09
$[$Ca/Fe$]$[i]{} +0.03 0.04 +0.02 0.14 $-$0.01 0.04 +0.23 0.06 +0.26 0.09 +0.14 0.17 0.00 0.14
$[$Sc/Fe$]$[ii]{} +0.15 0.15 +0.03 0.24 $-$0.07 0.20 $-$0.05 0.10 $-$0.12 0.18 +0.29 0.20
$[$Ti/Fe$]$[i]{} +0.29 0.10 +0.17 0.22 +0.20 0.08 +0.03 0.07 +0.19 0.06 +0.34 0.10 $-$0.05 0.07
$[$Ti/Fe$]$[ii]{} +0.33 0.14 +0.01 0.11 +0.12 0.19
$[$V/Fe$]$[i]{} +0.29 0.14 +0.27 0.08 +0.32 0.12 $-$0.20 0.09 +0.06 0.19
$[$Cr/Fe$]$[i]{} +0.15 0.18 +0.04 0.34 $-$0.06 0.01 0.00 0.04 +0.04 0.09 $-$0.04 0.19
$[$Mn/Fe$]$[i]{} +0.07 0.09 $-$0.04 0.12 +0.18 0.07 $-$0.37 0.07
$[$Co/Fe$]$[i]{} +0.14 0.07 +0.07 0.07 +0.10 0.13
$[$Ni/Fe$]$[i]{} +0.13 0.07 +0.09 0.04 +0.04 0.02 +0.10 0.05 +0.01 0.07 $-$0.04 0.08 $-$0.19 0.05
$[$Y/Fe$]$[i]{} $-$0.05 0.24 $-$0.26 0.05 $-$0.20 0.20
$[$Zr/Fe$]$[i]{} $-$0.40 0.19 $-$0.33 0.12 $-$0.51 0.10
$[$Ba/Fe$]$[ii]{} +0.17 0.13 +0.19 0.06 +0.07 0.05 +0.14 0.07 +0.20 0.28
$[$Eu/Fe$]$[ii]{} +0.38 0.11 +0.21 0.24 +0.16 0.10
------------------- --------- ---------- --------- ---------- --------- ---------- --------- ---------- --------- ---------- --------- ---------- --------- ----------
\(1) This paper\
(2) Carretta et al. (2001)\
(3) Cohen et al. (1999)\
(4) McWilliam & Rich (1994)\
(5) Sbordone et al. (2005)\
\[t:meanabund\]
Abundances for other elements
-----------------------------
Table \[t:abundgc6441\] lists the abundances for the individual elements for stars member of , while Table \[t:abundfield\] is for field stars. For each star and element, we give the number of lines used in the analysis, the average abundance, and the $r.m.s.$ scatter of individual values. The Na abundances include corrections for departures from LTE, following the treatment by Gratton et al. (1999). Abundances for the odd elements of the Fe group (Sc, V, and Mn) were derived with consideration for the not negligible hyperfine structure of these lines (see Gratton et al. 2003 for more details). Finally, we note that telluric absorption lines were removed from the spectra in the region around the \[OI\] lines. No attempt was made to remove the strong auroral emission line; however, due to the combination of the Earth and stellar motions at the epoch of observations, the auroral emission line typically is about 0.7 Å blueward to the stellar line, so that it does not create problems at the resolution of the UVES spectra. The O abundances were derived from equivalent widths: however, they were later confirmed by spectral synthesis. We did not apply any correction for the blending with the Ni I line at 6300.339 Å, nor for formation of CO. The blending Ni I line is expected to contribute about 4 mÅ to the EW of the \[OI\] line, using the line parameters by Allende Prieto et al. (2001); this corresponds to correcting the O abundances about 0.05 dex downward. CO coupling should be strong at the low effective temperature of the program stars. Unluckily, the abundance of C is not determined. However, we expect that C is strongly depleted in stars near the tip of the red giant branch, with expected values of \[C/Fe\]$\sim -0.6$ (Gratton et al. 2000). If the original C abundance in unevolved stars follows the Fe one (as usually observed in metal-rich environments: see e.g. Gratton et al. 2000), then we expect typical values of $[$C/O$]\sim -0.8$ for stars in . Neglecting CO formation, we should have underestimated the O abundances from forbidden lines by $\sim 0.05$ dex. These two corrections should roughly compensate, however, they both are within the error bars of the present determinations.
The last two columns of Table \[t:abundgc6441\] give the average abundance for the cluster, as well as the $r.m.s.$ scatter of individual values around this mean value. In general, the values for the scatter agree fairly well with those estimated in our error analysis. Large scatters are however determined for the light elements O, Na, Mg, and Al; the elements participating in the so-called Na-O anticorrelation. Inspection of Table \[t:abundgc6441\] reveals that while four stars share a similar composition (high O and Mg abundances, low Na and Al ones), one star (\#7004463) has much lower O and Mg abundances, and higher Na and Al ones. Figures \[f:ona1\], \[f:ona2\], and \[f:ona3\] show the spectral region around the two \[OI\] lines and the Na doublet at 6154-60 Å in the spectra of the stars of NGC 6441. The weakness of the \[OI\] lines and the strength of the Na doublet in the spectrum of star \#7004463 is quite obvious (note that the five stars have all similar temperatures and reddenings). This star seems a typical representative of the O-poor, Na-rich stars often found among Globular cluster stars. Figure \[f:ona4\] shows the O-Na and Mg-Al anticorrelations from our data set. The curve plotted in the upper panel is from Fig. 5 in Carretta et al. (2006) and indicates the typical behavior of \[O/Fe\] with respect to \[Na/Fe\] as defined by a collection of stars in about 20 Globular clusters. More extensive data about the O-Na anticorrelation among stars in will be discussed separately, on the basis of the GIRAFFE spectra taken simultaneously with the UVES ones.
Comparison between and the field stars
--------------------------------------
The runs of the abundance ratios with respect to Fe are plotted against metal abundances in Figures \[f:6441a1\] and \[f:6441a2\]. In these plots, filled symbols are for member stars of , while open symbols are for field stars. These figures highlight that the field stars are a quite heterogeneous group. Three stars (\#6003734, 7005308, and 8003092) are moderately metal-poor, with a metal abundance only slightly larger than that of . They have a composition quite similar to that of the cluster, though the last one has a smaller excess of Si. The two first stars have rather large absolute values of the radial velocities: this suggests that they are old, the excess of Oxygen and $\alpha-$elements being possibly due to a reduced contribution by type-Ia SNe. Three other stars (\#7005341, 7004329, and 7004453) have almost solar abundances. They have a low absolute value of the radial velocity. Finally, star \#7004360 has a peculiar, high metal content. Our analysis yielded a significant excess of the $\alpha-$elements Mg and Si, but not of O, Ca, and Ti; also Na is overabundant, but not Al. However, given the difficulties related to the analysis of this spectrum, extremely line rich, we prefer not to comment any more on this star.
Comparison between field (likely bulge) and cluster stars may be interesting and meaningful, because most of the analysis concerns are reduced. In particular, deficiencies of model atmospheres or departures from LTE are expected to act similarly in stars having similar atmospheres. We may for instance compare the stars of with the group of the field “metal-poor” stars, that have similar overall metal abundances (see Table \[t:meanabund\]). The two sets of stars display a quite similar pattern of abundances, both characterized by an excess of $\alpha-$elements; also the apparent deficiency of Ca, Y, and Zr is similar. On this respect we notice that the three field stars (likely bulge) display similar Ca abundances (\[Ca/Fe\]$\sim -0.2$), and similar abundances for Y (\[Y/Fe\]$\sim -0.5$) and Zr (\[Zr/Fe\]$\sim
-0.8$), suggesting that the low Ca, Y and Zr abundances are artifacts of the analysis, rather than real features of these stars. We notice that the present analysis makes the same assumptions of Gratton et al. (2003), where we did not find a similar low Ca abundance in (local) metal-rich dwarfs. It seems more likely that the problem is limited to the analysis of bright giants. All Ca lines used here are strong, with $EWs\geq 100$ mÅ. They form at tiny optical depths, and are potentially sensitive to details of the model atmospheres as well as to departures from LTE. However, the differential comparison with the non metal rich bulge stars suggests that and the field metal-poor stars considered in this analysis have a small but clear excess of Ca, and roughly normal abundances of Y and Zr.
The main difference between the stars of and the field moderately metal-poor stars, concerns the elements involved in the p-capture process of O, Na, Mg, and Al. Even leaving aside the obvious case of the O-poor, Na-rich star \#7004463, the stars of are systematically more O and Mg poor, and Na and Al rich than the similar field stars. This suggests that even the more normal star might have been somewhat polluted by material processed through H-burning at high temperature. This is well confirmed by a comparison of the \[O/Na\] abundance ratios in with those typical of other Globular cluster stars (see Fig. 5 in Carretta et al. 2006). The other analyzed elements appear much more similar between and the moderately metal-poor field stars, although the abundance ratios to Fe are generally higher in the stars of by about 0.05-0.1 dex, which might be explained collectively by assuming some difference in Fe. This might suggest the existence of some subtle difference in the nucleosynthesis, although this result is at the limit of the observational and analysis uncertainty, and we do not give much weight to it.
Among the n-capture elements, stars in have \[Ba/Fe\] abundance ratios similar to that of field stars; the \[Eu/Fe\] abundance ratios are on average larger than the putative metal rich bulge stars (by about 0.2 dex). The larger \[Eu/Ba\] ratios may be again explained as due to larger contributions by core collapse SNe.
Comparison with other bulge objects
-----------------------------------
Comparisons with abundances from the literature must be considered with more caution, due to possible systematic differences in the analysis. In Table \[t:meanabund\] we give average abundances for two other bulge clusters analyzed with a similar method (NGC 6528: Carretta et al. 2001; : Cohen et al. 1999), Baade’s Window (McWilliam & Rich 1994), and for the cluster Terzan 7, likely belonging to the Sagittarius Dwarf Galaxy (Sbordone et al. 2005). The pattern of abundances seen in the various bulge clusters is quite similar. The main differences concern: (i) the light elements involved in the p-capture processes (O, Na, and Mg). However, they may result from small number statistics, given the large star-to-star spread observed within each cluster. (ii) Ca, for which we think our abundance is not representative of the cluster real abundance (the stars discussed here are much cooler than those analyzed in and NGC 6553). (iii) Mn, for which we derive a nearly solar ratio to Fe in NGC 6441, while Carretta et al. derived a low abundance in NGC 6528. This difference in Mn abundance might indicate a different chemical history. Mn is known to be under-abundant in quite metal-rich stars of the Sagittarius dwarf spheroidal (McWilliam et al. 2003); however the composition of (and ) is also clearly different from that of Terzan 7 and other stars in the Sagittarius Dwarf Galaxy, since in these last cases the $\alpha-$elements are not at all overabundant. All these facts suggest that various bulge clusters had different chemical histories, possibly being originated in different fragments of our Galaxy.
Conclusion
==========
This paper presents the first high resolution spectroscopic analysis of individual stars. A total of thirteen RGB stars were observed using FLAMES/UVES, only five of them, on the basis of their measured radial velocities, positions and chemical compositions are very likely members of . The iron abundance measured for the cluster stars is \[Fe/H\]=$-0.39\pm0.04\pm0.05$dex, somewhat higher than those reported in the literature (see e.g. Zinn & West 1984 and Armandroff & Zinn, 1988) which are however based on lower resolution data. In a more recent paper, Clementini et al. (2005) measured the metal abundance of RR Lyrae stars finding an average metallicity of \[Fe/H\]=$-0.41\pm0.06$dex which is perfectly consistent with our results.
Some of the results obtained seem to hint that was characterized by slightly different nucleosynthetic processes. In fact, in the cluster stars O and Mg are systematically lower and Na and Al higher than in the field stars, and the \[O/Na\] ratios measured in are different from those typical of other clusters. We find a very small metallicity spread among the stars in our sample which belong to the cluster, only marginally larger than the level expected on the basis of the observational error, suggesting an homogeneous composition. However, our sample is far too small to be representative of the cluster distribution and, on the other hand, Clementini et al. (2005), find a scatter as high as 0.3dex. The derived homogeneity could be just due to limited sampling and thus no definite conclusion can be drawn on the basis of the present data. The GIRAFFE/FLAMES data presented by Gratton et al. (2006 in preparation) as well as the near-IR NIRSPEC ones in Origlia et al. (2006 in preparation) will hopefully address this issue.
We wish to thank our anonymous referee for his/her comments and suggestions. This research has been funded by PRIN 2003029437 “Continuità e discontinuità nella formazione della nostra Galassia” by the Italian MIUR. This publication makes use of 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 National Aeronautics and Space Administration and the National Science Foundation
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[^1]: Based on data collected at the European Southern Observatory with the VLT-UT2, Paranal, Chile (ESO 073.D-0211)
[^2]: While the velocity difference might appear large we note that is well known to have an extremely large central velocity dispersion (around 17km s$^{-1}$ see e.g. Dubath et al. 1997).
[^3]: This combination of effective temperatures and surface gravity would also be unlikely by evolutionary considerations, since sub-giants are much warmer than $\sim 4000$ K for any reasonable chemical composition. Only pre-main sequence stars could occupy a similar location on the color-magnitude diagram
|
---
abstract: 'We study a model of the motion by mean curvature of an (1+1) dimensional interface in a 2D Brownian velocity field. For the well-posedness of the model we prove existence and uniqueness for certain degenerate nonlinear stochastic evolution equations in the variational framework of , replacing the standard coercivity assumption by a Lyapunov type condition. Ergodicity is established for the case of additive noise, using the lower bound technique for Markov semigroups by Komorowski, Peszat and Szarek [@KPS].'
address: 'Technische Universität Berlin, Institut für Mathematik Stra[ß]{}e des 17. Juni 136, D-10623 Berlin, Germany'
author:
- 'Abdelhadi Es–Sarhir'
- 'Max-K. von Renesse'
title: Ergodicity of Stochastic Curve Shortening Flow in the Plane
---
[^1]
Introduction
============
Motion by mean curvature is a well studied and rich object in geometric PDE theory for which a variety of methods have been developed (see e.g. [@MR1931534] for a survey). In physics it arises as sharp interface limit of the Allen-Cahn equation for the phase field of a binary alloy, describing the motion of the interface between the two phases. Stochastic mean curvature flow was derived heuristically in e.g., [@citeulike:2163102] as a refined model incorporating the influence of thermal noise. In the (d+1)-dimensional graph case the corresponding SPDE is of the form $$d u = \sqrt {1 + |\nabla u|^2}\, {\rm div}\bigl( \frac{\nabla
u}{\sqrt {1 + |\nabla u|^2}} \bigr)\, dt+ B(u, \nabla u) \delta W, \label{grafspde}$$ where $\delta$ stands for Stratonovich or Itô differential, depending on the model. The degeneracy of the drift operator makes a rigorous treatment of this family of models very difficult. Motivated by the deterministic theory Lions and Souganidis introduced a notion of stochastic viscosity solutions [@MR1659958; @MR1799099], but some technical details of this approach are still awaiting full justification [@MR1920103; @caruana-2009]. Existence of weak subsequential limits along tight approximations of stochastic mean curvature flow has been obtained by Yip [@MR1656479] and more recently by Röger and Weber [@webrog].\
In this paper we consider the special case of a (1+1)-dimensional graph interface in an e.g. 2D Brownian velocity field, corresponding to the equation $$d u = \frac{ \partial _x^2 u }{ 1 + (\partial _x u)^2} \, dt+ \sum\limits_{i=1}^{\infty} \phi_i(.,u(.))\,db_t^i. \label{themodel}$$ In the deterministic case this equation is also known as curve shortening flow. Note that the mild solution approach by da Prato-Zabzcyk [@MR1207136] is not applicable because equation is not semilinear, i.e. does not contain a dominating linear component. For the analysis of we first establish an abstract existence and uniqueness result in the classical variational SPDE framework of [@krylovrozovskii] for a certain class of nonlinear stochastic evolution equations, which are not coercive but satisfy an alternative Lyapunov condition. This is then applied to equation which is treated in the Gelfand triple $$H_0^{1} ([0,1]) \subset L^2([0,1]) \subset H^{-1}([0,1]),$$ although the operator $A: H_0^{1}([0,1])\to H^{-1}([0,1])$ $$A u = \frac{ \partial _x^2 u }{ 1 + (\partial _x u )^2}$$ fails to be coercive. By our method we prove well-posedness of , assuming $u_0 \in H^{1}_0$, $\phi_i \in
\Lip([0,1]\times \R)$, $\phi_i(0,.)=\phi_i(1,.)=0$ and, for some finite $\Lambda$, $$\sum_{i=1}^ \infty (\Lip(\phi_i))^2 \leq \Lambda ^2. \label{regcond}$$ The latter condition should be compared to the weaker assumption that for all $z_1, z_2 \in [0,1]\times \R$ $$\sum_{i=1}^ \infty (\phi_i(z_1)-\phi_i(z_2))^2 \leq \Lambda^2 |z_1-z_2|^2, \label{kunitaregcond}$$ which is well-known e.g. in the theory of isotropic flows, where it guarantees the existence of a forward stochastic flow $d\Phi = F(\Phi, dt)$ of homeomorphisms of $[0,1]\times \R_+$ driven by the martingale field $F(z,t) = \sum _{i=1}^\infty \phi_i (z) b^{i}_t $, cf. [@MR1070361 Theorem 4.5.1].
In fact, we show below that the SPDE with noise field satisfying only and initial condition $u_0 \in L^2([0,1])$ still admits a unique generalized solution which is defined by approximation. More precisely, we obtain a unique Markov process $(\hat u_t^x; x\in L^2([0,1]); t\geq 0)$ on $L^2([0,1])$, inducing a Feller semigroup on the space of bounded continuos functions on $L^2([0,1])$ as the unique generalized solution of . However, in view of the poor regularity of the operator $A$, a more explicit characterization of the $L^2([0,1])$-valued process $(\hat u
_t^x)_{t\geq0}$ by some SPDE or even just an associated Kolmogorov operator on smooth finitely based test functions does not seem to be available. This is very similar to the generalized solutions for abstract SPDE with only $m$-accretive drift operators obtained in [@MR1400370] by means of nonlinear semigroup theory. The advantage in the present case is, however, that the variational approach is embedded such that we know the solution $(\hat
u^x_t)_{t\geq 0}$ is strong if holds and the initial condition $x=u_0$ belongs to $H_0^1([0,1])$.
Finally we show the ergodicity of the generalized solution $(\hat u^x_t) $ of in the case of additive noise by verifying the conditions of a recent abstract result by Komorowski, Peszat and Szarek for Markov semigroups with the so-called *e*-property [@KPS Theorem 1]. We point out that [@KPS Theorem 3] does not apply in our sitation because the deterministic flow does not converge to equilibrium locally uniformly with respect to the initial condition. However, for the verification of the lower bound in our case we exploit the fact that the stochastic flow admits a Lyapunov function with compact sublevel sets.
Well-Posedness of certain non-coercive variational SPDE
========================================================
Strong solutions for a class of non-coercive SPDE with regular initial condition
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Although we are mainly interested in the example we shall formulate here a general existence and uniqueness result in the abstract variational framework of [@krylovrozovskii] for stochastic evolution equations, following with only a few changes the excellent presentation in [@Ro]. Let $$V \subset H$$ be a continuous and dense embedding of two separable Hilbert spaces with corresponding inner products $\langle
.,.\rangle_V$ and $\la .,.\ra_H$. Via the Riesz isomorphism on $H$, this induces the Gelfand triple $$V \subset H \subset V^*$$ such that in particular $$_{V^*}\langle u, v\rangle_V = \langle u, v\rangle _H \quad \forall\: u \in H, v\in V.$$ In addition we shall also assume that the inner product $\langle
.,.\rangle_V$ induces a closed quadratic form on $H$. This implies the existence of a densely defined selfadjoint operator $\L :
H\supset D(\L) \to H$ on $H$ such that $V= D(\sqrt \L)$, $\la u, v\ra_V = \la u, \L v\ra_H$ for $u \in V, v \in D(\L)$ and such that the closure of $\L: V\supset D(\L) \to V^*$, still denoted by $\L$, defines an isometry. Moreover we assume that $L$ has discrete spectrum with corresponding eigenbasis $(e_i)_{i\geq n}$, which will be the case if e.g. the embedding $V\subset H$ is compact.
Let $(W(t))_{t\geq 0}$ be a cylindrical white noise on some separable Hilbert space $(U, \langle .,.\rangle _U)$ defined on some probability space $(\Omega,\mathbb P, \mathcal F)$ and let $\mathcal F_t = \sigma(W_s, s\leq t)$ be the associated filtration. For $X=H$ resp. $X=V$ we denote by $L_2(U,X)$ the class of Hilbert-Schmidt mappings from $U$ to $X$, equipped with the Hilbert-Schmidt norm $\|M\|_{L_2(U,X)}^2 = \sum\limits_{i\geq 1}
\langle M u_i, M u_i\rangle_X$, where $(u_i)_{i\geq 1}$ is some orthonormal basis of $U$. Let $$A : V \to V^*, \quad \sigma: V \to L_2(U,V)$$ be measurable maps, then the existence and uniqueness result below applies to $H$-valued Itô-type stochastic differential equations of the form $$\label{sde0}
\left\{
\begin{array}{ll}
du(t)=Au(t)dt+\sigma(u(t))dW_t \\
u(0)=u_0 \in H.
\end{array}
\right.$$ Below we shall work under the following set of assumptions on the coefficients $A$ and $B$.
- (Hemicontinuity) For all $u,\:v,\:x \in V$ the map $$\R \ni \lambda \to \subvstern\langle A(u+\lambda v ), x\rangle_V$$ is continuous.
- (Weak monotonicity) There exists $c_1\in \R$ such that for all $u$, $v\in V$ $$2\ \subvstern \la Au-Av,u-v\ra_V +
\|\sigma(u)-\sigma(v)\|_{L_2(U,H)}^2 \leq c_1 \| u-v\|_H^2$$
- (Lyapunov condition) For $n \in \mathbb N$, the operator $A$ maps $H^n:=\mathop{\rm span}\{e_1, \dots, e_n\}\subset V$ into $V$ and there exists a constant $c_2 \in \R$ such that $$2\ \la Au,u\ra_V + \|\sigma(u)\|_{L_2(U,V)}^2 \leq c_2(1+ \| u
\|_V^2 )\quad \forall u \in H^n, n\in \mathbb N.$$
- (Boundedness) There exists a constant $c_3 \in \R$ such that $$\|A(u) \|\subvstern \leq c_3(1+\| u\|_V).$$
[Note that (H3) replaces the standard coercivity assumption in [@krylovrozovskii] $$2\ \subvstern \la Au,u\ra_V + \|\sigma(u)\|_{L_2(U,H)}^2 \leq c_2
\| u \|_H^2 -c_4 \|u\|_V^\alpha, \quad \forall v \in V \tag{A}$$ for some positive constant $c_4$ and $\alpha >1$. Both conditions (H3) and (A) yield the compactness of the Galerkin approximation in $V$. Condition (A) is used indirectly by applying the finite dimensional Itô formula to the square of the $H$-norm. In our case we use condition (H3) directly by application of the finite dimensional Itô formula to the squared $V$-norm functional.]{}
Basically, a solution to is a $V$-valued process such that the equation holds in $V^*$ in integral form, c.f. [@krylovrozovskii]. The following precise definition is taken from [@Ro].
\[Definition\] A continuous $H$-valued $(\mathcal{F}_t)$-adapted process $(u(t))_{t\in [0,T]}$ is called a solution of , if for its $dt\otimes \PP$-equivalence class $[u]$ we have $[u]\in
L^2([0,T]\times \Omega, dt\otimes \PP,V)$ and $\PP$-a.s. $$u(t)=u(0)+\int_0^tA(\bar{u}(s))\:ds+\int_0^t\sigma(\bar{u}(s))\:dW_s,\quad
t\in[0,T],$$ where $\bar{u}$ is any $V$-valued progressively measurable $dt\otimes\PP$-version of $[u]$.
Now we can state the main result of this section as follows.
\[result\] Assume that conditions (H1)-(H4) hold, then for any initial data $u_0\in V$, there exists a unique solution $u$ to in the sense of Definition \[Definition\]. Moreover, $$\E\left(\sup\limits_{t\in[0,T]}\|u(t)\|_H^2\right)<\infty.$$
The proof follows the standard path of spectral Galerkin approximation, the only difference towards [@krylovrozovskii; @Ro] is the compactness argument, c.f., lemma \[compactnesslemma\] below. To this aim let $(e_n)_{n\geq 1}$ be an orthonormal basis in $H$ of eigenfunctions for the operator $\L:\: H \supset D(\L) \to
H$. Clearly $(e_n)_{n\geq 1}\subset V$ and the set $\operatorname{span}\{e_n,\:\:n\geq 1\}$ is dense in $V$. Let $H_n{\mathrel{\mathrm{\raise0.1ex\hbox{:}\hbox{=}\strut}}}\operatorname{span}\{e_1,\cdots,e_n\}$ and define $P_n:\:V^{\ast}\rightarrow H_n$ by $$P_n y{\mathrel{\mathrm{\raise0.1ex\hbox{:}\hbox{=}\strut}}}\sum\limits_{i=1}^n\:_{V^{\ast}}\la
y,e_i\ra_{V}e_i,\quad y\in V^{\ast}.$$ Then we have $P_n|_H$ is just the orthogonal projection onto $H_n$ in $H$. We shall define the family of $n$-dimensional Brownian motions by setting $$W^n_t{\mathrel{\mathrm{\raise0.1ex\hbox{:}\hbox{=}\strut}}}\sum\limits_{i=1}^n\la W_t,f_i\ra_{U}
f_i=\sum\limits_{i=1}^n B^i(t)f_i,$$
where $(f_i)_{i\geq 1}$ is an orthonormal basis of the Hilbert space $U$. We now consider the $n$-dimensional SDE $$\label{n-sde}
\left\{
\begin{array}{ll}
du^n(t)=P_n A
u^n(t)dt+P_n\sigma(u^n(t))dW^n_t \\
u^n(0,x)=P_n u_0(x) ,
\end{array}
\right\},$$ which is identified with a corresponding SDE $dx(t)= b ^n (x(t)) dt
+ \sigma^n(x(t))d B^n_t$ in $\R^n$ via the isometric map $\R^n \to
H^n, x \to \sum _{i=1}^n x_i e_i$. By [@Ro remark 4.1.2] conditions (H1) and (H2) imply the continuity of the fields $x\to
b^n(x)\in \R^n$ and $x\to \sigma^n(x)\in \R^{n\times n}$. Moreover, assumption (H2) implies $$2 \langle b^n (x) -b^n(y), x-y\ra_{\R^n} + |\sigma^n(x) - \sigma^n(y)|_{L_2(\R^n,\R^n)}^2 \leq c_1 |x-y|^2, \quad \forall x,y \in \R^n$$ and, by the equivalence of norms on $\R^n$, (H3) gives the bound $$2\la b^n(x),x\ra + |\sigma^n(x)|_{L_2(\R^n,\R^n)} \leq c_5 (1+|x|^2),$$ for some $c_5 \in \R$. Hence, equation is a weakly monotone and coercive equation in $\R^n$ which has a unique globally defined solution, cf. [@Ro chapter 3].
\[compactnesslemma\] Let $u^n$ be the solution to equation , then for any $T>0$ we have $$\sup\limits_{0\leq t\leq T}\E\|u^n(t)\|^2_V\leq (c_2 T+\E\bigl(
\|u_0\|^2_V\bigr))e^{c_2T}.$$
Due to the definition of $P_n$ we may write $$\la u^n(t),e_i\ra=\la u^n(0),e_i\ra+\int_0^t\Big\la
\sum\limits_{k=1}^n\:_{V^{\ast}}\la
A(u^n(s)),e_k\ra_{V}e_k\:ds,e_i\Big\ra+\Big\la\int_0^tP_n\sigma(u^n(s))\:dW^n_s,e_i\Big\ra .$$
Hence, the Itô formula in $\R^n$ yields $$\begin{split}
\|u^n(t)\|^2_V&=\|u_0^n\|_V^2+2\int_0^t\la P_n
A(u^n(s)),u^n(s)\ra_{V}\:ds+\int_0^t\|P_n\sigma(u^n(s))\|^2_{L_2(U_n,V)}\:ds\\&+
M^n(t),\quad t\in[0,T],
\end{split}$$ $\PP$-a.s, where $U_n := {\rm span}\,\{f_1, f_2, \cdots, f_n\} \subset U$ and $$M^n(t){\mathrel{\mathrm{\raise0.1ex\hbox{:}\hbox{=}\strut}}}2\int_0^t\la u^n(s),P_n\sigma
(u^n(s))\:dW^n_s\ra_V,\quad t\in[0,T],$$ is a local martingale. We consider a sequence of $\mathcal{F}_t$- stopping times $\tau_j$ with $\tau_j\uparrow +\infty$ as $j\to
+\infty$ and such that $\|u^n(t\wedge \tau_j)(\omega)\|_V$ is bounded uniformly in $(t,\omega)\in [0,T]\times \Omega$, $M^n(t\wedge \tau_j)$, $t\in [0,T]$ is a martingale for each $j\in
\NN$. Then we have $$\begin{split}
\E\|u^n(t\wedge \tau_j)\|^2_V&=\E
\|u_0^n\|_V^2+2\int_0^{t}\E\ein_{[0,\tau_j]}\la P_n
A(u^n(s)),u^n(s)\ra_{V}\:ds\\&+\int_0^{t}\E\ein_{[0,\tau_j]}\|P_n\sigma(u^n(s))\|^2_{L_2(U_n,V)}\:ds.
\end{split} \label{normstep}$$
Now using the definition of the operators $A$ and $P_n$ we can write $$\begin{split}
\la P_n A(u^n(s)),u^n(s)\ra_{V}&=\Big\la
\sum\limits_{i=1}^n\:_{V^{\ast}}\la
A(u^n(s)),e_i\ra_{V}e_i,u^n(s)\Big\ra_{V}\\
&=\sum\limits_{i=1}^n\:_{V^{\ast}}\la A(u^n(s)),e_i\ra_{V}\la
e_i,u^n(s)\ra_{V}.
\end{split}$$
Since $u^n(t)\in H_n$ for $t\in [0,T]$ and $(e_n)_{n\geq
1}\subset V$ by assumption (H3) we can write $$_{V^{\ast}}\la A(u^n(s)),e_i\ra_{V}=\la A(u^n(s)),e_i\ra_H,$$ this yields $$\begin{aligned}
\la P_n A(u^n(s)),u^n(s)\ra_{V}&=\sum\limits_{i=1}^n\:\la
A(u^n(s)),e_i\ra_{H}\la e_i,u^n(s)\ra_{V}\\
&=\sum\limits_{i=1}^n\:\la A(u^n(s)),e_i\ra_{H}\lambda_i \la
e_i,u^n (s)\ra_H\end{aligned}$$ where $\{\lambda_i \geq 0\}$ are the eigenvalues of the operator $L$.
Therefore we have $$\la P_n A(u^n(s)),u^n(s)\ra_{V}=\la A(u^n(s)),u^n(s)\ra_{V}.$$
Hence, the operator $P_n$ may be dropped in the fist integral on the right hand side term of such that by the second part of assumption (H3) $$\begin{split}
\E\|u^n(t\wedge \tau_j)\|^2_V&\leq\E \|u_0^n\|_V^2+c_2
\int_0^{t}(1+\E\|u^n\|_{V}^2) ds.
\end{split}$$
Hence letting $j\to +\infty$ and using Fatou’s lemma we obtain $$\E\|u^n(t)\|^2_V\leq \E
\|u_0^n\|_V^2+c_2\int_0^{t}(1+\E\|(u^n(s))\|_{V}^2)\:ds.$$
Now Gronwall’s lemma yields $$\label{Gr}
\E\|u^n(t)\|^2_V\leq (c_2 T+\E
\|u_0^n\|_V^2)e^{c_2T}.$$
For the estimate of $\E \|u_0^n\|_V^2$, we use the definition of $P_n$ and write $$\begin{split}
\|u_0^n\|_V^2=\|P_n u_0\|^2_V&=\la
P_nu_0,P_nu_0\ra_V=\sum\limits_{i=1}^n\sum\limits_{j=1}^n
\:_{V^{\ast}}\la u_0,e_i\ra_{V}\la e_i,e_j\ra_{V}\:_{V^{\ast}}\la
u_0,e_i\ra_{V}\\
&=\sum\limits_{i=1}^n \lambda_i \la u_0,e_i\ra_{H}^2 \leq \sum\limits_{i=1}^\infty \lambda_i \la u_0,e_i\ra_{H}^2 =\|u_0\|_{V}^2.
\end{split}$$
From here all remaining arguments from [@Ro chapter 4] carry over without change in order to complete the proof of the theorem. To make the paper self-contained we briefly recall the main steps. Let $$K{\mathrel{\mathrm{\raise0.1ex\hbox{:}\hbox{=}\strut}}}L^2([0,T]\times \Omega,dt\otimes\PP,V)\quad \mbox{and} \quad
J{\mathrel{\mathrm{\raise0.1ex\hbox{:}\hbox{=}\strut}}}L^2([0,T]\times \Omega,dt\otimes\PP,L_2(U,H)).$$
Due to the bound (H4) and the reflexivity of $K$ we find a subsequence $n_k\to +\infty$ such that $ u^{n_k}\to \bar{u}$ weakly in $K$ and weakly in $L^2([0,T]\times
\Omega,\:dt\otimes \PP, H)$, $v^{n_k}{\mathrel{\mathrm{\raise0.1ex\hbox{:}\hbox{=}\strut}}}A(u^{n_k})\to v$ weakly in $K^{\ast}$ and $\theta^{n_k}{\mathrel{\mathrm{\raise0.1ex\hbox{:}\hbox{=}\strut}}}P_{n_k}\sigma(u^{n_k})\to \theta$ weakly in $J$. Passing to the limit in one obtains in $V^*$ $$\label{u-formula}
u(t){\mathrel{\mathrm{\raise0.1ex\hbox{:}\hbox{=}\strut}}}u_0+\int_0^tv(s)\dd s+\int_0^t \theta(s)\dd W(s),\quad
t\in[0,T],$$ and in particular $u=\bar{u}\:\:dt\otimes\PP$-a.e. Now the following Itô formula for $\|u_t\|_H $ is crucial (c.f.[@krylovrozovskii]).
\[itotheorem\] Let $u_0\in L^2(\Omega,\mathcal{F}_0,\PP,H)$ and $v\in
L^2([0,T]\times \Omega,dt\otimes\PP,V^{\ast})$, $\theta\in
L^2([0,T]\times \Omega,dt\otimes\PP,L_2(U,H))$, both progressively measurable. Define the continuous $V^{\ast}$-valued process $$u(t){\mathrel{\mathrm{\raise0.1ex\hbox{:}\hbox{=}\strut}}}u_0+\int_0^t v(s)\:ds+\int_0^t\theta(s)\:dW_s,\quad
t\in[0,T].$$
If for its $dt\otimes\PP$-equivalence class $[u]$ we have $[u]\in L^2([0,T]\times \Omega,dt\otimes\PP,V)$, then $u$ is an $H$-valued continuous $\mathcal{F}_t$-adapted process, $$\E\bigl(\sup\limits_{t\in[0,T]}\|u(t)\|_H^2\bigr)<\infty$$ and the following Itô-formula holds for the square of its $H$-norm $\PP$-a.s. $$\label{Ito-formula}
\|u(t)\|_H^2=\|u_0\|_H^2
+2\int_0^t\Big(_{V^{\ast}}\la
v(s),\bar{u}(s)\ra_{V}+\|\theta(s)\|^2_{L_2(U,H)}\Big)\:ds
+2\int_0^t\la
u(s),\theta(s)\:dW_s\ra
$$ for all $t\in[0,T]$, where $\bar u$ is any $V$-valued progressively measurable $dt\otimes\PP$-version of $[u]$.
In view of this implies that $u$ is continuous in $H$, $\E (\sup\limits_{0\leq t\leq T}\|u(t)\|_H^2 )<+\infty$ and $$\label{product}
\E
\left(e^{-{c_1}
t}\|u(t)\|_H^2\right)-\E\left(\|u_0\|^2_H\right)
=\E\bigl(\int_0^te^{-{c_1}s}(2\:_{V^{\ast}}\la
v(s),\bar{u}(s)\ra_V+\|\theta(s)\|^2_{L_2(U,H)}-{c_1}\|u(s)\|^2_H)\dd
s\bigr).
$$
An analogous formula holds true for $(u^{n_k}(t))_{t\geq 0}$. Hence, for $\Phi\in K$, using (H2), $$\begin{split}
\E&\left(e^{-{c_1}
t}\|u^{n_k}(t)\|_H^2\right)-\E\left(\|u^{n_k}_0\|^2_H\right)
\\& \leq \E\left(\int_0^te^{-{c_2}s}\Big(2\:_{V^{\ast}}\la
A(\Phi(s)),u^{n_k}(s)\ra_V+2\:_{V^{\ast}}\la
A(u^{n_k}(s))-A(\Phi(s)),\Phi(s)\ra_V\right.\\
&- \|\sigma(\Phi(s))\|^2_{L_2(U,H)}+2\la
\sigma(u^{n_k}(s)),\sigma(\Phi(s))\ra_{L_2(U,H)}-2 {c_2}\la
u^{n_k}(s),\Phi(s)\ra_H+{c_2}\|\Phi(s)\|_H^2\Big)\dd s\Big).
\end{split}$$ Letting $k\to +\infty$ one concludes that for every nonnegative $\psi\in
L^{\infty}([0,T],\R)$ $$\label{est-apres monotonie}
\begin{split}
&\liminf\limits_{k\to+\infty}\E\left(\int_0^T
\psi(t)(e^{-{c_1}t}\|u^{n_k}(t)\|_H^2-\|u_0^{n_k}\|_H^2)\dd t\right)\\
&\leq \E\left(\int_0^T
\psi(t)\left(\int_0^te^{-{c_1}s}\Big(2\:_{V^{\ast}}\la
A(\Phi(s)),\bar{u}(s)\ra_V+2\:_{V^{\ast}}\la
v(s)-A(\Phi(s)),\Phi(s)\ra_V\right.\right.\\
&- \|\sigma(\Phi(s))\|^2_{L_2(U,H)}+2\la
\theta(s),\sigma(\Phi(s))\ra_{L_2(U,H)}-2 {c_1}\la
u(s),\Phi(s)\ra_H+{c_1}\|\Phi(s)\|_H^2\Big)\dd s\Big).
\end{split}$$ On the other hand, due to the weak lower semicontinuity of the norm in $K$ $$\E\left(\int_0^T\psi(t)\| {u}(t)\|_H^2\dd t\right)\leq
\liminf\limits_{k\to+\infty}\left(\E\int_0^T
\psi(t)\|u^{n_k}(t)\|^2_H\dd t\right).$$
Combining this with and one obtains that $$\label{final}
\begin{split}
\E\bigl(\int_0^T
\psi(t) & \int_0^te^{-{c_1}s} (2\:_{V^{\ast}}\la
v(s)-A(\Phi(s)),\bar{u}(s)-\Phi(s)\ra_V
\\&
+\|\sigma(\Phi(s))-\theta(s)\|^2_{L_2(U,H)}-{c_1}\|u(s)-\Phi(s)\|_H^2)\dd
s\dd t\bigr)\leq 0.
\end{split}$$ Taking $\Phi=\bar{u}$ in we obtain $\theta=\sigma(\bar{u})$. By applying to $\Phi=\bar{u}-\varepsilon \tilde{\Phi} h$ for $\varepsilon>0$ and $\tilde{\Phi}\in L^{\infty}([0,T]\times \Omega,\R)$, $h\in V$ and dividing both sides by $\varepsilon$ and letting $\varepsilon\to 0$, by (H2) and Lebesgue’s theorem we get $$\E\left(\int_0^T \psi(t)\left(\int_0^te^{-{c_1}s}
\tilde{\Phi}(s)\Big(2\:_{V^{\ast}}\la v(s)-A(\bar{u}(s)),h\ra_V\dd
s\right)\dd t\right)\leq 0.$$
By the arbitrariness of $\psi$ and $\tilde{\Phi}$ we conclude that $v=A(\bar{u})$.\
As for the uniqueness consider two solutions $u^{(1)}$ and $u^{(2)}$ of with initial condition $u_0^{(1)} \in V$ and $u_0^{(2)}\in V$ respectively. Applying theorem \[itotheorem\] to $u = u^{(1)} - u^{(2)}$ together with condition (H2) and Gronwall’s lemma $$\E\|u^{(1)}(t)-u^{(2)}(t)\|_H ^2\leq \| u^{(1)}_0 -u^{(2)}_0\|_H^2e^{2c_1t}. \label{contractionest}$$ This implies uniqueness of the solution for given initial state. Theorem \[result\] is proved.
Generalized solutions for initial condition in $H$
--------------------------------------------------
By means of it is possible to construct a unique generalized solution to for initial condition in $u_0 \in H$. In particular this yields a unique Feller process on $H$ which extends the regular strong solutions of .
\[absttractgensol\] Assume (H1) - (H4) , then there exists a unique time homogeneous $H$-valued Markov process $(\hat u_t^x, t \geq 0, x \in H)$ such that $t\to \hat u_t^x$ solves the SPDE in the sense of definition $\eqref{Definition}$ whenever $x=u_0 \in V$. Moerover, $(\hat u^x_t)$ induces a Feller semigroup on $H$, i.e. the space $C_b(H)$ of bounded continuous function on $H$ is invariant under the the operation $\varphi \to P_t\varphi$, where $ P_t \varphi (x) = \E(\varphi (\hat u^x_t)) , x\in H$ for any $t \geq 0$.
For $x \in V \subset H$ define $t \to \hat u^x_t\in H$ as the unique solution to with initial condition $u_0 =x$. For arbitrary $x\in H$, choose a sequence $(x_k)_k $ in $ V$ such that $\| x_k - x \|_H \to 0$, then by the sequence of processes $(t\to \hat u _t^{x_k})_{k\in N}$ is Cauchy in $ C([0,\infty); L^2(\Omega,H))$ with respect to the topology of locally uniform convergence and define $(t \to \hat u_t^x)$ as the unique limit. For $\varphi \in C_b(H)$ define $P_t\varphi (x)$ as above, then obviously yields $$\E\|\hat u^{x}_t -\hat u^y_t\|_H^2 \leq e^{2c_1 t} \|x-y\|^2_H,\quad t\geq 0, \label{contracttwo}$$ which implies that $P_t \varphi \in C_b (H)$ for $\varphi \in
C_b(H)$. To prove that $(\hat u^x_t)^{x\in H}_{t \geq 0}$ is Markov, by the monotone class argument it suffices to show for all $x\in H$ $$\E\bigl(\psi (\hat u^x_t)\cdot \varphi_1 (\hat u^x_{s_1})\cdots \varphi_n (\hat u^x_{s_n})\bigr) = \E\bigl( P_{t-s_n} \psi (\hat u^x_{s_n})\cdot \varphi_1 (\hat u^x_{s_1})\cdots \varphi_n (\hat u^x_{s_n})\bigr), \label{markoveq}$$ for any $0\leq s_1 \leq s_2 \cdots \leq s_n < t $ and $\varphi_1, \dots, \varphi_n, \psi \in C_b(H)\cap \Lip(H)$. By we have $$|P_{t-s}\varphi (x) - P_{t-s}\varphi (y)| \leq e^{c_2(t-s)} \Lip (\varphi)\|x-y\|_H \quad \forall \, \varphi \in \Lip(H),$$ hence will be enough to show for $x \in V$, where it follows by standard arguments from the uniqueness of solutions of , their adaptedness to the filtration $\mathcal F_s$, $s\geq 0$, which for $s\leq t$ is independent of the sigma algebra of increments $\mathcal G_{s,t} =\sigma(W_\sigma-W_s; s\leq \sigma \leq t)$, c.f., [@Ro proposition 4.3.5]. This proves the existence of $(\hat u^x_t; t\geq 0; x\in H)$ as in the claim of the theorem. Trivially, uniqueness of $(\hat u_t^x)$ follows from which holds for any $H$-valued closure of solutions to equation .
Application: Stochastic Curve Shortening Flow in (1+1) Dimension
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Strong solutions for $u_0 \in H^{1,2}([0,1])$ and smooth noise
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Let us now show how we can treat the model rigorously in the case $d=1$, which is also known as curve shortening flow, using the results of the previous section. The simple but essential observation is that for $d=1$ the drift operator in the SPDE above may be written $$\label{drift} Au = \frac{\partial^2 _x u}{1+(\partial _x u )^2} =\partial_x
( \arctan(\partial_x u )),$$ which fits into our slightly modified framework. To this aim let $$H_0^{1} ([0,1]) \subset L^2([0,1]) \subset H^{-1}([0,1]),$$ be the Gelfand triple, which is induced from the Dirichlet Laplacian $L = \Delta$ on $L^2([0,1])$.\
For $u \in H^1_0([0,1])$, let $Au\in H^{-1}([0,1])$ be defined by $$_{H^{-1}_0}\langle Au, v\rangle_{H^{1}_0} = - \int_{[0,1]} \arctan(\px u) \px v dx, \quad \forall v \in H^{1},$$ which is clearly hemicontinuous in the sense of condition (H1), due to the continuity and uniform boundedness of $\zeta\rightarrow\arctan\zeta$. Trivially $A$ is also bounded in the sense of (H4) because $$\| A u \|_{H^{-1}([0,1])} = \sup_{v \in H^{1}_0([0,1]), \|v\|_{H^1_0}\leq 1} \int_{[0,1]} \arctan(\px u) \px v dx \leq (\frac \pi 2)^{1/2}. \label{exboundedness}$$
Moreover, by the monotonicity of $\arctan$ $$_{H^{-1}} \la A u -A v , u -v \ra_{H^{1}} = - \int_{[0,1]} (\arctan(\px u) -\arctan(\px v))(\px u-\px v) dx \leq 0 \label{arctanmon}.$$ The eigenvectors of $L = \Delta_0$ are $e_i= (x\to \sin(i 2\pi x)),
i\in \mathbb N$, hence $Au = \px^2u /(1+(\px u)^2) \in
H^1_0([0,1])$ for any $u \in H^n =\mathop{\rm span}
\{e_1,\dots,e_n\}\subset {H^1_0([0,1])}$. Moreover, $$\langle Au, u \rangle _{H^1_0}= -\int_{[0,1]} \frac{\px^2u }{1+(\px u)^2} \px^2 u(x) dx \leq 0 \quad \forall u\in H^n. \label{energyestimate}$$
Let $(\phi_i)_{i\in \mathbb N }$ denote a sequence of linear independent Lipschitz functions on $[0,1]\times \R$ such that $\phi(0,y) = \phi(1,y) =0$ for all $y \in \R$ and such that the stronger regularity assumption holds for the noise field, and let furthermore $U$ denote the Hilbert space obtained from the closure of the span of $\{\phi_i, i \in \mathbb N\}$ with respect to the inner product $\langle \sum\limits_{i=1}^n{\lambda_i \phi_i}, \sum_{j=1}^m{\eta_j \phi_j}\rangle_U := \sum_{i=1}^{n \wedge m} \lambda_i \eta_i$.\
Define the diffusion operator $B: {H^1_0([0,1])} \to L(U,L^2([0,1])$ by $$B(u)[\phi](x) = \phi(x,u(x)) \in L^2([0,1])$$ Note that $B(u)$ is in fact in $L_2(U, L^2([0,1]))$ since $$\begin{aligned}
\|B(u)(\phi_i)\|_{L^2([0,1])}^2 &= \int_{[0,1]} \phi_i(x,u(x))^2 dx = \int_{[0,1]} |\phi_i(x,u(x))-\phi_i(0,u(0))|^2 dx \\
& \leq (\Lip(\phi_i))^2 \int_{[0,1]} (x^2+u^2(x))dx = (\Lip(\phi_i))^2 (\frac 1 3 + \|u\|^2_{L^2([0,1])}), \end{aligned}$$ such that $$\| B(u)\|_{L_2(U, L^2([0,1]))}^2 = \sum_{i} \|B(u)(\phi_i)\|_{L^2([0,1])}^2 \leq (\frac 1 3 + \|u\|^2_{L^2([0,1])}) \cdot \Lambda^2 \label{exhilbertschmidt}$$
Moreover, $$\begin{aligned}
\| B(u)-B(v) \|_{L_2(U, L^2([0,1]))}^2 & = \smallsum_{i} \| B(u)[\phi_i]- B(v)[\phi_i] \|_{L^2([0,1])}^2 \nonumber \\
& = \smallsum _i \int_{[0,1]} (\phi_i(x,u(x)) - \phi_i(x,v(x)))^2 dx \nonumber \\
&\leq \Lambda^2 \|u-v\|_{L^2([0,1])}^2
\label{h2cond}.\end{aligned}$$ Similarly, $B(u)[\phi] \in {H^1_0([0,1])}$ for $u\in H^{1}_0([0,1])$, and by the chain rule for weakly differentiable functions, $$\begin{aligned}
\|B(u)(\phi_i)\|_{H^1_0([0,1])}^2 &= \int_{[0,1]} (\px \phi_i(x,u(x)))^2 dx \\ \nonumber
&\leq (\Lip(\phi_i))^2\int_{[0,1]}(1+ |\px u(x)|^2) dx = (\Lip(\phi_i))^2 (1 + \|u\|^2_{H^1([0,1])}), \end{aligned}$$ which yields $$\| B(u)\|_{L_2(U, H^1([0,1]))}^2 = \smallsum_{i} \|B(u)(\phi_i)\|_{H^1([0,1])}^2 \leq ( 1 + \|u\|^2_{H^1([0,1])}) \cdot \Lambda^2 \label{growthcond}$$
In view of – we conclude that the conditions (H1) – (H4) are satisfied in the given case with constants $c_1=c_2=\Lambda^2$ and $c_3=\sqrt{ \pi/2}$. Hence, by theorem \[result\] we arrive at the following result.
\[smcfresult\] Assume the regularity condition holds for the noise field, then for any $T>0$ there is a (up to $dt\otimes \mathbb
P$-equivalence in $[0,T]\times \Omega$) unique $H^{1}_0([0,1])$-valued process $(u_t)_{t\in [0,T]}$ solving the SPDE in the sense of definition \[Definition\].
Generalized Markovian solution in $L^2([0,1])$ for non-smooth noise
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Proposition \[absttractgensol\] readily yields generalized solutions for initial condition in $L^2([0,1])$ as follows.
\[markovsmcf1\] Under condition $\eqref{regcond}$ there is a unique $L^2([0,1])$-valued Markov process $(\hat u^x _t, t \geq 0,x\in L^2([0,1]))$ such that $t \to \hat u^x_t $ is a strong solution to the equation when $x=u_0 \in H^{1,2}_0([0,1])$. Moreover, $(\hat
u^x _t)_{t\geq 0}$ induces a Feller semigroup on $C_b(L^2([0,1]))$.
However, noticing that estimates and remain true under the weaker regularity condition , by similar arguments as in the proof of proposition \[absttractgensol\] we arrive at the following well-posedness result for the SPDE under the Kunita-type regularity condition .
\[markovsmcf2\] Under condition $\eqref{kunitaregcond}$ there is a unique $L^2([0,1])$-valued Markov process $(\hat u^x _t; t \geq 0,x\in L^2([0,1]))$ such that for $x \in H^1_0([0,1])$, $ (u^x_t)_{t\geq 0} $ is the limit, in the sense of locally uniform convergence on $C\bigl([0,\infty); L^2(\Omega,\mathbb P ; L^2([0,1]))\bigr)$, of the strong solutions to the SPDE $$d u^{(k)} = \frac{ \partial _x^2 u^{(k)} }{ 1 + (\partial _x u^{(k)})^2} \, dt+ \sum_{i}^k \phi_i(.,u^{(k)}(.))\,db_t^i, \quad u_0^{k} =x.$$ Moreover, $$\label{monotonie}
E\| \hat u^x_t - \hat u^y_t\|_{L^2([0,1])}^2 \leq e^{\Lambda^2t} \|
x - y\|_{L^2([0,1])}^2 \quad \forall x,y \in L^2([0,1]), t\geq 0.$$ In particular, the induced semigroup, $P_t \varphi (x) =
\E(\varphi(\hat u^x_t))$ for measurable $\varphi: H \to \R$, is Feller.
Ergodicity for Stochastic Curve Shorting Flow with Additive Noise
=================================================================
In this final section we show existence and uniqueness of an invariant measure for the generalized $L^2([0,1])$-valued solution $(\hat u^x_t; t\geq 0, x\in L^2([0,1]))$ obtained in proposition \[markovsmcf1\] for the SPDE in the additive noise case, i.e. when $$du = \frac{\partial_x ^2 u }{1+ (\partial_x u)^2} dt + Q dW_t, \quad u(0)=u_0 \in H^{1,2}_0([0,1]), \label{constantdriftspde}$$ where $W$ is cylindrical white noise on some abstract Hilbert space $U$ and $Q \in L_2(U, H^{1,2}_0([0,1]))$. As an example consider the case of $U=L^2([0,1])$ and $Q = (-\Delta)^{-\beta}$ for $\beta > 3/4$, with $\Delta$ being the Dirichlet Laplacian on $[0,1]$.
Note also that for additive noise the condition (H2) is satisfied with $c_1 =0$. As a consequence of , the Feller semigroup on $L^2$ induced from the generalized solutios $\hat u$ of by $P_t \varphi (x) = \E(\varphi (\hat u^x_t))$ has the so-called $\textit {e}$-property [@KPS], i.e. for all bounded Lipschitz continuous functions $\varphi : L^2 \mapsto \R$ $$|P_t\varphi(x) - P_t\varphi(y) | \leq \mbox{Lip}(\varphi) \|x - y \| \quad \forall x, y \in L^2. \label{eprop}$$
\[ergodicthm\] *Let $(P_t)_{t\geq 0}$ denote the Feller semigroup on $L^2([0,1])$ corresponding to the generalized solution to , then $(P_t)$ is ergodic, i. e. there is a unique $(P_t)$-invariant probablity measure $\mu$ on $L^2([0,1])$. In particular, $\lim_{t \to \infty} \frac 1 t \int_0^t \langle P_t \varphi ,\nu\rangle = \langle \varphi ,\mu\rangle$ for any Borel probability measure $\nu \in \mathcal M_1 (L^2([0,1]))$ and any bounded continuous $\varphi :L^2([0,1]) \mapsto \R$.*
Let $ \qt(x, \cdot ) {\mathrel{\mathrm{\raise0.1ex\hbox{:}\hbox{=}\strut}}}\frac{1}{T}\int_0^T\mu_{\hat u_t}\:dt $, where $\mu_{\hat u_t}$ denotes the distribution at time $t$ of the generalized solution $\hat u^x_t$ to with initial conditon $u_0=x\in L^2$.
\[tight\] For any $x \in L^2$ the family of measures $
\left\{\qt(x,\cdot),\: T\geq 1\right\}$ is tight on $L^2([0,1])$.
*Proof.* Assume first that $ \in H^{1,2}_0([0,1])$. In view of $$|\xi|-\alpha\leq\operatorname{arctg}\xi\cdot\xi\leq \beta+|\xi|,\quad \xi\in
\R,\quad \alpha,\:\beta>0$$ it holds that $$\begin{aligned}
_{H^{-1}}\la Av,v\ra_{H^1}&=-\int_0^1 \operatorname{arctg}(\partial _x v) \cdot \partial _x v \:dx\leq -\int_0^1|\partial _x v |\:dx+\alpha \nonumber \\
&\leq-c\|v\|_{W^{1,1}(0,1)}+\alpha
\label{recurrent}\end{aligned}$$ for some $c>0$, by Poincaré inequality.
Let now $t\to u_t$ be the solution to equation with regular initial condition $x=u_0 \in
H^{1,2}_0([0,1])$, then theorem \[itotheorem\] holds. Hence by the Itô formula for $\| u_t \|_{L^2([0,1]}^2$ and we have $$\begin{split}
\E\|u(t)\|^2&= \E\|u(0)\|^2+2\E\int_0^t\: _{V^{\ast}}\la
A(\ub(s)),\ub(s)\ra_{V}
ds+\E\int_0^t\|Q\|_{\mathcal{L}_{HS}(U,H)}^2\:ds\\
&\leq \E\|u(0)\|^2-c \:\E\int_0^t\|\ub(s)\|_{W^{1,1}(0,1)}+Dt
\end{split}$$ where $D{\mathrel{\mathrm{\raise0.1ex\hbox{:}\hbox{=}\strut}}}\alpha+\|Q\|_{\mathcal{L}_{HS}(U,H)}^2$. In particular, $$\E\left(\frac{1}{t}
\int_0^t\|\ub(s)\|_{W^{1,1}(0,1)}\:ds\right)\leq
\frac{1}{c}\Big(\E\|x\|^2+D\Big)\quad \forall t\geq 1.
\label{chebyshevbound}$$ Since the functional $L^2 ([0,1]) \ni u \to \| u \|_{W^{1,1}(0,1)}
\in \R \cup \{ \infty\}$ has compact sublevel sets in $L^2([0,1])$, the claim follows for regular initial condition $x=u_0 \in H^{1,2}_0([0,1])$.\
For the tightness of $\qt(x, .)$ with general $x \in L^2$ recall (e.g. [@Par Remark on p. 49]) that it is sufficient (and necessary) to find for arbitrary $\epsilon>0, \delta>0$ a finite union of $\delta$-balls $S_\delta = \bigcup_{\dotsk}
B_{\delta}(x_i) \subset L^2$ such that $$\qt(x, S_\delta ) > 1- \epsilon \quad \forall \,T>1.$$ To this aim choose $z \in B_{\delta \epsilon/4}(x)\cap
H^{1,2}_0(0,1)$ and a finite union of $\delta/2$-balls $S_{\delta/2}= \bigcup\limits_{\dotsk} B_{\delta/2}(x_i) $ such that $\qt(z,S_{\delta/2}) \geq 1 -\frac{\epsilon}{2}$. Let $S_\delta =
\bigcup\limits_{\dotsk} B_{\delta }(x_i) $ and choose a bounded Lipschitz function $\varphi $ on $L^2$ with $\chi_{S_{\delta/2}}
\leq \varphi \leq \chi_{S_\delta}$ and $\mbox{Lip}(\varphi) \leq
\frac 2 \delta$. Hence, using , for all $T>1$ $$\begin{aligned}
\qt(x, S_{\delta}) \geq \frac 1 T \int_0^T P_s \varphi (x) ds & \geq \frac 1 T \int_0^T P_s \varphi (z) ds - \frac 2 \delta \|x-z\| \\
&
\geq \qt(z, S_{\frac \delta 2}) - \frac{2\|x -z\|}{\delta}>
1-\epsilon. \tag*{$\Box$}\end{aligned}$$
\[detlemm\] For $x\in L^2(0,1)$, let $( v^ x(t))_{t\geq 0}$ the (generalized) solution of corresponding to $Q=0$. Then it holds $$\lim\limits_{t\to +\infty}\|v^x(t)\|=0.$$
First, we consider the case where the initial data $v_0\in
C_0^{\infty}(0,1)$ (space of $C^{\infty}$-differentiable function compactly supported in $[0,1]$. We set $M{\mathrel{\mathrm{\raise0.1ex\hbox{:}\hbox{=}\strut}}}\|v^\prime_0\|_{\infty}$ and define a function $h(t)$ with
- $h$ is of class $C^{\infty}(\R)$ and satisfies $$h(t)=\arctan t \quad \mbox{for}\:\:|t|\leq M$$ $$|h(t)|\leq |t|,\quad t\in \R.$$
- $h^\prime$ is a bounded function on $\R$ satisfies $\inf _{x\in \R}h^\prime(x)\geq \mu>0$ for a positive constant $\mu$.
- $h^{\prime\prime}$ is a bounded function on $\R$.
For $T>0$ fixed, consider the equation $$\label{h} \left\{
\begin{aligned}
dv(t) & = (h(v_x(t)))_x\,dt ,\\
v(0)& = v_0.\\
\end{aligned}
\right.$$ Following a similar argument as in [@taniguchi] and a maximum principle for uniformly parabolic equation we can prove that the classical solution $v$ of satisfies $$\sup\limits_{0\leq t\leq T}\|v_x\|_{\infty}\leq M.$$ Hence from the construction of $h$ we deduce that this solution is also the solution of with $Q=0$ corresponding to the initial data $v_0\in C^\infty(0,1)$. Now we remark that for the function $z\mapsto \arctan z$ we can write $$\arctan z=k(z)\cdot z\quad \mbox{for all $z\in \R$},$$
for some positive decreasing function $k$ on $\R$. Therefore by using the energy estimate for the function $v(t)$ we can write $$\begin{split}
\frac 12 \frac{d}{dt}\|v(t)\|^2&=-\la \arctan v_x(t), v_x(t)\ra_{L^2(0,1)}\\
&\leq -\inf\limits_{z\in B(0,M)}k(z)\:\|v_x(t)\|^2\\ &\leq
-\inf\limits_{z\in B(0,M)}k(z)\:\|v(t)\|^2.
\end{split}$$
Thus we obtain $$\|v(t)\|^2\leq e^{-2 t\inf\limits_{z\in B(0,M)}k(z)}\|v_0\|^2.$$
This implies the statement of the lemma for regular initial datum $v_0$. For general $v_0\in L^2(0,1)$ we proceed by approximation and let $v_0^n$ a sequence of functions in $C^\infty _0(0,1)$ which converges to $v_0$ in $L^2(0,1)$ for $n\to +\infty$. For $n\geq 0$ we denote by $v_n(t)$ the solution corresponding to the initial condition $v_0^n$. By using the fact that $v_n(t)\to 0$ as $t\to 0$ and a triangle inequality argument we deduce the statement of the lemma for general initial datum $v_0\in L^2(0,1)$.
\[stablem\] For $x\in L^2(0,1)$, let $( \hat{v}^ x(t))_{t\geq 0}$ the (generalized) solution of corresponding to $Q=0$. Then for every $x \in L^2, T>0$ and $\epsilon >0$, it holds that $$\PP(\|\hat u^x_T -\hat v^x_T\|< \epsilon ) >0.$$
*Proof.* First we suppose that $x\in V$ and denote by $(v^x_t)_{t\geq 0}$ the solution corresponding to with $Q=0$. We write $$z(t)= u(t)-v(t),\quad t\geq 0.$$
Then the process $z(t)_{t\geq 0}$ solves the equation
$$\left\{
\begin{array}{lll}
dz(t)=(A u(t)-Av(t))dt+Q dW_t\\
z(0)=0.
\end{array}
\right.$$
We set $$z(t)=y(t)+QW_t.$$ Then we have
$$dy(t)=(Au(t)-Av(t))\:dt.$$
Therefore,
$$\begin{split}
\frac 1 2 \frac{d}{dt}\|y(t)\|^2&= \subvstern\la
Au(t)-Av(t),y(t)\ra_{V}\:dt\\
&=\subvstern\la Au(t)-Av(t),z(t)\ra_V\:dt-\subvstern\la
Au(t)-Av(t),QW_t\ra_{V}\\
&\leq 2\Big(\frac{\pi}{2}\Big)^{\frac 12}\|QW_t\|_V\leq
2\Big(\frac{\pi}{2}\Big)^{\frac 12} \|QW_t\|_V.
\end{split}$$
Where we used the monotonicity of $A$ and to obtain the estimate in the last line. Thus we deduce for $0\leq
t\leq T$
$$\|y(t)\|\leq c\:T\sup\limits_{0\leq t\leq T}\|QW_t\|_V,$$ for some positive constant $c$. We now use the splitting of $z(\cdot)$ and the Poincaré inequality to obtain for $0\leq t\leq T$ $$\label{zestimate}
\|z(t)\|\leq (c\: T+\frac 1 2 )\sup\limits_{0\leq t\leq T}\|QW_t\|_V.$$
For the case where $x\in H$ we proceed by approximation and use the uniform bound to obtain the same estimate as in for the process $z(t)=\hat u^x(t)-\hat v^x(t)$, $x\in H$. Since $Q$ is a Hilbert-Schmidt operator from $U$ to $V$, $(QW_t)_{t\geq 0}$ is a continuous Gaussian random process with values in $V$. Hence, for all $\delta>0$ $$\PP\:\Big(\sup\limits_{0\leq t\leq T}\|QW_t\|_V<\delta\Big)>0.$$
Now let $\ve>0$ and take $\delta>0$ such that $(cT +1/2)\delta <\ve$. Then $$\PP\:\Big(\|z(t)\|<\ve\Big)>\PP\:\Big(\sup\limits_{0\leq t\leq
T}\|QW_t\|_V<\delta\Big)>0.\tag*{$\Box$}$$
\[lowerbd\] *For every $\delta >0$ and every $x\in L^2([0,1])$ it holds that $$\liminf_{T\to \infty } \qt(x, B_\delta(0)) >0.$$*
*Proof.* We proceed in three steps. Let $\delta >0$ and $x \in L^2([0,1])$ be given.\
**Step 1.** For $R>0$ let $C_R = \{ u \in L^2| u \in
W^{1,1}_0(0,1) , \|u\|_{1,1} \leq R\} $, which is a compact subset of $L^2([0,1])$. From and Chebychev’s inequality we deduce $$\qt (0,L^2([0,1])\setminus C_R) \leq \frac {c}{R} \quad \forall\: T >1.$$ Hence we may pick some $R>0$ such that $ \qt(0,C_R) >\frac 3 4 $ for all $T>1$. From now we omit the subscript $R$, i.e. $C=C_R$.\
**Step 2.** Claim: There is some $\epsilon_1 >0$, a $\gamma_1 >0$ and a finite sequence $T_1, \cdots, T_k $, $T_i >0$ such that $$\frac 1 k \sum _{i = 1, \dots, k} P_{T_i}(x,B_\delta(0)) >\gamma_1 \quad \forall\: x \in C_{\epsilon_1},$$ where $ C_{\epsilon_1} = \{ u \in L^2([0,1]) \,| \:\, d_{L^2}(u, C)
< \epsilon_1\}$ and $P_T(x,\cdot)$ the transition probability corresponding to $(\hat u^x(t))_{t\geq 0}$ at time $T$. In fact, by lemma \[detlemm\] for each $x \in L^2([0,1])$ there exists a $T_x$ and a $r_x>0$ such that $\hat v^x_{T_x} \in B_{\delta/4}(0) $. For $T>0$ and $\delta
>0$ let $$D(x, T, \delta){\mathrel{\mathrm{\raise0.1ex\hbox{:}\hbox{=}\strut}}}\PP\{ \| \hat v^x_T -\hat u^x_T\|_{L^2([0,1])} \leq \delta\} ,$$ which is strictly positive by lemma \[stablem\]. Hence it follows that $P_{T_x}(x,B_{\frac \delta 2}(0)) \geq D(x,T_x,\delta/4)=: \gamma_x >0$. Similarly as in the second part of proposition \[tight\] we may use to deduce that for each $x \in L^2([0,1])$ there exists $r_x>0$ such that $P_{T_x}(y,B_\delta(0)) > \gamma_x /2$ for all $y \in B_{r_x}(x)$. Since $C$ is compact we may select a finite sequence $(x_i, r_i)$, $\dotsk$, such that $C \subset \bigcup_\dotsk B(x_i, r_i)$. Setting $T_i := T_{x_i}$ the claim follows with $\epsilon_1 := \min_\dotsk r_i $ and $\gamma_1 := \min_\dotsk \gamma_i /2k$.
**Step 3:** Choose $\rho >0$ such that $$\qt(x, C_{\epsilon_1}) > \frac 1 2 \quad \forall\: x \in B_{\rho}(0).$$ This is possible by a similar argument as in the second part proposition \[tight\]. Finally, by analogous reasons as in step 2, we may find some $T_0>0$ and $\gamma_2>0$ such that $P_{T_0}(x,B_\rho(0))> \gamma_2$.\
Hence, $$\begin{aligned}
\liminf_T & \qt(x, B_\delta (0)) = \liminf_{T} \frac 1 T \int_0^T P_{s} (x, B_\delta (0)) ds\\
& = \liminf_{T} \frac 1 k \sum _{\dotsk} \frac 1 T \int_0^T P_{s+T_i +T_0} (x, B_\delta (0)) ds \\
& = \liminf_{T}\frac 1 k \sum _{\dotsk}\frac 1 T \int_0^T \int_{L^2([0,1])}\int_{L_2([0,1])} P_{T_i}(z,B_\delta(0)) P_{s}(y,dz) P_{T_0} (x, dy) ds \\
& \geq \liminf_{T} \frac 1 T \int_0^T \int_{B_\rho(0)}\int_{C_{\epsilon_1}} \frac 1 k \sum _{\dotsk} P_{T_i}(z,B_\delta(0)) P_{s}(y,dz) P_{T_0} (x, dy) ds\\
& \geq \gamma_1 \liminf_{T} \frac 1 T \int_0^T \int_{B_\rho(0)}
P_{s}(y,C_{\epsilon_1}) P_{T_0} (x, dy) ds \intertext{which, by
Fatou's lemma is bounded from below by }
& \geq \gamma_1 \int_{B_\rho(0)} \liminf_{T} \frac 1 T \int_0^T P_{s}(y,C_{\epsilon_1}) P_{T_0} (x, dy) ds\\
& = \gamma_1 \int_{B_\rho(0)} \liminf_{T} \qt(y,C_{\epsilon_1}) P_{T_0} (x, dy) ds \\
& > \frac 1 2 \gamma_1 P_{T_0}(x, B_\rho(0)) > \frac 1 2 \gamma_1
\gamma_2>0. \tag*{$\Box$}\end{aligned}$$ In view of and proposition \[lowerbd\], Theorem \[ergodicthm\] is now a consequence of [@KPS Theorem 1], where $\mathcal T = L^2([0,1])$ according to proposition \[tight\].
\#1[0=]{} \#1[0=]{}
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[^1]: The authors acknowledge support from the DFG Forschergruppe 718 “Analysis and Stochastics in Complex Physical Systems”.
|
---
abstract: |
In this paper we generalize the algebraic density property to not necessarily smooth affine varieties relative to some closed subvariety containing the singular locus. This property implies the remarkable approximation results for holomorphic automorphisms of the Andersén-Lempert theory. We show that an affine toric variety $X$ satisfies this algebraic density property relative to a closed ${\ensuremath{\mathrm{T}}}$-invariant subvariety $Y$ if and only if $X\setminus Y \neq
{\ensuremath{\mathrm{T}}}$. For toric surfaces we are able to classify those which posses a strong version of the algebraic density property (relative to the singular locus). The main ingredient in this classification is our proof of an equivariant version of Brunella’s famous classification of complete algebraic vector fields in the affine plane.
address:
- 'Mathematisches Institut, Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland.'
- 'Mathematisches Institut, Universität Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland.'
- 'Instituto de Matemática y Física, Universidad de Talca, Casilla 721, Talca, Chile.'
author:
- Frank Kutzschebauch
- Matthias Leuenberger
- Alvaro Liendo
bibliography:
- 'density.bib'
title: The algebraic density property for affine toric varieties
---
[^1]
Introduction
============
A remarkable property of the Euclidean space of dimension at least two, that to a great extent compensates for the lack of partition of unity for holomorphic automorphisms, was discovered by Andersén and Lempert in early 1990’s [@An90; @AnLe92], see also the work by Forstnerič and Rosay [@FoRo93]. Since then, the theory of Stein manifolds with very large holomorphic automorphism group is called Andersén-Lempert theory.
The property was formalized by Varolin who named it the density property (DP). A Stein manifold $X$ has the DP if the Lie algebra generated by complete holomorphic vector fields is dense (in the compact-open topology) in the space of all holomorphic vector fields on $X$. Recall that a vector field is called complete if its flow exits for all complex time and all initial conditions.
The DP allows to construct (global) automorphisms of $X$ with prescribed local properties. More precisely, any local phase flow on a Runge domain in $X$ can be approximated by (global) automorphisms. This has remarkable applications for geometric questions in complex analysis, we refer the reader to survey articles [@Ro99; @KK11; @Ku14] and the recent book [@Forst11]. For smooth affine algebraic varieties, the algebraic density property (ADP) was also introduced by Varolin. The ADP implies the DP, therefore it is commonly used as a tool to prove the DP.
In this paper we generalize the ADP to not necessarily smooth affine varieties relative to some closed subvariety containing the singular locus as follows: Let $X$ be an affine algebraic variety and let ${\ensuremath{X^{\mathrm{sing}}}}$ be the singular locus. We also let $Y\subseteq X$ be an algebraic subvariety of $X$ containing ${\ensuremath{X^{\mathrm{sing}}}}$ and let $I=I(Y)\subseteq {\ensuremath{\mathbb{C}}}[X]$ be the ideal of $Y$. Let ${\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,Y)$ be the ${\ensuremath{\mathbb{C}}}[X]$-module of vector fields vanishing in $Y$, i.e., ${\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,Y)=\{\partial \mid \partial({\ensuremath{\mathbb{C}}}[X])\subseteq I\}$. Let ${\ensuremath{\operatorname{Lie}_{\mathrm{alg}}}}(X,Y)$ be the Lie algebra generated by all the complete vector fields in ${\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,Y)$.
\[ADP\] We say that $X$ has the strong ADP relative to $Y$ if ${\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,Y) =
{\ensuremath{\operatorname{Lie}_{\mathrm{alg}}}}(X,Y)$. Furthermore, we say that $X$ has the ADP relative to $Y$ if there exists $\ell\geq 0$ such that $I^\ell{\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,Y)
\subseteq {\ensuremath{\operatorname{Lie}_{\mathrm{alg}}}}(X,Y)$. With this definition, the ADP relative to $Y$ with $\ell=0$ is just the strong ADP relative to $Y$. If we let $Y={\ensuremath{X^{\mathrm{sing}}}}$ we simply say that $X$ has the strong ADP or the ADP, respectively.
Except for the fact that we consider not necessarily smooth varieties, the strong ADP is a version of Varolin’s Definition 3.1 in [@varolin01] of DP for the Lie subalgebra of vector fields vanishing on $Y$. Whereas for $\ell>0$ our property is slightly weaker than Varolin’s definition since we generate the Lie subalgebra of vector fields vanishing on $Y$ of order at least $\ell$ using complete vector fields vanishing on $Y$ of possibly lower order than $\ell$. Still this version of the ADP has the same remarkable consequences as in Varolin version of ADP for the group of holomorphic automorphisms of $X$ fixing $Y$ pointwise (see Theorem \[AL-Theorem\]).
In this paper we investigate the ADP for toric varieties. Our first main result is the following theorem (see Theorem \[finalthm\]).
Let $X$ be an affine toric variety of dimension at least two and let $Y$ be a ${\ensuremath{\mathrm{T}}}$-invariant closed subvariety of $X$ containing ${\ensuremath{X^{\mathrm{sing}}}}$. Then $X$ has the ADP relative to $Y$ if and only if $X\setminus Y\neq {\ensuremath{\mathrm{T}}}$.
Recall that every smooth affine toric variety is isomorphic ${\ensuremath{\mathbb{C}}}^k\times ({\ensuremath{\mathbb{C}}}^*)^{n-k}$. A special case of our theorem where $X={\ensuremath{\mathbb{C}}}^n$ and $Y$ is the union of up to $n-1$ coordinate hyperplanes has been already proven by Varolin [@varolin01].
It is well known that every affine toric surface different from $\C^*\times\C$ or $\C^*\times\C^*$ is obtained as a quotient of $\C^2$ by the action of a cyclic group. Let $d>e$ be relatively prime positive integers. We denote by $V_{d,e}$ the toric surface obtained as the quotient of $\C^2$ by the $\Z_d$-action $\zeta\cdot(u,v) = (\zeta u, \zeta^e v)$, where $\zeta$ is a primitive $d$-th root of unity. The following theorem is our second main result (see Corollary \[Z-ADP\]).
$V_{d,e}$ has the strong ADP if and only if $e$ divides $d+1$ and $e^2 \neq d + 1$.
Furthermore, for every affine toric surface our methods allow to determine the values of $\ell$ from Definition \[ADP\] for which $I^\ell{\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,{\ensuremath{X^{\mathrm{sing}}}}) \subseteq {\ensuremath{\operatorname{Lie}_{\mathrm{alg}}}}(X,{\ensuremath{X^{\mathrm{sing}}}})$. The main ingredient in the proof of this theorem is an equivariant version of Brunella’s famous classification of complete algebraic vector fields in the affine plane (see [@Br3]) or, equivalently, classification of complete algebraic vector fields on affine toric surfaces (see Theorem \[thmlist\]). This result might be of independent interest.
Vector fields and the algebraic density property
================================================
In this section we prove a general method for establishing the ADP that we later will use to show the ADP for toric varieties.
Let $X$ be an affine algebraic variety and $Y$ be a subvariety containing ${\ensuremath{X^{\mathrm{sing}}}}$.
1. Let ${\ensuremath{\operatorname{Aut}}}(X,Y)$ be the subgroup of automorphism of $X$ stabilizing $Y$. We say that $X$ is homogeneous with respect to $Y$ if ${\ensuremath{\operatorname{Aut}}}(X,Y)$ acts transitively on $X\setminus Y$.
2. We also let $x_0\in {\ensuremath{X^{\mathrm{reg}}}}$. A finite subset $M$ of the tangent space $T_{x_0}X$ is called a generating set if the image of $M$ under the action of the isotropy group of $x_0$ in ${\ensuremath{\operatorname{Aut}}}(X,Y)$ generate the whole tangent space $T_{x_0}X$.
The following is our main tool to establish the ADP for toric varieties. It is a generalization of [@KaKu08 Theorem 1].
\[thm\] Let $X$ be an algebraic variety homogeneous with respect to some subvariety $Y\supseteq {\ensuremath{X^{\mathrm{sing}}}}$. Let also $L$ be a finitely generated submodule of the ${\ensuremath{\mathbb{C}}}[X]$-module ${\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,Y)$ of vector fields vanishing on $Y$. Assume that $L\subseteq{\ensuremath{\operatorname{Lie}_{\mathrm{alg}}}}(X,Y)$. If the fiber of $L$ over some $x_0\in X\setminus Y$ contains a generating set, then $X$ has the ADP relative to $Y$.
Let $\{\partial_i\}$ be a finite set of vector fields in $L$ such that $\{\partial_i[x_0]\}$ is a generating set. Let now $\{\beta_j\}\subseteq {\ensuremath{\operatorname{Aut}}}(X,Y)$ be a finite collection of automorphisms fixing $x_0$ such that $\{\beta_j^*(\partial_i)[x_0]\}$ span the tangent space at $x_0$. Since change of coordinates does not change completeness of a vector field, for $\beta\in {\ensuremath{\operatorname{Aut}}}(X,Y)$, the finitely generated module $L_\beta=\beta^*(L)$ is again contained in ${\ensuremath{\operatorname{Lie}_{\mathrm{alg}}}}(X,Y)$. By replacing $L$ with $\bigoplus_{j} L_{\beta_j}$, we can assume that $\{\partial_i[x_0]\}$ span the tangent space at $x_0$.
We let $A_1=\{x\in X\setminus Y\mid {\ensuremath{\operatorname{span}}}(\partial_i[x])\neq
T_{x}X\}$. We also let $A_1=\bigcup A_1^j$ be the decomposition of $A_1$ in irreducible components and we pick $x_j\in A_1^j$. Since $X$ is homogeneous with respect to $Y$, we can choose $\alpha_j\in
{\ensuremath{\operatorname{Aut}}}(X,Y)$ sending $x_0$ to $x_j$. We also put $\alpha_0=\operatorname{Id}$. Let now $$A_2=\big\{x\in X\setminus
Y\mid {\ensuremath{\operatorname{span}}}\{\alpha_j^*(\partial_i)[x]\mid \forall i,j\}\neq
T_{x}X\big\}\,.$$ By construction $\dim A_1>\dim A_2$ and so we can proceed by induction on dimension to obtain a finite collection of automorphisms $\alpha_j\in {\ensuremath{\operatorname{Aut}}}(X,Y)$ such that the collection $\{\alpha_j^*(\partial_i)[x]\}$ span the tangent space at every point $x\in X\setminus Y$.
We let $E=\bigoplus_{j} L_{\alpha_j}$. With the same argument as before, $E$ is a finitely generated ${\ensuremath{\mathbb{C}}}[X]$-submodule of ${\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,Y)$ contained in ${\ensuremath{\operatorname{Lie}_{\mathrm{alg}}}}(X,Y)$. By construction, we have that the fiber of $\widetilde{E}:={\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,Y)\big/E$ at every $x\in
X\setminus Y$ is trivial. Hence, the support of $\widetilde{E}$ is contained in $Y$.
We define $$J=\operatorname{Ann}_{{\ensuremath{\mathbb{C}}}[X]}\widetilde{E}:=\left\{f\in {\ensuremath{\mathbb{C}}}[X]\mid
fa=0\mbox{ for all } a\in \widetilde{E}\right\}\,.$$ By construction $J\widetilde{E}=0$. This yields $J{\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,Y)\subseteq E$. Furthermore, by [@Har77 Ch. II Ex 5.6] we have that $V(J)\subseteq Y$. Recall that $I$ is the ideal of $Y$ and let $J'=J\cap I$ so that $V(J')=Y$. Let now $a_i$ be a finite set of generators of $i$. Since $\operatorname{rad}(J')=I$, we have that there exists $\ell_i$ such that $a_i^{\ell_i}\in J$ for all $i$. Letting $\ell=1+\sum_i(\ell_i-1)$ we obtain $$I^\ell\subseteq J'\subseteq J\quad\mbox{and so}\quad
I^\ell{\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,Y)\subseteq J{\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,Y)\subseteq E\subseteq
{\ensuremath{\operatorname{Lie}_{\mathrm{alg}}}}(X,Y)\,.$$ Hence the theorem follows.
The algebraic density property for affine toric varieties
=========================================================
We first recall the basic facts from toric geometry that will be needed in this section. They can be found in any text about toric geometry such as [@Fu93; @Oda88; @CLS].
Let $M$ and $N$ be mutually dual lattices of rank $n$ with duality pairing $M\times N\rightarrow {\ensuremath{\mathbb{Z}}}$, where $(m,p)\mapsto \langle
m,p\rangle=p(m)$. We also let $M_{\ensuremath{\mathbb{Q}}}=M\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{Q}}}$ and $N_{\ensuremath{\mathbb{Q}}}=N\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{Q}}}$. Letting ${\ensuremath{\mathrm{T}}}$ be the algebraic torus ${\ensuremath{\mathrm{T}}}={\ensuremath{\operatorname{Spec}}}{\ensuremath{\mathbb{C}}}[M]=N\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{C}}}^*$. A toric variety is a normal variety endowed with an effective action of ${\ensuremath{\mathrm{T}}}$ having an open orbit. Since the ${\ensuremath{\mathrm{T}}}$-action is effective, the open orbit is equal to ${\ensuremath{\mathrm{T}}}$.
It is well known that affine toric varieties can be described by means of strongly convex polyhedral cones (pointed cones) in the vector space $N_{\ensuremath{\mathbb{Q}}}$. Indeed, let $\sigma$ be a pointed cone in $N_{\ensuremath{\mathbb{Q}}}$, then $X_\sigma={\ensuremath{\operatorname{Spec}}}{\ensuremath{\mathbb{C}}}[\sigma^\vee\cap M]$ is an affine toric variety and every affine toric variety arises this way. Here ${\ensuremath{\mathbb{C}}}[\sigma^\vee\cap
M]$ is the semigroup algebra ${\ensuremath{\mathbb{C}}}[\sigma^\vee\cap M]=\bigoplus_{m\in
\sigma^\vee\cap M}{\ensuremath{\mathbb{C}}}\chi^m$. In the following, we denote $\sigma^\vee\cap M$ by $\sigma^\vee_ M$.
There is a one to one correspondence between the faces $\tau$ of the cone $\sigma$ and the orbits ${\ensuremath{\mathcal{O}}}(\tau)$ of the ${\ensuremath{\mathrm{T}}}$-action on $X_\sigma$ (usually called the Orbit-Cone correspondence). The dimension of an orbit is given by $\dim {\ensuremath{\mathcal{O}}}(\tau)={\ensuremath{\operatorname{rank}}}N-\dim \tau$ and its closure is given by $\overline{{\ensuremath{\mathcal{O}}}(\tau)}=\bigcup_\delta
{\ensuremath{\mathcal{O}}}(\delta)$ where $\delta$ runs over all faces of $\sigma$ containing $\tau$. The ideal $I(\tau)$ of an orbit closure $\overline{{\ensuremath{\mathcal{O}}}(\tau)}$ is given by $$I(\tau)=\bigoplus_{m\in \sigma^\vee_M\setminus
\tau^\bot}{\ensuremath{\mathbb{C}}}\chi^m\,$$ where $\tau^\bot\subseteq M_{\ensuremath{\mathbb{Q}}}$ is the orthogonal of $\tau$. Furthermore, the ideal of $X\setminus {\ensuremath{\mathrm{T}}}$ is $$I(X\setminus {\ensuremath{\mathrm{T}}})=\bigoplus_{m\in ({\ensuremath{\operatorname{rel.int}}}\sigma^\vee)\cap
M}{\ensuremath{\mathbb{C}}}\chi^m\,,$$ where ${\ensuremath{\operatorname{rel.int}}}$ denotes the relative interior.
As usual, we identify a ray $\rho\subseteq\sigma$ with its primitive vector. The set of all the rays of $\sigma$ is denoted by $\sigma(1)$. A cone $\sigma$ is called smooth if $\sigma(1)$ is part of a basis of the lattice $N$. Let $\tau\subseteq \sigma$ be any face. The orbit ${\ensuremath{\mathcal{O}}}(\tau)$ is contained in ${\ensuremath{X^{\mathrm{reg}}}}$ if and only if $\tau$ is smooth.
Let now $e\in M$ and $p\in N$. The linear map $$\partial_{e,p}:{\ensuremath{\mathbb{C}}}[M]\rightarrow{\ensuremath{\mathbb{C}}}[M],\quad \chi^m\mapsto
\langle m,p\rangle\cdot\chi^{m+e}$$ is a homogeneous derivation of the algebra ${\ensuremath{\mathbb{C}}}[M]$ and so it is a homogeneous vector field on ${\ensuremath{\mathrm{T}}}={\ensuremath{\operatorname{Spec}}}{\ensuremath{\mathbb{C}}}[M]$. By the exponential map, the tangent space of ${\ensuremath{\mathrm{T}}}=N\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{C}}}^*$ at the identity $\mathfrak{e}\in {\ensuremath{\mathrm{T}}}$ is isomorphic to $N\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{C}}}$ and the evaluation of the vector field $\partial_{e,p}$ at the smooth point $\mathfrak{e}$ is $\partial_{e,p}[\mathfrak{e}]=p$.
Let $\sigma\subseteq N_{\ensuremath{\mathbb{Q}}}$ be a pointed cone. The following proposition gives a description of all the homogeneous vector fields on $X_\sigma$. The first statement of the following result can be found in [@Dem70]. For the convenience of the reader we provide a short argument.
The homogeneous vector field $\partial_{e,p}$ on ${\ensuremath{\mathrm{T}}}$ extends to a homogeneous vector field in $X_\sigma$ if and only if
1. $e\in\sigma^\vee_M$, or
2. There exists $\rho_e\in \sigma(1)$ such that
1. $p\in {\ensuremath{\mathbb{Z}}}\rho_{e}$,
2. $\langle e,\rho_e\rangle=-1$, and
3. $\langle e,\rho\rangle\geq 0$ for all $\rho\in\sigma(1)\setminus \{\rho_e\}$.
Furthermore, $\partial_{e,p}$ is locally nilpotent if and only if it is of type II, and $\partial_{e,p}$ is semisimple if and only if it is of type I and $e=0$.
The vector field $\partial_{e,p}$ extends to $X_\sigma$ if and only if $\partial_{e,p}({\ensuremath{\mathbb{C}}}[\sigma^\vee_M])\subseteq
{\ensuremath{\mathbb{C}}}[\sigma^\vee_M]$. Since ${\ensuremath{\mathbb{C}}}[\sigma^\vee_M]$ is spanned by $\chi^m$ for all $m\in \sigma^\vee_M$, it is enough to show that $\partial_{e,p}(\chi^m)\in {\ensuremath{\mathbb{C}}}[\sigma^\vee_M]$. In combinatorial terms, this corresponds to the condition: $$\begin{aligned}
\label{eq:1}
\mbox{For every } m\in \sigma^\vee_M\setminus p^\bot,
\mbox{ we have }
\langle m+e, \rho\rangle\geq 0\mbox{ for all } \rho\in
\sigma(1)\,.
\end{aligned}$$
Assume first that $p$ is not proportional to any $\rho\in
\sigma(1)$. Then for every $\rho\in \sigma(1)$ there exists $m\in
\sigma^\vee_M$ such that $\langle \rho,m\rangle=0$ and $\langle
p,m\rangle\neq 0$. Hence, implies that $\langle
\rho,e\rangle\geq 0$ and so $\partial_{e,p}$ is of type I.
Assume now that there exists $\rho_e\in \sigma(1)$ such that $p\in
{\ensuremath{\mathbb{Z}}}\rho_{e}$. With the same argument as above we can show that $\langle \rho,e\rangle\geq 0$ for all $\rho\in
\sigma(1)\setminus\{\rho_e\}$. Let now $m\in \sigma^\vee_M$ such that $\langle \rho_e, m\rangle=1$. Then implies that $\langle \rho_e,m+e\rangle \geq 0$. This yields $\langle
\rho_e,e\rangle \geq -1$. If $\langle \rho_e,e\rangle=-1$ then $\partial_{e,p}$ is of type II. If $\langle \rho_e,e\rangle>-1$ then $\langle \rho_e,e\rangle\geq 0$ and $\partial_{e,p}$ is of type I.
To prove the second assertion, we let $\partial=\partial_{e,p}$ be a homogeneous vector field. A straightforward computation shows that $$\begin{aligned}
\label{eq:2}
\partial^{\ell+1}(\chi^m)=\langle m+\ell e,p\rangle
\cdot \partial^\ell(\chi^m)\cdot \chi^e\,.
\end{aligned}$$
Assume first that $\partial$ is of type I and that $e\in
\sigma^\vee_M\setminus \{0\}$. If $\langle e,p\rangle\neq 0$ then yields $$\partial^{\ell}(\chi^e)=\ell!\cdot\langle
e,p\rangle^\ell\cdot\chi^{\ell e}\neq 0\,,$$ and so $\partial$ is not locally finite since ${\ensuremath{\operatorname{span}}}\{\chi^{k e}\mid k\in {\ensuremath{\mathbb{Z}}}_{\geq
0}\}$ is not finite dimensional. If $\langle e,p\rangle=0$ then let $m\in \sigma_M^\vee$ be such that $\langle m,p\rangle\neq 0$. In this case implies $$\partial^{\ell}(\chi^m)=\langle
m,p\rangle^\ell\cdot\chi^{m+(\ell-1) e}\neq 0\,,$$ and again $\partial$ is not locally finite with a similar argument.
Assume now that $\partial$ is of type I and that $e=0$. The vector field $\partial$ is the infinitesimal generator of the algebraic ${\ensuremath{\mathbb{C}}}^*$-action on $X_\sigma$ given by the ${\ensuremath{\mathbb{Z}}}$-grading on ${\ensuremath{\mathbb{C}}}[\sigma_M^\vee]$ induced by the degree function $\deg(\chi^m)=\langle p,m\rangle$. Hence, the vector field $\partial$ is semisimple.
Finally, assume that $\partial$ is of type II. For every $m\in
\sigma^\vee_M$ we let $\ell=\langle m,\rho_e\rangle$. Now, $\partial_{e,p}$ is locally nilpotent since $\partial_{e,p}^{\ell+1}(\chi^m)=0$ by .
In the following corollary, we give an explicit description of the homogeneous complete vector fields on an affine toric variety.
\[cor-gl-int\] The vector field $\partial_{e,p}$ is complete if and only if it is of type II, or it is of type I and $\langle e,p\rangle=0$.
The vector fields of type II are locally nilpotent, hence complete. In the following, we assume that $\partial=\partial_{e,p}$ is of type I. First, assume that $\langle e,p\rangle=0$. Then $\partial=\chi^e\cdot \partial_{0,p}$ and since $\chi^e$ belongs to the kernel of $\partial_{0,p}$, we have that $\partial$ is complete.
Assume now that $\langle p,e\rangle\neq 0$. Let $I$ be the ideal of $X\setminus {\ensuremath{\mathrm{T}}}$, i.e., $$I=\bigoplus_{m\in {\ensuremath{\operatorname{rel.int}}}(\sigma^\vee)\cap M}{\ensuremath{\mathbb{C}}}\chi^m\,.$$ Since $e\in \sigma^\vee_M$, we have that $\partial(I)\subseteq I$. Hence, $X\setminus {\ensuremath{\mathrm{T}}}$ is invariant by $\partial_{e,p}$ and so ${\ensuremath{\mathrm{T}}}$ is also invariant by $\partial_{e,p}$. In the following, we show that $\partial$ is not complete when restricted to ${\ensuremath{\mathrm{T}}}$. Since $\lambda\partial$, $\lambda\in {\ensuremath{\mathbb{C}}}^*$ is complete if and only if $\partial$ is complete, we will assume that $p$ is a primitive vector in $N$ and $\langle e,p\rangle>0$.
Without loss of generality, we choose mutually dual bases of $N$ and $M$ such that $p=(1,0,\ldots,0)$ and $e=(e_1,\ldots,e_n)$, with $e_1>0$ and $n={\ensuremath{\operatorname{rank}}}N$. We will also denote $x_i=\chi^{\beta_i}$ the standard coordinates of the torus ${\ensuremath{\mathrm{T}}}$, where $\{\beta_i\mid
i=1,\ldots, n\}$ is the base of $N$. In this coordinates, the vector field $\partial$ restricted to ${\ensuremath{\mathrm{T}}}$ is given by $$\partial=x_1^{e_1+1}x_2^{e_2}\cdots
x_n^{e_n}\frac{\partial}{\partial x_1}\,,$$ which is not complete on ${\ensuremath{\mathrm{T}}}$ since $e_1>0$. Indeed the vector fields $x^n\partial/\partial x$ on $\C$ are not complete for $n\geq2$.
Remark that in Corollary \[cor-gl-int\] complete vector fields of type I are extensions of complete vector fields on the big torus ${\ensuremath{\mathrm{T}}}$ while complete vector fields of type II are locally nilpotent, hence not complete in ${\ensuremath{\mathrm{T}}}$. In the next lemma, we give a criterion for a homogeneous vector field to vanish in an orbit closure.
\[lm:vanish\] Let $\partial_{e,p}$ be a non-zero homogeneous vector field on $X_\sigma$ and let $\tau\subseteq
\sigma$ be a face. Then $\partial_{e,p}$ vanishes at the orbit closure $\overline{{\ensuremath{\mathcal{O}}}(\tau)}$ if and only if
1. $p\in{\ensuremath{\operatorname{Span}}}\tau$ or $\langle e,\rho\rangle>0$ for some $\rho\in
\tau(1)$.
2. $\langle e,\rho\rangle>0$ for some $\rho\in \tau(1)$.
The vector field $\partial_{e,p}$ does not vanish at the orbit closure $\overline{{\ensuremath{\mathcal{O}}}(\tau)}$ if and only if $\partial_{e,p}\left({\ensuremath{\mathbb{C}}}[\sigma_M^\vee]\right)\not\subseteq
I(\tau)$. In combinatorial terms this happens if and only if $$\begin{aligned}
\label{eq:4}
\mbox{there exists }m \in \sigma^\vee_M\setminus p^\bot \mbox{
such that } \langle m+e,\rho\rangle=0 \mbox{ for all } \rho\in
\tau(1)\,.
\end{aligned}$$
**Case of type I.** In this case, we have $e\in\sigma^\vee_M$ so $\langle m+e,\rho\rangle=0$ for all $\rho\in\tau(1)$ if and only if $\langle m,\rho\rangle=0$ and $\langle e,\rho\rangle=0$ for all $\rho\in\tau(1)$. This is the case if and only if $m\in\tau^\bot$ and $e\in\tau^\bot$. Such and $m\in\sigma^\vee_M\setminus p^\bot$ exists if and only if $\tau^\bot\not\subseteq p^\bot$, i.e., if and only if $p\notin {\ensuremath{\operatorname{Span}}}\tau$. Hence, we conclude that $\partial_{e,p}$ does not vanish at the orbit closure $\overline{{\ensuremath{\mathcal{O}}}(\tau)}$ if and only if $p\notin
{\ensuremath{\operatorname{Span}}}\tau$ and $\langle e,\rho\rangle=0$ for all $\rho\in \tau(1)$.
**Case of type II.** In this case we have that there exists $\rho_e\in \sigma(1)$ such that $p\in {\ensuremath{\mathbb{Z}}}\rho_e\setminus \{0\}$, $\langle e,\rho_e\rangle=-1$, and $\langle e,\rho\rangle\geq 0$ for all $\rho\in \sigma(1)\setminus \{\rho_e\}$.
Assume first that $\rho_e\notin \tau(1)$. An argument similar to case I yields that $\partial_{e,p}$ does not vanish at the orbit closure $\overline{{\ensuremath{\mathcal{O}}}(\tau)}$ if and only if $p\notin {\ensuremath{\operatorname{Span}}}\tau$ and $\langle e,\rho\rangle=0$ for all $\rho\in \tau(1)$. Since $\rho_e\notin \tau(1)$, we have that $p\notin {\ensuremath{\operatorname{Span}}}\tau$ and so the vector field $\partial_{e,p}$ does not vanish at the orbit closure $\overline{{\ensuremath{\mathcal{O}}}(\tau)}$ if and only if $\langle e,\rho\rangle=0$ for all $\rho\in \tau(1)$.
Assume now that $\rho_e\in \tau(1)$. If there exists $\rho\in
\tau(1)$ such that $\langle e,\rho\rangle>0$, then $\langle
m+e,\rho\rangle>0$ for all $m\in \sigma^\vee_M$ and so $\partial_{e,p}$ vanishes at the orbit $\overline{{\ensuremath{\mathcal{O}}}(\tau)}$ by . Assume $\langle e,\rho\rangle=0$ for all $\rho\in
\tau(1)\setminus \{\rho_e\}$ and let $m\in \sigma^\vee_M$ be such that $\langle m,\rho_e\rangle=1$ and $\langle m,\rho\rangle=0$ for all $\rho\in \tau(1)\setminus \{\rho_e\}$. We have $\langle
m,\rho_e\rangle\neq 0$ so $m\notin p^\bot$ and $\langle
m+e,\rho\rangle=0$ for all $\rho\in \tau(1)$. By , we conclude that $\partial_{e,p}$ does not vanish at the orbit closure $\overline{{\ensuremath{\mathcal{O}}}(\tau)}$.
The degree of a homogeneous locally nilpotent vector fields (of type II) is called a root of $\sigma$. The set of all roots of $\sigma$ is denoted by ${\ensuremath{\mathcal{R}}}(\sigma)$. For a root $e\in{\ensuremath{\mathcal{R}}}(\sigma)$, the ray $\rho_e$ is called the distinguished ray of $e$ and the ${\ensuremath{\mathbb{G}_{\mathrm{a}}}}$-action generated by the homogeneous locally nilpotent vector field $\partial_{e,\rho_e}$ is denoted by $H_e$.
In order to show the ADP for toric varieties, we need to show that $X_\sigma$ is homogeneous with respect to some closed subvariety $Y$. In [@AKZ10], the authors prove that $X_\sigma$ is homogeneous with respect to ${\ensuremath{X^{\mathrm{sing}}}}_\sigma$. In fact, they show that the group of special automorphisms acts infinite-transitively with respect to ${\ensuremath{X^{\mathrm{sing}}}}_\sigma$. In the following, we will show how their methods can be applied to show that $X_\sigma$ is homogeneous with respect to any ${\ensuremath{\mathrm{T}}}$-invariant closed subvariety $Y$.
\[rel-hom\] Let $\sigma\subseteq N_{\ensuremath{\mathbb{Q}}}$ be a pointed cone and let $Y$ be any ${\ensuremath{\mathrm{T}}}$-invariant closed subvariety of $X_\sigma$ containing ${\ensuremath{X^{\mathrm{sing}}}}_\sigma$. Then $X_\sigma$ is homogeneous relative to $Y$.
Using the ${\ensuremath{\mathrm{T}}}$-action and the Orbit-Cone correspondence, to prove the theorem it is enough to find, for every orbit ${\ensuremath{\mathcal{O}}}(\tau)$ in ${\ensuremath{X^{\mathrm{reg}}}}_\sigma$ different from the open orbit, an automorphism that
1. sends a point $x$ in ${\ensuremath{\mathcal{O}}}(\tau)$ into an orbit of higher dimension, and
2. leaves stable every orbit not containing ${\ensuremath{\mathcal{O}}}(\tau)$ in its closure.
Let $\rho_1,\ldots,\rho_\ell$ be the rays of $\tau$. In [@AKZ10 Lemma 2.3] and its proof, the authors show that for every smooth orbit ${\ensuremath{\mathcal{O}}}(\tau)$ there exists a root $e\in {\ensuremath{\mathcal{R}}}(\sigma)$ such that $$\begin{aligned}
\label{eq:3}
\langle \rho_1,e\rangle=-1,\ \langle \rho_2,e\rangle=\ldots=
\langle \rho_\ell,e\rangle=0,\mbox{ and } \langle \rho,e\rangle>0\
\mbox{ for all rays }\rho\not\notin\tau(1)\,.
\end{aligned}$$ Furthermore, they show that a generic automorphism $\alpha$ in the ${\ensuremath{\mathbb{G}_{\mathrm{a}}}}$-action $H_e$ corresponding to the root $e$ satisfies $(i)$.
Let ${\ensuremath{\mathcal{O}}}(\delta)$ be any orbit that does not contain ${\ensuremath{\mathcal{O}}}(\tau)$ in its closure. In combinatorial terms, this means that $\delta$ is a face of $\sigma$ that is not contained in $\tau$. We claim that $H_e$ leaves $\overline{{\ensuremath{\mathcal{O}}}(\delta)}$ point-wise invariant and so $\alpha$ satisfies $(ii)$ which proves the proposition.
In terms of the vector field $\partial_{e,\rho_e}$, our claim is equivalent to $\partial_{e,\rho_e}$ vanishes at $\overline{{\ensuremath{\mathcal{O}}}(\delta)}$. Since $\delta$ is not contained in $\tau$ there exists a ray $\rho$ of $\delta$ that is not a ray of $\tau$. By we have $\langle e,\rho\rangle>0$. Now the claim follows from Lemma \[lm:vanish\].
For our next theorem we need the following lemma that follows by direct computation.
\[commutator\] Let $\partial_{e_1,p_1}$ and $\partial_{e_2,p_2}$ be two homogeneous vector fields. Then $\left[\partial_{e_1,p_1},\partial_{e_2,p_2}\right]=\partial_{e,p}$, where $p=p_1(e_2)\cdot p_2-p_2(e_1)\cdot p_1$ and $e=e_1+e_2$.
\[finalthm\] Let $X$ be a affine toric variety of dimension at least two and let $Y$ be a ${\ensuremath{\mathrm{T}}}$-invariant closed subvariety of $X$ containing ${\ensuremath{X^{\mathrm{sing}}}}$. Then $X$ has the ADP relative to $Y$ if and only if $X\setminus Y\neq {\ensuremath{\mathrm{T}}}$.
Let $X=X_\sigma$ be the toric variety given by the pointed cone $\sigma\in N_{\ensuremath{\mathbb{Q}}}$ and let $X_\sigma\setminus Y\neq {\ensuremath{\mathrm{T}}}$. There is at least one codimension one ${\ensuremath{\mathrm{T}}}$-orbit not contained in $Y$. Assume it is ${\ensuremath{\mathcal{O}}}(\rho_1)$ for some ray $\rho_1\in \sigma(1)$. Let $e_1$ be a root with $\rho_1$ as distinguished ray. By , we can assume that $\langle e_1,\rho\rangle>0$ for all $\rho\in
\sigma(1)\setminus \{\rho_1\}$. By Lemma \[lm:vanish\], the locally nilpotent vector field $\partial_{e_1,\rho_1}$ vanishes at $Y$ and so $\partial_{e_1,\rho_1}\in {\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X_\sigma,Y)$.
Letting $e_2,e_3\in {\ensuremath{\operatorname{rel.int}}}(\sigma^\vee)\cap M$ be such that $e_3=e_1+e_2$, we let $$L={\ensuremath{\operatorname{Span}}}\left\{\partial_{e,p}\mid p\in N, e\in e_3+\sigma^\vee_M\right\}\,.$$
The set $L$ is contained in ${\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X_\sigma,Y)$ since $\partial_{e,p}\in L$ vanishes in $X_\sigma\setminus {\ensuremath{\mathrm{T}}}$. In fact, $L$ is a submodule of ${\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X_\sigma,Y)$ since for every $m\in
\sigma^\vee_M$ and every $\partial_{e,p}\in L$, we have $\chi^m\partial_{e,p}=\partial_{e+m,p}\in L$. Furthermore, the fiber over the identity $\mathfrak{e}\in {\ensuremath{\mathrm{T}}}\subseteq X_\sigma$ is given by $$\begin{aligned}
\label{spanning}
L_\mathfrak{e}={\ensuremath{\operatorname{Span}}}\{\partial_{e,p}[\mathfrak{e}]\mid \partial_{e,p}\in
L\}={\ensuremath{\operatorname{Span}}}\{p\mid \partial_{e,p}\in L\}=N\otimes_{\ensuremath{\mathbb{Z}}}{\ensuremath{\mathbb{C}}}=T_\mathfrak{e}X_\sigma\,,
\end{aligned}$$ and so $L_\mathfrak{e}$ contains a generating set. We claim that $L\subseteq {\ensuremath{\operatorname{Lie}_{\mathrm{alg}}}}(X_\sigma,Y)$. Hence $X_\sigma$ has the ADP relative to $Y$ by Theorem \[thm\] and Proposition \[rel-hom\].
By Corollary \[cor-gl-int\], the vector field $\partial_{e,p}$ is complete if $\langle e,p\rangle=0$. Hence, to prove our claim it is enough to show that for every $e\in e_3+\sigma^\vee_M$, there exists $p\in N$ such that $\langle e,p\rangle\neq 0$ and $\partial_{e,p}\in {\ensuremath{\operatorname{Lie}_{\mathrm{alg}}}}(X_\sigma,Y)$.
Indeed, let $e_4=e-e_1$ and choose $p_4$ be such that $\langle
e_4,p_4\rangle=0$ and $\langle e_1,p_4\rangle\neq 0$ which implies that $\partial_{e_4,p_4}$ belongs to ${\ensuremath{\operatorname{Lie}_{\mathrm{alg}}}}(X_\sigma,Y)$. This is possible since $e_4$ lies in ${\ensuremath{\operatorname{rel.int}}}\sigma^\vee$ and $e_1$ is a root of $\sigma^\vee$. By Lemma \[commutator\] we have $$\left[\partial_{e_1,\rho_1},\partial_{e_4,p_4}\right]=
\partial_{e,p}\quad\mbox{where}\quad p=\rho_1(e_4)\cdot
p_4-p_4(e_1)\cdot \rho_1\,.$$ A routine computation shows that $$\langle e,p\rangle=\
\langle e,\rho_1(e_4)\cdot p_4-p_4(e_1)\cdot \rho_1\rangle=\langle
e_1,p_4\rangle\neq 0\,,$$ proving the claim.
Assume now that $X\setminus Y= {\ensuremath{\mathrm{T}}}$. The converse of the theorem follows from the fact that for all affine toric varieties $X$ and all $\ell\in {\ensuremath{\mathbb{Z}}}_{>0}$ there is a vector field $\partial\in
I^\ell{\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,X\setminus {\ensuremath{\mathrm{T}}})\setminus {\ensuremath{\operatorname{Lie}_{\mathrm{alg}}}}(X,X\setminus {\ensuremath{\mathrm{T}}})$, where $I=I(X\setminus {\ensuremath{\mathrm{T}}})$. Indeed, Andersén [@Andersen00] proved that any complete algebraic vector field on ${\ensuremath{\mathrm{T}}}$ does preserve the Haar form $$\omega=\frac{dx_1}{x_1}\wedge\ldots\wedge \frac{dx_n}{x_n}\,.$$ Thus if we find $\partial$ in $I^\ell{\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,X\setminus {\ensuremath{\mathrm{T}}})$ whose restriction to ${\ensuremath{\mathrm{T}}}$ does not preserve $\omega$ we are done.
After a change of coordinates one can assume that $(1,0,\ldots,0)\in{\ensuremath{\operatorname{rel.int}}}\sigma^\vee$. Then $\partial=x_1^N\frac{\partial}{\partial x_1}$ is a regular vector field on $X$ contained in $I^\ell{\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,X\setminus {\ensuremath{\mathrm{T}}})$ for $N$ big enough which does not preserve $\omega$.
Lárusson proved in [@La11; @For13] that all smooth toric varieties are Oka-Forstnerič manifolds, however it is still unknown if they are elliptic, see [@Forst11; @Ku14] for definitions. The proof of Theorem \[finalthm\] can be adapted to prove the following: every smooth quasi-affine toric variety is elliptic (and thus an Oka-Forstnerič manifold). Indeed, the torus ${\ensuremath{\mathrm{T}}}$ is well known to be elliptic. Let $X_0$ be a smooth quasi-affine toric variety different from ${\ensuremath{\mathrm{T}}}$. Let also $X$ be an affine toric variety such that $X_0\subseteq X$ is an equivariant open embedding and let $Y=X\setminus X_0$. Now, Proposition \[rel-hom\] and imply that $X_0$ is elliptic [@Forst11 Example 5.5.13 (B)].
Classification of complete vector fields on affine toric surfaces
=================================================================
In this section we classify all complete algebraic vector fields on a given affine toric surface $X_\sigma$. The classification works essentially the same as the classification of complete vector fields on $\C^2$ done by Brunella [@Br3].
From now on we will use the fact that each affine toric surface different from $\C^*\times\C$ or $\C^*\times\C^*$ can be seen as the quotient of $\C^2$ by the action of a cyclic group. Let $d$ be the order of the group and let $e$ be a co-prime number $0<e<d$ and consider the action of $\Z_d$ given by $\zeta\cdot(u,v) = (\zeta u,
\zeta^e v)$ where $\zeta$ is a primitive $d$-th root of unity. We obtain the projection $\pi: \C^2 \rightarrow \C^2/\Z_d =: V_{d,e}$ onto our toric surface which is a ramified covering of $V_{d,e}$ ramified only over the unique singular point. Certainly each vector field on $X$ pulls back to an invariant vector field of $\C^2$ by using the fiber-wise isomorphism $D\pi$ on the tangent space. A complete vector field on $V_{d,e}$ will pull back to an invariant complete vector field on $\C^2$.
Let $f: \C^2 \rightarrow \C$ be a regular function on $\C^2$. The function $f$ is called $\Z_d$-preserved if the fibers of $f$ are sent to fibers of $f$ by the $\Z_d$-action. It is called $\Z_d$-homogeneous of degree $[i]\in\Z_d$ if $\zeta^*f(u,v)=
f(\zeta\cdot(u,v)) = \zeta^if(u,v)$ for all $(u,v)\in\C^2$. Let $A_{[i]}$ denote the space of $\Z_d$-homogeneous polynomials of degree $[i]$ then we obtain a decomposition of the ring of regular functions on $\C^2$ into $\Z_d$-homogeneous parts $\C[u,v] = A_{[0]}
\oplus \ldots \oplus A_{[d-1]}$. In particular $A_{[0]}$ is the ring of invariant functions $\C[u,v]^{\Z_d}=\C[V_{d,e}]$.
It is clear from the definition that $A_{[i]}$ is spanned by all monomials $u^mv^n$ with $[m+en]=[i]\in\Z_d$. Clearly invariant vector fields are of the form $f\partial/\partial u + g\partial/\partial v$ with $f\in A_{[1]}$ and $g\in A_{[e]}$. Moreover we have the following easy lemma:
\[lemhomog\] Let $f: \C^2 \rightarrow \C$ be a regular function then the following are equivalent:
1. $f$ is $\Z_d$-homogeneous,
2. $f$ is $\Z_d$-preserved with $f(0,0)=0$,
3. $f^{-1}(0)$ is $\Z_d$-invariant.
\(1) implies (2) since if $f$ is constant on a curve then also $\zeta^i \cdot f$ is constant and $f(0,0)=0$ follows directly from the homogeneity. The fiber $f^{-1}(0)$ contains the $\Z_d$-fixed point $(0,0)$ thus (3) follows from (2). If the zero fibers of $f$ and $\zeta^*f$ coincide then we have that $\zeta^*f(u,v) = a\cdot
f(u,v)$ for some $a\in\C^*$. By $f(u,v)=\zeta^{d*}f(u,v)=a^d f(u,v)$ we see that $a$ is a $d$-th root of unity and thus (3) implies (1).
The following lemma is the crucial step in the classification of invariant complete algebraic vector fields and hence of complete algebraic vector fields on the toric variety $V_{d,e}$. Recall that a rational first integral of a vector field is a rational function such that its fibers are tangential to the vector field.
\[lemhomfib\] Let $\partial$ be a $\Z_d$-invariant complete algebraic vector field on $\C^2$ then $\partial$ preserves either a $\Z_d$-homogeneous fibration $f: \C^2 \rightarrow \C$ with general fibers $\C$ or $\C^*$ or $\partial$ has a reduced rational first integral $g:\C^2
\dashrightarrow \C$.
By [@Br3] there is fibration $f:\C^2\rightarrow\C$ with $\C$ or $\C^*$ fibers which is preserved by the flow $\varphi^t$ of $\partial$. We may assume that $f(0,0)=0$. If $f$ is $\Z_d$-homogeneous then we are done. If $f$ is not $\Z_d$-homogeneous then we construct a rational first integral. The map $\varphi^t$ acts by multiplication with some $a_t$ on the set of fibers of $f$ parametrized by $\C$ so we have $f(\varphi^t(u,v))=a_tf(u,v)$ (indeed $(0,0)$ is a fixed point of $\varphi^t$). Since $\partial$ is invariant the same holds true for $g(u,v)=f(\zeta\cdot(u,v))$ and hence the rational map $f/g$ is a rational first integral for $\partial$. By Stein factorization $\partial$ has a reduced first integral. Recall that every rational function $\C^2\dashrightarrow \P^1$ can be decomposed into $F\circ\tilde f\C^2\dashrightarrow \P^1\rightarrow \P^1$ such that $\tilde f$ has connected regular fibers, or equivalently is reduced. This factorization is called Stein factorization.
The next step will be the classification of $\Z_d$-homogeneous fibrations with general fibers $\C$ or $\C^*$ and rational first integrals for invariant vector fields. The classification will be done up to equivariant automorphisms of $\C^2$ which will lead to a classification of the vector fields on $V_{d,e}$ up to automorphism of $V_{d,e}$ since equivariant automorphisms clearly project down to automorphims of the quotient. Equivariant automorphisms of $\C^2$ are given by invertible maps $(u,v)\mapsto (p(u,v),q(u,v))$ with $p\in
A_{[1]}$ and $q\in A_{[e]}$.
First we establish an equivariant version of the Abhyankar-Moh Theorem. We provide a proof using the classical verion of the theorem. See [@ArZa] for a different proof.
\[lemabhy\] Let $\C\cong L\subset \C^2$ be a line which is invariant by the group action. Then there is an equivariant automorphism of $\C^2$ mapping $L$ to $\lbrace u =0\rbrace$ or $\lbrace
v=0\rbrace$. Moreover a cross of two invariant lines can be mapped to $\lbrace uv=0\rbrace$.
By the classical Abhyankar-Moh Theorem we know that $L$ is given by a polynomial $p$ which is a component of an automorphism of $\C^2$. In order to find the other component of the automorphism we have to find an invariant section of the trivial line bundle given by $p$. We start with an arbitrary trivialization and get an invariant section taking the average over images of the zero section by the group action. Each image is another section because the action sends fibers of $p$ to fibers of $p$ since the zero fiber is invariant. We denote the polynomial giving this invariant section by $q$. The map given by $(p,q)$ is an automorphism of $\C^2$ since it is the composition of the trivialization we started with and the map $(u,v)\mapsto (u,v -s(u))$ where $s$ is the invariant section. Because the zero sets of $p$ and $q$ are invariant they are $\Z_d$-homogeneous by Lemma \[lemhomog\] and since they are the two components of an automorphism their homogeneity degrees coincide with $[1]$ and $[e]$ so either $(p,q)$ or $(q,p)$ is an equivariant automorphism and the claim follows. The second statement is trivial since there we already have an invariant section by assumption.
We get the following corollary as an immediate consequence, see also [@FlKaZa].
\[c-fiber\] Let $f:\C^2\rightarrow\C$ be a $\Z_d$-homogeneous fibration with $\C$ fibers and $f(0,0)=0$ then $f(u,v)=u$ or $f(u,v)=v$ up to equivariant automorphism of $\C^2$.
For the classification of $\Z_d$-homogeous fibration with $\C^*$ fibers we first state the non-equivariant version used in [@Br3], see also [@Su].
\[lemcstarfib\] Let $f:\C^2\rightarrow\C$ be a fibration with $\C^*$ fibers then $f(x,y)$ has one special fiber (say $f^{-1}(0)$) and it is isomorphic to $\C\cup\C^*$ or $\lbrace xy=0\rbrace$ and $f$ is up to automorphism of $\C^2$ of the form $f(x,y)=x^m(x^ly + p(x))^n$ or $f(x,y)=x^my^n$ for coprime $m,n\in\N$, $\deg p<l\geq1$ and $p(0)\neq 0$.
The equivariant version of this lemma is given by the two following lemmas.
\[cstar-fiber-1\] Let $f:\C^2\rightarrow \C$ be a $\Z_d$-homogeneous fibration with $\C^*$ fibers and $f^{-1}(0)\cong\C\cup\C^*$ then there are coprime $m,n\in\N$ and an invariant polynomial $p$ with $\deg p <l\geq1$ and $p(0)\neq 0$ such that up to equivariant automorphism $f(u,v)=u^m(u^lv + p(u))^n$ with $[l+e]=[0]$ or $f(u,v)=v^m(v^lu+p(v))^n$ with $[1+le]=[0]$.
By Lemma \[lemcstarfib\] we know that there exists a not necessary equivariant automorphism $(x(u,v),y(u,v))$ such that $f(x,y)$ is as in Lemma \[lemcstarfib\]. Clearly, the curve $\C\cong C \subset
f^{-1}(0)$ is invariant by the group action since it is the only fiber component isomophic to $\C$. By Lemma \[lemabhy\] we may assume the $C = \lbrace u=0\rbrace$ or $C=\lbrace v=0\rbrace$. In the first case this implies that, up to equivariant automorphism, $x(u,v)=au$ and $y(u,v)=bv + q(u)$ for some $a,b\in\C^*$ and $q\in\C[u]$ and hence $f$ is of the form $(au)^m((au)^l(bv+q(u)) +
p(u))^n$ with $\deg p<l$. Since $f$ is $\Z_d$-homogeneous we have $q\in A_{[e]}$ and $p\in A_{[l+e]}$ hence the map $(x(u,v),y(u,v))$ was equivariant after all and $f$ has the desired standard form up to equivariant automorphism. The equality $[l+e]=[0]$ follows from the fact that $p(0)\neq 0$. The case $C=\lbrace v=0\rbrace$ leads similarly to the second possibility.
\[cstar-fiber-2\] Let $f:\C^2\rightarrow \C$ be a $\Z_d$-homogeneous fibration with $\C^*$ fibers and $f^{-1}(0)\cong\lbrace uv=0\rbrace$ then there are coprime $m,n\in\N$ such that $f(u,v)=u^mv^n$ up to equivariant automorphism. If $d$ is divisible by 4 (say $d=4d'$) and $e=2d'+1$ then $f$ can also be of the form $f(u,v)=u^2-v^2$.
By Lemma \[lemcstarfib\] there is an automorphism $(x(u,v),y(u,v))$ such that $f=x^my^n$. Clearly the 0-fiber $\lbrace
x(u,v)=0\rbrace \cup \lbrace y(u,v)=0 \rbrace$ is invariant by the group action. If the two lines are invariant themselves then by Lemma \[lemabhy\] we may assume that they coincide with $\lbrace
uv=0\rbrace$ and hence we may assume $x(u,v)=au$ and $y(u,v)=bv$ for some $a,b\in\C^*$ and we are done. If the two lines are interchanged by the group action then we have $d=2d_0$ is even and $$x(\zeta\cdot(u,v))= ay(u,v) \quad \mathrm{and}\quad
y(\zeta\cdot(u,v)=bx(u,v)$$ for some $a,b \in \C^*$. After rescaling we may assume that $a=b$. The fibration $f=x^my^n$ is $\Z_d$-homogeneous so $x^my^n=const \cdot y^mx^n$ and hence $m=n=1$. Moreover we have $x(u,v)=x(\zeta^d\cdot(u,v))=a^dx(u,v)$ and hence $a=\zeta^i$ for some $i$. We see that the maps $P_\pm(u,v)=x(u,v)\pm y(u,v)$ are $\Z_d$-homogeneous and since they are the components of an automorphism of $\C^2$ we may assume that the functions $P_\pm$ coincides with the functions $u$ and $v$. Altogether we have $\frac{1}{4}f(u,v)=(u+v)(u-v)=u^2-v^2$ which is $\Z_d$-homogeneous only if $2e=2$ or $2e=2d_0+2$. In the first case $(x(u,v),y(u,v))$ is already equivariant so only the latter case remains. Since $d$ is even and thus $e=d_0+1$ is odd we have that $d_0=2d'$ is even.
\[lemfirstint\] Let $f: \C^2 \dashrightarrow \P^1$ be a reduced rational first integral of an invariant complete vector field $\partial$ on $\C^2$ then up to equivariant automorphism of $\C^2$ and Möbius transform of $\P^1$ the rational function $f$ is a $\Z_d$-homogeneous polynomial with $\C$ or $\C^*$ fibers or there are coprime $m,n\in\N$ such that $f(u,v)=u^m/v^n$.
A general fiber of $f$ is an orbit closure of the flow of $\partial$. Since $\partial$ is invariant the set of orbits is preserved by the $\Z_d$-action hence general fibers of $f$ are mapped to general fibers of $f$ by the action and the action induces a $\Z_d$-action on the base $\P^1$. Altogether this means that $f$ is $\Z_d$-preserved. If $f$ is not surjective then $f$ can be seen as a polynomial which is $\Z_d$-homogeneous by Lemma \[lemhomog\] and has general fibers isomomorphic $\C$ or $\C^*$ since they are orbit closures.
Now consider the case $f$ surjective. As mentioned in [@Br3] and [@Su] such a first integral is always of the form $f=x^m/y^n$ for some automorphism $(x(u,v),y(u,v))$. The $\Z_d$-action on the base $\P^1$ is either trivial (and hence $f$ is $\Z_d$-invariant) or it has exactly two fixed points (so two fibers of $f$ are $\Z_d$-invariant). In both cases there are two invariant fibers intersecting transversally (say the $0$- and the $\infty$-fiber). Indeed if $m=n=1$ all fibers intersect transversely and if $m\neq n$ all but one fiber intersect pairwise tangentially so this fiber is clearly invariant and it intersects all other fibers transversally. By Lemma \[lemabhy\] we may assume that these two fibers coincides with $\lbrace u = 0\rbrace$ and $\lbrace
v = 0\rbrace$ and hence $x(u,v)=au$ and $y(u,v)=bv$ or vice versa.
\[thmlist\] Let $\partial$ be a complete algebraic vector field on $\C^2$ which is invariant by the group action given by $\zeta\cdot(u,v) = (\zeta u,\zeta^e v)$ where $\zeta$ is a primitive $n$-th root of unity and $0<e<d$ coprime numbers. Then $\partial$ has, up to equivariant automorphism of $\C^2$, one of the forms in the following list.
1. 2. 1. $\displaystyle{ \partial=au\frac {\partial}{\partial u} +
((A(u^d)v+B(u^e))\frac{\partial}{\partial v}}$
2. $\displaystyle{ \partial=av\frac {\partial}{\partial v} + ((A(v^d)u+B(v^{e'}))\frac{\partial}{\partial u}}$
3. with $a\in\C$, $0<e'<d$ such that $[ee']=[1] \in \Z_d$ and $A,B\in \C[t]$.
4. 1. $\displaystyle{\partial=av\frac{\partial}{\partial v} +
A(u^mv^n)\left[nu\frac{\partial}{\partial u} -
mv\frac{\partial}{\partial v}\right]}$
2. If $d=4d'$ and $e=2d'+1$ then we also have
3. $\displaystyle{\partial=a(u+v)\left(\frac{\partial}{\partial u}
+ \frac{\partial}{\partial v}\right) +
A((u^2-v^2)^{2d'})\left[u\frac{\partial}{\partial v} +
v\frac{\partial}{\partial u}\right]}$
5. with $a\in\C$, $m,n\in\N$ with $[m+en]=[0]$ and $A\in\C[t]$ .
6. There are $a\in\C$, $m,n,l\in \N$ with $[m]=[0]$, $p\in
A_{[0]}$, $\deg p< l$, $p(0)\neq 0$ and $A\in\C[t]$ with the property that $$A(x^m(x^ly+p(x))^n)\cdot(mp(x)+nxp'(x))-ap(x) \in
x^l\cdot\C[x,y]$$ such that
7. 1. $\displaystyle{ \partial= a\left(v+\frac{p(u)}{u^l}\right)
\frac{\partial}{\partial v}+}$
2. $\displaystyle{A(u^m(u^lv +
p(u))^n)\cdot\left[nu\frac{\partial}{\partial u} - \left((m+nl)v
+ \frac{mp(u)+nup'(u)}{u^l}\right)\frac{\partial}{\partial
v}\right]}$
3. with $[l+e]=0$.
4. $\displaystyle{ \partial= a\left(u+\frac{p(v)}{v^l}\right)
\frac{\partial}{\partial u} + }$
5. $\displaystyle{A(v^m(v^lu +
p(v))^n)\cdot\left[nv\frac{\partial}{\partial v} - \left((m+nl)u
+ \frac{mp(v)+nvp'(v)}{v^l}\right)\frac{\partial}{\partial
u}\right]}$
6. with $[1+le]=0$.
By Theorem \[lemhomfib\] we know that the flow $\partial$ preserves fibers of a $\Z_d$-homogeneous $\C$- or $\C^*$-polynomial (which are described in Corollary \[c-fiber\] and Lemmas \[cstar-fiber-1\] and \[cstar-fiber-2\]) or it has a rational first integral (which may be assumed to be of the form $u^m/v^n$ by Lemma \[lemfirstint\]). Once we have a polynomial that is preserved by the flow we can check in Proposition 2 in [@Br3] how the vector field looks like. Since the vector fields need to be $\Z_d$-invariant some extra conditions are required. In the case of the rational first integral we have $\partial=nu\partial/\partial u +
mv\partial/\partial v$ which is already in the list.
The strong algebraic density property for affine toric surfaces
===============================================================
First we give a new concept of the ADP which was first introduced in [@KaKu14].
Let $\Gamma$ be a group acting on an smooth affine algebraic variety $X$. Then $X$ has $\Gamma$-ADP if the Lie algebra of all $\Gamma$-invariant algebraic vector fields coincides with the Lie algebra generated by all $\Gamma$-invariant complete algebraic vector fields.
As in the section above let $d,e\in\Z$ be two coprime numbers with $0<e<d$ and let $\zeta$ be a primitive $d$-th root of unity. Consider again the $\Z_d$-action on $\C^2$ given by $\zeta\cdot(u,v)=(\zeta u,
\zeta^e v)$. Moreover let $e'$ be the unique integer with $0<e'<d$ and $ee'=1$ mod $d$. It is clear that:
\[surf-ADP\] $V_{d,e}$ has the strong ADP if and only if $\C^2$ has the $\Z_d$-ADP.
Let us introduce the following subsets of ${\ensuremath{\mathbb{Z}}}^2$ $$\begin{aligned}
I & = & \lbrace (i,j)\in{\ensuremath{\mathbb{Z}}}_{\geq 0}^2: \ i+ej = 0 \ \mathrm{mod} \ d \rbrace,\\
J & = & \lbrace (i,j)\in I\setminus\lbrace(0,0)\rbrace: \ i<e \ \mathrm{and} \ j<e' \rbrace \subset I,\\\end{aligned}$$
$|J| \leq1 \Leftrightarrow e \ \vert \ d+1$.
If $e=1$ then also $e'=1$ and thus $J=\emptyset$. If $e,e'>1$ then $
|J| \geq 1$ since $(e-1,e'-1)\in J$. Assume $ee'=d+1$, $i<e$ and $j<e'$, then we have $i+je<e+d<2d$ and the equality $[i+je]=[0] \in
\Z_d$ implies $i+je=d$. Similarly we get $ie'+j=d$ and thus there is a unique solution for $(i,j)$ and hence $|J| =1$. If $ee'\geq 2d+1$ then we get another solution of $[i+je]=[0]$ in $J$. Indeed, choose $l\in\N$ such that $0<d-le<e$ then $(d-le,l)\neq(e-1,e'-1)$ lies in $J$, since $le<d$ implies $0<l<e'-1$.
Let us introduce the following notation:
$$\begin{aligned}
\mathrm{VF}^{(i,j)} & = & \left\lbrace u^i v^j \left(au\frac{\partial}{\partial u} + bv\frac{\partial}{\partial v}\right): \ a,b\in \C \right\rbrace,\\
\mathrm{CVF}^{(i,j)} & = & \left\lbrace au^i v^j \left(ju\frac{\partial}{\partial u} - iv\frac{\partial}{\partial v}\right): \ a\in \C \right\rbrace
\subset\mathrm{VF}^{(i,j)}, \\
\mathrm{LND}_u^k &=& \left\lbrace av^{ke'}\frac{\partial}{\partial u}: \ a\in\C\right\rbrace,\\
\mathrm{LND}_v^k& = &\left\lbrace au^{ke}\frac{\partial}{\partial v}: \ a\in\C\right\rbrace.\\\end{aligned}$$
Remark that $\mathrm{CVF}^{(i,j)}$ corresponds to the subset of complete vector fields in $\mathrm{VF}^{(i,j)}$ by Corollary \[cor-gl-int\]. We have the decomposition of $\Z_d$-invariant vector fields in homogeneous vector fields given by: $$\mathrm{VF}_{\mathrm{alg}}^{\Z_d}(\C^2) = \bigoplus_{(i,j)\in
I}\mathrm{VF}^{(i,j)} \oplus
\bigoplus_{k\in\N}\left(\mathrm{LND}_u^k \oplus
\mathrm{LND}_v^k\right).$$ We define the subspace $S$ of $\mathrm{VF}_{\mathrm{alg}}^{\Z_d}(\C^2)$. $$S = \bigoplus_{(i,j)\in J}\mathrm{CVF}^{(i,j)} \oplus \bigoplus_{(i,j)\in I\setminus J}\mathrm{VF}^{(i,j)} \oplus
\bigoplus_{k\in\N}\left(\mathrm{LND}_u^k \oplus \mathrm{LND}_v^k\right)\,.$$
The following is our main result in this section.
\[thminvliealg\] For the Lie algebra $\mathrm{Lie}_{\mathrm{alg}}^{\Z_d}(\C^2)$ generated by all $\Z_d$-invariant complete algebraic vector fields on $\C^2$ we have: $$\mathrm{Lie}_{\mathrm{alg}}^{\Z_d}(\C^2)=
\begin{cases}
S & e = e'\\
S \oplus \langle \partial \rangle & e \neq e'
\end{cases}$$ for any $\partial\in \mathrm{VF}^{(e-1,e'-1)}\setminus
\mathrm{CVF}^{(e-1,e'-1)}$. In particular the codimension of the inclusion $\mathrm{Lie}_{\mathrm{alg}}^{\Z_d}(\C^2) \subseteq
\mathrm{VF}_{\mathrm{alg}}^{\Z_d}(\C^2)$ is $|J|$ if $e=e'$ and $|J| -1$ otherwise.
Remark that $\dim_{\ensuremath{\mathbb{C}}}\mathrm{CVF}^{(i,j)}=1$ and $\dim_{\ensuremath{\mathbb{C}}}\mathrm{VF}^{(i,j)}=2$ as a vector space. Hence, in the case where $e\neq e'$ we have that $\mathrm{VF}^{(e-1,e'-1)}\subseteq
\mathrm{Lie}_{\mathrm{alg}}^{\Z_d}(\C^2)$. We postpone the proof of this theorem to the end of this section.
The theorem immediately shows in which cases $\C^2$ has $\Z_d$-ADP or, equivalently, $V_{d,e}$ has the strong ADP. It also allows in each particular case to determine the values of $\ell$ from Definition \[ADP\] for which $I^\ell{\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,{\ensuremath{X^{\mathrm{sing}}}}) \subseteq
{\ensuremath{\operatorname{Lie}_{\mathrm{alg}}}}(X,{\ensuremath{X^{\mathrm{sing}}}})$.
\[Z-ADP\] Let $V_{d,e}$ be a toric surface.
1. $V_{d,e}$ has the strong ADP if and only if if and only if $e\
\vert \ d+1$ and $e^2 \neq d + 1$.
2. $V_{d,e}$ has the ADP and an upper bound for the minimal $\ell$ such that $I^\ell{\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,{\ensuremath{X^{\mathrm{sing}}}}) \subseteq
{\ensuremath{\operatorname{Lie}_{\mathrm{alg}}}}(X,{\ensuremath{X^{\mathrm{sing}}}})$ is $e+e'-2$.
The next lemma shows what is happening if we take the Lie bracket of two complete homogeneous vector fields.
\[lembrackets\] Let $\partial_1\in \mathrm{CVF}^{(i,j)}$, $\partial_2 \in
\mathrm{CVF}^{(i',j')}$, $\partial_3\in\mathrm{LND}_u^k$ and $\partial_4 \in
\mathrm{LND}_v^{k'}$, then
1. $[\partial_1,\partial_2] \in \mathrm{CVF}^{(i+i',j+j')}$,
2. $[\partial_1,\partial_3] \in \mathrm{VF}^{(i-1,j+ke')}\setminus \mathrm{CVF}^{(i-1,j+ke')}$,
3. $[\partial_1,\partial_4] \in \mathrm{VF}^{(i+k'e,j-1)}\setminus
\mathrm{CVF}^{(i+k'e,j-1)}$,
4. $[\partial_3,\partial_4] \in \mathrm{VF}^{(k'e-1,ke'-1)}$. Furthermore, $[\partial_3,\partial_4] \in \mathrm{CVF}^{(k'e-1,ke'-1)}$ if and only if $ek'= e'k$.
All four statements follow by direct computation using Corollary \[cor-gl-int\] and Lemma \[commutator\].
The next two lemmas show $\mathrm{Lie}(S)=\mathrm{Lie}_{\mathrm{alg}}^{\Z_d}(\C^2)$, each of them showing one inclusion.
$S\subset \mathrm{Lie}_{\mathrm{alg}}^{\Z_d}(\C^2)$
Take $(i,j)\in I \setminus J$, then either $(i-e,j+1) \in I$ or $(i+1,j-e')\in I$. In the first case pick $\partial\in
\mathrm{CVF}^{(i-e,j+1)}$ and $\delta\in\mathrm{LND}_v^1$ and by Lemma \[lembrackets\] we have $[\partial,\delta]\in\mathrm{VF}^{(i,j)}\setminus
\mathrm{CVF}^{(i,j)}$ and thus $\mathrm{VF}^{(i,j)}\subset
\mathrm{Lie}_{\mathrm{alg}}^{\Z_d}(\C^2)$. The second case works similarly.
$\lbrace$[invariant complete algebraic vector fields]{}$\rbrace \subset \mathrm{Lie}(S)$.
Let $L$ be the set of vector fields appearing in the list of Theorem \[thmlist\]. We will first show that $L\subset S$. Let $\partial \in L$ and $\partial = \sum \partial_{i,j}$ its decomposition into homogeneous parts with respect to the standard grading on $\C^2$. We directly see that all homogeneous parts of vector fields (1) and (2a) are complete. For the vector fields (2b) and (3) we claim that $\partial_{i,j}=0$ whenever $(i,j)\in J$. Indeed, assume that $\partial_{i,j}\neq 0$ with $(i,j)
\neq (0,0)$ and $\partial_{i,j}$ is not an LND. Then in case (2b) we have $e=e'=2d'+1$, $i+j\geq 4d'$ and $i\neq j$ since for every monomial $\mathfrak{m}$ of the polynomial $A$ we have $\deg_u\mathfrak{m}-\deg_v\mathfrak{m}$ is a multiple of 4. Hence, either $i>e$ or $j>e'$. In case (3a) under the same assumptions we have $i> m+nl-l\geq m\geq d> e$. Similarly, in case (3b) we have $j>
m+ln -l\geq m\geq d> e'$.
In order to conclude the proof we only need to show that for a vector field $\delta \in \mathrm{Lie}(S)$ and an equivariant automorphism $\phi$ the vector field $\phi_*\delta \in
\mathrm{Lie}(S)$. By Lemma 4.10 in [@ArZa] $\phi$ is a composition of equivariant Jonquières automorphisms or more precisely it is a composition of linear equivariant automorphisms and flow maps of the vector fields $u^{ke}\partial/\partial v$ and $
v^{ke'} \partial/\partial u$ (which are contained in $S$). First we show that for any linear automorphism $\phi$ we have $\phi_*\delta
\in \mathrm{Lie}(S)$. For $e=1$ this statement is true for obvious reasons, indeed here we already have $\mathrm{Lie}(S)=\mathrm{Lie}_{\mathrm{alg}}^{\Z_d}(\C^2)$. For $e
\neq 1$ all equivariant linear automorphisms are of the form $(u,v)\mapsto (au,bv)$ so they act by homothety on homogeneous vector fields of $\mathrm{Lie}(S)$. Now, if $\phi^t$ is the flow of the LND $\partial$ then $\phi^t_*\delta \in$ Lie($\partial,\delta$) for all $t$, since the Taylor expansion of $\phi^t_*\delta$ gives $\phi^t_*\delta = \delta + t [\partial, \delta] + \frac{1}{2} t^2
[\partial, [\partial, \delta]] + \ldots + \frac{1}{n !} t^n
[\partial, \ldots [\partial, \delta]] \ldots]$ which is a finite sum since $\partial$ is an LND and hence its flow is algebraic in $t$. Since $\partial\in S$ the claim follows.
It is left to show that $\mathrm{Lie}(S)=S$ if $e=e'$ and $\mathrm{Lie}(S)=S\oplus \langle \partial \rangle$ if $e\neq e'$ for any $\partial\in \mathrm{VF}^{(e-1,e'-1)}\setminus
\mathrm{CVF}^{(e-1,e'-1)}$. Let $(i,j)\in J$, then we need to show that $\mathrm{VF}^{(i,j)}\nsubseteq\mathrm{Lie}(S)$ unless $e\neq
e'$ and $(i,j)=(e-1,e'-1)$. Assume $\mathrm{VF}^{(i,j)}\subset\mathrm{Lie}(S)$, then Lemma \[lembrackets\] implies the existence of $\partial\in\mathrm{VF}^{(i,j)}\setminus \mathrm{CVF}^{(i,j)}$ such that $\partial=[\partial_1,\partial_2]$ for some $\partial_1\in\mathrm{LND}_u^1$ and $\partial_2\in\mathrm{LND}_v^1$, $e\neq e'$ and $(i,j)=(e-1,e'-1)$.
Implications of the algebraic density property for the holomorphic automorphism group
=====================================================================================
We start with the obvious holomorphic version of Definition \[ADP\]. Let $X$ be a Stein space and let ${\ensuremath{X^{\mathrm{sing}}}}$ be the singular locus. We also let $Y\subseteq X$ be closed analytic subvariety of $X$ containing ${\ensuremath{X^{\mathrm{sing}}}}$ and let $\II=I(Y)\subseteq {\ensuremath{\mathcal{O}}}(X)$ be the ideal of $Y$. Let ${\ensuremath{\operatorname{VF}_{\mathrm{hol}}}}(X,Y)$ be the ${\ensuremath{\mathcal{O}}}(X)$-module of holomorphic vector fields vanishing in $Y$ i.e., ${\ensuremath{\operatorname{VF}_{\mathrm{hol}}}}(X,Y)=\{\partial
\mid \partial({\ensuremath{\mathcal{O}}}(X))\subseteq \II\}$. Let ${\ensuremath{\operatorname{Lie}_{\mathrm{hol}}}}(X,Y)$ be the Lie algebra generated by all the complete vector fields in ${\ensuremath{\operatorname{VF}_{\mathrm{hol}}}}(X,Y)$.
We say that $X$ has the strong density property (DP) relative to $Y$ if ${\ensuremath{\operatorname{Lie}_{\mathrm{hol}}}}(X,Y)$ is dense in ${\ensuremath{\operatorname{VF}_{\mathrm{hol}}}}(X,Y)$ in the compact-open topology. Furthermore, we say that $X$ has the DP relative to $Y$ if there exists $\ell\geq 0$ such that $\II^\ell{\ensuremath{\operatorname{VF}_{\mathrm{hol}}}}(X,Y)$ is contained in the closure of ${\ensuremath{\operatorname{Lie}_{\mathrm{hol}}}}(X,Y)$. With this definition, the DP relative to $Y$ with $\ell=0$ is just the strong DP relative to $Y$.
\[GAGA\] Let $X$ be an affine algebraic variety and let $Y$ be a subvariety containing ${\ensuremath{X^{\mathrm{sing}}}}$. Then the ADP for $X$ relative to $Y$ implies the DP for $X$ relative to $Y$.
The proposition follows from the fact that $I^\ell{\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,Y)$ is dense in $\II^\ell{\ensuremath{\operatorname{VF}_{\mathrm{hol}}}}(X,Y)$. Indeed, by Theorem A of Cartan, there are finitely many global sections $s_1,\ldots, s_N$ of $I^\ell{\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,Y)$ that generate the stalk at every point. A standard application of Theorem B of Cartan implies that any holomorphic section $s_h\in\II^\ell{\ensuremath{\operatorname{VF}_{\mathrm{hol}}}}(X,Y)$ over an ${\ensuremath{\mathcal{O}}}(X)$-convex compact $K\subseteq X$ can be written as $s_h=f_1s_1+\ldots+f_Ns_N$ with $f_i\in {\ensuremath{\mathcal{O}}}(K)$. By approximating the functions $f_i$ by global functions in ${\ensuremath{\mathbb{C}}}[X]$, this implies $I^\ell{\ensuremath{\operatorname{VF}_{\mathrm{alg}}}}(X,Y)$ is dense in $\II^\ell{\ensuremath{\operatorname{VF}_{\mathrm{hol}}}}(X,Y)$.
\[AL-Theorem\] Let $X$ be a Stein space with the DP relative to a closed analytic subvariety $Y$ containing ${\ensuremath{X^{\mathrm{sing}}}}$. Let $\Omega$ be an open subset of $X$. Suppose that $ \Phi : [0,1] \times \Omega \to
X$ is a $C^1$-smooth map such that
(i) $\Phi_t : \Omega \to X$ is holomorphic and injective for every $ t\in [0,1]$,
(ii) $\Phi_0 : \Omega \to X$ is the natural embedding of $\Omega$ into $X$,
(iii) $\Phi_t (\Omega)$ is a Runge subset of $X$ for every $t\in
[0,1]$, and
(iv) \[fixing\] $\Phi_t (\Omega)$ fixes $Y$ up to order $\ell$, where $\ell$ is such that $\II^\ell{\ensuremath{\operatorname{VF}_{\mathrm{hol}}}}(X,Y)$ is contained the closure of ${\ensuremath{\operatorname{Lie}_{\mathrm{hol}}}}(X,Y)$.
Then for each $\epsilon >0 $ and every compact subset $K \subset
\Omega$ there is a continuous family, $\alpha: [0, 1] \to
{\ensuremath{\operatorname{Aut}}}_{hol} (X)$ of holomorphic automorphisms of $X$ fixing $Y$ pointwise such that $$\alpha_0 = id \, \, \, {\rm and} \, \,
\,\vert \alpha_t - \Phi_t \vert_K <\epsilon {\rm \ \ for\ \ every\
\ } t \in [0,1]$$
Point $(iv)$ in the assumptions of the theorem means the following: Consider the time dependent vector field $V(x,t_0)=\left.\frac{d}{dt}\right|_{t=t_0}\Phi_t(\Phi_{t_0}^{-1}(x))$. The isotopy $\Phi_t (\Omega)$ fixes $Y$ up to order $\ell$ if $V(x,t_0)$ is a section of $\II^\ell{\ensuremath{\operatorname{VF}_{\mathrm{hol}}}}(X,Y)$ over $\Phi_{t_0}(\Omega)$ for all $t_0$.
The map $\Phi_{t_0}$ is the $t_0$-map of the time dependent vector field $V(x,t)$. It can be approximated by dividing the time interval into small pieces and integrating the time independent vector fields over each piece. By assumption, each of those time independent fields is a section in $\II^\ell{\ensuremath{\operatorname{VF}_{\mathrm{hol}}}}(X,Y)(\Phi_{t_0}(\Omega))$. Since the sheaf $\II^\ell{\ensuremath{\operatorname{VF}_{\mathrm{hol}}}}(X,Y)$ is coherent, a similar use of Theorem A and B of Cartan as in the proof of Proposition \[GAGA\] leads to the fact that these time independent vector fields in the Runge domain $\Phi_{t_0}(\Omega)$ can be approximated by global vector fields in $\II^\ell{\ensuremath{\operatorname{VF}_{\mathrm{hol}}}}(X,Y)$. By assumption, these vector fields can be approximated by Lie combinations of complete vector fields vanishing in $Y$ (not necessarily in $\II^\ell{\ensuremath{\operatorname{VF}_{\mathrm{hol}}}}(X,Y)$). Now the standard use of Euler’s method gives the desired conclusion.
If $Y\cap \Phi_t(\Omega)=\emptyset$ for all $t\in [0,1]$, then condition in Theorem \[AL-Theorem\] is trivially satisfied.
Any smooth point in an affine toric variety $X$ of dimension $n\geq
2$ different from the torus has an open neighborhood in the Euclidean topology biholomorphic to ${\ensuremath{\mathbb{C}}}^n$.
Let $x\in X$. Take a Runge neighborhood $U$ of $x$ biholomorphic to the unit ball sending $x$ to zero and let $\Phi_t$ be the map $\left(1-\frac{t}{2}\right)z$ in the unit ball. Since $X$ has the DP relative to ${\ensuremath{X^{\mathrm{sing}}}}$, Theorem \[AL-Theorem\] implies that these contractions can be approximated by holomorphic automorphisms $\alpha_t$ of $X$ (fixing ${\ensuremath{X^{\mathrm{sing}}}}$ pointwise). The automorphism $\alpha_1$ has an attractive fixed point near $x$. The bassin of attraction of this point is biholomorphic to ${\ensuremath{\mathbb{C}}}^n$ [@RoRu88]. Since the holomorphic automorphism group of $X$ is transitive on $X\setminus {\ensuremath{X^{\mathrm{sing}}}}$, the claim follows.
[^1]: [*2000 Mathematics Subject Classification*]{}: 32M05; 32M25; 14M25.\
: Density property, affine toric varieties, locally nilpotent derivations, holomorphic automorphisms, Lie algebras.\
The first and second authors were partially supported by Schweizerischer Nationalfond Grant 200021-140235/1 and the third author was supported by Fondecyt project 11121151
|
---
abstract: |
Let $K$ be a field and $G$ be a finite group. Let $G$ act on the rational function field $K(x(g):g\in G)$ by $K$-automorphisms defined by $g\cdot x(h)=x(gh)$ for any $g,h\in G$. Denote by $K(G)$ the fixed field $K(x(g):g\in G)^G$. Noether’s problem then asks whether $K(G)$ is rational over $K$. Let $p$ be an odd prime and let $G$ be a $p$-group of exponent $p^e$. Assume also that [(i)]{} char $K = p>0$, or [(ii)]{} char $K \ne p$ and $K$ contains a primitive $p^e$-th root of unity. In this paper we prove that $K(G)$ is rational over $K$ for the following two types of groups: [(1)]{} $G$ is a finite $p$-group with an abelian normal subgroup $H$ of index $p$, such that $H$ is a direct product of normal subgroups of $G$ of the type $C_{p^b}\times (C_p)^c$ for some $b,c:1\leq
b,0\leq c$; [(2)]{} $G$ is any group of order $p^5$ from the isoclinic families with numbers $1,2,3,4,8$ and $9$.
address: 'Faculty of Mathematics and Informatics, Constantin Preslavski University, Universitetska str. 115, 9700 Shumen, Bulgaria'
author:
- 'Ivo M. Michailov'
title: 'Noether’s problem for $p$-groups with an abelian subgroup of index $p$'
---
Introduction {#1}
============
Let $K$ be a field and $G$ be a finite group. Let $G$ act on the rational function field $K(x(g):g\in G)$ by $K$-automorphisms defined by $g\cdot x(h)=x(gh)$ for any $g,h\in G$. Denote by $K(G)$ the fixed field $K(x(g):g\in G)^G$. [*Noether’s problem*]{} then asks whether $K(G)$ is rational ($=$ purely transcendental) over $K$. It is related to the inverse Galois problem, to the existence of generic $G$-Galois extensions over $k$, and to the existence of versal $G$-torsors over $k$-rational field extensions (see [@Sw; @Sa1] and [@GMS 33.1, p.86]).
The following well-known theorem gives a positive answer to the Noether’s problem for abelian groups over a field $K$ which contains enough roots of unity.
\[section\]
\[t1.1\] [(Fischer ]{}[@Sw Theorem 6.1][)]{} Let $G$ be a finite abelian group of exponent $e$. Assume that [(i)]{} either char $K
= 0$ or char $K > 0$ with char $K\nmid e$, and [(ii)]{} $K$ contains a primitive $e$-th root of unity. Then $K(G)$ is rational over $K$.
Swan’s paper [@Sw] also gives a survey of many results related to the Noether’s problem for abelian groups. In the same time, just a handful of results about Noether’s problem are obtained when the groups are non-abelian.
We are going to list several results obtained recently by Kang:
\[t1.1\][Theorem]{}
\[t1.2\] [(]{}[@Ka1 Theorem 1.5][)]{} Let $G$ be a metacyclic $p$-group with exponent $p^e$, and let $K$ be any field such that [(i)]{} char $K = p$, or [(ii)]{} char $K \ne p$ and $K$ contains a primitive $p^e$-th root of unity. Then $K(G)$ is rational over $K$.
\[t1.1\][Theorem]{}
\[t1.21\] [(]{}[@Ka2 Theorem 1.8][)]{} Let $n\geq 3$ and let $G$ be a non-abelian $p$-group of order $p^n$ such that $G$ contains a cyclic subgroup of index $p^2$. Assume that $K$ is any field satisfying that either [(i)]{} char $K = p>0$, or [(ii)]{} char $K \ne p$ and $K$ contains a primitive $p^{n-2}$-th root of unity. Then $K(G)$ is rational over $K$.
\[t1.1\][Theorem]{}
\[t1.22\] [(]{}[@Ka3 Cor. 3.2][)]{} Let $K$ be a field and $G$ be a finite group. Assume that [(i)]{} $G$ contains an abelian normal subgroup $H$ so that $G/H$ is cyclic of order $n$, [(ii)]{} $\mathbb Z[\zeta_n]$ is a unique factorization domain, and [(iii)]{} $\zeta_{e'}\in K$ where $e'={\rm lcm}\{{{\rm ord}}(\tau),\exp(H)\}$ and $\tau$ is some element of $G$ whose image generates $G/H$. If $G\rightarrow {{\rm GL}}(V)$ is any finite-dimensional linear representation of $G$ over $K$, then $K(V)^G$ is rational over $K$.
The reader is referred to [@CK; @HuK] for other previous results of Noether’s problem for $p$-groups. It is still an open problem whether Theorem \[t1.22\] could be extended for other similar types of meta-abelian groups with a cyclic quotient. Notice the condition that $\mathbb Z[\zeta_n]$ is a unique factorization domain is satisfied only for $45$ integers $n$, listed in [@Ka3 Theorem 1.5] (the proof is given by Masley and Montgomery [@MM]).
The purpose of this paper is to extend the above results for a certain class of $p$-groups with an abelian subgroup of index $p$. However, we should not ”over-generalize” the above Theorems, because Saltman and Bogomolov prove the following results.
\[t1.1\][Theorem]{}
\[t1.3\] [(Saltman ]{}[@Sa2][)]{} For any prime number $p$ and for any infinite field $K$ with char $K \ne p$ (in particular, $K$ may be an algebraically closed field), there is a meta-abelian $p$-group $G$ of order $p^9$ such that $K(G)$ is not rational over $K$.
\[t1.1\][Theorem]{}
\[t1.31\] [(Bogomolov ]{}[@Bo][)]{} Let $p$ be any prime number, $k$ be any algebraically closed field with char $k\ne p$. There is a group $G$ with order $p^6$ such that $K(G)$ is not rational over $K$.
Moreover, Hoshi, Kang and Kunyavskii [@HKK Theorem 1.12] proved recently that if $G$ is a group of order $p^5$ which belongs to the isoclinism family $\Phi_{10}$, then $K(G)$ is not rational over $K$.
Let $G$ be a group of order $p^n$ for $n\geq 2$ with an abelian subgroup $H$ of order $p^{n-1}$. Bender [@Be2] determined some properties of these groups.
We introduce some notations now. The cyclic group of order $n$ we denote by $C_n$. The subgroups $G_{(0)}=G$ and $G_{(i)}=[G,G_{(i-1)}]$ for $i\geq 1$ are called the lower central series of $G$. Denote $H^p=\{h^p:h\in H\}$. Assume that $H$ is decomposed as a product of abelian groups in the following way: $H\simeq (C_p)^k\times C_{p^{i_1}}\times
C_{p^{i_2}}\times\cdots\times C_{p^{i_t}}$ for $1<i_1\leq
i_2\leq\cdots\leq i_t$, $k\geq 1$ and $k+i_1+i_2+\cdots +i_t=n-1$. Let $\alpha_j$ the generator of the factor $C_{p^{i_j}}$ (for $1\leq
j\leq t$). Choose arbitrary $\alpha\in G$ such that $\alpha\notin
H$. Define the groups $H_j=\langle\alpha^{-x}\alpha_j\alpha^x:0\leq
x\leq p-1\rangle$ for $1\leq j\leq t$. In the following Proposition we find a necessary and sufficient condition for the decomposition of $H$ as a direct product of normal subgroups of $G$ of the type $C_{p^b}\times (C_p)^c$.
\[t1.1\][Proposition]{}
\[prop\] Let $p$ be an odd prime, let $n\geq 2$ and let $G$ be a group of order $p^n$ with an abelian subgroup $H$ of order $p^{n-1}$. Assume that $H$ has at least one cyclic factor of order $p$. Then $H$ is a direct product of normal subgroups of $G$ of the type $C_{p^b}\times
(C_p)^c$ for some $b,c:1\leq b,0\leq c$, if and only if the following two conditions are satisfied:
1. The $p$-th lower central subgroup $G_{(p)}$ is trivial;
2. For any $j:1\leq j\leq t$ we have $H_j\cap H^p=H_j^p$.
In the following main result we investigate Noether’s problem for these $p$-groups. The key idea to prove our result is to find a faithful $G$-subspace $W$ of the regular representation space $\bigoplus_{g\in G} K\cdot x(g)$ and to show that $W^G$ is rational over $K$. The subspace $W$ is obtained as an induced representation from $H$. We apply a method from [@Ka1] to find this representation and then by various linearization methods we reduce the problem to verifying some combinatorial identities.
\[t1.1\][Theorem]{}
\[t1.4\] Let $p$ be an odd prime. Let $G$ be a group of order $p^n$ for $n\geq 2$ with an abelian normal subgroup $H$ of order $p^{n-1}$, and let $G$ be of exponent $p^e$. Assume that $H=H_1\times
H_2\times\cdots\times H_s$ for some $s\geq 1$ where $H_j\simeq
C_{p^{i_j}}\times (C_p)^{k_j}$ and $H_j$ is normal in $G$ for $1\leq
j\leq s, 0\leq k_j,1\leq i_1\leq i_2\leq\cdots\leq i_s$. Assume also that [(i)]{} char $K = p>0$, or [(ii)]{} char $K \ne p$ and $K$ contains a primitive $p^e$-th root of unity. Then $K(G)$ is rational over $K$.
The rationality problem for all $p$-groups of order $\leq p^4$ was solved by Chu and Kang.
\[t1.1\][Theorem]{}
\[t1.6\] [(]{}[@CK Theorem 1.6][)]{} Let $G$ be a $p$-group of order $\leq p^4$, and let $G$ be of exponent $p^e$. Assume that [(i)]{} char $K = p>0$, or [(ii)]{} char $K \ne p$ and $K$ contains a primitive $p^e$-th root of unity. Then $K(G)$ is rational over $K$.
Bender classified in [@Be1] the groups of order $p^5$ which contain an abelian subgroup of order $p^4$. Their number is $p+46$ (if $p-1\ne 3k$) and $p+48$ (if $p-1=3k$). By studying the classification of all groups of order $p^5$ made by James in [@Ja], we see that the non-abelian groups with an abelian subgroup of order $p^4$ and that are not direct products of smaller groups belong to the isoclinic families with numbers $2,3,4$ and $9$. In our second main result we give a positive answer for all groups from these families. As we show in our proof, most of these groups satisfy the conditions of Theorem \[t1.4\]. For the remaining groups we show in a unified way that the answer to Noether’s problem is indeed affirmative.
\[t1.1\][Theorem]{}
\[t1.5\] Let $p$ be an odd prime and let $G$ be any group of order $p^5$ from the isoclinic families (described by James [@Ja]) with numbers $1,2,3,4,8$ and $9$. Assume that $G$ is of exponent $p^e$. Assume also that [(i)]{} char $K = p>0$, or [(ii)]{} char $K \ne p$ and $K$ contains a primitive $p^e$-th root of unity. Then $K(G)$ is rational over $K$.
Notice that Theorem \[t1.5\] is a generalization of [@HKK Theorem 4.1], where is shown that $K(G)$ is *retract* rational over $K$ for any group $G$ that belongs to the isoclinism family $\Phi_i$ where $1\leq i\leq 4$ or $8\leq i\leq 9$.
We organize this paper as follows. In Section \[2\] we recall some preliminaries which will be used in the proof of Theorem \[t1.4\]. We prove Proposition \[prop\] in Section \[3\]. The proofs of Theorems \[t1.4\] and \[t1.5\] are given in Sections \[4\] and \[5\], respectively.
Generalities {#2}
============
We list several results which will be used in the sequel.
\[section\]
\[t2.1\] [(]{}[@HK Theorem 1][)]{} Let $G$ be a finite group acting on $L(x_1,\dots,x_m)$, the rational function field of $m$ variables over a field $L$ such that
(i)
: for any $\sigma\in G, \sigma(L)\subset L;$
(ii)
: the restriction of the action of $G$ to $L$ is faithful;
(iii)
: for any $\sigma\in G$, $$\begin{pmatrix}
\sigma(x_1)\\
\vdots\\
\sigma(x_m)\\
\end{pmatrix}
=A(\sigma)\begin{pmatrix}
x_1\\
\vdots\\
x_m\\
\end{pmatrix}
+B(\sigma)$$ where $A(\sigma)\in{{\rm GL}}_m(L)$ and $B(\sigma)$ is $m\times 1$ matrix over $L$. Then there exist $z_1,\dots,z_m\in L(x_1,\dots,x_m)$ so that $L(x_1,\dots,x_m)^G=L^G(z_1,\dots,z_m)$ and $\sigma(z_i)=z_i$ for any $\sigma\in G$, any $1\leq i\leq m$.
\[t2.1\][Theorem]{}
\[t2.2\] [(]{}[@AHK Theorem 3.1][)]{} Let $G$ be a finite group acting on $L(x)$, the rational function field of one variable over a field $L$. Assume that, for any $\sigma\in G,\sigma(L)\subset L$ and $\sigma(x)=a_\sigma x+b_\sigma$ for any $a_\sigma,b_\sigma\in L$ with $a_\sigma\ne 0$. Then $L(x)^G=L^G(z)$ for some $z\in L[x]$.
\[t2.1\][Theorem]{}
\[t2.3\] [(Kuniyoshi ]{}[@CK Theorem 1.7][)]{} If $char K=p>0$ and $G$ is a finite $p$-group, then $K(G)$ is rational over $K$.
Finally, we give a Lemma, which can be extracted from some proofs in [@Ka2; @HuK].
\[t2.1\][Lemma]{}
\[l2.7\] Let $\langle\tau\rangle$ be a cyclic group of order $n>1$, acting on $L(v_1,\dots,v_{n-1})$, the rational function field of $n-1$ variables over a field $L$ such that $$\begin{aligned}
\tau&:&v_1\mapsto v_2\mapsto\cdots\mapsto v_{n-1}\mapsto (v_1\cdots
v_{n-1})^{-1}\mapsto v_1.\end{aligned}$$ If $L$ contains a primitive $n$th root of unity $\xi$, then $L(v_1,\dots,v_{n-1})=L(s_1,\dots,s_{n-1})$ where $\tau:s_i\mapsto
\xi^is_i$ for $1\leq i\leq n-1$.
Define $w_0=1+v_1+v_1v_2+\cdots+v_1v_2\cdots
v_{n-1},w_1=(1/w_0)-1/n,w_{i+1}=(v_1v_2\cdots v_i/w_0)-1/n$ for $1\leq i\leq n-1$. Thus $L(v_1,\dots,v_{n-1})=L(w_1,\dots,w_n)$ with $w_1+w_2+\cdots+w_n=0$ and $$\begin{aligned}
\tau&:&w_1\mapsto w_2\mapsto\cdots\mapsto w_{n-1}\mapsto w_n\mapsto
w_1.\end{aligned}$$ Define $s_i=\sum_{1\leq j\leq n}\xi^{-ij}w_j$ for $1\leq i\leq n-1$. Then $L(w_1,\dots,w_n)=L(s_1,\dots,s_{n-1})$ and $\tau:s_i\mapsto
\xi^is_i$ for $1\leq i\leq n-1$.
\[t2.1\][Theorem]{}
\[t2.4\] [(]{}[@KP Theorem 1.3][)]{} Let $K$ be any field, and let $H$ and $G$ be finite groups. If $K(H)$ is rational (resp. stably rational, retract rational) over $K$, so is $K(H\times G)$ over $K(G)$. In particular, if both $K(H)$ and $K(G)$ are rational (resp. stably rational, retract rational) over $K$, so is $K(H\times G)$ over K.
Proof of Proposition \[prop\] {#3}
=============================
*I. ’If’ part.* Assume that the conditions (1) and (2) from the statement are satisfied. Put $\beta_1=\alpha_1$. Since $G_{(p)}=\{1\}$, there exist $\beta_1,\dots,\beta_k\in H$ for some $k:1\leq k\leq p$ such that $[\beta_j,\alpha]=\beta_{j+1}$, where $1\leq j\leq k-1$ and $\beta_k\ne 1$ is central. If $\langle\beta_1\rangle$ is a normal subgroup of $G$, then for $\mathcal H=(C_p)^kH_2\cdots H_t$ we have $H\cong\langle\beta_1\rangle\times\mathcal H$. In this way, we see that without a loss of generality, we may assume that $\langle\beta_1\rangle$ is not normal in $G$, and in particular $\beta_1$ is not central (i.e., $k\geq 2$).
We are going to show now that the order of $\beta_2$ is not greater than $p$.
From $[\beta_j,\alpha]=\beta_{j+1}$ it follows the well known formula $$\label{e3.1}
\alpha^{-p}\beta_1\alpha^p=\beta_1\beta_2^{\binom{p}{1}}\beta_3^{\binom{p}{2}}\cdots
\beta_p^{\binom{p}{p-1}}\beta_{p+1},$$ where we put $\beta_{k+1}=\cdots=\beta_{p+1}=1$. Since $\alpha^p$ is in $H$, we obtain the formula $$\beta_2^{\binom{p}{1}}\beta_3^{\binom{p}{2}}\cdots
\beta_k^{\binom{p}{k-1}}=1.$$ Hence $(\beta_2\cdot\prod_{j\ne
2}\beta_j^{a_j})^p=1$ for some integers $a_j$. This identity clearly is impossible if the order of $\beta_2$ is greater than $p$.
It is not hard to conclude now that $H_1\simeq C_{p^{i_1}}\times
(C_p)^{k_1}$ for some $k_1\geq 1$. Note that the elements of $H_1$ are not $p$-th powers of the elements from $H_2\cdots H_t$, since $H_1\cap H^p=H_1^p$. Furthermore, we can adjust the generators of $H_2,\dots,H_t$ so that $H_1\cap(H_2\cdots H_t)=\{1\}$. For example, if we assume that $[\alpha,\alpha_1]=[\alpha,\alpha_2]$, we can define $\alpha_2'=\alpha_2\alpha_1^{-1}$ and get $[\alpha,\alpha_2']=1$. Define $\mathcal
H_2=\langle\alpha^{-x}\alpha_2'\alpha^x\rangle$. Clearly, $H=(C_p)^kH_1\mathcal H_2\cdots H_t$ and $H_1\cap \mathcal
H_2=\{1\}$. With similar changes of the generators we can treat the more general case $[\alpha^x,\alpha_1]=[\alpha^y,\alpha_2]^z$. Proceeding by induction we will obtain a decomposition $H=(C_p)^k\mathcal H_1\mathcal H_2\cdots \mathcal H_t$ such that $\mathcal H_j\cap(\mathcal H_{j+1}\cdots\mathcal H_t)=\{1\}$ for any $j$. Therefore $H=N_1\times\cdots \times N_r\times\mathcal
H_1\times\cdots\times\mathcal H_t$ where $N_1,\dots,N_r$ are normal groups of the type $(C_p)^a$.
*II. ’Only if’ part.* Assume that $H=N_1\times\cdots \times
N_r\times\mathcal H_1\times\cdots\times\mathcal H_t$ where $N_1,\dots,N_r$ are normal groups of the type $(C_p)^a$ and $\mathcal H_1,\dots,\mathcal H_t$ are normal groups of the type $C_{p^b}\times (C_p)^c$.
Suppose that $G_{(p)}\ne\{1\}$. Then we can assume that there exist $\beta_1,\dots,\beta_{p+1}\in\mathcal H_1$ such that $[\beta_j,\alpha]=\beta_{j+1}$, where $1\leq j\leq p$ and $\beta_{p+1}\ne 1$. (We again assume that $\beta_1$ is the generator of the factor of the type $C_{p^b}$.) From the identity it follows that $$\beta_2^{\binom{p}{1}}\beta_3^{\binom{p}{2}}\cdots
\beta_p^{\binom{p}{p-1}}\beta_{p+1}=1.$$ Hence $\beta_2$ will have an order bigger than $p$, which is a contradiction. Therefore, $G_{(p)}=\{1\}$.
Now, suppose that $H_j\cap H^p\ne H_j^p$. Let us consider first the particular case $H_j\leq \mathcal H_j$. Then there exists $x\in\mathbb Z$ such that $[\alpha_j,\alpha^x]\notin
\langle\alpha_j^p\rangle$ and $[\alpha_j,\alpha^x]\in H_j\cap
H^p=\mathcal H_j^p$. Hence $[\alpha_j,\alpha^x]$ is of order not less than $p^2$, so $\mathcal H_j$ can not be of the type $C_{p^b}\times (C_p)^c$, which is a contradiction.
Finally, consider the general case when $\alpha_j$ is decomposed in an unique way as a product of elements from some normal factors $N_i$ and $\mathcal H_k$ ($1\leq i\leq r,1\leq k\leq t$). Then there exists $x\in\mathbb Z$ such that $[\alpha_j,\alpha^x]=\beta^p$ for some $\beta\in H$, where $\beta\notin H_j^p$. From the uniqueness of the decompositions of $[\alpha_j,\alpha^x]$ and $\beta$ as products of elements from the normal factors, it follows that we can reduce this general case to the particular case we just considered. We are done.
Proof of Theorem \[t1.4\] {#4}
=========================
If char $K=p>0$, we can apply Theorem \[t2.3\]. Therefore, we will assume that char $K\ne p$.
Recall that $H=H_1\times H_2\times\cdots\times H_s$, where $H_j\simeq C_{p^{i_j}}\times (C_p)^{k_j}$. Denote by $\beta_1$ the generator of the direct factor $C_{p^{i_1}}$ and put $k=k_1$. Then there exist $\beta_2,\dots,\beta_k\in H_1$ such that $[\beta_j,\alpha]=\beta_{j+1}$, where $1\leq j\leq k-1$ and $\beta_k\ne 1$ is central.
We divide the proof into several steps. We are going now to find a faithful representation of $G$.
*Step 1.* Let $V$ be a $K$-vector space whose dual space $V^*$ is defined as $V^*=\bigoplus_{g\in G}K\cdot x(g)$ where $G$ acts on $V^*$ by $h\cdot x(g)=x(hg)$ for any $h,g\in G$. Thus $K(V)^G=K(x(g):g\in G)^G=K(G)$.
Define $X_1,X_2,\dots,X_k\in V^*$ by $$X_j=\sum_{\ell_1,\dots,\ell_k}x\left(\prod_{m\ne
j}\beta_m^{\ell_m}\right),$$ for $1\leq j\leq k$. Note that $\beta_j\cdot X_i=X_i$ for $j\ne i$. Let $\zeta_{p^{i_1}}\in K$ be a primitive $p^{i_1}$-th root of unity and let $\zeta$ be a primitive $p$-th root of unity. Define $Y_1,Y_2,\dots,Y_k\in V^*$ by $$Y_1=\sum_{r=0}^{p^{i_1}-1}\zeta_{p^{i_1}}^{-r}\beta_1^r\cdot X_1,~
Y_j=\sum_{r=0}^{p-1}\zeta^{-r}\beta_j^r\cdot X_j$$ for $2\leq j\leq k$. [^1]
It follows that [$$\begin{aligned}
\beta_1\ :\
&Y_1\mapsto\zeta_{p^{i_1}} Y_1,~ Y_i\mapsto Y_i,\ \text{for}\ i\ne
1,\\
\beta_j\ :\
&Y_j\mapsto\zeta Y_j,~ Y_i\mapsto Y_i,\ \text{for}\ i\ne j\
\text{and}\ 2\leq j\leq k.\end{aligned}$$]{} Thus $V_1=\bigoplus_{1\leq j\leq k}K\cdot Y_j$ is a representation space of the subgroup $H_1$. In the same way we can construct a representation space $V_j$ of the group $H_j$ for any $j:2\leq j\leq
s$. Therefore, $\bigoplus_{1\leq j\leq s}V_j$ is a representation space of the subgroup $H$.
Define $x_{ji}=\alpha^i\cdot Y_j$ for $1\leq j\leq k,0\leq i\leq
p-1$. Recall that $[\beta_j,\alpha]=\beta_{j-1}$. Hence $$\alpha^{-i}\beta_j\alpha^i=\beta_j\beta_{j+1}^{\binom{i}{1}}\beta_{j+2}^{\binom{i}{2}}\cdots
\beta_k^{\binom{i}{k-j}}.$$
It follows that [$$\begin{aligned}
\beta_1\ :\
&x_{1i}\mapsto\zeta_{p^{i_1}} x_{1i},~ x_{ji}\mapsto
\zeta^{\binom{i}{j-1}} x_{ji},\ \text{for}\ 2\leq j\leq k\
\text{and}\ 0\leq i\leq p-1,\\
\beta_j\ :\ &x_{\ell i}\mapsto x_{\ell i},~ x_{mi}\mapsto
\zeta^{\binom{i}{m-j}} x_{mi},\ \text{for}\ 1\leq \ell\leq j-1,j\leq
m\leq k\ \text{and}\ 0\leq i\leq p-1,\\
\alpha\ :\ &x_{j0}\mapsto x_{j1}\mapsto\cdots\mapsto x_{jp-1}\mapsto
\zeta_{p^{a_j}}^{b_j}x_{j0},\ \text{for}\ 1\leq j\leq k,\end{aligned}$$]{} where $a_j,b_j$ are some integers such that $0\leq b_j< p^{a_j}\leq
p^{i_1}$.
Clearly, $W_1=\bigoplus_{j,i}K\cdot x_{ij}\subset V^*$ is the induced $G$-subspace obtained from $V_1$. In the same way we can construct the induced subspaces $W_j$ obtained from $V_j$. We find that $W=\bigoplus_{1\leq j\leq s}W_j$ is a faithful $G$-subspace of $V^*$. Thus, by Theorem \[t2.1\] it suffices to show that $W^G$ is rational over $K$.
Next, we will consider the action of $G$ on $W_1$.
*Step 2.* For $1\leq j\leq k$ and for $1\leq i\leq p-1$ define $y_{ji}=x_{ji}/x_{ji-1}$. Thus $W_1=K(x_{j0},y_{ji}:1\leq j\leq
k,1\leq i\leq p-1)$ and for every $g\in G$ $$g\cdot x_{j0}\in K(y_{ji}:1\leq j\leq k,1\leq i\leq p-1)\cdot
x_{j0},\ \text{for}\ 1\leq j\leq k$$ while the subfield $K(y_{ji}:1\leq j\leq k,1\leq i\leq p-1)$ is invariant by the action of $G$, i.e., [$$\begin{aligned}
\beta_1\ :\ &y_{1i}\mapsto
y_{1i},~ y_{ji}\mapsto \zeta^{\binom{i-1}{j-2}} y_{ji},\ \text{for}\
2\leq j\leq k\
\text{and}\ 1\leq i\leq p-1,\\
\beta_j\ :\ &y_{\ell i}\mapsto y_{\ell i},~ y_{mi}\mapsto
\zeta^{\binom{i-1}{m-j-1}} y_{mi},\ \text{for}\ 1\leq \ell\leq
j,j+1\leq
m\leq k\ \text{and}\ 1\leq i\leq p-1,\\
\alpha\ :\ &y_{j1}\mapsto y_{j2}\mapsto\cdots\mapsto y_{jp-1}\mapsto
\zeta_{p^{a_j}}^{b_j}(y_{j1}\cdots y_{jp-1})^{-1},\ \text{for}\
1\leq j\leq k.\end{aligned}$$]{} From Theorem \[t2.2\] it follows that if $K(y_{ji}:1\leq j\leq
k,1\leq i\leq p-1)^{G}$ is rational over $K$, so is $K(x_{j0},y_{ji}:1\leq j\leq k,1\leq i\leq p-1)^{G}$ over $K$.
Since $K$ contains a primitive $p^e$-th root of unity $\zeta_{p^e}$ where $p^e$ is the exponent of $G$, $K$ contains as well a primitive $p^{{a_j}+1}$-th root of unity, and we may replace the variables $y_{ji}$ by $y_{ji}/\zeta_{p^{{a_j}+1}}^{b_j}$ so that we obtain a more convenient action of $\alpha$ without changing the actions of $\beta_j$’s. Namely we may assume that $$\begin{aligned}
\label{e4.1}
\alpha\ :\ &y_{j1}\mapsto y_{j2}\mapsto\cdots\mapsto y_{jp-1}\mapsto
(y_{j1}y_{j2}\dots y_{jp-1})^{-1}\ \text{for}\ 1\leq j\leq k.\end{aligned}$$
Define $u_{k1}=y_{k1}^p,u_{ki}=y_{ki}/y_{ki-1}$ for $2\leq i\leq
p-1$. Then $K(y_{ji},u_{ki}:1\leq j\leq k-1,1\leq i\leq
p-1)=K(y_{ji}:1\leq j\leq k,1\leq i\leq
p-1)^{\langle\beta_{k-1}\rangle}$. From Theorem \[t2.2\] it follows that if $K(y_{ji},u_{ki}:1\leq j\leq k-1,2\leq i\leq
p-1)^{G}$ is rational over $K$, so is $K(y_{ji},u_{ki}:1\leq j\leq
k-1,1\leq i\leq p-1)^{G}$ over $K$. We have the following actions [$$\begin{aligned}
\beta_j\ :\ &u_{ki}\mapsto
\zeta^{\binom{i-2}{k-j-2}} u_{ki},\
\text{for}\ 2\leq i\leq p-1\ \text{and}\ 1\leq j\leq k-2,\\
\alpha\ :\ &u_{k2}\mapsto u_{k3}\mapsto\cdots\mapsto u_{kp-1}\mapsto
(u_{k1}u_{k2}^{p-1}u_{k3}^{p-2}\cdots u_{kp-1}^2)^{-1}\mapsto
u_{k1}u_{k2}^{p-2}u_{k3}^{p-3}\cdots u_{kp-2}^2u_{kp-1}.\end{aligned}$$]{} For $2\leq i\leq p-1$ define $$v_{ki}=u_{ki}y_{k-1i}^{-1}y_{k-2i}y_{k-3i}^{-1}\cdots
y_{4i}^{(-1)^k}y_{3i}^{(-1)^{k+1}}y_{2i}^{(-1)^{k+2}},$$ and put $v_{k1}=u_{k1}$.
With the aid of the well known property $\binom{n}{m}-\binom{n-1}{m}=\binom{n-1}{m-1}$, it is not hard to verify the following identity [$$\begin{aligned}
\
&\binom{i-2}{k-j-2}-\binom{i-1}{k-j-2}+\binom{i-1}{k-j-3}-\binom{i-1}{k-j-4}+\cdots\\
&\cdots+(-1)^{k-j-1}\binom{i-1}{2}+(-1)^{k-j}\binom{i-2}{1}=0.\end{aligned}$$]{} It follows that [$$\begin{aligned}
\beta_j\ :\
&v_{ki}\mapsto v_{ki},\
\text{for}\ 1\leq i\leq p-1\ \text{and}\ 1\leq j\leq k-2,\\
\alpha\ :\ &v_{k2}\mapsto v_{k3}\mapsto\cdots\mapsto v_{kp-1}\mapsto
A_k\cdot(v_{k1}v_{k2}^{p-1}v_{k3}^{p-2}\cdots v_{kp-1}^2)^{-1},\end{aligned}$$]{} where $A_k$ is some monomial in $y_{ji}$ for $2\leq j\leq k-1,1\leq
i\leq p-1$.
It is obvious that we can proceed in the same way defining elements $v_{k-1i},\dots,v_{2i}$ such that $\beta_j$ acts trivially on all $v_{mi}$’s and the action of $\alpha$ is given by $$\begin{aligned}
\label{e4.2}
\alpha\ :\ &v_{m1}\mapsto v_{m1}v_{m2}^p,~ v_{m2}\mapsto
v_{m3}\mapsto\cdots\mapsto v_{mp-1}\mapsto
A_m\cdot(v_{m1}v_{m2}^{p-1}v_{m3}^{p-2}\cdots v_{mp-1}^2)^{-1},\end{aligned}$$ where $A_m$ is some monomial in $v_{2i},\dots,v_{m-1i}$ for $3\leq
m\leq k$ and $A_2=1$. In this way we obtain that $K(y_{1i},v_{ji})=K(y_{ji})^{H_1}$.
We will ”linearize” the action applying repeatedly Kang’s argument from [@Ka2 Case 5, Step II]. (Note that the linearization of $\alpha$ on $y_{1i}$’s follows from Lemma \[l2.7\].)
*Step 3.* We write the additive version of the multiplication action of $\alpha$ in formula , i.e., consider the $\mathbb Z[\pi]$-module $M=\bigoplus_{1\leq m\leq k}(\oplus_{1\leq
i\leq p-1}\mathbb Z\cdot v_{mi})$ corresponding to , where $\pi=\langle\alpha\rangle$. Denote the submodules $M_j=\bigoplus_{1\leq m\leq j}(\oplus_{1\leq i\leq p-1}\mathbb
Z\cdot v_{mi})$ for $1\leq j\leq k$. Thus $\alpha$ has the following additive action [$$\begin{aligned}
\alpha\ :\ &v_{j1}\mapsto v_{j1}+pv_{j2},~\\
&v_{j2}\mapsto
v_{j3}\mapsto\cdots\mapsto v_{jp-1}\mapsto
A_j-v_{j1}-(p-1)v_{j2}-(p-2)v_{j3}-\cdots -2v_{jp-1},\end{aligned}$$]{} where $A_j\in M_{j-1}$.
By Lemma \[l2.7\], $M_1$ is isomorphic to the $\mathbb
Z[\pi]$-module $N=\oplus_{1\leq i\leq p-1}\mathbb Z\cdot s_i$ where $s_1=v_{12},s_i=\alpha^{i-1}\cdot v_{12}$ for $2\leq i\leq p-1$, and $$\begin{aligned}
\alpha\ :\ &s_1\mapsto s_2\mapsto\cdots\mapsto s_{p-1}\mapsto
-s_1-s_2-\cdots-s_{p-1}\mapsto s_1.\end{aligned}$$
Let $\Phi_p(T)\in\mathbb Z[T]$ be the $p$-th cyclotomic polynomial. Since $\mathbb Z[\pi]\simeq\mathbb Z[T]/T^p-1$, we find that $\mathbb Z[\pi]/\Phi_p(\alpha)\simeq \mathbb Z[T]/\Phi_p(T)\simeq
\mathbb Z[\omega]$, the ring of $p$-th cyclotomic integer. As $\Phi_p(\alpha)\cdot x=0$ for any $x\in N$, the $\mathbb
Z[\pi]$-module $N$ can be regarded as a $\mathbb Z[\omega]$-module through the morphism $\mathbb Z[\pi]\to\mathbb
Z[\pi]/\Phi_p(\alpha)$. When $N$ is regarded as a $\mathbb
Z[\omega]$-module, $N\simeq\mathbb Z[\omega]$ the rank-one free $\mathbb Z[\omega]$-module.
We claim that $M$ itself can be regarded as a $\mathbb
Z[\omega]$-module, i.e., $\Phi_p(\alpha)\cdot M=0$.
Return to the multiplicative notations in Step 2. Note that all $v_{ji}$’s are monomials in $y_{ji}$’s. The action of $\alpha$ on $y_{ji}$ given in formula satisfies the relation $\prod_{0\leq m\leq p-1}\alpha^m(y_{ji})=1$ for any $1\leq j\leq
k,1\leq i\leq p-1$. Using the additive notations, we get $\Phi_p(\alpha)\cdot y_{ji}=0$. Hence $\Phi_p(\alpha)\cdot M=0$.
Define $M'=M/M_{k-1}$. It follows that we have a short exact sequence of $\mathbb Z[\pi]$-modules $$\label{e4.3}
0\to M_{k-1}\to M\to M'\to 0.$$ Since $M$ is a $\mathbb Z[\omega]$-module, is a short exact sequence of $\mathbb Z[\omega]$-modules. Proceeding by induction, we obtain that $M$ is a direct sum of free $\mathbb
Z[\omega]$-modules isomorphic to $N$. Therefore, $M\simeq\oplus_{1\leq j\leq k}N_j$, where $N_j\simeq N$ is a free $\mathbb Z[\omega]$-module, and so a $\mathbb Z[\pi]$-module also (for $1\leq j\leq k$).
Finally, we interpret the additive version of $M\simeq\oplus_{1\leq
j\leq k}N_j\simeq N^k$ it terms of the multiplicative version as follows: There exist $w_{ji}$ that are monomials in $v_{ji}$ for $1\leq j\leq k,1\leq i\leq p-1$ such that $K(w_{ji})=K(v_{ji})$ and $\alpha$ acts as $$\begin{aligned}
\alpha\ :\ &w_{j1}\mapsto w_{j2}\mapsto\cdots\mapsto w_{jp-1}\mapsto
(w_{j1}w_{j2}\dots w_{jp-1})^{-1}\ \text{for}\ 1\leq j\leq k.\end{aligned}$$ According to Lemma \[l2.7\], the above action can be linearized. Since $H\simeq H_1\times\cdots \times H_s$ for some normal subgroups $H_j$ of $G$, we obtain that $W^H$ is a $K$-free compositum of fields having a linear action of $\alpha$. Therefore, $W^G$ is rational over $K$. We are done.
Proof of Theorem \[t1.5\] {#5}
=========================
All groups from family 1 are abelian, so we may apply Theorem \[t1.1\].
For the groups that are direct product of smaller groups (contained in families 2 and 3) we may apply Theorems \[t1.6\] and \[t2.4\].
The groups $\Phi_2(41),\Phi_2(32)a_1,\Phi_2(32)a_2$ and $\Phi_8(32)$ are metacyclic, so we may apply Theorem \[t1.2\].
The group $\Phi_2(311)b$ contains a normal subgroup $\langle\alpha_1,\gamma\rangle\simeq C_p\times C_{p^3}$. The group $\Phi_2(311)c$ contains a normal subgroup $\langle\alpha_2,\alpha\rangle\simeq C_p\times C_{p^3}$. The group $\Phi_2(221)c$ contains a normal subgroup $\langle\gamma,\alpha_1,\alpha^p\rangle\simeq C_{p^2}\times
(C_p)^2$. The group $\Phi_2(221)d$ contains a normal subgroup $\langle\alpha_1,\alpha_2,\alpha^p\rangle\simeq C_{p^2}\times
(C_p)^2$. For all these groups we may apply Theorem \[t1.4\].
The group $\Phi_3(311)a$ contains a normal subgroup $\langle\alpha^p,\alpha_1,\alpha_2\rangle\simeq C_{p^2}\times
(C_p)^2$. The group $\Phi_3(311)b_r$ contains a normal subgroup $\langle\alpha_1,\alpha_2\rangle\simeq C_{p^3}\times C_p$. The groups $\Phi_3(221)a$ and $\Phi_3(221)b_r$ contain a normal subgroup $\langle\alpha_1,\alpha_2,\alpha^p\rangle\simeq C_{p^2}\times
(C_p)^2$. The group $\Phi_3(2111)c$ contains a normal subgroup $\langle\gamma,\alpha_1,\alpha_2\rangle\simeq C_{p^2}\times
(C_p)^2$. The group $\Phi_3(2111)d$ contains a normal subgroup $\langle\alpha_1,\alpha_2,\alpha_3,\alpha^p\rangle\simeq (C_p)^4$. The group $\Phi_3(2111)e$ contains a normal subgroup $\langle\alpha_1,\alpha_2,\alpha_3\rangle\simeq C_{p^2}\times
(C_p)^2$. We may again apply Theorem \[t1.4\].
In the same way, we see that Theorem \[t1.4\] may be applied for the groups $\Phi_4(221)a$, $\Phi_4(221)b$, $\Phi_4(2111)a$, $\Phi_4(2111)b,\Phi_4(2111)c$ and $\Phi_4(1^5)$, because each group contains a normal subgroup $H\simeq (C_p)^4$ or $C_{p^2}\times
(C_p)^2$. Similarly, for $p\geq 5$ each group from family $9$ contains a normal subgroup $\langle\alpha_1,\alpha_2,\alpha_3,\alpha_4\rangle\simeq (C_p)^4$. For $p=3$ each group from family $9$ contains a normal subgroup $\langle\alpha_1,\alpha_2,\alpha_3,\alpha_4\rangle\simeq C_9\times
C_9$.
Thus it remains to consider the groups $\Phi_4(221)c$, $\Phi_4(221)d_r$, $\Phi_4(221)e$, $\Phi_4(221)f_0$, $\Phi_4(221)f_r$ (for any odd $p$) and the groups from family $9$ for $p=3$.
Let $G$ be any of these groups and let $H=\langle\alpha_1,\alpha_2\rangle\simeq C_{p^2}\times C_{p^2}$. Let $V$ be a $K$-vector space whose dual space $V^*$ is defined as $V^*=\bigoplus_{g\in G}K\cdot x(g)$ where $G$ acts on $V^*$ by $h\cdot x(g)=x(hg)$ for any $h,g\in G$. Thus $K(V)^G=K(x(g):g\in
G)^G=K(G)$. Let $\zeta_{p^2}$ be a primitive $p^2$-th root of unity, and let $\zeta=\zeta_{p^2}^p$. We may find $Y_1,Y_2\in V^*$ such that [$$\begin{aligned}
\alpha_1\ :\
&Y_1\mapsto\zeta_{p^2} Y_1,~ Y_2\mapsto Y_2,\\
\alpha_2\ :\
&Y_1\mapsto Y_2,~ Y_2\mapsto\zeta_{p^2} Y_2.\end{aligned}$$]{} Thus $V_1=\bigoplus_{1\leq j\leq 2}K\cdot Y_j$ is a representation space of the subgroup $H$. Define $x_i=\alpha^i\cdot
Y_1,y_i=\alpha^i\cdot Y_2$ for $0\leq i\leq p-1$. Note that for the groups from the family $\Phi_4$ we have the relations $\alpha_j\alpha^i=\alpha^i\alpha_j\beta_j^i$ for $0\leq i\leq
p-1,1\leq j\leq 2$. From now on we are going to consider each group individually.
*Case I.* $G=\Phi_4(221)d_r$. Let $\ell$ be an integer such that $k\ell\equiv 1\pmod{p}$. Hence $\beta_1=\alpha_1^{\ell p}$. We have the actions [$$\begin{aligned}
\alpha_1\ :\
&x_i\mapsto\zeta_{p^2}^{1+i\ell p} x_i,~ y_i\mapsto y_i,\\
\alpha_2\ :\
&x_i\mapsto x_i,~ y_i\mapsto\zeta_{p^2}^{1+ip} y_i,\\
\alpha\ :\ &x_0\mapsto x_1\mapsto\cdots\mapsto x_{p-1}\mapsto x_0,\\
&y_0\mapsto y_1\mapsto\cdots\mapsto y_{p-1}\mapsto y_0,\end{aligned}$$]{} where $0\leq i\leq p-1$. We find that $W=(\bigoplus_i K\cdot
x_i)\bigoplus(\bigoplus_i K\cdot y_i)\subset V^*$ is a faithful $G$-subspace of $V^*$. Thus, by Theorem \[t2.1\] it suffices to show that $W^G$ is rational over $K$.
For $1\leq i\leq p-1$ define $u_i=x_i/x_{i-1},v_i=y_i/y_{i-1}$ and $\zeta=\zeta_{p^2}^p$. Thus $W=K(x_0,y_0,u_i,v_i:1\leq i\leq p-1)$ and for every $g\in G$ $$g\cdot x_0\in K(u_i:1\leq i\leq p-1)\cdot x_0,\ \text{and}\ g\cdot
y_0\in K(v_i:1\leq i\leq p-1)\cdot y_0,$$ while the subfield $K(u_i,v_i:1\leq i\leq p-1)$ is invariant by the action of $G$, i.e., [$$\begin{aligned}
\alpha_1\ :\
&u_i\mapsto\zeta^\ell u_i,~ v_i\mapsto v_i,\\
\alpha_2\ :\
&u_i\mapsto u_i,~ v_i\mapsto\zeta v_i,\\
\alpha\ :\ &u_1\mapsto u_2\mapsto\cdots\mapsto u_{p-1}\mapsto (u_1u_2\cdots u_{p-1})^{-1},\\
&v_1\mapsto v_2\mapsto\cdots\mapsto v_{p-1}\mapsto (v_1v_2\cdots
v_{p-1})^{-1}.\end{aligned}$$]{} From Theorem \[t2.2\] it follows that if $K(u_i,v_i:1\leq i\leq
p-1)^{G}$ is rational over $K$, so is $K(x_0,y_0,u_i,v_i:1\leq i\leq
p-1)^{G}$ over $K$.
Define $U_1=u_1^p,U_i=u_i/u_{i-1}$ and $V_1=v_1^p,V_i=v_i/v_{i-1}$ for $2\leq i\leq p-1$. Then $K(U_i,V_i:1\leq i\leq
p-1)=K(u_i,v_i:1\leq i\leq p-1)^{\langle\alpha_1,\alpha_2\rangle}$ and the action of $\alpha$ on $U_i$ and $V_i$ is [$$\begin{aligned}
\alpha\ :\ &U_1\mapsto U_1U_2^p,\\
&U_2\mapsto U_3\mapsto\cdots\mapsto U_{p-1}\mapsto
(U_1U_2^{p-1}U_3^{p-2}\cdots U_{p-1}^2)^{-1}\mapsto\\
&\mapsto U_1U_2^{p-2}U_3^{p-3}\cdots U_{p-2}^2U_{p-1}\mapsto U_2,\\
&V_1\mapsto V_1V_2^p,\\
&V_2\mapsto V_3\mapsto\cdots\mapsto V_{p-1}\mapsto
(V_1V_2^{p-1}V_3^{p-2}\cdots V_{p-1}^2)^{-1}\mapsto\\
&\mapsto V_1V_2^{p-2}V_3^{p-3}\cdots V_{p-2}^2V_{p-1}\mapsto V_2.\end{aligned}$$]{} Define $W_1=U_2,W_i=\alpha^i\cdot U_2$ and $Z_1=V_2,Z_i=\alpha^i\cdot V_2$ for $2\leq i\leq p-1$. Now the action of $\alpha$ is [$$\begin{aligned}
\alpha\ :\ &W_1\mapsto W_2\mapsto\cdots\mapsto W_{p-1}\mapsto
(W_1W_2\cdots W_{p-1})^{-1},\\
&Z_1\mapsto Z_2\mapsto\cdots\mapsto Z_{p-1}\mapsto (Z_1Z_2\cdots
Z_{p-1})^{-1}.\end{aligned}$$]{} Since $U_1=(W_{p-1}W_1^{p-1}W_2^{p-2}\cdots W_{p-2}^2)^{-1}$ and $V_1=(Z_{p-1}Z_1^{p-1}Z_2^{p-2}\cdots Z_{p-2}^2)^{-1}$, we get that $K(U_1,\dots,U_{p-1},V_1,\dots,V_{p-1})=K(W_1,\dots,W_{p-1},Z_1,\dots,Z_{p-1})$. From Lemma \[l2.7\] it follows that the action of $\alpha$ on $K(W_1,\dots,W_{p-1},Z_1,\dots,Z_{p-1})$ can be linearized. It remains to apply Theorem \[t1.1\].
*Case II.* $G=\Phi_4(221)c$. The actions of $\alpha_1$ and $\alpha_2$ on $u_i$ and $v_i$ (we keep the notations from Case I) are [$$\begin{aligned}
\alpha_1\ :\
&u_i\mapsto\zeta u_i,~ v_i\mapsto v_i,\\
\alpha_2\ :\ &u_i\mapsto u_i,~ v_i\mapsto\zeta v_i.\end{aligned}$$]{} These actions are the same as in Case I, so we may apply the same proof.
*Case III.* $G=\Phi_4(221)f_0$. Let $\mu$ be an integer such that $\mu\nu\equiv 1\pmod{p}$. Then $\beta_1=\alpha_2^{\mu p}$. The actions of $\alpha_1$ and $\alpha_2$ on $u_i$ and $v_i$ are [$$\begin{aligned}
\alpha_1\ :\
&u_i\mapsto u_i,~ v_i\mapsto\zeta^\mu v_i,\\
\alpha_2\ :\ &u_i\mapsto\zeta u_i,~ v_i\mapsto v_i.\end{aligned}$$]{} These actions are almost the same as in Case I, so we may apply the same proof.
*Case IV.* $G=\Phi_4(221)e$. Let $s$ be an integer such that $(-1/4)s\equiv 1\pmod{p}$. Then $\beta_2=\alpha_1^{sp}$ and $\beta_1=\alpha_2^p\alpha_1^{-sp}$. The actions of $\alpha_1$ and $\alpha_2$ on $u_i$ and $v_i$ are [$$\begin{aligned}
\alpha_1\ :\
&u_i\mapsto\zeta^{-s} u_i,~ v_i\mapsto\zeta v_i,\\
\alpha_2\ :\ &u_i\mapsto\zeta^s u_i,~ v_i\mapsto v_i.\end{aligned}$$]{} For $1\leq i\leq p-1$ define $w_i=u_iv_i^s$. We have now [$$\begin{aligned}
\alpha_1\ :\
&w_i\mapsto w_i,~ v_i\mapsto\zeta v_i,\\
\alpha_2\ :\ &w_i\mapsto\zeta^s w_i,~ v_i\mapsto v_i.\end{aligned}$$]{} We may apply again the proof of Case I.
*Case V.* $G=\Phi_4(221)f_r$. The proof is the same as Case IV.
*Case VI.* $G=\Phi_9(1^5)$ for $p=3$. Calculations show that we have the following relations: $\alpha_1^3=\alpha_3^{-1}\alpha_4,\alpha_2^3=\alpha_4^{-1},\alpha_3^3=\alpha_4^3=1,
\alpha_1\alpha=\alpha\alpha_1\alpha_2,\alpha_1\alpha^2=\alpha^2\alpha_1^{-2}\alpha_2^{-1},
\alpha_2\alpha=\alpha\alpha_1^{-3}\alpha_2^{-2},\alpha_2\alpha^2=\alpha^2\alpha_2\alpha_1^3$. Hence we obtain the actions [$$\begin{aligned}
\alpha_1\ :\
&x_0\mapsto\zeta_9 x_0,~ x_1\mapsto\zeta_9 x_1,~ x_2\mapsto\zeta_9^{-2} x_2,~ y_0\mapsto y_0,~ y_1\mapsto\zeta_9 y_1,~ y_2\mapsto\zeta_9^{-1} y_2,\\
\alpha_2\ :\
&x_0\mapsto x_0,~ x_1\mapsto\zeta^{-1} x_1,~ x_2\mapsto\zeta x_2,~ y_0\mapsto\zeta_9 y_0,~ y_1\mapsto\zeta_9^{-2} y_1,~ y_2\mapsto\zeta_9 y_2,\\
\alpha\ :\ &x_0\mapsto x_1\mapsto x_2\mapsto x_0,\\
&y_0\mapsto y_1\mapsto y_2\mapsto y_0,\end{aligned}$$]{} where $\zeta_9$ is a primitive $9$-th root of unity and $\zeta=\zeta_9^3$ is a primitive $3$-th root of unity. We find that $W=(\bigoplus_i K\cdot x_i)\bigoplus(\bigoplus_i K\cdot y_i)\subset
V^*$ is a faithful $G$-subspace of $V^*$. Thus, by Theorem \[t2.1\] it suffices to show that $W^G$ is rational over $K$.
For $1\leq i\leq 2$ define $u_i=x_i/x_{i-1},v_i=y_i/y_{i-1}$. Thus $W=K(x_0,y_0,u_i,v_i:1\leq i\leq 2)$ and for every $g\in G$ $$g\cdot x_0\in K(u_i:1\leq i\leq 2)\cdot x_0,\ \text{and}\ g\cdot
y_0\in K(v_i:1\leq i\leq 2)\cdot y_0,$$ while the subfield $K(u_i,v_i:1\leq i\leq 2)$ is invariant by the action of $G$, i.e., [$$\begin{aligned}
\alpha_1\ :\
&u_1\mapsto u_1,~ u_2\mapsto\zeta^{-1} u_2,~ v_1\mapsto\zeta_9 v_1,~ v_2\mapsto\zeta_9^{-2} v_2,\\
\alpha_2\ :\
&u_1\mapsto\zeta^{-1} u_1,~ u_2\mapsto\zeta^{-1} u_2,~ v_1\mapsto\zeta^{-1} v_1,~ v_2\mapsto\zeta v_2,\\
\alpha\ :\ &u_1\mapsto u_2\mapsto (u_1u_2)^{-1},\\
&v_1\mapsto v_2\mapsto (v_1v_2)^{-1}.\end{aligned}$$]{} From Theorem \[t2.2\] it follows that if $K(u_i,v_i:1\leq i\leq
2)^{G}$ is rational over $K$, so is $K(x_0,y_0,u_i,v_i:1\leq i\leq
2)^{G}$ over $K$.
Define $w_1=v_1^3,w_2=v_2/v_1$. Then $K(u_1,u_2,w_1,w_2)=K(u_1,u_2,v_1,v_2)^{\langle\alpha_1^3\rangle}$, and the actions of $\alpha_1,\alpha_2$ and $\alpha$ on $K(u_1,u_2,w_1,w_2)$ are [$$\begin{aligned}
\alpha_1\
:\
&u_1\mapsto u_1,~ u_2\mapsto\zeta^{-1} u_2,~ w_1\mapsto\zeta w_1,~ w_2\mapsto\zeta^{-1} w_2,\\
\alpha_2\ :\
&u_1\mapsto\zeta^{-1} u_1,~ u_2\mapsto\zeta^{-1} u_2,~ w_1\mapsto w_1,~ w_2\mapsto\zeta^{-1} w_2,\\
\alpha\ :\ &u_1\mapsto u_2\mapsto (u_1u_2)^{-1},\\
&w_1\mapsto w_2^3w_1,~ w_2\mapsto (w_1w_2^2)^{-1}.\end{aligned}$$]{} Define $V_1=w_2,V_2=(w_1w_2^2)^{-1}$. Then $K(u_1,u_2,V_1,V_2)=K(u_1,u_2,w_1,w_2)$, and the actions of $\alpha_1,\alpha_2$ and $\alpha$ on $K(u_1,u_2,V_1,V_2)$ are [$$\begin{aligned}
\alpha_1\ :\
&u_1\mapsto u_1,~ u_2\mapsto\zeta^{-1} u_2,~ V_1\mapsto\zeta^{-1} V_1,~ V_2\mapsto\zeta V_2,\\
\alpha_2\ :\
&u_1\mapsto\zeta^{-1} u_1,~ u_2\mapsto\zeta^{-1} u_2,~ V_1\mapsto\zeta^{-1} V_1,~ V_2\mapsto\zeta^{-1} V_2,\\
\alpha\ :\ &u_1\mapsto u_2\mapsto (u_1u_2)^{-1},\\
&V_1\mapsto V_2\mapsto (V_1V_2)^{-1}.\end{aligned}$$]{} Define $U_1=u_1^3,U_2=u_2/u_1,W_1=V_1/u_1,W_2=V_2/u_2$. Then $K(U_1,U_2,W_1,W_2)=K(u_1,u_2,V_1,V_2)^{\langle\alpha_2\rangle}$, and the actions of $\alpha_1$ and $\alpha$ on $K(U_1,U_2,W_1,W_2)$ are [$$\begin{aligned}
\alpha_1\ :\
&U_1\mapsto U_1,~ U_2\mapsto\zeta^{-1} U_2,~ W_1\mapsto\zeta^{-1} W_1,~ W_2\mapsto\zeta^{-1} W_2,\\
\alpha\ :\ &U_1\mapsto U_2^3U_1,~ U_2\mapsto (U_1U_2^2)^{-1},\\
&W_1\mapsto W_2\mapsto (W_1W_2)^{-1}.\end{aligned}$$]{} Define $\tilde u_1=U_2,\tilde u_2=(U_1U_2^2)^{-1},\tilde
v_1=W_1/\tilde u_1,\tilde v_2=W_2/\tilde u_2$. Then $K(\tilde
u_1,\tilde u_2,\tilde v_1,\tilde v_2)=K(U_1,U_2,W_1,W_2)$, and the actions of $\alpha_1$ and $\alpha$ on $K(\tilde u_1,\tilde
u_2,\tilde v_1,\tilde v_2)$ are [$$\begin{aligned}
\alpha_1\ :\
&\tilde u_1\mapsto\zeta^{-1} \tilde u_1,~ \tilde u_2\mapsto\zeta^{-1} \tilde u_2,~ \tilde v_1\mapsto\tilde v_1,~ \tilde v_2\mapsto\tilde v_2,\\
\alpha\ :\ &\tilde u_1\mapsto \tilde u_2\mapsto (\tilde u_1\tilde u_2)^{-1},\\
&\tilde v_1\mapsto \tilde v_2\mapsto (\tilde v_1\tilde v_2)^{-1}.\end{aligned}$$]{} Define $\tilde U_1=\tilde u_1^3, \tilde U_2=\tilde u_2/\tilde u_1$. Then $K(\tilde U_1,\tilde U_2,\tilde v_1,\tilde v_2)=K(\tilde
u_1,\tilde u_2,\tilde v_1,\tilde v_2)^{\langle\alpha_1\rangle}$, and the action of $\alpha$ on $K(\tilde U_1,\tilde U_2,\tilde v_1,\tilde
v_2)$ is [$$\begin{aligned}
\alpha\ :\ &\tilde
U_1\mapsto \tilde U_2^3\tilde U_1,~ \tilde U_2\mapsto (\tilde
U_1\tilde
U_2^2)^{-1},\\
&\tilde v_1\mapsto \tilde v_2\mapsto (\tilde v_1\tilde v_2)^{-1}.\end{aligned}$$]{} Finally, define $\tilde w_1=\tilde U_2,\tilde w_2=(\tilde U_1\tilde
U_2^2)^{-1}$. Hence $\alpha(\tilde w_2)=(\tilde w_1\tilde
w_2)^{-1}$. According to Lemma \[l2.7\] the action of $\alpha$ on $K(\tilde w_1,\tilde w_2,\tilde v_1,\tilde v_2)$ can be linearized.
*Case VII.* $G=\Phi_9(2111)a$ for $p=3$. As we have shown in the proof of Theorem \[t1.4\], when $\alpha^p\in H$, we can easily find proper substitutions so that we may assume $\alpha^p=1$. Apply Case VI.
*Case VIII.* $G=\Phi_9(2111)b_r$ for $p=3$. Calculations show that we have the following relations: $\alpha_1^3=\alpha_3^{-1}\alpha_4^{-1},\alpha_2^3=\alpha_4^{-1},\alpha_3^3=\alpha_4^3=1,
\alpha_1\alpha=\alpha\alpha_1\alpha_2,\alpha_1\alpha^2=\alpha^2\alpha_1^{-2}\alpha_2^{-4},
\alpha_2\alpha=\alpha\alpha_1^{-3}\alpha_2^4,\alpha_2\alpha^2=\alpha^2\alpha_2^4\alpha_1^3$. Hence we obtain the actions [$$\begin{aligned}
\alpha_1\ :\
&x_0\mapsto\zeta_9 x_0,~ x_1\mapsto\zeta_9 x_1,~ x_2\mapsto\zeta_9^{-2} x_2,~ y_0\mapsto y_0,~ y_1\mapsto\zeta_9 y_1,~ y_2\mapsto\zeta_9^{-4} y_2,\\
\alpha_2\ :\
&x_0\mapsto x_0,~ x_1\mapsto\zeta^{-1} x_1,~ x_2\mapsto\zeta x_2,~ y_0\mapsto\zeta_9 y_0,~ y_1\mapsto\zeta_9^4 y_1,~ y_2\mapsto\zeta_9^4 y_2,\\
\alpha\ :\ &x_0\mapsto x_1\mapsto x_2\mapsto x_0,\\
&y_0\mapsto y_1\mapsto y_2\mapsto y_0.\end{aligned}$$]{} We find that $W=(\bigoplus_i K\cdot x_i)\bigoplus(\bigoplus_i K\cdot
y_i)\subset V^*$ is a faithful $G$-subspace of $V^*$. Thus, by Theorem \[t2.1\] it suffices to show that $W^G$ is rational over $K$.
For $1\leq i\leq 2$ define $u_i=x_i/x_{i-1},v_i=y_i/y_{i-1}$. Thus $W=K(x_0,y_0,u_i,v_i:1\leq i\leq 2)$ and for every $g\in G$ $$g\cdot x_0\in K(u_i:1\leq i\leq 2)\cdot x_0,\ \text{and}\ g\cdot
y_0\in K(v_i:1\leq i\leq 2)\cdot y_0,$$ while the subfield $K(u_i,v_i:1\leq i\leq 2)$ is invariant by the action of $G$, i.e., [$$\begin{aligned}
\alpha_1\ :\
&u_1\mapsto u_1,~ u_2\mapsto\zeta^{-1} u_2,~ v_1\mapsto\zeta_9 v_1,~ v_2\mapsto\zeta_9^4 v_2,\\
\alpha_2\ :\
&u_1\mapsto\zeta^{-1} u_1,~ u_2\mapsto\zeta^{-1} u_2,~ v_1\mapsto\zeta v_1,~ v_2\mapsto v_2,\\
\alpha\ :\ &u_1\mapsto u_2\mapsto (u_1u_2)^{-1},\\
&v_1\mapsto v_2\mapsto (v_1v_2)^{-1}.\end{aligned}$$]{} From Theorem \[t2.2\] it follows that if $K(u_i,v_i:1\leq i\leq
2)^{G}$ is rational over $K$, so is $K(x_0,y_0,u_i,v_i:1\leq i\leq
2)^{G}$ over $K$.
Define $w_1=v_1^3,w_2=v_2/v_1,V_1=w_2,V_2=(w_1w_2^2)^{-1}$. Then $K(u_1,u_2,V_1,V_2)=K(u_1,u_2,v_1,v_2)^{\langle\alpha_1^3\rangle}$, and the actions of $\alpha_1,\alpha_2$ and $\alpha$ on $K(u_1,u_2,V_1,Vw_2)$ are [$$\begin{aligned}
\alpha_1\ :\
&u_1\mapsto u_1,~ u_2\mapsto\zeta^{-1} u_2,~ V_1\mapsto\zeta V_1,~ V_2\mapsto V_2,\\
\alpha_2\ :\
&u_1\mapsto\zeta^{-1} u_1,~ u_2\mapsto\zeta^{-1} u_2,~ V_1\mapsto\zeta^{-1} V_1,~ V_2\mapsto\zeta^{-1} V_2,\\
\alpha\ :\ &u_1\mapsto u_2\mapsto (u_1u_2)^{-1},\\
&V_1\mapsto V_2\mapsto (V_1V_2)^{-1}.\end{aligned}$$]{} Define $U_1=u_1^3,U_2=u_2/u_1,W_1=V_1/u_1,W_2=V_2/u_2$. Then $K(U_1,U_2,W_1,W_2)=K(u_1,u_2,V_1,V_2)^{\langle\alpha_2\rangle}$, and the actions of $\alpha_1$ and $\alpha$ on $K(U_1,U_2,W_1,W_2)$ are [$$\begin{aligned}
\alpha_1\ :\
&U_1\mapsto U_1,~ U_2\mapsto\zeta^{-1} U_2,~ W_1\mapsto\zeta W_1,~ W_2\mapsto\zeta W_2,\\
\alpha\ :\ &U_1\mapsto U_2^3U_1,~ U_2\mapsto (U_1U_2^2)^{-1},\\
&W_1\mapsto W_2\mapsto (W_1W_2)^{-1}.\end{aligned}$$]{} Define $\tilde u_1=U_2,\tilde u_2=(U_1U_2^2)^{-1},\tilde
v_1=W_1\tilde u_1,\tilde v_2=W_2\tilde u_2$. Then $K(\tilde
u_1,\tilde u_2,\tilde v_1,\tilde v_2)=K(U_1,U_2,W_1,W_2)$, and the actions of $\alpha_1$ and $\alpha$ on $K(\tilde u_1,\tilde
u_2,\tilde v_1,\tilde v_2)$ are [$$\begin{aligned}
\alpha_1\ :\
&\tilde u_1\mapsto\zeta^{-1} \tilde u_1,~ \tilde u_2\mapsto\zeta^{-1} \tilde u_2,~ \tilde v_1\mapsto\tilde v_1,~ \tilde v_2\mapsto\tilde v_2,\\
\alpha\ :\ &\tilde u_1\mapsto \tilde u_2\mapsto (\tilde u_1\tilde u_2)^{-1},\\
&\tilde v_1\mapsto \tilde v_2\mapsto (\tilde v_1\tilde v_2)^{-1}.\end{aligned}$$]{} Define $\tilde U_1=\tilde u_1^3, \tilde U_2=\tilde u_2/\tilde u_1$. Then $K(\tilde U_1,\tilde U_2,\tilde v_1,\tilde v_2)=K(\tilde
u_1,\tilde u_2,\tilde v_1,\tilde v_2)^{\langle\alpha_1\rangle}$, and the action of $\alpha$ on $K(\tilde U_1,\tilde U_2,\tilde v_1,\tilde
v_2)$ is [$$\begin{aligned}
\alpha\ :\ &\tilde
U_1\mapsto \tilde U_2^3\tilde U_1,~ \tilde U_2\mapsto (\tilde
U_1\tilde
U_2^2)^{-1},\\
&\tilde v_1\mapsto \tilde v_2\mapsto (\tilde v_1\tilde v_2)^{-1}.\end{aligned}$$]{} Finally, define $\tilde w_1=\tilde U_2,\tilde w_2=(\tilde U_1\tilde
U_2^2)^{-1}$. Hence $\alpha(\tilde w_2)=(\tilde w_1\tilde
w_2)^{-1}$. According to Lemma \[l2.7\] the action of $\alpha$ on $K(\tilde w_1,\tilde w_2,\tilde v_1,\tilde v_2)$ can be linearized.
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[^1]: If $\beta_1$ is central, i.e., $k=1$ we simply do not have $\beta_j$ and $Y_j$ for $j\geq 2$. The proof remains valid, however.
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---
abstract: 'Amorphous solids are critical in the design and production of nanoscale devices, but under strong confinement these materials exhibit changes in their mechanical properties which are not well understood. Phenomenological models explain these properties by postulating an underlying defect structure in these materials but do not detail the microscopic properties of these defects. Using machine learning methods, we identify mesoscale defects that lead to shear banding in polymer nanopillars well below the glass transition temperature as a function of pillar diameter. Our results show that the primary structural features responsible for shear banding on this scale are fluctuations in the diameter of the pillar. Surprisingly, these fluctuations are quite small compared to the diameter of the pillar, less than half of a particle diameter in size. At intermediate pillar diameters, we find that these fluctuations tend to concentrate along the minor axis of shear band planes. We also see the importance of mean “softness” as a classifier of shear banding grow as a function of pillar diameter. Softness is a new field that characterizes local structure and is highly correlated with particle-level dynamics such that softer particles are more likely to rearrange. This demonstrates that softness, a quantity that relates particle-level structure to dynamics on short time and length scales, can predict large time and length scale phenomena related to material failure.'
author:
- 'Robert J. S. Ivancic'
- 'Robert A. Riggleman'
title: Identifying structural signatures of shear banding in polymer nanopillars
---
{width="9cm"} \[fig:TOC\]
There are numerous applications where amorphous organic materials are used in highly confined geometries, including as polymer photoresists in semiconductor manufacturing[@stoykovich_directed_2005], the active layers in organic light-emitting diodes[@zhang_efficient_2014; @kim_multicolored_2015], and in polymer nanocomposites at high loadings of nanoparticles[@huang_polymer_2015; @hor_nanoporous_2017]. In many of these applications, in particular semiconductor manufacturing, the mechanical properties of the confined material are of utmost importance. Generally speaking, amorphous materials have many unique mechanical properties including high strength, high stiffness, and low mechanical dissipation[@ashby_engineering_2012; @jones_engineering_2012; @trexler_mechanical_2010; @jaeger_granular_1996; @bansal_handbook_2013; @parmenter_mechanical_1998; @landel_mechanical_1993]. These properties make them desirable in a number of engineering applications; however, their use is hindered by their tendency to fail in a brittle manner[@robertson_diamond-like_2002; @ashby_metallic_2006; @herrmann_nanoparticle_2007; @zhang_using_2013; @hiemenz_principles_1997]. A hallmark of these catastrophic failure modes is shear banding, the localization of shear strain to a narrow region which develops during deformation [@bigoni_nonlinear_2012; @manning_strain_2007]. Shear banding has been experimentally observed in many types of amorphous materials including: granular materials [@fenistein_kinematics:_2003; @tsai_granular_2005], bubble rafts [@lauridsen_shear-induced_2002; @kabla_local_2003], complex fluids [@mair_observation_1996; @makhloufi_rheo-optical_1995], and metallic glasses [@lu_deformation_2003; @johnson_deformation_2002].
Although shear banding has been extensively studied in the bulk using phenomenological models, a microscopic theory of shear banding has proven elusive. The phenomenological models that describe shear banding can broadly be classified into two types. Solid mechanics models postulate some constitutive relations about how a material behaves at each point in space. In these theories, a shear band forms when a small region of the material has a perturbed set of constitutive relations causing it to shear more easily[@rice_initiation_1973; @rice_localization_1976; @bordignon_strain_2015]. Similarly, mean-field models, including shear transformation zones[@falk_dynamics_1998; @langer_shear-transformation-zone_2008], soft glassy rheology[@sollich_rheology_1997], and others[@zheng_mesoscopic_2009], hypothesize mesocale “configurational soft spots"[@manning_strain_2007], regions that are more likely to yield under shear stress, and these regions propagate to form a shear band. While these two types of theories have significantly different starting points, they both predict that shear bands form from mesoscale defects in a solid but provide few details as to the nature of these defects. Although some indirect estimates of their volume are available [@pan_correlation_2009; @choi_estimation_2012], the microscopic structure that underlies these defects is unknown [@schuh_mechanical_2007]. Moreover, it is unclear whether bulk defects are the primary cause of shear banding in confined materials. Previous work has shown that the location of strain localization is somehow quenched into the molecular structure when forming a glass[@shavit_strain_2014], suggesting that the local structure could play a key role.
In this study, we examine a large set of molecular dynamics simulations of amorphous oligomeric nanopillars that are strained to failure. Using a novel machine learning method, we detect mesoscale structural defects which lead to shear band formation. We systematically vary the pillar diameter in these systems from $12.5$ – $100$ monomer diameters to understand how these defects vary as the system becomes less confined and more bulk-like. From this defect structure, we make quantitative predictions about where shear bands will form. Our machine learning approach allows us to look at a broad array of structural features and perform an unbiased selection of those which correlate with shear banding at each pillar diameter. Here, we pay special attention to another machine-learned microscopic structural quantity, “softness," which is strongly predictive of particle-level rearrangements in disordered materials [@cubuk_identifying_2015]. Softer particles have structures which make them more likely to rearrange than harder (less soft) particles. This quantity has been implicated in the understanding of aging glasses [@schoenholz_relationship_2017] and the universal yield strain in bulk disordered materials [@cubuk_structure-property_2017], but the connection between softness and mesoscale phenomena such as shear banding has not been explored.
We find that small fluctuations in the diameter of the pillar, less than $\frac{1}{2}$ of a particle diameter in size, are most predictive of where shear bands will form in these pillars regardless of the diameter of the pillar. This is surprising as these surface fluctuations are not mechanically induced (from dust for example) but come about from the thermalization of the pillars themselves. We also find that our coarse grained softness features become more important for distinguishing whether a plane will shear band as pillar diameter increases. Planes that are softer than average are more likely to shear band. To ensure the density features are not sufficient to predict shear banding alone, we verify that these softness features do better than random chance at identifying shear bands even in the absence of correlations with other density features.
The importance of these results is twofold. First, they suggest that small surface defects induced during the thermalization of nanoscale amorphous components may play a major role in their mechanical properties up to the micron scale. Indeed, these results suggest that focusing on manufacturing processes that lead to smooth surfaces as opposed to hard interiors will yield stronger nanoscale materials. Second, more fundamentally, they suggest that softness may be the microscopic origin of mesoscale configurational soft spots in the bulk. This connection is non-trivial as we are relating a structural quantity (that is associated with local, short-time scale dynamics) to shear band formation, a non-local, long-time scale event. Even more interesting, we find that we do not need to know the dynamical nature of these defects as we approach the shear banding event. Knowing their configuration prior to deformation is sufficient. This suggests that at temperatures well below the glass transition temperature these defects are locked in place.
Results
=======
Our polymer model is a modified coarse-grained oligomer with five Lennard-Jones interaction sites per chain, and the monomers of each chain are connected with stiff harmonic bonds. The Lennard-Jones potential used in this work is modified to promote shear banding and fracture at temperatures far below the glass transition $T_g$. We prepare nanoscale cylinders by equilibrating our system at temperatures above $T_g$ in a simulation box that is periodic along the length before slowly quenching to $T = 0.05$, which is far below our simulated glass transition temperature $T_g = 0.38$. We note that all quantities are reported in reduced Lennard-Jones units, and complete details of the model are provided in the methods below.
Figure \[fig:MechanicalProperties\] shows that the mechanical properties of our pillars depends strongly on the pillar diameter. To deform our samples, we applied a uniaxial strain to the $\hat{z}$ axis at an engineering strain rate of $\dot{\epsilon} = 2.5 \times 10^{-5}$ at $T = 0.05$. We plot engineering stress-strain curves averaged over all configurations at each pillar diameter in Figure \[fig:MechanicalProperties\]a. We find that both the Young’s modulus, which was determined by linear fits to the initial ($\epsilon \leq 0.005$) stress-strain response, and the strength (stress maximum) of our pillars increases with pillar diameter. Both material properties increase by more than $50$ percent as the pillar diameter increases from $D=12.5$ to $D=100$ as shown in Figure \[fig:MechanicalProperties\]b. The overall trends with sample dimension are in good qualitative agreement with experiments on thin polymer films as a function of film thickness [@stafford_buckling-based_2004; @liu_directly_2015].
![ **Characterization of basic mechanical properties of oligomer nanopillars** **(a)** Stress-strain curves averaged over all configurations found for each nanopillar diameter when deformed under uxiaxial tension at an engineering strain rate $\dot \epsilon = 2.5 \times 10^{-5}$ at $T = 0.05$. The curves are vertically shifted for clarity. **(b)** Young’s modulus (navy squares) and the strength (red diamonds) of the nanopillars as a function of the pillar diameter. **(c)** The von Mises strain field of a single $D = 50$ pillar calculated by comparing the rearrangements surrounding each particle after a strain of $\epsilon = 5.5\%$, and **(d)** the von Mises strain field averaged over $50$ $D = 50$ pillars in the isoconfigurational ensemble. []{data-label="fig:MechanicalProperties"}](0.pdf){width="4in"}
The strain in our samples strongly localizes into a shear band as our deformations reach the yield point. To understand how deformation effects the strain field within our pillars, we examine the von Mises shear strain rate around each particle, denoted as $J_{2}$ which is a common metric in numerical studies of shear banding [@adibi_surface_2015; @li_deformation-driven_2015; @shimizu_theory_2007]. Figure \[fig:MechanicalProperties\]c shows the von Mises strain rate field of a single $D = 50$ pillar, and this field exhibits an unambiguous shear band plane of high von Mises shear strain rate. At this low temperature, all of our samples at any pillar diameter exhibit a strong strain localization.
A key point we wish to address with our study is whether the location where a material fails is dictated by the local structure, and if so, we further wish to identify the structural motifs that promote strain localization and shear banding. To first test whether the local structure plays a role in the localization of a shear band, we employ the isoconfigurational ensemble [@widmer-cooper_central_2009], which is a technique that played a key role in demonstrating that there exists an interplay between local structure and dynamic heterogeneities in supercooled liquids. By beginning a series of simulations with the same monomer positions, but with momenta re-drawn from the Maxwell-Boltzmann distribution, we can examine whether the location of the shear band in our pillar is caused by random thermal fluctuations or the material structure. If we begin with the same configuration used to generate the strain field in Figure \[fig:MechanicalProperties\]c and run 50 deformation trajectories with randomly initialized momenta, the average strain field $\langle J_{2,j}\rangle$ field for each particle $j$ is shown in Figure \[fig:MechanicalProperties\]d. Clearly the strain tends to localize in one of two locations, while if the location of the shear band were random, we would expect a more uniform distribution. These results indicate that the local structure that is frozen when the sample is quenched plays an important role in determining the shear band location, consistent with prior work . Furthermore, this tendency for strain to localize is robust across all studied pillar diameters.
Having established that the local structure dictates where shear bands will form using the isoconfigurational ensemble, in order to guide the development of mesoscale and constitutive models it is essential to determine the nature of the structural variables that lead to strain localization. As a result, our next goal is to identify which structural motifs (e.g., the local density in the center of the pillar, or perhaps the local roughness on the surface) are associated with shear band formation. We approach this problem as one of classification in which we want to distinguish between two sets of planes: those that are likely to shear band and those that are not; these sets will be called “weak" and “strong" planes respectively. Thus, we aim to create an independent function for each pillar diameter, called a “classifier", that can classify a plane into the weak or strong category at each pillar diameter based on its structure alone. Using specific classifiers for each pillar diameter allows for the possibility that the features which determine shear banding vary with pillar diameter. To develop our classifier, we build a “training set" of planes: one population that does shear band, and a second population that does not shear band, which are defined based on the largest and smallest average von Mises shear strain rate in a pillar, respectively. These planes are selected from a set of $50$ or more independent pillar thermalizations and deformations at each pillar diameter.
To solve this classification problem, each candidate plane is characterized by $M$ “structure functions", which encode the density and local softness distribution as a function of radial position in the plane, distance away from the plane, or angular slices through the major and minor plane axes. Each plane $i$ is assigned a vector $\boldsymbol{p}_i$ with $M$ elements that each correspond to a distinct structure function. A linear support vector machine (SVM) finds the best hyperplane to separate shear band and non-shear band structure vectors in $\mathbb{R}^{M}$. We define the “weakness" of a plane $i$, $W_{i}$, to be the shortest signed distance from $\boldsymbol{p}_{i}$ to this hyperplane in $\mathbb{R}^{M}$. Larger values of plane weakness indicate planes that are structurally similar to shear banding planes while smaller values of $W_i$ indicate little structural similarity to shear banding planes. This hyperplane is then employed to determine the plane weakness of any plane at a given pillar diameter. We normalize our hyperplane so that the distribution of plane weakness has a standard deviation of $1$. Our SVM method was implemented using scikit-learn [@pedregosa_scikit-learn:_2011], and recursive feature elimination allows us to ensure that our models are not overfit[@guyon_gene_2002].
![ **Performance of plane weakness as structural indicator of shear banding planes.** **(a)** Test set accuracy (navy squares) and expected percentage of shear bands that are weak (red diamonds) at all pillar diameters. **(b)** The probability that a plane will shear band as a function of its weakness at pillar diameters $D = 12.5$ and $D = 100$. Solid lines are exponential fits to the data. **(c)** A snapshot of an undeformed $D = 50$ pillar where each monomer $j$ is colored by $P_j$. Error bars in the above fits are calculated using a binomial confidence interval. []{data-label="fig:Performance"}](1.pdf){width="4.5in"}
Figure \[fig:Performance\]a demonstrates that our classifiers are able to distinguish shear banding planes from non-shear banding planes at each pillar diameter. The test set accuracy gives an unbiased estimate of the percentage of shear band and non-shear band planes that are correctly classified. At each pillar diameter over $85\%$ of planes are correctly classified, which is $8$ standard errors above random ($50\%$) proving that we do better than chance at distinguishing between shear band and non-shear band planes. The second metric, $P\left(W>0|\text{SB}\right)$, provides the probability that a shear band plane ($\text{SB}$) is classified as weak ($W>0$). We find that over $90\%$ of shear band planes are weak at each pillar diameter. These results show that our linear SVMs correctly classify the vast majority of shear band planes as weak.
Now we consider the predictive nature of plane weakness’ magnitude rather than its sign alone. We plot the probability a plane will shear band for a given plane weakness, $P\left(\text{SB}|W\right)$, in Figure \[fig:Performance\]b for the $D = 12.5$ and $D = 100$ pillars. We see an exponential increase by more than $2$ decades over the range $W = 0$ to $W = 3$ in the probability of shear banding, and the trends are remarkably similar across pillar diameter, despite the fact that each diameter is characterized by a distinct classifier. This plot explicitly demonstrates that the probability of a shear banding is a function of magnitude, not just the sign, of plane weakness. As a plane becomes weaker as quantified by the local structure through $W_i$, it is more likely to shear band.
We next investigate whether there are spatial correlations in plane weakness that lead to regions in our sample that are more (or less) likely to shear band. To do so, we begin with $P(SB | W_i)$, the probability that plane $i$ of given weakness will shear band, and map it to the particles near the plane to estimate the probability that particle $j$ will be in a shear band, $$P_{j} =
\frac{\sum_{i} P\left(\text{SB}|W_{i}\right) \Theta_{ij}^{P}\left( 0, \xi_{h} \right)}
{\sum_{i}\Theta_{ij}^{P}\left( 0, \xi_{h} \right)}.
\label{eq:7}$$ Here, the sum is over all planes, $\Theta_{ij}^{P} \left(h, \xi_{h} \right) = e^{-\left(\lvert h_{ij} \rvert-h \right)^{2}/\xi_{h}^{2}}$ is a weighting function that controls the spatial extent of the mapping from plane $i$ to particle $j$, $h_{ij}$ is the distance between plane $i$ and particle $j$ and $\xi_h = 1/2$ is a parameter that controls the decay length of $\Theta_{ij}^P$. The map of $P_j$ for all particles is shown for a $D = 50$ pillar in Figure \[fig:Performance\]c, and this is the same pillar configuration shown in Figures \[fig:MechanicalProperties\]c and \[fig:MechanicalProperties\]d. Evidently, spatial correlations exist in plane weakness leading to two large defect regions in the pillar where the particles are more likely to be involved in a shear band. The locations of high average von Mises shear strain rate seen in Figure \[fig:MechanicalProperties\]d show striking similarities with regions of high $P_j$ in Figure \[fig:Performance\]c. The Pearson correlation between these two plots is $0.52$, and the probability that there is no correlation between these fields is less than $10^{-6}$. This strong correlation demonstrates that plane weakness predicts not only the planes that are likely to fail but also the spatial regions that are likely to fail in a pillar. This distinction is important as it indicates that plane weakness is a *direct* structural measure of these regions as opposed to an *indirect* quantity that is only useful in plane space. We emphasize that what makes this result remarkable is that we are predicting the location of shear bands, a strongly nonlinear phenomenon, from the initial configuration prior to any deformation and then finding these results directly compare to the actual locations of failure.
Taken together the results in Figure \[fig:Performance\] demonstrate the structural origin of shear banding in glassy polymer nanopillars. This leads to the question: which plane structures cause shear banding? Since the plane weakness $W_i$ is defined as the signed normal distance to a hyperplane in a space defined by our structure functions, a natural approach to determining the importance of various structure functions would be to consider the magnitude of the projection of the hyperplane normal onto each structure function axis. This approach, however, would assume that each structure function is independent and would not account for correlations in the structural information encoded between structure functions. In other words, it would assume our structure functions form an orthogonal basis in the high-dimensional space, which is clearly false in our case. As a result of multicollinearity and our fitting proceedure, slight differences in sampled data may lead to large differences in the perceived importance of various structure functions.
Instead, we will say that a structure function is important if varying that structure function is likely to cause a large variance in plane weakness. A metric for this is called the Feature Importance Ranking Measure (FIRM) [@zien_feature_2009]. A structure function’s FIRM score is the percentage of the variance in plane weakness that can be described by the variance in that structure function if correlations with other structure functions are included. As such, FIRM scores range between $0$, where the variance in plane weakness is not described by a given structure function, and $1$, where the variance of plane weakness is entirely described by variance of a given structure function. In the event that our structure functions are uncorrelated, FIRM simplifies to the projection of the structure function onto the hyperplane normal.
![ **Plots of structure functions averaged over all (blue diamonds) and weak (red squares) planes with corresponding FIRM scores (black circles).** The left hand axis corresponds to the average of the set of structure functions. The right hand axis corresponds to the FIRM score of the given structure function. The graphics depicted to the right of the plots illustrate the region over which each structure function is calculated. The green plane represents the plane of consideration while the magenta regions represent the region over which the density function is calculated. All functions are plotted for the $D = 100$ pillar. The functions these plots show are: **(a)** $\langle \tilde{G}_{R} \left( i; 3.00, 0.5, R \right) \rangle$, **(b)** $\langle \tilde{G}_{h} \left( i; 0.5, h \right) \rangle$, **(c)** $\langle \Gamma_{R} \left( i; 3.00, 0.5, R \right) \rangle$ and **(d)** $\langle \Gamma_{h} \left( i; 0.5, h \right) \rangle$ for $h \leq 1.5$. Here, a tilde above the function indicates that it has been normalized by the maximum of the given structure function set averaged over all planes. []{data-label="fig:FIRM"}](2.pdf){width="5.836in"}
Figure \[fig:FIRM\] plots several of the structure functions along with their FIRM scores to demonstrate the relative importance of different structural variations to shear banding for pillars with $D = 100$. The structure function characterizing the density as a function of radial position in a given plane is shown in Figure \[fig:FIRM\]a for shear-banding and all planes, where each point in the curve corresponds to a different structure function. In general, we see that average radius of a shear banding plane is slightly smaller than the average plane. What is surprising about this feature is how small the fluctuation in the radius is, less than $\frac{1}{2}$ of a particle diameter. This length scale is nearly constant at all pillar diameters (See supporting information). The FIRM score for the density variations is also the highest near the surface, indicating that the variations in the density near the cylinder surface can be used to explain a large fraction of the variations in the plane weakness. In contrast, the density further away from the interface (where $R \approx 48$) is a less important indicator, as shown by the FIRM scores that decrease below 0.1 for $R \lesssim 48$. Remarkably, these fluctuations are not due to any mechanical scraping of the surface of the pillars but arise from the thermal fluctuations in the formation of our pillars alone.
The remaining panels in Figure \[fig:FIRM\] show the importance of some other families of structure functions that we have employed in our machine learning approach. Figure \[fig:FIRM\]b shows the importance of the total density in a plane a distance $h$ away from the test plane. Intuitively, this function is very important for small $h$ (FIRM score above 0.8) where it characterizes the density close to the plane, and this function becomes decreasingly important as $h$ increases. This provides further confirmation of our previous results revealing the most important feature is a slight undercoordination of the shear band plane due to these small surface fluctuations. We also see that these surface defects are quite long ranged along the surface of the pillar, approximately 18 particle diameters for the $D = 100$ pillar. The length scale of these surface defects grows sub-linearly with pillar diameter, which suggests that surface defects may become less important as the pillar diameter increases. This is in qualitative agreement with capillary-wave model (CWM) theory for planar liquid-vapor interfaces which suggests that this length scale should increase with the system’s interfacial area as these fluctuations can better explore large wavelength modes [@bedeaux_correlation_1985] (See supporting information). This suggests that these surface fluctuations are trapped during the quench of our pillars.
As described above, the softness of a particle has been shown to be intimately related to the tendency for an individual particle to rearrange under mechanical deformation or thermal relaxation [@cubuk_structure-property_2017; @schoenholz_structural_2016; @schoenholz_relationship_2017; @sussman_disconnecting_2017]. A natural question to ask is whether the softness of the particles associated with a given plane is in any way indicative of the tendency of that plane to shear band and lead to failure. In Figure \[fig:FIRM\]c, we plot the structure functions characterizing the average softness as a function of radial position in the pillars. The shear banding planes tend to have smaller values of softness near their surface compared to average planes, suggesting that shear band planes are harder near the surface. Now, we plot the structure functions that describe the average softness as a function of distance away from a test plane, $h$, in Figure \[fig:FIRM\]d. We note that shear band planes have larger values of softness for small $h$ than non-shear band planes. However, given the relatively small FIRM score for each of these softness-based structure functions, we find that softness is not as predictive of the structural variations in shear banding planes, and the other structure functions, such as the radial density shown in Figure \[fig:FIRM\]a, are better able to distinguish shear-banding planes.
The results described above in Figure \[fig:FIRM\] suggest that different families of structure functions can have varying amounts of overall importance, and a natural question to ask is how the importance of groups of structure functions might change with pillar diameter. However, the FIRM score in its current implementation is restricted to single structure function characterizations[@zien_feature_2009]; so to address this short-coming, in this work we extend FIRM to analyze the importance of multiple structure functions simultaneously. Our approach, the Multiple Feature Importance Ranking Measure (MFIRM), describes the percentage of the variance in plane weakness that can be ascribed to the variance in a given set of structure functions if we take correlations into account, and we use this metric to distinguish the importance of families of structure functions (e.g., surface density fluctuations, angular density fluctuations, etc.). MFIRM then enables us to examine how the importance of families of structure functions changes with pillar diameter and assess whether we approach a limit where the bulk-response dominates the behavior.
![ **Importance of sets of structure functions in shear band prediction.** Plots of the MFIRM scores the plane, radial, and angular structure functions along the major and minor axes of each plane weighted by **(a)** the local density and **(b)** the mean softness as a function of pillar diameter $D$. These plots explain the percentage of the variance in plane weakness explained by each of these sets of features respectively. []{data-label="fig:MFIRM"}](3.pdf){width="2in"}
Figure \[fig:MFIRM\]a considers the MFIRM score of each family of functions weighted by the density at each pillar diameter $D$. The most striking feature of this plot is the large MFIRM scores of the radial and plane density structure functions which correspond to the sets of structure functions plotted in Figures \[fig:FIRM\]a and \[fig:FIRM\]b respectively. These structure functions account for more than $90$ percent of the variance in plane weakness at all pillar diameters. We note that it is possible to have multiple feature sets with high scores due to the correlation between the families of structure functions, an issue we account for below. The second important feature of Figure \[fig:MFIRM\]a is the increasing MFIRM scores for angular density structure functions, which examine the density in angular slices along the minor and major axes of the ellipsoidal plane, with increasing pillar diameter. These scores explain around $70$ percent of the variance in plane weakness by $D = 25$, however these structure functions are unimportant for our smallest nanopillar. The MFIRM scores of the families of softness-based structure functions at each pillar diameter are shown in Figure \[fig:MFIRM\]b. These softness-based structure functions measure mean softness in the same regions defined by the corresponding density structure functions above. Interestingly, the percentage of the variance in plane weakness these structure functions can explain increases with the pillar diameter, suggesting that softness functions become increasingly important as $D$ increases. We observe the two largest increases in MFIRM occur in the radial and minor angular mean softness structure functions. These sets of functions increase from accounting for $13$ and $7$ percent of the variance in plane weakness at $D = 12.5$ to $39$ and $31$ percent of the variance in plane weakness respectively.
The correlation (multi-collinearity) between structure functions makes it difficult to disentangle whether these high MFIRM scores represent a single underlying important variable (the radial fluctuations in the plane) or if the large MFIRM scores are a result of many such important variables. To ascertain which scenario is at play, we adopt the following approach. First, we hypothesize a set of structure functions that we believe may represent an underlying variable other than the radial fluctuations in the plane. Then, we fit this set of structure functions to the radial and plane density structure functions for all planes at a given pillar diameter using least squares multiple linear regression. We interpret this fit as a function that provides the expected value of the set of structure functions given a plane’s radial and plane density structure functions which clearly measure these radial fluctuations in the plane. We next calculate the residuals between the actual and expected structure function values. We call these residuals the “fluctuations" away from the structure function set’s expected value. We then train a new machine learning hyperplane based exclusively on these fluctuations to obtain plane weakness, thus creating a metric that distinguishes between shear band and non-shear band planes based exclusively on these fluctuations. If a set of structure functions contains latent variables that are not described by the radial and plane density structure function model, then the fluctuations captured by these structure functions should be predictive of shear banding. Because much of the strength of plane weakness is attributable to these radial fluctuations (large MFIRM scores), we do not expect these models to be especially predictive. However, we may conclude that the more predictive these fluctuations are the greater the strength of the underlying latent variables that are not degenerate with the radial and plane density structure functions alone. In general, we denote these models based on fluctuations away from the radial and plane density functions as “fluctuation models".
![ **Fluctuation models for various sets of structure functions.** **(a)** The test set accuracy of the fluctuation models based on all angular density structure functions, the average softness structure functions as a function of radial position, and the average softness of planes $h \leq 1.5$ from the plane of consideration against all pillar diameters, $D$. Plots of the residuals of the angular density structure functions along the **(b)** minor, $\langle r_{A, m} \left( i; 3.00, 48.6, \theta_{c} \right) \rangle$, and **(c)** major, $\langle r_{A, M} \left( i; 3.00, 48.6, \theta_{c} \right) \rangle$, for the $D = 50$ pillars. **(d)** Plots of the residuals of the plane softness structure functions ($\langle \rho_{h} \left( i; 0.5, h \right) \rangle$) for the $D = 100$ pillars. For plots of residuals listed above, the FIRM score corresponds to the given fluctuation model, not the plane weakness measure found using all structure functions. A tilde above the residual function indicates that the residuals have been normalized by the maximum of the corresponding *original* structure function set averaged over all planes. []{data-label="fig:FluctuationModels"}](4.pdf){width="5.836in"}
We use test set accuracy as a metric of the predictive strength of various fluctuation models, and the results of this analysis are shown in Figure \[fig:FluctuationModels\]a. The fluctuation models based on the fluctuations of all of the angular density structure functions do no better than chance ($P = 50\%$) at $D = 12.5$ and $D = 100$ but do exhibit some predictive power at intermediate pillar diameters. To understand the predictive nature of these fluctuatons, we denote these residuals of the minor and major angular density structure functions as $r_{A,m}$ and $r_{A,M}$ respectively. FIRM scores listed describe the percentage of variance in the *fluctuation model* that is described by each residual. Here we see the minor angular structure functions in Figure \[fig:FluctuationModels\]b are quite undercoordinated and become increasingly more so with larger angular resolution. In contrast, the major angular structure functions in Figure \[fig:FluctuationModels\]c are overcoordinated compared to the average plane. This suggests that the undercoordination experienced by shear band planes at these intermediate pillar diameters, between $25$ and $50$ particle diameters, typically occurs along its minor axis. As the pillar diameter grows, the size of these fluctuations decrease as a percentage the plane’s radius. This leads to a decrease in the importance of these fluctuations at large pillar diameters. In small pillars, shear banding is entirely controlled by density fluctuations in pillar planes rather than the geometry of these fluctuations.
Next, we turn to fluctuation models based on the fluctuations of the radial softness structure functions. [*A priori*]{}, we might expect fluctuation models based on these structure functions to be the most predictive of all the mean softness models due to these structure functions’ large MFIRM scores relative to the other softness-based structure functions. Instead, Figure \[fig:FluctuationModels\]a shows that these models have test set accuracies of just higher than chance, approximately $55$ percent. Because these structure functions have such high MFIRM scores but are not very predictive on their own, these structure functions must be highly correlated with the plane or radial structure functions. Because this effect is not independent of radial fluctuations in the pillar diameter, we presume that much of this effect is due to enhanced surface mobility, which is commonly found in glassy materials with free surfaces [@paeng_direct_2011; @zhang_long-range_2016; @sussman_disconnecting_2017]. Particles near the surface are more mobile, potentially allowing them to explore phase space locally [@lyubimov_orientational_2015] and leading to harder structures due to a slower effective quench rate [@schoenholz_relationship_2017]. Thus, shear band planes which tend to have smaller local radii are likely to have harder particles at small $R$ than the average plane. Figure \[fig:FIRM\]c also supports this idea as we find that on both on average and in shear band planes, softness decreases as we approach the surface of the pillar.
Finally, we examine fluctuation models based on the fluctuations of the plane softness averaged over the entire plane. For simplicity of interpretation, we restrict our analysis to the mean softness of planes that are local to the test plane, $h \leq 1.5$. Although the plane softness structure functions have the smallest MFIRM scores out of all of the sets of structure functions we have examined, their fluctuation models obtain large test set accuracies ($P = 0.71 \pm 0.04$) at large pillar diameters. This indicates that they must measure some latent variable not covered by the simple model involving only the plane and radial density; i.e., the specific packing in the shear band plane becomes increasingly important as the pillar diameter increases. To understand this latent variable, we plot the residuals $\rho_h$ of the plane softness structure functions in Figure \[fig:FluctuationModels\]d for the $D=100$ pillar. Here, we see that shear band planes are softer than the average plane in the pillar ($h = 0$). This effect is apparently important since the FIRM scores suggest that the variance of each of the first three structure functions accounts for approximately $70$ percent of the variance in the fluctuation model. We find that the mean softness of shear band planes decreases sharply at $h = 1.5$, and adding additional plane softness or angular softness structure functions to this model does not improve its accuracy (See supporting information).
Taken together, our analysis of the fluctuation models suggests that as we approach the large pillar limit, the only latent variable that is predictive of shear banding and not accounted for by the plane’s radial fluctuations is the mean softness. This is interesting as the importance of these radial fluctuations is decreasing with increasing pillar diameter as shown by the MFIRM scores of the radial and plane density structure functions in Figure \[fig:MFIRM\]a. Therefore, we expect softness, a microscopic structural quantity to play a major role in the macroscopic dynamics, and the identification of such a structural quantity is a key step for the development of mesoscale and constitutive models for the dynamics of materials [@ottinger_beyond_2005].
CONCLUSION
==========
In summary, our results show that the mesocopic structure of planes can be used to predict shear banding in amorphous solids. This structure can be quantified by plane weakness. According to our analysis, the main component of plane weakness for submicroscopic pillars are small, less than $\frac{1}{2}$ of a particle diameter, radial fluctuations on the exterior of the plane. These fluctuations come from the thermalization of the pillar alone and are not artificially induced. This provides valuable insight about manufacturing strong nanoscale components: to strengthen nanoscale components we may neglect bulk effects and focus on developing components that are smooth on the atomistic level. Even in pristine lab environments, surface defects large enough to cause shear banding may arise in the melt of a material.
As pillar diameter increases, this variable becomes less important and is replaced by other structure functions. In particular, we find that the mean softness local to a plane is an increasingly important predictor of shear banding with increased pillar diameter and is the dominant predictor outside of the radial fluctuations at the largest pillar diameter considered. This observation links the machine learned quantity softness to mesoscale theories such as Shear Transformation Zone (STZ) theory which hypothesize mesocale “configurational soft spots", regions that are more likely to yield under shear stress [@manning_strain_2007]. This link is non-trivial as softness is constructed as a measure of short, local particle motions while shear bands are by definition long timescale, non-local events. Moreover, because we are only using configurational information prior to deformation to predict shear bands, we have shown that at temperatures well below the glass transition that these defects can be considered to be frozen in place, i.e. we do not need to consider thermal fluctuations to build a mesoscale model that predicts mechanical behavior so long as such behavior occurs well below $T_g$ even when the constituent pieces of a material are atomic in nature.
METHODS
=======
Simulation Model
================
We simulate a coarse grained bead-spring polymer with chains of length $N = 5$. The bonded interactions are taken through a harmonic bonding potential, $$\label{eq:1}
U_{jk}^{b}=\frac{k_h}{2}\left(r_{jk} - d \right)^2,$$ where $r_{jk}$ is the radial distance between monomers $j$ and $k$ and $k_h = 2000 \epsilon / d^2$. Here, $d$ and $\epsilon$ are the length and energy scales of our simulations respectively. The non-bonded interactions are taken using a modified 12-6 Lennard-Jones (LJ) potential, $$\label{eq:2}
U_{jk}^{nb}=4 \epsilon \left[ \left( \frac{\sigma}{r_{jk}-\Delta} \right)^{12} - \left( \frac{\sigma}{r_{jk}-\Delta} \right)^{6} \right].$$ We choose $\Delta = 0.75 d$ and $\sigma = d - \Delta / 2^{1/6}$. This gives our potential shorter range and higher curvature while restricting the minimum to reside at the same location as the standard LJ potential where $\Delta = 0$. This modification promotes brittle fracture at low temperatures as is expected in experiments. In the text, we present our findings in units reduced by $d, \epsilon$ and the monomer mass $m$. This study was completed using the LAMMPS [@plimpton_fast_1995] simulation package with a simulation timestep of $0.0006636$. The pillars are aligned along the $\hat{z}$ axis and periodic in this direction, and surfaces in the radial direction are free. We hold the length of our pillars fixed at $L = 200$ particle diameters and vary the diameter of our pillars to be nominally $D = 12.5$, $25$, $50$, and $100$ particle diameters. We generate $N_{\text{pillar}} = 100$ independent pillar configurations for the three smallest pillar diameters and $N_{\text{pillar}} = 50$ independent pillar configurations for the largest diameter pillars.
Using a cooling rate of $5 \times 10^{-5}$, we find the glass transition temperature of the pillars to be $T_g = 0.38$ by identifying the intersection of linear fits of the density as a function of temperature in the supercooled and glassy states. Pillars were thermalized at $T = 0.5$ within a cylindrical, harmonic confining wall which is fixed to ensure the density of the monomers is $\rho = 0.3$. The pillars were cooled at a rate of $5 \times 10^{-4}$ to a temperature of $T = 0.05$. This caused the pillar diameter to contract away from the confining wall as the density of monomers rose to $\rho = 1.0$ below $T_g$.
Development of Softness Field
-----------------------------
The softness field used in this study was first characterized in Ref. . We repeat relevant details here for completeness. This field is developed in a similar way to plane weakness. We first characterize the local structure around each particle $j$, using a set of “local structure functions":
$$\Psi_{R} (j;\mu, L) = \sum_{k} \mathrm{e}^{(r_{jk}-\mu)^2/L^2}
\label{eqM:1}$$
$$\Psi_{A} (j;\xi,\lambda,\zeta) = \sum_{k,l} \mathrm{e} ^{\left(r_{jk}^2+r_{kl}^2+r_{jl}^2\right)/\xi^2}\left(1+\lambda \cos \theta_{jkl} \right)^\zeta
\label{eqM:2}$$
where $\mu$, $L$, $\xi$, $\lambda$, and $\zeta$ are parameters that characterize the members of each family of structure functions. Here, $r_{jk}$ is the distance between particles $j$ and $k$. The variable $\theta_{jkl}$ is the angle made between particles $j$, $k$, and $l$. The summations are performed for all particles within a radius $R_c^S$. Our results are insensitive to changes in $R_c^S$ so long as we include the first few neighboring shells [@cubuk_identifying_2015]. In this work, we set $R_c^S = 2.5$. The parameter sets that we used to characterize the local environment may be found in the supporting information.
Next we need to develop a training set of rearranging and non-rearranging particles. To create this set, we ran additional independent molecular dynamics simulations in which we thermalized and strained pillars at several temperatures: $T = 0.05$, $0.1$, $0.15$, $0.2$, $0.25$, $0.275$, $0.3$, and $0.325$. These pillars all had a nominal diameter of $D = 50$ and had a length along their $\hat{z}$ axis of $100$. Because the deformation of the pillars causes affine transformations of particle configurations which do not necessarily correspond to rearrangements, we quantify rearrangements of particle $j$ using:
$$D^{2}_{\text{min}} (j;t) = \frac{1}{N_j} \sum_{k}^{N_j} [\boldsymbol{r}_{jk}(t+\Delta t)-\boldsymbol{\Lambda}_{j}(t) \boldsymbol{r}_{jk}(t) ]^{2}
\label{eqM:3}$$
which measures the non-affine motion of particle $j$ at time $t$. Here $\boldsymbol{r}_{jk}$ is the vector between particles $j$ and $k$ and $\boldsymbol{\Lambda}_{j}(t)$ is the best fit local gradient tensor about particle $j$ which minimizes the quantity [@falk_dynamics_1998]. Summations are performed over all $N_j$ particles within a cutoff radius of $2.5$ particle diameters. We chose $\Delta t$ to correspond to a strain of $0.00166$. We say that a particle $j$ at time $t$ rearranges if $D^{2}_{\text{min}}(j;t) > 0.1$. This value was chosen by using the same method as in Ref. . Additionally, we confine our rearranging and non-rearranging sets of particles to be selected from a region $8$ particle diameters from the center of the pillar and in the elastic regime of strain to avoid rearrangements caused by zero-modes on the surface of the pillar and particles in the shear band respectively. At each temperature, we chose $N_r = 700$ randomly rearranging particles, and $N_n = 700$ non-rearranging particles to be in our training set. We say that a particle is non-rearranging if it has the one of the lowest $N_n$ values of $D^{2}_{\text{min}}$ averaged over a relaxation time [@schoenholz_structural_2016].
We then use a linear support vector machine (SVM) to calculate the hyperplane that best separates points corresponding to rearranging particles from points corresponding to non-rearranging particles. It is not possible to specify a hyperplane that completely separates rearranging particles from non-rearranging ones. Thus, the SVM is designed to penalize particles whose classification is incorrect. This misclassification penalty is controlled by the parameter $C$ where larger $C$ values correspond to fewer incorrect classifications. This parameter was chosen to be $C=0.1$ by k-folds cross validation. We find that more than $93\%$ of rearrangements occur on particles with softness $S>0$ by nested cross validation [@cawley_over-fitting_2010]. As with plane weakness, SVM algorithm was implemented using the scikit-learn package [@pedregosa_scikit-learn:_2011]. For the purposes of this study, we normalize our softness field to have zero mean and unit variance at each pillar diameter. This leads to an easier interpretation of our softness based results as the number of standard deviations away from $0$.
Description of Structure Functions
----------------------------------
Shear bands are expected to form along approximately $45\degree$ planes in the pillars. We partition our pillars into $N_{\text{plane}}=7200$ $45\degree$–planes with $200$ partitions in the $\hat{z}$ axis and $36$ partitions in the $\hat{\theta}$ direction, along the polar angle. We seek to mathematically encode the structure of these planes. To do this, we divise a set of “structure functions" that describe the local structure of the pillar around each of plane. We define these functions to respect the symmetries of the elliptical prism that characterizes each plane in the pillar. These functions come in two categories with three families each. The first category is the density structure functions:
$$G_{h} \left( i; \xi_{h}, h \right) = \frac{1}{D^{2}}
\sum_{j} \Theta_{ij}^{P} \left(h, \xi_{h} \right)
\label{eq:3}$$
$$G_{R} \left( i; \xi_{h}, L_{R}, R \right) =
\frac{1}{R} \sum_{j}
\Theta_{ij}^{P} \left(0, \xi_{h} \right)
e^{-d_{ij} \left( R \right)^{2} / L_{R}^{2}}
\label{eq:4}$$
$$G_{A, a} \left( i; \xi_{h}, \xi_{R}, \theta_c \right) =
\frac{1}{D^{2}} \sum_{j}
\Theta_{ij}^{P} \left( 0, \xi_{h} \right)
\Theta_{ij}^{E} \left( \xi_{R} \right)
\cos \left( \theta_{ij}^{a} \right)^{\zeta \left( \theta_c \right)}
\label{eq:5}$$
where each structure function is for a plane $i$ and sums are performed over all particles $j$ whose contribution to the sum is greater than $0.1$ for numerical efficiency. Here, $L_{R}$, $\xi_{h}$, $\xi_{R}$, $h$, and $R$ are parameters that characterize these functions. The function $d_{ij}\left( R \right)$ is the distance in plane $i$ that particle $j$ is away from an ellipse that is centered on the $\hat{z}$ axis and has a minor axis of length $R$. This distance is found numerically using the algorithm in Ref. . The ellipse is defined by the equation $ (x^{M})^{2} / 2 + (x^{m})^{2} = R^{2}$ where $x^{M}$ and $x^{m}$ are the in plane distances along the major and minor axes respectively. The function $\Theta_{ij}^{E}\left( \xi_{R} \right) = e^{-\left( (x^{M}_{ij})^{2} / 2 + (x^{m}_{ij})^{2} \right) / \xi_{R}^{2}}$ is a soft step function for particles within an ellipse with a minor axis of length $\xi_R$. The variable $\theta^{a}_{ij}$ is the angle between the $a$ axis of plane $i$ and particle $j$ where $a$ is either the major $(M)$ or minor $(m)$ axis. Here, $\zeta \left( \theta_c \right) = \frac{-1}{\log_{2} \left( \cos(\theta_{c}) \right)}$.
These families correspond to simple physical quantities in the following way. Eq \[eq:3\] is proportional to the density of particles a distance $h$ away from plane $i$ in a plane of thickness $\xi_{h}$. Eq \[eq:4\] is proportional to the density of particles in an elliptical shell of width $L_R$ and thickness $\xi_{h}$ that has a minor axis of length of $R$ and is centered on plane $i$. Finally, $\zeta \left( \theta_{c} \right)$ is defined so that the $\cos \left( \theta_{c} \right)^{\zeta \left( \theta_{c} \right)} = \frac{1}{2}$ allowing us to interpret of this term as another soft step function with a cutoff angle of $\theta_{c}$. Thus, Eq \[eq:5\] is proportional to the density of particles in pie slices that have width $\theta_c$ and width of $\xi_{R}$ and depth of $\xi_{h}$ along the major and minor axes of plane $i$. We call these families of structure functions the plane density, radial density, and angular density structure functions respectively.
The other category is the softness structure functions. These come in three families, $\Gamma_{h} \left( i; \xi_{h}, h \right)$, $\Gamma_{R} \left( i; \xi_{h}, L_{R}, R \right)$, and $\Gamma_{A, a} \left( i; \xi_{h}, \xi_{R}, \theta_c \right)$, and measure the mean softness of the regions that correspond to the density structure functions, $G_{h} \left( i; \xi_{h}, h \right)$, $G_{R} \left( i; \xi_{h}, L_{R}, R \right)$, and $G_{A, a} \left( i; \xi_{h}, \xi_{R}, \theta_c \right)$ respectively. We define these functions specifically as:
$$\Gamma_{h} \left( i; \xi_{h}, h \right) = \frac{
\sum_{j} S_{j} \Theta_{ij}^{P} \left(h, \xi_{h} \right)}{
\sum_{j} \Theta_{ij}^{P} \left(h, \xi_{h} \right)}
\label{eqM:4}$$
$$\Gamma_{R} \left( i; \xi_{h}, L_{R}, R \right) =
\frac{\sum_{j} S_{j} \Theta_{ij}^{P} \left(0, \xi_{h} \right)
e^{-d_{ij} \left( R \right)^{2} / L_{R}^{2}}}
{\sum_{j} \Theta_{ij}^{P} \left(0, \xi_{h} \right)
e^{-d_{ij} \left( R \right)^{2} / L_{R}^{2}}}
\label{eqM:5}$$
$$\Gamma_{A, a} \left( i; \xi_{h}, \xi_{R}, \theta_c \right) =
\frac{\sum_{j} S_{j}
\Theta_{ij}^{P} \left( 0, \xi_{h} \right)
\Theta_{ij}^{E} \left( \xi_{R} \right)
\cos \left( \theta_{ij}^{a} \right)^{\zeta \left( \theta_c \right)}}{\sum_{j}
\Theta_{ij}^{P} \left( 0, \xi_{h} \right)
\Theta_{ij}^{E} \left( \xi_{R} \right)
\cos \left( \theta_{ij}^{a} \right)^{\zeta \left( \theta_c \right)}}
\label{eqM:6}$$
where each function is for a plane $i$ and sums are performed over all interior particles $j$. For this study, we define the interior of the pillar as all particles greater than $3.5$ particle diameters from the pillar’s surface. Summations are restricted to interior particles because the structures which cause rearrangements in the bulk, where the softness field was developed, are likely to be different than the structures on the surface of the pillars that lead to rearrangements. For numerical efficiency, we further restrict the summation so that a term only contributes to either sum if the product of that term’s functions (excluding $S_{j}$) is greater than $0.1$. We call these structure functions the plane, radial, and angular softness structure functions respectively.
Training and Parameter Selection
--------------------------------
For each pillar, we describe every 45–plane prior to deformation with $M=612$ structure functions (See supporting information). At each pillar diameter, we standardize each structure function by subtracting the mean and dividing by the standard deviation. We then assign each plane $i$ a vector, $\boldsymbol{p}_{i} \in \mathbb{R}^{M}$ where each orthogonal component of the vector is one of the standardized structure functions. We call these the “structure vectors", $\{\boldsymbol{p}_{1}, ..., \boldsymbol{p}_{N}\}$ where $N = N_{\text{plane}} \times N_{\text{pillar}}$.
We now evaluate the local von Mises shear strain rate between the unstretched pillar configuration and the pillar configuration at a strain of $\epsilon = 5.5 \%$ with a cut-off radius of $2.5$ particle diameters. At all pillar diameters, we see strain localization for this strain. For each pillar we evaluate the quantity,
$$\langle J_{2} \rangle_{j} =
\frac{\sum_{i} J_{2,i} \Theta_{ij}^{P}\left( 0, \xi_{h} \right)}
{\sum_{i}\Theta_{ij}^{P}\left( 0, \xi_{h} \right)}.
\label{eq:8}$$
where $J_{2,i}$ is the local von Mises strain rate of particle $i$ where the summation runs over the interior of the pillar. Here, we take $\xi_{h} = 2$. We pick the planes with the maximum and minimum values of $\langle J_{2} \rangle_{j}$ as shear band and non-shear band planes for each pillar studied. This yields a training set with $2 N_{\text{pillar}}$ elements at each pillar diameter.
Two choices are made in the development of our linear SVM used to generate plane weakness. First, we must decide which features to allow in our linear SVM. We limit the features used in our fit in order to prevent overfitting our model to noise in our data. Second, the SVM method typically incorporates a misclassification penalty $C$, as described in the Development of Softness Field section, which must be chosen as well. We want to make both of these choices so that our model best generalizes to new planes. To find the optimal features, we use recursive feature elimination (RFE) [@guyon_gene_2002]. The RFE algorithm starts with the initial $M$ structure functions and prunes the least important structure function at each step.
We use stratified 3-fold cross validation with a grid-search technique on the training set to determine the $C$ value and the step on which to terminate the RFE algorithm, $m$. Here, we select a set of possible $C$ values ranging from $10^{-4}$–$10^{0}$. Then, we partition the training set into $k = 3$ “folds" with equal numbers of shear band and non-shear band planes in each. We denote $1$ of these folds the “validation set". For each $C$, we eliminate structure functions using the RFE algorithm on planes from the other $2$ folds until $1$ structure function remains. At each step of the algorithm, we record the percentage of correctly classified planes in the validation set due to a linear SVM developed using the other $2$ folds. This process is repeated with each of the $3$ folds being used as validation sets. We randomly shuffle the planes between the $3$ folds and repeat this procedure $10$ times to ensure that our parameter selection is independent of how the folds are selected. We say the $m$ and $C$ values which best generalize to new data are those which produce the highest average percentage of correctly classified planes across all folds and re-shufflings. To determine the structure functions used in the final model, we run the RFE algorithm at the most generalize-able $C$ for the $m$ steps on entire training set. We train a final linear SVM with the remaining $M' = M - m$ structure functions at the most generalize-able $C$. To embed this hyperplane into $\mathbb{R}^{M}$, we simply add $0$’s to the plane normal at every location in which a feature was removed.
Measures of Binary Classification Performance
---------------------------------------------
We want to obtain an unbiased estimate of how well our model will generalize to data outside of the training set. We use nested stratified k-folds cross validation to do this [@cawley_over-fitting_2010]. We partition our data into $k = 10$ folds with an equal number of shear band and non-shear band planes in each. We retain one of these folds as a “test set". Then, we perform feature selection and train a linear SVM using the planes from the other folds. For the resulting linear SVM, we measure the percentage of correctly classified planes in the test set. We repeat the process using each of the other folds as test sets. To ensure that our results do not depend on how the folds are chosen, we randomly shuffle the planes between the folds and repeat the process $10$ times. We average our results to obtain the “test set accuracy" of our classifier. Because the test set accuracy was obtained by using planes that were not used in fitting our classifier, it is a good measure of how well our classifier will generalize to planes outside of the training set. Similarly, we obtain the expected percent shear bands on weak planes, $P(W>0|SB)$, by looking at the average percentage of correctly classified shear band planes in the test sets.
Development of MFIRM
--------------------
MFIRM is an extension of FIRM [@zien_feature_2009] which allows us to calculate the importance of a set of $N$ structure functions. Let
$$\boldsymbol{f}: \mathbb{R}^M \xrightarrow{} \mathbb{R}^{N}
\label{eqM:7}$$
be a function which projects the orthogonal components which correspond to the set of structure functions from the original vector space of all structure functions to a new vector space with only the structure functions of which we wish to find the importance. The expected plane weakness given a set of values of the selected features $\boldsymbol{t} \in \mathbb{R}^N$ is:
$$q_{\boldsymbol{f}} \left( \boldsymbol{t} \right) = \langle W \left(\boldsymbol{p} \right) | \boldsymbol{f} \left( \boldsymbol{p} \right) = \boldsymbol{t} \rangle
\label{eqM:8}$$
The MFIRM score of this set of features then corresponds to the standard deviation of $q_{\boldsymbol{f}} \left( \boldsymbol{t} \right)$:
$$Q_{\boldsymbol{f}} = \sqrt{
\int d \boldsymbol{t}
\left(q_{\boldsymbol{f}} \left( \boldsymbol{t} \right) - \langle q_{\boldsymbol{f}} \rangle \right)^2
P \left( \boldsymbol{f} \left( \boldsymbol{p} \right) = \boldsymbol{t} \right)
}
\label{eqM:9}$$
where $P \left( \boldsymbol{f} \left( \boldsymbol{p} \right) = \boldsymbol{t} \right)$ is the probability density of obtaining selecting the structure function values $\boldsymbol{t}$ and $\langle q_{\boldsymbol{f}} \rangle$ is the expected value of $q_{\boldsymbol{f}} \left( \boldsymbol{t} \right)$.
In general, this quantity is quite difficult to calculate as $P \left( \boldsymbol{f} \left( \boldsymbol{p} \right) = \boldsymbol{t} \right)$ is unknown. To simplify calculation, we assume the distribution of structure functions is normally distributed with a mean of $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$. The mean may be partitioned into $\boldsymbol{\mu_\text{f}}$ and $\boldsymbol{\mu_\text{l}}$ for which correspond to the sets of structure functions that we wish to know the importance of and leftover structure functions that are not in that set. Similarly, we may partition the covariance matrix as well,
$$\boldsymbol{\Sigma} =
\left(\begin{array}{@{}ccc}
\boldsymbol{\Sigma}_{\text{ll}} & \boldsymbol{\Sigma}_{\text{lf}} \\
\boldsymbol{\Sigma}_{\text{fl}} & \boldsymbol{\Sigma}_{\text{ff}}
\end{array}\right).
\label{eqM:10}$$
Then, via the properties of the conditional distributions of the multivariate normal distribution, we find
$$q_{\boldsymbol{f}} \left( \boldsymbol{t} \right) - \langle q_{\boldsymbol{f}} \rangle = \boldsymbol{n}_{\text{l}}^{T} \boldsymbol{\Sigma}_{\text{lf}} \boldsymbol{\Sigma}_{\text{ff}}^{-1} \left(\boldsymbol{t}-\boldsymbol{\mu_\text{f}}\right)
+
\boldsymbol{n}_{\text{f}}^{T}
\left(\boldsymbol{t}-\boldsymbol{\mu_\text{f}}\right),
\label{eqM:11}$$
where $\boldsymbol{n}_{\text{f}}$ and $\boldsymbol{n}_{\text{l}}$ is the partitioned normal of plane weakness. The superscript $T$’s denote transposition. Then, we may use the quadratic form expectation to show that Eq. \[eqM:9\] is
$$Q_{\boldsymbol{f}} = \sqrt{\boldsymbol{v}^{T} \Sigma_{\text{ff}} \boldsymbol{v}},
\label{eqM:12}$$
where $\boldsymbol{v}^{T} = \boldsymbol{n}_{\text{l}}^{T} \boldsymbol{\Sigma}_{\text{lf}} \boldsymbol{\Sigma}_{\text{ff}}^{-1}+\boldsymbol{n}_{\text{f}}^{T}$. If the structure functions are not normally distributed, this quantity provides a second-order approximation of MFIRM. Because plane weakness is normalized to have a standard deviation of $1$, $Q_{\boldsymbol{f}}$ may be readily interpreted as the percentage of variance in plane weakness that can be described by a given set of features. For models which are not normalized, we can normalize by the standard deviation in the measure to obtain the same interpretation.
This research was supported by the National Science Foundation through award CMMI-1536914 and the University of Pennsylvania Materials Research Science and Engineering Center (MRSEC) (DMR-1720530), including its computational facilities. Other computational facilities employed were provided by XSEDE through allocation TG-DMR150034.
The following files are available free of charge.
- SupportingInformation.pdf: figures and information referenced in text
|
---
abstract: |
Key establishment is the basic necessary tool in the network security, by which pairs in the network can establish shared keys for protecting their pairwise communications. There have been some key agreement or predistribution schemes with the property that the key can be established without the interaction ([@Blom84; @BSHKY92; @S97]). Recently the hierarchical cryptography and the key management for hierarchical networks have been active topics(see [@BBG05; @GHKRRW08; @GS02; @HNZI02; @HL02; @Matt04]. ). Key agreement schemes for hierarchical networks were presented in [@Matt04; @GHKRRW08] which is based on the Blom key predistribution scheme(Blom KPS, \[1\]) and pairing. In this paper we introduce generalized Blom-Blundo et al key predistribution schemes. These generalized Blom-Blundo et al key predistribution schemes have the same security functionality as the Blom-Blundo et al KPS. However different and random these KPSs can be used for various parts of the networks for enhancing the resilience. We also present key predistribution schemes from a family hyperelliptic curves. These key predistribution schemes from different random curves can be used for various parts of hierarchical networks. Then the non-interactive, identity-based and dynamic key predistributon scheme based on this generalized Blom-Blundo et al KPSs and hyperelliptic curve KPSs for hierarchical networks with the following properties are constructed.\
1)$O(A_KU)$ storage at each node in the network where $U$ is the expansion number and $A_K$ is the number of nodes at the $K$-th level of the hierarchical network;\
2)Strongly resilience to the compromising of arbitrary many leaf and internal nodes;\
3)Information theoretical security without random oracle.\
author:
- Hao Chen
- 'Hao Chen [^1]'
title: 'Strongly Resilient Non-Interactive Key Predistribution For Hierarchical Networks'
---
Introduction
=============
Key establishment is basic tool for secure communication in networks, two nodes in networks can have agreed shared key that is only known to them, thus allowing the shared key for protecting their communications. In many environment there is significant advantage to non-interactive key agreement schemes which need not to use any communication between nodes. The Diffie-Hellman type key agreement protocol(see [@BM03]) is non-interactive, but some known public keys are needed which is a impractical for large networks. Recently key agreement using key predistribution schemes have been presented for very large networks such as, hierarchical networks and wireless sensor networks([@EG02; @CPS03; @Matt04; @DDHVKK05; @GHKRRW08; @BST08]).\
The key predistribution scheme(KPS) was proposed by R.Blom in Eurocrypt 84 ([@Blom84]). It was extended by C. Blundo et al in Crypto 92 [@BSHKY92]. This cryptographic primitive has been a basic ingredient in the security of wireless sensor networks(see [@DF06; @DDHVKK05]) and hierarchical systems(see [@Matt04; @GHKRRW08]). However in the Blom and Blundo et al KPS, the size of the finite field in the KPS has to be larger than the number of users. The unique form of Blom-Blundo et al KPS has no flexibility in practical application. These are real drawbacks.\
In a HIERARCHICAL networks with $n$ nodes, the root authority only needs to distribute the secret information to a small number of large organizations or group leaders, and then each of these can further distribute the secret information to smaller and smaller units(see [@Matt04; @GHKRRW08]). In this way we can think the nodes are arranged on a tree, the root of tree distributes the secret information of its children nodes and then each of these distributes secret information to its children nodes... each node only get its secret information from its parent node. Finally the leaf nodes get their secret information from their parent nodes. Each pair of nodes at the same level (including the leaf nodes and internal nodes) can compute their shared key by the secret information and the identities of themselves and their parents. This would help for group level authentication and confidentiality in the whole hierarchical network. The expansion number $U$ is the maximal number of children nodes.\
In the application such as tactical networks, mobile ad-hoc networks, it is more reasonable to assume a [*Hierarchical*]{} network structure than a central trusted authority (see [@Matt04; @RM05; @GHKRRW08]). On the other hand, the using of Hierarchical network structure can reduce the workload of of the TAs. The Hierarchical identity based encryption (HIBE) was studied in [@HL02; @GS02; @BBG05]. In [@CHK03], HIBE was used for the construction of forward secure encryption. The hierarchical key agreement has been studied in [@Matt04; @GHKRRW08].\
In previous constructed key agreement schemes in [@Matt04] and [@GHKRRW08], every node in the hierarchical network needs the storage of $\frac{\prod{(t_i+1)(t_i+2)}}{2}$ elements of the base field for resisting the compromising of $t_i$ nodes at $i$-th level of the hierarchical networks. It will grows exponentially when the number of levels in the hierarchical network tends to the infinity. The KAS in [@GHKRRW08] can only resist the attack of compromising arbitrary many leaf nodes. The security of KAS in [@GHKRRW08]was proved with the random oracle model. The identity based key agreement scheme of [@GHKRRW08] is dynamic, nodes can be added at each level of the hierarchy without changing the information of other nodes.\
In this paper we construct generalized Blom-Blundo et al key predistribution schemes and key predistribution schemes from a family of hyperelliptic curves. New random polynomials are introduced in the functions computing shared keys in these generalized Blom-Blundo et al key predistribution schemes. Hyerelliptic curve KPSs are constructed from different random curves. These new randomness and flexibility of our key predistribution schemes can be used to construct strongly resilient key predistribution schemes for hierarchical networks with low storage, communication and computation cost. The size of the base field of our new key predistribution schemes depends only on the expansion number $U$ of the hierarchical network and the storage of every node is $O(A_KU)$, where $A_K$ is the number of nodes at $K$-th level of the hierarchical network. Moreover the constructed hierarchical network key predistribution schemes are dynamic and non-interactive. Our key predistribution schemes for hierarchical networks can resist the compromising of arbitrary many of nodes with very low storage at every node.\
Blom-Blundo et al KPS
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Now we recall the definition of KPS by following the presentation in the paper of Stinson [@S97]. Suppose we have a Trusted Authority (TA) and a set of users ${\bf
U}=\{1,...,n\}$. Let $2^{{\bf U}}$ be the set of all subsets of the user set ${\bf U}$. ${\bf P} \subset
2^{{\bf U}}$ will denote the collection of all privileged subsets to which the TA is distributing keys. ${\bf F}$ will denote the collection of all possible coalitions(forbidden subsets) against which each key is remain secure. In the [*Key Predistribution Scheme*]{}, at the set up stage, each user $i$ get its secret information $u_i$ from the TA, where $u_i$ is taken in a finite dimensional linear space over $GF(q)$. Once the secret information $u_i$, $i=1,...,n$ , is given to each user, in the computation stage, for any privileged subset $T \in {\bf P}$, the users in the privileged subset $T$ can compute the shared key $k_T \in GF(q)$ for their communications. No forbidden subset $J \in {\bf F}$ disjoint from $T$ can get any information of the key $k_T$. This is called $({\bf P},{\bf
F})$-KPS. When ${\bf P}$ consists of all subsets of ${\bf U}$ with $t$ elements and ${\bf F}$ consists of subsets with at most $w$ elements, we call it $t$-variable and $w$-secure KPS. Thus a $t$-variable and $w$-secure KPS can be used to get the shared keys of any subset with $t$ users, which is secure against the attack of any $w$ users.\
Generally the KPS is required information theoretically secure against the attack of the coalition of users, for the more formal presentation we refer to [@Blom84; @BSHKY92; @S97].\
The secret information $u_i$, $i=1,...,n$, is in the finite dimensional linear space over the finite field $GF(q)^h$, where $q$ is a prime power. Thus the storage is $hlog_2(q)$ bits. The shared key $k_T$, for each privileged subset $T \in {\bf
P}$, is in $GF(q)$. In the computation stage, each user $i$ in $T$ computes $k_T$ from its secret information $u_i$ and the IDs of other users in the set $T$. Only the arithmetic in $GF(q)$ is involved. We call $GF(q)$ the base field of the KPS.\
The first KPS proposed in [@Blom84] is a $2$-variable and $w$-secure KPS, and it was generalized in [@BSHKY92] to a $t$-variable and $w$-secure KPS. Let $q$ be a prime power satisfying $ q \geq n$. Each user $i$ is assigned to an element $e_i \in GF(q)$ as its identity. The TA takes a random $t$ variable symmetric polynomial in $GF(q)[x_1,...,x_t]$ of the form $f(x_1,...,x_t)=\Sigma_{j_1=1}^{w+1}\cdots
\Sigma_{j_t=1}^{w+1} a_{j_1 \cdots j_t} x_1^{j_1}\cdots x_t^{j_t} $ with coefficients $a_{j_1 \cdots j_t}$ in $GF(q)$ where $a_{j_1...j_t}=a_{j_{i_1}...j_{i_t}}$, that is, $f(x_1,...,x_t)=\Sigma_{j_1=1}^{w+1}\cdots
\Sigma_{j_t=1}^{w+1} a_{j_1 \cdots j_t} x_1^{j_1}\cdots x_t^{j_t} \in
GF(q)[x_1,...,x_t]$ and $f(x_1,...,x_t)=f(x_{i_1},...,x_{i_t})$( $\{i_1,...,i_t\}$ is an arbitrary permutation of $\{1,...,t\}$). This polynomial is only known to the TA. The symmetric $(t-1)$ polynomial $f(e_i, x_2,...,x_t)$ is given to the user $i$, $i=1,...,n$, as its secret information. For any privileged subset $T=\{e_{i_1},...,e_{i_t}\}$, each user in this subset $T$ can compute the shared key $k_T=f(e_{i_1},...,e_{i_t})$.\
In the case $t=2$, this is just the KPS in [@Blom84]. The bit length of secret information stored by each user in Blom-Blundo et al KPS is $\displaystyle{w+t-1 \choose t-1} \cdot log_2(q)$.\
Generalized Blom-Blundo et al key predistribution schemes
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In this section we present the generalized $2$-variable and $w$-secure Blom-Blundo et al KPS, which can be extended easily to $t$-variable and $w$-secure KPS.\
Let $GF(q)$ be a fixed finite field, there are at least $\frac{q^t-\Sigma_{d|t}q^d}{t}$ distinct degree $t$ irreducible polynomials in $GF(q)[x]$ Set $P(x)=p_1(x) \cdots p_h(x)$, where $p_i$’s are degree $t$ irreducible polynomial in $GF(q)[x]$. This is a degree $H=ht$ polynomial in $GF(q)[x]$ which is not zero at any element in $GF(q)$. Set $u(x)=\frac{f(x)}{P(x)}$, where $f(x)$ is a degree $w$ polynomial. Because $P(x) \neq 0$ for any $x \in GF(q)$, thus $u(x)$ is defined for any $x \in GF(q)$. Let $u_1=\frac{f_1}{P},...,u_{w+1}=\frac{f_{w+1}}{P}$, where $f_1,...,f_{w+1}$ is a base of the linear space of all polynomials in $GF(q)[x]$ with degree less than or equal to $w$, be a base of the linear space of all these functions, for example $u_1(x)=\frac{1}{P(x)},u_2(x)=\frac{x}{P(x)},...,u_{w+1}(x)=\frac{x^{w}}{P(x)}$.\
Suppose $H \geq w$ the $2$-variable and $w$-secure KPS associated with $P(x)$ on the set of $q$ users defined over $GF(q)$ can be constructed as follows. The elements in $GF(q)$ are assigned to the users as their IDs. The TA takes a random $F(P,Q)=\Sigma_{i=1,j=1}^{w+1} a_{ij}u_i(P) u_j(Q)$, where $a_{ij}=a_{ji}$(then $F(P,Q)=F(Q,P)$) where $P,Q \in GF(q)$. The function $F(P=e_i,Q)$, as a function of $Q$, where $e_i \in GF(q)$, can be given to the user $e_i$ as its secret information. The shared key of the users with IDs $e_i$ and $e_j$ is $F(P=e_i,Q=e_j)$. The bit length of the secret information stored by each user is $(H+w+2)log_2(q)$. Here $(H+1)log_2 q$ bits are used to store the polynomial $P(x)$.\
[**Theorem 1.**]{} [*Suppose $H \geq w$ the above KPS is $w$-secure.*]{}\
[**Proof.**]{} We take the matrix of $w+1$ rows and $q$ columns with the entry at $i$ row and $j$ column is $u_i(x_j)$, where $x_j$ is the $j$-th element in $GF(q)$. This is actually a rank $w+1$ matrix. Actually any linear combination of $w+1$ rows ${\bf v_{1}},...,{\bf v_{w+1}}$ of this matrix can not be zero at more than $w$ positions, since the function $c_1 u_1+ \cdots + c_{w+1}u_{w+1}=\frac{c_1f_1+\cdots+c_{w+1}f_{w+1}}{P}$ cannot have more than $w$ zero points. Then the $w$-security of the above KPS follows from the same argument as in \[1\].\
The functions in the generalized Blom-Blundo et al KPSs have poles at the extension fields of $GF(q)$. If the polynomials $P$’s are distinct, these poles are distinct elements in the extension fields. Thus it is impossible for these functions in $KPS(P_{random})$’s have an monic polynomial relation. That is, it is impossible to express the symmetric function used in one generalized Blom Blundo et al KPS as the polynomials of symmetric functions of other different generalized Blom Blundo et al KPSs.\
The $t$-variable version of the generalized Bom-Blundo et al KPSs will not be used in the hierarchical network key predistribution schemes given in section V. We include the construction here for the convenience of the readers. The $t$-variable and $w$-secure generalized Blom-Blundo et al KPS associated with $P(x)$ on the set of $q$ users defined over $GF(q)$ can be constructed as follows. The elements in $GF(q)$ are assigned to the users as their IDs. The TA takes a random $F(P_1,...,P_t)=\Sigma_{i_1 \cdots i_t} a_{i_1\cdots i_t} u_{i_1}(P_1)\times \cdots \times u_{i_t}(P_t)$, where $a_{i_1 \cdots i_t}$ are symmetric about its subindices (then $F$ is symmetric about its variables) where $P_1,...,P_t \in GF(q)$. The $t-1$ variable function $F(P_=x,P_2,...,P_t)$ can be given to the user with $ID=x$ as its secret information. The shared key of the $t$ users with IDs $e_1,...,e_t$ is $F(e_1,...,e_t)$. The bit length of the secret information stored by each user is $\displaystyle{t+w-1 \choose t-1}log_2(q)+(H+1)log_2q$. Here $(H+1)log_2q$ bits are used for the storage of the polynomial $P(x)$.\
The proof of the $w$-security of this $t$-variable KPS is directly since any $w+1$ columns of the matrix in Theorem 1 are linearly independent.\
Then how many different such KPSs can we have? We know there are at least ${\bf B_H}=\Sigma_{t|H}(\frac{q^t-\Sigma_{d|t}q^d}{t})^{\frac{H}{t}}$ polynomials $P(x)$ from the above argument corresponding to at least ${\bf B}$ such KPSs. When $w$ is a prime number ${\bf B_H}=\frac{q^H-q}{H}$. This is quite large when both $q$ and $H$ satisfying $q>H$ tends to the infinity. Thus there are sufficiently such different $KPS(P)$’s for the randomness we need in the design of KPS for the wireless sensor networks. Generally this number can be computed by zeta functions associated with the rational curve(see \[16\]).\
When $f_1=1,...,f_{w+1}=x^{w}$ in the above generalized $2$-variable and $w$-secure KPS, we have the shared key is computed by the function $\Sigma_{i=0,j=0}^w, a_{ij} \frac{x^i}{P(x)} \frac{y^j}{P(y)}=\frac{\Sigma_{i,j=0}^w a_{ij}x^iy^j}{P(x)P(y)}$. There are at least ${\bf B_H}=\Sigma_{t|H}(\frac{q^t-\Sigma_{d|t}q^d}{t})^{\frac{H}{t}}$ possible polynomials $P(x) \in GF(q)[x]$ in the computation of the shared keys. Hence the shared keys can be adjusted by these polynomials. So the randomness we needed in the design of KPS comes from these polynomials $P \in GF(q)[x]$.\
How can we use these irreducible polynomials in the implementation of the generalized Blom-Blundo et al KPSs? From the theory of finite fields, there are an enumeration of irreducible polynomials of arbitrary fixed degree. For these low degrees, some tables of irreducible polynomials over $GF(2)$ and $GF(3)$ were listed in the standard textbooks of finite fields. It can be used for the implementation of generalized Blom-Blundo et al KPSs for which we take $h=\frac{w}{t}$ large positive integer and $t$ small positive integer.\
[**Example 1.**]{} Let $p(x)=1+2x+x^3 \in GF(9)[x]$. It is to check $p(x)$ is an irreducible polynomial in $GF(3)[x]$ and thus irreducible in $GF(9)[x]$, since the root is in $GF(27)$ and the intersection of $GF(9)$ and $GF(27)$ is $GF(3)$. Set $f_1(x)=\frac{1}{p(x)}, f_2(x)=\frac{x}{p(x)},f_3(x)=\frac{x^2}{p(x)},f_4(x)=\frac{x^3}{p(x)}$. We can have a $2$-variable and $3$-secure $KPS(p)$ on the set of $9$ players by taking random function $F(x,y)=\Sigma_{i=1,j=1}^4a_{ij} f_if_j=\Sigma_{i=0,j=0}^3 a_{ij} \frac{x^i}{p(x)} \frac{y^j}{p(y)}$, where $a_{ij}=a_{ji}$ are random elements in $GF(9)$.\
[**Example 2.**]{} Let $p(x)=x^7+x+1 \in GF(2)[x]$. This is an irreducible polynomial in $GF(2)[x]$. It is easy to check $p(x)$ is also irreducible in $GF(2^{11})[x]$, otherwise the intersection of $GF(128)$ and $GF(2^{11})$ is bigger than $GF(2)$. If $7h \leq 2^{11}=2048$, the functions $f_1=\frac{1}{p(x)^h}, f_2=\frac{x}{p(x)^h},...,f_{7h}=\frac{x^{7h}}{p(x)^h}$ can be used to get a $2$-variable and $7h$-secure generalized Blom-Blundo et al KPS. The setup server takes a random symmetric function $F(x,y)=\Sigma_{i=0}^{7h} a_{ij} \frac{x^i}{p(x)^h} \cdot \frac{y^j}{p(x)^h}$ where $a_{ij}=a_{ji}$ are random elements in $GF(2^{11})$. The setup server then predistributes $F(e_i, y)$ to the sensor node with $ID=e \in GF(2^{11})$ as its secret information. The shared key of two sensor nodes with IDs $e,e' \in GF(2^{10})$ is $F(e,e')$. This generalized Blom-Blundo KPS can be used for at most $2^{11}=1024$ sensor nodes. Since $7$ is a prime number $\frac{2^{7}-2}{7}=18$, we have at least $18$ distinct degree $7$ irreducible polynomials in $GF(2)[x]$. These polynomials are also irreducible in $GF(2^{11})[x]$. If $7h \leq 2048$, we can have at least $(18^{h}$ distinct $2$-variable and $7h$-secure KPSs on the set of $2048$ sensor nodes. All these distinct KPSs have the same security functionality as $2$-variable and $7h$-secure Blom-Blundo et al KPS. Thus these distinct generalized Blom-Blundo et al KPSs can be used for the various parts of the wireless sensor networks.\
The generalized Blom-Blundo et al key predistribution schemes can be used for disigning strongly resilient wireless sensor networks KPSs(see [@CWSN]).
Random key predistribution schemes from hyperelliptic curves
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Key predistribution schemes from a family of hyperelliptic curves
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Let $q$ be an odd prime power, $X_a$ be the hyperelliptic curve $y^2=x^{q}+q+a$ defined over $GF(q^2)$, where $a \in GF(q)$ is an arbitrary element in $GF(q) \subset GF(q^2)$. The genus of this curve is $\frac{q-1}{2}$(see \[14\]). For each $x \in GF(q^2)$, $x^q+x=Tr_{GF(q^2)/GF(q)}(x)$ is an element in $GF(q)$. Thus $x^q+x+a \in GF(q)$. It is easy to show that each element in $GF(q) \subset GF(q^2)$ is a square element, thus we have $2q^2$ affine $GF(q^2)$ rational points on $X_a$, and one $GF(q^2)$ rational point $Q$ at the infinity. $x$ has a $2$-th pole at the point $Q$ and $y$ has a $q$-th pole at the infinity. Let $L(uQ)$ be the linear space of rational functions on the hyperelliptic curve with only pole at the point $Q$ and the pole order not bigger than $u$. It is known that $\{x^iy^j|2i+qj \leq u\}$, under the reduction $y^2=x^q+x+a$, is a base of the function space $L(uQ)$ if $u \geq 2g-1=q-2$, which is a $u-g+1$ dimensional space over $GF(q)$. For example when $u=2q$, then $\{1,x,...,x^{\frac{q+1}{2}},y,yx,...,yx^{\frac{q-1}{2}}\}$ is a base of $L((2q)Q)$(see \[14\]).\
Suppose $q \geq 5$. We have a key predistribution scheme over $GF(q^2)$ on the set of $2q^2$ users, the TA can take $X_a$ for a random $a \in GF(q)$ and a random function $F(P_1,P_2)=\Sigma_{i,j=1}^{w+\frac{q+1}{2}} a_{ij} f_i(P_1) f_j(P_2) \in L((w+q-1)Q) \otimes L((w+q-1)Q)$, where $(P_1, P_2) \in X_a \times X_a$. Here $a_{ij}$ is symmetric about $i$ and $j$, $f_1,...,f_{w+\frac{q+1}{2}}$ is a base of $L((w+q-1)Q)$ of the form $x^h_1 y^{h_2}$. Then $F(P_1=W,P_2) \in L((w+q-1)Q)$ is given to the user with the $ID=W$ as its secret information. For the users with $ID=W$ and $ID=W'$, the shared key between them is $F(W,W') \in GF(q^2)$. It is clear that in this $(2,w)$ KPS over $GF(q^2)$ on the set of $2q^2$ users the storage of secret information of each user is $2(w+\frac{q-1}{2}) log_2(q)$ bits.\
[**Theorem 2.**]{} [*The above key predistribution scheme is $w$-secure.*]{}\
[**Proof.**]{} We consider the $(w+\frac{q+1}{2}) \times (2q^2)$ matrix by evaluating the $w+\frac{q+1}{2}$ base functions of $L((w+q-1)Q)$ at the $2q^2$ points described as above. This is actually the generator matrix of the algebraic geometric code(see \[14\]). It is well-known the minimum Hamming distance of the dual code is at least $w+2$(see \[14\]). Thus any $w+1$ columns of the above matrix are linear independent vectors in $GF(q^2)^{w+\frac{q+1}{2}}$. From the construction of Blom key predistribution scheme in \[1\](also see [@DDHVKK05] pages 236-237), the above construction is a $w$-secure key predistribution scheme on $2q^2$ users.\
In this family of key predistribution schemes $KPS(a)$ on the set of $2q^2$ users, where $a$ is the parameter of curve equation, the shared keys are computed in a field with $q^2$ elements. The randomness of of these KPSs are from random curves instead of polynomials in the generalized Blom KPSs.\
Though we need not to use the $t$-variable case in section V for the key predistribution schemes of hierarchical networks the construction is included here for the convenience of the readers. The above $2$-variable and $w$-secure KPS can be extended to $t$-variable and $w$-secure KPS as follows. the TA can take $X_a$ for a random $a \in GF(q)$ and a random function $F(P_1,,...,P_t)=\Sigma_{i_1...i_t=1}^{w+\frac{q+1}{2}} a_{i_1...i_t} f_{i_1}(P_1)\times \cdots \times f_{i_t}(P_{i_t}) \in L((w+q-1)Q) \otimes \cdots \otimes L((w+q-1)Q)$, where $(P_{i_1},..., P_{i_t}) \in X_a \times \cdots \times X_a$. Here $a_{i_1...i_t}=a_{j_1..j_t}$, where $j_1...j_t$ is an arbitrary permutation of $i_1...i_t$, and $f_1,...,f_{w+\frac{q+1}{2}}$ is a base of $L((w+q-1)Q)$ of the form $x^h_1 y^{h_2}$. Then $F(P_{i_1}=W,P_{i_2},...,P_{i_t})$ of $t-1$ variables is given to the user with the $ID=W$ as its secret information. For the users with $ID_1=W_1,...,ID_t=W_t$, the shared key for them is $F(W_1,...W_t) \in GF(q^2)$. It can be proved similarly as above that this $t$-variable and $w$-secure KPS over $GF(q^2)$ on the set of $2q^2$ users. The storage of secret information of each user is $2\displaystyle{t+w+\frac{q-3}{2} \choose t-1}log_2(q)$ bits. The detailed construction and the proof will be included in our future paper \[9\].\
Implementation
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In the key predistribution schemes from hyperelliptic curve $X_a$ where $a$ can take any element in $GF(q)$, the TA can assign the coordinates of the $GF(q)$ rational points of the hyperelliptic curve $X_a, a \in GF(q)$ to the $2q^2$ users as their IDs. Then the TA can fix a base of the function space $L((w+q-1)Q)$ as above. The process of these key predistribution schemes is the same as in Blom KPS, the only difference is the polynomials and the elements of the finite field are replaced by rational functions in $L((w+q-1)Q)$ and $GF(q^2)$ rational points of the curve. It should be noted that the same monimial base as above can be used for arbitrary curve $X_a, a \in GF(q)$, in the process of the computation of the shared keys, the reduction used on the curve $X_a$ is $y^2=x^q+x+a$. The parameter $a$ playes the critical role in the computation of shared keys in the hyperelliptic curve key preditribution schemes. Here $(w+\frac{q+1}{2})log_2(q)$ bits of secret information need to be stored by each user.\
Strongly resilient key predistribution schemes for hierarchical networks
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Let $R$ be the root authority, it has at most $A_1$ children nodes $R_1$,...,$R_{A_1}$, each $R_i$ has $A_{(i)}$ children nodes, $R_{i1}...R_{iA_{(i)}}$. $A_2=\Sigma A_{(i)}$ is number of all nodes at the 3rd level. We assume the hierarchical system has $L+1$ levels. The node at the $K$ level is denoted by $R_{i_1i_2..i_{K-1}}$, which has $A_{(i_1i_2...i_{K-1})}$ children nodes. Here $i_j$ is its number at the $j$-th level. Let $A_K=\Sigma A_{(i_1i_2...\i_{K-1})}$ is the number of all nodes at the $K+1$-th level. We assume $A_{(i_1...i_K)} \leq U$ for any possible subindices, that is, for each node, it has at most $U$ children nodes. $U$ is called the expansion number.\
Generalized Blom-Blundo et al key predistribution schemes for hierarchical networks
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We fix a prime power $q \geq 2U$ and a positive integer $t$ such that $\frac{q^t-\Sigma_{d|t}q^d}{t} \geq q$ and $2U-1=th$ for some positive integer $h$. We consider the $q$ irreducible polynomials of degree $t$ $P_1,...,P_{q} \in GF(q)[x]$ and a one-to-one correspondence between $P_{\alpha_1},...,P_{\alpha_q}$ and the elements $\alpha_1,...,\alpha_q$ of $GF(q)$ will be used. For any parent node $R_{i_1...i_{K-2}}$ at the $K-1$-th level, each child node $R_{i_1...i_{K-2}j}$ at the $K$ level is assigned an element in $GF(q)$ as its ID. There are at least $(\frac{q^{t}-\Sigma_{d|t} q^d }{ t} )^h>q$ different $(2,2U-2)$ curve-KPS on the set of $2U$ users defined over $GF(q)$. The KPS associated with the polynomial $P_{\alpha_i}^h$ is denoted by $KPS(P_{\alpha_i})$\
The root authority $R$ uses the random $KPS(P_s)$, where $s$ is a random element in $GF(q)$, to give the secret information to each of its child node $R_i$, where $ i \leq A_1$. The bit length of the secret information is $2(U-1) log_2(q)$. For each node $R_i$ at the 2nd level, $R_i$ randomly picks up $KPS(P_{s_i})$, where $s_i \in GF(q)$ is random element in $GF(q)$, to give each of its child node the secret information. For any two $R_{i_1j_1}$ and $R_{i_2j_2}$ at the 3rd level, $R_{i_1}$ and $R_{i_2}$ at the 2nd level can have a shared key $s_{i_1i_2}$ in $GF(q)$ from the $KPS(P_s)$, then $R_{i_1}$ and $R_{i_2}$ use $KPS(P_{s_{i_1i_2}})$ to give secret information to their children nodes $R_{i_1j}$’s and $R_{i_2j}$’s. When $R_{ij_1}$ and $R_{ij_2}$ want to find their shared key, they can use $KPS(P_{s_i}$, and when $R_{i_1j_1}$ and $R_{i_2j_2}$ want to find their shared key, they can use $KPS(P_{s_{i_1i_2}})$. This process can proceed to all the levels. That is, $R_{i_1...i_w}$ randomly picks up $KPS(P_{s_{i_1...i_w}})$ for the shared key among its children nodes, and $R_{i_1...i_w}$ and $R_{i_1'...i_w'}$ use their shared key $s_{i_1i_1'...i_wi_w'}$ to fix a $KPS(P_{s_{i_1i_1'...i_wi_w'}})$, then this KPS is used for the shared key between the children nodes of $R_{i_1...i_w}$ and $R_{i_1'...i_w'}$.\
The bit length stored in each node at the $K+1$-th level is $2A_K(2U-1)log_2(q)$ and the computation of the shared key is mainly the $(2U-1)$ times of multiplications of the finite field $GF(q)$.\
Hyperelliptic curve key predistribution schemes for hierarchical networks
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We denote the generalized $(2,2U-2)$ curve-KPS on the set of $2U \leq 2q^2$ users defined over $GF(q^2)$ from the hyperellptic curve $X: y^2=x^q+x+a$ as in section 3.1 as $KPS(a)$ with parameter $a$ from the finite field $GF(q)$. We take a finite field $GF(q^2)$ satisfying $2U \leq 2q^2$. The root authority $R$ uses the random $KPS(a)$, that is $a$ is randomly picked up from the finite field $GF(q)$, to give the secret information to each of its child node $R_i$. The bit length of the secret information is $2(2U+\frac{q-3}{2}) log_2(q^2)$. For each node $R_i$ at the 2nd level, $R_i$ randomly picks up $KPS(P_{a_i})$, where $a_i \in GF(q)$, to give each of its child node the secret information. For any two $R_{i_1j_1}$ and $R_{i_2j_2}$ at the 3rd level, $R_{i_1}$ and $R_{i_2}$ at the 2nd level can have a shared key $s_{i_1i_2}$ in $GF(q^2)$ from the $KPS(a)$, then $R_{i_1}$ and $R_{i_2}$ use $KPS(P_{s_{i_1i_2}^{q+1}})$, to give secret information to their children nodes $R_{i_1j}$’s and $R_{i_2j}$’s. It should be noted $s_{i_1i_2}^{q+1} \in GF(q)$ since $s_{i_1i_2} \in GF(q^2)$. When $R_{ij_1}$ and $R_{ij_2}$ want to find their shared key, they can use $KPS(P_{a_i}$, and when $R_{i_1j_1}$ and $R_{i_2j_2}$ want to find their shared key, they can use $KPS(P_{s_{i_1i_2}^{q+1}})$. This process can proceed to all the levels. That is, $R_{i_1...i_w}$ randomly picks up $KPS(P_{s_{i_1...i_w}})$ , where $s_{i_1...i_w} \in GF(q)$, for the shared key among its children nodes. The nodes $R_{i_1...i_w}$ and $R_{i_1'...i_w'}$ use their shared key $s_{i_1i_1'...i_wi_w'}$ to fix a $KPS(P_{s_{i_1i_1'...i_wi_w'}^{q+1}})$, then this KPS is used for the shared key between the children nodes of $R_{i_1...i_w}$ and $R_{i_1'...i_w'}$.\
The field size in this hyperelliptic curve-KAS for the hierarchical system has to satisfy $q^2 \geq \frac{U}{2}$, which is much weaker than the previous KAS.\
The bit length of the secret information stored in each node at the $K+1$-th level is $2A_K(2U+\frac{q-3}{2}) log_2(q^2)$ and at most $4U+q-3$ times of multiplications of the field $GF(q^2)$ are used for computing the shared key.\
Key predistribution schemes for dynamic hierarchical networks
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In the above hierarchical KAS, when $q \geq 2A_{(i_1...i_{K-1})}$ is valid in genus 0 KPS and $q^2 \geq A_{(i_1...i_{K-1})}$ in hyperelliptic curve KPS, nodes can be added by the parent node $R_{i_1...i_{K-1}}$ to the hierarchy. That is, if we choose $q$ with suitable large size, the hierarchical nodes can added by the parent node without change the settings of other nodes.
Information theoretical security
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Because the number of children nodes of each node $A_{(i_1i_2...i_K)} \leq U$ and we use $(2,2U-2)$ KPS, the adversary compromising less than $2U$ nodes cannot get the full information of the KPS used, if the adversary compromise all children nodes (at the $K+1$-th level) of the nodes $R_{i_1...i_{K-1}}$ and $R_{i_1'...i_{K-1}'}$, the KPS used can be deleted and all the children nodes in the further levels of the nodes $R_{i_1...i_{K-1}}$ and $R_{i_1'...i_{K-1}'}$ and themselves can be deleted without any impact on the key agreement scheme of the other nodes, since we use the [**RANDOM**]{} KPS associated with random polynomials or from random curves for the key predistribution for the un-compromised nodes and their children nodes. The point here is, after deleting the compromising nodes, their children nodes and their parent nodes, the secret information stored in un-compromised nodes is random and the shared keys of the un-compromised nodes are uniformly distrubited random variables from the view of the compromised nodes.\
Conclusion
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In this paper the generalized Blom-Blundo eta la key predistribution schemes and key predistribution schemes from hyperelliptic curves have been constructed. This kind of KPSs is flexible and can be used to construct hierarchical network key predistribution schemes. The size of shared keys only depends on the expansion numbers of nodes. These hierarchical network KPSs are identity based and dynamic. They are more efficient than the previously known hierarchical key agreement schemes and information theoretical secure against the compromising of arbitrary many internal and leaf nodes. The storage of each node is linear about the number of nodes at each level.\
[**Acknowledgment:**]{} The work was supported by the National Natural Science Foundation of China Grant 10871068.\
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[^1]: Hao Chen is with the Software Engineering Institute, East China Normal University, Shanghai 200062, China. EMAIL: [email protected]
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abstract: |
Let $(K, v)$ be a Henselian valued field with a residue field $\widehat K$, and let $p$ be a prime number. This paper determines the Brauer $p$-dimension of $K$, provided that $p \neq {\rm
char}(\widehat K)$ and $\widehat K$ is a $p$-quasilocal field which is properly included in its maximal $p$-extension. When $\widehat K$ is a local field with char$(\widehat K) \neq p$, it fully describes index-exponent relations in the $p$-component of the Brauer group Br$(K)$. The same goal is achieved in case $(K, v)$ is maximally complete, char$(K) = p$ and $\widehat K$ is a local field.
address: |
Institute of Mathematics and Informatics\
Bulgarian Academy of Sciences\
1113 Sofia, Bulgaria
author:
- 'I.D. Chipchakov'
title: 'On index-exponent relations over Henselian fields with finite or local residue fields'
---
**Introduction**
================
Let $E$ be a field, $\mathbb P$ the set of prime numbers, and for each $p \in \mathbb P$, let $E(p)$ be the maximal $p$-extension of $E$ in a separable closure $E _{\rm sep}$, and $r _{p}(E)$ the rank of the Galois group $\mathcal{G}(E(p)/E)$ as a pro-$p$-group (put $r
_{p}(E) = 0$, if $E(p) = E$). Denote by $s(E)$ the class of finite-dimensional associative central simple $E$-algebras, and by $d(E)$ the subclass of division algebras $D \in s(E)$. For each $A
\in s(E)$, let $[A]$ be the equivalence class of $A$ in the Brauer group Br$(E)$, and $D _{A}$ a representative of $[A]$ lying in $d(E)$. The existence of $D _{A}$ and its uniqueness, up-to an $E$-isomorphism, is established by Wedderburn’s structure theorem (cf. [@P], Sect. 3.5), which implies the dimension $[A\colon E]$ is a square of a positive integer deg$(A)$ (the degree of $A$). It is known that Br$(E)$ is an abelian torsion group, so it decomposes into the direct sum of its $p$-components Br$(E) _{p}$, taken over $\mathbb P$ (see [@P], Sects. 3.5 and 14.4). The Schur index ind$(D) = {\rm deg}(D _{A})$ and the exponent exp$(A)$, i.e. the order of $[A]$ in Br$(E)$, are invariants of both $D _{A}$ and $[A]$. Their general relations and behaviour under scalar extensions of finite degrees are described as follows (cf. [@P], Sects. 13.4, 14.4 and 15.2):
(1.1) (a) exp$(A) \mid {\rm ind}(A)$ and $p \mid {\rm exp}(A)$, for each $p \in \mathbb P$ dividing ind$(A)$. For any $B \in s(E)$ with ind$(B)$ prime to ind$(A)$, ind$(A \otimes _{E} B) = {\rm
ind}(A).{\rm ind}(B)$; if $A$, $B \in d(E)$, then the tensor product $A \otimes _{E} B$ lies in $d(E)$;
\(b) ind$(A)$ and ind$(A \otimes _{E} R)$ divide ind$(A \otimes _{E}
R)[R\colon E]$ and ind$(A)$, respectively, for each finite field extension $R/E$ of degree $[R\colon E]$.
As shown by Brauer (see, e.g., [@P], Sect. 19.6), (1.1) (a) determines all generally valid relations between Schur indices and exponents. It is known, however, that, for a number of special fields $E$, the pairs ind$(A), {\rm exp}(A)$, $A \in s(E)$, are subject to much tougher restrictions than those described by (1.1) (a). The Brauer $p$-dimensions Brd$_{p}(E)$ of $E$, $p \in \mathbb
P$, and their supremum Brd$(E)$, the Brauer dimension of $E$, contain essential information about these restrictions. The field $E$ is said to be of Brauer $p$-dimension Brd$_{p}(E) = n$, where $n
\in \mathbb Z$, if $n$ is the least integer $\ge 0$ for which ind$(D) \le {\rm exp}(D) ^{n}$ whenever $D \in d(E)$ and $[D] \in
{\rm Br}(E) _{p}$; if no such $n$ exists, we put Brd$_{p}(E) =
\infty $. In view of (1.1), Brd$(E) \le 1$ if and only if ind$(D) =
{\rm exp}(D)$, for each $D \in d(E)$; Brd$_{p}(E) = 0$, for a given $p$, if and only if Br$(E) _{p} = \{0\}$. The absolute Brauer $p$-dimension abrd$_{p}(E)$ of $E$ is defined as the supremum Brd$_{p}(R)\colon \ R \in {\rm Fe}(E)$, where Fe$(E)$ is the set of finite extensions of $E$ in $E _{\rm sep}$. For example, when $E$ is a global or local field, Brd$_{p}(E) = {\rm abrd}_{p}(E) = 1$, for all $p \in \mathbb P$, and there exist $Y _{n} \in d(E)$, $n \in
\mathbb N$, with ind$(Y _{n}) = n$, for each $n$ (see [@We], Ch. XII, Sect. 2, and Ch. XIII, Sects. 3 and 6).
The main purpose of this paper is to determine Brd$_{p}(K)$ and to describe index-exponent relations over Br$(K) _{p}$, provided that $(K, v)$ is a Henselian (valued) field with a local residue field $\widehat K$, and $p \in \mathbb P$ is different from char$(\widehat
K)$ (for the case of a global field $\widehat K$, see [@Ch5], Sect. 5). Our main result, presented by the following theorem, concerns the case where the value group $v(K)$ is $p$-indivisible, i.e. its quotient group $v(K)/pv(K)$ is nontrivial:
\[theo1.1\] Assume that $(K, v)$ is a Henselian field, such that $\widehat K$ is a local field and [Brd]{}$_{p}(K) < \infty $, for some $p \in \mathbb P$ not equal to [char]{}$(\widehat K)$. Let $\varepsilon _{p}$ be a primitive $p$-th root of unity in $\widehat
K _{\rm sep}$, $\tau (p)$ the dimension of $v(K)/pv(K)$ as a vector space over the field $\mathbb F _{p}$ with $p$ elements, $m _{p} =
{\rm min}\{\tau (p), r _{p}(\widehat K)\} > 0$, and in case $\varepsilon _{p} \in \widehat K$, put $r _{p} ^{\prime }(\widehat
K) = r _{p}(\widehat K) - 1$ and $m _{p} ^{\prime } = {\rm
min}\{\tau (p), r _{p} ^{\prime }(\widehat K)\}$. For each $n \in
\mathbb N$, let $\mu (p, n) = nm _{p}$, if $\varepsilon _{p} \notin
\widehat K$, and $\mu (p, n) = nm _{p} ^{\prime } + \nu _{n}(m _{p}
- m _{p} ^{\prime } + [(\tau (p) - m _{p})/2])$, if $\varepsilon
_{p} \in \widehat K$ and $\nu _{n} = {\rm min}\{n, \nu \}$, $\nu $ being the greatest integer for which $\widehat K$ contains a primitive $p ^{\nu }$-th root of unity. Then [Brd]{}$_{p}(K) = \mu
(p, 1)$; also, for a pair $(k, n) \in \mathbb N ^{2}$, there exists $D _{k,n} \in d(K)$ with [ind]{}$(D _{k,n}) = p ^{k}$ and [exp]{}$(D _{k, n}) = p ^{n}$ and only if $n \le k \le \mu (p, n)$.
In addition to Theorem \[theo1.1\], we find Brd$_{p}(K)$ and describe index-exponent pairs of $p$-algebras over $K$, provided that $(K, v)$ is a maximally complete field, char$(K) = p$ and $\widehat K$ is a local field. This is obtained in Section 3 as a consequence of a complete description of index-exponent pairs of $p$-algebras over maximally complete fields of characteristic $p$ with perfect residue fields (see Propositions \[prop3.4\], \[prop3.5\] and Corollary \[coro3.6\]). The proof of Theorem \[theo1.1\] itself is based on the fact that local fields are primarily quasilocal (abbr, PQL), i.e. they are $p$-quasilocal fields with respect to every $p \in \mathbb P$. As a matter of fact, local fields are quasilocal, i.e. their finite extensions are PQL (see [@S1], Ch. XIII, Sect. 3). When $E$ is a field with $r
_{p}(E) > 0$, for a fixed $p \in \mathbb P$, we say that $E$ is $p$-quasilocal, if the relative Brauer group Br$(E ^{\prime }/E)$ equals the group $_{p}{\rm Br}(E) = \{b \in {\rm Br}(E)\colon \ pb =
0\}$, for every degree $p$ extension $E ^{\prime }$ of $E$ in $E(p)$. The formula for Brd$_{p}(K)$ given by Theorem \[theo1.1\] is deduced from a more general result applying to any $p$-quasilocal $\widehat K$ with $p \neq {\rm char}(\widehat K)$ and $r
_{p}(\widehat K) > 0$, for some $p \in \mathbb P$. This result is contained in Theorem \[theo4.1\] and its proof relies on the inequality Brd$_{p}(\widehat K) \le 1$, and on the following relations between finite extensions of $\widehat K$ in $\widehat
K(p)$ and algebras $\Delta _{p} \in d(\widehat K)$ of $p$-primary dimensions (see [@Ch2], I, Theorems 3.1 and 4.1 (iii)):
(1.2) (i) A field $L _{p} ^{\prime } \in I(\widehat K(p)/\widehat
K)$ is embeddable in $\Delta _{p}$ as a $\widehat K$-subalgebra if and only if $[L _{p} ^{\prime }\colon \widehat K] \mid {\rm
ind}(\Delta _{p})$.
\(ii) A finite extension $L _{p}$ of $\widehat K$ in $\widehat K(p)$ is a splitting field of $\Delta _{p}$, i.e.
$[\Delta _{p}] \in {\rm Br}(L _{p}/\widehat K)$, if and only if $[L
_{p}\colon \widehat K] < \infty $ and ind$(\Delta _{p}) \mid [L
_{p}\colon \widehat K]$.
The description of index-exponent relations over Br$(K) _{p}$, under the hypotheses of Theorem \[theo1.1\], is based on the knowledge of the structure of the (continuous) character group $C(\widehat
K(p)/\widehat K)$ of $\mathcal{G}(\widehat K(p)/\widehat K)$ as an abstract abelian group (see (6.3) and Remark \[rema6.2\]). As shown in Sections 5 and 6, this approach leads to a full description of index-exponent relations over Br$(K) _{p}$ whenever $(K, v)$ is a Henselian field, such that $\widehat K$ is $p$-quasilocal and the group $\mu _{p}(\widehat K)$ of roots of unity in $\widehat K$ of $p$-primary degrees is nontrivial. The imposed conditions on $\widehat K$ and $\mu _{p}(\widehat K)$ enable one not only to determine the structure of $C(\widehat K(p)/\widehat K)$ (see (5.1) (a), (6.1) (a), Remark \[rema5.3\] and Proposition \[prop5.4\]). They also make it possible to use it in our proofs in conjunction with the presentability of cyclic $\widehat K$-algebras of degree $p$ as symbol algebras, following from Kummer theory (these algebras are defined, for example, in [@P], Sect. 15, and in [@JW], respectively). When Br$(\widehat K) _{p} \neq \{0\}$, we rely at crucial points on the fact (see [@Ch6], Theorem 3.1) that the canonical correspondence of the set of finite abelian extensions of $\widehat K$ in $\widehat K(p)$ into the set of norm subgroups of $\widehat K ^{\ast }$ is injective and maps field compositums into group intersections, and field intersections into subgroup products.
The basic notation and terminology used and conventions kept in this paper are standard, like those in [@Ch2] and [@Ch4]. For a Henselian field $(K, v)$, $K _{\rm ur}$ denotes the compositum of inertial extensions of $K$ in $K _{\rm sep}$; the notions of an inertial, a nicely semi-ramified (abbr, NSR), an inertially split, and a totally ramified (division) $K$-algebra, are defined in [@JW]. Valuation-theoretic preliminaries used in the sequel are included in Section 2. By a Pythagorean field, we mean a formally real field whose set of squares is additively closed. As usual, $[r]$ stands for the integral part of any real number $r \ge 0$. Given a field extension $\Lambda /\Psi $, $I(\Lambda /\Psi )$ denotes the set of its intermediate fields. Throughout this paper, Galois groups are viewed as profinite with respect to the Krull topology, and by a profinite group homomorphism, we mean a continuous one. The reader is referred to [@L], [@Efr2], [@JW], [@P] and [@S2], for missing definitions concerning field extensions, orderings and valuation theory, simple algebras, Brauer groups and Galois cohomology.
**Preliminaries on Henselian fields and valued extensions**
===========================================================
Let $(K, v)$ be a Krull valued field with a residue field $\widehat
K$ and a (totally ordered) value group $v(K)$. We say that $(K, v)$ is Henselian, if $v$ extends uniquely, up-to an equivalence, to a valuation $v _{L}$ on each algebraic extension $L/K$. This occurs, for example, if $(K, v)$ is maximally complete, i.e. it has no valued extension $(K ^{\prime }, v ^{\prime })$, such that $K ^{\prime }
\neq K$, $\widehat K ^{\prime } = \widehat K$ and $v ^{\prime }(K
^{\prime }) = v(K)$. When $(K, v)$ is Henselian, we denote by $\widehat L$ the residue field of $(L, v _{L})$ and put $v(L) = v
_{L}(L)$, for each algebraic extension $L/K$. It is well-known that $\widehat L/\widehat K$ is an algebraic extension and $v(K)$ is a subgroup of $v(L)$. Moreover, Ostrowski’s theorem states the following (cf. [@Efr2], Theorem 17.2.1):
(2.1) If $L/K$ is finite and $e(L/K)$ is the index of $v(K)$ in $v(L)$, then
$[\widehat L\colon \widehat K]e(L/K)$ divides $[L\colon K]$ and $[L\colon K][\widehat L\colon \widehat K]
^{-1}e(L/K) ^{-1}$ is not divisible by any $p \in \mathbb P$, $p
\neq {\rm char}(\widehat K)$; when char$(\widehat K) \dagger
[L\colon K]$, $[L\colon K] = [\widehat L\colon \widehat K]e(L/K)$.
It is known (cf. [@Sch], Ch. 2, Sect. 7) that, for any Henselian field $(K, v)$, each $\Delta \in d(K)$ has a unique, up-to an equivalence, valuation $v _{\Delta }$ which extends $v$ and has an abelian value group $v(\Delta )$. The group $v(\Delta )$ is totally ordered and includes $v(K)$ as an ordered subgroup of index $e(\Delta /F) \le [\Delta \colon K]$; the residue division ring $\widehat {\Delta }$ of $(\Delta , v _{\Delta })$ is a $\widehat
K$-algebra with $[\widehat {\Delta }\colon \widehat K] \le [\Delta
\colon K]$. More precisely, by Ostrowski-Draxl’s theorem [@Dr2], $e(\Delta /K)[\widehat \Delta \colon \widehat K] \mid [\Delta \colon
K]$, and if char$(\widehat K)$ $\dagger $ ind$(\Delta )$, then $[\Delta \colon K] = e(\Delta /K)[\widehat \Delta \colon \widehat
K]$. Note that (2.1) and the Henselity of $(K, v)$ imply the following:
(2.2) The quotient groups $v(K)/pv(K)$ and $v(L)/pv(L)$ are isomorphic, if $p \in \mathbb P$ and $[L\colon K] < \infty $. When char$(\widehat K) \dagger [L\colon K]$, the natural embedding of $K$ into $L$ induces canonically an isomorphism $v(K)/pv(K) \cong
v(L)/pv(L)$.
A finite extension $R$ of $K$ is said to be inertial, if $[R\colon
K] = [\widehat R\colon \widehat K]$ and $\widehat R$ is separable over $\widehat K$. We say that $R/K$ is totally ramified, if $[R\colon K] = e(R/K)$; $R/K$ is called tamely ramified, if $\widehat R/\widehat K$ is separable and char$(\widehat K) \dagger
e(R/K)$. The properties of $K _{\rm ur}/K$ used in the sequel are essentially the same as those presented on page 135 of [@JW], and restated in [@Ch4], (3.3). Here we recall some results on central division $K$-algebras (most of which can be found in [@JW]):
(2.3) (a) If $D \in d(K)$ and char$(\widehat K) \dagger {\rm
ind}(D)$, then $[D] = [S \otimes _{K} V \otimes _{K} T]$, for some $S$, $V$, $T \in d(K)$, such that $S/K$ is inertial, $V/K$ is NSR, $T/K$ is totally ramified, $T \otimes _{K} K _{\rm ur} \in d(K _{\rm
ur})$, exp$(T \otimes _{K} K _{\rm ur}) = {\rm exp}(T)$, and $T$ is a tensor product of totally ramified cyclic $K$-algebras (see also [@Dr2], Theorem 1);
\(b) The set IBr$(K)$ of Brauer equivalence classes of inertial $K$-algebras $S ^{\prime } \in d(K)$ is a subgroup of Br$(K)$ canonically isomorphic to Br$(\widehat K)$; Brd$_{p}(\widehat K) \le
{\rm Brd}_{p}(K)$, $p \in \mathbb P$, and equality holds when $p
\neq {\rm char}(\widehat K)$ and $v(K) = pv(K)$;
\(c) With assumptions and notation being as in (a), if $T \neq K$, then $K$ contains a primitive root of unity of degree exp$(T)$; in addition, if $T _{n} \in d(K)$ and $[T _{n}] = n[T]$, for some $n
\in \mathbb N$, then $T _{n}/K$ is totally ramified;
Statement (2.3) can be supplemented as follows (see, e.g., [@Ch5], Sect. 4):
(2.4) If $D$, $S$, $V$ and $T$ are related as in (2.3) (a), then:
\(a) IBr$(K)$ contains the class $n[D]$, for a given $n \in \mathbb
N$, if and only if $n$ is divisible by exp$(V)$ and exp$(T)$;
\(b) $D/K$ is inertial if and only if $V = T = K$; $D/K$ is inertially split, i.e. $[D] \in {\rm Br}(K _{\rm ur}/K)$, if and only if $T = K$;
\(c) exp$(D) = {\rm lcm}({\rm exp}(S), {\rm exp}(V), {\rm exp}(T))$.
Our next result provides lower and upper bounds on Brd$_{p}(K)$, under the hypothesis that $(K, v)$ is a Henselian field with Brd$_{p}(\widehat K) < \infty $, for some $p \neq {\rm
char}(\widehat K)$. This result can be stated as follows (cf. [@Ch5], Theorem 2.3):
\[theo2.1\] Let $(K, v)$ be a Henselian field with a residue field $\widehat K$ satisfying the condition [Brd]{}$_{p}(\widehat
K) < \infty $, for some $p \in \mathbb P$ different from [char]{}$(\widehat K)$. Let also $\tau (p)$ be the dimension of $v(K)/pv(K)$ over the field $\mathbb F _{p}$, $\varepsilon _{p}$ a primitive $p$-th root of unity in $\widehat K _{\rm sep}$, and $m
_{p} = {\rm min}\{\tau (p), r _{p}(\widehat K)\}$. Then:
[(a)]{} [Brd]{}$_{p}(K) = \infty $ if and only if $m _{p} =
\infty $ or $\tau (p) = \infty $ and $\varepsilon _{p} \in \widehat
K$;
[(b)]{} [max]{}$({\rm Brd}_{p}(\widehat K) + [\tau (p)/2],
[(\tau (p) + m _{p})/2]) \le {\rm Brd}_{p}(K) \le {\rm
Brd}_{p}(\widehat K) + [(\tau (p) + m _{p})/2]$, provided that $\tau
(p) < \infty $ and $\varepsilon _{p} \in \widehat K$;
[(c)]{} When $m _{p} < \infty $ and $\varepsilon _{p} \notin
\widehat K$, $m _{p} \le {\rm Brd}_{p}(K) \le {\rm Brd}_{p}(\widehat
K) + m _{p}$.
When $(K, v)$ is Henselian with Brd$_{p}(\widehat K) < {\rm
Brd}_{p}(K) = \infty $, for some $p \in \mathbb P$, $p \neq {\rm
char}(\widehat K)$, index-exponent relations over Br$(K) _{p}$ are fully described by the following consequence of Theorem \[theo2.1\], obtained in [@Ch5], Sect. 4:
\[coro2.2\] Let $(K, v)$ be a Henselian field with [Brd]{}$_{p}(\widehat K) < \infty $ and [Brd]{}$_{p}(K) = \infty $, for some $p \neq {\rm char}(\widehat K)$. Then the following alternative holds:
[(a)]{} For each pair $(k, n) \in \mathbb N ^{2}$ with $n \le k$, there exists $D _{k,n} \in d(K)$, such that [ind]{}$(D _{k,n}) = p
^{k}$ and [exp]{}$(D _{k,n}) = p ^{n}$;
[(b)]{} $p = 2$ and $\widehat K$ is a Pythagorean field; such being the case, the group [Br]{}$(K) _{2}$ has period $2$, and there are $D _{m} \in d(K)$, $m \in \mathbb N$, with [ind]{}$(D
_{m}) = 2 ^{m}$.
We conclude these preliminaries with a lemma that is implicitly used in the proofs of the main results of the following Section.
\[lemm2.3\] Let $(K, v)$ be a valued field with [char]{}$(K) =
p > 0$ and $v(K) \neq pv(K)$, and let $\pi $ be an element of $K
^{\ast }$ of value $v(\pi ) \notin pv(K)$. Assume that $G$ is a finite abelian $p$-group of order $p ^{t}$. Then there exists a Galois extension $M$ of $K$ in $K(p)$, such that $\mathcal{G}(M/K)
\cong G$, $v$ is uniquely extendable to a valuation $v _{M}$ of $M$, up-to an equivalence, and $v(\pi ) \in p ^{t}v _{M}(M)$; in particular, $v _{M}(M)/v(K)$ is cyclic and $(M, v _{M})/(K, v)$ is totally ramified.
First we prove the existence of a sequence $L _{m} ^{\prime }$, $L
_{m}$, $m \in \mathbb N$, of Galois extensions of $K$ in $K(p)$ satisfying the following conditions, for each $m$:
(2.5) (a) $L _{m} ^{\prime }/K$ is a $\mathbb Z _{p}$-extension, i.e. $\mathcal{G}(L _{m} ^{\prime }/K)$ is isomorphic to the additive group $\mathbb Z _{p}$ of $p$-adic integers; $L _{m} \in
I(L _{m} ^{\prime }/K)$ and $[L _{m}\colon K] = p$;
\(b) The compositums $M _{m} = L _{1} \dots L _{m}$ and $M _{m}
^{\prime } = L _{1} ^{\prime } \dots L _{m} ^{\prime }$ are Galois extensions of $K$, such that $[M _{m}\colon K] = p ^{m}$ and $\mathcal{G}(M _{m} ^{\prime }/K) \cong \mathbb Z _{p} ^{m}$;
\(c) Every finite extension $M$ of $K$ in $M _{m} ^{\prime }$ has a unique valuation $v _{M}$ extending $v$, up-to an equivalence, $(M,
v _{M})/(K, v)$ is totally ramified, and $v(\pi ) \in p ^{t}v
_{M}(M)$, where $p ^{t} = [M\colon K]$.
One may assume without loss of generality that $v(\pi ) < 0$. Let $\mathbb F$ be the prime subfield of $K$, $(K _{v}, \bar v)$ a Henselization of $(K, v)$, $\rho (K _{v}) = \{u ^{p} - u\colon \ u
\in K _{v}\}$, $\omega $ the valuation of the field $\Phi $ induced by $v$ and for each $m \in \mathbb N$, let $L _{m}$ and $\Phi _{m}$ be the root fields in $K _{\rm sep}$ over $K$ and $\Phi $, respectively, of the polynomial $f _{m}(X) = X ^{p} - X - \pi _{m}$, where $\pi _{m} = \pi ^{1+qm}$. Identifying $K _{v}$ with its $K$-isomorphic copy in $K _{\rm sep}$, take a Henselization $(\Phi
_{\omega }, \bar \omega )$ of $(\Phi , \omega )$ as a valued subfield of $(K _{v}, \bar v)$ (this is possible, by [@Efr2], Theorem 15.3.5), and denote by $\Psi _{m} = \Phi _{1} \dots \Phi
_{m}$ and $M _{m} = L _{1} \dots L _{m}$, for each index $m$. It is well-known that $(K _{v}, \bar v)/(K, v)$ and $(\Phi _{\omega }, \bar
\omega )/(\Phi , \omega )$ are immediate and $\rho (K _{v})$ is an $\mathbb F$-subspace of $K _{v}$, and it is easily verified that $\bar v(u ^{\prime }) \in pv(K)$ whenever $u ^{\prime } \in \rho (K _{v})$ and $\bar v(u ^{\prime }) < 0$. This implies that the cosets $\pi _{m} + \rho (K _{v})$, $m \in \mathbb N$, are linearly independent over $\mathbb F$, so the Artin-Schreier theorem (cf. [@L], Ch. VIII, Sect. 6) implies the following, for each $m \in \mathbb N$:
(2.6) $L _{m}/K$, $L _{m}K _{v}/K _{v}$, $\Phi _{m}/\Phi $ and $\Phi
_{m}\Phi _{\omega }/\Phi _{\omega }$ are degree $p$ cyclic extensions; $M _{m}/K$, $M _{m}K _{v}/K _{v}$, $\Psi _{m}/\Phi $ and $\Psi _{m}\Phi _{\omega }/\Phi _{\omega }$ are abelian of degree $p
^{m}$.
Note further that, by Witt’s lemma (cf. [@Dr1], Sect. 15, Lemma 2), for any $m \in \mathbb N$, there is a $\mathbb Z
_{p}$-extension $\Phi _{m} ^{\prime }$ of $\Phi $ in $K _{\rm sep}$, such that $\Phi _{m} \in I(\Phi _{m} ^{\prime }/\Phi )$. Hence, by Galois theory, $L _{m} ^{\prime } = \Phi _{m} ^{\prime }K$ is a $\mathbb Z _{p}$-extension of $K$. We show that $M _{m}$ and the field $M _{m} ^{\prime } = L _{1} ^{\prime } \dots L _{m} ^{\prime
}$, $m \in \mathbb N$, have the properties required by (2.5). Note first that $M _{m} ^{\prime } = \Psi _{m} ^{\prime }K$, where $\Psi
_{m} ^{\prime } = \Phi _{1} ^{\prime } \dots \Phi _{m} ^{\prime }$. Also, it follows from (2.6) and Galois theory that $[\Psi _{0}\Phi
_{\omega }\colon \Phi _{\omega }] = p$ and $\Psi _{0} \in I(\Psi
_{m}/\Phi )$; for any degree $p$ extension $\Psi _{0}$ of $\Phi $ in $\Psi _{m} ^{\prime }$. Hence, $\Psi _{0}\Phi _{\omega }/\Phi
_{\omega }$ is totally ramified. Let now $\Psi $ be a finite extension of $\Phi $ in $\Psi _{m} ^{\prime }$. Observing that $\widehat \Phi $ is a finite field and $(\Phi _{\omega }, \bar
\omega )$ is a Henselian discrete valued field, one obtains that each $\Phi ^{\prime }_{\omega } \in {\rm Fe}(\Phi _{\omega })$ is defectless [@TY], Proposition 2.2, and contains as a subfield an inertial lift of $\widehat \Phi ^{\prime }_{\omega }$ over $\Phi
_{\omega }$. Therefore, Galois theory and our observations on $\Psi
_{0}$ indicate that $\Psi \Phi _{\omega }/\Phi _{\omega }$ is totally ramified and $[\Psi K\colon K] = [\Psi \Phi _{\omega }\colon
\Phi _{\omega }] = [\Psi \colon \Phi ]$. This implies $\Psi /\Phi $ is totally ramified, which means that $\Psi /\Phi $ possesses a primitive element $\theta $ whose minimal polynomial $f _{\theta
}(X)$ over $K$ is Eisensteinian relative to $\omega $ (cf. [@FV], Ch. 2, (3.6), and [@L], Ch. XII, Sects. 2, 3 and 6). Let $\theta _{0}$ be the free term of $f _{\theta }(X)$. As $\pi \in
\Phi $, $v(\pi ) \notin pv(K)$ and $\Psi /\Phi $ is a Galois extension, this implies $\theta $ is a primitive element of $\Psi
K/K$, $p ^{m}w(\theta ) = v(\theta _{0}) = \omega (\theta _{0})$ and $v(\pi ) \in p ^{m}w(M _{m})$, for any valuation $w$ of $\Psi K$ extending $v$. The obtained result proves the uniqueness of $w$, up-to an equivalence. It is now easy to see that $\Psi _{m} ^{\prime
} \cap K _{v} = \Phi $, so it follows from Galois theory that the mapping of $I(\Psi _{m} ^{\prime }/\Phi )$ into $I(M _{m} ^{\prime
}/K)$, by the rule $\Psi ^{\prime } \to \Psi ^{\prime }K$, is bijective with $\mathcal{G}(\Psi ^{\prime }K/K) \cong
\mathcal{G}(\Psi ^{\prime }/\Phi )$, for each $\Psi ^{\prime } \in
I(\Psi _{m} ^{\prime }/\Phi )$. This completes the proof of (2.5) and Lemma \[lemm2.3\].
**Brauer $p$-dimensions of Henselian fields of characteristic $p$**
===================================================================
In this Section we consider index-exponent relations of $p$-algebras over Henselian fields of characteristic $p$. For this purpose, we need the following lemma whose applicability is guaranteed by Lemma \[lemm2.3\]:
\[lemm3.1\] Assume that $(K, v)$ is a valued field with [char]{}$(K) = p > 0$ and $v(K) \neq pv(K)$, $\tau (p)$ is the $\mathbb
F _{p}$-dimension of $v(K)/pv(K)$, and $L$ is a finite abelian extension of $K$ in $K(p)$ satisfying the following conditions:
[(a)]{} $[L\colon K] = p ^{m}$, the period of $\mathcal{G}(L/K)$ is equal to $p ^{m'}$, and $\mathcal{G}(L/K)$ has a minimal system of $t$ generators;
[(b)]{} $L$ has a unique, up-to an equivalence, valuation $v _{L}$ extending $v$, and the group $v _{L}(L)/v(K)$ is cyclic of order $p
^{m}$.
Then there exists $T \in d(K)$, such that [exp]{}$(T) = p ^{m'}$ and $T$ possesses a maximal subfield $K$-isomorphic to $L$, except, possibly, in the case where $\tau (p) < \infty $ and $p ^{t-\tau
(p)} \ge [\widehat K\colon \widehat K ^{p}]$.
It is clear from Galois theory and the structure of finite abelian groups that $L = L _{1} \dots L _{t}$ and $[L\colon K] = \prod
_{j=1} ^{t} [L _{j}\colon K]$, for some cyclic extensions $L
_{j}/K$, $j = 1, \dots , t$. Put $\pi _{0} = \pi $ and suppose that there exist elements $\pi _{j} \in K ^{\ast }$, $j = 1, \dots , t$, and an integer $\mu $ with $0 \le \mu \le t$, such that the cosets $v(\pi _{i}) + pv(K)$, $i = 0, \dots , \mu $, are linearly independent over $\mathbb F _{p}$, and in case $\mu < t$, $v(\pi
_{u}) = 0$ and the residue classes $\hat \pi _{u}$, $u = \mu + 1,
\dots , t$, generate an extension of $\widehat K ^{p}$ of degree $p
^{t-\mu }$. Fix a generator $\lambda _{j}$ of $\mathcal{G}(L
_{j}/K)$, for $j = 1, \dots , t$, denote by $T$ the $K$-algebra $\otimes _{j=1} ^{t} (L _{j-1}/K, \lambda _{j-1}, \pi _{j})$, where $\otimes = \otimes _{K}$, and put $T ^{\prime } = T \otimes _{K} K
_{v}$. We show that $T \in d(K)$ (whence exp$(T) = {\rm
per}(\mathcal{G}(L/K)$ and ind$(T) = p ^{m}$). Clearly, there is a $K _{v}$-isomorphism $T ^{\prime } \cong \otimes _{j=1} ^{t} (L
_{j-1} ^{\prime }/K _{v}, \lambda _{j-1} ^{\prime }, \pi _{j})$, where $\otimes = \otimes _{K _{v}}$ and $\lambda _{j-1} ^{\prime }$ is the unique $K _{v}$-auto-
morphism of $L _{j-1} ^{\prime }$ extending $\lambda _{j-1}$, for each $j$. Therefore, it suffices for the proof of Lemma \[lemm3.1\] to show that $T ^{\prime } \in d(K _{v})$. Since, by the proof of Lemma \[lemm2.3\], $K _{v}$ and $L ^{\prime } = LK
_{v}$ are related as in our lemma, this amounts to proving that $T
\in d(K)$, for $(K, v)$ Henselian. Suppose first that $m = 1$. As $L
_{1}/K$ is totally ramified, it follows from the Henselity of $v$ that $v(l) \in pv(L _{1})$, for every element $l$ of the norm group $N(L _{1}/K)$. One also sees that if $l \in N(L _{1}/K)$ and $v
_{L}(l) = 0$, then $\widehat K ^{p}$ contains the residue class $\hat l$. These observations prove that $\pi _{1} \notin N(L
_{1}/K)$, so it follows from [@P], Sect. 15.1, Proposition b, that $T _{2} \in d(K)$. Henceforth, we assume that $m \ge 2$ and view all value groups considered in the rest of the proof as (ordered) subgroups of a fixed divisible hull of $v(K)$. Let $L
_{0}$ be the degree $p$ extension of $K$ in $L _{t}$, and $R _{j} =
L _{0}L _{j}$, $j = 1, \dots , t$. Put $\rho _{t} = \lambda _{t}
^{p}$, and in case $t \ge 2$, denote by $\rho _{j}$ the unique $L
_{0}$-automorphism of $R _{j}$ extending $\lambda _{j}$, for $j = 1,
\dots , t - 1$. Then the centralizer $C$ of $L _{0}$ in $T$ is $L
_{0}$-isomorphic to $\otimes _{j=1} ^{t} (R _{j}/L _{0}, \rho _{j},
\pi _{j})$, where $\otimes = \otimes _{L _{0}}$. Therefore, using (2.1) and Lemma \[lemm2.3\], one easily obtains that it suffices to prove that $T \in d(K)$ in the case where $C \in d(L _{0})$.
Denote by $w$ the valuation of $C$ extending $v _{L _{0}}$, and by $\widehat C$ its residue division ring. It follows from the Ostrowski-Draxl theorem that $w(C)$ equals the sum of $v(L)$ and the group generated by $[L _{i'}\colon K] ^{-1}v(\pi _{i'})$, $i
^{\prime } = 1, \dots , \mu $. Similarly, it is proved that $\widehat C/\widehat K$ is a purely inseparable field extension unless $\widehat C = \widehat K$. Moreover, one sees that $\widehat
C \neq \widehat K$ if and only if $\mu < t$, and when this is the case, $[\widehat C\colon \widehat K] = \prod _{u=\mu +1} ^{t} [L
_{u}\colon K]$ and $\widehat C = \widehat K(\eta _{\mu +1}, \dots ,
\eta _{t})$, where $\eta _{u}$ is a root of $\hat \pi _{u}$ of degree $[L _{u}\colon K]$, for each index $u$. In view of (2.1) and well-known general properties of purely inseparable finite extensions (cf. [@L], Ch. VII, Sect. 7), these results show that $v(\pi _{t}) \notin pw(C)$, if $\mu = t$, and $\hat \pi _{t} \notin
\widehat C ^{p}$, otherwise. Observe now that, by the Skolem-Noether theorem (cf. [@P], Sect. 12.6), there exists a $K$-isomorphism $\bar \rho _{t}$ of $C$ extending $\lambda _{t}$, and it is induced by an inner $K$-automorphism of $T$. This implies $w(c) = \bar \rho
_{t}(c)$, for each $c \in C$, the products $c ^{\prime } = \prod
_{\kappa =0} ^{p-1} \bar \rho _{t} ^{\kappa }(c)$, $c \in C$, have values $w(c ^{\prime }) \in pw(C)$, and $\hat c ^{\prime } \in
\widehat C ^{p}$, if $w(c) = 0$. Therefore, $c ^{\prime } \neq \pi
_{t}$, for any $c \in C$, so it follows from [@A2], Ch. XI, Theorems 11 and 12, that $T \in d(K)$. Let now $\Lambda $ be the fixed field of $\mathcal{G}(L/K) ^{p}$. Then [@P], Sect. 15.1, Corollary b, indicates that the class $p[D] \in {\rm Br}(K)$ is represented by a crossed product of $\Lambda /K$, defined similarly to $D$. Since $\Lambda /K$ and $\pi $ are related like $L/K$ and $\pi $, it is now easy to prove, proceeding by induction on $m'$, that exp$(D) = p ^{m'}$, as claimed.
Our next result is of independent interest. It reduces the study of Brauer $p$-dimensions of finitely-generated transcendental extensions of a field $E$ to the special case where $p \neq {\rm char}(E)$ (see [@Ch2], for more details).
\[prop3.2\] Let $E$ be a field with [char]{}$(E) = p > 0$ and $[E\colon E ^{p}]
= p ^{\nu } < \infty $, and $F/E$ a finitely-generated extension of transcendency degree $n > 0$. Then $n + \nu - 1 \le {\rm Brd}_{p}(F)
\le {\rm abrd}_{p}(F) \le n + \nu $, and when $n + \nu \ge 2$, $(p
^{t}, p ^{s})\colon t, s \in \mathbb N, s \le t \le (n + \nu - 1)s$, are index-exponent pairs over $F$.
Our assumptions ensure that $[F _{1}\colon F _{1} ^{p}] = p ^{n+\nu
}$, for every finite extension $F _{1}/F$, so it follows from [@Ch2], Lemma 4.1, and Albert’s theory of $p$-algebras (cf. [@A2], Ch. VII, Theorem 28) that Brd$_{p}(F) \le {\rm
abrd}_{p}(F) \le n + \nu $. At the same time, it is easy to see that if $S$ is a subset of $F$ consisting of $n$ algebraically independent elements over $E$, then any ordering on $S$ gives rise to a valuation $v$ of $F$, such that $v(F) = \mathbb Z ^{n}$, $v$ induces on $E$ the trivial valuation, and $\widehat F$ is a finite extension of $E$. Therefore, $[\widehat F\colon \widehat F ^{p}] = p ^{\nu }$ (cf. [@L], Ch. VII, Sect. 7) and $v(F)/pv(F)$ is of order $p ^{n}$, which enables one to deduce the remaining assertions of Proposition \[prop3.2\] from Lemma \[lemm3.1\].
\[rema3.3\] It is known (see [@PY], (3.19), or [@JW], Corollary 6.10) that if $(K, v)$ is a Henselian field and $T \in
d(K)$ is a tame algebra, in the sense of [@PY] or [@JW], then the period per$(T/K)$ of the group $v(T)/v(K)$ divides exp$(T)$. At the same time, by Lemma \[lemm3.1\] with its proof, $(K, v)$ can be chosen so that there exist $T _{n} \in d(K)$, $n \in \mathbb N$, such that ind$(T _{n}) = {\rm per}(T _{n}/K) = {\rm exp}(T _{n}/K) ^{n}$ and $[T _{n}\colon K] = [\widehat T _{n}\colon \widehat K]e(T
_{n}/K)$, for each $n$.
The following two results fully describe index-exponent pairs of $p$-algebras of maximally complete fields of characteristic $p$ with perfect residue fields.
\[prop3.4\] Let $(K, v)$ be a valued field of characteristic $p
> 0$. Suppose that $v(K)/pv(K)$ is infinite or $[\widehat K\colon
\widehat K ^{p}] = \infty $, where $\widehat K ^{p} = \{\hat a
^{p}\colon \ \hat a \in \widehat K\}$. Then $(p ^{k}, p ^{n})\colon
(k, n) \in \mathbb N ^{2}, n \le k$, are index-exponent pairs over $K$.
Lemma \[lemm3.1\], [@Ch5], Remark 4.3, and our assumptions show that there exist $D _{n} \in d(K)$, $n \in \mathbb N$, such that exp$(D _{n}) = p$, ind$(D _{n}) = p ^{n}$ and $D _{n}$ is a tensor product of degree $p$ cyclic $K$-algebras, for each index $n$. Hence, by [@Ch3], Lemma 5.2, it is sufficient to prove that $(p ^{n}, p ^{n})$, $n \in \mathbb N$, are index-exponent pairs over $K$. Note again that, by Witt’s lemma, for any cyclic extension $L$ of $K$ in $K(p)$, we have $L \in I(L ^{\prime }/K)$, for some $\mathbb Z _{p}$-extension $L ^{\prime }$ of $K$ in $K(p)$. Let $\sigma $ be a topological generator of $\mathcal{G}(L ^{\prime
}/K)$, and for each $n \in \mathbb N$, let $L _{n}$ be the extension of $K$ in $L ^{\prime }$ of degree $p ^{n}$, and $\sigma _{n}$ the automorphism of $L _{n}$ induced by $\sigma $. Clearly, $L _{n}/K$ is cyclic and $\sigma _{n}$ is a generator of $\mathcal{G}(L
_{n}/K)$. Now choose $L ^{\prime }$ so that $(L _{1}/K, \sigma _{1},
c) \cong D _{1}$, for some $c \in K ^{\ast }$. Then, by [@P], Sect. 15.1, Corollary a, the cyclic $K$-algebras $\Delta _{n} = (L
_{n}/K, \sigma _{n}, c)$, $n \in \mathbb N$, satisfy ind$(\Delta
_{n}) = {\rm exp}(\Delta _{n}) = p ^{n}$, which completes our proof.
\[prop3.5\] Let $(K, v)$ be a maximally complete field with [char]{}$(K) = p > 0$ and $[K\colon K ^{p}] = p ^{n}$, for some $n \in
\mathbb N$. Then $n - 1 \le {\rm Brd}_{p}(K) \le n$. Moreover, if $\widehat K$ is perfect, then:
[(a)]{} Brd$_{p}(K) = n - 1$ if and only if $n > r _{p}(\widehat
K)$;
[(b)]{} $(p ^{k}, p ^{s})\colon (k, s) \in \mathbb N ^{2}$, $s \le
k \le {\rm Brd}_{p}(K)s$, are index-exponent pairs over $K$.
[(c)]{} [abrd]{}$_{p}(K) = n - 1$ if and only if the Sylow pro-$p$-subgroups of the absolute Galois group $\mathcal{G}_{\widehat K}$ are trivial or isomorphic to $\mathbb Z
_{p}$.
Our assumptions show that $[K\colon K ^{p}] = [\widehat K\colon
\widehat K ^{p}]e(K/K ^{p})$ (cf. [@W], Theorem 31.21), so it follows from Lemma \[lemm3.1\] and Albert’s theory of $p$-algebras [@A2], Ch. VII, Theorem 28, that $n - 1 \le {\rm Brd}_{p}(K) \le
n$, as claimed. In the rest of the proof, we assume that $\widehat
K$ is perfect. Suppose first that $r _{p}(\widehat K) \ge n$. Then one obtains from Galois theory and Witt’s lemma that $\mathbb Z _{p}
^{n}$ is realizable as a Galois group over $\widehat K$. Hence, by well-known properties of the natural bijection $I(K _{\rm ur}/K) \to
I(\widehat K _{\rm sep}/\widehat K)$, there is a Galois extension $U
_{n}$ of $K$ in $K _{\rm ur}$ with $\mathcal{G}(U _{n}/K) \cong
\mathbb Z _{p} ^{n}$. This implies each finite abelian $p$-group $H$ that can be generated by $n$ elements is isomorphic to $\mathcal{G}(U _{H}/K)$, for some Galois extension $U _{H}$ of $K$ in $K _{\rm ur}$. Observing also that $v(K)/pv(K)$ has order $p
^{n}$, and applying [@JW], Exercise 4.3, one proves the existence of an NSR-algebra $N _{H} \in d(K)$ possessing a maximal subfield $K$-isomorphic to $U _{H}$. This result shows that Brd$_{p}(K) = n$, and reduces the rest of our proof to the special case where $n > r _{p}(\widehat K)$. Then it follows from [@AJ], Theorem 3.3, and [@Ch2], Lemma 4.1, that Brd$_{p}(K) \le n - 1$, which completes the proof of Proposition \[prop3.5\] (a). The validity of Proposition \[prop3.5\] (b) is proved as in the case of $n \le {\rm Brd}_{p}(K)$, using Lemma \[lemm3.1\] instead of [@JW], Exercise 4.3. Note finally that $(L, v _{L})$ is maximally complete and $[L\colon L ^{p}] = p ^{n}$, for every $L \in
{\rm Fe}(K)$ (cf. [@W], Theorem 31.22, and [@L], Ch. VII, Sect. 7). In view of Proposition \[prop3.5\] (a), this enables one to deduce Proposition \[prop3.5\] (c) from [@Wh], Theorem 2, Galois cohomology and Nielsen-Schreier’s formula for open subgroups of free pro-$p$-groups (cf. [@S2], Ch. I, 4.2, and Ch. II, 2.2).
Our next result complements Theorem \[theo1.1\] as follows:
\[coro3.6\] Assume that $(K, v)$ is a maximally complete field, [char]{}$(K) = p > 0$, $\widehat K$ is a local field and $\tau
(p)$ is defined as in Theorem \[theo2.1\]. Then:
[(a)]{} [Brd]{}$_{p}(K) = \infty $ if and only if $\tau (p) =
\infty $; when this holds, $(p ^{k}, p ^{n})$ is an index-exponent pair over $K$, for each $(k, n) \in \mathbb N ^{2}$ with $k \ge n$;
[(b)]{} [Brd]{}$_{p}(K) = \tau (p)$, provided that $\tau (p) <
\infty $; in this case, $(p ^{k}, p ^{n})$ is an index-exponent pair over $K$, where $(k, n) \in \mathbb N ^{2}$, if and only if $n \le k
\le n\tau (p)$.
Let $\omega $ be the natural discrete valuation of $\widehat K$, and $\widehat K _{\omega }$ its residue field. It is known (cf. [@Efr2], Sect. 5.2) that $K$ is endowed with a valuation $w$ (a refinement of $v$), such that $w(K) = v(K) \oplus \omega (\widehat
K)$, $\omega (\widehat K)$ is an isolated subgroup of $w(K)$, $v$ and $\omega $ are canonically induced by $w$ and $\omega (\widehat
K)$ upon $K$ and $\widehat K$, respectively, and the residue field $\widehat K _{w}$ of $(K, w)$ is isomorphic to $\widehat K _{\omega
}$. Observing further that, by theorems due to Krull and Hasse-Schmidt-MacLane (cf. [@Efr2], Theorems 12.2.3, 18.4.1, and [@W], Theorem 31.24 and page 483), $(\widehat K, \omega )$ is maximally complete and $(K, w)$ has a maximally complete valued extension $(K ^{\prime }, w ^{\prime })$ with $\widehat K ^{\prime }
= \widehat K _{w}$ and $w ^{\prime }(K ^{\prime }) = w(K)$, one concludes that $(K ^{\prime }, w ^{\prime }) = (K, w)$. Since $\widehat K _{w}$ is perfect and $r _{p}(\widehat K _{w}) = 1$, this allows one to deduce Corollary \[coro3.6\] from Propositions \[prop3.4\] and \[prop3.5\].
When $(K, v)$ is a Henselian field, such that char$(K) = p > 0$, $v(K)$ is a non-Archimedean group, $v(K)/pv(K)$ is finite and $[\widehat K\colon \widehat K ^{p}] = p ^{\nu } < \infty $, there is, generally, no formula for Brd$_{p}(K)$ involving only invariants of $\widehat K$ and $v(K)$. We illustrate this fact in case $v(K) =
\mathbb Z ^{t}$, for any integer $t \ge 2$.
Let $Y _{0}$ be a field with char$(Y _{0}) = p$ and $[Y _{0}\colon Y
_{0} ^{p}] = p ^{\nu }$, and let $Y _{t} = Y _{0}((T _{1})) \dots
((T _{t}))$ be the iterated formal Laurent power series field in $t$ variables over $Y _{0}$. It is known (see [@BK], page 2 and further references there) that there exists a sequence $X _{n} \in Y
_{t-1}$, $n \in \mathbb N$, of algebraically independent elements over the field $Y _{t-2}(T _{t-1})$, where $Y _{t-2} = Y _{0}((T
_{1})) \dots ((T _{t-2}))$ in case $t \ge 3$. Put $F _{n} = Y
_{t-2}(T _{t-1}, X _{1}, \dots X _{n})$, for each $n \in \mathbb N$, $F _{\infty } = \cup _{n=1} ^{\infty } F _{n}$, and $\mathbb N
_{\infty } = \mathbb N \cup \{\infty \}$. For any $n \in \mathbb N
_{\infty }$, denote by $F _{n} ^{\prime }$ the separable closure of $F _{n}$ in $Y _{t-1}$, and by $v _{n}$ the valuation of the field $K _{n} = F _{n} ^{\prime }((T _{t}))$ induced by the natural $\mathbb Z ^{t}$-valued valuation of $Y _{t}$ trivial on $Y _{0}$. It is easily verified that $(K _{n}, v _{n})$ is Henselian with $v
_{n}(K _{n}) = \mathbb Z ^{t}$ and $\widehat K _{n} = Y _{0}$, for every index $n$. Note also that $[F _{\infty } ^{\prime }\colon F
_{\infty } ^{\prime p}] = \infty $, so it follows from Proposition \[prop3.4\], applied to the valuation of $K _{n}$ induced by the natural discrete valuation of $Y _{t}$ trivial on $Y _{t-1}$, that Brd$_{p}(K _{\infty }) = \infty $. When $n \in \mathbb N$, we have $[K _{n}\colon K _{n} ^{p}] = p ^{\nu + t+n} = p[F _{n} ^{\prime
}\colon F _{n} ^{\prime p}]$, which enables one to deduce from Lemma \[lemm3.1\], [@Ch3], Lemma 4.1, and the theory of $p$-algebras [@A2], Ch. VII, Theorem 28 (see also [@L], Ch. VII, Sect. 7) that $\nu + t + n - 1 \le {\rm Brd}_{p}(K _{n}) \le \nu + n + t$.
**The Brauer $p$-dimension of a Henselian field with a $p$-quasilocal residue field**
=====================================================================================
Let $(K, v)$ be a Henselian field with a $p$-quasilocal field $\widehat K$ and $r _{p}(\widehat K) > 0$. Then Brd$_{p}(\widehat K)
_{p} \le 1$, so it follows from Theorem \[theo2.1\] that Brd$_{p}(K) = \infty $ if and only if $m _{p} = \infty $ or $\tau
(p) = \infty $ and $\varepsilon _{p} \in \widehat K$. When Brd$_{p}(K) = \infty $, index-exponent relations over Br$(K) _{p}$ are described by Corollary \[coro2.2\] and the characterization of formally real $2$-quasilocal fields, provided by [@Ch2], I, Lemma 3.5. When Brd$_{p}(K) < \infty $, Brd$_{p}(K)$ is determined as follows:
\[theo4.1\] In the setting of Theorem \[theo2.1\], let $\widehat K$ be a $p$-quasilocal field, $m _{p} > 0$ and [Brd]{}$_{p}(K) < \infty $. Then:
[(a)]{} ${\rm Brd}_{p}(K) = u _{p}$, where $u _{p} =
[(\tau (p) + m _{p})/2]$, if $\varepsilon _{p} \in \widehat K$ and $\widehat K$ is nonreal; $u _{p} = m _{p}$, if $\varepsilon _{p}
\notin \widehat K$;
[(b)]{} [Br]{}$(K) _{2}$ is a group of period $2$ and [Brd]{}$_{2}(K) = 1 + [\tau (2)/2]$, provided that $\widehat K$ is formally real and $p = 2$.
Suppose first that $\widehat K$ is formally real and $p = 2$. Then, by [@Ch2], I, Lemma 3.5, $\widehat K$ is Pythagorean, $\widehat
K(2) = \widehat K(\sqrt{-1})$ and Br$(\widehat K) _{2}$ is a group of order $2$. Therefore, $r _{2}(\widehat K) = 1$ and $r
_{2}(\widehat K(\sqrt{-1})) = 0$, so it follows from the Merkur’ev-Suslin theorem (see [@MS], (16.1)), that Br$(\widehat
K(\sqrt{-1})) _{2} = \{0\}$. Note also that $K$ is Pythagorean, which implies $2{\rm Br}(K) = \{0\}$ (cf. [@La], Theorem 3.16, and [@Efr1], Theorem 3.1). These observations and [@Ch5], Corollary 6.2, prove Theorem \[theo4.1\] (b). We turn to the proof of Theorem \[theo4.1\] (a), so we assume that $p > 2$ or $\widehat
K$ is a nonreal field. Our argument relies on the following results concerning inertial algebras $I \in d(K)$ with $[I] \in {\rm Br}(K)
_{p}$, and inertial extensions $U$ of $K$ in $K(p)$:
(4.1) (a) ind$(I) = {\rm exp}(I)$ and $I$ is a cyclic $K$-algebra;
\(b) $[I] \in {\rm Br}(U/K)$ if and only if ind$(U) \mid [U\colon
K]$; $U$ is embeddable in $I$ as a $K$-subalgebra if and only if $[U\colon K] \mid {\rm ind}(I)$;
\(c) ind$(I \otimes _{K} I ^{\prime })$ equals ind$(I)$ or ind$(I
^{\prime })$, if $I ^{\prime } \in d(K)$, $I ^{\prime }/K$ is NSR, and $[I ^{\prime }] \in {\rm Br}(K) _{p}$.
Statements (4.1) can be deduced from (1.2), (2.3) (b) and [@JW], Theorems 3.1 and 5.15. They imply in conjunction with [@Ch5], Lemma 4.1, that ind$(W) \mid {\rm exp}(W) ^{m _{p}}$, for each $W
\in d(K)$ inertially split over $K$. At the same time, it follows from [@Ch4], (3.3), and [@Mo], Theorem 1 (see also [@JW], Exercise 4.3), that there is an NSR-algebra $W ^{\prime }
\in d(K)$ with ind$(W ^{\prime }) = p ^{m _{p}}$ and exp$(W ^{\prime
}) = p$. Observe now that, by (2.3) (c), $d(K)$ consists of inertially split $K$-algebras in case $\varepsilon _{p} \notin
\widehat K$ or $\tau (p) = 1$. In view of (4.1) and [@JW], Theorem 4.4 and Lemma 5.14, this yields Brd$_{p}(K) = m _{p}$, so it remains for us to prove Theorem \[theo4.1\], under the extra hypothesis that $\varepsilon _{p} \in \widehat K$ and $\tau (p) \ge
2$. It is easily obtained from [@Mo], Theorem 1, and [@Ch5], Lemmas 4.1 and 4.2, that there exists $\Delta \in d(K)$ with exp$(\Delta ) = p$ and ind$(\Delta ) = p ^{\mu (p)}$, where $\mu (p)
= [(m _{p} + \tau (p))/2]$. This means that Brd$_{p}(K) \ge \mu
(p)$, so we have to prove that Brd$_{p}(K) \le \mu (p)$. Note first that $2 \le m _{p} \le r _{p}(\widehat K)$, provided that Br$(\widehat K) _{p} \neq \{0\}$. Assuming the opposite and taking into account that $\varepsilon _{p} \in \widehat K$, one obtains from the other conditions on $\widehat K$ that it is a nonreal field with $r _{p}(\widehat K) = 1$. Hence, by [@Wh], Theorem 2, $\widehat K(p)/\widehat K$ is a $\mathbb Z _{p}$-extension, i.e. $\mathcal{G}(\widehat K(p)/\widehat K)$ is isomorphic to the additive group $\mathbb Z _{p}$ of $p$-adic integers. In view of [@MS], (11.5) and (16.1), and Galois cohomology (cf. [@S2], Ch. I, 4.2), this requires that Br$(\widehat K) _{p} = \{0\}$. As $\tau (p) \ge 2$, the obtained contradiction proves the claimed inequalities. Now take an algebra $D \in d(K)$ so that exp$(D) = p
^{n}$, for some $n \in \mathbb N$. Suppose that $S$, $V$ and $T \in
d(K)$ are related with $D$ as in (2.3) (a), and fix $\Theta \in
d(K)$ so that $[\Theta ] = [V \otimes _{K} T]$. To prove that ind$(D) \mid p ^{n\mu (p)}$ we need the following statements:
(4.2) (a) If $n = 1$, then $S$, $V$ and $T$ can be chosen so that $V
\otimes _{K} T = \Theta $, and $S = K$ or $V = K$.
\(b) If $n \ge 2$, then there exists a totally ramified extension $Y$ of $K$ in $K(p)$, such that $[Y\colon K] \mid p ^{\mu (p)}$ and either exp$(D \otimes _{K} Y) \mid p ^{n-1}$, or
exp$(D \otimes _{K} Y) = {\rm exp}(S _{Y}) = p ^{n}$, $[Y\colon K]$ divides $p ^{[\tau (p)/2]}$ and exp$(V _{Y} \otimes _{Y} T _{Y})$ divides $p ^{n-1}$, where $S _{Y}, V _{Y}, T _{Y} \in d(Y)$ are attached in accordance with (2.3) (a) to the underlying division algebra $D _{Y}$ of $D \otimes _{K} Y$.
Statement (4.2) (a) can be deduced from (4.1), [@Ch5], (4.8), and well-known properties of cyclic algebras (cf. [@P], Sect. 15.1, Proposition b). Since (4.2) (a) implies the assertion of Theorem \[theo4.1\] (a) in the case of $n = 1$, we assume further that $n \ge 2$. The conclusion of (4.2) (b) is obvious, if exp$(\Theta ) \mid p ^{n-1}$ (one may put $Y = K$). Therefore, by (2.4) (c), it suffices to prove (4.2) (b) under the hypothesis that exp$(\Theta ) = p ^{n}$. Take $D _{n-1} \in d(K)$ so that $[D
_{n-1}] = p ^{n-1}[D]$ and attach to it a triple $S _{n-1}$, $V
_{n-1}$, $T _{n-1} \in d(K)$ in agreement with (4.2) (a). Then $V
_{n-1} \otimes _{K} T _{n-1}$ contains as a maximal subfield an abelian and totally ramified extension $Y$ of $K$. Identifying $Y$ with its $K$-isomorphic copy in $K(p)$, and using (2.4) (a), one sees that it has the properties required by (4.2) (b).
We continue with the proof of Theorem \[theo4.1\] (a). For any associative algebra $B$, denote by $Z(B)$ its centre. It is known (cf. [@JW], Corollary 6.8) that if $J \in d(K)$ is inertial over $K$ and $J ^{\prime } \in d(K)$ is a representative of $[J \otimes
_{K} \Theta ]$, then $v(J ^{\prime }) = v(\Theta )$, $Z(\widehat J
^{\prime }) = Z(\widehat \Theta )$ and $[\widehat J ^{\prime }] =
[\widehat J \otimes _{\widehat K} \widehat \Theta ] \in {\rm
Br}(Z(\widehat \Theta ))$. Note also that the period of the group $v(J ^{\prime })/v(K)$ divides exp$(J ^{\prime })$, by [@PY], (3.19) (see also [@JW], Corollary 6.10). These results imply in conjunction with (4.1) (a), (b) and the Ostrowski-Draxl theorem the following assertions:
(4.3) (a) If exp$(\Theta ) \mid p ^{n-1}$, then ind$(D) \mid p.{\rm
ind}(S _{0} \otimes _{K} V \otimes _{K} T)$, for some $S _{0} \in
d(K)$ inertial over $K$ with exp$(S _{0}) \mid p ^{n-1}$;
\(b) If exp$(\Theta ) \mid p ^{n-1}$ and ind$(D) > {\rm ind}(I
\otimes _{K} V \otimes _{K} T)$ whenever $I \in d(K)$, $I/K$ is inertial and exp$(I) \mid p ^{n-1}$, then $[Z(\widehat D)\colon
\widehat K] = p ^{k}$ and $[\widehat D\colon Z(\widehat D)] = p
^{2n-2k}$, for some integer $k$ with $0 \le k < n$, where $Z(\widehat D)$ is the centre of $\widehat D$; in particular, ind$(D) ^{2} \mid p ^{2n+(n-1)\tau (p)} \mid p ^{nm _{p} +
(n-1)\tau (p)}$.
Now fix an extension $Y/K$ and $Y$-algebras $D _{Y}$, $S _{Y}$, $V
_{Y}$, $T _{Y}$ as in (4.2) (b), and take $\Theta _{Y} \in d(Y)$ so that $[\Theta _{Y}] = [V _{Y} \otimes _{Y} T _{Y}]$. Arguing by induction on $n$, observing that, by (1.1) (b), ind$(D) \mid {\rm
ind}(D _{Y})[Y\colon K]$, and in case exp$(D _{Y}) = p ^{n}$, applying (4.3) to $D _{Y}$, $V _{Y}$, $T _{Y}$ and $\Theta _{Y}$, instead of $D$, $V$, $T$ and $\Theta _{Y}$, respectively, one concludes that ind$(D) ^{2} \mid p ^{n(m _{p}+\tau (p))}$. Thus Theorem \[theo4.1\] is proved.
\[rema4.2\] Theorem \[theo4.1\] (a) retains its validity, if $(K, v)$ is a Henselian field with $\tau (p) < \infty $, $r _{p}(\widehat K) = 0$ and $\mu _{p}(\widehat K) \neq \{1\}$. Then it follows from [@MS], (16.1), that Brd$_{p}(\widehat K) = 0$, so Theorem \[theo2.1\] (b) implies Brd$_{p}(K) = [\tau (p)/2]$.
Our next objective in the present paper is to describe index-exponent relations over Br$(K) _{p}$, provided that $(K, v)$ is a Henselian field, $\widehat K$ is $p$-quasilocal, $\mu
_{p}(\widehat K) \neq \{1\}$ and Brd$_{p}(K) < \infty $, for some $p
\in \mathbb P$. In this Section, we consider only the case where $\widehat K$ is formally real and $p = 2$. Then $d(K)$ contains the symbol $K$-algebra $A _{-1}(-1, -1; K)$, and it follows from [@Ch5], Lemma 4.2, that if $\tau (2) \ge 2$, then there exist $D _{n} \in d(K)$, $n = 1, \dots , [\tau (2)/2]$, totally ramified over $K$ with exp$(D _{n}) = 2$ and ind$(D _{n}) = 2 ^{n}$, for each $n$. Since $A _{-1}(-1, -1; K)/K$ is inertial, this implies together with [@Mo], Theorem 1, that $A _{-1}(-1, -1; K) \otimes _{K} D
_{n} \in d(K)$ (and ind$(A _{-1}(-1, -1; K) \otimes _{K} D _{n}) = 2
^{n+1}$), for $n = 1, \dots , [\tau (2)/2]$. In view of (2.3) (b) and Theorem \[theo4.1\] (b), these results prove that if $0 \le \tau
(2) < \infty $, then $(1, 1)$ and $(2 ^{n}, 2)$, $n = 1, \dots , 1 +
[\tau (2)/2]$, are all index-exponent pairs over Br$(K) _{2}$.
**Henselian fields $(K, v)$ with $p$-quasilocal $\widehat K$ satisfying $r _{p}(\widehat K) = \infty $**
========================================================================================================
This Section provides a description of index-exponent relations over Br$(K) _{p}$, for a Henselian field $(K, v)$, such that $\widehat K$ is $p$-quasilocal, $\mu _{p}(\widehat K) \neq \{1\}$ and $r
_{p}(\widehat K) = \infty $. Our main result concerning this case can be stated as follows:
\[prop5.1\] Under the hypotheses of Theorem \[theo4.1\], suppose that $r _{p}(\widehat K) = \infty $ and $\varepsilon _{p}
\in \widehat K$. Then:
[(a)]{} There exists a sequence $U _{n}$, $n \in \mathbb N$, of degree $p$ extensions of $K$ in $K _{\rm ur}$, such that $[U _{1}
\dots U _{n}\colon K] = p ^{n}$ and $U _{n} \in I(U _{n} ^{\prime
}/K)$, where $U _{n} ^{\prime }$ is a $\mathbb Z _{p}$-extension of $K$ in $K _{\rm ur}$, for each index $n$;
[(b)]{} When $0 < \tau (p) < \infty $ and $(n, k) \in \mathbb N
^{2}$, $(p ^{k}, p ^{n})$ is realizable as an index-exponent pair over $K$ if and only if $n \le k \le \tau (p)n$.
(a): The assertion follows at once from Kummer theory, if $\mu
_{p}(\widehat K)$ is infinite. We show that it also holds in the special case where Br$(\widehat K) _{p} = \{0\}$. Indeed, it follows from [@P], Sect. 15.1, Proposition b, that then $\varepsilon
_{p}$ lies in the norm group $N(L ^{\prime }/\widehat K)$, for every cyclic extension $L ^{\prime }$ of $\widehat K$ in $\widehat K(p)$; hence, by Albert’s theorem (cf. [@A1], Ch. IX, Sect. 6), there is a cyclic extension $L _{1} ^{\prime }$ of $\widehat K$ in $\widehat
K(p)$, such that $L ^{\prime } \in I(L _{1} ^{\prime }/\widehat K)$ and $[L _{1} ^{\prime }\colon L ^{\prime }] = p$. This observation proves that $L ^{\prime } \in I(L _{1}/\widehat K)$, for some $\mathbb Z _{p}$-extension $L _{1}$ of $\widehat K$ in $\widehat
K(p)$. In view of general properties of the natural bijection of $I(K _{\rm ur}/K)$ upon $I(\widehat K _{\rm sep}/\widehat K)$, the obtained result shows that each cyclic extension $U$ of $K$ in $K(p)
\cap K _{\rm ur}$ lies in $I(U ^{\prime }/K)$, for some $\mathbb Z
_{p}$-extension $U ^{\prime }$ of $K$ in $K _{\rm ur}$. It remains for us to consider the case where Br$(\widehat K) _{p} \neq \{0\}$ and $\mu _{p}(\widehat K)$ is finite of order $p ^{\nu }$. Let $\delta _{\nu }$ be a primitive $p ^{\nu }$-th root of unity in $\widehat K$, $D(\widehat K(p)/\widehat K)$ the maximal divisible subgroup of $C(\widehat K(p)/\widehat K)$, and $d(p)$ the dimension of $_{p}{\rm Br}(\widehat K)$ as an $\mathbb F _{p}$-vector space. It is known (see, e.g., [@Ka], Ch. 7, Sect. 5) that $C(\widehat
K(p)/\widehat K)$ is an abelian torsion $p$-group. Our starting point are the following assertions:
(5.1) (a) $C(\widehat K(p)/\widehat K)$ is isomorphic to the direct sum $D(\widehat K(p)/\widehat K) \oplus \mu _{p}(\widehat K)
^{d(p)}$, where $\mu _{p}(\widehat K) ^{d(p)}$ is a direct sum of isomorphic copies of $\mu _{p}(\widehat K)$, indexed by a set of cardinality $d(p)$.
\(b) A cyclic extension $M$ of $\widehat K$ in $\widehat K(p)$ lies in $I(M _{\infty }/\widehat K)$, for some $\mathbb Z _{p}$-extension $M _{\infty }$ of $\widehat K$ in $\widehat K(p)$ if and only if there is $M ^{\prime } \in I(\widehat K(p)/\widehat K)$, such that $M ^{\prime }/\widehat K$ is cyclic, $M \in I(M ^{\prime }/\widehat
K)$ and $[M ^{\prime }\colon M] = p ^{\nu }$; this is the case if and only if $\delta _{\nu } \in N(M/\widehat K)$.
Statement (5.1) (a) is contained in [@Ch2], II, Lemma 2.3, the former part of (5.1) (b) is implied by (5.1) (a) and Galois theory, and the latter one follows from Albert’s theorem referred to. Let now $M _{\lambda }$ be an extension of $\widehat K$ generated by a $p$-th root $\eta _{\lambda } \in \widehat K(p)$ of an element $\lambda \in \widehat K ^{\ast } \setminus \widehat K ^{\ast p}$. Then $M _{\lambda }/\widehat K$ is cyclic, $[M _{\lambda }\colon
\widehat K] = p$ and $\mathcal{G}(M _{\lambda }/\widehat K)$ contains a generator $\sigma _{\lambda }$, such that the cyclic $\widehat K$-algebra $(M _{\lambda }/\widehat K, \sigma _{\lambda },
\delta _{\nu })$ is isomorphic to the symbol $\widehat K$-algebra $A
_{\varepsilon _{p}}(\lambda , \delta _{\nu }; \widehat K)$. It is well-known that $A _{\varepsilon _{p}}(\lambda , \delta _{\nu };
\widehat K)$ and $A _{\varepsilon _{p}}(\delta _{\nu }, \lambda ;
\widehat K)$ are inversely-isomorphic $\widehat K$-algebras. Together with [@P], Sect. 15.1, Proposition b, this implies $\delta _{\nu } \in N(M _{\lambda }/\widehat K)$ if and only if $\lambda \in N(M _{\delta _{\nu }}/\widehat K)$. Hence, by (5.1) (b), the assertion of Proposition \[prop5.1\] (a) is equivalent to the one that $\widehat K ^{\ast p}$ is a subgroup of $N(M _{\delta
_{\nu }}/\widehat K)$ of infinite index. Obviously, $\widehat K
^{\ast p} \subseteq N(M _{\mu }/\widehat K)$, for an arbitrary $\mu
\in \widehat K ^{\ast } \setminus \widehat K ^{\ast p}$, so it suffices to show that the group $N(M _{\mu }/\widehat K)/\widehat K
^{\ast p}$ is infinite. Fix $\mu ^{\prime } \in \widehat K ^{\ast }
\setminus \widehat K ^{\ast p}$ so that $M _{\mu '} \neq M _{\mu }$. Then $\widehat K ^{\ast }/N(M _{\mu '}/\widehat K) \cong $ $_{p}{\rm
Br}(\widehat K)$, by (1.2) and [@P], Sect. 15.1, Proposition b, and $N(M _{\mu }/\widehat K)N(M _{\mu '}/\widehat K) = \widehat K
^{\ast }$, by [@Ch2], I, Lemma 4.3. Since, by [@Ch6], Theorem 3.1, $N(M _{\mu }/\widehat K) \cap N(M _{\mu '}/\widehat K)
= N(M _{\mu }M _{\mu '}/\widehat K)$, this yields $\widehat K ^{\ast
}/N(M _{\mu '}/\widehat K) \cong N(M _{\mu }/\widehat K)/N(M _{\mu
}M _{\mu '}/\widehat K)$, $\widehat K ^{\ast p} \le N(M _{\mu }M
_{\mu '}/\widehat K)$ and $N(M _{\mu }/\widehat K)/N(M _{\mu }M
_{\mu '}/\widehat K) \cong (N(M _{\mu }/\widehat K)/\widehat K
^{\ast p})/(N(M _{\mu }M _{\mu '}/\widehat K)/\widehat K ^{\ast
p})$; in particular, $_{p}{\rm Br}(\widehat K)$ is a homomorphic image of $N(M _{\mu }/\widehat K)/\widehat K ^{\ast p}$. Thus it turns out that if $d(p) = \infty $, i.e. $_{p}{\rm Br}(\widehat K)$ is infinite, then $N(M _{\mu }/\widehat K)/\widehat K ^{\ast p}$ is infinite as well. Observe now that $r _{p}(\widehat K) = \infty $ if and only if $\widehat K ^{\ast }/\widehat K ^{\ast p}$ is infinite (cf. [@S2], Ch. I, 4.1). As the groups $(\widehat K ^{\ast
}/\widehat K ^{\ast p})/(N(M _{\mu }/\widehat K)/\widehat K ^{\ast
p})$, $\widehat K ^{\ast }/N(M _{\mu }/\widehat K)$ and $_{p}{\rm
Br}(\widehat K)$ are isomorphic, this implies $N(M _{\mu }/\widehat
K)/\widehat K ^{\ast p}$ is infinite in case $d(p) < \infty $, so Proposition \[prop5.1\] (a) is proved.
(b): It follows from Proposition \[prop5.1\] (a) and Galois theory that, for each finite abelian $p$-group $G$, there exists a Galois extension $U _{G}$ of $K$ in $K _{\rm ur}$ with $\mathcal{G}(U
_{G}/K) \cong G$. When $G$ can be generated by at most $\tau (p)$ elements, one obtains from [@Mo], Theorem 1, that there is an NSR-algebra $D _{G} \in d(K)$ possessing a maximal subfield $K$-isomorphic to $U _{G}$. It is therefore clear that there exist $D
_{k,n} \in d(K)\colon (k, n) \in \mathbb N ^{2}, n \le k \le \tau
(p)n$, such that $D _{k,n}/K$ is NSR, ind$(D _{k,n}) = p ^{k}$ and exp$(D _{k,n}) = p ^{n}$. This proves Proposition \[prop5.1\] (b), since Theorem \[theo4.1\] and the condition $r _{p}(\widehat K) =
\infty $ yield Brd$_{p}(K) = \tau (p)$.
It is well-known that Henselian discrete valued fields with quasifinite residue fields are quasilocal (cf. [@S1], Ch. XIII, Sect. 3). Our next result shows that the conditions of Proposition \[prop5.1\] (b) are fulfilled, if char$(\widehat K) = 0$ and $\widehat K$ possesses a Henselian discrete valuation $\omega $ with an infinite quasifinite residue field of characteristic $p$.
\[prop5.2\] Let $(E, \omega )$ be a Henselian discrete valued field of zero characteristic with $\widehat E$ quasifinite of characteristic $p$. Then:
[(a)]{} $r _{p}(E) = \infty $, provided that $\widehat E$ is infinite;
[(b)]{} $C(E(p)/E)$ is divisible if and only if $\mu _{p}(E) =
\{1\}$.
(b): Let $\varepsilon $ be a primitive $p$-th root of unity in $E
_{\rm sep}$. It is well-known that $[E(\varepsilon )\colon E] \mid p
- 1$ (cf. [@L], Ch. VIII, Sect. 3). Note also that Br$(E
^{\prime }) \cong \mathbb Q/\mathbb Z$, for every $E ^{\prime } \in
{\rm Fe}(E)$; in particular, this ensures that the scalar extension map Br$(E) \to {\rm Br}(E ^{\prime })$ is surjective. These observations, combined with (1.1) (b) and [@P], Sect. 15.1, Proposition b, imply that if $L$ is a cyclic $p$-extension of $E$ in $E _{\rm sep}$, then $L(\varepsilon ) ^{\ast } = L ^{\ast
}N(L(\varepsilon )/E(\varepsilon ))$. When $\varepsilon \notin E$, this indicates that $\varepsilon \in N(L(\varepsilon )/E(\varepsilon
))$, which enables one to deduce from [@FSS], Theorem 3, that $C(E(p)/E)$ is divisible. Suppose now that $\mu _{p}(E) \neq \{1\}$ and denote by $\Gamma _{p}$ the extension of $E$ generated by all roots of unity in $E _{\rm sep}$ of $p$-primary degrees. It is well-known that $\mathbb Z[X]$ contains the cyclotomic polynomial $\Phi _{p^{n}}(X)$ of order $p ^{n}$ (and degree $p ^{n-1}(p - 1)$), and the polynomial $\Phi _{p^{n}}(X + 1)$ is Eisensteinian over $\mathbb Z$ relative to $p$. This implies that $p ^{n-1}(p -
1)\omega _{\Gamma _{p}}(\varepsilon _{n}) = \omega (p)$, where $\varepsilon _{n} \in \Gamma _{p}$ is a primitive $p ^{n}$-th root of unity. As $\omega $ is discrete and $\omega (p) \neq 0$, the noted equalities prove that $\mu _{p}(E)$. In view of (5.1) (a) and the nontriviality of Br$(E) _{p}$, the obtained result ensures that $C(E(p)/E) \neq pC(E(p)/E)$.
(a): Assume that $\widehat E$ is infinite, fix a uniform element $\pi \in E$ and take elements $a _{n} \in E$, $n \in \mathbb N$, so that $\omega (a _{n}) = 0$ and the residue classes $\hat a _{n}$, $n
\in \mathbb N$, are linearly independent over the prime subfield of $\widehat E$. It is easily verified that the cosets $(1 + a _{n}\pi
)E ^{\ast p}$, $n \in \mathbb N$, are linearly independent over $\mathbb F _{p}$. This means that $E ^{\ast p}$ is a subgroup of $E
^{\ast }$ of infinite index. At the same time, it is clear from local class field theory that if $L _{j}$, $j = 1, \dots , n$, are cyclic extensions of $E$ in $E(p)$ of degree $p$, then $E ^{\ast p}$ is included in $N(L _{1} \dots L _{n}/E)$, which in turn is a subgroup of $E ^{\ast }$ of index equal to $[L _{1} \dots L
_{n}\colon E]$. Finally, it follows from the quasilocal property of $E$ that if $a \in E ^{\ast } \setminus E ^{\ast p}$, $D \in d(E)$ and ind$(D) = p$, then there exists a cyclic extension $Y$ of $E$ in $E(p)$, such that $[Y\colon E] = p$ and $D \cong (Y/E, \tau , a)$, for some generator $\tau $ of $\mathcal{G}(Y/E)$. Hence, by [@P], Sect. 15.1, Proposition b, $a \notin N(Y/E)$, which means that $E ^{\ast p}$ equals the intersection of the norm groups of cyclic degree $p$ extensions of $E$. Now the equality $r _{p}(E) =
\infty $ becomes obvious, so Proposition \[prop5.2\] is proved.
\[rema5.3\] Assume that $(K, v)$ is Henselian field with $p$-quasilocal $\widehat K$ and $\mu _{p}(\widehat K) \neq \{1\}$. Then it follows from [@Ch2], II, Lemma 2.3, that $C(\widehat
K(p)/\widehat K)$ is divisible if and only if Br$(\widehat K) _{p} =
\{0\}$ or $\mu _{p}(\widehat K)$ is infinite. When this holds, one obtains by the method of proving Proposition \[prop5.1\] (b) that if $0 < {\rm Brd}_{p}(K) < \infty $ and $(k, n) \in \mathbb N ^{2}$, then $(p ^{k}, p ^{n})$ is an index-exponent pair over $K$ if and only if $n \le k \le {\rm Brd}_{p}(K)n$. Conversely, it is well-known that, for any divisible abelian torsion $p$-group $\Pi $, there exists a field $E _{\Pi }$, such that $\mu _{p}(E _{\Pi })
\neq \{1\}$, Br$(E _{\Pi }) _{p} = \{0\}$ and $C(E _{\Pi }(p)/E
_{\Pi }) \cong \Pi $.
It is worth noting in connection with (5.1) (a) that the $\mathbb F
_{p}$-dimension $d(p)$ of $_{p}{\rm Br}(E)$ is perhaps the most important invariant of a $p$-quasilocal field $E$ with $r _{p}(E) >
0$. This is illustrated, e.g., by [@F], Theorem 23.1, and [@Ch2], I, Theorem 3.1 and Lemma 3.5, which show that $d(p)$ fully determines the structure of Br$(E) _{p}$. Also, it follows from [@Ch6], Theorem 3.1, that if $d(p) > 0$, then for each finite extension $M$ of $E$ in $E(p)$, $E ^{\ast }/N(M/E)$ is isomorphic to the direct sum $\mathcal{G}(M/E) ^{d(p)}$ of isomorphic copies of $\mathcal{G}(M/E)$, taken over a set of cardinality $d(p)$. When $d(p) = 0$, we have $E ^{\ast } = N(R/E)$, for all $R \in I(E(p)/E) \cap {\rm Fe}(E)$ (cf. [@Ch2], I, Lemma 4.2 (ii)). These results attract interest in the fact that each divisible abelian torsion $p$-group $T _{p}$ is isomorphic to Br$(E(T _{p})) _{p}$, for some $p$-quasilocal field $E(T _{p})$. In view of [@Ch2], I, Theorem 3.1 and Lemma 3.5, this property of $T _{p}$ can be obtained as a consequence of the following result (see [@Ch7], Theorem 1.2 and Proposition 6.4):
(5.2) An abelian torsion group $T$ is isomorphic to Br$(E(T))$, for some PQL-field $E(T)$ if and only if it satisfies one of the following two conditions:
\(a) $T$ is divisible; when this holds, $E(T)$ is necessarily nonreal. Moreover, for a given field $E _{0}$, $E(T)$ can be defined so as to be a quasilocal field and an extension of $E _{0}$, such that $E _{0}$ is algebraically closed in $E(T)$ and the scalar extension map Br$(E(T)) \to {\rm Br}(\Lambda )$ is surjective, for each $\Lambda \in {\rm Fe}(E(T))$;
\(b) The $p$-components $T _{p}$ are divisible, for every $p \in
\mathbb P \setminus \{2\}$, and the group $T _{2}$ is of order $2$; such being the case, $E(T)$ is formally real.
Statement (5.2) is a refinement of [@VdBSch], Theorem 3.9, which in turn generalizes [@GS], Example 2.1 (cf. also [@VdBSch], Theorem 3.8, [@BeFr], Theorem 4, and [@Ch7], Theorem 1.2 (i), for more details). When $T$ is divisible, $E _{0}$ is a field of at most countable cardinality $d(0)$, and $t$ is an infinite cardinal number such that $t \ge d(p)$, for all $p \in \mathbb P
\cup \{0\}$, the quasilocal field $E(T)$ in (5.2) (a) can be chosen among those extensions of $E _{0}$ of transcendency degree $t$, which satisfy $r _{p}(E(T)) = t$, $p \in \mathbb P$ (see [@Ch7], Remark 5.4). At the same time, the condition that $E _{0}$ is algebraically closed in $E$ ensures that $\mu _{p}(E) = \mu _{p}(E
_{0})$, for each $p \in \mathbb P$. In addition, it is a well-known consequence of Galois theory and the irreducibility of cyclotomic polynomials over the field $\mathbb Q$ of rational numbers that every subgroup $\Gamma $ of $\mathbb Q/\mathbb Z$ is isomorphic to the group $\mu (\Phi _{\Gamma })$ of roots of unity in some algebraic extension $\Phi _{\Gamma }$ of $\mathbb Q$. Therefore, applying (5.2) (a) to the case of $T = T _{p}$, for a given $p \in
\mathbb P$, and using (5.1) (a) as well as the structure and the injectivity of divisible abelian torsion $p$-groups (cf. [@F], Theorems 23.1 and 24.5), one proves the following assertion:
\[prop5.4\] Let $W$ be an abelian torsion $p$-group, for some $p
\in \mathbb P$, and let $D(W)$ be the maximal divisible subgroup of $W$. Suppose that $W$ contains infinitely many elements of order $p$. Then there is a $p$-quasilocal field $F _{W}$ with $\mu _{p}(F
_{W}) \neq \{1\}$ and $C(F _{W}(p)/F _{W}) \cong W$, if and only if, $W/D(W)$ is embeddable as a subgroup in $D(W)$, and in case $W \neq
D(W)$, it decomposes into the direct sum of cyclic groups of order $p ^{n}$, for some $n \in \mathbb N$.
**Proof of Theorem \[theo1.1\]**
================================
Our first result completes the description of index-exponent relations over Br$(K) _{p}$, for a Henselian field $(K, v)$ with a $p$-quasilocal $\widehat K$ and $\mu _{p}(\widehat K) \neq \{1\}$.
\[prop6.1\] With assumptions and notation being as in Theorem \[theo4.1\], let [Brd]{}$_{p}(\widehat K) \neq 0$, $\varepsilon
_{p} \in \widehat K$, $\mu _{p}(\widehat K)$ be a finite group of order $p ^{\nu }$, $2 \le r _{p}(\widehat K) = r < \infty $, $r
^{\prime } = r - 1$, $m ^{\prime } = {\rm min}\{\tau (p), r ^{\prime
}\}$, and for each $n \in \mathbb N$, let $\nu _{n} = {\rm min}\{n,
\nu \}$ and $\mu (p, n) = nm ^{\prime } + \nu _{n}(m _{p} - m
^{\prime} + [(\tau (p) - m _{p})/2])$. If $(k, n) \in \mathbb N
^{2}$, then $(p ^{k}, p ^{n})$ is an index-exponent pair over $K$ if and only if $n \le k \le \mu (p, n)$.
First we prove the following assertions:
(6.1) (a) $C(\widehat K(p)/\widehat K) \cong \mathbb Z(p ^{\infty })
^{r'} \oplus \mu _{p}(\widehat K)$ and $\mathcal{G}(\widehat
K(p)_{\rm ab}/\widehat K) \cong \mathbb Z _{p} ^{r'} \times \mu
_{p}(\widehat K)$, where $\mathbb Z(p ^{\infty })$ is the quasicyclic $p$-group and $\widehat K(p)_{\rm ab}$ is the compositum of finite abelian extensions of $\widehat K$ in $\widehat K(p)$;
\(b) Statement (5.1) (b) retains validity in the setting of Proposition \[prop6.1\].
The inequality $2 \le r$ and the $p$-quasilocality of $\widehat K$ ensure that $\widehat K$ is nonreal and Br$(\widehat K) _{p}$ is divisible (cf. [@Ch2], I, Theorem 3.1 and Lemma 3.5). As $\varepsilon _{p} \in \widehat K$ and $r < \infty $, they also imply $\mathcal{G}(\widehat K(p)/\widehat K)$ is a Demushkin group, in the sense of [@Lab] and [@S2], and Br$(\widehat K) _{p} \cong
\mathbb Z(p ^{\infty })$ (see [@Ch6], Proposition 5.1 and Corollary 5.3). Therefore, (6.1) (a) can be deduced from [@Ch2], II, Lemma 2.3, and general properties of the natural bijection $I(K
_{\rm ur}/K) \to I(\widehat K _{\rm sep}/\widehat K)$. As to (6.1) (b), it follows from (6.1) (a) and Albert’s theorem.
We continue with the proof of Proposition 5.1. Statement (2.3) (b), the isomorphism Br$(\widehat K) _{p} \cong \mathbb Z(p ^{\infty })$, and the equality Brd$_{p}(\widehat K) = 1$ imply that $(p ^{m}, p
^{m})$, $m \in \mathbb N$, are index-exponent pairs over both $\widehat K$ and $K$. In view of Theorem \[theo4.1\], this proves Proposition \[prop6.1\] in the special case where $\tau (p) = 1$. Henceforth, we assume that $\tau (p) \ge 2$. Suppose first that $n
\in \mathbb N$ and $n \le \nu $. Then, by Theorem \[theo4.1\], ind$(\Delta _{n}) \mid p ^{\mu (p,n)}$, for each $\Delta _{n} \in
d(K)$ with exp$(\Delta _{n}) = p ^{n}$. Using [@Mo], Theorem 1, and the natural bijection between $I(Y/K)$ and the set of subgroups of $v(Y)/v(K)$, for any finite abelian tamely and totally ramified extension $Y/K$ (cf. [@Sch], Ch. 3, Theorem 2), one obtains that, for each $k \in \mathbb N$ with $n \le k \le \mu (p, n)$, there exist an NSR-algebra $V _{n,k} \in d(K)$ and a totally ramified $T _{n,k} \in d(K)$, such that $V _{n,k} \otimes _{K} T
_{n,k} \in d(K)$, exp$(V _{n,k} \otimes _{K} T _{n,k}) = p ^{n}$ and ind$(V _{n,k} \otimes _{K} T _{n,k}) = p ^{k}$. These observations and the former part of (1.1) (a) prove Proposition \[prop6.1\] when $n \le \nu $. The rest of the proof is carried out by induction on $n \ge \nu $. The basis of the induction is provided by Theorem \[theo4.1\], which allows us to assume that $n > \nu $ and ind$(X)
\mid p ^{\mu (p,(n-1))}$ whenever $X \in d(K)$ and exp$(X) \mid p
^{n-1}$. Fix an algebra $D \in d(K)$ so that exp$(D) = p ^{n}$ and attach to $D$ algebras $S$, $V$, $T \in d(K)$ as in (2.3) (a). Clearly, if exp$(V) \mid p ^{n-1}$, then exp$(V \otimes _{K} T) \mid
p ^{n-1}$, so (4.3) and the inductive hypothesis imply ind$(D) \mid
p ^{1+\mu (p,(n-1))} \mid p ^{\mu (p,n)}$, as claimed. In view of (2.4), it remains to consider the case where exp$(V) = p ^{n}$. Let $\Sigma $, $D _{\nu } \in d(K)$ satisfy $[\Sigma ] = [S \otimes _{K}
V]$ and $[D _{\nu }] = p ^{\nu }[D] (= p ^{\nu }[\Sigma ])$. Then, by (2.4) (c), exp$(\Sigma ) = p ^{n}$, and it follows from (4.1) and [@P], Sect. 15.1, Corollary b and Proposition b, that $\Sigma
/K$ is NSR. Also, exp$(D _{\nu }) \mid p ^{n-\nu }$, and (2.3) (c) and [@P], Sect. 15.1, Corollary b, imply $D _{\nu }/K$ is NSR. In particular, $D _{\nu }$ contains as a maximal subfield an inertial extension $U _{\nu }$ of $K$, and by [@JW], Theorem 4.4, $U _{\nu }/K$ is abelian and $\mathcal{G}(U _{\nu }/K)$ has a system of $\tau (p)$ generators. Moreover, it follows from (6.1), Galois theory and [@P], Sect. 15.1, Corollary b, that $U
_{\nu }$ has a $K$-isomorphic copy from $I(U _{\nu } ^{\prime }/K)$, for the Galois extension $U _{\nu } ^{\prime }$ of $K$ in $K _{\rm
ur}$ with $\mathcal{G}(U _{\nu } ^{\prime }/K) \cong \mathbb Z _{p}
^{r'}$. Therefore, $\mathcal{G}(U _{\nu }/K)$ has a system of $r
^{\prime }$ generators, so [@JW], Theorem 4.4 (or [@Ch5], Lemma 4.1), leads to the following conclusion:
(6.2) ind$(D _{\nu }) \mid p ^{(n-\nu )m'}$ and $D _{\nu }$ contains as a maximal subfield a $K$-isomorphic copy of a totally ramified extension $\Phi _{\nu }$ of $K$ in $K(p)$.
Statement (6.2) shows that $[D _{\nu }] \in {\rm Br}(\Phi _{\nu
}/K)$, $[\Phi _{\nu }\colon K] = {\rm ind}(D _{\nu })$ and $\widehat
\Phi _{\nu } = \widehat K$. Hence, exp$(D \otimes _{K} \Phi _{\nu })
\mid p ^{\nu }$ and $r _{p}(\widehat \Phi _{\nu }) = r _{p}(\widehat
K)$, so it follows from (2.2) and Theorem \[theo4.1\] that ind$(D
\otimes _{K} \Phi _{\nu }) \mid p ^{\nu \mu (p)}$, where $\mu (p) =
[(m _{p} + \tau (p))/2]$. As $\mu (p, n) = (n - \nu )m ^{\prime } +
\nu \mu _{p}$, it is now easy to see that ind$(D) \mid p ^{\mu
(p,n)}$, as required. Suppose finally that $(k, n) \in \mathbb N
^{2}$ and $n \le k \le \mu (p, n)$. Then [@JW], Exercise 4.3, [@Mo], Theorem 1, the above-noted properties of $U _{\nu }
^{\prime }$, and those of intermediate fields of an abelian tamely and totally ramified finite extension of $K$, imply the existence of $D _{k,n} \in d(K)$ with ind$(D _{k,n}) = p ^{k}$ and exp$(D _{k,n})
= p ^{n}$. Moreover, one can ensure that $D _{k,n} \cong N _{k,n}
\otimes _{K} D _{k,n} ^{\prime }$, for some $N _{k,n}$, $D _{k,n}
^{\prime } \in d(K)$, such that $N _{k,n}$ is NSR and $D _{k,n}
^{\prime }$ is totally ramified over $K$. Proposition \[prop6.1\] is proved.
. As noted in Section 1, $\widehat K$ is quasilocal, and by assumption, it is complete with respect to a discrete valuation $\omega $ whose residue field $\widehat K _{\omega }$ is finite. This implies $(\widehat K, \omega )$ is Henselian, $\mu _{p}(\widehat K)$ is finite, and in case $p \neq {\rm char}(\widehat K _{\omega })$, $\varepsilon _{p} \in \widehat K$ if and only if $p$ divides the order $o(\widehat K _{\omega } ^{\ast })$ of $\widehat K _{\omega }
^{\ast }$. Put $r = r _{p}(\widehat K)$, and denote by $\widehat
K(p)_{\rm ab}$ the compositum of abelian finite extensions of $\widehat K$ in $\widehat K(p)$. It is known (see [@Ko], Sect. 10.1 and Theorem 10.5) that if $\varepsilon _{p} \notin \widehat K$, then
(6.3) (a) $\mathcal{G}(\widehat K(p)/\widehat K) \cong \mathbb Z
_{p}$, provided that $p \neq {\rm char}(\widehat K _{\omega })$;
\(b) When char$(\widehat K) = 0$ and char$(\widehat K _{\omega }) =
p$, $\mathcal{G}(\widehat K(p)/\widehat K)$ is a free pro-$p$-group, and $\mathcal{G}(\widehat K(p)_{\rm ab}/\widehat K) \cong \mathbb Z
_{p} ^{r}$; in addition, $\widehat K$ is a finite extension of the field $\mathbb Q _{p}$ of $p$-adic numbers and $r = [\widehat
K\colon \mathbb Q _{p}] + 1$.
Note further that, by Theorem \[theo4.1\], Brd$_{p}(K) = m _{p}$, and by (2.3) (c), every $D \in d(K)$ is inertially split over $K$. These results enable one to deduce the assertion of Theorem \[theo1.1\] (in case $\varepsilon _{p} \notin \widehat K$) from (6.3), [@JW], Exercise 4.3, and [@Mo], Theorem 1, by the method of proving Proposition \[prop5.1\] (b).
Let now $\varepsilon _{p} \in \widehat K$. Then Theorem \[theo4.1\] yields Brd$_{p}(K) = \mu (p, 1)$, and Proposition \[prop6.1\] implies that if $(k, n) \in \mathbb N ^{2}$, then $(p
^{k}, p ^{n})$ is an index-exponent pair over $K$ if and only if $n
\le k \le \mu (p, n)$. This completes our proof.
\[rema6.2\] In the setting of Theorem \[theo1.1\], with its proof, if $\varepsilon _{p} \in \widehat K$, then $r = r
_{p}(\widehat K)$ is determined as follows: [(i)]{} $r = 2$, if $p
\neq {\rm char}(\widehat K _{\omega })$; [(ii)]{} when $p = {\rm
char}(\widehat K _{\omega })$ and char$(\widehat K) = 0$, $\widehat
K/\mathbb Q _{p}$ is a finite extension and $r = [\widehat K\colon
\mathbb Q _{p}] + 2$ (see [@Ko], Sect. 10.1, and [@Lab], Sect. 5). For a $p$-quasilocal field $E$ with
Br$(E) _{p} \neq \{0\}$, $\mu _{p}(E) \neq \{1\}$ and $3 \le r
_{p}(E) < \infty $, it is an open problem whether there exists a local field $L _{E}$, such that $\mathcal{G}(L _{E}(p)/L _{E}) \cong
\mathcal{G}(E(p)/E)$.
\[coro6.3\] Assume that $(K, v)$ is a Henselian field, such that $\widehat K$ is a local field, and let $\omega $ be the usual discrete valuation of $\widehat K$. Denote by $\widehat K _{\omega
}$ the residue field of $(\widehat K, \omega )$, and suppose that $\tau (p)$ is defined as in Theorem \[theo1.1\], for each $p \in
\mathbb P$, $p \neq {\rm char}(\widehat K)$. Then [abrd]{}$_{p}(K)
= 1 + [\tau (p)/2]$, provided that $p \neq {\rm char}(\widehat K
_{\omega })$; [abrd]{}$_{p}(K) = {\rm max}\{1, \tau (p)\}$, if [char]{}$(\widehat K) = 0$ and [char]{}$(\widehat K _{\omega })
= p$.
In view of (1.1) (b) and [@Ch4], (3.3), it suffices to consider only the special case where $\mu _{p}(\widehat K) \neq \{1\}$. Then our conclusion follows from Remark \[rema6.2\] and the fact that $\widehat K(p)/\widehat K$ is an infinite extension.
\[rema6.4\] Note that the conclusions of Theorem \[theo1.1\] hold, if the assumption on $\widehat K$ is replaced by the milder one that $\widehat K$ has a Henselian discrete valuation $\omega $ with a quasifinite residue field $\widehat K _{\omega }$. In the first place, then $\widehat K$ is quasilocal and Theorem \[theo4.1\] (a) applies to every $p \in \mathbb P$, $p \neq {\rm
char}(K)$. Secondly, it is known (e.g., [@FV], Ch. 2, (3.5)) that if $p \neq {\rm char}(\widehat K _{\omega })$, then $r
_{p}(\widehat K) \le 2$, and equality holds if and only if $\mu
_{p}(\widehat K) \neq \{1\}$. Moreover, $\mathcal{G}(\widehat K
_{\rm ab}(p)/\widehat K) \cong \mathbb Z _{p} \times \mu
_{p}(\widehat K)$, provided that $\mu _{p}(\widehat K) < \infty $; $\mathcal{G}(\widehat K _{\rm ab}(p)/\widehat K) \cong \mathbb Z
_{p}^{2}$, otherwise. Thereby, index-exponent relations over Br$(K)
_{p}$ are described, in the former case, as in Theorem \[theo1.1\], and in the latter one, by Remark \[rema5.3\]. Finally, if char$(\widehat K) = 0$ and $p = {\rm char}(\widehat K
_{\omega })$, then it follows from Theorem \[theo4.1\] (a) and Proposition \[prop5.2\] that Brd$_{p}(K) \le \tau (p)$, and equality holds in case $\widehat K _{\omega }$ is infinite. In addition, using (5.1) (a) and Proposition \[prop5.2\], one obtains as in the proof of Proposition \[prop5.1\] (b) that $(p ^{k}, p
^{n})\colon k, n \in \mathbb N, n \le k \le n{\rm Brd}_{p}(K)$, are index-exponent pairs over $K$ unless $r _{p}(\widehat K) \le \tau
(p)$ and $\mu _{p}(\widehat K) \neq \{1\}$. The same holds, by the proof of Corollary \[coro3.6\], if $(K, v)$ is maximally complete and char$(K) = p > 0$.
Assuming that $(K, v)$ is a Henselian field and $\widehat K$ is a local field, summing-up (1.1) (a), Theorem \[theo1.1\], Corollary \[coro2.2\], and the latter part of (2.3) (b), and using the equalities Brd$_{p}(\widehat K) = 1$, $p \in \mathbb P$, together with (6.3) and Remark \[rema6.2\], one obtains a complete description of the restrictions on index-exponent pairs over $K$ not divisible by char$(\widehat K)$. In view of Remark \[rema6.4\], the divisibility restriction can be removed, if $(K, v)$ is maximally complete.
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---
abstract: 'A new algorithm for deciding the satisfiability of polynomial formulas over the reals is proposed. The key point of the algorithm is a new projection operator, called sample-cell projection operator, custom-made for Conflict-Driven Clause Learning (CDCL)-style search. Although the new operator is also a CAD (Cylindrical Algebraic Decomposition)-like projection operator which computes the cell (not necessarily cylindrical) containing a given sample such that each polynomial from the problem is sign-invariant on the cell, it is of singly exponential time complexity. The sample-cell projection operator can efficiently guide CDCL-style search away from conflicting states. Experiments show the effectiveness of the new algorithm.'
author:
- Haokun Li
- Bican Xia
bibliography:
- 'samplecad.bib'
title: 'Solving Satisfiability of Polynomial Formulas By Sample-Cell Projection '
---
Introduction
============
The research on SMT (Satisfiability Modulo Theories) [@deMoura+Dutertre+Shankar:cav2007; @DBLP:journals/cacm/MouraB11; @DBLP:series/faia/2009-185] in recent years brings us many popular solvers such as Z3 [@DBLP:conf/tacas/MouraB08], CVC4 [@BCD+11], Yices [@Dutertre:cav2014], MathSAT5 [@mathsat5], etc. Nevertheless, in theory and practice, it is important to design efficient SMT algorithms and develop tools (or improve existing ones) for many other theories, [*e.g.*]{} string [@DBLP:conf/cav/LiangRTBD14], linear arithmetic [@DBLP:conf/cav/DutertreM06; @DBLP:conf/cade/JovanovicM12] and non-linear arithmetic [@DBLP:journals/corr/abs-1905-09227; @DBLP:conf/smt/KorovinKS14] over the reals. A straightforward idea is to integrate Conflict-Driven Clause Learning (CDCL)-style search with theory solvers [@DBLP:series/faia/2009-185]. For example, integrating CDCL-style search with a theory solver for determining whether a basic semialgebraic set is empty can solve satisfiability in the theory of non-linear arithmetic over the reals.
It is well-known that the problem whether a basic semialgebraic set is empty is decidable due to Tarski’s decision procedure [@10.1007/978-3-7091-9459-1_3]. Tarski’s algorithm cannot be a theory solver in practice because of its very high complexity. Cylindrical algebraic decomposition (CAD) algorithm [@DBLP:journals/cca/Collins76] is a widely used theory solver in practice though it is of doubly exponential time complexity. The idea of CAD algorithm is to decompose ${\mathbb{R}}^n$ into cells such that each polynomial from the problem is sign-invariant in every cell. A key concept in CAD algorithm is the projection operator. Although many improved projection operators have been proposed [@DBLP:conf/issac/Hong90; @DBLP:journals/jsc/McCallum88; @10.1007/978-3-7091-9459-1_12; @brown_improved_2001; @Han_Dai_Xia_2014; @Dai_Han_Hong_Xia_2015; @Xia_Yang_2016], the CAD method is still of doubly exponential time complexity. The main reason is that in order to carry enough information, projection of variables causes the number of polynomials grows rapidly. So the cost of simply using CAD as a theory solver is unacceptable.
Jovanovic and de Moura [@DBLP:conf/cade/JovanovicM12] eased the burden of using CAD as a theory solver by modifying the CDCL-style search framework. They changed the sequence of search states by adding variable assignments to the sequence. The benefit of this is that they can use real-root isolation, which is of polynomial time complexity, to check consistency of literals for there will be only one unassigned variable in the literals of the current state. When a conflict of literals is detected, they explain the conflict by applying CAD to a polynomial set called conflicting core to find the cell where the sample of assignments belongs. But even using CAD only when explaining conflicts is a huge computational cost, as CAD is of doubly exponential time complexity. Furthermore, CAD will produce all cells in ${\mathbb{R}}^n$ other than the only one we need, making computation waste.
In this paper, we propose a new custom-made CAD-like projection operator, called sample-cell projection operator. It only processes the cell containing a given sample, which is exactly what conflict explanation needs. The idea of our operator is trying to project polynomials related to the target cell and ignore irrelevant polynomials. We integrate our sample-cell projection operator with Jovanovic’s improved CDCL-style search framework. The new operator can efficiently guide CDCL-style search away from conflicting states. It is proved that the new algorithm is of singly exponential time complexity. We have implemented a prototype solver LiMbS which is base on Mathematica 12. Experiments show the effectiveness of the new algorithm.
The rest of this paper is structured as follows: Section \[sec:pre\] introduces the background knowledge and notation. Section \[sec:sample\] defines sample-cell projection and presents the details of our approach. Section \[sec:cdcl\] describes the CDCL-style search framework which we adopt. We evaluate our approach on many well-known examples and analyze its performance in Section \[sec:exp\]. The paper is concluded in Section \[sec:conclusion\].
Notation {#sec:pre}
========
Let ${\mathbb{R}}$ denote the field of real numbers, ${\mathbb{Z}}$ denote the ring of integers and ${\mathbb{Q}}$ denote the field of rational numbers. Unless stated otherwise, we assume that all polynomials in this paper are in ${\mathbb{Z}}[\bar{x}]$, the ring of multivariate polynomials in variables $\bar{x}$ with integer coefficients.
For a polynomial $f\in {\mathbb{Z}}[\bar{y},x]$: $$f(\bar{y},x)=a_mx^m+a_{m-1}x^{m-1}+\ldots+a_1x+a_0$$ where $a_m\neq0$ and $a_i\in {\mathbb{Z}}[\bar{y}]$ for $i=0,...,m$, the [*degree*]{} of $f$ with respect to (w.r.t.) $x$ is $m$, denoted by ${\mathtt{deg}}(f,x)$. The [*leading coefficient*]{} of $f$ w.r.t. $x$ is $a_m$, denoted by ${\mathtt{lc}}(f,x)$ and the [*leading term*]{} of $f$ w.r.t. $x$ is $a_mx^{m}$, denoted by ${\mathtt{lt}}(f,x)$. Let $${\mathtt{coeff}}(f, x)=\{a_i|0\leq i \leq m \land a_i\neq 0\}$$ denote the [*set of coefficients*]{} of $f$ w.r.t. $x$ and ${\mathtt{var}}(f)=\{\bar{y},x\}$ denote the variables appearing in $f$. Suppose $g\in {\mathbb{Z}}[\bar{y},x]$: $$g(\bar{y},x)=b_nx^{n}+b_{n-1}x^{n-1}+\ldots+b_1x+b_0$$ where $b_n\neq 0$ and $b_i\in {\mathbb{Z}}[\bar{y}]$ for $i=0,...,n$ . Let ${\mathtt{res}}(f,g,x)$ denote the Sylvester [*resultant*]{} of $f$ and $g$ w.r.t. $x$, [*i.e.*]{} the determinant of the following matrix $$\left(\begin{array}{cccccccc}
a_m & a_{m-1}& a_{m-2}& \ldots& a_0 & 0 & \ldots & 0 \\
0 & a_m &a_{m-1} & \ldots& a_1 & a_0 & \ldots & 0 \\
\vdots & \vdots &\ddots & \ddots&\ddots &\ddots& \ddots & \vdots \\
0 & 0 &\ldots & a_m &a_{m-1}&\ldots& \ldots & a_0 \\
b_n & b_{n-1}& b_{n-2}& \ldots& b_0 & 0 & \ldots & 0 \\
0 & b_n &b_{n-1} & \ldots& b_1 & b_0 & \ldots & 0 \\
\vdots & \vdots &\ddots & \ddots&\ddots &\ddots& \ddots & \vdots \\
0 & 0 &\ldots & b_n &b_{n-1}&\ldots& \ldots & b_0 \\
\end{array}\right)$$ which has $n$ rows of $a_i$ and $m$ rows of $b_j$. The discriminant of $f$ w.r.t. $x$ is $${\mathtt{disc}}(f,x)=(-1)^\frac{m(m-1)}{2}{\mathtt{res}}(f,f',x).$$
An [*atomic polynomial constraint*]{} is $f\triangleright0$ where $f$ is a polynomial and $\triangleright\in \{\geq,>,=\}$. A [*polynomial literal*]{} (simply [*literal*]{}) is an atomic polynomial constraint or its negation. For a literal $l$, ${\mathtt{poly}}(l)$ denotes the polynomial in $l$ and ${\mathtt{var}}(l)={\mathtt{var}}({\mathtt{poly}}(l))$. A [*polynomial clause*]{} is a disjunction $l_1\lor\cdots\lor l_s$ of literals. Sometimes, we write a clause as $\lnot(\bigwedge_i l_i)\lor \bigvee_j l_j$. A [*polynomial formula*]{} is a conjunction of clauses. An [*extended polynomial constraint*]{} $l$ is $x\triangleright {\mathtt{Root}}(f,k)$ where $\triangleright\in \{\geq,>,=\}$, $f\in{\mathbb{Z}}[\bar{y},u]$ with $x\not\in {\mathtt{var}}(f)$ and $k (0\leq k\leq {\mathtt{deg}}(f,u))$ is a given integer. Notice the variable $u$ is an exclusive free variable that cannot be used outside the ${\mathtt{Root}}$ object. For a formula $\phi$, $\phi[a/x]$ denote the resulting formula via substituting $a$ for $x$ in $\phi$. For variables $\bar{x}=(x_1,\ldots,x_r)$ and $\bar{a}=(a_1,\ldots,a_r)\in{\mathbb{R}}^r$, a mapping $\alpha$ which maps $x_i$ to $a_i$ for $i=1,...,r$ is called a [*variable assignment*]{} of $\bar{x}$ and $\bar{a}$ is called a [*sample*]{} of $\alpha$ or a [*sample*]{} of $\bar{x}$ in ${\mathbb{R}}^r$. We denote $\phi[a_1/x_1,\ldots,a_r/x_r]$ by $\alpha(\phi)$. If $\alpha(\phi)=0$, we say $\phi$ vanishes under $\alpha$ or vanishes under $\bar{a}$. Suppose an extended polynomial constraint $l$ is of the form $x\triangleright {\mathtt{Root}}(f,k)$ and $\alpha$ is a variable assignment of $(\bar{y},x)$. If $\beta_k$ is the $k$th real root of $\alpha(f)$, $\alpha(l)$ is defined to be $\alpha(x)\triangleright \beta_k$. If $\alpha(f)$ has less than $k$ real roots, $\alpha(l)$ is defined to be [False]{}.
Sample-Cell Projection {#sec:sample}
======================
In this section, we first introduce some well-known concepts and results concerning CAD and then define the so-called sample-cell projection operator.
Let $f$ be an analytic function defined in some open set $U$ of $K^n$ where $K$ is a field. For a point $p\in U$, if $f$ or some partial derivative (pure and mixed) of $f$ of some order does not vanish at $p$, then we say that $f$ has [*order*]{} $r$ where $r$ is the least non-negative integer such that some partial derivative of total order $r$ does not vanish at $p$. Otherwise, we say $f$ has infinite order at $p$. The order of $f$ at $p$ is denoted by ${\mathtt{order}}_pf$. We say $f$ is [*order-invariant*]{} in a subset $S\subset U$ if ${\mathtt{order}}_{p_1}f={\mathtt{order}}_{p_2}f$ for any $p_1, p_2\in S$. Obviously, if $K={\mathbb{R}}$ and the analytic function $f$ is order-invariant in $S$, then $f$ is sign-invariant in $S$.
An $r$-variable polynomial $f(\bar{x},x_r)$ where $\bar{x}=(x_1,\ldots,x_{r-1})$ is said to be [*analytic delineable*]{} on a connected $s$-dimensional submanifold $S\subset {\mathbb{R}}^{r-1}$ if
1. The number $k$ of different real roots of $f(a,x_r)$ is invariant for any point $a\in S$. And the trace of the real roots are the graphs of some pairwise disjoint analytic functions $\theta_1<\ldots<\theta_k$ from $S$ into ${\mathbb{R}}$ ([*i.e.*]{} the order of real roots of $f(a,x_r)$ is invariant for all point $a\in S$);
2. There exist positive integers $m_1,\ldots,m_k$ such that for every point $a\in S$, the multiplicity of the real root $\theta_i(a)$ of $f(a,x_r)$ is $m_i$ for $i=1,...,k$.
Especially, if $f$ has no zeros in $S\times {\mathbb{R}}$, then $f$ is delineable on $S$ with $k=0$. The analytic functions $\theta_i$’s are called the [*real root functions*]{} of $f$ on $S$, the graphs of the $\theta_i$’s are called the [*$f$-sections*]{} over $S$, and the connected regions between two consecutive $f$-sections (for convenience, let $\theta_0=-\infty$ and $\theta_{k+1}=+\infty$) are called [*$f$-sectors*]{} over $S$. Each $f$-section over $S$ is a connected $s$-dimensional submanifold in ${\mathbb{R}}^r$ and each $f$-sector over $S$ is a connected $(s+1)$-dimensional submanifold in ${\mathbb{R}}^r$.
\[thm:mc\] Let $r\geq 2$ and $f(\bar{x},x_r)$ be a polynomial in ${\mathbb{R}}[\bar{x},x_r]$ of positive degree where $\bar{x}=(x_1,...,x_{r-1})$. Let $S$ be a connect submanifold of ${\mathbb{R}}^{r-1}$ where $f$ is degree-invariant and does not vanish identically. Suppose that ${\mathtt{disc}}(f,x_r)$ is a nonzero polynomial and is order-invariant in $S$. Then $f$ is analytic delineable on $S$ and is order-invariant in each $f$-section over $S$.
Suppose $a=(\bar{a},a_n)=(a_1,\ldots,a_n)$ is a sample of $(\bar{x},x_n)$ in ${\mathbb{R}}^n$ and $F=\{f_1(\bar{x},x_n)$ $,\ldots , f_r(\bar{x},x_n)\}$ is a polynomial set in ${\mathbb{Z}}[\bar{x},x_n]$ where $\bar{x}=(x_1,\ldots,x_{n-1})$. Consider the real roots of polynomials in $\{f_1(\bar{a},x_n),\ldots,f_r(\bar{a},x_n)\}\setminus \{0\}$. Denote the $k$th real root of $f_i(\bar{a},x_n)$ by $\theta_{i,k}$. We define two concepts: the [*sample polynomials set*]{} of $a$ in $F$ (denoted by ${\mathtt{s\_poly}}(F,x_n,a)$) and the [*sample interval*]{} of $a$ in $F$ (denoted by ${\mathtt{s\_interval}}(F,x_n,a)$) as follows.
If there exists $\theta_{i,k}$ such that $\theta_{i,k}=a_n$ then $$\begin{aligned}
{\mathtt{s\_poly}}(F,x_n,a)&=\{f_i\},\\
{\mathtt{s\_interval}}(F,x_n,a)&=(x_n={\mathtt{Root}}(f_i(\bar{x},u),k));\end{aligned}$$
If there exist two consecutive real roots $\theta_{i_1,k_1}$ and $\theta_{i_2,k_2}$ such that $\theta_{i_1,k_1}<a_n<\theta_{i_2,k_2}$ then $$\begin{aligned}
{\mathtt{s\_poly}}(F,x_n,a)&=\{f_{i_1},f_{i_2}\},\\
{\mathtt{s\_interval}}(F,x_n,a)&={\mathtt{Root}}(f_{i_1}(\bar{x},u),k_1)<x_n<{\mathtt{Root}}(f_{i_2}(\bar{x},u),k_2);\end{aligned}$$
If there exists $\theta_{i',k'}$ such that $a_n>\theta_{i',k'}$ and for all $\theta_{i,k}$ $\theta_{i',k'}\geq\theta_{i,k}$ then $$\begin{aligned}
{\mathtt{s\_poly}}(F,x_n,a)&=\{f_{i'}\},\\
{\mathtt{s\_interval}}(F,x_n,a)&=x_n>{\mathtt{Root}}(f_{i'}(\bar{x},u),k');\end{aligned}$$
If there exists $\theta_{i',k'}$ such that $a_n<\theta_{i',k'}$ and for all $\theta_{i,k}$ $\theta_{i',k'}\leq\theta_{i,k}$then $$\begin{aligned}
{\mathtt{s\_poly}}(F,x_n,a)&=\{f_{i'}\},\\
{\mathtt{s\_interval}}(F,x_n,a)&=x_n<{\mathtt{Root}}(f_{i'}(\bar{x},u),k').\end{aligned}$$ Specially, if every polynomial in $\{f_1(\bar{a},x_n),\ldots,f_r(\bar{a},x_n)\}\setminus \{0\}$ does not have any real roots, define $$\begin{aligned}
{\mathtt{s\_poly}}(F,x_n,a)&=\emptyset,\\
{\mathtt{s\_interval}}(F,x_n,a)&=\text{{\tt True}}.\end{aligned}$$
\[ex:s\] Let $F=\{f_1,f_2,f_3\}$ where $f_1=y+0.5x-10$, $f_2=y+0.01(x-9)^2-7$, $f_3=y-0.03x^2-1$ and $A=(4,9),B=(4,6.75),C=(4,4),D=(4,1)$. We have (see Figure \[fig:1\]) $$\begin{array}{ll}
{\mathtt{s\_poly}}(F,y,A)=\{f_1\}, &{\mathtt{s\_interval}}(F,y,A)=y>{\mathtt{Root}}(f_1(x,u),1),\\
{\mathtt{s\_poly}}(F,y,B)=\{f_2\}, &{\mathtt{s\_interval}}(F,y,B)=y={\mathtt{Root}}(f_2(x,u),1),\\
{\mathtt{s\_poly}}(F,y,C)=\{f_2,f_3\},&{\mathtt{s\_interval}}(F,y,C)=\begin{array}{rl}
& y>{\mathtt{Root}}(f_3(x,u),1) \\
\wedge & y<{\mathtt{Root}}(f_2(x,u),1)
\end{array},\\
{\mathtt{s\_poly}}(F,y,D)=\{f_3\},&{\mathtt{s\_interval}}(F,y,D)=y<{\mathtt{Root}}(f_3(x,u),1).\\
\end{array}$$
(-0.5865830202854977,-0.6301322314049569) rectangle (12.486294515401967,10.113669421487607); plot(,[0-0.01\*(()-9)\^(2)+7]{}); plot(,[0-0.5\*()+10]{}); plot(,[0.03\*()\^(2)+1]{});
(5,-0.4) node\[scale=3\] [$x$]{} ; (-0.4,5) node\[scale=3\] [$y$]{} ; (1.4062674680691189,5.97677986476334) node\[scale=3\] [$f_2$]{} ; (1.0745740045078913,9.093459053343354) node\[scale=3\] [$f_1$]{}; (1.0062674680691189,1.5978918106686724) node\[scale=3\] [$f_3$]{}; (4,9) circle (2.5pt); (4.161726521412478,9.45251239669422) node\[scale=2\] [$A$]{}; (4.002201149761497,6.7502200665255465) circle (2.5pt); (4.161726521412478,7.198567993989485) node\[scale=2\] [$B$]{}; (4,4) circle (2.5pt); (4.161726521412478,4.44875582268971) node\[scale=2\] [$C$]{}; (4,1) circle (2.5pt); (4.161726521412478,1.4434966190833982) node\[scale=2\] [$D$]{};
Additionally, for a polynomial $$h=c_mx_n^{d_m}+c_{m-1}x_n^{d_{m-1}}+\ldots+c_{0}x_n^{d_0}$$ where $d_m>d_{m-1}>\cdots>d_0$, $c_i\in {\mathbb{R}}[\bar{x}]$ and $c_i\neq0$ for $i=0,...,m$. If there exists $j\ge 0$ such that $c_j(\bar{a})\neq 0$ and $c_i(\bar{a})=0$ for any $i>j$, then the [*sample coefficients*]{} of $h$ at $(\bar{a},a_n)$ is defined to be $\{c_m,c_{m-1},\ldots,c_j\}$, denoted by ${\mathtt{s\_coeff}}(h,x_n,(\bar{a},a_n))$. Otherwise ${\mathtt{s\_coeff}}(h,x_n,(\bar{a},a_n))=\{c_m,\ldots,c_0\}$.
\[def:projsc\] Suppose $\bar{a}$ is a sample of $\bar{x}$ in ${\mathbb{R}}^n$ and $F=\{f_1,\ldots , f_r\}$ is a polynomial set in ${\mathbb{Z}}[\bar{x}]$ where $\bar{x}=(x_1,\ldots,x_n)$. The [*sample-cell projection*]{} of $F$ on $x_n$ at $\bar{a}$ is $$\begin{split}
{\mathtt{Proj}}_{sc}(F,x_n,\bar{a})= \bigcup_{f\in F}&{\mathtt{s\_coeff}}(f,x_n,\bar{a})\cup\\
\bigcup_{f\in F}&\{{\mathtt{disc}}(f,x_n)\}\cup\\
\bigcup_{\begin{subarray}{c}f\in F,g\in\\ {\mathtt{s\_poly}}(F,x_n,\bar{a}),\\f\neq g\end{subarray}}&\{{\mathtt{res}}(f,g,x_n)\}
\end{split}$$
- If $f\in F$ and $x_n\not\in {\mathtt{var}}(f)$, $f$ is obviously an element of ${\mathtt{Proj}}_{sc}(F, x_n, \bar{a})$.
- Computing ${\mathtt{Proj}}_{sc}(F,x_n,\bar{a})$ will produce $O(rn+3r)$ elements, so the time complexity of projecting all the variables by recursively using ${\mathtt{Proj}}_{sc}$ is $O((n+3)^nr)$.
Now we prove the property of the new projection operator. A set of polynomials in ${\mathbb{Z}}[\bar{x}]$ is said to be a [*squarefree basis*]{} if the elements of the set have positive degrees, and are primitive, squarefree and pairwise relatively prime. For a connected submanifold $S$ of ${\mathbb{R}}^{n-1}$, we denote by $S\times {\mathtt{s\_interval}}(F,x_n,\bar{a})$ $$\left\{(\alpha_1,\ldots,\alpha_n)\in {\mathbb{R}}^n~|~~\begin{array}{l}
(\alpha_1,\ldots,\alpha_{n-1})\in S \\
\wedge~ {\mathtt{s\_interval}}(F,x_n,\bar{a})[\alpha_1/x_1,\ldots,\alpha_n/x_n]
\end{array} \right\}.$$
\[th:sc\] Let $F$ be a finite squarefree basis in ${\mathbb{Z}}[\bar{x}]$ where $\bar{x}=(x_1,\ldots,x_n)$ and $n\geq 2$. Let $\bar{a}=(a_1,\ldots,a_n)$ be a sample of $\bar{x}$ in ${\mathbb{R}}^n$ and $S$ be a connected submanifold of ${\mathbb{R}}^{n-1}$ such that $(a_1,\ldots,a_{n-1})\in S$. Suppose that each element of ${\mathtt{Proj}}_{sc}(F,x_n,\bar{a})$ is order-invariant in $S$. Then each element in $F$ either vanishes identically on $S$ or is analytic delineable on $S$, each section over $S$ of the element of $F$ which do not vanish identically on $S$ is either equal to or disjoint with $S\times {\mathtt{s\_interval}}(F,x_n,\bar{a})$, and each element of $F$ either vanishes identically on $S$ or is order-invariant in $S\times {\mathtt{s\_interval}}(F,x_n,\bar{a})$.
For any $f\in F$, if $f$ vanishes identically on $S$, there is nothing to prove. So we may assume that any element in $F$ does not vanish identically on $S$.
For any $f\in F$ such that $f\not\in{\mathtt{s\_poly}}(F,x_n,\bar{a})$, let $f'=f\cdot~\prod_{g\in {\mathtt{s\_poly}}(F,x_n,\bar{a})} g$. Notice that $f'$ is degree-invariant on $S$ (each element of ${\mathtt{s\_coeff}}(f,x_n,\bar{a})$ is order-invariant, hence sign-invariant in $S$). And we have $$\begin{split}
{\mathtt{disc}}(f',x_n)={\mathtt{disc}}(f,x_n)\cdot\prod_{g\in {\mathtt{s\_poly}}(F,x_n,\bar{a})} {\mathtt{disc}}(g,x_n)\cdot\\
\prod_{g\in {\mathtt{s\_poly}}(F,x_n,\bar{a})}{\mathtt{res}}(f,g,x_n)\cdot\\
\prod_{\begin{subarray}{c}g_1\in {\mathtt{s\_poly}}(F,x_n,\bar{a}),\\ g_2\in {\mathtt{s\_poly}}(F,x_n,\bar{a}),\\g_1\neq g_2\end{subarray}}{\mathtt{res}}(g_1,g_2,x_n).
\end{split}$$ It follows from this equality that ${\mathtt{disc}}(f',x_n)\neq 0$ (because $f_i$’s are squarefree and pairwise relatively prime). Obviously, each factor of ${\mathtt{disc}}(f',x_n)$ is a factor of ${\mathtt{Proj}}_{sc}(F,x_n,\bar{a})$, so ${\mathtt{disc}}(f',x_n)$ is order-invariant in $S$. By Theorem \[thm:mc\], $f'$ is analytic delineable on $S$ and is order-invariant in each $f'$-section over $S$. So $f$ and $g\in {\mathtt{s\_poly}}(F,x_n,\bar{a})$ are order-invariant in each $f'$-section over $S$. It follows that the sections over $S$ of $f$ and $g$ are pairwise disjoint. Therefore, $f$ and $g\in {\mathtt{s\_poly}}(F,x_n,\bar{a})$ are analytic delineable on $S$, every section of them is either equal to or disjoint with $S\times {\mathtt{s\_interval}}(F,x_n,\bar{a})$, and $f$ and $g$ are order-invariant in $S\times {\mathtt{s\_interval}}(F,x_n,\bar{a})$. $\blacksquare$
\[rm:sc\] Notice that when $f$ vanishes identically on $S$, $f$ isn’t always order-invariant in $S\times {\mathtt{s\_interval}}(F,x_n,\bar{a})$. This is avoidable by changing the ordering of variables and is negligible when the satisfiability set of formulas is full-dimensional. We find a way to handle this rare case: either to determine whether the coefficients of $f$ have finitely many common zeros, or to enlarge $F$ by adding partial derivatives of $f$ whose order is less than ${\mathtt{order}}(f)$ and one non-zero partial derivative whose order is exactly equal to ${\mathtt{order}}(f)$.
When integrating the new projection operator with the CDCL-type search (see Section \[sec:cdcl\]), we need a traditional CAD projection operator [@DBLP:journals/jsc/McCallum88; @10.1007/978-3-7091-9459-1_12].
Suppose $F=\{f_1,\ldots , f_r\}$ is a polynomial set in ${\mathbb{Z}}[\bar{x}]$ where $\bar{x}=(x_1,\ldots,x_n)$. The McCallum projection of $F$ on $x_n$ is $${\mathtt{Proj}}_{mc}(F)=\bigcup_{f\in F}\{{\mathtt{coeff}}(f),{\mathtt{disc}}(f,x_n)\}\cup\bigcup_{\begin{subarray}{c}f\in F,g\in F,\\f\neq g\end{subarray}} {\mathtt{res}}(f,g,x_n)$$
Notice that ${\mathtt{coeff}}$ can be replaced by ${\mathtt{s\_coeff}}$ when we have a sample of $n-1$ dimension.
\[[@10.1007/978-3-7091-9459-1_12], Theorem 1\]\[th:mc\] Let $F$ be a finite squarefree basis in ${\mathbb{Z}}[\bar{x}]$ where $\bar{x}=(x_1,\ldots,x_n)$ and $n\geq 2$ and $S$ be a connected submanifold of ${\mathbb{R}}^{n-1}$ such that each element of ${\mathtt{Proj}}_{mc}(F,x_n)$ is order-invariant in $S$. Then each element in $F$ either vanishes identically on $S$ or is analytic delineable on $S$, the sections over $S$ of the elements of $F$ which do not vanish identically on $S$ are pairwise disjoint, and each element of $F$ which does not vanish identically on $S$ is order-invariant in every such section.
Now, let us use the following definition to describe the procedure of calculating sample cells. We denote by ${\mathtt{factor}}(A)$ the set of irreducible factors of all polynomials in $A$.
Suppose $a=(a_1,\ldots,a_{n-1})$ is a sample of $(x_1,\ldots,x_{n-1})$ in ${\mathbb{R}}^{n-1}$ and $F=\{f_1,\ldots , f_r\}$ is a polynomial set in ${\mathbb{Z}}[\bar{x}]$ where $\bar{x}=(x_1,\ldots,x_{n})$. The [*sample cell*]{} of $F$ at $a$ is $${\mathtt{s\_cell}}(F,a)={\mathtt{s\_interval}}(F_1,\alpha_1)\land\cdots\land{\mathtt{s\_interval}}(F_{n-1},\alpha_{n-1})$$ where $\alpha_{n-1}=a$, $F_{n-1}={\mathtt{factor}}({\mathtt{Proj}}_{mc}({\mathtt{factor}}(F)))$, $\alpha_i=(a_1,\ldots,a_i)$, and $F_i={\mathtt{factor}}({\mathtt{Proj}}_{sc}(F_{i+1},x_{i+1},\alpha_{i+1}))$ for $i=1,\ldots,{n-2}$.
- It is a standard way to use ${\mathtt{factor}}$ to ensure that every $F_i$ is a finite squarefree basis.
- Notice that the complexity of computing sample cell ${\mathtt{s\_cell}}$ depends on $\sum_{i=1}^{n-1}|F_i|$ where $|F_i|$ means the number of polynomials in $F_i$. From the recursive relationship $|F_{n-1}|=O(r^2+rn)$, $|F_i|<(3+i+1)|F_{i+1}|,i=1,\ldots,n-2$, it is not hard to know that the complexity of computing ${\mathtt{s\_cell}}$ is $O((r^2+rn)(2+n)^{n-1})$.
Let $F=\{f_1(\bar{x},x_n),\ldots,f_r(\bar{x},x_n)\}$ be a polynomial set and $a\in {\mathbb{R}}^{n-1}$, where $\bar{x}=(x_1,\ldots,x_{n-1})$. If $$\forall b\in {\mathbb{R}}\;\bigvee_{i=1}^{r} f_i(a,b)\rhd_i 0,$$ where $\rhd_i\in \{>,\geq,=\}$, then $$\forall \alpha \in {\mathtt{s\_cell}}(\{f_1,\ldots,f_r\},a)\forall b \in {\mathbb{R}}\; \bigvee_{i=1}^{r} f_i(\alpha,b)\rhd_i 0.$$
It is a direct corollary of Theorem \[th:mc\] and Theorem \[th:sc\].
Suppose $f=ax^2+bx+c$ and $\alpha=(1,1,1)$ is a sample of $(a,b,c)$. Then $$\begin{aligned}
&F_3={\mathtt{factor}}({\mathtt{Proj}}_{mc}(\{f\},x))={\mathtt{factor}}(\{b^2-4ac,a\})=\{b^2-4ac,a\},\\
&F_2={\mathtt{factor}}({\mathtt{Proj}}_{sc}(\{b^2-4ac,a\},c))={\mathtt{factor}}(\{1,a,-4a\})=\{a\},\\
&F_1={\mathtt{factor}}({\mathtt{Proj}}_{sc}(\{a\},b))=\{a\}.\end{aligned}$$ So $${\mathtt{s\_cell}}(\{f\},a)=c>{\mathtt{Root}}(b^2-4au,1)\land a>{\mathtt{Root}}(u,1),$$ and after simplification $${\mathtt{s\_cell}}(\{f\},a)=c>\frac{b^2}{4a}\land a>0.$$
CDCL-style search framework {#sec:cdcl}
===========================
In this section, we introduce a search framework combined with the new projection operator proposed in the previous section. The main notation and concepts about the search framework are taken from Section 3 of [@DBLP:conf/cade/JovanovicM12] and Section 26.4.4 of [@DBLP:series/faia/2009-185]. Let $\bar{x}=(x_1,\ldots,x_n)$ and ${\mathtt{level}}(x_i)=i$. For a polynomial $f$, a literal $l$ and a clause $c$, we define ${\mathtt{level}}(f)=\max(\{{\mathtt{level}}(a)|a\in {\mathtt{var}}(f))\}$, ${\mathtt{level}}(l)={\mathtt{level}}({\mathtt{poly}}(l))$ and ${\mathtt{level}}(c)=\max(\{{\mathtt{level}}(l)|l \in c\})$. We describe the search framework by transition relations between search states as in [@DBLP:conf/cade/JovanovicM12].
The [*search states*]{} are indexed pairs of the form $M \| \zeta$, where $\zeta$ is a finite set of polynomial clauses and $M$ is a sequence of literals and variable assignments. Every literal is marked as a decision or a propagation literal. We denote a [*propagation literal*]{} $l$ by $c\rightarrow l$ if $l$ is propagated from $c$ and denote a [*decision literal*]{} $l$ by $l^\bullet$. We denote by $x_i\mapsto a_i$ a variable assignment. Let ${\mathtt{level}}(x_i\mapsto a_i)={\mathtt{level}}(x_i)$ and $v[M]=\{x_i\mapsto a_i|(x_i\mapsto a_i)\in M\}$. For a set $L$ of literals, $v[M](L)$ means the resulting set of $L$ after applying the assignments of $v[M]$.
Next, we introduce transition relations between search states. Transition relations are specified by a set of transition rules. In the following, we use simple juxtaposition to denote the concatenation of sequences ([*e.g.*]{}, $M,M'$). We treat a literal or a variable assignment as one-element sequence and denote the empty sequence as $\emptyset$. We say the sequence $M$ is ordered when the sequence is of the form $$M=[N_1,x_1\mapsto a_1,\ldots,N_{k-1},x_{k-1}\mapsto a_{k-1},N_k]$$ where $N_j$ is a sequence of literals and each literal $l\in N_j$ satisfies ${\mathtt{level}}(l)=j$. Notice that $N_j$ might be $\emptyset$. We define ${\mathtt{level}}(M)=k$ even if $N_k=\emptyset$. We use ${\mathtt{sample}}(M)$ to denote the sample $(a_1,\ldots,a_{k-1})$ of $(x_1,\ldots,x_{k-1})$ in $M$ and ${\mathtt{feasible}}(M)$ to denote the feasible set of $v[M](N_k)$. For a new literal $l$ with $x_k\in {\mathtt{var}}(l)$, we say $l$ is consistent with $M$ if ${\mathtt{feasible}}([M,l])\neq\emptyset$. If $l$ is not consistent with $M$, we define ${\mathtt{core}}(l,M)$ to be a minimal set of literals $L$ in $M$ such that $v[M](L\cup\{l\})$ does not have a solution for $x_k$.
Since there is only one unassigned variable $x_k$ in the polynomials in $N_k$, so ${\mathtt{feasible}}(M)$ can be easily calculated by real-root isolation.
Suppose $l$ is a literal and $M$ is an ordered sequence which satisfies ${\mathtt{level}}(M)={\mathtt{level}}(l)$ and $\lnot l$ is not consistent with $M$. Define the [*explain clause*]{} of $l$ with $M$ as $${\mathtt{explain}}(l,M)=\lnot({\mathtt{s\_cell}}(F,{\mathtt{sample}}(M))\land {\mathtt{core}}(\lnot l,M))\lor l,$$ where $F=\{{\mathtt{poly}}(l')|l'\in {\mathtt{core}}(\lnot l,M)\}\cup \{{\mathtt{poly}}(l)\}$.
Meanwhile, we define the [*state value*]{} of a literal $l$ as
$${\mathtt{value}}(l,M)=\left\{\begin{array}{lcl}
v[M](l) & &{\mathtt{level}}(l)<k, \\
\text{{\tt True}} & &l\in M, \\
\text{{\tt False}} & &\lnot l\in M, \\
\text{{\tt undef}} & &\text{otherwise}.
\end{array}
\right.$$ And for a clause $c$, $${\mathtt{value}}(c,M)=\left\{\begin{array}{lcl}
\text{{\tt True}} & &\exists l\in c ({\mathtt{value}}(l,M)=\text{{\tt True}}), \\
\text{{\tt False}} & &\forall l\in c ({\mathtt{value}}(l,M)=\text{{\tt False}}), \\
\text{{\tt undef}} & &\text{otherwise}.
\end{array}
\right.$$ Specially, ${\mathtt{value}}(\emptyset,M)=\text{{\tt False}}$.
A set of rules for transition relations between search states are defined as follows where $c$ is a clause and $l$ is a literal.
Decide-Literal
: $$M\|\zeta,c\Longrightarrow M,l^\bullet\|\zeta,c$$ if $l,l'\in c$, ${\mathtt{value}}(l,M)={\mathtt{value}}(l',M)=\text{{\tt undef}}$, ${\mathtt{level}}(c)={\mathtt{level}}(M)$ and $l$ is consistent with $M$.
Boolean-Propagation
: $$M\|\zeta, c\lor l\Longrightarrow M,c\lor l\rightarrow l\|\zeta,c\lor l$$ if ${\mathtt{value}}(c,M)=\text{{\tt False}},{\mathtt{value}}(l,M)=\text{{\tt undef}}$, ${\mathtt{level}}(c\lor l)={\mathtt{level}}(M)$ and $l$ is consistent with $M$.
Lemma-Propagation
: $$M\|\zeta\Longrightarrow M,{\mathtt{explain}}(l,M)\rightarrow l\|\zeta$$ if $l\in \zeta$ or $\lnot l \in \zeta$, ${\mathtt{value}}(l,M)=\text{{\tt undef}}$, ${\mathtt{level}}(l)={\mathtt{level}}(M)$ and $\lnot l$ is not consistent with $M$.
Up-Level
: $$M\|\zeta\Longrightarrow M,x\mapsto a\|\zeta$$ if $\forall c\in \zeta\;({\mathtt{level}}(c)\neq{\mathtt{level}}(M)\lor {\mathtt{value}}(c,M)=\text{{\tt True}})$, ${\mathtt{level}}(x)={\mathtt{level}}(M)$ and $a\in {\mathtt{feasible}}(M)$.
Sat
: $$M\|\zeta\Longrightarrow(\text{sat},v[M])$$ if ${\mathtt{level}}(M)>n$.
Conflict
: $$M\|\zeta\Longrightarrow M\|\zeta\not\vdash c$$ if ${\mathtt{level}}(c)={\mathtt{level}}(M)$ and ${\mathtt{value}}(c,M)=\text{{\tt False}}$.
backtrack-Propagation
: $$M,E\rightarrow l\|\zeta \not\vdash c\Longrightarrow M\|\zeta\not\vdash R$$ if $\lnot l\in c,{\mathtt{value}}(c,[M,E])=\text{{\tt False}}$ and $R={\mathtt{resolve}}(c,E,l)$[^1].
backtrack-Decision
: $$M,l^\bullet\| \zeta\not\vdash c \Longrightarrow M\|\zeta,c$$ if $\lnot l\in c$.
Skip
: $$\begin{aligned}
M,l^\bullet\| \zeta\not\vdash c &\Longrightarrow M\|\zeta\not\vdash c\\
M,E\rightarrow l\| \zeta\not\vdash c &\Longrightarrow M\|\zeta\not\vdash c
\end{aligned}$$
if $\lnot l\not\in c$ .
Down-Level
: $$\begin{array}{ll}
M,x\mapsto a\|\zeta\not\vdash c \Longrightarrow M\|\zeta \not \vdash c,&\text{ if }{\mathtt{value}}(c,M)=\text{{\tt False}}, \\
M,x\mapsto a\|\zeta\not\vdash c \Longrightarrow M\|\zeta,c,&\text{ if }{\mathtt{value}}(c,M)=\text{{\tt undef}}.
\end{array}$$
Unsat
: $$M\|\zeta \not \vdash c \Longrightarrow \text{unsat}$$ if ${\mathtt{value}}(c,M)=\text{{\tt False}}$ and no assignment or decide literal in $M$.
Forget
: $$M\|\zeta,c\Longrightarrow M\|\zeta$$ if $c$ is a learnt clause.
Note that in this framework we rely on the rule [*lemma-propagation*]{} to guide the search away from conflicting states. When applying lemma-propagation, the most important thing is the explain clause. We cannot simply use the conflicting core as the explain clause, as this will cause explain to be an incorrect lemma because it ignores assignments. Using full CAD to calculate explain is also costly. Thanks to the sample cell calculated by the novel sample-cell projection operator, we can now efficiently calculate an effective explain to achieve our purpose.
Given a polynomial formula $\zeta$ with finitely many clauses, any transition starting from the initial state $\emptyset\|\zeta$ will terminate either in a state $(sat,v)$, where the assignment $v$ satisfies the formula $\zeta$, or in the $unsat$ state. In the later case, $\zeta$ is unsatisfiable in ${\mathbb{R}}$.
By Theorem 1 in [@DBLP:conf/cade/JovanovicM12], if there is a finite set such that all the literals returned every time by calling ${\mathtt{explain}}$ are always contained in the set, then the above theorem holds. On the other hand, it is not hard to see that all literals that may be generated by ${\mathtt{s\_cell}}$ are determined by finitely many polynomials and their real roots and thus finite. That completes the proof. $\blacksquare$
Experiments {#sec:exp}
===========
In order to better demonstrate the effectiveness of our algorithm, we have implemented a prototype solver LiMbS[^2] which is base on Mathematica 12. The solver is a clean translation of the algorithm in this paper. Our solver is compared to the following solvers that have been popular in SMT nonlinear competition: Z3 (4.8.7-1), CVC4 (1.6-2), Yices (2.6.1) and MathSAT5 (5.6.0).
All tests were conducted on 6-Core Intel Core [email protected] with 32GB of memory and ARCH LINUX SYSTEM (5.5.4-arch1-1). The timeout is set to be 5 hours.
The examples listed below, which we collect from several related papers, are either special or cannot be well-solved by existing SMT solvers. All results are listed in Table \[tb:exp\].
([**Han**]{}\_$n$)[@Dai_Han_Hong_Xia_2015] Decide whether $$\exists x_1,\ldots,\exists x_n \;(\sum_{i=1}^nx_i^2)^2-4(\sum_{i=1}^nx_i^2x_{i+1}^2)<0$$ where $x_{n+1} = x_{1}$.
$({\mathbf P})$\
$\exists a,\exists b,\exists c,\exists d,\exists e,\exists f (a^2 b^2 e^2+a^2 b^2 f^2+a^2 b^2-a^2 b c d e-a^2 b d f+a^2 c^4 f^4+2 a^2 c^4 f^2+a^2 c^4-3 a^2 c^3 e f^3-3 a^2 c^3 e f+3 a^2 c^2 e^2 f^2+a^2 c^2 e^2+a^2 c^2 f^2+a^2 c^2-a^2 c e^3 f-a^2 c e f-a b^2 d e-2 a b c^3 f^4-4 a b c^3 f^2-2 a b c^3+4 a b c^2 e f^3+4 a b c^2 e f+a b c d^2-2 a b c e^2 f^2+a b c f^2+a b c-a b e f+2 a c^3 d f^3+2 a c^3 d f-4 a c^2 d e f^2-2 a c^2 d e+2 a c d e^2 f+b^2 c^2 f^4+2 b^2 c^2 f^2+b^2 c^2-b^2 c e f^3-b^2 c e f-2 b c^2 d f^3-2 b c^2 d f+2 b c d e f^2+c^2 d^2 f^2+c^2 d^2+c^2 f^2+c^2-c d^2 e f-c e f<0)$
[@Hong91comparisonof]
Hong\_$n$
: $$\exists x_1,\ldots,\exists x_n\;\sum_{i=1}^nx_i^2<1\land\prod_{i=1}^n x_i>1$$
Hong2\_$n$
: $$\exists x_1,\ldots,\exists x_n\;\sum_{i=1}^nx_i^2<2n\land\prod_{i=1}^n x_i>1$$
([**C**]{}\_$n$\_$r$) Whether the distance between the ball $B_r(\bar{x})$ and the complement of $B_8(\bar{x})$ is less than $\frac{1}{1000}$? $$\exists_{i=1}^{n} x_i,\exists_{i=1}^{n} y_i \;\sum_{i=1}^nx_i^2<r\land\sum_{i=1}^ny_i^2>8^2\land \sum_{i=1}^n(x_i-y_i)^2<\frac{1}{1000^2}$$
ans LiMbS Z3 CVC4 MathSAT5 Yices
----------- ------- -------- ------- --------- ---------- -------
Han\_3 SAT 0.01s 0.01s 0.01s 0.01s 0.01s
Han\_4 UNSAT 0.08s 0.01s $>5$h $>5$h 0.01s
Han\_5 UNSAT 1.26s $>5$h $>5$h $>5$h $>5$h
Han\_6 UNSAT 60s $>5$h $>5$h $>5$h $>5$h
P SAT 1.06s 0.05s $>5$h $>5$h $>5$h
Hong\_10 UNSAT 222s 2058s 0.01s 0.10s $>5$h
Hong\_11 UNSAT 806s 6357s 0.01s 0.10s $>5$h
Hong2\_11 SAT 30.43s 1997s 0.01s $>5$h 0.01s
Hong2\_12 SAT 563s 6693s 0.01s $>5$h 0.01s
C\_3\_1 UNSAT 0.44s $>5$h 0.62s 5811s $>5$h
C\_3\_32 UNSAT 0.48s $>5$h unknown $>5$h $>5$h
C\_3\_63 UNSAT 0.48s $>5$h unknown $>5$h $>5$h
C\_3\_64 SAT 0.02s 4682s unknown $>5$h $>5$h
C\_4\_1 UNSAT 1.31s $>5$h 2.28s $>5$h $>5$h
C\_4\_32 UNSAT 1.42s $>5$h unknown $>5$h $>5$h
C\_4\_63 UNSAT 1.42s $>5$h unknown $>5$h $>5$h
C\_4\_64 SAT 0.02s $>5$h unknown $>5$h $>5$h
C\_5\_1 UNSAT 5.48s $>5$h 19.33s $>5$h $>5$h
C\_5\_32 UNSAT 5.73s $>5$h unknown $>5$h $>5$h
C\_5\_63 UNSAT 5.68s $>5$h unknown $>5$h $>5$h
C\_5\_64 SAT 0.02s $>5$h unknown $>5$h 1.75s
: Comparison with other solvers on 21 examples\[tb:exp\]
Our solver LiMbs solves all the $21$ examples shown in Table \[tb:exp\]. LiMbs is faster than the other solvers on 15 examples. Only LiMbs can solve 9 of the examples within a reasonable time while other solvers either run time out or return unknown state. From this we can see that our algorithm has great potential in solving satisfiability of polynomial formulas, especially considering that our prototype solver is a small program with less than 1000 lines of codes. For Hong$\_n$ and Hong2$\_n$, though our solver is much faster than Z3, CVC4 is the one that performs best. We note that the examples of Hong$\_n$ and Hong2$\_n$ are all symmetric. This reminds us it is worth exploiting symmetry to optimize our solver’s performance.
Conclusions {#sec:conclusion}
===========
A new algorithm for deciding the satisfiability of polynomial formulas over the reals is proposed. The key point is that we design a new projection operator, the sample-cell projection operator, which can efficiently guide CDCL-style search away from conflicting states. Preliminary evaluation of the prototype solver LiMbS shows the effectiveness of the new algorithm.
We will further develop our algorithm, looking into problems with symmetry, equations or other special structures. We also hope to develop an easy-to-use, robust and concise open-source algorithm framework based on our prototype solver to achieve a wider range of applications.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work was supported partly by NSFC under grants 61732001 and 61532019.
[^1]: ${\mathtt{resolve}}(c_1\lor l,c_2\lor \lnot l,l)=c_1\lor c_2.$
[^2]: https://github.com/lihaokun/LiMbS
|
---
abstract: 'We prove a Feynman path integral formula for the *unitary group* $ \exp(-itL_{v,\theta})$, $t\geq 0$, associated with a discrete magnetic Schrödinger operator $L_{v,\theta}$ on a large class of weighted infinite graphs. As a consequence, we get a new Kato-Simon estimate $$|\exp(-itL_{v,\theta})(x,y)|\leq \exp(-tL_{-\mathrm{deg},0})(x,y),$$ which controls the unitary group uniformly in the potentials in terms of a Schrödinger semigroup, where the potential $\mathrm{deg}$ is the weighted degree function of the graph.'
address:
- 'Batu Güneysu, Institut für Mathematik, Humboldt-Universität zu Berlin, 12489 Berlin, Germany'
- 'M. Keller: Institut für Mathematik, Universität Potsdam, 14476 Potsdam, Germany'
author:
- Batu Güneysu
- Matthias Keller
title: Feynman path integrals for magnetic Schrödinger operators on infinite weighted graphs
---
Introduction
============
While the Schrödinger *semigroup* $\exp(-tH_{v,\theta})$, $t\geq 0$, associated to an electric potential $v$ and a magnetic potential $\theta$ on the Euclidean ${\mathbb{R}}^d$ or a general Riemannian manifold is given by a well-defined Brownian motion path integral formula [@simon; @batu], the Feynman-Kac-Ito formula, it is well-known that there cannot hold an analogous formula for the unitary Schrödinger group $\exp(-itH_{v,\theta})$, $t\geq 0$. For example, it can be proven [@RS2] that there cannot exist a complex measure $\mu$ on the space of continuous paths $[0,\infty)\to {\mathbb{R}}^m$ such that the finite dimensional distributions of $\mu$ are given by the integral kernel $\exp(-itH_{0,0})(x,y) $ of $\exp(-itH_{0,0})$, showing that there cannot even exist[^1] a path integral formula in the literal sense for the unitary group of the Laplace operator $H_{0,0}=-\Delta$ in ${\mathbb{R}}^d$. On the other hand, it is expected from some simple heuristics [@RS2] that the divergences of the Feynman path integral for $\exp(-itH_{v,\theta})$ actually stem from local singularities, so that in principle one can expect a well-defined Feynman path integral formula to hold true if one replaces the Riemannian manifold with an infinite weighted graph and considers the corresponding discrete magnetic Schrödinger operators thereon. *The main result of this paper shows that indeed such a path integral integral formula holds true in a very general setting.*
To this end, given a weighted graph $(X,b,m)$, possibly non-locally finite, a magnetic potential $\theta:\{b>0\}\to {\mathbb{R}}$ and an electric potential $v:X\to{\mathbb{R}}$, we use quadratic form methods to define a natural self-adjoint realization $L_{v,\theta}$ of the formal magnetic Schrödinger operator $${\widetilde{L}}_{v,\theta}f(x) = \frac{1}{m(x)}\sum_{y \in X} b(x,y)\big (f(x)-\exp(i\theta(x,y) )f(y)\big)+v(x)f(x)$$ in the complex Hilbert space $\ell^2(X,m)$. Operators of this type appear naturally in a gauge theoretic discretization procedure for continuum magnetic Schrödinger operators (cf. Remark \[pamy\] below), and have been used in the tight binding approximation in solid state physics [@Ha]. The most prominent example of such an operator is certainly the Harper operator [@Ha], whose spectral theory has been subject to the famous ten Martini problem [@martini].\
Our main result, Theorem \[main\], is the following Feynman path integral formula for the integral kernel $\exp(-itL_{v,\theta})(x,y)$ of $\exp(-itL_{v,\theta})$: defining a random variable $$\begin{aligned}
{\mathscr{A}}_t(v,\theta|\mathbb{X}):=i\int_0^t \theta( d \mathbb{X}_s)-i\int_{0}^{t}(v(\mathbb{X}_s)+\mathrm{deg}(\mathbb{X}_s))ds+\int^t_0\mathrm{deg}(\mathbb{X}_s)ds:\Omega\longrightarrow {\mathbb{C}}\end{aligned}$$ on the space $\Omega$ of explosive $X$-valued right-continuous jump paths, one has
$$\begin{aligned}
\label{fki22}
\exp(-itL_{v,\theta})(x,y)=\frac{1}{m(y)} \int_{\{\mathbb{X}_t=y\}\cap\{N_t(\mathbb{X})<\infty\}}i^{N_t(\mathbb{X})} \exp({\mathscr{A}}_t(v,\theta|\mathbb{X})) d\mathbb{P}_x,\end{aligned}$$
where
- $\mathbb{P}_x$, $x\in X$, denotes the Markov family of probability measures on $\Omega$ which is induced by $L_{v,\theta}|_{v=0,\theta=0}$ by the theory of regular Dirichlet forms
- $\mathbb{X}_t(\omega):=\omega(t)$ is the coordinate process on $\Omega$
- $N_t(\mathbb{X})\in {\mathbb{N}}\cup \{\infty\}$ is the number of jumps of $\mathbb{X}$ until the the time $t$
- $\int_0^t \theta( d \mathbb{X}_s):\Omega\to{\mathbb{R}}$ is the line integral of $\theta$ along the paths of $\mathbb{X}$
- $\mathrm{deg}:X\to [0,\infty)$ the weighted degree function on $(X,b,m)$.
To the best of our knowledge this formula is even conceptually entirely new, in the sense that the only previously established case was $v=0$, $\theta =0$ on the unweighted standard lattice in ${\mathbb{Z}}^d$ (cf. [@carmona]). The assumptions of Theorem \[main\] are satisfied, if e.g. the electric potential $v$ is bounded from below and $\mathrm{deg}$ is bounded, noting that, however, Theorem \[main\] can deal with much more general situations than the latter. We expect the Feynman path integral formula (\[fki22\]) to have several important spectral theoretic and geometric consequences: For example, by comparing (\[fki22\]) with the usual Feynman-Kac formula $$\exp(-tL_{-\mathrm{deg},0})(x,y)=\frac{1}{m(y)} \int_{\{\mathbb{X}_t=y\}\cap\{N_t(\mathbb{X})<\infty\}} \exp\Big(\int^t_0\mathrm{deg}(\mathbb{X}_s)\Big) d\mathbb{P}_x,$$ for the Schrödinger semigroup $\exp(-tL_{-\mathrm{deg},0})$, $t\geq 0$, one immediately gets the Kato-Simon type inequality $$|\exp(-itL_{v,\theta})(x,y)|\leq \exp(-tL_{-\mathrm{deg},0})(x,y),$$ which controls the underlying unitary magnetic Schrödinger group uniformly in both potentials in terms of the geometry of $(X,b,m)$. The latter inequality is expected to be of a fundamental importance in the context of discrete Kato-Strichartz estimates on general weighted graphs (cf. [@jacob] for a very recent study of such estimates for the unweighted standard lattice in ${\mathbb{Z}}^d$). Another interesting direction could be dictated by the following observation: Given another magnetic potential $\theta\rq{}:\{b>0\}\to {\mathbb{R}}$ and another electric potential $v\rq{}:X\to {\mathbb{R}}$ it is straightforward to derive the following explicit Feynman path integral formula for the composition $\exp(-itL_{v,\theta})\exp(itL_{v\rq{},\theta\rq{}})$
$$\begin{aligned}
&\big[\exp(-itL_{v,\theta})\exp(itL_{v',\theta'}) \big](x,y)\\
&=\int_{ \{ \omega:N_t(\mathbb{X}(\omega))<\infty\}}i^{N_t(\mathbb{X}(\omega))} \exp\big({\mathscr{A}}_t(v,\theta|\mathbb{X}(\omega))\big) m(\omega(t))^{-1}\Psi_t(v\rq{},\theta\rq{},y,\omega) d{\mathbb{P}}^x(\omega),\end{aligned}$$
where the random variable $\Psi_t(v\rq{},\theta\rq{},y,\cdot):\Omega\to{\mathbb{C}}$ is given by $$\Psi_t(v\rq{},\theta\rq{},y,\omega ):=\int_{\{\omega\rq{}:\omega'(t)=\omega(t)\}\cap\{\omega\rq{}:N_t(\mathbb{X}(\omega\rq{}))<\infty\}}\overline{i^{N_t(\mathbb{X}')} \exp\big({\mathscr{A}}_t(v',\theta'|\mathbb{X}(\omega\rq{}))\big)}d{\mathbb{P}}^y(\omega\rq{}).$$ We believe that this result, which is again conceptually completely new, will turn out to be very useful in the context of scattering theory (cf. [@jacob] for scattering theory results on the unweighted standard lattice in ${\mathbb{Z}}^d$).
Main results
============
Weighted graphs {#s:graphs}
---------------
Let $b$ be a graph over the countable set $ X $, i.e., $$b:X\times X \longrightarrow [0,\infty) \text{ is symmetric with $b(x,x)=0$ and $\sum_{y \in X}b(x,y) < \infty$ for all $x \in X$.}$$
Then, the elements of $X$ are called the *vertices* of $(X,b)$ and all $(x,y) \in X\times X$ with $b(x,y) >0$ are called the *edges* of $(X,b)$, where given $x\in X$ every $y\in X$ with $b(x,y)>0$ is called a *neighbor of $x$* and we write $ x\sim y $. The graph $(X,b)$ is called *locally finite*, if every vertex has only a finite number of neighbors. Furthermore, a *path* on the graph $(X,b)$ is a (finite or infinite) sequence of pairwise distinct vertices $(x_{j})$ such that $x_j \sim x_{j+1}$ for all $j$, and $X$ is called *connected*, if for any $x,y\in X$ there is a finite path $(x_j)^n_{j=0}$ such that $x_0=x$ and $x_n=y$. We equip $X$ with the discrete topology, so that any function $m: X \to (0,\infty)$ gives rise to a Radon measure of full support on $X$ by setting $m(A) := \sum_{x \in A}m(x)$. Then, we say $ b $ is a graph over the measure space $(X,m)$ and call the triple $(X,b,m)$ a *weighted graph*. For $x\in X$, we denote the *weighted degree function* by $$\text{deg}(x):= \frac{1}{m(x)}\sum_{y \in X} b(x,y).$$
Self-adjoint realizations of magnetic Schrödinger operators
-----------------------------------------------------------
In the following, we understand all spaces of functions to be complex-valued, and $i:=\sqrt{-1}$. Let ${C}(X)$ be the linear space of functions on $X$ and ${C}_c(X)$ its subspace of functions with finite support. We denote the standard scalar product and norm on $\ell^2(X,m)$ with ${\langle \cdot,\cdot\rangle}$ and ${\left\Vert \cdot\right\Vert}$. A *magnetic potential* on $(X,b)$ is an antisymmetric function $$\theta: \{b>0\}\longrightarrow {\mathbb{R}}\quad\mbox{such that}\quad\theta(x,y) = - \theta(y,x),\;x,y\in X.$$ Any function $v: X \to {\mathbb{R}{\hspace{0.5pt}}}$ will be simply called a *electric potential* on $X$.
We define a symmetric densely defined sesqui-linear form in the complex Hilbert space $\ell^2(X,m)$ with domain of definition ${C}_c(X)$ by $$\begin{aligned}
\label{dert}
Q_{v,\theta}^{(c)}(f,g):= & \frac{1}{2}\sum_{x\sim y} b(x,y)\Big(f(x)-\exp(i \theta(x,y) )f(y)\Big)\overline{\Big(g(x)-\exp(i \theta(x,y)) g(y)\Big)}\\
&\quad+\sum_{x\in X}v(x)f(x)\overline{g(x)}m(x){\nonumber}.$$
\[pamy\] Although not obvious, the above definition of $Q_{v,\theta}^{(c)}$ actually reflects a natural discretization procedure. To see that, one has to take the $U(1)$ gauge theory behind magnetic Schrödinger operators in ${\mathbb{R}}^d$ into account: Assume $\tilde{\theta}$ is a $C^1$ and real-valued $1$-form on ${\mathbb{R}}^d$ (= a magnetic potential) and $\tilde{v}:{\mathbb{R}}^d\to {\mathbb{R}}$ is continuous (= an electric potential). Then $\tilde{\theta}$ induces the metric covariant derivative $\nabla^{\tilde{\theta}}:= d+i\tilde{\theta}$ on the trivial complex line bundle ${\mathbb{R}}^d\times {\mathbb{C}}\to {\mathbb{R}}^d$ over ${\mathbb{R}}^d$, and one can define a symmetric sesquilinear form in $L^2({\mathbb{R}}^d)$ by $$\begin{aligned}
\label{pawy}
Q^{(c)}_{\tilde{v},\tilde{\theta}}= \frac{1}{2}\int \sum^{d}_{j=1}\nabla^{\tilde{\theta}}_{\partial_j}f\cdot\overline{\nabla^{\tilde{\theta}}_{\partial_j}g } \ dx+\int \tilde{v}f \overline{g} \ dx, \quad f,g\in C^{\infty}_c({\mathbb{R}}^d).\end{aligned}$$ The starting point for a discretization of the above sesquilinear form is simply to drop the limit and to set $\delta=1$ in the formula (cf. formula (7.66) in [@thalmaier]) $$\nabla^{\tilde{\theta}}_{\pm\partial_j}f(x)=\lim_{\delta\to 0}{\frac}{1}{\delta}\exp\Big(i\int_{\gamma^{\delta}_{x,x\pm e_j}}\tilde{\theta}\Big)f(x\pm e_j)-{\frac}{1}{\delta}f(x), \quad f \in C^{\infty}({\mathbb{R}}^d),$$ where $$\gamma^{\delta}_{x,x\pm e_j}:[0,\delta]\longrightarrow {\mathbb{R}}^d, \quad \gamma^{\delta}_{x,x\pm e_j}(t):={\frac}{1}{\delta}(\delta-t)x+{\frac}{t}{\delta}(x\pm e_j)$$ is the straight line which starts from $x$ and ends in $x\pm e_j$ at the time $\delta>0$. Note that $\exp\left(i\int_{\gamma^{\delta}_{x,x\pm e_j}}\tilde{\theta}\right)$ is precisely the (inverse) parallel transport along $\gamma^{\delta}_{x,x\pm e_j}$ with respect to the $U(1)$ covariant derivative $\nabla^{\tilde{\theta}}$. Then $\theta(x,y):=\int_{\gamma^1_{x,y}}\tilde{\theta}$ defines a magnetic potential on ${\mathbb{Z}}^d$ with its standard unweighted graph structure $b_{{\mathbb{Z}}^d}(x,y)=1$ if $|x-y|_{{\mathbb{R}}^d}=1$, and $b_{{\mathbb{Z}}^d}(x,y)= 0$ else. Note that $b_{{\mathbb{Z}}^d}(x,y)>0$ if and only of $y$ is of the form $x\pm e_j$. With $$\begin{aligned}
&\nabla^{\theta} f(x,y):= \exp\left(i\theta(x,y)\right)f(y)-f(x), \quad (x,y)\in \{b_{{\mathbb{Z}}^d}>0\},\\
& v(x):=\tilde{v}(x), \quad x\in {\mathbb{Z}}^d,\end{aligned}$$ we arrive at the sesquilinear form $$\begin{aligned}
Q_{v,\theta}^{(c)}(f,g):=&\frac{1}{2}\sum_{x\in X}\sum_{y:|x-y|_{{\mathbb{R}}^d}=1} \nabla^{\theta} f(x,y) \overline{\nabla^{\theta} g(x,y)}+\sum_{x\in X}v(x)f(x)\overline{g(x)}\\
&= \frac{1}{2}\sum_{ |x-y|_{{\mathbb{R}}^d}=1} \Big(f(x)-\exp(i \theta(x,y) )f(y)\Big)\overline{\Big(g(x)-\exp(i \theta(x,y)) g(y)\Big)}\\
&\quad+\sum_{x\in X}v(x)f(x)\overline{g(x)}\end{aligned}$$ in $\ell^2({\mathbb{Z}}^d)$ which is precisely of the type (\[dert\]) for $X={\mathbb{Z}}^d$, $b=b_{{\mathbb{Z}}^d}$, $m\equiv 1$, and in addition formally of the type (\[pawy\]). The above discretization procedure could be summarized as follows: one replaces the covariant derivative (an infinitesimal object) by its parallel transport. Field theoretic variants of this procedure are standard in lattice gauge theory[^2].
After discussing how the form $ Q^{(c)}_{v,\theta} $ arises from a discretization procedure, we continue by introducing the associated formal operator. Let $$\widetilde{C}(X):= \Big\{f\in {C}(X):\sum_{y\in X} b(x,y)|f(y)| < \infty \text{ for all }x\in X\Big\},$$ and we define the formal difference operator ${\widetilde{L}}_{v,\theta}:\widetilde{C}(X) \to {C}(X)$ by $${\widetilde{L}}_{v,\theta}f(x) = \frac{1}{m(x)}\sum_{y \in X} b(x,y)\big (f(x)-\exp(i\theta(x,y) )f(y)\big)+v(x)f(x).$$ The form $Q_{v,\theta}^{(c)}$ and the operator ${\widetilde{L}}_{v,\theta}$ are related by Green’s formula: for all $f\in \widetilde{C}(X)$, $g \in {C}_c(X)$, one has $$\begin{aligned}
\sum_{x\in X}&{\widetilde{L}}_{v,\theta}f(x)\overline{g(x)}m(x) = \sum_{x\in X} f(x)\overline{{\widetilde{L}}_{v,\theta}g(x)}m(x)\\
&=\frac{1}{2}\sum_{x,y \in X} b(x,y) \Big(f(x)-\exp(i \theta(x,y) )f(y)\Big)\overline{\Big(g(x)-\exp(i \theta(x,y))g(y)\Big)} \\
&\quad+ \sum_{x\in X}v(x) f(x)\overline{g(x)} m(x).{\nonumber}\end{aligned}$$ Moreover, if ${\widetilde{L}}_{v,\theta}[{C}_c(X)]\subseteq \ell^{2}(X,m)$, then for all $f,g\in C_{c}(X)$ one has $$\begin{aligned}
Q^{(c)}_{v,\theta}(f,g)=\langle {\widetilde{L}}_{v,\theta} f, g\rangle=\langle f,{\widetilde{L}}_{v,\theta} g\rangle.\end{aligned}$$
If $Q^{(c)}_{v,\theta}$ is bounded from below and closable, we denote its closure by $Q_{v,\theta}$ and the corresponding self-adjoint operator by $L_{v,\theta}$, referred to as the *magnetic Schrödinger operator induced by $(\theta,v)$*. From the Green’s formula it is obvious that $ L_{v,\theta} $ is a restriction of $ {\widetilde{L}}_{v,\theta} $, i.e., $$\begin{aligned}
L_{v,\theta} ={\widetilde{L}}_{v,\theta} \mbox{ on } {\mathrm{dom}}(L_{v,\theta} ).\end{aligned}$$ Likewise, the strongly continuous unitary group of operators $$\begin{aligned}
\exp(-itL_{v,\theta})\in {\mathscr{L}}(\ell^2(X,m)), \quad {t\in{\mathbb{R}}},\end{aligned}$$ defined by the spectral calculus, is called *magnetic Schrödinger group* induced by $\theta$ and $v$. The importance of this group for quantum mechanics is that for every $\psi\in {\mathrm{dom}}(L_{v,\theta})$ the function $t\mapsto \psi(t):=\exp(-itL_{v,\theta})\psi$ is the unique strong $C^1$-map ${\mathbb{R}}\to \ell^2(X,m)$ which satisfies the Schrödinger equation $$(d/dt) \psi (t) = -iL_{v,\theta}\psi(t),\quad \psi(0)=\psi.$$
The following remark (cf. Lemma 2.3, Lemma 2.11 and Theorem 2.12 in [@GKS]) addresses some functional analytic subtleties of these operators:
1\. If $X$ is locally finite, then one has $\widetilde{C}(X)={C}(X)$, however, in general, $\widetilde{C}(X)$ does not include $\ell^{2}(X,m)$.\
2. The condition ${\widetilde{L}}_{v,\theta}[{C}_c(X)]\subseteq \ell^{2}(X,m)$ for some (or equivalently all) $( v,\theta)$ is equivalent to $$\sum_{y\in X}\frac{ b(x,y)^2}{m(y)}<\infty\quad\text{ for all $x\in X$}.$$ We refer the reader to [@GKS; @Mi1; @Mi2] for essential self-adjointness results under the assumption ${\widetilde{L}}_{v,\theta}[{C}_c(X)]\subseteq \ell^{2}(X,m)$.\
3. If $Q^{(c)}_{v,0}$ is bounded from below then $Q^{(c)}_{v,\theta}$ is automatically closable for all magnetic potentials $ \theta $. If ${\widetilde{L}}_{v,\theta}[{C}_c(X)]\subseteq \ell^{2}(X,m)$ then ${\widetilde{L}}_{v,\theta}$ is a symmetric operator on $ C_{c}(X)\subseteq \ell^{2}(X,m) $ and, hence, if $ Q_{v,\theta}^{(c)} $ is bounded below, then $ Q_{v,\theta}^{(c)} $ is closable and $ L_{v,\theta} $ is a restriction of ${\widetilde{L}}_{v,\theta}$.
Stochastic processes on weighted graphs {#proc}
=======================================
Given a Hausdorff space $Y$ we denote its Alexandrov compactification by $\hat{Y}=Y\cup \{\infty_Y\}$ if $Y$ is noncompact and locally compact and $\hat{Y}=Y$ if $Y$ is compact. Let us introduce the probabilistic framework: Let us denote with $\Omega$ the space of right-continuous paths $\omega:[0,\infty)\to \hat{X}$ having left limits, which is equipped with its Borel-sigma-algebra $\mathcal{F}$. The latter is filtered by the filtration $\mathcal{F}_*$ generated by the coordinate process $$\mathbb{X}: [0,\infty)\times \Omega \longrightarrow \hat{X},\quad \mathbb{X}_t(\omega):=\omega(t).$$ For every subset $W\subset X$, let $$\tau_W:= \inf\{s\geq 0: \ \mathbb{X}_s\in X\setminus W\}:\Omega\longrightarrow [0,\infty]$$ be the first exit time of $\mathbb{X}$ from $W$. Note that $$\{t<\tau_W\}=\{\mathbb{X}_s\in W\text{ for all $s\in [0,t]$}\}\quad\text{ for all $t\geq 0$.}$$
For the sake of brevity we write $$Q^{(c)}:=Q^{(c)}_{v,\theta}|_{v=0,\theta=0},\quad Q:=Q_{v,\theta}|_{v=0,\theta=0} ,\quad L:= L_{v,\theta}|_{v=0,\theta=0}$$ for the underlying free forms and operator, respectively. An essential property of $Q$ is that it is a regular symmetric Dirichlet form in $\ell^2(X,m)$. It follows automatically from Fukushima’s theory that for every $x\in X$ there exists a unique probability measure $\mathbb{P}_x$ on $(\Omega,\mathcal{F})$ such that for all finite sequences $0=t_0<t_1<\dots < t_l$ and all $x=x_0,x_1,\dots, x_l\in \hat{X}$ one has $$\mathbb{P}_x\{\mathbb{X}_{t_1}=x_1,\dots, \mathbb{X}_{t_l}=x_l\}= \exp(-\delta_0L)(x_{0},x_1)m(x_1) \cdots \exp(-\delta_{l-1}L)(x_{l-1},x_l)m(x_l),$$ where $\delta_j:=t_{j+1}-t_{j}$, and where $\exp(- tL)(\bullet,\bullet)$ is extended to $\hat{X}\times \hat{X}$ according to $$\begin{aligned}
&\exp(- tL)(y,\infty_X):=0,\quad \exp(- tL)(\infty_X,\infty_X)=1,\\
& \exp(- tL)(\infty_X,y):= 1-\sum_{z\in X} \exp(- tL)(z,y)m(z),\quad y\in X .\end{aligned}$$
In addition, Fukushima’s result entails that these measures are concentrated on paths having $\infty_X$ as a cemetery, $$\begin{aligned}
\label{inti}\mathbb{P}_x\Big( \{\tau_X=\infty\}\cup \{ \text{$\tau_X<\infty$ and $\mathbb{X}_t=\infty_X$ for all $t\in [\tau_X,\infty)$} \} \Big)=1,\end{aligned}$$ and that $$\mathscr{M}:=(\Omega,\mathcal{F},\mathcal{F}_*,\mathbb{X},(\mathbb{P}_x)_{x\in X})$$ is a reversible strong Markov process. For every $n\in{\mathbb{N}}_{\geq 0}$ let $\tau_n:\Omega \to [0,\infty]$ denote the $n$-th jump time of $\mathbb{X}$ (with $\tau_0:=0$), an $\mathcal{F}_*$-stopping time. Let $$N(\mathbb{X}):[0,\infty)\times \Omega\longrightarrow \hat{{\mathbb{N}}},\quad N_t(\mathbb{X}):=\text{ number of jumps of ${\mathbb{X}}|_{[0,t]}$} ,$$ an $\mathcal{F}_*$-adapted process. We then define $$\tau:=\lim_{n\to\infty}\tau_n:\Omega \longrightarrow [0,\infty],$$ another $\mathcal{F}_*$-stopping time.
\[error\] a) For all $t\geq 0$, $x,y\in X$ one has $$\begin{aligned}
\label{inti2}&\mathbb{P}_x\{1_{\{N_t(\mathbb{X})<\infty\} }=1_{\{t<\tau_X\}}\} =1,\\
\label{inti3}& \mathbb{P}_x\left({\{N_t(\mathbb{X}) = 0\}} \right)=\exp(-t\mathrm{deg}(x)),\\
\label{inti4}&\mathbb{P}_x\{ \mathbb{X}_{\tau_n}\sim\mathbb{X}_{\tau_{n+1}}\text{ \emph{for all $n\in{\mathbb{N}}$}}\}=1,\\
\label{inti5}&\mathbb{P}_x(N_t(\mathbb{X}) = 1,\mathbb{X}_{\tau_1}= y)/t\to b(x,y)/m(x)\quad\text{ as $t\searrow 0$.}\end{aligned}$$ b) Let $f \in {C}_c(X)$, $t>0$, and let the function $\varphi_{t,f}:X\to {\mathbb{C}}$ be defined by $$\varphi_{t,f}(x) := \frac{1}{t}\mathbb{E}_x\left[1_{\{2\leq N_t(\mathbb{X})<\infty\}} f(\mathbb{X}_t) \right].$$ Then, for all $x\in X$, one has $\varphi_{t,f}(x)\to 0$ as $t\searrow 0$.
a\) To see (\[inti2\]), note that the inclusion $ \{N_{t}<\infty\}\subset \{t<\tau_{X}\} $ $\mathbb{P}_x$-a.s., is immediate since the form $Q$ does not have a killing term. Using now the formula $$\exp(-tL)f(x)=\mathbb{E}_x\left[1_{\{t<\tau_{X}\}} f(\mathbb{X}_t) \right],$$ which has been shown in [@GKS], we get $$0=\mathbb{E}_x\left[(1_{\{t<\tau_{X}\}}-1_{\{N_t(\mathbb{X})<\infty\}} )f(\mathbb{X}_t) \right],$$ for all $f\in \ell^2(X,m)$. Letting $f$ tend to $1$ from below and using $$1_{\{t<\tau_{X}\}}-1_{\{N_t(\mathbb{X})<\infty\}}\geq 0\quad\text{$\mathbb{P}_x$-a.s.},$$ we arrive at $$0=\mathbb{E}_x\left[1_{\{t<\tau_{X}\}}-1_{\{N_t(\mathbb{X})<\infty\}} \right]=\mathbb{E}_x\left[\left|1_{\{t<\tau_{X}\}}-1_{\{N_t(\mathbb{X})<\infty\}} \right|\right],$$ so that $$1_{\{t<\tau_{X}\}}=1_{\{N_t(\mathbb{X})<\infty\}}\quad\text{$\mathbb{P}_x$-a.s.}.$$ The properties (\[inti3\]), (\[inti4\]), (\[inti5\]) and b) have been shown in [@GKS].
Given a magnetic potential $\theta$ on $(X,b)$, the stochastic line integral of $\mathbb{X}$ along $\theta$ is defined by $$\begin{aligned}
&\int_0^{\bullet} \theta( d \mathbb{X}_s): [0,\infty)\times \Omega\longrightarrow {\mathbb{R}}\quad \int_0^{t} \theta( d \mathbb{X}_s)
:=\sum_{n= 1}^{N_t(\mathbb{X})}\theta(\mathbb{X}_{\tau_{n-1}}, \mathbb{X}_{\tau_{n}}),\end{aligned}$$ where $\int_0^{t} \theta( d \mathbb{X}_s)$ is set $0$ if $N_t(\mathbb{X})=0$, or if $N_t(\mathbb{X})=\infty$, or if $1\leq N_t(\mathbb{X})<\infty$ and $b(\mathbb{X}_{\tau_{n-1}},\mathbb{X}_{\tau_{n}})=0$ for some $n=1,\dots,N_t(\mathbb{X})$. For an electric potential $v$ on $X$ we also have the usual Riemannian integral $$\begin{aligned}
&\int_0^{\bullet} v( \mathbb{X}_s) ds: [0,\infty)\times \Omega\longrightarrow {\mathbb{R}},\quad \int_0^{t} v (\mathbb{X}_s)ds
=\sum_{n= 1}^{N_t(\mathbb{X})+1} v(\mathbb{X}_{\tau_{n-1}})(\tau_{n}-\tau_{n-1}),\end{aligned}$$ where $\int_0^{t} v (\mathbb{X}_s)ds$ is set $0$, if $N_t(\mathbb{X})=\infty$. Clearly these processes are $\mathcal{F}_*$-adapted.
The Feynman path integral formula {#haup}
=================================
Statement
---------
We recall that by the countability of $X$, for every bounded operator $A$ in $\ell^2(X,m)$ there is a uniquely determined map $$A(\cdot,\cdot): X\times X\longrightarrow {\mathbb{C}}$$ which satisfies $$Af(x)=\sum_{y\in X}A(x,y)f(y)m(y)\quad\text{for all $f\in \ell^2(X,m)$, $x\in X$}.$$ In fact, with $\delta_x\in C_c(X)$ the usual delta-function centered at $x$, one has $$\begin{aligned}
\label{l2}
A(x,y)=m(x)^{-1}\overline{A^*\delta_x(y)},\quad\text{ so that $A(x,\cdot)\in \ell^2(X,m)$ for all $x$}.\end{aligned}$$ In addition, it holds that $$\begin{aligned}
\label{l3}
A^*(x,y)=\overline{A(y,x)},\quad\text{ so that also $A(\cdot,x)\in \ell^2(X,m)$ for all $x$,}\end{aligned}$$ and one has the composition formula $$\begin{aligned}
\label{compo}
[AB](x,y)=\sum_{z\in X} A(x,z)B(z,y) m(z)\end{aligned}$$ for the integral kernel of a composition.
Given a magnetic potential $\theta$ on $(X,b)$ and an electric potential $v$ on $X$ we define an $\mathcal{F}_*$-adapted process $${\mathscr{A}}(v,\theta|\mathbb{X}):\Omega\times [0,\infty)\longrightarrow {\mathbb{C}}$$ by setting $$\begin{aligned}
{\mathscr{A}}_t(v,\theta|\mathbb{X}):=i\int_0^t \theta( d \mathbb{X}_s)-i\int_{0}^{t}(v(\mathbb{X}_s)+\mathrm{deg}(\mathbb{X}_s))ds+\int^t_0\mathrm{deg}(\mathbb{X}_s)ds,\quad t\geq 0.\end{aligned}$$
Here comes our main result:
\[main\]*(Feynman path integral formula)* Let $\theta$ be a magnetic potential on $(X,b)$ and let $v$ be an electric potential on $X$ such that $Q^{(c)}_{v,\theta}$ and $Q^{(c)}_{-\mathrm{deg},0}$ are semi-bounded from below and closable. Then for all $t\geq 0$, $x,y\in X$, one has $$\begin{aligned}
\label{fki}
\exp(-itL_{v,\theta})(x,y) = {\frac}{1}{m(y)}\mathbb{E}_x\left[1_{\{\mathbb{X}_t=y\}\cap\{N_t(\mathbb{X})<\infty\}}i^{N_t(\mathbb{X})}\exp({\mathscr{A}}_t(v,\theta|\mathbb{X}))\right].
\end{aligned}$$
The proof of Theorem \[main\] is given in the next two sections. One first proves the Feynman path integral formula on finite subgraphs, and then uses an exhaustion argument which relies on a new result from Mosco convergence theory (cf. Theorem \[mosco.char\]) which is proved in the appendix.
Note that by (\[inti\]) and Lemma \[error\], for all $x,y\in X$, $t\geq 0$, we actually have $$\begin{aligned}
1_{\{\mathbb{X}_t=y\}\cap\{N_t(\mathbb{X})<\infty\}}&=1_{\{\mathbb{X}_t=y\}}=1_{\{\mathbb{X}_t=y\}\cap\{N_t(\mathbb{X})<\infty\} \cap \{t<\tau_X\}} \\
&=1_{\{\mathbb{X}_t=y\}\cap \{N_t(\mathbb{X})<\infty,
\>b(\mathbb{X}_{\tau_{n-1}},\mathbb{X}_{\tau_{n}})>0\text{ for all $n=0,\dots,N_t(\mathbb{X})$} \} \cap \{t<\tau_X\}},\quad\text{$\mathbb{P}_x$-a.s. }\end{aligned}$$
Clearly, $Q^{(c)}_{-\mathrm{deg},0}$ is closable and bounded from below whenever $ \deg $ is a bounded function (in which case the form $Q^{(c)}_{-\mathrm{deg},0}$ is bounded as a sum of two bounded forms, namely, $Q^{(c)}_{0,0}$ and the form induced by $ -\deg $, cf. [@HKLW] for the boundedness of $Q^{(c)}_{0,0}$). The boundedness of $ \deg $ also implies the stochastic completess of $(X,b,m)$. The boundedness of $ \deg $ is certainly a natural assumption in the context of solid state physics.\
In the case of locally finite graphs the operator $ \widetilde{L}_{-\deg,0} $ is a well-defined symmetric operator with domain of definition $ C_{c}(X) $, and is therefore closable. The semiboundedness of $\widetilde{L}_{-\deg,0}$ has been investigated by Golénia in [@Gol], where the author examines whether the weighted adjacency matrix of a locally finite graph is unbounded from below. Indeed, the quadratic form of the adjacency matrix acting on the finitely supported functions is exactly $Q^{(c)}_{-\mathrm{deg},0}$. A negative result in this context is that, in case the edge weights $ b $ are unbounded, it follows that $Q^{(c)}_{-\mathrm{deg},0}$ is unbounded from below.
As it should be, the Feynman path integral formula also holds for negative times in the following sense: For all $t\geq 0$, $x,y\in X$, we have, using (\[l3\]), $$\begin{aligned}
&\exp(-i(-t)L_{v,\theta})(x,y)=\exp(itL_{v,\theta})(x,y)=\overline{\exp(-itL_{v,\theta})(y,x)}\\
&={\frac}{1}{m(x)}\mathbb{E}_y\left[1_{\{\mathbb{X}_{t}=x\}\cap \{N_t(\mathbb{X})<\infty\}}\overline{i^{N_t(\mathbb{X})} \exp({\mathscr{A}}_t(v,\theta|\mathbb{X}))} \right].\end{aligned}$$ The above result for magnetic Schrödinger groups should be compared with the Feynman-Kac-Ito formula [@GKS] for magnetic Schrödinger semigroups: The latter states that if $Q^{(c)}_{v,\theta}$ and $Q^{(c)}_{v,0}$ are bounded from below and closable, then for all $t\geq 0$, $x,y\in X$ one has $$\begin{aligned}
\exp(-tL_{v,\theta})(x,y) = {\frac}{1}{m(y)}\mathbb{E}_x\left[1_{\{\mathbb{X}_t=y\}} \exp\left(i\int_0^t \theta( d \mathbb{X}_s)- \int_{0}^{t} v(\mathbb{X}_s) ds\right)\right].\end{aligned}$$ It would also be interesting to see to what extent the Feynman path integral formula can be generalized to the setting of covariant Schrödinger operators in weighted graphs (a generalized Feynman-Kac-Ito formula [@GKS] for covariant Schrödinger semigroups has been established in [@GMT] and used in [@semic] to calculate semiclassical limits).
An immediate but nevertheless important consequence of the above formulae is the following very surprising Kato-Simon domination for magnetic Schrödinger groups in terms of certain geometric Schrödinger semigroups:
\[apisss\] Under the assumptions of Theorem \[main\], for all $t\geq 0$, $x,y\in X$ one has $$|\exp(-itL_{v,\theta})(x,y)|\leq \exp(-tL_{-\mathrm{deg},0})(x,y).$$
This estimate is to be compared with the domination result for magnetic Schrödinger semigroups, which reads $$|\exp(- tL_{v,\theta})(x,y)|\leq \exp(-tL_{v,0})(x,y),$$ provided $Q^{(c)}_{v,\theta}$ and $Q^{(c)}_{v,0}$ are semi-bounded from below and closable. We expect that Corollary \[apisss\] should play an important role in the derivation of Kato-Strichartz estimates on general weighted graphs (cf. Theorem 1.1 in [@jacob] for the unweighted standard lattice in ${\mathbb{Z}}^d$.)\
As a byproduct of our proof of the Feynman path integral formula, we also get the following result for possibly infinite subgraphs: To this end, if $W\subset X$ is any possibly infinite subset, we define $Q^{(c,W)}_{v,\theta}$ to be the restriction of $Q^{(c)}_{v,\theta}$ to $ C_{c}(W) $. Then, taking the closure in $$\ell^2(W,m):=\ell^2(W,m|_W)$$ yields a closed form $Q^{(W)}_{v,\theta}$ with associated operator $L^{(W)}_{v,\theta}$.
Under the assumptions of Theorem \[main\], let $W\subset X$ be an arbitrary subset. Then for all $t\geq 0$, $x,y\in W$ one has the following Feynman path integral formula, $$\begin{aligned}
\exp(-itL^{(W)}_{v,\theta})(x,y)= {\frac}{1}{m(y)}\mathbb{E}_x\left[1_{\{\mathbb{X}_t=y\}\cap \{N_t(\mathbb{X})<\infty\}\cap \{t<\tau_W\}}i^{N_t(\mathbb{X})} \exp({\mathscr{A}}_t(v,\theta|\mathbb{X}))\right].
\end{aligned}$$
In view of the definition of the wave operators from time dependent scattering theory, we expect that the following formula will play an important role in the context of scattering theory (cf. [@jacob] for some scattering results on the unweighted standard lattice in ${\mathbb{Z}}^d$):
\[scatt\] Let $\theta$, $\theta'$ be magnetic potentials on $(X,b)$ and let $v,v'$ be electric potentials on $X$ and assume that $Q^{(c)}_{v,\theta}$, $Q^{(c)}_{v',\theta'}$ and $Q^{(c)}_{-\mathrm{deg},0}$ are semi-bounded from below and closable. Then for all $t\geq 0$, $x,y\in X$, the integral kernel of $\exp(-itL_{v,\theta})\exp(itL_{v',\theta'})$ is given by $$\begin{aligned}
&\big[\exp(-itL_{v,\theta})\exp(itL_{v',\theta'}) \big](x,y)\\
&= \mathbb{E}_x\left[1_{\{ N_t(\mathbb{X})<\infty\}}i^{N_t(\mathbb{X})} {\mathscr{A}}_t(v,\theta|\mathbb{X})m({\mathbb{X}}_t)^{-1}\mathbb{E}_y\left[1_{\{\mathbb{X}'_t=\mathbb{X}_t\}\cap\{N_t(\mathbb{X}')<\infty\}}\overline{i^{N_t(\mathbb{X}')} \exp\big({\mathscr{A}}_t(v',\theta'|\mathbb{X}')\big)}\right]\right],\end{aligned}$$ where $\mathbb{X}'$ denotes an independent copy of $\mathbb{X}$.
Note that, explicitly, the formula from Proposition \[scatt\] reads as follows: $$\begin{aligned}
&\big[\exp(-itL_{v,\theta})\exp(itL_{v',\theta'}) \big](x,y)\\
&=\int_{ \{ \omega:N_t(\mathbb{X}(\omega))<\infty\}}i^{N_t(\mathbb{X}(\omega))} \exp\big({\mathscr{A}}_t(v,\theta|\mathbb{X}(\omega))\big) m(\omega(t))^{-1}\\
&\quad\times\int_{\{\omega\rq{}:\>\omega'(t)=\omega(t)\}\cap\{\omega\rq{}:N_t(\omega'(t))<\infty\}}\overline{i^{N_t(\mathbb{X}(\omega\rq{}))} \exp\big({\mathscr{A}}_t(v',\theta'|\mathbb{X}(\omega\rq{}))\big)}d{\mathbb{P}}^y(\omega\rq{})\>\> d{\mathbb{P}}^x(\omega).\end{aligned}$$
Using the composition formula (\[compo\]) and setting $$h(z):=\exp(itL_{v',\theta'})(z,y)=m(z)^{-1}\mathbb{E}_y\left[1_{\{\mathbb{X}'_t=z\}\cap\{N_t(\mathbb{X})<\infty\}}\overline{i^{N_t(\mathbb{X}')} \exp({\mathscr{A}}_t(v',\theta'|\mathbb{X}'))}\right].$$ for fixed $y$, we have $h\in \ell^2(X,m)$ by (\[l3\]) and $$\begin{aligned}
\big[\exp(-itL_{v,\theta})\exp(itL_{v',\theta'}) \big](x,y)&=\exp(-itL_{v,\theta})h(x)\\
&=\mathbb{E}_x\left[1_{\{N_t(\mathbb{X})<\infty\}}i^{N_t(\mathbb{X})} \exp({\mathscr{A}}_t(v,\theta|\mathbb{X}))h(\mathbb{X}_t)\right]\\
&=\mathbb{E}_x\left[1_{\{ N_t(\mathbb{X})<\infty\}}i^{N_t(\mathbb{X})} \exp({\mathscr{A}}_t(v,\theta|\mathbb{X}))m({\mathbb{X}}_t)^{-1}\right.\\
&\quad\times\left.\mathbb{E}_y\left[1_{\{\mathbb{X}'_t=\mathbb{X}_t\}\cap\{N_t(\mathbb{X}')<\infty\}}\overline{i^{N_t(\mathbb{X}')} \exp\big({\mathscr{A}}_t(v',\theta'|\mathbb{X}')\big)}\right]\right],\end{aligned}$$ completing the proof.
Proof of the Feynman path integral formula for finite subgraphs
---------------------------------------------------------------
Let $\theta$ be a magnetic potential and $v$ be an electric potential on $X$.
\[p:finite\] Let $W\subseteq X$ be finite. Then for all $f\in \ell^{2}(W,m)$, $x\in W$, $t\ge0$, one has $$\begin{aligned}
\exp(-itL^{(W)}_{v,\theta})f(x)=\mathbb{E}_x\left[1_{\{t<\tau_W\}}i^{N_t(\mathbb{X})} \exp({\mathscr{A}}_t(v,\theta|\mathbb{X}))f(\mathbb{X}_t)\right].\end{aligned}$$
The proof of the proposition above is based on three auxiliary lemmas.
\[l:semigroup\] Let $W \subseteq X$ be finite. Then, $(U_t(v,\theta,W))_{t\ge0}$ defined for $f\in\ell^{2}(W,m)$ by $$\begin{aligned}
U_t(v,\theta,W)f(x):=\mathbb{E}_x\left[1_{\{t<\tau_W\}} i^{N_t(\mathbb{X})}\exp({\mathscr{A}}_t(v,\theta|\mathbb{X})) f(\mathbb{X}_t)\right],\quad x\in W,t\ge0,\end{aligned}$$ is a strongly continuous semigroup of bounded operators on $\ell^{2}(W,m)$.
The asserted boundedness is trivial and the semigroup property follows from the strong Markov property of $\mathbb{X}$. By the semigroup property it is enough to check strong continuity at $t=0$, which can be easily checked using the boundedness of the integrand and the right continuity of $\mathbb{X}$.
\[generator\] Let $W\subseteq X$ be finite. Then, for all $f\in \ell^2(W,m)$ and $x\in W$, one has $$\begin{aligned}
\lim_{t\searrow0}\frac{U_t(v,\theta,W)f(x)-f(x)}{t}=-i L^{(W)}_{v,\theta}f(x).\end{aligned}$$
We fix an arbitrary $x\in W$ and compute $$\begin{aligned}
\lefteqn{\frac{U_t(v,\theta,W)f(x)-f(x)}{t} } \nonumber \\&=\frac{\mathbb{E}_x\left[1_{\{N_t(\mathbb{X}) = 0\}}\exp({\mathscr{A}}_t(v,\theta|\mathbb{X})) f(x)\right] -f(x)}{t}
&+\frac{\mathbb{E}_x\left[1_{\{N_t(\mathbb{X}) = 1,\mathbb{X}_{\tau_{1}}\in W\}}i \exp({\mathscr{A}}_t(v,\theta|\mathbb{X})) f(\mathbb{X}_t) \right]}{t} \\
&\quad + \psi_t(x)\end{aligned}$$ The error term $\psi_t(x)$ satisfies $|\psi_t(x)| \leq \varphi_{t,|f|}(x)$ with $\varphi_{t,|f|}$ defined in Lemma \[error\] b), therefore $\psi_t(x) \to 0$ as $t \searrow 0$. For the first term of the right hand side of the equality, we have, using $$\mathbb{E}_x\left[1_{\{N_t(\mathbb{X}) = 0\}} \right]=\mathbb{P}_x\left({\{N_t(\mathbb{X}) = 0\}} \right)=\exp(-t\mathrm{deg}(x)) ,$$ the convergence
$$\begin{aligned}
\lefteqn{\frac{\mathbb{E}_x\left[1_{\{N_t(\mathbb{X}) = 0\}} \exp({\mathscr{A}}_t(v,\theta|\mathbb{X})) f(x)\right] -f(x)}{t}}\\
&=\frac{\mathbb{E}_x\left[1_{\{N_t(\mathbb{X}) = 0\}}\exp\big(-t i( v(x)+\mathrm{deg}(x))+t\mathrm{deg}(x)\big) f(x)\right] -f(x)}{t}\\
&\to -i( v(x)+\mathrm{deg}(x)f(x).\end{aligned}$$
as $t\searrow 0$. Turning to the second term of the right hand side of the equation, setting $$a(y):=-i(v(y)+\mathrm{deg}(y))+\mathrm{deg}(y) ,\quad y\in W ,$$ we obtain
$$\begin{aligned}
\mathbb{E}_x&\left[1_{\{N_t(\mathbb{X}) = 1,\mathbb{X}_{\tau_{1}}\in W\}}i \exp({\mathscr{A}}_t(v,\theta|\mathbb{X})) f(\mathbb{X}_t)\right]\\
&=i\sum_{y \in W}\exp(i \theta(x,y))f(y) \underbrace{\mathbb{E}_x\left[1_{\{N_t(\mathbb{X}) = 1,\mathbb{X}_{\tau_1}= y\}} \exp\Big(-\tau_1a(x) - (t-\tau_1)a(y)\Big) \right]}_{=:\rho_t(x,y)}.\end{aligned}$$
Setting $$C := 2\max \{\mathrm{deg}(x)\mid x\in W\}$$ and using $\tau_1 \leq t$ on $\{N_t(\mathbb{X}) = 1\}$, we get $$\exp(-tC)\mathbb{P}_x(N_t(\mathbb{X}) = 1,\mathbb{X}_{\tau_1}= y) \leq| \rho_t(x,y)|\leq \exp(tC)\mathbb{P}_x(N_t(\mathbb{X}) = 1,\mathbb{X}_{\tau_1}= y).$$ Since by Lemma \[error\] (a) (\[inti5\]) $$\mathbb{P}_x(N_t(\mathbb{X}) = 1,\mathbb{X}_{\tau_1}= y)/t\to b(x,y)/m(x)$$ this shows that $$\rho_t(x,y)/t\to b(x,y)/m(x)$$ as $t\searrow 0$. As $W$ is finite, we conclude
$$\frac{1}{t}\mathbb{E}_x\left[ 1_{\{N_t(\mathbb{X}) = 1,\mathbb{X}_{\tau_{1}}\in W\}}i \exp({\mathscr{A}}_t(v,\theta|\mathbb{X})) f(\mathbb{X}_t) \right] \longrightarrow\frac{i}{m(x)}\sum_{y \in W} b(x,y)\exp(i \theta(x,y))f(y)\quad\text{as $t\searrow0$},$$ so, we infer $$\frac{U_t(v,\theta,W)f(x)-f(x)}{t} \longrightarrow - iL^{(W)}_{v,\theta}f(x)\>\>\text{ as $t\searrow0$.}$$
With these preparations we can now prove Theorem \[p:finite\].
For finite $W\subseteq X$, we have $\ell^{2}(W,m)=C_{c}(W)$. In particular, $L_{v,\theta}^{(W)}$ is a finite dimensional operator and the convergence $$-iL_{v,\theta}^{(W)}=\lim_{t\searrow 0}\frac{1}{t}\left(U_t(v,\theta,W)-\mathrm{id}\right)$$ from Lemma \[generator\] holds in the $\ell^{2}(W,m)$ sense. Therefore, the generator of the strongly continuous semigroup $(U_t(v,\theta,W))_{t\geq 0}$ is given by $L_{v,\theta}^{(W)}$. It follows that $\exp(-itL_{v,\theta}^{(W)})=U_t(v,\theta,W)$ for all $t\ge0$.
Proof of Theorem \[main\] in the general case
---------------------------------------------
For any subset $W\subseteq X$ we have a canonically given inclusion operator $$\iota_W: \ell^2(W,m)\hookrightarrow \ell^2(X,m),$$ which comes from extending functions to zero away from $W$, and its adjoint will be denoted with $\pi_W:=\iota^*_W$. Note that $\pi_W$ is given by the restriction map $f\mapsto f|_W$. The following geometric approximation is based on the Mosco convergence of the quadratic forms and will allow us to extend the Feynman path integral from finite to arbitrary graphs:
\[approx1\] Suppose $Q^{(c)}_{v,\theta}$ is semi-bounded and closable and let $(X_n)_{n\in{\mathbb{N}}}$ be an exhausting sequence for $X$, that is, $X_{n}\subseteq X_{n+1}$ for all $n$ and $X=\bigcup_{n\in{\mathbb{N}}}X_{n}$. Then, for all $t\geq 0$, one as $$\iota_{X_n} \exp(-itL_{v,\theta}^{(X_n)})\pi_{X_n} \to \exp(-itL_{v,\theta}) \text{ strongly in } \ell^2(X,m) \mbox{ as }n\to\infty.$$
By Theorem \[mosco.char\] it suffices to show that the forms $Q^{(X_n)}_{v,\theta}$ converge to $Q_{v,\theta}$ as $n\to\infty$ in the generalized Mosco sense. But this has been shown in [@GKS].
Let $f\in\ell^2(X,m)$, $x\in X$, $t>0$. We are going to prove $$\exp(-itL_{v,\theta})f (x) = \mathbb{E}_x\left[1_{\{t<\tau, N_t(\mathbb{X})<\infty\}} i^{N_t(\mathbb{X})}\exp({\mathscr{A}}_t(v,\theta|\mathbb{X})) f(\mathbb{X}_t)\right].$$ In view of (\[inti2\]), this clearly implies $$\begin{aligned}
&\mathbb{E}_x\left[1_{\{t<\tau, N_t(\mathbb{X})<\infty\}} i^{N_t(\mathbb{X})}\exp({\mathscr{A}}_t(v,\theta|\mathbb{X})) f(\mathbb{X}_t)\right]\\
&=\sum_{y\in X} \mathbb{E}_x\left[1_{\{{\mathbb{X}}_t=y, N_t(\mathbb{X})<\infty\}} i^{N_t(\mathbb{X})}\exp({\mathscr{A}}_t(v,\theta|\mathbb{X})) f(\mathbb{X}_t)\right],\end{aligned}$$ proving the asserted formula. Let $(X_n)$ be an exhausting sequence for $X$. Then, Proposition \[approx1\] implies the pointwise convergence $$\exp(-itL_{v,\theta})f (x) = \lim_{n \to \infty} \iota_{X_n} \exp(-itL^{(X_n)}_{v,\theta})\pi_{X_n} f (x).$$ Combining this with Proposition \[p:finite\], it remains to prove the equation $$\begin{aligned}
&\lim_{n\to \infty} \mathbb{E}_x\left[1_{\{t<\tau_{X_n}, N_t(\mathbb{X})<\infty\}} i^{N_t(\mathbb{X})}\exp({\mathscr{A}}_t(v,\theta|\mathbb{X}) )\pi_{X_n} f(\mathbb{X}_t)\right] \\
&= \mathbb{E}_x\left[1_{\{t<\tau, N_t(\mathbb{X})<\infty\}} i^{N_t(\mathbb{X})}\exp({\mathscr{A}}_t(v,\theta|\mathbb{X})) f(\mathbb{X}_t)\right]. \end{aligned}$$ This however follows from dominated convergence, as we have $$\left|1_{\{t<\tau_{X_n}, N_t(\mathbb{X})<\infty\}} i^{N_t(\mathbb{X})}\exp({\mathscr{A}}_t(v,\theta|\mathbb{X}))\pi_{X_n} f(\mathbb{X}_t)\right|\leq 1_{\{t<\tau_{X }, N_t(\mathbb{X})<\infty\}} \exp(\int^t_0 \mathrm{deg}(\mathbb{X}_s) ds) | f(\mathbb{X}_t)|$$ and $$\mathbb{E}_x\left[1_{\{t<\tau_{X}\}} \exp\left(\int^t_0 \mathrm{deg}(\mathbb{X}_s) ds\right)| f(\mathbb{X}_t)|\right]
=\exp(-t L_{-\mathrm{deg},0} )|f|(x)<\infty$$ by the usual Feynman-Kac formula [@GKS].
Mosco-convergence
=================
Let ${\mathscr{H}}_k $, $k \in{\mathbb{N}}$, and $ {\mathscr{H}}$ be Hilbert spaces. Suppose $q_k$ and $q$ are densely defined closed symmetric sesquilinear forms on ${\mathscr{H}}_k$ and ${\mathscr{H}}$, respectively, which are bounded below by a constant $c> -\infty$ which is *uniform* in $k$. Each $q_k$ is understood to be defined on the whole space ${\mathscr{H}}_k$ by the convention $q_k(u) = \infty$ whenever $u \in {\mathscr{H}}_k \setminus \mathrm{Dom}(q_k)$. Furthermore, we suppose that there exist bounded operators $\iota_k:{\mathscr{H}}_k \to {\mathscr{H}}$ such that $\pi_k := \iota_k ^*$ is a left inverse of $\iota_k$, that is $${\langle \pi_kf, f_k\rangle} = {\langle f,\iota_kf_k\rangle}\text{ and } \pi_k \iota_kf_k = f_k, \text{ for all } f\in {\mathscr{H}}, f_k\in {\mathscr{H}}_k.$$ Moreover, we assume that $\pi_k$ satisfies $$\sup_{k\in{\mathbb{N}}}\|\pi_k\|< \infty \text{ and } \lim_{k\to \infty} \|\pi_kf\| = \|f\|.$$
\[mosco\] In the above situation, we say that $q_k$ is *Mosco convergent* to $q$ as $k\to\infty$ *in the generalized sense*, if the following conditions hold:
- If $u_k \in {\mathscr{H}}_k$, $u \in {\mathscr{H}}$ and $\iota_ku_k \to u$ weakly in ${\mathscr{H}}$, then $$\liminf_{k \to \infty}\left(q_k(u_k) + c\|u_k\|_k^2\right) \geq q(u) + c\|u\|^2.$$
- For every $u \in {\mathscr{H}}$ there exist $u_k \in {\mathscr{H}}_k$, such that $\iota_k u_k \to u$ in ${\mathscr{H}}$ and $$\limsup_{k \to \infty}\left(q_k(u_k) + c \|u_k\|^2\right) \leq q(u) + c \|u\|^2.$$
We denote by $L_k$ the self-adjoint operator corresponding to $q_k$ and let $L$ be the self-adjoint operator corresponding to $q$ which are both bounded from below by $ c $. We will need the following generalization of the characterization of Mosco convergence from [@CKK] (see also the appendix of [@GKS]). Given an interval $I\subset {\mathbb{R}}$, we denote by $ C_{b}(I) $ the bounded continuous functions on $ I $ and by $ C_{\infty}(I) $ the space of continuous functions that become arbitrarily small outside of every compact set of $I$.
\[mosco.char\] If $q_k$ is Mosco convergent to $q$ as $k\to\infty$ in the generalized sense, then one has $\iota_k \psi(L_k)\pi_k \to \psi(L)$ as $k\to\infty$ strongly for every $\psi\in C_b({\mathbb{R}})$.
As the proof below shows, the fact that one can take $C_{\infty}({\mathbb{R}})$ in the above statement is a rather simple consequence of known results, the Stone-Weierstrass Theorem and the spectral calculus. The point of Theorem \[mosco.char\] is that one can even take $C_b({\mathbb{R}})$, which plays an essential role in this paper, for we are interested in operators of the form $\exp(-itL)$.
It is well-known that Mosco convergence is equivalent to $$\begin{aligned}
\label{popo}
\iota_k \exp(-aL_k)\pi_k \to \exp(-aL)\quad\text{ as $k\to\infty$ },\end{aligned}$$ strongly and locally uniformly in $a\geq 0$ (cf. Theorem 3.8 in [@CKK] and the appendix of [@GKS]). We are going to prove that the latter semigroup convergence implies $\iota_k \psi(L_k)\pi_k \to \psi(L)$ as $k\to\infty$ strongly for every $\psi\in C_b([c,\infty))$, proving the claim, as the spectra of $L$ and $L_k$ are subsets of $[c,\infty)$. To this end, we are going to follow the proof of Theorem VIII.20 in [@RS1] (which treats the case ${\mathscr{H}}_k={\mathscr{H}}$ and $\pi_k=\mathrm{id}_{{\mathscr{H}}}$).
Step 1: The claim holds for all $\psi\in C_{\infty}([c,\infty))$.\
Proof: Let us denote with ${\mathscr{A}}$ the space of complex linear combinations of functions of the form $x\mapsto \exp(-a x)$ on $[c,\infty)$, where $a,b\geq 0$. Then ${\mathscr{A}}$ is a separating unital \*-subalgebra of $C_{\infty}([c,\infty))$, thus dense in $C_{\infty}([c,\infty))$ by Stone-Weierstrass. Given an arbitrary $\varepsilon >0$, and $\psi\in C_{\infty}([c,\infty))$ we thus find $\psi_{\varepsilon }\in {\mathscr{A}}$ with $\left\|\psi-\psi_{\varepsilon}\right\|_{\infty}<\varepsilon $. By the spectral calculus we have $$\begin{aligned}
\label{kla}
\left\|\psi(L_k)-\psi_{\varepsilon}(L_k)\right\| <\varepsilon ,\quad \left\|\psi(L )-\psi_{\varepsilon}(L )\right\|<\varepsilon .\end{aligned}$$ for the operator norms. Let now $f\in {\mathscr{H}}$. Then we can estimate as follows, $$\begin{aligned}
&\left\| \big(\iota_k \psi(L_k)\pi_k - \psi(L)\big)f\right\|\\
&=\left\| \Big(\iota_k \psi(L_k)\pi_k - \psi(L) + \iota_k \psi_{\varepsilon}(L_k)\pi_k-\iota_k\psi_{\varepsilon}(L_k)\pi_k +\psi_{\varepsilon}(L)- \psi_{\varepsilon}(L)\Big)f\right\|\\
&\leq \left\| \iota_k \psi(L_k)\pi_k f-\iota_k \psi_{\varepsilon}(L_k)\pi_kf \right\|+\left\| \psi_{\varepsilon}(L)f- \psi(L)f \right\|+\left\| \iota_k \psi_{\varepsilon}(L_k)\pi_kf- \psi_{\varepsilon}(L)f\right\|.\end{aligned}$$ The last summand is $<\varepsilon$ for large $k$ by (\[popo\]) (which clearly extends from exponentials to ${\mathscr{A}}$), the second summand is $<\varepsilon\left\|f\right\|$ for all $k$ by (\[kla\]), and the first summand is $$< \sup_k\left\|\pi_k\right\|^2 \left\|f\right\|\varepsilon$$ for all $k$ by (\[kla\]), completing the proof of Step 1.
Step 2: The claim holds for all $\psi\in C_{b}([c,\infty))$.\
Proof: Fix $f\in {\mathscr{H}}$ and $\varepsilon>0$. For every $l\in {\mathbb{N}}$ set $g_l(x):=\mathrm{e}^{-x^2/l}$, a function in $C_{\infty}([c,\infty))$. Since $g_l(x)\to 1$ from below, the spectral calculus implies $g_l(B)\to \mathrm{id}$ strongly as $l\to\infty$ for every self-adjoint operator $B$. We can thus fix an $l$ such that $$\left\|f-g_l(L)f\right\|<\varepsilon.$$ Furthermore, let us set $$C_1:=\max\Big(\left\|\psi(L)\right\|,\sup_k\left\|\psi(L_k)\right\|\Big)\leq \left\|\psi\right\|_{\infty},\quad C_2:=\sup_k \left\|\pi_k\right\|=\sup_k \left\|\iota_k\right\|.$$ Then for large $k$ we can estimate as follows: $$\begin{aligned}
\lefteqn{\left\|\iota_k\psi(L_k)\pi_kf-\psi(L)f\right\|}\\
&\leq \left\|\psi(L) g_l(L)f -\psi(L)f\right\|+\left\|\iota_k\psi(L_k)g_l(L_k)\pi_kf- \psi(L) g_l(L)f\right\|\\
&\quad\>\>+\left\| \iota_k\psi(L_k)\pi_kf-\iota_k\psi(L_k)g_l(L_k)\pi_kf\right\|\\
&\leq C_1\varepsilon+\varepsilon+C_{1}C_2\left\| \pi_kf-g_l(L_k)\pi_kf\right\|\\
& \leq C_1\varepsilon+\varepsilon+C_{1}C_2\left\| \pi_kf-\pi_kg_l(L)f\right\|+C_{1}C_{2}\left\|g_l(L_k)\pi_kf+\pi_kg_l(L)f\right\|\\
&\leq C_1\varepsilon+\varepsilon+C_{1}C_2^2\left\| f-g_l(L)f\right\|+C_{1}C_2\left\|\pi_k\iota_kg_l(L_k)\pi_kf -\pi_kg_l(L)f\right\|\\
&\leq C_1\varepsilon+\varepsilon+C_{1}C_2^2\varepsilon+C_{1}C_2^2\left\|\iota_kg_l(L_k)\pi_kf -g_l(L)f\right\|\\
&\leq C_1\varepsilon+\varepsilon+C_{1}C_2^2\varepsilon+C_{1}C_2^2\varepsilon,\end{aligned}$$ where we have used $\psi g_l\in C_{\infty}([c,\infty))$ and step 1 for the second step, $\pi_k\iota_k=\mathrm{id}_{{\mathscr{H}}_k}$ for the fifth step, and $ g_l\in C_{\infty}([c,\infty))$ and Step 1 for the last step. This completes the proof.
[**Acknowledgements:**]{} The authors would like to thank Burkhard Eden, Evgeny Korotyaev, Ognjen Milatovic and Matthias Staudacher for very helpful discussions.
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[^1]: For the sake of completeness we remark that there exist several well-defined substitute results that mimick a path integral for $\exp(-itH_{v,\theta})$. For example, one can use white noise analysis [@hida], an infinite dimensional distribution theory to derive such a formula in the Euclidean ${\mathbb{R}}^d$ case, at least under some strong assumptions on the potentials.
[^2]: The authors would like to thank Burkhard Eden and Matthias Staudacher in this context.
|
---
abstract: 'Detection of hydroacoustic transmissions is a key enabling technology in applications such as depth measurements, detection of objects, and undersea mapping. To cope with the long channel delay spread and the low signal-to-noise ratio, hydroacoustic signals are constructed with a large time-bandwidth product, $N$. A promising detector for hydroacoustic signals is the normalized matched filter (NMF). For the NMF, the detection threshold depends only on $N$, thereby obviating the need to estimate the characteristics of the sea ambient noise which are time-varying and hard to estimate. While previous works analyzed the characteristics of the normalized matched filter (NMF), for hydroacoustic signals with large $N$ values the expressions available are computationally complicated to evaluate. Specifically for hydroacoustic signals of large $N$ values, this paper presents approximations for the probability distribution of the NMF. These approximations are found extremely accurate in numerical simulations. We also outline a computationally efficient method to calculate the receiver operating characteristic (ROC) which is required to determine the detection threshold. Results from an experiment conducted in the Mediterranean sea at depth of 900 m agree with the analysis.'
author:
- |
Roee Diamant\
Department of Marine Technology, The University of Haifa\
Email: [email protected]
title: Computationally Efficient Calculations of Target Performance of the Normalized Matched Filter Detector for Hydrocoustic Signals
---
Underwater acoustics; Matched filter; Detection.
Introduction {#sec:intro}
============
Underwater acoustics can fulfil the needs of a multitude of underwater applications. This include: oceanographic data collection, warning systems for natural disasters (e.g., seismic and tsunami monitoring), ecological applications (e.g., pollution, water quality and biological monitoring), military underwater surveillance, assisted navigation, industrial applications (offshore exploration), to name just a few [@Akyildiz:2006]. Detection of hydroacoustic signals is characterized by a target probability of false alarm and probability of detection. The detection is performed for a buffer of samples, $y(t)$, recorded from the channel (usually in a sliding time window fashion). In this paper, the focus is on detection of signals of known structure. The applications in mind are active sonar systems, acoustic localization systems (e.g., ultra-short baseline), and acoustic systems used for depth estimation, ranging, detection of objects, and communications.
In this paper, we focus on the first step in the detection chain, namely, a binary hypothesis problem where the decoder differentiate between a *noise-only* hypothesis and a *signal exist* hypothesis. The former is when the sample buffer, $y(t)$, consists of ambient noise, and the latter is the case where the sample buffer also includes a distinct received hydroacoustic signal. Without channel state information, the most common detection scheme is the matched filter [@Burdic:2002], which is optimal in terms of the signal-to-noise ratio (SNR) in case on an additive white Gaussian channel. The matched filter detector is a constant false alarm rate (CFAR) test, and its detection threshold is determined only by the target false alarm probability (cf. [@kazakos:1990]). Due to the (possibly) large dynamic range of the detected signal [@Bumiller:2007], and for reasons of template matching [@Conte:1998], the matched filter is often normalized by the noise covariance matrix. This normalization is often referred to as adaptive normalized matched filter (ANMF) and is the preferred choice in several tracking applications such as gradient descent search, active contour models, and wavelet convolution [@Lewis:1995].
To estimate the noise covariance matrix, several noise-only training signals are required [@Kelly:1989]. Since this limits the application, and since the noise may be time-varying, various ANMF detectors have been developed. Based on the noise texture model, [@Younsi:2011] suggested a maximum likelihood estimator for the noise covariance matrix. Alternatively, in [@Robey:1992] an iterative procedure is performed where first the covariance matrix is assumed known and the test statistics for a signal vector is calculated. Next, using these statistics and additional noise-only vectors, the noise covariance matrix is estimated and is substituted back into the test statistics. In [@Kraut:2001], an adaptive matched subspace detector is developed and its statistical behavior is analyzed to adapt the detector to unknown noise covariance matrices in cases where the received signal is distorted compared to transmitted one.
The above normalization methods of the matched filter require an estimation of the covariance matrix of the ambient noise. As shown in [@Scharf:1971], mismatch in this estimation effects detection performance and target false alarm and detection rates may not be satisfied. Since in underwater acoustics the noise characteristics are often fast time varying [@Burdic:2002], an alternative detection scheme is to normalize the matched filter with the power of $y(t)$ [@Scharf:2000], [@Burdic:2002]. We refer to this scheme as the *normalized matched filter* (NMF), as opposed to the ANMF. The NMF detector does not require estimation of the noise covariance matrix. Instead, its detection threshold depends only on the time-bandwidth product, $N$, of the expected signal. For underwater applications which require detection at target performance in various noise conditions, the NMF may be a suitable choice.
In [@Rangaswamy:2002], a low-rank NMF is suggested, where the linear matched filter is normalized by the power of the transmitted signal and a projection of the detected one. The projection is made according to the estimated noise covariance matrix, and the result is a simplified test which is proportional to the output of the standard colored-noise matched filter. A modification of the matched filter is proposed in [@Bumiller:2007] for the case of a multipath channel. The works in [@Bumiller:2007] and [@Rangaswamy:2002] include analysis for the false alarm and detection probabilities of the NMF. This analysis is either a modification of a similar study of the NMF or is based on semi-analytic matrix representation.
Due to low signal-to-noise ratio and the existence of narrow band interferences, hydroacoustic signals are constructed with a large time-bandwidth product of typical values $N>50$ [@Mason:2008; @Rouseff:2009; @Walree:2013]. While the NMF has been analyzed before, for large $N$ the available expressions are computationally complicated to evaluate. Consequently, it is difficult to evaluate the receiver operating characteristic (ROC), which is required to determine the detection threshold. As a result, most underwater applications avoid using the NMF as a detector. Considering this problem and based on the probability distribution of the NMF and its moments, in this paper computationally efficient approximations for the probability of false-alarm and for the probability of detection for signals of large $N$ are offered. This leads to a practical scheme for the evaluation of the ROC. Simulation results show that the developed expressions are extremely accurate in the large $N$ limit. To test the correctness of the analysis in real environment, results from a sea experiment are reported. The experiment was conducted in the Mediterranean sea to detect chirp signals reflected from the sea bottom at depth of 900 m.
The reminder of this paper is organized as follows. The system model is presented in Section \[sec:model\]. In Section \[sec:distribution\], we derive the probability distribution of the NMF and give expressions for the probability of false alarm and for the probability of detection. Next, performance evaluation in numerical simulation (Section \[sec:simulation\]) and results from the sea experiment (Section \[sec:experiement\]) are presented in Section \[sec:performance\]. Finally, conclusions are drawn in Section \[sec:Conclusions\]. The notations used in this paper are summarized in Table \[l:notation\].
------------------------ ----------------------------------------------------------
Notation Explanation
\[0.5ex\] $y(t)$ Received data from the channel
$s(t),s_k$ transmitted signal and its $k$th sample, respectively
$n(t),n_k$ Channel ambient noise and its $k$th sample, respectively
$\sigma^2$ variance of channel ambient noise
$T$ duration of signal
$W$ Bandwidth of signal
$N$ product of bandwidth and duration of signal
$\M$ output of normalised matched filter
$x=\cos(\theta_{N-2})$ Output of NMF for $y(t)=n(t)$
$x_T=\cos(\theta_{T})$ Detection threshold
$\cos(\phi)$ Output of NMF for $y(t)=s(t)+n(t)$
$P_{\mathrm{FA}}$ probability of false alarm
$P_{\mathrm{D}}$ probability of detection
\[1ex\]
------------------------ ----------------------------------------------------------
: List of major notations[]{data-label="l:notation"}
System Model {#sec:model}
============
The goal of this paper is to offer a computational efficient determination of the detection threshold of the NMF for signals with large $N$ property. Since the NMF is executed at the very first step of the reception chain, the receiver poses no information of the channel or range to transmitter. Therefore, only an additive noise of unknown variance can be assumed for the system model.
For a received signal, $y(t)$, we consider a binary detection test of hypotheses, $$\begin{aligned}
&H_0&: \ y(t)=n(t)\;,\nonumber\\
&H_1&: \ y(t)=s(t)+n(t)\;.
\label{e:hypo}\end{aligned}$$ In (\[e:hypo\]), $s(t)$ is an hydroacoustic signal of bandwidth $W$, duration $T$, and $n(t)$ is an additive noise. Let us define the time-bandwidth product $N=WT$. We assume that $N$ is large (values exceeding 50 are enough). In our analysis we consider the case of real signals. However, as demonstrated in Section \[sec:performance\], the analysis holds for the case of complex signals.
We are interested in the following quantity (referred to as the NMF), $$\begin{aligned}
\M&=&\frac{\left|\int s(t)y(t)dt\right|}{\sqrt{\int s^2(t)dt\int y^2(t)dt}}\nonumber\\
&=&\frac{\left|\sum\limits_{k=1}^N s_ky_k\right|}{\sqrt{\sum\limits_{k}s_k^2\sum\limits_{l}y_l^2}}\;,
\label{e:NMF}\end{aligned}$$ where $s_k$ and $y_k$ are the $k$th sample of $s(t)$ and $y(t)$, respectively, and $y(t)$ is sampled equally at the Nyquist rate. For a detection scheme which uses correlator (\[e:NMF\]) as its detection metric, the objective is to develop computational efficient expression for the probability of false alarm and for the probability of detection. Both figures are required to determine the detection threshold through the ROC.
The strong assumption in this paper is of i.i.d zero-mean Gaussian noise $n(t)$ with variance $\sigma^2$. As discussed in Section \[sec:simulation\], effect of mismatch in the noise model is shown negligible. However, the case of coloured noise can be treated by including a trivial whitening mechanism in the filtering process. Namely (\[e:NMF\]) becomes, $$\frac{\sum\limits_{j,k}^N s_jw_{j,k}y_k}{\sqrt{\sum\limits_{j,k}s_jw_{j,k}s_k\sum\limits_{j',k'}y_{j'}w_{j',k'}y_{k'}}}\;,
\label{e:NMF2}$$ where $w$ is the inverse correlation-matrix satisfying $\sum\limits_{k}w_{j,k}E\left[n_k,n_l\right]=\delta_{j,k}$ and $\delta$ is the Kronecker delta function. The following results can therefore be generalized without the need of significant modifications.
Probability Distribution Analysis {#sec:distribution}
=================================
In this section, we formulate the probability distribution of the NMF and for large $N$, give approximations for the probability of false alarm and for the probability of detection.
Probability of False Alarm {#sec:pfa}
--------------------------
![Spherical coordinates of received signal $s(t)$ and noise $n(t)$.[]{data-label="f:1"}](Figure1.eps){width="4.0in"}
Let $\vec{\ve{s}}$, $\vec{\ve{n}}$ be N-dimensional space vectors whose elements are $s_k$ and $n_k$, respectively. It is easier to manage the following analysis using spherical coordinates. To this end, we set $\vec{\ve{s}}$ along the polar-axis (see Fig. \[f:1\]), such that $\rho^2=\sum\limits_{k}n_k^2$. The assumption of i.i.d Gaussian noise leads to the probability density function $$P\left(\rho,\phi,\theta_1,\ldots,\theta_{N-2}\right)\partial\rho\partial\phi\prod\limits_{k=1}^{N-2}\partial\theta_k=\left(2\pi\sigma^2\right)^{-\frac{N}{2}}e^{-\frac{1}{2\sigma^2}\sum\limits_{i}n_i^2}\prod\limits_{l}\partial n_l\;.
\label{e:distibute}$$ Then, for a noise-only signal, i.e., $y(t)=n(t)$, the NMF is given by the angle $\theta_{N-2}$ between vectors $\vec{\ve{s}}$ and $\vec{\ve{n}}$, such that $$\M=\frac{\vec{s}\cdot\vec{n}}{|\vec{s}||\vec{n}|}=\cos{\theta_{N-2}}\;.
\label{e:NMF3}$$ To find the probability of false alarm, we first need to evaluate the distribution $P(\theta_{N-2})$. Then, given a detection threshold $x_T$, we obtain $$\hat{P}_{\mathrm{fa}}=\int\limits_0^{x_T}P(\theta_{N-2})d\theta_{N-2}\;.
\label{e:pfa_general}$$
Let the volume-element, $dV=\prod\limits_{l}\partial n_l$, be expressed in terms of the solid angle $d\Omega$ such that for $0\leq\phi\leq2\pi,0\leq\theta_k\leq\pi$, $$\begin{aligned}
dV&=&\rho^{N-1}d\rho d\Omega\nonumber\\
&=&\rho^{N-1}\partial\rho\partial\phi\prod\limits_{k=1}^{N-2}\partial\theta_k\sin^k\left(\theta_k\right)\;.
\label{e:omega}\end{aligned}$$ Then, by integrating (\[e:distibute\]) over all angular variables, except for the polar-angle $\theta_{N-2}$, one immediately obtains $$P(\rho,\theta_{N-2})\approx C_{N,1}\rho^{N-1}e^{-\frac{\rho^2}{2\sigma^2}}\sin^{N-2}(\theta_{N-2}), \ 0\leq\theta_{N-2}\leq\pi,0\leq\rho<\infty\;,
\label{e:distibute2}$$ where $C_{N,1}$ is a constant. Further integration over $\rho$ leads to $$P(\theta_{N-2})=C_{N,2}\sin^{N-2}(\theta_{N-2})\;,\label{e:distibute3_a}
$$ and $C_{N,2}$ is a constant. For convenience, denote $x=\cos(\theta_{N-2})$. Expression (\[e:distibute3\_a\]) implies that all the odd moments of $x$ vanish identically, whereas even moments are given by $$E\left[x^{2p}\right]=\frac{\Gamma\left(\frac{N}{2}\right)}{\sqrt(\pi)}\frac{\Gamma\left(p+\frac{1}{2}\right)}{\Gamma\left(p+\frac{N}{2}\right)}, \ p=0,1,2,\ldots\;.
\label{e:moment}$$ In particular, $$E\left[x^{2}\right]=\frac{1}{N}\;.
\label{e:moment2}$$ The result in (\[e:moment2\]) can be obtained directly by the method described in \[sec:appendix\], which confirms the above analysis.
By (\[e:distibute3\_a\]) and (\[e:moment2\]), when $N>>1$ the distribution $P(\theta_{N-2})$ approaches the Gaussian limit with the variance being $\frac{1}{N}$. Then, the probability of false alarm is approximated by $$\hat{P}_{\mathrm{fa}}=\frac{1}{2}\mathrm{erfc}\left(x_T\sqrt{\frac{N}{2}}\right)\;,
\label{e:pfa2}$$ However, since usually $P_{\mathrm{fa}}<<1$, unless $N$ is huge such that $P_{\mathrm{fa}}N>>1$ expression (\[e:pfa2\]) is not accurate enough. Instead, the accurate term for the probability of false alarm is $$P_{\mathrm{fa}}=1-B\left(x_T^2,\frac{1}{2},\frac{N-1}{2}\right)\;,
\label{e:pfa}$$ where $$B(a,b,z)=\int\limits_0^a t^{b-1}(1-t)^{z-1}dt$$ denotes the (tabulated) regularized incomplete beta function. Note that (\[e:pfa\]) does not require calculation of the noise characteristics.
Probability of Detection {#sec:pd}
------------------------
### Exact Term
Suppose $y(t)=s(t)+n(t)$, and mark $s^2$ as the energy of the received signal. Setting $\vec{\ve{s}}$ along the polar-axis (see Fig. \[f:1\]) we have $\vec{\ve{y}}\cdot\vec{\ve{s}}=R\cos(\phi)$ with $\M=\cos(\phi)$. Therefore, changing variables $(\rho,\theta_{N-2})$ into $(R,\phi)$ in (\[e:distibute2\]) we obtain $$P(R,\phi)\approx R^{N-1}\sin^{N-2}(\phi)e^{-\frac{(R\cos(\phi)-s)^2+R^2\sin^2(\phi)}{2\sigma^2}}, \ -\leq\phi\leq\pi,0\leq R<\infty\;.
\label{e:pd_distribution}$$ Integrating over $R$, $P(\phi)$ can be written in terms of the parabolic-cylinder function, $$D_p(z)=\frac{1}{\pi}\int\limits_0^{\pi}\sin\left(p\alpha)-z\sin(\alpha)\right)d\alpha\;,$$ i.e., $$P(\phi)=\pi^{-\frac{1}{2}}2^{1-\frac{N}{2}}\frac{\Gamma(N)}{\gamma\left(\frac{N-1}{2}\right)}e^{-s^2\frac{\frac{1}{2}-\frac{1}{4}\cos^2(\phi)}{\sigma^2}}\sin^{N-2}(\phi)D_{N}\left(-s\frac{\cos(\phi)}{\sigma}\right)\;.
\label{e:pd_distribution2}$$ Alternatively, by the definition of $D_p(z)$, $$\begin{split}
P(\phi)=\frac{e^{-\frac{s^2}{2\sigma^2}}\sin^{N-2}(\phi)}{\sqrt{\pi}\Gamma\left(\frac{N-1}{2}\right)}\cdot\left[\Gamma\left(\frac{N}{2}\right)F\left(\frac{N}{2},\frac{1}{2},\frac{s^2\cos^2(\phi)}{2\sigma^2}\right)+\right. \\
\left.\Gamma\left(\frac{N+1}{2}\right)\sqrt{2}s\frac{\cos(\phi)}{\sigma}F\left(\frac{N+1}{2},\frac{3}{2},\frac{s^2\cos^2(\phi)}{2\sigma^2}\right)
\right]\;,
\end{split}
\label{e:pd_distribution3}$$ where $$F(a,b,z)=\frac{\Gamma(b)}{\Gamma(b-a)\Gamma(a)}\int\limits_0^1 e^{zt}t^{a-1}(1-t)^{b-a-1}dt$$ is the confluent hypergeometric function. Note that as $\frac{s}{\sigma}\rightarrow0$, (\[e:pd\_distribution3\]) is reduced back to (\[e:distibute3\_a\]). The average NMF, derived from (\[e:pd\_distribution3\]), is given by the Kummer function, $$E\left[\cos(\phi)\right]=\frac{\Gamma\left(\frac{N+1}{2}\right)}{\Gamma\left(\frac{N+2}{2}\right)}\frac{se^{-\frac{s^2}{2\sigma^2}}}{\sqrt{2\sigma^2}}F\left(\frac{N+1}{2},\frac{N+2}{2},\frac{s^2}{2\sigma^2}\right)\;.
\label{e:MF_average}$$
The probability of detection for the detection threshold, $x_T$, can be found by $$P_D=\int_{0}^{x_T}P(\phi)d\phi\;.
\label{e:pd}$$ Unfortunately, for large $N$ direct numerical calculation of $P_D$ is bound to fail. This is because $P(\phi)$ contains infinitely many terms which oscillate rapidly as $N>>1$. It is therefore important to obtain asymptotic expressions for $P(\phi)$ in the large-$N$ limit.
### Approximated Solution
When both $N$ and $\frac{s}{\sigma}$ are large compared to unity, $P(\phi)$ can be approximated using the asymptotic form of $D_p(z)$ [@Gradshteyn:2004], $$D_p(z)\approx e^{-\frac{z^2}{4}}z^p\left(1+{\cal O}\left(\frac{z}{p}\right)\right)\;,
\label{e:pd_approx}$$ applicable for $z>>1$ and $|z|>>|p|$ (i.e., for large SNR). However, this may not be applicable to all considered cases. Instead, the *correct* asymptotic can be found by expanding $P(\phi)$ around its saddle-point.
To that end, let us go back to expression (\[e:pd\_distribution\]). Denoting $R\rightarrow \tilde{R}\frac{\sqrt{N}}{\sigma}$, and introducing $\gamma=\frac{s}{2\sigma\sqrt{N}}$, (\[e:pd\_distribution\]) takes the form $$P_{\gamma}(\tilde{R},\phi)\approx\left(\tilde{R}\sin^2(\phi)\right)^{-1}e^{Ng}\;,
\label{e:pd_distribution5}$$ where $g=\ln(\tilde{R})+\ln(\sin(\phi))-\frac{1}{2}\tilde{R}^2+2\tilde{R}\gamma\cos(\phi)$. Note that $\gamma$ is a function of $\frac{s}{\sigma}$ (which corresponds to the SNR). Since $P_{\gamma}=0$ at the end points $(\tilde{R},\phi)=(0,0)$ and $(\tilde{R},\phi)=(\infty,\pi)$, the large-$N$ behaviour of this function is dominated by Gaussian fluctuations around some saddle-points in the complex $(\tilde{R}\times\phi)$-hyper plane. The saddle-points equations are then $$\begin{aligned}
\frac{\partial g}{\partial\tilde{R}}&=&\tilde{R}^{-1}-\tilde{R}+2\gamma\cos(\phi)=0\;,\label{e:saddle_a}\nonumber\\
\frac{\partial g}{\partial\phi}&=&\cot(\phi)-2\tilde{R}\gamma\sin(\phi)=0\;,
\label{e:saddle_b}\end{aligned}$$ with fluctuations determined by the following Hessian (also known by the name ”Fisher information matrix”) $$H=\left(\begin{array}{cc}\frac{\partial^2g}{\partial\tilde{R}^2} & \frac{\partial^2g}{\partial\tilde{R}\partial\phi} \\ \frac{\partial^2g}{\partial\phi\partial\tilde{R}} & \frac{\partial^2g}{\partial\phi^2}\end{array}\right)=-\left(\begin{array}{cc}\tilde{R}^{-2}+1 & 2\gamma\sin(\phi) \\ 2\gamma\sin(\phi) & 2\tilde{R}\gamma\cos(\phi+\frac{cos(\phi)}{\sin^2(\phi)})\end{array}\right)\;.
\label{e:hassian}$$ Equation (\[e:saddle\_b\]) is solved by the quartet $$\begin{aligned}
R_c&=&\frac{2\gamma^2\pm\sqrt{4\gamma^4+4\gamma^2+1}}{\sqrt{4\gamma^2+1}}\;,\\
\sin(\phi_c)&=&\pm\frac{1}{\sqrt{4\gamma^2+1}}\;.
\label{e:hassian2}\end{aligned}$$ Fortunately, only one of these solutions (the one for which $R_c,\phi_c\geq0$) is reachable by a continuous deformation of the contour of integration. Substituting back into (\[e:hassian\]), one obtains $$|g''|=-\frac{\partial^2g}{\partial\phi_c^2}=1+4\gamma^2+\frac{4\gamma^4}{1+4\gamma^2}\left(2\gamma^2+\sqrt{4\gamma^4+4\gamma^2+1}\right)\;.
\label{e:hassian3}$$ To the leading order in powers of $N^{-1}$ and for arbitrary values of $\gamma\geq0$, $$P_{\gamma}(\phi)\approx\sqrt{\frac{N|g''|}{2\pi}}e^{-\frac{N}{2}|g''|(\phi-\phi_c)^2}, \ N>>1\;.
\label{e:pd_distribution4}$$
For $\gamma<<1$ (i.e., small SNR), we get $\phi_c\approx\left(\frac{\pi}{2}-2\gamma\right)$ and $|g''|\approx1$. Therefore, (\[e:pd\_distribution4\]) implies that the NMF maintains good deflection as long as $\gamma N>1$, which is similar to other compressing filters. (Note that under this condition, the variance of $\phi$ is smaller than the SNR separation). In the opposite limit, as $\gamma$ increases, $\phi_c\approx2\gamma^{-1}$ approaches towards the edge-point $\phi=0$. At the same time, however, $|g''|\rightarrow4\gamma^4$ and $\frac{\mathrm{var}(\phi)}{\phi_c^2}\approx\frac{N\gamma^2}{-1}<<1$. Thus, when $\phi_c\rightarrow0$, the Gaussian lube shrinks thereby avoiding any significant deformations due to edge-effects. As a result the probability of detection, $P_d$, can be evaluated as $$P_D=\frac{1}{2}\mathrm{erfc}\left((\phi_c-\theta_T)\sqrt{\frac{N|g''|}{2}}\right), \ \phi_c<\theta_T<\frac{\pi}{2}\;.
\label{e:pd2}$$ This approximation introduces a relative error of the order ${\cal O}\left(N^{-1}\right)$ in the estimation of $P_D$. It follows from (\[e:pd2\]) that, for a fixed $\frac{s}{\sigma}$, as the number of samples $N$ is increased, $P_d$ is saturated.
![ROC curves for $N=100$. Contour lines represents $s$ / $\sigma$ values. Logarithm base 10.[]{data-label="f:2"}](Figure2.eps){width="4.0in"}
Having expressions (\[e:pfa2\]), (\[e:hassian3\]), and (\[e:pd2\]), one can construct the ROC in the large-$N$ limit. First, the detection threshold is obtained by inverting (\[e:pfa2\]). Next, $|g''|$ is calculated with the help of (\[e:pd2\]). Finally, the required $s/\sigma$ ratio is determined by solving (\[e:hassian3\]) for $\gamma$. The resulting ROC curves for $N=100$ are shown in Fig. \[f:2\].
Performance Evaluation {#sec:performance}
======================
To evaluate the accuracy of the expressions for $P_{\mathrm{fa}}$ and $P_D$, results from numerical simulations and from a sea experiment are now presented. To that end, the above analysis is compared with empirical measurements of the probability of false alarm, $\hat{P}_{\mathrm{fa}}$, and the probability of detection, $\hat{P}_D$. This is performed by counting the number of occurrences for which $\M>\cos(\theta_T)$ when $y(t)=n(t)$ and when $y(t)=s(t)+n(t)$, respectively. Unless stated otherwise, we determine the detection threshold based on a target $P_{\mathrm{fa}}=10^{-4}$, i.e., a CFAR detector. For efficiency, the sample buffer $y(t)$ and the reference signal $s(t)$ are downscaled baseband converted. As a byproduct, this verifies that our analysis above for real signals holds also for complex ones after a factor adjustment.
Simulations {#sec:simulation}
-----------
The numerical simulations include transmission of a linear frequency modulation (LFM) chirp signal. The duration of the signal is set for $T_s=50$ msec, and its bandwidth varies with the considered $N$. Compliance with the system model, apart from the ambient noise, no channel distortion is used. The effect of the channel on performance is shown for the sea experiment discussed further below.
![Probability of false alarm as a function of detection Threshold.[]{data-label="f:3"}](Figure3.eps){width="4.0in"}
In Fig. \[f:3\], results for the probability of false alarm are shown. Good match between the analysis ($p_{fa}$) and the empirical ($\hat{p}_{fa}$) results is observed. The results show the strong dependency between threshold $\theta_T$ from (\[e:pfa2\]) and the compression ratio $N$. That is, for the same target probability of false alarm the threshold level dramatically decreases as $N$ increases. Fig. \[f:3\] also shows results of $\hat{P}_{\mathrm{fa}}$ for two $s$ / $\sigma$ ratios. As expected, the probability of false alarm does not depend on the SNR, i.e., the NMF detector is indeed a CFAR test.
![Probability of detection as a function of $s$ / $\sigma$. Target $P_{\mathrm{fa}}=10^{-4}$.[]{data-label="f:4"}](Figure4.eps){width="4.0in"}
In Fig. \[f:4\], approximation (\[e:pd2\]) is verified for several compression ratios $N$. Analysis $p_{d}$ and the empirical $\hat{p}_{d}$ results are given. Results are shown as a function of $s$ / $\sigma$. One can observe that $P_D$ increases with $N$. This is because of the dependency of threshold $x_T$ in $N$. However, for high levels of $N$, $P_D$ saturates. Fig. \[f:4\] shows that for small values of $N$, there is only a rough match between the analytic approximation $P_D$ and the empirical measurement $\hat{P}_D$. However, for higher values of $N$ and starting from $N=50$, an excellent match is observed.
Next, the effect of a mismatch in the noise model is considered. Results for three different additive noise models are shown. Together with the i.i.d. zero-mean Gaussian noise ($\hat{P}_{d}$), the considered noise cased include a strong single carrier interference at the centre frequency ($\hat{P}_{d}^{\mathrm{CW \ noise}}$), and an ambient noise recorded during the sea experiment at different time windows ($\hat{P}_{d^{\mathrm{Exp \ noise}}}$). An example of the noise recorded in the experiment is shown in Fig. \[f:5a\], and the empiric pdf evaluated from all noise instances recoded during the experiment is shown in Figure \[f:5b\]. One can observe the strong random transients in the experiment noise resulting wideband interferences, and the noise pdf appears to be similar to a Laplace Gaussian. The three noise components are normalized to test performance at different $s/\sigma$ ratios. Fig. \[f:6a\] shows results for the probability of false alarm, where $\hat{p}_{fa}$ represents results for i.i.d. Gaussian noise, $\hat{p}_{fa}^{\mathrm{CW \ noise}}$ represents results for noise with a CW interference, $\hat{p}_{fa}^{\mathrm{Exp \ noise}}$ represents results for noise recorded from the sea experiment, and Target $p_{fa}$ is the target false alarm probability. To reduce number of required simulation runs, the case considered is of a target false alarm of $10^{-3}$. Target false alarm probability is mostly achieved for the three noise models. Moreover, compliance with the results from Fig. \[f:3\], this match is observed for small and large values of $N$. In Fig. \[f:6b\], results are shown for the probability of detection for target false alarm of $10^{-4}$. Small differences are observed between the approximated analysis and the results for the recorded noise. A more significant effect is shown for the CW noise, where detection rate is better than the analysis. This is because, for a signal of large $N$, the NMF filters out narrowband interferences.
![Time-frequency response of sample buffer recorded during the sea experiment (depth roughly 900m).[]{data-label="f:7"}](Figure7.eps){width="4.0in"}
The results from the simulations verify the correctness of (\[e:pfa2\]). Furthermore, for hydroacoustic signals where the common case is $N>50$, approximation (\[e:pd2\]) predicts the probability of detection and thus the ROC. In addition, the results show that the analysis holds for an ambient noise consisting a single carrier interference and for the realistic case of noise recordings from a sea experiment.
Sea Experiment {#sec:experiement}
--------------
To test the applicability of the system model and to demonstrate the accuracy of expressions (\[e:pfa2\]) and (\[e:pd2\]) in an actual sea channel, a sea experiment was conducted. The experimental setting was about 10 km off the shores of Haifa, Israel, at depth of roughly 900 m. The set included a surface vessel from which a transmitting projector and a receiving hydrophone have been deployed. The experiment included 600 hydroacoustic transmissions. Each transmission consisted of two linear frequency modulation chirp signals, spaced by a time gap of 100 msec, whose carrier frequency was 50 kHz, bandwidth 10 kHz, and whose duration varied between: 10msec, 50msec, and 100msec. The tested $N$ values were therefore 100, 500, and 1000. Several $s/\sigma$ values were tested by changing the amplification level of the transmitted signals. Transmissions were made at depth of 10 m, and signals were received at depth of 100 m. This depth difference allowed sufficient separation between the receptions of the direct path, surface path, and bottom path, while allowing the use of a narrow voltage range to reduce quantisation errors.
![Output of the NMF for the latest arrival of the sampled buffer from Fig. \[f:8\].[]{data-label="f:8"}](Figure8.eps){width="4.0in"}
Following each transmission, a sampled buffer of 2 sec was collected from the channel. A time-frequency response of one of these sample buffers is shown in Fig. \[f:7\]. The reception of the two strong direct path is noticeable at time instances 4.1 sec and 5.3 sec, respectively. Strong surface reverberations are also observed. The much weaker signal reflected from the sea bottom can be seen at roughly 5.3 sec. For the latest arrival of the signals shown in Fig. \[f:7\], the output of the NMF is shown in Fig. \[f:8\]. There exists a time difference of roughly 0.025 sec (or 38 m) between the first arrival and the last arrival. Since the transmitted signal is wideband (relative to the carrier frequency), this figure represents the length of the channel impulse response. For each detected signal, the $s/\sigma$ ratio was evaluated by estimating the arrival time of the received signal and measuring the signal energy and the noise level.
Detection of the two chirp signals was performed using the NMF detector with target $P_{\mathrm{fa}}=10^{-4}$ and detection threshold (\[e:pfa2\]). Accurate detection was verified by comparing the time difference of arrival of each two local maxima of the NMF response with the expected time gap between the two transmitted chirp signals (100 msec). On the other hand, miss detection was declared when the output of the NMF did not exceed threshold. False alarm was determined for cases when the NMF response exceeded the threshold at wrong timing.
The decoding of the sea experiment achieved no false alarm. Considering the time window used for the sample buffer, this outcome corresponds to zero false alarms for roughly 2500 trials. The results for the detection rates are given in Table \[t:ExpPdRes\] alongside the predicted approximation (\[e:pd2\]). Since for high values of $N$ the probability of detection changes little with $N$ (see Fig. \[f:4\]), results for $N\geq500$ are accumulated. In addition, for clarify, the measured $s/\sigma$ levels are quantized. The results in Table \[t:ExpPdRes\] show that, compliance with the analysis, no miss detection was found at $s/\sigma$ levels above 15 dB. However, for lower $s$ / $\sigma$ ratios, detection performances are below the expected level. This is explained by the effect of the non-linear channel (especially the Doppler shift phenomena and non-resolved multipath), which distorts the received signal and thus reduces the output of the NMF. Nonetheless, the performance gap is minor, and the results of the sea experiment mostly agree with the analysis.
[ | P[1.5cm]{} ’ P[1.5cm]{} ’ P[3.5cm]{} ’ P[3.5cm]{} |]{}
------------------------------------------------------------------------
height 1pt @a xhline
**$N$** & **$s/\sigma$ \[dB\]** & **Detection Rate (experiment)** & **Probability of Detection (analysis)**\
------------------------------------------------------------------------
height 1pt @a xhline
& $\leq0$ & 0/57 (0)& 0.0022\
& 5 & 3/17 (0.17)& 0.21\
& 10 & 29/32 (0.9)& 1\
& 15 & 26/26 (1)& 1\
& $\geq$20 & 68/68 (1)& 1\
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height 1pt @a xhline
& $\leq$0 & 1/51 (0.01)& 0.0021\
& 5 & 5/22 (0.22)& 0.235\
& 10 & 35/38 (0.92)& 1\
& 15 & 37/37 (1)& 1\
& $\geq$20 & 52/52 (1)& 1\
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height 1pt @a xhline
Conclusions {#sec:Conclusions}
===========
This paper focused on detection of hydroacoustic signals, where the time-bandwidth product, $N$, is large. The goal was to develop a computationaly efficient method for the determination of the detection threshold of the normalized matched filter (NMF). This detector is a CFAR scheme used when the noise covariance matrix is fast time-varying and is hard to estimate. The probability distribution and the moments of the NMF were derived. Then, both the exact finite-$N$ distribution (\[e:pd\_distribution3\]) and the large-$N$ limit (\[e:pd\_distribution4\]) were studied. For the case of large $N$ value, computational efficient expressions for the probability of false-alarm (\[e:pfa2\]) and approximation for the probability of detection (\[e:pd2\]) were developed. Using this analysis, a practical scheme was provided for calculating the receiver operating characteristic (ROC). Numerical simulations showed that the developed expressions are extremely accurate. Furthermore, the performance of the NMF detector was demonstrated in a sea experiment conducted at depth of 900 m off the coast of Haifa, Israel. The result of this work may serve as the basis for using the NMF as a practical detector for hydroacoustic signals.
Second Moment of a NMF {#sec:appendix}
======================
In this section expression for the second moment of the NMF (\[e:NMF\]) are developed for the case of $y(t)=n(t)$. For a sampled noise signal with a sampling period $\Delta$ and number of samples $N$, $$\M^2=\frac{\left(\sum\limits_{k=1}^{N}n_ks_k\Delta\right)^2}{\left(\sum\limits_{k=1}^{N}n_k^2\Delta\right)\cdot\left(\sum\limits_{k=1}^{N}s_k^2\Delta\right)}\;.
\label{e:var_sample}$$ Denote $\tilde{n}_k=\frac{n_k}{\sigma}$, $\tilde{s}_k=s_k\sqrt{\frac{\Delta}{E_s}}$, where $\sigma^2$ is the variance of $n(t)$ and $E_s$ is the energy of $s(t)$. Then, $$\M^2=\frac{\left(\sum\limits_{k=1}^{N}\tilde{n}_k\tilde{s}_k\Delta\right)^2\sigma^2\frac{E_s}{\Delta}}{\left(\sum\limits_{k=1}^{N}\tilde{n}_k^2\Delta\right)\cdot\left(\sum\limits_{k=1}^{N}\tilde{s}_k^2\Delta\right)\sigma^2\frac{E}{\Delta}}\;.
\label{e:var_sample2}$$ Clearly, $E\left[\M^2\right]$ does not depend on $\sigma$ or $E_s$. In the following, $s_k$ and $n_k$ are therefore refer to as normalized variables. The second moment of the sampled $\M$ is $$E\left[\M^2\right]=E\left[\frac{\sum\limits_{k,l}^{N}s_ks_l\cdot n_kn_l}{\left(\sum\limits_{m=1}^{N}n_m^2\right)}\right]\;.
\label{e:var_sample3}$$ To simplify (\[e:var\_sample3\]), one can use the connection $$\frac{1}{X}=\int_{0}^{\infty}e^{-\lambda X}d\lambda\;,
\label{e:con}$$ such that $$E\left[\M^2\right]=\sum\limits_{k,l}^{N}s_ks_l\cdot E\left[\int_{0}^{\infty}n_kn_le^{-\lambda\sum\limits_{m}n_m^2}d\lambda\right]\;.
\label{e:var_sample4}$$ Since $n_k$ is Gaussian, so is the integral in (\[e:var\_sample4\]) and $$E\left[\M^2\right]=\sum\limits_{k}^{N}s_k^2\cdot E\left[\int_{0}^{\infty}n_k^2e^{-\lambda\sum\limits_{m}n_m^2}d\lambda\right]\;.
\label{e:var_sample5}$$ Consider $N=1$. Here, $$E\left[\M^2\right]=\int_{-\infty}^{\infty}\frac{dn}{\sqrt{2\pi}}\int_{0}^{\infty}n^2e^{-n^2\left(\lambda+\frac{1}{2}\right)}d\lambda=
\frac{\Gamma\left(\frac{3}{2}\right)}{\sqrt{2\pi}}\int_{\frac{1}{2}}^{\infty}\frac{da}{a^{\frac{3}{2}}}=1\;,
\label{e:var_example}$$ where $a\def\lambda+\frac{1}{2}$ is used. The result in (\[e:var\_example\]) is a good sanity check since for the case of a single sample, the variance of the NMF is 1. For a general $N$, $$E\left[\M^2\right]=\sum\limits_{k}s_k^2\cdot\frac{1}{\left(2\pi\right)^{\frac{N}{2}}}\Gamma\left(\frac{3}{2}\right)\pi^{\frac{N-1}{2}}\int_{\frac{1}{2}}^{\infty}\frac{da}{a^{\frac{3}{2}}a^{\frac{N-1}{2}}}=
\frac{\sqrt{\pi}}{2^{\frac{N}{2}}}\cdot\frac{1}{N}\pi^{-\frac{1}{2}}2^{\frac{N}{2}}=\frac{1}{N}\;.
\label{e:var_sample6}$$ By (\[e:var\_sample6\]), the variance of $\M$ for the case of noise-only signal is inverse proportional to $N$.
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abstract: 'The partition function of the Penner matrix model for both positive and negative values of the coupling constant can be explicitly written in terms of the Barnes $G$ function. In this paper we show that for negative values of the coupling constant this partition function can also be represented as the product of an holomorphic matrix integral by a nontrivial oscillatory function of $n$. We show that the planar limit of the free energy with ’t Hooft sequences does not exist. Therefore we use a certain modification that uses Kuijlaars-McLaughlin sequences instead of ’t Hooft sequences and leads to a well-defined planar free energy and to an associated two-dimensional phase space. We describe the different configurations of complex saddle points of the holomorphic matrix integral both to the left and to the right of the critical point, and interpret the phase transitions in terms of processes of gap closing, eigenvalue tunneling, and Bose condensation.'
address:
- 'Departamento de Física Teórica II, Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain'
- 'Departamento de Física Teórica II, Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain'
- 'Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, 11510 Puerto Real, Spain'
author:
- Gabriel Álvarez
- Luis Martínez Alonso
- Elena Medina
title: Phase space and phase transitions in the Penner matrix model with negative coupling constant
---
Introduction
============
The analysis of matrix models in the large-$n$ limit is a lively area of research for understanding relevant aspects of the phase structure of gauge theories [@HO11; @JA13; @HO15; @BUI16; @AL16]. In particular non-perturbative effects attract a great deal of attention. They were first studied in the critical case with the double-scaling limit method [@DA91; @DA93] but it was soon realized that they are also worth studying in matrix models away from criticality [@BO00; @MA08; @EY09; @MAR09; @MA08b; @SC14; @SC10; @PA10; @MA14].
The aim of the present paper is to analyze the phase space and the phase transitions of the Penner matrix model for negative values of the coupling constant in the planar limit. Several of the results obtained reproduce aspects of gauge theories which are usually described by unitary matrix models [@HO11; @HO15; @BUI16; @AL16; @GR80; @WA80]. The discussion is based on several important properties of (non-classical) generalized Laguerre polynomials which, in particular, determine the exact solvability of the Penner matrix model and the complete explicit description of its planar limit.
The standard Hermitian Penner matrix model [@PE88] is defined by the following formal partition function $$\label{mmp}
\mathbf{ Z}_{n}(g)
=
\int_{H_n} \frac{\rmd X}{N_n}
\exp\left[\frac{1}{g}\Tr\left( \sqrt{g} X+\log (1- \sqrt{g} X ) \right)\right], \quad g>0,$$ where the integration runs over the set $H_n$ of $n\times n$ Hermitian matrices $X$, and the normalization constant $N_n$ is $$\label{mmg}
N_n
=
\int_{H_n} \rmd X \exp\left(-\frac{1}{2} \Tr X^2\right)=2^{n/2} \pi^{n^2/2}.$$ The connection of the Penner model to continuum limits of theories of random discrete surfaces and to the $c=1$ string theory is a consequence of the application of the double-scaling limit to the large-$n$ expansion of $\mathbf{ Z}_{n}(g)$ (see references [@DI90; @DI91] and the comment following equation (\[sas\]) of appendix A of the present paper). However, the double-scaling limit is performed around a singular point of the large-$n$ expansion, which exists only for negative values of the coupling constant $g$ [@CP91; @MA06]. This is our motivation to study the Penner model and its large-$n$ limit in the region $g<0$.
The formal integral (\[mmp\]) was interpreted in references [@PE88; @MU98] as the limit of a sequence of matrix integrals corresponding to truncations of the Taylor series of the exponent, and leads to a well-defined topological expansion of the free energy. Remarkably [@MU98], the same asymptotic expansion can be obtained from the eigenvalue integral $$\label{eig}
Z_n(g)
=
C_n(g)\frac{1}{n!}
\left(
\prod_{i=1}^n \int_0^{\infty} \rmd x_i\,\rme^{-W_-(x_i)/g}
\right)
\Delta(\mathbf{x})^2,$$ where $$\label{cn}
C_n(g)
=
\frac{\rme^{n/g} g^{-n^2/2}}{(2\pi)^{n/2} \prod_{k=1}^{n-1} k!},$$ $\Delta(\mathbf{x})$ is the Vandermonde determinant $$\label{van}
\Delta(\mathbf{x}) = \prod_{j<k}(x_j-x_k),$$ and $$\label{pot0}
W_-(x) = x-\ln x.$$ In turn, the partition function $Z_n(g)$ is closely related to the family of classical Laguerre polynomials $L^{(\alpha)}_n(z)$ ($\alpha>0$) (see for instance [@PA10; @DI90; @DI91; @CP91; @MA06; @DE02; @AL14]), and by applying the method of orthogonal polynomials the eigenvalue integral (\[eig\]) can be written in terms of the Barnes $G$ function as [@PA10; @AL14] $$\label{ane0}
Z_n(g)
=
\frac{(\rme g) ^{n/g} g^{n^2/2}}{(2\pi)^{n/2}}
\frac{G(1+n+\frac{1}{g} )}{G(1+\frac{1}{g})}, \quad g>0.$$ Note that $|Z_n(g)|$ is well defined by the former expression not only for $g>0$ but also for all $g$ except $g=0$ and the zeros of the $G$ function in the denominator, i.e., for $g=-1/(k+1)$, $(k=1,2,3,\ldots)$. Therefore, using the asymptotic expansions of the $G$ function quoted in appendix A, we can find the asymptotic expansion of the free energy $$\label{lef}
\mathcal{F}_n(g) = - \frac{\ln|{Z}_n(-g)|}{n^2},\quad g>0.$$
However, to study spectral aspects like the possible existence and qualitative behavior of the asymptotic eigenvalue density, it is useful to have an eigenvalue integral representation of $Z_n(g)$ for $g<0$ (putting directly $g<0$ in equation (\[eig\]) leads to a divergent integral). This integral representation can be obtained by analytic continuation and is the subject of section \[sec:hooft\].
Section \[sec:Barnes\] is devoted to the large-$n$ expansion of the free energy (\[lef\]). The usual or ’t Hooft large-$n$ expansion is carried out with what in effect is a sequence of coupling constants $g_n$ such that $$\label{tsec}
g_n n = t = {\rm constant}.$$ We show that for these ’t Hooft sequences the asymptotic expansion of the free energy $\mathcal{F}_n(g)$ is the sum of an oscillatory contribution and a perturbative contribution. The perturbative contribution has essentially the same form as the large-$n$ expansion of the Penner model with positive coupling constant, and therefore it also provides a generating function for the virtual Euler characteristics of the spaces of Riemann surfaces with a finite number of punctures [@PE88]. However, we have not found the oscillatory contribution in the literature (for instance, it is missing in references [@CP91; @MA06]). Because of the zeros of the Barnes $G(1-1/g)$ quoted above, the free energy is singular at the value $t=1$ of the ’t Hooft parameter.
We also show that the oscillatory contribution does not converge in the planar limit for $t\neq 1$ with ’t Hooft sequences. In the context of generalized Laguerre polynomials, Kuijlaars and McLaughlin [@KU01; @KU04] introduced coupling constant sequences, which hereafter we will call KM sequences, determined by the two conditions $$\label{km}
\lim_{n\rightarrow \infty}g_n n = t,$$ (note that this condition is trivially satisfied by the ’t Hooft sequences), and the existence of the limit $$\label{eq:l}
l = \lim_{n\rightarrow \infty} |\sin(\pi/g_n)|^{1/n}.$$ In reference [@AL15] we showed that the eigenvalue integral that gives the analytic continuation of (\[eig\]) to negative values of $g$ (except for a prefactor discussed in section \[sec:hooft\]), does not have a planar limit with ’t Hooft sequences in the strong-coupling region $1<t<\infty$, but has a well defined planar limit with KM sequences, essentially because the condition (\[eq:l\]) permits the handling of the nonvanishing oscillatory contribution. In section \[sec:Barnes\] we also show that the planar limit of (\[lef\]) with ’t Hooft sequences also does not exist in the weak-coupling region $0<t<1$, but does exit with KM sequences. This result permits us to discuss the phase transitions in the $(t,l)$ plane and in particular the phase transitions at $t=1$.
In section \[sec:saddle\] we use the eigenvalue integral to discuss the associated Coulomb gas of eigenvalues and to describe the processes of gap-closing and Bose condensation of the asymptotic eigenvalue distribution. In section \[sec:summary\] we mention some matrix models of physical interest which share similar properties and which can be studied with the same ideas. The paper ends with two appendixes: in appendix A we collect some results on the Barnes $G$ function, and in appendix B we apply these results to derive the celebrated topological expansion of the standard Hermitian Penner model, which is compared in section \[sec:Barnes\] to the perturbative contribution of the corresponding expansion for the free energy (\[lef\]).
The partition function of the Penner model for negative values of the coupling constant \[sec:hooft\]
=====================================================================================================
To perform the analytic continuation of the partition function (\[eig\]) to negative values of $g$ we first write it in the form $$\label{eig1}
Z_n(g)
=
\frac{C_n(g)}{(1-\rme^{2\pi\rmi/g})^{n}}\,\frac{1}{n!} \left(\prod_{i=1}^n \int_{\Gamma}
\rmd z_i \,\rme^{-W_-(z_i)/g}\right)
\Delta({\bf z})^2,$$ where the contour $\Gamma$ is illustrated in figure \[fig:path\].
![Integration contour $\Gamma$ for the partition function (\[eig1\]).\[fig:path\]](figure1.pdf){width="8cm"}
Then we rotate counterclockwise the path $\Gamma$ to the path $\Gamma_{\theta}$ illustrated in figure \[fig:thetapath\], while simultaneously change the determination of $W_-(z)$ to $$\label{cw}
W_\theta(z) = z-\log_{\theta} z,$$ where $\log_{\theta} z= \ln|z|+\rmi \arg z$ with $\theta \leq \arg z< \theta+2\pi$. With these integration contour $\Gamma_{\theta}$ and determination $W_\theta(z)$, the rotated integral converges in the half-plane $\theta-\pi/2<\arg g<\theta+\pi/2$, and that the rotated and unrotated integrals are equal in $\theta<\arg g<\pi/2$ and $\theta-\pi/2<\arg g<0$.
![Rotated integration contour $\Gamma_{\theta}$ for the analytic continuation of the partition function (\[eig1\]).\[fig:thetapath\]](figure2.pdf){width="8cm"}
Thus, setting $\theta=\pi$, we find that the analytic continuation $\mathcal{Z}_n(g)=Z_n^{({\rm cont})}(-g)$ for $g>0$ reads $$\label{eig2}
\mathcal{Z}_n(g)
=
\frac{C_n(-g)}{(1-\rme^{-2\pi\rmi/g})^{n}}\frac{1}{n!}
\left(\prod_{i=1}^n\int_{\Gamma_{\pi}}\rmd z_i\,\rme^{(z_i-\log_{\pi} z_i)/g}\right)\Delta({\bf z})^2.$$ Finally we change variables $z_i \to -z_i$, and taking into account that $\log_{\pi} (-z)=\log_0 z+\rmi \pi$ it follows that $$\label{eig3}
\mathcal{Z}_n(g)
=
\frac{C_n(-g) (-1)^n\rme^{-\rmi n\frac{\pi}{g}}}{(1-\rme^{-2\pi\rmi/g})^{n}}
\frac{1}{n!}\left(\prod_{i=1}^n\int_{\Gamma}\rmd z_i \rme^{- (z_i+\log z_i)/g}\right)\Delta({\bf z})^2,$$ or, equivalently, $$\label{rel0}
\mathcal{Z}_n(g) = C_n(-g) (-2\rmi)^{-n} (\sin( \pi/g))^{-n}\, \mathcal{Z}^{(0)}_n(g),$$ where $\mathcal{Z}^{(0)}_n(g)$ is the holomorphic matrix integral [@AL15] $$\label{rel0h}
\mathcal{Z}^{(0)}_n(g)
=
\frac{1}{n!}
\left(
\prod_{i=1}^n \int_{\Gamma} \rmd z_i \,\rme^{-W_+(z_i)/g}
\right)
\Delta(\mathbf{z})^2,$$ and $$\label{pee}
W_+(z) = z+\log z.$$ We stress that the partition function $\mathcal{Z}_n(g)$ given by equation (\[eig2\]) in effect defines the Penner model for negative values of the coupling constant. Note in particular the factorization into a prefactor (the first fraction) and the eigenvalue integral $\mathcal{Z}^{(0)}_n(g)$ studied in reference [@AL15].
As a check of this analytic continuation process, we recall that using generalized Laguerre polynomials $L^{(-1/g)}_n(z /g)$ [@BE80; @DI95] it was proved in reference [@AL15] that $\mathcal{Z}^{(0)}_n(g)$ can be expressed in terms of the Barnes function as $$\label{znp00}
\mathcal{Z}^{(0)}_n(g)=g^{n(n-\frac{1}{g})}(1-\rme^{-2\pi\rmi/g})^n
\frac{G(1+n)G\left(1+n-\frac{1}{g} \right)}{G\left(1-\frac{1}{g} \right)}.$$ Hence, since $$\label{c}
C_n(-g)=\frac{\rme^{-n/g}(-g)^{-n^2/2}}{(2\pi)^{n/2}G(1+n)},$$ we finally get $$\label{ane10}
\mathcal{Z}_n(g)
=
\frac{ (-\rme g)^{-n/g} (-g)^{n^2/2}}{(2\pi)^{n/2}}
\frac{G(1+n-\frac{1}{g} )}{G(1-\frac{1}{g})}, \quad g>0,$$ which is the same expression obtained by replacing $g\to -g$ in equation (\[ane0\]).
Hereafter we will shorten the full qualification “Penner model with negative coupling constant” to simply “Penner model” when we refer to the model specified by the partition function $\mathcal{Z}_n(g)$.
The large-$n$ limit of the free energy \[sec:Barnes\]
=====================================================
The large-$n$ limit with ’t Hooft sequences
-------------------------------------------
Let us analyze the large-$n$ limit of the partition function $\mathcal{Z}_n(g)$ with ’t Hooft sequences $g_n=t/n$. From (\[ane10\]) and applying the reflection formula (\[bneg\]) to $\ln\left| G\left(1+n-1/g\right) \right|$ and to $\ln \left|G\left(1-1/g\right) \right|$ we have $$\begin{aligned}
\nonumber \mathcal{F}_n(g_n) & = & - \frac{\ln|\mathcal{Z}_n(g_n)|}{n^2}\\ \label{ane2mn}
& = & -\frac{1}{2n}\ln\left(\frac{\pi}{2}\right)+\left(\frac{1}{ng_n}-\frac{1}{2}\right)\ln g_n
+ \ln\left|\sin\left(\frac{\pi}{g_n}\right)\right|^{1/n}\nonumber\\
& & {}+\frac{1}{ng_n} -\frac{1}{n^2}\ln\left| G\left(1-n+\frac{1}{g_n}\right) \right|
+\frac{1}{n^2}\ln \left|G\left(1+\frac{1}{g_n}\right) \right|.\end{aligned}$$ Since $1+1/g_n >0$, the term $\ln |G(1+1/g_n)|$ in (\[ane2mn\]) has a Stirling expansion of the type (\[id1\]). However, the sign of the argument in $\ln |G(1-n+1/g_n)|$ depends on the value of $t$. Indeed, for $0<t<1$ and large $n$ we have $$\label{tm}
\ln \Big |G\left(1-n+\frac{1}{g_n}\right)\Big |= \ln\Big |G\left(1+x \right)\Big |, \quad x=n \left(\frac{1}{t}-1\right)\rightarrow +\infty,$$ so that this function has also a Stirling expansion of the type (\[id1\]). However, for $1<t<\infty$ and large $n$ we have $$\label{tm2}
\ln \Big |G\left(1-n+\frac{1}{g_n}\right)\Big| = \ln\Big| G\left(1-x \right)\Big |, \quad x=n \left(1-\frac{1}{t}\right)\rightarrow +\infty,$$ and we have to use the expansion (\[nxb\]) of Appendix A. Therefore, in this region the free energy can be decomposed into a sum of an oscillatory contribution and a perturbative contribution $$\label{des}
\mathcal{F}_n(t)=\mathcal{F}^{(\rm osc)}_n(t)+\mathcal{F}^{(\rm per)}_n(t),$$ where $$\label{nopf}
\mathcal{F}^{(\rm osc)}_n(t)
=
\left\{\begin{array}{ll}\ln \left|2\sin( \pi n/t) \right|^{1/n}, & \mbox{for $0<t<1$,} \\
\frac{1}{t}\ln \left|2\sin(\pi n/t) \right|^{1/n}+\frac{1}{2\pi n^2}{\rm Cl}_2\left(2\pi n/t\right),
& \mbox{for $1<t<\infty$,}\end{array}\right.$$ and in both cases, i.e., for $t\neq 1$, the perturbative contribution is $$\begin{aligned}
\label{top3}
\nonumber \mathcal{F}_n^{{\rm (per)}}(t) &\approx& -\left(\frac{(t-1)^2}{2t^2} \ln |1-t|-\frac{3}{4}+\frac{1}{2 t}\right)
+\frac{1}{12 n^2}\ln|1-t|
\\
& -&\sum_{k=2}^\infty \frac{B_{2k}}{2k(2k-2)} n^{-2k}t^{2k-2}
\left( (1-t)^{2-2k}-1 \right), \quad n\rightarrow \infty. \end{aligned}$$ The perturbative contribution $ \mathcal{F}_n^{{\rm (per)}}(t) $ coincides with the standard large-$n$ expansion for the Penner model with negative coupling constant [@CP91; @MA06]. Comparing the expression (\[top3\]) with the topological expansion (\[top\])–(\[chii\]) of the Penner model with positive coupling constant, it is clear that $\mathcal{F}_n^{{\rm (per)}}(t)$ also represents a generating function for the virtual Euler characteristics. To the best of our knowledge, the oscillatory contribution $\mathcal{F}^{(\rm osc)}_n(t)$ has not been considered in the literature. Note also that the value $t=1$ is a critical value for both contributions.
The planar limit
----------------
The planar limit of the free energy $$\label{fee}
\mathcal{F}= \lim_{n\rightarrow \infty} \mathcal{F}_n,$$ is not well-defined with the standard ’t Hooft sequences (\[tsec\]). Indeed, from the expression (\[nopf\]) of $\mathcal{F}^{(\rm osc)}_n(t)$ it is clear that the existence of $\mathcal{F}$ requires the existence of the limit $l$ defined in equation (\[eq:l\]).
Thus, instead of restricting to sequences satisfying the ’t Hooft coupling, we use KM sequences. From equations (\[des\])–(\[top3\]) it follows that for $t\neq 1$ $$\label{nopfp}
\mathcal{F}= \left\{\begin{array}{cc}\ln l -\frac{(t-1)^2}{2t^2} \ln |t-1|+\frac{3}{4}-
\frac{1}{2 t},
& \mbox{for $0<t<1$,} \\ & \\ \frac{1}{t}\,\ln l -\frac{(t-1)^2}{2t^2} \ln |t-1|+\frac{3}{4}-
\frac{1}{2 t},
& \mbox{for $1<t<\infty$.} \end{array}\right.$$ It is illustrative to clarify the origin of the different terms in equation (\[nopfp\]) with respect to the factors in the equation (\[rel0\]). From equation (\[c\]) and taking into account that $$\label{asb}
\frac{1}{n^2}\ln |G(1+n)|
=
\frac{1}{2}\ln n-\frac{3}{4}+ O(1/n),\quad n\rightarrow \infty,$$ it follows that $$\label{pers012}
\mathcal{F} = \ln l+\frac{1}{t}+\frac{1}{2}\ln t+\frac{3}{4}+ \mathcal{F}^{(0)},$$ where $ \mathcal{F}^{(0)}$ is the planar free energy for $\mathcal{Z}^{(0)}_n$ [@AL15] $$\label{llz}
\fl \mathcal{F}^{(0)} = H(t-1)\left(\frac{1}{t}-1 \right) \ln l
-\frac{1}{2}\ln t+\frac{3}{2}\left(\frac{t-1}{t}\right)
- \frac{(t-1)^2}{2t^2} \ln |t-1|,$$ where $H(x)$ is the Heaviside step function. It is now clear that the term $\ln l$ of $\mathcal{F}$ in the weak-coupling case comes only from the factor $|\sin(\pi/g)|^{-n}$ in (\[rel0\]), while the term $(\ln l )/t$ of $\mathcal{F}$ in the strong-coupling case is a result of the contributions from the two factors $|\sin(\pi/g)|^{-n}$ and $|\mathcal{Z}^{(0)}_n|$ in (\[rel0\]).
The phase space of the Penner model for negative values of the coupling constant
--------------------------------------------------------------------------------
The $l$-dependence of the planar free energy means that from the point of view of KM sequences, the phase space of the Penner model in the planar limit is the set $$\label{pes}
\mathcal{P} = \{(t,l)\in \mathbb{R}^2\, | \, 0<t<\infty,\; 0\leq l\leq 1\}.$$ In this way KM sequences reveal a fine phase space structure (see figure \[fig:path4\]), where both the weak- and strong-coupling phases depend on the additional parameter $l$. The points $(t,l)=(1,l)$ for $0<l<1$ represent first-order phase transitions $$\label{first}
\mathcal{F}\Big|_{t=1-0}
=
\mathcal{F}\Big|_{t=1+0},\quad \frac{\partial \mathcal{F}}{\partial t}\Big|_{t=1-0}
=
\frac{\partial \mathcal{F}}{\partial t}\Big|_{t=1+0}+\ln l,$$ while the point $(t,l)=(1,1)$ represents a continuous phase transition in which $ \mathcal{F}$ and $\partial\mathcal{F}/\partial t$ are continuous at $t=1$ but $\partial^2 \mathcal{F}/\partial t^2$ diverges. Furthermore, the line $l=0$ represents a singular phase with infinite free energy.
It is possible to characterize wide classes of KM sequences for all $0<t<\infty$ and $0\leq l\leq 1$, as for example the one-parameter family $$g_n = \frac{1}{[n/t]+c\,l^n}, \quad c\neq 0,$$ where $[x]$ denotes the integer part of $x$. They also arise as subsequences of ’t Hooft sequences $g_n=t/n$. For example, given an irreducible fraction $t=p/q$ $(p>1)$, the subsequences $g_{np}$ and $g_{np+1}$ of the ’t Hooft sequence are KM sequences with $l=0$ and $l=1$, respectively. For irrational $t$ we have that the sequence $\{n/t\}$, where $\{x\}$ denotes the fractional part of $x$ is a dense subset of the interval $[0,1]$, so that all subsequences of the ’t Hooft sequence $g_n=t/n$ such that $\{k_n/t\}\rightarrow x$ for some $0<x<1$ are KM sequences with $l=1$. The value $l=1$ represents the generic case of KM sequences (see remark 1.3 in [@KU04]).
A deeper understanding of the phase space of the Penner model is obtained from the analysis of the asymptotic eigenvalue (saddle point) distribution.
![Phase space of the non-Hermitian Penner model with negative coupling constant for KM sequences.\[fig:path4\]](figure3.pdf){width="8cm"}
Large-$n$ saddle points and Coulomb gas {#sec:saddle}
=======================================
The KM sequences originated in the theory of large-$n$ asymptotics of zeros of generalized Laguerre polynomials [@KU01; @KU04] $$\label{lagp}
L^{(\alpha)}_n(z)
=
\sum_{k=0}^n\left(\begin{array}{c}n+\alpha \\n-k\end{array}\right)\frac{(-z)^k}{k!}, \quad \alpha<0.$$ In [@AL15] we showed how these zeros determine the saddle points of the holomorphic matrix integral $\mathcal{Z}^{(0)}_n(g)$ for KM sequences in the strong-coupling case. In order to describe the phase transitions at $t=1$, we will extend here the analysis of [@AL15] by considering the weak-coupling case too.
Complex saddle points
---------------------
The saddle point equations for $\mathcal{Z}^{(0)}_n(g)$ are $$\label{sa0}
\frac{1}{g_n}\left(1+ \frac{1}{z_i^{(n)}}\right)
+
\sum_{j\neq i}\frac{2}{z_j^{(n)}-z_i^{(n)}}
=
0,\quad i=1,\ldots,n,$$ and their solutions are given by [@AL15] $$\label{sopl}
z_i^{(n)} = g_n\,l^{(\alpha_n,n)}_i,\quad i=1,\ldots,n,$$ where $l^{(\alpha_n,n)}_i$ are the zeros of the Laguerre polynomials $L^{(\alpha_n)}_n(z)$ with $$\label{zeroo}
\alpha_n=-1- \frac{1}{g_n}.$$
Let us denote by $\rho(z)$ the asymptotic eigenvalue (saddle point) distribution for the Penner model and by $\rho_L(z)$ the asymptotic zero distribution of the scaled Laguerre polynomials $L^{(\alpha_n)}_n(nz)$. The form of $\rho_L(z)$ has been completely characterized in the large-$n$ limit $$\label{clag}
n\rightarrow \infty,\quad \frac{\alpha_n}{n}\rightarrow A = \mbox{fixed},$$ for all real values of $A$ in references [@KU01; @KU04; @MA01; @DI11]. Using (\[sopl\]) and (\[zeroo\]) we can immediately translate the properties of $\rho_L(z)$ into properties of $\rho(z)$, since $$\label{tra}
t = -\frac{1}{A}, \quad \rho(z)=\frac{1}{t}\rho_L\left(\frac{z}{t}\right).$$
It turns out that the zeros of $L^{(\alpha_n)}_n(nz)$ cluster along certain curves $\gamma_L$ in the complex plane. More concretely, if we denote $$\label{end}
a_{\pm} = A+2\pm 2\sqrt{A+1},$$
1. For $-\infty< A<-1$ (the weak-coupling region $0<t<1$) the curve $\gamma_L$ is a simple open arc with endpoints $a_-$ and $a_+=\overline{a}_-$ symmetric with respect to $ \mathbb{R}$, which as $A\rightarrow -1$ closes and becomes the Szegő curve $$\label{zego}
|z\,\rme^{1-z}|=1,\quad |z|\leq 1.$$ This process is shown in figure \[fig:path1\]. The corresponding zero density is $$\label{cl0}
\rho_L(z) = \frac{1}{2\pi}\left|\frac{\sqrt{(z-a_-)(z-a_+)}}{z}\right|.$$
2. For $-1< A<0$ (the strong-coupling region $1<t<\infty$), and if the limit $$\label{eq:llll}
l=\lim_{n\rightarrow \infty} |\sin(\alpha_n \pi)|^{1/n}= \lim_{n\rightarrow \infty} |\sin(\pi/g_n)|^{1/n},$$ exists, then $\gamma_L$ is of the form $$\label{ge2}
\gamma_L=C_l \cup [a_-,a_+],$$ where
1. For $l\neq 0$ the curve $C_l \subset \mathbb{C}\setminus (\{0\}\cup [a_-,+\infty))$ is a simple closed curve encircling $0$ once, which is determined by the implicit equation $$\label{cl1}
{\rm Re}\,\int_{a_-}^z \frac{\sqrt{(z'-a_-)(z'-a_+)}}{z'}{\rm d}z'=-\log l.$$ The corresponding zero density is $$\label{cl2}
\rho_L(z)=\frac{1}{2\pi}\left|\frac{\sqrt{(z-a_-)(z-a_+)}}{z}\right|.$$
2. For $l=0$ $$\label{ge20}
\gamma_L=\{0\}\cup [a_-,a_+],$$ and the zero density is $$\label{cl3}
\rho_L(z)
=
A\, \delta(z)+\frac{1}{2\pi}\left|\frac{\sqrt{(x-a_-)(x-a_+)}}{x}\right|\chi_{[a_-,a_+]},$$
where $\chi_{[a_-,a_+]}$ is the characteristic function of the real interval $[a_-,a_+]$. The filling fraction of the zero density on $C_l$ is equal to $|A|$. The value $l=1$ is the generic case (see remark 1.3 in reference [@KU04]), and it follows from equation (\[cl1\]) that it is the only situation in which the loop $C_l$ and the interval $[a_-,a_+]$ intersect (at the point $a_-$). It should be noticed that the form of $\gamma_L$ depends not only on the value of $A$ but also on $l$.
Coulomb gas, gap closing, eigenvalue tunneling and Bose condensation
--------------------------------------------------------------------
In this section we use the electrostatic interpretation wherein the eigenvalue density $\rho(z)$ is thought of as a unit normalized positive charge density for a Coulomb gas in the external electrostatic potential $$\label{uve}
V(z) = x+\ln |z|,\quad (x=\re z).$$ The electrostatic energy of the Coulomb gas, $$\label{efe}
\mathcal{E}(t)=\frac{1}{t}\int_{\gamma} V(z) \rho(z)|{\rmd}z|
-
\int_{\gamma} |\rmd z| \int_{\gamma} |\rmd z'| \ln |z-z'| \rho(z) \rho(z'),$$ is given [@AL15] by $$\label{ee}
\mathcal{E}(t)=-\frac{1}{2}\ln t-\frac{(t-1)^2}{2t^2} \ln |t-1|+\frac{3}{2}\left(1-\frac{1}{t}\right).$$ We remark that $\mathcal{E}(t)$ is independent of $l$.
![Eigenvalue distribution in the gap-closing phase transition for $n=80$.\[fig:path1\]](figure4.pdf){width="8cm"}
The form of the support of $\rho(z)$ is independent of $l$ in the weak-coupling phase, but it depends on $l$ in the strong-coupling phase. Thus for $1<t<\infty$ the support of $\rho(z)$ consists of two pieces: an $l$-dependent closed loop $\gamma_1$ around the origin with filling fraction $1/t$, and an $l$-independent interval $\gamma_2=[ta_-,ta_+]$ on the positive real axis with filling fraction $1-1/t$.
The points of the weak-coupling phase are electrostatic stable equilibrium configurations of $\rho(z)$. However, points of the strong-coupling phase represent stable electrostatic equilibrium configurations only for $l=1$ [@AL15]. Indeed, for strong coupling the effective potential of the charge distribution corresponding to $\rho(z)$ $$\label{loge0}
V_{\rm eff}(z) = V(z) - 2 t \int_{\gamma} \ln |z-z'| \rho(z') |\rmd z'|,$$ is constant on the two pieces of the support of $\gamma$ [@AL15], but with values $$\label{une}
V_{\rm eff}\big |_{\gamma_1} = - t\ln l+V_{\rm eff}\big |_{\gamma_2},
\quad
V_{\rm eff}\big |_{\gamma_2}=(2t-1) -t \ln t - (t-1) \ln (t-1),$$ which coincide for $l=1$ only.
Finally, note the following two types of phase transitions:
1. In the weak-coupling phase the limit $t\rightarrow 1$ and constant $l\neq 0$ represents a gap-closing (confining) transition in which the open arc formed by the support of $\rho(z)$ closes and becomes the Szegő curve (figure \[fig:path1\]). In the strong-coupling phase this limit represents a gap-opening (deconfining) transition mediated by an eigenvalue tunneling process in which the charge $1-1/t$ located on the interval $\gamma_2$ tunnels to the higher-potential points of the loop $\gamma_1$.
2. As $l\rightarrow 0$ from the strong-coupling phase with constant $t$ the loop $\gamma_1$ of the support of $\rho(z)$ shrinks to the point $z=0$ forming a condensate of charge $1/t$ (figure \[fig:bose\]). This represents a process of Bose condensation in which the electrostatic energy $\mathcal{E}$ (\[ee\]) remains finite and constant. However, the planar free energy diverges ($\mathcal{F}\rightarrow -\infty$) as a consequence of the contribution to $\mathcal{F}$ of the oscillatory factor in (\[rel0\]).
![Eigenvalue distribution in Bose condensation for $n=80$.\[fig:bose\]](figure5.pdf){width="8cm"}
Outlook\[sec:summary\]
======================
There are several relevant non-Hermitian matrix models which have a phase space in the large-$n$ limit with properties similar to the Penner model with negative coupling constant. We will briefly mention some examples.
Multi-Penner models of the form $$\label{mp}
W(z) = -\sum_{i=1}^k \mu_i \log (z-q_i),$$ have been introduced to characterize the correlation functions of the $d=2$ conformal $A_1$ Toda field theory [@SC10; @DI09]. The simplest case is the double Penner model [@SC10] $$\label{jac}
W(z) = -\mu_+\log (1-z)-\mu_- \log(z+1),$$ which is connected to the theory of Jacobi polynomials $$\label{j1}
P^{(\alpha,\beta)}_n(z)
=
2^{-n}
\sum_{k=0}^n
\left(\begin{array}{c}n+\alpha \\n-k\end{array}\right)
\left(\begin{array}{c}n+\beta \\k\end{array}\right)(z-1)^k(z+1)^{n-k},$$ with $$\label{j2}
\alpha=\frac{\mu_+}{g},\quad \beta=\frac{\mu_-}{g}.$$ The Hermitian matrix model corresponds to the classical Jacobi polynomials with $\alpha,\beta>-1$. Its large-$n$ limit with ’t Hooft sequences is well defined, and the eigenvalue distribution is determined by the asymptotic zero distribution of the Jacobi polynomials on the real interval $[-1,1]$. However, there are non-classical cases for which the existence of the planar limit requires a special formulation based on KM sequences [@MA05]. For example, if we take a large-$n$ limit with sequences $n g_n\rightarrow t>0$ such that $$\label{j3}
n\rightarrow \infty,\quad \frac{\alpha_n}{n}\rightarrow A
=
\frac{\mu_+}{t},\quad \frac{\beta_n}{n}\rightarrow B
=
\frac{\mu_-}{t},$$ where $$\label{j4}
-1<A<0<B,$$ or, equivalently, $$\label{j5}
\mu_+<0,\quad \mu_->0,\quad t>|\mu_+|.$$ Then the existence of a well-defined asymptotic zero distribution requires the existence of the limit [@MA05] $$\label{j6}
l= \lim_{n\rightarrow \infty}|\sin( \pi A n)|^{1/n}.$$ The corresponding zero distribution of non-classical Jacobi polynomials exhibits properties like gap-closing processes [@MA05], similar to the generalized Laguerre polynomials. Therefore it would be interesting to analyze the non-Hermitian version of the double Penner model following the scheme used in the present paper for the Penner model.
Families of Laguerre polynomials also appear in certain matrix models used to study the low-energy limit in Quantum Chromodynamics, e.g., the chiral Gaussian Unitary matrix model (also called the Wishart-Laguerre ensemble) [@VE05; @AK16] $$\label{ch1}
Z_n^{(N_f,\nu)}
=
\frac{1}{n!}\left( \prod_{i=1}^n \int_0^{\infty}
\rmd x_i \, x_i^{\nu}\rme^{-x_i}\prod_{f=1}^{N_f}(x_i+m_f^2)\right)\Delta(\mathbf{x})^2,$$ where $N_f>0$ is the number of fermionic quark flavors, $m_f$ $(f=1,\ldots,N_f)$ are the corresponding masses, and $\nu\geq 0$ is the topological charge of the physical sector under consideration.
This partition function can be determined in terms of Laguerre polynomials (see [@AK16] and the references therein). For example, it is clear that the reduced case with $m_f=0$ for all $f$ gives $$\label{ch1r}
\mathcal{Z}_n^{(N_f,\nu)}
=
\frac{1}{n!}\left( \prod_{i=1}^n \int_0^{\infty}
\rmd x_i \, x_i^{\nu+N_f}\rme^{-x_i}\right)\Delta(\mathbf{x})^2,$$ which is directly associated to the family of Laguerre polynomials $L^{(\nu+N_f)}_n(x)$. Consequently, non-Hermitian versions of (\[ch1r\]), like those arising from (\[ch1\]) with bosonic quarks ($N_f<0$) [@SP06], will involve families of generalized Laguerre polynomials and may exhibit non-perturbative effects similar to the Penner model with negative coupling constant.
Appendix A. The Barnes $G$ function {#appendix-a.-the-barnes-g-function .unnumbered}
===================================
This Appendix collects the properties that we need about the Barnes $G$ function [@BA00; @OL10; @AD01]. The Barnes $G$ function is the entire function defined by the canonical product $$G(1+z) = (2\pi)^{z/2}
\rme^{-\frac{1}{2}(z+z^2(1+\gamma))}
\prod_{k=1}^\infty \left(1+\frac{z}{k}\right)^k \rme^{-z+z^2/2k}.$$ The Stirling-like asymptotic expansion of $G(1+z)$ for $z=x$ positive and large is $$\begin{aligned}
\ln G(1+x) & \sim & \frac{1}{2}x^2 \ln x-\frac{3}{4}x^2+\frac{x}{2} \ln(2\pi)-\frac{1}{12}\ln x \nonumber\\
& & {}+ \zeta'(-1) + \widetilde{\varphi}(x),\qquad\mathrm{as}\quad x\to \infty,
\label{id1}\end{aligned}$$ where $\widetilde{\varphi}(x)$ is the asymptotic negative power series $$\label{sas}
\widetilde{\varphi}(x)
=
\sum_{m=2}^{\infty}\frac{B_{2m}}{2m(2m-2)}\frac{1}{x^{2m-2}}.$$ Incidentally, we mention here that the asymptotic series $\widetilde{\varphi}(x)$ provides the connection between Penner models and string theory [@GR90], because $\widetilde{\varphi}(\rmi\mu)$ determines the genus expansion $\widetilde{F}_{c=1}(\mu)$ of the free energy of the $c=1$ string theory at the self-dual radius. The sector of validity of equation (\[id1\]) is often quoted as $-\pi<\arg z<\pi$ or, more precisely, $|\arg z|\leq \pi-\delta$ with $\delta >0$ (see equation 5.17.5 in [@OL10]). We want to emphasize that the expansion (\[id1\]) is asymptotic in this sector only in the sense of Poincaré.
To evaluate the $G$ function for large negative real values of the argument we use the reflection formula (see equation (6) of [@AD01]) to obtain $$\begin{aligned}
\label{bneg}
\ln |G(1-x)| = \ln G(1+x) + x \ln\left| \frac{\sin(\pi x)}{\pi} \right| + \frac{1}{2\pi}\cltwo(2\pi x),\end{aligned}$$ where $\mathrm{Cl}_2(x)$ is the Clausen function $$\label{cla}
\cltwo(x) = - \int_0^x \ln\left| 2 \sin \frac{\tau}{2} \right |\rmd\tau
= \sum_{m=1}^{\infty}\frac{\sin(m x)}{m^2}.$$ Note that equation (6) in [@AD01] is limited to $0<x<1$.
Thus, from (\[id1\]) and (\[bneg\]) we obtain $$\begin{aligned}
\label{nxb}
\fl & &\ln|G(1-x)|- x \ln\left| \frac{\sin(\pi x)}{\pi} \right| -\frac{1}{2\pi} \cltwo(2\pi x)
\sim \nonumber\\
& & \qquad\frac{1}{2}x^2 \ln x- \frac{3}{4}x^2 +\frac{x}{2}\ln(2\pi) -\frac{1}{12}\ln x+\zeta'(-1)
+
\widetilde{\varphi}(x),\nonumber\\
& & \qquad\qquad x\rightarrow\infty. \end{aligned}$$ The second term in the left-hand side of (\[nxb\]) cancels the singularities of $\ln|G(1-x)|$ at positive integer values of $x$, and therefore the left-hand side of (\[nxb\]) has a well-defined asymptotic expansion. However, in order to have a well-defined large-$n$ free energy of the Penner model with negative coupling constant, we need to control the behavior of $\ln|G(1-x)|$ as $x$ becomes large. Obviously the terms $x \ln\left| \sin(\pi x)/\pi \right|$ and $\cltwo(2\pi x)/2\pi$ in (\[nxb\]) are the origin of the $\mathcal{F}^{(\rm osc)}$ in (\[nopf\]) and $l$-dependent contributions in (\[nopfp\]) arising in the planar free energy of the Penner model. Therefore we have to restrict the way in which the coupling constant $g$ tends to zero (and consequently $x$ to infinity) in such a way that these terms give well-defined contributions to the planar limit. This is the ultimate reason for requiring the existence of the limit (\[eq:l\]).
Appendix B. The topological expansion of the Penner model with positive coupling constant {#appendix-b.-the-topological-expansion-of-the-penner-model-with-positive-coupling-constant .unnumbered}
=========================================================================================
The large $n$ expansion of the Penner model with positive coupling constants for ’t Hooft sequences can be readily obtained from equations (\[ane0\]) and (\[id1\]), $$\begin{aligned}
\label{top}
F_n(t) & = & - \frac{\ln|Z_n(g)|}{n^2} \nonumber\\
& \approx & -\left(\frac{(t+1)^2}{2t^2} \ln (1+t)-\frac{3}{4}-\frac{1}{2 t}\right)
+ \frac{1}{12 n^2}\ln (1+t) \nonumber\\
& & {}-\sum_{k=2}^\infty \frac{B_{2k}}{2k(2k-2)} n^{-2k}t^{2k-2}\left( (1+t)^{2-2k}-1 \right)\nonumber\\
& = &-\sum_{k \geq 0}^{\infty}n^{-2k}\sum_{s>0, 2-2k-s<0}
\frac{(-1)^s (2k+s-3)! (2k-1)}{(2k)! s!}B_{2k}\,t^{2k+s-2}\nonumber\\
& & \quad n\rightarrow \infty.\end{aligned}$$ Alternatively, the standard perturbative method applied to (\[ane0\]) leads to a topological expansion of the form [@PE88; @MU98] $$\label{top2}
F_n(t)\approx -\sum_{k \geq 0}^{\infty}n^{-2k}\sum_{ s>0, 2-2k-s<0}\chi_{k,s}\,t^{2k+s-2},$$ where $\chi_{k,s}$ is the virtual Euler characteristic of the space of Riemann surfaces of genus $k$ with a finite number $s$ of punctures. Then equations (\[top\]) and (\[top2\]) imply [@PE88] $$\label{chii}
\chi_{k,s} = \frac{(-1)^s(2k+s-3)!(2k-1)}{(2k)!s!}B_{2k}.$$
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Prof. A. Martínez Finkelshtein for calling our attention to many nice results on zero asymptotics of Laguerre and Jacobi polynomials. The financial support of the Spanish Ministerio de Economía y Competitividad under Project No. FIS2015-63966-P is gratefully acknowledged.
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|
---
abstract: |
We address the classic problem of stability and asymptotic stability in the sense of Lyapunov of the equilibrium point of autonomic differential equations using discrete approach. This new approach includes a consideration of a family of hypersurfaces instead of the Lyapunov functions, and conditions on the right part of the differential equation instead of conditions on a Lyapunov function along trajectories of the equation.
In this paper we generalize results of [@SharkoYu2005; @SharkoYu2010].
address: 'Inst. of Math., NAS of Ukraine, Kiev'
author:
- 'E. Polulyakh'
- 'V. Sharko'
- 'I. Vlasenko'
bibliography:
- 'V-P-Sh-en-4.bib'
title: Discrete conditions of Lyapunov stability
---
lyapunov stability ,discrete conditions
93D05
Introduction.
=============
Consider a system of differential equations $$\label{eq_system_1}
d{\bf x}/dt = {\overrightarrow{f}}({\bf x}), \;\;\;\; x(0) = x_0,$$ which is defined in a neighbourhood $G$ of $x_0$ from ${\mathbb R}^n$ and such that $x_0$ is its equilibrium point.
Suppose that ${\overrightarrow{f}}({\bf x})$ is a $C^1$-smooth function. Denote by $x_p(t)$ the solution of the system where $p$ is a point such that $x_p(0)=p$.
\[defn\_lse\] The equilibrium $x_o$ of the above system is said to be *Lyapunov stable*, if, for every $\epsilon > 0$, there exists $\delta = \delta(\epsilon) > 0$ such that, if $\|x_p(0)-x_0\| < \delta$, then $\|x_p(t)-x_o\| < \epsilon$, for every $t \geq 0$.
The classic method to prove that an equilibrium of the system is Lyapunov stable is to build a Lyapunov function for that system, [@RushAbetsLalua80].
We present a new practical method of proving the Lyapunov stability of a equilibrium point using a sequence of nested hypersurfaces. A Lyapunov function always exists in the case when the equilibrium is asymptotically stable, see [@Wilson67], or in the case of orbital stability, see [@Sharkovsky70]. However, it is not the case when the equilibrium is Lyapunov stable but not asymptotically stable. The paper [@Polulyakh2012] presents an example of a dynamical system such that its critical point is Lyapunov stable but no Lyapunov function exist in a neighbourhood of that critical point. Still, the method of sequences of nested hypersurfaces presented in this paper works for the system in [@Polulyakh2012]. Sequences of nested hypersurfaces can be naturally viewed as a generalization of Lyapunov functions. A Lyapunov funcfion naturally provides a continuous foliation of its level surfaces and scalar product with its gradient vector field. A countable subsequence of level surfaces naturally act as a sequence of nested hypersurfaces in proposed method. To further provide a link between Lyapunov functions and sequences of nested hypersurfaces we define $L$-functions which act as generators of sequences of nested hypersurfaces and use them to prove the existence theorem for the sequences of nested hypersurfaces and study stability of critical points of gradient systems.
The approach that uses a discrete sequence of nested hypersurfaces instead of a Lyapunov function was introduced in the papers [@SharkoYu2005; @SharkoYu2010]. Our article generalizes the results of these papers.
Authors want to express their gratitude to A. N. Sharkovsky for his valuable remarks and to S. I. Maksimenko for his help with the article.
Sequences of converging nested hypersurfaces.
=============================================
Denote by ${\mathbb R}^n$ an $n$-dimensional Euclidian space. Let $\rho$ be the standard metrics on this space. For a bounded set $A \subset {\mathbb R}^n$ we write $${\mathop{\mathrm{diam}}\nolimits}(A) = \sup_{\mathbf{x}, \mathbf{y} \in A} \rho(\mathbf{x}, \mathbf{y}) \,.$$
\[defn\_surf\]Let $H^{n-1} \subset {\mathbb R}^n$ be a connected closed hypersurface (smooth compact submanifold of dimension $n-1$ which has empty boundary). Let us say that $H^{n-1}$ [**bounds**]{} a point $p$ in ${\mathbb R}^n$ if $p \notin H^{n-1}$ and any path $\gamma$ from $p$ to $\mathbf{x} \in {\mathbb R}^n$ intersects $H^{n-1}$ when $\rho(p, \mathbf{x}) > {\mathop{\mathrm{diam}}\nolimits}(H^{n-1})$.
Note that a connected closed hypersurface in the Euclidian space is always oriented and splits the space on two components. One of them is bounded, and the other is not [@two_components]. Let us call the bounded component of the complement the *internal component*. It follows that a hypersurface has two different normal vector fields of unit length, one of which is directed towards the internal component, and the other is directed towards the other component.
\[defn\_nested\_seq\] We call a sequence of connected closed hypersurfaces $H^{n-1}_i$ **nested** if $H^{n-1}_{i+1}$ is contained in the internal component of the complement ${\mathbb R}^n \setminus H^{n-1}_{i}$ for every $i \in {\mathbb N}$.
Let $H^{n-1} \subset {\mathbb R}^n$ be a hypersurface that bounds a point $p$ in ${\mathbb R}^n$. Denote by $$d (p, H^{n-1}) = \max_{y\in H^{n-1}} \left \{||p-y||\right\}$$ a Hausdorff distance between a point $p$ and $H^{n-1}$.
\[defn\_surf\_family\_converge\] A sequence of nested hypersurfaces $\{H^{n-1}_i\}$ such that each $H^{n-1}_i$ bounds a point $p$ is said **to converge to** $p$ in ${\mathbb R}^n$ if $d (p, H^{n-1}_i)\to 0$ as $i \to \infty$.
\[defn\_surf\_family\] Let $x_0 \in {\mathbb R}^n$. Let $H^{n-1}_i \subset {\mathbb R}^n$, $i \in {\mathbb N}$, be a sequence of hypersurfaces that bound $x_0$ in ${\mathbb R}^n$.
A sequence of hypersurfaces $H^{n-1}_i$ is said to be *a sequence of nested hypersurfaces that converge to $x_0$ in ${\mathbb R}^n$* if the following conditions hold:
- hypersurfaces $H^{n-1}_i$ are connected;
- hypersurfaces $H^{n-1}_i$ are mutually disjoint;
- each $H^{n-1}_i$ bounds the compact set $K_i$ such that
- $K_i \supset K_j \ni x_0$ when $i < j$;
- $\bigcap_i K_i = \{x_0\}$.
Let us consider an autonomous system of differential equations . For the sake of convenience choose new coordinates that make the equilibrium point $x_0$ the origin.
By $\{{\bf H}_i^{n-1}\}$ denote a sequence of nested hypersurfaces that converge to origin. Let $\vec{N_i}({\bf x})$ be the vector fields of unit length on each ${\bf H}_i^{n-1}$ such that vectors of $\vec{N_i}({\bf x})$ are orthogonal to ${\bf H}_i^{n-1}$ and direct towards the internal component. Naturally, the system generates the smooth vector field ${\overrightarrow{f}}({\bf x})$ on each $\{{\bf H}_i^{n-1}\}$. Denote by $S_i({\bf x})$ the scalar product $<\vec{N_i}({\bf x}), {\overrightarrow{f}}({\bf x})>$. The function $S_i({\bf x})$ is defined for each hypersurface ${\bf H}_i^{n-1}$ and shows how integral trajectories of the system intersect ${\bf H}_i^{n-1}$.
We prove here the following theorem.
\[th:lyapunov\_from\_ge0\] If there exists a sequence $\{{\bf H}_i^{n-1}\}$ such that $\forall i$ we have $ S_i({\bf x}) \ge 0$ then the origin is stable in the sense of Lyapunov for the system .
The proof of Theorem \[th:lyapunov\_from\_ge0\] is given in next section.
This theorem is a strong version of results in [@SharkoYu2005; @SharkoYu2010], where the approach to study stability using a discrete set of nested hypersurfaces was first used.
Let us recall some definitions.
A *regular value* of a smooth function $F$ is a value such that the differential of $F$ is non-zero in every preimage of this value.
The maximal connected subsets (ordered by inclusion) of a nonempty topological space are called the *connected components* of the space.
\[defn\_quasi\_isolated-point\] Suppose $z = F({\bf x})$ is a continuous function defined in a domain $G\subset {\mathbb R}^n$. A point ${\bf y} \in G$ is *quasi-isolated* for the function $z = F({\bf x})$, if $\{{\bf y}\}$ is a connected component of the set $F^{-1}(F({\bf y}))$.
\[proposition:local extremum\] Let $z = F({\bf x})$ be a continuous function defined in a domain $G \subset {\mathbb R}^n (n\geq 2)$ and ${\bf y} \in G$. Then ${\bf y} $ is a local maximum or local minimum for $F$ if and only if $y$ is an isolated point of the set $F^{-1}(F({\bf y}))$.
[*Proof.*]{} Necessity is obvious.
Sufficiency is a consequence of the following arguments. Consider a domain $U \subset G $, such that $F^{-1}(F({\bf y}))\cap U = {\bf y}$. Let ${\bf y_1} $ and ${\bf y_2} $ be two points in $G $ such that $ F({\bf y_1}) > F({\bf y}) > F({\bf y_2})$. Consider a continuous path $\gamma (t)$ in $U $, such that $\gamma (0) = {\bf y_2}$, $\gamma (1) = {\bf y_1}$ and $\gamma(t) \cap {\bf y} = {\varnothing}$. Let $ F({\gamma (t)})$ be the restriction of the function $z = F({\bf x})$ to the path $\gamma (t)$. Since $ F(\gamma (1)) > F(\gamma (0))$ and the function $F(\gamma (t))$ is continuous, then there must be a $t_0$, such that $ F({\gamma (t_0)})= F({\bf y})$. But this is impossible due to the selection of the path $\gamma (t)$.
Therefore for any point ${\bf y_1}\in U$ we have that either $ F({\bf y_1}) > F({\bf y})$ and so $\mathbf{y}$ is a local minimum, or $F({\bf y_1}) < F({\bf y})$ and then $\mathbf{y}$ is a local maximum. $\square$
\[proposition:quasi\_isolated-point is critical point\] Suppose $z = F({\bf x})$ is a $C^r$-smooth function defined in the domain $G\subset {\mathbb R}^n$ and point ${\bf y} \in G$ is quasi-isolated for the function $z = F({\bf x})$. Then ${\bf y}$ is a critical point of the function $z = F({\bf x})$.
[*Proof.*]{} Suppose $z$ is not critical for $F$. Then there exist change of coordinates in a neighborhood $U \cap G$ of the point ${\bf y} $, such that our function will be linear function in $U$. But this contradicts to the assumption ${\bf y} $ is a quasi-isolated point. $\square$
\[defn\_L\_function\] Suppose $z = F({\bf x})$ is a $C^r$-smooth function defined in a domain $ G\subset {\mathbb R}^n$ and ${\bf y} \in G$. Let $ F({\bf y}) = a$. The function $z = F({\bf x})$ is called an $\bf {L}$-function for the point ${\bf y}$ if there exists a sequence $(a_i)$ of regular values of $z = F({\bf x})$ with the following properties:
- $a_i \rightarrow a$ when $i \rightarrow \infty$;
- for each $i$ there exists a connected component ${\bf
H}_{i}^{n-1}$ of the set $F^{-1}(a_i)$ such that ${\bf H}_{i}^{n-1}$ is a smooth hypersurface that bounds the point ${\bf y}$;
- diameters of ${\bf H}_{i}^{n-1}$ tend to $0$ when $i \rightarrow \infty$.
\[proposition: critical point of L\] Let $F$ be an $\bf {L}$-function for a point ${\bf y} \in G$. Then ${\bf y}$ is the critical point of the function $z = F({\bf x})$.
[*Proof.*]{} Suppose the opposite. Let the point ${\bf y}$ be a regular point of the function $z = F({\bf x})$. Then there exist change of coordinates in a neighborhood $U$ of the point ${\bf y} $, such that our function will be linear function in $U$. This contradicts to the existence of smooth hypersurfaces ${\bf H}_{i}^{n-1}$ that bound the point ${\bf y}$. $\square$
Note that if $z = F({\bf x})$ is a smooth $\bf {L}$-function, than it defines not only the sequence of hypersurfaces ${\bf H}_{i}^{n-1}$ but also a gradient vector field $\overrightarrow{{\mathop{\mathrm{grad}}\nolimits}}F({\bf x})$ which can act as an orthogonal vector field $\vec{N_i}({\bf x})$ in the definition of the function $S_i({\bf x})$ introduced above. On each $H^{n-1}_i$ all vectors $\overrightarrow{{\mathop{\mathrm{grad}}\nolimits}}F({\bf x})$ are either directed towards the internal component or they have the opposite direction. Let us assign $\varepsilon(H^{n-1}_i) = +1$ if $\overrightarrow{{\mathop{\mathrm{grad}}\nolimits}}F({\bf x})$ is directed towards the internal component on $H^{n-1}_i$ and $\varepsilon(H^{n-1}_i) = -1$ otherwise.
By $\tilde S_i({\bf x})$ denote the following function $<\varepsilon(H^{n-1}_i) \overrightarrow{{\mathop{\mathrm{grad}}\nolimits}}F({\bf x}), {\overrightarrow{f}}({\bf x})>$.
Let $z = F({\bf x})$ be an $\bf {L}$-function. Fix a sequence of hypersurfaces $\{ {\bf H}_{i}^{n-1} \}$ satisfying Definition \[defn\_L\_function\].
If $\tilde S_i({\bf x})\ge 0$ for any $i$ in every point ${\bf x}$ of submanifold ${\bf H}_{i}^{n-1}\in \{ {\bf H}_{i}^{n-1} \}$ then the origin is stable in the sense of Lyapunov for the system .
[*Proof.*]{} Observe that $\varepsilon(H^{n-1}_i) \overrightarrow{{\mathop{\mathrm{grad}}\nolimits}}F({\bf x})$ is the normal vector field on ${\bf H}_{i}^{n-1}$ and is directed towards its internal component for every $i$. Stability in the sense of Lyapunov for the system is the straightforward consequence of Theorem \[th:lyapunov\_from\_ge0\]. $\square$
Proof of theorem \[th:lyapunov\_from\_ge0\]. {#sec:lyapunov_from_ge0}
============================================
In order to prove theorem \[th:lyapunov\_from\_ge0\] we first need to prove some auxiliary statements.
Suppose $K^n \subset {\mathbb R}^n$ is a compact manifold with a $C^1$-smooth closed boundary hypersurface ${\bf H}^{n-1}$ and with an interior $W = {\mathop{\mathrm{Int}}\nolimits}K^n$. Let $\vec{N}({\bf x})$ be the unit normal vector field on ${\bf H}^{n-1}$ directed towards interior of $K^n$. Let also ${\overrightarrow{f}}({\bf x})$ be a vector field defined in a neighbourhood of $K^n$ such that $S({\bf x})\ge 0$ on ${\bf H}^{n-1}$, where $S({\bf x})$ is the scalar product $<\vec{N}({\bf x}), {\overrightarrow{f}}({\bf x})>$.
According to Long Tubular Flow Theorem (see [@PalisDiMelo]), for any arc of a trajectory of the vector field ${\overrightarrow{f}}({\bf x})$ which is compact and not closed there exists a $C^1$-smooth flow-box containing that arc. Consider a flow-box of an arc of a trajectory of the vector field ${\overrightarrow{f}}({\bf x})$. The boundary of the flow-box consists of three parts: two bases (parts of the boundary that act as cross-sections of the the flow) and the side part that consists of flow lines.
![a flow-box $B$ that intersects ${\bf H}^{n-1}$.[]{data-label="fig:flowbox"}](flowbox)
\[lm:flow\_box\_measure\_0\] Let $B$ be a flow-box of an arc of a trajectory of the vector field ${\overrightarrow{f}}({\bf x})$. Suppose that $B$ intersects ${\bf H}^{n-1}$ so that the bases of $B$ does not intersect ${\bf H}^{n-1}$. (See fig. \[fig:flowbox\]). Let $T$ be the set of points of the hypersurface ${\bf H}^{n-1}$ where flow lines are tangent to $H^{n-1}$. Let also $p_1$ and $p_2$ be projections of $T$ along the flow lines on the bases of $B$. Then the Lebesgue measures of $T_1 = p_1(T)$, $T_2 = p_2(T)$ in the corresponding bases are 0.
[*Proof.*]{} We can completely ignore the tangent points that belong to the part of the boundary of $B$ that consists of flow lines because projections of those points on the cross-section bases of $B$ have zero Lebesgue measure. Consider the set $T_0 = T \cap {\mathop{\mathrm{Int}}\nolimits}B$. The intersection of ${\mathop{\mathrm{Int}}\nolimits}B$ and the hypersurface ${\bf H}^{n-1}$ is open in ${\bf H}^{n-1}$. Hence, the intersection ${\mathop{\mathrm{Int}}\nolimits}P\cap {\bf H}^{n-1}$ is a submanifold in ${\bf H}^{n-1}$. Projection $p_1$ of the set $T_0$ along flow lines on a cross-section base is a smooth map. The smoothness of $p_1$ is the same as the smoothness of the flow-box $B$. Therefore, the set of singularities of $p_1$ has zero Lebesgue measure according to the Sard’s Theorem [@Sternberg]. The same arguments hold for $p_2$.
It is obvious that the set of singularities of $p_1$ (respectively, of $p_2$) coinsides with the intersection of the set of tangent points of hypersurface ${\bf H}^{n-1}$ to the flow lines with the interior of the flow-box. By definition, the flow-box $B$ is diffeomorphic to a cylinder $C=D^{n-1}\times I$ in ${\mathbb R}^n$ by a diffeomorphism $h$ such that $h$ maps flow lines into coordinate lines and cross-section bases of $B$ into discs on coordinate hyperplanes. Suppose $p_2$ is the projection of the set $h\left({\bf H}^{n-1}\cap {\mathop{\mathrm{Int}}\nolimits}P\right)\subset C\subset {\mathbb R}^n$ along the coordinate lines on a base of the cylinder $C$. Since $p_1= h^{-1}\circ p_2 \circ h$, then $h$ maps the set of singularities of $p_1$ into the set of singularities of $p_2$. But $p_2$ is a coordinate projection in ${\mathbb R}^n$. Therefore, the set of singularities of $p_2$ is the set of points of the image of ${\bf H}^{n-1}$ in $C$ that are tangent to the coordinate lines of ${\mathbb R}^n$. $h$ preserves transversality because it is is diffeomorphism. Hence, the set of singularities of $p_1$ that has zero Lebesgue measure is the set of points of ${\bf H}^{n-1}$ tangent to the flow lines. This completes the proof. $\square$
\[th:lyapunov\_boundness\_surface\]
If $S({\bf x})\ge 0$ on ${\bf H}^{n-1}$ then any trajectory of the vector field ${\overrightarrow{f}}({\bf x})$ does not leave the manifold $K^n$.
Assume the converse. Then there exists an integral trajectory $\xi$ of the vector field ${\overrightarrow{f}}({\bf x})$ such that $\xi$ leaves $K^n$. In other words, $\xi \cap K^n\not={\varnothing}$ and $\xi \cap \left( {\mathbb R}^n \setminus K^n \right)\not={\varnothing}$. Since ${\bf H}^{n-1}$ is a boundary of $K^n$, $\xi \cap {\bf H}^{n-1}\not={\varnothing}$ as well. Note that $\xi$ can not be an equilibrium point of the vector field ${\overrightarrow{f}}({\bf x})$, because in this case $\xi$ is just a point and can go nowhere. Therefore, ${\overrightarrow{f}}({\bf x})$ is a non-zero vector field along $\xi$ and in a neighbourhood of $\xi$.
Choose a flow-box $B$ of the trajectory $\xi$ such that the “out” base of $B$ (i. e. the part of $\partial P$ the trajectories of $B$ are going out through) does not intersect ${\bf H}^{n-1}$. It is always possible because $\xi$ leaves $K^n$. Denote the “out” base of $B$ by $B_{\rm\textsc{out}}$ and the “in” base of $B$ by $B_{\rm\textsc{in}}$. Then, the “out” base $B_{\rm\textsc{out}}$ must be outside $K^n$. (See fig. \[fig:lemma\]).
![Illustration to the proof of Lemma\[th:lyapunov\_boundness\_surface\].[]{data-label="fig:lemma"}](lemma32)
Since ${\overrightarrow{f}}({\bf x})$ is a non-zero vector field along $\xi$, then $\xi$ should travel some time either in $W$ or on ${\bf H}^{n-1}$ before leaving $K^n$. In the former case, $B_{\rm\textsc{in}}$ can be chosen to be inside of $W$ similarly to $B_{\rm\textsc{out}}$. Consider the latter case. Pick a point $s\in \xi \cap {\bf H}^{n-1}$. Choose $B$ to be long enough to contain $s$ inside. $s\in {\bf H}^{n-1}$ and ${\bf H}^{n-1}=\partial K^n$. Hence, in any neighbourhood of $s$ there is an open set of points that belongs to $W={\mathop{\mathrm{Int}}\nolimits}K$. But ${\mathop{\mathrm{Int}}\nolimits}B$ is also the neighbourhood of $s$ in ${\mathbb R}^n$. Therefore, we can always adjust $B_{\rm\textsc{in}}$ to contain the open in $B_{\rm\textsc{in}}$ set of points that belong to $W$.
As a result, in both cases $B_{\rm\textsc{in}}$ can be at least chosen to have an open (in $B_{\rm\textsc{in}}$) non-empty intersection with $W$.
Since the intersection $B_{\rm\textsc{in}} \cap W$ is open in $B_{\rm\textsc{in}}$ there exists a point $p_1\in B_{\rm\textsc{in}}$ such that a whole neighbourhood $V(p_1)\ni p_1$ is also contained in $B_{\rm\textsc{in}}\cap W$. Choose a new flow-box $B^1\subset B$ for the trajectory $\xi_1$ of the point $p_1$ with bases $B^1_{\rm\textsc{in}}$ and $B^1_{\rm\textsc{out}}$ such that $B^1_{\rm\textsc{in}}\subset V(p_1)\subset B_{\rm\textsc{in}}\cap W$ and $B^1_{\rm\textsc{out}} \subset B_{\rm\textsc{out}}$. By construction, both bases of $B^1$ do not intersect ${\bf H}^{n-1}$.
Applying Lemma \[lm:flow\_box\_measure\_0\], we conclude that the set $T
\subset B^1_{\rm\textsc{in}}$ of points whose trajectories always intersect ${\bf H}^{n-1}$ transversally have the full Lebesgue measure in $B^1_{\rm\textsc{in}}$. Consider a point $p_2\in T$ such that the trajectory $\xi_2$ of $p_2$ always intersects ${\bf H}^{n-1}$ transversally. By construction, $\xi_2$ intersects $B^1_{\rm\textsc{in}}$ inside of $W$ and intersects $B^1_{\rm\textsc{out}}$ outside of $K$. Also, $\xi_2$ intersects ${\bf H}^{n-1}$ in a finite number of points because the intersection is always transversal. In those points $S({\bf x})\not= 0$ due to transversality. But the hypersurface ${\bf H}^{n-1}$ divides ${\mathbb R}^n$. Then $\xi_2$ can only leave $K$ at a point of intersection with ${\bf
H}^{n-1}$. Furthermore, this point of intersection is transversal due to the choice of $\xi_2$. But then $S(p_2)< 0$ that contradicts the conditions of the theorem. This contradiction proves the theorem.$\square$ Using Lemma \[th:lyapunov\_boundness\_surface\] we can finish the proof of Theorem \[th:lyapunov\_from\_ge0\].
**Proof.** By assumption, $ S({\bf x})\ge 0$ on every hypersurface ${\bf H}_{i}^{n-1}$. According to Lemma \[th:lyapunov\_boundness\_surface\], for every hypersurface ${\bf H}_{i}^{n-1}$ any trajectory ${\bf
X}(t)$, $t\ge 0$, of the system that begins in a point ${\bf x}\in {\bf H}_{i}^{n-1}$ does not leave the manifold $K_{i}$ whose boundary is ${\bf H}_{i}^{n-1}$. As a consequence, any other trajectory that starts at a point of $K_{i}$ can not leave $K_{i}$, because otherwise that trajectory would intersect ${\bf H}_{i}^{n-1}$.
As diameters of ${\bf H}_{i}^{n-1}$ tend to $0$ when $i \rightarrow \infty$ then stability of the origin in the sense of Lyapunov directly follows from the definition of Lyapunov stability. $\square$
On existence of sequences of nested hypersurfaces.
==================================================
The theorem below shows how to build sequences of nested hypersurfaces converging to a point using a smooth enough function on an open domain $G \subset {\mathbb R}^n$.
\[th\_surf\_family\] Let $G$ be a domain in ${\mathbb R}^n$. Assume that $F \in C^n(G)$.
Then $F$ is an $\bf{L}$-function for $x_0 \in G$ if and only if $x_0$ is a quasi-isolated point of $F$.
The proof of this theorem is given in section \[sec:th\_surf\_family\_proof\].
\[corr:non-isolated\_crit\_point\] Suppose $z = F({\bf x})$ is a $C^n$-smooth function defined in a domain $G\subset {\mathbb R}^n$. Denote by $\Sigma(F({\bf x}))$ the set of critical points of the function $z = F({\bf x})$.
Let ${\bf y} \in G$ be a quasi-isolated point for the function $z = F({\bf x})$. Assume also that $\mathbf{y}$ is not an isolated point of the level set $F^{-1}(F({\bf y}))$.
Then $\mathbf{y} \in \Sigma(F({\bf x}))$ and $\mathbf{y}$ is not an isolated point of the set $\Sigma(F({\bf x}))$.
[*Proof.*]{} Observe that $\mathbf{y}$ is a critical point of $F$ according to Proposition \[proposition:quasi\_isolated-point is critical point\].
Let $\mathbf{y}$ be an isolated critical point. Then there exists an $\varepsilon > 0$ such that the open ball $U = \{ \mathbf{x} \in {\mathbb R}^n \,|\, \rho(\mathbf{x}, \mathbf{y}) < \varepsilon \} \subset G$ does not contain other critical points of $F$.
Since $\mathbf{y}$ is quasi-isolated, it follows from Theorem \[th\_surf\_family\] that $F$ is an $\bf{L}$-function for $\mathbf{y}$. Hence there exist a regular value $a$ of $F$ and a hypersurface $H^{n-1} \subset U \cap F^{-1}(a)$, such that $\mathbf{y}$ is contained in the inner component $W$ of ${\mathbb R}^n \setminus H^{n-1}$. It is easy to see that ${\mathop{\overline{W}}\nolimits} \subset U$. Observe also that $\mathbf{y} \notin F^{-1}(a)$ because $\mathbf{y}$ is critical point of $F$ and can not be contained in a regular level set.
The compact set ${\mathop{\overline{W}}\nolimits}$ has the interior $W$ and the frontier $H^{n-1}$. The function $F$ is continuous on ${\mathop{\overline{W}}\nolimits}$, so it achieves its maximum and minimum values on ${\mathop{\overline{W}}\nolimits}$. Let $$M = \max_{\mathbf{x} \in {\mathop{\overline{W}}\nolimits}} F(\mathbf{x})\,, \quad
m = \min_{\mathbf{x} \in {\mathop{\overline{W}}\nolimits}} F(\mathbf{x})\,.$$
If $m = M$, then $F$ is constant on ${\mathop{\overline{W}}\nolimits}$, which is impossible, since $H^{n-1} \subset F^{-1}(a)$ and $\mathbf{y} \notin F^{-1}(a)$. Therefore, either $m \neq a$, or $M \neq a$.
Let us suppose that $m \neq a$.
Obviously, ${\varnothing}\neq F^{-1}(m) \cap {\mathop{\overline{W}}\nolimits} \subset W$ and $(F^{-1}(m) \cap {\mathop{\overline{W}}\nolimits}) \subset \Sigma(F({\bf x}))$ since every point of this set is a local minimum of $F$. Therefore, if $F(\mathbf{y}) \neq m$ then $U$ contains other critical points of $F$ distinct from $\mathbf{y}$. If $F(\mathbf{y}) = m$, then the set $F^{-1}(m) \cap {\mathop{\overline{W}}\nolimits} = F^{-1}(F(\mathbf{y})) \cap W$ is the subset of $\Sigma(F({\bf x}))$ and also contains more than one point because $\mathbf{y}$ is not an isolated point of its level set. This contradicts to our initial assumption that $\mathbf{y}$ is an isolated critical point of $F$.
The case $M \neq a$ is considered similarly.
From the arbitrariness in the choice of $\varepsilon > 0$ we conclude that $\mathbf{y}$ is not an isolated point of the set $\Sigma(F({\bf x}))$. $\square$
With the help of technique from [@Dancer] it can be proved that Corollary \[corr:non-isolated\_crit\_point\] is valid for $F \in C^r(G)$, $r \geq 2$. But this is outside the scope of the current discussion.
Now we shall derive some consequences from Theorem \[th\_surf\_family\]
Let $G$ be a domain in ${\mathbb R}^n$ and let $F \in C^2(G)$. Consider the *gradient system* of $F$ on $G$ $$\frac{dx}{dt} = - {\mathop{\mathrm{grad}}\nolimits}F(x)$$ where $${\mathop{\mathrm{grad}}\nolimits}F(x) = \left( \frac{\partial F}{\partial x_1}, \ldots, \frac{\partial F}{\partial x_n} \right)^T \,.$$
\[theorem\_gradient\_system\] Let $G$ be a domain in ${\mathbb R}^n$. Suppose $F \in C^n(G)$ and $x_0 \in G$ be a connected component of the level set $F^{-1}(F(x_0))$.
Then either the gradient system of $F$ or of $-F$ on $G$ is Lyapunov stable in $x_0$.
It follows from Theorem \[th\_surf\_family\] that we can select a sequence $\{a_i\}_{i \in {\mathbb N}}$ of regular values of $F$ that converges to $F(x_0)$ and a sequence $\{H^{n-1}_i\}_{i \in {\mathbb N}}$ of nested connected hypersurfaces that converge to $x_0$ and such that each $H^{n-1}_i$ is a connected component of $F^{-1}(F(a_i))$.
Let $\vec{N_i}({\bf x})$, ${\bf x} \in H^{n-1}_i$, be a unique normal vector field of unit length on $H^{n-1}_i$ such that it directs towards the internal component of the complement ${\mathbb R}^n \setminus H^{n-1}_i$. It is obvious that vectors ${\mathop{\mathrm{grad}}\nolimits}F({\bf x})$ and $\vec{N_i}({\bf x})$ are collinear for all ${\bf x} \in H^{n-1}_i$, $i \in {\mathbb N}$.
Since each $a_i$ is regular value of $F$, then ${\mathop{\mathrm{grad}}\nolimits}F({\bf x}) \neq 0$ for all ${\bf x} \in H^{n-1}_i$, $i \in {\mathbb N}$. Therefore, $S({\bf x}) = <-{\mathop{\mathrm{grad}}\nolimits}F({\bf x}), \vec{N_i}({\bf x})> \neq 0$ for all ${\bf x} \in H^{n-1}_i$, $i \in {\mathbb N}$. Function $S({\bf x})$ is nonzero and continuous on each connected set $H^{n-1}_i$, consequentely it is sign-definite on each $H^{n-1}_i$.
Let us consider two subsequences of $\{H^{n-1}_i\}_{i \in {\mathbb N}}$ $$\begin{aligned}
S_{+} & = \{H^{n-1}_i \;|\; S({\bf x}) > 0 \textrm{ on } H^{n-1}_i \}\,, \\
S_{-} & = \{H^{n-1}_i \;|\; S({\bf x}) < 0 \textrm{ on } H^{n-1}_i \}\,.
\end{aligned}$$ At least one of these subsequences contains an infinite number of elements.
If $|S_{+}| = \infty$, then we can take $S_{+}$ and apply Theorem \[th:lyapunov\_from\_ge0\] to the system $\frac{d {\bf x}}{dt} = - {\mathop{\mathrm{grad}}\nolimits}F({\bf x})$.
If $|S_{-}| = \infty$, then we take $S_{-}$ and observe that $<-{\mathop{\mathrm{grad}}\nolimits}(-F({\bf x})), \vec{N_i}({\bf x})> = -S({\bf x}) > 0$ on every element $H^{n-1}_i$ of $S_{-}$. Therefore, applying Theorem \[th:lyapunov\_from\_ge0\] to the system $\frac{d {\bf x}}{dt} = - {\mathop{\mathrm{grad}}\nolimits}(-F({\bf x}))$ we conclude that this system is Lyapunov stable in $x_0$.
Let $G$ be a domain in ${\mathbb R}^{2n}$ and let $F \in C^2(G)$ where $F = F(y, z)$ with $y, z \in {\mathbb R}^n$. Consider the *Hamiltonian system* with $n$ degrees of freedom on $G$ $$\begin{aligned}
\frac{dy}{dt} & = \frac{\partial F}{\partial z} \\
\frac{dz}{dt} & = -\frac{\partial F}{\partial y} \,,
\end{aligned}$$ where $$\begin{aligned}
\frac{\partial F}{\partial y} & = \left( \frac{\partial F}{\partial y_1}, \ldots, \frac{\partial F}{\partial y_n} \right)^T \\
\frac{\partial F}{\partial z} & = \left( \frac{\partial F}{\partial z_1}, \ldots, \frac{\partial F}{\partial z_n} \right)^T \,.
\end{aligned}$$
\[theorem\_Hamiltonian\_system\] Consider a domain $G \in {\mathbb R}^{2n}$. Let $F \in C^{2n}(G)$ and $x_0 \in G$ be a connected component of the level set $F^{-1}(F(x_0))$.
Then the corresponding Hamiltonian system on $G$ is Lyapunov stable in $x_0$.
Let us denote $${\bf x} = (y, z) \,, \quad {\overrightarrow{f}}({\bf x}) = \left( \left(\frac{\partial F}{\partial z}\right)^T, \left(-\frac{\partial F}{\partial y}\right)^T \right)^T \,.$$ In this notation our system has the form .
We make use of Theorem \[th\_surf\_family\] and select a sequence $\{H^{n-1}_i\}_{i \in {\mathbb N}}$ of nested connected hypersurfaces that converges to $x_0$ and such that each $H^{n-1}_i$ is contained in a level set of $F$. Let $\vec{N_i}({\bf x})$, ${\bf x} \in H^{n-1}_i$, be a normal vector field of unit length on $H^{n-1}_i$ such that it directs towards the internal component of the complement ${\mathbb R}^{2n} \setminus H^{n-1}_i$.
It is known that the trajectories of Hamiltonian system lie on the level surfaces of $F$. Therefore $S_i({\bf x}) = \;<\vec{N_i}({\bf x}), {\overrightarrow{f}}({\bf x})>\; = 0$ for every ${\bf x} \in H^{n-1}_i$, $i \in {\mathbb N}$, and we are in the conditions of Theorem \[th:lyapunov\_from\_ge0\]. Applying it we conclude that $x_0$ is stable in the sense of Lyapunov for our Hamiltonian system.
So, it turns out that in order to check Lyapunov stability of gradient or Hamiltonian systems at a critical point $x_0 \in G$ of a function $F \in C^n(G)$ it suffices to verify that this point is the connected component of its level set $F^{-1}(F(x_0))$.
Proof of Theorem \[th\_surf\_family\]. {#sec:th_surf_family_proof}
======================================
Before we proceed with the proof of theorem, let us consider some necessary auxiliary statements.
On closed hypersurfaces in ${\mathbb R}^{n}$.
---------------------------------------------
Let $X$ be a metric space and $A \subset X$. We denote by $LC(A)$ a set of all $x \in {\mathop{\overline{A}}\nolimits}$ with the following property: there exists an open neighbourhood $G$ of $x$ of an arbitrary small diameter such that $G \cap A$ is connected.
Let $\rho$ be a metrics in ${\mathbb R}^{n}$. We designate by $U_{\varepsilon}(A)$ the $\varepsilon$-neighbourhood of a set $A \subset {\mathbb R}^{n}$: $$U_{\varepsilon}(A) = \{ x \in {\mathbb R}^n \,|\, \inf_{y \in A}(\rho(x, y) < \varepsilon \} \,.$$
\[lemma\_connected\_collar\] Let $W$ be a domain in ${\mathbb R}^{n}$. Suppose the frontier $R = {\mathop{\mathrm{Fr}}\nolimits}(W)$ is connected and $R \subset LC(W)$.
Then $W_{\varepsilon} = W \cap U_{\varepsilon}(R)$ is connected for every $\varepsilon > 0$.
Fix $\varepsilon > 0$.
Let $x_1$, $x_2 \in W_{\varepsilon}$. Let $x^0_1$ and $x^0_2$ be the closest points of $R$ to $x_1$ and $x_2$ accordingly. Let also $\gamma_1, \gamma_2 : {\mathbb R}\to {\mathbb R}^n$ be continuous curves which comply with the correlations
- $\gamma_1(0) = x_1$, $\gamma_1(1) = x^0_1$, $\gamma_2(0) = x_2^0$, $\gamma_2(1) = x_2$;
- $\gamma_1[0, 1) \cup \gamma_2(0, 1] \subset W_{\varepsilon}$.
We can take for instance $\gamma_1(t) = (1-t) x_1 + t x_1^0$, $\gamma_2(t) = (1-t) x_2^0 + t x_2$, $t \in I$.
We use the inclusion $R \subset LC(W)$ from condition of lemma and choose for every $x \in R$ an open neighbourhood $G(x)$ in ${\mathbb R}^n$ which is contained in $U_{\varepsilon}(x)$ and such that the set $V(x) = G(x) \cap W$ is connected. Then $V(x) \subset W_{\varepsilon}$, $x \in R$.
It is known (see [@Kuratowski]) that for an open cover of a connected space we can connect every pair of points of this space by a finite chain which consists of elements of this cover. Thus, with a pair of points $x_1^0$, $x_2^0 \in R$ one can associate a finite set of points $y_1, \ldots, y_s$ such that $x_1^0 \in G(y_1)$, $x_2^0 \in G(y_s)$ and $$G(y_i) \cap G(y_{i+1}) \cap R \neq {\varnothing}\text{ for all } i \in \{1, \ldots, s-1\} \,.$$
Since $x_1^0 \in G(y_1)$, we have that $\gamma_1[0, 1) \cap G(y_1) \neq {\varnothing}$. And from $\gamma_1[0, 1) \subset W_{\varepsilon}$ it follows that $\gamma_1[0, 1) \cap V(y_1) \neq {\varnothing}$. Similarly, $\gamma_2(0, 1] \cap V(y_s) \neq {\varnothing}$.
Let $i \in \{1, \ldots, s-1\}$. The nonempty set $G(y_i) \cap G(y_{i+1}) \cap R$ is contained in ${\mathop{\mathrm{Fr}}\nolimits}W$. Consequently, its neighbourhood $G(y_i) \cap G(y_{i+1})$ in ${\mathbb R}^n$ intersects $W$ and $V(y_i) \cap V(y_{i+1}) \neq {\varnothing}$.
Thus, all members of the union $$R = \gamma_1[0, 1) \cup V(y_1) \cup \ldots \cup V(y_s) \cup \gamma_2(0, 1]$$ are connected and each pair of adjacent sets in this sequence have a common point. Therefore, $R$ is connected set. Furthermore, by construction it lies in $W_{\varepsilon}$ and contains points $x_1 = \gamma_1(0)$ and $x_2 = \gamma_2(1)$.
From the arbitrariness of a choice of $x_1$, $x_2 \in W_{\varepsilon}$ it follows that the set $W_{\varepsilon}$ is connected.
\[corr\_conectedness\] Let $N$ be a connected closed hypersurface in ${\mathbb R}^n$. Let $W$ be a connected component of the complement ${\mathbb R}^n \setminus N$.
Then the intersection $W_{\varepsilon} = W \cap U_{\varepsilon}(N)$ is connected for every $\varepsilon > 0$.
A closed hypersurface in ${\mathbb R}^n$ splits ${\mathbb R}^n$ (see [@two_components]), therefore ${\mathbb R}^n \setminus (W \cup N) \neq {\varnothing}$.
Let us consider following functions. $$\begin{aligned}
\chi(x) & = &
\left\{
\begin{aligned}
1, &\quad\text{when } x \in W \,, \\
-1, &\quad\text{otherwise} \,,
\end{aligned}
\right.
\\
\Phi(x) & = & \rho(x, N) \,,\\
\Psi(x) & = & \chi(x) \cdot \Phi(x) \,.\end{aligned}$$ Obviously, $\Phi$ is continuous in ${\mathbb R}^n$ and both $\chi$ and $\Psi$ are continuous at all points of the open set $({\mathbb R}^n \setminus N) \subseteq ({\mathbb R}^n \setminus {\mathop{\mathrm{Fr}}\nolimits}W)$.
It is clear that $N = \Psi^{-1}(0)$. Furthermore, $\Phi(x) = |\Psi(x)|$, $x \in {\mathbb R}^n$. So, $\Psi^{-1}(-\varepsilon, \varepsilon) = \Phi^{-1}(-\varepsilon, \varepsilon)$ is open for every $\varepsilon > 0$. Hence $\Psi$ is also continuous at every $x \in N$.
Let us examine two subsets of $N$. $$\begin{gathered}
N_1 = \{ x \in N \,|\, \exists\, \varepsilon > 0 : \Psi(y) \leq 0 \quad \forall\, y \in U_{\varepsilon}(x) \} \, \cup \\
\cup \{ x \in N \,|\, \exists\, \varepsilon > 0 : \Psi(y) \geq 0 \quad \forall\, y \in U_{\varepsilon}(x) \} \,,\end{gathered}$$ $$N_2 = \{ x \in N \,|\, \forall\, \varepsilon > 0 \;\exists\, y_1, y_2 \in U_{\varepsilon}(x) : \Psi(y_1) \Psi(y_2) < 0 \} \,.$$
Relations $N_1 \cap N_2 = {\varnothing}$ and $N_1 \cup N_2 = N$ are obviously fulfilled.
By definition of hypersurface for every point $x \in {\mathbb R}^n$ there are a neighbourhood $V_x$ in ${\mathbb R}^n$ and a diffeomorphism $\psi_x : V_x \to {\mathbb R}^n$, such that $\psi_x(V) = {\mathbb R}^n$, $\psi_x(V \cap N) = {\mathbb R}^{n-1} \times \{0\}$.
By construction the sign of $\Psi$ is fixed on each connected component of the complement ${\mathbb R}^n \setminus N$. Therefore every $x \in N$ is contained in one of the sets $N_1$ or $N_2$ together with its neighbourhood $V_x \cap N$, and both $N_1$ and $N_2$ are open in $N$. Moreover, if $x \in N_2$ then exactly one of two components of $V_x \setminus N$ belongs to $W$. Consequently, $N_2 \subset LC(W)$.
According to the condition of this corollary $N$ is connected. Hence either $N_1 = {\varnothing}$ or $N_2 = {\varnothing}$.
It is clear that ${\mathop{\overline{W}}\nolimits} \subseteq (W \cup N)$. Moreover ${\mathop{\mathrm{Fr}}\nolimits}{\mathop{\overline{W}}\nolimits} \neq {\varnothing}$ since ${\mathbb R}^n \setminus (W \cup N) \neq {\varnothing}$. It is straightforward that ${\mathop{\mathrm{Fr}}\nolimits}{\mathop{\overline{W}}\nolimits} = N_2$. So $N_2 \neq {\varnothing}$, hence $N_1 = {\varnothing}$ and $N = N_2 \subset LC(W)$. Finally since ${\mathop{\mathrm{Fr}}\nolimits}{\mathop{\overline{W}}\nolimits} \subseteq {\mathop{\mathrm{Fr}}\nolimits}W \subseteq N$, it follows that $N = {\mathop{\mathrm{Fr}}\nolimits}W$ and we can apply lemma \[lemma\_connected\_collar\].
\[lemma\_complements\_intersection\] Let $N_1$ and $N_2$ be closed hypersurfaces in ${\mathbb R}^n$ such that $N_1 \cap N_2 = {\varnothing}$.
Let $V_1$ and $V_2$ be connected components of the sets ${\mathbb R}^n \setminus N_1$ and ${\mathbb R}^n \setminus N_2$, respectively.
Then $V_1 \cap V_2$ is connected.
Suppose $x_1$, $x_2 \in V_1 \cap V_2$. Let us verify that the set $V_1 \cap V_2$ contains a connected subset which includes $x_1$ and $x_2$.
It is known that open connected subsets of ${\mathbb R}^n$ are arcwise connected. So, $V_1$ and $V_2$ are arcwise connected sets.
Let $\gamma : I \to {\mathbb R}^n$ be a continuous path which connects $x_1$ to $x_2$ in $V_1$, i. e. $\gamma(0) = x_1$, $\gamma(1) = x_2$ and $\gamma(I) \subset V_1$.
Since $N_2$ is a closed hypersurface (i. e. compact and borderless), it has a finite number of connected components. Let us designate them by $N_2^{1}, \ldots, N_2^{m}$.
Denote $\tau_1' = \inf \{ t \in I \,|\, \gamma(t) \in N_2 \}$. Since $N_2$ is closed in ${\mathbb R}^n$, we have that $\gamma(\tau_1') \in N_2$. Moreover $\tau_1' > 0$, as $x_1 = \gamma(0) \in V_1 \cap V_2$. Let $\gamma(\tau_1') \in N_2^{\sigma(1)}$. Write $\tau_1'' = \sup \{ t \in I \,|\, \gamma(t) \in N_2^{\sigma(1)} \}$. Then
- $\gamma(\tau_1'') \in N_2^{\sigma(1)}$,
- $\tau_1'' < 1$, since $x_2 = \gamma(1) \in V_1 \cap V_2$,
- $\gamma(t) \notin N_2^{\sigma(1)}$ for all $t > \tau_1''$.
If $\gamma(\tau_1'', 1] \cap N_2 \neq {\varnothing}$, there exists $\tau_2' = \inf \{ t > \tau_1'' \,|\, \gamma(t) \in N_2 \}$. Observe that $\tau_2' > \tau_1''$. Indeed, the compacts $N_2^{\sigma(1)}$ and $N \setminus N_2^{\sigma(1)}$ have disjoint neighbourhoods, therefore there exists $\varepsilon > 0$, such that $\gamma(t) \in N_2^{\sigma(1)}$ as soon as correlations $t \in \gamma^{-1}(N_2)$ and $|t - \tau_1''| < \varepsilon$ are fulfilled.
As above it is verified that $\gamma(\tau_2') \in N_2^{\sigma(2)}$ for a certain $\sigma(2) \in \{1, \ldots, m\}$. By definition we have $\gamma(\tau_1'', \tau_2') \cap N_2 = {\varnothing}$.
Denote $\tau_2'' = \sup \{ t \in I \,|\, \gamma(t) \in N_2^{\sigma(2)} \}$. Then
- $\gamma(\tau_2'') \in N_2^{\sigma(2)}$,
- $\tau_2'' < 1$,
- $\gamma(t) \notin N_2^{\sigma(1)} \cup N_2^{\sigma(2)}$ for every $t > \tau_2''$.
Suppose that we have already constructed numbers $$\label{eq_parameters}
0 < \tau_1' \leq \tau_1'' < \tau_2' \leq \tau_2'' < \cdots < \tau_k' \leq \tau_k'' < 1 \,,$$ such that
- $\bigl( \gamma[0, \tau_1') \cup \gamma(\tau_1'', \tau_{2}') \cup \cdots \cup \gamma(\tau_{k-1}'', \tau_{k}') \bigr) \cap N_2 = {\varnothing}$;
- $\gamma(\tau_i')$, $\gamma(\tau_i'') \in N_2^{\sigma(i)}$, $i \in \{1, \ldots, k\}$;
- $\gamma(t) \notin N_2^{\sigma(1)} \cup \cdots \cup N_2^{\sigma(i)}$ when $t > \tau_i''$, $i \in \{1, \ldots, k\}$.
Let $\gamma(\tau_k'', 1] \cap N_2 \neq {\varnothing}$. Designate $\tau_{k+1}' = \inf \{ t \in (\tau_k'', 1] \,|\, \gamma(t) \in N_2 \}$.
Since $\gamma(\tau_k'') \in N_2^{\sigma(k)}$ and compacts $$\bigcup_{i = 1}^k N_2^{\sigma(i)} \quad \text{and} \quad N_2 \setminus \Bigl( \bigcup_{i = 1}^k N_2^{\sigma(i)} \Bigr)$$ do not intersect, we obtain that $\tau_{k+1}' > \tau_k''$. It is also clear that $\gamma(\tau_k'', \tau_{k+1}') \cap N_2 = {\varnothing}$.
As the set $N_2$ is compact, we get that $\gamma(\tau_{k+1}') \in N_2$. So there exists $\sigma(k+1)$, such that $\gamma(\tau_{k+1}') \in N_2^{\sigma(k+1)}$. Denote $\tau_{k+1}'' = \sup \{ t \in I \,|\, \gamma(t) \in N_2^{\sigma(k+1)} \}$. Then
- $\gamma(\tau_{k+1}'') \in N_2^{\sigma(k+1)}$;
- $\tau_{k+1}'' < 1$;
- $\gamma(t) \notin N_2^{\sigma(1)} \cup \cdots \cup N_2^{\sigma(i)}$ whenever $t > \tau_i''$, $i \in \{1, \ldots, k+1\}$.
Consequently, the sequence $$0 < \tau_1' \leq \tau_1'' < \tau_2' \leq \tau_2'' < \cdots < \tau_{k+1}' \leq \tau_{k+1}'' < 1$$ complies with properties which are similar to (a)–(c).
Observe that it follows from (b) and (c) that all numbers $\sigma(i)$ are distinct, $\sigma(i) \in \{1, \ldots, m\}$, $i \in \{1, \ldots, k\}$. Therefore, if the sequence complies with the properties (a)–(c), then $k \leq m$. Consequently, there exists $k \leq m$, such that if the sequence satisfies to (a)–(c), then it also meets the following property $$\gamma(\tau_k'', 1] \cap N_2 = {\varnothing}\,.$$
As a matter of convenience we reindex connected components of $N_2$ in order to satisfy equalities $\sigma(i) = i$, $i \in \{1, \ldots, k\}$. Then the sequence meets the following properties:
- $\bigl( \gamma[0, \tau_1') \cup \gamma(\tau_1'', \tau_{2}') \cup \cdots \cup \gamma(\tau_{k-1}'', \tau_{k}') \cup \gamma(\tau_k'', 1] \bigr) \cap N_2 = {\varnothing}$;
- $\gamma(\tau_i')$, $\gamma(\tau_i'') \in N_2^i$, $i \in \{1, \ldots, k\}$;
- $\gamma(t) \notin N_2^{1} \cup \cdots \cup N_2^{i}$ when $t > \tau_i''$, $i \in \{1, \ldots, k\}$.
Notice that all the sets $N_1$, $N_2^1, \ldots, N_2^m$ are disjoint compacts. So there exists an $\varepsilon > 0$ complying with the following equalities: $$\begin{aligned}
U_{\varepsilon}(N_1) \cap U_{\varepsilon}(N_2^i) & = {\varnothing}\,, \quad i \in \{1, \ldots, m\} \,; \\
U_{\varepsilon}(N_2^i) \cap U_{\varepsilon}(N_2^j) & = {\varnothing}\,, \quad i \neq j\,,\quad i, j \in \{1, \ldots, m\} \,.\end{aligned}$$
Let $W_i$, $i \in \{1, \ldots, m\}$, be a connected component of ${\mathbb R}^n \setminus N_2^i$, such that $W_i \cap V_2 \neq {\varnothing}$. It is easy to see that $V_2 \subseteq W_i$ for every $i$. Thus, the component $W_i$ is uniquely determined for each $i$, and moreover $x_1, x_2 \in W_i$, $i \in \{1, \ldots, m\}$.
Observe that the sets $$\begin{aligned}
K_i' & = \gamma[0, \tau_1') \cup \Bigl( \bigcup_{j < i} N_2^j \Bigr) \cup \Bigl( \bigcup_{2 \leq j \leq i} \gamma(\tau_{j-1}'', \tau_j') \Bigr) \,, \\
K_i'' & = \gamma(\tau_i'', 1]\end{aligned}$$ are connected for all $i \in \{1, \ldots, m\}$, since all sets $N_2^j$ are so and conditions (b$'$) are fulfilled.
It is also true that $K_i' \cup K_i'' \subset W_i$, $i \in \{1, \ldots, k\}$. In fact, on one hand it follows from (a$'$) that $K_i' \cap N_2^i = {\varnothing}$, and (c$'$) implies $K_i'' \cap N_2^i = {\varnothing}$; on the other hand, $x_1 = \gamma(0) \in K_i' \cap W_i$ and $x_2 = \gamma(1) \in K_i'' \cap W_i$.
It follows from what has been said that $$\gamma(\tau_{i-1}'', \tau_i') \cup \gamma(\tau_i'', \tau_{i+1}') \subset W_i \,, \quad i \in \{1, \ldots, k\} \,.$$ Together with the condition (b$'$) this results in the inequalities $$\label{eq_intervals}
\gamma(\tau_{i-1}'', \tau_i') \cap W_{i, \varepsilon} \neq {\varnothing}\,, \quad
\gamma(\tau_{i}'', \tau_{i+1}') \cap W_{i, \varepsilon} \neq {\varnothing}\,, \quad
i \in \{1, \ldots, k\} \,.$$ We designated here $\tau_0'' = 0$, $\tau_{k+1}' = 1$, $W_{i, \varepsilon} = W_i \cap U_{\varepsilon}(N_2^i)$, $i \in \{1, \ldots, k\}$.
Now Corollary \[corr\_conectedness\] and correlations imply that the set $$K = \gamma[0, \tau_1') \cup W_{1, \varepsilon} \cup \gamma(\tau_1'', \tau_2') \cup \ldots \cup \gamma(\tau_{k-i}'', \tau_k') \cup W_{k, \varepsilon} \cup \gamma(\tau_k'', 1]$$ is connected. Indeed, all sets in this union are connected, and from it follows that every two adjacent sets in this chain have a common point.
In addition, by virtue of choice of $\varepsilon > 0$ we have $$W_{i, \varepsilon} \cap (N_1 \cup N_2) = {\varnothing}\,, \quad i \in \{1, \ldots, m\} \,.$$ Therefore, it follows from the choice of $\gamma$ and from conditions (a$'$) that $K \cap (N_1 \cup N_2) = {\varnothing}$.
So, we have constructed a connected set $K$, which contains points $x_1 = \gamma(0)$ and $x_2 = \gamma(1)$ and does not intersect surfaces $N_1$ and $N_2$.
>From the arbitrariness in the choice of points $x_1, x_2 \in V_1 \cap V_2$ we conclude that the set $V_1 \cap V_2$ is connected.
\[corr\_intersection\] Under the condition of Lemma \[lemma\_complements\_intersection\] the set $V_1 \cap V_2$ is the connected component of the complement ${\mathbb R}^n \setminus (N_1 \cup N_2)$.
By Lemma \[lemma\_complements\_intersection\] the set $V_1 \cap V_2$ is connected. Moreover, it is easy to see from condition of Lemma \[lemma\_complements\_intersection\] that this set does not intersect $N_1 \cup N_2$. Therefore, there is a component $W$ of the complement ${\mathbb R}^n \setminus (N_1 \cup N_2)$ which contains $V_1 \cap V_2$.
Suppose that Corollary is invalid. Then $W \neq (V_1 \cap V_2)$ and there exists $x \in W \cap (({\mathbb R}^n \setminus V_1) \cup ({\mathbb R}^n \setminus V_2))$.
Let $x \in ({\mathbb R}^n \setminus V_1)$. Since $W \cap V_1 \supset W \cap (V_1 \cap V_2) = V_1 \cap V_2 \neq {\varnothing}$, the set $W \cup V_1$ is connected. By construction $W \cap N_1 = {\varnothing}$, so $W \cup V_1 \subset ({\mathbb R}^n \setminus N_1)$. In this case $(W \cup V_1) \supsetneq V_1$, as $x \in W \setminus V_1$ by our hypothesis. We obtain a contradiction to the condition of Lemma \[lemma\_complements\_intersection\] which says that $V_1$ is the connected component of the complement ${\mathbb R}^n \setminus N_1$. Consequently, our supposition is false and $x \in V_1$.
The inclusion $x \in V_2$ is proved similarly.
Therefore, $W = V_1 \cap V_2$. Corollary is proved.
\[corr\_component\] Let $x_0 \in {\mathbb R}^n$. Let $N$ be a closed hypersurface in ${\mathbb R}^n$ and $W$ be the component of the complement ${\mathbb R}^n \setminus N$, such that $x_0 \in W$. Suppose the set $W$ is bounded.
Then there exists a connected component $N_0$ of $N$, such that the connected component $W_0$ of ${\mathbb R}^n \setminus N_0$ which contains $x_0$ is bounded.
Since $N$ is compact, and so bounded in ${\mathbb R}^n$, we can assume without loss of generality that $N$ is contained in the unit ball $B$ which has its center at the origin.
Denote by $S$ the unit sphere ${\mathop{\mathrm{Fr}}\nolimits}B$. It is clear that $S \subset {\mathbb R}^n \setminus N$. Boundedness of $W$ means that $x_0$ and the connected set $S$ belong to distinct components of the complement ${\mathbb R}^n \setminus N$.
Let $N_1, \ldots, N_m$ be the connected components of $N$. Let $W_i$ be the component of the complement ${\mathbb R}^n \setminus N_i$, such that $x_0 \in W_i$, $i \in \{1, \ldots, m\}$.
Suppose that the conclusion of the Corollary is false. This is equivalent to the claim that $S \subset W_i$ for all $i \in \{1, \ldots, m\}$. By the sequential application of Lemma \[lemma\_complements\_intersection\] and of Corollary \[corr\_intersection\] to the pairs of sets $$N'(i) = \bigcup_{j=1}^i N_j \,, \quad N''(i) = N_{i+1} \,, \quad i \in \{1, \ldots, m-1\} \,,$$ we verify that the set $W_1 \cap \cdots \cap W_m \subset {\mathbb R}^n \setminus N$ is connected. Therefore, both $x_0$ and $S$ are contained in the same connected component of the complement ${\mathbb R}^n \setminus N$. But this contradicts to the condition of Corollary.
Proof of Theorem \[th\_surf\_family\]
-------------------------------------
Without loss of generality we can assume that $F(x_0) = 0$.
We shall use the following designations throughout the proof: $$B_{\delta} = \{ y \in {\mathbb R}^n \,|\, \rho(y, x_0) \leq \delta \}\,.$$ $$\mathring{B}_{\delta} = {\mathop{\mathrm{Int}}\nolimits}(B_{\delta}) = \{ y \in {\mathbb R}^n \,|\, \rho(y, x_0) < \delta \}\,.$$
[**Necessity.**]{} Let $F$ be an $\bf {L}$-function for the point $x_0$.
Let us select a sequence $\{a_i\}_{i \in {\mathbb N}}$ of regular values of $F$ and a sequence $\{ H_{i}^{n-1} \subset F^{-1}(a_i)\}_{i \in {\mathbb N}}$ of connected components of level sets of $F$ such that they comply with Definition \[defn\_L\_function\].
Denote by $C$ the connected component of $F^{-1}(0)$ which contains $x_0$.
Suppose that $C \neq \{x_0\}$ contrary to the statement of Theorem. Then there exists $x_1 \in C$, $x_1 \neq x_0$. Denote $\delta = \rho(x_0, x_1)/2$.
Since the sequence $\{H_{i}^{n-1}\}$ converges to $x_0$, there is an index $M \in {\mathbb N}$ such that $H_M^{n-1} \subset B_\delta$.
On one hand, by definition the point $x_0$ is contained in the bounded component of the complement ${\mathbb R}^n \setminus H_M^{n-1}$. However, it is straightforward that $x_1$ is contained in the unbounded component of this complement by the choice of $H_M^{n-1}$. Consequently, the connected set $C$ must intersect $H_M^{n-1}$.
It easily follows from the relation $H_M^{n-1} \cap C \neq {\varnothing}$ that $H_M^{n-1} \cup C \subset F^{-1}(0)$. The set $H_M^{n-1} \cup C$ is connected and does not coincide with $H_M^{n-1}$ since $\{x_0, x_1\} \subset C \setminus H_M^{n-1}$. This contradicts to the choice of $H_M^{n-1}$ being the connected component of the corresponding level set of $F$.
The contradiction obtained proves that $C = \{x_0\}$ and $x_0$ is the quasi-isolated point of $F$.
[**Sufficiency.**]{} Let $x_0$ be a quasi-isolated point of $F$.
It follows from Sard’s Theorem [@Sard] that the set of regular values of $F$ is residual and everywhere dense. So there exists a decreasing sequence of positive real numbers $\{\varepsilon_i\}_{i \in {\mathbb N}}$, which complies with the following properties:
- $\lim_{i \to \infty} \varepsilon_i = 0$;
- all numbers $\pm\varepsilon_i$ are regular values of $F$.
Denote by $U^i$ the connected component of the open set $Q^i = \{ x \in G \,|\, -\varepsilon_i < F(x) < \varepsilon_i \}$, such that $x_0 \in U^i$. Obviously, $U^j \subset U^i$ when $i < j$.
We fix $\delta > 0$ small enough to satisfy the inclusion $B_\delta \subset G$. Let us denote by $U^i_{\delta}$ the component of $U^i \cap B_{\delta}$, such that $x_0 \in U^i_{\delta}$. Let also $$K^i_{\delta} = {\mathop{\overline{U^i_{\delta}}}\nolimits} \,, \quad i \in {\mathbb N}\,.$$ It obviously follows from the continuity of $F$ that $$K^i_{\delta} \subset \{ x \in G \,|\, -\varepsilon_i \leq F(x) \leq \varepsilon_i \}\,.$$ We have also $U^j_{\delta} \subset U^i_{\delta}$ and $K^j_{\delta} \subset K^i_{\delta}$ for $i < j$ by construction.
Let us consider the set $$K_{\delta} = \bigcap_{i \in {\mathbb N}} K^i_{\delta} \,.$$
On one hand $x_0 \in K_{\delta} \subseteq F^{-1}(0)$, since $\varepsilon_i \to 0$ when $i \to \infty$.
On the other hand, $\{K^i_{\delta}\}$ is the sequence of embedded connected compacts. Hence, $K_{\delta}$ is connected.
Thus, from the condition of Theorem we conclude that $K_{\delta} = \{x_0\}$.
Then there exists an $m \in {\mathbb N}$, such that $U^r_{\delta} \subset K^r_{\delta} \subset \mathring{B}_{\delta}$ for all $r \geq m$.
Recall that by construction the set $U^i$ is connected component of the open subset $Q^i$ of $G$. The space ${\mathbb R}^n$ is locally connected, therefore $U^i$ is open in ${\mathbb R}^n$ (see [@Kuratowski]). Consequently, $U^i_{\delta}$ is open in $B_{\delta}$. In fact, the space $B_{\delta}$ is locally connected, as it is the homeomorphic image of closed disk. The set $U^i_{\delta}$ is a connected component of $U^i \cap B_{\delta}$, so $U^i \cap B_{\delta}$ is open in $B_{\delta}$.
In this way, if the inclusion $U^i_{\delta} \subset \mathring{B}_{\delta} = {\mathop{\mathrm{Int}}\nolimits}(B_{\delta})$ is valid, then the set $U^i_{\delta}$ is open in ${\mathbb R}^n$.
On the other hand, it is known (see [@Kuratowski]), that if arbitrary sets $A$ and $B$ satisfy the inclusion $A \subset B \subset {\mathop{\overline{A}}\nolimits}$, then connectedness of $A$ results in the connectedness of $B$. So, the set $K^i_{\delta} \cap U^i$ is connected. Indeed, the set $U^i_{\delta}$ is connected and $U^i_{\delta} \subset K^i_{\delta} \cap U^i \subset {\mathop{\overline{U^i_{\delta}}}\nolimits} = K^i_{\delta}$.
Consequently, $K^i_{\delta} \cap U^i = U^i_{\delta}$, since $K^i_{\delta} \cap U^i \subseteq K^i_{\delta} \cap (U^i \cap B_{\delta})$ and $U^i_{\delta}$ is a connected component of $U^i \cap B_{\delta}$.
Thus, the compact $K^i_{\delta}$ complies with the correlation $U_i \setminus U^i_{\delta} = U_i \setminus K^i_{\delta}$, so the set $U_i \setminus U^i_{\delta}$ is open in ${\mathbb R}^n$.
From what has been said we conclude that $U^r_{\delta} = U^r$ for $r \geq m$. Indeed, $U^r = U^r_{\delta} \sqcup (U^r \setminus U^r_{\delta})$, moreover in our case both the sets $U^r_{\delta} \ni x_0$ and $(U^r \setminus U^r_{\delta})$ are open. It now follows from the connectedness of $U^r$ that $(U^r \setminus K^r_{\delta}) = {\varnothing}$.
Let $r \geq m$.
Consider the set $N_r = {\mathop{\mathrm{Fr}}\nolimits}U^r$. The set $U^r$ is open, so $N_r \cap U^r = {\varnothing}$. On the other hand, $U^r$ is the connected component of $Q^r = \{ x \in M^n \,|\, -\varepsilon_r < F(x) < \varepsilon_r \}$, therefore ${\mathop{\overline{U^r}}\nolimits} \cap Q^r = K^r_{\delta} \cap Q^r = U^r$. It implies that $N_r \subseteq F^{-1}(-\varepsilon_r) \cup F^{-1}(\varepsilon_r)$.
Both $-\varepsilon_r$ and $\varepsilon_r$ are regular values of $F$. Hence by the implicit function theorem (see [@PalisDiMelo]) for every point $x \in N_r$ there exist an open neighbourhood $V_x$ in ${\mathbb R}^n$ and a diffeomorphism $\psi_x : V_x \to (-1, 1)^n$, which satisfy to the following conditions:
- $\psi_x^{-1} \bigl( (-1, 1)^{n-1} \times \{0\} \bigr) \subset F^{-1}(F(x))$;
- $\psi_x^{-1} \bigl( (-1, 1)^{n-1} \times (-1, 0) \bigr) \subset \{ y \in {\mathbb R}^n \,|\, |F(y)| < \varepsilon_r \}$;
- $\psi_x^{-1} \bigl( (-1, 1)^{n-1} \times (0, 1) \bigr) \subset \{ y \in {\mathbb R}^n \,|\, |F(y)| > \varepsilon_r \}$.
So, it follows from the properties of $U^r$ that $$V_x \cap N_r = \psi_x^{-1} \bigl( (-1, 1)^{n-1} \times \{0\} \bigr) \,, \quad x \in N_r.$$ the space $N_r$ is locally diffeomorphic to ${\mathbb R}^{n-1}$ at every point. Therefore, $N_r$ is the closed hypersurface in ${\mathbb R}^n$.
Applying Corollary \[corr\_component\] to $N_r$, we conclude that there exists a component $N_r^0$ of $N_r$, which complies with the following property: if a component $W_r$ of the complement ${\mathbb R}^n \setminus N_r^0$ contains $x_0$, then $W_r$ is limited.
We have proved already that $N_r \subset K^r_{\delta} \subset \mathring{B}_{\delta}$, hence $W_r \subset \mathring{B}_{\delta}$ and ${\mathop{\overline{W_r}}\nolimits} \subset B_{\delta}$.
Let $B_{\delta_1} = B_{\delta}$. We already found $m_1 = m \in {\mathbb N}$, such that there exists a component $N_{m_1}^0$ of the hypersurface $N_{m_1}$, which has the following property: if a component $W_{m_1}$ of the complement ${\mathbb R}^n \setminus N_{m_1}^0$ contains $x_0$, then its closure $K_{m_1} = {\mathop{\overline{W}}\nolimits}_{m_1}$ belongs to $B_{\delta_1}$.
Let us assume now that for some $k \in {\mathbb N}$ the following objects are already determined:
- a sequence of positive numbers $\delta_1, \ldots, \delta_k$, such that $\delta_{i+1} < \delta_i/2$, $i \in \{1, \ldots, k-1\}$;
- a family $B_{\delta_1}, \ldots, B_{\delta_k}$ of neighbourhoods of $x_0$, $B_{\delta_i} = \{ y \in {\mathbb R}^n \,|\, \rho(y, x_0) \leq \delta_i \} \subset G$, $i \in \{1, \ldots, k\}$;
- a sequence of numbers $m_1 < \cdots < m_k$;
- a collection of connected components $N_{m_i}^0$ of hypersurfaces $N_{m_i} \subseteq F^{-1}(-\varepsilon_{m_i}) \cup F^{-1}(\varepsilon_{m_i})$, $i \in \{1, \ldots, k\}$;
- closures $K_{m_i}$ of components $W_{m_i}$ of the complements $M^n \setminus N_{m_i}^0$, $i \in \{1, \ldots, k\}$.
Assume that these objects are interconnected by the correlations $$\begin{gathered}
B_{\delta_i} \supset K_{m_i} \supset W_{m_i} \ni x_0 \,, \quad i \in \{1, \ldots, k\} \,, \\
W_{m_i} \supset B_{\delta_{i+1}} \,, \quad i \in \{1, \ldots, k-1\} \,.\end{gathered}$$
Let us fix $\delta_{k+1} \in (0, \delta_k/2)$, such that $B_{\delta_{k+1}} = \{ y \in {\mathbb R}^n \,|\, \rho(y, x_0) \leq \delta_{k+1} \} \subset W_{m_k} \subset G$.
By repeating the argument above, we obtain $m_{k+1} > m_k$, such that $x_0 \in U^{m_{k+1}} \subset {\mathop{\overline{U^{m_{k+1}}}}\nolimits} \subset {\mathop{\mathrm{Int}}\nolimits}B_{\delta_{k+1}}$. Then the set $N_{m_{k+1}} = {\mathop{\mathrm{Fr}}\nolimits}U^{m_{k+1}}$ is a closed hypersurface in ${\mathbb R}^n$ and $N_{m_{k+1}} \subset {\mathop{\mathrm{Int}}\nolimits}B_{\delta_{k+1}}$. There exists also a component $N_{m_{k+1}}^0$ of this hypersurface, which comply with the following property: if a component $W_{m_{k+1}}$ of the complement ${\mathbb R}^n \setminus N_{m_{k+1}}^0$ contains $x_0$, then its closure $K_{m_{k+1}}$ lies in $B_{\delta_{k+1}}$.
If we continue this process by induction, we will construct a sequence of closed connected hypersurfaces $\{ N_{m_{i}}^0,\; i \in {\mathbb N}\}$, a sequence of components $\{ W_{m_i} \}$ of the sets ${\mathbb R}^n \setminus N_{m_i}^0$, and a sequence of their closures $\{ K_{m_i} = {\mathop{\overline{W_{m_i}}}\nolimits} \}$, which are interconnected by the correlations $$\label{eq_seq_of_domains}
G \supset B_{\delta_i} \supset K_{m_i} \supset W_{m_i} \supset B_{\delta_{i+1}} \ni x_0 \,, \quad i \in {\mathbb N}\,.$$
All $B_{\delta_i}$ are closed $n$-disks, therefore all sets $K_{m_i}$ are compact. By construction hypersurfaces $N_{m_i}^0$ are boundaries of the sets $K_{m_i}$. It follows from relations that $K_{m_i} \supset K_{m_j} \ni x_0$ when $i < j$, so $x_0 \in \bigcap_{i \in {\mathbb N}} K_{m_{i}}$. It is obvious that $\lim_{i \to \infty} \delta_i = 0$. Therefore, the family of sets $\{B_{\delta_i}\}$ forms the basis of neighbourhoods of the point $x_0$ in ${\mathbb R}^n$. So, $\{x_0\} = \bigcap_{i \in {\mathbb N}} K_{m_{i}}$. Finally, hypersurfaces $N_{m_i}^0$ are disjoint, since $N_{m_i}^0 \subset F^{-1}(-\varepsilon_{m_i}) \cup F^{-1}(\varepsilon_{m_i})$ and by construction $|\varepsilon_{m_i}| \neq |\varepsilon_{m_j}|$ for $i \neq j$.
Thus, the family $\{H_i^{n-1} = N_{m_i}^0\}_{i \in {\mathbb N}}$ of connected closed hypersurfaces meets all conditions of Definition \[defn\_L\_function\].
Theorem is proved.
|
---
abstract: 'A new approach to the parameterization of pion form factors is presented and for illustration applied to the pion vector form factor. It has the correct analytic structure, is at low energies consistent with recent high accuracy analyses of $\pi\pi$ scattering phase shifts and, at high energies, maps smoothly onto the well–known, successful isobar model.'
address: |
Institut für Kernphysik, Institute for Advanced Simulation and Jülich Center for Hadron Physics,\
Forschungszentrum Jülich, D–52425 Jülich, Germany
author:
- 'C. Hanhart'
title: A new Parameterization for the Pion Vector Form Factor
---
Pion form factor ,Omnès representation ,11.55.Bq ,13.40.Gp
Introduction
============
In recent years the knowledge about the low energy two–pion system has improved significantly, both experimentally as well as theoretically: for the low partial waves phase shift parameterizations of high accuracy exist from different dispersive analyses, either involving data only [@madrid_new], or involving both data as well as constraints from chiral symmetry [@bern]. The analyses are based on Roy or Roy-type equations that respect analyticity as well as crossing symmetry. Especially, left–hand cuts are included without approximation.
In contrast to this, pion form factors or production reactions are often modeled either by sums of Breit-Wigners or improved versions thereof [@GS; @HL] or by the K–matrix formalism. In case of overlapping resonances unitarity gets violated by the former ansatz. The K–matrix provides a clear improvement compared to the Breit-Wigner parameterization, since two–body unitarity is built in. However, in general analyticity is violated. On the one hand, in the standard treatment not the full dispersive corrections are considered (in the expressions for the self energies only the imaginary parts and their analytic continuation are being kept and not the full expressions — c.f. Eq. (\[Ldef\]) below), although some works include them (see, e.g., Ref. [@anisovich2011]), on the other hand, the left hand cuts are not treated properly — if they are included at all, in order to fit the scattering amplitudes, they are often in the same way included in the production amplitude although there the left hand cuts are different or, as in case of form factors, even absent.
To be specific, in this work we focus on form factors and scattering with the goal to present simple formulas that allow for a data analysis that is consistent with analyticity and unitarity, however, without the necessity to solve dynamical equations. In addition, we present formulas that, by construction, in the low energy regime map smoothly and consistently onto what can be derived from the high accuracy analyses mentioned above and thus for the scattering even include the proper left hand cuts.
As an example and for demonstration we apply the formalism in this paper to the pion vector form factor, related to $\pi\pi$ scattering in the $p$–wave. The experimental situation for the $\rho$ resonances beyond the $\rho(770)$ is at present not very clear: different experiments find indications for different resonances — for a summary of the current situation see ’note on the $\rho(1450)$ and the $\rho(1700)$’ in the Review of Particle Physics [@PDG]. The formalism presented here could be an important step forward to clarify the situation for it allows for a simultaneous, consistent analysis of various channels/observables. To parameterize the vector form factor beyond $s=1$ GeV$^2$ all studies agree on the need to include at least two resonances in addition to the $\rho(770)$, which is elastic. Thus, if we include one inelastic channel the pion vector form factor is parameterized in terms of in total 9 parameters — 8 for the resonances and 1 additional parameter for the $\rho$-$\omega$ mixing (see Sec. \[sec:derivation\]). The number of parameters needed for each resonance agrees to standard parameterizations. The advantage of the parameterization presented here is that we do not need to approximate the left hand cuts, the consistency with low energy phase shift is ensured by construction and the connection between scattering and form factors is properly implemented. We found that once simultaneously to the form factor also the data on $e^+e^-\to$non 2$\pi$ in the isovector state was fitted, the inclusion of not only an additional resonance but also of the $\pi\omega$ cahnnel together with a direct $\pi\pi\to \pi\omega$ coupling became necessary. This model, with in total 17 parameters, allowed for an acceptable fit to the data.
The paper is structured as follows: the most important formulas are motivated and presented in Sec. \[sec:sum\]. Their more detailed derivation is given in Sec. \[sec:derivation\]. Results are presented in Sec. \[sec:results\] and the paper closes with a short summary in Sec. \[sec:summary\]. The two-potential formalism which forms the basis for the derivation is introduced in the Appendix.
Summary of most important results {#sec:sum}
=================================
![Diagrammatic representation for the various ingredients of the formalism. \[fig:diags\]](BS-eq.eps){width="0.7\linewidth"}
When it comes to the interplay of resonance contributions and background terms the case of only one scattering channel is especially simple for here it is possible to give a closed form expression for the form factor $\mathcal F$ solely in terms of the elastic scattering phase shift — the so–called Omnès solution [@Omnes:1958hv]. It is derived from a dispersion relation using the fact that $$\mbox{disc}({\mathcal F}(s))=2i\sigma T(s)^*{\mathcal F}(s) \ ,
\label{ima}$$ where $T$ denotes the on–shell elastic scattering amplitude, and $\sigma=\sqrt{1-4m^2/s}$ the two–body phase space — for simplicity we assume the scattering particles to have equal mass $m$. Here ’disc’ denotes the discontinuity of the form factor defined via $$\mbox{disc}({\mathcal F}(s))={\mathcal F}(s+i\epsilon)-{\mathcal F}(s-i\epsilon)=2i{\rm Im}({\mathcal F}) \ .$$ If the elastic phase shift is $\delta(s)$, with $$T(s)=\frac1{\sigma} \sin (\delta(s))e^{i\delta(s)} \ ,$$ then the form factor reads in the absence of bound states $$\label{elastFdef}
{\mathcal F}(s)=\xi(s)\Omega[\delta](s)P_A(s)$$ with the Omnès function $$\Omega[\delta](s)=
\exp\left\{ \frac{s}{\pi}\int_{4m^2}^\infty \frac{ds'}{s'}
\frac{\delta(s')}{s'-s}\right\} \ ,
\label{omnes}$$ where the centrifugal barrier is introduced via the factor $\xi(s)=\sqrt{s-s_{{\rm thr}}}^L$ for $L$–waves, where $s_{{\rm thr}}$ denotes the location of the threshold and the function $P_A(s)$ is a polynomial. Its degree may be fixed by the large $s$ behavior of the form factors. For values of $s<1$ GeV$^2$ this methodology was used by various authors for the pion vector form factor, see, e.g. Refs. [@HL; @gassermeissner; @guerrero; @pich; @yndurain; @oller; @ours]. Here we will present a formalism that provides expressions that smoothly map onto Eq. (\[omnes\]) at lower energies while being an analytically improved version of the isobar model at higher energies.
Eqs. (\[ima\]) and (\[omnes\]) apply only if the interactions are purely elastic — for the latter it is even necessary that they are elastic up to infinite energies. Clearly this is not realistic. However, experimental data show that at higher energies inelasticities are typically accompanied by resonances. We therefore split the full, partial wave projected, interaction potential $V$ into two pieces $$V(s)_{ij} = \tilde V(s)_{ij} + V_R(s)_{ij} \ ,
\label{vdef}$$ where $i$ and $j$ denote the channels. The crucial feature for this approach is that the potential $\tilde V$ needs to be specified at no point. All what is needed are the corresponding phase shifts $\tilde \delta$. We now $postulate$ the following properties:
- the potential $\tilde V$ is purely elastic, such that $\tilde V$ is non–vanishing only for $i=j=1$;
- deviations in the $\pi\pi$ phase–shifts from $\tilde \delta$ come either from $s$–channel resonances or via the coupling to inelastic channels;
- all long ranged forces are in the elastic $\pi\pi$ interactions $\tilde V$; all interactions in other channels are regarded as short ranged.
These are the model dependent assumptions of this approach (note, in case of the traditional isobar model one needs to assume that $all$ interactions are mediated by $s$–channel resonances). Based on these Eqs. (\[ima\]) and (\[omnes\]) can be easily generalized to multiple channels providing a convenient parameterization for both scattering as well as production amplitudes.
![Fits result for the pion $p$–wave phase shift (left panel) and inelasticity (right panel). The red solid (green dashed) line denotes the result of the fit \#2 (\#1). The dot–dashed line in the left panel refers to the input phase $\tilde \delta$. Data are from Ref. [@hyams] (solid dots — only data below 1.4 GeV are shown [@WO]), Ref. [@hyams1] (solid squares for solution $(---)$; solid triangles for solution $(-+-)$), and Ref. [@propo] (open dots) and Ref. [@madrid_new] (turquoise band). \[fig:phases\]](phases_eta.eps){width="0.8\linewidth"}
The full scattering $T$–matrix appears as the solution of a Bethe–Salpeter equation with input potential $V$ defined in Eq. (\[vdef\]). Using the two potential formalism (see Appendix) it is straightforward to derive the decomposition
$$T(s)_{ij}= \delta_{ij}\delta_{1i}\tilde T(s) + T_R(s)_{ij} = \delta_{ij}\delta_{1i}\tilde T(s) + \Gamma_{\rm
out}(s)_i t_R(s)_{ij}\Gamma_{\rm in}(s)^\dagger_j \ ,
\label{Tsplit}$$
where $\tilde T$ is the purely elastic scattering $T$–matrix that derives from the potential $\tilde V$. For the vertex functions one has from time reversal invariance $\Gamma_{\rm out}=\Gamma_{\rm in}^\dagger$. They are, by assumption, non–trivial only in the $\pi\pi$ channel, where elastic scattering is mediated by $\tilde T$. As a result $\Gamma_{\rm out}$ is diagonal. We may therefore write for any given partial wave $$\tilde T = \frac{1}{\sigma_1}e^{i\tilde\delta}\sin (\tilde\delta)$$ and $$\Gamma_{\rm out}(s)_i = \xi_i(s)\Omega[\tilde \delta](s) \ \mbox{for i=1 (}\pi\pi\mbox{--channel); otherwise } \Gamma_{\rm
out}(s)_i= \xi_i(s) \ ,
\label{Gammaccdef}$$ where $\xi_i(s)\sqrt{s-s_{{\rm thr} \, i}}^L$ is the multichannel version of the centrifugal barrier factor given above. The resonance $T$ matrix $t_R$ may be written as $$t_R(s)_{ij} = \left[ 1 - V_R(s)\Sigma(s)\right]^{-1}_{ik}V_R(s)_{kj}
\label{tRdef}$$ with the resonance potential (note: not all resonances are in $V_R$; elastic resonances may be included in $\tilde T$ — as the $\rho(770)$ in the example below) $$\bar V_R(s)_{ij} = -\sum_{l=1}^n \frac{g_i^{(l)}g_j^{(l)}}{s-m_{(l)}^2} \ ;
\ \ V_R(s) = \bar V_R(s)- \bar V_R(0)+\gamma_{j}\delta_{i1}+\gamma_{i}\delta_{j1} \ .
\label{VRdef}$$ In Fig. \[fig:diags\] a graphic representation of the various quantities is given. The potential is subtracted at $s=0$ to ensure that the phase of the full $T$ matrix at low energies agrees to the input phase $\tilde \delta$. Clearly, the procedure does not guarantee a priori that the phase of the full $T$–matrix is close to that of $\tilde T$ in the whole range where $\tilde T$ is well determined, however, in practice this is indeed the case: as can be seen from the left panel of Fig. \[fig:phases\], at energies below 1 GeV all curves shown, including that for the input phase shift, are indistinguishable.
The term containing the parameter $\gamma_i$, with $\gamma_1=0$, is included to allow for a direct transition of the $\pi\pi$ channel (channel number 1) to inelastic channels. For the pion vector form factor the fit to data requires such a coupling to the $\pi\omega$ channel — in effect it provides a direct $\rho \pi \omega$ coupling, which from phenomenology is known to be significant [@GMSW].
The self energy loop $\Sigma_i$ in channel $i$ appearing in Eq. (\[tRdef\]) and Fig. \[fig:diags\] may be expressed via a twice subtracted dispersion relation (see Sec. \[sec:derivation\] and Appendix) — one finds $$\Sigma_i(s) = \frac{s^2}{\pi}\int_{s_{\rm thr}}^\infty \frac{ds'}{s'{}^2}
\frac{\sigma_i(s')\left|\Gamma_i(s')\right|^2}{s'-s-i\epsilon} \ .
\label{Ldef}$$
For given interactions $T_{ij}$ it is straight forward to calculate the form factors ${\mathcal F}_i$. As shown in the next section, one finds
$${\mathcal F}(s)_i = \Gamma_{\rm out}(s)_{i}\left[ 1 - V_R(s)\Sigma(s)\right]^{-1}_{ik}M_k \ ,
\label{FFstructure}$$
where the $M_i$ denote point like source terms for the production of particles into channel $i$. It may be written as $$M_k = c_k - \sum_{l=1}^n \frac{g_i^{(l)}\alpha^{(l)}}{s-m_{(l)}^2} \ .
\label{Mdef}$$ Since we assume that all interactions but those in the two–pion channel are driven by resonances, we choose $c_k=c\delta_{1k}$, with some constant $c$. In case of the pion vector form factor, discussed in detail below, charge conservation demands $c=1$. In addition, to ensure the proper normalization of the form factor and to suppress the influence of the higher resonances on low energies, we use a photon–resonance coupling linear in $s$[^1]. In practice this means replacing $\alpha^{(l)}$ in Eq. (\[Mdef\]) by $s\alpha^{(l)}$. The ingredients of the formalism are illustrated in Fig. \[fig:diags\].
All parameters introduced — $\kappa$, $g_i^{(l)}$, $\alpha^{(l)}$, $m_{(l)}$, $\gamma_i$ — are real as long as all (relevant) channels are treated explicitly as a consequence of time reversal invariance. Thus, in case of two channels, for each inelastic resonance we need to include 4 real parameters. Each additional channel adds in one more parameter per resonance. For the predominantly elastic resonances, which are included in the vertex functions $\Gamma$, the only free parameters are their couplings to the inelastic channels introduced via the $\gamma_i$.
Eq. (\[FFstructure\]) is the central result of our paper. If all vertex functions were chosen to be constant, it would reduce to the famous $P$–vector formalism [@ian]. However, since in our case $\Gamma_1$, which enters explicitly in the expression for ${\mathcal F}_1$ as well as through $\Sigma_1$, is non–trivial, Eq. (\[FFstructure\]) provides a generalization to the conventional treatment.
Derivation of the Formalism {#sec:derivation}
===========================
In this section the expressions presented in the previous section are derived. It is based on the two–potential formalism used, e.g., in Ref. [@twopotform] to control the Coulomb-nuclear interference and rederived in the notation used here in the Appendix. The important aspect to observe is that, although the derivation starts from a potential $\tilde V$, introduced in Eq. (\[vdef\]), the final expressions can all be expressed in terms of the corresponding scattering phase $\tilde \delta$.
![Real and imaginary parts for the self energies $\Sigma_i(s)$ defined in Eq. (\[Ldef\]). Left panel: self energy for the elastic channel, $\Sigma_1(s)$ — the solid (dashed) line shows the imaginary (real) part. Right panel: self energies for the inelastic channels. The solid (dashed) line shows the imaginary (real) part for the 4$\pi$ channel, while the dashed–dotted (double dashed–dotted) line shows the imaginary (real) part for the $\omega \pi$ channel. \[fig:selfenergies\]](Lfunctions.eps){width="0.8\linewidth"}
The two–potential formalism splits the $T$–matrix into two pieces, the elastic $T$–matrix $\tilde T$ and the resonance $T$–matrix $T_R$ (c.f. Eq. (\[Tsplit\])). The latter quantity is the solution of a Bethe–Salpeter equation derived from the resonance potential $V_R$, however, here the intermediate state free propagators, used in the standard treatment, are to be replaced by interacting propagators. This is achieved via inclusion of the vertex functions $\Gamma$ on the external legs as well as a proper modification of the intermediate meson loops connecting two insertions of the resonance potential — the so–called resonance self–energies $\Sigma_i$. It is straightforward to show that the imaginary parts of the self–energies read (see Appendix) $$\mbox{Im}(\Sigma_i(s))=\sigma_i |\Gamma_i(s)|^2 \ .
\label{imL}$$ Thus the self energies can be calculated from a properly subtracted dispersion integral. Since $\Sigma(s_0)$ and $\partial \Sigma/\partial s(s_0)$ at some $s_0$ can be absorbed into the resonance masses and wave function renormalization constants (or, more appropriate for the present context: in the effective coupling constants of the resonances) we here use a twice subtracted version — c.f. Eq. (\[Ldef\]). The self energy in the 2-$\pi$ channel needs as input only the vertex function $\Gamma_1(s)$ discussed in the previous paragraph. The resulting $\Sigma_1(s)$ is shown in the left panel Fig. \[fig:selfenergies\].
To parameterize the inelastic channels we use for $i=2$ a structureless 4$\pi$ channel via the phase space factor $\sigma_2=\sqrt{1-16m_\pi^2/s}^7$, which provides the proper scaling of the four–body phase space near the threshold. For the barrier factor, appearing in Eqs. (\[Gammaccdef\]), we use $\xi_2= \sqrt{s-16m_\pi^2}$. The resulting self energy $\Sigma_2(s)$ is shown in the right panel Fig. \[fig:selfenergies\]. Although the $\bar KK$ channel contributes significantly to isoscalar $\pi\pi$ interactions, it gives negligible contributions in the isovector channel [@Simon]. Therefore in fit \#2 we include as additional inelastic channel ($i=3$) the $\pi\omega$ channel. For a discussion on the possible role of the $\pi\omega$ channel on the $\pi\pi$ inelasticity see Ref. [@costa77; @bastianneu]. We take $\sigma_3=\lambda(s,m_\omega^2,m_\pi^2)^{1/2}/s$ and $\xi_3=\lambda(s,m_\omega^2,m_\pi^2)^{1/2}$. The resulting self energy $\Sigma_3(s)$ is also shown in the right panel Fig. \[fig:selfenergies\]. The linear rise of the imaginary part of both $\Sigma_2$ and $\Sigma_3$ comes from the centrifugal barrier terms $\xi_2$ and $\xi_3$, respectively.
To calculate the form factor we may write (using the notation of the Appendix) $${\mathcal F} = M + TGM \ ,
\label{Fdefeq}$$ where $G_i$ is the operator representation for the integration over all intermediate $n$–body states of channel $i$ and the production vertices $M_i$ were defined in Eq. (\[Mdef\]). Inserting Eq. (\[Tsplit\]) into Eq. (\[Fdefeq\]), we get $${\mathcal F} = M+\tilde TGM + T_RGM = \Gamma_{\rm out}(1+t_R\Gamma_{\rm
in}^\dagger G )M \ .$$ To proceed we may use the definition of the self energy, $\Sigma_i=\Gamma_{{\rm in}\, i}^\dagger G_i$, to write $$t_R\Sigma = \left[1-V_R\Sigma\right]^{-1} V_R\Sigma = -1+\left[1-V_R\Sigma\right]^{-1} \ .$$ Here we needed to assume that the range of interactions in the production vertex and in the vertex functions of the resonances is similar in all channels, for only then the same loop integral $\Sigma_i$ can be used as self energy contribution for the resonances as well as convolution integral of $M_i$ and the resonance potential. From this we get $${\mathcal F} = \Gamma_{\rm out} \left[1-V_R\Sigma\right]^{-1}M \ ,$$ which agrees to Eq. (\[FFstructure\]).
It is important to observe that the expression given in Eq. (\[FFstructure\]) is consistent with the coupled channel version of the unitarity relation for form factors, Eq. (\[ima\]), since
$$\begin{aligned}
\nonumber
\mbox{disc}({\mathcal F}) &=& \mbox{disc}(\Gamma_{\rm out})\left[1-V_R\Sigma\right]^{-1}M
+\Gamma_{\rm out}^*\mbox{disc}(\left[1-V_R\Sigma\right]^{-1})M \\ \nonumber
&=&2i\tilde T^* \sigma \Gamma_{\rm out}\left[1-V_R\Sigma\right]^{-1}M
+\Gamma_{\rm
out}^*\left[1-V_R\Sigma^*\right]^{-1}V_R\mbox{disc}(\Sigma)\left[1-V_R\Sigma\right]^{-1}M
\\
&=& 2i\underbrace{\left( \tilde T^*+\Gamma_{\rm
out}^*\left[1-V_R\Sigma^*\right]^{-1}V_R\Gamma_{\rm
out}^*\right)}_{T^*}\sigma \underbrace{\Gamma_{\rm
out}\left[1-V_R\Sigma\right]^{-1}M}_{{\mathcal F}} $$
where in the intermediate step the unitarity relation for the vertex function, Eq. (\[ima\]), and the self energy, Eq. (\[imL\]), were used.
An interesting observable is the ratio $r$ of the total cross section for $e^+e^-$ annihilation into hadronic states with $I=1$ other than $\pi^+\pi^-$ over $\sigma_{e^+e^-\to \pi^+\pi^-}$ — a compilation of this quantity can be found in Ref. [@Simon]. In this ratio the unitarization effects in the resonance $T$–matrix cancel largely. For example, in case of only one inelastic channel we get $$r = \left|\left(\frac{\sigma_2\Gamma_{{\rm out}\ 2}}{\sigma_1\Gamma_{{\rm
out}\ 1}}\right)\frac{(1-V_{R \, 11}\Sigma_1)M_2+V_{R \, 12}\Sigma_1 M_1}
{(1-V_{R \, 22}\Sigma_2)M_1+V_{R \, 12}\Sigma_2M_2}\right|^2 \ ,$$ clearly being very directly sensitive to the resonance parameters.
![Fit result for the pion vector form factor $F_V={\mathcal F}_1/\xi_1$. Left panel: For the neutral channel. The red solid (green dashed) line denotes the result of fit \#1 (\#2). The black dot–dashed line shows the form factor derived from the Omnès function only. Data are from the reaction $e^+e^-\to \pi^+\pi^-$ presented in Refs. [@babarFF; @KLOE; @Na7FF]. Right panel: For the charged channel. Data are from Belle [@BelleFF] and CLEO [@cleo]. \[fig:pionFF\]](pionFF_fitcomp.eps "fig:"){width="0.49\linewidth"} ![Fit result for the pion vector form factor $F_V={\mathcal F}_1/\xi_1$. Left panel: For the neutral channel. The red solid (green dashed) line denotes the result of fit \#1 (\#2). The black dot–dashed line shows the form factor derived from the Omnès function only. Data are from the reaction $e^+e^-\to \pi^+\pi^-$ presented in Refs. [@babarFF; @KLOE; @Na7FF]. Right panel: For the charged channel. Data are from Belle [@BelleFF] and CLEO [@cleo]. \[fig:pionFF\]](pionFF_fitcomp_taudata.eps "fig:"){width="0.49\linewidth"}
The vector form factor in the two pion channel is directly accessible from two reactions, namely from $e^+e$–annihilation and from $\tau$ decays. In the former case, in addition to what was discussed so far, the isospin violating mechanism of $\rho$-$\omega$ mixing needs to be included. It is visible as a striking narrow structure close to the peak of the corresponding form factor (c.f. inlay in Fig. \[fig:pionFF\]). The inclusion of this mixing in the present formalism is straight forward — we here use a slightly modified version to what is used in Ref. [@yndurain], namely, for the neutral, $\pi^+\pi^-$, channel $$F_1(s) \longrightarrow
F_1(s)\left(1+\kappa\frac{s}{s-m_\omega^2+im_\omega\Gamma_\omega}\right) .$$ We here use $m_\omega=0.7826$ GeV and $\Gamma_\omega=0.0085$ GeV. The strength parameter $\kappa$ is part of the fit. The only difference between the form factors in the neutral (from $e^+e^-$ annihilation) and in the charged (from $\tau$ decays) channel is that in the latter we choose $\kappa=0$.
Results for the pion phases, inelasticities and form factors in the $p$-wave {#sec:results}
============================================================================
To be specific we now focus on the pion vector form factor. The resonance parameters will be determined by a fit to data on the pion vector form factor as well as inclusive data on inelastic channels. In addition we need as input the elastic phase shifts $\tilde \delta$, which largely fix the properties of the $\rho(770)$.
For our work we will use for energies below $s_{\rm cut}=1.4^2$ GeV$^2$ the central values for the phase shift provided in Ref. [@madrid_new] — see Eqs. (A7) and (A8) therein. For energies above this value we smoothly extrapolate the phase shift to a value of $\pi$ via $$\tilde \delta (s)=\pi + (\tilde \delta(s_{\rm
cut})-\pi)\left(\frac{\Lambda^2-s_{\rm cut}}{\Lambda^2-s}\right) \ .$$ It turns out that for $\Lambda\geq 2$ GeV the results are basically insensitive to the actual value used [^2]. We thus chose $\Lambda = 10$ GeV in what follows. The asymptotic value $\pi$ for the phase ensures that the vertex function $\Gamma_1(s)$ decreases as $1/s$ as demanded for the vector form factor. The resulting elastic $\pi\pi$ phase shifts shown as the black dot–dashed line in the left panel of Fig. \[fig:phases\]. The form factor from the Omnés function alone is shown as the black dot–dashed line in Figs. \[fig:pionFF\]. It provides an acceptable description of the data up to $s=1$ GeV$^2$, although there are some deviations visible (see inlay in the left panel). At higher energies a significant deviation becomes visible.
fit $\kappa \times 10^3$ $m_{(1)}$ $m_{(2)}$ $m_{(3)}$ $\gamma_3$ $g^{(1)}_1$ $g^{(1)}_2$ $g^{(1)}_3$ $g^{(2)}_1$ $g^{(2)}_2$ $g^{(2)}_3$ $g^{(3)}_1$ $g^{(3)}_2$ $g^{(3)}_3$ $\alpha^{(1)}$ $\alpha^{(2)}$ $\alpha^{(3)}$
----- ---------------------- ----------- ----------- ----------- ------------ ------------- ------------- ------------- ------------- ------------- ------------- ------------- ------------- ------------- ---------------- ---------------- ----------------
\#1 -1.95(4) 1.5(1) 2.2(1) – 0 0.020(2) 0.65(1) – 0.25(2) -1.8(2) – – – – 3.1(2) 1.4(1) –
\#2 -1.93(3) 1.2(1) 1.6(1) 4.7(4) -0.06(1) 0.0(7) 1.3(2) 0.5(1) 0.10(2) 0.0(4) -0.8(2) 0.5(1) -1(1) -12(2) 0.0(1) -0.4(2) 5.2(2)
: Parameter values for the mixing parameter $\kappa$ and the resonance parameters for the two fits performed. The uncertainties listed refer to the statistical uncertainty of the fit only. Masses and $\gamma_3$ are given in GeV and GeV$^{-2}$, respectively; all other couplings are dimensionless.\[tab:parameters\]
![Results for the ratio $r=\sigma_{e^+e^-\to ({\rm non 2 }
\pi)}^{\rm I=1}/\sigma_{e^+e^-\to \pi^+\pi^-}$. The red solid (green dashed) line denotes the result of the fit \#2, (\#1). Data are from the compilation of Ref. [@Simon]. The small step at $s=0.6$ GeV$^2$ results from $\rho$-$\omega$ mixing present in the denominator of $r$. \[fig:FFnon2pi\]](FFnon2pioverFF2pi.eps){width="0.7\linewidth"}
Two different fits were performed using the MINUIT package of the CERN library: the goal of fit \#1 is to find the minimal parameter set necessary to get an acceptable description of the pion vector form factor. In line with previous studies we find that at least two resonances and one inelastic channel are needed. In addition we may chose all $\gamma_i=0$ (c.f. Eq. (\[VRdef\])). The fit result is shown as the dashed green line in Fig. \[fig:pionFF\] for the pion vector form factor and in Fig. \[fig:phases\] for $p$–wave phases and inelasticity. However, it turned out that with this parameter set it was impossible to simultaneously also describe $r$ — see green dashed line in Fig. \[fig:FFnon2pi\]. Therefore we performed a second fit, fit \#2, with the goal to get a decent description of both $F_V$ as well as $r$. This called for an inclusion of 2 inelastic channels — in addition to the 4$\pi$ channel already included in fit \#1 we now also include the $\pi\omega$ channel — 3 resonances as well as a non–vanishing value of $\gamma_3$. Fit \#2 has 17 adjustable parameters. The fit results are shown as the red solid lines in Figs. \[fig:phases\], \[fig:pionFF\] and \[fig:FFnon2pi\] for phases and inelasticities, the pion vector form factor from $e^+e^-$ annihilation and $\tau$ decays, and the ratio $r$, respectively. Note, in case of the pion vector form factor we fit to the BaBar data on $e^+e^-$ annihilation only, for it extends to higher energies [^3]. Especially, the $\tau$ data are not included in the fit — the result comes out as a prediction. The parameters determined in both fits are given in Tab. \[tab:parameters\]. It is important to note that for the pion vector form factor alone both fits are of similar quality: for fit \#1 and \#2 we have $\chi^2/d.o.f$=1.3 and 1.2, respectively. However, only fit \#2 provides an acceptable description for $r$. This result nicely illustrates that one should not analyze the pion vector form factor without looking at the non 2$\pi$ channels at the same time — to our knowledge in this work a combined analysis was performed for the first time. Note that at $s=2$ GeV$^2$ Ref. [@Simon] reports a value of $r\sim 26$, which shows that already at this relatively low energy the $2\pi$ channel provides only a small fraction of the $e^+e^-$ annihilation rate in the $I=1$ channel.
The $s$ dependence of the non 2$\pi$ data shown in Fig. \[fig:FFnon2pi\] calls at least for two inelastic channels, since there is a change in slope visible at around $s\sim 0.9$. In fit \#2 this is accounted for by the inclusion of the $\pi\omega$ channel. However, even this three channel fit is still too simplified, for there should be not only correlations amongst the 4 pions in channel 2 included, e.g. from $a_1\pi$, $\rho\rho$, and $\rho \sigma$. In addition there are also channels like $\eta\pi\pi$. The data included in the current study does not allow one to disentangle these and therefore to improve the description of $r$ further. What is necessary is an inclusion of the large number of exclusive measurements available from $e^+e^-$ annihilation. We leave this to a future study.
It is important to stress that the mass parameters given in the table are bare parameters that get renormalized by the self energies. It is therefore possible that the resonance poles come out similar in both fits, for the unitarization effects are different. However, we postpone the determination of pole positions and residues, which requires the evaluation of the elastic $T$–matrix $\tilde T$ in the complex plane, to a later work.
![Left panel: Fit result for the pion vector form factor at space–like energies. The lines are the same as in Fig. \[fig:pionFF\]. Data are from Ref. [@spacelikecollection; @Na7FF]. Right panel: phase difference between $\psi$, the phase of the form factor, and the scattering phase shift $\delta$. The data shown indicate the upper bound of the phase shift difference presented in Ref. [@Simon]. \[fig:pionFFsmalls\]](piFF_spacelike.eps "fig:"){width="0.4\linewidth"} ![Left panel: Fit result for the pion vector form factor at space–like energies. The lines are the same as in Fig. \[fig:pionFF\]. Data are from Ref. [@spacelikecollection; @Na7FF]. Right panel: phase difference between $\psi$, the phase of the form factor, and the scattering phase shift $\delta$. The data shown indicate the upper bound of the phase shift difference presented in Ref. [@Simon]. \[fig:pionFFsmalls\]](phasedifference.eps "fig:"){width="0.4\linewidth"}
At small, space like energies the results of the two fits give identical results, c.f. left panel of Fig. \[fig:pionFFsmalls\]. Both lead to a pion radius slightly enhanced compared to what comes from the Omnès function itself, shown by the dot–dashed line: through the inclusion of the high-lying resonances the mean square charge radius of the pion increases by nearly 10% from 0.40 fm$^2$ to 0.44 fm$^2$. The latter value is consistent with the values extracted in Refs. [@eta2pipigamma; @gilberto]. For the curvature we find $c_V=3.9$ GeV$^{-4}$, in line with Refs. [@ours; @ana]. The effect of the higher resonances is quantitatively in line with expectations from dimensional analysis that predicts an effect on the mean square radius of order of the square of the inverse resonance mass $\sim 0.02$ fm$^2$. At higher space like energies the results move appart from each other. While fit \# 2 is consistent with the largely model independent bounds for the form factor derived in Ref. [@ananew], fit \# 1 (the form factor from the Omnes function) is at most marginally consistent (inconsistent).
The left panel of Fig. \[fig:phases\] nicely illustrates that within the formalism presented the high accuracy phase shifts up to 1 GeV are reproduced very well. One also finds that the phase shifts for the full model largely agree to the input phase in the whole energy range considered (this is not the case for the phase of the form factor, as discussed in the next paragraph) as well as to the data of Ref. [@hyams; @hyams1] [^4]. This happens, since the resonance couplings to the $\pi\pi$ channel are rather small — the resonances show up prominently in the form factor only due to large couplings to the photon. Especially, the present model can not account for the significant inelasticity visible in the data of Ref. [@hyams1] — the according to Ref. [@WOnew] preferred solution $(-+-)$ shows an inelasticity of the 0.8 already at 1 GeV. At this point in time it is not possible to decide whether this failure is an indication of a short coming of the model used here, or of the data of Refs. [@hyams; @hyams1]. What might support the latter conjecture is that the data on $\eta$ of Refs. [@hyams; @hyams1] are in disagreement with both the analysis of Ref. [@propo] as well as that of Ref. [@madrid_new] at around 1 GeV.
In the elastic regime the phase of the form factor has to agree to the phase of elastic scattering — a fact known as Watson theorem. At higher energies this connection is lost. In the right panel of Fig. \[fig:pionFFsmalls\] we show the difference between $\psi$, the phase of the form factor, and the scattering phase shift $\delta$. Also shown in the panel is the allowed upper bound of the phase shift difference given in Ref. [@Simon]. As one can see our amplitudes largely exhaust the range allowed by unitarity. This reflects again the fact that in the present formalism all resonances besides the $\rho(770)$ couple to elastic scattering only weakly.
Summary and Outlook {#sec:summary}
===================
In this paper a formalism was presented that allows for a simultaneous description of both $\pi\pi$ scattering data as well as form factors without the need to model the low energy regime: at low energies $\pi\pi$ phases can be used as input directly. At higher energies the formalism maps smoothly onto the well known $N/D$ method which is similar to the $K$–matrix approach, however, with improved analytical properties. As an example in this paper the formalism was applied to pion pairs in the $p$-wave. An excellent description is found for the pion vector form factor in both the neutral channel — from $e^+e^-$ annihilations, with $\rho$-$\omega$ mixing included — as well as the charged channel — from $\tau$ decays. In addition we also found a qualitative agreement with data on the non 2$\pi$ channels from $e^+e^-$ annihilations — these data were studied within a dynamical model here for the first time.
We found, however, that within the given formalism it was not possible to to describe the behavior of the inelasticity given in Refs. [@hyams; @hyams1]. At this point in time we are not able to judge if this deviation indicates a short coming of the model or points at a problem in the data. However, the observation that the values of $\eta$ of Refs. [@hyams; @hyams1] are in disagreement with the analyses of Refs. [@madrid_new; @propo] at $s\sim
1$ GeV$^2$ might indicate that there is a problem in the data of Refs. [@hyams; @hyams1] also at higher energies.
The formalism described here can be applied to all partial waves, especially also the isoscalar $s$–wave. Here, however, it is less clear what to use for the elastic phase shift $\tilde \delta$, since the pronounced structure from the $f_0(980)$, which also couples strongly to $\bar KK$, shows up already short after the phase reached 90$^0$. We leave this study to a future work.
[**Acknowledgment**]{}
I thank Simon Eidelman, Martin Hoferichter, Bastian Kubis, Ulf-G. Meißner and Juan M. Nieves for useful and inspiring discussions and comments to the manuscript and Wolfgang Ochs for useful remarks about the data of Refs. [@hyams; @hyams1].
The two potential formalism
===========================
Let us assume that there is a sensible way to split the scattering potential into two pieces (as in the main text the potentials $\tilde V$ and $V_R$, the vertex function $\Gamma$ as well as the T-matrices are matrices in channel space, while the form factor $F$ and the production vertices $M$ are vectors in channel space) $$V=\tilde V+V_R \ .$$ We will show in this Appendix that the full $T$–matrix can be split accordingly. In operator form the Bethe–Salpeter equation for the $T$ matrix may be written as $$T=V+VGT = \tilde V + V_R + (\tilde V+V_R)GT \ .$$ Here $G_i$ denotes the operator for the integral over the $n$–particle intermediate state of channel $i$, e.g. for the two–$\pi$ intermediate state we have $$VGV\propto \frac1{i}\int \frac{d^4k}{(2\pi)^4}V(k,..)
\frac1{k^2-m^2+i\epsilon}
\frac1{(k-P)^2-m^2+i\epsilon}V(k,..) \ ,$$ where $P$ denotes the total 4–momentum of the system, $P^2=s$. Note, not all arguments of the potential $V$ are shown explicitly. Introducing $\tilde T$ as the solution of $$\tilde T=\tilde V+\tilde V G\tilde T$$ and the dressed vertex functions (to simplify notations, in the appendix we do not show the centrifugal barrier factors $\xi$ explicitly) $$\Gamma_{\rm out} = 1+\tilde TG \quad \mbox{and} \quad \Gamma_{\rm in}^\dagger = 1+G\tilde T$$ we get $$T=\tilde T+T_R=\tilde T+V_R\Gamma_{\rm in}^\dagger+(V_RG+\tilde VG)T_R \ .$$ Due to time reversal invariance we have $\Gamma_{\rm out}=\Gamma_{\rm in}^\dagger$. Since disc($G$)=$2i\sigma$ and disc$(\tilde T)=2i\sigma \tilde T^*\tilde T$ one has $$\mbox{disc}(\Gamma_{\rm out}(s))=\mbox{disc}(\tilde T)G+\tilde
T^*\mbox{disc}(G)=\sigma \tilde T(s)^*\Gamma_{\rm out}(s) \ ,$$ thus $\Gamma_{\rm out}$ holds the unitarity relation for a form factor in a channel where the interactions are given by $\tilde T$.
We may therefore define $$T_R=\Gamma_{\rm out}t_R \Gamma_{\rm in}^\dagger$$ and derive with $\Sigma=G\Gamma_{\rm out}$ and $\tilde VG\Gamma_{\rm out}=\Gamma_{\rm out}-1$ $$t_R=V_R+V_R\Sigma t_R \quad \longrightarrow \quad t_R = \left[1-V_R\Sigma \right]^{-1} V_R \ .$$ From the definition above the discontinuity of the self–energy $\Sigma$ is found to be $$\mbox{disc}(\Sigma)=\mbox{disc}(G)\Gamma_{\rm out}+G^*\mbox{disc}(\Gamma_{\rm out})
=2i\underbrace{\left(1+G^*\tilde T^*\right)}_{\Gamma_{\rm
out}^*}\sigma\Gamma_{\rm out} \ ,$$ which was used to derive Eq. (\[Ldef\]).
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[^1]: On the Lagrangian level this means a coupling of the resonance to the photon field via $F^{\mu\nu}\partial_\mu V_\nu$ with $F^{\mu\nu}$ for the electro–magnetic field strength tensor and $V_\nu$ for the resonance field.
[^2]: This is correct up to an unphysical pole located at $s=-\Lambda^2$, which, however, does not influence visibly the amplitude for $s>-\Lambda^2$.
[^3]: In this exploratory study we regard this as appropriate since in this work we do not perform an uncertainty estimate of the parameters extracted.
[^4]: Only two of the 4 solutions presented in that paper are given, for the other two are in strong discrepancy with the phase shifts of Ref. [@madrid_new].
|
---
abstract: |
A *network creation game* simulates a decentralized and non-cooperative building of a communication network. Informally, there are $n$ players sitting on the network nodes, which attempt to establish a reciprocal communication by activating, incurring a certain cost, any of their incident links. The goal of each player is to have all the other nodes as close as possible in the resulting network, while buying as few links as possible. According to this intuition, any model of the game must then appropriately address a balance between these two conflicting objectives. Motivated by the fact that a player might have a strong requirement about its centrality in the network, in this paper we introduce a new setting in which if a player maintains its (either *maximum* or *average*) distance to the other nodes within a given associated *bound*, then its cost is simply equal to the *number* of activated edges, otherwise its cost is unbounded. We study the problem of understanding the structure of associated pure Nash equilibria of the resulting games, that we call <span style="font-variant:small-caps;">MaxBD</span> and <span style="font-variant:small-caps;">SumBD</span>, respectively. For both games, we show that computing the best response of a player is an -hard problem. Next, we show that when distance bounds associated with players are *non-uniform*, then equilibria can be arbitrarily bad. On the other hand, for <span style="font-variant:small-caps;">MaxBD</span>, we show that when nodes have a *uniform* bound $R$ on the maximum distance, then the *Price of Anarchy* (PoA) is lower and upper bounded by $2$ and $O\left(n^{\frac{1}{\lfloor\log_3
R\rfloor+1}}\right)$ for $R \ge 3$ (i.e., the PoA is constant as soon as the bound on the maximum distance is $\Omega(n^{\epsilon})$, for some $\epsilon>0$), while for the interesting case $R=2$, we are able to prove that the PoA is $\Omega(\sqrt{n})$ and $O(\sqrt{n \log n} )$. For the uniform <span style="font-variant:small-caps;">SumBD</span> we obtain similar (asymptotically) results, and moreover we show that the PoA becomes constant as soon as the bound on the average distance is $n^{\omega\left(\frac{1}{\sqrt{\log n}}\right)}$.
author:
- Davide Bilò
- Luciano Gualà
- Guido Proietti
title: 'Bounded-Distance Network Creation Games[^1]'
---
Game Theory, NP-hardness, Nash Equilibria, Network Creation Game.
Introduction
============
Communication networks are rapidly evolving towards a model in which the constituting components (e.g., routers and links) are activated and maintained by different owners, which one can imagine as players sitting on the network nodes. When these players act in a selfish way with the final intent of creating a connected network, the challenge is exactly to understand whether the pursuit of individual profit is compatible with the attainment of an equilibrium status for the system (i.e., a status in which players are not willing to move from), and how the social utility for the system as a whole is affected by the selfish behavior of the players. This task, which involves both computational and economical issues of the system, is exactly the aim of a research line which started with the seminal paper of Fabrikant *et al.* [@FLM03], where the by now classic *network creation game* (NCG) was initially formalized and investigated.
#### Definition of the NCG.
In its original formulation, the NCG is defined as follows: We are given a set of $n$ players, say $V$, where the strategy space of player $v \in V$ is the power set $2^{V \setminus\{v\}}$. Given a combination of strategies $S=(S_v)_{v \in V}$, let $G(S)$ denote the underlying undirected graph whose node set is $V$, and whose edge set is $E(S)=\{\cup_{v \in V} (v \times S_v)\}$. Then, the *cost* incurred by player $v$ under $S$ is
$$\label{cost} \mathit{cost}_v(S) = \alpha \cdot |S_v| + \sum_{u \in V}
d_{G(S)}(u,v)$$
where $d_{G(S)}(u,v)$ is the distance between nodes $u$ and $v$ in $G(S)$. Thus, the cost function implements the inherently antagonistic goals of a player, which on the one hand attempts to buy as little edges as possible, and on the other hand aims to be as close as possible to the other nodes in the outcoming network. These two criteria are suitably balanced in (\[cost\]) by making use of the parameter $\alpha \geq 0$. Consequently, the *Nash Equilibria*[^2] (NE) space of the game is heavily influenced by $\alpha$, and the corresponding characterization must be given as a function of it. The state-of-the-art for the *Price of Anarchy* (PoA) of the game, that we will call henceforth <span style="font-variant:small-caps;">Sum</span>NCG, is summarized in [@MS10], where the most recent progresses on the problem have been reported.
#### Further NCG models.
A criticism made to the classic NCG model is that the parameter $\alpha$ is in a sense exogenous to the system. Moreover, usage and building cost are summed up together in the player’s cost, and this mixing is reflected in the social cost of the resulting network. As a consequence, we have that in this game the PoA alone does not say so much about the structural properties of the network, such as density, diameter, or routing cost. This gave rise to a sequence of new NCG models. A first natural variant of <span style="font-variant:small-caps;">Sum</span>NCG was introduced in [@DHM07], where the authors redefined the player cost function as follows
$$\label{max} \mathit{cost}_v(S) = \alpha \cdot |S_v| + \max \{d_{G(S)}(u,v):u \in V\}.$$
This variant, named <span style="font-variant:small-caps;">Max</span>NCG, received further attention in [@MS10], where the authors improved the PoA of the game on the whole range of values of $\alpha$. However, <span style="font-variant:small-caps;">Max</span>NCG still incorporates in its definition the parameter $\alpha$. In an effort of defining new parameter-free models, in [@LPR08] the authors proposed an interesting variant in which a player $v$, when forming the network, has a limited *budget* $b_v$ to establish links to other players. This way, the player cost function restricts to the usage cost, namely either the maximum or the total distance to other nodes. In particular, in [@LPR08] the authors focused on the latter measure. For this *bounded-budget* version of the game, that we call <span style="font-variant:small-caps;">Sum</span>BB, they showed that determining the existence of NE is -hard. On a positive side, they proved that for uniform budgets, say $k$, <span style="font-variant:small-caps;">Sum</span>BB always admits a NE, and that its *Price of Stability* (PoS) is 1, while its PoA is $\Omega\left(\sqrt{\frac{n/k}{\log_k
n}}\right)$ and $O\left(\sqrt{\frac{n}{\log_k n}}\right)$. Notice that in <span style="font-variant:small-caps;">Sum</span>BB, links are seen as directed. Thus, a natural extension of the model was given in [@EFM11], were the undirected case was considered. For this, it was proven that both <span style="font-variant:small-caps;">Max</span>BB and <span style="font-variant:small-caps;">Sum</span>BB always admit a NE. Moreover, the authors showed that the PoA for <span style="font-variant:small-caps;">Max</span>BB and <span style="font-variant:small-caps;">Sum</span>BB is $\Omega(\sqrt{\log n})$ and $2^{O(\sqrt{\log
n})}$, respectively, while in the special case in which the budget is equal to 1 for all the players, the PoA is $O(1)$ for both versions of the game.
In all the above models it must be noticed that, as stated in [@FLM03], for a player it is -hard to find a best response once that the other players’ strategies are fixed. To circumvent this problem, in [@ADH10] the authors proposed a further variant, called *basic NCG* (BNCG), in which given some existing network, the only improving transformations allowed are *edge swaps*, i.e., a player can only modify a *single* incident edge, by either replacing it with a new incident edge, or by removing it. This naturally induces a weaker concept of equilibrium for which a best response of a player can be computed in polynomial time. In this setting, the authors were able to give, among other results, an upper bound of $2^{O(\sqrt{\log
n})}$ for the PoA of <span style="font-variant:small-caps;">Sum</span>BNCG, and a lower bound of $\Omega(\sqrt{n})$ for the PoA of <span style="font-variant:small-caps;">Max</span>BNCG. However, as pointed out in [@MS10], the fact that now an edge has not a specific owner, prevents the possibility to establish any implications on the PoA of the classic NCG, since a NE in a BNCG is not necessarily a NE of a NCG.
Finally, another NCG model which is barely related to the NCG model we study in this paper has been addressed in [@BS08].
#### Our results.
In this paper, we propose a new NCG variant that complements the model proposed in [@EFM11]. More precisely, we assume that the cost function of each player only consists of the number of bought edges (without any budget on them), but with the additional constraint that a player $v$ needs to connect to the network by staying within a given (either *maximum* or *average*) *distance*, say (either $R_v$ or $D_v$), to the set of players. Our model is motivated by the fact that in a realistic scenario, a player might have a strong objective about its centrality in the created network, and this can only be guaranteed by means of our approach.
For this bounded-distance version of the NCG, we address the problem of understanding the structure of the NE associated with the two variants of the game, that we denote by <span style="font-variant:small-caps;">MaxBD</span> and <span style="font-variant:small-caps;">SumBD</span>. To this respect, we first show that both games can have an unbounded PoA as soon as players hold at least two different distance bounds. Moreover, in both games, computing a best response for a player is -hard. These bad news are counterbalanced by the positive result we get for *uniform* distance bounds. In this case, first of all, the PoS for <span style="font-variant:small-caps;">MaxBD</span> is equal to 1, while for <span style="font-variant:small-caps;">SumBD</span> is at most equal to 2. Then, as far as the PoA is concerned, let $R$ and $D$ denote the uniform bound on the maximum and the average distance, respectively. We show that
1. for <span style="font-variant:small-caps;">MaxBD</span>, the PoA is lower and upper bounded by $2$ and $O\left(n^{\frac{1}{\lfloor\log_3
R\rfloor+1}}\right)$ for $R \ge 3$, while for $R=2$ is $\Omega(\sqrt{n})$ and $O(\sqrt{n \log n} )$; thus, the PoA is constant as soon as $R=\Omega(n^{\epsilon})$, for some $\epsilon>0$;
2. for <span style="font-variant:small-caps;">SumBD</span>, the PoA is lower bounded by $2-\epsilon$, for any $\epsilon > 0$, as soon as $D \geq 2-3/n$, while it is upper bounded as reported in Table \[UBsum\].
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
[$D$]{} $\in [2,3)$ $\ge 3$ and $O(1)$ $\omega(1) \cap O\left(3^{\sqrt{\log n}}\right)$ $\omega\left(3^{\sqrt{\log n}}\right) \cap n^{O\left(\frac{1}{\sqrt{\log n}}\right)}$ $n^{\omega\left(\frac{1}{\sqrt{\log n}}\right)}$
--------- --------------------------- ----------------------------------- -------------------------------------------------- --------------------------------------------------------------------------------------- --------------------------------------------------
PoA $O\left(\sqrt{n \log {n}} $O\left(n^{\frac{1}{\lfloor\log_3 $2^{O\left(\sqrt{\log n}\right)}$ $O\left(n^{\frac{1}{\lfloor\log_3 $O(1)$
\right)$ D/4\rfloor+2}}\right)$ D/4\rfloor+2}}\right)$
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Obtained PoA upper bounds for <span style="font-variant:small-caps;">SumBD</span>.[]{data-label="UBsum"}
The paper is organized as follows. After giving some basic definitions in Section \[sct:definitions\], we provide some preliminary results in Section \[sct:preliminary\]. Then, we study upper and lower bounds for <span style="font-variant:small-caps;">MaxBD</span> and <span style="font-variant:small-caps;">SumBD</span> in Sections \[sct:max\] and \[sct:sum\], respectively. Finally, in Section \[sct:conclusions\] we conclude the paper by discussing some intriguing relationships of our games with the famous graph-theoretic *degree-diameter* problem.
Problem Definition {#sct:definitions}
==================
#### Graph terminology.
Let $G=(V,E)$ be an undirected (simple) graph with $n$ vertices. For a graph $G$, we will also denote by $V(G)$ and $E(G)$ its set of vertices and its set of edges, respectively. For every vertex $v \in V$, let $N_G(v):=\{u \mid u\in V\setminus\{v\}, (u,v)\in E\}$. The [*minimum degree*]{} of $G$ is equal to $\min_{v \in V}|N_G(v)|$.
We denote by $d_G(u,v)$ the [*distance*]{} in $G$ from $u$ to $v$. The *eccentricity* of a vertex $v$ in $G$, denoted by $\varepsilon_G(v)$, is equal to $\max_{u \in V} d_G(u,v)$. The [*diameter*]{} and the *radius* of $G$ are equal to the maximum and the minimum eccentricity of its nodes, respectively. A node is said to be a *center* of $G$ if $\varepsilon_G(v)$ is equal to the radius of $G$. We define the *broadcast cost* of $v$ in $G$ as $B_G(v)=\sum_{u \in V}d_G(u,v)$, while the *average distance* from $v$ to a node in $G$ is denoted by $D_G(v)=B_G(v)/n$.
A *dominating set* of $G$ is a subset of nodes $U \subseteq
V$ such that every node of $V \setminus U$ is adjacent to some node of $U$. We denote by $\gamma(G)$ the cardinality of a minimum cardinality dominating set of $G$. Moreover, for any real $k \ge
1$, the $k$th power of $G$ is defined as the graph $G^k=(V,E(G^k))$ where $E(G^k)$ contains an edge $(u,v)$ if and only if $d_G(u,v) \le k$.
Let $U\subseteq V$ be a set of vertices, we denote by $G[U]$ the subgraph of $G$ induced by $U$. Let $F\subseteq \{(u,v)\mid u,v \in V, u\neq v\}$. We denote by $G+F$ (resp., $G-F$) the graph on $V$ with edge set $E\cup F$ (resp., $E\setminus F$). When $F=\{e\}$ we will denote $G+\{e\}$ (resp., $G-\{e\}$) by $G+e$ (resp., $G-e$). For two graphs $G_1$ and $G_2$, we denote by $G_1\cup G_2$ the graph with $V(G_1\cup G_2)=V(G_1)\cup V(G_2)$, and $E(G_1\cup G_2)=E(G_1)\cup
E(G_2)$.
#### Problem statements.
The *bounded maximum distance* NCG (<span style="font-variant:small-caps;">MaxBD</span>) is defined as follows: Let $V$ be a set of $n$ nodes, each representing a selfish player, and for any $v \in V$, let $R_v >0$ be an integer representing a bound on the eccentricity of $v$. The strategy of a player $v$ consists of a subset $S_v \subseteq V \setminus \{v\}$. Denoting by $S$ the strategy profile of all players, let $G(S)$ be the undirected graph with node set $V$, and with edge set $E(S)=\{\cup_{v \in V} (v \times S_v)\}$. When $u \in S_v$, we will say that $v$ is buying the edge $(u,v)$, or that the edge $(u,v)$ is bought by $v$. Then, the cost of a player $v$ in $S$ is $\mathit{cost}_v(S)=|S_v|$ if $\varepsilon_{G(S)}(v) \le R_v$, $+\infty$ otherwise.
The *bounded average distance* NCG (<span style="font-variant:small-caps;">SumBD</span>) is defined analogously, with a bound $D_v$ on the average distance, and cost function $\mathit{cost}_v(S)=|S_v|$ if $D_{G(S)}(v) \le D_v$, $+\infty$ otherwise. In the rest of the paper, depending on the context, we will interchangeably make use of the bound on the broadcast cost $B_v=D_v \cdot n$ when referring to <span style="font-variant:small-caps;">SumBD</span>.
In both variants, we say that a node $v$ is *within the bound in $S$* if $\mathit{cost}_v(S)<+\infty$. We measure the overall quality of a graph $G(S)$ by its social cost $\mathit{SC}(S)=\sum_{v \in V} \mathit{cost}_v(S)$. A graph $G(S)$ minimizing $\mathit{SC}(S)$ is called *social optimum*.
We use the Nash Equilibrium (NE) as solution concept. More precisely, a NE is a strategy profile $S$ in which no player can decrease its cost by changing its strategy assuming that the strategies of the other players are fixed. When $S$ is a NE, we will say that $G(S)$ is *stable*, and that a graph $G$ is stable if there exists a strategy profile $S$ such that $G=G(S)$. Notice that in both games, when $S$ is a NE, all nodes are within the bound and, since every edge is bought by a single player, $\mathit{SC}(S)$ coincides with the number of edges of $G(S)$.
We conclude this section by recalling the definition of the two measures we will use to characterize the NE space of our games, namely the *Price of Anarchy* (PoA) [@FLM03] and the *Price of Stability* (PoS) [@ADT03], which are defined as the ratio between the highest (respectively, the lowest) social cost of a NE and the cost of a social optimum.
Preliminary results {#sct:preliminary}
===================
First of all, observe that for <span style="font-variant:small-caps;">MaxBD</span> it is easy to see that a stable graph always exists. Indeed, if there is at least one node having distance bound 1, then the graph where all 1-bound nodes buy edges towards all the other nodes is stable. Otherwise, any spanning star is stable. Notice that any spanning star is stable for <span style="font-variant:small-caps;">SumBD</span> as well, but only when all vertices have a bound of at least $2n-3$, while the problem of understanding whether a NE always exists for the remaining values is open. From these observations, we can derive the following negative result:
The PoA of <span style="font-variant:small-caps;">MaxBD</span> and <span style="font-variant:small-caps;">SumBD</span> (with distance bounds $B_v \ge 2n-3$) is $\Omega(n)$, even for only two distance-bound values.
We will define a graph $G$ with $\Omega(n^2)$ edges, and we will prove that $G$ is stable for both versions of the game. Then, we will show that in both cases the cost of the social optimum is $n-1$.
The graph $G$ is defined as follows. We have a clique of $k$ nodes. For each node $v$ of the clique, we add four nodes $v^1_1,v^1_2,v^2_1,v^2_2$ and four edges $(v^1_2,v^1_1),
(v^1_1,v),(v^2_2,v^2_1), (v^2_1,v)$. Clearly, $G$ has $n=5k$ nodes and $\Omega(n^2)$ edges. Now, consider a strategy profile $S$ with $G=G(S)$ and such that (i) every edge is bought by a single player, and (ii) the edges $(v^j_2,v^j_1), (v^j_1,v)$ are bought by $v^j_2$ and $v^j_1$, respectively, $j=1,2$. Now, we show that $S$ is a NE, once we have defined suitable bounds for the players.
Let us consider <span style="font-variant:small-caps;">MaxBD</span> first. We set the bound of every node of the clique to 3, while all the other nodes have bound 5. Trivially, all nodes are within the bound. Moreover, a node $v^j_i$ is buying only one edge and, since the removal of such edge disconnects the graph, $v^j_i$ cannot decreases its cost. Let $v$ be a node of the clique, and assume that $v$ is buying $h$ edges in $S$. Let $S'$ be a strategy profile where $v$ switches its strategy $S_v$ with $S'_v$ and such that $|S'_v|<h$. Since $h
\le n-1$, it must exist a vertex $u$ of the clique such that $u
\in S_v$ and $u,u^1_1,u^1_2,u^2_1,u^2_2 \notin S'_v$, from which we have that $v$ cannot be within the bound in $S'$ since $d_{G(S')}(v,u^2_2)
> 3$.
Concerning <span style="font-variant:small-caps;">SumBD</span>, we set the bound of each node $v$ of the clique to $\sum_{u \in V}d_G(v,u)=11k-5 > 2n-3$, while we assign to all the other nodes bound $n^2$. Similar arguments used for <span style="font-variant:small-caps;">MaxBD</span> can be used to show that $S$ is a NE for <span style="font-variant:small-caps;">SumBD</span> as well.
To conclude the proof, observe that any star (with cost $n-1$) is a social optimum for the two instances of <span style="font-variant:small-caps;">MaxBD</span> and <span style="font-variant:small-caps;">SumBD</span> given above.
Given the above bad news, from now on we focus our attention on the *uniform* case of the games, i.e., that in which all the bounds on the distances are the same, say $R$ and $D$ (i.e., $B=D \cdot n$) for the maximum and the average version, respectively. Similarly to other NCGs, also here we have the problem of computing a best response for a player, as stated in the following theorem.
\[th:best response\] Computing the best response of a player in <span style="font-variant:small-caps;">MaxBD</span> and <span style="font-variant:small-caps;">SumBD</span> is -hard.
Let us consider <span style="font-variant:small-caps;">MaxBD</span> first. The reduction is from the -hard *minimum dominating set* problem which, given a graph $G'=(V',E')$, asks for finding a dominating set of $G'$ of minimum cardinality, say $\gamma(G')$. Let $N=|V'|$. We build a graph $G$ with $n=N+2N(R-2)
+ 1$ vertices as follows: We have an isolated vertex $u$, a copy of $G'$, and two paths of length $R-2$ appended to every vertex $v \in V'$. Now, let $S$ be the strategy profile such that $G=G(S)$. Clearly, $\mathit{cost}_u(S)=+\infty$, and it is easy to see that $u$ has a strategy yielding a cost of $k$ if and only if $\gamma(G') \le k$.
Now, for <span style="font-variant:small-caps;">SumBD</span>, we sketch a reduction from the *$k$-median* problem. Let $G'=(V',E')$ be an instance of the $k$-median problem which, given a value $\beta$, asks for finding a subset $U \subseteq V$ of size $k$ such that $\sum_{v \in V}
\min_{u \in U}d_G(u,v) \le \beta$. This problem is -hard even when $G'$ is an unweighted graph [@KH79]. Let $G$ be the graph defined as $G'$ with an additional isolated node $u$, and let $S$ be a strategy profile such that $G=G(S)$, and let $B=\beta+N$, where $N=|V'|$. It is easy to see that $u$ has a strategy yielding a cost of $k$ if and only if $G'$ has a $k$-median of cost at most $\beta$.
On the other hand, a positive result which clearly implies that SUMBD always admits a pure NE is the following.
\[th:PoS\] The PoS of <span style="font-variant:small-caps;">MaxBD</span> is 1, while for <span style="font-variant:small-caps;">SumBD</span> is at most 2.
Concerning <span style="font-variant:small-caps;">MaxBD</span>, when $R=1$ the complete graph is a social optimum as well as the only stable graph. For $R>1$, let $T$ be a spanning star with center $c \in V$ and edges $(c,v)$, $v
\in V\setminus \{c\}$. Clearly, $T$ is a social optimum, and the strategy profile $S$ in which $S_v=\{c\}$ for every $v \in
V\setminus \{c\}$, and $S_c=\emptyset$, is a NE.
Concerning <span style="font-variant:small-caps;">SumBD</span>, observe that such a $T$ is an optimum as well as stable when $B
\ge 2n-3$. Now assume that $B = n-1+k$ with $0 \le k \le n-2$. We will define a graph $G$ with a number of edges that is at most twice the number of edges of the optimum, and we show that it is stable. Let $h,t \geq 0$ be s.t. $n=(k+1)h+t$. We partition $V$ into $h$ groups of $k+1$ nodes, say $V_1,\dots,V_h$ and, when $t
\neq 0$, an additional group $V_0$ of $t$ vertices. The edge set of $G$ is defined as $\{(u,v) \mid u \in V_i, v \in V_j, i \neq j
\}$. Let $S$ be a strategy profile such that $G=G(S)$ with the constraint that every node in $V_0$ buys no edge in $S$. Clearly, every node in $G$ is within the bound. Moreover, observe that in order to be within the bound, each node $v$ must have degree at least $n-1-k$. Now, since every node not in $V_0$ has degree exactly $n-1-k$, and since nodes in $V_0$ buy no edges, then $G(S)$ is stable. To bound the social cost of $G$, notice that the cost of the optimum, say <span style="font-variant:small-caps;">Opt</span>, is at least $\frac{n(n-1-k)}{2}$. Let us consider the case in which $k < n/2$. Then, $\textsc{Opt} \ge n^2/4$, while $\mathit{SC}(S) \le n^2/2
\le 2 \cdot \textsc{Opt}$. On the other hand, when $k \ge n/2$, we have only two groups, one with $t=n-k-1$ nodes and the other with $n-t$. Then, we have $\mathit{SC}(S)=t(n-t) \le t \, n \le 2 \cdot
\textsc{Opt}$.
We conclude this section by providing a lemma which will simplify the exposition of the remaining results.
\[lm:aux\] Let $G(S)$ be a stable graph and let $H$ be a subgraph of $G(S)$. If for each node $v$ there exists a set $E_v$ of edges (all incident to $v$) such that $v$ is within the bound in $H+E_v$, then $\mathit{SC}(S) \le |E(H)|+ \sum_{v \in V}|E_v|$.
Let $k_v$ be the number of edges of $H$ that $v$ is buying in $S$. If $v$ buys $E_v$ additionally to its $k_v$ edges, then $v$ will be within the bound. Hence, since $S$ is a NE, we have that $\mathit{cost}_v(S) \le k_v+|E_v|$, from which it follows that: $$\mathit{SC}(S)=\sum_{v \in V}\mathit{cost}_v(S) \le \sum_{v \in V}k_v + \sum_{v \in
V}|E_v|= |E(H)|+\sum_{v \in V} |E_v|.$$
PoA for <span style="font-variant:small-caps;">MaxBD</span> {#sct:max}
===========================================================
Upper bounds
------------
\[lm:PoA le gamma\] Let $G(S)$ be a NE, and let $\gamma$ be the cardinality of a minimum dominating set of $G(S)^{R-1}$, then $\mathit{SC}(S) \le
(\gamma+1)(n-1)$.
Let $U$ be a minimum dominating set of $G(S)^{R-1}$, with $\gamma=|U|$. It is easy to see that there is a spanning forest $F$ of $G(S)$ consisting of $\gamma$ trees $T_1, \dots,
T_{\gamma}$, such that every $T_j$ contains exactly one vertex in $U$, and when we root $T_j$ at such vertex the height of $T_j$ is at most $R-1$.
For a node $v \in V$, let $E_v=\{(v,u) \mid u \in
U\setminus\{v\}\}$. Clearly, $v$ is within the bound in $F+E_v$, hence by using Lemma \[lm:aux\], we have $$\mathit{SC}(S) \le |E(F)|+\sum_{u \in U}|E_u|+\sum_{v \in V\setminus
U}|E_v| = n-\gamma +(\gamma-1) \gamma+
\gamma(n-\gamma)\le(\gamma+1)(n-1).$$
Let $G(S)$ be a NE and let $v$ be a node of $G(S)$. Since $v$ is within the bound, the neigborhood of $v$ in $G$ is a dominating set of $G^{R-1}$. Therefore, thanks to Lemma \[lm:PoA le gamma\] we have proved the following corollary.
\[lm:PoA le delta\] Let $G(S)$ be a NE, and let $\delta$ be the minimum degree of $G(S)$, then $\mathit{SC}(S) \le (\delta+1)(n-1)$.
We are now ready to prove our upper bound to the PoA for the game.
\[th:UB PoA Max\] The PoA of <span style="font-variant:small-caps;">MaxBD</span> is $O(n^{\frac{1}{\lfloor\log_3
R\rfloor+1}})$ for $R \ge 3$, and $O( \sqrt{n \log n} )$ for $R=2$.
Let $G$ be a stable graph, and let $\gamma$ be the size of a minimum dominating set of $G^{R-1}$. We define the *ball* of radius $k$ centered at a node $u$ as $\beta_k(u)=\{v \mid d_G(u,v) \le
k\}$. Moreover, let $\beta_k=\min_{u \in V} |\beta_k(u)|$. The idea is to show that in $G$ the size of any ball increases quite fast as the radius of the ball increases.
For any $k \ge 1$, we have $\beta_{3k+1} \ge \min \{n,\gamma
\beta_k\}$.
Consider the ball $\beta_{3k+1}(u)$ centered at any given node $u$, and assume that $|\beta_{3k+1}(u)| \le n$. Let $T$ be the maximal set of nodes at distance exactly $2k+1$ from $u$ and subject to the distance between any pair of nodes in $T$ being at least $2k+1$. We claim that for every node $v \notin
\beta_{3k+1}(u)$, there is a vertex $t \in T$ with $d_G(t,v)<d_G(u,v)$. Indeed, consider the node $t'$ in the shortest path between $v$ and $u$ at distance exactly $2k+1$ from $u$. If $t' \in T$ the claim trivially holds, otherwise consider the node $t \in T$ that is closest to $t'$. From the maximality of $T$ we have that $d_G(t,v) \le d_G(t,t')+d_G(t',v) \le
2k+d_G(u,v)-(2k+1) < d_G(u,v)$.
As a consequence, we have that $T\cup\{u\}$ is a dominating set of $G^{R-1}$, and hence $|T|+1 \ge \gamma$. Moreover, all the balls centered at nodes in $T\cup\{u\}$ with radius $k$ are all pairwise disjoint. Then: $$|\beta_{3k+1}(u)|\ge |\beta_k(u)|+\sum_{t \in T} |\beta_k(t)| \ge
\gamma \beta_k.$$
Now, observe that since the neighborhood of any node is a dominating set of $G^{R-1}$, we have that $\beta_1 \ge \gamma$. Then, after using the above claim $x$ times, we obtain $$\beta_{\frac{3^{x+1}-1}{2}} \ge \min \{n,\gamma^{x+1}\}.$$
Let us consider the case $R\ge 3$ first. Let $U$ be a maximal independent set of $G^{R-1}$. Since $U$ is also a dominating set of $G^{R-1}$, it holds that $|U| \ge \gamma$. We consider the $|U|$ balls centered at nodes in $U$ with radius given by the value of the parameter $x=\lfloor\log_3 R-1\rfloor$. Every ball has radius at most $(R-1)/2$ and since $U$ is an independent set of $G^{R-1}$, all balls are pairwise disjoint and hence we have $n
\ge |U| \gamma^{\lfloor\log_3 R-1\rfloor + 1} \ge
\gamma^{\lfloor\log_3 R\rfloor+1}$. As a consequence, we obtain $\gamma \le n^{\frac{1}{\lfloor\log_3 R\rfloor+1}}$, and the claim now follows from Lemma \[lm:PoA le gamma\].
Now assume $R=2$. We use the bound given in [@AS92] to the size $\gamma(G)$ of a minimum dominating set of a graph $G$ with $n$ nodes and minimum degree $\delta$, namely $\gamma(G) \le
\frac{n}{\delta +1}H_{\delta+1}$, where $H_i=\sum_{j=1}^i 1/j$ is the $i$-th harmonic number. Hence, since a social optimum has cost $n-1$, from Lemma \[lm:PoA le gamma\] and Corollary \[lm:PoA le delta\], we have $\frac{\mathit{SC}(S)}{n-1} \le \min\{\delta +
1, \frac{n}{\delta+1} H_{\delta+1}+1 \}$, for any stable graph $G(S)$ with minimum degree $\delta$. The claim follows.
Lower bounds
------------
We first prove a simple constant lower bound for any value of $R=o(n)$, and then we show an almost tight lower bound of $\Omega(\sqrt{n})$ for $R=2$. We postpone to the concluding section a discussion on the difficulty of finding better lower bounds for large values of $R$.
For any $\epsilon>0$ and for $1<R=o(n)$, the PoA for <span style="font-variant:small-caps;">MaxBD</span> is at least $2-\epsilon$.
Assume we are given a set of $n=2R+h$ vertices $\{u_1,\dots,u_{2R}\}\cup\{v_1,\dots,v_h\}$. The strategy profile $S$ is defined as follows. Vertex $u_j$ buys a single edge towards $u_{j+1}$, for each $j=1,\dots,2R-1$, and every $v_i$ buys two edges towards $u_1$ and $u_{2R}$. It is easy to see that $G(S)$ has diameter $R$ and is stable. The claim follows from the fact that $SC(S)$ goes to $2(n-1)$ as $h$ goes to infinity and the fact that, as observed in Section \[sct:preliminary\], a spanning star (having social cost equal to $n-1$) is a social optimum.
We close this section by providing a much stronger lower bound for the special case in which $R=2$. Before stating the theorem, we give some additional notation. Let $v$ be a player, $S$ be a strategy profile, and $\ell$ a positive integer. We define $N_{S}^\ell(v)=\{u \mid u\in V, d_{G(S)}(u,v)\leq \ell\}$ and $\bar N_{S}^\ell(v)=V\setminus N_S^\ell(v)$. We will omit the superscript $\ell$ when $\ell=1$. Moreover, we denote by $\gamma(S,v,\ell)$ the size of a minimum cardinality set $X\subseteq V$ of vertices that [*dominates $\bar
N_{S}^\ell(v)$ in $G(S)^{\ell-1}$*]{}, i.e., for every vertex $u \in
\bar N_{S}^\ell(v)$ there exists a vertex $x \in X$ such that $d_{G(S)}(x,u)\leq \ell-1$, i.e., $(x,u)\in E(G(S)^{\ell-1})$. Finally, denote by $S_{\neg v}$ the strategy profile where each player but $v$ plays the same strategy as in $S$, while $v$ buys no edge, i.e., the strategy of $v$ is $\emptyset$. The following proposition, whose proof is straightforward, provides exact bounds to the cost incurred by each player in every connected graph, and will be used in the proof of the theorem.
\[prop: bounds for S\_v max game\] Let $G(S)$ be a connected graph. The cost incurred by each player $v$ in $S$ in <span style="font-variant:small-caps;">MaxBD</span> with bound $R$ is $|S_v|\geq\gamma(S_{\neg v},v,R)$. Moreover, a player $v$ is in equilibrium in $S$ iff $|S_v|=\gamma(S_{\neg v},v,R)$.
Let $S'$ be a strategy profile for a set of players $V$. A strategy profile $S$ for $V$ [*extends*]{} $S'$ if $S'_v\subseteq S_v$ for every $v \in V$. Let $S$ and $S'$ be two strategy profiles for a set of players $V$ such that $S$ extends $S'$, For every $v \in V$ let $N(S',S,v)=N_S(v)\setminus N_{S'}(v)$ and let $S^{v,S'}$ be the strategy profile such that $S^{v,S'}_v=S'_v\cup N(S',S,v)$ and $S^{v,S'}_u=S'_u\cup \{v' \mid v' \in S_u, v'\neq v\}$ for each $u \in V, u \neq v$. Observe that $G(S^{v,S'})=G(S)$.
The following proposition will be also used in the proof of the theorem.
\[prop: sufficient conditions for NE\] Let $V$ be a set of players and let $S,S'$ be two strategy profiles for $V$ such that $S$ extends $S'$. If every player $v$ is in equilibium in $S^{v,S'}$, then $G(S)$ is a stable graph.
For the sake of contradiction, assume that every player $v \in V$ is in equilibium in $S^{v,S'}$ but $G(S)$ is not stable. Then there exists a player $u$ and a strategy profile $S''$ such that (i) $S''_v=S_v$ for every $v \in V, v\neq u$, (ii) the eccentricity of $u$ in $G(S'')$ is less than or equal to $R$, and (iii) $|S''_u|<|S_u|$. Let $X=\{x\mid x \in S''_u, x \not\in S_u\}$ and let $Y=\{y\mid y \in S_u, y\not\in S''_u\}$. By (iii) we have $|X|<|Y|$. Let $\bar S$ be the strategy profile such that $\bar S_u=(S^{u,S'}_u\setminus Y)\cup X$ and $\bar S_v=S^{u,S'}_v$ for every $v \in V,v\neq u$. Clearly, $G(S'')=G(\bar S)$ and thus, by (ii) the eccentricity of $u$ in $G(\bar S)$ is less than or equal to 2. Furthermore, $|X|<|Y|$ implies $|\bar S_u|\leq |S^{u,S'}_u|$ and therefore $u$ is not in equilibrium in $S^{u,S'}$.
We are now ready to prove the following.
\[th:PoA R=2\] The PoA of <span style="font-variant:small-caps;">MaxBD</span> for $R=2$ is $\Omega(\sqrt{n})$.
Let $p\geq 3$ be a prime number. We provide a graph $G'$ of diameter 2 containing $O(p^2)$ vertices and $\Omega(p^3)$ edges and show that there exists a strategy profile $S$ such that $G(S)=G'$ and $G(S)$ is stable. $G'$ contains two vertex-disjoint rooted trees $T$ and $T'$ as subgraphs. $T$ is a complete $p$-ary tree of height 2. We denote by $r$ the root of $T$, by $C=\{c_0,\ldots,c_{p-1}\}$ the set of children of $r$, and by $V_i=\{v_{i,0},\ldots,v_{i,p-1}\}$ the set of children of $c_i$. $T'$ is a star with $p^2$ leaves rooted at the center $r'$. The leaves of $T'$ are partitioned in $p$ groups each having exactly $p$ vertices. For every $i=0,\ldots,p-1$, we denote by $U_i=\{u_{i,0},\ldots,u_{i,p-1}\}$ the set of vertices of group $i$. $G'=(V,E)$ has vertex set $V=V(T)\cup V(T')$ and edge set (see also Figure \[fig:lower bound sqrt n\]) $$\begin{aligned}
E &=& E(T)\cup E(T')\cup\{(r,r')\}\\
&\cup &\big\{(c,c')\mid c,c'\in C, c \neq c'\big\}\\
&\cup& \bigcup_{i=0}^{p-1}\big\{(u,u')\mid u,u'\in U_i, u\neq u'\big\}\\
&\cup &\big\{(u_{i,j},v_{i',j'})\mid i,i',j,j'\in[p-1],
j+i'i\equiv j' \pmod{p}\big\}.\end{aligned}$$
We claim that the diameter of $G'$ is 2. The eccentricity of $r$ is 2 as $T$ has height 2, $T'$ has height 1, and $G'$ contains the edge $(r,r')$. Observe that the subgraphs of $G'$ induced by $C$ and $U_i$, for all $i\in[p-1]$, are all cliques of $p$ vertices. Furthermore, by construction, there is an edge linking each vertex $u\in \bar U$ with some $v \in V_i$, for every $i\in[p-1]$, and thus $V_i$ dominates $\bar U$. Therefore, the eccentricity of each vertex in $C$ is 2. As a consequence, to prove that $G'$ has diameter 2, it is enough to prove that
1. $U_i$ dominates $\bar V$, for every $i\in[p-1]$ (so as each vertex in $\bar U$ would have eccentricity 2),
2. for every pair $v \in V_i$ and $v'\in V_{i'}$, $i,i'\in[p-1],i\neq
i'$, there is a vertex $u \in \bar U$ such that $(v,u),(v',u)\in
E$ (so as each vertex in $\bar V$ would have eccentricity 2).
To prove (i), simply observe that for every $i',j'\in[p-1]$, there always exists a $j\in[p-1]$ such that $j+i'i\equiv j' \pmod{p}$, and thus, $(u_{i,j},v_{i',j'}) \in E$. To prove (ii), observe that for every $v_{i,j},v_{i',j'}\in \bar V$, with $i\neq i'$, there always exists $i'',j''\in[p-1]$ such that $j''+ii''\equiv j
\pmod{p}$ and $j''+i'i''\equiv j' \pmod{p}$ as $p$ is a prime number (simply choose $i''$ such that $i''(i-i')\equiv(j-j') \pmod
p$). Therefore, $(v_{i,j},u_{i'',j''}),(v_{i',j'},u_{i'',j''})\in
E$.
To complete the proof, it remains to show that there exists a strategy profile $S$ such that $G(S)=G'$ and $G(S)$ is stable. Let $\bar V$ and $\bar U$ be the set of leaves of $T$ and $T'$, respectively. Let $S'$ be a strategy profile where:
- each vertex in $\bar V\cup\{r\}$ buys all edges incident to it (thus, each vertex in $\bar V\cup\{r\}$ buys exactly $p+1$ edges),
- each vertex in $\bar U$ buys the edge towards $r'$,
- each of the remaining vertices buys no edge, i.e., $S'_v=\emptyset$ for every $v \in C\cup\{r'\}$.
Let $S$ be any strategy profile that extends $S'$ such that $G(S)=G'$. Observe that
- $r'$ buys no edge in $S^{r,S'}$,
- each vertex $v$ in $C\cup \bar U$ buys exactly $p$ edges in $S^{v,S'}$,
- each vertex $v$ in $\bar V\cup\{r\}$ buys exactly $p+1$ edges in $S^{v,S'}$.
First of all observe that $$|\bar N_{S^{v,S'}_{\neg v}}^2(v)\cap \bar V|=
\begin{cases}
0 & \text{if $v=r'$;}\\
p^2 & \text{if $v=r$;}\\
p^2-1 & \text{if $v \in \bar V$;}\\
p(p-1) & \text{otherwise.}
\end{cases}$$
Let $\hat V\subseteq \bar V$ be such that $|\hat
V|\in\{p^2,p^2-1,p(p-1)\}$, and let $X$ be a set of vertices that dominates $\hat V$ in $G(S)$.
$|X|\geq \lceil\frac{|\hat
V|}{p}\rceil$ where equality holds only if $X\subseteq C$ or $X\subseteq\bar U$.
Let $X$ be a set of vertices that dominates $\hat V$ in $G(S)$ and observe that $X\subseteq \hat V \cup
C\cup\bar U$. As any vertex of $G(S)$ can dominate at most $p$ vertices of $\hat V$, we have that $|X|\geq \lceil\frac{|\hat
V|}{p}\rceil$.
Now we prove that if $|X|=\lceil\frac{|\hat
V|}{p}\rceil$ then either $X\subseteq C$ or $X\subseteq\bar U$. Indeed, any set $X'$ dominating $\hat V$ in $G(S)$ and containing any vertex of $\hat V$ has size $|X'|\geq 1+\lceil\frac{|\hat
V|-1}{p}\rceil >\lceil\frac{|\hat V|}{p}\rceil$, where the first inequality holds because any vertex in $\hat V$ dominates only itself while the second inequality holds by the choice of $|\hat
V|$ and because $p\geq 3$. Furthermore, any set $X'$ containing $0<k<\lceil\frac{|\hat
V|}{p}\rceil$ vertices of $C$ and $\lceil\frac{|\hat
V|}{p}\rceil-k$ vertices of $\bar U$ can dominate at most $pk+p(\lceil\frac{|\hat V|}{p}\rceil-k)-k(\lceil\frac{|\hat
V|}{p}\rceil-k)=p\lceil\frac{|\hat V|}{p}\rceil-k\lceil\frac{|\hat
V|}{p}\rceil+k^2\leq p \lceil\frac{|\hat
V|}{p}\rceil-\lceil\frac{|\hat V|}{p}\rceil + 1\leq|\hat V|-1$ vertices of $\hat V$ (and so $X'$ cannot dominate $\hat V$), where the last inequality holds by the choice of $|\hat V|$ and because $p\geq 3$.
As a consequence of the above claim, if $|X|=\lceil\frac{|\hat V|}{p}\rceil$, i.e., either $X
\subseteq C$ or $X \subseteq \bar{U}$, then either $X$ does not dominate $r$ in $G(S)$ or $X$ does not dominate $r'$ in $G(S)$. This implies that, $$\gamma(S^{v,S'}_{\neg v},v,2)=
\begin{cases}
0 & \text{if $v=r'$;}\\
p & \text{if $v \in C\cup\bar U$;}\\
p+1 & \text{otherwise.}
\end{cases}$$ Since $|S^{v,S'}_v|=\gamma(S^{v,S'}_{\neg v},v,2)$ for each player $v\in V$, from Proposition \[prop: bounds for S\_v max game\] we have that $v$ is in equilibrium w.r.t. $S^{v,S'}$. Therefore, by Proposition \[prop: sufficient conditions for NE\], $G(S)$ is stable.
<span style="font-variant:small-caps;">SumBD</span> {#sct:sum}
===================================================
Upper bounds
------------
For <span style="font-variant:small-caps;">SumBD</span>, we start by giving an upper bound to the PoA similar to the one obtained for <span style="font-variant:small-caps;">MaxBD</span>. For the remaining of this section we use $D$ to denote the average bound of every node, namely $D=B/n$.
\[th:UB sum\] The PoA of <span style="font-variant:small-caps;">SumBD</span> is $O(n^{\frac{1}{\lfloor\log_3
D/4\rfloor+2}})$ for $D\ge 3$, and $O\left( \sqrt{n \log {n}}
\right)$, when $2 \le D < 3$.
Let $G=G(S)$ be a stable graph, and let $\rho=SC(S)/(n-1)$. Remind that the ball of radius $k$ centered at a node $u$ is defined as $\beta_k(u)=\{v \mid d_G(u,v) \le k\}$. Moreover, let $\beta_k=\min_{u \in V} |\beta_k(u)|$. We have the following
For any $k \ge 1$, we have $\beta_{3k+2} \ge \min \{n/2+1,\lfloor
\rho \rfloor \beta_k\}$.
Consider the ball $\beta_{3k+2}(u)$ centered at any given node $u$, and assume that $|\beta_{3k+2}(u)| \le n/2$. Let $T$ be the maximal set of nodes at distance exactly $2k+2$ from $u$ and subject to the distance between any pair of nodes in $T$ being at least $2k+1$. We claim that for every node $v \notin
\beta_{3k+2}(u)$, there is a vertex $t \in T$ with $d_G(t,v)\le
d_G(u,v) -2$. Indeed, consider the node $t'$ in the shortest path between $v$ and $u$ at distance exactly $2k+2$ from $u$. If $t'
\in T$ the claim trivially holds, otherwise consider the node $t
\in T$ that is closest to $t'$. From the maximality of $T$ we have that $d_G(t,v) \le d_G(t,t')+d_G(t',v) \le 2k+d_G(u,v)-(2k+2) \le
d_G(u,v)-2$.
Let $H$ be the forest consisting of the following disjoint trees. For every node $t\in T \cup\{u\}$, let $U_t$ be the nodes that are closer to $t$ than any other $t' \in T \cup\{u\}$, and let $F_t$ be the subtree of the shortest path tree of $G$ rooted at $t$ spanning $U_t$. As a consequence, since $u$ is within the bound in $G$, it is easy too see that every vertex $x$ is within the bound in $H \cup
\{(x,t) \mid t \in (T \cup\{u\})\setminus \{x\}\}$. Hence, From Lemma \[lm:aux\], we have that $\rho < |T|+1$ and hence $|T|+1
\ge \lfloor \rho \rfloor$. Now, all the balls centered at nodes in $T\cup\{u\}$ with radius $k$ are all pairwise disjoint. Then: $$|\beta_{3k+2}(u)|\ge |\beta_k(u)|+\sum_{t \in T} |\beta_k(t)| \ge
\lfloor \rho \rfloor \beta_k.$$
Now, observe that $\beta_1 \ge \lfloor \rho \rfloor$. Then, after using the above claim $x$ times, we obtain $$\beta_{2 \, 3^{x}-1} \ge \min \{n/2+1,\lfloor \rho
\rfloor^{x+1}\}.$$
Let us consider the case $R\ge 3$ first. Let $U$ be a maximal independent set of $G^{D-1}$. Since $U$ is also a dominating set of $G^{D-1}$, it holds that $|U| \ge \lfloor \rho \rfloor$. We consider the $|U|$ balls centered at nodes in $U$ with maximal radius at most $(D-2)/2$. Since $U$ is an independent set of $G^{D-1}$, all balls are pairwise disjoint and hence we have $n
\ge |U| \lfloor \rho \rfloor^{\lfloor\log_3 D/4\rfloor + 1} \ge
\lfloor \rho \rfloor^{\lfloor\log_3 D/4\rfloor + 2}$. As a consequence, we obtain $\lfloor \rho \rfloor \le
n^{\frac{1}{\lfloor\log_3 D/4\rfloor}}$, and the claim follows.
Now assume $2\le D < 3$. To use the same argument used for <span style="font-variant:small-caps;">MaxBD</span>, it suffices to prove that for any stable graph $G(S)$ with minimum degree $\delta$, it holds that $\frac{SC(S)}{n-1} \le \min\{\delta+1, O(\gamma(G^{D-1})) \}$. The upper bound $\frac{SC(S)}{n-1}=O(\gamma(G^{D-1}))$ can be proved by using the same arguments used in the proof of Lemma \[lm:PoA le gamma\] where we exchange the role of $R$ with $D$. Now, we prove that $\frac{SC(S)}{n-1} \le \delta +1$. Let $v$ be a node with degree $\delta$, and let $N_{G(S)}(v)=\{u_1,\dots,u_{\delta}\}$. Consider a shortest path tree $T$ of $G(S)$ rooted at $v$. Clearly, $v$ is within the bound in $T$, and if we define $E_x=\{(x,u_j) \mid 1 \le j \le \delta
\}$ for any $x \neq v$, we have $B_{T+E_x}(x) \le B_{G(S)}(v)\le
B$. Hence, from Lemma \[lm:aux\], if follows that $\mathit{SC}(S)\le |E(T)| + (n-1)\delta \le (\delta+1) (n-1)$.
From the above result, it follows that the PoA becomes constant when $D=\Omega(n^{\epsilon})$, for some $\epsilon>0$. We now show how to lower such a threshold to $D=2^{\omega(\sqrt{\log n})} = n^{\omega\left(\frac{1}{\sqrt{\log n}}\right)}$ (and we also improve the upper bound when $D=\omega(1) \cap o(3^{\sqrt{\log n}})$).
\[lm:Bcost>B-n\] Let $G(S)$ be stable and let $v$ be a node such that $B_{G(S)}(v) \le B-n$, then $\mathit{SC}(S) \le 2(n-1)$.
Let $T$ be the shortest path of $G$ rooted at $v$. The claim immediately follows from Lemma \[lm:aux\] by observing that $v$ is within the bound in $T$ and every other node $u$ is within the bound in $T+(u,v)$.
Notice that the above Lemma shows that when a stable graph $G$ has diameter at most $D-1$ then the social cost of $G$ is at most twice the optimum. Now, the idea is to provide an upper bound to the diameter of any stable graph $G$ as function of $\delta$, where $\delta$ is minimum degree of $G$. Then we combine this bound with Lemma \[lm:Bcost>B-n\] in order to get a better upper bound to PoA for interesting ranges of $D$.
The proof of the following theorem follows the schema of that of Theorem 9 in [@ADH10].
\[th:diam SUM\] Let $G$ be stable with minimum degree $\delta$. Then the diameter of $G$ is $2^{O(\sqrt{\log n})}$ if $\delta=2^{O(\sqrt{\log
n})}$, and $O(1)$ otherwise.
We start by proving two lemmas:
Let $G$ be stable with minimum degree $\delta$. Then either $G$ has diameter at most $2 \log n$ or, for every node $u$, there is a node $x$ with $d_G(u,x)\le \log n$ such that (i) $x$ is buying $\delta /c$ edges (for some constant $c>1$), and (ii) the removal of these edges increases the sum of distances from $x$ by at most $2n(1+\log n)$.
Assume that the diameter of $G$ is greater than $2 \log n$ and consider a node $u$. Let $U_j$ be the set of nodes at distance exactly $j$ from $u$ and let $n_j=|U_j|$. Moreover, denote by $T$ the shortest path tree of $G$ rooted at node $u$. Let $i$ be the minimum index such that $n_{i+1}< 2n_i$ ($i$ must exist since the height of $T$ is greater than $\log n$). Consider the set of edges $F$ of $G$ having both endpoints in $U_{i-1}\cup U_i \cup U_{i+1}$ and that do not belong to $T$. Then, $|F| \ge \delta n_i /2 -
3n_i$. Moreover, we have that $n_{i-1}+n_i+n_{i+1} \le 1/2 \,
n_i+n_i+2n_i=7/2 n_i$. As a consequence, there is a vertex $x \in
U_{i-1}\cup U_i \cup U_{i+1}$ which is buying at least $\frac{n_i
/2 - 3n_i}{7/2 n_i} \ge \delta /c$ edges of $F$, for some constant $c>1$. Moreover, when $x$ removes these edges, the distance to any other node $y$ increases by at most $2(1+\log n)$ because $d_T(x,y) \le 2(1+\log n)$. The claim follows.
\[lem:add delta/c’ edges\] In any stable graph $G$, there is a constant $c'>1$ the addition of $\delta/c'$ edges all incident to a node $u$ decreases the sum of distances from $u$ by at most $5n \log n$.
If $G$ has diameter at most $2 \log n$, then the claim trivially holds. Otherwise, let $x$ be the node of the previous Lemma and let $c'$ be such that $\delta/c' \le \delta/c-1$. Moreover, assume by contradiction that the sum of distances from $u$ decreases by more than $5n\log n$ when we add to $G$ the following set of edges $F=\{(u,v_1),\dots,(u,v_h)\}$, with $h=\delta/c'$. Then, let $F'=\{(x,v_j) \mid j=1,\dots,h \}$. We argue that $x$ can improves his cost by saving at least an edge as follows: $x$ deletes its $\delta/c$ edges and adds $F'$. Indeed, the sum of distances from $x$ increases by at most $2n(1+\log n) \le 4n \log n$ and decreases by at least $5n \log n - n\log n$, since for every node $y$ such that the shortest path in $G+F$ from $u$ to $y$ passes through $x$, we have that $d_G(u,y)-d_{G+F}(u,y) \le \log n$. Hence, $x$ is still within the bound in $G+F'$ and is saving at least one edge: a contradiction.
Recall that the ball of radius $k$ centered at a node $u$ is defined as $\beta_k(u)=\{v \mid d_G(u,v) \le k\}$. Moreover, let $\beta_k=\min_{u \in V} |\beta_k(u)|$. We claim that $$\label{eq:ricorrenza potenziata}
\beta_{4k} \ge \min\{n/2+1, \frac{k \delta}{20 c \log n} \beta_k
\},$$
for some constant $c >1$. To prove that, consider the ball $\beta_{4k}(u)$ centered at any given node $u$, and assume that $|\beta_{4k}(u)| \le n/2$. Let $T$ be the maximal set of nodes at distance exactly $2k+1$ from $u$ and subject to the distance between any pair of nodes in $T$ being at least $2k+1$. It is easy to see that, from the maximality of $T$, for every node $v \notin \beta_{3k}$ there is a node $t \in T$ such that $d_{G}(v,t)\le d_G(u,v)-k$. We assumed that at least $n/2$ nodes have distance more than $3k$. This implies that there must be a set $T'\subseteq T$ of size $\delta/c$ such that at least $n
\delta/2|T|$ such vertices $v$ whose distance is at most $d(u,v)-k$ from some node in $T'$. If we add $\delta/c$ edges from $u$ to nodes in $T'$, the sum of distances from $u$ decreases by at least $(k-1)n/2|T| \ge kn/4|T|$. By Lemma \[lem:add delta/c’ edges\] this improvement is at most $5n \log n$. As a consequence we have that $|T| \ge \delta k /(20 c \log n)$. Moreover, all the balls centered at nodes in $T$ are disjoint, and this proves (\[eq:ricorrenza potenziata\]). Now, the claim follows by solving the recurrence (\[eq:ricorrenza potenziata\]).
By using the above theorem along with Lemma \[lm:Bcost>B-n\], and observing that if $G(S)$ is stable and has minimum degree $\delta$, then $\frac{SC(S)}{n-1} \le \delta +1$, as shown in the proof of Theorem \[th:UB sum\], we have:
\[th:UB sum2\] The PoA of <span style="font-variant:small-caps;">SumBD</span> is $2^{O(\sqrt{\log n})}$ if $D=\omega(1)$, and $O(1)$ if $D=2^{\omega(\sqrt{\log n})}$.
Then, by combining the results of Theorems \[th:UB sum\] and \[th:UB sum2\], we get the bounds reported in Table \[UBsum\].
Lower bounds
------------
We can finally prove the following theorem.
\[th:LB sum\] For any $\epsilon >0$ and for $2n-3\leq B=o(n^2)$, the PoA of <span style="font-variant:small-caps;">SumBD</span> is at least $2-\epsilon$.
To prove the theorem, we use the following scheme. First, for every integer $k\geq 2$, we provide a family ${\cal G}_k$ of graphs that are stable when $B\in\big[\lambda(k,n),\lambda'(k,n)\big)$, where $n$ is the size of the graph and $\lambda(k,n)$ and $\lambda'(k,n)$ are functions that depend on $k$ and $n$. We also prove that the social cost of infinitely many graphs in ${\cal G}_k$ is at least $2-\epsilon$ far from the social cost of an optimum, for every $k=o(n)$. Then, we show that $\lambda(2,n)\leq 2n-3$, $\lambda(k+1,n)\leq
\lambda'(k,n)$, and $\lambda(\Omega(n),n)=\Omega(n^2)$.
Family ${\cal G}_k$ contains a graph $G_{k,h}$ for every positive integer $h$. More precisely, $G_{k,h}$ has $n_{k,h}=(h+1)k$ vertices and $m_{k,h}=2kh$ edges. Therefore, the lower bound of $2-\epsilon$ for the PoA when $k=o(n)$ follows by choosing $h\geq
\frac{2}{\epsilon}-1$. For the rest of the proof, we assume that $h$ is an arbitrary, but fixed, positive integer. Moreover, with a little abuse of notation, we will drop the subscript $h$ from $G_{k,h}$ and $n_{k,h}$ and we will also drop the parameter $n_k=n_{k,h}$ as argument of the two functions $\lambda$ and $\lambda'$.
The graph $G_k$ is a highly symmetric graph consisting of $k$ players $\{u_0,\ldots,u_{k-1}\}$ which buy no edge, and, for every $i=0,\ldots,k-1$, there are $h$ copies of a player (we denote by $v_i$ any of such players) each buying exactly two edges: one towards $u_{i}$ and one towards $u_{i+1\bmod k}$. Observe that $G_k$ has diameter $k$.
The broadcast cost of each player $v_i$ is exactly $\lambda(k)$ while the broadcast cost of each player $u_j$ is equal to $\bar
\lambda(k)$. It is easy to see that $\lambda(2)=2n_2-4,\bar
\lambda(2)=n_2$.
Moreover, one can observe that for every $k\geq 2$ $$\lambda(k+1)=\lambda(k)+n_k+
\begin{cases}
h & \text{if $k+1$ is even;}\\
1 & \text{if $k+1$ is odd,}
\end{cases}$$
as well as
$$\bar \lambda(k+1)=\bar \lambda(k)+n_k+
\begin{cases}
1 & \text{if $k+1$ is even;}\\
h & \text{if $k+1$ is odd.}
\end{cases}$$
As each player $v_i$ owns exactly two edges, the only strategy $v_i$ has to connect to $G_k-v_i$ with exactly one edge, is that of connecting either to some $v_{i}'$ or to some $u_j$. Therefore, a lower bound on the broadcast cost of player $v_i$ if he uses only one edge to connect to $G_k-v_i$ is $\lambda'(k)=
\min\{\lambda(k),\bar\lambda(k)\}+n_k-1-k$, as $G_k$ has diameter $k$. Therefore, we have that $G_k$ is stable for every $B\in\Big[\max\{\lambda(k),\bar\lambda(k)\},\lambda'(k)\Big)$. In what follows, we show that $\max\{\lambda(k),\bar\lambda(k)\}=\lambda(k)$, thus proving that $B\in\big[\lambda(k),\lambda'(k)\big)$, as well as $\lambda'(k)=\bar \lambda(k)+n_k-1-k$.
Indeed, for every $k\geq 2$, and using $n_{k+1}=n_k+h+1$, we have that $$\lambda(k+2)=\lambda(k)+2n_{k+1} \text{\,\,\,\, and \,\,\,\,}
\bar\lambda(k+2)=\bar\lambda(k)+2n_{k+1}.$$ Furthermore, using the relations $n_{k+1}=n_k+h+1, n_k=(h+1)k$, and the formulas above, $\lambda(3)=2n_3-3,
\bar\lambda(3)=\frac{5}{3}n_3-1$. Therefore, $\bar \lambda(2)\leq
\lambda(2)$ and $\bar \lambda(3)\leq \lambda(3)$. As a consequence, for every $k\geq 2$, $\bar
\lambda(k+2)=\bar\lambda(k)+2n_{k+1}\leq
\lambda(k)+2n_{k+1}=\lambda(k+2)$. Therefore $\max\{\lambda(k),\bar\lambda(k)\}=\lambda(k)$.
To complete the proof, it remains to show that $\lambda(2)\leq
2n_2-3$, $\lambda(k+1)\leq \lambda'(k)$, and $\lambda(\Omega(n))=\Omega(n^2)$. We already proved that $\lambda(2)=2n_2-4\leq 2n_2-3$. Moreover, for $n_k=2k$, i.e., $k=n_k/2$, we have that $G_k$ is a cycle of $2k$ vertices, and thus the broadcast cost of any vertex is $\Omega(k^2)=\Omega(n^2)$. Finally, using induction, and observing that $\lambda(3)\leq \lambda'(2)$, we can prove that $\lambda(k+1)\leq \lambda'(k)$. Indeed, if $k+1$ is even, then $$\begin{aligned}
\lambda(k+1) &=& \lambda(k)+n_k+h\leq \lambda'(k-1)+n_k+h\\
&=&\bar\lambda(k-1)+n_{k-1}+n_{k}-1-k+h\\
&=&\bar\lambda(k)+n_k-1-k=\lambda'(k),\end{aligned}$$ while, if $k+1$ is odd, then $$\begin{aligned}
\lambda(k+1) &=& \lambda(k)+n_k+1\leq \lambda'(k-1)+n_k+1\\
&=&\bar\lambda(k-1)+n_{k-1}+n_{k}-1-k+1\\
&=&\bar\lambda(k)+n_k-1-k=\lambda'(k).\end{aligned}$$
Concluding remarks {#sct:conclusions}
==================
In this paper, we have introduced a new NCG model in which the emphasis is put on the fact that a player might have a strong requirement about its centrality in the resulting network, as it may well happen in decentralized computing (where, for instance, the bound on the maximum distance could be used for synchronizing a distributed algorithm). We developed a systematic study on the PoA of the two (uniform) games <span style="font-variant:small-caps;">MaxBD</span> and <span style="font-variant:small-caps;">SumBD</span>, which, however, needs to be continued, since a significant gap between the corresponding lower and upper bounds is still open. In particular, it is worth to notice that finding a better upper bound to the PoA would provide a better estimation about how much dense a network in equilibrium can be.
Actually, in an effort of reducing such a gap, we focused on <span style="font-variant:small-caps;">MaxBD</span>, and we observed the following fact: Recall that a graph is said to be *self-centered* if every node is a center of the graph (thus, the eccentricity of every node is equal to the radius of the graph, which then coincides with the diameter of the graph). An interesting consequence of Lemma \[lm:PoA le gamma\] is that only stable graphs that are self-centered can be dense, as one can infer from the following
Let $G(S)$ be a NE for <span style="font-variant:small-caps;">MaxBD</span> such that $G(S)$ is not self-centered. Then, $\mathit{SC}(S) \le 2(n-1)$.
Let $v$ be a node with minimum eccentricity. It must be that $\varepsilon_{G(S)}(v) \le R - 1$. Then, $U=\{v\}$ is a dominating set of $G^{R-1}$, and Lemma \[lm:PoA le gamma\] implies the claim.
Thus, to improve the lower bound for <span style="font-variant:small-caps;">MaxBD</span>, one has to look to self-centered graphs. Moreover, if one wants to establish a lower bound of $\rho$, then a stable graph of minimum degree $\rho-1$ (from Corollary \[lm:PoA le delta\]) is needed. Starting from these observations, we investigated the possibility to use small and suitably dense self-centered graphs as *gadgets* to build lower bound instances for increasing values of $R$. To illustrate the process, see Figure \[fig:pallone\], where using a self-centered cubic graph of diameter 3 and size 20, we have been able to obtain a lower bound of 3 (it is not very hard to see that the obtained graph is in equilibrium).
Interestingly enough, the gadget is a famous extremal (i.e., maximal w.r.t. node addition) graph arising from the study of the *degree-diameter* problem, namely the problem of finding a largest size graph having a fixed maximum degree and diameter (for a comprehensive overview of the problem, we refer the reader to [@degreediam]). More precisely, the gadget is a graph of largest possible size having maximum degree $\Delta=3$ and diameter $R=3$. In fact, this seems not to be coincidental, since also *Moore graphs* (which are extremal graphs for $R=2$ and $\Delta=2,3,7,57$), and the extremal graph for $R=4$ and $\Delta=3$ (see [@degreediam]), can be shown to be in equilibrium, and then they can be used as gadgets (clearly, the lower bounds implied by Moore graphs for $R=2$ are subsumed by our result in Theorem \[th:PoA R=2\]). Notice that from this, it follows that we actually have a lower bound of $3$ for the PoA of <span style="font-variant:small-caps;">MaxBD</span> also for $R=4$. So, apparently there could be some strong connection between the equilibria for <span style="font-variant:small-caps;">MaxBD</span> and the extremal graphs w.r.t. to the degree-diameter problem, and we plan in the near future to explore such intriguing issue.
[99]{}
-[mat.upc.es/grup\_de\_grafs/grafs/taula\_delta\_d.html/]{}, Universitat Politècnica de Catalunya, Barcelona, Spain.
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[^1]: This work was partially supported by the PRIN 2008 research project COGENT (COmputational and GamE-theoretic aspects of uncoordinated NeTworks), funded by the Italian Ministry of Education, University, and Research.
[^2]: In this paper, we only focus on *pure* strategies Nash equilibria.
|
---
abstract: |
We consider the continuity equation $\partial_t \mu_t + \operatorname{div}({{\bm b}}\mu_t) = 0$, where $\{\mu_t\}_{t \in {\mathbb{R}}}$ is a measurable family of (possibily signed) Borel measures on ${\mathbb{R}}^d$ and ${{\bm b}}\colon {\mathbb{R}}\times {\mathbb{R}}^d \to {\mathbb{R}}^d$ is a bounded Borel vector field (and the equation is understood in the sense of distributions). If the measure-valued solution $\mu_t$ is non-negative, then the following *superposition principle* holds: $\mu_t$ can be decomposed into a superposition of measures concentrated along the integral curves of ${{\bm b}}$. For smooth ${{\bm b}}$ this result follows from the method of characteristics, and in the general case it was established by L. Ambrosio. A partial extension of this result for signed measure-valued solutions $\mu_t$ was obtained in [@AB], where the following problem was proposed: does the superposition principle hold for signed measure-valued solutions in presence of unique flow of homeomorphisms solving the associated ordinary differential equation? We answer to this question in the negative, presenting two counterexamples in which uniqueness of the flow of the vector field holds but one can construct non-trivial signed measure-valued solutions to the continuity equation with zero initial data.\
<span style="font-variant:small-caps;">Keywords</span>: *continuity equation, measure-valued solutions, uniqueness, Superposition Principle.*\
<span style="font-variant:small-caps;">MSC (2010): 34A12, 35A30, 49Q20.</span>
address:
- 'Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, CH-4051, Basel, Switzerland.'
- 'Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700; RUDN University, 6 Miklukho-Maklay St, Moscow, 117198; Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St, Moscow, 119991;'
author:
- Paolo Bonicatto
- 'Nikolay A. Gusev'
bibliography:
- 'biblio.bib'
title: 'Non-uniqueness of signed measure-valued solutions to the continuity equation in presence of a unique flow'
---
Acknowledgements {#acknowledgements .unnumbered}
================
The first author acknowledges ERC Starting Grant 676675 FLIRT. The work of the second author was supported by the “RUDN University Program 5-100” and RFBR Grant 18-31-00279.
|
---
abstract: 'We study a Lorentz invariant pairing mechanism that arises when two relativistic spin-1/2 fermions are subjected to a Dirac string coupling. In the weak coupling regime, we find remarkable analogies between this relativistic bound system and the well known superconducting Cooper pair. As the coupling strength is raised, quenched phonons become unfrozen and dynamically contribute to the gluing mechanism, which translates into novel features of this relativistic superconducting pair.'
author:
- 'A. Bermudez$^1$ and M.A. Martin-Delgado$^1$'
title: 'A Lorentz Invariant Pairing Mechanism: Relativistic Cooper Pairs'
---
Introduction {#sectionI}
============
A large class of superconducting materials can be accurately described by the BCS theory [@BCS], which is based upon two major contributions. First, Frölich showed how the coupling between electrons and crystal phonons leads to an effective attractive interaction between the electrons [@frohlich]. Inspired by this result, Cooper discussed how any attractive interaction can bind a couple of electrons which lay around a filled Fermi sea [@Cooper]. This bound system, known as a Cooper pair, is responsible for several intriguing properties displayed by superconductors, which can be described as a many-body coherent state where electrons above the Fermi surface are bounded in pairs.
The oversimplified picture developed by Cooper captures the essence of the underlying physical phenomena occurring in superconducting solids. In the same spirit, we study a simple model of two relativistic fermions with an effective attractive interaction. In order to maintain the similarities with the Cooper problem, we must fulfill the two following requirements:
**Phonon gluing mechanism:** In a relativistic scenario, the simplest phonon-like coupling is modeled by a Dirac string coupling mechanism where the vibrations of the string describe the lattice phonons (see fig. \[feynman\_diagram\] left). This interaction is introduced by a non-minimal coupling procedure in the free Dirac equation $$\label{1_body_dirac_oscillator} \ii \hbar \frac{\partial
\ket{\Psi}}{\partial t}=\left(c\boldsymbol{\alpha}(\textbf{p}-\ii
m\omega\beta\textbf{r})+mc^2\beta\right)\ket{\Psi},$$ where $\ket{\Psi}$ stands for the Dirac 4-component spinor, $\textbf{p}$ represents the momentum operator, and $c$ the speed of light. Here $\beta=\text{diag}(\mathbb{I}_2,-\mathbb{I}_2),\alpha_j=\text{off-diag}(\sigma_j,\sigma_j)$ are the Dirac matrices in the standard representation with $\sigma_j$ as the usual Pauli matrices [@greiner_book]. This Dirac string coupling $\textbf{p}\to\textbf{p}-\ii
m\omega\beta\textbf{r}$ was introduced in [@imc67; @moshinsky] as a relativistic extension of the harmonic oscillator, usually coined as the Dirac oscillator, where $\omega$ represents the oscillator’s frequency. In our picture, this frequency effectively describes the lattice vibrations and its coupling to the fermionic degrees of freedom.
-- --
-- --
**Two-fermion binding:** Regardless of the coupling strength, we shall show that such an effective string coupling binds relativistic fermions in pairs (see fig. \[feynman\_diagram\] right). A Lorentz invariant extension of the Dirac string Hamiltonian in Eq. to two-fermion systems is possible [@barut; @moshinsky_2Body; @moshinsky_meson; @moshinsky_pp+], which in the center of mass reference frame reads as follows $$\label{2body_DO} H_{\text{3D}}=\frac
{c}{\sqrt{2}}(\boldsymbol{\alpha}_1-\boldsymbol{\alpha}_2)(\textbf{p}-\ii
m \omega\beta_{12}\textbf{r})+mc^2(\beta_1+\beta_2),$$ where $\boldsymbol{\alpha}_1=\boldsymbol{\alpha}\otimes\mathbb{I}_4$, $\boldsymbol{\alpha}_2=\mathbb{I}_4\otimes\boldsymbol{\alpha}$, $\beta_1=\beta\otimes\mathbb{I}_4$, $\beta_2=\mathbb{I}_4\otimes\beta$, and $\beta_{12}=\beta\otimes\beta$ represent the generalization of the Dirac matrices in the two-body Hilbert space. Here $\textbf{p}:=(\textbf{p}_1-\textbf{p}_2)/\sqrt{2}$, and $\textbf{r}:=(\textbf{r}_1-\textbf{r}_2)/\sqrt{2}$ stand for the relative momentum and position operators.
In this work, we study the binding properties of this two-body relativistic Hamiltonian, and discuss under which circumstances an analogy to Cooper pairs can be performed. Phonons in this relativistic system are dynamical and always provide a pairing mechanism, as we shall see. Thus, there is no need to invoke a many-body effect through the Pauli principle as in the original Cooper pair scenario. In fact, there are real materials which deviate from standard BCS theory. In BCS, phonons are quenched and their effect appears as a pairing energy scale, but they are not explicit in the Hamiltonian. There is an extension of the BCS theory that accounts for the effects of dynamical phonons, known as the Migdal-Eliashberg theory [@migdal; @eliashberg; @carbotte]. Our relativistic fermionic pairing mechanism is thus closer to this latter treatment.
We shall restrict ourselves to a two-dimensional system, where an exact solution is derived and several interesting properties can be neatly discussed. Two spatial dimensions are also natural for other superconducting materials like the cuprates [@bednorz_muller]. In this case, the Dirac matrices reduce to the usual Pauli matrices $\alpha_x=\sigma_x,\alpha_y=\sigma_y,\beta=\sigma_z$, and the relativistic 1-body state $|\Psi\rangle$ can be described by a 2-component spinor. The 2-body relativistic Hamiltonian in two dimensions can be written as follows $$\label{2D_2body_DO}
\begin{split}
H_{\text{2D}}= & \frac
{c}{\sqrt{2}}(\sigma_x\otimes\mathbb{I}_2-\mathbb{I}_2\otimes\sigma_x)(p_x-\ii
m \omega\sigma_z\otimes\sigma_zx)\\
+&\frac
{c}{\sqrt{2}}(\sigma_y\otimes\mathbb{I}_2-\mathbb{I}_2\otimes\sigma_y)(p_y-\ii
m \omega\sigma_z\otimes\sigma_zy)\\
+&mc^2(\sigma_z\otimes\mathbb{I}_2+\mathbb{I}_2\otimes\sigma_z).
\end{split}$$
Energy spectrum and eigenstates {#sectionII}
===============================
In two dimensions, chiral creation-annihilation operators which carry dual aspects of a left- or right-handed symmetry are defined as follows $$\label{circular_operators}
\begin{array}{c}
a_r:=\frac{1}{\sqrt{2}}(a_x - \ii a_y),\hspace{2ex}a_r^\dagger:=\frac{1}{\sqrt{2}}(a_x^\dagger + \ii a_y^\dagger) , \\
a_l:=\frac{1}{\sqrt{2}}(a_x + \ii a_y), \hspace{2ex}a_l^\dagger:=\frac{1}{\sqrt{2}}(a_x^\dagger - \ii a_y^\dagger) , \\
\end{array}$$ where $ a_x^\dagger,a_x, a_y^\dagger, a_y$, are the usual creation-annihilation operators of the harmonic oscillator $a^{\dagger}_i=\frac{1}{\sqrt{2}}\left(\frac{1}{\tilde{\Delta}}r_i
- \ii \frac{\tilde{\Delta}}{\hbar}p_i\right)$, and $\tilde{\Delta}=\sqrt{\hbar/m\omega}$ is related to the ground state width. Using these operators, the relativistic Hamiltonian in Eq. takes a simpler and amenable form $$\label{dirac_hamiltonian_matrix}
H_{\text{2D}}=\left[%
\begin{array}{cccc}
\Delta & g^* a_l^{\dagger} & g a_l^{\dagger} & 0 \\
g a_l & 0 & 0 & g^* a_r \\
g^* a_l & 0 & 0 & g a_r \\
0 & g a_r^{\dagger} & g^* a^{\dagger}_r & -\Delta \\
\end{array}%
\right],$$ where $\Delta:=2mc^2$ stands for the system rest mass, $g:=\ii
mc^2\sqrt{2\zeta}$ is a coupling parameter, and $\zeta:=\hbar\omega/mc^2$ controls the strength of the effective interaction. Considering the two-body spinorial basis $\{\ket{\uparrow\uparrow},\ket{\uparrow\downarrow},\linebreak
\ket{\downarrow\uparrow},\ket{\downarrow\downarrow}\}$, we can understand the Dirac string coupling as a four-level system depicted in fig. \[niveles\].
[niveles\_cooper\_bis.eps]{} (46,38)[[$\ket{\uparrow\uparrow}$]{}]{} (46,2)[[$\ket{\downarrow\downarrow}$]{}]{} (11,23)[[$\ket{\uparrow\downarrow}$]{}]{} (82,23)[[$\ket{\downarrow\uparrow}$]{}]{}
We now proceed to describe the energy spectrum of the 2-body interacting relativistic system, in terms of the phonon Fock states $$\label{chiral_Fock}
\ket{n_r,n_l}:=\frac{1}{\sqrt{n_r!n_l!}}(a_r^{\dagger})^{n_r}(a_l^{\dagger})^{n_l}\ket{\text{vac}},$$ where $n_r,n_l=0,1...$ specify the number of right- and left-handed phonons coupling the two-fermion system. One immediately observes that the Hilbert space can be divided in a series of invariant subspaces $\mathcal{H}=\bigoplus_{n_r,n_l=0}^{\infty}\mathcal{H}_{n_rn_l}$, where each subspace can be described by $\mathcal{H}_{n_rn_l}:=\mathcal{H}'_{n_r,n_l}\bigoplus\mathcal{H}''_{n_r,n_l}$. These subspaces are spanned by $$\label{hilbert_subspaces}
\begin{split}
\mathcal{H}'_{n_r,n_l}=&\text{span}\{\ket{+}\ket{n_r,n_l}\},\\
\mathcal{H}''_{n_r,n_l}=&\text{span}\{\ket{\uparrow\uparrow}\ket{n_r,n_l+1},\ket{-}\ket{n_r,n_l},\ket{\downarrow\downarrow}\ket{n_r+1,n_l}\},
\end{split}$$ where the states $\ket{-}:=(\ket{\uparrow\downarrow}-\ket{\downarrow\uparrow})/\sqrt{2}$ , and $\ket{+}:=(\ket{\uparrow\downarrow}+\ket{\downarrow\uparrow})/\sqrt{2}$ are maximally entangled unpolarized Bell states. In particular, $\mathcal{H}'_{n_r,n_l}$ describes a zero-energy subspace $E_{+,n_r,n_l}=0$. The Hamiltonian in the remaining subspaces $\mathcal{H}''_{n_r,n_l}$ can be expressed as follows $$\label{dirac_hamiltonian_matrix_3}
H_{\text{2D}}^{n_r n_l}=\Delta\left[%
\begin{array}{ccc}
1& -\ii\sqrt{\zeta(n_l+1)} & 0 \\
\ii\sqrt{\zeta(n_l+1)} & 0 & -\ii\sqrt{\zeta(n_r+1)} \\
0 & \ii\sqrt{\zeta(n_r+1)} & -1\\
\end{array}%
\right],$$ where the 2-body interaction couples three different levels and can be exactly diagonalized. Using Cardano-Vietta solution to third order polynomials, we obtain the following energies $$\label{energy_level}
\begin{split}
\frac{E_{1n_rn_l}}{\Delta}:=&\sqrt{\frac{4\left[1+\zeta(n_r+n_l+2)\right]}{3}}\cos\Theta,\\
\frac{E_{2n_rn_l}}{\Delta}:=&\sqrt{\frac{4[1+\zeta(n_r+n_l+2)]}{3}}\cos\left(\Theta+\frac{2\pi}{3}\right),\\
\frac{E_{3n_rn_l}}{\Delta}:=&\sqrt{\frac{4[1+\zeta(n_r+n_l+2)]}{3}}\cos\left(\Theta+\frac{4\pi}{3}\right),\\
\end{split}$$ where $$\Theta:=\frac{1}{3}\text{arccos}\left[\frac{27(n_l-n_r)\zeta}{2[3(1+\zeta(n_r+n_l+2))]^{3/2}}\right].$$ These eigenstates are represented for different values of the coupling strength $\zeta$ in Fig. \[energy\_level\_figure\], where the chiral quantum numbers have been set to $n_r=n_l+1$.
[NR\_energy\_levels\_nr\_2\_nl\_1.eps]{} (-2,39)[[$E/mc^2$]{}]{} (50,2)[[$1/\zeta$]{}]{} (60,12)[[$\textcolor[rgb]{0.00,0.51,0.00}{E_2}$]{}]{} (60,35)[[$\textcolor[rgb]{0.00,0.00,0.63}{E_+}$]{}]{} (60,43)[[$\textcolor[rgb]{0.62,0.62,0.00}{E_3}$]{}]{} (60,63)[[$\textcolor[rgb]{0.69,0.00,0.00}{E_1}$]{}]{}
In this figure we observe two different regimes:
**Weak Coupling regime $\zeta\ll1$:** In this case, the low energy properties can be accurately described by a two-level system. This feature will turn out to be crucial for the analogies of the system to a non-relativistic Cooper pair discussed in section \[sectionIII\].
**Strong Coupling regime $\zeta\gg1$:** In this case, the four levels become essential in order to describe the low energy excitations. Consequently, the description becomes more involved but also gives a richer structure that may show certain novel properties with respect to non-relativistic Cooper pairs that are described in section \[sectionIV\].
Once the eigenvalues have been obtained, we may derive the corresponding eigenstates, which we list below $$\label{eigenstates}
\begin{split}
\ket{E_{+,n_r,n_l}}:=\ket{+,&n_r,n_l},\\
\ket{E_{j,n_r,n_l}}:=\frac{1}{\Omega_j}[&\alpha_j\ket{-}\ket{n_r,n_l}+\ii\beta_j\ket{\uparrow\uparrow,n_r,n_l+1}\\
+&\ii\delta_j\ket{\downarrow\downarrow,n_r+1,n_l}],
\end{split}$$ where we have defined the following parameters $$\label{norm_constants}
\begin{split}
&\alpha_j:=\Delta^2-E_{jn_rn_l^2}^2,\\
&\beta_j:=\Delta(\Delta+E_{jn_rn_l})\sqrt{\zeta(n_l+1)},\\
&\delta_j:=\Delta(\Delta-E_{jn_rn_l})\sqrt{\zeta(n_r+1)},\\
&\Omega_j:=\sqrt{\alpha_j^2+\beta_j^2+\delta_j^2}.\\
\end{split}$$ Here the indexes $j=1,2,3$ correspond to the three different eigenvalues . Let us mention that the total angular momentum $J_z:= S_z + L_z$ is conserved. Thus, the eigenstates have well-defined angular momenta, namely, $\hbar(n_r-n_l)$. Finally, we must consider the consequences of fermion indistinguishability. The Symmetrization postulate states that a system of identical fermions must be described in terms of antisymmetrical states, which establishes the following constraint $$\label{antisymmetry}
\text{P}_{21}\ket{\Psi(1,2)}=-\ket{\Psi(1,2)},$$ where $\text{P}_{21}$ stands for the permutation operator that swaps the fermion labels $1\leftrightarrow2$. Considering the eigenstates in Eq. under the permutation operator, we obtain the following expressions $$\label{permutation_eigenstates}
\begin{split}
\text{P}_{21}\ket{E_{+,n_r,n_l}}&=(-1)^{n_r+n_l}\ket{E_{+,n_r,n_l}},\\
\text{P}_{21}\ket{E_{j\hspace{0.5ex},n_r,n_l}}&=(-1)^{n_r+n_l+1}\ket{E_{j,n_r,n_l}}.
\end{split}$$ Since these expressions must satisfy the antisymmetric condition in Eq., the number of chiral quanta are constrained as follows $$\begin{split}
\ket{E_{+,n_r,n_l}}&\Rightarrow n_r+n_l=2k+1\hspace{2ex}:
k=0,1,2...\\
\ket{E_{j\hspace{0.5ex},n_r,n_l}}&\Rightarrow
n_r+n_l=2k\hspace{6ex}:
k=0,1,2...\\
\end{split}$$ Due to the indistinguishability of the relativistic fermions, the eigenstates $\ket{E_{+,n_r,n_l}}$ must contain an odd number of chiral quanta, whereas $\ket{E_{j,n_r,n_l}}$ are restricted to even number of chiral quanta.
We have thus derived a complete solution of the relativistic Dirac equation for two bodies interacting via a Dirac string coupling. Therefore, this 2-fermion system belongs to the small class of exactly solvable few-body relativistic systems. In sections \[sectionIII\] and \[sectionIV\] we show that this relativistic interaction does indeed lead to the formation of bound pairs, both in the weak and strong coupling regimes. Furthermore, we present a detailed study of the similar properties that the relativistic bound pair shares with the well-known non-relativistic Cooper pair. As we will see, there are profound analogies in the weak coupling regime, whereas novel properties are found in the strong coupling limit.
Weak Coupling Regime {#sectionIII}
====================
The standard description of Cooper pairs in superconducting solids is usually performed in a weak coupling regime, where a slightly phonon-mediated attractive interaction binds electron which lay close to the Fermi surface. We shall consider that the two-body Hamiltonian in Eq. effectively describes the gluing mechanism above the Fermi sea, and therefore a weak coupling regime is obtained when $\zeta\ll1$.
In this weak coupling regime, we have seen in Fig. \[energy\_level\_figure\] that the low-lying excitations can be entirely described by a two-level system. This situation is schematically described fig. \[2\_niveles\], where we see how spin-polarized levels become decoupled from those responsible of the low-energy properties. In this situation, we can obtain an effective Hamiltonian for the low energy sector, by adiabatic elimination.
[niveles\_cooper\_weak\_coupling\_bis.eps]{} (46,65)[[$\ket{\uparrow\uparrow}$]{}]{} (46,4)[[$\ket{\downarrow\downarrow}$]{}]{} (11,38)[[$\ket{\uparrow\downarrow}$]{}]{} (82,38)[[$\ket{\downarrow\uparrow}$]{}]{}
Let us consider an arbitrary state $\ket{\Psi(t)}\in\mathcal{H}_{n_rn_l}$ $$\begin{split}
\ket{\psi(t)}=&c_{\uparrow\uparrow}(t)\ket{\uparrow\uparrow,n_r,n_l+1}+c_{\uparrow\downarrow}(t)\ket{\uparrow\downarrow,n_r,n_l}\\
&+c_{\downarrow\uparrow}(t)\ket{\downarrow\uparrow,n_r,n_l}+c_{\downarrow\downarrow}(t)\ket{\downarrow\downarrow,n_r+1,n_l},
\end{split}$$ whose dynamical evolution, described by the Dirac Hamiltonian , can be represented as $$\label{Hamiltonian_dynamics}
\ii\hbar\frac{\dd}{\dd t}\left[%
\begin{array}{c}
c_{\uparrow\uparrow}(t) \\
c_{\uparrow\downarrow}(t) \\
c_{\downarrow\uparrow}(t) \\
c_{\downarrow\downarrow}(t) \\
\end{array}%
\right]=\left[%
\begin{array}{cccc}
\Delta & g^* a_l^{\dagger} & g a_l^{\dagger} & 0 \\
g a_l & 0 & 0 & g^* a_r \\
g^* a_l & 0 & 0 & g a_r \\
0 & g a_r^{\dagger} & g^* a^{\dagger}_r & -\Delta \\
\end{array}%
\right]\left[%
\begin{array}{c}
c_{\uparrow\uparrow}(t) \\
c_{\uparrow\downarrow}(t) \\
c_{\downarrow\uparrow}(t) \\
c_{\downarrow\downarrow}(t) \\
\end{array}%%
\right].$$ In the weak coupling limit, the transitions to the spin-polarized $\{\ket{\uparrow\uparrow,n_r,n_l+1},\ket{\downarrow\downarrow,n_r+1,n_l}\}$ upper and lower levels can be considered negligible. Therefore, the level population does not evolve under the action of the two-body interaction $\frac{\dd c_{\uparrow\uparrow}}{dt}=\frac{d
c_{\downarrow\downarrow}}{\dd t}=0$, and we can adiabatically eliminate these two levels. The latter conditions substituted in Eq. , give rise to the following relations $$\begin{split}
c_{\uparrow\uparrow}&=\ii\sqrt{\frac{\zeta(n_l+1)}{2}}(c_{\uparrow\downarrow}-c_{\downarrow\uparrow}),\\
c_{\downarrow\downarrow}&=\ii\sqrt{\frac{\zeta(n_r+1)}{2}}(c_{\uparrow\downarrow}-c_{\downarrow\uparrow}),
\end{split}$$ and an effective two-level dynamics $$\label{two_level_Hamiltonian_dynamics}
\ii\hbar\frac{\dd}{\dd t}\left[%
\begin{array}{c}
c_{\uparrow\downarrow}(t) \\
c_{\downarrow\uparrow}(t) \\
\end{array}%
\right]=\frac{\Delta\zeta}{2}(n_r-n_l)\left[%
\begin{array}{cc}
\hspace{1.5ex}1 & -1 \\
-1 & \hspace{1.5ex}1 \\
\end{array}%
\right]\left[%
\begin{array}{c}
c_{\uparrow\downarrow}(t) \\
c_{\downarrow\uparrow}(t) \\
\end{array}%
\right].$$ In this sense, we can integrate out the high-frequency modes by projecting onto the effective spin-unpolarized invariant subspace spanned by $\mathcal{H}^{\text{eff}}_{n_rn_l}:=\text{span}\{\ket{\uparrow\downarrow,n_r,n_l},\ket{\downarrow\uparrow,n_r,n_l}\}$, by means of an orthogonal projector $$\mathcal{P}^{\text{eff}}_{n_rn_l}:=\ket{\uparrow\downarrow,n_r,n_l}\bra{\uparrow\downarrow,n_r,n_l}+\ket{\downarrow\uparrow,n_r,n_l}\bra{\downarrow\uparrow,n_r,n_l}$$ The $z$-component of the orbital angular momentum operator $L_z=\hbar(a_r^{\dagger}a_r-a_l^{\dagger}a_l)$ constrained to this invariant subspace becomes $
\mathcal{P}^{\text{eff}}_{n_rn_l}L_z\mathcal{P}^{\text{eff}}_{n_rn_l}=\hbar(n_r-n_l)\mathbb{I}_2,
$ which allows us to rewrite Eq. as an effective two-level Hamiltonian $$\label{raman_DO}
H_{\text{eff}}:=\omega L_z \left[%
\begin{array}{cc}
\hspace{1.5ex}1 & -1 \\
-1 & \hspace{1.5ex} 1 \\
\end{array}%
\right]=\hbar\omega(a_r^{\dagger}a_r-a_l^{\dagger}a_l)\left[%
\begin{array}{cc}
\hspace{1.5ex}1 & -1 \\
-1 &\hspace{1.5ex} 1 \\
\end{array}%
\right].$$ This effective interaction in the weak coupling regime is represented in Fig. \[2\_niveles\], where the allowed transitions can take on two different channels via the consecutive creation-annihilation of right- or left-handed phonons. This process can be understood as an instance of a superexchange coupling between the spins $\ket{\uparrow\downarrow}\longleftrightarrow\ket{\downarrow\uparrow}$ driven by a second order two-phonon process where a chiral phonon is virtually created and then annihilated. There exist two different exchange paths, as seen in fig. \[2\_niveles\], depending on the left- or right-handed chiralities of the virtual phonons involved in the process.
This effective Hamiltonian can be exactly diagonalized yielding the eigenvalues $$E_{+n_rn_l}^{\text{eff}}:=0,\hspace{2ex}E_{-n_rn_l}^{\text{eff}}:=2\hbar\omega(n_r-n_l),$$ with the following associated eigenstates $$\label{NR_eigenstates}
\begin{split}
\ket{E_{+n_rn_l}^{\text{eff}}}:=\ket{+,n_r,n_l}&\Rightarrow
n_r+n_l=2k+1\hspace{2ex}:
k=0,1,2...\\
\ket{E_{-n_rn_l}^{\text{eff}}}:=\ket{-,n_r,n_l}&\Rightarrow
n_r+n_l=2k\hspace{6ex}:
k=0,1,2...\\
\end{split}$$ where the anti-symmetric character of the fermionic states has already been considered. Therefore, the low-lying solution in the weak coupling regime can be described by the maximally entangled Bell states in the spin degree of freedom, and rotational Fock states in the orbital degree of freedom.
Furthermore, these states describe a bound fermion pair. In order to show that such binding occurs, we must show that the inter-particle distance only attains finite values. Let us introduce the square-distance operator $ \Gamma:=x^2+y^2, $ where $x:=(x_1-x_2)/\sqrt{2}$ and $y:=(y_1-y_2)/\sqrt{2}$ denote the space coordinate operators for the relative fermionic distance. The expectation values in the weak-coupling eigenstates are $$\label{two-level-confinement}
\langle\Gamma\rangle_{\pm}=\frac{\tilde{\Delta}^2}{\sqrt{2}}(1+n_r+n_l),$$ which is always finite. We observe the crucial property that this system shares with a non-relativistic Cooper pair, namely, the pair of relativistic fermions are bounded in pairs even for a weak attraction $\zeta\ll 1$.
Another fundamental property that occurs in standard Cooper pairs is the presence of an energy gap between the paired energy level and the Fermi surface. This energy gap is responsible of the stability of Cooper pairs with respect to free fermion pairs and is proportional to the lattice Debye frequency $\Delta
E\sim\hbar\omega_D$. In the relativistic regime, we observe that the energy gap with respect to the displaced Fermi surface ( i.e. $\epsilon_F'=0$ ) is $$\begin{split}
&\Delta E_{+n_rn_l}=0, \\
&\Delta E_{-n_rn_l}=2\hbar\omega(n_r-n_l),
\end{split}$$ and therefore the only stable pair (i.e. $\Delta E<0$) is that described by the spin-singlet state when $n_l\geq n_r$.
In this sense we obtain a spin-singlet bound pair which clearly resembles the situation in standard Cooper pairs where the fermions are also in the singlet state. Furthermore, we can observe from this discussion that the relativistic gap is proportional to the Dirac string frequency $\Delta
E\sim\hbar\omega$, which plays the role of the usual Debye frequency in superconducting materials.
Finally, to take this comparison further, we should study the properties of the stable pair eigenstates in Eq. and compare them to the non-relativistic Cooper pair features.
**Spin degrees of freedom:** In BCS theory, Cooper pairs display a singlet state in the spin degree of freedom. We observe in Eq. that the stable bound fermionic pair state has also a spin-singlet component.
**Orbital degrees of freedom:** In BCS theory, Cooper pairs display a spherically symmetrical wave function with an onion-like layered structure. We directly observe from fig. \[fock\_states\_1\_profile\] that relativistic bound pair probability distribution $\rho^{\text{eff}}_{-n_rn_l}(r)$ display a similar spherically symmetric onion-like structure.
[densidad\_11\_weak\_coupling\_bis\_2.eps]{} (18,12)[[$y/\tilde{\Delta}$]{}]{} (75,9)[[$x/\tilde{\Delta}$]{}]{} (-2,41)[[$\rho^{\text{eff}}_{-11}$]{}]{}
[densidad\_22\_weak\_coupling\_bis\_2.eps]{} (18,12)[[$y/\tilde{\Delta}$]{}]{} (75,9)[[$x/\tilde{\Delta}$]{}]{} (-2,41)[[$\rho^{\text{eff}}_{-22}$]{}]{}
In this section we have discussed a relativistic pairing mechanism in a weak coupling regime. We have discussed in detail several analogies with a non-relativistic Cooper pair that naturally arise in this weak coupling limit. Remarkably, we obtain binding regardless of the interaction strength, which is a fundamental property of BCS systems. Additionally, we have shown how the relativistic energy gap scales with the string frequency in the same manner as the Cooper pair gap scales with the phonon Debye frequency. In this regard, we may conclude that the string interaction plays the role of the lattice phonons that mediate the effective attraction between fermions in the BCS theory. Furthermore, we have also compared the nature of the relativistic pair eigenstates with the Cooper pair wave functions. We have seen that the relativistic bound pair is also described by a spin-singlet state and a spherically symmetric onion-like state in the orbital degrees of freedom. All these similarities allow us to state that this fermionic pairing mechanism can be interpreted as a relativistic Cooper pair, since we recover most of the usual BCS properties in the weak coupling regime. Nonetheless, this Relativistic Cooper pair can also be studied in the strong coupling regime, where novel properties with respect to the usual Cooper pair in BCS theory arise. As we describe below, when the Dirac string interaction becomes strong enough, phonons contribute dynamically to the gluing mechanism.
Strong Coupling Regime {#sectionIV}
======================
In this section we study the pairing properties of the two-body relativistic system in the strong coupling regime $\zeta\gg1$. In this limit we must consider the complete four-level structure of the system ( see fig. \[niveles\] ), and the energy spectrum becomes clearly more involved in Eq. (see fig. \[energy\_levels\_strong\_coupling\]).
[energy\_levels\_xi\_5\_strong\_coupling\_2.eps]{} (2,31)[[$\frac{E}{mc^2}$]{}]{} (17,10)[[$n_l$]{}]{} (65,8)[[$n_r$]{}]{} (58,20)[[$E_{2n_rn_l}$]{}]{} (58,38)[[$E_{1n_rn_l}$]{}]{} (58,30)[[$E_{3n_rn_l}$]{}]{} (70,25)[[$E_{+n_rn_l}$]{}]{} (85,65)[[$E_{+n_rn_l}$]{}]{} (85,75)[[$E_{3n_rn_l}$]{}]{}
In fig. \[energy\_levels\_strong\_coupling\] we have represented the different energies for an interaction strength $\zeta=5$ which lays in the strong coupling regime. We clearly see how two levels $E_{2,n_r,n_l},E_{3,n_r,n_l}$ become stable pairs with a certain gap $\Delta E_{2,n_r,n_l}<\Delta E_{3,n_r,n_l}<0$. Therefore the strong coupling gives raise to a couple of stable bound fermionic states, namely, $$\label{cooper_pair_strong_coupling}
\begin{split}
\ket{E_{2,n_r,n_l}}:=\frac{1}{\Omega_2}[&\alpha_2\ket{-}\ket{n_r,n_l}+\ii\beta_2\ket{\uparrow\uparrow,n_r,n_l+1}+\\
+&\ii\delta_2\ket{\downarrow\downarrow,n_r+1,n_l}];\\
\ket{E_{3,n_r,n_l}}:=\frac{1}{\Omega_3}[&\alpha_3\ket{-}\ket{n_r,n_l}+\ii\beta_3\ket{\uparrow\uparrow,n_r,n_l+1}+\\
+&\ii\delta_3\ket{\downarrow\downarrow,n_r+1,n_l}].
\end{split}$$
The spatial probability distribution $\rho_{jn_rn_l}(r)$ of these stable fermionic pairs has been represented in fig. \[fock\_states\_2\_profile\] in the case of $n_r=n_l=1$. We can clearly observe that the density profile preserves the spherically symmetric onion-like structure. Nonetheless, noteworthy differences arise with respect to the weak coupling regime ( compare to the top fig. \[fock\_states\_1\_profile\]).
[densidad\_211\_strong\_coupling\_bis\_2.eps]{} (18,12)[[$y/\tilde{\Delta}$]{}]{} (75,9)[[$x/\tilde{\Delta}$]{}]{} (-2,41)[[$\rho_{211}$]{}]{}
[densidad\_311\_strong\_coupling\_bis\_2.eps]{} (18,12)[[$y/\tilde{\Delta}$]{}]{} (75,9)[[$x/\tilde{\Delta}$]{}]{} (-2,41)[[$\rho_{311}$]{}]{}
Furthermore, these two stable states form a fermionic bound pair since the inter-particle distance is finite $$\label{binding_distance_strong_coupling}
\begin{split}
\langle\Gamma\rangle_{2\hspace{0.5ex}}&=\tilde{\Delta}^2\left[(1+n_r+n_l)\alpha_2^2+(\beta_2^2+\delta_2^2)(2+n_r+n_l)\right],\\
\langle\Gamma\rangle_{3\hspace{0.5ex}}&=\tilde{\Delta}^2\left[(1+n_r+n_l)\alpha_3^2+(\beta_3^2+\delta_3^2)(2+n_r+n_l)\right].
\end{split}$$ We may conclude that the Dirac string pairing mechanism leads to bound pairs in the strong coupling regime, which display substantial differences with respect to the weakly coupled bound states in Eq. . It follows from Eqs. that the bound pairs are not in a singlet state but rather in a linear superposition of different spin singlet and triplet states entangled with different orbital Fock states. In this regard, the relativistic pairing mechanism does not induce an anti-ferromagnetic ordering any longer, and certain spin-polarization may arise depending on the value of the coupling strength $\zeta$.
It is also instructive to compare the orbital degrees of freedom of bound pairs in the weak and strong coupling limits. The weakly coupled states in Eq. are in orbital Fock states, which represent a certain number of vibrational phonons which are frozen in this limit. On the other hand, strongly coupled states in Eqs. cannot be described by a single Fock state, and therefore the vibrational phonons acquire a dynamical behavior $\ket{n_r+1,n_l}\leftrightarrows\ket{n_r,n_l}\leftrightarrows\ket{n_r,n_l+1}$, which is a clear sign of strong coupling in superconductors. We may conclude that the Dirac string phonons, responsible of the gluing mechanism, become unfrozen as the coupling becomes stronger and contribute to the effective attraction in a dynamic phenomenon. This is reminiscent of a $(s,p)$-wave symmetry of a SC order parameter. Similar types of superconducting states appear in some quantum liquids like superfluid ${\rm He}^3$: the so-called A- and B-phases exhibit different patterns of spin-orbit symmetry breaking [@helium-3; @leggett]. Layered materials like the ruthenates also exhibit unusual symmetry properties like triplet superconductivity [@maeno; @rice_sigrist; @baskaran].
We also observe that the strong pairing mechanism leads to a couple of possible stable bound pairs , whereas the weak coupling only produces one stable bound pair. Furthermore, the energy gap displayed by the bound pairs also depends on the strength of the coupling. In the weak coupling regime, we have already seen that the energy gap scales as $\Delta
E_-\sim\hbar\omega$, whereas the scaling in the strongly interaction limit does not present such a simple scaling (see fig. \[energy\_levels\_strong\_coupling\]).
Conclusions
===========
We have studied the relativistic pairing mechanism of the two-body Dirac oscillator in two dimensions, where a Dirac string coupling leads to fermionic bound pairs. We have described two different regimes where binding occurs regardless of the interaction strength.
In a weak coupling regime, the fermionic pair bears a strong resemblance to the usual Cooper pair in BCS theory. We remarkably found a similar scaling of the energy gap, which allows us to identify the Dirac string frequency $\omega$ with the lattice phonon Debye frequency $\omega_{D}$ in superconducting materials. Additionally, we found that the relativistic bound pair eigenstates are also in a spin singlet state, and present a spherically symmetric onion-like structure in the probability distribution. All these remarkable analogies suggest to interpret this two-body Dirac oscillator as an instance of a relativistic Cooper pair. Nevertheless, there may also be other types of relativistic binding mechanisms yielding also the formation of Cooper pairs.
On the other hand, a strong interaction leads to remarkable differences with respect to BCS Cooper pairs. In this case, more than one bound pair can be built, which in any case is not in a singlet state but rather in a linear superposition of singlet and triplet states. Furthermore, the gluing phonons become unfrozen as the coupling strength is raised and dynamically contribute to the pairing mechanism.
[*Acknowledgements*]{} We acknowledge financial support from a FPU grant of the MEC (A.B.), DGS grant under contract FIS2006-04885, CAM-UCM grant under ref. 910758. (A.B., M.A.M.D,), and the ESF Science Programme INSTANS 2005-2010 (M.A.M.D.).
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---
abstract: |
We present a study on the performance of Wang-Landau algorithm in a lattice model of liquid crystals which is a continuous lattice spin model. We propose a novel method of the spin update scheme in a continuous lattice spin model. The proposed scheme reduces the autocorrelation time of the simulation and results in faster convergence.\
Keywords : [Monte Carlo methods, Computational techniques, Phase transitions]{}
author:
- Suman Sinha
title: 'Performance of Wang-Landau algorithm in lattice model of liquid crystals'
---
Introduction {#intro}
============
The Wang-Landau (WL) algorithm [@wl], introduced in 2001, has received much attention and has been applied to a wide range of problems [@yo; @oty; @sdp; @rp; @jp; @yfp; @jsm; @cp; @lwll; @wl1; @swl]. In most of these investigations, the authors have applied the WL algorithm to systems with discrete energy levels. However, relatively fewer papers have so far appeared on lattice models with continuous energy spectrum [@yfp; @pcabd; @zstl; @mgr; @br; @sr]. Techniques, in general, to improve the algorithm for different problems have also been proposed [@yk; @sbm; @sbml; @bab; @yp; @sdp2; @tht; @dtwwtsc; @vmmb; @zb; @td; @lol; @ed; @ml; @bp; @zs]. The review [@lte] illustrates the versatile applications of the WL algorithm in protein folding, fluid simulations, systems with first order phase transitions and other systems with rough energy terrain. Some authors find its applications in performing numerical integration [@td; @lwll].
The WL algorithm allows us to calculate the density of states (DOS) as a function of energy or the joint density of states (JDOS) as a function of energy and a second variable [@zstl]. For a macroscopic system, the DOS $\Omega(E_i)$ (where $i=1,2,\cdots ,n$, $n$ being the bin index) is a large number and it is convenient to work with its logarithm $g(E_i)={\tt \ln} ~\Omega(E_i)$. Since the DOS is independent of temperature and contains complete information about the system, the task is to determine it as accurately as possible. The next step involves the determination of partition function ($\beta = 1/T$, Boltzmann constant has been set to unity) at any temperature ($T$) by the standard Boltzmann reweighting procedure. Once the partition function is known, the model is essentially “solved” since most thermodynamic quantities at any temperature can be calculated from it. The algorithm is implemented by performing an one-dimensional random walk that produces a “flat” histogram in the energy space. For a continuous model, one needs to use a discretization scheme to divide the energy range of interest into a number of bins which label the macrostates of the system. In the WL algorithm, these macrostates are sampled with a probability which is proportional to the reciprocal of the current DOS. The estimate for the DOS is improved at each step of the random walk using a carefully controlled modification factor $f$ to produce a result that converges to the true DOS quickly. A histogram record $H(E_i)$ of all states visited is maintained throughout the simulation. When $g(E_i)$ corresponding to a certain macrostate is modified as $g(E_i) \rightarrow
g(E_i)+{\tt \ln}f$, the corresponding $H(E_i)$ is modified as $H(E_i) \rightarrow H(E_i)+1$. In the original proposal of WL algorithm, an iteration is said to be complete when the histogram satisfies a certain “flatness” condition. This means that $H(E_i)$, for all values of $i$, has attained $90\%$ (or some other preset value) of the average histogram. In the following iteration, $f$ is reduced in some fashion, the $H(E_i)$’s are reset to zero and the process is continued till ${\tt \ln}f$ is as small as $10^{-8}$ or $10^{-9}$. Since the history of the entire sampling process determines the DOS, the WL algorithm is non-Markovian besides being multicanonical in nature.
In course of the random walk in a WL simulation, the fluctuations of energy histogram, for a given modification factor $f$, initially grows with time and then saturates to a certain value. Zhou and Bhatt [@zb] carried out a mathematical analysis of the WL algorithm. They provided a proof of the convergence of the iterative procedure and have shown that the fluctuations in histogram, proportional to $1/ \sqrt {\ln f}$ for a given $f$, cause statistical errors which can be reduced by averaging over multiple simulations. They have also shown that the correlation between adjacent records in the histogram introduces a systematic error which is reduced at smaller $f$. The prediction in Ref. [@zb] has been numerically verified by different authors independently [@lol; @sr]. Although to obtain a flat histogram is the initial motivation behind the WL algorithm, Ref. [@zb] concluded that flatness is not a necessary criterion to achieve convergence and suggested that one should instead focus on the fluctuations of the histogram rather than the “flatness”. They had shown that $1/ \sqrt {\ln f}$ visits on each macroscopic state is enough to guarantee the convergence. In fact, fluctuations in the histogram is intrinsic to WL algorithm. These fluctuations lead to a statistical error in the DOS which scales as $\sqrt {\ln f}$, for a given $f$. The iterative WL algorithm partially reduces this statistical fluctuations by decreasing $f$ monotonically. However Ref. [@bp] clearly illustrates that even if $f$ is reduced to a very small value according to the original prescription, the statistical error stops to decrease at a certain point. In practice there always exists a systematic error in the simulation which is a function of $f$ and the correlation between adjacent records in the histogram. Ref. [@zb] observed that this systematic error decreases when either $f$ or the correlation decreases. In this context, we refer to the work of Morozov and Lin [@ml] who presented a study on the estimations of accuracy and convergence of the Wang-Landau algorithm on a two level system with a significant efficiency improvement in [@ml2]. The WL algorithm compares $\Omega (E_i)$ and $\Omega (E_f)$, i.e, DOS before and after an attempted move, but it does not require $E_i$ to be close to $E_f$. This is why Ref. [@zb] suggested the use of cluster algorithms that allow “nonlocal” moves in the parameter space. The Ref. [@zs] rightly pointed out that the update schemes for the underlying model certainly have an effect on the outcome. In the present paper we suggest a method for the spin update scheme of a lattice model with continuous energy spectrum, which reduces the autocorrelation time by an appreciable amount compared to the conventional spin update scheme. The suggested spin update method to obtain a less correlated configuration has also the advantage that this method is free from tuning any adjustable parameter. The method is described in Section \[ct\]. We also investigate the growth of the histogram fluctuations in the one-dimensional Lebwohl-Lasher (LL) model, described in Section \[model\], to check if the nature of the dependence of the maximum of the histogram fluctuations on the modification factor $f$ is model independent or not. We mention in passing that Ref. [@lol] suggested the model-independent nature of the maximum of the histogram fluctuations by performing simulations on two discrete Ising models and concluded that many more simulations on different models are needed to confirm this universality nature. Ref. [@sr] confirmed this universality behavior for two continuous lattice spin models with spin dimensionality two. We have found that for the present model (spin dimensionality three), the fluctuations in the energy histogram, after an initial increase, saturates to a value which is inversely proportional to $\sqrt {\ln f}$ and confirm that this feature is generic to the WL algorithm. In the second part of the work, we have carried out the WL simulation with the proposed spin update scheme to estimate the canonical averages of various thermodynamic quantities for lattices of reasonably large size where minimum number of visits to each macrostate are $1/ \sqrt{\ln f}$. Results obtained from our simulation are compared with the exact results available for the model.
The rest of the paper is arranged as follows. In Section \[model\], we have described the model. The computational techniques are discussed in Section \[ct\]. Section \[rd\] presents our results and discussions. Section \[conclu\] draws the conclusions.
Model
=====
For the purpose of investigation, we have chosen an one-dimensional array of three-dimensional spins ($d=1,l=3$, where $d$ is the space dimensionality and $l$ is the spin dimensionality) interacting with nearest neighbors (nn) via a potential $$V_{ij}=-P_2(\tt \cos~ \theta_{ij})
\label{eqn1}$$ where $P_2$ is the second Legendre polynomial and $\theta_{ij}$ is the angle between the nearest neighbor spins $i$ and $j$ (the coupling constant in the interaction has been set to unity). The spins are three-dimensional and headless, i.e, the system has the $O(3)$ as well as the local $Z_2$ symmetry, characteristic of a nematic liquid crystal. The model, known as the Lebwohl-Lasher (LL) model [@ll], is the lattice version of the Maier-Saupe (MS) model [@ms] which describes a nematic liquid crystal in the mean field approximation. Being a low-dimensional model with nn interaction, the $1d$ LL model does not exhibit any finite temperature phase transition. This model has been solved exactly by Vuillermot and Romerio [@vr] in $1973$, using a group theoretical method. The results obtained in [@vr] are quoted below. The partition function $Z_N(\widetilde K)$ for the $N$-particle system is given by $$Z_N(\widetilde K)=\widetilde K^{N/2} {\tt exp}\left[\frac {2}{3}N\widetilde K \right]D^N(\widetilde K^{1/2})
\label{eqn2}$$ where $\widetilde K=3/2T$ is a dimensionless quantity. $D$ is the Dawson function [@as] given by $$D(x)={\tt exp}(-x^2)\int_0^x e^{u^2}du
\label{eqn3}$$ The dimensionless internal energy $U_N(\widetilde K)$, entropy $S_N(\widetilde K)$ and the specific heat $C_N(\widetilde K)$ are given by $$\frac{2U_N(\widetilde K)}{N}=1+\frac{3\widetilde K^{-1}}{2}-\frac{3}{2} \widetilde K^{-1/2}D^{-1}
\left (\widetilde K^{1/2}\right)
\label{eqn4}$$ $$\begin{gathered}
\frac{S_N(\widetilde K)}{N}=\frac{1}{2}+\widetilde K-\frac{1}{2}\widetilde K^{1/2}D^{-1}
\left (\widetilde K^{1/2}\right) \\
+\ln \left[\widetilde K^{-1/2}D\left(\widetilde K^{1/2}\right)\right]
\label{eqn5}\end{gathered}$$ $$\begin{gathered}
\frac{2C_N(\widetilde K)}{N}=1-\widetilde K^{3/2}\left[\frac{\widetilde K^{-1}}{2}-1\right]D^{-1}
\left (\widetilde K^{1/2}\right) \\
-\frac{1}{2} \widetilde K D^{-2}\left(\widetilde K^{1/2}\right)
\label{eqn6}\end{gathered}$$ We decided to choose this model to test the performance of WL algorithm using the suggested spin update scheme so that a comparison can be made with the exact results available for the model.
Computational Techniques {#ct}
========================
In the first part of this Section, we will describe the computational techniques used to determine the fluctuations in the energy histogram. In the later part of this Section, we will discuss the method for the new spin update scheme.
Let us first explain the notations and symbols relevant to the present work. The saturation value of the energy histogram fluctuation in the $k^{th}$ iteration is represented by $\beta_k$. Let $f_k$ be the modification factor for the $k^{th}$ iteration. One usually starts with a modification factor $f=f_1 \geq 1$ and uses a sequence of decreasing $f_k$’s ($k=1,2,3,\cdots$) defined in some manner. One Monte Carlo (MC) sweep is taken to be completed when the number of attempted single spin moves equals the number of spins in the system. The error in the DOS after the $n^{th}$ iteration is directly related to $\beta_i$ for $i > n$, the saturation values of the fluctuations. In the WL algorithm the logarithm of the DOS after $n$ iterations is given by $$g_n(E_i)=\sum_{k=1}^n H_k(E_i)\ln (f_k)
\label{eqn7}$$ where $H_k(E_i)$ is the accumulated histogram count for the $i^{th}$ energy bin during the $k^{th}$ iteration. In order to get an idea of the fluctuations in the histogram and its growth with the number of MC sweeps, we subtract the minimum of the histogram count $h_k^j$ which occurs in the histogram after the $j^{th}$ MC sweep has been completed during the $k^{th}$ iteration, i.e., we consider the quantity $$\widetilde H_k^j(E_i)=H_k^j(E_i)-h_k^j
\label{eqn8}$$ It may be noted that $h_k^j$ does not refer to any particular bin and may occur in any of the visited bins. The quantity $\widetilde H_k^j(E_i)$ is now summed over all bins to give $\Delta H_k^j$. $$\Delta H_k^j=\sum_i \widetilde H_k^j(E_i)
\label{eqn9}$$ $\Delta H_k^j$ is thus a measure of the fluctuations which occurs in the $j^{th}$ MC sweep during $k^{th}$ iteration and is a sort of average over all macrostates or bins. $\Delta H_k^j$ fluctuates with $j$ because of statistical errors and its mean value taken over $j$ is nothing but $\beta_k$. The error of the logarithm of the DOS, summed over all energy levels or bins, after the completion of $n$ iterations is therefore given by [@lol] $$\eta_n=\sum_{k=n+1}^{\infty} \beta_k \ln (f_k)
\label{eqn10}$$ Eq. (\[eqn10\]) means that the error depends only on the fluctuations in histogram and the sequence of modification factors. When the values of $f_k$ are predetermined, the fluctuations in histogram, i.e., $\Delta H_k^j$, becomes the only determining factor for the error. For this reason the observable $\Delta H_k^j$, defined by Eq. (\[eqn9\]), is considered to be a good measure of the fluctutations in histogram. However, we point out that because of the summation over the index $i$ in Eq. (\[eqn9\]), the nature of the distribution of the errors over the energy bins is not reflected in the summed quantity $\Delta H_k^j$. What we get instead is an error which has been summed over all the energy bins. Since the predicted value of the error $\eta_n$ is of the order of $\sqrt {\ln f_n}$ [@zb], one expects that the histogram saturation value $\beta_n$, for the $n^{th}$ iteration, should be proportional to $1/ \sqrt {\ln f_n}$.
Proposal for a novel spin update method
---------------------------------------
Now we discuss the method to generate a subsequent less-correlated spin configuration. In the conventional spin update method for a continuous lattice spin model, the orientation of each spin $\vec s$ is stored in terms of the direction cosines $(l_1, l_2, l_3)$. To generate a new configuration (microstate), a spin is selected at random and each direction cosine of it is updated as $l_i \rightarrow l_i +p*x_i$ for ($i=1,2,3$) where the parameter “p” denotes the amplitude of the random angular displacements, chosen such that approximately half of the configurations are accepted and half rejected [@lg] and $x_i$ is a random number between $-1$ to $1$. We have seen for a number of continuous lattice spin models that the results for the thermodynamic quantities become very sensitive to the value of the parameter “p”. “p” is generally taken such that $p < 1$ and the choice of “p” also depends on the systems we are working on. The reason for taking $p < 1$ is that small values of “p” correspond to small changes in the direction of the spin, i.e., the energy cost of an attempted move will be small. However, this is not the only form of update, nor is it known whether this is the most efficient form. The thing is, there is quite a lot of flexibility about the choice of the new state for the spins. A good discussion of it may be found in Ref. [@nb].
In the present work, we propose a novel protocol to generate a less-correlated spin configuration in the following manner. We take a random unit vector $\vec r$ and a spin update $\vec s \rightarrow \vec s^{~\prime}$ is defined as $\vec s^{~\prime}=\vec s - 2\left(\vec s \cdot \vec r\right)\vec r$ where $(\vec s \cdot \vec r)$ is the dot product of $\vec s$ and $\vec r$. This represents a reflection with respect to the hyperplane orthogonal to $\vec r$ and this is an idempotent operation. The idea came from Wolff [@wlf]. One may think of a linear transformation $R(\vec r)$ such that $\vec s^{~\prime}=R(\vec r) \vec s$. This linear transformation has the property $$R(\vec r)^2=1$$ i.e., idempotent and $$[R(\vec r)\vec s_1]\cdot[R(\vec r)\vec s_2]=\vec s_1 \cdot \vec s_2$$ i.e., the Hamiltonian (\[eqn1\]) is invariant under global R transformations. This spin update method reduces the autocorrelation time to a considerable amount and consequently systematic error decreases. Moreover, defining a spin update in that way, the algorithm becomes free from tuning any adjustable parameter even while simulating a lattice spin model with continuous energy spectrum. This spin update method has resulted in efficient simulation of continuous lattice spin models with $XY$ symmetry [@sr3; @sr4].
The energy of the $1d$ LL model is a continuous variable and it can have any value between $-L$ to $L/2$ where $L$ is the system size. To discretize the system, we have chosen an energy range ($-L,0$) and divided this energy range into a number of bins (macrostates) each having a width, say $w$. In the present work, the bin width is taken to be $0.2$.
Results and discussions {#rd}
=======================
-- --
-- --
We have determined for the lattice model we have defined, the dependence of the quantity $\Delta H_k^j$, given by Eq. (\[eqn9\]), on $j$, the number of MC sweeps for a given iteration denoted by $k$. For the purpose of testing the fluctuations in histogram, we have taken linear spin chains of length $L=80$ and $160$. Nearest neighbor interactions along with periodic boundary conditions were always used. The starting value of the modification factor $\ln f_1$ was taken to be $0.1$ and the sequence $\ln f_{n+1}=
(\ln f_n)/10^{1/4}$ was chosen and for the purpose of determination of fluctuations, the minimum $\ln f$ used was $10^{-5}$. Clearly, the chosen sequence of $f$ is to ensure that it gets reduced by a factor of $10$ after four iterations. We have determined the quantity $\Delta H_k^j$ defined by Eq. (\[eqn9\]) at intervals of $10^3$ MC sweeps and the maximum number of sweeps chosen for a given value of $f$ is such that the saturation of the histogram is clearly evident. The system energy is always considered up to $E=0$. The lower limit of the energy for $L=80$ is taken to be $-78$ and for $L=160$, it is $-158$, while the corresponding ground state energies are $-80$ and $-160$. Thus the visited energy range goes to a sufficiently low value to cover the entire range of interest, though the small cut near the ground state is necessary, as configurations near the minimum energy take a very long time to be sampled during the random walk.
In Fig. \[hisfluc\], we have plotted the fluctuations in the histogram $\Delta H_k^j$ against the number of MC sweeps $j$ for four values of the modification factor $f$. The plots shown are for $L=160$ lattice and for $\ln f$ equal to $10^{-2}$, $10^{-3}$, $10^{-4}$ and $10^{-5}$. We did not go to values of $\ln f$ less than $10^{-5}$ as it takes a very large CPU time. Averages were taken over hundred independent simulations to improve the statistics and accuracy. Similar plots are also taken for the $L=80$ lattice. It is evident from Fig. \[hisfluc\] that $\Delta H_k^j$ increases initially and then saturates and as $f$ gets smaller, the saturation value as well as the number of MC sweeps necessary to reach the saturation ($\tt MCS_{sat}$) increases. Fig. \[lnfvsmcs\] explicitly reveals this fact.
The standard error calculated from the hundred independent simulations are also shown in Fig. \[hisfluc\]. In Fig. \[slope\], we have plotted the logarithm of saturation value $\beta_k$, i.e., $\ln(\beta_k)$ vs $\ln (\ln f)$ for system sizes $L=80$ and $L=160$.
From this figure, it is clear that $$\beta_k \propto (\ln f)^{\alpha}
\label{eqn11}$$ where the index $\alpha=-0.50133 \pm 0.007$ for $L=80$ and $\alpha=-0.50844 \pm 0.005$ for $L=160$ respectively. This is in agreement with the prediction of Zhou and Bhatt [@zb]. Certainly, this result is not new. It confirms the previous results that the values of the slope is generic to the WL algorithm, in this case, it is a continuous lattice spin model with spin dimensionality three.
Now we present the results of various thermodynamic quantities obtained from the simulation. In Fig. \[engy\], we have plotted the average energy per spin against temperature ($T$) for $L=220$. The results have been compared with the exact values of this observable obtained from Ref. [@vr].
The specific heat, calculated as fluctuations of the energy, has been plotted against $T$ in Fig. \[cv\] for $L=220$ and compared with the exact results. In the inset of Fig. \[cv\], the percentage error ($\epsilon$) in the $C_v$ near the peak in comparison with the exact results is shown. Percentage error is a measure of how inaccurate (or accurate) a measurement is and is defined by the formula $\frac{\tt measured~ value - \tt actual~ value}{\tt actual~ value} \times 100 \%$. Exact results show that the specific heat peak is maximum at a temperature $T_{max}^{ex}=0.24$ and from our simulation we obtain the temperature at which the peak of the specific heat is maximum is $T_{max}^{sim}=0.2351$ for $L=220$. This implies that the percentage error in temperature at which the peak of the specific heat is maximum is $2.04 \%$. Fig. \[entpy\] shows the variation of entropy per particle for $L=220$ and the exact results are also shown in the same plot.
The attention is now focused on the autocorrelation time of the simulation. The autocorrelation function for an observable $O(t)$ is given by $$\chi(t)=\int \mathrm{d}t^{\prime}(O(t^{\prime})-\langle O \rangle)
(O(t^{\prime}+t)-\langle O \rangle)
\label{autocorfn}$$ where $O(t)$ is the instantaneous value of the observable at time $t$ and $\langle O \rangle$ is the average value. The integrand in the above equation actually measures the correlation between the fluctuation of the observables at two different times, one an interval $t$ later than the other. So $\chi(t)$ will take a nonzero value if on the average the fluctuations are correlated, otherwise it is zero. Thus when $t$ is just a single MC step apart, we will have a large positive autocorrelation. For large $t$, $\chi(t)$ will be zero and the measurements are totally uncorrelated. The autocorrelation is expected to fall off exponentially at long times thus: $$\chi(t) \sim {\tt e}^{-t/\tau}
\label{autocor}$$ where $\tau$ is a measure of autocorrelation time of our simulation. At time $t=\tau$, the autocorrelation function, which is a measure of the similarity of the two states, is only a factor of $1/{\tt e}$ down from its maximum value at $t=0$. We have estimated the autocorrelation time both for the simulations with the conventional spin update method and the proposed spin update method. The autocorrelation time is calculated following the method proposed by Madras and Sokal [@madso]. In the conventional spin update scheme, when we flip a single spin in each update, the total energy can only change by a small amount every time. In the proposed spin update scheme, the change in total energy is greater compared to that in the conventional scheme. As the WL algorithm does not require $E_i$ to be close to $E_f$, but compares only $\Omega (E_i)$ and $\Omega (E_f)$, the convergence becomes faster with the proposed scheme than with the conventional scheme.
We have found that the autocorrelation time ($\tau$) exhibits a power law scaling with system size, i.e., $$\tau \propto L^z
\label{eqn12}$$ The scaling exponent ($z$) is determined from a linear fit of the plot $\ln \tau$ versus $\ln L$. The logarithm of the autocorrelation time for both the conventional and the proposed spin update scheme has been plotted against $\ln L$ for $\ln f=0.01$ in Fig. \[autocor\]. The scaling exponent for the proposed spin update scheme ($z_{\tt new}$) is found to be $z_{\tt new}=1.36351 \pm 0.024$ while that for the conventional spin update scheme ($z_{\tt old}$) is found to be $z_{\tt old}=1.57591 \pm 0.013$. The proposed spin update scheme significantly decreases the scaling exponent.
We would like to point out that the autocorrelation time ($\tau$) increases rapidly as the modification factor ($f$) becomes smaller and for a larger system size, the calculation of $\tau$, specially for smaller $f$, becomes very much costly in terms of CPU time. The autocorrelation time for a number of modification factors $f$ for $L=200$ for both the proposed and the conventional spin update schemes is listed in Table \[table1\] and plotted in Fig. \[lnfvstau\].
$\ln f$ $\tau_{\tt new}$ $\tau_{\tt old}$
---------- ------------------ ------------------
$1.0$ $28954$ $78966$
$0.1$ $31226$ $96902$
$0.01$ $67642$ $234502$
$0.001$ $173250$ $638082$
$0.0001$ $246118$ $1359694$
: Autocorrelation time (in units of MC sweep) for different $\ln f$ for $L=200$.
\[table1\]
Conclusions {#conclu}
===========
To summarize, we have tested the performance of the WL algorithm in a continuous lattice spin model, namely, the $1d$ LL model which describes a nematic liquid crystal in the mean field approximation. The results obtained from our simulation are compared with the exact results available for this model. It has been observed that the results obtained tally accurately with the exact results. We focus on the fluctuations of histogram and replace the “flatness” criterion with that of minimum histogram. We have found that in this continuous lattice model, the fluctuations in the energy histogram, after an initial accumulation stage, saturates to a value that is proportional to $1/\sqrt{\ln f}$ where $f$ is the modification factor in the WL algorithm and confirm that this behavior is generic to the WL algorithm. We also present a novel method for spin update scheme to obtain a subsequent configuration which is less-correlated than the previous method. The proposed spin update scheme makes the WL “driver” to move from one sampling point to the next faster. As a result, the autocorrelation time between successive moves decreases and the convergence becomes faster. It may be noted that the WL algorithm only asks for the next sampling point (say $X$) with probability distribution $P(X) \propto \Omega(X)/\Omega (\bar X)$ where $\Omega (X)$ and $\Omega (\bar{X})$ are the exact and the estimated DOS respectively. A previous study [@sbml] suggested that $N$-fold way updates yields better performance in flat-histogram sampling. However, Dayal $et. al$ [@dtwwtsc] argued that the performance is limited by the added expense of the CPU time needed to implement the $N$-fold way updates. The proposed method is simple to implement and has also the merit that it makes us free from tuning any adjustable parameter while simulating a continuous lattice spin model. Although the method has been applied to a liquid crystalline system in the present work, the method can, in general, be applied to any lattice spin model with continuous energy spectrum. This method has resulted in efficient simulation of continuous models with $XY$ symmetry [@sr3; @sr4]. Finally, we stress that the focus in this paper is to test the performance of the WL algorithm in continuous lattice spin models with the proposed spin update scheme. We hope that this spin update method will be of general interest in the area of research in Monte Carlo simulations of continuous lattice spin models.
Acknowledgements
================
I wish to thank Prof. S. K. Roy for fruitful discussions and critical reading of the manuscript. This work is supported by the UGC Dr. D. S. Kothari Post Doctoral Fellowship under grant No. F-2/2006(BSR)/13-398/2011(BSR). Part of the computations of this work has been done using the computer facilities of the TCMP Division of Saha Institute of Nuclear Physics, Kolkata, India. I thankfully acknowledge the unanimous referee for a number of suggestions in improving the manuscript.
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|
---
abstract: 'Cardinal scores (numeric ratings) collected from people are well known to suffer from miscalibrations. A popular approach to address this issue is to assume simplistic models of miscalibration (such as linear biases) to de-bias the scores. This approach, however, often fares poorly because people’s miscalibrations are typically far more complex and not well understood. In the absence of simplifying assumptions on the miscalibration, it is widely believed by the crowdsourcing community that the only useful information in the cardinal scores is the induced ranking. In this paper, inspired by the framework of Stein’s shrinkage, empirical Bayes, and the classic two-envelope problem, we contest this widespread belief. Specifically, we consider cardinal scores with arbitrary (or even adversarially chosen) miscalibrations which are only required to be consistent with the induced ranking. We design estimators which despite making no assumptions on the miscalibration, strictly and uniformly outperform all possible estimators that rely on only the ranking. Our estimators are flexible in that they can be used as a plug-in for a variety of applications, and we provide a proof-of-concept for A/B testing and ranking. Our results thus provide novel insights in the eternal debate between cardinal and ordinal data.'
author:
- |
\
Jingyan Wang and Nihar B. Shah\
\
School of Computer Science\
Carnegie Mellon University\
`{jingyanw,nihars}@cs.cmu.edu`
bibliography:
- 'references.bib'
title: |
**Your 2 is My 1, Your 3 is My 9:\
Handling Arbitrary Miscalibrations in Ratings**
---
Introduction
============
*“A raw rating of 7 out of 10 in the absence of any other information is potentially useless.”* [@mitliagkas2011]
*“The rating scale as well as the individual ratings are often arbitrary and may not be consistent from one user to another.”* [@ammar2012aggregation]
Consider two items that need to be evaluated (for example, papers submitted to a conference) and two [[[reviewer]{}]{}s]{}. Suppose each [reviewer]{}is assigned one distinct item for evaluation, and this assignment is done uniformly at random. The two [[[reviewer]{}]{}s]{}provide their evaluations (say, in the range $[0,1]$) for the respective item they [evaluate]{}, from which the better item must be chosen. However, the [[[reviewer]{}]{}s]{}’ rating scales may be miscalibrated. It might be the case that the first [reviewer]{}is lenient and always provides scores in $[0.6,1]$ whereas the second [reviewer]{}is more stringent and provides scores in the range $[0,0.4]$. Or it might be the case that one [reviewer]{}is moderate whereas the other is extreme – the first [reviewer]{}’s 0.2 is equivalent to the second [reviewer]{}’s 0.1 whereas the first [reviewer]{}’s 0.3 is equivalent to the second [reviewer]{}’s 0.9. More generally, the miscalibration of the reviewers may be arbitrary and unknown. Then is there any hope of identifying the better of the two items with any non-trivial degree of certainty?
A variety of applications involve collection of human preferences or judgments in terms of cardinal scores (numeric ratings). A perennial problem with eliciting cardinal scores is that of miscalibration – the systematic errors introduced due to incomparability of cardinal scores provided by different people (see [@griffin2008calibration] and references therein).
This issue of miscalibration is sometimes addressed by making simplifying assumptions about the form of miscalibration, and post-hoc corrections under these assumptions. Such models include one-parameter-per-reviewer additive biases [@paul1981calibration; @baba2013quality; @ge13bias; @mackay2017calibration], two-parameters-per-reviewer scale-and-shift biases [@paul1981calibration; @roos2011calibrate] and others [@flach2010kdd]. The calibration issues with human-provided scores are often significantly more complex causing significant violations to these simplified assumptions (see [@griffin2008calibration] and references therein). Moreover, the algorithms for post-hoc correction often try to estimate the individual parameters which may not be feasible due to low sample sizes. For instance, John Langford notes from his experience as the program chair of the ICML 2012 conference [@langford2012icml]:
*“We experimented with reviewer normalization and generally found it significantly harmful.”*
This problem of low sample size is exacerbated in a number of applications such as A/B testing where every [reviewer]{} [[[evaluate]{}]{}s]{}only one item, thereby making the problem underdetermined even under highly restrictive models.
It is commonly believed that when unable or unwilling to make any simplifying assumptions on the bias in cardinal scores, the only useful information is the ranking of the scores [@rokeach1968values; @freund2003boosting; @harzing2009rating; @mitliagkas2011; @ammar2012aggregation; @negahban2012ranking]. This perception gives rise to a second approach towards handling miscalibrations – that of using only the induced ranking or otherwise directly eliciting a ranking and not scores from the use. As noted by Freund et al. [@freund2003boosting]:
*“\[Using rankings instead of ratings\] becomes very important when we combine the rankings of many viewers who often use completely different ranges of scores to express identical preferences.”*
These motivations have spurred a long line of literature on analyzing data that takes the form of partial or total rankings of items [@cook2007proposal; @baskin2009recommender; @ammar2012aggregation; @negahban2012ranking; @rajkumar2015ranking; @shah2016estimation; @shah18simple].
In this paper, we contest this widely held belief with the following two fundamental questions:
In the absence of simplifying modeling assumptions on the miscalibration, is there any estimator (based on the scores) that can outperform estimators based on the induced rankings?
If only one evaluation per [reviewer]{}is available, and if each [reviewer]{}may have an arbitrary (possibly adversarially chosen) miscalibration, is there hope of estimation better than random guessing?
We show that the answer to both questions is “Yes”. One need not make simplifying assumptions about the miscalibration and yet guarantee a performance superior to that of any estimator that uses only the induced rankings.
In more detail, we consider settings where a number of people provide cardinal scores for one or more from a collection of items. The calibration of each [reviewer]{}is represented by an unknown monotonic function that maps the space of true values to the scores given by this [reviewer]{}. These functions are arbitrary and may even be chosen adversarially. We present a class of estimators based on cardinal scores given by the [[[reviewer]{}]{}s]{}which *uniformly* outperforms any estimator that uses only the induced rankings. A compelling feature of our estimators is that they can be used as a plug-in to improve ranking-based algorithms in a variety of applications, and we provide a proof-of-concept for two applications: A/B testing and ranking.
The techniques used in our analyses draw inspiration from the framework of Stein’s shrinkage [@stein1956inadmissibility; @james1961estimation] and empirical Bayes [@robbins1956empirical]. Moreover, our setting with $2$ reviewers and $2$ papers presented subsequently in the paper carries a close connection to the classic two-envelope problem (for a survey of the two-envelope problem, see [@gnedin2016survey]), and our estimator in this setting is similar in spirit to the randomized strategy [@cover1987envelope] proposed by Thomas Cover. We discuss connections with the literature in more detail in Section \[sec:two\_envelope\].
Our work provides a new perspective on the eternal debate between cardinal scores and ordinal rankings. It is often believed that ordinal rankings are a panacea for the miscalibration issues with cardinal scores. Here we show that ordinal estimators are not only inadmissible, they are also strictly and uniformly beaten by our cardinal estimators. Our results thus uncover a new point on the bias-variance tradeoff for this class of problems: Estimators that rely on simplified assumptions about the miscalibration incur biases due to model mismatch, whereas the absence of such assumptions in our work eliminates the modeling bias. Moreover, in this minimal-bias regime, our cardinal estimators incur a strictly smaller variance as compared to estimators based on ordinal data alone.
Finally, a note qualifying the scope of the problem setting considered here. In applications such as crowdsourced microtasks where workers often spend very little time answering every question, the cardinal scores elicited may not necessarily be consistent with the ordinal rankings, and moreover, ordinal rankings are often easier and faster to provide. These differences cease to exist in a variety of applications such as peer-review or in-person laboratory A/B tests which require the [[[reviewer]{}]{}s]{}to spend a non-trivial amount of time and effort in the review process, and these applications form the motivation of this work.
Preliminaries {#sec:prelim}
=============
Consider a set of ${n}$ items denoted as $\{1,\ldots,{n}\}$ or $[{n}]$ in short.[^1] Each item ${i}\in [{n}]$ has an unknown value ${x}_{i}\in {{\mathbb{R}}}$. For ease of exposition, we assume that all items have distinct values. There are ${m}$ [[[reviewer]{}]{}s]{}$\{1,\ldots,{m}\}$ and each [reviewer]{}[[[evaluate]{}]{}s]{}a subset of the items. The calibration of any [reviewer]{}${j}\in [{m}]$ is given by an unknown, strictly-increasing function ${f}_{j}: {{\mathbb{R}}}\rightarrow {{\mathbb{R}}}$. (More generally, our results hold for any non-singleton intervals on the real line as the domain and range of the calibration functions). When [reviewer]{}${j}$ evaluates item ${i}$, the reported score is ${f}_{j}({x}_{i})$. We make no other assumptions on the calibration functions ${f}_1,\ldots,{f}_{m}$. We use the notation ${\succ}$ to represent a relative order of any items, for instance, we use “$1 {\succ}2$” to say that item $1$ has a larger value (ranked higher) than item $2$. We assume that ${m}$ and ${n}$ are finite.
Every [reviewer]{}is assigned one or more items to [evaluate]{}. We denote the assignment of items to [[[reviewer]{}]{}s]{}as ${A}= ({S}_1, \ldots, {S}_{m})$, where ${S}_{j}\subseteq [{n}]$ is the set of items assigned to [reviewer]{}${j}\in [{m}]$. We use the notation ${\Pi}$ to represent the set of all permutations of ${n}$ items. We let ${{\pi}^*}\in {\Pi}$ denote the ranking of the ${n}$ items induced by their respective values $({x}_1,\ldots,{x}_{n})$, such that ${x}_{{{\pi}^*}(1)} > {x}_{{{\pi}^*}(2)} > \cdots > {x}_{{{\pi}^*}({n})}$. The goal is to estimate this ranking ${{\pi}^*}$ from the evaluations of the [[[reviewer]{}]{}s]{}. We consider two types of settings: an ordinal setting where estimation is performed using the rankings induced by each [reviewer]{}’s reported [[[score]{}]{}s]{}, and a cardinal setting where the estimation is performed using the [[[reviewer]{}]{}s]{}’ [[[score]{}]{}s]{}(which can have an arbitrary miscalibration and only need to be consistent with the rankings). Formally:
**Ordinal:** Each [reviewer]{}${j}$ reports a total ranking among the items in ${S}_{j}$, that is, the ranking of the items induced by the values $\{{f}_{{j}}({x}_{i})\}_{{i}\in {S}_{j}}$. An ordinal estimator observes the assignment ${A}$ and the rankings reported by all [[[reviewer]{}]{}s]{}.
**Cardinal:** Each [reviewer]{}${j}$ reports the [[[score]{}]{}s]{}for the items in ${S}_{j}$, that is, the values of $\{{f}_{{j}}({x}_{i})\}_{{i}\in {S}_{j}}$. A cardinal estimator observes the assignment ${A}$ and the [[[score]{}]{}s]{}reported by all [[[reviewer]{}]{}s]{}.
Observe that the setting described above considers “noiseless” data, where each [reviewer]{}reports either the [[[score]{}]{}s]{}$\{{f}_{j}({x}_{i})\}$ or the induced rankings. We provide an extension to the noisy setting in Appendix \[app:noisy\].
In order to compare the performance of different estimators, we use the notion of *strict uniform dominance*. Informally, we say that one estimator strictly uniformly dominates another if it incurs a strictly lower risk for all possible choices of the miscalibration functions and the item values.
In more detail, suppose that you wish to show that an estimator ${\widehat{{\pi}}}_1$ is superior to estimator ${\widehat{{\pi}}}_2$ with respect to some metric for estimating ${{\pi}^*}$. However, there is a clever adversary who intends to thwart your attempts. The adversary can choose the miscalibration functions of all [[[reviewer]{}]{}s]{}and the values of all items, and moreover, can tailor these choices for different realizations of ${{\pi}^*}$. Formally, the adversary specifies a set of values $\{{{f}^{{\pi}}}_1,\ldots,{{f}^{{\pi}}}_{m}, {{x}^{{\pi}}}_{1},\ldots,{{x}^{{\pi}}}_{{n}}\}_{{\pi}\in {\Pi}}$. The only constraints in this choice are that the miscalibration functions ${{f}^{{\pi}}}_1,\ldots,{{f}^{{\pi}}}_{m}$ must be strictly monotonic and that the item values ${{x}^{{\pi}}}_{1},\ldots,{{x}^{{\pi}}}_{{n}}$ should induce the ranking ${\pi}$. In the sequel, we consider two ways of choosing the true ranking ${{\pi}^*}$: In one setting, ${{\pi}^*}$ can be chosen by the adversary, and in the second setting ${{\pi}^*}$ is drawn uniformly at random from ${\Pi}$. Once this ranking ${{\pi}^*}$ is chosen, the actual values of the miscalibration functions and the item values are set as ${{f}^{{{\pi}^*}}}_1,\ldots,{{f}^{{{\pi}^*}}}_{m}$ and ${{x}^{{{\pi}^*}}}_{1},\ldots,{{x}^{{{\pi}^*}}}_{{n}}$. The items are then assigned to [[[reviewer]{}]{}s]{}according to the (possibly random) assignment ${A}$. The [[[reviewer]{}]{}s]{}now provide their ordinal or cardinal evaluations as described earlier, and these evaluations are used to compute and evaluate the two estimators ${\widehat{{\pi}}}_1$ and ${\widehat{{\pi}}}_2$. We say that estimator ${\widehat{{\pi}}}_1$ strictly uniformly dominates ${\widehat{{\pi}}}_2$, if ${\widehat{{\pi}}}_1$ is always guaranteed to incur a strictly smaller (expected) error than ${\widehat{{\pi}}}_2$. Formally:
\[def:uniformly\_better\] Let ${\widehat{{\pi}}}_1$ and ${\widehat{{\pi}}}_2$ be two estimators for the true ranking ${{\pi}^*}$. Estimator ${\widehat{{\pi}}}_1$ is said to strictly uniformly dominate estimator ${\widehat{{\pi}}}_2$ with respect to a given loss ${L}: {\Pi}\times {\Pi}\rightarrow {\mathbb{R}}$ if $$\begin{aligned}
{\mathbb{E}}[{L}({{\pi}^*}, {\widehat{{\pi}}}_1)] < {\mathbb{E}}[{L}({{\pi}^*}, {\widehat{{\pi}}}_2)] \qquad \text{ for all permissible~~} \{{{f}^{{\pi}}}_1,\ldots,{{f}^{{\pi}}}_{m}, {{x}^{{\pi}}}_{1},\ldots,{{x}^{{\pi}}}_{{n}}\}_{{\pi}\in {\Pi}}.
\label{eq:uniformly_better}\end{aligned}$$ The expectation is taken over any randomness in the assignment ${A}$ and the estimators. If the true ranking ${{\pi}^*}$ is drawn at random from a fixed distribution, then the expectation is also taken over this distribution; otherwise, inequality must hold for all values of ${{\pi}^*}$.
Note that strict uniform dominance is a stronger notion than comparing estimators in terms of their minimax (worst-case) or average-case risks. Moreover, if an estimator ${\widehat{{\pi}}}_2$ is strictly uniformly dominated by some estimator ${\widehat{{\pi}}}_1$, then the estimator ${\widehat{{\pi}}}_2$ is inadmissible.
Finally, for ease of exposition, we focus on the 0-1 loss in the main text: $$\begin{aligned}
{L}({\pi}^*, {\pi}) = {{\mathbbm{1}}{\{{\pi}^* \ne {\pi}\}}},\end{aligned}$$ where we use the standard notation ${{\mathbbm{1}}{\{{A}\}}}$ to denote the indicator function of an event ${A}$, where ${{\mathbbm{1}}{\{{A}\}}} = 1$ if the event ${A}$ is true, and $0$ otherwise. Extensions to other metrics of Kendall-tau distance and Spearman’s footrule distance are provided in Appendix \[app:other\_metrics\].
Main results
============
In this section we present our main theoretical results. All proofs are provided in Section \[app:proofs\].
A canonical setting {#sec:canonical}
-------------------
We begin with a canonical setting that involves two items and two [[[reviewer]{}]{}s]{}(that is, ${n}=2$, ${m}= 2$), where each [reviewer]{}[[[evaluate]{}]{}s]{}one of the two items. Our analysis for this setting conveys the key ideas underlying our general results. These ideas are directly applicable towards designing uniformly superior estimators for a variety of applications, and we subsequently demonstrate this general utility with two applications.
In this canonical setting, each of the two [[[reviewer]{}]{}s]{}[[[evaluate]{}]{}s]{}one of the two items chosen uniformly at random without replacement, that is, the assignment ${A}$ is chosen uniformly at random from the two possibilities $({S}_1 = 1, {S}_2 = 2)$ and $({S}_1 = 2, {S}_2 = 1)$. Since each [reviewer]{}is assigned only one item, the ordinal data is vacuous. Then the natural ordinal baseline is an estimator which makes a guess uniformly at random: $$\begin{aligned}
{{\widehat}{{\pi}}_{{\text{can}}}}({A}, \{\}) = \begin{cases}
1 {\succ}2 & \text{with probability } 0.5\\
2 {\succ}1 & \text{with probability } 0.5.
\end{cases}\end{aligned}$$
In the cardinal setting, let ${y}_1 $ denote the [score]{}reported for item $1$ by its respective [reviewer]{}, and let ${y}_2$ denote the [score]{}for item $2$ reported by its respective [reviewer]{}. Since the calibration functions are arbitrary (and may be adversarial), it appears hopeless to obtain information about the relative values of ${x}_1$ and ${x}_2$ from just this data. Indeed, as we show below, standard estimators such as the sign test — ranking the items in terms of their [reviewer]{}-provided scores — provably fail to achieve this goal. More generally, the following theorem holds for the class of all deterministic estimators, that is, estimators given by deterministic mappings from $\{{A},{y}_1,{y}_2\}$ to the set $\{1 {\succ}2, 2 {\succ}1\}$.
\[thm:canonical\_det\_fails\] No deterministic (cardinal or ordinal) estimator can strictly uniformly dominate the random-guessing estimator ${{\widehat}{{\pi}}_{{\text{can}}}}$.
This theorem demonstrates the difficulty of this problem by ruling out all deterministic estimators. Our original question then still remains: is there any estimator that can strictly uniformly outperform the random-guessing ordinal baseline?
We show that the answer is yes, with the construction of a randomized estimator for this canonical setting, denoted as ${{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}$. This estimator is based on a function ${w}:{[0,\infty)}\rightarrow [0,1)$ which may be chosen as any arbitrary strictly-increasing function. For instance, one could choose ${w}({x}) = \frac{{x}}{1+{x}}$ or ${w}$ as the sigmoid function. Given the scores ${y}_1,{y}_2$ reported for the two items, let ${\widehat{{i}}^{(1)}}\in \operatorname*{argmax}_{{i}\in \{1,2\}} {y}_{i}$ denote the item which receives the higher score, and let ${\widehat{{i}}^{(2)}}$ denote the remaining item (with ties broken uniformly). Then our randomized estimator outputs: $$\begin{aligned}
\label{eq:canonical_estimator}
{{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}({A},{y}_1,{y}_2) = & \begin{cases}{\widehat{{i}}^{(1)}}{\succ}{\widehat{{i}}^{(2)}}& \text{with probability } \frac{1 + {w}(\abs{{y}_1-{y}_2})}{2}\\
{\widehat{{i}}^{(2)}}{\succ}{\widehat{{i}}^{(1)}}& \text{otherwise}.
\end{cases}\end{aligned}$$ Note that the the output of this estimator is independent of the assignment ${A}$, so in the remainder of this paper we also denote this estimator as ${{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}({y}_1, {y}_2)$.
The following theorem now proves that our proposed estimator indeed achieves the stated goal.
\[thm:canonical\_ours\] The randomized estimator ${{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}$ strictly uniformly dominates the random-guessing baseline ${{\widehat}{{\pi}}_{{\text{can}}}}$.
While this result considers a setting with “noiseless” observations (that is, where ${y}= {f}({x})$), in Appendix \[app:noisy\] we show that the guarantee for ${{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}$ continues to hold when the observations are noisy.
Having established the positive result for this canonical setting, we now discuss some connections and inspirations in the literature.
### Connections to the literature {#sec:two_envelope}
The canonical setting has a close connection to the randomized version of the two-envelope problem [@cover1987envelope]. In the two-envelope problem, there are two arbitrary numbers. One of the two numbers is observed uniformly at random, and the other remains unknown. The goal is to estimate which number is larger. This problem can also be viewed from a game-theoretic perspective [@gnedin2016survey] as ours, where one player picks an estimator and the other player picks the two values. Cover [@cover1987envelope] proposed a randomized estimator whose probability of success is strictly larger than $0.5$ uniformly across all arbitrary pairs of numbers. The proposed estimator samples a new random variable ${Z}$ whose distribution has a probability density function ${p}$ with ${p}({z}) > 0$ for all ${z}\in {\mathbb{R}}$. Then if the observed number is smaller than ${Z}$, the estimator decides that the observed number is the smaller number; if the observed number is larger than ${Z}$, the estimator decides that the observed number is the larger number.
Our canonical setting can be reduced to the two-envelope problem as follows. Consider the two values ${f}_1({x}_1) - {f}_2({x}_2)$ and ${f}_1({x}_2) - {f}_2({x}_1)$. Since the two items are assigned to the two [[[reviewer]{}]{}s]{}uniformly at random, we observe one of these two values uniformly at random. By the assumption that ${f}_1$ and ${f}_2$ are monotonically increasing, we know that these two values are distinct, and furthermore, ${f}_1({x}_1) - {f}_2({x}_2) > {f}_1({x}_2) - {f}_2({x}_1)$ if and only if ${x}_1 > {x}_2$. Hence, the relative ordering of these two values is identical to the relative ordering of ${x}_1$ and ${x}_2$, reducing our canonical setting to the two-envelope problem. Our estimator ${{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}$ also carries a close connection to Cover’s estimator to the two-envelope problem. Specifically, Cover’s estimator can be equivalently viewed as being designated by a “switching function” [@mcdonnell2009switch]. This switching function specifies the probability to “switch” (that is, to guess that the unobserved value is larger), and is a monotonically-decreasing function in the observed value. The use of the monotonic function ${w}$ in our estimator in is similar in spirit.
The two-envelope problem can also be alternatively viewed as a secretary problem with two candidates. Negative results have been shown regarding the effect of cardinal vs. ordinal data when there are more than two candidates [@silverman1992googol; @gnedin1994googol], and positive result has been shown on extensions of the secretary problem to different losses [@gnedin1996exchangeable].
Our original inspiration for our proposed estimator arose from Stein’s phenomenon [@stein1956inadmissibility] and empirical Bayes [@robbins1956empirical]. This inspiration stems for the fact that the two items are not to be estimated in isolation, but in a joint manner. That said, a significant fraction of the work (e.g., [@robbins1956empirical; @stein1956inadmissibility; @james1961estimation; @baranchik1970minimax; @bock1975minimax; @tian2017population]) in these areas is based on deterministic estimators. In comparison, our negative result for all deterministic estimators (Theorem \[thm:canonical\_det\_fails\]) and the positive result for our randomized estimator (Theorem \[thm:canonical\_ours\]) provide interesting insights in this space.
A/B testing {#sec:abtest}
-----------
We now demonstrate how to use the result in the canonical setting as a plug-in for more general scenarios. Specifically, we construct simple extensions to our canonical estimator, as a proof-of-concept for the superiority of cardinal data over ordinal data in A/B testing (this section) and ranking (Section \[sec:ranking\]). A/B testing is concerned with the problem of choosing the better of two given items, based on multiple evaluations of each item, and is used widely for the web and e-commerce (e.g. [@kohavi2009guide]). In many applications of A/B testing, the two items are rated by disjoint sets of individuals (for example, when comparing two web designs, each user sees one and only one design). It is therefore important to take into account the different calibrations of different individuals, and this problem fits in our setting with ${n}= 2$ items and ${m}$ [[[reviewer]{}]{}s]{}. For simplicity, we assume that ${m}$ is even. We consider the assignment obtained by assigning item $1$ to some ${{m}/2}$ [[[reviewer]{}]{}s]{}chosen uniformly at random (without replacement) from the set of ${m}$ [[[reviewer]{}]{}s]{}, and assigning item $2$ to the remaining ${{m}/2}$ [[[reviewer]{}]{}s]{}.[^2]
As in the canonical setting we studied earlier, in the absence of any direct comparison between the two items, a natural ordinal estimator in the A/B testing setting is a random guess: $$\begin{aligned}
{{\widehat}{{\pi}}_{{\text{ab}}}}({A}, \{\}) = \begin{cases}
1 {\succ}2 & \text{with probability } 0.5\\
2 {\succ}1 & \text{with probability } 0.5.
\end{cases}\end{aligned}$$
For concreteness, we consider the following method of performing the random assignment of the two items to the ${m}$ [[[reviewer]{}]{}s]{}. We first perform a uniformly random permutation of the ${m}$ [[[reviewer]{}]{}s]{}, and then assign the first ${{m}/2}$ [[[reviewer]{}]{}s]{}in this permutation to item $1$; we assign the last ${{m}/2}$ [[[reviewer]{}]{}s]{}in this permutation to item $2$. We let ${y}_{1}^{(1)}, \ldots, {y}_{1}^{({{m}/2})}$ denote the scores given by the ${{m}/2}$ [[[reviewer]{}]{}s]{}to item $1$, and let ${y}_{2}^{(1)}, \ldots, {y}_{2}^{({{m}/2})}$ denote the scores given by the ${{m}/2}$ [[[reviewer]{}]{}s]{}assigned to item $2$. Namely, the [[[reviewer]{}]{}s]{}(in the permuted order) provide the scores $[{y}_1^{(1)}, \ldots, {y}_1^{({{m}/2})}, {y}_2^{(1)}, \ldots, {y}_2^{({{m}/2})}]$. Now consider the following standard (deterministic) estimators:
*Sign estimator:* The sign estimator outputs the item which has more pairwise wins:\
$\sum_{{j}=1}^{{{m}/2}} {{\mathbbm{1}}{\{{y}_1^{({j})} > {y}_2^{({j})}\}}} {\mathrel{\mathop\gtrless\limits^{1 \succ 2}_{2 \succ 1}}} \sum_{{j}=1}^{{{m}/2}} {{\mathbbm{1}}{\{{y}_2^{({j})} > {y}_1^{({j})}\}}}$.
*Mean estimator:* The mean estimator outputs the item with the higher mean [score]{}:\
$\operatorname{mean}({y}_1^{(1)}, \ldots, {y}_1^{({{m}/2})}) {\mathrel{\mathop\gtrless\limits^{1 \succ 2}_{2 \succ 1}}} \operatorname{mean}({y}_2^{(1)}, \ldots, {y}_2^{({{m}/2})})$.
*Median estimator:* The median estimator outputs the item with the higher median [score]{}(upper median if there are multiple medians)[^3]: $\operatorname{median}({y}_1^{(1)}, \ldots, {y}_1^{({{m}/2})}) {\mathrel{\mathop\gtrless\limits^{1 \succ 2}_{2 \succ 1}}} \operatorname{median}({y}_2^{(1)}, \ldots, {y}_2^{({{m}/2})})$.
In each estimator, ties are assumed to be broken uniformly at random.
We now show that despite using the [[[score]{}]{}s]{}given by all ${m}$ [[[reviewer]{}]{}s]{}, where ${m}$ can be arbitrarily large, these natural estimators fail to uniformly dominate the naïve random-guessing ordinal estimator.
\[thm:abtest\_det\_examples\_fails\] For any (even) number of [[[reviewer]{}]{}s]{}, none of the sign, mean, and median estimators can strictly uniformly dominate the random-guessing estimator ${{\widehat}{{\pi}}_{{\text{ab}}}}$.
The negative result of Theorem \[thm:abtest\_det\_examples\_fails\] demonstrates the challenges even when one is allowed to collect an arbitrarily large number of [[[score]{}]{}s]{}for each item. Intuitively, the more [[[reviewer]{}]{}s]{}there are, the more miscalibration functions they introduce. Even if the statistics used by these estimators converge as the number of the [[[reviewer]{}]{}s]{}${m}$ grows large, these values are not guaranteed to be informative towards comparing the values of the items due to the miscalibrations.
The failure of these standard estimators suggests the need of a novel approach towards this problem of A/B testing under arbitrary miscalibrations. To this end, we build on top of our canonical estimator ${{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}$ from Section \[sec:canonical\], and present a simple randomized estimator ${{\widetilde}{{\pi}}^{{\text{our}}}_{{\text{ab}}}}$ as follows:
For every ${j}\in [{{m}/2}]$, use the canonical estimator ${{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}$ on the ${j}^{th}$ pair of [[[score]{}]{}s]{}$({y}_1^{({j})}, {y}_2^{({j})})$ and obtain the estimate ${r}_{j}{:=}{{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}({y}_1^{({j})}, {y}_2^{({j})}) \in \{1 {\succ}2, 2 {\succ}1\}$.
Set the output ${{\widetilde}{{\pi}}^{{\text{our}}}_{{\text{ab}}}}$ as the outcome of the majority vote among the estimates $\{{r}_{j}\}_{{j}\in [{{m}/2}]}$ with ties broken uniformly at random.
The following theorem now shows that the results for the canonical setting from Section \[sec:canonical\] translate to this A/B testing application.
\[thm:abtest\_ours\] For any (even) number of [[[reviewer]{}]{}s]{}, the estimator ${{\widetilde}{{\pi}}^{{\text{our}}}_{{\text{ab}}}}$ strictly uniformly dominates the random guessing estimator ${{\widehat}{{\pi}}_{{\text{ab}}}}$.
This result thus illustrates the use of our canonical estimator ${{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}$ as a plug-in for A/B testing. So far we have considered settings where there are only two items and where each [reviewer]{}is assigned only one item, thereby making the ordinal information vacuous. We now turn to an application that is free of these restrictions.
Ranking {#sec:ranking}
-------
It is common in practice to estimate the partial or total ranking for a list of items by soliciting ordinal or cardinal responses from individuals. In conference reviews or peer-grading, each reviewer is asked to rank [@douceur2009paper; @shah2013case; @shah2017design] or rate [@ge13bias; @piech2013tuned; @shah2017design] a small subset of the papers, and this information is subsequently used to estimate a partial or total ranking of the papers (or student homework). Other applications for aggregating rankings include voting [@young1988condorcet; @procaccia2016vote], crowdsourcing [@shah2016estimation; @shah18simple], recommendation systems [@freund2003boosting] and meta-search [@dwork2001rank].
Formally, we let ${n}> 2$ denote the number of items and ${m}$ denote the number of [[[reviewer]{}]{}s]{}. For simplicity, we focus on a setting where each [reviewer]{}reports noiseless evaluations of some pair of items, and the goal is to estimate the total ranking of all items. We consider a random design setup where the pairs compared are randomly chosen and randomly assigned to [[[reviewer]{}]{}s]{}. We assume $1<{m}< \binom{{n}}{2}$ so that the problem does not degenerate. Each [reviewer]{}[[[evaluate]{}]{}s]{}a pair of items, and these pairs are drawn uniformly without replacement from the ${{n}\choose 2}$ possible pairs of items. We let ${A}= ({S}_1,\ldots,{S}_{m})$ denote these ${m}$ pairs of items to be [evaluated]{}by the ${m}$ respective [[[reviewer]{}]{}s]{}, where ${S}_{j}\in [{n}]\times[{n}]$ denotes the pair of items [evaluated]{}by [reviewer]{}${j}\in [{m}]$. For each pair ${S}_{j}= ({i}, {i}')$, denote the cardinal evaluation as ${y}({S}_{j}) = ({f}_{{j}}({x}_{i}), {f}_{{j}}({x}_{{i}'}))$, and the ordinal evaluation as the induced ranking ${b}({S}_{j}) \in \{{i}{\succ}{i}', {i}' {\succ}{i}\}$. Denote the set of ordinal observations as ${\mathcal{B}}= \{{b}({S}_{j})\}_{{j}=1}^{m}$, and the set of cardinal observations as ${\mathcal{Y}}= \{{y}({S}_{j})\}_{{j}=1}^{m}$. The input to an ordinal estimator is the ordinal information ${\mathcal{B}}$. The input to a cardinal estimator is the [reviewer]{}assignment ${A}$ and the set of cardinal observations ${\mathcal{Y}}$. Finally, let ${\mathcal{G}}( {\mathcal{B}})$ denote a directed acyclic graph (DAG) with nodes comprising the ${n}$ items and with an edge from any node ${i}$ to any other node ${i}'$ if and only if $\{{i}{\succ}{i}'\} \in {\mathcal{B}}$. A topological ordering on ${\mathcal{G}}$ is any total ranking of its vertices which does not violate any pairwise comparisons indicated by ${\mathcal{B}}$.
Deduce the ordinal observations ${\mathcal{B}}$ from the cardinal observations ${\mathcal{Y}}$. Compute a topological ordering ${\widehat{{\pi}}}$ on the graph ${\mathcal{G}}({\mathcal{B}})$, with ties broken in order of the indices of the items. ${t}\leftarrow 1$. Output ${{{\widetilde}{{\pi}}}_{{\text{rank}}}^{{\text{our}}}}({A}, {\mathcal{Y}}) = {\widehat{{\pi}}}$.
We now present our (randomized) cardinal estimator ${{{\widetilde}{{\pi}}}_{{\text{rank}}}^{{\text{our}}}}({A}, {\mathcal{Y}})$ in Algorithm \[alg:sort\_cardinal\]. In words, this algorithm start from any topological ordering of the items as the initial estimate of the true ranking. Then the algorithm scans one-by-one over the pairs with adjacent items in the initial estimated ranking. If a pair can be flipped (that is, if the ranking after flipping this pair is also a topological ordering), we uniformly sample a pair of [[[score]{}]{}s]{}for these two items from the cardinal observations ${\mathcal{Y}}$, and use the randomized estimator ${{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}$ to determine the relative order of the pair. After ${{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}$ is called, the positions of this pair are finalized. We remove all [[[score]{}]{}s]{}of these two [[[reviewer]{}]{}s]{}from future use, and jump to the next pair that does not contain these two items.
The following theorem now presents the main result of this section.
\[thm:sort\_ours\_uniform\_prior\] Suppose that the true ranking ${{\pi}^*}$ is drawn uniformly at random from the collection of all possible rankings, and consider any ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}}}$ for ${{\pi}^*}$. Then the cardinal estimator ${{{\widetilde}{{\pi}}}_{{\text{rank}}}^{{\text{our}}}}$ strictly uniformly dominates the ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}}}$.
We note that Algorithm \[alg:sort\_cardinal\] runs in polynomial time (in the number of items ${n}$) because the two major operations of this estimator – finding a topological ordering, and checking if a ranking is a topological ordering on the DAG – can be implemented in polynomial time [@dasgupta2008algorithms]. Theorem \[thm:sort\_ours\_uniform\_prior\] thus demonstrates again the power of the canonical estimator ${{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}$ as a plug-in component to be used in a variety of applications. An extension of our results to the setting where ${{\pi}^*}$ can be arbitrary (adversarially chosen) is presented in Appendix \[app:ranking\_arbit\].
Simulations
===========
We now experimentally evaluate our proposed estimators for A/B testing and ranking. Since the performance of the ordinal estimators vary significantly in different problem instances, we use the notion of “relative improvement”. The relative improvement ${\rho_{{\widehat}{{\pi}}}({\widetilde}{{\pi}})}$ of an estimator ${\widetilde}{{\pi}}$ as compared to a baseline estimator ${\widehat}{{\pi}}$ is defined as: $
{\rho_{{\widehat}{{\pi}}}({\widetilde}{{\pi}})} = \frac{{\mathbb{E}}[{L}({{\pi}^*}, {\widehat}{{\pi}})] - {\mathbb{E}}[{L}({{\pi}^*}, {\widetilde}{{\pi}})]}{{\mathbb{E}}[{L}({{\pi}^*}, {\widehat}{{\pi}})]} \times 100\%.$ A positive value of the relative improvement ${\rho_{{\widehat}{{\pi}}}({\widetilde}{{\pi}})}$ indicates the superiority of estimator ${\widetilde}{{\pi}}$ over the estimator ${\widehat}{{\pi}}$. A relative improvement of zero indicates an identical performance of the two estimators. In our proposed estimators, the function ${w}$ is set as ${w}({x}) = \frac{{x}}{1 + {x}}$.
A/B testing {#sec:exp_abtest}
-----------
We now present simulations to evaluate various points on the bias-variance tradeoff. For A/B testing, we compare our estimator ${{\widetilde}{{\pi}}^{{\text{our}}}_{{\text{ab}}}}$ with other standard estimators — the sign, mean and median estimators introduced in Section \[sec:abtest\]. The item values ${x}_1$ and ${x}_2$ are chosen independently and uniformly at random from the interval $[0,1]$. The calibration functions are linear and given by:
*One biased [reviewer]{}:* One [reviewer]{}gives an abnormally (high or low) [score]{}. Formally, ${f}_{j}({x}) = {x}$ for $ {j}\in [{m}-1]$, and ${f}_{m}({x}) = {x}+ {m}$.\[item:one\_biased\_grader\]
*Incremental biases:* Calibration functions of reviewers are shifted from each other. Formally, ${f}_{j}({x}) = {x}+ {j}$ for ${j}\in [{m}]$.\[item:incremental biases\]
*Incremental biases with one biased [reviewer]{}:* A combination of setting \[item:one\_biased\_grader\] and setting \[item:incremental biases\]. Formally, ${f}_{j}({x}) = {x}+ ({j}-1)$ for ${j}\in [{m}- 1]$, and ${f}_{m}({x}) = {x}+ \frac{{m}({m}-1)}{2}$.
We simulate and compute the relative improvement of the different estimators as compared to the random-guessing estimator ${{\widehat}{{\pi}}_{{\text{ab}}}}$. The results are shown in Figure \[fig:abtest\]. While the performance of the estimators vary with respect to each other, our estimator consistently beats the baseline whereas every other estimator fails. Our estimator thus indeed operates at a unique point on the bias-variance tradeoff with a low (zero) bias and a variance strictly smaller than the ordinal estimators, whereas all other estimators incur a non-zero error due to bias.
Ranking {#sec:exp_sort}
-------
![\[fig:sort\] Relative improvement in Kendall-tau distance of our ranking estimator ${{{\widetilde}{{\pi}}}_{{\text{rank}}}^{{\text{our}}}}$ as compared to an optimal ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}}}$ for ranking. Each point is an average over $100$ trials, where in each trial the quantities ${\mathbb{E}}[{L}({{\pi}^*}, {{{\widetilde}{{\pi}}}_{{\text{rank}}}^{{\text{our}}}})]$ and ${\mathbb{E}}[{L}({{\pi}^*}, {{\widehat}{{\pi}}_{{\text{rank}}}})]$ are approximated by an empirical average over $1000$ samples. ](sort_kt_vary_n.pdf){width="0.8\linewidth"}
Next, we evaluate the performance of our ranking estimator ${{{\widetilde}{{\pi}}}_{{\text{rank}}}^{{\text{our}}}}$ when the true ranking ${{\pi}^*}$ is drawn from a uniform prior. We compare this estimator with an optimal ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}}}$ which outputs a topological ordering with ties broken in order of the indices of the items (this ordinal estimator is optimal regardless of the tie-breaking strategy).
For any number of items ${n}$, we generate the values ${x}_1,\ldots,{x}_{n}$ of the items i.i.d. uniformly from the interval $[0, {n}]$. We set ${m}= \floor{\frac{1}{2}{{n}\choose 2}}$. We assume that the ${j}^{th}$ [reviewer]{}has a linear calibration function ${f}_{j}({x}) = {k}_{j}{x}+ {b}_{j}$, where we sample ${k}_{j}$ and ${b}_{j}$ i.i.d. uniformly from the interval $[0, 1]$.
We have previously proved that our estimator ${{{\widetilde}{{\pi}}}_{{\text{rank}}}^{{\text{our}}}}$ based on cardinal data can strictly uniformly outperform the optimal ordinal estimator for the 0-1 loss. We use these simulations to evaluate the efficacy of our approach for a different loss function – Kendall-tau distance. Specifically, Figure \[fig:sort\] compares these two estimators in terms of Kendall-tau distance (Appendix \[app:other\_metrics\] provides a formal definition of this distance and associated theoretical results). We observe that our estimator ${{{\widetilde}{{\pi}}}_{{\text{rank}}}^{{\text{our}}}}$ is able to consistently yield improvements even for this loss. The reason that the improvement becomes smaller when the number of items is large is that by flipping pairs, our estimator only modifies the ranking in the neighborhood of the initial estimate. We strongly believe that it should be possible to design better estimators for the large ${n}$ regime using the tools developed in this paper. Having met our stated goal of outperforming ordinal estimators to handle arbitrary miscalibrations, we leave this interesting problem for future work.
Tradeoff between estimation under perfect calibration vs. miscalibration
------------------------------------------------------------------------
In this section, we present a preliminary experiment showing the tradeoff between estimation under perfect calibration (all reviewers reporting the true values of the papers) and estimation under miscalibration. For simplicity, we consider the canonical setting from Section \[sec:canonical\]. We evaluate the performance of our estimator under two scenarios: (1) perfect calibration, where ${f}_{j}({x}) = {x}$ for each ${j}\in \{1, 2\}$; (2) miscalibration with one biased reviewer, where ${f}_1({x}) = {x}$ and ${f}_2({x}) = {x}+ 1$. We consider the function ${w}$ in our estimator as ${w}({x}) = \frac{{\gamma}{x}}{1 + {\gamma}{x}}$, where ${\gamma}\in \{2^k {\mid}-10 \le k \le 10, k \in {\mathbb{Z}}\}$. We sample ${x}_1$ and ${x}_2$ uniformly at random from the interval $[0, 1]$.
![\[fig:canonical\] Relative improvement of our canonical estimator ${{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}$ under perfect calibration and under miscalibration of one biased reviewer, with ${w}({x}) = \frac{{\gamma}{x}}{1 + {\gamma}{x}}$ and ${\gamma}\in \{2^k {\mid}-10 \le k \le 10, k \in {\mathbb{Z}}\}$, where ${\gamma}$ increases from left to right in the plot. Each point is an average over $5\times 10^5$ trials. The error bars are too small to display.](canonical_relative_curve_labeled.pdf){width="0.4\linewidth"}
Figure \[fig:canonical\] shows the relative improvement of our estimator over the random-guessing baseline under perfect calibration and under miscalibration, where ${\gamma}$ increases from left to right. Let us focus on a few regimes in this plot. First, on the left end of the curve, when ${\gamma}$ is close to $0$, we have ${w}({x})$ close to $0$. The estimator is close to random-guessing. At the other extreme, on the right end of the curve, when ${\gamma}$ goes to infinity, we have ${w}({x})$ close to $1$. The estimator always outputs the item with the higher score, and hence gives perfect estimation under perfect calibration. Under miscalibration, this estimator always chooses the biased reviewer giving the higher score and hence performs the same as random guess. Past the maximum point of the function at approximately $(25\%, 9\%)$ when ${\gamma}= 1$, the value of the curve starts decreasing, suggesting a tradeoff of estimation accuracy under perfect calibration and under miscalibration. It is clear that points to the left of the maximum point are not Pareto-efficient, since there exist other points with the same accuracy under miscalibration but improved accuracy under perfect calibration.
We thus see that robustness under arbitrary miscalibration comes at a cost of lower accuracy under perfect calibration. Establishing a formal understanding of this tradeoff and designing estimators that are provably Pareto-efficient are important open problems.
Proofs {#app:proofs}
======
In this section, we present the proofs of our theoretical results.
For notational simplicity, we use “$1 {\prec}2$” to denote that item $1$ has a smaller value than item $2$. Since the items have distinct values, we have $1{\prec}2$ if and only if $2 {\succ}1$. For the 0-1 loss ${L}({{\pi}^*}, {\widehat{{\pi}}}) = {{\mathbbm{1}}{\{{\widehat{{\pi}}}\ne {{\pi}^*}\}}}$, we call the expected loss ${\mathbb{E}}[{L}({{\pi}^*}, {\widehat{{\pi}}})] = {\mathbb{P}}({\widehat{{\pi}}}\ne {{\pi}^*})$ as the “probability of error” of any estimator ${\widehat{{\pi}}}$, and ${\mathbb{P}}({\widehat{{\pi}}}= {{\pi}^*})$ as the “probability of success”. For the canonical setting and A/B testing, the probability of success of random guessing is $0.5$. To show that some estimator ${\widehat{{\pi}}}$ strictly uniformly dominates random guessing for the canonical setting or A/B testing, we only need to show that the probability of success of this estimator is strictly higher than $0.5$, or equivalently, the probability of error of of this estimator is strictly lower than $0.5$.
Proof of Theorem \[thm:canonical\_det\_fails\]
----------------------------------------------
We prove that no deterministic cardinal estimator can strictly uniformly dominate the random-guessing estimator ${{\widehat}{{\pi}}_{{\text{can}}}}$, which implies the negative result for any deterministic ordinal estimator.
Recall the notation ${\widehat{{i}}^{(1)}}= \operatorname*{argmax}_{{i}\in \{1, 2\}} {y}_{i}$ as the item receiving the higher [score]{}(with ties broken uniformly at random), and the notation ${\widehat{{i}}^{(2)}}$ as the remaining item. First, we consider a deterministic estimator that always outputs ${\widehat{{i}}^{(1)}}$ as the item whose value is greater. We call this estimator the “sign estimator”, denoted ${{\widehat{{\pi}}}_\text{sign}}$: $$\begin{aligned}
{{\widehat{{\pi}}}_\text{sign}}({A}, {y}_1, {y}_2) = ({\widehat{{i}}^{(1)}}{\succ}{\widehat{{i}}^{(2)}}).\end{aligned}$$
The proof consists of two steps. (1) We show that the sign estimator does not strictly uniformly dominate random guess. (2) Building on top of (1), we show that more generally, no deterministic estimator strictly uniformly dominates random guess.
**Step 1:** The sign estimator does not strictly uniformly dominate random guess.
We construct the following counterexample such that the probability of error of the sign estimator is $0.5$. We construct [reviewer]{}calibration functions such that their ranges are disjoint, that is, one [reviewer]{}always gives a higher [score]{}than the other [reviewer]{}, regardless of the items they are assigned. Then the relative ordering of the two [[[score]{}]{}s]{}does not convey any information about the relative ordering of the two items, and we show that in this case, the sign estimator has a probability of error of $0.5$. Concretely, let the item values be bounded as ${x}_1, {x}_2\in (0, 1)$, and let the calibration functions be ${f}_1({x}) = {x}$ and ${f}_2({x}) = {x}+ 1$. Then the [score]{}given by [reviewer]{}$2$ is higher than the [score]{}given by [reviewer]{}$1$ regardless of the item values they are assigned. The sign estimator always observes ${y}_1 < {y}_2$, and outputs the item assigned to [reviewer]{}$2$ as the larger item. The assignment is either ${A}= ({S}_1 = 1, {S}_2 = 2)$ or $({S}_1 = 2, {S}_2 = 1)$ with probability $0.5$ each. Under assignment $({S}_1 = 1, {S}_2 = 2)$, the sign estimator outputs $1 {\prec}2$. Under assignment $({S}_1 = 2, {S}_2 = 1)$, the sign estimator outputs $1{\succ}2$. Under one (and exactly one) of the two assignments, the output of the sign estimator is correct. Hence, the probability of error of the sign estimator is $0.5$.
**Step 2:** No deterministic estimator strictly uniformly dominates random guess.
Let ${\mathcal{A}}$ be the set of the two assignments, ${\mathcal{A}}= \{({S}_1 = 1, {S}_2 = 2), ({S}_1 = 2, {S}_2 = 1)\}$. A deterministic estimator ${{\widehat{{\pi}}}_\text{det}}: {\mathcal{A}}\times {\mathbb{R}}\times {\mathbb{R}}\rightarrow \{1 {\succ}2, 1{\prec}2\}$ is a deterministic function that takes as input the assignment and the [[[score]{}]{}s]{}for the two items, and outputs the relative ordering between the two items. Step 1 has shown that the sign estimator does not strictly uniformly dominate random guess. Hence, we only need to prove that any deterministic estimator ${{\widehat{{\pi}}}_\text{det}}$ that is different from the sign estimator does not strictly uniformly dominate random guess. For this deterministic estimator ${{\widehat{{\pi}}}_\text{det}}$, there exist some input values $({a}, \widetilde{{y}}_1, \widetilde{{y}}_2)$ such that the output of this deterministic estimator differs from the sign estimator. If the two estimators ${{\widehat{{\pi}}}_\text{sign}}$ and ${{\widehat{{\pi}}}_\text{det}}$ only differ at points where $\widetilde{{y}}_1 = \widetilde{{y}}_2$, then we can use the same counterexample in Step 1 to show that the probability of error of this deterministic estimator is $0.5$. It remains to consider the case when $\widetilde{{y}}_1 \ne \widetilde{{y}}_2$. Without loss of generality, assume $\widetilde{{y}}_1 > \widetilde{{y}}_2$. Then consider the following counterexample. Let ${x}_1 > {x}_2$. Let ${f}_1, {f}_2$ be strictly-increasing functions such that ${f}_1({x}_1) = {f}_2({x}_1) = \widetilde{{y}}_1, {f}_1({x}_2) = {f}_2({x}_2) = \widetilde{{y}}_2$. Regardless of the [reviewer]{}assignment, the [score]{}${y}_1$ for item $1$ is $\widetilde{{y}}_1$, and the [score]{}${y}_2$ for item $2$ is $\widetilde{{y}}_2$. The item receiving a higher [score]{}is always ${\widehat{{i}}^{(1)}}= \operatorname*{argmax}_{{i}\in\{1, 2\}}{y}_{i}= 1$, so the sign estimator ${{\widehat{{\pi}}}_\text{sign}}$ always outputs $1{\succ}2$. Under assignment ${a}$, the deterministic estimator differs from the sign estimator, so the deterministic estimator gives the incorrect output $(1 {\prec}2)$. The assignment ${a}$ happens with probability $0.5$, so the probability of error of this deterministic estimator is at least $0.5$.
The two steps above complete the proof that there exists no deterministic estimator that strictly uniformly dominates random guess.
Proof of Theorem \[thm:canonical\_ours\]
----------------------------------------
In what follows, we prove that the probability of success of our estimator is strictly greater than $0.5$ under arbitrary item values ${x}_1, {x}_2$ and arbitrary calibration functions ${f}_1, {f}_2$. We start with re-writing our estimator in into an alternative and equivalent expression, and then prove the result on this new expression of our estimator.
Recall that ${\widehat{{i}}^{(1)}}= \operatorname*{argmax}_{{i}\in \{1, 2\}} {y}_{i}$ denotes the item receiving the higher [score]{}, and ${\widehat{{i}}^{(2)}}$ denotes the remaining item (with ties broken uniformly). Depending on the relative ordering of ${y}_1$ and ${y}_2$, we can split into the following three cases:
\[eq:canonical\_rewrite\_three\_cases\] $$\begin{aligned}
{{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}({A}, {y}_1, {y}_2 {\mid}{y}_1 > {y}_2) & =
\begin{cases}
1 {\succ}2 & \text{with probability $\frac{1 + {w}({y}_1 - {y}_2)}{2}$}\\
2 {\succ}1 &\text{otherwise,}
\end{cases}\\
{{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}({A}, {y}_1, {y}_2 {\mid}{y}_1 < {y}_2) & =
\begin{cases}
1 {\succ}2 & \text{with probability $\frac{1 - {w}({y}_2 - {y}_1)}{2}$}\\
2 {\succ}1 &\text{otherwise,}
\end{cases}\\
{{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}({A}, {y}_1, {y}_2 {\mid}{y}_1 = {y}_2) & =
\begin{cases}
1 {\succ}2 & \text{with probability $\frac{1}{2}$}\\
2 {\succ}1 &\text{otherwise.}
\end{cases}\end{aligned}$$
Recall that the function ${w}$ is from ${[0,\infty)}$ to $[0,1)$. Now we define the following auxiliary function ${\widetilde{{w}}}: {\mathbb{R}}\rightarrow (0, 1)$: $$\begin{aligned}
\label{eq:envelope_monotonic_function}
{\widetilde{{w}}}({x}) = \begin{cases}
\frac{1 + {w}({x})}{2} & \text{if } {x}> 0\\
\frac{1}{2} & \text{if }{x}= 0\\
\frac{1 - {w}(-{x})}{2} & \text{otherwise.}
\end{cases}\end{aligned}$$
Combining and , we have $$\begin{aligned}
\label{eq:canonical_rewrite}
{{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}({A}, {y}_1, {y}_2) =
\begin{cases}
1 {\succ}2 & \text{with probability } {\widetilde{{w}}}({y}_1 - {y}_2)\\
2 {\succ}1 & \text{otherwise.}
\end{cases}\end{aligned}$$
Without loss of generality, assume ${x}_1 > {x}_2$. The assignment is either ${a}{:=}({S}_1 = 1, {S}_2 = 2)$ or ${a}' {:=}({S}_1 = 2, {S}_2 = 1)$ with probability $0.5$ each. Thus, the estimator observes $\{{y}_1 = {f}_1({x}_1), {y}_2 = {f}_2({x}_2)\}$ under assignment ${a}$, or $\{{y}_1 = {f}_2({x}_1), {y}_2 = {f}_1({x}_2)\}$ under assignment ${a}'$. The probability of success of our estimator ${{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}$ is:
$$\begin{aligned}
{\mathbb{P}}({{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}= {{\pi}^*}) = & \sum_{\widetilde{{a}}\in \{{a}, {a}'\}} {\mathbb{P}}({{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}={{\pi}^*}{\mid}{A}= \widetilde{{a}}) {\mathbb{P}}({A}= \widetilde{{a}})\nonumber\\
\stackrel{{\text{(i)}\xspace}}{=} & \frac{1}{2}{\widetilde{{w}}}({f}_1({x}_1) - {f}_2({x}_2)) + \frac{1}{2} {\widetilde{{w}}}({f}_2({x}_1) - {f}_1({x}_2)) \nonumber\\
= & \frac{1}{2}\left[{\widetilde{{w}}}({f}_1({x}_1) - {f}_2({x}_2)) + {\widetilde{{w}}}({f}_2({x}_1) - {f}_1({x}_2))\right]\nonumber\\
\stackrel{{\text{(ii)}\xspace}}{=} & \frac{1}{2}\left[1 + {\widetilde{{w}}}({f}_1({x}_1) - {f}_2({x}_2)) - {\widetilde{{w}}}( {f}_1({x}_2) - {f}_2({x}_1))\right],\label{eq:canonical_probability_success_intermediate}\end{aligned}$$
where step is true by plugging in , and step is true because ${\widetilde{{w}}}({x}) + {\widetilde{{w}}}(-{x}) = 1$ by the definition of the function ${\widetilde{{w}}}$ in .
By the monotonicity of the functions ${f}_1$ and ${f}_2$, and by the assumption that ${x}_1 > {x}_2$, we have ${f}_1({x}_1) + {f}_2({x}_1) > {f}_1({x}_2) + {f}_2({x}_2)$, and therefore ${f}_1({x}_1) - {f}_2({x}_2) > {f}_1({x}_2) - {f}_2({x}_1)$. Since ${w}(0) \ge 0$ and ${w}$ is monotonically increasing on ${[0,\infty)}$, it is straightforward to verify that ${\widetilde{{w}}}$ is monotonically increasing on ${\mathbb{R}}$. Hence, we have $$\begin{aligned}
\label{eq:canonical_func_redefine_relation}
{\widetilde{{w}}}({f}_1({x}_1) - {f}_2({x}_2)) > {\widetilde{{w}}}({f}_1({x}_2) - {f}_2({x}_1)).\end{aligned}$$
Combining and , we have $$\begin{aligned}
{\mathbb{P}}({{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}= {{\pi}^*}) > \frac{1}{2}.\end{aligned}$$
Proof of Theorem \[thm:abtest\_det\_examples\_fails\]
-----------------------------------------------------
We construct a counterexample on which the mean, median and sign estimators have a probability of error of $0.5$. In this counterexample, let the item values be bounded as ${x}_1, {x}_2\in (0, 1)$, and let the ${m}$ [reviewer]{}calibration functions be as follows:
$$\begin{aligned}
{f}_{j}({x}) =
\begin{dcases}
{x}+ ({j}-1) & \text{if } 1\le {j}\le {m}-1\\
{x}+ \frac{{m}({m}-1)}{2} & \text{if } {j}={m}.
\end{dcases}\label{eq:counterexample_calibration}\end{aligned}$$
In these calibration functions, the [score]{}provided by each [reviewer]{}is the sum of the true value of the item assigned to this [reviewer]{}, and a bias term specific to this [reviewer]{}. The analysis is performed separately for the three estimators. At a high level, the analysis for the mean estimator uses the fact that one [reviewer]{}(specifically, [reviewer]{}${m})$ has a significantly greater bias than the rest of the [[[reviewer]{}]{}s]{}. The analysis for the median and the sign estimators uses the fact that the ranges of these calibration functions are disjoint.
**Mean estimator:** Recall that each [reviewer]{}is assigned one of the two items. Given any assignment, consider the item assigned to [reviewer]{}${m}$. Trivially, the sum of the [[[score]{}]{}s]{}for this item must be strictly greater than ${f}_{m}(0) = \frac{{m}({m}-1)}{2}$. Now consider the remaining item (not assigned to [reviewer]{}${m}$). The sum of the [[[score]{}]{}s]{}for the remaining item can be at most $\sum_{{j}=1}^{{m}-1} {f}_{j}(1) = \sum_{{j}=1}^{{m}-1} {j}= \frac{{m}({m}-1)}{2}$.
From these two bounds on the sum of the [[[score]{}]{}s]{}, an item has a greater sum of [[[score]{}]{}s]{}if and only if [reviewer]{}${m}$ is assigned to this item. By symmetry of the assignment, [reviewer]{}${m}$ is assigned to either item with probability $0.5$. With the true ranking being either $1 {\succ}2$ or $1{\prec}2$, the mean estimator makes an error in one of the two assignments, and this assignment happens with probability $0.5$. Hence, the mean estimator makes an error with probability $0.5$.\
**Median estimator and sign estimator:** For the median estimator and the sign estimator, we first present an alternative view on the assignment, which is used for the analysis of both estimators. Recall that the assignment specifies ${{m}/2}$ [[[reviewer]{}]{}s]{}to [evaluate]{}item $1$, drawn uniformly at random without replacement, and the remaining ${{m}/2}$ [[[reviewer]{}]{}s]{}to item $2$. Equivalently, we can view this assignment as comprising the following two steps. (1) We sample uniformly at random a permutation of the ${m}$ [[[reviewer]{}]{}s]{}, denoted as a list $({j}_1, \ldots, {j}_{m})$. Define ${R}$ and ${R}'$ as the first half and second half of the [[[reviewer]{}]{}s]{}in the list, ${R}= ({j}_1, \ldots, {j}_{{\frac{{m}}{2}}})$ and ${R}' = ({j}_{{\frac{{m}}{2}}+ 1}, \ldots, {j}_{m})$. (2) We draw uniformly at random one of the two items, and assign the list ${R}$ of [[[reviewer]{}]{}s]{}to this item. Then assign the list ${R}'$ of [[[reviewer]{}]{}s]{}to the remaining item. For each ${k}\in [{{m}/2}]$ , call reviewers $\{{j}_{k}, {j}_{{\frac{{m}}{2}}+{k}}\}$ as the ${k}^{th}$ pair of [[[reviewer]{}]{}s]{}.
For the median estimator and the sign estimator, we prove that given any arbitrary lists of [[[reviewer]{}]{}s]{}${R}$ and ${R}'$ in Step (1) of the assignment, the randomness in Step (2) yields the probability of error of the two estimators as $0.5$.
Recall that the item values are bounded as ${x}_1, {x}_2 \in (0, 1)$. Since the biases of any two [[[reviewer]{}]{}s]{}differ by at least $1$ in Eq. , any [reviewer]{}${j}$ gives a higher [score]{}than any other [reviewer]{}${j}'$ if and only if ${j}< {j}'$, independent of the item values and the assignment. Formally, for any ${x}, {x}'\in (0, 1)$, and any ${j}, {j}' \in [{m}]$, we have $$\begin{aligned}
{f}_{j}({x}) < {f}_{{j}'}({x}')\quad \text{ if and only if } \quad {j}< {j}'.\label{eq:median_odd}\end{aligned}$$
The remaining analysis is performed separately for the median estimator and the sign estimator.
*Median estimator:* Denote ${j}_1^{\text{med}}$ and ${j}_2^{\text{med}}$ as the indices of the [[[reviewer]{}]{}s]{}providing the (upper) median [[[score]{}]{}s]{}in the set ${R}_1$ and ${R}_2$, respectively. From , we have $$\begin{aligned}
\label{eq:median_indices}
\begin{split}
{j}_1^{\text{med}}= & \operatorname{median}({j}_1, \ldots, {j}_{\frac{{m}}{2}})\\ {j}_2^{\text{med}}= & \operatorname{median}({j}_{{\frac{{m}}{2}}+ 1}, \ldots, {j}_{m}).
\end{split}\end{aligned}$$ Also from , the higher [score]{}in the two [[[score]{}]{}s]{}given by [reviewer]{}${j}_1^{\text{med}}$ and ${j}_2^{\text{med}}$ is the [reviewer]{}with the larger index, $\max\{{j}_1^{\text{med}}, {j}_2^{\text{med}}\}$. In Step (2) of the assignment, [reviewer]{}${j}_1^{\text{med}}$ is assigned to item $1$ or item $2$ with equal probability. Hence, the probability of error of the median estimator is $0.5$. This proves the claim that the (upper) median estimator does not strictly uniformly dominates random guess.\
We now comment on using the median function defined as the lower median, or as the mean of the two middle values. For the lower median, the same argument as above applies. Now consider the median defined as the mean of the two middle values. When ${{m}/2}$ is odd, Eq. still holds, and the argument as above still applies. When ${{m}/2}$ is even, the median value may not be equal to any [[[score]{}]{}s]{}from the [[[reviewer]{}]{}s]{}. We construct a counterexample where the item values are still bounded as ${x}_1, {x}_2\in (0, 1)$, and the calibration functions as follows: $$\begin{aligned}
{f}_{j}({x}) =
{x}+ 2^{{j}}\qquad \text{for every } {j}\in [{m}].\end{aligned}$$
With these calibration functions, for any ${x}, {x}', {x}'', {x}'''\in (0, 1)$, and any ${j}, {j}', {j}'', {j}'''\in [{m}]$, we have $$\begin{aligned}
{f}_{j}({x}) + {f}_{{j}'}({x}') < {f}_{{j}''}({x}'')+ {f}_{{j}'''}({x}''') \quad\text{ if and only if } \quad \max\{{j}, {j}'\} < \max\{{j}'', {j}'''\}.\end{aligned}$$
Using this fact, we can show that the output of this median estimator only depends on [reviewer]{}indices and the realization of Step (2), independent of the item values. The probability of error of this median estimator is also $0.5$.\
*Sign estimator:* Denote ${a}$ as the assignment that [[[reviewer]{}]{}s]{}in ${R}$ are assigned to item $1$, and denote ${a}'$ as the assignment that [[[reviewer]{}]{}s]{}in ${R}$ are assigned to item $2$. For each ${k}\in [{{m}/2}]$, define ${v}_{k}\in\{0, 1\}$ as the binary value of whether the higher [score]{}in the ${k}^{th}$ pair of [[[score]{}]{}s]{}comes from item $1$, under assignment ${a}$. Set ${v}_{k}= 1$ if the higher [score]{}comes from item $1$ and ${v}_{k}=0$ otherwise. Define ${v}'_{k}\in \{0, 1\}$ similarly under assignment ${a}'$. Set ${v}'_{k}= 1$ if the higher [score]{}comes from item $1$, and ${v}'_{k}=0$ otherwise. Inequality implies that ${v}_{k}+ {v}'_{k}= 1$ for any ${k}\in [{{m}/2}]$. Define ${v}= \sum_{{k}=1}^{{{m}/2}} {v}_{k}$ as the count of pairwise wins for item $1$ under assignment ${a}$, and define ${v}'$ similarly. Then we have $$\begin{aligned}
{v}+ {v}' = {\frac{{m}}{2}}.\label{eq:sign_estimator_distinct_output}\end{aligned}$$
The sign estimator outputs the item with more pairwise wins. That is, the sign estimator outputs item 1 under assignment ${a}$ if ${v}> {m}/4$, outputs item 1 under assignment ${a}'$ if ${v}' > {m}/4$, and outputs one of the two items uniformly at random if ${v}={m}/4$ or ${v}' = {m}/4$. When ${v}={v}' = {{m}/4}$, then under either assignment, the sign estimator has a tie, and hence outputs one of the two items uniformly at random. The probability of error of the sign estimator is $0.5$. Otherwise, we have ${v}\ne {{m}/4}$. By , we have either ${v}> {m}/ 4 > {v}'$ or ${v}' > {m}/ 4 > {v}$. The sign estimator gives different outputs under the two assignments, out of which one and only one output is correct. The probability of error of the sign estimator is $0.5$.
Proof of Theorem \[thm:abtest\_ours\]
-------------------------------------
Recall that a subset of ${{m}/2}$ [[[reviewer]{}]{}s]{}, drawn uniformly at random without replacement, are assigned to item $1$, and the remaining ${{m}/2}$ [[[reviewer]{}]{}s]{}are assigned to item $2$. We provide an alternative and equivalent view of the assignment as the following two steps:
We sample two [[[reviewer]{}]{}s]{}, uniformly at random without replacement, as the first pair of [[[reviewer]{}]{}s]{}for the two items, and call them $\{{j}_1, {j}_1'\}$. Then sample two [[[reviewer]{}]{}s]{}, uniformly at random without replacement, from the remaining $({m}-2)$ [[[reviewer]{}]{}s]{}as the second pair of [[[reviewer]{}]{}s]{}for the two items, and call them $\{{j}_2, {j}'_2\}$. Continue until all ${m}$ [[[reviewer]{}]{}s]{}are exhausted, and call the subsequent pairs of [[[reviewer]{}]{}s]{}$\{{j}_3, {j}'_3\}, \ldots, \{{j}_{{{m}/2}}, {j}'_{{{m}/2}}\}$.
Within each pair, assign the pair of [[[reviewer]{}]{}s]{}to the two items uniformly at random. That is, for each ${k}\in [{{m}/2}]$, assign [reviewer]{}${j}_{k}$ to one of the two items uniformly at random, and assign [reviewer]{}${j}_{k}'$ to the remaining item. The assignments are independent across pairs.
Consider any arbitrary values of items ${x}_1, {x}_2\in {\mathbb{R}}$. Given any arbitrary realization of Step (1) of the assignment procedure described above, we apply Theorem \[thm:canonical\_ours\] and show that on each pair of [[[reviewer]{}]{}s]{}, the canonical estimator gives the correct output with probability strictly greater than $0.5$. Then we show that combining the ${{m}/2}$ pairs by majority-voting yields probability of success strictly greater than $0.5$.
Denote ${\lambda}({x}_1, {x}_2, \{{f}, {f}'\})$ as the probability that our canonical estimator in Eq. gives the correct output comparing items of values ${x}_1, {x}_2$ under [reviewer]{}calibration functions ${f}, {f}'$. In Step (2) of the assignment procedure described above, for any ${k}\in [{{m}/2}]$, consider the ${k}^{th}$ pair of [[[reviewer]{}]{}s]{}, $\{{j}_{k}, {j}_{k}'\}$. Suppose that the calibration functions of these two [[[reviewer]{}]{}s]{}are denoted as $\{{f}, {f}'\}$. By Theorem \[thm:canonical\_ours\], since the two [[[reviewer]{}]{}s]{}are assigned to the two items uniformly at random, we have $$\begin{aligned}
\label{eq:abtest_ours_pair_success}
{\lambda}\left({x}_1, {x}_2, \{{f}, {f}'\}\right) > \frac{1}{2}\qquad \text{for all permissible } {f}, {f}'.\end{aligned}$$
Let ${{\lambda}_{\text{min}}}$ denote the probability of success of our canonical estimator when run on the worst pair of calibration functions among all pairs of [[[reviewer]{}]{}s]{}$$\begin{aligned}
{{\lambda}_{\text{min}}}= \min_{{f}, {f}' \in \{{f}_1, \ldots, {f}_{m}\}} {\lambda}({x}_1, {x}_2, \{{f}, {f}'\}) \stackrel{{\text{(i)}\xspace}}{>} \frac{1}{2},\end{aligned}$$ where inequality is true because of Eq. , and because the number of [[[reviewer]{}]{}s]{}${m}$ is finite.
Now assume that we are given any arbitrary realization of Step (1) of the assignment. For each ${k}\in [{{m}/2}]$, define ${V}_{k}\in \{0, 1\}$ as the indicator variable of the correctness of our canonical estimator on the ${k}^{th}$ pair of [[[score]{}]{}s]{}. We set ${V}_{k}= 1$ if the canonical estimator gives the correct output on the ${k}^{th}$ pair, and $0$ otherwise. Then ${V}_{k}$ is a Bernoulli random variable with mean ${\lambda}({x}_1, {x}_2, \{{f}_{{j}_{k}}, {f}_{{j}_{k}'}\}) \ge {{\lambda}_{\text{min}}}$. Moreover, since Step (2) of the assignment is performed independently across all pairs, the variables $\{{V}_{j}\}_{{j}=1}^{k}$ are independent given the item values and Step (1) of the assignment.
Let ${V}= \sum_{{j}=1}^{{{m}/2}} {V}_{j}$ be the number of pairs for which the canonical estimator ${{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}$ gives the correct output. Define a binomial random variable ${B}$ with ${k}$ trials and the success probability parameter ${{\lambda}_{\text{min}}}$. Then the random variable ${V}$ stochastically dominates the random variable ${B}$. Recall that our estimator breaks ties uniformly at random. The probability of success of our estimator with the majority-voting procedure is then bounded as $$\begin{aligned}
{\mathbb{P}}[{V}> \frac{{k}}{2}] + \frac{1}{2}{\mathbb{P}}[{V}= \frac{{k}}{2}] = & \frac{1}{2}\left({\mathbb{P}}[{V}> \frac{{k}}{2}] + {\mathbb{P}}[{V}\ge \frac{{k}}{2}]\right)\\
\ge & \frac{1}{2}\left({\mathbb{P}}[{B}> \frac{{k}}{2}] + {\mathbb{P}}[{B}\ge \frac{{k}}{2}]\right)\\
= & {\mathbb{P}}[{B}> \frac{{k}}{2}] + \frac{1}{2}{\mathbb{P}}[{B}= \frac{{k}}{2}]\\
\stackrel{{\text{(i)}\xspace}}{>} & \frac{1}{2},\end{aligned}$$ where inequality is true because the success probability parameter ${{\lambda}_{\text{min}}}$ of the binomial variable is strictly greater than $\frac{1}{2}$.
We complete the proof that the probability of success of our estimator is strictly greater than $0.5$ uniformly on any item values ${x}_1, {x}_2$ and any permissible calibration functions $\{{f}_{j}\}_{{j}=1}^{m}$.
Proof of Theorem \[thm:sort\_ours\_uniform\_prior\]
---------------------------------------------------
We first provide a high-level description of the proof. We call a pair of items “flippable”, if Algorithm \[alg:sort\_cardinal\] uses the canonical estimator to decide the relative ordering of this pair (that is, the if-condition in Line \[line:if\_flippable\] in Algorithm \[alg:sort\_cardinal\] is true). Note that a “flippable” pair may or may not be flipped by the algorithm, as the outcome depends on the output of the canonical estimator. In Theorem \[thm:canonical\_ours\], we show that our canonical estimator ${{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}$ predicts the relative ordering of a pair of items correctly with probability strictly greater than $0.5$. The main idea of the proof is to apply Theorem \[thm:canonical\_ours\] to each flippable pair. Then we show that an improvement on the probability of correctness on these flippable pairs translates to an improvement on the probability of success of exact recovery.
Theorem \[thm:canonical\_ours\] requires that within each pair, the two [[[reviewer]{}]{}s]{}are assigned the two items uniformly at random. To be able to apply this theorem, we separate the different sources of randomness in the joint procedure of the assignment and the algorithm. We derive an equivalent algorithm by re-ordering the steps of Algorithm \[alg:sort\_cardinal\], so that in this equivalent algorithm, given any flippable pair of items and two [[[reviewer]{}]{}s]{}evaluating this pair, the last sources of randomness comes from the random assignment of the two [[[reviewer]{}]{}s]{}to the two items within this pair.
We introduce some additional notation for our re-ordered algorithm. Recall the notation of ${A}= ({S}_1, \ldots, {S}_{m})$ for the [reviewer]{}assignment, where ${S}_{j}$ is a pair of items assigned to [reviewer]{}${j}$ for each ${j}\in [{m}]$. Denote ${\mathcal{Q}}= \{{\widetilde{{S}}}_{j}\}_{{j}=1}^{m}$ as the same ${m}$ pairs of items, but the [reviewer]{}assigned to each pair ${\widetilde{{S}}}_{j}$ is unspecified. Now we present an equivalent joint procedure of the assignment and the cardinal estimator ${{{\widetilde}{{\pi}}}_{{\text{rank}}}^{{\text{our}}}}$ in Algorithm \[alg:sort\_cardinal\_alternative\]. In what follows, we provide a high-level summary of Algorithm \[alg:sort\_cardinal\_alternative\]:
Sample pairwise comparisons ${\mathcal{Q}}= \{{\widetilde{{S}}}_{j}\}_{{j}=1}^{m}$ uniformly at random from all ${{n}\choose 2}$ pairs. Obtain the ordinal comparisons ${\mathcal{B}}$.\[line\_alt:assignment\_sample\_pairs\] Compute a topological ordering ${\widehat{{\pi}}}$ on the graph ${\mathcal{G}}({\mathcal{B}})$, with ties broken in order of the indices of the items.\[line\_alt:initial\_guess\] ${t}\leftarrow 1$.\[line\_alt:estimator\_store\_start\] ${{\mathcal{Q}}_\text{avail}}\leftarrow {\mathcal{Q}}$. ${\texttt{flippable\_positions}}\leftarrow [\,]$. ${\texttt{{{reviewer\xspace}}\_indices}}\leftarrow [\,]$. \[line\_alt:estimator\_store\_end\] For each pair $[{\widetilde{{S}}}_{{t}}, {\widetilde{{S}}}_{{t}+ 1}]$ in [`reviewer_indices`]{}, sample uniformly at random without replacement a pair of [[[reviewer]{}]{}s]{}$\{{j}_{t}, {j}_{{t}+1}\}$.\[line\_alt:assignment\_assign\_graders\] Output ${\widehat{{\pi}}}$.\[line\_alt:estimator\_flip\_end\]
*Line \[line\_alt:assignment\_sample\_pairs\]-\[line\_alt:initial\_guess\]:* We sample ${m}$ pairwise comparisons of the items, drawn uniformly at random without replacement from the ${{n}\choose 2}$ pairs. Obtain an initial estimate ${\widehat{{\pi}}}$ of the ranking, by computing a topological ordering on the graph ${\mathcal{G}}({\mathcal{B}})$. \[enum:assignment\_sample\_pairs\]
*Line \[line\_alt:estimator\_store\_start\]-\[line\_alt:estimator\_store\_end\]:* Store the positions of all flippable pairs (if any) determined by Algorithm \[alg:sort\_cardinal\]. If an item is included in some flippable pair, then this item is matched to a distinct pairwise comparison in ${\mathcal{Q}}$. Store the matching between the items in flippable pairs and the pairwise comparisons.\[enum:estimator\_store\]
*Line \[line\_alt:assignment\_assign\_graders\]:* For the two pairwise comparisons associated with each pair of flippable items, sample two [[[reviewer]{}]{}s]{}uniformly at random without replacement to [evaluate]{}the two comparisons.\[enum:assignment\_assign\_graders\]
1. *Line \[line\_alt:estimator\_flip\_start\]-\[line\_alt:assignment\_within\_pair\]:* Within each flippable pair, assign the two [[[reviewer]{}]{}s]{}to the two items uniformly at random.\[enum:assignment\_within\_pair\]
2. *Line \[line\_alt:estimator\_flip\_substart\]-\[line\_alt:estimator\_flip\_end\]:* Run the canonical estimator on each flippable pair, and flip the pair if the canonical estimator decides to do so (Line \[line\_alt:canonical\_call\]-\[line\_alt:canonical\_call\_end\]). After all flippable pairs are examined, output the final ranking ${\widehat{{\pi}}}$.\[enum:estimator:flip\]
We now briefly discuss the equivalence of Algorithm \[alg:sort\_cardinal\_alternative\] to Algorithm \[alg:sort\_cardinal\]. We first discuss the equivalence of the assignment procedures in the two algorithms, and then the estimation aspect in the next paragraph. The assignment consists of Steps \[enum:assignment\_sample\_pairs\], \[enum:assignment\_assign\_graders\] and \[enum:assignment\_within\_pair\]. Recall that the assignment in Algorithm \[alg:sort\_cardinal\] samples ${m}$ pairwise comparisons, uniformly at random without replacement, to assign to the ${m}$ [[[reviewer]{}]{}s]{}. In Algorithm \[alg:sort\_cardinal\_alternative\], this assignment is decomposed into the choice of pairwise comparisons, the choice of a pair of [[[reviewer]{}]{}s]{}to two pairwise comparisons in each flippable pair, and the assignment within each flippable pair, corresponding to Steps \[enum:assignment\_sample\_pairs\], \[enum:assignment\_assign\_graders\] and \[enum:assignment\_within\_pair\], respectively. Note that only the selected pairwise comparison for each item within some flippable pair is used for Algorithm \[alg:sort\_cardinal\_alternative\], so we do not need to specify the assignment of the [[[reviewer]{}]{}s]{}for the rest of the comparisons. This re-ordering of the assignment is equivalent to Algorithm \[alg:sort\_cardinal\].
The cardinal ranking estimator consists of the rest of the steps, namely Steps \[enum:estimator\_store\] and \[enum:estimator:flip\]. In the original presentation of the estimator in Algorithm \[alg:sort\_cardinal\], the estimator scans through the items, identifies flippable pairs, calls the canonical estimator on each flippable pair, and flips the pairs accordingly. Note that the identification of flippable pairs does not need the assignment of [[[reviewer]{}]{}s]{}or the [[[score]{}]{}s]{}from the [[[reviewer]{}]{}s]{}, so Algorithm \[alg:sort\_cardinal\_alternative\] first scans through the items and identifies all flippable pairs, without using the choice of the [[[reviewer]{}]{}s]{}in the assignment or using the [[[score]{}]{}s]{}from the [[[reviewer]{}]{}s]{}. Then Algorithm \[alg:sort\_cardinal\_alternative\] calls the canonical estimator on each flippable pair once the choice of the [[[reviewer]{}]{}s]{}and the [[[score]{}]{}s]{}are determined, and flips each pair based on the corresponding output from the canonical estimator. Note that when checking for a flippable pair (the if-condition in Line \[line:if\_flippable\] in Algorithm \[alg:sort\_cardinal\] and Line \[line\_alt:if\_flippable\] in Algorithm \[alg:sort\_cardinal\_alternative\]), Algorithm \[alg:sort\_cardinal\] checks whether the flipped ranking ${{\widehat{{\pi}}}_{\text{flip}}}$ is a topological ordering, where the previous flippable pairs in ${{\widehat{{\pi}}}_{\text{flip}}}$ may have already been flipped. In Algorithm \[alg:sort\_cardinal\_alternative\], the previous flippable pairs are identified but are not flipped. However, whether the flipped ranking ${{\widehat{{\pi}}}_{\text{flip}}}$ is a topological ordering is independent of whether the previous flippable pairs in ${{\widehat{{\pi}}}_{\text{flip}}}$ are flipped. Hence, the identification of the flippalbe pairs is equivalent in the two algorithms. The re-ordering of the steps of the cardinal estimator ${{{\widetilde}{{\pi}}}_{{\text{rank}}}^{{\text{our}}}}$ is valid.
Having now established the equivalence of Algorithm \[alg:sort\_cardinal\_alternative\] to Algorithm \[alg:sort\_cardinal\], we now prove Theorem \[thm:sort\_ours\_uniform\_prior\] with respect to Algorithm \[alg:sort\_cardinal\_alternative\]. Let us denote ${{{\widetilde}{{\pi}}}_{{\text{rank}}}^{\text{eq}}}$ as the cardinal estimator in Algorithm \[alg:sort\_cardinal\_alternative\]. Denote $\operatorname{topo}({\mathcal{B}})$ as the set of all topological orderings on the directed graph ${\mathcal{G}}({\mathcal{B}})$ induced by the set of ordinal observations ${\mathcal{B}}$. We denote a random variable ${T}({\mathcal{B}}) {:=}\abs{\operatorname{topo}({\mathcal{B}})}$ as the number of such topological orderings. Note that the definition of flippable pairs carries over from Algorithm \[alg:sort\_cardinal\] to Algorithm \[alg:sort\_cardinal\_alternative\]. We denote a random variable ${L}$ as the number of flippable pairs in Algorithm \[alg:sort\_cardinal\_alternative\].
Let us first consider the probability of success of the ordinal estimator. The following lemma describes the posterior distribution of the true ranking conditioned on the set of ordinal observations ${\mathcal{B}}$. Using this posterior distribution, the optimal ordinal estimators and their probability of success are derived.
\[lem:ranking\] (a) Given any possible set of ordinal observations ${\beta}$, the posterior distribution of the true ranking ${{\pi}^*}$ is uniformly distributed over the ${T}({\beta})$ topological orderings: $$\begin{aligned}
{\mathbb{P}}({{\pi}^*}={\pi}{\mid}{\mathcal{B}}= {\beta}) = \begin{cases}
\frac{1}{{T}({\beta})} & \text{if } {\pi}\in \operatorname{topo}({\beta})\\
0 & \text{otherwise}.\label{eq:posterior}
\end{cases}\end{aligned}$$
\(b) Any ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}}^\text{opt}}$ is optimal for the 0-1 loss, if and only if given any set of ordinal observations ${\beta}$, the output of this ordinal estimator belongs to the ${T}({\beta})$ topological orderings with probability $1$, that is, if and only if $$\begin{aligned}
{\mathbb{P}}({{\widehat}{{\pi}}_{{\text{rank}}}^\text{opt}}({\beta})\in \operatorname{topo}({\beta}) {\mid}{\mathcal{B}}= {\beta}) = 1 \qquad \text{for all possible set of ordinal observations }{\beta}.\label{eq:ranking_ordinal_optimality_condition}\end{aligned}$$
Moreover, conditioned on the set of ordinal observations ${\beta}$, the probability of success of any optimal ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}}^\text{opt}}$ is $$\begin{aligned}
{\mathbb{P}}({{\widehat}{{\pi}}_{{\text{rank}}}^\text{opt}}= {{\pi}^*}{\mid}{\mathcal{B}}= {\beta}) = \frac{1}{{T}({\beta})}.\label{eq:ranking_success_optimal_ordinal}\end{aligned}$$
See Section \[app:proof\_lemma\_ranking\] for the proof of the lemma.
Now consider the probability of success of our cardinal estimator ${{{\widetilde}{{\pi}}}_{{\text{rank}}}^{\text{eq}}}$ from Algorithm \[alg:sort\_cardinal\_alternative\]. We write the probability of success of our cardinal estimator as $$\begin{aligned}
{\mathbb{P}}({{{\widetilde}{{\pi}}}_{{\text{rank}}}^{\text{eq}}}={{\pi}^*}) = & \sum_{\beta}\sum_{\ell}{\mathbb{P}}({{{\widetilde}{{\pi}}}_{{\text{rank}}}^{{\text{our}}}}= {{\pi}^*}{\mid}{\mathcal{B}}={\beta}, {L}= {\ell}){\mathbb{P}}({\mathcal{B}}={\beta}, {L}= {\ell}),\label{eq:ranking_cardinal_success}\end{aligned}$$ where ${\beta}$ is summed over all possible sets of ordinal observations, and ${\ell}$ is summed from $0$ to $\floor{{n}/2}$.
We consider each term ${\mathbb{P}}({{{\widetilde}{{\pi}}}_{{\text{rank}}}^{\text{eq}}}= {{\pi}^*}{\mid}{\mathcal{B}}={\beta}, {L}= {\ell})$ separately for each ${\beta}$ and ${\ell}$. We prove that for any ${\beta}$ and any ${\ell}$, the probability of success of our cardinal estimator is greater than or equal to the probability of success of any optimal ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}}^\text{opt}}$. We also show that the probability of success is strictly greater for some ${\beta}$ and ${\ell}$. We discuss the following two cases separately, depending on the number of flippable pairs.
**Case 1:** ${\ell}= 0$.
We have the number of flippable pairs ${L}= 0$ either if there is a unique topological ordering on the graph ${\mathcal{G}}({\mathcal{B}})$, or if in each pair of adjacent items that can be flipped without violating pairwise comparisons, at least one item in this pair does not have any [score]{}. Note that these two conditions are fully determined by the set of ordinal observations. Hence, conditioned on the set of ordinal observations ${\mathcal{B}}= {\beta}$, the event of ${L}= 0$ is fully determined, and is independent of everything else given ${\mathcal{B}}$.
The initial estimated ranking of the cardinal estimator is a topological ordering (Line \[line\_alt:initial\_guess\] of Algorithm \[alg:sort\_cardinal\_alternative\]). Since there is no flippable pair, the cardinal estimator simply outputs this topological ordering. For any set of ordinal observations ${\beta}$ such that ${\mathbb{P}}({\mathcal{B}}= {\beta}, {L}= 0) > 0$, we have $$\begin{aligned}
{\mathbb{P}}({{{\widetilde}{{\pi}}}_{{\text{rank}}}^{\text{eq}}}= {{\pi}^*}{\mid}{\mathcal{B}}={\beta}, {L}= 0) \stackrel{{\text{(i)}\xspace}}{=} & {\mathbb{P}}({{{\widetilde}{{\pi}}}_{{\text{rank}}}^{\text{eq}}}= {{\pi}^*}{\mid}{\mathcal{B}}={\beta})\nonumber\\
\stackrel{{\text{(ii)}\xspace}}{=} & {\mathbb{P}}({{\widehat}{{\pi}}_{{\text{rank}}}^\text{opt}}= {{\pi}^*}{\mid}{\mathcal{B}}= {\beta}),\label{eq:ranking_cardinal_flippable_pair_zero}\end{aligned}$$ where ${{\widehat}{{\pi}}_{{\text{rank}}}^\text{opt}}$ denotes any optimal ordinal estimator. Here in , equality is true because the event ${L}= 0$ is fully determined by ${\mathcal{B}}$, and equality is true because this cardinal estimator that simply outputs a topological ordering is equivalent to an ordinal estimator that outputs the same topological ordering. From , this ordinal estimator is one optimal ordinal estimator.
**Case 2:** ${\ell}> 0$.
In this case, Algorithm \[alg:sort\_cardinal\_alternative\] identifies at least one flippable pair. The probability of success of our cardinal estimator is $$\begin{aligned}
{\mathbb{P}}({{{\widetilde}{{\pi}}}_{{\text{rank}}}^{\text{eq}}}= {{\pi}^*}{\mid}{\mathcal{B}}= {\beta}, {L}= {\ell}) = & \sum_{{\pi}\in {\Pi}}{\mathbb{P}}({{{\widetilde}{{\pi}}}_{{\text{rank}}}^{\text{eq}}}= {\pi}{\mid}{{\pi}^*}= {\pi}, {\mathcal{B}}= {\beta}, {L}= {\ell}) {\mathbb{P}}({{\pi}^*}= {\pi}{\mid}{\mathcal{B}}= {\beta}, {L}= {\ell}) \nonumber\\
\stackrel{{\text{(i)}\xspace}}{=} & \sum_{{\pi}\in {\Pi}}{\mathbb{P}}({{{\widetilde}{{\pi}}}_{{\text{rank}}}^{\text{eq}}}= {\pi}{\mid}{{\pi}^*}= {\pi}, {\mathcal{B}}= {\beta}, {L}= {\ell}) {\mathbb{P}}({{\pi}^*}= {\pi}{\mid}{\mathcal{B}}= {\beta}) \nonumber\\
\stackrel{{\text{(ii)}\xspace}}{=} & \frac{1}{{T}({\beta})}\sum_{{\pi}\in \operatorname{topo}({\beta})}{\mathbb{P}}({{{\widetilde}{{\pi}}}_{{\text{rank}}}^{\text{eq}}}= {\pi}{\mid}{{\pi}^*}= {\pi}, {\mathcal{B}}= {\beta}, {L}= {\ell}),\label{eq:ranking_cardinal_flippable_pair_at_least_one_intermediate}\end{aligned}$$ where equality is true because ${L}$ is independent of ${{\pi}^*}$ conditioned on ${\mathcal{B}}$. Equality is true by plugging in .
In Algorithm \[alg:sort\_cardinal\_alternative\], Lines \[line\_alt:assignment\_sample\_pairs\]-\[line\_alt:assignment\_assign\_graders\] fully determine the number of the flippable pairs, their positions, and the two [[[reviewer]{}]{}s]{}[evaluating]{}each flippable pair. In Lines \[line\_alt:estimator\_flip\_start\]-\[line\_alt:estimator\_flip\_end\], within each flippable pair, the algorithm first assigns uniformly at random one [reviewer]{}to one item and the remaining [reviewer]{}to the remaining item, and then calls the canonical estimator to output the relative ordering of this pair. Conditioned on the randomness in Lines \[line\_alt:assignment\_sample\_pairs\]-\[line\_alt:assignment\_assign\_graders\] of Algorithm \[alg:sort\_cardinal\_alternative\], we now apply Theorem \[thm:canonical\_ours\] to each flippable pair. Since the [reviewer]{}assignment within each flippable pair (Line \[line\_alt:assignment\_within\_pair\]) is uniformly at random, by Thoerem \[thm:canonical\_ours\], the probability that the canonical estimator outputs the correct relative ordering of each flippable pair is strictly greater than $\frac{1}{2}$. Since the assignment within each flippable pair is independent across pairs, the probability that the canonical estimator outputs the correct relative ordering of all ${\ell}$ flippable pairs is strictly greater than $\frac{1}{2^{\ell}}$.
Recall that the initial estimated ranking of our cardinal estimator is a topological ordering. Consider all total rankings that are identical to this initial ranking, except for (possibly) the relative ordering of the ${\ell}$ flippable pairs. Since the items in the flippable pairs are disjoint, there are $2^{\ell}$ such total rankings. By definition, a pair is flippable only if the total ranking after this pair is flipped is still a topological ordering. Hence, all these $2^{\ell}$ total rankings are topological orderings on the graph ${\mathcal{G}}({\mathcal{B}})$. In , the summation of ${\pi}$ is over all topological orderings. In particular, this summation includes these $2^{\ell}$ total rankings. On each of these $2^{\ell}$ total rankings, the output of our ranking estimator ${{{\widetilde}{{\pi}}}_{{\text{rank}}}^{\text{eq}}}$ is correct if and only if the canonical estimator outputs the correct relative orderings for the ${\ell}$ flippable pairs, which happens with probability strictly greater than $\frac{1}{2^{\ell}}$. Hence, we bound as
$$\begin{aligned}
{\mathbb{P}}({{{\widetilde}{{\pi}}}_{{\text{rank}}}^{\text{eq}}}= {{\pi}^*}{\mid}{\mathcal{B}}= {\beta}, {L}= {\ell}) > \frac{1}{{T}({\beta})} \cdot 2^{\ell}\cdot \frac{1}{2^{\ell}} = & \frac{1}{{T}({\beta})}\nonumber\\
\stackrel{{\text{(i)}\xspace}}{=} & {\mathbb{P}}({{\widehat}{{\pi}}_{{\text{rank}}}^\text{opt}}= {{\pi}^*}{\mid}{\mathcal{B}}= {\beta}),\label{eq:ranking_cardinal_flippable_pair_at_least_one}\end{aligned}$$
where ${{\widehat}{{\pi}}_{{\text{rank}}}^\text{opt}}$ denotes any optimal ordinal estimator. Equality is true because of in Lemma \[lem:ranking\].
Plugging and into , we have $$\begin{aligned}
{\mathbb{P}}({{{\widetilde}{{\pi}}}_{{\text{rank}}}^{\text{eq}}}= {{\pi}^*}) \ge {\mathbb{P}}({{\widehat}{{\pi}}_{{\text{rank}}}^\text{opt}}= {{\pi}^*}) \qquad \text{for any optimal ordinal estimator } {{\widehat}{{\pi}}_{{\text{rank}}}^\text{opt}},\label{eq:ranking_cardinal_uniform_dominance_equal_to}\end{aligned}$$ and a strict inequality holds in if there exists some ${\beta}$ and some ${\ell}> 0$, such that $$\begin{aligned}
{\mathbb{P}}({\mathcal{B}}= {\beta}, {L}= {\ell}) > 0.\label{eq:ranking_flippable_nonzero_probability}\end{aligned}$$
It remains to find some ${\beta}$ and some ${\ell}> 0$ such that is true. We construct such ${\beta}$ and ${\ell}> 0$ as follows. Consider the true ranking $1{\succ}2 {\succ}\cdots {\succ}{n}$, which happens with a strictly positive probability as the prior distribution of the true ranking is uniform. Conditioned on this true ranking, consider the event that the sampled pairwise comparisons in ${\mathcal{Q}}$ do not include a direct comparison between items $1$ and $2$, but both item $1$ and item $2$ have at least one [score]{}each (from comparisons with at least one of the remaining $({n}- 2)$ items). Recall that the number of pairs satisfies $1 < {m}< {{n}\choose 2}$, so such a set ${\mathcal{Q}}$ of pairwise comparisons arises with a strictly positive probability. Let ${\beta}$ be the set of ordinal observations derived from the true ranking and the set ${\mathcal{Q}}$ of pairwise comparisons described as above. With this ${\beta}$, item $1$ and item $2$ are the first two items in the initial ranking of the topological ordering, they can be flipped, and they both have some [[[score]{}]{}s]{}. Hence, item $1$ and item $2$ form a flippable pair, and we have ${L}> 0$. Hence, with this construction of ${\beta}$, we have $$\begin{aligned}
\sum_{{\ell}= 1}^{\floor{{n}/ 2}} {\mathbb{P}}({\mathcal{B}}= {\beta}, {L}= {\ell}) > 0.\end{aligned}$$ Thus there exists some ${\ell}> 0$ such that ${\mathbb{P}}({\mathcal{B}}= {\beta}, {L}= {\ell}) > 0$. Hence, Equation is true, implying the strictly inequality in . Consequently, the probability of success of our cardinal ranking estimator ${{{\widetilde}{{\pi}}}_{{\text{rank}}}^{\text{eq}}}$ is strictly uniformly greater than the probability of success of any optimal ordinal estimator. By the equivalence of Algorithm \[alg:sort\_cardinal\_alternative\] and Algorithm \[alg:sort\_cardinal\], this result also holds for the original cardinal estimator ${{{\widetilde}{{\pi}}}_{{\text{rank}}}^{{\text{our}}}}$, completing the proof.
### Proof of Lemma \[lem:ranking\] {#app:proof_lemma_ranking}
We first prove part (a) of the lemma. By Bayes rule, for any ranking ${\pi}\in {\Pi}$ and any possible set of ordinal observations ${\beta}$, we have $$\begin{aligned}
{\mathbb{P}}({{\pi}^*}={\pi}{\mid}{\mathcal{B}}= {\beta}) = \frac{{\mathbb{P}}( {\mathcal{B}}= {\beta}{\mid}{{\pi}^*}= {\pi}){\mathbb{P}}({{\pi}^*}={\pi})}{{\mathbb{P}}({\mathcal{B}}={\beta})}.\label{eq:lem_bayes_rule}\end{aligned}$$
Given the set of ordinal observations ${\beta}$, the denominator in is independent of ${\pi}$. Since the prior of the true ranking is uniform, in the numerator we have ${\mathbb{P}}({{\pi}^*}= {\pi}) =\frac{1}{{n}!}$, independent of ${\pi}$. Now it remains to consider the term ${\mathbb{P}}({\mathcal{B}}= {\beta}{\mid}{{\pi}^*}= {\pi})$ in the numerator. Recall the notation of the random variable ${\mathcal{Q}}$ as the set of pairwise comparisons in the ordinal observations (but ${\mathcal{Q}}$ does not include the results of the relative orderings of these pairs). We write the term ${\mathbb{P}}({\mathcal{B}}= {\beta}{\mid}{{\pi}^*}= {\pi})$ as $$\begin{aligned}
{\mathbb{P}}({\mathcal{B}}= {\beta}{\mid}{{\pi}^*}= {\pi}) = & \sum_{q}{\mathbb{P}}({\mathcal{B}}= {\beta}{\mid}{\mathcal{Q}}= {q}, {{\pi}^*}= {\pi}){\mathbb{P}}({\mathcal{Q}}= {q}{\mid}{{\pi}^*}= {\pi})\nonumber\\
\stackrel{{\text{(i)}\xspace}}{=} & \sum_{q}{\mathbb{P}}({\mathcal{B}}= {\beta}{\mid}{\mathcal{Q}}= {q}, {{\pi}^*}= {\pi}){\mathbb{P}}({\mathcal{Q}}= {q}),\label{eq:lem_bayes_rule_conditional}\end{aligned}$$ where ${q}$ is summed over all possible sets of ${m}$ pairwise comparisons. Equality is true because the sampling of the set of pairwise comparisons ${\mathcal{Q}}$ is independent of the true ranking ${{\pi}^*}$.
Recall that the set of ordinal observations ${\beta}$ includes the pairwise comparisons and results of the relative orderings of these pairwise comparisons, whereas ${q}$ only includes the pairwise comparisons themselves, so ${\beta}$ fully determines ${q}$. For this term to be non-zero, the set of pairwise comparisons indicated by ${\beta}$ and the set of pairwise comparisons indicated by ${q}$ need to be identical. Hence, there is only one ${q}$ in the summation of consistent with ${\beta}$, and we denote $\widetilde{{q}}$ as the set of pairs consistent with ${\beta}$. Then reduces to $$\begin{aligned}
{\mathbb{P}}({\mathcal{B}}= {\beta}{\mid}{{\pi}^*}= {\pi}) = & {\mathbb{P}}({\mathcal{B}}= {\beta}{\mid}{\mathcal{Q}}= \widetilde{{q}}, {{\pi}^*}= {\pi}){\mathbb{P}}({\mathcal{Q}}= \widetilde{{q}}),\label{eq:lem_bayes_rule_conditional_intermediate}\end{aligned}$$
In , the second term ${\mathbb{P}}({\mathcal{Q}}= \widetilde{{q}})$ is independent of ${\pi}$. Now consider the first term ${\mathbb{P}}({\mathcal{B}}= {\beta}{\mid}{\mathcal{Q}}= \widetilde{{q}}, {{\pi}^*}= {\pi})$. If ${\pi}$ is a topological ordering on ${\mathcal{G}}({\beta})$, then by definition, the relative orderings on the pairs $\widetilde{{q}}$ induced by the ranking ${\pi}$ is the set of ordinal observations ${\beta}$. If ${\pi}$ is not a topological ordering, then by definition, the relative orderings induced by the ranking ${\pi}$ violates at least one relative ordering in ${\beta}$. Hence,
$$\begin{aligned}
{\mathbb{P}}({\mathcal{B}}= {\beta}{\mid}{\mathcal{Q}}= \widetilde{{q}}, {{\pi}^*}= {\pi}) =
\begin{cases}
1 & \text{if } {\pi}\in \operatorname{topo}({\beta})\\
0 & \text{otherwise}.
\end{cases}\label{eq:lem_bayes_rule_topological_ordering}\end{aligned}$$
Combining the law of total probability with , and , the posterior distribution of the true ranking is $$\begin{aligned}
\label{eq:lemma_posterior}
{\mathbb{P}}({{\pi}^*}= {\pi}{\mid}{\mathcal{B}}={\beta}) = \begin{cases}
\frac{1}{{T}({\beta})} & \text{if }{\pi}\in \operatorname{topo}({\beta})\\
0 & \text{otherwise.}
\end{cases}\end{aligned}$$
Conditioned on the set of ordinal observations ${\beta}$, the posterior distribution of the true ranking is uniform over all topological ordering on the graph ${\mathcal{G}}({\beta})$. This completes the proof for part (a) of the lemma.\
For part (b) of the lemma, we condition on any possible set of ordinal observations ${\beta}$. On the input ${\beta}$, the probability of success of any (possibly-randomized) ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}}}$ is: $$\begin{aligned}
{\mathbb{P}}({{\widehat}{{\pi}}_{{\text{rank}}}}({\beta}) = {{\pi}^*}{\mid}{\mathcal{B}}= {\beta})
= & \sum_{{\pi}\in {\Pi}} {\mathbb{P}}({{\widehat}{{\pi}}_{{\text{rank}}}}({\beta}) = {\pi}{\mid}{{\pi}^*}= {\pi}, {\mathcal{B}}= {\beta}) {\mathbb{P}}({{\pi}^*}= {\pi}{\mid}{\mathcal{B}}= {\beta})\nonumber\\
\stackrel{{\text{(i)}\xspace}}{=} & \frac{1}{{T}({\beta})}\sum_{{\pi}\in \operatorname{topo}({\beta})} {\mathbb{P}}({{\widehat}{{\pi}}_{{\text{rank}}}}({\beta}) = {\pi}{\mid}{{\pi}^*}= {\pi}, {\mathcal{B}}= {\beta})\nonumber\\
\stackrel{{\text{(ii)}\xspace}}{=} & \frac{1}{{T}({\beta})} \sum_{{\pi}\in \operatorname{topo}({\beta})}{\mathbb{P}}({{\widehat}{{\pi}}_{{\text{rank}}}}({\beta}) = {\pi})\nonumber\\
\stackrel{{\text{(iii)}\xspace}}{\le} & \frac{1}{T({\beta})},\label{eq:lem_success_probability_inequality}\end{aligned}$$ where equality is true by plugging in . Equality is true because the output of the ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}}}({\beta})$ on the input ${\beta}$ only depends on its internal randomness, and hence independent of the ${{\pi}^*}$ and ${\mathcal{B}}$. Inequality is true by the law of total probability. In particular, the equality sign in holds if and only if the output of the ordinal estimator is always a topological ordering consistent with ${\beta}$, that is, if and only if $$\begin{aligned}
{\mathbb{P}}({{\widehat}{{\pi}}_{{\text{rank}}}}({\beta})\in \operatorname{topo}({\beta}) {\mid}{\mathcal{B}}= {\beta}) = 1. \label{eq:lemma_optimal_condition}\end{aligned}$$ Taking an expectation over all possible ordinal observations ${\beta}$, we have $$\begin{aligned}
{\mathbb{P}}({{\widehat}{{\pi}}_{{\text{rank}}}}({\mathcal{B}}) = {{\pi}^*}) = \sum_{{\beta}} {\mathbb{P}}({{\widehat}{{\pi}}_{{\text{rank}}}}({\beta}) = {{\pi}^*}{\mid}{\mathcal{B}}= {\beta}) {\mathbb{P}}({\mathcal{B}}={\beta}).\label{eq:lem_success_probability_conditional}\end{aligned}$$
Combining with the condition for the equality sign in , an ordinal estimator is optimal if and only if Eq. holds on all possible ordinal observations ${\beta}$ where ${\mathbb{P}}({\mathcal{B}}= {\beta}) > 0$. This completes the proof for part (b) of the lemma.
Discussion
==========
Breaking the barrier of using only ranking data in the presence of arbitrary (and potentially adversarial) miscalibrations, we show that cardinal scores can yield strict and uniform improvements over rankings. This result uncovers a novel, strictly-superior point on the tradeoff between cardinal scores and ordinal rankings, and provides a new perspective on this eternally-debated tradeoff. Our estimator allows for easily plugging into a variety of algorithms, thereby yielding it a wide applicability.
The results of this paper lead to several useful open problems. First, while our estimators indeed uniformly outperform ordinal estimators, in the future, a more careful design in our estimators (e.g. how to choose the function ${w}$ in the canonical estimator, and how to design better estimators for A/B testing and ranking) may yield even better results. Second, it is of interest to obtain statistical bounds on the relative errors of the cardinal and ordinal estimators in terms of the unknown miscalibration functions. Third, a promising direction of future research is to design estimators that achieve the guarantees of our proposed estimator under arbitrary/adversarial miscalibrations while simultaneously being able to adapt and yield stronger guarantees when the calibration functions follow one of the popular simpler models of miscalibration (à la “win-win” models and estimators in prior work [@shah2017learning Part I] [@heckel2016active; @shah2016permutation; @shah2017stochastically; @shah2018low; @shah18simple]). Fourth, although we consider the rating scales as continuous intervals, it is not hard to see that our results extend to discrete scales (but with the strict inequality in Equation sometimes replaced by a non-strict inequality to account for ties). Using our results to guide the choice of the scale used for elicitation is an open problem of interest. And finally, practical applications such as peer-review do not suffer from the problem of miscalibration in isolation. It is a useful and challenging open problem to address miscalibration simultaneously with other issues such as noise [@stelmakh2018forall], subjectivity [@noothigattu2018choosing], strategic behavior [@xu2018strategyproof] and others.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported in parts by NSF grants CRII: CIF: 1755656 and CCF: 1763734. The authors thank Bryan Parno for very useful discussions on biases in conference peer review. The authors thank Pieter Abbeel for pointing out the related work on the two-envelope problem.
\
\
[**Appendix: Extensions**]{}\
We now present three extensions of our problem setting and results from the main text.
Noisy data {#app:noisy}
==========
In this section, we show that even when the [[[score]{}]{}s]{}given by the reviewers are noisy, our estimator in continues to strictly uniformly dominate random guessing in the canonical setting (Section \[sec:canonical\]). We focus on the canonical estimator.
In the noisy setting, when [reviewer]{}${j}\in [{m}]$ [[[evaluate]{}]{}s]{}item ${i}\in [{n}]$, the reported [score]{}is $$\begin{aligned}
{f}_{j}({x}_{i}) + {\epsilon}_{{i}{j}},\end{aligned}$$ where ${\epsilon}_{{i}{j}}$ is a noise term. We assume that the noise terms $\{{\epsilon}_{{i}{j}}\}_{{i}\in [{n}], {j}\in [{m}]}$ are drawn i.i.d. from an unknown distribution. In this setting of noisy reported [[[score]{}]{}s]{}, we modify Definition \[def:uniformly\_better\] of strict uniform dominance, and let the expectation include the randomness in the noise.
The following theorem establishes the strict uniform dominance in the noisy setting for the cardinal estimator ${{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}$ in (cf. Theorem \[thm:canonical\_ours\] for the noiseless setting).
\[cor:noisy\] The canonical estimator ${{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}$ strictly uniformly dominates the random-guessing estimator ${{\widehat}{{\pi}}_{{\text{can}}}}$ in the presence of noise.
Observe that this result is quite general, since the noise distribution can be arbitrary and unknown. The remainder of this section is devoted to the proof of Theorem \[cor:noisy\].
Proof of Theorem \[cor:noisy\] {#app:proof_noisy}
------------------------------
The proof is a slight modification to the proof of Theorem \[thm:canonical\_ours\], so we only highlight the difference.
Recall that ${\epsilon}_{{i}{j}}$ denotes the noise in the reported [score]{}of [reviewer]{}${j}\in \{1, 2\}$ for item ${i}\in \{1, 2\}$. In Eq. from the proof of Theorem \[thm:canonical\_ours\], we replace all the noiseless terms ${f}_{j}({x}_{i})$ by the noisy terms ${f}_{j}({x}_{i}) + {\epsilon}_{{i}{j}}$ for each ${i}\in \{1, 2\}$ and ${j}\in \{1, 2\}$. Using the fact that the noise terms are independent of everything else, and taking an expectation over all the noise terms, we have $$\begin{aligned}
{\mathbb{P}}({{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}= {{\pi}^*}) = & \frac{1}{2}{\mathbb{E}}_{{\epsilon}_{11}, {\epsilon}_{12}, {\epsilon}_{21}, {\epsilon}_{22}} \left[1 + {\widetilde{{w}}}(({f}_1({x}_1) + {\epsilon}_{11}) - ({f}_2({x}_2) + {\epsilon}_{22})) - {\widetilde{{w}}}( ({f}_1({x}_2) + {\epsilon}_{21}) - ({f}_2({x}_1) + {\epsilon}_{12}))\right]\nonumber\\
= & \frac{1}{2}{\mathbb{E}}_{{\epsilon}_{11}, {\epsilon}_{12}, {\epsilon}_{21}, {\epsilon}_{22}} \left[1 + {\widetilde{{w}}}({f}_1({x}_1) - {f}_2({x}_2) + {\epsilon}_{11} - {\epsilon}_{22}) - {\widetilde{{w}}}( {f}_1({x}_2) - {f}_2({x}_1) +{\epsilon}_{21} - {\epsilon}_{12})\right]\nonumber\\
\stackrel{{\text{(i)}\xspace}}{=} & \frac{1}{2}{\mathbb{E}}_{{\epsilon}_{1}, {\epsilon}_{2}}\left[1 + {\widetilde{{w}}}({f}_1({x}_1) - {f}_2({x}_2) + {\epsilon}_{1} - {\epsilon}_{2}) - {\widetilde{{w}}}( {f}_1({x}_2) - {f}_2({x}_1) +{\epsilon}_{1} - {\epsilon}_{2})\right],\label{eq:noisy_probability_success_intermediate}\end{aligned}$$ where step uses linearity of expectation with a change of variable names, as the noise terms ${\epsilon}_{ij}$ are i.i.d.
Without loss of generality, assume ${x}_1 > {x}_2$. Recall from the proof of Theorem \[thm:canonical\_ours\] that ${f}_1({x}_1) - {f}_2({x}_2) > {f}_1({x}_2) - {f}_2({x}_1)$, and therefore we have the deterministic inequality $$\begin{aligned}
{f}_1({x}_1) - {f}_2({x}_2) + {\epsilon}_1 - {\epsilon}_2 > {f}_1({x}_2) - {f}_2({x}_1) + {\epsilon}_1 - {\epsilon}_2,\quad \text{for any ${\epsilon}_1,{\epsilon}_2\in {\mathbb{R}}$.}\end{aligned}$$
Using the monotonicity of ${\widetilde{{w}}}$, we have $$\begin{aligned}
\label{eq:noisy_func_redefine_relation}
{\widetilde{{w}}}({f}_1({x}_1) - {f}_2({x}_2) + {\epsilon}_1 - {\epsilon}_2)) > {\widetilde{{w}}}({f}_1({x}_2) - {f}_2({x}_1) + {\epsilon}_1 - {\epsilon}_2).\end{aligned}$$
Taking an expectation over ${\epsilon}_1$ and ${\epsilon}_2$ in and combining with gives $$\begin{aligned}
{\mathbb{P}}({{\widetilde}{{\pi}}_{{\text{can}}}^{{\text{our}}}}= {{\pi}^*}) > \frac{1}{2}.\end{aligned}$$
Ranking under Kendall-tau and Spearman’s footrule distance {#app:other_metrics}
==========================================================
In addition to the 0-1 exact recovery loss considered in Theorem \[thm:sort\_ours\_uniform\_prior\], Kendall-tau distance and Spearman’s footrule distance are also common metrics for ranking. Recall that a ranking of ${n}$ items is defined by a function ${\pi}:[{n}]\rightarrow [{n}]$, such that ${\pi}({t})$ is the index of the ${t}^{th}$ ranked item for each ${t}\in [{n}]$. Equivalently, we can define a ranking by the function ${\sigma}:[{n}] \rightarrow [{n}]$, such that ${\sigma}({i})$ is the rank of each item ${i}\in [{n}]$. With this notation, we have the relation ${\sigma}= {\pi}^{-1}$.
The Kendall-tau distance and the Spearman’s footrule distance are usually defined in terms of the ranking ${\sigma}$. Hence for consistency with these definitions, throughout this section we focus on the rankings as defined by ${\sigma}$ (instead of ${\pi}$ as done throughout the remainder of the paper). Kendall-tau distance and Spearman’s footrule distance between any two rankings ${\sigma}_1$ and ${\sigma}_2$ of ${n}$ items are defined as: $$\begin{aligned}
\text{Kendall-tau distance: } \qquad & {{L}_{\text{KT}}}({\sigma}_1, {\sigma}_2) = \sum_{\substack{ {i}\in [{n}], {i}' \in [{n}]:\\ {\sigma}_1({i}) < {\sigma}_1({i}')}} {{\mathbbm{1}}{\{{\sigma}_2({i}) > {\sigma}_2({i}')\}}}\\
\text{Spearman's footrule distance: }\qquad & {{L}_{\text{SF}}}({\sigma}_1, {\sigma}_2) = \sum_{{i}\in [{n}]}\abs{{\sigma}_1({i}) - {\sigma}_2({i})}.\end{aligned}$$
The following theorem states that given any arbitrary ordinal estimator, there exists a cardinal estimator that performs strictly uniformly better than this ordinal estimator, simultaneously on Kendall-tau distance and Spearman’s footrule distance (cf. Theorem \[thm:sort\_ours\_uniform\_prior\] for 0-1 loss).
\[thm:other\_metrics\] Suppose that the true ranking ${{\sigma}^*}$ is drawn uniformly at random from the collection of all possible rankings. For any arbitrary ordinal estimator ${{\widehat}{{\sigma}}_{{\text{rank}}}}$, there exists a cardinal estimator with access to one call to the ordinal estimator ${{\widehat}{{\sigma}}_{{\text{rank}}}}$ that strictly uniformly dominates the ordinal estimator ${{\widehat}{{\sigma}}_{{\text{rank}}}}$ with respect to Kendall-tau distance and Spearman’s footrule distance. The computatinal complexity of this cardinal estimator is polynomial in the number of items ${n}$, in addition to the time taken by one call to the ordinal estimator ${{\widehat}{{\sigma}}_{{\text{rank}}}}$.
This result demonstrates the generality of our results in the main text with respect to various (not only 0-1) loss functions. The remainder of this section is devoted to the proof of Theorem \[thm:other\_metrics\].
Proof of Theorem \[thm:other\_metrics\] {#app:proof_other_metrics}
---------------------------------------
We first present the construction of a cardinal estimator ${{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}$, which has access to one call to any arbitrary ordinal estimator ${{\widehat}{{\sigma}}_{{\text{rank}}}}$. For any ${i}, {i}' \in [{n}]$ with ${i}\ne {i}'$, we call the pair of items $({i}, {i}')$ “[topologically-identical]{}” under the set of ordinal observations ${\mathcal{B}}$, if the following conditions are met. There is no direct comparison between item ${i}$ and item ${i}'$ in ${\mathcal{B}}$. For any item ${\ell}\not\in \{{i}, {i}'\}$, the set ${\mathcal{B}}$ includes a comparison between item ${i}$ and item ${\ell}$, if and only if the set ${\mathcal{B}}$ includes a comparison between item ${i}'$ and item ${\ell}$. Moreover, if two comparisons $({i}, {\ell})$ and $({i}', {\ell})$ are in the set ${\mathcal{B}}$, their comparison results are the same, that is, ${{\mathbbm{1}}{\{{i}{\succ}{\ell}\}}} = {{\mathbbm{1}}{\{{i}'{\succ}{\ell}\}}}$. Note that it is possible that item ${i}$ is compared to no item in ${\mathcal{B}}$ (and hence item ${i}'$ is also compared to no item).
For any item ${i}\in [{n}]$ and any possible set of ordinal observations ${\mathcal{B}}$, we define the following sets: $$\begin{aligned}
{{V}^+}({i}, {\mathcal{B}}) = & \{{\ell}\in [{n}], {\ell}\ne {i}\mid \text{there exists a directed path from item ${\ell}$ to item ${i}$ in the graph ${\mathcal{G}}({\mathcal{B}})$}\}\\
{{V}^-}({i}, {\mathcal{B}}) = & \{{\ell}\in [{n}], {\ell}\ne {i}\mid \text{there exists a directed path from item ${i}$ to item ${\ell}$ in the graph ${\mathcal{G}}({\mathcal{B}})$}\}.\end{aligned}$$ In words, ${{V}^+}({i}, {\mathcal{B}})$ is the set of items that are ranked higher than item ${i}$ according to the set of ordinal observations ${\mathcal{B}}$, and ${{V}^-}({i}, {\mathcal{B}})$ is the set of items that are ranked lower than item ${i}$. For any [topologically-identical]{}pair $({i}, {i}')$, we have ${{V}^+}({i}, {\mathcal{B}})= {{V}^+}({i}', {\mathcal{B}})$ and ${{V}^-}({i}, {\mathcal{B}})= {{V}^-}({i}', {\mathcal{B}})$, so we denote ${{V}^+}({i}, {i}', {\mathcal{B}}) {:=}{{V}^+}({i}, {\mathcal{B}})$ and ${{V}^-}({i}, {i}', {\mathcal{B}}) {:=}{{V}^-}({i}, {\mathcal{B}})$. Now we present a cardinal estimator ${{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}$ in Algorithm \[alg:sort\_kendall\_tau\].
In words, Algorithm \[alg:sort\_kendall\_tau\] obtains an initial estimated ranking ${\widehat{{\sigma}}_\text{init}}$ by making one call to the given ordinal estimator ${{\widehat}{{\sigma}}_{{\text{rank}}}}$. Then Algorithm \[alg:sort\_kendall\_tau\] identifies two items that are [topologically-identical]{}. If such a [topologically-identical]{}pair $({i}, {i}')$ exists, we perform the following two steps on this [topologically-identical]{}pair:
*Line \[line:kt\_rearrange\_start\]-\[line:kt\_rearrange\_end\]:* Using the set of ordinal observations ${\mathcal{B}}$, we obtain a new ranking ${\widehat{{\sigma}}}$ by re-arranging the items in the initial estimated ranking ${\widehat{{\sigma}}_\text{init}}$. In this new ranking ${\widehat{{\sigma}}}$, we keep all items outside ${{V}^+}{\cup}{{V}^-}{\cup}\{{i}, {i}'\}$ in the same positions as they are in ${\widehat{{\sigma}}_\text{init}}$. We re-arrange the items in ${{V}^+}{\cup}{{V}^-}{\cup}\{{i}, {i}'\}$, so that they occupy the remaining positions; the set ${{V}^+}$ is ranked higher than items $\{{i}, {i}'\}$, and the set ${{V}^-}$ is ranked lower than items $\{{i}, {i}'\}$. Moreover, the relative ranking of items within each set (${{V}^+}$, ${{V}^-}$ or $\{{i}, {i}'\}$) is preserved. That is, if ${\ell}, {\ell}'\in {V}$ with some ${V}\in \{{{V}^+}, {{V}^-}, \{{i}, {i}'\}\}$, we have ${\widehat{{\sigma}}}({\ell}) < {\widehat{{\sigma}}}({\ell}')$ if and only if ${\widehat{{\sigma}}_\text{init}}({\ell}) < {\widehat{{\sigma}}_\text{init}}({\ell}')$.
*Line \[line:kt\_canonical\_start\]-\[line:kt\_canonical\_end\]:* We sample uniformly at random a [score]{}for each item in the [topologically-identical]{}pair $({i}, {i}')$. Based on this pair of [[[score]{}]{}s]{}, we call the canonical estimator to decide the relative ordering of these two items. Depending on the outcome of the canonical estimator, we keep the relative ordering of these two items unchanged, or flip the two items accordingly.
This completes the description of the cardinal estimator ${{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}$.
Deduce the ordinal observations ${\mathcal{B}}$ from the cardinal observations ${\mathcal{Y}}$. Compute an initial estimated ranking ${\widehat{{\sigma}}_\text{init}}={{\widehat}{{\sigma}}_{{\text{rank}}}}({\mathcal{B}})$.\[line\_kt:call\_ordinal\] Output ${{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}({A}, {\mathcal{Y}}) = {\widehat{{\sigma}}}$.\[line:kt\_output\]
We now show that the cardinal estimator ${{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}$ takes polynomial time in the number of items ${n}$, in addition to the time taken by one call to the given ordinal estimator ${{\widehat}{{\sigma}}_{{\text{rank}}}}$. To check if a pair of items $({i}, {i}')$ is [topologically-identical]{}, it takes polynomial time to go through the pairwise comparisons in ${\mathcal{B}}$. Hence, it takes polynomial time to identify a [topologically-identical]{}pair (or determine that such a pair does not exist). For any [topologically-identical]{}pair, in the re-arranging step, the set ${{V}^-}({i}, {i}', {\mathcal{B}})$ can be found by a graph traversal from node ${i}$. The set ${{V}^+}({i}, {i}', {\mathcal{B}})$ can be found by a graph traversal from node ${i}$ on the graph ${\mathcal{G}}({\mathcal{B}})$ but with all edges reversed. Both traversals take polynomial time. Hence, Algorithm \[alg:sort\_kendall\_tau\] takes polynomial time, in addition to one call to the ordinal estimator ${{\widehat}{{\sigma}}_{{\text{rank}}}}$.\
We now present the proof for the uniform strict dominance of the cardinal estimator ${{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}$ over the given ordinal estimator ${{\widehat}{{\sigma}}_{{\text{rank}}}}$. Given any two rankings ${\sigma}_1, {\sigma}_2$ and any two items $({i}, {i}')$, we denote ${\alpha}({\sigma}_1, {\sigma}_2, {i}, {i}') {:=}{{\mathbbm{1}}{\{{{\mathbbm{1}}{\{{\sigma}_1({i}) > {\sigma}_1({i}')\}}} \ne {{\mathbbm{1}}{\{{\sigma}_2({i}) > {\sigma}_2({i}')\}}}\}}}$ as Kendall-tau distance between ${\sigma}_1$ and ${\sigma}_2$ contributed by the pair of items $({i}, {i}')$. Then we can write Kendall-tau distance between ${\sigma}_1, {\sigma}_2$ as $$\begin{aligned}
{{L}_{\text{KT}}}({\sigma}_1, {\sigma}_2) = & \sum_{\substack{ {i}\in[{n}], {i}'\in [{n}] :\\ {\sigma}_1({i}) < {\sigma}_2({i}')}} {{\mathbbm{1}}{\{{\sigma}_2({i}) > {\sigma}_2({i}')\}}}\nonumber\\
= & \sum_{1\le {i}< {i}'\le {n}} {{\mathbbm{1}}{\{{{\mathbbm{1}}{\{{\sigma}_1({i}) > {\sigma}_1({i}')\}}} \ne {{\mathbbm{1}}{\{{\sigma}_2({i}) > {\sigma}_2({i}')\}}}\}}}\nonumber\\
= & \sum_{1 \le {i}< {i}' \le {n}}{\alpha}({\sigma}_1, {\sigma}_2, {i}, {i}').\label{eq:kt_rewrite}\end{aligned}$$
For Spearman’s footrule dsitance, for each item ${i}\in [{n}]$, we call the term $\abs{{\sigma}_1({i}) - {\sigma}_2({i})}$ as Spearman’s footrule distance between ${\sigma}_1$ and ${\sigma}_2$ contributed by item ${i}$.
We analyze Step (1) of re-arranging the items and Step (2) of evoking the canonical estimator separately. The following rearrangement inequality is used for analyzing both steps. For any ${a}_1, {a}_2, {b}_1, {b}_2\in {\mathbb{R}}$ where ${a}_1 < {a}_2$ and ${b}_1 < {b}_2$, it is straightforward to verify that $$\begin{aligned}
\abs{{a}_1 - {b}_2} + \abs{{a}_2 - {b}_1} \ge \abs{{a}_1 - {b}_1} + \abs{{a}_2 - {b}_2}.\label{eq:rearrangement}\end{aligned}$$
We first analyze the re-arranging step in Line \[line:kt\_rearrange\_start\]-\[line:kt\_rearrange\_end\] of Algorithm \[alg:sort\_kendall\_tau\]. We denote the random variable ${\widehat{{\sigma}}_{\text{re}}}$ as the estimated ranking after the re-arranging step (that is, the value of the quantity ${\widehat{{\sigma}}}$ after Line \[line:kt\_rearrange\_end\] of Algorithm \[alg:sort\_kendall\_tau\]). The re-arranged ranking ${\widehat{{\sigma}}_{\text{re}}}$ is a deterministic function of the initial ranking ${\widehat{{\sigma}}_\text{init}}$. The following lemma proves a deterministic result about this re-arranging step.
\[lemma\_rearrange\] For any true ranking ${{\sigma}^*}$, any set of ordinal observations ${\mathcal{B}}$ consistent with the true ranking, and any initial estimated ranking ${\widehat{{\sigma}}_\text{init}}$, the re-arranged ranking ${\widehat{{\sigma}}_{\text{re}}}$ yields smaller or equal loss compared to the initial ranking ${\widehat{{\sigma}}_\text{init}}$, regarding Kendall-tau distance and Spearman’s footrule distance. That is,
$$\begin{aligned}
{{L}_{\text{KT}}}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}) & \le {{L}_{\text{KT}}}({\widehat{{\sigma}}_\text{init}}, {{\sigma}^*})\label{eq:kt_kt_step_rearranging}\\
{{L}_{\text{SF}}}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}) & \le {{L}_{\text{SF}}}({\widehat{{\sigma}}_\text{init}}, {{\sigma}^*}).\label{eq:kt_sf_step_rearranging}\end{aligned}$$
The lemma is proved at the end of this section.\
Now we turn to analyze the second step of calling the canonical estimator on the [topologically-identical]{}pair. This step starts from the re-arranged ranking ${\widehat{{\sigma}}_{\text{re}}}$. Denote ${{E}}$ as the event that Algorithm \[alg:sort\_kendall\_tau\] identifies some [topologically-identical]{}pair (that is, Line \[line:kt\_rearrange\_start\]-\[line:kt\_break\] of Algorithm \[alg:sort\_kendall\_tau\] is executed). Then ${{E}}^c$ denotes the event that no [topologically-identical]{}pair is found. If there exists no [topologically-identical]{}pairs, then the second step in Line \[line:kt\_canonical\_start\]-\[line:kt\_canonical\_end\] of Algorithm \[alg:sort\_kendall\_tau\] is never executed. Trivially, the final output ${{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}$ is identical to the re-arranged ranking ${\widehat{{\sigma}}_{\text{re}}}$. We have
$$\begin{aligned}
{\mathbb{E}}[{{L}_{\text{KT}}}({{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}, {{\sigma}^*}) {\mid}{{E}}^c] & = {\mathbb{E}}[{{L}_{\text{KT}}}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}) {\mid}{{E}}^c]\label{eq:kt_kt_conditioned_no_twin}\\
{\mathbb{E}}[{{L}_{\text{SF}}}({{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}, {{\sigma}^*}) {\mid}{{E}}^c] & = {\mathbb{E}}[{{L}_{\text{SF}}}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}) {\mid}{{E}}^c]\label{eq:kt_sf_conditioned_no_twin}.\end{aligned}$$
It remains to consider the case when the event ${{E}}$ is true. We start by showing that the event ${{E}}$ happens with non-zero probability. Consider any arbitrary true ranking ${{\sigma}^*}$. Under this true ranking, denote the top item as ${i}^{(1)}$, and denote the second-ranked item as ${i}^{(2)}$. Conditioned on this true ranking, consider the set of pairwise comparisons ${\mathcal{Q}}$ such that the set ${\mathcal{Q}}$ includes comparisons between item ${i}^{(1)}$ and a subset of $\min\{\floor{{m}/ 2}, {n}- 2\}$ items from $[{n}]\setminus \{{i}^{(1)}, {i}^{(2)}\}$. Assume that ${\mathcal{Q}}$ also includes comparisons between item ${i}^{(2)}$ and the identical subset of items from $[{n}] \setminus \{{i}^{(1)}, {i}^{(2)}\}$. The rest of the comparisons can be arbitrary between the $({n}- 2)$ items in $[{n}] \setminus \{{i}^{(1)}, {i}^{(2)}\}$. Recall that $1 < {m}< {{n}\choose 2}$, so such a set ${\mathcal{Q}}$ arises with non-zero probability. Hence, the event ${{E}}$ happens with non-zero probability.
Note that the set of ordinal observations ${\mathcal{B}}$ fully determines the [topologically-identical]{}pair (if any) selected by Algorithm \[alg:sort\_kendall\_tau\]. Since the event ${{E}}$ happens with non-zero probability, there exists ${\beta}$ such that ${\mathbb{P}}({\mathcal{B}}= {\beta}, {{E}}) > 0$. We condition on the event ${{E}}$ and any set of ordinal observations ${\beta}$ such that ${\mathbb{P}}({\mathcal{B}}= {\beta}, {{E}}) > 0$. We denote the two items in the [topologically-identical]{}pair selected by the algorithm as items $({i}({\beta}), {i}'({\beta}))$ (or items (${i},{i}')$ in short). In what follows, we consider Kendall-tau distance and Spearman’s footrule separately.\
**Kendall-tau distance:** For each ${\ell}, {\ell}'\in [{n}]$ with ${\ell}\ne {\ell}'$, we consider Kendall-tau distance contributed by the pair $({\ell}, {\ell}')$ . Recall that conditioned on the event event ${E}$ and the set of ordinal observations ${\beta}$, the only pair that can be flipped by Algorithm \[alg:sort\_kendall\_tau\] is $({i}({\beta}), {i}'({\beta}))$. We only need to consider the pairs $({\ell}, {\ell}')$ such that the relative ordering of $({\ell}, {\ell}')$ can be potentially changed by flipping the pair $({i}, {i}')$. We consider the following two cases separately.
*Case 1: We consider Kendall-tau distance contributed by the pair $({i}, {i}')$ itself. That is, $\{{\ell}, {\ell}'\} = \{{i},{i}'\}$.*
Consider the ranking ${\widehat{{\sigma}}_{\text{re}}}$ from the re-arranging step. We have $$\begin{aligned}
{\mathbb{E}}[{\alpha}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}, {i}, {i}') {\mid}{\mathcal{B}}= {\beta}, {{E}}] = & \sum_{\sigma}{\mathbb{E}}[{\alpha}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}, {i}, {i}') {\mid}{{\sigma}^*}= {\sigma}, {\mathcal{B}}= {\beta}, {{E}}]\cdot {\mathbb{P}}({{\sigma}^*}= {\sigma}{\mid}{\mathcal{B}}= {\beta}, {{E}})\nonumber\\
\stackrel{{\text{(i)}\xspace}}{=} & \sum_{\sigma}{\mathbb{E}}[{\alpha}({\widehat{{\sigma}}_{\text{re}}}, {\sigma}, {i}, {i}') {\mid}{{\sigma}^*}= {\sigma}, {\mathcal{B}}= {\beta}, {{E}}]\cdot{\mathbb{P}}({{\sigma}^*}= {\sigma}{\mid}{\mathcal{B}}= {\beta})\nonumber\\
\stackrel{{\text{(ii)}\xspace}}{=} & \frac{1}{{T}({\beta})}\sum_{{\sigma}\in \operatorname{topo}({\beta})} {\mathbb{E}}[{\alpha}({\widehat{{\sigma}}_{\text{re}}}, {\sigma}, {i}, {i}') {\mid}{{\sigma}^*}= {\sigma}, {\mathcal{B}}= {\beta}, {{E}}].\label{eq:kt_kt_case_one_intermediate}\end{aligned}$$ where equality is true because ${{\sigma}^*}$ is independent of ${{E}}$ conditioned on ${\mathcal{B}}$. Equality is true because of in Lemma \[lem:ranking\].
Recall that the initial ranking ${\widehat{{\sigma}}_\text{init}}$ is obtained by calling the (possibly randomized) ordinal estimator ${{\widehat}{{\sigma}}_{{\text{rank}}}}$ taking input ${\mathcal{B}}$, and the re-arranged ranking ${\widehat{{\sigma}}_{\text{re}}}$ is fully determined by ${\widehat{{\sigma}}_\text{init}}$. Hence, we further write as $$\begin{aligned}
{\mathbb{E}}[{\alpha}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}, & {i}, {i}') {\mid}{\mathcal{B}}= {\beta}, {E}] \nonumber\\
= & \frac{1}{{T}({\beta})}\sum_{{\widehat{{\sigma}}}} \sum_{{\sigma}\in \operatorname{topo}({\beta})} {\mathbb{E}}[{\alpha}({\widehat{{\sigma}}}, {\sigma}, {i}, {i}') {\mid}{\widehat{{\sigma}}_{\text{re}}}= {\widehat{{\sigma}}}, {{\sigma}^*}= {\sigma}, {\mathcal{B}}= {\beta}, {{E}}] \cdot {\mathbb{P}}({\widehat{{\sigma}}_{\text{re}}}= {\widehat{{\sigma}}}{\mid}{{\sigma}^*}= {\sigma}, {\mathcal{B}}= {\beta}, {{E}})\nonumber\\
\stackrel{{\text{(i)}\xspace}}{=} & \frac{1}{{T}({\beta})}\sum_{{\widehat{{\sigma}}}} \sum_{{\sigma}\in \operatorname{topo}({\beta})} {\mathbb{E}}[{\alpha}({\widehat{{\sigma}}}, {\sigma}, {i}, {i}') {\mid}{\widehat{{\sigma}}_{\text{re}}}= {\widehat{{\sigma}}}, {{\sigma}^*}= {\sigma}, {\mathcal{B}}= {\beta}, {{E}}] \cdot {\mathbb{P}}({\widehat{{\sigma}}_{\text{re}}}= {\widehat{{\sigma}}}{\mid}{\mathcal{B}}= {\beta}),\label{eq:kt_kt_half_expression}\end{aligned}$$ where equality is true, because ${{\widehat}{{\sigma}}_{{\text{rank}}}}$ is independent of the true ranking ${{\sigma}^*}$ and the event ${{E}}$ conditioned on ${\mathcal{B}}$. Hence, ${\widehat{{\sigma}}_{\text{re}}}$ is independent of the true ranking ${{\pi}^*}$ and the event ${{E}}$ conditioned on ${\mathcal{B}}$.
Define the set ${\Omega}_{{i}{\succ}{i}'}\subseteq\operatorname{topo}({\beta})$ as the collection of topological orderings where ${i}$ is ranked higher than ${i}'$. Define the set ${\Omega}_{{i}{\prec}{i}'}\subseteq \operatorname{topo}({\beta})$ as the collection of topological orderings where ${i}$ is ranked lower than ${i}$. Then $\{{\Omega}_{{i}{\succ}{i}'}, {\Omega}_{{i}{\prec}{i}'}\}$ is a partition of the collection of all topological orderings, $\operatorname{topo}({\beta})$. Given that the pair $({i}, {i}')$ is [topologically-identical]{}, for any ranking ${\sigma}\in \operatorname{topo}({\beta})$, we can flip items $({i}, {i}')$, and the flipped ranking is still a topological ordering. Flipping the items $({i}, {i}')$ defines a bijection between the set ${\Omega}_{{i}{\succ}{i}'}, {\Omega}_{{i}{\prec}{i}'}$, so we have $\abs{{\Omega}_{{i}{\prec}{i}'}} = \abs{ {\Omega}_{{i}{\prec}{i}'}}$. Any ranking ${\widehat{{\sigma}}_{\text{re}}}$ is correct on one and only one of the sets ${\Omega}_{{i}{\succ}{i}'}$ and ${\Omega}_{{i}{\prec}{i}'}$, and hence the re-arranged ranking ${\widehat{{\sigma}}_{\text{re}}}$ is correct on exactly half of the topological orderings. For any ${\widehat{{\sigma}}}$, we have $$\begin{aligned}
\sum_{{\sigma}\in \operatorname{topo}({\beta})} {\mathbb{E}}[{\alpha}({\sigma}, {\pi}, {i}, {i}') {\mid}{\widehat{{\sigma}}_{\text{re}}}= {\widehat{{\sigma}}}, {{\sigma}^*}= {\sigma}, {\mathcal{B}}= {\beta}, {{E}}] = \frac{1}{2}.\label{eq:kt_kt_half}\end{aligned}$$
Combining with yields $$\begin{aligned}
{\mathbb{E}}[{\alpha}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}, {i}, {i}') {\mid}{\mathcal{B}}= {\beta}, {{E}}] = \frac{1}{2}.\end{aligned}$$
Now consider the cardinal estimator. Similar to the proof of Theorem \[thm:sort\_ours\_uniform\_prior\], we have $$\begin{aligned}
{\mathbb{E}}[{\alpha}({{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}, {{\sigma}^*}, {i}, {i}') {\mid}{\mathcal{B}}= {\beta}, {{E}}] < \frac{1}{2}.\end{aligned}$$
Consequently, in Case 1, we have $$\begin{aligned}
{\mathbb{E}}[{\alpha}({{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}, {{\sigma}^*}, {i}, {i}') {\mid}{\mathcal{B}}= {\beta}, {{E}}] < {\mathbb{E}}[{\alpha}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}, {i}, {i}') {\mid}{\mathcal{B}}= {\beta}, {{E}}].\label{eq:kt_kt_case_one}\end{aligned}$$
*Case 2: Consider any pair $({\ell}, {\ell}')$ that is not identical to the pair $({i}, {i}')$. Since the relative ordering of the pair $({\ell}, {\ell}')$ is changed by flipping the pair $({i}, {i}')$, then one item has to be either ${i}$ or ${i}'$. Without loss of generality, assume ${\ell}\not\in \{{i}, {i}'\}$ and ${\ell}'\in \{{i}, {i}'\}$. We consider pairs in the form of $({\ell}, {i})$ and $({\ell}, {i}')$.*
If the position of ${\ell}$ is not in between item ${i}$ and item ${i}'$ in the ranking ${\widehat{{\sigma}}_{\text{re}}}$ (that is, if ${\widehat{{\sigma}}_{\text{re}}}({\ell}) < \min\{{\widehat{{\sigma}}_{\text{re}}}({i}), {\widehat{{\sigma}}_{\text{re}}}({i}')\}$ or ${\widehat{{\sigma}}_{\text{re}}}({\ell}) > \max\{{\widehat{{\sigma}}_{\text{re}}}({i}), {\widehat{{\sigma}}_{\text{re}}}({i}')\}$), then flipping the pair $({i}, {i}')$ does not change the relative ordering of the pair $({\ell}, {i})$ or $({\ell}, {i}')$. Now we restrict our attention to item ${\ell}$ ranked in between item ${i}$ and item ${i}'$ in the ranking ${\widehat{{\sigma}}_{\text{re}}}$. Moreover, if the position of ${\ell}$ is not in between the positions of item ${i}$ and item ${i}'$ in the true ranking (that is, if ${{\sigma}^*}({\ell}) < \min\{{{\sigma}^*}({i}), {{\sigma}^*}({i}')\}$ or ${{\sigma}^*}({\ell}) > \max\{{{\sigma}^*}({i}), {{\sigma}^*}({i}')\}$), then whether flipping the pair $({i}, {i}')$ or not, one and only one of the two comparisons $({\ell}, {i})$ and $({\ell}', {i})$ is correct. Hence, we only need to consider each item ${\ell}$ ranked between the two items ${i}$ and ${i}'$, in both the re-arranged ranking ${\widehat{{\sigma}}_{\text{re}}}$ and the true ranking ${{\sigma}^*}$. For each such item ${\ell}$, for any re-arranged ranking ${\widehat{{\sigma}}_{\text{re}}}$, we have the determinisitc equality $$\begin{aligned}
\begin{split}
& {\alpha}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}, {\ell}, {i}) + {\alpha}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}, {\ell}, {i}') = 2{\alpha}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}, {i}, {i}')\\
& {\alpha}({{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}, {{\sigma}^*}, {\ell}, {i}) + {\alpha}({{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}, {{\sigma}^*}, {\ell}, {i}') = 2{\alpha}({{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}, {{\sigma}^*}, {i}, {i}')
\end{split}\label{eq:kt_kt_deterministic_equality_between}\end{aligned}$$ Combining and , for each item ${\ell}$ ranked in between item ${i}$ and item ${i}'$ in both the re-arranged ranking ${\widehat{{\sigma}}_{\text{re}}}$ and the true ranking ${{\sigma}^*}$, we have $$\begin{aligned}
{\mathbb{E}}[{\alpha}({{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}, {{\sigma}^*}, {\ell}, {i}) + {\alpha}({{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}, {{\sigma}^*}, & {\ell}, {i}') {\mid}{\mathcal{B}}= {\beta}, {{E}}] \nonumber\\ &< {\mathbb{E}}[{\alpha}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}, {\ell}, {i}) + {\alpha}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}, {\ell}, {i}') {\mid}{\mathcal{B}}= {\beta}, {{E}}].\label{eq:kt_kt_case_two}\end{aligned}$$ Combining the expression of Kendall-tau distance in with the two cases in and of which the relative ordering of some pair $({\ell}, {\ell}')$ is changed, we have $$\begin{aligned}
{\mathbb{E}}[{{L}_{\text{KT}}}({{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}, {{\sigma}^*}) {\mid}{\mathcal{B}}= {\beta}, {{E}}] < {\mathbb{E}}[{{L}_{\text{KT}}}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}) {\mid}{\mathcal{B}}= {\beta}, {{E}}].\end{aligned}$$
Recall that ${\mathbb{P}}({\mathcal{B}}= {\beta}, {{E}}) > 0$ for some ${\beta}$. Taking an expectation over ${\mathcal{B}}$ yields $$\begin{aligned}
{\mathbb{E}}[{{L}_{\text{KT}}}({{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}, {{\sigma}^*}) {\mid}{{E}}] < {\mathbb{E}}[{{L}_{\text{KT}}}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}){\mid}{{E}}].\label{eq:kt_kt_conditioned_twin}\end{aligned}$$
Combining and yields $$\begin{aligned}
{\mathbb{E}}[{{L}_{\text{KT}}}({{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}, {{\sigma}^*})] < {\mathbb{E}}[{{L}_{\text{KT}}}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*})].\label{eq:kt_kt_step_canonical}\end{aligned}$$
Finally, combining with inequality for the re-arranging step completes the proof for Kendall-tau distance.\
**Spearman’s footrule distance:** We condition on the event ${{E}}$ and any set of ordinal observations ${\beta}$ such that ${\mathbb{P}}({\mathcal{B}}= {\beta}, {{E}}) > 0$. Since only one pair $({i}({\beta}), {i}'({\beta}))$ can be flipped by Algorithm \[alg:sort\_kendall\_tau\], we only need to consider Spearman’s footrule distance contributed by these two items. Consider any ranking ${\sigma}_{{i}{\succ}{i}'}\in {\Omega}_{{i}{\succ}{i}'}$. Let ${\sigma}_{{i}{\prec}{i}'}$ be the ranking obtained by flipping items $({i}, {i}')$ in ${\sigma}_{{i}{\succ}{i}'}$. Then we have ${\sigma}_{{i}{\prec}{i}'}\in {\Omega}_{{i}{\prec}{i}'}$. For any such pair $\{{\sigma}_{{i}{\succ}{i}'}, {\sigma}_{{i}{\prec}{i}'}\}$, we have
$$\begin{aligned}
{\mathbb{P}}({{\sigma}^*}\in \{{\sigma}_{{i}{\succ}{i}'}, {\sigma}_{{i}{\prec}{i}'}\}, {\mathcal{B}}= {\beta}, {{E}}) = & {\mathbb{P}}({{\sigma}^*}\in \{{\sigma}_{{i}{\succ}{i}'}, {\sigma}_{{i}{\prec}{i}'}\} {\mid}{\mathcal{B}}= {\beta}, {{E}})\cdot {\mathbb{P}}({\mathcal{B}}= {\beta}, {{E}})\nonumber\nonumber\\
= & {\mathbb{P}}({{\sigma}^*}\in \{{\sigma}_{{i}{\succ}{i}'}, {\sigma}_{{i}{\prec}{i}'}\}{\mid}{\mathcal{B}}= {\beta}, {{E}})\cdot {\mathbb{P}}({\mathcal{B}}= {\beta}, {{E}})\nonumber\\
= & {\mathbb{P}}({{\sigma}^*}\in \{{\sigma}_{{i}{\succ}{i}'}, {\sigma}_{{i}{\prec}{i}'}\}{\mid}{\mathcal{B}}= {\beta})\cdot {\mathbb{P}}({\mathcal{B}}= {\beta}, {{E}})\label{eq:kt_sf_strict_nonzero_probability_intermediate}\\
\stackrel{{\text{(i)}\xspace}}{>} & 0\label{eq:kt_sf_strict_nonzero_probability},\end{aligned}$$
where inequality is true, because the two terms in are both non-zero. The first term in is non-zero by the fact that ${\sigma}_{{i}{\succ}{i}'}, {\sigma}_{{i}{\prec}{i}'}$ are topological orderings, and by in Lemma \[lem:ranking\]. The second term in is non-zero, because by construction we find ${\beta}$ such that the second term ${\mathbb{P}}({\mathcal{B}}= {\beta}, {{E}}) > 0$.
Now we analyze the Spearman’s footrule distance conditioned on the event ${{\sigma}^*}\in \{{\sigma}_{{i}{\succ}{i}'}, {\sigma}_{{i}{\prec}{i}'}\}$. Using the argument deriving , we can further derive $$\begin{aligned}
{\mathbb{E}}[{\alpha}({{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}, {{\sigma}^*}, {i}, {i}') {\mid}{{\sigma}^*}\in \{{\sigma}_{{i}{\succ}{i}'}, {\sigma}_{{i}{\prec}{i}'}\} &, {\mathcal{B}}= {\beta}, {{E}}] \nonumber \\
< & {\mathbb{E}}[{\alpha}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}, {i}, {i}') {\mid}{{\sigma}^*}\in \{{\sigma}_{{i}{\succ}{i}'}, {\sigma}_{{i}{\prec}{i}'}\}, {\mathcal{B}}= {\beta}, {{E}}].\label{eq:kt_sf_pair}\end{aligned}$$ By the rearrangement inequality , if the relative ordering of the pair $({i}, {i}')$ is correct, then Spearman’s footrule distance does not increase compared to the ranking with the relative ordering of $({i}, {i}')$ incorrect. Eq. implies that conditioned on ${\beta}$, the event ${{E}}$ and the event of ${{\sigma}^*}\in \{{\sigma}_{{i}{\succ}{i}'}, {\sigma}_{{i}{\prec}{i}'}\}$, the probability that the cardinal estimator ${{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}$ gives the correct relative ordering of the pair $({i}, {i}')$ is higher than the probability that ${\widehat{{\sigma}}_{\text{re}}}$ gives the correct relative ordering. Hence, for any set of ordinal observations ${\beta}$ and any pair $\{{\sigma}_{{i}{\succ}{i}'}, {\sigma}_{{i}{\prec}{i}'}\}$ of the true rankings, we have $$\begin{aligned}
{\mathbb{E}}[{{L}_{\text{SF}}}({{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}, {{\sigma}^*}){\mid}{{\sigma}^*}\in \{{\sigma}_{{i}{\succ}{i}'}, {\sigma}_{{i}{\prec}{i}'}\}, {\mathcal{B}}= {\beta}, {{E}}] \le {\mathbb{E}}[{{L}_{\text{SF}}}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}) {\mid}{{\sigma}^*}\in \{{\sigma}_{{i}{\succ}{i}'}, {\sigma}_{{i}{\prec}{i}'}\}, {\mathcal{B}}= {\beta}, {{E}}].\label{eq:kt_sf_equal}\end{aligned}$$
Note that directly applying the re-arrangement inequality does not translate the strict inequality from to . The reason is that correctly ordering a [topologically-identical]{}pair does not guarantee strictly smaller Spearman’s footrule distance. For example, if item ${i}$ and item ${i}'$ are the top-$2$ items in the true ranking, but are the bottom-$2$ items in ${\widehat{{\sigma}}_{\text{re}}}$. Then the relative ordering of the pair $({i}, {i}')$ does not change the Spearman’s footrule distance. In the rearrangement inequality , strictly inequality holds if ${a}_1 \le \{{b}_1, {b}_2\} \le {a}_2$. Hence, we find one pair of true rankings $\{{\sigma}^*_{{i}{\succ}{i}'}, {\sigma}^*_{{i}{\prec}{i}'}\}$ such that one of the following is true: $$\begin{aligned}
\label{eq:kt_sf_condition_strict_inequality}
\begin{split}
& {{\sigma}^*}_{{i}{\succ}{i}'}({i}) \le \{{\widehat{{\sigma}}_{\text{re}}}({i}), {\widehat{{\sigma}}_{\text{re}}}({i}')\} \le {{\sigma}^*}_{{i}{\succ}{i}'}({i}')\\
\text{or }\qquad & {{\sigma}^*}_{{i}{\succ}{i}'}({i}') \le \{{\widehat{{\sigma}}_{\text{re}}}({i}), {\widehat{{\sigma}}_{\text{re}}}({i}')\} \le {{\sigma}^*}_{{i}{\succ}{i}'}({i}).
\end{split}\end{aligned}$$ Then strictly inequality in holds on the pair $\{{\sigma}^*_{{i}{\succ}{i}'}, {\sigma}^*_{{i}{\prec}{i}'}\}$. Now we provide the construction of this pair $\{{\sigma}^*_{{i}{\succ}{i}'}, {\sigma}^*_{{i}{\prec}{i}'}\}$.
We start by constructing a topological ordering ${\sigma}({i}, {i}', {\beta})$ (or ${\sigma}$ in short) as follows. We topologically sort the items in ${{V}^+}{:=}{V}({i}, {i}', {\beta})$ and place them as the top $\abs{{{V}^+}}$ items in ${\sigma}$. We topologically sort the items in ${{V}^-}{:=}{{V}^-}({i}, {i}, {\beta})$ and place them as the bottom $\abs{{{V}^-}}$ items. Arbitrarily choose one item from $\{{i}, {i}'\}$ and place it at the position $(\abs{{{V}^+}} + 1)$, and place the remaining item from the pair $\{{i}, {i}'\}$ at the position $({n}- \abs{{{V}^-}})$. Topologically sort the rest of the items, and place them in the remaining positions in ${\sigma}$.
We now prove that the ranking ${\sigma}$ is a valid topological ordering. Assume for contradiction that ${\sigma}$ is not a valid topological ordering. Then there exists a pair $({\ell}, {\ell}')$ that violates some pairwise comparison in ${\mathcal{B}}$. Denote ${V}^c =[{n}] \setminus ({{V}^+}{\cup}{{V}^-}{\cup}\{{i}, {i}'\})$. Within each set ${{V}^+}$, ${{V}^-}$ or ${V}^c$, the items are ordered by a topological ordering. Moreover, there is no direct comparison between item ${i}$ and item ${i}'$, so items $\{{i}, {i}'\}$ can be ranked with either ${i}{\succ}{i}'$ or ${i}{\prec}{i}'$. Hence, ${\ell}$ and ${\ell}'$ cannot belong to the same set of ${{V}^+}, {V}, {V}^c$ or $\{{i}, {i}'\}$. By the definition of the sets ${{V}^+}$ and ${{V}^-}$, in the true ranking ${{V}^+}$ should be ranked higher than $\{{i}, {i}'\}$, and ${{V}^-}$ should be ranked lower than $\{{i}, {i}'\}$. In our ranking ${\sigma}$, we also rank ${{V}^+}$ higher than $\{{i}, {i}'\}$, and ${{V}^-}$ lower than $\{{i}, {i}'\}$. Hence, if both item ${\ell}$ and item ${\ell}'$ are in ${{V}^+}{\cup}{{V}^-}{\cup}\{{i}, {i}'\}$, the relative ordering between $({\ell}, {\ell}')$ must be consistent with ${\mathcal{B}}$. Then at least one item from the pair $({\ell}, {\ell}')$ must be in ${V}^c$. Without loss of generality, assume ${\ell}'\in {V}^c$. Since ${\ell}$ and ${\ell}'$ cannot belong to the same set, we have ${\ell}\not\in {V}^c$. If ${\ell}\in \{{i}, {i}'\}$, since the pair $({\ell}, {\ell}')$ violates some pairwise comparison, the items $({\ell}, {\ell}')$ are compared in ${\mathcal{B}}$, that is, ${\ell}'$ is compared to either ${i}$ or ${i}'$. By the definition of the sets ${{V}^+}$ and ${{V}^-}$, it must be true that ${\ell}'\in {{V}^+}$ or ${\ell}'\in {{V}^-}$, contradicting the assumption that ${\ell}'\in {V}^c$. If ${\ell}\in {{V}^+}$, by construction the ranking ${\sigma}$ ranks ${\ell}$ higher than ${\ell}'$. Since the pair $({\ell}, {\ell}')$ violates some pairwise comparison, the set ${\mathcal{B}}$ must include the pairwise comparison ${\ell}' {\succ}{\ell}$. By the definition of ${{V}^+}$, since ${\ell}\in {{V}^+}$, there exists a path from ${\ell}$ to ${i}$. Concatenating the pairwise comparison ${\ell}' {\succ}{\ell}$ with the path from ${\ell}$ to ${i}$, we have a path from ${\ell}'$ to ${i}$. Hence, ${\ell}'\in {{V}^+}$, contradicting the assumption that ${\ell}'\in {V}^c$. Similarly, ${\ell}\in {{V}^-}$ gives a contradiction. Hence, in the ranking ${\sigma}$ there exists no pair of items violating pairwise comparisons in ${\mathcal{B}}$. By definition, the ranking ${\sigma}$ is a topological ordering. The ranking ${\sigma}$ places items $\{{i}, {i}'\}$ at positions $\{\abs{{{V}^+}} + 1, {n}-\abs{{{V}^-}}\}$. In Algorithm \[alg:sort\_kendall\_tau\], the re-arranged ranking ${\widehat{{\sigma}}_{\text{re}}}$ places the set ${{V}^+}$ before items $\{{i}, {i}'\}$, and the set ${{V}^-}$ after items $\{{i}, {i}'\}$. Hence, we have either ${\sigma}({i}) \le \{{\widehat{{\sigma}}_{\text{re}}}({i}), {\widehat{{\sigma}}_{\text{re}}}({i}')\} \le {\sigma}({i}')$ or ${\sigma}({i}') \le \{{\widehat{{\sigma}}_{\text{re}}}({i}), {\widehat{{\sigma}}_{\text{re}}}({i}')\} \le {\sigma}({i})$.
Recall that when constructing ${\sigma}$, we arbitrarily place an item from the set $\{{i}, {i}'\}$ at position $(\abs{{{V}^+}} + 1)$, and the remaining item from $\{{i}, {i}'\}$ at position $({n}-\abs{{{V}^-}})$. Denote ${\sigma}^*_{{i}{\succ}{i}'}$ as the topological ordering with item ${i}$ in position $(\abs{{{V}^+}} + 1)$. Denote ${\sigma}^*_{{i}{\prec}{i}'}$ as the topological ordering with item ${i}'$ in position $(\abs{{{V}^+}} + 1)$. For any possible ${\widehat{{\sigma}}_{\text{re}}}$, one of the conditions in holds on the pair $\{{\sigma}^*_{{i}{\succ}{i}'}, {\sigma}^*_{{i}{\prec}{i}'}\}$, and hence strict inequality in holds for the pair $\{{\sigma}^*_{{i}{\succ}{i}'}, {\sigma}^*_{{i}{\prec}{i}'}\}$.
Eq. implies that the event ${{\sigma}^*}\in \{{\sigma}^*_{{i}{\succ}{i}'}, {\sigma}^*_{{i}{\prec}{i}'}\}$ arises with non-zero probability. Taking an expectation over all possible pairs $\{{\sigma}_{{i}{\succ}{i}'}, {\sigma}_{{i}{\prec}{i}'}\}$ in , and using the strict inequality for the pair $\{{\sigma}^*_{{i}{\succ}{i}'}, {\sigma}^*_{{i}{\prec}{i}'}\}$ yields $$\begin{aligned}
& {\mathbb{E}}[{{L}_{\text{SF}}}({{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}, {{\sigma}^*}) {\mid}{\mathcal{B}}= {\beta}, {{E}}] < {\mathbb{E}}[{{L}_{\text{SF}}}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}) {\mid}{\mathcal{B}}= {\beta}, {{E}}].\end{aligned}$$
Taking an expectation over the set of ordinal observations ${\mathcal{B}}$ yields $$\begin{aligned}
& {\mathbb{E}}[{{L}_{\text{SF}}}({{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}, {{\sigma}^*}) {\mid}{{E}}] < {\mathbb{E}}[{{L}_{\text{SF}}}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*}) {\mid}{{E}}].\label{eq:kt_sf_conditioned_twin}\end{aligned}$$
Combining with inequality for the re-arranging step yields $$\begin{aligned}
{\mathbb{E}}[{{L}_{\text{SF}}}({{{\widetilde}{{\sigma}}}_{{\text{rank}}\text{-metric}}^{{\text{our}}}}, {{\sigma}^*})] < {\mathbb{E}}[{{L}_{\text{SF}}}({\widehat{{\sigma}}_{\text{re}}}, {{\sigma}^*})].\label{eq:kt_sf_step_canonical}\end{aligned}$$
Finally, combining with inequality for the re-arranging step completes the proof for Spearman’s footrule.\
We make a comment about having multiple [topologically-identical]{}pairs. Notice that in Algorithm \[alg:sort\_kendall\_tau\], we only find one [topologically-identical]{}pair, and then break out of the for-loops. Alternatively, we can identify and flip multiple disjoint [topologically-identical]{}pairs in a similar fashion as in Algorithm \[alg:sort\_cardinal\]. This is still a valid algorithm, because each step of processing one [topologically-identical]{}pair does not increase Kendall-tau distance or Spearman’s footrule distance.
It remains to prove Lemma \[lemma\_rearrange\].
Proof of Lemma \[lemma\_rearrange\]
-----------------------------------
Consider any two items ${\ell}, {\ell}'\in [{n}]$, such that ${\ell}{\succ}{\ell}'$ in the true ranking ${{\sigma}^*}$. Let ${\widehat{{\sigma}}}_1$ be an arbitrary ranking. Let ${\widehat{{\sigma}}}_2$ be a ranking where all items are ranked the same as in ${\widehat{{\sigma}}}_1$, except that the positions of items ${\ell}$ and ${\ell}'$ are flipped as compared to ${\widehat{{\sigma}}}_1$. The remainder of the proof is broken into two parts.\
*Part 1: If the relative ordering of a pair is inconsistent with the relative ordering indicated by the true ranking, then flipping this pair does not increase Kendall-tau distance or Spearman’s footrule distance.*
Specifically, we claim that if ${\ell}{\prec}{\ell}'$ in ${\widehat{{\pi}}}_1$, then ${\widehat{{\sigma}}}_2$ has a smaller or equal loss than ${\widehat{{\sigma}}}_1$, with respect to Kendall-tau distance and Spearman’s footrule distance. We discuss the two distance metrics separately.\
**Kendall-tau distance:** First, consider Kendall-tau distance contributed by the pair $({\ell}, {\ell}')$. We have ${\ell}{\prec}{\ell}'$ in ${\widehat{{\sigma}}}_1$ and ${\ell}{\succ}{\ell}'$ in ${\widehat{{\sigma}}}_2$. Since we have ${\ell}{\succ}{\ell}'$ in the true ranking, the relative ordering of this pair is correct in ${\widehat{{\sigma}}}_2$, and incorrect in ${\widehat{{\sigma}}}_1$. Hence, $$\begin{aligned}
0 = {\alpha}({\widehat{{\sigma}}}_2, {{\sigma}^*}, {\ell}, {\ell}') < {\alpha}({\widehat{{\sigma}}}_1, {{\sigma}^*}, {\ell}, {\ell}') = 1.\label{eq:kt_lem_kt_one}\end{aligned}$$
Denote ${{\ell}_{\text{mid}}}$ as any item ranked in between ${\ell}$ and ${\ell}'$ in ${\widehat{{\sigma}}}_1$ (or equivalently, in ${\widehat{{\sigma}}}_2$). In the rest of the pairs that are not $({\ell}, {\ell}')$, the flip only changes the relative ordering of each pair of the form $({\ell}, {{\ell}_{\text{mid}}})$ or $({\ell}', {{\ell}_{\text{mid}}})$. If in the true ranking ${{\sigma}^*}$, item ${{\ell}_{\text{mid}}}$ is ranked higher than both $({\ell}, {\ell}')$, or ranked lower than both $({\ell}, {\ell}')$, then the sum of the contributions to Kendall-tau distance by the pair $({\ell}, {{\ell}_{\text{mid}}})$ and the pair $({\ell}', {{\ell}_{\text{mid}}})$ is the same in ${\widehat{{\sigma}}}_1$ and ${\widehat{{\sigma}}}_2$: $$\begin{aligned}
{\alpha}({\widehat{{\sigma}}}_2, {{\sigma}^*}, {\ell}, {{\ell}_{\text{mid}}}) + {\alpha}({\widehat{{\sigma}}}_2, {{\sigma}^*}, {\ell}', {{\ell}_{\text{mid}}}) = 1 = {\alpha}({\widehat{{\sigma}}}_1, {{\sigma}^*}, {\ell}, {{\ell}_{\text{mid}}}) + {\alpha}({\widehat{{\sigma}}}_1, {{\sigma}^*}, {\ell}', {{\ell}_{\text{mid}}}).\label{eq:kt_lem_kt_two}\end{aligned}$$
Otherwise ${{\ell}_{\text{mid}}}$ is ranked in between ${\ell}$ and ${\ell}'$ in the true ranking ${{\sigma}^*}$, then we have $$\begin{aligned}
0 = {\alpha}({\widehat{{\sigma}}}_2, {{\sigma}^*}, {\ell}, {{\ell}_{\text{mid}}}) + {\alpha}({\widehat{{\sigma}}}_2, {{\sigma}^*}, {\ell}', {{\ell}_{\text{mid}}}) < {\alpha}({\widehat{{\sigma}}}_1, {{\sigma}^*}, {\ell}, {{\ell}_{\text{mid}}}) + {\alpha}({\widehat{{\sigma}}}_1, {{\sigma}^*}, {\ell}', {{\ell}_{\text{mid}}}) = 2.\label{eq:kt_lem_kt_three}\end{aligned}$$
Combining the expression of Kendall-tau distance with , and yields $$\begin{aligned}
{{L}_{\text{KT}}}({\widehat{{\sigma}}}_2, {{\sigma}^*}) < {{L}_{\text{KT}}}({\widehat{{\sigma}}}_1, {{\sigma}^*}).\end{aligned}$$
**Spearman’s footrule distance:** By flipping the positions of the items $({\ell}, {\ell}')$, only Spearman’s footrule distance contributed by these two items has changed. Recall that the condition for flipping the pair $({\ell}, {\ell}')$ requires ${\ell}{\prec}{\ell}'$ in ${\widehat{{\pi}}}_1$ and ${\ell}{\succ}{\ell}'$ in ${{\sigma}^*}$. Applying the rearrangement inequality with ${a}_1 = {\widehat{{\sigma}}}_1({\ell}'), {a}_2 = {\widehat{{\sigma}}}_1({\ell}), {b}_1 = {{\sigma}^*}({\ell}), {b}_2 = {{\sigma}^*}({\ell}')$, we have $$\begin{aligned}
\abs{{\widehat{{\sigma}}}_1({\ell}') - {{\sigma}^*}({\ell}') } + \abs{{\widehat{{\sigma}}}_1({\ell}) - {{\sigma}^*}({\ell})} \ge & \abs{{\widehat{{\sigma}}}_1({\ell}') - {{\sigma}^*}({\ell}) } + \abs{{\widehat{{\sigma}}}_1({\ell}) - {{\sigma}^*}({\ell}')}\nonumber\\
= & \abs{{\widehat{{\sigma}}}_2({\ell}) - {{\sigma}^*}({\ell}) } + \abs{{\widehat{{\sigma}}}_2({\ell}') - {{\sigma}^*}({\ell}')}.\label{eq:kt_lem_sf}\end{aligned}$$
Combining with the definition of Spearman’s footrule distance yields $$\begin{aligned}
{{L}_{\text{SF}}}({\widehat{{\sigma}}}_2, {{\sigma}^*}) \le {{L}_{\text{SF}}}({\widehat{{\sigma}}}_1, {{\sigma}^*}).\end{aligned}$$
This completes Part 1 of the proof.\
*Part 2: The re-arranging step in Algorithm \[alg:sort\_kendall\_tau\] is equivalent to a sequence of pair flips.*
With Part 1 in place, we now explain the rest of the proof. For any arbitrary [topologically-identical]{}pair of items $({i}, {i}')$ and any arbitrary set of ordinal observations ${\mathcal{B}}$, denote the sets ${V}_1 {:=}{{V}^+}({i}, {i}', {\mathcal{B}}), {V}_2 {:=}\{{i}, {i}'\}, {V}_3 {:=}{{V}^-}({i}, {i}', {\mathcal{B}})$. We consider the following procedure. We start by setting ${\widehat{{\sigma}}}_1$ as the initial estimated ranking ${\widehat{{\sigma}}_\text{init}}$. We identify one pair $({\ell}, {\ell}')$ (if any) such that the following three conditions are met. First, we have ${\ell}\in {V}_{j}, {\ell}' \in {V}_{{j}'}$ with ${j}< {j}'$. Second, we have ${\ell}{\prec}{\ell}'$ in ${\widehat{{\sigma}}}_1$. Third, there is no item in ${V}_1 {\cup}{V}_2 {\cup}{V}_3$, whose position is in between ${\ell}$ and ${\ell}'$ in the ranking ${\widehat{{\sigma}}}_1$. If such a pair is found, we flip the positions of ${\ell}$ and ${\ell}'$, and update ${\widehat{{\sigma}}}_1$ to be this new ranking. Repeat this procedure until no such pair can be found.
Now we show that this procedure is equivalent to the re-arranging step in Line \[line:kt\_rearrange\_start\]-\[line:kt\_rearrange\_end\] of Algorithm \[alg:sort\_kendall\_tau\]. This procedure properly terminates, because each pair of items $({\ell}, {\ell}')$ can be swapped at most once, and there is a finite number of pairs. When the procedure terminates, the ranking is identical to the re-arranged ranking ${\widehat{{\sigma}}}$ after Line \[line:kt\_rearrange\_end\] of Algorithm \[alg:sort\_kendall\_tau\]. To see this claim, we first note that this procedure has never moved items outside ${V}_1 {\cup}{V}_2 {\cup}{V}_3$, so we only need to concern about the items in ${V}_1 {\cup}{V}_2 {\cup}{V}_3$ and their positions. For each pair $({\ell}, {\ell}')$ to be flipped, the procedure specifies that ${\ell}$ and ${\ell}'$ belong to two different sets from ${V}_1, {V}_2$ and ${V}_3$. Moreover, by the condition on the pair $({\ell}, {\ell}')$, there cannot be any item in ${V}_1 {\cup}{V}_2 {\cup}{V}_3$ that is ranked in between ${\ell}$ and ${\ell}'$. Hence, the relative ordering of the items within each set of ${V}_1, {V}_2$ or ${V}_3$ is unchanged, consistent with the ranking specified in Line \[line:kt\_rearrange\_placement\_one\] and Line \[line:kt\_rearrange\_placement\_two\] of Algorithm \[alg:sort\_kendall\_tau\]. Moreover, the re-arranging step in Algorithm \[alg:sort\_kendall\_tau\] ranks all items in ${V}_1$ before all items in ${V}_2$, and all items in ${V}_2$ before all items in ${V}_3$. Assume that the final output of the procedure is a different ranking from the re-arranging step in Algorithm \[alg:sort\_kendall\_tau\], then we can find a pair $({\ell}, {\ell}')$ that can be flipped, contradicting the fact that no such pairs can be found at the termination of the procedure. Hence, the procedure and the re-arranging step in Algorithm \[alg:sort\_kendall\_tau\] are equivalent. Applying Part 1 to each flip in this procedure completes the proof of the lemma.
Ranking under arbitrary true ranking {#app:ranking_arbit}
====================================
Theorem \[thm:sort\_ours\_uniform\_prior\] in Section \[sec:ranking\] compared our cardinal estimator with arbitrary ordinal estimators under a uniform prior over the true ranking. In this section, we present a result for ranking under any arbitrary true ranking. This setting is more similar to our results on the canonical setting (Theorem \[thm:canonical\_ours\]) and A/B testing (Theorem \[thm:abtest\_ours\]) in the main text. When the true ranking is arbitrary, a minimax-optimal ordinal estimator outputs uniformly at random a topoglocial ordering consistent with the pairwise comparisons. We denote this optimal ordinal estimator as ${{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}$.
Given this ordinal estimator, we then construct a cardinal estimator ${{\widetilde}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}^{{\text{our}}}}$ by simply setting the initial estimate ${\widehat{{\pi}}}= {{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}({\mathcal{B}})$ in Line 2 of Algorithm \[alg:sort\_cardinal\] (instead of executing the current Line 2). The following theorem states the desired result for strict uniform dominance of this cardinal estimator over the optimal ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}$.
\[cor:sort\_ours\] When the true ranking is arbitrary, the cardinal estimator ${{\widetilde}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}^{{\text{our}}}}$ strictly uniformly dominates the minimax-optimal ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}$.
Importantly, we can think of this cardinal estimator as a post-processing step which builds on the output of the optimal ordinal estimator. This cardinal estimator takes polynomial time in the number of items ${n}$, in addition to the time taken by one call to the ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}$.
We prove Theorem \[cor:sort\_ours\] in the remainder of this section.
Proof of Theorem \[cor:sort\_ours\]
-----------------------------------
The proof is a slight modification to the proof of Theorem \[thm:sort\_ours\_uniform\_prior\], so we only highlight the difference. First, we consider the probability of success of the optimal ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}$ that outputs one of the topological orderings uniformly at random:
$$\begin{aligned}
{\mathbb{P}}({{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}({\beta}) = {{\pi}^*}{\mid}{\mathcal{B}}= {\beta}) = & \sum_{{\pi}\in \operatorname{topo}({\beta})} {\mathbb{P}}({\pi}= {{\pi}^*}{\mid}{{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}= {\pi}, {\mathcal{B}}= {\beta}){\mathbb{P}}({{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}= {\pi}{\mid}{\mathcal{B}}= {\beta})\nonumber\\
\stackrel{{\text{(i)}\xspace}}{=} & \frac{1}{{T}({\beta})}\sum_{{\pi}\in \operatorname{topo}({\beta})} {\mathbb{P}}({\pi}= {{\pi}^*}{\mid}{{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}= {\pi}, {\mathcal{B}}= {\beta}),\label{eq:cor_arbitrary_ordinal}\end{aligned}$$
where equality is true because the ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}$ uniformly at random outputs one of the topological orderings consistent with ${\beta}$.
Now we consider each term ${\mathbb{P}}({\pi}= {{\pi}^*}{\mid}{{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}= {\pi}, {\mathcal{B}}= {\beta})$ in . The quantities ${{\pi}^*}$ and ${\pi}$ are both deterministic. Trivially, we have $$\begin{aligned}
{\mathbb{P}}({\pi}= {{\pi}^*}{\mid}{{\widehat}{{\pi}}_{{\text{rank}}}}= {\pi}, {\mathcal{B}}= {\beta})=
\begin{cases}
1 & \text{if } {\pi}= {{\pi}^*}\\
0 & \text{otherwise}.
\end{cases}\label{eq:cor_arbitrary_ordinal_term}\end{aligned}$$
Combining and with the fact that the true ranking ${{\pi}^*}$ must be a topological ordering consistent with ${\beta}$, we have $$\begin{aligned}
{\mathbb{P}}({{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}({\beta}) = {{\pi}^*}{\mid}{\mathcal{B}}= {\beta}) = & \frac{1}{{T}({\beta})}.\label{eq:ranking_ordinal_arbitrary}\end{aligned}$$
Now consider the cardinal estimator ${{\widetilde}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}^{{\text{our}}}}$. When the number of flippable pairs is zero, the cardinal estimator behaves equivalently as the ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}$. Following a similar argument as Case 1 in the proof of Theorem \[thm:sort\_ours\_uniform\_prior\], for any set of ordinal observations ${\beta}$, we have (cf. Equation in the proof of Theorem \[thm:sort\_ours\_uniform\_prior\]): $$\begin{aligned}
{\mathbb{P}}({{\widetilde}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}^{{\text{our}}}}{\mid}{\mathcal{B}}= {\beta}, {L}= 0) = {\mathbb{P}}({{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}= {{\pi}^*}{\mid}{\mathcal{B}}= {\beta}).\label{eq:ranking_cardinal_arbitrary_flippable_pair_zero}\end{aligned}$$
Denote ${{\widehat{{\pi}}}_{\text{init}}}$ as the initial estimated ranking obtained by calling the ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}$. When the number of flippable pairs is ${L}= {\ell}> 0$, the probability of success of the cardinal estimator is $$\begin{aligned}
{\mathbb{P}}({{\widetilde}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}^{{\text{our}}}}= {{\pi}^*}& {\mid}{\mathcal{B}}= {b}, {L}= {\ell}) \nonumber \\
= & \sum_{{\pi}\in \operatorname{topo}({\beta})}{\mathbb{P}}({{\widetilde}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}^{{\text{our}}}}= {{\pi}^*}{\mid}{\mathcal{B}}= {\beta}, {L}= {\ell}, {{\widehat{{\pi}}}_{\text{init}}}= {\pi}){\mathbb{P}}({{\widehat{{\pi}}}_{\text{init}}}= {\pi}{\mid}{\mathcal{B}}= {\beta}, {L}= {\ell})\nonumber\\
\stackrel{{\text{(i)}\xspace}}{=} & \frac{1}{{T}({\beta})}\sum_{{\pi}\in \operatorname{topo}({\beta})}{\mathbb{P}}({{\widetilde}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}^{{\text{our}}}}= {{\pi}^*}{\mid}{\mathcal{B}}= {\beta}, {L}= {\ell}, {{\widehat{{\pi}}}_{\text{init}}}= {\pi}),\label{eq:ranking_cardinal_arbitrary_flippable_pair_at_least_one_intermediate}\end{aligned}$$ where equality is true because the ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}$ outputs a topological ordering uniformly at random.
The remaining argument is similar to Case 2 in the proof of Theorem \[thm:sort\_ours\_uniform\_prior\], so we only outline the main steps. Consider all total rankings that are identical to the true ranking ${{\pi}^*}$, except for (possibly) the relative ordering of the ${\ell}$ flippable pairs. There are $2^{\ell}$ such total rankings, and all these $2^{\ell}$ total rankings are topological orderings on the graph ${\mathcal{G}}({\mathcal{B}})$. In , the summation of ${\pi}$ is over all topological orderings. In particular, this summation includes these $2^{\ell}$ total rankings. Recall that the cardinal estimator ${{\widetilde}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}^{{\text{our}}}}$ is obtained by replacing Line \[line\_alt:initial\_guess\] of Algorithm \[alg:sort\_cardinal\] by calling the ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}$. To be able to apply Theorem \[thm:canonical\_ours\], we obtain a cardinal estimator ${{\widetilde}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}^{\text{eq}}}$ by replacing Line \[line\_alt:initial\_guess\] of Algorithm \[alg:sort\_cardinal\_alternative\] by calling the ordinal estimator ${{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}$. This estimator ${{\widetilde}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}^{\text{eq}}}$ is equivalent to the original estimator ${{\widetilde}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}^{{\text{our}}}}$. When the initial estimated ranking ${{\widehat{{\pi}}}_{\text{init}}}$ is any of the $2^{\ell}$ total rankings, the probability that the cardinal estimator ${{\widetilde}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}^{\text{eq}}}$ gives the correct output is strictly greater than $\frac{1}{2^{\ell}}$. Hence, we bound as (cf. Equation in the proof of Theorem \[thm:sort\_ours\_uniform\_prior\]): $$\begin{aligned}
{\mathbb{P}}({{\widetilde}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}^{\text{eq}}}= {{\pi}^*}{\mid}{\mathcal{B}}= {b}, {L}= {\ell}) > \frac{1}{{T}({\beta})}\cdot 2^{\ell}\cdot \frac{1}{2^{\ell}} = \frac{1}{{T}({\beta})} \stackrel{{\text{(i)}\xspace}}{=} {\mathbb{P}}({{\widehat}{{\pi}}_{{\text{rank}}\text{-}{\text{unif}}}}= {{\pi}^*}{\mid}{\mathcal{B}}= {\beta}).\label{eq:ranking_cardinal_arbitrary_flippable_pair_at_least_one}\end{aligned}$$ where equality is true from .
Having established and , the rest of the argument follows the proof of Theorem \[thm:sort\_ours\_uniform\_prior\].
[^1]: We use the standard notation of $[\kappa]$ to denote the set $\{1,\ldots,\kappa\}$ for any positive integer $\kappa$.
[^2]: Our results also hold in the following settings: (a) Each [reviewer]{}is assigned one of the two items independently and uniformly at random. (b) [Reviewers]{}are grouped (in any arbitrary manner) into ${{m}/2}$ pairs, and within each pair, the two [[[reviewer]{}]{}s]{}are assigned one distinct item each uniformly at random.
[^3]: For values $a_1 \ge \cdots \ge a_n$, we define the median function as the upper median, $\operatorname{median}(a_1, \ldots, a_n) = a_{\floor{(n +1)/ 2}}$. Theorem \[thm:abtest\_det\_examples\_fails\] also holds instead for the lower median $a_{\floor{(n+2) / 2}}$, and the median defined as the mean of the two middle values, $(a_{\floor{(n+1) / 2}} + a_{\floor{(n+2) / 2}}) / 2$.
|
---
abstract: 'Although electron energy loss near edge structure analysis provides a tool for experimentally probing unoccupied density of states, a detailed comparison with simulations is necessary in order to understand the origin of individual peaks. This paper presents a [density functional theory based]{} technique for predicting the N K-edge for ternary (quasi-binary) nitrogen alloys by adopting a core hole approach, a methodology that has been successful for binary nitride compounds. It is demonstrated that using the spectra of binary compounds for optimising the core hole charge ($0.35{\ensuremath\,\mathrm{e}}$ for cubic Ti$_{1-x}$Al$_x$N and $0.45{\ensuremath\,\mathrm{e}}$ for wurtzite Al$_x$Ga$_{1-x}$N), the predicted spectra evolutions of the ternary alloys agree well with the experiments. The spectral features are subsequently discussed in terms of the electronic structure and bonding of the alloys.'
author:
- 'D. Holec'
- 'R. Rachbauer'
- 'D. Kiener'
- 'P.D. Cherns'
- 'P.M.F.J. Costa'
- 'C. McAleese'
- 'P.H. Mayrhofer'
- 'C.J. Humphreys'
title: 'Towards predictive modelling of near-edge structures in electron energy loss spectra of AlN based ternary alloys'
---
Introduction
============
Wurtzite aluminium nitride (w-AlN), gallium nitride (w-GaN) and their ternary alloy w-Al$_x$Ga$_{1-x}$N are important materials for devices such as light emitting diodes (LEDs) and laser diodes (LDs) [@jain00]. Cubic titanium nitride (c-TiN) and in particular its alloy with AlN, cubic Ti$_{1-x}$Al$_x$N, are widely used hard coatings due to their high hardness, corrosion and oxidation resistance [@mayrhofer06b]. In both ternary alloys, a crucial requirement for getting the optimal application-tailored properties is an accurate control of both the composition *and* structure of the alloy.
Electron energy loss spectroscopy (EELS) is a powerful technique to microanalyse compositional features [@egerton96]. The spectra can be recorded with very high spatial resolution, thus taking advantage of the high sensitivity of EELS to local changes in the electronic structure of materials [@botton96; @rez08]. Its subset, electron energy loss near edge structure (ELNES) reflects the density of unoccupied states and provides thus an experimental probe for this part of the electronic structure of materials. However, the interaction of high energy electrons with lattice atoms does not always have a straightforward interpretation. In order to understand the experimental data, measured EELS spectra need to be compared in detail with their calculated counterparts.
[X-ray absorbtion near-edge structure (XANES) is a closely related technique to ELNES. The edge shapes are very similar [@keast01; @rez08], it also allows for studying polarisation effects [@katsikini98], and it has been successfully applied to diluted alloys [@ciatto05]. Therefore, the here proposed methodology is highly relevant also for XANES.]{}
Recent developments of theoretical methods for solid state physics have provided the EELS community with increasingly reliable and comprehensive tools to simulate ELNES. A particularly important example is the Telnes program which is distributed as a part of Wien2k [@wien2k], an all-electron full-potential linearised augmented plane-wave (FP-LAPW) code. It has been suggested in the literature [see, e.g. @hebert07] that a calculation including a core hole provides a better description of the excitation process by means of the standard (ground state) density functional theory (DFT). To create a core hole, one takes an electron (or a fraction of it) from its ground state position (N $1s$-state in the case of N K-edge) and puts it in the lowest unoccupied state above the Fermi level (N $2p$) or adds it as a background charge to keep the total charge of the cell neutral [@hebert07]. This can be easily realised in Wien2k where all electrons are accessible and explicitly treated. Recently, the core hole calculations have become routinely available also for pseudopotential codes [@seabourne09; @mizoguchi09].
A considerable effort has been spent on studying ELNES of binary nitrides with respect to their crystal structures (wurtzite, rock salt, zinc blende) [@lawniczak-jablonska97; @lawniczak-jablonska00; @mizoguchi03; @mizoguchi04; @sennour03; @gao04], polarisation effects [@lawniczak-jablonska97; @lawniczak-jablonska00; @keast02; @radtke03; @gao04], doping [@serin98], and stoichiometry [@mirguet06; @lazar08; @kwak10]. The literature is vast and provides a good background for understanding the origin of peaks in ELNES in terms of bonding, and thus establishes a solid basis for a finger-print identification of materials and their properties.
Nonetheless, there are only a few reports on the compositional dependence of ELNES of ternary alloys. @keast03 measured N K-edge ELNES of In$_x$Ga$_{1-x}$N alloys and correlated their findings with DFT calculations using, however, extremely small supercells. @mackenzie05 used ordered structures and averaging of the boundary binaries spectra to get a guess on the shape of Ti$_{1-x}$Al$_x$N N K-edge. @holec08 showed that using small ordered cells failed to reproduce the experimental spectrum of an Al$_x$Ga$_{1-x}$N alloy. To date, a systematic study showing the effect of alloying on the ELNES shape (“evolution” of the edge), as well as discussing the computational methodology in a recipe-like form, is missing. The present paper is aiming to fill this gap by facilitating a [*semi-empirical approach*]{} [in which]{} the calculation parameters (e.g. the core hole charge) are first [*adjusted*]{} to reproduce the spectra of the boundary binary systems, and subsequently used to [*predict*]{} the N K-edge evolution of ternary alloys.
Methodology
===========
Calculation details
-------------------
The individual structures are modelled with supercells constructed using a special quasi-random structure (SQS) approach [@wei90]. All alloys considered in this paper are quasi-binary which means that mixing of elements (either Ti and Al or Al and Ga) takes place only on one sublattice (bigger atoms in Fig. \[fig:structures\]); the other sublattice is fully occupied with N atoms. $3\times3\times2$ (36 atoms) and $2\times2\times2$ (32 atoms) supercells were used for the cubic B1 and wurtzite B4 modifications, respectively. The short range order parameters were optimised for pairs up to the fourth order, triplets up to the third order and quadruplets up to the second order [@wei90; @holec10-diss]. [More details about the cells and the process of their generation can be found in Ref. .]{}
![Crystal structures investigated in this work: (a) cubic B1 (NaCl prototype), (b) wurtzite B4 (ZnS prototype), and (c) five-coordinated hexagonal [B$_{\mathrm{k}}$]{}structure (BN prototype).[]{data-label="fig:structures"}](1.eps){width="8cm"}
The cubic and wurtzite structures of the Ti$_{1-x}$Al$_x$N alloy were optimised with respect to volume (lattice constants) as well as to internal (local) relaxations, for which we used the VASP code [@vasp2; @vasp1] together with the projector augmented wave pseudopotentials [@vasp-pot] employing the generalised gradient approximation (GGA) as parametrised by [@wang91]. We used $500{\ensuremath\,\mathrm{eV}}$ for the plane-wave cut-off energy and a minimum of $\approx600\,k$-points$\cdot$atom (usually more). Such parameters guarantee the calculation accuracy in the order of meV/atom. The obtained equilibrium lattice parameters (see Tab. \[tab:lattice\]) and energies are in good agreement with those previously published [@mayrhofer06a; @alling07; @alling08].
structure $a\ [\mbox{\AA}]$ $c\ [\mbox{\AA}]$ $B_0\ [\mbox{GPa}]$
----- ---------------------- ------------------- ------------------- ---------------------
TiN B1 4.256 292
[B$_{\mathrm{k}}$]{} 3.524 4.227 238
AlN B1 4.070 253
B4 3.129 5.016 197
GaN B4 3.216 5.238 169
: Optimised lattice parameters and bulk moduli of the binary compounds used in this work.[]{data-label="tab:lattice"}
TiN is mechanically unstable in the four co-ordinated wurtzite (B4) structure (Fig. \[fig:structures\]b) and relaxes into a five-coordinated [B$_{\mathrm{k}}$]{}structure (Fig. \[fig:structures\]c). The reason for this is that the presence of $d$ electrons favours a different hybridisation scheme ($sp^3d$) than the tetrahedral $sp^3$ (see Refs. ). Consequently, the wurtzite variant of the Ti$_{1-x}$Al$_x$N alloy becomes unstable around $x\approx0.6$ whereas for $x\lessapprox0.6$ the [B$_{\mathrm{k}}$]{}structure is obtained [@holec08; @holec_inprep]. This is, however, the composition where also the phase transition to a lower energy, experimentally observed cubic variant happens [@mayrhofer06a]. Therefore, in the following we present only results for the high Al containing w-Ti$_{1-x}$Al$_x$N.
As for the Al$_x$Ga$_{1-x}$N alloy, we optimised the crystal lattices only for the boundary binary compounds (Tab. \[tab:lattice\]), AlN and GaN, and then used Vegard’s rule to obtain the lattice constants for the intermediate composition. This is justified by the work of @Dridi03 who showed for this alloy that the lattice parameters (unlike the band gap) exhibit a linear dependence on the composition.
Electronic properties and the ELNES spectra were calculated using the Wien2k code [@wien2k] employing the GGA–PBE parametrisation [@perdew96] of the exchange-correlation potential. An equivalent of $\approx900$ $k$-points within the whole first Brillouin zone of the unit cell, the expansion of the spherical harmonics up to $l=10$ inside the non-overlapping muffin tin (MT) spheres, and $R_{\mathrm{MT}}k_{\max}=7$ were used[^1]. The MT radii were automatically set by structGen (a part of the Wien2k package) to values $\approx1.70$–$1.80$, $\approx1.95$–$2.00$, $\approx1.85$–$1.95$ and $\approx1.90$–$1.95{\ensuremath\,\mathrm{a.u.}}$ for N, Ti, Al, and Ga atom, respectively. The spin polarisation effects were not taken into account. The core holes were implemented by reducing the N $1s$ core level occupation on a specific site and putting the corresponding charge in the background in order to keep the cell neutral [@hebert07]. ELNES was calculated using the Telnes program, a part of Wien2k. [This was repeated for all N sites in the supercell. The spectrum representing the particular alloy composition was obtained by averaging this set of N K-edges.]{}
Experiment
----------
In order to experimentally confirm the *ab initio* predicted ELNES spectra, two material systems, Ti$_{1-x}$Al$_x$N and Al$_x$Ga$_{1-x}$N were investigated.
The Ti$_{1-x}$Al$_x$N samples were deposited in Leoben using the plasma-assisted unbalanced magnetron sputtering technique [see @rachbauer10]. The variation of the Al mole fraction $x$ in Ti$_{1-x}$Al$_x$N was achieved by using powder metallurgically produced targets (PLANSEE AG, 99% purity), with Ti/Al ratios of 1, 0.5 and 0.33, and manual placing of additional Ti or Al platelets ($\varnothing 5\times3{\ensuremath\,\mathrm{mm}}$) in the racetrack of the targets, respectively. TEM sample preparation was performed by Ar-ion thinning in a Gatan precision ion polishing system (PIPS) at $4$ and $2.2{\ensuremath\,\mathrm{keV}}$ in plan view. The EELS measurements were carried out on a Cs corrected Jeol 2100F operated at $200{\ensuremath\,\mathrm{kV}}$ and equipped with a Gatan Tridiem GIF camera using nanobeam diffraction mode. This ensures high signal to noise ratios and makes it possible to acquire information from individual grains, necessary for the investigation of the polycrystalline Ti$_{1-x}$Al$_x$N films. Thus, several different grains were measured for each alloy composition to rule out possible orientation effects. The spectra were recorded with a dispersion of $0.3{\ensuremath\,\mathrm{eV/channel}}$ and the energy resolution, measured by the full-width at half-maximum of the zero-loss peak, was $1.5$–$1.8{\ensuremath\,\mathrm{eV}}$. The convergence and collection semi-angles during analysis were $5{\ensuremath\,\mathrm{mrad}}$ and $>8{\ensuremath\,\mathrm{mrad}}$, respectively.
The Al$_x$Ga$_{1-x}$N films were epitaxially grown at Cambridge by $6\times2$-inch Thomas Swan Close-Coupled Showerhead metalorganic vapour-phase epitaxy (MOVPE) system [@mcaleese04]. A standard two-step growth method was used to deposit a $5{\ensuremath\,\mathrm{\mu m}}$ thick GaN pseudo-substrate on $(0001)$ sapphire [@datta04] on which the Al$_x$Ga$_{1-x}$N layers were grown at $1020{\ensuremath\,\mathrm{^\circ C}}$. The different compositions were obtained by varying the flow rate for Al and Ga precursors. The EELS spectra were obtained on a FEI Tecnai F20 microscope equipped with a Schottky FEG source, Gatan Imaging Filter and operated at $200{\ensuremath\,\mathrm{kV}}$. [The electron beam was parallel to the $\langle0001\rangle$ direction.]{}
Prior to the comparison with the *ab initio* calculations, all measured spectra were corrected for the dark current and the channel-to-channel gain variation. The pre-edge background was extrapolated using a power-law function and subtracted from the original data [@egerton96].
Results
=======
Binary compounds {#binaries}
----------------
The strategy adopted in this paper is to find calculation parameters that reproduce the N K-edge ELNES for the binary compounds as closely as possible, and subsequently use these settings for [*predicting*]{} the ELNES evolution of the alloys. Figure \[fig:binaries\] shows how the edge shape of cubic and wurtzite/hexagonal AlN and TiN changes with increasing core hole charge from $0{\ensuremath\,\mathrm{e}}$ (ground state) to $1{\ensuremath\,\mathrm{e}}$ (final excited state – full core hole). The effect is stronger for AlN than for TiN. This is due to fast core hole screening in TiN originating from its metallic character [@rez08]. Nevertheless, some small changes can still be observed, e.g. the peak broadening for the cubic modification or the disappearing high-energy shoulder of the main peak for the hexagonal TiN with increasing core hole charge. @lazar08 arrived at the same conclusion for c-TiN based on the comparison of their calculations with experimental measurements.
![N K-edge onset (energy difference between the initial core state and lowest unoccupied state) as a function of the core hole charge for the two allotropes of each AlN and TiN. The inset shows zoomed-in region around the experimentally observed N K-edge onset energy.[]{data-label="fig:edge-onset"}](3.eps)
The strong effect of the core hole charge on the AlN N K-edge shape has been discussed in the literature [@mizoguchi09; @holec08]. A detailed analysis of the relative peak positions and intensities for the wurtzite AlN revealed that a core hole charge $\approx0.5$–$0.6{\ensuremath\,\mathrm{e}}$ reproduces the experimental ELNES the best [@holec10-diss]. Such a comparison could still be misleading as the experimental results also depend strongly on the acquisition conditions [@radtke03]. Since the spectra around $0.5{\ensuremath\,\mathrm{e}}$ (Slater’s transition state) [for each allotrope look akin]{} and are almost equally resembling the experiment, another criterion was adopted here. The edge onset, measured as the energy between the initial core state and the lowest unoccupied state [@rez08], is plotted as a function of the core hole charge in Fig. \[fig:edge-onset\]. This dependence is rather strong, and using the experimental value for the edge onset allows an optimal value of the core hole charge to be estimated. Taking $402{\ensuremath\,\mathrm{eV}}$ for w-AlN [@mizoguchi03; @mizoguchi09a or this work] and $397{\ensuremath\,\mathrm{eV}}$ for c-TiN [@rashkova07; @craven95 or this work] yields $0.45{\ensuremath\,\mathrm{e}}$ and $0.35{\ensuremath\,\mathrm{e}}$, respectively, which are the values used in this work.
Ti$_{1-x}$Al$_x$N alloy
-----------------------
The calculated evolution of the N K-edge for Ti$_{1-x}$Al$_x$N is shown by solid lines in Fig. \[fig:TiAlN\] for the cubic and wurtzite modifications. The raw ELNES was broadened with a Gaussian having $1{\ensuremath\,\mathrm{eV}}$ FWHM. Moreover, the curves of pure AlN ($x=1$) were shifted along the energy axis (as labelled in Fig. \[fig:TiAlN\]) to account for the abrupt change in the Fermi energy due to the development of the band gap (no Ti $d$-states present).
The compositional step used for the cubic alloy is $\Delta x=0.167$ (Fig. \[fig:TiAlN\]a). Three developments of the main peaks are predicted: (i) the double-maxima at $0$–$5{\ensuremath\,\mathrm{eV}}$ above $E_F$ disappears with increasing AlN content, (ii) the peak at $\approx13{\ensuremath\,\mathrm{eV}}$ for TiN gradually moves to $\approx9{\ensuremath\,\mathrm{eV}}$ for AlN and at the same time its intensity increases, and (iii) a small hump at $\approx32{\ensuremath\,\mathrm{eV}}$ for TiN broadens to an almost undetectable background at $x=0.5$. At the same time, a small hump develops at $\approx26{\ensuremath\,\mathrm{eV}}$ with increasing AlN content. The origins of these composition-induced peak variations can be tracked down to the changes in bonding in the alloy as is discussed later in section \[sec:origin\_of\_peaks\].
The situation is more complicated for the w-Ti$_{1-x}$Al$_x$N alloy (Fig. \[fig:TiAlN\]b, compositional step $\Delta x=0.125$). The calculated spectra suggest that the characteristic triple-peak character of the w-AlN N K-edge is levelled out with the addition of only $0.125$ mole fraction of TiN. Additionally, the spectra do not show any clear trends in the peak development as in the cubic case, apart from the peak at $\approx20{\ensuremath\,\mathrm{eV}}$, whose position is not influenced by the composition. This is most likely connected with local relaxations taking place similar to those reported for Nb$_{1-x}$Al$_x$N [@holec10]. In particular, N sites near Al tend to have the four-coordinated (wurtzite-like) neighbourhood, while in the vicinity of Ti atoms, five-coordinated local neighbourhoods are preferred (hexagonal [B$_{\mathrm{k}}$]{}-like). The structure is therefore much more sensitive to the actual arrangement of atoms in the supercell which is reflected, e.g. in much bigger scatter of the “optimised” lattice constants for different arrangements of atoms in the SQS (with a constant composition $x$) [@holec_inprep].
To compare the calculated and measured N K-edge evolutions (Fig. \[fig:TiAlN\]c), a larger spectrometer broadening parameter of $1.5{\ensuremath\,\mathrm{eV}}$ was used. As a consequence, the double-maximum of the first peak in c-Ti$_{1-x}$Al$_{x}$N at $\approx0$–$5{\ensuremath\,\mathrm{eV}}$ above the $E_F$ “smears out” and the measured shape is obtained. It is therefore concluded that the fine double-maximum character is not resolved due to experimental limitations. The experimental spectra were smoothed and normalised to fit the intensity of the highest peak of the simulated pattern in each individual case; no other operation was performed on them. The theoretical spectra were, on the other hand, shifted by the calculated energy of the core-holed core level. The spectra thus obtained exhibit a very good correlation between experiment and theory.
Al$_x$Ga$_{1-x}$N alloy
-----------------------
As another example, the semiconducting wurtzite solid solution of AlN and GaN is chosen to demonstrate the ability of the current approach to predict ELNES. In contrast to the meta-stable Ti$_{1-x}$Al$_x$N, the wurtzite Al$_x$Ga$_{1-x}$N mixture is stable in this modification for all concentrations $x$. Therefore, no local distortions as in the case of the w-Ti$_{1-x}$Al$_x$N alloy are expected, which results in a gradual N K-edge evolution as shown in Fig. \[fig:AlGaN\]a. The intensities of the first and third peaks of the triple-peak shape characteristic for w-AlN decrease with decreasing AlN mole fraction, and become shoulders around a central peak, a shape typical for GaN. The peak at $\approx24{\ensuremath\,\mathrm{eV}}$ above the $E_F$ in the w-AlN spectrum gradually moves to $\approx26{\ensuremath\,\mathrm{eV}}$ for w-GaN with increasing GaN content. The edge onset moves towards the Fermi level reflecting the narrowing band gap from $4.2{\ensuremath\,\mathrm{eV}}$ (AlN) to $1.7{\ensuremath\,\mathrm{eV}}$ (GaN)[^2]. This gradual tranformation of the spectra shape is traced down to the changes in electronic structure, see section \[sec:origin\_of\_peaks\]. The predicted evolution of the N K-edge is again confirmed by the experimental observations (Fig. \[fig:AlGaN\]b).
Discussion
==========
Shape and evolution of the N K-edge ELNES {#sec:shape}
-----------------------------------------
The spectra of the binary TiN, AlN and GaN systems have been extensively discussed in the literature both from the experimental and theoretical perspective. Despite that, several issues remain unclear, in particular the edge onset energy:
its value varies in the literature within the range of several eVs [@serin98; @sennour03; @mizoguchi04; @mackenzie05; @holec08; @mizoguchi09; @mizoguchi09a]. Consequently, we used our own measurements to calibrate the calculations instead of taking spectra from the literature. The lineshapes of binary w-AlN, c-TiN, and w-GaN resemble those previously published for the same materials.
The edge evolution for c-Ti$_{1-x}$Al$_{x}$N exhibits the same trends as the one previously published by @mackenzie05 (which, however, provided only one intermediate composition). @gago09 used XANES to measure the N K-edge of Ti$_{1-x}$Al$_x$N experimentally. Their XANES spectra have all the main features of our experimental as well as calculated N K-edge ELNES. Also in the case of w-Al$_x$Ga$_{1-x}$N, the calculated evolution of N K-edge ELNES correlates with the here presented experimental data as well with those published previously [@radtke04; @holec08][^3].
@mackenzie05 tried to model the edge with a linear interpolation of the boundary binary spectra. Having in mind the problems with the accurate edge onset determination and the lack of cubic AlN (in the B1 structure) for getting a reliable binary spectrum, this approach is questionable. To demonstrate this further, Figs. \[fig:TiAlN\]a and \[fig:AlGaN\]a include the linear interpolations of the binary spectra (dashed lines). Although this may serve as a first (and quick) guess on what the evolution should look like, in many cases the relative intensities and/or positions of the peaks are not predicted correctly.
@craven95 showed using several binary transition metal nitrides (TMN) that the spacing between the double-maximum peaks increases with increasing lattice parameter. This is not predicted for the c-Ti$_{1-x}$Al$_x$N alloy (Fig. \[fig:TiAlN\]) where the lattice parameter decreases from $4.25{\ensuremath\,\mathrm{\mbox{\AA}}}$ for TiN ($x=0$) to $4.07{\ensuremath\,\mathrm{\mbox{\AA}}}$ for AlN ($x=1$) [@mayrhofer06a], but the peak spacing is almost unaffected. The reason for this is that the bonding of various TMN is similar and the peaks follow small shifts of the density of states associated with the varying valence configuration. On the contrary, the peak shifts in the Ti$_{1-x}$Al$_x$N evolution result in the first place from the changing character of bonding (see section \[sec:origin\_of\_peaks\]). Consequently, when the ELNES of the two boundary binary systems are similar, the simple approach of interpolating between the binary ELNES spectra [@mackenzie05] is expected to give acceptable results, see e.g. Ti$_{1-x}$V$_x$N [@mackenzie97] or In$_{1-x}$Ga$_x$N [@keast02]. The extremely small cells (1 In and 1 Ga atom for In$_{0.5}$Ga$_{0.5}$N) used in the latter reference satisfactorily reproduced the N K-edge evolution, and a much more computationally expensive approach (using the averaging of several N core-holed sites in SQSs as in the present paper) employed by @holec10-diss is not necessary.
On the other hand, when the spectrum evolution is pronounced, small ordered structures do not provide reliable predictions. This has been shown by @mackenzie05 for the case of c-Ti$_{1-x}$Al$_x$N and by @holec08 for w-Al$_x$Ga$_{1-x}$N. In such cases, the approach adopted here is essential.
Electronic origin of the peaks {#sec:origin_of_peaks}
------------------------------
There exists extensive literature on the origin of peaks for semiconducting III-N binaries. As summarised by @mizoguchi09a using the overlap population analysis, the main peak structure (up to $\approx10{\ensuremath\,\mathrm{eV}}$ above the edge onset) reflects the anti-bonding N–cation interaction while the later peak ($20$–$30{\ensuremath\,\mathrm{eV}}$ above the edge onset) corresponds to cation–cation (mostly anti-bonding) interactions. The difference between the AlN and GaN ELNES shapes can be traced down to the presence of the valence $d$-electrons in GaN which cause (slight) redistribution of the valence density of states, and consequently also the unoccupied density of states. The electronic structure of the valence band of InN is similar to that of GaN thus resulting in a similar ELNES spectrum.
The ground state projected density of states (PDOS) in Fig. \[fig:TiAlN\_el\] helps to understand the meaning and evolution of peaks in ELNES of the c-Ti$_{1-x}$Al$_x$N alloy. The site and symmetry projected DOS were obtained by averaging the corresponding quantities over all sites occupied with the same specie. The final states of the N K-edge transition are unoccupied N $p$-states which clearly correlate with the ELNES. Based on the PDOS overlaps it can be concluded that the double-maximum structure just above the edge onset arises from the N $p$–Ti $d$-states interaction which agrees with the findings of @tsujimoto05 and @lazar08 for binary c-TiN. The second peak at around $10$–$12{\ensuremath\,\mathrm{eV}}$ above $E_F$ have the strongest contribution from the N $p$–Al $p$-states interaction, the only exception being pure TiN (no Al present) where a small peak in Ti $d$-states at the same position can be detected. The different interactions contributing to this delayed peak are responsible for a sharper maximum with clear shoulders in the case of pure TiN while resulting in a rather broad (and nearly symmetric) shape when Al is present, see Fig. \[fig:TiAlN\_el\]. The peak position changes by almost $1{\ensuremath\,\mathrm{eV}}$ upon adding $x=0.167$ mole fraction of AlN to TiN, while further increase of AlN content results in only a small shift of the peak ($0.5{\ensuremath\,\mathrm{eV}}$ for increasing $x$ from $0.167$ to $0.667$). This can serve as an example why the simple interpolation between the properties of binary compounds (as suggested by @mackenzie05) does not work.
![Bader analysis of a charge transfer (absolute values) on atoms of individual species.[]{data-label="fig:bader"}](7.eps)
The bonding of the c-Ti$_{1-x}$Al$_x$N alloy consists of a mixture of covalent and ionic type. The bonding of TiN has been discussed many times in the literature [see, e.g. a review by @schwarz87]. It was concluded that the covalent part is realised by the $sp^3d^2$ orbitals (interaction of Ti $4s$ and $3d$-states with N $2p$-states). Additionally, the interaction between Ti $3d$ orbitals with the $t_{2g}$ symmetry causes a non-zero DOS at the Fermi level resulting in the metallic character of the compound. The influence of Al on bonding in the alloy was discussed by several authors [@mayrhofer06a; @alling07; @rovere10], generally showing a gradual weakening of the $sp^3d^2$ hybridisation (which is reflected by the decreased intensity of the first double-maximum peak in the N K-edge ELNES, see Figs. \[fig:TiAlN\] and \[fig:TiAlN\_el\]). Additionally, the Bader analysis [@bader90] as implemented in Wien2k shows that there is a significantly increased charge transfer from metallic sites to N resulting in a stronger ionic bonding with increasing AlN mole fraction (see Fig. \[fig:bader\]). It is interesting to note that the different Al sites “provide” on average always almost the same charge to be transferred on N sites (practically no scatter around the mean values shown by the triangles in Fig. \[fig:bader\]), but the charge transferred from Ti sites is much more influenced by the alloy composition. This is likely to be due to different degrees of hybridisation between Ti and N atoms depending on the actual neighbourhood of N atoms (i.e. second-order neighbours of Ti sites).
Influence of the local environment
----------------------------------
There is some controversy in the literature on how big the supercells should be in order to suppress the mutual interactions between core holes. For example, @mizoguchi04 and @tanaka09 claimed that cells with more than $100$ atoms are needed while $32$ atom cells were found sufficient by @lazar04 for GaN and by @holec08 for w-AlN. To address this issue we plotted the individual spectra resulting from core hole being placed on various N sites, and sorted them according to the number of the nearest neighbours of each specie (in total 4 for the tetrahedrally coordinated wurtzite structure (Fig. \[fig:loc\_env\]a) and 6 for the octahedrally coordinated cubic structure (Fig. \[fig:loc\_env\]b,c)) surrounding the particular N site with the core hole. [The numbers of spectra corresponding to individual local environments (i.e. the nearest neighbour configuration) results from their real counts in the used supercells. Although configurations of the nearest neighbours of some N sites are the same, the higher-order nearest neighbour configurations differ which is why small variations between individual spectra labelled with the same local environment are obtained. The thick lines on top of each panels in Fig. \[fig:loc\_env\] were obtained by averaging all the curves underneath (their number is the same as the number of N sites in the supercell), and thus account for the statistical distribution of various local environments of N atoms.]{}
The graphs clearly demonstrate the huge differences between spectra depending on the local environment of the N site where the excitation takes place. At the same time one can see that almost doubling the number of atoms in the supercell (from 36 in Fig. \[fig:loc\_env\]b to 64 in Fig. \[fig:loc\_env\]c) does not alter the N K-edge significantly. The small changes might be due to having insufficiently big cell in the case of 36 atoms, but also could be due to a non-representative (i.e. not SQS-like) cell for the bigger structure which is suggested, e.g. by not having the 2Al, 4Ti local environment present. In summary, Fig. \[fig:loc\_env\] demonstrates that (i) the local environment influences the final shape of the edge much more drastically than the actual cell size (provided the cell is big enough to model the “randomness” of an alloy), and (ii) the similarity of the curves from Al-, Ti-, Ga-rich local neighbourhoods to those of pure AlN, TiN and GaN, respectively, gives some grounds for the interpolation approach (see section \[sec:shape\]).
Energy of the edge onset
------------------------
The edge onset energy is another important feature of the edge; for example, @mizoguchi03 predicted a range of $\approx2{\ensuremath\,\mathrm{eV}}$ for the N K-edge onset of AlN depending on the crystal structure thus having a potential to distinguish between these allotropes. It is, however, not straightforward to define the edge energy, in particular due to the [ambiguity]{} in the background subtraction as well as due to the background noise itself. Consequently, we have chosen the energy of the first inflection point above the edge threshold for the comparison between experiment and theory (the edge onset is about $2$–$4{\ensuremath\,\mathrm{eV}}$ below). It is not surprising that we obtained excellent agreement for the binary systems ($<0.1{\ensuremath\,\mathrm{eV}}$ for GaN and $\approx0.35{\ensuremath\,\mathrm{eV}}$ for AlN) since for these systems the edge onset energy was used as a fitting parameter for the core hole charge (see section \[binaries\]). However, both the experiment and the theory suggest that the energy of the inflection point does not vary too much with the composition. The variations within $0.2{\ensuremath\,\mathrm{eV}}$ (theory) and $0.4{\ensuremath\,\mathrm{eV}}$ (experiment) can be regarded as the accuracy of the present approach (due to, e.g. the background subtraction on the experimental side or the supercell design/size on the theoretical side).
A common method calculating the excitation energy is to calculate the difference between total energies of the initial ground state and the final state with a full core hole [see for example @mizoguchi03; @rashkova07; @tanaka09] which, in principle, follows the excitation process[^4]. This results in values of $384.5{\ensuremath\,\mathrm{eV}}$ and $368.2{\ensuremath\,\mathrm{eV}}$ for w-AlN and c-TiN, respectively, which are hugely underestimated as compared with the experimental values of $402{\ensuremath\,\mathrm{eV}}$ and $397{\ensuremath\,\mathrm{eV}}$. The reason for this is that the excited electron was put as the background charge (Fig. \[fig:excitation\]b) rather than in the unoccupied states (Fig. \[fig:excitation\]c). When excited to the unoccupied states values of $406.4{\ensuremath\,\mathrm{eV}}$ and $404.6{\ensuremath\,\mathrm{eV}}$ for the w-AlN and c-TiN are obtained which are much closer to the experimental values. @rashkova07 showed that a further improvement (towards the experimental values) could be obtained by performing spin polarised calculations. Nevertheless, the edge shapes as well as the energies of the initial core levels are almost identical using both approaches (background charge vs. valence band) thus yielding comparable results (except for the edge onset), which is in agreement with @hebert07.
When the edge onset is calculated as the total energy difference between the ground and full core hole states, its value is a given number without any degree of freedom for adjustments. The corresponding ELNES shape then should be calculated with exactly $0.5{\ensuremath\,\mathrm{e}}$ core hole [@paxton00]. This could be useful when estimating, e.g. ELNES of experimentally inaccessible phases. However, when one uses the core hole charge as a fitting parameter (as in this paper) then it is well justified that also the edge onset is not a unique number but instead a function of the core hole charge. For evaluation of the energy difference between the core state and the lowest unoccupied state, however, the approach with background charge is more appropriate since, for example, in the case of the conductive TiN it allows to get the energy of the lowest (originally) unoccupied state (compare Figs. \[fig:excitation\]b and c).
Conclusions
===========
In this paper we have demonstrated a semi-empirical approach for predictive calculations of the N K-edge ELNES of various classes of alloys (cubic vs. wurtzite, metallic vs. semiconducting). We used fractional core holes with charges, carefully adjusted according to the edge shape and the onset energy, to reproduce experimental spectra. Subsequently we utilised these to model the ELNES spectra of alloys using the special quasi-random supercells. We introduced core holes on all individual N sites, and by averaging these spectra we achieved a representative alloy spectrum. A comparison with the experimental measurements (for c-Ti$_{1-x}$Al$_x$N and w-Al$_x$Ga$_{1-x}$N systems) yielded an excellent agreement on both the edge shapes (including peak positions and relative intensities) as well as the edge onset energies. Finally, we related the individual peaks in the N K-edge ELNES to various interactions between cations and N atoms demonstrating that the decrease in intensity of the N K-edge structure $\approx3{\ensuremath\,\mathrm{eV}}$ above the edge onset reflects a weakening of the $sp^3d^2$ hybridisation with increasing Al content in Ti$_{1-x}$Al$_x$N.
Acknowledgements
================
Financial support by the START Program (Y371) of the Austrian Science Fund (FWF) and by the UK Engineering and Physical Sciences Research Council (EPSRC) is greatly acknowledged.
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[^1]: Those are the standard parameters. However, we checked their values for convergence.
[^2]: DFT is known to underestimate the values of band gap with respect to the experimental measurements.
[^3]: It is worth noting that the experimental spectra shown by @holec08 do not have subtracted background which, unfortunately, led to incorrect edge shapes and conclusions about the core hole charge.
[^4]: Still, this is not fully justified as DFT is a [*ground state*]{} theory (Fig. \[fig:excitation\]a).
|
---
author:
- |
B. K. Jain and Swapan Das\
Nuclear Physics Division, Bhabha Atomic Research Centre,\
Mumbai 400 085, India\
\
Bijoy Kundu\
Institute of Physics, Sachivalaya Marg,\
Bhubaneswar 751 005, India
title: 'Exploring vector meson masses in nuclear collisions [^1]'
---
psfig.tex plus 12pt minus 1pt plus 1pt 0.22in
PS. [postscript]{}
**Abstract**
The formalism developed earlier by us for the propagation of a resonance in the nuclear medium in proton-nucleus collisions has been modified to the case of vector boson production in heavy-ion collisions. The first part of the talk describes this formalism. The formalism includes coherently the contribution to the observed di-lepton production from the decay of a vector boson inside as well as outside the nuclear medium. The calculated invariant rho mass distributions are presented for the $\rho $-meson production using optical potentials estimated within the VDM and the resonance model.
In the second part of the talk we write a formalism for coherent rho production in proton nucleus collisions and explore the sensitivity of the (p,p$^\prime \rho ^0$) reaction cross section to medium mass modification of the rho meson.
Introduction
============
The modification of hadron masses in the nuclear medium is an issue of much interest currently [@hdms]. Since in QCD the hadrons are excitations of the vacuum, it is natural that these excitations can get affected by the proximity of other hadrons. Experimentally the masses of unstable hadrons are explored by producing them in nuclear reactions and then measuring the invariant masses of their leptonic decay products [@dre]. The medium modification of the unstable hadron from these data is, normally, inferred by adding incoherently the decay of the hadron inside and outside the nuclear medium [@incoh]. Recently formalisms have been developed by us [@jnku] and Boreskov et al. [@bor] for the propagation of resonances produced in proton-nucleus collisions. These formalisms incorporate the interaction of the resonance with the nuclear medium. The invariant mass spectrum of the measured decay products in these formalisms is obtained by adding coherently the contribution of the resonance decay inside and outside the nuclear medium. The formalism of Jain et al. [@jnku] also includes the interaction of the resonance decay products with the nuclear medium if the latter are hadrons. In the first part of the present talk we give the modification of our earlier formalism to heavy-ion collisions, and apply it to the propagation and decay of rho-meson. Our aim is to see as how the medium modification of $\rho $-meson shows up in the di-lepton invariant mass distribution using a proper quantum mechanical description for the rho propagation in the nucleus. We do not compare our calculated results with the existing experimental data on rho-meson production in high energy heavy-ion collisions, because these data contain contribution from several other di-lepton processes than that considered in the present paper.
In the second part of the talk we write a formalism for coherent rho production in proton nucleus collisions and explore the sensitivity of the (p,p$^\prime \rho ^0$) reaction cross section to medium mass modification of the rho meson.
Heavy-ion collisions
====================
Formalism
---------
Let us suppose that two heavy nuclei, one the projectile $A$ and another the target $B$, collide at high energies. We assume that one nucleon in the projectile and one in the target collide and a resonance $R$ is produced at the collision point. This resonance then moves along the beam direction, which is taken as the z-axis, and decays at some subsequent point. Denoting by ${\bf
r}$ the relative coordinate between the target and the projectile, and by ${\bf (r_{B}, r_{A})}$ the intrinsic coordinates of the target and projectile nucleons, respectively, the resonance coordinate is written as, $$\begin{aligned}
{\bf r_R} = {\bf r_A} + \frac {B} {A+B} {\bf r}.\end{aligned}$$ With this definition the ratio of the cross sections for the resonance production in AB and NN collisions, for an inclusive situation where the state of the (A+B) system is not identified, can be written as, $$\begin{aligned}
\frac {\Delta \sigma _R^{AB}}{\Delta \sigma _R^{NN}} =[K.F.] \int d {\bf b}
\int dz \int d {\bf r_A} \rho_A ({\bf r_A}) \rho_B ({\bf r + r_A})
|G({\bf r_R;k_R,\mu})|^2,
\label{sigma}\end{aligned}$$ where $\rho_x$ are the nuclear densities. \[K.F.\] is the relevant kinematical factor. The function $G({\bf r_R;k_R,\mu})$ physically gives the probability amplitude for finding the decay products of the resonance in the detector with the total momentum ${\bf k_R}$ and the invariant mass $\mu $, if the resonance is produced at a point ${\bf r_R}$ in the nucleus (for details see Ref. [@jnku; @bor]. In terms of the resonance propagator $G({\bf
r^{'}_R,r_R})$, the function $G({\bf r_R;k_R,\mu})$ is defined as $$\begin{aligned}
G({\bf r_R;k_R,\mu}) = \int d{\bf r^{'}_R } exp(-i {\bf k_R.r^{'}_R})
G({\bf r^{'}_R,r_R}),\end{aligned}$$ where $G({\bf r^{'}_R,r_R})$ satisfies $$\begin{aligned}
[\nabla^2 + E^2 -m^2_R +i \Gamma_R m_R -\Pi_R] G({\bf r^{'}_R,r_R})=
\delta ({\bf r^{'}_R-r_R}).\end{aligned}$$ Here $\Pi_R$ is the self energy of the resonance in the medium and $\Gamma_R$ is its free space decay width.
In the eikonal approximation we can write, $$\begin{aligned}
G({\bf r_R;k_R,\mu})=exp(-i {\bf k_R.r_R})\phi_R({\bf r_R;k_R,\mu}),\end{aligned}$$ where $\phi_R$ is a slowly varying modulating function. With this, and using the Eqs. (3,4), $\phi_R$ approximately works out to $$\begin{aligned}
\phi({\bf r_R;k_R,\mu}) &=& \frac {1} {2ik_R} \int dz^{'}_R exp[\frac {1}{2ik_R}
(\mu^2 -m^2_R+i\Gamma_R m_R)(z_R-z^{'}_R)] \nonumber \\
& &\times exp[\frac {-i}{v_R} \int ^{z'_R}_{z_R}
V_R (b_R,z{''}) d z^{''}_R].\end{aligned}$$ Here we have written $$\begin{aligned}
\Pi_R= 2 E_R V_R,\end{aligned}$$ where $V_R$ is the optical potential of the resonance, R, in the nuclear medium. In general, it is complex. Its real part, as we shall see later, is related to the mass shift of the resonance and the imaginary part gives the collision broadening of the resonance in the medium.
For a nucleus with a sharp surface, function $\phi({\bf
r_R;k_R,\mu})$ splits into a sum of two terms, one corresponding to the decay of the resonance inside the nucleus and another to the decay outside the nucleus, i.e. $$\begin{aligned}
\phi({\bf r_R;k_R,\mu}) = \phi _{in} ({\bf r_R}) + \phi_{out} ({\bf r_R})\end{aligned}$$ with $$\begin{aligned}
\phi_{in}({\bf r_R}) = \frac{1}{2ik_R} \int_{z_R}^{\sqrt (R^2-b^2)} d z'_R
\phi_R
({\bf b_R}; z_R,z'_R),\end{aligned}$$ and $$\begin{aligned}
\phi_{out}({\bf r_R}) = \frac{1}{2ik_R} \int_{\sqrt (R^2-b^2)}^{\infty} d z'_R
\phi_R
({\bf b_R}; z_R,z'_R).\end{aligned}$$ Here $$\begin{aligned}
\phi_R({\bf b_R}; z_R,z'_R) &=& exp[\frac {1}{2ik_R}
(\mu^2 -m^2_R+i\Gamma_R m_R)(z_R-z^{'}_R)] \nonumber \\
& & \times exp[\frac {-i}{v_R}
\int ^{z'_R}_{z_R}
V_R (b_R,z{''}_R) d z^{''}_R].\end{aligned}$$ After a little bit of manipulations, the final expressions for $\phi_{in}$ and $\phi_{out}$ work out to , $$\begin{aligned}
\phi_{in}({\bf r_R}; k_R,\mu) = \frac {G_0^*} {2 m_R} [1 - exp(\frac {i}
{v_R G_0^*} [L(b_R) - z_R])],
\label{phiin}\end{aligned}$$ $$\begin{aligned}
\phi_{out}({\bf r_R}; k_R,\mu) = \frac {G_0} {2 m_R} [ exp(\frac {i}
{v_R G_0^*} [L(b_R) - z_R])],
\label{phiout}\end{aligned}$$ where $v_R$ is the speed of the resonance and $ L (= \sqrt{(R^2
-b ^2_R)})$ is the length from the production point to the surface of the nucleus. $G_{0}$ and $G_{0}^{*}$ in Eqs. (12) and (13) are the free and the in-medium resonance propagators. Their forms are $$\begin{aligned}
G_0= \frac{2 m_R} {\mu^2 - m^2_R +i \Gamma _R m_R},\end{aligned}$$ $$\begin{aligned}
G_0^*= \frac{2 m_R} {\mu^2 - m^{*2}_R +i \Gamma _R^* m_R},\end{aligned}$$ with $$\begin{aligned}
m_R^*\approx& m_R + \frac{E_R}{m_R} U_R .\end{aligned}$$ $$\begin{aligned}
\Gamma ^* _R = \Gamma_R + \frac{E_R}{m_R} |2 W_R|.\end{aligned}$$
It may be mentioned that, for a nucleus with no sharp surface approximation the expression given in Eq. (6) can be used directly to evaluate the function $\phi({\bf r_R;k_R,\mu})$.
In the above we have written, $$\begin{aligned}
V_R= U_R + i W_R.\end{aligned}$$ These potentials, as given in Eqs. (16,17) give a measure of the mass and width modification of the resonance in the nuclear medium. Their values are an open question and a subject of much research internationally. In one approach they can be treated as completely unknown quantities and data on appropriate experiments can be used to extract their values. This exercise would be of use if the theoretical formalism used describes the reaction dynamics correctly and the data do not have much uncertainty. Alternatively, they can be estimated in a particular model and the ensuing values can be used to make an estimate of the cross section for the rho production. In literature, various efforts [@rhop; @kon] have been made to estimate $V_R$ using the high energy ansatz, i.e. $$\begin{aligned}
U_R = -\alpha [\frac{1}{2} v_R \sigma _ T^{RN} \rho_0]
\label{uurr}\end{aligned}$$ and $$\begin{aligned}
W_R= -[\frac{1}{2} v_R \sigma _ T^{RN} \rho_0],
\label{vvrr}\end{aligned}$$ where $\alpha$ is the ratio of the real to the imaginary part of the elementary RN scattering amplitude and $\sigma _ T^{RN}$ is the total cross section for it. $\rho_0$ is the typical nuclear density. A detailed calculation for the rho-meson has been done on these lines by Kondrayuk et al. [@kon] which give these potentials as a function of momentum. They use VDM at high enegies and resonance model at low energies to generate the $\rho
$N scattering parametrs. We have used these values for our calculations in the present paper. Some representative values of the self energies required in our calculations are given in Table 1.
[ccc]{} $v/c$ & U(MeV) & W(MeV)\
0.04 & -20.4 & -40.8\
0.6 & 37.9 & -50.6\
0.9 & 25.8 & -54.7\
Results and Discussion
----------------------
Examining Eqs. (9-11) $\phi({\bf r_R;k_R,\mu})$ in above we find that the cross sections for the decay of the resonance in the nucleus depends upon the length of the nuclear medium, the speed ($v_R$), free decay width and self energy of the resonance. To represent the effect of all these quantities, we present results for the decay of the rho-meson for different values of $v_R$ and for two sets of nuclear systems, viz. $Pb+Pb$ and $S+Au$. The free width of the rho-meson is taken equal to 150 MeV. The optical potentials, which we need at several $\rho $ momenta, are taken, as mentioned above, from Kondratyuk et al. [@kon]. Nuclear densities are taken from Ref. [@at].
Denoting the ratio $\frac {1}{[K.F.]} \frac {\Delta \sigma
_R^{AB}}{\Delta \sigma _R^{NN}}$ in Eq. (2) as $|\Phi|^2$, we plot $|\Phi|^2$ as a function of the invariant mass, $\mu$, of the decay products of the $\rho$ meson. Figures 1-4 show the invariant mass spectra of the $\rho$ meson in $Pb+Pb$ and $S+Au$ collisions at rho velocities of $0.6c$ and $0.9c$.
Here $c$ is the speed of light. The solid curve in all the figures gives the coherently summed cross-section from the decay of $\rho $-meson inside and outside the nuclear medium. The dashed curve gives the same added incoherently. The individual contributions corresponding to the inside and the outside decay are given by the dash-dot and dash-dot-dot curves respectively. We observe two things. One, the coherent and the incoherent cross-sections are different and second, this difference increases with the increase in the rho-meson speed. At 0.6c speed, while the coherent and incoherent curves differ only in the peak cross sections, at 0.9c speed their shape and peak cross sections both are different. We also observe that the difference is larger for the smaller system like S on Au. If we compare the mass shift seen in our calculations (Figs. 1-4) with those indicated in the high energy heavy ion collisions, our shifts are small and are in the opposite direction. To explore as what kind of optical potentials would produce a shift as large as those indicated experimentally we calculated $|\Phi|^2$ for Pb+Pb at 0.9c for three arbitrarily chosen values of $U_R$, viz. -40, -80 and -120 MeV. These results are shown in Fig. 5.
We find that only with -120 MeV value the distribution starts having features resembling those indicated in heavy-ion experiments. But this value, compared with the value around +25 MeV coming from the high energy ansatz of Kondratyuk et al. (see Table 1) is very large and is of opposite sign.
Proton-nucleus collisions
=========================
The cross section for a coherent rho production reaction, (p,p$^\prime \rho^0$), is given by $$d \sigma = [PS] S(m^2) <|T_{coh}|^2>,$$ where the phase-space factor, \[$PS$\], is written as $$[PS]=\frac{\pi m^2_p m_A}{(2\pi)^6} \frac{k_{p^\prime}k^2_{\rho}}
{k_p[k_\rho(E_i-E_{p^\prime})-{\bf (k_p-k_{p^\prime})}.{\hat{k_\rho}}E_\rho]}
dm^2dE_{p^\prime}d\Omega_{p^\prime}d\Omega_\rho.$$ $S(m^2)$ is the free space rho mass distribution function, which is given by $$S(m^2)=\frac{1}{\pi} \frac{m_\rho \Gamma_\rho}
{[(m^2-m_\rho^2)^2 + m_\rho^2 \Gamma_\rho^2]},$$ with $m_\rho =770$ MeV and $\Gamma_\rho =150$ MeV.
The T-matrix, $T_{coh}$, is given by $$T_{coh}=(\chi^{(-)*}_{\bf k_{p^\prime}}, \Psi^{(-)*}_{\bf k_\rho}
<p^\prime,\rho^0|{\cal L}_{\rho NN}|p>\chi^{(+)}_{\bf k_p}),$$ where $\chi $’s denote the distorted waves for the incoming and outgoing protons. However, in the energy region of interest for rho production the distortion effects are mainly absorptive. Therefore, the proton distorted waves in above can be replaced by plane waves for the present purpose.
$\Psi^{(-)*}_{\bf k_\rho}$ is the $\rho $-meson scattering wave function with asymptotic momentum ${\bf k_\rho }$. It has the form $$\Psi^{(-)*}_{\bf k_\rho }=e^{-i{\bf k_\rho \cdot r}}+ \Psi_{scat.}^*$$ In the absence of any dispersive nuclear distortion of p and p$^\prime$, the first term in this equation does not contribute to $T_{coh}$ because ${\bf k_\rho} \not = {\bf k_p}-{\bf
k_{p^\prime}}$. This in other words means that the rho meson produced at the proton vertex is off-shell. It can not be seen in the detector without incorporating the medium effects on it. $
\Psi_{scat. }$ is the part of the wave function which include these effects. If we associate a self energy $\Pi (=2\omega V$, where $V$ is the corresponding optical potential) with the $\rho
$-meson, $\Psi_{scat.}$ is given by $$\Psi^*_{scat.} = \chi^{(-)*}_{\bf k_\rho} V G_\rho (t),$$ where $\chi^{(-)*}_{\bf k_\rho}$ is the scattering solution of the potential $V$. $G_\rho(t)$ is the $\rho$-meson propagator, and is given by $$G_\rho (t) = -\frac{2\omega}{m^2_\rho -t-i\omega\Gamma_\rho}.$$ $t(=\omega^2-{\bf q}^2)$ is the four-momentum transfer squared to $\rho$-meson at the production vertex.
For the $\rho $ production Lagrangian in Eq. (24) we have taken $${\cal L}_{\rho NN}=\frac{fF(t)}{m_\rho}
N^{\dag} {\bf (\sigma x q) \tau}N \cdot
{\bf \rho},$$ with $\rho$NN coupling constant, $f$, equal to 7.81, and the off-shell extrapolation form factor as $$F(t)=\frac{\Lambda^2-m_\rho^2}{\Lambda^2-t},$$ with $\Lambda$= 2 GeV/c.
With the above formalism we calculate the $\rho $ production cross section for the $^{12}$C target nucleus. The only quantity required for the calculation is the description of the optical potential, $V$, of the $\rho$-meson. The values of the optical potential are fixed using the same prescription as given earlier for the heavy ion reactions. Some representative values required by us are given in Table 2. The radial shape of the optical potential is approximated by the radial density distribution of the $^{12}$C nucleus.
k$_\rho$ (MeV/c) 30 50 75 100 500
------------------ -------- -------- -------- -------- --------
U (MeV) -20.75 -9.27 -0.37 5.90 33.49
W (MeV) -26.51 -30.46 -34.79 -42.09 -46.73
: Optical potentials for certain values of the $\rho$ momentum.
In Fig. \[kk1\] we plot the calculated outgoing proton energy spectrum for p$^\prime $ going very near to the forward direction against the energy transfer $\omega $(=T$_p$-T$_{p^\prime}$). This energy transfer and the corresponding momentum transfer [**q**]{}(=${\bf k_p}$ - ${\bf k_{p^\prime}}$) are shared between the rho-meson and the recoiling nucleus through the interaction of the rho-meson with the target nucleus. The beam energy is taken equal to 1.5 GeV. We see in the figure that the calculated distribution has a broad peak. The peak cross section is around 0.34 $\mu$b/MeV.
In Fig. \[kk2\] we show the angular distribution of the above rho-mesons at the peak position in Fig \[kk1\]. It is observed that most of the rho-meson flux gets emitted in the forward direction only. Very little is seen beyond 15$^0$ or so.
Above results are given for a certain choice of the $\rho$-meson optical potential. However, they would be sensitive to the change in this potential. In Fig. \[jj1\] we have investigated this sensitivity. The optical potential for this purpose has been taken purely real, and different values for it are fixed through different mass-shifts of the $\rho$-meson in the medium using the relation given in Eq. (16). In Fig. \[jj1\] we show the calculated proton energy spectrum for $\Delta \mbox{m(=m-m}^*)$ taken equal to 50, 100 and 150 MeV. On x-axis, instead of $\omega
$, we have $\frac{\omega }{\Delta m}$. This is done because the essential parameter determining the dynamics of the rho-meson in the potential is likely to be the rho energy relative to the depth of the potential. We observe that
1. the magnitude of the cross sections increases with the increase in the strength of the potential.
2. In addition to the broad peak, we see a sharp peak in the small energy region of the rho-meson. The position of this peak on the $\frac{\omega} {\Delta m}$ scale is around 3.7 for $\Delta$m=100 and 150 MeV. For $\Delta $m=50 MeV, this peak is not seen in the results because by then the cross section becomes too small.
On examining the phase-shifts of the scattered wave function of rho-meson in the potential, we find that the sharp peak is like a shape elastic resonance seen in the elastic scattering experiments.
Of course, when the $\rho$-potential is made complex, as is in Fig. \[kk1\] the sharp peak disappears.
To summarize, we find that in proton scattering on nuclei a measurable cross section exists for $\rho $ meson production due to coherent effect of the target nucleus. The actual magnitude of the cross section depends sensitively on the strength of the $\rho $-meson optical potential, which is related to the rho-mass modification in the nuclear medium. The cross section increases with the increase in the potential strength. The angular distribution of the emitted rho-meson is such that most of them go in a forward cone of about 15$^0$. For a purely real potential a sharp peak appears in the proton energy spectrum in the region of the small rho-meson energy.\
\
The authors acknowledge many useful discussions they had with Shashi Phatak and A. B. Santra, and thank them for the same.
[100]{}
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[^1]: Talk delivered by B. K. Jain at National Seminar on Nuclear Physics, July 26-29, 1999, Institute of Physics, Bhubaneswar, India
|
---
author:
- Joscha Prochno
- Christoph Thäle
- Nicola Turchi
bibliography:
- 'chapter\_bibliography.bib'
title: |
Geometry of $\ell_p^n\,\text{-balls}$:\
Classical results and recent developments
---
[^1]
[^2]
Introduction
============
The geometry of the classical $\ell_p$ sequence spaces and their finite-dimensional versions is nowadays quite well understood. It has turned out that it is often a probabilistic point of view that shed (new) light on various geometric aspects and characteristics of these spaces and, in particular, their unit balls. In this survey we want to take a fresh look at some of the classical results and also on some more recent developments. The probabilistic approach to study the geometry of $\ell_p^n$-balls will be an asymptotic one. In particular, our aim is to demonstrate the usage of various limit theorems from probability theory, such as laws of large numbers, central limit theorems or large deviation principles. While the law of large numbers and the central limit theorem are already part of the – by now – classical theory (see, e.g., [@SS1991; @S1998; @S]), the latter approach via large deviation principles was introduced only recently in the theory of asymptotic geometric analysis by Gantert, Kim and Ramanan in [@GKR2017]. Most of the results we present below are not new and we shall always give precise references to the original papers. On the other hand, we provide detailed arguments at those places where we present generalizations of existing results that cannot be found somewhere else. For some of the other results the arguments are occasionally sketched as well.
Our text is structured as follows. In Section \[sec2:Preliminaries\] we collect some preliminary material. In particular, we introduce our notation (Section \[subsec21:Notation\]), the class of $\ell_p^n$-balls (Section \[subsec22:LpBalls\]), and also rephrase some background material on Grassmannian manifolds (Section \[subsec23:Grassmannians\]) and large deviation theory (Section \[subsec24:LargeDeviations\]). In Section \[sec3:PMonConvexBodies\] we introduce a number of probability measures that can be considered in connection with a convex body. We do this for the case of $\ell_p^n$-balls (Section \[subsec31:PMonLpBalls\]), but also more generally for symmetric convex bodies (Section \[subsec32:ConeMeasureSymmConvBody\]). The usage of the central limit theorem and the law of large numbers in the context of $\ell_p^n$-balls is demonstrated in Section \[sec4:CLTandLLN\]. We rephrase there some more classical results of Schechtman and Schmuckenschläger (Section \[subsec41:ClassiclSchechtSchmuck\]) and also consider some more recent developments (Section \[subsec42:RecentDevelopmentsMultivariateCLT+OutlookMatrix\]) including applications of the multivariate central limit theorem. We also take there an outlook to the matrix-valued set-up. The final Section \[sec5:LargeDeviations\] is concerned with various aspects of large deviations. We start with the classical concentration inequalities of Schechtman and Zinn (Section \[subsec51:ClassicalConcentrationIneq\]) and then describe large deviation principles for random projections of $\ell_p^n$-balls (Section \[subsec52:RecentDevelopmentsLDPs\]).
Preliminaries {#sec2:Preliminaries}
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In this section we shall provide the basics from both asymptotic geometric analysis and probability theory that are used throughout this survey article. The reader may also consult [@IsotropicConvexBodies; @AsymptoticGeometricAnalysisBookPart1; @DZ; @Kallenberg; @dH] for detailed expositions and additional explanations when necessary.
Notation {#subsec21:Notation}
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We shall denote with ${{\mathbb}{N}}=\{1,2,\ldots\} $, ${{\mathbb}{R}}$ and ${{\mathbb}{R}}^+ $ the set of natural, real and real non-negative numbers, respectively. Given $n\in{{\mathbb}{N}}$, let ${{\mathbb}{R}}^n$ be the $n\text{-dimensional} $ vector space on the real numbers, equipped with the standard inner product denoted by $\langle\cdot\,,\cdot\rangle$. We write ${\mathcal{B}}({{\mathbb}{R}}^n) $ for the $\sigma\text{-field}$ of all Borel subsets of ${{\mathbb}{R}}^n $. Analogously, for a subset $S\subseteq {{\mathbb}{R}}^n$, we denote by ${\mathcal{B}}(S)\coloneqq\{A\cap S:A\in{\mathcal{B}}({{\mathbb}{R}}^n)\}$ the corresponding trace $\sigma\text{-field}$ of ${\mathcal{B}}({{\mathbb}{R}}^n)$. Given a set $A$, we write $\# A$ for its cardinality. For a set $A\subseteq{{\mathbb}{R}}^n$, we shall write ${\boldsymbol{1}}_A\colon{{\mathbb}{R}}^n\to\{0,1\} $ for the indicator function of $A$. Given $A\in{\mathcal{B}}({{\mathbb}{R}}^n)$, we write ${\abs}{A} $ for its $n$-dimensional Lebesgue measure and frequently refer to this as the volume of $A$. Given sets $I\subseteq{{\mathbb}{R}}^+$ and $A\subseteq{{\mathbb}{R}}^n$, we define the set $I A $ as follows, $$IA\coloneqq\{rx\in{{\mathbb}{R}}^n:r\in I, x\in A \}.$$ If $I=\{r\}$, we also write $r A$ instead of $\{r\}A $. Note that ${{\mathbb}{R}}^+\! A$ is usually called the cone spanned by $A$.
We say that $K\subseteq{{\mathbb}{R}}^n$ is a convex body if it is a convex, compact set with non-empty interior. We indicate with $\partial K $ its boundary.
Fix now a probability space $(\Omega,\mathcal{F},{\mathbf{P}}) $. We will always assume that our random variables live in this probability space. Given a random variable $X\colon\Omega\to{{\mathbb}{R}}^n$ and a probability measure $Q $ on ${{\mathbb}{R}}^n$, we write $X\sim Q $ to indicate that $Q $ is the probability distribution of $X$, namely, for any $A\in{\mathcal{B}}({{\mathbb}{R}}^n)$, $${\mathbf{P}}(X\in A)=\int_{{{\mathbb}{R}}^n}{\boldsymbol{1}}_A(x){\mathop{}\!\mathrm{d}}Q(x).$$ We write $\operatorname{\mathbf{E}}$ and $\operatorname{\mathbf{Var}}$ to denote the expectation and the variance with respect to the probability ${\mathbf{P}}$, respectively.
Given a sequence of random variables $(X_n)_{n\in{{\mathbb}{N}}}$ and a random variable $Y$ we write $$X_n{\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{d}}Y, \qquad X_n{\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{{\mathbf{P}}}}Y, \qquad X_n{\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{\text{a.s.}}}Y,$$ to indicate that $(X_n)_{n\in{{\mathbb}{N}}}$ converges to $Y$ in distribution, probability or almost surely, respectively, as $n\to\infty$.
We write $N\sim \mathcal{N}(0,\Sigma)$ and say that $N$ is a centred Gaussian random vector in ${{\mathbb}{R}}^n$ with covariance matrix $ \Sigma$, i.e., its density function w.r.t. the Lebesgue measure is given by $$f(x)={\frac{1}{\sqrt{(2\pi)^n\det\Sigma}}}\exp\Bigl(-\frac{1}{2}\big\langle x, \Sigma^{-1} x\big\rangle\Bigr),\qquad x\in{{\mathbb}{R}}^n.$$ For $\alpha,\theta>0$, we write $X\sim\Gamma(\alpha,\vartheta) $ (resp. $X\sim\beta(\alpha,\vartheta)$) and say that $X$ has a Gamma distribution (resp. a Beta distribution) with parameters $\alpha$ and $\vartheta $ if the probability density function of $X$ w.r.t. to the Lebesgue measure is proportional to $x\mapsto x^{\alpha-1}e^{-\vartheta x}{\boldsymbol{1}}_{[0,\infty)}(x)$ (resp. $x\mapsto x^{\alpha-1}(1-x)^{\vartheta-1}{\boldsymbol{1}}_{[0,1]}(x)$). We also say that $X$ has a uniform distribution on $[0,1]$ if $X\sim\mathrm{Unif}([0,1])\coloneqq\beta(1,1)$ or an exponential distribution with parameter $1$ if $X\sim\exp(\vartheta)\coloneqq\Gamma(1,\vartheta)$.
The following properties of the aforementioned distributions are of interest and easy to verify by direct computation: $$\begin{aligned}
&\text{if }X\sim\Gamma(\alpha,\vartheta) \text{ and }Y\sim\Gamma(\tilde\alpha,\vartheta)\text{ are independent, then }\frac{X}{X+Y}\sim\beta(\alpha,\tilde\alpha)\,,\label{eq:Gammaproperty1}\\
&\text{if }X\sim\mathrm{Unif}([0,1]),\text{ then } X^k\sim\beta(1/k,1)\label{eq:Gammaproperty2}\,,
\end{aligned}$$ for any $\alpha,\tilde\alpha,\vartheta,k\in(0,\infty) $.
Given a real sequence $(a_n)_{n\in{{\mathbb}{N}}}$, we write $a_n\equiv a$ if $a_n=a$ for every $n\in{{\mathbb}{N}}$. If $(b_n)_{n\in{{\mathbb}{N}}}$ is a positive sequence, we write $a_n=\mathcal{O}(b_n)$ if there exists $C\in(0,\infty) $ such that $\abs{a_n}\le C b_n$ for every $n\in{{\mathbb}{N}}$, and $a_n=o(b_n)$ if $\lim_{n\to\infty} (a_n/b_n)=0 $.
The $\ell_p^n$-balls {#subsec22:LpBalls}
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For $n\in{{\mathbb}{N}}$, let $x=(x_1,\ldots,x_n)\in{{\mathbb}{R}}^n$ and define the $p\text{-norm}$ of $x$ via $$\label{eq:pnorm}
\norm{x}_p\coloneqq
\begin{dcases}
\Bigl(\sum_{i=1}^n\abs{x_i}^p\Bigr)^{1/p}&{\quad\text{if }\,}p\in[1,\infty),\\
\max_{1\leq i \leq n}\abs{x_i}&{\quad\text{if }\,}p=\infty.
\end{dcases}$$ The unit ball ${\mathbb{B}_p^n}$ and sphere ${\mathbb{S}_p^{n-1}}$ with respect to this norm are defined as $${\mathbb{B}_p^n}\coloneqq\{x\in{{\mathbb}{R}}^n:\pnorm{x}\le 1\}\qquad\text{and}\qquad {\mathbb{S}_p^{n-1}}\coloneqq\{x\in{{\mathbb}{R}}^n:\pnorm{x}= 1\}=\partial{\mathbb{B}_p^n}.$$ As usual, we shall write $\ell_p^n$ for the Banach space $({{\mathbb}{R}}^n,\pnorm{\cdot})$. The exact value of ${\abs{{\mathbb{B}_p^n}}}$ is known since Dirichlet [@D] and is given by $${\abs}{{\mathbb{B}_p^n}}=\frac{(2\Gamma(1+1/p))^n}{\Gamma(1+n/p)}.$$ The interested reader may consult [@P] for a modern computation. The volume-normalized ball shall be denoted by $\mathbb D_p^n$ and is given by $$\mathbb{D}_p^n=\frac{{\mathbb{B}_p^n}}{\abs{{\mathbb{B}_p^n}}^{1/n}}.$$ For convenience, in what follows we will use the convention that in the case $p=\infty$, $1/p\coloneqq0 $. It is worth noticing that the restriction on the domain of $p$ is due to the fact that an analogous definition of $\pnorm{\,\!\cdot\,\!}$ for $p<1$ does only result in a quasi-norm, meaning that the triangle inequality does not hold. As a consequence, ${\mathbb{B}_p^n}$ is convex if and only if $p\ge1$. Although a priori many arguments of this survey do not rely on $\pnorm{\cdot}$ being a norm, we restrict our presentation to the case $p\ge 1$, since it is necessary in some of the theorems.
Grassmannian manifolds {#subsec23:Grassmannians}
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The group of $(n\times n)\text{-orthogonal}$ matrices is denoted by $\mathbb O(n)$ and we let $\mathbb{SO}(n)$ be the subgroup of orthogonal $n\times n$ matrices with determinant $1$. As subsets of ${{\mathbb}{R}}^{n^2}$, $\mathbb{O}(n)$ and $\mathbb{SO}(n)$ can be equipped with the trace $\sigma\text{-field}$ of ${\mathcal{B}}({{\mathbb}{R}}^{n^2})$. Moreover, both compact groups $\mathbb O(n)$ and $\mathbb{SO}(n)$ carry a unique Haar probability measure which we denote by $\eta$ and $\tilde{\eta}$, respectively. Since $\mathbb{O}(n)$ consists of two copies of $\mathbb{SO}(n)$, the measure $\eta$ can easily be derived from $\tilde{\eta}$ and vice versa. Given $k\in\{0,1,\ldots,n\}$, we use the symbol ${\mathbb{G}}^n_k$ to denote the Grassmannian of $k$-dimensional linear subspaces of ${{\mathbb}{R}}^n$. We supply ${\mathbb{G}}_{k}^n$ with the metric $$d(E,F)\coloneqq\max\Bigl\{\adjustlimits{\sup}_{x\in B_E}{\inf}_{y\in B_F} \norm{x-y}_2,\adjustlimits{\sup}_{y\in B_F}{\inf}_{x\in B_E} \norm{x-y}_2\Bigr\},\qquad E,F\in {\mathbb{G}}_k^n,$$ where $B_E$ and $B_F$ stand for the Euclidean unit balls in $E$ and $F$, respectively. The Borel $\sigma$-field on ${\mathbb{G}}_k^n$ induced by this metric is denoted by ${\mathcal{B}}({\mathbb{G}}_k^{n})$ and we supply the arising measurable space ${\mathbb{G}}_k^n$ with the unique Haar probability measure $\eta_k^n$. It can be identified with the image measure of the Haar probability measure $\tilde{\eta}$ on $\mathbb{SO}(n)$ under the mapping $\mathbb{SO}(n)\to{\mathbb{G}}_k^n,\, T\mapsto TE_0$ with $E_0\coloneqq\mathrm{span}(\{e_1,\ldots,e_k\})$. Here, we write $e_1\coloneqq(1,0,\ldots,0),e_2\coloneqq(0,1,0,\ldots,0),\ldots,e_n\coloneqq(0,\ldots,0,1)\in{{\mathbb}{R}}^n$ for the standard orthonormal basis in ${{\mathbb}{R}}^n$ and $\mathrm{span}(\{e_1,\ldots,e_k\})\in {\mathbb{G}}_k^n$, $k\in\{1,\ldots,n\}$, for the $k$-dimensional linear subspace spanned by the first $k$ vectors of this basis.
Large deviation principles {#subsec24:LargeDeviations}
--------------------------
Consider a sequence $(X_n)_{n\in{{\mathbb}{N}}}$ of i.i.d integrable real random variables and let $$S_n\coloneqq{\frac{1}{n}}\sum_{i=1}^n X_i$$ be the empirical average of the first $n$ random variables of the sequence. It is well known that the law of large numbers provides the asymptotic behaviour of $S_n$, as $n$ tends to infinity. In particular, the strong law of large numbers says that $$S_n{\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{\text{a.s.}}}\operatorname{\mathbf{E}}[X_1].$$ If $X_1$ has also positive and finite variance, then the classical central limit theorem states that the fluctuations of $S_n$ around $\operatorname{\mathbf{E}}[X_1]$ are normal and of scale $1/\sqrt{n}$. More precisely, $$\sqrt{n}(S_n-\operatorname{\mathbf{E}}[X_1]){\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{d}}\mathcal{N}(0,\operatorname{\mathbf{Var}}[X_1]).$$ One of the important features of the central limit theorem is its universality, i.e., that the limiting distribution is normal independently of the precise distribution of the summands $X_1,X_2,\ldots$. This allows to have a good estimate for probabilities of the kind $${\mathbf{P}}(S_n>x),\qquad x\in{{\mathbb}{R}},$$ when $n$ is large, but fixed. However, such estimate can be quite imprecise if $x$ is much larger than $\operatorname{\mathbf{E}}[X_1]$. Moreover, it does not provide any rate of convergence for such tail probabilities as $n$ tends to infinity for fixed $x$.
In typical situations, if $S_n$ arises as a sum of $n$ independent random variables $X_1,\ldots,X_n$ with finite exponential moments, say, one has that $${\mathbf{P}}(S_n>x)\approx e^{-n {\mathcal{I}}(x)}, \qquad x>\operatorname{\mathbf{E}}[X_1]$$ if $n\to\infty$, where ${\mathcal{I}}$ is the so-called rate function. Here $\approx$ expresses an asymptotic equivalence up to sub-exponential functions of $n$. For concreteness, let us consider two examples. If ${\mathbf{P}}(X_1=1)={\mathbf{P}}(X_1=0)=1/2$, then $${\mathcal{I}}(x) =\begin{cases}
x\log x +(1-x)\log(1-x)+\log 2 & {\quad\text{if }\,}x\in[0,1],\\
+\infty &{\quad\text{otherwise}},
\end{cases}$$ which describes the upper large deviations. If on the other hand $X_1\sim\mathcal N(0,\sigma^2)$, then the rate function is given by $${\mathcal{I}}(x) = \frac{x^2}{2\sigma^2}, \qquad x\in{{\mathbb}{R}}.$$ Contrarily to the universality shown in the central limit theorem, these two examples already underline that the function ${\mathcal{I}}$ and thus the decay of the tail probabilities is much more sensitive and specific to the distribution of $X_1$.
The study of the atypical situations (in contrast to the typical ones described in the laws of large numbers and the central limit theorem) is called Large Deviations Theory. The concept expressed heuristically in the examples above can be made formal in the following way. Let ${\mathbf{X}}\coloneqq(X_n)_{n\in{{\mathbb}{N}}}$ be a sequence of random vectors taking values in ${{\mathbb}{R}}^d$. Further, let $s\colon{{\mathbb}{N}}\to[0,\infty]$ be a non-negative sequence such that $s(n)\uparrow \infty$ and assume that ${\mathcal{I}}\colon{{\mathbb}{R}}^d\to[0,\infty]$ is a lower semi-continuous function, i.e., all of its lower level sets $\{x\in{{\mathbb}{R}}^d:{\mathcal{I}}(x) \leq\ell \}$, $\ell\in[0,\infty]$, are closed. We say that ${\mathbf{X}}$ satisfies a large deviation principle (or simply LDP) with speed $s(n)$ and rate function ${\mathcal{I}}$ if and only if $$\label{eq:LDPdefinition}
-\inf_{x\in A^\circ}{\mathcal{I}}(x) \leq\,\liminf_{n\to\infty}\;\!{\frac{1}{s(n)}}\log {\mathbf{P}}(X_n\in A)
\leq\limsup_{n\to\infty}{\frac{1}{s(n)}}\log{\mathbf{P}}(X_n\in A)\leq-\inf_{x\in\overline{A}}{\mathcal{I}}(x)
$$ for all $A\in{\mathcal{B}}({{\mathbb}{R}}^d)$. Moreover, ${\mathcal{I}}$ is said to be a good rate function if all of its lower level sets are compact. The latter property is essential to guarantee the so-called exponential tightness of the sequence of measures.
The following result, known as Cramér’s Theorem, guarantees an LDP for the empirical average of a sequence of i.i.d. random vectors, provided that their common distribution is sufficiently nice (see, e.g. [@Kallenberg Theorem 27.5]).
\[thm:Cramér\] Let $(X_n)_{n\in{{\mathbb}{N}}}$ be a sequence of i.i.d. random vectors in ${{\mathbb}{R}}^d$ such that the cumulant generating function of $X_1$, $$\Lambda(u)\coloneqq \log\operatorname{\mathbf{E}}\bigl[\exp\scalar{X_1}{u}\bigr]\,,\qquad u\in{{\mathbb}{R}}^d,$$ is finite in a neighbourhood of $0\in{{\mathbb}{R}}^d$. Let $\mathbf{S}\coloneqq({\frac{1}{n}}\sum_{i=1}^n X_i)_{n\in{{\mathbb}{N}}}$ be the sequence of the sample means. Then $\mathbf{S}$ satisfies an LDP with speed $n$ and good rate function ${\mathcal{I}}=\Lambda^*$, where $$\Lambda^*(x)\coloneqq\sup_{u\in{{\mathbb}{R}}^d}\bigl(\scalar{x}{u}-\Lambda(u) \bigr),\qquad x\in{{\mathbb}{R}}^d,$$ is the Fenchel-Legendre transform of $\Lambda $.
Cramér’s Theorem is a fundamental tool that allows to prove an LDP if the random variables of interest can be transformed into a sum of independent random variables.
Sometimes there is the need to ‘transport’ a large deviation principle from one space to another by means of a continuous function. This can be done with a device known as the contraction principle and we refer to [@DZ Theorem 4.2.1] or [@Kallenberg Theorem 27.11(i)].
\[prop:contraction principle\] Let $d_1,d_2\in{{\mathbb}{N}}$ and let $F:{{\mathbb}{R}}^{d_1}\to{{\mathbb}{R}}^{d_2}$ be a continuous function. Further, let ${\mathbf{X}}\coloneqq(X_n)_{n\in{{\mathbb}{N}}}$ be a sequence of ${{\mathbb}{R}}^{d_1}\text{-valued}$ random vectors that satisfies an LDP with speed $s(n)$ and rate function ${\mathcal{I}}_{\mathbf{X}}$. Then the sequence $\mathbf{Y}\coloneqq(F(X_n))_{n\in{{\mathbb}{N}}}$ of ${{\mathbb}{R}}^{d_2}\text{-valued}$ random vectors satisfies an LDP with the same speed and with good rate function ${\mathcal{I}}_\mathbf{Y}={\mathcal{I}}_{\mathbf{X}}\circ F^{-1}$, i.e., ${\mathcal{I}}_\mathbf{Y}(y)\coloneqq\inf\{{\mathcal{I}}_{\mathbf{X}}(x):F(x)=y\}$, $y\in{{\mathbb}{R}}^{d_2}$, with the convention that ${\mathcal{I}}_\mathbf{Y}(y)=+\infty$ if $F^{-1}(\{y\})=\emptyset$.
While this form of the contraction principle is sufficient to analyse the large deviation behavior for $1$-dimensional random projections of $\ell_p^n$-balls, a refinement to treat the higher-dimensional cases is needed. To handle this situation, the classical contraction principle can be extended to allow a dependency on $n$ of the continuous function $F$. We refer the interested reader to [@DZ Corollary 4.2.21] for the precise statement.
Probability measures on convex bodies {#sec3:PMonConvexBodies}
=====================================
There is a variety of probability measures that can be defined on the family of $\ell_p^n$-balls or spheres. We shall present some of them and their key properties below.
Probability measures on an $\ell_p^n\,\text{-ball}$ {#subsec31:PMonLpBalls}
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One can endow ${\mathbb{B}_p^n}$ with a natural volume probability measure. This is defined as follows, $$\label{eq:puniform}
\nu_p^n(A)\coloneqq\frac{\abs{A\cap{\mathbb{B}_p^n}}}{\abs{{\mathbb{B}_p^n}}},$$ for any $A\in{\mathcal{B}}({{\mathbb}{R}}^n)$. We also refer to $\nu_p^n$ as the uniform distribution on ${\mathbb{B}_p^n}$.
As far as ${\mathbb{S}_p^{n-1}}$ is concerned, there are two probability measures that are of particular interest. The first is the so-called surface measure, which we denote by $\sigma_p^n$, and which is defined as the normalised $(n-1)$-dimensional Hausdorff measure. The second, $\mu_p^n$, is the so-called cone (probability) measure and is defined via $$\label{eq:pconemeasure}
\mu_p^n(A)\coloneqq\frac{{\abs}{[0,1]A}}{{\abs}{{\mathbb{B}_p^n}}},\qquad A\in{\mathcal{B}}({\mathbb{S}_p^{n-1}}).$$ In other words, $\mu_p^n(A)$ is the normalised volume of the cone that intersects ${\mathbb{S}_p^{n-1}}$ in $A$, intersected with ${\mathbb{B}_p^n}$. The cone measure is known to be the unique measure that satisfies the following polar integration formula for any integrable function $f$ on ${{\mathbb}{R}}^n$ (see, e.g., [@NR Proposition 1]) $$\label{eq:polarintegration}
\int_{{{\mathbb}{R}}^n} f(x){\mathop{}\!\mathrm{d}}x= n\,{\abs}{{\mathbb{B}_p^n}}\int_0^\infty r^{n-1}\int_{{\mathbb{S}_p^{n-1}}} f(rz){\mathop{}\!\mathrm{d}}\mu_p^n(z){\mathop{}\!\mathrm{d}}r.$$ In particular, whenever $f$ is $p\text{-radial}$, i.e., there exists a function $g$ defined on ${{\mathbb}{R}}^+$ such that $f(x)=g(\pnorm{x})$, then $$\label{eq:pradial}
\int_{{{\mathbb}{R}}^n}g(\pnorm{x}){\mathop{}\!\mathrm{d}}x= n\,{\abs}{{\mathbb{B}_p^n}} \int_0^\infty r^{n-1}g(r){\mathop{}\!\mathrm{d}}r.$$ The relation between $\sigma_p^n$ and $\mu_p^n$ has been deeply investigated. It is known, for example, that they coincide whenever $p\in\{1,2,\infty\}$ (see, e.g., [@RR]). In the other cases, Naor [@N] provided a bound on the total variation distance of these two measures.
Let $\sigma_p^n$ and $\mu_p^n$ be the surface probability and cone probability measure on ${\mathbb{S}_p^{n-1}}$, respectively. Then $$\begin{aligned}
d_{\mathrm{TV}}(\sigma_p^n,\mu_p^n)&\coloneqq\sup\Big\{\abs{\sigma_p^n(A)-\mu_p^n(A)}:A\in{\mathcal{B}}({\mathbb{S}_p^{n-1}})\Big\}
\le C\Bigl(1-{\frac{1}{p}}\Bigr)\abs[\bigg]{1-\frac{2}{p}}\frac{\sqrt{np}}{n+p},
\end{aligned}$$ where $C\in(0,\infty)$ is an absolute constant.
In particular, the above proposition ensures that for $p$ fixed, such a distance decreases to $0$ not slower than $n^{-1/2}$.
An important feature of the cone measure is described by the following probabilistic representation, due to Schechtman and Zinn [@SZ1] (independently discovered by Rachev and Rüschendorf [@RR]). We will below present a proof in a more general set-up.
\[thm:cone\] Let $n\in{{\mathbb}{N}}$ and $p\in[1,\infty]$. Let $(Z_i)_{i\in{{\mathbb}{N}}}$ be independent and $p\text{-generalized}$ Gaussian random variables, meaning absolutely continuous w.r.t. to the Lebesgue measure on ${{\mathbb}{R}}$ with density $$\label{eq:pGauss}
f_p(x)\coloneqq
\begin{dcases}
{\frac{1}{2p^{1/p}\Gamma(1+1/p)}}e^{-\abs{x}^p/p}&{\quad\text{if }\,}p\in[1,\infty)\,,\\
{\frac{1}{2}}{\boldsymbol{1}}_{[0,1]}(\abs{x})&{\quad\text{if }\,}p=\infty.
\end{dcases}$$ Consider the random vector $Z\coloneqq(Z_1,\ldots,Z_n)\in{{\mathbb}{R}}^n$ and let $U\sim{\rm Unif}([0,1])$ be independent of $Z_1,\ldots,Z_n$. Then $$\frac{Z}{\pnorm{Z}}\sim\mu_p^n\qquad\text{and}\qquad U^{1/n}\frac{Z}{\pnorm{Z}}\sim\nu_p^n.$$ Moreover, $Z/\pnorm{Z}$ is independent of $\pnorm{Z}$.
It is worth noticing that in [@SZ1] the density used by the authors for $Z_1$ is actually proportional to $ x\mapsto\exp(-\abs{x}^p)$. As will become clear later, this difference is irrelevant as far as the conclusion of the theorem is concerned.
Indeed, although the statement of reflects the focus of this survey on the $\ell_p^n\text{-balls} $ and the literature on the topic, its result is not strictly dependent on the particular choice of $f_p$ in . In fact, it is not even a prerogative of the $\ell_p^n\text{-balls} $, as subsequently explained in .
The cone measure on a symmetric convex body {#subsec32:ConeMeasureSymmConvBody}
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Consider a symmetric convex body $K\subseteq{{\mathbb}{R}}^n$, meaning that if $x\in K$ then also $-x\in K$. Define the functional $\norm{\cdot}_K\colon{{\mathbb}{R}}^n\to[0,\infty) $ by $$\norm{x}_K\coloneqq\inf\{r>0:x\in r K\}.$$ The functional $\norm{\cdot}_K$ is known as the Minkowski functional associated with $K$ and, under the aforementioned conditions on $K$, defines a norm on ${{\mathbb}{R}}^n $. We will also say that $\norm{x}_K $ is the $K\text{-norm}$ of the vector $x\in{{\mathbb}{R}}^n$. Whenever a function on ${{\mathbb}{R}}^n$ is dependent only on $\norm{\cdot}_K $, we say that it is a $K\text{-radial}$ function. Analogously, we call a probability measure ${K\text{-radial}}$ when its distribution function is ${K\text{-radial}}$. We will also write ${p\text{-radial}}$ meaning ${\mathbb{B}_p^n}\text{-radial} $.
In analogy with Equations and , it is possible to define a uniform probability measure $\nu_K$ on $K$ and a cone measure $\mu_K $ on $\partial K $, respectively, as $$\nu_K(A)\coloneqq\frac{{\abs}{A\cap K}}{{\abs}{K}}\qquad\text{and}\qquad
\mu_K(B)\coloneqq\frac{{\abs}{[0,1]B}}{{\abs}{K}},$$ for any $A\in{\mathcal{B}}({{\mathbb}{R}}^n)$ and $B\in{\mathcal{B}}(\partial K)$.
Note that $\mu_K$, as a ratio of volumes, is invariant under a simultaneous transformation of both the numerator and the denominator. In particular, for any $I\in{\mathcal{B}}({{\mathbb}{R}}^+)$, such that ${\abs}{I}>0 $, it holds $$\label{eq:anyI}
\mu_K(B)=\frac{{\abs}{IB}}{{\abs}{I\partial K}},$$ for any $B\in{\mathcal{B}}(\partial K)$ (note that $K=[0,1]\partial K$). This fact will be used in the proof of the following generalization of to arbitrary symmetric convex bodies.
\[prop:SZGeneralK\] Let $K\subseteq{{\mathbb}{R}}^n$ be a symmetric convex body. Suppose that there exists a continuous function $f\colon[0,\infty)\to[0,\infty)$ with the property $\int_{{{\mathbb}{R}}^n} f(\norm{x}_K){\mathop{}\!\mathrm{d}}x=1$ such that the distribution of a random vector $Z$ on ${{\mathbb}{R}}^n$ is given by $${\mathbf{P}}(Z\in A)=\int_A f(\norm{x}_K){\mathop{}\!\mathrm{d}}x,$$ for any $A\in{\mathcal{B}}({{\mathbb}{R}}^n)$. Also, let $U\sim{\rm Unif}([0,1])$ be independent of $Z$. Then, $$\label{eq:Kcone}
\frac{Z}{\norm{Z}_K}\sim\mu_K\qquad\text{and}\qquad U^{1/n}\frac{Z}{\norm{Z}_K}\sim\nu_K.$$ In addition, $Z/\norm{Z}_K$ is independent of $\norm{Z}_K$.
The proof of is based on the following polar integration formula, which generalizes . It says that for measurable functions $h:{{\mathbb}{R}}^n\to[0,\infty)$, $$\label{eq:PolarInterationGeneralK}
\int_{{{\mathbb}{R}}^n}h(x){\mathop{}\!\mathrm{d}}x = n{\abs}{K}\int_0^{\infty}r^{n-1}\int_{\partial K}h(rz)\,{\mathop{}\!\mathrm{d}}\mu_K(z){\mathop{}\!\mathrm{d}}r.$$ By the usual measure-theoretic standard procedure to prove it is sufficient to consider functions $h$ of the form $h(x)={\bf 1}_A(x)$, where $A=(a,b)E$ with $0<a<b<\infty$ and $E$ a Borel subset of $\partial K$. However, in this case, the left-hand side is just ${\abs}{A}$, while for the right-hand side we obtain, by definition of the cone measure $\mu_K$, $$\begin{aligned}
n{\abs}{K}\int_0^\infty r^{n-1}{\boldsymbol{1}}_{(a,b)}(r)\int_{\partial K}{\boldsymbol{1}}_{E}(z)\,{\mathop{}\!\mathrm{d}}\mu_K(z){\mathop{}\!\mathrm{d}}r = n{\abs}{K}\int_a^b r^{n-1}{\mathop{}\!\mathrm{d}}r\,\frac{{\abs}{[0,1]E}}{{\abs}{K}} = (b^n-a^n){\abs}{[0,1]E},\end{aligned}$$ which is clearly also equal to ${\abs}{A}$.
Let ${\varphi}:{{\mathbb}{R}}^n\to{{\mathbb}{R}}$ and $\psi:{{\mathbb}{R}}\to{{\mathbb}{R}}$ be non-negative measurable functions. Applying the polar integration formula, , yields $$\begin{split}
\operatorname{\mathbf{E}}\Bigl[{\varphi}\Bigl(\frac{Z}{\norm{Z}_K}\Bigr)\psi(\norm{Z}_K)\Bigr] &= \int_{{{\mathbb}{R}}^n}{\varphi}\Bigl(\frac{x}{\norm{x}_K}\Bigr)\psi(\norm{x}_K)f(\norm{x}_K){\mathop{}\!\mathrm{d}}x\\
&=n{\abs}{K}\int_0^\infty \psi(r)f(r)r^{n-1}{\mathop{}\!\mathrm{d}}r\,\int_{\partial K}{\varphi}(z){\mathop{}\!\mathrm{d}}\mu_K(z).
\end{split}$$ By the product structure of the last expression this first shows the independence of $Z/\norm{Z}_K$ and $\norm{Z}_K$. Moreover, choosing $\psi\equiv 1$ we see that $$\operatorname{\mathbf{E}}{\varphi}\Bigl(\frac{Z}{\norm{Z}_K}\Bigr) = n{\abs}{K}\int_0^\infty f(r)r^{n-1}{\mathop{}\!\mathrm{d}}r\,\int_{\partial K}{\varphi}(z){\mathop{}\!\mathrm{d}}\mu_K(z) = \int_{\partial K}{\varphi}(z){\mathop{}\!\mathrm{d}}\mu_K(z)$$ by definition of $f$. This proves that $Z/\norm{Z}_K\sim\mu_K$. That $U^{1/n}\frac{Z}{\norm{Z}_K}\sim\nu_K$ finally follows from the fact that $U^{1/n}\sim\beta(n,1)$, which has density $r\mapsto nr^{n-1}$ for $r\in(0,1)$.
The main reason why the theory treated in this survey is restricted to $\ell_p^n\text{-balls} $, and not to more general convex bodies $K$, is that $\ell_p^n\text{-balls} $ are a class of convex bodies whose Minkowski functional is of the form $$\label{eq:niceMinkowski}
\norm{x}_K=F\Bigl(\sum_{i=1}^n f_i(x_i)\Bigr)$$ for certain functions $f_1,\ldots, f_n$ and invertible positive function $F$. This is necessary for $Z$ to have independent coordinates. Indeed, in this case one can assign a joint density on $Z$ that factorizes into its components, like for example (omitting the normalizing constant), $$e^{-F^{-1}(\norm{x}_K)}=e^{-\sum_{i=1}^n f_i(x_i)}=\prod_{i=1}^{n}e^{- f_i(x_i)},$$ which ensures the independence of the coordinates $Z_i$ of $Z$.
Already for slightly more complicated convex bodies than $\ell_p^n\text{-balls} $, no longer holds. For example, considering the convex body defined as $$\mathbb{B}^2_{1,2}\coloneqq\{x\in{{\mathbb}{R}}^2: \abs{x_1}+x_2^2\le 1\}.$$ It can be computed that $\norm{x}_{\mathbb{B}^2_{1,2}}=\abs{x_1}/2+\sqrt{x_1^2/4+x_2^2}$, which is not of the form .
On the other hand, the coordinate-wise representation of the density of $Z$ in the precise form given by , is also convenient to explicitly compute the distribution of some functionals of $Z$, as we will see in the following section.
A different probabilistic representation for $p\text{-radial}$ probability measures
-----------------------------------------------------------------------------------
Another probabilistic representation for a $p\text{-symmetric}$ probability measure on ${\mathbb{B}_p^n}$ has been given by Barthe, Guédon, Mendelson and Naor [@BGMN] in the following way,
\[thm:W\]
Let $Z$ be a random vector in ${{\mathbb}{R}}^n$ defined as in $\Cref{thm:cone}$. Let $W$ be a non-negative random variable with probability distribution ${\mathbf{P}}_W$ and independent of $Z$. Then $$\frac{Z}{(\pnorm{Z}^p+W)^{1/p}}\sim {\mathbf{P}}_W(\{0\})\,\mu_p^n+\mathrm{H}_W(\cdot)\,\nu_p^n,$$ where $\mathrm{H}_W\colon{\mathbb{B}_p^n}\to{{\mathbb}{R}}$, $\mathrm{H}_W(x)=h(\pnorm{x})$, with $$h(r)={\frac{1}{\Gamma(1+n/p)(1-r^p)^{1+n/p}}}\int_{(0,\infty)} s^{n/p}e^{s r^p/(r^p-1)}{\mathop{}\!\mathrm{d}}{\mathbf{P}}_W(s).$$
Note that all the distributions obtainable from are $p\text{-radial}$, especially the $p\text{-norm}$ of $Z/(\pnorm{Z}^p+W)^{1/p}$ is $$R=\Bigl(\frac{\pnorm{Z}^p}{\pnorm{Z}^p+W}\Bigr)^{1/p}.$$ Moreover, some particular choices of $W$ in lead to interesting distributions:
1. When $W\equiv 0$ we recover the cone measure of ;
2. For $\alpha>0$, choosing $W\sim\Gamma(\alpha,1)$ results in the density proportional to $x\mapsto (1-\pnorm{x}^p)^{\alpha-1}$ for $\pnorm{x}\le1 $.
3. As a particular case of the previous one, when $W\sim\exp(1)=\Gamma(1,1) $, then $H_W\equiv 1$ and $$\frac{Z}{(\pnorm{Z}^p+W)^{1/p}}\sim\nu_p^n.$$ This is not in contrast with . Indeed, it is easy to compute that $$\pnorm{Z}^p\sim\Gamma(n/p,1).$$ In view of the properties and , this implies $$\frac{\pnorm{Z}^p}{\pnorm{Z}^p+W}\sim\beta(n/p,1)\sim U^{p/n}.$$ As a consequence of this fact, the orthogonal projection of the cone measure $\mu_p^{n+p}$ on $\partial \mathbb{B}_p^{n+p}$ onto the first n coordinates is $\nu_p^n $. Indeed, if $W=\sum_{i=n+1}^{n+p}\abs{Z_i}^p$, then $W\sim\exp(1)$, while $$\frac{Z}{(\pnorm{Z}^p+W)^{1/p}}=\frac{(Z_1,\ldots,Z_n)}{(\sum_{i=1}^{n+p}\abs{Z_i}^p)^{1/p}}$$ is the required projection. We refer to [@BGMN Corollaries 3-4] for more details in this direction.
Central limit theorems & Laws of large numbers {#sec4:CLTandLLN}
==============================================
The law of large numbers and the central limit theorem are arguably among the most prominent limit theorems in probability theory. Thanks to the probabilistic representation for the various geometric measures on $\ell_p^n$-balls described in Section \[subsec31:PMonLpBalls\], both of these limit theorems can successfully applied to deduce information about the geometry of $\ell_p^n$-balls. This – by now classical – approach will be described here, but we will also consider some more recent developments in this direction as well as several generalizations of known results.
Classical results: Limit theorems à la Schechtman-Schmuckenschläger {#subsec41:ClassiclSchechtSchmuck}
-------------------------------------------------------------------
The following result on the absolute moments of a $p\text{-generalized}$ Gaussian random variable is easy to derive by direct computation, and therefore we omit its proof, which the reader can find in [@KPT Lemma 4.1]
\[lem:MOMENTSpGenGAUSSIAN\] Let $p\in(0,\infty]$ and let $Z_0$ be a $p\text{-generalized}$ Gaussian random variable (i.e., its density is given by ). Then, for any $q\in[0,\infty]$, $$\operatorname{\mathbf{E}}\bigl[\abs{Z_0}^q\bigr]=
\begin{dcases}
\frac{p^{q/p}}{q+1}\frac{\Gamma\bigl(1+\frac{q+1}{p} \bigr)}{\Gamma\bigl(1+\frac{1}{p} \bigr)}=: M_p(q)&{\quad\text{if }\,}p<\infty, \\
{\frac{1}{q+1}}=: M_\infty(q)&{\quad\text{if }\,}p=\infty.
\end{dcases}$$
For convenience, we will also indicate $m_{p,q}\coloneqq M_p(q)^{1/q}$ and $$C_p(q,r)\coloneqq\operatorname{\mathbf{Cov}}(\abs{Z_0}^q,\abs{Z_0}^r)=M_p(q+r)-M_p(q)M_p(r).$$ We use the convention that $M_\infty(\infty)=C_\infty(\infty,\infty)=C_\infty(\infty,q)=0$. The next theorem is a version of the central limit theorem in [@S Proposition 2.4].
\[thm:clt-qnorm-1dim\] Let $0<p,q<\infty$, $p\neq q$ and $X\sim\nu_p^n$. Then $$\sqrt n\Bigl(n^{1/p-1/q}\frac{\norm{X}_q}{m_{p,q}}-1 \Bigr){\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{d}}N,$$ where $N\sim\mathcal{N}\bigl(0,\sigma^2_{p,q}\bigr)$ and $$\label{eq:sigma}
\begin{split}
\sigma^2_{p,q}&\coloneqq\frac{C_p(q,q)}{q^2 M_p(q)^2}-\frac{2C_p(p,q)}{pqM_p(q)}+\frac{C_p(p,p)}{p^2}
\end{split}$$
Note that, since $M_p(p)=1$, then $ \sigma^2_{p,p}=0$. In fact, in such a case $$\sqrt{n}(\pnorm{X}-1){\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{d}}0,$$ and a different normalization than $\sqrt{n} $ is needed to obtain a non-degenerate limit distribution. Moreover, $\sigma_{p,q}^2>0 $ whenever $p\neq q$.
For our purposes, it is convenient to define the following quantities $$k_{p,n}\coloneqq n^{1/p}{\abs}{{\mathbb{B}_p^n}}^{1/n}, \qquad k_{q,n}\coloneqq n^{1/q}{\abs}{\mathbb{B}_q^n}^{1/n}$$ and $$A_{p,q,n}\coloneqq\frac{k_{p,n}}{m_{p,q}k_{q,n}}.$$ It is easy to verify with Sterling’s approximation that, for any $p,q>0$, $A_{p,q,n}=A_{p,q}+\mathcal{O}(1/n)$ for $ A_{p,q}\in(0,\infty)$, as $n\to\infty$.
With this definition in mind, we exploit to prove a result on the volume of the intersection of $\ell_p^n\text{-balls}$. This can be regarded as a generalization of the main results in Schechtman and Schmuckenschläger [@SS1991], and Schmuckenschläger [@S1998; @S].
\[cor:intersectballs\] Let $0<p,q<\infty$ and $p\neq q$. Let $r\in[0,1]$ and $(t_n)_{n\in{{\mathbb}{N}}}\subseteq{{\mathbb}{R}}^+$ be such that $$\lim_{n\to\infty}\sqrt{n}(t_nA_{p,q}-1)=\Phi_{p,q}^{-1}(r),$$ where $\Phi_{p,q}:[-\infty,+\infty]\to[0,1] $ is the distribution function of $N\sim\mathcal{N}(0,\sigma^2_{p,q})$ and $\sigma^2_{p,q}$ is defined in , i.e., $$\Phi_{p,q}(x)\coloneqq {\frac{1}{\sqrt{2\pi\sigma_{p,q}^2}}}\int_{-\infty}^xe^{-s^2/(2\sigma_{p,q}^2)}{\mathop{}\!\mathrm{d}}s.$$ Then $$\lim_{n\to\infty}{\abs}[\big]{\mathbb{D}_p^n\cap t_n\mathbb{D}_q^n}=r.$$ In particular, when $t_n\equiv t$, then $$\lim_{n\to\infty}{\abs}[\big]{\mathbb{D}_p^n\cap t\,\mathbb{D}_q^n}=
\begin{cases}
0&{\quad\text{if }\,}t<1/A_{p,q},\\
1/2&{\quad\text{if }\,}t=1/A_{p,q},\\
1&{\quad\text{if }\,}t> 1/A_{p,q}.
\end{cases}$$
First of all, note that, since $A_{p,q,n}=A_{p,q}+\mathcal{O}(1/n)$, then $$\lim_{n\to\infty}\sqrt{n}(t_nA_{p,q,n}-1)=\lim_{n\to\infty}\sqrt{n}(t_nA_{p,q}-1),$$ provided that the latter exists in $[-\infty,\infty]$, as per assumption. In particular, taking the limit on both sides of the following equality, $${\mathbf{P}}\bigl(\norm{X}_q\le t_n k_{p,n}k_{q,n}^{-1}n^{1/p-1/q} \bigr)={\mathbf{P}}\bigl(\sqrt{n}(n^{1/p-1/q}m_{p,q}^{-1}\norm{X}_q-1)\le \sqrt{n}(t_n A_{p,q,n}-1) \bigr),$$ we get, because of , $$\lim_{n\to\infty}{\mathbf{P}}\bigl(\norm{X}_q\le t_n k_{p,n}k_{q,n}^{-1}n^{1/p-1/q} \bigr)
={\mathbf{P}}\bigl(N\le \Phi_{p,q}^{-1}(r)\bigr)
=r.$$ On the other hand, it is true that the following chain of equalities hold: $$\begin{split}
{\mathbf{P}}\bigl(\norm{X}_q\le t_n k_{p,n}k_{q,n}^{-1}n^{1/p-1/q} \bigr)
&=\frac{{\abs}{z\in{\mathbb{B}_p^n}:z\in t_n k_{p,n}k_{q,n}^{-1}n^{1/p-1/q}\mathbb{B}_q^n}}{{\abs}{{\mathbb{B}_p^n}}}\\
&={\abs}[\big]{z\in{\abs}{{\mathbb{B}_p^n}}^{-1/n}{\mathbb{B}_p^n}:z\in t_n k_{p,n}k_{q,n}^{-1}n^{1/p-1/q}{\abs}{{\mathbb{B}_p^n}}^{-1/n}\mathbb{B}_q^n}\\
&={\abs}{z\in \mathbb{D}_p^n:z\in t_n\mathbb{D}_q^n}\\
&={\abs}{ \mathbb{D}_p^n\cap t_n \mathbb{D}_q^n},
\end{split}$$ which concludes the main part of proof. For the last observation, note that for any $t$ constant, either $\sqrt{n}(t A_{p,q}-1)\equiv 0$ or it diverges.
Recent developments {#subsec42:RecentDevelopmentsMultivariateCLT+OutlookMatrix}
-------------------
### The multivariate CLT
We present here a multivariate central limit theorem that recently appeared in [@KPT]. It constitutes the multivariate generalization of . Similar to the classical results of Schechtman and Schmuckenschläger [@SS1991], and Schmuckenschläger [@S1998; @S] this was used to study intersections of (this time multiple) $\ell_p^n$-balls. In part $1.$, we replace the original assumption $X\sim\nu_p^n$ of [@KPT] to a more general one, that appears naturally from the proof. Part $2.$ is substantially different and cannot be generalized with the same assumption.
\[thm:multCLT\] Let $n,k\in{{\mathbb}{N}}$ and $p\in[1,\infty]$.
1. Let $X$ be a continuous $p\text{-radial}$ random vector in ${{\mathbb}{R}}^n$ such that $$\label{eq:Rconv}
\sqrt{n}\big(1-\pnorm{X}\big){\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{{\mathbf{P}}}}0.$$ Fix a $k\text{-tuple}$ $(q_1,\ldots,q_k)\in([1,\infty)\setminus\{p\})^k$. We have the multivariate central limit theorem $$\sqrt{n}\Bigl(n^{1/p-1/q_1}\frac{\norm{X}_{q_1}}{m_{p,q_1}}-1,\ldots, n^{1/p-1/q_k}\frac{\norm{X}_{q_k}}{m_{p,q_k}}-1\Bigr){\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{d}}N,$$ where $N=(N_1,\ldots,N_k)\sim\mathcal{N}(0,\Sigma)$, with covariance matrix $\Sigma=(c_{i,j})_{i,j=1}^k $ whose entries are given by $$\label{eq:multisigma}
c_{i,j}\coloneqq
\begin{dcases}
{\frac{1}{q_iq_j}}\biggl( \frac{\Gamma(\frac{1}{p})\Gamma(\frac{q_i+q_j+1}{p})}{\Gamma(\frac{q_i+1}{p})\Gamma(\frac{q_j+1}{p})}-1\biggr)-{\frac{1}{p}}&{\quad\text{if }\,}p<\infty,\\
{\frac{1}{q_i+q_j+1}}&{\quad\text{if }\,}p=\infty.
\end{dcases}$$
2. Let $X\sim\nu_p^n$. If $p<\infty$, then we have the non-central limit theorem $$\frac{n^{1/p}}{(p\log n)^{1/p-1}}\norm{X}_\infty-A_n^{(p)}{\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{d}}G,$$ where $$A_n^{(p)}\coloneqq p\log n -\frac{1-p}{p}\log (p\log n)+\log(p^{1/p}\Gamma(1+1/p))$$ and $G$ is a Gumbel random variable with distribution function ${{\mathbb}{R}}\ni t\mapsto e^{-e^{-t}}$.
Note that the assumptions of include the cases $X\sim\nu_p^n$ and $ X\sim\mu_p^n$. In fact, condition is just the quantitative version of the following concept: to have Gaussian fluctuations it is necessary that the bigger $n$ gets, the more the distribution of $X$ is concentrated in near $\partial{\mathbb{B}_p^n}$. It is relevant to note that also keeps open the possibility for a non-trivial limit distribution when rescaling $(1-\pnorm{X})$ with a sequence that grows faster than $\sqrt{n}$. This would yield a limit-theorem for $\pnorm{X}$. For example, when $X\sim\nu_p^n$, we already noted that $\pnorm{X}{\stackrel{d}{=}}U^{1/n}$, so that $$n(1-\pnorm{X}){\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{d}}E\sim\exp(1).$$ On the other hand, when $X\sim\mu_p^n$, then $1-\pnorm{X}\equiv 0 $ .
We only give a proof for the first part of the theorem, the second one can be found in [@KPT].
Let first $p\in[1,\infty)$. Consider a sequence of independent $p\text{-generalized}$ Gaussian random variables $(Z_j)_{j\in{{\mathbb}{N}}}$, also independent from every $X$. Set $ Z=(Z_1,\ldots,Z_n)$. For any $n\in {{\mathbb}{N}}$ and $i\in\{1,\ldots, k\}$, consider the random variables $$\xi_n^{(i)}\coloneqq{\frac{1}{\sqrt{n}}}\sum_{j=1}^n\bigl(\abs{Z_j}^{q_i}-M_p(q_i)\bigr)\qquad\text{and}\qquad\eta_n\coloneqq{\frac{1}{\sqrt{n}}}\sum_{j=1}^n\bigl(\abs{Z_j}^{p}-1\bigr).$$ According to the classical multivariate central limit theorem, we get $$(\xi_n^{(1)},\ldots,\xi_n^{(k)},\eta_n){\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{d}}(\xi^{(1)},\ldots,\xi^{(k)},\eta)\sim\mathcal{N}(0,\widetilde\Sigma)$$ with covariance matrix given by $$\widetilde\Sigma=\begin{pmatrix}
C_p(q_1,q_1) & \cdots & C_p(q_1,q_k) & C_p(q_1,p) \\
\vdots & \!\ddots & \vdots & \vdots \\
C_p(q_k,q_1) & \cdots & C_p(q_k,q_k) & C_p(q_k,p) \\
\vspace{-10pt} \\
C_p(p,q_1) & \cdots & C_p(p,q_k) & C_p(p,p)
\end{pmatrix}$$ Using and the aforementioned definitions we can write, for $i\in\{1,
\ldots,k\} $, $$\begin{split}
\norm{X}_{q_i}&{\stackrel{d}{=}}\frac{\pnorm{X}\norm{Z}_{q_i}}{\pnorm{Z}}\\
&=\pnorm{X}\frac{(nM_p(q_i)+\sqrt{n}\xi_n^{(i)})^{1/q_i}}{(n+\sqrt{n}\eta_n)^{1/p}}\\
&=\pnorm{X}\frac{(nM_p(q_i))^{1/q_i}}{n^{1/p}}F_i\Bigl( \frac{\xi_n^{(i)}}{\sqrt{n}},\frac{\eta_n}{\sqrt{n}}\Bigr)\\
&=\pnorm{X}\,n^{1/q_i-1/p}m_{p,q}F_i\Bigl( \frac{\xi_n^{(i)}}{\sqrt{n}},\frac{\eta_n}{\sqrt{n}}\Bigr)\\
&=(\pnorm{X}-1)n^{1/q_i-1/p}m_{p,q}F_i\Bigl( \frac{\xi_n^{(i)}}{\sqrt{n}},\frac{\eta_n}{\sqrt{n}}\Bigr)+n^{1/q_i-1/p}m_{p,q}F_i\Bigl( \frac{\xi_n^{(i)}}{\sqrt{n}},\frac{\eta_n}{\sqrt{n}}\Bigr)
\end{split}$$ where we defined the function $F_i\colon{{\mathbb}{R}}\times ({{\mathbb}{R}}\setminus \{-1\} )\to{{\mathbb}{R}}$ as $$F_i(x,y)\coloneqq\frac{(1+x/M_p(q_i))^{1/q_i}}{(1+y)^{1/p}}.$$ Note that $F_i$ is continuously differentiable around $(0,0)$ with Taylor expansion given by $$F_i(x,y)=1+\frac{x}{q_iM_p(q_i)}-\frac{y}{p}+\mathcal{O}(x^2+y^2).$$ Since, for the law of large numbers, $\xi_n^{(i)}/\sqrt{n}{\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{\text{a.s.}}}0 $ and $\eta_n/\sqrt{n}{\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{\text{a.s.}}}0 $, the previous equation means that there exists a random variable $C$, independent of $n$, such that $$\abs[\Big]{F_i\Bigl( \frac{\xi_n^{(i)}}{\sqrt{n}},\frac{\eta_n}{\sqrt{n}}\Bigr)-\Bigl(1+\frac{1}{q_iM_p(q_i)}\frac{\xi_n^{(i)}}{\sqrt{n}}-\frac{1}{p}\frac{\eta_n}{\sqrt{n}}\Bigr)}\le C\frac{(\xi_n^{(i)})^2+\eta_n^2}{n}.$$ In particular, $$\begin{split}
\sqrt{n}(&\pnorm{X}-1)\Bigl(1+\frac{1}{q_iM_p(q_i)}\frac{\xi_n^{(i)}}{\sqrt{n}}-\frac{1}{p}\frac{\eta_n}{\sqrt{n}}-C\frac{(\xi_n^{(i)})^2+\eta_n^2}{n}\Bigr)\\
&+\Bigl(\frac{1}{q_iM_p(q_i)}\xi_n^{(i)}-\frac{1}{p}\eta_n-C\frac{(\xi_n^{(i)})^2+\eta_n^2}{\sqrt{n}}\Bigr)\\
&\qquad\qquad\le\sqrt{n}\Bigl(n^{1/p-1/q_i}\frac{\norm{X}_{q_i}}{m_{p,q_i}}-1\Bigr)\\
&\qquad\qquad\qquad\qquad
\le\sqrt{n}(\pnorm{X}-1)\Bigl(1+\frac{1}{q_iM_p(q_i)}\frac{\xi_n^{(i)}}{\sqrt{n}}-\frac{1}{p}\frac{\eta_n}{\sqrt{n}}+C\frac{(\xi_n^{(i)})^2+\eta_n^2}{n}\Bigr)\\
&\quad\qquad\qquad\qquad\qquad\qquad+\Bigl(\frac{1}{q_iM_p(q_i)}\xi_n^{(i)}-\frac{1}{p}\eta_n+C\frac{(\xi_n^{(i)})^2+\eta_n^2}{\sqrt{n}}\Bigr)
\end{split}$$ Note that the first summand of both bounding expressions tends to $0$ in distribution by assumption , while the second converges in distribution to $\frac{1}{q_iM_p(q_i)}\xi^{(i)}-\frac{1}{p}\eta$. This implies that $$\sqrt{n}\Bigl(n^{1/p-1/q_i}\frac{\norm{X}_{q_i}}{m_{p,q_i}}-1\Bigr){\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{d}}\frac{1}{q_iM_p(q_i)}\xi^{(i)}-\frac{1}{p}\eta\eqqcolon N_i,$$ where $N_i$ is a centered Gaussian random variable. To obtain the final multivariate central limit theorem, we only have to compute the covariance matrix $ \Sigma$. For $\{i,j\}\subseteq\{1,\ldots,k \} $, its entries are given by $$\begin{split}
c_{i,j}&=\operatorname{\mathbf{Cov}}\Bigl(\frac{\xi^{(i)}}{q_iM_p(q_i)}-\frac{\eta}{p}, \frac{\xi^{(j)}}{q_jM_p(q_j)}-\frac{\eta}{p}\Bigr)\\
&=\frac{\operatorname{\mathbf{Cov}}(\xi^{(i)},\xi^{(j)})}{q_i q_jM_p(q_i)M_p(q_j)}-\frac{1}{p}\Bigl(\frac{\operatorname{\mathbf{Cov}}(\xi^{(i)},\eta)}{q_iM_p(q_i)}+\frac{\operatorname{\mathbf{Cov}}(\eta,\xi^{(j)})}{q_jM_p(q_j)}\Bigr)+\frac{\operatorname{\mathbf{Cov}}(\eta,\eta)}{p^2}\\
&=\frac{C_p(q_i,q_j)}{q_i q_jM_p(q_i)M_p(q_j)}-\frac{1}{p}\Bigl(\frac{C_p(q_i,p)}{q_iM_p(q_i)}+\frac{C_p(q_j,p)}{q_jM_p(q_j)}\Bigr)+\frac{C_p(p,p)}{p^2},
\end{split}$$ and this can be made explicit to get . The remaining case of $p=\infty$ can be repeated using the aforementioned conventions on the quantities $M_\infty$ and $C_\infty$.
From the proof is evident that in the case when $\sqrt{n}(\pnorm{X}-1)$ converges in distribution to a random variable $F$, independence yields, for every $i\in\{1,\ldots,k\}$, the convergence in distribution $$\sqrt{n}\Bigl(n^{1/p-1/q_i}\frac{\norm{X}_{q_i}}{m_{p,q_i}}-1\Bigr){\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{d}}F+ N_i$$ in which case the limiting random variable is not normal in general. Analogously, if there exists a sequence $(a_n)_{n\in{{\mathbb}{N}}}$, $a_n=o(\sqrt{n})$ and a random variable $F$ such that $$a_n(\pnorm{X}-1){\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{d}}F,$$ then the previous proof, with just a change of normalization, yields the limit theorem $$a_n\Bigl(n^{1/p-1/q}\frac{\norm{X}_{q}}{m_{p,q}}-1\Bigr){\xrightarrow[{\mathpalette{{\raisebox{3pt}{$}}$}}{n\to\infty}]{d}}F$$ for every $q\in[1,\infty)$, as $n\to\infty$.
In analogy to , one can prove in a similar way the following result concerning the simultaneous intersection of several dilated $\ell_p\text{-balls} $. In particular, we emphasize that the volume of the simultaneous intersection of three balls $\mathbb{D}_p^n\cap t_1\mathbb{D}_{q_1}^n\cap t_2\mathbb{D}_{q_2}$ is *not* equal to $1/4$ if these balls are in ‘critical’ position, as one might conjecture in view of .
Let $n,k\in{{\mathbb}{N}}$ and $p\in[1,\infty]$. Fix a $k\text{-uple}$ $(q_1,\ldots,q_k)\in([1,\infty)\setminus\{p\})^k$ . Let $t_1,
\ldots,t_k$ be positive constants and define the sets $I_\star\coloneqq\{i\in\{1,\ldots,k\}:A_{p,q_i}t_i\star 1\}$, where $\star$ is one of the symbols $>$, $=$ or $<$. Then, $$\lim_{n\to\infty}{\abs}{\mathbb{D}_p^n\cap t_1\mathbb{D}_{q_1}^n\cap\cdots\cap t_k\mathbb{D}_{q_k}^n}=
\begin{cases}
1 &{\quad\text{if }\,}\# I_>=k,\\
{\mathbf{P}}(N_i\le 0:i\in I_=) &{\quad\text{if }\,}\# I_=\ge 1\text{ and } \# I_<=0,\\
0 &{\quad\text{if }\,}\# I_<\ge 1,
\end{cases}$$ where $N=(N_1,\ldots,N_k)$ is as in .
### Outlook – the non-commutative setting {#subsubsec: outlook non-commutative schechtman-schmuckenschlaeger}
Very recently, Kabluchko, Prochno and Thäle obtained in [@KPT2018] a non-commutative analogue of the classical result by Schechtman and Schmuckenschläger [@SS1991]. Instead of considering the family of $\ell_p^n$-balls, they studied the volumetric properties of unit balls in classes of classical matrix ensembles.
More precisely, we let $\beta\in\{1,2,4\}$ and consider the collection $\mathscr H_n(\mathbb{F}_\beta)$ of all self-adjoint $n\times n$ matrices with entries from the (skew) field $\mathbb{F}_\beta$, where $\mathbb{F}_1={{\mathbb}{R}}$, $\mathbb{F}_2=\mathbb C$ or $\mathbb{F}_4=\mathbb H$ (the set of Hamiltonian quaternions). By $\lambda_1(A),\ldots,\lambda_n(A)$ we denote the (real) eigenvalues of a matrix $A$ from $\mathscr H_n(\mathbb{F}_\beta)$ and consider the following matrix analogues of the classical $\ell_p^n$-balls discussed above: $$\mathbb{B}_{p,\beta}^n\coloneqq\Bigl\{A\in \mathscr H_n(\mathbb{F}_\beta):\sum_{j=1}^n\abs{\lambda_j(A)}^p \leq 1\Bigr\},\qquad \beta \in\{1,2,4 \}\quad\text{and}\quad p\in[1,\infty],$$ where we interpret the sum in brackets as $\max\{\lambda_j(A):j=1,\ldots,n\}$ if $p=\infty$. As in the case of the classical $\ell_p^n$-balls we denote by $\mathbb D_{p,\beta}^n$, $\beta\in\{1,2,4\}$ the volume normalized versions of these matrix unit balls. Here the volume can be identified with the $(\beta\frac{n(n-1)}{2}+\beta n)$-dimensional Hausdorff measure on $\mathscr H_n(\mathbb{F}_\beta)$.
\[thm:ApplInto\] Let $1\leq p, q <\infty$ with $p\neq q$ and $\beta\in\{1,2,4\}$. Then $$\lim_{n\to\infty}{\abs}{\mathbb D^n_{p,\beta}\cap t\, \mathbb D^n_{q,\beta}}=
\begin{cases}
0 &{\quad\text{if }\,}t < e^{\frac{1}{2p} - \frac{1}{2q}} \bigl(\frac{2p}{p+q}\bigr)^{1/q},\\
1 &{\quad\text{if }\,}t > e^{\frac{1}{2p} - \frac{1}{2q}} \bigl(\frac{2p}{p+q}\bigr)^{1/q}\,.
\end{cases}$$
To obtain this result, one first needs to study the asymptotic volume of the unit balls of $\mathscr H_n(\mathbb{F}_\beta)$. This is done by resorting to ideas from the theory of logarithmic potentials with external fields. The second ingredient is a weak law of large numbers for the eigenvalues of a matrix chosen uniformly at random from $\mathbb B_{p,\beta}^n$. For details we refer the interested reader to [@KPT2018].
Large deviations vs. large deviation principles {#sec5:LargeDeviations}
===============================================
The final section is devoted to large deviations and large deviation principles for geometric characteristics of $\ell_p^n$-balls. We start by presenting some classical results on large deviations related to the geometry of $\ell_p^n$-balls due to Schechtman and Zinn. Its LDP counterpart has entered the stage of asymptotic geometry analysis only recently in [@KPT]. We then continue by presenting a large deviation principle for $1$-dimensional random projections of $\ell_p^n$-balls of Gantert, Kim and Ramanan [@GKR2017]. Finally, we present a similar result for higher-dimensional projections as well.
Classical results: Large deviations à la Schechtman-Zinn {#subsec51:ClassicalConcentrationIneq}
--------------------------------------------------------
We start by rephrasing the large deviation inequality of Schechtman and Zinn [@SZ1]. It is concerned with the $\ell_q$-norm of a random vector in an $\ell_p^n$-balls. The proof that we present follows the argument of [@N].
\[thm:SZInequality\] Let $1\le p< q\le\infty$ and $X\sim\nu_p^n$ or $X\sim\mu_p^n$. Then there exists a constant $c\in(0,\infty)$, depending only on $p$ and $q$, such that $${\mathbf{P}}(n^{1/p-1/q}\norm{X}_q>z)\le\exp(-c\,n^{p/q} z^p),$$ for every $z>1/c$.
We sketch the proof for the case that $X\sim\mu_p^n$. Let $Z_1,\ldots,Z_n$ be $p\text{-generalized}$ Gaussian random variables and put $S_r\coloneqq\abs{Z_1}^r+\ldots+\abs{Z_n}^r$ for $r\geq 1$. Now observe that by the exponential Markov inequality and , for $t>0$, $$\begin{aligned}
{\mathbf{P}}(n^{1/p-1/q}\norm{X}_q>z) &={\mathbf{P}}\Bigl(\frac{S_q^{p/q}}{S_p}>\frac{z^p}{n^{1-p/q}}\Bigr)\\ &\leq\exp\Bigl(-\frac{tz^p}{n^{1-p/q}}\Bigr)\operatorname{\mathbf{E}}\exp\Bigl(t\frac{S_q^{p/q}}{S_p}\Bigr)\\
&\leq\exp\Bigl(-\frac{tz^p}{n^{1-p/q}}\Bigr)\operatorname{\mathbf{E}}\exp\Bigl(t\frac{S_q^{p/q}}{\operatorname{\mathbf{E}}S_p}\Bigr),\end{aligned}$$ where we also used the independence property in in the last step. Next, we observe that $\operatorname{\mathbf{E}}S_p=n$ by . Moreover from [@N Corollary 3] it is known that there exists a constant $c\in(0,\infty)$ only depending on $p$ and $q$ such that $$\operatorname{\mathbf{E}}\exp\bigl(tS_q^{p/q}\bigr) \leq n^{1-p/q}\bigl(1-ct\bigr)^{-n^{p/q}}$$ as long as $0<t<1/c$. Thus, choosing $t=\frac{n}{c}-\frac{n}{z^p}$ we arrive at $${\mathbf{P}}(n^{1/p-1/q}\norm{X}_q>z) \leq n^{1-p/q}\Bigl(\frac{ez^p}{c}\Bigr)^{n^{p/q}}\exp(-cn^{p/q}z^p).$$ This implies the result.
Recent developments {#subsec52:RecentDevelopmentsLDPs}
-------------------
### The LDP counterpart to Schechtman-Zinn
After having presented the classical Schechtman-Zinn large deviation inequality, we turn now to a LDP counterpart. The next result is a summary of the results presented in from [@KPT Theorems from 1.2 to 1.5]. The speed and the rate function in its part 4 resembles the right hand side of the inequality in .
Let $n\in{{\mathbb}{N}}$, $p\in[1,\infty] $, $q\in[1,\infty)$ and $X\sim\nu_p^n$. Define the sequence $$\norm{{\mathbf{X}}}\coloneqq(n^{1/p-1/q}\qnorm{X})_{n\in{{\mathbb}{N}}}.$$
1. If $q<p<\infty$, then $\norm{{\mathbf{X}}}$ satisfies an LDP with speed $n$ and good rate function $${\mathcal{I}}_{\norm{{\mathbf{X}}}}(x)=\begin{cases}
\inf\{ {\mathcal{I}}_1(x_1)+{\mathcal{I}}_2(x_2):x=x_1x_2, x_1\ge 0, x_2\ge 0\}&{\quad\text{if }\,}x\ge 0,\\
+\infty&{\quad\text{otherwise}}.
\end{cases}$$ Here $$\label{eq:rate1}
{\mathcal{I}}_1(x)=\begin{cases}
-\log(x)&{\quad\text{if }\,}x\in(0,1],\\
+\infty&{\quad\text{otherwise}},
\end{cases}$$ and $${\mathcal{I}}_2(x)=\begin{cases}
\inf\{\Lambda^*(y,z):x=y^{1/q}z^{-1/p}, y\ge 0, z\ge 0 \}&{\quad\text{if }\,}x\ge 0\\
+\infty&{\quad\text{otherwise}},
\end{cases}$$ where $\Lambda^* $ is the Fenchel-Legendre transform of the function $$\Lambda(t_1,t_2)\coloneqq\log\int_0^{+\infty} {\frac{1}{p^{1/p}\Gamma(1+1/p)}}e^{t_1 s^q+(t_2-1/p) s^p }{\mathop{}\!\mathrm{d}}s,\qquad (t_1,t_2)\in{{\mathbb}{R}}\times\Bigl(-\infty,{\frac{1}{p}}\Bigr).$$
2. If $q<p=\infty$, then $\norm{{\mathbf{X}}}$ satisfies an LDP with speed $n$ and good rate function $${\mathcal{I}}_{\norm{{\mathbf{X}}}}(x)=\begin{cases}
\Psi^*(x)&{\quad\text{if }\,}x\ge 0,\\
+\infty&{\quad\text{otherwise}},
\end{cases}$$ where $\Psi^*$ is the Fenchel-Legendre transform of the function $$\Psi(t)\coloneqq\int_0^1 e^{t s^q}{\mathop{}\!\mathrm{d}}s,\qquad t\in{{\mathbb}{R}}.$$
3. If $p=q$, then $\norm{{\mathbf{X}}}$ satisfies an LDP with speed $n$ and good rate function ${\mathcal{I}}_1 $ defined in .
4. If $p<q$, then $\norm{{\mathbf{X}}} $ satisfies an LDP with speed $n^{p/q}$ and good rate function $${\mathcal{I}}_{\norm{{\mathbf{X}}}}(x)=\begin{dcases}
{\frac{1}{p}}\bigl(x^q-m_{p,q}^q\bigr)^{p/q}&{\quad\text{if }\,}x\ge m_{p,q},\\
+\infty&{\quad\text{otherwise}}.
\end{dcases}$$
### LDPs for projections of $\ell_p^n$-balls – $1$-dimensional projections
We turn now to a different type of large deviation principles. More precisely, we consider random projections of points uniformly distributed in an $\ell_p^n$-ball or distributed according to the corresponding cone probability measure onto a uniform random direction. The following result is a summary of from [@GKR2017 Theorems 2.2,2.3]. The proof of the first part follows rather directly from Cramér’s theorem () and the contraction principle (), the second part is based on large deviation theory for sums of stretched exponentials.
Let $n\in{{\mathbb}{N}}$ and $p\in[1,\infty) $. Let $X\sim\nu_p^n $ or $X\sim\mu_p^n $ and $\Theta\sim\sigma_2^n $ be independent random vectors. Consider the sequence $${\mathbf{W}}\coloneqq(n^{1/p-1/2}\scalar{X}{\Theta})_{n\in{{\mathbb}{N}}}.$$
1. If $p\ge 2$, then $ {\mathbf{W}}$ satisfies an LDP with speed $n$ and good rate function $${\mathcal{I}}_{\mathbf{W}}(w)=\inf\{\Phi^*(\tau_0,\tau_1,\tau_2):w=\tau_0^{-1/2}\tau_1^{\vphantom{1/2}}\tau_2^{-1/p}, \tau_0>0, \tau_1\in{{\mathbb}{R}}, \tau_2>0\},$$ where $\Phi^* $ is the Fenchel-Legendre transform of $$\Phi(t_0,t_1,t_2)\coloneqq\log\int_{{\mathbb}{R}}\int_{{\mathbb}{R}}e^{t_0 z^2+t_1 zy+t_2\abs{z}^p }f_2(z)f_p(y){\mathop{}\!\mathrm{d}}z{\mathop{}\!\mathrm{d}}y,\qquad t_0,t_1,t_2\in{{\mathbb}{R}}.$$
2. If $p< 2$, then $ {\mathbf{W}}$ satisfies an LDP with speed $n^{2p/(2+p)}$ and good rate function $${\mathcal{I}}_{\mathbf{W}}(w)=\frac{2+p}{2p}\abs{w}^{2p/(2+p)}.$$
Let us sketch the proof for the case that $p>2$, by leaving out any technical details. For this, let $Z_1,\ldots,Z_n$ be $p$-generalized Gaussian random variables, $G_1,\ldots,G_n$ be Gaussian random variables and $U$ be a uniform random variable over $[0,1]$. Also assume that all the aforementioned random variables are independent. Also put $Z\coloneqq(Z_1,\ldots,Z_n)$ and $G\coloneqq(G_1,\ldots,G_n)$. When $X\sim\mu_p^n$, by , we can state that for each $n\in{{\mathbb}{N}}$ the target random variable $n^{1/p-1/2}\scalar{X}{\Theta}$ has the same distribution as $$\label{eq:coneX}
n^{1/p-1/2}\frac{\sum\limits_{i=1}^nG_iZ_i}{\norm{G}_2\pnorm{Z}} = \frac{{\frac{1}{ n}}\sum\limits_{i=1}^nG_iZ_i}{\Bigl( {\frac{1}{n}}\sum\limits_{i=1}^n\abs{G_i}^2\Bigr)^{1/2}\Bigl(\frac{1} {n}\sum\limits_{i=1}^n\abs{Z_i}^p\Bigr)^{1/p}}.$$ Note that $\Phi$ is finite whenever $p<2$, $t_0<1/2$, $t_1\in{{\mathbb}{R}}$ and $t_2<1/p$. Then, Cramér’s theorem () shows that the ${{\mathbb}{R}}^3\text{-valued}$ sum $${\frac{1}{n}}\sum_{i=1}^n\bigl(\abs{G_i}^2,G_iZ_i,\abs{Z_i}^p\bigr)$$ satisfies an LDP with speed $n$ and rate function $\Phi^*$. Applying the contraction principle () to the function $F(x,y,z)=x^{-1/2}yz^{-1/p}$ yields the LDP for ${\mathbf{W}}$ with speed $n$ and the desired rate function ${\mathcal{I}}_{\mathbf{W}}$. Once the LDP is proven for the cone measure, it can be pushed to the case of the uniform measure. By , multiplying the expression in by $U^{1/n}$, we obtain a random variable distributed according to $ \nu_p^n$. It is proven in [@GKR2017 Lemma 3.2] that multiplying by $U^{1/n}$ every element of the sequence $\mathbf{W}$, we obtain a new sequence of random variables that also satisfies an LDP with the same speed and the same rate function as $\mathbf{W}$. On the other hand, when $p<2$, $\Phi(t_0,t_1,t_2)=\infty$ for any $t_1\neq 0$, hence suggesting that in this case the LDP could only occur at a lower speed than $n$.
### LDPs for projections of $\ell_p^n$-balls – the Grassmannian setting
Finally, let us discuss projections to higher dimensional subspaces, generalizing thereby the set-up from the previous section. We adopt the Grassmannian setting and consider the $2$-norm of the projection to a uniformly distributed random subspace in the Grassmannian $\mathbb{G}_n^k$ of $k$-dimensional subspaces of ${{\mathbb}{R}}^n$ of a point uniformly distributed in the $\ell_p^n$-unit ball. Since we are interested in the asymptotic regime where $n\to\infty$, we also allow the subspace dimension $k$ to vary with $n$. However, in order to keep our notation transparent, we shall nevertheless write $k$ instead of $k(n)$. The next result is the collection of [@APT Theorems 1.1,1.2].
Let $n\in{{\mathbb}{N}}$. Fix $p\in[1,\infty]$ and a sequence $k=k(n)\in\{1,\ldots,n-1\}$ such that the limit $\lambda\coloneqq\lim_{n\to\infty} (k/n)$ exists. Let $P_E X$ be the orthogonal projection of a random vector $X\sim\nu_p^n$ onto a random independent linear subspace $E\sim\eta_k^n$. Consider the sequence $${\boldsymbol{\norm{P_EX}}}\coloneqq(n^{1/p-1/2}\norm{P_E X }_2)_{n\in{{\mathbb}{N}}}.$$
1. If $p\ge 2$, then ${\boldsymbol{\norm{P_EX}}}$ satisfies an LDP with speed $n$ and good rate function $${\mathcal{I}}_{{\boldsymbol{\norm{P_EX}}}}(y)\coloneqq\begin{dcases}
\inf_{x> y}\Big[{\frac{\lambda}{2}}\log\Bigl(\frac{\lambda x^2}{ y^2}\Bigr)+\frac{1-\lambda}{2}\log\Bigl(\frac{1-\lambda}{ 1-y^2x^{-2}}\Bigr)+\mathcal{J}_p(x)\Bigr] &\!\!\!\!{\quad\text{if }\,}y>0,\\
\mathcal{J}_p(0) &\!\!\!\!{\quad\text{if }\,}y=0,\, \lambda\in(0,1],\\
\inf\limits_{x\geq 0}\mathcal{J}_p(x) &\!\!\!\!{\quad\text{if }\,}y=0,\, \lambda=0,\\
+\infty &\!\!\!\!{\quad\text{if }\,}y<0\,,
\end{dcases}$$ where we use the convention $0\log 0\coloneqq 0$ and for $p\neq\infty$ we have $$\mathcal{J}_p(y)\coloneqq\inf_{\substack{x_1, x_2>0\\ x_1^{1/2}x_2^{-1/p}=y}}\mathcal{I}_p^*(x_1,x_2),\qquad y\in{{\mathbb}{R}}\,,$$ and $\mathcal{I}_p^*(x_1,x_2)$ is the Fenchel-Legendre transform of $${\mathcal{I}}_p(t_1,t_2)\coloneqq\log\int_{{{\mathbb}{R}}}e^{t_1x^2+t_2\abs{x}^p}f_p(x){\mathop{}\!\mathrm{d}}x,\qquad (t_1,t_2)\in{{\mathbb}{R}}\times\Bigl(-\infty,{\frac{1}{p}}\Bigr).$$ For $p=\infty$, we write $\mathcal{J}_\infty(y)\coloneqq{\mathcal{I}}_\infty^*(y^2)$ with ${\mathcal{I}}_\infty^*$ being the Fenchel-Legendre transform of ${\mathcal{I}}_\infty(t)\coloneqq\log\int_0^1e^{tx^2}{\mathop{}\!\mathrm{d}}x$.
2. If $p<2$ and $\lambda>0$, then ${\boldsymbol{\norm{P_EX}}}$ satisfies and LDP with speed $n^{p/2}$ and good rate function $${\mathcal{I}}_{{\boldsymbol{\norm{P_EX}}}}(y)\coloneqq\begin{dcases}
{\frac{1}{p}}\Bigl(\frac{y^2}{\lambda}-m\Bigr)^{p/2}&{\quad\text{if }\,}y\ge\sqrt{\lambda m_p}\,,\\
+\infty &{\quad\text{otherwise}},
\end{dcases}$$ where $m_p\coloneqq p^{p/2}\Gamma(1+3/p)/(3\Gamma(1+1/p))$.
Let us emphasize that the proof of this theorem is in some sense similar to its $1$-dimensional counterpart that we have discussed in the previous section. However, there are a number of technicalities that need to be overcome when projections to high-dimensional subspaces are considered. Among others, one needs a new probabilistic representation of the target random variables. In fact, the previous theorem heavily relies on the following probabilistic representation, proved in [@APT Theorem 3.1] for the case $X\sim\nu_n^p$. We shall give a proof here for a more general set-up, which might be of independent interest.
Let $n\in{{\mathbb}{N}}$, $k\in\{1,\ldots,n \}$ and $p\in[1,\infty]$. Let $X$ be a continuous $p\text{-radial}$ random vector in ${{\mathbb}{R}}^n$ and $E\sim\eta_k^n$ be a random $k\text{-dimensional}$ linear subspace. Let $Z=(Z_1,\ldots,Z_n)$ and $G=(G_1,\ldots,G_n)$ having i.i.d. coordinates, distributed according to the densities $f_p$ and $f_2$, respectively. Moreover, let $X$, $E$, $Z$ and $G$ be independent. Then $$\norm{P_E X}_2{\stackrel{d}{=}}\pnorm{X}\frac{\norm{Z}_2}{\pnorm{Z} }\frac{\norm{(G_1,\ldots,G_k)}_2}{\norm{G}_2}.$$
Fix a vector $x\in{{\mathbb}{R}}^n$. By construction of the Haar measure $\eta_k^n$ on $\mathbb{G}_k^n$ and uniqueness of the Haar measure $\eta$ on $\mathbb{O}(n)$, we have that, for any $t\in{{\mathbb}{R}}$, $$\begin{split}
\eta_k^n (E\in \mathbb{G}_k^n: \norm{P_E x}_2\geq t)&=\eta(T\in \mathbb O(n): \norm{P_{T E_0} x}_2\geq t)\\
&=\eta(T\in \mathbb O(n): \norm{P_{E_0} Tx}_2\geq t)\\
&=\eta\bigl(T\in \mathbb O(n): \norm{x}_2\norm[\big]{P_{E_0} T(x/\norm{x}_2)}_2\geq t\bigr),
\end{split}$$ where $E_0\coloneqq\mathrm{span}(\{e_1,\ldots,e_k\})$. Again, by the uniqueness of the Haar measure $\sigma_2^n$ on $\mathbb{S}_2^{n-1}$, $T(x/\norm{x}_2)\sim\sigma_2^n$, provided that $T\in\mathbb{O}(n)$ has distribution $\eta$. Thus, $$\eta\Bigl(T\in \mathbb O(n): \norm{x}_2\norm[\Big]{P_{E_0} T\Bigl(\frac{x}{\norm{x}_2}\Bigr)}_2\geq t\Bigr)=\sigma_2^n(u\in \mathbb{S}_2^{n-1}: \norm{x}_2\norm{P_{E_0} u}_2\geq t)\,.$$ By , $G/\norm{G}_2\sim\sigma_2^n$. Thus, $$\sigma_2^n (u\in \mathbb{S}_2^{n-1}: \norm{x}_2\norm{P_{E_0} Tu}_2\geq t)={\mathbf{P}}\Bigl(\norm{x}_2\,\frac{\norm{P_{ E_0}G}_2 }{\norm{G}_2}\geq t\Bigr).$$ Therefore, if $E\in {\mathbb{G}}^n_k$ is a random subspace independent of $X$ having distribution $\eta_k^n$, and $G$ is a standard Gaussian random vector in ${{\mathbb}{R}}^n$ that is independent of $X$ and $E$, we have that $${\mathbf{P}}_{(X,E)}\bigl((x,F)\in{{\mathbb}{R}}^n \times{\mathbb{G}}_k^n:\norm{P_F x}_2\geq t\bigr)=
{\mathbf{P}}_{(X,G)}\Bigl((x,g)\in{{\mathbb}{R}}^n\times{{\mathbb}{R}}^n:\norm{x}_2\,\frac{\norm{P_{ E_0}g}_2 }{\norm{g}_2}\geq t\Bigr).$$ Here, ${\mathbf{P}}_{(X,E)}$ denotes the joint distribution of the random vector $(X,E)\in{{\mathbb}{R}}^n\times{\mathbb{G}}_{k}^n$, while ${\mathbf{P}}_{(X,G)}$ stands for that of $(X,G)\in{{\mathbb}{R}}^n\times{{\mathbb}{R}}^n$. By , $X$ has the same distribution as $\pnorm{X}Z/\pnorm{Z}$. Therefore, $$\begin{split}
&{\mathbf{P}}_{(X,G)}\Bigl((x,g)\in{{\mathbb}{R}}^n\times{{\mathbb}{R}}^n:\norm{x}_2\,\frac{\norm{P_{ E_0}g}_2 }{\norm{g}_2}\geq t\Bigr)\\
&={\mathbf{P}}_{(X,Z,G)}\Bigl((x,z,g)\in{{\mathbb}{R}}^n\times{{\mathbb}{R}}^n\times{{\mathbb}{R}}^n:\pnorm{x}\frac{\norm{z}_2}{\pnorm{z}}\frac{\norm{P_{ E_0}g}_2 }{\norm{g}_2}\geq t\Bigr)
\end{split}$$ with ${\mathbf{P}}_{(X,Z,G)}$ being the joint distribution of the random vector $(X,Z,G)\in{{\mathbb}{R}}^n\times{{\mathbb}{R}}^n\times{{\mathbb}{R}}^n$. Consequently, we conclude that the two random variables $\norm{P_E X}_2$ and $\pnorm{X}\frac{\norm{Z}_2}{\pnorm{Z} }\frac{\norm{P_{E_0} G}_2}{\norm{G}_2}$ have the same distribution.
Let us remark that in his PhD thesis, Kim [@K] was recently able to extend the results from [@APT] and [@GKR2017] to more general classes of random vectors under an asymptotic thin-shell-type condition in the spirit of [@ABP2003] (see [@K Assumption 5.1.2]). For instance, this condition is satisfied by random vectors chosen uniformly at random from an Orlicz ball.
### Outlook – the non-commutative setting {#outlook-the-non-commutative-setting}
The body of research on large deviation principles in asymptotic geometric analysis, which we have just described above, is complemented by another paper of Kim and Ramanan [@KimRamanan], in which they proved an LDP for the empirical measure of an $n^{1/p}$ multiple of a point drawn from an $\ell_p^n$-sphere with respect to the cone or surface measure. The rate function identified is essentially the so-called relative entropy perturbed by some $p$-th moment penalty (see [@KimRamanan Equation (3.4)]).
While this result is again in the commutative setting of the $\ell_p^n$-balls, Kabluchko, Prochno, and Thäle [@KPT2018b] recently studied principles of large deviations in the non-commutative framework of self-adjoint and classical Schatten $p$-classes. The self-adjoint setting is the one of the classical matrix ensembles which has already been introduced in Subsection \[subsubsec: outlook non-commutative schechtman-schmuckenschlaeger\] (to avoid introducing further notation, for the case of Schatten trace classes we refer the reader to [@KPT2018b] directly). In the spirit of [@KimRamanan], they proved a so-called Sanov-type large deviations principles for the spectral measure of $n^{1/p}$ multiples of random matrices chosen uniformly (or with respect to the cone measure on the boundary) from the unit balls of self-adjoint and non self-adjoint Schatten $p$-classes where $0< p \leq +\infty$. The good rate function identified and the speed are quite different in the non-commutative setting and the rate is essentially given by the logarithmic energy (which is the negative of Voiculescu’s free entropy introduced in [@V1993]). Interestingly also a perturbation by a constant connected to the famous Ullman distribution appears. This constant already made an appearance in the recent works [@KPT2018; @KPT2018a], where the precise asymptotic volume of unit balls in classical matrix ensembles and Schatten trace classes were computed using ideas from the theory of logarithmic potentials with external fields.
The main result of [@KPT2018b] for the self-adjoint case is the following theorem, where we denote by $\mathcal M({{\mathbb}{R}})$ the space of Borel probability measures on ${{\mathbb}{R}}$ equipped with the topology of weak convergence. On this topological space we consider the Borel $\sigma$-algebra, denoted by $\mathcal B(\mathcal M({{\mathbb}{R}}))$.
\[thm:sanov type ldp schatten\] Fix $p\in(0,\infty)$ and $\beta\in\{1,2,4\}$. For every $n\in{{\mathbb}{N}}$, let $Z_n$ be a random matrix chosen according to the uniform distribution on $\mathbb{B}_{p,\beta}^n$ or the cone measure on its boundary. Then the sequence of random probability measures $$\mu_n= \frac{1}{n}\sum_{i=1}^n \delta_{n^{1/p}\lambda_i(Z_n)},
\qquad n\in{{\mathbb}{N}},$$ satisfies an LDP on $\mathcal M({{\mathbb}{R}})$ with speed $n^2$ and good rate function $\mathcal I:\mathcal M({{\mathbb}{R}}) \to [0,+\infty]$ defined by $$\label{eq:J_def_rate_funct}
\mathcal I(\mu) =
\begin{cases}
- \frac \beta 2 \int_{{{\mathbb}{R}}}\int_{{{\mathbb}{R}}} \log\abs{x-y} \, \mu({\mathop{}\!\mathrm{d}}x)\,\mu({\mathop{}\!\mathrm{d}}y) + \frac{\beta}{2p} \log\Bigl(\frac{\sqrt{\pi}p \Gamma(\frac{p}{2})}{2^p\sqrt{e}\Gamma(\frac{p+1}{2})}\Bigr) &{\quad\text{if }\,}\int_{{{\mathbb}{R}}}\abs{x}^p\mu({\mathop{}\!\mathrm{d}}x) \leq 1,\\
+\infty &{\quad\text{if }\,}\int_{{{\mathbb}{R}}}\abs{x}^p\mu({\mathop{}\!\mathrm{d}}x) > 1.
\end{cases}$$
Let us note that the case $p=+\infty$ as well as the case of Schatten trace classes is also covered in that paper (see [@KPT2018b Theorems 1.3 and 1.5]). The proof of Theorem \[thm:sanov type ldp schatten\] requires to control *simultaneously* the deviations of the empirical measures and their $p$-th moments towards arbitrary small balls in the product topology of the weak topology on the space of probability measures and the standard topology on ${{\mathbb}{R}}$. It is then completed by proving exponential tightness. Moreover, they also use the probabilistic representation for random points in the unit balls of classical matrix ensembles which they have recently obtained in [@KPT2018a]. We close this survey by saying that as a consequence of the LDP in Theorem \[thm:sanov type ldp schatten\], they obtained that the spectral measure of $n^{1/p} Z_n$ converges weakly almost surely to a non-random limiting measure given by the Ullman distribution, as $n\to\infty$ (see [@KPT2018b Corollary 1.4] for the self-adjoint case and [@KPT2018b Corollary 1.6] for the non-self-adjoint case).
Joscha Prochno: Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Austria
*E-mail address:*
Christoph Thäle: Fakultät für Mathematik, Ruhr-Universität Bochum, Germany
*E-mail address:*
Nicola Turchi: Fakultät für Mathematik, Ruhr-Universität Bochum, Germany
*E-mail address:*
[^1]: *2010 Mathematics Subject Classification.* 46B06, 47B10, 60B20, 60F10.
[^2]: *Key words and phrases.* Asymptotic geometric analysis, $\ell_p^n\text{-balls}$, central limit theorem, law of large numbers, large deviations, polar integration formula.
|
---
author:
- 'Bartley D. Richardson'
- 'Benjamin J. Radford'
- 'Shawn E. Davis'
- Keegan Hines
- David Pekarek
bibliography:
- 'BDRRef\_no\_order.bib'
title: Anomaly Detection in Cyber Network Data Using a Cyber Language Approach
---
[^1]
Introduction
============
As the amount of cyber data continues to grow, cyber network defenders are faced with increasing amounts of data they must analyze to ensure the security of their networks. In addition, new types of attacks are constantly being created and executed globally. Current rules-based approaches are effective at characterizing and flagging known attacks, but they typically fail when presented with a new attack or new types of data. By comparison, unsupervised machine learning offers distinct advantages by not requiring labeled data to learn from large amounts of network traffic. In this paper, we present a natural language-based technique (suffix trees) as applied to cyber anomaly detection. We illustrate one methodology to generate a language using cyber data features, and our experimental results illustrate positive preliminary results in applying this technique to flow-type data. As an underlying assumption to this work, we make the claim that malicious cyber actors leave observables in the data as they execute their attacks. This work seeks to identify those artifacts and exploit them to identify a wide range of cyber attacks without the need for labeled ground-truth data.
Previous Work
=============
Previous work has investigated network data for pattern-of-life and anomaly detection that informs the approaches taken in this paper. In terms of pattern-of-life, work focuses on identifying and classifying users within a network [@gu2015novel; @sharafuddin2010know; @verde2014no; @abt2014small]. Clustering for anomaly detection [@leung2005unsupervised; @portnoy2001intrusion; @munz2007traffic] is a common technique for network data due to the fact that most network datasets are unlabeled and contain no ground truth. The focus of these works is largely centered around intrusion detection and make the assumption that intrusions should be anomalous relative to the network as a whole. Likewise, there exists previous work around the probabilistic suffix tree (PST) and using it to model and predict protein families [@bejerano2001variations] as well as determine anomalous user behavior from event logs [@liu2013incorporating].
Data Sources and Experimental Configuration
===========================================
Our data for this effort consists of network traffic data collected in a traditional compute environment. Specifically, we utilize the University of New Brunswick Information Security Centre of Excellence (ISCX) Intrusion Detection Evaluation DataSet [@ids2012]. The reason we selected this dataset is that it contains labeled data for known attacks, and this permits us to produce ROC curves and calculate AUC in order to evaluate the effectiveness of the technique. We concentrate on Bro netflow data. By default, Bro creates separate log files for different actions. For example, the DNS log file contains DNS resolution requests and associated metadata while the HTTP log file contains, among other things, GET and POST requests (with the associated URLs) for activity over HTTP and HTTPS ports. The Bro CONN (connection) log contains high-level metadata for other log types, including: IP addresses, ports used, bytes transferred, packets transferred, duration, TCP flags, and protocol. Bro netflow is aggregated to the session level and is bidirectional, enabling it to represent many PCAP frames in a single row of information. The relationship between PCAP and flow-type data is illustrated in Figure \[fig1\]. The ISCX dataset contains 2,028,053 labeled Netflow records, with 96.6% of them labeled normal and the remaining 3.4% labeled as attack. The volume of traffic is shown Figure \[fig2\]. Statistical analysis of real network traffic is used to create agent-based background traffic in a testbed environment, and the attacks are real and planned by a white hat team based on the architecture of the testbed environment.
![Flow-type data (top) compared with PCAP data (bottom). Flow data is aggregated and contains information across multiple PCAP frames, and raw PCAP data provides access to the application payload that may not be represented in aggregated flow data.[]{data-label="fig1"}](Picture1.png){width=".95\columnwidth"}
![Volume of attack (top) and normal (bottom) traffic in the ISCX dataset[]{data-label="fig2"}](Picture2.png){width=".95\columnwidth"}
Our analytics run on a cloud compute environment using Cloudera CDH v5.11.0 and a heavily modified Spark v.2.1.0.cloudera1. The physical machine contains 6.15TB of addressable RAM with 420 VCores. Actual physical servers include: 8x24 cores, 22x40 cores, 1x12 core, and 1x4 cores. HDFS is currently configured for a total capacity of 260.4TiB. Our analytics utilize Spark MLLib heavily, and input/output is managed via Hive tables in HDFS. It is important to note that this is a research cloud and is not a production environment.
Creating Sequences of Cyber Data
================================
Before we can apply the analytic to the cyber data, we must transform it into a sequence of activity (i.e., create the communication language). Figure \[fig3\] illustrates this process in detail. In general, some feature engineering is performed *a priori* to sequence creation. Various combinations of protocol, port, bytes, packets, and other features can be encoded into discrete tokens, and sequences of these tokens effectively compress the communication between two networked computers. For this work, we focus on proto-bytes (a protocol identifier with the sum of the bytes transferred) and proto-density (a protocol identifier with the sum of bytes/packet transferred). In order to keep the vocabulary at a manageable size, we also bin the quantitative features in some way. One method that has shown promise is to take the floor of the log2 feature value. This allows us to keep some sense of magnitude (e.g., bytes, KB, MB, GB) while reducing the number of tokens produced.
![Construction of cyber language sequences from flow data[]{data-label="fig3"}](Picture3.png){width=".95\columnwidth"}
Another issue to consider when creating sequences of cyber data is the sequence length. Sequence length directly relates to time and how long the communication remains open. Various ways to sessionize exist, including by hour, day, week, and after 30 minutes of no activity between two IP addresses. For this work, we construct sequences that terminate after an hour. This has the added benefit of keeping most sequences to a relatively similar length, so there is not the issue of normalizing all sequences to account for widely varying lengths. These sequences are created using a parallelized Spark-based approach that can construct multiple types of sequences relatively quickly.
Modeling Cyber Data using a Probabilistic Suffix Tree
=====================================================
Creating the PST model is relatively straightforward. The cyber language sequences are fed into a slightly modified PST code model that distributes the learning across a Spark cluster. Typical starting hyperparemeter values set the depth of the tree to 14, the minimum probability to 0.0001, the probability threshold (specifying the minimum probability necessary for inclusion of the suffix in the tree) to 0.0005, and the two smoothing parameters at $\tau=10$ and $\epsilon=0.0$.
After creating the sequences and the PST model, we then score each sequence using the model. The overall process is shown in Figure \[fig4\]. Each sequence receives a likelihood score (a probability between 0 and 1), and we flag for investigation those sequences that receive a non-zero likelihood score below a set limit. These sequences represent those less likely to exist in the data (i.e., anomalous sequences).
![Analytic flow for creating PST models from cyber network data[]{data-label="fig4"}](Picture4.png){width=".95\columnwidth"}
To build intuition, we present Figure \[fig5\]. In this figure, the application of PST modeling to English words is on the left while the cyber application is on the right. In the traditional application, we seek to quantify a word’s conformity to traditional spelling patterns. Notice that words like “actions” and “stations” are more likely (therefore further to the right on the histogram) while words like “chutzpah” and “syzygy” are less likely (further to the left). In our application to cyber data via construction of a cyber language, we seek to add interpretability to findings using similar intuition. Our application necessitates an additional step to transform the data and sequence the tokens. Instead of analyzing English spelling patterns, we are quantifying spelling patterns of our tokenized sequences representing machine-to-machine communication. The underlying assumption is that sequences of less-likely spellings are anomalous to the network environment where they are observed, and these events warrant increased scrutiny by a cyber expert.
![Application of PST to natural language (left) and cyber data (right)[]{data-label="fig5"}](Picture5.png){width=".95\columnwidth"}
Experimental Results
====================
This section presents experimental results of the PST approach to cyber anomaly detection on the ISCX dataset. Figure \[fig6\] shows the results for two different types of tokens and their respective ROC curves. We experimented with two main types of tokenization including proto-density binned in buckets of 10 (left) and proto-bytes binned using log2 (right). For the ISCX dataset, using proto-bytes as a feature significantly outperformed using proto-density.
![The effect of tokenization on the performance of the PST analytic[]{data-label="fig6"}](Picture6.png){width=".95\columnwidth"}
Another factor in PST performance is the tuning of the analytic hyperparameters. As implemented, the PST has several hyperparameters. The tree depth specifies the maximum depth of the model generated while the probability threshold value is used to determine if a sequence is significant and is a candidate to add to the PST. Raising the probability threshold makes the PST more restrictive. The other parameters (tau, epsilon, and probability minimum) are essentially used together to remove useless nodes from the PST model and as a smoothing factor. Figure \[fig7\] illustrates the effect of tuning these PST hyperparameters to a specific dataset.
![The effect of tuning PST hyperparameters to increase the AUC[]{data-label="fig7"}](Picture7.png){width=".95\columnwidth"}
By tuning the PST hyperparameters, we are able to increase the AUC from 0.545 (shown on the right side of Figure \[fig6\]) to 0.748. It should also be noted that the shape of the ROC curve is of interest from the view of a cyber analyst. By noting the sharp rise in the ROC curve at the beginning, we observe that the results presented to a cyber analyst (assuming this same ordering) are less likely to be false positives (i.e., less likely to degrade trust in the system). In an operational environment, we would typically not have ground truth labels for our data. It is important to build trust in the system by presenting minimal false positives to the cyber security analyst.
Conclusions and Future Work
===========================
This work demonstrates that there is a method to view and interpret cyber communications as a language and that applying language-based analytic techniques to this new synthetic language has potential. We have shown that there is value in viewing network traffic as a language, and that PSTs can be used to characterize that language. By selectively engineering the input features and tuning the PST hyperparameters, we can substantially increase the AUC while maintaining a favorable ROC curve shape.
Future work in this area includes how to best retain the labels when aggregating flows into sessions. While we have characterized on attack vs. non-attack (normal), there are various types of attacks that exist and generating more nuanced labels than these binary indicators would be useful. Additional research and experimentation should be devoted to the correct evaluation criteria for results. Do cyber analysts care about the overall AUC or, perhaps more importantly, the true positive rate for the first $n$ predicted anomalies/attacks? In addition, experiments that evaluate how generalizable the results from applying this methodology on the ISCX data are necessary. One method to do this is to use additional labeled datasets from ISCX and then generate model-fit comparisons on labeled and unlabeled data to show correlation.
[^1]: This research was developed with funding from the Defense Research Projects Agency (DARPA). The views, opinions and/or findings expressed are those of the authors and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government.
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abstract: |
We consider a $C^\infty$ boundary $b\Om\subset \C^n$ which is $q$-convex in the sense that its Levi-form has positive trace on every complex $q$-plane. We prove that $b\Om$ is tangent of infinite order to the complexification of each of its submanifolds which is complex tangential and of finite bracket type. This generalizes Diederich-Fornaess [@DF78] from pseudoconvex to $q$-convex domains. We also readily prove that the rows of the Levi-form are $\frac12$-subelliptic multipliers for the $\dib$-Neumann problem on $q$-forms (cf. Ho [@H91]). This allows to run the Kohn algorithm of [@K79] in the chain of ideals of subelliptic multipliers for $q$-forms. If $b\Om$ is real analytic and the algorithm stucks on $q$-forms, then it produces a variety of holomorphic dimension $q$, and in fact, by our result above, a complex $q$-manifold which is not only tangent but indeed contained in $b\Om$. Altogether, the absence of complex $q$-manifolds in $b\Om$ produces a subelliptic estimate on $q$-forms.
32F10, 32F20, 32N15, 32T25
author:
- Stefano Pinton and Giuseppe Zampieri
title: 'complex manifolds in $Q$-convex boundaries'
---
ł Ł[L]{} \[section\] \[Thm\][Corollary]{} \[Thm\][Proposition]{} \[Thm\][Lemma]{} \[Thm\][Definition]{} \[Thm\][Remark]{} \[Thm\][Example]{} \[Thm\][Examples]{} \#1[\[\#1\]]{}
alpha[\[1\]]{}
Complex $q$-manifolds in the boundary and the Kohn algorithm on $q$-forms
=========================================================================
Let $\Om$ be a smooth domain in $\C^n$ defined by $r=0$ with $\di r\neq 0$, and $M$ a smooth CR submanifold of $b\Om$ of CR dimension $q$ and CR codimension $p$. We assume that $M$ is “complex tangential" to $b\Om$ in the sense that $$\Label{1.1}
TM\subset T^\C b\Om.$$ Condition is familiar in the ambient of peak-interpolation sets. If $M$ is minimal in the sense of Tumanov, it is endowed with a “wedge complexification" of dimension $q+p$, that is, a complex $(q+p)$-manifold $\mathcal W$ of wedge type with edge $M$ (cf. [@T88]). When $b\Om$ is pseudoconvex, then $\mathcal W\subset b\Om$; this refines Bedford-Fornaess [@BF81] which is in turn a development of Diederich-Fornaess [@DF78]. In fact, according to [@T88], $\mathcal W $ is made out of analytic discs attached to $M$. The pseudoconvexity of $b\Om$ brings the discs inside $\bar\Om$ and their complex tangency to $b\Om$, which follows from , brings them in $b\Om$. For the last implication, we have just to apply Hopf Lemma to a plurisubharmonic Hölder exhaustion function of $\Om$ of type $-(-r)^\eta$, for $\eta$ close to 1, restricted to each disc. We weaken the hypothesis of pseudoconvexity and assume that $b\Om$ is $q$-convex, that is, for a choice of the Hermitian metric, the trace of the Levi form $L_{b\Om}=\di\dib r|_{T^\C b\Om}$ is positive on every complex $q$-plane of $T^\C b\Om$, the complex tangent bundle to $b\Om$.
We strengthen the hypothesis of minimality and assume that $M$ is of “finite bracket type", that is, the subsequent brackets of $C^\infty$ vector fields with values in $T^\C M$ generate the whole tangent bundle $TM$. Note that when $M$ is real analytic, finite type and minimality coincide. Let $b\Om$ be $q$-convex and let $M\subset b\Om$ be complex tangential and of finite bracket type. Then $\mathcal W $ is tangent to $b\Om$ of infinite order along $M$. The proof follows in Section \[s2\].
The holomorphic dimension of a variety $V\subset b\Om$ at $z_o$ is defined by $$\Label{1.2}
\T{hol dim}_{z_o}V=\underset {U_{z_o}}\sup\underset{z\in U\cap V}\inf\dim_\C(T^\C V\cap\T{Ker}L_{b\Om}),$$ for $U_{z_o}$ ranging through the family of neighborhoods of $z_o$. Remark that $TV\cap Ker L_{b\Om}$ is involutive; moving from $z_o$ to a nearby point where the real and the CR ranks are constant, we may apply Frobenious Theorem and produce a foliation by smooth leaves of CR-dimension $q$. We select a leaf $M$, denote by $\L$ the Lie span of $T^\C M$, and observe that $\L\subset Ker L_{b\Om}\subset T^\C b\Om$. By redefining $z_o$, if necessary, we may assume that $\L=TM$; thus $M $ is complex tangential and of finite type. Altogether, we have obtained (i) Let $b\Om$ be $q$-convex and let $V\subset b\Om$ have holomorphic dimension $q$ at $z_o$. Then, there is $M\subset V$ of $CR$-dimension $\ge q$ whose wedge complexification $\mathcal W $ is tangent of infinite order to $b\Om$.
\(ii) If, moreover, $b\Om$ and $V$ are real analytic, then $\mathcal W $ is contained in $b\Om$ and is a (complex) manifold not just a wedge manifold.
Our purpose is now to run the Kohn algorithm in a $q$-convex domain and to show that, when it goes through, it produces a subelliptic estimate for $q$-forms. This requires a minor effort in adapting the proof by Kohn [@K79] in which the domain is pseudoconvex in the usual sense.
We choose an orthonormal basis $\om_1,,...,\om_n=\di r$ of $(1,0)$ forms, and the dual basis $L_j$ of $(1,0)$ vector fields. In this basis, we denote by $(r_{ij})$ the matrix of $\di\dib r$ and by $u=\sumJ u_J\bar \om_J$ an antiholomorphic $q$ form with summation being taken over ordered multiindices $|J|=q$. The form is assumed to belong to the domain $D_{\dib^*}$ of $\dib^*$ that is, to satisfy $u_J|_{b\Om}\equiv0$ when $n\in J$; we denote by $C^\infty_c(\bar\Om\cap U)^{q}$ the space of $q$-forms with support in a neighborhood $U$ of a boundary point $z_o\in b\Om$ with smooth coefficients up to $b\Om$. We also denote by $|||\cdot|||_\epsilon$ the [*tangential*]{} Sobolev norm (cf. [@K79]). Let $b\Om$ be $q$-convex; then $$\Label{2.1}
\begin{split}
\sumK\sum_i|||\sum_j r_{ij}\bar u_{jK}|||^2_{\frac12}&\leq \NO{\dib u}+\NO{\dib^* u}+\NO{u}
\\
&\T{ for any $u\in D_{\dib^*}\cap C^\infty_c(\bar\Om\cap U)^{k},\,\,k\ge q$}.
\end{split}$$ We express by saying that each row of $\di\dib r$ is a $\frac12$-subelliptic row-multiplier on $k$-form. We use the notation $Q(u,u)$ for the energy of the $\dib$-Neumann problem, that is, the term in the right of . We show that for any $v\in C^\infty_c(U'\cap \bar \Om)^k$, for $U'\supset\supset U$, and for any derivative $D$, we have $$\Label{2.3}
\begin{split}
\Big|\sumK\sumij\int_\Om r_{ij}u_{iK}D\bar v_{jK}dV\Big|^2&\simleq Q(u,u)+\sum_j\NO{\bar L_j(v)}
\\
&+\sumK\sumij\int_{b\Om}r_{ij}v_{iK}\bar v_{jK} dS.
\end{split}$$ For $D=L_k$, follows from Schwartz inequality. For $ D=\bar L_k,\,\,k<n$, it follows from integration by parts, Schwartz inequality, and basic estimate for $u$. Finally, for $D=\bar L_n$, we write $$\Label{2.4}
\begin{split}
\sumK\sumij\int_\Om r_{ij}u_{iK}\bar L_n\bar v_{jK}dV&=\sumK\sum_{i,j<n}\int_{b\Om} r_{ij}u_{iK}\bar v_{jK}dS
\\
&+O((\sum_j\no{\bar L_j u})\no{v}).
\end{split}$$ Using again Schwartz inequality on $b\Om$ for the positive $2$-form $\sumK\sumij r_{ij}u_{iK}\bar u_{jK}$ over $k$-vectors $u$, we get $$\sumK\Big|\sum_{ij}r_{ij}u_{iK}\bar v_{jK} dV\Big|\le \Big(\sumK\sumij r_{ij}u_{iK}\bar u_{jK}\Big)^{\frac12}\Big(\sumK\sumij r_{ij}v_{iK}\bar v_{jK}\Big)^{\frac12},$$ and this yields from and the basic estimate. We use now for $v_{jK}=\sum_i r_{ij}u_{iK}$. Reasoning as in [@K79] p. 97, we get $$\Label{2.5}
\Big |\sumK\underset{ijk}\sum\int_\Om r_{ij}u_{iK}D(r_{kj}\bar u_{kK})dV\Big |\simleq Q(u,u).$$ Using the microlocal factorization $\Lambda^1=\Lambda^{\frac12}\Lambda^{\frac12}$ for the tangential standard elliptic psedodifferential operator of order 1 (together with the fact that the different derivatives $D$’s represent the full $\Lambda^1$), we get from .
We recall briefly the Kohn’s algorithm. We define, in a neighborhood of $z_o$, the chain of ideals $I_1^q\subset I_2^q\subset...I^q_h$ and of modules $M_1^q\subset M_2^q\subset...M^q_h$, starting from $$\begin{cases}
M^q_1=\{\di r,\,\di_i\dib r\}_{i=1,...,n}
\\
I^q_1=\sqrt{r,det_{n-q+1}M^q_1}^\R\quad \T{where $\sqrt{\cdot}^\R$ denotes the real radical},
\end{cases}$$ and, inductively, $$\begin{cases}
M^q_h=\{M^q_{h-1},\di I^q_{h-1}\},
\\
I^q_h=\sqrt{I^q_{h-1},\,\T{det}_{n-q+1}M^q_h}^\R.
\end{cases}$$ By Proposition \[p2.1\], and by Garding inequality, $M_1^q$ is made out of $\frac12$-subelliptic row multipliers (that is, holds) and $I^q_1$ is an ideal of $\frac12$-subelliptic multipliers over $q$-forms. By [@K79] Proposition 4.7, the full chain of $M^q_h$’s (resp. $I^q_h$’s) is made out of subelliptic row multipliers (resp. function multipliers). The proof of this point remains unchanged from pseudoconvex to $q$-convex domains.
We take our conclusions. If $1\in I^q_h$ for some $h$, then we have a subelliptic estimate (for some $\epsilon$ depending on the number $h$ of steps and on the operation of radical) on $q$-forms and, in fact, on $k$-forms for any $k\ge q$. If, instead, $I_{h+1}^q=I^q_h$ (and $I^q_h$ does not capture $1$), this reveals under the extra assumption $b\Om\in C^\om$, that $V=V(I_h^q)$, the zero-set of $I^q_h$, has holomorphic dimension $\ge q$. By Corollary \[c1.1\] this implies the existence of a complex $q$-manifold in $b\Om$. Putting alltogether, we get the proof of Assume that in a neighborhood of $z_o$, $b\Om$ is real analytic, $q$-convex, and contains no germ of holomorphic manifold of dimension $\ge q$. Then a subelliptic estimate in degree $k\ge q$ for the $\dib$-Neumann problem holds in a neighborhood $U$ of $z_o$, that is, for some $\epsilon$ we have $$\NO{u}_\epsilon\simleq Q(u,u)\quad\T{for any $u\in D_{\dib^*}\cap C^\infty_c(\bar \Om\cap U)^k$}.$$ In $\C^3$, consider the domain $\Om$ defined by $$x_3>-|z_1|^2|z_2|^2+(\frac14|z_1|^4+\frac34|z_2|^4).$$ Here $b\Om$ is real analytic, there are no complex 2-manifolds at $0$ but just the complex curve defined by $z_1=z_2$. Also, if we compute the Levi form of $b\Om$ in the metric in which $\pi_z^{-1}(1,0,0)$ and $\pi_z^{-1}(0,1,0)$ (for $\pi_z:\,T_zb\Om\to \C^2\times\R$ being the projection along the $x_3$-axis) is an orthonormal system for $T^\C_z b\Om$, we have $$L_{b\Om}=
\left[
\begin{matrix} -|z_2|^2+|z_1|^2&-\bar z_1z_2
\\
-z_1\bar z_2&-|z_1|^2+3|z_2|^2
\end{matrix}\right].$$ It follows $$\T{trace}\,L_{b\Om}=2|z_2|^2\ge0.$$ Thus we have a subelliptic estimate in degree $2$ according to Theorem \[t2.1\]. Note that this example could not be explained neither by usual pseudoconvexity nor by [*strong $2$-pseudoconvexity*]{}. In fact $$\begin{split}
\T{det}\,L_{b\Om}&=(|z_1|^2-|z_2|^2)(3|z_2|^2-|z_1|^2)-|z_1|^2|z_2|^2
\\
&=-|z_1|^4-3|z_2|^4+3|z_1|^2|z_2|^2
\\
&\le 0.
\end{split}$$ Thus,
- $b\Om$ is not pseudoconvex (because $\T{det}\,L_{b\Om}\le 0$ implies that there are eigenvalues of opposite sign),
- $b\Om$ does not satisfy $Z(2)$ (in the sense of [@FK72]) because there are no positive eigenvalues at $0$.
Proof of Theorem \[t1.1\]
==========================
We adapt the proof of [@DF78] Proposition 3 to the new situation in which $b\Om$ is no more pseudoconvex but just $q$-convex. We move to a nearby point that we still denote by $z_o$ at which the “multitype" in the sense of (i)–(v) below is minimal (in the lessicographic order). We observe that the wedge complexification $\mathcal W$ can be (non-uniquely) continued to a smooth manifold without boundary $W$ of real dimension $2(q+p)$. Since $\mathcal W$ is holomorphic, then $W$ is “approximatly holomorphic" at $M$. By a linear unitary coordinate change we can assume that $z_o=0$, $T_{z_o}M=\C^{q}\times\R^p\times \{0\}$ and $T_{z_o}W =\C^{q}\times\C^p\times \{0\}$ and $T_{z_o}b\Om=\C^{n-1}\times i\R$. We observe that the projection $\pi$ along the $z_n$-axis is transversal to $W $; thus $\pi(W) $ and $\pi^{-1}\pi(W) $ are real manifolds of dimension $2(q+p)$ and $2(q+p+1)$ respectively. We use the notation $t:=n-(p+q+1)$. We suppose that $\pi^{-1}\pi( W)$ is defined by real equations $\mu_j(z')=0,\,\,j=1,...,2t$ such that, putting $f_j=:\mu_j+i\mu_{t+j},\,\,j\le t$, we have $\dib f_j=O^\infty_M$, and $W$ is graphed over $\pi( W)$ by $z_n=h+ig$ with $\dib(h+ig)=O^\infty_{M}$; here $O^\infty_{M}$ denotes a zero of infinite order at $M$. Clearly $M$ is defined by $x_n-h=0,\,\,y_n-g=0,\,\,\rho=0,\,\,\mu=0$ (where by $\rho$ and $\mu$ we denote the full set of the $\rho_j$’s and $\mu_j$’s). We consider the Hermitian metric on $\C^n$ in which $\Om$ is $q$-convex and the induced Euclidian metric on $\R^{2n}$. In this metric, we choose an orthonormal basis $\{X_{0,i}\}_{i=1}^{p_0}$ of $T^\C M$ and a completion to a full basis of $TM$ $$\Label{nova}
\{X_{0,i}\}_{i=1}^{p_0},\,\{X_{1,i}\}_{i=1}^{p_1},...,\{X_{s,i}\}_{i=1}^{p_s}\quad\T{with $p_0=2q$ and $\sum_{j=1}^sp_j=p$}.$$ We may assume that
- any $j$-iterated bracket of the $X_{0,i}$’s is in the span of the $X_{h,i}$’s for $h\le j$,\
- $X_{j,i}=[X_{0,\nu},X_{j-1,\mu}]$ modulo $\T{Span} \{X_{j',i}\}_{j'\le j-1}$ for suitable $X_{0,\nu}\in\T{Span}\{X_{0,i}\}$ and $X_{j-1,\mu}\in \T{Span}\{X_{h,i}\}_{h\le j-1}$ when $j\ge 1$.
This is an immediate consequence of Jacobi identity. We put $\mathcal L^0=T^\C M$, write, inductively, $\L^j=\T{Span}\{\L^{j-1},[X_{0,\nu},X_{j-1,\mu}]\}_{\nu,\mu}$ and decompose $$TM=\L^0\oplus \frac{\L^1}{\L^0}\oplus...\oplus \frac{\L^{s}}{\L^{s-1}}.$$ We can assume that our linear unitary tranformation gives $\frac{\L^j}{\L^{j-1}}\Big|_{z_o}=\{0\}\times\R^{p_j}\times\{0\}$. Also, we can choose our basis so that, in addition to (i)–(ii) we also have
- each group $\{X_{j,i}\}_{i=1,...,p_j}$ is orthogonal one to another for different $j$.\
- in a basis $z_{0,1},...z_{0,q},z_{1,1},...z_{1,p_1},...$ of $\C^{q+p}$ we have $X_{0,i}|_{z_o}=\di_{x_i}$, $X_{0,i+q}|_{z_o}=\di_{y_i},\,\,i\le q$, and $X_{j,i}|_{z_o}=\di_{x_{j,i}}$ for $j\ge1$,\
- $M$ is the intersection of $W $ with the set defined by $\rho_{j,i}=0,\,\,j\ge1$, where the $\rho_{j,i}$’s are functions on $\pi(W )$ with $\T{Span}\{\Re \di\rho_{j,i}\}=\T{Span}\{\Re\di y_{j,i}\}$ and with $\langle\di\rho_{h,l},L_{j,i}\rangle=0$ for any $h\ge j+1$.
Note that, in particular, (v) implies that $\di\dib\rho_h(L_{j,i},\bar L_{j',i'})=0$ for any $j,\,j'\le h-2$.
We identify the $X_{j,i}\in TM$ to the real or imaginary parts of vector fields $L_{j,i}\in \C(TM+JTM)\cap T^{1,0}\C^n$ defined by $$\Label{supernova}
\{L_{0,i}:=X_{0,i}+iX_{0,q+i}\}_{i=1}^{q},\,\{L_{1,i}:=X_{1,i}+iJX_{1,i}\}_{i=1}^{p_1},...,\{L_{s,i}:=X_{s,i}+iJX_{s,i}\}_{i=1}^{p_s}.$$ Since $\C(TM+JTM)\cap T^{1,0}\C^n\subset T^{1,0}b\Om|_M$, we extend the $L=L_{j,i}$ from $M$ to the whole $b\Om$ as sections of $T^{1,0} b\Om$ keeping unchanged their notation. We can also arrange that the $L_{j,i}$ are extended from $M$ to $W$ so that $\langle\di \mu_j,L\rangle=O_M^\infty,\,\,j=1,...,2t$. For that, we extend them with the request $\langle \di f_j,L\rangle\equiv0,\,\,j=1,...,t$; since $\langle \dib f_j,L\rangle=O^\infty_M$, the conclusion follows remembering that the $\mu$’s are the real and imaginary parts of the $f$’s. By (iii) above, and by the fact that $\L^0$ is invariant under $J$, we have that the $L_{j,i}$, $j\ge1$, are orthogonal to $\C\L^0$; this stays true also outside $M$ for the extended vector fields. Moreover, possibly after renormalization, the $L_{0,i}$ can be chosen so that they form an orthonormal system.
Recall that for the equation $z_n=h+ig$ of $W$, we have supposed $\dib( h+ig)=O^\infty_M$ and thus, in particular, $\di\dib h=O^\infty_M$. Thus, if $b\Om$ is graphed by $x_n=h+\sigma$ (which serves as a definition of $\sigma$), we have $L_{b\Om}=\di\dib \sigma|_{T^{1,0}b\Om}+O^\infty_M$. We also denote by $r:=x_n-(h+\sigma)$ a definig function for $b\Om$. Note that $\sigma=0$ on $M$; we want to prove that $$\sigma=O(\rho^\infty)\quad\T{when $y_n-g=0$ and $\mu=0$,}$$ and hence $W$ is tangent of infinite order to $b\Om$ along $M$. We expand $$\Label{expansion}
\sigma =\underset{|I|=k}\sum a_I\rho^I+O(\rho^{k+1})+\mathcal E+\mathcal E_1,$$ where $I$ is a multi bi-index in the $(j,i)$’s and where $\mathcal E=O(y_n-g)$ and $\mathcal E_1=O(\mu)$. We observe that $$\Label{star}
\di\dib \mathcal E(L,\bar L)=O(\langle dz_n,L\rangle \langle\di\rho,L\rangle)+|\langle dz_n,L\rangle|^2+O(y_n-g).$$ In particular, recalling that $\langle \di r,L_{j,i}\rangle=0$ and $\langle \di\rho,L_{0,i}\rangle=O(\rho)$, we have for $y_n-g=0$ $$\Label{vanishing}
\di\dib\mathcal E(L_{j,i},\bar L_{j',i'})=O(\rho^{k-1}),\quad \di\dib\mathcal E(L_{j,i},\bar L_{0,i'})=O(\rho^{k}),\quad \di\dib\mathcal E(L_{0,i},\bar L_{0,i'})=O(\rho^{k+1}).$$
As for $\di\dib \mathcal E_1$, recalling also $\langle \di\mu,L\rangle=O^\infty_M$, we have $$\Label{superstar}
\begin{split}
\di\dib\mathcal E_1(L,\bar L)&\sim \di\dib\mu(L,\bar L)+|\langle \di\mu,L\rangle|(|\langle\di\rho,L\rangle|+|\langle dz_n,L\rangle|)+O(\mu)
\\
&=O^\infty_M+O(\rho^\infty)+O(\mu).
\end{split}$$ For this reason, when evaluating $\di\dib\sigma$ on $L$ as above, we can assume without loss of generality that $\mathcal E_1=0$ in . We call $k$ the first integer for which there is in a non-trivial occurence $a_{I}$ for $|I|=k$; we wish to show that $k$ cannot exist finite. First, the inclusion $TW |_M\subset T^\C b\Om|_M$ implies $k\ge 2$. We first show that $k$ cannot be odd. In fact, by a choice of $L=X+iJX,\,X\in\L^j,\,j\ge1$ such that $\di\dib\sigma(L,\bar L)$ is obtained by differentiating two factors once, we get $$\Label{3.2}
\di\dib\sigma(L,\bar L)=\underset{|I'|=k-2}\sum a_{I'}\rho^{I'}+O(\rho^{k-1})+\di\dib\mathcal E(L,\bar L),$$ with $a_{I'}\neq0$ for at least one $I'$. By the first of , the last term in can be neglected. Thus the form in the right of , having odd order, it changes sign. On the other hand $$\Label{3.3}
\di\dib\sigma(L_{0,i},\bar L_{0,i'})=O(\rho^{k-1}).$$ Define a $q$-plane by $Q_{q}:=\T{Span}\{L,L_{0,i}\}_i$ (for any choice of $q-1$ between the indices $i$); we can conclude that $\T{trace}_{Q_{q}}\di\dib \sigma\quad\T{changes sign}$, which violates the $q$-convexity of $b\Om$. Thus $k$ cannot be odd.
We show that $k$ cannot be even, either. We first remove any possible term with a factor of $\rho_{1,i}$ in the homogeneous expression of degree $k$ of $\sigma$, that is, $\sum a_{(1,i)I'}\rho^{(1,i)I'}$. We have $$\Label{3.4}
\di\dib \sigma|_{\C\L^0}=\underset{|I'|=k-1}\sum\Big(\sum_ia_{(1,i)I'}\di\dib\rho_{1,i}|_{\C\L^0}\Big)\rho^{I'}+O(\rho^k)+\di\dib\mathcal E|_{\C\L^0}.$$ By the third of , the last term in can be neglected. If, for some $|I'_o|=k-1$, we have $\T{trace}_{\C\L^0}(\sum_ia_{(1,i)I'_o}\di\dib\rho_{1,i})\neq0$, then $\T{trace}_{\C\L^0}(\di\dib \sigma)$ changes sign since the $\rho^{I'}$ vary independently.
Otherwise, assume $$\Label{3.5}
\T{trace}_{\C\L^0}\Big(\sum_ia_{(1,i)I'}\di\dib\rho_{1,i}\Big)=0\quad\T{ for any $I'$}.$$ Recall that the commutators of the $L_{0,i}'s$ span a space of dimension $p_1$; by Cartan formula, this is equivalent as to saying that the Levi matrices $\di\dib\rho_{1,i}|_{\C\L^0},\,\,i=1,...,p_1$ are independent. Thus, from $\sum_ia_{(1,i)I'_o}\rho_{1,i}\neq0$ for some $I'_o$, we get for some vector of $\C\L^0$, say $L_{0,1}$, $$\Label{3.6}
\sum_ia_{(1,i)I'_o}\di\dib\rho_{1,i}(L_{0,1},\bar L_{0,1})\neq0.$$ Define $L_t=\frac{(1-t)L_{0,1}+t^2L_{1,i}}{c_t}$ (any $i$) where $c_t$ is a factor which normalizes $|U_t|=1$. We deform $\C\L^0$ to $$Q_q=\T{Span}\{L_t,L_{0,2},...,L_{0,q}\}.$$ Combination of and yields $$\T{trace}_{Q_q}(\sum_ia_{(1,i)I'_o}\di\dib\rho_{1,i})=tc_{I'_o}\quad\T{for $c_{I'_o}\neq0$.}$$ Then, using , we have for the trace of the full $\sigma=\sum_{|I|\geq k}a_I\rho^I$ $$\Label{3.6,5}
\T{trace}_{Q_q}\sigma=tc_{I'_o}\rho^{I'_o}+\underset{\underset{|I'|=k-1}{I'\neq I'_o}}\sum c_{I'}\rho^{I'}
+t^4O(\rho^{k-2})+t^2O(\rho^{k-1})+O(\rho^k)+\di\dib\mathcal E|_{Q_q};$$ observing that by we have $\di\dib\mathcal E|_{Q_q}=t^4O(\rho^{k-1})+t^2O(\rho^k)+O(\rho^{k+1})$, we see that this term can be neglected. By taking restriction to a suitable region of the plane $\R^r\times \R$ of $(\rho_{j,i},t)$, all terms in the right of are negligeable comparing to the first: thus, again, $\T{trace}_{Q_q}(\di\dib\sigma)$ changes sign.
At last, we have to consider the case when $\sum_{|I|=k}a_I\rho^I$ contains factors $\rho_{j_o,i}$ which start from $j_o>1$. For fixed $h$, each group of matrices $\di\dib\rho_{h,i},\,\,i=1,...,p_h$, are independent. Thus, for a pair of vectors, say $L_{0,1}\in\C\L^0$ and $L_{j_o-1,1}\in \C\L^{j_o-1}$, and for some $|I'_o|=k-1$, we have $
(\sum_ia_{(j_o,i)I'_o}\di\dib\rho_{j_o,i})(L_{0,1},\bar L_{j_o-1,1})\neq0.
$ But then, under the choice $L:=\frac{t^{-1}L_{0,1}+L_{j_o-1,1}}{c_t},\,\,t<<1$, (for a normalization factor $c_t$) we have $$\Label{3.7}
(\sum_ia_{(j_o,i)I'}\di\dib\rho_{j_o,i})(L,\bar L)=c_{I'_o}\neq0.$$ We then complete $L$ by $q-1$ vectors in $\C\L^0$ to an orthonormal basis of a $q$-space $Q_q$ thus obtaining $$\Label{3.8}
\T{trace}_{Q_q}(\di\dib\sigma)=tc_{I'_o}\rho^{I'_o}+t\underset{\underset{|I'|=k-1}{I'\neq I'_o}}\sum c_{I'}\rho^{I'}+t^2O(\rho^{k-1})+O(\rho^k),$$ where $tO(\rho^{k-1})$ comes from differentiation once with respect to $L$ different terms $\rho_{j_o,i}$ in $(k+1)$-powers and where we have controlled the term $\di\dib\mathcal E$ by $t^2O(\rho^{k-1})+O(\rho^k)+O(\rho^{k+1})$. Again, we can make negleageable in the right of the terms which follow the first and conclude that the trace changes sign, a contradiction.
In conclusion, $k$ cannot exist neither odd nor even and therefore $\sigma$ vanishes of infinite order along $M$.
$\Box$
[BZZ07]{} —Complex manifolds in pseudoconvex boundaries, [*Duke Math. J.*]{} [**38**]{} n. 1 (1981), 279–288 —Pseudoconvex domains with real analytic boundary, [*Annals of Math.*]{} [**107**]{} 3 (1978), 371–384 —The Neumann problem for the Cauchy-Riemann complex, [*Ann. Math. Studies, Princeton Univ. Press, Princeton N.J.*]{} [**75**]{} (1972) —$\dib$-problem on weakly $q$-convex domains, [*Math. Ann.*]{} [**290**]{} (1991), 3–18 —Subellipticity of the $\bar\partial$-Neumann problem on pseudoconvex domains: sufficient conditions, [*Acta Math.*]{} [**142**]{} (1979), 79–122 —Extending CR functions on a manifold of finite type over a wedge, [*Mat. Sb.*]{} [**136**]{} (1988), 129–140
|
---
abstract: 'We report Hubble Space Telescope Cosmic Origins Spectrograph far-ultraviolet and Arecibo Telescope H[i]{} 21cm spectroscopic studies of six damped and sub-damped Lyman-$\alpha$ absorbers (DLAs and sub-DLAs, respectively) at $z \lesssim 0.1$, that have yielded estimates of their H[i]{} column density, metallicity and atomic gas mass. This significantly increases the number of DLAs with gas mass estimates, allowing the first comparison between the gas masses of DLAs and local galaxies. Including three absorbers from the literature, we obtain H[i]{} masses $\approx (0.24 - 5.2) \times 10^9 \: {\rm M}_\odot$, lower than the knee of the local H[i]{} mass function. This implies that massive galaxies do not dominate the absorption cross-section for low-$z$ DLAs. We use Sloan Digital Sky Survey photometry and spectroscopy to identify the likely hosts of four absorbers, obtaining low stellar masses, $\approx 10^7-10^{8.7} M_\odot$, in all cases, consistent with the hosts being dwarf galaxies. We obtain high H[i]{} 21cm or CO emission line widths, $\Delta V_{20} \approx 100-290$ km s$^{-1}$, and high gas fractions, $f_{\rm HI} \approx 5-100$, suggesting that the absorber hosts are gas-rich galaxies with low star formation efficiencies. However, the H[i]{} 21cm velocity spreads ($\gtrsim 100$ km s$^{-1}$) appear systematically larger than the velocity spreads in typical dwarf galaxies.'
author:
- |
Nissim Kanekar$^1$[^1], Marcel Neeleman$^2$, J. Xavier Prochaska$^2$, and Tapasi Ghosh$^3$\
$^1$National Centre for Radio Astrophysics, Tata Institute of Fundamental Research, Ganeshkhind, Pune - 411007, India\
$^2$UCO/Lick Observatory, University of California – Santa Cruz, Santa Cruz, CA 95064, USA\
$^3$Arecibo Observatory, Arecibo, PR 00612, USA
bibliography:
- 'ms.bib'
date: 'Accepted yyyy month dd. Received yyyy month dd; in original form yyyy month dd'
title: 'The gas and stellar mass of low-redshift damped Lyman-$\alpha$ absorbers'
---
\[firstpage\]
galaxies: evolution — galaxies: high-redshift — quasars: absorption lines
Introduction {#sec:intro}
============
Absorption-selected galaxy samples, based on the presence of strong Lyman-$\alpha$ absorption in quasar spectra, are not biased towards high-luminosity objects and hence provide a view of “normal” galaxies at high redshifts. The highest [H[i]{}]{} column density ([$N_{\rm HI}$]{}) systems, the damped and sub-damped Lyman-$\alpha$ absorbers (DLAs and sub-DLAs, respectively) have [$N_{\rm HI}$]{} values similar to those in nearby gas-rich galaxies, and have hence been of much interest in studies of galaxy evolution [e.g. @wolfe05].
Absorption spectroscopy has yielded much information on DLAs, including their metallicities [e.g. @prochaska03a; @rafelski12], gas temperatures [e.g. @kanekar03; @kanekar14], and molecular fractions [e.g. @ledoux03; @noterdaeme08]. However, despite many searches, the galaxy counterparts of only a dozen DLAs and sub-DLAs, mostly targetted due to an atypically high metallicity, have so far been detected in optical/ultraviolet emission at $z \gtrsim 2$ [e.g. @fynbo11; @fynbo13]. Typical high-$z$ DLAs appear to have low in-situ star formation rates (SFRs), $\lesssim 0.3 \; {M_{\odot}}$ yr$^{-1}$ [@fumagalli15]. The situation is somewhat better at low redshifts, $z < 1$, with estimates of the SFR, stellar mass, etc. available for $\approx 25$ absorbers [e.g. @peroux12].
Our knowledge of the gas content of the absorbers is even worse than that of the stellar content. The radio [H[i]{} 21cm]{} hyperfine and CO rotational transitions are the main probes of atomic and molecular gas in nearby galaxies. Unfortunately, few DLAs are known at low redshifts, $z \lesssim 0.2$, where the weak [H[i]{} 21cm]{} line is detectable with today’s radio telescopes. [H[i]{} 21cm]{} emission has hence only been detected in one DLA, at $z \approx 0.009$ towards SBS 1543+593 [@bowen01b; @chengalur02], and one sub-DLA, at $z \approx 0.006$ towards PG 1216+069 [@briggs06; @chengalur15], with a few non-detections yielding limits on the [H[i]{}]{} mass [@mazumdar14]. In the case of molecular gas, there is so far only a single CO detection, at $z=0.101$ towards PKS 0439$-$433 [@neeleman16b]. And, while the recent detection of C[ii]{}-158$\mu$m emission in two $z \approx 4$ DLAs [@neeleman17] provides an exciting new tool to identify high-$z$ DLA host galaxies, this transition does not provide information on the gas content of the absorbers.
A detailed comparison between the stellar and gas properties of absorption-selected galaxies requires a large absorber sample at low redshifts, $z \lesssim 0.2$. The excellent far-ultraviolet (FUV) sensitivity of the Cosmic Origins Spectrograph (COS) onboard the Hubble Space Telescope (HST) now allows the detection of Lyman-$\alpha$ absorption at very low redshifts. We have hence analysed the HST data archive [@neeleman16], to identify low-$z$ absorbers suitable for follow-up studies to characterize the host galaxies. We have now used the Arecibo Telescope to carry out a search for [H[i]{} 21cm]{} emission from a set of low-$z$ absorbers identified in our survey. In this [*Letter*]{}, we present the $N_{\rm HI}$ values, metallicities, and atomic gas and stellar masses for six systems at $z < 0.1$.
------------ --------------- ---------------------------------- ---------------------------------- ---------------------------------- ---------------------------------- ---------------------------------- --------------------------- -------------------
QSO $z_{\rm DLA}$ ${\ensuremath{N_{\rm HI}}}$ N(O[i]{}) N(Si[ii]{}) N(S[ii]{}) N(Fe[ii]{}) \[M/H\] M$^a$
${\times 10^{20}}$ [cm$^{-2}$]{} ${\times 10^{14}}$ [cm$^{-2}$]{} ${\times 10^{14}}$ [cm$^{-2}$]{} ${\times 10^{14}}$ [cm$^{-2}$]{} ${\times 10^{14}}$ [cm$^{-2}$]{}
J0930+2845 $0.0228$ $5.6 \pm 1.2$ $> 13.8$ $> 1.0$ $< 13.5$ $< 4.4$ \[$-2.26$ , $-0.77$\]$^b$ S[ii]{}, Si[ii]{}
J0951+3307 $0.0054$ $10.0 \pm 2.5$ $> 17.8$ $> 7.9$ $14.5 \pm 3.0$ $> 6.9$ $-0.99 \pm 0.14$ S[ii]{}
J1415+1634 $0.0077$ $0.5 \pm 0.1$ $3.02 \pm 0.28$ $0.35 \pm 0.02$ $<1.7$ $< 0.62$ $-1.91 \pm 0.11$$^c$ O[i]{}
J1512+0128 $0.0295$ $2.5 \pm 0.6$ $> 25.1$ $4.27 \pm 0.88$ $<17.4$ $< 2.7$ $-1.29 \pm 0.13$ Si[ii]{}
J1553+3548 $0.0829$ $0.5 \pm 0.1$ $> 3.1$ $1.32 \pm 0.09$ $<3.1$ $1.15 \pm 0.19$ $-1.35 \pm 0.16$ Si[ii]{}
J1619+3342 $0.0963$ $4.0 \pm 1.4$ $> 6.3$ $>0.91$ $4.57 \pm 0.53$ $1.00 \pm 0.09$ $-1.09 \pm 0.16$ S[ii]{}
------------ --------------- ---------------------------------- ---------------------------------- ---------------------------------- ---------------------------------- ---------------------------------- --------------------------- -------------------
$^a$The element used in the metallicity estimate of the previous column. $^b$The allowed metallicity range; see main text for discussion. $^c$Ionization corrections have not been included, but are expected to be small, $\lesssim 0.3$ dex [e.g. @battisti12].\
Observations, data analysis and spectra {#sec:data}
=======================================
The HST observations {#sec:sample}
--------------------
The details of our analysis of the HST archival data on quasars observed with COS, the Space Telescope Imaging Spectrograph, or the Faint Object Spectrograph, are presented in @neeleman16. Standard pipelines were used to produce the final spectrum for each quasar. The search for Lyman-$\alpha$ absorption followed the approach of @prochaska05, with minor modifications [see @neeleman16]. For each absorber, [$N_{\rm HI}$]{} was estimated using a custom IDL Voigt-profile fitting program, simultaneously fitting both the Voigt profile and the quasar continuum. The metal column densities were derived using the apparent optical depth method [@savage91], and then used to infer the gas metallicity [@rafelski12].
Our search yielded 15 DLAs and sub-DLAs at $z \lesssim 0.2$; note that this is [*not*]{} an unbiased sample (see Section 4). We focus here on the six absorbers for which we were able to obtain [H[i]{} 21cm]{} spectroscopy. Four systems are DLAs, with [$N_{\rm HI}$]{} $\geq 2 \times 10^{20}$ [cm$^{-2}$]{}, and two are sub-DLAs, both with [$N_{\rm HI}$]{} $=5 \times 10^{19}$ [cm$^{-2}$]{}. Their Lyman-$\alpha$ absorption profiles and the Voigt profile fits to estimate the [H[i]{}]{} column density are shown in Fig. \[fig:lya\], and their redshifts, [H[i]{}]{} and metal column densities, and metallicities are listed in Table \[tab:data1\].
The Arecibo observations {#sec:arecibo}
------------------------
     
We used the Arecibo L-Band-wide receiver over April–July 2015 and May 2016 in proposal A2940 (PI: Kanekar) to search for redshifted [H[i]{} 21cm]{} emission from 11 DLAs and sub-DLAs at $z \lesssim 0.2$, observable with the Arecibo Telescope [e.g. @meiring11; @neeleman16]. Observations of five targets were affected by radio frequency interference (RFI); it was not possible to obtain clean spectra for these systems, which will not be discussed further. Bandwidths of 6.25 MHz, 12.5 MHz, 25 MHz and 50 MHz were simultaneously used for the observations, centred at the expected redshifted [H[i]{} 21cm]{} line frequency, and sub-divided into 2048 or 8192 channels, with the WAPP backend. Position switching (On/Off) was used for bandpass calibration, while the flux density scale was calibrated using a noise diode. The total on-source time was $\approx 1-3$ hours per source.
All the Arecibo data were analysed in IDL, using standard procedures. While the data on all six targets were relatively free of RFI, detailed flagging was necessary for the two systems at $z \approx 0.1$ (towards J1553+3548 and J1619+3342) to obtain clean spectra. The root-mean-square (RMS) noise values on the final spectra range from $0.33 - 0.68$ mJy per 12.2 kHz channel.
The final [H[i]{} 21cm]{} spectra for the six DLAs and sub-DLAs are shown in Fig. \[fig:hi\], in order of increasing right ascension. We obtained five detections of [H[i]{} 21cm]{} emission (with $> 5\sigma$ significance), out to $z = 0.0829$. For four of the six systems, the spectra are presented after Hanning-smoothing to, and resampling at, a resolution of 12.2 kHz ($\approx 2.6$ [kms$^{-1}$]{} at the respective line frequencies). For the fifth detection, at $z=0.0829$ towards J1553+3548, we further boxcar-smoothed the spectrum by 5 channels; the spectrum of this source in Fig. \[fig:hi\] is at a resolution of $61$ kHz ($\approx 13.9$ [kms$^{-1}$]{} at the redshifted [H[i]{} 21cm]{} line frequency). Finally, in the case of the sole non-detection, for the $z = 0.0963$ DLA towards J1619+3342, the spectrum shown in Fig. \[fig:hi\] has been box-car smoothed to, and resampled at, a resolution of $100$ [kms$^{-1}$]{}. Table \[tab:data2\] provides details of the results of the [H[i]{} 21cm]{} observations.
Results
=======
[|c|c|c|c|c|c|c|c|c|c|]{} QSO & $z_{\rm DLA}$ & ${\ensuremath{N_{\rm HI}}}$ & \[M/H\] & M$_*$ & ${\int S_{\rm HI} {\rm dV}}$$^\ast$ & ${M_{\rm HI}}$$^a$$^\ast$ & ${\Delta V_{\rm 20}}^b$ & ${\Delta V_{\rm 90}}^c$ & $f_{\rm HI}^d$\
& & ${\times 10^{20}}$ [cm$^{-2}$]{}& & ${\times 10^{7}}\; {M_{\odot}}$ & Jy [kms$^{-1}$]{} & ${\times 10^{9}}\; {M_{\odot}}$ & [kms$^{-1}$]{}& [kms$^{-1}$]{}&\
J0930+2845 & $0.0228$ & $5.6 \pm 1.2$ & \[$-2.26$ , $-0.77$\] & $6.9^{+4.6}_{-0.5}$ & $0.729 \pm 0.073$ & $1.75 \pm 0.18$ & $165$ & $153$ & $25$\
J0951+3307 & $0.0054$ & $10.0 \pm 2.5$ & $-0.99 \pm 0.14$ & $4.0$ & $1.83 \pm 0.18$ & $0.237 \pm 0.024$ & $130$ & $94$ & $6$\
J1415+1634 & $0.0077$ & $0.5 \pm 0.1$ & $-1.91 \pm 0.11$ & $1.0 - 2.6$ & $3.53 \pm 0.35$ & $0.965 \pm 0.097$ & $100$ & $118$ & $37-97$\
J1512+0128 & $0.0295$ & $2.5 \pm 0.6$ & $-1.29 \pm 0.13$ & $-$ & $1.27 \pm 0.13$ & $5.15 \pm 0.52$ & $230$ & $267$ & $-$\
J1553+3548 & $0.0829$ & $0.5 \pm 0.1$ & $-1.35 \pm 0.16$ & $23.4^{+34.6}_{-3.9}$ & $0.120 \pm 0.016$ & $3.93 \pm 0.61$ & $270$ & $74$ & $17$\
J1619+3342 & $0.0963$ & $4.0 \pm 1.4$ & $-1.09 \pm 0.16$ & $-$ & $< 0.014^e$ & $< 1.9^e$ & $-$ & $22$ & $-$\
\
PG 1216+069 & $0.0063$ & $0.21 \pm 0.01$ & $-1.60 \pm 0.10^f$ & $-$ & $0.178^h$ & $0.032^h$ & $100$ & $120$ & $-$\
SBS 1543+543 & $0.0096$ & $2.6 \pm 0.2$ & $-0.41 \pm 0.06^f$ & $4.5^g$ & $ 3.9^h$ & $1.5^h$ & $75$ & $128$ & $33$\
PKS 0439$-$433 & $0.1011$ & $0.43 \pm 0.03$ & $+0.28 \pm 0.08^f$ & $1023 \pm 47^g$ & $< 0.064^{e,h}$ & $< 3.1^{e,h}$ & $290^i$ & $109$ & $< 0.3$\
$^\ast$The quoted errors are the sums in quadrature of the measurement errors and $\approx 10$% systematic errors on the flux density scale.\
$^a$The [H[i]{}]{} masses assume a $\Lambda$-cold-dark-matter cosmology, with $H_0 = 67.8$ [kms$^{-1}$]{} Mpc$^{-1}$, $\Omega_m = 0.308$, and $\Omega_\Lambda = 0.692$ [@planck16].\
$^b$${\Delta V_{\rm 20}}$ is the velocity width between 20% points of the peak [H[i]{} 21cm]{} flux density.\
$^c$${\Delta V_{\rm 90}}$ is the velocity width containing 90% of the equivalent width of the low-ionization metal absorption lines.\
$^d$The gas fraction, $f_{\rm HI}$, is defined here as the ratio of the [H[i]{}]{} mass to the stellar mass [@huang12].\
$^e$These are $3\sigma$ limits on the integrated [H[i]{} 21cm]{} line flux density and the [H[i]{}]{} mass, assuming a Gaussian line profile with an FWHM of $100$ [kms$^{-1}$]{}.\
References for literature absorbers: $^f$Metallicity: @dutta15, @tripp05, @bowen05; $^g$Stellar mass: @christensen14, @rosenberg06; $^h$[H[i]{} 21cm]{} data: @bowen01b, @kanekar01e, @chengalur15.\
$^i$The value of ${\Delta V_{\rm 20}}$ for the $z=0.1010$ sub-DLA towards PKS 0439$-$433 is for the CO J=1-0 line [@neeleman16b].
[**J0930+2845**]{} $\bm{(z = 0.0228)}$: The $z = 0.0228$ DLA towards J0930+2845 has [$N_{\rm HI}$]{} $=(5.6 \pm 1.2) \times 10^{20}$ [cm$^{-2}$]{}. Unfortunately, most of the metal lines covered by the HST-COS spectrum for this DLA are either saturated or not detected: we obtain log\[N(Si[ii]{})/[cm$^{-2}$]{}\]$> 14.00$, log\[N(O[i]{})/[cm$^{-2}$]{}\] $>15.25$, and log\[N(S[ii]{})/[cm$^{-2}$]{}\] $< 15.13$. Combining the Si[ii]{} lower limit and the S[ii]{} upper limit yields the metallicity range \[M/H\] $=[-2.26,-0.77]$. The [H[i]{}]{} mass of the galaxy is ${M_{\rm HI}}= (1.75 \pm 0.18) \times 10^9 \; {M_{\odot}}$, while the velocity spread between 20% points is ${\Delta V_{\rm 20}}\approx 165$ km s$^{-1}$ [see also @haynes11]. No spectroscopically confirmed galaxy is known at or near the DLA redshift. The nearest galaxy, located at RA=142.50583$^\circ$, Dec=28.81740$^\circ$ (at an impact parameter of $\approx 7.9''$, i.e. $\approx 3.6$ kpc, to the quasar line-of-sight) is clearly detected in Sloan Digital Sky Survey (SDSS) images, with a photometric redshift of $z_{\rm phot} = 0.0207$ [@brescia14], consistent with the DLA redshift. Applying the [kcorrect]{} software [@blanton07] to the SDSS photometry of this galaxy yields a stellar mass of $\approx 6.9^{+4.6}_{-0.5} \times 10^7 \; {M_{\odot}}$.
[**J0951+3307**]{} $\bm{(z = 0.0054)}$: The $z=0.0054$ DLA towards J0951+3307 has [$N_{\rm HI}$]{} $=(1.0 \pm 0.25) \times 10^{21}$ [cm$^{-2}$]{}, the highest of our sample. The Arecibo [H[i]{}]{} spectrum yields a low [H[i]{}]{} mass for this DLA, ${M_{\rm HI}}=(2.37 \pm 0.24) \times 10^8 \;
{M_{\odot}}$, consistent with a dwarf galaxy. However, the [H[i]{} 21cm]{} line is quite wide, with ${\Delta V_{\rm 20}}\approx 130$ km s$^{-1}$. We measure log\[N(S[ii]{})/[cm$^{-2}$]{}\] $=(15.16 \pm 0.09)$, implying \[M/H\] $=-0.99 \pm 0.14$. A well-known galaxy, UGC5282 (at $ z= 0.0052$ and at an impact parameter of $\approx 11.2''$, i.e. $\approx 1.2$ kpc), is the likely DLA host; this has been earlier detected in [H[i]{} 21cm]{} emission by @schneider90, with an integrated [H[i]{} 21cm]{} line flux density of $(2.25 \pm 0.5)$ Jy [kms$^{-1}$]{}, consistent within the errors with our estimate of $(1.83 \pm 0.01)$ Jy [kms$^{-1}$]{}. @ann15 used the SDSS photometry of UGC5282 to infer a stellar mass of $\approx 10^{7.6} \; {M_{\odot}}$, after correcting for the spatial extent of the galaxy. We obtain a gas-to-stars ratio of $M_{\rm gas}/M_{\rm stars} \approx 6$, not atypical for dwarf galaxies [e.g. @begum08].
[**J1415+1634**]{} $\bm{(z = 0.0077)}$: The $z = 0.0077$ sub-DLA towards J1415+1634 has [$N_{\rm HI}$]{} $=(5.0 \pm 1.2) \times 10^{19}$ [cm$^{-2}$]{}. The Arecibo [H[i]{} 21cm]{} spectrum yields an [H[i]{}]{} mass of $(9.65 \pm 0.97) \times 10^{8} \; {M_{\odot}}$, with ${\Delta V_{\rm 20}}\approx 100$ [kms$^{-1}$]{}. We obtain log\[N(O[i]{})/[cm$^{-2}$]{}\] $=14.48 \pm 0.04$ from the HST-COS spectrum, yielding \[M/H\] $=-1.91 \pm 0.11$. This is a lower limit to \[M/H\], as ionization corrections may be significant for this sub-DLA. The host galaxy is likely to be UGC9126 (at $z = 0.007576$), at an impact parameter of $\approx 80.1''$, i.e. $\approx 12.7$ kpc, to the quasar sightline. @wong06 estimate its integrated [H[i]{} 21cm]{} line flux density to be $\approx 5.2$ Jy [kms$^{-1}$]{}, $\approx 1.5$ times larger than our estimate for the absorber host. This is likely to be due to the relatively large impact parameter; note that the Arecibo primary beam has an FWHM of $\approx 3.4'$. Finally, the stellar mass of UGC9126 has been estimated to be quite low, $\approx (10^7 - 10^{7.42}) \; {M_{\odot}}$ by @chang15, via a fit to its optical and mid-infrared spectral energy distribution. With $M_{\rm gas}/M_{\rm stars} \approx 30-100$, the host of the $z=0.0077$ sub-DLA appears to be an extremely gas-rich galaxy.
[**J1512+0128**]{} $\bm{(z = 0.0295)}$: We obtain [$N_{\rm HI}$]{} $=(2.5 \pm 0.6) \times 10^{20}$ [cm$^{-2}$]{}for the $z = 0.0295$ DLA towards J1512+0128, and ${M_{\rm HI}}=(5.15 \pm 0.52) \times 10^9 \; {M_{\odot}}$, the highest in our sample. The [H[i]{} 21cm]{} velocity spread is ${\Delta V_{\rm 20}}\approx 230$ [kms$^{-1}$]{}, consistent with that expected from a massive galaxy. We derive a metallicity of \[M/H\] $=-1.29 \pm 0.13$ from the Si[ii]{} line detected in the HST-COS spectrum. The absorber appears to be part of a galaxy group: six SDSS galaxies are present within $\approx 3'$ of the quasar line-of-sight and with redshifts within $0.005$ of the absorber redshift. The nearest identified galaxies with confirmed spectroscopic redshifts are part of a triple system (UZC-CG 236), with an impact parameter of $\approx 0.97'$ (i.e. $\approx 35$ kpc) to the quasar line-of-sight. However, it is possible that the absorber might arise in a fainter galaxy with a lower impact parameter.
[**J1553+3548**]{} $\bm{(z = 0.0829)}$: This system is also a sub-DLA, with [$N_{\rm HI}$]{} $=(5.0 \pm 1.2) \times 10^{19}$ [cm$^{-2}$]{}. We obtain ${M_{\rm HI}}= (3.93 \pm 0.61) \times 10^9 \; {M_{\odot}}$ and ${\Delta V_{\rm 20}}\approx 270$ [kms$^{-1}$]{} from the [H[i]{} 21cm]{} spectrum. The Si[ii]{} line detected in the HST-COS spectrum yields \[M/H\] $=-1.09 \pm 0.11$, before ionization corrections. @battisti12 estimate the correction for this ion to lie in the range $[-0.38,-0.14]$; this would imply \[M/H\] $=-1.35 \pm 0.16$. We also detect C[ii]{}\*$\lambda$1335 absorption in this absorber, yielding log\[N(C[ii]{}\*)/[cm$^{-2}$]{}\]=$13.41 \pm 0.14$. A galaxy at low impact parameter ($\approx 8.8''$) is seen in the SDSS imaging. We used the Low Resolution Imaging Spectrograph on the Keck telescope to obtain a spectrum of this galaxy (which will be discussed elsewhere) and measured $z=0.0827$ from the H$\beta$ and H$\alpha$ lines, indicating that this object is likely to be the DLA host. We infer a stellar mass of $23.4^{+34.6}_{-3.9} \times 10^7 \; {M_{\odot}}$, applying [kcorrect]{} to the SDSS photometry of the galaxy.
[**J1619+3342**]{} $\bm{(z = 0.0963)}$: We were unable to detect [H[i]{} 21cm]{} emission from the $z = 0.0963$ DLA towards J1619+3342, obtaining the $3\sigma$ upper limit ${M_{\rm HI}}\leq 1.9 \times 10^9 \; {M_{\odot}}$. We obtain [$N_{\rm HI}$]{} $=(4.0 \pm 1.4) \times 10^{20}$ [cm$^{-2}$]{} from the HST-COS spectrum, and a metallicity of \[M/H\]=$-1.09 \pm 0.16$ from the S[ii]{} line [see also @meiring11; @battisti12]. There are no galaxies at the DLA redshift in the SDSS spectroscopic catalog and no obvious candidate hosts close to the quasar line-of-sight. This system, the only absorber of our sample that does not have an [H[i]{} 21cm]{} emission detection, also has no candidate optical counterpart, suggesting that it is likely to be an optically faint, low-mass galaxy.
Discussion and Summary {#sec:discuss}
======================
We emphasize at the outset that our six absorbers are [*not*]{} an unbiased sample: J0951+3307 and J1415+1634 were targeted with the HST due to the presence of a low-mass, gas-rich galaxy within 15 kpc of the quasar sightline, while J1512+0128 is part of the GASS sample [@borthakur15]. Caution must hence be taken when extending our results to general DLA samples.
Our observations have yielded [H[i]{}]{} masses or [H[i]{}]{} mass limits for six new DLAs and sub-DLAs. There are now nine such absorbers with estimates of the atomic gas mass (see Table \[tab:data2\]). Five systems are DLAs, with ${\ensuremath{N_{\rm HI}}}= (2.5-10) \times 10^{20}$ [cm$^{-2}$]{}, while four are sub-DLAs, with ${\ensuremath{N_{\rm HI}}}= (2-7) \times 10^{19}$ [cm$^{-2}$]{}. Table \[tab:data2\] also lists the derived [$N_{\rm HI}$]{} values, metallicities, stellar masses, the [H[i]{} 21cm]{} (or CO) velocity spread between 20% points (${\Delta V_{\rm 20}}$), the velocity spread of low-ionization metal absorption lines (${\Delta V_{\rm 90}}$), and the gas fraction $f_{\rm HI} = M_{\rm HI}/M_*$ of the above nine systems.
The [H[i]{}]{} masses of the nine absorbers of our sample lie in the range $3.3 \times 10^7 - 5.2 \times
10^9 \; {M_{\odot}}$, significantly lower, in all cases, than the knee of the local [H[i]{}]{} mass function [M\*$_{\rm HI} \approx 10^{10} \; {M_{\odot}}$; e.g. @martin10]. This is consistent with expectations for the relatively low [H[i]{}]{} column densities of most of the nine absorbers [e.g. @zwaan05]. While the low [H[i]{}]{} mass of the high-${\ensuremath{N_{\rm HI}}}$ ($\approx 10^{21}$ [cm$^{-2}$]{}) DLA towards J0951+3307 may appear surprising, this sightline was selected for the presence of a low-mass, gas-rich dwarf close to the quasar sightline. Overall, it appears that massive galaxies do not dominate the cross-section for damped or sub-damped absorption at low redshifts [see also @zwaan05].
While the angular resolutions of our Arecibo data ($\approx 3.4'$, i.e. $\approx 23-370$ kpc at the absorber redshift) are too coarse to allow a direct identification of the absorber host galaxies, we have used SDSS spectroscopy and photometry to identify candidate host galaxies for five of our six absorbers, and to estimate the stellar mass for four systems. Including systems from the literature, five of the six absorbers with stellar mass estimates have very low stellar masses ($\lesssim 2 \times 10^8 {M_{\odot}}$) and high gas fractions ($f_{\rm HI} \approx 5-100$), amongst the highest gas fractions of galaxies in the local Universe [e.g. @huang12]. Only two of these systems were targetted due to their gas richness. Further, even the sixth absorber (at $z=0.101$ towards PKS0439$-$433) has a high H$_2$ mass, M$_{\rm H_2} = 4.2 \times 10^9 \, {M_{\odot}}$ [@neeleman16b], yielding a gas fraction (including molecular gas) of $ 0.42 \lesssim f_{\rm HI+H_2} \lesssim 0.7$, quite high for late-type disk galaxies. The metallicities of most absorbers are relatively low, with seven of the nine systems having \[M/H\] $\lesssim -1$; this is surprising for objects in the nearby Universe [although consistent with metallicity evolution in DLAs; @rafelski12]. Most of the absorbers of the sample thus appear to arise in gas-rich galaxies, with low star formation activity. However, the [H[i]{} 21cm]{} velocity spreads are too large to be explained by an origin in individual dwarf galaxies. It is possible that the large velocity widths arise from [H[i]{} 21cm]{} emission from multiple faint galaxies in the relatively large Arecibo beam; we are now investigating this with interferometric [H[i]{} 21cm]{} studies.
In conclusion, we have used a combination of HST-COS FUV spectroscopy, Arecibo [H[i]{} 21cm]{} emission spectroscopy and SDSS photometry to estimate or constrain the [H[i]{}]{} column density, the metallicity, the atomic gas mass, and the stellar mass for a sample of six DLAs and sub-DLAs at $z \lesssim 0.1$. We obtain [H[i]{}]{} masses $\approx (0.2 - 5) \times 10^9 {M_{\odot}}$, stellar masses $\lesssim 5 \times 10^8 ~{M_{\odot}}$, low metallicities, $\lesssim 0.1$ solar, and high gas fractions, $f_{\rm HI} \equiv {M_{\rm HI}}/{M_*} \gtrsim 5$ (amongst the highest in the nearby Universe), for the absorbers of our sample. The large velocity spreads of the [H[i]{} 21cm]{} and CO emission lines (${\Delta V_{\rm 20}}\approx 100-290$ [kms$^{-1}$]{}), the high gas fractions ($f_{\rm HI} \approx 5-100$), and the low metallicities and stellar masses suggest that the absorbers are gas-rich galaxies with a low star formation efficiency.
Acknowledgments {#acknowledgments .unnumbered}
===============
It is a pleasure to thank Chris Salter and Phil Perrilat for much help with the Arecibo observations. NK acknowledges support from the Department of Science and Technology via a Swarnajayanti Fellowship (DST/SJF/PSA-01/2012-13). The Arecibo Observatory is operated by SRI International under a cooperative agreement with the National Science Foundation (AST-1100968), and in alliance with Ana G. Méndez-Universidad Metropolitana, and the Universities Space Research Association.
\[lastpage\]
[^1]: E-mail: [email protected]
|
---
address: ' Institut de Théorie des Phénomènes Physiques, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland'
author:
- 'F. L. Bezrukov'
title: The Standard model Higgs as the inflaton
---
Introduction
============
This talk is based on the recent work [@Bezrukov:2007ep], and closely follows it. Note, that the expression for the inflationary potential presented here differs from the one presented in the original work—both expressions coincide in the region relevant for inflation, while the expression given here has a wider range of validity (down to the Standard Model regime).
The fact that our universe is almost flat, homogeneous and isotropic is often considered as a strong indication that the Standard Model (SM) of elementary particles is not complete. Indeed, these puzzles, together with the problem of generation of (almost) scale invariant spectrum of perturbations, necessary for structure formation, are most elegantly solved by inflation [@Starobinsky:1979ty; @Starobinsky:1980te; @Mukhanov:1981xt; @Guth:1980zm; @Linde:1981mu; @Albrecht:1982wi]. The majority of present models of inflation require an introduction of an additional scalar—the “inflaton”. Inflaton properties are constrained by the observations of fluctuations of the Cosmic Microwave Background (CMB) and the matter distribution in the universe. Though the mass and the interaction of the inflaton with matter fields are not fixed, the well known considerations prefer a heavy scalar field with a mass $\sim
\unit[10^{13}]{GeV}$ and extremely small self-interacting quartic coupling constant $\lambda \sim 10^{-13}$ for realization of the chaotic inflationary scenario [@Linde:1983gd]. This value of the mass is close to the GUT scale, which is often considered as an argument in favour of existence of new physics between the electroweak and Planck scales.
It was recently demonstrated in [@Bezrukov:2007ep] that the SM itself can give rise to inflation, provided non-minimal copling of the Higgs field with gravity. The spectral index and the amplitude of tensor perturbations can be predicted and be used to distinguish this possibility from other models for inflation; these parameters for the SM fall within the $1\sigma$ confidence contours of the WMAP-5 observations [@Komatsu:2008hk].
To explain our main idea, let us consider the Lagrangian of the SM non-minimally coupled to gravity, $$\label{main}
L_{\mathrm{tot}}= L_{\mathrm{SM}} - \frac{M^2}{2} R -\xi H^\dagger HR
\;,$$ where $L_{\mathrm{SM}}$ is the SM part, $M$ is some mass parameter, $R$ is the scalar curvature, $H$ is the Higgs field, and $\xi$ is an unknown constant to be fixed later. The third term in (\[main\]) is in fact required by the renormalization properties of the scalar field in a curved space-time background [@Birrell:1982ix], so, in principle, it should be added to the usual SM Lagrangian with some constant. Here, we will analyse the situation with large non-minimal coupling parameter $\xi\gg1$, but still not too large for the non-minimal term to contribute significantly to the Plank mass in the SM regime ($H\sim v$), i.e. $\sqrt{\xi} \lll 10^{17}$. Thus, we have $M\simeq M_P=(8\pi G_N)^{-1/2}=\unit[2.4\times 10^{18}]{GeV}$.
It is well known that inflation has interesting properties in models of this type [@Spokoiny:1984bd; @Futamase:1987ua; @Salopek:1988qh; @Fakir1990; @Kaiser:1994wj; @Kaiser:1994vs; @Komatsu:1999mt]. However, in these works the scalar was not identified with the Higgs field of the SM. Basically, most attempts were made to identify the inflaton field with the GUT Higgs field. In this case one naturally gets into the regime of induced gravity (where, unlike this paper, $M=0$ and $M_P$ is generated from the non-minimal coupling term by the Higgs vacuum expectation value). In this case the Higgs field decouples from the other fields of the model [@vanderBij:1993hx; @CervantesCota:1995tz; @Bij1995], which is generally undesirable. Here we demonstrate, that when the SM Higgs boson is coupled non-minimally to gravity, the scales for the electroweak physics and inflation are separate, the electroweak properties are unchanged, while for much larger field values the inflation is possible.
The paper is organised as follows. We start from discussion of inflation in the model, and use the slow-roll approximation to find the perturbation spectra parameters. Then we will argue in Section \[sec:radcorr\] that quantum corrections are unlikely to spoil the classical analysis we used in Section \[sec:cmb\]. We conclude in Section \[sec:concl\].
Inflation and CMB fluctuations {#sec:cmb}
==============================
Let us consider the scalar sector of the Standard Model, coupled to gravity in a non-minimal way. We will use the unitary gauge $H=h/\sqrt{2}$ and neglect all gauge interactions for the time being, they will be discussed later in Section \[sec:radcorr\]. Then the Lagrangian has the form: $$\label{eq:1}
S_{J} =\int d^4x \sqrt{-g} \Bigg\{
- \frac{M^2+\xi h^2}{2}R
\\
+ \frac{{\partial_\mu}h\partial^\mu h}{2}
-\frac{\lambda}{4}\left(h^2-v^2\right)^2
\Bigg\}
\;.
$$ This Lagrangian has been studied in detail in many papers on inflation [@Salopek:1988qh; @Fakir1990; @Kaiser:1994vs; @Komatsu:1999mt], we will reproduce here the main results of [@Salopek:1988qh; @Kaiser:1994vs]. Compared to [@Bezrukov:2007ep] we present a better approximation for the inflationary potential here. To simplify the formulae, we will consider only $\xi$ in the region $1\ll\sqrt{\xi}\lll10^{17}$, in which $M \simeq M_P$ with very good accuracy.
It is possible to get rid of the non-minimal coupling to gravity by making the conformal transformation from the Jordan frame to the Einstein frame $$\label{eq:2}
\hat{g}_{\mu\nu} = \Omega^2 g_{\mu\nu}
\;,\quad
\Omega(h)^2 = 1 + \frac{\xi h^2}{M_P^2}
\;.$$ This transformation leads to a non-minimal kinetic term for the Higgs field. So, it is convenient to make the change to the new scalar field $\chi$ with $$\label{eq:3}
\frac{d\chi}{dh}=\frac{\sqrt{\Omega^2+\frac{3}{2}M_P^2\left(\frac{d(\Omega^2)}{dh}\right)^2}}{\Omega^2}
=\frac{\sqrt{1 + (\xi+6\xi^2)\frac{h^2}{M_P^2}}}{1 + \xi\frac{ h^2}{M_P^2}}
\;.$$ Finally, the action in the Einstein frame is $$\label{eq:4}
S_E =\int d^4x\sqrt{-\hat{g}} \Bigg\{
- \frac{M_P^2}{2}\hat{R}
+ \frac{{\partial_\mu}\chi\partial^\mu \chi}{2}
- U(\chi)
\Bigg\}
\;,$$ where $\hat{R}$ is calculated using the metric $\hat{g}_{\mu\nu}$ and the potential is $$\label{eq:5}
U(\chi) =
\frac{1}{\Omega(h(\chi))^4}\frac{\lambda}{4}\left(h(\chi)^2-v^2\right)^2
\;.$$ For small field values $h,\chi<M_P/\xi$ the change of variables is trivial, $h\simeq\xi$ and $\Omega^2\simeq1$, so the potential for the field $\chi$ is the same as that for the initial Higgs field and we get into the SM regime. For $h,\chi\gg M_P/\xi$ the situation changes a lot. In this limit the variable change (\[eq:3\]) is [^1] $$\label{eq:hlarge}
\Omega(h)^2\simeq \exp\left(\frac{2\chi}{\sqrt{6}M_P}\right)
\;.$$ The potential for the Higgs field is exponentially flat for large $\xi$ and has the form $$\label{eq:6}
U(\chi) = \frac{\lambda M_P^4}{4\xi^2}
\left(
1-\exp\left(
-\frac{2\chi}{\sqrt{6}M_P}
\right)
\right)^{2}
\;.$$ The full effective potential in the Einstein frame is presented in Fig. \[fig:Ueff\]. It is the flatness of the potential at $\chi\gtrsim M_P$ which makes the successful (chaotic) inflation possible.
Basically, there are two distinct scales—for low field values $h,\chi\ll
M_P/\xi$ we have the SM, for high field values $h\gg M_P/\sqrt{\xi}$ ($\chi>M_P$) we have inflation with exponentially flat potential (\[eq:6\]) and the Higgs field is decoupled from all other SM fields (because $\Omega\propto
h$, see Section \[sec:radcorr\]). In the intermediate region $M_P/\xi\ll h\ll M_P/\sqrt{\xi}$ ($M_P/\xi\ll\chi<M_P$) the coupling with other particles is not suppressed ($\Omega\sim 1$), while the potential and change of variables are still given by (\[eq:6\]) and (\[eq:hlarge\]).
![The allowed WMAP region for inflationary parameters ($r$, $n$). The green boxes are our predictions supposing 50 and 60 e-foldings of inflation. Black and white dots are predictions of usual chaotic inflation with $\lambda\phi^4$ and $m^2\phi^2$ potentials, HZ is the Harrison-Zeldovich spectrum.[]{data-label="fig:wmap"}](potential){width="\textwidth"}
![The allowed WMAP region for inflationary parameters ($r$, $n$). The green boxes are our predictions supposing 50 and 60 e-foldings of inflation. Black and white dots are predictions of usual chaotic inflation with $\lambda\phi^4$ and $m^2\phi^2$ potentials, HZ is the Harrison-Zeldovich spectrum.[]{data-label="fig:wmap"}](wmap05-e){width="86.50000%"}
Analysis of the inflation in the Einstein frame [^2] can be performed in the standard way using the slow-roll approximation. The slow roll parameters (in notations of [@Lyth:1998xn]) can be expressed analytically as functions of the field $h(\chi)$ using (\[eq:3\]) and (\[eq:5\]) (we give here the expressions for the case [^3] $h^2\gtrsim M_P^2/\xi\gg v^2$, $\xi\gg1$, exact expressions can be found in [@Kaiser:1994vs]), $$\begin{aligned}
\label{eq:7}
\epsilon & =& \frac{M_P^2}{2}\left(\frac{dU/d\chi}{U}\right)^2
\simeq\frac{4 M_P^4 }{3
\xi^2h^4}
\;, \\
\eta & = & M_P^2\frac{d^2U/d\chi^2}{U}
\simeq \frac{4 M_P^4}{3 \xi^2 h^4 }\left(1-\frac{\xi h^2}{M_P^2}\right)
\;, \\
\zeta^2 &= & M_P^4\frac{(d^3U/d\chi^3)dU/d\chi}{U^2}
\simeq \frac{16 M_P^6 }{9\xi^3 h^6}\left(\frac{\xi h^2}{M_P^2}-3\right)
\;.\end{aligned}$$ Slow roll ends when $\epsilon\simeq1$, so the field value at the end of inflation is $h_{\mathrm{end}}\simeq(4/3)^{1/4}M_P/\sqrt{\xi}\simeq1.07M_P/\sqrt{\xi}$. The number of e-foldings for the change of the field $h$ from $h_0$ to $h_{\mathrm{end}}$ is given by $$\label{eq:8}
N = \int_{h_{\mathrm{end}}}^{h_0}
\frac{1}{M_P^2}\frac{U}{dU/dh}\left(\frac{d\chi}{dh}\right)^2dh
\simeq \frac{3}{4}\frac{h_0^2-h_{\mathrm{end}}^2}{M_P^2/\xi}
\;.$$ We see that for all values of $\sqrt{\xi}\lll10^{17}$ the scale of the Standard Model $v$ does not enter in the formulae, so the inflationary physics is independent on it.
After end of the slow roll the $\chi$ field enters oscillatory stage with diminishing amplitude. After the oscillation amplitude falls below $M_P/\xi$, the situation returns to the SM one, so at this moment the reheating is imminent due to the SM interactions, which guarantees the minimum reheating temperature $T_{\mathrm{reh}}\gtrsim
(\frac{15\lambda}{8\pi^2 g^*})^{1/4}\frac{M_P}{\xi}\simeq\unit[1.5\times
10^{13}]{GeV}$, where $g^*=106.75$ is the number of degrees of freedom of the SM. Careful analysis may give a larger temperature generated during the decay of the oscillating $\chi$ field, but definitely below the energy scale at the end of the inflation $T_{\mathrm{reh}}<(\frac{2\lambda}{\pi^2
g^*})^{1/4}\frac{M_P}{\sqrt{\xi}}\simeq\unit[2\times10^{15}]{GeV}$.
As far as the reheating mechanism and the universe evolution after the end of the inflation is fixed in the model, the number of e-foldings for the the COBE scale entering the horizon can be calculated (see [@Lyth:1998xn]). Here we estimate it as $N_{\mathrm{COBE}}\simeq62$ (exact value depends on the detailed analysis of reheating, which will be done elsewhere). The corresponding field value is $h_{\mathrm{COBE}}\simeq9.4M_P/\sqrt{\xi}$. Inserting (\[eq:8\]) into the COBE normalization $U/\epsilon=(0.027M_P)^4$ we find the required value for $\xi$ $$\label{eq:9}
\xi \simeq \sqrt{\frac{\lambda}{3}}\frac{N_{\mathrm{COBE}}}{0.027^2}
\simeq 49000\sqrt{\lambda}
= 49000\frac{m_H}{\sqrt{2}v}
\;.$$ Note, that if one could deduce $\xi$ from some fundamental theory this relation would provide a connection between the Higgs mass and the amplitude of primordial perturbations.
The spectral index $n_s=1-6\epsilon+2\eta$ calculated for $N=60$ (corresponding to the scale $k=0.002/\mathrm{Mpc}$) is $n_s\simeq1-8(4N+9)/(4N+3)^2\simeq0.97$. The tensor to scalar perturbation ratio [@Komatsu:2008hk] is $r=16\epsilon\simeq192/(4N+3)^2\simeq0.0033$. The predicted values are well within one sigma of the current WMAP measurements [@Komatsu:2008hk], see Fig. \[fig:wmap\].
Radiative corrections {#sec:radcorr}
=====================
An essential point for inflation is the flatness of the scalar potential in the region of the field values $h\sim10M_P/\sqrt{\xi}$ ($\chi\sim 6 M_P$). It is important that radiative corrections do not spoil this property. Of course, any discussion of quantum corrections is flawed by the non-renormalizable character of gravity, so the arguments we present below are not rigorous.
There are two qualitatively different type of corrections one can think about. The first one is related to the quantum gravity contribution. It is conceivable to think [@Linde:1987yb] that these terms are proportional to the energy density of the field $\chi$ rather than its value and are of the order of magnitude $U(\chi)/M_P^4
\sim \lambda/\xi^2$. They are small at large $\xi$ required by observations. Moreover, adding non-renormalizable operators $h^{4+2n}/M_P^{2n}$ to the Lagrangian (\[eq:1\]) also does not change the flatness of the potential in the inflationary region.[^4]
Other type of corrections is induced by the fields of the Standard Model coupled to the Higgs field. In one loop approximation these contributions have the structure $$\Delta U \sim \frac{m^4(\chi)}{64\pi^2} \log\frac{m^2(\chi)}{\mu^2}~,
\label{1loop}$$ where $m(\chi)$ is the mass of the particle (vector boson, fermion, or the Higgs field itself) in the background of field $\chi$, and $\mu$ is the normalization point. Note that the terms of the type $m^2(\chi)
M_P^2$ (related to quadratic divergences) do not appear in scale-invariant subtraction schemes that are based, for example, on dimensional regularisation (see a relevant discussion in [@Shaposhnikov:2006xi; @Meissner:2006zh; @Shaposhnikov:2007nj; @Meissner:2007xv]). The masses of the SM fields can be readily computed [@Salopek:1988qh] and have the form $$m_{\psi,A}(\chi) = \frac{m(v)}{v}\frac{h(\chi)}{\Omega(\chi)}
\;,\quad
m^2_H(\chi) = \frac{d^2U}{d\chi^2}$$ for fermions, vector bosons and the Higgs (inflaton) field. It is crucial that for large $\chi$ these masses approach different constants (i.e. the one-loop contribution is as flat as the tree potential) and that (\[1loop\]) is suppressed by the gauge or Yukawa couplings in comparison with the tree term. In other words, one-loop radiative corrections do not spoil the flatness of the potential as well. This argument is identical to the one given in [@Salopek:1988qh].
Conclusions {#sec:concl}
===========
Non-minimal coupling of the Higgs field to gravity leads to the possibility of chaotic inflation in SM. Specific predictions for the primordial perturbation spectrum are obtained. Specifically, very small amount of tensor perturbations is expected, which means that future CMB experiments measuring the B-mode of the CMB polarization (PLANCK) can distinguish between the described scenario from other models (based, e.g. on inflaton with quadratic potential).
At the same time, we expect that the Higgs potential does not enter into the string coupling regime, nor generates another vacuum up to the scale of at least $M_P/\xi\sim\unit[10^{14}]{GeV}$, so we expect the Higgs mass to be in the window $\unit[130]{GeV}<M_H<\unit[190]{GeV}$ (see, eg. [@Pirogov:1998tj]), otherwise the inflation would be impossible.
The inflation mechanism we discussed has in fact a general character and can be used in many extensions of the SM. Thus, the $\nu$MSM of [@Asaka:2005an; @Asaka:2005pn; @Bezrukov:2005mx; @Asaka:2006ek; @Shaposhnikov:2006xi; @Shaposhnikov:2006nn; @Asaka:2006rw; @Asaka:2006nq; @Bezrukov:2006cy; @Gorbunov:2007ak; @Shaposhnikov:2008pf; @Laine:2008pg] (SM plus three light fermionic singlets) can explain simultaneously neutrino masses, dark matter, baryon asymmetry of the universe and inflation without introducing any additional particles (the $\nu$MSM with the inflaton was considered in [@Shaposhnikov:2006xi]). This provides an extra argument in favour of absence of a new energy scale between the electroweak and Planck scales, advocated in [@Shaposhnikov:2007nj].
Acknowledgements {#acknowledgements .unnumbered}
================
The author thank M. Shaposhnikov, S. Sibiryakov, V. Rubakov, G. Dvali, I. Tkachev, O. Ruchayskiy, H.D. Kim, P. Tinyakov, and A. Boyarsky for valuable discussions. This work was supported by the Swiss National Science Foundation.
References {#references .unnumbered}
==========
[10]{}
F.L. Bezrukov and M. Shaposhnikov, Phys. Lett. B659 (2008) 703. A.A. Starobinsky, JETP Lett. 30 (1979) 682. A.A. Starobinsky, Phys. Lett. B91 (1980) 99. V.F. Mukhanov and G.V. Chibisov, JETP Lett. 33 (1981) 532. A.H. Guth, Phys. Rev. D23 (1981) 347. A.D. Linde, Phys. Lett. B108 (1982) 389. A. Albrecht and P.J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220. A.D. Linde, Phys. Lett. B129 (1983) 177. WMAP, E. Komatsu et al., (2008), arXiv:0803.0547 \[astro-ph\]. N.D. Birrell and P.C.W. Davies, Quantum Fields in Curved Space (Cambridge, UK: Univ. Pr., 1982).
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N. Makino and M. Sasaki, Prog. Theor. Phys. 86 (1991) 103. R. Fakir, S. Habib and W. Unruh, Astrophys. J. 394 (1992) 396. D.H. Lyth and A. Riotto, Phys. Rept. 314 (1999) 1. A.D. Linde, Phys. Lett. B202 (1988) 194. M. Shaposhnikov and I. Tkachev, Phys. Lett. B639 (2006) 414. K.A. Meissner and H. Nicolai, Phys. Lett. B648 (2007) 312. M. Shaposhnikov, (2007), arXiv:0708.3550 \[hep-th\]. K.A. Meissner and H. Nicolai, (2007), arXiv:0710.2840 \[hep-th\]. Y.F. Pirogov and O.V. Zenin, Eur. Phys. J. C10 (1999) 629. T. Asaka, S. Blanchet and M. Shaposhnikov, Phys. Lett. B631 (2005) 151. T. Asaka and M. Shaposhnikov, Phys. Lett. B620 (2005) 17. F. Bezrukov, Phys. Rev. D72 (2005) 071303. T. Asaka, M. Shaposhnikov and A. Kusenko, Phys. Lett. B638 (2006) 401. M. Shaposhnikov, Nucl. Phys. B763 (2007) 49. T. Asaka, M. Laine and M. Shaposhnikov, JHEP 06 (2006) 053. T. Asaka, M. Laine and M. Shaposhnikov, JHEP 0701 (2007) 091. F. Bezrukov and M. Shaposhnikov, Phys. Rev. D75 (2007) 053005. D. Gorbunov and M. Shaposhnikov, (2007), arXiv:0705.1729 \[hep-ph\]. M. Shaposhnikov, (2008), arXiv:0804.4542 \[hep-ph\]. M. Laine and M. Shaposhnikov, (2008), arXiv:0804.4543 \[hep-ph\].
[^1]: The following two formulae have wider validity range than those in [@Bezrukov:2007ep], which are valid only for $h\gg M_P/\sqrt{\xi}$.
[^2]: The same results can be obtained in the Jordan frame [@Makino1991; @Fakir:1992cg].
[^3]: These formulas are valid up to the end of the slow roll regime $h_\mathrm{end}$, while the formulas (10) and (11) in [@Bezrukov:2007ep] are applicable only for the earlier inflationary stages, $h^2\gg M_P^2/\xi$, which is sufficient to calculate primordial spectrum parameters $n_s$ and $r$.
[^4]: Actually, in the Jordan frame, we expect that higher-dimensional operators are suppressed by the effective Planck scale $M_P^2+\xi h^2$.
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abstract: 'Cascading failures and epidemic dynamics, as two successful application realms of network science, are usually investigated separately. How do they affect each other is still one open, interesting problem. In this letter, we couple both processes and put them into the framework of interdependent networks, where each network only supports one dynamical process. Of particular interest, they spontaneously form a feedback loop: virus propagation triggers cascading failures of systems while cascading failures suppress virus propagation. Especially, there exists crucial threshold of virus transmissibility, above which the interdependent networks collapse completely. In addition, the interdependent networks will be more vulnerable if the network supporting virus propagation has denser connections; otherwise the interdependent systems are robust against the change of connections in other layer(s). This discovery differs from previous framework of cascading failure in interdependent networks, where better robustness usually needs denser connections. Finally, to protect interdependent networks we also propose the control measures based on the identification capability. The larger this capability, more robustness the interdependent networks will be.'
author:
- 'Dawei Zhao$^{1\ast}$, Zhen Wang$^{2}$, Gaoxi Xiao$^{3,4}$, Bo Gao$^{5}$, and Lianhai Wang$^{1}$'
bibliography:
- 'reference.bib'
title: The robustness of interdependent networks under the interplay between cascading failures and virus propagation
---
Introduction
============
During the past years, complex networks have proven to be a successful tool in describing a large variety of real-world complex systems, ranging from biological, technological, social to information, engineering, and physical systems [@newman2010networks; @barrat2008dynamical]. The investigations of the structure and dynamics of complex networks have triggered enormous interests, and a lot of remarkable results have been achieved [@newman2002random; @pastor2015epidemic; @cohen2010complex; @song2005self; @wang2013impact; @wang2011coveting; @perc2015double; @song2006origins]. However, vast majority of existing works mainly focus on single networks that are isolated from each other, despite of the fact that many real-world networks usually interact with and depend on each other. In 2010, Buldyrev et al. [@buldyrev2010catastrophic] proposed a new model of networks, so-called interdependent networks, and developed theoretical framework to study the cascading failures of interdependent systems caused by random node removal. Surprisingly, they found that systems made of interdependent networks would be intrinsically more fragile than each isolated component. After that, much research attention moves to more complicated yet more realistic multilayer networks, mainly including interdependent networks [@son2012percolation; @parshani2011inter; @gao2012networks; @wang2014self; @dong2013robustness], interconnected networks [@saumell2012epidemic; @dickison2012epidemics; @zhao2013identifying; @radicchi2013abrupt; @de2014navigability] and multiplex networks [@nicosia2013growing; @zhao2014immunization; @zhao2014multiple; @boccaletti2014structure].
Cascade of failures, as one of the hottest research topics in network science, has attracted great attentions after the seminal idea of Buldyrev [@son2012percolation; @parshani2011inter; @gao2012networks; @wang2014self; @dong2013robustness; @crucitti2004model; @wang2007high; @huang2011robustness; @shao2011cascade; @parshani2010interdependent; @schneider2013towards]. For example, Parshani et al. explored the influence of degree correlation on the robustness of interdependent networks to cascading failures, and found that the systems become more robust when they share higher inter-similarity [@parshani2011inter]. By mapping the random attack to the targeted attack problem, Huang et al. evaluated the cascading failures in interdependent networks under an initial targeted attack [@huang2011robustness]. Shao et al. developed a theoretical framework for understanding the cascade process of failures in interdependent networks with a random number of support and dependence relationships [@shao2011cascade]. Refs. [@parshani2010interdependent; @schneider2013towards] showed that when a small fraction of autonomous nodes were properly selected, the nature of the percolation transition changed from discontinuous to continuous fashions and the cascading failures could be largely suppressed.
Epidemic dynamic [@pastor2015epidemic], as another rapidly developing research area in network science, is broadly used to mimic many real propagation processes, such as disease in human contact networks [@salathe2010high; @wang2014spatial], information and rumor in social networks [@min2014layer], and virus in computer or communication networks [@zhao2013efficient; @gao2013modeling]. Understanding the epidemic spreading processes is thus crucial for developing efficient methods to either prevent propagation of disease, rumor and virus, or accelerate information dissemination. At present, the most popular models to describe the propagation of epidemic include susceptible-infected (SI) model, susceptible-infected-susceptible (SIS) model, and susceptible-infected-recovered (SIR) model [@pastor2015epidemic]. Like other dynamic processes upon networks [@boccaletti2014structure], the recent concerns of epidemic spreading also extend from single networks to multilayer networks [@saumell2012epidemic; @dickison2012epidemics; @zhao2014multiple; @min2014layer].
In spite of great progress of recent years, cascading failures and epidemic dynamics are usually considered as two irrelevant research topics and studied separately. However, in many real world systems, cascading failures and epidemic dynamics often influence and interact with each other. For example, the virus propagation on communication network not only causes node failure or load redistribution of communication network, but also triggers the collapse of other related networks like power grid due to the interdependency relationships between them, thus resulting in cascading failures in interdependent systems. The cascade of failure on other networks in turn enables more nodes or fragmentation to be removed in communication network, which thus suppresses the propagation of the virus. In particular, if the virus is not completely suppressed, it will lead to new cash of nodes and thus triggers successive cascading failures in the interdependent networks. Compared with the existing researches on the robustness of interdependent networks, the above case introduces a novel and much more severe attack method for interdependent networks.
Aim to this issue, here we develop a new framework where the virus propagation could induce cascading failures and cascading failures are able to suppress virus propagation (i.e. forming feedback between cascade dynamics and epidemic dynamics). By means of numerous simulations, we will investigate the interplay of both processes in interdependent scale-free (SF) networks [@barabasi1999emergence], and explore the robustness of interdependent networks under this novel setup.
![A schematic illustration of CF-VP model in interdependent networks. Green nodes represent the functional nodes, while grey nodes represent the removed nodes. Each stage is composed of two substages (see the number in the brackets): disease spreading process and cascade process. Besides, red $\lambda$ means the successful propagation of virus, while black is the opposite case.[]{data-label="fig.1"}](fig1.eps)
Model
=====
Before defining the detailed model, we first survey the general cascading failures of interdependent networks and the SIR dynamics which we use as the paradigmatic example for the collapse process of interdependent systems under attack of the spread of virus.
The general cascading failures in interdependent networks were first proposed in Ref. [@buldyrev2010catastrophic], where there are two networks A and B with the same size of $N$ nodes, then both of them are coupled via one-to-one interdependence. If node A$_i$ (B$_i$, $i = 1,2,\ldots,N$) stops function owing to attack or failure, its inter-layer counterpart B$_i$ (A$_i$) becomes nonfunctional as well. When some nodes on network A (hereafter A-nodes) are removed, the nodes of network B (hereafter B-nodes) that connect to the nonfunctional A-nodes will also be removed (because of the dependence between both networks), which further prunes connections of these B-nodes with the giant component of network B. Subsequently, A-nodes that connect to the non functional B-nodes will stop function and cut their connections with the new giant component of network A (only the nodes that belong to the giant component of network remain functional). These cascade processes repeat until no A-nodes and B-nodes could be removed.
![Fraction $f_I$ of infected nodes versus the time stage. The solid and dash lines represent results on network A of interdependent systems and single-layer networks, respectively. The interdependent networks are SF networks with size $N=2,000$ and same average degree $\langle k_A\rangle =\langle k_B\rangle=$ 4 (black), 6 (blue), 8 (green), and 10 (red). The transmissibility probability is $\lambda=0.5$.[]{data-label="fig.2"}](fig2.eps)
SIR model [@pastor2015epidemic], as one of the most fundamental and important paradigms of epidemic dynamics, classifies the network nodes into three states: susceptible (S), infected (I), or recovered or removed (R). Susceptible nodes are free of epidemic and can get infection via direct contacts with infected counterparts. Infected nodes are assumed to carry the disease and pass it towards susceptible nodes. Recovered (removed) state means the nodes recovered (died) from the disease so that these nodes neither diffuse the infection nor be infected again. In addition, classic SIR model considers discrete time process: at each time step, the infected node can infect its susceptible neighbors with transmission rate $\lambda$, and then becomes recovered or removed state with recovery rate $\delta$.
Now, we turn to our model: cascading failure and disease spreading are coupled via interdependent networks, namely, the interplay between cascading failures and virus propagation (CF-VP for short) in interdependent networks. Given the same interdependent networks as Ref. [@buldyrev2010catastrophic], we give two additional hypotheses: 1) only one network (e.g. network A) supports the propagation of virus; 2) the time scale of cascading failures is much smaller than that of virus propagation, so that virus propagation can repeat until there is no failure node in the systems. Moreover, our model also considers discrete time-step: each time stage contains the virus propagation process and one general cascading failure process. Initially, one random chosen A-node is infected by the virus, then the infected node propagates the virus: it infects its susceptible neighbors with probability $\lambda$, and then becomes removed state with probability $\delta$ (without loss of generality, we use $\delta$ = 1). In particular, if the removed nodes are assumed to be nonfunctional, a general cascading failure process will be triggered in interdependent networks and more nodes may be pruned. If there still exist infected nodes in the networks after the cascading process, a new virus propagation process and the subsequent triggered cascading failure will repeat until no infected nodes exist in the network.
To get a better understanding, Fig. \[fig.1\] provides a schematic example for this novel CF-VP model. Assume A$_2$ being initially infected, it can infect its neighbors with probability $\lambda$. After the spreading process of stage 1, A$_2$ and A$_3$ become removed node and infected node respectively. Due to interdependence, node B$_2$ will be nonfunctional, which subsequently causes node B$_1$ to be removed since it does not belong to the giant component of network B. Similarly, A$_1$ is removed because of the removal of B$_1$. The first stage ends. Now there exists a new infected node A$_3$, it can bring infection to its neighbors A$_4$ and A$_5$. Since A$_3$ becomes nonfunctional soon, new cascade process is triggered: B$_3$ is removed due to losing dependent counterpart; A$_4$ is removed because of separation from the giant component of network A, which in turn causes B$_4$ nonfunctional. In stage 3, even if A$_5$ fails to infect its neighbor, itself and its partner B$_5$ will also be nonfunctional due to the state transition I$\rightarrow$R of A$_5$. Because no giant component exists, A$_6$ and B$_6$ are finally removed and the systems are completely collapsed. From this illustration, it is clear that the virus propagation causes cascading failures, while the cascading failures suppress the virus propagation: S-state node A$_1$ and I-state node A$_4$ are isolated owing to the cascading failures.
Results
=======
![The size $G$ of the remaining giant component of network A versus the transmissibility probability $\lambda$. The interdependent networks are SF networks with average degree $\langle k_B\rangle =8$, $\langle k_A\rangle=$ 4 (squares), 6 (triangles), 8 (circles), 10 (diamonds), and 16 (stars), respectively. The inset features how the threshold $\lambda_c$ changes as a function of $\langle k_A\rangle$.[]{data-label="fig.3"}](fig3.eps)
Results of computation simulations are obtained on interdependent scale-free (SF) networks with average degree $\langle k_A\rangle$ and $\langle k_B\rangle$ of networks A and B. In each CF-VP process, we assume that, initially, only one randomly chosen node is infected on network A. What we are interested is the robustness of interdependent networks against CF-VP process, which is measured by the size $G$ of the remaining giant component of network A when CF-VP ends. Here, it is worth mentioning that CF-VP model finally generates the identical size of the remaining giant component on networks A and B.
![The size $G$ of the remaining giant component of network A versus transmissibility probability $\lambda$. The interdependent networks are SF networks with average degree $\langle k_A\rangle =8$, $\langle k_B\rangle=$ 4 (squares), 6 (triangles), 8 (circles), 10 (diamonds), and 16 (stars), respectively. The inset features how the threshold $\lambda_c$ changes as a function of $\langle k_B\rangle$.[]{data-label="fig.4"}](fig4.eps)
Fig. \[fig.2\] shows the evolution of virus on network A with the proposed CF-VP model, which is featured by the solid lines. To take a direct comparison, we also add the traditional case of virus propagation on single-layer networks (dash lines, i.e. without the interplay of cascading failure and virus propagation). It is obvious that though the fraction of infected nodes in both scenarios is almost identical at the early stages, the following trend becomes greatly different. Comparing with traditional case, CF-VP model not only makes infection reach an peak faster, but also impedes the total infection risk. In fact, it is easy to elucidate these phenomena. At the early stages, only a small fraction of nodes are infected and removed, the interdependent networks are not broken and most nodes are still functional. Thus, the virus propagates on network A almost as on single-layer networks. But with continual propagation of virus, the triggered cascading failures cause more nodes be removed and make interdependent networks collapse into the unconnected fragments. In particular, many infected and susceptible nodes are also removed due to the cascading failures, which in turn leads to the effective suppression of virus propagation (also see Fig. \[fig.1\]). With the CF-VP framework, the role of feedback loop becomes clear: virus propagation induces cascading failure, while cascading failure suppresses virus propagation.
Besides, another interesting observation from Fig. \[fig.2\] is that, similar to traditional case, the spreading scale of virus is larger in network A with denser connections (i.e. the larger the average degree, more obvious the infection peak will be), which makes the total transmission become easier. Due to feedback loop (refer to Fig. \[fig.1\]), this should in turn cause larger-scale cascading failures in interdependent systems and make systems more vulnerable to CF-VP model, which we will systematically discuss in what follows.
To explore the influence of CF-VP model on the robustness of interdependent networks, we focus on two opposite cases. The first case is to fix the average degree of network B ($\langle k_B\rangle$) yet vary the average degree of network A ($\langle k_A\rangle$); another case is to fix $\langle k_A\rangle$ yet vary $\langle k_B\rangle$ (Indeed, there exists the third case: keep $\langle k_A\rangle$ and $\langle k_B\rangle$ equal, i.e. $\langle k_A\rangle=\langle k_B\rangle$, and simultaneous changing, like Fig. \[fig.2\]. But here we do not plot the curves of this case, which will be explained soon). Interestingly, such a change that seems trivial will lead to greatly different outcomes. First, irrespective of which case, increasing $\lambda$ makes $G$ become smaller, namely, fast propagation of virus will trigger larger, stronger crash of systems. In particular, there exists the critical threshold of virus transmissibility, $\lambda_c$, above which the remaining giant component will be null. From Fig. \[fig.3\], we can see that $\lambda_c$ becomes smaller with the increment of $\langle k_A\rangle$, which means that denser connections of network A where virus spreading takes place will accelerate the propagation of virus and thus the progress of cascading failures in systems. At variance, the change of $\langle k_B\rangle$ has no obvious impact on the threshold $\lambda_c$ (see Fig. \[fig.4\]), i.e. the crash trend is nearly identical if only $\langle k_B\rangle$ changes. Combining these observations, a significant finding poses itself: the interdependent systems will be more vulnerable only if the network layer supporting virus propagation has denser connections (i.e. larger average degree); otherwise the interdependent systems are robust against the change of connections in other layer(s). Along this discovery, it now becomes easy to understand that simultaneously changing $\langle k_A\rangle$ and $\langle k_B\rangle$ will generate the same results as Fig. \[fig.3\]. In addition, this discovery also differs from previous framework of cascading failure in interdependent networks \[11\], where better robustness usually needs denser connections. Thus, our outcomes, to some extent, prove the necessity and significance of feedback loop when designing the interdependent networks.
Control Strategy
================
Up to now, it has been very clear that in interdependent networks virus propagation on one layer could lead to continuous cascading failures and fragmentation of systems. Along this line, the most intuitive method of protecting interdependent networks is to control the spread of virus when it appears. In reality, it seems hard to timely restrain the spreading of virus (especially the emerging virus) by using the well-known pre-immunization strategies [@pastor2002immunization; @cohen2003efficient], due to the absence of effective antivirus programs [@gao2013modeling]. However, in CF-VP model it seem feasible to identify the infected neighbor based on knowledge and abnormal behavior of infected nodes. Here we consider such a control strategy: after the emergence of virus, susceptible node $i$ can identify one infected neighbor with probability $q_i$, and then prunes its connection with this neighbor. This strategy not only isolates the health nodes from their infected neighbors, but also decreases the average degree of network layer which supports the virus propagation (see Fig. \[fig.3\] for its impact). With respect to the identification capability $q_i$, we consider two following cases.
![The size $G$ of the remaining giant component of network A versus identification probability $q$ for deterministic adaptive isolation case (a) and degree-based adaptive isolation where $\sigma=0.3$ (b). The interdependent networks are SF networks with average degree $\langle k_B\rangle = 8$, $\langle k_A\rangle=$ 4 (squares), 6 (triangles), 8 (circles), 10 (diamonds) and 16 (stars), respectively. The transmissibility probability is $\lambda=0.5$.[]{data-label="fig.5"}](fig5-1.eps)
![The size $G$ of the remaining giant component of network A versus identification probability $q$ for deterministic adaptive isolation case (a) and degree-based adaptive isolation where $\sigma=0.3$ (b). The interdependent networks are SF networks with average degree $\langle k_B\rangle = 8$, $\langle k_A\rangle=$ 4 (squares), 6 (triangles), 8 (circles), 10 (diamonds) and 16 (stars), respectively. The transmissibility probability is $\lambda=0.5$.[]{data-label="fig.5"}](fig5-2.eps)
1\) Deterministic adaptive isolation: $q_i=q_j=q$ for $i\neq j$. That is, all of nodes have the same ability to idenfify infected neighbors.
2\) Degree-based adaptive isolation: $\{q_1,q_2,\cdots, q_N\}$ following gauss distribution. That is, $\{q_1,q_2,\cdots, q_N\}\sim N(q, \sigma)$, where $q$ and $\sigma$ are mean and standard deviation respectively. Moreover, if $k_i \geq k_j$, we assume $q_i\geq q_j$, which means that large-degree nodes have higher ability to identify infected neighbors. Considering that $q_i$ must be between 0 and 1, we assign $q_i$ as
$$q_i=
\left\{\begin{array}{ll} 0,\ \emph{\emph{if}}\ \ q_i<0,\\
q_i,\ \ \emph{\emph{if}}\ \ 0\leq q_i\leq 1\\ 1,\ \ \emph{\emph{if}}\ q_i>1,\end{array}\right.$$
Subsequently, we explore how the control measures improve the robustness of interdependent networks under CF-VP model, where we still use two opposite cases as Figs. \[fig.3\] and \[fig.4\]. Fig. \[fig.5\] first shows the impact of isolation strategies when $\langle k_B\rangle$ is fixed and $\langle k_A\rangle$ changes. It is clear that the size $G$ of remaining giant component increases with $q$, which indicates the robustness of interdependent networks can be significantly improved by increasing nodes’ identification capability, regardless of which strategy. With large identification probability, the infection source(s) can be controlled and isolated earlier. The removal of these infected nodes further makes the cascading process become slow, which also decreases the possibility of infection outbreak. This thus validates the importance of feedback loop in the coupled disease-cascading model once again. Moreover, another similar phenomenon in Figs. \[fig.5\] (a) and (b) is that network A possessing large average degree needs larger $q$ to maintain the equivalent robustness with the case of small $\langle k_A\rangle$, which in fact is consistent with the prediction of Fig. \[fig.3\]: larger $\langle k_A\rangle$ usually enables systems to become more vulnerable, thus requiring more powerful protection. Except for similarity, we can also notice that degree-based adaptive isolation performs much better than deterministic adaptive isolation. This actually agrees with our intuition, because (as single-layer networks) large-degree nodes play a more significant role in the propagation of virus than small-degree nodes. If there exist infected nodes among the neighborhood of large-degree nodes, they can easily prune the connections with infected neighbor(s) due to large identification ability. With fast removal of infection sources, cascading will be controlled better (i.e. larger $G$ for the same $q$ value).
![The size $G$ of the remaining giant component of network A versus identification probability $q$ for deterministic adaptive isolation case (a) and degree-based adaptive isolation where $\sigma=0.3$ (b). The interdependent networks are SF networks with average degree $\langle k_A\rangle = 8$, $\langle k_B\rangle=$ 4 (squares), 6 (triangles), 8 (circles), 10 (diamonds) and 16 (stars), respectively. The transmissibility probability is $\lambda=0.5$.[]{data-label="fig.6"}](fig6-1.eps)
![The size $G$ of the remaining giant component of network A versus identification probability $q$ for deterministic adaptive isolation case (a) and degree-based adaptive isolation where $\sigma=0.3$ (b). The interdependent networks are SF networks with average degree $\langle k_A\rangle = 8$, $\langle k_B\rangle=$ 4 (squares), 6 (triangles), 8 (circles), 10 (diamonds) and 16 (stars), respectively. The transmissibility probability is $\lambda=0.5$.[]{data-label="fig.6"}](fig6-2.eps)
We now turn to another case: fixing $\langle k_A\rangle$ yet varying $\langle k_B\rangle$ and study how the isolation strategies improve the robustness of interdependent networks. As reflected in Fig. \[fig.4\], this case has no impact on the system crash. Though the size $G$ of remaining giant component enhances with identification capability $q$ and degree-based adaptive isolation performs better, only changing $\langle k_B\rangle$ will generate nearly identical results with each isolation strategy (i.e. the overlapped curves in Fig. \[fig.6\]). This is because, for each $q$ value, the isolation probability of infected neighbors on network supporting virus propagation is the same, irrespective of average degree in other network. Combining Figs. \[fig.5\] and \[fig.6\], it seems to indicate that the best way of controlling system crash is to eradicate the infection sources in epidemic layer, which is specially useful for this layer with denser connections.
Conclusion
==========
In this letter, we have developed a toy model (CF-VP model) in which virus propagation and cascading failure are coupled on interdependent networks, where each network layer sustains one dynamic process. For both processes, they spontaneously form a novel feedback loop: virus propagation triggers continuous cascading failures and even complete fragmentation if transmissibility probability is above a threshold; while cascading failures will break the connections of networks and thus suppresses virus propagation. Of note, if the network layer supporting epidemic spreading has denser connections, interdependent systems will be more vulnerable, which is opposite to the observation of traditional cascading fashion in interdependent networks [@buldyrev2010catastrophic]. To protect interdependent networks, we further propose the control measures based on the capability to identify the infected neighbor. Interestingly, the larger the identification capability (especially for larger-degree node), more robustness the interdependent networks will be.
In spite of simplicity, our model describing the interplay between cascading failures and virus propagation in interdependent networks seems reasonable and as well easily justifiable with realistic situations. For example, Internet and some social online networks could be encapsulated into the framework of multilayer networks. But how they influence each other will be a long-term question. This work may provide some new insights into understanding the interplay and proposing the protection measures. Besides, another point that deserves our attention is to consider theoretical analysis framework, which may validate the present simulation findings. Except for interplay between dynamical processes, the co-evolution between dynamics and interdependent network topology is also worth of our endeavors in future.
This work is supported by the Shandong Province Outstanding Young Scientists Research Award Fund Project (Grant No. BS2015DX006), the Shandong Academy of Sciences Youth Fund Project (Grant No. 2016QN003), the Inner Mongolia Colleges and Universities Scientific and Technological Research Projects (Grant no. NJZY132), and the National Natural Science Foundation of China (Grant Nos. 61572297, 31560622, 31260538, 30960246).
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abstract: 'Spin waves are collective excitations of magnetic systems. An attractive setting for studying long-lived spin-wave physics is the quantum Hall (QH) ferromagnet, which forms spontaneously in clean two-dimensional electron systems at low temperature and in a perpendicular magnetic field. We used out-of-equilibrium occupation of QH edge channels in graphene to excite and detect spin waves in magnetically ordered QH states. Our experiments provide direct evidence for long distance spin wave propagation through different ferromagnetic phases in the N=0 Landau level, as well as across the insulating canted antiferromagnetic phase. Our results will enable experimental investigation of the fundamental magnetic properties of these exotic two-dimensional electron systems.'
author:
- 'Di S. Wei'
- Toeno van der Sar
- Seung Hwan Lee
- Kenji Watanabe
- Takashi Taniguchi
- 'Bertrand I. Halperin'
- Amir Yacoby
bibliography:
- 'Bibliography.bib'
title: Electrical generation and detection of spin waves in a quantum Hall ferromagnet
---
Quantum Hall (QH) ferromagnetism arises from the interaction of electrons in massively degenerate, quantized energy levels known as Landau levels (LLs) [@Girvin2000]. When disorder is low enough for Coulomb interactions to manifest, the electrons in partially filled LLs spin-polarize spontaneously to minimize their exchange energy, with the single-particle Zeeman effect dictating their polarization axis [@Young2012; @Sondhi1993]. In graphene, these phenomena give rise to ferromagnetic phases when the N=0 LL is at quarter- and three-quarter-filling [@Alicea2006; @Yang2006; @Zhang2006; @Nomura2008; @Goerbig2011]. Such QH ferromagnets have an insulating topological bulk and spin-polarized edge states. Furthermore, a canted antiferromagnetic (CAF) state is believed to emerge at one-half filling, with a canting angle determined by the competing valley anisotropy and Zeeman energy [@Kharitonov2012; @Young2014]. Spin waves, also known as magnons, are the lowest energy excitation in both the QH ferromagnet and CAF [@Girvin2000; @Green2002; @Takei2016], and could provide crucial information about these topologically non trivial magnetic states.
In our experimental setup, we generate magnons by creating an imbalance of chemical potential between two edge states of opposite spin that run along the boundary of a QH magnet. If this imbalance is smaller than the energy required for generating magnons in the QH magnet (and there are no thermal magnons already present in the system), scattering between these two edge states is forbidden because the change in angular momentum of a scattered electron cannot be absorbed by the system. Indeed, previous measurements have shown that oppositely spin-polarized edge channels do not equilibrate as long as the imbalance is small [@Amet2014; @Wei2017]. However, we find edge channel equilibration commences when the imbalance exceeds the minimum energy required for exciting magnons in the QH ferromagnet. Because the magnetization of the QH ferromagnet is extremely dilute, there are negligible demagnetizing fields and the minimum energy to excite magnons is given by the Zeeman energy $E_\mathrm{Z}=g\mu_B B$ [@Girvin2000; @Kittel1948], where $g$ is the electron g-factor, $\mu_B$ is the Bohr magneton, and $B$ is the external magnetic field. Although magnon generation does not directly affect the conductance of the system, the reverse process of magnon absorption by far-away edge states does, allowing us to detect the propagation of magnons electrically, in close analogy to the conventional detection of magnons in insulators via the inverse spin Hall effect [@Kajiwara2010; @Cornelissen2015; @Chumak2015; @Wesenberg2017].
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To demonstrate spin wave propagation, we begin with a dual-gated monolayer graphene device (device $1$) where the central region can be tuned to a different filling factor than the adjacent regions (Fig. 1A). Connecting the two leads is a chiral edge state that carries spin-polarized electrons aligned with the magnetic field, which we call spin-up. We tune the central region to a three-quarters-filled LL ($\nu = 1$), whereas the outer regions are tuned to a non-magnetic fully filled LL ($\nu = 2$). We apply a source-drain voltage $V_\mathrm{dc}$ to induce a difference in chemical potential $\mu = -e V_\mathrm{dc}$ between the edge channels emerging from the two contacts, where $e$ is the electron charge. Once $|\mu| \geq E_\mathrm{Z}$, an electron traveling in a high-energy (“hot”), spin-down edge state can relax into a low-energy (“cold”), spin-up edge state by emitting a magnon into the ferromagnetic bulk (Fig. 1, B and C). Because equilibration must occur close to the ferromagnetic bulk in order to launch magnons, the edge states must equilibrate over short length scales at localized “hot spots” where the hot and cold edges meet. This makes graphene an ideal platform to observe this phenomenon, where edge state equilibration can occur over length scales $< 1 \mu$m [@Amet2014; @Williams2007; @Ozyilmaz2007] (See [@WeiSupp2018] for further discussion). Because only spin-down angular momentum can be propagated into the spin-up bulk, magnon generation occurs at the location denoted by an encircled minus sign when $\mu \geq E_\mathrm{Z}$ (Fig. 1B) and at the location denoted by an encircled plus sign when $\mu \leq -E_\mathrm{Z}$ (Fig. 1C). These magnons propagate through the insulating QH ferromagnet and can be absorbed by the reverse process between other edge channels (Fig. 1B-C), which causes a deviation in the conductance from a well-quantized $\nu = 1$ QH state.
When we measure the conductance of the graphene device (Fig. 1E, atomic force microscopy image in fig. S3) as a function of $V_\mathrm{dc}$, we find that the $\nu = 1$ QH ferromagnet remains precisely quantized at the expected value of $e^2/h$, and then changes once the applied bias reaches the Zeeman threshold ($V_\mathrm{dc}= \pm V_\mathrm{EZ}= \mp E_\mathrm{Z}/e$), as expected from our model (Fig. 1F). Interestingly, we find that thanks to contact doping [@WeiSupp2018; @Giovannetti2008] we can tune the entire device to $\nu = 1$ and find the same phenomenon of conductance deviation at the Zeeman threshold (fig. S4).
{width="100.00000%"}
{width="95.00000%"}
By tilting the external magnetic field with respect to the sample-plane normal axis, we verify that the change in conductance occurs when the applied chemical potential exceeds the bare Zeeman energy $E_\mathrm{Z} = g \mu_\mathrm{B} B_\mathrm{T}$ ($g$=2), which is given by the total field $B_\mathrm{T}$ (Fig. 1G – sample is tuned entirely to $\nu=1$). In contrast, previous transport studies of spin and valley excitations in graphene and GaAs have only found excitations related to the exchange energy gap [@Young2012; @Sondhi1993; @Schmeller1995], which depends on the component of the field perpendicular to the sample plane ($B_\perp$). Our tilted-field measurements therefore corroborate our magnon-based interpretation of the observed change in sample conductance. All subsequent experiments described in this work are done at perpendicular field.
The conductance change at $E_\mathrm{Z}$ can either be positive or negative, depending on the number of magnons absorbed at each contact. To examine this, we use different sets of leads in the same device (Fig. 2A, device 2) to perform two-terminal conductance measurements. We start with leads L$_2$ and L$_1$ in Fig. 2B. We label the amount of redistributed chemical potential at each of the absorption sites $\varepsilon_{i}$, with $i$ indexing the absorption site (note that $\varepsilon_{i}$ = 0 for $-E_\mathrm{Z} < \mu < +E_\mathrm{Z}$), where $\varepsilon_{i}$ is proportional to the number of magnons absorbed at site $i$. Absorption at $\varepsilon_1$ and $\varepsilon_2$ have opposite effects on the conductance, as magnon absorption transfers chemical potential from the outer edge to the inner edge. Therefore, for $\mu \geq E_\mathrm{Z}$, magnon absorption at $\varepsilon_1$ decreases the particle current ($I_\mathrm{P} = - I/e$ where $I$ is the charge current) whereas magnon absorption at $\varepsilon_2$ increases $I_\mathrm{P}$ (Fig. 2B). For $\mu \leq -E_\mathrm{Z}$, the hot and cold reservoirs are reversed, and we now consider the change to the negative particle current $-I_\mathrm{P}$. Although $\varepsilon_1$ still decreases the particle current, $I_\mathrm{P}$ is now negative, and so $\varepsilon_1$ actually increases the magnitude of the particle current ($|-I_\mathrm{P}|$); similarly, for $\mu \leq -E_\mathrm{Z}$, $\varepsilon_2$ decreases $|-I_\mathrm{P}|$ (Fig. 2C). We can quantify this using current conservation to formulate the differential conductance as a function of $\varepsilon_i$ and $\mu$:
$$\label{eq1}
\frac{\mathrm{d}I}{\mathrm{d}V} = \frac{\mathrm{d}I_\mathrm{P}}{\mathrm{d}\mu}
= \frac{1}{R_\mathrm{Q}} \Big( 1+\frac{\mathrm{d}\varepsilon_2}{\mathrm{d}\mu}-\frac{\mathrm{d}\varepsilon_1}{\mathrm{d}\mu} \Big)$$
where $R_\mathrm{Q} =h/e^2$ is the resistance quantum, $V = V_\mathrm{ac} + V_\mathrm{dc}$, and we have neglected contact resistance (see [@WeiSupp2018] for a derivation of Eq. 1, which takes contact resistance into account). We find that the conductance decreases at negative bias and increases at positive bias (Fig. 2D) – indicating that $\varepsilon_1 > \varepsilon_2$ for both positive and negative bias. This implies that more magnons are absorbed at $\varepsilon_1$ than at $\varepsilon_2$. Because our contacts have all been fabricated identically, we conclude this is because $\varepsilon_1$ is closer to magnon generation than $\varepsilon_2$ (for both positive and negative bias, see Fig. 2, A and B). Using different sets of contacts and top gates (Fig. 2, E to H) we can change the relative distances of $\varepsilon_i$ to the locations of magnon generation. We confirm that for each configuration, the conductance values after $E_\mathrm{Z}$ correspond to a greater number of magnons absorbed at the site closer to magnon generation.
This change to the conductance is not a consequence of QH breakdown. Conductance deviations after the Zeeman threshold that depend on the sign of $V_\mathrm{dc}$ are not explained by any current breakdown theories [@Nachtwei1999]. Additionally, we find that the threshold voltage bias does not depend on the lead configuration (Fig. 2), the size of the $\nu = 1$ region (fig. S4), or the density of the $\nu = 1$ region (fig. S6) – which is all inconsistent with trivial QH breakdown, but consistent with our magnon model. In total, we have measured this $\nu = 1$ conductance deviation occurring at the Zeeman energy for eight devices of widely varying geometries (figs. S3, S4, and S11).
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Thus far we have established that we are able to generate and absorb magnons at current carrying contacts. If these chargeless excitations propagate through the insulating bulk, we also expect to see signatures of magnon propagation and absorption via non-local voltage measurements (d$V_\mathrm{NL}$/d$V$—referred to as nonlocal signal $S_\mathrm{NL}$), away from the source-drain current. To measure $S_\mathrm{NL}$ we use L$_3$ and L$_2$ in device 2 as source-drain contacts, and use contacts L$_4$ and L$_5$ as voltage probes (Fig. 3A). These contacts are separated from the source-drain contacts by a top gate (TG2) which we tune between $\nu_\mathrm{TG2}$ = -2 and $\nu_\mathrm{TG2}$ = 2, where all other regions are tuned to $\nu=1$. The conductance between L$_3$ and L$_2$ drops at $V_\mathrm{EZ}$ in accordance with our model (Fig. 3B), whereas magnon generation is largely unaffected by TG2 (fig. S7, A). At $\nu_\mathrm{TG2}$ = 1 we measure a change in $S_\mathrm{NL}$ at $\pm V_\mathrm{EZ}$ due to the relative absorption at each magnon absorption site ($\varepsilon_i$).
The sign of S$_\mathrm{NL}$ indicates that there is more magnon absorption at sites closer to where magnon generation occurs. Through current conservation ([@WeiSupp2018]) we find that the measured differential voltage (unitless) is:
$$\label{eq2}
\frac{\mathrm{d} V_\mathrm{NL}}{\mathrm{d}V} = \Big( \frac{\mathrm{d} \varepsilon_4}{\mathrm{d} \mu} - \frac{\mathrm{d} \varepsilon_5}{\mathrm{d} \mu} \Big )$$
The site labeled by $\varepsilon_4$ is closer to magnon generation than $\varepsilon_5$ for both negative and positive bias, so $|\mathrm{d}\varepsilon_4| > |\mathrm{d}\varepsilon_5|$. However, the differential change in voltage ($\mathrm{d}\varepsilon_i/\mathrm{d}\mu$) is negative for $V_\mathrm{dc} \geq V_\mathrm{EZ}$ and positive for $V_\mathrm{dc} \leq -V_\mathrm{EZ}$, corresponding to an overall negative value for $S_\mathrm{NL}$ at $V_\mathrm{dc} \geq V_\mathrm{EZ}$ and a positive value at $V_\mathrm{dc} \leq -V_\mathrm{EZ}$ (Fig. 3C).
The device geometry used for our non-local measurements allows us to tune TG2 away from $\nu_\mathrm{TG2} = 1 $, and thereby examine magnon transmission through different filling factors. We make two surprising observations. We observe that when $\nu_\mathrm{TG2} = -1 $ the signal $s_\mathrm{NL}$ is almost identical signal to when $\nu_\mathrm{TG2} = 1 $ (Fig. 3C and fig. S8). This signal arises in the absence of any charge leakage across the $\nu_\mathrm{TG2} = -1$ region (fig. S9), so that changes in $S_\mathrm{NL}$ can be attributed to magnon transport through the $\nu_\mathrm{TG2} = -1$ ferromagnet. This suggest that there is neither spin nor valley mismatch between the ferromagnetic states on either side of the boundary. We therefor propose an ordering of the LLs that does not require a spin or valley flip for magnons to travel across the interface between $\nu_\mathrm{BG} = 1$ and $\nu_\mathrm{TG2} = -1$ (Fig. 3D; see [@WeiSupp2018] for a theoretical discussion.)
In addition, we unexpectedly find that $S_\mathrm{NL}$ is suppressed at $\pm V_\mathrm{EZ}$ when $\nu_\mathrm{TG2} = 0$. For non-magnetic regions such as $\nu_\mathrm{TG2}=2$, it is expected that magnons will be blocked from passing through, as experimentally confirmed in Fig. 3C (the non-local signal occurring at the transition between $\nu = 1$ and $\nu =2$ is explained in Fig. S7E). However, $\nu = 0$ is purportedly a canted antiferromagnet which is theoretically capable of hosting even zero-energy magnons [@Takei2016]. It appears that the probability for an incident magnon to be transmitted across the junction between the $\nu = 0$ and $\nu = 1$ regions is very small for energies close to $E_\mathrm{Z}$. This may be caused by, in part, the mismatch in propagation velocities in the two phases, or a barrier due to the complex nature of the interface region. Close to the boundary with a $\nu = 1$ phase, the ground state of the $\nu = 0$ phase may not have canted spins but may instead be in an aligned antiferromagnet state, where spins are parallel to the magnetic field on one sublattice and antiparallel on the other. Eventually, far from the boundary, we may expect the local spin arrangement to rotate into the CAF orientation (Fig. 4B). In the transition region, the minimum magnon energy will be larger than $E_\mathrm{Z}$ due to effects of the valley-dependent interaction terms [@Kharitonov2012], which were initially responsible for the antiferromagnet arrangement to be favored over the ferromagnetic arrangement. In order to cross from the $\nu = 1$ region to the CAF region, a magnon with energy close to $E_\mathrm{Z}$ would have to tunnel through the barrier region, and we would expect the transmission rate to be low. If the magnons have enough energy to overcome this barrier, they should be able to more easily enter the CAF region. Fig. 4C shows that we can experimentally exceed this barrier, where we see non-local signals at higher $|V_\mathrm{dc}|$ with signs in agreement with our magnon model. The onset of this magnon signal is unaffected by any charge transport across the $\nu_\mathrm{TG2} = 0$ region (fig. S10). Closely examining the signal at $\nu_\mathrm{TG2} = 0$, we see signals commencing at $\pm V_\mathrm{EZ}$ which we attribute to tunneling events across this $\nu_\mathrm{BG} = 1/\nu_\mathrm{TG2} = 0$ barrier (Fig. 4D).
Note that all non-local signals (occurring at $\nu_\mathrm{TG2}$ =-1, 0, and 1) appear only in a finite band of $V_\mathrm{dc}$. This suppression of the differential voltage signal indicates that either magnon generation is suppressed, or alternatively, that the differently-spaced contacts begin to see identical amounts of magnon absorption once the system has reached a certain magnon density threshold. We further speculate that this cut-off could be related to the magnon bandwidth, but leave this to a future investigation.
The experiments presented here introduce a method of using magnons to probe the SU(4) spin and valley anisotropies of graphene QH systems, whichc can be used to probe highly correlated states such as the fractional QH regime [@MacDonald1998], or the quantum-spin Hall phase of monolayer graphene [@Young2014]. Owing to the theoretical prediction for spin superfluidity in the CAF state [@Takei2016], this study paves the way for exploring and realizing dissipationless spin waves in a Bose-Einstein condensate (BEC) of magnons. Such condensates should result in a coherent precession of the spin in the QH magnet, which may be probed through emitted microwave radiation. Furthermore, coherent spin waves associated with a BEC may be able to propagate long distances with negligible dissipation, which could be tested by careful length dependence measurements.
Methods
=======
Sample Fabrication
------------------
All devices consist of graphene encapsulated by two layers of hexagonal boron nitride (hBN) on doped Si chips with a 285 nm layer of SiO$_2$ that acts as a dielectric for the Si back gate. Graphene is mechanically exfoliated from bulk graphite obtained from NGS Naturgraphit GmbH using 1009R tape from Ultron Systems and subsequently encapsulated in hexagonal boron nitride (hBN) using a dry transfer process [@Wang2013]. Before the first metal deposition step, we annealed the devices in vacuum at $500^{\circ}$C to improve device quality. We then created top gates using electron-beam lithography and thermal evaporation of Cr/Au. We etched the devices into the desired geometry by reactive ion etching in $\mathrm{O}_2/\mathrm{CHF}_3$ using a PMMA/HSQ bilayer of resist (patterned by electron-beam lithography) as the etch mask. To fabricate edge-contacts to the graphene we etched through the entire hBN/graphene stack. We then created edge contacts by thermally evaporating Cr/Au while rotating the sample using a tilted rotation stage.
Measurement
-----------
Our measurements were performed in a Leiden dry dilution refrigerator with a base temperature of 20 mK. Measurements of differential conductance were performed using a lock-in amplifier with an a.c. excitation voltage of 50 $\mu$V at 17.77 Hz. All measurements of differential conductance were corrected for contact/line resistances, which were independently determined by lining up the robust $\nu = 2$ QH conductance plateau with $2e^2/h$.
Acknowledgements
================
General
-------
We thank A. H. Macdonald, J. D. Sanchez-Yamagishi, S. L. Tomarken, and S. P. Harvey for helpful discussions and feedback, X. Liu for fabrication help, and P. Kim for providing the transfer setup. **Funding**: Supported by the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant GBMF4531; the U.S. Department of Energy, Basic Energy Sciences Office, Division of Materials Sciences and Engineering under award DE-SC0001819 (D.S.W. and T.v.d.S.); NSF Graduate Research Fellowship grant DGE1144152 (D.S.W.); the STC Center for Integrated Quantum Materials, NSF grant DMR-1231319 (B.I.H.); and the Elemental Strategy Initiative conducted by MEXT, Japan, and JSPS KAKENHI grant JP15K21722 (K.W. and T.T.). Nanofabrication was performed at the Center for Nanoscale Systems at Harvard, supported in part by NSF NNIN award ECS-00335765. **Author contributions**: D.S.W., T.v.d.S., B.I.H., and A.Y. conceived and designed the experiments; D.S.W. fabricated the devices; D.S.W. and A.Y. performed the experiments; D.S.W., T.v.d.S., S.H.L., B.I.H., and A.Y. analyzed the data and wrote the paper; and K.W. and T.T. synthesized the hexagonal boron nitride crystals. **Competing interests**: The authors declare no competing financial interests. **Data and materials availability**: All measured data are available in the supplementary materials.
Correspondence
==============
Correspondence and requests for materials should be addressed to A.Y. (email: [email protected]).
Supplementary Information
=========================
**Supplementary Note 1. Equilibration of edge states at ‘hot spots’**\
One question that arises from this study is why, after decades of experimental investigation into QH ferromagnets, has this phenomenon not been observed in GaAs quantum wells? We posit that this is due to the readiness of edge states in graphene to equilibrate over small length scales due to the sharp confining potentials. Because there is a limited spatial range over which the ‘hot spot’ magnon generation can occur adjacent to the $\nu = 1$ ferromagnetic bulk, spin-flip induced edge equilibration must occur over short lengths. Past studies have shown that in graphene edges of the same spin are able to fully equilibrate over length scales $< 1$ $\mu$m [@Amet2014; @Williams2007; @Ozyilmaz2007], while similar studies done in GaAs found typical lengths of around tens of microns, and sometimes up to 200 $\mu$m [@VanWees1991; @Alphenaar1990; @Komiyama1989]. In order to experimentally verify that magnons are generated at these corner ‘hot spots’, we have fabricated a device with gated corners showing conductance changes at $E_\mathrm{Z}$ in accordance with our magnon model (Fig. S11).
The difference in experimentally-determined equilibration lengths between graphene and GaAs is likely due to the sharper confining potentials in graphene, which allow for small spatial edge channel separation—increasing the likelihood of inter-channel scattering. Additionally, a smooth potential may allow for edge reconstruction [@Chklovskii1992], which if present, could also affect the inter-channel scattering rate and limit magnon generation. Experiments in GaAs based systems have shown that edge recondsturction plays an important role due to the smooth confining potential [@Ahlswede2002]. The graphene devices investigated in this work have gate electrodes located at just a few tens of nanometers distance away, likely limiting the amount of edge reconstruction. Furthermore, the close proximity of the metal gates to the graphene may screen the electric fields that cause edge reconstruction, which is another potential difference with GaAs-based systems [@Li2013].
However, interestingly, we note that although an electrostatically-defined confinement potential is able to suppress the magnon signal, it does not eliminate it completely (Fig. S11). This suggests that strong edge disorder is not required for the $\nu$ = 2 and $\nu$ = 1 edge channels to equilibrate, and that it is possible that magnons could be generated in GaAs devices with a sharp electrostatic confinement potential. We note, however, that while magnon generation may be possible, magnon propagation may not be as efficient in GaAs systems due to large spin-orbit coupling [@Muller1992] and more nuclear spins [@Burkard1999] relative to graphene, which could facilitate magnon dissipation. Such dissipative processes would be important because, as we describe in the main text, magnon generation itself does not affect the sample conductance – only when magnons are able to propagate and are absorbed in by electrons in other edge channels do we detect a change in sample conductance.
Additionally, we note that even when we do not explicitly add an extra edge state near the contacts (by gating the side regions to $\nu$ = 2), contact doping of the graphene by the Cr/Au leads [@Giovannetti2008] introduces additional spin-down edge states—which also leads to magnon generation at EZ.\
**Supplementary Note 2. Calculating $V_\mathrm{dc}$ necessary to exceed $V_\mathrm{EZ}$ given a finite contact resistance**\
In a two-terminal measurement, the applied bias voltage ($V_\mathrm{dc}$) drops over both the contact resistances at both the source and the drain. The d.c. current is therefore
$$\label{eqS1.1}
I_\mathrm{dc} = \frac{V_\mathrm{dc}}{2R_\mathrm{C}+R_\mathrm{Q}}$$
where $R_\mathrm{Q}$ is the quantum resistance of an edge channel, and where $R_\mathrm{C}$ includs both the contact resistance at each lead as well as the filtering on the lines. The filtering on each line is 4.5k$\Omega$, and the contact resistance at each lead of a typical device is about 500$\Omega$.
The actual d.c. voltage that drops over the edge channel ($V_\mathrm{dc}'$) is therefore given by
$$\label{eqS1.2}
V_\mathrm{dc}' = I_\mathrm{dc} R_\mathrm{Q}$$
Therefore:
$$\label{eqS1.3}
V_\mathrm{dc} = \frac{V_\mathrm{dc}' \Big( 2R_\mathrm{C}+R_\mathrm{Q} \Big)}{R_\mathrm{Q}}$$
In our figures, we use ‘$V_\mathrm{EZ}$’ ($V_\mathrm{EZ}$ = -$E_\mathrm{Z}/e$) to denote the bias at which -$eV_\mathrm{dc}'$ reaches the Zeeman energy ($E_\mathrm{Z} = g \mu_B B_T$).\
**Supplementary Note 3. Circuit analysis for two-terminal conductance measurement.**\
![**Schematic of a two-terminal device where a voltage is sourced at the left contact and drained at the right.** The negative and positive signs denote magnon generation for negative and positive $V_\mathrm{dc}$ respectively. $\varepsilon_1$ and $\varepsilon_2$ denote locations where magnon absorption occurs. Arrows indicate how the chemical potential redistributes after magnon generation, and the chemical potential of the edge states after magnons are absorbed are labeled. the electrochemical potential applied by the voltage source is defined as $\mu$. $\mu_1$ and $\mu_2$ are the chemical potential reservoirs connected to the source and drain via a contact resistance $R_\mathrm{C}$ (assumed to be identical for both contacts).[]{data-label="fig:Supp1"}](FigS1.png){width="50.00000%"}
In Fig. S1 The (particle) current conservation equation at the source reservoir (labeled $\mu_1$) is:
$$\label{eqS2.1}
\frac{2 \mu_1}{R_\mathrm{Q}} = \frac{\mu_2 -\varepsilon_2}{R_\mathrm{Q}} + \frac{\mu_1 +\varepsilon_1}{R_\mathrm{Q}} + \frac{\mu -\mu_1}{R_\mathrm{C}}$$
where $R_\mathrm{Q}$ is the resistance quantum. As described in the main text, $\varepsilon_i$ denotes the chemical potential redistributed between edge states at the $i$th contact. Additionally, although there are indeed also spin-flips occurring at the negative-bias magnon generation location when positive bias magnons are being generated (and for the reverse case), we ignore these in our analysis because they do not contribute to changes in the conductance.
The equation at the drain contact is:
$$\label{eqS2.2}
\frac{\mu_2 +\varepsilon_2}{R_\mathrm{Q}} + \frac{\mu_1 -\varepsilon_1}{R_\mathrm{Q}} = \frac{2\mu_2}{R_\mathrm{Q}} + \frac{\mu_2}{R_\mathrm{C}}$$
Solving for $\mu_2$ we find
$$\label{eqS2.3}
\mu_2 = \frac{\mu +\varepsilon_2 - \varepsilon_1}{2+\frac{R_\mathrm{Q}}{R_\mathrm{C}}}$$
Using the chemical potential of the voltage source as $\mu = -eV_\mathrm{dc}$, and the charge current $I = -eI_\mathrm{P}$ (where $I_\mathrm{P}$ is defined as $\frac{1}{e^2}$ $\frac{\mu_2}{R_\mathrm{C}}$ to normalize the units) the differential conductance measured is
$$\label{eqS2.4}
\begin{split}
\frac{\mathrm{d}I}{\mathrm{d}V_\mathrm{dc}} & = \frac{\mathrm{d}I_\mathrm{P}}{\mathrm{d}\mu} = \frac{\big(\mathrm{d}\mu_2/R_\mathrm{C} \big)}{\mathrm{d}\mu}
\end{split}$$
$$\label{eqS2.4}
\begin{split}
\frac{\mathrm{d}I}{\mathrm{d}V_\mathrm{dc}} = \frac{1}{2 R_\mathrm{C}+R_\mathrm{Q}} \Big( 1+\frac{\mathrm{d}\varepsilon_2}{\mathrm{d}\mu}-\frac{\mathrm{d}\varepsilon_1}{\mathrm{d}\mu} \Big)
\end{split}$$
This becomes Equation 1 (main text) in the absence of contact resistance ($R_\mathrm{C}$ =0).\
**Supplementary Note 4. Circuit analysis for non-local voltage measurements.**\
![**Schematic circuit diagram of the multi-terminal device** Device 2 - optical micrograph shown in Fig. 2A) that is used to measure $S_\mathrm{NL}$. $\mu_1$, $\mu_2$, $\mu_3$, $\mu_4$ and $\mu_5$ are the chemical potential reservoirs connected to each contact (L$_1$-L$_5$) by a contact resistance $R_\mathrm{C}$. The electrochemical potential applied by the voltage source is defined as $\mu$. Voltage is sourced at L$_3$ and drained from L$_2$. L$_1$ is floating. $S_\mathrm{NL}$ is measured between L$_4$ and L$_5$. The negative and positive signs denote magnon generation for negative and positive $V_\mathrm{dc}$ respectively. $\varepsilon_1$, $\varepsilon_2$, $\varepsilon_3$, $\varepsilon_4$, and $\varepsilon_5$ label locations of magnon absorption. []{data-label="fig:Supp1"}](FigS2.png){width="50.00000%"}
In Fig. S2 the chemical potential of the edge states after magnons are absorbed are labeled as $\mu_{i} - \varepsilon_{i}$ and $\mu_i +\varepsilon_{i}$ where ‘$i$’ denotes the contact where the chemical potential originates. We calculate the conductance expected after the Zeeman energy has been reached. This device has 5 contacts in total. We write a current conservation equation at each contact:
$$\label{eqS3.1}
\mathrm{L}_1: \mu_2 - \varepsilon_2+\mu_1 + \varepsilon_1 = 2 \mu_1$$
$$\label{eqS3.2}
\mathrm{L}_2: \mu_3 - \varepsilon_3+\mu_2 + \varepsilon_2 = 2 \mu_2 + \mu_2 \frac{R_\mathrm{Q}}{R_\mathrm{C}}$$
$$\label{eqS3.3}
\mathrm{L}_3: (\mu - \mu_3) \frac{R_\mathrm{Q}}{R_\mathrm{C}} + \mu_3 + \varepsilon_3 + \mu_4 - \varepsilon_4 = 2 \mu_3$$
$$\label{eqS3.4}
\mathrm{L}_4: \mu_5 - \varepsilon_5+\mu_4 + \varepsilon_4 = 2 \mu_4$$
$$\label{eqS3.5}
\mathrm{L}_5: \mu_1 - \varepsilon_1+\mu_5 + \varepsilon_5 = 2 \mu_5$$
Solving for $\mu_2$ we find
$$\label{eqS3.6}
\mu_2 = \frac{\mu +\varepsilon_2 - \varepsilon_3}{2+\frac{R_\mathrm{Q}}{R_\mathrm{C}}}$$
Therefore,
$$\label{eqS3.7}
\begin{split}
\frac{\mathrm{d}I}{\mathrm{d}V_\mathrm{dc}} = \frac{\mathrm{d}I_\mathrm{P}}{\mathrm{d}\mu} = \frac{\mathrm{d}\big(\mu_2/R_\mathrm{C} \big)}{\mathrm{d}\mu} = \frac{1}{2 R_\mathrm{C}+R_\mathrm{Q}} \Big( 1+\frac{\mathrm{d}\varepsilon_2}{\mathrm{d}\mu}-\frac{\mathrm{d}\varepsilon_3}{\mathrm{d}\mu} \Big)
\end{split}$$
The non-local voltage measured is: $$\label{eqS3.8}
S_\mathrm{NL} = \frac{\mathrm{d}V_\mathrm{NL}}{\mathrm{d}V} = \Big( \frac{\mathrm{d}\varepsilon_4}{\mathrm{d}\mu}-\frac{\mathrm{d}\varepsilon_5}{\mathrm{d}\mu} \Big)$$
By defining $S_\mathrm{NL}$ as the difference between two voltage probes, any edge current which reaches the two voltage probes should not affect the measurement — although we do see some small background voltage which is explained in Fig. S7, B.
A similar circuit analysis can be done for any of the configurations found in the main text or supplementary materials.\
\
**Supplementary Note 5. Theoretical Notes.**\
We first note that the energy levels shown in the Figure 3D are only schematic. The actual Landau levels will be broadened due to electron-electron interactions and, perhaps, disorder. The curves represent more accurately the energy in the middle of the Landau level, and the ordering of the levels is more meaningful than the actual energies.
Our ordering of levels was guided by the following observations. For a uniform graphene system at $\nu$ = 0, it is believed that the valley anisotropy energy is large compared to the Zeeman energy, and that the ground state is a canted antiferromagnet state [@Young2014; @Kharitonov2012]. In this half-filled N=0 Landau level, there is one electron per flux quantum on each sublattice, with spins oriented predominantly in opposite directions. In the absence of Zeeman coupling the antiferromagnetic axis could point equally well in any direction, with no difference in the energy [@Jung2009]. In the presence of the Zeeman field, there is a small energy gain for the antiferromagnetic axis to line up in the x-y plane, perpendicular to the Zeeman field, allowing the spins on both sublattices to cant slightly in the direction of the Zeeman field. The energy gain for this is of order $E_Z^2$/$E_A$, where $E_A$ is the valley anisotropy energy.
For a general filling fraction in the range $-1 < \nu < 0$, if one calculates the ground state energy in a restricted Hartree-Fock approximation, which assumes that only two of the possible spin-valley states are occupied by electrons, one generally finds that one valley, say the K valley, has one electron per flux quantum, while the other valley has occupancy $1 + \nu < 1$. For fillings very close to $\nu$ = 0, the system may remain in a canted configuration, but for $|\nu|$ exceeding a critical value, of order $E_Z/E_A$, it will be more favorable for the antiferromagnetic axis to align in the z-direction, so that the majority spin is fully aligned with the Zeeman field. (See, e.g. the discussion in [@Abanin2013]) Similarly, for $0 < \nu < 1$, we would find the antiferromagnetic spin axis to be aligned with the magnetic field, except for a small region close to $\nu = 0.$
In a situation where the electron density varies rapidly in space, the spin and valley orientationsshould be determined by the dominant exchange energy, arising from the long-range part of theCoulomb interaction, which is indifferent to the specific orientation of the occupied levels in spin-valley space, but disfavors any rapid changes or discontinuities in the occupations. In a boundary between $\nu =-1$ and $\nu = 1$, we are forced to have two discontinuities in occupancy, but we can avoid any other discontinuities, if we choose to fill the levels in the order suggested in Fig. 3D. Moreover, it is likely that in a relatively steep boundary, the canted orientation will be completely suppressed, and that spins will remain quantized along the z-axis. We have seen in a previous study an absence of mixing between spin states at a $\nu = 1/ \nu = -1$ interface, supporting our assumption of spin alignment in the present case [@Wei2017].
By contrast, when the filling fraction is $\nu = 0$ under the center of our gate, it is likely that the system will assume the canted orientation near the center of the gate. At the same time, there should be a strip on either side of the gate, where the filling fraction is intermediate between $\nu = 1$ and $\nu = 0$, where the antiferromagnetic axis is in the z-direction. An interval where the filling fraction is between $\nu = 1$ and $\nu = -1$ , with spin axes parallel to z, will act as a barrier, to a spin wave incident from a region where $\nu = 1$, as the energy at the bottom of the spin wave band will be raised by an amount of order the valley anisotropy energy (This should be small compared to the Coulomb exchange energy, but larger than the Zeeman energy) [@Pientka2017]. In the case where the filling under the gate is $\nu = -1$, we would expect the barrier regions at the two sides to be relatively thin, and it is plausible that the spin waves can tunnel rather easily through the barrier region. When the filling at the gate center is ν = 0, we would expect the barriers to be much thicker, and tunneling through the barriers should be reduced accordingly.
In a bulk region where the filling is very close to ν = 1, we expect that the unoccupied spin state will have its spin opposite to the magnetic field, but it will have no particular preference for either the K or K’ valley or an arbitrary linear combination of them. Different valley polarizations may be selected near the physical boundaries of the sample, but we expect that the valley orientation in the vicinity of a gate where the charge density varies rapidly should be determined by energy considerations under the gate. It should cost relatively little energy for the valley orientation to vary smoothly between the sample edges and the gate, and we would not expect spin-wave propagation to be affected by such variations.
Our analysis, based on a Hartree-Fock approximation, ignores correlation effects, which can lead to fractional quantized Hall states, varying spin polarization, and transitions between different spin states in uniform graphene sample [@Abanin2013; @Feldman2013]. However, we would not expect such correlation effects to be important in the present case, where the charge density varies considerably on a sub-micron scale.
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---
abstract: |
We present a new bound for the minimum distance of a general primary linear code. For affine variety codes defined from generalised $C_{ab}$ polynomials the new bound often improves dramatically on the Feng-Rao bound for primary codes [@AG; @geithom]. The method does not only work for the minimum distance but can be applied to any generalised Hamming weight.\
**Keywords:** Affine variety code, $C_{ab}$ curve, Feng-Rao bound, footprint bound, generalised $C_{ab}$ polynomial, generalised Hamming weight, minimum distance, one-way well-behaving pair, order domain conditions.\
**MSC:** 94B65, 94B27, 94B05.
author:
- 'Olav Geil[^1]'
- 'Stefano Martin[^2]'
bibliography:
- 'bibfile.bib'
title: 'An improvement of the Feng-Rao bound for primary codes'
---
Introduction
============
In this paper we present an improvement to the Feng-Rao bound for [*primary*]{} codes [@AG; @geithom; @agismm]. Our method does not only apply to the minimum distance but estimates any generalised Hamming weight. In the same way as the Feng-Rao bound for primary codes suggests an improved code construction our new bound does also. The new bound is particular suited for affine variety codes for which it often improves dramatically on the Feng-Rao bound. Interestingly, for such codes it can be viewed as a simple application of the footprint bound from Gröbner basis theory. We pay particular attention to the case of the affine variety being defined by a bivariate polynomial that, in the support, has two univariate monomials of the same weight and all other monomials of lower weight. Such polynomials can be viewed as a generalisation of the polynomials defining $C_{ab}$ curves and therefore we name them [*[generalised $C_{ab}$ polynomials]{}*]{}. We develop a method for constructing generalised $C_{ab}$ polynomials with many zeros by the use of $({\mathbb{F}}_{p^m},{\mathbb{F}}_p)$-polynomials, that are polynomials returning values in ${\mathbb{F}}_p$ when evaluated in ${\mathbb{F}}_{p^m}$ (see, [@redei Chap. 1]). Here, $p$ is any prime power and $m$ is an integer larger than $1$. With this method in hand we can design long affine variety codes for which our bound produces good results. The new bound of the present paper is closely related to an improvement of the Feng-Rao bound for [*dual*]{} codes that we presented recently in [@geilmartin2013further]. Recall from [@agismm] that the usual Feng-Rao bound for primary and dual codes can be viewed as consequences of each other. This result holds when one uses the concept of well-behaving pairs or one-way well-behaving pairs. For weakly well-behaving pairs a possible connection is unknown. In a similar way as the proof from [@agismm] breaks down for weakly well-behaving, it also breaks down when one tries to establish a connection between the new bound from the present paper and the new bound from [@geilmartin2013further]. We shall leave it as an open problem to decide if the two bounds are consequences of each other or not.\
In the first part of the paper we concentrate solely on affine variety codes. For such codes the new method is intuitive. We start by formulating in Section \[sectwo\] our new bound at the level of affine variety codes and explain how it gives rise to an improved code construction ${\widetilde{E}}_{imp}(\delta)$. Then we continue in Section \[secthree\] by showing how to construct generalised $C_{ab}$ polynomials with many zeros. In Section \[secfour\] we give a thorough treatment of codes defined from so-called optimal generalised $C_{ab}$ polynomials demonstrating the strength of our new method. In Section \[secfourandaquarter\] we show how to improve the improved code construction ${\widetilde{E}}_{imp}(\delta)$ even further. This is done for the case of the affine variety being the Klein quartic. Having up till now only considered the minimum distance, in Section \[secfourandahalf\] we explain how to deal with generalised Hamming weights. Then we turn to the level of general primary linear codes lifting in Section \[secfive\] our method to a bound on any primary linear code. In Section \[secsix\] we recall the recent bound from [@geilmartin2013further] on dual codes, and in Section \[seccomp\] we discuss the relation between this bound and the new bound of the present paper. Section \[seceight\] is the conclusion.
Improving the Feng-Rao bound for primary affine variety codes
=============================================================
\[sectwo\] Affine variety codes were introduced by Fitzgerald and Lax in [@lax] as follows. For $q$ a prime power consider an ideal $I \subseteq
{\mathbb{F}}_q[X_1, \ldots , X_m]$ and define $$I_q=I + \langle X_1^q-X_1, \ldots , X_m^q-X_m\rangle,\label{eqiq}$$ $$R_q={\mathbb{F}}_q[X_1, \ldots , X_m]/I_q.$$ Let $\{P_1, \ldots , P_n\}={\mathbb{V}}_{\mathbb{F}_q}(I_q)$ be the corresponding variety over ${\mathbb{F}}_q$. Here, $P_i \neq P_j$ for $i \neq j$. Define the ${\mathbb{F}}_q$-linear map ${\mbox{ev}} : R_q
\rightarrow {\mathbb{F}}_q^n$ by ${\mbox{ev}}(A+I_q)=(A(P_1), \ldots ,
A(P_n))$. It is well-known that this map is a vector space isomorphism.
\[defaffcode\] Let $L$ be an ${\mathbb{F}}_q$ vector subspace of $R_q$. Define $C(I,L)={\mbox{ev}}(L)$ and $C^\perp(I,L)=\big(C(I,L)\big)^\perp$.
We shall call $C(I,L)$ a primary affine variety code and $C^\perp(I,L)$ a dual affine variety code. For the case of primary affine variety codes both the Feng-Rao bound and the bound of the present paper can be viewed as consequences of the footprint bound from Gröbner basis theory as we now explain.
\[deffoot\] Let $J \subseteq k[X_1, \ldots ,X_m]$ be an ideal and let $\prec$ be a fixed monomial ordering. Here, $k$ is an arbitrary field. Denote by ${\mathcal{M}}(X_1, \ldots , X_m)$ the monomials in the variables $X_1,\ldots , X_m$. The footprint of $J$ with respect to $\prec$ is the set $$\begin{aligned}
\Delta_{\prec}(J)&=&\{ M \in {\mathcal{M}}(X_1, \ldots , X_m) \mid M
{\mbox{ is not }}\nonumber \\
&&{\mbox{ \ \ \ \ \ the leading monomial of any polynomial in }} J\}.\nonumber\end{aligned}$$
\[probasis\] Let the notation be as in Definition \[deffoot\]. The set $\{M+J \mid M \in \Delta_\prec (J)\}$ constitutes a basis for $k[X_1, \ldots , X_m]/J$ as a vector space over $k$.
See [@clo Pro. 4, Sec. 5.3].
We shall make extensive use of the following incidence of the footprint bound (for a more general version, see [@geil2000footprints]).
\[thefoot\] Let $F_1, \ldots , F_s \in {\mathbb{F}}_q[X_1, \ldots , X_m]$. For any monomial ordering $\prec$ the variety ${\mathbb{V}}_{\mathbb{F}_q}(\langle F_1, \ldots ,F_s\rangle)$ is of size equal to $\# \Delta_\prec (\langle F_1, \ldots , F_s, X_1^q-X_1,
\ldots , X_m^q-X_m\rangle )$.
Follows from Proposition \[probasis\] and the fact that the map ${\mbox{ev}}$ is a bijection.
We next recall the interpretation from [@bookAG] of the Feng-Rao bound for primary affine variety codes.
\[deftksh1\] A basis $\{B_1+I_q, \ldots , B_{\dim (L)}+I_q\}$ for a subspace $L \subseteq R_q$ where ${\mbox{Supp}}(B_i) \subseteq \Delta_\prec (I_q)$ for $i=1,
\ldots , \dim(L)$ and where ${\mbox{lm}}(B_1) \prec \cdots \prec
{\mbox{lm}}(B_{\dim(L)})$, is said to be well-behaving with respect to $\prec$. Here, ${\mbox{lm}}(F)$ means the leading monomial of the polynomial $F$.
For fixed $\prec$ the sequence $({\mbox{lm}}(B_1), \ldots
,{\mbox{lm}}(B_{\dim(L)}))$ is the same for all choices of well-behaving bases of $L$. Therefore the following definition makes sense.
\[deffirkant\] Let $L$ be a subspace of $R_q$ and define $$\Box_\prec(L)=\{ {\mbox{lm}}(B_1), \ldots
,{\mbox{lm}}(B_{\dim(L)})\},$$ where $\{B_1+I_q, \ldots , B_{\dim(L)}+I_q\}$ is any well-behaving basis for $L$.
The concept of one-way well-behaving plays a crucial role in the Feng-Rao bound as well as in our new bound. It is a relaxation of the well-behaving property and the weakly well-behaving property (see [@bookAG; @geithom] for a reference) and therefore it gives the strongest bounds.
\[defaffowb\] Let ${\mathcal{G}}$ be a Gröbner basis for $I_q$ with respect to $\prec$. An ordered pair of monomials $(M_i,M_j)$, $M_i, M_j \in
\Delta_\prec(I_q)$ is said to be one-way well-behaving (OWB) if for all $H \in {\mathbb{F}}_q[X_1, \ldots , X_m]$ with ${\mbox{Supp}}(H)
\subseteq \Delta_\prec(I_q)$ and ${\mbox{lm}}(H)=M_i$ it holds that $${\mbox{lm}}(M_iM_j {\mbox{ rem }} {\mathcal{G}})={\mbox{lm}}(HM_j
{\mbox{ rem }} {\mathcal{G}}).$$ Here, $F {\mbox{ rem }} {\mathcal{G}}$ means the remainder of $F$ after division with ${\mathcal{G}}$ (see [@clo Sec. 2.3] for the division algorithm for multivariate polynomials).
As noted in [@bookAG] the concept of OWB is independent of which Gröbner basis ${\mathcal{G}}$ is used as long as $I_q$ and $\prec$ are fixed. We are now ready to describe the Feng-Rao bound for primary affine variety codes. We include the proof from [@bookAG Th.4.9].
\[theseven\] Let ${\mathcal{G}}$ be a Gröbner basis for $I_q$ with respect to $\prec$. Consider a non-zero word $\vec{c}$ and let $A$ be the unique polynomial such that ${\mbox{Supp}}(A) \subseteq \Delta_\prec(I_q)$ and $\vec{c}={\mbox{ev}}(A)$. Let ${\mbox{lm}}(A)=P$. We have $$\begin{aligned}
w_H(\vec{c})&\geq & \# \{ K \in \Delta_\prec(I_q) \mid \exists N \in
\Delta_\prec (I_q) {\mbox{ such that }} \nonumber \\
&&{\mbox{ \ \ \ }} (P,N) {\mbox{ is OWB and }} {\mbox{lm}}(PN {\mbox{
rem }} {\mathcal{G}})=K\}.\label{eqafffr}\end{aligned}$$ A bound on the minimum distance of $C(I,L)$ is found by taking the minimum of (\[eqafffr\]) when $P$ runs through $\Box_\prec(L)$.
From Corollary \[thefoot\] we know that $$\begin{aligned}
w_H(\vec{c}) &=& n - \#
\Delta_\prec (I_q+\langle A \rangle )\nonumber \\
&=&\#\Delta_{\prec}(I_q)-\#
\Delta_\prec(I_q+\langle A \rangle )\nonumber \\
&=&\# \bigg(\Delta_\prec(I_q) \backslash \Delta_\prec(I_q+\langle A
\rangle)\bigg).\label{eqerher}\end{aligned}$$ If $N,K \in \Delta_\prec
(I_q)$ satisfy that $(P,N)$ is OWB and ${\mbox{lm}}(PN {\mbox{ rem }}
{\mathcal{G}} )=K$ then $K \in \Delta_\prec(I_q) \backslash
\Delta_\prec(I_q +\langle A \rangle)$. Hence, $$\begin{aligned}
w_H(\vec{c})&\geq&\# \{ K \in \Delta_\prec(I_q) \mid \exists N \in \Delta_\prec (I_q)
\nonumber \\
&&{\mbox{ \ \ \ \ \ \ \ \ such that }} (P,N) {\mbox{ is OWB and }}
{\mbox{lm}}(PN {\mbox{ rem }} {\mathcal{G}}) = K\}.\nonumber\end{aligned}$$
The Feng-Rao bound is particular suited for affine varieties which satisfy the order domain conditions [@bookAG Def. 4.22]. For other varieties it does not seem to produce very good results. The new bound of the present paper solves this problem for affine varieties which satisfy the first half of the order domain conditions. This gives a lot of freedom as the latter set of varieties is much larger than the former. In its most general form the order domain conditions involves a weighted degree monomial ordering with weights in ${\mathbb{N}}_0^r \backslash \{ \vec{0}\}$, $r$ a positive integer (see [@bookAG Def.4.21]). Here, for simplicity we shall only consider weights in $\mathbb{N}$.
\[defwdeg\] Let $w(X_1), \ldots , w(X_m) \in {\mathbb{N}}$ and define the weight of $X_1^{i_1}\cdots X_m^{i_m}$ to be the number $w(X_1^{i_1} \cdots X_m^{i_m})=i_1 w(X_1)+ \cdots +i_m w(X_m)$. The weighted degree ordering $\prec_w$ on ${\mathcal{M}}(X_1, \ldots ,X_m)$ is the ordering with $X_1^{i_1} \cdots X_m^{i_m} \prec_w X_1^{j_1} \cdots
X_m^{j_m}$ if either $w(X_1^{i_1} \cdots X_m^{i_m}) < w(X_1^{j_1} \cdots
X_m^{j_m})$ holds or $w(X_1^{i_1} \cdots X_m^{i_m}) = w(X_1^{j_1} \cdots
X_m^{j_m})$ holds but $X_1^{i_1} \cdots X_m^{i_m} \prec^\prime X_1^{j_1} \cdots
X_m^{j_m}$. Here, $\prec^\prime$ is some fixed monomial ordering. When $\prec^\prime$ is the lexicographic ordering $\prec_{lex}$ with $X_m \prec_{lex} \cdots \prec_{lex} X_1$ we shall call $\prec_w$ a weighted degree lexicographic ordering.
We now state the order domain conditions which play a central role in the present paper.
\[defupher\] Consider an ideal $J \subseteq k[X_1, \ldots , X_m]$ where $k$ is a field. Let a weighted degree ordering $\prec_w$ be given. Assume that $J$ possesses a Gröbner basis ${\mathcal{F}}$ with respect to $\prec_w$ such that:
- Any $F \in {\mathcal{F}}$ has exactly two monomials of highest weight.
- No two monomials in $\Delta_{\prec_w}(J)$ are of the same weight.
Then we say that $J$ and $\prec_w$ satisfy the order domain conditions.
In the following we restrict to weighted degree orderings where $\prec^\prime =\prec_{lex}$. That is, $\prec_w$ shall always be a weighted degree lexicographic ordering.
\[exord\] Consider $I=\langle X^2+X-Y^3 \rangle \subseteq
{\mathbb{F}}_4[X,Y]$ and $I_4$ accordingly (see (\[eqiq\])). Choosing $X=X_1$, $Y=X_2$, $w(X)=3$ and $w(Y)=2$ we see that the order domain conditions are satisfied. By inspection we have $$\Delta_{\prec_w}(I_4)=\{1, Y, X,
Y^2, XY, Y^3, XY^2, XY^3\}$$ with corresponding weights $\{0, 2, 3, 4,
5, 6, 7, 9\}$. Consider a word $\vec{c}={\mbox{ev}}(A+I_4)$ where $A=a_1
1+a_2Y+a_3X$, $a_1, a_2 \in {\mathbb{F}}_4$ and $a_3 \in {\mathbb{F}}_4 \backslash \{0\}$. By Corollary \[thefoot\] the length is $n=8$. We now estimate the Hamming weight $w_H(\vec{c})=\# \big( \Delta_{\prec_w} (I_4)
\backslash \Delta_{\prec_w}(I_4 + \langle A \rangle)\big)$ (see (\[eqerher\])). The following elements in $\Delta_{\prec_w}(I_4)$ do not belong to $\Delta_{\prec_w}(I_4 +\langle A\rangle)$. Namely, ${\mbox{lm}}(A
\cdot 1)=X$, ${\mbox{lm}}(A
\cdot Y)=XY$, ${\mbox{lm}}(A
\cdot Y^2)=XY^2$, ${\mbox{lm}}(A
\cdot Y^3)=XY^3$, and ${\mbox{lm}}(A
\cdot X {\mbox{ rem }} X^2+X-Y^3)=Y^3$. Observe that the last calculation holds due to the fact that $X^2+X-Y^3$ contains exactly two monomials of the highest weight. We have shown that the Hamming weight of $\vec{c}$ is at least $5$. With the proof of Theorem \[theseven\] in mind an equivalent formulation of the above is to observe that $(X,1)$, $(X,Y)$, $(X,Y^2)$, $(X,Y^3)$, and $(X,X)$ are OWB. Another equivalent method is guaranteed by the condition that $\Delta_{\prec_w}(I)$ does not contain two monomials of the same weight. This implies that rather than counting the above OWB pairs we only need to observe that $w(\Delta_{\prec_w}(I_4)) \cap \big(
w(X)+w(\Delta_{\prec_w}(I_4)) \big)=\{3,5,6,7,9\}$. Again, a set of size $5$.
The following Proposition (corresponding to [@bookAG Pro. 4.25]) summarises how the Feng-Rao bound is supported by the order domain condition.
\[proptotwo\] Assume $I \subseteq {\mathbb{F}}_q[X_1, \ldots , X_m]$ and $\prec_w$ satisfy the order domain conditions. Consider $I_q=I+\langle
X_1^q-X_1, \ldots , X_m^q-X_m\rangle$. A pair $(P,N)$ where $P, N \in
\Delta_{\prec_w}(I_q)$ is OWB if $w(P)+w(N) \in w(\Delta_{\prec_w}
(I_q))$.
The order domain conditions historically [@handbook; @pellikaan2001existence; @AG; @bookAG] were designed to support the Feng-Rao bounds and therefore it is not surprising that the bound does not work very well without them. The improvement to the Feng-Rao bound that we introduce below allows us to consider relaxed conditions in that we can produce good estimates in the case that the order domain condition (C1) is satisfied but (C2) is not. The following example illustrates the idea in our improvement to Theorem \[theseven\].
\[exmot\] Consider $I=\langle X^4+X^2+X-Y^6-Y^5-Y^3\rangle \subseteq
{\mathbb{F}}_8[X,Y]$. Let $\prec_w$ be the weighted degree lexicographic ordering (Definition \[defwdeg\]) given by $X=X_1$, $Y=X_2$, $w(X)=3$ and $w(Y)=2$. From [@salazar Sec. 3] and [@geilmartin2013further Sec.4.2] we know that the variety ${\mathbb{V}}_{\mathbb{F}_8}(I_8)$ is of size $32$. Combining this observation with Corollary \[thefoot\] we see that $$\Delta_{\prec_w} (I_8) = \{ X^\alpha Y^\beta \mid 0 \leq \alpha < 4, 0
\leq \beta < 8\}.$$ By inspection we see that some weights appear twice in $\Delta_{\prec_w}(I_8)$, some only once. Consider $\vec{c}={\mbox{ev}}(A+I_8)$ where ${\mbox{lm}}(A)=X^3$. That is, $$\begin{aligned}
A&=&a_1 1+a_2Y+a_3 X+a_4 Y^2 + a_5
XY+a_6 Y^3+a_7 X^2\nonumber \\
&&+ a_8 XY^2+a_9 Y^4+a_{10}
X^2Y+a_{11}XY^3+a_{12}X^3.\nonumber\end{aligned}$$ Here, $a_i \in {\mathbb{F}}_8$, $i=1, \ldots , 12$ and $a_{12} \neq 0$. Note that $A$ has two monomials of the highest weight if $a_{11} \neq 0$, namely $X^3$ and $XY^3$. Following the proof of Theorem \[theseven\] we consider $P=X^3$ and look for $N,K \in
\Delta_{\prec_w}(I_8)$ such that $(P,N)$ is OWB and ${\mbox{lm}}(PN
{\mbox{ rem }} {\mathcal{G}})=K$. We have the following possible choices of $(N,K)$, namely $(1,X^3)$, $(Y,X^3Y)$, $(Y^2,X^3Y^2), \ldots
, (Y^7,X^3Y^7),(X^3,X^2Y^6),(X^3Y,X^2Y^7)$. From this we conclude that $w_H(\vec{c}) \geq
10$.\
Note that $X^3 \cdot X {\mbox{ rem }} {\mathcal{G}}=Y^6$. However, $(X^3,X)$ is not OWB as $$XY^3 \prec_w X^3 {\mbox{ but }} XY^3 \cdot X {\mbox{ rem }} {\mathcal{
G}} = X^2Y^3 \succ_w Y^6. \label{eqsnabel}$$ Our improved method consists in considering separately two different cases: $XY^3 \in {\mbox{Supp}}(A)$ and $XY^3 \notin
{\mbox{Supp}}(A)$.\
Assume $a_{11}\neq
0$. Following (\[eqsnabel\]) we see that ${\mbox{lm}}(A \cdot X
{\mbox{ rem }} {\mathcal{G}})=X^2Y^3$. In a similar way we derive ${\mbox{lm}}(A \cdot XY
{\mbox{ rem }} {\mathcal{G}})=X^2Y^4$ and ${\mbox{lm}}(A \cdot XY^2
{\mbox{ rem }} {\mathcal{G}})=X^2Y^5$. From this we conclude $$\Delta_{\prec_w}(I_q+\langle A \rangle)\subseteq \{ X^\alpha Y^\beta
\mid 0 \leq \alpha < 3, 0 \leq \beta <8, {\mbox{ and if }} \alpha=2
{\mbox{ then }} \beta < 3\}$$ and therefore that $w_H(\vec{c}) \geq n-\#
\Delta_{\prec_w}(I_8+\langle A \rangle)=32-19=13$.\
Assume $a_{11}=0$. This means that we do not have to worry about (\[eqsnabel\]) and consequently ${\mbox{lm}}(A\cdot X {\mbox{ rem }} {\mathcal{G}} )=Y^6$ holds. In a similar way we derive ${\mbox{lm}}(A\cdot X^2 {\mbox{ rem }} {\mathcal{G}} )=XY^6$, ${\mbox{lm}}(A\cdot XY {\mbox{ rem }} {\mathcal{G}} )=Y^7$, and ${\mbox{lm}}(A\cdot X^2Y {\mbox{ rem }} {\mathcal{G}} )=XY^7$. We conclude that $$\Delta_{\prec_w}(I_q+\langle A \rangle ) \subseteq \{X^\alpha Y^\beta
\mid 0 \leq \alpha < 3, 0 \leq \beta < 6\}$$ and therefore from the proof of Theorem \[theseven\] we have that $w_H(\vec{c}) \geq n-\# \Delta_{\prec_w}(I_8+\langle A \rangle )=32-18=14$.\
In conclusion $w_H(\vec{c}) \geq \min \{13,14\}=13$.
With Example \[exmot\] in mind we now improve upon Theorem \[theseven\].
\[defstrong\] Let ${\mathcal{G}}$ be a Gröbner basis for $I_q$ with respect to a fixed arbitrary monomial ordering $\prec$. Write $\Delta_\prec(I_q) =\{ M_1, \ldots , M_n\}$ with $M_1
\prec \cdots \prec M_n$. Let ${\mathcal{I}}=\{ 1, \ldots , n\}$ and consider ${\mathcal{I}}^\prime \subseteq {\mathcal{I}}$. An ordered pair of monomials $(M_i,M_j)$, $ 1 \leq i, j \leq n$ is said to be strongly one-way well-behaving (SOWB) with respect to ${\mathcal{I}}^\prime$ if for all $H$ with ${\mbox{Supp}}(H) \subseteq
\{M_s \mid s \in {\mathcal{I}}^\prime \}$, $M_i \in {\mbox{Supp}}(H)$ it holds that ${\mbox{lm}}(M_iM_j {\mbox{ rem }}
{\mathcal{G}})={\mbox{lm}}(H M_j {\mbox{ rem }} {\mathcal{G}})$.
In the following, when writing $\Delta_\prec(I_q) =\{ M_1, \ldots , M_n\}$, we shall always assume that $M_1
\prec \cdots \prec M_n$ holds.\
Consider a non-zero codeword $\vec{c}={\mbox{ev}}(A+I_q)$, where $A=\sum_{s=1}^i a_sM_s$, $i \geq 2$, $a_s \in {\mathbb{F}}_q$ for $s=1,
\ldots ,i$ and $a_i \neq 0$. Let $v$ be an integer $1 \leq v <
i$. We consider $v+1$ different cases that cover all possibilities:\
$a_{i-1}\neq 0$.\
$a_{i-1}= 0$, $a_{i-2}\neq 0$.\
$\ \vdots$\
$a_{i-1}=a_{i-2}= \cdots
=a_{i-v+1}= 0$, $a_{i-v}\neq 0$.\
$a_{i-1}=\cdots =
a_{i-v}=0$.\
For each of the above $v+1$ cases we shall estimate $n-\#
\Delta_{\prec}(I_q+\langle A \rangle )$. Then the minimal obtained value constitutes a lower bound on $w_H(\vec{c})$. Note that in Example \[exmot\] we used $v=1$.
\[thenewbound\] Let $\prec$ be a fixed arbitrary monomial ordering. Consider $\vec{c}={\mbox{ev}}(\sum_{s=1}^i a_s M_s +I_q)$, $a_s \in {\mathbb{F}}_q$, $s=1, \ldots , i$, and $a_i \neq
0$. Let $v$ be an integer $0 \leq v < i$. We have $$w_H(\vec{c}) \geq \min \{ \#{\mathcal{L}}(1), \ldots ,
\#{\mathcal{L}}(v+1)\}$$ where for $t=1, \ldots , v$ we define ${\mathcal{L}}(t)$ as follows: $$\begin{aligned}
{\mathcal{L}}(1)&=& \big\{ K \in \Delta_\prec(I_q) \mid \exists M_j \in \Delta_\prec(I_q)
{\mbox{ such that either }}\nonumber \\
&&(M_i,M_j) {\mbox{ is OWB and }} {\mbox{lm}}(M_iM_j {\mbox{ rem }} {\mathcal{G}})=K
{\mbox{ or }}
\nonumber \\
&&(M_{i-1},M_j) {\mbox{ is SOWB with respect to }} \{1, \ldots ,
i\}\nonumber \\
&&{\mbox{ and }} {\mbox{lm}}(M_{i-1}M_j {\mbox{ rem }} {\mathcal{G}})=K\big\},
\nonumber \\
\ \nonumber \\
{\mathcal{L}}(2)&=& \big\{ K \in \Delta_\prec(I_q) \mid \exists M_j \in \Delta_\prec(I_q)
{\mbox{ such that either }}\nonumber \\
&&(M_i,M_j) {\mbox{ is SOWB with respect to }} \{ 1, \ldots , i-2,i\} \nonumber \\
&&{\mbox{ and }} {\mbox{lm}}(M_iM_j {\mbox{ rem }} {\mathcal{G}})=K
{\mbox{ or }}
\nonumber \\
&&(M_{i-2},M_j) {\mbox{ is SOWB with respect to }} \{1, \ldots ,i-2,
i\}\nonumber \\
&&{\mbox{ and }} {\mbox{lm}}(M_{i-2}M_j {\mbox{ rem }} {\mathcal{G}})=K\big\},
\nonumber \\
\ \nonumber \\
&&{\mbox{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }} \vdots \nonumber \\
\ \nonumber \\
{\mathcal{L}}(v)&=&\big\{ K \in \Delta_\prec(I_q) \mid \exists M_j \in \Delta_\prec(I_q)
{\mbox{ such that either }}\nonumber \\
&&(M_i,M_j) {\mbox{ is SOWB with respect to }} \{ 1, \ldots , i-v,i\} \nonumber \\
&&{\mbox{ and }} {\mbox{lm}}(M_iM_j {\mbox{ rem }} {\mathcal{G}})=K
{\mbox{ or }}
\nonumber \\
&&(M_{i-v},M_j) {\mbox{ is SOWB with respect to }} \{1, \ldots ,i-v,
i\}\nonumber \\
&&{\mbox{ and }} {\mbox{lm}}(M_{i-v}M_j {\mbox{ rem }} {\mathcal{G}})=K\big\},
\nonumber \end{aligned}$$ Finally, $$\begin{aligned}
{\mathcal{L}}(v+1)&=& \big\{ K \in \Delta_\prec(I_q) \mid \exists M_j \in \Delta_\prec(I_q)
{\mbox{ such that }} (M_i,M_j) {\mbox{ is SOWB}}
\nonumber \\
&& {\mbox{ with respect to }} \{1, \ldots ,
i-v-1,i\} {\mbox{ and }} {\mbox{lm}}(M_iM_j {\mbox{ rem }} {\mathcal{G}})=K \big\}.
\nonumber \end{aligned}$$ Given a code $C(I,L)$ write $\Box_\prec(L)=\{M_{i_1}, \ldots ,
M_{i_{\dim (L)}}\}$. A lower bound on the minimum distance is obtained by repeating the above calculation for each $i \in \{i_1, \ldots , i_{\dim (L) }\}$. For each choice of $i$ an appropriate value $v$ is chosen.
If $v=0$ then only the last set is present and this set equals the set in (\[eqafffr\]). For $v>0$ the $v+1$ expressions correspond to the $v+1$ cases described prior to the theorem (in the same order). The proof technique resembles the arguments used in Example \[exmot\].
\[remtksh2\] Consider an ideal $I\subseteq {\mathbb{F}}_q[X_1, \ldots , X_m]$ and a corresponding weighted degree lexicographic ordering $\prec_w$ such that the order domain condition (C1) is satisfied but (C2) is not. Let ${\mathcal{F}}$ be a Gröbner basis for $I$ with respect to $\prec_w$. Assume Theorem \[thenewbound\] is used to estimate the Hamming weight of $\vec{c}={\mbox{ev}}(A+I_q)$ where ${\mbox{lm}}(A)=M_i$. A natural choice of $v$ is the unique non-negative integer which satisfies $w(M_i)=w(M_{i-1})= \cdots =w(M_{i-v}) > w(M_{i-v-1})$. To see why this choice of $v$ is natural, note that when reducing $A M_j$ modulo ${\mathcal{F}}$ the weight of the leading monomial remains the same. Hence, the leading monomial of $AM_j {\mbox{ rem }}
{\mathcal{F}}$ can not be equal to $M_t M_j {\mbox{ rem }}
{\mathcal{F}}$ for $t \leq i-v-1$. On the other hand as illustrated in Example \[exmot\] this may happen when $t \geq i-v$. For $I$ and $\prec_w$ such that both order domain conditions are satisfied the above choice of $v$ is $v=0$ and Theorem \[thenewbound\] therefore simplifies to the usual Feng-Rao bound Theorem \[theseven\] in this case.
Theorem \[thenewbound\] can be applied to any code $C(I,L)$. However, it is not clear if there is any advantage in considering other choices of $L$ than $L={\mbox{Span}}_{\mathbb{F}_q} \{ {\mbox{ev}}(M_{i_1}+I_q), \ldots ,
{\mbox{ev}}(M_{i_k}+I_q)\}$. When $i_1=1, \ldots , i_k=k$ we shall denote the corresponding code by $E(k)$. Observe that Theorem \[thenewbound\] suggests an improved code construction as follows.
\[defimpcode\] Fix non-negative numbers $v_1, \ldots , v_n$ and calculate for each $M_i$, $i=1, \ldots , n$ the number in Theorem \[thenewbound\] where $v=v_i$. Call these number ${\widetilde{\sigma}}(i)$, $i=1, \ldots , n$. We define ${\widetilde{E}}_{imp}(\delta)$ to be the code with $L={\mbox{Span}}_{\mathbb{F}_q}\{ {\mbox{ev}}(M_i+I_q) \mid
{\widetilde{\sigma}}(i) \geq \delta \}$.
The minimum distance of ${\widetilde{E}}_{imp}(\delta)$ satisfies $d({\widetilde{E}}_{imp}(\delta)) \geq \delta$.
The above improved code construction is in the spirit of Feng and Rao’s work. When improved codes are constructed on the basis of the Feng-Rao bound, Theorem \[theseven\], rather than on the basis of the improved bound of the present paper, Theorem \[thenewbound\], the notation used is ${\widetilde{E}}(\delta )$ (see [@bookAG Def. 4.38]). In Section \[secfourandaquarter\] we shall see that one can sometimes derive even further improved codes from Theorem \[thenewbound\] than ${\widetilde{E}}_{imp}(\delta)$.\
We conclude this section by noting that in a straight forward manner one can enhance the above bound to deal also with generalised Hamming weights. We postpone the discussion of the details to Section \[secfourandahalf\].
Generalised $C_{ab}$ polynomials {#secthree}
================================
As mentioned in the previous section good candidates for our new bound are affine variety codes where the order domain condition (C1) is satisfied, but the order domain condition (C2) is not. A particular simple class of curves that satisfy the order domain conditions are the well-known $C_{ab}$ curves. They were introduced by Miura in [@miura1993algebraic; @miura1; @miura1998linear] to facilitate the use of the Feng-Rao bound for dual codes. In this section we introduce generalised $C_{ab}$ polynomials which corresponds to allowing the same weight to occur more than once in the footprint (condition (C2)). It should be stressed that we make no assumption that generalised $C_{ab}$ polynomials are irreducible as it has no implication for our analysis.\
From [@miura1998linear App.B and the lemma at p. 1416] we have a complete characterisation of $C_{ab}$ curves. We shall adapt the description in [@matsumoto1998c_ab] which is an English translation of Miura’s results. From [@matsumoto1998c_ab Th.1] we have:
\[thetksh3\] Let $\bar{k}$ be the algebraic closure of a perfect field $k$, ${\mathcal{X}}
\subseteq \bar{k}^2$ be a possibly reducible affine algebraic set defined over $k$, $x,y$ the coordinate of the affine plane $\bar{k}^2$, and $a$, $b$ relatively prime positive integers. The following two conditions are equivalent:
- ${\mathcal{X}}$ is an absolutely irreducible algebraic curve with exactly one $k$ rational place $Q$ at infinity, and the pole divisors of $x$ and $y$ are $bQ$ and $aQ$, respectively.
- ${\mathcal{X}}$ is defined by a bivariate polynomial of the form $$\alpha_{a,0}x^a+\alpha_{0,b}y^b+\sum_{ib+ja<ab}\alpha_{i,j}x^iy^j,\label{eqryutaroh1}$$ where $\alpha_{i,j} \in k$ for all $i,j$ and $\alpha_{a,0}$, $\alpha_{0,b}$ are non-zero.
The definition of $C_{ab}$ curves given in the literature is that of (\[eqryutaroh1\]). We recall the following result from [@miura1998linear]. We adapt the description from [@matsumoto1998c_ab Cor. 3].
\[propquotient\] Let $F(X,Y) \in k[X,Y]$ be a polynomial of the form (\[eqryutaroh1\]), $Q$ a unique place at infinity of the $C_{ab}$ curve defined by $F(X,Y)$. Then $$\{X^iY^j+\langle F(X,Y)\rangle \mid
0 \leq i \leq a-1, 0 \leq j
\}$$ is a $k$-basis for $k[X,Y]/\langle F(X,Y)\rangle$ and the elements in the basis have pairwise distinct discrete valuations at $Q$. If the $C_{ab}$ curve is non-singular, then $$k[X,Y]/\langle F(X,Y) \rangle
={\mathcal{L}}( \infty Q)$$ and a basis of ${\mathcal{L}}(mQ)$ is $$\{X^iY^j+\langle F(X,Y)\rangle \mid 0 \leq i \leq a-1,
0 \leq j,
ai+bj \leq m\}$$ for any non-negative integer $m$.
Let $w(X)$ and $w(Y)$, respectively, be minus the discrete valuation of $x$ at $Q$ and minus the discrete valuation of $y$ at $Q$, respectively. Consider the corresponding weighted degree lexicographic ordering with $X=X_1$ and $Y=X_2$. If we combine (\[eqryutaroh1\]) with the first part of Proposition \[propquotient\] we see that $C_{ab}$ curves satisfy the order domain conditions. Observe, that we can consider the related affine variety codes $C(I,L)$ and $C^\perp(I,L)$ regardless of the curve being non-singular or not. This point of view is taken in [@handbook Sec. 4.2]. If the curve is non-singular the corresponding affine variety code description does not have an algebraic geometric code counterpart. We now introduce generalised $C_{ab}$ polynomials.
\[defpab\] Let $w(X)=\frac{b}{\gcd (a,b)}$ and $w(Y)=\frac{a}{\gcd (a,b)}$ where $a$ and $b$ are two different positive integers. Given a field $k$, let $F(X,Y)=X^a+\alpha Y^b + R(X,Y)\subseteq k[X,Y]$, $\alpha \in k
\backslash \{ 0 \}$, be such that all monomials in the support of $R$ have smaller weight than $w(X^a)=w(Y^b)=\frac{ab}{\gcd(a,b)}$. Then $F(X,Y)$ is called a generalized $C_{ab}$ polynomial.
Miura in [@miura1993algebraic Sec. 4.1.4] treated the curves related to irreducible generalized $C_{ab}$ polynomials. Besides that we do not require the generalized $C_{ab}$ polynomials to be irreducible, our point of view is different from Miura’s as we will use for the code construction the algebra ${\mathbb{F}}_q[X,Y]/\langle F(X,Y)\rangle$. For generalized $C_{ab}$ polynomials this algebra does not in general equal a space ${\mathcal{L}}(m_1P_1+\cdots +m_sP_s)$, $P_1, \ldots, P_s$ being rational places. We mention that the variations of $C_{ab}$ curves considered by Feng and Rao in [@FR1] is different from Definition \[defpab\].\
For the code construction we would like to have generalised $C_{ab}$ polynomials with many zeros and at the same time to have a variety of possible $a,b$ to choose from, as these parameters turn out to play a crucial role in our bound for the minimum distance. As we shall now demonstrate there is a simple technique for deriving this when the field under consideration is not prime. The situation is in contrast to $C_{ab}$ curves for which it is only known how to get many points for restricted classes of $a$ and $b$. Our method builds on ideas from [@salazar] and [@miura1993algebraic Sec. 5].\
Let $p$ be a prime power and $q=p^m$ where $m\geq 2$ is an integer. The technique that we shall employ involves letting $F(X,Y)=G(X)-H(Y)$ where both $G$ and $H$ are $({\mathbb{F}}_{p^m},{\mathbb{F}}_p)$-polynomials.
\[deftksh4\] Let $m$ be an integer, $m \geq 2$. A polynomial $F(X) \in
{\mathbb{F}}_{p^m}[X]$ is called an $({\mathbb{F}}_{p^m},{\mathbb{F}}_p)$-polynomial if $F(\gamma ) \in
{\mathbb{F}}_p$ holds for all $\gamma \in {\mathbb{F}}_{p^m}$.
An obvious characterisation of $({\mathbb{F}}_{p^m},{\mathbb{F}}_p)$-polynomials is that $F(X)=(X^{p^m}-X)Q(X)+F^\prime (X)$, where $F^\prime(X)$ is an $({\mathbb{F}}_{p^m},{\mathbb{F}}_p)$-polynomial of degree less than $p^m$. Here, we used the convention that $\deg (0) =-\infty$. By Fermat’s little theorem the set of $({\mathbb{F}}_{p^m},{\mathbb{F}}_p)$-polynomials of degree less than $p^m$ constitutes a vector space over ${\mathbb{F}}_p$. Clearly, one could derive a basis by Lagrange interpolation. For our purpose, however, it is interesting to know what are the possible degrees of the polynomials in the vector space.
\[propduderdu\] Let $C_{i_1}, \ldots , C_{i_t}$ be the different cyclotomic cosets modulo $p^m-1$ (multiplication by $p$). Here, for $s=1, \ldots , t$ it is assumed that $i_s$ is chosen as the smallest element in the given coset. For $s=1,
\ldots , t$, $F_{i_s}(X)=\sum_{l \in C_{i_s}}X^l$, is an $({\mathbb{F}}_{p^m},{\mathbb{F}}_p)$-polynomial. Furthermore, the polynomial $X^{p^m-1}$ is an $({\mathbb{F}}_{p^m},{\mathbb{F}}_p)$-polynomial.\
For all the polynomials $F$ in the proposition we have $F^p=F$.
The set $\{F_{i_1}, \ldots , F_{i_t},X^{p^m-1}
\}$ contains two of the most prominent $({\mathbb{F}}_{p^m},{\mathbb{F}}_p)$-polynomials, namely the trace polynomial $F_1(X)=X^{p^{m-1}}+X^{p^{m-2}}+ \cdots + X^p+X$ and the norm polynomial $X^{(p^m-1)/(p-1)}$. Note that the norm polynomial equals $F_{(p^m-1)/(p-1)}$ if $p>2$. For $p=2$ it equals $X^{p^m-1}$. Observe also that except for the constant polynomial $F_0=1$, the trace polynomial is of lowest possible degree.\
From [@hernando2011dimension Prop. 3.2] we have:
\[prohern\] A polynomial $F(X)\in {\mathbb{F}}_{p^m}[X]$ is an $({\mathbb{F}}_{p^m},{\mathbb{F}}_p)$-polynomial of degree less than $p^m-1$ if and only if $$F(X)= F_1(H(X)) {\mbox{ rem }} (X^{p^m-1}-1)$$ for some $H(X) \in {\mathbb{F}}_{p^m}[X]$.
From Proposition \[propduderdu\] and Proposition \[prohern\] we conclude:
Let $F(X)$ be an $({\mathbb{F}}_{p^m},{\mathbb{F}}_p)$-polynomial of degree less than $p^m$. Then $\deg(F) \in \{\deg(F_{i_1}), \ldots ,
\deg(F_{i_t}),p^m-1\}$.
We now return to the question of designing generalised $C_{ab}$ polynomials $F(X,Y)=G(X)-H(Y)$ with many zeros. One way of doing this is to choose $G(X)$ to be the trace polynomial [@salazar Sec. 3]. As is well-known this polynomial maps exactly $p^{m-1}$ elements from ${\mathbb{F}}_{p^m}$ to each value in ${\mathbb{F}}_p$. Hence, such a polynomial $F(X,Y)$ must have $p^{2m-1}$ zeros. However, there are other polynomials in the above set with properties similar to the trace polynomial.
\[proptracelike\] Consider the polynomials $F_{i_s}$, $s=1, \ldots , t$ related to a field extension ${\mathbb{F}}_{p^m}/{\mathbb{F}}_p$, $m\geq 2$ (Proposition \[propduderdu\]). We have $\gcd (i_s ,p^m-1)=1$ if and only if for each $\eta \in {\mathbb{F}}_p$ there exists exactly $p^{m-1}$ $\gamma \in {\mathbb{F}}_{p^m}$ such that $F_{i_s}(\gamma)=\eta$.
We have $F_{i_s}(X)=F_1(X^{i_s}) {\mbox{ mod }} (X^{q^m-1}-1)$, where $F_1(X)$ is the trace polynomial. Under the condition that $\gcd (i_s,
p^m-1)=1$ the monomial $X^{i_s}$ defines a bijective map from ${\mathbb{F}}_{p^m} \rightarrow
{\mathbb{F}}_{p^m}$. This proves the “only if” part. We leave the “if” part for the reader.
\[extksh5\] Consider first the field extension ${\mathbb{F}}_8/{\mathbb{F}}_2$. The non-trivial cyclotomic cosets modulo $7$ are $C_1=\{1,2,4\}$, and $C_3=\{3,6,5\}$. From this we find the following $({\mathbb{F}}_8,{\mathbb{F}}_2)$-polynomials: $F_1(X)=X^4+X^2+X$, $F_3(X)=X^6+X^5+X^3$, and $X^7$. The first two polynomials have the property described in Proposition \[proptracelike\]. This is a consequence of $7$ being a prime.\
Consider next the field extension ${\mathbb{F}}_{16}/{\mathbb{F}}_2$. The non-trivial cyclotomic cosets modulo $15$ are $C_1=\{1,2,4,8\}$, $C_3=\{3,6,12,9\}$, $C_5=\{5,10\}$, $C_7=\{7,14,13,11\}$. Hence, we get the following $({\mathbb{F}}_{16},{\mathbb{F}}_2)$-polynomials $F_1(X)=X^8+X^4+X^2+X$, $F_3(X)=X^{12}+X^9+X^6+X^3$, $F_5(X)=X^{10}+X^5$, $F_7(X)=X^{14}+X^{13}+X^{11}+X^7$, and $X^{15}$. The polynomials with the property described in Proposition \[proptracelike\] are $F_1(X)$, $F_7(X)$.\
Consider finally the field extension ${\mathbb{F}}_{32}/{\mathbb{F}}_2$. Observe that $31$ is a prime. Hence, all the polynomials $F_{i_s}$, $i_s >0$, have the property of Proposition \[proptracelike\]. These are $F_1(X)=X^{16}+X^8+X^4+X^2+X$, $F_3(X)=X^{24}+X^{17}+X^{12}+X^6+X^3$, $F_5(X)=X^{20}+X^{18}+X^{10}+X^9+X^5$, $F_7(X)=X^{28}+X^{25}+X^{19}+X^{14}+X^7$, $F_{11}(X)=X^{26}+X^{22}+X^{21}+X^{13}+X^{11}$, and $F_{15}(X)=X^{30}+X^{29}+X^{27}+X^{23}+X^{15}$.
Codes from optimal generalised $C_{ab}$ polynomials {#secfour}
===================================================
In this section we consider codes from generalised $C_{ab}$ polynomials over ${\mathbb{F}}_q$ with $n=aq$ zeros. These polynomials are optimal in the sense that a bivariate polynomial with leading monomial $X^a$ can have no more zeros over ${\mathbb{F}}_q$, as is seen from the footprint bound Corollary \[thefoot\]. Hence, we shall call them [*[optimal generalised $C_{ab}$ polynomials]{}*]{}. We list a couple of properties of optimal generalised $C_{ab}$ polynomials $F(X,Y)=X^a+\alpha Y^b+R(X,Y)$. It holds that $a<b$ and that $\{F(X,Y), Y^{q}-Y\}$ constitutes a Gröbner basis ${\mathcal{G}}$ for $I_q=\langle F(X,Y), X^q-X, Y^q-Y\rangle$ with respect to $\prec_w$. Here, and in the remaining part of the section, $\prec_w$ is the weighted degree lexicographic ordering in Definition \[defwdeg\] with weights as in Definition \[defpab\] and with $X=X_1$, $Y=X_2$. Furthermore, $\{M_1, \ldots , M_n\}=\Delta_{\prec_w}(I_{q})=\{ X^{i_1}Y^{i_2} \mid 0
\leq i_1 < a, 0 \leq i_2 < q\}$. Recall, that we assume $M_1 \prec_w \cdots \prec_w M_n$.\
From the previous section we have a simple method for constructing optimal generalised $C_{ab}$ polynomials over ${\mathbb{F}}_q={\mathbb{F}}_{p^m}$, where $p$ is a prime power and $m$ is an integer greater or equal to $2$. The method consists in letting $F(X,Y)=G(X)-H(Y)$ where $G(X)$ is the trace polynomial and $H(Y)$ is an arbitrary non-trivial $({\mathbb{F}}_{p^m},{\mathbb{F}}_p)$-polynomial. We stress that the results of the present section hold for any optimal generalised $C_{ab}$ polynomial over arbitrary finite field ${\mathbb{F}}_q$. The main result of the section is:
\[teoalpha\] Let $I_{q}$ be defined from an optimal generalised $C_{ab}$ polynomial and let the weights $w(X)$ and $w(Y)$ be as in Definition \[defpab\]. Consider $\vec{c} = \mathrm{ev}(\sum_{s=1}^i a_s M_s +
I_q),$ $a_s \in {{\mathbb{F}}}_q$, $s=1,\ldots,i$ and $a_i \neq 0$. Write $M_i = X^{\alpha_1} Y^{\alpha_2}$ and $T=\alpha_1 {\mbox{ rem }} w(Y)$. We have that $$w_H(\vec{c}) \geq (a-\alpha_1)(q-\alpha_2) + \epsilon \mbox{ where}$$ $$\epsilon = \begin{cases}
0 & \mbox{if }q-b \leq \alpha_2 < q \\
T(q-\alpha_2-b) & \mbox{if }0 \leq \alpha_1 \leq a-w(Y)\\
& \mbox{and }0 \leq \alpha_2 < q-b \\
\alpha_1(q-\alpha_2-b) & \mbox{if }a-w(Y) < \alpha_1 < a \mbox{ and }\\
& q-w(X)-\alpha_1\frac{b-w(X)}{a-w(Y)} < \alpha_2 < q-b \\
T(q-\alpha_2-w(X)) & \mbox{if }a-w(Y) < \alpha_1 < a \mbox{ and }\\
& 0 \leq \alpha_2 \leq q-w(X)-\alpha_1\frac{b-w(X)}{a-w(Y)}. \end{cases}$$
The proof of Theorem \[teoalpha\] calls for a definition and some lemmas. Recall from Theorem \[thenewbound\] that we need to estimate the size of the sets ${\mathcal{L}}(u)$, $u=1, \ldots ,
v+1$. For this purpose we introduce the following related sets:
\[defiB\] Let the notation be as in Definition \[defpab\] and Theorem \[teoalpha\]. For arbitrary $\alpha_1, \alpha_2$, $0 \leq
\alpha_1 <a$, $0\leq \alpha_2 <q$ we define $$\begin{array}{l}
B_1(X^{\alpha_1}Y^{\alpha_2}) = \{X^{\gamma_1} Y^{\gamma_2}\mid \alpha_1 \leq \gamma_1 < a, \alpha_2 \leq \gamma_2 < q\}, \\
\ \\
B_2(X^{\alpha_1}Y^{\alpha_2}) = \\
\ \\
{\mbox{ \ \ \ }} \left\{ \begin{array}{ll}
\bigg\{X^{\gamma_1}Y^{\gamma_2}\mid \alpha_1-T \leq \gamma_1 < \alpha_1, \\
{\mbox{ \ \hspace{4cm} }} \alpha_2+b \leq \gamma_2 < q \bigg\} & \begin{array}{l} {\mbox{\ if \ }} T \neq 0 \\
{\mbox{\ and \ }} 0 \leq \alpha_2 < q-b
\end{array}\\
\ \\
\emptyset & {\mbox{\hspace{1.3mm} otherwise, \ }}
\end{array} \right.\\
\ \\
\mbox{and for $u =1,\ldots,\gcd(a,b)$}\\
\ \\
B_3(X^{\alpha_1}Y^{\alpha_2},u) = \\
\ \\
{\mbox{ \ \ \ }} \left\{ \begin{array}{ll}
\bigg\{X^{\gamma_1}Y^{\gamma_2}\mid a-w(Y)u \leq \gamma_1 < \alpha_1,\\
{\mbox{ \ \hspace{0.7cm}}} \alpha_2+w(X)u \leq \gamma_2 < q\bigg\} & \begin{array}{l} {\mbox{ \ if \ }} a-w(Y) < \alpha_1 < a \\
{\mbox{ \ and \ }} 0 \leq \alpha_2 < q-b
\end{array} \\
\ \\
\emptyset & {\mbox{\ {\hspace{2.3mm}} otherwise.}}
\end{array} \right.\\
\end{array}$$
\[remarkB\] Note that $w(X)\gcd(a,b) = b$ and $w(Y)\gcd(a,b) = a$, thus: $$B_3(X^{\alpha_1}Y^{\alpha_2},\gcd(a,b)) = \bigg\{X^{\gamma_1}Y^{\gamma_2} \mid 0 \leq \gamma_1 < \alpha_1,\alpha_2+b \leq \gamma_2 < q\bigg\}.$$ Furthermore for any choice of $u \in \{1,\ldots,\gcd(a,b)\}$ and $M \in \Delta_\prec(I_q)$ we have that $B_1(M) \cap B_2(M) = B_1(M) \cap B_3(M,u) = \emptyset$. If $B_3(M,u) \neq \emptyset$ then $B_2(M)\subseteq B_3(M,u)$.
Before continuing with the lemmas we illustrate Definition \[defiB\] with an example.
\[exB\] Consider an optimal generalised $C_{ab}$ polynomial $F(X,Y)=X^9-Y^{12}+R(X,Y) \in {\mathbb{F}}_{27}[X,Y]$. We have $a=9$, $b=12$, $w(X)=4$, $w(Y)=3$, and $\Delta_{\prec_w}(I_q)=\{X^{i_1}Y^{i_2} \mid 0 \leq i_1 < 9, 0 \leq i_2 < 27\}$.\
We first treat the case $X^{\alpha_1}Y^{\alpha_2}=X^5Y^{16}$. We have $\alpha_2 \geq q-b$, thus $B_2(X^{\alpha_1}Y^{\alpha_2})=B_3(X^{\alpha_1}Y^{\alpha_2},u)=\emptyset$ for any $u$. For an illustration see Figure \[figure1a\].\
Now consider the case $X^{\alpha_1}Y^{\alpha_2}=X^5Y^{4}$. We have $\alpha_2 < q-b$ and $T = 2 \neq 0$ and therefore $B_2(X^{\alpha_1}Y^{\alpha_2})$ is non-empty. Because $T=2$, the width of $B_2(X^{\alpha_1}Y^{\alpha_2})$ is $2$. Turning to $B_3(X^{\alpha_1}Y^{\alpha_2},u)$ we see that $\alpha_1<a-w(Y)$ and therefore the sets $B_3(X^{\alpha_1}Y^{\alpha_2},u)$’s are empty. See Figure \[figure1a\] for an illustration.\
![Left part: $X^{\alpha_1}Y^{\alpha_2}=X^5Y^{16}$. Only $B_1$ present. Right part: $X^{\alpha_1}Y^{\alpha_2}=X^5Y^{4}$. Light grey area is $B_1$, medium grey area is $B_2$. $B_3$ is not present.[]{data-label="figure1a"}](case1b.jpg){width="90.00000%"}
Consider next the case $X^{\alpha_1}Y^{\alpha_2}=X^8Y^{3}$. We have $\alpha_2 < q-b$ and $\alpha_1 > a-w(Y)$ and therefore $B_2(X^{\alpha_1}Y^{\alpha_2})$ and $B_3(X^{\alpha_1}Y^{\alpha_2},u)$ for $u=1,2,3$ are non-empty. The situation regarding $B_2(X^{\alpha_1}Y^{\alpha_2})$ is similar to the case $X^5Y^4$. The set $B_3(X^{\alpha_1}Y^{\alpha_2},u)$ can be thought of as an improvement to $B_2(X^{\alpha_1}Y^{\alpha_2})$. We see that $\gamma_1$ runs from $a-w(Y)u$ to $\alpha_1$ and $\gamma_2$ from $\alpha_2+w(X)u$ to $q$. For an illustration see Figure \[figure2a\].\
![In both parts $X^{\alpha_1}Y^{\alpha_2}=X^8Y^{3}$. Left part: Light grey area is $B_1$, medium grey area is $B_2$, and dark grey area plus medium grey area correspond to $B_3(X^{\alpha_1}Y^{\alpha_2},1)$. Right part: Light grey area is $B_1$, medium grey area is $B_2$, and dark grey area plus medium grey area correspond to $B_3(X^{\alpha_1}Y^{\alpha_2},3)$.[]{data-label="figure2a"}](case3b.jpg){width="90.00000%"}
\[propBset\] Consider $\vec{c}={\mbox{ev}}(\sum_{s=1}^i a_s M_s +I_q)$, $a_s \in {\mathbb{F}}_q$, $s=1, \ldots , i$, and $a_i \neq 0$. Let $M_i = X^{\alpha_1} Y^{\alpha_2}$ and $v= \alpha_1 {\mbox{ div }} w(Y)$ (that is, $v$ satisfies $\alpha_1 = w(Y)v+T$, where $T = \alpha_1 {\mbox{ rem }} w(Y)$). It holds that:
- $B_1(X^{\alpha_1}Y^{\alpha_2}) \subseteq {\mathcal{L}}(u) \mbox{ for }u =1,\ldots,v+1$.
- $B_2(X^{\alpha_1}Y^{\alpha_2}) \subseteq {\mathcal{L}}(u) \mbox{ for }u =1,\ldots,v+1$.
- $B_3(X^{\alpha_1}Y^{\alpha_2},\gcd(a,b)) \subseteq {\mathcal{L}}(v+1)$.
- $B_3(X^{\alpha_1}Y^{\alpha_2},u) \subseteq {\mathcal{L}}(u) \mbox{ for }u =1,\ldots,v$.
\
\
Assume $M_l = X^{\gamma_1}Y^{\gamma_2} \in
B_1(X^{\alpha_1}Y^{\alpha_2})$. We have $\alpha_1 \leq \gamma_1 < a$ and $\alpha_2 \leq \gamma_2 < q$. Choosing $M_j =
X^{\gamma_1-\alpha_1}Y^{\gamma_2-\alpha_2}$ we get $\mbox{lm}(M_{i}M_j
{\mbox{ rem }} {\mathcal{G}}) = M_l$. Let $i' \in \{1,\ldots,i-1\}$, then by the properties of a monomial ordering $M_{i'}M_j \prec_w M_iM_j$ holds. This means that $(M_i,M_j)$ is SOWB with respect the set $\{1,\ldots,i\}$. Thus $M_l \in {\mathcal{L}}(u)$ for $u =1,
\ldots,v+1$.\
\
If $T=0$ or $q-b \leq \alpha_2 < q$ then the result follows trivially.\
Assume $T \neq 0$ and $0 \leq \alpha_2 < q-b$. Let $M_l = X^{\gamma_1}Y^{\gamma_2} \in B_2(X^{\alpha_1}Y^{\alpha_2})$. We have $\alpha_1-T \leq \gamma_1 < \alpha_1$ and $\alpha_2 + b \leq \gamma_2 < q$. Choosing $M_j = X^{\gamma_1-\alpha_1+a}Y^{\gamma_2-\alpha_2-b}$ (which belongs to $\Delta_{\prec_w}(I_q)$ by the definition of $B_2$) we get $${\mbox{lm}}(M_{i}M_j {\mbox{\ rem\ }} {\mathcal{G}})={\mbox{lm}}(M_{i}M_j -X^{\gamma_1}Y^{\gamma_2-b}F(X,Y))=X^{\gamma_1}Y^{\gamma_2}.$$ We want to prove that $(M_i,M_j)$ is SOWB with respect the set $\{1,\ldots,i\}$. We consider $M_{i'}$ with $i' \in
\{1,\ldots,i-1\}$. If $w(M_{i'}) < w(M_i)$ then the proof follows from $w(M_{i'}M_j) < w(M_{i}M_j)$ using the fact that reducing modulo $F$ does not change the weight of the leading monomial. If $w(M_{i'}) = w(M_i)$ then there exists an integer $z$ with $\alpha_1 - zw(Y) \geq 0$ such that $M_{i'} = X^{\alpha_1 - zw(Y)}Y^{\alpha_2 + zw(Y)}$. Therefore $\gamma_1 -zw(Y) \geq 0.$\
Now $M_{i^\prime}M_j=X^{a+\gamma_1-zw(Y)}Y^{\gamma_2-b+zw(X)}$ and therefore $${\mbox{lm}}(M_{i^\prime}M_j {\mbox{ rem }} {\mathcal{G}}) = {\mbox{lm}}(M_{i^\prime}M_j - X^{\gamma_1-zw(Y)}Y^{\gamma_2-b+zw(X)}F(X,Y))$$ $$= X^{\gamma_1-zw(Y)}Y^{\gamma_2+zw(X)} \prec_w X^{\gamma_1}Y^{\gamma_2}.$$ Again we employed the fact that reducing modulo $F$ does not change the weight of the leading monomial. We conclude that ${\mbox{lm}}(M_{i^\prime}M_j {\mbox{\ rem\ }} {\mathcal{G}}) \prec_w
X^{\gamma_1}Y^{\gamma_2}$ and that $(M_i,M_j)$ is SOWB with respect the set $\{1,\ldots,i\}$. Thus $M_l \in {\mathcal{L}}(u)$ for $u
=1, \ldots,v+1$.\
\
If $0 \leq \alpha_1 \leq a-w(Y)$ or $q-b \leq \alpha_2 < q$ then the result follows trivially.\
Assume $a-w(Y) < \alpha_1 < a$ and $0 \leq \alpha_2 < q-b$, then $v=\gcd (a,b)-1$. Let $M_l =
X^{\gamma_1}Y^{\gamma_2} \in
B_3(X^{\alpha_1}Y^{\alpha_2},\gcd(a,b))$. We have $0 \leq \gamma_1 <
\alpha_1$ and $\alpha_2+b \leq \gamma_2 < q$. Choosing $M_j =
X^{\gamma_1-\alpha_1+a}Y^{\gamma_2-\alpha_2-b}$ we get $\mbox{lm}(M_iM_j \mbox{ rem } {\mathcal{G}}) = M_l$. We want to prove that $(M_i,M_j)$ is SOWB with respect the set $\{1,\ldots,i-v-1\}$. We consider $M_{i'}$ with $i' \in \{1,\ldots,i-1\}$. If $w(M_{i'}) <
w(M_i)$ the proof follows because $w(M_{i'}M_j) <
w(M_{i}M_j)$ using the fact that reducing modulo $F$ does not change the weight of the leading monomial. As $v = \gcd(a,b)-1$ there does not exists any $i' \in \{1,\ldots,i-v-1,i\}$ such that $w(M_{i'}) = w(M_{i})$. From this it follows that $(M_i,M_j)$ is SOWB with respect the set $\{1,\ldots,i-v-1\}$ and thus $M_l \in {\mathcal{L}}(v+1)$.\
\
If $q-b \leq \alpha_2 < q$ or $0 \leq \alpha_1 \leq a-w(Y)$ then the result follows trivially.\
Assume $a-w(Y) < \alpha_1 < a$ and $0 \leq \alpha_2 < q-b$, then $v=\gcd (a,b)-1$. Let $M_l =
X^{\gamma_1}Y^{\gamma_2} \in B_3(X^{\alpha_1}Y^{\alpha_2},u)$. We have $a-w(Y)u \leq \gamma_1 < \alpha_1$ and $\alpha_2+w(X)u \leq \gamma_2 <
q$. By the definition of $\prec_w$ and the form of $\Delta_{\prec_w}(I_q)$ we have that $M_{i-u}=X^{\alpha_1-w(Y)u}Y^{\alpha_2+w(X)u}$. Choosing $M_j =
X^{\gamma_1-\alpha_1+w(Y)u}Y^{\gamma_2-\alpha_2-w(Y)u}$ we get $\mbox{lm}(M_{i-u}M_j {\mbox{ rem }} {\mathcal{G}}) = M_l$. Note that $M_{i-u}$ and $M_j$ are in $\Delta_{\prec_w}(I_q)$ because $v=\gcd(a,b)-1$, $a-w(Y) < \alpha_1 < a$ and $0 \leq \alpha_2 <
q-b$. We want to prove that $(M_i,M_j)$ is SOWB with respect the set $\{1,\ldots,i-u,i\}$. We consider $M_{i'}$ with $i' \in
\{1,\ldots,i-1\}$. If $w(M_{i'}) < w(M_i)$ then the proof follows from $w(M_{i'}M_j) < w(M_{i}M_j)$ using the fact that reducing modulo $F$ does not change the weight of the leading monomial. The monomials $M_{i'}$ which satisfy $w(M_{i'}) =
w(M_{i-u})$ are $M_i$ and $M_{i-z}$ for $z=u,\ldots,v$. However, $M_iM_j {\mbox{ rem }} {\mathcal{G}} \prec_w M_{i-u}M_j {\mbox{ rem }} {\mathcal{G}}$ because $\gamma_1+w(Y)u > a$ and $M_{i-t}M_j \prec_w M_{i-u}M_j$ for any $t=u+1,\ldots,v$ due to the properties of a monomial ordering. From this it follows that $(M_i,M_j)$ is SOWB with respect the set $\{1,\ldots,i-u,i\}$ and thus $M_l \in {\mathcal{L}}(u)$, for $u=1, \ldots , v$.
\[corB\] Consider $\vec{c}={\mbox{ev}}(\sum_{s=1}^i a_s M_s +I_q)$, $a_s \in {\mathbb{F}}_q$, $s=1, \ldots , i$, and $a_i \neq 0$. Write $M_i = X^{\alpha_1} Y^{\alpha_2}$. For $u =1,\ldots,v+1$, with $v=\alpha_1 {\mbox{ div }} w(Y)$, we have that: $$B_1(X^{\alpha_1}Y^{\alpha_2}) \cup B_2(X^{\alpha_1}Y^{\alpha_2}) \subseteq {\mathcal{L}}(u),$$ $$B_1(X^{\alpha_1}Y^{\alpha_2}) \cup B_3(X^{\alpha_1}Y^{\alpha_2},u) \subseteq {\mathcal{L}}(u).$$
The lemma follows directly from Remark \[remarkB\] and Lemma \[propBset\].
It is not hard to compute the cardinality of the sets $B_1$, $B_2$ and $B_3$. For $u =1,\ldots,\gcd(a,b)$, we have that: $$\# B_1(X^{\alpha_1}Y^{\alpha_2}) = (a-\alpha_1)(q-\alpha_2),$$ $$\# B_2(X^{\alpha_1}Y^{\alpha_2}) = \begin{cases}
\alpha_1(q-\alpha_2-b) &\mbox{ if }0 \leq \alpha_2 < q-b \\
0 & \mbox{ otherwise,} \end{cases}$$ $$\# B_3(X^{\alpha_1}Y^{\alpha_2},u) = \begin{cases}
(w(Y)u-a+\alpha_1)(q-\alpha_2-w(X)u) & \mbox{ if }0 \leq \alpha_2 < q-b \mbox{ and } \\
& a-w(Y) < \alpha_1 < a \\
0 & \mbox{ otherwise.} \end{cases}$$
Thus, for $u =1,\ldots,v+1$ by Lemma \[corB\] we get: $$\#{\mathcal{L}}(u) \geq (a-\alpha_1)(q-\alpha_2) +
\begin{cases}
\alpha_1(q-\alpha_2-b) & \mbox{ if }0 \leq \alpha_2 < q-b \\
0 & \mbox{ otherwise}
\end{cases}$$ And if $a-w(Y) < \alpha_1 < a$: $$\# {\mathcal{L}}(u) \geq (a-\alpha_1)(q-\alpha_2) +
\begin{cases}
(w(Y)u-a+\alpha_1)(q-\alpha_2-w(X)u) & \mbox{ if }0 \leq \alpha_2 < q-b \\
0 & \mbox{ otherwise.}
\end{cases}$$ Now we can prove Theorem \[teoalpha\].
Let $v = \alpha_1 {\mbox{ div }} w(Y)$. If $0 \leq \alpha_1 \leq a-w(Y)$ then we obtain $$\begin{aligned}
w_H(\vec{c})
& \geq & \min \{\# {\mathcal{L}}(1), \ldots , \# {\mathcal{L}}(v+1)\} \\
& \geq & (a-\alpha_1)(q-\alpha_2) +\\
&&
\begin{cases}
\alpha_1(q-\alpha_2-b) & \mbox{ if }0 \leq \alpha_2 < q-b \\
0 & \mbox{ otherwise.}
\end{cases} \\\end{aligned}$$ If $a-w(Y) < \alpha_1 < a$, then $v=\gcd (a,b)-1$ and we obtain $$\begin{aligned}
w_H(\vec{c}) & \geq & \min\{ \#
{\mathcal{L}}(1),\ldots , \#
{\mathcal{L}}(v+1) \}\\
& \geq & (a-\alpha_1)(q-\alpha_2) +\\
&&
\begin{cases}
\min\{
(w(Y)u-a+\alpha_1)(q-\alpha_2-w(X)u)
\mid u=1,\ldots ,v+1 \} & \mbox{ if }0 \leq \alpha_2 < q-b \\
0 & \mbox{ otherwise.}
\end{cases} \\\end{aligned}$$ The function $f(u)=(w(Y)u-a+\alpha_1)(q-\alpha_2-w(X)u)$ is a concave parabola, thus we have minimum in $u=1$ or $u=v+1=\gcd(a,b)$. By inspection $f(1)=(w(Y)-a+\alpha_1)(q-\alpha_2-w(X))=T(q-\alpha_2-w(X))$ and $f(\gcd(a,b))=(w(Y)\gcd(a,b)-a+\alpha_1)(q-\alpha_2-w(X)\gcd(a,b))=\alpha_1(q-\alpha_2-b)$. We therefore get the biimplication: $$\begin{aligned}
&&f(1) \leq f(\gcd(a,b)), \\
&\Updownarrow \\
&&\alpha_2 \leq q-w(X)-\alpha_1\frac{b-w(X)}{a-w(Y)},\end{aligned}$$ and the theorem follows.
If for codes from optimal generalised $C_{ab}$ polynomials rather than applying Theorem \[thenewbound\] we apply the usual Feng-Rao bound (Theorem \[theseven\]) then the $\epsilon$ in Theorem \[teoalpha\] should be replaced with: $$\begin{cases}
0 & \mbox{if }q-b \leq \alpha_2 < q \\
T(q-\alpha_2-b) & \mbox{and }0 \leq \alpha_2 < q-b. \\
\end{cases}$$ We see that our new bound improves the Feng-Rao bound by $$\begin{cases}
0 & \mbox{if }q-b \leq \alpha_2 < q \\
& \mbox{or }0 \leq \alpha_1 \leq a-w(Y)\\
(\alpha_1-T)(q-\alpha_2-b) & \mbox{if }a-w(Y) < \alpha_1 < a \mbox{ and }\\
& q-w(X)-\alpha_1\frac{b-w(X)}{a-w(Y)} < \alpha_2 < q-b \\
T(b-w(X)) & \mbox{if }a-w(Y) < \alpha_1 < a \mbox{ and }\\
& 0 \leq \alpha_2 \leq q-w(X)-\alpha_1\frac{b-w(X)}{a-w(Y)}. \end{cases}$$
It is possible to show that Theorem \[teoalpha\] is the strongest possible result one can derive from Theorem \[thenewbound\] regarding the minimum distance of codes from optimal generalised $C_{ab}$ polynomials.
In the following we apply Theorem \[teoalpha\] in a number of cases where $F(X,Y)=G(X)-H(Y) \in {\mathbb{F}}_{p^m}[X,Y]$ with $G(X)$ being the trace polynomial and $H(Y)$ being an $({\mathbb{F}}_{p^m},{\mathbb{F}}_p)$-polynomial of another degree. Recall from the discussion at the beginning of the section that these are optimal generalised $C_{ab}$ polynomials. The strength of our new bound Theorem \[thenewbound\] and Theorem \[teoalpha\] lies in the cases where $a$ and $b$ are not relatively prime, as for $a$ and $b$ relatively prime it reduces to the usual Feng-Rao bound for primary codes (see the last part of Remark \[remtksh2\]). The well-known norm-trace polynomial corresponds to choosing $H(Y)$ to be the norm polynomial. This gives $a=p^{m-1}$ and $b=(p^m-1)/(p-1)$ which are clearly relatively prime. The related codes, which are called norm-trace codes, are one-point algebraic geometric codes. As a measure for how good is our new code constructions it seems fair to compare the outcome of Theorem \[teoalpha\] for the cases of $\gcd(a,b)>1$ with the parameters of the one-point algebraic geometric codes from norm-trace curves over the same alphabet. The two corresponding sets of ideals have the same footprint $\Delta_{\prec_w}(I_q)$ and consequently the corresponding codes are of the same length. We remind the reader that it was shown in [@geil2003codes] that the Feng-Rao bound gives the true parameters of the norm-trace codes.
\[exqis8\] In this example we consider optimal generalised $C_{ab}$ polynomials derived from $({\mathbb{F}}_8,{\mathbb{F}}_2)$-polynomials. The trace polynomial $G(X)$ is of degree $a=4$ and from Example \[extksh5\] we see that besides the norm polynomial which is of degree $b=7$ we can choose $H(Y)$ as $F_3(Y)=Y^6+Y^5+Y^3$ which is of degree $b=6$. The corresponding codes are of length $n=32$ over the alphabet ${\mathbb{F}}_8$. In Figure \[grafo4678\] below we compare the parameters of the related two sequences of improved codes ${\widetilde{E}}_{imp}(\delta)$ (Definition \[defimpcode\]). For few choices of $\delta$ the norm-trace code is the best, but for many choices of $\delta$, from $(a,b)=(4,6)$ we get better codes. We note that the latter sequence of codes contains two non-trivial codes that has the best known parameters according to the linear code bound at [@tysker], namely $[n,k,d]$ equal to $[32,2,28]$ and $[32,15,12]$.
![Improved codes from Example \[exqis8\]. A $\circ$ corresponds to $(a,b)=(4,6)$, and an $\ast$ corresponds to $(a,b)=(4,7)$ (the norm-trace codes).[]{data-label="grafo4678"}](a4b67q8.jpg){width="60.00000%"}
\[exqis16\] In this example we consider optimal generalised $C_{ab}$ polynomials derived from $({\mathbb{F}}_{16},{\mathbb{F}}_2)$-polynomials. The trace polynomial $G(X)$ is of degree $a=8$ and from Example \[extksh5\] we see that besides the norm polynomial which is of degree $b=15$ we can choose $H(Y)$ to be of degree $10$, $12$ and $14$. The corresponding codes are of length $n=128$ over the alphabet ${\mathbb{F}}_{16}$. In Figure \[grafo8101516\] below we compare the parameters of the related two sequences of improved codes ${\widetilde{E}}_{imp}(\delta)$ when $b=10$ and when $b=15$ (the norm-trace codes). For most choices of $\delta$ from $(a,b)=(8,10)$ we get the best codes. The norm-trace codes are never strictly best.
![Improved codes from Example \[exqis16\]. A $\circ$ corresponds to $(a,b)=(8,10)$, and an $\ast$ corresponds to $(a,b)=(8,15)$ (the norm-trace codes).[]{data-label="grafo8101516"}](a8b1015q16.jpg){width="80.00000%"}
\[exqis32\] In this example we consider optimal generalised $C_{ab}$ polynomials derived from $({\mathbb{F}}_{32},{\mathbb{F}}_2)$-polynomials. The trace polynomial $G(X)$ is of degree $a=16$ and from Example \[extksh5\] we see that besides the norm-polynomial which is of degree $b=31$ we can choose $H(Y)$ to be of degree $20$, $24$, $26$, $28$ and $30$. The corresponding codes are of length $n=512$ over the alphabet ${\mathbb{F}}_{32}$. In Figure \[ZOOM1grafo1620263132\] below we compare the parameters of the related three sequences of improved codes ${\widetilde{E}}_{imp}(\delta)$ when $b=20$, $b=26$ and when $b=31$ (the norm-trace codes). For no choices of $\delta$ the norm-trace codes are strictly best (this holds for all values of $k/n$). For some choices $b=20$ gives the best codes for other choices the best parameters are found by choosing $b=26$.
![Improved codes from Example \[exqis32\]. A $\circ$ corresponds to $(a,b)=(16,20)$, an $\ast$ to $(a,b)=(16,26)$, and finally a $+$ corresponds to $(a,b)=(16,31)$ (the norm-trace codes).[]{data-label="ZOOM1grafo1620263132"}](ZOOM3a16b20,26,31q32.jpg){width="100.00000%"}
\[exqis64\] In this example we consider optimal generalised $C_{ab}$ polynomials derived from $({\mathbb{F}}_{64},{\mathbb{F}}_2)$-polynomials. The trace polynomial $G(X)$ is of degree $a=32$ and by studying cyclotomic cosets we see that as an alternative to the norm polynomial which is of degree $b=63$ we can for instance choose an $H(Y)$ of degree $42$. The corresponding codes are of length $n=2048$ over the alphabet ${\mathbb{F}}_{64}$. In Figure \[grafo32426364\] below we compare the parameters of the related two sequences of improved codes ${\widetilde{E}}_{imp}(\delta)$ when $b=42$ and when $b=63$ (the norm-trace codes). As is seen the first codes outperforms the last codes for all parameters.
![Improved codes from Example \[exqis64\]. The upper curve corresponds to $(a,b)=(32,42)$, the lower curve to $(a,b)=(32,63)$ (the norm-trace codes)[]{data-label="grafo32426364"}](a32b4263q64.jpg){width="80.00000%"}
A new construction of improved codes {#secfourandaquarter}
====================================
In Definition \[defimpcode\] we presented a Feng-Rao style improved code construction ${\widetilde{E}}_{imp}(\delta)$. As shall be demonstrated in this section it is sometimes possible to do even better. Recall that the idea behind Theorem \[thenewbound\] is to consider case 1 up till case v+1 as described prior to the theorem. Consider a general codeword $$\vec{c} ={\mbox{ev}}\big(\sum_{s=1}^i a_s M_s +I_q\big) \in C(I,L)$$ $a_i \neq 0$, where $L$ is some fixed known subspace of ${\mathbb{F}}_q^n$. From $L$ we might [*[a priori]{}*]{} be able to conclude that certain $a_s$s equal zero for all codewords as above. This corresponds to saying that [*[a priori]{}*]{} we might know that some of the cases case 1 up to case v do not happen. Clearly we could then leave out the corresponding sets in Theorem \[thenewbound\]. This might result in a higher estimate on $w_H(\vec{c})$. We illustrate the phenomenon with an example in which we also show how to derive improved codes based on this observation.
\[exklein\] In this example we consider the Klein quartic $X^3Y+Y^3+X \in
{\mathbb{F}}_8[X,Y]$. Let $w(X)=2$ and $w(Y)=3$. The ideal $I=\langle X^3Y+Y^3+X\rangle \subseteq {\mathbb{F}}_8[X,Y]$ and the corresponding weighted degree lexicographic ordering $\prec_w$ satisfy order domain condition (C1) but not (C2) (as usual, in the definition of $\prec_w$ we choose $X=X_1$ and $Y=X_2$). Hence, it makes sense to apply Theorem \[thenewbound\]. The footprint of $I_8=\langle X^3Y+Y^3+X,X^8+X,Y^8+Y\rangle$ is (for a reference see [@bookAG Ex. 4.19] and [@FR1 Ex. 3.3]): $$\begin{aligned}
\Delta_{\prec_w}(I_8)&=&\{ 1, X, Y, X^2, XY, Y^2,X^3,X^2Y, XY^2,
X^4,Y^3,X^2Y^2,\\
&&X^5,XY^3,Y^4,X^6,X^2Y^3,XY^4,X^7,Y^5,X^2Y^4,Y^6\}\end{aligned}$$ written in increasing order with respect to $\prec_w$. Consider $$\vec{c}={\mbox{ev}}\big( a_1
1+a_2X+a_3Y+a_4X^2+a_5XY+a_6Y^2+a_7X^3+I_8\big),$$ $a_7 \neq 0$. We have $w(X^3)=w(Y^2)>w(XY)$. Hence, by Remark \[remtksh2\] we choose $v=1$.\
By inspection the set corresponding to case 1 is $${\mathcal{L}}(1)=\{X^3,X^4,X^5,X^6,X^7,X^2Y^4\}.$$ (Note that $X^2Y^4$ belongs to ${\mathcal{L}}(1)$ of the following reason: We have ${\mbox{lm}}(X^3X^5 {\mbox{ rem }} X^8+X)=X$ and ${\mbox{lm}}(Y^2X^5 {\mbox{ rem }} X^3Y+Y^3+X)=X^2Y^4$, and from $w(Y^2X^5)=w(X^2Y^4)>w(X)$ we conclude that $(Y^2,X^5)$ is SOWB with respect to $\{1,2,3,4,5,6,7\}$.) The set corresponding to case 2 is $${\mathcal{L}}(2)=\{X^3,X^4,Y^3,X^5,XY^3,Y^4,X^6,X^2Y^3,XY^4,X^7,Y^5,X^2Y^4,Y^6\}.$$ If we know [*[a priori]{}*]{} that $a_6=0$ then we can conclude from the above that $w_H(\vec{c}) \geq \# {\mathcal{L}}(2)=13$. Without such an information we can only conclude $$w_H(\vec{c}) \geq \min
\{\#{\mathcal{L}}(1),\#{\mathcal{L}}(2)\}=6.$$ It can be shown using Theorem \[thenewbound\] that ${\widetilde{E}}_{imp}(11)=C(I,L)$ where $$L={\mbox{ev}} \big( {\mbox{Span}}_{\mathbb{F}_8}\{1+I_8,
X+I_8,Y+I_8,X^2+I_8,XY+I_8,Y^2+I_8\}\big).$$ That is, a code with parameters $[n,k,d]$ equal to $[22,6, \geq
11]$.\
If instead we choose $${\widetilde{L}}={\mbox{ev}} \big( {\mbox{Span}}_{\mathbb{F}_8}\{1+I_8,
X+I_8,Y+I_8,X^2+I_8,XY+I_8,X^3+I_8\}\big)$$ then we do not need to consider the case 1 described above. By inspection the code parameters $[n,k,d]$ of $C(I,{\widetilde{L}})$ are $[22,
6, \geq 12]$.
Generalised Hamming weights {#secfourandahalf}
===========================
As mentioned at the end of Section \[sectwo\] it is possible to lift Theorem \[thenewbound\] to also deal with generalised Hamming weights. Recall that these parameters are important in the analysis of the wiretap channel of type II as well as in the analysis of secret sharing schemes based on coding theory, see [@wei], [@luo] and [@kurihara].
Let $C \subseteq {\mathbb{F}}_q^n$ be a code of dimension $k$. For $t=1, \ldots , k$ the $t$th generalised Hamming weight is $$d_t(C)=\min \{ \# {\mbox{Supp}} \, D \mid D {\mbox{ \ is a subspace of
\ }} C {\mbox{ \ of dimension \ }}
t\}.$$ Here, ${\mbox{Supp}} \, D$ means the entries for which some word in $D$ is different from zero.
Clearly, $d_1$ is nothing but the usual minimum distance. In Proposition \[protemmeliglang\] below we lift Theorem \[thenewbound\] to deal with the second generalised Hamming weight. From this the reader can understand how to treat any generalised Hamming weight.
\[protemmeliglang\] Let $D \subseteq {\mathbb{F}}_q^n$ be a subspace of dimension 2. Write $D={\mbox{Span}}_{\mathbb{F}_q} \{
{\mbox{ev}}(\sum_{s=0}^{i_1}a_s
M_s),{\mbox{ev}}(\sum_{s=0}^{i_2}b_s M_s)\}$. Here, $\Delta_\prec(I_q)=\{M_1, \ldots , M_n\}$, $a_s \in
{\mathbb{F}}_q$, $b_s \in
{\mathbb{F}}_q$ with $a_{i_1} \neq 0$ and $b_{i_2} \neq
0$. Without loss of generality we may assume $i_1 \neq i_2$. Let $v_1$ and $v_2$ be integers satisfying $0\leq v_1 < i_1$ and $0 \leq v_2
<i_2$. We have $$\# {\mbox{Supp}}(D) \geq \min \{ \# {\mathcal{L}}(z_1,z_2) \mid 1
\leq z_1 \leq v_1+1, 1 \leq z_2 \leq v_2+1 \}.$$ The above sets are defined as follows: For $z=1, \ldots , v_1$ $$\begin{aligned}
&&{\mathcal{L}}(z,v_2+1)=\nonumber \\
&& \big\{ K \in \Delta_\prec(I_q) \mid \exists M_j \in \Delta_\prec(I_q)
{\mbox{ such that either }} (M_{i_1},M_j) {\mbox{ is SOWB}}
\nonumber \\
&& {\mbox{ with respect to }} \{1, \ldots ,
i_1-z,i_1\} {\mbox{ and }} {\mbox{lm}}(M_{i_1}M_j {\mbox{ rem }} {\mathcal{G}})=K
\nonumber \\
&&{\mbox{ or }}
\nonumber \\
&&(M_{i_1-z},M_j) {\mbox{ is SOWB with respect to }} \{1, \ldots ,
i_1-z,i_1\}\nonumber \\
&&{\mbox{ and }} {\mbox{lm}}(M_{i_1-z}M_j {\mbox{ rem }} {\mathcal{G}})=K
\nonumber \\
&&{\mbox{ or }}
\nonumber \\
&&(M_{i_2},M_j) {\mbox{ is SOWB with respect to }} \{1,
\ldots , i_2-v_2-1\} \nonumber \\
&&{\mbox{ and }} {\mbox{lm}}(M_{i_2}M_j {\mbox{ rem }} {\mathcal{G}})=K\big\}.
\nonumber\end{aligned}$$ For $z=1, \ldots , v_2$, ${\mathcal{L}}(v_1+1,z)$ is defined in a similar way.\
For $z_1=1, \ldots , v_1$ and $z_2=1, \ldots ,
v_2$ we have $$\begin{aligned}
&&{\mathcal{L}}(z_1,z_2)=\nonumber \\
&& \big\{ K \in \Delta_\prec(I_q) \mid \exists M_j \in \Delta_\prec(I_q)
{\mbox{ such that for some $u \in \{1,2\}$}}\nonumber\\
&& (M_{i_u},M_j) {\mbox{ is SOWB with respect to }} \{1, \ldots ,
i_u-z_u,i_u\} \nonumber \\
&&{\mbox{ and }} {\mbox{lm}}(M_{i_u}M_j {\mbox{ rem }} {\mathcal{G}})=K
\nonumber \\
&&{\mbox{ or }}
\nonumber \\
&&(M_{i_u-z_u},M_j) {\mbox{ is SOWB with respect to }} \{1, \ldots ,
i_u-z_u,i_u\}\nonumber \\
&&{\mbox{ and }} {\mbox{lm}}(M_{i_u-z_u}M_j {\mbox{ rem }} {\mathcal{G}})=K\big\},
\nonumber \end{aligned}$$ and finally $$\begin{aligned}
&&{\mathcal{L}}(v_1+1,v_2+1)=\nonumber \\
&& \big\{ K \in \Delta_\prec(I_q) \mid \exists M_j \in \Delta_\prec(I_q)
{\mbox{ such that }} (M_{i_u},M_j) {\mbox{ is SOWB}}
\nonumber \\
&& {\mbox{ with respect to }} \{1, \ldots ,
i_u-v_u-1\} {\mbox{ and }} {\mbox{lm}}(M_{i_u}M_j {\mbox{ rem }}
{\mathcal{G}})=K \nonumber \\
&& {\mbox{ for some }} u \in \{1,2\}\big\}.
\nonumber \end{aligned}$$ The second generalised Hamming weight of $C(I,L)$ is found by repeating the above process for all possible choices of $i_1<i_2$ corresponding to the cases that $D \subseteq C(I,L)$.
The proof is a straight forward enhancement of the proof for Theorem \[thenewbound\].
For the choice of $v_1$ and $v_2$ in Proposition \[protemmeliglang\] we refer to Remark \[remtksh2\]. Admittedly, the proposition is rather technical. Nevertheless even its generalisation to higher generalised Hamming weights can often be quite manageable. We shall comment further on this in Section \[seccomp\].
Formulation at linear code level {#secfive}
================================
As mentioned in the introduction the Feng-Rao bound for primary codes in its most general form is a bound on any linear code described by means of a generator matrix. All other versions of the bound, such as the order bound for primary codes and the Feng-Rao bound for primary affine variety codes, can be viewed as corollaries to it. Below we reformulate the new bound in Theorem \[thenewbound\] at the linear code level.\
Let $n$ be a positive integer and $q$ a prime power. Consider a fixed ordered triple $({\mathcal{U}},{\mathcal{V}},{\mathcal{W}})$ where ${\mathcal{U}}=\{\vec{u}_1, \ldots ,
\vec{u}_n\}$, ${\mathcal{V}}=\{\vec{v}_1, \ldots , \vec{v}_n\}$, and ${\mathcal{W}}=\{\vec{w}_1, \ldots , \vec{w}_n\}$ are three (possibly different) bases for ${\mathbb{F}}_q^n$ as a vector space over ${\mathbb{F}}_q$. We shall always denote by ${\mathcal{I}}$ the set $\{1, \ldots , n\}$.
\[def1\] Consider a basis ${\mathcal{A}}=\{\vec{a}_1, \ldots , \vec{a}_n\}$ for ${\mathbb{F}}_q^n$ as a vector space over ${\mathbb{F}}_q$. We define a function $\bar{\rho}_{\mathcal{A}}: {\mathbb{F}}_q^n \rightarrow \{0, 1, \ldots ,
n\}$ as follows. For $\vec{c} \in {\mathbb{F}}_q \backslash \{
\vec{0} \}$ we let $\bar{\rho}_{\mathcal{A}}(\vec{c})=i$ if $\vec{c} \in
{\mbox{Span}}_{\mathbb{F}_q}\{\vec{a}_1, \ldots , \vec{a}_i\} \backslash
{\mbox{Span}}_{\mathbb{F}_q} \{\vec{a}_1, \ldots , \vec{a}_{i-1}\}$. Here, we used the notion ${\mbox{Span}}_{\mathbb{F}_q}\, \emptyset = \{\vec{0}\}$. Finally, we let $\bar{\rho}_{\mathcal{A}}(\vec{0})=0$.
The component wise product plays a crucial role in the linear code enhancement of Theorem \[thenewbound\].
\[deftksh6\] The component wise product of two vectors $\vec{u}$ and $\vec{v}$ in ${\mathbb{F}}_q^n$ is defined by $(u_1, \ldots , u_n)\ast (v_1,
\ldots, v_n)=(u_1v_1, \ldots , u_nv_n)$.
\[def2\] Let $({\mathcal{U}},{\mathcal{V}},{\mathcal{W}})$ and ${\mathcal{I}}$ be as above. Consider ${\mathcal{I}}^\prime
\subseteq {\mathcal{I}}$. An ordered pair $(i,j) \subseteq
{\mathcal{I}}^\prime \times {\mathcal{I}}$ is said to be one-way well-behaving (OWB) with respect to ${\mathcal{I}}^\prime$ if $\bar{\rho}_{\mathcal{W}}(\vec{u}_{i^\prime} \ast
\vec{v}_j) < \bar{\rho}_{\mathcal{W}}(\vec{u}_{i} \ast
\vec{v}_j)$ holds for all $i^\prime \in {\mathcal{I}}^\prime$ with $i^\prime <
i$.
The following theorem is a first generalisation of the Feng-Rao bound for primary codes. The generalisation compared to the usual Feng-Rao bound [@AG; @geithom] is that we allow ${\mathcal{I}}^\prime$ to be different from $\{1, \ldots
,\# {\mathcal{I}}^\prime\}$. This is in the spirit of Section \[secfourandaquarter\].
\[the2\] Consider $\vec{c}=\sum_{s=1}^t a_s \vec{u}_{i_s}$ with $a_s
\in {\mathbb{F}}_q$, $s=1, \ldots , t$, $a_t
\neq 0$ and $i_1< \cdots <i_t$. We have $$\begin{array}{r} w_H(\vec{c}) \geq
\# \big\{ l\in {\mathcal{I}} \mid \exists j \in {\mathcal{I}} {\mbox{ \ such that \
}} \bar{\rho}_{\mathcal{W}}(\vec{u}_{i_t}\ast \vec{v}_j)=l,
{\mbox{ \ \hspace{1cm} \ }} \\
(i_t,j) {\mbox{ \ is OWB with respect to \ }} \{ i_1,
\ldots , i_t\} \big\}.
\end{array} \label{eqth2}$$
Let $l_1 < \cdots < l_\sigma$ be the indexes $l$ counted in (\[eqth2\]). Denote by $j_1,\ldots , j_\sigma$ the corresponding $j$-values from (\[eqth2\]). By assumption $\vec{c} \ast
\vec{v}_{j_1}, \ldots , \vec{c} \ast \vec{v}_{j_\sigma}$ are linearly independent and therefore $${\mbox{Span}}_{\mathbb{F}_q}
\{\vec{c} \ast
\vec{v}_{j_1}, \ldots , \vec{c} \ast \vec{v}_{j_\sigma}\}= \vec{c}
\ast {\mbox{Span}}_{\mathbb{F}_q}
\{ \vec{v}_{j_1}, \ldots , \vec{v}_{j_\sigma}\}$$ is a vector space of dimension $\sigma$. The theorem follows from the fact that $\vec{c} \ast {\mathbb{F}}_q^n$ is a vector space of dimension $w_H(\vec{c})$ containing the above space.
A slight modification of Definition \[def2\] and the above proof allows for further improvements.
\[deftksh7\] Let ${\mathcal{I}}^\prime \subseteq {\mathcal{I}}$. A pair $(i,j) \in {\mathcal{I}}^\prime \times {\mathcal{I}}$ is called strongly one-way well-behaving (SOWB) with respect to ${\mathcal{I}}^\prime$ if $\bar{\rho}_{\mathcal{W}}(\vec{u}_{i^\prime}\ast\vec{v}_j)<\bar{\rho}_{\mathcal{W}}(\vec{u}_i\ast
\vec{v}_j)$ holds for all $i^\prime \in {\mathcal{I}}^\prime \backslash \{ i
\}$.
The following theorem is the linear code interpretation of Theorem \[thenewbound\]. Besides working for a larger class of codes, it is slightly stronger in that we formulate it in such a way that it supports the technique explained in Section \[secfourandaquarter\]. Concretely, what makes it stronger than Theorem \[thenewbound\] is the presence of the set $\hat{\mathcal{I}}$.
\[the4\] Consider a non-zero codeword $\vec{c}=\sum_{t=1}^i a_t \vec{u}_t$, $a_t \in {\mathbb{F}}_q$ for $t=1,\ldots , i$, $a_i \neq 0$. Let $v$ be an integer $0 \leq v <i$. Assume that for some set $\hat{\mathcal{I}}\subseteq \{1, \ldots , i-1\}$ we know [*[a priori]{}*]{} that $a_x=0$ when $x \in \hat{\mathcal{I}}$. Let $z_1<
\cdots <z_s$ be the numbers in $\{z \in \{i-v, \ldots , i-1\} \mid z \notin
\hat{\mathcal{I}}\}$. Write ${\mathcal{I}}^\ast=\{ z \in \{1, \ldots ,
i-v-1\}\mid z \notin \hat{\mathcal{I}}\}$. We have $$w_H(\vec{c})\geq \min \{{\mathcal{L}}^\prime(1), \ldots ,
{\mathcal{L}}^\prime(s+1) \}$$ where for $t=1, \ldots ,s$ we define ${\mathcal{L}}^\prime(t)$ as follows: $$\begin{aligned}
{\mathcal{L}}^\prime(1)&=&\{ l \in {\mathcal{I}} \mid \exists z \in \{ z_s, i\} {\mbox{
\ and \ }} j
\in {\mathcal{I}} {\mbox{ \ such that \ }} \bar{\rho}_{\mathcal{W}}(\vec{u}_z \ast
\vec{v_j})=l\nonumber \\
&&(z,j) {\mbox{ \ is SOWB with respect to
\ }} {\mathcal{I}}^\ast \cup \{z_1, \ldots , z_s,i\} \big\}, \nonumber \\
{\mathcal{L}}^\prime(2)&=&\big\{ l \in {\mathcal{I}} \mid \exists z \in \{z_{s-1},i\}
{\mbox{ \ and \ }} j \in {\mathcal{I}} {\mbox{ \ such that \ }}
\bar{\rho}_{\mathcal{W}}(\vec{u}_{z} \ast \vec{v}_j) =l\nonumber \\
&&(z,j) {\mbox{ \ is SOWB
with respect to \ }} {\mathcal{I}}^\ast \cup \{z_1, \ldots , z_{s-1},i\}\big\},\nonumber \\
&&{\mbox{ \ {\hspace{3cm} }}}\vdots \nonumber \\
{\mathcal{L}}^\prime(s)&=&\big\{ l \in {\mathcal{I}} \mid \exists z \in \{z_{1},i\}
{\mbox{ \ and \ }} j \in {\mathcal{I}} {\mbox{ \ such that \ }}
\bar{\rho}_{\mathcal{W}}(\vec{u}_{z} \ast \vec{v}_j) =l\nonumber \\
&& (z,j) {\mbox{ \ is SOWB
with respect to \ }} {\mathcal{I}}^\ast \cup
\{z_1,i\}\big\}.\nonumber\end{aligned}$$ Finally, $$\begin{aligned}
{\mathcal{L}}^\prime(s+1)&=&\big\{ l \in {\mathcal{I}} \mid \exists j \in {\mathcal{I}} {\mbox{ \ such that \ }}
\bar{\rho}_{\mathcal{W}}(\vec{u}_{i} \ast \vec{v}_j) =l \nonumber \\
&&(i,j) {\mbox{ \ is OWB
with respect to \ }} {\mathcal{I}}^\ast \cup \{i\}\big\}.\nonumber \end{aligned}$$ To establish a lower bound on the minimum distance of a code $C$ we repeat the above process for each $i \in
\bar{\rho}_{\mathcal{U}}(C)$. For each such $i$ we choose a corresponding $v$, defining an $s$, and we determine the sets ${\mathcal{L}}^\prime(1), \ldots , {\mathcal{L}}^\prime(s+1)$ and calculate their cardinalities. The smallest cardinality found when $i$ runs through $\bar{\rho}_{\mathcal{U}}(C)$ serves as a lower bound on the minimum distance.
The proof is a direct translation of the proof of Theorem \[thenewbound\].
\[remtksh8\] For $v=0$ Theorem \[the4\] reduces to Theorem \[the2\]. For higher values of $v$ Theorem \[the4\] is at least as strong as Theorem \[the2\] and sometimes stronger. In the same way as Theorem \[thenewbound\] was lifted in Section \[secfourandahalf\] to deal with generalised Hamming weights one can lift Theorem \[the2\] and Theorem \[the4\].
A related bound for dual codes {#secsix}
==============================
In the recent paper [@geilmartin2013further] we presented a new bound for dual codes. This bound is an improvement to the Feng-Rao bound for such codes as well as an improvement to the advisory bound from [@salazar]. The new bound of the present paper can be viewed as a natural counter part to the bound from [@geilmartin2013further], the one bound dealing with primary codes and the other with dual codes.
\[def3\]Consider an ordered triple of bases $({\mathcal{U}},{\mathcal{V}},{\mathcal{W}})$ for ${\mathbb{F}}_q^n$ and ${\mathcal{I}}$ as in Section \[secfive\]. We define $m : {\mathbb{F}}_q^n \backslash \{ \vec{0}\} \rightarrow
{\mathcal{I}}$ by $m(\vec{c})=l$ if $l$ is the smallest number in ${\mathcal{I}}$ for which $\vec{c} \cdot \vec{w}_l \neq 0$. (Here, and in the following the symbol $\cdot$ means the usual inner product).
\[definvolved\] Consider numbers $1 \leq l, l+1, \ldots , l+g \leq n$. A set ${\mathcal{I}}^\prime\subseteq {\mathcal{I}}$ is said to have the $\mu$-property with respect to $l$ with exception $\{l+1, \ldots , l+g\}$ if for all $i
\in {\mathcal{I}}^\prime$ a $j\in {\mathcal{I}}$ exists such that
- $\bar{\rho}_{\mathcal{W}}(\vec{u}_i \ast \vec{v}_j)=l$, and
- for all $i^\prime \in {\mathcal{I}}^\prime$ with $i^\prime < i$ either $\bar{\rho}_{\mathcal{W}}(\vec{u}_{i^\prime}\ast
\vec{v}_j)< l$ or $\bar{\rho}_{\mathcal{W}}(\vec{u}_{i^\prime}\ast
\vec{v}_j) \in \{l+1, \ldots , l+g\}$ holds.
Assume next that $l+g+1 \leq n$. The set ${\mathcal{I}}^\prime$ is said to have the relaxed $\mu$-property with respect to $(l,l+g+1)$ with exception $\{l+1, \ldots , l+g\}$ if for all $i \in
{\mathcal{I}}^\prime$ a $j \in {\mathcal{I}}$ exists such that either conditions $(1a)$ and $(1b)$ above hold or
- $\bar{\rho}_{\mathcal{W}}(\vec{u}_i \ast \vec{v}_j)=l+g+1$, and
- $(i,j)$ is OWB with respect to ${\mathcal{I}}^\prime$, and
- no $i^\prime \in {\mathcal{I}}^\prime$ with $i^\prime < i$ satisfies $\bar{\rho}_{\mathcal{W}}(\vec{u}_{i^\prime} \ast \vec{v}_j)=l$.
The new bound from [@geilmartin2013further Th. 19] reads:
\[thenew\]\[thestrongone\] Consider a non-zero codeword $\vec{c}$ and let $l=m(\vec{c})$. Choose a non-negative integer $v$ such that $l+v\leq n$. Assume that for some indexes $x \in \{l+1, \ldots , l+v\}$ we know [*a priori*]{} that $\vec{c} \cdot \vec{w}_x=0$. Let $l^\prime_1< \cdots < l^\prime_s$ be the remaining indexes from $\{l+1, \ldots , l+v\}$. Consider the sets ${\mathcal{I}}^\prime_0, {\mathcal{I}}_1^\prime, \ldots ,
{\mathcal{I}}_s^\prime$ such that:
- ${\mathcal{I}}^\prime_0$ has the $\mu$-property with respect to $l$ with exception $\{ l+1, \ldots , l+v\}$.
- For $i=1, \ldots , s$, ${\mathcal{I}}^\prime_i$ has the relaxed $\mu$-property with respect to $(l, l^\prime_i)$ with exception $\{l+1, \ldots , l^\prime_i -1\}$.
We have $$w_H(\vec{c}) \geq \min \{ \# {\mathcal{I}}_0^\prime, \#
{\mathcal{I}}_1^\prime, \ldots , \# {\mathcal{I}}_s^\prime\}. \label{eqcirkel}$$ To establish a lower bound on the minimum distance of a code $C$ we repeat the above process for each $l\in m(C)$. For each such $l$ we choose a corresponding $v$, we determine sets ${\mathcal{I}}^{\prime}_i$ as above and we calculate the right side of (\[eqcirkel\]). The smallest value found when $l$ runs through $m(C)$ constitutes a lower bound on the minimum distance.
If we compare Theorem \[thestrongone\] with Theorem \[the4\] we see that to some extend they have the same flavor. Besides that one deals with dual codes and the other with primary codes another difference is that we in Theorem \[thestrongone\] has the freedom to choose appropriate sets ${\mathcal{I}}^\prime_0, \ldots , {\mathcal{I}}^\prime_s$ whereas the sets ${\mathcal{L}}^\prime(1), \ldots , {\mathcal{L}}^\prime (s+1)$ in Theorem \[the4\] are unique. In [@geilmartin2013further] it was also shown how to lift Theorem \[thestrongone\] to deal with generalised Hamming weights. Similar remarks as above hold for the two bounds when applied to such parameters.
A comparison of the new bounds for primary and dual codes {#seccomp}
=========================================================
Recall that it was shown in [@agismm] how the Feng-Rao bound for primary codes and the Feng-Rao bound for dual codes can be viewed as consequences of each other. This result holds when the bound is equipped with one of the well-behaving properties WB or OWB. Regarding the case where WWB is used a possible connection is unknown. In a similar fashion as the proof in [@agismm] breaks down if one uses WWB it also breaks down when one tries to prove a correspondence between Theorem \[the4\] and Theorem \[thestrongone\]. We leave it as an open research problem to decide if a general connection exists or not.\
In Section \[secfour\] we analysed the performance of primary affine variety codes coming from optimal generalised $C_{ab}$ polynomials. Using the method from Section \[secsix\] one can make a similar analysis for the corresponding dual codes producing similar code parameters. As an alternative, below we explain how to derive this result directly from what we have already shown regarding primary codes from optimal generalised $C_{ab}$ polynomials.\
Recall that for optimal generalised $C_{ab}$ polynomials $\Delta_{\prec_w}(I_q)$ is a box: $$\Delta_{\prec_w}(I_q)=\{M_1, \ldots ,
M_n\}=\{X^{\alpha_1}Y^{\alpha_2} \mid 0 \leq \alpha_1 < a, 0 \leq
\alpha_2 < q\}.$$ This fact gives us the following crucial implication (as usual we assume $M_1 \prec_w \cdots \prec_w M_n$): $$M_i=X^{\alpha_1}Y^{\alpha_2} \Rightarrow
M_{n-i+1}=X^{a-1-\alpha_1}Y^{q-1-\alpha_2}, {\mbox{ for }} i=1, \ldots
, n.\label{eqimp}$$ Consider codewords $\vec{c}={\mbox{ev}}\big( \sum_{s=1}^i a_s M_s+I_q\big)$, $a_s \in {\mathbb{F}}_q$, $a_i \neq 0$, and $\vec{c}^\prime
\in {\mathbb{F}}_q^n$ such that $m(\vec{c}^\prime)=n-i+1$. Let $v$ be an integer, $0 \leq v < i$. Recall that in Section \[secfour\] we determined ${\mathcal{L}}(u)$, $u=1, \ldots , v+1$. If we use Theorem \[thestrongone\] with $\{l+1, \ldots ,
l+v\}=\{l_1^\prime,\ldots ,
l_s^\prime\}$ (no [*[a priori]{}*]{} knowledge) then we can choose $${\mathcal{I}}_0^\prime=\{n-l+1 \mid M_l \in {\mathcal{L}}(v+1)\}$$ and for $u=1, \ldots , v$ $${\mathcal{I}}_u^\prime= \{ n-l+1 \mid M_l \in {\mathcal{L}}(u) \}.$$ For $S \subseteq \{1,\ldots , n\}$ define $\bar{S}=\{1, \ldots , n\}
\backslash \{n-s+1 \mid s \in S\}$. Consider $$L={\mbox{Span}}_{\mathbb{F}_q}\{{\mbox{ev}}(M_s+I_q) \mid s \in
S\},$$ $$\bar{L}={\mbox{Span}}_{\mathbb{F}_q}\{{\mbox{ev}}(M_s+I_q) \mid s \in
\bar{S} \}.$$ As $\#{\mathcal{I}}_0^\prime=\# {\mathcal{L}}(v+1)$ and for $u=1,
\ldots , v$, $\#{\mathcal{I}}_u^\prime = \#{\mathcal{L}}(u)$ we conclude that Theorem \[thestrongone\] produces the same estimate for the minimum distance of $C^\perp(I,\bar{L})$ as Theorem \[thenewbound\] produces for the minimum distance of $C(I,L)$. However, we do not in general have $C(I,L)=C^\perp(I,\bar{L})$ and therefore the above analysis does not imply that Theorem \[thenewbound\] is a consequence of Theorem \[thestrongone\] even in the case of optimal generalised $C_{ab}$ polynomials.\
The above correspondence regarding the minimum distance immediately carries over to the generalised Hamming weights. In [@geilmartin2013further Sec.4] we implemented the enhancement of Theorem \[thestrongone\] to generalised Hamming weights for a couple of concrete dual affine variety codes coming from optimal generalised $C_{ab}$ polynomials. As a consequence of (\[eqimp\]) the estimates found in [@geilmartin2013further Sec. 4] for $C^\perp(I,\bar{L})$ also hold for $C(I,L)$. This demonstrates the usefulness of the method described in Section \[secfourandahalf\].\
We conclude the section by demonstrating that $d\big( C(I,L) \big) =d
\big( C^\perp (I, \bar{L})\big)$ does not hold for all generalised $C_{ab}$ polynomials.
\[exdualprimary\] In this example we consider the generalised $C_{ab}$ polynomial $F(X,Y)=G(X)-H(Y)\in {\mathbb{F}}_{32}[X,Y]$ where $G(X)=X^{20}+X^{18}+X^{10}+X^9+X^5$ and $H(Y)=Y^{26}+Y^{22}+Y^{21}+Y^{13}+Y^{11}$. Observe that both $G$ and $H$ are $({\mathbb{F}}_{32},{\mathbb{F}}_2)$-polynomials and that $G$ satisfies the condition in Proposition \[proptracelike\] ensuring that for each $\eta \in {\mathbb{F}}_2$ there exists exactly $2^{4}=16$ $\gamma \in {\mathbb{F}}_{32}$ such that $G(\gamma)=\eta$. In particular $F(X,Y)$ has exactly $512$ zeros in ${\mathbb{F}}_{32}$. As $a=\deg G > 16$ $\{ F(X,Y),X^{32}-X,Y^{32}-Y\}$ cannot be a Gröbner basis with respect to $\prec_w$ (it would violate the footprint bound, Corollary \[thefoot\]). Applying Buchberger’s algorithm we find a Gröbner basis and from that the corresponding footprint $$\begin{aligned}
\Delta_{\prec_w}(I_{32})&=& \{X^{\alpha_1} X^{\alpha_2} \mid 0 \leq
\alpha_1 < 12, 0 \leq \alpha_2 < 32\} \nonumber \\
&&\cup {\mbox{ }} \{X^{\alpha_1} X^{\alpha_2} \mid 12 \leq \alpha_1 < 20, 0 \leq
\alpha_2 < 16\} .\nonumber\end{aligned}$$ Recall the improved construction ${\widetilde{E}}_{imp}(\delta)$ of primary affine variety codes as introduced in Definition \[defimpcode\]. In a similar way, as Theorem \[thenewbound\] gives rise to the above Feng-Rao style improved primary codes, Theorem \[thestrongone\] gives rise to improved dual codes. These codes were named ${\widetilde{C}}_{fim}(\delta)$ in [@geilmartin2013further Rem. 20]. In a computer experiment we calculated the parameters of these codes. In Figure \[figfaktisk\] we plot the derived relative parameters. As is seen for some designed distances $\delta$, ${\widetilde{E}}_{imp}(\delta)$ has the highest dimension. For other designed distances $\delta$, ${\widetilde{C}}_{fim}(\delta)$ is of highest dimension.
![Improved codes from Example \[exdualprimary\]. A $\circ$ corresponds to ${\widetilde{E}}_{imp}(\delta)$, and an $\ast$ corresponds to ${\widetilde{C}}_{fim}(\delta)$.[]{data-label="figfaktisk"}](ZOOMstaircase.jpg){width="65.00000%"}
Conclusion {#seceight}
==========
In this paper we proposed a new bound for the minimum distance and the generalised Hamming weights of general linear code for which a generator matrix is known. We demonstrated the usefulness of our bound in the case of affine variety codes where only the first of the two order domain conditions is satisfied. For this purpose we introduced the concept of generalised $C_{ab}$ polynomials. We touched upon the connection to a bound for dual codes introduced in the recent paper [@geilmartin2013further], but leave an investigation of a possible general relation between the two bounds for further research. It is an interesting question if there exists examples where our new method improves on the Feng-Rao bound for one-point algebraic geometric codes. This would require that we do not choose $v$ as in Remark \[remtksh2\] and that we make extensive use of the polynomials $X_i^q-X_i$. Also this question is left for further research. The usual Feng-Rao bound for primary codes comes with a decoding algorithm that corrects up to half the estimated minimum distance [@agismm]. This result holds when the bound is equipped with the well-behaving property WB. For the case of WWB or OWB no decoding algorithm is known. Finding a decoding algorithm that corrects up to half the value guaranteed by Theorem \[thenewbound\] would impose the missing decoding algorithms mentioned above.\
Part of this research was done while the second listed author was visiting East China Normal University. We are grateful to Professor Hao Chen for his hospitality. The authors also gratefully acknowledge the support from the Danish National Research Foundation and the National Science Foundation of China (Grant No. 11061130539) for the Danish-Chinese Center for Applications of Algebraic Geometry in Coding Theory and Cryptography. The authors would like to thank Diego Ruano, Peter Beelen and Ryutaroh Matsumoto for pleasant discussions.
[^1]: [email protected]
[^2]: [email protected]
|
---
abstract: 'We propose two interferometric schemes to experimentally detect the onset of pair condensation in a two spin-component Fermi gas. Two atomic wave-packets are coherently extracted from the gas at different positions and are mixed by a matter-wave beam splitter: we show that the spatial long range order of the atomic pairs in the gas then reflects in the atom counting statistics in the output channels of the beam splitter. Alternatively, the same long range order is also shown to create a matter-wave grating in the overlapping region of the two extracted wave-packets, grating that can be revealed by a light scattering experiment.'
author:
- Iacopo Carusotto
- Yvan Castin
title: Atom interferometric detection of the pairing order parameter in a Fermi gas
---
The experimental possibility of controlling at will the scattering length $a$ between two spin components of fermionic atoms [*via*]{} a Feshbach resonance has opened the way to a comprehensive study of the pairing transition in a degenerate Fermi gas [@Feshbach; @MolecBEC; @AtomicBCS]. The weakly interacting limits are well understood theoretically: the phase transition is the Bose-Einstein condensation (BEC) of diatomic molecules ($a=0^+$) or the BCS transition due to pairing in momentum space ($a=0^-$). But one can now investigate experimentally the theoretically challenging crossover region, including the unitary limit $|a|=\infty$ [@CrossoverTheory].
While the standard techniques used for atomic BECs have allowed to detect and characterize a molecular BEC [@MolecBEC], a debate is still in progress about experimental signatures of pair condensation for a negative scattering length. Several proposals have been put forward [@TheoryObsBCS]; none of them was proved to demonstrate the existence of long range order in the pairing parameter. First experimental evidences of a condensation of fermionic pairs in the crossover regime have been recently presented [@AtomicBCS], based on a fast ramping of the magnetic field to convert pairs on the $a<0$ side into bound molecules on the $a>0$ side, and on the observation of the Bose condensed fraction of the resulting gas of dimers. This method is expected to work only when the fermionic pairs are small enough, that is in the vicinity of the Feshbach resonance, $k_F |a|>1$ where $k_F$ is the Fermi momentum.
In this paper, we propose a more direct and general way of proving the condensation of pairs, by a measurement of the pairing order parameter, which is not restricted to the small pair regime $k_F |a|>1$. This proposal is the fermionic analog of the atom interferometric measurement of the first order coherence function $G^{(1)}$ of a Bose gas [@Esslinger]. More subtle schemes than the observation of the mean atomic density have however to be introduced as there is no long range first order coherence for fermions. Their experimental implementation would constitute a remarkable transposition of quantum optics techniques to a fermionic matter field.
In the current theories of the superfluid state in fermionic systems [@BCS; @SupercondBooks; @CrossoverTheory], the onset of pair condensation is defined by a non-zero long-distance limit $x_{AB}\equiv
|{ {\bf x}}_A-{ {\bf x}}_B|\rightarrow
+\infty$ of the pair coherence function $$\label{eq:g1pair}
G{^{(1)}}_{\rm pair}({ {\bf x}}_A,{ {\bf x}}_B)=\left\langle
{\hat\Psi^\dagger}_\uparrow({ {\bf x}}_A)\,{\hat\Psi^\dagger}_\downarrow({ {\bf x}}_A)
{\hat\Psi}_\downarrow({ {\bf x}}_B)\,{\hat\Psi}_\uparrow({ {\bf x}}_B)
\right\rangle,$$ this function then factorizing in the product of the order parameter in ${ {\bf x}}_B$ and the complex conjugate of the order parameter in ${ {\bf x}}_A$. In the following, we shall propose two distinct methods to measure $G{^{(1)}}_{\rm pair}({ {\bf x}}_A,{ {\bf x}}_B)$, both relying on the coherent extraction of two atomic wave-packets in ${ {\bf x}}_{A,B}$ and their subsequent beating. The first method is based on a two-atom interferometric technique inspired by two-photon techniques [@Mandel]: it relies on atom counting in the two output channels of a matter-wave beam splitter. The second method is based on the coherence properties of light elastically scattered off the matter-wave interference pattern of the two overlapping wave-packets.
Consider a gas of spin-1/2 fermionic atoms at thermal equilibrium in a trap. At the time $t=0$, the trap potential is suddenly switched off and the atom-atom interactions brought to a negligible strength, so that the subsequent propagation is the one of a free atomic field. At the same time, a suitable short pulse of spin-independent optical potential is applied (Fig.\[fig:setup\]) to the atoms situated in regions of size $\ell_u$ around the points ${ {\bf x}}_A$ and ${ {\bf x}}_B$ so to impart them a momentum kick of respectively ${ {\bf k}}_0\pm{ {\bf k}}_1$ by means of Bragg processes and to produce wave packets which are a coherent copy of the field in the trap, but for a shift in momentum space. ${ {\bf k}}_0$ is taken orthogonal to ${ {\bf x}}_A-{ {\bf x}}_B$, while ${ {\bf k}}_1$ is parallel to it. The magnitude $\hbar k_1$ of the counter-propagating momentum kicks is taken larger than the momentum width $\Delta p$ of the gas, which is on the order of the Fermi wavevector $k_F$ in the resonance region ($|a|=+\infty$) and in the weakly interacting BCS regime ($a<0$), or on the order of $\hbar/a$ in the case of a molecular condensate ($a>0$). The size of the extraction region $\ell_u$ is taken much smaller than the distance $x_{AB}$ between the extraction points. This latter is taken as macroscopic, that is much larger than any other length scale of the problem, e.g. the Fermi distance $1/k_F$ and the Cooper-pair size $\ell_{\rm BCS}$.
In Heisenberg picture the field operator at the end of the optical pulse can be related to the initial one by [^1]: $$\begin{gathered}
\label{eq:extracted}
{\hat\Psi}_\sigma({ {\bf x}},\Delta t) =
u({ {\bf x}}-{ {\bf x}}_A)\,e^{i({ {\bf k}}_0+{ {\bf k}}_1)\cdot({ {\bf x}}-{ {\bf x}}_A)} {\hat\Psi}_\sigma({ {\bf x}})+ \\
+ u({ {\bf x}}-{ {\bf x}}_B)\,e^{i({ {\bf k}}_0-{ {\bf k}}_1)\cdot({ {\bf x}}-{ {\bf x}}_B)}\,{\hat\Psi}_\sigma({ {\bf x}})+{\hat\Psi}_\sigma^{\rm bg}({ {\bf x}}).\end{gathered}$$ The atoms which are left in their original momentum state as well as the ones having received a different momentum kick during the extraction process are included in the background field ${\hat\Psi}_\sigma^{\rm bg}$: as they spatially separate during the evolution, they will be omitted in the discussion [^2]. We shall assume for simplicity that the extraction function $u({ {\bm \xi}})$ is a Gaussian, $u({ {\bm \xi}})=u_0\,e^{-{ {\bm \xi}}^2/2\ell_u^2}$ of size $\ell_u$; its peak amplitude $u_0$ is of modulus less than one.
This out-coupling scheme produces two atomic wave-packets traveling with momentum ${ {\bf k}}_0\pm{ {\bf k}}_1$ starting from respectively ${ {\bf x}}_{A,B}$. At a time $t_1=mx_{AB}/2\hbar\,|{ {\bf k}}_1|$, the two wave-packets superimpose around ${ {\bf X}}$. As mentioned in the introduction, the mean density profile in the overlap region does not show fringes so that more elaborate manipulations have to be performed onto the atoms in order to measure the pair coherence function $G{^{(1)}}_{\rm
pair}({ {\bf x}}_A,{ {\bf x}}_B)$.
[**Atom-number correlations**]{}: At $t=t_1$, the two overlapping wave-packets of momentum respectively ${ {\bf k}}_0\pm{ {\bf k}}_1$ can be coherently mixed by a spin-insensitive 50-50 matter-wave beam splitter, with reflection and transmission amplitudes of momentum-independent phase difference $\phi$. Such a beam splitter may be realized[@Bragg] by applying a pulse of sinusoidal optical potential $U({ {\bf x}},t)=4\hbar\Omega(t)\,
\sin^2({ {\bf k}}_1\cdot{ {\bf x}}+\phi/2)$ [^3].
![First proposed set-up: atoms are extracted by a Bragg process from the gas at points ${ {\bf x}}_{A,B}$, using pairs of laser beams of wavevectors $-{ {\bf k}}_0$, ${ {\bf k}}_1$ and $-{ {\bf k}}_0$, $-{ {\bf k}}_1$ respectively; at their overlap position ${ {\bf X}}$, the two atomic wave-packets are coherently mixed by a laser standing wave acting as a 50-50 beam-splitter with adjustable phase shift $\phi$; the number of atoms in each wave-packet is measured at the final positions ${ {\bf X}}_\pm$.[]{data-label="fig:setup"}](setup1.eps){width="6.5cm"}
At a time $t_2$ after the splitting procedure, the two emerging wave-packets of momentum ${ {\bf k}}_0\pm{ {\bf k}}_1$ will be again spatially separated and centered at ${ {\bf X}}_\pm={ {\bf X}}+\hbar ({ {\bf k}}_0\pm{ {\bf k}}_1)(t_2-t_1)/m$. The total field can be written as ${\hat\Psi}_{\sigma}({ {\bf x}},t_2) = {\hat\Psi}^+_{\sigma}({ {\bf x}},t_2)+{\hat\Psi}^-_{\sigma}({ {\bf x}},t_2)+{\hat\Psi}^{\rm bg}({ {\bf x}},t_2)$ where the contribution of each packet is $$\begin{gathered}
{\hat\Psi}^{\pm}_{\sigma}({ {\bm \xi}}+{ {\bf X}}_\pm,t_2) =\frac{e^{i({ {\bf k}}_0\pm{ {\bf k}}_1){ {\bm \xi}}}\,e^{i\theta}}{\sqrt{2}}\,
\int\!d{ {\bm \xi}}'\,{\mathcal R}({ {\bm \xi}},{ {\bm \xi}}';t_2)\times \\
\times \,u({ {\bm \xi}}')\,
\Big[{\hat\Psi}_\sigma({ {\bf x}}_{A,B}+{ {\bm \xi}}')\,+i\,e^{\pm i\phi}\,
{\hat\Psi}_\sigma({ {\bf x}}_{B,A}+{ {\bm \xi}}')\Big].\end{gathered}$$ ${\mathcal R}({ {\bm \xi}},{ {\bm \xi}}';t)$ is the free-particle propagator and $\theta$ is an irrelevant propagation phase which depends on the details of the beam splitting procedure. The unitarity of ${\mathcal R}$ ensures that the results to come do not depend on $t_2$.
The operator $\hat{N}_\sigma^\pm$ giving the number of atoms with spin $\sigma$ in the wave-packet $\pm$ is obtained by integration of ${\hat\Psi^\dagger}_{\sigma}({ {\bf x}},t_2){\hat\Psi}_{\sigma}({ {\bf x}},t_2)$ over the spatial extension of the packet $\pm$ at time $t_2$. The operator giving the atom number difference between the two wave-packets is then $\hat{D}_\sigma=\hat{N}_\sigma^+-\hat{N}_\sigma^-$. Its expectation value $\big\langle \hat{D}_\sigma\big\rangle$ involves the first-order coherence function $\big\langle{\hat\Psi^\dagger}_\sigma({ {\bf x}}_A+{ {\bm \xi}})\,{\hat\Psi}_\sigma({ {\bf x}}_B+{ {\bm \xi}})\big\rangle$ of the initially trapped atoms, and therefore vanishes for a macroscopic distance $x_{AB}\gg \hbar/\Delta p$: we shall now take $\langle \hat{N}_\sigma^+\rangle =\langle \hat{N}_\sigma^-\rangle =
N_\sigma$. Information on the pair coherence function $G_{\rm pair}^{(1)}$ is obtained from the correlation between the two spin components: $$\begin{gathered}
\label{eq:up_down}
C_{{\uparrow}{\downarrow}}=\langle \hat{D}_{\uparrow}\,\hat{D}_{\downarrow}\rangle=
\int\!d{ {\bm \xi}}\,d{ {\bm \xi}}'\,|u({ {\bm \xi}})|^2\,|u({ {\bm \xi}}')|^2\, \\
\Big[-e^{2i\phi}\,\Big\langle\,{\hat\Psi^\dagger}_{\uparrow}({ {\bf x}}_A+{ {\bm \xi}})\,{\hat\Psi^\dagger}_{\downarrow}({ {\bf x}}_A+{ {\bm \xi}}')\,
{\hat\Psi}_{\downarrow}({ {\bf x}}_B+{ {\bm \xi}}')\,{\hat\Psi}_{\uparrow}({ {\bf x}}_B+{ {\bm \xi}})\,\Big\rangle\\
+\Big\langle\,{\hat\Psi^\dagger}_{\uparrow}({ {\bf x}}_A+{ {\bm \xi}})\,{\hat\Psi^\dagger}_{\downarrow}({ {\bf x}}_B+{ {\bm \xi}}')\,
{\hat\Psi}_{\downarrow}({ {\bf x}}_A+{ {\bm \xi}}')\,{\hat\Psi}_{\uparrow}({ {\bf x}}_B+{ {\bm \xi}})\,\Big\rangle+\textrm{h.c.}\Big].\end{gathered}$$ From an experimental measurement of the $\phi$ dependence of $C_{{\uparrow}{\downarrow}}$, it is therefore possible to determine whether the system has long range order or not.
An explicit calculation of $C_{{\uparrow}{\downarrow}}$ as a function of the energy gap $\Delta$ can be performed by using the zero temperature BCS theory [@BCS; @SupercondBooks], with predictions that are accurate in the weakly interacting limit only. In the large $x_{AB}$ limit, only the $\phi$ dependent part of Eq.(\[eq:up\_down\]) has a non-zero value: $C_{{\uparrow}{\downarrow}}=C_{{\uparrow}{\downarrow}}^{(0)}\,\cos(2\phi)$. In the local density approximation, and assuming for simplicity that the mean densities are the same in the two extraction points and in the two spin states, we find an analytical expression for a wide extraction region $\ell_u\gg\ell_{\rm BCS}$, both in the weakly interacting BCS regime $$\label{eq:corr_expl}
C_{{\uparrow}{\downarrow}}^{(0)}=
-\frac{3\pi}{8\,\sqrt{2}}\,|u_0|^2\,\frac{\Delta}{E_F}\,N_\sigma$$ and in the regime of a molecular condensate $$C_{{\uparrow}{\downarrow}}^{(0)} = -\frac{1}{\sqrt{2}}\,|u_0|^2\, N_\sigma.$$ $C_{{\uparrow}{\downarrow}}$ is obtained by averaging over many realizations of the whole experimental procedure starting from a trapped gas in the same initial conditions, so that a knowledge of the signal-to-noise ratio is relevant. We estimate the noise by the standard deviation of $\hat{D}_\sigma$: from Wick’s theorem and to leading order in $u_0$, $\big\langle \hat{D}_\sigma^2 \big\rangle \simeq 2N_\sigma$ which shows that the shot-noise [@WallsMilburn] in the initial extraction process is the dominant source of noise. The number of realizations over which to average therefore scales as $(N_\sigma/C_{\uparrow\downarrow}^{(0)})^2$, which is on the order of $1$ in the BEC limit and on the order of $(E_F/\Delta)^2$ in the BCS limit.
[**Light scattering off the matter-wave grating**]{}: Information on the pairing coherence function $G{^{(1)}}_{\rm pair}$ of the trapped gas can also be obtained by means of light-scattering off the matter-wave interference pattern formed by the overlapping wave-packets at $t=t_1$ around point ${ {\bf X}}$, which is taken in what follows as the origin of the coordinates. As already mentioned, the mean density does not show fringes. On the other hand, fringes appear in the opposite spin density-density correlation function ${\mathcal G}{^{(2)}}_{\uparrow\downarrow}({ {\bf x}},{ {\bf x}}')=
\big\langle
{\hat\Psi^\dagger}_{\uparrow}({ {\bf x}},t_1)\,{\hat\Psi^\dagger}_{{\downarrow}}({ {\bf x}}',t_1)
{\hat\Psi}_{{\downarrow}}({ {\bf x}}',t_1)\,{\hat\Psi}_{{\uparrow}}({ {\bf x}},t_1)\big\rangle$. As a guideline, one performs an explicit calculation for BCS theory: one finds that, as soon as a condensate of pairs is present, fringes appear as a function of the center of mass coordinates $({ {\bf x}}+{ {\bf x}}')/2$ of a pair, with a sinusoidal oscillation of wavevector $4{ {\bf k}}_1$. Their amplitude is proportional to the product of the in-trap anomalous averages in ${ {\bf x}}_A$ and ${ {\bf x}}_B$ and extends up to relative distances $|{ {\bf x}}-{ {\bf x}}'|$ of the order of the Cooper-pair size $\ell_{\rm BCS}$. This matter-wave grating is not easily detected in position space since its spatial period is smaller than the mean interatomic distance. We therefore switch to Fourier space: $$\tilde{{\mathcal G}}{^{(2)}}_{\uparrow\downarrow}({ {\bf q}},{ {\bf q}}')
\equiv \int\!d{ {\bf x}}\,d{ {\bf x}}'\,e^{-i({ {\bf q}}\cdot{ {\bf x}}+{ {\bf q}}'\cdot{ {\bf x}}')}\,{\mathcal
G}{^{(2)}}_{{\uparrow}{\downarrow}}({ {\bf x}},{ {\bf x}}').$$ Since fringes show up on the center of mass coordinate with a wavevector $4{ {\bf k}}_1$ we limit ourselves to the region ${ {\bf q}}'= { {\bf q}}\simeq -2{ {\bf k}}_1$ [^4]. Taking into account the free expansion during $t_1$, one then obtains: $$\begin{gathered}
\tilde{{\mathcal G}}{^{(2)}}_{\uparrow\downarrow}({ {\bf q}},{ {\bf q}})=
e^{i\hbar\,\Delta q^2 t_1/m}
\int\!d{ {\bm \xi}}\,d{ {\bm \xi}}'\, e^{-i({ {\bf q}}+2{ {\bf k}}_1)\cdot({ {\bm \xi}}+{ {\bm \xi}}')} \\
\times u^*({ {\bm \xi}})u^*({ {\bm \xi}}')
u({ {\bm \xi}}-{ {\bf b}}) u({ {\bm \xi}}'-{ {\bf b}})\times \\
\Big\langle\,{\hat\Psi^\dagger}_{\uparrow}({ {\bf x}}_A+{ {\bm \xi}})\,{\hat\Psi^\dagger}_{\downarrow}({ {\bf x}}_A+{ {\bm \xi}}')\,
{\hat\Psi}_{\downarrow}({ {\bf x}}_B+{ {\bm \xi}}'-{ {\bf b}})\,{\hat\Psi}_{\uparrow}({ {\bf x}}_B+{ {\bm \xi}}-{ {\bf b}})
\Big\rangle
{\label{eq:G2q}}\end{gathered}$$ where ${ {\bf b}}=\hbar({ {\bf q}}+2{ {\bf k}}_1)t_1/m$ and $\Delta q=|{ {\bf q}}+2{ {\bf k}}_1|$. Remarkably, for ${ {\bf q}}=-2{ {\bf k}}_1$ one recovers the factor in front of $e^{2i\phi}$ in Eq.(\[eq:up\_down\]).
This Fourier component of ${\mathcal G}{^{(2)}}_{{\uparrow}{\downarrow}}$ is detectable in the angular patterns of the elastic light scattering from the atomic cloud. The incoming laser intensity has to be weak enough to avoid saturation of the atomic transition. Optical pumping processes have to be negligible during the whole measurement time: the mean number of scattered photons per atom has to be much less than one not to wash out the information on the internal atomic state. The imaging sequence is assumed to take place in a short time so that the positions of the atoms can be safely considered as fixed. For each realization of the whole experiment, a different distribution of the atomic positions is obtained, and consequently a different angular pattern for the elastic scattering. Information on the density-density correlation function will be obtained by taking the average over many different realizations.
![Second proposed set-up: atoms are extracted from the cloud at ${ {\bf x}}_{A,B}$ and create a matter-wave grating when the wave-packets overlap at ${ {\bf X}}$; one detects this grating by shining a pair of counter-propagating, $\sigma_\pm$ polarized laser beams on the overlap region and by beating the two resulting backscattered light beams on a beam splitter: as function of the mixing phase $\phi_{\rm mix}$, the beating intensity $I(\phi_{\rm mix})$ averaged over many realizations presents fringes revealing the pair long range order. Dashed (dot dashed) lines: $\sigma_+$ ($\sigma_-$) polarization.[]{data-label="fig:setup2"}](setup2.eps){width="6.5cm"}
We consider here the simple case when the laser field frequency is close to resonance with the transition from the $F_g=1/2$ ground state to a $F_e=1/2$ hyperfine component of the excited state. In this case, $\sigma_\pm$ polarized light interacts only with atoms respectively in the ${\downarrow},{\uparrow}$ spin state. The geometry adapted to get information on the condensation of pairs is shown in Fig.\[fig:setup2\]: a pair of mutually coherent laser beams with a common intensity $I_{\rm inc}$ is sent on the atomic cloud with opposite circular polarizations $\sigma_{\pm}$ and opposite wavevectors $\pm { {\bf k}}_1$. We look at the mutual coherence of two back-scattered beams in opposite directions $\pm{ {\bf k}}_{\rm sc}$ [^5], with opposite circular polarizations. Within the Born approximation (valid if the cloud is optically dilute and optically thin), the amplitudes in Fourier space of the scattered light on the circular polarizations $\sigma_\pm$ are related to the ones of the incoming field by $$E_{\pm}^{\rm sc}(\pm{ {\bf k}}_{\rm sc})=
A\,{\hat \rho}_{{\downarrow},{\uparrow}}(\pm{ {\bf q}},t_1)\,E_\pm^{\rm inc}(\pm{ {\bf k}}_1),$$ where $A$ is a factor depending on the dipole moment of the transition and on the atom-laser detuning, and ${\hat \rho}_{\sigma}({ {\bf q}},t_1)$ is the Fourier component at the transferred wavevector ${ {\bf q}}={ {\bf k}}_{\rm sc}-{ {\bf k}}_1$ of the density operator $\hat{\Psi}^\dagger_\sigma({ {\bf x}},t_1)\hat{\Psi}_\sigma({ {\bf x}},t_1)$ at time $t_1$. This mutual coherence is quantified by the correlation function $$\begin{gathered}
I_{-+}/I_{\rm inc}=\big\langle
\left[E^{\rm sc}_{-}(-{ {\bf k}}_{\rm sc})\right]^\dagger\,
E^{\rm sc}_{+}({ {\bf k}}_{\rm sc}) \big\rangle/I_{\rm inc}\\
=|A|^2\,\big\langle
\left[{\hat \rho}_{{\uparrow}}(-{ {\bf q}},t_1)\right]^\dagger
\,
{\hat \rho}_{{\downarrow}}({ {\bf q}},t_1)
\big\rangle
= |A|^2
\tilde{{\mathcal G}}{^{(2)}}_{\uparrow\downarrow}({ {\bf q}},{ {\bf q}}).\end{gathered}$$ By using [(\[eq:G2q\])]{}, one indeed sees that the correlation function $I_{-+}$ can reveal the pair long range order. In the weakly interacting BCS regime as well as in the one of a molecular condensate, it has a simple expression for ${ {\bf q}}\simeq -2{ {\bf k}}_1$: $$I_{-+}=-\frac{|A|^2}{2}
\,C^{(0)}_{\uparrow\downarrow}
\,e^{-\ell_I^2\Delta q^2/2} I_{\rm inc}.
{\label{eq:Iud}}$$ As a function of $\Delta q$, it has a narrow peak with a height proportional to the correlation function $C_{\uparrow\downarrow}$ of the first proposed set-up and with a width $1/\ell_I$ such that $\ell_I^2=\ell_u^2+(\hbar t_1/m\ell_u)^2$. Experimentally, this can be determined by beating the two scattered beams: as a function of the mixing phase $\phi_{\rm mix}$, the resulting intensity presents oscillations with an amplitude equal to $2\,\big|I_{-+}\big|$ on a background of value $\simeq 2 |A|^2 N_\sigma I_{\rm inc}$.
In conclusion, we have proposed two possible ways of detecting a long-range pairing order in a degenerate Fermi gas by measuring the coherence function of the pairs via matter-wave interferometric techniques. This has the advantage over other techniques of directly measuring the order parameter without relying on a microscopic description of the many-body state, so that it applies in an unambiguous way both in the weakly interacting and the strongly interacting regimes. More generally, the proposed scheme is an application of quantum optics techniques to Fermi fields, a line of research expected to open new possibilities in the experimental manipulation and characterization of fermionic systems.
We acknowledge helpful discussions with C. Salomon and the members of his group, with J. Dalibard, Z. Hadzibabic and L. Carr. Laboratoire Kastler Brossel is a Unité de Recherche de l’École Normale Supérieure et de l’Université Paris 6, associée au CNRS.
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[^1]: We assume that $\Delta p |{ {\bf k}}_0\pm{ {\bf k}}_1|\Delta t/m\ll 1$, where $\Delta p$ is the initial momentum spread of the gas.
[^2]: Calculation of expectation values are performed by putting the observables in normal order and using the fact that ${\hat\Psi}^{\rm bg}_\sigma$, when acting on the state vector of the system, gives zero for ${ {\bf x}}$ in one of the two considered wave-packets.
[^3]: In order to avoid scattering into higher, non-resonant, momentum states, the duration $\tau$ of the optical potential pulse has to be longer than the inverse of the atomic recoil frequency $\omega_R$. In order for the reflection and transmission amplitude to have an equal modulus and a constant relative phase within the initial momentum width $\Delta p$ of the gas, $\big|{ {\bf k}}-({ {\bf k}}_0\pm{ {\bf k}}_1)\big|\lesssim
\Delta p/\hbar$, $\tau$ has to be short enough for $\tau^{-1}\gg \hbar k_1 \Delta p/m$. The two conditions are compatible since we assumed that $\Delta p\ll \hbar k_1$.
[^4]: More precisely, one assumes $\hbar qt_1/m \gg \ell_u$ and $\hbar |{ {\bf q}}-2{ {\bf k}}_1| t_1/m \gg \ell_u$.
[^5]: In practice, the scattering wavevectors ${ {\bf k}}_{\rm sc}$ and ${ {\bf k}}_{\rm sc}'$ can be considered as opposite if $|{ {\bf k}}_{\rm sc}+{ {\bf k}}_{\rm sc}'|<\ell_{\rm BCS}^{-1},
m \ell_{\rm BCS}/\hbar t_1$.
|
---
abstract: |
We prove that the cartesian product of octahedra $B_{1,\infty}^{n,m}=B_1^n\times\ldots\times B_1^n$ ($m$ octahedra) is badly approximated by half–dimensional subspaces in mixed–norm: $d_{N/2}(B_{1,\infty}^{n,m},\ell_{2,1}^{n,m}){\geqslant}cm$, $N=mn$. As a corollary the orders for linear widths of Hölder–Nikolskii classes $H^r_p(\mathbb
T^d)$ in the $L_q$ metric are obtained for $(p,q)$ in a certain set (a domain in the parameter space).
author:
- 'Yu.V. Malykhin, K.S. Ryutin'
title: 'Product of octahedra is badly approximated in the $\ell_{2,1}$–metric'
---
We consider the space ${\mathbb R}^N$, with $N=mn$.The set of coordinates $\{1,\ldots,N\}$ is split into $m$ blocks $\Delta_1,\ldots,\Delta_m$ of cardinality $n$; where $\Delta_s=\{(s-1)n+1,\ldots,sn\}$. For a vector $x\in{\mathbb R}^N$ we denote by $x(i)$ its $i$-th coordinate, and $x[s]$ — its restriction on the $s$-block: $x[s]=(x(i))_{i\in\Delta_s}$.
We equip ${\mathbb R}^k$ with usual norms $\|x\|_p=(\sum_{j=1}^k |x_j|^p)^{1/p},$ for $1{\leqslant}p <\infty$; $\|x\|_\infty=\max_{j=1}^k |x_j|$, and ${\mathbb R}^{N}$ — with mixed norms $$\|x\|_{p,q} := \|x\|_{\ell_{p,q}^{n,m}} := \|y\|_{\ell_q^m},\quad\mbox{where
}y=(\|x[s]\|_{\ell_p^n})_{s=1}^m.$$ Let $B_{p,q}^{n,m}$ be the unit ball in $\ell_{p,q}^{n,m}$ space, and $B_p^k$, as usual, the unit ball of $\ell_p^k$. So, $B_{1,\infty}^{n,m}$ is the cartesian product of $m$ octahedra $B_1^n$. Throughout the paper we write $\|x\|_p:=\|x\|_{\ell_p^N}$ and $|x|:=\|x\|_2$ for brevity.
Let us recall the definition of the Kolmogorov width of the subset $M$ in the normed space $X$: $$d_n(M,X) = \inf_{\substack{L\subset X,\\\dim L{\leqslant}n}} \sup_{x\in M}\inf_{y\in
L} \|x-y\|_X,$$ and also the Gelfand width $$d^n(M,X) = \inf_{\substack{L\subset X,\\\operatorname{codim}L{\leqslant}n}} \sup_{x\in M\cap
L}\|x\|_X.$$ We make use of the standard duality [@IT; @LGM] $$d_n(B_X,Y) = d^n(B_{Y^*},X^*),$$ for $X=({\mathbb R}^N,\|\cdot\|_X)$ and $Y=({\mathbb R}^N,\|\cdot\|_Y)$ — finite dimensional normed spaces with balls $B_X$ and $B_Y$, accordingly.
Throughout the paper with $c,c_1,c_2,\ldots,C,C_1,C_2,\ldots$ we denote different absolute positive constants (whose value may depend on the formula). The dependence on some parameters will be stated explicitly.
It is well known, that the calculation of widths of Sobolev classes is often reduced to analogous calculation for $B_p^N$ sets in $\ell_q^N$. While studying widths of Hölder-Nikolskii classes for functions of several variables E.M. Galeev gave a lower estimate via widths of $B_{p,q}^{n,m}$. In particular, in [@Gal90] he proved the following inequality: $$d_{N/2}(B_{1,\infty}^{n,m},\ell_{2,q}^{n,m}){\geqslant}c_qm^{1/q},\quad 1<q{\leqslant}2.$$ (We certainly have simple upper estimate $d_k(B_{1,\infty}^{n,m},\ell_{2,q}^{n,m}){\leqslant}m^{1/q}$ for all $k$ and $1{\leqslant}q{\leqslant}\infty$.) Galeev asked about the $q=1$–case; A.D. Izaak [@Iz94] obtained the estimate $$\label{izaak}
d_{N/2}(B_{1,\infty}^{n,m},\ell_{2,q}^{n,m}){\geqslant}cm\frac{\sqrt{\log\log m}}{\log
m}.$$ Both proofs by Galeev and Izaak are based on the E.D. Gluskin [@Gl87] method, that can not give the true order for $q=1$.
In this paper we establish the following result:
For all $m,n\in\mathbb N$ the following inequality holds $$d_{\lfloor mn/{2}\rfloor}(B_{1,\infty}^{n,m},\ell_{2,1}^{n,m}){\geqslant}cm,$$ with $c>0$ — an absolute constant.
There are some corollaries of the theorem. For example, for the width of the generalized octahedron $V_m^N=B_\infty^N\cap mB_1^N$ we have an estimate $d_{N/2}(V_m^N,\ell_q^N){\geqslant}cm^{1/q}$, $q\in[1,2]$, with some absolute $c$ (in Gluskin [@Gl87] it depends on $q$), and also $$cm^{1/q_2-1/p_2}{\leqslant}d_{N/2}(B_{p_1,p_2}^{n,m},\ell_{q_1,q_2}^{n,m}) {\leqslant}m^{1/q_2-1/p_2},\quad p_1{\leqslant}q_1{\leqslant}2,\; p_2{\geqslant}q_2.$$
We obtain from theorem the following orders for the linear widths of the Hölder–Nikolskii classes $H^r_p=H^r_p(\mathbb T^d)$, $0<r_1=\ldots=r_{l+1}<r_{l+2}=\ldots=r_d$ in $L_q$ space, $1<p,q<\infty$ (for necessary definitions see e.g. [@Gal90; @Gal96]): $$\lambda_N(H^r_p,L_q)\asymp \begin{cases}
\left(\frac{\log^l N}{N}\right)^{r_1-1/2+1/q}\log^{l/q}N,&\quad
\frac1p+\frac1q<1,\;p{\leqslant}2,\;r_1>1-\frac1q,\\
\left(\frac{\log^l N}{N}\right)^{r_1-1/p+1/q}\log^{l/q}N,&\quad
2{\leqslant}p<q,\;r_1>\frac1p-\frac1q.
\end{cases}$$ Upper and lower bounds for $\lambda_N(H^r_p,L_q)$ were obtained in [@Gal96; @Iz96], using the inequality (\[izaak\]) and, so, differed by a power of $\log\log N$. Substituting the estimate given by our Theorem into Galeev’s [@Gal96] proof, we get the true order for $\lambda_N(H^r_p,L_q)$.
It seems interesting to obtain from the Theorem appropriate corollaries for the widths of Besov classes.
Let us recall that $N=mn$; throughout the paper we suppose that $n$ and $m$ are large and $N$ is a multiple of $4$. In our proof we will use the well–known identity $d_k(B_1^N,\ell_2^N)=\sqrt{1-k/N}$. In dual terms for $k=3N/4$ is means that $$\label{gen}
\forall L\subset{\mathbb R}^N,\;\dim L{\geqslant}N/4\quad\Rightarrow\quad \exists x\in
L\;\;|x|=1,\|x\|_\infty{\geqslant}1/2.$$
We will also need the next
Let $\mathcal X_1,\ldots,\mathcal X_m$ be finite sets of vectors in ${\mathbb R}^d$, with $\sum_{x\in\mathcal X_s}|x|^2{\leqslant}1$ for all $s$. Then there exists some vector ${\varepsilon}\in\{0,1,-1\}^d$, $\|{\varepsilon}\|_1{\geqslant}cd$, such that $$\max_{s=1,\ldots,m}\left(\sum_{x\in\mathcal X_s}\langle
{\varepsilon},x\rangle^2\right)^{1/2} {\leqslant}C\log^{1/2}\left(\frac{m}{d}+2\right).$$
This lemma generalizes Theorem 3 by Gluskin [@Gl88], which dealt with vectors $x_1,\ldots,x_m$ (i.e. single element sets $\mathcal X_s = \{x_s\}$). Gluskin strengthens and develops the preceding result by B.S. Kashin [@K85]; the approaches to vector balancing problems introduced in these papers were later improved by several authors. The proof of lemma 1 from known results on vector balancing and properties of gaussian measure on ${\mathbb R}^d$ will be given at the end of the paper.
By duality, $d_{N/2}(B_{1,\infty}^{n,m},\ell_{2,1}^{n,m})=d^{N/2}(B_{2,\infty}^{n,m},\ell_{\infty,1}^{n,m})$. We have to find in any subspace $L\subset{\mathbb R}^N$ of the half–dimension ($N/2$) a vector $x$, with $\|x\|_{\infty,1}{\geqslant}cm$, $\|x\|_{2,\infty}{\leqslant}C$. These requirements will be fulfilled provided
- $\ell_2$-norm of any block is bounded: $|x[s]|{\leqslant}C$;
- there are ${\geqslant}cm$ blocks $s$, such that $|x(i_s)|{\geqslant}1/4$ for some $i_s\in\Delta_s$.
The required vector will be obtained as a certain linear combination of vectors $x_1,\ldots,x_l\in L$, where subspaces $Z_j\subset Z_{j-1}\subset L$ and vector $x_j\in Z_j$ will be produced on the $j$-th step of the construction desribed below. On the first step we let $Z_1:=L$ and take any vector $x_1$ given by (\[gen\]). We have $x_1\in L$, $|x_1|=1$ and $\|x_1\|_\infty{\geqslant}1/2$. So, $|x_1(i_1^*)|{\geqslant}1/2$ for some coordinate $i_1^*$; we call this coordinate “large”; if there are several such coordinates (certainly, no more than $4$), we choose only one of them. We denote by $i_j^*$ the large coordinate produced on the $j$-th step.
The numbers $(x_k(i))_{\substack{1{\leqslant}k{\leqslant}j\\1{\leqslant}i{\leqslant}N}}$ obtained after the first $j$ steps are written into an array where $k$ is the row number and $i$ — the column number.
After $j$-th step we introduce the set of “vanishing” coordinates $\Lambda_j\subset\{1,\ldots,N\}$. This set consist of:
- coordinates $i$, with a large sum of squares (in fixed coordinate): $x_1(i)^2+\ldots+x_j(i)^2{\geqslant}1/n$;
- all coordinates of all blocks $s$, that contain some “large” coordinate: $i_k^*\in\Delta_s$, $1{\leqslant}k{\leqslant}j$;
- all coordinates from any block $s$, with a large sum of squares (in block): $|x_1[s]|^2 + \ldots + |x_j[s]|^2 {\geqslant}1$.
At step $(j+1)$ we apply (\[gen\]) to the subspace $Z_{j+1}:=\{x\in L\colon
x(i)=0\;\forall i\in\Lambda_j\}$ and obtain a vector $x_{j+1}$.
We can apply (\[gen\]) if $|\Lambda_j|{\leqslant}N/4$. Let us show that this holds for $j{\leqslant}cm$. Really, at each step only one new block with a large coordinate is introduced, therefore we get no more than $jn{\leqslant}cN<N/12$ vanishing coordinates. Then, since the sum (on all $k{\leqslant}j$ and $i{\leqslant}N$) of squares of numbers $x_k(i)$ does not exceed $j$, we have not more than $j/(1/n)=jn<N/12$ columns with sum of squares ${\geqslant}1/n$. Similarly we can upper estimate the number of vanishing coordinates from blocks with large sum of squares (in block).
So, we will make $l$ steps, $l\asymp m$ (i.e. $c_1m{\leqslant}l{\leqslant}c_2m$) and will obtain vectors $x_1,\ldots,x_l$. All numbers $x_j(i)$ can be divided into three groups:
- “large”: $x_j(i_j^*)$;
- “intermediate”: $x_j(i)$, such that $\sum_{k{\leqslant}j}x_k(i)^2{\geqslant}1/n$;
- “small”: all other numbers.
We note that (according to our construction) if some number $x_j(i)$ is large or intermediate then on all steps after $j$-th the $i$-th coordinate vanishes: $x_{j+1}(i)=\ldots=x_l(i)=0$. Let $x_j$ be written as $v_j+w_j$, where $v_j$ consist of all large and intermediate numbers and $w_j$ of small.
We will obtain the required vector $x$ as a (balanced) sum $$x=\sum_{j=1}^l {\varepsilon}_j x_j = \sum_{j=1}^l {\varepsilon}_j v_j + \sum_{j=1}^l {\varepsilon}_j w_j =
v + w,$$ where ${\varepsilon}=({\varepsilon}_j)_{j=1}^l\in\{0,1,-1\}^l$ will be given by lemma. It is clear that $x\in L$. We have $|x[s]|{\leqslant}|v[s]|+|w[s]|$. Since for any $i$ the column $(v_j(i))_{j=1}^l$ contains only one nonzero number we get $$|v[s]|^2 = \sum_{i\in\Delta_s} v(i)^2 = \sum_{i\in\Delta_s} (\sum_{j=1}^l
{\varepsilon}_j v_j(i))^2 {\leqslant}\sum_{i\in\Delta_s}\sum_{j=1}^l v_j(i)^2 =
\sum_{j=1}^l|v_j[s]|^2 {\leqslant}C.$$ The last inequality holds because our construction and (iii) imply even an estimate $\sum_{j=1}^l|x_j[s]|^2 {\leqslant}C$.
It is clear that $|v(i_j^*)|=|v_j(i_j^*)|{\geqslant}1/2$, if $|{\varepsilon}_j|=1$. All large coordinates are from different blocks by our construction, so $\|v\|_{\infty,1}{\geqslant}\frac12\|{\varepsilon}\|_1$. In order to finish the proof we need to find vector ${\varepsilon}$, $\|{\varepsilon}\|_1{\geqslant}cm$, such that 1) $\ell_2$-norm of $w$ in any block is bounded by some absolute constant 2) $|w(i)|<1/4$ for all $i\in
I:=\{i_1^*,\ldots,i_l^*\}$.
Let us consider columns $f_i := (w_j(i))_{j=1}^l\in{\mathbb R}^l$; we have $|f_i|{\leqslant}1/\sqrt{n}$ because of (i). We apply the lemma to the following sets of vectors: for each block $s=1,\ldots,m$ we introduce the set $\{f_i\colon i\in\Delta_s\}$, and we also add single–point sets $\{f_i/|f_i|\}$ for $i\in I$. Lemma gives us some vector ${\varepsilon}$, such that $|w[s]|^2 = \sum_{i\in\Delta_s}\langle {\varepsilon},f_i\rangle^2 {\leqslant}C$, and also $|w(i)| = |\langle{\varepsilon},f_i\rangle| {\leqslant}C|f_i|{\leqslant}C/\sqrt{n} < 1/4$ for $i\in I$. The proof is finished.
Let us prove lemma 1. Let $\xi=(\xi_1,\ldots,\xi_d)$ denote the standard gaussian vector in ${\mathbb R}^d$, $\gamma_d$ — be the gaussian measure in ${\mathbb R}^d$: $\gamma_d(A)=\mathsf P(\xi\in
A)=\int_A(2\pi)^{-d/2}\exp(-|x|^2/2)\,dx$, $A\subset{\mathbb R}^d$. Let $\Psi(t) := \gamma_1(-t,t)$. We recall well-known estimates [@Gl88],[@lifsh]: $\Psi(t){\geqslant}1-e^{-t^2/2}{\geqslant}\exp(-2\exp(-t^2/2))$, $t{\geqslant}1$.
We will use the following results.
Let $K\subset {\mathbb R}^d$ be a convex closed centrally–symmetric set, $w\in{\mathbb R}^d$, and $P=\{x:|\langle w,x\rangle|{\leqslant}1\}$ — the strip of the same gaussian measure $\gamma_n(K)=\gamma_n(P)$. Then the inequality $\gamma_n(tK){\geqslant}\gamma_n(tP)$ holds for $t{\geqslant}1$ and the opposite inequality holds for $t\in(0,1)$.
The proof was obtained by R. Latala and K. Oleszkiewicz [@latol].
For any convex centrally–symmetric sets $K_1,K_2\subset {\mathbb R}^d$ holds $\gamma_d(K_1\cap K_2){\geqslant}\gamma_d(K_1)\gamma_d(K_2)$.
This theorem was proved in a recent paper by T. Royen [@roy]; its particular case when $K_1$ is an ellipsoid (it was proved in [@H99]) is sufficient for our purposes.
Besides it, we will need the following particular case of lemma 3.2 from [@Gi97]:
For any convex centrally–symmetric set $V\subset{\mathbb R}^d$, such that $\gamma_d(V){\geqslant}2^{-d/7}$, and any vectors $u_1,\ldots,u_d\in B_2^d$ there exists vector ${\varepsilon}\in\{0,1,-1\}^d$, $\|{\varepsilon}\|_1{\geqslant}d/2$, such that $\sum_{i=1}^d{\varepsilon}_iu_i\in 4V$.
We will use it when $\{u_1,\ldots,u_d\}$ are standard basis vectors in ${\mathbb R}^d$, and get a vector ${\varepsilon}\in 4V$. Let $$h_s(v):=(\sum_{x\in\mathcal X_s}\langle v,x\rangle^2)^{1/2},\quad
\mathcal E_s:=\{v\in{\mathbb R}^d\colon h_s(v){\leqslant}1\}.$$ For a standard gaussian vector $\mathsf{E}h_s(\xi)^2{\leqslant}\sum_{x\in\mathcal X_s}|x|^2 {\leqslant}1$, so we get the estimate $$\label{ebound}
\gamma_d(2\mathcal E_s) = 1-\mathsf P(h_s(\xi)>2){\geqslant}1-\frac{\mathsf{E}h_s(\xi)^2}{4}{\geqslant}3/4 > \Psi(1).$$ By $S$–inequality, using (\[ebound\]), we get $$\gamma_d(2t\mathcal E_s) {\geqslant}\Psi(t) {\geqslant}\exp(-2\exp(-t^2/2)),\quad t{\geqslant}1.$$
Let $V=\bigcap_{s=1}^m(2t\mathcal E_s)$; applying the Gaussian Correlation Conjecture for ellipsoids, we obtain $$\gamma_d(V){\geqslant}\prod_{s=1}^m\gamma_d(2t\mathcal E_s) {\geqslant}\exp(-2m\exp(-t^2/2)).$$ For $t\asymp \log^{1/2}(m/d+2)$ we have $\gamma_d(V){\geqslant}2^{-d/7}$. Therefore, we can apply proposition 1 to $V$, which gives us the required vector ${\varepsilon}$.
[XXXX]{}
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abstract: 'We present a study of the morphology and the thermoelectric properties of short-pulse laser-sintered (LS) nanoparticle (NP) thin films, consisting of $\rm{SiGe}$ alloy NPs or composites of $\rm{Si}$ and $\rm{Ge}$ NPs. Laser-sintering of spin-coated NP films in vacuum results in a macroporous percolating network with a typical thickness of . The Seebeck coefficient is independent of the sintering process and typical for degenerate doping. The electrical conductivity of LS films rises with increasing temperature, best described by a power-law and influenced by two-dimensional percolation effects.'
author:
- 'B. Stoib'
- 'T. Langmann'
- 'S. Matich'
- 'T. Antesberger'
- 'N. Stein'
- 'S. Angst'
- 'N. Petermann'
- 'R. Schmechel'
- 'G. Schierning'
- 'D. E. Wolf'
- 'H. Wiggers'
- 'M. Stutzmann'
- 'M. S. Brandt'
title: 'Laser-sintered thin films of doped SiGe nanoparticles'
---
Nanostructured thermoelectric materials have attracted substantial interest due to the prospect of an increased power factor, a reduced thermal conductivity $\kappa$ and the resulting enhancement of the energy conversion efficiency[@Dresselhaus2007; @Snyder2008; @Minnich2009; @Kanatzidis2010]. On the one hand, a reduction in $\kappa$ can be obtained by classical alloying and the introduction of larger mass fluctuations such as inclusions or precipitates and crystallographic discontinuities as typically found in nanocrystalline materials. Also porosity[@Song2004; @Tang2010] and surface modification[@Boukai2008] can reduce $\kappa$. The variety of length scales present should best be tuned to efficiently affect the propagation of the whole spectrum of phonons while, at the same time, maintaining good electronic properties. On the other hand, an enhancement of the power factor[@Heremans2011; @Zebarjadi2011] can be achieved by a reduction of dimensionality, bandgap engineering, modulation doping or strain. Classical thermoelectric materials such as SiGe[@Wood1988; @Slack1991] profit from nanostructuring[@Zhu2009] as well as materials such as pure Si, which has recently been reconsidered for thermoelectric applications[@Hochbaum2008; @Boukai2008; @Tang2010; @Petermann2011]. Besides their technical relevance, group-IV semiconductors with their well known properties are an ideal model system to study the benefits of nanostructures on the overall thermoelectric performance. In particular thin films are highly suited for such fundamental investigations since they allow access to many analytical methods and provide a range of potential parameters for optimization such as the compatibility factor in segmented thermoelectrics[@Seifert2010]. In this work we report on laser-sintered (LS) thin layers of $\rm{SiGe}$ nanoparticles (NPs), the morphology of the thin films obtained and the influence of porosity on the homogeneity of electrical conduction. Both the Seebeck coefficient $S$ and the temperature-dependent macroscopic electrical conductivity $\sigma$ are determined and compared to bulk samples produced by current-assisted sintering (CAS) of the same initial NPs[@Stein2011].
The NPs were synthesized by plasma-assisted decomposition[@Knipping2004] of silane and germane. Decomposition of both gases in a single microwave plasma leads to a homogeneous mixing of $\rm{Si}$ and $\rm{Ge}$ within each NP[@Stein2011] (henceforth called alloy NPs, denoted $\rm{Si}_{x}\rm{Ge}_{y}$). Separate growth of $\rm{Si}$ NPs and $\rm{Ge}$ NPs in two microwave plasmas and their subsequent mixing in the gas phase leads to a homogeneous mixing of the two sorts of NPs (henceforth called NP composites, denoted $\rm{Si}\text{-}\rm{Ge}\,\text{x:y}$). The Si, Ge as well as alloy NPs obtained are highly crystalline and spherical with a mean diameter of . For all samples in this work, 1% per volume of phosphine was admixed to the silane source as dopant. A native oxide shell is formed due to an unavoidable exposure to air upon removal of the NPs from the plasma reactor. Previous studies on pure $\rm{Si}$ NPs have shown that the overall $\rm{P}$ incorporation efficiency is almost 100%. However, about 95% of the dopants accumulate in the outer shell of the NP which is oxidized upon exposure to air, while only 5% remain within the core and are electrically active[@Stegner2009]. An analogous secondary ion mass spectroscopy of the $\rm{Si}_{80}\rm{Ge}_{20}$ NP films studied here suggests that P accumulation at the NP surface is reduced in alloy NPs, with approximately 40% of the $\rm{P}$ atoms incorporated in the un-oxidized core.
The NP powders were dispersed in ethanol (5% by weight). Spin coating at on flexible polyimid foils (Kapton^^ HN ) resulted in homogeneous layers of typically thickness. Etching in an aqueous solution of 5% hydrofluoric acid (HF) for , rinsing in deionized water and flushing with dry $\rm{N}_2$ efficiently removed the native oxide. The films were transferred to a vacuum chamber with a borosilicate window within . Laser-sintering was performed at a base pressure of less than $\unit{5\cdot 10^{-5}}{mbar}$ with a frequency-doubled $Q$-switched Nd:YAG laser at $\lambda=\unit{532}{\nano\meter}$, a pulse length of and a repetition rate of . The pulse energy could be adjusted up to for a beam diameter of . For all NP layers studied in this work an average energy density of led to the best results in terms of $\sigma$. The sintering was performed at constant energy density, moving the sample under the beam at a constant speed of . The advantages of this “moving spot” approach are twofold: After HF etching the NPs are partly H-terminated[@Stegner2008] and residual solvent is contained inside the macroporous film. The overall Gaussian beam shape in combination with the movement of the sample through the beam leads to a pre-heating which drives out hydrogen and solvents which otherwise would have left the film explosive, leading to ruptures in the morphology. Furthermore, this approach averages out inhomogeneities in the laser beam which cannot be removed optically.
Secondary electron micrographs (SEM) were obtained using a Hitachi S3200N. Scanning transmission electron micrographs (STEM) were obtained using a Zeiss NVision 40 equipped with a segmented annular as well as an on-axis detector. Since the minimum angle for the annular detector was , gray-levels cannot only be ascribed to a compositional contrast[@Pennycook1989]. For thermoelectrical measurements silver contacts were sputtered onto the samples. Temperature-dependent conductivity measurements were preformed in two-point coplanar geometry in vacuum applying a ramp speed of . To minimize the influence of adsorbate desorption on $\sigma$ we only show cooling curves. $I\text{-}V$ characteristics demonstrated ohmic behaviour at all temperatures. Profilometry was used to determine the average macroscopic height $h$ of the films with width $w$ and length $l$. Only these quantities were used to calculate a macroscopic $\sigma$, not accounting for porosity. Seebeck measurements were performed using a direct method applying two typeK thermocouples[@Brandt1998]. The thermovoltage $U$ was picked up between the alumel legs and corrected for $S_{alumel}=\unit{-19}{\micro\volt\per\kelvin}$ for the whole temperature range investigated here[@Bernhard2004]. $S$ was deduced from the slope of $\Delta U$ vs. $\Delta T$, maintaining a constant mean temperature. For all measurements $\Delta T$ was kept smaller than . Bulk CAS samples produced of the same initial NP powders as LS samples were measured with a commercial setup (ZEM-3 by Ulvac Technologies, Inc.) as well as with the apparatus just described to check for the consistency of the measurement setups.
![(a) Secondary electron microscopy (SEM) top view of a laser-sintered (LS) $\rm{Si}\text{-}\rm{Ge}~\text{91:09}$ thin film showing the porosity of the meander structure typical for all samples investigated in this work. (b) Side view of the LS film illustrating the partial substrate coverage. (c) Superposition of the SEM image of a LS film of $\rm{Si}_{80}\rm{Ge}_{20}$ NPs as the basis of a resistor network simulation and a false color image of the current density, as obtained by this simulation. The electric field is applied between top and bottom of the image. (d) and (e) Cross-sectional scanning transmission electron micrographs (STEM) of a LS $\rm{Si}\text{-}\rm{Ge}~\text{60:40}$ composite thin film. Circles indicate inclusions.[]{data-label="fig:morphology"}](morphology.png){width="8.5cm"}
Figure \[fig:morphology\](a) shows a top view SEM image of the general sample morphology of LS thin films which forms connected, meander-like structures constituting a macroporous network covering the substrate to typically 60-80%[^1]. The meander structure is similar for alloy and composite films studied here. The side view in Fig. \[fig:morphology\](b) shows the partial coverage of the substrate by the meander network as well as its varying height of $\unit{200\text{-}300}{\nano\metre}$, which is a characteristic length of the macroporous meander structure. The side view suggests that in-plane electrical transport is governed by an effectively two-dimensional network, with a percolation threshold $p_{th}$ of $\approx 50\text{-}60\,\%$ (depending on the assumed lattice) and a slow increase of conductivity above $p_{th}$[@Last1971]. To study the effect of substrate coverage and porosity on the conductance we applied a random resistor network simulation[@Knudsen2006] on a square lattice where we exemplarily converted a series of typical SEM pixel images into networks of equal resistors. A pixel belonging to the meander layer is assigned to be conductive while a pixel belonging to pore space is nonconductive. Assuming a certain external current entering the resulting resistor network on one side and leaving on the opposite side, the local potentials are calculated. The local current densities are then obtained from the corresponding potential differences. Figure \[fig:morphology\](c) shows that electrical conduction is mainly carried by a few percolating conduction paths. In the example shown, the macroscopic resistivity is increased by a factor of 14 compared to a fully conductive layer without meanders and holes. Although alloys and composites form similar meander structures on the scale and have similar grain sizes of $\approx\unit{100\text{-}150}{\nano\meter}$ (see Fig. \[fig:morphology\](d) and (e)) they differ on a tens of scale. Whereas LS alloys form smooth meanders, LS composites show grainy features of size on the surface as well as within grains (circles). We ascribe this to inclusions of mostly as-grown NPs, unaffected by LS.
![Seebeck coefficient $S$ of samples of four $\rm{SiGe}$ mixtures prepared by laser-sintering (LS) and current-assisted sintering (CAS). $S$ is largely independent of the sintering process, showing the behaviour of degenerately doped $\rm{SiGe}$.[]{data-label="fig:seebeck"}](seebeck.pdf)
In Fig. \[fig:seebeck\] the Seebeck coefficient of LS $\rm{Si}_{80}\rm{Ge}_{20}$ and $\rm{Si}_{95}\rm{Ge}_{05}$ alloys and LS $\rm{Si}\text{-}\rm{Ge}~\text{91:09}$ and $\rm{Si}\text{-}\rm{Ge}~\text{60:40}$ composites is shown together with the results of the corresponding CAS samples of the same initial raw materials. The results obtained by the two Seebeck setups agree quantitatively, demonstrating the high reproducibility of the measurments. All samples show a negative $S$ whose absolute values increase with increasing temperature. This can be understood due to the heavy $\rm{P}$ doping above the metal-insulator transition[@Dismukes1964; @Conwell1956]. For all samples the measured data can well be extrapolated to $S=\unit{0}{\micro\volt\per\kelvin}$ at $T=\unit{0}{\kelvin}$, as expected for degenerate doping. Taking into account the differences in the microstructure of CAS and LS samples, it is encouraging to see that the Seebeck coefficients $S$ of alloys and composites treated by these two sintering methods agree well. Assuming Si-like effective masses[@Green1990] $m^{*}$, the n-type carrier concentration $n$ can be evaluated according to Ref. via $S=\frac{8 \pi^2 k_{B}^2}{3 e h^2} m^{*} T (\frac{\pi}{3n})^{2/3}$, with the Boltzmann constant $k_{B}$, the Planck constant $h$ and the elemental charge $e$. For all LS samples $n\approx \unit{4 \text{-} 9 \cdot 10^{19}}{\centi\meter^{-3}}$.
![Comparison of the macroscopic electrical conductivity $\sigma$ of LS thin films and CAS samples. Data of LS films were not corrected for porosity. While CAS samples show classical metallic behaviour, the LS films exhibit a power-law $\sigma \propto T^{\sim 1.2}$.[]{data-label="fig:conductivity"}](conductivity.pdf)
Figure \[fig:conductivity\] shows the $T$-dependence of the macroscopic electrical conductivity $\sigma$ of LS and CAS samples. In the case of degenerate doping the number of charge carriers should be temperature-independent with the conductivity limited by the mobility. While CAS samples show a flat behaviour, the LS samples show an increasing conductivity with temperature. Furthermore, LS samples differ from CAS samples in the absolute value of $\sigma$, which can at least partly be ascribed to percolation effects in LS samples, discussed in conjunction with Fig. \[fig:morphology\](c), which were not corrected in Fig. \[fig:conductivity\]. Empirically, the observed temperature dependence of $\sigma$ for LS samples can be described by a power-law with $\sigma \propto T^{\sim 1.2}$.
To discuss this finding we additionally present a more systematic study of the influence of doping in a simplified system, namely LS films of pure Si NPs. Due to availability, those films were doped with B, which should not alter the principal findings. A doping series of LS Si NP films is shown in Fig. \[fig:siliconboron\]. Presented in an Arrhenius plot power-law dependencies $\sigma\propto T^\alpha$ appear bent (dashed lines). Experimental data of samples with different doping levels ranging from $\unit{3 \cdot 10^{18}}{\centi\meter^{-3}}$ to $\unit{8 \cdot 10^{19}}{\centi\meter^{-3}}$ show a correlation of doping level and the apparent exponent $\alpha$. Typical for undoped Si, a nominally undoped film shows an activated exponential behaviour with $E_{a}\approx\unit{580}{\milli e \volt}$. This exponential behaviour is also observed for the least doped film at higher temperatures, indicating a continous change from power-law to exponentially activated behaviour. Since LS Si NP films have a similar morphology as the samples discussed above, percolation also affects their absolute value of $\sigma$. As a polycrystalline reference without porosity but with comparable p-type doping, a sample prepared by aluminum-induced layer exchange[@Antesberger2008] (ALILE) is shown. The $T$-dependence of $\sigma$ is similar to LS samples but the absolute values of $\sigma$ are higher for the ALILE sample, confirming the influence of percolation on LS samples.
Our model to explain the temperature dependence is that the transport is governed by an interplay of doping density within the grains and a distribution of barrier heights across the grains. For high doping and low temperature, a certain subensemble of the lowest barriers can be regarded as conductive due to screening by free charge carriers. With increasing temperature more conductive paths are available because also grain boundaries with higher barriers become transparent for charge carriers. We believe that this increased number of possible paths is responsible for the bending in an Arrhenius plot. We note that in recent years a power-law of the electrical conductivity was reported in a number nano-scale systems (see, e.g., Ref. and references therein).
![Study of the influence of B doping in macroporous laser-sintered (LS) Si films. Dashed lines indicate power-laws. The conductivity obtained on a continuous poly-Si film fabricated via aluminum-induced layer exchange (ALILE) is shown as a reference.[]{data-label="fig:siliconboron"}](siliconboron.pdf)
Reducing $\kappa$ via nanostructuring while maintaining/enhancing the power factor are mostly contradicting goals. Introducing disorder via alloying, fine graining, inclusions and porosity easily will lower $\kappa$. We showed that the bottom-up approach of laser-sintering SiGe nanoparticles leads to promising morphological properties, with alloy disorder on the atomic scale and structural features with sizes in the (inclusions), (grains) and (macroporosity) ranges. The fact that the Seebeck coefficient is in agreement with other SiGe systems with similar doping and that the electrical conductivity rises with temperature leads to the conclusions that LS samples already show high quality within the grains and that transport across grain boundaries is largely affected by potential barriers with a wide distribution of barrier heights, depending on the local defect and doping density. The discrepancy to CAS samples suggests a different chemical environment of the grain boundary. Thus, future experiments including H-passivation, furnace or microwave post-treatments will be helpful to understand the transport mechanism and to find strategies for further improvement. Furthermore, overcoming the limitation of two-dimensional percolation can be achieved by filling the remaining pores with a second layer. First studies already showed that the layer deposition sequence of spin-coating, etching and laser-sintering can be applied several times. The meander-like structures of different layers produced are partially sintered together, increasing the degree of interconnection but still maintaining a high degree of porosity.
This work was supported by Deutsche Forschungsgemeinschaft via priority program SPP 1386.
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12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [ ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} , ed., @noop [**]{} (, ) @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{} @noop [****, ()]{}
[^1]: LS under ambient conditions led to rather dewetted isolated drops. No sufficient electrical conduction could be established.
|
---
abstract: |
A family $\mathcal{F}$ of subsets of a set $X$ is $t$-intersecting if $\vert A_i \cap A_j \vert \geq t$ for every $A_i, \; A_j \in
\mathcal{F}$. We study intersecting families in the Hamming geometry. Given $X=\mathbb{F}_q^3$ a vector space over the finite field $\mathbb{F}_q$, consider a family where each $A_i$ is an extended ball, that is, $A_i$ is the union of all balls centered in the scalar multiples of a vector. The geometric behavior of extended balls is discussed. As the main result, we investigate a “large" arrangement of vectors whose extended balls are “highly intersecting". Consider the following covering problem: a subset $\mathcal{H}$ of $\mathbb{F}_q^3$ is a short covering if the union of the all extended balls centered in the elements of $\mathcal{H}$ is the whole space. As an application of this work, minimal cardinality of a short covering is improved for some instances of $q$.\
**MSC(2010):** 05D05, 05B40, 11T99.\
**Keywords:** Intersecting family, ball, Hamming distance, finite field, extremal problem.
author:
- |
Anderson N. Martinhão[^1] [^2] and Emerson L. Monte Carmelo[^3] [^4]\
[Universidade Estadual de Maringá]{}\
[Departamento de Matemática]{}\
title: Intersecting families of extended balls in the Hamming spaces
---
Introduction {#intro}
============
Intersecting family
-------------------
A family $\mathcal{F}=\{A_1, \ldots, A_m \}$ of subsets of an underlying set $X$ is *$t$-intersecting* if $\vert A_i \cap
A_j \vert \geq t$ for any $i \neq j$. A classical class of problems in extremal combinatorics deals with the computation of the maximum cardinality of a $t$-intersecting family under certain constraints. Typically, the imposed conditions are $X=\{1, \ldots, n\}$ and $\vert A_i \vert =k$ for any $1 \leq i \leq m$. The solution for $t=1$ is called the Erdős-Ko-Rado theorem [@EKR]. The complete solution for arbitrary $t$ was proved by Katona [@K]. Algebraic versions of the Erdős-Ko-Rado theorem have been investigated for intersecting chains of boolean algebra by Erdős et al. [@ESS], and for subspaces of a finite vector space by Czabarka [@Cz].
On the other hand, the characterization of the extremal families was obtained by Ahlswede and Khachatrian [@AKh] in connection with the diameter problem in Hamming spaces. Interplays between extremal combinatorics and geometry in Hamming spaces present several difficult problems (see [@AK; @Ko], for instance), some of them are motivated by applications to information theory (see [@B; @Ho]). These contributions have investigated union or intersection of suitable arrangements of balls and their relationships with lines, hyper-planes, or other geometric configurations.
In this work, we investigate intersecting families under a new perspective: each $A_i$ is an arrangement of balls, as described below.
Extended ball
-------------
Let $X=\mathbb{F}_q^3$ be the vector space over the finite field $\mathbb{F}_q$, where $q$ denotes a prime power. Recall that the *Hamming distance* between two vectors $u=(u_1, u_2, u_3)$ and $v=(v_1, v_2, v_3)$ is $d(u,v)= \vert \{ i \ : \ u_i \neq v_i \}
\vert$. The *ball of center* $u$ and *radius* $1$ is denoted by $B(u)=\{ v \in \mathbb{F}_q^3 \ : \ d(u,v) \leq 1 \}$.
If each $A_i$ is a ball and $m=\vert \mathcal{F} \vert$ is “sufficiently large", then clearly there are two disjoint balls.
Consider a variant induced by a geometric change: each center $u$ is “replaced" by a line. More precisely, given a vector $u$ in $\mathbb{F}_q^3$, the [*extended ball*]{} (along the line induced by $u$) is defined as $$\label{e0}
E(u)=\bigcup_{\lambda \in \mathbb{F}_q} B(\lambda u).$$
The covering problem induced by extended balls in an arbitrary space $\mathbb{F}_q^n$ is called [*short covering*]{}, motivated by the fact that short covering might provide us a way to store non-linear codes using less memory than the classical ones. Applications to the classical numbers $K_{q}(n,R)$ (from covering codes) appear in [@otavio] and some of its references. On theoretical viewpoints, results on short coverings have been obtained from distinct tools: graph theory [@anderson], linear algebra [@otavio], ring theory [@Nakaoka; @YSBY].
In this paper we investigate how the extended balls intersect one another. However, a new obstacle arises here: the cardinality of $E(u)$ is not invariant, for example, $\vert E((0,0,0)) \vert= \vert
B((0,0,0)) \vert =3q-2$ but $\vert E((1,1,1)) \vert = 3q^2-2q$, see [@carmelo].
Consider the family $ \mathcal{G}=\{ E(u) : \; u \in \mathbb{F}_q^3
\}$. The “bound" $ \vert {E}(u) \cap {E}(v) \vert \geq \vert B(0)
\vert =3q-2$ shows us that $\mathcal{G}$ is a trivial $3q-2$ intersecting family. This bound can not be improved for all space, because it is sharp for the case $u=(0,0,1)$ and $v=(1,2,0)$ in $\mathbb{F}_5^3$, for instance.
The main statement
------------------
As an attempt to improve the trivial bound above, we investigate a “large" subfamily such that each pair of extended balls has intersection with “bigger size". More precisely, given a family $\mathcal{F}$, let us introduce $$\theta(\mathcal{F})= \max \{ \; t\; : \; \mathcal{F} \mbox { is
$t$-intersecting} \}.$$ This parameter is closely related to concepts from extremal set theory. A family $\mathcal{F}$ is a [*weak $\Delta$-system*]{} if there is $\lambda$ such that $\vert A_i \cap A_j \vert =\lambda$ for any $i \neq j$, introduced by Erdős et.al [@EMR]. Note that $\theta(\mathcal{F}) \geq \lambda$. The [*intersection structure*]{} of a family $\mathcal{F}$ is the set $$I(\mathcal{F})=\{ A_i \cap A_j \; : \; i \neq j \},$$ which was studied by Talbot[@Talbot], for instance. The min-max property $ \theta(\mathcal{F})= min\{ \vert C \vert \;: \; C \in
I(\mathcal{F})\}$ holds.
Take $\mathcal{H} = \{\widetilde{E}(u) \ : \ u \in \mathbb{F}_q^3,
\; u\neq(0,0,0) \}$ as an example. What about $\theta(\mathcal{H})$? We will see that $\theta(\mathcal{H})=0$ as an immediate consequence of Theorem \[bolas\] . The behavior of these intersections is more curious when restricted to the following environmental $$\mathcal{D}_q= \{ (u_1,u_2,u_3) \in \mathbb{F}_q^3 \ : \ u_1,u_2,u_3
\mbox{ are pairwise distinct and non-zero} \}.$$ Indeed, this computation depends on the arithmetic form of $q$; more precisely:
\[dream\] Given a prime power $q$, let $\mathcal{E} = \{ \widetilde{E}(u) \ : \ u \in \mathcal{D}_q \}$. Thus $$\theta(\mathcal{E})= \left\{
\begin{array}{ll}
2(q-1) & \mbox{ if } q-1 \not\equiv 0 \,(mod)\, 3\\
0 & \mbox{ if } q-1 \equiv 0 \, (mod)\, 3.
\end{array}
\right.$$
As a consequence, Theorem \[dream\] reveals a high degree of intersection if $3$ does not divide $q-1$: $ \vert {E}(u) \cap
{E}(v) \vert \geq \vert \widetilde{E}(u) \cap \widetilde{E}(v) \vert
+ \vert B(0) \vert \geq 5q+4$ for any $u$ and $v$ in $\mathcal{D}_q$.
An application to short coverings
---------------------------------
As a complement of this work, the impact of the previous results into the short covering problem is discussed. The set $\mathcal{H} \subset \mathbb{F}_q^3$ is a *short covering of $\mathbb{F}_q^3$* if $$\label{e1}
\bigcup_{h \in \mathcal{H}} E(h)= \mathbb{F}_q^3.$$ What is the minimum number $c(q)$ of extended balls that cover the whole space $\mathbb{F}_q^3$? The only known values are: $c(2)=1$, $c(3)=3$, $c(4)=3$, and $c(5)=4$. The best known bounds for $q \geq
7$ are described below, according to [@anderson] and its references. $$\label{bestinfsup}
\left\lceil \frac{q+1}{2} \right\rceil \leq c(q) \leq \left\{
\begin{array}{l}
(q+3)/2, \mbox{ if } q \equiv 3 \ (mod \ 4)\\
(q+5)/2, \mbox{ if } q \equiv 1 \ (mod \ 4)\\
3(q+4)/4, \mbox{ if } q \mbox{ is even}.
\end{array} \right.$$ In particular, $4 \leq c(7) \leq 5$, $5 \leq c(8) \leq 9$, and $5
\leq c(9) \leq 7$.
\[anderson\] The values $c(7)=5$, $c(8)=6$, and $c(9)=6$ hold.
This work is structured as follows. The geometry of the substructures $ \widetilde{B}(u)={B}(u) \cap \mathcal{D}_q$ and $
\widetilde{E}(u)=E(u)\cap \mathcal{D}_q$ play a central role in our research. Information on these sets and their cardinalities can be derived from a group of transformations described in Section \[Bq\]. In contrast to the classical ball, the cardinality of $\widetilde E(u)$ vanishes according to certain parameters, as stated in Theorem \[bolas\]. A harder problem is studied in Section \[Intersection\], namely, the intersection of extended balls restricted to the set $\mathcal{D}_q$. Theorem \[dream\] is proved in Section \[prova\]. The lower bounds on $c(q)$ are obtained in Section \[lower bounds\]. On the other hand, optimal upper bounds are constructed in Section \[bounds\].
Extended balls {#Bq}
==============
Preliminaries: extended balls in $\mathbb{F}_q^3$
-------------------------------------------------
What kind of application preserves the cardinality of $E(u)$? In order to answer this question, we review briefly a well-known action on groups. We recommend the book [@Ts] for further details.
Given a prime power $q$, $L_q$ denotes the group of non-singular linear operators of $\mathbb{F}_q$. Let $L_q^{3}$ be the direct product $L_q \oplus L_q \oplus L_q$. As usual, $S_3$ denotes the symmetric group of degree $3$. A natural action of $S_3$ on the group $L_q^{3}$ is obtained by permutation of coordinate. This action induces the *wreath product* of $\mathbb{F}^{*}_q$ by $S_3$, $$S_{3} \ltimes L_q^{3} = \{ (\varphi,\sigma) \, : \, \varphi \in S_3
\mbox{ and } \sigma \in L_q^{3} \},$$ The cardinality of $E(u)$ vanishes according to the weight of $u$. Recall that the *weight* of a vector $u=(u_1,u_2,u_3)$ denotes the number $\omega(u)=\vert \{ i \ : \ u_i \neq 0 \}\vert $.
[@carmelo]\[grazieli\] If the vectors $u$ and $v$ are in the same orbit of $\mathbb{F}^3_q$ by the action $S_3 \ltimes L_q^{3}$, then the cardinalities of the sets $E(u)$ and $E(v)$ are equal. In particular, $\vert E(u)\vert
=\vert E(v) \vert$ whenever $\omega(u) =\omega(v)$.
Extended balls in $\mathcal{D}_q$
---------------------------------
This subsection provides information on the sets: $$\widetilde{B}(u)={B}(u) \cap \mathcal{D}_q \;\;\; \mbox{ and } \;\;\;\
\widetilde{E}(u)=E(u) \cap \mathcal{D}_q.$$
We begin with a version of Lemma \[grazieli\] for extended balls restricted to $\mathcal{D}_q$. Consider the following subgroup of $L_q^{3}$ $$K=\{(\sigma ,\sigma ,\sigma ) \ : \ \sigma \in L_q \}=\{
(u_1,u_2,u_3) \mapsto (\lambda u_1, \lambda u_2, \lambda u_3) \ : \
\lambda \in \mathbb{F}_{q}^{*} \}.$$ The commutative property $(\varphi,\sigma) \cdot
(\psi,\tau)=(\psi,\tau)\cdot(\varphi,\sigma)$ in $S_3 \ltimes K$ yields that $S_3 \ltimes K$ can be regard as the direct product $S_3 \times K$.
\[grazieli2\] The standard action of $S_3 \times K$ on $\mathbb{F}_{q}^{3}$ preserves certain cardinalities of extended balls restricted to the set $\mathcal{D}_q$:
1. $ \widetilde{E}(u^{\varphi}) = (\widetilde{E}(u))^{\varphi} $ for any $u$ in $\mathbb{F}_{q}^{3}$ and any $ \varphi \in S_3 $.
2. $ \widetilde{E}(u^{\sigma}) = \widetilde{E}(u) $ for any $u$ in $\mathbb{F}_{q}^{3}$ and $ \sigma \in K $.
3. If $u$ and $v$ are in the same orbit of $\mathbb{F}_q^3$ by the action $S_3 \times K$, then $ \vert \widetilde{E}(u) \vert = \vert
\widetilde{E}(v) \vert $.
The proofs are straightforward.
What is the cardinality of $\widetilde{E}(u)$? Of course, $\widetilde{E}(0)=\emptyset$. In order to compute $\vert \widetilde{E}(u) \vert,$ we can assume that the first non-zero coordinate of $u$ is $1$, by Lemma \[grazieli2\]. The cardinality of $\widetilde{E}(u) $ depends on the weight of $u=(u_1,u_2,u_3)$ and the parameter $\delta(u)=\vert
\{u_1,u_2,u_3 \} \vert$, as described in the next statement.
\[bolas\] Let $u$ be a vector in $\mathbb{F}_q^3$.
1. If $\omega(u)=1$ and $\delta(u)=2$, then $\vert \widetilde{E}(u)\vert =0$.
2. If $\omega(u)=2$ and $\delta(u)=2$, then $\vert \widetilde{E}(u)\vert =0$.
3. If $\omega(u)=2$ and $\delta(u)=3$, then $\vert \widetilde{E}(u)\vert = (q-1)(q-3)$.
4. If $\omega(u)=3$ and $\delta(u)=1$, then $\vert \widetilde{E}(u) \vert=0.$
5. If $\omega(u)=3$ and $\delta(u)=2$, then $\vert \widetilde{E}(u) \vert=(q-1)(2q-6)$.
6. If $\omega(u)=3$ and $\delta(u)=3$, then $\vert \widetilde{E}(u) \vert=(q-1)(3q-11)$.
[**Part 1**]{}: if $\omega(u)=1$ and $\delta(u)=2$. Every scalar multiple of $u$ contains at least two $0$. These multiples are not able to cover any vector in $\mathcal{D}_q$.
[**Part 2**]{}: if $\omega(u)=2$ and $\delta(u)=2$. By Lemma \[grazieli2\], we can assume $u=(0,1,1)$ without loss of generality. Suppose for a contradiction that $v=(v_1,v_2,v_3) \in
\widetilde{E}(u)$, that is, there is $\lambda \in \mathbb{F}_q^{*}$ such that $d((0,\lambda,\lambda),(v_1,v_2,v_3)) \leq 1$. The condition $(v_1,v_2,v_3) \in \mathcal{D}_q$ implies that $v_1 \neq
0$ and consequently $d((v_1,v_2,v_3),(0,\lambda,\lambda)) = 1$. Since the vectors $(v_1,v_2,v_3)$ and $(0,\lambda,\lambda)$ differ in the first coordinate, the absurd $v_2=v_3=\lambda$ is obtained.
[**Part 3**]{}: if $\omega(u)=2$ and $\delta(u)=3$. Again by Lemma \[grazieli2\], we can assume $u=(0,1,z)$, with $z \in
\mathbb{F}_q^{*}$ and $z \neq 1$. Let us prove first the following statement.
*Claim 1*: $\widetilde{B}(\lambda u) \cap
\widetilde{B}(\lambda' u) = \emptyset$ for distinct $\lambda,
\lambda' \in \mathbb{F}_q^{*}$.
Indeed, if $v=(v_1,v_2,v_3) \in \widetilde{B}(\lambda u)\cap
\widetilde{B}(\lambda' u)$, then $\lambda u$ and $v$ differ in the first coordinate, and $\lambda' u$ and $v$ differ in the first coordinate too, because $v_1 \neq 0$. Hence $v$ assumes both forms $v=(v_1,\lambda,\lambda z)$ and $v=(v_1,\lambda',\lambda' z)$, and $\lambda=\lambda'$. This leads an absurd.
If $v=(v_1,v_2,v_3) \in \widetilde{B}(\lambda u)$, then $v$ assumes the form $v=(v_1,\lambda,\lambda z)$, where $v_1 \in
\mathbb{F}_q^{*}$, $v_1 \neq \lambda$ and $v_1 \neq \lambda z $. Each one of these $(q-1)$ scalar multiples covers exactly $(q-3)$ vectors of $\mathcal{D}_q$. Claim 1 states that these $q-1$ sets are pairwise disjoint, hence their union yields $\vert \widetilde{E}(u)
\vert =(q-1)(q-3)$.
Before proving the remaining parts, we need the following statement.
*Claim 2*: Let $\omega(u)=3$. For distinct $\lambda, \lambda'
\in \mathbb{F}_q^{*}$, we claim that ${B}(\lambda u) \cap
{B}(\lambda' u)=\emptyset$. In particular, $$\label{isa}
\vert \widetilde{E}(u) \vert= (q-1) \vert \widetilde{B}(u) \vert.$$
Suppose for a contradiction that ${B}(\lambda u) \cap {B}(\lambda'
u)\neq \emptyset$, thus the vectors $\lambda u$ and $\lambda' u$ agree in at least one coordinate, say $\lambda u_i=\lambda' u_i$. Since $u_i \neq 0$, the condition $\lambda=\lambda'$ holds, which leads an absurd.
[**Part 4**]{}: if $\omega(u)=3$ and $\delta(u)=1$. Here $E(u) \cap
\mathcal{D}_q = \emptyset$, because each scalar multiple of $u$ has three coincident coordinates too. There is not a vector of $\mathcal{D}_q$ which is covered by some multiple of $u$.
[**Part 5**]{}: if $\omega(u)=3$ and $\delta(u)=2$. By Lemma \[grazieli2\], we can assume $u=(1,1,z)$ for some $z \in
\mathbb{F}_q^{*}$ and $z \neq 1$. Suppose that $v=(v_1,v_2,v_3) \in
\widetilde{B}(u)$. Since $v$ has three distinct coordinates and $u$ has two coincident coordinates, we obtain $d(u,v) = 1$. The vector $v$ assumes one of the forms: $(v_1,1,z)$ or $(1,v_2,z)$. Since $v_1,v_2 \in \mathbb{F}_q^{*}$ and $\{ v_1,v_2\} \cap \{1, z\} = \emptyset$, we can choose $v_1$ and $v_2$ from $q-3$ distinct ways, thus $\vert
\widetilde{B}(u) \vert=2q-6$. Equation (\[isa\]) implies $\vert
\widetilde{E}(u) \vert = (q-1)(2q-6)$.
[**Part 6**]{}: if $\omega(u)=3$ and $\delta(u)=3$. We can choose $u=(1,y,z)$, where $1$, $y \neq 0$, and $z \neq 0$ are pairwise distinct. Let $v=(v_1,v_2,v_3)$ be a vector in $\widetilde{B}(u)$. If $d(u,v)=0$ then $v=u$. If $d(u,v)=1$, then $v$ assumes one of the forms: $(v_1,y,z)$, $(1,v_2,z)$, or $(1,y,v_3)$. Since $\{v_1,v_2,v_3\} \cap \{0,1,y,z\}=\emptyset$, each variable $v_1$, $v_2$, and $v_3$ can be chosen from $q-4$ possibilities. The additive principle yields $\vert \widetilde{B}(u) \vert=3q-11$ and Equation (\[isa\]) concludes the counting.
Intersection of extended balls in $\mathcal{D}_q$ {#Intersection}
=================================================
Let us now focus on the behavior of $\widetilde{E}(u) \cap
\widetilde{E}(v)$, where $u,v$ are arbitrary vectors in $\mathbb{F}_q^3$. Obviously $\vert \widetilde{E}(u)\cap
\widetilde{E}(v) \vert = 0$ whenever $\vert \widetilde{E}(u) \vert =
0$ or $\vert \widetilde{E}(v) \vert = 0$. On the other hand, if $\vert \widetilde{E}(u) \cap \widetilde{E}(v) \vert \neq 0$, Theorem \[bolas\] implies that $u$ and $v$ must be of two types:
- [**Type I**]{}: vector of weight two with three distinct coordinates.
- [**Type II**]{}: vector of weight three.
It is well-known that the additive group $\mathbb{Z}_{q-1}$ and the multiplicative group $\mathbb{F}_q^{*}$ are isomorphic by the relation $\overline{a} \mapsto \xi^a$, where $\xi$ denotes an arbitrary generator of $\mathbb{F}_{q} ^{*}$. As usual, the class $\overline{a} \in \mathbb{Z}_{q-1}$ is simply denoted by $a$, where $0 \leq a \leq q-1$ and the multiplication follows the rule $ \xi^a
\xi^b=\xi^{a+b}=\xi^c, $ where $c=a+b$ in $\mathbb{Z}_{q-1}$.
\[cachorro\] *We illustrate here that the cardinality $\widetilde{E}(u) \cap
\widetilde{E}(v) $ can vary widely, even though $u$ and $v$ are vectors of the type I or II.*
Intersection of extended balls reduced to intersection of balls {#aux}
---------------------------------------------------------------
\[uva\] *Let $u=(u_1,u_2,u_3)$ and $v=(v_1,v_2,v_3)$ be arbitrary vectors in $\mathbb{F}_q^3$. The set $B(u) \cap B(v)$ varies according to the distance of the vectors, as follows:*
*Case 1: if $d(u,v)=0$. Clearly $u=v$ and $B(u) \cap
B(v)=B(u)$.*
*Case 4: if $d(u,v)=3$, then clearly $B(u) \cap B(v) =
\emptyset$.*
It is a little surprising that the computation of $\vert \widetilde{E}(u) \cap \widetilde{E}(v) \vert $ under the condition $u,v \in \mathcal{D}_q$ can be reduced to the cardinality of suitable intersections of balls. For this purpose, denote $\lambda
Z=\{\lambda z \, : \, z \in Z\}.$
\[banana000\] Let $u$, $v$ be two vectors in $\mathbb{F}_q^3$, and $ \mu \in
\mathbb{F}_q$.
1. For every $\lambda \in \mathbb{F}_q^{*}$, $$\widetilde{B}(\lambda u) \cap
\widetilde{B}(\mu v) = \lambda [\widetilde{B}(u) \cap
\widetilde{B}(\lambda^{-1}\mu v)].$$
2. If the family $\{ \widetilde{B}(\lambda u) : \, \lambda \in \mathbb{F}_q^{*}\}$ is a partition of $\widetilde{E}(u)$ and the family $\{ \widetilde{B}(\lambda v) : \, \lambda \in
\mathbb{F}_q^{*}\}$ is a partition of $\widetilde{E}(v)$, then $$\vert \widetilde{E}(u) \cap \widetilde{E}(v) \vert =
(q-1)\sum_{\mu\in\mathbb{F}_q^{*}}\vert
\widetilde{B}(u)\cap\widetilde{B}(\mu v)\vert.$$
**Part 1**: Note that $w \in \widetilde{B}(\lambda u)\cap
\widetilde{B}(\mu v)$ if and only if there are scalars $\alpha ,
\beta \in \mathbb{F}_q$ and canonical vectors $e_i, \; e_j $ such that $w=\lambda u + \alpha e_i$ and $w=\mu v + \beta e_j$. These equalities are equivalent to $\lambda^{-1}w = u + \lambda^{-1}\alpha
e_i$ and $\lambda^{-1}w= \lambda^{-1}\mu v + \lambda^{-1}\beta e_j$, that is, $\lambda^{-1} w \in \widetilde{B}(u) \cap
\widetilde{B}(\lambda^{-1} \mu v)$.
**Part 2**: Part 1 and the fact that $\lambda (\cup Z_i)=\cup
(\lambda Z_i)$ produce $$\begin{aligned}
\nonumber \widetilde{E}(u) \cap \widetilde{E}(v) & = &
\bigcup_{\lambda \in \mathbb{F}_q^{*}}
\bigcup_{\mu \in \mathbb{F}_q^{*}}\widetilde{B}(\lambda u)\cap \widetilde{B}(\mu
v) \nonumber\\
& = & \bigcup_{\lambda \in \mathbb{F}_q^{*}} \bigcup_{\mu \in \mathbb{F}_q^{*}} \lambda[\widetilde{B}(u)\cap \widetilde{B}(\lambda^{-1} \mu v)] = \nonumber \\
& = & \bigcup_{\lambda \in \mathbb{F}_q^{*}} \lambda
\left[ \bigcup_{\mu \in \mathbb{F}_q^{*}}\widetilde{B}(u)\cap \widetilde{B}(\lambda^{-1} \mu v)\right]
=\nonumber\\
& = & \bigcup_{\lambda \in \mathbb{F}_q^{*}} \lambda
\left[ \bigcup_{\mu \in \mathbb{F}_q^{*}}\widetilde{B}(u)\cap \widetilde{B}(\mu
v)\right]. \label{abacate}\end{aligned}$$ Since the sets in $\{ \widetilde{B}(\lambda u) : \, \lambda \in
\mathbb{F}_q^{*}\}$ are pairwise disjoint, we claim that $$\label{abacaxi}
\left\vert \bigcup_{\lambda \in \mathbb{F}_q^{*}}
\lambda \left[ \bigcup_{\mu \in \mathbb{F}_q^{*}}\widetilde{B}(u)\cap \widetilde{B}(\mu v)\right] \right\vert =
\sum_{\lambda \in \mathbb{F}_q^{*}} \left\vert \lambda\left[ \bigcup_{\mu \in \mathbb{F}_q^{*}}\widetilde{B}(u)\cap \widetilde{B}(\mu
v)\right]\right\vert.$$ Indeed, if there is $x \in \lambda \left[ \bigcup_{\mu \in
\mathbb{F}_q^{*}}\widetilde{B}(u)\cap \widetilde{B}(\mu v)\right]
\cap \lambda' \left[ \bigcup_{\mu \in
\mathbb{F}_q^{*}}\widetilde{B}(u)\cap \widetilde{B}(\mu v)\right]$ for $\lambda \neq \lambda'$, then in particular, $x \in
\widetilde{B}(\lambda u) \cap \widetilde{B}( \lambda' u)
=\emptyset$, an absurd.
For $\lambda \in \mathbb{F}_q^{*}$, $$\label{amora}
\left\vert \lambda\left[
\bigcup_{\mu \in \mathbb{F}_q^{*}}\widetilde{B}(u)\cap
\widetilde{B}(\mu
v)\right]\right\vert =
\left\vert \bigcup_{\mu \in\mathbb{F}_q^{*}}\widetilde{B}(u)\cap
\widetilde{B}(\mu v)\right\vert=\sum_{\mu \in \mathbb{F}_q^{*}}
\vert \widetilde{B}(u) \cap \widetilde{B}(\mu v) \vert,$$ since $\widetilde{B}(u)\cap\widetilde{B}(\mu v) \subset
\widetilde{B}(\mu v)$ for all $\mu \in \mathbb{F}_q^{*}$, and the sets in $\{ \widetilde{B}(\lambda v) : \; \lambda \in
\mathbb{F}_q^{*}\}$ are pairwise disjoint. From (\[abacate\]), (\[abacaxi\]) and (\[amora\]), we conclude $$\vert \widetilde{E}(u) \cap \widetilde{E}(v) \vert = \sum_{\lambda
\in \mathbb{F}_q^{*}} \left( \sum_{\mu \in \mathbb{F}_q^{*}} \vert
\widetilde{B}(u) \cap \widetilde{B}(\mu v) \vert \right) = (q-1)
\sum_{\mu \in \mathbb{F}_q^{*}} \vert \widetilde{B}(u) \cap
\widetilde{B}(\mu v) \vert.$$
An auxiliary parameter
----------------------
*Given arbitrary vectors $u$ and $v \in \mathbb{F}_q^3$, define $$\rho_q(u,v) =\left\{
\begin{array}{ll}
0 & \mbox{ if } \vert \widetilde{E}(u) \vert = 0$ { \mbox or } $\vert \widetilde{E}(v)\vert = 0\\
\sum_{\mu\in\mathbb{F}_q^{*}}\vert
\widetilde{B}(u)\cap\widetilde{B}(\mu v)\vert & \mbox{ otherwise.}
\end{array}
\right.$$*
\[banana001\] For arbitrary vectors $u,$ $v$ in $\mathbb{F}_q^3$, $$\vert \widetilde{E}(u) \cap \widetilde{E}(v) \vert =
\rho_q(u,v)(q-1).$$
The statement is obvious when $\vert \widetilde{E}(u) \vert = 0$ or $\vert \widetilde{E}(v) \vert = 0$. Otherwise, $\vert
\widetilde{E}(u) \vert \neq 0$, $\vert \widetilde{E}(v) \vert \neq
0$, and both vectors are of type I or II. Hence $\widetilde{B}(\lambda u) \cap \widetilde{B}(\lambda' u) =
\emptyset$ and $\widetilde{B}(\lambda v) \cap \widetilde{B}(\lambda'
v) = \emptyset$ for all $\lambda \neq \lambda'$ in $\mathbb{F}_q^{*}$, according to Claims 1 and 2 of the proof in Theorem \[bolas\]. Lemma \[banana000\] concludes the statement.
\[ozzy\] *Example \[cachorro\] and Corollary \[banana001\] illustrate a few sharp values: $$\begin{array}{llll}
\rho_5((1,4,2),(1,3,4))=3, & \rho_5((1,2,4),(1,4,2))=2, & \rho_5((1,0,2),(1,2,3))=1, \\
\rho_4((1,\xi^{1},\xi^{2}),(1,\xi^{2},\xi^{1}))=0, &
\rho_7((1,2,4),(1,4,2))=0.
\end{array}$$*
The computation of $\rho_q(u,v)$
--------------------------------
In this subsection we are concerned with the computation of $\rho_q(u,v)$ for arbitrary vectors $u,v \in \mathcal{D}_q$.
*Let $u=(2,0,5)$ and $v=(6,7,9)$ be vectors in $\mathbb{Z}_{11}^3$. Since $u$ is a vector of the type I and $v$ is a vector of type II, Corollary \[banana001\] yields $\rho_{11}(u,v) = \sum_{\mu \in \mathbb{Z}_{11}^{*}} \vert
\widetilde{B}(u) \cap \widetilde{B}(\mu v) \vert$. Clearly $\vert
\widetilde{B}(u) \cap \widetilde{B}(\mu v) \vert = 0$ when $d(u,\mu
v) = 3$. The scalars $\mu$ which satisfy $d(u,\mu v) \leq 2$ are: $0$, $3$, and $4$. Thus $$\rho_{11}(u,v) = \vert \widetilde{B}(u) \cap \widetilde{B}(0) \vert
+ \vert \widetilde{B}(u) \cap \widetilde{B}(3 v) \vert + \vert
\widetilde{B}(u) \cap \widetilde{B}(4 v) \vert.$$ A simple inspection reveals that $ \widetilde{B}(u) \cap
\widetilde{B}(0) = \emptyset$, $\widetilde{B}(u) \cap
\widetilde{B}(3 v)=\{ (2,10,5)\}$, and $\widetilde{B}(u) \cap
\widetilde{B}(4 v) = \{ (2,6,5)\}.$ Therefore $\rho_{11}((2,0,5),(6,7,9))=2$.*
The example above illustrates a curious but important fact: in order to compute $\rho_q(u,v),$ we do not have to verify $\vert
\widetilde{B}(u) \cap \widetilde{B}(\mu v) \vert$ for all $\mu \in
\mathbb{F}_q^{*}$. Indeed, it is sufficient to evaluate $\vert
\widetilde{B}(u) \cap \widetilde{B}(\mu v) \vert$ for at most three instances of $\mu \in \mathbb{F}_q^{*}$, according to the next result.
\[estranged\] Given $u=(\xi^{a},\xi^{b},\xi^{c})$ and $v=(\xi^{d},\xi^{e},\xi^{f})$ in $\mathcal{D}_q$, $$\rho_q(u,v) = \vert \widetilde{B}(u) \cap \widetilde{B}(\xi^{a-d}v)
\vert + \vert \widetilde{B}(u) \cap \widetilde{B}(\xi^{b-e}v) \vert
+ \vert \widetilde{B}(u) \cap \widetilde{B}(\xi^{c-f}v) \vert.$$
Since $u$ and $v$ are vectors of type II, Corollary \[banana001\] implies $$\rho_q(u,v) = \sum_{\mu\in\mathbb{F}_q^{*}}\vert
\widetilde{B}(u)\cap\widetilde{B}(\mu v)\vert.$$ We analyze now the contribution of each scalar ${\mu}$. For a scalar $\mu$ such that $d(u,\mu v)=3$, Remark \[uva\] implies $\vert B(u)
\cap B(\mu v) \vert = 0$, and consequently $\vert \widetilde{B}(u)
\cap \widetilde{B}(\mu v) \vert = 0$. It remains the case where $d(u,\mu v) \leq 2$, which produces the following possibilities for $\mu $: $\xi^{a-d}$, $\xi^{b-e}$, and $\xi^{c-f}$.
\[icarus\] Let $q$ be a prime power and $u,v \in \mathcal{D}_q$. The following characterization holds:
1. $\rho_q(u,v)=0$ if and only if $u=\lambda (1,\xi^{a},\xi^{b})$, $v=\mu (1,\xi^{b},\xi^{a})$ for some $\lambda,\mu \in \mathbb{F}_q^{*}$, where $a$, $b$ are distinct, non-zero, $2a=b$, and $2b=a$.
2. $\rho_q(u,v) \geq 2$, otherwise.
[**Part 1**]{}: We can assume without lost of generality that $u$ and $v$ have the first coordinate equal to $1$, that is, $u=(1,\xi^{a},\xi^{b})$, $v=(1,\xi^{b},\xi^{a})$, where $a$, $b$ are distinct, non-zero, $2a=b$ and $2b=a$. By Lemma \[estranged\], $$\rho_q(u,v) = \vert \widetilde{B}(u) \cap \widetilde{B}(v) \vert +
\vert \widetilde{B}(u) \cap \widetilde{B}(\xi^{a-b}v) \vert + \vert
\widetilde{B}(u) \cap \widetilde{B}(\xi^{b-a}v) \vert.$$ The characterization in Remark \[uva\] shows us that $$\begin{aligned}
B(u) \cap B(v) & = & \{ (1,\xi^{b},\xi^{b}),(1,\xi^{a},\xi^{a}) \} ,\\
B(u) \cap B(\xi^{a-b} v) & = & \{(\xi^{a-b},\xi^{a},\xi^{b}),(1,\xi^{a},\xi^{2a-b}) \},\\
B(u) \cap B(\xi^{b-a} v) & = &
\{(\xi^{b-a},\xi^{a},\xi^{b}),(1,\xi^{2b-a},\xi^{b}) \}.\end{aligned}$$ Since $2a=b$ and $2b=a$, we obtain $\widetilde{B}(u) \cap
\widetilde{B}(v) = \emptyset$, $\widetilde{B}(u) \cap
\widetilde{B}(\xi^{a-b}v) = \emptyset$, and $\widetilde{B}(u) \cap
\widetilde{B}(\xi^{b-a}v)= \emptyset$. Therefore $\rho_q (u,v) = 0$.
[**Part 2**]{}: It is enough to prove that $\rho_q(u,v) \geq 2$ for the following situations:
- $u=(1,\xi^a,\xi^b)$, $v=(1,\xi^c,\xi^d)$,
- $u=(1,\xi^a,\xi^b)$, $v=(1,\xi^a,\xi^c)$,
- $u=(1,\xi^a,\xi^b)$, $v=(1,\xi^c,\xi^b)$,
- $u=(1,\xi^a,\xi^b)$, $v=(1,\xi^c,\xi^a)$,
- $u=(1,\xi^a,\xi^b)$, $v=(1,\xi^b,\xi^a)$, with $2a \neq
b$ or $2b \neq a$,
where the elements $a,b,c,d \in \mathbb{Z}_{q-1}$ are pairwise distinct and non-zero.
[*Item (i)*]{}: Here $ \rho_q(u,v) = \vert \widetilde{B}(u) \cap
\widetilde{B}(v) \vert + \vert \widetilde{B}(u) \cap
\widetilde{B}(\xi^{a-c}v) \vert + \vert \widetilde{B}(u) \cap
\widetilde{B}(\xi^{b-d}v) \vert$. Since $d(u,v)=2$, it follows by Remark \[uva\] that $$B(u) \cap B(v) = \{ (1,\xi^{c},\xi^{b}),
(1,\xi^{a},\xi^{d}) \},$$ which is a subset of $\mathcal{D}_q$, thus $\rho_q (u,v)\geq 2$.
[*Item (ii)*]{}: Note that $\rho_q(u,v) = \vert \widetilde{B}(u)
\cap \widetilde{B}(v) \vert + \vert \widetilde{B}(u) \cap
\widetilde{B}(\xi^{b-c}v) \vert$. Apply Remark \[uva\] when $d(u,v)=1$. Since $$B(u) \cap B(v) = \{ (1,\xi^{a},x) \ : \ x \in \mathbb{F}_q \} ,\\$$ is a subset of $\mathcal{D}_q$, $\vert \widetilde{B}(u) \cap
\widetilde{B}(v) \vert =q-3$ holds, and $\rho_q(u,v) \geq q-3$ follows as a consequence.
[*Item (iii)*]{} This case can be proved as an immediate consequence of item (ii) and the concept of $\mathbb{F}_q$-equivalence.
[*Item (iv)*]{} Lemma \[estranged\] implies $$\rho_q(u,v) = \vert
\widetilde{B}(u) \cap \widetilde{B}(v) \vert + \vert
\widetilde{B}(u) \cap \widetilde{B}(\xi^{a-c}v) \vert + \vert
\widetilde{B}(u) \cap \widetilde{B}(\xi^{b-a}v) \vert.$$ We consider two cases.
Case 1: if $a-c \neq b-a$. Since $d(u,v)=2$, $d(u,\xi^{a-c}v)=2$, and $d(u,\xi^{b-a}v)=2$, Remark \[uva\] gives us $$\begin{aligned}
B(u) \cap B(v) & = & \{ (1,\xi^{c},\xi^{b}),(1,\xi^{a},\xi^{a}) \} ,\\
B(u) \cap B(\xi^{a-c} v) & = & \{(\xi^{a-c},\xi^{a},\xi^{b}),(1,\xi^{a},\xi^{2a-c}) \},\\
B(u) \cap B(\xi^{b-a} v) & = &
\{(\xi^{b-a},\xi^{a},\xi^{b}),(1,\xi^{b+c-a},\xi^{b}) \}.\end{aligned}$$ We still need to analyze more two subcases. It is easy to check that $\vert \widetilde{B}(u) \cap \widetilde{B}(v) \vert =1$ for all subcases below.
Subcase 1.1: If $2a \neq c$, then $(1,\xi^{a},\xi^{2a-c}) \in
\mathcal{D}_q$ and $\vert \widetilde{B}(u) \cap
\widetilde{B}(\xi^{a-c} v) \vert \geq 1$.
Subcase 1.2: If $2a = c$, then $(\xi^{b-a},\xi^{a},\xi^{b}) \in
\mathcal{D}_q$ and $\vert \widetilde{B}(u) \cap
\widetilde{B}(\xi^{b-a} v) \vert \geq 1$.
Therefore $\rho_q(u,v) \geq 2$ in both subcases.
Case 2: if $a-c = b-a$. Here $d(u,\xi^{a-c}v)=1$. By Remark \[uva\], $$B(u) \cap B(\xi^{a-c} v) = \{(x,\xi^{a},\xi^{b}) \ : \ x \in
\mathbb{F}_q \}.$$ Thus $\vert \widetilde{B}(u) \cap \widetilde{B}(v) \vert =1$ and $\vert \widetilde{B}(u) \cap \widetilde{B}(\xi^{a-c}v) \vert = q-3$, which implies $\rho_q(u,v) \geq q-3$.
[*Item (v)*]{} In this case, $$\rho_q(u,v) = \vert \widetilde{B}(u)
\cap \widetilde{B}(v) \vert + \vert \widetilde{B}(u) \cap
\widetilde{B}(\xi^{a-b}v) \vert + \vert \widetilde{B}(u) \cap
\widetilde{B}(\xi^{b-a}v) \vert.$$ We divide the proof into two cases.
Case 1: if $a-b \neq b-a$. Since $d(u,\xi^{a-b}v)=2$ and $d(u,\xi^{b-a}v)=2$, Remark \[uva\] implies $$\begin{aligned}
B(u) \cap B(\xi^{a-b} v) & = & \{(\xi^{a-b},\xi^{a},\xi^{b}),(1,\xi^{a},\xi^{2a-b}) \},\\
B(u) \cap B(\xi^{b-a} v) & = &
\{(\xi^{b-a},\xi^{a},\xi^{b}),(1,\xi^{2b-a},\xi^{b}) \}.\end{aligned}$$
Subcase 1.1: if $2a \neq b$. The vectors $(1,\xi^{a},\xi^{2a-b})$ and $(\xi^{b-a},\xi^{a},\xi^{b})$ belong to $\mathcal{D}_q$. Hence $\vert \widetilde{B}(u) \cap \widetilde{B}(\xi^{a-b}v) \vert \geq 1$ and $\vert \widetilde{B}(u) \cap \widetilde{B}(\xi^{b-a}v) \vert
\geq 1$. It means that $\rho_q(u,v) \geq 2$.
Subcase 1.2: if $2b \neq a$. The vectors $(\xi^{a-b},\xi^{a},\xi^{b})$ and $(1,\xi^{2b-a},\xi^{b})$ belong to $\mathcal{D}_q$. Since $\vert \widetilde{B}(u) \cap
\widetilde{B}(\xi^{a-b}v) \vert \geq 1$ and $\vert \widetilde{B}(u)
\cap \widetilde{B}(\xi^{b-a}v) \vert \geq 1$, the bound $\rho_q(u,v)
\geq 2$ holds.
Case 2: if $a-b = b-a$. Because $d(u,\xi^{a-b}v)=1$, Remark \[uva\] implies $$B(u) \cap B(\xi^{a-b} v) = \{(x,\xi^{a},\xi^{b}) \ : \ x \in
\mathbb{F}_q \}.$$ It is easy to check that $\vert \widetilde{B}(u) \cap
\widetilde{B}(\xi^{a-b}v) \vert = q-3$, and $\rho_q(u,v) \geq q-3$ follows.
Proof of Theorem \[dream\] {#prova}
==========================
\[problemaextremal\] *Given a prime power $q$, define $$\rho(q) = \min \{ \rho_q(u,v) \ : \ u, v \in \mathcal{D}_q \}.$$*
*We obtain immediately from Example \[ozzy\] that $\rho(4)=0$, $\rho(5) \leq 2$ and $\rho(7)=0$.*
The parameter $\rho(q)$ is completely determined, according to the next statement.
\[forever\] For a prime power $q$, $$\rho(q) =\left\{
\begin{array}{l}
0 \mbox{ if } 3 \mbox{ divides } q-1,\\
2 \mbox{ otherwise.}
\end{array}
\right.$$
[**Part 1**]{}: If $3$ divides $q-1$, then there is a non-zero $k \in
\mathbb{Z}$ such that $3k=q-1$, that is, $3k=0$ in the ring $\mathbb{Z}_{q-1}$. Since the vectors $u=(1,\xi^{k},\xi^{-k})$ and $v=(1,\xi^{-k},\xi^{k})$ satisfy the hypothesis of Proposition \[icarus\], the value $\rho_q (u,v)=0$ holds.
[**Part 2**]{}: If $3$ does not divide $q-1$, then there are distinct numbers $a,b \in \mathbb{Z}_{q-1}^{*}$ such that $2a=b$ and $2b=a$. An application of Proposition \[icarus\] yields $\rho_q (u,v) \geq
2$ for all $u,v \in \mathcal{D}_q$, that is, $\rho(q) \geq 2$.
Choose an element $a \in \mathbb{Z}_{q-1}$ such that $a \neq 0$, $2a
\neq 0$ and $3a \neq 0$. We consider the vectors $u=(1,\xi^{a},\xi^{2a})$ and $v=(1,\xi^{2a},\xi^{a})$. Thus $\rho_q(u,v) = \vert \widetilde{B}(u) \cap \widetilde{B}(v) \vert +
\vert \widetilde{B}(u) \cap \widetilde{B}(\xi^{-a}v) \vert + \vert
\widetilde{B}(u) \cap \widetilde{B}(\xi^{a}v) \vert $, and by Remark \[uva\], $$\begin{aligned}
B(u) \cap B(v) & = & \{ (1,\xi^{2a},\xi^{2a}),(1,\xi^{a},\xi^{a}) \} ,\\
B(u) \cap B(\xi^{-a} v) & = & \{ (\xi^{-a},\xi^{a},\xi^{2a}),(1,\xi^{a},1) \},\\
B(u) \cap B(\xi^{a} v) & = &
\{(\xi^{a},\xi^{a},\xi^{2a}),(1,\xi^{3a},\xi^{2a}) \}.\end{aligned}$$ Clearly, the vectors $(1,\xi^{2a},\xi^{2a})$, $(1,\xi^{a},\xi^{a})$, $(1,\xi^{a},1)$ and $(\xi^{a},\xi^{a},\xi^{2a})$ do not belong to $\mathcal{D}_q$. Both vectors $(\xi^{-a},\xi^{a},\xi^{2a})$ and $(1,\xi^{3a},\xi^{2a})$ belong to $\mathcal{D}_q$. Hence $\vert
\widetilde{B}(u) \cap \widetilde{B}(v) \vert = 0$, $\vert
\widetilde{B}(u) \cap \widetilde{B}(\xi^{-a} v) \vert = 1$, $\vert
\widetilde{B}(u) \cap \widetilde{B}(\xi^{a} v) \vert = 1$, and consequently $\rho_q (u,v) = 2$.
: Since $\mathcal{E} = \{ \widetilde{E}(u) \ : \ u \in \mathcal{D}_q
\}$, Corollary \[banana001\] reveals that $(q-1)\rho(q)$ is the maximum $t$ such that the family $\mathcal{E}$ is $t$-intersecting. Thus the computation of $\theta(\mathcal{E})$ is reduced to Theorem \[forever\].
Lower bounds of short coverings {#lower bounds}
===============================
Necessary conditions for a short covering
-----------------------------------------
Some necessary conditions for a short covering with “few vectors" are established here. For sake this purpose, let $\pi_j(u_1,u_2,u_3)=u_j$ denote the $j$-th canonical projection of $\mathbb{F}_q^3$ into $\mathbb{F}_q$, where $1 \leq j \leq 3$. The symbol $*$ represents an arbitrary element in $\mathbb{F}_q$.
\[caracterizacao\] Given a prime power $q \geq 7$, let $m = \lceil (q+1)/2 \rceil$. Suppose that $\mathcal{H}=\{ h_1,\ldots,h_m \}$ is a short covering of $\mathbb{F}_q^3$. The following conditions hold:
1. There must be at least a vector in $\mathcal{H}$ with weight 3.
2. For each coordinate $j$, $1 \leq j \leq 3$, there must be at least a vector $h_k \in \mathcal{H}$ such that $\pi_j(h_k)=0$.
3. The set $\mathcal{H}$ is $\mathbb{F}_q$-equivalent to one of the sets: $$\begin{aligned}
\mathcal{H}_1 & = & \{ (1,1,1),(0,*,*),(*,0,*),(*,*,0),h_5,\ldots,h_m \},\\
\mathcal{H}_2 & = & \{ (1,1,1),(0,*,*),(*,0,0),h_4,\ldots,h_m \}.\end{aligned}$$
**Part 1**: If each vector in $\mathcal{H}$ has weight at most $2$, Theorem \[bolas\] yields $\vert \widetilde{E}(h_i) \vert \leq
(q-1)(q-3)$ for every $i$, $1 \leq i \leq m$. Thus the set $\mathcal{H}$ is able to cover at most $m(q-1)(q-3)$ vectors of $\mathcal{D}_q$. Because $\vert \mathcal{D}_q
\vert=(q-1)(q-2)(q-3)$, the set $\mathcal{H}$ is not a short covering of $\mathbb{F}_q^3$, when $q \geq 7$.
**Part 2**: Assume without loss of generality that $\omega(h_1)=3$. We also suppose $h_1=(1,1,1)$, by $\mathbb{F}_q$-equivalence. Consider the plane $$\Pi_1= \{ (0,u_2,u_3) \ : \ u_2,u_3 \in \mathbb{F}_q \}$$ and its subset $\mathcal{X}_1=\{ (0,u_2,u_3) \in \Pi_1 \ : \ u_2
\neq u_3 \mbox{ and } u_2,u_3 \neq 0 \}$. The heart of the proof consists in checking that $\mathcal{H}$ is not able to cover (shortly) all the plane $\Pi_1$. Since $E(h_1) \cap \mathcal{X}_1=
\emptyset$, the whole set $\mathcal{X}_1$ must be covered by $\{h_2,\ldots,h_m\}$. Suppose for a contradiction that $\pi_1(h_2)
\neq 0, \ldots, \pi_1(h_m) \neq 0$. Each one of the vectors in $\{h_2,\ldots,h_m\}$ covers at most $q-1$ vectors of $\mathcal{X}_1$, thus $$\vert [ E(h_2)\cup \cdots \cup E(h_m)] \cap
\mathcal{X}_1 \vert \leq (m-1)(q-1).$$ From the fact that $m = (q+1)/2$ if $q$ is odd and $m = (q+2)/2$ if $q$ is even, $$(m-1)(q-1) = \frac{q}{2}(q-1)<(q-1)(q-2)=|\mathcal{X}_1|$$ holds for every $q \geq 5$. Hence $\mathcal{X}_1 \not\subset E(h_2)
\cup \cdots \cup E(h_m)$. The statement for the case $j=1$ is proved. The argument for $j=2$ and $j=3$ follows analogously.
**Part 3**: It is a consequence of both Parts 1 and 2. There is a vector $h_1$ in $\mathcal{H}$ with $\omega(h_1)=3$. We also assume $h_1=(1,1,1)$, by $\mathbb{F}_q$-equivalence. The Part 2 implies that for each coordinate $j$, $1 \leq j \leq 3$, there must be at least a vector $h_k \in \mathcal{H}$ such that $\pi_j(h_k)=0$. Thus there are three vectors of type $(0,*,*)$, $(*,0,*)$, $(*,*,0)$ in $\mathcal{H}$ or there are two vectors of the type $(0,*,*)$, $(*,0,0)$ in $\mathcal{H}$. The first case yields that $\mathcal{H}$ and $\mathcal{H}_1$ are $\mathbb{F}_q$-equivalent, while the second case implies that $\mathcal{H}$ and $\mathcal{H}_2$ are $\mathbb{F}_q$-equivalent.
Sketch
------
The rest of this section is concerned with the computation of lower bounds on $c(q)$, where $7 \leq q \leq 9$. The condition $c(q) > m$ corresponds to the statement: neither of the $ \binom{q^{3}}{m}$ $m$-subsets of $\mathbb{F}_{q}^3$, $\mathcal{H}$ satisfies the equation (\[e1\]).
Since the search space is often huge and the extended balls are highly intersecting, it is not so accurate checking Eq. (\[e1\]) straightforwardly. A sharp approach essentially analyzes the behavior of the extended balls in $\mathcal{D}_q$. A little more precise, the idea is described briefly as follows.
Given $q$, suppose by absurd that there is a short covering $\mathcal{H}=\{h_1,\ldots,h_m \}$ of $\mathbb{F}_q^3$ with $m=
\lceil (q+1)/2 \rceil$ vectors. Theorem \[caracterizacao\] states that there are only two possibilities for $\mathcal{H}$. Since $\mathcal{H}$ is also a short covering of the subset $\mathcal{D}_q$, the condition $\mathcal{D}_q \subset \cup_{i=1}^{m}
\widetilde{E}(h_i)$ holds. On the other hand, if we show that $$\label{e2}
\left\vert \bigcup_{i=1}^{m} \widetilde{E}(h_i) \right\vert <
(q-1)(q-2)(q-3)$$ then an absurd raises: $\mathcal{D}_q$ is not contained in $\cup_{i=1}^{m} \widetilde{E}(h_i)$. For the cases $q=7$ and $q=9$, the stronger condition $$\label{e3}
\sum_{i=1}^{m} \vert \widetilde{E}(h_i) \vert < (q-1)(q-2)(q-3)$$ is sufficient to show that (\[e2\]) is valid.
New lower bounds
----------------
\[inferior7\] We obtain $c(7) \geq 5$.
Suppose by a contradiction that $\mathcal{H}=\{ h_1,\ldots,h_4 \}$ is a short covering of $\mathbb{F}_7^3$. Theorem \[caracterizacao\] states that there are only two forms for $\mathcal{H}$, namely: $$\begin{aligned}
\mathcal{H}_1 & = & \{ (1,1,1),(0,*,*),(*,0,*),(*,*,0) \},\\
\mathcal{H}_2 & = & \{ (1,1,1),(0,*,*),(*,0,0),(*,*,*) \}.\end{aligned}$$
If $\mathcal{H}=\mathcal{H}_1$, then Theorem \[bolas\] yields $\vert \widetilde{E}(h_1) \vert =0$ and $\vert \widetilde{E}(h_i)
\vert \leq 24$ for $i \in \{2,3,4\}$. Thus $\mathcal{H}_1$ covers at most $72$ vectors in $\mathcal{D}_7$.
Otherwise, $\mathcal{H}=\mathcal{H}_2$. Theorem \[bolas\] implies that $\vert \widetilde{E}(h_1) \vert =0$, $\vert \widetilde{E}(h_2)
\vert \leq 24$, $\vert \widetilde{E}(h_3) \vert =0$ and $\vert
\widetilde{E}(h_4) \vert \leq 60$. Hence $\mathcal{H}_2$ covers at most $84$ vectors in $\mathcal{D}_7$.
Since $\vert \mathcal{D}_7 \vert=120$, the inequality (\[e3\]) holds. We conclude that neither $\mathcal{H}_1$ nor $\mathcal{H}_2$ could cover all the space $\mathcal{D}_7$. Thus $c(7) \geq 5$.
\[inferior9\] The lower bound $c(9) \geq 6$ holds.
Suppose for a contradiction that ${\mathcal{H}}=\{ h_1,\ldots,h_5
\}$ is a short covering of $\mathbb{F}_9^3$. From Theorem \[caracterizacao\], the set $\mathcal{H}$ can be $\mathbb{F}_q$-equivalent to only two forms: $$\begin{aligned}
\mathcal{H}_1 & = & \{ (1,1,1),(0,*,*),(*,0,*),(*,*,0),(*,*,*) \},\\
\mathcal{H}_2& = & \{ (1,1,1),(0,*,*),(*,0,0),(*,*,*),(*,*,*) \}.\end{aligned}$$
If $\mathcal{{H}}=\mathcal{{H}}_1$, then Theorem \[bolas\] yields $\vert \widetilde{E}(h_1) \vert=0$, $\vert \widetilde{E}(h_i) \vert
\leq 48$ for $i \in \{ 2,3,4 \}$, and $\vert \widetilde{E}(h_5)
\vert \leq 128$. The set $\mathcal{H}_1$ covers at most $272$ vectors of $\mathcal{D}_9$.
If $\mathcal{{H}}=\mathcal{{H}}_2$, Theorem \[bolas\] produces $\vert \widetilde{E}(h_1) \vert =0$, $\vert \widetilde{E}(h_2) \vert
\leq 48$, $\vert \widetilde{E}(h_3) \vert =0$ and $\vert
\widetilde{E}(h_i) \vert \leq 128$ for $i \in \{4,5\}$. Hence $\mathcal{H}_2$ covers at most $304$ vectors of $\mathcal{D}_9$.
Since $\vert \mathcal{D}_9 \vert = 336$, the inequality (\[e3\]) is satisfied here; neither $\mathcal{H}_1$ nor $\mathcal{{H}}_2$ is a short covering of $\mathcal{{D}}_9$.
The argument for $c(8) >5 $ is a little more intricate than the previous ones, because the inequality (\[e3\]) does not hold for all candidates $\mathcal{H}$. The search space corresponds to the family of all $5$-subsets of $\mathbb{F}_{8}^3$, with $\binom{8^{3}}{5}\simeq 2.8 \times 10^{11}$ candidates. Theorem \[bolas\] is not enough powerful to deal with all candidates. Therefore we shall apply Theorem \[dream\] too.
\[inferior8\] The bound $c(8) \geq 6$ holds.
Suppose for a contradiction that $\mathcal{H}=\{ h_1,\ldots,h_5 \}$ is a short covering of $\mathbb{F}_8^3$. Now, by Theorem \[caracterizacao\], we can assume that $\mathcal{{H}}$ has one of two possible forms: $$\begin{aligned}
\mathcal{{H}}_1 & = & \{ (1,1,1),(0,*,*),(*,0,*),(*,*,0),(*,*,*) \},\\
\mathcal{{H}}_2 & = & \{ (1,1,1),(0,*,*),(*,0,0),(*,*,*),(*,*,*) \}.\end{aligned}$$
An application of Theorem \[bolas\] to the case $\mathcal{H}=\mathcal{H}_1$ yields $\vert \widetilde{E}(h_1) \vert
=0$, $\vert \widetilde{E}(h_i) \vert \leq 35$ for $i \in \{2,3,4\}$, and $\vert \widetilde{E}(h_5) \vert \leq 91$. Hence $\mathcal{H}_1$ covers at most $196$ vectors in $\mathcal{D}_8$. Since $\vert
\mathcal{D}_8 \vert=210$, the inequality (\[e3\]) holds, and thus $\mathcal{H}_1$ could not be a short covering of $\mathcal{D}_8$.
On the other hand, computation of the bounds from Theorem \[bolas\] for the case $\mathcal{H}=\mathcal{H}_2$ shows us $\vert
\widetilde{E}(h_1) \vert =0$, $\vert \widetilde{E}(h_2) \vert \leq
35$, $\vert \widetilde{E}(h_3) \vert =0$, and $\vert
\widetilde{E}(h_i) \vert \leq 91$ for $i \in \{4,5\}$. Hence $\mathcal{H}_2$ covers at most $217$ vectors in $\mathcal{D}_8$.
*Claim*: The vectors $h_4$ and $h_5$ belong to $\mathcal{D}_8.$
Indeed, if $h_4 \not\in \mathcal{D}_8$, Theorem \[bolas\] implies $\vert \widetilde{E}(h_4) \vert \leq 70$. Hence $\mathcal{H}_2$ covers at most $196$ vectors of $\mathcal{D}_{8}$, and $\mathcal{H}_2$ is not a short covering of $\mathcal{D}_{8}$, since $\vert \mathcal{D}_8 \vert=210$. If $h_5$ does not satisfy these conditions, we obtain an absurd analogously.
By Theorem \[dream\], $\vert \widetilde{E}(h_4) \cap
\widetilde{E}(h_5) \vert \geq 14$, and thus $\mathcal{H}_2$ is a short covering of at most $203$ vectors of $\mathcal{D}_8$, thus the inequality (\[e2\]) holds.
Construction of short coverings {#bounds}
===============================
From actions of groups to coverings
-----------------------------------
A systematical way of finding good short coverings is described in [@irene Theorem 1] on the basis of invariant sets under suitable actions. An adaptation of this method is described below.
The standard action of $G=S_3 \times K$ on $\mathbb{F}_{q}^{3}$ plays a central role in our results. The set $$\mathcal{A}_{q}=\{(u_1,u_2,u_3) \in \mathbb{F}_q^3 \ : \ u_1,u_2,u_3
\mbox{ are pairwise distinct} \}.$$ is invariant by the action of the direct group $S_{3} \times K$, which has two orbits, namely, $\{u \in \mathcal{A}_q \ : \
d(u,0)=3\}$ and $\{u\in \mathcal{A}_q \ : \ d(u,0)=2\}$.
\[metodo\] Let $N$ be a subgroup of $S_{3}$ and choose a subset $\mathcal{L}$ of $\mathbb{F}_{q}^{3}$ which is invariant under the action of $N$, that is, $\mathcal{L}^{N}=\mathcal{L}$. Let $\mathcal{O}$ denote the family of all orbits of the action of $N \times K$ under $\mathcal{A}_q$. Suppose that each orbit of the action of $S_{3}\times K$ on $\mathcal{A}_q$ contains an element $u$ which can be written as $u=\lambda h+ \mu e_{j}$ for some $h\in \mathcal{L}$, $\lambda $, $\mu \in \mathbb{F}_{q}$ and $j\in
\{1,2,3\}$. Thus the set $\mathcal{L} \cup \{(1,1,1)\}$ is a short covering of $\mathbb{F}%
_{q}^{3}.$
Take an arbitrary vector $v$ in $\mathbb{F}_{q}^{3}$. We analyze two cases.
Case 1: the case where $v\in \mathbb{F}_{q}^{3}\setminus
\mathcal{A}_q$. Here $v$ has at least two coincident coordinates, say $v=(\lambda,\lambda,\mu )$. Thus $v=\lambda
(1,1,1)+(\mu-\lambda) e_{3}$, that is, $v \in E((1,1,1))$.
Case 2: $v\in \mathcal{A}_q$. We show that $\cup _{h\in
\mathcal{L}}{E}(h)$ contains $v$. Since $v\in \mathcal{A}_q$, there is a vector $u\in \mathcal{O}$ such that $u=v^{\sigma \gamma }$. By hypothesis, $u=\lambda h+\mu e_{j}$, where $\lambda \neq 0$. By applying $\sigma $, we obtain $u^{\sigma }=\lambda' h+\mu' e_{j}$. By applying $\varphi $, we have $v=u^{\sigma \varphi}=\lambda'
h^{\varphi }+\mu' e_{j}^{\varphi }$. Because $\mathcal{L}$ is an invariant set under the action of $\varphi $, the required statement is obtained.
*The theorem above is optimal for small instances $q=3,4,5,7$. For example, the sharp bound $c(5)\leq 4$ can be reproved as follows. Choose $\mathcal{L}=\{(0,2,3),(3,0,2),(2,3,0)\}$. Because $\mathcal{L}$ is invariant under the action of the 3-cycle $\varphi:
(u_1,u_2,u_3) \mapsto (u_2,u_3,u_1),$ take $N= <\varphi>$ the subgroup generated by $\varphi$. The action of $G=<\varphi> \times
K$ on $\mathcal{A}_5$ produces five orbits. Since the stabilizer of a vector $u$ is the trivial subgroup, each orbit $u^{G}$ has twelve elements. Moreover, the representatives are covered by $\mathcal{L}$, as described below $$\begin{array}{ll}
(0,1,2)=3(0,2,3)+3e_{3} & (0,1,3)=3(0,2,3)+4e_{3} \\
(0,1,4)=3(0,2,3) & (1,2,3)=1(0,2,3)+1e_{1} \\
(1,3,2)=4(0,2,3)+1e_{1}.&
\end{array}$$ The bound follows from Theorem \[metodo\].*
\[superior8\] The upper bound $c(8) \leq 6$ holds.
Choose the vectors $$\begin{array}{lll}
h_1= (1,1,1) , & h_2= (0,0,\xi^1) , & h_3=(1,\xi^1,0) , \\
h_4= (1,\xi^2,\xi^3) , & h_5=(1,\xi^3,\xi^2) , &
h_6=(\xi^6,\xi^5,1),
\end{array}$$ where $\xi$ denotes a generator of the group $\mathbb{F}_8^{*}$. We claim that $ \mathcal{H}=\{ h_1, \ldots, h_6 \}$ is a short covering of $\mathbb{F}_8^3$. In order to apply Theorem \[metodo\], choose the group $G$ isomorphic to $K$.
Take an arbitrary vector $u=(u_1,u_2,u_3) \in \mathbb{F}_8^3$. We analyze now the case where $u=(u_1,u_2,u_3) \in \mathbb{F}_8^3
\setminus \mathcal{D}_8$. If there are at least two coincident coordinates in $u$, then $u$ is covered by $h_1$. If zero appears as a coordinate of $u$, then $u$ is covered (shortly) by $h_2$ or $h_3$.
Otherwise, $u$ belongs to $\mathcal{D}_8$. The representatives of the orbits on $\mathcal{D}_8$ can be chosen as $u=(1,u_2,u_3)$, where $u_2,u_3\in \{ \xi^1,\xi^2,\xi^3,\xi^4,\xi^5,\xi^6\}$. For the cases where $u_2\in \{\xi^1,\xi^2,\xi^3 \}$ or $u_3 \in
\{\xi^2,\xi^3 \}$, a simple look shows us that $u$ is covered (shortly) by the vectors $h_3$, $h_4$ or $h_5$. It remains to analyze the cases $u_2 \not\in \{\xi^1,\xi^2,\xi^3 \}$, and $u_3
\not\in \{\xi^2,\xi^3 \}$, that is, we still need to show that $\mathcal{H}$ is a short covering of the representatives $$(1,\xi^4,\xi^1), \ (1,\xi^4,\xi^5), \ (1,\xi^4,\xi^6), \
(1,\xi^5,\xi^1), \ (1,\xi^5,\xi^4), \ (1,\xi^5,\xi^6), \
(1,\xi^6,\xi^1), \ (1,\xi^6,\xi^4), \ (1,\xi^6,\xi^5).$$ From the equalities $$\xi^2(1,\xi^2,\xi^3) = (\xi^2,\xi^4,\xi^5), \ \mbox{ and } \
\xi^3(1,\xi^2,\xi^3) = (\xi^3,\xi^5,\xi^6),$$ the vector $h_4=(1,\xi^2,\xi^3)$ covers $(1,\xi^4,\xi^5)$, and $(1,\xi^5,\xi^6)$. The equality $$\xi^2(1,\xi^3,\xi^2) = (\xi^2,\xi^5,\xi^4)$$ implies that $h_5=(1,\xi^3,\xi^2)$ covers $(1,\xi^5,\xi^4)$. From $$\xi^1(\xi^6,\xi^5,1) = (1,\xi^6,\xi^1), \mbox{ and } \
\xi^6(\xi^6,\xi^5,1) = (\xi^5,\xi^4,\xi^6),$$ the vector $h_6= (\xi^6,\xi^5,1)$ covers $(1,\xi^4,\xi^1)$, $(1,\xi^6,\xi^1)$, $(1,\xi^6,\xi^4)$, $(1,\xi^6,\xi^5)$, $(1,\xi^5,\xi^1)$, and $(1,\xi^4,\xi^6)$.
Therefore $\mathcal{H}$ is a short covering of $\mathbb{F}_8^3$.
\[superior9\] We obtain $c(9) \leq 6$.
Consider the vectors $$\begin{array}{llllll}
h_1=(1,1,1), & h_2=(1,0,0), & h_3=(0,1,\xi^4),\\
h_4= (1,\xi^2,\xi^4), & h_5=(1,\xi^4,\xi^2), & h_6(1,\xi^6,\xi^6),
\end{array}$$ where $\xi$ denotes an generator of $\mathbb{F}_9^{*}$. Let us show that $\mathcal{H}=\{ h_1,\ldots,h_6\}$ is a short covering of $\mathbb{F}_9^3$.
Let $u$ be an arbitrary vector in $\mathbb{F}_9^3$. If $u \in
\mathbb{F}_9^3 \setminus \mathcal{D}_9$, then $u$ has at least two coincident coordinates, or zero appears at least in one coordinate of $u$. In the first case, $u$ is covered by $(1,1,1)$, and $u$ is covered by $(1,0,0)$ and $(0,1,\xi^4)$ in the second.
We suppose now $u=(u_1,u_2,u_3)\in \mathcal{D}_9$. Assume that $u=(1,u_2,u_3)$, where $u_2,u_3\in \{
\xi^1,\xi^2,\xi^3,\xi^4,\xi^5,\xi^6,\xi^7\}$. If $u_2\in
\{\xi^2,\xi^4,\xi^6 \}$ or $u_3 \in \{\xi^2,\xi^4,\xi^6 \}$, then $u$ is covered by the vectors $(1,\xi^2,\xi^4)$, $(1,\xi^4,\xi^2)$ or $(1,\xi^6,\xi^6)$.
We need to show that $\mathcal{H}$ is a short covering of the vectors below: $$\label{bloco9}
\begin{array}{llllllllllllll}
(1,\xi^1,\xi^3), & (1,\xi^1,\xi^5), & (1,\xi^1,\xi^7), &
(1,\xi^3,\xi^1), & (1,\xi^3,\xi^5), & (1,\xi^3,\xi^7), \\
(1,\xi^5,\xi^1), & (1,\xi^5,\xi^3), & (1,\xi^5,\xi^7), &
(1,\xi^7,\xi^1), & (1,\xi^7,\xi^3), & (1,\xi^7,\xi^5).
\end{array}$$ Note that $$\begin{array}{lllllllllllllll}
\xi^1(0,1,\xi^4) = (0,\xi^1,\xi^5), & \xi^3(0,1,\xi^4) =
(0,\xi^3,\xi^7) \\
\xi^5(0,1,\xi^4)=
(0,\xi^5,\xi^1), & \xi^7(0,1,\xi^4) = (0,\xi^7,\xi^3), &
\end{array}$$ thus $h_3=(0,1,\xi^4)$ covers $(1,\xi^1,\xi^5)$, $(1,\xi^3,\xi^7)$, $(1,\xi^5,\xi^1)$, and $(1, \xi^7, \xi^3)$. The equalities $$\begin{array}{llll}
\xi^1(1,\xi^2,\xi^4) = (\xi^1,\xi^3,\xi^5), & \xi^3(1,\xi^2,\xi^4) =(\xi^3,\xi^5,\xi^7), \\
\xi^5(1,\xi^2,\xi^4) = (\xi^5,\xi^7,\xi^1), & \xi^7(1,\xi^2,\xi^4) = (\xi^7,\xi^1,\xi^3)
\end{array}$$ imply that $h_4=(1,\xi^2,\xi^4)$ covers $(1,\xi^3,\xi^5)$, $(1,\xi^5,\xi^7)$, $(1,\xi^7,\xi^1)$, and $(1,\xi^1,\xi^3)$. Since $$\begin{array}{llllllllll}
\xi^1(1,\xi^4,\xi^2) = (\xi^1,\xi^5,\xi^3), & \xi^3(1,\xi^4,\xi^2)
=(\xi^3,\xi^7,\xi^5),\\
\xi^5(1,\xi^4,\xi^2)=(\xi^5,\xi^1,\xi^7) , & \xi^7(1,\xi^4,\xi^2) = (\xi^7,\xi^3,\xi^1),
\end{array}$$ the vector $h_5=(1,\xi^4,\xi^2)$ covers $(1,\xi^5,\xi^3)$, $(1,\xi^7,\xi^5)$, $(1,\xi^1,\xi^7)$ and $(1,\xi^3,\xi^1)$.
Because the vectors in (\[bloco9\]) are covered by $(0,1,\xi^4)$, $(1,\xi^2,\xi^4)$, or $(1,\xi^4,\xi^2),$ we conclude that $\mathcal{H}$ is a short covering of $\mathbb{F}_{9}^3$.
Conclusion
==========
We conclude this work with the following contribution to the computation of the function $c$.
It is an immediately consequence of the results in this work. In fact, the lower bound $c(7) \geq 5$ follows from Proposition \[inferior7\], while the upper bound $c(7) \leq 5$ follows from the inequality (\[bestinfsup\]). The value $c(8)=6$ is consequence of the Propositions \[inferior8\], and \[superior8\]. We obtain from Propositions \[inferior9\], and \[superior9\] that $c(9)=6$.
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[^1]: The first author is supported by Capes
[^2]: E-mail: [email protected]
[^3]: The second author is supported in part by CNPq.
[^4]: E-mail: [email protected]
|
---
abstract: 'We have experimentally demonstrated the absence of spectroscopic resonance shifts in a mixture of two interacting Fermi gases. This result is linked to observations in an ultracold gas of thermal bosons. There, the measured resonance shift due to interstate collisions is independent of the coherence in the system, and twice that expected from the equilibrium energy splitting between the two internal states in a fully decohered cloud. We give a simple theoretical explanation of these observations, which elucidates the effect of coherent radiation on an incoherent mixture of atoms.'
author:
- 'Martin W. Zwierlein, Zoran Hadzibabic, Subhadeep Gupta, and Wolfgang Ketterle'
date: 'June 24, 2003'
title: 'Spectroscopic insensitivity to cold collisions in a two-state mixture of fermions'
---
‘= ‘= ‘= ‘= ‘= ‘= ‘= ‘= ‘= ‘= ‘= ‘= ‘= ‘= ‘=
The coherence properties of light and matter are intimately connected with the quantum statistics of the constituent particles. One quantitative measure of the coherence in a system is the two-particle correlation function at zero distance, ${g^{(2)}}$, which measures the probability that two particles are simultaneously detected. Intensity fluctuations in the incoherent light emitted by a light bulb lead to photon “bunching", making this probability twice higher than in the coherent light of a laser. Identical fermions on the other hand exhibit “anti-bunching", making such a probability zero.
Interactions in ultracold atomic gases crucially depend on the value of ${g^{(2)}}$ [@kett97]. The reason is that s-wave scattering relies on particles overlapping in real space. The interaction energy in a many-body system is determined by coherent collisions, for which the outgoing and the incoming two-particle states are identical. Under this constraint, the two colliding particles can at most do two things - either preserve their momenta, or exchange them. We can thus distinguish four cases: (1) Two identical bosons in a thermal gas can collide in both ways, corresponding to ${g^{(2)}}=2$. (2) Two atoms in a Bose-Einstein condensate (BEC) have the same momenta and cannot undergo the exchange interaction. Here ${g^{(2)}}=1$. (3) Two distinguishable particles, fermions or bosons, also cannot exchange their momenta because that would make the outgoing state different from the incoming one. Again, ${g^{(2)}}=1$. (4) Two identical fermions cannot collide at all, so ${g^{(2)}}=0$. In all cases, the mean-field energy of a particle with mass $m$ is given by ${g^{(2)}}(4\pi \hbar^2 / m)
a n$, where $a$ is the s-wave scattering length, and $n$ is the density of atoms it interacts with.
Mean field energies and therefore ${g^{(2)}}$ can be measured spectroscopically. In experiments on ultracold hydrogen, mean-field shifts of the 1S-2S two-photon transition were used to prove the existence of a BEC [@frie98]. However, quantitative interpretation of the shifts led to a vivid theoretical discussion about the coherence related “factors of 2" [@cote99; @okte99; @okte02; @peth01]. More recently, Harber [*et al.*]{} performed Ramsey spectroscopy in a two-component, thermal gas of $^{87}$Rb bosons to measure ${g^{(2)}}$ in the interstate collisional shift [@harb02]. Their measurements yielded ${g^{(2)}}=2$, independent of the degree of coherence between the two states. The spectroscopic results thus seemed to correspond to the case of all particles being in an identical coherent superposition of the two internal states, even though the binary mixture was partially decohered and should have had a mean-field energy corresponding to $1<{g^{(2)}}<2$. The authors commented on this mystery [@harb02ICAP]: “it is a pleasure to note that a two-level system can still yield surprises, 75 years after the advent of quantum mechanics." The mystery can be formally resolved using a quantum Boltzmann equation [@okte02spin; @will02; @brad02; @fuch02; @fuch03].
Here, we experimentally address the relation between coherence and spectroscopic measurements in a binary mixture of ultracold [*fermions*]{}. We demonstrate that shifts of spectroscopic lines are absent even in a fully decohered binary mixture, in which the particles are distinguishable, and the many-body mean-field energy in the system has developed. We theoretically show that this is a direct consequence of the coherent nature of the RF [*excitation*]{}, which, in general, leads to a final state with ${g^{(2)}}$ different from the initial state.
Our calculation intuitively explains both our results for fermions, and the results for bosons of ref. [@harb02]. In a recent paper [@gupt03], we demonstrated the absence of mean-field “clock-shifts" in a coherent two-state superposition of [[$^6$Li]{}]{} fermions. In this case, RF spectroscopy was performed on a gas prepared purely in one internal state. Since an RF pulse acts as a rotation in the two-state Hilbert space, all the atoms stayed in an identical (superposition) state and could not interact. As long as the fermionic atoms were indistinguishable, ${g^{(2)}}=0$, and the resonance was thus found to be unperturbed at $\nu_0 = \frac{E_{12}}{h}$, where $E_{12}$ is the energy difference between the internal states $|1\rangle$ and $|2\rangle$.
However, once decoherence sets in, for example due to inhomogeneous magnetic fields across the cloud, the spatial overlap between atoms in different states grows and mean-field energy density builds up: $$\begin{aligned}
{\cal E}_{\rm int}({\bf r}) = {g^{(2)}}V_{12}\, n_1({\bf r})\, n_2({\bf r}),\quad V_{12} =
\frac{4\pi\hbar^2}{m}a_{12},\label{eq:meanfield}\end{aligned}$$ where $n_1$ and $n_2$ are the local densities of particles in states ${|1\rangle}$ and ${|2\rangle}$, and $a_{12}$ is the interstate s-wave scattering length. Here decoherence means that off-diagonal matrix elements of the density matrix have vanished locally. As a result, everywhere in the sample, atoms are no longer in one pure state, but occupy two orthogonal states, and s-wave collisions are no longer suppressed by the Pauli principle. In a fully decohered cloud, we have a binary mixture of two distinct species of atoms, with a mean-field energy density ${\cal E}_{\rm int} = V_{12} n_1 n_2$. This interaction changes the equilibrium energy level of atoms in state ${|1\rangle}$ (${|2\rangle}$) according to $\delta \mu_{1,2} = V_{12} n_{2,1}$. The difference in equilibrium mean-field energy of the two states is then $$\begin{aligned}
\Delta E_{\rm int} = \delta\mu_2 - \delta\mu_1 = {V_{12}}(n_1 - n_2).
\label{eq:energydiff}\end{aligned}$$ This suggests [@gupt03; @harb02; @harb02ICAP] that in a decohering sample, the resonant frequency for population transfer between the two states gradually changes from $\nu_{12} = \nu_0$ to $\nu_{12}
= \nu_0 + \frac{1}{h}\Delta E_{\rm int}$. Here, we show both experimentally and theoretically that this conclusion is wrong, and that the spectroscopic resonance frequency $\nu_{12}$ is always the unperturbed frequency $\nu_0$.
Our experimental setup was described in [@gupt03; @hadz03]. About $10^7$ fermionic [[$^6$Li]{}]{} atoms were confined in an optical dipole trap at a temperature of $35\, {\rm \mu K}$. The two-level system under consideration is formed by the two lowest ground state hyperfine levels, $|1\rangle$ and $|2\rangle$, corresponding to $|F,m_F\rangle = |1/2,1/2\rangle$ and $|1/2,-1/2\rangle$ in the low field basis, respectively. A DC magnetic field of $B = 320 \rm G$ was applied to the sample in order to tune the interstate scattering length $a_{12}$ to a large value of $\sim -300 a_0$, where $a_0$ is the Bohr radius [@gupt03].
We created a superposition of atoms in states ${|1\rangle}$ and ${|2\rangle}$ using a non-adiabatic RF sweep around the energy splitting of $74
\rm MHz$. As the sample decohered, efficient evaporative cooling set in, confirming a large elastic scattering length. After 1 second, we were left with a fully decohered mixture at a mean density $n =
5 \times 10^{13} {\rm cm}^{-3}$. The rate of the RF sweep was adjusted so that after decoherence and cooling, $80\%$ of the atoms were in state ${|2\rangle}$. The mean-field interaction should thus have increased the energy splitting of the two levels by $h \delta\nu = \delta\mu_2 -
\delta\mu_1 = {V_{12}}(n_1 - n_2) \approx h \times 10 \,{\rm kHz}$. Our experiments involving a third state [@gupt03] have confirmed the presence of such energy shifts, and prove that full decoherence has been reached.
Rabi spectroscopy in the interacting binary mixture was performed by applying $200 \mu {\rm s}$ RF pulses of different frequencies, and recording the final populations in the two states by simultaneous absorption imaging (Fig. \[fig:result\]). In order to eliminate the systematic uncertainty in the value of $\nu_0$, we performed a second experiment with the population ratios of states ${|1\rangle}$ and ${|2\rangle}$ reversed. According to Eq. \[eq:energydiff\], one would expect an opposite shift of the resonance.
Within our precision, no interaction shift of the resonance frequency was observed. Comparing the expected difference in mean-field shifts for the two experiments, $2 \delta
\nu = 20\, {\rm kHz}$, with the measured line separation of $(34\,\pm\, 146)\,\rm Hz$, we arrive at an apparent value for ${g^{(2)}}= 0.002(7)$. This demonstrates the universal absence of a resonance shift in a very cold two-level Fermi gas, independent of the coherence in the system.
![Absence of mean-field shift of an RF transition in a binary Fermi system. The resonance curves were measured for fully decohered 80%/20% two-state mixtures of fermions. The measured frequency difference between the two lines is $(34
\pm 146)\,$Hz, even though Eq. \[eq:energydiff\] would predict a splitting of 20 kHz.[]{data-label="fig:result"}](fig1.eps){width="0.8\columnwidth"}
Evidently, RF spectroscopy does not measure the expected difference in thermodynamic chemical potentials for the two states. Experiments with thermal bosons have posed a similar puzzle [@harb02]. Here we explain that this is a direct consequence of the coherent nature of the RF excitation.
In Fig. \[fig:bloch\], the average properties of the many-body state at a specific point ${\bf r}$ in the trap are described by the three coordinates of the local spin-1/2 Bloch vector ${\mathbf m}({\bf r}) = m_z({\bf r}) \hat{{\mathbf e}}_z + {\mathbf m}_\perp({\bf r})$. In the following, we omit the label ${\bf r}$. $m_z = \frac{n_2 - n_1}{2}$ represents the population difference in the two states, whereas the transverse component ${\mathbf m}_\perp$ is a measure of the coherence in the system. The length of the Bloch vector measures the purity of the mixture and hence the entropy of the system. Fully decohered statistical mixtures do not have off-diagonal matrix elements of the density matrix and are represented by vectors with ${\mathbf m}_\perp = 0$, with state A being the special case of a pure state. In Fig. \[fig:bloch\]a, state B is created by applying an RF pulse on a pure sample A. In this case, there is no interaction energy in the system during the RF pulse, and no frequency shift is expected [@gupt03]. State C is formed through subsequent decoherence of state B. States B and C have the same number of particles in ${|1\rangle}$ and ${|2\rangle}$, but in C the mean-field has fully developed.
Our experiment is performed on a C-like state (Fig. \[fig:bloch\]b). Here we explain why Eq. \[eq:energydiff\] still does not give the correct resonance frequency for an infinitesimal transfer of atoms between ${|1\rangle}$ and ${|2\rangle}$. The key point is that even though the sample is fully decohered, the applied RF pulse re-introduces coherence into the system. According to Eq. (6) below, this will change the value of ${g^{(2)}}$. Let us consider two fully decohered states, C and E. Eq. \[eq:energydiff\] correctly gives the energy of the transformation C$\rightarrow$E. However, these two states have different entropies, as indicated by Bloch vectors of different lengths. An RF pulse is a unitary transformation of the system, and must preserve entropy. The true effect of the RF pulse is thus to change the relative populations of ${|1\rangle}$ and ${|2\rangle}$ by tilting the Bloch vector away from the $z$ axis, into state D. It is the energy of [*this*]{} transformation, C$\rightarrow$D, that needs to be calculated in order to find the correct resonant RF frequency.
![Bloch sphere representation of RF transitions. a) An RF pulse rotates a pure state A into B. The superposition state decoheres into a “ring" distribution, represented by its average, C. b) A second RF pulse transforms the fully decohered state C into a partially coherent state D. The final state E is reached only after further decoherence. c) Transfers A$\rightarrow$B and C$\rightarrow$D are coherent and reversible. B$\rightarrow$C and D$\rightarrow$E are irreversible.[]{data-label="fig:bloch"}](fig2.eps){width="\columnwidth"}
In the case of fermions with short-range (delta function) interactions, we can prove very generally that the resonance frequency will always be $\nu_0$, by showing that the interaction hamiltonian is invariant under rotations of the Bloch vector. The interstate s-wave interaction at point ${\bf r}$ is described by the second-quantized hamiltonian density $$H_{\rm int}({\bf r}) = {V_{12}}\,\psi^\dagger_1({\bf r})\, \psi^\dagger_2({\bf r})\,
\psi^{ }_2({\bf r})\, \psi^{ }_1({\bf r}) . \label{eq:Hint}$$ Under a general rotation, described by polar angles $\theta,
\phi$, the field operators $\psi^\dagger_{1,2}$ transform according to: $$\begin{aligned}
\psi^\dagger_{1 \theta, \phi} &= &\cos{\mbox{$\frac{\theta}{2}$}} \; {\rm e}^{-{\rm i} \phi / 2} \;\psi^\dagger_1
+ \sin{\mbox{$\frac{\theta}{2}$}} \;{\rm e}^{{\rm i} \phi / 2} \;\psi^\dagger_2\nonumber\\
\psi^{\dagger}_{2 \theta, \phi} &= -
&\sin{\mbox{$\frac{\theta}{2}$}} \;{\rm e}^{-{\rm i} \phi / 2}\;
\psi^\dagger_1 + \cos{\mbox{$\frac{\theta}{2}$}} \;{\rm e}^{{\rm i} \phi / 2} \; \psi^\dagger_2
\label{eq:rotfield}\end{aligned}$$ Using the standard fermionic anticommutation relations ($\psi_1
\psi_2 = -\psi_2 \psi_1, \psi_1 \psi_1 = 0$ etc. ), it is easy to show that: $$\begin{aligned}
H^{\theta, \phi}_{\rm int} = {V_{12}}\psi^\dagger_{1 \theta, \phi} \psi^\dagger_{2 \theta, \phi} \psi^{ }_{2 \theta, \phi} \psi^{ }_{1 \theta, \phi} \stackrel{!}{=} H_{\rm int}
$$ We therefore see that an RF-induced rotation on the Bloch sphere commutes with the interaction hamiltonian, and hence does not change the energy of the many-body state. It is then obvious that the resonant frequency will always be $\nu_0$, independent of the coherence of the system.
We now present a more general calculation of the mean-field frequency shifts, which holds for both fermions and bosons. To reduce complexity and concentrate on the only controversial case of interstate interactions, we consider a fictitious boson with no intrastate interactions ($a_{11}=a_{22}=0$). The (local) mean-field expectation value of the hamiltonian density in Eq. \[eq:Hint\] is [@fett71] $$\begin{aligned}
\label{eq:Eint}
{\cal E}_{\rm int}({\bf r}) = \langle H_{\rm int} \rangle &=& {V_{12}}(n_{1} n_{2} + \epsilon \;n_{12} n_{21}),\nonumber\\
g^{(2)} &=& 1 + \epsilon \frac{n_{12} n_{21}}{n_1 n_2},\end{aligned}$$ where $n_1=\langle \psi_1^\dagger\, \psi^{ }_1 \rangle$ and $n_2=\langle \psi_2^\dagger\, \psi^{ }_2 \rangle$ are the local densities in the two states, we have introduced “coherences" $n_{12}=\langle \psi^\dagger_1\, \psi^{ }_2 \rangle$ and $n_{21}=\langle
\psi^\dagger_2\, \psi^{ }_1 \rangle$, and $\epsilon=\pm 1$ for bosons/fermions. In a fully coherent sample $n_{12}n_{21} = n_1
n_2$ and ${g^{(2)}}= 1+\epsilon$. As decoherence sets in, ${g^{(2)}}$ increases (decreases) from 0 (2) to 1 for fermions (bosons). For the most general case of a partially decohered sample, we can rewrite Eq. \[eq:Eint\] in terms of the (local) Bloch vector, using $n_{1,2} = \frac{n}{2} \mp m_z$, $n_{12} = m_x + i\, m_y = n^*_{21}$, and $n_{12} n_{21} = m^2_x + m^2_y = m^2_\perp$, where $n$ is the total particle density. This gives $$\begin{aligned}
{\cal E}_{\rm int} &=& {V_{12}}\frac{n^2}{4} + \epsilon {V_{12}}\left|{\mathbf m}\right|^2 - (1 + \epsilon) \,{V_{12}}\,m^2_z.\label{eq:EintBloch}\end{aligned}$$
Two samples with same numbers of atoms in states ${|1\rangle}$ and ${|2\rangle}$, but different levels of coherence, have the same $m_z$, but different $|{\mathbf m_{\perp}}|$ (e.g. states D and E in Fig. \[fig:bloch\]b). Again we see that two such samples indeed have different interaction energies.
Now, let us evaluate the effect of coherence on the resonant RF frequency. A coherent RF excitation preserves entropy ($|{\mathbf m}|={\rm const.}$), and the total density $n$. In an infinitesimal tilt of the Bloch vector, the density of atoms transferred from ${|1\rangle}$ to ${|2\rangle}$ is $d n_2 = - d n_1 = d m_z$. Therefore, the change of interaction energy per transferred particle, and thus the shift in the resonant frequency $\Delta \nu$, comes out to be $$\begin{aligned}
\Delta \nu = \frac{1}{h}\frac{\partial {\cal E}_{\rm int}}{\partial m_z}\bigg|_{n,\,|{\mathbf m}|} = \frac{1}{h}
(1+\epsilon){V_{12}}\,\; (n_1 - n_2) \label{freqshift}.\end{aligned}$$
In analogy with a spinning top which precesses in the gravitational field, the resonant frequency for an infinitesimal tilt of the Bloch vector is also equal to the frequency of its free precession. In the traditional language of atomic physics, this analogy just reiterates that Rabi [@gupt03] and Ramsey [@harb02] spectroscopy measure the same characteristic frequency of the system. The striking result is that in contrast to the interaction energy (Eqs. \[eq:Eint\], \[eq:EintBloch\]), the precession of the Bloch vector, or equivalently the RF frequency shift (Eq. \[freqshift\]), [*does not*]{} depend on the level of coherence in the sample. Remarkably, the final state may have a value of ${g^{(2)}}$ different from the initial state, such that the energy difference per transferred particle is [*independent*]{} of the initial ${g^{(2)}}$. Equation \[freqshift\] explains both our measurements with fermions, and the experiment with thermal bosons of ref. [@harb02].
![Mean-field represented as effective magnetic field. a) Fermions: The exchange and direct interaction add up to form a magnetic field aligned with the average spin (${V_{12}}<0$ in the drawing). The net torque vanishes and the Bloch vector ${\mathbf m}$ precesses at the unperturbed frequency $\nu_0$. b) Bosons: The exchange interaction has opposite sign to that in fermions. It exerts a torque on the average spin equal to the torque induced by the direct interaction, as can be seen by comparing the two cross products with ${\mathbf m}$. The Bloch vector thus precesses at $\nu_0$ plus twice the frequency shift due to direct interaction.[]{data-label="fig:meanfield"}](fig3.eps){width="0.8\columnwidth"}
In order to further elucidate the role of coherences in the precession of the Bloch vector, we employ the interpretation of the mean-field energy as the interaction of the average spin with an effective magnetic field [@fuch02; @fuch03]. Using Eq. \[eq:EintBloch\], we obtain ${\cal E}_{\rm int} = {\rm const.} - \frac{1}{2}{\mathbf B}_{\rm eff}\cdot{\mathbf m}$ [@selfint] with $$\begin{aligned}
{\mathbf B}_{\rm eff} = 2 {V_{12}}\, (m_z \hat{{\mathbf e}}_z - \epsilon\, {\mathbf m}_\perp) \label{eq:Bint}.\end{aligned}$$ In this picture, the precession of the spin due to interactions is driven by the torque ${\mathbf B}_{\rm eff} \times {\mathbf m}$. The magnetic field along the $z$ axis is induced by the direct interaction, and has the same sign for fermions and bosons (Fig. \[fig:meanfield\]). The transverse magnetic field comes from the exchange interaction, and has opposite signs for fermions and bosons. For fermions, ${\mathbf B}_{\rm eff}$ is parallel to ${\mathbf m}$ (Eq. \[eq:Bint\]) and hence does not cause any precession. Equivalently, the direct and exchange interaction exert torques equal and opposite to each other. For bosons, the two contributions add up to yield exactly twice the precession frequency given by the direct interaction alone. During decoherence, the exerted torque shrinks in proportion to the decaying transverse spin. Therefore, the precession frequency remains constant, no matter how small the coherences are.
In conclusion, we have demonstrated the absence of the mean-field shift of RF transitions in a fully decohered, interacting binary mixture of fermions. This was explained by proving the invariance of the interaction energy under coherent Hilbert space rotations. This result is relevant for the potential use of a fermionic atom supplying the frequency standard in an atomic or optical clock, since it implies a robust elimination of the systematic errors due to density dependent frequency shifts. Previously, the absence of such clock shifts was explained by the absence of mean-field energy in a purely coherent superposition state [@gupt03]. Now we have shown that there is no spectroscopic shift even after decoherence has led to measurable mean-field energies. Further, we have presented a simple theoretical framework for calculating the precession frequency of the Bloch vector which describes an arbitrary spin state of either fermions or bosons. This resolves “The Mystery of the Ramsey Fringe that Didn’t Chirp" [@harb02ICAP] with a simple and intuitive picture.
We thank Claudiu Stan and Christian Schunck for experimental assistance, and Michele Saba for critical reading of the manuscript. This work was supported by the NSF, ONR, ARO, and NASA.
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---
abstract: 'We propose a new approach to study the Kronecker coefficients by using the Schur-Weyl duality between the symmetric group and the partition algebra. We explain the limiting behavior and associated bounds in the context of the partition algebra. Our analysis leads to a uniform description of the Kronecker coefficients when one of the indexing partitions is a hook or a two-part partition.'
address:
- 'Institut de Mathématiques de Jussieu, 5 rue du Thomas Mann, 75013, Paris, France'
- 'Centre for Mathematical Science, City University London, Northampton Square, London, EC1V 0HB, England.'
- 'Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755, USA '
author:
- 'C. Bowman'
- 'M. De Visscher'
- 'R. Orellana'
bibliography:
- 'book3.bib'
title: |
The partition algebra and the\
Kronecker coefficients
---
Introduction {#introduction .unnumbered}
============
A fundamental problem in the representation theory of the symmetric group is to describe the coefficients in the decomposition of the tensor product of two Specht modules. These coefficients are known in the literature as the [*Kronecker coefficients*]{}. They are labelled by triples of partitions. Finding a formula or combinatorial interpretation for these coefficients has been described by Richard Stanley as ‘one of the main problems in the combinatorial representation theory of the symmetric group’. This question has received the attention of Littlewood [@littlewood], James [@jk Chapter 2.9], Lascoux [@Lascoux], Thibon [@Thibon], Garsia and Remmel [@GR], Kleshchev and Bessenrodt [@BK] amongst others and yet a combinatorial solution has remained beyond reach for over a hundred years.
Murnaghan discovered an amazing limiting phenomenon satisfied by the Kronecker coefficients; as we increase the length of the first row of the indexing partitions the sequence of Kronecker coefficients obtained stabilises. The limits of these sequences are known as the [*reduced Kronecker coefficients*]{}.
The novel idea of this paper is to study the Kronecker and reduced coefficients through the Schur–Weyl duality between the symmetric group, $\mathfrak{S}_n$, and the partition algebra, $P_r(n)$. The key observation being that the tensor product of Specht modules corresponds to the restriction of simple modules in $P_r(n)$ to a Young subalgebra. The combinatorics underlying the representation theory of both objects is based on partitions. The duality results in a Schur functor, ${\rm F} : \mathfrak{S}_n{\text{-mod}} \to P_r(n){\text{-mod}}, $ which acts by *first row removal* on the partitions labelling the simple modules. We exploit this functor along with the following three key facts concerning the representation theory of the partition algebra: (a) it is semisimple for large $n$ (b) it has a stratification by symmetric groups (c) its non-semisimple representation theory is well developed.
We interpret the Kronecker and reduced Kronecker coefficients and the passage between them in terms of the representation theory of the partition algebra. The limiting phenomenon discovered by Murnaghan and some associated bounds (due to Brion) are then naturally explained by the fact that $P_r(n)$ is semisimple for large enough $n$.
Closed formulas for Kronecker coefficients have only been obtained for triples of partitions with (i) one 2-part partition (ii) two hook partitions and (iii) a hook and a 2-part partition. We give a unified simple approach which covers all these cases and generalises further to triples with one hook partition.
Our approach brings forward a general tool to study these coefficients and provides a natural framework for the study of the outstanding problems in the area. In particular, one should notice that our proofs are surprisingly elementary.
The paper is organised as follows. In Sections 1 and 2 we recall the combinatorics underlying the representation theories of the symmetric group and partition algebra. In Section 3 we show how to pass the Kronecker problem through Schur–Weyl duality and phrase it as a question concerning the partition algebra. We then summarise results concerning the Kronecker and reduced Kronecker coefficients that have a natural interpretation in this setting. Section 4 contains a description of the restriction of a standard module for $P_r(n)$ to a Young subalgebra, giving a new representation theoretic interpretation of [@rosa Lemma 2.1]. In Section 5 we specialise to hook and two-part partitions and obtain closed positive formulas in these cases. Section 6 contains an extended example.
Symmetric group combinatorics
=============================
The combinatorics underlying the representation theory of the symmetric group, $\mathfrak{S}_n$, is based on partitions. A *partition* $\lambda$ of $n$, denoted $\lambda\vdash n$, is defined to be a weakly decreasing sequence $\lambda=(\lambda_1,\lambda_2,\dots,\lambda_\ell)$ of non-negative integers such that the sum $|\lambda|=\lambda_1+\lambda_2+\dots +\lambda_\ell$ equals $n$. The [*length*]{} of a partition is the number of nonzero parts, we denote this by $\ell(\lambda)$. We let $\Lambda_n$ denote the set of all partitions of $n$.
With a partition, $\lambda$, is associated its *Young diagram*, which is the set of nodes $$[\lambda]=\left\{(i,j)\in{{\mathbb Z}}_{>0}^2\ \left|\ j\leq \lambda_i\right.\right\}.$$ Given a node specified by $i,j\geq1$, we say the node has *content* $j-i$. We let ${\operatorname{ct}}(\lambda_i)$ denote the content of the last node in the $i$th row of $[\lambda]$, that is ${\operatorname{ct}}(\lambda_i)=\lambda_i-i$.
Over the complex numbers, the irreducible *Specht* modules, ${\textbf{\rm \textbf{S}}}(\lambda)$, of $\mathfrak{S}_n$ are indexed by the partitions, $\lambda$, of $ n$. An explicit construction of these modules is given in [@jk].
The classical Littlewood–Richardson rule
----------------------------------------
The Littlewood–Richardson rule is a combinatorial description of the restriction of a Specht module to a Young subgroup of the symmetric group. Through Schur–Weyl duality, the rule also computes the decomposition of a tensor product of two simple modules of ${\mathrm{GL}}_n(\mathbb{C})$. The Littlewood–Richardson rule is the most famous algorithm for decomposing tensor products and has been generalised in several directions.
The following is a simple restatement of this rule as it appears in [@jk Section 2.8.13].
For $\lambda\vdash r_1$, $\mu\vdash r_2$ and $\nu \vdash r_1+r_2$, $$\begin{aligned}
{\textbf{\rm \textbf{S}}}(\nu) \!\!\downarrow_{\mathfrak{S}_{r_1} \times \mathfrak{S}_{r_2}}^{\mathfrak{S}_{r_1+r_2}} \cong \bigoplus_{ \lambda\vdash r_1, \mu\vdash r_2}c^\nu_{\lambda, \mu} {\textbf{\rm \textbf{S}}}(\lambda)\boxtimes{\textbf{\rm \textbf{S}}}(\mu)\end{aligned}$$ where the $c^\nu_{\lambda, \mu}$ are the Littlewood–Richardson coefficients (defined below).
The Littlewood–Richardson coefficient $c^\nu_{\lambda, \mu}$ is zero, unless $\lambda \subseteq \nu$, and, otherwise may be calculated as follows. For each node $(i,j)$ of $\mu$, take a symbol $u_{i,j}$. Begin with the diagram $\lambda$ and:
1. Add to it all symbols $u_{1,j}$ (corresponding to the first row of nodes of $\mu$) in such a way as to produce the diagram of a partition and to satisfy (3).
2. Next add all symbols $u_{2,j}$ (corresponding to the second row of nodes of $\mu$) following the same rules. Continue this process with all rows of $\mu$.
3. The added symbols must satisfy: $(a)$ for all $i$, if $y <j$, $u_{i,y}$ is in a later column than $u_{i,j}$; and $(b)$ for all $j$, if $x<i$ $u_{x,j}$ is in an earlier row than $u_{i,j}$.
By transitivity of induction we have that the Littlewood–Richardson rule determines the structure of the restriction of a Specht module to any Young subgroup. Of particular importance in this paper is the three-part case $$\begin{aligned}
{\textbf{\rm \textbf{S}}}(\nu)\!\! \downarrow_{\mathfrak{S}_{r_1} \times \mathfrak{S}_{r_2}\times\mathfrak{S}_{r_3} }^{\mathfrak{S}_{r_1+r_2+r_3}}
&\cong\bigoplus_{\begin{subarray}{c} {{\xi\vdash r_1+r_2} }
\\ {{\nu\vdash r_3} } \end{subarray}}
( c^\nu_{\xi, \eta}{\textbf{\rm \textbf{S}}}(\xi)\boxtimes{\textbf{\rm \textbf{S}}}(\eta))\!\!\downarrow^{\mathfrak{S}_{r_1+r_2} \times \mathfrak{S}_{r_3}}_{\mathfrak{S}_{r_1}\times\mathfrak{S}_{r_2}\times\mathfrak{S}_{r_3}\ }\\
&
\cong\bigoplus_{\begin{subarray}{c} {{\lambda\vdash r_1, } }
\\ {{\mu\vdash r_2 } }, {{\nu\vdash r_3} } \end{subarray}}
\left( \sum_{\xi\vdash r_1+r_2}c^\xi_{\lambda,\mu} c^\nu_{\xi,\eta}\right){\textbf{\rm \textbf{S}}}(\lambda)\boxtimes{\textbf{\rm \textbf{S}}}(\mu)\boxtimes{\textbf{\rm \textbf{S}}}(\eta).\end{aligned}$$ We therefore set $c^\nu_{\lambda,\mu,\eta}=\sum_{\xi}c^\xi_{\lambda,\mu} c^\nu_{\xi,\eta}$.
Tensor products of Specht modules of the symmetric group {#murg}
--------------------------------------------------------
In this section we define the Kronecker coefficients and the [reduced Kronecker coefficients]{} as well as set some notation. Let $\lambda$ and $\mu$ be two partitions of $n$, then $${\textbf{\rm \textbf{S}}}(\lambda)\otimes {\textbf{\rm \textbf{S}}}(\mu) = \bigoplus_{\nu\vdash n} g_{\lambda,\mu}^\nu {\textbf{\rm \textbf{S}}}(\nu),$$ the coefficients $g_{\lambda , \mu}^\nu$ are known as the [*Kronecker coefficients*]{}. These coefficients satisfy an amazing stability property illustrated in the following example.
We have the following tensor products of Specht modules: $$\begin{aligned}
{\textbf{\rm \textbf{S}}}(1^2) \otimes {\textbf{\rm \textbf{S}}}(1^2) &= {\textbf{\rm \textbf{S}}}(2) \\
{\textbf{\rm \textbf{S}}}(2,1) \otimes {\textbf{\rm \textbf{S}}}(2,1) &= {\textbf{\rm \textbf{S}}}(3) \oplus {\textbf{\rm \textbf{S}}}(2,1) \oplus {\textbf{\rm \textbf{S}}}(1^3) \\
{\textbf{\rm \textbf{S}}}(3,1) \otimes {\textbf{\rm \textbf{S}}}(3,1) &= {\textbf{\rm \textbf{S}}}(4) \oplus {\textbf{\rm \textbf{S}}}(3,1)\oplus {\textbf{\rm \textbf{S}}}(2,1^2) \oplus {\textbf{\rm \textbf{S}}}(2^2)
\intertext{at which point the product stabilises, i.e. for all $n\geq4$, we have}
{\textbf{\rm \textbf{S}}}(n-1,1) \otimes {\textbf{\rm \textbf{S}}}(n-1,1) &= {\textbf{\rm \textbf{S}}}(n) \oplus {\textbf{\rm \textbf{S}}}(n-1,1)\oplus {\textbf{\rm \textbf{S}}}(n-2,1^2) \oplus {\textbf{\rm \textbf{S}}}(n-2,2).
$$
Let $\lambda = (\lambda_1,\lambda_2, \ldots,\lambda_{\ell} )$ be a partition and $n$ be an integer, define $\lambda_{[n]}=(n-|\lambda|, \lambda_1,\lambda_2, \ldots,\lambda_{\ell})$. Note that all partitions of $n$ can be written in this form.
For $\lambda_{[n]}, \mu_{[n]}, \nu_{[n]} \in \Lambda_n$ we let $$g_{\lambda_{[n]},\mu_{[n]}}^{\nu_{[n]}} = \dim_{\mathbb{C}}({\operatorname{Hom}}_{\mathfrak{S}_n}({\textbf{\rm \textbf{S}}}(\lambda_{[n]}) \otimes {\textbf{\rm \textbf{S}}}(\mu_{[n]}) , {\textbf{\rm \textbf{S}}}(\nu_{[n]}))),$$ denote the multiplicity of ${\textbf{\rm \textbf{S}}}(\nu_{[n]})$ in the tensor product ${\textbf{\rm \textbf{S}}}(\lambda_{[n]}) \otimes {\textbf{\rm \textbf{S}}}(\mu_{[n]})$. Murnaghan showed (see [@murn; @murn2]) that if we allow the first parts of the partitions to increase in length then we obtain a limiting behaviour as follows. For $\lambda_{[N]}, \mu_{[N]}, \nu_{[N]} \in \Lambda_N$ and $N$ sufficiently large we have that $$g_{\lambda_{[N+k]},\mu_{[N+k]}}^{\nu_{[N+k]}}=\overline{g}_{\lambda,\mu}^\nu$$ for all $k\geq 1$; the integers $\overline{g}_{\lambda,\mu}^\nu$ are called the *reduced Kronecker coefficients*. Bounds for this stability have been given in [@brion; @vallejo; @klyachko; @rosa].
The reduced Kronecker coefficients are also the structural constants for a linear basis for the polynomials in countably many variables known as the *character polynomials*, see [@mac].
The partition algebra
=====================
The partition algebra was originally defined by Martin in [@marbook]. All the results in this section are due to Martin and his collaborators, see [@mar1] and references therein.
Definitions
-----------
For $r\in \mathbb{Z}_{>0}$, $\delta\in\mathbb{C}$, we let $P_r(\delta)$ denote the complex vector space with basis given by all set-partitions of $\{1,2,\ldots, r, \bar{1},\bar{2}, \ldots, \bar{r}\}.$ A part of a set-partition is called a *block*. For example, $$d=\{\{1, 2, 4, \bar{2}, \bar{5}\}, \{3\}, \{5, 6, 7, \bar{3}, \bar{4}, \bar{6}, \bar{7}\}, \{8, \bar{8}\}, \{\bar{1}\}\},$$ is a set-partition (for $r=8$) with 5 blocks.
A set-partition can be represented *uniquely* by an $(r,r)$-*partition diagram* consisting of a frame with $r$ distinguished points on the northern and southern boundaries, which we call vertices. We number the northern vertices from left to right by $1,2,\ldots, r$ and the southern vertices similarly by $\bar{1},\bar{2},\ldots, \bar{r}$. Any block in a set-partition is of the form $A\cup B$ where $A=\{i_1<i_2<\ldots <i_p\}$ and $B=\{\bar{j_1}<\bar{j_2}<\ldots <\bar{j_q}\}$ (and $A$ or $B$ could be empty). We draw this block by putting an arc joining each pair $(i_l, i_{l+1})$ and $(\bar{j_l}, \bar{j }_{l+1})$ and if $A$ and $B$ are non-empty we draw a strand from $i_1$ to $\bar{j_1}$, that is we draw a single propagating line on the leftmost vertices of the block. Blocks containing a northern and a southern vertex will be called *propagating blocks*; all other blocks will be called *non-propagating blocks*. For $d$ as in the example above, the [partition diagram]{} of $d$ is given by: $$\begin{tikzpicture}[scale=0.6]
\draw (0,0) rectangle (8,3);
\foreach \x in {0.5,1.5,...,7.5}
{\fill (\x,3) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope} \draw (0.5,3) -- (1.5,0);
\draw (7.5,3) -- (7.5,0);
\draw (4.5,3) -- (2.5,0);
\draw (0.5,3) arc (180:360:0.5 and 0.25);
\draw (1.5,3) arc (180:360:1 and 0.25);
\draw (4.5,0) arc (0:180:1.5 and 1);
\draw (5.5,0) arc (0:180:1 and .7);
\draw (3.5,0) arc (0:180:.5 and .25);
\draw (6.5,0) arc (0:180:0.5 and 0.5);
\draw (4.5,3) arc (180:360:0.5 and 0.25);
\draw (5.5,3) arc (180:360:0.5 and 0.25);
\end{scope}
\end{tikzpicture}$$
We can generalise this definition to $(r,m)$-partition diagrams as diagrams representing set-partitions of $\{1, \ldots , r, \bar{1}, \ldots, \bar{m}\}$ in the obvious way.
We define the product $x \cdot y$ of two diagrams $x$ and $y$ using the concatenation of $x$ above $y$, where we identify the southern vertices of $x$ with the northern vertices of $y$. If there are $t$ connected components consisting only of middle vertices, then the product is set equal to $\delta^t$ times the diagram with the middle components removed. Extending this by linearity defines a multiplication on $P_r(\delta)$.
**Assumption:** *We assume throughout the paper that $\delta\neq 0$.*
The following elements of the partition algebra will be of importance. $$\begin{aligned}
s_{i,j}=\begin{minipage}{34mm}\begin{tikzpicture}[scale=0.5]
\draw (0,0) rectangle (6,3);
\foreach \x in {0.5,1.5,...,5.5}
{\fill (\x,3) circle (2pt);
\fill (\x,0) circle (2pt);}
\draw (4.5,3.5) node {$j$};
\draw (4.5,-0.5) node {$\overline{j}$};
\draw (1.5,3.5) node {$i$};
\draw (1.5,-0.5) node {$\overline{i}$};
\begin{scope} \draw (0.5,3) -- (0.5,0);
\draw (5.5,3) -- (5.5,0);
\draw (1.5,3) -- (4.5,0);
\draw (4.5,3) -- (1.5,0);
\draw (2.5,3) -- (2.5,0);
\draw (3.5,3) -- (3.5,0);
\end{scope}
\end{tikzpicture}\end{minipage} \quad
e_{l}= \frac{1}{\delta}\begin{minipage}{34mm} \ \begin{tikzpicture}[scale=0.5]
\draw (0,0) rectangle (6,3);
\foreach \x in {0.5,1.5,...,5.5}
{\fill (\x,3) circle (2pt);
\fill (\x,0) circle (2pt);}
\draw (2.5,3.5) node {$l$};
\draw (2.5,-.5) node {$\overline{l}$};
\begin{scope} \draw (0.5,3) -- (0.5,0);
\draw (3.5,0) arc (0:180:.5 and .5);
\draw (4.5,0) arc (0:180:.5 and .5);
\draw (5.5,0) arc (0:180:.5 and .5);
\draw (4.5,3) arc (180:360:0.5 and 0.5);
\draw (3.5,3) arc (180:360:0.5 and 0.5);
\draw (2.5,3) arc (180:360:0.5 and 0.5);
\draw (1.5,3) -- (1.5,0);
\end{scope}
\end{tikzpicture}\end{minipage}\end{aligned}$$
In particular, note that $e_r$ is the idempotent corresponding to the set-partition $\{1,\bar{1}\}\{2,\bar{2}\}\cdots \{r-1,\overline{r-1}\}\{r\}\{\bar{r}\}$.
Filtration by propagating blocks and standard modules
-----------------------------------------------------
Fix $\delta\in\mathbb{C}^{\times}$ and write $P_r=P_r(\delta)$. Note that the multiplication in $P_r$ cannot increase the number of propagating blocks. More precisely, if $x$, respectively $y$, is a partition diagram with $p_x$, respectively $p_y$, propagating blocks then $x \cdot y$ is equal to $\delta^t z$ for some $t\geq0$ and some partition diagram $z$ with $p_z\leq \min\{p_x, p_y\}$. This gives a filtration of the algebra $P_r$ by the number of propagating blocks. This filtration can be realised using the idempotents $e_l$. We have $$\mathbb{C}\cong P_r e_1 P_r \subset \ldots \subset P_re_{r-1}P_r \subset P_re_rP_r \subset P_r.$$ It is easy to see that $$\label{bob}
e_rP_r e_r \cong P_{r-1}$$ and that this generalises to $P_{r-l}\cong e_{r-l+1}P_re_{r-l+1}$ for $1\leq l\leq r$. We also have $$\label{bob2} P_r /(P_r e_rP_r )\cong \mathbb{C}\mathfrak{S}_r.$$ Using equation (\[bob2\]), we get that any $\mathbb{C}\mathfrak{S}_r$-module can be *inflated* to a $P_r$-module. We also get from equations (\[bob\]) and (\[bob2\]), by induction, that the simple $P_r$-modules are indexed by the set $\Lambda_{\leq r}=\bigcup_{0\leq i\leq r}\Lambda_i$.
For any $\nu \in \Lambda_{\leq r}$ with $\nu \vdash r-l$, we define a $P_r$-module, $\Delta_r(\nu)$, by $$\Delta_r(\nu)=P_re_{r-l+1} \otimes_{P_{r-l}}{\textbf{\rm \textbf{S}}}(\nu).$$ (Here we have identified $P_{r-l}$ with $e_{r-l+1}P_re_{r-l+1}$ using the isomorphism given in equation (\[bob\]).)
For $\delta \not\in \{0,1,\ldots, 2r-2\}$ the algebra $P_r(\delta)$ is semisimple and the set $\{\Delta_r(\nu):\nu \in \Lambda_{\leq r}\}$ forms a complete set of non-isomorphic simple modules.
In general, the algebra $P_r(\delta)$ is quasi-hereditary with respect to the partial order on $\Lambda_{\leq r}$ given by $\lambda < \mu$ if $|\lambda| > |\mu|$ (see [@mar1]). The modules $\Delta_r(\nu)$ are the standard modules, each of which has a simple head $L_r(\nu)$, and the set $\{L_r(\nu): \nu\in \Lambda_{\leq r}\}$ forms a complete set of non-isomorphic simple modules.
We now give an explicit description of the standard modules. We set $V(r,r-l)$ to be the span of all $(r,r-l)$-partition diagrams having precisely $(r-l)$ propagating blocks. This has a natural structure of a $(P_r(\delta), \mathfrak{S}_{r-l})$-bimodule. It is easy to see that, as vector spaces, we have $$\Delta_r(\nu) \cong V(r,r-l)\otimes_{\mathfrak{S}_{r-l}} {\textbf{\rm \textbf{S}}}(\nu).$$ The action of $P_r(\delta)$ is given as follows. Let $v$ be a partition diagram in $V(r,r-l)$, $x\in {\textbf{\rm \textbf{S}}}(\nu)$ and $X$ be an $(r,r)$-partition diagram. Concatenate $X$ and $v$ to get $\delta^t v'$ for some $(r,r-l)$-partition diagram $v'$ and some non-negative integer $t$. If $v'$ has fewer than $(r-l)$ propagating blocks then we set $X(v\otimes x)=0$. Otherwise we set $X(v\otimes x)=\delta^t v'\otimes x$. Note that in this case we have $v'=v''\sigma$ for a unique $v''\in V(r,r-l)$ with non-crossing propagating lines and a unique $\sigma\in \mathfrak{S}_{r-l}$ and we have $\delta^{t}v'\otimes x = \delta^tv''\otimes \sigma x$.
Non-semisimple representation theory of the partition algebra {#comb}
-------------------------------------------------------------
We assume that $\delta=n\in\mathbb{Z}_{>0}$ (as otherwise the algebra is semisimple).
Let $\mu\subset \lambda$ be partitions. We say that $(\mu,\lambda)$ is an $n$-pair, and write $\mu\hookrightarrow_n \lambda$, if the Young diagram of $\lambda$ differs from the Young diagram of $\mu$ by a horizontal row of boxes of which the last (rightmost) one has content $n-|\mu|$.
For example, $((2,1), (4,1))$ is a $6$-pair. We have that $6-|\mu|=3$ and the Young diagrams (with contents) are as follows: $$\young(01,{$-1$}) \ \subset \ \young(0123,{$-1$})$$ note that they differ by $\young(23)$.
Recall that the set of simple (or standard) modules for $P_r(n)$ are labelled by the set $\Lambda_{\leq r}$. This set splits into $P_r(n)$-blocks. The set of labels in each block forms a maximal chain of $n$-pairs $$\lambda^{(0)}\hookrightarrow_n \lambda^{(1)} \hookrightarrow_n \lambda^{(2)} \hookrightarrow_n \ldots \hookrightarrow_n \lambda^{(t)}.$$ Moreover, for $1\leq i\leq t$ we have that $\lambda^{(i)}/\lambda^{(i-1)}$ consists of a strip of boxes in the $i$th row. Now we have an exact sequence of $P_r(n)$-modules $$0\rightarrow \Delta_r( \lambda^{(t)})\rightarrow \ldots \rightarrow \Delta_r(\lambda^{(2)})\rightarrow \Delta_r(\lambda^{(1)})\rightarrow \Delta_r(\lambda^{(0)})\rightarrow L_r(\lambda^{(0)})\rightarrow 0$$ with the image of each homomorphism being simple. Each standard module $\Delta_r(\lambda^{(i)})$ (for $0\leq i \leq t-1$) has Loewy structure $$\begin{array}{c} L_r(\lambda^{(i)})\\ L_r(\lambda^{(i+1)}) \end{array}$$ and so in the Grothendieck group we have $$\label{alt}
[L_r(\lambda^{(i)})]=\sum_{j=i}^{t} (-1)^{j-i} [\Delta_r(\lambda^{(j)})].$$ Note that each block is totally ordered by the size of the partitions.
\[singular\] Let $\nu\in \Lambda_{\leq r}$ and assume that $\nu_{[n]}$ is a partition. Then we have that (i) $\nu$ is the minimal element in its $P_r(n)$-block, and\
(ii) $\nu$ is the unique element in its block if and only if $n+1-\nu_1 > r$.
\(i) Observe that for $\nu_{[n]}$ to be a partition we must have $n-|\nu|\geq \nu_1$. This implies that ${\operatorname{ct}}(\nu_1)=\nu_1 -1\leq n-|\nu| -1$. So we have $\nu \hookrightarrow_n \mu$ for some partition $\mu$ with $\mu/\nu$ being a single strip in the first row. Thus we have $\nu=\nu^{(0)}$ and $\mu=\nu^{(1)}$.\
(ii) Now as $\nu^{(1)}/\nu$ is a single strip in the first row with last box having content $n-|\nu|$, we have that $|\nu^{(1)}/\nu|=n-|\nu|+1-\nu_1$ and thus $|\nu^{(1)}|=n+1-\nu_1$. Thus if $n+1-\nu_1>r$ then $\nu^{(1)}\notin \Lambda_{\leq r}$ and we have that $\nu$ is the only partition in its $P_r(n)$-block.
Schur–Weyl duality
===================
Classical Schur–Weyl duality is the relationship between the general linear and symmetric groups over tensor space. To be more specific, let $V_n$ be an $n$-dimensional complex vector space and let $V^{\otimes r}_n$ denote its $r$th tensor power.
We have that the symmetric group $\mathfrak{S}_r$ acts on the right by permuting the factors. The general linear group, ${\mathrm{GL}}_n$, acts on the left by matrix multiplication on each factor. These two actions commute and moreover ${\mathrm{GL}}_n$ and $\mathfrak{S}_r$ are full mutual centralisers in ${\operatorname{End}}(V^{\otimes r}_n)$.
The partition algebra, $P_r(n)$, plays the role of the symmetric group, $\mathfrak{S}_r$, when we restrict the action of ${\mathrm{GL}}_n$ to the subgroup of permutation matrices, $\mathfrak{S}_n$.
Schur-Weyl duality between $\mathfrak{S}_n$ and $P_r(n)$
--------------------------------------------------------
Let $V_n$ denote an $n$-dimensional complex space. Then $\mathfrak{S}_n$ acts on $V_n$ via the permutation matrices. $$\label{Vaction}
\sigma\cdot v_i = v_{\sigma(i)}\qquad \mbox{ for } \sigma \in \mathfrak{S}_n.$$ Notice that we are simply restricting the ${\mathrm{GL}}_n$ action in the classical Schur-Weyl duality to the permutation matrices. Thus, $\mathfrak{S}_n$ acts diagonally on the basis of simple tensors of $V_n^{\otimes r}$ as follows $$\sigma\cdot (v_{i_1}\otimes v_{i_2} \otimes \cdots \otimes v_{i_r}) = v_{\sigma(i_1)}\otimes v_{\sigma(i_2)} \otimes \cdots \otimes v_{\sigma(i_r)}.$$
For each $(r,r)$-partition diagram $d$ and each integer sequence $i_1\ldots, i_r, i_{\bar 1}, \ldots, i_{\bar r}$ with $1\leq i_j, i_{\bar j}\leq n$, define $$\label{eq:diagrammatrix}
\phi_{r,n}(d)_{i_{\bar 1}, \ldots, i_{\bar r}}^{i_1,\ldots, i_r} = \begin{cases} 1 & \mbox{ if $i_t= i_s$ whenever vertices $t$ and $s$ are connected in $d$} \\ 0 & \mbox{ otherwise.}
\end{cases}$$ A partition diagram $d\in P_r(n)$ acts on the basis of simple tensors of $V_n^{\otimes r}$ as follows $$\Phi_{r,n}(d) (v_{i_{1}} \otimes v_{i_{2}} \otimes \cdots \otimes v_{i_{r}}) = \sum_{i_{\bar 1}, \ldots i_{\bar r}} \phi_{r,n}(d)_{i_{\bar 1}, \ldots, i_{\bar r}}^{i_1,\ldots, i_r} v_{i_{\bar 1}} \otimes v_{i_{\bar 2}} \otimes \cdots \otimes v_{i_{\bar r}}.$$
\[SWduality\] $\mathfrak{S}_n$ and $P_r(n)$ generate the full centralisers of each other in ${\operatorname{End}}(V_n^{\otimes r})$.
1. $P_r(n)$ generates ${\operatorname{End}}_{\mathfrak{S}_n}(V_n^{\otimes r})$, and when $n\geq 2r$, $P_r(n) \cong {\operatorname{End}}_{\mathfrak{S}_n}(V_n^{\otimes r})$.
2. $\mathfrak{S}_n$ generates ${\operatorname{End}}_{\mathfrak{S}_n}(V_n^{\otimes r})$.
We will denote $E_r(n)={\operatorname{End}}_{\mathfrak{S}_n}(V_n^{\otimes r})$.
We have a decomposition of $V_n^{\otimes r}$ as a $(\mathfrak{S}_n,P_r(n))$-bimodule $$V_n^{\otimes r} = \bigoplus {\textbf{\rm \textbf{S}}}(\lambda_{[n]})\otimes L_r(\lambda)$$ where the sum is over all partitions $\lambda_{[n]}$ of $n$ such that $|\lambda| \leq r$.
Using [@goodwall Theorem 9.2.2] we have, for $\lambda_{[n]}, \mu_{[n]},\nu_{[n]} \vdash n$ with $\lambda \vdash r$ and $\mu \vdash s$, $$\label{gw}{\operatorname{Hom}}_{\mathfrak{S}_n}({\textbf{\rm \textbf{S}}}(\nu_{[n]}), {\textbf{\rm \textbf{S}}}(\lambda_{[n]})\otimes {\textbf{\rm \textbf{S}}}(\mu_{[n]}))$$ $$\begin{aligned}
&& \cong \left\{
\begin{array}{ll} {\operatorname{Hom}}_{E_r(n)\otimes E_s(n)}(L_r(\lambda) \boxtimes L_s(\mu), L_{r+s}(\nu)\!\downarrow_{E_r(n)\otimes E_s(n)}) & \mbox{if $\nu \in \Lambda_{\leq r+s}$}\\
0 & \mbox{otherwise.}
\end{array}\right.\end{aligned}$$
Kronecker product via the partition algebra {#3.2}
-------------------------------------------
Going back to the formula in (\[gw\]) we need to consider $L_{r+s}(\nu)\!\downarrow_{E_r(n)\otimes E_s(n)}$. Now $L_{r+s}(\nu )$ is a simple $P_{r+s}(n)$-module annihilated by $\ker \Phi_{r+s,n}$ and hence also by $\ker \Phi_{r,n} \otimes \ker \Phi_{s,n}$. Thus $L_{r+s}(\nu )\!\!\downarrow_{P_r(s)\otimes P_s(n)}$ is semisimple and has the same simple factors as $L_{r+s}(\nu)\!\!\downarrow_{E_r(s)\otimes E_s(n)}$.
Now combining (\[gw\]) with (\[alt\]) we have the following result.
\[transfer\] Let $\lambda_{[n]}, \mu_{[n]}, \nu_{[n]}\vdash n$ with $\lambda \vdash r$ and $\mu \vdash s$. Then we have $${g}_{\lambda_{[n]} , \mu_{[n]} }^{\nu_{[n]}} =
\left\{
\begin{array}{ll} \sum\limits_{i=0}^t (-1)^i [\Delta_{r+s}(\nu^{(i)})\!\downarrow_{P_r(n)\otimes P_s(n)}:L_r(\lambda)\boxtimes L_s(\mu)] & \mbox{if $\nu\in \Lambda_{\leq(r+s)}$}\\ 0 & \mbox{otherwise} \end{array}
\right.$$ where $\nu = \nu^{(0)} \hookrightarrow_n \nu^{(1)} \hookrightarrow_n \ldots \hookrightarrow \nu^{(t)}$ is the $P_{r+s}(n)$-block of $\nu$.
For sufficiently large values of $n$ the partition algebra is semisimple. Therefore Theorem \[transfer\] reproves the limiting behaviour of tensor products observed by Murnaghan (see Section \[murg\]). It also offers the following concrete representation theoretic interpretation of the $\overline{g}_{\lambda , \mu}^{\nu}$.
Let $\lambda \vdash r$ and $\mu \vdash s$ and suppose $|\nu|\leq r+s$. Then we have $$\overline{g}_{\lambda , \mu }^{\nu} = [\Delta_{r+s}(\nu)\!\downarrow_{P_r(n)\otimes P_s(n)}:L_r(\lambda)\boxtimes L_s(\mu)] .$$
We recover the Murnaghan–Littlewood Theorem as follows. Let $\lambda,\mu,\nu$ be partitions and suppose that $|\lambda|+|\mu|=|\nu|$. Then we have that $\Delta_{r+s}(\nu)={\textbf{\rm \textbf{S}}}(\nu)$, $\Delta_{r}(\lambda)={\textbf{\rm \textbf{S}}}(\lambda)$ and $\Delta_s(\mu)={\textbf{\rm \textbf{S}}}(\mu)$ and so we have $$\overline{g}^\nu_{\lambda,\mu}=c^\nu_{\lambda,\mu}.$$
\[corol\] We have that $\overline{g}_{\lambda,\mu}^\nu={g}_{ \lambda_{[n]},\mu_{[n]} }^{ \nu_{[n]} }$ if $$n \geq {\rm min}\{|\lambda|+|\mu|+\nu_1,|\lambda|+|\nu|+\mu_1, |\nu|+|\mu|+\lambda_1\}.$$
When $n\geq |\lambda|+|\mu|+\nu_1$ we have that $\Delta_{r+s}(\nu)=L_{r+s}(\nu)$ by Proposition \[singular\]. The result now follows as $$g^{\nu_{[n]}}_{\lambda_{[n]}, \mu_{[n]}}=
g^{\mu_{[n]}}_{\lambda_{[n]}, \nu_{[n]}}=
g^{\lambda_{[n]}}_{\nu_{[n]}, \mu_{[n]}}.\qedhere$$
Corollary \[corol\] gives a new proof of Brion’s bound [@brion] for the stability of the Kronecker coefficients using the partition algebra.
The Kronecker coefficients as a sum of reduced Kronecker coefficients {#3.3}
---------------------------------------------------------------------
In [@rosa] a formula is given for writing the Kronecker coefficients as a sum of reduced Kronecker coefficients. We shall now interpret this formula in the Grothendieck group of the partition algebra by showing that it coincides with the formula in Theorem \[transfer\].
Let $\nu_{[n]}$ be a partition of $n$. We make the convention that $\nu_0=n-|\nu|$ is the $0$th row of $\nu_{[n]}$. For $i\in \mathbb{Z}_{\geq 0}$ define ${\nu_{[n]}^{\dagger i}}$ to be the partition obtained from $\nu_{[n]}$ by adding 1 to its first $i-1$ rows and erasing its $i$th row. In particular we have ${\nu_{[n]}^{\dagger 0}}=\nu$.
\[rossss\] Let $\lambda_{[n]}, \mu_{[n]}, \nu_{[n]}\vdash n$. Then $$g^{\nu_{[n]}}_{\lambda_{[n]},\mu_{[n]}} = \sum_{i=0}^{l} (-1)^i \overline{g}^{\nu_{[n]}^{\dagger i}}_{\lambda,\mu}$$ where $l=\ell(\lambda_{[n]})\ell(\mu_{[n]})-1$.
Relating this to the partition algebra, we have the following.
Let $ \nu_{[n]}\vdash n$ and let $ \nu= \nu^{(0)} \hookrightarrow_n \nu^{(1)}\hookrightarrow_n \ldots $ be a chain of $n$-pairs. Then the partitions $${\nu_{[n]}^{\dagger i}}= \nu^{(i)}$$ for all $ i \geq 0$.
The $i=0$ case is clear from the definitions. We proceed by induction. Assume that $${\nu_{[n]}^{\dagger i}}= \nu^{(i)}.$$ Then $( \nu^{(i)} )_1= n- |\nu| + 1$, $( \nu^{(i)} )_j= \nu_{j-1} + 1$ for $j\leq i$, and $( \nu^{(i)} )_j= \nu_{j}$ for $j>i$. Therefore $$\begin{aligned}
|\nu^{(i)}|=n- |\nu|+1 +\sum_{j \neq i}\nu_j + i-1 = n- \nu_i+i\end{aligned}$$
We have that $ \nu^{(i+1)} / \nu^{(i)}$ is a skew partition consisting of a strip in the $(i+1)$th row. By definition of an $n$-pair the content, ${\operatorname{ct}}(\nu^{(i+1)}_{i+1})$, of the last node is $n-| \nu^{(i)}|$. Therefore $$\begin{aligned}
{\operatorname{ct}}(\nu^{(i+1)}_{i+1}):= \nu^{(i+1)}_{i+1} - (i+1)= n- (n- \nu_i+i) = \nu_i - i\end{aligned}$$ and $\nu^{(i+1)}_{i+1}=\nu_i+1$, therefore ${\nu_{[n]}^{\dagger (i+1)}}= \nu^{(i+1)}$.
In Theorem \[transfer\], $t$ is chosen so that $|\nu^{(t)}| \leq |\lambda|+|\mu|$ and $|\nu^{(t+1)}| > |\lambda|+|\mu|$. So Theorem \[transfer\] and \[rossss\] seem to give a different number of terms in the sum. For example consider $$g^{(2)}_{(1^2),(1^2)}=1 \quad g^{(1^2)}_{(1^2),(1^2)}=0$$ these are given as a sum of one, respectively two terms in Theorem \[transfer\], both cases have four terms in Theorem \[rossss\]. Now consider $$\lambda_{[n]}=\mu_{[n]}=\nu_{[n]}=(10,10,10)$$ then $\ell(\lambda_{[n]})\ell(\mu_{[n]})=9$. We have $\nu_{[n]}^{\dagger 8}=(11^3,1^5)$ with $|\nu_{[n]}^{\dagger 8}|=38$. But $r+s=40$, so we have two more terms in Theorem \[transfer\], corresponding to $\nu^{(9)}=(11^3, 1^6)$ and $\nu^{(10)}=(11^3, 1^7)$. However, we can show that in fact the two theorems give the same sum.
First assume that $\ell(\lambda_{[n]})\ell(\mu_{[n]})-1>t$, then for all $i>t$ we have $$\bar{g}_{\lambda , \mu}^{\nu_{[n]}^{\dagger i}}=0$$ as $|\nu_{[n]}^{\dagger i}|>|\lambda|+|\mu|$. And so the two sums coincide.
Now assume that $\ell(\lambda_{[n]})\ell(\mu_{[n]})-1 <t$. Then for all $i>\ell(\lambda_{[n]})\ell(\mu_{[n]})-1$, we have $$i\geq \ell(\lambda_{[n]})\ell(\mu_{[n]}) \geq |\lambda_{[n]}\cap (\mu_{[n]})'|.$$ where $(\mu_{[n]})'$ denotes the conjugate partition of $\mu_{[n]}$. (To see this observe that the Young diagram of $\lambda_{[n]}\cap (\mu_{[n]})'$ fits in a rectangle of size $\ell(\lambda_{[n]}) \times \ell(\mu_{[n]})$). Now we have $$\ell(\nu^{(i)})\geq i \geq |\lambda_{[n]}\cap (\mu_{[n]})'|.$$ But this implies that $\bar{g}_{\lambda, \mu}^{\nu^{(i)}}=0$ by [@Dvir].
The restriction of a standard module to a Young subalgebra
===========================================================
In this section we compute the restriction of a standard module to a Young subalgebra of the partition algebra.
Set $m=r+s$ for some $r,s\geq 1$ and fix $\delta\in \mathbb{C}^\times$. We write $P_r=P_r(\delta)$, $P_s=P_s(\delta)$ and $P_m=P_m(\delta)$. We view $P_r\otimes P_s$ as a subalgebra of $P_m$ by mapping each $d\otimes d'$, where $d$ (resp. $d'$) is an $(r,r)$- (resp. $(s,s)$-) partition diagram, to the $(m,m)$-partition diagram obtained by putting $d$ and $d'$ side by side, with $d$ to the left of $d'$.
We wish to understand the restriction of $\Delta_m(\nu)$ to the subalgebra $P_r\otimes P_s$. Let $\mathfrak{D}_r$ denote the diagonal copy of $\mathfrak{S}_r$ in $\mathfrak{S}_r\times \mathfrak{S}_r$. We will need the following lemmas.
\[brauerlemma\] Let $V(2r,0)_r$ be the subspace of $V(2r,0)$ spanned by all $(2r,0)$-partition diagrams having precisely $r$ blocks of the form $\{i,j\}$ with $1\leq i\leq r$ and $r+1\leq j\leq 2r$. Then $V(2r,0)_r$ is a $\mathfrak{S}_r\times \mathfrak{S}_r$-module and we have $$V(2r,0)_r \cong \mathbb{C}\!\uparrow_{\mathfrak{D}_r}^{\mathfrak{S}_r \times \mathfrak{S}_r} \cong \bigoplus_{\lambda\vdash r} {\textbf{\rm \textbf{S}}}(\lambda)\boxtimes {\textbf{\rm \textbf{S}}}(\lambda).$$ The structure as a $P_r\times P_r$-module is (trivially) obtained by inflation.
A basis for $V(2r,0)_r$ is given by the set $\{(1,\sigma)v_0 : \sigma \in \mathfrak{S}_r\}$ where $v_0$ is the diagram given in Figure \[fig1\].
(0,0) rectangle (8,3); in [0.5,1.5,...,7.5]{} [(,3) circle (2pt); ]{}
(4,3.5) – (4,-.5);
(0.5,3) – (4.5,3); (2.5,3) arc (180:360:1.5 and .5); (3.5,3) arc (180:360:.5 and .25); (0.5,3) arc (180:360:3.5 and 1); (1.5,3) arc (180:360:2.5 and 0.75);
Now, the map $$f: V(2r,0)_r \to \mathbb{C}\!\uparrow_{\mathfrak{D}_r}^{\mathfrak{S}_r\times \mathfrak{S}_r}$$ given by $$f((1,\sigma)v_0)=(1,\sigma)$$ gives the required isomorphism.
\[partitionlemma\] Let $V(2r,r)_r$ be the subspace of $V(2r,r)$ spanned by all $(2r,r)$-partition diagrams having precisely $r$ propagating blocks of the form $\{i,j,\bar{k}\}$ with $1\leq i\leq r$ and $r+1\leq j\leq 2r$. Then, for any $\mu\vdash r$ we have that $V(2r,r)_r\otimes_{\mathfrak{S}_r}{\textbf{\rm \textbf{S}}}(\mu)$ is a $\mathfrak{S}_r\times \mathfrak{S}_r$-module and we have $$V(2r,r)_r\otimes_{\mathfrak{S}_r}{\textbf{\rm \textbf{S}}}(\pi)\cong \mathbf{S}(\pi) \!\uparrow_{\mathfrak{D}_r}^{\mathfrak{S}_r\times \mathfrak{S}_r}\cong\bigoplus_{\rho,\sigma} g_{\rho,\sigma}^\pi {\textbf{\rm \textbf{S}}}(\rho)\boxtimes {\textbf{\rm \textbf{S}}}(\sigma).$$ The structure as a $P_r\times P_r$-module is (trivially) obtained by inflation.
Let $X(\pi)$ be a basis for $\mathbf{S}(\pi)$. A basis for $V(2r,r)_r\otimes_{\mathfrak{S}_r} \mathbf{S}(\pi)$ is given by the set $$\{(1,\sigma)v_1\otimes x : \sigma \in \mathfrak{S}_r, x \in X(\pi) \}$$ where $v_1$ is given in Figure \[fig2\].
(0,0) rectangle (8,3); in [0.5,1.5,...,7.5]{} [(,3) circle (2pt); ]{} (0,0) rectangle (8,3); in [0.5,1.5,...,3.5]{} [ (,0) circle (2pt); ]{}
(4,4) – (4,-1);
(0.5,3) – (0.5,0); (1.5,3) – (1.5,0); (2.5,3) – (2.5,0); (3.5,3) – (3.5,0); (0.5,3) – (4.5,3); (2.5,3) arc (180:360:1.5 and .5); (3.5,3) arc (180:360:.5 and .25); (0.5,3) arc (180:360:3.5 and 1); (1.5,3) arc (180:360:2.5 and 0.75); (0.3,-.3) rectangle (3.7,-1); (4.3,3.3) rectangle (7.7,4); (2,-0.65) node [$ x$]{}; (6,3.65) node [$\sigma $]{};
Now the map $$g: V(2r,r)_r \otimes_{\mathfrak{S}_r} \mathbf{S}(\pi) \to \mathbf{S}(\pi)\!\uparrow_{\mathfrak{D}_r}^{\mathfrak{S}_r\times \mathfrak{S}_r}$$ given by $$g(1,\sigma)v_1 \otimes x = (1,\sigma)\otimes x$$ gives the required isomorphism.
\[partition\] Write $m=r+s$ and let $\nu\vdash m-l$, $\lambda\vdash r-l_r$ and $\mu\vdash s-l_s$ for some non-negative integers $l, l_r, l_s$. Then $\Delta_m(\nu)\!\!\downarrow_{P_r\otimes P_s}$ has a filtration by standard modules with multiplicities given by $$[\Delta_m(\nu)\!\downarrow_{P_r\otimes P_s} \, : \, \Delta_r(\lambda)\boxtimes \Delta_s(\mu)]=\sum_{\begin{subarray}{c} l_1, l_2 \\ l_1+2l_2=l-l_r-l_s \end{subarray}} \sum_{\begin{subarray}{c} \alpha\vdash r-l_r-l_1-l_2 \\ \beta\vdash s-l_s-l_1-l_2 \\ \pi,\rho,\sigma\vdash l_1 \\ \gamma\vdash l_2 \end{subarray}} c_{\alpha ,\beta ,\pi}^\nu c_{\alpha , \rho ,\gamma}^\lambda c_{\gamma ,\sigma , \beta}^{\mu}g_{\rho,\sigma}^\pi .$$
Note that these multiplicities are well-defined, this follows by the general theory of quasi-hereditary algebras, see [@dr1].
Recall that $\Delta_m(\nu)=V(m,m-l)\otimes_{\mathfrak{S}_{m-l}}{\textbf{\rm \textbf{S}}}(\nu)$. Now, we say that a block of a diagram in $V(m,m-l)$ is a crossing block if it contains at least one vertex in $\{1, \ldots, r\}$ and one vertex in $\{r+1, \ldots , r+s\}$. It is clear that when we apply a diagram in $P_r\otimes P_s$ to a diagram in $V(m,m-l)$ we cannot increase the number of crossing blocks. Thus we get a filtration of $\Delta_m(\nu)\!\downarrow_{P_r\otimes P_s}$ with subquotients isomorphic to $$V(m,m-l)_c\otimes_{\mathfrak{S}_{m-l}} {\textbf{\rm \textbf{S}}}(\nu)$$ where $V(m,m-l)_c$ denotes the span of all diagrams in $V(m,m-l)$ having precisely $c$ crossing blocks. Now each subquotient splits as $$V(m,m-l)_c\otimes_{\mathfrak{S}_{m-l}}{\textbf{\rm \textbf{S}}}(\nu) = \bigoplus_{\begin{subarray}{c} p_r, p_s, p_c, n_c \\ p_r+p_s+p_c=m-l \\ p_c+n_c=c\end{subarray}} V(m,m-l)_{p_r, p_s, p_c, n_c}\otimes_{\mathfrak{S}_{m-l}}{\textbf{\rm \textbf{S}}}(\nu)$$ where $V(m,m-l)_{p_r, p_s, p_c, n_c}$ is the span of all diagrams in $V(m,m-l)_c$ having precisely $p_r$ (resp. $p_s$) propagating blocks containing only vertices in the set $\{1, \ldots , r\}$ (resp. $\{r+1. \ldots , r+s\}$) and some $\bar{j}$ (for some $1\leq j\leq m-l$), precisely $p_c$ propagating crossing blocks and precisely $n_c$ non-propagating crossing blocks. This is illustrated in Figure \[fig4\].
(0,0) rectangle (16,3); in [0.5,1.5,...,15.5]{} [(,3) circle (2pt); ]{} [(2,0) circle (2pt); ]{} [(5,0) circle (2pt); ]{} [(8,0) circle (2pt); ]{} [(11,0) circle (2pt); ]{} [(14,0) circle (2pt); ]{}
(10,3.5) – (10,.7);
(1.5,3) – (2,0); (7.5,3) – (5,0); (9.5,3) – (8,0); (15.5,3) – (11,0); (14.5,3) – (14,0); (1.5,3) arc (180:360:.5 and .3); (9.5,3) arc (180:360:.5 and .5); (8.5,3) arc (180:360:2.5 and 2); (7.5,3) arc (180:360:.5 and .5); (2.5,3) arc (180:360:.5 and .5); (4.5,3) arc (180:360:3.5 and 1.5); (5.5,3) arc (180:360:3.5 and 1);
To see this, note that multiplication by elements of $P_r\otimes P_s$ on this subquotient preserves both the total number of propagating blocks and the total number of crossing blocks.
Now we have $$\begin{aligned}
&V(m,m-l)_{p_r,p_s,p_c,n_c}\otimes_{\mathfrak{S}_{m-l}}{\textbf{\rm \textbf{S}}}(\nu) \\
=& V(m,m-l)_{p_r,p_s,p_c,n_c}\otimes_{\mathfrak{S}_{p_r}\times \mathfrak{S}_{p_c}\times \mathfrak{S}_{p_s}}{\textbf{\rm \textbf{S}}}(\nu)\!\downarrow_{\mathfrak{S}_{p_r}\times \mathfrak{S}_{p_c}\times \mathfrak{S}_{p_s}}\\
=& V(m,m-l)_{p_r,p_s,p_c,n_c}\otimes_{\mathfrak{S}_{p_r}\times \mathfrak{S}_{p_c}\times \mathfrak{S}_{p_s}} \bigoplus_{\begin{subarray}{c} \alpha\vdash p_r\\\beta\vdash p_s \\ \pi\vdash p_c\end{subarray}} c_{\alpha , \beta ,\pi}^\nu {\textbf{\rm \textbf{S}}}(\alpha)\boxtimes {\textbf{\rm \textbf{S}}}(\pi)\boxtimes {\textbf{\rm \textbf{S}}}(\beta).\end{aligned}$$ The key point of the proof is that $$V(m,m-l)_{p_r, p_s, p_c, n_c}\otimes_{\mathfrak{S}_{p_r, p_c, p_s}} {\textbf{\rm \textbf{S}}}(\alpha)\boxtimes {\textbf{\rm \textbf{S}}}(\pi)\boxtimes {\textbf{\rm \textbf{S}}}(\beta) \tag{$\dagger$} \label{dagger}$$ is isomorphic, as a $P_r \times P_s$-module, to $$(V(r, p_r+p_c+n_c)\boxtimes V(s, p_s+p_c+n_c)) \otimes_{\mathfrak{S}_{p_r+p_c+n_c, p_s+p_c+n_c}}$$ $$({\textbf{\rm \textbf{S}}}(\alpha) \boxtimes (V(2p_c, p_c)_{p_c}\otimes_{\mathfrak{S}_{p_c}}{\textbf{\rm \textbf{S}}}(\pi)) \boxtimes V(2n_c, 0)_{n_c} \boxtimes {\textbf{\rm \textbf{S}}}(\beta))\!\uparrow_{\mathfrak{S}_{p_r,p_c,n_c,n_c,p_c,p_s}}^{\mathfrak{S}_{p_r+p_c+n_c, n_c+p_c+p_s}}.$$ This follows by ‘decomposing the diagrams’, this is best illustrated by an example. Under this isomorphism, the element $w \otimes (x_1 \otimes x_2 \otimes x_3)$ where $w$ is as in Figure \[fig4\] and $x_i \in {\textbf{\rm \textbf{S}}}(\alpha)$ is mapped to the element in Figure \[fig5\].
(0,0) rectangle (16,3); (0,-.5) rectangle (16,-3.5); in [0.5,1.5,...,15.5]{} [(,3) circle (2pt); (,0) circle (2pt); (,-0.5) circle (2pt); ]{}
(10,3.5) – (10,-0.5);
(1.5,3) – (1.5,0); (10.5,3) – (10.5,0); (13.5,3) – (13.5,0); (7.5,3) – (7.5,0); (9.5,3) – (9.5,0); (15.5,3) – (15.5,0); (14.5,3) – (14.5,0); (1.5,-2) – (1.5,-0.5); (1.5,-3) – (1.5,-3.5); (7.5,-2) – (7.5,-0.5); (7.5,-3) – (7.5,-3.5); (9.5,-2) – (9.5,-0.5); (9.5,-3) – (9.5,-3.5); (4.5,3) – (4.5,0); (14.5,-0.5) – (15.5,-2); (14.5,-3) – (14.5,-3.5); (15.5,-3) – (15.5,-3.5); (15.5,-0.5) – (14.5,-2); (5.5,3) – (5.5,0); (11.5,3) – (11.5,0); (12.5,3) – (12.5,0); (1.5,3) arc (180:360:.5 and .4); (9.5,-.5) arc (180:360:.5 and .4); (7.5,-.5) arc (180:360:3 and 1.2); (7.5,3) arc (180:360:.5 and .5); (2.5,3) arc (180:360:.5 and .5); (4.5,-.5) arc (180:360:3.5 and 1); (5.5,-.5) arc (180:360:3.5 and 0.6); (1,-2) rectangle (2,-3); (7,-2) rectangle (10,-3); (14.2,-2) rectangle (15.8,-3); (1.5,-2.5) node [$ x_1$]{}; (8.5,-2.5) node [$ x_2$]{}; (15,-2.5) node [$ x_3$]{};
By application of Lemmas \[brauerlemma\] and \[partitionlemma\], equation (\[dagger\]) is isomorphic to $$(V(r, p_r+p_c+n_c)\boxtimes V(s, p_s+p_c+n_c)) \otimes_{\mathfrak{S}_{p_r+p_c+n_c, p_s+p_c+n_c}}$$ $$\left(\oplus_{\gamma\vdash n_c}\oplus_{\rho,\sigma\vdash p_c}g_{\rho,\sigma}^\pi
{\textbf{\rm \textbf{S}}}(\alpha) \boxtimes {\textbf{\rm \textbf{S}}}(\rho) \boxtimes {\textbf{\rm \textbf{S}}}(\gamma)\boxtimes {\textbf{\rm \textbf{S}}}(\gamma)\boxtimes {\textbf{\rm \textbf{S}}}(\sigma) \boxtimes {\textbf{\rm \textbf{S}}}(\beta)\right)\! {\large \uparrow}_{\mathfrak{S}_{p_r,p_c,n_c,n_c,p_c,p_s}}^{\mathfrak{S}_{p_r+p_c+n_c, n_c+p_c+p_s}}$$ which by the Littlewood–Richardson rule is isomorphic to $$\oplus_{\gamma\vdash n_c}\oplus_{\rho,\sigma\vdash p_c}g_{\rho,\sigma}^\pi (V(r, p_r+p_c+n_c)\boxtimes V(s, p_s+p_c+n_c)) \otimes_{\mathfrak{S}_{p_r+p_c+n_c, p_s+p_c+n_c}}$$ $$\oplus_{\begin{subarray}{c} \lambda\vdash p_r+p_c +n_c \\ \mu\vdash p_s+p_c+n_c \end{subarray}} c_{\alpha , \rho ,\gamma}^\lambda c_{\gamma ,\sigma ,\beta}^\mu {\textbf{\rm \textbf{S}}}(\lambda)\boxtimes {\textbf{\rm \textbf{S}}}(\mu).$$ We can now rewrite this as a product of standard modules as follows: $$\bigoplus_{\begin{subarray}{c} {\gamma\vdash n_c} \\ {\rho,\sigma\vdash p_c} \end{subarray}}
\bigoplus_{\begin{subarray}{c} \lambda\vdash p_r+p_c+n_c \\ \mu\vdash p_s+p_c+n_c \end{subarray}} g_{\rho,\sigma}^\pi c_{\alpha , \rho ,\gamma}^\lambda c_{\gamma , \sigma ,\beta}^\mu\Delta_r(\lambda)\boxtimes \Delta_s(\mu).$$ Now noting that $p_r+p_s+p_c=m-l$, $p_r+p_c+n_c=r-l_r$ and $p_s+p_c+n_c=s-l_s$ and writing $l_1=p_c$ and $l_2=n_c$, we get that $l-l_r-l_s=l_1+l_2$ and the result follows.
In [@rosa Lemma 2.1] a formula is given for writing the reduced Kronecker coefficients as a sum of Kronecker coefficients and Littlewood–Richardson coefficients. An immediate corollary of the above theorem is an interpretation of this formula in the setting of the partition algebra.
\[bigthm\] Let $\lambda,\mu,\nu$ be any partitions with $|\lambda|=r$, $|\mu|=s$ and $|\nu|=r+s-l$. Then the reduced Kronecker coefficient $\overline{g}_{\lambda , \mu}^{\nu}$ is given by $$\overline{g}^\nu_{\lambda,\mu}=
\sum_{\begin{subarray}{c}l_1, l_2 \\
l=l_1+2l_2\end{subarray}}
\sum_{\begin{subarray}{c}{\alpha \vdash r-l_1-l_2}
\\ {\beta\vdash s-l_1-l_2} \end{subarray}}
\sum_{\begin{subarray}{c}
{\pi,\rho,\sigma \vdash l_1} \\{\gamma\vdash l_2}
\end{subarray}}
c_{\alpha ,\beta,\pi}^{\nu } c_{\alpha ,\rho,\gamma}^{\lambda } c_{ \gamma, \sigma, \beta}^{\mu } g_{\rho,\sigma}^\pi$$
This follows from Theorems \[transfer\] and \[partition\], noting that for $|\lambda|=r$ and $|\mu|=s$, $\Delta_r(\lambda)=L_r(\lambda)$ and $\Delta_r(\mu)=L_r(\mu)$.
Hooks and two-part partitions
=============================
We now consider the case where one of the partitions in a Kronecker coefficient is either a hook or two-part partition. The first positive closed formula for the two-part partition case was due to Ballantine and Orellana [@BallantineOrellanaEJC]. Blasiak [@blasiak] has recently given a combinatorial interpretation of the one hook case. The result below provides positive closed formulas for $g_{\lambda[n],\mu[n]}^{\nu_{[n]}}$ in the case that $\nu_{[n]}$ is a two-part or hook partition; in the hook case, this is the first such closed formula, in the two-part case our formula simplifies that of [@BallantineOrellanaEJC]. These formulas reveal a distinct symmetry between the two cases.
\[bigthmspecialcase2\] Let $\lambda_{[n]}, \mu_{[n]}, \nu_{[n]}$ be partitions of $n$ with $|\lambda|=r$, $|\mu|=s$ and $|\nu|=r+s-l$.
\(i) Suppose $\nu_{[n]}=(n-k,k)$ is a two-part partition. Then we have $${g}^{(n-k,k)}_{\lambda_{[n]},\mu_{[n]}}=
\sum_{\begin{subarray}{c}l_1, l_2 \\
l=l_1+2l_2\end{subarray}}
\sum_{\begin{subarray}{c}
{ \sigma \vdash l_1} \\{\gamma\vdash l_2}
\end{subarray}}
c_{(r-l_1-l_2) ,\sigma,\gamma}^{\lambda } c_{ \gamma, \sigma,(s-l_1-l_2)}^{\mu }$$ for all $n \geq {\rm min}\{|\lambda|+\mu_1+k, |\mu|+\lambda_1+k\}$.
\(ii) Suppose $\nu_{[n]}=(n-k,1^k)$ is a hook partition. Then we have $${g}^{(n-k,1^k)}_{\lambda_{[n]},\mu_{[n]}}=
\sum_{\begin{subarray}{c}l_1, l_2 \\
l=l_1+2l_2\end{subarray}}
\sum_{\begin{subarray}{c}
{ \sigma \vdash l_1} \\{\gamma\vdash l_2}
\end{subarray}}
c_{(1^{r-l_1-l_2 }),\sigma,\gamma}^{\lambda } c_{ \gamma, \sigma',(1^{s-l_1-l_2})}^{\mu }$$ for all $n \geq {\rm min}\{|\lambda|+|\mu| +1, |\mu|+\lambda_1+k, |\lambda|+\mu_1+k\}$ and where $\sigma'$ denotes the transpose of $\sigma$.
Our assumption on $n$ implies that ${g}^{\nu_{[n]}}_{\lambda_{[n]},\mu_{[n]}}= \overline{g}^\nu_{\lambda,\mu}$ by Corollary \[corol\].
The result follows from Corollary \[bigthm\], noting that $c_{\alpha,\beta,\pi}^{(k)}$ (respectively $c_{\alpha,\beta,\pi}^{(1^k)}$) is zero unless $\alpha=(r-l_1-l_2)$, $\beta=(s-l_1-l_2), \pi = (l_1)$ (respectively $\alpha=(1^{r-l_1-l_2})$, $\beta=(1^{s-l_1-l_2}), \pi = (1^{l_1})$) in which case it is equal to 1 and $g^{(l_1)}_{\rho,\sigma}$ (respectively $g^{(1^{l_1})}_{\rho,\sigma}$) is zero unless $\rho=\sigma$ (respectively $\rho=\sigma'$), in which case it is equal to 1.
In [@BallantineOrellanaEJC] they compute the Kronecker coefficients $$g_{(n-k,k),\lambda_{[n]}}^{\mu_{[n]}} = g_{\lambda_{[n]},\mu_{[n]}}^{(n-k,k)}$$ when $n-|\lambda|-\lambda_1\geq 2k$, equivalently $$n \geq |\lambda|+\lambda_1+2k.$$ Noting that $k=|\mu|$ and for $\overline{g}^\nu_{\lambda,\mu}\neq0$, we must have that $|\mu|\leq |\lambda|+|\nu|$, we see that Corollary \[bigthmspecialcase2\] improves this bound (as $|\mu|+\lambda_1+k \leq |\lambda|+\lambda_1+2k)$.
Example
========
In this section, we shall compute the tensor square of the Specht module, ${\textbf{\rm \textbf{S}}}(n-1,1)$ for $n\geq 2$, labelled by the first non-trivial hook, via the partition algebra. We have that $${\operatorname{Hom}}_{\mathfrak{S}_n}({\textbf{\rm \textbf{S}}}(\nu_{[n]}), {\textbf{\rm \textbf{S}}}(n-1,1)\otimes {\textbf{\rm \textbf{S}}}(n-1,1))
\cong{\operatorname{Hom}}_{P_1(n)\otimes P_1(n)}(L_1(1) \otimes L_1(1), L_{2}(\nu)\!\!\downarrow)$$ if $\nu \in\Lambda_{\leq 2}$ and zero otherwise. Therefore, it is enough to consider the restriction of simple modules from $P_2(n)$ to the Young subalgebra $P_1(n) \otimes P_1(n)$.
The partition algebra $P_2(n)$ is a 15-dimensional algebra with basis: $$\begin{aligned}
\begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (0.5,2) -- (0.5,0);
\draw (1.5,2) -- (1.5,0);
\end{scope}
\end{tikzpicture}\end{minipage}
\begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (1.5,2) -- (0.5,0);
\draw (1.5,0) -- (0.5,2);
\end{scope}
\end{tikzpicture}\end{minipage}
\begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (0.5,2) -- (0.5,0);
\draw (0.5,2) arc (180:360:0.5 and 0.3);
\draw (0.5,0) arc (180:360:0.5 and -0.3);
\end{scope}
\end{tikzpicture}\end{minipage}
\begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (0.5,2) -- (0.5,0);
\draw (0.5,0) arc (180:360:0.5 and -0.3);
\end{scope}
\end{tikzpicture}\end{minipage}
\begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (1.5,2) -- (0.5,0);
\draw (0.5,0) arc (180:360:0.5 and -0.3);
\end{scope}
\end{tikzpicture}\end{minipage}
\begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope} \draw (0.5,2) -- (1.5,0);
\draw (0.5,2) arc (180:360:0.5 and 0.3);
\end{scope}
\end{tikzpicture}\end{minipage}
\begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope} \draw (0.5,2) -- (1.5,0);
\end{scope}
\end{tikzpicture}\end{minipage}
\begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope} \draw (1.5,2) -- (1.5,0);
\end{scope}
\end{tikzpicture}\end{minipage}
\\ \begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope} \draw (0.5,2) -- (0.5,0);
\end{scope}
\end{tikzpicture}\end{minipage}
\begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope} \draw (0.5,2) -- (0.5,0);
\draw (0.5,2) arc (180:360:0.5 and 0.3);
\end{scope}
\end{tikzpicture}\end{minipage}
\begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope} \draw (1.5,2) -- (0.5,0);
\end{scope}
\end{tikzpicture}\end{minipage}
\begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope} \draw (0.5,2) arc (180:360:0.5 and 0.3);
\draw (0.5,0) arc (180:360:0.5 and -0.3);
\end{scope}
\end{tikzpicture}\end{minipage}
\begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope} \draw (0.5,0) arc (180:360:0.5 and -0.3);
\end{scope}
\end{tikzpicture}\end{minipage}
\begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (0.5,2) arc (180:360:0.5 and 0.3);
\end{scope}
\end{tikzpicture}\end{minipage}
\begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\end{scope}
\end{tikzpicture}\end{minipage}
\end{aligned}$$ and multiplication defined by concatenation. For example: $$\begin{aligned}
\begin{minipage}{10mm}
\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (1.5,2) -- (0.5,0);
\draw (1.5,0) -- (0.5,2);
\end{scope}
\draw (0,0) rectangle (2,-2);
\foreach \x in {0.5,1.5}
{\fill (\x,-2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (0.5,-2) -- (0.5,0);
\draw (0.5,-2) arc (180:360:0.5 and -0.3);
\end{scope}
\end{tikzpicture}\end{minipage} =
\begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (1.5,2) -- (0.5,0);
\draw (0.5,0) arc (180:360:0.5 and -0.3);
\end{scope}
\end{tikzpicture}\end{minipage} ,
\quad\quad
\begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (0.5,2) -- (0.5,0);
\end{scope}
\draw (0,0) rectangle (2,-2);
\foreach \x in {0.5,1.5}
{\fill (\x,-2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (0.5,0) -- (1.5,-2);
\draw (0.5,2) arc (180:360:0.5 and 0.3);
\end{scope}
\end{tikzpicture}\end{minipage} = n\ \!
\ \begin{minipage}{10mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (0.5,2) -- (1.5,0);
\draw (0.5,2) arc (180:360:0.5 and 0.3);
\end{scope}
\end{tikzpicture}\end{minipage} \end{aligned}$$
There are four standard modules corresponding to the partitions of degree less than or equal to $2$; these are obtained by inflating the Specht modules from the symmetric groups of degree $0,1,2$. These modules have bases:
$$\begin{array}{llll}
&\Delta_2(2)= {\rm Span}_{\mathbb{C}} \left\{ \, \begin{minipage}{8.5mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (0.5,2) -- (0.5,0);
\draw (1.5,2) -- (1.5,0);
\end{scope}
\end{tikzpicture}\end{minipage} +
\begin{minipage}{8.5mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (1.5,2) -- (0.5,0);
\draw (1.5,0) -- (0.5,2);
\end{scope}
\end{tikzpicture}\end{minipage}\, \right\}
&\Delta_2(1^2) = {\rm Span}_{\mathbb{C}}
\left\{ \, \begin{minipage}{8.5mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (0.5,2) -- (0.5,0);
\draw (1.5,2) -- (1.5,0);
\end{scope}
\end{tikzpicture}\end{minipage} -
\begin{minipage}{8.5mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (1.5,2) -- (0.5,0);
\draw (1.5,0) -- (0.5,2);
\end{scope}
\end{tikzpicture}\end{minipage} \,\right\} \\
\end{array}$$
$$\begin{array}{llll}
&\Delta_2(1) = {\rm Span}_{\mathbb{C}} \left\{\, \begin{minipage}{8.5mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (0.5,2) -- (0.5,0);
\draw (0.5,2) arc (180:360:0.5 and 0.3);
\end{scope}
\end{tikzpicture}\end{minipage} ,
\begin{minipage}{8.5mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope} \draw (1.5,2) -- (0.5,0);
\end{scope}
\end{tikzpicture}\end{minipage} ,
\begin{minipage}{8.5mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (0.5,2) -- (0.5,0);
\end{scope}
\end{tikzpicture}\end{minipage}\, \right\}
&\Delta_2(\emptyset) = {\rm Span}_{\mathbb{C}}\left\{\,\begin{minipage}{8.5mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (0.5,2) arc (180:360:0.5 and 0.3);
\draw (0.5,0) arc (180:360:0.5 and -0.3);
\end{scope}
\end{tikzpicture}\end{minipage} ,
\begin{minipage}{8.5mm}\begin{tikzpicture}[scale=0.4]
\draw (0,0) rectangle (2,2);
\foreach \x in {0.5,1.5}
{\fill (\x,2) circle (2pt);
\fill (\x,0) circle (2pt);}
\begin{scope}
\draw (0.5,0) arc (180:360:0.5 and -0.3);
\end{scope}
\end{tikzpicture}\end{minipage}\,\right\}
\end{array}$$
The action of $P_2(n)$ is given by concatenation. If the resulting diagram has fewer propagating lines than the original, we set the product equal to zero. The algebra $P_1(n)\otimes P_1(n)$ is the 4-dimensional subalgebra spanned by the diagrams with no lines crossing an imagined vertical wall down the centre of the diagram. The restriction of the standard modules to this subalgebra is as follows: $$\Delta_2(2)\!\!\downarrow_{P_1\otimes P_1}\cong \Delta_1(1) \boxtimes \Delta_1(1),\quad \Delta_2(1^2)\!\!\downarrow_{P_1\otimes P_1}\cong \Delta_1(1) \boxtimes \Delta_1(1),$$ $$\Delta_2(1)\!\!\downarrow_{P_1\otimes P_1}\cong \Delta_1(1) \boxtimes \Delta_1(1) \oplus \Delta_1(\emptyset) \boxtimes \Delta_1(1) \oplus \Delta_1(1) \boxtimes \Delta_1(\emptyset),$$ $$\Delta_2(\emptyset)\!\!\downarrow_{P_1\otimes P_1}\cong \Delta_1(1) \boxtimes \Delta_1(1) \oplus \Delta_1(\emptyset) \boxtimes \Delta_1(\emptyset) .$$ In particular, note that $\bar{g}_{(1),(1)}^\nu = [\Delta_2(\nu)\!\!\downarrow_{P_1\otimes P_1}:\Delta_1(1)\boxtimes\Delta_1(1)]=1$ for $\nu = \emptyset,1,1^2,2$.
The partition algebra $P_2(n)$ is semisimple for $n>2$. For $\nu=\emptyset, (1), (1^2)$ or $(2)$ we have that $\nu_{[n]}=(n), (n-1,1), (n-2,1^2),$ or $(n-2,2)$ and $\nu_{[n]}$ is a partition for $n\geq 0,2,3,4$ respectively. Therefore the Kronecker coefficients $$g^{\nu_{[n]}}_{(n-1,1),(n-1,1)}$$ stabilise for $n\geq 4$ and are non-zero for $n\geq 4$ if and only $\nu_{[n]}$ is one of the partitions above.
Now consider the case $n=2$. Neither $\nu=(1^2),$ nor $(2)$ correspond to partitions of 2, we therefore consider $\nu=\emptyset$ and $(1)$. We have that $(1) \subset (2)$ is the unique $2$-pair of partitions of degree less than or equal to 2 (see Section 3.3). Therefore the only standard $P_2(2)$-module which is not simple is $\Delta_2(1)$ and we have an exact sequence $$0\to L_2(2) \to \Delta_2(1) \to L_2(1) \to 0.$$ Thus in the Grothendieck group we have that $[L_2(1)]=[\Delta_2(1)]-[\Delta_2(2)]$. Hence, we have that $[L_2(1)\!\!\downarrow_{P_1(2)\otimes P_1(2)}:L_1(1) \boxtimes L_1(1)]=0$. We conclude that $g_{(1^2), (1^2)}^{(1^2)}=0$ and $g_{(1^2),(1^2)}^{(2)}=1$ as expected.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors wish to thank David Speyer for pointing out a crucial error in an earlier version of Theorem \[partition\] and Corollary \[bigthm\]. M. De Visscher and R. Orellana thank Georgia Benkart, Monica Vazirani and Stephanie van Willigenburg and the Banff International Research Station for providing support and a stimulating environment during the Algebraic Combinatorixx workshop where this project started. C. Bowman and R. Orellana are grateful for the financial support received from the ANR and NSF grants ANR-10-BLAN-0110 and DMS-1101740, respectively.
|
---
abstract: 'As is well-known the Complex Langevin (CL) method sometimes fails to converge or converges to the wrong limit. We identified one reason for this long ago: insufficient decay of the probability density either near infinity or near poles of the drift, leading to boundary terms that spoil the formal argument for correctness. To gain a deeper understanding of this phenomenon, we analyze the emergence of such boundary terms thoroughly in a simple model, where analytic results can be compared with numerics. We also show how some simple modification stabilizes the CL process in such a way that it can produce results agreeing with direct integration. Besides explicitly demonstrating the connection between boundary terms and correct convergence our analysis also suggests a correctness criterion which could be applied in realistic lattice simulations.'
author:
- Manuel Scherzer
- Erhard Seiler
- Dénes Sexty
- 'Ion-Olimpiu Stamatescu'
title: Complex Langevin and boundary terms
---
Introduction
============
It has been known for a long time that the Complex Langevin (CL) method for simulating systems with complex action may fail by either not converging or by converging to the wrong limit. These failures were traced either to insufficient decay of the probability distribution in the complexified configuration space [@Aarts:2009uq; @Aarts:2011ax; @Nishimura:2015pba] (at infinity or at poles of the drift force), or to failure of ergodicity [@Aarts:2017vrv; @Seiler:2017wvd]. Recently Salcedo [@salc2] has formulated interesting criteria for failure that at first sight seem to be unrelated to the ones identified by us. The most interesting ones derive support properties of the equilibrium measure which are shown in these cases to be in conflict with the correct expectation values.
In this note we will focus on one such example and show explicitly that the problems are due to slow decay, leading to the appearance of boundary terms in an integration by parts, spoiling the formal proof of correctness. We stress that we are here concerned with the behavior at large non-compact dimensions. The effects of non-holomorphicity have been shown, e.g. in random matrix models [@Mollgaard:2013qra; @Bloch:2017sex] to lead to wrong convergence and were specifically addressed in [@Aarts:2017vrv] both in simple models and in QCD.
Here we consider a complex density (x)=(-S(x)), periodic with period $2\pi$ and extending to an entire analytic function without zeroes.
The complex Langevin equation (CLE) in the form used here is $$\begin{aligned}
dx=&K_x dt +dw,\notag\\ dy=&K_y dt\,,
\label{cle2} \end{aligned}$$ where $dw$ is the Wiener process normalized as dw\^2= 2 dt and the drift is given by $$\begin{aligned}
K_x=& -{\rm Re}\,S'(x+iy) ,\cr K_y=& -{\rm Im}\,S'(x+iy)\,. \end{aligned}$$ The long time asymptotic average of a generic observable ${{\cal O}}$ is denoted by ${\langle}{{{\cal O}}} {\rangle}_\infty$; we say that the CL process yields correct results if this agrees with the ‘correct’ expectation value of the same observable defined as [[[O]{}]{}]{} \_c = dx [[[O]{}]{}]{}(x) (x), i. e. [[[O]{}]{}]{} \_= [[[O]{}]{}]{} \_c. In [@Aarts:2009uq; @Aarts:2011ax] correctness was derived from the consideration of CL expectation values at [finite]{} Langevin time; it was shown that correctness is assured if a certain quantity $F_{{\cal O}}(t,\tau)$ is independent of an interpolation parameter $\tau\in[0,t]$, i. e. \[interpol\] F\_[[O]{}]{}(t,) = 0.
Here $F_{{\cal O}}(t,\tau)$ interpolates between the ‘correct’ time evolution $F_{{\cal O}}(t,t)={\langle}{{\cal O}}(t){\rangle}_0$ (defined in Section \[boundary\] and analyzed in Appendix \[app2\]) and the time evolution of the expectation of ${{\cal O}}$ under the Langevin process $F_{{\cal O}}(t,0)= {\langle}{{\cal O}}{\rangle}_t $. The key points are that (\[interpol\]) implies
[[O]{}]{}\_t= [[O]{}]{}(t)\_0 t > 0 and hence, \_[t]{} F\_[[O]{}]{}(t,t)=[[O]{}]{}\_c.
The left hand side of (\[interpol\]), by using integration by parts, is found to be equal to a boundary term; explicitly F\_[[O]{}]{}(t,)= \_[Y]{}B\_[[O]{}]{}(Y;t,) , where &B\_[[O]{}]{}(Y;t,)\
&dx , \[boundt\]
$P(x,y;t)$ is the time evolved probability density under the Langevin evolution and ${{\cal O}}(t)\equiv{{\cal O}}(z;t)$ is the $L_c$ evolved observable (see Section \[boundary\] and appendix \[app2\]).
This form of the boundary term makes clear that correctness requires sufficient decay of the product $K_y P{{\cal O}}$ for all Langevin times $t$.
The model
=========
The model studied here is defined by the complex density = \[model\] and has been studied already in 2007 by Stamatescu [@nucunotes] and in 2008 by Berges and Sexty [@Berges:2007nr]. The ‘correct’ expectation values of exponentials (‘modes’) are dx &(ikx)(x)=\
=&(-i)\^k 0. \[rhoexp\] It was found in [@nucunotes; @Berges:2007nr] that the CL process does not reproduce the correct EV’s which, however, can be regained by a certain reweighting procedure (with different observables requiring sometimes different reweightings).
The remarkable fact found by Salcedo [@salc2] is that the static probability distribution $P(x,y)\equiv P(x,y;\infty)$ for this model can be written down explicitly by solving the time independent Fokker-Planck equation (FPE); it is P(x,y)=. \[P\] It is the only non-Gaussian example known to us for which a solution of the static FPE has been found in analytic form. Three features of this solution are remarkable:
\(1) $P$ is independent of $x$,
\(2) $P$ is independent of $\beta$,
\(3) $P$ decays as $\exp(-2|y|)$ for large $|y|$; this decay is not sufficient to make the integrals of the modes (ik(x+iy)),|k|2 absolutely convergent, in other words, already here we are faced with slow decay.
Complex Langevin results
------------------------
But first let us demonstrate that (\[P\]) is indeed the distribution produced by running a CL simulation for a long time. The drift force is &K\_x=[[Re]{}]{}=-(x)(y)\
&K\_y=[[Im]{}]{}=(x)(y), \[drift\] for the Langevin process Eq. (\[cle2\]).
In Fig. \[1dhisto\] we show the histogram of the converged marginal distribution $P_y(y;t)=\int dx P(x,y;t)$ in log scale for $\beta=1$, overlaid with the distribution (\[P\]). The histogram is obtained from one long trajectory (Langevin time $t \approx$ 125000). The agreement over about 6 orders of magnitude is convincing. The distribution can also be seen to be independent of $x$, cf. also Fig. \[marginalx\].
We also show in Fig. \[marginal\] the histograms of $P_y(y;t)$ for various shorter times and $\beta=0.1$, illustrating the convergence as $t\to\infty$.
![Comparison of the analytic expression for the marginal distribution $P_y(y;t)$ (\[P\]) (red) with the histogram of a CL simulation with $\beta=1$.[]{data-label="1dhisto"}](CLE_Erhard){width="1\columnwidth"}
![The marginal distributions $P_y(y;t)$ obtained by numerically solving the FPE for $\beta=0.1$. The ordering of times corresponds to decreasing maxima; note that for $t=200$ no difference is visible between the FPE and the analytical solution.[]{data-label="marginal"}](linplot_Py_max_smear){width="1\columnwidth"}
As noted by Salcedo [@salc2], it is obvious that the distribution $P$ (\[P\]) cannot reproduce the correct expectation values Eq. \[rhoexp\] of the observables ${{\cal O}}_k = \exp(ikx)$, because it is independent of $x$, entailing dxP(x,y) e\^[ikx]{}=0, and its slow decay makes the expectation values of ${{\cal O}}_k$ ill-defined for $|k|\ge 2$. In Table \[salc1\] we collect a few CL results, together with the exact expectation values determined by $\rho$ for $\beta=1$. The simulation used 100 independent trajectories with randomly chosen starting points on the real axis, running for a Langevin time of $t\approx 2500$, where measurements were taken after every time step, typically $5\times 10^{-6}$. The CL values for $|k|> 2$ are completely submerged by noise, as expected. For $k=\pm 2$ we find a value close to $1$. It should be remarked that the CL process for $k=2$ evaluates a conditionally convergent integral, so also the measuring schedule plays a role; for instance measuring after every time increment of $0.01$ yields very noisy results, consistent with both $0$ and $1$. Evaluating the second mode with a fixed cutoff in $y$, we find $0$.
The Schwinger-Dyson equations (SDE) ike\^[ikz]{}+e\^[i(k+1)z]{}- e\^[i(k-1)z]{}=0, arising from the identity \_[-]{}\^ (x) [[O]{}]{}’(x)dx= - \_[-]{}\^’(x) [[O]{}]{}(x)dx would be satisfied for $k=0,\pm 1$ if the modes $\pm 1$ are $0$ and the modes $\pm 2$ are $1$, even though these values are not the ‘correct’ ones.
${\langle}{{\cal O}}{\rangle}$ ${\langle}{\text{e}\,}^{ix}{\rangle}$ ${\langle}{\text{e}\,}^{-ix}{\rangle}$ ${\langle}{\text{e}\,}^{2ix}{\rangle}$ ${\langle}{\text{e}\,}^{-2ix}{\rangle}$
-------------------------------- --------------------------------------- ---------------------------------------- ---------------------------------------- -----------------------------------------
CL 0.004(3) 0.002(3) 1.027(22) 1.001(20)
correct -0.575081$i$ -0.575081$i$ -0.150162 -0.150162
$e^{t L_c}{{\cal O}}$ -0.575081$i$ -0.575081$i$ -0.150162 -0.150162
: CLE (real part, imaginary part negligible) and correct results for model (\[model\]) with $\beta=1$. Last line: ‘correct evolution’ for $t=20$ (see Appendix \[app2\]).[]{data-label="salc1"}
So the CL results, where they are defined, are incorrect, but mostly – for $|k|\ge 3$, they are completely undefined due to uncontrollable fluctuations.
The last row in the table gives the correct results from the $L_c$ evolved observables, as will be explained in the next section. Notice that the correct results of course also satisfy the SDE, but these equations, having the structure of a two-step recursion, have a two-parameter family of solutions [@Berges:2006xc; @Pehlevan:2007eq; @Aarts:2011ax].
A puzzle
--------
The remaining question is: how can CL fail for the first mode, i. e. observables ${{\cal O}}_{\pm 1}\equiv \exp(\pm i(x+iy))$? ${{\cal O}}_{\pm 1}P$ as well as $KP$ decay exponentially in $y$.
Actually the densities of ${{\cal O}}_{\pm 1}$ and $K$, if considered not as functions of $y$, but as functions of their actual value decay only power-like (see Eq. \[sigma\_anal\]). Nagata et al [@Nagata:2016vkn] gave an argument that correctness requires exponential decay of the distribution of $K$ and checked their criterion successfully for various cases; so by this criterion correctness is not to be expected here, corroborating the criterion. We will, however, formulate a different criterion in Section \[boundary\], which directly relates to the (non-)occurrence of boundary terms.
The CL simulation produces for ${{\cal O}}_{\pm 1}$ well converged, yet incorrect results, close to 0 (consistent with (\[P\]) but inconsistent with (\[rhoexp\])).
The resolution lies in the nonvanishing boundary terms arising in the time dependent expectation values and persisting for arbitrarily large times; this is the mechanism described in [@Aarts:2009uq; @Aarts:2011ax]. In the following section we will analyze those boundary terms in detail.
Boundary terms for finite Langevin time {#boundary}
=======================================
The formal argument for correctness [@Aarts:2009uq; @Aarts:2011ax] is revisited in Appendix \[app1\]. It requires the choice of an initial distribution $P(x,y;0)$; in the following we will choose for simplicity P(x,y;0)= (y). \[init\] The identity (\[interpol\]) follows by integrating by parts, assuming that there are no boundary terms, and using the Cauchy-Riemann equations.
In order to check for the appearance of boundary terms as in (\[boundt\]), we need the $L_c$ evolution of the observables (see below and Appendix \[app2\]) and the time evolution of the probability density $P$ by solving the FPE with the initial condition (\[init\]).
Indirect evidence for boundary terms {#indirect}
------------------------------------
In [@Aarts:2011ax] we found numerically for a somewhat different model that the $L_c$ evolved observables ${{\cal O}}(x+iy;t)$ grow in the $y$ direction as an iterated exponential. The same can be seen here, but we will not go into this. This growth makes the appearance of boundary terms already plausible.
In the following we show explicitly that Eq. (\[interpol\]) is numerically satisfied for short times (up to $t\approx 20$), choosing $\beta=0.1$.
The $L_c$ evolution of an observable ${{\cal O}}$ is defined by the differential equation \_t [[O]{}]{}\_k(z;t)&= L\_c [[O]{}]{}\_k(z;t)(t0),\
[[O]{}]{}\_k(z;0)&=(ikz); \[obsevol\] with L\_c =\_z. We compare (see (\[fttau\]) for the definition of $F_{{\cal O}}$) dx dy& P(x,y;0)[[O]{}]{}(x+iy;t) \
&F\_[[O]{}]{}(t,t) [[[O]{}]{}]{}(t)\_0 \[lhs\] with
dxdy &P(x,y;t)[[O]{}]{}(x+iy;0)[[O]{}]{}(0)\_t\
&[[O]{}]{}\_tF\_[[O]{}]{}(t,0). \[rhs\] Here $P(x,y;t)$ is the solution of the [*real*]{} FokkerPlanck equation (FPE) \[realFPE\] P(x,y;t)= L\^T P(x,y;t), with L\^T=\_x-\_y K\_y, and initial condition (\[init\]), which describes the time evolution of the probability density under the CL process.
For our model the FPE is (\[realFPE\]) with the drift force (\[drift\]). Eq.(\[realFPE\]) is solved numerically as well; some details are found in Appendix \[FPE\].
Fig. \[corr\] compares (\[lhs\]) and (\[rhs\]) for the Fourier modes [[O]{}]{}\_k(z)=(ikz). for $k=1,2,3$, Langevin times $t$ between $0$ and $50$ and $\beta=0.1$.
It is seen that the left hand side (\[lhs\]) reaches its asymptotic value already for quite short Langevin times (around $t\approx 7$). This is in accordance with the value of the smallest nonzero eigenvalue $\lambda_1\approx -1$ of $L_c$(cf. Eq.(\[eigenval\])). For this value of $\beta$ also the right hand side does the same; for $t\lessapprox 20$ there is no difference visible between the left and the right hand sides (dashed and solid curves). This indicates that any boundary terms are negligible there.
So there is a ‘plateau’ corresponding to the correct value in the solid curve, and the boundary term starts picking up around $t=20$.
![Comparison of expectation values using the FPE evolution of $P$ (solid lines) Eq. (\[rhs\]) with the $L_c$ evolution of the observables (dashed lines – Eq. (\[lhs\])) for $\beta=0.1$. Note that for times up to about $20$ the dashed and solid lines are practically indistinguishable.[]{data-label="corr"}](corr.pdf){width="1\columnwidth"}
In Fig. \[regul\] we show (in black) the evolution of the first mode up to time $t=200$. It is seen that after the plateau it converges to zero, the value corresponding to the stationary solution (\[P\]) of the FPE. We will return to this figure in Section 5.
![FPE evolution of $O_1$ at $\beta=0.1$ (black). For comparison we show the evolution with regularization $K_{R,y} = -s\, y$, $s=0.1$ (red), see Section 4.[]{data-label="regul"}](s_0_1_max_smear){width="1\columnwidth"}
Direct study of the boundary terms
----------------------------------
We next study explicitly the evolution of the boundary term Eq. (\[boundt\]) for the modes $k=1,2,3$ and $\tau=0$ with Langevin time $t$. As explained in Appendix \[app1\] the definition of this term implies a certain order of limits: Integrate by parts restricted to $|y|\le Y$, send $t\to\infty$ and then $Y\to \infty$ (notice that this does not require a separate simulation but a certain processing of the data). We obtain for the $k$th mode in our model:
&B\_k(Y;t,0)= F\_k(t,) \_[=0]{}\
&= \_[-]{}\^ dx (x)(Y) \^[ikx]{}\
&. \[boundt1\] We first note that we can take the limit $t\to\infty$ of this expression, using the fact that $P(x,y;t)$ indeed converges to Eq. (\[P\]), which was verified before. We obtain &B\_k(Y;,0)=\
& -2\_[-]{}\^ dx. \[bound\_tanh\] For $k=\pm 1$ this can be evaluated to B\_[1]{}(Y;,0)=(Y), (converging to $\mp i\beta/2$ for $Y\to\infty$), whereas for $|k|>1$ we obtain $0$.
In Fig. \[boundterm\] we compare $B_1$ determined numerically for Langevin times up to $t=200$ with the asymptotic value at $t=\infty$ for $\beta=0.1$. $Y$ was chosen to be $5$ which is close to the asymptotic value $Y=\infty$ ($\tanh(5.)=0.99991$). We see here directly that the boundary term stays very small up to $t\lessapprox 20$, then picks up and approaches the analytically determined value $-i\beta/2$. For the value $\beta=0.1$ it also follows closely the difference between the first mode shown in Fig. \[regul\] and the correct value, but this cannot remain true for larger $\beta$.
We also checked the cases $k=2,3$ and found that $B_k$ also starts out very small up to about $t=20$, then increases and for large $t$ seems to go to the asymptotic value $0$ determined above. But one has to keep in mind that for $|k|\ge 2$ we are for $Y\to\infty$ evaluating a conditionally convergent integral; the CL process or equivalently the FPE evaluates that integral in a different way and may therefore produce different results. For $k=1$, however, there is no such subtlety and the boundary term $B_1(\infty;t,0)$ agrees with the slope of $F_k(t,\tau)$ at $\tau=0$.
![Numerical evolution via FPE of the imaginary part of the boundary term Eq. (\[boundt1\]) for $\beta=0.1$ (top) and $\beta=0.5$ (bottom), $k=1$ and $Y=5$.[]{data-label="boundterm"}](bound_lin_max_smear_beta_one "fig:"){width="1\columnwidth"} ![Numerical evolution via FPE of the imaginary part of the boundary term Eq. (\[boundt1\]) for $\beta=0.1$ (top) and $\beta=0.5$ (bottom), $k=1$ and $Y=5$.[]{data-label="boundterm"}](bound_lin_max_smear_beta_five "fig:"){width="1\columnwidth"}
In Fig. \[boundtermY\] we also show the boundary term $B_1$ for different values of the cutoff $Y$, showing the fast approach to the asymptotic value. Note that in the lower panel we show the boundary term as measured using the CLE alone, without making use of the Fokker-Planck evolution, which would be prohibitively costly in a lattice model.
![Top panel: numerical evolution via FPE of the imaginary part of the boundary term Eq. (\[boundt1\]) for $k=1$, $\beta=0.1$ and different values of $Y$ vs. $t$. Bottom panel: the same boundary term evaluated via Langevin simulation at asymptotic $t$ vs. $Y$.[]{data-label="boundtermY"}](bound_lin_max_smear "fig:"){width="1\columnwidth"} ![Top panel: numerical evolution via FPE of the imaginary part of the boundary term Eq. (\[boundt1\]) for $k=1$, $\beta=0.1$ and different values of $Y$ vs. $t$. Bottom panel: the same boundary term evaluated via Langevin simulation at asymptotic $t$ vs. $Y$.[]{data-label="boundtermY"}](CLE_DENES_BOUNDARY "fig:"){width="1\columnwidth"}
So we established implicitly and explicitly that boundary terms appear appreciably only after some Langevin time. Nonvanishing boundary terms at any $t>0$ invalidates the argument for correctness.
It can also seen by inspection of Eq. (\[boundt1\]) that the presence of the observable ${{\cal O}}_1$ is essential; the distribution of the drift force alone goes to zero. Quite generally it is the product of observable, drift and probability $P$ that decides about the presence or absence of boundary terms.
Boundary terms and skirts {#bc_skirts}
-------------------------
Thinking now of $Y$ not as a cutoff, but as a variable, and denoting it by $y$ again, we see that the the first term of the boundary term $B_1(y;\infty,0)$, considered as a function of $y$ Eq. (\[boundt1\]) is just the probability density of the observable v(y)dx K\_y(x,y) [[O]{}]{}\_1(x-iy) \~\^[2y]{} for large $y$. The nonvanishing of the boundary term is the fact that \_[y]{} v(y) P\_y(y)0. On the other hand the distributions of $v$ itself has a density $p(v)$, related to $P$ by p(v) = P\_y(y(v)). This can easily be worked out, but the point is that for large $y$ p(v)\~v\^[-2]{}, which shows that there is no finite expectation value of $v$ since $v\,p(v) $ is not integrable.
In other words: [*a ‘skirt’ in the distribution of $K_y{{\cal O}}$ falling off like the power $-2$ or more slowly corresponds to a nonvanishing (possibly diverging) boundary term*]{}. Note, however, such a simple reasoning is only possible because here $P$ is independent of $x$.
The interpolating function
--------------------------
So far we have only compared $F_k(t,0)$ and $F_k(t,t)$. But it is instructive also to look at the interpolating function $F_k(t,\tau)$ F\_[[O]{}]{}(t,)P(x,y;t-) [[O]{}]{}(x+iy;)dxdy, \[fttau\] which should be independent of $\tau$ for the correctness argument to hold. This is shown in Fig. \[fttaufig\] for $k=1$ for $\beta=0.1$ and $\beta=0.5$.
![The interpolating function $F_1(t,\tau)$ defined in (\[fttau\]) for the first mode; $\beta=0.1$ (top) and $\beta=0.5$ (bottom) for various values of $t$; the small circles denote the beginning and end of the respective curves. []{data-label="fttaufig"}](Fttau_max_smear_one "fig:"){width="1\columnwidth"} ![The interpolating function $F_1(t,\tau)$ defined in (\[fttau\]) for the first mode; $\beta=0.1$ (top) and $\beta=0.5$ (bottom) for various values of $t$; the small circles denote the beginning and end of the respective curves. []{data-label="fttaufig"}](Fttau_max_smear_five "fig:"){width="1\columnwidth"}
Again it is seen that for $\beta=0.1$, $t\lessapprox 20$ the curves are flat, indicating the absence of any appreciable boundary terms. For $t>20$ a $\tau$ dependence develops, being maximal near $\tau=0$. This is understandable from what we have seen: the FPE evolution of $P$ proceeds up to time $t-\tau$, which allows for the boundary terms to arise. On the other hand, for $\tau\gtrapprox 7\;$ ${{\cal O}}_k(z;\tau)$ has practically reached its asymptotic limit (cf. Appendix \[app2\]), in which only the constant mode survives; this constant can be pulled outside the integral defining $F$, so that for $t,\tau > 7 $ [[O]{}]{}\_k(z;) \_[-]{}\^dx’(x’)[[O]{}]{}\_k(x’) = [[O]{}]{}\_k\_c and F\_k(t,)dxdy P(x,y;t)[[O]{}]{}\_k\_c = [[O]{}]{}\_k\_c. i.e. the correct value (where we used the fact that the density $P$ is always normalized).
At small $t,\tau $ flat curves for $F_1(t,\tau)$ indicate that CL gives the correct values, however these are dependent on the initial condition if the process did not yet thermalize. This is seen in Fig. \[fttaufig\], bottom plot, for $\beta=0.5$.
Notice that the slope of $F_k(t,\tau)$ appears maximal near $\tau=0$ for large $t$. Therefore the estimation of noxious boundary terms as defined in (\[boundt1\]) is relevant for judging the asymptotic correctness of the CL procedure – cf. Figs. \[boundterm\], \[fttaufig\].
Plots similar to Fig. \[fttaufig\] appeared in [@Aarts:2011ax] for a different model.
Evolution of some marginal distributions
----------------------------------------
For $\beta=0.1$ we saw clearly the evolution first apparently converging to the correct value and then departing from it (the ‘plateau’ in Fig. \[corr\]). Similar behavior in Langevin time was observed in a real time $SU(2)$ lattice simulation [@Berges:2006xc]. This is also reflected in some marginal distributions.
![The marginal distribution $P_x(x;t)$ obtained from solving the Fokker-Planck equation for $\beta=0.1$.[]{data-label="marginalx"}](linplot_Px_max_smear){width="1\columnwidth"}
In Fig. \[marginalx\] we show the evolution of $P_x(x;t)=\int dy
P(x,y;t)$ for $\beta=0.1$. It starts out flat, corresponding to our choice of initial condition; at $t=10$ and $t=20$ it shows maximal structure, while for larger $t$ it approaches a flat distribution again, in agreement with (\[P\]).
The distribution of the first mode also show a similar behavior. Of interest is the imaginary part. Its density is (u;t)dx dy P(x,y;t) ((x)e\^[-y]{}-u). \[sigmadef\] We present in Fig. \[sigmau\] histograms for $\sigma(u;t)$, obtained from the numerical solution of the FPE; for the limiting distribution $P(x,y;\infty)=1/(4\pi\cosh^2(y)$ we can evaluate (\[sigmadef\]) analytically:
![Evolution of the distribution of the first mode $\sigma(u)$. Top panel: times from $t=1$ to 200 in linear scale; bottom panel: times from $t=50$ to 200 in log scale. Again note that for $t=200$ no difference is visible between the numerical results and the analytic expression.[]{data-label="sigmau"}](sigma_lin_u_max_smear "fig:"){width="1\columnwidth"} ![Evolution of the distribution of the first mode $\sigma(u)$. Top panel: times from $t=1$ to 200 in linear scale; bottom panel: times from $t=50$ to 200 in log scale. Again note that for $t=200$ no difference is visible between the numerical results and the analytic expression.[]{data-label="sigmau"}](sigma_log_u_max_smear "fig:"){width="1\columnwidth"}
(u;)&=\_[-1]{}\^1 dt\
&=. \[sigma\_anal\] Fig. \[sigmau\] shows first the development of an asymmetric structure with two maxima, whereas for larger $t$ one sees clearly the approach to the symmetric analytic result (\[sigma\_anal\]).
In this context it may be of interest to compare with the criterion of [@Nagata:2016vkn]. In Fig. \[histodrift\] we show the distribution of the drift itself for various Langevin times in double logarithmic scale. The decay always seems power-like, albeit with a very high power for short times. This would indicate, according to [@Nagata:2016vkn], that even for the small times where the CL results seem to be correct (but not necessarily converged) there might be a tiny boundary term making the results incorrect by an invisible amount.
![Histograms of the drift for different Langevin times $t$ and $\beta=0.1$.[]{data-label="histodrift"}](drift_histogram){width="1\columnwidth"}
Illustration of the effect of the boundary terms in a regularized model {#sect_reg}
=======================================================================
In the preceding section we described how the boundary terms accumulate in the Langevin (and Fokker-Plank) evolution, spoiling the proof of convergence such that the process would lead to wrong results.
Here we want to explicitly see the effect of those terms by considering a ‘regularization’ of the model using a damping term in the action, $S_R = \frac{s}{2}\,x^2$, which leads to a modification of the drift by $K_R(z)=- s\,z$ (a similar regularization has been used in [@Loheac:2017yar]; we thank J. Drut and A. C. Loheac for making us aware of this). The philosophy of this regularization is very similar to that of dynamical stabilization [@Attanasio:2018rtq]. In both cases, and different from modifications using symmetries, such as in the gauge cooling paradigm, the dynamics is really changed, but in a way intended to be controllable.
For $s=0$ we regain the original model Eq. (\[model\]) (including its problems) while for $s > 0$ we should observe an interplay between the original tendency to build boundary terms and their damping in the modified model, allowing us to estimate the effect of these terms. This particular modification leads to loss of periodicity in $x$ which becomes noncompact at $s>0$. The CLE process was allowed to drift unbounded in the full $z$ plane and the exact integral was correspondingly done in the infinite interval. The following plots show $O_1= {{\rm Im}}{\langle}e^{iz} {\rangle}$.
![Comparison of the $O_1$ expectation values from FPE (solid line) and CLE and from regularized CLE, vs $t$. [Top]{}: $\beta=0.1,\, s=0$ and $0.1$, respectively. [ Bottom]{}: $\beta=0.5,\, s=0$ and $0.4$, respectively.[]{data-label="f.regt"}](fts_0-01_dm_one.pdf "fig:"){width="1\columnwidth"} ![Comparison of the $O_1$ expectation values from FPE (solid line) and CLE and from regularized CLE, vs $t$. [Top]{}: $\beta=0.1,\, s=0$ and $0.1$, respectively. [ Bottom]{}: $\beta=0.5,\, s=0$ and $0.4$, respectively.[]{data-label="f.regt"}](fts_0-04_dm_five.pdf "fig:"){width="1\columnwidth"}
We see from Fig. \[f.regt\] that the regularization stabilizes the expectation values in the CLE evolution. When the non-regularized data show a plateau at the correct value for intermediary $t$ the regularization extends this plateau into the asymptotic region ($\beta=0.1$ case). When a plateau is missing the regularization still stabilises the expectation value (EV) but at a value shifted from the correct one ($\beta=0.5$ case), since now a larger $s$ is needed to counteract the boundary terms.
Note that an alternative regularization of the process itself is to modify [*only the imaginary drift*]{} by a damping term $K_{R,y}=-s\,y$. This leads to similar results (see Fig. \[regul\], here from the FPE evolution), and has the advantage that periodicity in $x$ is preserved. We preferred the action variant, however, also since it allows us to obtain exact correct results for the regularized model by simple numerical integration.
In Fig. \[f.regs\] we show the $s$ dependence in CLE for the regularized model for the same values of $\beta$. The plots suggest an extrapolation toward the exact expectation value (EV) for $s \rightarrow 0$, however this might not be simply linear, but depend on the particular regularization, $\beta$, etc. Therefore we mean this discussion not yet as a direct cure but mainly as illustration of the effects of the boundary terms on the EV’s. For $k=1$, e.g., these effects can be estimated from the distance between the CLE regularized data and the exact values from the numerical integration of the regularized model: As can be seen from the figure at small $s$ these effects are still present while gradually vanishing with increasing $s$.
![Dependence of $O_1$ on the regularization parameter $s$ of the CLE simulation at $\beta =0.1$ (top) and $0.5$ (bottom). The solid lines show the exact correct values from numerical integration.[]{data-label="f.regs"}](fs_reg_dm_one.pdf "fig:"){width="1\columnwidth"} ![Dependence of $O_1$ on the regularization parameter $s$ of the CLE simulation at $\beta =0.1$ (top) and $0.5$ (bottom). The solid lines show the exact correct values from numerical integration.[]{data-label="f.regs"}](fs_reg_dm_five.pdf "fig:"){width="1\columnwidth"}
![Histogram of the drift for $\beta=0.1$ and various Langevin times, comparing $s=0$ and $s=0.1$.[]{data-label="histodrifts"}](drifthistb01sexp){width="1\columnwidth"}
Finally it is again instructive to look at the histograms of the drift itself, as advocated by [@Nagata:2016vkn], to see the effect of the regularization. This is shown in Fig. \[histodrifts\]. One can see that the distribution seems to show power-like decay $s=0$ whereas for $s=0.1$ the decay appears to be exponential. This supports the criterion of [@Nagata:2016vkn], because for the value of $\beta=0.1$ used here, already $s=0.1$ suffices to bring the CL results into agreement with the correct results of the regularized as well as the unregularized model, which are indistinguishable in this case, as shown in Fig. \[f.regs\].
Conclusions
===========
We have in great detail analyzed a simple example in which the CL fails, establishing very explicitly that the failure is due to boundary terms spoiling the correctness argument, as argued already long ago [@Aarts:2009uq; @Aarts:2011ax]. The absence of such boundary terms requires that the product of observable, drift force and probability distribution (${{\cal O}}K P$) goes to zero in the noncompact (imaginary) directions. The relation between boundary terms and ‘skirts’, i. e. decay of distributions was addressed in Subsection \[bc\_skirts\], making clear that possible skirts in the distribution of the product ${{\cal O}}K P$ and not just $KP$ are relevant. Remarkably, the criterion proposed by [@Nagata:2016vkn] does not involve the observable.
Generally the $\tau-$dependent boundary term (\[boundtgeneral\]) cannot be estimated in a realistic (lattice) calculation. Fortunately, however, the considerations in this paper suggest that relevant for the correctness of the asymptotic (large $t$) EV’s is the boundary term at $\tau=0$, $B_k(t,0)$, as defined in (\[boundt1\]). This term appears to approximately maximize $B_k(t,\tau)$ and it stabilizes at large $t$; it is accessible in principle to online monitoring using the CLE alone, and may provide a correctness criterion for the EV’s obtained in the CL simulation.
As remarked before, uncovering the boundary term requires a certain processing of the data obtained in the simulation, implying essentially sampling first at a fixed value of a quantity specifying the boundary in the non-compact directions (in a lattice gauge theory for instance the unitarity norm or some other related quantity) before taking the other limits. This however does not require a separate simulation.
[*Acknowledgments:*]{} M. S., E. S. and I.-O. S. gratefully acknowledge kind support from DFG under Grant Sta 283/16-2. D. S. gratefully acknowledges funding by the DFG grant Heisenberg Programme (SE 2466/1-2). The authors acknowledge support by the High Performance and Cloud Computing Group at the Zentrum für Datenverarbeitung of the University of Tübingen, the state of Baden-Württemberg through bwHPC and the German Research Foundation (DFG) through grant no INST 37/935-1 FUGG.
The argument for correctness revisited {#app1}
======================================
We briefly revisit the formal proof of correctness for the CL method for our simple periodic one-dimensional models, spelling out the conditions needed for it to work as well as the mechanisms that may lead either to no convergence or ‘wrong convergence’ of the CL process (cf. [@Aarts:2011ax; @Aarts:2017vrv; @Seiler:2017wvd]).
$P(x,y;t)$ is the time dependent probability distribution corresponding to the CL process, determined by the [*real*]{} Fokker-Planck equation (\[realFPE\]).
We also consider the time evolution of the complex density $\rho(x;t)$ by the [*complex*]{} Fokker-Planck equation \[complexFPE\] (x;t)= L\_c\^T (x;t), where now the complex Fokker-Planck operator $L_c^T$ is \[fpc0\] L\_c\^T = \_x . The initial conditions for (\[realFPE\]) and (\[complexFPE\]) are required to be consistent, i.e. P(x,y;0)&=(x;0)(y)0\
(x;0)dx&=1. The crucial point is that one can now, under conditions to be spelled out below, show that dx[[O]{}]{}(x) (x;t)= dx dy [[O]{}]{}(x+iy) P(x,y;t). \[correctness\] If in addition the operator $L^T_c$ has spectrum in the left half plane with 0 a nondegenerate eigenvalue, if follows that \_[t]{}dx[[O]{}]{}(x) (x;t)= dx[[O]{}]{}(x) (x) and by (\[correctness\]) \_[t]{} dx dy &[[O]{}]{}(x+iy) P(x,y;t)\
&=dx[[O]{}]{}(x) (x). By our choice of initial conditions, (\[correctness\]) holds for $t=0$. For $t>0$ we consider $F_{{\cal O}}(t,\tau)$ defined in Eq. (\[fttau\]), which interpolates between the two sides of (\[correctness\]): F\_[[O]{}]{}(t,)P(x,y;t-) [[O]{}]{}(x+iy;)dxdy, with ${{\cal O}}(x+iy;t)$ defined by solving the differential equation (\[obsevol\]) $L_c$, the complex Langevin operator, is the transpose of $L^T_c$: L\_c =\_z. We call the solution of Eq. (\[obsevol\]) the ‘$L_c$ evolved’ observable.
The interpolating property follows from
F\_[[O]{}]{}(t,0)&= dx dy [[O]{}]{}(x+iy) P(x,y;t)= [[O]{}]{}\_t\
F\_[[O]{}]{}(t,t)&=dx[[O]{}]{}(x,t) (x;0)= [[O]{}]{}(t)\_c, see Section \[indirect\], Eqs. (\[lhs\]) and (\[rhs\]). The first equality is obvious, the second one follows by integration by parts in $x$; because of periodicity there are no boundary terms. (\[correctness\]) would follow if we could prove \[correctinter\] & F\_[[O]{}]{}(t,) =\
& -(L\^T P(x,y;t-))[[O]{}]{}(x+iy;)dxdy\
& + P(x,y;t-) L\_c[[O]{}]{}(x+iy;) dxdy=0. This would again follow from integration by parts, provided there are no boundary terms. For the term $\partial_x^2$ of both $L^T$ and $L_c$ this is obvious because of periodicity, so we can drop these terms, obtaining $$\begin{aligned}
&\frac{\partial}{\partial \tau} F_{{\cal O}}(t,\tau) =\nonumber\\
&\int {{\cal O}}(x+iy;\tau)(\partial_x K_x +\partial_y K_y) P(x,y;t-\tau)
dxdy\notag\\
& - \int P(x,y;t-\tau) S'(x+iy)\partial_x{{\cal O}}(x+iy;\tau) dx dy\,.
\label{diff1}\end{aligned}$$ In [@Aarts:2009uq] we argued that ${{\cal O}}(x+iy;\tau)$ is holomorphic for any $\tau$, i.e. it obeys the Cauchy Riemann equations \_y [[O]{}]{}(x+iy;)= i \_x [[O]{}]{}(x+iy;). This allows us to write the second term of the right hand side of (\[diff1\]) as P(x,y;t-) (K\_x\_x+K\_y \_y) [[O]{}]{}(x+iy;) dx dy. \[diff0\] Again the part involving $\partial_x$ can be canceled against the corresponding term in the first term of (\[diff0\]) using integration by parts in $x$, so we only have to consider $$\begin{aligned}
&\frac{\partial}{\partial \tau} F_{{\cal O}}(t,\tau) =\nonumber\\
&\int \left(\partial_y K_y P(x,y;t-\tau)\right)
{{\cal O}}(x+iy;\tau)dxdy\notag\\
& +\int P(x,y;t-\tau) K_y \partial_y{{\cal O}}(x+iy;\tau) dx dy\,.
\label{diff2}\end{aligned}$$ We have to interprete this as a the limit $Y\to\infty$ of the integral restricted to $|y|\le Y$. For finite $Y$ (\[diff2\]), since the integrand is a total derivative, this is given by the boundary term &B\_[[O]{}]{}(Y;t,)\
&dx. \[boundtgeneral\] Evaluating this term at $\tau =0$ leads then for our model to (\[boundt1\]) where we can then take the large $t$ limit to obtain for our model, obtaining (\[bound\_tanh\]). This form of the boundary term makes clear that correctness requires sufficient decay of the products $K_y P{{\cal O}}$.
Notice that if we take the $t \rightarrow \infty$ limit directly in (\[correctinter\]) the first term vanishes by stationarity and the second one leads to the ‘Correctness Conditions’ (CC) defined in [@Aarts:2011ax] and is approximately zero by stationarity. Hence, it might appear that the boundary term vanishes and we might erroneously conclude correctness of the results. Therefore the CC, while expressing convergence and being necessary for correctness, are not sufficient.
The correct evolution {#app2}
=====================
What was called the ‘correct time evolution’ ${\langle}{{\cal O}}(t){\rangle}_0 $ of the expectation value of ${{\cal O}}$ is simply the expectation value in the starting probability density $P(x,y;0)$ of the $L_c$ evolved observable ${{\cal O}}$, see Eq.(\[lhs\]). To analyze this we rewrite the Langevin operator $L_c$ in the basis of Fourier modes: L\_c (ikx) &= -k\^2 (ikx)\
&- k (i(k+1)x)\
&+ k (i(k-1)x) or equivalently, for a general observable [[O]{}]{}(x)=\_k a\_k (ikx) (L\_c a)\_k=&-k\^2 a\_k - (k-1) a\_[k-1]{}\
+& (k+1)a\_[k+1]{}. \[obsevol2\] So $L_c$ is represented on the Fourier coefficients by the sparse infinite matrix with elements (L\_c)\_[kl]{}=&-k\^2\_[kl]{}- (k-1)\_[k-1,l]{}\
&+ (k+1)\_[k+1,l]{}. It is easy to compute numerically the action of $\exp(t L_c)$ on observables of the form ${{\cal O}}_k=\exp(ikx)$; cutting off the modes at $|k|\ge K$ with $K=50$ and $K=150$, and for $t=50$, gave identical results, with only the constant mode surviving. Its value agrees to at least 5 digits with \_[t]{} (t L\_c) [[O]{}]{}\_k= dx (x)[[O]{}]{}\_k(x) i.e. the correct expectation value.
We also checked, using Mathematica, that the eigenvalues of the truncated matrix $(L_c)_{kl}$ have negative real part except for the unique zero eigenvalue corresponding to $a_k\propto\delta_{k0}$. All nonzero eigenvalues are real and doubly degenerate. The one with the smallest modulus determines the approach to the infinite time limit; it depends only weakly on $\beta$, e. g. &\_1=-0.998333(=0.1);\
&\_1= -0.832189 (=1) \[eigenval\] It is easy to show that by a similarity transformation $L_c$ can be transformed into the [*dissipative*]{} operator -H &= (S/2) L (-S/2)\
&= -\^2 \^2(x)-(x). Dissipativity means $-H-H^*\le 0$, which is obvious. For such operators general theorems guarantee that the spectrum is contained in the left half of the complex plane (see for instance [@davies]). It is also not hard to see that there is exactly one vector with eigenvalue zero.
Remarks on the numerical solution of the FPE {#FPE}
============================================
The real Fokker-Planck equation in our case is $$\begin{aligned}
&\frac{\partial P(x,y;t)}{\partial t} =
\left[\partial_{x}\left(\partial_x - K_x\right) - \partial_y K_y\right]
P(x,y;t) \nonumber \\ =
& [\partial_{x}^2 + \beta(-2\sin x \sinh
y + \cos x \sinh y \partial_{x} \nonumber\\
&- \sin x \cosh y
\partial_{y})]P(x,y;t)\, .
\label{modelFPE}\end{aligned}$$
Discretizing in $x$ and $y$ using symmetric derivatives yields $$\begin{aligned}
&P(x, y; t+dt) =\nonumber\\
& \frac{1}{dx^2} \left(P(x_+, y; t) -2P(x,y;t) +
P(x_-,y;t)\right)\nonumber\\
-& 2\beta \sin x \sinh y P(x,y;t)\nonumber \\
+& \frac{\beta}{2dx} \cos x \sinh y
\left(P(x_+,y;t)-P(x_-,y;t)\right)\nonumber \\
-& \frac{\beta}{2dy} \sin x \cosh y \left(P(x,y_+;t)-P(x,y_-;t)\right)\, ,\end{aligned}$$ where we defined $x_{\pm}=x\pm dx$ and similarly for $y$. In case of a regularization term in the $y$-drift $K_y\rightarrow K_y-s_y
y$ (see Section \[sect\_reg\]), additional terms occur $$\begin{aligned}
&P(x, y; t+dt) \rightarrow P(x, y; t+dt) + s_y P(x,y;t) \nonumber \\
&+ s_y \frac{y}{2 dy}\left(P(x,y_+;t)-P(x,y_-;t)\right)\, .\end{aligned}$$
We solved the Fokker-Planck equation on an $x$-$y$-grid with parameters $dt=10^{-6}$, $dx=0.005=dy$, a cutoff in $y$-direction of $Y=5$ was found to be sufficient (compare , $\tanh(5)\approx\tanh(\infty)$), and a cutoff in $x$-direction of $X=3.14$, which is due to the $2\pi$ periodicity of the problem. Boundary conditions in $x$ and $y$ were both chosen to be periodic. Initial condition were chosen according to , however the $\delta$-function was smeared out slightly to avoid numerical issues; so we actually used $$P(x,y;0)=\frac{1}{2\pi\sqrt{2\pi \sigma_{y}^2}}e^{-\frac{y^2}{2\sigma_y^2}}\, ,$$ where we chose $\sigma_y=0.1$. Note that using this discretization it is hard to resolve the higher modes. This can be done more easily when solving the Fokker-Planck equation in Fourier space, where it is given by $$\begin{aligned}
&P(k,y;t+dt)= -k^2P(k,y;t) \nonumber\\
&-\frac{i\beta}{2}\sinh(y)\left(k_- P(k_+,y;t)+
k_+P(k_-,y;t)\right)\nonumber\\
&+\frac{i\beta}{4dy}\cosh(y)\left(P(k_+,y_+;t)-P(k_-,y_+;t)\right)\nonumber\\
&+\frac{i\beta}{4dy}\cosh(y)\left(-P(k_+,y_-;t)+P(k_-,y_-;t)\right)\, ,\end{aligned}$$ where $k_\pm=k\pm 1$ and similarly for $y$. Here we chose $dt=0.5\times 10^{-5}$, $k\in\{-19,\ldots,20\}$, $dy=\sqrt{dt}$, $Y\approx 2.8$ with antiperiodic boundary conditions in $k$ for the imaginary part of $P(k,y;t)$ and periodic boundary conditions for the real part in $k$ and for $y$. After $t\sim 30$ or so the result strongly depends on the choice of discretization. Hence, we use the $k$-$y$ discretization to resolve the plateaus in the higher modes and the $x$-$y$-discretization for everything else.
The solution to the Fokker-Planck equation shows that for $\beta=0.1$ the evolution initially follows the correct evolution. This is suggested by Figs. \[fttaufig\] and \[boundterm\]. By looking at $P(x,y;t)$ in Figs. \[marginal\] and \[marginalx\] and the histogram of the first mode in Fig. \[sigmau\], one can see that initially nontrivial structures occur. Those die out and everything approaches the asymptotic solution, which yields the wrong results. This strengthens the argument that until $t\sim 20$ or so CLE yields the correct solution but then the occurrence of boundary terms leads to wrong convergence.
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|
---
author:
- 'Jorick S. Vink'
title: 'Fast & slow winds from supergiants and Luminous Blue Variables'
---
Introduction {#s_intro}
============
Mass loss is an important driver of massive star evolution (Chiosi & Maeder 1986; Langer 2012). This is thought to occur via stationary stellar winds on & off the main sequence (Vink & Gräfener 2012; Groh et al. 2014), and possibly also in eruptive mode during Luminous Blue Variable (LBV) events (Shaviv 2000; Smith & Owocki 2006; Owocki 2015). Although detailed stationary wind models of LBVs have been constructed by Vink & de Koter (2002) using the Abbott & Lucy (1985) Monte Carlo radiative transfer approach, the predicted mass-loss rates () were semi-empirical in nature, and assumed terminal wind velocities (). Dynamical modelling has yet to be explored. In this paper, we will employ the dynamically-consistent approach of Müller & Vink (2008) to predict velocity structures and mass-loss rates for a range of OB supergiants and LBV models, as well as their metallicity ($Z$) dependence.
Pauldrach & Puls (1990) first encountered the bi-stability jump in modelling the wind of the LBV PCygni. Lamers et al. (1995) subsequently found observational evidence for the bi-stability jump through a drop in wind velocities by a factor of two for a sample of supergiants around spectral type B1 – at an effective temperature of about 21000 K (see also Crowther et al. 2006). It it was originally assumed that the jump was caused by the optical depth of the Lyman continuum, until Vink et al. (1999) showed that the recombination of the main line-driving element iron (Fe) caused an increased amount of line acceleration from Fe [iii]{} – and an increase in the mass-loss rate by a factor of five. As these models were semi-empirical in nature, the drop in terminal wind velocity has yet to be theoretically modeled.
The issue of whether the mass-loss rate increases at the bi-stability location, as predicted, or whether it drops instead as suggested by empirical results (Trundle et al. 2004; Crowther et al. 2006; Benaglia et al. 2007; Markova & Puls 2008; Morford et al. 2016), remains unresolved, and may depend on the question whether the discrepancy may be attributed to macro-clumping, as Petrov et al. (2014) showed that the H$\alpha$ line changes its character completely, from an optically thin to an optically thick line below the bi-stability jump.
In the current paper, we present new dynamically consistent mass-loss predictions on both sides of the bi-stability jump, and in turns our that we are indeed able to confirm the observed drop in terminal wind velocities. Moreover, we predict an even [*stronger*]{} jump in the mass-loss rate by a [*factor of 10*]{} than the factor of $\sim$5 that we found originally, with relevant consequences for massive star evolution, including the efficiency of bi-stability braking (Vink et al. 2010), the possible formation of B\[e\] supergiant disks (Lamers & Pauldrach 1991), the slow winds in High-Mass X-ray binaries (HMXBs), LBVs as supernova SN progenitors (Trundle et al. 2008; Groh & Vink 2011), and very massive stars (VMS) as the possible origin of observed chemical anti-correlations in globular clusters (Vink 2018).
In Sects. \[s\_model\] we briefly describe the Monte Carlo modelling and physical assumptions. In Sect. \[s\_res\] mass-loss rates and wind terminal velocities are presented for a canonical 60 supergiant across the temperature regime of the bi-stability jump, whilst Sect.\[s\_lbv\] describes similar results for LBVs, characterized by a larger Eddington $\Gamma$ parameter. The $Z$ dependence is discussed in Sect.\[s\_lbvz\], before ending with a summary in Sect. \[s\_sum\].
Physical assumptions and Monte Carlo modelling {#s_model}
==============================================
We simultaneously predict mass-loss rates and wind velocity structures on the basis of the line-driven wind model of Lucy & Solomon (1970) and Castor, Abbott & Klein (1975; CAK) including multi-line scattering physics (Abbott & Lucy 1985). In the first part of the paper, we expand on the supergiant results of Vink et al. (1999), and we subsequently move on to LBV models, improving the Vink & de Koter (2002) results. The dynamical improvements are based on the Müller & Vink (2008) approach.
The underlying model atmosphere is the Improved Sobolev Approximation code [isa-wind]{} (de Koter et al. 1993) in which the effects of the diffuse radiation field are included in the line resonance zones. It computes H, He, C, N, O, S, Si, and Fe explicitly in non-LTE, but as we only found minor differences when treating Fe in the modified nebular approximation (Schmutz 1991), we decided to treat Fe approximately. [isa-wind]{} treats the star (“core”) and wind (“halo”) in a unified manner, i.e. there is no core-halo approximation. The temperature is calculated using radiative equilibrium in an extended grey LTE atmosphere, and is not allowed to drop below a value of half the effective temperature.
In the Monte Carlo part the lines are described in the Sobolev approximation, which is an excellent approximation in the outer parts of the winds, where velocity gradients are substantial. This may provide confidence in our aim of predicting the outer wind dynamics and terminal wind velocity correctly. However, if subtle non-Sobolev effects in the inner wind are relevant, this may have relevant implications for the predicted values of our mass-loss rates (see Krticka & Kubat 2017). Observational and theoretical line transitions have been adopted from Kurucz as previously (Kurucz & Bell 1995).
The abundances are taken from Anders & Grevesse (1989). Although it has been argued that the overall solar metallicity is smaller than it was thought to be about a decade ago, the Fe abundance does not appear to have changed during this time. Although the effect of a smaller overall solar metallicity might lead to lower mass-loss predictions, given the dominance of Fe in setting the mass-loss rate, the expected differences may turn out to be relatively small. In any case, the prime reason to keep the abundances the same as in previous (Vink et al. 1999; 2000; 2001) predictions is that this approach allows for more straightforward comparisons.
Our 1D wind models are spherically symmetric and homogeneous, although wind clumping (micro-clumping) may result in a downward adjustment of [*empirical*]{} mass-loss rates, by a factor of $\simeq$3 (Hillier 1991; Moffat & Robert 1994; Davies et al. 2007; Puls et al. 2008; Hamann et al. 2008; Sundqvist et al. 2014; Ramírez-Agudelo et al. 2017), whilst it may also affect the driving itself (Muijres et al. 2011).
As LBVs find themselves in close proximity to the observed Humphreys-Davidson limit, which is thought to be associated with the theoretical Eddington limit, additional physics may lead to the development of porous structures (van Marle et al. 2008; Gräfener et al. 2012; Jiang et al. 2015). Porosity effects on mass-loss predictions were also investigated by Muijres et al. (2011), where it was noted that it is unlikely that predictions would change dramatically. However, porosity (or macro-clumping) may have important implications on observational indicators (Oskinova et al. 2007; Surlan et al. 2013; Sunqvist et al. 2014; Petrov et al. 2014).
A dynamically consistent bi-stability jump {#s_res}
==========================================
[llll|l|ccc]{}\
& $\log L$ & & $Z/{\mbox{$Z_{\odot}$}}$ & & & $\log {\mbox{$\dot{M}$}}$ & $\beta$\
\[\] & \[\] & (kK) & & \[\] & \[\]& \[\] &\
\
60 & 6.0 & 40 & 1.0 & 1049 & 3512 & $-$5.47 & 0.99\
& & 37.5 & & 983 & 2989 & $-$5.38 & 1.02\
& & 35 & & 918 & 3139 & $-$5.51 & 1.14\
& & 32.5 & & 852 & 2766 & $-$5.48 & 1.10\
& & 30 & & 787 & 2381 & $-$5.53 & 1.08\
& & 27.5 & & 721 & 2884 & $-$5.83 & 1.28\
& & 25 & & 655 & 2743 & $-$5.87 & 1.40\
& & 22.5 & & 590 & 1837 & $-$5.25 & 1.10\
& & 20 & & 524 & 818 & $-$4.74 & 0.81\
& & 17.5 & & 459 & 362 & $-$4.48 & 0.69\
\
Table \[tab:results\] lists the Monte Carlo predictions for our canonical 60 supergiant over a range of effective temperatures. The stellar parameters are identical to model series \#10 of Vink et al. (2000). They are listed in columns (1) - (5), whilst predicted wind terminal velocities, new mass-loss rates, and wind acceleration parameter $\beta$ from $v(r) = {\mbox{$\varv_{\infty}$}}(1 - r/R)^\beta$ are listed in columns (6) - (8). The mass-loss predictions are also shown in Fig. \[f\_mdot\], and the predicted terminal wind velocities are displayed in Fig. \[f\_vinf\].
Figure 1 shows that the mass-loss rates increase dramatically [*by an order of magnitude*]{} between 25000 and 20000 K. Moreover, the terminal wind velocity is found to drop significantly over the same temperature range (Fig.2). This second result is new, whilst the first result is more dramatic than the factor of five found from the semi-empirical Vink et al. (1999) models, although the overall mass-loss rates are in reasonable agreement with the Vink et al. (1999; 2000; 2001) rates.
Figure 2 shows relatively low terminal wind velocities on the cool side of the bi-stability jump down to 400 km/s, instead of values in the range 2000-3500 km/s for hotter objects. As the stellar escape velocity also drops at lower due to the larger stellar radii, it is more insightful to consider the ratio over instead. This ratio is plotted in Fig.3. It is seen that the ratio drops rather steeply from values larger than 3 at hot temperatures to values below 1 on the cool side of the bi-stability jump.
Note that observationally, Crowther et al. (2006) found somewhat smaller values for this ratio with an average value of 3.4 on the hot side of the bi-stability range, and somewhat higher values on the cool side (an average value of 1.9). This raises the question whether our predicted wind velocities are too high on the hot side of the bi-stability jump, and too low on the cool side, which would have consequences for our predicted mass-loss rates as well. For this reason it is helpful to consider the wind momentum efficiency number $\eta$ which is defined as the ratio between the wind momentum per unit time ($\dot{M} {\mbox{$\varv_{\infty}$}}$) over the momentum of the radiation field per unit time ($L/c$), or $\eta = \dot{M} {\mbox{$\varv_{\infty}$}}/ (L/c)$, and displayed in Fig.\[f\_eta\]. Similar to the earlier computations of Vink et al. (2000), the $\eta$ behaviour shows an increase in the wind efficiency by a factor 2-3, from 25000 K onwards, now peaking at 20,000 K. This bi-stability temperature is in agreement with the peak temperature of our alternative [cmfgen]{} approach of Petrov et al. (2016), as well as the observed bi-stability location around spectral type B1 (Lamers et al. 1995; Crowther et al. 2006).
As the mass-loss rate discrepancy is considered to be unresolvable at the current time until appropriate atmosphere modelling including macro-clumping becomes a reality, the issue of the wind terminal velocity deserves extra attention, as the terminal wind velocity is generally considered to be the more robust empirical wind parameter. Their values are generally derived from the maximum blue-shifted absorption in resonance lines in the ultraviolet part of the spectrum. Systematic errors may work in both directions, and it may be difficult to see how empirical values could be underestimated on the hot side of the bi-stability jump, whilst being overestimated on the cool side of the jump.
However, it is not inconceivable that this would indeed be the case when the wind physics completely changes at 21000 K. It is for instance well possible that for the faster winds on the hot side of the bi-stability jump the measured lines have not yet reached their predicted terminal wind speeds yet, resulting in an underestimation of , whilst the slower winds on the cool side of the jump may be overestimated due to an increased wind turbulence. This is of course rather speculative, and this would require further investigation to find out whether the outflow speeds on the cool side of the jump are indeed as fast as derived empirically, or as slow as predicted by our Monte Carlo modelling.
A stronger bi-stability jump may have important consequences for bi-stability braking (Vink et al. 2010; Markova et al. 2014; Keszthelyi et al. 2017) as well as the formation of dense disks around B\[e\] supergiants via the rotationally-induced bi-stability jump mechanism of Lamers & Pauldrach (1991). The basic idea of this model is that the cooler stellar equator has a higher mass flux and lower wind velocity than the hotter pole, due to the Von Zeipel effect. In the updated computations of Pelupessy et al. (2000) that employed the wind parameters of Vink et al. (1999) this resulted in a density contrast between the equator and the pole of a factor of 10. At the time this was deemed insufficient to explain the dense disks of B\[e\] supergiants, which lead Curé et al. (2005) to combine the bi-stability mechanism with slow wind solutions at very high rotation speeds.
Our new bi-stability models here predict both an order of magnitude increase in the mass-loss rate, as well as an order of magnitude drop in the wind velocity. In the 2D models of Pelupessy et al. (2000) and Müller & Vink (2014) this implies a density contrast between the stellar equator and the pole by a factor of 100, which may be sufficient to explain the disk density of B\[e\] supergiants (Kraus 2017). Future multi-dimensional computations are needed to find out if the disk can remain this high density in the presence of non-radial line forces (Owocki et al. 1996), whilst we also require an explanation for the outflow speeds of just tens of km/s (Kraus et al. 2010; Cidale et al. 2012; Kraus et al. 2016).
Another interesting puzzle regarding wind velocities in massive stars involves the obscured supergiant HMXB IGR J17252-3616 uncovered by [Integral]{}. Manousakis et al. (2012) performed hydro-dynamical modelling, which appears similar to the unobscured classical system Vela X-1, but the authors could only explain the obscured system with a slow wind of order 500 km/s. A possible explanation for the low terminal velocity in IGR J17252-3616 and other HMXBs, such as Vela X1 (see Sander et al. 2018) would be that the supergiant donor star might be located on the cool side of the bi-stability jump where the outflow velocity is found to be lower. However, more work is needed to study the ionization and wind physics in the donor stars of HMXBs.
Mass loss from Luminous Blue Variables {#s_lbv}
======================================
For the second part of the paper we extend the supergiant computations to models characteristic for LBVs. The defining property of LBVs is their S Doradus variability over timescales of years (Humphreys & Davidson 1994; Vink 2012), which might be explained by stellar radius inflation (Gräfener et al. 2012). This property enables individual stars to cross the bi-stability jump on relatively short evolutionary timescales.
We adopt similar hydrogen (X=0.38) and helium (Y=0.60) fractions as we adopted in Vink & de Koter (2002), and as a representative model set we chose a similar set of mass, luminosity, and associated Eddington $\Gamma$ factors as in that paper. For the first model set in Table 2 we chose a much lower mass (of 35) for the same luminosity of ${\mbox{$\log (L/L_{\odot}$)}}= 6.0$ as in the previous section, resulting in an Eddington factor of 0.5, whilst the next series of models were chosen to have masses of 25 and 23 solar masses – for the same fixed luminosity – resulting in Eddington factors of 0.7 and 0.8 respectively. Note that these Eddington $\Gamma$ values only include the opacity of electron scattering, and for a discussion on the total opacity, we refer the reader to Vink et al. (2011). The relevance of the Eddington parameter for mass-loss rates was highlighted for VMS in the VLT Flames Tarantula Survey (Bestenlehner et al. 2014).
[lllll|l|ccc]{}\
& $\log L$ & $\Gamma$ & & $Z/{\mbox{$Z_{\odot}$}}$ & & & $\log {\mbox{$\dot{M}$}}$ & $\beta$\
\[\] & \[\] & & (kK) & & \[\] & \[\] & \[\] &\
\
35 & 6.0 & 0.5 & 27.5 & 1.0 & 551 & 1811 & $-$5.39 & 1.29\
& & & 25 & & 501 & 1863 & $-$5.45 & 1.44\
& & & 22.5 & & 451 & 1645 & $-$5.18 & 1.22\
& & & 21 & & 421 & 961 & $-$4.74 & 0.97\
& & & 20 & & 401 & 663 & $-$4.52 & 0.84\
& & & 19 & & 380 & 457 & $-$4.38 & 0.76\
& & & 17.5 & & 350 & 253 & $-$4.22 & 0.69\
25 & 6.0 & 0.7 & 27.5 & 1.0 & 465 & 1259 & $-$4.95 & 1.43\
& & & 25 & & 423 & 1412 & $-$5.01 & 1.68\
& & & 22.5 & & 381 & 1289 & $-$4.87 & 1.46\
& & & 21 & & 355 & 802 & $-$4.54 & 1.09\
& & & 20 & & 339 & 557 & $-$4.35 & 0.93\
& & & 19 & & 322 & 370 & $-$4.17 & 0.82\
& & & 17.5 & & 296 & 237 & $-$4.13 & 0.70\
23 & 6.0 & 0.8 & 27.5 & 1.0 & 446 & 1102 & $-$4.83 & 1.53\
& & & 25 & & 406 & 1313 & $-$4.87 & 1.87\
& & & 22.5 & & 365 & 1155 & $-$4.74 & 1.49\
& & & 21 & & 341 & 718 & $-$4.44 & 1.10\
& & & 20 & & 325 & 506 & $-$4.26 & 0.96\
& & & 19 & & 308 & 321 & $-$4.14 & 0.82\
& & & 17.5 & & 284 & 199 & $-$4.10 & 0.70\
The results from Table 2 are plotted in Figs. (5) - (9), showing the mass-loss rates, wind terminal velocities, and wind velocity over escape velocity ratio, the wind structure parameter $\beta$, and wind efficiency number $\eta$, respectively. The results are qualitatively similar to the earlier 60 supergiant model. Note that the jumps for these LBV models are steeper than for the supergiant model, and that in contrast to previous globally consistent predictions in Vink & de Koter (2002) the jump for LBVs has now shifted to the correct effective temperature of approximately 21000 K. The reason for the onset of the bi-stability jump at lower temperature than previously is thought to be that for the stars on the hot side of the jump the winds are now thinner (due to the lower mass-loss rates and higher wind velocities), and that the lower density causes the Fe [iv]{} to [iii]{} recombination of Vink et al. (1999) to occur at lower effective temperature.
An additional finding is that we find the size of the bi-stability jump, perhaps unexpectedly, to be [*less*]{} pronounced at higher $\Gamma$ values. We interpret this to be the result of saturation. Higher $\Gamma$ models already show higher mass-loss rates than lower $\Gamma$ models on the hot side of the bi-stability jump (as well as associated lower terminal wind velocities), so there is less of an opportunity to increase dramatically on the cool side of the bi-stability jump.
The absolute values of the terminal wind velocities are now down to 200 km/s, which is in the correct range for S Doradus type LBVs (Vink 2012). These slow winds are consistent with LBVs being the direct progenitors of Type IIb and IIn supernovae inferred from the the slow outflows (Kotak & Vink 2006; Trundle et al. 2008; Smith et al. 2008; Groh & Vink 2011; Groh 2014; Gräfener & Vink 2016), and inconsistent with the fast outflows expected from Wolf-Rayet stars (Soderberg et al. 2006; Gal-Yam et al. 2014).
Note that the steepest part of the increase in the mass-loss rate is seen in the temperature range 22- 20 kK, i.e. at [*lower*]{} values of than suggested in the Vink et al. 2000/2001 mass-loss recipe. These lower values of the bi-stability are in good accord with both the observed drop in terminal wind velocity around spectral type B1 (Lamers et al. 1995; Crowther et al. 2006), as well as [cmfgen]{} modelling by Petrov et al. (2014; 2016). This means that our earlier attribution of the bi-stability jump temperature offset to the use of the modified nebular approximation was not correct. The reason for the discrepancy was the semi-empirical nature of the earlier approach instead.
Figure \[f\_lbv-beta\] shows the behaviour of the wind structure parameter $\beta$ as a function of temperature. Whilst values on both the cool and the hot side of the bi-stability jump are in the range 0.7 - 1.5, and in general accord with earlier CAK-type models (Pauldrach et al. 1986; Müller & Vink 2008; Muijres et al. 2012; Krticka et al. 2016), [*at*]{} the bi-stability jump the $\beta$ parameter peaks at larger values of order 1.5 - 2. We attribute this to several ionization stages of the driving ions to be playing a role, and in more sophisticated modelling a $\beta$ law would not be appropriate to describe the dynamical wind behaviour in detail (see e.g. Sander et al. 2018). We wish to emphasize that for models on the cool side of the bi-stability jump below 20 kK the derived $\beta$ values are no larger than for O star models, i.e. in the range 0.7 - 1. This contrasts with the high $\beta$ values of up to 2 - 3 derived from empirical H$\alpha$ modelling (Trundle et al. 2004; Crowther et al. 2006), which may be artificially large due to the neglect of optically thick (macro) clumping in the atmosphere modelling (Petrov et al. 2014).
Finally, we note that we did not converge dynamical models at values in the range of the [*second*]{} bi-stability jump around 10000 K (Lamers et al. 1995; Petrov et al. 2016).
Metallicity Dependence of the bi-stability jump for LBVs {#s_lbvz}
========================================================
As a next step in our modelling we vary the metal contents $Z$, in order to investigate if the size of the bi-stability jump is expected to be different in lower $Z$ galaxies, which are also representative of massive stars at earlier Cosmic times. For this purpose we zoom in on the relevant range for the bi-stability jump over a $Z$ range varying from solar, to values as low as 1% solar, with results listed in Table3 and plotted in Figs. 10 & 11.
[llll|ccc]{}\
& $\log L$ & $Z/{\mbox{$Z_{\odot}$}}$ & & & $\log {\mbox{$\dot{M}$}}$ & $\beta$\
\[\] & \[\] & & (kK) & \[\] & \[\] &\
\
23 & 6.0 & $1/3$ & 25 & 1004 & $-$5.11 & 1.53\
& & & 21 & 636 & $-$4.72 & 1.10\
& & & 19 & 277 & $-$4.35 & 0.78\
23 & 6.0 & $1/10$ & 25 & 788 & $-$5.39 & 1.32\
& & & 21 & 577 & $-$5.06 & 1.07\
& & & 19 & 206 & $-$4.67 & 0.70\
23 & 6.0 & $1/33$ & 25 & 696 & $-$5.76 & 1.14\
& & & 21 & 495 & $-$5.41 & 1.00\
& & & 19 & 226 & $-$5.13 & 0.76\
23 & 6.0 & $1/100$ & 25 & 577 & $-$5.95 & 0.92\
& & & 21 & 407 & $-$5.82 & 0.96\
& & & 19 & 253 & $-$5.57 & 0.75\
As expected mass-loss rates generally drop with lower $Z$ (Fig.10), whilst terminal wind velocities display opposite behaviour with (Fig.11). At values below the bi-stability jump, terminal velocities seem to converge to similar values for all metallicities (see also Vink 2018). The size of the bi-stability jump does indeed appear to be a function of $Z$, with larger $Z$ giving rise to a larger bi-stability jump, due to an increase in the role of Fe in the line driving at higher $Z$ (Vink et al. 2001).
Kalari et al. (2018) recently investigated the incidence of S Doradus variability amongst normal B supergiants in the low metallicity environment of the Small Magellanic Cloud (SMC), finding a surprisingly low number of S Dor variables in the SMC. This may be related to our finding of lower $Z$ leading to a smaller bi-stability jump.
Summary {#s_sum}
=======
We presented mass-loss predictions from Monte Carlo radiative transfer models for early-type supergiants and LBVs, and we found that:
- [The previously discovered observed drop in terminal wind velocities at spectral type B1 is confirmed by our dynamically consistent supergiant models.]{}
- [The bi-stability jump in mass-loss rate is stronger than was derived in previous Monte Carlo modelling.]{}
- [This would imply that within the rotationally induced bi-stability model of Pelupessy et al. (2000) for B\[e\] supergiants, the expected density contrast between the hotter pole and cooler equator could increase by up to one order of magnitude – to a factor 100 – which may be sufficient to account for the disk densities of B\[e\] supergiants, although the disk velocity structure would still need to be explained]{}.
- [Our wind predictions may have relevance for the slow wind inferred for the HMXB IGR J17252-3616, or other HMXBs.]{}
- [The temperature of the bi-stability jump is now at the observed location of 21000 K, in agreement with [cmfgen]{} models. This boosts confidence in the applicability of the modified nebular approximation.]{}
- [The bi-stability jump is larger at [*lower*]{} Eddington $\Gamma$ parameter.]{}
- [The bi-stability jump is larger at higher metallicity.]{}
I would like to thank the anonymous referee for a constructive report. And I acknowledge hospitality of the Kavli Institute for Theoretical Physics (KITP), Santa Barbara, and to Ed van den Heuvel for introducing me to the problem of the slow wind in IGR J17252-3616 during my stay at KITP, which was supported by the National Science Foundation under Grant No. NSF PHY11-25915
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|
---
abstract: 'The bondage number $b(G)$ of a graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination number. Let $G$ be embeddable on a surface whose Euler characteristic $\chi$ is as large as possible, and assume $\chi\leq0$. Gagarin–Zverovich and Huang have recently found upper bounds of $b(G)$ in terms of the maximum degree $\Delta(G)$ and the Euler characteristic $\chi$. In this paper we prove a better upper bound $b(G)\leq\Delta(G)+\lfloor t\rfloor$ where $t$ is the largest real root of the cubic equation $z^3 + z^2 + (3\chi - 8)z + 9\chi - 12=0$; this upper bound is asymptotically equivalent to $b(G)\leq\Delta(G)+1+\lfloor \sqrt{4-3\chi} \rfloor$. We also establish further improved upper bounds for $b(G)$ when the girth, order, or size of the graph $G$ is large compared with $|\chi|$.'
address:
- |
Department of Mathematics and Statistics\
University of Nebraska at Kearney\
Kearney, NE 68849, USA
- |
Department of Mathematics\
Texas State University\
San Marcos, TX 78666, USA
author:
- Jia Huang
- Jian Shen
title: New upper bounds for the bondage number of a graph in terms of its maximum degree and Euler characteristic
---
[^1]
Introduction
============
The graphs considered in this paper are all finite, undirected, and without loops or multiple edges. Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The *order* and *size* of $G$ are $|V(G)|$ and $|E(G)|$, respectively. A graph with order $1$ is called *trivial* and a graph with size $0$ is called *empty*. The *degree* $d(v)$ of a vertex $v$ in $G$ is the cardinality $|N(v)|$ of the set $N(v)$ of all neighbors of $v$ in $G$. The maximum and minimum vertex degrees of $G$ are denoted by $\Delta(G)$ and $\delta(G)$.
A *dominating set* of a graph $G$ is a subset $D\subseteq V$ of vertices such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The minimum cardinality of a dominating set is called the *domination number* of the graph $G$. The concept of domination in graphs has many applications in a wide range of areas. The *bondage number* $b(G)$ of a graph $G$, introduced in [@BHNS; @FJKR], is the smallest number of edges whose removal from $G$ results in a graph with larger domination number. It measures to some extent the reliability of the domination number of the graph $G$, as edge removal from $G$ corresponds to link failure in a communication network whose underlying structure is given by the graph $G$. One can check that the bondage number $b(G)$ is well defined for any nonempty graph $G$.
In general it is NP-hard to determine the bondage number $b(G)$ (see Hu and Xu [@HuXu]). There have been studies on its upper and lower bounds, such as the following results.
\[lem:HR\] For any edge $uv$ in a graph $G$ one has $$b(G) \leq d(u) + d(v)-1-|N(u) \cap N(v)|.$$ In particular, $b(G)\leq \Delta(G) + \delta(G)-1$.
\[thm:ad\] For any connected graph $G$ one has a sharp bound $|E(G)| \geq |V(G)|(b(G) + 1)/4$.
The *average degree* of a graph $G$ is defined as $ad(G):=2|E(G)|/|V(G)|$. It follows from Lemma \[lem:HR\] and Theorem \[thm:ad\] that $b(G)\leq b'(G)$ where $b'(G)$ is an integer defined as $$\label{eq:b'}
b'(G) = \min\left\{ \min_{uv\in E(G)} d(u) + d(v)-1-|N(u) \cap N(v)|,\ 2 \lfloor ad(G) \rfloor-1 \right\}.$$ Any upper bound for $b'(G)$ is automatically an upper bound for the bondage number $b(G)$.
There are two conjectures on upper bounds of $b(G)$, which are still open.
\[conj1\] For any graph $G$ we have $b(G)\leq \frac32 \Delta(G)$.
\[conj2\] If $G$ is a planar graph then we have $b(G)\leq \Delta(G)+1$.
The best upper bound known so far for the bondage number of a planar graph is the following.
For any planar graph $G$, $b(G)\leq\min\{\Delta(G)+2,8\}$.
Carlson and Develin [@CarlsonDevelin] provided a simpler proof for the above theorem, which was further extended by Gagarin and Zverovich [@GZ1] to establish a nice upper bound for arbitrary graphs, a step forward towards Conjecture \[conj1\]. The main idea is to embed graphs on surfaces, which we outline next. The readers are referred to Mohar and Thomassen [@MoharThomassen] for more details on graph embedding.
Throughout this paper a *surface* means a connected compact Hausdorff topological space which is locally homeomorphic to an open disc in $\mathbb R^2$. According to the classification theorem for surfaces [@MoharThomassen Theorem 3.1.3], any surface $S$ is homeomorphic to either $S_h$ ($h\geq0$) which is obtained from a sphere by adding $h$ handles, or $N_k$ ($k\geq1$) which is obtained from a sphere by adding $k$ crosscaps. In the former case $S$ is an *orientable surface of genus $h$*, and in the latter case $S$ is a *non-orientable surface of genus $k$*. For instance, the torus, the projective plane, and the Klein bottle are homeomorphic to $S_1$, $N_1$, and $N_2$, respectively. The *Euler characteristic* of the surface $S$ is defined as $$\chi(S)=\begin{cases}
2-2h, & {\rm if}\ S\cong S_h,\\
2-k, & {\rm if}\ S\cong N_k.
\end{cases}$$
Any graph $G$ can be embedded on some surface $S$, that is, it can be drawn on $S$ with no crossing edges; in addition, one can take the surface $S$ to be either orientable or non-orientable. Denote by $\chi(G)$ the largest integer $\chi$ for which $G$ admits an embedding on a surface $S$ with $\chi(S)=\chi$. For example, $\chi(G)=2$ means $G$ is planar, while $\chi(G)=1$ means $G$ is non-planar but can be embedded on the projective plane.
Suppose that $G$ is a connected graph which admits an embedding on a surface $S$ whose Euler characteristic $\chi$ is as large as possible, i.e. $\chi(S)=\chi(G)$. By Mohar and Thomassen [@MoharThomassen §3.4], this embedding of $G$ on $S$ can be taken to be a *2-cell embedding*, meaning that all faces are homeomorphic to an open disk. In this case one has the following result.
(c.f. [@MoharThomassen]) Suppose that a graph $G$ with vertex set $V(G)$ and edge set $E(G)$ admits a $2$-cell embedding on a surface $S$, and let $F(G)$ be the set of faces in this embedding. Then $$|V(G)|-|E(G)|+|F(G)|=\chi(S).$$
One needs to rewrite Euler’s formula in a different form in order to apply it to obtain upper bounds for the bondage number. Every edge $uv$ in the 2-cell embedding of $G$ on $S$ appears on the boundary of either two distinct faces $F\ne F'$ or a unique face $F=F'$; in the former case $uv$ occurs exactly once on the boundary of each of the two faces $F$ and $F'$, while in the latter case $uv$ occurs exactly twice on the boundary of the face $F=F'$. Let $f(uv)$ and $f'(uv)$ be the number of edges on the boundary of $F$ and $F'$, whether or not $F$ and $F'$ are distinct. For instance, a path $P_n$ with $n$ vertices is embedded on a sphere with only one face, and for any edge in $P_n$ we have $f(uv)=f'(uv)=2(n-1)$. We may assume that $f(uv)$ and $f'(uv)$ are at least $3$—otherwise one has $G=P_2$ which is a trivial case. There is a weight associated to the edge $uv$, that is, $$w(uv)=\frac{1}{d(u)}+\frac{1}{d(v)}-1+\frac{1}{f(uv)}
+\frac{1}{f'(uv)}-\frac{\chi}{|E(G)|}.$$ This weight is called the *curvature* of $uv$ by Gagarin and Zverovich [@GZ1]. One can rewrite Euler’s formula as $$\label{eq:weight}
\sum_{uv\in E(G)} w(uv)=|V(G)|-|E(G)|+|F(G)|-\chi=0.$$
The above equation was used by Carlson and Develin [@CarlsonDevelin] in two special cases $\chi=2$ and $\chi=0$, and later used by Gargarin and Zverovich in full generality to establish the following result.
\[thm:GZ\] Let $G$ be a graph embeddable on an orientable surface of genus $h$ and a non-orientable surface of genus $k$. Then $b(G)\leq\min\{\Delta(G)+h+2,\Delta(G)+k+1\}$.
According to Theorem \[thm:GZ\], if $G$ is planar ($h=0$, $\chi=2$) or can be embedded on the real projective plane ($k=1$, $\chi=1$), then $b(G)\leq\Delta(G)+2$. For larger values of $h$ and $k$, it was mentioned in [@GZ1] that improvements of Theorem \[thm:GZ\] can be achieved by adjusting its proof, and an explicit improvement of Theorem \[thm:GZ\] was obtained by the first author of this paper.
\[thm:H\] Let $G$ be a graph embedded on a surface whose Euler characteristic $\chi$ is as large as possible. If $\chi\leq 0$ then $b(G)\leq\Delta(G)+\lfloor r\rfloor$, where $r$ is the largest real root of the following cubic equation in $z$: $$z^3+2z^2+(6\chi-7)z+18\chi-24 = 0.$$ A weaker but asymptotically equivalent upper bound is $b(G)\leq\Delta(G)+\lceil\sqrt{12-6\chi}-1/2\rceil$.
In Section \[sec:arbitrary\] we will prove an upper bound which is not only stronger than the above result but also asymptotically better as $\chi\to -\infty$. We will also establish further upper bounds in Section \[sec:girth\], Section \[sec:order\], and Section \[sec:size\] for the bondage number $b(G)$ when the girth, order, or size of the graph $G$ is large, which improve another result of the first auther [@Huang] and the following result of Gagarin and Zverovich [@GZ2 Corollary 17, Corollary 19].
\[thm:GZ2\] Let G be a connected graph $2$-cell embeddable on an orientable surface of genus $h\geq1$ and a non-orientable surface of genus $k\geq1$. Then
- $b(G) \leq \Delta(G) + \lceil \ln^2 h \rceil + 3$ if $n \geq h$,
- $b(G) \leq \Delta(G) + \lceil \ln h \rceil + 3$ if $n \geq h^{1.9}$,
- $b(G) \leq \Delta(G) + 4$ if $n \geq h^{2.5}$,
- $b(G) \leq \Delta(G) + \lceil \ln^2 k \rceil + 2$ if $n \geq k/6$,
- $b(G) \leq \Delta(G) + \lceil \ln k \rceil + 3$ if $n \geq k^{1.6}$,
- $b(G) \leq \Delta(G) + 3$ if $n \geq k^2$.
Gagarin and Zverovich [@GZ2] also obtained some constant upper bounds for the bondage number of graphs embedded on surfaces. It seems interesting to look for further improvements of these constant bounds in the future.
The general case {#sec:arbitrary}
================
In this section we establish an upper bound for the bondage number $b(G)$ of a graph $G$ with $\chi(G)\leq 0$, which improves the previously known upper bounds for $b(G)$ in such cases.
\[lem:main\] Let $G$ be a connected graph with $b'(G)\geq\Delta(G)+z$ for some integer $z\geq0$. Let $uv$ be an arbitrary edge of $G$ and write $c(u,v)=|N(u)\cap N(v)|$. Then $\min\{d(u),d(v)\}\geq z+1+c(u,v)$ and $$\label{eq:edge}
|E(G)|\geq |V(G)|(2z+2+c(u,v))/4 \geq (2z+2+c(u,v))^2/4.$$
Assume $d(u)\leq d(v)$, without loss of generality. By (\[eq:b’\]), one has $$\Delta(G)+z\leq b'(G)\leq d(u)+d(v)-1-c(u,v).$$ Thus $$d(u)\geq \Delta(G)-d(v)+z+1+c(u,v)\geq z+1+c(u,v).$$ It follows that $d(v)\geq d(u)\geq z+1+c(u,v)$, which implies that $v$ has at least $z$ neighbors that are not contained in $\{u\}\cup N(u)$. Hence $$|V(G)|\geq 1+(z+1+c(u,v))+z=2z+2+c(u,v).$$ By (\[eq:b’\]), one also has $$4|E(G)|/|V(G)|-1 \geq b'(G)\geq \Delta(G)+z\geq d(u)+z\geq 2z+1+c(u,v).$$ Thus (\[eq:edge\]) holds.
\[lem:roots\] Let $z\geq0$ and $\chi\leq0$. Then the following inequalities $$\label{ineq1}
A(z)=z^2-2z+2\chi-3>0,$$ $$\label{ineq2}
B(z)=20z^3+4z^2+3(16\chi-41)z+96\chi-126>0,$$ $$\label{ineq3}
C(z)=z^3 + z^2 + (3\chi - 8)z + 9\chi - 12>0$$ hold if and only if $z$ is larger than the largest real root $t=t(\chi)$ of $C(z)$. In addition, $t\geq3$ is a decreasing function of $\chi\leq0$.
We first show that the largest real root $t=t(\chi)$ of $C(z)$ is larger than or equal to all the real roots of $A(z)$ and $B(z)$, by using the intermediate value theorem and the limits $$\lim_{z\to\infty}A(z)=
\lim_{z\to\infty}B(z)=
\lim_{z\to\infty}C(z)=\infty.$$
The polynomial $A(z)$ has two roots $$z_1=1+\sqrt{4-2\chi}>0 \quad \text{and} \quad
z_2=1-\sqrt{4-2\chi}<0.$$ Substituting $z_1$ in $C(z)$ gives $$C(z_1)= (\chi + 1)\sqrt{4-2\chi} + 4\chi - 2$$ which is negative if $\chi\leq -1$ and is $0$ if $\chi=0$. By the intermediate value theorem, $C(z)$ has a real root larger than or equal to $z_1$. Thus $t\geq z_1>z_2$.
Next consider $B(z)$. If $\chi=0$ then $B(z)=(5z-14)(2z+3)^2$, $C(z)=(z-3)(z+2)^2$, and thus $t=3$ is larger than the two roots $14/5$ and $-3/2$ of $B(z)$. Assume $\chi\leq -1$ below. Then $B(3)=240\chi+81<0$. Applying the intermediate value theorem to $B(z)$ gives the existence of real root(s) of $B(z)$ in $(3,\infty)$; let $z_3$ be the largest one. Then $$\begin{aligned}
B(z_3)-16C(z_3) &=& 4z_3^3-12z_3^2+5z_3+66-48\chi \\
&=& z_3(2z_3-1)(2z_3-5) + 66-48\chi >0\end{aligned}$$ which implies $C(z_3)<0$. Again by the intermediate value theorem, $C(z)$ has a root larger than $z_3$, and thus $t>z_3>3$.
Therefore $t=t(\chi)\geq3$ is larger than or equal to all the real roots of $A(z)$ and $B(z)$ for all $\chi\leq 0$. It follows that $A(z)$, $B(z)$, and $C(z)$ are all positive whenever $z>t$; otherwise the intermediate value theorem would imply that $A(z)$, $B(z)$, or $C(z)$ has a root larger than $t$, a contradiction.
Conversely, assume that $A(z)$, $B(z)$, and $C(z)$ are all positive and we need to show that $z>t$. Suppose to the contrary that $0\leq z\leq t$. It is clear that $z\ne t$ since $C(z)>0=C(t)$. Thus $z<t$ and there exists a point $s$ in $(z,t)$ such that $$C'(s)=3s^2 + 2s + 3\chi - 8 <0$$ by the mean value theorem. Then $$C'(s)-3A(s) = 8s-3\chi+1>0$$ implies $A(s)<0$. We have seen that the upward parabola $A(z)$ has two roots $z_1>0$ and $z_2<0$. Since $A(z)>0$ and $z\geq0$, one has $z>z_1$. Then $s>z$ implies $A(s)>0$, a contradiction. Hence $z>t$.
Finally, we show that $t(\chi)\geq 3$ is a decreasing function of $\chi\leq 0$. For any $\epsilon>0$, one has $$C(z;\chi)-C(z;\chi-\epsilon) = (3z+9)\epsilon.$$ This implies $$C(t(\chi);\chi-\epsilon)=-(3t(\chi)+9)\epsilon<0.$$ By the intermediate value theorem, $C(z;\chi-\epsilon)$ has a real root larger than $t(\chi)$, and thus its largest real root $t(\chi-\epsilon)$ is also larger than $t(\chi)$.
\[lem1\] Let $G$ be a connected graph embedded on a surface whose Euler characteristic $\chi$ is as large as possible. Assume $\chi\leq 0$ and let the largest real root of $z^3 + z^2 + (3\chi - 8)z + 9\chi - 12$ be $z=t=t(\chi)$. Then $b'(G)\leq\Delta(G)+\lfloor t\rfloor$.
Since $b'(G)$ is an integer, it suffices to show that $b'(G)<\Delta(G)+z$ for any integer $z>t$. Suppose to the contrary that $b'(G)\geq \Delta(G)+z$ for some integer $z>t$. By Lemma \[lem:roots\], the inequalities (\[ineq1\]), (\[ineq2\]), and (\[ineq3\]) all hold since $z>t$. Let $uv$ be an arbitrary edge in $G$. Assume $d(u)\leq d(v)$ and $f(uv)\leq f'(uv)$, without loss of generality. Let $c(u,v)=|N(u)\cap N(v)|$. By Lemma \[lem:main\], one has $$d(v)\geq d(u) \ge z+1+c(u,v),$$ $$|E(G)|\geq (2z+2+c(u,v))^2/4.$$
If $c(u,v)=0$ then $d(v)\geq d(u)\geq z+1$, $|E(G)|\geq (z+1)^2$, and $f'(uv)\geq f(uv)\geq4$. Thus $$\begin{aligned}
w(uv) &\leq & \frac 2{z+1}+\frac14+\frac14 -1
-\frac{\chi}{(z+1)^2} \\
&=& -\frac{z^2-2z+2\chi-3}{2(z+1)^2} <0\end{aligned}$$ where the last inequality follows from (\[ineq1\]).
If $c(u,v)=1$ then $d(v)\geq d(u)\geq z+2$, $|E(G)|\geq (2z+3)^2/4$, $f'(uv)\geq 4$, and $f(uv)\geq 3$. Thus $$\begin{aligned}
w(uv) & \leq & \frac2{z+2}+\frac14+\frac13-1-
\frac{4\chi}{(2z+3)^2} \\
&=& -\frac{20z^3+4z^2+3(16\chi-41)z+96\chi-126}
{12(z+2)(2z+3)^2}<0\end{aligned}$$ where the last inequality follows from (\[ineq2\]).
If $c(u,v)\geq 2$ then $d(v)\geq d(u)\geq z+3$, $|E(G)|\geq (z+2)^2$, and $f'(uv)\geq f(uv)\geq 3$. Thus $$\begin{aligned}
w(uv)&\leq& \frac2{z+3}+\frac13+\frac13-1
-\frac{\chi}{(z+2)^2}\\
&=&-\frac{z^3 + z^2 + (3\chi - 8)z + 9\chi - 12}{3(z+3)(z+2)^2}<0\end{aligned}$$ where the last inequality follows from (\[ineq3\]).
Therefore $w(uv)<0$ for all edges $uv$ in $G$. This contradicts Equation (\[eq:weight\]). It follows that $b'(G)<\Delta(G)+z$ for any integer $z>t$, which implies $b'(G)\leq\Delta(G)+\lfloor t\rfloor$.
\[thm1\] Let $G$ be a graph embedded on a surface whose Euler characteristic $\chi$ is as large as possible. Assume $\chi\leq 0$ and let the largest real root of $z^3 + z^2 + (3\chi - 8)z + 9\chi - 12$ be $z=t=t(\chi)$. Then $b(G)\leq\Delta(G)+\lfloor t\rfloor$.
If $G$ is connected then Lemma \[lem:HR\], Theorem \[thm:ad\], and Lemma \[lem1\] imply $b(G)\leq b'(G)\leq \Delta(G)+\lfloor t\rfloor$. If $G$ has multiple components $G_1,\ldots,G_\ell$, then $\chi\leq\chi_i=\chi(G_i)$ for all $i$, since an embedding of $G$ on a surface $S$ automatically includes an embedding of $G_i$ on $S$. By definition, $b(G)=\min\{b(G_1),\ldots,b(G_\ell)\}$. If $\chi_i>0$ for some $i$ then Theorem \[thm:GZ\] implies $$b(G)\leq b(G_i)\leq \Delta(G_i)+2 <\Delta(G)+ 3\leq \Delta(G)+\lfloor t \rfloor.$$ Assume $\chi_i\leq 0$ for all $i=1,\ldots,\ell$. By Lemma \[lem:roots\], $\chi\leq\chi_i$ implies $t(\chi_i)\leq t(\chi)$. Hence $$b(G)\leq b(G_i)\leq\Delta(G_i)+\lfloor t(\chi_i)\rfloor
\leq \Delta(G)+\lfloor t(\chi)\rfloor.$$ This completes the proof.
\[cor1\] Let $G$ be a graph embedded on a surface whose Euler characteristic $\chi$ is as large as possible. If $\chi\leq 0$ then $b(G)\leq\Delta(G)+1+\lfloor \sqrt{4-3\chi} \rfloor$.
If $\chi=0$ then $1+\sqrt{4-3\chi}=3=t(\chi=0)$. If $\chi\leq-1$ then $1+\sqrt{4-3\chi}>t(\chi)$ by Lemma \[lem:roots\] and the following inequialities: $$\begin{aligned}
A(1+\sqrt{4-3\chi}) & = & -\chi >0, \\
B(1+\sqrt{4-3\chi}) &=& (25-12\chi)\sqrt{4-3\chi} +31 - 48\chi > 0,\\
C(1+\sqrt{4-3\chi}) &=& \sqrt{4-3\chi}-2 >0.\end{aligned}$$ Hence the result follows immediately from Theorem \[thm1\].
Computations in give the following formula $$t(\chi) = \frac 13\left(D + (25-9\chi)/D -1 \right)$$ where $D=\left( 9\sqrt{9\chi^3 + 69\chi^2 - 125\chi} - 108\chi + 125\right)^{\frac 13}$. Note that this formula works in $\mathbb C$, the field of complex numbers. Athough Theorem \[thm1\] is stronger than Corollary \[cor1\], asymptotically they are equivalent by the following limit $$\lim_{\chi\to-\infty} t(\chi)/(1+\sqrt{4-3\chi})=1.$$ One can check that Theorem \[thm1\] and Corollary \[cor1\] are both stronger than the previous result of Theorem \[thm:H\]. Asymptotically, the improvement is by a factor of $\sqrt 2$, as shown by the limit $$\lim_{\chi\to-\infty} \frac{1/2+ \sqrt{12-6\chi}}{1+\sqrt{4-3\chi}} =\sqrt 2.$$
We give a table below to compare Theorem \[thm1\] with Theorem \[thm:H\] for $-21\leq \chi\leq0$. $$\begin{tabular}{|c|ccccccccccc|}
\hline
$\chi$ & 0 & -1 & -2 & -3 & -4 & -5 & -6 & -7 & -8 & -9 & -10 \\
\hline
$\lfloor r \rfloor$ & 3 & 3 & 4 & 5 & 5 & 6 & 6 & 7 & 7 & 8 & 8\\
\hline
$\lfloor t\rfloor$ & 3 & 3 & 4 & 4 & 4 & 5 & 5 & 5 & 6 & 6 & 6\\
\hline\hline
$\chi$ &-11&-12&-13& -14 & -15 & -16 & -17 & -18 & -19 & -20 & -21\\
\hline
$\lfloor r\rfloor$ & 8 & 9 & 9 & 9 & 10 & 10 & 10 & 11 & 11 & 11 &11\\
\hline
$\lfloor t \rfloor$ &7 & 7 & 7 & 7 & 7 & 8 & 8 & 8 & 8 & 8 & 9\\
\hline
\end{tabular}$$
We do not know whether the upper bound for $b(G)$ given in Theorem \[thm1\] is sharp for all graphs $G$ with $\chi(G)\leq 0$, or whether the weaker result for $b'(G)$ given by Lemma \[lem1\] is sharp for connected graphs $G$ with $\chi(G)\leq 0$.
Graphs with large girth {#sec:girth}
=======================
Our result can be further improved when the graph $G$ has large *girth* $g(G)$, defined as the length of the shortest cycle in $G$. If $G$ has no cycle then we set $g(G)=\infty$ by convention, and in such case one has $b(G)\leq 2$ by [@BHNS].
\[thm:girth\] Let $G$ be a graph embedded on a surface whose Euler characteristic $\chi$ is as large as possible. If $\chi\leq0$ and $g=g(G)<\infty$, then $b(G)\leq \Delta(G)+\lfloor s \rfloor$ where $s$ is the larger root of the quadratic polynomial $A(z)=(g-2)z^2-4z+\chi g-g-2$, i.e. $$s=\frac{2+\sqrt{g^2-g(g-2)\chi}}{(g-2)}.$$
Assume $G$ is connected for the same reason as argued in the proof of Theorem \[thm1\]. It suffices to show that $b(G)<\Delta(G)+z$ for any positive integer $z$ satisfying $A(z)>0$.
Suppose to the contrary that $b(G)\geq \Delta(G)+z$ for some positive integer $z$ satisfying $A(z)>0$. Let $uv$ be an arbitrary edge of $G$. By Lemma \[lem:main\], $\min\{d(u),d(v)\}\geq z+1$ and $|E(G)|\geq (z+1)^2$. One also has $f(uv)\geq g$ and $f'(uv)\geq g$ by definition. Hence $$\begin{aligned}
w(uv) &\leq & \frac 2{z+1}+\frac2g-1-\frac{\chi}{(z+1)^2} \\
&=& -\frac{(g-2)z^2-4z+\chi g-g-2}{g(z+1)^2}<0\end{aligned}$$ where the last inequality follows from $A(z)>0$. This contradicts Equation (\[eq:weight\]).
The first author [@Huang Proposition 10] showed that, with the same conditions as Theorem \[thm:girth\], $$b(G)\leq \Delta(G)+ \left\lfloor \frac{\sqrt{8g(2-g)\chi+(3g-2)^2}-(g-6)}{2(g-2)} \right\rfloor.$$ It is not hard to check that Theorem \[thm:girth\] improves this result.
One has $$b(G)\leq \Delta(G)+
\begin{cases}
2, & {\rm if}\ \chi=0,\ g\geq 5,\\
1, & {\rm if}\ \chi=0,\ g\geq 7,\\
2, & {\rm if}\ \chi=-1,\ g\geq 5,\\
3, & {\rm if}\ \chi=-2,\ g\geq 4, \\
2, & {\rm if}\ \chi=-2,\ g\geq 6.
\end{cases}$$
\[cor:girth\] Let $G$ be a triangle-free graph embedded on a surface whose Euler characteristic $\chi$ is as large as possible. If $\chi\leq0$ then $b(G)\leq \Delta(G)+1+ \lfloor \sqrt{4-2\chi} \rfloor$.
One can check that the upper bound for $b(G)$ provided by the previous proposition is a decreasing function of $g\geq3$. Hence taking $g=4$ gives the desired result.
We do not know whether the upper bounds for $b(G)$ given by Theorem \[thm:girth\] and Corollary \[cor:girth\] are sharp, but one can check that they are actually upper bounds for $b'(G)$ as long as $G$ is connected. When is $G$ connected and triangle-free, Corollary \[cor:girth\] implies $$\label{eq:b'g4}
b'(G)\leq \Delta(G)+1+\sqrt{4-2\chi}$$ This bound is indeed sharp. For example, let $G$ be the complete bipartite graph $K_{n,n}$, which is triangle-free. One sees that $b'(G)=2n-1$ and $\Delta(G)=n$. One also has $\chi(G)={(4n-n^2)}/2$ by Ringel [@Ringel1; @Ringel2]. Hence the quality in (\[eq:b’g4\]) holds for $G=K_{n,n}$. On the other hand, for $G=K_{n,n}$ with $n\geq 2$ one can check that $\gamma(G)=2$ and $b(G)=n<2n-1$. So it is not clear whether the upper bound $b(G)\leq \Delta(G)+1+\sqrt{4-2\chi}$ is sharp for triangle-free graphs.
Connected graphs with large order {#sec:order}
=================================
The order $|V(G)|$ of a connected nontrivial graph $G$ has the following lower bound in terms of its Euler characteristic $\chi(G)$.
\[prop:order\] Let $G$ be a connected graph with $n=|V(G)|\geq 2$ embedded on a surface whose Euler characteristic $\chi$ is as large as possible. Then $n\geq(3+\sqrt{17-8\chi})/2$.
In this section we assume $|V(G)|\geq -\chi$ and obtain asymptotically better upper bounds for the bondage number $b(G)$.
Let $\chi\leq0$ and $n\geq1$. Then the following inequalities $$\label{ineq4}
A(z)=nz-3n+4\chi>0,$$ $$\label{ineq5}
B(z)=10nz^2 - (13n - 48\chi)z - 42n + 96\chi>0,$$$$\label{ineq6}
C(z)=nz^2 - (n - 6\chi)z - 6n + 18\chi>0$$ are all valid if and only if $
z> 1/2 - {3\chi}/{n} + \sqrt{25/4 - 21\chi/n + 9\chi^2/n^2}.
$
The graph of $A(z)$ is an upward straight line with a unique root $a = 3 - 4\chi/n\geq 3$. The graph of $B(z)$ is an upward parabola with two roots $$b = \frac{1}{20n}(13n - 48\chi + \sqrt{\beta}),\quad b' = \frac{1}{20n}(13n - 48\chi - \sqrt{\beta})$$ where $\beta = (43n)^2 - 5088n\chi + (48\chi)^2$. One sees that $$43n-48\chi \leq \sqrt\beta \leq 43n-2544\chi/43.$$ Hence $b'<0<b\leq 2.8-5.4\chi/n$. The graph of $C(z)$ is an upward parabola with two roots $$c=\frac{1}{2n}(n - 6\chi + \sqrt{\gamma}),\quad c'=\frac{1}{2n}(n - 6\chi - \sqrt{\gamma})$$ where $\gamma=25n^2 - 84n\chi + 36\chi^2$. One sees that $\gamma\geq 5n-6\chi$. Hence $c'<0< c$ and $$c\geq 3-6\chi/n \geq \max\{a,b\}.$$ Hence $A(z)$, $B(z)$, and $C(z)$ are all positive if and only if $z>c$.
\[thm:order\] Let $G$ be a connected graph embedded on a surface whose Euler characteristic $\chi$ is as large as possible. Let $n=|V(G)|$ and assume $\chi\leq0$. Then $b(G) \leq b'(G)\leq \Delta(G) + \lfloor c \rfloor$ where $$c = 1/2 - {3\chi}/{n} + \sqrt{25/4 - 21\chi/n + 9\chi^2/n^2}.$$
It suffices to show that $b'(G)< \Delta(G)+z$ for any integer $z>c$. Suppose to the contrary that $b'(G)\geq \Delta(G)+z$ for some integer $z>c$. Let $uv$ be an arbitrary edge in $G$. Assume $d(u)\leq d(v)$ and $f(uv)\leq f'(uv)$, without loss of generality. Let $|N(u)\cap N(v)|=c(u,v)$. By Lemma \[lem:main\], $d(u)\geq z+1+c(u,v)$ and $|E(G)|\geq n(2z+2+c(u,v))/4$.
If $c(u,v)=0$ then $d(v)\geq d(u)\geq z+1$, $f'(uv)\geq f(uv)\geq4$, and $|E(G)|\geq n(z+1)/2$. Thus $$\begin{aligned}
w(uv) &\leq & \frac 2{z+1}+\frac14+\frac14 -1
-\frac{2\chi}{n(z+1)} \\
&=& -\frac{nz-3n+4\chi}{2n(z+1)} <0\end{aligned}$$ where the last inequality follows from (\[ineq4\]).
If $c(u,v)=1$ then $d(v)\geq d(u)\geq z+2$, $f'(uv)\geq 4$, $f(uv)\geq 3$, and $|E(G)|\geq n(2z+3)/4$. Thus $$\begin{aligned}
w(uv) & \leq & \frac2{z+2}+\frac14+\frac13-1-
\frac{4\chi}{n(2z+3)} \\
&=& -\frac{10nz^2 - (13n - 48\chi)z - 42n + 96\chi}
{12n(z+2)(2z+3)}<0\end{aligned}$$ where the last inequality follows from (\[ineq5\]).
If $c(u,v)\geq 2$ then $d(v)\geq d(u)\geq z+3$, $f'(uv)\geq f(uv)\geq 3$, and $|E(G)|\geq n(z+2)/2$. Thus $$\begin{aligned}
w(uv)&\leq& \frac2{z+3}+\frac13+\frac13-1
-\frac{2\chi}{n(z+2)}\\
&=&-\frac{nz^2 - (n - 6\chi)z - 6n + 18\chi}{3n(z+2)(z+3)}<0\end{aligned}$$ where the last inequality follows from (\[ineq6\]).
Therefore $w(uv)<0$ for all edges $uv$ in $G$. This contradicts Equation (\[eq:weight\]).
\[cor:order\] Let $G$ be a connected graph embedded on a surface whose Euler characteristic $\chi$ is as large as possible. Suppose that $\chi\leq0$ and $n=|V(G)|$. Then
- $b(G)\leq b'(G)\leq \Delta(G)+9$ if $n\geq-\chi$,
- $b(G)\leq b'(G)\leq \Delta(G)+6$ if $n\geq -2\chi$,
- $b(G)\leq b'(G)\leq \Delta(G)+5$ if $n\geq -3\chi$,
- $b(G)\leq b'(G)\leq \Delta(G)+4$ if $n\geq -4\chi$,
- $b(G)\leq b'(G)\leq \Delta(G)+3$ if $n\geq -8\chi$.
Suppose that $n\geq -d \chi$ for some $d>0$. Then $$c = \frac 12 - \frac{3\chi}{n} + \sqrt{ \frac{25}{4} - \frac{21\chi}{n} + \frac{9\chi^2}{n^2} } \leq \frac 12+\frac3d + \sqrt{\frac{25}4+\frac{21}{d}+\frac{9}{d^2}}$$ and Theorem \[thm:order\] implies that $$b(G)\leq \Delta(G)+\lfloor c \rfloor \leq \Delta(G) + \left\lfloor \frac 12+\frac3d + \sqrt{\frac{25}4+\frac{21}{d}+\frac{9}{d^2}} \right\rfloor.$$ Taking $d=1,2,3,4,8$ gives the desired upper bounds.
Theorem \[thm:order\] and Corollary \[cor:order\] asymptotically improve a result of Gagarin and Zverovich [@GZ2 Corollary 17, Corollary 19] (see Theorem \[thm:GZ2\]).
Connected graphs with large size {#sec:size}
================================
Using Euler’s formula, Proposition \[prop:order\], and the fact that $|F(G)|\geq1$, one obtains a lower bound $$|E(G)| \geq \frac52-\chi+\frac12\sqrt{17-8\chi(G)}$$ for the size of a connected nontrivial graph $G$ in terms of its Euler characteristic $\chi(G)$. In this section we assume $|E(G)|>-3\chi(G)$ and obtain better upper bounds for the bondage number $b(G)$.
Let $\chi\leq0$ and $m>-3\chi$. Then the following inequalities $$\label{ineq7}
A(z)=(m +2\chi)z - 3m +2\chi >0,$$ $$\label{ineq8}
B(z)=(5m + 12\chi)z - 14m + 24\chi >0,$$$$\label{ineq9}
C(z)= (m + 3\chi)z - 3m + 9\chi >0$$ are all valid if and only if $z>3 - 18\chi/(m+3\chi)$.
Since $m>-3\chi$, one sees that $A(z)$, $B(z)$, and $C(z)$ are all upward straight lines whose roots are$$a=3 - \frac{8\chi}{m + 2\chi},\quad b = \frac{14}5 - \frac{288\chi}{5(5m+12\chi)},\quad c = 3 - \frac{18\chi}{m+3\chi}.$$ One can check that $$c-a = \frac{-2\chi(5m+6\chi)}{(m+2\chi)(m+3\chi)}>0,$$ $$c-b = \frac{(m+3\chi)(m-12\chi)-18m\chi}{(5m+12\chi)(m+3\chi)}>0.$$ Hence $A(z)$, $B(z)$, and $C(z)$ are all positive if and only if $z>c$.
\[thm4\] Let $G$ be a connected graph embedded on a surface whose Euler characteristic $\chi$ is as large as possible. Suppose that $m=|E(G)|>-3\chi\geq0$. Then $b(G) \leq b'(G) \leq \Delta(G) + \lfloor c \rfloor$ where $c = 3 - 18\chi/(m+3\chi)$.
It suffices to show that $b'(G)< \Delta(G)+z$ for any integer $z>c$. Suppose to the contrary that $b'(G)\geq \Delta(G)+z$ for some integer $z>c$. Let $uv$ be an arbitrary edge in $G$. Assume $d(u)\leq d(v)$ and $f(uv)\leq f'(uv)$, without loss of generality. Let $|N(u)\cap N(v)|=c(u,v)$. By Lemma \[lem:main\], one has $d(u)\geq z+1+c(u,v)$.
If $c(u,v)=0$ then $d(v)\geq d(u)\geq z+1$ and $f'(uv)\geq f(uv)\geq4$. Thus $$\begin{aligned}
w(uv) &\leq & \frac 2{z+1}+\frac14+\frac14 -1 -\frac\chi m \\
&=& -\frac{(m +2\chi)z - 3m +2\chi}{2m(z + 1)} <0\end{aligned}$$ where the last inequality follows from (\[ineq7\]).
If $c(u,v)=1$ then $d(v)\geq d(u)\geq z+2$, $f'(uv)\geq 4$, $f(uv)\geq 3$. Thus $$\begin{aligned}
w(uv) & \leq & \frac2{z+2}+\frac14+\frac13-1 -\frac\chi m\\
&=& -\frac{(5m + 12\chi)z - 14m + 24\chi}{12m(z + 2)} <0\end{aligned}$$ where the last inequality follows from (\[ineq8\]).
If $c(u,v)\geq 2$ then $d(v)\geq d(u)\geq z+3$, and $f'(uv)\geq f(uv)\geq 3$. Thus $$\begin{aligned}
w(uv)&\leq& \frac2{z+3}+\frac13+\frac13-1 - \frac\chi m\\
&=& -\frac{(m + 3\chi)z - 3m + 9\chi}{3m(z + 3)} <0\end{aligned}$$ where the last inequality follows from (\[ineq9\]).
Therefore $w(uv)<0$ for any edge $uv$ in $G$. This contradicts Equation (\[eq:weight\]).
Let $G$ be a connected graph embedded on a surface whose Euler characteristic $\chi$ is as large as possible. Suppose that $\chi\leq 0$ and $m=|E(G)|$. Then
- $b(G) \leq b'(G) \leq \Delta(G)+8$ if $m> -6\chi$,
- $b(G)\leq b'(G) \leq \Delta(G)+7$ if $m> -6.6\chi$,
- $b(G)\leq b'(G) \leq \Delta(G)+6$ if $m> -7.5\chi$,
- $b(G)\leq b'(G) \leq \Delta(G)+5$ if $m> -9\chi$,
- $b(G)\leq b'(G) \leq \Delta(G)+4$ if $m> -12\chi$,
- $b(G)\leq b'(G) \leq \Delta(G)+3$ if $m> -21\chi$.
This follows immediately from the above theorem.
[30]{}
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[^1]: The authors thank the anonymous referee for providing helpful suggestions.
|
---
abstract: 'We apply Nelson’s technique of constructing Euclidean fields to the case of classical scalar fields on curved spaces. It is shown how to construct a transfer matrix and, for a class of metrics, the basic spectral properties of its generator are investigated. An application concerning decoupling of non-convex disjoint region is given.'
address: 'University of Houston, 4800 Calhoun Rd, Houston TX 77204-5508, USA'
author:
- 'E. Prodan'
title: Transfer matrices for scalar fields on curved spaces
---
\[theorem\][Acknowledgement]{} \[theorem\][Algorithm]{} \[theorem\][Axiom]{} \[theorem\][Claim]{} \[theorem\][Conclusion]{} \[theorem\][Condition]{} \[theorem\][Conjecture]{} \[theorem\][Corollary]{} \[theorem\][Criterion]{} \[theorem\][Definition]{} \[theorem\][Example]{} \[theorem\][Exercise]{} \[theorem\][Lemma]{} \[theorem\][Notation]{} \[theorem\][Problem]{} \[theorem\][Proposition]{} \[theorem\][Remark]{} \[theorem\][Solution]{} \[theorem\][Summary]{}
Introduction
============
We start our construction from the ideas comprised in Nelson$%
\prime $s axioms [@Ne] for scalar Euclidean-Markoff quantum fields. Here, the Markoff property of certain projectors is one of the basic ingredient in defining the transfer matrix of whom generator is identified with the Hamiltonian of Wightman quantum scalar field. We found that these ideas can be used in the same way at the non-quantum level. In the case of the scalar fields on Riemannian manifolds, for an arbitrary direction, we construct a propagator by using the Markoff property. In the stationary case it becomes a semigroup which can be considered as the transfer matrix of the system and, further, it can be used in introducing a Hamiltonian. We will show that the propagator is exponentially bounded by using Agmon$\prime $s [@Ag] results in exponential decay of solutions of second-order elliptic equations. An application concerning the decoupling (in the sense of [@Si1]) of two disjoint non-convex regions is given.
Introductory definitions and results
====================================
Let us consider the Riemannian manifold $\left( R^{n+1},g\right) $ and the Laplace-Beltrami operator on it, $\Delta $. For a point in $R^{n+1}$ we use the notation $\left( t,x\right) $. Let $E_{m}\left( t,x;s,y\right) $ be the kernel of $\left( \Delta +m^{2}\right) ^{-1}$ on $L^{2}\left( R^{n+1},%
\sqrt{g}dtdx\right) $. As in [@Di], we will not consider the additional term $%
%TCIMACRO{\dfrac{1}{6}}%
%BeginExpansion
{\displaystyle{1 \over 6}}%
%EndExpansion
\rho .$ One defines the space $N\subset {\cal D}^{\prime }\left(
R^{n+1}\right) $, $f\in N$ if : $$\left\| f\right\| _{N}^{2}=%
%TCIMACRO{\dint}%
%BeginExpansion
\displaystyle\int %
%EndExpansion
_{R^{n+1}}%
%TCIMACRO{\dint}%
%BeginExpansion
\displaystyle\int %
%EndExpansion
_{R^{n+1}}\bar{f}\left( t,x\right) E_{m}\left( t,x;s,y\right) f\left(
s,y\right) \sqrt{g\left( t,x\right) }\sqrt{g\left( s,y\right) }%
dtdxdsdy<\infty \text{,}$$ and, for each $\sigma \in R$, let $N_{\sigma }\subset D^{\prime }\left(
R^{n}\right) $ be the space: $g\in N_{\sigma }$ if $$\left\| g\right\| _{N_{\sigma }}^{2}=%
%TCIMACRO{\dint}%
%BeginExpansion
\displaystyle\int %
%EndExpansion
_{R^{n}}\bar{g}\left( x\right) E_{m}\left( \sigma ,x;\sigma ,y\right)
g\left( y\right) \sqrt{g\left( \sigma ,x\right) }\sqrt{g\left( \sigma
,y\right) }dxdy<\infty \text{.}$$ We will consider that, as in the Euclidean case, the space $L^{2}\left(
R^{n},d\mu _{\sigma }\right) \subset N_{\sigma }$, where $d\mu _{\sigma
}\left( x\right) =\sqrt{g\left( \sigma ,x\right) }d^{n}x$ and that it is dense in $N_{\sigma }$ for each $\sigma \in R$. Now, let $\hat{E}_{\sigma
}:N_{\sigma }\rightarrow L^{2}\left( R^{n},d\mu _{\sigma }\right) $ be the operator corresponding to the kernel $E_{m}\left( \sigma ,x;\sigma ,y\right)
$. Then $\hat{E}_{\sigma }^{1/2}$ defines an isometry from $N_{\sigma }$ to $%
L^{2}\left( R^{n},d\mu _{\sigma }\right) $ and let $\left( \hat{E}_{\sigma
}^{1/2}\right) ^{\dagger }:L^{2}\left( R^{n},d\mu _{\sigma }\right)
\rightarrow $ $N_{\sigma }$ be its adjoint. The following are true: $$\hat{E}_{\sigma }^{1/2}\circ \left( \hat{E}_{\sigma }^{1/2}\right) ^{\dagger
}=1_{L^{2}\left( R^{n},d\mu _{\sigma }\right) }\text{ and }\left( \hat{E}%
_{\sigma }^{1/2}\right) ^{\dagger }\circ \hat{E}_{\sigma
}^{1/2}=1_{N_{\sigma }}\text{.}$$ With our assumptions, $\hat{E}_{\sigma }^{1/2}\left( N_{\sigma }\right)
=L^{2}\left( R^{n},d\mu _{\sigma }\right) \subset N_{\sigma }$, the operator $\hat{E}_{\sigma }^{1/2}$ is bounded on $N_{\sigma }$. Moreover, one can view $\left( \hat{E}_{\sigma }\right) ^{\dagger }$ as a dense defined unbounded operator on $N_{\sigma }$, in fact, it is the inverse operator of $%
\hat{E}_{\sigma }$.
For $\sigma \in R$, let $j_{\sigma }$ be the operator $j_{\sigma
}:N_{\sigma }\rightarrow N$, $\left( j_{\sigma }\psi \right) \left(
t,x\right) =\psi \left( x\right) \delta \left( t-\sigma \right) $ and $%
j_{\sigma }^{\ast }$ be its adjoint. If $\Lambda $ is a closed subset of $%
R^{n+1}$ we denote by $N_{\Lambda }$ the subspace of $N$ which comprises all distributions with support in $\Lambda $. The orthogonal projection of $N$ in $N_{\Lambda }$ will be denoted by $e_{\Lambda }$. Following [@Si2] we have:
The operators $j_{\sigma }$ are isometries and $j_{\sigma }^{*}j_{\sigma
}=1_{N_{\sigma }}$, $j_{\sigma }j_{\sigma }^{*}=e_{\sigma }$, where $%
e_{\sigma }$ denotes the projector corresponding to the subset of $R^{n+1}$, $t=\sigma $.
Then we define the operators: $$U_{\sigma ,\sigma ^{\prime }}:N_{\sigma ^{\prime }}\rightarrow N_{\sigma }%
\text{, }U_{\sigma ,\sigma ^{\prime }}=j_{\sigma }^{\ast }\circ j_{\sigma
^{\prime }}\text{.} \label{!}$$ We will derive in the following that $U_{\sigma ,\sigma ^{\prime }}$ are propagators in the sense of [@Si3]. This will follow from the Markoff property of the projectors $e_{\sigma }$.
Let $A$, $B$ and $C$ be closed subsets in $R^{n+1}$ such that $C$ separates $%
A$ and $B$. Then $e_{A}\circ e_{C}\circ e_{B}=e_{A}\circ e_{B}$.
This is the consequence of the fact that $E_{m}$ is the kernel of a local operator. The proof is identic with that of [@Si2].
The basics properties of $U_{\sigma ,\sigma ^{\prime }}$ operators are stated in the following proposition.
The family of operators $U_{\sigma ,\sigma ^{\prime }}$, $\sigma $, $\sigma
^{\prime }\in R$ has the following properties:1) $U_{\sigma ,\sigma ^{\prime }}\circ U_{\sigma ^{\prime },\sigma ^{\prime
\prime }}=U_{\sigma ,\sigma ^{\prime \prime }}$2) $U_{\sigma ,\sigma }=1_{N_{\sigma }}$3) $\left\| U_{\sigma ,\sigma ^{\prime }}\right\| \leqslant 1$.
1\) Using the Markoff property we have: $$e_{\sigma }\circ e_{\sigma ^{\prime }}\circ e_{\sigma ^{\prime \prime
}}=e_{\sigma }e_{\sigma ^{\prime \prime }}\Leftrightarrow j_{\sigma }\circ
j_{\sigma }^{*}\circ j_{\sigma ^{\prime }}\circ j_{\sigma ^{\prime
}}^{*}\circ j_{\sigma ^{\prime \prime }}\circ j_{\sigma ^{\prime \prime
}}^{*}=j_{\sigma }\circ j_{\sigma }^{*}\circ j_{\sigma ^{\prime \prime
}}\circ j_{\sigma ^{\prime \prime }}^{*}.$$ By composition with $j_{\sigma ^{\prime \prime }}$ at the right, we have $$j_{\sigma }\circ \left( j_{\sigma }^{*}\circ j_{\sigma ^{\prime }}\circ
j_{\sigma ^{\prime }}^{*}\circ j_{\sigma ^{\prime \prime }}-j_{\sigma
}^{*}\circ j_{\sigma ^{\prime \prime }}\right) =0.$$ From the definition of $U_{\sigma ,\sigma ^{\prime }}$ and since $j_{\sigma
} $ are isometries, we conclude $U_{\sigma ,\sigma ^{\prime }}U_{\sigma
^{\prime },\sigma ^{\prime \prime }}=U_{\sigma ,\sigma ^{\prime \prime }}$.2) It follows from proposition 1.1 and definition of $U_{\sigma ,\sigma
^{\prime }}$.3) Because $j_{\sigma }^{*}$ and $j_{\sigma }$ are isometries, the property results immediately.
Exponential bounds on propagators
=================================
To improve our estimates on the propagators $U_{\sigma ,\sigma
^{\prime }}$ we need a supplementary condition on the metric $g$. We say that an application $Q:R^{n+1}\rightarrow M\left( n+1,n+1\right) $ has stable positivity if there exists $\varepsilon >0$ such that for any application $\delta :R^{n+1}\rightarrow M\left( n+1,n+1\right) $ with $%
\left| \delta \left( x\right) ^{ij}\right| \leqslant \varepsilon $ the matrices $Q\left( x\right) -\delta \left( x\right) $ are positive defined for any $x\in R^{n+1}$. The following result is a direct application of Agmon theory [@Ag] of exponentially decay of solutions of elliptic second order operators.
If the metric $g$ has stable positivity then for any $f\in N_{\sigma
^{\prime }}$: $$%TCIMACRO{\dint}%
%BeginExpansion
\displaystyle\int %
%EndExpansion
_{T_{0}}^{\infty }d\sigma \left\{ e^{\omega \sigma }\left\| \hat{E}_{\sigma
}^{1/2}\circ U_{\sigma ,\sigma ^{\prime }}f\right\| _{N_{\sigma }}\right\}
^{2}<\infty \text{,}$$ provided $\omega <%
%TCIMACRO{\dfrac{m}{\sqrt{\sup g^{11}}}}%
%BeginExpansion
{\displaystyle{m \over \sqrt{\sup g^{11}}}}%
%EndExpansion
$.
Starting from $$\begin{array}{l}
\left\langle u,U_{\sigma ,\sigma ^{\prime }}f\right\rangle _{N_{\sigma
}}=\left\langle u,\hat{E}_{\sigma }\circ U_{\sigma ,\sigma ^{\prime
}}f\right\rangle _{L^{2}\left( R^{n},d\mu _{\sigma }\right) } \\
=\int_{R^{n}}\bar{u}\left( x\right) \left[ \int_{R^{n}}E_{m}\left( \sigma
,x;\sigma ^{\prime },y\right) f\left( y\right) d\mu _{\sigma ^{\prime
}}\left( y\right) \right] d\mu _{\sigma }\left( x\right)
\end{array}$$ for $u\in N_{\sigma }$ and $f\in N_{\sigma ^{\prime }}$, it follows that $%
\varphi \left( \sigma ,x\right) =\left( \hat{E}_{\sigma }\circ U_{\sigma
,\sigma ^{\prime }}f\right) \left( x\right) $ is a solution of $$\left( \Delta +m^{2}\right) \varphi \left( \sigma ,x\right) =0$$ for $\sigma >\sigma ^{\prime }$. Let $\rho _{m}\left( \cdot \,;\,\cdot
\right) $ denotes the distance corresponding to the metric $g_{m}=mg$. The metric $g$ has stable positivity so, there is an $\varepsilon \in R_{+}$ such that $\rho _{m}\left( \sigma _{0},x_{0};\sigma ,x\right) >%
%TCIMACRO{\dfrac{\varepsilon }{m}}%
%BeginExpansion
{\displaystyle{\varepsilon \over m}}%
%EndExpansion
\left| \sigma -\sigma _{0}\right| $. For $\Omega =\left\{ \left( \sigma
,x\right) :\sigma >T_{0}\right\} $, $T_{0}\in R_{+}$ and for some positive $%
\lambda $: $$\begin{array}{l}
\int_{\Omega }\left| \varphi \left( \sigma ,x\right) \right| ^{2}e^{-\lambda
\rho _{m}\left( T_{0},x_{0};\sigma ,x\right) }\sqrt{g\left( \sigma ,x\right)
}d\sigma d^{n}x \\
=\int_{T_{0}}^{\infty }d\sigma \left\langle \hat{E}_{\sigma }\circ U_{\sigma
,\sigma ^{\prime }}f,\hat{E}_{\sigma }\circ U_{\sigma ,\sigma ^{\prime
}}f\right\rangle _{L^{2}\left( R^{n},d\mu _{\sigma }\right) }e^{-\lambda
\frac{\varepsilon }{m}\left( \sigma -T_{0}\right) } \\
<ct.\int_{T_{0}}^{\infty }d\sigma \left\langle U_{\sigma ,\sigma ^{\prime
}}f,\hat{E}_{\sigma }\circ U_{\sigma ,\sigma ^{\prime }}f\right\rangle
_{L^{2}\left( R^{n},d\mu _{\sigma }\right) }e^{-\lambda \frac{\varepsilon }{m%
}\left( \sigma -T_{0}\right) } \\
=ct.\int_{T_{0}}^{\infty }d\sigma \left\| U_{\sigma ,\sigma ^{\prime
}}f\right\| _{N_{\sigma }}^{2}e^{-\lambda \frac{\varepsilon }{m}\left(
\sigma -T_{0}\right) }<\infty \text{.}
\end{array}$$ So we are in the conditions of the main theorem of [@Ag]. It follows that: $$\begin{array}{l}
\int_{\Omega }d\sigma d^{n}x\sqrt{g\left( \sigma ,x\right) }\left| \varphi
\left( \sigma ,x\right) \right| ^{2}\left( m^{2}-g\left( \nabla h\left(
\sigma ,x\right) ,\nabla h\left( \sigma ,x\right) \right) \right)
e^{2h\left( \sigma ,x\right) } \\
\leqslant \frac{2\left( 1+2d\right) }{d^{2}}m^{2}\int_{\Omega \setminus
\Omega _{d}}\left| \varphi \left( \sigma ,x\right) \right| ^{2}e^{2h\left(
\sigma ,x\right) }\sqrt{g\left( \sigma ,x\right) }dx\text{,}
\end{array}$$ where $d$ is a positive number and $\Omega _{d}=\left\{ \left( \sigma
,x\right) \in \Omega :\rho _{m}\left( \left( \sigma ,x\right) ,\left\{
\infty \right\} \right) >d\right\} $. Here $$\rho _{m}\left( \left( \sigma ,x\right) ,\left\{ \infty \right\} \right)
=\sup \left\{ \rho _{m}\left( \left( \sigma ,x\right) ,\Omega \setminus
K\right) :K\text{ is a compact subset of }\Omega \right\} \text{.}$$ The function $h$ is any function which satisfies the condition $g\left(
\nabla h\left( \sigma ,x\right) ,\nabla h\left( \sigma ,x\right) \right)
<m^{2}$. We choose $h\left( \sigma ,x\right) =\omega \sigma $ with $\omega <%
%TCIMACRO{\dfrac{m}{\sqrt{\sup g^{11}}}}%
%BeginExpansion
{\displaystyle{m \over \sqrt{\sup g^{11}}}}%
%EndExpansion
$. The above inequality becomes $$\begin{array}{l}
\int_{\Omega }d\sigma d^{n}x\sqrt{g\left( \sigma ,x\right) }\left| \varphi
\left( \sigma ,x\right) \right| ^{2}e^{2\omega \sigma } \\
<\frac{2\left( 1+2d\right) }{d^{2}}\frac{m^{2}}{m^{2}-\omega ^{2}}%
\int_{\Omega \setminus \Omega _{d}}d\sigma dx\sqrt{g\left( \sigma ,x\right) }%
\left| \varphi \left( \sigma ,x\right) \right| ^{2}e^{2\omega \sigma }\text{.%
}
\end{array}$$ If for any point $\left( \sigma ,x\right) \in \Omega $ there is a geodesic which starts in $\left( \sigma ,x\right) $ and ends in the hyperplane $%
\sigma =T_{0}$ then $\Omega \setminus \Omega _{d}\subset \left\{ \left( \tau
,x\right) :0<\sigma \leqslant T\right\} $ with $T$ sufficiently large but finite. In conclusion $$\begin{array}{l}
\int_{\Omega }d\sigma d^{n}x\sqrt{g\left( \sigma ,x\right) }\left| \varphi
\left( \tau ,x\right) \right| ^{2}e^{2\omega \sigma } \\
=\int_{T_{0}}^{\infty }d\sigma e^{2\omega \sigma }\left\langle \hat{E}%
_{\sigma }\circ U_{\sigma ,\sigma ^{\prime }}f,\hat{E}_{\sigma }\circ
U_{\sigma ,\sigma ^{\prime }}f\right\rangle _{L^{2}\left( R^{n},\mu _{\sigma
}\right) }<\infty \text{,}
\end{array}$$ or $$%TCIMACRO{\dint}%
%BeginExpansion
\displaystyle\int %
%EndExpansion
_{T_{0}}^{\infty }d\sigma e^{2\omega \sigma }\left\langle \hat{E}_{\sigma
}\circ U_{\sigma ,\sigma ^{\prime }}f,\hat{E}_{\sigma }\circ U_{\sigma
,\sigma ^{\prime }}f\right\rangle _{L^{2}\left( R^{n},\mu _{\sigma }\right)
}<\infty \text{,}$$ which implies $$%TCIMACRO{\dint}%
%BeginExpansion
\displaystyle\int %
%EndExpansion
_{T_{0}}^{\infty }d\sigma \left\{ e^{\omega \sigma }\left\| \hat{E}_{\sigma
}^{1/2}\circ U_{\sigma ,\sigma ^{\prime }}f\right\| _{N_{\sigma }}\right\}
^{2}<\infty \text{.}$$
The stationary case
===================
We consider in this section that there is a coordinate system such that the metric $g$ is independent of first coordinate. In this case, the spaces $N_{\sigma }$ and the operators $\hat{E}_{\sigma }^{1/2}$ are identically and will be denoted by $N_{0}$ and $\hat{E}_{0}^{1/2}$ respectively. Thus, the operators $U_{\sigma ,\sigma ^{\prime }}$ are defined on the same Hilbert space and depend only on the difference $\sigma
-\sigma ^{\prime }:$ $U_{\sigma ,\sigma ^{\prime }}=U_{\sigma -\sigma
^{\prime }}$. The family of operators $\left\{ U_{\tau }\right\} _{\tau \in
R_{+}}$ forms a semigroup. Using the results about existence and properties of the generators of semigroups [@X], we can obtain bounds directly on the transfer matrix $U_{\tau }$.
The semigroup $\left\{ U_{\tau }\right\} _{\tau \in R_{+}}$ is exponentially bounded: $\left\| U_{\tau }\right\| _{N_{0}}<e^{-\tau \omega }$ provided $%
\omega <%
%TCIMACRO{\dfrac{m}{\sqrt{sup\,g^{11}}}}%
%BeginExpansion
{\displaystyle{m \over \sqrt{sup\,g^{11}}}}%
%EndExpansion
$.
Because we have found estimates on $\hat{E}_{0}^{1/2}\circ U_{\tau }$, we will consider the operators $\tilde{U}_{\tau }=\hat{E}_{0}^{1/2}\circ
U_{\tau }\circ \left( \hat{E}_{0}^{1/2}\right) ^{\dagger }$, well defined on $L^{2}\left( R^{n},d\mu _{0}\right) $. Using the fact that $L^{2}\left(
R^{n},d\mu _{0}\right) $ is dense in $N_{0}$ we can extend these operators by continuity on the space $N_{0}$. In this way we have build the semigroup $%
\left\{ \tilde{U}_{\tau }\right\} _{\tau \in R_{+}}$ which satisfies the estimates of the precedent section: $$%TCIMACRO{\dint}%
%BeginExpansion
\displaystyle\int %
%EndExpansion
_{T_{0}}^{\infty }d\tau \left\{ e^{\omega \tau }\left\| \tilde{U}_{\tau
}\right\| _{N_{0}}\right\} ^{2}<\infty \text{,}$$ for some $T_{0}>0$. So $\left\{ \tilde{U}_{\tau }\right\} _{\tau \in R_{+}}$ is exponentially bounded and in consequence [@X], if $\tilde{K}$ is its generator ($\tilde{U}_{\tau }=e^{-\tau \tilde{K}}$) the resolvent set of $%
\tilde{K}$ satisfies: $$\left\{ z\in C\shortmid
%TCIMACRO{\func{Re}}%
%BeginExpansion
\mathop{\rm Re}%
%EndExpansion
z\in (-\infty ,\omega )\right\} \subset \rho \left( \tilde{K}\right) \text{.}$$ If $K$ is the generator of $\left\{ U_{\tau }\right\} _{\tau \in R_{+}}$ then, on ${\cal D}\left( K\right) $ we have: $$K=\left( \hat{E}_{0}^{1/2}\right) ^{\dagger }\circ \tilde{K}\circ \hat{E}%
_{0}^{1/2}$$ by using the reciprocal formula $$U_{\tau }=\left( \hat{E}_{0}^{1/2}\right) ^{\dagger }\circ \tilde{U}_{\tau
}\circ \hat{E}_{0}^{1/2}\text{,} \label{!!}$$ valid on $N_{0}$. If the operator $$\left( \hat{E}_{0}^{1/2}\right) ^{\dagger }\circ \left( \tilde{K}-z\right)
^{-1}\circ \hat{E}_{0}^{1/2} \label{!!!!}$$ is well defined, even on a dense subset of $N_{0}$, then $K-z$ is inversable. From 20 it follows that, if $\left( \tilde{K}-z\right) ^{-1}$ exists, then: $$\left( \tilde{K}-z\right) ^{-1}\left( L^{2}\left( R^{n},d\mu _{0}\right)
\right) \subset L^{2}\left( R^{n},d\mu _{0}\right) \text{,}$$ and in consequence $\left( \hat{E}_{0}^{1/2}\right) ^{\dagger }\circ \left(
\tilde{K}-z\right) ^{-1}\circ \hat{E}_{0}^{1/2}$ is well defined on the entire $N_{0}$. Will follow that $\rho \left( \tilde{K}\right) \subset \rho
\left( K\right) $ and this ends the proof.
If the metric is symmetric at transformation $x^{1}\rightarrow
-x^{1}$, the transfer matrix generator is self-adjoint and it can be considered as the Hamiltonian of the scalar field.
Application
===========
Our application is for the Euclidean case. The results concerning decoupling of different regions in quantum Euclidean fields are based primarily on estimates of $\left\| e_{\Lambda _{1}}e_{\Lambda _{2}}\right\|
_{N}$, where $\Lambda _{1}$, $\Lambda _{2}$ are two disjoint regions. Let us consider the two dimensional case. The most difficult case is when $\Lambda
_{1}$, $\Lambda _{2}$ are not convex and there is no possibility of drawing a straight line between the two subsets. We can sharpen the existent estimates [@Si2] for these cases by using the previous results. The idea is to make a change of coordinates such that for the new coordinates, lines like $\sigma =ct.$ separate the two sets and they are as closed as possible to the boundaries of $\Lambda _{1}$, $\Lambda _{2}$. Then we can use the exponential bounds of the previous section to evaluate $\left\| e_{\Lambda
_{1}}e_{\Lambda _{2}}\right\| _{N}$. More precisely:
Let $\Lambda _{1}$, $\Lambda _{2}$ two regions in $R^{2}$ such that the construction of the coordinates \[c\] to be possible (after a rotation if necessary). Then $$\left\| e_{\Lambda _{1}}\circ e_{\Lambda _{2}}\right\| _{N}\leqslant
e^{-m\left| \beta -\alpha \right| \min \left| \cos \theta \right| }\text{,}$$ where $\theta $ and $\left| \beta -\alpha \right| $ will be defined during the proof.
Let $\left( t,x\right) $ denotes the original coordinates in which the metric is diagonal. Let $\gamma :R\rightarrow R^{2}$ be a curve which separates $\Lambda _{1}$, $\Lambda _{2}$ and $\gamma \left( 0\right) =\left(
t=0,x=0\right) $. We define a new coordinate system $\left( \sigma ,\xi
\right) $ by $$\left\{
\begin{array}{c}
t\left( \sigma ,\xi \right) =\sigma +\gamma ^{1}\left( \xi \right) \\
x\left( \sigma ,\xi \right) =\gamma ^{2}\left( \xi \right)
\end{array}
\right. \label{c}$$ In the new coordinates, the metric is $$g^{\prime }\left( \sigma ,\xi \right) =\left(
\begin{array}{cc}
1 &
%TCIMACRO{\dfrac{d\gamma ^{1}}{d\xi } }%
%BeginExpansion
{\displaystyle{d\gamma ^{1} \over d\xi }}%
%EndExpansion
\\
%TCIMACRO{\dfrac{d\gamma ^{1}}{d\xi } }%
%BeginExpansion
{\displaystyle{d\gamma ^{1} \over d\xi }}%
%EndExpansion
& \left(
%TCIMACRO{\dfrac{d\gamma ^{1}}{d\xi }}%
%BeginExpansion
{\displaystyle{d\gamma ^{1} \over d\xi }}%
%EndExpansion
\right) ^{2}+\left(
%TCIMACRO{\dfrac{d\gamma ^{2}}{d\xi }}%
%BeginExpansion
{\displaystyle{d\gamma ^{2} \over d\xi }}%
%EndExpansion
\right) ^{2}
\end{array}
\right)$$ so we are in the conditions of the last section. Using the Markoff property, $$\left\| e_{\Lambda _{1}}\circ e_{\Lambda _{2}}\right\| _{N}=\left\|
e_{\Lambda _{1}}\circ e_{\alpha }\circ e_{\beta }\circ e_{\Lambda
_{2}}\right\| _{N}\leqslant \left\| e_{\alpha }\circ e_{\beta }\right\| _{N}%
\text{,}$$ where the lines $\sigma =\alpha $, $\sigma =\beta $ separate $\Lambda _{1}$ and $\Lambda _{2}$ exactly in the order they appear in the above relation (in the sense that $\sigma =\alpha $ separates $\Lambda _{1}$ by $\sigma
=\beta $ etc.). Further $$\left\| j_{\alpha }\circ j_{\alpha }^{\dagger }\circ j_{\beta }\circ
j_{\beta }^{\dagger }\right\| _{N}=\left\| j_{\alpha }\circ U_{\alpha -\beta
}\circ j_{\beta }^{\dagger }\right\| _{N}=\left\| U_{\alpha -\beta }\right\|
_{N_{0}}.$$ The element $\left( g^{\prime }\right) ^{11}$ is given by $\left( g^{\prime
}\right) ^{11}=%
%TCIMACRO{\dfrac{1}{\cos ^{2}\theta }}%
%BeginExpansion
{\displaystyle{1 \over \cos ^{2}\theta }}%
%EndExpansion
$, where $\theta $ is the angle between the tangent to the curve $\gamma $ and the $x$ axis. Using the bounds of the last section we have $$\left\| e_{\Lambda _{1}}\circ e_{\Lambda _{2}}\right\| _{N}\leqslant
e^{-m\left| \beta -\alpha \right| \min \left| \cos \theta \right| }.$$ Performing first a rotation, one can choose the best values for $\left|
\beta -\alpha \right| $ and $\min \left| \cos \theta \right| $.
Conclusions
===========
Our primary goal was to define the transfer matrix for scalar fields on curved spaces and to investigate the basic spectral properties of its generator. Even though the generator is not self-adjoint in the general case, this approach allows us to investigate this problem by using at least two new tools besides the methods of Green functions. One is the perturbations of hypercontractive semigroups [@Si4] and the other is the adiabatic theorem.
Now it is straightforward to quantize the field by defining the Markoff field over the space $N$. For the stationary, symmetric at time reflection case (static), we think that one has now all elements to construct the physical field (for example that proposed in [@Di]) by following Nelson reconstruction method and holomorphic continuation of the transfer matrix. Note that, acording to results of [@Si3], the holomorphic continuation of the transfer matrix to real time is still possible, in the stationary case without symmetry at time reflection, as long the spectrum of the generator belongs to the real axis. Of course, one has to check that the results of [@Ne2] (sistematized in [@Si2]), which are the core of the reconstruction theorem, are still valid. For the general case, we think that the adiabatic theorem, especially the adiabatic reduction theory [@Nen], may play an important role in defining the physical quantum field by following Nelson$\prime $s approach.
Nelson E., J. Func. Anal. [**12**]{}, 97 (1973)
Agmon S., [*Lectures on exponential decay of solutions of second-order elliptic equations*]{}, Princeton: Princeton Univ. Press (1982)
Guerra F., Rosen L., Simon B., Ann. Math. [**101**]{}, 111 (1975)
Dimock J., J. Math. Phys. [**20**]{}, 2549 (1979)
Simon B., [*The*]{} $P\left( \phi \right) _{2}$ [*Euclidean (Quantum) Field Theory*]{}, Princeton: Princeton University Press (1974)
Reed M., Simon B., [*Methods of Modern Mathematical Physics*]{}, Vol. 2, New York: Academic Press (1975)
Neerven J., [*The asymptotic behavior of semigroups of linear operators*]{}, Basel, Boston: Birkhauser (1996)
Simon B., Hoegh-Krohn R., J. Func. Anal. [**9**]{}, 121 (1972)
Nelson E., J. Func.[* *]{}Anal. [**11**]{}, 211 (1972)
Nenciu G., Commun. Math. Phys. [**152**]{}, 479 (1993)
|
---
abstract: 'The core idea of metric-based few-shot image classification is to directly measure the relations between query images and support classes to learn transferable feature embeddings. Previous work mainly focuses on image-level feature representations, which actually cannot effectively estimate a class’s distribution due to the scarcity of samples. Some recent work shows that local descriptor based representations can achieve richer representations than image-level based representations. However, such works are still based on a less effective instance-level metric, especially a symmetric metric, to measure the relations between query images and support classes. Given the natural asymmetric relation between a query image and a support class, we argue that an asymmetric measure is more suitable for metric-based few-shot learning. To that end, we propose a novel *Asymmetric Distribution Measure (ADM)* network for few-shot learning by calculating a joint local and global asymmetric measure between two multivariate local distributions of queries and classes. Moreover, a task-aware *Contrastive Measure Strategy (CMS)* is proposed to further enhance the measure function. On popular *mini*ImageNet and *tiered*ImageNet, we achieve $3.02\%$ and $1.56\%$ gains over the state-of-the-art method on the $5$-way $1$-shot task, respectively, validating our innovative design of asymmetric distribution measures for few-shot learning.'
author:
- |
Wenbin Li^$1$^,Lei Wang^$2$^,Jing Huo^$1$^,Yinghuan Shi^$1$^,Yang Gao^$1$^,Jiebo Luo^$3$^\
^$1$^Nanjing University, China,^$2$^University of Wollongong, Australia,\
^$4$^University of Rochester, USA\
[email protected], [email protected], {huojing, syh, gaoy}@nju.edu.cn, [email protected]
- |
Wenbin Li^$1$^,Lei Wang^$2$^,Jing Huo^$1$^,Yinghuan Shi^$1$^,Yang Gao^$1$^,Jiebo Luo^$3$^\
^$1$^Nanjing University, China,^$2$^University of Wollongong, Australia\
^$4$^University of Rochester, USA\
bibliography:
- 'ijcai20.bib'
title: 'Asymmetric Distribution Measure for Few-shot Learning'
---
Introduction
============
Few-shot learning for image classification has gained considerable attention in recent years [@vinyals2016matching; @finn2017model; @sung2018learning; @lee2019meta], which attempts to learn a classifier with good generalization capacity for new unseen classes with only a few samples. Because of the scarcity of data, it is almost impossible to directly train a conventional supervised model (*e.g.,* a convolutional neural network) from scratch by only using the few available samples. Therefore, transfer learning shall be a natural way to learn transferable knowledge to boost the target few-shot classification. Along this way, a variety of methods have been proposed, which can be roughly divided into three categories: [*data-augmentation based*]{} methods [@antoniou2017data; @schwartz2018delta; @xian2019f], [*meta-learning based*]{} methods [@ravi2016optimization; @jamal2019task; @lee2019meta] and [*metric-based*]{} methods [@vinyals2016matching; @sung2018learning; @li2019CovaMNet]. Metric-based few-shot learning methods have achieved significant successes and attracted increasing attention due to their simplicity and effectiveness. In this work, we focus on this kind of methods.
The basic idea of metric-based few-shot learning methods is to learn a transferable deep embedding network by directly measuring the relations between query images and support classes. Thus, two key issues are involved in such a kind of methods, *i.e.,* feature representations and relation measure. For feature representations, traditional methods such as ProtoNet [@snell2017prototypical] and RelationNet [@sung2018learning] generally adopt image-level global feature representations for both query images and support classes. However, due to the scarcity of samples in each class, such image-level global features are not effective in representing the underlying distribution of each class. Recently, CovaMNet [@li2019CovaMNet] and DN4 [@li2019DN4] introduce deep local descriptors into few-shot learning and attempt to utilize the distribution of local descriptors to represent each support class, which have been verified to be more effective than using the image-level global features.
On the relation measure, the existing methods including the above methods usually adopt an instance-level metric, where the query image is taken as one single instance (*i.e.,* an image-level feature representation) or a set of instances (*i.e.,* a set of local feature descriptors). For example, in ProtoNet, the Euclidean distance is chosen to calculate the distance between a query instance and the prototype (*i.e.,* the mean vector) of each support class. Also, CovaMNet proposes a covariance metric function to measure a local similarity between each local descriptor of a query image and a support class. Next, it aggregates all the local similarities to obtain a global similarity as the relation between this query image and this class.
However, these existing methods have only considered the distributions of the support classes while neglecting the natural distributions of the query images. Moreover, the instance-level metric they employ can only capture a kind of local relations (*i.e.,* local similarities) between the query images and support classes. We argue that the distributions of the query images are equally important and a *distribution-level measure* shall be designed to capture the global-level relations between the queries and classes. More importantly, we observe that the existing methods usually adopt a symmetric metric function (*i.e.,* $M(a,b)=M(b,a)$) to calculate the symmetric relations between queries and classes. For instance, both the Euclidean distance used in ProtoNet and the cosine similarity adopted in CovaMNet and DN4 are symmetric functions. However, we highlight that there is an asymmetric relation between the query images and a certain class. In particular, when each image is represented by a set of deep local descriptors, the distribution of the descriptors in one query image is only comparable to part of the distribution of the descriptors extracted from a support class. Therefore, we argue that an asymmetric measure is more suitable for the metric-based few-shot learning to capture the asymmetric relations.
To this end, we develop a novel *Asymmetric Distribution Measure (ADM)* network for metric-based few-shot learning. First, we represent each image as a set of deep local descriptors (instead of a single image-level global feature representation) and consider characterizing both query images and support classes from the perspective of local descriptor based distributions (*i.e.,* mean vector and covariance matrix). Second, we employ an asymmetric Kullback–Leibler (KL) divergence measure to align a query distribution with a support class distribution to capture the global distribution-level asymmetric relations. Third, to further improve the metric space by taking the context of the task into consideration, we propose a task-aware *Contrastive Measure Strategy (CMS)*, which can be used as a plug-in to any measure functions. Finally, inspired by the successful image-to-class measure (an asymmetric measure as a whole) introduced in DN4 which mainly captures the asymmetric relations via individual local descriptor based cosine similarity measures, we combine the whole distribution based KL divergence measure with the image-to-class measure together to simultaneously capture the global and local relations.
The main contributions of this work are as follows:
- We propose a pure distribution based method for metric-based few-shot learning and show that an asymmetric measure is more suitable for this kind of few-shot learning methods.
- We simultaneously combine the global relations (*i.e.,* the KL divergence measure) and the local relations (*i.e.,* the image-to-class measure) together to measure the complete asymmetric distribution relations between queries and classes.
- We propose an adaptive fusion strategy to adaptively integrate the global and local relations.
- We design a task-aware contrastive measure strategy (CMS) as a plug-in to further enhance the adopted measure functions.
Related Work
============
We first briefly review the metric-based few-shot learning methods in the literature, and then introduce related work that inspired our work in this paper.
The first metric-based few-shot learning method was proposed in [@koch2015siamese], which adopted a Siamese neural network to learn transferable and discriminative feature representations. In [@vinyals2016matching], a Matching Net which directly compares the query images with the support classes was presented, where a subsequently widely used episodic training mechanism was also proposed. After that, [@snell2017prototypical] proposed a ProtoNet, which represents a support class by a prototype, *i.e.,* the mean vector of all sample in this class. Then a specific metric, *i.e.,* Euclidean distance, was used to perform the final classification. Recently, based on ProtoNet, an infinite mixture prototypes (IMP) network was proposed [@allen2019infinite], where each support class was represented by a set of adaptive prototypes. In addition, to avoid choosing a specific metric function, RelationNet [@sung2018learning] proposed to learn a metric through a deep convolutional neural network to measure the similarity between queries and support classes.
The above methods are all based on image-level feature representations. Due to the scarcity of samples in each class in few-shot learning, the distribution of each class cannot be reliably estimated in a space of image-level features. Thus, some recent work, such as CovaMNet [@li2019CovaMNet] and DN4 [@li2019DN4] shows that the rich local features (*i.e.,* deep local descriptors) can achieve better representations than the image-level features, because the local features can be taken as a natural data augmentation operation. CovaMNet employs the second-order covariance matrix of the extracted deep local descriptors to represent each support class and designs a covariance-based metric to measure the similarities between query images and support classes. Different from CovaMNet, DN4 argues that the pooling of local features into a compact image-level representation will lose considerable discriminative information. Therefore, DN4 proposes to directly use the raw local descriptor sets to represent both query images and support classes, and then employs a cosine-based image-to-class measure to perform the relation measure.
Inspired by CovaMNet and DN4, our ADM also takes the rich deep local descriptors to represent an image. Compared with CovaMNet, the key difference is that CovaMNet only considers the distributions of the support classes but neglect the distributions of the query images, while we consider the both. Another important difference is that both CovaMNet and DN4 employ a cosine similarity function (*i.e.,* an instance-level metric) to calculate a series of local relations between a query image and a certain class. In contrast, our ADM can capture the complementary global relations by using an extra distribution-level measure. In addition, we observe that the relations between query images and a certain class are actually asymmetric, *i.e.,* a query image is only commensurate with an element in an image class when it is viewed as a set. Therefore, we argue that an asymmetric measure shall be considered for metric-based few-shot learning to reflect this property.
{width="0.8\linewidth"}
Preliminary
===========
**Problem formulation.** Under the few-shot setting, there are usually three sets of data, *i.e.,* a support set $\mathcal{S}$, a query set $\mathcal{Q}$ and an auxiliary set $\mathcal{A}$. In particular, $\mathcal{S}$ and $\mathcal{Q}$ share the same label space, which are corresponding to the training and test sets respectively in the general classification task. If $\mathcal{S}$ contains $C$ classes with $K$ (*e.g.,* $1$ or $5$) samples per class, we call this classification task $C$-way $K$-shot. However, $\mathcal{S}$ only has a few samples in each class, making it almost impossible to train a deep neural network effectively. Therefore, the auxiliary set $\mathcal{A}$ is generally introduced to learn transferable knowledge to tackle this problem. Also, $\mathcal{A}$ enjoys more classes and more samples per class than $\mathcal{S}$, but has a disjoint label space from $\mathcal{S}$.
**Episodic training.** To learn a classifier that can generalize well, an episodic training mechanism [@vinyals2016matching] is normally adopted in the training stage of the metric-based few-shot learning methods. Specifically, in each episode, a new task simulating the target few-shot task is randomly constructed from $\mathcal{A}$. Each simulated task consists of two subsets, $\mathcal{A_{S}}$ and $\mathcal{A_{Q}}$, which are akin to $\mathcal{S}$ and $\mathcal{Q}$, respectively. At each iteration, one episode (task) is adopted to train the current model. Basically, tens of thousands of episodes (tasks) will be randomly sampled to train this model. Once the training process is completed, we can predict the labels of $\mathcal{Q}$ using the trained model based on $\mathcal{S}$.
Methodology
===========
As illustrated in Figure \[fig:model\], our ADM model mainly consists of three components: a feature embedding module, a joint asymmetric measure module, and a classifier module. The first module learns feature embeddings and produces rich deep local descriptors for an input image. Next, the distributions of query images and support classes can be represented at the level of deep local descriptors. The second module defines a joint asymmetric distribution measure between the query distribution and the support class distribution by considering both the asymmetric local relations and the asymmetric global relations. As for the last module, we adaptively fuse the local and global relations together by a jointly learned weight vector, and then adopt a non-parametric nearest neighbor classifier as the final classifier. These three modules are jointly trained from scratch in an end-to-end manner.
Feature Embedding with Local Descriptors
----------------------------------------
As have been shown by some recent work [@li2019CovaMNet; @li2019DN4], local descriptor based feature representations are much richer than image-level features and can alleviate the scarcity issue of samples in few-shot learning. Following these work, we employ the rich and informative local descriptors to represent each image as well.
To this end, we design a feature embedding module $f_\varphi(\cdot)$, which can extract rich deep local descriptors for input images. Specifically, given an image $X$, $f_\varphi(X)$ will be a $c \times h \times w$ three-dimensional (3D) tensor, which can be seen as a set of $c$-dimensional local descriptors $$\label{function_1}
f_\varphi(X)=[\bm{x}_1,\ldots,\bm{x}_n]\in \mathbb{R}^{c \times n}\,,$$ where $\bm{x}_i$ is the $i$-th local descriptor and $n=h\times w$ is the total number of local descriptors for image $X$. These local descriptors can be seen as the local representations of the spatial local patches in this image. Basically, for each query image, we use the extracted $n$ local descriptors to estimate its distribution in the space of $\mathbb{R}^c$. As for each support class, all the local descriptors of all the images in this class will be used together to estimate its distribution in the space of $\mathbb{R}^c$. Since the local descriptors can capture the local subtle information, they can benefit more for the final image recognition.
Our Asymmetric Distribution Measure (ADM) {#SubSec:Model}
-----------------------------------------
[ **Kullback–Leibler divergence based distribution measure.** ]{} Assuming that the distributions of local descriptors extracted from an image or a support class are multivariate Gaussian, a query image’s distribution can be denoted by $Q=\mathcal{N}(\bm{\mu}_Q,\bm{\Sigma}_Q)$, and a certain support class’s distribution can be expressed by $S=\mathcal{N}(\bm{\mu}_S,\bm{\Sigma}_S)$, where $\bm{\mu}\in \mathbb{R}^c$ and $\bm{\Sigma}\in \mathbb{R}^{c \times c}$ indicate the mean vector and covariance matrix of a specific distribution, respectively. Thus, Kullback-Leibler (KL) divergence [@duchi2007derivations] between $Q$ and $S$ can be defined as: $$\label{function_2}
\begin{split}
D_\text{KL}(Q\|S) &= \frac{1}{2}\Big(\text{trace}(\bm{\Sigma}^{-1}_S\bm{\Sigma}_Q)+\ln\big( \frac{\det\bm{\Sigma}_S}{\det\bm{\Sigma}_Q}\big) \\
&+ (\bm{\mu}_S-\bm{\mu}_Q)^\top\bm{\Sigma}^{-1}_S(\bm{\mu}_S-\bm{\mu}_Q)-c \Big)\,,
\end{split}$$ where $\text{trace}(\cdot)$ is the trace operation of matrix, $\ln(\cdot)$ denotes logarithm with the base of $e$, and $\det$ indicates the determinant of a square matrix. As seen, Eq.(\[function\_2\]) takes both the mean and covariance into account to calculate the distance between two distributions.
Typically, since the KL divergence measure is asymmetric, $D_\text{KL}(Q\|S)$ mainly matches the distribution of $Q$ to the one of $S$, which is essentially different from $D_\text{KL}(S\|Q)$. One important advantage of using Eq.(\[function\_2\]) is that it can naturally capture the asymmetric relations between query images to support classes, forcing the query images to be close to the corresponding true class when used in our network training.
To further show the advantage of using an asymmetric measure, we purposely introduce a symmetric distribution metric function, *e.g.,* $2$-Wasserstein distance [@olkin1982distance], whose formulation is defined as follows, $$\label{function_3}
\begin{split}
D_\text{wass}(Q,S)^2 & = \left\|\bm{\mu}_Q-\bm{\mu}_S\right\|_2^2+ \\
&\text{trace}\Big(\bm{\Sigma}_Q+\bm{\Sigma}_S-2\big(\bm{\Sigma}_Q^{\frac{1}{2}}\bm{\Sigma}_S\bm{\Sigma}_Q^{\frac{1}{2}}\big)^{\frac{1}{2}}\Big)\,,
\end{split}$$ However, due to the square root of matrices, the calculation of the above distance function is time consuming and the optimization of this function is difficult. Therefore, in the literature [@berthelot2017began; @he2018wasserstein], an approximation function is normally employed $$\label{function_4}
D_\text{wass}(Q,S)^2 =\left\|\bm{\mu}_Q-\bm{\mu}_S\right\|_2^2+\left\|\bm{\Sigma}_Q-\bm{\Sigma}_S\right\|_F^2\,,$$ where the first term calculates the squared Euclidean distance between two mean vectors and the second term is a squared Frobenius norm of the difference between two covariance matrices. The comparison and analysis between $2$-Wasserstein distance and KL divergence will be detailed in Section \[ablation\_study\].
[ **Image-to-Class based distribution measure.** ]{} The above KL divergence measure can capture the global distribution-level relations between a query image and support classes. Nevertheless, the local relations are not taken into consideration yet. According to a deep analysis of DN4 [@li2019DN4], we observe that there may be two implicit reasons of the success of DN4. One reason is that the local descriptor based measure (*i.e.,* local relations) it used enjoys a stronger generalization ability than the image-level feature based measure. The other key reason is that the image-to-class measure used in DN4 is asymmetric on the whole, which aligns well with our argument of the necessity of the asymmetric measure. Therefore, such an asymmetric image-to-class measure is also introduced into our model to capture the local-level relations between queries and support classes. However, the difference in our work lies that the indispensable global relations are also complemented by an asymmetric distribution-level measure (*i.e.,* KL divergence).
To be specific, given a query image $Q$ and a support class $S$, which will be represented as $f_\varphi(Q)=[\bm{x}_1,\ldots,\bm{x}_n]\in \mathbb{R}^{c \times n}$ and $f_\varphi(S)=[f_\varphi(X_1),\ldots,f_\varphi(X_K)]\in \mathbb{R}^{c \times nK}$, respectively, where $K$ is the number of shots in $S$. Thus, the image-to-class (I2C) similarity measure can be formulated as $$\label{function_5}
\begin{split}
D_\text{I2C}(Q,S) & = \sum_{i=1}^{n}\text{Topk}\Big(\frac{f_\varphi(Q)^\top\cdot f_\varphi(S)}{\|f_\varphi(Q)^\top\|_F\cdot\|f_\varphi(S)\|_F}\Big)\,,
\end{split}$$ where $\text{Topk}(\cdot)$ means selecting the $k$ largest elements in each row of the correlation matrix between $Q$ and $S$, *i.e.,* [$\frac{f_\varphi(Q)^\top\cdot f_\varphi(S)}{\|f_\varphi(Q)^\top\|_F\cdot\|f_\varphi(S)\|_F}$]{}. Typically, $k$ is set as $1$ in our work.
[ **Classification with an adaptive fusion strategy.** ]{} Since two types of relations have been calculated, *i.e.,* global-level relations calculated by the KL divergence measure and local-level relations produced by the I2C measure, a fusion strategy shall be designed to integrate these two parts. To tackle this issue, we adopt a learnable $2$-dimensional weight vector $\bm{w}=[w_1, w_2]$ to implement this fusion. It is worth noting that because the KL divergence indicates dissimilarity rather than similarity, we use the negative of this divergence to obtain a similarity. Specifically, the final fusion similarity between a query $Q$ and a class $S$ can be defined as follows $$\label{function_6}
\begin{split}
D(Q,S) & = -w_1\cdot D_\text{KL}(Q\|S) + w_2\cdot D_\text{I2C}(Q,S)\,.
\end{split}$$ As seen in Figure \[fig:model\], for a $5$-way $1$-shot task and a specific query $Q$, the outputs of the I2C branch and KL branch are a $5$-dimensional similarity vector, respectively. Next, we concatenate these two vectors together to get a $10$-dimensional vector. And then, we apply a 1D convolution layer with the kernel size of $1\times 1$ along with a dilation value of $5$. In this way, we can obtain a weighted $5$-dimensional similarity vector by learning a $2$-dimensional weights $\bm{w}$. Additionally, a Batch Normalization layer is also added before the 1D convolution layer to balance the scale of the two parts of similarities. Finally, a non-parametric nearest neighbor classifier is performed to obtain the final classification results.
--------------------------------------- ------------ -------------------- --------------------------------------------------- --------------------------------------------------- --------------------------------------------------- ---------------------------------------------------
1-shot 5-shot 1-shot 5-shot
**ProtoNet**$^\ddag$ \[NeurIPS 2017\] Symmetric Instance-level [ $48.45 \scriptstyle\small \pm 0.96$]{} [ $66.53 \scriptstyle\small \pm 0.51$]{} [ $48.58 \scriptstyle\small \pm 0.87$]{} [ $69.57 \scriptstyle\small \pm 0.75$]{}
**RelationNet** \[CVPR 2018\] Symmetric Instance-level [ $50.44 \scriptstyle\small \pm 0.82$]{} [ $65.32 \scriptstyle\small \pm 0.70$]{} [ $54.48 \scriptstyle\small \pm 0.93$]{} [ $71.31 \scriptstyle\small \pm 0.78$]{}
**Wasserstein** (Ours) Symmetric Distribution-level [ $50.27 \scriptstyle\small \pm 0.62$]{} [ $67.50 \scriptstyle\small \pm 0.52$]{} [ $52.76 \scriptstyle\small \pm 0.71$]{} [ $73.58 \scriptstyle\small \pm 0.57$]{}
**Wass-CMS** (Ours) Symmetric Distribution-level [ $50.80 \scriptstyle\small \pm 0.64$]{} [ $68.36 \scriptstyle\small \pm 0.50$]{} [ $53.48 \scriptstyle\small \pm 0.68$]{} [ $73.95 \scriptstyle\small \pm 0.56$]{}
**KL** (Ours) Asymmetric Distribution-level [ $\mathbf{52.94 \scriptstyle\small \pm 0.63}$]{} [ $\mathbf{69.38 \scriptstyle\small \pm 0.51}$]{} [ $\mathbf{55.59 \scriptstyle\small \pm 0.70}$]{} [ $\mathbf{74.21 \scriptstyle\small \pm 0.56}$]{}
**KL-CMS** (Ours) Asymmetric Distribution-level [ $\mathbf{53.10 \scriptstyle\small \pm 0.62}$]{} [ $\mathbf{69.73 \scriptstyle\small \pm 0.50}$]{} [ $\mathbf{56.54 \scriptstyle\small \pm 0.70}$]{} [ $\mathbf{74.83 \scriptstyle\small \pm 0.56}$]{}
--------------------------------------- ------------ -------------------- --------------------------------------------------- --------------------------------------------------- --------------------------------------------------- ---------------------------------------------------
Our Contrastive Measure Strategy (CMS)
--------------------------------------
To make the distribution measure more discriminative, we further propose an alternative task-aware *Contrastive Measure Strategy (CMS)* by introducing additional contrastive information. Specifically, for a specific support set $\mathcal{S}=\{S_1,\cdots,S_C\}$, where $C$ is the number of classes in $\mathcal{S}$, we construct a *distribution-level triplet* $\langle Q,S_i,S_i'\rangle$. In this triplet, $Q$ denotes a query distribution, $S_i$ is one class distribution we want to match $Q$ with, and $S_i'$ indicates the entire distribution of the remaining classes $S_j|_{j=1}^C(j\neq i)$. In this way, we can define the contrastive KL divergence measure as follows $$D_\text{KL}^\text{con}(Q\| S_i) = D_\text{KL}(Q\| S_i) - D_\text{KL}(Q\|S_i')\,.$$
The advantage of using the above contrastive measure function over merely using $D_\text{KL}(Q\|S_i)$ in Eq.(\[function\_2\]) is that the context of the entire support classes is taken into consideration. In this way, we can take a whole view of the entire task when measuring the relation between $Q$ and each individual class $S_i$, making the measure function more discriminative. This will be experimentally demonstrated shortly.
Experiments
===========
In this section, extensive experiments on two benchmark datasets are conducted, including an ablation study.
Datasets
--------
All experiments are conducted on two popular few-shot learning benchmarks, *i.e.,* *mini*ImageNet [@vinyals2016matching] and *tiered*ImageNet [@ren2018meta].
***mini*ImageNet.** This dataset is widely used in few-shot learning, which is a small subset of ImageNet [@deng2009imagenet]. It contains $100$ classes with $600$ images in each class. We use the same splits as in [@ravi2016optimization], which takes $64$, $16$ and $20$ classes for training, validation and test, respectively.
***tiered*ImageNet.** Similar to *mini*ImageNet, it is also a mini-version of ImageNet. However, they are different in two aspects. The first is that *tiered*ImageNet has a larger number of classes ($608$ classes) and more images for each class ($1281$ images per class). The other difference is that *tiered*ImageNet has a hierarchical structure of categories. Specifically, there are $34$ categories at the top hierarchy and they are split into $20$ training categories ($351$ classes), $6$ validation categories ($97$ classes) and $8$ test categories ($160$ classes). On this dataset, we strictly follow the splits used in [@ren2018meta].
For both *mini*ImageNet and *tiered*ImageNet, the resolution of all the images is resized to $84\times 84$.
=5.5pt
---------------------- ------------ ------------ -------- -------- --------------------------------------------------- --------------------------------------------------- --------------------------------------------------- ---------------------------------------------------
1-shot 5-shot 1-shot 5-shot
Meta LSTM ICLR’17 *Conv-32F* Meta - [ $43.44 \scriptstyle\small \pm 0.77$]{} [ $60.60 \scriptstyle\small \pm 0.71$]{} - -
MAML ICML’17 *Conv-32F* Meta - [ $48.70 \scriptstyle\small \pm 1.84$]{} [ $63.11 \scriptstyle\small \pm 0.92$]{} [ $51.67 \scriptstyle\small \pm 1.81$]{} [ $70.30 \scriptstyle\small \pm 1.75$]{}
SNAIL ICLR’18 *Conv-32F* Meta - 45.10 55.20 - -
MTL CVPR’19 *Conv-32F* Meta - [ $45.60 \scriptstyle\small \pm 1.80$]{} [ $61.20 \scriptstyle\small \pm 0.90$]{} - -
TAML-Entropy CVPR’19 *Conv-32F* Meta - [ $49.33 \scriptstyle\small \pm 1.80$]{} [ $66.05 \scriptstyle\small \pm 0.85$]{} - -
MetaOptNet-RR CVPR’19 *Conv-64F* Meta - [ $52.87 \scriptstyle\small \pm 0.57$]{} [ $69.51 \scriptstyle\small \pm 0.48$]{} [ $54.63 \scriptstyle\small \pm 0.67$]{} [ $72.11 \scriptstyle\small \pm 0.59$]{}
**Matching Nets** NeurIPS’16 *Conv-64F* Metric 113 kB [ $43.56 \scriptstyle\small \pm 0.84$]{} [ $55.31 \scriptstyle\small \pm 0.73$]{} - -
**ProtoNet**$^\ddag$ NeurIPS’17 *Conv-64F* Metric 113 kB [ $48.45 \scriptstyle\small \pm 0.96$]{} [ $66.53 \scriptstyle\small \pm 0.51$]{} [ $48.58 \scriptstyle\small \pm 0.87$]{} [ $69.57 \scriptstyle\small \pm 0.75$]{}
**RelationNet** CVPR’18 *Conv-64F* Metric 228 kB [ $50.44 \scriptstyle\small \pm 0.82$]{} [ $65.32 \scriptstyle\small \pm 0.70$]{} [ $54.48 \scriptstyle\small \pm 0.93$]{} [ $71.31 \scriptstyle\small \pm 0.78$]{}
**IMP** ICML’19 *Conv-64F* Metric 113 kB [ $49.6 \scriptstyle\small \pm 0.8$]{} [ $68.1 \scriptstyle\small \pm 0.8$]{} - -
**CovaMNet** AAAI’19 *Conv-64F* Metric 113 kB [ $51.19 \scriptstyle\small \pm 0.76$]{} [ $67.65 \scriptstyle\small \pm 0.63$]{} [ $54.98 \scriptstyle\small \pm 0.90$]{} [ $71.51 \scriptstyle\small \pm 0.75$]{}
**DN4** CVPR’19 *Conv-64F* Metric 113 kB [ $51.24 \scriptstyle\small \pm 0.74$]{} [ $\mathbf{71.02 \scriptstyle\small \pm 0.64}$]{} [ $53.37 \scriptstyle\small \pm 0.86$]{} [ $74.45 \scriptstyle\small \pm 0.70$]{}
**KL** Ours *Conv-64F* Metric 113 kB [ $52.94 \scriptstyle\small \pm 0.63$]{} [ $69.38 \scriptstyle\small \pm 0.51$]{} [ $55.59 \scriptstyle\small \pm 0.70$]{} [ $74.21 \scriptstyle\small \pm 0.56$]{}
**KL-CMS** Ours *Conv-64F* Metric 113 kB [ $\mathbf{53.10 \scriptstyle\small \pm 0.62}$]{} [ $69.73 \scriptstyle\small \pm 0.50$]{} [ $\mathbf{56.54 \scriptstyle\small \pm 0.70}$]{} [ $\mathbf{74.83 \scriptstyle\small \pm 0.56}$]{}
**ADM** Ours *Conv-64F* Metric 113 kB [ $\mathbf{54.26 \scriptstyle\small \pm 0.63}$]{} [ $\mathbf{72.54 \scriptstyle\small \pm 0.50}$]{} [ $\mathbf{56.01 \scriptstyle\small \pm 0.69}$]{} [ $\mathbf{75.18 \scriptstyle\small \pm 0.56}$]{}
---------------------- ------------ ------------ -------- -------- --------------------------------------------------- --------------------------------------------------- --------------------------------------------------- ---------------------------------------------------
Network Architecture
--------------------
It can be easily verified that adopting a deeper network for embedding or using pre-trained weights will provide higher accuracy. Following the previous works [@snell2017prototypical; @sung2018learning; @li2019CovaMNet; @li2019DN4], we adopt the same embedding network with four convolutional blocks, *i.e.,* *Conv-64F*, to make a fair comparison with other methods. Specifically, the first two blocks each contains a convolutional layer (with $64$ filters of size $3\times 3$), a batch-normalization layer, a Leaky ReLU layer and a max pooling layer. The last two blocks adopt the same architecture but without pooling layers. The reason for only using two pooling layers is that we need richer local descriptors to represent the distributions of both queries and classes. For example, in a $5$-way $1$-shot setting, when the size of the input image is $84 \times 84$, we can only obtain $25$ local descriptors for each image (class) by adopting four pooling layers. It is clearly insufficient to represent a distribution with a feature dimensionality of $64$. In contrast, using the adopted network architecture with two pooling layers, we obtain $441$ local descriptors for each image (class).
Implementation Details
----------------------
Both $5$-way $1$-shot and $5$-way $5$-shot classification tasks are conducted to evaluate our methods. We use $15$ query images per class in each single task ($75$ query images in total) in both training and test stages. In particular, we employ the episodic training mechanism [@vinyals2016matching] to train our models from scratch without pre-training. In the training stage, we use the Adam algorithm [@kingma2014adam] to train all the models for $40$ epoches. In each epoch, we randomly construct $10000$ episodes (tasks). Also, the initial learning rate is set as $1 \times 10^{-3}$ and multiplied by $0.5$ every $10$ epoches. During test, $1000$ tasks are randomly constructed to calculate the final results, and this process is repeated five times. The top-1 mean accuracy is taken as the evaluation criterion. At the same time, the $95\%$ confidence intervals are also reported.
Comparison Methods
------------------
Since our methods belong to the metric-based few-shot learning methods, we will mainly compare our methods with metric-based methods, such as Matching Net [@vinyals2016matching], ProtoNet [@snell2017prototypical], RelationNet [@sung2018learning], IMP [@allen2019infinite], CovaMNet [@li2019CovaMNet] and DN4 [@li2019DN4]. Moreover, representative meta-learning based few-shot learning methods are also listed for reference, including Meta LSTM [@ravi2016optimization], MAML [@finn2017model], SNAIL [@mishra2017simple], MTL [@sun2019meta], TAML-Entropy [@jamal2019task], and MetaOptNet-RR [@lee2019meta]. Note that meta-learning based methods are essentially different from metric-based methods at two aspects. The first aspect is that an additional parameterized meta-learner is usually learned in meta-learning based methods while the metric-based methods do not have. The second aspect is that during test, meta-learning based methods will fine-tune the model (or classifier) to obtain the final classification results while metric-based methods do not need fine-tuning.
Most results of these compared methods are quoted from their original work or the relevant reference. Some methods are not in the same setting with our method, such as ProtoNet, so we use the results of their modified versions to ensure fair comparison. For some recent meta-based methods, such as SNAIL, MTL and TAML-Entropy, we only report their results with a similar embedding network, *e.g.,* *Conv-32F*, which has the same architecture with *Conv-64F* but has $32$ filters in each convolutional block.
Ablation Study {#ablation_study}
--------------
In this section, we first verify the validity of our argument on asymmetric measure for metric-based few-shot learning. Next, based on two distribution-level measure functions, we evaluate the effectiveness of the proposed CMS strategy. Specifically, both the $2$-Wasserstein distance (Wasserstein for short) and KL divergence (KL for short) are performed on the *mini*ImageNet and *tiered*ImageNet datasets. Also, the contrastive versions using our proposed CMS are named as Wass-CMS and KL-CMS, respectively. Moreover, two instance-level symmetric metric based methods, *i.e.,* ProtoNet and RelationNet, are picked as baselines.
As seen in Table \[Ablation\_Study\], compared to symmetric metric based methods, such as ProtoNet, RelationNet and Wasserstein, the proposed asymmetric measure can obtain superior results. For example, on the *mini*ImageNet, KL gains $4.49\%$, $2.50\%$ and $2.67\%$ over these methods on the $1$-shot task, respectively. This verifies that an asymmetric measure is more suitable for metric-based few-shot learning.
We can also see that the proposed CMS strategy can indeed improve the performance of distribution-based measure functions, especially on the $1$-shot setting. For instance, on the *tiered*ImageNet, Wass-CMS achieves $0.72\%$ improvement over Wass, and KL-CMS obtains $0.95\%$ improvement over KL on the $1$-shot task. This shows that the task-aware CMS strategy does enhance the distribution-based measure functions, thanks to taking a whole view of the entire task.
Comparison with the State of the Art
------------------------------------
Experimental results on the comparison with the state-of-the-art methods are reported in Table \[Table2\_SOTA\], where two types of few-shot learning methods (*i.e.,* both meta-learning based and metric-based) are compared. Since our methods are metric-based methods, we will mainly compare our methods with other metric-based ones. Moreover, the total number of parameters of each method is also shown in the fifth column.
From Table \[Table2\_SOTA\], it can be seen that the proposed ADM (without CMS) outperforms all the other metric-based and meta-learning based methods on both $1$-shot and $5$-shot settings. For example, on the *mini*ImageNet, our ADM obtains $10.7\%$, $5.81\%$, $3.82\%$, $4.66\%$, $3.07\%$ and $3.02\%$ improvements over Matching Nets, ProtoNet, RelationNet, IMP, CovaMNet and DN4 on the $1$-shot task, respectively. Moreover, on the *tiered*ImageNet, our ADM achieves $5.61\%$, $3.87\%$, $3.67\%$, $0.73\%$ improvements over ProtoNet, RelationNet, CovaMNet and DN4 on the $5$-shot task, respectively. This verifies the effectiveness and superiority of our proposed ADM, owing to the integration of both local and global asymmetric relations.
The proposed KL and KL-CMS are also very competitive with the state-of-the-art methods. Specifically, on the $1$-shot setting, KL and KL-CMS can obtain significantly improvements over the existing metric-based methods. For instance, on the *mini*ImageNet, KL/KL-CMS gains $9.38\%/9.54\%$, $4.49\%/4.65\%$, $2.5\%/2.66\%$, $3.34\%/3.5\%$, $1.75\%/1.91\%$ and $1.7\%/1.86\%$ improvements over Matching Nets, ProtoNet, RelationNet, IMP, CovaMNet and DN4, respectively. It verifies that such kind of distribution-based asymmetric measure is more suitable for metric-based few-shot learning.
Conclusion
==========
In this study, we provide a new perspective for metric-based few-shot learning by considering the asymmetric nature of the similarity measure and design a novel *Asymmetric Distribution Measure (ADM)* network to address this task. Furthermore, to make full use of the context of the entire task, we propose a *Contrastive Measure Strategy (CMS)* to learn a more discriminative distribution metric space. Extensive experiments on two benchmark datasets verify the effectiveness and advantages of both local asymmetric relations and global asymmetric relations in metric-based few-shot learning.
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---
abstract: 'Low temperature susceptibility and specific heat for single crystals of Pr$_{1-x}$La$_x$Os$_4$Sb$_{12}$ (0 $\leq x$ $\leq 1$) are reported. La-doping leaves CEF energies of Pr essentially unchanged. The average $T_c$ is only weakly affected by the La-substitution and varies approximately linearly between the end-compounds. The second superconducting transition disappears between $x$=0.05 and 0.1. The discontinuity in $C/T$ at $T_c$, on the other hand, is drastically reduced from about 1000 mJ/K$^2$mol for $x$=0 to 200 mJ/K$^2$mol for $x$=0.2. $\Delta C/T_c$ decreases further for alloys corresponding to $x\geq 0.6$ to the value of a conventional superconductor LaOs$_4$Sb$_{12}$. This behavior implies non-single impurity origin of the heavy fermion state in PrOs$_4$Sb$_{12}$. We argue that critical quadrupolar fluctuations are responsible for this heavy fermion state.'
author:
- 'C.R. Rotundu, P. Kumar, and B. Andraka'
title: ' A Collective Heavy Fermion State and Superconductivity in Pr$_{1-x}$La$_x$Os$_4$Sb$_{12}$: Specific Heat and Susceptibility Study.'
---
Lately, there has been great interest in the first discovered Pr-based heavy fermion superconductor[@Bauer], PrOs$_4$Sb$_{12}$. The significance of this system stems from several reasons. Firstly, a large discontinuity in $C/T$ at $T_c$ and correspondingly large temperature slope of the upper critical field ($dH_{c2}/dT$) unambiguously prove that Pr can support heavy fermion state(s). Secondly, the paramagnetic heavy fermion state seems to be unconventional; i.e., not of a magnetic Kondo effect origin. It has been postulated that the J=4 multiplet of the Pr$^{3+}$ ion is split by the crystalline electric field (CEF) such that the ground state is a nonmagnetic $\Gamma_3$ doublet. Since the $\Gamma_3$ doublet carries no magnetic dipole moment but a quadrupolar electric moment, a quadrupolar Kondo effect, a mechanism[@Cox] never established experimentally to be relevant for heavy fermions, has been proposed as the source of heavy electrons and superconductivity in PrOs$_4$Sb$_{12}$. However, our recent magnetic field investigation[@Rotundu] provided support for a complementary CEF configuration, with a singlet ($\Gamma_1$) ground state and the first excited state being a triplet ($\Gamma_5$), at about 8 K. This model is also consistent with inelastic[@Maple] and elastic neutron diffraction[@Kohgi], specific heat[@Bauer; @Aoki] and magnetic susceptibility data. Thus, the formation of heavy electrons requires participation of the excited crystal field levels. The qudrupolar interactions are also expected to be strong in this $\Gamma_1$ - $\Gamma_5$ model. Upon application of the magnetic field, the singlet crosses the lowest level of the split triplet (at about 8-9 T), forming a pseudodoublet possessing a quadrupolar electric moment. Antiferroquadrupolar order is observed in PrOs$_4$Sb$_{12}$ in magnetic fields[@Kohgi; @Aoki], between about 4.5 and 14 T. The strength of quadrupolar interactions peaks at the crossing field. Thus, the $\Gamma_1$ - $\Gamma_5$ CEF model does not apriori preclude the possibility of the superconductivity mediated by quadrupolar interactions.
One of the strongest arguments for the unconventional (anisotropic) superconductivity is provided by the observation of two superconducting transitions in some PrOs$_4$Sb$_{12}$ samples[@Vollmer]. And yet, the results of the $\mu SR$ experiment[@MacLaughlin] performed on samples with a single $T_c$ suggest isotropic superconducting order parameter. To shed further light on the character of the superconductivity, we have performed a systematic study of the specific heat on the La-doped samples. Since La has no f-electrons, substituting La for Pr should have a strong effect on the superconductivity in the quadrupolar scenario.
All single-crystalline samples used in this investigation were by the Sb-flux method[@Bauer2]. La and Pr were premelted several times in an arc-melter to improve the homogeneity of samples. The results of the X-ray diffraction analysis were consistent with single phase materials. We have detected a monotonic, but very small, increase of the lattice constant with the La-content. These very small changes (0.03 % between the end compounds; on the border of sensitivity of our technique) are in an agreement with previously reported[@Braun] an almost non-existent lanthanide contraction in ternary skutterudites containing Sb, of a general form LnT$_4$Sb$_{12}$, where T and Ln are transition element and light lanthanide, respectively. For most of the concentrations studied, we have performed measurements, both specific heat and susceptibility, on two different, randomely selected crystals. Within the uncertainty of these measurements, we have observed very good reproducibility for crystals with the same nominal concentration.
![ Magnetic susceptibility of Pr$_{1-x}$La$_x$Os$_4$Sb$_{12}$ between 1.8 and 10 K, measured in the field of 0.5 T.[]{data-label="fig1"}](fig1.ps){width="0.8\linewidth"}
Figure 1 shows the susceptibility for Pr$_{1-x}$La$_x$Os$_4$Sb$_{12}$ samples at temperatures 1.85 to 10 K obtained in the field of 0.5 T and normalized to a mole of Pr. All curves show a low temperature maximum that we believe is due to excitations between the singlet and triplet CEF states. This maximum is at approximately 3.4 K and shifts only slightly to higher temperatures with x, such that it is around 4.3 K for x=0.6. Thus, these results indicate that the crystalline electric field scheme of Pr remains unchanged and that average separation between the lowest CEF levels increases only slightly with the La-doping. The maximum value of the susceptibility, however, does not stay constant across the system. Initially, it decreases continuously from about 100 memu/Pr mol for the pure material to about 50 memu/Pr mol for x=0.4 followed by smaller changes for higher concentrations of La. Note that because of the small size of the crystals (mass 1 to 4 mg), the sample signal was comparable to the background (sample holder) below 10 K and was much smaller than the background at room temperature. Therefore, we do not attempt to analyze the data above 10 K. Also, the discrepancy in the 1.8 K susceptibility for the three highest La-content compositions is within the absolute error bar. We observe a broadening of the maximum, particularly for x=0.6 and 0.8 compositions. Such a broadening in mixed alloys is expected because of the La/Pr disorder leading to some distribution of Pr-ligand ion distances and, consequently, to smearing out of sharp CEF levels in the undoped material. Susceptibility curves corresponding to different x’s have a clear tendency to converge at some higher temperatures. However, the very large initial drop in the lowest temperature susceptibility (1.8 K) is difficult to reconcile with the disorder only. Quite possibly, some characteristic electronic energy (inversely proportional to temperature, analogous to a Kondo temperature) increases sharply upon substituting La for Pr.
![ $C/T$ versus $T$ for PrOs$_4$Sb$_{12}$ samples from two different batches. The sample labeled “dirty” was made of recycled Os. The arrow indicates the second superconducting anomaly.[]{data-label="fig2"}](fig2.ps){width="0.8\linewidth"}
The specific heat shown in Figs. 2 and 3 was measured on the same crystals as the susceptibility presented in Fig. 1. From the total specific that corresponding to LaOs$_4$Sb$_{12}$ was subtracted and the result divided by $(1-x)$. LaOs$_4$Sb$_{12}$ has been approximated by the following expression proposed by Bauer et al.[@Bauer], $C$ = 36 $T$ + 1.18 $T^3$; where $C$ is in mJ/K mol. Note that the cubic coefficient is significantly smaller than the phonon term derived by Vollmer et al.[@Vollmer] for PrOs$_4$Sb$_{12}$. Also, the linear term is smaller than the value reported by Sugawara, et al[@Sugawara]. These discrepancies are of some significance when analyzing the data for the largest x-values, particularly for $x=0.8$. Fig. 2 shows such normalized f-electron specific heat divided by temperature ($C/T$) for two crystals of PrOs$_4$Sb$_{12}$ from two different batches obtained in the same manner but with somewhat different purity of starting Os. The crystal labeled “dirty” was grown from recycled Os, which had not been analyzed for impurities. Within the experimental resolution, $C/T$ results are identical. Also, these results are consistent with two superconducting transitions reported by Vollmer et al. The positions of the smaller, higher temperature, peaks are marked with an arrow in Fig. 2. Some smearing out of this higher temperature peak could be due to the measurement method itself, which integrates the specific heat at any temperature $T$ over 0.03 $T$ interval. Two superconducting transitions can still be resolved for $x=0.05$ but not for higher concentrations of La (Fig. 3). Thus, for consistency of the analysis, we treat this double structures in $x=0$ and 0.05 as a single transition. The average $T_c$ for the pure material, approximated using an equal area construction method, is 1.83 K, in a good agreement with the result of Bauer et al.[@Bauer] that used a similar procedure. On the other hand, the equal area construction is not reliable for the extraction of $\Delta C/T$ at $T_c$ in the case of a double transition Therefore, we estimate this discontinuity as the measured difference between the maximum value of $C/T$ and the value of $C/T$ just above the transition. This $\Delta C/T$ for $x=0$ is about 1000 mJ/K$^2$mol and among the highest reported, confirming the heavy fermion state. ($\Delta C/T$ associated with the lower temperature, more pronounced transition, is at least 500 mJ/K$^2$mol.) The specific heat near its local maximum around 3 K is about 6.9 J/K mol. This value is in an excellent agreement with that reported by Aoki et al.[@Aoki] Note further that this value is significantly larger than that expected for the Schottky maximum corresponding to excitations between the doublet and triplet (about 5.1 J/K mol) and it is smaller than that for the excitations between the singlet and triplet states (8.5 J/Kmol)[@Gopal]. However, we do expect some hybridization between the f-electrons of Pr and ligand states leading to the reduction of ionic properties of Pr. Extrapolated values of $T_c$, $\Delta C/T$, and $C/T$ at its maximum (2.2 - 2.5 K) are also shown in Table 1.
Figure 3 displays the f-electron contribution to $C/T$ versus $T$ for the remaining Pr$_{1-x}$La$_x$Os$_4$Sb$_{12}$ alloys. There are several important contributions to the uncertainty of the data, such as the aforementioned phonons, normal electrons, and the addenda that become more critical for larger values of x. Therefore, we estimate this uncertainty in the f-electron contribution to $C/T$ to increase from about 10 % for $x=0$ to 25 % for $x=0.8$. Note further that our procedure of calculating the f-electrons specific heat by subtracting the normal state specific heat of LaOs$_4$Sb$_{12}$ is incorrect for the superconducting state and can lead to unphysical negative values at low temperatures. The Schottky maximum, which is seen near 2.2 K in $C/T$, is indeed smeared out for $x>0$ and moved to slightly higher temperatures, as expected from the susceptibility data. The $C/T$ value at this shallow maximum is reduced by a factor of 2 between the pure compound and alloys corresponding to $x=0.6$ and 0.4 (see also Table 1). Thus, the size of the anomaly in $C/T$(at 2.2 - 2.5 K) scales roughly with the corresponding low temperature maximum in the low temperature susceptibility.
![ $C/T$ versus $T$ for Pr$_{1-x}$La$_x$Os$_4$Sb$_{12}$, $x>0$. []{data-label="fig3"}](fig3.ps){width="0.8\linewidth"}
As it can be inferred from Fig. 3, the superconducting transition is only moderately suppressed by the La substitution. This is more clearly shown in Fig 4 and Table 1. We include the published result[@Bauer2; @Sugawara] for LaOs$_4$Sb$_{12}$, material that is also superconducting below about 0.74 K. $T_c$ varies approximately linearly between the end-compounds. Thus, these results seem to imply that La impurities are not strong pair-brakers and there is a smooth evolution between the superconducting states of PrOs$_4$Sb$_{12}$ and LaOs$_4$Sb$_{12}$. Interestingly, the width of the superconducting transition in $C/T$, measured in a consistent manner for all samples, is the same for $x=0$ and 0.05 and is reduced by a factor of 3 for all the remaining alloys. This is quite an unexpected result; the transition is sharp for all concentrations and becomes even narrowor for mixed alloys. The sharpness of the transition for all concentrations could be due to very small variation of the lattice constant. The reduction of the width is because of the disapearance of one of the superconducting anomalies. Although additional studies are needed on samples with very small amount of La, a closer inspection of superconductive anomalies for $x=0$, 0.02 (not shown), and 0.05 suggests that the La-doping suppresses mainly the lower temperature anomaly in $C/T$.
This approximately linear variation of the average $T_c$ on $x$ in Pr$_{1-x}$La$_x$Os$_4$Sb$_{12}$ is unusual for heavy fermion alloys. For instance, UBe$_{13}$ that shows a number of striking similarities to PrOs$_4$Sb$_{12}$ has its superconductivity suppressed by just 3 % of La[@Ahlheim]. This very different sensitivity of $T_c$ on La-impurities between PrOs$_4$Sb$_{12}$ and canonical heavy fermion superconductors cannot be accounted for by vastly different coherence lengths. In fact, the coherence lengths of the UBe$_{13}$ and PrOs$_4$Sb$_{12}$ are almost identical, 140 (ref.[@Maple2]) and 120 $\r{A}$ (ref.[@Bauer]), respectively. A somewhat stronger supression of $T_c$ in PrOs$_4$Sb$_{12}$ was found for Ru impurities replacing Os[@Fredrick]. In addition, Pr(Os$_{1-x}$Ru$_x$)$_4$Sb$_{12}$ shows a shallow minimum in $T_c$ near $x=0.6$. However, even in this case the rate of the reduction of $T_c$ is small in comparisson with majority of Ce- and U-based heavy fermions and considering very different paramagnetic states of PrOs$_4$Sb$_{12}$ and PrRu$_4$Sb$_{12}$. PrRu$_4$Sb$_{12}$ is quite an ordinary metal[@Takeda] with a different CEF scheme of Pr and relevant CEF energies much larger than those for PrOs$_4$Sb$_{12}$.
Despite the weak dependence of $T_c$ in PrOs$_4$Sb$_{12}$ on La impurities, the discontinuity in $C/T$ (and $C$) at $T_c$ is strongly reduced with $x$ (Table 1). An apparent increase of $\Delta C/T_c$ between $x=0.6$ and 0.8 in Fig. 3 is because of the normalization by the Pr concentration used in this figure. Such a normalization becomes incorrect for sufficiently large $x$ -values, since LaOs$_4$Sb$_{12}$ itself is superconducting and thus displays this discontinuity. Fig. 4 shows both $T_c$ and unnormalized $\Delta C/T_c$ versus $x$. $\Delta C/T_c$ decreases from approximately 1000 mJ/K$^2$mol ($x=0$) to about 460, and 210 mJ/K$^2$mol when only 10, and 20 % of La is substituted for Pr, respectively. Lack of normalization to a mole of Pr can not account for this dramatic reduction. Alloys corresponding to $x\geq 0.6$ have $\Delta C/T_c$ approximately equal to that of a conventional superconductor LaOs$_4$Sb$_{12}$. In BCS-type superconductors, $\Delta C/T_c$ is simply related to the electronic specific heat coefficient $\gamma$. There is no such a direct relationship in heavy fermion metals exhibiting multiple superconducting transitions. In some thoriated UBe$_{13}$ samples[@Ott], showing two superconducting transitions, $\Delta C/T_c$ is enhanced by a factor of 2 with respect to UBe$_{13}$. This enhancement seems to be due to some collective excitations present in the normal state and condensing at $T_c$. Thus, similar fluctuations can be responsible for the enhanced $\Delta C/T_c$ in the pure PrOs$_4$Sb$_{12}$ over the alloys with a single transition. On the other hand, this strong variation of $\Delta C/T_c$ on $x$ among alloys with a single transition ($x\geq 0.1$) can be associted with the suppression of $\gamma$ upon dilution with La. Therefore, this behavior clearly implies non-single impurity origin of the heavy fermion state in PrOs$_4$Sb$_{12}$. The electronic specific heat coefficient is reduced to that of LaOs$_4$Sb$_{12}$ by diluting Pr with La.
![ Superconducting transition temperature $T_c$ (left scale) and $\Delta C/T_c$ (right scale) versus concentration $x$ for Pr$_{1-x}$La$_x$Os$_4$Sb$_{12}$, where 0 $\leq x$ $\leq 1$. $\Delta C$ is a discontinuity in the specific heat at $T_c$.[]{data-label="fig4"}](fig4.ps){width="0.8\linewidth"}
Proving or disproving single-ion behavior using just one kind of alloying is, in general, a difficult tusk because alloying can alter parameters of a single-ion Hamiltonian as well. However, in the Pr$_{1-x}$La$_x$Os$_4$Sb$_{12}$ case, the relevant single-ion parameters seem to be unaffected or only slightly affected by alloying, probably due to the peculiarity of the crystal structure. The lattice constant, CEF scheme of Pr, and the lowest CEF energy show very weak sensitivity to the La-doping. On the other hand, the alloying is expected to reduce qudrupolar fluctuations shown to be strong in PrOs$_4$Sb$_{12}$. As it has been discussed, magnetic fields 4.5 T and larger induce AFQ state. Such a state is known to be very susceptible to alloying. For instance, in PrPb$_3$, undergoing AFQ ordering in zero field at 0.4 K, just 2 % of La substituted for Pr supresses the long range order completely[@Kawae]. Our, preliminary investigation of Pr$_{0.8}$La$_{0.2}$Os$_4$Sb$_{12}$ in magnetic fields, has not found any evidence of a field-induced AFQ order above 0.4 K. Thus, the heavy fermion state in PrOs$_4$Sb$_{12}$ seems to correlate with the field-induced AFQ order, or proximity to AFQ order.
-------- -------------------------- ------------------------------ ----------------------------
La $x$ $T_c$ (K) $\Delta C/T_c$ (mJ/K$^2$mol) $C/T$ max (mJ/K$^2$Pr mol)
0 1.83 1000 2900
0.05 1.77 640 2700
0.1 1.73 460 2400
0.2 1.66 210 1600
0.4 1.45 120 1700
0.6 1.20 70 1500
0.8 0.93 80 1900
1 0.74[@Bauer2; @Sugawara] 84[@Sugawara] -
-------- -------------------------- ------------------------------ ----------------------------
: Specific heat parameters for Pr$_{1-x}$La$_x$Os$_4$Sb$_{12}$ alloys, $0\leq\,x\leq\,1$.[]{data-label="table1"}
Interestingly, there is no correlation between the heavy fermion character measured by $\Delta C/T_c$ and the average $T_c$. Our results argue against the same mechanism responsible for the heavy fermion state and enhanced value of $T_c$ in PrOs$_4$Sb$_{12}$. Changes in the phonon spectrum are most probably behind the variation of $T_c$ in Pr$_{1-x}$La$_x$Os$_4$Sb$_{12}$. In fact, Vollmer et al. derive a much lower Debye temperature for PrOs$_4$Sb$_{12}$ than that for LaOs$_4$Sb$_{12}$, from the specific heat data at tempertures to 10 K. Although the average $T_c$ seems to be uncorrelated with the Sommerfeld coefficient, the superconducting state in PrOs$_4$Sb$_{12}$ is clearly affected by the heavy fermion state. In particular, a second superconducting transition and spontaneous magnetic field show up[@Aoki2] below $T_c$ in Pr$_{1-x}$La$_x$Os$_4$Sb$_{12}$ alloys with a heavy fermion normal state. This superconductivity provides also a proof of the heavy fermion state in PrOs$_4$Sb$_{12}$.
This work has been supported by the U.S. Department of Energy, Grant No. DE-FG02-99ER45748, National Science Foundation, DMR-0104240. We thank G.R. Stewart and Y. Takano for stimulating discussions.
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---
abstract: 'Cosmological phase transitions (CPTs), such as the Grand Unified Theory (GUT) and the electroweak (EW) ones, play a significant role in both particle physics and cosmology. In this letter, we propose to probe the first-order CPTs, by detecting gravitational waves (GWs) which are generated during the phase transitions through the cosmic microwave background (CMB). If happened around the inflation era, the first-order CPTs may yield low-frequency GWs due to bubble dynamics, leaving imprints on the CMB. In contrast to the nearly scale-invariant primordial GWs caused by vacuum fluctuation, these bubble-generated GWs are scale dependent and have non-trivial B-mode spectra. If decoupled from inflaton, the EWPT during inflation may serve as a probe for the one after reheating where the baryon asymmetry could be generated via EW baryogenesis (EWBG). The CMB thus provides a potential way to test the feasibility of the EWBG, complementary to the collider measurements of Higgs potential and the direct detection of GWs generated during EWPT.'
author:
- Hongliang Jiang
- Tao Liu
- Sichun Sun
- Yi Wang
title: 'Echoes of Inflationary First-Order Phase Transitions in the CMB'
---
Introduction
============
Phase transitions in particle physics have deep implications in cosmology. One famous example is the invention of inflation theory, which was originally motivated by addressing the missing magnetic monopole problem produced during the GUT phase transition [@Guth:1980zm]. Another example is related to cosmic baryon asymmetry. If the electroweak phase transition (EWPT) is of first order, the baryon asymmetry could be generated during the phase transition, with CP-violating Higgs couplings [@Kuzmin:1985mm].
With the discovery of the Higgs particle, the questions about the EWPT have intensified. Though highly challenging, we expect to be able to probe the first-order EWPT by measuring Higgs self-interaction at High-Luminosity LHC or at future colliders (see, e.g., [@Noble:2007kk; @Dolan:2012rv; @Papaefstathiou:2012qe; @Baglio:2012np; @Liu:2014rva; @Barr:2014sga; @He:2015spf; @Huang:2015tdv]). More generally, the cosmological phase transitions (CPTs) of first order are implemented via bubble nucleation. The expanding bubbles may collide with each other or stir up turbulence and sound wave in the thermal plasma, yielding gravitational waves (GWs) in spacetime [@Witten:1984rs; @Kamionkowski:1993fg; @Hindmarsh:2013xza; @Hindmarsh:2015qta]. Particularly, if the phase transitions are of EW scale or PeV scale and happened after reheating [@Kosowsky:1991ua; @Ignatius:1993qn; @Huber:2008hg; @Cline:2006ts], the produced GWs are characterized by a frequency $\gtrsim 10^{-4}$Hz [@Grojean:2006bp] that direct detection experiments, like Advanced LIGO [@Harry:2010zz], Advanced Virgo [@Aasi:2013wya] and LISA [@AmaroSeoane:2012km], are currently looking for or will look for. In this letter, we propose a new approach of probing the first-order CPTs, by detecting the bubble-generated GWs through the cosmic microwave background (CMB).
The CMB temperature and polarization fluctuations provide us rich information about the primordial universe [@Ade:2015xua; @Array:2015xqh]. Cosmic inflation [@Guth:1980zm] is the leading paradigm to seed those CMB fluctuations. The potential role of the CMB in probing the first-order CPTs was ignored. Because if the CPTs happened after reheating, the produced GWs have a characteristic frequency far beyond the scope of CMB.
One fact often ignored before about inflation is that an inflationary universe may undergo some thermal phase transition due to temperature decreasing, if the initial temperature of the universe is above the inflationary Hubble scale. In some physical contexts, such as the GUT and the recently proposed cosmological relaxation models aimed to solve the hierarchy problem [@Graham:2015cka], such phase transitions could be generic. These phase transitions, if of first order, may lead to entirely different cosmological consequences. Particularly, they are different from old inflation scenarios, e.g. [@Guth:1980zm; @Turner:1992tz], where inflation was driven by a first-order phase transition and thus the phase transition can not finish. In our scenarios, inflation occurred inside and outside the bubbles simultaneously, enabling the sub-horizon bubbles to collide with each other and hence generate GWs during inflation. Since the phase transitions happened during inflation, the produced GWs can be characterized by scales comparable to the size of the current universe with a scale dependent power spectrum. Thus they can be observed in the CMB potentially.
More explicitly, the temperature contribution from radiation drops quickly at the beginning of inflation, leaving only the Gibbons-Hawking temperature $T_{GH}=H/(2\pi)$ [@Gibbons:1977mu]. $T_{GH}$ can vary between $10^{14}$GeV to $10^{-24}$GeV (see Fig. \[HubbleScale\]). The vast span of the unknown energy scale of inflation may encompass rich physics. Among different models of inflation with low energy scales [@Dimopoulos:2004yb; @Allahverdi:2006iq]: $H \sim T_{GH} \leq 10^2$GeV, the EWPT will happen during the drastic decreasing of temperature at the beginning of inflation (denoted as “EWPTa” in Fig. \[temperatureEvolution\]). Note, for $H \sim 10^2 $GeV, the energy density is still as high as $\rho \sim (10^{10} \text{GeV})^4$, ensuring that inflation can happen. Moreover, the GUT phase transition, if exists, will happen in almost all the inflation scenarios.
![ In various inflation models, Hubble constant during inflation can take different values from $10^{-24}$ GeV up to $10^{14}$GeV. The upper bound is set by the latest experiment [@Array:2015xqh]. When the Hubble constant is below $10^{-14}$ GeV, the universe can not reheat above $100$GeV later, where the EW baryogenesis can be hardly achieved. Below $10^{-24}$ GeV, the reheated universe is too cool to have big bang nucleosynthesis (for a recent bound after *Planck* 2015, see [@deSalas:2015glj]). []{data-label="HubbleScale"}](EWHubble.png){width="7cm"}
![ An exemplifying thermal temperature versus time relation in the early universe. After inflation reaches the attractor phase, the temperature comes mainly from the curvature contribution of the de-Sitter space, which is comparable with the Hubble constant. In addition to the EWPT after reheating, two more EWPTs are proposed, namely EWPTa during inflation, and EWPTb when the universe gets heated up during reheating. Here $10^{-14}{\rm GeV}< H < \Lambda_{\rm EW}$ is assumed. EWPTa is the one relevant to discussions below.[]{data-label="temperatureEvolution"}](temperature.png){width="7cm"}
Gravitational wave spectra {#Sec_GW_Spectrum}
==========================
We derive the power spectrum $P_\gamma$ coming from scale dependent GWs produced in de-Sitter space here. During inflation, the action of GWs is $$S=\frac{M_p^2}{8} \int d\tau d^3 \bm x\; a^2\Big[(h_{ij}\rq{})^2-(\nabla h_{ij})^2 \Big]~,$$ where the prime denotes the derivative with respect to conformal time $\tau$ and $a(\tau)\approx -1/H\tau$ is the scale factor. We introduce the polarization tensors $\epsilon^+_{ij}, \epsilon^\times_{ij}$ and decompose the gravitational fields: $$\begin{aligned}
h_{ij}(\bm{k}) = \frac{\sqrt2}{M_p} \left[ \gamma_+(\bm{k}) \epsilon^+_{ij}(\bm{k})
+ \gamma_\times (\bm{k}) \epsilon^\times_{ij}(\bm{k}) \right]~.\end{aligned}$$ Then $\gamma_s$ can be quantized as $
\gamma_s(\bm k,\tau)= v_k(\tau)a_{\bm k s}+v^*_k(\tau)a^\dagger_{-\bm k s }~,
$ where $a_{\bm k s}$ and $a_{-\bm ks}^\dagger$ are creation and annihilation operators. Solving the equation of motion, we can get the mode function $v_k$: $$v_{k}(\tau)=\frac{H}{\sqrt{2k^3}}\Big(c_1( k) (1+ik\tau)e^{-ik\tau}+c_2( k) (1-ik\tau)e^{ik\tau} \Big) ~,$$ where the coefficients $c_1(k),c_2(k)$ are subject to the consistency condition of quantization $ |c_1|^2-|c_2|^2=1 $.
The energy density of GWs is $$\rho_{GW}=\int \frac{ dk}{k} \frac{ k^3}{2\pi ^2} \Big( |\dot{v}_{k} |^2 +\frac{k^2}{a^2}|v_{k} |^2 \Big)~.$$ The gravitational energy spectrum is thus $$\begin{aligned}
\Omega_{GW}(k,\tau)&=&k\frac{d\rho_{GW} }{dk}/ \rho_{\text{tot}} \\
&=&\frac{1 }{3H^2 M_p^2} \frac{ k^3}{2\pi ^2}
\frac{|v_{k}\rq{}(\tau)|^2 +k^2|v_{k}(\tau)|^2 }{a(\tau)^2}~,\end{aligned}$$ where we assume that the universe is spatially flat, meaning that $\rho_{\text{tot}}=\rho_{\text{critical}}=3H^2M_p^2$. Particularly note that during inflation $\rho_{\text{tot}}=\rho_{\text{inflaton}}+\rho_{\text{rad}}+\rho_{\text{higgs}}$.
We can also calculate the power spectrum of GWs: $$P_\gamma(k,\tau)=\frac{4k^3}{\pi^2 M_p^2} |v_k(\tau)|^2 ~.$$ This power spectrum of GWs contributes to both CMB temperature fluctuations and polarizations.
Both power spectrum and energy spectrum depend on the unknown functions $c_1(k),c_2(k)$. We are particularly interested in the power spectrum at the time $\tau_{\text{obs}}\rightarrow 0$ when the modes exit the horizon and do not evolve anymore. While in general the GWs generated at time $\tau_*$ may be either sub-horizon or super-horizon. The relation between them yields: $$P_\gamma( \tau_{\text{obs}} )=24H^2 \Big( \frac{a(\tau_*)}{k} \Big)^2
\frac{k^2|v_k(\tau_{\text{obs}})|^2}{k^2|v_k(\tau_*)|^2+|v_k\rq(\tau_*)|^2} \Omega_{GW}( \tau_*)~.$$
We consider the classical limit $c_1\approx c_2 \gg 1$. Inserting the mode functions, we can get the relations for sub-horizon and super-horizon modes respectively, $$\label{P_and_Omega}
P_{\gamma}(k,\tau_{\text{obs}}\rightarrow0)=
\begin{cases}
24 \Big( \frac{a(\tau_*)H}{k} \Big)^4 \Omega_{GW}(k,\tau_*) , & |k\tau_*| \gg 1
\\
24 \Big( \frac{a(\tau_*)H}{k} \Big)^2 \Omega_{GW}(k,\tau_*) , & |k\tau_*| \ll 1
\end{cases} ~.$$ We will mainly discuss the GWs generated by sub-horizon bubbles below and leave the super-horizon case to the final discussion.
Gravitational waves by the bubbles {#Sec_Bubble_GW_Spectrum}
==================================
The sub-horizon case in our inflationary scenarios is similar to the previous semi-numerical studies on the EW phase transitions such as [@Huber:2008hg; @Kamionkowski:1993fg]. Usually the phase transition is a rapid process compared to the Hubble time and thus the effect of expansion of the universe can be ignored even during inflation. The only difference is that the total energy density $\rho_{\text{tot}}$ is higher with the contribution from the inflaton to drive inflation. Sub-horizon bubbles can stir up turbulence [@Kamionkowski:1993fg] and sound wave [@Hindmarsh:2013xza; @Hindmarsh:2015qta] in thermal plasma and thus generate GWs. For illustration, however, we focus on the GWs generated via bubble collisions, and apply “envelope approximation” to bubble walls only, with thermal effects neglected (similar to the physics of “runaway bubbles in vacuum” . For a review, see, e.g., [@Caprini:2015zlo]). It is straightforward to generalize the discussions to the GWs caused by sound wave and turbulence in thermal transitions.
Based on the similarity between inflationary and late phase transitions, one immediately obtains original energy spectrum [@Kamionkowski:1993fg; @Huber:2008hg; @Weir:2016tov] $$\begin{aligned}
\Omega_{GW}(k)&=&\Omega_{GW}^{\text{crit}} \frac{(a+b) k_{\text{crit}} ^b k^a}{ b k_{\text{crit}} ^{a+b} +a k^{a+b} }~,
\\
\Omega_{GW}^{\text{crit}} &=& \frac{0.11 v_b^3 }{0.42+v_b^2} \kappa^2 \Big(\frac{H}{\beta}\Big)^2
\Big( \frac{\rho_{\text{higgs}} }{\rho_{\text{tot}}}\Big)^2 ~,
\end{aligned}$$ where $a,b$ are exponents parameterizing the scale dependence of the spectrum. Notice that for usual late phase transition we have $\rho_{\text{tot}} \rightarrow \rho_{\text{higgs}}+\rho_{\text{rad}}$, where we can recover the standard results in the literature.
The critical point $k_{\text{crit}}$ is also the peak momentum of energy spectrum, which is given by $
k_{\text{crit}}/[2\pi a(\tau_* )\beta ] = 0.62/(1.8-0.1v_b+v_b^2).
$ Here a relativistic expansion rate $v_b\approx 1$ is typically expected, yielding $a\approx 2.8$, $b\approx 1$, as discussed in [@Huber:2008hg]. The parameter $\beta^{-1}$ is approximately the duration of the phase transition. In sub-horizon case we have $H/\beta <1$ although the specific values strongly depend on the shapes of the scalar potentials [@Kehayias:2009tn]. $\kappa$ denotes the efficiency of converting vacuum energy into the bubble wall kinetic energy instead of thermal energy [@Kamionkowski:1993fg; @Huber:2008hg]. It depends on $\alpha= \rho_{\text{higgs}}/\rho_{\text{rad}}$, the ratio between the Higgs vacuum energy density and the radiation energy density, as well as the bubble nucleation rate.
We can then finally determine the power spectrum of GW generated by bubbles. Using Eq. (\[P\_and\_Omega\]), we arrive at: $$\label{final_powerS}
P_{\gamma}(k,\tau_{\text{obs}}\rightarrow 0 )=P_{\gamma}^{\text{crit}} \Big( \frac{k_{\text{crit}}}{k}\Big)^4\frac{(a+b) k_{\text{crit}} ^b k^a}{ b k_{\text{crit}} ^{a+b} +a k^{a+b} }~,$$ where $P_{\gamma}^{\text{crit}}$ is the power spectrum at the critical point: $
P_{\gamma}^{\text{crit}} =24 \Big(\frac{a(\tau_*)H}{k_{\text{crit}}}\Big)^4 \Omega_{GW}^{\text{crit}}~.
$ An estimation yields: $P_{\gamma}^{\text{crit}}\sim\Big(\frac{H}{\beta}\Big)^6
\Big( \frac{\rho_{\text{higgs}} }{\rho_{\text{tot}}}\Big)^2$ for the sub-horizon case. The power spectrum is scale dependent and provides us a way to probe phase transition parameters.
![CMB temperature power spectrum. The gray points and error bars are from *Planck* 2015 while the black curve is the best fit of *Planck* 2015. The green, blue and red curves represent power spectra (including the GWs) in three model-independent benchmarks, which could be projected to various scenarios in particle physics. Note that for the red curve, its deviation from the standard one is magnified by 30 times (The unit of $k$ is Mpc$^{-1}$ here and below.).[]{data-label="CMB_TT"}](SubH_CMB.png){width="48.50000%"}
\[t\] ![B-mode power spectrum from phase transition bubbles. For comparison, the primordial GWs from quantum vacuum fluctuations are showed in dashed line with tensor-to-scalar ratio $r=0.07$. The black symbols at the right-upper corner represent the CMB component bandpowers obtained from BICEP2 & [*Keck Array*]{} experiments, with error bars denoting 68% credible intervals and downward triangles indicating the 95% upper bound [@Array:2015xqh]. The colored solid lines correspond to the benchmarks defined in Fig. \[CMB\_TT\]. The two dotted lines denoted effective noise level of two representative experiments POLARBEAR [@Ade:2014afa] and SPIDER [@Crill:2008rd], respectively, based on a Fisher forecast analysis as done in [@Ma:2010yb; @Wang:2014oka; @Creminelli:2015oda].[]{data-label="CMB_Bmode"}](SubH_Bmode.png "fig:"){width="48.50000%"}
\[t\] ![Constraints of *Planck*$ 2015+$BICEP2/Keck data for $\{P_{\gamma}^{\text{crit}}$, $k_{\text{crit}}$, $k_{\text{cutoff}}/k_{\text{crit}}\}$. The one-parameter panels show the parameter likelihood. In the two-parameter panels, dark yellow and yellow are the marginalized 1$\sigma$ and 2$\sigma$ contours, respectively. The colored points correspond to the benchmarks defined in Fig. \[CMB\_TT\]. The black, orange, purple dashed lines in the left-bottom panel corresponds to $ \Big ( \frac{\rho_{\rm higgs}}{\rho_{\rm tot}} \Big)^2 =1, 10^{-4}$ and $10^{-8}$, respectively.[]{data-label="param_constrain"}](parameter.pdf "fig:"){width="48.50000%"}
Imprints in the CMB {#Sec_CMBimprint}
===================
The CMB spectrum can be obtained by inputting the power spectrum into the [@Blas] where the transfer function is calculated. As we see from Eq. (\[final\_powerS\]), the power spectrum diverges as $k\rightarrow 0$. This divergence is unphysical as the GW generating formulae break down at super horizon scale. We can introduce the horizon scale as a natural cut-off. The power spectrum $P_{\gamma}(k,\tau_{\text{obs}}\rightarrow 0 )$ of the GW generated by bubbles therefore are described by three free parameters: $P_{\gamma}^{\text{crit}}$, $k_{\text{crit}}$, $k_{\text{cutoff}}$. The physical $k_{\text{cutoff-physical}}$ is given by $k_{\text{cutoff-physical}}=k_{\text{cutoff}}/a_* \sim H $, and the physical critical momentum is related to the bubble size via $k_{\text{crit} }/a_*\sim R_b^{-1}$. The comoving momenta are $$k_{\text{crit}} \sim \frac{1}{v_b} \frac{\beta}{H} e^{N_*} k_0, \quad
k_{\text{cutoff}} \sim e^{N_*} k_0,$$ yielding $ k_{\text{cutoff}}/k_{\text{crit}} \sim H/\beta$. Here $N_*$ is the e-folds of phase transition counting from the time that the largest mode $k_0$ exits the horizon. We choose the scale factor today $a_0$ as one, thus the largest physical mode today is $k_0=0.0002 \text{Mpc}^{-1}$ as the inverse of observed universe size. Approximately we can find $k_0 $ corresponding to the position at CMB multipole $\ell_0\sim 2$. Then we arrive at relations: $\ell_{\text{crit}} \sim 2e^{N_*}\beta/(v_b H ) ,\ell_{\text{cutoff}} \sim 2 e^{N_*}$.
We plot the CMB spectrum with the GW contributions: $P_{\gamma}^{\text{crit}}\sim 10^{-10} -10^{-11}$ in Fig. \[CMB\_TT\] and Fig. \[CMB\_Bmode\]. The phase transition happened at early time of inflation era generates a new peak on CMB temperature spectrum. The generic peak positions are at $\ell<200$. For $\ell>200$, GW modes have already entered the horizon at recombination and thus are subject to rapid decay, posing a greater challenge in experiments. This also implies that the observable phase transitions should happen soon after the largest mode exits horizon with $N_*\lesssim 5$. The amplitudes of those peaks encode the information about the energy scale of inflation and the phase transitions.
Fig. \[CMB\_Bmode\] shows the tensorial B-mode spectra, which have not been observed yet in experiment. For primordial GWs caused by vacuum fluctuations, B-mode spectrums are known to have the recombination peak at $\ell \sim 100$ and the reionization bump at $\ell< 10$. For $10 \lesssim \ell \lesssim 100 $, the spectrum roughly scales like $\ell^2$ due to the dominant contribution from recombination [@Pritchard:2004qp]. For our scenarios, the power spectrums roughly scale like $k^{-1}$ when $k_{\text{cutoff}}<k<k_{\text{crit}}$ and like $k^{-5}$ when $k>k_{\text{crit}}$. Therefore, if $10 \lesssim \ell_{\text{cutoff}} \lesssim 100 $ and $\ell_{\text{crit}}\gtrsim 100$, B-mode spectrum will scale like $\ell$ for $\ell<100$. Thus we expect a peak to appear near the recombination peak. For small enough $\ell_{\text{cutoff}} \lesssim 10$, the reionization bump and the primordial GW peak in our scenario may lead to more complicated multiple dependence, e.g. as the blue curves shown in Fig. \[CMB\_Bmode\]. The ongoing and future CMB experiments are expected to extend the BICEP2 sensitivities to large angular scales. In Fig. \[CMB\_Bmode\] we also show the effective noise level of two representative experiments POLARBEAR [@Ade:2014afa] and SPIDER [@Crill:2008rd], respectively, based on a Fisher forecast analysis as done in [@Ma:2010yb; @Wang:2014oka; @Creminelli:2015oda]. The effective noise includes contributions from instrumental noise, residual foreground contamination, and the gravitational lensing (without delensing). The dash-dotted part ($l<20$) of the POLARBEA curve represents its limitation in probing large angular scales due to its relatively small survey areas in sky as a ground-based experiment. In Fig. \[param\_constrain\], we show the constraints of the *Planck*2015 $+$BICEP2/Keck data for the three parameters $\{P_{\gamma}^{\text{crit}}$, $k_{\text{crit}}$, $k_{\text{cutoff}}/k_{\text{crit}}\}$, based on a Bayesian analysis using [@Lewis:2002ah]. The constraints are more sensitive to $P_{\gamma}^{\text{crit}}$ and $k_{\text{cutoff}}/k_{\text{crit}}$, compared to $k_{\rm crit}$, as they characterize the overall magnitude of the GWs power spectrum (see Eq. (\[final\_powerS\])). In the two-parameter panels, the three benchmark scenarios defined in Fig. \[CMB\_TT\] are also marked with colored points: green, blue and red. At $2\sigma$ C.L., the green one has been excluded, whereas the blue and red ones are marginally allowed and safe, respectively. In the left-bottom panel, the region below the black dashed line is theoretically forbidden, because of the requirement $$\begin{aligned}
P_{\gamma}^{\text{crit}}\sim\Big(\frac{H}{\beta}\Big)^6
\Big( \frac{\rho_{\text{higgs}} }{\rho_{\text{tot}}}\Big)^2 \lesssim \Big(\frac{H}{\beta}\Big)^6 \sim \Big(\frac{k_{\text{cutoff}}}{k_{\text{crit}}} \Big)^6 \ .\end{aligned}$$ The lower bound for $P_{\gamma}^{\text{crit}}$ is model-dependent. Below are two concrete examples in particle physics:
- GUT scenarios: The GUT phase transition happens around $10^{16}$ GeV [@Georgi:1974yf], which can be of strong first-order in some generic scenarios [@Guth:1979bh]. To avoid re-introducing the problem of magnetic monopoles, the GUT phase transition can only happen before or soon after the begin of inflation, with $\frac{\Lambda_{\rm GUT}^2}{M_p} \lesssim H \lesssim 10^{14} {\rm GeV} < \Lambda_{\rm GUT}$. Thus the projection of the allowed parameter space in the $\log_{10}P_{\gamma}^{\text{crit}} - \log_{10}\frac{k_{\text{cutoff}}}{k_{\text{crit}}}$ plane is along the black-dashed line (see Fig. \[param\_constrain\]), yielding $P_{\gamma}^{\text{crit}}\sim \Big(\frac{H}{\beta}\Big)^6 $. The inflation is of high scale here, potentially generating detectable scale-invariant GWs via vacuum fluctuation. If both the vacuum and bubble GWs are detected, we may infer that the PT is at GUT scale instead of EW scale.
- EW scenarios: A first-order EWPT can be achieved in various theories, e.g. [@Huet:1995sh; @Trodden:1998ym; @Morrissey:2012db; @Cohen:1993nk; @Apreda:2001us; @Kang:2004pp]. Unlike what happens to the GUT scenario, the inflation needs to be low-scale here. The theoretically allowed range for $P_{\gamma}^{\text{crit}}$ is broad, say, $\sim \mathcal O(10^{-60}-1) \Big(\frac{k_{\text{cutoff}}}{k_{\text{crit}}}\Big)^6$, extending from the black-dashed line to above (see Fig. \[param\_constrain\]). If taking $(H/\beta)^6 \sim 10^{-6}, \rho_\text{higgs} \sim (10^3\text{GeV})^4$, and $H \sim 10^{-11} \text{GeV} $, then we have $P_{\gamma}^{\text{crit}}\sim 10^{-10}$. This possibility has been excluded, if $N_* \lesssim 5$ or $k_{\rm crit}< 0.3$, as is indicated by the benchmark point in green in Fig. \[param\_constrain\]. The baryon asymmetry in the universe today can not be directly connected to the first-order EWPT during inflation, due to inflationary dilution. However, if decoupled from inflaton, the effective Higgs potential is sensitive to the temperature of thermal plasma only, subjecting to a negligible curvature correction of order $\mathcal {O}\Big(\frac{H}{\Lambda_{\rm EW}}\Big)$. The EWPT during inflation (“EWPTa” in Fig. \[temperatureEvolution\]) thus may serve as a probe for the one after reheating and hence to test the feasibility of the EWBG.
Discussions {#Sec_conclusion}
===========
In this letter we propose an indirect approach to probe the first-order CPTs, such as the GUT phase transition and the EWPT, by detecting the GWs through the CMB. These GWs are generated during inflation via bubble collisions or by the turbulence caused by bubble motion in the thermal plasma. Generically different from the primordial GWs caused by vacuum fluctuation during inflation, which have been extensively studied in the past decades, the bubble-generated GWs are scale dependent, potentially yielding non-trivial temperature and B-mode spectra in the CMB. Therefore, these GWs (or the first-order CPTs under exploration) represent a new class of physics targets, characterized by a scale-dependent power spectrum, for the ongoing and future CMB experiments to explore. The large-scale scalar power spectrum caused by the first-order CPTs during inflation might be suppressed. This is due to the relatively large values of the slow-roll parameter when the thermal radiation is diluted. The further discussion on the density fluctuation depends on the details of the specific inflation models. The GWs and density fluctuations generated by super-horizon bubbles share some features in physics discussed above, which may leave imprints on the CMB as well. It remains interesting to investigate reheating in more details, especially in the EW secnarios. We expect preheating [@Traschen:1990sw; @Kofman:1997yn] may provide efficient reheating, if inflaton and Higgs fields are decoupled. We leave the detailed study to a future work.
**Acknowledgments.** The authors would like to thank Andrew Cohen, Lam Hui, Yin-Zhe Ma, Ann E. Nelson, Henry Tye and Lian-Tao Wang for useful discussions. This work was supported by the CRF Grants of the Government of the Hong Kong SAR under HKUST4/CRF/13G.
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---
abstract: 'We theoretically investigate the RKKY exchange coupling between two ferromagnets (FM) separated by a thin topological insulator film (TI). We find an unusual dependence of the RKKY exchange coupling $\Phi_\mathrm{ex}$ on the TI thickness ($t_{TI}$). First, when $t_{TI}$ decreases, the coupling amplitude increases at first and reaches its maximum value at some critical thickness, below which the amplitude turns to diminish. This trend is attributed to the hybridization between surfaces of the TI film, which opens a gap below critical thickness and thus turns the surfaces into insulating state from semi-metal state. In insulating phase, diamagnetism induced by the gap-opening compensates paramagnetism of Dirac state, resulting in a diminishing magnetic susceptibility and RKKY coupling. For typical parameters, the critical thickness in $\mathrm{Bi_2Se_3}$ thin film is estimated to be in the range of 3-5 nm.'
author:
- Cong Son Ho
- 'Mansoor B. A. Jalil'
title: 'Effect of surface hybridization on RKKY coupling in ferromagnet/topological insulator/ferromagnet trilayer system'
---
Introduction
============
In the past decade, topological insulators (TI) have emerged as one of the most attractive topics, in both theory and experiment [@Kane:prl05; @Bem:sci06; @Xia:nat09; @Mell:nat14; @Wang:prl15; @Fan:nat14]. Topological insulators possess surface states with strong spin-momentum locking [@Kane:prl05; @Bem:sci06; @Xia:nat09], which translates to a large spin-dependent effects such as spin-orbit torques [@Mell:nat14; @Wang:prl15; @Fan:nat14] and quantum anomalous Hall effects [@Chang:sci13; @Qi:prl16]. Moreover, the spin-locked topological surface states of TI are robust under effects of the time-reversal symmetric impurity scattering. These factors enable TI to be one of the most promising candidates for spintronic devices [@ga:prl10].
At the same time, ferromagnetic hetero-structures (FMs) are also integral spintronic elements that have been used as a platform for practical spintronic devices, such as spin Hall [@Jung:nat12; @Wunder:nat09; @Wunder:sci10; @Seki:nat08] and spin-orbit-based memories [@Jalil:srep14; @Kent:nat15], topological Hall-based sensors [@Ni:ieee16]. When two ferromagnets form a spin-valve structure where they are separated by a metal spacer, there is a interlayer exchange coupling between the FMs mediated by the itinerant electron in the metal spacer, which is known as RKKY coupling [@bruno1995theory; @bruno1993interlayer]. The RKKY coupling oscillates with spacer thickness between ferromagnetic and antiferromagnetic values, and typically its amplitude decreases with increasing thickness. Recently, the RKKY coupling has been studied in TI systems [@liu2009magnetic; @li2015magnetic], where the magnetic moments can be in the same surface [@liu2009magnetic] or belong to different surfaces [@li2015magnetic] of the TI film. In the latter case [@li2015magnetic], by assuming unchanged surface states as the thickness changes, the interlayer was shown to behave in the same way as in FM/metal/FM system, i.e., oscillatory damping with increasing TI thickness. However, in TI films which are very thin, coupling between surfaces becomes significantly dependent on the film thickness [@Linder:prb09; @Lu:prb10] and it can critically modify the surface states and their magnetic property [@Zyu:prb11].
In this work, we study the interlayer coupling between two FMs separated by a thin topological insulator film [@luo2013massive; @wang2015electrically; @li2015magnetic], where thickness-dependence of the surface states is considered. We find that, besides the oscillatory behavior, the coupling amplitude is not monotonically dependent on the thickness, but there is a critical value of thickness at which the coupling amplitude reaches its peak. Below the critical value, the coupling does not increase but rather decreases. These trends are attributed to the phase transition at the critical thickness [@Zyu:prb11] and strong surface hybridization in thin TI films.
Theory
======
We consider a trilayer system comprising of a thin topological insulator (TI) film sandwiched by two ferromagnetic films (see Fig. \[Fig1\]). In this structure, both top and bottom Diract surface of the TI film are active, and they are independently coupled to top and bottom FMs, respectively, via $sd$ coupling. Note that in a super-thin film, the top FM can also couple to the bottom TI surface and vice versa, however, for the simplicity such couplings are neglected. The system is described by model Hamiltonian
![(Color online) Schematic diagram of ferromagnet/topological insulator/ferromagnet (FM/TI/FM) trilayer. $t_{TI}$ is the thickness of the TI thin film. Bottom panel shows the energy bands for various value of the inter-surface coupling $\left(\mathrm{\Delta }\right)$, with $J$ being the value of the sd coupling between electron spin and FM. \[Fig1\]](Diagram.pdf){width="40.00000%"}
$$\label{GrindEQ__1_}
\mathcal{H}=\left[ \begin{array}{cc}
h_{TI}+J{\bf{m}}_{\bf{1}}\cdot\hat{\sigma } & \mathrm{\Delta }I_2 \\
\mathrm{\Delta }I_2 & -h_{TI}+J{\bf{m}}_2\cdot\hat{\sigma } \end{array}
\right],$$
where $h_{TI}={\hbar v}_F\left(~\hat{z}\times \hat{\sigma }\right)\cdot\bf{k}$ describes the effective Hamiltonian of the (top) topological surface state of the TI film, with $v_F$ being the Fermi velocity, $\hat{\sigma}$ is the vector of the Pauli matrices, and $\hat{z}$ being the normal unit vector. Whereas, the bottom surface state, which has opposite helicity, is then described by $h_{TI}(-{\bf{k}})=-h_{TI}(\bf{k})$. The magnetizations are assumed to align along in-plane directions to assure that the gap opening is solely induced by the surface hybridization, see later. The off-diagonal elements describe the hybridization between top and bottom TI surfaces (inter-surface coupling), which is quantified by the tunneling element $\mathrm{\Delta }$. In free-standing TI films, the hybridization opens a gap of $2\mathrm{\Delta }$ in the TI surface states. In previous work [@Zyu:prb11], it has been shown that the gap can be close or open by applying an appropriate in-plane magnetic field. In our case, the proximity coupling ($J\mathbf{m}$) can take the role of driving field to adjust the gap, see Fig. \[Fig1\].
In thick TI films, the hybridization coupling is vanishingly small, meanwhile in the limit of thin TI films, it can be related to the TI thickness $t_{TI}$ as [@Linder:prb09; @Lu:prb10] $$\label{GrindEQ__2_}
\mathrm{\Delta }\approx \frac{{\pi }^2B_1}{t^2_{TI}},$$ with $B_1$ is a material-dependent parameter. With typical value of $B_1=0.1\ \mathrm{eV}\ \mathrm{nm}^2$ in $\mathrm{Bi_2Se_3}$ [@Zhang:nat10; @Liu:prb10], the tunneling element is $0.05\ \mathrm{eV}$ for 5 nm film, and can be up to 0.25 eV for 2nm thin film [@Peng:nat10].
Eigenenergies of are given by $$\label{GrindEQ__3_}
E_{s\tau}=s \sqrt{U+\tau V} ,$$ with $$\begin{aligned}
U=J^2+\Delta^2+v_F^2\hbar^2k^2+v_F\hbar({\bf{m}}_1-{\bf{m}}_2)\cdot({\bf{k}}\times\hat{z}),\nonumber\\
V=J\sqrt{v_F^2\hbar^2[({\bf{m}}_1+{\bf{m}}_2)\cdot({\bf{k}}\times\hat{z})]^2+\Delta^2({\bf{m}}_1+{\bf{m}}_2)^2},\nonumber\end{aligned}$$ where $s,\tau =\pm 1$ representing spin and hyperbola indices, respectively. The band diagrams are shown in Fig. \[Fig1\].
To derive the interlayer exchange coupling between the ferromagnets, we apply the RKKY formula given by [@bruno1995theory; @bruno1993interlayer] $$\label{GrindEQ__4_}
\Phi_\mathrm{ex}=-N\int^{k_{F\bot}}_{-k_{F\bot}}{dq_ze^{iq_zt_{TI}}\chi \left(q_{\parallel }=0,q_z\right)}.$$ In the above, $k_{F\bot}$ is the Fermi momentum in the direction perpendicular to the film, which has typical value of the order of $1/a_0$, with $a_0\approx$ 1 nm being the thickness of a quintuple layer. $\chi (\bf{q})$ is the $\bf{q}$-dependent magnetic susceptibility of the TI film. $N=\frac{1}{2}{\left(\frac{A}{V_0}\right)}^2\left(\frac{S^2a^2_0}{2\pi V_0}\right)$, where $A$ is the contact potential between electron spin and ferromagnets, $S$ is the spin of the FM spin, $V_0$ is the atomic volume.
The magnetic susceptibility of the TI thin film is given by the Kubo’s formula [@white2007quantum] $$\label{GrindEQ__5_}
{\chi }_\mathrm{spin}({\bf{q}})=\frac{{g^2_s\mu}^2_B}{2}\sum_{m^>,n^<}{\int{\frac{d\bf{k}}{(2\pi)^2}\frac{f_0\left(E_{n,\bf{k}}\right)-f_0\left(E_{m,\bf{k}\bf{-}\bf{q}}\right)}{E_{n,\bf{k}}-E_{m,\bf{k}\bf{-}\bf{q}}+i0^+ }}},$$ in which $m^>$ ($n^<$) are for occupied (empty) bands, $g_s=2$ is the g-factor of electron spin, $f_0\left(E_{n,k}\right)$ is the Fermi distribution function corresponding to eigen-energy branch $E_{n,k}$. In addition, the orbital angular momentum can also contribute to the total susceptibility, however, for the simplicity we will ignore this contribution.
Results
=======
Substituting the energy in Eq. to and assuming that the Fermi energy $E_F=0$, the magnetic susceptibility at temperature $T=0$ can be evaluated as $$\label{X1}
\chi_\mathrm{spin}=\frac{{\mu }^2_B}{2{\pi }^2}\frac{\left(J^2+{\mathrm{\Delta }}^2+3{\left|J^2-\mathrm{\Delta }^2\right|}\right)}{\left(J+\mathrm{\Delta }+\left|J-\mathrm{\Delta }\right|\right){\left|J^2-{\mathrm{\Delta }}^{\mathrm{2}}\right|}},$$ for parallel magnetizations $\bf{m}_1\|\bf{m}_2$, and $$\label{X2}
\chi_\mathrm{spin}=\frac{\mu^2_B}{\pi^2}\frac{1}{\sqrt{J^2+\Delta^2}},$$ for the opposite case $\bf{m}_1\|-\bf{m}_2$.
![(Color online) The spin susceptibility of the TI film as a function of the film thickness. The singularity occurs at a critical thickness where the bands are met (see Fig. \[Fig1\]). Other parameters: $J=0.1$ eV, $B_1= 0.1\ \mathrm{eV\ nm^2}$ \[Fig2\]](suscep.pdf){width="50.00000%"}
In derivation of the above equations, we have assumed strong exchange limit, i.e., $J\gg \hbar v_Fk_F$. Otherwise, since the interlayer exchange coupling is quadratic in the proximity coupling between electron spin and FM spin (see Eq. \[GrindEQ\_\_4\_\]), the weak FM/TI coupling limit is not interesting. From Eq. \[X1\], it is obvious that the spin susceptibility encounters singularity when $\Delta=J$, see Fig. \[Fig2\]. In general, any singularity in the susceptibility relates to a second order phase transition [@white2007quantum], which in this case is the transition between insulating state ($\Delta>J$) and metallic state ($\Delta<J$) at the singularity point [@Zyu:prb11]. The diminishing susceptibility below the critical thickness is possibly due to the emergence of diamagnetism in the insulating surface state [@Zyu:prb11]. On the other hand, if the magnetizations are in the anti-parallel configuration, the band gap is always open and the surface states are always in the insulating phase, the singularity is thus avoided.
From Eqs. and , the interlayer exchange coupling is readily obtained as $$\label{GrindEQ__7_}
\Phi_\mathrm{ex}={-I_0\frac{\left(J^2+{\mathrm{\Delta }}^2+3{\left|J^2-\mathrm{\Delta }^2\right|}\right)}{\left(J+\mathrm{\Delta }+\left|J-\mathrm{\Delta }\right|\right){\left|J^2-{\mathrm{\Delta }}^{\mathrm{2}}\right|}}}{\frac{2{\mathrm{sin} \left(k_{F\bot} t_{TI}\right)}}{t_{TI}}},$$ for parallel magnetizations $\bf{m}_1\|\bf{m}_2$, and $$\Phi_\mathrm{ex}=-I_0\frac{1}{\sqrt{J^2+\Delta^2}}{\frac{2{\mathrm{sin} \left(k_{F\bot} t_{TI}\right)}}{t_{TI}}},$$ for the opposite case $\bf{m}_1\|-\bf{m}_2$, where $I_0=N \frac{\mu^2_B}{\pi^2}$.
![(Color online) The interlayer exchange coupling ($\Phi_\mathrm{ex}$) as a function of the TI thickness ($t_{TI})$ at $T=0$ and $E_F=0$. The exchange coupling is oscillatory with a maximum amplitude at $t_{TI}\approx 3\ \mathrm{nm}$. Other parameters: $J=0.1$ eV, $B_1= 0.1\ \mathrm{eV\ nm^2}$, $k_{F\bot}=5\ \mathrm{nm^{-1}}$. \[Fig3\]](Exchange.pdf){width="50.00000%"}
First, we see that the exchange coupling has the oscillatory nature of RKKY-type coupling (see Fig. \[Fig3\]). However, its amplitude does not monotonically decrease with increasing thickness, but reaches its peak at a thickness $t_{crit}=\pi\sqrt{B_1/J}$ such that $\mathrm{\Delta }=J$. This trend is in accordance with the behavior of the spin susceptibility discussed above. The compensation between paramagnetic Dirac surfaces and diamagnetic gapped-surfaces leads to the diminishing magnetic susceptibility, and so the diminishing interlayer exchange coupling. For typical parameters of $\mathrm{Bi_2Se_3}$ thin films, the thickness for maximum interlayer exchange coupling is estimated to be in the range of 3-5 nm.
Conclusion
==========
In this work, we have shown that topological insulator films as spacers in spin-valve structures play a fruitful role in mediating the exchange coupling between two ferromagnets. In one hand, the TI film provides electrons for mediating the coupling, which induce stronger coupling for thinner film as the magnetic moment transfer rate is stronger. On the other hand, thin TI film will encounter gap-opening in its surface states due to the hybridization, which in turn suppresses the spin susceptibility and RKKY coupling. Therefore, our work provides a guide to optimize TI-based spin valves for spintronics application.
We acknowledge the financial support of MOE Tier II grant MOE2013-T2-2-125 (NUS Grant No. R-263-000-B10-112), and the National Research Foundation of Singapore under the CRP Programs “Next Generation Spin Torque Memories: From Fundamental Physics to Applications” NRF-CRP12-2013-01 and “Non-Volatile Magnetic Logic and Memory Integrated Circuit Devices” NRF-CRP9-2011-01.
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abstract: 'Machine reading comprehension with unanswerable questions is a new challenging task for natural language processing. A key subtask is to reliably predict whether the question is unanswerable. In this paper, we propose a unified model, called U-Net, with three important components: answer pointer, no-answer pointer, and answer verifier. We introduce a universal node and thus process the question and its context passage as a single contiguous sequence of tokens. The universal node encodes the fused information from both the question and passage, and plays an important role to predict whether the question is answerable and also greatly improves the conciseness of the U-Net. Different from the state-of-art pipeline models, U-Net can be learned in an end-to-end fashion. The experimental results on the SQuAD 2.0 dataset show that U-Net can effectively predict the unanswerability of questions and achieves an F1 score of 71.7 on SQuAD 2.0.'
author:
- |
Fu Sun, Linyang Li, Xipeng Qiu[^1], Yang Liu\
Shanghai Key Laboratory of Intelligent Information Processing, Fudan University\
School of Computer Science, Fudan University\
Liulishuo Silicon Valley AI Lab\
{fsun17,lyli15,xpqiu}@fudan.edu.cn, [email protected]
bibliography:
- 'nlp.bib'
title: 'U-Net: Machine Reading Comprehension with Unanswerable Questions'
---
Introduction
============
Machine reading comprehension (MRC) is a challenging task in natural language processing, which requires that machine can read, understand, and answer questions about a text. Benefiting from the rapid development of deep learning techniques and large-scale benchmarks [@hermann2015teaching; @hill2015goldilocks; @rajpurkar2016squad], the end-to-end neural methods have achieved promising results on MRC task [@seo2016bidirectional; @fusionnet; @drqa; @clark2017simple; @hu2017reinforced]. The best systems have even surpassed human performance on the Stanford Question Answering Dataset (SQuAD) [@rajpurkar2016squad], one of the most widely used MRC benchmarks. However, one of the limitations of the SQuAD task is that each question has a correct answer in the context passage, therefore most models just need to select the most relevant text span as the answer, without necessarily checking whether it is indeed the answer to the question.
To remedy the deficiency of SQuAD, @squad2.0 developed SQuAD 2.0 that combines SQuAD with new unanswerable questions. Table \[tab:example\] shows two examples of unanswerable questions. The new dataset requires the MRC systems to know what they don’t know.
Article: Endangered Species Act
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Paragraph: “... Other legislation followed, including the Migratory Bird Conservation Act of 1929, a prohibiting the hunting of right and gray whales, and the . These had a low cost to society—the species were relatively rare—and little was raised.”
Question 1: “Which laws faced significant ?”
Plausible Answer:
Question 2: “What was the name of the ?”
Plausible Answer:
: Unanswerable Questions from SQUAD 2.0 [@squad2.0].[]{data-label="tab:example"}
To do well on MRC with unanswerable questions, the model needs to comprehend the question, reason among the passage, judge the unanswerability and then identify the answer span. Since extensive work has been done on how to correctly predict the answer span when the question is answerable (e.g., SQuAD 1.1), the main challenge of this task lies in how to reliably determine whether a question is not answerable from the passage.
There are two kinds of approaches to model the answerability of a question. One approach is to directly extend previous MRC models by introducing a no-answer score to the score vector of the answer span [@levy2017zero; @clark2017simple]. But this kind of approaches is relatively simple and cannot effectively model the answerability of a question. Another approach introduces an answer verifier to determine whether the question is unanswerable [@hu2018read; @tan2018know]. However, this kind of approaches usually has a pipeline structure. The answer pointer and answer verifier have their respective models, which are trained separately. Intuitively, it is unnecessary since the underlying comprehension and reasoning of language for these components is the same.
In this paper, we decompose the problem of MRC with unanswerable questions into three sub-tasks: answer pointer, no-answer pointer, and answer verifier. Since these three sub-tasks are highly related, we regard the MRC with unanswerable questions as a multi-task learning problem [@multiDBLP:journals/ml/Caruana97] by sharing some meta-knowledge.
We propose the U-Net to incorporate these three sub-tasks into a unified model: 1) an answer pointer to predict a candidate answer span for a question; 2) a no-answer pointer to avoid selecting any text span when a question has no answer; and 3) an answer verifier to determine the probability of the “unanswerability” of a question with candidate answer information. Additionally, we also introduce a universal node and process the question and its context passage as a single contiguous sequence of tokens, which greatly improves the conciseness of U-Net. The universal node acts on both question and passage to learn whether the question is answerable. Different from the previous pipeline models, U-Net can be learned in an end-to-end fashion. Our experimental results on the SQuAD 2.0 dataset show that U-Net effectively predicts the unanswerability of questions and achieves an F1 score of 72.6.
The contributions of this paper can be summarized as follows.
- We decompose the problem of MRC with unanswerable questions into three sub-tasks and combine them into a unified model, which uses the shared encoding and interaction layers. Thus, the three-tasks can be trained simultaneously in an end-to-end fashion.
- We introduce a universal node to encode the common information of the question and passage. Thus, we can use a unified representation to model the question and passage, which makes our model more condensed.
- U-Net is very easy to implement yet effective.
Proposed Model
==============
![Architecture of the U-Net.[]{data-label="fig:overview"}](overview){width="0.9\linewidth"}
Formally, we can represent the MRC problem as: given a set of tuples $(Q,P,A)$, where $Q = (q_1,q_2,\cdots,q_m)$ is the question with $m$ words, $P=(p_1,p_2,\cdots,p_n)$ is the context passage with $n$ words, and $A = p_{r_s:r_e}$ is the answer with $r_s$ and $r_e$ indicating the start and end points, the task is to estimate the conditional probability $P(A|Q,P)$.
The architecture of our proposed U-Net is illustrated in Figure \[fig:overview\].
U-Net consists of four major blocks: Unified Encoding, Multi-Level Attention, Final Fusion, and Prediction. As shown in Figure \[fig:overview\], we first combine the embedded representation of the question and passage with a universal node $u$ and pass them through a BiLSTM to encode the whole text. We then use the encoded representation to do the information interaction. Then we use the encoded and interacted representation to fuse the full representation and feed them into the final prediction layers to do the multi-task training.
We will describe our model in details in the following.
(A) Unified Encoding {#sub:Encoder layer}
--------------------
#### Embedding
Following the successful models on SQuAD 1.1, we first embed both the question and the passage with the following features. Glove embedding [@pennington2014glove] and Elmo embedding [@elmo_] are used as basic embeddings. Besides, we use POS embedding, NER embedding, and a feature embedding that includes the exact match, lower-case match, lemma match, and a TF-IDF feature [@drqa]. Now we get the question representation $Q = \bq_{i=1}^m$ and the passage representation $P = \bp_{i=1}^n$, where each word is represented as a $d$-dim embedding by combining the features/embedding described above.
#### Universal Node
We create a universal node $u$, which is a key factor in our model and has several roles in predicting the unanswerability of question $Q$.
We expect this node to learn universal information from both passage and question. This universal node is added and connects the passage and question at the phase of embedding, and then goes along with the whole representation, so it is a key factor in information representation. Since the universal node is in between and later shared between passage and question, it has an abstract semantic meaning rather than just a word embedding.
Also, the universal node is later shared in the attention interaction mechanism and used in both the answer boundary detection and classification tasks, so this node carries massive information and has several important roles in our whole model construction.
The universal node $u$ is first represented by a $d$-dim randomly-initialized vector. We concatenated question representation, universal node representation, passage representation together as: $$\begin{aligned}
V=[Q,\bu, P] &= [\bq_1, \bq_2 \dots \bq_m, \bu, \bp_1, \bp_2, \cdots, \bp_n],\end{aligned}$$ $V\in \mathbb{R}^{d \times(m+n+1)}$ is a joint representation of question and passage.
#### Word-level Fusion
Then we first use two-layer bidirectional LSTM (BiLSTM) [@hochreiter1997long] to fuse the joint representation of question, universal node, and passage. $$\begin{aligned}
H^l &= \mathrm{BiLSTM}(V),\\
H^h &= \mathrm{BiLSTM}(H^l),\end{aligned}$$ where $H^l$ is the hidden states of the first BiLSTM, representing the low-level semantic information, and $H^h$ is the hidden states of the second BiLSTM, representing the high-level semantic information.
Finally, we concatenate $H^l$ and $H^h$ together and pass them through the third BiLSTM and obtain a full representation $H^f$. $$\begin{aligned}
H^f &= \mathrm{BiLSTM}([H^l;H^h]).\end{aligned}$$
Thus, $H=[H^l;H^h;H^f]$ represents the deep fusion information of the question and passage on word-level. When a BiLSTM is applied to encode representations, it learns the semantic information bi-directionally. Since the universal node $u$ is between the question and passage, its hidden states $\bh_{m+1}$ can learn both question and passage information. When the passage-question pair was encoded as a unified representation and information flows via the BiLSTM, the universal node has an important role in information representation.
(B) Multi-Level Attention {#sub:attention_layer}
-------------------------
To fully fuse the semantic representation of the question and passage, we use the attention mechanism [@bahdanau2014neural] to capture their interactions on different levels. We expected that we could simply use self-attention on the encoded representation $H$ for interaction between question and passage, which contains both bi-attention [@seo2016bidirectional] and self-attention [@selfmatchDBLP:conf/acl/WangYWCZ17] of the question and passage. But we found that it performed slightly worse than the traditional bi-directional attention with the universal node included. Therefore, we use a bi-directional attention between the question and passage.
We first divide $H$ into two representations: attached passage $H_q$ and attached question $H_p$, and let the universal node representation $\bh_{m+1}$ attached to both the passage and question, i.e., $$\begin{aligned}
H_q &= [\bh_1, \bh_2, \cdots, \bh_{m+1}],\\
H_p &= [\bh_{m+1}, \bh_{m+2}, \cdots, \bh_{m+n+1}],\end{aligned}$$ Note $\bh_{m+1}$ is shared by $H_q$ and $H_p$. Here the universal node works as a special information carrier, and both passage and question can focus attention information on this node so that the connection between them is closer than a traditional bi-attention interaction.
Since both $H_q=[H^l_q;H^h_q;H^f_q]$ and $H_p=[H^l_p;H^h_p;H^f_p]$ are concatenated by three-level representations, we followed previous work FusionNet [@fusionnet] to construct their iterations on three levels.
Take the first level as an example. We first compute the affine matrix of $H^l_q$ and $H_p^l$ by $$\begin{aligned}
S &= \Big(\mathrm{ReLU}(W_1 H_q^l)\Big)\tran \mathrm{ReLU}(W_2 H^l_p),\end{aligned}$$ where $S\in \mathbb{R}^{(m+1) \times(n+1)}$; $W_1$ and $W_2$ are learnable parameters. Next, a bi-directional attention is used to compute the interacted representation $\widehat{H_q^l}$ and $\widehat{H_p^l}$. $$\begin{aligned}
\widehat{H_q^l} &= H^l_p \times \mathrm{softmax}(S\tran),\\
\widehat{H_p^l} &= H^l_q \times \mathrm{softmax}(S),\end{aligned}$$ where $\mathrm{softmax}(\cdot)$ is column-wise normalized function.
We use the same attention layer to model the interactions for all the three levels, and get the final fused representation $\widehat{H^l}, \widehat{H^h}, \widehat{H^f}$ for the question and passage respectively.
Note that while dealing with the attention output of the universal node, we added two outputs from passage-to-question attention and question-to-passage attention. So after the interaction, the fused representation $\widehat{H^l}, \widehat{H^h}, \widehat{H^f}$ still have the same length as the encoded representation $H^l$, $H^h$ and $H^f$.
(C) Final Fusion
----------------
After the three-level attentive interaction, we generate the final fused information for the question and passage. We concatenate all the history information: we first concatenate the encoded representation $H$ and the representation after attention $\widehat{H}$ (again, we use $H^l, H^h, H^f$, and $\widehat{H^l}, \widehat{H^h}, \widehat{H^f}$ to represent 3 different levels of representation for the two previous steps respectively).
Following the success of DenseNet [@densenetDBLP:journals/corr/HuangLW16a], we concatenate the input and output of each layer as the input of the next layer.
First, we pass the concatenated representation $H$ through a BiLSTM to get $H^A$. $$\begin{aligned}
H^A &= \mathrm{BiLSTM}\Big([H^l; H^h; H^f; \widehat{H^l}; \widehat{H^h}; \widehat{H^f}]\Big),
\end{aligned}$$ where the representation $H^A$ is a fusion of information from different levels.
Then we concatenate the original embedded representation $V$ and $H^A$ for better representation of the fused information of passage, universal node, and question. $$\begin{aligned}
A &= [V; H^A].\end{aligned}$$
Finally, we use a self-attention layer to get the attention information within the fused information. The self-attention layer is constructed the same way as [@attentionisallyouneed]: $$\begin{aligned}
\widehat{A} = A \times \mathrm{softmax}(A\tran A),\end{aligned}$$ where $\widehat{A}$ is the representation after self-attention of the fused information $A$. Next we concatenated representation $H^A$ and $\widehat{A}$ and pass them through another BiLSTM layer: $$\begin{aligned}
O=\mathrm{BiLSTM}[H^A; \widehat{A}].\end{aligned}$$
Now $O$ is the final fused representation of all the information. At this point, we divide $O$ into two parts: $O^P$, $O^Q$, representing the fused information of the question and passage respectively. $$\begin{aligned}
O^P &= [\bo_1, \bo_2, \cdots, \bo_{m}],\\
O^Q &= [\bo_{m+1}, \bo_{m+2}, \cdots, \bo_{m+n+1}],\end{aligned}$$ Note for the final representation, we attach the universal node only in the passage representation $O^P$. This is because we need the universal node as a focus for the pointer when the question is unanswerable. These will be fed into the next decoder prediction layer.
(D) Prediction {#sub:prediction}
--------------
The prediction layer receives fused information of passage $O^P$ and question $O^Q$, and tackles three prediction tasks: (1) answer pointer, (2) no-answer pointer and (3) answer verifier.
First, we use a function shown below to summarize the question information $O^Q$ into a fixed-dim representation $\bc_q$.
$$\begin{aligned}
\bc_q &= \sum_i \frac{\exp(W_q^\top o^Q_i)}{\sum_j \exp(W^\top o^Q_j)} o^Q_i,\label{eq:c_q}\end{aligned}$$
where $W_q$ is a learnable weight matrix and $o_i^Q$ represents the $i_{th}$ word in the question representation. Then we feed $\bc_q$ into the answer pointer to find boundaries of answers [@matchDBLP:journals/corr/WangJ16a], and the classification layer to distinguish whether the question is answerable.
#### (i) Answer Pointer
We use this answer pointer to detect the answer boundaries from the passage when the question is answerable (i.e., the answer is a span in the passage). This layer is a classic pointer net structure [@pointernet]. We use two trainable matrices $W_s$ and $W_e$ to estimate the probability of the answer start and end boundaries of the $i_{th}$ word in the passage, $\alpha_i$ and $ \beta_i$. $$\begin{aligned}
\alpha_i &\propto \exp(\bc_q W_s o^P_i), \\
\beta_i &\propto \exp(\bc_q W_eo^P_i),\end{aligned}$$
Note here when the question is answerable, we do not consider the universal node in answer boundary detection, so we have $i >0$ ($i=0$ is the universal node in the passage representation). The loss function for the answerable question pairs is: $$\begin{aligned}
\mathcal{L}_{A} = -\big(\log \alpha_a +\log \beta_b \big),\end{aligned}$$ where $a$ and $b$ are the ground-truth of the start and end boundary of the answer.
#### (ii) No-Answer Pointer
Then we use the same pointer for questions that are not answerable. Here the loss $\mathcal{L}_{NA}$ is:
$$\begin{aligned}
\mathcal{L}_{NA} = -\big(\log \alpha_0 +\log \beta_0 \big),\end{aligned}$$
$\alpha_0$ and $\beta_0$ correspond to the position of the universal node, which is at the front of the passage representation $O_p$. For this scenario, the loss is calculated for the universal node.
Additionally, since there exits a plausible answer for each unanswerable question in SQuAD 2.0, we introduce an auxiliary *plausible answer pointer* to predict the boundaries of the plausible answers. The plausible answer pointer has the same structure as the answer pointer, but with different parameters. Thus, the total loss function is: $$\begin{aligned}
\mathcal{L}_{NA} = -\big(\log \alpha_0 +\log \beta_0 \big) -\big(\log \alpha'_{a^*} +\log \beta'_{b^*} \big),\end{aligned}$$ where $\alpha'$ and $\beta'$ are the output of the plausible answer pointer; $a^*$ and $b^*$ are the start and end boundary of the unanswerable answer.
The no-answer pointer and plausible answer pointer are removed at test phase.
#### (iii) Answer Verifier {#sub:Classification}
We use the answer verifier to distinguish whether the question is answerable.
Answer verifier applies a weighted summary layer to summarize the passage information into a fixed-dim representation $\bc_q$ (as shown in Eq.).
And we use the weight matrix obtained from the answer pointer to get two representations of the passage. $$\begin{aligned}
\bc_s &= \sum_i \alpha_i \cdot o^P_i\\
\bc_e &= \sum_i \beta_i \cdot o^P_i\end{aligned}$$
Then we use the universal node $\bo_{m+1}$ and concatenate it with the summary of question and passage to make a fixed vector $$\begin{aligned}
F= [\bc_q ; \bo_{m+1} ; \bc_s ; \bc_e].\end{aligned}$$ This fixed $F$ includes the representation $\bc_q$ representing the question information, and $\bc_s$ and $\bc_e$ representing the passage information. Since these representations are highly summarized specially for classification, we believe that this passage-question pair contains information to distinguish whether this question is answerable. In addition, we include the universal node as a supplement. Since the universal node is pointed at when the question is unanswerable and this node itself already contains information collected from both the passage and question during encoding and information interaction, we believe that this node is important in distinguishing whether the question is answerable.
Finally, we pass this fixed vector $F$ through a linear layer to obtain the prediction whether the question is answerable. $$\begin{aligned}
p^{c} = \sigma(W_f\tran F)\end{aligned}$$ where $\sigma$ is a sigmoid function, $W_f$ is a learnable weight matrix.
Here we use the cross-entropy loss in training. $$\begin{aligned}
\mathcal{L}_{AV} = -\Big(\delta \cdot\log {p^c} + (1-\delta)\cdot(\log{(1-p^c)})\Big),\end{aligned}$$ where $\delta\in \{0,1\}$ indicates whether the question has an answer in the passage.
Compared with other relatively complex structures developped for this MRC task, our U-Net model passes the original question and passage pair through embedding and encoding, which then interacts with each other, yielding fused information merged from all the levels. The entire architecture is very easy to construct. After we have the fused representation of the question and passage, we pass them through the pointer layer and a fused information classification layer in a multi-task setup.
Training
========
We jointly train the three tasks by combining the three loss functions. The final loss function is: $$\begin{aligned}
\mathcal{L} = \delta\mathcal{L}_{A}+ (1-\delta) \mathcal{L}_{NA} + \mathcal{L}_{AV},\end{aligned}$$ where $\delta\in \{0,1\}$ indicates whether the question has an answer in the passage, $\mathcal{L}_{A}$, $\mathcal{L}_{NA}$ and $\mathcal{L}_{AV}$ are the three loss functions of the answer pointer, no-answer pointer, and answer verifier.
Although the three tasks could have different weights in the final loss function and be further fine-tuned after joint training, here we just consider them in the same weight and do not fine-tune them individually.
At the test phase, we first use the answer pointer to find a potential answer to the question, while the verifier layer judges whether the question is answerable. If the classifier predicts the question is unanswerable, we consider the answer extracted by the answer pointer as plausible. In this way, we get the system result.
--------------------------------------------- ------ ------ ---------- ----------
\*[**Model**]{}
EM F1 EM F1
**End-to-end Model**
BNA$^\star$ [@squad2.0] 59.8 62.6 59.2 62.1
DocQA [@squad2.0] 65.1 67.6 63.4 66.3
FusionNet++ [@fusionnet] - - 66.6 69.6
SAN [@sanDBLP:journals/corr/abs-1712-03556] - - 68.6 71.4
VS$^3$-Net - - 68.4 71.3
U-Net 70.3 74.0 **69.2** **72.6**
**Ensemble Model**
FusionNet++ (ensemble) - - 70.3 72.6
SAN (ensemble) - - 71.3 73.7
U-Net (ensemble) - - **71.5** **75.0**
**Pipeline Model**
RMR+ELMo+Verifier [@hu2018read] 72.3 74.8 71.7 74.2
Human 86.3 89.0 86.9 89.5
--------------------------------------------- ------ ------ ---------- ----------
Experiment
==========
Datasets {#sub:Datasets}
--------
Recently, machine reading comprehension and question answering have progressed rapidly, owing to the computation ability and publicly available high-quality datasets such as SQuAD. Now new research efforts have been devoted to the newly released answer extraction test with unanswerable questions, SQuAD 2.0 [@squad2.0]. It is constructed by combining question-answer pairs selected from SQuAD 1.0 and newly crafted unanswerable questions. These unanswerable questions are created by workers that were asked to pose questions that cannot be answered based on the paragraph alone but are similar to the answerable questions. It is very difficult to distinguish these questions from the answerable ones. We evaluate our model using this data set. It contains over 100,000+ questions on 500+ wikipedia articles.
Implementation Details {#sub:Implementation Details}
----------------------
We use Spacy to process each question and passage to obtain tokens, POS tags, NER tags and lemmas tags of each text. We use 12 dimensions to embed POS tags, 8 for NER tags [@drqa]. We use 3 binary features: exact match, lower-case match and lemma match between the question and passage [@featDBLP:journals/corr/LeeKP016]. We use 100-dim Glove pretrained word embeddings and 1024-dim Elmo embeddings. All the LSTM blocks are bi-directional with one single layer. We set the hidden layer dimension as 125, attention layer dimension as 250. We added a dropout layer over all the modeling layers, including the embedding layer, at a dropout rate of 0.3 [@dropoutDBLP:journals/jmlr/SrivastavaHKSS14]. We use Adam optimizer with a learning rate of 0.002 [@admaDBLP:journals/corr/KingmaB14].
During training, we omit passage with over 400 words and question with more than 50 words. For testing, when the passage has over 600 words and the question is over 100 words, we simply label these questions as unanswerable.
Main Results {#sub:Main Results}
------------
Our model achieves an F1 score of 74.0 and an EM score of 70.3 on the development set, and an F1 score of 72.6 and an EM score of 69.2 on Test set[^2], as shown in Table \[tab:main\]. Our model outperforms most of the previous approaches. Comparing to the best-performing systems, our model has a simple architecture and is an end-to-end model. In fact, among all the end-to-end models, we achieve the best F1 scores. We believe that the performance of the U-Net can be boosted with an additional post-processing step to verify answers using approaches such as [@hu2018read].
Ablation Study {#sub:Ablation Study}
--------------
We also do an ablation study on the SQuAD 2.0 development set to further test the effectiveness of different components in our model. In Table \[tab:configuration\], we show four different configurations.
[**Configuration**]{} [**EM**]{} [**F1**]{} [**$\Delta$EM**]{} [**$\Delta$ F1**]{}
----------------------------- ------------ ------------ -------------------- ---------------------
U-Net 70.3 74.0 - -
no node U 67.9 71.4 -2.4 -2.6
no share U 69.7 73.5 -0.6 -0.5
no concatenate P & Q 69.0 72.8 -1.3 -1.2
no plausible answer pointer 69.6 72.9 -0.7 -1.1
no classification 63.5 68.5 -6.8 -5.5
Self-Attn Only 69.7 73.5 -0.5 -0.5
: Comparison of different configurations for our U-Net model. []{data-label="tab:configuration"}
First, we remove the universal node $U$. We let the negative examples focus on the plausible answer spans instead of focusing on the universal node $U$. This results in a loss of 2.6% F1 score on the development set, showing that the universal node $U$ indeed learns information about whether the question is answerable.
We also tried to make the universal node $U$ only attached to the passage representation when passing the attention layer. Our results showed that when node $U$ is shared, as it is called ‘universal’, it learns information interaction between the question and passage, and when it is not shared, the performance slightly degraded.
As for the approaches to encode the representations, we pass both the question and passage through a shared BiLSTM. To test the effectiveness of this, we ran the experiment using separate BiLSTMs on embedded question and passage representations. Results show that the performance dropped slightly, suggesting sharing BiLSTM is an effective method to improve the quality of the encoder.
After removing the plausible answer pointer, the performance also dropped, indicating the plausible answers are useful to improve the model even though they are incorrect.
After removing the answer verifier, the performance dropped greatly, indicating it is vital for our model.
Lastly, we run a test using a more concise configuration. In the second block (multi-level attention) of the U-Net, we do not split the output of the encoded presentation and let it pass through a self-attention layer. The bidirectional attention is removed. In this way, our model uses only one unified representation of the question and passage at all time. We simply pass this representation layer by layer to get the final result. Compared to the bi-attention model, the F1-score decreases 0.5%.
Multi-task Study {#sub:multi-task Study}
----------------
We also run an experiment to test the performance of our multi-task model. We select different losses that participate in the training procedure to observe the performance affected by answer boundary detect or classification.
Table \[tab:multi-task\] shows the performance. Here we use $EM^*$ and $F1^*$ to represent the EM and F1 score when the classification is not part of the task, which makes it very much like the task in SQuAD 1.1.
[**Loss**]{} [**EM$^*$**]{} [**F1$^*$**]{} [**Classification Acc.**]{}
-------------------- ---------------- ---------------- -----------------------------
$\mathcal{L}$ 75.3 84.8 80.2
$\mathcal{L}_{AV}$ - - 67.1
$\mathcal{L}_A$ 77.2 85.1 -
: Multi-task performance on the development set. []{data-label="tab:multi-task"}
To test our classifier performance, we do not use backward propagation over the loss of answer boundary detection and simply run a classification task. Results (the first two rows in Table \[tab:multi-task\]) show that there is a large gain when using the multi-task model. The answer boundary detection task helps the encoder learn information between the passage and question and also feed information into the universal node, therefore we can use a summarized representation of the passage and question as well as the universal node to distinguish whether the question is answerable, i.e., help improve classification.
For the answer boundary detection task, we find that the multi-task setup (i.e., the classification layer participates in the training process) does not help its performance. Since the classifier and pointer layer shared the encoding process, we originally expected that classification information can help detect answer boundaries. But this is not the case. We think this is also reasonable since distinguishing whether the question is answerable is mainly focusing on the interactions between the passage-question pair, so once the question is predicted as answerable or not, it has nothing to do with the answer boundaries. This is consistent with how human-beings do this classification task.
We also run the test over SQuAD 1.1 development test to evaluate the performance. Due to a condensed structure, our model achieves an $F1^*$ score of less than 86%, which is not a very competitive score on SQuAD 1.1 test. But as shown above, our model achieves a good score in SQuAD 2.0 test, which shows this model has the potential to achieve higher performance by making progress on both the answer detection and classification tasks.
Overall, we can conclude that our multi-task model works well since the performance of unanswerability classification improves significantly when the answer pointer and answer verifier work simultaneously.
Study on the Different Thresholds of Unanswerability Classification {#sub: More }
-------------------------------------------------------------------
The output $b$ of the answer verifier is the probability of a question being unanswerable. The smaller the output, the lower the probability of unanswerability is. In SQuAD 2.0, the proportions of unanswerable questions are different in the training and test sets. The default threshold $0.5$ is optimized on the training set, but not suitable for the test set. Therefore, it is reasonable to set a proper threshold to manually adapt to the test set.
As mentioned in SQuAD 2.0 paper [@squad2.0], different thresholds for answerability prediction result in fluctuated scores between answerable and unanswerable questions. Here we show the variation of the F1 score with different thresholds in Figure \[fig:F1-t\]. The threshold between $[0,1]$ is used to decide whether a question can be answered. When the threshold is set to $0$, all questions are considered as answerable.
table \[x index=0, y index=1\] [L-F1.txt]{}; table \[x index=0, y index=3\] [L-F1.txt]{}; table \[x index=0, y index=2\] [L-F1.txt]{};
As we can see, when the threshold is set to 0.5, F1 score of answerable questions is similar to that of unanswerable questions. When we increase the threshold (i.e., more likely to predict the question as unanswerable), performance for answerable questions degrades, and improves for unanswerable questions. This is as expected. We can see that the overall $F1$ score is slightly better, which is consistent with the idea from SQuAD 2.0. In addition, we find that for larger thresholds, the variance between $EM$ and $F1$ is narrowed since $EM$ and $F1$ scores for unanswerable questions are the same.
Finally, we set the threshold to be $0.7$ for the submission system to SQuAD evaluation.
Related Work
============
End-to-end Models for MRC
-------------------------
Currently, end-to-end neural network models have achieved great successes for machine reading comprehension [@seo2016bidirectional; @kumar2015ask; @sukhbaatar2015end; @cui2016attention; @xiong2016dynamic; @dhingra2016gated; @shen2016reasonet; @hu2017reinforced; @wang2018multi]. Most of these models consist of three components: encoder, interaction, and pointer. The BiLSTM is widely used for encoding the embedded representation. For the interaction, bidirectional attention mechanism is very effective to fuse information of the question and passage. Finally, a pointer network [@pointernet] is used to predict the span boundaries of the answer. Specifically, in SQuAD test [@rajpurkar2016squad], there are approaches to combine match-LSTM and pointer networks to produce boundaries of the answer and employ variant bidirectional attention mechanism to match the question and passage mutually.
In our model, we learn from previous work and develop a condensed end-to-end model for the SQuAD 2.0 task. Different from the previous models, we use a unified representation to encode the question and passage simultaneously, and introduce a universal node to encode the fused information of the question and passage, which also plays an important role to predict the unanswerability of a question.
MRC with Unanswerable Questions
-------------------------------
MRC with unanswerable questions is a more challenging task. Previous work @levy2017zero [@clark2017simple] has attempted to normalize a no-answer score depending on the probability of all answer spans and still detect boundaries at the same time. But the scores of the answer span predictions are not very discriminative in distinguishing whether the question is answerable. Therefore, this kind of approaches, though relatively simple, cannot effectively deal with the answerability of a question.
@hu2018read [@tan2018know] introduced an answer verifier idea to construct a classification layer. However, this kind of approaches usually has a pipeline structure. The answer pointer and answer verifier have their respective models that are trained separately.
#### Multi-task models
Different from existing work, we regard the MRC with unanswerable questions as a multi-task learning problem [@multiDBLP:journals/ml/Caruana97] by sharing some meta-knowledge. Intuitively, answer prediction and answer verification are related tasks since the underlying comprehension and reasoning of language for these components is the same. Therefore, we construct a multi-task model to solve three sub-tasks: answer pointer, no-answer pointer, and answer verifier.
Conclusion and Future Work
===========================
In this paper, we regard the MRC with unanswerable questions as multi-task learning problems and propose the U-Net, a simple end-to-end model for MRC challenges. U-Net has good performance on SQuAD 2.0. We first add a universal node to learn a fused representation from both the question and passage, then use a concatenated representation to pass through encoding layers. We only treat question and passage differently during attention interactions. In the rest blocks of U-Net, we still use the unified representation containing both the question and passage representation. Finally, we train the U-Net as a multi-task framework to determine the final answer boundaries as well as whether the question is answerable. Our model has very simple structure yet achieves good results on SQuAD 2.0 test.
Our future work is to reconstruct the structure of U-Net by replacing the current multi-level attention block with a simpler self-attention mechanism, which we believe can capture the question and passage information, and intuitively is also coherent with the rest of our U-Net model. In addition, we will improve the answer boundary detection performance based on some of the previous successful models. Since our model actually does not achieve very competitive performance in the boundary detection task yet still has a good overall performance on SQuAD 2.0 test, we are optimistic that our U-Net model is potentially capable of achieving better performance. Furthermore, our model has a simple structure and is easy to implement, therefore we believe that our model can be easily modified for various datasets.
Acknowledgement
================
We would like to thank Robin Jia, Pranav Rajpurkar for their help with SQuAD 2.0 submissions.
[^1]: Corresponding Author.
[^2]: <https://rajpurkar.github.io/SQuAD-explorer/>
|
---
abstract: 'Spin-dependent and spin-independent quark light-cone momentum distributions and structure functions are calculated for the nucleon. We utilize a modified Nambu$-$Jona-Lasinio model in which confinement is simulated by eliminating unphysical thresholds for nucleon decay into quarks. The nucleon bound state is obtained by solving the Faddeev equation in the quark-diquark approximation, where both scalar and axial-vector diquark channels are included. We find excellent agreement between our model results and empirical data.'
author:
- 'I.C. Cloët'
- 'W. Bentz'
- 'A.W. Thomas'
title: 'Nucleon quark distributions in a covariant quark-diquark model'
---
Introduction
============
The discovery in the late 1980’s by the European Muon Collaboration (EMC) that the fraction of the spin of the proton carried by the quarks is unexpectedly small [@Ashman:1987hv], caused much excitement in the nuclear and particle physics communities. The “proton spin crisis” prompted many new experiments, leading to major new insights into the spin structure of the proton. However, a thorough theoretical understanding of the non-perturbative parton distributions still remains an exciting challenge.
In this paper we calculate the spin-independent and spin-dependent quark distributions in the framework developed by Bentz, Thomas and collaborators, in which proper-time regularization is applied to the Nambu$-$Jona-Lasinio (NJL) model [@Nambu:1961tp] in order to simulate the effects of confinement [@Mineo:2003vc]. This model is attractive because of its covariance and the transparent description of spontaneous chiral symmetry breaking. We construct the nucleon as a bound state solution of the relativistic Faddeev equation [@Huang:1993yd; @Ishii:1995bu; @Oettel:2000jj; @Maris:2003vk] in the quark-diquark approximation [@Bentz:2001vc], where both scalar and axial-vector diquark channels are included. This quark-diquark description of the single nucleon has the further advantage that it can be extended to finite baryon density [@Cloet:2005rt]. We pay special attention to the polarized structure of the nucleon, comparing our results for the quark distributions with the empirical parameterizations.
Quark distributions
===================
The spin-dependent quark light-cone momentum distribution in the nucleon, at leading twist, is defined by Eq. (\[eqn:defQD\]), where $\psi_q$ is the quark field of flavour $q$, $x$ is the Bjorken scaling variable and the subscript $c$ reminds us that only connected matrix elements are included. $$\begin{gathered}
\D f_q(x) = p_- \, \int \frac{d\xi^-}{2\,\pi}\,e^{i\,x\,p^+\,\xi^-} \\
\langle p,s \lvert \ol{\psi}_q(0)\,\g^+\g_5 \psi_q(\xi^-)\rvert p,s\rangle_c.
\label{eqn:defQD}\end{gathered}$$ We normalize the nucleon state vector according to non-covariant light-cone normalization: $\langle p,s \lvert \ol{\psi}_q \,\g^+ \psi_q \rvert p,s\rangle_c = 3$. The spin-independent distribution, $f_q(x)$, is defined by the replacement $\gamma^+ \gamma_5 \to \gamma^+$ in Eq. (\[eqn:defQD\]). To determine the quark distributions in this model, it is convenient to express Eq. (\[eqn:defQD\]) in the form [@Jaffe:1985je; @Barone:2001sp] $$\begin{gathered}
\D f_q(x) = -i\int \frac{d^4 k}{\lf(2\pi\rg)^4} \\
\delta\!\lf(x - \frac{k_-}{p_-}\rg) \textrm{Tr}\biggl(\g^+\g_5\,M_q\lf(p,k\rg)\biggr),
\label{eqn:def2}\end{gathered}$$ where $M_q\lf(p,k\rg)$ is the quark two-point function in the bound nucleon. Hence, within any model that describes the nucleon as a bound state of quarks, the distribution functions can be associated with a straightforward Feynman diagram calculation.
The Feynman diagrams considered here are given in Fig. \[fig:feydiagrams\], where in our model the resulting distributions have no support for negative $x$. Therefore this is essentially a valence quark picture. By separating the isospin factors, the spin-dependent $u$ and $d$ distributions in the proton can be expressed as $$\begin{aligned}
\label{eqn:delu}
\D u_v(x) &= \D f^s_{q/N}(x) + \frac{1}{2}\,\D f^s_{q(D)/N}(x) + \frac{1}{3}\, \D f^a_{q/N}(x)\no \\
&\hs{6mm} + \frac{5}{6}\,\D f^a_{q(D)/N}(x) + \frac{1}{2\sqrt{3}} \D f_{q(D)/N}^m (x), \\
\label{eqn:deld}
\D d_v(x) &= \frac{1}{2}\,\D f^s_{q(D)/N}(x) + \frac{2}{3}\,\D f^a_{q/N}(x) \no \\
&\hs{6mm} + \frac{1}{6}\,\D f^a_{q(D)/N}(x) - \frac{1}{2\sqrt{3}} \D f_{q(D)/N}^m (x).\end{aligned}$$ The superscripts $s$, $a$ and $m$ refer to the scalar, axial-vector or mixing terms, respectively, the subscript $q/N$ implies a quark diagram and similarly $q(D)/N$ a diquark diagram. Because the scalar diquark has spin zero, we have $\D f^s_{q(D)/N}(x)=0$ and hence the polarization of the $d$-quark arises exclusively from the axial-vector and the mixing terms. Similar expressions hold for the spin-independent distributions, but in that case there is no mixing contribution ($f_{q(D)/N}^m=0$) [@Mineo:2002bg].
![Feynman diagrams representing the quark distributions in the nucleon, needed in the evaluation of Eq. (\[eqn:defQD\]). The single line represents the quark propagator and the double line the diquark $t$-matrix. The shaded oval denotes the quark-diquark vertex function and the operator insertion has the form $\gamma^+\gamma_5\,\delta\!\!\left(x - \frac{k_-}{p_-}\right)\frac{1}{2}\left(1 \pm \tau_z\right)$ for the spin-dependent distribution and $\g^+\g_5 \to \g^+$ for the spin-independent case.[]{data-label="fig:feydiagrams"}](Diagrams.eps){width="\columnwidth"}
Importantly, in this covariant framework, the Ward identities corresponding to number and momentum conservation are satisfied, guaranteeing the validity of the baryon number and momentum sum rules [@Mineo:1999eq; @Mineo:2003vc].
The nucleon in the NJL model
============================
The NJL model is a chiral effective quark theory that is characterized by a 4-Fermi contact interaction of the form, ${\cal L}_I = \sum_i G_i \,\lf(\ol{\psi}\,\G_i\,\psi\rg)^2$, where the $\G_i$ represent matrices in Dirac, colour and flavour space and $G_i$ are coupling constants [@Nambu:1961tp]. Applying Fierz transformations, the interaction Lagrangian can be decomposed into various interacting $q \bar{q}$ and $qq$ channels. Writing only those terms relevant to this discussion, we have $${\cal L} = \ol{\psi}\lf(i\!\! \not\!\pl - m\rg)\psi + {\cal L}_{I,\pi} + {\cal L}_{I,s} + {\cal L}_{I,a},$$ where $m$ is the current quark mass. The interaction terms are given by $$\begin{aligned}
&\hs{-1.5mm}{\cal L}_{I,\pi} = \frac{1}{2}\, G_\pi \lf(\lf(\ol{\psi}\psi\rg)^2 -
\lf(\ol{\psi}\,\g_5\vec{\tau}\,\psi \rg)^2\rg), \\
&\hs{-1.5mm}{\cal L}_{I,s} = G_s \Bigl(\ol{\psi}\,\g_5 C \tau_2 \beta^A\, \ol{\psi}^T\Bigr)
\Bigl(\psi^T\,C^{-1}\g_5 \tau_2 \beta^A\, \psi\Bigr),\\
&\hs{-1.5mm}{\cal L}_{I,a} = G_a \Bigl(\ol{\psi}\,\g_\mu C \tau_i\tau_2 \beta^A\, \ol{\psi}^T\Bigr)
\Bigl(\psi^T\,C^{-1}\g^{\mu} \tau_2\tau_i \beta^A\, \psi\Bigr),
\label{eqn:Lint}\end{aligned}$$ where $\beta^A = \sqrt{\frac{3}{2}}\,\lambda^A~(A=2,5,7)$ are the colour $\ol{3}$ matrices and $C = i\g_2\g_0$. The familiar term ${\cal L}_{I,\pi}$ generates the constituent quark mass, $M$, via the gap equation and the pion as a $q \bar{q}$ bound state. The terms ${\cal L}_{I,s}$ and ${\cal L}_{I,a}$ represent the interactions in the scalar ($J^{\pi} = 0^+, T = 0, \text{colour}\,\ol{3}$) and axial-vector ($J^{\pi} = 1^+, T = 1, \text{colour}\,\ol{3}$) diquark channels and are used to construct the nucleon as a quark-diquark bound state. The couplings $G_\pi$, $G_s$ and $G_a$ are related to the original couplings, $G_i$, via the Fierz transformation, but we use them here as free parameters which will be fixed by the properties of the pion and the nucleon.
Solving the appropriate Bethe-Salpeter equations, the standard NJL results for the diquark $t$-matrices are obtained [@Ishii:1995bu; @Mineo:2002bg]. As explained in Ref. [@Bentz:2001vc], these can be accurately approximated by the forms $$\begin{aligned}
\label{taus}
\tau_{s}(q) &= 4i \, G_s\, - \frac{i g_s} {q^2 - M_s^2}, \\
\tau_a^{\mu\nu}(q) &= 4i G_a\, g^{\mu\nu} - \frac{i g_a}{q^2 - M_a^2}
\left(g^{\mu\nu} - \frac{q^{\mu}q^{\nu}}{M_a^2} \right),
\label{taua}\end{aligned}$$ which we also use here. The masses of the diquarks $M_s, \, M_a$ and their couplings to the quarks $g_s, \, g_a$ are defined as the poles and residues of the appropriate full diquark $t$-matrices.
The nucleon (quark-diquark) $t$-matrix satisfies the Faddeev equation T = Z + Z\_NT = Z + T\_NZ, \[eqn:t\] where $Z$ is the quark exchange kernel and $\Pi_N$ the product of a quark propagator and a diquark $t$-matrix. In the non-covariant light-cone normalization used already in Eq. (\[eqn:defQD\]), the quark-diquark vertex function, $\Gamma_N$, is defined by the behaviour of $T$ near the pole $$\begin{aligned}
T \stackrel{p_+ \to \varepsilon_p}{\longrightarrow}
\frac{\Gamma_N\,\overline{\Gamma}_N}{p_+ - \varepsilon_p},
\label{pole}\end{aligned}$$ where $\varepsilon_p = \frac{M_N^2}{2p_-}$ is the light-cone energy. Substituting this result into Eq. (\[eqn:t\]) gives the homogeneous Faddeev equations for the vertex functions $$\begin{aligned}
\label{eqn:f}
\G_N = Z\,\Pi_N \, \G_N, \quad \text{and} \quad
\ol{\G}_N = \ol{\G}_N\, \Pi_N \,Z.\end{aligned}$$
For this investigation we restrict ourselves to the static approximation, where we neglect the momentum dependence of the quark exchange kernel, $Z$. Including both scalar and axial-vector diquark channels, $Z$ takes the following form in the colour singlet and isospin-$\tfrac{1}{2}$ channel: $$\begin{aligned}
Z = \frac{3}{M} \begin{pmatrix} 1 & \sqrt{3}\g_{\mu'}\g_5 \\
\sqrt{3}\g_5\g^{\mu} & -\g_{\mu'}\g^{\mu} \end{pmatrix}.\end{aligned}$$ The quantity $\Pi_N$ effectively becomes the quark-diquark bubble graph: $$\begin{aligned}
\Pi_N(p) &= \int \frac{d^4k}{(2\pi)^4}\, \tau(p-k)\, S(k),\end{aligned}$$ where $$\begin{aligned}
\tau(q) = \begin{pmatrix} \tau_s(q) & 0 \\ 0 & \tau_a^{\mu\nu}(q) \end{pmatrix}.
\label{tau}\end{aligned}$$ The eigenfunction of the kernel $K \equiv Z\,\Pi_N$, in Eq. (\[eqn:f\]), has the following form, up to normalization: $$\begin{aligned}
\G(p,s) = \begin{bmatrix} \a_1 \\ \a_2\,\frac{p^{\mu}}{M_N}\,\g_5\ + \a_3\,\g^\mu\g_5 \end{bmatrix}u_N(p,s),
\label{eqn:nvertex}\end{aligned}$$ where the upper and lower component refer to the scalar and axial-vector diquark channels, respectively and $u_N$ is a free Dirac spinor with mass $M_N$. We choose the normalization $\ol{u}_N u_N = 1 = \ol{\G}\G$.[^1] Inserting this form into Eq. (\[eqn:f\]) gives three homogeneous equations for the $\alpha$’s and the nucleon mass $M_N$ is determined by the requirement that the eigenvalue of $K$, in Eq. (\[eqn:f\]), equal 1.
The normalization of the vertex function follows from the definition given in Eq. (\[pole\]), we obtain $$\begin{aligned}
\Gamma_N(p,s) &= \sqrt{-Z_N\frac{M_N}{p_-}}\,\G(p,s),\end{aligned}$$ where $$\begin{aligned}
Z_N = \frac{p_-}{M_N}\,\frac{-1}{\G(p)\,\frac{\pl \Pi_N(p)}{\pl p_+}\,\G(p)}.\end{aligned}$$
As with any non-renormalizable theory a regularization prescription must be specified to fully define the model. We choose the proper-time regularization scheme [@Schwinger:1951nm; @Ebert:1996vx; @Hellstern:1997nv; @Bentz:2001vc], where loop integrals of products of propagators are evaluated by introducing Feynman parameters, Wick rotating and making the denominator replacement $$\frac{1}{X^n}
\longrightarrow
\frac{1}{(n-1)!}\,\int_{1/(\La_{UV})^2}^{1/(\La_{IR})^2}\,d\tau\,\tau^{n-1}\,e^{-\tau\,X},$$ where $\La_{IR}$ and $\La_{UV}$ are, respectively, ultraviolet and infrared cutoffs. The former has the effect of eliminating unphysical thresholds for hadron decay into quarks, hence simulating an important aspect of confinement [@Hellstern:1997nv].
Results
=======
The parameters of the model are $\La_{IR}$, $\La_{UV}$, $m$, $G_\pi$, $G_s$ and $G_a$. The infrared scale is expected to be of order $\La_{QCD}$ and we set it to $\La_{IR} = 0.28\,$GeV. This is slightly larger than our previous work [@Mineo:2003vc], because our studies of the saturation properties of nuclear matter favour this [@Cloet:2005rt]. The parameters $m$, $\La_{UV}$ and $G_\pi$ are determined by requiring $M=400\,$MeV via the gap equation, $f_\pi=93\,$MeV from the familiar one loop pion decay diagram and $m_\pi=140\,$MeV from the pole of the $q \overline{q}$ $t$-matrix in the pion channel. This gives $m=15.3\,$MeV, $\La_{UV} = 0.66\,$GeV and $G_\pi = 17.81\,$GeV$^{-2}$. The couplings $G_s$ and $G_a$ are determined by reproducing the nucleon mass $M_N = 940\,$MeV as the solution of Eq. (\[eqn:f\]) and satisfying the Bjorken sum rule within our model, where $g_A = 1.267$. We obtain $G_s = 8.41~$GeV$^{-2}$ and $G_a = 1.36~$GeV$^{-2}$. With these model parameters the diquark masses are $M_s= 0.65\,$GeV and $M_a=1.2\,$GeV and the coefficients in the nucleon vertex function, Eq.(\[eqn:nvertex\]), are $\lf(\a_1,\a_2,\a_3\rg) = \lf(-0.35,-0.0088,0.47\rg)$.
To compare the predictions of the model with experimental data as well as the empirical parameterizations, it is necessary to determine the model scale, $Q_0^2$. We do this by optimizing $Q_0^2$ such that the spin-independent distribution, $u_v(x)$, best reproduces the empirical parameterization after $Q^2$ evolution. We find a model scale of $Q_0^2 = 0.16~$GeV$^2$, which is typical of valence dominated models [@Mineo:2002bg; @Mineo:1999eq; @Schreiber:1991tc].
![Spin-independent valence $u$ and $d$ distributions multiplied by Bjorken $x$. There are three curves for each quark flavour, with the lower curve of each type representing the $d$ distribution. The dotted line is the model prediction at the NJL scale of $Q_0^2 = 0.16~\text{GeV}^2$ and the solid line is the result after QCD evolution to the scale $Q^2 = 5.0~\text{GeV}^2$. The dashed line is the empirical parametrization of Ref. [@Martin:2002dr], at the scale $Q^2 = 5.0~\text{GeV}^2$.[]{data-label="fig:SI_xup"}](SI_xupdown.ps){width="\columnwidth"}
![Spin-dependent valence $u$ and $d$ distributions multiplied by Bjorken $x$. The curves are as in Fig. \[fig:SI\_xup\], with empirical parameterizations taken from Ref. [@Hirai:2003pm].[]{data-label="fig:SD_xup"}](SD_xupdown.ps){width="\columnwidth"}
Results for the spin-independent and spin-dependent valence $u$ and $d$ distributions are presented in Figs. \[fig:SI\_xup\] and \[fig:SD\_xup\], respectively. We show the predictions at the model scale and after QCD evolution[^2] to $Q^2 = 5\,$GeV$^2$, where they are compared to empirical parameterizations. We find excellent agreement between the model results and the parameterizations. In comparison with the pure scalar model [@Mineo:2003vc; @Cloet:2005tq], the agreement has improved substantially, especially for the spin-dependent case.
Our model results for the first polarized moments are $\D \, u_v = 0.924$ and $\D \, d_v = -0.343$ which agree with the values $\D \, u_v = 0.926 \pm 0.014$ and $\D \, d_v = -0.341 \pm 0.018$ determined from the axial coupling constants of octet baryons discussed in Ref. [@Goto:1999by]. This emphasizes the importance of including axial-vector diquark correlations, since the pure scalar model would give a vanishing $\D \,d_v$ and a somewhat smaller $\D \,u_v$. The spin sum in our model is $\D \S = 0.581$, which is smaller than the result of the pure scalar model, but still somewhat larger than the accepted value of $\D \S = 0.213 \pm 0.138$ [@Hirai:2003pm]. This discrepancy primarily reflects the absence of the $U$(1) axial anomaly [@Altarelli:1988nr; @Efremov:1988zh].
The behaviour of structure function and hence quark distribution ratios at large $x$ has been an area of considerable debate [@Melnitchouk:1995fc; @Zheng:2004ce] and is one of the regions where perturbative QCD (pQCD) offers firm predictions [@Farrar:1975yb]. Experimentally, the ratio $d(x)/u(x)$ is surprisingly poorly known [@Botje:1999dj]. In the limit $x \to 1$ it is thought to lie somewhere between $0$, the prediction based on scalar diquark dominance [@Close:1973xw] and $\tfrac{1}{5}$, the pQCD result [@Farrar:1975yb]. Analysis in Ref. [@Melnitchouk:1995fc] favours the pQCD prediction. The same predictions also hold for the spin-dependent ratio, $\Delta d(x)/\Delta u(x)$, as $x$ approaches 1.
![Mixed flavour ratios for spin-independent and spin-dependent distributions. There are two curves for each ratio, with the lower curves the polarized result. The curves are as in Fig. \[fig:SI\_xup\], with spin-independent parameterizations taken from Ref. [@Martin:2002dr] and the spin-dependent from Ref. [@Hirai:2003pm][]{data-label="fig:SD_SI_du"}](Down_Up.ps){width="\columnwidth"}
In Fig. \[fig:SD\_SI\_du\] we plot our results for the ratios $d_v(x)/u_v(x)$ and $\D d_v(x)/\D u_v(x)$, together with the ratios of the empirical distributions. The $x \to 1$ limit of the spin-independent ratio is in agreement with the pQCD result. The spin-dependent ratio, however, approaches $\sim -\tfrac{1}{16}$, the opposite sign to the pQCD prediction. Although the empirical parameterizations are constrained to give $0$ for these ratios as $x \to 1$, we note that the systematic errors in both empirical ratios are very large in the region $x \gtrsim 0.5$ [@Blumlein:2002be; @Martin:2002aw; @Martin:2003sk; @Hirai:2003pm].
![Single flavour ratios $\lf(\D q + \D \ol{q}\rg)/\lf(q + \ol{q}\rg)$ where $q \in \lf(u,d\rg)$, at the scale $Q^2 = 5.0\,\text{GeV}^2$. The lower curve represents the $d$-quark distribution. The experimental results are from Hall A at Jefferson Lab [@Zheng:2004ce] (solid squares) and Hermes [@Ackerstaff:1999ey] (solid stars).[]{data-label="fig:SD_SI_du_total"}](Del_Down_Up.ps){width="\columnwidth"}
It is important to note that the pQCD predictions for the mixed flavour ratios are somewhat model dependent, as assumptions have to be made about the relative strengths of the $u$ and $d$ contributions to the nucleon wavefunction. A more rigorous pQCD prediction, relying only on helicity conservation, is possible for the single flavour ratios $\D u(x)/u(x)$ and $\D d(x)/d(x)$. Perturbative QCD predicts that both these ratios should approach 1 for large $x$, which would require a change of sign in the $\D d$ distribution.
In Fig. \[fig:SD\_SI\_du\_total\] we plot our results for the ratios $\lf(\D q + \D \ol{q}\rg)/\lf(q + \ol{q}\rg)$ where $q \in \lf(u,d\rg)$. Since we wish to compare these ratios directly to recent experimental data, we include sea quark distributions generated through the $Q^2$ evolution. In the $x \to 1$ limit our model ratios approach $\approx 0.8$ for the $u$-quark and $\approx -0.25$ for the $d$-quark. This seeming contradiction to pQCD has also been suggested by recent experiments by the Jefferson Lab Hall A collaboration [@Zheng:2004ce; @Zheng:2003un], with our predictions consistent with their experimental results. This data is also shown in Fig. \[fig:SD\_SI\_du\_total\].
![Polarized structure functions $g_{1p}$ and $g_{1n}$ at $Q^2 = 5\,$GeV$^
2$. The solid line is the model prediction, with the lower curve corresponding to $g_{1n}$. The shaded areas represent the empirical parameterizations with uncertainties of Ref. [@Blumlein:2002be], at the same scale. The experimental data, with $1 \leq Q^2 \leq 10\,$GeV$^2$, is from SMC [@Adeva:1998vw] (open stars), SLAC E143 [@Abe:1998wq] (open circles) and JLab [@Zheng:2004ce] (solid squares).[]{data-label="fig:g1pg1n"}](g1.ps){width="\columnwidth"}
In Fig. \[fig:g1pg1n\] we give our results for the spin-dependent structure functions $g_{1p}(x)$ and $g_{1n}(x)$. The parameterizations of Ref. [@Blumlein:2002be] are also included as the shaded areas, which indicate the empirical uncertainties. Our results compare well with the empirical parameterizations, agreeing within uncertainties for the region $x \gtrsim 0.25$. Comparison with experiment is also favorable, although the experimental determination for $g_{1n}(x)$ is less certain.
Model results for the asymmetries $A_{1p}(x)$ and $A_{1n}(x)$ are given in Fig. \[fig:A1pA1n\], where we see good agreement in the valence region. However, for small $x$, $A_{1p}(x)$ is slightly too large, because of the enhancement of $g_{1p}(x)$ in the same region. This is most likely associated with the omission of the effects of the axial anomaly in the present work. It is also clear from the experimental data that the uncertainties in these ratios at large $x$, are still significant.
![Structure function ratios $A_{1p}$ and $A_{1n}$, at the scale $Q^2 = 5\
,$GeV$^2$. The solid line is the model prediction, with the lower line corresponding to $A_
{1n}$. The experimental points represent data with $1 \leq Q^2 \leq 10\,$GeV$^2$ from Hermes [@Airapetian:1998wi] (closed star) , SMC [@Adeva:1998vw] (open star), E143 [@Abe:1998wq] (open circles) and JLab [@Zheng:2004ce] (solid squares). We do not plot the E155 [@Anthony:2000fn] results as they are similar to those from E143.[]{data-label="fig:A1pA1n"}](A1.ps){width="\columnwidth"}
Conclusion
==========
Using a covariant quark-diquark model for the nucleon, including both scalar and axial-vector diquark channels, we calculated the spin-independent and spin-dependent quark light-cone momentum distributions and structure functions. A key feature of the framework is that it produces quark distributions that have the correct support and obey the number and momentum sum rules. The model also incorporates important aspects of confinement by eliminating unphysical thresholds for nucleon decay into quarks.
Highlights of our results are obtaining values for the polarized first moments of the quark distributions $\D u_v = 0.924$ and $\D d_v = -0.343$, in agreement with those obtained from axial couplings of octet baryons. We also obtain excellent agreement with empirical parameterizations of the valence quark distributions. We paid special attention to the single flavour ratios $\lf(\D q + \D \ol{q}\rg)/\lf(q + \ol{q}\rg)$ and the asymmetries $A_{1p}$ and $A_{1n}$, finding good agreement with recent experimental results from JLab.
These results indicate that diquark correlations are an essential feature of the non-perturbative structure of the nucleon. In particular, the admixture of axial-vector diquarks, though small, is essential to obtain the observed agreement with empirical data.
Finally, we would like to mention that a very important advantage of this covariant quark-diquark model is that it can be readily extended to the case of finite nucleon density. The results presented in this paper strongly suggest that this model should provide a reliable basis from which to begin investigation of the medium modifications of both spin-independent and spin-dependent structure functions.
Acknowledgments {#acknowledgments .unnumbered}
===============
IC and WB thank W. Melnitchouk and H. Mineo for interesting and helpful discussions. This work was supported by the Australian Research Council and DOE contract DE-AC05-84ER40150, under which SURA operates Jefferson Lab, and by the Grant in Aid for Scientific Research of the Japanese Ministry of Education, Culture, Sports, Science and Technology, Project No. C2-16540267.
[40]{}
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[^1]: The conjugate vertex function, $\ol{\G}$, which is a left eigenfunction of $\ol{K} \equiv \Pi_N\,Z$, is obtained by taking the ordinary hermitian conjugate of $\Gamma$ and introducing a minus sign for the axial-vector components.
[^2]: We utilize the computer program of Ref. [@Miyama:1995bd] for the spin-independent case and of Ref. [@Hirai:1997gb] for the spin-dependent case. We choose DGLAP evolution with $N_f=3$, $\Lambda_{\text{QCD}}=250~$MeV in the $\overline{\text{MS}}$ renormalization scheme up to NLO.
|
---
abstract: 'Zoom-whirl behavior has the reputation of being a rare phenomenon. The concern has been that gravitational radiation would drain angular momentum so rapidly that generic orbits would circularize before zoom-whirl behavior could play out, and only rare highly tuned orbits would retain their imprint. Using full numerical relativity, we catch zoom-whirl behavior despite dissipation. The larger the mass ratio, the longer the pair can spend in orbit before merging and therefore the more zooms and whirls seen. Larger spins also enhance zoom-whirliness. An important implication is that these eccentric orbits can merge during a whirl phase, before enough angular momentum has been lost to truly circularize the orbit. Waveforms will be modulated by the harmonics of zoom-whirls, showing quiet phases during zooms and louder glitches during whirls.'
author:
- 'James Healy${}^{1}$, Janna Levin${}^{2,3}$, and Deirdre Shoemaker${}^{4}$'
bibliography:
- 'gr.bib'
- 'nr.bib'
title: 'Zoom-Whirl Orbits in Black Hole Binaries'
---
Kepler’s laws describe closed elliptical planetary motions. A small relativistic correction accounts for the tiny, anomalous precession of Mercury’s perihelion. If the sun is replaced by a black hole, the geodesic motions can zoom and whirl in an extreme form of precession – whirling around the center of mass in nearly circular inspiral before zooming out along elliptical leaves. Zoom-whirl behavior is characteristic of strong relativity and could potentially be detected in the harmonics of the gravitational waves generated.
There are astrophysical settings that could populate eccentric merges, such as dense galactic nuclei [@O'Leary:2008xt] or globular clusters [@Kocsis:2006hq; @wen2003]. Consequently, it is an important astrophysical question to ask: Can zoom-whirl behavior, an intrinsically eccentric phenomenon, survive the dissipative drain of gravitational radiation? In this article we report on results of numerical relativity that show zoom-whirl behavior in comparable mass binaries, answering this question in the affirmative.
As there is no analytic description of the curved spacetime around two black holes, we rely either on analytic approximations or on numerical relativity to describe comparable black hole pairs. Zoom-whirl behavior has been studied in extreme-mass ratio inspirals [@2002PhRvD..66d4002G; @2004PhRvD..69h2005B; @Burko:2006ua; @Haas:2007kz] and was recently found in an analytic approximation, specifically conservative Post-Newtonian (PN) approximations to black hole binaries [@levin2008:2; @grossman2008]. Now we find zoom-whirl orbits in full numerical relativity of spinning pairs. Zoom-whirl behavior has already been found in numerical relativity for equal-mass, nonspinning binaries in [@pretorius2007]. In that work, the initial conditions were carefully tuned to find a special orbit, the separatrix between bound orbital motion and plunge. The separatrix is studied in detail in the PN approximation in [@grossman2008] and an analytic solution for the separatrix in Kerr systems was found in [@Levin:2008yp; @PerezGiz:2008yq].
In this Letter we show that zoom-whirl behavior in spinning pairs is a common feature of eccentric orbits [@Levin:2008yp; @levin2008:2; @grossman2008], despite dissipation. In particular, zoom-whirl orbits happen well away from the separatrices and so do not in general require fine tuning of initial conditions (see also [@sperhake-09; @gold]). Due to the computational expense of running these simulations, a full scan of parameters is not possible. To focus our investigation, we rely on analytic approximations to estimate good initial conditions and we then run full numerical simulations to easily locate zoom-whirl inspirals. The further utility of the analytic estimates is the transparency of interpretation.
The anatomy of zoom-whirl behavior was quantified in Ref. [@levin2008] where it was shown that every orbit of a given $L$ can be described by one number that specifies the precession of the orbit per radial cycle from apastron to apastron. The amount by which an orbit will precess, that is, overshoot the previous apastron is $$\Delta \phi_{precess}=2\pi q\quad {\rm where}\quad q=w+\frac{v}{z} \quad .$$ A perfectly periodic orbit looks like a closed 1-leaf clover or 2-leaf clover or 3-leaf clover, or $z$-leaf clover. And each periodic orbit corresponds to a rational $q$ made up of $w$ integer number of nearly circular whirls close to perihelion per leaf in the $z$-leaf clover. The vertex $v$ is more subtle and indicates the order in which the $z$-leaves are traced out. So, a simple 3-leaf clover for instance is a $q=1/3$ ($w=0,v=1,z=3$) and precesses past the previous apastra by $\Delta \phi= 2\pi q=2\pi/3$ per radial cycle. A 3-leaf clover that skips a leaf in the pattern each time corresponds to $q=2/3$ ($w=0,v=2,z=3$) and precesses by $\Delta \phi= 2\pi q=4\pi/3$ per radial cycle. Therefore, $q$ quantifies zoom-whirl behavior, the integer part signals the whirls per leaf and the fractional part signals the number and order of the leaves.
Another way to interpret the number is as the ratio of frequencies, $q={\omega_\phi}/{\omega_r}$, where $\omega_\phi$ is the average of the angular frequency per radial cycle and $\omega_r=2\pi/T_r$ where $T_r$ is the time between apastra. For periodic orbits the frequencies are rationally related and the orbit will eventually close. Whirls accumulate near perihelion simply because the angular velocity is greatest on closest approach and the circumference smallest.
Generic orbits are not periodic and do not correspond to rational $q$. However, any generic orbit can be approximated by a nearby periodic, just as any irrational number can be approximated by a nearby rational number. Kepler’s ellipse corresponds to $q=0$ since it does not precess at all. Mercury’s precessing orbit corresponds to $q\sim 10^{-7}$ since it precesses very little. Technically, zoom-whirl behavior corresponds to $q>1$, so there is at least one whirl, although we will generally be interested in any substantial precession, say $q>1/4$.
[*If*]{} angular momentum is fixed, $q$ decreases monotonically with decreasing energy. However angular momentum decreases, along with energy, as gravitational radiation is emitted and the evolution of $q$ will not necessarily be monotonic. In the case of comparable mass black hole pairs, $q$ will change quickly due to the rapid losses to gravitational waves as evident in the simulations.
The range of zoom-whirl behavior is most easily targeted with an effective potential picture. In Ref. [@levin2008; @grossman2008] an effective potential method was used to describe the center-of-mass motion of black hole binaries in the conservative 3PN Hamiltonian description with spin-orbit coupling included. There it was shown that at the turning points of the orbital motion, the Hamiltonian itself is an effective potential and the motion can be read off as easily as interpreting a ball rolling along hills.
![The 3PN effective potential. Top: $L_{IBCO}\sim 4.4$. The straight line corresponds to the orbit $r_a=30$. Bottom: $L\sim 4.1$, for which the orbit with $r_a=30M$ is the separatrix.[]{data-label="Vm.25Sm8"}](Vm.333Sm3Libco.eps "fig:"){width="50mm"} ![The 3PN effective potential. Top: $L_{IBCO}\sim 4.4$. The straight line corresponds to the orbit $r_a=30$. Bottom: $L\sim 4.1$, for which the orbit with $r_a=30M$ is the separatrix.[]{data-label="Vm.25Sm8"}](m.333Sm3r30.eps "fig:"){width="30mm"} ![The 3PN effective potential. Top: $L_{IBCO}\sim 4.4$. The straight line corresponds to the orbit $r_a=30$. Bottom: $L\sim 4.1$, for which the orbit with $r_a=30M$ is the separatrix.[]{data-label="Vm.25Sm8"}](Vm.333Sm3sep.eps "fig:"){width="50mm"} ![The 3PN effective potential. Top: $L_{IBCO}\sim 4.4$. The straight line corresponds to the orbit $r_a=30$. Bottom: $L\sim 4.1$, for which the orbit with $r_a=30M$ is the separatrix.[]{data-label="Vm.25Sm8"}](m.333Sm3sep.eps "fig:"){width="30mm"}
Consider the effective potential on the top left of Fig. \[Vm.25Sm8\] from the conservative 3PN Hamiltonian [@buonanno2006; @damoureob2001; @levin2008; @grossman2008]. There is clearly a minimum of that potential which defines a stable circular orbit. An orbit energetically above the stable circle will execute elliptical precessions as shown on the top right. Note these precessions are much more extreme than Mercury’s with $q> \frac{1}{3}$. It looks like a precession around a three leaf clover.
The aperiodic orbit will eventually fill out an annulus.
Also evident is a maximum of the potential, an unstable circular orbit. For this angular momentum the unstable circular orbit is marginally, energetically bound since $V_{\rm eff}$ just skims zero there, and has been called an innermost bound circular orbit (IBCO). An orbit at rest at infinity would asymptotically approach this circular orbit. And though initially of eccentricity one, $e=1$, this orbit whirls an infinite number of times as it climbs the potential toward the unstable circle at the top. This separatrix is the infinite whirl orbit, $q=\infty$, and is also known as a homoclinic orbit of eccentricity one.
For angular momenta below this critical value, the unstable circular orbit marches down in energy. For each unstable circle, there is a corresponding separatrix of decreasing eccentricity. As an example, on the bottom of Fig. \[Vm.25Sm8\], the separatrix between bound motion and plunge has apastron $r_a=30M$ and asymptotically approaches the unstable circular orbit in the infinite whirl limit. So this is the $q=\infty$ orbit for $L\sim 4.1$, where throughout we measure angular momentum in units of $\mu M$ and $M=m_1+m_2$ while $\mu=m_1 m_2/M$. The homoclinic separatrix is plotted alongside the corresponding effective potential on the bottom right of Fig. \[Vm.25Sm8\].
We expect to see zoom-whirl behavior until the unstable circular orbit and the stable circular orbit merge at the ISCO (innermost stable circular orbit), the separatrix with $e=0$. Quasi-circular inspiral plunges at the ISCO and so this inflection point in the effective potential has received preferential attention. However, orbits that maintain an eccentricity during inspiral will merge by rolling over the top of the potential, through a homoclinic separatrix, behavior we observe in our simulations.
Roughly then, for given external parameters ($m_2/m_1,\bsone,\bstwo$), zoom-whirl behavior should be sought in the range $L_{ISCO}<L<L_{IBCO}$. One more initial condition needs to be specified to define an orbit and that could be either the energy or the apastron. We’ll choose to fix the apastron. We’ll fill in the details for a fiducial example and then flush out more general trends we have observed.
The procedure is the following: (1) choose external parameters namely mass ratio and spins ($m_2/m_1,\bsone,\bstwo$), (2) render the effective potential for the 3PN Hamiltonian with spin-orbit couping, (3) glean the range of $L$ for which there will be zoom-whirl behavior and (4) run simulations for this range of initial conditions.
For our fiducial example we take mass ratio $m_2/m_1=1/3$ and spin magnitudes $S_1/m_1^2=S_2/m_2^2=0.3$ with spins antialigned with the orbital angular momentum. Antialigned spins, like aligned spins, will retain the orbital motion in the equatorial plane making it easier to distinguish whirl precession. Other than the choice to retain equatorial motion for transparency, there is nothing special about the fiducial configuration. We consider a set of orbits all of which begin at apastron $r_a=30M$. Since the simulations are so costly, we tighten the range to the more realistic values of angular momenta below $L_{IBCO}$ and above the value of the angular momentum at which $r=r_a$ is the apastron of a homoclinic orbit. Specifically, we look between the values represented by the top and bottom of Fig. \[Vm.25Sm8\]; $4.1<L<4.4$.
There are a few reasons why the actual range for numerical relativity will be offset from this 3PN prediction. For one, dissipation ensures that $L$ changes as the orbit evolves. This is equivalent to the effective potential dropping as the orbit evolves. For another, the 3PN approximation is by definition not exact and its predictions are not to be overstated in this strong-field regime. Finally, although spin-orbit coupling is included in our analysis, spin-spin coupling terms are not included since they introduce angular dependencies that spoil the effective potential [@levin2008; @grossman2008]. Since spin-spin terms are small, their effect will presumably result in a small perturbation to our 3PN effective potential. Our range is intended only as a guide and apparently does well enough, as we’ll see.
In between this range, all orbits with our external parameters $(m_2/m_1=1/3,S_1/m_1^2=S_1/m_2^2=0.3)$, antialigned, and an apastron of $r_a=30M$, will show zoom-whirl behavior with $1/3<q<\infty$. This is the story told by the conservative dynamics. To test the survival of zoom-whirl orbits under the dissipative effects of radiation reaction we turn to full numerical relativity now.
The runs in Fig. \[NR\] were simulated using the Georgia Tech `MayaKranc` code that uses the same computational infrastructure and methodology as in previous studies [@2008arXiv0802.2520W; @2007ApJ...661..430H; @2007CQGra..24...33H; @2009arXiv0905.3914H; @2009PhRvL.102d1101H], namely a Baumgarte-Shapiro-Shibata-Nakamura code with moving puncture gauge conditions [@campanelli2006; @baker2005] using the `Kranc` code generator [@Husa:2004ip]. The black-hole encounters are initiated with Bowen-York initial data [@Bowen:1980yu]. The black holes are located on the $x$-axis: bh$_\pm$ are located at $x_+ = 7.5 M$ and $x_- = -22.5 M$ where $m_+ =3 M/4$ and $m_-= M/4$. The spatial finite differencing is sixth order. We used eleven levels of refinement with Carpet [@Schnetter-etal-03b], a mesh refinement package in Cactus [@cactus_web]. The finest resolution is $M/143$ and the outer boundary is at $287M$. The total initial orbital angular momentum, is varied between the values of $3.9$ and $4.1$. The range is shifted from the conservative PN value as expected, yet zoom-whirl behavior was comfortably found given that initial prediction.
Fig. \[NR\] shows three different orbits and their waveforms corresponding to different initial values of $L$. The largest value of $L$ shows zoom-whirl behavior characterized by $q\sim 1/2$ before merging. The middle value of $L$ has a $q\sim 1$ while the smallest value of $L$ we show is very close to the separatrix and whirls nearly twice. We caution that we are only loosely reading off these values since $q$ is changing so rapidly during inspiral.
As expected, the pairs merge by rolling over the top of the effective potential (Fig. \[Vm.25Sm8\]). We emphasize that the nearly circular pattern of the whirl phase is not equivalent to full circularization of the orbit. In other words, quite importantly, the orbits shown in Fig. \[NR\] do not merge through the ISCO but rather roll over the top of the potential, merging through a whirl. To compare with the language in a previous paper [@Hinder:2007qu], some of those eccentric orbits were circular because they merged through a whirl phase and not because they merged through the ISCO. This could be relevant for initial conditions for the ringdown phase.
The waveforms show distinctly quiet phases during the highly elliptical zooms followed by louder glitches during the nearly circular whirls. The distinctive spikes of zoom-whirls are directly related to their rational number $q$. This suggests that zoom-whirl orbits could be dug out of the data using algorithms suited for burst searches, perhaps in conjunction with more targeted template searches.
Fig. \[relsep\] plots the relative orbital separation of the black holes versus phase for our three cases of $L$. The plot illustrates the rapid merger of the black holes as $L$ decreases. The figure also shows that inspiral ends and plunge begins in the simulations for radii $\sim 5M$. In line with the numerical results, the effective potential picture in Fig. \[Vm.25Sm8\] predicts whirls around periastron, $\sim 5M$, which is much less than the PN predicted value of the ISCO, $r_{ISCO}\sim 8.8M$, for a mass ratio of 1/3 and these spins. The figure therefore confirms that the zoom-whirl pair merge near a whirl phase and not near the ISCO. The top line, corresponding to $L=4.1$, shows the three zooms before merger that we see in frame 1 of Fig. \[NR\]. The bottom line, corresponding to $L=3.915$ and frame 3 of Fig. \[NR\] is almost homoclinic with no zooms, precessing around the unstable circular orbit before merging. In general, we conclude from the PN approximation that zoom-whirl behavior has better odds at surviving dissipation and leaving a mark in the gravitational waveform for (1) more disparate values of $m_2/m_1$ because of the slower dissipation time, (2) larger spin magnitudes, all else being equal, because of the greater frame dragging effect, (3) larger initial apastra because of the longer orbital lifetime. There is less zoom-whirl behavior in equal-mass nonspinning pairs, hence the importance of sticking closer to the separatrices [@pretorius2007]. Yet more generally, zoom-whirl orbits ought to extend down below the separatrices. In the future, we intend to extend this work by pushing the numerical simulations to large mass ratio and extracting a measure of $\dot q$.
For any zoom-whirl pair, the glitchy waveforms followed by longer quiescent phases are highly distinctive and beg for tailored search algorithms. Furthermore, the loud glitches could nudge these highly precessing orbits into an optimistic position for early direct detection.
[**Acknowledgements:**]{} The authors wish to acknowledge NSF grants AST-0908365, PHY-0925345, PHY-0653303 and TG-PHY060013N that supported this work. We also thank Ian Hinder, Frank Herrmann,Tanja Bode and Pablo Laguna for contributions to the `MayaKranc` code.
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abstract: 'Recently, many deep neural network (DNN) based modulation classification schemes have been proposed in the literature. We have evaluated the robustness of two famous such modulation classifiers (based on the techniques of convolutional neural networks and long short term memory) against adversarial machine learning attacks in black-box settings. We have used Carlini & Wagner (C-W) attack for performing the adversarial attack. To the best of our knowledge, the robustness of these modulation classifiers have not been evaluated through C-W attack before. Our results clearly indicate that state-of-art deep machine learning based modulation classifiers are not robust against adversarial attacks.'
author:
- Muhammad Usama
- Junaid Qadir
- 'Ala Al-Fuqaha'
title: 'Black-box Adversarial ML Attack on Modulation Classification'
---
Introduction
============
Machine learning (ML) especially deep ML schemes have beaten human-level performance in many computer vision, language, and speech processing tasks which were considered impossible a decade ago. This success of ML schemes has inspired the ideas of self-driving networks [@feamster2017and] and knowledge defined networking [@mestres2017knowledge] where ML schemes are profoundly utilized to ensure automation and control of networking tasks such as dynamic resource allocation, modulation classification, network traffic classification, etc.
Despite the success of ML in different modern communication and data networking applications, there are some pitfalls in the fundamental assumptions of ML schemes which can be exploited by the adversaries to craft adversarial examples in order to compromise the ML-based system. An *adversarial example* is defined as an input to the ML model specially crafted by an adversary by adding a small imperceptible perturbation to the input sample to compromise the performance of the ML model. Mathematically, an adversarial example $x^*$ can be formed by adding a typically-imperceptible perturbation $\delta$ to the legitimate test example $x$ of the deployed trained classifier $f(.)$. The perturbation $\delta$ is computed by approximating the following nonlinear optimization problem provided in equation 1 where $t$ is the targeted class in case of a targeted attack or any other wrong class is the case of untargeted attack.
$$\label{eq1}
x^* = x + \arg \underset{\eta{_x}}{\text{min}} \{\|\eta\|: f(x + \eta) = t\}$$
Adversarial examples are possible because of two major faulty assumptions in ML schemes. *Firstly*, the underlying data distribution experienced during the training phase of the ML model will also be encountered in the testing phase. This data stationarity is not valid for most of the real world cases and the void created by following this assumption is exploited by the adversary for crafting the adversarial examples. *Secondly*, most of the ML schemes are based on the empirical risk minimization (ERM), which is an approximation of the actual unknown probability distribution. The ERM has an associated error with it which can be exploited by the adversary to make an adversarial example.
Adversarial attacks can be classified broadly into *white-box* and *black-box* attacks based on the knowledge of the adversary about the deployed ML model. In a *white-box attack*, it is assumed that adversary has complete knowledge (hyperparameters, test data, etc.) of the deployed model whereas in a *black-box attack* no such knowledge is assumed and it is assumed that the adversary can only act as a standard user and query the system for a response.
In this paper, we have taken modulation classification (which is an important component of modern communication and data networks) as a proxy of functional areas of cognitive self-driving networks. We have performed a black-box adversarial attack on DNN-based modulation classification to highlight the brittleness of ML schemes utilized in cognitive self-driving networks.
Related Work
============
There does not exist much literature on adversarial attacks on modulation classification. Recently, Sadeghi et al. [@sadeghi2018adversarial] used a variant of fast gradient sign method (FGSM) attack [@goodfellow2014explaining] on modulation classification on CNN-based modulation classification to highlight the threat of the adversarial examples. FGSM is an adversarial sample crafting algorithm where the adversarial perturbation is calculated by taking a gradient step in the direction of the sign of the gradient of test example. Kokalj et al. [@kokalj2019adversarial] also crafted the adversarial examples for modulation classification by using the FGSM perturbation generation algorithm. Most of the available results on the application of the adversarial attacks are reported by using the FGSM attack.
A shortcoming with the FGSM attack is its lack of optimality in adversarial perturbation generation as FGSM was designed to quickly craft adversarial examples irrespective of the optimality and the size of the perturbation in the test example. To overcome the lack of optimality and to highlight that optimal adversarial example for modulation classification can be crafted we have used Carlini & Wagner (C-W) attack [@carlini2017towards] where the adversarial examples are crafted using the following optimization process provided in equation \[eq3\].
$$\label{eq3}
\begin{split}
\underset{\eta}{\text{minimize}} \quad \|\eta\|{_\mathcal{P}} + c . g(x{^*}) \\
\text{such that} \quad x{^*} \in [0, 1]{^n}
\end{split}$$
{width="50.00000%"}
Black-box Adversarial Attack Procedure
======================================
In this section, we will provide our black-box adversarial attack procedure (illustrated in Figure 1). The steps followed are: **1)** the adversary queries the deployed modulation classifier with test examples; **2)** the deployed modulation classifier provides a labeled response to the adversary considering the adversary as a normal user; **3)** the adversary stores the query-response pair in a database (which is later used as a substitute dataset for training a surrogate DNN); **4)** once sufficient data is collected in the adversarial database, the adversary constructs a fully connected DNN model and trains it for suitable classification performance; **5)** once the surrogate DNN is trained, the adversary launches a C-W attack on the surrogate DNN for crafting adversarial examples that compromises the performance of the surrogate DNN model; **6)** adversarial examples that compromises the performance of surrogate DNN-model are then transferred to black-box DL-based modulation classifier which according to the transferability property of adversarial examples will compromise the performance of DL-based modulation classifier.
Since we are performing this experiment in lab settings, we have opted for training two modulation classifiers based on CNN and LSTM and then considered them as black-box models. We have used highly-cited GNU radio ML RML2016.10a dataset [@o2016radio] which provides 11 digital and analog modulation schemes on the SNR ranging from -20 dB to 18dB. We have used only 10% of the test examples to construct the surrogate classifier and then performed C-W attack the performance of the surrogate DNN model before and after the attack is provided in Figure 1. Once the adversarial attack on surrogate DNN is completed, we have transferred the adversarial examples that evaded the surrogate DNN to black-box modulation classifier by leveraging the transferability property of adversarial ML. The performance impact of the adversarial attack is provided in Figures 1 and 2. A clear drop in the accuracy of the modulation classifier after the adversarial attack highlights that our method of performing black-box adversarial attack has successfully compromised the performance crafted adversarial examples.
Conclusions
===========
In this paper, we have highlighted the lack of robustness in deep learning based modulation classification by performing a black-box adversarial attack on CNN and LSTM based modulation classifiers. We have used a surrogate deep neural network for crafting adversarial examples and then showed that adversarial examples crafted for modulation classification are transferable to other deep learning based models. We have achieved a 60% performance drop in both CNN and LSTM based modulation classification.
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\#1 \#1[\#1]{}\#1 \#1 \#1 \#1 \#1[\#1]{} \#1[\#1]{}
[carlini2017towards]{} . . In . IEEE, .
[feamster2017and]{} . . ().
[goodfellow2014explaining]{} . . ().
[kokalj2019adversarial]{} . . ().
[mestres2017knowledge]{} . . , (), .
[o2016radio]{} . . In , Vol. .
[sadeghi2018adversarial]{} . . , (), .
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[**Explicit Evaluations of Matrix-variate Gamma**]{}\
.2cm[**and Beta Integrals in the Real and Complex Cases**]{}\
.3cm[A.M. Mathai]{}\
.2cm[Director, Centre for Mathematical Sciences India]{}\
.1cm[\[Arunapuram P.O., Palai, Kerala-686574, Kerala, India\]]{}\
[email protected] , Phone:91+9495427558\
.2cm[and]{}\
.2cm[Emeritus Professor of Mathematics and Statistics, McGill University Canada;\
[email protected]]{}\
.1cm[\[805 Sherbrooke Street West, Montreal, Quebec, Canada, H3A2K6\]]{}
.5cm[**Abstract**]{} .3cm Matrix transformations in terms of triangular matrices is the easiest method of evaluating matrix-variate gamma and beta integrals in the real and complex cases. Here we give several procedures of explicit evaluation of gamma and beta integrals in the general real and complex situations. The procedure also reveals the structure of these matrix-variate integrals. Apart from the evaluation of matrix-variate gamma and beta integrals, the procedure can also be applied to evaluate such integrals explicitly in similar situations. Various methods described here will be useful to those who are working on integrals involving real-valued scalar functions of matrix argument in general and gamma and beta integrals in particular.
.3cm[**Keywords**]{}.3cm Matrix-variate gamma integral, matrix-variate beta integrals, explicit evaluations, real and complex cases, partitioned matrices and determinants.
.3cmMathematics Subject Classification: 15B57, 30E20,60B20, 62E15
.5cm[**1..3cm Introduction**]{} .3cm First we consider matrix-variate gamma integrals in the real case, then we look at matrix-variate type-1 beta integrals in the real case. The procedure is parallel in the matrix-variate type-2 beta integrals. Then we look at all these in the complex domain.
.3cm[**1.1..3cm Real matrix variate gamma integral**]{} .3cm Matrix-variate gamma integral is a very popular integral in many areas. A particular case is the most popular Wishart density in multivariate statistical analysis. Let $X$ be a $p\times p$ real symmetric and positive definite matrix of mathematical or random variables. Consider the real-valued function of matrix argument $$f(X)=C~|X|^{\alpha-\frac{p+1}{2}}{\rm e}^{-{\rm tr}(BX)}\eqno(1.1)$$where $C$ is a constant, $|(\cdot)|$ denotes the determinant of $(\cdot)$ and ${\rm tr}(\cdot)$ denotes the trace of the matrix $(\cdot)$. All matrices appearing in this article are $p\times p$ unless stated otherwise. When $X$ is real and positive definite, $X>O$, then $f(X)$ in (1.1) represents a real matrix-variate gamma density when $C=\frac{|B|^{\alpha}}{\Gamma_p(\alpha)}$ where $B>O$ is a constant matrix and $$\Gamma_p(\alpha)=\pi^{\frac{p(p-1)}{4}}\Gamma(\alpha)\Gamma(\alpha-\frac{1}{2})...\Gamma(\alpha-\frac{p-1}{2}),
~\Re(\alpha)>\frac{p-1}{2}.\eqno(1.2)$$When $B$ is of the form $B={\frac{1}{2}}V^{-1}, ~V=V'>O$, where a prime denotes the transpose, then $f(X)$ in (1.1) is the Wishart density in multivariate statistical analysis, which is the central density in the area, see for example, Anderson \[1\], Kshirsagar \[2\], Srivastava and Khatri \[3\]. The real matrix-variate gamma integral is $$\Gamma_p(\alpha)=\int_{X>O}|X|^{\alpha-\frac{p+1}{2}}{\rm e}^{-{\rm tr}(X)}{\rm d}X\eqno(1.3)$$where ${\rm d}X$ is the wedge product of the $p(p+1)/2$ differentials ${\rm d}X=\prod_{i\ge j}\wedge{\rm d}x_{ij}$. For evaluating the integral in (1.3), the standard technique used is to write $X=TT'$ where $T$ is a lower or upper triangular matrix with positive diagonal elements. Then the integral on the right of (1.3) will split into conventional integrals on individual scalar variables. When $T$ is lower triangular then the integral over $t_{ij},i>j$ gives $\sqrt{\pi}$ and there are $p(p-1)/2$ such factors giving $\pi^{\frac{p(p-1)}{4}}$. The integral over $t_{jj}>0$ gives $\Gamma(\alpha-\frac{j-1}{2}),j=1,...,p$ and the product thus gives $\Gamma_p(\alpha)$ on the left of (1.3). .2cm When Wishart density is derived, starting from samples from a Gaussian population, the basic technique is the triangularization process. Can we evaluate the integral on the right of (1.3) by using conventional methods, or by direct evaluation? We will look into this problem by using the technique of partitioned matrices. Let us partition $$X=\left[ \begin{matrix}X_{11}&X_{12}\\
X_{21}&X_{22}
\end{matrix}\right]$$where let $X_{22}=x_{pp}$ so that $X_{21}=(x_{p1},...,x_{pp-1}), X_{12}=X_{21}'$. Then $$|X|^{\alpha-\frac{p+1}{2}}=|X_{11}|^{\alpha-\frac{p+1}{2}}[x_{pp}-X_{21}X_{11}^{-1}X_{12}]^{\alpha-\frac{p+1}{2}}$$by using partitioned matrix and determinant. Note that when $X$ is positive definite, that is, $X>O$, then $X_{11}>O,x_{pp}>0$ and the quadratic form $X_{21}X_{11}^{-1}X_{12}>0$. Note that $$[x_{pp}-X_{21}X_{11}^{-1}X_{12}]^{\alpha-\frac{p+1}{2}}=x_{pp}^{\alpha-\frac{p+1}{2}}[1-x_{pp}^{-\frac{1}{2}}
X_{21}X_{11}^{-\frac{1}{2}}X_{11}^{-\frac{1}{2}}X_{12}x_{pp}^{-\frac{1}{2}}]^{\alpha-\frac{p+1}{2}}.$$Let $Y=x_{pp}^{-\frac{1}{2}}X_{21}X_{11}^{-\frac{1}{2}}$ then ${\rm d}Y=x_{pp}^{-\frac{p-1}{2}}|X_{11}|^{-\frac{1}{2}}{\rm d}X_{21}$ for fixed $X_{11},x_{pp}$, see Mathai (\[4\], Theorem 1.18.) The integral over $x_{pp}$ gives $$\int_0^{\infty}x_{pp}^{\alpha+\frac{p-1}{2}-\frac{p+1}{2}}{\rm e}^{-x_{pp}}{\rm d}x_{pp}=\Gamma(\alpha),~\Re(\alpha)>0.$$Let $u=YY'$. Then from Theorem 2.16 and Remark 2.13 of \[4\] and after integrating out over the Stiefel manifold we have $${\rm d}Y=\frac{\pi^{\frac{p-1}{2}}}{\Gamma(\frac{p-1}{2})}u^{\frac{p-1}{2}-1}{\rm d}u.$$(Note that $n$ in Theorem 2.16 corresponds to $p-1$ and $p$ corresponds to $1$). Then the integral over $u$ gives $$\int_0^1u^{\frac{p-1}{2}-1}(1-u)^{\alpha-\frac{p+1}{2}}{\rm d}u=\frac{\Gamma(\frac{p-1}{2})\Gamma(\alpha-\frac{p-1}{2})}{\Gamma(\alpha)},~\Re(\alpha)>\frac{p-1}{2}.$$Now, collecting all the factors, we have $$\begin{aligned}
|X_{11}|^{\alpha+\frac{1}{2}-\frac{p+1}{2}}\Gamma(\alpha)&\frac{\pi^{\frac{p-1}{2}}}{\Gamma(\frac{p-1}{2})}
\frac{\Gamma(\frac{p-1}{2})\Gamma(\alpha-\frac{p-1}{2})}{\Gamma(\alpha)}\nonumber\\
&=|X_{11}^{(1)}|^{\alpha+\frac{1}{2}-\frac{p+1}{2}}
\pi^{\frac{p-1}{2}}\Gamma(\alpha-\frac{p-1}{2})\nonumber\end{aligned}$$for $\Re(\alpha)>\frac{p-1}{2}$. Note that $|X_{11}^{(1)}|$ is $(p-1)\times (p-1)$ and $|X_{11}|$ after the completion of the first part of the operations is denoted by $|X_{11}^{(1)}|$, and the exponent is changed to $\alpha+\frac{1}{2}-\frac{p+1}{2}$. Now repeat the process by separating $x_{p-1,p-1}$, that is by writing $$X_{11}^{(1)}=\left[\begin{matrix}X_{11}^{(2)}&X_{12}^{(2)}\\
X_{21}^{(2)}&x_{p-1,p-1}
\end{matrix}\right].$$Here $X_{11}^{(2)}$ is of order $(p-2)\times (p-2)$ and $X_{21}^{(2)}$ is of order $1\times (p-2)$. As before, let $u=YY', Y=x_{p-1,p-1}^{-\frac{1}{2}}X_{21}^{(2)}[X_{11}^{(2)}]^{-\frac{1}{2}}.$ Then ${\rm d}Y=x_{p-1,p-1}^{-\frac{p-2}{2}}|X_{11}^{(2)}|^{-\frac{1}{2}}{\rm d}X_{21}^{(2)}.$ Integral over the Stiefel manifold gives $\frac{\pi^{\frac{p-2}{2}}}{\Gamma(\frac{p-2}{2})}u^{\frac{p-2}{2}-1}{\rm d}u$ and the factor containing $(1-u)$ is $(1-u)^{\alpha+\frac{1}{2}-\frac{p+1}{2}}$ and the integral over $u$ gives $$\int_0^1u^{\frac{p-2}{2}-1}(1-u)^{\alpha+\frac{1}{2}-\frac{p+1}{2}}{\rm d}u=\frac{\Gamma(\frac{p-2}{2})\Gamma(\alpha-\frac{p-2}{2})}{\Gamma(\alpha)}.$$Intgral over $v=x_{p-1,p-1}$ gives $$\int_0^1v^{\alpha+\frac{1}{2}+\frac{p-2}{2}-\frac{p+1}{2}}{\rm e}^{-v}{\rm d}v=\Gamma(\alpha),~\Re(\alpha)>0.$$Taking all product we have $$|X_{11}^{(2)}|^{\alpha+1-\frac{p+1}{2}}\pi^{\frac{p-2}{2}}\Gamma(\alpha-\frac{p-2}{2}),~\Re(\alpha)>\frac{p-2}{2}.$$Successive evaluations by using the same procedure gives the exponent of $\pi$ as $\frac{p-1}{2}+\frac{p-2}{2}+...+\frac{1}{2}=\frac{p(p-1)}{4}$ and the gamma product is $\Gamma(\alpha-\frac{p-1}{2})\Gamma(\alpha-\frac{p-2}{2})...\Gamma(\alpha)$ and the final result is $\Gamma_p(\alpha)$. Hence the result is verified.
.3cm[**1.2..3cm Evaluation of matrix-variate gamma in the complex case**]{}
.3cm In the complex case, the matrices and gamma will be denoted with a tilde. In the complex case, all matrices appearing in the integrals will be $p\times p$ hermitian positive definite unless stated otherwise, denoted by $\tilde{X}>O$. Our integral of interest is $$\tilde{\Gamma_p}(\alpha)=\int_{\tilde{X}>O}|{\rm det}(\tilde{X})|^{\alpha-p}{\rm e}^{-{\rm tr}(\tilde{X})}{\rm d}\tilde{X}.\eqno(1.4)$$One standard procedure to evaluate the integral in (1.4) is to write the hermitian positive definite matrix as $\tilde{X}=\tilde{T}\tilde{T}^{*}$ where $\tilde{T}$ is a lower triangular matrix with real and positive diagonal elements $t_{jj}>0,j=1,...,p$, where \* indicates the conjugate transpose. Then the Jacobian can be seen to be the following, see also (\[4\], Theorem 3.7): $${\rm d}\tilde{X}=2^p\{\prod_{j=1}^pt_{jj}^{2(p-j)+1}\}{\rm d}\tilde{T}\eqno(1.5)$$and then $$\begin{aligned}
{\rm tr}(\tilde{X})&={\rm tr}(\tilde{T}\tilde{T}^{*})\nonumber\\
&=t_{11}^2+...+t_{pp}^2+|\tilde{t_{21}}|^2+...+|\tilde{t_{p1}}|^2+...+|\tilde{t_{pp-1}}|^2\nonumber\end{aligned}$$and $${\rm tr}(\tilde{X}){\rm d}\tilde{X}=2^p\{\prod_{j=1}^pt_{jj}^{2\alpha-2j+1}\}{\rm d}\tilde{T}.$$Now, integrating out over $\tilde{t_{jk}}$ for $j>k$ $$\int_{\tilde{t_{jk}}}{\rm e}^{-|\tilde{t_{jk}}|^2}{\rm d}\tilde{t_{jk}}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}{\rm e}^{-(t_{jk1}^2+t_{jk2}^2)}{\rm d}t_{jk1}\wedge{\rm d}t_{jk2}=\pi$$and
$$\prod_{j>k}\pi=\pi^{\frac{p(p-1)}{2}}.$$Now, $$2\int_0^{\infty}t_{jj}^{2\alpha-2j+1}{\rm e}^{-t_{jj}^2}{\rm d}t_{jj}=\Gamma(\alpha-j+1),~\Re(\alpha)>j-1,$$for $j=1,...,p.$ Now the product of all these gives $$\pi^{\frac{p(p-1)}{2}}\Gamma(\alpha)\Gamma(\alpha-1)...\Gamma(\alpha-p+1)=\tilde{\Gamma_p}(\alpha),~\Re(\alpha)>p-1$$and hence the result is verified.
.3cm[**1.3..3cm An alternate method based on partitioned matrix**]{} .3cm Let us separate $x_{pp}$. When $\tilde{X}$ is $p\times p$ hermitian positive definite then all its diagonal elements are real and positive. That is, $x_{jj}>0,j=1,...,p$. Let $$\tilde{X}=\left[\begin{matrix}\tilde{X_{11}}&\tilde{X_{12}}\\
\tilde{X_{21}}&x_{pp}
\end{matrix}\right]$$where $\tilde{X_{11}}$ is $(p-1)\times (p-1)$ and $$|{\rm det}(\tilde{X})|^{\alpha-p}=|{\rm det}(\tilde{X_{11}})|^{\alpha-p}|x_{pp}-\tilde{X_{21}}\tilde{X_{11}}^{-1}\tilde{X_{12}}|^{\alpha-p}$$and $${\rm tr}(\tilde{X})={\rm tr}(\tilde{X_{11}})+x_{pp}.$$Then
$$|x_{pp}-\tilde{X_{21}}\tilde{X_{11}}^{-1}\tilde{X_{12}}|^{\alpha-p}=x_{pp}^{\alpha-p}|1-x_{pp}^{-\frac{1}{2}}
\tilde{X_{21}}\tilde{X_{11}}^{-\frac{1}{2}}\tilde{X_{11}}^{-\frac{1}{2}}\tilde{X_{12}}x_{pp}^{-\frac{1}{2}}|^{\alpha-p}.$$Put
$$\tilde{Y}=x_{pp}^{-\frac{1}{2}}\tilde{X_{21}}\tilde{X_{11}}^{-\frac{1}{2}}\Rightarrow {\rm d}\tilde{Y}=x_{pp}^{-(p-1)}|{\rm det}(\tilde{X_{11}})|^{-1}{\rm d}\tilde{X_{21}}$$from (\[4\], Theorem 3.2(c)). Now, the integral over $x_{pp}$ gives
$$\int_0^{\infty}x_{pp}^{\alpha-p+(p-1)}{\rm e}^{-x_{pp}}{\rm d}x_{pp}=\Gamma(\alpha),~\Re(\alpha)>0.$$Let $u=\tilde{Y}\tilde{Y}^{*}$. Then ${\rm d}\tilde{Y}=u^{p-2}\frac{\pi^{p-1}}{\Gamma(p-1)}{\rm d}u$ by using Corollaries 4.5.2 and 4.5.3 of \[4\]. Note that $u$ is real and positive. Integral over $u$ gives
$$\int_0^{\infty}u^{(p-1)-1}(1-u)^{\alpha-(p-1)-1}{\rm d}u=\frac{\Gamma(p-1)\Gamma(\alpha-(p-1))}{\Gamma(\alpha)},\Re(\alpha)>p-1.$$Taking the product we have $$\begin{aligned}
|{\rm det}(\tilde{X}_{11}^{(1)})|^{\alpha+\frac{1}{2}-p}\Gamma(\alpha)&\frac{\pi^{p-1}}{\Gamma(p-1)}\frac{\Gamma(p-1)
\Gamma(\alpha-(p-1))}{\Gamma(\alpha)}\nonumber\\
&=\pi^{p-1}\Gamma(\alpha-(p-1))|{\rm det}(\tilde{X}_{11}^{(1)})|^{\alpha+\frac{1}{2}-p}\nonumber\end{aligned}$$ where $\tilde{X}_{11}^{(1)}$ indicates $\tilde{X}_{11}$ after the first set of integrations. Now for the second stage, separate $x_{p-1,p-1}$ and the first $(p-2)\times (p-2)$ block may be denoted by $\tilde{X}_{11}^{(2)}$. Now proceed as before to get $|{\rm det}(\tilde{X}_{11}^{(2)})|^{\alpha+1-p}\pi^{p-2}\Gamma(\alpha-(p-2))$. Proceeding like this we have the exponent of $\pi$ as $(p-1)+(p-2)+...+1=p(p-1)/2$ and the gamma product will be $\Gamma(\alpha-(p-1))\Gamma(\alpha-(p-2))...\Gamma(\alpha)$ for $\Re(\alpha)>p-1$. That is, $$\pi^{\frac{p(p-1)}{2}}\Gamma(\alpha)\Gamma(\alpha-1)...\Gamma(\alpha-(p-1))=\tilde{\Gamma_p}(\alpha).$$ .3cm[**2..3cm Evaluation of Matrix-variate Beta Integrals**]{} .3cm Here we will consider a direct way of evaluating matrix-variate type-1 and type-2 beta integrals in the real and complex cases.
.3cm[**2.1..3cm Evaluation of matrix-variate type-1 beta integral in the real case**]{}
.3cm The real matrix-variate type-1 beta function is denoted by $$B_p(\alpha,\beta)=\frac{\Gamma_p(\alpha)\Gamma_p(\beta)}{\Gamma_p(\alpha+\beta)},~\Re(\alpha)>p-1,~\Re(\beta)>p-1$$and it has the following type-1 beta integral representation: $$B_p(\alpha,\beta)=\int_{O<X<I}|X|^{\alpha-\frac{p+1}{2}}|I-X|^{\beta-\frac{p+1}{2}}{\rm d}X,$$for $\Re(\alpha)>\frac{p-1}{2},~\Re(\beta)>\frac{p-1}{2}$ where $X$ is real symmetric and positive definite $p\times p$ matrix. The standard derivation of this integral is from the properties of real matrix-variate gamma integrals by making suitable transformations, see for example, \[4\]. Is it possible to evaluate the integral directly and show that it is equal to $\frac{\Gamma_p(\alpha)\Gamma_p(\beta)}{\Gamma_p(\alpha+\beta)}$, where, for example, $$\Gamma_p(\alpha)=\pi^{\frac{p(p-1)}{4}}\Gamma(\alpha)\Gamma(\alpha-\frac{1}{2})...\Gamma(\alpha-\frac{p-1}{2}),
~\Re(\alpha)>\frac{p-1}{2}?$$For evaluating real matrix-variate gamma integral an easy method is to make the transformation $X=TT'$ where $T$ is a lower triangular matrix with positive diagonal elements. Even if this transformation is applied here, the integral does not simplify due to the presence of the factor $|I-X|^{\beta-\frac{p+1}{2}}$. Hence we will try to evaluate the integral by using a partitioning of the matrices and then integrating step by step. Let $X=(x_{ij})$ be a $p\times p$ matrix. Let us separate $x_{pp}$. This can be done by partitioning $|X|$ and $|I-X|$. That is, let $$X=\left[\begin{matrix}X_{11}&X_{12}\\
X_{21}&X_{22}\end{matrix}\right]$$where $X_{11}$ is the $(p-1)\times (p-1)$ leading submatrix, $X_{21}$ is $1\times (p-1)$, $X_{22}=x_{pp}$ and $X_{12}=X_{21}'$. Then $|X|=|X_{11}|[x_{pp}-X_{21}X_{11}^{-1}X_{12}]$ and $$|X|^{\alpha-\frac{p+1}{2}}=|X_{11}|^{\alpha-\frac{p+1}{2}}[x_{pp}-X_{21}X_{11}^{-1}X_{12}]^{\alpha-\frac{p+1}{2}}\eqno(1)$$ $$|I-X|^{\beta-\frac{p+1}{2}}=|I-X_{11}|^{\beta-\frac{p+1}{2}}[(1-x_{pp})-X_{21}(I-X_{11})^{-1}X_{12}]^{\beta-\frac{p+1}{2}}
\eqno(2)$$From (1) we have $x_{pp}>X_{21}X_{11}^{-1}X_{12}$ and from (2) we have $x_{pp}<1-X_{21}(I-X_{11})^{-1}X_{12}$. That is, $X_{21}X_{11}^{-1}X_{12}<x_{pp}<1-X_{21}(I-X_{11})^{-1}X_{12}$. Let $y=x_{pp}-X_{21}X_{11}^{-1}X_{12}\Rightarrow {\rm d}y={\rm d}x_{pp}$ for fixed $X_{21},X_{11}$. Also, $0<y<b$ where $$\begin{aligned}
b&=1-X_{21}X_{11}^{-1}X_{12}-X_{21}(I-X_{11})^{-1}X_{12}\nonumber\\
&=1-X_{21}X_{11}^{-\frac{1}{2}}(I-X_{11})^{-\frac{1}{2}}(I-X_{11})^{-\frac{1}{2}}X_{11}^{-\frac{1}{2}}X_{12}\nonumber\\
&=1-WW',~W=X_{21}X_{11}^{-\frac{1}{2}}(I-X_{11})^{-\frac{1}{2}}.\nonumber\end{aligned}$$The second factor on the right in (2) becomes $$[b-y]^{\beta-\frac{p+1}{2}}=b^{\beta-\frac{p+1}{2}}[1-\frac{y}{b}]^{\beta-\frac{p+1}{2}}.$$Put $u=\frac{y}{b}$ for fixed $b$. Then the factors containing $u$ and $b$ become $b^{\alpha+\beta-(p+1)+1}u^{\alpha-\frac{p+1}{2}}(1-u)^{\beta-\frac{p+1}{2}}$. Integral over $u$ gives $$\int_0^1u^{\alpha-\frac{p+1}{2}}(1-u)^{\beta-\frac{p+1}{2}}{\rm d}u=\frac{\Gamma(\alpha-\frac{p-1}{2})\Gamma(\beta-\frac{p-1}{2})}{\Gamma(\alpha+\beta-(p-1))},$$for $~\Re(\alpha)>\frac{p-1}{2},
~\Re(\beta)>\frac{p-1}{2}.$ Let $W=X_{21}X_{11}^{-\frac{1}{2}}(I-X_{11})^{-\frac{1}{2}}$ for fixed $X_{11}$. Then ${\rm d}X_{21}=|X_{11}|^{\frac{1}{2}}|I-X_{11}|^{\frac{1}{2}}{\rm d}W$ from Theorem 1.18 of \[4\], where $X_{11}$ is $(p-1)\times (p-1)$. Put $v=WW'$ and integrate out over the Stiefel manifold by using Theorem 2.16 and Remark 2.13 of \[4\]. Then we have $${\rm d}W=\frac{\pi^{\frac{p-1}{2}}}{\Gamma(\frac{p-1}{2})}v^{\frac{p-1}{2}-1}{\rm d}v.$$Now the integral over $b$ becomes $$\begin{aligned}
\int b^{\alpha+\beta-p}{\rm d}X_{21}&=\int_0^1v^{\frac{p-1}{2}-1}(1-v)^{\alpha+\beta-p}{\rm d}v\nonumber\\
&=\frac{\Gamma(\frac{p-1}{2})\Gamma(\alpha+\beta-(p-1))}{\Gamma(\alpha+\beta-\frac{p-1}{2})},~\Re(\alpha+\beta)>p-1.\nonumber\end{aligned}$$Now, multiplying all the factors together we have $$|X_{11}^{(1)}|^{\alpha+\frac{1}{2}-\frac{p+1}{2}}|I-X_{11}^{(1)}|^{\beta+\frac{1}{2}-\frac{p+1}{2}}\pi^{\frac{p-1}{2}}
\frac{\Gamma(\alpha-\frac{p-1}{2})\Gamma(\beta-\frac{p-1}{2})}{\Gamma(\alpha+\beta-\frac{p-1}{2})}$$for $\Re(\alpha)>\frac{p-1}{2},~\Re(\beta)>\frac{p-1}{2}$. Here $X_{11}^{(1)}$ indicates the $(p-1)\times(p-1)$ leading submatrix at the end of the first set of operations. At the end of the second set of operations we will denote the $(p-2)\times (p-2)$ leading submatrix by $X_{11}^{(2)}$, and so on. The second step of operations starts by separating $x_{p-1,p-1}$ and writing
$$|X_{11}^{(1)}|=|X_{11}^{(2)}|[x_{p-1,p-1}-X_{21}^{(2)}[X_{11}^{(2)}]^{-1}X_{12}^{(2)}]$$where $X_{21}^{(2)}$ is $1\times (p-2)$. Now, proceed as in the first sequence of steps to obtain the final factors of the following form: $$|X_{11}^{(2)}|^{\alpha+1-\frac{p+1}{2}}|I-X_{11}^{(2)}|^{\beta+1-\frac{p+1}{2}}\pi^{\frac{p-2}{2}}
\frac{\Gamma(\alpha-\frac{p-2}{2})\Gamma(\beta-\frac{p-2}{2})}{\Gamma(\alpha+\beta-\frac{p-2}{2})}$$for $\Re(\alpha)>\frac{p-2}{2},~\Re(\beta)>\frac{p-2}{2}$. Proceeding like this the exponent of $\pi$ at the end will be of the form $$\frac{p-1}{2}+\frac{p-2}{2}+...+\frac{1}{2}=\frac{p(p-1)}{4}.$$The gamma product will be of the form $$\frac{\Gamma(\alpha-\frac{p-1}{2})\Gamma(\alpha-\frac{p-2}{2})...\Gamma(\alpha)\Gamma(\beta-\frac{p-1}{2})...\Gamma(\beta)}
{\Gamma(\alpha+\beta-\frac{p-1}{2})...\Gamma(\alpha+\beta)}.$$These gamma products, together with $\pi^{\frac{p(p-1)}{4}}$ can be written as $\frac{\Gamma_p(\alpha)\Gamma_p(\beta)}{\Gamma_p(\alpha+\beta)}=B_p(\alpha,\beta)$ and hence the result. Thus, it is possible to evaluate the type-1 real matrix-variate beta integral directly to obtain the beta function in the real matrix variate case. .2cm A similar procedure can yield the real matrix-variate beta function from the type-2 real matrix-variate beta integral of the form $$\int_{X>O}|X|^{\alpha-\frac{p+1}{2}}|I+X|^{-(\alpha+\beta)}{\rm d}X$$for $X=X'>O$ and $p\times p,~\Re(\alpha)>\frac{p-1}{2},~\Re(\beta)>\frac{p-1}{2}$. The procedure for the evaluation will be parallel.
.3cm[**2.2..3cm Evaluation of matrix-variate type-1 beta integral in the complex case**]{} .3cm The integral representation for $B_p(\alpha,\beta)$ in the complex case is the following:
$$\int_{O<\tilde{X}<I}|{\rm det}(\tilde{X})|^{\alpha-p}|{\rm det}(I-\tilde{X})|^{\beta-p}{\rm d}\tilde{X}=\tilde{B}_p(\alpha,\beta)$$for $\Re(\alpha)>p-1,~\Re(\beta)>p-1$ where ${\rm det}(\cdot)$ denotes the determinant of $(\dot)$ and $|{\rm det}(\cdot)|$ denotes the absolute value of the determinant of $(\cdot)$. Here $\tilde{X}=(\tilde{x}_{ij})$ is a $p\times p$ hermitian positive definite matrix and hence all the diagonal elements are real and positive. As in the real case, let us separate $x_{pp}$ by partitioning:
$$\tilde{X}=\left[\begin{matrix}\tilde{X}_{11}&\tilde{X}_{12}\\
\tilde{X}_{21}&\tilde{X}_{22}\end{matrix}\right]\mbox{ as well as }I-\tilde{X}=\left[\begin{matrix}I-\tilde{X}_{11}&-\tilde{X}_{12}\\
-\tilde{X}_{21}&I-\tilde{X}_{22}\end{matrix}\right].$$Then the absolute value of the determinants are of the form:
$$|{\rm det}(\tilde{X})|^{\alpha-p}=|{\rm det}(\tilde{X}_{11})|^{\alpha-p}|x_{pp}-\tilde{X}_{21}\tilde{X}_{11}^{-1}\tilde{X}_{12}^{*}|^{\alpha-p}\eqno(a)$$where \* indicates conjugate transpose, and $$|{\rm det}(I-\tilde{X})|^{\beta-p}=|{\rm det}(I-\tilde{X}_{11})|^{\beta-p}|(1-x_{pp})-\tilde{X}_{21}(I-\tilde{X}_{11})^{-1}\tilde{X}_{12}^{*}|^{\beta-p}.\eqno(b)$$Note that when $\tilde{X}$ and $I-\tilde{X}$ are hermitian positive definite then $\tilde{X}_{11}^{-1}$ and $(I-\tilde{X}_{11})^{-1}$ are also hermitian positive definite. Further, the hermitian forms $\tilde{X}_{21}\tilde{X}_{11}^{-1}\tilde{X}_{12}^{*}$ and $\tilde{X}_{21}(I-\tilde{X}_{11})^{-1}\tilde{X}_{12}^{*}$ remain real and positive. From (a) and (b) it follows that
$$\tilde{X}_{21}\tilde{X}_{11}^{-1}\tilde{X}_{12}^{*}<x_{pp}<1-\tilde{X}_{21}(I-\tilde{X}_{11})^{-1}\tilde{X}_{12}^{*}.$$Since hermitian forms are real, the lower and upper bounds of $x_{pp}$ are real. Let $$\tilde{W}=\tilde{X}_{21}\tilde{X}_{11}^{-\frac{1}{2}}(I-\tilde{X}_{11})^{-\frac{1}{2}}$$for fixed $\tilde{X}_{11}$. Then $${\rm d}\tilde{X}_{21}=|{\rm det}(\tilde{X}_{11})|^{-1}|{\rm det}(I-\tilde{X}_{11})|^{-1}{\rm d}\tilde{W}$$and $|{\rm det}(\tilde{X})|^{\alpha-p}, |{\rm det}(I-\tilde{X}_{11})|^{\beta-p}$ change to $|{\rm det}(\tilde{X}_{11})|^{\alpha+1-p}, |{\rm det}(I-\tilde{X}_{11})|^{\beta+1-p}$ respectively. Then we can write $$\begin{aligned}
|(1-x_{pp})&-\tilde{X}_{21}\tilde{X}_{11}^{-1}\tilde{X}_{12}^{*}-\tilde{X}_{21}(I-\tilde{X}_{11})^{-1}\tilde{X}_{12}^{*}|^{\beta-p}\nonumber\\
&=(b-y)^{\beta-p}=b^{\beta-p}[1-\frac{y}{b}]^{\beta-p}.\nonumber\end{aligned}$$Put $u=\frac{y}{b}$. Then the factors containing $u$ and $b$ will be of the form $u^{\alpha-p}(1-u)^{\beta-p}b^{\alpha+\beta-2p+1}$ and the integral over $u$ gives
$$\int_0^1u^{\alpha-p}(1-u)^{\beta-p}{\rm d}u=\frac{\Gamma(\alpha-(p-1))\Gamma(\beta-(p-1))}{\Gamma(\alpha+\beta-2(p-1))},$$for $\Re(\alpha)>p-1,\Re(\beta)>p-1$. Let $v=\tilde{W}\tilde{W}^{*}$ and integrate out over the Stiefel manifold by using Corollaries 4.5.2 and 4.5.3 of \[4\]. Then
$${\rm d}\tilde{W}=\frac{\pi^{p-1}}{\Gamma(p-1)}v^{(p-1)-1}{\rm d}v.$$The integral over $b$ gives $$\begin{aligned}
\int b^{\alpha+\beta-2p+1}{\rm d}\tilde{X}_{21}&=\int_0^1v^{(p-1)-1}(1-v)^{\alpha+\beta-2p+1}{\rm d}v\nonumber\\
&=\frac{\Gamma(p-1)\Gamma(\alpha+\beta-2p+2)}{\Gamma(\alpha+\beta-p+1)},\nonumber\end{aligned}$$ for $\Re(\alpha)>p-1,\Re(\beta)>p-1$. Now, taking the product of all factors we have
$$|{\rm det}(\tilde{X}_{11})|^{\alpha+1-p}|{\rm det}(I-\tilde{X}_{11})|^{\beta+1-p}\pi^{p-1}\frac{\Gamma(\alpha-p+1)\Gamma(\beta-p+1)}{\Gamma(\alpha+\beta-p+1)}$$for $\Re(\alpha)>p-1,\Re(\beta)>p-1$. Separate $x_{p-1,p-1}$ from $\tilde{X}_{11}$ and $I-\tilde{X}_{11}$ and continue the process. Then at the end, the exponent of $\pi$ will be $(p-1)+(p-2)+...+1=\frac{p(p-1)}{2}$ and the gamma product will be $$\frac{\Gamma(\alpha-(p-1))\Gamma(\alpha-(p-2))...\Gamma(\alpha)\Gamma(\beta-(p-1))...\Gamma(\beta)}
{\Gamma(\alpha+\beta-(p-1))...\Gamma(\alpha+\beta)}.$$These factors, together with $\pi^{\frac{p(p-1)}{2}}$ give $$\frac{\tilde{\Gamma_p}(\alpha)\tilde{\Gamma_p}(\beta)}{\tilde{\Gamma_p}(\alpha+\beta)}=\tilde{B_p}(\alpha,\beta),
\Re(\alpha)>p-1,\Re(\beta)>p-1.$$The procedure for evaluating a type-2 matrix-variate beta integral by the method of partitioning is parallel and hence it will not be detailed here.
.3cm[**3..3cm General Partitions**]{} .3cm In section 2 we have considered integrating one variable at a time by suitably partitioning the matrices. Is it possible to have a general partitioning and integrate a block of variables at a time, rather than integrating out individual variables? Let us consider the real matrix-variate gamma integral first. Let
$$X=\left[\begin{matrix}X_{11}&X_{12}\\
X_{21}&X_{22}\end{matrix}\right],~X_{11}\mbox{ is }p_1\times p_1\mbox{ and }X_{22}\mbox{ is }p_2\times p_2$$so that $X_{12}$ is $p_1\times p_2$ and $X_{21}=X_{12}'$ and $p_1+p_2=p$. Without loss of generality, let us assume that $p_1\ge p_2$. Then the determinant can be partitioned as follows: $$\begin{aligned}
|X|^{\alpha-\frac{p+1}{2}}&=|X_{11}|^{\alpha-\frac{p+1}{2}}|X_{22}-X_{21}X_{11}^{-1}X_{12}|^{\alpha-\frac{p+1}{2}}\nonumber\\
&=|X_{11}|^{\alpha-\frac{p+1}{2}}|X_{22}|^{\alpha-\frac{p+1}{2}}|I-X_{22}^{-\frac{1}{2}}X_{21}X_{11}^{-1}X_{12}
X_{22}^{-\frac{1}{2}}|^{\alpha-\frac{p+1}{2}}.\nonumber\end{aligned}$$Put $$Y=X_{22}^{-\frac{1}{2}}X_{21}X_{11}^{-\frac{1}{2}}\Rightarrow {\rm d}Y=|X_{22}|^{-\frac{p_1}{2}}|X_{11}|^{-\frac{p_2}{2}}{\rm d}X_{21}$$for fixed $X_{11}$ and $X_{22}.$ $$|X|^{\alpha-\frac{p+1}{2}}=|X_{11}|^{\alpha+\frac{p_2}{2}-\frac{p+1}{2}}|X_{22}|^{\alpha+\frac{p_1}{2}-\frac{p+1}{2}}
|I-YY'|^{\alpha-\frac{p+1}{2}}.$$The Jacobian above is available from Theorem 1.18 of \[4\]. Let $S=YY'$. Then integrating out over the Stiefel manifold we have
$${\rm d}Y=\frac{\pi^{\frac{p_1p_2}{2}}}{\Gamma_{p_2}(\frac{p_1}{2})}|S|^{\frac{p_1}{2}-\frac{p_2+1}{2}}{\rm d}S,$$see Theorem 2.16 and Remark 2.13 of \[4\]. Now, integral over $S$ gives
$$\int_{O<S<I}|S|^{\frac{p_1}{2}-\frac{P_2+1}{2}}|I-S|^{\alpha-\frac{p_1}{2}-\frac{p_2+1}{2}}{\rm d}S=\frac{\Gamma_{p_2}(\frac{p_1}{2})\Gamma_{p_2}(\alpha-\frac{p_1}{2})}{\Gamma_{p_2}(\alpha)},$$for $\Re(\alpha)>\frac{p_1-1}{2}$. Collecting all the factors, we have
$$|X_{11}|^{\alpha-\frac{p_1+1}{2}}|X_{22}|^{\alpha-\frac{p_2+1}{2}}\pi^{\frac{p_1p_2}{2}}\frac{\Gamma_{p_2}(\alpha
-\frac{p_1}{2})}{\Gamma_{p_2}(\alpha)}.$$From here one can also observe that the original determinant splits into functions of $X_{11}$ and $X_{22}$. This also shows that if we are considering a real matrix-variate gamma density then the diagonal blocks $X_{11}$ and $X_{22}$ are statistically independently distributed, where $X_{11}$ will have a $p_1$-variate gamma distribution and $X_{22}$ has a $p_2$-variate gamma distribution. Observe that ${\rm tr}(X)={\rm tr}(X_{11})+{\rm tr}(X_{22})$ and hence the integral over $X_{22}$ gives $\Gamma_{p_2}(\alpha)$ and the integral over $X_{11}$ gives $\Gamma_{p_1}(\alpha)$. Hence the total integral is available as
$$\Gamma_{p_1}(\alpha)\Gamma_{p_2}(\alpha)\pi^{\frac{p_1p_2}{2}}\frac{\Gamma_{p_2}(\alpha-\frac{p_1}{2})}
{\Gamma_{p_2}(\alpha)}
=\Gamma_p(\alpha)$$since $\pi^{\frac{p_1p_2}{2}}\Gamma_{p_1}(\alpha)\Gamma_{p_2}(\alpha-\frac{p_1}{2})=\Gamma_p(\alpha)$. .2cm Hence it is seen that instead of integrating out variables one at a time we could have also integrated out blocks of variables at a time and could have verified the result. Similar procedure works for real matrix-variate type-1 and type-2 beta, and matrix-variate gamma, type-1 and type-2 beta in the complex domain also.
.3cm[**3.1..3cm Methods avoiding integration over the Stiefel manifold**]{}
.3cm The general method of partitioning described above involves the integration over the Stiefel manifold as an intermediate step. We will consider another procedure which will avoid integration over Stiefel manifold. Let us consider the real gamma case first. Again, we start with the decomposition $$|X|^{\alpha-\frac{p+1}{2}}=|X_{11}|^{\alpha-\frac{p+1}{2}}|X_{22}-X_{21}X_{11}^{-1}X_{12}|^{\alpha-\frac{p+1}{2}}.\eqno(3.1)$$Instead of integrating out $X_{21}$ or $X_{12}$ let us integrate out $X_{22}$. Let $X_{11}$ be $p_1\times p_1$ and $X_{22}$ be $p_2\times p_2$ with $p_1+p_2=p$. In the above partitioning we require that $X_{11}$ be nonsingular. But when $X$ is positive definite, both $X_{11}$ and $X_{22}$ will be positive definite, thereby nonsingular also. From the second factor in (3.1), $X_{22}>X_{21}X_{11}^{-1}X_{12}$ from $X_{22}-X_{21}X_{11}^{-1}X_{12}$ being positive definite. We will try to integrate out $X_{22}$ first. Let $U=X_{22}-X_{21}X_{11}^{-1}X_{12}$ so that ${\rm d}U={\rm d}X_{22}$ for fixed $X_{11}$ and $X_{12}$. Since ${\rm tr}(X)={\rm tr}(X_{11})+{\rm tr}(X_{22})$ we have $${\rm e}^{-{\rm tr}(X_{22})}={\rm e}^{-{\rm tr}(U)-{\rm tr}(X_{21}X_{11}^{-1}X_{12})}.$$Integrating out $U$ we have $$\int_{U>O}|U|^{\alpha-\frac{p+1}{2}}{\rm e}^{-{\rm tr}(U)}{\rm d}U=\Gamma_{p_2}(\alpha-\frac{p_1}{2}),~\Re(\alpha)>\frac{p-1}{2}$$since $\alpha-\frac{p+1}{2}=\alpha-\frac{p_1}{2}-\frac{p_2+1}{2}$. Let $$Y=X_{21}X_{11}^{-\frac{1}{2}}\Rightarrow {\rm d}Y=|X_{11}|^{-\frac{p_2}{2}}{\rm d}X_{21}$$for fixed $X_{11}$. Then $$\int_{X_{21}}{\rm e}^{-{\rm tr}(X_{21}X_{11}^{-1}X_{12})}{\rm d}X_{21}=|X_{11}|^{\frac{p_2}{2}}\int_{Y}{\rm e}^{-{\rm tr}(YY')}{\rm d}Y.$$But ${\rm tr}(YY')$ is the sum of squares of the $p_1p_2$ elements in $Y$ and each integral is of the form $\int_{-\infty}^{\infty}{\rm e}^{-z^2}{\rm d}z=\sqrt{\pi}$. Hence $$\int_{Y}{\rm e}^{-{\rm tr}(YY')}{\rm d}Y=\pi^{\frac{p_1p_2}{2}}.$$Now we can integrate out $X_{11}$. $$\begin{aligned}
\int_{X_{11}>O}|X_{11}|^{\alpha+\frac{p_2}{2}-\frac{p+1}{2}}&{\rm e}^{-{\rm tr}(X_{11})}{\rm d}X_{11}\nonumber\\
&=\int_{X_{11}>O}|X_{11}|^{\alpha-\frac{p_1+1}{2}}{\rm e}^{-{\rm tr}(X_{11})}{\rm d}X_{11}\nonumber\\
&=\Gamma_{p_1}(\alpha).\nonumber
\end{aligned}$$Hence we have the following factors: $$\pi^{\frac{p_1p_2}{2}}\Gamma_{p_2}(\alpha-\frac{p_1}{2})\Gamma_{p_1}(\alpha)=\Gamma_p(\alpha)$$since $$\frac{p_1(p_1-1)}{4}+\frac{p_2(p_2-1)}{4}+\frac{p_1p_2}{2}=\frac{p(p-1)}{4},~p=p_1+p_2$$and $$\begin{aligned}
\Gamma_{p_1}(\alpha)\Gamma_{p_2}(\alpha-\frac{p_1}{2})&=\Gamma(\alpha)\Gamma(\alpha-\frac{1}{2})
...\Gamma(\alpha-\frac{p_1-1}{2})\Gamma_{p_2}(\alpha-\frac{p_1}{2})\nonumber\\
&=\Gamma(\alpha)...\Gamma(\alpha-\frac{p_1+p_2-1}{2}).\nonumber
\end{aligned}$$Hence the result. In this procedure we did not have to go through integration over the Stiefel manifold and we did not have to assume that $p_1\ge p_2$. We could have integrated out $X_{11}$ first if needed. In this case, expand $$|X|^{\alpha-\frac{p+1}{2}}=|X_{22}|^{\alpha-\frac{p+1}{2}}|X_{11}-X_{12}X_{22}^{-1}X_{21}|^{\alpha-\frac{p+1}{2}}.$$Then proceed as before by integrating out $X_{11}$ first. Then we end up with $$\pi^{\frac{p_1p_2}{2}}\Gamma_{p_1}(\alpha-\frac{p_2}{2})\Gamma_{p_2}(\alpha)=\Gamma_p(\alpha), p=p_1+p_2.$$ .3cm[**Note:**]{}.3cm If we are considering a real matrix-variate gamma density, such as the Wishart density, then from the above procedure observe that after integrating out $X_{22}$ the only factor containing $X_{21}$ is the exponential function, which has the structure of a matrix-variate Gaussian density. Hence for given $X_{11}$, $X_{21}$ is matrix-variate Gaussian distributed. Similarly, for given $X_{22}$, $X_{12}$ is matrix-variate Gaussian distributed. Further, the diagonal blocks $X_{11}$ and $X_{22}$ are independently distributed. .3cm The same procedure as above goes through for the evaluation of gamma integrals in the complex domain also. Since the steps are parallel they will not be detailed here.
.3cm[**Acknowledgement**]{} .3cm The author would like to thank the Department of Science and Technology, Government of India, for the financial assistance for this work under project number SR/S4/MS:287/05 and the Centre for Mathematical Sciences for the facilities.
.3cm
References
.3cm\[1\] T.W. Anderson, An Introduction to Multivariate Statistical Analysis, Wiley, New York, 1971. .2cm\[2\] A.M. Kshirsagar, Multivariate Analysis, Marcel Dekker, New York, 1972. .2cm\[3\] M.S. Srivastava and C.G., An Introduction to Multivariate Statistics, North Holland, New York, 1979. .2cm\[4\] A.M. Mathai, Jacobians of Matrix Transformations and Functions of Matrix Argument, World Scientific Publishing, New York, 1997.
|
24.5cm =1.5pc
[**Clock-transport synchronisation for neutrino time-of-flight measurements**]{}
J.H.Field
*Département de Physique Nucléaire et Corpusculaire, Université de Genève*
*24, quai Ernest-Ansermet CH-1211Genève 4.*
E-mail: [email protected]
The aims of the present paper are twofold, firstly to present an analysis of clock transport synchronisation as a possible calibration method for time-of-flight measurements in two existing neutrino beams [@MINOS; @CNGS] and one proposed one [@LBNE] and secondly to point out the interest of such measurements, at the level of 1ns accuracy, as a test of the Sagnac effect for neutrinos —that the measured velocity is not expected to be exactly equal to the speed, $c$, of light in vacuum [@Kuhn; @JHFSagnac]. The clock transport synchronisation method is not proposed as a stand-alone one but rather as a cross-check of continous methods such as GPS Common-View GPS [@GPSCV] as used for the published neutrino time-of-flight measurements or Two Way Satellite Time Transfer [@TWSTT] which have a comparable precision ($\simeq 1$ns or better) than the method proposed here.
In the region of the Earth, the proper time interval, $d\tau$, of a clock is related to the interval of ‘coordinate time’, $dt$, by the Schwartzschild metric equation [@Schwartzs; @Weinb]: $$d \tau = \left[1+\frac{2 \phi_{{\rm E}}}{c^2}-\frac{1}{c^2}\left(\frac{v_r^2}{1+\frac{2 \phi_{{\rm E}}}{c^2}}
+v_{\theta}^2 +v_{\phi}^2\right)\right]^{\frac{1}{2}}dt$$ where $v_r$, $v_{\theta}$ and $v_{\phi}$ are components of the velocity of the clock in a spherical polar coordinate system with origin at the centre of the Earth, and polar axis in the south-north direction, fixed in the Earth Centered Inertial (ECI) frame. The ECI frame is a non-rotating inertial frame instantaneously comoving with the centroid of the Earth. The cordinate axes of the ECI frame therefore have fixed directions relative to the Celestial sphere. The quantity $\phi_{{\rm E}}$ is the gravitational potential of the Earth, which, on the assumption of a spherical Earth is given as: $\phi_{{\rm E}} =-G M_{{\rm E}}/r$ at distance $r$ from the centre of the Earth, where $M_{{\rm E}}$ is the mass of the Earth. As can be seen by setting $\vec{v} = 0$, $\phi_{{\rm E}} = 0$ in (1), coordinate time is the proper time of any clock at rest in the ECI frame sufficiently far from the Earth that all gravitational effects of the latter may be neglected. In applying (1) it is assumed that the spatial variation of the gravitational fields of the Sun, Moon and other members of the Solar System, that give rise to tidal effects, may be neglected, as well as the rotation of the Earth around the Sun that changes the angular velocity of a rotating, Earth-fixed, frame relative to the ECI frame. These are the same approximations that are made in applications of (1) to clocks in the satellites of the GPS system [@AshbyPT; @AshbyLRR].
For a clock at rest on the Surface of the Earth $v_r = v_{\theta} = 0$ so that (1) simplifies to: $$d \tau = \left[1+\frac{1}{c^2}\left(2 \phi_{{\rm E}}
-\Omega^2 R_{xy}^2\right)\right]^{\frac{1}{2}}dt$$ where $R_{xy}$ is the distance of the clock from the axis of rotation of the Earth and $\Omega$ is the angular velocity of the latter relative to the ECI frame. Due to its rotation, the Earth has the form of an oblate spheroid, the surface of which is known as the geoid. Locally, in the oceans, the plane of the geoid is parallel to the surface of the sea.
It was shown by Cocke [@Cocke] that the distortion of the Earth from a spherical shape modifies the external gravitational potential of the Earth in such a way that, to a very good approximation, the quantity $2 \phi_{{\rm E}}-\Omega^2 R_{xy}^2$ in (2) is constant for all points on the geoid so that identical clocks at the poles or on the Equator will be observed, from the ECI frame, to run at the same rate[^1].
If a clock moves, at fixed $r$, relative to the surface of the Earth, with speed $\vec{v}$, the elapsed proper time interval $\Delta \tau$ is given by integrating (1): $$\Delta \tau = \int\left[1+\frac{1}{c^2}\left(2 \phi_{{\rm E}}-(\vec{v}_{\Omega}+\vec{v})^2
\right)\right]^{\frac{1}{2}} dt
\simeq\int\left[1+\frac{1}{c^2}\left(\phi_{{\rm E}}
-\frac{1}{2}(\vec{v}_{\Omega}+\vec{v})^2
\right)\right]dt$$ where $\vec{v}_{\Omega}$ is the speed, in the ECI frame, of an object, at the same position as the clock, but at rest relative to the Earth, and where, in the last member, only terms linear in $\phi_{{\rm E}}/c^2$ and quadratic in $v/c$ are retained.
As suggested by Hafele [@HN; @HAJP] the correctness of the relation (3) was verified in an experiment performed by Hafele and Keating (HK) [@HK] in 1972. An array of caesium-beam atomic clocks was flown around the Earth in commercial airliners in West to East (W-E) and East to West (E-W) directions. They were compared, before and after the flights, with reference Earth-bound clocks at the U.S. Naval Observatory. In agreement with prediction, time interval differences, relative to the reference clocks, of 273ns for the W-E and -59ns for the E-W flights were observed [@HK].
In the application of Eq. (3) suggested, in the present work, to synchronise clocks for time-of-flight measurements, comparisons are made between the epochs recorded by similar clocks (i.e. ones running at the same frequency), one at the position of the source, and the other at the position of the detector, of the particles. If the clocks are placed on the geoid they will be observed from the ECI frame to run at the same rate. As will be shown, the clocks can be synchronised by comparing their epochs with that of a third similar clock that is transported between them. The single fixed reference clock of the HK experiment is therefore replaced by two similar clocks at different, but known, spatial locations. In order to make this comparison the proper time intervals $\Delta \tau_0$, $\Delta \tau$ recorded by the Earth-fixed and moving clocks, respectively, for the same interval, $\Delta t$, of coordinate time are calculated using Eq. (3) taking account of the variation of the values of $\phi_{{\rm E}}$, $\vec{v}_{\Omega}$ and $\vec{v}$ along the path followed by the moveable clock. For the purposes of the calculation presented here it is assumed that the Earth is spherical; in actual experimental applications the actual shape of the geoid [@AshbyLRR] should be taken into account.
In a practical realisation of the synchronisation method suggested here the clock might be equipped with a GPS receiver and associated hardware to record the precise position of the clock in the Earth-fixed frame at known epochs during the transport so that the integral in Eq. (3) can be accurately evaluated. In the present work the dependence of $\Delta \tau$ on the parameters of the clock trajectory is studied within a simple model where only the mean speed and altitude of the transport are considered. This is sufficient to quantitatively demonstrate the predictions for $\Delta \tau$ given by different choices of transport trajectory for the three neutrino time-of-flight experiments that are considered.
Considering the specific case of neutrino time-of-flight measurements in the CNGS neutrino beam [@CNGS; @OPERA] the proper time intervals recorded by reference clocks at CERN ($\Delta \tau_0^{{\rm C}}$) and Gran Sasso ($\Delta \tau_0^{{\rm GS}}$), assumed to lie on the geoid, are given by (3), to first order in $\phi_{{\rm E}}$ and second order in $\beta = v_{\Omega}/c$ as $$\begin{aligned}
\Delta \tau_0 & = & \Delta \tau_0^{{\rm C}} =\int\left[1+\frac{1}{c^2}\left(\phi_{{\rm E}}^{{\rm C}}
-\frac{1}{2}(\vec{v}_{\Omega}^{{\rm C}})^2
\right)\right]dt \nonumber \\
& = & \Delta \tau_0^{{\rm GS}} = \int\left[1+\frac{1}{c^2}\left(\phi_{{\rm E}}^{{\rm GS}}
-\frac{1}{2}(\vec{v}_{\Omega}^{{\rm GS}})^2
\right)\right]dt
\end{aligned}$$ since $\phi_{{\rm E}} -(\vec{v}_{\Omega})^2/2$ is invariant on the geoid. The proper time interval $\Delta \tau$ recorded by a clock transported from CERN (C) to Gran Sasso (GS) during the interval $\Delta t$ of coordinate time is, from (3) and (4): $$\Delta \tau = \Delta \tau_0 + \int_{{\rm C}}^{{\rm GS}}
\left[ \frac{G M_{{\rm E}} h}{c^2R^2}
-\frac{1}{2c^2}[(\vec{v}_{\Omega})^2-(\vec{v}_{\Omega}^{{\rm C}})^2 +v^2+ 2\vec{v}_{\Omega}\cdot \vec{v}
]\right]dt$$ where $h$ is the altitude above the Earth’s surface. Using a mean value approximation: $$\int f(t)dt \simeq \langle f(t) \rangle \int dt$$ to evaluate the integral in (5), and assuming transport at constant $v$, gives: $$\frac{\Delta \tau-\Delta \tau_0}{\Delta t} \simeq \frac{\Delta \tau-\Delta \tau_0}{\Delta \tau_0}
\equiv \frac{\delta \tau}{\Delta \tau_0}
= \frac{G M_{{\rm E}}\langle h \rangle}{c^2R^2}
-\frac{1}{2c^2}[\langle (\vec{v}_{\Omega})^2\rangle-(\vec{v}_{\Omega}^{{\rm C}})^2
+v^2+ 2\langle \vec{v}_{\Omega} \rangle \cdot \vec{v}]$$ Since the values of $\delta \tau$ in (6) are of the order of a few nanoseconds precise modelling of the altitude and speed of the travelling clock in order to evaluate the integral in (5) is not required. For example a 5$\%$ uncertainy in the evaluation of the integral gives only $0.05$ns uncertainty in $\delta \tau$ when $\delta \tau = 1$ ns.
Introducing a primed, Earth-centered, Earth-fixed, polar coordinate system with north-pointing polar axis, $\phi' = 0$ at the prime meridian and $\lambda'$ the angle of latitude, the coordinates of CERN are [@MJDM]: $\lambda'_{{\rm C}} = 46.05^{\circ}$, $\phi'_{{\rm C}} = 6.08^{\circ}$ and of Gran Sasso are: $\lambda'_{{\rm GS}} = 42.3^{\circ}$, $\phi'_{{\rm GS}} = 13.58^{\circ}$. If $\hat{\imath}'$,$\hat{\jmath}'$ and $\hat{k}'$ are unit vectors along the $x'$ ($\lambda' =0$, $\phi' = 0$), $y'$($\lambda' =0$, $\phi' = 90^{\circ}$) and $z'$ axes of the corresponding right-handed Cartesian coordinate system, the spatial separation of the neutrino source at CERN from the OPERA detector at Gran Sasso is $d =\sqrt{(\Delta x')^2 +(\Delta y')^2+(\Delta z')^2}$ where: $$\begin{aligned}
\Delta x' & = & R(\cos\lambda'_{{\rm GS}}\cos\phi'_{{\rm GS}}-
\cos\lambda'_{{\rm C}}\cos\phi'_{{\rm C}}) = 0.02936R \\
\Delta y' & = & R(\cos\lambda'_{{\rm GS}}\sin\phi'_{{\rm GS}}-
\cos\lambda'_{{\rm C}}\sin\phi'_{{\rm C}}) = 0.1003R \\
\Delta z' & = & R(\sin\lambda'_{{\rm GS}}- \sin\lambda'_{{\rm C}}) = -0.0476R
\end{aligned}$$ The velocity vector $\vec{v}$ parallel to the neutrino beam is then: $$\vec{v} = v(0.2557\hat{\imath}'+0.8734\hat{\jmath}'-0.4144\hat{k}') \equiv v \hat{v}$$ Also $$\langle \vec{v}_{\Omega} \rangle = \Omega R(-\hat{\imath}'\langle \cos\lambda'\sin \phi'\rangle
+ \hat{\jmath}'\langle \cos\lambda'\cos \phi'\rangle)
= \Omega R(-0.1236\hat{\imath}'
+ 0.7047\hat{\jmath}')$$ so that $$\langle \vec{v}_{\Omega} \rangle \cdot \vec{v} = 0.5839\Omega R v$$ and $$\begin{aligned}
\langle (\vec{v}_{\Omega})^2\rangle & = & \Omega^2 R^2 \langle \cos^2\lambda'\rangle
= 0.5146 \Omega^2 R^2 \\
(\vec{v}_{\Omega}^{{\rm C}})^2 & = & \Omega^2 R^2 \cos^2\lambda'_{{\rm C}}
= 0.4815 \Omega^2 R^2
\end{aligned}$$ Substituting (12)-(14) in (6) with $\Omega R = 465$ m/s gives: $$\delta \tau = \frac{0.00407}{v}\left[19.57 \langle h \rangle -7157-v^2-543v\right]~{\rm ns~~~CERN-CNGS}$$ with $\langle h \rangle$ in m and v in m/s. Here the relation $$\Delta t \simeq \Delta \tau_0 = \frac{d C_{{\rm GS}}}{v} = \frac{730.53 \times 10^{3} \times 1.0022}{v} =
\frac{732.14 \times 10^{3}}{v}$$ is used. The straight line distance $d$ from CERN to the OPERA detector at Gran Sasso [@MJDM] is corrected by the factor $C_{{\rm GS}}$ which accounts for clock transport along a Grand Circle.
A similar calculation for the MINOS experiment [@MINOS] in the Fermilab (FNAL) NuMI beam with $d = 732.7$ km, $C_{{\rm GS}}= 1.0022$, gives: $$\delta \tau = \frac{0.00408}{v}\left[19.57 \langle h \rangle +11222-v^2+280v\right]~{\rm ns~~~FNAL-MuMI}$$ Here FNAL is assumed to be at latitude: $\lambda'_{{\rm F}} = 41.85^{\circ}$, longitude: $\phi'_{{\rm F}} = -88.29^{\circ}$ and the MINOS detector in the Soudan mine at $\lambda'_{{\rm S}} = 47.81^{\circ}$, $\phi'_{{\rm S}} = -92.24^{\circ}$.
For the proposed long-baseline FNAL-Homestake neutrino beam [@LBNE] LBNE, with $d = 1298$ km, $C_{{\rm GS}}= 1.0069$ it is found that: $$\delta \tau = \frac{0.00726}{v}\left[19.57 \langle h \rangle +4670-v^2+657v\right]~{\rm ns~~~FNAL-LBNE}$$ where the Homestake mine at Lead, South Dakota, has coordinates: $\lambda'_{{\rm H}} = 44.33^{\circ}$, $\phi'_{{\rm H}} = -103.83^{\circ}$ [@Homestake].
The procedure to synchronise a clock, $A$, at the source laboratory, registering an epoch $\tau_{A}$ with clock $B$, at the detector laboratory, registering an epoch $\tau_{B}$ is as follows. Before the clock transport, the difference, $\Delta \tau_{A}$, between the epochs registered by clock $A$ and the moveable clock $M$ are recorded at some instant 1. $\Delta \tau_{A} =\tau_{A}^{(1)}-\tau_{M}^{(1)}$. After the clock transport the difference between the epochs of $B$ and $M$ is recorded at some instant 2: $\Delta \tau_{B} =\tau_{B}^{(2)}-\tau_{M}^{(2)}$. In order to synchronise clock $B$ with clock $A$ the quantity $ \Delta \tau \equiv \Delta \tau_{B} - \Delta \tau_{A}+\delta \tau$ (see Fig. 1) must be subtracted from the epoch registered by clock $B$. Note that if all three clocks have the same frequency when at rest in the same frame of reference the differences of epochs $\Delta \tau_{A}$ ($\Delta \tau_{B}$) may be recorded at any time before (after) the transport of $M$, so that the instants 1 and 2 are arbitary.
$\langle h \rangle$ (km) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.36551
-------------------------- ------- ------- ------- ------- ------- ------- ------- ------- ---------
$v_0$ (km/h) 305 283 260 234 205 171 129 63 0
$\delta \tau_0$ (ns) -2.90 -2.85 -2.80 -2.74 -2.67 -2.60 -2.50 -2.35 -2.21
: [*Values of $v_0$ and $\delta \tau_0$ at which $\delta \tau$ has a maximum value, as a function of $\langle h \rangle$, for the CERN-CGNS neutrino beam.*]{}
$v~(km/h)$ 10 50 100 200 400 600 800 1000
------------------------- ------- ------- ------- ------ ------ ------ ------ ------
$\langle h \rangle$ (m)
0 -12.7 -4.4 -3.4 -3.0 -2.9 -3.1 -3.2 -3.4
200 -7.0 -3.2 -2.8 -2.7 -2.8 -3.0 -3.2 -3.4
400 -1.2 -2.1 -2.2 -2.4 -2.6 -2.9 -3.1 -3.3
600 4.5 -0.92 -1.7 -2.1 -2.5 -2.8 -3.0 -3.3
800 10.2 0.22 -1.1 -1.8 -2.4 -2.7 -3.0 -3.2
1000 16.0 1.4 -0.51 -1.5 -2.2 -2.6 -2.9 -3.2
: [*Values of $\delta \tau$ in ns as a function of $v$ and $\langle h \rangle$ for the CERN-CNGS neutrino beam.*]{}
Curves of $\delta \tau$ versus $v$ for various values of $\langle h \rangle$ as calculated by (15), (16) and (17) are shown in Figs. 2, 3 and 4 respectively. A feature of all three neutrino beams is the existence, for some values of $\langle h \rangle$, of ‘magic’ values $v_{{\rm M}}$ of the velocity for which $\delta \tau$ vanishes (given by the intersection of the curves with the dashed line) and so gives the possibility of a simple and accurate synchronisation procedure, as if the moving clock was equivalent to the ground-based ones. Magic velocity values correspond to exact cancellation of the special-relativistic time dilation effect, which slows the rate of the moving clock as observed in the ECI frame, and the general-relativistic gravitational blue shift which increases its rate. Curves of $v_{{\rm M}}$ versus $\langle h \rangle$ for all three beams are shown in Fig. 5. For values of $v$: 50 km/h $< v <$ 1000 km/h corresponding to road, rail or air transport $\delta \tau$ takes, in all cases, absolute values of a few nanoseconds or less.
The geometry of the CERN-CNGS beam is such that, for small values of $\langle h \rangle$, the curves of $\delta \tau$ versus $v$ have maximum values. The corresponding velocity $v_0$ and time interval $\delta \tau_0$ are given by the solution of the equation $d(\delta \tau)/dv = 0$ as: $$\begin{aligned}
v_0 & = & \left[\langle (\vec{v}_{\Omega})^2\rangle-(\vec{v}_{\Omega}^{{\rm C}})^2
-\frac{2 G M_{{\rm E}}\langle h \rangle}{R^2}\right]^{\frac{1}{2}} \\
\delta \tau_0 & = & -\frac{d C_{{\rm GC}}}{c^2}(v_0+\hat{v} \cdot \langle \vec{v}_{\Omega}\rangle)
\end{aligned}$$ The largest value of $\langle h \rangle$ for which a maximium value of $\delta \tau$ exists is $~\simeq 365$ m corresponding to $v_0 = 0$. The dependence of $v_0$ and $\delta \tau_0$ on $\langle h \rangle$ is given in Table 1. Over the range 0 m $<\langle h \rangle <$ 300 m, $\delta \tau_0$ varies by only $\pm 0.2$ns about an average value of 2.7ns. As shown in Table 2 the feature of a weak dependence of $\delta \tau$ on both $v$ and $\langle h \rangle$ persists for 365 m $ < \langle h \rangle < $ 1000 m where the curves are monotonic. For example, for $v = $500 km/h and $\langle h \rangle =$ 500m one percent uncertainties in $v$ and $\langle h \rangle$ give only $\simeq 0.24~\%$ and $\simeq 0.09~\%$ uncertainties, respectively in $\delta \tau$, suggesting that this quantity may be determined with a precision of one tenth of a nanosecond or better. In order to achieve this precision over the typical beam length of $\simeq 1000$ km, with 100 km/h $< v <$ 1000 km/h requires a frequency stability of the clock in the range $3\times 10^{-14}$—$3\times 10^{-13}$/h, compatible with the typical frequency stability of a precision atomic clock of 1ns/day.
Since the Great Circle route from CERN to Gran Sasso passes over the Alps a possible strategy to keep $h$ below 1 km to take advantage of the almost constant value of $\delta \tau$ as a function of $v$ and $\langle h \rangle$, as shown in Table 2, is to transport the clocks by air above the Rhone to the Mediterranean, then by a Grand Circle route to Gran Sasso. An alternative strategy is a Grand Circle route directly over the Alps at an altitude of 5-10km at or near the magic value of $v$, again enabling a precise synchronisation of the local clocks at CERN and Gran Sasso.
For the FNAL neutrino beams: $$\begin{aligned}
\langle (\vec{v}_{\Omega})^2\rangle-(\vec{v}_{\Omega}^{{\rm F}})^2 & = & -11.22\times 10^3
~~~~{\rm MuMI} \nonumber \\
& = & -4.67\times 10^3
~~~~{\rm LBNE} \nonumber
\end{aligned}$$ Inspection of Eq. (18) shows that no real value of $v_0$ exists for these beams; the curves of $\delta \tau$ versus $v$ are therefore monotonic for all values of $\langle h \rangle$. As can be seen in Figs. 3 and 4 magic values of $v$ are obtainable by conventional air transport at speeds of 1000-2000 km/h for the MuMI beam but not for LBNE that requires speeds in excess of 2000 km/h.
It is interesting to compare the uncertainty in the clock synchronisation constant $\delta \tau$ with the variation of neutrino times-of-flight due to the Sagnac effect [@Kuhn; @JHFSagnac; @Sagnac; @Post], i.e. the effect of the motion of the neutrino target, due to the Earth’s rotation, on the observed time-of-flight. For neutrinos with energy much greater than their mass, the relative speed $c_r$ of the neutrino and the target is, at lowest order in the velocity $\vec{v}_{\Omega}^{{\rm T}}$ of the target, given by [@Kuhn; @JHFSagnac]: $$c_r = c -\hat{v}\cdot\vec{v}_{\Omega}^{{\rm T}}$$ where $\hat{v}$ is a unit vector in the direction from the neutrino source to the target. For the three neutrino beams considered above it is found that: $$\begin{aligned}
\frac{c_r-c}{c} & = & -9.05 \times 10^{-7}~~~~{\rm CNGS} \nonumber \\
& = & 4.65 \times 10^{-7}~~~~{\rm MuMI} \nonumber \\
& = & 1.10 \times 10^{-6}~~~~{\rm LBNE} \nonumber
\end{aligned}$$ The corresponding differences of time-of-flight compared with propagation at speed $c$ are: 2.2 ns [@Kuhn], -1.14 ns and -4.74 ns respectively, to be compared with the possible uncertainty in $\delta \tau$ of the order of a nanosecond or less. Especially the proposed FNAL-LBNE beam [@LBNE] is well adapted to a precise test of the Sagnac effect for neutrinos.
In the case of four experiments which have published neutrino time-of-flight measurements: MINOS [@MINOS], OPERA[@OPERA], ICARUS[@ICARUS] and LVD [@LVD] the dominant systematic errors were related to local measurements of the times of production and detection of the neutrinos rather than errors in the synchronisation of the precision master clocks near to the source and target. In the LVD measuremet that has the smallest quoted systematic uncertainty of 3.3 ns on the time-of-flight measurement, only 1 ns is assigned to GPS synchronisation. In order to be sensitive to the Sagnac effect more accurate local timimg of neutrino production and detection events than hitherto achieved is required in addition to synchronisation uncertainty at the 1 ns level or better.
[**Acknowledgements**]{} The author would like to thank two anonymous referees for pointing out several shortcomings in a previous version of this paper and for suggestions to improve its clarity.
[99]{} MINOS Collaboration, P.Adamson [*et al*]{} Phys. Rev. [**D56**]{} 072005 (2007). Ed. K. Elsener, ‘The CERN Neutrino beam to Gran Sasso (Conceptual Technical Design)’, CERN 98-02, INFN/AE-98/05. V. Papadimitriou, ‘Status of the LBNE Neutrino Beamline’, arXiv pre-print: http://xxx.lanl.gov/abs/1112.0720v1. Cited 4 Dec 2011. M.G. Kuhn, ‘The influence of Earth rotation in neutrino speed measurements between CERN and OPERA detector’, arXiv pre-print: http://xxx.lanl.gov/abs/1110.03920v2. Cited 20 Oct 2011. J.H. Field, ‘The Sagnac effect and transformations of relative velocities between inertial frames’, December 2011. Submitted to General Relativity and Gravitation. Available at: http://www.relativity-myths.org.uk/jhfield/. D.W. Allan and M.A. Weiss, ‘Accurate Time and Frequency Transfer During Common-View of a GPS Satellite’, 34th Annual Frequency Control Symposium pp. 334-346 May 1980. NIST, Time and Frequency Division ‘Two Way Time Transfer’ http://tf.nist.gov/time/twoway.htm. K. Schwartzschild, Sitzungberichte Prüssiche Akademie der Wissenschaften p.198 (1916). S. Weinberg, ‘Gravitation and Cosmology, Principles and Applications of the General Theory of Relativity’, (John Wiley, New York, 1972) Ch 8 Section 2. N. Ashby, ‘Relativity and the global positioning system’, Physics Today, May 2002, p.41. N. Ashby, Living Reviews in Relativity [**6**]{} 1 (2003). W.J. Cocke, Phys. Rev. Lett. [**16**]{} 66 (1966). A. Einstein, Ann. Physik [**17**]{}, 891 (1905). English translation by W. Perrett and G.B. Jeffery in ‘The Principle of Relativity’ (Dover, New York, 1952) , p.37, or in ‘Einstein’s Miraculous Year’ (Princeton University Press, Princeton, New Jersey, 1998) p.123. A. Harvey and E. Schuking, ‘A small Puzzle from 1905’ Physics Today, March 2005, p.34 and comments in Physics Today, September 2005 p.12. J.C. Hafele, Nature [**227**]{} 270 (1970). J.C. Hafele, Am. J. Phys. [**40**]{} 81 (1972). J.C. Hafele and R.E. Keating, Science [**177**]{} 166, 168 (1972). G. Colosimo [*et al*]{}, ‘Determination of the CNGS global geodesy’, Opera public note 132 v2. Cited 10 Oct 2011; http://operaweb.lngs.infn.it/Opera/publicnotes/note132.pdf. B.T. Cleveland [*et al*]{}, Astrophys. J. [**496**]{} 505 (1998). G. Sagnac, Compt. Rend. [**157**]{} 708, 1410 (1913). E.J. Post, Rev. Mod. Phys. [**39**]{} No 2 475 (1967). OPERA Collaboration, T. Adam [*et al*]{}, ‘Measurement of the neutrino velocity with the OPERA detector in the CNGS beam’, arXiv pre-print: http://xxx.lanl.gov/abs/1109.48973v4. Cited 12 Jul 2012. ICARUS Collaboration, M. Antonello [*et al*]{}, Phys. Lett. B [**713**]{} 17 (2012). LVD Collaboration, N. Yu Agafonova [*et al*]{}, ‘Measurement of the velocity of neutrinos from the CNGS beam with the Large Volume Detector.’ arXiv pre-print: http://xxx.lanl.gov/abs/1208.1392v1. Cited 7 Aug 2012.
[^1]: Not as stated in Einstein’s 1905 special relativity paper [@Ein1], slower at the equator. See [@HSPT] for a discussion of this point.
|
---
author:
- |
Cupjin Huang and Yaoyun Shi\
\
Department of Electrical Engineering and Computer Science\
University of Michigan, Ann Arbor, MI 48109, USA\
`cupjinh,[email protected]`
bibliography:
- 'qhash.bib'
title: Quantum hashing is maximally secure against classical leakage
---
|
---
abstract: 'I examine the topic of training scientific generalists. To focus the discussion, I propose the creation of a new graduate program, analogous in structure to existing MD/PhD programs, aimed at training a critical mass of scientific researchers with substantial intellectual breadth. In addition to completing the normal requirements for a PhD, students would undergo an intense, several year training period designed to expose them to the core vocabulary of multiple subjects at the graduate level. After providing some historical and philosophical context for this proposal, I outline how such a program could be implemented with little institutional overhead by existing research universities. Finally, I discuss alternative possibilities for training generalists by taking advantage of contemporary developments in online learning and open science.'
author:
- |
<span style="font-variant:small-caps;">Gopal P. Sarma</span>\
[*[email protected]*]{}
title: '**Should we train scientific generalists?**'
---
In the age of highly specialized science, the generalist is a long forgotten job description. We have come to assume that the role played by those intellectual titans of earlier eras, such as Da Vinci, Aristotle, or Gauss, to name just a few, is an impossibility given the massive explosion of scientific knowledge of recent decades and centuries.\
There is a factual reality to this sentiment that is uncontroversial. Certainly, as a percentage of existing knowledge, one could not conceivably attain the breadth of understanding that one might have in previous centuries. However, it does seem worth considering if a more modest goal could be achieved which would serve an important stabilizing role for modern science and engineering. That goal would be to train a critical mass of scientific generalists, researchers, who in addition to the specialized training of an ordinary graduate program, would also have broad exposure to multiple subjects at the graduate level.\
While the need for specialization might have been something of an inevitability, it is also worth considering that there may be negative ramifications to this kind of stratification of knowledge. With so much to know, how can we be confident that we are allocating our intellectual capital efficiently? How can we be confident in our collective understanding of global trends in science?\
There is no doubt that in the coming years, data analytics of the scientific corpus will play a significant role in contributing to the creation of precisely such a global view of the scientific enterprise. The digitization of journals, the availability of open API’s for accessing scientific meta-data, and the integration of reference management with social networking are all poised to transform our understanding of the scientific process at a high-level. However, it seems naive to imagine that data mining techniques alone will allow us to conceive of and test the most important hypotheses about the global structure and dynamics of science without some amount of guiding intuition. To complement and maximally take advantage of the availability of massive data sets about science, as well as the computational tools to analyze those data sets, we need a critical mass of scientific generalists whose training has been designed to encourage hypothesis generation about the scientific process itself.\
Furthermore, another major trend in contemporary science is the move towards ambitious scientific agendas of substantially larger scope and project size [@nielsen]. Whereas the pioneering theories of earlier eras were often crafted by solitary thinkers working in isolation, today’s breakthroughs frequently come about from large international collaborations involving hundreds or thousands of people and research budgets in the billion dollar range. In this context, the question of how to ideally train an individual scientist might be re-conceptualized as the question of how to train a scientific team member. Scientific generalists could be pivotal members of such large collaborations and play critical organizational and leadership roles.\
There are certainly scientific generalists today, although they are perhaps not thought of in this way. I would broadly (and informally) categorize them into thee types:
- **The organic academic generalist**\
This is someone who has led a traditional academic career on the tenure track, and whose research has naturally led to developing significant breadth in multiple topics. Certainly many fields have researchers in this category.\
- **The academic-industrial wanderer**\
This is someone who has left academia, or possibly had extended post-doctoral or research scientist appointments in subjects different from their PhD, and ultimately came back to academia, or led significant efforts at major industrial research laboratories. For example, the growth of computational biology and theoretical neuroscience has been driven by many theoretical physicists who have gone on to do post-doctoral training in the biological sciences, or for example, physicists from the world of quantitative finance, who have returned to academia armed with a new set of skills quite different from their PhD training.\
- **The autodidact**\
The widespread availability of advanced scientific materials via the Internet has resulted in an organic trend towards the creation of generalists simply by lowering the barrier to accessing knowledge from a wide variety of fields, scientific or otherwise. Certainly, there are many brilliant scientists in industry and elsewhere who do not have PhD’s and it is not uncommon these days to encounter truly first class thinkers on a variety of topics who are largely self-taught.
The question that motivates this essay is the following: should there be another category of generalist who has been trained from outset to play a different role in the modern scientific enterprise than researchers who set out to be specialists?\
As a means to stimulate discussion, but as an idea unto its own as well, I propose the following: the creation of a new graduate program, roughly analogous in structure to an MD/PhD, where in addition to the normal research requirements for completing a PhD, students complete 5 or more qualifying examinations in subjects of their choosing. For adequately prepared students, I believe that after completion of the requirements for their home department in their first or second year, students would be able pass 4 additional qualifying examinations over the course of 2-3 years, after which they would resume their PhD research and complete their degree.[^1]\
The choice of the qualifying examination as the focal point for this program is that it encapsulates the basic vocabulary of a field, the core knowledge required to conduct in depth research. The aim of this program is emphatically *not* to train researchers who have in depth, specialized knowledge of 5 different subjects– that would be an unreasonable, if not outright impossible goal. Rather, the aim is to train students who understand the culture, the basic tools, and broad perspectives of multiple subjects, so that they can contribute to strengthening the very foundations of the scientific establishment.\
Certainly, universities who undertake the process of creating such a program might choose to begin with a fewer number of qualifying examinations. I chose this number because it would allow for individual students to engage with multiple, quite different subjects over the course of their graduate education, and because 6-8 month blocks per subject would create a program roughly on par with the length of an MD/PhD. Part of the value in creating such a program would be the message and the vision it would send to younger students who are aspiring to life-long careers as scientists. Just as undergraduates who aspire to careers as physician-scientists must adequately prepare themselves with appropriate exposure to both research and clinical work, aspiring scientific generalists would need to prepare themselves with advanced coursework of sufficient breadth to tackle the challenging initial years of this graduate program.\
For an ambitious program such as this one to maximally benefit both the student and the scientific establishment at large, there would need to be a strong culture to support those students who choose to undergo such a rigorous and extended training. In particular, in order for the knowledge gained by these students to develop into something much more rich and robust than a massive list of facts and problem solving techniques from 5 different subjects, they would need to be part of a mentoring program in which the process of learning each of these different subjects was accompanied by historical and philosophical discussion. During each qualifying examination block, students would ideally also attend regular seminars in the department, and perhaps nominally be affiliated with a research group and attend group meetings. There would need to be a culture among the students and faculty mentors that supported reflection about problem solving strategies, about the structural differences between the vocabulary and subject matter across different fields. Ultimately, these observations and insights, whether in raw or more developed form would need to be communicated more broadly.\
One possibility might be to accompany the qualifying examination process with a historical essay exploring some topic of interest to the student in consultation with a faculty mentor. For instance, a student whose PhD was in theoretical condensed matter physics and who passed examinations in physics, mathematics, chemistry, biology, and computer science might write an in depth essay on the emergence of quantitative methods in the study of natural selection. A mathematics PhD student specializing in stochastic analysis and who passed qualifying examinations in mathematics, physics, statistics, computer science, and economics, might write about the contributions that mathematical finance pioneer Fisher Black made to macroeconomics.\
While this program may seem daunting, I would like to emphasize that individuals who pursue MD/PhD degrees and ultimately become board certified in a medical specialty need to pass a similar array of hurdles– in addition to PhD requirements for their research training they have to pass multiple level of board examinations to become licensed physicians.\
It is also important to keep in mind that the training program described here is a graduate level training program, and consequently, should be thought of as being the first step in a career-long trajectory. A person who completed this program is no more a mature scientific generalist than a person who completes an ordinary PhD program is a mature specialist. In order for the subsequent phases of growth and development to take place, there would need to be a supporting infrastructure overseeing the post-doctoral period of the students’ training. Furthermore, it is certainly possible, and would be expected even, that a subset of students who successfully complete this training would simply choose to pursue tenure track jobs in their area of specialty. Again, the MD/PhD is something of a guide– certainly many dual-degree graduates become purely clinical practitioners or pure researchers and do not actively build careers bridging the two. Students who pursue the more traditional routes will not be at a disadvantage and one would hope that the unique and rigorous educational experience they went through would inform the remainder of their scientific careers both as researchers and as teachers. But for those that wish to mature into the novel role of scientific generalists that I am proposing, there would need to be special post-doctoral programs providing generous several year funding that would give them the freedom to develop their vision. For the initial batch of students, there would inevitably be some amount of trial and error while both students and faculty developed an understanding of the strengths and weaknesses of the program.\
While one can only speculate about the contributions graduates of this program would ultimately come to make, I close by suggesting a few possibilities. We might imagine tenured professorships for generalists who have smaller research groups than they would otherwise have, but who are active members of several different groups led by other faculty. In addition to playing a critical organizational role, these faculty members would bring their considerable technical expertise and scientific breadth to each group in the capacity of something along the lines of a scientific consultant.\
Venture capital might be another place where scientific generalists could have a significant impact, playing the role of bridge builders between academia and industry, and perhaps actively managing their own portfolios and overseeing scientific startup incubators.\
One of the most important roles generalists could play would be to aid in the development of younger scientific institutions, particularly in the developing world. The specific aim of this program is to train scientists who have significant exposure to the cultural elements of advanced science in multiple disciplines, whose training allowed them to be both scientists as well as participatory anthropologists of the scientific process. For both younger universities in the developed world, as well as new institutions in the developing world, scientific generalists could be critical leaders and agenda setters, and perhaps, will be in a position to identify important research trajectories, or important cultural elements for executing those trajectories, that existing institutions have overlooked.\
It would be incomplete and short-sighted to discuss novel educational initiatives without considering important contemporary developments in online education and open science. Furthermore, given that another major contemporary theme in graduate education is the over production of PhD’s relative to the availability of faculty positions, it would be understandable if a lengthy and extremely demanding variant of the PhD program is difficult to mobilize. One possibility to balance these different factors would be to create an open system for crediting a student for having passed a qualifying examination. Just as universities (and private companies) now offer certificates for coursework completed in a non-degree granting context, an open certification for anyone who is able to pass a qualifying examination would be a valuable credential that an individual could earn to demonstrate competency at the beginning graduate level. For this certification to be available across all disciplines would be one step towards many different forms of educational innovation in the research world, including the training of scientific generalists.\
Before closing, let me re-examine the choice of the qualifying examination as the focal point for this particular proposal and consider alternatives. Although the qualifying examination is an important rite of passage in graduate education, many will correctly point out that it is hardly something that contributes to depth of research maturity. This is certainly a valid point, and in response, I would argue that the purpose of the program is not to train individuals who have achieved the maturity of the best specialists in multiple subjects, but rather individuals who can appreciate and communicate the knowledge of specialists, and who therefore would make strong collaborators, bridge builders, program managers, and journal editors etc. The purpose of organizing a program for training generalists around the qualifying examination is in a large part analogous to why we have such examinations in the first place- they do contribute to some amount of intellectual and technical maturity and are an important experience to have early in one’s education. Furthermore, it is a simply stated idea that would require little institutional overhead, and would circumvent the inevitably controversial process of otherwise designing a curriculum.[^2]\
It is not difficult to imagine alternatives, however. One possibility would be a several year post-doctoral program where fellows rotated through several different laboratories and research groups in succession. Or, in the spirit of the newly emerging trend of “hacker schools” and data-science boot camps, we could imagine creating analogously structured mini-courses designed by experts in the field targeted at advanced graduates whose training was in another field entirely. Indeed, many academic research areas have highly topic specific summer schools and winter schools and one could imagine a several year post-doctoral program built around a handful of different sessions spread across multiple subjects. Perhaps there should be a component both at the beginning of the PhD, like the hybrid qualifying examination system I outline above, as well as a post-doctoral component. Ultimately, it is difficult to imagine that some trial and error would not be required in the design of such a program. In addition, one thing is certain- successfully executing a program like this would require an organization to support students’ growth for many years, and given the fundamentally experimental nature of such an effort, several years longer than we are accustomed to supporting graduate students. Perhaps then, the ideal path forward would be to set into motion multiple efforts aimed at the common goal of training scientific generalists, so that over time, we can learn from our successes and mistakes. To do so, of course, would require long-term institutional efforts to scientifically investigate the efficacy and impact of different training programs.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I would like to thank Aaswath Raman, Doug Bemis, Venkatesh Narayanamurti, and Rob Spekkens for insightful discussions.
[99]{} Michael Nielsen, *Reinventing Discovery: The New Era of Networked Science*, Princeton 2011.
Hendrik Bode, Frederick Mosteller, John Tukey, and Charles Winsor, “The Education of a Scientific Generalist,” *Science* June 3, Vol. 109, 1949.
Computation & Neural Systems Program. “100 Questions.” California Institute of technology. Accessed October 10, 2014. <http://www.cns.caltech.edu/academics/100questions.html>
[^1]: Over 60 years ago, in the essay “The Education of a Scientific Generalist,” Hendrik Bode, Frederick Mosteller, John Tukey, and Charles Winsor argued for a program of similar breadth, but at the undergraduate level [@tukey]. One wonders what their reaction would be to the current proposal given the enormous growth of science in the intervening decades.
[^2]: It is worth mentioning that even within the same subject, there are many different types of qualifying examinations. In the context of this essay, perhaps Caltech’s Computation and Neural Systems program (CNS) provides a possible template. The model they employ is to give students a list of 100 questions that they use as preparatory material in the year leading up to an oral qualifying examination with 5 faculty members [@caltech100]. In a sense, the program is aimed at training “generalists” within the computational and neurobiological sciences. It seems natural to ask if this model could be extended to incorporate other subjects as well. That is, what if a list of 500 questions were to be assembled spanning multiple subjects and a set of oral qualifying examinations were conducted by faculty spanning a number of different departments? Or a few thousand questions from which a student selected some subset to prepare?
|
---
author:
- |
Thang Doan[^1]\
Desautels Faculty of Management\
McGill University\
`[email protected]`\
Bogdan Mazoure$^*$\
Department of Mathematics & Statistics\
McGill University\
`[email protected]`\
Clare Lyle\
School of Computer Science\
McGill University\
`[email protected]`\
title: 'GAN Q-learning'
---
=1
Introduction
============
Related Work
============
GAN Q-learning
==============
Convergence
===========
Experiments
===========
Discussion
==========
Acknowledgments
===============
We would like to thank Marc G. Bellemare from Google Brain for helpful advice throughout this paper.
[^1]: These authors contributed equally.
|
---
abstract: 'We discuss a generic model of Bayesian inference with binary variables defined on edges of a planar graph. The Loop Calculus approach of [@06CCa; @06CCb] is used to evaluate the resulting series expansion for the partition function. We show that, for planar graphs, truncating the series at single-connected loops reduces, via a map reminiscent of the Fisher transformation [@61Fis], to evaluating the partition function of the dimer matching model on an auxiliary planar graph. Thus, the truncated series can be easily re-summed, using the Pfaffian formula of Kasteleyn [@61Kas]. This allows to identify a big class of computationally tractable planar models reducible to a dimer model via the Belief Propagation (gauge) transformation. The Pfaffian representation can also be extended to the full Loop Series, in which case the expansion becomes a sum of Pfaffian contributions, each associated with dimer matchings on an extension to a subgraph of the original graph. Algorithmic consequences of the Pfaffian representation, as well as relations to quantum and non-planar models, are discussed.'
address:
- '$^1$Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545'
- '$^2$Department of Chemistry, Wayne State University, 5101 Cass Ave,Detroit, MI 48202'
author:
- 'Michael Chertkov$^1$, Vladimir Y. Chernyak$^2$ and Razvan Teodorescu$^1$'
title: Belief Propagation and Loop Series on Planar Graphs
---
Introduction
============
[*Bayesian Inference*]{} can be seen both as a sub-field of Information Theory and of general Statistical Inference [@03Mac]. A typical problem in this field is: given observed noisy data and known statistical model of a noisy communication channel (transition probability), as well as a [*prior distribution*]{} for the input (a pre-image), find the most likely pre-image, or compute the [*a posteriori*]{} marginal probability for some part of the pre-image.
This field is also deeply related to [*Combinatorial Optimization*]{}, which is a branch of optimization in Computer Science, related to operations research, algorithm theory and complexity theory [@98PS]. A typical problem in [*Combinatorial Optimization*]{} is: solve, approximate or count (exactly or approximately) instances of problems by exploring the exponentially large space of solutions. In many emerging applications (in magnetic and optical recording, micro-fabrication, chip design, computer vision, network routing and logistics), the data are structured in a two-dimensional grid (array). Moreover, data associated with an element of the grid are often binary and correlations imposed by the problem are local, so that only nearest neighbors on the grid are correlated. Such problems are typically stated in terms of binary statistical models on planar graphs.
In this paper, we discuss a generic problem of Bayesian inference defined on a planar graph. We focus on the problem of weighted counting, or (from the perspective of statistical physics) we aim to calculate the partition function of an underlying statistical model. As the seminal work of Onsager [@44Ons] on the two-dimensional Ising model and its combinatorial interpretation by Kac and Ward [@52KW] have shown, the planarity constraint dramatically simplifies statistical calculations. By contrast, three-dimensional statistical models are much more challenging, and no exact results are known.
Building on the work of physicists, specifically on results of Fisher [@61Fis; @66Fis] and Kasteleyn [@61Kas; @63Kas], Barahona [@82Bar] has shown that calculating the partition function of the spin glass Ising model on an arbitrary planar graph is [*easy*]{}, as the number of operations required to evaluate the partition function scales algebraically, $O(N^3)$, with the size of the system. To prove this, the partition function of the spin-glass Ising model was reduced to a dimer model on an auxiliary graph, and the partition function was expressed as the Pfaffian of a skew-symmetric matrix defined on the graph. The polynomial algorithm was later used in simulations of spin glasses [@93SK]. However, Barahona also added a grain of salt to the exciting positive result, showing that generic planar binary problem is difficult [@82Bar; @Jerrum]. Specifically, evaluating two-dimensional spin glass Ising model in a magnetic field is NP-hard, i.e. it is a task of likely exponential complexity.
When an exact computational algorithm of polynomial complexity is not available, efficient approximations become relevant. Typically, the approximation is built around a tractable case. One such approximate algorithm built around the Fisher-Kasteleyn Pfaffian formula was recently suggested by Globerson and Jaakkola in [@06GJ]. Although this approximation (coined “planar-graph decomposition") gives a provable upper bound for the partition function for some special graphical models, it constitutes just heuristics, i.e. it suffers from lack of error-control and the inability of gradual error-reduction.
Controlling errors in approximate evaluations of the partition function of a graphical model is generally difficult. However, one recent approach, developed by two of us and called Loop Calculus [@06CCa; @06CCb], offers a new method. Loop Calculus allows to express explicitly the partition function of a general statistical inference problem via an expansion (the Loop Series), where each term is explicitly expressed via a solution of the Belief Propagation [@63Gal; @68Gal; @88Pea], or Bethe-Peierls [@82Bax; @35Bet; @36Pei] (BP) equations. This brought new significance to the BP concept, which previously was seen as just heuristics.
The BP equations are tractable for any graph; generally, the number of terms in the Loop Series is exponentially large, so direct re-summation is not feasible. However, since any individual term in the series can be evaluated explicitly (once the BP solution is known), the Loop Series representation offers a possibility for correcting the bare BP approximation perturbatively, accounting for loop contributions one after another sequentially. This scheme was shown to work well in improving BP decoding of Low-Density Parity Check codes in the error-floor regime, where the number of important loop contributions to the Loop Series is (experimentally) small, and the most important loop contributions (comparable by absolute value to the bare BP one) have a simple, single-connected structure [@06CCc; @07Che]. In spite of this progress, the question remained: what to do with other truly difficult cases when the number of important loop corrections is not small, and when the important corrections are not necessarily single-connected? In general, we still do not know how to answer these questions, while a partial answer for the important class of planar models is provided in this paper.
Brief Description of Our Results
--------------------------------
In this manuscript we show that, for any graph (planar or not), the partial sum of the loop series over single-connected loops reduces to evaluation of the full partition function of an auxiliary dimer-matching model on an extended, regular degree-3 graph. Weights of dimers calculated on the extended graph are expressed explicitly via solution of the respective BP equations. The dimer weights can be positive or negative. In general, summing the single-connected partition is not tractable. However, in the planar case, it reduces (through manipulations reminiscent of the Fisher-Kasteleyn transformations) to a Pfaffian defined on the extended graph, which is also planar by construction. Thus, we find a big class of planar graphical models which are computationally tractable by reduction (via a BP/gauge transformation) to a loop series including only single-connected loops, and summable into a Pfaffian. Moreover, we find that the partition function of the entire Loop Series is generally reducible to a weighted Pfaffians series, where each higher-order Pfaffian is associated with a sum of dimer configurations on a modified subgraph of the original graph. Each term in the Pfaffian series is computationally tractable via the Belief Propagation solution on the original graph.
The material in the manuscript is organized as follows. A formal definition of the model is given in Section \[subsec:Model\] and a brief description of Loop Calculus [@06CCa; @06CCb] forms Section \[subsec:LoopCalc\]. Some introductory material on the graphical transformations is also given in \[sec:Graph\]. Section \[sec:SingleLoops\] is devoted to re-summation of the single-connected loops in the Loop Series (we called it single connected partition). Section \[subsec:ToDimer\] introduces graphical transformation from the original graph ${\cal G}$ to the extended graph ${\cal G}_e$, reminiscent the Fisher transformation [@61Fis; @66Fis]. This allows to restate the single-connected loop partition of the Loop Series on the original graph in terms of a sum over dimer configurations on the extended graph. Subsection \[subsec:Pfaff\] adapts the Kasteleyn transformation [@61Kas; @63Kas] to our case, thus expressing the partition function of the single-connected series as a Pfaffian of a matrix defined on the extended graph. Section \[sec:Easy\] describes a set of graphical models reducible under Belief Propagation gauge (transformation) to a Loop Series which is computationally tractable. Section \[sec:Full\] describes the representation of the Loop Series for planar graphs in terms of the Pfaffian Series, where each Pfaffian sums dimer matchings on a graph extended from a subgraph of ${\cal G}$, with the later correspondent to exclusion of an even set of vertices from ${\cal G}$. Grassmann representations, as well as fermionic models are discussed in Section \[sec:Grass\]: a general set of Grassmann models on super-spaces is given in Section \[subsec:GrassGen\], while Section \[subsec:comments-fermions\] addresses the relation between binary models and integrable hierarchies. A brief list of future research topics is given in Section \[sec:Con\].
Vertex-function Model {#subsec:Model}
---------------------
We introduce an undirected graph ${\cal G}=({\cal V},{\cal E})$ consisting of vertices ${\cal V}=(a=1,\cdots,N)$ and edges ${\cal
E}$. This study focuses mainly on planar graphs, like those emerging in communication or logistics networks connecting or relating nearest neighbors on a 2d mesh or terrain. However, the material discussed in the present and the following Subsections is general, and applies to any graph, planar or not. A binary variable, $\sigma_{ab}=\pm 1$, which we will also be calling a spin, is associated with any edge $(a,b)\in{\cal E}$. The graphical model is defined in terms of the probability function $$\begin{aligned}
p(\vec{\sigma})=Z^{-1}\prod_{a\in{\cal V}}f_a(\vec{\sigma}_a),
\label{P_sigma}\end{aligned}$$ for a spin configuration $\vec{\sigma}\equiv \{ \sigma_{ab}=\pm 1| \forall
(a,b)\in{\cal E} \}$. In (\[P\_sigma\]), $\vec{\sigma}_a=(\sigma_{ab}| \forall b,\mbox{ s.t. }(a,b)\in{\cal
E})$ is the vector built from all edge variables associated with the given vertex $a$. $f_a$’s are positive and otherwise we will assume no restrictions on the factor functions. $Z$ is the normalization factor, the so-called partition function of the graphical model.
We refer to (\[P\_sigma\]) as “vertex-function" models, according to statistical physics notation [@82Bax]. In the information theory, they are known as Forney-style graphical models [@01For; @01Loe].
We will assume in the following that the degree of connectivity of any vertex in the graph is three. Note that this is not a restrictive condition, as the $n$-th order vertices, correspondent to $n$-spin interactions with $n>3$, can always be represented in terms of a product of triplet terms. Then the $n$-th degree vertex can be transformed into a planar graph consisting of degree three vertices. We discuss transformations to the triplets, in general but also on some examples (Ising Model and Parity Check Decoding of a linear code), in \[sec:Graph\].
Loop Calculus {#subsec:LoopCalc}
-------------
Loop Calculus [@06CCa; @06CCb] gives an explicit expression for $Z$ through the Loop Series: $$\begin{aligned}
&& Z=Z_0 \cdot z,\,\, z\equiv \left(1+\sum_{\it C}
\prod\limits_{a\in{\it C}}\mu_{a,\bar{a}_{\it C}}\right),\,\,
\mu_{a,\bar{a}_{\it C}}\equiv\frac{\tilde{\mu}_{a,\bar{a}_{\it C}}}
{\prod\limits_{b\in {\it C}}^{(a,b)\in {\it C}}\sqrt{1-m_{ab}({\it C})}} \label{Zseries}\\
&& m_{ab}=\sum_{\sigma_{ab}}\sigma_{ab}b_{ab}(\sigma_{ab}),\,\,
\tilde{\mu}_{a,\bar{a}_{\it C}}=\sum_{\vec{\sigma}_a}\prod_{b\in \bar{a}_{\it C}}(\sigma_{ab}-m_{ab})
b_a(\vec{\sigma}_a),
\label{mu_ab_a}\end{aligned}$$ where ${\it C}$ can be any allowed generalized loop on the graph ${\cal G}$, i.e. ${\it C}$ is a subgraph of ${\cal G}$ which does not contain any vertices of degree one; $\bar{a}_{\it C}$ is a set of vertices of graph ${\cal G}$ which are also contained in the generalized loop ${\cal C}$ (by construction $\bar{a}$ consists of two or three elements); and $b_a(\vec{\sigma}_a)$ and $b_{ab}(\sigma_{ab})$ are beliefs associated with vertex $a$ and edge $(ab)$. The beliefs are defined via message variables $\eta_{ab}\neq\eta_{ba}$ $$\begin{aligned}
&& \forall\ (a,b)\in{\cal E}:\quad
b_{ab}(\sigma_{ab})=\frac{\exp\left((\eta_{ab}+\eta_{ba})\sigma_{ab}\right)}{
2\cosh\left(\eta_{ab}+\eta_{ba}\right)},\label{bab}\\
&& \forall\ a\in{\cal V}:\quad
b_a(\vec{\sigma}_a)=\frac{f_a(\vec{\sigma}_a)\exp\left(\sum_b^{(a,b)\in{\cal
E}}\eta_{ab}\sigma_{ab}\right)}{\sum_{\vec{\sigma}_a}
f_a(\vec{\sigma}_a)\exp\left(\sum_c^{(a,c)\in{\cal E}}\eta_{ac}\sigma_{ac}\right)},
\label{ba}\end{aligned}$$ solving the following system of the Belief Propagation (BP) equations $$\!\!\!\!\!\!\!\!\!\!\!\!\forall\ (a,b)\in{\cal E}:\quad
\sum_{\vec{\sigma}_a}
f_a(\vec{\sigma}_a)\exp\left(\sum_b^{(a,b)\in{\cal
E}}\eta_{ab}\sigma_{ab}\right)\left(\sigma_{ab}-\tanh\left(\eta_{ab}+\eta_{ba}\right)\right)=0.
\label{bb}$$ The bare (BP) partition function $Z_0$ in Eq. (\[Zseries\]) has the following expression in terms of the message variables: $$Z_0=\frac{\prod_a\sum_{\vec{\sigma}_a\in{\cal V}}f_a\left(\vec{\sigma}_a\right)
\exp\left(\sum_{(a,b)\in{\cal E}} \eta_{ab}\sigma_{ab}\right)}
{\prod_{(a,b)\in{\cal E}}\left [ 2\cosh\left ( \eta_{ab}+\eta_{ba}\right)\right ]}.\label{Z0}$$
BP equations (\[bb\]) are interpreted as conditions on the gauge transformations, leaving the partition function of the model invariant. These equations may allow multiple solutions, related to each other via respective gauge transformations. The multiple solutions correspond to multiple extrema of the Bethe Free Energy and Loop Series can be constructed around any of the BP solutions. [^1]
Re-summation of the Single-connected Partition {#sec:SingleLoops}
==============================================
In the following we will show how to re-sum a part of the Loop Series accounting for all the single-connected loops, i.e. subgraphs of ${\cal G}$ with all vertices of degree two $$\begin{aligned}
Z_s=Z_0 \cdot z_s,\quad z_s=1+\sum_{{\it C}\in{\cal G}}^{\forall a\in {\it
C},\ |\delta(a)|_{\it C}=2}r_{\it C}, \label{sub_graph}\end{aligned}$$ where $|\delta(a)|_{\it C}$ stands for the number of neighbors of $a$ within ${\it C}$. The evaluation will consist of the following two steps:
- Show that $z_s$ is equal to the partition function of the dimer-matching model on an auxiliary graph, ${\cal G}_e$. The graph will be constructed from the original ${\cal G}$ by a transformation reminiscent of the Fisher’s trick, introduced in [@61Fis; @66Fis; @82Bar] to streamline reduction of Ising model to the dimer-matching model;
- Use the Pfaffian formula of Kasteleyn [@61Kas; @63Kas; @82Bar] to reduce $z_s$ to a Pfaffian of a skew-symmetric matrix defined on ${\cal G}_e$. Note that complexity of the Pfaffian evaluation is $N^3$, where $N$ is the size of ${\cal G}$.
Note: while A) is valid for any graphical model, B) applies only to the planar case.
Transformation to Dimer Matching Problem {#subsec:ToDimer}
----------------------------------------
![Left panel: Transformation from a vertex of ${\cal G}$ to respective three-vertex of the extended graph ${\cal G}_e$. Right panel: maps from the colorings of a vertex of ${\cal G}$ to coloring of the respective $3$-vertex of ${\cal G}_e$. Notice, that the coloring of the external edges of ${\cal G}_e$ are reversed in comparison with the coloring of original edges on ${\cal G}$.[]{data-label="trans"}](g1.pdf "fig:"){width="6cm"} ![Left panel: Transformation from a vertex of ${\cal G}$ to respective three-vertex of the extended graph ${\cal G}_e$. Right panel: maps from the colorings of a vertex of ${\cal G}$ to coloring of the respective $3$-vertex of ${\cal G}_e$. Notice, that the coloring of the external edges of ${\cal G}_e$ are reversed in comparison with the coloring of original edges on ${\cal G}$.[]{data-label="trans"}](g2.pdf "fig:"){width="6cm"}
Following the construction of Fisher [@61Fis; @66Fis], we expand each vertex of ${\cal G}$ into a three-vertex of the extended graph ${\cal G}_e$, according to the scheme shown in the left panel of Figure 1. Consider a vertex $a$ of ${\cal G}$ and assume that $b,c,d$ are three neighbors of $a$ on ${\cal G}$. For each vertex $a$, there are three $\mu_{a;\bar{a}_{\it C}}$ contributions of degree two within a generalized loop ${\it C}$, i.e. with $|\delta \bar{a}_{\it C}|=2$, which can possibly contribute to the single-connected partition $r_s$: $\mu_{a;bc},\mu_{a;bd},\mu_{a;cd}$. We associate the three weights with internal edges of the respective three-vertex of ${\cal G}_e$, while the weights of all the external edges of the three-vertex are equal to unity. Then any coloring of the original graph, marking a single connected loop of ${\cal G}$, is in the one-to-one correspondence to a dimer-matching (which we also call coloring) of ${\cal G}_e$. The weights and coloring assignments are illustrated on an example at the left panel of Figure 1. An example of transformation mapping a single-connected-loop on ${\cal G}$ respective dimer on ${\cal G}_e$ is shown in Figure \[trans\_example\].
This map from the single-connected loops to dimers leads to the following representation for the single-connected partition $z_s$ $$\begin{aligned}
z_s=\sum_{\vec{\pi}}\prod_{(a,b)\in{\cal G}_e}\left(\mu_{ab}\right)^{\pi_{ab}}
\prod_a \delta\left(\sum_b^{(a,b)\in{\cal G}_e}\pi_{ab},1\right),
\label{zsd}\end{aligned}$$ where the dimer-weights on ${\cal G}_e$ are defined according to the simple rules explained in the previous paragraph. One finds that the right hand side of (\[zsd\]) is nothing but the partition function of a dimer-matching problem on ${\cal G}_e$.
![Example of ${\cal G}$ (upper left) to ${\cal G}_e$ (lower right) map. Single connected loop of ${\cal G}$ (shown in red) is in one-to-one correspondence with a valid dimer matching of ${\cal G}_e$, where dimers are also shown in red.[]{data-label="trans_example"}](g3.pdf){width="6cm"}
Pfaffian Expression for the Partition Function {#subsec:Pfaff}
----------------------------------------------
![$\vec{p}$ orientation (left panel) and respective dimer (matching) configurations (right panel) correspondent to example of ${\cal G}_e$ described by Eq. (\[Pfaf-eq\]).[]{data-label="Pfaf-fig"}](g4.pdf "fig:"){width="6cm"} ![$\vec{p}$ orientation (left panel) and respective dimer (matching) configurations (right panel) correspondent to example of ${\cal G}_e$ described by Eq. (\[Pfaf-eq\]).[]{data-label="Pfaf-fig"}](g5.pdf "fig:"){width="6cm"}
Kasteleyn has shown in [@61Kas; @63Kas] (see also [@82Bar]) that $z_s$ is equal to a Pfaffian (the square root of determinant) of a skew-symmetric matrix $\hat{A}=-\hat{A}^{t}$ of size $N_a\times N_a$, where $N_a$ is the number of vertices in ${\cal G}_a$. Each element of the matrix with $a>b$ (ordering is arbitrary, but it is fixed once and forever) is $A_{ab}=p_{ab}z_{ab}$, where $p_{ab}=\pm 1$. There are many possible choices of $\vec{p}=(\pi_{ab}=\pm 1|(a,b)\in{\cal G}_e)$ which guarantee the Pfaffian relation: $z_s=\sqrt{\mbox{det}\hat{A}}$. A simple constructive way of choosing such a valid $\vec{p}$ is to relate it to orientation of edges in a directed version of ${\cal
G}_a$, built according to the following “odd-face" rule: number of clockwise-oriented segments of any internal face of ${\cal G}_e$ should be negative. [^2] Example of a valid orientation is shown in Figure \[trans\_example\] and the respective expressions are $$\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
z_{s;{\it example}}=\mu_{12}\mu_{34}+\mu_{14}\mu_{23}=
\sqrt{\mbox{Det}\hat{A}},\quad
\hat{A}=\left(\begin{array}{cccc} 0 & -\mu_{12} & 0 &
-\mu_{14}\\ \mu_{12} & 0 & \mu_{23} & -\mu_{24}\\
0 & -\mu_{23} & 0 & \mu_{34}\\
\mu_{14} & \mu_{24} & -\mu_{34} & 0\end{array}\right)
\label{Pfaf-eq}$$
Since calculating the determinant requires $\sim N_a^3$ operations, one finds that re-summation of all the single-connected loops in the Loop Series expression for the partition function of a planar graphical model can be done efficiently in $O(N_a^3)$ steps.
Tractable Problems Reducible to Single-Connected Partition {#sec:Easy}
==========================================================
In the case of a general vertex-function graphical model, the BP-gauge transformations, described by the set of BP equations (\[bb\]), result in exact cancelation in the Loop Series of all the subgraphs containing at least one vertex of degree one within the subgraph. Thus, for the graph with all vertices of degree three, any vertex contributing a generalized loop (subgraph) should be of degree two or three within the subgraph. As shown in the previous Section, if one ignores generalized loops with vertices of degree three and the original graph is planar, the resulting sub-series (single-connected partition) is computationally tractable, i.e. the number of operations required to evaluate the single-connected partition is cubic in the system size (not exponential !).
In this Section we discuss the class of planar models whose Loop Series do not contain any generalized loops with vertices of degree three. According to Section \[sec:SingleLoops\], these models are tractable.
Indeed, it is known that BP Eqs. (\[bb\]) have at least one solution for the set of messages $\{\eta\}$ on any graph and for any factor functions defined on the vertices of the graph. The aforementioned requirement for the generalized loop not to contain any vertex of degree three translates into the following set of additional equations $$\!\!\!\!\!\!\!\!\forall\ a\in{\cal G}:\quad
\sum_{\vec{\sigma}_a}
f_a(\vec{\sigma}_a)\prod_{b}^{(a,b)\in{\cal E}}\left(\exp\left(\eta_{ab}\sigma_{ab}\right)
\left(\sigma_{ab}-\tanh\left(\eta_{ab}+\eta_{ba}\right)\right)\right)=0.
\label{b3}$$ Considered together, the set of Eqs. (\[bb\],\[b3\]) is overdefined, i.e. it cannot be solved in terms of $\eta$ variables for any values of the factor functions. However, if one allows flexibility in the factor functions, and, in fact, considers Eqs. (\[bb\],\[b3\]) as a set of conditions on both the messages $\{\eta\}$ and the factor functions $\{f\}$, one arrives at a big set of possible solutions.
Therefore, Eqs. (\[bb\],\[b3\]) define a big set of models reducible via BP transformations to a tractable Loop Series consisting only of single connected loops.
Moreover, the relations we established may be reversed. One may start from an arbitrary Loop Series consisting of only single connected loops, apply an arbitrary gauge transformation leaving the Loop Series invariant (these transformations are not necessarily of BP type), and arrive at a graphical model with some set of factor functions. At first sight, the resulting graphical model might not look tractable, but it actually is, by construction.
Loop Series as a Pfaffian Series {#sec:Full}
================================
Let us notice that the general planar problem (e.g. spin glass in a magnetic field) is NP-hard [@82Bar], and it is thus not surprising that full re-summation does not allow expression in terms of a single Pfaffian (or a determinant).
On the other hand, we already found that a part of the Loop Series, specifically its single-connected partition, reduces to a computationally tractable Pfaffian. This suggests to represent the full Loop Series as a sum over terms, each representing a set of triplets (fully colored vertices of degree tree on ${\cal G}$): $$z=\sum_{\Psi} z_\Psi\prod_{a\in\Psi}^{|\bar{a}|=3}\mu_{a;\bar{a}},\label{ser_trip}$$ where $\Psi$ is either the empty set or any set of even nodes on ${\cal G}$; $\mu_{a;\bar{a}}=\mu_{a;bcd}$ are the weights from Eq. (\[Zseries\]) associated with the triplet $(a;b,c,d)$, such that $(a,b),(a,c),(a,d)\in{\cal E}$; and $z_\Psi$ is the sum over all generalized loops (proper Loop Series colorings, i.e. subgraphs) of ${\cal G}$ such that all nodes of $\Psi$ are fully colored (all edges adjusted to the nodes belong to the generalized loop), while any other vertices of ${\cal G}$ are not colored or only partially colored. Thus, the first term in Eq. (\[ser\_trip\]), where $\Psi$ is the empty set, represents the single-connected partition, $z_s$.
We show here that not only the first term in Eq. (\[ser\_trip\]), associated with $\Psi=\varnothing$, but any term $z_\Psi$ in Eq. (\[ser\_trip\]) is computationally tractable, being equal to a Pfaffian of a matrix defined on ${\cal G}_e$.
Indeed, it is straightforward to verify that the generalized loops associated with the given set of triplets (fully colored vertices) from the set $\Psi$ are in one-to-one correspondence with the set of dimer matchings on ${\cal G}_{e;\Psi}$, which is a subgraph of ${\cal G}_e$ with all $3$-vertices correspondent to $\Psi$, and external edges connected to the vertices, completely removed. Notice that some vertices of ${\cal G}_{e;\Psi}$ are of degree two. (These are vertices neighboring the removed triplets of $\Psi$.)
An example of a ${\cal G}_{e;\Psi}$ construction is given in Figure \[fig:Psi\]. One associates weights to the edges of ${\cal G}_{e;\Psi}$ in exactly the same way as for the single-connected partition: the weights of all the external edges of $3$-vertices of ${\cal G}_{e;\Psi}$ are equal to unity, while the internal edges are associated with the respective values $\mu_{a;bc}$, defined in Eq. (\[mu\_ab\_a\]).
![Two generalized loops (shown on the top) of an exemplary ${\cal G}$ correspondent to the same configuration of triplets $G$, $|G|=2$, and their respective dimer configurations on ${\cal G}_{e;\Psi}$ (shown on the bottom).[]{data-label="fig:Psi"}](g10.pdf "fig:"){width="6cm"} ![Two generalized loops (shown on the top) of an exemplary ${\cal G}$ correspondent to the same configuration of triplets $G$, $|G|=2$, and their respective dimer configurations on ${\cal G}_{e;\Psi}$ (shown on the bottom).[]{data-label="fig:Psi"}](g11.pdf "fig:"){width="6cm"}
For any of ${\cal G}_{e;\Psi}$ one constructs the skew-symmetric $\hat{A}_\Psi$ matrix according to the Kasteleyn rule for the dimer-matching model described in Section \[subsec:ToDimer\]. As before, the dimensionality of the matrix is $|{\cal G}_{e;\Psi}|\times |{\cal G}_{e;\Psi}|$ and each element of the matrix is the product of the respective dimer weight and orientation sign. Notice that the choice of signs for the elements of $\hat{A}_\Psi$ depends on the set of “excluded" triplets $\Psi$, and thus $\hat{A}_\Psi$ is not simply a minor of the original matrix $\hat{A}$, the one corresponding to the single-connected partition (without exclusion). Thus, $$\begin{aligned}
z_{\Psi}=\mbox{Pf}\left(\hat{A}_{\Psi}\right)=
\sqrt{\mbox{Det}\left(\hat{A}_{\Psi}\right)}.
\label{zdm2}\end{aligned}$$ Eqs. (\[ser\_trip\],\[zdm2\]) describe the Pfaffian series representation for the Loop Series of the planar problem.
Fermion Representation and Models {#sec:Grass}
=================================
Any Pfaffian in Eq.(\[zdm2\]) allows a compact representation in terms of Grassmann variables [@87Ber]. Indeed, let us associate a Grassmann (anti-commuting or fermionic) variable $\theta_a$ with each vertex of ${\cal G}_e$. The Grassmann variables satisfy $$\begin{aligned}
\forall (a,b)\in{\cal G}_e:\quad
\theta_a\theta_b+\theta_b\theta_a=0, \label{Grass}\end{aligned}$$ and commute with ordinary $c$-numbers. One also introduces the Berezin integration rules over the Grassmann variables $$\begin{aligned}
\int d\theta=0,\quad\int \theta d\theta=1.\label{Ber}\end{aligned}$$ This translates into the following rule of Gaussian integration over the Grassmann variables: $$\begin{aligned}
\int
\exp\left(-\frac{1}{2}\vec{\theta}^{t}\hat{A}\vec{\theta}\right)d\vec{\theta}=\mbox{Pf}(\hat{A})=
\sqrt{\mbox{det}(\hat{A})}, \label{BerGauss}\end{aligned}$$ where $\vec{\theta}$ is the vector of the Grassmann variables over the entire graph, $\vec{\theta}=\left(\theta_i|i\in{\cal
G}_a\right)$ and $\hat{A}$ is an arbitrary skew-symmetric matrix on the graph. For example, applying this formula to the first term of the Pfaffian series (\[ser\_trip\]) one derives $$\begin{aligned}
z_{\vec{0}}=\int
\exp\left(-\frac{1}{2}\vec{\theta}^{t}\hat{A}\vec{\theta}\right)d\vec{\theta}.
\label{Grassmann0}\end{aligned}$$ In general, any term in the Pfaffian series of Eq. (\[ser\_trip\]) can be represented as a Gaussian Grassmann integrable, however with different Gaussian kernels, not reducible simply to minors of $\hat{A}$.
Graphical Models on Super-Spaces {#subsec:GrassGen}
--------------------------------
In this Subsection we first consider graphical models on spaces generalizing the $2$-point (binary) set to super-spaces containing commuting and anti-commuting parts. The models will be defined on arbitrary (non necessarily planar) graphs. Then, we return to the simple example (\[Grassmann0\]) of pure dimer model with the Grassmann (anticommuting) variables defined on vertices of ${\cal
G}_e$, to see that the model can be restated as the vertex-function Grassmann model on the original graph ${\cal G}$.
The general class of vertex-function models can be introduced as follows. For our graph, ${\cal G}$, consider a set of spaces $\{M_{a\alpha}|a\in\alpha\}$, i.e., we associate a space with any edge, $\alpha$, together with a vertex, $a$, that belongs to the edge. For simplicity we assume the spaces to be identical, i.e., $M_{a\alpha}\cong M$ for all $a\in\alpha$. The basic variables are $\sigma_{a\alpha}\in M_{a\alpha}$. We also introduce the notation (all products below are cartesian) $$\begin{aligned}
\label{define-spaces} M_{a}=\prod_{\alpha\ni a}M_{a\alpha}, \;\;
M_{\alpha}=\prod_{a\in\alpha}M_{a\alpha}, \;\; {\cal
M}=\prod_{a\alpha}^{a\in\alpha}M_{a\alpha}=\prod_{a}M_{a}=\prod_{\alpha}M_{\alpha};
\;\; \\
\vec{\sigma}_{a}\in M_{a}, \;\; \vec{\sigma}_{\alpha}\in
M_{\alpha}, \;\; \vec{\sigma}\in{\cal M}.\end{aligned}$$ Note that any $M_{\alpha}$ is a two-component cartesian product. The vertex-function model is determined by a set of vertex functions $f_{a}(\vec{\sigma}_{a})$ defined on $M_{a}$ and a set of integration measures $d\mu_{\alpha}(\vec{\sigma}_{\alpha})$ on $M_{\alpha}$. The model partition function is $$\begin{aligned}
\label{define-Z-continuous} Z=\int_{{\cal
M}}\prod_{\alpha}d\mu_{\alpha}(\vec{\sigma}_{\alpha})\prod_{a}f_{a}(\vec{\sigma}_a).\end{aligned}$$ For the particular case when measures have supports restricted to the diagonals $M\cong \Delta_{\alpha}\subset M_{\alpha}\cong M\times
M$, i.e. ${\rm supp}\mu_{\alpha}\subset\Delta_{\alpha}$, we can consider the basic variables that belong to the diagonals. This corresponds to a more conventional formulation of the vertex-function models with the variables residing on edges. Note that the models introduced allow for loop-tower calculus [@07CC], formulated in terms of fixing a proper gauge. The BP gauge fixing for a general vertex-function model described by Eq. (\[define-Z-continuous\]) is nothing more than choosing basis sets in the vector spaces (maybe infinite-dimensional) of functions in $M_{a\alpha}$. A standard binary model, defined in Eq. (\[P\_sigma\]), corresponds to the choice $M=\{0,1\}$ of the basic space to be a $2$-point set. Vertex models with $q$-ary alphabet, e.g. discussed in [@07CC], are described by $M=\{0,1,\ldots,q-1\}$. Continuous models are obtained if $M$ is chosen to be a manifold of dimension $m$. The continuous case can be extended to the choice of $M$ to be a supermanifold $M$ of dimension $(m_{+},m_{-})$ that contains $m_{-}$ Grassmann (anticommuting) coordinates and whose substrate $\bar{M}\subset M$ is an $m_{+}$-dimensional manifold. Note that a manifold can be considered as a supermanifold with zero odd dimension $m_{-}=0$. In the remainder of this Subsection we will be dealing with an opposite case of the zero even dimension $m_{+}=0$, specifically with the purely Grassmann case of the $(0,1)$ supermanifold.
Eq. (\[Grassmann0\]) is the partition function of a model stated in terms of Grassmann variables defined on the vertices of ${\cal G}_e$. The extended graph ${\cal G}_e$ is constructed from the original graph ${\cal G}$ so that a vertex of ${\cal G}$ extends into a triangle with three vertices of degree three (see the left panel of Figure \[trans\]). Therefore, the three Grassmann variables in (\[Grassmann0\]) are associated with a vertex of ${\cal G}$. Then, Eq. (\[Grassmann0\]) defined on ${\cal G}_e$ allows an obvious reformulation in the vertex-function form (\[define-Z-continuous\]) on ${\cal G}$, where $\vec{\sigma}_{a}$ represents the three Grassmann variables that reside on the vertices of ${\cal G}_{e}$, obtained by expanding the vertex $a$ of the original graph. The dimer weights for the three edges of ${\cal G}_{e}$ associated with the extended vertex of ${\cal G}$ are encoded in the Gaussian function $f_{a}(\vec{\sigma}_{a})$. The dimer weight associated with an edge of ${\cal G}_{e}$ that represents and edge $\alpha$ of the original graph ${\cal G}$ is encoded in the integration measure $d\mu_{\alpha}(\vec{\sigma}_{\alpha})$.
Also notice that the vertex-function Grassmann model on a planar graph ${\cal G}$ can be restated as a model on the triangulated graph, dual to ${\cal G}$, with complex fermion (Grassmann) variables associated with the edges of the dual graph and functions associated with a face (elementary triangle) of the dual graph (Figure \[fig:Triang\] illustrates the duality transformation). One interesting conclusion here is that the sequence of transformations discussed above leads us from a special binary model on a planar graph ${\cal G}$ to a Gaussian fermion (Grassmann) model on the dual graph, thus representing an instance of the disorder operator approach of Kadanoff-Ceva [@71KC] developed originally for the Ising model on a square lattice.
Comments on Relation to Quantum Algorithms and Integrable Hierarchies {#subsec:comments-fermions}
---------------------------------------------------------------------
A mapping of a classical inference problem onto finding an expectation value in a corresponding quantum model takes on a natural interpretation as a [*quantum*]{} algorithm. This can be tried by using the theory of the infinite Kadomtsev-Petviashvilii (KP) hierarchy, specifically its fermionic formulation [@SMJ]. Consider $1D$ lattice fermions $\psi_{k},\psi_{k}^{*}$ with $k\in{\mathbb Z}$ and introduce the population $\widehat{n}_{k}=\psi_{k}^{*}\psi_{k}$ and shift operators $\widehat{H}_{k}=\sum_{j\in{\mathbb
Z}}\psi_{k+j}^{*}\psi_{j}$. Let $|0\rangle$ denote the standard many-particle vacuum state where all single-fermion orbitals with $k\leq 0$ are occupied, and $|W\rangle$ is some uncorrelated (i.e. represented by a single Slater determinant) many-particle state, which is sufficiently close to $|0\rangle$. Introducing ${\bm
t}=t_{1},t_{2},\ldots$, $\bar{{\bm
t}}=\bar{t}_{0}, \bar{t}_{1},\bar{t}_{2},\ldots$, and ${\bm\xi}=\ldots,\xi_{-1},\xi_{0},\xi_{1},\ldots$ we consider an expectation value $$\begin{aligned}
\label{define-Z} Z_{W}({\bm t},\bar{{\bm t}},{\bm\xi})=\langle 0|e^{\sum_{k> 0}t_{k}\widehat
{H}_{k}}e^{\sum_{k\in{\mathbb Z}}\xi_{k}\widehat{n}_{k}}e^{\sum_{k\leq
0}\bar{t}_{-k}\widehat{H}_{k}}|W\rangle\end{aligned}$$ The approach is based on mapping the partition function of a classical inference problem on a graph onto a calculation of an expectation value represented by Eq. (\[define-Z\]). We have established such a mapping for some simple Grassmannian models on planar graphs [@08TCC], where all the details on the suggested approach will be presented. Note that in the case ${\bm\xi}=0$ and $\bar{{\bm t}}=0$ the expectation value $Z_{W}({\bm
t},0,0)=\tau_{W}({\bm t})$ is related to the $\tau$-function of the KP integrable hierarchy.
Future Challenges {#sec:Con}
=================
We conclude with a brief and incomplete discussion of future challenges and opportunities raised by this study.
- We plan to extend the study looking at new approximate schemes for intractable planar problems. One new direction, suggested in Section \[sec:Easy\], consists of exploring the vicinity of the computationally tractable models reducible via the BP-gauge transformation to the series of single-connected loops. It is also of great interest to explore the vicinity of integrable tractable models mentioned in \[subsec:comments-fermions\].
- Perturbative exploration of a larger set of intractable non-planar problems which are close, in some sense, to planar problems, constitutes another interesting extension of the research. Here, one would aim to blend the aforementioned planar techniques with planar (or similar) decomposition techniques, e.g. these of the type discussed in [@06GJ].
- One important component of our analysis consisted in the Pfaffian re-summation of the single-connected loop (dimer) contributions, which is a special feature of the graph planarity. On the other hand, it is known that the planarity is equivalent to the graph being minor-excluded with respect to $K_5$ and $K_{3,3}$ subgraphs. Therefore, one wonders if there exists a generalization of the Pfaffian reduction to partition functions of models from other and/or broader graph-minor classes defined within the graph-minor theory [@05Lov]?Likewise, comparing with previous studies of the non-planar/non-spherical cases, based on the dimer approach [@n1; @n2; @n3].
- Extending the Loop Series analysis of the binary planar problem to the q-ary case seems feasible via the Loop Tower construction of [@07CC]. This research should be of a special interest in the context of recently proposed polynomial quantum algorithm for calculating partition function of the Potts model [@07AAEL]. Besides, recent progress [@DiFrancesco; @Kenyon] shows that a Kasteleyn-type approach is extendable to a $q$-ary case, leading to the concept of “heaps of dimers", and (in the continuum limit) to fascinating connections with special, highly symmetric complex surfaces, known as Harnack curves.
- One would also be interested to study how (and if) phase transitions in the disorder-averaged planar ensembles, e.g. analyzed in [@Sherrington; @83DD; @Picco; @Hastings; @Tsvelik; @Efetov; @E-A; @FH; @SKP], are related to distribution of parameters characterizing computational tractability (complexity) of the models.
- In [@Pascal], the problem of finding all pseudo-codewords in a finite cycle code (corresponding to the type of graphical model discussed in this paper), was addressed by constructing a generating function known as graph zeta function [@Stark]. The interesting fact discovered in [@Pascal] is that this generating function of pseudo-codewords has a determinant formulation, based on a discrete graph operator. Hence, one may anticipate an existence of yet uncovered relation between the graph zeta function and a Pfaffian-Loop resummation of related graphical models.
Acknowledgments
===============
Research of M.C. and R.T. was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE C52-06NA25396, and specifically the LDRD Directed Research grant on [*Physics of Algorithms*]{}. M.C. also acknowledges support of the Weston Visiting Professorship program at the Weizmann Institute of Science, where he started to work on the manuscript. V.Y.C. acknowledges support through the start-up funds from Wayne State University.
Graphical Transformations {#sec:Graph}
=========================
In this Appendix we discuss graphical transformations reducing any binary problem to the vertex-function model described by Eq. (\[P\_sigma\]), where all vertices are of degree three. Our main focus here is on the planar graphs, and on the graphical transformations preserving planarity. However some of the transformations and considerations discussed below apply to an arbitrary graph.
![Transformation from binary variable on a vertex, $\sigma_1$, to set of variables, $\sigma_{12},\sigma_{13},\sigma_{14}$ on respective edges. []{data-label="fig:VtoE"}](g6.pdf){width="6cm"}
Often the original binary model is not represented in the vertex-function form. Some or all binary variables describing a problem may actually be assigned to vertices of a graph, then respective functions are associated with edges and not vertices. Obviously, one can also reformulate the model reducing it to the vertex (canonical for our purposes) form. The transformation is illustrated in Figure \[fig:VtoE\]. Algebraic form of the transformation shown in the Figure reads, $\sum_{\sigma_1}f_{12}(\sigma_1,\sigma_2)f_{13}(\sigma_1,\sigma_3)f_{14}(\sigma_1,\sigma_4)=
\sum_{\sigma_{12},\sigma_{13},\sigma_{14}}\chi(\sigma_{12},\sigma_{13},\sigma_{14})
f_{12}(\sigma_{12},\sigma_2)f_{13}(\sigma_{13},\sigma_3)f_{14}(\sigma_{14},\sigma_4)$, where $\chi(\sigma_{12},\sigma_{13},\sigma_{14})$ is the characteristic function equal to unity if all variables $\sigma_{12},\sigma_{13},\sigma_{14}$ are equal each other and equal to zero otherwise.
![Transformation which allows reduction of an $N$-vertex to two $(N-1)$-vertices. It is assumed that (1) number of nodes in the gray area is not large, i.e. $O(N)$, (2) the new graph (on the right) is planar, (3) ordering (say clockwise) of the external nodes is preserved. The number of parameters characterizing the $N$-vertex is $2^N$ or smaller, thus the number of parameters characterizing the two $(N-1)$ vertices and vertices from the gray area is sufficient, i.e. $>2\cdot2^{N-1}$, to parameterize the original vertex.[]{data-label="fig:NN-1"}](g7.pdf){width="6cm"}
Next, let us notice that, given a vertex-function model (\[P\_sigma\]) with the degree of connectivity higher than three, one can always perform a sequence of transformations reducing the degree of connectivity of all the nodes in the resulting graphical model to three. An elementary graphical transformation of the kind is illustrated in Figure \[fig:NN-1\]. It is assumed that the transformation is applied sequentially to vertices of degree larger than three till none of these are left. The end result is that: (a) there are no vertices of degree larger than three left within the graph; (b) the increase in the total number of vertices is polynomial; (c) if the original graph is planar the resulting graph is also planar.
The set of transformations just described is general, and thus often inefficient, in the sense that knowing specific form of the factor functions one can practically always do a more efficient, customized and simpler reduction. Below we will illustrate this point on examples.
Ising Model {#subsec:Ising11}
-----------
![Planar triangulated graph (black) and its dual (red).[]{data-label="fig:Triang"}](g8.pdf){width="6cm"}
The spin glass Ising model is usually defined in terms of $\sigma_i=\pm 1$ variables associated with vertices of the graph $$\begin{aligned}
p({\bm \sigma})=Z^{-1}\exp\left(\sum_{(i,j)}J_{ij}\sigma_i\sigma_j\right),
\label{Ising}\end{aligned}$$ where summation under the exponential on the r.h.s. goes over all edges of the graph, and $J_{ij}$ associated with an edge can be positive or negative. Obviously one can apply the vertex-to-edges transformation, explained in Figure \[fig:VtoE\], to restate the spin glass Ising model as a vertex-function model. However, in this case one can also do a simpler transformation to the dual graph. Let us consider a planar triangulated graph $\Gamma$ shown in black in Figure \[fig:Triang\]. All vertices of the respective dual graph, $\Gamma_d$, shown in red in Figure \[fig:Triang\], have degree of connectivity three. We assume that the spin glass Ising model is defined on the planar triangulated graph $\Gamma$. Defining a new variable $\sigma_{ab}$ on an edge of $\Gamma_d$ as the product of two variables of the original graph $\sigma_{ab}\equiv\sigma_i\sigma_j$ connected by an edge $(i,j)$ of $\Gamma$ crossing the edge $(a,b$ of $\Gamma_d$, one finds that the sum on the r.h.s. of Eq. (\[Ising\]), rewritten in terms of the new variables, becomes, $\sum_{(a,b)\in\Gamma_d}J_{ab}\sigma_{ab}$. However, the new variables, $\sigma_{ab}$ are not independent, but rather related to each other via a set of local constraints, $\forall a\in\Gamma_d$: $\prod_b^{(a,b)\in\Gamma_d}\sigma_{ab}=1$. Then, Eq. (\[Ising\]) restated in terms of the new variables on the dual graph gets the following compact vertex-style form $$\begin{aligned}
p({\bm \sigma}_d)=Z^{-1}\exp\left(\sum_{(a,b)\in\Gamma_d}J_{ab}\sigma_{ab}\right)
\prod_{a\in\Gamma}\delta\left(\prod_b^{(a,b)\in\Gamma_d}\sigma_{ab},1\right).
\label{Isingg}\end{aligned}$$ One interesting observation is that the allowed configurations of ${\bm\sigma}_d\equiv\left(\sigma_{ab}|(a,b)\in\Gamma_d\right)$ on the dual graph correspond exactly to the single-connected loops on $\Gamma_d$, where the loops are built from the excited, $\sigma_{ab}=-1$, edges. Therefore, and in accordance with discussion of Section \[sec:SingleLoops\], calculation of the partition function for the spin glass Ising is reduced to evaluation of the respective Pfaffian, which is the task of a polynomial complexity. Notice also that adding a magnetic field (linear in $\sigma$) term in the expression under the exponent on the r.h.s. of Eq. (\[Isingg\]) will raise the complexity level to exponential.
Parity-Check Based Error-Correction {#subsec:Parity}
-----------------------------------
![An illustrative example of a Tanner graph (left), as well as check-vertex (A) and bit-vertex transformations. []{data-label="fig:LDPC"}](g9.pdf){width="6cm"}
Consider a linear code with the code-book defined in terms of the bi-partite Tanner graph, $G=(V_b,V_c,{\cal E})$ consisting of $N=|V_b|$ bits and $M=|V_c|$ parity checks, and the set of edges ${\cal E}$ relating bits to checks and checks to bits. Then a message $\vec{\sigma}=(\sigma_i=0,1|i=1,\cdots N)$ is a codeword of the code if it satisfies all the parity checks, i.e. $\forall
\alpha=1,\cdots M:\ \prod_i^{(i,\alpha)\in{\cal E}}\sigma_i=+1$. Assuming that all the codewords are equally probable originally, and that the white channel transform a bit $\sigma$ of the original codeword into the signal $x$ with the probability $p(x|\sigma)$, one finds that the probability for $\vec{\sigma}$ to be a codeword resulted in the measurement $\vec{x}$ is $$\begin{aligned}
p(\vec{\sigma}|\vec{x})=\frac{1}{Z}e^{\sum_{i\in V_b}\sigma_i
h_i}\prod_{\alpha\in V_c}\delta\left(\prod_i^{(i,\alpha)\in{\cal
E}}\sigma_i,+1\right),\,\,\, h_i\equiv\frac{1}{2}\ln\frac{p(x_i|+1)}{p(x_i|-1)},
\label{LDPC}\end{aligned}$$ where, as usual, the partition function $Z$ is fixed by the normalization condition, $\sum_{\vec{\sigma}}p(\vec{\sigma}|\vec{x})=1$.
Eq. (\[LDPC\]) represents an example of a mixed graphical model, with variables $\sigma_i$ defined on bit-vertices, the parity-check functions defined on check-vertices and the channel functions (carrying the dependencies on the log-likelihoods $h_i$) also associated with the bit-vertices. In this case transformation to the vertex-style model is done by direct application of the vertex-to-edges procedure of Figure \[fig:VtoE\] to all the bit-vertices of $G$. Then, the vertex-style version of Eq. (\[LDPC\]) becomes $$\begin{aligned}
&& p(\vec{\sigma}|\vec{x})=Z^{-1}\prod_{\alpha\in V_c}
f_\alpha(\vec{\sigma})\prod_{i\in
V_b}f_i(\vec{\sigma}_i),\label{LDPC1}\\
&& \forall i:\ \ \vec{\sigma}_i\equiv (\sigma_{i\alpha}=\pm 1|(i,\alpha)\in{\cal
E}),\\
&&f_i(\vec{\sigma}_i)=
\left\{\begin{array}{cc}
\exp(h_i\sigma_{i\alpha}), & \forall \alpha,\beta \mbox{ s.t. } (i,\alpha),(i,\beta)\in{\cal E}:\ \
\sigma_{i\alpha}=\sigma_{i\beta},\\
0, & \mbox{otherwise}\end{array}\right.,\label{fi}\\
&&
\forall\alpha:\ \
\vec{\sigma}_\alpha\equiv(\sigma_{i\alpha}|(i,\alpha)\in{\cal E}),\quad
f_\alpha(\vec{\sigma}_\alpha)=\delta\left(\prod_i^{(i,\alpha)\in{\cal
E}}\sigma_{i\alpha},+1\right).
\label{f_alpha}\end{aligned}$$
In general, degree of connectivity of bit-vertices and check-vertices may be arbitrary. Direct application of the general procedure explained above (see Figure \[fig:VtoE\] and discussion therein) allows to reduce all the higher-degree nodes to a larger set of nodes of degree three. However, a simpler dendro-reduction is possible both for the bit-vertices and check-vertices. The dendro trick (e.g. discussed in [@07CS] for complexity reduction of a Linear Programming decoding of LDPC codes) is schematically illustrated in the two right panels of Figure \[fig:LDPC\], where respective algebraic relations are $$\begin{aligned}
&& \mbox{(A)}:\quad \delta\left(\prod_{i=1}^6\sigma_i,+1\right)= \label{LDPC-A} \\
&&
\sum_{\sigma_{12},\sigma_{34},\sigma_{56}=\pm 1}\delta(\sigma_1\sigma_2\sigma_{12},+1)
\delta(\sigma_3\sigma_4\sigma_{34},+1)\delta(\sigma_5\sigma_6\sigma_{56},+1)
\delta(\sigma_{12}\sigma_{34}\sigma_{56},+1), \nonumber \\
&& \mbox{(B)}:\quad \delta\left(\sigma_1,\cdots,\sigma_6\right)= \label{LDPC-B} \\
&&
\sum_{\sigma_{12},\sigma_{34},\sigma_{56}=\pm 1}\delta(\sigma_1,\sigma_2,\sigma_{12})
\delta(\sigma_3,\sigma_4,\sigma_{34})\delta(\sigma_5,\sigma_6,\sigma_{56})
\delta(\sigma_{12},\sigma_{34},\sigma_{56}), \nonumber\end{aligned}$$ and $\delta(\sigma_1,\cdots,\sigma_6)$ is equal to unity if all arguments are the same, and it is zero otherwise.
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[^1]: See [@06CCa; @06CCb; @06CCc] for a detailed discussion of this and other related features of BP equations as gauge fixing conditions.
[^2]: Except, possibly, the external face.
|
---
abstract: 'We present the LAUE project devoted to develop an advanced technology for building a high focal length Laue lens for soft gamma–ray astronomy (80-600 keV). The final goal is to develop a focusing optics that can improve the current sensitivity in the above energy band by 2 orders of magnitude.'
author:
- |
E. Virgilli, F. Frontera, V. Valsan, V. Liccardo, E. Caroli, J.B. Stephen, F. Cassese, L. Recanatesi, M. Pecora, S. Mottini, P. Attiná, B. Negri *Physics Department, University of Ferrara - Italy*;\
*IASF-INAF via P.Gobetti, Bologna - Italy*;\
*Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 2, France*;\
*DTM, Modena, Via Tacito, I-41100 Modena, Italy*;\
*Thales Alenia Space-Italy, Milan, Italy*;\
*Thales Alenia Space-Italy, Turin, Italy*;\
*ASI, Agenzia Spaziale Italiana, Viale Liegi 26, I-00198 Roma, Italy*.
bibliography:
- 'lauepaperFF.bib'
title: 'The LAUE project for broadband gamma-ray focusing lenses'
---
INTRODUCTION {#sec:intro}
============
The astrophysical importance of the X–ray broad band (0.1–200 keV and beyond) has been demonstrated by missions like $Beppo$SAX, XTE, and INTEGRAL. This band has been shown to be crucial to get a complete physical description of the astrophysical sources, like to establishing the geometry of the systems, the physical phenomena occurring in the emission region and the radiation production mechanisms. Furthermore, it enables us to distinguish the contribution of thermal emission phenomena from those due to the presence of high energy non thermal plasma and/or magnetic fields. The focusing telescopes in the soft gamma–ray band are nowadays taking key relevance to overcome the sensitivity limits of the current generation of gamma–ray telescopes which see the sky through mechanical telescopes(see, e.g., Ref. ) or coded mask systems (see, e.g., Ref. ).
Low energy focusing telescopes up to $\sim$ 70–80 keV have been successfully built (e.g. ASTRO-H \[\] and NuSTAR \[\]), but beyond 100 keV an efficient way to focus photons appears to be the use of diffraction techniques from crystals. The use of gamma–ray focusing optics is also crucial for improving the angular resolution, the best being now obtained with coded mask telescopes (about $\sim 15$ arcmin of the mask in the case of INTEGRAL/ISGRI).
Open issues that can be settled only with deep soft $\gamma$-ray observations
=============================================================================
The astrophysical issues that are expected to be solved only with focusing telescopes that cover the hard X–ray band from 80 to 511 keV and beyond are manifold. A discussion of them is extensively given in Ref. \[\]. Here we summarize some of them.
Soft Gamma Ray Repeaters (SGRs) and Anomalous X-ray Pulsars (AXPs) have raised many questions related to the role of their strong magnetic field in the high energy emission. Furthermore, it is still not clear the belonging of these two types of source to the same class. The origin of their high energy component ($>$100 keV) is still not understood. To clarify the emission above this limit we need a better sensitivity than that of the current instruments at E $>$100 keV.
High energy emission mechanisms in compact galactic objects and AGNs is also still not well understood. The emission region can be investigated measuring the high energy cutoff and its relation to the power-law photon index of the energy spectrum. Much more sensitive observations are needed, for both AGNs and compact galactic sources.
Furthermore, AGNs physics is in the focus of astrophysicists also to establish the CXB origin. Most models assume a combination of unobscured, Compton thin and Compton thick radio-quiet AGN populations with different photon index distributions and fixed high energy spectral cutoff (E$_c$). This assumption could be rejected or confirmed, with a large sampling and well studied (in terms of sensitivity) AGNs.
Positron production occurs in a variety of cosmic explosion and acceleration sites, and the observation of the characteristic 511 keV annihilation line provides a powerful tool to probe plasma composition, temperature, density and ionization degree. Compact objects - both galactic and extragalactic - are believed to release a significant number of positrons, leading to 511 keV gamma-ray line emission in the inevitable process of annihilation. A recent SPI/INTEGRAL all-sky map of galactic e$^-$/e$^+$ annihilation radiation shows an asymmetric distribution of 511 keV emission arond the Galactic Center that has been interpreted as a signature of low mass X-ray binaries with strong emission at photon energies $>$20 keV (hard LMXBs). Much more sensitive observations are needed to study the annihilation line origin, sources and their nature.
A gamma–ray telescope with a passband from 800 to 900 keV could study on fascinating class of events: the explosion of Type Ia supernovae (SNe Ia). These explosions are the major contributors to the production of heavy elements. Hence they are a critical component for the understanding of the matter life cycle in the Universe and of the chemical evolution of galaxies. Because Laue lens telescopes allow the direct observation of radioactive isotopes that power the observable light curves and spectra, gamma-ray observations of SNe Ia can be performed with this type of instrument to get a breakthrough on the detailed physical understanding of SNe Ia. This is important for its own sake, but it is also necessary to constrain systematic errors when using high-z SNe Ia to determine cosmological parameters. A sensitivity of $10^{-6}$ photons cm$^{-2}$ s$^{-1}$ to broaden gamma-ray lines allows observations of supernovae up to distances of 50-100 Mpc. Within this distance it is expected that there will always be a type Ia SN in the phase of gamma-ray line emission, starting shortly after explosion and lasting several months.
The LAUE project: a focusing lens for soft gamma-rays {#sec:laue}
=====================================================
The main goal of the LAUE project is to develop an advanced technology for building a Laue lens with broad energy band (70/100–600 keV) and long focal length (up to 100 m), for space astrophysics. The project also faces an aged and difficult issue: the development of crystals suitable for a lens. The project is supported by ASI, the Italian Space Agency.
The adopted lens assembly technology is new and it is the result of the experience gained thus far. In the previous attempt (HAXTEL project \[\]) most of the mechanical errors and uncertanties in the lens assembly were related to the use of pins directly glued on each crystal, as a reference for the X-ray beam direction.
The gluing of pins introduces uncertainties and these is summed to the second phase of the process, of counter-mask positioning and gluing phase on the carbon fiber support.
The new adopted technology consists in the positioning of the crystal tiles on the lens frame under the control of a gamma–ray beam. Each lens crystal is correctly oriented when it focuses the beam photons in the lens focal plane. This position will be kept steady by gluing each crystal upon the lens frame. The lens frame is kept fixed during the lens assembling.
This method would allow to minimize uncertainty effects in orienting the crystal tiles in the lens and would increase the crystal assembling rate in the lens with respect to that of the technology adopted so far \[\]. The main contractor of the LAUE project, DTM Technologies (Modena), is responsible for the mechanical device (a robot) able to handle and set each crystal tile with the correct orientation for the diffraction.
The lens is assumed to be made of a number of petals. One of these petal will be developed as a result of the LAUE project. The crystal production task for the petal is shared by two partners of the project: the Sensor and Semiconductor Laboratory (LSS) of the University of Ferrara and the CNR/IMEM, Parma.
The lens assembly apparatus is installed in the LArge Italian X-ray Facility (LARIX) located in a tunnel (see Fig. \[fig:tunnel\]) of the Physics Department of the University of Ferrara. For the lens assembling, it makes use of an X-ray generator with a fine focus of 0.2 mm radius with a maximum voltage of 320 kV and a maximum power of $\sim$ 1800 W. The photons coming from the X-ray tube are first collimated, then they pass through a 20 m long under–vacuum pipeline. The X–ray entrance and exit windows of the pipeline are made of carbon fiber 3 mm thick. The final beam collimation is performed at the exit window of the pipeline, in a clean room (class 10$^5$, US FED STD 209E Cleanroom Standards) in which the crystal assembly apparatus is located. The clean room is furthermore endowed with a thermal control (within $1\,^{\circ}{\rm C}$ accuracy) and an hygrometric control (relative humidity $\Phi$ = 60% within an error of 10%).
![Layout of the LARIX facility in which the petal will be built and tested.[]{data-label="fig:tunnel"}](tunnel.eps)
The final collimator aperture can be remotely adjusted in two orthogonal directions for divergence control. The beam through the final collimator is used to establish the crystal orientation in the lens.
The control of the single crystal focusing point is performed by means of two focal plane detectors: an X-ray imaging detector with spatial resolution of 300 $\mu$m and a cooled HPGe spectrometer with a 200 eV spectral resolution. Both are located on a rail and can be moved back and forth along the beam axis. Due to the maximum voltage of the available X–ray generator, the passband of the lens petal that will be produced is 80–300 keV.
Crystals adopted for the lens petal
-----------------------------------
After a development phase, for the lens petal that will be built with bent crystal tiles of Germanium (220), Silicon (111) and Gallium Arsenide (220) will be used.
The crystal cross section has been chosen to be 30 $\times$ 10 mm$^{2}$, with the longer side radially placed on the lens frame. The main advantages of the rectangular shape, together with the radial dispositon, concerns the focusing effect provided by bent crystals, which only acts in the radial direction. In such a way, a shorter tangential dimension provides a smaller defocusing factor, being proportional to the tile size. On the other hand, a bigger radial dimension makes the total number of crystal smaller, reducing the error budget potentially caused by each crystal misalignment contribution.
The thickness $t$ of the crystal tiles, for each type of crystal material, is a compromise between the need of a high effective area and the current limitation in the thickness imposed by the current status of the bending technology adopted. Moreover, the focal spot dimension is also linearly proportional to the thickness and an increase of this parameter causes a spreading of the PSF (for a 20 m focal length the longitudinal dimension of the PSF for a single crystal is roughly proportional to $t$/4).
Bent crystals can be obtained with different methods \[\]. For space applications, a bent crystal can be obtained by growing a two component crystal (e.g., Si and Ge) with a concentration of each material that changes along the growth axis. This method produces an intrinsic cuvature avoiding an external stress, even if the manufacturing turns out to be difficult and awkward.
It has been recently demonstrated that indentations on the surface normal to the diffraction planes bend the crystal \[\]. For the LAUE project, an excellent cilindrical or spherical shape with the desired curvature radius can be obtained, by finely tuning the parameters of the process (grooves number, width and depth of indentation, speed of the process). The bending has been successfully applied to Silicon and Germanium. Following the theory for bent crystals \[\], the reflectivity of bent crystals can be evaluated.
Configuration of the petal that will be built and its expected performance
==========================================================================
In Table \[tab:configuration\] the main properties of the petal have been reported. The petal is derived assuming a spherical lens equally divided into 20 petals. The energy passband is defined by the inner and outer radius of the lens. For the petal that will built, these values (see Table \[tab:configuration\]) are those allowed by the pipeline diameter within which the petal has to be inscribed to assemble the lens petal (see Fig. \[fig:petal\]). The best distribution crystal materials within each ring can performed by using a genetic code that distributes each crystal position of established cross section, available on each ring, to the crystal material that satisfies pre-defined criteria (e.g. maximization of the effective area and/or maximum allowed derivative of the effecttive area with photon energy, etc).
Materials and selected planes Si(111), Ge(220), GaAs(220)
---------------------------------- -----------------------------
Energy passband (keV) 80–300 keV
Focal length (m) 20
Petal inner/outer radius 41/93 cm
Crystal cross section (mm$^2$) 30 x 10
Number of crystals for the petal $\sim$350
Number of crystal rings 18
: Main properties of the petal that will be built with the LAUE project.[]{data-label="tab:configuration"}
![Sketch of the petal extracted from an entire Laue lens, that will be built and tested in the Ferrara LARIX facility. [*Left panel*]{}: Configuration in the assumption of a flat 1.5 $\times$ 1.5 cm$^2$ crystal cross section. Different colors give the different energies that will be reflected at 1st order diffraction. The energy scale of colors is also shown. The black circle shows the size of the beamline within the petal has to be inscribed. [*Right panel*]{}: The petal configuration in the case of rectangular tiles (10 $\times$ 1.5 cm$^2$) that increases the filling factor and decreases the required number of tiles.[]{data-label="fig:petal"}](quad.ps "fig:") ![Sketch of the petal extracted from an entire Laue lens, that will be built and tested in the Ferrara LARIX facility. [*Left panel*]{}: Configuration in the assumption of a flat 1.5 $\times$ 1.5 cm$^2$ crystal cross section. Different colors give the different energies that will be reflected at 1st order diffraction. The energy scale of colors is also shown. The black circle shows the size of the beamline within the petal has to be inscribed. [*Right panel*]{}: The petal configuration in the case of rectangular tiles (10 $\times$ 1.5 cm$^2$) that increases the filling factor and decreases the required number of tiles.[]{data-label="fig:petal"}](rect.ps "fig:")
The expected PSF of the assumed lens, in both cases of flat mosaic crystals and bent crystals, is shown in Fig. \[fig:spot.mosaic.bent.TOP.and.3D\]. The PSFs have been obtained with a Monte Carlo code assuming a single ring made either with Silicon bent crystals or flat mosaic crystals, respectively. Crystals are simulated to be perfectly aligned with respect to the theoretical diffraction angle. The same cross section of 1.5 $\times$ 1.5 cm$^2$ is assumed the same for both types of crystals which are considered to focus in a narrow band centred at 150 keV. Crystal orientation in the lens is that required by the Bragg diffraction.
As can be seen, the three-dimensional plots show that in the case of bent crystals the PSF is significantly narrower than that obtained in the case of flat mosaic crystals. The number of photons collected in the center of the focal spot is more than one order of magnitude higher in case of bent crystals.
![Simulated PSF for a ring made of curved crystals ([*left panel*]{}) and flat mosaic crystals ([*right panel*]{}).[]{data-label="fig:spot.mosaic.bent.TOP.and.3D"}](PSFmos_and_curved128px.TOP.ps "fig:") ![Simulated PSF for a ring made of curved crystals ([*left panel*]{}) and flat mosaic crystals ([*right panel*]{}).[]{data-label="fig:spot.mosaic.bent.TOP.and.3D"}](PSFimage_curved_V15x15.ps "fig:") ![Simulated PSF for a ring made of curved crystals ([*left panel*]{}) and flat mosaic crystals ([*right panel*]{}).[]{data-label="fig:spot.mosaic.bent.TOP.and.3D"}](PSFimage_mosaicV15x15.ps "fig:")
The difference between the two types of crystals (bent or flat mosaic) is better shown if the percentage of enclosed photons as a function of the distance from the focus is plotted (see Fig. \[fig:enclosed.bent.mosaic\]).
![Radial profile of the enclosed photons for perfectly oriented crystals in the lens for both cases of bent crystals and flat mosaic crystals.[]{data-label="fig:enclosed.bent.mosaic"}](Flat.eps)
The better focusing capability of bent crystals is evident, for a bent crystal the 50% of the total photons are enclosed at a radius of 3.75 mm, the radius at the same percentage is 5.82 for a flat mosaic crystal.
With the same Monte Carlo code, the effect of a misalignment has been investigated for bent crystals (see Fig. \[fig:PSFmisaligned.bent\]) in which each tile has been simulated to be affected by misalignment angles with gaussian distribution around the angle of perfect position on the lens, with a spread of 30 arcsec ($left~panel$) and 60 arcsec ($right~panel$). In both cases the effect on the number of collected photons at peak drastically decrease of a factor 75 for the former case, and of a factor 125 for the latter, with respect to the peak counts collected in the perfect aligned case.
The effect also strongly depends on the crystal cross section adopted (see Fig. \[fig:cumulative.aligned.misaligned\]). These results show the importance of a negligible misalignments in the crystal orientation. We expect, from the adopted lens assembly technique a crystal misalignments not larger than 10 arcsec.
![Simulated PSF for a ring made of curved crystals with a misalignment of 0.5 arcmin (left) and 1 arcmin (right).[]{data-label="fig:PSFmisaligned.bent"}](curved128px_0.5arcmin_misal.ps "fig:") ![Simulated PSF for a ring made of curved crystals with a misalignment of 0.5 arcmin (left) and 1 arcmin (right).[]{data-label="fig:PSFmisaligned.bent"}](curved128px_1.0arcmin_misal.ps "fig:")
![Cumulative distribution curves of the focused photons in the case of bent crystals for a perfect alignment, Gaussian misalignment of crystals with a $\sigma$ of 30 arcsec and 60 arcsec (fwhm) with respect to the orientation required by the Bragg diffraction. [*Left panel*]{}: tile cross section of 1.5 $\times$ 1.5 cm$^2$. [*Right panel*]{}: tile cross section of 2 $\times$ 0.5 cm$^2$.[]{data-label="fig:cumulative.aligned.misaligned"}](cumulativecurved1.5x1.5_PERFECT_0.5_1arcmin.eps "fig:") ![Cumulative distribution curves of the focused photons in the case of bent crystals for a perfect alignment, Gaussian misalignment of crystals with a $\sigma$ of 30 arcsec and 60 arcsec (fwhm) with respect to the orientation required by the Bragg diffraction. [*Left panel*]{}: tile cross section of 1.5 $\times$ 1.5 cm$^2$. [*Right panel*]{}: tile cross section of 2 $\times$ 0.5 cm$^2$.[]{data-label="fig:cumulative.aligned.misaligned"}](cumulativecurved2x0.5_PERFECT_0.5_1arcmin.eps "fig:")
Conclusions
===========
From the already gained experience on Laue lenses (see, e.g., ), we have started a new project, LAUE, supported by the Italian Space Agency (ASI), devoted to the development of an advanced lens assembling technology that we expect it eventually will allow to accurately build Laue lenses for space astrophysics. The expected accuracy in the lens assembling would allow to build lenses even with very long focal lengths (up to 100 m), a goal never achieved so far. In addition to the develoment of the lens assembling technology the LAUE project is facing the crystal production of proper crystals for Laue lenses. Bent crystals appear the most suitable.
As a demonstration of the validity of the adopted technology, within the LAUE project, in the LARIX facility of the University of Ferrara, we will build a lens petal of 20 m focal length. Results of the petal lens built will be reported in one of the next SPIE conferences.
The LAUE project is the result of big efforts made by a large number of organizations and people. We would like to thank all of them, as the success of the petal assembling means a step forward on the building of the whole lens. We aknowledge the ASI Italian Space Agency for its support to the LAUE project.
|
---
abstract: 'Transiting exoplanets provide access to data to study the mass-radius relation and internal structure of extrasolar planets. Long-period transiting planets allow insight into planetary environments similar to the Solar System where, in contrast to hot Jupiters, planets are not constantly exposed to the intense radiation of their parent stars. Observations of secondary eclipses additionally permit studies of exoplanet temperatures and large-scale exo-atmospheric properties. We show how transit and eclipse probabilities are related to planet-star system geometries, particularly for long-period, eccentric orbits. The resulting target selection and observational strategies represent the principal ingredients of our photometric survey of known radial-velocity planets with the aim of detecting transit signatures (TERMS).'
author:
- 'K. von Braun$^1$, S. R. Kane$^1$, S. Mahadevan$^2$, G. Laughlin$^3$, A. Howard$^4$, & D. R. Ciardi$^1$'
title: 'System Geometries and Transit$/$Eclipse Probabilities'
---
Transit/Eclipse Probabilities {#sec:probabilities}
=============================
The geometric probabilities with which an existing planet will transit its parent star ($P_t$) or be eclipsed by its parent star ($P_e$) are given by the following equations (Barnes 2007; Burke 2008; Kane & von Braun 2008, 2009; von Braun & Kane 2010) $$P_t = \frac{(R_p + R_\star)(1 + e \cos (\pi/2 - \omega))}{a (1 - e^2)}
\label{eq:transit_prob}$$ and $$P_e = \frac{(R_{p} + R_\star)(1 + e \cos (3\pi/2 - \omega))}{a (1 - e^2)},
\label{eq:eclipse_prob}$$ where $R_{p}$ and $R_\star$ are planetary and stellar radii, respectively, and $a$, $e$, and $\omega$ are the orbital semi-major axis, eccentricity, and argument of periastron. When averaging over all possible values of $\omega$, $P_t$ increases with increasing eccentricity (left panel in Fig. \[fig:probabilities\]). For fixed eccentricities, equations \[eq:transit\_prob\] and \[eq:eclipse\_prob\] show that $P_e$ is highest for $\omega = 3\pi / 2$, whereas $P_t$ is highest for $\omega = \pi / 2$ (right panel in Fig. \[fig:probabilities\]).
TERMS
=====
The Transit Ephemerides Refinement and Monitoring Survey (TERMS) takes advantage of the combinations of (measured) $e$ and $w$ of planets detected by the radial velocity (RV) method to search for transiting signatures among these planets (Kane et al. 2009). In the process, we use concurrently obtained RV data to minimize the size of the transit window for our photometric observations.
Barnes, J. W. 2007, , 119, 986 Burke, C. J. 2008, , 679, 1566 Butler, R. P., et al. 2006, , 646, 505 Kane, S. R., & von Braun, K. 2008, , 689, 492 Kane, S. R., & von Braun, K. 2009, , 121, 1096 Kane, S. R., et al. 2009, , 121, 1386 von Braun, K. & Kane, S. R. 2010, ASP Conf. Series, 430, 551
|
---
abstract: 'We propose the first multi-frame video object detection framework trained to detect great apes. It is applicable to challenging camera trap footage in complex jungle environments and extends a traditional feature pyramid architecture by adding self-attention driven feature blending in both the spatial as well as the temporal domain. We demonstrate that this extension can detect distinctive species appearance and motion signatures despite significant partial occlusion. We evaluate the framework using $500$ camera trap videos of great apes from the Pan African Programme containing $180K$ frames, which we manually annotated with accurate per-frame animal bounding boxes. These clips contain significant partial occlusions, challenging lighting, dynamic backgrounds, and natural camouflage effects. We show that our approach performs highly robustly and significantly outperforms frame-based detectors. We also perform detailed ablation studies and a validation on the full ILSVRC 2015 VID data corpus to demonstrate wider applicability at adequate performance levels. We conclude that the framework is ready to assist human camera trap inspection efforts. We publish key parts of the code as well as network weights and ground truth annotations with this paper.'
author:
- |
Xinyu Yang\
Dept of Computer Science\
University of Bristol, UK\
[[email protected]]{}
- |
Majid Mirmehdi\
Dept of Computer Science\
University of Bristol, UK\
[[email protected]]{}
- |
Tilo Burghardt\
Dept of Computer Science\
University of Bristol, UK\
[[email protected]]{}
bibliography:
- 'references\_.bib'
title: |
Great Ape Detection in Challenging Jungle Camera Trap Footage\
via Attention-Based Spatial and Temporal Feature Blending
---
Introduction {#intro}
=============
{width="237pt" height="307pt"}
\[fig:arch\]
The problem of visually identifying the presence and locations of animal species filmed in natural habitats [@kuhlburghardt2013] is of central importance for automating the interpretation of large-scale camera trap imagery. This is particularly challenging in scenarios where lighting is difficult, backgrounds are non-static, and major occlusions, image noise, as well as animal camouflage effects occur: filming great apes via camera traps in jungle environments constitutes one such setting. There, animals appear uniformly dark and blend into the forest during eating, playing or moving in groups sometimes behind trees or thicket. An animal’s visual presence and location is thus often only determinable by linking selective spatial and temporal information about species-typical appearance and locomotion across video snippets.
In this paper, we address this specific video object detection challenge by proposing a blended detection framework integrated into a feature pyramid network (FPN) architecture [@He2016DeepRecognition; @Lin2017FeatureDetection] as illustrated in Figure \[fig:arch\]. We introduce two extra components to a traditional detection backbone: a Spatial Context Module (SCM) and a Temporal Context Module (TCM). Each of these modules is driven by a self-attention mechanism tasked to learn how to emphasise most relevant elements of a feature given its context. In particular, these attention components are effective in learning how to ‘blend’ spatially and temporally distributed visual cues in order to reconstruct object locations under dispersed partial information; be that due to occlusion or lighting.
Whilst the self-attention concept has been used recently in various application contexts [@Wang2018Non-localNetworks; @Cao2019GCNet:Beyond], we tailor it here to spatio-temporal video object detection and propose a flexible component setup that can be utilized as an add-on to different backbone networks. We show that the approach is beneficial in scenarios where distinctive species appearance and motion signatures are only partly accessible and intermittently dispersed across the spatial and temporal domain. Figure \[fig:dataset\] exemplifies such scenarios on the Pan Africa camera trap data used in this work. This dataset contains $500\times$ $15s$ video clips with the resolution of $720\times 404$ and was collected by the Pan African Programme ‘The Cultured Chimpanzee’ (see Acknowledgements). It was subsequently labelled for this paper with accurate bounding box ground truth for all animals in each frame.
![ []{data-label="fig:dataset"}](images/ICCVW_figure2.pdf){width="237pt" height="188pt"}
**Contributions.** ***(1)*** FPN-integrated blended detection framework driven by spatial and temporal self-attention components; ***(2)*** Cross-dataset ablation study of the framework quantifying details for various backbones and setups; ***(3)*** Comprehensive animal bounding box annotations in 180K frames of the Pan Africa dataset.
Related Work
============
Object detection in video is usually implemented via video feature extraction [@Simonyan2014Two-StreamVideos; @Carreira2017QuoDataset; @Karpathy2014Large-scaleNetworks], frame feature aggregation [@Chen2018OptimizingLattice; @Zhu2017DeepRecognition; @Zhu2017Flow-GuidedDetection], or detection post-processing [@han2016Seq-NMSDetection; @Kang2018T-CNN:Videos; @Kang2016ObjectNetworks]. The task is distinct from object tracking [@Zhu2018Distractor-awareTracking; @Li2018SiamRPN++:Networks; @Bertinetto2016Fully-convolutionalTracking] since the object instances to follow are not given – they may appear or disappear at any point and need to be localised and determined as absent or present on a per-frame basis.
Similar to other video tasks, video object detection relies on extracting relevant information from the spatio-temporal corpus formed by frame sequences. For instance, C3D [@Tran2015LearningNetworks] and I3D [@Carreira2017QuoDataset] explore 3D convolutional neural networks (CNNs) to generate mixed motion-RGB features from frame sequences. In contrast, dual-stream ConvNets such as Siamese ConvNet [@Simonyan2014Two-StreamVideos] and TSN [@Wang2016TemporalRecognition] apply separate networks for RGB and motion processing where optical flow is often used as a pre-computed input. Frame feature aggregation, as used in DFF [@Zhu2017DeepRecognition], FGFA [@Zhu2017Flow-GuidedDetection], and ST-Lattice [@Chen2018OptimizingLattice], deals with motion implicitly by training networks that fuse frame-level spatial features to explore the temporal structure of videos. Detection post-processing, such as used in T-CNN [@Kang2018T-CNN:Videos] and Seq-NMS [@han2016Seq-NMSDetection], ignores explicit temporal feature construction altogether. Operating closer to a traditional tracking paradigm, these methods instead link detection results from individual frames into tracklets by optimising overall trajectory scores.
Before focusing on the most relevant details of some of the above methods, we will first review the foundations on which many of the approaches rest, that is: how to extract detection-relevant features from single frames.
Single Image Object Detection
-----------------------------
Object detection fundamentally requires two conceptual tasks to be solved: localisation and classification of content of interest. Region proposal based methods such as F-RCNN [@Ren2017FasterNetworks.] or Cascade-RCNN [@Cai2018CascadeDetection] pre-process images first to generate class-agnostic regions of interest (ROIs) before classifying these and regressing associated bounding boxes. F-RCNN, in particular, uses a region proposal network (RPN) built on top of a backbone shared with the classification component. More recently, to improve the quality of detection, Cascade-RCNN appends two more stages based on F-RCNN and replaces ROI pooling with ROI alignment making the framework more robust.
In contrast to all region proposal based methods, single-shot detectors infer class probability and bounding box offsets within a single feed forward network. This approach is usually simpler in design and faster at runtime as impressively shown, for instance, by YOLO [@Redmon2016YouDetection; @Redmon2017YOLO9000:Stronger] and SSD [@Liu2016SSD:Detector]. Darknet-based YOLO [@Redmon2016YouDetection] in particular regresses anchor boxes and box scores directly as well as class probabilities. Due to its versatility and fast performance, YOLO has formed the detection backbone for successfully constructed primate face detectors [@Brust2017TowardsWild] for single images in the past. Similarly to the YOLO approach, SSD predicts a fixed number of bounding boxes. However, by applying detection at different scales, SSD has been shown to adjust better to different object sizes and aspect ratios. More recently, by addressing the class imbalance problem of all of the previous single-shot methods, RetinaNet [@Lin2017FocalDetection] replaces the cross-entropy loss with the focal loss for classification, which focusses the training on hard examples.
Video Object Detection
----------------------
Whilst single image object detectors are directly applicable to video in a frame-by-frame manner, they ignore – by definition – temporal cues. Yet, these are often vital for detection under challenging conditions as for the case at hand. We will next briefly review the key ideas behind most relevant recent works such as FGFA [@Zhu2017Flow-GuidedDetection], D&T [@Feichtenhofer2017DetectDetect], T-CNN [@Kang2018T-CNN:Videos], and Seq-NMS [@han2016Seq-NMSDetection]. The fundamental task for all these methods is to integrate information from the temporal domain directly into detection decisions.
Linking single frame detections across the temporal dimension as done by T-CNN [@Kang2018T-CNN:Videos] constitutes possibly the simplest form of temporal domain exploration. T-CNN essentially runs region-based detectors per frame and enforces motion-based propagation to adjacent frames. This classical tracking paradigm thereby extends detections into tubelets, which after re-scoring and suppression of overlaps yield the final set of detected objects. In contrast, D&T [@Feichtenhofer2017DetectDetect] interlinks single image detection and tracking in a unified approach using ROI pooling on both detection-based feature maps and tracking-based correlation feature maps, where a specific correlation layer is introduced to produce the latter.
Seq-NMS [@han2016Seq-NMSDetection] follows a similar paradigm constructing sequences along temporally close, high confidence bounding boxes in consecutive frames. Their governing metric for sequence association maximises overall confidence and IOU scores, and sequence-based non-maximum suppression is utilised to fuse or filter out overlapping tracklets. Whilst temporal consistency can be extrapolated this way, the spatial distortion effects across the temporal domain are not accounted for.
To address this, FGFA [@Zhu2017Flow-GuidedDetection] performs optical flow guided spatial warping before aggregating features. The resulting features are subsequently fused temporally by weighted element-wise addition where weights are determined by the optical flow field. These descriptors contain rich spatio-temporal information that have been shown to address problems such as blurred object detection and partially occluded object detection.
Taking closest inspiration from the temporal cue aggregation mechanism used in FGFA, we propose an attention-based spatial *and* temporal feature blending framework that can be used as an add-on to existing (and future) feature pyramid networks as they are in standard use for object detection today [@Lin2017FocalDetection]. Attention-based spatial blending was successfully used before in GCNet [@Cao2019GCNet:Beyond], but attention-based spatio-temporal blending is – to the best of our knowledge – novel as a core concept for video object detection.
We will now describe how such information blending across space and time can be implemented, and how it is beneficial to addressing the problem of great ape detection.
\[fig:concept\]
Detection via Spatial & Temporal Blending
=========================================
**Task.** Our task is to generate frame-level bounding box detections $\mathbf{D}=\{\mathbf{D}_1, \dots, \mathbf{D}_t, \dots\}$ of great apes across a video $\mathbf{X}=\{\mathbf{X}_1, \dots,\mathbf{X}_t,\dots\}$, where $\mathbf{D}_t$ are the detection results for frame $\mathbf{X}_t$ at time step $t$. To predict the bounding box detections $\mathbf{D}_t$, the network can utilise both previous *and* future frames, $i.e.$ a snippet $\{\mathbf{X}_{t-\tau}, \dots, \mathbf{X}_t, \dots, \mathbf{X}_{t+\tau}\}$ of length $T$.
**Overall Concept.** As shown in Figure \[fig:arch\], our integrated architecture extends a standard feature pyramid network by two extra components: a Spatial Context Module (SCM) and a Temporal Context Module (TCM). Each of these modules is driven by a self-attention mechanism that learns how to emphasise the most relevant elements of a feature given its context. Both components follow a principled workflow similar to the one described by Cao et al [@Cao2019GCNet:Beyond] and visualised in Figure \[fig:concept\]. Essentially, after grouping inputs along the dimensions of attentional interest (spatial or temporal), features are embedded into a lower dimensional space and a self-attention map of the feature is created. This map is then applied back onto the features in order to ‘blend’ it and emphasise elements important to the detection process whilst suppressing other content. Critically, these components are trainable as part of the network and can be rolled out across space and time so that dispersed species-distinctive information can be selected from within the spatio-temporal volume.
For a given spatial or temporal module and position $i$ in the input feature $x_i$, the context-enhanced and ‘blended’ output feature $z_i$ can in its simplest form be expressed as: $$\label{equ1}
z_i=x_i+f\bigg(\frac{1}{M}\sum^{M}_{j=1} s\Big(e_i(x_i),e_j(x_j)\Big) e_i(x_i)\bigg) ~, \vspace{-4pt}$$ where $x_i$ is [the]{} $i$th descriptor of the residual feature map, $e(\cdot)$ is the embedding function, $f(\cdot)$ is the transform function, $s(\cdot)$ is the correlation function, $M$ is the number of positions in the feature, $e.g.$ $M=HWT$ for [a]{} video [sequence]{} [($H$ denotes height, $W$ denotes width, $T$ denotes snippet length)]{}, and $j$ enumerates all the positions across the context. The embedding function $e(\cdot)$ and transfer function $f(\cdot)$ are implemented via $1\times 1$ convolution kernels without bias using learnable weights $w$.
**Spatial Module.** As depicted in Figure \[fig:arch\] (green), we use a simplified non-local attention component with a SoftMax function applied across the embedded feature vector as the SCM. The module output $z_i^{SCM}$ is:
$$z_{i}^{SCM}=x_{i} \oplus w^3 \sum_{j=1}^{M}\left(\begin{small}\frac{\exp{(w^1x_j)}}{\sum^{M}_{a=1}\exp{(w^1x_a)}}\end{small}\otimes(w^2x_j) \right) ~, \vspace{-3pt}$$
where $j$ enumerates all context locations, $x_i$ represents the incoming features, $\oplus$ denotes element-wise broadcast addition, $\otimes$ denotes multiplication of tensor elements, $w^{1}$ and $w^{2}$ describe the different learnable parameters of linear embedding functions $e(\cdot)$, and $w^{3}$ represent the parameters of a linear transform function $f(\cdot)$.
**Temporal Module.** Figure \[fig:arch\] (red) visualises the temporal module in detail, which follows the principles layed out in Equation (\[equ1\]) in general terms. In particular, the TCM module is constructed to apply self-attention across a short-term temporal context where, for each input frame $X_t$, a feature $x_t$ from the previous layer is first embedded by a linear function $e(\cdot)$ with weights $w^4_{t}$. Subsequently, its temporal correlation function $s(\cdot)$ is modelled by a global SoftMax $\mathcal{C}(\cdot)$ across a temporal context of $T=2\tau+1$ selected nearby frames. For each feature position $i \in M$ the global SoftMax is defined as: $$\mathcal{C}(x_{t,i} ; w^4)=\frac{\exp{(w^{4}_{t}x_{t,i})}}{\sum_{m \in T}\exp{(w^4_{m}x_{m,i})}} ~, \vspace{-2pt}$$ where frame $m$ enumerates all $T$ frame positions, and $w_{t}$ is the linear embedding parameter for time step $t$. We then use the mean value of all positional temporal attention to normalise the term yielding a temporal attention map for time step $t$ as: $$\hat{x}_{t,i}=\frac{1}{HW}\mathcal{C}(x_{t,i} ; w^4)\sum_{j=1}^{HW}\mathcal{C}(x_{t,j} ; w^4) ~.\vspace{-3pt}$$ In order to visually illustrate this concept, two examples of temporal attention maps projected back into the image domain are depicted in Figure \[fig:attention maps\]. The maps highlight distinctive, dispersed features for context-aware inference revealing target object locations despite heavy partial occlusion.
These maps are subsequently applied back onto the original feature by matrix multiplication $\otimes$, element-wise broadcast addition $\oplus$, and two linear transformations. One transform learns adaptive weights (*i.e.* $w^5_{m}$) to ‘blend’ between original and attention-adapted features, the other learns weights (*i.e.* *$w^6_{m}$*) to ‘blend’ across the temporal domain. Finally, an additive connection to the original inputs $x_{t,i}$ is made to ensure learning stability yielding: $$z_{t,i}^{TCM}=x_{t,i}+\sum_{m \in T}w^6_{m}(x_{m,i} \oplus w^5_{m}\sum_{j=1}^{HW}\hat{x}_{m,j}\otimes x_{m,j}) ~.\vspace{-3pt}$$
{width="237pt" height="180pt"}
\[fig:attention maps\]
Overall Network Architecture
-----------------------------
**Backbones.** [We follow the layout of RetinaNet [@Lin2017FocalDetection]]{} and Cascade [@Cai2018CascadeDetection] as the [two major]{} state-of-the-art architectures in our study. Figure \[fig:arch\] (blue) visualises a ResNet50 architecture as an example backbone [@He2016DeepRecognition] where we extract FPN levels from network layers $P_3$ to $P_7$. Across different scales of the FPN, anchors cover area sizes of $32^2$ to $512^2$ distributed over pyramid levels from $P_3$ to $P_7$, respectively. We embed our TCM and SCM after the last layer of the backbone for a maximal receptive field during blending.
**Detection Head.** The RetinaNet based detection head contains two streams, *i.e.* a localisation stream and classification stream. The classification subnet (coloured grey in Figure \[fig:arch\]) predicts a likelihood vector of class membership for each anchor box. This network is fully convolutional and operates on top of the $5$ FPN layer outputs $P_3$ to $P_7$. It is composed of $4$ cascaded $3 \times 3$ convolutional layers, each with $256$ channels, and followed by a ReLU activation. The final layer uses $3 \times 3$ convolutions with a Sigmoid function as activation for producing the final classification output. In the localisation subnet (coloured black in Figure \[fig:arch\]) the final layer yields $4\times$ channel outputs describing bounding box locations. We assign initial anchor areas ranging from $32^2$ to $512^2$ elements corresponding to the pyramid levels $P_3$ to $P_7$, respectively. Subsequent rescaling uses the ratios of \[$2^0$, $2^{1/3}$, $2^{2/3}$\] and transforms anchors with three aspect ratios \[1:2, 1:1, 2:1\]. In total there are 9 anchors per level. Focal loss from [@Lin2017FocalDetection] is adopted to tackle the foreground and background imbalance problem.
We also implemented a stronger, cascaded detection head [@Cai2018CascadeDetection] for maximum detection quality. The cascaded detection head contains four stages, one for an RPN, which regresses class-agnostic regions of interest, and three for detection with different IoU ground truth assignments parameterized as $[0.5,0.6,0.7]$. Note that the three cascaded detection heads are fully convolutional and ROI alignment is used for ROI feature pooling.
Training Details
----------------
We train the network end-to-end in two separate stages (see Figure \[fig:training\]). First, we pre-train our model on the ImageNet VID dataset for 14 epochs, and then the entire model is trained with synchronised batch normalisation and [training data augmentation (sequence-based random brightness, horizontal flip, central crop)]{} for $14$ epochs on the $400$ video training portion of the Pan Africa dataset. The remaining $100$ clips are used for testing.
During the various experiments, we use short-term video snippets and sparse sampling of between $T=3$ to $T=8$ frames [whose shorter size is ]{}resized [to at least $512$ and up to $800$]{} pixels horizontally keeping the aspect ratio constant by padding if necessary. During training, each mini-batch contains $8$ snippets, that is $1$ per deployed Nvidia Tesla P100 GPU. The learning rate is warmed up for the first $500$ iterations of each training phase from $0.002$ to $0.01$ and decreased $10$ times at epoch $6$ and $11$, respectively.
We use SGD with the momentum of $0.9$ as the optimizer and utilise ImageNet pre-trained weights to initialise the backbone. FPN layers and all other convolutional layers are initialised by normal distributions.
Experimental Results
====================
We evaluate the proposed system both quantitatively and qualitatively, and perform detailed ablation studies. We also perform a validation on the full ILSVRC 2015 VID data corpus to demonstrate wider applicability.
Quantitative Evaluation
-----------------------
Using the test portion of the Pan Africa dataset, we first evaluate single frame detectors as our baselines. For evaluation, we compute the average precision (AP) as the area under precision-recall curve and then report the mean of AP (mAP) for the classes in the validation set.
We re-implemented RetinaNet [@Lin2017FocalDetection] with ResNet50 and ResNet101 baselines and, as shown in Table \[tab:chimpresult\], these two architectures achieve 80.79% and 85.25% mAP, respectively. Whilst adding a SCM component to the ResNet50 RetinaNet setup improves performance slightly to 81.21%, the current state-of-the-art Cascade-RCNN [@Cai2018CascadeDetection] outperforms this setup and older single frame baselines, achieving 88.31% on the Pan Africa test data. The basic addition of a TCM component, on the other hand, produces significant performance improvements by up to $10.02\%\uparrow$ for ResNet50 RetinaNet, and still $2.86\%\uparrow$ for Cascade-RCNN.
{width="237pt" height="111pt"}
\[fig:training\]
[width=,center]{}
[cc|cc|c]{}
& $T_{train}$ & $T_{test}$ & mAP(%)\
-----------
Res50
RetinaNet
-----------
: **Pan Africa Performance Results**. Boosts in mean average precision (mAP) on the Pan Africa test dataset when applying the proposed TCM & SCM components with various state-of-the-art FPN architectures.
&
----------------------------------
baseline[@Lin2017FocalDetection]
+SCM
+TCM
+SCM+TCM
----------------------------------
: **Pan Africa Performance Results**. Boosts in mean average precision (mAP) on the Pan Africa test dataset when applying the proposed TCM & SCM components with various state-of-the-art FPN architectures.
&
----
\\
\\
7
7
----
: **Pan Africa Performance Results**. Boosts in mean average precision (mAP) on the Pan Africa test dataset when applying the proposed TCM & SCM components with various state-of-the-art FPN architectures.
&
----
\\
\\
21
21
----
: **Pan Africa Performance Results**. Boosts in mean average precision (mAP) on the Pan Africa test dataset when applying the proposed TCM & SCM components with various state-of-the-art FPN architectures.
&
-----------
80.79
81.21
90.02
**90.81**
-----------
: **Pan Africa Performance Results**. Boosts in mean average precision (mAP) on the Pan Africa test dataset when applying the proposed TCM & SCM components with various state-of-the-art FPN architectures.
\
-----------
Res101
RetinaNet
-----------
: **Pan Africa Performance Results**. Boosts in mean average precision (mAP) on the Pan Africa test dataset when applying the proposed TCM & SCM components with various state-of-the-art FPN architectures.
&
----------------------------------
baseline[@Lin2017FocalDetection]
+SCM+TCM
----------------------------------
: **Pan Africa Performance Results**. Boosts in mean average precision (mAP) on the Pan Africa test dataset when applying the proposed TCM & SCM components with various state-of-the-art FPN architectures.
&
----
\\
5
----
: **Pan Africa Performance Results**. Boosts in mean average precision (mAP) on the Pan Africa test dataset when applying the proposed TCM & SCM components with various state-of-the-art FPN architectures.
&
----
\\
21
----
: **Pan Africa Performance Results**. Boosts in mean average precision (mAP) on the Pan Africa test dataset when applying the proposed TCM & SCM components with various state-of-the-art FPN architectures.
&
-----------
85.25
**90.21**
-----------
: **Pan Africa Performance Results**. Boosts in mean average precision (mAP) on the Pan Africa test dataset when applying the proposed TCM & SCM components with various state-of-the-art FPN architectures.
\
---------
ResX101
Cascade
---------
: **Pan Africa Performance Results**. Boosts in mean average precision (mAP) on the Pan Africa test dataset when applying the proposed TCM & SCM components with various state-of-the-art FPN architectures.
&
------------------------------------
baseline[@Cai2018CascadeDetection]
+SCM+TCM
------------------------------------
: **Pan Africa Performance Results**. Boosts in mean average precision (mAP) on the Pan Africa test dataset when applying the proposed TCM & SCM components with various state-of-the-art FPN architectures.
&
----
\\
3
----
: **Pan Africa Performance Results**. Boosts in mean average precision (mAP) on the Pan Africa test dataset when applying the proposed TCM & SCM components with various state-of-the-art FPN architectures.
&
----
\\
21
----
: **Pan Africa Performance Results**. Boosts in mean average precision (mAP) on the Pan Africa test dataset when applying the proposed TCM & SCM components with various state-of-the-art FPN architectures.
&
-----------
88.31
**91.17**
-----------
: **Pan Africa Performance Results**. Boosts in mean average precision (mAP) on the Pan Africa test dataset when applying the proposed TCM & SCM components with various state-of-the-art FPN architectures.
\[tab:chimpresult\]
The training process for a full SCM+TCM setup, as quantified earlier in Figure \[fig:training\] for instance, furthermore reveals that the generalisation gap of such a model (mAP distance between blue and green curves) narrows significantly for the PanAfrica dataset compared to the ImageNet VID pre-training in late stages. Narrow generalisation gaps indicate that a model is particularly capable of carrying over learned inference strategies to unseen instances of the particular dataset. For the case at hand, this and the improved overall mAP indicate that fine-tuning is indeed successful.
We also validated a basic ResNet50 RetinaNet version that uses SCM+TCM on the full ILSVRC 2015 VID corpus to show that our rather simple and flexible SCM+TCM network extension can achieve [strong]{} results also in this general detection setting at $63.85\%$ mAP. This ranks within the top three of the original mAP based competition [@ImageNet].
{width="240pt" height="260pt"}
\[fig:results\]
Qualitative Observations on Pan Africa Data
-------------------------------------------
The Pan Africa dataset contains many scenes where illumination, occlusions, noise or animal camouflage effects make it challenging to recognise animals (as seen before in Figure \[fig:dataset\]). We found that the SCM+TCM setup consistently improves detection robustness compared to baselines in such cases. These improvements make a significant contribution to the overall quantitative results reported before. Figure \[fig:results\] provides examples of successful cases where per-frame accurate animal detection is achieved by the SCM+TCM components in the presence of partial occlusion or challenging lighting. However, as depicted in the bottom examples of Figure \[fig:results2\], a number of particular animal appearances remain challenging to detect despite the availability of spatial and temporal context information.
{width="240pt" height="202pt"}
\[fig:results2\]
Ablation Study
--------------
We conduct an ablation study on both the Pan Africa and the ILSVRC 2015 VID datasets in order to quantify in detail the impact of the key system parameters. All of the results are reported based on ResNet50+SCM+TCM setup with a RetinaNet detection head.
**SCM+TCM Insertion Point.** Considering arguments in [@Cao2019GCNet:Beyond], we acknowledge that the embedding position of our module along the original backbone network is important. To determine the best insertion point, we test three possible positions for insertion into ResNet50: after the $3 \times 3$ convolution of the last ResNet block; after the last $1 \times 1$ convolution before the residual addition; and after the residual addition. As shown in Table \[tab:ablation1\](a) and Table \[tab:ablation2\](a), we find that the last option is superior and can gain $6.01\%\uparrow$ mAP on Pan Africa and $6.42\% \uparrow$ mAP on VID improvement when compared with insertion after the last $1 \times 1$ convolution. This indicates that the final residual addition in the base network *does* provide useful extra information.
**Temporal Frame Support.** Different choices on supporting frames $T$ are ablated in Table \[tab:ablation1\](b) and \[tab:ablation2\](b). Results confirm that wider windows for temporal integration do indeed benefit detection, however, particularly during training, GPU sizes limit possible choices of $T$. During testing, longer exposures are possible. Quantifying the effect of varying test exposures we find that, for $T_{test}=21$ frames compared to $T_{test}=5$, there is a $1.96\% \uparrow$ and $3.69\% \uparrow$ improvement for fixed $T_{train}=7$ on the Pan Africa and ILSVRC 2015 VID datasets, respectively.
**Embedding Strategy in the TCM.** We found that when applying an embedding strategy in the TCM where only the current main frame features (indicated as black arrows of the TCM in Figure \[fig:arch\]) are backpropagated and reference features are used, but not backprogagated, then a marginally improved overall performance can be observed (marked as *Main & Refs* in contrast to *Positional* in Tables \[tab:ablation1\] and \[tab:ablation2\]).
[.5]{}
[ccccc]{} Embedding & position &$T_{train}$ & $T_{test}$ & mAP(%)\
Positional & after add & 8 & 8 & 88.21\
Positional & after 1x1 & 8 & 8 & 82.20\
Positional & after 3x3 & 8 & 8 & **87.75**\
[.5]{}
[ccccc]{} Embedding & position & $T_{train}$ & $T_{test}$ & mAP(%)\
Main & Refs & after add & 7 & 21 & **90.81**\
Main & Refs & after add & 7 & 5 & 88.85\
Main & Refs & after add & 3 & 5 & 87.76\
[.5]{}
[ccccc]{} Embedding & position & $T_{train}$ & $T_{test}$ & mAP(%)\
Positional & after add & 7 & 21 & 88.61\
Main & Refs & after add & 7 & 21 & **90.81**\
[.5]{}
[ccccc]{} Embedding & position & $T_{train}$ & $T_{test}$ & mAP(%)\
Positional & after add & 8 & 8 & 56.25\
Positional & after 1x1 & 8 & 8 & 49.83\
Positional & after 3x3 & 8 & 8 & **58.20**\
[.5]{}
[ccccc]{} Embedding & position & $T_{train}$ & $T_{test}$ & mAP(%)\
Positional & after add & 8 & 8 & 56.25\
Positional & after add & 5 & 8 & 54.38\
Main & Refs & after add & 7 & 21 & **63.85**\
Main & Refs & after add & 7 & 5 & 60.16\
[.5]{}
[ccccc]{} Embedding & position & $T_{train}$ & $T_{test}$ & mAP(%)\
Positional & after add & 7 & 21 & 59.25\
Main & Refs & after add & 7 & 21 & **63.85**\
\[tab:ablation2\]
Conclusion and Implications
===========================
In this paper we proposed the first multi-frame video object detection framework trained and evaluated for detecting great apes utilising their full body morphology. We demonstrated that the framework is applicable to challenging camera trap footage taken in complex jungle environments. We introduced two self-attention driven feature blending components operating in both the spatial and the temporal domains to facilitate detection under heavy partial occlusions and challenging lighting variations.
We showed that this novel and flexible extension performs robustly at $91.17\%$ mAP on a real world Pan Africa camera trap $500$ video dataset, which we labelled accurately with animal ground truth annotations for $180$K frames. We conducted detailed ablation studies on our method and showed that the setup significantly outperforms state-of-the-art frame based detectors. For general evaluation beyond the task at hand, we also performed a validation on the ILSVRC 2015 VID data corpus to demonstrate [significant]{} performance on non-specialised video object detection.
We note that currently ecological camera trap studies are widely conducted by manual inspection, although great ape face detection [@Brust2017TowardsWild; @loos2013automated] has been used for ecological surveys before [@crunchant2017automated] and DrivenData [@Competition:Factorization] hosted a recent challenge to classify jungle camera trap clips by species, without detecting animals and their location in frames explicitly.
The presented system, in contrast, provides explicit animal locations and is independent of visibility constraints regarding the animal’s face. It adds a new capability of detection and localisation of animals partly occluded by vegetation at adequate performance levels. Whilst tests against other current video detection frameworks are outstanding and will form part of our future work, we conclude that the presented system is ready to assist human camera trap inspection efforts.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank the entire team of the Pan African Programme: ‘The Cultured Chimpanzee’ and its collaborators for allowing the use of their data for this paper. Please contact the copyright holder Pan African Programme at http://panafrican.eva.mpg.de to obtain the dataset. Particularly, we thank: H Kuehl, C Boesch, M Arandjelovic, and P Dieguez. We would also like to thank: K Zuberbuehler, K Corogenes, E Normand, V Vergnes, A Meier, J Lapuente, D Dowd, S Jones, V Leinert, E Wessling, H Eshuis, K Langergraber, S Angedakin, S Marrocoli, K Dierks, T C Hicks, J Hart, K Lee, and M Murai. Thanks also to the team at https://www.chimpandsee.org. The work that allowed for the collection of the dataset was funded by the Max Planck Society, Max Planck Society Innovation Fund, and Heinz L. Krekeler. In this respect we would also like to thank: Foundation Ministère de la Recherche Scientifique, and Ministère des Eaux et Forêts in Co´te d’Ivoire; Institut Congolais pour la Conservation de la Nature and Ministère de la Recherche Scientifique in DR Congo; Forestry Development Authority in Liberia; Direction des Eaux, Forêts Chasses et de la Conservation des Sols, Senegal; and Uganda National Council for Science and Technology, Uganda Wildlife Authority, National Forestry Authority in Uganda.
|
---
abstract: 'Weak gravitational, electromagnetic, neutrino and scalar fields, considered as perturbations on Kerr background satisfy Teukolsky Master Equation. The two non-trivial equations obtained after separating the variables are the polar angle equation and the radial equation. We solve them by transforming each one into the form of a confluent Heun equation. The transformation depends on a set of parameters, which can be chosen in a such a way, so the resulting angular and radial equations separately have simple polynomial solutions for neutrino, electromagnetic, and gravitational perturbations, provided some additional conditions are satisfied. Remarkably there exists a class of solutions for which these additional conditions are the same for both the angular and the radial equations for spins $|s|=1/2$ and $|s|=1$. As a result the additional conditions fix the dependence of the separation constant on the angular frequency but the frequency itself remains unconstrained and belongs to a continuous spectrum.'
author:
- |
Roumen S. Borissov [^1], Plamen P. Fiziev [^2]\
Physics Department, Sofia University “St. Kliment Ohridski",\
5 James Bourchier Blvd., 1164 Sofia, Bulgaria
title: Exact Solutions of Teukolsky Master Equation with Continuous Spectrum
---
1.5truecm
Keywords: Teukolsky master equation, Heun equation, Heun functions, perturbations of black holes.
Introduction
============
Fields of various types – scalar, neutrino, electromagnetic, and gravitational – have been extensively studied as perturbations to known solutions of Einstein’s equations for configurations with spherical and cylindrical symmetry. The fields considered are weak in the sense that we can neglect the influence of their stress-energy tensor on the background metric. Regge and Wheeler [@ReggeWheeler] and Zerilli [@Zerilli; @Johnston] were the first to study the linear response of Schwarzschild solution of Einstein’s equations to perturbations. In order to study Kerr metric perturbations Teukolsky [@Teukolsky]-[@Teukolsky3] analyzed the components of Weyl tensor, using Newman-Penrose formalism [@NP]. (For a detailed extended presentation see [@Chandra].). As a result one obtains the Teukolsky Master Equation, which describes the dynamics of various fields of different spins as perturbations to Kerr metric. In recent years there is an increased interest on the subject [@Andersson]-[@Hod], mostly aimed at studying the quasi-normal modes. Another problem analyzed via Teukolsky’s equations is related to the decaying of the various fields present during a gravitational collapse at very late times at large distances – the so called late-time tails. All these investigations however, are performed via indirect, approximate methods [@Andersson; @Kokkotas],[@Hod]-[@Leaver1].
On the other hand, already for quite some time it has been recognized in the literature [@Marc1; @Marc2; @GE; @STU] that Regge-Wheeler and Teukolsky’s equations can be transformed into the form of a confluent Heun equation [@Heun]-[@Fiziev6]. The reason there has not been much attention paid to the Heun-type solutions is that they are not completely analyzed and, in general, difficult to work with. Some basic classes of exact solutions to Rege-Wheeler equation in terms of special solutions to the confluent Heun equation – the so-called confluent Heun functions (see the Appendix), were described recently and were used for finding solutions to a number of physical problems [@Fiziev1]-[@Fiziev3].
Following the articles [@Fiziev1]-[@Fiziev3] we continue with the application of the confluent Heun functions to Teukolsky’s equations. The first results, presented in [@PF1]-[@PFDS4], were very encouraging and drew special attention to the solutions in terms of the confluent Heun polynomials [@Heun]-[@Fiziev6] (see the Appendix). It should be emphasized that long time ago in [@Marc1; @Marc2] it was recognized by Baldin, Pons, and Marcilhacy that the conditions for polynomial solutions to Heun equations lead to polynomial solutions to Teukolsky’s equations in a generalized sense, i.e. polynomials multiplied by non-polynomial factors which are elementary functions. Having in mind the general description of all 256 classes of factorized solutions to Teukolsky Master Equation [@Fiziev4; @Fiziev5] we intend to focus on the mathematical properties of some of them and study various physical applications.
The general description of all polynomial solutions of Teukolsky Master Equation was given for the first time in [@Fiziev4; @Fiziev5]. These fall into two different classes. For the first class, the first polynomial condition (\[alphaint\]), called the $\delta_N$-condition in [@Fiziev6], [@Fiziev4; @Fiziev5], is automatically satisfied. For waves of spin $|s|$ this condition fixes only the degree $(N+1)=2|s|$ of the second polynomial condition $\Delta_{N+1}=0$. For the second class of polynomial solutions the $\delta_N$-condition is fulfilled only for certain complex frequencies $\omega_N$ which belong to definite equidistant discrete spectra. For the two classes the second polynomial condition $\Delta_{N+1}=0$ defines an algebraic equation of degree $2|s|$ for the second separation constant: $E_m=E_m(\omega)$, and $E_m=E_m(\omega_N)$, correspondingly.
Here we are considering only polynomial solutions of the first class. Thus an independent derivation of the specific relations, valid only for the first class of polynomial solutions becomes possible. It is based on a direct check of the two necessary conditions (\[alphaint\]) and (\[conditpoly\]), which together are sufficient to ensure the polynomial character of the solutions (See the Appendix.).
Below we present an independent derivation of the first class polynomial solution both for Teukolsky’s angular and radial equations using the notations of reference [@Ron]. This notation has some advantages since it simplifies significantly the form of the $\delta_N$-condition. The correspondence between the notation of [@Ron] and the notation used in [@DDLMRR1; @DDLMRR2], [@Fiziev6],[@Fiziev1]-[@Fiziev5] and in the computer application Maple is described in section 9.4 of the Appendix.
In the present paper we consider a specific type of evolution of week fields with spin $|s|=1/2, 1\,\,\text{and}\,\, 2$ on Kerr background. The solutions studied here are double polynomial solutions that describe one-way waves of corresponding spins, the so-called total transmission modes. These are factorized solutions to Teukolsky Master Equation, in which the solutions both of the angular and the radial equations (of the same spin weight $s$) belong to the corresponding first classes of polynomial solutions, introduced in [@Fiziev4; @Fiziev5]. Here we show that these solutions yield a complex one-parameter continuous spectrum of the frequency $\omega$ and derive the explicit form of the separation constant $E$ in the various cases. Finally we discus some overall solutions of Teukolsky Master Equation, constructed making use only of these continuous spectrum solutions.
To the best of our knowledge this is the first time when for Teukolsky Master Equation exact solutions with continuous spectrum are presented for a specific boundary problem. An interesting observation is that continuous spectrum emerges only for neutrino and electromagnetic waves, because of the simultaneous fulfillment of the polynomial condition both for the angular and the radial Teukolsky equations. We have to stress that such a simultaneous fulfillment is not in place for gravitational waves. The physical consequences of this mathematical result may be deep and very important. Its roots can be traced back to some results, originally obtained in [@Teukolsky3] and developed further in [@Chandra]. We present here the mathematical basis, needed for further developments in this direction.
In the next section we start by reminding the procedure for separation of the variables in Teukolsky Master Equation via factorization of the solutions and the corresponding basic results. In section 3 we present the general scheme for transforming Teukolsky’s radial equation (TRE) into the one of the many known “canonical” forms of the confluent Heun equation [@Heun]-[@Fiziev6][^3], namely into the so-called non-symmetrical canonical form. We show that for specific values of the indices of the regular singular points [@Ince] the first condition for polynomial solution to TRE in the form of a confluent Heun equation is automatically attained. We impose the second condition for having a polynomial solution and obtain the value of the separation parameter $E$ as function of the frequency $\omega$. In section 4 we continue by presenting the transformation to non-symmetrical canonical Heun form of Teukolky’s angular equation (TAE) and again show that for corresponding specific choice of the indices of its regular singular points we obtain again a polynomial solution. In order to achieve this result we derive the explicit form of the second polynomial condition and arrive at the result that in some cases it is the same as for the radial equation. Thus we find a simultaneous fulfillment of the polynomial conditions for the angular and for the radial Teukolsky’s equations for perturbations with spin $|s|= 1/2$ and $1$. In section 5 we discuss why such a simultaneous fulfillment of the polynomial conditions is not possible for gravitational waves ($|s|=2$). In section 6 we present the basic properties of the overall solutions to the Teukolsky Master Equation constructed only from the factorized solutions with continuous spectrum. We show that these solutions describe one-way collimated waves, which may be regular along the rotational axes, despite the singular character of the polynomial solutions of the angular Teukolsky equation. In mathematical sense these solutions form a natural orthogonal basis of singular functions for integral representation of physically meaningful solutions. In the conclusion we give a brief summary and ideas for future studies on the matter.
In an Appendix some basic properties of the confluent Heun equation and its solutions and different forms are presented for the reader’s convenience.
Spin weight $s$ fields on Kerr background
=========================================
In this section we present some basic results of Teukolsky’s approach [@Teukolsky]-[@Teukolsky3] to the perturbations of spin $|s|$ of Kerr vacuum solution for the metric of a rotating black hole. In Boyer-Lindquist coordinates the metric is given by [@Chandra], [@MTW]:
$$\begin{aligned}
\label{KNMETRIC}
ds^2 &=& {\left( 1- \frac{2Mr}{\Sigma} \right)} dt^2 + \frac{{
4aMr\sin^2\theta }}{\Sigma} dt d\phi - { {\Sigma} \over {\Delta} }
dr^2\nonumber \\ &&-\,\Sigma d\theta^2 -
{\left[r^2+a^2+\frac{2Ma^2r\sin^2\theta}{\Sigma}\right]}
\sin^2\theta d\phi^2\,.\end{aligned}$$
Here $M$ is the Keplerian mass of the rotating black hole and $a$ is its angular momentum per unit mass. Also, ${\Delta}$ and ${\Sigma}$ are defined in the usual way: $$\begin{aligned}
\label{DEL}
\Delta \equiv r^2 - 2Mr + a^2, \qquad \Sigma \equiv r^2 +
a^2\cos^2\theta\,.\end{aligned}$$ The dynamics of a massless field $\Psi = \Psi(t,r,\theta,\phi)$ with spin weight [*s*]{} is described by Teukolsky Master Equation:
$$\begin{aligned}
\nonumber
&& \left[\frac{\left(\ r^2+a^2 \right)^2}{\Delta} - a^2
\sin^2\theta \right]{\partial^2\Psi \over \partial t^2} + {\frac
{4Mar}{\Delta}}{\partial^2\Psi \over \partial t
\partial\phi} + {\left[ {a^2 \over \Delta} - {1 \over
\sin^2\theta }\right]}{\partial^2\Psi \over \partial\phi^2} -
\\ \nonumber
&& - \Delta^{-s}\frac{\partial}{\partial
r}\left(\Delta^{s+1}\frac{\partial\Psi}{\partial r}\right) - {1
\over \sin \theta} {\partial \over \partial \theta} \left(\sin
\theta {\partial\Psi \over \partial \theta}\right)- 2s{\left[
\frac{a\left( r-M\right)}{\Delta} +
\frac{i\cos\theta}{\sin^2\theta}
\right]\frac{\partial\Psi}{\partial\phi}} - \\ \nonumber \\ && -
2s{\left[ \frac{M\left( r^2-a^2\right)}{\Delta} - r -ia\cos\theta
\right]\frac{\partial\Psi}{\partial t}} + \left( s^2\cot^2\theta -
s \right)\Psi =0\,.\label{TME}\end{aligned}$$
In the above equation we have the following expressions for $\Psi$:
- For $s=1/2$, $\Psi=\chi_{0}$ and for $s=-1/2$, $\Psi=\rho^{-1}\chi_{1}$, where $\chi_{0}$ and $\chi_{1}$ represent the two components of the neutrino spinor in Newman-Penrose formalism.
- For $s=1$, $\Psi=\varphi_{0}$ and for $s=-1$, $\Psi=\rho^{-2}\varphi_{2}$, where $\varphi_{0}$ and $\varphi_{2}$ are Maxwell tensor tetrad components in Newman-Penrose formalism.
- In the gravitational case for $s=2$, $\Psi=\psi_{0}$ and for $s=-2$, $\Psi=\rho^{-4}\psi_{4}$, where $\psi_{0}$ and $\psi_{4}$ are Weyl tensor Newman-Penrose tetrad components.
In all cases $\rho=-1/(r-ia\cos\theta )$ and the choice of the Kinnersley tetrad is assumed. In order to separate the variables following [@Teukolsky]-[@Teukolsky3] we set
$$\label{Fourier}
\Psi(t,r,\theta,\phi)= {1 \over 2\pi} \int_{-\infty +
i\epsilon}^{\infty +i\epsilon} \sum_{m=-\infty}^{\infty}
\Psi_{m}(\omega; t,r,\theta,\phi) d\omega\,,$$
where $\Psi_{m}(\omega; t,r,\theta,\phi)= e^{im\phi} e^{-i\omega
t} S_{m}(\omega;\theta)R_{m}(\omega;r)$, and $\epsilon$ is a parameter defining the contour of integration. The azimuthal number has values $m = 0,\pm 1,\pm 2, \dots$ for integer spin, or $m = \pm 1/2,\pm 3/2, \dots$ for half-integer spin [@Gold; @Dolan1; @Dolan2; @Fiziev5].
Thus we are looking for an integral representation with a [*factorized*]{} kernel $\Psi_{m}(\omega; t,r,\theta,\phi)$. For the unknown factors $S_{m}(\omega;\theta)$ and $R_{m}(\omega;r)$ we obtain respectively Teukolsky’s angular equation (TAE): $$\begin{aligned}
\label{angular}
&& {1 \over \sin \theta}{\partial \over \partial\theta} \left(
\sin \theta \frac{\partial }{\partial\theta}
\right)S_{m}(\omega;\theta) + \\ \nonumber && +
\left(a^2\omega^2\cos^2\theta - {2a\omega s \cos\theta}-
\frac{m^2+s^2+2ms\cos\theta}{\sin^2\theta} + E_{m}
\right)S_{m}(\omega;\theta) = 0\,,\end{aligned}$$ and Teukolsky’s radial equation (TRE) $$\label{radial}
\Delta^{-s} {d \over dr} \left( \Delta^{s+1} \frac{d }{d r}
\right)R_{m}(\omega;r)+ \left(\frac{K^2-2is\left( r-M
\right)K}{\Delta} + 4is\omega r - \lambda_{m}
\right)R_{m}(\omega;r) = 0\,,$$ where $K \equiv \left( r^2 +a^2 \right)\omega - am$ and $\lambda_{m} \equiv E_{m} + a^2\omega^2-2am\omega -s(s+1)$ is the separation “constant"[^4]. These are the two equations, some special solutions of which we will study in detail in our paper.
Solutions to TRE
================
Transforming TRE into the non-symmetrical canonical form of Heun equation
-------------------------------------------------------------------------
Both equations and are second order ordinary differential equations with two regular singular points at finite value of the independent variable and one irregular singularity at infinity. The most general equation with such properties is the confluent Heun equation [@Ron]. Thus both TRE and TAE are specific cases of confluent Heun equations. At this point we continue by transforming the radial equation into one of the forms of the confluent Heun equation, the non-symmetrical canonical form . In order to do so we apply the so-called [*s*]{}-homotopic transformation to by setting[^5] $$\label{anzatzR}
R_{m}(\omega;r) = (r-r_{+})^{\xi}(r-r_{-})^{\eta}e^{\zeta
r}H(r)\,,$$ where $r_{+}$ and $r_{-}$ are the event and Cauchy horizons of a Kerr black hole defined by $r_{\pm} = M \pm \sqrt{M^2 - a^2}$ and $\xi$, $\eta$ (the indices of the regular singularities) and $\zeta$ are parameters to be determined. By substituting into and after some straightforward algebra we arrive at the following equation for $H(r)$:
$$\begin{aligned}
\nonumber
&&{d^2 H(r) \over dr^2} + \left( \frac{2\xi+s+1}{r-r_{+}} +
\frac{2\eta+s+1}{r-r_{-}} +2\zeta\right) {d H(r) \over dr} +
\\ \nonumber \\ \nonumber &&\frac{1}{(r-r_{+})(r-r_{-})}\{
\left[4\omega^2 M + 2\zeta(\xi+\eta + s+1)+2is\omega \right]r +
\\ \nonumber \\ \nonumber &&+(\xi + \eta)^2 + (\xi +
\eta)+2s(\xi+\eta)-2\zeta (\xi r_{-} + \eta r_{+}) - 2s\zeta M
-2\zeta M- \\ \nonumber \\ \nonumber &&-2a\omega m +4\omega^2 M^2
-2is\omega M - \lambda_{m} \}H(r) = 0\,.\end{aligned}$$
In order to obtain this form of the equation we had to fix the parameters $\xi$, $\eta$ and $\zeta$. For them we end up with quadratic equations and thus with pairs of possible expressions, namely:
$$\begin{aligned}
\label{valuesxi} \xi_{1} =
ia_{+} (\omega - m\Omega_{+}) , \qquad \xi_{2} = -s - ia_{+}
(\omega - m\Omega_{+})\,,\end{aligned}$$
$$\begin{aligned}
\label{valueseta} \eta_{1} = -s +
ia_{-} (\omega - m\Omega_{-}), \qquad \eta_{2} = - ia_{-} (\omega
- m\Omega_{-})\,,\end{aligned}$$
and
$$\begin{aligned}
\label{valueszeta} \zeta_{1} = i\omega ,
\qquad \zeta_{2} = - i\omega\,,\end{aligned}$$
where we have set
$$\begin{aligned}
\label{defaOmega}
a_{\pm}= {2Mr_{\pm} \over r_{+} - r_{-}}, \qquad \Omega_{\pm}= {a
\over 2Mr_{\pm} } \,.\end{aligned}$$
Any one of the eight possible triplets of expressions from , , and leads to a Heun equation in the desired form with different parameters. In order to determine these parameters it is necessary to perform one more step and to introduce dimensionless variables instead of $r$. In order to keep the symmetries in the problem manifest it is best to set different variables for positive and for negative spin weights, namely for $s=1/2, 1, 2$ and for $s=-1/2, -1, -2$ we will have respectively: $$\begin{aligned}
\nonumber
{}_{+}z=\frac{r_{+}-r}{r_{+}-r_{-}}, \qquad
{}_{-}z=\frac{r-r_{-}}{r_{+}-r_{-}}.\end{aligned}$$ Note that from now on in the paper we will denote the sign of the spin weight (and when there might be a confusion the spin weight itself) by a subscript to the left of the variable. Also always the signs “+”, “-”, or $\pm$ used as subscripts to the right from the variables will be related to the two singularities in the radial equation, namely to the event and the Cauchy horizons. After performing these adjustments we end up with a Heun equation in the non-symmetrical standard form : $$\begin{aligned}
\label{eqHrlast}
{{d^2H}\over{d({}_{\pm}z)^2}}+\left(4({}_{\pm}p)+{{{}_{\pm}\gamma}\over{{}_{\pm}z}}+
{{{}_{\pm}\delta}\over{{}_{\pm}z-1}}\right){{dH}\over{d({}_{\pm}z)}}+{4({}_{\pm}\alpha)
({}_{\pm}p)({}_{\pm}z) -({}_{\pm}\sigma)
\over{({}_{\pm}z)(({}_{\pm}z)-1)}}H=0\end{aligned}$$ with the parameters given by the following expressions: ${}_{\pm}p= \mp (r_{+}-r_{-}) \zeta/2$, ${}_{+}\gamma =
{}_{-}\delta = 2\xi +s +1$, ${}_{-}\gamma = {}_{+}\delta = 2\eta +
s + 1$,
$$\begin{aligned}
\label{valuesHalpha}
{}_{\pm}\alpha = 2M\omega^2 \zeta^{-1} +\xi + \eta + s + 1 +
is\omega \zeta^{-1}\,,\end{aligned}$$
and $$\begin{aligned}
\nonumber
&& {}_{\pm}\sigma = - 2\zeta r_{\pm} \left( {2M\omega^2 \over
\zeta} +\xi + \eta + s + 1 + {is\omega \over \zeta} \right) - [
(\xi + \eta)^2 + (\xi + \eta)+ 2s(\xi+\eta) - \\ \nonumber \\
\nonumber && \qquad - 2\zeta (\xi r_{-} + \eta r_{+}) - 2s\zeta M
-2\zeta M -2a\omega m +4\omega^2 M^2 -2is\omega M - \lambda_{m}]
\,,\end{aligned}$$ for any choice of a triplet $\xi, \eta, \zeta$. In general there exist pairs of solutions to Heun equation expressed as appropriate power series about each one of the singularities. The solutions of the equation can be expressed with the use of the so-called Frobenius solutions $Hc^{(r)}({}_{\pm}p,{}_{\pm}\alpha,{}_{\pm}\gamma,{}_{\pm}\delta,
{}_{\pm}\sigma;{}_{\pm}z)$. There exists a multitude of other solutions to Heun equation which can be obtained from these Frobenius solution by interchanging the finite singular points or performing appropriate [*s*]{}-homotopic transformations (See the Apendix).
Polynomial solutions
--------------------
Thus we have completed the transformation of TRE to the non-symmetrical form of the confluent Heun equation and we can formally identify the solutions which are given by specific Heun functions. There exists though a special case, in which the confluent Heun equation admits polynomial solutions (See the Appendix). This special case depends on the values of ${}_{\pm}\alpha$ in and another, more involved condition on the parameters in the Heun equation. If the parameters ${}_{\pm}\alpha$ are equal to negative integer numbers or zero and if we can solve the second condition then we will have polynomial solutions to Heun equation. At this point we can make the following observation regarding the possible values of ${}_{\pm}\alpha$ in : In the cases of positive spin weights if we pick the values $\xi_{2}$, $\eta_{1}$, and $\zeta_{2}$ from , , and , then we obtain ${}_{+}\alpha=1-2s$, which equals $0$ for neutrino, $-1$ in the electromagnetic case and $-3$ in the gravitational case. Similarly, for $s=-1/2$, $s=-1$, or $s=-2$ we have to choose $\xi_{1}$, $\eta_{2}$, and $\zeta_{1}$ and we will get ${}_{-}\alpha=1+2s$, which again gives $\alpha=0,-1$ or $-3$ for the neutrino, for the electromagnetic, and for gravitational case respectively. Thus we have identified particular combinations of the values of the parameters $\xi$, $\eta$, and $\zeta$ for which we may expect to find polynomial solutions of the radial equation for $|s|=1/2, 1, 2$ since the necessary condition $\alpha=-N$, $N$-integer, is satisfied. Also, it can be seen easily that for $s=0$ we obtain ${}_{\pm}\alpha=1$ so there is no polynomial solution for scalar fields. The second condition will depend on the value of $N$. It is an algebraic equation of order $N+1$ and leads to polynomial solutions to Heun equations of order $N$. Thus we expect that in the neutrino case the polynomial solutions, if they exist, are simply constants, in the electromagnetic case - linear functions, and in the gravitational case the solutions are cubic polynomials. We will consider here in more detail only the electromagnetic case. The results for $s=\pm 1/2$ can be easily obtained using the same procedure. The polynomial solutions to TRE with spin $|s|=2$ were described long time ago in quite a different setting by Chandrasekhar [@Chandra84]. For a more recent treatment see [@Brink].
Polynomial solution to TRE for spin $|s|= 1$
--------------------------------------------
In order to find a polynomial solution to TRE we have to impose in addition to the fact that ${}_{\pm}\alpha=-N$ the condition . In this case we are looking for an expansion about the singular point at infinity. The second polynomial condition for electromagnetic perturbations has the form $g^{(r)}_{0}g^{(r)}_{1}=h^{(r)}_{1}f^{(r)}_{0}$ with the coefficients from the three-term relation given by $g^{(r)}_{0}=-\sigma -4p+\gamma+\delta,
g^{(r)}_{1}=-\sigma, f^{(r)}_{0}= -4p, h^{(r)}_{1}= -\gamma$. Provided we make the above mentioned choices rendering ${}_{\pm}\alpha=-1$ we obtain the following values for the remaining parameters in the Heun equation: ${}_{\pm}p = (i/2)
\omega (r_{+}-r_{-}), {}_{\pm}\gamma = \pm 2i a_{-} (\omega -
m\Omega_{-}), {}_{\pm}\delta = \mp 2ia_{+} (\omega - m\Omega_{+}),
{}_{\pm}\sigma = {}_{\pm}E_{m} +a^2\omega^2 -2a\omega m \mp
2i\omega r_{\pm} $. This second polynomiality condition essentially fixes ${}_{\pm}\sigma$. It leads to a quadratic equation for ${}_{\pm}\sigma$ with solutions ${}_{\pm} \sigma_{1}
= \mp 2i\omega r_{\pm} + 2\sqrt{a\omega (a\omega -m)}$ and ${}_{\pm} \sigma_{2} = \mp 2i\omega r_{\pm} - 2\sqrt{a\omega
(a\omega -m)}$ (recall that the $\pm$ sign to the left of $\sigma$ refers to the sign of the spin weight, while the subscripts $1$ and $2$ number the solutions to the quadratic equation).
The first important result from solving the second condition for having polynomial solutions is that we find the dependence of the separation “constant" $E_{m}$ on the frequency $\omega$, which is the same for both $s=+1$ and for $s=-1$ [@Fiziev4; @Fiziev5] and is given by $$\begin{aligned}
&& \nonumber {}_{\pm1} E_{m}(a\omega)_{1} = -a^2\omega^2 +
2a\omega m + 2\sqrt{a\omega (a\omega -m)}
\\ && \nonumber {}_{\pm1} E_{m}(a\omega)_{2} = -a^2\omega^2 + 2a\omega m - 2\sqrt{a\omega
(a\omega -m)} \,.\end{aligned}$$
By returning to the $r$ variable we arrive at the following expressions for the polynomial solutions of the Heun equation for $s=1$:
$$\begin{aligned}
\label{Hemplusr12}
({}_{+1}H_{m}(\omega;r))_{1,2}={1 \over r_{+}-r_{-}} \left(- r \pm
{i \over \omega} \sqrt{a\omega (a\omega - m)}\right) \,.\end{aligned}$$
Similarly, for $s=-1$ we have: $$\begin{aligned}
\label{Hemminusr12}
({}_{-1}H_{m}(\omega;r))_{1,2}={1 \over r_{+}-r_{-}} \left(r \pm
{i \over \omega} \sqrt{a\omega (a\omega - m)}\right)\,.\end{aligned}$$ Putting together with the specific values of $\xi$, $\eta$, and $\zeta$ and the polynomial solutions of Heun equations, the solutions to the TRE for $s=1$ and for $s=-1$ can be written (modulo normalizing constants) respectively as:
$$\begin{aligned}
\label{finalRplus}
&& ({}_{+1}R_{m}(\omega;r))_{1,2} =
{e^{-i\omega r_{*}} \over \Delta}
{\left({r-r_{+} \over r-r_{-}}\right)^{{ima \over r_{+} - r_{-}}}}
({}_{+1}H_{m}(\omega;r))_{1,2}
\,,\end{aligned}$$
and $$\begin{aligned}
\label{finalRminus}
&& ({}_{-1}R_{m}(\omega;r))_{1,2} =
e^{i\omega r_{*}}
{\left({r-r_{+} \over r-r_{-}}\right)^{-{ ima \over r_{+} -
r_{-}}}}
({}_{-1}H_{m}(\omega;r))_{1,2}
\,,\end{aligned}$$ where $$\nonumber
r_{*}=r + a_{+}\ln|r-r_{+}| - a_{-}\ln|r-r_{-}|$$ is the “tortoise" coordinate and $a_{\pm}$ are defined in . Note that these exact solutions for $|s|=1$ are presented here for the first time in explicit form. A similar form of polynomial solutions to TRE in the case $|s|=2$ can be found in [@Brink].
Using the orthogonality relations of Heun polynomials [@Ron] it can be shown that in terms of the intermediate variables ${}_{s}z = {}_{\pm}z$ we have:
$$\label{ortho}
\int_{-\infty}^{0}\Delta^{s}({}_{s}R_{m}(\omega;{}_{s}z))_{j}
({}_{s}R_{m}(\omega;{}_{s}z))_{l} d({}_{s}z) =0 , \qquad j \neq l
\qquad j,l=1,2 \,,$$
where $\Delta$ is the standard factor from the Kerr metric, defined in . The behavior of these solutions at infinity and at the event horizon can be readily determined form and . First, both solutions in with $s=1$ have the behavior:
$$\begin{aligned}
\label{Hemplusinfty}
({}_{+1}R_{m}(\omega;r))_{1,2} \sim \left\{
\begin{array}{lcr}
r^{-1}e^{-i\omega r_{*}} \; & r \rightarrow \infty & (r_{*}
\rightarrow \infty) \\
\\ \Delta^{-s} e^{-i\varpi r_{*}} \; & r \rightarrow r_{+}
& (r_{*} \rightarrow -\infty)
\end{array}\right.,\end{aligned}$$
while both solutions in with $s=-1$ behave like
$$\begin{aligned}
\label{Hemminusinfty}
({}_{-1}R_{m}(\omega;r))_{1,2} \sim \left\{
\begin{array}{lcr}
r^{-(2s+1)} e^{i\omega r_{*}} \; & r \rightarrow \infty & (r_{*} \rightarrow
\infty) \\
\\e^{i\varpi r_{*}} \; & r \rightarrow r_{+}
& (r_{*} \rightarrow -\infty)
\end{array}\right.,\end{aligned}$$
where in the expressions for the behavior at $r_{+}$ we have set $\varpi = \omega - m\Omega_{+}$. Thus we found exact solutions to TRE, the nature of which depends on the relative sign between $\omega$ and $\varpi$, in agreement with the general analysis in [@Teukolsky]-[@Teukolsky2]. When $\omega$ and $\varpi$ have the same sign then the solutions we found describe one-way waves traveling from $r_{+}$ to infinity or in the opposite direction. The solutions with $\omega$ and $\varpi$ with opposite signs describe either waves traveling towards $r_{+}$ and towards infinity or leaving from $r_{+}$ and coming from infinity.
Polynomial solution for spin $|s|= {1 / 2}$
-------------------------------------------
This case follows along the same lines like the electromagnetic perturbations but is simpler since $\alpha=0$. This leads to $c^{(r)}_{1}=0$ in which corresponds to a constant solution to the Heun equation. This implies that $\sigma=0$ and thus determines the dependence of the separation constant $E$ from $\omega$. The result is [@Fiziev4; @Fiziev5]
$$\nonumber
{}_{\pm {1 \over2}}E_{m}(a\omega)= - a^2\omega^2 + 2a\omega m - {1
\over 4}.$$
The solutions to TRE for $s=\pm {1 \over 2}$ are presented here for the first time:
$$\label{Rneuplus}
{}_{1 \over 2}R_{m}(\omega;r) = {e^{-i\omega r_{*}} \over
\sqrt{\Delta}}
{\left({r-r_{+} \over r-r_{-}}\right)^{{ima \over r_{+} - r_{-}}}}
{}_{1 \over 2} H_{m}(\omega)\,$$
$$\label{Rneuminus}
{}_{-{1 \over 2}}R_{m}(\omega;r) = e^{i\omega r_{*}}
{\left({r-r_{+} \over r-r_{-}}\right)^{-{ima \over r_{+} -
r_{-}}}} {}_{-{1 \over 2}}H_{m}(\omega)\,,$$
where ${}_{1 \over 2}H_{m}(\omega)$ and ${}_{-{1 \over
2}}H_{m}(\omega)$ are constants in $r$. Note that these solutions satisfy the same orthogonality relations as those in and have the same behavior at infinity and at $r_{+}$ as those given in and .
Solutions to TAE
================
Transformation of TAE into the form of a Heun equation
------------------------------------------------------
In order to transform TAE into the form of a Heun equation we follow the same procedure as with the radial one. Following [@Fackerell]-[@Berti] we set
$$\label{substang}
S_{m}(\omega;u) =
(1-u)^{\mu_{1}}(1+u)^{\mu_{2}}e^{\nu u}T_{m}(\omega;u)\,,$$
where $u =\cos\theta$ and $\mu_{1}$ and $\mu_{2}$ are the indices of the regular singular points at $\theta=0$ and $\theta=\pi$. We plug into and obtain an equation for $T_{m}(\omega;u)$. Imposing the condition that the equation for $T_{m}(\omega;u)$ has the form leads to a system of quadratic algebraic equations for the parameters $\mu_{1}$, $\mu_{2}$, and $\nu$. Their solutions are given by $$\begin{aligned}
\label{uvwvalues}
\mu_{1}=\pm \frac{m+s}{2} , \qquad \mu_{2}=\pm \frac{m-s}{2}, \qquad \nu=\pm
a\omega \,.\end{aligned}$$ For these values of $\mu_{1}$, $\mu_{2}$, and $\nu$ the equation for $T_{m}(\omega;u)$ is given by
$$\begin{aligned}
\label{eqHa}
&&{d^2 T_m(\omega;u) \over du^2}+ \left( \frac{2\mu_{1}+1}{u-1} +
\frac{2\mu_{2}+1}{u+1} +2\nu \right) {d T_m(\omega;u) \over du} +
\\ \nonumber && \frac{1}{(u-1)(u+1)}\{ 2\nu \left(
\mu_{1}+\mu_{2}+1+{a\omega s \over \nu }\right)u - \\ \nonumber \\
\nonumber && - \left[ E_{m} + a^2\omega^2 -(\mu_{1}+\mu_{2})^2 -
(\mu_{1}+\mu_{2}) -2\nu ( \mu_{1}-\mu_{2})\right]\}T_m(\omega;u) =
0 \,.\end{aligned}$$
In order to complete the transformation we introduce new independent variables (the reason for the specific form of this transformation will become clear latter) $u \mapsto {}_{+}x$ for $s=+1/2,1,2$ and $u \mapsto {}_{-}x$ for $s=-1/2,-1,-2$, where ${}_{\pm}x=(1\pm u)/2$. It is important to notice that these two independent variables are related by the transformation $\theta
\mapsto \pi-\theta$. Thus we arrive at the following specific non-symmetric canonical form of a confluent Heun equation: $$\label{eqHalast}
{d^2 T_m(\omega;{}_{\pm}x) \over d{({}_{\pm}x)}^2}+ \left(
\frac{{}_{\pm}\gamma}{{{}_{\pm}x}} +
\frac{{}_{\pm}\delta}{{{}_{\pm}x}-1} +4({}_{\pm}p) \right) {d
T_m(\omega;{{}_{\pm}x}) \over d({}_{\pm}x)} +
\frac{4({}_{\pm}p)({}_{\pm}\alpha) ({}_{\pm}x) -
({}_{\pm}\sigma)}{{{}_{\pm}x}({{}_{\pm}x}-1)}T_m(\omega;{{}_{\pm}x})
= 0 \,,$$ with the identification $ {}_{\pm}p= \pm \nu$, ${}_{+}\gamma =
{}_{-}\delta = 2\mu_{2}+1 $, ${}_{-}\gamma = {}_{+}\delta =
2\mu_{1}+1$, and for both signs we have
$$\begin{aligned}
\nonumber
&& {}_{\pm} \alpha =
\mu_{1}+\mu_{2}+1+{a\omega s \over \nu } \\ \nonumber \\
\label{paramH} && {}_{\pm} \sigma = - 2\nu ({}_{\pm} \alpha) +
E_{m} + a^2\omega^2 -(\mu_{1}+\mu_{2})^2 - (\mu_{1}+\mu_{2}) -2\nu
(\mu_{1}-\mu_{2}) \,.\end{aligned}$$
The solutions of equation can be expressed with the use of the Frobenius solutions $Hc^{(a)}(p,\alpha,\gamma,\delta,\sigma;{}_{\pm}x)$ and all other solutions obtained from it by interchanging the finite singular points or performing appropriate [*s*]{}-homotopic transformations (See the Apendix). Thus we obtained the needed solutions to the Heun equation obtained from TAE[^6]. In the general case these are given by two sets of infinite series about each one of the regular singularities. For more detailed description of all local solutions to the angular Teukolsky equation see [@Fiziev4; @Fiziev5]. With these we can build the solutions to TAE.
For each choice of a triple $\mu_{1}$, $\mu_{2}$, and $\nu$ we get a different solution to TAE. There are two special cases though: First, we can choose $\mu_{1}$, $\mu_{2}$, and $\nu$ in such a way so to obtain solutions that are regular at both $\theta=0$ and $\theta=\pi$. This chice leads to a well studied Sturm-Liouville eigenvalue problem [@Fackerell]-[@Berti] for the spin-weighted spheroidal wave functions $S_{lm}(\omega;\theta)$ [@AS] and the separation constant $E_{lm}=E_{lm}(a\omega)$, which for fixed $s$, $m$, and $a\omega$ are labeled by an additional integer $l$. The eigenfunctions $S_{lm}(\omega;\theta)$ are complete and orthogonal on $0\leq \theta \leq \pi$ for each set $s$, $m$, and $a\omega$. In the case $s=0$, $S_{lm}(\omega;\theta)$ are the spheroidal wave functions. When $a\omega=0$, the eigenfunctions are the spin-weighted spherical harmonics ${}_{s}Y^{m}_{l}={}_{s}S^{m}_{l}(\theta)e^{im\phi}$. In the general case, as it is shown for example in [@Seidel] the function $T_m(\omega;z)$ can be expanded as a series of Jacobi polynomials in the case of an integer spin, and to their spin-weighted generalizations for half-integer spins [@Gold; @Dolan1; @Dolan2], and one can obtain the separation constant $\lambda_{lm}$ as a power series in $a\omega$.
The second special case corresponds to polynomial solutions to TAE [@Fiziev4; @Fiziev5]. Let us choose $\mu_{1}= -(m+s)/2,
\mu_{2}=(m-s)/2, \nu= - a\omega$. This choice leads in to $\alpha = 1-2s$ , which for $s=1/2$, $s=1$ and $s=2$ is zero or a negative integer number. This means that the condition , which is necessary for having a polynomial solution (a constant in the case $\alpha=0$) of the confluent Heun equation is met for these values of the spin weight $s$. The case of zero spin is again excluded by this condition. If instead we have $s=-1/2$, $s=-1$ or $s=-2$, we must choose $\mu_{1}= (m+s)/2, \mu_{2}= -(m-s)/2, \nu= a\omega$, and this will give us $\alpha = 2s+1$, which in this case again is zero or a negative integer number. We have to emphasize that this result comes at a price - depending on the specific values of $m$ and $s$ either one or the other of the pre-factors in diverge at the corresponding singular point. This means that for the cases of neutrino, electromagnetic and gravitational perturbations we eventually may write down the solutions to TAE into the form of diverging at some of the singularities pre-factor multiplied by a polynomial expression, simultaneously regular at both regular singular points. Again, the neutrino case is relatively simple and can be easily deduced from the electromagnetic one, which we will present in detail.
Polynomial solutions for perturbations with spin $|s|= 1$
---------------------------------------------------------
Since is satisfied we can obtain a polynomial solution of by imposing as an additional (already sufficiency) condition . In our case translates into an $\omega$ dependence of the separation constant $E_{m}$, which enters in the parameter $\sigma$ from .
In this case $\alpha=-1$, so we will have $c^{(a)}_{2}=0$ in . Because of the specific choice we made when introducing ${}_{\pm}x$ in this case we have for both positive and negative spin weights the same values of the parameters in the Heun equation: $p= -a\omega, \alpha=-1, \gamma = m, \delta = -m,
\sigma = E_{m} + a^2\omega^2 +2a\omega -2a\omega m$. This means that we obtain the same Heun equation for both $s=1$ and $s=-1$ with Frobenius solutions $Hc^{(a)}(p,\alpha,\gamma,\delta,\sigma;{}_{\pm}x)$. The only difference between the two cases is in the definition of the independent variable ${}_{\pm}x=(1\pm \cos\theta)/2$. Thus in this section we will look for a solution only for the case $s=1$ and will obtain the solution for $s=-1$ by performing in the final results the transformation $\theta \mapsto \pi - \theta$. In terms of the coefficients from the three term relation the sufficiency condition for $m\neq 0$ has the form $g^{(a)}_{0}g^{(a)}_{1}=h^{(a)}_{1}f^{(a)}_{0}$, where $g^{(a)}_{0}=-\sigma, g^{(a)}_{1}= 4a\omega -\sigma, f^{(a)}_{0}=
- m, h^{(a)}_{1}= 4a\omega $. Since this condition leads to a quadratic equation for $ \sigma$ we obtain pairs of solutions. For both $s=+1$ and $s=-1$ we get the same expressions $\sigma_{1} =
2a\omega + 2\sqrt{a\omega (a\omega -m)}$ and $\sigma_{2} =
2a\omega - 2\sqrt{a\omega (a\omega -m)}$. Returning back to the original variables of the angular equation we can write down the following expressions [@PFDS3; @PFDS4], for the solutions for $s=+1$ (for $m \neq 0$):
$$\label{emSplusfinal}
({}_{+1}S_{m}(\omega;\theta))_{1,2} ={e^{-a\omega \cos\theta}
\over \sin\theta} \left(\cot{\theta \over 2}\right)^{m}
({}_{+1}T_{m}(\omega;\theta))_{1,2}\,,$$
and we have restored the pre-subscripts denoting the two different spin weights. The polynomial parts of the solutions for $m \neq 0$ are: $$\nonumber
({}_{+1}T_{m}(\omega;\theta))_{1} = 1 - \frac{2a\omega +
2\sqrt{a\omega(a\omega - m)}}{m}\cos^{2}{\theta \over 2}\,,$$
$$\nonumber
({}_{+1}T_{m}(\omega;\theta))_{2} = 1 - \frac{2a\omega -
2\sqrt{a\omega(a\omega - m)}}{m}\cos^{2}{\theta \over 2} \,.$$
In the case $m=0$ it is easy to solve the equation directly and arrive at the following overall solutions: $$\begin{aligned}
\nonumber
({}_{+1}S_{0}(\omega;\theta))_{1} =
e^{-a\omega\cos\theta}\tan{\theta \over 2}, \qquad
({}_{+1}S_{0}(\omega;\theta))_{2} =
e^{-a\omega\cos\theta}\cot{\theta \over 2} \,.\end{aligned}$$ The behavior of the solutions to TAE we found at the two singularities $\theta=0$ and $\theta=\pi$ can be easily obtained from the expressions above. Each one is divergent either at the one or at the other singularity. The exact expressions for $m \neq
0$ are:
$$\begin{aligned}
\label{Semplus}
({}_{+1}S_{m}(\omega;\theta))_{1,2} \sim \left\{
\begin{array}{lcr}
\theta^{-(m+1)} & \mbox{at}\;\theta \rightarrow 0 &
\mbox{for}\; m\geq1 \\ \\ (\pi - \theta)^{-(m+1)} &
\mbox{at}\;\theta \rightarrow \pi & \mbox{for}\; m\leq1
\end{array}\right..\end{aligned}$$
For $m=0$ we have $({}_{+1}S_{0}(\omega;\theta))_{1} \sim
\theta^{-1}$ at $\theta \rightarrow 0$ and $({}_{+1}S_{0}(\omega;\theta))_{2} \sim (\pi - \theta)^{-1}$ at $\theta \rightarrow \pi$. If we perform the transformation $\theta
\mapsto \pi - \theta$ we obtain the solutions for $s=-1$:
$$\label{emSminusfinal}
({}_{-1}S_{m}(\omega;\theta))_{1,2} = {e^{a\omega \cos\theta}
\over \sin\theta} \left(\cot{(\pi - \theta) \over 2}\right)^{m}
({}_{-1}T_{m}(\omega;\theta))_{1,2}\,,$$
where for $m \neq 0$:
$$\nonumber
({}_{-1}T_{m}(\omega;\theta))_{1} = 1 - \frac{2a\omega +
2\sqrt{a\omega(a\omega - m)}}{m}\sin^{2}{\theta \over 2}\,,$$
$$\nonumber
({}_{-1}T_{m}(\omega;\theta))_{2} = 1 - \frac{2a\omega -
2\sqrt{a\omega(a\omega - m)}}{m}\sin^{2}{\theta \over 2}\,,$$
and for $m=0$:
$$\begin{aligned}
\nonumber
({}_{-1}S_{0}(\omega;\theta))_{1} =
e^{a\omega\cos\theta}\cot{\theta \over 2}, \qquad
({}_{-1}S_{0}(\omega;\theta))_{2} =
e^{a\omega\cos\theta}\tan{\theta \over 2} \,.\end{aligned}$$
It can be shown that because of the orthogonality properties of Heun polynomials [@Ron], the following relations hold for the functions $({}_{\pm1}S_{m}(\omega;x))_{j}$ separately for $s=1$ and for $s=-1$:
$$\begin{aligned}
\label{ortho2}
\int_{0}^{1}
({}_{\pm1}S_{m}(\omega;{}_{\pm1}x))_{j}({}_{\pm1}S_{m}(\omega;{}_{\pm1}x))_{l}
d({}_{\pm1}x) = 0 \, \qquad j \neq l \, \qquad j,l=1,2.\end{aligned}$$
The second condition for a polynomial solution of the radial equation leads to the expressions for the separation “constant" $E_{m}$ for both $s=+1$ and for $s=-1$. For any $m$ we have [@Fiziev4; @Fiziev5] $$\begin{aligned}
&& \nonumber {}_{\pm1} E_{m}(a\omega)_{1} = -a^2\omega^2 +
2a\omega m + 2\sqrt{a\omega (a\omega -m)}
\\ && \nonumber {}_{\pm1} E_{m}(a\omega)_{2} = -a^2\omega^2 + 2a\omega m - 2\sqrt{a\omega
(a\omega -m)} \,.\end{aligned}$$ Surprisingly, these expressions are the same both for the TRE and TAE (see Section 3.2).
Hence, the second condition does not produce an additional relation between $E_{m}$ and $\omega$. The main consequence from this result is that we can express the separation constant $E_{m}$ as a function of $\omega$ but the (complex) frequency itself remains unconstrained. As a result we obtain a continuous spectrum in $\omega$ for the solutions of Teukolsky’s angular and radial equations and .
Thus, the surprising phenomenon of simultaneous fulfilment of the polynomial conditions for angular and radial equations with spin $|s|=1$ is related to the existence of the continuous spectrum of Teukolsky Master Equation in the specific boundary problem under consideration.
Polynomial solutions for perturbations with spin $|s|= {1/2}$
-------------------------------------------------------------
Again we have $\sigma=0$ and this determines the dependence of the separation constant $E$ from $\omega$. The result again is [@Fiziev4; @Fiziev5]
$$\nonumber
{}_{\pm {1 \over2}}E_{m}(a\omega)= - a^2\omega^2 + 2a\omega m - {1
\over 4}.$$
Like in the electromagnetic case both conditions for polynomiality are the same. Thus in this case again we obtain a continuous spectrum. The solutions to TAE for $s=\pm {1 \over 2}$, presented here for the first time, are:
$$\label{Sneuplus}
{}_{ {1 \over2}} S_{m}(\omega;\theta) = \left( \cot{\theta \over
2} \right)^{m} {e^{-a\omega \cos\theta} \over \sqrt{\sin\theta}}
\,,$$
$$\label{Sneuminus}
{}_{-{1 \over2}}S_{m}(\omega;\theta) = \left( \tan{\theta \over 2}
\right)^{m} {e^{a\omega \cos\theta} \over \sqrt{\sin\theta}}\,.$$
These solutions satisfy the same orthogonality relations as in . The behaviour at $\theta = 0$ and at $\theta =
\pi$ can be easily deduced from and .
Nonexistence of gravitational one-way waves of continuous spectrum
==================================================================
The derivation of the polynomial solutions for perturbation with spin $|s|=2$ can be found in [@Chandra84; @Brink]. Our analysis in the gravitational case follows the same lines as in the electromagnetic one and reproduces the results of these articles. As it was already discussed, with appropriate choice of the powers in and we achieve $\alpha=-3$ for both TRE and TAE. When we impose the second polynomiality condition we obtain quartic equations for the values of the separation “constant". The important difference is that unlike in the electromagnetic and the neutrino cases, the Heun polynomial conditions are different for the angular and for the radial equations. The proof based on Taylor series expansions of the roots of quartic equations can be found in [@Fiziev4; @Fiziev5]. This result is consistent with the observation by Teukolsky and Press [@Teukolsky2], developed further by Chandrasekar [@Chandra], that the difference between the Starobinsky’s constant for the angular and for the radial equations in the gravitational case is equal to $(12M\omega)^2$. Thus in the gravitational case we have two independent conditions, relating $E_{m}$ and $\omega$ which leads to a discrete spectrum of $\omega$ labeled by $m$ and some additional indexes for the different solutions of the quartic equations.
Overall solutions to Teukolsky’s Master Equation for spin $|s|= 1$
==================================================================
Polynomial in both $r$ and $\cos\theta$, diverging at $\theta=0$ and $\theta=\pi$ solutions
-------------------------------------------------------------------------------------------
At this point we are ready to return to the original physical fields. For $s=1$ and $s=-1$ we have respectively $\Psi=\varphi_{0}$ and $\Psi=\rho^{-2}\varphi_{2}$. With the solutions we found we can write:
$$\begin{aligned}
\nonumber
&& ((\varphi_{0})_{m}(\omega;t,r,\theta,\phi))_{1,2} \sim
{e^{-a\omega \cos\theta} e^{im\phi} \over \sin\theta}
\left(\cot{\theta \over 2}\right)^{m} \times \\ \nonumber
\\ && \nonumber \qquad \times {e^{-i\omega (r_{*}+t)} \over \Delta}
\exp{\left({i m a \over r_{+} - r_{-}}\ln\left\arrowvert{r-r_{+}
\over r-r_{-}}\right\arrowvert\right)}
({}_{+}T_{m}(\omega;\theta))_{1,2} ({}_{+}H_{m}(\omega;r))_{1,2}
\,,\end{aligned}$$
and $$\begin{aligned}
\nonumber
&& ((\varphi_{2})_{m}(\omega;t,r,\theta,\phi))_{1,2} \sim
{e^{a\omega \cos\theta}e^{im\phi} \over \sin\theta}
\left(\tan{\theta \over 2}\right)^{m} \times \\ \nonumber
\\ && \nonumber \qquad \times {e^{i\omega (r_{*}-t)} \over
(r-ia\cos\theta)^2}
\exp{\left(-{ima \over r_{+} - r_{-}}\ln\left\arrowvert{r-r_{+}
\over r-r_{-}}\right\arrowvert\right)}
({}_{-}T_{m}(\omega;\theta))_{1,2} ({}_{-}H_{m}(\omega;r))_{1,2}
\,.\end{aligned}$$ These expressions can be written in a more compact form if we introduce the Kerr coordinates with the relations:
$$\begin{aligned}
\nonumber
&& \tilde{V}=t+r_{*}, \qquad \qquad \qquad \qquad \qquad \qquad
\tilde{U}=t-r_{*} \\ \nonumber && {}_{+}\tilde{\phi}=\phi+{a \over
r_{+} - r_{-}}\ln\left\arrowvert{r-r_{+} \over
r-r_{-}}\right\arrowvert, \qquad {}_{-}\tilde{\phi}=\phi-{a \over
r_{+} - r_{-}}\ln\left\arrowvert{r-r_{+} \over
r-r_{-}}\right\arrowvert \,.\end{aligned}$$
Thus the final expressions for the solutions to Teukolsky Master Equation will be
$$\label{finaloverall}
((\varphi_{0})_{m}(\omega;t,r,\theta,\phi))_{1,2} \sim
{e^{-i\omega \tilde{V}} e^{-a\omega\cos\theta} \over (r^2-2Mr+a^2)
\sin\theta} ({}_{+}W)^{m} ({}_{+}T_{m}(\omega;\theta))_{1,2}
({}_{+}H_{m}(\omega;r))_{1,2}\,$$
and $$\nonumber
((\varphi_{2})_{m}(\omega;t,r,\theta,\phi))_{1,2} \sim
{e^{-i\omega \tilde{U}}e^{a\omega\cos\theta} \over
(r-ia\cos\theta)^2 \sin\theta} ({}_{-}W)^{m}
({}_{-}T_{m}(\omega;\theta))_{1,2} ({}_{-}H_{m}(\omega;r))_{1,2}
\,,$$ where we have introduced the following expressions $$\begin{aligned}
\label{Wpm}
{}_{+}W =e^{i({}_{+}\tilde{\phi})} \cot\left({{}_{+}\theta / 2}\right), \qquad
{}_{-}W = e^{i({}_{-}\tilde{\phi})} \cot\left({{{}_{-}\theta}/ 2}\right)
\,\end{aligned}$$ to denote the stereographic projections of a unit sphere parameterized by angles ${}_{+}\tilde{\phi},\,
{}_{+}\theta=\theta$ for $s=1$ and ${}_{-}\tilde{\phi},\,
{}_{-}\theta=\pi - \theta$ for $s=-1$ on the complex planes $\mathbb{}{C}_{{}_\pm W}$. The first formula in describes stereographic projection from the North pole and the second one – from the South one. Note that after the transition from real variables $({}_{\pm}\theta, {}_{\pm}\tilde{\phi})$ to the complex one ${}_{+}W$ one must introduce an additional phase factor $\exp{(-is{}_{\pm}\tilde{\phi})}$ in the spin-weighted spheroidal harmonics, due to the back rotation of the basis (See the paper by Goldberg et al. in [@Gold].). In the case of spin $1/2$ the introduction of such a factor $\exp{(\mp {i \over 2
}{}_{\pm}\tilde{\phi})}$ is equivalent to a transition in what follows from half-integer to integer values of the azimuthal number $m$ and a replacement $m \rightarrow \pm 1/2$.
The two expressions in together with their behavior at $r
\rightarrow \infty $ and at the regular singularities of TRE and TAE provide us with a basis from which we can build solutions with specific boundary conditions. The basis describes wave collimated along the poles $\theta=0$ and $\theta=\pi$. Depending on the values of the parameters $\omega$ and $m$, when $\omega$ and $\varpi = \omega - m\Omega_{+}$ have the same sign, the basis describes waves having the same direction of propagation at both $r_{+}$ and at infinity. Otherwise, when $\omega$ and $\varpi =
\omega - m\Omega_{+}$ have opposite signs, the waves have opposite directions of propagation at $r_{+}$ and at infinity. One possible application of this basis is to try to explain the Central Engine of the Gamma Ray Bursts (GRB), discussed in [@PFDS3] and [@PFDS4].
Polynomial in $r$ regular at $\theta=0$ and $\theta=\pi$ solutions
------------------------------------------------------------------
We could combine the polynomial solutions to TRE with the spin-weighted spheroidal harmonics representing the regular at $\theta=0$ and $\theta=\pi$ solution. The result will be a basis of waves which have the same properties in radial direction as those discussed above but will not be collimated along the axes of rotation. Possible application of the basis in this form is to study the influence of rotating gravitational field for formation and evolution of the Supernovae outbursts [@Fiziev4; @Fiziev5].
Overall solutions to Teukolsky Master Equation for $|s|=1/2$, diverging at $\theta=0$ and $\theta=\pi$
======================================================================================================
Combining the results from , , , and we can write down the expressions for the neutrino components $\chi_{0}$ and $\chi_{1}$ as:
$$\label{neufinal}
(\chi_{0})_{m}(\omega;t,r,\theta,\phi) \sim { e^{-i\omega
\tilde{V}} e^{-a\omega\cos\theta} \over \sqrt{r^2-2Mr+a^2}
\sqrt{\sin \theta}}
({}_{+}W)^{m} \,,$$
$$\nonumber
(\chi_{1})_{m}(\omega;t,r,\theta,\phi) \sim { e^{-i\omega
\tilde{U}} e^{a\omega\cos\theta} \over (r-ia\cos\theta) \sqrt{\sin
\theta}}
({}_{-}W)^{m} \,.$$
Using these solutions, which to the best of our knowledge are published for the first time, we will show how we can build regular with respect to the $\theta$ solutions. The general form of the one-way solutions, based on is
$$\label{neusum}
\chi_{0}(\omega;t,r,\theta,\phi) = { e^{-i\omega \tilde{V}}
e^{-a\omega\cos\theta} \over \sqrt{r^2-2Mr+a^2} \sqrt{\sin
\theta}} \sum_{m=-\infty}^{\infty} A_{m}(\omega)
({}_{+}W)^{m} \,.$$
The physical model is determined by the amplitudes $A_{m}(\omega)$. Physically sound solutions correspond to amplitudes $A_{m}(\omega)$, which lead to a finite result after performing the summation. Here we consider only a formal example proving the existence of proper choice of the amplitudes giving finite results. Let us write down the part of $\chi_{0}$ containing the potentially singular factor as $$(\sin\theta)^{-1/2}\sum_{m=-\infty}^{\infty} A_{m}(\omega)
\left({}_{+}W\right)^{m}=\sqrt{{1 \over 2}\left(\left|{}_{+}W\right|
+ \left|{}_{+}W\right|^{-1}\right)}
\sum_{m=-\infty}^{\infty} A_{m}(\omega) \left({}_{+}W\right)^{m}\,,$$ where $\left|{}_{+}W\right|=\cot{\theta \over 2}\geq 0$. At this point we proceed by choosing in an appropriate way functions $f(\omega, {}_{+}W)=\sum_{m=-\infty}^{\infty} A_{m}(\omega)
\left({}_{+}W\right)^{m}$ thus defining the amplitudes. For example we can choose $f(\omega, {}_{+}W)=\big({}_{+}W +
({}_{+}W)^{-1} + \text{const}\big)^{-1}$ or more generally $f(\omega,{}_{+}W)=\big(P(\omega,{}_{+}W) +
Q(\omega,{}_{+}W^{-1})\big)^{-1}$, where $P$ and $Q$ are arbitrary polynomials of degree not less than one. Then the amplitudes $A_{m}(\omega)$ are the coefficients in the Laurent series of the functions $f(\omega,{}_{+}W)$ with respect to ${}_{+}W$. With this choice it is clear that there are no singularities in the limits $\left|{}_{+}W\right| \to 0$ and $\left|{}_{+}W\right| \to
\infty$. It can be shown that choosing the polynomials $P$ and $Q$ properly we preserve the collimation of the neutrino waves.
We can go one step further and formally perform the integration in . Thus we arrive at the following general solutions to Teukolsky’s Master Equation for neutrino waves with spin weights $s=1/2$ and $s=-1/2$ respectively:
$$\nonumber
\chi_{0}(t,r,\theta,\phi) = { F_{0}(\tilde{V}-ia\cos\theta,
{}_{+}W) \over \sqrt{\Delta} \sqrt{\sin \theta}} \,,$$
$$\nonumber
\chi_{1}(t,r,\theta,\phi) = { F_{1}(\tilde{U}+ia\cos\theta,
{}_{-}W) \over \sqrt{\sin \theta}} \,,$$
where $F_{0}$ and $ F_{1}$ are arbitrary functions of their respective variables. The fact that arbitrary $F_{0}$ and $ F_{1}$ satisfy Teukolsky’s Master Equation can be verified directly. The exact forms of $F_{0}$ and $ F_{1}$ are to be determined by the specific physical situations. The above formal examples demonstrate mathematical technics, which make possible the application of singular solutions to Teukolsky’s Master Equation for description of physical reality.
Conclusion
==========
In the paper we presented an approach for solving Teukolsky Master Equation based on the use of the confluent Heun equation. After separating the variables we showed that both TRE and TAE can be transformed into the non-symmetric canonical form of the confluent Heun equation. The transformation depends on a set of parameters which when properly chosen lead to polynomial solutions to both Heun equations related to TRE and to TAE. The surprising result (which simply means that there should be a deeper physical explanation we do not understand yet) is that for neutrino and for electromagnetic perturbations we find solutions which have continuous spectrum, but for gravitational perturbations this does not happen. The richness of the results we found give us the opportunity to construct different types of solutions in accordance with the specific boundary problem we want to study.
There are many possible directions we intend to pursue. One possibility is to see if indeed we can explain some of the basic features of the GRB’s and of Supernovae using the basis we found. Related problem is to investigate further the stability of Kerr black holes.
1truecm [***Acknowledgements***]{} .3truecm
The authors would like to thank Dimo Arnaudov and Denitsa Staicova for valuable comments and suggestions.
P.F. is grateful to Professor Saul Teukolsky for his comments on the problems, related with present article, and especially for raising the question about the use of singular solutions of Teukolsky’s Master Equation.
This article was partially supported by the Foundation “Theoretical and Computational Physics and Astrophysics” and by the Bulgarian National Scientific Fund under contracts DO-1-895 and DO-02-136.
Author Contributions
====================
In 2008 R.B. joined the comprehensive program for of the use of Heun’s functions, developed by P.F. since 2005. He made calculations for Teukolsky radial and angular equations for electromagnetic case (i.e. for spin-weight $1$) in the notations of reference [@Ron], which simplify the polynomial conditions and discovered that these conditions coincide for both Teukolsky radial and angular equations. R.B. confirmed the results for spin-weight $1/2$, obtained previously by P.F. in Maple-notation and wrote down the overall solutions in section 7. The text of the present article was written by R.B.
P.F. developed the program for of the use of Heun’s functions for solution of Regge-Wheeler and Teukolsky equations since 2005. He found all specific basic classes of solutions to these equations, in particular, all polynomial solutions for different integer and half-integer spin-weights, as well as justification of some properties of the confluent Heun functions. P.F. explained the relation of the coinciding polynomial conditions for electromagnetic and neutrino perturbations with a novel continuous spectrum of Teukolsky master equation and discovered the method of deriving regular solutions of this equation using proper superposition of the singular polynomial solutions. He formulated the possible astrophysical applications of the obtained mathematical results. P.F. is responsible for the references, some corrections and editing of the text.
Appendix: Heun Equation and Heun Functions
==========================================
In this appendix we remind the reader some basic information about the confluent Heun equation and its solutions, following the notations of the reference [@Ron].
In subsection 9.1 we give a brief description of the non-symmetrical canonical form of confluent Heun equation and its solutions. In subsection 9.2 a basic information about local Frobenius and Tomé solutions around regular and irregular singular points is presented. In subsection 9.3 we remind the basic information about polynomial solutions in notations of [@Ron]. In subsection 9.4 we describe the correspondence between these notations and the notations, used in the basic articles [@DDLMRR1], [@DDLMRR2] on the modern general theory of all kinds of Heun equations and the properties of their solutions. The last notations are used in [@Fiziev6], [@Fiziev1]-[@Fiziev5]. At present these conventions become more popular, because they are used in the computer package Maple, based on the articles [@DDLMRR1], [@DDLMRR2]. This package is still the only one for analytical and numerical computer calculations with Heun equations and Heun functions. In Maple’s Help one can find an available and rich collection of relations and properties of Heun functions of all kinds.
Non-symmetrical canonical form of confluent Heun equation and its solutions
---------------------------------------------------------------------------
The general Heun equation is a second order ODE of Fuchsian type with four regular singular points. In the present paper we have to solve the confluent Heun equation (CHE) for different cases. It is relatively well studied [@Heun]-[@Fiziev6], but there still exist essential gaps in the theory. CHE can be obtained from the general Heun equation by coalescing of two of the singular points by redefining certain parameters and taking the appropriate limits. In this way two regular singular points coalesce into one irregular (in general) point. The solutions of the confluent Heun equation are relatively well-studied special functions, already included in modern computer package Maple. These functions represent non-trivial generalization of known hypergeometric functions, yet have richer properties, because confluent Heun equation has one more singular point than the hypergeometric. One of the canonical forms of the confluent Heun equation is the so called non-symmetric canonical form [@Ron]: $$\begin{aligned}
{{d^2H}\over{dz^2}}+\left(4p+{{\gamma}\over{z}}+
{{\delta}\over{z-1}}\right){{dH}\over{dz}}+{4\alpha pz -\sigma
\over{z(z-1)}}H=0.\label{H}\end{aligned}$$ The only regular Frobenius’ type solution to about the regular singular point $z=0$ is denoted by $Hc^{(a)}(p,\alpha,\gamma,\delta,\sigma;z)$. It is defined for non-integral values of $(1-\gamma)$ in the domain $|z|<1$ by the condition $$\label{Hca}
Hc^{(a)}(p,\alpha,\gamma,\delta,\sigma;0)=1.$$ In [@Ron] it is called the “angular solution” of the confluent Heun equation.
Another solution, is the Tomé’s type asymptotical solution $Hc^{(r)}(p,\alpha,\gamma,\delta,\sigma;z)$. It is defined for complex $p=|p|e^{i\varphi}$ in the domain $|z|>1$ by the condition: $$\label{ Hcr}
\lim_{|z|\rightarrow \infty}
{z^{\alpha}Hc^{(r)}(p,\alpha,\gamma,\delta,\sigma; -|z| e^{-i\varphi})}=1.$$ In [@Ron] it is called the “radial solution” of the confluent Heun equation. Different pairs of local solutions can be constructed using the combinations of four known independent transformations of the parameters, which preserve the chosen canonical form of the Heun Equation. For example by interchanging the regular singular points $z_{1}=0$ and $z_{2}=1$: $$\nonumber
z \mapsto 1-z,$$ one obtains the following new solutions: $$\begin{aligned}
\nonumber
&& Hc^{(a)}(-p,\alpha,\delta,\gamma,\sigma+4p\alpha;1-z) \\ &&
Hc^{(r)}(-p,\alpha,\delta,\gamma,\sigma+4p\alpha;1-z)\,.\end{aligned}$$
All possible sets of local solutions to Regge-Wheeler and Teukolsky equations were described for the first time in [@Fiziev4; @Fiziev5].
Power-series solutions of the confluent Heun equation
-----------------------------------------------------
### Taylor series expansion about the regular singularity $z=0$
If we expand the solution $Hc^{(a)}(p,\alpha,\gamma,\delta,\sigma;z)$ as a power series
$$\label{series}
Hc^{(a)}(p,\alpha,\gamma,\delta,\sigma;z)=\sum_{k=0}^{\infty}c^{(a)}_{k}z^k$$
then we get a three-term recurrence relation for the coefficients $c^{(a)}_{k}$:
$$\begin{aligned}
\label{3termrelation1}
&& f^{(a)}_{k}c^{(a)}_{k+1}+g^{(a)}_{k}c^{(a)}_{k}+h^{(a)}_{k}c^{(a)}_{k-1}=0
\\ \nonumber && c_{-1}=0, \qquad c_{0}=1\,,\end{aligned}$$
where
$$\begin{aligned}
\nonumber
&& g^{(a)}_{k}=k(k-4p+\gamma+\delta-1)-\sigma \\ \nonumber &&
f^{(a)}_{k}=-(k+1)(k+\gamma) \\ \nonumber &&
h^{(a)}_{k}=4p(k+\alpha-1) \,.\end{aligned}$$
The radius of convergence of the series is equal to unity, which is the distance to the next regular singular point [@Ron].
### Laurent series expansion about the singular point at infinity
Another power series can be constructed at infinity. In general this series is not convergent but only asymptotic. For the function $Hc^{(r)}(p,\alpha,\gamma,\delta,\sigma;z)$ we will have the expansion
$$\label{seriesinf}
Hc^{(r)}(p,\alpha,\gamma,\delta,\sigma;z)=z^{-\alpha}\sum_{k=0}^{\infty}c^{(r)}_{k}z^{-k}.$$
The three-term recurrence relation for the coefficients $c^{(r)}_{k}$ reads
$$\begin{aligned}
\label{3termrelation2} &&
f^{(r)}_{k}c^{(r)}_{k+1}+g^{(r)}_{k}c^{(r)}_{k}+h^{(r)}_{k}c^{(r)}_{k-1}=0
\\ \nonumber && c_{-1}=0, \qquad c_{0}=1\,,\end{aligned}$$
with the following expressions[^7] for $f^{(r)}_{k}$, $g^{(r)}_{k}$, and $h^{(r)}_{k}$:
$$\begin{aligned}
\nonumber
&& g^{(r)}_{k}=(\alpha+k)(\alpha+k+4p-\gamma-\delta+1)-\sigma \\
\nonumber && f^{(r)}_{k}=-4p(k+1) \\ \nonumber &&
h^{(r)}_{k}=-(k+\alpha-1)(\alpha+k-\gamma) \,.\end{aligned}$$
It is easy to show that in general the series expansion diverges [@Ron].
Polynomial solutions of the confluent Heun equation
---------------------------------------------------
Let us consider the case in which the parameter $\alpha$ has a fixed negative integral value
$$\label{alphaint}
\alpha = - N,\qquad N\in \mathbb{N}.$$
In this case the coefficient $h^{(a)}_{N+1}$ vanishes for both expansions above. If we impose in addition the second condition that
$$\label{conditpoly}
c^{(a)}_{N+1}=0$$
then the recurrence relation breaks down and we obtain a polynomial of $N$-th order instead of the infinite series. Since the coefficients $g^{(a)}_{k}$ are linear functions of the parameter $\sigma$ the equation $c^{(a)}_{N+1}=0$ is an algebraic equation of $(N+1)$-th order and thus it has $(N+1)$ zeros $\sigma_{0}, \sigma_{1}, \dots, \sigma_{N}$ [@Ron; @Fiziev6].
In [@Fiziev4; @Fiziev5] one can find an explicit representation of the coefficient $c^{(a)}_{N+1}=0$ in form of a specific determinant $\Delta_{N+1}$. This form is most convenient for practical calculations.
Correspondence between the notations of present article and the notations, used in computer package Maple
---------------------------------------------------------------------------------------------------------
The computer package Maple uses the conventions of the basic articles [@DDLMRR1], [@DDLMRR2]. In the Maple notation $\text{HeunC}(\alpha,\beta,\gamma,\delta,\eta,z)$ for the solution (\[Hca\]) the parameters $\alpha,\beta,\gamma,\delta,\eta$ are related in the following way with the parameters of the non-symmetrical canonical form of confluent Heun equation [@Ron], used in the present article, too: $$\begin{aligned}
\label{Maple} \alpha_{{}_{Maple}}=4p,\,\,\,
\beta_{{}_{Maple}}=\gamma-1,\,\,\, \gamma_{{}_{Maple}}=\delta-1,
\\ \delta_{{}_{Maple}}=4p\alpha-2p(\gamma+\delta),\,\,\,
\eta_{{}_{Maple}}=2p\gamma-{\frac{\gamma\delta-1} 2}-\sigma.
\nonumber\end{aligned}$$ In [@Fiziev6; @Fiziev4; @Fiziev5] a modified Maple-like parametrization of the confluent Heun equation is used: $$\begin{aligned}
\label{DHeunC}
{{d^2H}\over{dz^2}}+\left(\alpha+{{\beta+1}\over{z}}+{{\gamma+1}\over{z-1}}\right){{dH}\over{dz}}+
\left( {\mu\over z}+{\nu\over{z-1}} \right)H = 0.\end{aligned}$$ The equation (\[DHeunC\]) has a uniform shape. This uniform parametrization simplifies the explicit expressions for the coefficients in the series (\[series\]) and (\[seriesinf\]). For the parameters $\mu$ and $\nu$ one obtains the following relations with the parametrization, used in present article: $$\begin{aligned}
\label{mu_nu}
\mu=\sigma,\,\,\,\mu+\nu=4p\alpha.
\nonumber\end{aligned}$$
The first polynomial condition (\[alphaint\]) in Maple notations, as well as in the above uniform parametrization, reads: $${\frac{\delta} {\alpha}}+{\frac{\beta+\gamma}{2}}+N+1=0.$$ It yields discrete values $\delta=-\alpha\left({{1} \over
{2}}(\beta+\gamma)+N+1\right)$ of the Maple parameter $\delta$. Hence, the name $\delta_N$-condition [@Fiziev6],[@Fiziev4; @Fiziev5].
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[^1]: E-mail:[email protected], Phone: +359-2-816-1819, Fax:+359-2-962-4951
[^2]: E-mail:[email protected], Phone: +359-2-962-4951, Fax:+359-2-962-4951
[^3]: In the literature there still does not exist a commonly accepted standard form both of the five classes of Heun equations and of the corresponding Heun functions. Different possible forms are in use, since in different applications the authors prefer the form, which is most suitable for their specific needs.
[^4]: In the literature often another form of the separation constant is used, namely $A_{m}=E_{m}-s(s+1)$ so $\lambda_{m}$ can also be written as $\lambda_{m} \equiv A_{m} + a^2\omega^2-2am\omega $.
[^5]: This anzatz was used for the first time for analytical and numerical studies of the problem at hand by Leaver [@Leaver1], [@Leaver; @Leaver2].
[^6]: The Tomé asymptotic solution at infinity $Hc^{(r)}(p,\alpha,\gamma,\delta,\sigma;x)$ is not of physical significance in the case of angular equation (\[eqHa\]), because here we are interested only in solutions on the interval $u\in (-1,1)$.
[^7]: Note that these expressions are somewhat different from those in [@Ron].
|
---
abstract: 'We have demonstrated for the first time that an array of nanoantennas (central nanotips inside sub-micron pits) on an aluminum surface, fabricated using a specific double-pulse femtosecond laser irradiation scheme, results in a 28-fold enhancement of the non-linear (three-photon) electron photoemission yield, driven by a third intense IR femtosecond laser pulse. The supporting numerical electrodynamic modeling indicates that the electron emission is increased not owing to a larger effective aluminum surface, but due to instant local electromagnetic field enhancement near the nanoantenna, contributed by both the tip’s lightning rod effect and the focusing effect of the pit as a microreflector and annular edge as a plasmonic lens.'
author:
- 'M. A. Gubko$^{1}$, W. Husinsky$^{2}$, A. A. Ionin$^{1}$, S. I. Kudryashov$^{1\ast}$, S. V. Makarov$^{1,2\ast}$,C.S.R. Nathala$^{2}$, A. A. Rudenko$^{1,}$ L. V. Seleznev$^{1}$, D. V. Sinitsyn$^{1}$, I.V. Treshin$^{1}$'
date: date
title: Enhancement of ultrafast electron photoemission from metallic nanoantennas excited by a femtosecond laser pulse
---
Strong-field plasmonics, involving excitation of plasmons (collective free-electrons oscillations) in different nanoobjects by intense femtosecond (fs) laser pulses, is of high interest for basic and applied research. Surface-plasmon-enhanced multi-photon photoelectric emission [@01], high-harmonic generation [@02], electron acceleration [@03; @04] and x-ray enhancement [@05] were demonstrated using such nanostructures as diffractive gratings [@03], plasmonic bow-tie nanoantennas[@02; @04], spherical [@06] and nonspherical [@04; @05] metallic nanoparticles.
One of the most popular plasmonic elements is a metallic nanotip, providing strong optical field enhancement via the lightning rod effect. The nanometer-long decay length of the evanescent field corresponds to its strong gradients, which can be used for nanoscale acceleration of photo-emitted electrons in different regimes (multiphoton [@07], above-threshold [@08] or optical-field [@09] regimes). Interestingly, that the strong gradient of localized evanescent field can suppress the quiver motion of the electrons in the oscillating laser electric field [@10]. Such a strong-field steering of electrons in the vicinity of nanostructures with large local field enhancement and steep field gradients leads to emission of highly-directed, confined coherent electron wavepackets [@07; @09; @10; @11]. Generally, such a pulsed electron nanoemitter, triggered by femtosecond laser irradiation, could serve as an efficient source for time-resolved nanoscale imaging. For instance, ultrashort electron pulses were employed for time-domain visualization of metal melting [@12] and ionization dynamics of H${}_{2}$ [@13].
Fabrication of plasmonic nanotips usually faces problems of long fabrication cycle, chemical treatment and production costs. To provide more efficient fabrication ways, tight focusing of single nanosecond [@14] and femtosecond [@15] laser pulses into diffraction-limited spots was tested to produce one nanotip per shot. However, femtosecond laser irradiation makes it possible and realistic to easily fabricate huge arrays of nanostructures (down to the sub-100-nm scale) via intense surface plasmons polaritons (SPPs) excitation, where only weak laser beam focusing on the surface is required [@16]. Such a method for surface nanograting formation was i.e. successfully used for surface-plasmon-enhanced photoelectron emission [@01]. In the same manner, also an array of nanotips can be easily fabricated by means of fs-laser beam weak focusing on a metallic surface [@17].
In this Letter, we report a simple, double-pulse fs-laser fabrication scheme to produce an array of nanoantennas (nanotips inside sub-micron pits) on an aluminum surface and demonstrate their strongly enhanced non-linear electron photoemission, excited by a single fs-laser pulse, in comparison to flat and randomly nanostructured aluminum surfaces. These observations are supported by numerical electrodynamic modeling, indicating high local electromagnetic (EM) field enhancement in the nanoantennas.
In our experiments 100-fs, 744-nm linearly-polarized Ti:sapphire laser pulses with a maximum pulse energy of 6 mJ in the TEM${}_{00}$-mode were focused by a silica lens (focal distance of 11 cm) onto a 4-mm-thick aluminum sample mounted vertically on an *X*-*Y-Z* motorized translation stage. The mechanically polished and ultrasonically cleaned sample was located several mm above the focal plane to obtain a large spot diameter *D${}_{1/e}$*${}\thickapprox$ 180 mm. The nanostructured samples surfaces were characterized using field-emission scanning electron microscopy (FE-SEM).
To measure photoelectron emission, a stationary collecting aluminum electrode (anode) with a 2-mm aperture was mounted at a distance of 1 mm away from the sample surface and a positive voltage of 150 V was applied to extract the emitted electrons (the scheme was described elsewhere [@18]). The fs-laser pump pulses were focused on the target surface through the anode aperture. The extracting field ($\thickapprox$1 kV/cm) in this scheme is two or three orders of magnitude higher than values typical for high-vacuum schemes, where the field values must not exceed $\thickapprox$1–10 V/cm to prevent secondary electron emission, since at atmospheric pressure emitted electrons become attached to oxygen molecules on a nanosecond time scale. Then, the resulting negatively charged ions slowly move in the applied electric field on a sub-millisecond time scale, inducing an image current (potential) in the collector, which was registered using a M$\Omega$-input of a digital oscilloscope. The high extracting electric fields eliminate the space-charge effect even at intense electron emission at fs-laser fluences even as high as several J/cm${}^{2}$ [@18].
Nanoantennas fabrication on an aluminum surface was performed by two fs-laser pulses at the same peak fluence *F${}_{0}$* $\thickapprox$ 0.85 J/cm${}^{2}$ (slightly below the spallative ablation threshold *F${}_{spal}$* $\thickapprox$ 0.7 J/cm${}^{2}$ [@19]), following with a delay of a few seconds between them [@17]. After the first laser pulse an irregular array of round spallative pits with a surface density $\sim$10${}^{7}$ cm${}^{-2}$ appeared on the surface (Fig. 1a) at local fluences *F* $>$ *F${}_{spal}$* along an outer border of a macroscopic spallation crater. Their edges have widths of about $\Delta\thickapprox$ 100 nm, their bottom is semispherical appearing, in average, 100 nm below the initial surface level (Fig. 1b). The average diameter of the pits depends on local laser fluence, but usually amounts to 1.3 $\mu$m. They result from intense sub-surface nanovoid generation (homogeneous nucleation) in the melted surface layer [@20; @21] at fs-laser fluences slightly lower than the spallation threshold *F${}_{s}$*.
\[ptbh\]
[Fig1.eps]{}
Such pits with prominent edges respond to EM fields in the optical range as plasmonic nanolenses [@22], providing excitation and sub-diffraction focusing of SPPs in their centers. The focusing in plasmonic lenses exposed by fs-laser pulses at *F${}_{0}$* $\thickapprox$ 0.85 J/cm${}^{2}$ results within each pit in the formation of a single nanojet (Fig. 1c), related to material expulsion and its ultrafast cooling [@17] expected for much higher fs-laser fluences, exceeding the threshold *F${}_{frag}$* $\thickapprox$1.4 J/cm${}^{2}$ for supercritical hydrodynamic (fragmentation) [@19].
To evaluate the optical field enhancement in such a nanoantenna (a nanojet in a microscale pit), we performed numerical modeling by solving Maxwell’s equations using finite-elements method (COMSOL). EM intensity distribution was calculated for a plane EM wave ($\lambda$ = 744 nm reaching a nanotip in a pit at normal incidence (Fig. 2a) with the geometrical parameters: *H* = 550 nm, *h* = 100 nm, *R${}_{0}$* = 650 nm, *R* = 100 nm, *r* = 20 nm, $\Delta$ = 100 nm (notations see in Fig. 2a), taken from Fig. 1c. The dielectric function of unexcited aluminum at the 744-nm wavelength equals $\mathit{\varepsilon=}-68.9\mathit{+i}39.9$ [@23].
\[ptbh\]
[Fig2.eps]{}
This modeling has revealed an intensity enhancement up to 56 times outside and 5.5 times inside the peak of the nanotip (Fig. 2b,c). The enhancement factor inside the nanotip is the ratio between the maximal laser intensity values under the nanotip surface and under the flat metallic surface. The model calculation takes into account all possible interference effects, and, consequently, the enhancement is attributed not only to local phenomena such as the “lighting rod” effect, but also to EM wave reflection from the semi-spherical surface of the pit and SPP excitation from its edges. Calculation of the field for a nanotip on a flat aluminum surface resulted in a corresponding local field enhancement factor 2 times lower (as compared to a nanotip in a pit) outside the nanotip and 1.3 times lower inside. Hence, this proves that the pit works like a reflector in a parabolic antenna, which focuses the incident EM waves onto the nanotip. Additionally, in our case such pits have sharp edges, providing SPPs excitation and focusing to the nanotip.
To study the electron emissivity of the fabricated array of the fs-laser-induced nano-tips in the micro-craters, we measured the photoemission of electrons from the nano-structured surface in the appropriate intensity regime ($\thickapprox$1-10 TW/cm${}^{2}$), where the micro-craters and nano-tips are typically formed, and compared the yield from a polished surface with the yield from surfaces with laser-induced random nanostructures.
\[ptbh\]
[Fig3.eps]{}
In Fig. 3 the enhancement of the electron photoemission is shown versus *N* at two fluences *F${}_{0}$* $\thickapprox$ 0.85 J/cm${}^{2}$ and *F${}_{0}$* $\thickapprox$ 0.5 J/cm${}^{2}$. At the highest fluence *F${}_{0}$* $\thickapprox$ 0.85 J/cm${}^{2}$, the electron yield enhancement is characterized by a maximum of $\thickapprox$ 28 at *N* = 3 and subsequent decrease for increasing laser exposure *N* $>$ 3, since the nanotips are destroyed in the next shot, leaving nanopits underneath them within the sub-micron pits (Fig. 3d). The succeeding multi-shot fs-laser exposure results in a random structure of ablative nanoparticles (Fig. 3e), providing the saturated electron yield enhancement about 20.
Similar multi-shot electron yield enhancement is achieved for smaller fluence *F${}_{0}$* $\thickapprox$ 0.5 J/cm${}^{2}$, which does not produce high-fluence nanotips, but just lower-fluence nanopits (Fig. 3g). Eventually, multi-shot irradiation in this fluence regime leads in similar random nanorelief (Fig. 3i) through cumulative random formation of multiple overlapping surface nanopits via the sub-surface cavitation mechanism \[20,21\]. As a result, for large *N* the surface is again covered by nanoparticles (Fig. 3i) due to enhanced local ablation in the nanopits (Fig. 3h). In this case, the electron yield enhancement factor grows monotonically up to the almost same saturation level of $\thickapprox$ 20.
The fs-laser induced electron emission enhancement factor of nearly 30 achieved for the nanoantennas has a straightforward explanation in terms of the local field enhancement in the nanofeature. For that purpose, we have obtained the experimental dependence of the electron emission yield on fluence for the flat Al surface. The variation of electron emission yield as a function of *F${}_{0}$* is represented by the consequent cubic and linear dependences for *F${}_{0}$* $<$ 1.5 J/cm${}^{2}$ and *F${}_{0}$* $>$ 1.5 J/cm${}^{2}$, respectively (Fig. 3j). This indicates that, for the incident fs-laser fluence *F${}_{0}$* $\thickapprox$ 0.85 J/cm${}^{2}$ the electron emission yield from the reference flat surface exhibits in Fig. 3j magnitudes of 0.2-0.4 arbitrary units within the third-power region of its fluence dependence. Following the local intensity enhancement of 5.5 inside the nanotip, the effective fluence becomes equal to 4-5 J/cm${}^{2}$, corresponding to the electron emission yield values of 5-8 arbitrary units within the linear region of its fluence dependence (Fig. 3j). As a result, we would expect an enhancement of the photoemission yield due to the nanotips in range 15 – 40. However, the surface after the second fs-pulse is covered by nanotips only in part (less than 10% of the irradiated surface). In this case total electron photoemission yield has contribution from excitation of SPPs outside the pits, where they interfere with the incident laser field and each other on relatively large area. Such interference SPP-light is the main origin of the yield enhancement in case of *N* = 3 at *F${}_{0}$* $\thickapprox$ 0.5 J/cm${}^{2}$ (Fig. 3g), where the rare sub-wavelength pits play role of SPPs sources. It should be noted, that in comparison with random nanostructures, a surface with nanotips has evidently a smaller density of nanoelements, but a higher electron emission yield, indicating even stronger local EM field enhancement on individual nanotips.
Moreover, another important characteristic of the nanoantennas is their large ($\thickapprox$50) electrodynamical enhancement of optical intensity outside the nanotip, which is significantly higher than the internal enhancement factor of $\thickapprox$5 inside (Fig. 2). Such discrepancy between both enhancements results from their different electrical field polarizations. Particularly, the internal electric field inside the nanotip appears as a mostly longitudinal one with the predominating *E${}_{z}$*-component (Fig. 2c), as compared to the external electric field near the nanotip apex with nearly equal *E${}_{x}$*- and *E${}_{z}$*-components (Fig. 2b). The EM wave reflected for the pit bottom at the almost normal incidence angle contributes presumably its transversal component (*E${}_{x}$$>$* *E${}_{z}$*) to the nanotip apex field. Hence, the internal field inside the nanotip apex is contributed by SPP waves with the predominating *E${}_{z}$*-component, which are rather inefficiently excited at the pit edges.
As a result of such a high external EM field enhancement, such nanoantenna design, accompanied by the related chemical enhancement effect of electronic structure of noble metals, can be very promising for diverse nanophotonic applications, such as surface-enhanced absorption [@24], Raman scattering [@25] and luminescence [@26].
In conclusion, we have demonstrated for the first time that an array of laser-induced metallic nanotips within semispherical sub-micron pits provides 28-fold enhancement of ultrafast electron photoemission. Numerical calculations of the intensity distribution near a nanotip have proven that such an assembly works like nanoantennas with microreflectors, yielding in high EM field concentration near the peak of the nanotip. Comparative study of electron emission from the nanotips versus other types of laser-induced nanotopologies showed that the nanotips provide the highest enhancement, despite relatively low surface density. The experiments were carried out at intensities higher than the damage threshold for the nanotips to show that their simple way of fabrication opens a possibility of their using in high-fluence ($>$ 1 J/cm${}^{2}$) regime.
This work was partly supported by Die Österreichische Forschungsförderungsgesellschaft FFG project SLFNM 834325, Russian Foundation for Basic Research (projects nos. 11-02-01202-a, 11-08-01165-a, 12-02-13506 ofi\_m\_RA, 12-20-33045 mol-a\_ved, 13-02-00971-a, 14-02-00460-a, 14-02-00748-a, and 14-02-00881-a), and by RAS Presidium’s programs (nos. 13 and 24).
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---
abstract: 'In this work, we present a class of new efficient models for water flow in shallow unconfined aquifers, giving an alternative to the classical but less tractable 3D-Richards model. Its derivation is guided by two ambitions: any new model should be low cost in computational time and should still give relevant results at every time scale. We thus keep track of two types of flow occurring in such a context and which are dominant when the [*ratio*]{} thickness over longitudinal length is small: the first one is dominant in a small time scale and is described by a vertical 1D-Richards problem; the second one corresponds to a large time scale, when the evolution of the hydraulic head turns to become independent of the vertical variable. These two types of flow are appropriately modelled by, respectively, a one-dimensional and a two-dimensional system of PDEs boundary value problems. They are coupled along an artificial level below which the Dupuit hypothesis holds true ([*i.e.*]{} the vertical flow is instantaneous) in a way ensuring that the global model is mass conservative. Tuning the artificial level, which even can depend on an unknown of the problem, we browse the new class of models. We prove using asymptotic expansions that the 3D-Richards problem and each model of the class behaves the same at every considered time scale (short, intermediate and large) in thin aquifers. The results are illustrated by numerical simulations, showing especially that the new models results fit well with the ones obtained with the original 3d-Richards problem even in non-thin aquifers.'
address:
- 'Univ. Littoral Côte d’Opale, EA 2797 - LMPA, F- 62228 Calais, France'
- 'La Rochelle Université, MIA, EA 3165, F-17031 La Rochelle, France'
author:
- Christophe Bourel
- Catherine Choquet
- Carole Rosier
- Munkhgerel Tsegmid
title: Modelling of shallow aquifers in interaction with overland water
---
Fluid flow modelling; Saturated and unsaturated porous media; Numerical simulations; Asymptotic analysis; Vertical Richards equations; Dupuit Hypothesis.
Introduction
============
Contamination of soil and groundwater is a major concern that affects all populated areas. Many works are thus developed for studying the vulnerability of aquifers with regard to agricultural, industrial, or sewage pollutions. There is an abundant literature on each of the involved processes (geological, physical, chemical...), so that we can consider that the corresponding model is already available. Nevertheless there is a so wide variety of processes (chemical, hydrogeological, anthropic) acting in a so wide range of temporal and geometrical length scales that the assembly of the corresponding model bricks, if considered like toolboxes of a software, is, at best, computationally expensive.
In this multi-scale context, a particularly interesting issue is a proper and tractable model for the exchanges between the overland and the underground waters. Indeed, the challenge consists in capturing very different physical phenomena, the fast and essentially vertical leakage coming from the surface through an unsaturated soil and the slow and essentially horizontal displacement in the saturated part of the aquifer, that are classically modelled by mathematical systems with very different structures. The question is all the more important that an accurate study of the interaction between the water table and the overland water is essential for many concerns, concerns that disallow the use of classical time upscaling processes. It is in particular crucial for studying the transport of chemical components in the aquifer. Indeed, it turns out that many chemical reactions occur in the first meters of the subsoil, where oxygen is still very present. As a byproduct, the chemical species that reach the water table are not necessarily the same than those that have left the surface, and there is a large range of kinetics reaction times to handle with. There is actually no scale separation.
In the present paper, we focus on the hydrogeological question. We thus consider the displacement of a wetting phase (water) in the presence of a non-wetting fluid (air) in a porous medium. Assuming that the air present in the unsaturated zone has infinite mobility allows to use a model for immiscible fluid flow simplified by the Richards hypothesis. The saturation is thus considered as a monotone function depending of the pressure head and the so-called Richards model consists in a nonlinear three-dimensional equation of degenerate parabolic type. All the existing simplified models for the fluid displacement in aquifers are motivated by the characteristics of the flow in their saturated part. A form of stratification enables the definition of interfaces and the slowness of the natural dynamics ensures that these interfaces have a smooth and stable behaviour. Moreover the flows are essentially orthogonal to the walls (Dupuit’s hypothesis). These points allow the vertical integration of the Richards equation in the saturated area and lead to the use of a family of 2D models developed since the 60’s (see e.g. the works of Jacob Bear, [@Bear1972; @Bear1987]). A main weakness of the approach by vertical integration lies in its justification. It is only valuable for very precise length and time scales, the time scale in particular being completely different of the typical durations of chemical reactions (see once again [@Bear1972] for empirical and qualitative arguments, see [@Pan] for asymptotic computations). However, such 2D models are widely used, even out of their validity range and even if it turns out to be especially difficult to properly couple them with the flow in the unsaturated part of the underground. Only numerical attempts were done in this direction. We mention [@Kong] where the integrated model is directly coupled with a surface model (see also the references therein). The unsaturated area of the aquifer is taken into account in [@pikul1974numerical] using a $1D$-Richards equation coupled with a simplified model in the saturated part. However, the study is purely numerical and the model is not mathematically justified. In [@abbott1986introduction], the latter kind of model is integrated into a computational code called “SHE” (for “European Hydrological System” and later became SHETRAN) in the case where the water table remains away from the ground level. See also [@Yak], [@Paulus].
To the best of our knowledge, there is no mathematical justification for any “Dupuit-Richards” model specifying the hypotheses as well as the scales that allow its derivation from a more complete model (such as the $3D$-Richards one).
Notice finally that the coupling of the surface and underground flows turns out to be more tractable when handling with a Richards equation (see e.g. [@Ern] or [@Ber] and [@Bern] where the surface behavior is reduced to a Signorini boundary condition).
The goal of this work is to provide a simple model exploiting the low thickness of a confined or unconfined aquifer. In summary it consists in coupling purely vertical models (describing the flow at a small time scale) with a horizontal model (describing the flow at a long time scale). Clearly, given its construction, the model is simpler to manipulate numerically since the original $3D$ problem is replaced by the coupling of a $2D$ problem with several independent $1D$-problems (which can be solved in parallel). Significant time savings are expected in the numerical processing.
This work could be viewed as another attempt using the numerically pragmatical methodology of [@abbott1986introduction] and leading to a “Dupuit-Richards” model. Yet, our approach is quite different. First, we actually derive a class of models, each of them being characterised by the definition of some virtual interface which does not necessarily coincide with the water table (especially when trying to optimize the error). It follows that a model of this class does not necessarily contains a Dupuit component. The position of the virtual interface may even be an unknown of our model. Next, we aim at describing the flow in a large range of time scales, and, more precisely without any assumption of scale separation. The idea consists in always capturing both the fast and slow components of the flow given by Richards $3D$ equations, whatever the time scale. Their coupling is done through flux terms ensuring that the model is mass conservative (and thus avoiding the criticism done in [@Vachaud]). Finally, the large validity range of the new class of models is justified by an asymptotic study. But, as already mentioned, no time scale separation is assumed in the present paper so that we adopt a new methodology for the asymptotic arguments. Let ${\varepsilon}>0$ describe the ratio of the aquifer’s deepness over its characteristic horizontal length. Assume that ${\varepsilon}$ is small. The usual approach would consist in choosing a reference time for the study, introducing an asymptotic expansion of the solution of the 3D-Richards system and using the scale separation for identifying the equations governing the main order terms of this ansatz. This is the classical process for deriving an effective model. Here the asymptotic analysis is not used for deriving an effective model for a given reference time. Rather, it is used for proving that each model of our new class and the 3D-Richards equation are associated with the same effective problem for any time scale. Basically:
1. At short times, the horizontal flow is very small and the vertical one satisfies a $1D$-Richards problem.
2. At non-short times, the vertical flow appears instantaneous. The corresponding pressure profile satisfies the stationary $1D$-Richards problem. Then the hydraulic head $H$ does not depend on the vertical variable $z$. This corresponds to the so-called Dupuit hypothesis.
3. At large times, the horizontal flux is non-zero. It is ruled by a $2D$-horizontal diffusion equation where the conductivity is the vertical average of the permeability tensor on the *whole* depth of the aquifer.
The paper is organised as follows: In Section \[sec\_intro\], we describe the geometry of the problem, the physical parameters and unknowns. The classical 3D-Richards model is recalled. The main result and numerical simulations are given in Section \[sec\_main\]. Namely, we present the systems coupling the vertical and the horizontal flows and we comment on the model. Finally, the formal asymptotic analysis of our models and of the 3D-Richards model are performed and compared in Section \[formal\_asymptotic\].
Description of the problem {#sec_intro}
==========================
This section is devoted to the description of the domain of study, of the physical parameters and of the unknowns which are chosen for characterising the flow through the Richards model.
Geometry
--------
The aquifer corresponds to a cylindrical domain $\Omega\subset{\mathbb{R}}^3$. For the sake of the simplicity, we assume vertical walls. The projection of $\Omega$ on any horizontal plane is an open domain $\Omega_x\subset{\mathbb{R}}^2$ with boundary $\partial\Omega_x$. The lower and upper bases of $\Omega$ are respectively the graphs of real-valued functions ${ h_{\mathrm{bot}} }$ and ${ h_{\mathrm{soil}} }$ such that $$\label{soilbot}
{ h_{\mathrm{soil}} }(x)>{ h_{\mathrm{bot}} }(x)\ ,\qquad \forall x\in\Omega_x .$$ In summary the domain is given by: $$\label{omega}
\Omega=\big\{(x,z)\in\Omega_x\times{\mathbb{R}}\quad |\quad z\in\big]{ h_{\mathrm{bot}} }(x) ,{ h_{\mathrm{soil}} }(x)\big[ \big\} .$$ We split the boundary $\partial\Omega$ of $\Omega$ in three parts (bottom, top and vertical) $$\begin{gathered}
\partial\Omega = { \Gamma_{\mathrm{bot}} }\sqcup{{ \Gamma_{\mathrm{soil}} }}\sqcup{ \Gamma_{\mathrm{ver}} }\, ,
\\
{ \Gamma_{\mathrm{bot}} }:=\big\{(x,z)\in\Omega\ |\ z={ h_{\mathrm{bot}} }(x) \big\}\ , \quad {{ \Gamma_{\mathrm{soil}} }}:=\big\{(x,z)\in\Omega\ |\ z={ h_{\mathrm{soil}} }(x)\big\}\ , \quad { \Gamma_{\mathrm{ver}} }:=\big\{(x,z)\in\Omega\ |\ x\in\partial\Omega_x\big\}.\end{gathered}$$ In the present paper, as already mentioned, we derive a class of models that are characterised by the position $h$ of some virtual interface in the reservoir. For our construction, this function has to take its values in the semi-open interval $[{ h_{\mathrm{bot}} },{ h_{\mathrm{soil}} })$. For numerical implementation, an easy recipe consists in replacing the condition $h < { h_{\mathrm{soil}} }$ by $h \le { h_{\mathrm{soil}} }-\delta$ where $\delta $ is an arbitrary small positive real number. We thus introduce the auxiliary function ${ h_{\mathrm{max}} }$ defined by $$\label{hamax}
{ h_{\mathrm{max}} }={ h_{\mathrm{soil}} }-\delta , \quad 0<\delta \ll 1.$$
\[fig.geometry\]
![ Bidimensional representation of the cylindrical geometry of the problem: $\Omega_x\subset{\mathbb{R}}$ is an interval. \[pres\_geo\]](presentation_geometrie.pdf)
Three-dimensional Richards equation {#sec.model3d}
-----------------------------------
We aim at deriving alternatives to the Richards equation. Let us briefly describe this classical model. In this paper we limit our study to a one-phase incompressible fluid which accordingly admits a constant density $\rho\in {\mathbb{R}}^{*}_+$. First, in multiphase systems, observations have shown that an increase of the saturation of the non-wetting phase leads to an increase of the capillary pressure. The Richards model is moreover based on the assumption that the air pressure in the underground equals the atmospheric pressure, thus is not an unknown of the problem. One thus assumes that the saturation and the relative conductivity of the soil are given as *functions* of the fluid pressure $P$, denoted respectively by $s=s(P)$ and $k_r=k_r(P)$. There is a large choice of available models for $s$ and $k_r$. The most classical examples for an air-water system are the van Genuchten model [@van1980closed], with no-explicit dependance on the bubbling pressure but with fitting parameters, and the Brooks and Corey model [@brooks1964hydraulic], that we use in the simulations below: $$\label{brooks}
s(P)=\begin{cases}
\left({P_{s}}/P\right)^\lambda&\text{if }P<{P_{s}}\\
1&\text{if }P\ge{P_{s}}\\
\end{cases},\qquad
k_r(P)=\begin{cases}
\left({P_{s}}/P\right)^\gamma&\text{if }P<{P_{s}}\\
1&\text{if }P\ge{P_{s}}\\
\end{cases},$$ where $\lambda>0$, $\gamma=2+3\lambda$ and ${P_{s}}<0$. Notice that our model would easily adapt to hysteretic soil properties ([@Pham], [@Schw]). Since these methods, as of today, do not permit three-dimensional calculations, we guess that our 1D-2D models are even more interesting for their implementation than the 3D-Richards model. The important point is that these models are such that $$\label{sat}
s(P)=1 \quad\Longleftrightarrow\quad P\ge{P_{s}}{\qquad\text{and}\qquad }k_r(P)=1 \quad\Longleftrightarrow\quad P\ge{P_{s}}.$$ In particular, the water pressure is greater than the bubbling pressure ${P_{s}}$ if and only if the soil is completely saturated (${P_{s}}$ being a fixed real number). The graphs of the functions $s$ and $k_r$ given by the Brooks-Corey model used below for the numerical simulations are represented in Figure \[saturation\_profile\_figure\] (the parameters are given at the beginning of Subsection \[numeric\]).
The soil transmission properties are characterised by the porosity function, $\phi=\phi(x,z)\in(0,1)$, and the permeability tensor, $K_0(x,z)$. The latter is a $3\times3$ symmetric positive definite tensor which describes the conductivity of the *saturated* soil at the position $(x,z)\in\Omega$. We introduce $K_{xx}\in \mathcal{M}_{22}({\mathbb{R}})$, $K_{zz}\in {\mathbb{R}}^*$ and $K_{xz}\in\mathcal{M}_{21}({\mathbb{R}})$ such that $$\label{K0}
K_0=\begin{pmatrix}
K_{xx}&K_{xz}\\
K_{xz}^T&K_{zz}
\end{pmatrix}.$$
(-4.5,0) – (1.7,0) node\[below left\] [$P$]{}; (0,-0.5) – (0,1.5) node\[left\] [$s(P)$]{}; plot (,[(-1.5/)\^3]{}); plot (,[1]{}); (-1.5,0)node\[below\] [$P_s$]{} – (-1.5,1) ;
(-4.5,0) – (1.7,0) node\[below left\] [$P$]{}; (0,-0.5) – (0,1.5) node\[left\] [$k_r(P)$]{}; plot (,[(-1.5/)\^11]{}); plot (,[1]{}); (-1.5,0)node\[below\] [$P_s$]{} – (-1.5,1) ;
The fluid is characterised by its pressure $P$ and its velocity $v$ solving the following Richards problem: $$\label{richards3d}
\begin{cases}
{\displaystyle}\phi { {\frac{\partial s(P)}{\partial t} }}+\operatorname{div}(v)=0&\text{in }{ ]0,T[ }{\times }\Omega\\
{\displaystyle}v = -k_r(P)\,K_0\,\Big( \frac{1}{\rho g}\nabla P+e_3 \Big)&\text{in }{ ]0,T[ }{\times }\Omega\\
\alpha\, P + \beta\, v\cdot n = F&\text{on }{ ]0,T[ }{\times }{ \Gamma_{\mathrm{soil}} }\\
v\cdot n = 0&\text{on }{ ]0,T[ }{\times }({ \Gamma_{\mathrm{bot}} }\cup{ \Gamma_{\mathrm{ver}} })
\end{cases}$$ where $g$ is the gravity constant and $e_3$ is the unitary vertical vector pointing up. The first equation describes the mass conservation of the constant density fluid in the case of an incompressible soil. The second equation is the Darcy’s law associated with the nonlinear anisotropic conductivity $k_r(P)\,K_0$. The boundary condition $v\cdot n=0$ on ${ \Gamma_{\mathrm{bot}} }$ corresponds to the impermeable layer at the bottom of the aquifer. The same is assumed on ${ \Gamma_{\mathrm{ver}} }$ to simplify the presentation. The condition at the soil level ${{ \Gamma_{\mathrm{soil}} }}$ is a Robin condition associated with given $(\alpha,\beta)\in({\mathbb{R}}_+)^2\setminus \{0,0\}$ and $F:{{ \Gamma_{\mathrm{soil}} }}\to {\mathbb{R}}$.
\[rem\_behavior\] In Section \[formal\_asymptotic\] we investigate the behavior of the flow described by the 3D-Richards equations in the case of a thin aquifer and for various time scales. Let us summarise the conclusions of this asymptotic analysis. They might shed light on the comments about our models in the next section.
1. At any time scale, the dominant flow is the one in the vertical direction (see for example in which the horizontal diffusion term appears multiplied by the small parameter ${\varepsilon}$).
2. In the short time scale ($T\sim 1$), the horizontal flow is very small and the vertical one solves a classical 1D-Richards problem.
3. In non-short time scales ($T\sim {\varepsilon}^{-1}$ or $T\sim {\varepsilon}^{-2}$), the vertical flow appears as being instantaneous. The corresponding pressure profile satisfies a stationary 1D-Richards problem. Then the pressure is $P=\rho\,g(H-z)$ where the hydraulic head $H$ does not depend on the vertical variable $z$. The velocity is horizontal. This corresponds to the so-called Dupuit hypothesis.
4. In the long time scale ($T\sim {\varepsilon}^{-2}$), the horizontal flow is non-zero and it is ruled by a 2D-horizontal diffusion equation where the conductivity is the vertical average of the permeability tensor on the *whole* depth of the aquifer, from ${ h_{\mathrm{bot}} }$ to ${ h_{\mathrm{soil}} }$.
Main result and numerical simulations {#sec_main}
======================================
Models coupling vertical 1d-Richards flow and Dupuit horizontal flow
--------------------------------------------------------------------
Each of our models splits the description of the flow into two subregions of $\Omega$ (possibly time-dependent). These zones are defined by a function ${ h}={ h}(t,x)$ such that ${ h_{\mathrm{bot}} }\leq{ h}<{ h_{\mathrm{soil}} }$: $$\begin{gathered}
{ {\Omega_{{ h}}^- }}(t):= \big\{ (x,z)\in\Omega\ |\ z<{ h}(x,t) \big \}\quad\text{and}\quad{ {\Omega_{{ h}}^+ }}(t):= \big\{ (x,z)\in\Omega\ |\ z>{ h}(x,t) \big \} ,
\label{omplusmoins}
\\
{ {\Gamma_{{ h}}} }:=\big\{ (x,z)\in\Omega\ |\ z={ h}(x,t) \big \} .
\label{gamaha}\end{gathered}$$ We emphasise that choosing the level ${ h}$ corresponds to the specification of one of the models of our class. The function ${ h}$ can even be an unknown of our problem, more precisely depending of an unknown of the problem (see condition below).
On the other hand we introduce the following tensor $M_0$ which will act as an effective permeability tensor: $$\label{M0}
M_0=\begin{pmatrix}
S_0&0\\0&0
\end{pmatrix},
\qquad
S_0=K_{xx}-\frac{1}{K_{zz}}K_{xz}K_{zx} .$$ The $2{\times }2$ matrix $S_0$ is the Schur complement of the block $K_{zz}$ in the tensor $K_0$. Since $K_0$ is a symetric positive definite matrix (see just before ), the same holds for $S_0$. We then introduce the averaged conductivity tensor $\tilde K$ defined in ${ ]0,T[ }{\times }\Omega_x$ for any function $\tilde H=\tilde H(t,x)$ by $$\label{averaged_conductivity}
\tilde K(\tilde H)(t,x)=\int_{{ h_{\mathrm{bot}} }(x)}^{{ h_{\mathrm{soil}} }(x)}k_r\big(\rho\,g(\tilde H(t,x)-z)\big)\,M_0(x,z)\,dz.$$
Finally, for the 2D part of the model, we introduce the notations $\nabla_x=(\partial_{x_1}, \partial_{x_2}, 0)^T$ for the horizontal gradient and $\operatorname{div}_x(v)=\nabla_x\cdot v=\partial_{x_1}v_1+\partial_{x_2}v_2$ for the horizontal divergence of $v\in {\mathbb{R}}^3$.
#### The model.
Our coupled model consists in finding the pressure $P$, the velocity $v$ and the auxiliary unknowns $u$, $w$, $\tilde H$ and ${ h}$ such that:
- In ${ {\Omega_{{ h}}^+ }}(t)$ the following 1D-Richards equation holds $$\label{coupled_transition}
\begin{cases}
{\displaystyle}\phi { {\frac{\partial s(P)}{\partial t} }}+{ {\frac{\partial }{\partial z} }}\big(u\cdot e_3\big)=0 & \text{for \ }t\in]0,T[\ ,\quad (x,z)\in{ {\Omega_{{ h}}^+ }}(t)\\
\alpha\, P + \beta\, u\cdot e_3=F &\text{for \ }(t,x)\in]0,T[{\times }{ \Gamma_{\mathrm{soil}} }\\
P\big(t,x,{ h}(t,x)\big)=\rho\,g\big(\tilde H(t,x)-{ h}(t,x) \big)&\text{for \ }(t,x)\in]0,T[{\times }\Omega_x\\
P(0,x,z)={ P_{\mathrm{init}} }(x,z)&\text{for \ } (x,z)\in{ {\Omega_{{ h}}^+ }}(0)\\
\end{cases}$$
- In ${ {\Omega_{{ h}}^- }}(t)$ the pressure $P$ satisfies $$\label{coupled_watertable}
{\displaystyle}P(t,x,z)=\rho\,g\big( \tilde H(t,x)-z \big) \qquad \text{for \ }t\in[0,T[\ ,\quad (x,z)\in{ {\Omega_{{ h}}^- }}(t)\\$$
- The hydraulic head solves in $\Omega_x$ $$\label{coupled_hydraulic}
\begin{cases}
\operatorname{div}_x\Big(\tilde K(\tilde H)\,\nabla_x\tilde H \Big)=(u\cdot e_3)\big|_{{ {\Gamma_{{ h}}^+} }}&\text{for \ }(t,x)\in]0,T[{\times }\Omega_x
\\
{\displaystyle}\tilde K(\tilde H)\,\nabla_x \tilde H\cdot n=0&\text{for \ }(t,x)\in]0,T[{\times }\partial\Omega_x\\
\tilde H(0,x)={ H_{\mathrm{init}} }(x)&\text{for \ } x\in\Omega_x
\end{cases}$$ where $(u\cdot e_3)\big|_{{ {\Gamma_{{ h}}^+} }}$ denotes the trace of $u\cdot e_3$ on ${ {\Gamma_{{ h}}} }$ from above.
- The level $z=h$ below which we consider the vertical flow to be instantaneous is set such that $$\label{coupled_interface}
{ h_{\mathrm{bot}} }(x) \le { h}(t,x)\le\max\Bigl\{\min\Bigl\{ \tilde H(t,x)-\frac{{P_{s}}}{\rho\,g},{ h_{\mathrm{max}} }(x) \Bigr\},{ h_{\mathrm{bot}} }(x)\Bigr\}, \quad (t,x)\in[0,T[{\times }\Omega_x.$$
- The velocity $v$ is defined in $\Omega$ by $$\label{coupled_velocity}
\begin{cases}
{\displaystyle}v = u + w &\text{for \ }t\in]0,T[\ ,\quad (x,z)\in\Omega\\
{\displaystyle}u = -k_r(P)\,\Big(\frac{1}{\rho\,g}{ {\frac{\partial P}{\partial z} }}+1\Big)\,K_0\,e_3 &\text{for \ }t\in]0,T[\ ,\quad (x,z)\in\Omega\\
{\displaystyle}w = -k_r\big( \rho\,g( \tilde H-z ) \big)\,M_0\nabla_x\tilde H& \text{for \ }t\in]0,T[\ ,\quad (x,z)\in\Omega
\end{cases}$$
The coupled model [–]{}depends on the definition of the function $h$. Although all intermediate choices respecting are allowed, we will focus in the next on the two extremal choices $$\begin{aligned}
&h(t,x)= { h_{\mathrm{bot}} }(x),\label{h_hbot}\\
&h(t,x)= \max\Bigl\{\min\Big\{ \tilde H(t,x)-\frac{{P_{s}}}{\rho\,g},{ h_{\mathrm{max}} }(x) \Big\},{ h_{\mathrm{bot}} }(x)\Bigr\}:=h_s(t,x),
\label{h_H}
\end{aligned}$$ and on the intermediate one $$\label{h_inter}
h(t,x)= \max\Bigl\{\min\Big\{ \tilde H(t,x)-\frac{{P_{s}}+R}{\rho\,g},{ h_{\mathrm{max}} }(x) \Big\},{ h_{\mathrm{bot}} }(x)\Bigr\},$$ where $R$ is some positive function possibly depending on $\tilde H$.
The class of models [–]{}is an alternative to the 3D-Richards problem for describing the flow in a shallow aquifer in a large range of time scales. This model is designed to fulfill the two following properties:
- to be simpler to handle numerically than the 3D-Richards model
- to behave like the 3D-Richards model for any time scale when the [*ratio*]{} ${\varepsilon}$ of the deepness over the horizontal length of the aquifer is small[^1]. For example the behaviors presented in Remark \[rem\_behavior\] are respected.
The first property holds for [–]{}since the 3D original Richards problem is replaced by the coupling of a 2D-problem with a lot of independent 1D-problems which can be solved in parallel. Significant time savings are expected in the computations. The second property is justified in Section \[formal\_asymptotic\]. The idea is to study the limit ${\varepsilon}\to0$ of the solution of the 3D-Richards equations and to derive formally the associated effective problem. The same asymptotic analysis is performed for the coupled models [–]{}and shows that the corresponding effective problems are exactly the same for every considered time scale and for every choice of $h$ satisfying .
\[rem\_coupling\_time\_scales\] It is natural to think that it is possibly not so useful to couple two phenomena which does not hold at the same time scale, since by essence they can not interact with each other. But the notion of time *scale* is senseless for a fixed physical situation and we just employ this term to enlighten the interpretations. The notion of scale has a precise sense when a sequence of problems is considered, for example parametrised by a small parameter ${\varepsilon}$ tending to zero with the reference time of study depending on ${\varepsilon}$. This is what we do in Section \[formal\_asymptotic\] where ${\varepsilon}$ is the [*ratio*]{} deepness/length of the aquifer. This limit process shows that the two kinds of flow appear at different time scales and then do not interact with each other. Nevertheless, the coupled problem [–]{}*is not* an effective problem and holds without time scale separation assumption. The *depth / width* ratio of the aquifer is then a fixed positive number given by the geometry of the aquifer. In particular, “short” and “long” time scales flows can interact without either being negligible or instantaneous.
The remainder of this subsection is devoted to comments on the new models [–]{}. Before splitting those comments according to the choice of the function $h$, we prove that the model is always mass conservative.
#### Mass conservation.
Let ${ M_{\mathrm{tot}} }(t)$ the total mass of the water contained in domain $\Omega$ at time $t$. We denote by ${ {M_{{ h}}^+ }}$ (resp. ${ {M_{{ h}}^- }}$) the mass of the water filling the domain ${ {\Omega_{{ h}}^+ }}$ (resp. ${ {\Omega_{{ h}}^- }}$). We have $$\begin{gathered}
{ {M_{{ h}}^+ }}(t)=\rho\int_{\Omega_x}\int_{{ h}(t,x)}^{{ h_{\mathrm{soil}} }}\phi\,s(P)\,dz\,dx,
\qquad
{ {M_{{ h}}^- }}(t)=\rho\int_{\Omega_x}\int_{{ h_{\mathrm{bot}} }(x)}^{{ h}(t,x)}\phi\,dz\,dx ,
\label{mass_DR}
\\
{ M_{\mathrm{tot}} }(t)={ {M_{{ h}}^+ }}(t)+{ {M_{{ h}}^- }}(t) .
\label{mass_conservation}\end{gathered}$$
\[prop\_mass\] The total mass satisfies for all $t\in(0,T)$: $${ {\frac{\partial }{\partial t} }}{ M_{\mathrm{tot}} }=-\rho\int_{\Omega_x} (u\cdot e_3)|_{{ \Gamma_{\mathrm{soil}} }} \, dx .$$
By using relation and it comes $$\label{proof_mass_1}
{ {\frac{\partial }{\partial t} }}{ M_{\mathrm{tot}} }=\rho\int_{\Omega_x}\int_{{ h_{\mathrm{bot}} }(x)}^{{ h}(t,x)}\phi{ {\frac{\partial s(P)}{\partial t} }}\,dz\,dx+\rho\int_{\Omega_x}\int_{{ h}(t,x)}^{{ h_{\mathrm{soil}} }(x)}\phi{ {\frac{\partial s(P)}{\partial t} }}\,dz\,dx = \rho\int_{\Omega_x}\int_{{ h}(t,x)}^{{ h_{\mathrm{soil}} }(x)}\phi{ {\frac{\partial s(P)}{\partial t} }}\,dz\,dx,$$ where the first equality is due to $s(P)=1$ in $]{ h_{\mathrm{bot}} }(x),{ h}(t,x)]$ (indeed $P\ge {P_{s}}$ by and ). Thanks to the first equation in we deduce $$\label{proof_mass_2}
\int_{\Omega_x}\int_{{ h}(t,x)}^{{ h_{\mathrm{soil}} }(x)}\phi{ {\frac{\partial s(P)}{\partial t} }}\,dz\,dx = \int_{\Omega_x} (u\cdot e_3)|_{{ {\Gamma_{{ h}}^+} }} \,dx-\int_{\Omega_x} (u\cdot e_3)|_{{ \Gamma_{\mathrm{soil}} }} \,dx.$$ Finally by and after an integration by parts $$\label{proof_mass_3}
\int_{\Omega_x} (u\cdot e_3)|_{{ {\Gamma_{{ h}}^+} }} \,dx=\int_{\partial\Omega_x} \tilde K(\tilde H)\nabla_x\tilde H \cdot n =0 .$$ The result is obtained by plugging and in .
Comments on the model in the case (\[h\_hbot\]) {#subsec_comments_bot}
-----------------------------------------------
In this case, we have $h={ h_{\mathrm{bot}} }$, then ${ {\Omega_{{ h}}^+ }}=\Omega$, ${ {\Omega_{{ h}}^- }}=\emptyset$ and ${ {\Gamma_{{ h}}} }={ \Gamma_{\mathrm{bot}} }$ (see ). The coupled model [–]{}reduces in: finding the pressure $P$, the velocity $v$ and the auxiliary unknowns $u$, $w$ and $\tilde H$ such that: $$\label{coupled_velocity_h_hbot}
\begin{cases}
{\displaystyle}v = u + w &\text{for \ }t\in]0,T[\ ,\quad (x,z)\in\Omega\\
{\displaystyle}u = -k_r(P)\,\Big(\frac{1}{\rho\,g}{ {\frac{\partial P}{\partial z} }}+1\Big)\,K_0\,e_3 &\text{for \ }t\in]0,T[\ ,\quad (x,z)\in\Omega\\
{\displaystyle}w = -k_r\big( \rho\,g( \tilde H-z ) \big)\,M_0\nabla_x\tilde H& \text{for \ }t\in]0,T[\ ,\quad (x,z)\in\Omega
\end{cases}$$ $$\label{coupled_transition_h_hbot}
\begin{cases}
{\displaystyle}\phi { {\frac{\partial s(P)}{\partial t} }}+{ {\frac{\partial }{\partial z} }}\big(u\cdot e_3\big)=0 & \text{for \ }t\in]0,T[\ ,\quad (x,z)\in\Omega\\
\alpha\, P + \beta\, u\cdot e_3=F &\text{for \ }(t,x,z)\in]0,T[{\times }{ \Gamma_{\mathrm{soil}} }\\
P=\rho\,g\big(\tilde H-{ h_{\mathrm{bot}} }\big)&\text{for \ }(t,x,z)\in]0,T[{\times }{ \Gamma_{\mathrm{bot}} }\\
P(0,x,z)={ P_{\mathrm{init}} }(x,z)&\text{for \ } (x,z)\in\Omega\\
\end{cases}$$ $$\label{coupled_hydraulic_h_hbot}
\begin{cases}
-\operatorname{div}_x\big(\tilde K(\tilde H)\,\nabla_x\tilde H \big)=-(u\cdot e_3)|_{{ \Gamma_{\mathrm{bot}} }}&\text{for \ }(t,x)\in]0,T[{\times }\Omega_x
\\
{\displaystyle}\tilde K(\tilde H)\,\nabla_x \tilde H\cdot n=0&\text{for \ }(t,x)\in]0,T[{\times }\partial\Omega_x\\
\tilde H(0,x)={ H_{\mathrm{init}} }(x)&\text{for \ } x\in\Omega_x
\end{cases}$$ This setting corresponds to the simplest form of the model [–]{}since is a classical boundary value problem. Nevertheless the simulations below illustrate that it is not the better form of approximation for the 3D-Richards equation.
#### Velocity of the flow.
The velocity $v$ of the flow turns out to be the superposition of the two velocities $u$ and $w$ which respectively describe the fast and slow components of the flow. Actually $u$ (resp. $w$) is the dominant component of the flow in the short time scale (resp. large time scale).
#### Fast component of the flow: globally vertical.
The unknown $u$ represents the velocity associated with the pressure $P$ by the one dimensional Darcy’s law given in the second equation of . This one is deduced from the 3D law (see the second equation of ) by neglecting the horizontal components of the gradient of the pressure $P$. By construction the field $u$ is vertical if the conductivity tensor $K_0$ introduced in is such that $K_{xz}=0$ but it may admit a non-zero horizontal component in the anisotropic case.
Furthermore the mass conservation equation holds. The pressure $P$ then satisfies the following vertical Richards equation where the horizontal variable $x\in \Omega_x$ appears only as a parameter: $$\label{richards_vertical}
\phi { {\frac{\partial s(P)}{\partial t} }}-{ {\frac{\partial }{\partial z} }}\Bigl(k_r(P)\,K_{zz}\,\Big(\frac{1}{\rho\,g}{ {\frac{\partial P}{\partial z} }}+1\Big) \Bigr)=0 \qquad\text{in }]0,T[{\times }\Omega.$$ The original 3D-Richards problem reduces to the latter equation when the horizontal diffusion terms are neglected. In the short-time scale indeed, those turn to be non-dominant in shallow aquifers as announced in Remark \[rem\_behavior\] and shown in Section \[formal\_asymptotic\].
The boundary condition on ${ \Gamma_{\mathrm{soil}} }$ remains the same than in the 3D-Richards problem. But on the bottom ${ \Gamma_{\mathrm{bot}} }$, the structure of the boundary condition changes and becomes of Dirichlet type, namely ${P }\big(t,x,{ h_{\mathrm{bot}} }(t,x)\big)=\rho\,g\big(\tilde H(t,x)-{ h_{\mathrm{bot}} }(t,x) \big)$. In fact, even if this Dirichlet condition holds, we do not allow the water flowing out the aquifer through the bottom boundary. Indeed the possibly non-zero flux $(u\cdot e_3)|_{{ \Gamma_{\mathrm{bot}} }}$ appears as a source term in the first equation of , so that, as proved in Proposition \[mass\_conservation\], the coupled model is globally mass-conservative. The particular value ${P }=\rho\,g(\tilde H-{ h_{\mathrm{bot}} })$ for the bottom Dirichlet condition, has been chosen so that the fast and slow flows are correctly coupled. This point is further explained in the next paragraph.
#### Slow component of the flow: globally horizontal.
On the one hand, introduce the auxiliary pressure $Q$, $$Q:=\rho\,g(\tilde H-z),$$ for which $\tilde H$ plays the role of the hydraulic head. Since $\tilde H$ does not depend on $z$, we have $(\rho\,g)^{-1}\partial_zQ+1=0$. The first consequence is that the unknown $w$ satisfies (see ) $$w=-k_r(Q)\,M_0\nabla_x\tilde H.$$ We recover here the velocity associated to $Q$ by the classical Darcy’s law for the conductivity $k_r(Q)\,M_0$. The second consequence is that $Q$ is ruled by $${ {\frac{\partial }{\partial z} }}\Big(k_r(Q)\,K_{zz}\,\Big(\frac{1}{\rho\,g}{ {\frac{\partial Q}{\partial z} }}+1\Big) \Big)=0 \qquad\text{in }]0,T[{\times }\Omega ,$$ that is the stationary version of equation .
On the other hand, we expect $P$ to solve the same stationary problem when the duration of the experiment and when the boundary conditions allow the 1D-Richards problem to reach its stationary state. Notice that such a vertical affine profile is also expected in the 3D-Richards model in any non-short time scale (see Remark \[rem\_behavior\] and Section \[formal\_asymptotic\]). When this situation occurs, the hydraulic head $H:=P/\rho g+z$ is constant with respect to $z$. The Dirichlet boundary condition on ${ h_{\mathrm{bot}} }$ in then implies that $$H(t,x,z)=H\big(t,x,{ h_{\mathrm{bot}} }(x)\big)=\frac{P\big(t,x,{ h_{\mathrm{bot}} }(x)\big)}{\rho g}+{ h_{\mathrm{bot}} }(x)=\tilde H(t,x).$$ Accordingly, in any non-short time scale, we get $H\simeq\tilde H$ and then $P\simeq Q$ in $\Omega$. This is the reason of the particular choice $P=\rho\,g(\tilde H-{ h_{\mathrm{bot}} })$ for the Dirichlet boundary condition on ${ h_{\mathrm{bot}} }$ in . Roughly speaking, the couple $(Q,w)$ characterizes the flow in a long-time experiment in which the vertical flow seems instantaneous with respect to the horizontal one.
Unlike the velocity $u$, the field $w$ is horizontal both in the isotropic and anisotropic cases due to the definition of the tensor $M_0$. The computations leading to the definition of $M_0$ are done in Section \[formal\_asymptotic\]. Let us give here some qualitative arguments. For large times, $w$ is the main order term of the flow which turns out to be horizontal. The velocity $w$ is also related to some hydraulic head, say $L$, by the classical Darcy’s law $w=-k_r\,K_0\nabla L$ (as in the Richards equation ; see ). But since $w$ is horizontal we have $$0=w\cdot e_3=-k_r\,K_0\nabla L\cdot e_3 = -k_r\,K_{zx}\nabla_x L -k_r\,K_{zz}{ {\frac{\partial L}{\partial z} }}\quad \text{and then}\quad{ {\frac{\partial L}{\partial z} }}=-k_r\,\frac{K_{zx}}{K_{zz}} \nabla_x L$$ if $K_{zz} \ne 0$ as assumed in this paper, otherwise the question is trivial. Accordingly, in the expression of $w=-k_r\,K_0\nabla L$, only the term $\nabla_x L$ appears and it follows $w=-k_r\,M_0\nabla_x L$. Notice that the tensor $M_0$ reduces to $K_{xx}$ in the isotropic case $K_{xz}=K_{zx}=0$.
Moreover $w$ depends on $z$ only through the term $k_r(\rho\,g(\tilde H-z))M_0$ which decreases to $0$ when $z$ increases above $\tilde H -{P_{s}}/\rho g$. This decrease is fast in general depending on the soil characteristic function $k_r$. Then, roughly speaking, the horizontal component of the flow is maximum in the saturated part and almost vanishing in the unsaturated one far from the capillary fringe.
The evolution of the “stationary pressure” $Q$ is ruled by the first equation of . This is an horizontal mass-conservation equation associated with the average velocity $\tilde w:=-\tilde K(\tilde H)\nabla_x \tilde H=\int_{{ h_{\mathrm{bot}} }}^{{ h_{\mathrm{soil}} }}w\,dz$. The right-hand side is the source term computed from the 1D-Richards problem and which transfers the mass from the vertical description to the horizontal one.
Notice that in this model -, the Dupuit hypothesis is not considered. We precise this point in the next Subsection.
Comments on the model in the cases (\[h\_H\]) and (\[h\_inter\]) {#subsec_comments_inter}
----------------------------------------------------------------
Now we come back to the model [–]{}in which we set the virtual interface $h$ by $$h(t,x)= \max\Big\{\min\Big\{ \tilde H(t,x)-\frac{{P_{s}}+R}{\rho\,g},{ h_{\mathrm{max}} }(x) \Big\},{ h_{\mathrm{bot}} }(x)\Big\},
\label{h_R_choice}$$ for a given non-negative function $R$ possibly depending on $\tilde H$. In the numerical simulations at the end of this section, we consider the constant cases $R=0$, corresponding to , and $R=3$. Choosing could be guessed as the most intuitive choice since it means in general splitting the domain along the water table, thus separating the flows in the saturated and in the unsaturated areas. But simulations show that it is not necessary the optimal choice for the quality of the 3D-Richards approximation.
#### Velocity of the flow.
As previously, the velocity $v$ of the flow results from the contribution of a fast component $u$ and of a slow one $w$. The set ${ {\Omega_{{ h}}^- }}$ is no more empty in general and an additional brick is introduced in the model for describing the flow in this area. We start by giving some properties of the interface ${ {\Gamma_{{ h}}} }$.
#### Interface discriminating the flow behaviors.
As seen in , the sets ${ {\Omega_{{ h}}^- }}(t)$ and ${ {\Omega_{{ h}}^+ }}(t)$ are characterised by ${ h}$. In view of the constraint , the condition $$\label{bound.ha}
{ h_{\mathrm{bot}} }(x) \le { h}(t,x)\le{ h_{\mathrm{max}} }(x)$$ holds for all $(t,x)\in{ ]0,T[ }{\times }\Omega_x$. Due to and the pressure at the level $z={ h}(t,x)$ satisfies for all $(t,x) \in{ ]0,T[ }{\times }\Omega_x$: $$\label{phazero}
P\big(t,x,{ h}(t,x)\big)\begin{cases}
={P_{s}}+R\qquad \text{if } \quad { h_{\mathrm{bot}} }(x)<{ h}(t,x)<{ h_{\mathrm{max}} }(x) ,\\
\ge {P_{s}}+R\qquad \text{if } \quad { h}(t,x)={ h_{\mathrm{max}} }(x), \\
\le {P_{s}}+R\qquad \text{if } \quad { h}(t,x)={ h_{\mathrm{bot}} }(x).
\end{cases}$$ In particular, thanks to and since $R\ge0$ we get $$\label{saturated}
s\big(P(t,x,z)\big)=1\qquad \text{if } \quad { h_{\mathrm{bot}} }(x)< z\le{ h}(t,x) ,$$ which means that the set ${ {\Omega_{{ h}}^- }}(t)$ contains a saturated part of the aquifer for any choice of $R\ge0$. More precisely, the soil is fully saturated in ${ {\Omega_{{ h}}^- }}(t)$ for every $t\in{ ]0,T[ }$ if $R>0$, and if $R=0$, that is for , ${ {\Omega_{{ h}}^- }}$ can be interpreted as the water table (see Remark \[rem\_hsat\] below for precisions).
By construction ${ h}(t,x)\le{ h_{\mathrm{max}} }$ so that the interval $]{ h}(t,x),{ h_{\mathrm{soil}} }(x)[$ remains non-empty for all $(t,x)\in{ ]0,T[ }{\times }\Omega_x$. Then we do not have to explicit a direct coupling of the flow in ${ {\Omega_{{ h}}^- }}$ with the one in the overland. The coupling between ${ {\Omega_{{ h}}^- }}$ and ${ {\Omega_{{ h}}^+ }}$ is sufficient.
#### Fast component of the flow: globally vertical, a part being instantaneous.
We start by remarking that, as in the previous case, the velocity $u$ is related to $P$ by the vertical Darcy’s law . Moreover the same 1D-Richards equation holds, but now, only in the upper part of the aquifer. In particular, in the short-time scale, the dominant vertical flow in ${ {\Omega_{{ h}}^+ }}(t)$ remains well described.
The main difference between cases $h={ h_{\mathrm{bot}} }$ and $h\neq { h_{\mathrm{bot}} }$ is related to the vertical flow in the saturated area ${ {\Omega_{{ h}}^- }}(t)$. Indeed, the pressure profile now holds in ${ {\Omega_{{ h}}^- }}$ and in particular $u$ is zero in ${ {\Omega_{{ h}}^- }}$. As said before, this affine profile is expected in the non-short time scale when the vertical flow appears instantaneous. Hence, the model [–]{}describes precisely the vertical flow in ${ {\Omega_{{ h}}^+ }}$ and assumes that this flow is instantaneous in ${ {\Omega_{{ h}}^- }}$. Such an assumption is classical in models of saturated shallow aquifers and is known as the Dupuit hypothesis. Then, the model [–]{}in the cases can be seen as the coupling of a Dupuit horizontal flow in a saturated part at the bottom of the aquifer with many vertical 1D-Richards flows for a precise description of the leaking fluxes from the overland to the water table.
Notice that, even if $h\ne{ h_{\mathrm{bot}} }$, the model [–]{}does approximate the 3D-Richards problem at every time scale when the ration ${\varepsilon}=$*deepness/horizontal length* tends to zero. Indeed, Proposition \[prop\_richards\] below holds for any choice of function $h$ such that is satisfied. This is explained by the following points in short times:
- From the 3D-Richards problem, we expect a vertical description given by the 1D-Richards in the whole $\Omega$, with a vanishing flux at the bottom of the domain (see ).
- From our model, we get 1D-Richards only in ${ {\Omega_{{ h}}^+ }}$ with a zero flux in ${ {\Omega_{{ h}}^- }}$ (see proof of the short-time scale near equation ) *and* the continuity of the pressure.
In fact, these problems are exactly the same.
The field $u$ is non-singular thanks to the continuity condition satisfied by $P$ on ${ {\Gamma_{{ h}}} }$ (see and ). As for $h={ h_{\mathrm{bot}} }$, the particular value of the Dirichlet condition on ${ {\Gamma_{{ h}}} }$ has been chosen for a proper coupling of the fast and slow components of the flow. This is further developped in the next paragraph. However if $u\cdot e_3$ has a trace on the boundary ${ {\Gamma_{{ h}}} }$ of ${ {\Omega_{{ h}}^+ }}$, this one is non-zero in general whereas $u\cdot e_3=0$ in ${ {\Omega_{{ h}}^- }}$. This is a notable difference with the case $h={ h_{\mathrm{bot}} }$.
#### Slow component of the flow.
Again, we introduce the auxiliary pressure $Q=\rho\,g(\tilde H-z)$ and we remark that now $P=Q$ in ${ {\Omega_{{ h}}^- }}(t)$ (even for short times). The fact that $P\simeq Q$ in the whole $\Omega$ for any non-short times comes, as in the case $h={ h_{\mathrm{bot}} }$, from the Dirichlet condition $P=\rho\,g(\tilde H-z)$ which holds on ${ {\Gamma_{{ h}}} }$.
The evolution of $(Q,w)$ is characterized by the evolution of $\tilde H$ given in . In this case where ${ {\Omega_{{ h}}^- }}(t)$ in non-empty in general, we can explicit a little more the dynamic of $\tilde H$. This is detailed in the next paragraph.
#### Evolution of the hydraulic head.
Rewrite the problem using the first equation of averaged on $[{ h},{ h_{\mathrm{soil}} }]$: $$-\operatorname{div}_x\Big(\tilde K(\tilde H)\,\nabla_x\tilde H \Big) = -u\big|_{{ \Gamma_{\mathrm{soil}} }}\cdot e_3-\int_{h(t,x)}^{{ h_{\mathrm{soil}} }(x)} \phi\,{ {\frac{\partial s({P })}{\partial t} }} \,dz \qquad\text{in }{ ]0,T[ }{\times }\Omega_x.
\label{alternate_dupuit_-1}$$ Since $s({P })=1$ for $z\in[{ h_{\mathrm{bot}} },h]$, we get $$\label{alternate_dupuit_0}
-\operatorname{div}_x\Big(\tilde K(\tilde H)\,\nabla_x\tilde H \Big) = -u\big|_{{ \Gamma_{\mathrm{soil}} }}\cdot e_3-{ {\frac{\partial }{\partial t} }}\int_{{ h_{\mathrm{bot}} }(x)}^{{ h_{\mathrm{soil}} }(x)} \phi\,s({P }) \,dz \qquad\text{in }{ ]0,T[ }{\times }\Omega_x ,$$ or equivalently by using the Leibniz rule in and $s(P)_{\vert z=h}=1$: $$\label{alternate_dupuit}
\phi|_{{ {\Gamma_{{ h}}} }} { {\frac{\partial h}{\partial t} }}-\operatorname{div}_x\Big(\tilde K(\tilde H)\,\nabla_x\tilde H \Big) \\
=-u\big|_{{ \Gamma_{\mathrm{soil}} }}\cdot e_3-{ {\frac{\partial }{\partial t} }}\left(\int_{h(t,x)}^{{ h_{\mathrm{soil}} }(x)} \phi\,s({P }) \,dz \right)\qquad\text{in }{ ]0,T[ }{\times }\Omega_x.$$ The hydraulic head $\tilde H$ is characterized by the latter equation completed by the limit conditions in . This problem is a non-linear degenerate diffusion equation. Indeed, the diffusion tensor $\tilde K(\tilde H)$ vanishes when $\tilde H$ tends to $-\infty$. If moreover holds, in view of , the time derivative can be expressed as $${ {\frac{\partial h}{\partial t} }}=C(\tilde H)\,{ {\frac{\partial \tilde H}{\partial t} }}\qquad\text{with}\qquad C(\tilde H)=\begin{cases}
1& \text{if }\tilde H-{P_{s}}/\rho\, g\in]{ h_{\mathrm{bot}} },{ h_{\mathrm{max}} }[\\
0&\text{if not}.
\end{cases}$$
The right-hand side of the first equation in plays the role of a *source term* and represents for each $x\in \Omega$ the evolution of the amount of water which flows in or out the column $]{ h}(t,x),{ h_{\mathrm{soil}} }(x)[$ through its lower boundary ${ h}(t,x)$. As we have shown in Proposition \[prop\_mass\] above, this source term ensures the mass conservation in the coupled model [–]{}. Of course this term also depends (non linearly) on the solution $\tilde H$. However this dependence is more easy to handle than the one given in the first equation of . In particular, the expression is well adapted to the numerical implementation of the coupled problem [–]{}.
Notice that the level $z=h_s$, defined in , represents the interface between the saturated and unsaturated part of the aquifer according to the auxiliary pressure $Q:=\rho\,g(\tilde H(t,x)-z)$. In particular $Q(t,x,h_s(t,x))={P_{s}}$ if $h_s(t,x)\in({ h_{\mathrm{bot}} }(x),{ h_{\mathrm{soil}} }(x))$ (regardless of the choice of $R\ge0$ in ). The conductivity tensor $\tilde K(\tilde H)$ defined in can be then decomposed into two parts: $$\label{conductivity_split}
\tilde K(\tilde H)(t,x) =\tilde C_0+ \int_{h_s(t,x)}^{{ h_{\mathrm{soil}} }(x)}k_r(Q)\,M_0(x,z)\,dz$$ where $\tilde C_0$ is the averaged conductivity of the saturated soil, [*i.e.*]{} $$\tilde C_0 =\int_{{ h_{\mathrm{bot}} }(x)}^{h_s(t,x)}M_0(x,z)\,dz.$$ In classical models for the saturated part of an aquifer obtained by vertical integration under the Dupuit’s assumption, the definition of the effective conductivity (see for example [@Bear1987]) reduces to $\tilde C_0$ instead of $\tilde K(\tilde H)$, the latter being a little greater. The quantity $\tilde C_0$ takes into account the horizontal flow in the saturated part but it ignores the (little) one in the unsaturated part, in particular close to the interface $z=h_s$ where the capillary effects lead to a non-negligible saturation. In practice, the smaller $h_s$, the more significant is the difference $\tilde K(\tilde H)-\tilde C_0$. In particular, if a part of the bottom of the aquifer is not saturated, that is $h_s={ h_{\mathrm{bot}} }$, considering only the vanishing conductivity $\tilde C_0$ whereas $\tilde K(\tilde H)$ remains positive is physically incorrect.
Numerical simulations {#numeric}
---------------------
In this section we compare numerically the original 3D-Richards model and the coupled model [–]{}for several choices of $h$ satisfying .
#### Physical parameters and geometry.
All the simulations are done with the following set of data. Denoting $I_3$ the $3{\times }3$ identity matrix we set: $$s(P)=({P_{s}}/P)^{\lambda}, \quad k_r(P)=({P_{s}}/P)^{2+3\lambda}, \quad ({P_{s}},\lambda)=(-1.5,3), \quad
\rho=1 , \quad \phi=0.1 , \quad K_0=0.1\,I_3 .$$ To lighten the numerical results, we consider the simplified 2D aquifer $\Omega=]-5,0[{\times }\Omega_x$, $\Omega_x=]0,L_x[$. In the experiments illustrated in Figures \[fig\_richards\_2d\] and \[fig\_compare\], the horizontal length is $L_x=28$. In those of Figure \[fig\_error\], $L_x \in [21,393]$. The parameter $\delta$ in is chosen as small as possible, that is equal to the size of one vertical mesh. We assume an impermeable layer at the bottom and the top of the aquifer.
#### Visualisation.
For the visualization of the results, we introduce a function ${ h_{\mathrm{sat}} }$ representing in a lot of cases the top level of the saturated region at the bottom of the aquifer ([*i.e.*]{} the water table). Let ${ h_{\mathrm{sat}} }={ h_{\mathrm{sat}} }(t,x)$ and the set ${ {\Omega_{{ h_{\mathrm{sat}} }}^- }}(t)$ be defined for a given pressure $P=P(t,x,z)$ by $$\label{hsat}
{ h_{\mathrm{sat}} }(t,x):=\sup I_{t,x},\quad I_{t,x}:=\big\{ z\in[{ h_{\mathrm{bot}} }(x),{ h_{\mathrm{max}} }(x)]\ |\ P(t,x,z')>{P_{s}}\ ,\ \forall z'\in[{ h_{\mathrm{bot}} }(x),z[ \big\} ,$$ $$\label{Omsat}
{ {\Omega_{{ h_{\mathrm{sat}} }}^- }}(t):=\bigl\{(x,z)\in\Omega\ |\ z<{ h_{\mathrm{sat}} }(t,x)\bigr\}.$$ By construction and if $P$ is continuous we have $$P\big(t,x,{ h_{\mathrm{sat}} }(t,x)\big) \begin{cases}
={P_{s}}&\text{if }{ h_{\mathrm{bot}} }<{ h_{\mathrm{sat}} }<{ h_{\mathrm{max}} }\\
\ge{P_{s}}&\text{if }{ h_{\mathrm{sat}} }={ h_{\mathrm{max}} }\\
\le{P_{s}}&\text{if }{ h_{\mathrm{sat}} }={ h_{\mathrm{bot}} }\end{cases}$$ and $P(t,x,z)\ge{P_{s}}$ for all $z\in]{ h_{\mathrm{bot}} },{ h_{\mathrm{sat}} }]$. In particular the soil is fully saturated in ${ {\Omega_{{ h_{\mathrm{sat}} }}^- }}(t)$ for every $t\in{ ]0,T[ }$.
\[rem\_hsat\] Notice that the set ${ {\Omega_{{ h_{\mathrm{sat}} }}^- }}$ does not coincide with *the* saturated region of the soil at the bottom of the aquifer. Indeed a saturated region just over $z={ h_{\mathrm{sat}} }$ is possible for example if $P\ge {P_{s}}$ also in $\Omega\setminus{ {\Omega_{{ h_{\mathrm{sat}} }}^- }}$. The interface $z={ h_{\mathrm{sat}} }$ then describes
- either the interface between the saturated part at the bottom of the aquifer and the unsaturated part above in the simplest setting,
- or a level between two saturated part when for example a saturated front flow down and reach ${ {\Omega_{{ h_{\mathrm{sat}} }}^- }}$,
- or the bottom of the aquifer when ${ h_{\mathrm{sat}} }={ h_{\mathrm{bot}} }$, that is when there is no saturated part at the bottom,
- or the maximum allowed height ${ h_{\mathrm{sat}} }={ h_{\mathrm{max}} }$ when, roughly speaking, the water table overflows.
Of course here, since ${ h_{\mathrm{sat}} }(t,x)\le { h_{\mathrm{soil}} }-\delta$ by , the set ${ {\Omega_{{ h_{\mathrm{sat}} }}^- }}$ cannot reach the soil level ${ h_{\mathrm{soil}} }$. In this sense ${ {\Omega_{{ h_{\mathrm{sat}} }}^- }}$ does not represent the physical water table which possibly touches the soil level. We only have done this choice for the definition of ${ h_{\mathrm{sat}} }$ to recover the unknown $h$ in the maximal case and thus to facilitate the visualisation.
#### Numerical scheme.
For the numerical approximation of the problem [–]{}we use mass-conservative fully implicit time schemes associated with finite elements methods in space for both horizontal and vertical directions. The schemes for and differ:
- In the case , we solve directly equation in which the right-hand side $(u \cdot e_3)|_{{ {\Gamma_{{ h}}^+} }}$ is seen as a Dirichlet to Neumann operator depending on $\tilde H$ and obtained by solving the 1D-vertical Richards equations. This non-linear term is treated with a Newton method.
- In the case , the nonlinear coupling between the 1D-vertical Richards equations and the 1D-horizontal diffusion equation is performed by using a Picard’s fixed-point method at each time step. This one alternatively solves (for an explicit $\tilde H$ and $h$) and (for an explicit right-hand side).
In any case all the 1D-Richards equations remain independent at the discrete level and can be solved in parallel.
#### Reference flowing experiment.
\(a) at (0.6,0.15) [Impermeable rock]{};
(0,0) – (0.6,0) – (0.6,0.3) – (0,0.3) – cycle;
\(a) at (0.6,0.0) [${ h_{\mathrm{sat}} }$]{}; (0.0,0) – (0.48,0) ;
\(a) at (0.6,0) [${ h_{\mathrm{soil}} }$, ${ h_{\mathrm{bot}} }$]{}; (0,0) – (0.5,0) ;
(0,0) – (10,0) – (10,0.5) – (0,0.5) – cycle; (0,0) – (10,0) – (10,0.5) – (0,0.5) – cycle;
(0,0) node\[below\] [$s=0$]{}; (5,0) node\[below\] [$s=0.5$]{}; (10,0) node\[below\] [$s=1$]{};
At time $t=0$, we consider a setting where the function ${ h_{\mathrm{sat}} }$ introduced in corresponds to the height of the water table. To show the influence of the deepness of the saturated area, we choose a function ${ h_{\mathrm{sat}} }(0,\cdot)$ which goes smoothly from $-4.5$ on the left part of the aquifer to $-2.5$ on the right one: $${ h_{\mathrm{sat}} }(0,x)=\begin{cases}
-4.5 + 2\,e^{-(\frac{15}{L_x})^2(x-0.55\,L_x)^2}&\text{in }[0,0.55L_x],\\
-2.5&\text{in }]0.55L_x,L_x].
\end{cases}$$ The initial pressure $P$ is defined by $P(0,x,z)=\rho\,g\,({ h_{\mathrm{sat}} }(0,x)-z)+{P_{s}}$ for all $(x,z)$ except near two rectangular regions above $z={ h_{\mathrm{sat}} }$ where the pressure goes smoothly to the saturation value ${P_{s}}$, corresponding to an infiltration process. These rectangles are $$\label{rectangles}
R_1=]L_x/10,3L_x/10[{\times }\left]-3.5,-1.7\right[{\quad\text{and}\quad }R_2=]7L_x/10,9L_x/10[{\times }\left]-2,-0.2\right[.$$ This initial situation is drawn in the first picture of Figure \[fig\_richards\_2d\]. In every picture the gray scale corresponds to the saturation value, the maximal darkness corresponding to $s\simeq1$.
The total time of the experiment is $4$ days. The solution of the classical Richards problem at time $0$, $10$, $20$ and $96$ hours respectively, is drawn in Figure \[fig\_richards\_2d\]. The graph of the visualization function ${ h_{\mathrm{sat}} }$ defined in is also plotted. Its evolution will be used for comparing the original Richards model with the coupled model [–]{}.
At time $t=10$ the water initially in rectangles $R_1$ and $R_2$ started to flow down. In the right part, some water coming from $R_2$ have reached the saturated water table inducing an increase of its level. In the mean time, we see in the middle of the domain $\Omega_x$ that the water moves to the left and that the function ${ h_{\mathrm{sat}} }$ is smoother than the initial one.
At time $t=20$ the water initially in rectangle $R_1$ has continued to flow down and is about to reach the water table. It is important to notice that this flow was essentially along the vertical direction. In particular the water front which is very close to ${ h_{\mathrm{sat}} }$ is approximately horizontal as in the initial situation.
After some time almost all the water initially located in the rectangle supplies have reached the water table. Then the interface ${ h_{\mathrm{sat}} }$ becomes flat and is associated with a pressure admitting the stationary profile $P(t,x,z)={P_{s}}+\rho\,g\big({ h_{\mathrm{sat}} }(t,x)-z\big)$.
#### Comparison of the models.
\(a) at (0.6,0.15) [Impermeable rock]{};
(0,0) – (0.6,0) – (0.6,0.3) – (0,0.3) – cycle;
\(a) at (0.6,0.04) [${ h_{\mathrm{soil}} }$, ${ h_{\mathrm{bot}} }$]{}; (0,0) – (0.5,0);
\(a) at (0.6,0.03) [${ { h_{\mathrm{sat}} }^{ 2d} }$]{};
(0,0) – (0.5,0) ;
\(a) at (0.6,0.04) [${ { h_{\mathrm{sat}} }^{ a} }$]{}; (0,0) – (0.48,0);
\(a) at (0.6,0) [${ { h_{\mathrm{sat}} }^{ b} }$]{};
(0,0) – (0.48,0) ;
\(a) at (0.6,0) [${ { h_{\mathrm{sat}} }^{ c} }$]{};
(0,0) – (0.48,0) ;
table\[x=X,y=coupled\_Pbubp2,col sep=comma\] [annim\_10110.txt]{}; table\[x=X,y=coupled\_classic, col sep=comma\] [annim\_10110.txt]{}; table\[x=X,y=bot, col sep=comma\] [annim\_10110.txt]{}; table\[x=X,y=2d, col sep=comma\] [annim\_10110.txt]{}; coordinates [(0,-5) (28.56,-5) (28.56,-5.2) (0,-5.2) ]{}; coordinates [(-0.5,0.4) ]{}; coordinates [(29.3,-5.4) ]{};
coordinates [(0,0) (28.58,0) ]{}; coordinates [(0,-5) (28.58,-5) ]{};
(1,4) node [$t=10$]{}; (-0.3,2.5) node [$z$]{} ;
table\[x=X,y=coupled\_Pbubp2,col sep=comma\] [annim\_10255.txt]{}; table\[x=X,y=coupled\_classic, col sep=comma\] [annim\_10255.txt]{}; table\[x=X,y=bot, col sep=comma\] [annim\_10255.txt]{}; table\[x=X,y=2d, col sep=comma\] [annim\_10255.txt]{}; coordinates [(0,-5) (28.56,-5) (28.56,-5.2) (0,-5.2) ]{}; coordinates [(-0.5,0.4) ]{}; coordinates [(29.3,-5.4) ]{};
coordinates [(0,0) (28.58,0) ]{}; coordinates [(0,-5) (28.58,-5) ]{};
(1,4) node [$t=24$]{};
table\[x=X,y=coupled\_Pbubp2,col sep=comma\] [annim\_10500.txt]{}; table\[x=X,y=coupled\_classic, col sep=comma\] [annim\_10500.txt]{}; table\[x=X,y=bot, col sep=comma\] [annim\_10500.txt]{}; table\[x=X,y=2d, col sep=comma\] [annim\_10500.txt]{}; coordinates [(0,-5) (28.56,-5) (28.56,-5.2) (0,-5.2) ]{}; coordinates [(-0.5,0.4) ]{}; coordinates [(29.3,-5.4) ]{};
coordinates [(0,0) (28.58,0) ]{}; coordinates [(0,-5) (28.58,-5) ]{};
(1,4) node [$t=48$]{}; (3.2,-0.3) node [$x$]{} ; (-0.3,2.5) node [$z$]{} ;
table\[x=X,y=coupled\_Pbubp2,col sep=comma\] [annim\_10998.txt]{}; table\[x=X,y=coupled\_classic, col sep=comma\] [annim\_10998.txt]{}; table\[x=X,y=bot, col sep=comma\] [annim\_10998.txt]{}; table\[x=X,y=2d, col sep=comma\] [annim\_10998.txt]{}; coordinates [(0,-5) (28.56,-5) (28.56,-5.2) (0,-5.2) ]{}; coordinates [(-0.5,0.4) ]{}; coordinates [(29.3,-5.4) ]{};
coordinates [(0,0) (28.58,0) ]{}; coordinates [(0,-5) (28.58,-5) ]{};
(1,4) node [$t=96$]{}; (3.2,-0.3) node [$x$]{} ;
In this part we compare the solution of the classical Richards model with the one obtained by using the coupled model [–]{}. We test three particular choices for the function $h$ satisfying : the minimal one , the maximal one and an intermediate one given by for $R=3$. All data remain the same as in the previous paragraph. In this paper, we focus on the evolution of the functions ${ h_{\mathrm{sat}} }$ defined by . As indicated in Remark \[rem\_hsat\], this function roughly represents the upper level of the water table. In the following we denote by ${ { h_{\mathrm{sat}} }^{ 2d} }$ the level coming from the reference 2d-Richards model and we denote by ${ { h_{\mathrm{sat}} }^{ a} }$, ${ { h_{\mathrm{sat}} }^{ b} }$ and ${ { h_{\mathrm{sat}} }^{ c} }$ the ones coming from the model [–]{}with the function $h$ given respectively by , and .
The functions ${ { h_{\mathrm{sat}} }^{ 2d} }$, ${ { h_{\mathrm{sat}} }^{ a} }$, ${ { h_{\mathrm{sat}} }^{ b} }$ and ${ { h_{\mathrm{sat}} }^{ c} }$ are plotted in Figure \[fig\_compare\] at time $t \in \{10,24,48,96\}$ (in hours). We of course do not plot the initial situation which is the same for each model and is the one of the reference test case described in the previous paragraph. The curve ${ { h_{\mathrm{sat}} }^{ 2d} }$ is the reference one and is plotted with a black solid line in Figure \[fig\_compare\].
Bear in mind that the function $h$ characterizes the level below which the vertical flow is assumed to be instantaneous (instead of being described by the 1D-Richards equation). In every case, the horizontal flow is ruled by equation .
- In the case , $h={ h_{\mathrm{bot}} }$. The vertical flow is described by the 1D-Richards model in the whole domain, even in the saturated part below the level $z={ { h_{\mathrm{sat}} }^{ a} }$. The horizontal flow in this case seems to be slower than the one given by the Richards model (compare the gray dot-dashed line with the black solid one in Figure \[fig\_compare\]).\
Roughly the idea is that in this case the water have to travel along the whole vertical direction before reaching the level $z=h={ h_{\mathrm{bot}} }$. Then the flux $(u\cdot e_3)|_{{ \Gamma_{\mathrm{bot}} }}$ at the bottom of the aquifer takes a lot of time to increase when the water coming from rectangles $R_1$ and $R_2$ reaches the water table. This flux being the source term in equation , the function $\tilde H$ increases with some delay and the corresponding horizontal flow is slower.
- In the case , $h={ { h_{\mathrm{sat}} }^{ b} }$. This case is opposite of the previous one in the sense that the vertical flow in the whole saturated zone ${ {\Omega_{{ h_{\mathrm{sat}} }}^- }}$ is considered to be instantaneous. Then, when the water coming from rectangles $R_1$ and $R_2$ reaches the water table, the flux $(u\cdot e_3)|_{{ {\Gamma_{{ h}}} }}$ increases very quickly. So does the corresponding hydraulic head $\tilde H$ and the horizontal flow is very and even too fast (see the black dotted line compared to the black solid line in Figure \[fig\_compare\]).
- In the case for $R=3$, ${ h_{\mathrm{bot}} }\le h\le{ { h_{\mathrm{sat}} }^{ c} }$. The corresponding flow should exhibit an intermediate behavior between the too previous ones. Here, the value $R=3$ was chosen so that ${ { h_{\mathrm{sat}} }^{ c} }$ is very close to the reference one ${ { h_{\mathrm{sat}} }^{ 2d} }$ (see the gray dashed line).
Notice that in every situation, the error between ${ { h_{\mathrm{sat}} }^{ 2d} }$ and ${ h_{\mathrm{sat}} }^\kappa$, $\kappa\in\{a,b,c\}$, is smaller in the left part of the domain than in the right one. This is due to the fact that the saturated zone is thiner in this region. For a very thin saturated region, considering an instantaneous vertical flow or the one given by the vertical 1D-Richards problem gives similar results. Conversely, the thicker the saturated water table is, the more the results issued from the two extremal situations and differ from the reference one. Basically, ${ h_{\mathrm{sat}} }^b$ is expected to move too fast while ${ h_{\mathrm{sat}} }^a$ moves too slowly. In this kind of deep situation and if the [*ratio*]{} between the deepness and the length of the aquifer is not so small, one of the intermediate choices is obviously more appropriate.
#### Error made by the coupled model versus the ratio deepness/largeness.
In the previous simulations, where $\Omega=]0,28[{\times }]-5,0[$, the [*ratio*]{} ${\varepsilon}$=*deepness/length* of the aquifer is such that $1/{\varepsilon}=5.6$. It is important to notice that even in this case of large ratio ${\varepsilon}$ the error between the original Richards model and the coupled model [–]{}in the case is particularly small (see the dashed plot in Figure \[fig\_compare\]). This supports the fact that the coupled model may be considered for approaching the Richards model also in an aquifer which is not so shallow. This guess is confirmed by the results plotted in Figure \[fig\_error\]. The evolution of the error $\|{ { h_{\mathrm{sat}} }^{ 2d} }-{ h_{\mathrm{sat}} }^\kappa\|_{L^1({ ]0,T[ }{\times }\Omega_x)}$ for $\kappa\in \{a,b,c\}$ is drawn in terms of the ratio $1/{\varepsilon}$.
As expected all the errors decrease with ${\varepsilon}$. Moreover, the intermediate case is always the best, mainly in the case of a “large” value of ${\varepsilon}$. After comes the maximal choice. The worst choice is the maximal one but with an error which decreases a lot with ${\varepsilon}$.
The accuracy of the model depends on the choice of $R$ in , e.g. for minimizing the error $\|{ { h_{\mathrm{sat}} }^{ 2d} }-{ h_{\mathrm{sat}} }^\kappa\|_{L^p({ ]0,T[ }{\times }\Omega_x)}$. This optimization process is postponed to a forthcoming work.
table\[x=eps,y=err\_bot,col sep=comma\] [compar\_error.txt]{}; table\[x=eps,y=err\_coupled\_classic,col sep=comma\] [compar\_error.txt]{}; table\[x=eps,y=err\_coupled\_Pbubp3,col sep=comma\] [compar\_error.txt]{};
Formal asymptotic expansion {#formal_asymptotic}
===========================
In this section, the 3D-Richards problem and the coupled model [–]{}are compared using asymptotic analysis arguments. We prove that these models behave the same, whatever the time scale, when the [*ratio*]{} between the characteristic deepness and the length of the shallow aquifer tends to zero.
Dimensionless form of the 3D-Richards and coupled problems
----------------------------------------------------------
Introduce a fixed dimensionless reference domain ${\overline}\Omega$ of type and a dimensionless real number ${\overline}T>0$. Fix ${\overline}\Omega_x$, ${ {\overline}h_{\mathrm{soil}} }$ and ${ {\overline}h_{\mathrm{bot}} }$ such that $${\overline}\Omega=\Big\{({\overline}x,{\overline}z)\in{\overline}\Omega_x\times{\mathbb{R}}\quad |\quad {\overline}z\in\big]{ {\overline}h_{\mathrm{bot}} }({\overline}x) ,{ {\overline}h_{\mathrm{soil}} }({\overline}x)\big[ \Big\} .$$ To obtain a rescaled version of equations and [–]{}in the domain $]0,{\overline}T[{\times }{\overline}\Omega$, we introduce positive reference numbers $L_x$, $L_z$, $T$. Then, keeping the same notations as in Section \[sec\_main\], we have:
- The physical variables are given by $$x =L_x\,{\overline}x,\quad z =L_z \,{\overline}z , \quad t = \frac{T}{{\overline}T}\,{\overline}t .$$
- The corresponding physical domain $\Omega$ is given as in with $$\Omega_x = L_x\, {\overline}\Omega_x,\quad{ h_{\mathrm{soil}} }(x) =L_z\,{ {\overline}h_{\mathrm{soil}} }({\overline}x), \quad { h_{\mathrm{bot}} }(x)= L_z\,{ {\overline}h_{\mathrm{bot}} }({\overline}x).$$
- The unknowns are such that $$\begin{gathered}
{\overline}P({\overline}t,{\overline}x, {\overline}z) = P(t,x,z),\quad {\overline}v({\overline}t,{\overline}x, {\overline}z) = v(t,x,z),\quad {\overline}u({\overline}t,{\overline}x, {\overline}z) = u(t,x,z),\quad {\overline}w({\overline}t,{\overline}x, {\overline}z) = w(t,x,z),
\\
L_z {\overline}H({\overline}t,{\overline}x) =\tilde H(t,x),\quad L_z {\overline}h({\overline}t,{\overline}x) =h(t,x) .\end{gathered}$$
- The reference subdomains are $${ {\Omega_{{ {\overline}h}}^- }}({\overline}t)=\big\{({\overline}x,{\overline}z)\in{\overline}\Omega_x\times{\mathbb{R}}\ |\ {\overline}z\in\big]{ {\overline}h_{\mathrm{bot}} }({\overline}x) , { {\overline}h}({\overline}t,{\overline}x)\big[ \big\},
\qquad
{ {\Omega_{{ {\overline}h}}^+ }}({\overline}t)=\big\{({\overline}x,{\overline}z)\in{\overline}\Omega_x\times{\mathbb{R}}\ |\ {\overline}z\in\big]{ {\overline}h}({\overline}t,{\overline}x) ,{ {\overline}h_{\mathrm{soil}} }({\overline}x)\big[ \big\} .$$
- The reference boundaries are $ { {\overline}\Gamma_{\mathrm{bot}} }:=\{( {\overline}x,{\overline}z)\in{\overline}\Omega\ |\ {\overline}z={ {\overline}h_{\mathrm{bot}} }({\overline}x) \}$, $ {{\overline}\Gamma_{\mathrm{soil}} }:=\{({\overline}x,{\overline}z)\in{\overline}\Omega\ |\ {\overline}z={ {\overline}h_{\mathrm{soil}} }({\overline}x)\}$ and $ { {\overline}\Gamma_{\mathrm{ver}} }:=\{({\overline}x,{\overline}z)\in{\overline}\Omega\ |\ {\overline}x\in\partial{\overline}\Omega_x\}$.
- The reference exterior normals are $${\overline}n({\overline}x,{\overline}z)=\begin{cases}
{\displaystyle}\left(e_3-\frac{L_z}{L_x}\nabla_{{\overline}x}{ {\overline}h_{\mathrm{soil}} }({\overline}x)\right)\left(\frac{L_z^2}{L_x^2}|\nabla_{{\overline}x}{ {\overline}h_{\mathrm{soil}} }({\overline}x)|^2+1\right)^{-1/2}&\text{on }{{\overline}\Gamma_{\mathrm{soil}} }\\
{\displaystyle}\left(\frac{L_z}{L_x}\nabla_{{\overline}x}{ {\overline}h_{\mathrm{bot}} }({\overline}x)-e_3\right)\left(\frac{L_z^2}{L_x^2}|\nabla_{{\overline}x}{ {\overline}h_{\mathrm{bot}} }({\overline}x)|^2+1\right)^{-1/2}&\text{on }{ {\overline}\Gamma_{\mathrm{bot}} }\\
n(x,z)&\text{on }{ {\overline}\Gamma_{\mathrm{ver}} }\end{cases}$$ where the vector ${\overline}n$ is horizontal and does not change during the rescaling.
- The saturation and relative conductivity satisfy $$\label{ovskr}
s({\overline}P) = s(P),\qquad k_r({\overline}P) = k_r(P).$$ It means that the reference saturation and relative permeability are of order one. Indeed $P$ and ${\overline}P$ take the same values, independently of the scale change.
- For the conductivities, we set $$\begin{gathered}
{\overline}K_0({\overline}x,{\overline}z)=K_0( x, z),\qquad {\overline}M_0({\overline}x,{\overline}z)=M_0( x, z),
\label{ovK0}
\\
{\overline}K({\overline}H)({\overline}t,{\overline}x) = L_z\int_{{ {\overline}h_{\mathrm{bot}} }({\overline}x)}^{{ {\overline}h_{\mathrm{soil}} }({\overline}x)} k_r\big(\rho\,g({\overline}H({\overline}t,{\overline}x)-{\overline}z\big){\overline}M_0\,d{\overline}z .
\label{ovKtilde}\end{gathered}$$ We choose for the sake of simplicity in the presentation. Indeed, we could also introduce $K$ and $M$ such that $K {\overline}K_0({\overline}x, {\overline}z)=K_0(x,z)$ and $M {\overline}M_0({\overline}x, {\overline}z)=M_0(x,z)$ and then perform the same study assuming that $K/L_x=\mathcal{O}({\varepsilon})$, $M/L_x=\mathcal{O}({\varepsilon})$ and $K/L_z=\mathcal{O}(1)$.
- The source term is $${\overline}F({\overline}t,{\overline}x) = F(t,x)$$
#### Dimensionless Richards problem.
Introducing the latter quantities in , we get the following set of rescaled equations: $$\label{rescaled_mass_richard}
\frac{{\overline}T}{T}\phi\,{ {\frac{\partial s({\overline}P)}{\partial {\overline}t} }}+\frac1{L_x}\operatorname{div}_{{\overline}x}({\overline}v) + \frac1{L_z}{ {\frac{\partial {\overline}v}{\partial {\overline}z} }}=0\quad \text{in }]0,{\overline}T[{\times }{\overline}\Omega ,$$ $$\label{rescaled_momentum}
{\overline}v = -k_r({\overline}P)\,{\overline}K_0\, \Big( \frac1{L_x}\frac{1}{\rho g}\nabla_{{\overline}x} {\overline}P + \Big(\frac1{L_z}\frac{1}{\rho g}{ {\frac{\partial {\overline}P}{\partial {\overline}z} }}+1\big)e_3 \Big)\quad \text{in }]0,{\overline}T[{\times }{\overline}\Omega,$$ $$\label{rescaled_bottom}
{\overline}v\cdot \Big(\frac{L_z}{L_x}\nabla_{{\overline}x}{ {\overline}h_{\mathrm{bot}} }-e_3\Big) = 0 \quad \text{on }]0,{\overline}T[{\times }{ {\overline}\Gamma_{\mathrm{bot}} },$$ $$\label{rescaled_soil}
\alpha\, {\overline}P\,\Big(\frac{L_z^2}{L_x^2}\|\nabla_{{\overline}x}{ {\overline}h_{\mathrm{soil}} }\|^2+1\Big)^{1/2} + \beta\, {\overline}v\cdot \Big(e_3-\frac{L_z}{L_x}\nabla_{{\overline}x}{ {\overline}h_{\mathrm{soil}} }\Big)= {\overline}F\, \Big(\frac{L_z^2}{L_x^2}\|\nabla_{{\overline}x}{ {\overline}h_{\mathrm{soil}} }\|^2+1\Big)^{1/2} \quad \text{on }]0,{\overline}T[{\times }{{\overline}\Gamma_{\mathrm{soil}} },$$ $$\label{rescaled_vert}
{\overline}v\cdot {\overline}n= 0 \quad \text{on }]0,{\overline}T[{\times }{ {\overline}\Gamma_{\mathrm{ver}} }.$$
Since the aquifer is assumed to be very thin with respect to its horizontal width, the quantity $L_z/L_x$ is very small. We choose to consider an aquifer with a fixed height of order $L_z=1$ and a large horizontal dimension $L_x=1/{\varepsilon}$ for ${\varepsilon}\ll1$. We get
- the mass conservation equation which depends on the time scaling choice $T$: $$\label{rescaled_mass}
{\displaystyle}\frac{{\overline}T}{T}\phi\,{ {\frac{\partial s({\overline}P)}{\partial {\overline}t} }}+{\varepsilon}\operatorname{div}_{{\overline}x}({\overline}v) + { {\frac{\partial {\overline}v\cdot e_3}{\partial {\overline}z} }}=0\qquad\text{in }]0,{\overline}T[\times{\overline}\Omega$$
- associated with the following Darcy’s law and boundary conditions: $$\label{rescaled}
\begin{cases}
{\displaystyle}{\overline}v = -k_r({\overline}P)\,{\overline}K_0\, \left( \frac{{\varepsilon}}{\rho g}\nabla_{{\overline}x} {\overline}P + \Big(\frac{1}{\rho g}{ {\frac{\partial {\overline}P}{\partial {\overline}z} }}+1\big)e_3 \right)&\text{in } ]0,{\overline}T[\times{\overline}\Omega\\
{\displaystyle}\alpha\,{\overline}P\,\Big({\varepsilon}^2\,\|\nabla_{{\overline}x}{ {\overline}h_{\mathrm{soil}} }\|^2+1 \Big)^{1/2} +\beta\,{\overline}v\cdot \Big(e_3-{\varepsilon}\,\nabla_{{\overline}x}{ {\overline}h_{\mathrm{soil}} }\Big) =\Big({\varepsilon}^2\,\|\nabla_{{\overline}x}{ {\overline}h_{\mathrm{soil}} }\|^2+1 \Big)^{1/2} \, {\overline}{ F }&\text{on }]0,{\overline}T[\times{{\overline}\Gamma_{\mathrm{soil}} }\\
{\overline}v\cdot {\overline}n = 0 &\text{on }]0,{\overline}T[\times{ {\overline}\Gamma_{\mathrm{ver}} }\\
{\overline}v\cdot \Big({\varepsilon}\nabla_{{\overline}x}{ {\overline}h_{\mathrm{bot}} }-e_3\Big) = 0&\text{on }]0,{\overline}T[\times{ {\overline}\Gamma_{\mathrm{bot}} }\end{cases}$$
#### Dimensionless coupled Dupuit-Richards model.
By introducing the same parameter ${\varepsilon}\ll1$, the rescaled coupled problem [–]{}reads:
- The velocity problem: $$\label{dimensionless_velocity}
\begin{cases}
{\displaystyle}{\overline}v ={\overline}u + {\overline}w &\text{for \ }{\overline}t\in]0,{\overline}T[\ ,\quad ({\overline}x,{\overline}z)\in{\overline}\Omega\\
{\displaystyle}{\overline}u = -k_r({\overline}P)\,\Big(\frac{1}{\rho\,g}{ {\frac{\partial {\overline}P}{\partial {\overline}z} }}+1\Big)\, {\overline}K_0\,e_3 &\text{for \ }{\overline}t\in]0,{\overline}T[\ ,\quad ({\overline}x,{\overline}z)\in{\overline}\Omega\\
{\displaystyle}{\overline}w = -{\varepsilon}\,k_r\big( \rho\,g( {\overline}H-{\overline}z ) \big)\, {\overline}M_0\nabla_{{\overline}x}{\overline}H& \text{for \ }{\overline}t\in]{\overline}0,{\overline}T[\ ,\quad ({\overline}x,{\overline}z)\in{\overline}\Omega
\end{cases}$$
- The 1D-Richards equation in the transition zone: $$\label{dimensionless_transition}
\begin{cases}
{\displaystyle}\phi\frac{{\overline}T}{T} { {\frac{\partial s({\overline}P)}{\partial t} }}+{ {\frac{\partial }{\partial {\overline}z} }}\big({\overline}u\cdot e_3\big)=0 & \text{for \ }{\overline}t\in]0,{\overline}T[\ ,\quad ({\overline}x,{\overline}z)\in{ {\Omega_{{ {\overline}h}}^+ }}({\overline}t)\\
\alpha\,{\overline}P+\beta\, {\overline}u\cdot e_3={\overline}F &\text{for \ }({\overline}t,{\overline}x)\in]0,{\overline}T[{\times }{{\overline}\Gamma_{\mathrm{soil}} }\\
{\overline}P\big({\overline}t,{\overline}x,{ {\overline}h}({\overline}t,{\overline}x)\big)=\rho\,g\big({\overline}H({\overline}t,{\overline}x)-{ {\overline}h}({\overline}t,{\overline}x) \big)&\text{for \ }({\overline}t,{\overline}x)\in]0,{\overline}T[{\times }{\overline}\Omega_x \\
{\overline}P(0,{\overline}x,{\overline}z)={\overline}{ P_{\mathrm{init}} }({\overline}x,{\overline}z)&\text{for \ } ({\overline}x,{\overline}z)\in{ {\Omega_{{ {\overline}h}}^+ }}(0)\\
\end{cases}$$
- The pressure problem in the water table: $$\label{dimensionless_watertable}
{\displaystyle}{\overline}P({\overline}t,{\overline}x,{\overline}z)=\rho\,g\big( {\overline}H({\overline}t,{\overline}x)-{\overline}z \big) \qquad \text{for \ }{\overline}t\in[0,{\overline}T[\ ,\quad ({\overline}x,{\overline}z)\in{ {\Omega_{{ {\overline}h}}^- }}({\overline}t)\\$$
- The hydraulic head problem: $$\label{dimensionless_hydraulic}
\begin{cases}
{\varepsilon}^2\,\operatorname{div}_{{\overline}x}\Big({\overline}K({\overline}H)\,\nabla_{{\overline}x}{\overline}H \Big)={\overline}u\big|_{{ {\Gamma_{{ h}}^+} }}\cdot e_3&\text{for \ }({\overline}t,{\overline}x)\in]0,{\overline}T[{\times }{\overline}\Omega_x
\\
{\displaystyle}{\overline}K({\overline}H) \nabla_{{\overline}x} {\overline}H\cdot {\overline}n=0&\text{for \ }({\overline}t,{\overline}x)\in]0,{\overline}T[{\times }\partial{\overline}\Omega_x \\
{\overline}H(0,{\overline}x)={\overline}{ H_{\mathrm{init}} }({\overline}x)&\text{for \ } {\overline}x\in{\overline}\Omega_x
\end{cases}$$ Equivalently, by using , the first equation of admits the formulation: for $({\overline}t,{\overline}x)\in]0,{\overline}T[{\times }{\overline}\Omega_x$ $$\label{dimensionless_hydraulic_alternate}
{\varepsilon}^2\,\operatorname{div}_{{\overline}x}\Big({\overline}K({\overline}H)\,\nabla_{{\overline}x}{\overline}H \Big) \\
={\overline}u\big|_{{{\overline}\Gamma_{\mathrm{soil}} }}\cdot e_3+\frac{{\overline}T}{T}{ {\frac{\partial }{\partial {\overline}t} }}\left(\int_{{ {\overline}h_{\mathrm{bot}} }({\overline}x)}^{{\overline}{ h_{\mathrm{soil}} }({\overline}x)} \phi\,s({\overline}{P }) \,d{\overline}z \right)$$
- The definition of the interface separating the two different kind of flows: $$\label{dimensionless_interface}
{\displaystyle}{ {\overline}h_{\mathrm{bot}} }({\overline}x)\leq { {\overline}h}({\overline}t,{\overline}x)\leq\max\left\{\min\Big\{ {\overline}H({\overline}t,{\overline}x)-\frac{{P_{s}}}{\rho\,g}, { {\overline}h_{\mathrm{max}} }({\overline}x) \Big\}, { {\overline}h_{\mathrm{bot}} }({\overline}x)\right\}\qquad \text{for \ }({\overline}t,{\overline}x)\in[0,{\overline}T[{\times }{\overline}\Omega_x\\$$
Effective problems
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We are interested in the asymptotic behavior of the flow, thus of the models, for both short, intermediate and large times. For the asymptotic analysis, the question is related to the behavior of the dimensionless models above. More precisely, we want to describe the effective flow obtained for the short time $T={\overline}T$, the intermediate time $T={\varepsilon}^{-1}{\overline}T$ and the long time scales $T={\varepsilon}^{-2}{\overline}T$.
#### Asymptotic expansion.
We introduce the following formal asymptotics for the pressure and the velocity: $$\label{asymptotic_richards}
{\displaystyle}{\overline}P_{{\varepsilon}}^\gamma={\overline}P_0^\gamma+{\varepsilon}\,{\overline}P_1^\gamma+{\varepsilon}^2\,{\overline}P_2^\gamma+\dots\qquad {\overline}v_{{\varepsilon}}^\gamma={\overline}v_0^\gamma+{\varepsilon}\,{\overline}v_1^\gamma+{\varepsilon}^2\,{\overline}v_2^\gamma+\dots$$ We emphasize that no arbitrary scaling is imposed, in particular we do not suppose as in [@Jazar2014] that the vertical velocity is much smaller than the horizontal one when the ratio $\epsilon$ is very small. We assume also the existence of formal asymptotics for the auxiliary variables appearing in [–]{}$$\label{asymptotic_coupled}
\begin{cases}
\begin{array}{ll}
{\displaystyle}{\overline}u_{{\varepsilon}}^\gamma={\overline}u_0^\gamma+{\varepsilon}\,{\overline}u_1^\gamma+{\varepsilon}^2\,{\overline}u_2^\gamma+\dots& {\overline}w_{{\varepsilon}}^\gamma={\overline}w_0^\gamma+{\varepsilon}\,{\overline}w_1^\gamma+{\varepsilon}^2\,{\overline}w_2^\gamma+\dots\\
{\displaystyle}{\overline}H_{{\varepsilon}}^\gamma={\overline}H_0+{\varepsilon}\,{\overline}H_1^\gamma+{\varepsilon}^2\,{\overline}H_2^\gamma+\dots \qquad& {\displaystyle}{\overline}h_{{\varepsilon}}^\gamma={\overline}h_0^\gamma+{\varepsilon}\,{\overline}h_1^\gamma+{\varepsilon}^2\,{\overline}h_2^\gamma+\dots,
\end{array}
\end{cases}$$ and for the flux at the soil level $$\label{asymptotic_source}
{\displaystyle}{\overline}{ F }_{{\varepsilon}}={\overline}{ F }_0+{\varepsilon}\,{\overline}{ F }_1+{\varepsilon}^2\,{\overline}{ F }_2+\dots.$$ Moreover, since $s$ and $k_r$ are $\mathcal{C}^\infty$ by part functions, we write $$\label{asymptotic_sat}
\begin{cases}
\begin{array}{ll}
{\displaystyle}s({\overline}P_{{\varepsilon}}^\gamma)= s({\overline}P_0^\gamma)+{\varepsilon}({\overline}P_1^\gamma+{\varepsilon}\,{\overline}P_2^\gamma+\dots) s'({\overline}P_0^\gamma)+\frac{{\varepsilon}^2}{2}({\overline}P_1^\gamma+{\varepsilon}\,{\overline}P_2^\gamma+\dots)^2 s''({\overline}P_0^\gamma)+\dots\\
{\displaystyle}k_r({\overline}P_{{\varepsilon}}^\gamma)= k_r({\overline}P_0^\gamma)+{\varepsilon}({\overline}P_1^\gamma+{\varepsilon}\,{\overline}P_2^\gamma+\dots) k_r'({\overline}P_0^\gamma)+\frac{{\varepsilon}^2}{2}({\overline}P_1^\gamma+{\varepsilon}\,{\overline}P_2^\gamma+\dots)^2 k_r''({\overline}P_0^\gamma)+\dots\\
\end{array}
\end{cases}$$
#### Effective problems at the main order.
Let us introduce the following effective problems:
- related to the short time scale ($T={\overline}T$), $$\label{effective_short}
\begin{cases}
\phi\,{\displaystyle}{ {\frac{\partial s({\overline}P_0)}{\partial {\overline}t} }}+ { {\frac{\partial {\overline}v_0\cdot e_3}{\partial {\overline}z} }}=0&\text{in }]0,{\overline}T[\times\Omega\\
{\displaystyle}{\overline}v_0 = - k_r({\overline}P_0) \Big(\frac{1}{\rho g}{ {\frac{\partial {\overline}P_0}{\partial {\overline}z} }}+1\Big){\overline}K_0\,e_3&\text{in }]0,{\overline}T[\times\Omega \\
\alpha\,{\overline}P_0+\beta\, {\overline}v_0\cdot e_3 = {\overline}{ F }_0 &\text{on }]0,{\overline}T[\times{{\overline}\Gamma_{\mathrm{soil}} }\\
{\overline}v_0\cdot e_3 = 0&\text{on }]0,{\overline}T[\times{ {\overline}\Gamma_{\mathrm{bot}} }\end{cases}$$
- related to the non-short cases ($T={\varepsilon}^{-1}{\overline}T$ or $T={\varepsilon}^{-2}{\overline}T$ ), $$\label{effective_instant}
\begin{cases}
{\displaystyle}{\overline}P_0(t,x,z)=\rho\,g\big({\overline}H_0(t,x)-{\overline}z\big)&\text{in }]0,{\overline}T[\times{\overline}\Omega\\
{\displaystyle}{\overline}v_0=0&\text{in }]0,{\overline}T[\times{\overline}\Omega
\end{cases}$$
- related to the non-short cases ($T={\varepsilon}^{-1}{\overline}T$ or $T={\varepsilon}^{-2}{\overline}T$ ) if $\alpha \neq0$ $$\label{effective_alpha_0}
{\overline}H_0({\overline}t,{\overline}x)=\frac{{\overline}F_0({\overline}t,{\overline}x)}{\alpha\,\rho\,g}+{ {\overline}h_{\mathrm{soil}} }({\overline}t,{\overline}x)\qquad\text{in } ]0,{\overline}T[\times{\overline}\Omega_x$$
- related to the intermediate time scale ($T={\varepsilon}^{-1}{\overline}T$) if $\alpha = 0$ (and then $\beta\neq0$) $$\label{effective_inter}
\rho\,g\Big(\int_{{ {\overline}h_{\mathrm{bot}} }}^{{ {\overline}h_{\mathrm{soil}} }} \phi\, s'({\overline}P_0)\,dz\Big){ {\frac{\partial {\overline}H_0}{\partial t} }} =-\frac{{\overline}F_1}{\beta}\qquad \text{in } ]0,{\overline}T[\times{\overline}\Omega_x$$
- related to the long time scale ($T={\varepsilon}^{-2}{\overline}T$) if $\alpha = 0$ $$\label{effective_long}
\begin{cases}
{\displaystyle}-\operatorname{div}_x\big({\overline}K({\overline}H_0)\,\nabla_x{\overline}H_0\big)=-\frac{{\overline}{ F }_2}{\beta}-{ {\frac{\partial }{\partial {\overline}t} }}\Big(\int_{{ {\overline}h_{\mathrm{bot}} }}^{{ {\overline}h_{\mathrm{soil}} }}\phi\, s({\overline}P_0) \,d{\overline}z\Big)&\text{in }]0,{\overline}T[\times{\overline}\Omega_x\\
{\overline}K({\overline}H_0)\, \nabla_{{\overline}x}{\overline}H_0\cdot {\overline}n = 0 &\text{on }]0,{\overline}T[\times{ {\overline}\Gamma_{\mathrm{ver}} }\end{cases}$$ and concerning the first order of the velocity $$\label{effective_v1}
{\overline}v_1=-{\overline}k_r({\overline}P_0)\,{\overline}M_0\,\nabla_{{\overline}x}{\overline}H_0\qquad\text{in }]0,{\overline}T[\times{\overline}\Omega$$
\[prop\_richards\] Let $({\overline}P_{{\varepsilon}}^\gamma,{\overline}v_{{\varepsilon}}^\gamma)$ be the solution of the rescaled 3D-Richards problem – *or* of the rescaled coupled model [–]{} for $T={\varepsilon}^{-\gamma}{\overline}T$ and $\gamma\in\{0,1,2\}$. Assume that – hold true, then
- $({\overline}P_{0}^0,{\overline}v_{0}^0)$ satisfies .
- $({\overline}P_{0}^1,{\overline}v_{0}^1)$ satisfies and if $\alpha\neq0$, or and with the compatibility condition ${\overline}F_0=0$ if $\alpha=0$.
- $({\overline}P_{0}^2,{\overline}v_{0}^2)$ satisfies and if $\alpha\neq0$, or and with the compatibility condition ${\overline}F_0={\overline}F_1=0$ if $\alpha=0$. Moreover ${\overline}v_1^2$ satisfies if $\alpha=0$.
We emphasize that the intermediate variable ${\overline}h$ which characterizes the coupled model - does not appear in any of the main order effective problems -. This agrees with the fact that the whole class of models given by [–]{}for *any* $h$ satisfying can approximate the reference Richards model.
Proof of Proposition \[prop\_richards\] for the Richards model
--------------------------------------------------------------
The proof of Proposition \[prop\_richards\] consists in substituting the formal asymptotic expansion – in the rescaled 3D-Richards problem –. A cascade of equations follows by identifying the powers of ${\varepsilon}$. Then we characterize the main order terms in the expansion . In order to reduce ratings in this section, we do not write the exponent $\gamma$ on the variables name.
#### General relations.
Let us start by obtaining the first relations holding in every time scale (i.e. for all $\gamma\in\{0,1,2\}$). By plugging the asymptotic expansion in the first equation of we get the following relations holding in $]0,{\overline}T[\times\Omega$ $$\label{velocity_gen}
\begin{cases}
{\displaystyle}{\overline}v_0=-k_r({\overline}P_0)\,\Big(\frac{1}{\rho\,g}{ {\frac{\partial {\overline}P_0}{\partial {\overline}z} }}+1\Big)\,{\overline}K_0\,e_3 ,\\
{\displaystyle}{\overline}v_1=-\frac{k_r({\overline}P_0)}{\rho\,g}\,{\overline}K_0\,\Big(\nabla_{{\overline}x}{\overline}P_0+{ {\frac{\partial {\overline}P_1}{\partial {\overline}z } }}\,e_3\Big)-k_r'({\overline}P_0)\,{\overline}P_1\,\Big(\frac{1}{\rho\,g}{ {\frac{\partial {\overline}P_0}{\partial {\overline}z} }}+1\Big)\,{\overline}K_0\,e_3 .
\end{cases}$$ The same process in the three last equations of yields the following relations in $]0,{\overline}T[$:
- on ${{\overline}\Gamma_{\mathrm{soil}} }$ $$\label{boundary_gen_soil}
\begin{cases}
{\displaystyle}\alpha\,{\overline}P_0+\beta\, {\overline}v_0\cdot e_3 = {\overline}{ F }_0 , \qquad
{\displaystyle}\alpha\,{\overline}P_1+\beta\, \big( {\overline}v_1\cdot e_3-{\overline}v_0\cdot \nabla_x{ {\overline}h_{\mathrm{soil}} }\big) = {\overline}{ F }_1 ,\\
{\displaystyle}\alpha\,\left( {\overline}P_2+\frac12\|\nabla_x{ {\overline}h_{\mathrm{soil}} }\|^2 \,{\overline}P_0\right)+\beta\, \big({\overline}v_2\cdot e_3- {\overline}v_1\cdot \nabla_x{ {\overline}h_{\mathrm{soil}} }\big)= \frac12\|\nabla_x{ {\overline}h_{\mathrm{soil}} }\|^2\,{\overline}{ F }_0+{\overline}{ F }_2 ;
\end{cases}$$
- on ${ {\overline}\Gamma_{\mathrm{bot}} }$, for all $k\in {\mathbb{N}}^* $ $$\label{boundary_gen_bottom}
{\displaystyle}{\overline}v_0\cdot e_3 =0 ,\qquad
{\displaystyle}{\overline}v_{k-1}\cdot \nabla_{{\overline}x}{ {\overline}h_{\mathrm{bot}} }={\overline}v_{k}\cdot e_3 ;$$
- on ${ {\overline}\Gamma_{\mathrm{ver}} }$, for all $k\in {\mathbb{N}}$ $$\label{boundary_gen_ver}
{\overline}v_k\cdot {\overline}n = 0 .$$
#### Short time case.
We prove the first claim of Proposition \[prop\_richards\] which is associated with the short characteristic time scale $T={\varepsilon}^{-\gamma}{\overline}T$ for $\gamma=0$. The equation here reads $$\label{rescaled_short}
\phi\,{\displaystyle}{ {\frac{\partial s({\overline}P)}{\partial {\overline}t} }}+{\varepsilon}\operatorname{div}_{{\overline}x}({\overline}v) + { {\frac{\partial {\overline}v\cdot e_3}{\partial {\overline}z} }}=0 .$$ Some computations show that the main order terms in the latter equation give $$\phi\,{\displaystyle}{ {\frac{\partial s({\overline}P_0)}{\partial {\overline}t} }}+ { {\frac{\partial {\overline}v_0\cdot e_3}{\partial {\overline}z} }}=0\qquad\text{in } ]0,{\overline}T[ \times {\overline}\Omega .$$ The latter equation completed with the first equations of , and gives exactly the system . The first claim of Proposition \[prop\_richards\] is proven.
#### Intermediate time case.
In this part, we prove the second claim of Proposition \[prop\_richards\] which is associated with the intermediate time scale $T={\varepsilon}^{-\gamma}{\overline}T$ for $\gamma=1$. Equation of is now $$\label{rescaled_inter}
{\varepsilon}\phi\,{\displaystyle}{ {\frac{\partial s({\overline}P)}{\partial {\overline}t} }}+{\varepsilon}\operatorname{div}_{{\overline}x}({\overline}v) + { {\frac{\partial {\overline}v\cdot e_3}{\partial {\overline}z} }}=0 .$$ We introduce the asymptotic expansion in the previous equation and we identify the main order terms. We obtain $$\label{inter_order_0}
{\displaystyle}\ { {\frac{\partial {\overline}v_0\cdot e_3}{\partial {\overline}z} }}=0\qquad \text{on }]0,{\overline}T[\times{\overline}\Omega.$$ This constant vertical velocity is actually zero due to . Moreover, with the first equation of and since $k_r$ and $({\overline}K_0)_{33}$ are non-vanishing (${\overline}K_0$ is positive definite), we get in $]0,{\overline}T[{\times }{\overline}\Omega$ $$\label{velocity_0}
{ {\frac{\partial {\overline}P_0}{\partial {\overline}z} }}+\rho\,g=0 {\qquad\text{and}\qquad }{\overline}v_0=0.$$ The existence of ${\overline}H_0={\overline}H_0(t,x)$ such that $$\label{P0order0inter}
{\overline}P_0(t,x,z)=\rho\,g\big({\overline}H_0(t,x)-{\overline}z\big)\qquad\text{in } ]0,{\overline}T[{\times }{\overline}\Omega$$ follows. Next, since ${\overline}v_0=0$, the first equation of is $$\label{P0F0_order1_richards}
\alpha\,{\overline}P_0={\overline}F_0\qquad \text{on }{{\overline}\Gamma_{\mathrm{soil}} }.$$ We now have to differentiate the computations depending on whether $\alpha=0$ or not.
If $\alpha\neq0$, then for all $({\overline}t,{\overline}x)\in ]0,{\overline}T[{\times }{\overline}\Omega_x$ we have ${\overline}P_0({\overline}t,{\overline}x,{ {\overline}h_{\mathrm{soil}} }({\overline}t,{\overline}x))={\overline}F_0({\overline}t,{\overline}x)/\alpha$. Accordingly, thanks to , it holds $${\overline}H_0({\overline}t,{\overline}x)=\frac{{\overline}F_0({\overline}t,{\overline}x)}{\alpha\,\rho\,g}+{ {\overline}h_{\mathrm{soil}} }({\overline}t,{\overline}x).$$ This ends the proof of the second claim of Proposition \[prop\_richards\] in the case $\alpha\neq0$.
If $\alpha=0$ (then $\beta\neq0$), equation only implies that ${\overline}{ F }_0=0$. In particular, ${\overline}H_0$ remains as a degree of freedom and we have to exploit the next order terms in the asymptotic expansion for the closure of the effective problem. Identifying the coefficients associated with ${\varepsilon}^1$ in equation we have $$\label{inter_order_1}
{\displaystyle}\phi\,{ {\frac{\partial s({\overline}P_0)}{\partial {\overline}t} }}+ { {\frac{\partial {\overline}v_1\cdot e_3}{\partial {\overline}z} }}=0\qquad\text{in } ]0,{\overline}T[ \times {\overline}\Omega.$$ To eliminate ${\overline}v_1$, we integrate vertically on $]{ {\overline}h_{\mathrm{bot}} },{ {\overline}h_{\mathrm{soil}} }[$ the equation above. After using the fact that $\partial_t (s({\overline}P_0))=\rho\,g\,s'({\overline}P_0)\,\partial_t {\overline}H_0$ (consequence of ) we have $$\label{inter_order_11}
\rho\,g\Big(\int_{{ {\overline}h_{\mathrm{bot}} }}^{{ {\overline}h_{\mathrm{soil}} }} \phi\, s'({\overline}P_0)\,dz\Big){ {\frac{\partial {\overline}H_0}{\partial t} }} +({\overline}v_1|_{{ {\overline}h_{\mathrm{soil}} }}-{\overline}v_1|_{{ {\overline}h_{\mathrm{bot}} }})\cdot e_3 =0.$$ Thanks to the second equations of and in the case where $\alpha=0$ and ${\overline}v_0=0$, it follows: $${\overline}v_1\cdot e_3 = {\overline}{ F }_1/\beta\quad \text{on } {{\overline}\Gamma_{\mathrm{soil}} }{\qquad\text{and}\qquad }{\overline}v_1\cdot e_3= 0\quad \text{on } { {\overline}\Gamma_{\mathrm{bot}} }.$$ Accordingly, equation becomes $$\label{inter_order_12}
\rho\,g\Big(\int_{{ {\overline}h_{\mathrm{bot}} }}^{{ {\overline}h_{\mathrm{soil}} }} \phi\, s'({\overline}P_0)\,dz\Big){ {\frac{\partial {\overline}H_0}{\partial t} }} =-\frac{{\overline}F_1}{\beta}.$$ Finally, collecting equations and we get ${\overline}v_0=0$ and $$\label{inter_final}
\begin{cases}
{\displaystyle}{\overline}P_0({\overline}t,{\overline}x,{\overline}z)=\rho\,g\big({\overline}H_0({\overline}t,{\overline}x)-{\overline}z\big)&\text{in }]0,{\overline}T[\times{\overline}\Omega\\
{\displaystyle}\rho\,g\Big(\int_{{ {\overline}h_{\mathrm{bot}} }}^{{ {\overline}h_{\mathrm{soil}} }} \phi\, s'({\overline}P_0)\,dz\Big){ {\frac{\partial {\overline}H_0}{\partial t} }} =-\frac{{\overline}F_1}{\beta}
&\text{in }]0,{\overline}T[\times{\overline}\Omega_x
\end{cases}$$ which correspond to the second claim of Proposition \[prop\_richards\] in the case $\alpha=0$.
#### Long time case.
In this part, we prove the third claim of Proposition \[prop\_richards\] which is associated with the intermediate time scale $T={\varepsilon}^{-\gamma}{\overline}T$ for $\gamma=2$. Equation takes the form $$\label{rescaled_long}
{\varepsilon}^2\phi\,{\displaystyle}{ {\frac{\partial s({\overline}P)}{\partial {\overline}t} }}+{\varepsilon}\operatorname{div}_{{\overline}x}({\overline}v) + { {\frac{\partial {\overline}v\cdot e_3}{\partial {\overline}z} }}=0 .$$ We substitute the asymptotic expansion in the previous equation. The main order part of the equation is $\partial_z({\overline}v_0\cdot e_3)=0$ which leads, as before, to for some function ${\overline}H_0$ which does not depends on ${\overline}z$. The same relation holds and the characterization of ${\overline}H_0$ depends on the values of $\alpha$. As before, if $\alpha\neq0$ we have .
It remains to deal with the case $\alpha=0$ and to exhibit the equations of system . In this case, the compatibility condition $F_0=0$ holds as before because of . The characterization of ${\overline}H_0$ needs to go at the next order in the asymptotic expansion. In equation we get $$\label{long_order_1}
0=\operatorname{div}_{{\overline}x}({\overline}v_0) + { {\frac{\partial {\overline}v_1\cdot e_3}{\partial {\overline}z} }}={ {\frac{\partial {\overline}v_1\cdot e_3}{\partial {\overline}z} }}$$ where the second equality is due to ${\overline}v_0=0$. Moreover, the second equations of and for $k=1$ lead to (since $\alpha=0$) $$\label{long_order_1bis}
\beta\,{\overline}v_1\cdot e_3 = {\overline}{ F }_1\quad\text{on }{{\overline}\Gamma_{\mathrm{soil}} }{\qquad\text{and}\qquad }{\overline}v_1\cdot e_3= 0\quad\text{on }{ {\overline}\Gamma_{\mathrm{bot}} }.$$ Then, the vertical component of the velocity (which is constant by ) ${\overline}v_1\cdot e_3$ is zero. Moreover the second compatibility condition ${\overline}F_1=0$ holds true thanks to . Using the second equation of and bearing in mind that $(\rho\,g)^{-1}\partial_z {\overline}P_0+1=0$, we obtain $$\label{anisotropic_velocity}
{\overline}v_1=-\frac{k_r({\overline}P_0)}{\rho\,g}\,{\overline}K_0\,\Big(\nabla_{{\overline}x}{\overline}P_0+{ {\frac{\partial {\overline}P_1}{\partial {\overline}z} }}\,e_3\Big).$$ Since ${\overline}v_1\cdot e_3=0$, using the same notation for ${\overline}K_0$ than in , we compute $\partial_z {\overline}P_1$ by $${ {\frac{\partial {\overline}P_1}{\partial {\overline}z} }} = -\frac{1}{{\overline}K_{zz}}{\overline}K_0\nabla_{{\overline}x}{\overline}P_0\cdot e_3.$$ Finally, substitution in the equation above with the relation ${\overline}P_0=\rho\,g({\overline}H_0-z)$ give $$\label{long_order_1final}
{\overline}v_1=- k_r({\overline}P_0)\,{\overline}M_0\,\nabla_{{\overline}x}{\overline}H_0\quad\text{with}\quad {\overline}M_0=\begin{pmatrix}
I_2 &-\frac{{\overline}K_{xz}}{{\overline}K_{zz}}\\
0&0
\end{pmatrix}{\overline}K_0=\begin{pmatrix}
{\overline}S_0&0\\
0&0
\end{pmatrix}$$ where $I_2$ is the $2d$ identity matrix and ${\overline}S_0={\overline}K_{xx}-K_{zz}^{-1}{\overline}K_{xz}{\overline}K_{zx}$.
On the other hand, the equation for $k=1$ leads to ${\overline}v_1\cdot{\overline}n=0$ on ${ {\overline}\Gamma_{\mathrm{ver}} }$. Since $ k_r({\overline}P_0)$ does not vanish, we obtain the last equation of . After identifying the coefficients associated with ${\varepsilon}^2$ in equation we get $$\label{long_order_2}
{\displaystyle}\phi\,{ {\frac{\partial s({\overline}P_0)}{\partial {\overline}t} }}+\operatorname{div}_{{\overline}x}({\overline}v_1) + { {\frac{\partial {\overline}v_2\cdot e_3}{\partial {\overline}z} }}=0.$$ Taking into account , and the fact that $\alpha=F_0=0$, the third equation of and the second equations of for $k=2$ become $$\label{long_order_2boundary}
{\displaystyle}{\overline}v_2\cdot e_3-{\overline}v_1\cdot \nabla_{{\overline}x}{ {\overline}h_{\mathrm{soil}} }= {\overline}{ F }_2/\beta ,\qquad
{\displaystyle}{\overline}v_2\cdot e_3- {\overline}v_1\cdot \nabla_{{\overline}x}{ {\overline}h_{\mathrm{bot}} }= 0 \qquad \text{on }{ {\overline}\Gamma_{\mathrm{bot}} }.$$ To eliminate $v_2$ in system –, we integrate with respect to ${\overline}z$ on $[{ {\overline}h_{\mathrm{bot}} },{ {\overline}h_{\mathrm{soil}} }]$. Taking into account the boundary conditions on ${ {\overline}\Gamma_{\mathrm{bot}} }$ and ${{\overline}\Gamma_{\mathrm{soil}} }$ we obtain $${ {\frac{\partial }{\partial {\overline}t} }}\int_{{ {\overline}h_{\mathrm{bot}} }}^{{ {\overline}h_{\mathrm{soil}} }}\phi\, s({\overline}P_0) \,d{\overline}z+ \int_{{ {\overline}h_{\mathrm{bot}} }}^{{ {\overline}h_{\mathrm{soil}} }}\operatorname{div}_{{\overline}x} {\overline}v_1\,d{\overline}z+{\overline}v_1|_{{ {\overline}h_{\mathrm{soil}} }}\cdot \nabla_{{\overline}x}{ {\overline}h_{\mathrm{soil}} }+\frac{{\overline}{ F }_2}{\beta}-{\overline}v_1|_{{ {\overline}h_{\mathrm{bot}} }}\cdot \nabla_{{\overline}x}{ {\overline}h_{\mathrm{bot}} }=0.$$ We use the Leibniz rule in the second integral and we get $$\label{long_dupuit}
{ {\frac{\partial }{\partial {\overline}t} }}\int_{{ {\overline}h_{\mathrm{bot}} }}^{{ {\overline}h_{\mathrm{soil}} }}\phi\, s({\overline}P_0) \,d{\overline}z+ \operatorname{div}_{{\overline}x} \Big(\int_{{ {\overline}h_{\mathrm{bot}} }}^{{ {\overline}h_{\mathrm{soil}} }}{\overline}v_1\,d{\overline}z\Big) = -\frac{{\overline}F_2}{\beta}.$$ Using the averaged conductivity ${\overline}K$ defined in , we get, with the first equation of , $$\int_{{ {\overline}h_{\mathrm{bot}} }}^{{ {\overline}h_{\mathrm{soil}} }}{\overline}v_1\,d{\overline}z = -\int_{{ {\overline}h_{\mathrm{bot}} }}^{{ {\overline}h_{\mathrm{soil}} }}k_r({\overline}P_0)\, {\overline}M_0\,\nabla_{{\overline}x}{\overline}H_0=-{\overline}K({\overline}H_0)\,\nabla_{{\overline}x}{\overline}H_0.$$ The above equation associated with equation is exactly the system . This ends the proof of the last claim of Proposition .
Proof of Proposition \[prop\_richards\] for the coupled models
--------------------------------------------------------------
The strategy of the proof is exactly the same than in the previous subsection.
#### General relations.
Let $\gamma\in\{0,1,2\}$. Using the expansion –, we identify powers of ${\varepsilon}$ in all the equations in – that does not depend on the time scale $T$. We obtain from the second equation of $$\label{velocity_general1}
\begin{cases}
{\displaystyle}{\overline}u_0 = -k_r({\overline}P_0)\,\Big(\frac{1}{\rho\,g}{ {\frac{\partial {\overline}P_0}{\partial {\overline}z} }}+1\Big)\,{\overline}K_0\,e_3&\text{in }]0,{\overline}T[{\times }{\overline}\Omega ,\\
{\displaystyle}{\overline}u_1= -\frac{k_r({\overline}P_0)}{\rho\,g}\,{ {\frac{\partial {\overline}P_1}{\partial {\overline}z} }}\,{\overline}K_0\,e_3-k_r'({\overline}P_0)\,{\overline}P_1\,\left(\frac{1}{\rho\,g}{ {\frac{\partial {\overline}P_0}{\partial {\overline}z} }}+1\right)\,{\overline}K_0\,e_3 &\text{in }]0,{\overline}T[{\times }{\overline}\Omega ,
\end{cases}$$ from the third equation of $$\label{velocity_general2}
{\displaystyle}{\overline}w_0=0 ,\qquad
{\displaystyle}{\overline}w_1=-k_r\big(\rho\,g({\overline}H_0-{\overline}z)\big) {\overline}M_0\,\nabla_{{\overline}x} {\overline}H_0 \qquad \text{in }]0,{\overline}T[{\times }{\overline}\Omega ,$$ from the first equation of $$\label{velocity_global}
\begin{cases}
{\displaystyle}{\overline}v_0= {\overline}u_0+ {\overline}w_0= {\overline}u_0=-k_r({\overline}P_0)\,\Big(\frac{1}{\rho\,g}{ {\frac{\partial {\overline}P_0}{\partial {\overline}z} }}+1\Big)\,{\overline}K_0\,e_3&\text{in }]0,{\overline}T[{\times }{\overline}\Omega ,\\
{\displaystyle}{\overline}v_1= {\overline}u_1+ {\overline}w_1&\text{in }]0,{\overline}T[{\times }{\overline}\Omega .
\end{cases}$$ It follows from that for ${\overline}t\in]0,{\overline}T[$ and $({\overline}x,{\overline}z)\in\Omega^-_{{ {\overline}h}_0}({\overline}t)$ $$\label{pressure_general1}
{\displaystyle}{\overline}P_0({\overline}t,{\overline}x,{\overline}z)=\rho\,g\big( {\overline}H_0({\overline}t,{\overline}x)-{\overline}z \big), \quad
{\displaystyle}{\overline}P_k({\overline}t,{\overline}x,{\overline}z)=\rho\,g {\overline}H_k({\overline}t,{\overline}x) \ \forall k>0 .$$ Equation gives $$\label{h_general1}
{\displaystyle}{ {\overline}h_{\mathrm{bot}} }({\overline}x) \leq {\overline}{ h}_0({\overline}t,{\overline}x)\le\max\Big\{\min\Big\{ {\overline}H_0({\overline}t,{\overline}x)-\frac{{{\overline}P_{s}}}{\rho\,g},{ {\overline}h_{\mathrm{max}} }({\overline}x) \Big\},{ {\overline}h_{\mathrm{bot}} }({\overline}x) \Big\}\qquad\text{in }]0,{\overline}T[{\times }{\overline}\Omega_x .$$ For the boundary conditions, we infer from the second and third equations of and from the second equation of that, for all $k\in{\mathbb{N}}$, $$\label{boundary_general1}
\begin{cases}
{\displaystyle}\alpha\,{\overline}P_k+\beta\,{\overline}u_k\cdot e_3={\overline}F_k&\text{on } { ]0,{\overline}T[ }{\times }{{\overline}\Gamma_{\mathrm{soil}} },\\
{\displaystyle}{\overline}P_0\big({\overline}t,{\overline}x,{\overline}{ h}_0({\overline}t,{\overline}x)\big)=\rho\,g\big({\overline}H_0({\overline}t,{\overline}x)-{\overline}{ h}_0({\overline}t,{\overline}x) \big)&\text{for } {\overline}t\in]0,{\overline}T[,\quad {\overline}x\in { {\Gamma_{{ {\overline}h}}} }({\overline}t) ,\\
{\displaystyle}{\overline}K({\overline}H_0) \nabla_{{\overline}x}{\overline}H_0 \cdot{\overline}n=0&\text{on } { ]0,{\overline}T[ }{\times }{ {\overline}\Gamma_{\mathrm{ver}} }.
\end{cases}$$ By for $k=1$, $\partial_z {\overline}P_1=0$ on $\Omega^-_{{ {\overline}h}_0}({\overline}t)$. Then by and the first equation of $$\label{u1}
{\overline}u_1 = 0\qquad\text{in }\Omega^-_{{ {\overline}h}_0}({\overline}t).$$
#### Short time case.
In this part, $T={\overline}T$, that is $\gamma=0$. The first equations of and become $$\label{short_rescaled_coupled}
\begin{cases}
{\displaystyle}\phi { {\frac{\partial s({\overline}{P })}{\partial t} }}+{ {\frac{\partial }{\partial z} }}\big({\overline}{ {u }}\cdot e_3\big)=0\phantom{\int_\int} & \text{for \ }{\overline}t\in]0,{\overline}T[\ ,\quad ({\overline}x,{\overline}z)\in{ {\Omega_{{ {\overline}h}}^+ }}({\overline}t) ,\\
{\varepsilon}^2\operatorname{div}_x\big({\overline}K({\overline}H)\,\nabla {\overline}H\big) = ({\overline}u_0\cdot e_3)|_{{ {\Gamma_{{ {\overline}h}}} }} &\text{for \ }({\overline}t,{\overline}x)\in]0,{\overline}T[{\times }{\overline}\Omega_x .
\end{cases}$$ We identify the main order terms appearing when the asymptotics – are substituted in the previous equations: for ${\overline}t\in]0,{\overline}T[$ and $({\overline}x,{\overline}z)\in\Omega^+_{{\overline}{ h}_0}({\overline}t)$ $$\label{order_0_coupled_1}
\phi { {\frac{\partial s({\overline}P_0)}{\partial t} }}+{ {\frac{\partial }{\partial {\overline}z} }}\big({\overline}u_0\cdot e_3\big)=0,$$ $$\label{order_0_coupled_2}
({\overline}u_0 \cdot e_3)|_{\Gamma_{{\overline}{ h}_0}}=0\qquad\text{for \ }({\overline}t,{\overline}x)\in]0,{\overline}T[{\times }{\overline}\Omega_x.$$ From and we also compute ${\overline}u_0=0$ in $\Omega^-_{{\overline}{ h}_0}({\overline}t)$. In addition, from we get $s({\overline}P_0)=1$ in ${ {\Omega_{{ {\overline}h}}^- }}({\overline}t)$ so that $({\overline}P_0,{\overline}u_0)$ satisfies also in ${ {\Omega_{{ {\overline}h}}^- }}({\overline}t)$. The continuity of ${\overline}u_0\cdot e_3$ being ensured by , $({\overline}P_0,{\overline}u_0)$ satisfies in the whole $\Omega$. By using and we obtain the system and then the first claim of Proposition \[prop\_richards\] holds once again.
#### Intermediate time case.
In this part, $T={\varepsilon}^{-1}{\overline}T$, $\gamma=1$. The first equation of and the equation become $$\label{inter_rescaled_coupled}
\begin{cases}
{\displaystyle}\phi {\varepsilon}{ {\frac{\partial s({\overline}{P })}{\partial t} }}+{ {\frac{\partial }{\partial z} }}\big({\overline}{ {u }}\cdot e_3\big)=0\phantom{\int_\int} & \text{for \ }{\overline}t\in]0,{\overline}T[\ ,\quad ({\overline}x,{\overline}z)\in{ {\Omega_{{ {\overline}h}}^+ }}({\overline}t)\\
\begin{aligned}
&-{\varepsilon}^2\operatorname{div}_x\big({\overline}K({\overline}H)\,\nabla_x {\overline}H\big)
=-({\overline}u\cdot e_3)|_{{{\overline}\Gamma_{\mathrm{soil}} }}-{\varepsilon}{ {\frac{\partial }{\partial t} }}\Big(\int_{{ {\overline}h_{\mathrm{bot}} }(t,x)}^{{ {\overline}h_{\mathrm{soil}} }(x)} \phi\,s({\overline}{P }) \,dz \Big)
\end{aligned}&\text{for \ }({\overline}t,{\overline}x)\in]0,{\overline}T[{\times }{\overline}\Omega_x
\end{cases}$$ The corresponding main order relations are $$\label{ue3_soil}
{\overline}u_0\cdot e_3=0\qquad \text{on } ]0,{\overline}T[{\times }{{\overline}\Gamma_{\mathrm{soil}} }$$ and for ${\overline}t\in]0,{\overline}T[$ and $ ({\overline}x,{\overline}z)\in{ {\Omega_{{ {\overline}h}}^+ }}({\overline}t)$, $$\label{inter_order_0_coupled_1}
{ {\frac{\partial }{\partial z} }}\big({\overline}u_0\cdot e_3\big)=0 .$$ It follows that the constant vertical component of the velocity ${\overline}u_0\cdot e_3$ equals zero in ${ {\Omega_{{ {\overline}h}}^+ }}({\overline}t)$. We deduce from the first equation of that the pressure ${\overline}P_0$ is affine with respect to the $z$ variable with the slope $-\rho\,g$ in ${ {\Omega_{{ {\overline}h}}^+ }}({\overline}t)$. Accordingly, thanks to the first equation of and the continuity condition given in , the first equation of holds. Using relation we obtain the second equation of . Next, thanks to ${\overline}u_0=0$ and to the first equation of for $k=0$, we get $\alpha\,{\overline}P_0={\overline}F_0$.
If $\alpha\neq0$ then for all $(t,x)\in { ]0,T[ }{\times }{\overline}\Omega_x$ we have $P_0({\overline}t,{\overline}x,{ {\overline}h_{\mathrm{soil}} }({\overline}t,{\overline}x))={\overline}F_0({\overline}t,{\overline}x)/\alpha$. Accordingly, thanks to the first equation of , we have $${\overline}H_0({\overline}t,{\overline}x)=\frac{{\overline}F_0({\overline}t,{\overline}x)}{\alpha\,\rho\,g}+{ {\overline}h_{\mathrm{soil}} }({\overline}t,{\overline}x).$$ The second claim of Proposition \[prop\_richards\] in the case $\alpha\neq0$ is proved.
If $\alpha=0$, the compatibility condition ${\overline}F_0=0$ is imposed. After identifying the coefficients associated with ${\varepsilon}^1$ in the second equation of we have $$0=-({\overline}u_1\cdot e_3)|_{{{\overline}\Gamma_{\mathrm{soil}} }}-{ {\frac{\partial }{\partial t} }}\Big(\int_{{ {\overline}h_{\mathrm{bot}} }(x)}^{{ {\overline}h_{\mathrm{soil}} }(x)} \phi\,s({\overline}P_0) \,d{\overline}z \Big)$$ and, with the first equation of , $$\rho\,g\,\Big(\int_{{ {\overline}h_{\mathrm{bot}} }(x)}^{{ {\overline}h_{\mathrm{soil}} }(x)} \phi\,s'({\overline}P_0) \,d{\overline}z \Big){ {\frac{\partial {\overline}H_0}{\partial t} }}=-( {\overline}u_1\cdot e_3)|_{{{\overline}\Gamma_{\mathrm{soil}} }} .$$ The first equation of for $k=1$ implies, since $\alpha=0$, that $({\overline}u_1\cdot e_3)|_{{{\overline}\Gamma_{\mathrm{soil}} }}={\overline}F_1/\beta$. This ends the proof of the second claim of Proposition \[prop\_richards\] in the case $\alpha=0$.
#### Long time case.
In this part, $T={\varepsilon}^{-\gamma}{\overline}T$, $\gamma=2$. The first equation of and equation are now $$\label{long_rescaled_coupled}
\begin{cases}
{\displaystyle}\phi\, {\varepsilon}^2{ {\frac{\partial s({\overline}{P })}{\partial t} }}+{ {\frac{\partial }{\partial z} }}\big({\overline}{ {u }}\cdot e_3\big)=0 & \text{for \ }{\overline}t\in]0,{\overline}T[\ ,\quad ({\overline}x,{\overline}z)\in{ {\Omega_{{ {\overline}h}}^+ }}({\overline}t)\\
\begin{aligned}
&- {\varepsilon}^2\,\operatorname{div}_{{\overline}x}\Big({\overline}K({\overline}H)\,\nabla_{{\overline}x}{\overline}H \Big)
=-({\overline}u\cdot e_3)|_{{{\overline}\Gamma_{\mathrm{soil}} }}- {\varepsilon}^2{ {\frac{\partial }{\partial {\overline}t} }}\Big(\int_{{ {\overline}h_{\mathrm{bot}} }({\overline}x)}^{{\overline}{ h_{\mathrm{soil}} }({\overline}x)} \phi\,s({\overline}{P }) \,d{\overline}z \Big)
\end{aligned}&\text{for \ }({\overline}t,{\overline}x)\in]0,{\overline}T[{\times }{\overline}\Omega_x \\
\end{cases}$$ As in the intermediate time case, we substitute asymptotics – in the previous equations. Identifying the coefficients associated with ${\varepsilon}^n$ for $n\in\{1,2\}$, we get $\partial_z({\overline}u_n\cdot e_3)=0$ in ${ {\Omega_{{ {\overline}h}}^+ }}({\overline}t)$ and ${\overline}u_n\cdot e_3=0$ on ${{\overline}\Gamma_{\mathrm{soil}} }$. This leads to $$\label{u_10}
{\overline}u_0\cdot e_3={\overline}u_1\cdot e_3=0\qquad\text{on }{ {\Omega_{{ {\overline}h}}^+ }}({\overline}t).$$ By using the same arguments we obtain ${\overline}P_0 = \rho\,g({\overline}H_0-z)$ and ${\overline}v_0=0$ in whole $\Omega$. System is satisfied. The characterization of ${\overline}H_0$ depends on the values of $\alpha$. Similar arguments to those employed in the intermediate time case when $\alpha\neq0$ lead to .
It remains to deal with the case $\alpha=0$. In this case we first remark that the compatibility condition ${\overline}F_0=0$ holds (see for $k=0$). Furthermore, since ${\overline}P_0=\rho\,g({\overline}H_0+z)$ we get from and that ${\overline}u_1=0$ in $]0,{\overline}T[{\times }{\overline}\Omega$. Thus, using and we get ${\overline}v_1={\overline}w_1=-k_r({\overline}P_0) {\overline}M_0\,\nabla_x {\overline}H_0$. Moreover the first equation of for $k=1$ gives ${\overline}F_1=0$ (since $\alpha=0$). It remains to get the first relation of system . By plugging asymptotics – in the second equation of and by identifying the coefficients associated with ${\varepsilon}^2$ we get () $$\label{long_order_2_coupled}
-\operatorname{div}_x\big( {\overline}K({\overline}H_0)\,\nabla_x{\overline}H_0 \big) =- ({\overline}u_2\ cdot e_3)|_{{{\overline}\Gamma_{\mathrm{soil}} }}-{ {\frac{\partial }{\partial t} }}\Big(\int_{ { {\overline}h_{\mathrm{bot}} }(x)}^{{ {\overline}h_{\mathrm{soil}} }(x)} \phi\,s({\overline}P_0) \,dz \Big)
\qquad\text{for \ }({\overline}t,{\overline}x)\in]0,{\overline}T[{\times }{\overline}\Omega_x .$$ We end the proof by noting that, thanks to the equality $\alpha=0$ and the first equation of for $k=2$, we have $( {\overline}u_2\cdot e_3)|_{{{\overline}\Gamma_{\mathrm{soil}} }}={\overline}F_2/\beta$.
[10]{}
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[^1]: however the numerical simulations below show good results even for a [*ratio*]{} of order 0.1, which is not exceeded by the large majority of the unconfined aquifers.
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---
abstract: |
Consider an ephemeral sale-and-repurchase of a security resulting in the same position before the sale and after the repurchase. A sale-and-repurchase is a wash sale if these transactions result in a loss within $\pm 30$ calendar days. Since a portfolio is essentially the same after a wash sale, any tax advantage from such a loss is not allowed. That is, after a wash sale a portfolio is unchanged so any loss captured by the wash sale is deemed to be solely for tax advantage and not investment purposes.
This paper starts by exploring variations of the birthday problem to model wash sales. The birthday problem is: Determine the number of independent and identically distributed random variables required so there is a probability of at least $\frac{1}{2}$ that two or more of these random variables share the same outcome. This paper gives necessary conditions for wash sales based on variations on the birthday problem. Suitable variations of the birthday problem are new to this paper. This allows us to answer questions such as: What is the likelihood of a wash sale in an unmanaged portfolio where purchases and sales are independent, uniform, and random? Portfolios containing options may lead to wash sales resembling these characteristics. This paper ends by exploring the Littlewood-Offord problem as it relates capital gains and losses with wash sales.
author:
- 'Phillip G. Bradford[^1]'
title: '[**Foundations for Wash Sales**]{}'
---
Introduction {#Intro}
============
Wash sales impact a portfolio’s tax liabilities. Determining the likelihood of wash sales is also important for understanding investment strategies and for comparing actively and passively managed portfolios. Wash sales apply to investors, but not to market makers.
Taxes play a significant role in economics and finance. Taxes influence behavior, shape the engineering of financial transactions, and sometimes have unintended consequences. Therefore, thoughtful analysis is imperative for taxes. This paper adds firm mathematical foundations to aid the understanding of wash sale taxes.
The main goal of this paper is: To provide foundations for certain wash sales - in cases when they may occur as well as the capital gain implications. This may also help differentiate managed funds and unmanaged index funds in terms of wash sales.
Wash sales are sometimes created by the exercise of options, thus a portfolio manager may not be able to avoid a wash sale in some contexts. For example, suppose an in-the-money American-style put option is written in a portfolio. Provided this option remains in-the-money, it may be exercised by its holder[^2] at anytime up to its expiry. If the exercise of this put option replaces shares sold at a loss in the prior 30 days, then this is a wash sale. This option’s exercise is beyond control of the portfolio manager.
The foundations given here start with variations of the classical birthday problem from probability theory [@vM; @F; @CLRS]. This work has implications on wash sales. Also, the Littlewood-Offord problem [@LO; @E1945; @TV] is applied to understand capital gains for certain wash sales. The Littlewood-Offord problem is viewed from the perspective of the probabilistic method.
For convenience, let $[n] \equiv \{ \ 1, 2, \cdots, n \ \}$.
Suppose a security is sold at a loss on day $d_2$. This sale is a wash sale if substantially the same security is purchased within $\pm 30$ calendar days from $d_2$, see for example [@IRS550].
Consider three dates $d_1, d_2, d_3 : d_1 \leq d_2$ and $d_1 \leq d_3$ where $|d_3 - d_2| \leq 30$ calendar days.
Suppose $s$ shares of a security are purchased on date $d_1$ at price $p_1$.
At some later date $d_2$, $s$ shares are sold for price $p_2 < p_1$. Thus, the $s$ shares are sold at a loss. Then within $\pm 30$ days on date $d_3$, $s$ shares are repurchased for price $p_3$.
Since $|d_3 - d_2| \leq 30$ days, then the next adjustments must be made [@IRS550]:
1. The loss $p_1 - p_2$ is not permissible for taxes. That is, this loss may not be subtracted from profits or gains and it may not be used to get a lower tax rate.
2. The cost-basis of the shares repurchased on $d_3$ is set to $p_3 + (p_1 - p_2)$. The shares purchased on $d_3$ have the start of their holding period reset to $d_1$.
\[wash-sale-definition\]
Short positions may also be wash sales. For example, consider holding a short position of 100 shares of a security starting on date $d_1$ in a portfolio $\Pi$. Then suppose this short position is closed at a loss by purchasing 100 shares on day $d_2$. Once this position is closed on day $d_2$, then $\Pi$ contains no shares of this security. Next re-short another 100 shares of substantially the same security on $d_3$ where $|d_3 - d_2| \leq 30$ days. These transactions leave the portfolio the same while getting a tax advantage for the loss. This tax advantage is also disallowed by the wash sale rules.
Consider a wash sale as described by Definition \[wash-sale-definition\], where $(p_1 - p_2) + p_3 > p_1$ or in other words $p_3 > p_2$. Suppose the shares are sold at price $p_4 > (p_1 - p_2) + p_3 > p_1$ at the later date $d_4 \geq d_3$. In the case with the wash sale, there is a capital gain of $p_4 - \left[ (p_1 - p_2) + p_3 \right]$ which is smaller than the capital gain $p_4 - p_1$ if the wash sale had not occurred. Capital gains are taxable. A capital gain $p_4 - p_1$ is from the single purchase of the shares for price $p_1$ on $d_1$ and the single sale of the shares on date $d_4$ for price $p_4$, thus skipping the sale at a loss and repurchase.
This means such a wash sale gives $p_4 - p_1 - \left[ p_4 - \left[ (p_1 - p_2) + p_3 \right] \right]$ or $p_3 - p_2$ less taxable income than a single purchase of the security at price $p_1$ on date $d_1$ and a single sale for price $p_4$ on $d_4$. Of course, a wash sale’s loss is not allowed.
Wash sales may be avoided by restricting each security in a portfolio to be either purchased or sold only every 31 calendar days. This restriction may not be suitable for many portfolios. In a portfolio containing options, it may be impossible to maintain this restriction.
It has also been suggested, e.g. [@JM], wash sales may be avoided by purchasing or selling (moderately) correlated, but not substantially the same, securities. That is, if a security is sold at a loss then purchase a different but correlated security within 30 days maintaining some of a portfolio’s characteristics while keeping the tax advantage.
Historically many securities are assumed to only trade on about $n=252$ business days per year [@H]. Although reflecting on global markets one may assume there are $n=365$ trading days.
Background
----------
There has not been much research on wash sales, e.g., [@JM]. There is important work on taxation and its investment implications. Take, for example, [@C83; @C84] and [@DSZ].
The birthday problem is classical. According to a blog post by Pat B [@PWorld] the birthday problem may have originally been given by Harold Davenport as cited in [@BC] and later published by [@vM]. In any case, von Mises gave the first published version to the best of our knowledge.
Bounds of day counts for the birthday problems include [@N] who gives bounds for birthdays of distance $d$ for both linear years as well as cyclic years. In a cyclic year, 1-January is a single day from 31-December of the same year. Bounds for birthdays of distance $d$ for cyclic years are given by [@AM].
The birthday problem applied to boys and girls (random variables with different labels) are discussed in [@CN] as well as [@P]. That is, how many birthdays are shared by one or more boys and one or more girls? A comprehensive view is provided by [@NS] including stopping problems with the boy-girl birthday problem. Non-uniform bounds for online boy-girl birthday problems are given by [@GH] and [@S].
Tight bounded Poisson approximations for birthday problems are given by [@CDM]. Poisson approximations to the binomial distribution for the boy-girl birthday problem is given by [@P]. A Stein-Chen Poisson approximation is used by [@AGG] to solve variations of the standard birthday problem. Matching and birthday problems are given by [@DG]. Incidence variables are used to study birthday problems with Pareto-type distributions in [@BPSW].
Applications of the birthday problem include: computer security [@NS; @Stinson; @BPSW; @GH], public health and epidemiology [@SGMG], psychology, DNA sequence alignment, experiments, and games [@DM; @DG_m]. Summaries of work on the birthday problem are in [@DG_m], [@DM], and [@R].
Results on the expectation for getting $j$ different letter $k$-collisions are given by [@FGT]. Their results are expressed as truncated exponentials or gamma functions.
Structure of this Paper
-----------------------
Section \[bday\_problem\] reviews variants the birthday problem applied here. First the classical birthday problem is discussed. Next this section progresses through the $\pm d$ birthday problem. After the definition and key results are given about the $\pm d$ birthday problem, the boy-girl birthday problem is explored. Finally, the $\pm d$ boy-girl birthday problem is defined and several bounds are derived as they relate to a necessary condition for wash sales.
Subsection \[Wash-sale-example-1\] gives an example of wash sales based on boy-girl birthday collisions of a single day.
Section \[General-Wash-Sales\] generalizes results of the previous sections. In particular, it shows how to compute $B_d(n,b,g)$, the number of $b$ boys and $g$ girls that give a probability of $\frac{1}{2}$ or more where a boy and a girl have birthdays within $d$ days of each other over $n$ days.
Subsection \[Wash-sale-example-2\] gives an example of wash sales based on boy-girl birthday collisions over a range of $\pm d = 30$ days.
Finally, Section \[VarCGCL\] explores how wash sales impact capital gains and losses. Since wash sales are capital losses, they may offset capital gains. Several results, including the Littlewood-Offord problem, are applied to capital gains and losses as they may be impacted by wash sales.
The Birthday Problem and Wash Sales {#bday_problem}
===================================
The birthday problem is often applied to finding the probability of coincidences. So there is a rich literature on variations of the birthday problem [@DM; @DG_m]. Asset sales are often viewed as carefully selected. However, portfolios using American-style options may exhibit asset sales or purchases beyond the control of the portfolio managers.
[Given two random variables $X_1, X_2$ mapping respectively to $x_1, x_2$ in the same range $[n]$, then a [*birthday-collision*]{} is when $x_1 = x_2$. ]{}
To model random wash sales, this paper assumes independent identically distributed random variables. A common statement of the birthday problem is:
[Consider $n$ days in a year and $k$ independent identically distributed ([*iid*]{}) uniform random variables whose range is $[n]$ and $n \geq k$. What is the probability $B(n,k)$ of at least one birthday-collision among these $k$ random variables? ]{} \[standardBirthday\]
A key question is: Over $n$ consecutive days for what integer $k$ does $\displaystyle \operatorname*{arg\,min}_{k} \left\{ B(n,k) \geq \frac{1}{2} \right\}$ hold for $k$ iid uniform random variables? In other words, given $n$ days, what is the least $k$ iid uniform random variables so that $B(n,k) = \frac{1}{2}$ ?
Solutions to this basic variation of the birthday problem are well known. The probability $B(n,k)$ is the compliment of the probability of $k$ iid uniform random variables having no birthday-collisions. Therefore, if there are no birthday-collisions, then $k$ birthdays can be in ${n \choose k} \, k$! permutations out of all possible $n^k$ mappings of the $k$ random variables onto $[n]$. In other words, the ${n \choose k}$ subsets of $k$ distinct elements of $[n]$ is the exact number of subsets the $k$ variables may map to without a collision. These $k$ variables may be ordered in $k!$ permutations. That is,
$$\begin{aligned}
B(n,k) & = & 1-{n \choose k} \frac{k!}{n^k} \ \ = \ \ 1-\frac{n!}{(n-k)!}\cdot\frac{1}{n^k},\end{aligned}$$
for $n \geq k$ and $B(n,k) = 1$ otherwise.
Starting with $n$ and a probability $p = B(n,k)$, then computing $k$ is often done using the inequality $1 - x \leq e^{-x}$. In particular, the smallest $k$ giving a probability of $\frac{1}{2}$ that there is at least one birthday-collision requires $k$ to be roughly $\sqrt{2(\ln 2)n}$ or about $1.18\sqrt{n}$. See for example, [@vM; @MR95; @MU].
Another classical approach is to look at the random variable $X$ as the sum of all birthday-collisions of $k$ people over $n$ days, see for example [@P85; @BHS; @P; @DG]. A concise exposition is given in [@P85] which we follow. Presume the birthday day of person $i \in [k]$ is given by the random variable $Y_i \in [n]$. Since a potential birthday collision is a Bernoulli trial, so $X$ is binomially distributed. Thus, $X \in \{ \, 0, 1, 2, \cdots, {k \choose 2} \, \}$ where ${k \choose 2}$ is the maximum number of potential birthday-collisions. The expectation of the maximum number of birthday collisions possible is ${k \choose 2}$ with probability $\frac{1}{n} = {\mbox{\sf \hbox{I\kern-.15em P}}}[ Y_i =t | Y_j = t], \ \ t \in [n]$ where $\{ i,j \} \subseteq [k]$. The expected maximum number of birthday-collisions is $\frac{1}{n}{k \choose 2}$. If $n$ is sufficiently larger than $k$, then $X$ is approximately Poisson where $\lambda = \frac{1}{n}{k \choose 2}$. Thus, ${\mbox{\sf \hbox{I\kern-.15em P}}}[X \geq 1] \approx 1 - e^{-{k \choose 2}/n}$.
In the case of the $\pm d$ birthday problem, if two random variables $X_1, X_2$ map within $d$ days of each other, then this is a $\pm d$ birthday-collision [@N].
Two birthdays $x_1$ and $x_2$ of distance $|x_1 - x_2|$ demark a span of size $1 + |x_1 - x_2|$. For example, $|4\_{\mbox{\bf July}} - 3\_{\mbox{\bf July}}| = 1$, so these dates are in a $\pm d = \pm 2$ span, but not in a span of $\pm d = \pm 1$.
The next definition is based on [@N; @AM; @DM].
[Consider $n$ days in a year, spans of less than $\pm d$ days, and $k$ iid uniform random variables with range $[n]$: Then $B_d(n,k)$ is the probability at least two such random variables have a $\pm d$ [*birthday-collision*]{}. That is, these two random variables have ranges in [*less than*]{} $d$ days of each other. ]{}
In $n$ days with a $\pm d$ span, then $\displaystyle \operatorname*{arg\,min}_k \left\{ B_d(n,k) \geq \frac{1}{2} \right\}$ gives the smallest $k$ so there is a probability of at least $\frac{1}{2}$ where at least two such random variables are fewer than $d$ days from each other.
Let $i: k > i > 1$. Suppose birthdays are ordered as $x_1 \leq x_2 \leq \cdots \leq x_k$, then for a birthday $x_i$ its [*nearest birthday pairs*]{} are $(x_{i-1}, x_i)$ and $(x_i, x_{i+1})$. There are no birthdays between $x_{i-1}$ and $x_i$ and there are no birthdays between $x_i$ and $x_{i+1}$.
A [*block*]{} of days contains a single birthday on one of its end-points. The birthday $x_i$ is associated with two blocks: $(x_{i-1}, x_i]$ and $[x_i, x_{i+1})$.
The days between $x_1$ and $x_2$ form a block of size $|x_1 - x_2|$ since there are no birthdays between $x_1$ and $x_2$. Thus, two nearest birthday pairs contained in a span of $\pm d$ are separated by a block of size $d-1$.
Take $k$ iid uniform random variables and consider $\pm d$ birthday-collisions over $[n]$ days. Naus [@N] gives the next idea: If there are no $\pm d$ birthday-collisions, then there must be at least size $d-1$ blocks of no birthdays between each nearest birthday pair. This gives a total of $(k-1)(d-1)$ days with no birthdays in $k-1$ contiguous blocks of at least $d-1$ days each. Therefore, if there are no $\pm d$ birthday-collisions, then $k$ birthdays can be in ${n - (k-1)(d -1) \choose k} \, k!$ permutations out of all possible $n^k$ mappings of the $k$ random variables. Thus, to get the probability of at least one $\pm d$ birthday collision, take the compliment of the probability of having no $\pm d$ birthday-collisions. The next result follows.
[$$\begin{aligned}
B_d(n,k) & = & 1 - {n - (k-1)(d -1) \choose k} \ \frac{k!}{n^k} \ \ = \ \ 1 - \frac{(n - (k-1)(d -1))!}{(n - (k-1)(d -1) -k)!} \cdot \frac{1}{n^k},\end{aligned}$$ for $n \geq (k-1)(d-1) +k$ and $B_d(n,k) = 1$ otherwise. ]{} \[Naus\]
Using the bound $1 - x \leq e^{-x}$ on Naus’ result gives $k$ of about $0.83\sqrt{\frac{n}{d-4}}$, see [@N]. Also [@DM] approximate $k$ to about $1.2 \sqrt{\frac{n}{2d+1}}$ for the cyclic version.
Note, Theorem \[Naus\] with $d = 1$ gives the solution to the standard birthday problem of Definition \[standardBirthday\]. That is, a span of $d = 1$ and blocks of size $d - 1 = 0$.
The falling factorial is $$\begin{aligned}
m^{\underline{k}} & = & m(m-1) \cdots (m-k+1) \ \ = \ \ {m \choose k} k!\end{aligned}$$
In these terms, Theorem \[Naus\] may be expressed as $B_d(n,k) = 1 - \frac{(n - (k-1)(d -1))^{\underline{k}}}{n^k}$.
The next classic result is important.
[Let $m \geq k \geq 1$. The falling factorial $m^{\underline{k}}$ is the number of injective mappings of $k \geq 1$ elements to the range $[m]$. ]{} \[surjective\]
The next definition is based on [@CN; @NS; @CDM].
[Consider $n$ days in a year and two sets of distinctly labeled iid uniform random variables all with range $[n]$: $g$ of these variables are girls and $b$ of these variables are boys. Then $B(n,b,g)$ is the probability at least one girl and one boy have a birthday-collision. ]{}
For instance, in $n$ days, $\displaystyle \operatorname*{arg\,min}_{\substack{k = b+g\\b=g}} \left\{ B(n,b,g) \geq \frac{1}{2} \right\}$ gives the value $k = b+g$ and $b=g$ so there is a probability of $\frac{1}{2}$ where at least one girl and one boy have the same birthday.
Stirling numbers of the second kind [@GKP] count the number of non-empty partitions of a given set. For example given the set $[m]$, the number of partitions of $[m]$ into $i$ non-empty subsets is ${m \brace i}$.
Due to their nature, it is common to define Stirling numbers of the second kind recursively [@GKP]: ${m \brace i} = i {m-1 \brace i} + {m-1 \brace i-1}$ with the base cases ${m \brace 1} = 1$ and ${m \brace m} = 1$. Finally, ${m \brace m+i} = 0$ for any $i > 0$. As an example,
$$\begin{aligned}
\left\{ \begin{array}{c}
3\\
2
\end{array} \right\}
& = & \left| \Bigl\{ \bigl\{ \{ 1,2 \}, \{ 3 \} \bigr\}, \bigl\{ \{ 1,3 \}, \{ 2 \} \bigr\}, \bigl\{ \{ 1 \}, \{ 2, 3 \} \bigr\} \Bigr\} \right| \ \ = \ \ 3.\end{aligned}$$
The next classical equality counts the number of functions from $[n]$ elements to $[m]$ elements, $m \geq n$, $$\begin{aligned}
m^n & = & \displaystyle \sum_{i=1}^{n} \left\{ \begin{array}{c}
n\\
i
\end{array}
\right\} m^{\underline{i}} \label{EQ1}\end{aligned}$$
expressed as the number of non-empty $i$ partitions of the $[n]$ elements and the number of surjections from the $i$ partitions by Lemma \[surjective\].
[Consider $n$ days in a year and two sets of distinctly labeled iid uniform random variables all with range $[n]$: $g$ random variables are girls and $b$ random variables are boys. Then $B(n,b,g)$ is the probability at least one girl and at least one boy have a birthday-collision and $$\begin{aligned}
B( n, b, g ) & = & 1 - \frac{1}{n^{b+g}} \sum_{i=1}^{g} (n-i)^{b}
\left\{ \begin{array}{c}
g\\
i
\end{array} \right\}
n^{\underline{i}}.\end{aligned}$$ ]{} \[BG\]
The next Lemma is from [@CN; @W].
[Consider $n$ days in a year and two sets of distinctly labeled iid random variables all with range $[n]$: $g$ random variables are girls and $b$ random variables are boys. Then $B(n,b,g)$ is the probability that at least one girl and at least one boy have a birthday-collision and $$\begin{aligned}
B( n, b, g ) & = & 1 - \frac{1}{n^{b+g}} \sum_{i=1}^{g} \sum_{j=1}^{b}
\left\{ \begin{array}{c}
b\\
j
\end{array} \right\}
\left\{ \begin{array}{c}
g\\
i
\end{array} \right\}
n^{\underline{i+j}}.\end{aligned}$$ ]{}
Wash sale Example 1: Same Day Purchase and Sale {#Wash-sale-example-1}
-----------------------------------------------
Consider a portfolio $\Pi = \{ a_1, \cdots, a_k \}$ where $a_i: k \geq i \geq 1$ is asset (security) $i$ held in $\Pi$. At the end of business on day $\ell$, consider portfolio $\Pi_{\ell} = \{ a_{1,\ell}, \cdots, a_{k,\ell} \}$ the market value of asset $i$ in $\Pi_{\ell}$ is $|a_{i,\ell}|$ and the total value of $\Pi_{\ell}$ is $|\Pi_{\ell}| = \sum_{i=1}^{k} |a_{i,\ell}|$. Just before the start of each tax year, asset $i$ has market value $|a_{i,0}|$ and $\Pi$ has total market value $|\Pi_0|$. Assume each asset is sufficiently liquid so our purchases or sales do not impact its market price.
Suppose portfolio $\Pi$ has $T$ total iid uniform and random transactions during the business days of one calendar year. Assume trades are distributed on an asset-weighted basis from the initial weight of each asset in the portfolio just before the trading year commences. Thus, just prior to the first trading day and with no other information, asset $a_{i}$ is expected to have $t(i) = T \frac{|a_{i,0}|}{|\Pi_0|}$ trades in one year.
Take $t(i)$ transactions and define the independent Rademacher[^3] random variables $\eta_1, \cdots, \eta_{t(i)}$ representing buys or sells of portions of asset class $i$ in portfolio $\Pi$: $$\begin{aligned}
\eta_j & = & \left\{ \begin{array}{c}
+1 \ \ \mbox{ if transaction $j$ is a buy of asset } i \\
-1 \ \ \mbox{ if transaction $j$ is a sell of asset } i \\
\end{array} \right.\end{aligned}$$
for $j: t(i) \geq j \geq 1$. That is, the $b$ independent Rademacher random variables where $\eta_j = +1$ represent buys (boys) and the $g$ random variables where $\eta_j = -1$ represent sells (gals).
To apply a suitable version of Chernoff’s bound [@AS Appendix A] where ${\mbox{\sf \hbox{I\kern-.15em P}}}[\eta_j = +1] = {\mbox{\sf \hbox{I\kern-.15em P}}}[\eta_j = -1] = \frac{1}{2}$, then for any $c > 0$
$$\begin{aligned}
{\mbox{\sf \hbox{I\kern-.15em P}}}\left[ \left( \eta_1 + \cdots + \eta_{t(i)} \right) > c \right] & < & e^{-c^2/(2t(i))}.\end{aligned}$$
So, for example, take $c =1$, then $|b-g| \leq 1$ holds with high probability as $t(i)$ gets large. Of course, as $t(i)$ gets large, the likelihood of wash sales increases. That is, the total number of buys and sells is expected to converge to be about the same as the total number of transactions grows. However, along the way, the number of buys or sells may not be as balanced [@F; @NV].
Select the probabilities that the number of buys and sales are the same, given $t(i)$ total trades, in asset class $a_i$ are:
$t(i)$ 10 20 30 40 50
------------------ ------- ------- ------- ------- -------
$e^{-1/(2t(i))}$ 0.951 0.975 0.983 0.987 0.990
Let $h$ be half the total trades $t(i)$ in day $i$. That is, $h \leftarrow t(i)/2$. Assuming $n \in \{ 252, 365 \}$ trading days gives the probabilities of same-day girl-boy birthday collisions for a single asset-type as:
$h$ 1 5 10 15 20 25 30 35
------------------ -------- -------- -------- -------- -------- -------- -------- --------
$B( 252, h, h )$ 0.0040 0.0946 0.3280 0.5909 0.7957 0.9162 0.9717 0.9921
$B( 365, h, h )$ 0.0027 0.0663 0.2399 0.4605 0.6660 0.8196 0.9150 0.9650
In fact, $B( 252, 13, 13 ) = 0.4891$ and $B( 252, 14, 14 ) = 0.5410$. So, considering only equal numbers of sales and buys over $n=252$ days of the same asset type, 14 girls and 14 boys is the first case where there is greater than a 50% chance of a (same-day) boy-girl birthday collision.
Assuming the portfolio $\Pi$ already holds this single asset type, a boy-girl collision only is a necessary condition for a wash sale. A birthday collision must be accompanied by a sale at a loss and a repurchase of substantially the same security within 30 calendar days.
General Wash Sales {#General-Wash-Sales}
==================
Necessary conditions are given here for wash sales where a purchase and sale are within $\pm d$ calendar days. Since the purchase and sale are not known to be at a loss while keeping substantially the same portfolio before and after the $\pm d$ birthday collision.
[Consider $n$ days in a year, spans of $\pm d$ days, and two sets of distinctly labeled iid uniform random variables all with range $[n]$: $g$ random variables are girls and $b$ random variables are boys. Then $B_d(n,g,b)$ is the probability at least one girl and one boy are mapped to less than $d$ days of each other. ]{}
For example, starting with $n,d$ and $k=g+b$ and $g = b$, then $\displaystyle \operatorname*{arg\,min}_k \left\{ B_d(n,\textstyle\frac{k}{2}, \textstyle\frac{k}{2}) \geq \frac{1}{2} \right\}$ gives $k$ so there is a probability of $\geq \frac{1}{2}$ so at least one girl and one boy have $\pm d$-birthday collisions.
The next result is based on [@N], [@NS], [@CN], and [@W].
[Consider $n$ days in a year, a span of $\pm d$ days, and two sets of distinctly labeled iid uniform random variables all with range $[n]$: $g$ random variables are girls and $b$ random variables are boys. Then $B_d(n,g,b)$ is the probability at least one girl and one boy have a $\pm d$ birthday-collision and: $$\begin{aligned}
B_d(n,g,b) & = & 1 - \frac{1}{n^{b+g}} \sum_{i=1}^{b} \sum_{j=1}^{g} \left\{ \begin{array}{c}
b\\
i
\end{array} \right\}
\left\{ \begin{array}{c}
g\\
j
\end{array} \right\}
(n - (i+j-1)(d -1))^{\underline{i+j}}.\end{aligned}$$ ]{} \[BG\_Lower\]
This proof calculates the probability of not having no boy-girl $\pm d$ birthday collisions. That is, one minus the probability of no boy-girl $\pm d$ birthday collisions. This gives the probability of at least one boy-girl $\pm d$ birthday collision.
Given $n$ days, a $\pm d$ span, and iid uniform random variables separated into $g$ (girls) random variables and $b$ (boys) random variables. Then the total number unconstrained mappings of the $b$ and $g$ variables to $[n]$ is $n^{b+g}$ giving the denominator in front of the double sum.
The value $B_d(n,g,b)$ is not impacted if either any number of boys have the same birthday or separately any number of girls have the same birthday. Rather $B_d(n,g,b)$ is impacted by boy-girl collisions. Therefore, consider partitions of $b$ boys and $g$ girls. To prevent the girls’ partitions and boys’ partitions from colliding into $\pm d$ spans of the same range, count the number of places these $i$ and $j$ non-empty partitions may be mapped so there is no $\pm d > 1$ birthday-collision. By Lemma \[surjective\] there are $$\begin{aligned}
(n - (i+j-1)(d -1))^{\underline{i+j}}
& = & {n - (i+j-1)(d -1) \choose i+j} (i+j)!
\label{EQ7}\end{aligned}$$ injective functions to $[n]$ for sets of $i \in [b]$ boys and sets of $j \in [g]$ girls with $(i+j-1)$ blocks of $(d-1)$ contiguous days with no boy or girl in them.
Now, consider placing the $i$ and $j$ partitions in separate locations among the $(n - (i+j-1)(d -1))^{\underline{i+j}}$ function mappings to $[n]$. The $i$ partitions of $[b]$ where each partition is in a different location and $j$ partitions of $[g]$ where each partition is also in a different location by Equation \[EQ7\]. That is, given $i \in [b]$ and $j \in [g]$, then the product ${b \brace i}{g \brace j}$ is the total number of injective mappings of boys to $i$ non-empty partitions and independently the number of injective mappings of girls to $j$ non-empty partitions.
This completes the proof.
Wash sale Example 2: $d = \pm 30$ Calendar Days {#Wash-sale-example-2}
-----------------------------------------------
Start with the same setup as the previous wash sale example from subsection \[Wash-sale-example-1\].
Let $h$ be half the total trades $t(i)$ in day $i$. That is, $h \leftarrow t(i)/2$. Assuming $n \in \{ 252, 365 \}$ trading days and $d = \pm 30$ calendar days gives the probabilities of girl-boy $\pm 30$-day birthday-collisions for a single asset type is:
$h$ 1 2 3 4
----------------------- ------- ------- ------- --------- -- -- -- --
$B_{30}( 252, h, h )$ 0.220 0.819 0.994 0.99998
$B_{30}( 365, h, h )$ 0.155 0.667 0.953 0.99840
Consider only a single asset type. The intuition behind these probabilities is straight-forward. For instance, consider $n = 365$ days and to avoid boy-girl collisions each girl and boy must be separated by at least 30 days before and after their birthday from the other gender. So the $365$ days may be broken into about six blocks of about 60 days.
Wash Sale and Integral Capital Gains and Losses {#VarCGCL}
===============================================
Capital gains or capital losses may be rounded to the nearest integer for US tax calculations. Provided all trades are rounded. Rounding drops the cents portion for gains whose cents portion is 50-cents or below. Rounding adds a dollar to the dollar portion of gains whose cents portion is greater than 50 cents while dropping the cents portion. Losses work the same way. Gains and losses must all be rounded or none must be rounded. So, from here on, let all gains or losses be integers.
Long term capital gains and losses are aggregated and at the same time short term capital gains and losses are aggregated. At the end of the tax year the long term and short term aggregates are added together to get the final capital gain or loss for taxation.
The focus here is capital gains or losses for capital assets that may have wash sales. Wash sales are losses, but losses may offset gains. The study of options and their associated premiums is classical [@H] and we do not address it here. So, option premiums are ignored.
In a portfolio, individual capital gain values and individual capital loss values are usually distinct. Though rare, identical capital gains and capital losses are possible. Identical capital gains or losses are possible for portfolios built using options. We are ignoring option premiums. That is, asset purchases may be done via the exercise of cash-covered American-style put options. Also asset sales may be done via the exercise of American-style covered-call options. In these cases with options that become in-the-money, a portfolio manager has no control of the asset sales or purchases or timing of such trades. See Figure \[GainWashSale\].
Most often, put or call option strike prices are at discrete increments. For example, many put and call equity options have strike prices in \$5 or \$10 increments. Suppose a portfolio is built only using the exercise of American-style options. Many asset gains and losses may be for identical amounts. Of course, this depends on the size of the underlying positions or the number of options written. Options with the same expiry on identically sized underlying assets may have very different values [@H].
In such option-based portfolios assume uniform, independent, and random capital gains and capital losses. This may be modeled by the Littlewood-Offord Problem.
Definition \[LO\] is classical and extensive discussion may be found in the likes of [@TV2006; @TV]. It is based directly on [@LO; @TV2006; @TV]
[The integer Littlewood and Offord’s problem is given an integer multi-set $V = \{ v_1, v_2, \cdots, v_n \}$ where $v_i \geq 1, \ \forall i \in [n]$ and $S_v = \xi_1 v_1 + \xi_2 v_2 + \cdots + \xi_n v_n$ so each $\xi_i$ is such that ${\mbox{\sf \hbox{I\kern-.15em P}}}[\xi_i = -1] = {\mbox{\sf \hbox{I\kern-.15em P}}}[\xi_i = +1] = \frac{1}{2}$, for $i \in [n]$, then what is $\max_{x \in {{\sf\hbox{Z\kern-.40em Z}}}} {\mbox{\sf \hbox{I\kern-.15em P}}}\left[ S_v = x \right]$? ]{} \[LO\]
Assuming equal probability of gains and losses and no drift [@H]. Given an integer multi-set $V = \{ v_1, v_2, \cdots, v_n \}$ so $v_i \geq 1, \forall i \in [n]$. The multi-set $V$ represents capital gains and capital losses. Capital gains and capital losses are all from sales. The iid Rademacher random variables $\xi_i \in \{ +1, -1 \}$ determine if a $v_i$ is a capital gain or loss. All $v_i$ are positive since all the Rademacher variables have range $\{ -1, +1 \}$, see also [@E1945] and [@NV].
Over a tax year, the total capital gain or loss is $$\begin{aligned}
S_v & = & \xi_1 v_1 + \xi_2 v_2 + \cdots + \xi_n v_n.\end{aligned}$$
In an optimal solution of this version of the Littlewood-Offord problem, [@E1945] showed the $n$-element multi-set $V = \{ 1, 1, \cdots, 1 \}$ has $\max_{x \in {{\sf\hbox{Z\kern-.40em Z}}}} \left\{ {\mbox{\sf \hbox{I\kern-.15em P}}}\left[ S_v = x \right] \right\} = O\left( \frac{1}{\sqrt{n}} \right)$.
The next lemma’s proof follows immediately from the linearity of expectation given Rademacher random variables. See, for example, [@AS].
[Consider any integer multi-set $V = \{ v_1, v_2, \cdots, v_n \}$ where $v_i \geq 1, \forall i \in [n]$ and the random variable $S_v = \xi_1 v_1 + \xi_2 v_2 + \cdots + \xi_n v_n$, where ${\mbox{\sf \hbox{I\kern-.15em P}}}[\xi_i = -1] = {\mbox{\sf \hbox{I\kern-.15em P}}}[\xi_i = +1] = \frac{1}{2}$, for all $i \in [n]$, then ${\mbox{\sf {I\kern-.15em E}}}[S_v] = 0$. ]{} \[Mean0\]
For any Rademacher random variable $\xi_i$, it must be ${\mbox{\sf {I\kern-.15em E}}}[\xi_i] = 0$ and ${\mbox{\sf {I\kern-.15em E}}}[\xi_i^2] = 1$. Since $v_i$ is constant $\sigma^2_{\xi_i v_i} = {\mbox{\sf {I\kern-.15em E}}}[\xi_i^2 v_i^2] - {\mbox{\sf {I\kern-.15em E}}}[\xi_i v_i]^2 = v_i^2$. Thus, a proof of the next theorem follows since the variance of a sum of independent random variables is the sum of the variances.
[Consider any non-negative integer vector $v$ and the random variable $S_v = \xi_1 v_1 + \xi_2 v_2 + \cdots + \xi_n v_n$, where ${\mbox{\sf \hbox{I\kern-.15em P}}}[\xi_i = -1] = {\mbox{\sf \hbox{I\kern-.15em P}}}[\xi_i = +1] = \frac{1}{2}$, for all $i \in [n]$, then ${\mbox{\sf {I\kern-.15em E}}}[S_v^2] = v_1^2 + v_2^2 + \cdots + v_n^2$ and $\sigma_{S_v} = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$. ]{} \[SumSquareTheorem\]
Thus, the lowest variance, $\sigma_{S_v}^2$, for the integer Littlewood-Offord problem occurs exactly when $V = \{ 1, 1, \cdots, 1 \}$ and $|V| = n$. Assuming the $\xi_i, \ \forall i \in [n]$ are all Rademacher random variables, then ${\mbox{\sf \hbox{I\kern-.15em P}}}_{x \in {{\sf\hbox{Z\kern-.40em Z}}}} [S_v = x]$ is maximized [@TV2006; @TV; @NV] as $O(1/\sqrt{n})$ and $\sigma_{S_v} = \sqrt{n}$.
Theorem \[SumSquareTheorem\] implies the next corollary.
[Assume $1 = v_1 = v_2 = \cdots = v_n$ and $S_v = \xi_1 v_1 + \xi_2 v_2 + \cdots + \xi_n v_n$ where ${\mbox{\sf \hbox{I\kern-.15em P}}}[\xi_i = -1] = {\mbox{\sf \hbox{I\kern-.15em P}}}[\xi_i = +1] = \frac{1}{2}$, for all $i \in [n]$, then the standard deviation of $S_v$ is $\sigma_{S_v} = \sqrt{n}$. ]{} \[linearCorollary\]
Corollary \[linearCorollary\] highlights an exceptional case where all capital gains and capital losses are the same. Wash sales require the loss and gain to be from essentially the same security.
The generality of Theorem \[SumSquareTheorem\] asserts large variances too. Consider the set $V = \{ 2^0, 2^1, \cdots, 2^{n-1} \}$, then by Theorem \[SumSquareTheorem\], $\sigma_{S_v}^2 = \sum_{i=0}^{n-1} 2^{2i} = \frac{2^{2n} -1}{3}$. This last equality follows since the sum is a geometric series.
Consider a set or multi-set $V = \{ v_1, v_2, \cdots, v_n \}$ and let each element of the lists $H_1 = \langle \hat{\xi}_{1,1}, \hat{\xi}_{2,1}, \cdots, \hat{\xi}_{n,1} \rangle$ and $H_2 = \langle \hat{\xi}_{1,2}, \hat{\xi}_{2,2}, \cdots, \hat{\xi}_{n,2} \rangle$ be fixed values from $\{ -1, +1 \}$. The two sums of $V$,
$$\begin{aligned}
s_{v,1} & = & \hat{\xi}_{1,1} \, v_1 + \hat{\xi}_{2,1} \, v_2 + \cdots + \hat{\xi}_{n,1} \, v_n\\
s_{v,2} & = & \hat{\xi}_{1,2} \, v_1 + \hat{\xi}_{2,2} \, v_2 + \cdots + \hat{\xi}_{n,2} \, v_n,\end{aligned}$$
are [*distinct*]{} iff there is some $\hat{\xi}_{i,1} \neq \hat{\xi}_{i,2}$, for $i \in [n]$.
Given any multi-set of positive integers $V = \{ v_1, v_2, \cdots, v_n \}$, enumerate all $2^n$ distinct sums as $s_v[1] \geq s_v[2] \geq \cdots \geq s_v[2^n]$, for example, see Figure \[Sv-binomial\]. Given any set of positive integers $V = \{ v_1, v_2, \cdots, v_n \}$, where none of the $2^n$ distinct sums add to the same value gives $s_v[1] > s_v[2] > \cdots > s_v[2^n]$.
An important observation by [@E1945], is that for any fixed sum $s$ the values $s + v_{i}$ and $s - v_{i}$ differ by $2 v_{i}$. Next, this observation is used to show the set $V = \{ 2^0, 2^1, \cdots, 2^{n-1} \}$ has no distinct sums that add to the same value.
In particular, take any distinct sums $s_{v,1}$ and $s_{v,2}$ with associated fixed values $\hat{\xi}_{i,1} \in \{ -1, +1 \}$ and $\hat{\xi}_{i,2} \in \{ -1, +1 \}$, respectively, for all $i \in [n]$. Suppose, for the sake of a contradiction, that $s_{v,1} = s_{v,2}$. Building on Erdős’ observation, the values $s_{v,1}$ and $s_{v,2}$ may be written as $s_{v,1} = 2^{n} - 1 - 2 \, m_1$ where $m_1 = \sum_{i \in I_1} 2^{i-1}$ and $I_1 = \{ \, i : \hat{\xi}_{i,1} = -1 \, \}$ and likewise $s_{v,2} = 2^{n} - 1 - 2 \, m_2$ where $m_2 = \sum_{i \in I_2} 2^{i-1}$ and $I_2 = \{ \, i : \hat{\xi}_{i,2} = -1 \, \}$, for all $i \in [n]$. Finally, the uniqueness of binary-number representations means $m_1 = m_2$ which in turn means $\hat{\xi}_{i,1} = \hat{\xi}_{i,2}$, for all $i \in [n]$. So, in fact, the sums $s_{v,1}$ and $s_{v,2}$ are equal, giving a contradiction.
Thus, the set $V = \{ 2^0, 2^1, \cdots, 2^{n-1} \}$ satisfies the antecedent of the next theorem.
[Among all sets of distinct positive integers where no two distinct sums add to the same value, the set $V = \{ 2^0, 2^1, \cdots, 2^{n-1} \}$ has a minimal sum $s_v = v_1 + v_2 + \cdots + v_n = 2^{n} -1$. ]{} \[Distinct-Theorem\]
Suppose, for the sake of a contradiction, that $s_v = v_1 + v_2 + \cdots + v_n < 2^n -1$ for some set of distinct positive integers $V = \{ v_1, v_2, \cdots, v_n \}$ where no two distinct sums add to the same value.
Take the next enumeration of the $2^n$ distinct sums, $s_v[1] > s_v[2] > \cdots > s_v[2^n]$, and by our supposition, $2^n -2 \geq s_v[1]$ and $s_v[2^n] \geq -2^n +2$, so $s_v[1] - s_v[2^n] \leq 2^{n+1} - 4$.
Let $\{ \, s_{v,1}, s_{v,2} \} \subseteq \{ \, s_v[1], s_v[2], \cdots, s_v[2^n] \, \}$ where sum $s_{v,1}$ has the list of fixed values $H = \langle \hat{\xi}_{1,1}, \hat{\xi}_{2,1}, \cdots, \hat{\xi}_{n,1} \rangle$ so that $s_{v,1} = \langle v_1, v_2, \cdots, v_n \rangle \cdot H$, where $\cdot$ is the vector dot product. Likewise, the sum $s_{v,2}$ has the list of fixed values $\langle \hat{\xi}_{1,2}, \hat{\xi}_{2,2}, \cdots, \hat{\xi}_{n,2} \rangle$.
The difference of any two distinct sums $s_{v,1} - s_{v,2}$ must be even since any fixed values $\hat{\xi}_{i,1} \in \{ -1, +1 \}$ and $\hat{\xi}_{i,2} \in \{ -1, +1 \}$, for $i \in [n]$, are so that, $$\begin{aligned}
\hat{\xi}_{i,1} - \hat{\xi}_{i,2} \in \{ \ 0, -2, +2 \ \},\end{aligned}$$ giving $$\begin{aligned}
s_{v,1} - s_{v,2} & = & \sum_{i=1}^{n} v_i \left( \hat{\xi}_{i,1} - \hat{\xi}_{i,2} \right)\end{aligned}$$ which must be even.
Starting from $s_v[1]$ and going to $s_v[2^n]$ contains $2^n-1$ intervals. Since all $s_v[i]$, for $i \in [2^n]$, are different and their differences must be even so $s_v[1] - s_v[2^n]$ spans at least $2 (2^{n} -1) = 2^{n+1} -2$. That is, $s_v[1] - s_v[2^n] \geq 2^{n+1} -2$. This gives a contradiction of the assumption $s_v[1] - s_v[2^n] \leq 2^{n+1} - 4$, completing the proof.
Given a set of distinct positive integers $V$ where $|V| = n$, Theorem \[Distinct-Theorem\] indicates that $\max_{x \in {{\sf\hbox{Z\kern-.40em Z}}}} \left\{ {\mbox{\sf \hbox{I\kern-.15em P}}}[S_v = x] \right\} \leq \frac{1}{2^{n}}$. So in the case where all distinct sums of $V$ add to different values, erasing a wash sale loss may have a very large impact. In particular, the multi-set $V = \{ 1, 1, \cdots, 1 \}$ has largest loss $s_v[2^n] = -n$, where Theorem \[Distinct-Theorem\] indicates $V = \{ 2^0, 2^1, \cdots, 2^{n-1} \}$ has the largest loss $s_v[2^n] = -2^n +1$. In this case, when no distinct sums add to the same value, let $U = \{ 2^n - (2i -1) : i \in [2^{n}] \}$ giving $\max_{x \in U} \left\{ {\mbox{\sf \hbox{I\kern-.15em P}}}[S_v = x] \right\} = \frac{1}{2^{n}}$. Assuming wash sales occur with the same random and uniform probability among all losses, the expected disallowed loss is $\frac{2^n -1}{n}$. This is because all losses are of the form $-(2^{i-1})$, for $i \in [n+1]$, and by assumption these losses all have the same probability of occurring.
Since ${\mbox{\sf {I\kern-.15em E}}}[S_v] = 0$ by Lemma \[Mean0\], Littlewood-Offord results are useful for understanding likely values for $S_v$. That is, $\max_{x \in {{\sf\hbox{Z\kern-.40em Z}}}- \{ 0 \}} \left\{ {\mbox{\sf \hbox{I\kern-.15em P}}}[S_v = x] \right\}$ gives most likely capital gains or losses outside of the expected value ${\mbox{\sf {I\kern-.15em E}}}[S_v] = 0$. None of the $s_v$ values in Figure \[Sv-binomial\] are $0$, but if $V$ has an even number of $1$s, then the most common value is $0$.
The following tail bound is given by [@SJMS] where $\| v_1, v_2, \cdots, v_n \|_2 = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$,
$$\begin{aligned}
{\mbox{\sf \hbox{I\kern-.15em P}}}\left[ \sum_{i=1}^{n} \xi_i v_i > t \| v_1, v_2, \cdots, v_n \|_2 \right] & \leq & e^{-t^2/2}\\
{\mbox{\sf \hbox{I\kern-.15em P}}}\left[ S_v > t \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2} \right] & \leq & e^{-t^2/2}\\
{\mbox{\sf \hbox{I\kern-.15em P}}}\left[ S_v > t \sigma_{S_v} \right] & \leq & e^{-t^2/2}\end{aligned}$$
Since by Theorem \[SumSquareTheorem\], $\sigma_{S_v} = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$.
Suppose $V = \{ 1, 1, \cdots, 1 \}$ and $|V|$ is odd. Since no sum of $V$ is $0$, there are $\frac{n}{2}$ capital gains and $\frac{n}{2}$ capital losses. This means if $S_v = t \sigma$, then there are $\frac{n}{2} + \frac{t \sigma}{2}$ capital gains and $\frac{n}{2} - \frac{t \sigma}{2}$ capital losses. Losses are necessary for wash sales. Therefore, the bound ${\mbox{\sf \hbox{I\kern-.15em P}}}\left[ S_v > t \sigma_{S_v} \right] \leq e^{-t^2/2}$ gives the probability there are at least $\frac{t \sigma_{S_v}}{2}$ more gains than losses. That is, there are $\frac{t \sigma_{S_v}}{2}$ fewer opportunities for wash sales.
$V =\{ \, 1,1,1 \, \}$ $s_v$
------------------------ ---------------
$+1+1+1$ $s_v[1] = 3$
$-1+1+1$ $s_v[2] = 1$
$+1-1+1$ $s_v[3] = 1$
$+1+1-1$ $s_v[4] = 1$
$-1-1+1$ $s_v[5] = -1$
$-1+1-1$ $s_v[6] = -1$
$+1-1-1$ $s_v[7] = -1$
$-1-1-1$ $s_v[8] = -3$
Following Figure \[Sv-binomial\], given $|V| = n$ then $s_v[1] = n$ is the case with zero capital losses. Likewise, $s_v[2^n] = -n$ is the case with zero capital gains. By Lemma \[Mean0\], since ${\mbox{\sf {I\kern-.15em E}}}[S_v] = 0$ and $s_v[1] + \cdots + s_v[2^n] = 0$, thus $-n = s_v[2] + \cdots + s_v[2^n]$. Also suppose a single wash sale disallows a capital loss among all identical capital gains and losses. The single wash sale disallows a single capital loss giving the expected capital gain or loss: $$\begin{aligned}
\frac{(s_v[2] + 1) + (s_v[3] + 1) + \cdots + (s_v[2^n] +1)}{2^{n}-1}.\end{aligned}$$ The term $s_v[1]$ is excluded since it has no losses, hence no wash sales.
The boy-girl $\pm 30$ birthday problem gives a necessary condition for wash sales of substantially identical securities. Recall $B_{30}(252,g,b)$ is the probability of at least one boy-girl $\pm 30$ birthday collision, so $1- B_{30}(252,g,b)$ is the probability of no such birthday collision.
Given any number of boy-girl $\pm 30$ birthday collisions of the same security and suppose these birthday collisions produce at most a single wash sale. In this case let $G$ be a total taxable gain or loss where all gains and losses are the same. Suppose these gains and losses are all $1$. This gives,
$$\begin{aligned}
{\mbox{\sf {I\kern-.15em E}}}[ \, G \, ] & = & \left(1 - B_{30}(252,g,b)\right) \frac{s_v[1] + s_v[2] + \cdots + s_v[2^n]}{2^n} \\
& & \ + \ B_{30}(252,g,b) \frac{(s_v[2] + 1) + \cdots + (s_v[2^n] +1)}{2^n-1} \\
& = & \frac{1 - B_{30}(252,g,b)}{2^n-1}(0) + \frac{B_{30}(252,g,b)}{2^n-1}\left( 2^n-1 -n \right)\\
& = & B_{30}(252,g,b)\left( 1 - \frac{n}{2^n -1} \right).\end{aligned}$$
Conclusions and further directions
==================================
Wash sales may be modeled in a number of ways. These include variations of the birthday problem and the capital gains of portfolios and wash sales impact may be modeled using the Littlewood-Offord problem.
The $k$-armed bandit, see for example [@Robbins] or [@KV], etc., appears to apply to wash sales and the birthday problem. Robbins’ discussion of maximizing expected value of sums of random variables selected from different distributions is applicable to constructing portfolios by writing options.
Acknowledgement
===============
Thanks to Noga Alon for insightful comments.
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[^1]: University of Connecticut at Stamford, Department of Computer Science and Engineering, Stamford Connecticut. [email protected], [email protected]
[^2]: Options, like shares of stock, are fungible and there are specific option exercise assignment allocation methods used to allocate exercised options [@FINRA].
[^3]: We used Bernoulli random variables for $\{ 0,1 \}$ outcomes and we use Rademacher for $\{-1, +1 \}$ outcomes.
|
---
abstract: 'We investigate a new structure for machine learning classifiers applied to problems in high-energy physics by expanding the inputs to include not only measured features but also physics parameters. The physics parameters represent a smoothly varying learning task, and the resulting parameterized classifier can smoothly interpolate between them and replace sets of classifiers trained at individual values. This simplifies the training process and gives improved performance at intermediate values, even for complex problems requiring deep learning. Applications include tools parameterized in terms of theoretical model parameters, such as the mass of a particle, which allow for a single network to provide improved discrimination across a range of masses. This concept is simple to implement and allows for optimized interpolatable results.'
author:
- Pierre Baldi
- Kyle Cranmer
- Taylor Faucett
- Peter Sadowski
- Daniel Whiteson
bibliography:
- 'paramnn.bib'
title: 'Parameterized Machine Learning for High-Energy Physics'
---
Introduction
============
Neural networks have been applied to a wide variety of problems in high-energy physics [@Denby:1987rk; @Peterson:1993nk], from event classification [@Abreu:1992jp; @Kolanoski:1995zn] to object reconstruction [@Peterson:1988gs; @Aad:2014yva] and triggering [@Lonnblad:1990bi; @Denby:1990wb]. Typically, however, these networks are applied to solve a specific isolated problem, even when this problem is part of a set of closely related problems. An illustrative example is the signal-background classification problem for a particle with a range of possible masses. The classification tasks at different masses are related, but distinct. Current approaches require the training of a set of isolated networks [@Aaltonen:2012qt; @Chatrchyan:2012tx], each of which are ignorant of the larger context and lack the ability to smoothly interpolate, or the use of a single signal sample in training [@Aad:2014xea; @Chatrchyan:2012yca], sacrificing performance at other values.
In this paper, we describe the application of the ideas in Ref. [@cranmer2015] to a new neural network strategy, a [*parameterized neural network*]{} in which a single network tackles the full set of related tasks. This is done by simply extending the list of input features to include not only the traditional set of event-level features but also one or more [*parameters*]{} that describe the larger scope of the problem such as a new particle’s mass. The approach can be applied to any classification algorithm; however, neural networks provide a smooth interpolation, while tree-based methods may not.
A single parameterized network can replace a set of individual networks trained for specific cases, as well as smoothly interpolate to cases where it has not been trained. In the case of a search for a hypothetical new particle, this greatly simplifies the task – by requiring only one network – as well as making the results more powerful – by allowing them to be interpolated between specific values. In addition, they may outperform isolated networks by generalizing from the full parameter-dependent dataset.
In the following, we describe the network structure needed to apply a single parameterized network to a set of smoothly related problems and demonstrate the application for theoretical model parameters (such as new particle masses) in a set of examples of increasing complexity.
Network Structure & Training
============================
A typical network takes as input a vector of features, $\bar{x}$, where the features are based on event-level quantities. After training, the resulting network is then a function of these features, $f(\bar{x})$. In the case that the task at hand is part of a larger context, described by one or more parameters, $\bar{\theta}$. It is straightforward to construct a network that uses both sets of inputs, $\bar{x}$ and $\bar{\theta}$, and operates as a function of both: $f(\bar{x},\bar{\theta})$. For a given set of inputs $\bar{x}_0$, a traditional network evaluates to a real number $f(\bar{x}_0)$. A parameterized network, however, provides a result that is parameterized in terms of $\bar{\theta}$: $f(\bar{x}_0,\bar{\theta})$, yielding different output values for different choices of the parameters $\bar{\theta}$; see Fig. \[fig:networks\].
Training data for the parameterized network has the form $(\bar{x}, \bar{\theta}, y)_i$, where $y$ is a label for the target class. The addition of $\bar\theta$ introduces additional considerations in the training procedure. While traditionally the training only requires the conditional distribution of $\bar{x}$ given $\bar{\theta}$ (which is predicted by the theory and detector simulation), now the training data has some implicit prior distribution over $\bar\theta$ as well (which is arbitrary). When the network is used in practice it will be to predict $y$ conditional on both $\bar{x}$ and $\bar{\theta}$, so the distribution of $\bar\theta$ used for training is only relevant in how it affects the quality of the resulting parameterized network – it does not imply that the resulting inference is Bayesian. In the studies presented below, we simply use equal sized samples for a few discrete values of $\bar{\theta}$. Another issue is that some or all of the components of $\bar{\theta}$ may not be meaningful for a particular target class. For instance, the mass of a new particle is not meaningful for the background training examples. In what follows, we randomly assign values to those components of $\bar\theta$ according to the same distribution used for the signal class. In the examples studied below the networks have enough generalization capacity and the training sets are large enough that the resulting parameterized classifier performs well without any tuning of the training procedure. However, the robustness of the resulting parameterized classifier to the implicit distribution of $\bar\theta$ in the training sample will in general depend on the generalization capacity of the classifier, the number of training examples, the physics encoded in the distributions $p(\bar{x} | \bar{\theta}, y)$, and how much those distributions change with $\bar\theta$.
![ Left, individual networks with input features $(x_1,x_2)$, each trained with examples with a single value of some parameter $\theta=\theta_a,\theta_b$. The individual networks are purely functions of the input features. Performance for intermediate values of $\theta$ is not optimal nor does it necessarily vary smoothly between the networks. Right, a single network trained with input features $(x_1,x_2)$ as well as an input parameter $\theta$; such a network is trained with examples at several values of the parameter $\theta$. []{data-label="fig:networks"}](networks.pdf){width="40.00000%"}
Toy Example
===========
As a demonstration for a simple toy problem, we construct a parameterized network, which has a single input feature $x$ and a single parameter $\theta$. The network is trained using labeled examples where examples with label 0 are drawn from a uniform background and examples with label 1 are drawn from a Gaussian with mean $\theta$ and width $\sigma=0.25$. Training samples are generated with $\theta=-2,-1,0,1,2$; see Fig. \[fig:toy\]a.
As shown in Fig. \[fig:toy\], this network generalizes the solution and provides reasonable output [*even for values of the parameter where it was given no examples*]{}. Note that the response function has the same shape for these values ($\theta=-1.5,-0.5,0.5,1.5$) as for values where training data was provided, indicating that the network has successfully parameterized the solution. The signal-background classification accuracy is as good for values where training data exist as it is for values where training data does not.
![Top, training samples in which the signal is drawn from a Gaussian and the background is uniform. Bottom, neural network response as a function of the value of the input feature $x$, for various choices of the input parameter $\theta$; note that the single parameterized network has seen no training examples for $\theta=-1.5,-0.5,0.5,1.5$.[]{data-label="fig:toy"}](histogram_gaussian.pdf "fig:"){width="35.00000%"} ![Top, training samples in which the signal is drawn from a Gaussian and the background is uniform. Bottom, neural network response as a function of the value of the input feature $x$, for various choices of the input parameter $\theta$; note that the single parameterized network has seen no training examples for $\theta=-1.5,-0.5,0.5,1.5$.[]{data-label="fig:toy"}](parameterized_output_plot.pdf "fig:"){width="35.00000%"}
1D Physical Example
===================
A natural physical case is the application to the search for new particle of unknown mass. As an example, we consider the search for a new particle $X$ which decays to $t\bar{t}$. We treat the most powerful decay mode, in which $t\bar{t}\rightarrow W^+bW^-\bar{b}\rightarrow qq'b\ell\nu \bar{b}$. The dominant background is standard model $t\bar{t}$ production, which is identical in final state but distinct in kinematics due to the lack of an intermediate resonance. Figure \[fig:diag\] shows diagrams for both the signal and background processes.
![ Feynman diagrams showing the production and decay of the hypothetical particle $X\rightarrow t\bar{t}$, as well as the dominant standard model background process of top quark pair production. In both cases, the $t\bar{t}$ pair decay to a single charged lepton ($\ell$), a neutrino ($\nu$) and several quarks ($q,b$).[]{data-label="fig:diag"}](xttbar.pdf "fig:"){width="20.00000%"} ![ Feynman diagrams showing the production and decay of the hypothetical particle $X\rightarrow t\bar{t}$, as well as the dominant standard model background process of top quark pair production. In both cases, the $t\bar{t}$ pair decay to a single charged lepton ($\ell$), a neutrino ($\nu$) and several quarks ($q,b$).[]{data-label="fig:diag"}](smttbar.pdf "fig:"){width="20.00000%"}
We first explore the performance in a one-dimensional case. The single event-level feature of the network is $m_{WWbb}$, the reconstructed resonance mass, calculated using standard techniques identical to those described in Ref. [@Aad:2013dza]. Specifically, we assume resolved top quarks in each case, for simplicity. Events are are simulated at parton level with [madgraph]{}5 [@Alwall:2011uj], using [pythia]{} [@Sjostrand:2006za] for showering and hadronization and [delphes]{} [@deFavereau:2013fsa] with the ATLAS-style configuration for detector simulation. Figure \[fig:1dperf\]a shows the distribution of reconstructed masses for the background process as well as several values of $m_X$, the mass of the hypothetical $X$ particle. Clearly the nature of the discrimination problem is distinct at each mass, though similar to those at other masses.
In a typical application of neural networks, one might consider various options:
- Train a single neural network at one intermediate value of the mass and use it for all other mass values as was done in Refs. [@Aad:2014xea; @Chatrchyan:2012yca]. This approach gives the best performance at the mass used in the training sample, but performance degrades at other masses.
- Train a single neural network using an unlabeled mixture of signal samples and use it for all other mass values. This approach may reduce the loss in performance away from the single mass value used in the previous approach, but it also degrades the performance near that mass point, as the signal is smeared.
- Train a set of neural networks for a set of mass values as done in Refs. [@Aaltonen:2012qt; @Chatrchyan:2012tx]. This approach gives the best signal-background classification performance at each of the trained mass values, degrades for mass values away from the ones used in training, and leads to discontinuous performance as one switches between networks.
In contrast, we train a single neural network with an additional parameter, the true mass, as an input feature. For a learning task with $n$ event-level features and $m$ parameters, one can trivially reconcieve this as a learning task with $n+m$ features. Evaluating the network requires supplying the set of event-level features as well as the desired values of the parameters.
These neural networks are implemented using the multi-layer perceptron in PyLearn2, with outputs treated with a regressor method and logistic activation function. Input and output data are subject to preprocessing via a scikit-learn pipeline (i.e. MinMaxScaler transformation to inputs/outputs with a minimum and maximum of zero and one, respectively). Each neural network is trained with 3 hidden layers and using Nesterov’s method for stochastic gradient descent. Learning rates were initiated at 0.01, learning momentum was set to 0.9, and minibatch size is set to treat each point individually (i.e. minibatch size of 1). The training samples have approximately 100k examples per mass point.
The critical test is the signal-background classification performance. To measure the ability of the network to perform well at interpolated values of the parameter – values at which it has seen no training data – we compare the performance of a single fixed network trained at a specific value of $m_{X}^0$ to a parameterized network trained at the other available values other than $m_{X}^0$. For example, Fig. \[fig:1dperf\] compares a single network trained at $m_{X}^0=750$ GeV to a parameterized network trained with data at $m_{X}=500,1000,1250,1500$ GeV. The parameterized network’s input parameter is set to the true value of the mass ($m_X^0$, and it is applied to data generated at that mass; recall that it saw no examples at this value of $m_X^0$ in training. Its performance matches that of the single network trained at that value, validating the ability of the single parameterized network to interpolate between mass values without any appreciable loss of performance.
![ Top, distributions of neural network input $m_{WWbb}$ for the background and two signal cases. Bottom, ROC curves for individual fixed networks as well as the parameterized network evaluated at the true mass, but trained only at other masses. []{data-label="fig:1dperf"}](signal_background_histogram.pdf "fig:"){width="40.00000%"} ![ Top, distributions of neural network input $m_{WWbb}$ for the background and two signal cases. Bottom, ROC curves for individual fixed networks as well as the parameterized network evaluated at the true mass, but trained only at other masses. []{data-label="fig:1dperf"}](parameterized_vs_fixed_ROC_plot.pdf "fig:"){width="40.00000%"}
High-dimensional Physical Example
=================================
The preceding examples serve to demonstrate the concept in one-dimensional cases where the variation of the output on both the parameters and features can be easily visualized. In this section, we demonstrate that the parameterization of the problem and the interpolation power that it provides can be achieved also in high-dimensional cases.
We consider the same hypothetical signal and background process as above, but now expand the set of features to include both low-level kinematic features which correspond to the result of standard reconstruction algorithms, and high-level features, which benefit from the application of physics domain knowledge. The low-level features are roughly the four-vectors of the reconstructed events, namely:
- the leading lepton momenta,
- the momenta of the four leading jets,
- the $b$-tagging information for each jets
- the missing transverse momentum magnitude and angle
for a total of 21 low-level features; see Fig \[fig:llvar\]. The high-level features strictly combine the low-level information to form approximate values of the invariant masses of the intermediate objects. These are:
- the mass ($m_{\ell\nu}$) of the $W\rightarrow\ell\nu$,
- the mass ($m_{jj}$) of the $W\rightarrow qq'$,
- the mass ($m_{jjj}$) of the $t\rightarrow Wb\rightarrow bqq'$,
- the mass ($m_{j\ell\nu}$) of the $t\rightarrow Wb\rightarrow\ell\nu b$,
- the mass ($m_{WWbb}$) of the hypothetical $X\rightarrow t\bar{t}$,
for a total of five high-level features; see Fig \[fig:hlvar\].
![ Distributions of some of the low-level event features for the decay of $X\rightarrow t\bar{t}$ with two choices of $m_X$ as well as the dominant background process.[]{data-label="fig:llvar"}](var_xtt_l_pt.pdf "fig:"){width="20.00000%"} ![ Distributions of some of the low-level event features for the decay of $X\rightarrow t\bar{t}$ with two choices of $m_X$ as well as the dominant background process.[]{data-label="fig:llvar"}](var_xtt_j1_pt.pdf "fig:"){width="20.00000%"} ![ Distributions of some of the low-level event features for the decay of $X\rightarrow t\bar{t}$ with two choices of $m_X$ as well as the dominant background process.[]{data-label="fig:llvar"}](var_xtt_j2_pt.pdf "fig:"){width="20.00000%"} ![ Distributions of some of the low-level event features for the decay of $X\rightarrow t\bar{t}$ with two choices of $m_X$ as well as the dominant background process.[]{data-label="fig:llvar"}](var_xtt_met.pdf "fig:"){width="20.00000%"}
![Distributions of high-level event features for the decay of $X\rightarrow t\bar{t}$ with two choices of $m_X$ as well as the dominant background process; see text for definitions.[]{data-label="fig:hlvar"}](var_xtt_mjlv.pdf "fig:"){width="20.00000%"} ![Distributions of high-level event features for the decay of $X\rightarrow t\bar{t}$ with two choices of $m_X$ as well as the dominant background process; see text for definitions.[]{data-label="fig:hlvar"}](var_xtt_mjj.pdf "fig:"){width="20.00000%"} ![Distributions of high-level event features for the decay of $X\rightarrow t\bar{t}$ with two choices of $m_X$ as well as the dominant background process; see text for definitions.[]{data-label="fig:hlvar"}](var_xtt_mjjj.pdf "fig:"){width="20.00000%"} ![Distributions of high-level event features for the decay of $X\rightarrow t\bar{t}$ with two choices of $m_X$ as well as the dominant background process; see text for definitions.[]{data-label="fig:hlvar"}](var_xtt_mwwbb.pdf "fig:"){width="20.00000%"}
The parameterized deep neural network models were trained on GPUs using the Blocks framework [@van_merrienboer_blocks_2015; @bastien_theano:_2012; @bergstra_theano:_2010]. Seven million examples were used for training and one million were used for testing, with 50% background and 50% signal. The architectures contain five hidden layers of 500 hidden rectified linear units with a logistic output unit. Parameters were initialized from $\mathcal{N}(0,0.1)$, and updated using stochastic gradient descent with mini-batches of size 100 and 0.5 momentum. The learning rate was initialized to 0.1 and decayed by a factor of 0.89 every epoch. Training was stopped after 200 epochs.
The high dimensionality of this problem makes it difficult to visually explore the dependence of the neural network output on the parameter $m_{X}$. However, we can test the performance in signal-background classification tasks. We use three types of networks. A single parameterized network is trained using 7M training samples with masses $m_X=500,750,1000, 1250, 1500$ GeV and tested in a sample generated with $m_X=1000$ GeV; the performance is compared to a single fixed network trained with samples at $m_X=1000$ (with 7M training examples). In each case, we use approximately the same number of training and testing examples per mass point. Fig \[fig:dnn\_roc\] shows that the parameterized network matches the performance of the fixed network. A more stringent follow-up test removes the $m_X=1000$ sample from the training set of the parameterized network, so that this network is required to interpolate its solution. The performance is unchanged, demonstrating that the parameterized network is capable of generalizing the solution even in a high-dimensional example.
![ Comparison of the signal-to-background discrimination for four classes of networks for a testing sample with $m_X=1000$ GeV. A parameterized network trained on a set of masses ($m_X=500,750,1000,1250, 1500$) is compared to a single network trained at $m_X=1000$ GeV. The performance is equivalent. A second parameterized network is trained only with samples at $m_x=500,750,1250, 1500$, forcing it to interpolate the solution at $m_X =1000$ GeV. Lastly, a single non-parameterized network trained with all the mass points shows a reduced performance. The results are indistinguishable for cases where the networks use only low-level features (shown) or low-level as well as high-level features (not shown, but identical).[]{data-label="fig:dnn_roc"}](ROC_4nets.pdf){width="40.00000%"}
Conversely, Fig \[fig:vmass\] compares the performance of the parameterized network to a single network trained at $m_X=1000$ GeV when applied across the mass range of interest, which is a common application case. This demonstrates the loss of performance incurred by traditional approaches and recovered in this approach. Similarly, we see that a single network trained an unlabeled mixture of signal samples from all masses has reduced performance at each mass value tested.
In previous work, we have shown that deep networks such as these do not require the additional of high-level features [@Baldi:2014kfa; @Baldi:2014pta] but are capable of learning the necessary functions directly from the low-level four-vectors. Here we extend that by repeating the study above without the use of the high-level features; see Fig \[fig:dnn\_roc\]. Using only the low-level features, the parameterized deep network is achieves essentially indistinguishable performance for this particular problem and training sets of this size.
![ Comparison of the performance in the signal-background discrimination for the parameterized network, which learns the entire problem as a function of mass, and a single network trained only at $m_X=1000$ GeV. As expected, the AUC score decreases for the single network as the mass deviates from the value in the training sample. The parameterized network shows improvement over this performance; the trend of improving AUC versus mass reflects the increasing separation between the signal and background samples with mass, see Figs. \[fig:llvar\] and \[fig:hlvar\]. For comparison, also shown in the performance a single network trained with an unlabeled mixture of signal samples at all masses.[]{data-label="fig:vmass"}](nn_vmass.pdf){width="40.00000%"}
Discussion
==========
We have presented a novel structure for neural networks that allows for a simplified and more powerful solution to a common use case in high-energy physics and demonstrated improved performance in a set of examples with increasing dimensionality for the input feature space. While these example use a single parameter $\theta$, the technique is easily applied to higher dimensional parameter spaces.
Parameterized networks can also provide optimized performance as a function of nuisance parameters that describe systematic uncertainties, where typical networks are optimal only for a single specific value used during training. This allows statistical procedures that make use of profile likelihood ratio tests [@Cowan:2010js] to select the network corresponding to the profiled values of the nuisance parameters [@cranmer2015].
Datasets used in this paper containing millions of simulated collisions can be found in the UCI Machine Learning Repository [@Bache+Lichman:2013] at [archive.ics.uci.edu/ml/datasets/HEPMASS](archive.ics.uci.edu/ml/datasets/HEPMASS).
Acknowledgements
----------------
We thank Tobias Golling, Daniel Guest, Kevin Lannon, Juan Rojo, Gilles Louppe, and Chase Shimmin for useful discussions. KC is supported by the US National Science Foundation grants PHY-0955626, PHY-1205376, and ACI-1450310. KC is grateful to UC-Irvine for their hospitality while this research was initiated and the Moore and Sloan foundations for their generous support of the data science environment at NYU. We thank Yuzo Kanomata for computing support. We also wish to acknowledge a hardware grant from NVIDIA, NSF grant IIS-1550705, and a Google Faculty Research award to PB.
|
---
address: |
Lawrence Berkeley National Laboratory\
Berkeley, CA, 94720, USA\
E-mail: [email protected]
author:
- 'Spencer R. Klein'
title: 'The LPM Effect: Comparing SLAC E-146 Data with Experiment'
---
=cmr8
1.5pt
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
[LBNL - 42069]{} .01 in [August, 1998]{} .2 in
Presented at the *Fourth Workshop on Quantum Chromodynamics*
June 1-6, 1998, Paris, France.
Introduction
============
Originally, the LPM effect referred to the suppression of electron bremsstrahlung or pair production due to multiple scattering. More recently, it has been applied to the effects of the nuclear medium on quark and gluon interactions. This subject is of interest because calculations predict that quark or gluons should radiate increased energy in traversing a quark gluon plasma, compared to normal nuclear matter.
This talk will consider the electrodynamic version of the LPM effect. I will discuss recent experimental results from SLAC E-146, and compare these results with theory. This might seem like an odd choice for a conference devoted to quantum chromodynamics. However, electrodynamics can be an important point of comparison for the chromodynamics calculations. For electrodynamics, it is possible to identify a variety of different kinematic regimes, with significantly different photon spectral indices. And, the electrodynamics calculations can be tested experimentally, over a broad range of target thicknesses.
Suppression Mechanisms
======================
The LPM effect stems from the formation length, the distance over which an interaction such as pair production or bremsstrahlung occurs. For bremsstrahlung from an isolated atom, this distance is $$l_f = {2\hbar E(E-k) \over m^2 k}$$ where $E$ is the incoming electron energy, $k$ is the photon energy and $m$ the electron mass. This distance can be very long; for a 25 GeV electron emitting a 1 MeV photon, $l_f= 1$ mm! Classically, if the electron is disturbed while traversing this distance, then the emission can be disturbed. In field theoretical language, other interactions within $l_f$ can interfere with the bremsstrahlung, reducing its amplitude. In dense media, the Bethe-Heitler $1/k$ bremsstrahlung photon spectrum is suppressed by a factor $S$ [@LP]: $$S= {\sigma\over\sigma_{BH}} = \sqrt{kE_{LPM} \over E(E-k)}$$ where $E_{LPM}= m^4 X_0/E_s^2$ where $X_0$ is the radiation length of the material, and $E_s=m\sqrt{4\pi/\alpha}$.
In 1956, Migdal used the Boltzman transport equation to model multiple scattering, calculating the emission for each path [@Migdal]. He used a simple model for the potential, and found results good to logarithmic accuracy.
One limitation of Migdals result is that neglected surface effects, which are important for targets of finite thickness. This is important for QCD, where the target size is limited to a nuclear diameter. Gol’dman [@goldman] and Ternovskii [@ternovskii] extended Migdals calculation to include finite thickness targets. In the limit $T\ll l_f$, the Bethe-Heitler $1/k$ spectrum is recovered, albeit at a reduced intensity, proportional to $\ln{T}$. Since E-146, there have been several new calculations [@newwork]; since several of the authors are speaking here, I will not further discuss them further.
One interesting aspect of the QED is that it allows for a variety of different suppression mechanisms. In dielectric suppression[@dprl], the produced photon interacts with the electrons in the medium. This bulk interaction, mediated by forward Compton scattering, is best expressed in terms of the dielectric constant of the medium, $\epsilon(k) = 1 - (\hbar\omega_p)^2/k^2$, where $\omega_p$ is the plasma frequency. This interaction gives the photon an effective mass $\omega_pc^2$, which shortens the formation length, and produces a suppression that scales as $k^2$: $$S = { k^2 \over k^2 + (\gamma\hbar\omega_p)^2 }.
\label{sdiel}$$ where $\gamma=E/m$. A similar effect can occur when a radiated gluon undergoes further interactions. Although these interactions are included in current calculations, the specific effects of these diagrams have not been considered separately.
Another mechanism occurs when the $l_f$ is longer than the radiation length. Then, the nascent photon can interact before it is fully created. This limits $l_f$ to $X_0$, suppressing photon emission[@myreview]. Similarly, bremsstrahlung can also suppress pair production when a produced leptons radiates a photon in the formation zone. For electrodynamics, this effect only occurs at extremely high energies. For QCD however, the interaction length inside a nucleus can be smaller than $l_f$, so multiple interactions in a single formation length are likely. This higher order effect is not considered in current calculations. Unfortunately, this ’correction’ is likely to be very large, and numerical predictions of quark $dE/dx$ in nuclear media should be used with great caution. This problem will greatly complicate the interpretation of energy loss measurements planned for RHIC.
Suppression can also occur when bremsstrahlung or pair production occurs in an external magnetic field [@myreview]. In the absence of a bulk color magnetic field, this effect is unlikely to be important in QCD. These different suppression mechanisms are summarized in Table 1.
Region Dominant Mechanism Photon Spectrum Importance in QCD
----------------- --------------------- ----------------- -------------------
none - $k^{-1}$ ?
LPM Multiple Scattering $k^{-1/2}$ yes
Pair Production Pair Production $k^0$ very
Dielectric Compton Scattering $k$ ?
Magnetic Magnetic Field $k^{-1/3}$ no
: Bremsstrahlung photon spectral indices.
E-146 Data and Analysis
=======================
The SLAC E-146 collaboration has studied LPM [@prl] [@prd] and dielectric suppression [@dprl] by observing 200 keV to 500 MeV photons produced by 8 and 25 GeV electrons passing through a variety of targets. For most materials, two different thickness targets were studied. Since the experiment is well described elsewhere, here I will focus on the data and its implications for theory. The photon flux is histogrammed in logarithmic bins in $k$, $1/X_0 dN/d\log{k}$. This binning allowed the histograms to cover many orders of magnitude in $k$. It also flattened out the $1/k$ Bethe-Heitler spectrum.
Figures 1-3 show the E-146 data for carbon, uranium and thin gold targets. These targets cover a wide range in density, and also in $l_f/T$. Figure 1 compares the carbon ($T\gg l_f$) data with predictions based on Bethe-Heitler, LPM suppression only, and LPM plus dielectric suppression; both mechanisms are clearly required to match the data. However, the 25 GeV data shows a significant disagreement in the region $k\approx E^2/E_{LPM}$. Figure 2 shows uranium data, compared with curves based on LPM plus dielectric suppression. The data and theory agree for $k>10$ MeV, but the data rises above the theory for $k<10$ MeV. The difference can be attributed to the small thickness of the targets. For these target thicknesses, when $k< 10$ MeV, $T\approx l_f$, so surface effects are likely to be important. Several early calculations of the surface effects, shown in the figure, fail to reproduce the data. Newer calculations do appear to reproduce the surface terms, but are not easily comparable with data because they do not localize the emission, and hence cannot be easily included in a simulation.
Figure 3 shows the E-146 data for electrons passing through a 0.7 (23$\mu$m) thick gold target. Here, $l_f>T$ for $k< 7$ MeV. In this regime, the target interacts as a coherent whole, and the Bethe-Heitler $1/k$ spectrum is recovered, albeit at a reduced intensity[@ternovskii]. This flattening is also predicted by newer calculations, such as those by Blankenbecler and Drell. In this case, the target is thin enough that multiple interactions by a single electron are unlikely, and so the calculation can be directly compared with the data.
The systematic error for these measurements are small, ranging from 3.3% at the higher photon energies, up to 17% for $k< 5$ MeV in a 25 GeV beam. The systematic errors are smaller than the discrepancy between the carbon data and theory.
Conclusions
===========
I have discussed several mechanisms which can suppress electron bremsstrahlung. Different mechanisms are important in different kinematic regions. At current accelerators, for electrodynamics, suppression due to multiple scattering is the most important, followed by dielectric suppression.
At much higher electron energies, pair production can suppress bremsstrahlung in the regime $l_f > X_0$. In this regime, in fact, the concept of individual interactions in an electromagnetic shower break down, leaving an (so far) unsolvable complex Feynman diagram containing multiple steps in a shower. For colored interactions in dense media, $X_0$ is much smaller, and these higher order diagrams are likely to be very important, even at near-future colliders.
The accuracy of the electrodynamics calculations is shown by data from SLAC E-146. The data generally matches the theory to within 10%. The one exception to this is with the light (low $Z$) targets, where the data is somewhat below the theory around $k=E(E-k)/E_{LPM}$. The reason for this is unknown, but may stem from an inadequate treatment of atomic effects.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank my E-146 colleagues for their support. This work was supported by the U.S. D.O.E. under contract DE-AC03-76SF00098.
References {#references .unnumbered}
==========
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P. Anthony, , Phys. Rev. Lett. [**76**]{}, 3550 (1996).
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|
---
author:
- |
James M. Cline, Luis Valcárcel\
Physics Department, McGill University, 3600 University Street, Montréal, Québec, Canada H3A 2T8\
,
title: |
Asymmetrically warped compactifications\
and gravitational Lorentz violation
---
Introduction
============
The violation of Lorentz symmetry at high energies would be a revolutionary discovery, giving us important clues about the nature of physics beyond the standard model [@Coleman]-[@Kostelecky]. It has been argued that such violations are predicted by some theories of quantum gravity [@Smolin]. Although string theory does not necessarily predict Lorentz violation, it can do so via background fields which lead to noncommutative geometry [@Carroll], or which explicitly violate Lorentz invariance in the gravitational sector [@Frey].
Nevertheless, it is not easy to invent fundamental theories which break Lorentz symmetry in a plausible way. Therefore it is useful to consider new possibilities for breaking Lorentz symmetry. A novel approach was presented by Csaki [*et al.*]{} [@Csaki], in the context of the Randall-Sundrum (RS) solution to the hierarchy problem [@RS], involving two branes separated by an extra dimension with the geometry of 5D anti-de Sitter space. Although the original RS model had Lorentz symmetry, ref. [@Csaki] (hereafter called CEG) considered a variant in which the 5D bulk contains a black hole of charge $Q$ and mass $\mu$, giving a metric (the AdS Reissner-Nördstrom black hole solution) with the line element \[metric\] ds\^2 &=& -h(r) dt\^2 + [r\^2l\^2]{} dx\^[ 2]{} + [1h(r)]{} dr\^2;\
\[h\] h(r) &=& [r\^2l\^2]{} - [r\^2]{} + [Q\^2r\^4]{}. The extra dimension has coordinate $r$, and the length scale $l$ characterizes the curvature of the AdS geometry, which is determined by the negative bulk cosmological constant through $l^2 = -6/\Lambda$. (We work in units where the 5D gravitational constant $\kappa_5=1$.)
The presence of a bulk black hole (BH) can be physically motivated: it was argued in [@Hebecker] that emission of gravitational radiation from a brane in the early universe can lead to the formation of a BH in the bulk. Possible cosmological consequences of gravity traveling faster than light in this scenario were considered in references [@xxx]. However in the cosmological solutions, the brane moves away from the BH as the universe expands, and as a result the presence of the BH quickly becomes irrelevant for the brane observer: its effects are redshifted away. The presence of charge on the BH is motivated by the desire to have a solution in which the brane-BH distance remains constant, and therefore its effects can also be important at late times.
Along any 4D slice of constant $r$, the metric is Lorentz invariant, with speed of light given by $dx/dt = (l/r) h^{1/2}$. Thus an observer on a D-brane which is restricted to a particular value of $r$ does not see direct evidence of Lorentz violation in the matter sector. However gravity propagates in all dimensions, not just along the brane. A 4D observer would find that gravity propagates with a different speed than matter.
It might be thought that the weakness of gravitational interactions would render this mild Lorentz violation phenomenologically harmless; after all, the speed of gravity has not been measured. However, there are ways in which it would be manifested. One possibility is that self-energy diagrams with graviton loops would induce speed differences between observable particles, like photons and electrons (where by “speed" we mean the maximum speed, as the momentum becomes infinite). These kinds of constraints were considered in [@Burgess].
A more powerful constraint exists in the case where gravity travels slower than visible particles, especially protons [@Moore]. The emission of gravitational Čerenkov radiation would very efficiently damp ultra high-energy cosmic rays (UHECR’s) to levels below those which are observed. One of the main observations of the present paper is that this is precisely the situation in asymmetrically warped models in which we are assumed to be living on a negative tension brane, where the weak scale hierarchy can be solved by the warping. In CEG it was assumed that we live on a positive tension brane; in that the speed of gravity is faster than that of light, so that the present strong constraint is not relevant.
In CEG, the asymmetric effects of warping were treated perturbatively. In the present work we also extend their analysis by considering these effects in the nonperturbative regime, and for the Kaluza-Klein (KK) excitations of the graviton. These could be relevant sources of Lorentz violation in colliders, since in RS-like models the KK gravitons couple strongly to the standard model particles [@Rizzo].
We begin by reviewing the CEG model and their perturbative derivation of the speed of gravity, and we enumerate the most stringent experimental constraints on Lorentz violation in these models. In section 3 we point out that the perturbative treatment can be misleading if the observer is located on the positive tension brane, whereas it gives reliable predictions for observers on the negative tension brane. In section 4 we show that the KK excitations of the graviton have similar Lorentz-violating properties to the zero mode, considerably strengthening the Čerenkov bound introduced in section 2.
The Models; perturbative speed of gravity
=========================================
Model with two branes
---------------------
Similarly to the RS model, CEG cuts the bulk at two positions, call them $r_+$ and $r_-$, by the insertion of branes and the use of $Z_2$ orbifold symmetry to define the discontinuity in the derivatives of the metric functions at the branes. The parts of the solution $r>r_+$ and $r<r_-$ are discarded, and replaced by mirror copies of the kept part of the solution to create a solution with orbifold fixed points at $r_\pm$. Unlike RS, it is not possible to use conventional branes with equation of state $p=-\rho$ at both positions. There are four junction conditions which relate the energy densities $\rho_\pm$ and equations of state $p_\pm=w_\pm \rho_\pm$ to the black hole parameters: \[jc1\] [l\^2]{} &=& 3( 1+ )r\_+\^4 = 3( 1+ )r\_-\^4 ;\
[Q\^2l\^2]{} &=& 2( 1+ )r\_+\^6 = 2( 1+ )r\_-\^6 . \[jc2\] It is easy to see that taking $w_+=w_-=-1$ implies that $r_+=r_-$. Let us consider the physically interesting case of a large hierarchy, where r\_-/r\_+ \~10\^[-16]{}. Physical masses on the negative tension brane at $r_-$ (the “TeV brane”) are suppressed relative to their bare values by the factor $\epsilon$, giving a resolution of the weak scale hierarchy problem [@RS]. By solving the jump conditions - for $\epsilon$, we see that there are two ways of achieving a small value of $\epsilon$: either by (1) tuning the numerators $( 1+ \frac{w_+}{36}{\rho_+^2 l^2} )$, $(\! 1\!+\! \frac{1\!+\!3w_+}{72}{\rho_+^2 l^2})$ to be small, or (2) tuning the denominators $( 1+ \frac{w_-}{36}{\rho_-^2 l^2} )$, $(\! 1\!+\! \frac{1\!+\!3w_-}{72}{\rho_-^2 l^2})$ to be large.
1. In the first case, $l\rho_-/6$ and $w_- (l\rho_-/6)^2$ can naturally be of order unity, while $w_+$ and $\rho_+$ must be tuned to the values \[wec1\] w\_+ -1-2\^4( 1+ ) < -1; l\_+ = 6 + O(\^4), implying the relation \[murel1\] = c\_1[Q\^2r\_-\^2]{}; c\_1 = [ 3( 1+ ) 2( 1+ ) ]{} = O(1) . The inequality in is a violation of the weak energy condition (albeit a small one), which states that $p \ge -\rho$, and would thus require some exotic kind of stress energy on the Planck brane.
2. In the second case, we need \[wec2\] w\_- = O(\^[2]{});l\_- = O(\^[-6]{}) , which implies \[murel2\] = c\_2[Q\^2r\_+\^2]{}; c\_2 = [ 3( 1+ ) 2( 1+ ) ]{} = O(1) .
The relations , are useful for computing the speed of electromagnetic radiation (or any relativistic particles) on the branes. Working to first order in the Lorentz violating parameters $\mu$ and $Q^2$, the deviation in the speed relative to unity is \[cem\] c\_[em]{} = - [ l\^22 r\_\^4]{} + [Q\^2 l\^2 2 r\_\^6]{} = [l\^2Q\^22 r\_\^4]{} ( [1r\_\^2]{} - [c\_ir\_i\^2]{}) , where the choice $\pm$ refers to which brane one is on, and the label $i$ refers to the choice of tunings immediately above; hence $r_1 = r_-$ and $r_2 = r_+$.
Model with one brane and a horizon
----------------------------------
The preceding discussion assumed the existence of two branes bounding the extra dimensional space. It is also possible to eliminate the TeV brane altogether by not cutting out the small $r$ region containing the black hole. In that case one should choose $\mu$ and $Q^2$ so that $h(r)$ vanishes at least once between $r=0$ and $r=r_+$. This insures there is a horizon shielding the black hole, in accordance with the cosmic censorship hypothesis. In this scenario we would necessarily be living on the Planck brane at $r=r_+$. The parts of the jump conditions - involving $\rho_+$ and $w_+$ would still be valid. CEG showed that it is not possible to embed the brane and still have a horizon shielding the black hole unless the weak energy condition is again violated, $w_+ < -1$. This is most easily studied when the special relation $4\mu^3 l^2 = 27 Q^4$ holds; this is the limiting case where the two horizons of the AdS Reissner-Nördstrom black hole degenerate into a single horizon, located at $r_h/l = ( 2Q/l^2)^{1/3}/\sqrt{2}$. One can solve the jump conditions to find that the brane’s equation of state depends on its position, $r_+$, and is given by w\_+ = -1 -2([r\_hr\_+]{})\^4 + O([r\_hr\_+]{})\^6 . Hence the violation of the weak energy condition becomes small as the brane is moved farther away from the black hole. Ref. [@CF] showed that this violation could be moved into the bulk stress-energy tensor, but never eliminated, so long as the horizon exists. The deviation in the speed of light on the brane in this scenario is given by c\_[em]{} = -32 ([r\_hr\_+]{})\^4 + ([r\_hr\_+]{})\^6 . We mention this case for completeness, but we will not further consider how gravity propagates in this case.
The speed of gravity; experimental constraint
---------------------------------------------
Now we turn to the speed of gravity. Its determination is simplified by CEG’s observation that the dynamics of gravity are the same as those for a massless bulk scalar field, with action and equation of motion S &=& 12d\^[4]{}x dr g\^[AB]{}\_A\_B;\
\[eom\] [S]{} &=& -\_A ( g\^[AB]{}\_B) = 0 . The problem can be further simplified by splitting the metric into a pure AdS part and the perturbation involving the black hole parameters. In other words, one works perturbatively to first order in $\mu$ and $Q^2$. The solution for the $n$th KK mode is separable, $\phi_n(x^\mu,r) =
e^{i(\omega_n t - \vec q\cdot \vec x)}\phi_n(r)$, where $\phi_n(r)$ satisfies Neumann boundary conditions at the branes, since the jump conditions - have already been satisfied by the background solution. The dispersion relation of the graviton zero mode solution in the tower of KK excitations can be determined analytically (in the model with two branes) for a solution with 3-momentum $\vec q$: \[disp\] \_0 = |q| + \_0; \_0 = [|q| l\^24 \^2]{}( -2[r\_+\^4]{} + [Q\^2r\_+\^6]{} (1 + \^[-2]{}) ) . The deviation in the speed of gravity from 1 is given by \[dcgrav\] c\_[grav]{} = [\_0|q|]{} .
The relevant Lorentz-violating observable is the difference $\delta c_{\rm em} - \delta c_{\rm grav}$. In general, the sign of this difference depends on the value of the $O(1)$ constant $c_1$ or $c_2$. However for TeV brane observers, if the hierarchy is very large ($\epsilon\ll 1$), and if we exlude the case - which violates the weak energy condition, the difference is completely dominated by the term $l^2 Q^2/(2r_-^6)$ in , and leads to c\_[em]{} - c\_[grav]{} = [l\^2 Q\^22r\_-\^6]{} . Because the difference is positive, the stringent constraints due to gravitational Čerenkov radiation of UHECR protons is applicable [@Moore]: c\_[em]{} - c\_[grav]{} < 210\^[-15]{} (10\^[-19]{}) . where the less stringent limit assumes the UHECR’s are of galactic origin, and the more stringent one applies if they originally come from neutrinos originating from cosmological distances. Therefore we have the bound \[newbound\] [h(r\_-)h(r\_-)]{} 410\^[-15]{} (10\^[-19]{}) . Notice that the quantity on the left is the fractional perturbation to the metric function $h(r)$ due to the dominant Lorentz violating term, evaluated on the TeV brane: $h(r) \cong (r/l)^2 (1 + \delta h/h)$ It is therefore natural that this is the combination which is experimentally bounded. is one of the new constraints derived in this paper.
If we are willing to entertain the possibility of case 1, with exotic matter on the Planck brane which violates the weak energy condition, then the difference of speeds as measured on the TeV brane becomes dependent on the details of the TeV brane stress energy, c\_[em]{} - c\_[grav]{} = [l\^2 Q\^22r\_-\^6]{}(1 - c\_1) . If $c_1<1$, the Čerenkov bound again applies, in the obvious way. If $c_1>1$, gravitational Čerenkov radiation cannot be produced. Then the strongest bounds come from tests for deviations from general relativity in the parametrized post-Newtonian (PPN) formalism. A difference between the speed of light and that of gravity implies the existence of a preferred reference frame, the one in which we have been working, where the speeds of gravity and light are independent of direction. In a boosted frame, one of these would become anisotropic, depending upon whether the boost respects Lorentz invariance of the gravitational or the electromagnetic propagation. (One but not both can be preserved, due to the difference in speeds.) In such a situation, the parameter $\alpha_2$ of the PPN formalism is nonzero. Essentially, we have two metrics, one for photons and one for gravitons, so the effective theory is Rosen’s bimetric one [@Will]. Such theories predict a torque which would cause precession of the sun’s spin axis. The latter is closely aligned with the solar system’s planetary angular momentum vector. If we assume that the alignment is not coincidental, then precession due to PPN effects is constrained, and leads to the bound [@Nordtvedt] \[ssbound\] |\_2| = 2[|c\_[em]{} - c\_[grav]{}|c\_[em]{}]{} < 1.210\^[-7]{} , which carries over to the quantity ${l^2 Q^2\over r_-^6}|1 - c_1|$.[^1]
Nonperturbative analysis
========================
The above discussion assumed that the dispersion relation of the graviton gets modified in the simple way which was predicted by treating the Lorentz-violating part of the metric, $\delta h$, to first order in perturbation theory. However, at large graviton momenta, the perturbative prediction can be modified. To see this, one must examine the equation of motion . Let us rewrite it to first order in $\delta
h$, with $\phi_0^{(0)}$ and $\phi_0^{(1)}$ respectively denoting the zeroth and first order solutions, in powers of $\delta h$, for the Kaluza-Klein zero mode. We will also change coordinates to the form $r/r_+ = e^{-ky}$ with $k=l^{-1}$. The wave equation for the zero mode with 3-momentum $q$ is $$\begin{aligned}
{\partial_y^2}{\phi_0^{(1)}}(y)-4k\, \mathop{\rm sgn}(y)\,\partial_y{\phi_0^{(1)}}(y)
&=& e^{2k|y|}\left(-2q\,\omega_0^{(1)}+{q^2}
{\delta h\over h}(y)\right)\phi_0^{(0)}\\ &=& e^{2k|y|}\left(-2q\,\omega_0^{(1)}+{q^2\over k^2} \left(- \frac{{
\mu}}{{ r_+^4 }}e^{4k|y|}+ \frac{{ Q^2}}{{ r_+^6 }}e^{6k|y|}\right)\right)\phi_0^{(0)}.
\nonumber\end{aligned}$$ Regardless of how small $\delta h/h$ is, at sufficiently large momenta the source term $q^2(\delta
h/h)\,\phi^{(0)}$ becomes so large that it can no longer be reliably treated as a perturbation.
To explore this, we have numerically solved the full, unperturbed graviton wave equation to find the dispersion relation for the zero mode, $\omega_0(q)$, and compared this result to the perturbative prediction . To make the problem tractable, we have chosen definite values for the brane stress energy parameters. For the tuning of parameters, we adopt case 2 above, eqs. -, so that $c_2$ depends on the positive tension brane parameters. These we take to be $w_+=-1$, while leaving $\rho_+$ free to vary. The full wave equation can be written as $$\phi''(y) -4k\, \mathop{\rm sgn}(y)\,\left( \frac{1 -{\delta}\, e^{6k|y|}}{\hat h(y)} \right)\phi'(y)
+ e^{2k|y|}\left( \frac{\omega^2}{\hat h^2(y)}-\frac{q^2}{\hat h(y)} \right)\phi(y)=0,
\label{EOM}$$ where $$\hat h(y)=1+ \delta\, e^{4k|y|} \left( -3+2e^{2k|y|} \right);\quad \delta \equiv 1 -
{\rho_+^2 l^2\over 36} .$$ In this form, it is clear that the Lorentz-violating effects, coming from $\hat h - 1$, are maximized at the TeV brane, where $y = y_-$. For the purposes of comparing exact results to the perturbative ones, we quantify the perturbation by the parameter 2e\^[6ky\_-]{} . We expect the perturbative treatment to be valid when $\Delta \ll 1$.
With these definitions we can now compare the exact numerical results with the perturbative ones. The deviation of the dispersion relation of the zero mode, relative to its Lorentz-conserving value, is plotted as a function of momentum for several values of $\Delta$, and as a function of $\Delta$ for several momenta, in figure \[fig1\], where $\delta\omega$ and $q$ are given in units of $k = l^{-1}$. We have taken a very small hierarchy, $ky_- = 2$ for this illustration, and will comment on the extrapolation to larger values below. The interesting feature is that the true $\delta\omega(q)$ flattens to a constant value at large $q$, contrary to the perturbative expectation.
If we compare the speed of gravity to that of hypothetical photons which are trapped on the [*positive tension brane*]{}, a consequence of this behavior of the graviton dispersion relation is that the speed difference (with speed defined using the group velocity $v = d\omega/dq$) does not remain constant at large momenta, but rather vanishes as $q\to\infty$, as shown in figure \[fig2\]. At large momenta, gravitons tend toward the same speed as radiation on the Planck brane. This can be understood as follows. The graviton zero mode is localized on the Planck brane. At large $q^2$, the effect of the terms $( \frac{\omega^2}{\hat
h^2(y)}-\frac{q^2}{\hat h(y)})$ in the equation of motion is to localize it even more, thus driving the graviton to resemble more closely radiation which is trapped on the Planck brane. This trend becomes evident for momenta $q\gsim 1/l$, where the speed difference is significantly reduced relative to its maximum value at $q=0$.
As the hierarchy between the Planck brane and the negative tension brane is increased by letting $y_-$ go to larger values, the Lorentz violating effects seen by a Planck brane observer are suppressed, if the parameter $\Delta$ is held fixed. The magnitude of $c_{\rm grav} - c_{\rm em}$ scales like $e^{-2ky_-}$. This can be understood analytically, using the perturbative results of the previous section. One can show that, for the present choice of parameters, c\_[grav]{} - c\_[em]{} = 12((2ky\_-)-1)e\^[-4ky\_-]{} \~ +O(e\^[-2ky\_-]{}) . At the same time, the $q$-axis in figure \[fig2\] gets rescaled by a factor of $e^{-ky_-}$, such that the range of momenta with an appreciable deviation in $c_{\rm grav} - c_{\rm em}$ gets shifted to smaller physical values. In the hierarchy-solving case $e^{-ky_-}\sim 10^{16}$, this corresponds to the TeV scale.
On the other hand, an observer on the TeV brane will not be concerned with this small momentum-dependence in the graviton speed, because the difference between $c_{\rm grav}$ and $c_{\rm em}$ remains relatively large even at high momenta, as shown in figure \[fig3\]. Moreover, in contrast to the case of the Planck brane observer, the difference $c_{\rm grav}-c_{\rm em}$ remains constant as the hierarchy is increased, if $\Delta$ is held fixed. Again, this can be understood by computing the speed difference perturbatively, for the present choice of parameters, c\_[grav]{} - c\_[em]{} = -((2ky\_-)-1)e\^[-2ky\_-]{} \~ -O(1).
The conclusion of this section’s analysis is that we can trust the results of the perturbative treatment if we assume that observers are living on the negative tension brane, as one would expect if the hierarchy problem is being addressed. Only for Planck brane observers would it be important to distinguish the perturbative from the exact results.
Lorentz-violating KK modes
==========================
It would be interesting if Lorentz violating kinematics could be observed in the laboratory, in particle collider experiments. In section 2 we noted that the speed difference between gravity and light on the TeV brane should be less than a few parts in $10^7$. Such a small effect would probably be impossible to see at the LHC, even though KK gravitons interact strongly (with TeV-suppressed instead of Planck-suppressed couplings) with standard model particles and would be copiously produced, if sufficiently light.
Even if their Lorentz-violating properties cannot be directly detected, an interesting indirect effect is the Čerenkov emission of KK gravitons from UHECR’s, which would strengthen the bound mentioned in section 2. That bound conservatively counted only the damping due to emission of massless gravitons. But KK modes can also be emitted, so long as their mass obeys the inequality \[Cer\] [m\^2\_[KK]{}2p\^[2]{} ]{}< c\_[em]{} - c\_[grav]{}(p) , where $\vec p$ is the momentum of the UHECR. This is the condition for Čerenkov emission to be kinematically allowed. In the situation where the warp factor $\epsilon = 10^{-16}$, the KK mass gap is of order TeV. The highest energy cosmic ray which has been detected had energy $3\times 10^{11}$ GeV [@Bird]; thus for a speed difference that saturates the solar system bound , the number of relevant KK modes is of order $10^5$. It is therefore worth exploring whether the KK modes have similar Lorentz-violating kinematics as the graviton zero mode.
We have addressed this problem both numerically and analytically. The analytic approach is to solve the wave equation once again treating $\delta h$ to first order in perturbation theory, but now expanding around the zeroth order Lorentz-conserving KK wave function, $\phi_n(y) \cong \phi_n^{(0)}(y) + \phi_n^{(1)}(y)$, in order to find the corresponding energy eigenvalue, $\omega_n(q) \cong \omega_n^{(0)}(q) +
\omega_n^{(1)}(q)$. The solution is described in detail in the Appendix. We have also numerically solved the full wave equation using the shooting method, to find the dispersion relation $\omega_n(q)$. The deviation of $\omega_n(q)$ from the standard result $\omega=q$ is shown for the first two KK modes in figure \[fig4\]. The comparison between the perturbative and numerical results is shown there.
As we did for the KK zero mode, we can compute the difference in the speed of gravity relative to that of particles trapped on the positive tension brane. In order to highlight the differences which are due to Lorentz violation, we take the conventional trapped particles to have the same rest mass as that of the KK graviton to which it is being compared. The results, shown in figure \[fig5\] are qualitatively similar to those for the zero mode: gravity is faster than particles on the Planck brane, but the speed difference becomes smaller at higher momenta.
More interesting is the speed difference relative to particles on the TeV brane. Figure \[fig6\] shows that, again like the zero mode, KK gravitons are slower than same-mass particles on the TeV brane. Therefore our expectation that they are produced in gravi-Čerenkov radiation by particles with momenta satisfying is justified, and we should revise the bound. Adapting the result of [@Moore] to the tower of TeV-mass-gap gravitons whose couplings are only TeV-suppressed, the rate of energy loss of a UHECR with momentum $p$ and mass $m_p$ is given by \~[p\^2\^3]{} \_0\^[q\_[max]{}]{} dq q \_0\^[M\_[max]{}]{} dM \^4, where $q$ is the momentum of the emitted KK graviton, $M$ is its mass, and $\theta$ is the angle of emission of the Čerenkov radiation. The kinematic limits and $\theta$ (in the limit $\theta\ll 1$) are given by \^2 &=& (1-q/p)q\^[-2]{}(M\_[max]{}\^2 - M\^2)\
M\^2\_[max]{} &=& q\^2( 2c - [m\_p\^2p\^2(1-q/p)]{} )\
q\_[max]{} &=& p(1 - [m\_p\^22 p\^2c]{}) . Here $\delta c = c_{\rm em} - c_{\rm grav}$, which must be positive in order for Čerenkov emission to occur. Evaluating the integrals gives \~0.1 (c)\^[5/2]{} f(c)[p\^5\^3]{} , where the function $f$ is approximately 1 for $\delta c\gg {m_p^2\over 2p^2}$ and drops very rapidly to zero (like $(1-m_p^2/2 p^2\delta c )^{11/2}$) as $\delta c\to {m_p^2\over 2p^2}$. The bound on $\delta c$ comes from demanding that $dE/dt$ be less than $p/L$ for a cosmic ray which is propagating over a distance $L$. For the highest energy cosmic ray observed, which is identified with a proton, the bound is satisfied by taking $\delta c$ to be below the kinematic limit where $f=0$. This implies that c < [m\_p\^22p\^2]{} 10\^[-23]{} .
Conclusions
===========
Asymmetric warping can provide a plausible means of introducing Lorentz violation into a theory with extra dimensions, which is essentially a form of spontaneous breaking due to the gravitational background. Since at tree level the violation is confined to the gravitational sector, the effects can be sufficiently weak to be at the borderline of detection. In this paper we have explored some of the consequences of a theory where the asymmetric warping comes from the presence of a charged black hole in the extra dimension. Such a scenario can be compatible with the RS solution to the weak scale hierarchy problem. Unlike RS, in the black hole case it is necessary to allow for nonstandard equations of state for the tensions of at least one of the branes, though it is still possible to respect the weak energy condition. One issue we have not explored is the role of a stabilization mechanism such as that of Goldberger and Wise [@GW] for the size of the extra dimension. One might expect that the bulk is already stabilized once the brane energies and equations of state are fixed, since algebraically the ratio of brane positions $r_-/r_+$ is determined. However it is possible the radion mass$^2$ is negative, even though it is not zero. Even if it has the right sign, the distortions of the bulk geometry by a bulk scalar could have an effect on the propagation of gravity. It would also be interesting to know whether pure tension branes could be admitted with the addition of a bulk scalar, as it would be desirable to eliminate the need for an unusual equation of state. Finally, we did not consider how Lorentz violation is manifested in the single brane case, mentioned in section 2.2, although we expect it to be similar to the case of a Planck-brane observer in the two brane model. These subjects could merit further study.
We thank Guy Moore for very helpful discussions.
Perturbative KK mode solution
=============================
The zeroth order in Lorentz-violation KK graviton wave function (which like the zero mode obeys Neumann boundary conditions) is [@GW0] $$\phi_n^{(0)}(y)=\frac{e^{2k|y|}}{N_n} \left( J_2 \left(\frac{m_n e^{k|y|}}{k} \right) -
\frac{J_1\left(\frac{m_n}{k} \right)}{Y_1 \left(\frac{m_n}{k} \right)} Y_2 \left(\frac{m_n
e^{k|y|}}{k} \right) \right) ,$$ where $N_n$ is a normalization constant. The $m_i$ are determined by the boundary conditions; in the limit of a large hierarchy, they satisfy $J_1(m_n e^{ky_-}/k) = 0$.
The wave function correction $\phi_n^{(1)}(y)$ is difficult to compute, but as is familiar from perturbation theory in quantum mechanics, we don’t really need it if we are only interested in the correction to the eigenvalue, $\omega_n^{(1)}(q)$. In the quantum mechanical analogy, we require only the matrix element of the perturbation to the Hamiltonian. Defining $\psi_n(y)=e^{-2k|y|}\phi_n^{}(y)$, the equation of motion becomes $$H\psi_n(y)=-\frac{\omega_n^2e^{2k|y|}}{\hat h(y)}\psi_n(y) ,
\label{HermitianEOM}$$ where, to first order in the perturbation, $$\begin{aligned}
H &=& \partial_y(\hat h(y)\partial_y)-4k^2\hat h(y)+2k\mathop{\rm sgn}(y)\,\hat h'(y)-q^2e^{2k|y|}+4k\hat h(y)\delta(y)\\
&=&\left[\partial_y^2 -4k^2 -q^2e^{2k|y|}+4k \delta(y)\right] \nonumber\\
& &+ \left[\partial_y(\hat h^{(1)}(y)\partial_y)- \left(4k^2 - 4k\delta(y)\right)
\hat h^{(1)}(y) +2k\mathop{\rm sgn}(y)\,\hat h'^{(1)}(y)\right] . \\
&\equiv& H^{(0)}+ H^{(1)},\end{aligned}$$ with $\hat h^{(1)} =\hat h - 1$.
We take the inner product with $\psi_n^{(0)}$ and use the Hermicity of $H^{(0)}$ to cancel the terms involving $\psi_n^{(1)}$. This yields $$\omega_n^{(1)}=\frac{-\langle\psi_n^{(0)}|H^{(1)}| \psi_n^{(0)}\rangle +
(\omega_n^{(0)})^2\langle\psi_n^{(0)}|{\hat h^{(1)}}e^{2k|y|}|
\psi_n^{(0)}\rangle}{2\omega_n^{(0)}\langle\psi_n^{(0)}|{e^{2k|y|}}|
\psi_n^{(0)}\rangle}.$$ The inner product in the denominator is unity, with the appropriately normalized wave function. Putting in the explicit form of $H^{(1)}$, the expression becomes $$\begin{aligned}
\omega_n^{(1)}&=&-\frac{1}{\omega_n^{(0)}}\int_0^{y_-}\psi_n^{*(0)}\left(\partial_y(\hat h^{(1)}
\partial_y)-4k^2\hat h^{(1)}+2k\hat h'^{(1)}\right) \psi_n^{(0)}\,dy \nonumber \\
& & + \omega_n^{(0)}\int_0^{y_-}{\hat h^{(1)}} e^{2ky} |\psi_n^{(0)}|^2\,dy \\
&\equiv&-\frac{1}{\omega_n^{(0)}}A_n+ \omega_n^{(0)}B_n.
\label{eq:omega1}\end{aligned}$$ Since $\omega_n^{(0)}=\sqrt{m_n^2+q^2}$, and $m_n/k\propto e^{-ky_-}$, all the terms are roughly comparable for $q \ll m_n$. But, for large momentum $q$, only the second integral is important. In that case, $$\begin{aligned}
\omega_n^{(1)}&\approx& q\int_0^{y_-}|\psi_n^{(0)}|^2 {\hat h^{(1)}}e^{2ky} \,dy \\
&=& {q\over k^2}\int_+^{y_-}|\psi_n^{(0)}|^2 \left(- \frac{{ \mu}}{{ r_+^4 }}e^{6ky}+ \frac{{ Q^2}}{{ r_+^6 }}e^{8ky} \right) \,dy \\
&\cong& \frac{q}{k^2 N_n^2}\int_+^{y_-} J_2^2 \left(\frac{m_n e^{ky}}{k} \right) \left(- \frac{{ \mu}}{{ r_+^4 }}e^{6ky}+ \frac{{ Q^2}}{{ r_+^6 }}e^{8ky} \right) \,dy \end{aligned}$$ where the last approximation holds in the limit of a large hierarchy. The resulting integral can only be done numerically. However, our analysis is still useful since it tells us that the first correction term to $\omega_n$ is proportional to $q$ for large momentum, just as for the zero mode. Moreover we can show how the dispersion relation is expected to change. Squaring $\omega_n$ gives $$\omega_n^2=(\omega_n^{(0)}+\omega_n^{(1)}+\cdots)^2 \cong (m_n^2+q^2)(1+2B_n-2A_n)$$ to first order in the perturbation. Comparing with the usual dispersion relation, one sees that the limiting speed of the $n{\rm{th}}$ mode has been modified to $$c_{\rm{grav}_n}^2=1+2B_n ,
\label{eq:cgravPert}$$ and that the mass of the mode changes due to the presence of the black hole perturbation: $$M_n^2=m_n^2(1+2B_n)-2A_n.
\label{eq:FirstOrderMass}$$ In terms of these, the group velocity of the $n{\rm{th}}$ mode with momentum $q$ is $$v_{\rm{grav}_n} = \frac{\partial \omega_n}{\partial q} \cong
\frac{c_{\rm{grav}_n}^2 q}{\sqrt{M_n^2+c_{\rm{grav}_n}^2 q^2}}. \label{eq:VgravPert}$$
As a consistency check, it can be verified that this result reduces to the previous one for the zero mode case where it is possible to do the integrals.
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[^1]: We thank Guy Moore for discussions on this point.
|
---
abstract: 'Much research in systemic risk is focused on default contagion. While this demands an understanding of valuation, fewer articles specifically deal with the existence, the uniqueness, and the computation of equilibrium prices in structural models of interconnected financial systems. However, beyond contagion research, these topics are also essential for risk-neutral pricing. In this article, we therefore study and compare valuation algorithms in the standard model of debt and equity cross-ownership which has crystallized in the work of several authors over the past one and a half decades. Since known algorithms have potentially infinite runtime, we develop a class of new algorithms, which find exact solutions in finitely many calculation steps. A simulation study for a range of financial system designs allows us to derive conclusions about the efficiency of different numerical methods under different system parameters.'
author:
- |
Johannes Hain and Tom Fischer [^1] [^2]\
[University of Wuerzburg]{}
bibliography:
- 'literature.bib'
title: Valuation Algorithms for Structural Models of Financial Interconnectedness
---
[**Key words:**]{} Counterparty risk, financial interconnectedness, financial networks, numerical asset valuation, structural model, systemic risk.\
[**JEL Classification:**]{} G12, G13, G32, G33\
[**MSC2010:**]{} 91B24, 91B25, 91G20, 91G40, 91G50, 91G60
Introduction
============
Since the turn of the millennium, research interest in systemic financial risk has steadily grown, with a noticeable pick-up in the number of publications over the past five years. One main field of interest is default contagion, see for instance the works of @acemoglu13, @elliott13, @gai11 and @nier07, or @staum12 for a survey. Many publications regarding systemic risk refer to the seminal work of @eisenberg01, who were the first to structurally model financial systems in which firms can hold each other’s financial obligations as assets under the assumption of limited liability. However, the core idea behind the model of @eisenberg01, which can be interpreted as a multi-firm extension of the @merton74 model where cross-holdings of zero-coupon debt between members of the system is allowed, received somewhat less attention by the research community. The main difference between Eisenberg and Noe and the standard multi-firm Merton model is that prices at maturity are not trivially determined since the value of one firm’s equity or debt may depend on the value of the debt of any other firm in the system. @eisenberg01 gave conditions under which only one equilibrium solution exists at maturity. Together with a finite numerical algorithm that they provided, the model could not only be used for default contagion research, but also for risk-neutral no-arbitrage valuation under financial interconnectedness [see also @fischer14]. @elsinger09 generalized the Eisenberg and Noe setup by also including cross-holdings in equity, and by allowing a seniority structure of the liabilities. A numerical algorithm was provided, however, a finite number of steps to the equilibrium price vector could not be guaranteed anymore.
Already in [-@suzuki02], Teruyoshi Suzuki had – unbeknown to him – generalized the @eisenberg01 setup to the situation where debt of one single seniority and equity could be cross-owned within a financial system (Suzuki, 2002). Unlike @eisenberg01 and @elsinger09, @suzuki02, who had the clear intention of generalizing @merton74, provided a Picard Iteration as the numerical means of calculating price equilibria. In a further generalization of Suzuki’s and Elsinger’s work, @fischer14 extended the structural model of interconnectedness to the case where liabilities could be derivatives in the sense of a dependence on other liabilities or equities in the system. As in @suzuki02, the numerical procedure provided to solve the liquidation value equations at maturity was the Picard Iteration. However, unfortunately, a Picard Iteration cannot warrant an exact solution in finitely many steps.
So, while there exists a small but growing amount of research on the existence and the uniqueness of price equilibria in systems with financial interconnectedness, the provided algorithms mainly reflect the individual authors’ particular approach to the problem. For instance, there also exists a publication by @gourieroux13 which mentions that, in the Elsinger model with two debt seniorities but no equity cross-holdings, a simplex method can be applied. However, comparative studies of the different methods seems to be absent from the existing literature. Furthermore, at present, no numerical algorithm for the setup with cross-holdings of equity and one seniority class of debt [@suzuki02; @elsinger09; @gourieroux12] is known that reaches the exact solution in a finite number of calculation steps.
For these reasons, the article at hand has three main objectives. First, we want to provide an overview of the already existing valuation algorithms by unifying notation and by embedding them in one general model framework. Second, we provide a new type of algorithm which is a hybrid of Eisenberg and Noe’s and Elsinger’s approach that has improved convergence properties. Third, we introduce a whole range of algorithm versions which are based on the three different types – namely Picard, Elsinger, and Hybrid – which will reach the exact solution (if existing and unique) in a finite number of calculation steps. Introducing these new algorithms, we show that for the three types of algorithms there always exists an increasing and a decreasing version – depending on a properly chosen (and explicitly given) starting point. Furthermore, we use default set techniques and linearization techniques to achieve finiteness. A simulation study finally allows to draw some conclusions about the efficiency of the presented numerical algorithms with respect to model parameters such as system size.
The structure of this paper is as follows. In the next section, we will establish the model and necessary assumptions for a unique solution of the financial system. The existing valuation methods of @suzuki02 and @elsinger09 are presented in Section \[sec:iterative\_algo\], where a hybrid version of the algorithms of @eisenberg01 and @elsinger09 is developed as well. The algorithms of this section are all non-finite. In the fourth section, we introduce a new class of valuation algorithms based on default set techniques that find solutions in finite time. A simulation study in Section \[sec:simulation\] compares the runtimes of the different algorithms for different classes of financial systems. In Section \[sec:summary\], we conclude. A technical appendix follows.
Notation and Model Assumptions {#sec:notation}
==============================
For two matrices ${\mathbf{M}}= (M_{ij})_{i,j=1{,\ldots,}n}\in{\mathbb{R}}^{n\times n}$ and ${\mathbf{N}}= (N_{ij})_{i,j=1{,\ldots,}n}\in{\mathbb{R}}^{n\times n}$ we write ${\mathbf{M}}\ge{\mathbf{N}}$ if $M_{ij}\ge N_{ij}$ for all $i,j\in\{1{,\ldots,}n\}$ and ${\mathbf{M}}>{\mathbf{N}}$ if $M_{ij}>N_{ij}$ for at least one pair $(i,j)$. For two vectors ${\mathbf{u}}= (u_1{,\ldots,}u_n)^t\in{\mathbb{R}}^n$ and ${\mathbf{v}}= (v_1{,\ldots,}v_n)^t\in{\mathbb{R}}^n$ the definition of ${\mathbf{u}}\ge{\mathbf{v}}$ and ${\mathbf{u}}> {\mathbf{v}}$ is analogous to the conventions for matrices above. A matrix ${\mathbf{M}}\in{\mathbb{R}}^{n\times n}$ is said to be *left substochastic* if $M_{ij}\ge 0$ for all $i,j\in\{1{,\ldots,}n\}$ and if $\sum_{i=1}^nM_{ij}\le 1$ for all $j\in\{1{,\ldots,}n\}$. The symbol ${\mathbf{I}}_n$ is used for the $n\times n$-identity matrix and ${\mathbf{0}}_n$ is used for a (column) vector of length $n$ that contains only zeros. Additionally, ${\mathbf{0}}_{n\times n}$ stands for an $(n\times n)$-matrix with only zero entries. For a vector ${\mathbf{u}}\in{\mathbb{R}}^n$, the expression ${{\rm diag}}({\mathbf{u}}\le{\mathbf{0}}_n)$ stands for an $(n\times n)$-diagonal matrix where the $i$-th entry is 1 if $u_i\le0$ and 0 otherwise, i.e. $${{\rm diag}}({\mathbf{u}}\le{\mathbf{0}}_n) = \begin{cases} 1 ,& \text{for $i=j$ and $u_i\le0$}, \\
0 ,& \text{else}. \end{cases}$$ All operations such as the minimum, $\min\{\cdot\}$, the maximum, $\max\{\cdot\}$, or the positive part $(\cdot)^+$ are applied element-wise to vectors and matrices. The norm in this paper is the $\ell^1$-norm on ${\mathbb{R}}^n$ defined as $$\|{\mathbf{x}}\| := \|{\mathbf{x}}\|_1 = \sum_{i=1}^n |x_i| \quad \text{for ${\mathbf{x}}\in{\mathbb{R}}^n$}.$$ The corresponding norm for a left substochastic matrix ${\mathbf{M}}\in{\mathbb{R}}^{n\times n}$ is given by $$\|{\mathbf{M}}\| := \|{\mathbf{M}}\|_1 = \max_{\|{\mathbf{x}}\|=1}\|{\mathbf{M}}{\mathbf{x}}\|_1 = \max_j \sum_{i=1}^nM_{ij}
\le 1,$$ meaning that $\|{\mathbf{M}}\|$ is the maximum of the column sums. One easily can show that $\|{\mathbf{M}}{\mathbf{x}}\|\le\|{\mathbf{M}}\|\|{\mathbf{x}}\|$.
We consider a system of $n$ financial entities, and denote $\mathcal N=\{1{,\ldots,}n\}$. In the following these entities are simply called “firms”. Each firm owns exogenous assets, that are defined in the next step.
\[def:ex\_assets\] Let $a_i\ge0$ denote the market value of the *exogenous assets* held by firm $i$. As the name implies, these assets are priced outside the considered system in the sense that the capital structure of the $n$ firms has no influence on the pricing mechanism of such an asset. By ${\mathbf{a}}=(a_1{,\ldots,}a_n)^t\in({\mathbb{R}}_0^+)^n$ we denote the (column) vector of the exogenous assets.
Moreover, we assume that the firms have outstanding liabilities with a nominal value at maturity of $d_i$ for each firm $i$. These liabilities are summarized in the vector ${\mathbf{d}}\in({\mathbb{R}}_0^+)^n$. In our framework we assume that the entries of ${\mathbf{d}}$ are constant. Since it is assumed that the exogenous assets’ prices are given by the constant vector ${\mathbf{a}}$, the results of this paper presented in the Sections \[sec:notation\] to \[sec:def-set-algo\] also hold if ${\mathbf{d}}$ depends on ${\mathbf{a}}$, i.e. if ${\mathbf{d}}={\mathbf{d}}({\mathbf{a}})$. However, for the remainder, we will write ${\mathbf{d}}$ for convenience. This definition of the liability vector allows the interpretation that the $d_i$ are simple loans or zero coupon bonds since they are not derivatives that can depend on the other assets within the system. The case of constant liabilities is used in most existing publications in this field, see for example @suzuki02, @gourieroux12 and @elsinger09. The more general case in which ${\mathbf{d}}$ depends for example on the endogenous assets, is also treated in the literature [see @fischer14].
To take the interconnectedness of the firms into account, we allow that each firm can own a fraction of the liabilities of the other firms. To formalize these possible cross-holdings, we use ownership matrices.
\[def:xos\_matrix\] The left substochastic matrix ${\mathbf{M}^{{\mathbf{d}}}}\in{\mathbb{R}}^{n\times n}$ in which the entry $0\le M^{{\mathbf{d}}}_{ij}\le 1$ denotes the fraction that firm $i$ owns of the liability of firm $j$ is called *debt ownership matrix*. Since no firm is allowed to hold liabilities against themselves, we assume $M^{{\mathbf{d}}}_{ii}=0$ for all $i\in\mathcal N$. The entries $M^{{\mathbf{s}}}_{ij}$ of the (left substochastic) *equity ownership matrix* ${\mathbf{M}^{{\mathbf{s}}}}\in{\mathbb{R}}^{n\times n}$ are defined as the fraction that firm $i$ owns of firm $j$’s equity.
Note that the diagonal entries of ${\mathbf{M}^{{\mathbf{s}}}}$ must not be zero; that means it is allowed that firm $i$ holds its own shares in which case $M^{{\mathbf{s}}}_{ii}>0$. For the debt ownership matrix it is a common convention (cf. @eisenberg01 or @elsinger09) that $M^{{\mathbf{d}}}_{ii}=0$ since a firm cannot have debt obligations to itself. The tuple ${\mathcal{F}}=({\mathbf{a}},{\mathbf{d}},{\mathbf{M}^{{\mathbf{d}}}},{\mathbf{M}^{{\mathbf{s}}}})$ is in the following sometimes referred to as the *financial system*.
Associated with the liability vector, we consider the recovery claim vector ${\mathbf{r}}\in({\mathbb{R}}_0^+)^n$. The recovery claim vector represents the actual payments of the firms at maturity, i.e. in general we have ${\mathbf{r}}^k\le{\mathbf{d}}^k$ since default risk is present. The value of the debt claim that firm $i$ has against firm $j$ is hence given by $M^{{\mathbf{d}}}_{ij} r_j$ and the total value of firm $i$’s debt claim against the other members of the system is $\sum_{j=1}^nM_{ij}^{{\mathbf{d}}} r_j$. Furthermore, denote by ${\mathbf{s}}\in({\mathbb{R}}_0^+)^n$ the equity values of the $n$ firms which means that the total recovery value of all system-endogenous assets that firm $i$ owns is given by the $i$-th entry of $${\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}.$$
The basic assumption for the model is that equity is considered to be the residual claim which means that any outstanding liability has to be paid off completely before the shareholders receive a positive payment. Hence, the equity value $s_i$ of firm $i$ is positive if and only if firm $i$ can fully satisfy all their obligees. The Absolute Priority Rule immediately leads to the following *liquidation value equations* for the recovery claims and the equities [cf. @fischer14]: $$\begin{aligned}
\label{eq:liq_eq_debt}{\mathbf{r}}&= \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}\} \\
\label{eq:liq_eq_equity}{\mathbf{s}}&= ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}- {\mathbf{d}})^+,\end{aligned}$$ where the sum in contains no $(\cdot)^+$ since Theorem \[theo:unique\_fp\] will show that all solutions of this system are non-negative. A solution for the liquidation value equations and is hence a fixed point of the mapping $\Phi:({\mathbb{R}}_0^+)^{2n}\to ({\mathbb{R}}_0^+)^{2n}$, where ${\mathbf{R}}=({\mathbf{r}}^t,{\mathbf{s}}^t)^t\in({\mathbb{R}}_0^+)^{2n}$ and $$\label{eq:Phi}
\Phi({\mathbf{R}}) = \Phi\begin{pmatrix}{\mathbf{r}}\\ {\mathbf{s}}\end{pmatrix} =
\begin{pmatrix} \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}\} \\ ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}- {\mathbf{d}})^+ \end{pmatrix}.$$ We will sometimes refer to the debt component of ${\mathbf{R}}$ and mean in such cases the first $n$ components of ${\mathbf{R}}$ that represent the debt payments of the systems. The components $n+1$ to $2n$ of ${\mathbf{R}}$ we also call equity component for the same reasons. We are interested in finding the fixed points of $\Phi$, which we will also call solutions of the financial system ${\mathcal{F}}=({\mathbf{a}},{\mathbf{d}},{\mathbf{M}^{{\mathbf{d}}}},{\mathbf{M}^{{\mathbf{s}}}})$. Without further constraints it is possible that there exist several fixed points. To ensure that the solution is unique, we have to make an additional assumption in which we need another property of an ownership matrix.
An ownership matrix ${\mathbf{M}}\in{\mathbb{R}}^{n\times n}$ possesses the *Elsinger Property* if there exists no subset $\mathcal J \subset \mathcal N$ such that $$\sum_{i\in\mathcal J} M_{ij} = 1 \quad \text{for all $j\in\mathcal J$}.$$
The name of this property is chosen because @elsinger09 is, by the best knowledge of the authors, the first one to use this assumption in the context of ownership matrices and the valuation of systemic risk. For our model, we demand that the considered ownership matrices fulfill this property.
\[assu:holding\_mat\] The Elsinger Property holds for the debt and the equity ownership matrices ${\mathbf{M}^{{\mathbf{d}}}}$ and ${\mathbf{M}^{{\mathbf{s}}}}$ .
Note that the fact that ${\mathbf{M}^{{\mathbf{d}}}}$ and ${\mathbf{M}^{{\mathbf{s}}}}$ are holding matrices is equivalent with the existence of $({\mathbf{I}}_n-{\mathbf{M}^{{\mathbf{d}}}})^{-1}$ and $({\mathbf{I}}_n-{\mathbf{M}^{{\mathbf{s}}}})^{-1}$, as shown by @elsinger09. Moreover, Theorem \[theo:unique\_fp\] below will show that Assumption \[assu:holding\_mat\] ensures that there is only one fixed point of $\Phi$. To show this, we introduce the two vectors $$\label{eq:def_Rgreat}
{\mathbf{R}_{\rm great}}= \begin{pmatrix} {\mathbf{r}_{\rm great}}\\ {\mathbf{s}_{\rm great}}\end{pmatrix} = \begin{pmatrix} {\mathbf{d}}\\ ({\mathbf{I}}_n-{\mathbf{M}^{{\mathbf{s}}}})^{-1}({\mathbf{a}}+{\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}-{\mathbf{d}})^+ \end{pmatrix}$$ and $${\mathbf{R}_{\rm small}}= \begin{pmatrix} {\mathbf{r}_{\rm small}}\\ {\mathbf{s}_{\rm small}}\end{pmatrix} = \begin{pmatrix} \min\{{\mathbf{d}}, {\mathbf{a}}\} \\ ({\mathbf{a}}-{\mathbf{d}})^+ \end{pmatrix}$$ The vector ${\mathbf{R}_{\rm great}}$ assumes that the debt payments are fully recovered so that in the debt component ${\mathbf{r}}={\mathbf{d}}$. Note that even if for a fixed point ${\mathbf{R}}^*=\left(\begin{smallmatrix}{\mathbf{r}}^* \\ {\mathbf{s}}^*\end{smallmatrix}\right)$ of $\Phi$ it holds that ${\mathbf{r}}^*={\mathbf{d}}$, it must not necessarily hold that ${\mathbf{R}_{\rm great}}={\mathbf{R}}^*$. The second vector ${\mathbf{R}_{\rm small}}$ emerges when the liquidation equations and are applied and the ownership structure of liabilities and equities is completely ignored. In this case the term ${\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}$ that represents the income of each firm stemming from debt and equity cross-ownership is set to zero. Hence, the firms only have the exogenous assets ${\mathbf{a}}$ as an income. The vector ${\mathbf{R}_{\rm small}}$ results from applying the mapping $\Phi$ in equation to the vector ${\mathbf{0}}_{2n}$, i.e. $$\label{eq:Rsmall_zero}
\Phi({\mathbf{0}}_{2n}) = \Phi\begin{pmatrix}{\mathbf{0}}_n\\{\mathbf{0}}_n\end{pmatrix}
= \begin{pmatrix} \min\{{\mathbf{d}}, {\mathbf{a}}\} \\ ({\mathbf{a}}-{\mathbf{d}})^+ \end{pmatrix} = {\mathbf{R}_{\rm small}}.$$
\[lem:Phi\_self-mapping\] With the definitions above, it holds that $\Phi([{\mathbf{R}_{\rm small}}, {\mathbf{R}_{\rm great}}])\subset[{\mathbf{R}_{\rm small}}, {\mathbf{R}_{\rm great}}]$.
Assume ${\mathbf{R}}\in[{\mathbf{R}_{\rm small}}, {\mathbf{R}_{\rm great}}]$. Because of and the monotony of $\Phi$, we have that $\Phi({\mathbf{R}})\ge{\mathbf{R}_{\rm small}}$. By definition of $\Phi$ and ${\mathbf{R}_{\rm great}}$, only the lower $n$ lines of the vector inequality $\Phi({\mathbf{R}})\le{\mathbf{R}_{\rm great}}$ need to be shown. By Lemma \[lem:inv\_xos-mat\] and , it holds that $${\mathbf{s}_{\rm great}}- {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}_{\rm great}}= ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}- {\mathbf{d}})^+$$ and therefore $${\mathbf{s}_{\rm great}}\ge ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}_{\rm great}}- {\mathbf{d}})^+.$$
Before showing the importance of ${\mathbf{R}_{\rm great}}$ and ${\mathbf{R}_{\rm small}}$ as upper and lower bounds of the solution ${\mathbf{R}}^*$, we need to introduce the terms *default set* and *default matrix*. For ${\mathbf{r}}\ge{\mathbf{0}}_n$ and ${\mathbf{s}}\ge{\mathbf{0}}_n$ the set $$\label{eq:defi_def_set}
D({\mathbf{r}},{\mathbf{s}}) = \left\{i\in\mathcal N:a_i + \sum_{j=1}^n M^{{\mathbf{d}}}_{ij} r_j + \sum_{j=1}^n M^{{\mathbf{s}}}_{ij} s_j < d_i\right\}$$ is called *default set under ${\mathbf{r}}$ and ${\mathbf{s}}$* because – given ${\mathbf{r}}$ and ${\mathbf{s}}$ – the firms in $D({\mathbf{r}},{\mathbf{s}})$ are not able to fully satisfy their obligations and hence are in default. We say that firm $i$ is in default under ${\mathbf{r}}$ and ${\mathbf{s}}$ if $i\in D({\mathbf{r}},{\mathbf{s}})$. For ${\mathbf{R}}=({\mathbf{r}}^t,{\mathbf{s}}^t)^t$ we will sometimes abbreviate the default set as $D({\mathbf{R}})$. The *default matrix corresponding to ${\mathbf{r}}$ and ${\mathbf{s}}$*, ${\mathbf{\Lambda}}({\mathbf{r}}, {\mathbf{s}})\in{\mathbb{R}}^{n\times n}$, is defined as $$\label{eq:defi_def_mat}
{\mathbf{\Lambda}}({\mathbf{r}},{\mathbf{s}}) = {{\rm diag}}({\mathbf{a}}+{\mathbf{M}}^{{\mathbf{d}}}{\mathbf{r}}+ {\mathbf{M}}^{{\mathbf{s}}}{\mathbf{s}}-{\mathbf{d}}< {\mathbf{0}}_n)$$ and is the diagonal matrix with entry 1 for firms in default under ${\mathbf{r}}$ and ${\mathbf{s}}$ at the corresponding position and with the value 0 for firms not in default. With the new notation, we can show the crucial limiting property of ${\mathbf{R}_{\rm great}}$ and ${\mathbf{R}_{\rm small}}$.
\[prop:lim\_R\] Let ${\mathbf{R}}^*$ be a non-negative solution of the fixed point problem defined in . Then ${\mathbf{R}}^*\in[{\mathbf{R}_{\rm small}},{\mathbf{R}_{\rm great}}]$.
Because of and the monotony of $\Phi$, ${\mathbf{R}}^*\ge {\mathbf{R}_{\rm small}}$, so we only show the validity of the upper bound ${\mathbf{R}_{\rm great}}$. Since ${\mathbf{R}}^*$ is a fixed point of $\Phi$, we can write $$\Phi\begin{pmatrix}{\mathbf{r}}^* \\ {\mathbf{s}}^*\end{pmatrix} =
\begin{pmatrix} \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^* + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}^*\} \\ ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^* + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}^* - {\mathbf{d}})^+ \end{pmatrix}
= \begin{pmatrix}{\mathbf{r}}^* \\ {\mathbf{s}}^*\end{pmatrix} = {\mathbf{R}}^*.$$ Obviously, ${\mathbf{r}}^*\le{\mathbf{d}}={\mathbf{r}_{\rm great}}$, hence we reduce our considerations to the equity component of ${\mathbf{R}}^*$ which, together with ${\mathbf{\Lambda}}({\mathbf{r}}^*,{\mathbf{s}}^*)={\mathbf{\Lambda}}^*$, can be presented as $$\label{eq:fp_eq_comp}
{\mathbf{s}}^* = ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^* + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}^* - {\mathbf{d}})^+ = ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*) ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^* + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}^* - {\mathbf{d}}).$$ Because of $({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{s}}^* = {\mathbf{s}}^*$ we can reformulate into $$\begin{split}
{\mathbf{s}}^* &= ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*) {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}^* + ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*) ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^* - {\mathbf{d}}) \\
&= ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*) {\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{s}}^* + ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*) ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^* - {\mathbf{d}}).
\end{split}$$ Rearranging yields to $$\label{eq:bfs_inv}
{\mathbf{s}}^* = ({\mathbf{I}}_n - ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*))^{-1}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*)({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^* - {\mathbf{d}}).$$ Together with Lemma \[lem:inv\_aux\] in the Appendix, this leads to $$\begin{split}
{\mathbf{s}}^* &= ({\mathbf{I}}_n - ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*))^{-1}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*)({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^* - {\mathbf{d}}) \\
&\le ({\mathbf{I}}_n - ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*))^{-1}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*)({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}- {\mathbf{d}}) \\
&\le ({\mathbf{I}}_n - ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*))^{-1}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*)({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}- {\mathbf{d}})^+ \\
&\le ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*)({\mathbf{I}}_n-{\mathbf{M}^{{\mathbf{s}}}})^{-1}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*)({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}- {\mathbf{d}})^+ \\
&\le ({\mathbf{I}}_n-{\mathbf{M}^{{\mathbf{s}}}})^{-1}({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}- {\mathbf{d}})^+ \\
&= {\mathbf{s}_{\rm great}},
\end{split}$$ from which the assertion follows.
Using the results from above, we can now show that there is only one fixed point which is in the interval $[{\mathbf{R}_{\rm small}},{\mathbf{R}_{\rm great}}]$. The proof of the following theorem is given in the Appendix.
\[theo:unique\_fp\] Under Assumption \[assu:holding\_mat\] and for an arbitrary financial system ${\mathcal{F}}=({\mathbf{a}},{\mathbf{d}},{\mathbf{M}^{{\mathbf{d}}}},{\mathbf{M}^{{\mathbf{s}}}})$, there exists a unique fixed point of the mapping $\Phi$. The fixed point ${\mathbf{R}}^*$ is non-negative and ${\mathbf{R}}^*\in[{\mathbf{R}_{\rm small}},{\mathbf{R}_{\rm great}}]$.
In the sequel, we assume that Assumption \[assu:holding\_mat\] holds so that ${\mathcal{F}}$ has only one solution ${\mathbf{R}}^*\in({\mathbb{R}}_0^+)^{2n}$, i.e. $${\mathbf{R}}^*=\begin{pmatrix}{\mathbf{r}}^*\\{\mathbf{s}}^*\end{pmatrix} = \Phi \begin{pmatrix}{\mathbf{r}}^*\\{\mathbf{s}}^*\end{pmatrix} = \Phi({\mathbf{R}}^*).$$ The requirements of Assumption \[assu:holding\_mat\] are less strict than the assumption that both $\|{\mathbf{M}^{{\mathbf{d}}}}\|<1$ and $\|{\mathbf{M}^{{\mathbf{s}}}}\|<1$ that is used for example in @fischer14, @suzuki02 or @gourieroux12. This follows by the fact that in case of ${\mathbf{M}^{{\mathbf{d}}}}$ having the Elsinger Property, it must not necessarily hold that $\|{\mathbf{M}^{{\mathbf{d}}}}\|<1$ and hence $\Phi$ is no strict contraction anymore. However, the assumption still guarantees that the solution of the system is unique.
Non-Finite Algorithms {#sec:iterative_algo}
=====================
In this section, two existing solution algorithms that can be found in the literature are presented. One algorithm consists of the iterative use of the mapping $\Phi$ on a chosen starting vector and is given in the first subsection. A modification of this Picard Iteration is used in the work of @elsinger09, where for the determination of the equity component, a more sophisticated subalgorithm is used (Section \[subsec:elsinger\]). In the last subsection, a new algorithm is developed that combines the ideas of @elsinger09 and @eisenberg01 which results in a faster convergence of the procedure.
The Picard Algorithm
--------------------
The most intuitive way to calculate ${\mathbf{R}}^*$ for the system ${\mathcal{F}}$ consists of the iterative use of $\Phi$. It will be shown in this section that with an arbitrary starting vector ${\mathbf{R}}^0\in({\mathbb{R}}_0^+)^{2n}$, $$\label{eq:picard-iteration}
{\mathbf{R}}^* = \lim_{l\to\infty}\Phi^l({\mathbf{R}}^0) = \lim_{l\to\infty} \underbrace{\Phi\circ\ldots\circ\Phi}_{l}({\mathbf{R}}^0),$$ which is commonly known as the *Picard Iteration*. Since ${\mathbf{R}}^*\ge{\mathbf{0}}$, the range for the starting vector ${\mathbf{R}}^0$ can be reduced to only non-negative vectors. Beyond that, the search for an optimal starting point can be limited to the interval $[{\mathbf{R}_{\rm small}}, {\mathbf{R}_{\rm great}}]$, as shown in Theorem \[theo:unique\_fp\]. A direct consequence is that any iteration procedure that aims to calculate ${\mathbf{R}}^*$ should make sure that
no starting point of the iteration is chosen outside the interval $[{\mathbf{R}_{\rm small}},{\mathbf{R}_{\rm great}}]$ and that
every interim result of the procedure also needs to be in that interval.
Otherwise, the procedure is inefficient. For these reasons, we present an algorithm that can start with both, ${\mathbf{R}_{\rm great}}$ and ${\mathbf{R}_{\rm small}}$.
[1]{}\[Picard Algorithm\]\[alg:picard\]
1. For $k=0$, choose ${\mathbf{R}}^0\in[{\mathbf{R}_{\rm small}}, {\mathbf{R}_{\rm great}}]$ and ${\varepsilon}> 0$.
2. \[alg:fp\_picard\] For $k\ge 1$, determine ${\mathbf{R}}^k = \Phi({\mathbf{R}}^{k-1})$.
3. If $\|{\mathbf{R}}^{k-1}-{\mathbf{R}}^k\|<{\varepsilon}$, stop the algorithm. Else, set $k = k + 1$ and proceed with step \[alg:fp\_picard\].
We will use the two expressions Picard Iteration and Picard Algorithm synonymously for Algorithm \[alg:picard\]. In @suzuki02 and @fischer14 the Picard Iteration is the algorithm of choice to determine solutions of and .
\[prop:conv\_picard\] In case of ${\mathbf{R}}^0={\mathbf{R}_{\rm small}}$, Algorithm \[alg:picard\] generates a sequence of increasing vectors ${\mathbf{R}}^k$, and for ${\mathbf{R}}^0={\mathbf{R}_{\rm great}}$ a sequence of decreasing vectors. For all starting points, the algorithm converges to the solution ${\mathbf{R}}^*$.
Let ${\mathbf{R}}^0={\mathbf{R}_{\rm small}}$, then $$\Phi({\mathbf{R}_{\rm small}})=\begin{pmatrix}\min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}_{\rm small}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}_{\rm small}}\} \\ ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}_{\rm small}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}_{\rm small}}- {\mathbf{d}})^+\end{pmatrix}
\ge \begin{pmatrix} \min\{{\mathbf{d}}, {\mathbf{a}}\} \\ ({\mathbf{a}}- {\mathbf{d}})^+ \end{pmatrix} = {\mathbf{R}_{\rm small}}.$$ From the monotonicity of $\Phi$, it follows that for all iterates we have ${\mathbf{R}}^{k+1}\ge{\mathbf{R}}^k, k\ge 1$. For ${\mathbf{R}}^0={\mathbf{R}_{\rm great}}$, first check that because of ${\mathbf{s}_{\rm great}}=({\mathbf{I}}_n-{\mathbf{M}^{{\mathbf{s}}}})^{-1}({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}- {\mathbf{d}})^+$ and ${\mathbf{r}_{\rm great}}= {\mathbf{d}}$, $$\begin{split}
& ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}_{\rm great}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}_{\rm great}}- {\mathbf{d}})^+ \\
& \quad = \left({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}- {\mathbf{d}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}_{\rm great}}- {\mathbf{s}_{\rm great}}+ {\mathbf{s}_{\rm great}}\right)^+ \\
& \quad = \left({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}- {\mathbf{d}}- ({\mathbf{I}}_n-{\mathbf{M}^{{\mathbf{s}}}}){\mathbf{s}_{\rm great}}+ {\mathbf{s}_{\rm great}}\right)^+ \\
& \quad = \left(\underbrace{{\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}- {\mathbf{d}}- ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}- {\mathbf{d}})^+}_{\le {\mathbf{0}}_n} +\; {\mathbf{s}_{\rm great}}\right)^+ \\
& \quad \le ({\mathbf{s}_{\rm great}})^+ = {\mathbf{s}_{\rm great}}\end{split}$$ and thus $$\Phi({\mathbf{R}_{\rm great}})= \begin{pmatrix}\min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}_{\rm great}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{r}_{\rm great}}\} \\ ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}_{\rm great}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}_{\rm great}}- {\mathbf{d}})^+\end{pmatrix}
\le \begin{pmatrix} {\mathbf{d}}\\ {\mathbf{s}_{\rm great}}\end{pmatrix} = {\mathbf{R}_{\rm great}}.$$ Again it holds, due to the monotonicity of $\Phi$, that ${\mathbf{R}}^{k+1}\le{\mathbf{R}}^k, k\ge 1$. Hence for any ${\mathbf{R}}\in[{\mathbf{R}_{\rm small}}, {\mathbf{R}_{\rm great}}]$ it follows because of ${\mathbf{R}}\le{\mathbf{R}_{\rm great}}$ that $\Phi({\mathbf{R}})\le\Phi({\mathbf{R}_{\rm great}})\le{\mathbf{R}_{\rm great}}$ and with the same argumentation it follows that $\Phi({\mathbf{R}})\ge\Phi({\mathbf{R}_{\rm small}})\ge{\mathbf{R}_{\rm small}}$. This means that any series from the Picard Iteration with a starting point in the interval $[{\mathbf{R}_{\rm small}}, {\mathbf{R}_{\rm great}}]$ is bounded from above and from below. Since $\Phi$ is continuous, it follows that the series must converge to some $\widetilde {\mathbf{R}}$ such that $\Phi(\widetilde {\mathbf{R}})=\widetilde {\mathbf{R}}$. According to Theorem \[theo:unique\_fp\], there is only one fixed point, so it must hold that $\widetilde {\mathbf{R}}={\mathbf{R}}^*$.
Regarding the Picard Iteration with the starting points ${\mathbf{R}_{\rm small}}$ or ${\mathbf{R}_{\rm great}}$, it should be mentioned that besides @suzuki02 and @fischer14, also @shin06 considers a Picard iteration in a system valuation context. Shin’s model is one with debt cross-ownership and multiple seniorities, while equity cross-ownership is not considered. As such, the model is situated somewhere between the @eisenberg01 and the @elsinger09 framework. Instead of maturity values (as done here), Shin directly considers risk-neutral values at time 0, and takes for the start of the iteration procedure either a “conservative” viewpoint, where debt is assumed to have the value zero, or an “optimistic” viewpoint, where the value of debt is assumed to be the face value. Shin’s starting points therefore seem to be risk-neutral time zero equivalents to the here considered ${\mathbf{R}_{\rm small}}$ and ${\mathbf{R}_{\rm great}}$.
The Picard Iteration – or any other iterative algorithm in this section – might not reach the solution ${\mathbf{R}}^*$ in finitely many iteration steps. Examples of financial systems with this property can easily be constructed. From a computational or practical point of view this means that iterative algorithms like the Picard Iteration have the disadvantage that under some circumstances many iterations are needed to approach to ${\mathbf{R}}^*$ sufficiently close, which makes these algorithms somewhat inefficient. The Trial-and-Error Algorithms presented in Section \[sec:def-set-algo\] do not have this drawback since for these procedures it is assured that they will reach the solution in a finite number of steps.
The @elsinger09 Algorithm {#subsec:elsinger}
-------------------------
In @elsinger09, an algorithm for ${\mathbf{R}}^*$ is presented which differs from the Picard Iteration. This procedure consists of splitting the two components of ${\mathbf{R}}$, the equity and the debt component, and apply different computation methods on both components in each iteration step. For the equity component, a sub-algorithm is applied where the equity payments of the system are determined assuming a fixed amount of debt payments. Denote this vector of debt payments in the following by $\bar{\mathbf{r}}$, hence ${\mathbf{0}}_n\le\bar{\mathbf{r}}\le{\mathbf{d}}$. Aim of the sub-algorithm is to find a fixed point of the mapping $\Phi^{{\mathbf{s}}}:({\mathbb{R}}_0^+)^n\to({\mathbb{R}}_0^+)^n$ with $$\label{eq:phi_aux_eq}
\Phi^{{\mathbf{s}}}({\mathbf{s}}; \bar{\mathbf{r}}) = ({\mathbf{a}}+{\mathbf{M}^{{\mathbf{d}}}}\bar{\mathbf{r}}+ {\mathbf{M}}^{{\mathbf{s}}}{\mathbf{s}}- {\mathbf{d}})^+.$$ This mapping represents the equity component of $\Phi$ for a given debt payment of $\bar{\mathbf{r}}$. The fixed point of $\Phi^{{\mathbf{s}}}(\cdot;\bar{\mathbf{r}})$ is denoted by ${\mathbf{s}}(\bar{\mathbf{r}})$, i.e. $$\label{eq:phi_aux_eq_fp}
\Phi^{{\mathbf{s}}}({\mathbf{s}}(\bar{\mathbf{r}}); \bar{\mathbf{r}}) = ({\mathbf{a}}+{\mathbf{M}^{{\mathbf{d}}}}\bar{\mathbf{r}}+ {\mathbf{M}}^{{\mathbf{s}}}{\mathbf{s}}(\bar{\mathbf{r}}) - {\mathbf{d}})^+ = {\mathbf{s}}(\bar{\mathbf{r}}).$$ As shown in @elsinger09, this fixed point exists and is unique since ${\mathbf{M}^{{\mathbf{s}}}}$ has the Elsinger Property.
The following algorithm delivers for given $\bar{\mathbf{r}}$ a series of vectors ${\mathbf{w}}^k\in{\mathbb{R}}^n$ that converge to a vector whose positive part is the fixed point of . To explain this in more detail, first define for a given vector ${\mathbf{w}}\in{\mathbb{R}}^n$ the set $$P({\mathbf{w}}) = \left\{i\in\mathcal N: w_i \ge 0\right\}$$ and the matrix $${\mathbf{\Gamma}}({\mathbf{w}}) = {{\rm diag}}({\mathbf{w}}\ge {\mathbf{0}}_n)$$ as the corresponding diagonal matrix. Note that these definitions of $P({\mathbf{w}})$ and ${\mathbf{\Gamma}}({\mathbf{w}})$ slightly differ from the original ones in @elsinger09, where a strictly larger sign was used. By our definition of default in , a firm with zero equity value can still be not in default in the sense that all obligations can fully served. This situation is referred to as *borderline firms* (cf. Section \[subsec:trial\_error\_alg\_incr\]). However, this modification does not change the forthcoming theoretical results.
[2A]{}\[alg:equity\]
1. For $k=0$, set ${\mathbf{w}}^0={\mathbf{a}}+{\mathbf{M}^{{\mathbf{d}}}}\bar{\mathbf{r}}-{\mathbf{d}}$ and determine $P({\mathbf{w}}^0)$ and ${\mathbf{\Gamma}}({\mathbf{w}}^0)$.
2. \[alg:fp\_equity\] For $k\ge 1$, solve $\Psi_{{\mathbf{w}}^k}({\mathbf{w}})={\mathbf{w}}$ where $$\label{eq:psi_s}
\Psi_{{\mathbf{w}}^k}({\mathbf{w}}) = {\mathbf{w}}^0 + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^k){\mathbf{w}}$$ and denote the solution by ${\mathbf{w}}^{k+1}$, i.e. $\Psi_{{\mathbf{w}}^k}\left({\mathbf{w}}^{k+1}\right)= {\mathbf{w}}^{k+1}$. Determine $P({\mathbf{w}}^{k+1})$ and ${\mathbf{\Gamma}}({\mathbf{w}}^{k+1})$.
3. If $P({\mathbf{w}}^k)=P({\mathbf{w}}^{k+1})$, stop the algorithm. Else, set $k = k + 1$ and proceed with step \[alg:fp\_equity\].
Before the properties of Algorithm \[alg:equity\] are shown, we give some explanations for a better understanding of its functioning. The starting point is ${\mathbf{w}}^0$, which is the difference between ${\mathbf{a}}+{\mathbf{M}^{{\mathbf{d}}}}\bar{\mathbf{r}}$ and ${\mathbf{d}}$. The sum represents the firms incomes on their balance sheet that consists of the external assets and the payments due to cross-ownership of debt. Note that in this step the potential income from equity cross-ownership is ignored since ${\mathbf{M}^{{\mathbf{s}}}}$ does not appear. The idea is now as follows: The firms not in $P({\mathbf{w}}^0)$ are not able to fully satisfy their liabilities (assuming debt payments of $\bar{\mathbf{r}}$) and will be in default. On the other hand, the firms that are in $P({\mathbf{w}}^0)$ will be able to satisfy their obligees and can be regarded as solvent (again assuming debt payments of $\bar{\mathbf{r}}$), even though no intersystem payments due to equity cross-ownership are taken into account. As a consequence, the equity payments of the non-defaulting firms are added into the system via the product ${\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^0){\mathbf{w}}$. We can interpret the vector ${\mathbf{w}}^0$, as well as the other iterates ${\mathbf{w}}^k$, as pseudo equity vectors that give us information about solvent and defaulting firms under the current debt and equity payments. The fact that the entries of ${\mathbf{w}}^k$ can be negative prevents that they can be naturally interpreted as equity vectors which is why we use the term “pseudo”.
The difference compared to the Picard Algorithm is that a linear equation system is solved to achieve a new equity payment vector instead of applying $\Phi$ to $(\bar{\mathbf{r}}^t, ({\mathbf{w}}^k)^t)^t$. This is because for the fixed point of $\Psi_{{\mathbf{w}}^k}$ it holds together with that $$\label{eq:eq_alg_inv}
{\mathbf{w}}^{k+1} = ({\mathbf{I}}_n - {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^k))^{-1} {\mathbf{w}}^0$$ Note that the inverse matrix exists since ${\mathbf{M}^{{\mathbf{s}}}}$ and hence ${\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{s}}^k)$ have the Elsinger Property.
The vector ${\mathbf{w}}^1$ can be interpreted as an “updated” version of ${\mathbf{w}}^0$ since the equity of the non-defaulting firms that are in $P({\mathbf{w}}^0)$ is included in ${\mathbf{w}}^1$. Based on the updated vector ${\mathbf{w}}^1$ it might appear that some firms that are not in $P({\mathbf{w}}^0)$ have now non-negative entries in ${\mathbf{w}}^1$. This can be concluded from ${\mathbf{w}}^1\ge{\mathbf{w}}^0$ that we will show later. But these firms are now also able to contribute equity payments to the system. Consequently, the system has to be updated again by determining ${\mathbf{w}}^2$. The procedure continues until the set of defaulting firms stays the same from one iteration step to the next one.
\[prop:equ\_comp\_conv\] Given a fixed vector of debt payments $\bar{\mathbf{r}}\ge{\mathbf{0}}_n$:
1. Algorithm \[alg:equity\] generates an increasing sequence of vectors ${\mathbf{w}}^k$.
2. Let $1\le l\le n$ such that $$\label{eq:iter_eq_final}
l:=\min\{j\in\{0, 1 {,\ldots,}n\}:P({\mathbf{w}}^j)=P({\mathbf{w}}^{j+1})\}.$$ Then ${\mathbf{s}}(\bar{\mathbf{r}})=({\mathbf{w}}^{l+1})^+$ is the fixed point of the mapping $\Phi^{{\mathbf{s}}}(\cdot; \bar{\mathbf{r}})$.
3. Let $d_0 = |P({\mathbf{w}}^0)|\in\{0, 1 {,\ldots,}n\}$ be the number of firms with a positive entry in ${\mathbf{w}}^0$. The fixed point ${\mathbf{s}}(\bar{\mathbf{r}})$ is reached after no more than $n-d_0$ iteration steps.
<!-- -->
1. This part of the Proposition is shown by @elsinger09. We give a different version of the proof. Because of , the fact that ${\mathbf{\Gamma}}({\mathbf{w}}^0){\mathbf{w}}^0\ge{\mathbf{0}}_n$ and using the series representation of $({\mathbf{I}}_n - {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^0))^{-1}$ as shown in Lemma \[lem:inv\_xos-mat\] of the Appendix we get $$\begin{split}
{\mathbf{w}}^1 &= ({\mathbf{I}}_n - {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^0))^{-1} {\mathbf{w}}^0\\
&= ({\mathbf{I}}_n + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^0) + ({\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^0))^2 + \ldots){\mathbf{w}}^0 \\
&= {\mathbf{w}}^0 + {\mathbf{M}^{{\mathbf{s}}}}\underbrace{{\mathbf{\Gamma}}({\mathbf{w}}^0){\mathbf{w}}^0}_{\ge{\mathbf{0}}_n} +
{\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^0){\mathbf{M}^{{\mathbf{s}}}}\underbrace{{\mathbf{\Gamma}}({\mathbf{w}}^0){\mathbf{w}}^0}_{\ge{\mathbf{0}}_n}+\ldots \\
&\ge {\mathbf{w}}^0,
\end{split}$$ which is the induction start. For the induction step we assume ${\mathbf{w}}^k\ge{\mathbf{w}}^{k-1}$ and ${\mathbf{\Gamma}}({\mathbf{w}}^k)\ge{\mathbf{\Gamma}}({\mathbf{w}}^{k-1})$ following from it. We need to show that ${\mathbf{w}}^{k+1} \ge {\mathbf{w}}^k$, or, equivalently, ${\mathbf{w}}^{k+1} = {\mathbf{w}}^k + {\mathbf{e}}$ where ${\mathbf{e}}\ge {\mathbf{0}}_n$. Since ${\mathbf{\Gamma}}({\mathbf{w}}^k){\mathbf{w}}^k\ge{\mathbf{\Gamma}}({\mathbf{w}}^{k-1}){\mathbf{w}}^k$ and ${\mathbf{w}}^k={\mathbf{w}}^0+{\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^{k-1}){\mathbf{w}}^k$, it follows that $${\mathbf{u}}:= {\mathbf{w}}^0 + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^k){\mathbf{w}}^k - {\mathbf{w}}^k={\mathbf{M}^{{\mathbf{s}}}}({\mathbf{\Gamma}}({\mathbf{w}}^k)-{\mathbf{\Gamma}}({\mathbf{w}}^{k-1})){\mathbf{w}}^k \ge {\mathbf{0}}_n.$$ With this definition we have that $${\mathbf{w}}^k+ {\mathbf{e}}= {\mathbf{w}}^0 + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^k)({\mathbf{w}}^k+ {\mathbf{e}}) = {\mathbf{w}}^0 + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^k){\mathbf{w}}^k + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^k){\mathbf{e}}$$ and we can rearrange to $${\mathbf{e}}- {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^k){\mathbf{e}}= {\mathbf{w}}^0 + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^k){\mathbf{w}}^k - {\mathbf{w}}^k = {\mathbf{u}}\ge {\mathbf{0}}_n.$$ Solving this for ${\mathbf{e}}$ leads to $${\mathbf{e}}= ({\mathbf{I}}_n - {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^k))^{-1}{\mathbf{u}}\ge {\mathbf{0}}_n$$ from which follows that ${\mathbf{w}}^{k+1}\ge{\mathbf{w}}^k$.
2. First, we will show that once a “stable system” has been reached, i.e. for $k\ge0$ we have $P({\mathbf{w}}^k)=P({\mathbf{w}}^{k+1})$, the sequence ${\mathbf{w}}^k$ will be constant. Let $l$ be defined as above in . Note that such an $l$ exists since ${\mathbf{w}}^k\le{\mathbf{w}}^{k+1}$ and therefore $P({\mathbf{w}}^{k+1})\supseteq P({\mathbf{w}}^k)$ for all $k\ge0$ as shown above. Due to ${\mathbf{\Gamma}}({\mathbf{w}}^l) = {\mathbf{\Gamma}}({\mathbf{w}}^{l+1})$, it follows because of $$\Psi_{{\mathbf{w}}^l}({\mathbf{w}}) = {\mathbf{w}}^0 + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^l){\mathbf{w}}= {\mathbf{w}}^0 + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^{l+1}){\mathbf{w}}= \Psi_{{\mathbf{w}}^{l+1}}({\mathbf{w}})$$ that the two mappings $\Psi_{{\mathbf{w}}^l}$ and $\Psi_{{\mathbf{w}}^{l+1}}$ are the same and consequently ${\mathbf{w}}^{l+1}={\mathbf{w}}^{l+2}$. A direct consequence is that $P({\mathbf{w}}^{l+2}) = P({\mathbf{w}}^{l+1}) = P({\mathbf{w}}^l)$ which implies ${\mathbf{\Gamma}}({\mathbf{w}}^{l+2})={\mathbf{\Gamma}}({\mathbf{w}}^{l+1})={\mathbf{\Gamma}}({\mathbf{w}}^l)$. By iteration, all following vectors will be equal to ${\mathbf{w}}^{l+1}$.
What remains to be shown out is that the positive part of this iteration vector is the fixed point of the mapping $\Phi^{{\mathbf{s}}}(\cdot; \bar{\mathbf{r}})$. Since ${\mathbf{w}}^{l+1}$ is the fixed point of $\Psi_{{\mathbf{w}}^l}$ it holds that ${\mathbf{w}}^{l+1} = {\mathbf{w}}^0 + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^l){\mathbf{w}}^{l+1}$. This yields to $$\label{eq:fp_prop_w}
\begin{split}
\Phi^{{\mathbf{s}}}(({\mathbf{w}}^{l+1})^+; \bar{\mathbf{r}}) &= ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}\bar{\mathbf{r}}+ {\mathbf{M}^{{\mathbf{s}}}}({\mathbf{w}}^{l+1})^+ - {\mathbf{d}})^+ \\
&= ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}\bar{\mathbf{r}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^{l+1}){\mathbf{w}}^{l+1} - {\mathbf{d}})^+ \\
&= ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}\bar{\mathbf{r}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^l){\mathbf{w}}^{l+1} - {\mathbf{d}})^+ \\
&= ({\mathbf{w}}^0 + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{\Gamma}}({\mathbf{w}}^l){\mathbf{w}}^{l+1})^+ \\
&= ({\mathbf{w}}^{l+1})^+.
\end{split}$$
3. As shown, the series ${\mathbf{w}}^k$ increases which means that the firms in $P({\mathbf{w}}^0)$ will maintain their positive entries in every further iteration step. The same statement holds for every firm $i$ with $w_i^k < 0$ and $w_i^{k+1} \ge 0$ for any $k\ge 0$. Because of (ii) this means that the number of iteration steps would certainly be maximal, if in every iteration step the set $P({\mathbf{w}}^k)$ increased by one and if $|P({\mathbf{w}}^{l+1})|=n$. In that case we would therefore have $|P({\mathbf{w}}^{l+1})|-|P({\mathbf{w}}^0)| = n-d_0$ maximal possible iteration steps.
Using Algorithm \[alg:equity\] to get an equity vector for a given debt payment vector, we can now present the algorithm to calculate the solution ${\mathbf{R}}^*$. In the sequel, we will make use of the mapping $\Phi^{{\mathbf{d}}}:({\mathbb{R}}_0^+)^n\to({\mathbb{R}}_0^+)^n$ defined by $$\label{eq:phi_aux_debt}
\Phi^{{\mathbf{d}}}({\mathbf{r}}; \bar{\mathbf{s}}) = \min\{{\mathbf{d}}, {\mathbf{a}}+{\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}+{\mathbf{M}^{{\mathbf{s}}}}\bar{\mathbf{s}}\}$$ that represents the debt component of $\Phi$ for a given equity payment vector $\bar{\mathbf{s}}\ge{\mathbf{0}}_n$.
[3]{}\[Elsinger Algorithm\]\[alg:elsinger\] Set ${\varepsilon}>0$.
1. For $k=0$, choose ${\mathbf{r}}^0\in\{{\mathbf{r}_{\rm small}}, {\mathbf{r}_{\rm great}}\}$ and determine ${\mathbf{s}}({\mathbf{r}}^0)$ using Algorithm \[alg:equity\].
2. \[alg:fp\_elsinger\] For $k\ge1$, set ${\mathbf{r}}^k = \Phi^{{\mathbf{d}}}({\mathbf{r}}^{k-1}; {\mathbf{s}}({\mathbf{r}}^{k-1}))$ and calculate ${\mathbf{s}}({\mathbf{r}}^k)$ by Algorithm \[alg:equity\].
3. If $\left\|\begin{pmatrix}{\mathbf{r}}^{k-1}\\{\mathbf{s}}^{k-1}\end{pmatrix} - \begin{pmatrix}{\mathbf{r}}^k\\{\mathbf{s}}^k\end{pmatrix}\right\|<{\varepsilon}$, stop the algorithm. Else, set $k = k + 1$ and proceed with step \[alg:fp\_elsinger\].
The algorithm starts either assuming that all firms can fully deliver on their debt obligations (${\mathbf{r}}^0={\mathbf{r}_{\rm great}}={\mathbf{d}}$) or that all firms have only their exogenous assets for paying their obligations (${\mathbf{r}}^0={\mathbf{r}_{\rm small}}=\min\{{\mathbf{d}}, {\mathbf{a}}\}$). With this payment vector, the corresponding equity payments are obtained by using Algorithm \[alg:equity\]. In the next step the debt vector has to be adapted to the new equity payments which is done applying $\Phi^{{\mathbf{d}}}$ to the previous debt vector. The updated debt payment vector is then used for determining a new equity payment vector. This procedure continues until the iterates are sufficiently close to each other. Additional to the original algorithm first presented in @elsinger09, Algorithm \[alg:elsinger\] contains the second possible starting point ${\mathbf{r}_{\rm small}}$. We will show in the next proposition that if ${\mathbf{r}}^0={\mathbf{r}_{\rm small}}$ is chosen, the vector of debt and equity payments establish an increasing sequence and hence converges to the solution ${\mathbf{R}}^*$ from below, while for ${\mathbf{r}}^0={\mathbf{r}_{\rm great}}$, it converges from above.
\[prop:elsinger\_conv\] The Elsinger Algorithm delivers a series of decreasing vectors if ${\mathbf{r}}^0={\mathbf{r}_{\rm great}}$ and a series of increasing vectors if ${\mathbf{r}}^0={\mathbf{r}_{\rm small}}$. Both series converge to the fixed point of the mapping $\Phi$ in .
The decreasing part is shown in @elsinger09, we only have to show that the debt iterate in the algorithm therein is identical to ${\mathbf{r}}^k$ in Algorithm \[alg:elsinger\]. With our notation, the iterate of the debt component in @elsinger09 is defined as $$\label{eq:phi_elsinger}
{\mathbf{r}}^k = \min\{{\mathbf{d}}, ({\mathbf{w}}^*({\mathbf{r}}^{k-1}) + {\mathbf{d}})^+\},$$ where ${\mathbf{w}}^*({\mathbf{r}}^{k-1})$ is the solution of $$\label{eq:equity_fp_els}
{\mathbf{w}}= {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^{k-1} + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{w}}^+ - {\mathbf{d}}.$$ However, it follows from that for $\bar{\mathbf{r}}={\mathbf{r}}^{k-1}$, $$({\mathbf{w}}^{l+1})^+ = \Phi^{{\mathbf{s}}}\left(({\mathbf{w}}^{l+1})^+;{\mathbf{r}}^{k-1}\right) = \left({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^{k-1} + {\mathbf{M}^{{\mathbf{s}}}}({\mathbf{w}}^{l+1})^+ - {\mathbf{d}}\right)^+ = \left({\mathbf{w}}^*({\mathbf{r}}^{k-1})\right)^+,$$ where ${\mathbf{w}}^{l+1}$ is the result of Algorithm \[alg:equity\] with the debt payment vector ${\mathbf{r}}^{k-1}$, i.e. $({\mathbf{w}}^*({\mathbf{r}}^{k-1}))^+ = {\mathbf{s}}({\mathbf{r}}^{k-1})$. Because of , we have that $${\mathbf{w}}^*({\mathbf{r}}^{k-1}) = {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^{k-1} - {\mathbf{d}}+ {\mathbf{M}^{{\mathbf{s}}}}({\mathbf{w}}^*({\mathbf{r}}^{k-1}))^+,$$ from which follows with and ${\mathbf{a}}\ge{\mathbf{0}}_n$ that $$\begin{split}
{\mathbf{r}}^k &= \min\{{\mathbf{d}}, ({\mathbf{w}}^*({\mathbf{r}}^{k-1}) + {\mathbf{d}})^+\} \\
&= \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^{k-1} + {\mathbf{M}^{{\mathbf{s}}}}({\mathbf{w}}^*({\mathbf{r}}^{k-1}))^+\} \\
&= \Phi^{{\mathbf{d}}}({\mathbf{r}}^{k-1}; ({\mathbf{w}}^*({\mathbf{r}}^{k-1}))^+)\\
&= \Phi^{{\mathbf{d}}}({\mathbf{r}}^{k-1}; {\mathbf{s}}({\mathbf{r}}^{k-1})).
\end{split}$$
What remains to be shown is that for the starting point ${\mathbf{r}}^0={\mathbf{r}_{\rm small}}$ the generated series increases and converges to ${\mathbf{R}}^*$, which is done by induction. For the induction start check that $${\mathbf{r}}^0 = \min\{{\mathbf{d}}, {\mathbf{a}}\} \le \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^0 + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{r}}^0)\} = \Phi^{{\mathbf{d}}}({\mathbf{r}}^0;{\mathbf{s}}({\mathbf{r}}^0)) = {\mathbf{r}}^1.$$ As shown in @elsinger09, the result ${\mathbf{w}}^*({\mathbf{r}})$ of Algorithm \[alg:equity\] is increasing in ${\mathbf{r}}$ from which also follows that ${\mathbf{s}}({\mathbf{r}})$ is increasing in ${\mathbf{r}}$. Hence, ${\mathbf{s}}({\mathbf{r}}^0)\le{\mathbf{s}}({\mathbf{r}}^1)$ which completes the induction start. Assume for the induction step that ${\mathbf{r}}^{k-1}\le {\mathbf{r}}^k$ and consequently ${\mathbf{s}}({\mathbf{r}}^{k-1})\le {\mathbf{s}}({\mathbf{r}}^k)$. The next debt iterate emerges as $$\begin{split}
{\mathbf{r}}^{k+1} &= \min\{{\mathbf{d}},{\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^k + {\mathbf{M}^{{\mathbf{s}}}}({\mathbf{s}}({\mathbf{r}}^k))^+\} \\
&\ge \min\{{\mathbf{d}},{\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^{k-1} + {\mathbf{M}^{{\mathbf{s}}}}({\mathbf{s}}({\mathbf{r}}^{k-1}))^+\} \\
&= {\mathbf{r}}^k,
\end{split}$$ from which also follows that ${\mathbf{s}}({\mathbf{r}}^{k+1})\ge{\mathbf{s}}({\mathbf{r}}^k)$ and, hence, the increasing property of the series. For the convergence, check that ${\mathbf{s}}({\mathbf{r}}^k)\ge{\mathbf{0}}_n$ and it holds that $${\mathbf{s}}({\mathbf{r}}^k) = ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^k + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{r}}^k) - {\mathbf{d}})^+ \le {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{r}}^k) + ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^k - {\mathbf{d}})^+.$$ Because of ${\mathbf{r}}^k\le{\mathbf{r}}^*$, it follows after some rearrangements that $$\label{eq:els_algo_start_bound}
{\mathbf{s}}({\mathbf{r}}^k) \le ({\mathbf{I}}_n - {\mathbf{M}^{{\mathbf{s}}}})^{-1} ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^k - {\mathbf{d}})^+ \le ({\mathbf{I}}_n - {\mathbf{M}^{{\mathbf{s}}}})^{-1} ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^* - {\mathbf{d}})^+,$$ hence the series ${\mathbf{s}}({\mathbf{r}}^k)$ is bounded from above as well and therefore converges to some $\mathbf{s}^*$ from below. The fact that $\Phi^{{\mathbf{s}}}$ is continuous in $({\mathbf{r}}^t,{\mathbf{s}}^t)^t$ implies together with $\Phi^{{\mathbf{s}}}({\mathbf{s}}({\mathbf{r}}^k);{\mathbf{r}}^k)={\mathbf{s}}({\mathbf{r}}^k)$ that $\Phi^{{\mathbf{s}}}({\mathbf{s}}^*;{\mathbf{r}}^*)={\mathbf{s}}^*$. Thus $(({\mathbf{r}}^*)^t,({\mathbf{s}}^*)^t)^t$ solves . Similarly, we can argue that because of the continuity of $\Phi^{{\mathbf{d}}}$, $\Phi^{{\mathbf{d}}}({\mathbf{r}}^*;{\mathbf{s}}^*)={\mathbf{r}}^*$ from which follows that $(({\mathbf{r}}^*)^t,({\mathbf{s}}^*)^t)^t$ also solves and therefore must be the fixed point ${\mathbf{R}}^*$.
As described above, the Elsinger Algorithm determines the equity component of the iterates ${\mathbf{R}}^k$ in a different way than the Picard Iteration. An important consequence of this approach is that the iterates of the Elsinger Algorithm will for the decreasing version be in every step smaller than the iterates of the Picard Algorithm, as we will show in the next proposition. The same statement holds for the increasing version of both procedures, where the iterates from the Elsinger Algorithm will be greater than the iterates form the Picard Algorithm. Both procedures are difficult to compare concerning their total calculation effort due to different ways of obtaining the next equity iterate (cf. Section \[subsec:algo\_efficency\]). However, if we only take the number of needed iterations as a quality criterion, we can conclude that the Elsinger Algorithm converges faster to ${\mathbf{R}}^*$ than the Picard Iteration, no matter whether the algorithms start from the upper or the lower boundary.
\[prop:elsinger\_better\_picard\] Let ${\mathbf{R}}^k_{\rm P} = (({\mathbf{r}}^k_{\rm P})^t, ({\mathbf{s}}^k_{\rm P})^t)^t$ be the $k$-th iterate of the Picard Algorithm and ${\mathbf{R}}^k_{\rm E} = (({\mathbf{r}}^k_{\rm E})^t, ({\mathbf{s}}^k_{\rm E})^t)^t$ the corresponding iterate of the Elsinger Algorithm.
(i) For any iterate $k\ge 0$ it holds that ${\mathbf{R}}^k_{\rm P}\ge{\mathbf{R}}^k_{\rm E}$ if ${\mathbf{R}}^0_{\rm P} = {\mathbf{R}_{\rm great}}$ and ${\mathbf{R}}^0_{\rm E} =({\mathbf{r}_{\rm great}}^t, {\mathbf{s}}({\mathbf{r}_{\rm great}})^t)^t$. In case of ${\mathbf{R}}^0_{\rm P} = {\mathbf{R}_{\rm small}}$ and ${\mathbf{R}}^0_{\rm E} =({\mathbf{r}_{\rm small}}^t, {\mathbf{s}}({\mathbf{r}_{\rm small}})^t)^t$, we have that ${\mathbf{R}}^k_{\rm P}\le{\mathbf{R}}^k_{\rm E}$ for every iterate.
(ii) Let ${\mathbf{R}}^k$, $k\ge1$, be an iterate either of the Picard Algorithm with ${\mathbf{R}}^0={\mathbf{R}_{\rm great}}$ or of the Elsinger Algorithm with ${\mathbf{R}}^0 = ({\mathbf{r}_{\rm great}}^t, {\mathbf{s}}({\mathbf{r}_{\rm great}})^t)^t$. Then ${\mathbf{R}}^{k+1}_{\rm P}({\mathbf{R}}^k)\ge{\mathbf{R}}^{k+1}_{\rm E}({\mathbf{R}}^k)$. If the starting vector is either ${\mathbf{R}}^0={\mathbf{R}_{\rm small}}$ or ${\mathbf{R}}^0 = ({\mathbf{r}_{\rm small}}^t, {\mathbf{s}}({\mathbf{r}_{\rm small}})^t)^t$, it holds that ${\mathbf{R}}^{k+1}_{\rm P}({\mathbf{R}}^k) \le {\mathbf{R}}^{k+1}_{\rm E}({\mathbf{R}}^k)$.
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(i) The assertion is shown by induction. For $k=0$, suppose that the upper boundary is the starting vector for both algorithms. In Equation it was shown that $${\mathbf{s}}^0_{\rm E} = {\mathbf{s}}({\mathbf{d}}) \le ({\mathbf{I}}_n-{\mathbf{M}^{{\mathbf{s}}}})^{-1}({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}- {\mathbf{d}})^+ = {\mathbf{s}_{\rm great}}= {\mathbf{s}}^0_{\rm P}.$$ Since ${\mathbf{r}}^0_{\rm E}={\mathbf{r}}^0_{\rm P}={\mathbf{d}}$, the induction start is complete. Assume now, that for $k\ge1$ it holds that ${\mathbf{R}}^k_{\rm P}\ge{\mathbf{R}}^k_{\rm E}$. From Proposition \[prop:elsinger\_conv\], we know that ${\mathbf{R}}^{k+1}_{\rm E}\le{\mathbf{R}}^k_{\rm E}$. This leads to $${\mathbf{r}}^{k+1}_{\rm P} = \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^k_{\rm P} + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}^k_{\rm P}\} \ge
\min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^k_{\rm E} + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}^k_{\rm E}\} = {\mathbf{r}}^{k+1}_{\rm E}$$ and $$\begin{split}
{\mathbf{s}}^{k+1}_{\rm P} &= ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^k_{\rm P} + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}^k_{\rm P} - {\mathbf{d}})^+\\
&\ge ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^k_{\rm E} + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}^k_{\rm E} - {\mathbf{d}})^+ \\
&\ge ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^{k+1}_{\rm E} + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}^{k+1}_{\rm E} - {\mathbf{d}})^+ \\
&= {\mathbf{s}}^{k+1}_{\rm E}.
\end{split}$$ If the starting vector is the lower boundary and the series ${\mathbf{R}}^k_{\rm P}$ and ${\mathbf{R}}^k_{\rm E}$ are increasing, the argumentation is similar.
(ii) We prove the claim for the decreasing version of the algorithms, the proof for the reverse direction is similar. First, let ${\mathbf{R}}^k={\mathbf{R}}^k_{\rm P}$. The next iteration of the debt component is equal for both algorithms, i.e. ${\mathbf{r}}^{k+1}_{\rm P} = \Phi^{{\mathbf{d}}}({\mathbf{r}}^k;{\mathbf{s}}^k) = {\mathbf{r}}^{k+1}_{\rm E}$. For the equity component, it holds that ${\mathbf{s}}^{k+1}_{\rm P}=\Phi^{{\mathbf{s}}}({\mathbf{s}}^k;{\mathbf{r}}^k)$. The mapping $\Phi^{{\mathbf{s}}}(\cdot;{\mathbf{r}}^k)$ has a unique fixed point, that we denote by ${\mathbf{s}}({\mathbf{r}}^k)$ and that can be obtained via a Picard Iteration: $$\lim_{l\to\infty}\left(\Phi^{{\mathbf{s}}}\right)^l({\mathbf{s}}^k;{\mathbf{r}}^k) = {\mathbf{s}}({\mathbf{r}}^k).$$ The iterates obviously form a decreasing sequence so that $${\mathbf{s}}^{k+1}_{\rm P} \ge {\mathbf{s}}({\mathbf{r}}^k) \ge {\mathbf{s}}({\mathbf{r}}^{k+1}) = {\mathbf{s}}^{k+1}_{\rm E},$$ where the second inequality follows from the fact that ${\mathbf{s}}({\mathbf{r}})$ is increasing in ${\mathbf{r}}$ (cf. @elsinger09). If the $k$-th iterate is given by ${\mathbf{R}}^k={\mathbf{R}}^k_{\rm E}$, the arguments are analogous to the ones above.
A Hybrid Algorithm
------------------
To motivate the approach of the next algorithm, we have to compare the functioning of the Elsinger Algorithm and the Picard Iteration. The major difference between both iterations emerges in the calculation of the equity component. Suppose that we are in iteration step $k\ge0$ and want to calculate the next iteration of the equity component. We ignore for an instant that both algorithms deliver different iterates and assume that the $k$-th iterate is given by ${\mathbf{R}}^k=(({\mathbf{r}}^k)^t, ({\mathbf{s}}^k)^t)^t$. In the Elsinger Algorithm, ${\mathbf{r}}^{k+1}$ is calculated first and then ${\mathbf{s}}^{k+1}$ as the fixed point of $\Phi^{{\mathbf{s}}}(\cdot;{\mathbf{r}}^{k+1})$ so that it holds that ${\mathbf{s}}^{k+1}=\Phi^{{\mathbf{s}}}({\mathbf{s}}^{k+1};{\mathbf{r}}^{k+1})$. The Picard iterate, on the other side, can be written as ${\mathbf{s}}^{k+1}=\Phi^{{\mathbf{s}}}({\mathbf{s}}^k;{\mathbf{r}}^k)$ from which it becomes clear that the Picard Iteration neither uses the “updated” debt vector ${\mathbf{r}}^{k+1}$, nor does it solve a separate fixed point mapping to obtain ${\mathbf{s}}^{k+1}$.
The determination of the debt component ${\mathbf{r}}^{k+1}$, however, is comparable in both algorithms. Again starting with ${\mathbf{R}}^k$ we have that ${\mathbf{r}}^{k+1}=\Phi^{{\mathbf{d}}}({\mathbf{r}}^k;{\mathbf{s}}^k)$ for both procedures. An obvious extension of the Elsinger Algorithm would be to utilize the principle used for the equity component for the debt component as well. In the article of @eisenberg01, this concept is used for systems with no cross-ownership of equity, i.e. where ${\mathbf{M}^{{\mathbf{s}}}}={\mathbf{0}}_{n\times n}$. In this subsection we will generalize the results of this work and it will turn out that combining both ideas, the one of @elsinger09 and the one of @eisenberg01, will help to minimize the number of needed iteration steps of the global algorithm to find ${\mathbf{R}}^*$.
To explain this idea in more detail, say that for a debt payment vector $\bar{\mathbf{r}}\in[{\mathbf{r}_{\rm small}}, {\mathbf{r}_{\rm great}}]$ we have a corresponding equity vector ${\mathbf{s}}(\bar{\mathbf{r}})$, that is, a fixed point of the mapping $\Phi^{{\mathbf{s}}}(\cdot; \bar{\mathbf{r}})$ in . In the Elsinger Algorithm the next debt iterate emerges as $\Phi^{{\mathbf{d}}}(\bar{\mathbf{r}}; {\mathbf{s}}(\bar{\mathbf{r}}))$. Instead of using this iterate, our aim is now to find the fixed point of $\Phi^{{\mathbf{d}}}(\cdot; {\mathbf{s}}(\bar{\mathbf{r}}))$ as the new iterate. This can be done using the following Algorithm.
[4A]{}\[alg:debt\] Suppose ${\mathbf{s}}\ge{\mathbf{0}}_n$.
1. For $k=0$, set ${\mathbf{r}}^0=\bar{\mathbf{r}}$ and determine $D({\mathbf{r}}^0,{\mathbf{s}})$ and ${\mathbf{\Lambda}}({\mathbf{r}}^0,{\mathbf{s}})$.
2. \[alg:fp\_debt\] For $k\ge 1$, solve $\Theta_{{\mathbf{r}}^{k-1},{\mathbf{s}}}({\mathbf{r}}) = {\mathbf{r}}$ where $$\label{eq:fp_debt}
\begin{split}
\Theta_{{\mathbf{r}}^{k-1},{\mathbf{s}}}({\mathbf{r}}) &= {\mathbf{\Lambda}}({\mathbf{r}}^{k-1},{\mathbf{s}})\left({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}\left({\mathbf{\Lambda}}({\mathbf{r}}^{k-1},{\mathbf{s}}){\mathbf{r}}+
\left({\mathbf{I}}_n-{\mathbf{\Lambda}}({\mathbf{r}}^{k-1},{\mathbf{s}})\right){\mathbf{d}}\right) + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}\right) \\
&\qquad + \left({\mathbf{I}}_n - {\mathbf{\Lambda}}({\mathbf{r}}^{k-1},{\mathbf{s}})\right){\mathbf{d}}\end{split}$$
3. Denote the solution by ${\mathbf{r}}^k$, i.e. $\Theta_{{\mathbf{r}}^{k-1},{\mathbf{s}}}({\mathbf{r}}^k)={\mathbf{r}}^k$ and determine $D({\mathbf{r}}^k,{\mathbf{s}})$ and ${\mathbf{\Lambda}}({\mathbf{r}}^k,{\mathbf{s}})$.
4. If $D({\mathbf{r}}^k,{\mathbf{s}}) = D({\mathbf{r}}^{k-1},{\mathbf{s}})$, stop the algorithm. Else, set $k = k + 1$ and proceed with step \[alg:fp\_debt\].
The algorithm is identical to the one given in @eisenberg01 with the modification that some additional fixed payments due to equity cross-ownership are included. It solves for a fixed amount of equity payment ${\mathbf{s}}\ge{\mathbf{0}}_n$, i.e. is the fixed point of the mapping $\Phi^{{\mathbf{d}}}(\cdot, {\mathbf{s}})$, as we will show in the next proposition. Denote this fixed point by ${\mathbf{r}}^*({\mathbf{s}})$ for instance. In the Hybrid Algorithm following later on, ${\mathbf{r}}^*({\mathbf{s}})$ is used as the next iterate for the debt component. To see the difference between the calculation of the debt component in the Elsinger Algorithm, assume that an arbitrary debt payment vector ${\mathbf{r}}\in[{\mathbf{0}}_n,{\mathbf{d}}]$ is given and that the corresponding equity payment vector ${\mathbf{s}}({\mathbf{r}})$ is given too. The fixed point of the mapping $\Phi^{{\mathbf{d}}}(\cdot;{\mathbf{s}}({\mathbf{r}}))$ can on the one hand be obtained using Algorithm \[alg:debt\] above, but on the other hand, we could also use a Picard Iteration, since for any ${\mathbf{r}}\in[{\mathbf{0}}_n,{\mathbf{d}}]$ it holds that $${\mathbf{0}}_n \le \Phi^{{\mathbf{d}}}({\mathbf{r}};{\mathbf{s}}({\mathbf{d}})) = \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{d}})\} \le {\mathbf{d}}.$$ Starting with the vector ${\mathbf{r}}$, the fixed point ${\mathbf{r}}^*({\mathbf{s}})$ is given as $$\label{eq:picard_debt}
{\mathbf{r}}^*({\mathbf{s}}) = \lim_{l \to \infty}(\Phi^{{\mathbf{d}}})^l({\mathbf{r}};{\mathbf{s}}({\mathbf{d}})).$$ In the Elsinger Algorithm, however, the next iterate for the debt component is defined as $\Phi^{{\mathbf{d}}}({\mathbf{r}};{\mathbf{s}}({\mathbf{r}}))$ which is therefore the first iterate of the Picard Iteration in . Hence, one can say that using in the Elsinger Algorithm, a simple mapping is applied to obtain the next iterate, whereas in the Hybrid Algorithm, the fixed point of a mapping is determined. In Proposition \[prop:hybrid\_better\_elsinger\], we will show that when using the idea of the latter algorithm, the iterates ${\mathbf{R}}^k$ will always be closer to the searched solution ${\mathbf{R}}^*$.
\[prop:debt\_comp\_conv\] Let $\bar{\mathbf{r}}\in[{\mathbf{r}_{\rm small}}, {\mathbf{r}_{\rm great}}]$ be a debt payment vector and ${\mathbf{s}}\ge{\mathbf{0}}_n$ a vector of equity payments such that $$\label{eq:assu_debt_decr}
\Phi^{{\mathbf{d}}}(\bar{\mathbf{r}}; {\mathbf{s}}) \le \bar{\mathbf{r}}.$$
1. Algorithm \[alg:debt\] generates a well-defined decreasing sequence of vectors ${\mathbf{r}}^k$.
2. Let $1\le l\le n$ such that $$\label{eq:iter_debt_final}
l:=\min\{j\in\{0, 1 {,\ldots,}n\}:D({\mathbf{r}}^j, {\mathbf{s}})=D({\mathbf{r}}^{j+1}, {\mathbf{s}})\}.$$ Then ${\mathbf{r}}^*({\mathbf{s}})={\mathbf{r}}^{l+1}$ is the fixed point of the mapping $\Phi^{{\mathbf{d}}}(\cdot; {\mathbf{s}})$ defined in .
3. Let $d_0=|D(\bar{\mathbf{r}}, {\mathbf{s}})|$ be the number of firms in default under $\bar{\mathbf{r}}$ and ${\mathbf{s}}$. The fixed point ${\mathbf{r}}^*({\mathbf{s}})$ is reached after no more than $n-d_0$ iteration steps.
Since the equity vector ${\mathbf{s}}$ is considered as fixed we can modify the financial system ${\mathcal{F}}$ by setting $\tilde{\mathbf{a}}={\mathbf{a}}+{\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}$ and $\widetilde{\mathbf{M}}^{{\mathbf{s}}}={\mathbf{0}}_{n\times n}$. The new system $\tilde{\mathcal{F}}=(\tilde{\mathbf{a}},{\mathbf{d}},{\mathbf{M}^{{\mathbf{d}}}},\widetilde{\mathbf{M}}^{{\mathbf{s}}})$ is then a system without cross-ownership of equity. Such systems are considered in @eisenberg01.
1. The proof that the sequence ${\mathbf{r}}^k$ decreases is now equivalent to the proof given in @eisenberg01. A needed assumption in the proof therein is that $\bar{\mathbf{r}}$ is a so-called supersolution which is given because of $\Phi^{{\mathbf{d}}}(\bar{\mathbf{r}}; {\mathbf{s}}(\bar{\mathbf{r}})) \le \bar{\mathbf{r}}\le{\mathbf{d}}$. What we have to show to complete this part is that the fixed point of the mapping in exists and is unique, since their definition of a financial system, differs slightly from ours. Denote by ${\mathbf{\Lambda}}:={\mathbf{\Lambda}}({\mathbf{r}}^k,{\mathbf{s}})$ the diagonal matrix for ${\mathbf{r}}^k$. The next iterate ${\mathbf{r}}^{k+1}$ is according to given by $$\begin{split}
{\mathbf{r}}^{k+1} &= {\mathbf{\Lambda}}\left(\tilde{\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}({\mathbf{\Lambda}}{\mathbf{r}}^{k+1} + ({\mathbf{I}}_n-{\mathbf{\Lambda}}){\mathbf{d}}) \right) + ({\mathbf{I}}_n-{\mathbf{\Lambda}}){\mathbf{d}}\\
&= {\mathbf{\Lambda}}{\mathbf{M}^{{\mathbf{d}}}}{\mathbf{\Lambda}}{\mathbf{r}}^{k+1} + {\mathbf{\Lambda}}\left(\tilde{\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}){\mathbf{d}}\right) + ({\mathbf{I}}_n-{\mathbf{\Lambda}}){\mathbf{d}}\end{split}$$ and rearranging yields to $${\mathbf{r}}^{k+1} = \left({\mathbf{I}}_n-{\mathbf{\Lambda}}{\mathbf{M}^{{\mathbf{d}}}}{\mathbf{\Lambda}}\right)^{-1}\left({\mathbf{\Lambda}}\left(\tilde{\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}){\mathbf{d}}\right) + ({\mathbf{I}}_n-{\mathbf{\Lambda}}){\mathbf{d}}\right).$$ Note that ${\mathbf{M}^{{\mathbf{d}}}}$ has the Elsinger Property and, hence, so does ${\mathbf{\Lambda}}{\mathbf{M}^{{\mathbf{d}}}}{\mathbf{\Lambda}}$, which means that the inverse of ${\mathbf{I}}_n-{\mathbf{\Lambda}}{\mathbf{M}^{{\mathbf{d}}}}{\mathbf{\Lambda}}$ exists. This proves the uniqueness of ${\mathbf{r}}^{k+1}$.
2. The argumentation that the sequence converges and becomes constant in the end is analogous to part (ii) of the proof of Proposition \[prop:equ\_comp\_conv\]. Since ${\mathbf{r}}^k$ is decreasing, we have that $D({\mathbf{r}}^k,{\mathbf{s}})\subseteq D({\mathbf{r}}^{k+1},{\mathbf{s}})$ that means the number of firms in default increases. If $D({\mathbf{r}}^l, {\mathbf{s}})=D({\mathbf{r}}^{l+1}, {\mathbf{s}})$, then we also have that ${\mathbf{\Lambda}}({\mathbf{r}}^l,{\mathbf{s}})= {\mathbf{\Lambda}}({\mathbf{r}}^{l+1},{\mathbf{s}})$, from which follows that the mappings $\Theta_{{\mathbf{r}}^l,{\mathbf{s}}}$ and $\Theta_{{\mathbf{r}}^{l+1},{\mathbf{s}}}$ have the same fixed point. It must hold then that all consequent iterates are equal.
To show that ${\mathbf{r}}^{l+1}$ is the fixed point of $\Phi^{{\mathbf{d}}}(\cdot,{\mathbf{s}})$, first check that by definition of ${\mathbf{\Lambda}}({\mathbf{r}}^{l+1},{\mathbf{s}})$: $${\mathbf{r}}^{l+1} = {\mathbf{\Lambda}}({\mathbf{r}}^{l+1},{\mathbf{s}}){\mathbf{r}}^{l+1} + ({\mathbf{I}}_n - {\mathbf{\Lambda}}({\mathbf{r}}^{l+1},{\mathbf{s}})){\mathbf{d}}.$$ It then holds that $$\begin{split}
\Phi^{{\mathbf{d}}}({\mathbf{r}}^{l+1}; {\mathbf{s}}) &= \min\{{\mathbf{d}}, \tilde{\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^{l+1}\} \\
&= ({\mathbf{I}}_n - {\mathbf{\Lambda}}({\mathbf{r}}^{l+1},{\mathbf{s}})){\mathbf{d}}+ {\mathbf{\Lambda}}({\mathbf{r}}^{l+1},{\mathbf{s}})(\tilde{\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^{l+1}) \\
&= ({\mathbf{I}}_n - {\mathbf{\Lambda}}({\mathbf{r}}^l,{\mathbf{s}})){\mathbf{d}}\\
&\qquad + {\mathbf{\Lambda}}({\mathbf{r}}^l,{\mathbf{s}})\left(\tilde{\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}\left({\mathbf{\Lambda}}({\mathbf{r}}^{l+1},{\mathbf{s}}){\mathbf{r}}^{l+1} + ({\mathbf{I}}_n - {\mathbf{\Lambda}}({\mathbf{r}}^{l+1},{\mathbf{s}})){\mathbf{d}}\right) \right)\\
&= ({\mathbf{I}}_n - {\mathbf{\Lambda}}({\mathbf{r}}^l,{\mathbf{s}})){\mathbf{d}}+ {\mathbf{\Lambda}}({\mathbf{r}}^l,{\mathbf{s}})\left(\tilde{\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}\left({\mathbf{\Lambda}}({\mathbf{r}}^l,{\mathbf{s}}){\mathbf{r}}^{l+1} + ({\mathbf{I}}_n - {\mathbf{\Lambda}}({\mathbf{r}}^l,{\mathbf{s}})){\mathbf{d}}\right) \right)\\
&= {\mathbf{r}}^{l+1},
\end{split}$$ where the last equality follows from .
3. This part is similar to part (iii) of the proof of Proposition \[prop:equ\_comp\_conv\] with the reverse argumentation. The $d_0$ firms in default under the starting vector will stay in default since the series decreases. To achieve a maximum theoretical length of the algorithm, exactly one additional default step has to occur in every new iteration step. This results in no more than $n-d_0$ possible iteration steps.
The validity of the inequality in is crucial for the monotonicity of the iterates ${\mathbf{r}}^k$ produced by Algorithm \[alg:debt\]. However, there are situations in which a debt payment vector $\bar{\mathbf{r}}\in[{\mathbf{r}_{\rm small}},{\mathbf{r}_{\rm great}}]$ is given together with an arbitrary vector ${\mathbf{s}}\ge{\mathbf{0}}_n$ and where does not hold. Think of an algorithm to find ${\mathbf{R}}^*$ that starts with ${\mathbf{R}_{\rm small}}$. In this case the first debt iterate is ${\mathbf{r}_{\rm small}}=\min\{{\mathbf{d}},{\mathbf{a}}\}$ and the corresponding equity iterate is ${\mathbf{s}}({\mathbf{r}_{\rm small}})$. Applying $\Phi^{{\mathbf{d}}}$ on these vectors yields to $$\Phi^{{\mathbf{d}}}({\mathbf{r}_{\rm small}};{\mathbf{s}}({\mathbf{r}_{\rm small}})) = \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}_{\rm small}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{r}_{\rm small}})\} \ge \min\{{\mathbf{d}},{\mathbf{a}}\} = {\mathbf{r}_{\rm small}}$$ and to a violation of . Finding the next debt iterate as the fixed point of $\Phi^{{\mathbf{d}}}(\cdot;{\mathbf{s}}({\mathbf{r}_{\rm small}}))$ and applying Algorithm \[alg:debt\] to do so, can under certain circumstances lead to a non-monotone series, as one can simply verify by a self-chosen example. This makes it difficult to prove the convergence of such a series in general. Nevertheless, given a debt vector $\bar{\mathbf{r}}$ and ${\mathbf{s}}\ge{\mathbf{0}}_n$, we can still calculate the fixed point by avoiding Algorithm \[alg:debt\] and use a Picard-type algorithm instead.
[5A]{}\[Picard Iteration for the Debt Component\]\[alg:picard\_debt\] Suppose that ${\mathbf{s}}\ge{\mathbf{0}}_n$ and ${\varepsilon}> 0$.
1. For $k=0$, set ${\mathbf{r}}^0=\bar{\mathbf{r}}$.
2. \[alg:fp\_picard\_debt\] For $k\ge 1$, determine ${\mathbf{r}}^k = \Phi^{{\mathbf{d}}}({\mathbf{r}}^{k-1};{\mathbf{s}})$.
3. If $\|{\mathbf{r}}^{k-1}-{\mathbf{r}}^k\|<{\varepsilon}$, stop the algorithm. Else, set $k = k + 1$ and proceed with step \[alg:fp\_picard\_debt\].
\[prop:debt\_comp\_conv\_picard\] Algorithm \[alg:picard\_debt\] delivers a series of decreasing vectors ${\mathbf{r}}^k$ if $\Phi^{{\mathbf{d}}}(\bar{\mathbf{r}};{\mathbf{s}})\le \bar{\mathbf{r}}$ and a series of increasing vectors if $\Phi^{{\mathbf{d}}}(\bar{\mathbf{r}};{\mathbf{s}})\ge \bar{\mathbf{r}}$. Both series converge to the unique fixed point of $\Phi^{{\mathbf{d}}}(\cdot;{\mathbf{s}})$.
Assume that $\Phi^{{\mathbf{d}}}(\bar{\mathbf{r}};{\mathbf{s}})\ge \bar{\mathbf{r}}={\mathbf{r}}^0$. For the first iterate, it holds that ${\mathbf{r}}^1=\Phi^{{\mathbf{d}}}(\bar{\mathbf{r}};{\mathbf{s}})\ge{\mathbf{r}}^0$. Via induction, it follows that ${\mathbf{r}}^{k+1}\ge{\mathbf{r}}^k$ for all $k\ge1$. Because the monotone series ${\mathbf{r}}^k$ is bounded by ${\mathbf{d}}$, it must converge to some fixed point. Because of the fact that the Elsinger condition holds, it follows directly that this fixed must be unique (see @elsinger09, Theorem 3). The argumentation is similar if $\Phi^{{\mathbf{d}}}(\bar{\mathbf{r}};{\mathbf{s}})\le \bar{\mathbf{r}}$.
The Algorithms \[alg:debt\] and \[alg:picard\_debt\] both enable us to calculate a new debt iterate given an equity vector. Together with Algorithm \[alg:equity\] for the equity component, we can now combine both procedures in a common algorithm that searches for the fixed point ${\mathbf{R}}^*$.
[6]{}\[Hybrid Algorithm\]\[alg:comb\_method\] Set ${\varepsilon}> 0$.
1. For $k=0$, choose ${\mathbf{r}}^0\in\{{\mathbf{r}_{\rm great}},{\mathbf{r}_{\rm small}}\}$ and determine ${\mathbf{s}}({\mathbf{r}}^0)$ with Algorithm \[alg:equity\].
2. \[alg:comb\_method\_fp\] For $k\ge1$:
1. Determine ${\mathbf{r}}^k$ using Algorithm \[alg:debt\] if ${\mathbf{r}}^0={\mathbf{r}_{\rm great}}$ or using Algorithm \[alg:picard\_debt\] if ${\mathbf{r}}^0={\mathbf{r}_{\rm small}}$ in both cases with ${\mathbf{s}}={\mathbf{s}}({\mathbf{r}}^{k-1})$.
2. Determine ${\mathbf{s}}^k = {\mathbf{s}}({\mathbf{r}}^k)$ using Algorithm \[alg:equity\].
3. If $\left\|\begin{pmatrix}{\mathbf{r}}^{k-1}\\{\mathbf{s}}^{k-1}\end{pmatrix} - \begin{pmatrix}{\mathbf{r}}^k\\{\mathbf{s}}^k\end{pmatrix}\right\|<{\varepsilon}$, stop the algorithm. Else, set $k = k + 1$ and proceed with step \[alg:comb\_method\_fp\].
For given ${\mathbf{r}}= {\mathbf{r}}^k, k\ge 0$, the Hybrid Algorithm determines ${\mathbf{s}}^k={\mathbf{s}}({\mathbf{r}}^k)$ as the correct equity value that solves and for given ${\mathbf{s}}= {\mathbf{s}}({\mathbf{r}}^k), k\ge 0$, it determines the correct debt value ${\mathbf{r}}^{k+1}$ that solves . As such, conditional on the values determined in the previous step, the algorithm calculates an exact solution of either or in the next iteration step.
\[prop:comb\_meth\_conv\] The Hybrid Algorithm delivers a series of decreasing vectors if ${\mathbf{r}}^0={\mathbf{r}_{\rm great}}$ that converges to the fixed point ${\mathbf{R}}^*$. In case of ${\mathbf{r}}^0={\mathbf{r}_{\rm small}}$ the series is increasing with the same limit.
First, suppose that ${\mathbf{r}}^0={\mathbf{r}_{\rm great}}$. We will first show by induction that the series decreases. For the induction start note that $${\mathbf{r}}^1 = \Phi^{{\mathbf{d}}}({\mathbf{r}}^1;{\mathbf{s}}({\mathbf{r}}^0)) = \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^1 + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{r}}^0)\} \le {\mathbf{d}}= {\mathbf{r}}^0.$$ As mentioned in the proof of Proposition \[prop:elsinger\_better\_picard\], the equity vectors ${\mathbf{s}}({\mathbf{r}})$ are increasing in ${\mathbf{r}}$ which yields to ${\mathbf{s}}({\mathbf{r}}^1)\le {\mathbf{s}}({\mathbf{r}}^0)$. For the induction step, assume that for $k>1$ it holds that ${\mathbf{r}}^{k-1}\ge{\mathbf{r}}^k$ and consequently ${\mathbf{s}}({\mathbf{r}}^{k-1})\ge{\mathbf{s}}({\mathbf{r}}^k)$. Since ${\mathbf{r}}^k=\Phi^{{\mathbf{d}}}({\mathbf{r}}^k;{\mathbf{s}}({\mathbf{r}}^{k-1}))$ and because of $$\begin{split}
\Phi^{{\mathbf{d}}}({\mathbf{r}}^k;{\mathbf{s}}({\mathbf{r}}^k)) &= \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^k + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{r}}^k)\} \\
&\le \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^k + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{r}}^{k-1})\} \\
&= \Phi^{{\mathbf{d}}}({\mathbf{r}}^k;{\mathbf{s}}({\mathbf{r}}^{k-1}))\\
&= {\mathbf{r}}^k
\end{split}$$ the assumption is fulfilled. The next iterate ${\mathbf{r}}^{k+1}$ emerges from a decreasing sequence produced by applying Algorithm \[alg:debt\] beginning with $\bar{\mathbf{r}}={\mathbf{r}}^k$. Hence ${\mathbf{r}}^{k+1}\le{\mathbf{r}}^k$ and thus ${\mathbf{s}}({\mathbf{r}}^{k+1})\le{\mathbf{s}}({\mathbf{r}}^k)$. Next step is to show that the series converges to ${\mathbf{R}}^*$. We have that the two sequences $(({\mathbf{r}}^{k+1})^t, ({\mathbf{s}}({\mathbf{r}}^k))^t)^t$ and $(({\mathbf{r}}^k)^t, ({\mathbf{s}}({\mathbf{r}}^k))^t)^t$ are both decreasing in $({\mathbb{R}}_0^+)^{2n}$ and therefore converge to the same limit $(({\mathbf{r}}^*)^t, ({\mathbf{s}}^*)^t)^t\in({\mathbb{R}}_0^+)^{2n}$. Because of the continuity of $\Phi^{{\mathbf{d}}}$ and $\Phi^{{\mathbf{s}}}$ it must hold that $\Phi^{{\mathbf{d}}}({\mathbf{r}}^*,{\mathbf{s}}^*)={\mathbf{r}}^*$ and $\Phi^{{\mathbf{s}}}({\mathbf{s}}^*,{\mathbf{r}}^*)={\mathbf{s}}^*$. Thus, $(({\mathbf{r}}^*)^t, ({\mathbf{s}}^*)^t)^t$ solves and . The proof for ${\mathbf{r}}^0={\mathbf{r}_{\rm small}}$ is similar.
In Proposition \[prop:elsinger\_better\_picard\], we have shown that when using the Elsinger Algorithm, the iterates will always be nearer to the solution ${\mathbf{R}}^*$ than the corresponding iterates of the Picard Algorithm. This lead to the conclusion that the iteration number is minimized for the Elsinger Algorithm. The next Proposition shows the same when comparing the Elsinger and the Hybrid Algorithm and it will become clear that the Hybrid Algorithm will need less iteration steps to reach ${\mathbf{R}}^*$ than the Elsinger Algorithm.
\[prop:hybrid\_better\_elsinger\] As in Proposition \[prop:elsinger\_better\_picard\], we denote the iterates of the two algorithms with subscripts, where E stands for the Elsinger and H for the Hybrid Algorithm.
(i) For any iterate $k\ge 1$ it holds that ${\mathbf{R}}^k_{\rm E}\ge {\mathbf{R}}^k_{\rm H}$ if ${\mathbf{R}}^0 = ({\mathbf{r}_{\rm great}}^t, {\mathbf{s}}({\mathbf{r}_{\rm great}})^t)^t$ and ${\mathbf{R}}^k_{\rm E}\le {\mathbf{R}}^k_{\rm H}$ when $({\mathbf{r}_{\rm small}}^t, {\mathbf{s}}({\mathbf{r}_{\rm small}})^t)^t$ is the starting vector of both algorithms.
(ii) Let ${\mathbf{R}}^k$, $k\ge0$, be an iterate either of the Elsinger Algorithm or of the Hybrid Algorithm that started with ${\mathbf{R}}^0 = ({\mathbf{r}_{\rm great}}^t, {\mathbf{s}}({\mathbf{r}_{\rm great}})^t)^t$. Then ${\mathbf{R}}^{k+1}_{{\rm E}}({\mathbf{r}}^k)\ge {\mathbf{R}}^{k+1}_{{\rm H}}({\mathbf{r}}^k)$ for the next iterates which were calculated with either the Elsinger or the Hybrid Algorithm starting from ${\mathbf{R}}^k$. If ${\mathbf{R}}^0 = (({\mathbf{r}_{\rm small}}^t, {\mathbf{s}}({\mathbf{r}_{\rm small}})^t)^t$, it holds that ${\mathbf{R}}^{k+1}_{{\rm E}}({\mathbf{r}}^k) \le {\mathbf{R}}^{k+1}_{{\rm H}}({\mathbf{r}}^k)$.
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(i) Let ${\mathbf{R}}^0 = ({\mathbf{r}_{\rm great}}^t, {\mathbf{s}}({\mathbf{r}_{\rm great}})^t)^t$. From Proposition \[prop:comb\_meth\_conv\] we know that ${\mathbf{r}}^1_{\rm H}\le{\mathbf{r}}^0_{\rm H}={\mathbf{d}}$ which yields to $$\begin{split}
{\mathbf{r}}^1_{\rm E} = \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{d}})\} \ge
\min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^1_{\rm H} + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{d}})\} = {\mathbf{r}}^1_{\rm H}.
\end{split}$$ Further, since ${\mathbf{s}}({\mathbf{r}})$ is increasing in ${\mathbf{r}}$ (cf. Proposition \[prop:elsinger\_better\_picard\]), ${\mathbf{s}}^1_{\rm E} = {\mathbf{s}}({\mathbf{r}}^1_{\rm E}) \ge {\mathbf{s}}({\mathbf{r}}^1_{\rm H}) = {\mathbf{s}}^1_{\rm H}$, which completes the induction start. For the induction step, assume that it holds for $k>1$ that ${\mathbf{R}}^{k-1}_{\rm E}\ge{\mathbf{R}}^{k-1}_{\rm H}$. Because of Proposition \[prop:comb\_meth\_conv\], ${\mathbf{r}}^{k-1}_{\rm H}\ge {\mathbf{r}}^k_{\rm H}$ and thus $$\begin{split}
{\mathbf{r}}^k_{\rm E} &= \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^{k-1}_{\rm E} + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{r}}^{k-1}_{\rm E})\} \\
&\ge \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^{k-1}_{\rm H} + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{r}}^{k-1}_{\rm H})\} \\
&\ge \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^k_{\rm H} + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{r}}^{k-1}_{\rm H})\} \\
&= {\mathbf{r}}^k_{\rm H},
\end{split}$$ where we again used the fact that ${\mathbf{s}}({\mathbf{r}})$ is increasing in ${\mathbf{r}}$ from which follows that ${\mathbf{s}}({\mathbf{r}}^{k-1}_{\rm E})\ge{\mathbf{s}}({\mathbf{r}}^{k-1}_{\rm H})$ and also ${\mathbf{s}}({\mathbf{r}}^k_{\rm E})\ge{\mathbf{s}}({\mathbf{r}}^k_{\rm H})$. The proof when ${\mathbf{R}}^0 = ({\mathbf{r}_{\rm small}}^t, {\mathbf{s}}({\mathbf{r}_{\rm small}})^t)^t$ is completely analogous.
(ii) Let ${\mathbf{R}}^0 = ({\mathbf{r}_{\rm great}}^t, {\mathbf{s}}({\mathbf{r}_{\rm great}})^t)^t$ and ${\mathbf{R}}^k={\mathbf{R}}^k_{\rm E}$. Note that because of ${\mathbf{r}}^k_{\rm E}\le{\mathbf{r}}^{k-1}_{\rm E}$ is holds that $$\begin{split}
\Phi^{{\mathbf{d}}}({\mathbf{r}}^k_{\rm E}; {\mathbf{s}}({\mathbf{r}}^k_{\rm E})) &= \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^k_{\rm E} + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{r}}^k_{\rm E})\} \\
&\le \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^{k-1}_{\rm E} + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{r}}^{k-1}_{\rm E})\} \\
&= {\mathbf{r}}^k_{\rm E}.
\end{split}$$ Therefore, the assumption in is fulfilled which ensures that ${\mathbf{r}}^{k+1}_{\rm H}\le{\mathbf{r}}^k_{\rm E}$. For the next iterate it follows that $$\begin{split}
{\mathbf{r}}^{k+1}_{\rm E} &= \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^k_{\rm E} + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{r}}^k_{\rm E})\} \\
&\ge \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^{k+1}_{\rm H} + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{r}}^k_{\rm E})\} \\
&= {\mathbf{r}}^{k+1}_{\rm H},
\end{split}$$ which in turn implies ${\mathbf{s}}^{k+1}_{\rm E}\ge{\mathbf{s}}^{k+1}_{\rm H}$. On the other hand, starting with ${\mathbf{R}}^k={\mathbf{R}}^k_{\rm H}$ yields because of ${\mathbf{r}}^{k+1}_{\rm H}\le {\mathbf{r}}^k_{\rm H}$ to $$\begin{split}
{\mathbf{r}}^{k+1}_{\rm E} &= \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^k_{\rm H} + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{r}}^k_{\rm H})\} \\
&\ge \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^{k+1}_{\rm H} + {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{r}}^k_{\rm H})\} \\
&= {\mathbf{r}}^{k+1}_{\rm H}.
\end{split}$$ If follows from this results that ${\mathbf{s}}^{k+1}_{\rm E}\le{\mathbf{s}}^{k+1}_{\rm H}$. A similar argumentation together with Proposition \[prop:debt\_comp\_conv\_picard\] delivers the proof in case of ${\mathbf{R}}^0 = ({\mathbf{r}_{\rm small}}^t, {\mathbf{s}}({\mathbf{r}_{\rm small}})^t)^t$.
Of course, within an iteration step of the Hybrid Algorithm, potentially many linear equation systems have to be solved since for the debt component Algorithm \[alg:debt\] is applied, which results in higher computational costs. But if we ignore for a moment this circumstance it follows from Proposition \[prop:hybrid\_better\_elsinger\] that the convergence speed of the Hybrid Algorithm is higher than the one of the Elsinger Algorithm.
Finite Algorithms {#sec:def-set-algo}
=================
The Algorithms in the previous section all had the drawback that it could not be ensured that the solution ${\mathbf{R}}^*$ is reached in a finite number of iteration steps. In this section we will present two ways in which potentially infinite solution algorithms can be turned into procedures that reach the solution in finitely many steps. The common principle of these methods is to include the information which firms are in default under a current iterate ${\mathbf{R}}^k$. It turns out that this slight modification helps to overcome the disadvantage of potentially infinitely many iteration steps.
To guarantee that the forthcoming procedures are well-defined, we have to drop the Elsinger Property and demand a stricter property of the ownership matrices (see @fischer14).
\[assu:os\_mat\_norm\] For both the debt and the equity ownership matrices it holds that $\|{\mathbf{M}^{{\mathbf{d}}}}\|<1$ and $\|{\mathbf{M}^{{\mathbf{s}}}}\|<1$.
For the remainder of this section we suppose that Assumption \[assu:os\_mat\_norm\] holds. Note that Assumption \[assu:os\_mat\_norm\] implies Assumption \[assu:holding\_mat\], but not the other way round. The financial system therefore still has a unique solution under Assumption \[assu:os\_mat\_norm\].
\[def:sys\_sol\] Let ${\mathbf{R}}=({\mathbf{r}}^t,{\mathbf{s}}^t)^t\in({\mathbb{R}}_0^+)^{2n}$ be an arbitrary vector with corresponding default set $D({\mathbf{R}})$ and default matrix ${\mathbf{\Lambda}}={\mathbf{\Lambda}}({\mathbf{R}})$. The *pseudo solution $\widehat{\mathbf{R}}\in({\mathbb{R}}_0^+)^{2n}$ of and that belongs to $D({\mathbf{R}})$* is defined by $$\widehat{\mathbf{R}}= \begin{pmatrix} ({\mathbf{I}}_n-{\mathbf{\Lambda}}){\mathbf{d}}+ {\mathbf{\Lambda}}{\mathbf{x}}\\
({\mathbf{I}}_n-{\mathbf{\Lambda}}){\mathbf{x}}\end{pmatrix},$$ where ${\mathbf{x}}\in{\mathbb{R}}^n$ is the solution of the linear equation system ${\mathbf{A}}{\mathbf{x}}={\mathbf{b}}$ with $$\label{eq:sys_sol_A}
{\mathbf{A}}= {\mathbf{I}}_n - \left({\mathbf{M}^{{\mathbf{d}}}}{\mathbf{\Lambda}}+ {\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n - {\mathbf{\Lambda}})\right)\in{\mathbb{R}}^{n\times n}$$ and $$\label{eq:sys_sol_b}
{\mathbf{b}}= {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}){\mathbf{d}}- ({\mathbf{I}}_n-{\mathbf{\Lambda}}){\mathbf{d}}\in{\mathbb{R}}^n.$$
To motivate the definition of a pseudo solution, assume that it was known for each firm whether it was in default under the solution of and or not. Denote by $D^*\subseteq\mathcal N$ the set of firms that were in default under ${\mathbf{R}}^*$: $$D^* = D({\mathbf{r}}^*,{\mathbf{s}}^*) = \left\{i\in\mathcal N:a_i + \sum_{j=1}^n M^{{\mathbf{d}}}_{ij} r^*_j + \sum_{j=1}^n M^{{\mathbf{s}}}_{ij} s^*_j < d_i\right\}$$ and let ${\mathbf{\Lambda}}^*={\mathbf{\Lambda}}({\mathbf{r}}^*,{\mathbf{s}}^*)$ be the corresponding default matrix. We assume that the set was known even though this information is not available *a priori*. However, if we had this information, no iteration procedure would be needed to find the fixed point ${\mathbf{R}}^*$. We only had to compute the pseudo solution that belongs to $D^*$, as is shown in Proposition \[prop:def\_set\_sol\].
The reason why we have to restrict the following considerations to ownership matrices with a matrix norm smaller one is because we have to guarantee that ${\mathbf{x}}$ from Definition \[def:sys\_sol\] is uniquely defined. This can only be ensured if $\|{\mathbf{M}^{{\mathbf{d}}}}\|<1$ and $\|{\mathbf{M}^{{\mathbf{s}}}}\|<1$ since then $\|{\mathbf{M}^{{\mathbf{d}}}}{\mathbf{\Lambda}}+ {\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n - {\mathbf{\Lambda}})\|<1$ for any ${\mathbf{\Lambda}}$ as well. This in turn implies that ${\mathbf{A}}$ in is invertible. For ownership matrices ${\mathbf{M}^{{\mathbf{d}}}}$ and ${\mathbf{M}^{{\mathbf{s}}}}$ that have the Elsinger Property, the invertibility of ${\mathbf{A}}$ is obviously not always given.
\[prop:def\_set\_sol\] The pseudo solution belonging to $D^*$ is the solution ${\mathbf{R}}^*$ of the financial system $\mathcal F({\mathbf{a}}, {\mathbf{M}^{{\mathbf{d}}}}, {\mathbf{M}^{{\mathbf{s}}}}, {\mathbf{d}})$, i.e. $${\mathbf{R}}^* = \begin{pmatrix} ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{d}}+ {\mathbf{\Lambda}}^*{\mathbf{x}}\\
({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{x}}\end{pmatrix},$$ where ${\mathbf{\Lambda}}^*$ is the default matrix belonging to $D^*$ and ${\mathbf{x}}$ is the solution of the equation ${\mathbf{A}}{\mathbf{x}}={\mathbf{b}}$ defined in and .
According to the liquidation value equations in and , the vectors ${\mathbf{r}}^*$ and ${\mathbf{s}}^*$ are given as $$r_i^* = \begin{cases} d_i ,& \text{if $i\notin D^*$,} \\
a_i + \sum_{j=1}^nM^{{\mathbf{d}}}_{ij}r^*_j + \sum_{j=1}^nM^{{\mathbf{s}}}_{ij}s^*_j ,& \text{if $i\in D^*$} \end{cases}$$ and $$s_i^* = \begin{cases} a_i + \sum_{j=1}^nM^{{\mathbf{d}}}_{ij}r^*_j + \sum_{j=1}^nM^{{\mathbf{s}}}_{ij}s^*_j - d_i ,& \text{if $i\notin D^*$,} \\
0 ,& \text{if $i\in D^*$}. \end{cases}$$ In matrix notation this means in particular that $({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{r}}^*=({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{d}}$ and ${\mathbf{\Lambda}}^*{\mathbf{s}}^* = {\mathbf{0}}_n$ and thus $${\mathbf{R}}^* = \begin{pmatrix} ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{d}}+ {\mathbf{\Lambda}}^*{\mathbf{r}}^* \\
({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{s}}^* \end{pmatrix}.$$ For the firms in default we only have to calculate the debt payments and for the firms not in default we have to determine the equity value. The solution ${\mathbf{R}}^*$ does hence contain only $n$ unknown values and we only have to consider the two subsystems $${\mathbf{\Lambda}}^*{\mathbf{r}}^* = {\mathbf{\Lambda}}^*{\mathbf{a}}+ {\mathbf{\Lambda}}^*{\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^*+{\mathbf{\Lambda}}^*{\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}^*$$ and $$({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{s}}^* = ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*)({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^*+{\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}^* - {\mathbf{d}})$$ We can add the two equations and write the system more compact as: $${\mathbf{\Lambda}}^*{\mathbf{r}}^* + ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{s}}^* = {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^*+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}^* - ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{d}}.$$ Because of $({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{s}}^*={\mathbf{s}}^*$ we get $${\mathbf{\Lambda}}^*{\mathbf{r}}^* + ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{s}}^* = {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}^*+ {\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{s}}^* - ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{d}},$$ which leads after some rearrangements to $${\mathbf{\Lambda}}^*{\mathbf{r}}^* + ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{s}}^* - {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{\Lambda}}^*{\mathbf{r}}^* - {\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{s}}^* = {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{r}}^* - ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{d}}$$ that is equivalent to $$\left({\mathbf{I}}_n - \left({\mathbf{M}^{{\mathbf{d}}}}{\mathbf{\Lambda}}^* + {\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n - {\mathbf{\Lambda}}^*)\right)\right) ({\mathbf{\Lambda}}^*{\mathbf{r}}^* + ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{s}}^*)
= {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{d}}- ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{d}},$$ since ${\mathbf{\Lambda}}^*({\mathbf{I}}_n-{\mathbf{\Lambda}}^*)={\mathbf{0}}_{n\times n}$. Setting ${\mathbf{x}}={\mathbf{\Lambda}}^*{\mathbf{r}}^* + ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{s}}^*$ and with the notation of Definition \[def:sys\_sol\], the equation system becomes ${\mathbf{A}}{\mathbf{x}}={\mathbf{b}}$.
The main challenge in this solution approach is of course that the final default set $D^*$ is unknown. Algorithms that follow this idea to find ${\mathbf{R}}^*$ consequently have to find $D^*$ in a fast way. A naive strategy could be to check all possible default scenarios of the financial system, calculate the pseudo solution for the corresponding default set and check whether it actually is the fixed point of $\Phi$. However, there are $2^n$ possible scenarios that would have to be checked, which could be cumbersome for large $n$. Therefore, more efficient algorithms are needed that require less computation to find $D^*$. Some possible algorithms are presented in the next subsections .
Decreasing Trial-and-Error Algorithms {#subsec:trial_error_alg_decr}
-------------------------------------
The three algorithms \[alg:picard\], \[alg:elsinger\] and \[alg:comb\_method\] from Section \[sec:iterative\_algo\] can start with a vector ${\mathbf{R}}^0$ that was the upper boundary of the solution vector ${\mathbf{R}}^*$. The procedures in this subsection have in common that they also start with this upper boundary and calculate a corresponding default set. For every following iterate, the corresponding default set is determined as well. To avoid that every default set it is checked whether it actually is $D^*$ and whether the corresponding pseudo solution is the fixed point of $\Phi$, the algorithm will identify potential default sets to reduce the computational effort. If it turns out that the potential default set is $D^*$, the algorithm stops. Otherwise, the procedure continues until a new potential default set is found that has to be checked again, and so on. Due to these characteristics we name this type of algorithm *Trial-and-Error Algorithm*. The general procedure of algorithms of this type is similar.
[7]{}\[Decreasing Trial-and-Error Algorithm\]\[alg:trial\_error\_decr\] Set $l\ge2$ and $p=0$.
1. \[alg:t-e\_decr\_choice\] Choose either the Picard (Algorithm \[alg:picard\]), the Elsinger (Algorithm \[alg:elsinger\]) or the Hybrid Algorithm (Algorithm \[alg:comb\_method\]) which is used in the following to generate the next iterate.
2. If in Step \[alg:t-e\_decr\_choice\] the Picard Algorithm is chosen, set $d=-1$, ${\mathbf{R}}^0={\mathbf{R}_{\rm great}}$ and determine $D({\mathbf{R}}^0)$. Else, set $d=0$, ${\mathbf{R}}^0=\left(\begin{smallmatrix}{\mathbf{d}}\\ {\mathbf{s}}({\mathbf{d}})\end{smallmatrix}\right)$ and determine $D({\mathbf{R}}^0)$.
3. \[alg:t-e\_decr\_all\_def\] If $D({\mathbf{R}}^0)=\mathcal N$, set ${\mathbf{R}}^* = \left(\begin{smallmatrix}({\mathbf{I}}_n-{\mathbf{M}^{{\mathbf{d}}}})^{-1}{\mathbf{a}}\\{\mathbf{0}}_n\end{smallmatrix}\right)$ and stop the algorithm.
4. \[alg:t-e\_decr\_no\_def\] If the Elsinger or the Hybrid Algorithm is chosen in Step \[alg:t-e\_decr\_choice\] and if $D({\mathbf{R}}^0)=\emptyset$, set ${\mathbf{R}}^*={\mathbf{R}}^0$ and stop the algorithm.
5. \[alg:trial\_error\_fp\_decr\] Else, calculate for $k > p$ the iterates ${\mathbf{R}}^k$ starting with ${\mathbf{R}}^p$ using the algorithm chosen in step \[alg:t-e\_decr\_choice\] and the corresponding default sets $D({\mathbf{R}}^k)$ until $k=q$ with $$\label{eq:pot_def_set}
q = \min\{m>p: D({\mathbf{R}}^{m-l+1}) = \ldots = D({\mathbf{R}}^m)\ \text{and}\ |D({\mathbf{R}}^m)| > d\}$$ is reached. Determine the pseudo solution belonging to $D({\mathbf{R}}^q)$ and denote it by $\widehat{\mathbf{R}}^q$.
6. If $\Phi(\widehat{\mathbf{R}}^q) = \widehat{\mathbf{R}}^q$, stop the algorithm. Else, set $d = |D({\mathbf{R}}^q)|$ and $p=q$ and proceed with step \[alg:trial\_error\_fp\_decr\].
The Algorithms \[alg:picard\], \[alg:elsinger\] and \[alg:comb\_method\] in their decreasing versions produce decreasing sequences of iterates and thus increasing sequences of default sets, i.e. $D({\mathbf{R}}^k)\subseteq D({\mathbf{R}}^{k+1})$ for $k\ge0$. Algorithm \[alg:trial\_error\_decr\] means that one iterates and checks whether the default set has not changed compared to the previous default set. If the default set stays the same for the next $l-1$ consecutive iterations, this is an indication that the actual $D^*$ might have been reached. To check this, the pseudo solution is calculated and it is checked whether it solves and . If no solution has been found, one iterates again until a larger default sets stays identical for $l-1$ consecutive times, and the described procedure can be repeated. If a solution is reached, the procedure stops. Due to its described property, we call $l$ the *lag value*.
In the special case of $l=2$ this means that the pseudo solution is calculated if the default set stays the same from one iteration step to another. Obviously, choosing a higher lag value inspires more confidence in the potential default set since the longer the default set stays unchanged, the higher is the chance that it is the actual default set.
Depending on the choice of the algorithm in Step \[alg:t-e\_decr\_choice\] of the Decreasing Trial-and-Error Algorithm, we obtain three different versions of Algorithm \[alg:trial\_error\_decr\]:
- The *Decreasing Trial-and-Error Picard Algorithm* with ${\mathbf{R}}^0 = {\mathbf{R}_{\rm great}}$, where the iterates as given by ${\mathbf{R}}^k=\Phi({\mathbf{R}}^{k-1})$.
- The *Decreasing Trial-and-Error Elsinger Algorithm* with ${\mathbf{R}}^0 = (({\mathbf{r}_{\rm great}})^t, ({\mathbf{s}}({\mathbf{r}_{\rm great}}))^t)^t$, where ${\mathbf{s}}({\mathbf{r}_{\rm great}})$ is obtained via Algorithm \[alg:equity\] and the next iterates are obtained using Algorithm \[alg:elsinger\].
- The *Decreasing Trial-and-Error Hybrid Algorithm* with the same starting vector as in (ii), where the next iterates are obtained using Algorithm \[alg:comb\_method\].
The particular cases when $D({\mathbf{R}}^0)\in\{\emptyset,\mathcal N\}$ in the steps \[alg:t-e\_decr\_all\_def\] and \[alg:t-e\_decr\_no\_def\], deserve a separate mention since in such situations, no iteration is necessary and the solution ${\mathbf{R}}^*$ can be given explicitly under some circumstances. The justification of this phenomena is given in the following proposition.
\[prop:hybrid\_zero\_all\] For the Decreasing Trial-and-Error Hybrid Algorithm the following holds:
1. If $D({\mathbf{R}}^0)=\mathcal N$, then ${\mathbf{R}}^*=\left(\begin{smallmatrix}({\mathbf{I}}_n-{\mathbf{M}^{{\mathbf{d}}}})^{-1}{\mathbf{a}}\\{\mathbf{0}}_n\end{smallmatrix}\right)$, no matter which version of the algorithm is taken.
2. If $D({\mathbf{R}}^0)=\emptyset$ and either the Decreasing Trial-and-Error Elsinger Algorithm or the Decreasing Trial-and-Error Hybrid Algorithm is used, then ${\mathbf{R}}^0={\mathbf{R}}^*$.
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1. First, assume that the Picard Algorithm is chosen in Step \[alg:t-e\_decr\_choice\] of Algorithm \[alg:trial\_error\_decr\]. Because of $D({\mathbf{R}}^0)=\mathcal N$, it must hold that ${\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}_{\rm great}}< {\mathbf{d}}$ and also ${\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}< {\mathbf{d}}$. A consequence is that ${\mathbf{s}_{\rm great}}= ({\mathbf{I}}_n-{\mathbf{M}^{{\mathbf{s}}}})^{-1}({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}- {\mathbf{d}})^+={\mathbf{0}}_n$. From Proposition \[prop:lim\_R\] it follows that ${\mathbf{s}}^*={\mathbf{0}}$. For ${\mathbf{s}}={\mathbf{s}}^*={\mathbf{0}}$, Equation is now solved by ${\mathbf{r}}^*=({\mathbf{I}}_n - {\mathbf{M}^{{\mathbf{d}}}})^{-1}{\mathbf{a}}$, where Lemma \[lem:inv\_xos-mat\] proves that $({\mathbf{I}}_n - {\mathbf{M}^{{\mathbf{d}}}})^{-1}$ exists. If the Elsinger or the Hybrid Algorithm is chosen in Step \[alg:t-e\_decr\_choice\], we have that ${\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{d}}) < {\mathbf{d}}$. It follows from that ${\mathbf{s}}({\mathbf{d}})={\mathbf{s}}^*={\mathbf{0}}_n$ since ${\mathbf{s}}^*={\mathbf{s}}({\mathbf{r}}^*)\le{\mathbf{s}}({\mathbf{d}})$ because of ${\mathbf{r}}^*\le{\mathbf{d}}$ and the fact that ${\mathbf{s}}({\mathbf{r}})$ is increasing in ${\mathbf{r}}$. The solution of Equation is therefore the same as in the Picard case.
2. Now, ${\mathbf{R}}^0=\left(\begin{smallmatrix}{\mathbf{d}}\\ {\mathbf{s}}({\mathbf{d}})\end{smallmatrix}\right)$ and since $D({\mathbf{R}}^0)=\emptyset$, it holds that ${\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{d}})\ge{\mathbf{d}}$. This leads to $$\label{eq:fp_t-e_decr_no_def}
\Phi\begin{pmatrix} {\mathbf{d}}\\ {\mathbf{s}}({\mathbf{d}}) \end{pmatrix} =
\Phi\begin{pmatrix} \min\{{\mathbf{d}}, {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{d}})\} \\ ({\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{d}}) - {\mathbf{d}})^+ \end{pmatrix} =
\begin{pmatrix} {\mathbf{d}}\\ {\mathbf{s}}({\mathbf{d}}) \end{pmatrix},$$ which proves the claim.
Note that for the Decreasing Trial-and-Error Picard Algorithm, we cannot conclude that ${\mathbf{R}}^0={\mathbf{R}_{\rm great}}={\mathbf{R}}^*$ if $D({\mathbf{R}}^0)=\emptyset$. There are simple counterexamples for situations like this.
\[prop:conv\_def\_set\] Algorithm \[alg:trial\_error\_decr\] reaches the solution ${\mathbf{R}}^*$ of and in a finite number of iteration steps.
By definition of $D({\mathbf{R}}^k)$ in and since ${\mathbf{R}}^k$ converges to ${\mathbf{R}}^*$ from above for any of the three algorithms \[alg:picard\], \[alg:elsinger\] and \[alg:comb\_method\], there exists a $k^0\ge0$ such that $D({\mathbf{R}}^k)=D({\mathbf{R}}^*)=D^*$ for all $k\ge k^0$.
Increasing Trial-and-Error Algorithms {#subsec:trial_error_alg_incr}
-------------------------------------
In contrast to the decreasing algorithms presented in the subsection above, it is of course also possible to use an algorithm with the reverse direction, i.e. in which the series of produced iterates is increasing and in which the default sets are decreasing. The general form is very similar to Algorithm \[alg:trial\_error\_decr\].
[8]{}\[Increasing Trial-and-Error Algorithm\]\[alg:trial\_error\_incr\] Set $l\ge2$, $d=n+1$ and $p=0$.
1. Choose a starting vector ${\mathbf{R}}^0$ and determine $D({\mathbf{R}}^0)$.
2. \[alg:t-e\_incr\_no\_def\] If $D({\mathbf{R}}^0)=\emptyset$, set ${\mathbf{R}}^*=\left(\begin{smallmatrix}{\mathbf{d}}\\ {\mathbf{s}}({\mathbf{d}})\end{smallmatrix}\right)$ and stop the algorithm.
3. \[alg:trial\_error\_fp\_incr\] Else, calculate for $k > p$ the iterates ${\mathbf{R}}^k$ starting with ${\mathbf{R}}^p$ using one of the Algorithms \[alg:picard\], \[alg:elsinger\] or \[alg:comb\_method\] and the corresponding default sets $D({\mathbf{R}}^k)$ until $k=q$ with $$\label{eq:pot_def_set_incr}
q = \min\{m>p: D({\mathbf{R}}^{m-l+1}) = \ldots = D({\mathbf{R}}^m)\ \text{and}\ |D({\mathbf{R}}^m)| < d\}$$ is reached. Determine the pseudo solution belonging to $D({\mathbf{R}}^q)$ and denote it by $\widehat{\mathbf{R}}^q$.
4. If $\Phi(\widehat{\mathbf{R}}^q) = \widehat{\mathbf{R}}^q$, stop the algorithm. Else, set $d = |D({\mathbf{R}}^q)|$ and $p=q$ and proceed with step \[alg:trial\_error\_fp\_incr\].
The functioning of Algorithm \[alg:trial\_error\_incr\] is similar to the Decreasing Trial-and-Error Algorithms with the difference that the resulting sequence of default sets is obviously decreasing. As in Section \[subsec:trial\_error\_alg\_decr\], the way of choosing the calculation method to determine the next iterate, allows three different modifications:
- The *Increasing Trial-and-Error Picard Algorithm* with ${\mathbf{R}}^0={\mathbf{R}_{\rm small}}$ and ${\mathbf{R}}^k=\Phi({\mathbf{R}}^{k-1})$.
- The *Increasing Trial-and-Error Elsinger Algorithm* with ${\mathbf{R}}^0=(({\mathbf{r}_{\rm small}})^t, ({\mathbf{s}}({\mathbf{r}_{\rm small}}))^t)^t$, where ${\mathbf{s}}({\mathbf{r}_{\rm small}})$ is obtained via Algorithm \[alg:equity\] and the next iterates are obtained using Algorithm \[alg:elsinger\].
- The *Increasing Trial-and-Error Hybrid Algorithm* with the same starting vector as in (ii) and where the next iterates are obtained using Algorithm \[alg:comb\_method\]. Note that for the next debt iterate, Algorithm \[alg:picard\_debt\] is used instead of Algorithm \[alg:debt\] in the decreasing version.
The justification of the stopping criteria in Step \[alg:t-e\_incr\_no\_def\] of Algorithm \[alg:trial\_error\_incr\] is as follows. Suppose that the Picard version of the algorithm is chosen and that $D({\mathbf{R}}^0)=\emptyset$, which means that ${\mathbf{a}}+{\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}_{\rm small}}+{\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}_{\rm small}}\ge {\mathbf{d}}$. Since ${\mathbf{r}_{\rm small}}\le{\mathbf{d}}$ and ${\mathbf{s}_{\rm small}}\le{\mathbf{s}}({\mathbf{d}})$, it also holds that ${\mathbf{a}}+{\mathbf{M}^{{\mathbf{d}}}}{\mathbf{d}}+{\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}({\mathbf{d}}) \ge {\mathbf{d}}$ and ${\mathbf{s}}({\mathbf{d}})\ge{\mathbf{0}}_n$ following from this. With Equation , we see that ${\mathbf{R}}^*=\left(\begin{smallmatrix}{\mathbf{d}}\\ {\mathbf{s}}({\mathbf{d}})\end{smallmatrix}\right)$. Also note that, in contrast to Algorithm \[alg:trial\_error\_decr\], there is no stopping criteria in case of $D({\mathbf{R}}^0)=\mathcal N$. The reason is that in this case, no general statement about the structure of the solution ${\mathbf{R}}^*$ can be made, no matter which version of the algorithm is used. In particular, from $D({\mathbf{R}}^0)=\mathcal N$ it does not follow in general that $D({\mathbf{R}}^*)=\mathcal N$, since there are easily constructable counterexamples for this.
The reason why we distinguish between decreasing and increasing Trial-and-Error Algorithms is that Algorithm \[alg:trial\_error\_decr\] will always find the correct default set $D^*=D({\mathbf{R}}^*)$, and this in a finite number of iteration steps. For the Increasing Trial-and-Error Algorithms such a statement is not possible in general since there are some situations in which the default sets do not converge to $D^*$, no matter which lag value is chosen. Situations in which this “anomaly” occurs are always financial systems that contain a so-called *borderline firm*. The expression borderline is taken from @liu10 and denotes a firm $i\in\mathcal N$ in a financial system with fixed point ${\mathbf{R}}^*$ for which it holds that $r^*_i=d_i$ and $s^*_i=0$. In other words, borderline firms are just able to fully cover their liabilities, but have no remaining capital left in their balance sheet that can be furnished to their shareholders. By definition of a default set in , a borderline firm $i$ is not in default since $$0 = s_i = a_i + \sum_{j=1}^nM_{ij}^{{\mathbf{d}}}r^*_j + \sum_{j=1}^nM_{ij}^{{\mathbf{s}}}s^*_j - d_i$$ and therefore $i\notin D({\mathbf{R}}^*)$. However, when using an Increasing Trial-and-Error Algorithm it can happen for such a borderline firm $i$ that $i\in D({\mathbf{R}}^k)$ for every iterate ${\mathbf{R}}^k$, $k\ge0$. This means that the true default set $D^*$ will never be identified by the algorithm. There exist many examples of financial systems that have this property. To show that in such situations, the fixed point ${\mathbf{R}}^*$ can still be determined via the calculation of the pseudo solution, assume that the set $\mathcal B\subset \mathcal N$ contains an arbitrary selection of borderline firms. The common set of defaulting firms and the selected borderline firms is denoted by $\widetilde D$, i.e. $\widetilde D=D^* \cup \mathcal B$. The corresponding “default” matrices are given by $\widetilde{\mathbf{\Lambda}}=\widetilde{\mathbf{\Lambda}}(\widetilde D)$ and ${\mathbf{\Lambda}}^*={\mathbf{\Lambda}}^*(D^*)$, respectively. Following this notation, $\widetilde{\mathbf{A}}$ and ${\mathbf{A}}^*$ define the matrices from with the corresponding default matrix, and $\widetilde{\mathbf{b}}$ and ${\mathbf{b}}^*$ are defined analogously. Moreover, we define $\widetilde{\mathbf{\Lambda}}_{\mathcal B}=\widetilde{\mathbf{\Lambda}}-{\mathbf{\Lambda}}^*$ as the diagonal matrix that indicates only the selected borderline firms.
\[lem:pseudo\_equiv\] The vector ${\mathbf{x}}^*={\mathbf{\Lambda}}^*{\mathbf{r}}^* + ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{s}}^*$ solves the equation system ${\mathbf{A}}^*{\mathbf{x}}={\mathbf{b}}^*$ if and only if $\widetilde{\mathbf{x}}= {\mathbf{x}}^* + \widetilde{\mathbf{\Lambda}}_{\mathcal B}{\mathbf{d}}$ is the solution of $\widetilde{\mathbf{A}}{\mathbf{x}}=\widetilde{\mathbf{b}}$.
Without loss of generality, we assume that the first $n_1$ firms of the system are solvent, that the next $n_2-n_1$ firms are the selected borderline cases, and that the remaining firms are in default under ${\mathbf{R}}^*$. This means that $$\mathcal N = \{1{,\ldots,}n_1\} \cup \mathcal B \cup D^* = \{1{,\ldots,}n_1\} \cup \{n_1+1{,\ldots,}n_2\} \cup \{n_2+1{,\ldots,}n\}.$$ It follows from Proposition \[prop:def\_set\_sol\] that $${\mathbf{x}}^* = (x_1{,\ldots,}x_{n_1}, x_{n_1+1}{,\ldots,}x_{n_2}, x_{n_2+1}{,\ldots,}x_n)^t = (s^*_1{,\ldots,}s^*_{n_1}, 0{,\ldots,}0, r^*_{n_2+1}{,\ldots,}r^*_n)^t.$$ Further, note that $${\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{x}}^*={\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n-\widetilde{\mathbf{\Lambda}}){\mathbf{x}}^* = {\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n-\widetilde{\mathbf{\Lambda}})({\mathbf{x}}^* + \widetilde{\mathbf{\Lambda}}_{\mathcal B}{\mathbf{d}})$$ and $${\mathbf{M}^{{\mathbf{d}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{d}}- {\mathbf{M}^{{\mathbf{d}}}}\widetilde{\mathbf{\Lambda}}_{\mathcal B}{\mathbf{d}}= {\mathbf{M}^{{\mathbf{d}}}}({\mathbf{I}}_n-\widetilde{\mathbf{\Lambda}}){\mathbf{d}}$$ and that $${\mathbf{M}^{{\mathbf{d}}}}{\mathbf{\Lambda}}^*{\mathbf{x}}^*+{\mathbf{M}^{{\mathbf{d}}}}\widetilde{\mathbf{\Lambda}}_{\mathcal B}{\mathbf{d}}={\mathbf{M}^{{\mathbf{d}}}}\widetilde{\mathbf{\Lambda}}({\mathbf{x}}^* + \widetilde{\mathbf{\Lambda}}_{\mathcal B}{\mathbf{d}})$$ because of the structure of ${\mathbf{x}}^*$. By and , ${\mathbf{x}}^*$ solves ${\mathbf{A}}^*{\mathbf{x}}={\mathbf{b}}^*$ if and only if $$\begin{split}
{\mathbf{x}}^* &= {\mathbf{b}}^* + ({\mathbf{M}^{{\mathbf{d}}}}{\mathbf{\Lambda}}^* + {\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*)){\mathbf{x}}^* \\
&= {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{d}}- ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{d}}+ ({\mathbf{M}^{{\mathbf{d}}}}{\mathbf{\Lambda}}^* + {\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*)){\mathbf{x}}^* \\
&= {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{d}}- {\mathbf{M}^{{\mathbf{d}}}}\widetilde{\mathbf{\Lambda}}_{\mathcal B}{\mathbf{d}}- ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{d}}+
({\mathbf{M}^{{\mathbf{d}}}}{\mathbf{\Lambda}}^* + {\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n-{\mathbf{\Lambda}}^*)){\mathbf{x}}^* + {\mathbf{M}^{{\mathbf{d}}}}\widetilde{\mathbf{\Lambda}}_{\mathcal B}{\mathbf{d}}\\
&= {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}({\mathbf{I}}_n-\widetilde{\mathbf{\Lambda}}){\mathbf{d}}- ({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{d}}+ ({\mathbf{M}^{{\mathbf{d}}}}\widetilde{\mathbf{\Lambda}}+
{\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n-\widetilde{\mathbf{\Lambda}}))({\mathbf{x}}^* + \widetilde{\mathbf{\Lambda}}_{\mathcal B}{\mathbf{d}}).
\end{split}$$ Since $({\mathbf{I}}_n-{\mathbf{\Lambda}}^*){\mathbf{d}}=({\mathbf{I}}_n-\widetilde{\mathbf{\Lambda}}){\mathbf{d}}+ \widetilde{\mathbf{\Lambda}}_{\mathcal B}{\mathbf{d}}$, we can add $\widetilde{\mathbf{\Lambda}}_{\mathcal B}{\mathbf{d}}$ on both sides of the equation and obtain $${\mathbf{x}}^* + \widetilde{\mathbf{\Lambda}}_{\mathcal B}{\mathbf{d}}= {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}({\mathbf{I}}_n-\widetilde{\mathbf{\Lambda}}){\mathbf{d}}- ({\mathbf{I}}_n-\widetilde{\mathbf{\Lambda}}){\mathbf{d}}+ ({\mathbf{M}^{{\mathbf{d}}}}\widetilde{\mathbf{\Lambda}}+
{\mathbf{M}^{{\mathbf{s}}}}({\mathbf{I}}_n-\widetilde{\mathbf{\Lambda}}))({\mathbf{x}}^* + \widetilde{\mathbf{\Lambda}}_{\mathcal B}{\mathbf{d}}),$$ Therefore $\widetilde{\mathbf{x}}=(s^*_1{,\ldots,}s^*_{n_1}, d_{n_1+1}{,\ldots,}d_{n_2}, r^*_{n_2+1}{,\ldots,}r^*_n)^t$ is the solution of $\widetilde{\mathbf{A}}{\mathbf{x}}=\widetilde{\mathbf{b}}$ if and only if ${\mathbf{x}}^*$ solves ${\mathbf{A}}^*{\mathbf{x}}={\mathbf{b}}^*$.
The pseudo solution belonging to $D^*$ is the solution ${\mathbf{R}}^*$ of the system. A direct consequence of Lemma \[lem:pseudo\_equiv\] is that the pseudo solution of $\widetilde D$ is also equal to ${\mathbf{R}}^*$. Similar to the proof of Proposition \[prop:conv\_def\_set\], we can argue that the set $\widetilde D$ will be reached by the Increasing Trial-and-Error Algorithms in a finite number of steps. Note that this statement holds in particular for the Increasing Hybrid Trial-and-Error Algorithm, where Algorithm is used \[alg:picard\_debt\] to calculate the next debt iterate. Even though a Picard-typed procedure is used in this auxiliary algorithm, we can conclude together with Proposition \[prop:hybrid\_better\_elsinger\] and ${\varepsilon}>0$ that the number of iteration will still be finite. We summarize the findings in the next proposition.
\[prop:conv\_def\_set\_incr\] Algorithm \[alg:trial\_error\_incr\] reaches the solution ${\mathbf{R}}^*$ of and in a finite number of iteration steps.
Sandwich Algorithms
-------------------
A disadvantage of the Trial-and-Error Algorithms was that when a potential default set is reached, the only way to find out whether this default set is actually $D^*$, is to calculate the corresponding pseudo solution and check whether it is a fixed point of . The choice of a high lag value can of course increase the chance that $D^*$ is reached at the first trial, but there is no certainty.
Another way to find $D^*$ is to start an iteration simultaneously with the largest and smallest possible solution and use one of the Algorithms \[alg:picard\], \[alg:elsinger\] or \[alg:comb\_method\] to obtain the next iterate. For $k\ge0$ denote by $\overline{{\mathbf{R}}}^k$ the $k$-th iterate of the series that emerges when starting the algorithm with the maximum and by $\underline{{\mathbf{R}}}^k$ its counterpart when starting with the minimum possible solution. Depending which algorithm is chosen, the starting vector can either be $\overline{{\mathbf{R}}}^0={\mathbf{R}_{\rm great}}$ (Picard Iteration) or $\overline{{\mathbf{R}}}^0=({\mathbf{r}_{\rm great}}^t, ({\mathbf{s}}({\mathbf{r}_{\rm great}}))^t)^t$ (Elsinger and Hybrid Algorithm) when starting with the upper boundary. Analogously, we have $\underline{{\mathbf{R}}}^0={\mathbf{R}_{\rm small}}$ or $\underline{{\mathbf{R}}}^0=({\mathbf{r}_{\rm small}}^t, ({\mathbf{s}}({\mathbf{r}_{\rm small}}))^t)^t$ if the minimum possible solution is the starting point. By the Propositions \[prop:conv\_picard\], \[prop:elsinger\_conv\], \[prop:comb\_meth\_conv\] and by Equation , the iterative use of one of the mentioned algorithms entails that the default sets approach one another, i.e. for $k\ge0$ $$D(\underline{{\mathbf{R}}}^k) \supseteq D(\underline{{\mathbf{R}}}^{k+1}) \supseteq D^* \supseteq D(\overline{{\mathbf{R}}}^{k+1}) \supseteq D(\overline{{\mathbf{R}}}^k).$$ Let $$\label{eq:lag_sandwich}
l = \min\{k\ge0:D(\underline{{\mathbf{R}}}^k) = D(\overline{{\mathbf{R}}}^k)\}$$ be the first iteration step in which the default set for both starting vectors is the same. Then we must have that $D(\overline{{\mathbf{R}}}^l)=D^*$ and, by Proposition \[prop:def\_set\_sol\], determining the pseudo solution belonging to $D^*$ leads to ${\mathbf{R}}^*$. Because of its characteristics we call this algorithm the *Sandwich Algorithm*.
[9]{}\[Sandwich Algorithm\]\[alg:sandwich\]
1. Determine $\overline{{\mathbf{R}}}^0$ and $\underline{{\mathbf{R}}}^0$ as well as their corresponding default sets $D(\overline{{\mathbf{R}}}^0)$ and $D(\underline{{\mathbf{R}}}^0)$.
2. \[alg:fp\_sandwich\] For $k\ge 1$, calculate the iterates $\overline{{\mathbf{R}}}^k$ and $\underline{{\mathbf{R}}}^k$ using one of the Algorithms \[alg:picard\], \[alg:elsinger\] or \[alg:comb\_method\] and the corresponding default sets $D(\overline{{\mathbf{R}}}^k)$ and $D(\underline{{\mathbf{R}}}^k)$.
3. If $D(\overline{{\mathbf{R}}}^k)=D(\underline{{\mathbf{R}}}^k)$, stop the algorithm, set $D^*=D(\overline{{\mathbf{R}}}^k)$ and calculate the pseudo solution that belongs to $D^*$ following Definition \[def:sys\_sol\]. Else, set $k=k+1$ and go back to step \[alg:fp\_sandwich\].
As for the Trial-and-Error Algorithms in the sections above, the Sandwich Algorithm results in different versions:
1. The *Sandwich Picard Algorithm* with $\overline{{\mathbf{R}}}^0={\mathbf{R}_{\rm great}}$ and $\underline{{\mathbf{R}}}^0={\mathbf{R}_{\rm small}}$ and the use of Algorithm \[alg:picard\] in step \[alg:fp\_sandwich\].
2. The *Sandwich Elsinger Algorithm* with $\overline{{\mathbf{R}}}^0=({\mathbf{r}_{\rm great}}^t,({\mathbf{s}}({\mathbf{r}_{\rm great}}))^t)^t$ and $\underline{{\mathbf{R}}}^0=({\mathbf{r}_{\rm small}}^t,({\mathbf{s}}({\mathbf{r}_{\rm small}}))^t)^t$ and the use of Algorithm \[alg:elsinger\] in step \[alg:fp\_sandwich\].
3. The *Sandwich Hybrid Algorithm* with the same starting points as the Sandwich Elsinger Algorithm and the iterative use of Algorithm \[alg:comb\_method\] in step \[alg:fp\_sandwich\].
Recall the insights from Section \[subsec:trial\_error\_alg\_incr\], where it was shown that, under some circumstances, it may happen that the series of default sets $D(\underline{{\mathbf{R}}}^k)$ will never converge to the actual default set $D^*$. Situations in which this problem occurs always contain at least one firm that is on borderline in the solution ${\mathbf{R}}^*$. As a result of this behavior, the Sandwich Algorithm may not converge in the sense that the default sets $D(\overline{{\mathbf{R}}}^k)$ and $D(\underline{{\mathbf{R}}}^k)$ will never become identical. However, if we consider a stochastic setting and assume a distribution for the vector ${\mathbf{a}}$ of the exogenous assets’ prices which has a density with respect to the Lebesgue measure on $({\mathbb{R}}_0^+)^n$, then situations in which the convergence cannot be assured occur only with probability zero as the next Proposition shows. Note that this assumption is fulfilled in the usual $n$-firm Merton models where the individual $a_i$ are log-normally distributed.
\[prop:sandwich\] The Sandwich Algorithm generates a sequence of decreasing default sets $D(\underline{{\mathbf{R}}}^k)$ and a sequence of increasing default sets $D(\overline{{\mathbf{R}}}^k)$ that reach the default set $D^*$ of the solution ${\mathbf{R}}^*$ almost surely after finitely many steps. Thus, it reaches the solution ${\mathbf{R}}^*$ of almost surely after finitely many steps.
The increasing and decreasing property of the default sets follows directly from the Propositions \[prop:conv\_picard\], \[prop:elsinger\_conv\] and \[prop:comb\_meth\_conv\]. The two series of default sets of the algorithm both converge in finitely many iteration steps to $D^*$ if there is no firm in the financial system that is borderline. Lemma \[lem:boderline\_as\] in the Appendix shows that the probability for borderline firms in ${\mathbf{R}}^*$ is zero from which almost sure convergence follows.
By its nature, the Sandwich Algorithm converges to $D^*$ from both directions which doubles the computation and makes the algorithm somewhat inefficient from a computational point of view. On the other hand, the algorithm computes an exact solution in finitely many iteration steps without wasting time on “Trial-and-Error”. In contrast to the Trial-and-Error Algorithms, the drawback of the Sandwich Algorithm is that the convergence of the procedure cannot be ensured when borderline firms are present in the system. To overcome this problem, we recommend for practical purposes to apply the idea of a lag value in the Sandwich Algorithm as well.
[10]{}\[Modified Sandwich Algorithm\]\[alg:sandwich\_mod\] Set $l\ge2$.
1. Determine $\overline{{\mathbf{R}}}^0$ and $\underline{{\mathbf{R}}}^0$ as well as their corresponding default sets $D(\overline{{\mathbf{R}}}^0)$ and $D(\underline{{\mathbf{R}}}^0)$.
2. \[alg:fp\_sandwich\_mod\] For $k\ge 1$, calculate the iterates $\overline{{\mathbf{R}}}^k$ and $\underline{{\mathbf{R}}}^k$ using one of the Algorithms \[alg:picard\], \[alg:elsinger\] or \[alg:comb\_method\] and the corresponding default sets $D(\overline{{\mathbf{R}}}^k)$ and $D(\underline{{\mathbf{R}}}^k)$.
3. If $D(\overline{{\mathbf{R}}}^k)=D(\underline{{\mathbf{R}}}^k)$, stop the algorithm, set $D^*=D(\overline{{\mathbf{R}}}^k)$ and calculate the pseudo solution that belongs to $D^*$ following Definition \[def:sys\_sol\]. Else, if $k\ge l$ and $$\label{eq:lag_equal_sandwich}
|D(\underline{{\mathbf{R}}}^k)| - |D(\overline{{\mathbf{R}}}^k)| = \ldots = |D(\underline{{\mathbf{R}}}^{k-l+1})| - |D(\overline{{\mathbf{R}}}^{k-l+1})|,$$ calculate the pseudo solution belonging to $D(\overline{{\mathbf{R}}}^k)$ and stop the algorithm if it solves the Equations and . Else, set $k=k+1$ and go back to step \[alg:fp\_sandwich\_mod\].
The modification consists of interrupting the algorithm if the default sets $D(\overline{{\mathbf{R}}}^k)$ and $D(\underline{{\mathbf{R}}}^k)$ for both iteration directions are not identical but stay constant for $l$ consecutive times. If $l$ is chosen large enough (e.g. $l\ge5$) and holds, this is a strong indication that at least one firm in the system is borderline and that the convergence of both series is not given. In this situation, a check whether the default set has already been reached is suitable.
Simulation Study {#sec:simulation}
================
Aim of this section is to confirm the theoretical findings in the former sections by simulation. In particular, we focus our considerations on the following subjects:
(i) Investigate the trade-off when choosing a lag value $l$ in the Trial-and-Error Algorithms of Section \[subsec:trial\_error\_alg\_decr\]. The result of this part contains ‘optimal’ lag values for each algorithm and will be used in the next part.
(ii) Investigate the algorithm efficiency of all the presented algorithms in the Sections \[sec:iterative\_algo\] and \[sec:def-set-algo\]. For every different technique to calculate the next iterate (Picard, Elsinger, Hybrid), we compare the three types of algorithms (non-finite, Trial-and-Error, Sandwich) with each other.
Before presenting the results of our study, the used financial systems are further specified.
General Structure of the Financial Systems
------------------------------------------
For the system size $n$ we chose six different values, viz. $n\in\{5, 10, 25, 50, 100, 200\}$. A system with only 5 or 10 firms can be considered as relatively small whereas networks with $n=25$ or $n=50$ are regarded as medium-sized. Small systems are investigated for example in @gourieroux12, @rogers13 and @elsinger06a where the size was 5, 6 and 10 firms respectively. Examples of medium-sized systems are @acemoglu13 and @nier07 where the size was 20 and 25 firms respectively. Further, we added networks with 100 and 200 firms into our study to also include larger systems. Existing studies for such sizes are @elliott13, @cont11 and @gai11 that entailed networks with 100, 125 and 250 firms, respectively.
There are some empirical studies (@elsinger06b, @gai10) that investigated system sizes of about $n=1000$. We believe that for practical purposes such large systems are not of interest which is why we did not take values $n>200$ into account. However, it is expectable that the results obtained for our system sizes also hold for networks with more than 200 firms.
The next input parameters to define are the asset and debt values. To keep it simple, we assumed in every simulation scenario of this study for the exogenous assets a value of 1 for each firm, i.e. ${\mathbf{a}}= (1{,\ldots,}1)^t\in{\mathbb{R}}^n$. For firm $i$’s nominal debt value, we set a fixed value $d_i=d$ for all $i\in\mathcal N$ and added a random variation to each debt value in order to get differing setups which leads to $$\label{eq:sim_debt_value}
{\mathbf{d}}= (d_1{,\ldots,}d_n)^t + ({\varepsilon}_1{,\ldots,}{\varepsilon}_n)^t \in{\mathbb{R}}^n,$$ where the ${\varepsilon}_i$ are independently normally distributed with mean value 0 and standard deviation 0.5, i.e. ${\varepsilon}_i\sim N(0, 0.25)$. Note that in case of shocks with ${\varepsilon}_i < d_i$ we set $d_i=0$ to avoid negative liabilities.
When constructing an ownership matrix, the degree of ownership that is operationalized by the expression *integration* can provide some crucial information.
Consider a financial System $\mathcal F = ({\mathbf{a}}, {\mathbf{d}}, {\mathbf{M}^{{\mathbf{d}}}}, {\mathbf{M}^{{\mathbf{s}}}})$. The *debt integration level $\mu^{{\mathbf{d}}}$* is defined as the maximum column sum of ${\mathbf{M}^{{\mathbf{d}}}}$, i.e. $$\nu^{{\mathbf{d}}} = \max_{i\in\mathcal N} \sum_{j=1}^n M_{ij}^{{\mathbf{d}}} = \|{\mathbf{M}^{{\mathbf{d}}}}\|.$$ Analogously, $$\nu^{{\mathbf{s}}} = \max_{i\in\mathcal N} \sum_{j=1}^n M_{ij}^{{\mathbf{s}}} = \|{\mathbf{M}^{{\mathbf{s}}}}\|$$ is called the *equity integration level*.
The integration level is hence a measure of the extent of cross-ownership in either the debt or the equity component and its definition is based on the one given in the work of @elliott13. Because of Assumption \[assu:os\_mat\_norm\], it follows directly that $\nu^{{\mathbf{d}}}, \nu^{{\mathbf{s}}}\in[0,1)$.
The integration levels $\nu^{{\mathbf{d}}}$ and $\nu^{{\mathbf{s}}}$ obviously do not specify the single entries of the ownership matrices. For this purpose we will limit our consideration in the following to somewhat regular structures of the matrices.
An ownership matrix ${\mathbf{M}}$ is called
- a *ring ownership matrix* if in every column only one entry is larger than 0 and
- a *complete ownership matrix* if every entry, except for the diagonal entry, is larger than 0 and of the same size.
Further, let $\widetilde{\mathbf{M}}$ be a ring ownership matrix and $\widehat{\mathbf{M}}$ be a complete ownership matrix. A *$\lambda$-convex combination* of $\widetilde{\mathbf{M}}$ and $\widehat{\mathbf{M}}$ is defined as the matrix ${\mathbf{M}}$ with entries $$\label{eq:os_mat_convex}
M_{ij} = \lambda \widetilde M_{ij} + (1-\lambda) \widehat M_{ij}, \quad \lambda\in[0,1].$$
The concepts of ring and complete matrices and of convex combinations are originally used in @acemoglu13. If ${\mathbf{M}^{{\mathbf{d}}}}$ is a ring matrix this means that every firm has only one creditor within the system, and only one shareholder if ${\mathbf{M}^{{\mathbf{s}}}}$ is a ring matrix. Without loss of generality we assume that firm $i+1$ is the creditor (shareholder) of firm $i$ for $i=1{,\ldots,}n-1$ and that firm $n$ is the creditor (shareholder) of firm 1. When ${\mathbf{M}^{{\mathbf{d}}}}$ (${\mathbf{M}^{{\mathbf{s}}}}$) is a complete ownership matrix, debt (share) proportions are equally distributed between the $n-1$ firms. The lower $\lambda$ is chosen, the more equal are the entries of the corresponding convex combination.
For a system of size $n=4$ we assume a debt integration level of $\nu^{{\mathbf{d}}}=0.9$ and set $\lambda = 0.5$. The ring ownership matrix $\widetilde{\mathbf{M}}^{{\mathbf{d}}}$, the complete ownership matrix $\widehat{\mathbf{M}}^{{\mathbf{d}}}$ and the $\lambda$-convex combination matrix ${\mathbf{M}^{{\mathbf{d}}}}$ then are $$\widetilde{\mathbf{M}}^{{\mathbf{d}}} = \begin{pmatrix} 0 & 0 & 0 & .9 \\
.9 & 0 & 0 & 0 \\
0 & .9 & 0 & 0 \\
0 & 0 & .9 & 0 \\\end{pmatrix},\
\widehat{\mathbf{M}}^{{\mathbf{d}}} = \begin{pmatrix} 0 & .3 & .3 & .3 \\
.3 & 0 & .3 & .3 \\
.3 & .3 & 0 & .3 \\
.3 & .3 & .3 & 0 \\\end{pmatrix}\ \text{and}\
{\mathbf{M}^{{\mathbf{d}}}}= \begin{pmatrix} 0 & .15 & .15 & .6 \\
.6 & 0 & .15 & .15 \\
.15 & .6 & 0 & .15 \\
.15 & .15 & .6 & 0 \\\end{pmatrix}.$$
Consider two financial systems $\mathcal F = ({\mathbf{a}},{\mathbf{d}},{\mathbf{M}^{{\mathbf{d}}}},{\mathbf{M}^{{\mathbf{s}}}})$ and $\widetilde {\mathcal{F}}= (\widetilde{\mathbf{a}},\widetilde{\mathbf{d}},\widetilde{\mathbf{M}}^{{\mathbf{d}}},\widetilde{\mathbf{M}}^{{\mathbf{s}}})$ with corresponding integration levels $\nu^{{\mathbf{d}}}$, $\nu^{{\mathbf{s}}}$ and $\tilde\nu^{{\mathbf{d}}}$, $\tilde\nu^{{\mathbf{s}}}$. Due to the regular structure of the ownership matrices we can say that a system $\widetilde {\mathcal{F}}$ is more *debt-integrated* than ${\mathcal{F}}$, if and only if $\tilde\nu^{{\mathbf{d}}}>\nu^{{\mathbf{d}}}$. In the same way, we define that a system is more *equity-integrated*.
With the former definitions, the following parameters are needed for the simulation of a financial system: $n$, $d$, ${\nu^{{\mathbf{d}}}}$, ${\nu^{{\mathbf{s}}}}$ and $\lambda$, where we will use the same $\lambda$ to define the debt and the equity ownership matrix according to . A *simulated system* is the financial system ${\mathcal{F}}= ({\mathbf{a}},{\mathbf{d}},{\mathbf{M}^{{\mathbf{d}}}},{\mathbf{M}^{{\mathbf{s}}}})$, where the parameters $n$, $d$, ${\nu^{{\mathbf{d}}}}$, ${\nu^{{\mathbf{s}}}}$ and $\lambda$ are used to define ${\mathbf{a}}$, ${\mathbf{M}^{{\mathbf{d}}}}$ and ${\mathbf{M}^{{\mathbf{s}}}}$ and where the liabilities ${\mathbf{d}}$ are a realization of the random variable in .
Effect of the Lag Value {#subsec:effect_lag_value}
-----------------------
As mentioned in Section \[subsec:trial\_error\_alg\_decr\], the smaller the lag value $l$ is chosen in the Trial-and-Error Algorithms, the higher is the chance that the first possible default set is not the actual $D^*$. This results in unnecessary computation steps to reach the real default set. On the other hand, if $l$ is taken as very high, say $l=5$ or higher, there are many iteration steps in the algorithm that are possibly not needed. For this reason, we wanted to investigate this trade-off situation by determining the error rate for the first potential default set.
Assume that for a given parameters $n$, $d$, ${\nu^{{\mathbf{d}}}}$, ${\nu^{{\mathbf{s}}}}$ and $\lambda$, we have generated $N$ simulated systems. For every system we determine for the Trial-and-Error Picard (TP), the Trial-and-Error Elsinger (TE) and the Trial-and-Error Hybrid Algorithm (TH) for a lag value $l\ge 2$ the first potential default set $\bar D^j_{{\rm TP}}(l)$, $\bar D^j_{{\rm TE}}(l)$ and $\bar D^j_{{\rm TH}}(l)$ where $j=1{,\ldots,}N$. In case of the Trial-and-Error Picard Algorithm we define $${\varepsilon}^j_{{\rm IP}}(l) = \begin{cases} 1, & \text{if $\bar D^j_{{\rm IP}}(l) \not= D^*$} \\
0, & \text{else}, \end{cases}$$ and analogously ${\varepsilon}^j_{\rm TE}(l)$ and ${\varepsilon}^j_{\rm TH}(l)$ for the TE and TH Algorithm, respectively. The *error rate* for the TP Algorithm for the lag value $l$ is then given by $${\varepsilon}_{\rm TP}(l) = \frac 1N \sum_{j=1}^N {\varepsilon}^j_{\rm TP}(l) \in [0, 1].$$ In the same way the error rates ${\varepsilon}_{\rm TE}(l)$ and ${\varepsilon}_{\rm TH}(l)$ are defined.
For the investigation of the error rate we chose $d_i=d=1.5$ in as the debt value. The debt integration values where $\nu^{{\mathbf{d}}}\in\{0.9, 0.5, 0.1\}$, which we considered as systems with high, moderate and low debt cross-ownership. Similarly, we took $\nu^{{\mathbf{s}}}\in\{0.45, 0.25, 0.05\}$ for equity integration, where each value is half the associated debt integration. The justification for this approach is that equity cross-ownership is probably commonly less pronounced than debt cross-ownership. Further, we wanted to avoid possible cross-ownership entries lager that 0.5 since this would mean that a firm is owned by majority by another firm in the system. Each equity and debt integration value was combined with each other which results in 9 possible system settings. Beyond that, all 9 settings were investigated three times, where the structure parameter $\lambda$ took the three possible values $\lambda\in\{0, 0.5, 1\}$, i.e. systems with only ring ownership matrices, systems with only complete matrices and systems with a 0.5-convex combination were considered. In total, the combination of the parameters ${\nu^{{\mathbf{d}}}}$, ${\nu^{{\mathbf{s}}}}$ and $\lambda$ leads to 27 different settings and for every setting $N=1000$ simulated systems were generated. The error rates were calculated for the three algorithms for the lag values $l=2{,\ldots,}7$. Repetitive simulations with $N=1000$ showed that the error rates are fairly stable for different simulation runs which is why we viewed the number of 1000 repetitions as reliable. We used the Decreasing Trial-and-Error Algorithm defined in Algorithm \[alg:trial\_error\_decr\], but simulations with the Increasing Trial-and-Error Algorithm showed very similar results.
$l=2$ $l=3$ $l=4$ $l=5$ $l=6$ $l=7$
-- ---- -------- ------- ------- ------- ------- -------
TP 9.652 3.111 1.170 0.500 0.230 0.104
TE 3.452 0.718 0.159 0.037 0.022 0.007
TH 0.156 0.015 0.004 0 0 0
TP 4.574 1.452 0.529 0.196 0.089 0.048
TE 1.256 0.111 0.015 0.004 0 0
TH 0.233 0.004 0 0 0 0
TP 8.608 2.730 1.063 0.455 0.207 0.067
TE 2.819 0.348 0.041 0.004 0 0
TH 0.533 0 0 0 0 0
TP 10.596 3.122 1.111 0.482 0.189 0.078
TE 3.300 0.389 0.041 0.007 0 0
TH 0.537 0.011 0 0 0 0
TP 11.359 3.385 1.211 0.478 0.2 0.093
TE 3.233 0.411 0.044 0.007 0 0
TH 0.404 0.007 0 0 0 0
TP 11.485 3.230 1.111 0.419 0.167 0.063
TE 2.870 0.322 0.022 0.007 0 0
TH 0.267 0.007 0 0 0 0
TP 9.379 2.838 1.033 0.422 0.180 0.075
TE 2.822 0.383 0.054 0.011 0.004 0.001
TH 0.355 0.007 0.001 0 0 0
: Error rates ${\varepsilon}^k_{\rm TP}(l)$, ${\varepsilon}^k_{\rm TE}(l)$ and ${\varepsilon}^k_{\rm TH}(l)$ in percentage points for the Decreasing TP, TE and TH Algorithm for $l=2{,\ldots,}7$. Mean values over all three values of $\lambda$ and all values of $\nu^{{\mathbf{d}}}$ and $\nu^{{\mathbf{s}}}$ are shown for each combination. The last three rows of the table show the overall mean error rates over all considered system sizes.[]{data-label="tab:lag_sim_res_n"}
The first observation was that the structure of the ownership matrix, that is the choice of $\lambda$, had no severe effect on the error rates. This results in comparing the error rates for systems with given $n$, $\nu^{{\mathbf{d}}}$ and $\nu^{{\mathbf{s}}}$ for the three choices of $\lambda$. In case of $n=100$ where moderate debt and low equity cross-ownership was present ($\nu^{{\mathbf{d}}}=0.5$, $\nu^{{\mathbf{s}}}=0.05$) and a lag value $l=1$, the largest absolute difference is documented for the Trial-and-Error Picard Algorithm between complete and ring ownership matrices (9.6% to 3.2%). In the large majority of possible combinations, the difference was much smaller which is why we concluded that the ownership structure itself does not affect the error rate in an essential way. For this reason, we summarized the three values of $\lambda$ and calculated the mean error rates for every combination of $n$, $\nu^{{\mathbf{d}}}$ and $\nu^{{\mathbf{s}}}$ over all $\lambda$ for the further results.
In Table \[tab:lag\_sim\_res\_n\], the results of the simulation to investigate the effect of the system size are summarized. The error rates are calculated as the mean over all possible combinations for debt and equity integration for every value of $n$. We observe that the network size only slightly affects the error rates, since for the same lag value they are relatively close for all system sizes. The only exception are systems with $n=10$ firms, where the error rate is smaller compared to the other systems. In the last three rows of the table, an overall impression of the error rates shows that even for the TP Algorithm and a lag value of $l=2$, the error rate is not higher than 10% in total. We also detected that the error rates for the TE Algorithm are much smaller and even more so for the TH Algorithm. For increasing lag values, the error rates quickly diminish in size for all considered methods.
-- ---------------------- ------ ------- ------- ---------------------- -- ------ ------ ------ ------- ------
$\nu^{{\mathbf{d}}}$ $\nu^{{\mathbf{d}}}$
low mod. high low mod. high
low 2.60 4.65 7.64 4.96 low 0.22 0.54 1.41 0.72
mod. 8.53 19.64 35.83 21.33 mod. 1.72 5.58 14.62 7.31
high 1.31 2.47 1.75 1.84 high 0.14 0.73 0.57 0.48
4.11 8.73 15.14 9.38 0.69 2.29 5.53 2.84
$\nu^{{\mathbf{d}}}$ $\nu^{{\mathbf{d}}}$
low mod. high low mod. high
low 2.16 2.34 2.51 2.34 low 0.19 0.19 0.18 0.19
mod. 6.24 6.07 5.60 5.97 mod. 1.07 0.91 0.86 0.95
high 0.29 0.13 0.06 0.16 high 0.03 0.02 0 0.02
2.90 2.84 2.72 2.82 0.43 0.37 0.35 0.38
$\nu^{{\mathbf{d}}}$ $\nu^{{\mathbf{d}}}$
low mod. high low mod. high
low 0.02 0.13 0.24 0.13 low 0 0 0 0
mod. 0.22 0.96 1.59 0.92 mod. 0 0.01 0.04 0.02
high 0.04 0 0 0.01 high 0 0 0 0
0.09 0.36 0.61 0.35 0 0.01 0.01 0.01
-- ---------------------- ------ ------- ------- ---------------------- -- ------ ------ ------ ------- ------
: Error rates ${\varepsilon}^k_{\rm TP}(l)$, ${\varepsilon}^k_{\rm TE}(l)$ and ${\varepsilon}^k_{\rm TH}(l)$ in percentage points for the TP, TE and TH Algorithm for $l=2,3$. For the debt integration level $\nu^{{\mathbf{d}}}$, low, moderate and high integration is defined as 0.1, 0.5 and 0.9, respectively. For the equity integration level $\nu^{{\mathbf{s}}}$ the corresponding levels are defined as 0.05, 0.25 and 0.45. Mean values over all three values of $\lambda$ and all values of $n$ are given for each combination. Further, overall mean error rates for within each debt and equity integration level are shown in an additional column and row.[]{data-label="tab:lag_sim_res_alg"}
To assess the influence of the debt and the equity integration level, the mean error rate over all considered system sizes are taken and listed for each possible combination of the integration level, as shown in Table \[tab:lag\_sim\_res\_alg\]. If the debt integration level increases from low to moderate, the error rates increase as well. This can, for example, be seen when comparing the mean error rates for the debt integration levels 0.1 and 0.5 over all equity integration levels in the last column in Table \[tab:lag\_sim\_res\_alg\] for every lag value. For the Trial-and-Error Picard Algorithm, the error rate increases from 4.96% to 21.33%; for the other algorithms we observe similar results. A further increase of the debt integration from 0.5 to 0.9, however, has the reverse effect since the error rates decrease in this case. Again we take the TP Algorithm as an example where the error rates shrinks from 21.33% to 1.84%. A possible explanation for this behavior could be that for the TP Algorithm the number of needed iteration steps to converge to the solution was always highest for the combination $\nu^{{\mathbf{d}}}=0.5$ and $\nu^{{\mathbf{s}}}=0.45$ (data not shown). Hence, the convergence speed in these situations is very slow which explains why the first potential default set is often not the actual default set. When the mean error rates for every equity integration level averaged over all values of $\nu^{{\mathbf{d}}}$ is examined, we observed that, except for the TE Algorithm, the error rates increase for increasing integration levels. In case of the TP Algorithm, we have an increase from 4.11% to 8.73% to 15.14% for $\nu^{{\mathbf{s}}}=0.05$, $\nu^{{\mathbf{s}}}=0.25$ and $\nu^{{\mathbf{s}}}=0.45$, respectively. The error rates for the TE Algorithm stay approximately constant (2.90%, 2.84%, 2.72%).
Our overall conclusion of this part of the simulation study is that the choice of the algorithm and the lag value $l$ has the strongest effect on the error rate, i.e the error rates quickly decrease for greater lag values. The TP Algorithm has, as expected, the highest overall error rates, much higher than the TE and the TH Algorithm. What also affects the error rate is the integration level of the ownership matrices. For our simulation setting, it was the combination of moderate debt and high equity integration that yielded in the highest rates. The structure of the ownership matrices, on the other hand, had no influence on the error rate. Taking the overall mean error rates as the main reference, we can state that for the TE and the TH Algorithm a lag value of 2 is appropriate, since the corresponding error rates are with 2.82% and 0.36% very small. For the TP Algorithm, however, for $l=2$ we get an overall error rate of 9.38% which is why a lag value of 3 with an overall error rate of 2.84% seems more convenient for this procedure.
Comparison of Algorithm Efficiency {#subsec:algo_efficency}
----------------------------------
Searching for the most efficient algorithm, the main issue is to minimize the calculation effort to find the solution ${\mathbf{R}}^*$. In every iteration step of the algorithm, different kind of calculations are carried out for the different algorithms. We quantify the calculation costs with the Landau symbol (Big ${\mathcal{O}}$ notation), where for example ${\mathcal{O}}(n)$ means that the time $T(n)$ to compute a problem of size $n$ grows at the rate $n$. We distinguish between two different types of calculations in our considerations. For the first type, a mapping is applied to a given vector. This mapping can either be the mapping $\Phi$ in or the mappings $\Phi^{{\mathbf{d}}}$ and $\Phi^{{\mathbf{s}}}$ in and , respectively, whereas the second type, matrix multiplications are done. In all cases, the most expensive calculations are matrix multiplications, whereas the type embodies the solution of a linear equation system such as the ones defined in the Algorithms \[alg:equity\] and \[alg:debt\]. The computational costs of both types are between the range of ${\mathcal{O}}(n^2)$ and ${\mathcal{O}}(n^3)$ (cf. @dahlquist08).
Keeping the functioning of the algorithms in mind, the Elsinger and the Hybrid Algorithm seem to be less efficient than the Picard Algorithm, since in the latter one, no linear equation system has to be solved which results in smaller computational costs. However, as we have seen in the Propositions \[prop:elsinger\_better\_picard\] and \[prop:hybrid\_better\_elsinger\], in terms of iteration steps, the Hybrid Algorithm converges faster to the solution compared to the Elsinger Algorithm, which in turn converges faster to the solution than the Picard Iteration. Therefore, a typical trade-off-situation is given between computational costs and convergence speed of an algorithm. Note that for a sequence ${\mathbf{R}}^k$ that converges to a fixed point ${\mathbf{R}}^*$ the *convergence rate* is called *linear*, if there exists a $c\in(0,1)$ such that $$\|{\mathbf{R}}^* - {\mathbf{R}}^{k+1}\| \le c \|{\mathbf{R}}^* - {\mathbf{R}}^k\|$$ for all $k\ge0$. Since $\|\Phi({\mathbf{R}}^*)-\Phi({\mathbf{R}}^k)\|\le{I^{\max}}\|{\mathbf{R}}^*-{\mathbf{R}}^k\|$ with ${I^{\max}}=\max\{||{\mathbf{M}^{{\mathbf{d}}}}||,||{\mathbf{M}^{{\mathbf{s}}}}||\}$ (see Lemma 4.1 in @fischer14), linear convergence holds for the Picard Algorithm if instead of Assumption \[assu:holding\_mat\] the stronger assumption of matrix norms being strictly smaller than $1$ is made (which means that of all debt and equity a non-zero share is held by a system outsider). The properties of the Elsinger and the Hybrid Algorithms, however, made it impossible to prove linear convergence (or an even higher convergence rate). The next problem is that the total computational cost for one the algorithms is not determinable in general. For these reasons, the comparison of the different calculation techniques (Picard, Elsinger, Hybrid) on an analytical base seems impossible.
This is why we measured the time that was needed to execute an algorithm and considered this value as the primary outcome of our simulation. Though this measure strongly depends on the processor speed and memory capacity of the computer, it allows an objective comparison of the different algorithms. The simulations were conducted on a computer with 3.2 GHz and 4 GB RAM, the software used was R (@Rsoftware).
The parameters defining the financial systems are given in the following. Unlike to the simulation in Section \[subsec:effect\_lag\_value\], where a fixed debt value $d$ was used, we varied between five possible the debt values and chose $d\in\{1, 1.5, 2, 2.5, 3\}$. The set of equity and debt integration levels was extended to $\nu^{{\mathbf{s}}}\in\{0.025,0.1,0.175,0.25,0.325,0.4,0.475\}$ and $\nu^{{\mathbf{d}}} \in \{0.05,0.2,0.35,0.5,0.65,0.8,0.95\}$, hence seven possible integration level respectively. A result in Section \[subsec:effect\_lag\_value\] was that the structure of the ownership matrix does not influence the error rates, which is why we only took complete ownership matrices into account for this simulation, i.e. $\lambda=0$ in . Together with the six considered systems sizes ($n\in\{5, 10, 25, 50, 100, 200\}$), this new setting results in $6\cdot5\cdot7\cdot7=1470$ different settings. Again, for each setting, $N=1000$ simulated systems were generated for every parameter combination. For every simulated systems we applied all 15 algorithms presented in the former section and documented the runtime for every procedure to find the solution ${\mathbf{R}}^*$. We used the Picard, the Elsinger and the Hybrid Algorithm (Algorithms \[alg:picard\], \[alg:elsinger\] and \[alg:comb\_method\]) with both versions, i.e. the decreasing and the increasing version. Further, the Trial-and-Error versions were considered, again both the decreasing (Algorithm \[alg:trial\_error\_decr\]) and the increasing version (Algorithm \[alg:trial\_error\_incr\]) with lag values of $l=3$ for the Picard versions and $l=2$ for the Elsinger and the Hybrid versions. The choice of the lag value is a result of the simulations in Section \[subsec:effect\_lag\_value\]. Note that minimizing the error rate to an appropriate value is not necessarily equivalent to minimizing the runtime of the algorithms. For these reasons, we compared for every considered scenario the runtime of the Trial-and-Error Algorithms using lag values from $l=2$ to $l=5$. The results (not shown here) are that for the Elsinger and the Hybrid versions of the algorithms, the choice of $l=2$ does not only keep the error rate on a very low level, but also minimizes the runtime. For the Trial-and-Error Picard Algorithm, the simulation showed that the runtime is almost identical for $l=2$ and $l=3$. However, there is no clear tendency between those choices of $l$: for some parameter combinations $l=2$ lead to smaller runtimes and for some situations this was the case for $l=3$. Due to this indifference for the Trial-and-Error Picard Algorithm, we set $l=3$ for these procedures in accordance with the findings of Section \[subsec:effect\_lag\_value\]. At last, the three versions of the Sandwich Algorithm (Algorithm \[alg:sandwich\]) were also taken into account. The tolerance level in all algorithms was set to $\varepsilon=10^{-3}$.
-- ---------- ------ ------ ------ ------ ------- --------
5 10 25 50 100 200
Picard 1.81 1.90 2.27 3.01 6.19 21.16
Elsinger 1.86 2.08 2.90 5.71 24.38 175.40
Hybrid 1.65 1.84 2.58 5.24 23.06 164.03
1.78 1.94 2.58 4.65 17.87 120.20
Picard 1.29 1.30 1.64 2.81 8.77 45.21
Elsinger 1.32 1.39 1.91 3.95 17.14 119.38
Hybrid 1.57 1.67 2.24 4.49 19.04 130.88
1.39 1.45 1.93 3.75 14.98 98.49
Picard 1.16 1.28 1.79 3.31 10.73 55.08
Elsinger 1.33 1.54 2.41 5.53 26.18 192.65
Hybrid 1.49 1.70 2.55 5.57 25.44 182.91
1.32 1.51 2.25 4.80 20.78 143.55
Picard 1.47 1.54 1.92 2.99 8.13 37.56
Elsinger 1.54 1.69 2.41 4.97 21.84 156.44
Hybrid 1.59 1.74 2.44 5.01 21.93 154.55
1.53 1.66 2.26 4.32 17.30 116.19
-- ---------- ------ ------ ------ ------ ------- --------
: Mean runtime in seconds for every algorithm over all debt and equity integration values ($\nu^{{\mathbf{d}}}$ and $\nu^{{\mathbf{d}}}$) and all debt values ($d$) grouped by system size, algorithm and iteration type. For each of the three iteration types, the average runtime of the corresponding increasing and decreasing version was calculated, except for the Sandwich Algorithms. []{data-label="tab:res_sim_size"}
An important topic is of course the comparison of new developed techniques (Trial-and-Error, Sandwich) with the existing procedure. In Table \[tab:res\_sim\_size\], the mean runtimes are listed grouped by the size of the financial system as well as the algorithm and the iteration type. We summarized the decreasing and the increasing version of every algorithm respectively by calculating the mean runtime of both procedures. The runtimes of the decreasing versions were in most situations smaller than their counterparts. Ignoring the random structure of ${\mathbf{d}}$ for an instance and calculating the fixed point, it could be seen that in about 60 % of all considered scenarios, no firm was in default which explains the slight ‘overperformance’ of the decreasing algorithms. If we compare the mean runtimes over all iteration types, we find that for $n=5$ the Sandwich Algorithms have the best performance (1.32 s) compared to the Trial-and-Error and the non-finite methods (1.39 s and 1.78 s, respectively). For $n\ge5$, the fastest runtimes averaged over the iteration types are achieved for the Trial-and-Error procedures. Comparing the different iteration techniques with each other, we find that using the Picard Iteration technique results in a minimal computational effort for all considered system sizes. To be more specific, for small financial systems ($n=5,10$), the Sandwich Picard Algorithm shows the smallest runtime (1.16 s and 1.28 s, respectively), whereas for medium-sized systems ($n=25,50$), the Trial-and-Error Picard Algorithm performs best compared to the other algorithm types (1.64 s and 2.81 s, respectively). In case of large financial systems, i.e. $n=100,200$, the Picard Iteration in its non-finite form yields to lowest runtimes (6.19 s and 21.16 s, respectively). In general, it is clearly visible that Picard-typed algorithms have the best performance within every Algorithm type. The only exception of this trend can be found in the class of non-finite algorithm, where for $n=5$ and $n=10$ the Hybrid Algorithm showed a slightly lower runtime than the Picard Algorithm. In all other situations, however, the Picard Algorithm is superior to the other algorithms.
Beside the size $n$, we also investigated the influence of the other parameters that define the form of the financial system on the runtime. We observed that increasing debt values $d$ result in an increasing calculation effort, see Table \[tab:res\_sim\_debt\_int\] in the Appendix for a detailed overview. An exception of this tendency represents the Hybrid Algorithm, where the runtime for large $d$ begins to decrease again, no matter which algorithm type is considered. The reason for this behavior is that the runtime for the increasing versions of the Hybrid Algorithm becomes smaller for large $n$ and so does the average of the increasing and the decreasing version of the algorithm, that is shown in Table \[tab:res\_sim\_debt\_int\]. A possible explanation is that the Increasing Hybrid Algorithm uses a Picard-type technique to determine the next debt iterate (cf. Algorithm \[alg:picard\_debt\]). As shown above, the Picard iteration results, in particular for large $n$, in much better runtime performances.
If the debt integration level increases, we first observe a similar effect on the runtime as for the debt values, i.e. the higher the integration level, the higher the runtime. However, this monotonicity holds only up to $\nu^{{\mathbf{d}}}=0.5$ or $\nu^{{\mathbf{d}}}=0.65$ in most cases. For larger debt integration levels, the computational effort decreases again. The reason for this behavior could be that for small values of $\nu^{{\mathbf{d}}}$, it is very likely that many firms in the financial system are in default. In such situations, we observe that only few iteration steps are needed until ${\mathbf{R}}^*$ is reached. If the debt integration level is very high, the same effect establishes with the difference that many firms in the system are solvent. For medium debt integration levels this clear distinction for a firm between solvent and default disappears. A consequence is a higher number of needed iteration steps which also influences the runtime. Moreover, this interpretation is underlined by the fact that for small $\nu^{{\mathbf{d}}}$ (firms more likely in default), the increasing versions of the algorithms have a better performances, whereas this relationship inverts for large integration levels where the firms are more likely to be solvent. For increasing equity integration levels this effect is not visible, the runtime increases if $\nu^{{\mathbf{s}}}$ increases (results not shown). Since $\nu^{{\mathbf{s}}}\le 0.45$ the equity integration seems to have a less strong effect on the status of the firms in the system and therefore, the effect seen in the debt integration levels does probably not appear.
Summary {#sec:summary}
=======
In this article, we gave a survey of the existing algorithms (“Picard” and “Elsinger”) for the computation of equilibrium prices in a financial system in which cross-holdings of equity and debt are present. Moreover, we showed how the ideas of @elsinger09 and @eisenberg01 can be combined to get an iteration procedure (“Hybrid Algorithm”) that is in every iteration step closer to the solution ${\mathbf{R}}^*$ than the “Picard” and “Elsinger” algorithms. We developed new iteration methods based on the information of defaulting and solvent firms under a current payment vector. A consequence of these default set-based methods is that the exact solution of the system is reached in a finite number of steps, which could not be ensured for the existing iteration procedures. Using this new approach yields to two different concepts, that we called “Sandwich” algorithms and “Trial-and-Error” algorithms. While for the former type, a clear stopping criteria can be defined (at least almost surely), the latter algorithms have the drawback that every potential solution has to be checked for validity. In a simulation study, we showed that choosing an appropriate lag value $l$, the computational effort can be kept to a minimum.
Another simulation showed that essentially less iteration steps have to be performed when using the new default-set based techniques. However, the faster convergence concerning the number of iterations has its price: In the new algorithms other than the Picard type, potentially several linear equation systems have to be solved in every iteration step. This leads to a higher calculation effort for those methods and a result of the empirical investigation of the runtime for all algorithms was that in particular for large financial systems the computational costs then become higher than for algorithms where no linear equation systems have to be solved. Another result of the runtime analysis is that the most efficient iteration technique is of Picard type. In the majority of the considered settings those iteration techniques performed best, no matter which algorithm type (Non-finite, Trial-and-Error, Sandwich) was used. One of the main results is that the choice of the most efficient algorithm strongly depends on the size $n$ of the financial system. We observed that for small systems ($n=5,10$) the Sandwich Picard technique, for medium-sized systems ($n=25,50$) the Trial-and-Error Picard technique and for large systems ($n=100,200$) the simple non-finite Picard technique achieves the best results with regard to the minimization of the runtime. Regarding the choice of the tolerance level ${\varepsilon}$, smaller values than the used one of ${\varepsilon}=10^{-3}$ will strongly affect the results of the non-finite iteration techniques, since additional simulations (results not listed here) revealed that an increase of ${\varepsilon}$ will lead to a disproportionally strong increase of the needed iteration steps and therefore the runtime. One consequence could be that for large systems the non-finite Picard iteration would not be optimal anymore, since the finite algorithm techniques do not depend on ${\varepsilon}$. We are aware of this effect, but we think that, for practical purposes, a tolerance level of ${\varepsilon}=10^{-3}$ is sufficiently small.
The simulation in Section \[subsec:algo\_efficency\] contained only complete ownership matrices ($\lambda = 0$ in ). It is of potential interest, whether in case of ring ownership matrices or $\lambda$-convex combinations the results lead to the same conclusions. No matter which value of $\lambda$ is chosen, the entries of ${\mathbf{M}^{{\mathbf{d}}}}$ and ${\mathbf{M}^{{\mathbf{s}}}}$ still are uniquely determinable. A potential extension of this assumption would be to allow random ownership matrices based on a random network matrix as used for example in @elliott13. Besides these questions, the main focus for further research should be on generalizing the algorithms for systems with more than one seniority level for the liabilities. In @fischer14, this was done for the non-finite Picard Algorithm and in @elsinger09, an extension of the non-finite Elsinger Algorithm is discussed as well. It would be of interest whether and – if yes – how the Hybrid Algorithm can be generalized for a model that allows for a seniority structure of debt.
Appendix
========
Proofs and Auxiliary Results
----------------------------
\[non-expansive\_infinite\] Let $||\cdot||$ be a not necessarily strictly convex norm on $\mathbb{R}^n$, and let $\Phi$ be a map on a nonempty convex and compact set $C\subset \mathbb{R}^n$ which is non-expansive with respect to the norm-induced metric. The set of fixed points of $\Phi$ in $C$ is then nonempty, closed, and either a singleton, or uncountable.
Much-refined versions of this result are known (e.g. @bruck73). For convenience, a short proof is given. Non-expansiveness implies that $\Phi$ is (1-Lipschitz) continuous. The set of fixed points is hence closed, and the Brouwer–Schauder Fixed Point Theorem (e.g. @rudin91) provides the existence of at least one fixed point. Assume now that $\mathbf{x}, \mathbf{y}\in C$ are two distinct fixed points of $\Phi$. For $\mathbf{v}\in C$ and $\varepsilon > 0$, $B_{\varepsilon}(\mathbf{v})
=
\{\mathbf{w}\in C: ||\mathbf{w}-\mathbf{v}|| \leq \varepsilon\}$ is a non-empty, convex and compact subset of $C$. For $\lambda\in(0,1)$, the intersection $$C_{\lambda} = B_{\lambda||\mathbf{y}-\mathbf{x}||}(\mathbf{x})
\cap B_{(1-\lambda)||\mathbf{y}-\mathbf{x}||}(\mathbf{y})$$ is non-empty (as it contains $(1-\lambda)\mathbf{x}+\lambda\mathbf{y}$)), convex and compact, and it contains neither $\mathbf{x}$, nor $\mathbf{y}$. By the triangle inequality, $C_{\lambda_1}\cap C_{\lambda_2}=\emptyset$ for $\lambda_1 \neq \lambda_2$. Non-expansiveness implies that $\Phi(C_{\lambda})\subset C_{\lambda}$. By Brouwer–Schauder, there exists a fixed point of $\Phi$ in $C_{\lambda}$. Hence there exist uncountably many fixed points of $\Phi$ in $C$.
Let ${\mathbf{M}}\in{\mathbb{R}}^{n\times n}$ be an ownership matrix that has the Elsinger Property. Then $\rho({\mathbf{M}}) < 1$, where $$\rho({\mathbf{M}}) = \max\{|\lambda_i|: \text{$\lambda_i$ eigenvalue of ${\mathbf{M}}$}\}$$ is the spectral radius of ${\mathbf{M}}$.
A well known result (cf. [@rudin91]) is that $\rho({\mathbf{M}})\le\|{\mathbf{M}}\|\le 1$. In case of $\|{\mathbf{M}}\|<1$ there is nothing to show, so we assume that $\|{\mathbf{M}}\|=1$ which is no contradiction to the Elsinger Property of ${\mathbf{M}}$. We will show the claim by contradiction. To this end, assume that $\rho({\mathbf{M}})=1$. For the corresponding eigenvalue ${\mathbf{v}}$ is must hold that ${\mathbf{v}}\not={\mathbf{0}}$ and ${\mathbf{M}}{\mathbf{v}}=\rho({\mathbf{M}}){\mathbf{v}}={\mathbf{v}}$. We can formulate this equation alternatively as $$\label{eq:ev_zero}
({\mathbf{I}}_n-{\mathbf{M}}){\mathbf{v}}= {\mathbf{0}}_n.$$ Since ${\mathbf{M}}$ has the Elsinger Property, it follows by @elsinger09, Lemma 1, that $({\mathbf{I}}_n-{\mathbf{M}})$ is invertible. But that means that there exists no vector ${\mathbf{v}}\not={\mathbf{0}}$ such that is true. Hence, ${\mathbf{v}}={\mathbf{0}}$ which is a contradiction and from which follows that $\rho({\mathbf{M}})<1$.
\[lem:inv\_xos-mat\] Let ${\mathbf{M}}\in{\mathbb{R}}^{n\times n}$ be an ownership matrix that has the Elsinger property and for which $\rho({\mathbf{M}})<1$. Then $({\mathbf{I}}_n - {\mathbf{M}})^{-1}$ exists and can be obtained via the *Neumann expansion*: $$({\mathbf{I}}_n - {\mathbf{M}})^{-1} = \sum_{n=0}^{\infty} {\mathbf{M}}^n,$$ where ${\mathbf{M}}^0 = {\mathbf{I}}_n$. Consequently, the diagonal entries of $({\mathbf{I}}_n - {\mathbf{M}})^{-1}$ are greater than or equal to 1 and the other entries are all non-negative.
See [@rudin91].
*Proof* of **THEOREM \[theo:unique\_fp\]**: A proof of Theorem \[theo:unique\_fp\] is necessary because related proofs in @suzuki02, @gourieroux12 and @fischer14 rely on stronger matrix conditions than the Elsinger Property, while @elsinger09 considers an equation system which slightly differs from and . First, note that and can only have non-negative solutions. This is shown in Lemma 3.5 of @fischer14 under stricter matrix conditions, but because of Lemma \[lem:inv\_xos-mat\] of this paper, it is straightforward to see that the proof works in the same manner under the Elsinger Property. The interval $[{\mathbf{R}_{\rm small}},{\mathbf{R}_{\rm great}}]$ is convex and compact, and $\Phi({\mathbf{R}})$ is continuous in ${\mathbf{R}}$. By Lemma \[lem:Phi\_self-mapping\] and the Brouwer-Schauder Fixed Point Theorem, it follows that at least one solution exists. Furthermore, $\Phi$ as in Eq. is a non-expansive mapping. This follows from Lemma 4.1 of @fischer14, where a strict contraction property is shown under stricter matrix conditions, but again it is straightforward to see how the corresponding proof implies non-expansiveness under the Elsinger Property for all ownership matrices. Since it follows from Proposition \[prop:def\_set\_sol\] that there can be a maximum of $2^n$ possible solutions of and , uniqueness follows from Lemma \[lem:Phi\_self-mapping\] and Lemma \[non-expansive\_infinite\] in the Appendix.
\[lem:inv\_aux\] Let ${\mathbf{M}}\in{\mathbb{R}}^{n\times n}$ be an ownership matrix as in Lemma \[lem:inv\_xos-mat\], such that $\mathcal N_0\subset \mathcal N$, and the matrix ${\mathbf{\Lambda}}\in{\mathbb{R}}^{n\times n}$ be defined as $$\left({\mathbf{\Lambda}}\right)_{ij} = \begin{cases} 1 ,& \text{if $i=j$ and $i\in \mathcal N_0$,} \\
0 ,& \text{else.} \end{cases}$$ Then it holds that $$({\mathbf{I}}_n - {\mathbf{\Lambda}}{\mathbf{M}}{\mathbf{\Lambda}})^{-1} {\mathbf{\Lambda}}\le {\mathbf{\Lambda}}({\mathbf{I}}_n - {\mathbf{M}})^{-1}{\mathbf{\Lambda}}.$$
Note that $$({\mathbf{I}}_n-{\mathbf{\Lambda}})^k = ({\mathbf{I}}_n-{\mathbf{\Lambda}}) \quad \text{for $k\in{\mathbb{N}}$}$$ and that ${\mathbf{M}}^0 =({\mathbf{I}}_n-{\mathbf{\Lambda}})^0 = {\mathbf{I}}_n$. Using Lemma \[lem:inv\_xos-mat\] we have that $$\begin{split}
({\mathbf{I}}_n - {\mathbf{\Lambda}}{\mathbf{M}}{\mathbf{\Lambda}})^{-1} {\mathbf{\Lambda}}&= \left(\sum_{n=0}^{\infty}({\mathbf{\Lambda}}{\mathbf{M}}{\mathbf{\Lambda}})^n\right){\mathbf{\Lambda}}\\
&= ({\mathbf{I}}_n + {\mathbf{\Lambda}}{\mathbf{M}}{\mathbf{\Lambda}}+ {\mathbf{\Lambda}}{\mathbf{M}}{\mathbf{\Lambda}}{\mathbf{M}}{\mathbf{\Lambda}}+ {\mathbf{\Lambda}}{\mathbf{M}}{\mathbf{\Lambda}}{\mathbf{M}}{\mathbf{\Lambda}}{\mathbf{M}}{\mathbf{\Lambda}}+ \ldots){\mathbf{\Lambda}}\\
&= {\mathbf{\Lambda}}+ {\mathbf{\Lambda}}({\mathbf{M}}+ \underbrace{{\mathbf{M}}{\mathbf{\Lambda}}{\mathbf{M}}}_{\le {\mathbf{M}}^2} + \underbrace{{\mathbf{M}}{\mathbf{\Lambda}}{\mathbf{M}}{\mathbf{\Lambda}}{\mathbf{M}}}_{\le {\mathbf{M}}^3} + \ldots){\mathbf{\Lambda}}\\
&\le {\mathbf{\Lambda}}+ {\mathbf{\Lambda}}\left(\sum_{n=1}^{\infty}{\mathbf{M}}^n\right){\mathbf{\Lambda}}\\
&= {\mathbf{\Lambda}}\left(\sum_{n=0}^{\infty}{\mathbf{M}}^n\right){\mathbf{\Lambda}}\\
&= {\mathbf{\Lambda}}({\mathbf{I}}_n - {\mathbf{M}})^{-1}{\mathbf{\Lambda}}.
\end{split}$$
\[lem:boderline\_as\] Let the random variable ${\mathbf{a}}$ have a have a density with respect to the Lebesgue measure on $({\mathbb{R}}_0^+)^n$. The set of all $\mathbf{a}$ for which the system solution contains at least one borderline firm, i.e. one $i\in\mathcal N$ such that $r_i^* = d_i$ and $s_i^* = 0$, then has measure zero.
First, note that it suffices to show the claim for a set $A(I)$ of all $\mathbf{a}$ for which $r_i^*=d_i$ and $s_i^*=0$ for each $i \in I \subset \mathcal N$, since the number of subsets of $\{1,...,n\}$ is finite and a finite union of sets of Lebesgue measure zero has Lebesgue measure zero. We first show that $A(I)$ is a Borel set and hence Lebesgue measurable. For this, note that it is shown in @fischer14 that the mapping $\Psi : \mathbf{a} \mapsto \mathbf{R}^*(\mathbf{a})$ that maps any price vector of the exogenous assets onto the corresponding solution of and is Borel measurable. Let now $H(I)$ denote the $2(n-|I|)$-dimensional hyperplane in $\mathbb{R}^{2n}$ for which $$H(I) = \{({\mathbf{r}}^t, {\mathbf{s}}^t)^t\in {\mathbb{R}}^{2n}: r_i = d \text{ and } s_i=0 \text{ for all } i\in I\} .$$ Clearly, $H(I)$ is a Borel set. One obtains $$A(I) = \Psi^{-1}(H(I) \cap (\mathbb{R}^+_0)^{2n}) ,$$ which must be Borel-measurable, too. Observe now that if $\mathbf{a}_2 \gg \mathbf{a}_1$ ($\mathbf{a}_2$ strictly larger than $\mathbf{a}_1$ in all components), then $\Phi^n_{\mathbf{a}_2}(\mathbf{R}) \ge \Phi^n_{\mathbf{a}_1}(\mathbf{R})$ for any non-negative $\mathbf{R}$. Hence, by the Picard Iteration, $\mathbf{R}^*(\mathbf{a }_2) \geq \mathbf{R}^*(\mathbf{a}_1)$. From Eq. and , it follows that ${\mathbf{r}}+ {\mathbf{s}}= {\mathbf{a}}+ {\mathbf{M}^{{\mathbf{d}}}}{\mathbf{r}}+ {\mathbf{M}^{{\mathbf{s}}}}{\mathbf{s}}$ (see also @fischer14). Therefore, if $\mathbf{a}_2 \gg \mathbf{a}_1$ , then $r_i^*({\mathbf{a}}_2) + s_i^*({\mathbf{a}}_2) > r_i^*({\mathbf{a}}_1) + s_i^*({\mathbf{a}}_1)$ for all $i\in\mathcal N$, which is a contradiction to $\mathbf{a}_1, \mathbf{a}_2 \in A(I)$. Thus, since $r_i^*({\mathbf{a}}) + s_i^*({\mathbf{a}}) = d_i$ for ${\mathbf{a}}\in A(I)$ and $i\in I$, $\mathbf{a}_2 \gg \mathbf{a}_1$ can hold for no pair $\mathbf{a}_1, \mathbf{a}_2 \in A(I)$. This means that $A(I)$ bears some resemblance to a Pareto set. It follows that the set $A(I)$ intersects any straight line parallel to the vector $(1,\ldots,1)^t\in{\mathbb{R}}^n$ either once, or not at all. As such, and since the Lebesgue measure is rotation invariant, the problem reduces now to the one which is shown in Lemma \[lem:borel\_set\_measure\_zero\].
\[lem:borel\_set\_measure\_zero\] Let $Q$ be a Borel set in $\mathbb{R}^n$ such that $|Q_{\omega}|\leq 1$ for any $\omega\in\mathbb{R}^{n-1}$, where $Q_{\omega}=\{x\in\mathbb{R}: (x,\omega^t)^t\in Q\}$. Then $Q$ has Lebesgue measure zero.
Let $\lambda_{m}, m\in\mathbb N$, denote the Lebesgue measure on $\mathbb{R}^{m}$. For any Borel set $Q$, it follows from the definition of product measures (e.g. @billingsley95) and from $\lambda_{n}=\lambda_{1}\otimes\lambda_{n-1}$ that $$\lambda_{n}(Q) = \int \lambda_{1}(Q_\omega) d \lambda_{n-1}(\omega) .$$ Since $\lambda_{1}(Q_\omega)=0$, the result follows.
-- ------ ------ ------- ------- ------- ------- ------- ------- ------- -------
P E H TP TE TH SP SE SH
1 5.43 31.21 33.87 9.51 22.71 26.78 10.94 30.90 35.17
1.5 5.61 35.73 36.59 9.95 25.03 28.61 11.86 38.27 39.72
2 5.98 36.46 34.70 10.31 25.00 27.56 12.55 40.92 39.19
2.5 6.46 36.67 31.69 10.53 24.65 26.23 12.97 41.04 36.29
3 6.81 36.87 28.49 10.56 23.53 24.07 12.80 40.22 32.67
0.05 4.04 17.24 20.85 8.86 13.21 18.79 9.13 15.25 20.49
0.2 5.34 26.56 27.01 9.41 17.04 23.52 10.42 22.60 28.80
0.35 6.38 36.29 32.89 10.14 22.30 27.12 11.95 32.59 36.41
0.5 7.40 47.30 40.33 11.44 30.18 31.53 13.98 47.05 44.93
0.65 7.45 55.41 46.72 11.75 35.70 34.44 15.24 61.52 51.42
0.8 6.21 39.27 37.51 10.23 28.01 28.20 13.31 50.73 41.69
0.95 5.58 25.65 26.16 9.37 22.84 22.95 11.53 38.17 32.53
-- ------ ------ ------- ------- ------- ------- ------- ------- ------- -------
: Above: Mean runtime in seconds for every algorithm over all debt and equity integration values ($\nu^{{\mathbf{d}}}$ and $\nu^{{\mathbf{d}}}$) and system sizes ($n$) grouped by the debt values ($d$). Below: Mean runtime in seconds for every algorithm over all equity integration values ($\nu^{{\mathbf{s}}}$), debt values ($d$) and system sizes ($n$) grouped by the debt integration values ($\nu^{{\mathbf{d}}}$). []{data-label="tab:res_sim_debt_int"}
Additional Tables and Simulation Results
----------------------------------------
The notation in the tables in this section is as follows. The names of the algorithms in the table are composed out of their iteration type (“P” for Picard, “E” for Elsinger and “H” for Hybrid) and their direction (“D” for decreasing, “I” for increasing). If the prefix “D” or “I” is omitted, the mean value of the corresponding increasing and decreasing version is shown. The additional prefix “T” denotes the Trial-and-Error version and “S” denotes the Sandwich version of the algorithm.
----- ------- ------- ------- -------- -------- --------
5 10 25 50 100 200
DP 6 7 8 8 9 9
IP 8 9 10 10 11 11
DE 4 (5) 5 (6) 5 (8) 5 (9) 5 (10) 6 (12)
IE 4 (6) 5 (7) 6 (9) 6 (11) 7 (12) 7 (13)
DH 2 (5) 3 (7) 3 (8) 3 (9) 3 (11) 3 (11)
IH 2 (4) 3 (4) 3 (4) 3 (5) 3 (6) 3 (8)
DTP 2 (1) 2 (1) 2 (1) 3 (1) 3 (1) 3 (1)
ITP 2 (1) 3 (1) 3 (1) 4 (1) 4 (1) 5 (1)
DTE 1 (3) 1 (3) 1 (5) 1 (5) 2 (5) 2 (7)
ITE 1 (3) 1 (4) 2 (5) 2 (7) 2 (7) 3 (8)
DTH 1 (4) 1 (4) 1 (6) 1 (6) 2 (7) 2 (8)
ITH 1 (3) 1 (4) 2 (4) 2 (5) 2 (6) 2 (7)
SP 1 1 2 2 3 3
SE 0 (4) 1 (4) 1 (6) 1 (8) 2 (8) 2 (12)
SH 0 (4) 1 (4) 1 (5) 1 (6) 1 (8) 1 (7)
----- ------- ------- ------- -------- -------- --------
: Median of the iteration (calculation) steps for every algorithm over all debt and equity integration values ($\nu^{{\mathbf{d}}}$ and $\nu^{{\mathbf{d}}}$) and debt values ($d$) grouped by the system size ($n$). A calculation step is defined as the solution of a linear equation system, which is done for example in Algorithm \[alg:equity\] in every iteration step. An iteration step is defined as the step from the $k$-th iterate ${\mathbf{R}}^k$ to ${\mathbf{R}}^{k+1}$ for $k\ge0$, no matter which algorithm is used.[]{data-label="tab:res_sim_iter_calc"}
[^1]: Institute of Mathematics, University of Wuerzburg, Am Hubland, 97074 Wuerzburg, Germany. Tel: +49 931 31 84869. E-mail address of corresponding author: [[email protected]]{}
[^2]: The authors thank Roger Nussbaum for background information regarding a fixed point problem.
|
---
abstract: 'Recently, $\textrm{Cd}_3\textrm{As}_2$ has attracted intensive research interest as an archetypical Dirac semimetal, hosting three dimensional linear-dispersive electronic bands near the Fermi level. Previous studies have shown that single-crystalline $\textrm{Cd}_3\textrm{As}_2$ has an anomalously low lattice thermal conductivity, ranging from 0.3 W/mK to 0.7 W/mK at 300 K, which has been attributed to point defects. In this work, we combine first-principles lattice dynamics calculations and temperature-dependent high-resolution Raman spectroscopy of high-quality single-crystal thin films grown by molecular beam epitaxy to reveal the existence of a group of soft optical phonon modes at the Brillouin zone center of $\textrm{Cd}_3\textrm{As}_2$. These soft phonon modes significantly increase the scattering phase space of heat-carrying acoustic phonons and are the origin of the low lattice thermal conductivity of $\textrm{Cd}_3\textrm{As}_2$. Furthermore, we show that the interplay between the phonon-phonon Umklapp scattering rates and the soft optical phonon frequency explains the unusual non-monotonic temperature dependence of the lattice thermal conductivity of $\textrm{Cd}_3\textrm{As}_2$. Our results further suggest that the soft phonon modes are potentially induced by a Kohn anomaly associated with the Dirac nodes, in analogy to similar, nonetheless weaker, effects in graphene and Weyl semimetals.'
author:
- Shengying Yue
- 'Hamid T. Chorsi'
- Manik Goyal
- Timo Schumann
- Runqing Yang
- Tashi Xu
- Bowen Deng
- Susanne Stemmer
- 'Jon A. Schuller'
- Bolin Liao
bibliography:
- 'references.bib'
title: 'Soft phonons and ultralow lattice thermal conductivity in the Dirac semimetal $\textrm{Cd}_3\textrm{As}_2$'
---
Cadmium arsenide ($\textrm{Cd}_3\textrm{As}_2$) is a well-studied electronic material due to its semimetalicity and ultrahigh charge carrier mobility[@rosenberg1959cd3as2; @schumann2016molecular]. The research interest in $\textrm{Cd}_3\textrm{As}_2$ has been revived recently due to both theoretical prediction[@wang2013three] and experimental observations[@borisenko2014experimental; @liu2014stable; @jeon2014landau; @neupane2014observation] that $\textrm{Cd}_3\textrm{As}_2$ hosts three-dimensional linear Dirac bands[@crassee20183d]. Namely, the electronic structure of $\textrm{Cd}_3\textrm{As}_2$ is a three-dimensional analog to that of graphene. Extensive optical and electrical transport measurements have been conducted on $\textrm{Cd}_3\textrm{As}_2$ to probe the Dirac physics of its electronic states[@liang2015ultrahigh; @liang2017anomalous; @schumann2018observation]. In comparison, less studied aspects of $\textrm{Cd}_3\textrm{As}_2$ are its lattice dynamics and heat transport properties, which are essential for thermal management of actual electronic devices utilizing this material. Furthermore, combining the recently predicted large thermopower of topological semimetals[@skinner2018large] and the ultrahigh charge carrier mobility, $\textrm{Cd}_3\textrm{As}_2$ is also a promising candidate for thermoelectric energy conversion applications[@wang2018magnetic; @hosseini2016large], where a low thermal conductivity is required for high efficiency. Therefore, a thorough understanding of the lattice dynamics and thermal transport properties of $\textrm{Cd}_3\textrm{As}_2$ is of paramount importance for its practical applications. Moreover, how the lattice degree of freedom couples to topological properties of the electronic structure is of fundamental interest and remains largely unexplored[@song2016detecting; @rinkel2017signatures; @cheng2019large].
The thermal conductivity of crystalline $\textrm{Cd}_3\textrm{As}_2$ was first measured in the 1960s[@spitzer1966anomalous; @armitage1969thermal]. Subtracting the electronic contribution from the total thermal conductivity using the Wiedemann-Franz law, researchers found that $\textrm{Cd}_3\textrm{As}_2$ possessed an anomalously low lattice thermal conductivity (0.3 W/mK at 300 K reported by Spitzer et al.[@spitzer1966anomalous]). This value is lower than that of amorphous glass and many polymers and in sharp contrast to other crystalline materials with similar atomic mass and crystal structure. A recent study used a strong magnetic field up to 14 Tesla to freeze out electron transport and measured a lattice thermal conductivity of around 0.7 W/mK at 300 K[@wang2018magnetic]. More surprisingly, they found the lattice thermal conductivity of $\textrm{Cd}_3\textrm{As}_2$ increases with temperature above roughly 300 K, which runs counter to the familiar $1/T$ dependence of lattice thermal conductivity in crystalline materials due to phonon-phonon Umklapp scatterings. So far, the origin of the anomalously low lattice thermal conductivity of $\textrm{Cd}_3\textrm{As}_2$ and its unusual temperature dependence remains unclear. Recent measurements of another Dirac semimetal, $\textrm{Zr}\textrm{Te}_5$, also revealed highly anisotropic and very low thermal conductivity[@zhu2018record].
In this work, we combine first-principles lattice dynamics calculations and temperature-dependent high-resolution Raman measurement of high-quality $\textrm{Cd}_3\textrm{As}_2$ thin films grown by molecular beam epitaxy (MBE) to reveal the existence of a group of soft optical phonons in $\textrm{Cd}_3\textrm{As}_2$ with anomalously low frequencies at the Brillouin zone center. These soft optical phonons largely increase the scattering phase space of heat-carrying acoustic phonons and are responsible for the reduction of the lattice thermal conductivity. More interestingly, we find that the optical phonon frequency is highly sensitive to the smearing parameter of the electronic Fermi-Dirac distribution in our calculation. This is potentially a signature of Kohn anomaly associated with the Dirac nodes, as has been found in graphene[@piscanec2004kohn; @lazzeri2006nonadiabatic] and Weyl semimetal tantalum phosphide[@nguyen2019discovery]. As a result, the frequency of the soft optical phonon increases with temperature, which reduces phonon-phonon scattering and leads to an increasing lattice thermal conductivity. Our investigation quantitatively reproduces the low lattice thermal conductivity of $\textrm{Cd}_3\textrm{As}_2$ and qualitatively explains its anomalous temperature dependence.
![Crystal structure and calculated electronic bands of $\textrm{Cd}_3\textrm{As}_2$. (a) Schematic of the $\textrm{Cd}_3\textrm{As}_2$ conventional cell. (b) Structure of the 10-atom building block for the full cell shown in (a). (c) The first Brillouin zone of $\textrm{Cd}_3\textrm{As}_2$, where the location of the Dirac nodes and the position of the Kohn anomaly $\mathbf{q}_0$ are marked. (d) Calculated electronic band structure of $\textrm{Cd}_3\textrm{As}_2$. []{data-label="fig1"}](fig-1.jpg){width="\columnwidth"}
We used the Vienna ab-initio simulation package (VASP)[@vasp01; @vasp02] to calculate the electronic structure using density functional theory (DFT). The projector augmented wave (PAW)[@PAW01; @PAW02] method was adopted with the Perdew-Burke-Ernzerhof (PBE)[@PBE] exchange correlation functional. The spin-orbit coupling was included in the calculation as implemented in VASP[@steiner2016calculation]. The cutoff energy for the plane wave expansion was set to 500 eV. The Monkhorst-Pack k-mesh $\rm 6\times 6 \times 6$ including $\Gamma$ point was used to sample the whole Brillouin zone (BZ). The low-temperature phase of $\textrm{Cd}_3\textrm{As}_2$ below 748 K is body-centered tetragonal (bct) with the space group $\textrm{I}4_1/\textrm{acd}$ and has a conventional cell consisting of 160 atoms[@ali2014crystal] (the corresponding primitive cell includes 80 atoms), as shown in Fig. \[fig1\](a). The conventional cell can be constructed as a $2\times 2\times 4$ supercell of 10-atom substructures. Each substructure has an antifluorite-derived structure with arsenic atoms at the face-center locations and cadmium atoms that form a cube with two vacancies located diagonally on one surface of the cube[@ali2014crystal], as illustrated in Fig. \[fig1\](b). The arrangement of the cadmium vacancies in the substructures alternates in a fashion depicted in Fig. \[fig1\](a), where the red arrows mark the position of the cadmium vacancies. Before the electronic band structure calculations, the lattice structure was fully optimized with the Hellmann-Feynman force tolerance $\rm 0.001~ eV/\AA$. The lattice constants of the relaxed conventional cell are given in Fig. \[fig1\](a). The calculated electronic structure of $\textrm{Cd}_3\textrm{As}_2$ using the 80-atom primitive cell is shown in Fig. \[fig1\](d), where the Dirac cones are located along the $\Gamma$-$Z$ direction at $(0,0,\pm 0.043\ \textrm{\AA}^{-1}$), in agreement with previous reports[@ali2014crystal; @conte2017electronic; @crassee20183d]. The BZ shape, high-symmetry points and the locations of the Dirac cones are shown in Fig. \[fig1\](c).
{width="80.00000%"}
Next, we used first-principles calculation to examine the phonon dispersion relations of $\textrm{Cd}_3\textrm{As}_2$, which, to the best of our knowledge, have not been reported before. For this calculation, we obtained the harmonic interatomic force constants (IFCs) by employing the finite displacement method based on DFT force calculations[@esfarjani2011heat] implemented in VASP and then diagonalized the dynamical matrices to generate the phonon dispersion relations using PHONOPY[@phonopy]. To understand the effect of the short-range arrangement of cadmium vacancies, we calculated and compared the phonon dispersion relations of both the 80-atom bct primitive cell and a crystal constructed by the 10-atom substructure shown in Fig. \[fig1\](b) as its primitive cell. The calculated phonon dispersions along the $\Gamma$-Z direction are presented in Fig. \[fig2\]. The phonon dispersions along other high-symmetry lines in the BZ are given in the Supplementary Information. We notice that the phonon dispersions calculated using the primitive cell and the substructure agree with each other very well near the zone center considering zone folding. This can be explained by the fact that the long-wavelength phonons near the zone center should be less sensitive to the short-range arrangement of the cadmium vacancies. A prominent feature of the phonon dispersions is the presence of a group of soft optical phonon modes at the zone center below 1 THz. In particular, the frequency of the lowest optical branch (LOB) is highly sensitive to the smearing parameter $\sigma$ of the electronic Fermi-Dirac distribution used in the DFT calculation. $\sigma$ (in units of eV) represents the broadening of the electronic occupation around the Fermi level and a fictitious electronic temperature $T_{\textrm{el}}=\sigma / k_B$ can be defined, where $k_B$ is the Boltzmann constant. In $\textrm{Cd}_3\textrm{As}_2$, we found that the frequency of the LOB decreases monotonically with decreasing $\sigma$, approaching roughly 0.1 THz with $\sigma=0.01$ eV ($T_{\textrm{el}}=115$ K). In comparison, the well-known high-performance thermoelectric materials with soft optical phonons, such as $\textrm{Bi}_2\textrm{Te}_3$[@hellman2014phonon] and PbTe[@delaire2011giant], have their LOBs around 1 THz.
This strong dependence of optical phonon frequencies on $\sigma$ was also observed in graphene, where optical phonons at the $\Gamma$ and K points in the Brillouin zone showed similar behaviors[@piscanec2004kohn]. In graphene, this phenomenon was attributed to the Kohn anomaly[@kohn1959image]. In 1959, Kohn pointed out that, in a metal, the resonance between lattice vibrations and the Fermi surface can lead to nonanalytic points in the phonon dispersion. These anomalies should happen when $\mathbf{q}=2\mathbf{k}_\textrm{f}$, where $\mathbf{q}$ is the phonon wavevector and $\mathbf{k}_\textrm{f}$ is the Fermi wave vector[@kohn1959image]. Here, $\mathbf{q}=2\mathbf{k}_\textrm{f}$ is the maximum wave vector of a phonon that can be involved in a scattering event with two electronic states on the Fermi surface. In graphene, the Fermi surface is virtually two Dirac points located at K and $\textrm{K}'$, therefore Kohn anomaly can only happen at $\Gamma$ (intra-node scattering) and K points (inter-node scattering between K and $\textrm{K}'$). In a Dirac material, the size of the Fermi surface is highly sensitive to the electronic smearing parameter $\sigma$, which explains the strong dependence of the optical phonon frequency in graphene on $\sigma$. In $\textrm{Cd}_3\textrm{As}_2$, the two Dirac nodes are located at $(0,0,\pm 0.043\ \textrm{\AA}^{-1})$. Therefore, Kohn anomalies should potentially appear at $\Gamma$ (intra-node scattering) and $\mathbf{q}_0=(0,0,\pm 0.086\ \textrm{\AA}^{-1})$ (inter-node scattering), as marked in Fig. \[fig1\](c) and \[fig2\](b). In addition to the drastic phonon softening at $\Gamma$, a slight dip of one optical phonon branch at $(0,0,\pm 0.086\ \textrm{\AA}^{-1})$ is observed, again signaling the presence of Kohn anomaly. We emphasize here that the current calculation can only capture the Kohn anomaly due to static electronic screening of lattice vibrations, while a full treatment including dynamic screening effect will require the calculation of dynamic electron-phonon coupling that was done in the case of graphene[@lazzeri2006nonadiabatic] but is currently inaccessible for $\textrm{Cd}_3\textrm{As}_2$ given its complex structure.
To verify the existence of the group of low-frequency optical phonons at $\Gamma$, we conducted temperature-dependent Raman measurement of high-quality $\textrm{Cd}_3\textrm{As}_2$ thin films grown by MBE[@schumann2016molecular]. The $\textrm{Cd}_3\textrm{As}_2$ layer is $\sim$30 nm thick and grown on a (111) GaSb/GaAs substrate. Growth details and electrical characterization have been reported elsewhere[@schumann2016molecular]. We used unpolarized excitation with 488 nm wavelength and 65 mW power for the Raman measurement at 300 K, 120 K and 77 K. To rule out the influence of the substrate, we also measured the Raman spectrum of the substrate with the same configuration (provided in Supplementary Information). As compared to existing reports of Raman spectra of bulk $\textrm{Cd}_3\textrm{As}_2$[@jandl1984raman; @hosseini2016large; @sharafeev2017optical; @weszka1986some], our measurement, as shown in Fig. 3, revealed the existence of a low-frequency optical phonon mode near 20 $\textrm{cm}^{-1}$ at all three temperatures and another mode near 15 $\textrm{cm}^{-1}$ at 120 K and 77 K. The spectral region with smaller Raman shift is masked by a strong background signal due to quasielastic electronic scatterings[@sharafeev2017optical] and thus the LOB cannot be resolved. The frequencies of both resolved modes agree well with our first-principles calculation. In addition, we observed a decreasing trend of the frequency of the phonon mode near 20 $\textrm{cm}^{-1}$ as the temperature was lowered, which was in contrast to the opposite temperature dependence of optical phonons with higher frequencies[@sharafeev2017optical] caused by thermal expansion.
![The Raman spectrum of $\textrm{Cd}_3\textrm{As}_2$ measured at three different temperatures. The inset shows the enlarged low-frequency region and the arrows mark the Raman peaks resolved by the measurement.[]{data-label="fig3"}](fig-3.jpg){width="\columnwidth"}
{width="\textwidth"}
. \[fig4\]
Finally, we investigated how the existence of soft optical phonon modes affects the lattice thermal conductivity of $\textrm{Cd}_3\textrm{As}_2$. It is well understood that low-frequency optical phonon modes can significantly increase the available phase space for phonon-phonon scattering of heat-carrying acoustic phonons, which is responsible for the low lattice thermal conductivity of the best thermoelectric materials, such as $\textrm{Bi}_2 \textrm{Te}_3$[@hellman2014phonon; @lee2014resonant; @yue2018ultralow], PbTe[@delaire2011giant; @li2014phonon] and SnSe[@li2015orbitally]. In these materials, the soft optical phonon mode usually signals their proximity to a structural phase transition. Here, we expect the low-lying optical phonon modes in $\textrm{Cd}_3\textrm{As}_2$ will have a similar effect. Due to the complexity and low symmetry of the primitive cell of $\textrm{Cd}_3\textrm{As}_2$, it is computationally intractable to calculate the anharmonic force constants and phonon-phonon scattering rates using the full cell. To overcome this obstacle, we examined the thermal transport properties of the crystal constructed from the 10-atom substructures. The similarity between the dispersion relations of the long-wavelength phonons calculated using the full cell and the 10-atom substructure, as observed in Fig. \[fig2\](a), provides justification to this approach, since these long-wavelength phonons are major heat carriers in $\textrm{Cd}_3\textrm{As}_2$. In particular, we expect this approach can qualitatively capture the physics of the soft phonon modes and their impact on thermal transport. Another unique aspect is the strong dependence of the LOB frequency on the electronic smearing parameter $\sigma$, which implies that the actual phonon dispersion of $\textrm{Cd}_3\textrm{As}_2$ will be highly sensitive to temperature. To include this effect, we analyzed thermal transport properties based on phonon dispersion relations assuming different values of $\sigma$. We calculated the lattice thermal conductivity $\kappa_{\textrm{ph}}$ by solving phonon Boltzmann transport equation (BTE) iteratively using ShengBTE[@ShengBTE]. The anharmonic third-order IFCs were calculated using the finite displacement method[@bte; @3RD-IFCs]. A $2\times 2\times 2$ supercell was used for the calculation and the interactions between atoms were taken into account up to sixth nearest neighbors. The convergence of $\kappa_{\textrm{ph}}$ with the interaction distance between atoms was checked. The q-space (phonon momentum space) sampling grid was set to $\rm 12 \times 12 \times 12$. The grid density convergence of all cases were examined and all the $\kappa_{\textrm{ph}}$ reported here are converged values.
Our results are shown in Fig. \[fig4\](a). Here, the lattice temperature $T_{\textrm{ph}}$ determines the Bose-Einstein distribution of the phonons involved in the phonon-phonon scattering calculation while $\sigma$ dictates the broadening of the Fermi-Dirac distribution of electrons used in the DFT calculation. For a given $\sigma$, the lattice thermal conductivity decreases with an increasing lattice temperature, which is typical for crystalline materials limited by phonon-phonon Umklapp scattering. At 300 K, our calculated lattice thermal conductivity is in the range of 0.3 to 0.9 W/mK, in good agreement with experimental reports[@spitzer1966anomalous; @armitage1969thermal; @wang2018magnetic]. For a given lattice temperature, the calculated lattice thermal conductivity is lower with a smaller $\sigma$. This is expected as a smaller $\sigma$ leads to a lower LOB frequency, which in turn causes stronger scattering of low-frequency acoustic phonons. We confirm this hypothesis by calculating the phonon-phonon scattering rates at a fixed lattice temperature but with two different $\sigma$ values, as shown in Fig. \[fig4\](b). While the scattering rates of higher-frequency optical phonons above 2 THz are almost identical in the two cases, the scattering rates of low-frequency heat-carrying acoustic phonons are significantly enhanced with a smaller $\sigma$.
In order to make a direct connection to experimental results, both the phonon-phonon Umklapp scattering and the $\sigma$-dependent optical phonon frequency need to be taken into account. The counteraction of the two effects is expected to generate a non-monotonic temperature dependence of the lattice thermal conductivity in $\textrm{Cd}_3\textrm{As}_2$. A rigorous treatment would entail a dynamic calculation[@lazzeri2006nonadiabatic; @giustino2017electron] that is beyond the current scope. However, to qualitatively capture the interplay of the two effects, we can compare the lattice thermal conductivity values calculated when the fictitious electronic temperature $T_\textrm{el}$ determined by $\sigma$ is set equal to the lattice temperature $T_\textrm{ph}$. These values are marked in Fig. \[fig4\](a) with open circles and connected by a dotted line. Below 450 K, the temperature dependence of phonon-phonon Umklapp scattering is more prominent, and thus the lattice thermal conductivity decreases with temperature. Above 450 K, however, the increasing optical phonon frequency plays a more important role and the lattice thermal conductivity starts to increase with temperature. This physical picture provides a qualitative explanation for the increasing trend of the lattice thermal conductivity of $\textrm{Cd}_3\textrm{As}_2$ with temperature as observed experimentally[@wang2018magnetic], while the quantitative difference from the experimental result might be due to the simplified treatment used here.
In summary, we investigated the lattice dynamics and thermal transport of Dirac semimetal $\textrm{Cd}_3\textrm{As}_2$ using first-principles calculation and Raman measurement. We identified the existence of soft optical phonons likely due to Kohn anomaly associated with the Dirac nodes. Based on the observation of the soft modes, we explained the ultralow lattice thermal conductivity of $\textrm{Cd}_3\textrm{As}_2$ due to soft-mode-enhanced phonon-phonon scatterings. We further suggested that the interplay of phonon-phonon Umklapp scattering and the optical phonon frequency can potentially explain the anomalous temperature dependence of the lattice thermal conductivity of $\textrm{Cd}_3\textrm{As}_2$ as observed experimentally. Our work exemplifies the rich phonon physics in topological materials.
This work is based on research supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering through the Early Career Research Program under the award number DE-SC0019244. B.L. acknowledges the support provided by the Regents’ Junior Faculty Fellowship from the University of California, Santa Barbara (UCSB). M.G., T.S., and S.S. acknowledge support through a Vannevar Bush Faculty Fellowship program by the U.S. Department of Defense (Grant number N00014-16-1-2814).
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abstract: 'Despite the imperative importance in solar-cell efficiency, the intriguing phenomena at the interface between perovskite solar-cell and adjacent carrier transfer layers are hardly uncovered. Here we show that PbI$_2$/AI-terminated lead-iodide-perovskite (APbI$_3$; A=Cs$^+$/ methylammonium(MA)) interfaced with the charge transport medium of graphene or TiO2 exhibits the sizable/robust Rashba-Dresselhaus (RD) effect using density-functional-theory and ab initio molecular dynamics (AIMD) simulations above cubic-phase temperature. At the PbI$_2$-terminated graphene/CsPbI3(001) interface, ferroelectric distortion towards graphene facilitates an inversion breaking field. At the MAI-terminated TiO$_2$/MAPbI$_3$(001) interface, the enrooted alignment of MA$^+$ towards TiO$_2$ by short-strong hydrogen-bonding and the concomitant PbI$_3$ distortion preserve the RD interactions even above 330 K. The robust RD effect at the interface even at high temperatures, unlike in bulk, changes the direct-type band to the indirect to suppress recombination of electron and hole, thereby letting these accumulated carriers overcome the potential barrier between perovskite and charge transfer materials, which promotes the solar-cell efficiency.'
author:
- Chang Woo Myung
- Saqib Javaid
- 'Kwang S. Kim'
- Geunsik Lee
bibliography:
- 'myungbib.bib'
title: 'Rashba-Dresselhaus Effect in Inorganic/Organic Lead Iodide Perovskite Interfaces'
---
Solar energy is a highly efficient and eco-friendly energy source for future energy harvesting. In recent years, inorganic/organic hybrid halide perovskite solar cell (PSC) based on ABX3 (A = Cs$^+$, CH$_3$NH$_3$$^+$ (MA$^+$), CHN$_2$H$_4$$^+$ (FA$^+$); B = Pb$_2$$^+$; X = Cl$^-$, Br$^-$ or I$^-$) have shown rapid progress achieving over 22 $\%$[@1] solar cell efficiency which is considered to be most promising large-scale solar energy materials.[@2] PSC owns many interesting physical properties including giant dielectric screening,[@3] bottleneck of hot phonon relaxation process,[@4] multi-excitonic states,[@5] and polaron state.[@6] Owing to large electron-phonon coupling nature of PSC, Frohlich polaron state has been proved experimentally,[@7; @8] with its polaron radius being $\sim$ 4 unit cells.[@9] This quasiparticle state would explain its good carrier transport property even in the presence of impurities.[@10] Upon photoexcitation, electronic dielectric screening leaps orders of magnitude which help the exciton be dissociated with small binding energy.[@3; @11] Impact ionization of hot exciton with carriers is expected to be very high in perovskite nanocrystals giving rise to multi-exciton emission.[@12] These interesting characters of PSC materials are ideal for practical solar cells and light emitting diodes.[@13] Nevertheless, despite explosive discoveries in experiments, theoretical understandings underneath ongoing experiments are hardly made yet particularly regarding the Rashba-Dresselhaus (RD) effect at the interface between PSC and adjacent carrier transfer layers. The spin-orbit coupling (SOC) field, which is odd-in-k (momentum) and time reversal symmetric, in non-centrosymmetric crystals or at the interface of heterostrucures, gives rise to intriguing Rashba-Dresselhaus (RD) splitting. The effective low order perturbation terms of RD interactions are derived according to a given symmetry of the model. The lowest order Hamiltonian in $kp$ is [@14]
$$\begin{aligned}
H_{RD}(k) = \alpha_R (k_x \sigma_y + k_y \sigma_x ) + \alpha_D (k_x \sigma_x-k_y \sigma_y ), \alpha_{RD}= \frac{\Delta E}{2\Delta k}, (1)\end{aligned}$$
where k is momentum, $\sigma_{i= x, y, z}$ is the spin Pauli matrices and the strength of RD interactions is defined by coupling constant $\alpha_{RD}$. The RD interactions are universal so that many systems such as noncentrosymmetric crystals, heterojunction,[@15] metal surface,[@16] and graphene[@17] show a sizable energy splitting. $\alpha_{RD}$ varies depending on systems ranging from 0.067 $eV \cdot \AA$ (InAlAs/InGaAs)[@18] to 4.0 $eV \cdot \AA$ (Bi$_2$Se$_3$).[@19] Recently, it is realized that perovskite solar cell materials containing heavy elements like Pb or I show large RD coupling constants: $\alpha_{RD}$ 1.6 $eV \cdot \AA$ in 2D PSC (C$_6$H$_5$C$_2$H$_4$NH$_3$)$_2$PbI$_4$[@20] and $\alpha_{RD}$ 2.75-3.75 $eV \cdot \AA$ (in the original paper $\alpha_{RD}$ is 7-11 with different definition of $\alpha_{RD} = \frac{2\Delta E}{\Delta k}$)) in MAPbBr$_3$[@21] and 3D CsPbBr$_3$ nanocrystal.[@22] Previous studies on RD splitting in PSCs have focused on an inversion symmetry breaking in bulk phases with an artificial condition such as uniaxial pressure to trigger ferroelectricity in PSC.[@23] Recent work clarified the importance of the RD effect on 1s exciton state of PSC.[@24]
Meanwhile, an interesting aspect has been realized that dynamical Rashba splitting occurs in both centrosymmetric I4/mcm and non-centrosymmetric I4cm tetragonal phases simulated by Car-Parrinello molecular dynamics.[@25; @26] It has been proposed that on a large scale (> 8 $nm^3$) where the entropy of MA$^+$’s orientations is high, the RD effect might be quenched.[@26] An application to the spin filter device that makes the spin precess during the propagation in PSC has been proposed using RD interaction.[@27] A technological impact is that the RD interaction changes the direct-type band structure to the indirect one to suppress the recombination of carriers and to promote carrier accumulation at the barrier between PSC and charge transfer materials. Although the understanding of interface phenomena in solar cell device is crucial, until now there is no work related to RD interaction at the interface of PSC and other material layers including electron and hole transport materials and the impact of RD interaction on solar cell performance. Graphene is a fascinating material for various applications such as transistors, optoelectronics, nanoelectronics, medical application etc.[@28] Particularly multi-layer graphene has been proposed as an effective hole transfer material by its lower work functions close to the valence band maximum (VBM) of PSC.[@29] TiO$_2$ is widely used for electron transfer materials (ETM) because of its transparency, ideal band alignment and synergetic effect with PSC.[@30; @31; @32] However, the RD effect at the interface between PSC and hole/electron transfer materials has not been studied yet.
In this work, for the first time, we clarify an elusive aspect of PSC heterostructures using the first principles calculations and AIMD simulations accounting for the RD effect. We have carried out the calculations for graphene/cubic-CsPbI$_3$(001) as a prospect interface for improving carrier transport and TiO2/cubic-MAPbI$_3$(001) as a well-known electron transport layer for PSCs device. Although we observed RD interactions at both interfaces, their mechanisms are different in intriguing ways. At PbI$_2$-terminated Gr/CsPbI$_3$(001), the ferroelectric Pb-I distortion promotes significant Rashba interactions both at 0 K and above 600 K. On the other hand, at MAI-terminated TiO$_2$/MAPbI$_3$(001), the direction of organic MA$^+$ near-fixed by strong short hydrogen bonding (SSHB),[@33; @34] even above 330 K and the concomitant distortion of PbI3 sublattice promote the RD interactions. Here, we show that unlike bulk where high entropic disorder of MA would quench the RD effect, the interfacial RD effect is robust in thermal effects and is beneficial for solar cell efficiency.
![\[Figure. 1\] (a) 10 layers of $(\sqrt(2) \times \sqrt(2) \times 1)$ (a) PbI$_2$-terminated and (b) CsI-terminated cubic CsPbI$_3$ for (001) surface sandwiched by graphene (yellow) viewed from \[100\] after structural relaxation. The number denotes the layer number. Each PbI2 layer undergoes a ferroelectric distortion of Pb (red arrow) and I (green arrow) along \[001\] direction with its magnitude gradually increasing towards graphene. For the correct description of the system, the bulk region (dashed red box) retained to have the inversion symmetry by surrounding it with 5-6 layers. (c) A unit cell of Gr/CsPbI$_3$(001)-$(\sqrt(2) \times \sqrt(2) \times 1)$ viewed from \[001\]. (d) Schematic displacements of Pb and I atoms near surface along \[001\] direction, denoted as $\delta(Pb)$ and $\delta$(I), respectively.](fig1.png){width="8.6cm"}
We constructed an interface of graphene/CsPbI$_3$(001)/graphene modeled by a slab of 10(9) layers of $(\sqrt(2) \times \sqrt(2) \times 1)$ PbI$_2$-(CsI-)terminated cubic CsPbI$_3$ with the lattice mismatch 1.93 $\%$ between two systems (Fig. 1a, 1b and 1c). We confirmed that the surface dipole does not affect both the relaxed geometry and the corresponding electronic structures because the cation and anions (Pb$_2^+$I$^‒_2$ or Cs$^+$I$^‒$) at the termination are oriented parallel to surface. We adopt a symmetric slab sandwiched by graphene at each end to avoid any unphysical artifact. As reported from a previous LDA+D2 calculation of Gr/tetragonal-MAPbI$_3$,[@35] we observe a ferroelectric distortion driven by an attraction between graphene and cation Pb$^{2+}$ in the PbI$_2$-termination. A measured distance between PbI$_2$ layer and graphene (Fig. 1a) is $d_{PBE-D3}$ 3.21 $\AA$ at the PBE+D3 level and $d_{PBE-TS}$ 3.28 $\AA$ at the PBE+TS level of theory which is smaller than LDA+D2 $d_{LDA-D2}$ 3.45 $\AA$. As for CsI-termination, a distance between CsI layer and graphene (Fig. 1b) is $d_{PBE-TS}$ 3.4 $\AA$ slightly larger than PbI$_2$-terminated surface. Despite the ferroelectric distortion, $C4v$ point symmetry of cubic structure is conserved which is manifested as pure Rashba type splitting in contrast to TiO$_2$/MAPbI$_3$ interface that will be discussed later. Atomic displacements near surface in both cases (Fig. 1a, b) are only significant along \[001\] (Fig. 1d). However, we observe that the displacement directions of both cases are opposite to each other. For PbI$_2$-termination, the displacements are $\delta$(Pb) +0.4 $\AA$ and $\delta$(I) -0.6 $\AA$, while for MAI-termination, the displacements are $\delta$(Pb) -0.5 $\AA$ and $\delta$(I) +0.6 $\AA$ (Fig. 1d). The ferroelectric displacement gradually increases when approaching the interface with its maximum at the very interface. Bulk-like (or inversion symmetric) layers, 5 th and 6 th layer (Fig. 1a and b), is crucial for illustrating a reasonable band structure, unless the band gap closes because of ferroelectricity over the whole structure.[@36] The binding energy (BE) with graphene for PbI¬2-termination is 20.5 $meV$/atom, 3.3 times larger than CsI-termination (Table 1).
The electronic band structure of PbI$_2$-(MAI-)terminated Gr/CsPbI$_3$(001)/Gr with PBE+TS+SOC (Fig. 2) reveal some of interesting features. Due to sizable Rashba interactions, in both conduction band (CB) and valence band (VB), surface bands split by momentum $\delta k$ and energy $\delta E$. For the bulk cubic $Pm3m$ (centrosymmetric) CsPbI$_3$ crystal, strong SOC splits the conduction band into one j = 1/2 doublet and one j = 3/2 quartet. Because the valence band is s-like, there is no effective splitting in the highest valence band.[@37] However, at the Gr/CsPbI$_3$ interface, we note that s-like valence band at M experiences an asymmetric field with respect to the xy plane and its eigenstate is not s = 1/2 but j = 1/2 being mixed with pz state and other states of adjacent layers. This is manifested in the calculated band structure with non-vanishing Rashba splitting of the surface valence band. In CB, the energy splitting between $j_z$=-1/2 and $j_z$=1/2 is significant, $\delta E$(PbI$_2$-termination) 280 $meV$; this large barrier would hinder electrons to overcome the barrier from the CB extreme in order to directly recombine with the VB extreme.[@38] Effective Hamiltonian (eq. (1)) should preserve j=1/2 so that the eigenstate of Rashba split bands for both CB and VB are spanned by $\ket{j=1/2,j_z=\pm 1/2}$. Diagonalization of the Hamiltonian gives an entangled spin-orbital texture which resembles the surface state of topological insulator Bi$_2$Se$_3$ (Fig. S1), “spin-momentum locking”.[@39] Indeed, a recent experiment confirmed an emergence of spin-orbital chiral nature by observing circularly polarized light.[@40] An explicit calculation of direct transition amplitude from $\ket{j=1/2}$ conduction band to all valence band states $|p|^2 = \sum_{(i=x,y,z)} |p_i|^2$ (Fig. S2a) shows that a direct band-to-band transition is largely suppressed due to spin-orbit entanglement and 2D confinement of wavefunction with the ferroelectric distortion,. We also find that Rashba split band promotes the density of states (Fig. 2c).
![\[Figure. 2\] (a) Electronic band structure of 10(9) layers of PbI$_2$-terminated (CsI-terminated) Gr/CsPbI3(001)/Gr-$(1 \times 1)$ symmetric slab with the contributions from the topmost Pb (red), I (blue) and graphene (green). (b) Magnified 2D band of conduction band $\ket{j=1/2,j_z= \pm 1/2}$ around Kramer point M of BZ with the energy difference ($\Delta E$) between the minimum and the upper band and the momentum change ($\Delta k$) between the minimum and M. (c) Density of states in the vicinity of conduction band extremum for Pb p of bulk CsPbI$_3$ (blue) and topmost Pb p (red) of Gr/CsPbI$_3$(001)/Gr.](fig2.png){width="8.6cm"}
In Gr/CsPbI$_3$(001)/Gr, a surface or gap state shows a rather peculiar structure than usual semi-conductors in which gap states are mainly composed of surface states. For PbI$_2$-termination, while the lowest unoccupied surface state is at the CBM, the highest occupied surface state sits at 1 $eV$ below the VBM. Interestingly, the maximum of VB is composed of bulk state without Rashba splitting due to its centrosymmetry. The surface state shows the opposite trend in CsI-termination. The highest occupied surface state is the VBM, but the lowest unoccupied surface state is 0.5 $eV$ above the CBM. The origin of peculiar energy levels of Gr/CsPbI$_3$(001) can be explained by observing the ferroelectric displacement on CsPbI3 surface. It is found that both CsI- and PbI$_2$-terminated CsPbI$_3$(001) experience an intrinsic ferroelectric displacement resembling the relaxed hetero-interface with graphene (Fig. S3). Compared with unrelaxed and relaxed CsPbI$_3$ slabs, significant energy shifts of surface states are observed (Fig. S4). This natural ferroelectric surface distortion would hint the origin of recent observations of Rashba splitting in CsPbBr$_3$ nanocrystal.[@22] However, the role of graphene differs at each termination. At PbI$_2$ termination, graphene further promotes the ferroelectric distortion and the resulting $\alpha_{RD}$ is enhanced to 0.42(VB) and 1.17(CB) compared with 0.18(VB) and 1.00(CB) of the pristine slab. At CsI termination, graphene suppresses the ferroelectric distortion (Fig. 1b and Fig. S3b) and the RD effect is comparable or even less (Table 1).
The Rashba effect has been shown to exist in bulk MAPbI$_3$: the organic cation MA$^+$ breaks the inversion symmetry and distorts the PbI$_3$ sublattice.[@26; @27] However, this effect could be local due to orientational disorder of MA+ cations which significantly reduces the Rashba interaction parameter ($\alpha_{RD}$) at the length scale of 3 $nm$. [@25; @26] Previous work for pristine tetragonal MAPbI$_3$ slab has shown that surface reconstruction could lead to a large RD effect and the effect is more pronounced at PbI$_2$-termination.[@41] We calculated the band structure of rutile TiO$_2$/MAPbI$_3$ (001) interface for PbI$_2$-(MAI-) termination (Fig. 3a and 3b) and also the pristine cubic MAPbI3 slab for comparison (Fig. S5). Despite that the bare rutile (001) is not stable, this facet is favored in device configuration.[@42] As reported for tetragonal MAPbI3 slab, $\alpha_{RD}$ of CB and VB in the cubic MAPbI$_3$ slab scales by a factor of 2 (Table 1). Also our result is consistent with the previous work showing larger $\alpha_{RD}$ for PbI$_2$-terminated slab evidenced by the distortions of Pb-I bonds along \[001\] ($\theta_{Pb-I-Pb}$ 155.5(PbI$_2$-termination), $\theta_{I-Pb-I}$ 170.4 (MAI-termination)) (Fig. S5).[@25]
![\[Figure. 3\] The schematics of optimized (a) PbI$_2$-terminated and (b) MAI-terminated TiO$_2$/cubic MAPbI$_3$-(001) interface. Presence of PbO and TiI bonding in PbI$_2$ termination and SSHB in MAI-termination are highlighted by connecting lines of Pb(light red)-O(red) atoms ( 2.44 $\AA$), Ti(light blue)-I(purple) atoms ( 2.90 $\AA$) and H(light blue)-O(red) atoms ( 1.47 $\AA$). (c) Band structure for optimized TiO$_2$/MAPbI$_3$(001) interface calculated with PBE+TS+SOC. The blue(red) circles indicate the contribution of the surface I(Pb) interfaced to TiO$_2$ layer. ](fig3.png){width="8.6cm"}
At the PbI$_2$-termination,[@41] we observe the bonding between PbO$_2$ ($d \sim$ 2.44 $\AA$) and TiI ($d \sim$ 2.9 $\AA$).[@41] Regarding MAI-terminated interface, our previous work has shown that the orientational freedom of MA$^+$ is significantly curtailed at the TiO$_2$/MAPbI$_3$ interface due to the presence of SSHB between H+ of MA$^+$ and O$^-$ of TiO$_2$.[@43] Indeed, AIMD simulations indicate that the O-H bond at the interface remains intact even at 330 K within cubic phase. Moreover, the optimized MA+ orientation at the interface distorts the PbI3 sublattice by increasing the bond distance between Pb and apical I by 0.15 $\AA$. Since SSHB considerably reduces the orientational freedom of MA$^+$ at the interface, the Rashba effect at TiO$_2$/MAPbI$_3$ interface should be more robust than the bulk where it is found to be local or semi-local.[@44] Moreover, the near-pinning effect of MA-orientation at the interface was also confirmed by previous DFT/AIMD simulation studies on stable anatase (101) and (001) interfaces with PSC.[@32; @45] Therefore, the robust RD effect by SSHB is expected to be universal over various TiO2/MAPbI3 interface configurations.
In general, the VBM/CBM related states of TiO$_2$ and MAPbI$_3$ lie at high symmetry $\Gamma$ and M of Brillouin zone (BZ), respectively. The energy splitting is considerably larger for CB (Fig. 3c). This is anticipated as MAPbI$_3$ CB is dominated by heavier Pb states compared with valence states of relatively lighter I atoms. On the other hand, we note that the RD effect has been suppressed at PbI2-terminated interface because of stabilization of the surface via PbO$_2$ and TiI bonding with BE 15.6 $meV$/atom. We also observe that a significant distortion along \[001\] has been quenched ($\theta_{Pb-I-Pb}$ 172.1). Therefore, $\alpha_{RD}$ 0.08 (VB) and 0.17 (CB) in TiO$_2$-termination interface is significantly reduced compared to the pristine slab (Table 1). In MAI-terminated interface, SSHB results in enhancing the RD effect for $\alpha_{RD}$ being 0.29 (VBM) and 0.58 (CBM). A momentum (k-space) mismatch is also found between the VBM and CBM as indicated by different Δk for CBM and VBM which suppresses the recombination and increases the carrier lifetime.[@46] As in the case of Gr/CsPbI$_3$, a direct band-to-band recombination of MAPbI3 is significantly quenched being interfaced to TiO$_2$. (Fig. S2b) All the calculated RD interaction parameters $\alpha_{RD}$ are in Table 1.
![\[Figure. 4\] Schematic of electron transfer process with and without Rashba-Dresselhaus Effect at the interface between perovskite solar cell and electron transfer material. (a) Without the RD effect, the large potential barrier between PSC and ETM cannot be overcome by a rapid electron(red circle)-hole(red circle) pair recombination process. (b) With the RD effect, long lifetime of the electron-hole pair contributes to the accumulation of electrons to overcome the barrier.](fig4.png){width="8.6cm"}
The direct optical measurement and photoemission study have shown that a sizeable electron transfer barrier of 0.4 eV may exist at TiO$_2$/MAPbI$_3$ interface.[@47; @48] In the presence of such a large barrier, electron transfer process should be strongly impeded, resulting in accumulation of electrons at interface. In the case that the charge recombination process is favorable, this should result in considerable degradation in device performance. However, the presence of Rashba splitting at the interface (Fig. 3 and Table 1) reduces such recombination process until the potential barrier is overcome by electron accumulation (Fig. 4). Therefore, the presence of Rashba effect at the interface is critical for improving the electron transfer process at the TiO$_2$/MAPbI$_3$ interface.
Apart from 0 K DFT result, to elucidate the impact of thermal degrees of freedom, we average $\alpha_{RD}$ from NVT ensemble based on AIMD simulations around 600 K (for cubic CsPbI$_3$)[@49] and 330 K (for cubic MAPbI3).[@50] We choose the most pronounced case, PbI$_2$-termination for Gr/CsPbI$_3$/Gr and MAI-termination TiO$_2$/MAPbI$_3$. Initially we assumed that the RD effect is overestimated in the 0 K DFT. However, the thermal average of RD splitting is found to be comparable to 0 K DFT result (Table 1). Thermal average of $\alpha_{RD}$(600 K) for PbI2-terminated Gr/CsPbI$_3$ is 0.46($\pm$0.15) $eV \cdot \AA$ (VB) and 1.25($\pm$0.54) $eV \cdot \AA$ (CB). $\alpha_{RD}$(330 K) for MAI-terminated TiO$_2$/MAPbI$_3$, where the alignment of MA at the TiO$_2$ interface is kept fixed, is 0.26($\pm$0.07) eV·Å (VB) and 0.61($\pm$0.15) eV·Å (CB). In particular, the surface I’s configuration does not stray from 0 K configuration even at 330 K because of the enrooted MA (Fig. 3b). Therefore, VB has almost the same $\alpha_{RD}$ as that of 0 K DFT with a negligible standard deviation 0.07$eV \cdot \AA$.
In summary, we have shown two examples of inorganic/organic perovskite solar cell interfaces: Gr/CsPbI$_3$, a promising interface for efficient carrier transport and TiO2/MAPbI3, a common electron transport layer interface. We report the sizable Rashba interaction $\alpha_{RD}$(0 K) 1.17 (0.42) $eV \cdot \AA$ and $\alpha_{RD}$(600 K) 1.25 (0.46) $eV \cdot \AA$ for electron (hole) at PbI2-terminated Gr/CsPbI$_3$. The TiO$_2$/MAPbI$_3$ interface also shows a significant RD interaction $\alpha_{RD}$(0 K) 0.58 (0.29) $eV \cdot \AA$ and $\alpha_{RD}$(330 K) 0.61 (0.26) $eV \cdot \AA$ for electron (hole) carriers. The enrooted alignment of MA+ at the TiO$_2$ interface gives rise to a strong and firm RD effect even at high temperatures (> 330 K) with small variance. Because of large SOC nature and geometrical complexity of PSCs, its interface with other layers poses rich phenomena. A clever manipulation of such interfaces could accelerate further improvement of the PSC efficiency.
------------------------------ ------- ------ ------
CB VB
Gr/CsPbI$_3$/Gr (PbI$_2$) 20.54 0.42 1.17
Pristine CsPbI$_3$ (PbI$_2$) - 0.18 1.00
Gr/CsPbI$_3$/Gr (CsI) 6.23 0.59 0.50
Pristine CsPbI$_3$ (CsI) - 0.53 0.54
TiO$_2$/MAPbI$_3$ (PbI$_2$) 15.64 0.08 0.17
Pristine MAPbI$_3$ (PbI$_2$) - 0.78 0.81
TiO$_2$/MAPbI$_3$ (MAI) 10.33 0.29 0.58
Pristine MAPbI$_3$ (MAI) - 0.18 0.30
------------------------------ ------- ------ ------
: Binding energy (BE) and Rashba-Dresselhaus parameter $\alpha_{RD}$ for graphene-, TiO$_2$-interfaced cubic perovskite solar cell materials along BZ depending on the terminations, PbI$_2$ and AI where A refers to the A-site cation, Cs$^+$ or MA$^+$.
We used Vienna Ab initio Simulation Package (VASP)[@51] for non-collinear DFT calculations using PBE functional plus Tkatchenko-Scheffler(TS)[@52]/Grimme DFT-D3[@53] van der Waals correction with inclusion of spin-orbit coupling by switching off any presumed symmetry. Our previous work has shown that GGA+SOC results are consistent with that of higher level but computationally expensive HSE06[@54] +SOC calculations.[@36] For Gr/CsPbI$_3$(001) system, we used $(4 \times 4 \times 1)$ kmesh for sampling the Brillouin zone and 500 eV for the energy cutoff. As we checked the convergence of band gap with respect to the thickness of CsPbI$_3$ slab, the convergence has been met from 6 cubic CsPbI$_3$ layers. For TiO$_2$/MAPbI$_3$(001) system, we used $(6 \times 6 \times 1)$ kmesh with 520 eV energy cutoff. A supercell consists of 11 rutile TiO$_2$ layers and 3 cubic MAPbI$_3$ layers with (001) orientation for both, where the lattice mismatch using TiO$_2$(001)-$(\sqrt(2) \times \sqrt(2) \times 1)$ is as small as 3 $\%$. A vacuum size of 30 $\AA$ is included. We also used Quantum ESPRESSO package v.6.1[@55] with fully relativistic PAW PBE pseudopotential for Pb 6p6s5d, I 5p5s and Cs 6s5p5s at the energy cutoff of 40 Ry. Ab initio MD simulations with time step $\delta t$ = 0.5 $fs$ were performed with total duration of 12 $ps$ and 30 $ps$ at 600 K and 330 K for Gr/CsPbI$_3$ and TiO$_2$/MAPbI$_3$, respectively. We sampled NVT configuration using Nosé thermostat and discarded 3 $ps$ for the initialization.
C.W.M. conceived the idea, performed DFT and AIMD simulations and analyzed the data. S.J. helped in DFT calculation. All discussed and C.W.M., K.S.K. and G.L. wrote the manuscript. This work was supported by National Honor Scientist Program (2010-0020414) and Basic Science Research Program (2015R1C1A1A01055922) of NRF. Computation was supported by KISTI (KSC-2017-S1-0025, KSC-2017-C3-0081).
|
---
abstract: 'We collect a list of known four-dimensional Fano manifolds and compute their quantum periods. This list includes all four-dimensional Fano manifolds of index greater than one, all four-dimensional toric Fano manifolds, all four-dimensional products of lower-dimensional Fano manifolds, and certain complete intersections in projective bundles.'
address:
- |
Department of Mathematics\
Imperial College London\
180 Queen’s Gate\
London SW7 2AZ\
UK
- |
National Research University Higher School of Economics (HSE)\
Faculty of Mathematics and Laboratory of Algebraic Geometry\
7 Vavilova str., 117312, Moscow, Russia
author:
- Tom Coates
- Sergey Galkin
- Alexander Kasprzyk
- Andrew Strangeway
bibliography:
- 'bibliography.bib'
title: 'Quantum Periods for Certain Four-Dimensional Fano Manifolds'
---
Introduction
============
In this paper we take the first step towards implementing a program, laid out in [@ProcECM], to find and classify four-dimensional Fano manifolds using mirror symmetry. We compute quantum periods and quantum differential equations for many known four-dimensional Fano manifolds, using techniques described in [@QC105]. Our basic reference for the theory of Fano manifolds is the book by Iskovskikh–Prokhorov [@Iskovskikh--Prokhorov]. Recall that the *index* of a Fano manifold $X$ is the largest integer $r$ such that ${-K_X} = rH$ for some ample divisor $H$. A four-dimensional Fano manifold has index at most $5$ [@Shokurov]. Four-dimensional Fano manifolds with index $r>1$ have been classified. In what follows we compute the quantum periods and quantum differential equations for all four-dimensional Fano manifolds of index $r>1$, for all four-dimensional Fano toric manifolds, and for certain other four-dimensional Fano manifolds of index $1$.
Highlights {#highlights .unnumbered}
----------
We draw the reader’s attention to:
- §\[sec:double\_cover\], where new tools for computing Gromov–Witten invariants (twisted $I$-functions for toric complete intersections [@CCIT:toric_stacks_2] and an improved Quantum Lefschetz theorem [@Coates]) make a big practical difference to the computation of quantum periods. This should be contrasted with [@QC105 §19], where the new techniques were not available.
- §\[sec:null\_correlation\], which relies on a new construction of Szurek–Wiśniewski’s null-correlation bundle [@Szurek--Wisniewski] that may be of independent interest.
- The tables of regularized quantum period sequences in Appendix \[appendix:periods\].
- The numerical calculation of quantum differential operators in §\[sec:qdes\] and Appendix \[appendix:qdes\]. This suggests in particular that, for each four-dimensional Fano manifold $X$ with Fano index $r>1$, the regularized quantum differential equation of $X$ is either extremal or of low ramification.
- §\[sec:MW\^4\_17\] and §\[operator:MW\^4\_17\], which together give an example of a product such that the regularized quantum differential equation for each factor is extremal, but the regularized quantum differential equation for the product itself is not.
This paper is accompanied by fully-commented source code, written in the computational algebra system Magma [@Magma]. This will allow the reader to verify the calculations presented here, or to perform similar computations.
Methodology {#sec:methodology}
===========
The quantum period $G_X$ of a Fano manifold $X$ is a generating function $$\begin{aligned}
\label{eq:quantum_period}
G_X(t) = 1 + \sum_{d=1}^\infty c_d t^d && \text{$t \in {\mathbb{C}}$}\end{aligned}$$ for certain genus-zero Gromov–Witten invariants $c_d$ of $X$. A precise definition can be found in [@QC105 §B], but roughly speaking $c_d$ is the ‘virtual number’ of degree-$d$ rational curves $C$ in $X$ that pass through a given point and satisfy certain constraints on their complex structure. (The degree of a curve $C$ here is the quantity $\langle {-K_X}, C\rangle$.) The quantum period is discussed in detail in [@ProcECM; @QC105]; one property that will be important in what follows is that the regularized quantum period $$\begin{aligned}
\label{eq:regularized_quantum_period}
{\widehat{G}}_X(t) = 1 + \sum_{d=1}^\infty d! c_d t^d && \text{$t \in {\mathbb{C}}$, $|t| \ll \infty$}\end{aligned}$$ satisfies a differential equation called the *regularized quantum differential equation* of $X$: $$\begin{aligned}
\label{eq:regularized_QDE}
L_X {\widehat{G}}_X \equiv 0 && L_X = \sum_{m=0}^{m=N} p_m(t) D^m\end{aligned}$$ where the $p_m$ are polynomials and $D = t \frac{d}{dt}$. It is expected that the regularized quantum differential equation for a Fano manifold $X$ is *extremal* or *of low ramification*, as described in §\[sec:qdes\] below. This is a strong constraint on the Gromov–Witten invariants $c_d$ of $X$.
Quantum periods for a broad class of toric complete intersections can be computed using Givental’s mirror theorem [@Givental]:
\[thm:toric\_mirror\] Let $X$ be a toric Fano manifold and let $D_1,\ldots,D_N \in H^2(X;{\mathbb{Q}})$ be the cohomology classes Poincaré-dual to the torus-invariant divisors on $X$. The quantum period of $X$ is: $$G_X(t) = \sum_{\substack{
\beta \in H_2(X;{\mathbb{Z}}) : \\
\text{$\langle \beta, D_i \rangle \geq 0$ $\forall i$}
}}
\frac{
t^{\langle \beta,{-K_X}\rangle}
}{
\prod_{i=1}^N \langle \beta, D_i \rangle !
}$$
\[thm:toric\_ci\_mirror\] Let $Y$ be a toric Fano manifold, and let $D_1,\ldots,D_N \in H^2(Y;{\mathbb{Q}})$ be the cohomology classes Poincaré-dual to the torus-invariant divisors on $Y$. Let $X$ be the complete intersection in $Y$ defined by a regular section of $E = L_1 \oplus \cdots \oplus L_s$ where each $L_i$ is a nef line bundle, and let $\rho_i = c_1(L_i)$, $1 \leq i \leq s$. Suppose that the class $c_1(Y)-\Lambda$ is ample on $Y$, where $\Lambda = c_1(L_1) + \cdots + c_1(L_s)$. Then $X$ is Fano, and the quantum period of $X$ is: $$G_X(t) = e^{{-c} t} \sum_{\substack{
\beta \in H_2(Y;{\mathbb{Z}}) : \\
\text{$\langle \beta, D_i \rangle \geq 0$ $\forall i$}
}}
t^{\langle \beta,{-K_Y}- \Lambda\rangle}
\frac{
\prod_{j=1}^s \langle \beta, \rho_j \rangle !
}{
\prod_{i=1}^N \langle \beta, D_i \rangle !
}$$ where $c$ is the unique rational number such that the right-hand side has the form $1 + O(t^2)$.
An analogous mirror theorem holds for certain complete intersections in toric Deligne–Mumford stacks, but we will need only the case where the ambient stack is a weighted projective space:
\[thm:wps\_ci\_mirror\] Let $Y$ be the weighted projective space ${\mathbb{P}}(w_0,\ldots,w_n)$, let $X$ be a smooth Fano manifold given as a complete intersection in $Y$ defined by a section of $E = {\mathcal{O}}(d_1) \oplus \cdots \oplus {\mathcal{O}}(d_m)$, and let ${-k} = w_0 + \cdots + w_n - d_1 - \cdots - d_m$. Suppose that each $d_i$ is a positive integer, that ${-k}>0$, and that $w_i$ divides $d_j$ for all $i$, $j$ such that $0 \leq i \leq n$ and $1 \leq j \leq m$. Then the quantum period of $X$ is: $$G_X(t) = e^{{-c} t} \sum_{d=0}^\infty
t^{{-k} d}
\frac{
\prod_{j=1}^m (d d_j) !
}{
\prod_{i=1}^n (d w_i) !
}$$ where $c$ is the unique rational number such that the right-hand side has the form $1 + O(t^2)$.
The quantum period of a product is the product of the quantum periods:
\[thm:products\] Let $X$ and $Y$ be smooth projective complex manifolds. Then: $$G_{X \times Y}(t) = G_X(t)\, G_Y(t)$$
As we will see below, another powerful tool for computing quantum periods is the Abelian/non-Abelian Correspondence of Ciocan-Fontanine–Kim–Sabbah [@Ciocan-Fontanine--Kim--Sabbah]. We now proceed to the calculation of quantum periods.
Four-Dimensional Fano Manifolds of Index $5$ {#sec:index_5}
============================================
The only example here is ${\mathbb{P}}^4$ [@Kobayashi--Ochiai; @Kollar; @Serpico]. This is a toric variety. Theorem \[thm:toric\_mirror\] yields: $$\begin{aligned}
G_{{\mathbb{P}}^4}(t) = \sum_{d=0}^\infty \frac{t^{5d}}{(d!)^5}
&&
\text{[\hyperref[table:index_5]{regularized quantum period p.~\pageref*{table:index_5}}, \hyperref[operator:P4]{operator p.~\pageref*{operator:P4}}]}\end{aligned}$$
Four-Dimensional Fano Manifolds of Index $4$ {#sec:index_4}
============================================
The only example here is the quadric $Q^4 \subset {\mathbb{P}}^5$ [@Kollar; @Serpico]. This is a complete intersection in a toric variety. Theorem \[thm:toric\_ci\_mirror\] yields: $$\begin{aligned}
G_{Q^4}(t) = \sum_{d=0}^\infty \frac{(2d)!}{(d!)^6} t^{4d}
&&
\text{[\hyperref[table:index_4]{regularized quantum period p.~\pageref*{table:index_4}}, \hyperref[operator:Q4]{operator p.~\pageref*{operator:Q4}}]}\end{aligned}$$
Four-Dimensional Fano Manifolds of Index $3$ {#sec:index_3}
============================================
There are six examples [@Fujita:polarized_1; @Fujita:polarized_2; @Fujita:polarized_3; @Fujita:book; @Iskovskikh:Fano_1; @Iskovskikh:anticanonical; @Iskovskikh--Prokhorov], which are known as del Pezzo fourfolds:
- a sextic hypersurface ${\mathrm{FI}^{4}_{1}}$ in the weighted projective space ${\mathbb{P}}^5(1^4,2,3)$;
- a quartic hypersurface ${\mathrm{FI}^{4}_{2}}$ in the weighted projective space ${\mathbb{P}}^5(1^5,2)$;
- a cubic hypersurface ${\mathrm{FI}^{4}_{3}} \subset {\mathbb{P}}^5$;
- a complete intersection ${\mathrm{FI}^{4}_{4}} \subset {\mathbb{P}}^6$ of type $(2H) \cap (2H)$, where $H = {\mathcal{O}}_{{\mathbb{P}}^6}(1)$;
- a complete intersection ${\mathrm{FI}^{4}_{5}} \subset \operatorname{Gr}(2,5)$ of type $H \cap H$, where $H$ is the hyperplane bundle; and
- ${\mathrm{FI}^{4}_{6}} = {\mathbb{P}}^2 \times {\mathbb{P}}^2$.
The first four examples here are complete intersections in weighted projective spaces. Theorem \[thm:wps\_ci\_mirror\] yields: $$\begin{aligned}
& G_{{\mathrm{FI}^{4}_{1}}}(t) = \sum_{d=0}^\infty \frac{(6d)!}{(3d)!(2d)!(d!)^4} t^{3d} \\
& G_{{\mathrm{FI}^{4}_{2}}}(t) = \sum_{d=0}^\infty \frac{(4d)!}{(2d)!(d!)^5} t^{3d} \\
& G_{{\mathrm{FI}^{4}_{3}}}(t) = \sum_{d=0}^\infty \frac{(3d)!}{(d!)^6} t^{3d} \\
& G_{{\mathrm{FI}^{4}_{4}}}(t) = \sum_{d=0}^\infty \frac{(2d)!(2d)!}{(d!)^7} t^{3d}
$$ For ${\mathrm{FI}^{4}_{5}} \subset \operatorname{Gr}(2,5)$ we use the Abelian/non-Abelian Correspondence, applying Theorem F.1 in [@QC105] with $a=2$, $b=c=d=e=0$. This yields: $$G_{{\mathrm{FI}^{4}_{5}}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty
(-1)^{l+m}
t^{3l+3m}
\frac
{
(l+m)! (l+m)!
}
{
(l!)^5 (m!)^5
}
\big(1-5 (m-l)H_m \big)$$ where $H_m$ is the $m$th harmonic number. For ${\mathbb{P}}^2 \times {\mathbb{P}}^2$, combining Theorem \[thm:products\] with [@QC105 Example G.2] yields: $$G_{{\mathbb{P}}^2 \times {\mathbb{P}}^2}(t) =
\sum_{l=0}^\infty \sum_{m=0}^\infty \frac{t^{3l+3m}}{(l!)^3 (m!)^3}$$
------------------------- -- -- -- ---------------------------------------- -- --
${\mathrm{FI}^{4}_{1}}$ ${\mathrm{FI}^{4}_{4}}$
${\mathrm{FI}^{4}_{2}}$ ${\mathrm{FI}^{4}_{5}}$
${\mathrm{FI}^{4}_{3}}$ ${\mathbb{P}}^2 \times {\mathbb{P}}^2$
------------------------- -- -- -- ---------------------------------------- -- --
Four-Dimensional Fano Manifolds of Index $2$
============================================
Consider now a four-dimensional Fano manifold with index $r=2$ and Picard rank $\rho$.
The Case $\rho=1$ {#sec:index_2_rank_1}
-----------------
Four-dimensional Fano manifolds with index $r=2$ and Picard rank $\rho=1$ have been classified [@Mukai:PNAS; @Wilson], [@Iskovskikh--Prokhorov Chapter 5]. Up to deformation, there are 9 examples: the ‘linear unsections’ of smooth three-dimensional Fano manifolds with $\rho=1$, $r=1$, and degree at most $144$. We compute the quantum periods of these examples using the constructions in [@QC105 §§8–16], writing $V^4_k$ for a four-dimensional Fano manifold with $\rho=1$, $r=2$, and degree $16k$.
### $V^4_2$
\[sec:V\^4\_2\]
This is a sextic hypersurface in ${\mathbb{P}}^5(1^5,3)$. Proposition D.9 in [@QC105] yields: $$G_{V^4_2}(t) = \sum_{d=0}^\infty \frac{(6d)!}{(d!)^5(3d)!} t^{2d}$$
### $V^4_4$
\[sec:V\^4\_4\]
This is a quartic hypersurface in ${\mathbb{P}}^5$. Theorem \[thm:toric\_ci\_mirror\] yields: $$G_{V^4_4}(t) = \sum_{d=0}^\infty \frac{(4d)!}{(d!)^6} t^{2d}$$
### $V^4_6$
\[sec:V\^4\_6\]
This is a complete intersection of type $(2H) \cap (3H)$ in ${\mathbb{P}}^6$, where $H = {\mathcal{O}}_{{\mathbb{P}}^6}(1)$. Theorem \[thm:toric\_ci\_mirror\] yields: $$G_{V^4_6}(t) = \sum_{d=0}^\infty \frac{(2d)!(3d)!}{(d!)^7} t^{2d}$$
### $V^4_8$
\[sec:V\^4\_8\]
This is a complete intersection of type $(2H) \cap (2H) \cap (2H)$ in ${\mathbb{P}}^7$, where $H = {\mathcal{O}}_{{\mathbb{P}}^7}(1)$. Theorem \[thm:toric\_ci\_mirror\] yields: $$G_{V^4_8}(t) = \sum_{d=0}^\infty \frac{\big((2d)!\big)^3}{(d!)^8} t^{2d}$$
### $V^4_{10}$
\[sec:V\^4\_10\]
This is a complete intersection in $\operatorname{Gr}(2,5)$, cut out by a regular section of ${\mathcal{O}}(1) \oplus {\mathcal{O}}(2)$ where ${\mathcal{O}}(1)$ is the pullback of ${\mathcal{O}}(1)$ on projective space under the Plücker embedding. We apply Theorem F.1 in [@QC105] with $a=b=1$ and $c=d=e=0$. This yields: $$G_{V^4_{10}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty
(-1)^{l+m}
t^{2l+2m}
\frac
{
(l+m)! (2l+2m)!
}
{
(l!)^5 (m!)^5
}
\big(1-5 (m-l)H_m \big)$$ where $H_m$ is the $m$th harmonic number.
### $V^4_{12}$
\[sec:V\^4\_12\]
This is the subvariety of $\operatorname{Gr}(2,5)$ cut out by a regular section of $S^\star \otimes \det S^\star$, where $S$ is the universal bundle of subspaces on $\operatorname{Gr}(2,5)$. We apply Theorem F.1 in [@QC105] with $c = 1$ and $a = b = d = e = 0$. This yields: $$G_{V^4_{12}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty
({-1})^{l+m} t^{2l+2m}
\frac
{
(2l+m)! (l+2m)!
}
{
(l!)^5 (m!)^5
}
\big(1+(m-l)(H_{2l+m} + 2 H_{l + 2m}-5H_m) \big)$$
### $V^4_{14}$
\[sec:V\^4\_14\]
This is a complete intersection in $\operatorname{Gr}(2,6)$, cut out by a regular section of ${\mathcal{O}}(1)^{\oplus 4}$ where ${\mathcal{O}}(1)$ is the pullback of ${\mathcal{O}}(1)$ on projective space under the Plücker embedding. We apply Theorem F.1 in [@QC105] with $a=4$ and $b=c=d=e=0$. This yields: $$G_{V^4_{14}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty
(-1)^{l+m}
t^{2l+2m}
\frac
{
\big((l+m)!\big)^4
}
{
(l!)^6 (m!)^6
}
\big(1-6 (m-l)H_m \big)$$
### $V^4_{16}$
\[sec:V\^4\_16\]
This is the subvariety of $\operatorname{Gr}(3,6)$ cut out by a regular section of $\wedge^2 S^\star \oplus (\det S^\star)^{\oplus 2}$, where $S$ is the universal bundle of subspaces on $\operatorname{Gr}(3,6)$. We apply Theorem F.1 in [@QC105] with $a=2$, $b=c=d=0$, and $e=1$. This shows that the quantum period $G_{V^4_{16}}(t)$ is the coefficient of $(p_2-p_1)(p_3-p_1)(p_3-p_2)$ in the expression: $$\begin{gathered}
\sum_{l_1 = 0}^\infty
\sum_{l_2 = 0}^\infty
\sum_{l_3 = 0}^\infty
t^{2l_1 + 2 l_2 + 2 l_3}
{
\prod_{k=1}^{l_1+l_2+l_3} (p_1+p_2+p_3 + k)^2
\over
\prod_{j=1}^{j=3}
\prod_{k=1}^{k=l_j} (p_j + k)^6
}
\prod_{1 \leq i < j \leq 3} \prod_{k=1}^{l_i + l_j} (p_i + p_j + k) \\
\times \prod_{1 \leq i < j \leq 3} \big(p_j-p_i + (l_j-l_i) \big) \end{gathered}$$ (Since this expression is totally antisymmetric in $p_1$, $p_2$, $p_3$, it is divisible by $(p_2-p_1)(p_3-p_1)(p_3-p_2)$.)
### $V^4_{18}$
\[sec:V\^4\_18\]
This is the subvariety of $\operatorname{Gr}(5,7)$ cut out by a regular section of $\big( S \otimes \det S^\star\big) \oplus \det S^{\star}$, where $S$ is the universal bundle of subspaces on $\operatorname{Gr}(5,7)$. We apply Theorem F.1 in [@QC105] with $a=d=1$ and $b=c=e=0$. This shows that the quantum period $G_{V^4_{18}}(t)$ is the coefficient of $\prod_{1 \leq i < j \leq 5} (p_j-p_i)$ in the expression: $$\begin{gathered}
\sum_{l_1 = 0}^\infty
\sum_{l_2 = 0}^\infty
\sum_{l_3 = 0}^\infty
\sum_{l_4 = 0}^\infty
\sum_{l_5 = 0}^\infty
t^{2|l|}
{
\prod_{k=1}^{k=|l|} (p_1+p_2+\cdots + p_5 + k)
\over
\prod_{j=1}^{j=5}
\prod_{k=1}^{k=l_j} (p_j + k)^7
}
\prod_{j=1}^{j=5} \prod_{k=1}^{|l| - l_j} (p_1 + p_2 + \cdots + p_5 - p_j + k) \\
\times
\prod_{1 \leq i < j \leq 5} \big(p_j-p_i + (l_j-l_i) \big) \end{gathered}$$ where $|l| = l_1 + l_2 + \cdots + l_5$. (As above, antisymmetry implies that the long formula here is divisible by $\prod_{1 \leq i < j \leq 5} (p_j-p_i)$.)
The Case $\rho>1$ {#sec:index_2_higher_rank}
-----------------
Four-dimensional Fano manifolds with $\rho>1$ and $r=2$ have been classified by Mukai [@Mukai:PNAS; @Mukai:10] and Wiśniewski [@Wisniewski:classification]. There are 18 deformation families, as follows. We denote the $k$th such deformation family, as given in [@Iskovskikh--Prokhorov Table 12.7], by ${\mathrm{MW}^{4}_{k}}$.
### ${\mathrm{MW}^{4}_{1}}$
\[sec:MW\^4\_1\]
This is the product ${\mathbb{P}}^1 \times B^3_1$. Combining Theorem \[thm:products\] with [@QC105 Example G.1] and [@QC105 §3] yields: $$G_{{\mathrm{MW}^{4}_{1}}}(t) =
\sum_{l=0}^\infty \sum_{m=0}^\infty \frac{(6m)!}{(l!)^2 (m!)^3 (2m)! (3m)!} t^{2l+2m}$$
### ${\mathrm{MW}^{4}_{2}}$
\[sec:MW\^4\_2\]
This is the product ${\mathbb{P}}^1 \times B^3_2$. Combining Theorem \[thm:products\] with [@QC105 Example G.1] and [@QC105 §4] yields: $$G_{{\mathrm{MW}^{4}_{2}}}(t) =
\sum_{l=0}^\infty \sum_{m=0}^\infty \frac{(4m)!}{(l!)^2 (m!)^4 (2m)!} t^{2l+2m}$$
### ${\mathrm{MW}^{4}_{3}}$
\[sec:MW\^4\_3\]
This is the product ${\mathbb{P}}^1 \times B^3_3$. Combining Theorem \[thm:products\] with [@QC105 Example G.1] and [@QC105 §5] yields: $$G_{{\mathrm{MW}^{4}_{3}}}(t) =
\sum_{l=0}^\infty \sum_{m=0}^\infty \frac{(3m)!}{(l!)^2 (m!)^5} t^{2l+2m}$$
### ${\mathrm{MW}^{4}_{4}}$
\[sec:MW\^4\_4\] \[sec:double\_cover\]
This is a double cover of ${\mathbb{P}}^2 \times {\mathbb{P}}^2$, branched over a divisor of bidegree $(2,2)$. Consider the toric variety $F$ with weight data: $$\begin{array}{rrrrrrrl}
\multicolumn{1}{c}{x_0} &
\multicolumn{1}{c}{x_1} &
\multicolumn{1}{c}{x_2} &
\multicolumn{1}{c}{y_0} &
\multicolumn{1}{c}{y_1} &
\multicolumn{1}{c}{y_2} &
\multicolumn{1}{c}{w} & \\
\cmidrule{1-7}
1 & 1 & 1 & 0 & 0 & 0 & 1 & \hspace{1.5ex} L\\
0 & 0 & 0 & 1 & 1 & 1 & 1 & \hspace{1.5ex} M \\
\end{array}$$ and $\operatorname{\overline{Amp}}F = \langle L, L+M \rangle$. Let $X$ be a member of the linear system $|2L+2M|$ defined by the equation $w^2=f_{2,2}$, where $f_{2,2}$ is a bihomogeneous polynomial of degrees $2$ in $x_0$, $x_1$, $x_2$ and $2$ in $y_0$, $y_1$, $y_2$. Let $p \colon F \dashrightarrow {\mathbb{P}}^2 \times {\mathbb{P}}^2$ be the rational map which sends (contravariantly) the homogeneous co-ordinate functions $[x_0,x_1,x_2,y_0,y_1,y_2] $ on ${\mathbb{P}}^2_{x_0,x_1,x_2} \times {\mathbb{P}}^2_{y_0,y_1,y_2}$ to $[x_0,x_1,x_2,y_0,y_1,y_2]$. The restriction of $p$ to $X$ is a morphism, which exhibits $X$ as a double cover of ${\mathbb{P}}^2 \times {\mathbb{P}}^2$ branched over the locus $(f_{2,2}=0) \subset {\mathbb{P}}^2_{x_0,x_1,x_2} \times {\mathbb{P}}^2_{y_0,y_1,y_2}$. Thus $X = {\mathrm{MW}^{4}_{4}}$.
Recall the definition of the $J$-function $J_X(t,z)$ from [@Coates--Givental equation 11]. Recall from [@Coates] that there is a Lagrangian cone ${\mathcal{L}}_X \subset H^\bullet(X;\Lambda_X)\otimes {\mathbb{C}}(\!(z^{-1})\!)$ that encodes all genus-zero Gromov–Witten invariants of $X$, and a Lagrangian cone ${\mathcal{L}}_{{\boldsymbol{e}}}\subset H^\bullet(F;\Lambda_F)\otimes {\mathbb{C}}(\!(z^{-1})\!) \otimes {\mathbb{C}}(\lambda)$ that encodes all genus-zero $({{\boldsymbol{e}}},2L+2M)$-twisted Gromov–Witten invariants of $F$. Here $\Lambda_X$ and $\Lambda_F$ are certain Novikov rings and ${{\boldsymbol{e}}}$ is the total Chern class with parameter $\lambda$ (or, equivalently, ${{\boldsymbol{e}}}$ is the $S^1$-equivariant Euler class with respect to an action of $S^1$ described in [@Coates]; in this case one should regard $\lambda$ as the standard generator for the $S^1$-equivariant cohomology algebra of a point). The $J$-function $J_X$ is characterised by the fact that $J_X(t,{-z})$ is the unique point on ${\mathcal{L}}_X$ of the form ${-z} + t + O(z^{-1})$.
Let $p_1$, $p_2 \in H^2(F;{\mathbb{Q}})$ denote the first Chern class of $L$, $L+M$ respectively and let $P_1$, $P_2 \in H^2(X;{\mathbb{Q}})$ denote the pullbacks of $p_1$, $p_2$ along the inclusion map $i \colon X \to F$. Let $Q_1$, $Q_2$ denote the elements of the Novikov ring $\Lambda_X$ that are dual respectively to $P_1$, $P_2$, and note that $\Lambda_X$ and $\Lambda_F$ are canonically isomorphic (via $i_\star$). Theorem 21 in [@CCIT:toric_stacks_2] implies that: $$I(t_1,t_2,\lambda,z) = z e^{t_1 p_1/z} e^{t_2 p_2/z}
\sum_{l=0}^\infty \sum_{m=0}^\infty
\frac{
Q_1^l Q_2^m e^{l t_1} e^{m t_2}
\prod_{k=1}^{k=2m} (\lambda + 2p_2 + k z)
}{
\prod_{k=1}^{k=l} (p_1 + k z)^3
\prod_{k=1}^{k=m} (p_2 + k z)
}
\frac{
\prod_{k={-\infty}}^{k=0} (p_2-p_1+kz)^3
}{
\prod_{k={-\infty}}^{k=m-l} (p_2-p_1+kz)^3
}$$ satisfies $I(t_1,t_2,\lambda,{-z}) \in {\mathcal{L}}_{{\boldsymbol{e}}}$. Theorem 1.1 in [@Coates] gives that $i^\star {\mathcal{L}}_{{\boldsymbol{e}}}\big|_{\lambda = 0} \subset {\mathcal{L}}_X$, and therefore that: $$i^\star I(t_1,t_2,0,{-z}) \in {\mathcal{L}}_X$$ Since the hypersurface $X$ misses the locus $y_1 = y_2 = y_3 = 0$ in $F$, we have that $i^\star (p_2-p_1)^3 = 0$. Thus: $$i^\star I(t_1,t_2,0,z) =
z e^{t_1 P_1/z} e^{t_2 P_2/z}
\sum_{l=0}^\infty \sum_{m=l}^\infty
\frac{
Q_1^l Q_2^m e^{l t_1} e^{m t_2}
\prod_{k=1}^{k=2m} (2P_2 + k z)
}{
\prod_{k=1}^{k=l} (P_1 + k z)^3
\prod_{k=1}^{k=m} (P_2 + k z)
}
\frac{
1
}{
\prod_{k=1}^{k=m-l} (P_2-P_1+kz)^3
}$$ In particular, $i^\star I(t_1,t_2,0,{-z})$ has the form ${-z} + t_1 P_1 + t_2 P_2 + O(z^{-1})$ and, from the characterisation of $J_X$ discussed above, we conclude that $J_X(t_1P_1+t_2P_2,{-z}) = i^\star I(t_1,t_2,0,{-z})$.
To extract the quantum period $G_X$ from the $J$-function $J_X(t_1P_1+t_2P_2,z)$ we take the component along the unit class $1 \in H^\bullet(X;{\mathbb{Q}})$, set $z=1$, set $t_1 = t_2 = 0$, and set $Q_1 =1$, $ Q_2 =t^2$, obtaining: $$G_{{\mathrm{MW}^{4}_{4}}}(t) = \sum_{l=0}^\infty \sum_{m=l}^\infty \frac{(2m)!}{(l!)^3 m! ((m-l)!)^3} t^{2m}$$
### ${\mathrm{MW}^{4}_{5}}$
\[sec:MW\^4\_5\]
This is a divisor on ${\mathbb{P}}^2 \times {\mathbb{P}}^3$ of bidegree $(1,2)$. Theorem \[thm:toric\_ci\_mirror\] yields: $$G_{{\mathrm{MW}^{4}_{5}}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty \frac{(l+2m)!}{(l!)^3 (m!)^4} t^{2l+2m}$$
### ${\mathrm{MW}^{4}_{6}}$
\[sec:MW\^4\_6\]
This is the product ${\mathbb{P}}^1 \times B^3_4$. Combining Theorem \[thm:products\] with [@QC105 Example G.1] and [@QC105 §6] yields: $$G_{{\mathrm{MW}^{4}_{6}}}(t) =
\sum_{l=0}^\infty \sum_{m=0}^\infty \frac{(2m)!(2m)!}{(l!)^2 (m!)^6} t^{2l+2m}$$
### ${\mathrm{MW}^{4}_{7}}$
\[sec:MW\^4\_7\]
This is a complete intersection of two divisors in ${\mathbb{P}}^3 \times {\mathbb{P}}^3$, each of bidegree $(1,1)$. Theorem \[thm:toric\_ci\_mirror\] yields: $$G_{{\mathrm{MW}^{4}_{7}}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty \frac{(l+m)!(l+m)!}{(l!)^4 (m!)^4} t^{2l+2m}$$
### ${\mathrm{MW}^{4}_{8}}$
\[sec:MW\^4\_8\]
This is a divisor on ${\mathbb{P}}^2 \times Q^3$ of bidegree $(1,1)$. Theorem \[thm:toric\_ci\_mirror\] yields: $$G_{{\mathrm{MW}^{4}_{8}}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty \frac{(l+m)!(2m)!}{(l!)^3 (m!)^5} t^{2l+2m}$$
### ${\mathrm{MW}^{4}_{9}}$
\[sec:MW\^4\_9\]
This is the product ${\mathbb{P}}^1 \times B^3_5$. Combining Theorem \[thm:products\] with [@QC105 Example G.1] and [@QC105 §7] yields: $$G_{{\mathrm{MW}^{4}_{9}}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty \sum_{n=0}^\infty
(-1)^{m+n}
t^{2l+2m+2n}
\frac
{
\big((m+n)!\big)^3
}
{
(l!)^2 (m!)^5 (n!)^5
}
\big(1-5 (n-m)H_n \big)$$
### ${\mathrm{MW}^{4}_{10}}$
\[sec:MW\^4\_10\]
This is the blow-up of the quadric $Q^4$ along a conic that is not contained in a plane lying in $Q^4$. Consider the toric variety $F$ with weight data: $$\begin{array}{rrrrrrrl}
\multicolumn{1}{c}{s_0} &
\multicolumn{1}{c}{s_1} &
\multicolumn{1}{c}{s_2} &
\multicolumn{1}{c}{x} &
\multicolumn{1}{c}{x_3} &
\multicolumn{1}{c}{x_4} &
\multicolumn{1}{c}{x_5} & \\
\cmidrule{1-7}
1 & 1 & 1 & -1 & 0 & 0 & 0 & \hspace{1.5ex} L\\
0 & 0 & 0 & 1 & 1 & 1 & 1 & \hspace{1.5ex} M \\
\end{array}$$ and $\operatorname{\overline{Amp}}F = \langle L, M\rangle$. The morphism $F \to {\mathbb{P}}^5$ that sends (contravariantly) the homogeneous co-ordinate functions $[x_0,x_1,\dots,x_5]$ to $[xs_0, xs_1, x s_2, x_3, x_4, x_5]$ blows up the plane $\Pi = (x_0=x_1=x_2=0)$ in ${\mathbb{P}}^5$. Thus a general member of $|2M|$ on $F$ is the blow-up of $Q^4$ with centre a conic on $\Pi$. In other words, a general member of $|2M|$ on $F$ is ${\mathrm{MW}^{4}_{10}}$. We have:
- $-K_F=2L+4M$ is ample, so that $F$ is a Fano variety;
- ${\mathrm{MW}^{4}_{10}}\sim 2M$ is ample;
- $-(K_F+2M)\sim 2L+2M$ is ample.
Theorem \[thm:toric\_ci\_mirror\] yields: $$G_{{\mathrm{MW}^{4}_{10}}}(t) = \sum_{l=0}^\infty \sum_{m=l}^\infty \frac{(2m)!}{(l!)^3(m-l)!(m!)^3} t^{2l+2m}$$
### ${\mathrm{MW}^{4}_{11}}$
\[sec:MW\^4\_11\] \[sec:null\_correlation\]
This is the projective bundle ${\mathbb{P}}_{{\mathbb{P}}^3}({\mathcal{E}}^\vee)$, where ${\mathcal{E}}\to {\mathbb{P}}^3$ is the null-correlation bundle of Szurek–Wiśniewski [@Szurek--Wisniewski].
\[rem:slice\] For us ${\mathbb{P}}(E)$ denotes the projective bundle of lines in $E$, whereas in Szurek–Wiśniewski and Iskovskikh–Prokhorov, ${\mathbb{P}}(E)$ denotes the projective bundle of one-dimensional quotients. With our conventions, if $\pi \colon {\mathbb{P}}(E) \to X$ is a projective bundle then $E^\star = \pi_\star {\mathcal{O}}_{{\mathbb{P}}(E)}(1)$, and so a regular section $s \in \Gamma\big({\mathbb{P}}(E),{\mathcal{O}}_{{\mathbb{P}}(E)}(1)\big)$ vanishes on ${\mathbb{P}}(F^\star) \subset {\mathbb{P}}(E)$, where the vector bundle $F \to X$ is the cokernel of $s \colon {\mathcal{O}}_{{\mathbb{P}}(E)} \to E^\star$.
\[pro:better\_model\] Let $V = {\mathbb{C}}^4$, so that ${\mathbb{P}}(V) = {\mathbb{P}}^3$. Consider the partial flag manifold $\operatorname{Fl}_{1,2}(V)$ and the natural projections $$\label{eq:flag_diagram}
\begin{aligned}
\xymatrix{ & \operatorname{Fl}_{1,2}(V) \ar[dl]_{p_1} \ar[dr]^{p_2} \\ {\mathbb{P}}(V) && \operatorname{Gr}(2,V) }
\end{aligned}$$ Let $|L|$ denote the linear system defined by ${\mathcal{O}}(1)$ for the projective bundle $p_1$. Then a general element of $|L|$ is ${\mathbb{P}}({\mathcal{E}}^\vee)$, where ${\mathcal{E}}\to {\mathbb{P}}(V)$ is the null-correlation bundle.
The null-correlation bundle has rank $2$, and so the perfect pairing ${\mathcal{E}}\otimes {\mathcal{E}}\to \det {\mathcal{E}}$ gives canonical isomorphisms ${\mathcal{E}}^\vee \cong {\mathcal{E}}\otimes (\det {\mathcal{E}})^{-1}$ and ${\mathbb{P}}({\mathcal{E}}^\vee) \cong {\mathbb{P}}({\mathcal{E}})$. There is an exact sequence: $$\xymatrix{0 \ar[r] & {\mathcal{E}}(-1) \ar[r] & T_{{\mathbb{P}}(V)}(-2) \ar[r] & {\mathcal{O}}_{{\mathbb{P}}(V)} \ar[r] & 0}$$ and the map $s^\star \colon T_{{\mathbb{P}}(V)}(-2) \to {\mathcal{O}}_{{\mathbb{P}}(V)}$ therein defines a section $s \in \Gamma\big({\mathbb{P}}(T_{{\mathbb{P}}(V)}(-2)),{\mathcal{O}}_{{\mathbb{P}}(T_{{\mathbb{P}}(V)}(-2))}(1)\big)$. The construction in Remark \[rem:slice\] now exhibits ${\mathbb{P}}\big({\mathcal{E}}(-1)\big) \cong {\mathbb{P}}({\mathcal{E}})$ as the locus $(s=0)$ in ${\mathbb{P}}(T_{{\mathbb{P}}(V)}(-2))$. We will identify ${\mathbb{P}}(T_{{\mathbb{P}}(V)}(-2))$ with the partial flag manifold $\operatorname{Fl}_{1,2}(V)$.
For a vector bundle ${\mathcal{F}}\to X$ of rank $3$, the perfect pairing ${\mathcal{F}}\otimes \wedge^2 {\mathcal{F}}\to \det {\mathcal{F}}$ gives a canonical isomorphism ${\mathcal{F}}^\star \cong (\wedge^2 {\mathcal{F}}) \otimes (\det {\mathcal{F}})^{-1}$. Applying this with ${\mathcal{F}}\to X$ equal to $\Omega_{{\mathbb{P}}(V)}(2) \to {\mathbb{P}}(V)$ gives: $$T_{{\mathbb{P}}(V)}(-2) \cong \Omega_{{\mathbb{P}}(V)}^2(2)$$ where $\Omega_{{\mathbb{P}}(V)}^2 := \wedge^2 \Omega_{{\mathbb{P}}(V)}$. We thus need to identify ${\mathbb{P}}\big(\Omega^2_{{\mathbb{P}}(V)}(2)\big)$ with $\operatorname{Fl}_{1,2}(V)$.
The Plücker embedding $\operatorname{Gr}(2,V) \to {\mathbb{P}}\big(\wedge^2 V\big)$ maps a subspace $W \in \operatorname{Gr}(2,V)$ to the antisymmetric linear map $L_W \colon V^\star \to V$, well-defined up to scale, given by: $$L_W(f) = f(w_1) w_2 - f(w_2) w_1$$ where $\{w_1,w_2\}$ is a basis for $W$. The kernel of $L_W$ is the annihilator $W^\perp \subset V^\star$. If $f \not \in W^\perp$ then $\langle L_W(f) \rangle = \ker f \cap W$; this implies in particular that $\operatorname{rk}L_W = 2$. Thus the image of the Plücker embedding consists of (the lines spanned by) antisymmetric linear maps $L_W \colon V^\star \to V$ of rank $2$, and one can recover $W \in \operatorname{Gr}(2,V)$ from its image $\langle L_W \rangle$ by taking the annihilator of the kernel: $$W = \big(\ker L_W \big)^\perp$$ There is a canonical isomorphism $\operatorname{Ann}\colon \operatorname{Gr}(2,V) \to \operatorname{Gr}(2,V^\star)$ which maps $W \in \operatorname{Gr}(2,V)$ to $W^\perp$.
Recall that our goal is to identify ${\mathbb{P}}\big(\Omega^2_{{\mathbb{P}}(V)}(2)\big)$ with $\operatorname{Fl}_{1,2}(V)$. Let $q_1 \colon {\mathbb{P}}\big(\Omega^2_{{\mathbb{P}}(V)}(2)\big) \to {\mathbb{P}}(V)$ denote the projection. The Euler sequence: $$\xymatrix{ 0 \ar[r] & \Omega_{{\mathbb{P}}(V)} \ar[r] & \pi^\star V^\star (-1) \ar[r] & {\mathcal{O}}_{{\mathbb{P}}(V)} \ar[r] & 0}$$ gives, via [@Hartshorne II, Exercise 5.16]: $$\xymatrix{ 0 \ar[r] & \Omega^2_{{\mathbb{P}}(V)} \ar[r] & \pi^\star \big(\wedge^2 V^\star \big) (-2) \ar[r] & \Omega_{{\mathbb{P}}(V)} \ar[r] & 0}$$ and thus: $$\label{eq:Omega_2_sequence}
\xymatrix{ 0 \ar[r] & \Omega^2_{{\mathbb{P}}(V)}(2) \ar[r] & \pi^\star \big(\wedge^2 V^\star\big) \ar[r] & \pi^\star V^\star (1) \ar[r] & {\mathcal{O}}_{{\mathbb{P}}(V)}(2) \ar[r] & 0}$$ This defines a map $f \colon {\mathbb{P}}\big(\Omega^2_{{\mathbb{P}}(V)}(2)\big) \to {\mathbb{P}}\big(\wedge^2 V^\star\big)$. Consider the fiber of the sequence over $[v] \in {\mathbb{P}}(V)$. The map $\pi^\star \big(\wedge^2 V^\star \big) \to \pi^\star V^\star (1)$ here is given by contraction with $v$, and so non-zero elements of the kernel are antisymmetric linear maps $V \to V^\star$ of rank $2$. (They are antisymmetric, hence have rank $0$, $2$, or $4$; they are non-zero, hence are not of rank $0$; and they have the non-zero element $v$ in their kernel, hence are not of rank $4$.) In particular, we see that the image of $f$ lies in $\operatorname{Gr}(2,V^\star) \subset {\mathbb{P}}\big(\wedge^2 V^\star\big)$. Given $[x] \in {\mathbb{P}}\big(\Omega^2_{{\mathbb{P}}(V)}(2)\big)$, write $W_{[x]} \subset V^\star$ for the linear subspace defined by $f([x])$. Suppose that $[x] \in {\mathbb{P}}\big(\Omega^2_{{\mathbb{P}}(V)}(2)\big)$ lies over $[v] \in {\mathbb{P}}(V)$. Then, applying the discussion in the previous paragraph but with $V$ there replaced by $V^\star$, we see that $v \in W_{[x]}^\perp$. Thus, writing $q_2 \colon {\mathbb{P}}\big(\Omega^2_{{\mathbb{P}}(V)}(2)\big) \to \operatorname{Gr}(2,V)$ for the composition $$\xymatrix{ {\mathbb{P}}\big(\Omega^2_{{\mathbb{P}}(V)}(2)\big) \ar[r]^f & \operatorname{Gr}(2,V^\star) \ar[r]^\operatorname{Ann}& \operatorname{Gr}(2,V) }$$ we have that $q_1([x]) \subset q_2([x])$, i.e., that the diagram: $$\xymatrix{
& {\mathbb{P}}\big(\Omega^2_{{\mathbb{P}}(V)}(2)\big) \ar[dl]_{q_1} \ar[rd]^{q_2} \\
{\mathbb{P}}(V) && \operatorname{Gr}(2,V)}$$ coincides with the diagram . This identifies ${\mathbb{P}}\big(\Omega^2_{{\mathbb{P}}(V)}(2)\big)$ with the partial flag manifold $\operatorname{Fl}_{1,2}(V)$, and exhibits ${\mathbb{P}}({\mathcal{E}}^\vee)$ as an element of the linear system $|L|$ as claimed.
Abelianization: {#abelianization .unnumbered}
---------------
To compute the quantum period, we use the Abelian/non-Abelian Correspondence of Ciocan-Fontanine–Kim–Sabbah, as in [@QC105 §39]. Consider the situation as in §3.1 of [@Ciocan-Fontanine--Kim--Sabbah] with:
- $X = {\mathbb{C}}^{10}$, regarded as the space of pairs: $$\{(v,w) : \text{$v \in {\mathbb{C}}^2$ is a row vector, $w$ is a $2 \times 4$ complex matrix}\}$$
- $G = {{\mathbb{C}}^\times}\times \operatorname{GL}_2({\mathbb{C}})$, acting on $X$ as: $$(\lambda, g) \colon (v, w) \mapsto (\lambda v g^{-1}, gw)$$
- $T = ({{\mathbb{C}}^\times})^3$, the diagonal subtorus in $G$;
- the group that is denoted by $S$ in [@Ciocan-Fontanine--Kim--Sabbah] set equal to the trivial group;
- ${\mathcal{V}}$ equal to the representation of $G$ given by the determinant of the standard representation of the second factor $\operatorname{GL}_2({\mathbb{C}})$.
Then $X {/\!\!/}G$ is the partial flag manifold $\operatorname{Fl}=\operatorname{Fl}_{1,2}({\mathbb{C}}^4)$, whereas $X {/\!\!/}T$ is the toric variety with weight data: $$\begin{array}{rrrrrrrrrrl}
1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & \hspace{1.5ex} L_1\\
0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & -1& \hspace{1.5ex} L_2 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & \hspace{1.5ex} H
\end{array}$$ and $\operatorname{\overline{Amp}}=\langle L_1, L_2 , H \rangle$; that is, $X {/\!\!/}T$ is the projective bundle ${\mathbb{P}}({\mathcal{O}}(-1,0)\oplus {\mathcal{O}}(0,-1))$ over ${\mathbb{P}}^3\times {\mathbb{P}}^3$. The non-trivial element of the Weyl group $W={\mathbb{Z}}/2{\mathbb{Z}}$ exchanges the two factors of ${\mathbb{P}}^3\times {\mathbb{P}}^3$. The representation ${\mathcal{V}}$ induces the line bundle ${\mathcal{V}}_G = L$ over $X {/\!\!/}G = \operatorname{Fl}$, where $L$ was defined in the statement of Proposition \[pro:better\_model\], whereas the representation ${\mathcal{V}}$ induces the line bundle ${\mathcal{V}}_T = L_1+L_2$ over $X {/\!\!/}T$.
The Abelian/non-Abelian Correspondence: {#the-abeliannon-abelian-correspondence .unnumbered}
---------------------------------------
Let $p_1$, $p_2$, and $p_3 \in H^2(X{/\!\!/}T;{\mathbb{Q}})$ denote the first Chern classes of the line bundles $L_1$, $L_2$, and $H$ respectively. We fix a lift of $H^\bullet(X {/\!\!/}G;{\mathbb{Q}})$ to $H^\bullet(X {/\!\!/}T,{\mathbb{Q}})^W$ in the sense of [@Ciocan-Fontanine--Kim--Sabbah §3]; there are many possible choices for such a lift, and the precise choice made will be unimportant in what follows. The lift allows us to regard $H^\bullet(X {/\!\!/}G;{\mathbb{Q}})$ as a subspace of $H^\bullet(X {/\!\!/}T,{\mathbb{Q}})^W$, which maps isomorphically to the Weyl-anti-invariant part $H^\bullet(X {/\!\!/}T,{\mathbb{Q}})^a$ of $H^\bullet(X {/\!\!/}T,{\mathbb{Q}})$ via: $$\xymatrix{
H^\bullet(X {/\!\!/}T,{\mathbb{Q}})^W \ar[rr]^{\cup(p_2-p_1)} &&
H^\bullet(X {/\!\!/}T,{\mathbb{Q}})^a}$$ We compute the quantum period of ${\mathrm{MW}^{4}_{11}} \subset X {/\!\!/}G$ by computing the $J$-function of $\operatorname{Fl}= X {/\!\!/}G$ twisted, in the sense of [@Coates--Givental], by the Euler class and the bundle ${\mathcal{V}}_G$, using the Abelian/non-Abelian Correspondence.
Our first step is to compute the $J$-function of $X {/\!\!/}T$ twisted by the Euler class and the bundle ${\mathcal{V}}_T$. As in [@QC105 §D.1] and as in [@Ciocan-Fontanine--Kim--Sabbah], consider the bundles ${\mathcal{V}}_T$ and ${\mathcal{V}}_G$ equipped with the canonical ${{\mathbb{C}}^\times}$-action that rotates fibers and acts trivially on the base. Recall the definition of the twisted $J$-function $J_{{{\boldsymbol{e}}},{\mathcal{V}}_T}$ of $X {/\!\!/}T$ from [@QC105 §D.1]. We will compute $J_{{{\boldsymbol{e}}},{\mathcal{V}}_T}$ using the Quantum Lefschetz theorem; $J_{{{\boldsymbol{e}}},{\mathcal{V}}_T}$ is the restriction to the locus $\tau \in H^0(X {/\!\!/}T) \oplus H^2(X {/\!\!/}T)$ of what was denoted by $J^{S \times {{\mathbb{C}}^\times}}_{{\mathcal{V}}_T}(\tau)$ in [@Ciocan-Fontanine--Kim--Sabbah]. The toric variety $X {/\!\!/}T$ is Fano, so Theorem C.1 in [@QC105] gives: $$J_{X {/\!\!/}T}(\tau) = e^{\tau/z} \sum_{l, m, n \geq 0} {
Q_1^l Q_2^m Q_3^n
e^{l \tau_1} e^{m \tau_2} e^{m \tau_3}
\over
\prod_{k=1}^{k=l} (p_1 + k z)^4
\prod_{k=1}^{k=m} (p_2 + k z)^4
}
{
\prod_{k =-\infty}^{k=0} p_3-p_1 + k z
\over
\prod_{k=-\infty}^{k=n-l} p_3-p_1 + k z
}
{
\prod_{k = -\infty}^{k=0} p_3-p_2 + k z
\over
\prod_{k=-\infty}^{k= n-m} p_3-p_2 + k z
}$$ where $\tau = \tau_1 p_1 + \tau_2 p_2 + \tau_3 p_3$ and we have identified the group ring ${\mathbb{Q}}[H_2(X {/\!\!/}T;{\mathbb{Z}})]$ with ${\mathbb{Q}}[Q_1,Q_2,Q_3]$ via the ${\mathbb{Q}}$-linear map that sends $Q^\beta$ to $Q_1^{\langle \beta, p_1 \rangle} Q_2^{\langle \beta, p_2\rangle} Q_3^{\langle \beta, p_3 \rangle}$. The line bundles $L_1$, $L_2$, and $H$ are nef, and $c_1(X {/\!\!/}T) - c_1({\mathcal{V}}_T)$ is ample, so Theorem D.3 in [@QC105] gives: $$\begin{gathered}
J_{{{\boldsymbol{e}}},{\mathcal{V}}_T}(\tau) =
e^{\tau/z}
\sum_{l=0}^\infty \sum_{m=0}^\infty \sum_{n=0}^\infty
Q_1^l Q_2^m Q_3^n e^{l \tau_1} e^{m \tau_2} e^{m \tau_3}
{
\prod_{k=1}^{k=l+m} (\lambda + p_1 + p_2 + k z)
\over
\prod_{k=1}^{k=l} (p_1 + k z)^4
\prod_{k=1}^{k=m} (p_2 + k z)^4
} \times \\
{
\prod_{k =-\infty}^{k=0} p_3-p_1 + k z
\over
\prod_{k=-\infty}^{k=n-l} p_3-p_1 + k z
}
{
\prod_{k = -\infty}^{k=0} p_3-p_2 + k z
\over
\prod_{k=-\infty}^{k= n-m} p_3-p_2 + k z
}\end{gathered}$$
Consider now $\operatorname{Fl}= X {/\!\!/}G$ and a point $t \in H^\bullet(\operatorname{Fl})$. Recall that $\operatorname{Fl}={\mathbb{P}}(S)$ is the projectivization of the universal bundle $S$ of subspaces on $\operatorname{Gr}:= \operatorname{Gr}(2,4)$. Let $\epsilon_1 \in H^2(\operatorname{Fl};{\mathbb{Q}})$ be the pullback to $\operatorname{Fl}$ (under the projection map $p_2\colon\operatorname{Fl}\to \operatorname{Gr}$) of the ample generator of $H^2(\operatorname{Gr})$, and let $\epsilon_2 \in H^2(\operatorname{Fl};{\mathbb{Q}})$ be the first Chern class of ${\mathcal{O}}_{{\mathbb{P}}(S)}(1)$. Identify the group ring ${\mathbb{Q}}[H_2(\operatorname{Fl};{\mathbb{Z}})]$ with ${\mathbb{Q}}[q_1,q_2]$ via the ${\mathbb{Q}}$-linear map which sends $Q^\beta$ to $q_1^{\langle \beta,\epsilon_1 \rangle} q_2^{\langle \beta,\epsilon_2 \rangle}$. In [@Ciocan-Fontanine--Kim--Sabbah §6.1] the authors consider the lift $\tilde{J}^{S \times {{\mathbb{C}}^\times}}_{{\mathcal{V}}_G}(t)$ of their twisted $J$-function $J^{S \times {{\mathbb{C}}^\times}}_{{\mathcal{V}}_G}(t)$ determined by a choice of lift $H^\bullet(X {/\!\!/}G;{\mathbb{Q}}) \to H^\bullet(X {/\!\!/}T,{\mathbb{Q}})^W$. We restrict to the locus $t \in H^0(X {/\!\!/}G;{\mathbb{Q}}) \oplus H^2(X {/\!\!/}G;{\mathbb{Q}})$, considering the lift: $$\begin{aligned}
\tilde{J}_{{{\boldsymbol{e}}},{\mathcal{V}}_G}(t) := \tilde{J}^{S \times {{\mathbb{C}}^\times}}_{{\mathcal{V}}_G}(t) && t \in H^0(X {/\!\!/}G;{\mathbb{Q}}) \oplus H^2(X {/\!\!/}G;{\mathbb{Q}})\end{aligned}$$ of our twisted $J$-function $J_{{{\boldsymbol{e}}},{\mathcal{V}}_G}$ determined by our choice of lift $H^\bullet(X {/\!\!/}G;{\mathbb{Q}}) \to H^\bullet(X {/\!\!/}T,{\mathbb{Q}})^W$. Theorems 4.1.1 and 6.1.2 in [@Ciocan-Fontanine--Kim--Sabbah] imply that: $$\tilde{J}_{{{\boldsymbol{e}}},{\mathcal{V}}_G}\big(\varphi(t)\big) \cup (p_2 - p_1) = \Big[
\textstyle
\big(z {\partial \over \partial \tau_2}
-
z {\partial \over \partial \tau_1}
\big) J_{{{\boldsymbol{e}}},{\mathcal{V}}_T}(\tau) \Big]_{\tau=t, Q_1 = Q_2 = -q_1, Q_3=q_2}$$ for some function $\varphi:H^2(X {/\!\!/}G;{\mathbb{Q}}) \to H^\bullet(X {/\!\!/}G; \Lambda_G)$. Setting $t = 0$ gives: $$\begin{gathered}
\tilde{J}_{{{\boldsymbol{e}}},{\mathcal{V}}_G}\big(\varphi(0)\big) \cup (p_2 - p_1) =
\sum_{l=0}^\infty \sum_{m=0}^\infty \sum_{n=0}^\infty
(-1)^{l+m} q_1^{l+m} q_2^n
{
\prod_{k=1}^{k=l+m} (\lambda + p_1 + p_2 + k z)
\over
\prod_{k=1}^{k=l} (p_1 + k z)^4
\prod_{k=1}^{k=m} (p_2 + k z)^4
} \times \\
{
\prod_{k =-\infty}^{k=0} p_3-p_1 + k z
\over
\prod_{k=-\infty}^{k=n-l} p_3-p_1 + k z
}
{
\prod_{k = -\infty}^{k=0} p_3-p_2 + k z
\over
\prod_{k=-\infty}^{k= n-m} p_3-p_2 + k z
}
\big(p_2-p_1 + (m-l) z \big)\end{gathered}$$ For symmetry reasons the right-hand side here is divisible by $p_2-p_1$; it takes the form: $$(p_2-p_1)\Big(1 + O(z^{-2})\Big)$$ whereas: $$\tilde{J}_{{{\boldsymbol{e}}},{\mathcal{V}}_G}\big(\varphi(0)\big) \cup (p_2 - p_1) =
(p_2-p_1) \Big(1 + \varphi(0) z^{-1} + O(z^{-2}) \Big)$$ We conclude that $\varphi(0) = 0$. Thus: $$\begin{gathered}
\label{eq:almost_there}
\tilde{J}_{{{\boldsymbol{e}}},{\mathcal{V}}_G}(0) \cup (p_2 - p_1) =
\sum_{l=0}^\infty \sum_{m=0}^\infty \sum_{n=0}^\infty
(-1)^{l+m} q_1^{l+m} q_2^n
{
\prod_{k=1}^{k=l+m} (\lambda + p_1 + p_2 + k z)
\over
\prod_{k=1}^{k=l} (p_1 + k z)^4
\prod_{k=1}^{k=m} (p_2 + k z)^4
} \times \\
{
\prod_{k =-\infty}^{k=0} p_3-p_1 + k z
\over
\prod_{k=-\infty}^{k=n-l} p_3-p_1 + k z
}
{
\prod_{k = -\infty}^{k=0} p_3-p_2 + k z
\over
\prod_{k=-\infty}^{k= n-m} p_3-p_2 + k z
}
\big(p_2-p_1 + (m-l) z \big)\end{gathered}$$
To extract the quantum period $G_{{\mathrm{MW}^{4}_{11}}}$ from the twisted $J$-function $J_{{{\boldsymbol{e}}},{\mathcal{V}}_G}(0)$, we proceed as in [@QC105 Example D.8]: we take the non-equivariant limit, extract the component along the unit class $1 \in H^\bullet(X {/\!\!/}G;{\mathbb{Q}})$, set $z=1$, and set $Q^\beta = t^{\langle \beta, {-K} \rangle}$ where $K = K_{{\mathrm{MW}^{4}_{11}}}$. Thus we consider the right-hand side of , take the non-equivariant limit, extract the coefficient of $p_2-p_1$, set $z=1$, and set $q_1 = q_2 =2t$, obtaining: $$\begin{gathered}
G_{{\mathrm{MW}^{4}_{11}}}(t) =
\sum_{l=0}^\infty \sum_{m=0}^\infty \sum_{n=\max(l,m)}^\infty
(-1)^{l+m}
t^{2l+2m+2n}
\frac
{
(l+m)!
}
{
(l!)^4 (m!)^4 (n-l)! (n-m)!
}
\Big(1 + (m-l)(H_{n-m}-4H_m)\Big)
\\
+
\sum_{l=0}^\infty \sum_{m=l+1}^\infty \sum_{n=l}^{m-1}
(-1)^{l+n}
t^{2l+2m+2n}
\frac
{
(l+m)!
(m-n-1)!
}
{
(l!)^4 (m!)^4 (n-l)!
}
(m-l)\end{gathered}$$
The quantum period of ${\mathbb{P}}\big(\Omega^2_{{\mathbb{P}}(V)}(2)\big)$ can also be computed using Strangeway’s reconstruction theorem for the quantum cohomology of Fano bundles [@Strangeway Theorem 1]. Thus the quantum period of ${\mathrm{MW}^{4}_{11}}$ can be derived from this result together with the Quantum Lefschetz theorem. The Gromov–Witten invariants required as input to the reconstruction theorem can be computed via [@Strangeway Lemma 1], using Schubert calculus on $\operatorname{Gr}(2,4)$ and intersection numbers in ${\mathbb{P}}^3$.
### ${\mathrm{MW}^{4}_{12}}$
\[sec:MW\^4\_12\]
This is the blow-up of the quadric $Q^4$ along a line. Consider the toric variety $F$ with weight data: $$\begin{array}{rrrrrrrl}
\multicolumn{1}{c}{s_0} &
\multicolumn{1}{c}{s_1} &
\multicolumn{1}{c}{s_2} &
\multicolumn{1}{c}{s_3} &
\multicolumn{1}{c}{x} &
\multicolumn{1}{c}{x_4} &
\multicolumn{1}{c}{x_5} & \\
\cmidrule{1-7}
1 & 1 & 1 & 1 & -1 & 0 & 0 & \hspace{1.5ex} L\\
0 & 0 & 0 & 0 & 1 & 1 & 1 & \hspace{1.5ex} M \\
\end{array}$$ and $\operatorname{\overline{Amp}}F = \langle L, M\rangle$. The morphism $F \to {\mathbb{P}}^5$ that sends (contravariantly) the homogeneous co-ordinate functions $[x_0,x_1,\dots,x_5]$ to $[xs_0, xs_1, x s_2, x s_3, x_4, x_5]$ blows up the line $(x_0=x_1=x_2=x_3=0)$ in ${\mathbb{P}}^5$, and ${\mathrm{MW}^{4}_{12}}$ is the proper transform of a quadric containing this line. Thus ${\mathrm{MW}^{4}_{12}}$ is a member of $|L+M|$ in the toric variety $F$. We have:
- $-K_F=3L+3M$ is ample, so that $F$ is a Fano variety;
- ${\mathrm{MW}^{4}_{12}}\sim L+M$ is ample;
- $-(K_F+L+M)\sim 2L+2M$ is ample.
Theorem \[thm:toric\_ci\_mirror\] yields: $$G_{{\mathrm{MW}^{4}_{12}}}(t) = \sum_{l=0}^\infty \sum_{m=l}^\infty \frac{(l+m)!}{(l!)^4(m-l)!(m!)^2} t^{2l+2m}$$
### ${\mathrm{MW}^{4}_{13}}$
\[sec:MW\^4\_13\]
This is the projective bundle ${\mathbb{P}}_{Q^3}\big({\mathcal{O}}(1) \oplus {\mathcal{O}}\big)$ or, equivalently, a member of $|2L|$ in the toric variety $F$ with weight data: $$\begin{array}{rrrrrrrl}
\multicolumn{1}{c}{x_0} &
\multicolumn{1}{c}{x_1} &
\multicolumn{1}{c}{x_2} &
\multicolumn{1}{c}{x_3} &
\multicolumn{1}{c}{x_4} &
\multicolumn{1}{c}{u} &
\multicolumn{1}{c}{v} & \\
\cmidrule{1-7}
1 & 1 & 1 & 1 & 1 & 0 & -1 & \hspace{1.5ex} L\\
0 & 0 & 0 & 0 & 0 & 1 & 1 & \hspace{1.5ex} M \\
\end{array}$$ and $\operatorname{\overline{Amp}}F = \langle L, M\rangle$. We have:
- $-K_F=4L+2M$ is ample, that is $F$ is a Fano variety;
- ${\mathrm{MW}^{4}_{13}} \sim 2L$ is nef;
- $-(K_F+2L)\sim 2L+2M$ is ample.
The projection $[x_0:x_1:x_2:x_3:x_4:x_5:u:v] \mapsto [x_0:x_1:x_2:x_3:x_4:x_5]$ exhibits $F$ as the scroll ${\mathbb{P}}_{{\mathbb{P}}^4}\big({\mathcal{O}}(1) \oplus {\mathcal{O}}\big)$ over ${\mathbb{P}}^4$, and passing to a member of $|2L|$ restricts this scroll to $Q^3 \subset {\mathbb{P}}^4$. Theorem \[thm:toric\_ci\_mirror\] yields: $$G_{{\mathrm{MW}^{4}_{13}}}(t) = \sum_{l=0}^\infty \sum_{m=l}^\infty \frac{(2l)!}{(l!)^5 m! (m-l)!} t^{2l+2m}$$
### ${\mathrm{MW}^{4}_{14}}$
\[sec:MW\^4\_14\]
This is the product ${\mathbb{P}}^1 \times {\mathbb{P}}^3$. Combining Theorem \[thm:products\] with [@QC105 Example G.1] and [@QC105 §1] yields: $$G_{{\mathrm{MW}^{4}_{14}}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty \frac{t^{2l+4m}}{(l!)^2 (m!)^4}$$
### ${\mathrm{MW}^{4}_{15}}$
\[sec:MW\^4\_15\]
This is the projective bundle ${\mathbb{P}}_{{\mathbb{P}}^3}\big({\mathcal{O}}(1) \oplus {\mathcal{O}}(-1)\big)$, or in other words, the toric variety with weight data: $$\begin{array}{rrrrrrl}
\multicolumn{1}{c}{x_0} &
\multicolumn{1}{c}{x_1} &
\multicolumn{1}{c}{x_2} &
\multicolumn{1}{c}{x_3} &
\multicolumn{1}{c}{u} &
\multicolumn{1}{c}{v} \\
\cmidrule{1-6}
1 & 1 & 1 & 1 & 0 & -2 & \hspace{1.5ex} L\\
0 & 0 & 0 & 0 & 1 & 1 & \hspace{1.5ex} M\\
\end{array}$$ and $\operatorname{\overline{Amp}}F = \langle L, M\rangle$. Theorem \[thm:toric\_mirror\] yields: $$G_{{\mathrm{MW}^{4}_{15}}}(t) = \sum_{l=0}^\infty \sum_{m=2l}^\infty \frac{t^{2l+2m}}{(l!)^4 m! (m-2l)!}$$
### ${\mathrm{MW}^{4}_{16}}$
\[sec:MW\^4\_16\]
This is the product ${\mathbb{P}}^1 \times W^3$, where $W^3 \subset {\mathbb{P}}^2 \times {\mathbb{P}}^2$ is a divisor of bidegree $(1,1)$. Theorem \[thm:toric\_ci\_mirror\] yields: $$G_{{\mathrm{MW}^{4}_{16}}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty \sum_{n=0}^\infty \frac{(m+n)!}{(l!)^2 (m!)^3 (n!)^3} t^{2l+2m+2n}$$
### ${\mathrm{MW}^{4}_{17}}$
\[sec:MW\^4\_17\]
This is the product ${\mathbb{P}}^1 \times B^3_7$, where $B^3_7$ is the blow-up of ${\mathbb{P}}^3$ at a point. Note that $B^3_7$ is the projective bundle ${\mathbb{P}}_{{\mathbb{P}}^2}({\mathcal{O}}\oplus {\mathcal{O}}(-1))$. It follows that ${\mathrm{MW}^{4}_{17}}$ is the toric variety with weight data: $$\begin{array}{rrrrrrrl}
\multicolumn{1}{c}{x_0} &
\multicolumn{1}{c}{x_1} &
\multicolumn{1}{c}{y_0} &
\multicolumn{1}{c}{y_1} &
\multicolumn{1}{c}{y_2} &
\multicolumn{1}{c}{u} &
\multicolumn{1}{c}{v} \\
\cmidrule{1-7}
1 & 1 & 0 & 0 & 0 & 0 & 0 & \hspace{1.5ex} L\\
0 & 0 & 1 & 1 & 1 & 0 & -1 & \hspace{1.5ex} M\\
0 & 0 & 0 & 0 & 0 & 1 & 1 & \hspace{1.5ex} N\\
\end{array}$$ and $\operatorname{\overline{Amp}}F = \langle L, M, N\rangle$. Theorem \[thm:toric\_mirror\] yields: $$G_{{\mathrm{MW}^{4}_{17}}}(t) = \sum_{l=0}^\infty \sum_{m=0}^\infty \sum_{n=m}^\infty \frac{t^{2l+2m+2n}}{(l!)^2 (m!)^3 n! (n-m)!}$$
### ${\mathrm{MW}^{4}_{18}}$
\[sec:MW\^4\_18\]
This is the product ${\mathbb{P}}^1 \times {\mathbb{P}}^1 \times {\mathbb{P}}^1 \times {\mathbb{P}}^1$. Combining Theorem \[thm:products\] with [@QC105 Example G.1] yields: $$G_{{\mathrm{MW}^{4}_{18}}}(t) = \sum_{k=0}^\infty \sum_{l=0}^\infty \sum_{m=0}^\infty \sum_{n=0}^\infty \frac{t^{2k+2l+2m+2n}}{(k!)^2(l!)^2(m!)^2(n!)^2}$$
Four-Dimensional Fano Toric Manifolds {#sec:toric}
=====================================
Four-dimensional Fano toric manifolds were classified by Batyrev [@Batyrev] and Sato [@Sato]. [Ø]{}bro classified Fano toric manifolds in dimensions 2–8 [@Obro] and, to standardise notation, we will write ${\text{\rm B\O S}^{4}_{k}}$ for the $k$th four-dimensional Fano toric manifold in [Ø]{}bro’s list. ${\text{\rm B\O S}^{4}_{k}}$ is the $(23+k)$th Fano toric manifold in the Graded Ring Database [@GRDB], as the list there is the concatenation of [Ø]{}bro’s lists in dimensions 2–8. We can compute the quantum periods of the ${\text{\rm B\O S}^{4}_{k}}$ using Theorem \[thm:toric\_mirror\]; the first few Taylor coefficients of their regularized quantum periods can be found in the tables in the Appendix.
Product Manifolds and Other Index $1$ Examples {#sec:other_index_1}
==============================================
Quantum periods for one-, two- and three-dimensional Fano manifolds were computed in [@QC105]. Combining these results with Theorem \[thm:products\] allows us to compute the quantum period of any four-dimensional Fano manifold that is a product of lower-dimensional manifolds. Many of these examples have Fano index $r=1$.
In his thesis [@Strangeway:thesis], Strangeway determined the quantum periods of two four-dimensional Fano manifolds of index $r=1$ that have not yet been discussed. These manifolds arise as complete intersections in the $9$-dimensional projective bundle $F = {\mathbb{P}}\big(\Omega^2_{{\mathbb{P}}^4}(2) \big)$. Let $\pi \colon F \to {\mathbb{P}}^4$ denote the canonical projection, let $p \in H^2(F)$ be the first Chern class of $\pi^\star {\mathcal{O}}_{{\mathbb{P}}^4}(1)$, and let $\xi \in H^2(F)$ be the first Chern class of the tautological bundle ${\mathcal{O}}_F(1)$. The manifold $F$ is Fano of Picard rank $2$, with nef cone generated by $\{\xi,p\}$ and ${-K_F} = 6\xi + 2p$. Let: $$\begin{aligned}
& \text{${\mathrm{Str}}_1 \subset F$ denote a complete intersection of five divisors of type $\xi$} \\
& \text{${\mathrm{Str}}_2 \subset F$ denote a complete intersection of four divisors of type $\xi$ and a divisor of type $p$} \\
\intertext{We consider also:}
& \text{${\mathrm{Str}}_3 \subset F$, a complete intersection of four divisors of type $\xi$ and a divisor of type $\xi + p$} \end{aligned}$$ which was unaccountably omitted from [@Strangeway:thesis].
The manifolds ${\mathrm{Str}}_k$, $k \in \{1,2,3\}$, each have Picard rank two. To see this, observe that the ambient manifold $F$ is the blow-up of ${\mathbb{P}}^9$ along $\operatorname{Gr}(2,5)$, where $\operatorname{Gr}(2,5) \to {\mathbb{P}}^9$ is the Plücker embedding [@Strangeway]; the blow-up $F \to {\mathbb{P}}^9$ and the projection $\pi \colon F \to {\mathbb{P}}^4$ are the extremal contractions corresponding to the two extremal rays in ${\overline{\operatorname{NE}}}(F)$. Thus ${\mathrm{Str}}_1$ is the blow-up of ${\mathbb{P}}^4$ along an elliptic curve $E_5 \subset {\mathbb{P}}^4$ of degree $5$. Consider the five-dimensional Fano manifold $F_5$ given by the complete intersection of four divisors of type $\xi$ in $F$. Then $F_5$ is the blow-up of ${\mathbb{P}}^5$ along a del Pezzo surface $S_5$ of degree $5$; in particular, $F_5$ has Picard rank two. ${\mathrm{Str}}_3$ is an ample divisor (of type $\xi + p$) in $F_5$, so the Picard rank of ${\mathrm{Str}}_3$ is also two. The manifold ${\mathrm{Str}}_2$ is a divisor in $F_5$ of type $p$, and $F_5$ arises as the closure of the graph of the map ${\mathbb{P}}^5 \to {\mathbb{P}}^4$ given by the $5$-dimensional linear system of quadrics passing through $S_5$. This exhibits ${\mathrm{Str}}_2$ as the blow-up of a smooth four-dimensional quadric $Q^4$ along $S_5$, which implies that the Picard rank of ${\mathrm{Str}}_2$ is two.
We can compute the quantum periods of ${\mathrm{Str}}_k$, $k \in \{1,2,3\}$, by observing that a complete intersection in $F$ of five divisors of type $\xi$ and one divisor of type $p$ is a three-dimensional Fano manifold ${\mathrm{MM}^3_{2\text{--}17}}$, ‘unsectioning’ to compute the quantum period for $F$, and then applying the quantum Lefschetz theorem to compute the quantum periods for ${\mathrm{Str}}_1$, ${\mathrm{Str}}_2$, and ${\mathrm{Str}}_3$. Recall the definition of the $J$-function $J_X(t,z)$ from [@Coates--Givental equation 11]. The identity component of the $J$-function of ${\mathrm{MM}^3_{2\text{--}17}}$ is: $$\label{eq:equate_me_1}
e^{-q_1 - q_2}
\sum_{l_1,l_2,l_3\geq 0}
(-q_1)^{l_1+l_2} q_2^{l_3}
\frac
{
(l_1+l_2)! (l_1+l_3)!(l_2+l_3)! (l_1+l_2+l_3)!
}
{
(l_1!)^4 (l_2!)^4 (l_3!)^4 z^{l_1+l_2+l_3}
}
\Big(1 + (l_2-l_1)(H_{l_2+l_3} - 4H_{l_2})\Big)$$ where $q_1$, $q_2$ are generators of the Novikov ring for ${\mathrm{MM}^3_{2\text{--}17}}$ dual respectively to $\xi$ and $p$; see [@QC105 §34]. The identity component of the $J$-function of $F$ takes the form: $$\sum_{l=0}^\infty \sum_{m=0}^\infty c_{l,m} z^{-6l-2m} q_1^l q_2^m$$ for some coefficients $c_{l,m} \in {\mathbb{Q}}$. The Quantum Lefschetz theorem implies (cf. [@QC105 §D.1]) that the identity component of the $J$-function of ${\mathrm{MM}^3_{2\text{--}17}}$ is equal to: $$\label{eq:equate_me_2}
e^{-c_{1,0} q_1 -c_{0,1} q_2} \sum_{l=0}^\infty \sum_{m=0}^\infty (l!)^5 m! c_{l,m} z^{-l-m} q_1^l q_2^m$$ and it is known that $c_{1,0} = 1$ and $c_{0,1} = 0$ [@Strangeway §5.1]. Equating and determines the $c_{l,m}$: $$c_{l,m}=
\sum_{i=0}^{l} \sum_{j=0}^{m}
(-1)^{j+l}
\frac
{
(m+l-i-j)!(i+m-j)! (m+l-j)!
}
{
((l-i)!)^4 (i!)^4 ((m-j)!)^4j!m!(l!)^4
}
\Big(1 + (2i-l)(H_{i+m-j} - 4H_{i})\Big)$$ The Quantum Lefschetz theorem now gives that: $$\begin{aligned}
& G_{{\mathrm{Str}}_1}(t) = e^{-t}
\sum_{l=0}^\infty
\sum_{m=0}^\infty
(l!)^5 c_{l,m} t^{l+2m} \\
& G_{{\mathrm{Str}}_2}(t) =
\sum_{l=0}^\infty
\sum_{m=0}^\infty
(l!)^4 m! \, c_{l,m} t^{2l+m} \\
& G_{{\mathrm{Str}}_3}(t) =
e^{-t} \sum_{l=0}^\infty
\sum_{m=0}^\infty
(l!)^4 (l+m)! \, c_{l,m} t^{l+m} \end{aligned}$$
Numerical Calculations of Quantum Differential Operators {#sec:qdes}
========================================================
As discussed in §\[sec:methodology\], the regularized quantum period ${\widehat{G}}_X(t)$ of a Fano manifold $X$ satisfies a differential equation: $$\begin{aligned}
\label{eq:regularized_qde_again}
L_X {\widehat{G}}_X \equiv 0 && L_X = \sum_{k=0}^{k=N} p_k(t) D^k\end{aligned}$$ called the regularized quantum differential equation. Here the $p_m$ are polynomials and $D = t \frac{d}{dt}$. The regularized quantum differential equation for $X$ coincides with the (unregularized) quantum differential equation for an anticanonical Calabi–Yau manifold $Y \subset X$; the study of the regularized quantum period from this point of view was pioneered by Batyrev–Ciocan-Fontanine–Kim–van Straten [@BCFKvS:1; @BCFKvS:2]. The differential equation is expected to be Fuchsian, and the local system of solutions to $L_X f \equiv 0$ is expected to be of *low ramification* in the following sense.
Let $S \subset {\mathbb{P}}^1$ a finite set, and ${\mathbb{V}}\to {\mathbb{P}}^1 \setminus S$ a local system. Fix a basepoint $x \in {\mathbb{P}}^1 \setminus S$. For $s \in S$, choose a small loop that winds once anticlockwise around $s$ and connect it to $x$ via a path, thereby making a loop $\gamma_s$ about $s$ based at $x$. Let $T_s \colon {\mathbb{V}}_x \to {\mathbb{V}}_x$ denote the monodromy of ${\mathbb{V}}$ along $\gamma_s$. The *ramification* of ${\mathbb{V}}$ is: $$\operatorname{rf}({\mathbb{V}}) := \sum_{s \in S} \dim\Big({\mathbb{V}}_x/{{\mathbb{V}}_x}^{\!\!\!T_s}\Big)$$ The *ramification defect* of ${\mathbb{V}}$ is the quantity $\operatorname{rf}({\mathbb{V}}) - 2\operatorname{rk}({\mathbb{V}})$. Non-trivial irreducible local systems ${\mathbb{V}}\to {\mathbb{P}}^1 \setminus S$ have non-negative ramification defect; this gives a lower bound for the ramification of ${\mathbb{V}}$. A local system of ramification defect zero is called *extremal*.
The *ramification* (respectively *ramification defect*) of a differential operator $L_X$ is the ramification (respectively ramification defect) of the local system of solutions $L_X f \equiv 0$.
The *quantum differential operator* for a Fano manifold $X$ is the operator $L_X \in {\mathbb{Q}}[t]\langle D \rangle$ such that $L_X {\widehat{G}}_X \equiv 0$ which is of lowest order in $D$ and, among all such operators of this order, is of lowest degree in $t$. (This defines $L_X$ only up to an overall scalar factor, but this suffices for our purposes.)
Suppose that each of the polynomials $p_0,\ldots,p_N$ are of degree at most $r$, and write: $$\begin{aligned}
L_X = \sum_{k=0}^{k=N} p_k(t) D^k && p_k(t) = \sum_{l=0}^r a_{kl} t^l\end{aligned}$$ The differential equation $L_X {\widehat{G}}_X \equiv 0$ gives a system of linear equations for the coefficients $a_{kl}$ which, given sufficiently many terms of the Taylor expansion of ${\widehat{G}}_X$, becomes over-determined. Given *a priori* bounds on $N$ and $r$, therefore, we could compute the quantum differential operator $L_X$ by calculating sufficiently many terms in the Taylor expansion. In general we do not have such bounds, but nonetheless by ensuring the linear system for $(a_{kl})$ is highly over-determined we can be reasonably confident that the operator $L_X$ which we compute is correct. In addition, since $L_X$ is expected to correspond under mirror symmetry to a Picard–Fuchs differential equation for the mirror family, $L_X$ is expected to be of Fuchsian type. This is an extremely delicate condition on the coefficients $(a_{kl})$, and it can be checked by exact computation.
We computed candidate quantum differential operators $L_X$ for all four-dimensional Fano manifolds of Fano index $r>1$, and checked the Fuchsian condition in each case. The operators $L_X$, together with their ramification defects and the log-monodromy data $\{\log T_s : s \in S\}$ in Jordan normal form, can be found in Appendix \[appendix:qdes\]. In $24$ cases, the local system of solutions to the regularized quantum differential equation is extremal, and in the remaining $11$ cases it is of ramification defect $1$.
To compute the ramification of $L_X$, we follow Kedlaya [@Kedlaya §7.3]. This involves only linear algebra over a splitting field for $p_N(t)$—recall that every singular point of $L_X$ occurs at a root of $p_N(t)$—and thus can be implemented using exact (not numerical) computer algebra. For this we use Steel’s symbolic implementation of $\overline{{\mathbb{Q}}}$ in the computational algebra system Magma [@Magma; @Steel].
Source Code {#source-code .unnumbered}
-----------
This paper is accompanied by full source code, written in Magma. See the included file [README.txt]{} for usage instructions. The source code, but not the text of this paper, is released under a Creative Commons CC0 license [@CC0]: see the included file [COPYING.txt]{} for details. If you make use of the source code in an academic or commercial context, you should acknowledge this by including a reference or citation to this paper.
Acknowledgements {#acknowledgements .unnumbered}
----------------
We thank Alessio Corti for a number of very useful conversations. This research was supported by a Royal Society University Research Fellowship (TC); ERC Starting Investigator Grant number 240123; the Leverhulme Trust; grant MK-1297.2014.1; AG Laboratory NRU-HSE, RF government grant ag. 11.G34.31.0023; Grant of Leading Scientific Schools (N.Sh. 2998.2014.1); and EPSRC grant EP/I008128/1.
Regularized Quantum Period Sequences {#appendix:periods}
====================================
In this Appendix we record the description, degree, and Picard rank $\rho_X$ for each of the four-dimensional Fano manifolds $X$ considered in this paper, together with the first few Taylor coefficients $\alpha_d$ of the regularized quantum period: $${\widehat{G}}_X(t) = \sum_{d=0}^\infty \alpha_d t^d$$ The tables are divided by Fano index $r$. We include only coefficients $\alpha_d$ with $d \equiv 0 \bmod r$, since coefficients $\alpha_d$ with $d \not \equiv 0 \bmod r$ are zero. Notation is as follows:
- ${\mathbb{P}}^n$ denotes $n$-dimensional complex projective space;
- $Q^n$ denotes a quadric hypersurface in ${\mathbb{P}}^{n+1}$;
- ${\mathrm{FI}^{4}_{k}}$ is as in §\[sec:index\_3\] above;
- ${V^{4}_{k}}$ is as in §\[sec:index\_2\_rank\_1\] above;
- ${\mathrm{MW}^{4}_{k}}$ is as in §\[sec:index\_2\_higher\_rank\] above;
- ${\text{\rm B\O S}^{4}_{k}}$ is as in §\[sec:toric\] above;
- ${\mathrm{Str}}_{k}$ is as in §\[sec:other\_index\_1\] above;
- ${S^2_{k}}$ denotes the del Pezzo surface of degree $k$;
- $F_1$ denotes the Hirzebruch surface ${\mathbb{P}}\big({\mathcal{O}}_{{\mathbb{P}}^1}(-1) \oplus {\mathcal{O}}_{{\mathbb{P}}^1}\big)$;
- ${V^{3}_{k}}$ denotes the three-dimensional Fano manifold of Picard rank $1$, Fano index $1$, and degree $k$;
- ${B^{3}_{k}}$ denotes the three-dimensional Fano manifold of Picard rank $1$, Fano index $2$, and degree $8k$;
- ${\mathrm{MM}^3_{\rho\text{--}k}}$ denotes the $k$th entry in the Mori–Mukai list of three-dimensional Fano manifolds of Picard rank $\rho$ [@Mori--Mukai:Manuscripta; @Mori--Mukai:81; @Mori--Mukai:84; @Mori--Mukai:erratum; @Mori--Mukai:fanoconf]. We use the the ordering as in [@QC105], which agrees with the original papers of Mori–Mukai except when $\rho=4$.
We prefer to express manifolds as products of lower-dimensional manifolds where possible, so for example ${\text{\rm B\O S}^{4}_{122}}$ is the product ${\mathbb{P}}^1 \times {\mathbb{P}}^3$, but we refer to this space as ${\mathbb{P}}^1 \times {\mathbb{P}}^3$ rather than ${\text{\rm B\O S}^{4}_{122}}$. The tables for Fano index $r$ with $r \in \{2,3,4,5\}$ are complete. The table for $r=1$ is very far from complete.
=
It appears from Table 5 as if the regularized quantum period might coincide for the pairs $\{{\text{\rm B\O S}^{4}_{6}},{\text{\rm B\O S}^{4}_{41}}\}$ and $\{{\text{\rm B\O S}^{4}_{35}}, {\text{\rm B\O S}^{4}_{88}}\}$. This is not the case. The coefficients $\alpha_8$, $\alpha_9$ in these cases are:
$X$ $\alpha_8$ $\alpha_9$
------------------------------- ------------ ------------
${\text{\rm B\O S}^{4}_{6}}$ 14350 87360
${\text{\rm B\O S}^{4}_{35}}$ 32830 227640
${\text{\rm B\O S}^{4}_{41}}$ 10990 102480
${\text{\rm B\O S}^{4}_{88}}$ 32830 212520
Thus 10 terms of the Taylor expansion of the regularized quantum period suffice to distinguish all of the four-dimensional Fano manifolds considered in this paper.
Quantum Differential Operators for Four-Dimensional Fano Manifolds of Index $r>1$: Numerical Results {#appendix:qdes}
====================================================================================================
In this Appendix we record the quantum differential operators for all four-dimensional Fano manifolds of Fano index $r>1$. These were computed numerically, as described in §\[sec:qdes\], from $500$ terms of the Taylor expansion of the quantum period. They pass a number of strong consistency checks, and so we are reasonably confident that they are correct, but this has not been rigorously proven. We record also the local log-monodromies and ramification defect for the quantum local system, that is, for the local system of solutions to the regularized quantum differential equation. These are derived using exact computer algebra from the (numerically computed) operators $L_X$, as described in §\[sec:qdes\].
|
---
abstract: 'Massive and extended dark halos can inhibit the formation of long tidal tails in galaxy collisions. We examine this effect using an extensive survey of simulations with different dark halo potentials to constrain halo properties of interacting galaxies. These constraints are compared to other observational limits and theoretical predictions of halo structure. The dark halos predicted by $\Omega=1$ cosmological models like CDM are too massive and extended to produce the long tidal tails seen in nearby galaxy collisions. There is also a conflict with the halo potentials inferred from satellite kinematics; such halos would likewise inhibit tail formation in galaxy collisions.'
author:
- John Dubinski
- Lars Hernquist
- 'J. C. Mihos'
title: Constraining Dark Halo Potentials with Tidal Tails
---
Introduction
============
When disk galaxies collide, the strong mutual tidal fields of their interaction momentarily transform the disks into open, bisymmetric spirals and catapult the outer disk stars in each galaxy onto long, arcing trajectories commonly called tidal tails and bridges (Toomre & Toomre 1972; Wright 1972). Tidal bridges connect the merging pair and tidal tails continue to lengthen and thin out after a collision. Some tail stars may escape the system but most eventually fall back into the merger remnant. The currently interacting pairs in the Antennae (NGC 4038), the Mice (NGC 4676) and the merged NGC 7252 for example, all have tails which extend to distances $\sim$50-100 kpc from the galaxies (Hibbard 1995). The Superantennae (IRAS 19254-7245) is an extreme case in which the tails span 350 kpc from tip to tip (e.g., Mirabel, Lutz, & Maza 1991).
A tidal tail can be thought of as a single trajectory shaped by the potential of the interacting pair since it arises from stars in the disk with similar orbital properties. Tidal tails can extend far out into the dark halo, perhaps out to $>$200 kpc in 3 dimensions. Tails were therefore recognized as a possible probe of the dark matter distribution in galaxies (Faber and Gallagher 1979) and their formation in simulations with dark halos has been the subject of much numerical work during the past decade (e.g. Negroponte & White 1983; White 1982; Barnes 1988). The suggestion is that massive dark halos can inhibit the formation of tidal tails in two ways. First, the deeper potentials could lead to larger relative encounter velocities which are more impulsive and less resonant and therefore less effective at giving disk stars the velocity perturbation they need to be ejected as tails. Second, the deeper and steeper potential wells of more massive halos could trap tail stars by shortening their turn-around radius after ejection. Both effects would combine to make the maximum length of tidal tails smaller in the presence of a more massive dark halo. The observed [*maximum*]{} length of tails is therefore telling us something about the mass profile of the dark potential.
The length and kinematics of tidal tails depend on other factors besides the dark halo structure so simulation surveys are required to understand how they form in different circumstances. Other factors include the epoch of the collision, disk orientations, orbit orientation, orbital energy, galaxy mass ratio, and pericentric distance. The only way to study dark halo potential constraints is therefore to model the tail formation process as a function of dark halo parameters in galaxy collision N-body simulations (Dubinski, Mihos & Hernquist 1996 \[DMH96\]). A complete parameter survey is out of reach so in our studies we have focussed on the ideal case of colliding equal mass, co-planar disks in a direct encounter. This is the geometry of the first galaxy collision ever simulated, namely that done by Holmberg in 1941 ([*before*]{} the computer age!). These encounters are strongly resonant and are the most effective at transferring energy and angular momentum to the tails and so the lengths of tails in these collisions probably represent an upper limit for all collision geometries. We have also examined more specific geometries which would produce facsimiles of the Antennae and NGC 7252 using different mass models (DMH96; Mihos, Dubinski, Hernquist 1998).
Previous Results
================
In previous work, we simulated galaxy collisions using four mass models labelled A, B, C and D with progressively more massive and more extended halos (DMH96). The models are self-consistent realizations containing an exponential disk, a truncated King model bulge and King model dark halo, constructed using the method of Kuijken & Dubinski (1995). By construction, the disk and bulge mass distributions are held fixed and the inner rotation curve is kept nearly flat within 5 scale lengths in the model sequence. With this assumption, the mass ratio of dark matter to stars (disk + bulge) is 4:1, 8:1, 16:1 and 30:1 for models A, B, C, and D. Of course, more extended models with this mass ratio are also possible when dark halos have smaller circular velocities but we have only examined those with rotation curves which are approximately flat out to the halo truncation radius.
The results of this study are given in DMH96 and the main points are summarized here. Galaxy collisions with models A and B have no difficulty ejecting long tails ($l > 100$ kpc) while model C encounters eject tails of only moderate length ($l \sim 50-100$ kpc) and model D collisions produce short tails ($l < 50$ kpc). When these simulations are carried through to the final merger to a state similar to NGC 7252, models A and B still have long tails while all of the stars in the tails in model C and D galaxies have fallen back into the merger remnant. On these grounds, we can rule out galaxies with mass distributions like models C and D as precursors of the Antennae, the Mice and NGC 7252. Collision geometries that produce facsimiles of the Antennae and NGC 7252 worked well with the low mass models A and B but failed with models C and D. Also, kinematic comparisons of NGC 7252 facsimiles to the real observations also rule out model C and D even when we include an “HI gas” disk (represented by test particles) extending out to 10 scale lengths (MDH98). Since models C and D had dark halo mass distributions very similar to those predicted from CDM or other $\Omega=1$ cosmological models, we argued that a lower $\Omega$ cosmology might be required to explain the observations. Indeed, small $\Omega$ cosmologies produce concentrated dark halos that look more like those in models A and B (Navarro et al. 1997).
New Results
===========
How representative are these 4 models? They were constructed under the assumption that the flat inner rotation curve continues to large radii. Clearly, there are many other possibilities. For example, at large radii in an isothermal halo, $\phi \sim v_c^2 \log r$ so $\partial \phi/\partial r \sim
v_c^2/r$. If the disk contributes considerably more mass to the inner rotation curve the [*asymptotic*]{} value of $v_c$ may be smaller and so the outer potential will be shallower. Tidal tails may therefore be expelled to larger distances in this case since the potentials are not as effective a trap. This is just one example of how things can stray from our first 4 models.
The only way to understand the effects on tidal tail formation is to widen the survey to include other dark halo potentials. We have done just that by looking at 84 new models (Dubinski, Mihos, Hernquist \[DMH98\]). We have parameterized the dark halo using two models which have been motivated by cosmological simulations of dark halo formation: the Hernquist (H) profile (Hernquist 1990; Dubinski & Carlberg 1991) and the NFW profile (Navarro, Frenk & White 1996). Both profiles have two free parameters, a mass (or effective mass within some radius in the case of the NFW profile since total mass formally diverges) and a scale length. The H-profile is: $$\rho(r) = \frac{M_H r_H}{2 \pi} \frac{1}{r(r+r_H)^3},$$ where $M_H$ is the H mass and $r_H$ is the H scale length. The NFW profile is: $$\rho(r) = \frac{M_s}{4 \pi} \frac{1}{r(r+r_s)^2},$$ where $M_s$ is the NFW effective mass (mass within 5.3 $r_s$ ) and $r_s$ is the NFW scale length. Both profiles have peak circular velocities which we use to characterize them. For the H model $v_H^2 = 0.5 GM_H/r_H$ at $r=r_H$ and for the NFW model $v_s^2 = 0.46 G M_s/r_s$ at $r=2.16 r_s$. The velocity maxima of the two dark halo rotation curves coincide when we use the mapping $r_s = 0.46 r_H$ and $M_s = 0.54 M_H$. The NFW model is more extended than the H model and has a steeper potential at larger radii, however, within the region where tails form the potentials are very similar. We found that there was very little difference in the results for both models so we only present the results of the NFW models here, although a more extensive comparison will be given elsewhere (DMH98).
In our previous work, we simulated the collisions with self-consistent treecode N-body simulations. The large number of models in this extended study precludes the use of self-consistent simulations at the moment so we use instead a restricted, 3-body method somewhat similar to Toomre & Toomre’s (1972) original scheme. We can get away with this since the tidal tails are essentially a kinematic phenomenon. The technique will be described elsewhere in detail (DMH98) but in brief it works as follows. The trajectories of the two interacting galaxies are calculated assuming an interaction potential with zero total energy which accounts for the extended mass distribution and is somewhat different than the parabolic orbits of point masses used by Toomre & Toomre. The orbital decay of the galaxies is treated used Chandrasekhar’s dynamical friction formula (Binney & Tremaine 1987) using a value of the Coulomb logarithm, $\ln \Lambda = 2$, found to fit the observed orbital decay well in a small number of self-consistent simulations. Finally, a test-particle realization of the orbital distribution of the disk is generated self-consistently according to the total galaxy potential for the models using the technique of Kuijken & Dubinski (1995). The test particles are then integrated in the time-dependent gravitational field of the two rigid potentials moving along the pre-calculated collision trajectory. This technique produces tidal tail morphologies and kinematics remarkably similar to self-consistent simulations, so we apply it generally to models in our survey.
Halos are parameterized in our study by their scale length and circular velocity maximum. We have chosen six halo scale-lengths ranging from $r_s=1.2$ to 11.6 and seven circular velocities ranging from $v_s = 0.5$ to 1.0 for a total of 42 models in the NFW study. We also looked at a similar set of 42 H models. The disk exponential scale length and mass are both unity, so the disk’s peak circular velocity is $v_d = 0.62$ in these units. The bulge has a mass of $0.5 M_d$ and is the main contributor to the rotation curve within 1 scale length. This range of models broadly covers the inferred properties of dark halos around exponential galactic disks of Hubble type Sa-b similar to the Milky Way. Figure \[fig-2\] shows the rotation curves within 10 scale-lengths.
Figure \[fig-3\] presents the results of a series of planar, prograde galaxy collisions.
Each collision is shown at the same time, $t=30$, after pericentric passage, corresponding to 5 disk orbital times at one scale length or 500 million years for the Milky Way. The size of each box is 80 scale lengths or about 300 kpc for the Milky Way. The main results are that galaxies with low-mass, compact halos expel long tidal tails in collisions similar to those in observed interacting pairs while galaxies with high-mass extended halos expel very short tails unlike the Antennae or NGC 7525. The region where halos are similar to CDM predictions also produce relatively short tails in accord with our earlier conclusions. Galaxies with extended halos having smaller peak circular velocities (upper right corner) can also produce long tidal tails and bridges since the galaxies separate significantly after their encounter. The potential is shallow enough to allow long tails to fly off to large distances but the low central density of the halo does not provide enough dynamical friction to slow down the galaxies significantly during their encounter in contrast to the models in the lower left corner where the central halo density is much higher. The interacting pair Arp 295 with its wide separation and long tails and connecting bridge has a similar morphology to galaxies in the upper right corner.
Discussion and Conclusions
==========================
This more extensive survey confirms our previous conclusion that collisions of galaxies with dark halos like CDM halos are ineffective at making long tidal tails. Halos which are compact and centrally concentrated with a range of circular velocities all make long tidal tails but extended halos can only make long tails if they have a small peak velocity i.e. a smaller contribution than the disk and bulge to the inner rotation curve. The successful galaxy models with extended halos are probably most similar to maximal and Bottema (1997) disk/halo models which are preferred by some rotation curve analysts. The main cosmological implication is that CDM and critical density universe models are not favored because their dark halos inhibit the formation of long tidal tails. More compact dark halos are predicted in low density universes, however, similar to models in the lower left of Figure 2 (Navarro et al. 1997).
The disagreement with the relatively large mass estimates in the Milky Way and external galaxies made through studies of satellite kinematics is harder to understand (e.g. Zaritsky et al. 1989; Kochanek 1996; Zaritsky & White 1994; Zaritsky et al. 1997). Perhaps the difference is due to a selection effect – galaxies chosen for satellite kinematics studies are well-isolated while interacting galaxies are obviously in pairs and not isolated. The different halos may reflect some cosmological variance resulting from differences in environment or initial conditions. In any case, the lower bounds of the confidence limits on halo masses from satellites overlap with range of galaxy models that make long tidal tails. Clearly, the systematic errors of these mass estimates should be studied in more detail, perhaps through comparison to cosmological simulations with adequate resolution to resolve satellites within their galactic halos. Galaxy interaction studies should also take their initial conditions from cosmological models instead of the idealized models examined here. Perhaps a consistent picture will be found soon and the tails will stop wagging the dogs.
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**MASS–LUMINOSITY RELATION AND\
PULSATIONAL PROPERTIES OF WOLF–RAYET STARS**
**Yu.A. Fadeyev[^1]**
*Institute of Astronomy of the Russian Academy of Sciences, Moscow*\
Submitted 12 May 2008
Evolution of Population I stars with initial masses ranging within $70M_\odot\le\mzams\le 130M_\odot$ is considered under various assumptions on the mass loss rate $\dot M$. The mass–luminosity relation of W–R stars is shown to be most sensitive to the mass loss rate $\dmhe$ during the helium burning phase. Together with the mass–luminosity relation obtained for all evolutionary sequences several more exact relations are determined for the constant ratios $0.5\le \fhe\le 3$, where $\fhe = \dot M/\dmhe$. Evolutionary models of W–R stars were used as initial conditions in hydrodynamic computations of radial nonlinear stellar oscillations. The oscillation amplitude is larger in W–R stars with smaller initial mass $\mzams$ or with lower mass loss rate $\dot M$ due to higher surface abundances of carbon and oxygen. In the evolving W–R star the oscilation amplitude decreases with decreasing stellar mass $M$ and for $M < 10M_\odot$ the sufficiently small nonlinear effects allow us to calculate the integral of the mechanical work $W$ done over the pulsation cycle in each mass zone of the hydrodynamical model. The only positive maximum on the radial dependence of $W$ is in the layers with temperature of $T\sim 2\times 10^5\K$ where oscillations are excited by the iron Z–bump $\kappa$–mechanism. Radial oscillations of W–R stars with mass of $M > 10M_\odot$ are shown to be also excited by the $\kappa$–mechanism but the instability driving zone is at the bottom of the envelope and pulsation motions exist in the form of nonlinear running waves propagating outward from the inner layers of the envelope.
*Key words*: stars – variable and peculiar
PACS numbers: 97.10.Cv; 97.10.Me; 97.10.Sj; 97.30.Eh
Introduction {#introduction .unnumbered}
============
During the core helium burning the structure of Population I stars with initial mass of $\mzams\ge 30M_\odot$ weakly depends on the radial distribution of the mean molecular weight $\mu$. This is due to the fact that radiation dominates in the internal energy, so that the hydrostatic equilibrium is governed by the gravity and by the gradient of radiation pressure whereas the gas pressure is negligible. Moreover, the opacity is mostly due to the Tomphson scattering and radiative equilibrium above the convective core is almost independent of the radial distribution of $\mu$. Both these effects are strongest in stars with high effective temperature ($\Teff > 5\times 10^4\K$) and lead to the mass–luminosity relation of W–R stars (Langer, 1989; M. Beech, R. Mitalas, 1992).
Together with the mass–luminosity relation (Maeder, 1983; Doom et al., 1986; Maeder, 1987, Maeder, Meynet, 1987) W–R stars obey the mass–radius relation (Langer, 1989; Schaerer, Maeder, 1992) and therefore the sound travel time between the stellar center and the stellar surface can be considered approximately as a function of the stellar mass $M$. In recent years W–R stars were found to be unstable against radial oscillations (Glatzel et al., 1999; Fadeyev, Novikova, 2003; 2004; Dorfi et al., 2006; Fadeyev, 2007; Fokin, Tutukov, 2007; Fadeyev, 2008), so that one might assume that evolution of W–R stars is accompanined by decrease of the period of their radial oscillations $\Pi$. However one should bear in mind that if the pulsational instability is due to the iron Z–bump $\kappa$–mechanism (Dorfi et al., 2006; Fadeyev, 2008) then both the amplitude growth rate and the amplitude of radial oscillations depend on the chemical composition of the outer layers of the star. Such a dependence is still unclear and in the present paper we consider the pulsational properties of W–R stars with various surface abundances of helium, carbon and oxygen.
The chemical composition of outer layers of the evolving star depends on the both initial mass $\mzams$ and mass loss rate $\dot M$. In this paper we consider the Population I stars with initial masses ranging within $70M_\odot\le\mzams\le 130M_\odot$. To evaluate the effect of uncertainties in empirical estimates of mass loss rates the evolutionary computations were carried out under various assumptions on $\dot M$. Some stellar models corresponding to the W–R phase were used as initial conditions in hydrodynamic computations of nonlinear radial stellar oscillations. In the present paper we continue our previous studies (Fadeyev, 2007; 2008) but now we use more recent data on the mass loss rate during helium burning that are most important for the structure of W–R stars.
Method of computations {#method-of-computations .unnumbered}
======================
Calculations of stellar evolution and stellar pulsations were done with methods that in general were described in the previous paper (Fadeyev, 2007), so that below only a few important changes are noted.
The mass loss rate during hydrogen burning and at the initial phase of helium burning $\dmh$ was calculated according to Nieuwenhuijzen and de Jager (1990). The mass loss rate at the later stages of evolution $\dmhe$ was calculated using the empirical formula by Nugis and Lamers (2000) which takes into account both clumping of the stellar wind and dependence of the mass loss rate on surface abundances of helium and heavier elements. In computations of stellar evolutuion the transition from $\dmh$ to $\dmhe$ was done during the initial stage of helium burning when the effective temperature of the contracting star reaches $\Teff=10^4\K$. At this point of the evolutionary track the difference between $\dmh$ and $\dmhe$ is about a few percent. For example, for $\mzams=80M_\odot$ and $\dot M = \dmh$ the star reaches the effective temperature $\Teff = 10^4\K$ when its mass is about $M=37M_\odot$ and surface abundances of hydrogen and helium are $X(\hydr)\approx 0.1$ and $X(\hel)\approx 0.88$, respectively. The mass loss rates evaluated with formulae by Nieuwenhuijzen and de Jager (1990) and by Nugis and Lamers (2000) are $\dmh=9.3\times 10^{-5} M_\odot/\mathrm{yr}$ and $\dmhe=9.6\times 10^{-5} M_\odot/\mathrm{yr}$. It should be noted that during such a short phase of contraction of the star the evolutionary track crosses the H–R diagram so fast that variation of the threshold effective temperature by a factor of two around $\Teff=10^4\K$ does not affect perceptibly the later evolution of the stars.
Thus, in calculations of stellar evolution the mass loss rate was determined as $$\dot M =
\left\{
\begin{array}{ll}
\fh \dmh , & (\mbox{hydrogen burning and initial helium burning}) ,
\\[4pt]
\fhe \dmhe , & (\mbox{helium burning at\ } \Teff\ge 10^4\K) ,
\end{array}
\right.$$ where constant factors $1 < \fh < 2$ � $0.5 < \fhe < 3$ were introduced in order to evaluate the dependence of the results of evolution calculations on uncertainties in mass loss rates.
In previous papers (Fadeyev, 2007; 2008) the thermodynamic functions of the gas with temperature of $T < 10^7\K$ were calculated using the OPAL equation of state data (Rogers et al., 1996). Unfortunately, these data are not quite appropriate at late stages of evolution when abundances of elements heavier than helium substantially increase in the outer layers of the star. In the present study we computed the tables of thermodynamic quantities for about three dozen compositions, so that both stellar evolution and pulsational instability can be selfconsistently calculated up to the core helium exhaustion when the stellar matter consists mostly of carbon and oxygen. Computing the tables of thermodynamic quantities we assumed that heavy elements are carbon, nytrogen, oxygen and neon. ZAMS abundances of these elements were taken from Rodgers et al. (1996).
The thermonuclear reaction network was extended to several dozen isotopes from hydrogen ${}^1\mathrm{H}$ to nickel ${}^{56}\mathrm{Ni}$. However in comparison with previous computations the larger number of reactions did not affect significantly the results of evolution calculations. This is due to the fact that evolutionary calculations were terminated just after the helium exhaustion when the energy generation is due mostly to reactions of the tripple–alpha process and $\alpha({}^{12}\mathrm{C},\gamma){}^{16}\mathrm{O}$.
Mass–luminosity relation {#massluminosity-relation .unnumbered}
========================
Domination of radiation in the helium burning phase leads to convergence of the evolutionary tracks of massive stars on the H–R diagram. In the upper panel of Fig. 1 are shown the parts of two evolutionary tracks with initial masses $\mzams=80M_\odot$ and $\mzams=130M_\odot$, both of them being computed with $\fh=\fhe=1$. At luminosity $\log L/L_\odot = 5.9$ masses and effective temperatures of these stars ($M\approx 25M_\odot$, $\Teff\approx 8.2\times 10^4\K$) differ from one another by $2\%$ and $0.1\%$, respectively.
![Upper panel: the parts of evolutionary tracks on the H–R diagram for initial masses $\mzams=130M_\odot$ (solid line) and $\mzams=80M_\odot$ (dashed line) with $\fh=\fhe=1$. The circles indicate the models with central helium abundance $Y_c=0.9$ and $Y_c=0.1$. Lower panel: The parts of evolutionary tracks of the star $\mzams=100M_\odot$ with $\fh=\fhe=1$ (solid line), $\fh=2$, $\fhe=1$ (dashed line) and $\fh=1$, $\fhe=2$ (dotted line). []{data-label="fig1"}](fig1.ps){width="9cm"}
W–R stars with close values of the stellar mass $M$, luminosity $L$ and radius $R$ but with different initial masses $\mzams$ have significantly different radial distributions of the mean molecular weight. This is illustrated in the upper panel of Fig. 2 where for two models of W–R stars with mass of $M=25M_\odot$ ($\mzams=80M_\odot$ and $\mzams=130M_\odot$) the radial dependencies of the helium abundance $X(\hel)$ are plotted as a functions of the Lagrangean mass coordinate $M_r$. The lower helium abundance and the higher carbon abundance in outer layers of the star with larger initial mass $\mzams$ is due to the more efficient helium burning during the preceding stellar evolution. Homogeneity of the chemical composition within the large mass fraction of the star is due to the large extention of convective cores.
![ Relative mass abundance of helium $X(\hel)$ as a function of the Lagrangean mass coordinate $M_r$ in W–R stars with mass of $M=25M_\odot$. Upper panel: the case of mass loss $\fh=\fhe=1$. Numbers at the curves indicate the initial stellar mass $\mzams$. Lower panel: three different cases of mass loss for $\mzams=100M_\odot$. []{data-label="fig2"}](fig2.ps){width="7.2cm"}
Dependence of the evolution of W–R stars on the mass loss rate is illustrated in the lower panel of Fig. 1 where three evolutionary tracks of the star with initial mass of $\mzams=100M_\odot$ are plotted for three different cases of the mass loss: ($\fh=\fhe=1$), ($\fh=2$, $\fhe=1$) and ($\fh=1$, $\fhe=2$). Doubling the mass loss rate $\dmh$ during hydrogen burning leads to decrease of the luminosity of the W–R star with mass of $M=25M_\odot$ by about $\approx 3\%$, whereas the core helium abundance increases by $\Delta Y_c\approx 0.1$.
The most important role in the structure of W–R stars belongs to the mass loss rate during the helium burning. For example, doubling $\dmhe$ leads to decrease of the luminosity in the W–R star with mass of $M=25M_\odot$ by $\approx 9\%$ and at bthe same time to increase of the core helium abundance by $\Delta Y_c\approx 0.25$. Dependence of the radial distribution of helium abundance $X(\hel)$ on the mass loss rate $\dmhe$ is shown in the lower panel of Fig. 2. Moreover, increase of $\dmhe$ is accompanied by the slower growth of the central temperature, so that the end of core helium exhaustion occurs at the lower stellar mass and on the H–R diagram the evolutionary track extends to lower luminosities.
Convergence of the evolutionary tracks on the H–R diagram implies the existence of the correlation between the stellar mass $M$ and the stellar luminosity $L$. Fig. 3 shows the mass–luminosity diagram with two pairs of evolutionary tracks for two initial stellar masses ($\mzams=90M_\odot$, $\mzams=130M_\odot$) and two cases of mass loss: ($\fh=1$, $\fhe=1; \fh=2$, $\fhe=1$). At the final point of each track plotted in Fig. 3 the central helium abundance is $Y_c\approx 10^{-3}$. It is clearly seen that during the significant part of the helium burning phase the stellar mass and luminosity are nearly related by the power dependence. The position of the initial point of the power dependence changes with initial stellar mass and mass loss rate. For example, the central helium abundance at the initial point ranges from $Y_c=0.85$ for $\mzams=70M_\odot$ to $Y_c=0.94$ for $\mzams=130M_\odot$. For smaller $\mzams$ or larger $\dmh$ the helium burning occurs at the lower central temperature, so that on the mass–luminosity diagram the part of the track with power dependence moves to smaller values of $M$ and $L$.
![Mass–luminosity relation for $\fh=1$ and $\fh=2$ at initial masses $\mzams=130M_\odot$ (solid and dashed lines) and $\mzams=90M_\odot$ (dotted and dashed–dotted lines). For all models $\fhe=1$. []{data-label="fig3"}](fig3.ps){width="10cm"}
The parts of evolutionary tracks that can be approximated by the power dependence on the mass–luminosity diagram are displayed in Fig. 4. The maximum deviation of each track from the power dependence is $\dL=\max|\Delta\log L| \approx 0.01$ for $0.5\le\fhe\le 2$ and $\dL \approx 0.03$ for $\fhe=3$. The linear fit on the $(\log M,\log L)$ plane for models with $70M_\odot\le\mzams\le 130M_\odot$, $1\le\fh\le 2$ and $0.5\le\fhe\le 3$ shown in Fig. 4 is given by $$\label{wrml1}
\log L/L_\odot = 3.675 + 1.568 \log M/M_\odot .$$
![Mass–luminosity relation of W–R stars with $70M_\odot\le\mzams\le 130M_\odot$, $1\le\fh\le 2$ and $0.5\le\fhe\le 3$ The dashed line shows relation (ref[wrml1]{}). []{data-label="fig4"}](fig4.ps){width="10cm"}
The maximum deviation of the parts of evolutionary tracks from relation (\[wrml1\]) is $\dL\le 0.11$ and its large value is due to the dependence of the coefficients of this relation on the mass loss rate $\dmhe$. Therefore, more exact approximation can be obtained for fixed values of $\fhe$: $$\label{wrml2}
\log L/L_\odot = \left\{
\begin{array}{lll}
4.236 + 1.212 \log M/M_\odot , \quad & (\fhe = 0.5, & \dL\le 0.06) ,
\\[4pt]
4.099 + 1.282 \log M/M_\odot , & (\fhe = 1, & \dL\le 0.03) ,
\\[4pt]
3.821 + 1.454 \log M/M_\odot , & (\fhe = 2, & \dL\le 0.02) ,
\\[4pt]
3.632 + 1.580 \log M/M_\odot , & (\fhe = 3, & \dL\le 0.04) .
\end{array}
\right.$$ It should be noted that the mass interval within which formulae (\[wrml2\]) can be applied depends on the initial mass $\mzams$ as well as on parameters $\fh$ and $\fhe$. The stellar mass $M$ and the central helium abundance $Y_c$ corresponding to limits of these intervals are given in Table 1. The last column of Table 1 gives the evolution time $t_\mathrm{ev}$ for this interval.
$\fhe$ $\fh$ $\mzams/M_\odot$ $t_\mathrm{ev}$, $10^6$ yr�
-------- ------- ------------------ ------ ------ ----------------------------- ------ ------
0.5 1 80 36.2 24.1 0.85 0.02 0.29
120 50.1 28.9 0.93 0.05 0.27
2 80 31.1 21.0 0.86 0.02 0.31
120 38.4 23.8 0.92 0.03 0.30
1.0 1 80 36.1 16.6 0.88 0.09 0.28
120 50.1 18.7 0.95 0.14 0.26
2 80 31.0 14.4 0.90 0.08 0.30
120 38.2 16.1 0.94 0.12 0.28
2.0 1 80 35.6 8.5 0.88 0.08 0.36
120 49.9 8.2 0.95 0.05 0.41
2 80 31.0 7.7 0.92 0.08 0.38
120 38.3 7.8 0.95 0.07 0.40
3.0 1 80 36.2 4.9 0.90 0.01 0.54
120 50.4 4.7 0.96 0.01 0.58
2 80 31.2 4.5 0.94 0.01 0.59
120 38.5 4.6 0.96 0.02 0.58
: Limiting values of stellar mass $M$ and central helium abundance $Y_c$ in the mass–luminosity relations (\[wrml2\]).
When one considers the stellar mass $M$ as a parameter one should bear in mind that the stellar evolution becomes slower as $M$ decreases. In Fig. 5 for $\mzams=100M_\odot$ and $\fh=1$ are shown the plots of the life time $\Delta t_\mathrm{ev}$ within the mass interval $[M-\Delta M, M]$, where $\Delta M=0.5M_\odot$. The plots shown in Fig. 5 weakly depend on $\mzams$ and $\fh$. The growth of the life time $\Delta t_\mathrm{ev}$ with increasing $\fhe$ is due to the slower increase of the central temperature and slower conversion of helium into carbon. For initial stellar mass $\mzams=100M_\odot$ and mass loss rate during hydrogen burning $\fh=1$ the second half of the life time correspond to W–R masses $M<33M_\odot$, $M<25M_\odot$, $M<12M_\odot$ and $M<7M_\odot$ for $\fhe=0.5$, 1, 2 and 3, respectively.
![The life time $\Delta t_\mathrm{ev}$ of W–R stars within the mass interval $\Delta M=0.5M_\odot$ as a function of the stellar mass $M$. The initial mass is $\mzams=100M_\odot$ and $\fh=1$. The numbers at the curves indicate correspondig values of $\fhe$. []{data-label="fig5"}](fig5.ps){width="8.5cm"}
Nonlinear radial oscillations {#nonlinear-radial-oscillations .unnumbered}
=============================
During evolution of the W–R star the relative radius of the stellar core increases, so that both the sound travel time from the center to the surface and the period of radial oscillations gradually decrease. At the same time the instability excitation zone moves closer to the stellar surface and the amplitude of oscillations decreases due to diminishing mass of the driving zone. Decrease of the amplitude and the period of radial oscillations in evolving W–R stars is shown in Fig. 6 where for stars with initial masses $\mzams=90M_\odot$ and $\mzams=120M_\odot$ are given the plots of the maximum expansion velocity of the outer boundary of the hydrodynamic model $\umax$ expressed in units of the local escape velocity $\vesc$ and the period of radial oscillations $\Pi$.
![ The ratio of the maximum expansion velocity of the outer boundary $\umax$ to the local escape velocity $\vesc$ (upper panel) and the period of radial oscillations $\Pi$ in days (lower panel) as a function of the stellar mass $M$. In solid and dashed lines are shown the evoltuionary sequences with $\mzams=120M_\odot$ and $\mzams=90M_\odot$, respectively. []{data-label="fig6"}](fig6.ps){width="7.4cm"}
The growth time of radial oscillations is of the order of the stellar dynamic time scale, so that oscillations of W–R stars are stronly nonadiabatic. In contrast to many other radially pulsating stars oscillations of W–R stars cannot be described in terms of the standing wave since the kinetic energy of the pulsating envelope only once per period passes the minimum and the maximum. Radial oscillations of W–R stars should be considered as nonlinear running waves propagating from the envelope bottom to the outer boundary. That is why in W–R stars with mass of $M > 15M_\odot$ the pulsation constant ($Q\ge 0.1$ day) is significantly larger in comparison with pulsation constants of stars radially pulsating in the form of standing waves. The only exception is W–R stars with mass of $M < 10M_\odot$ where the small–amplitude radial oscillations can be approximately represented by nonadiabatic standing waves.
In the stellar mass range of $4.5M_\odot\le M\le 20M_\odot$ the pulsation constant of W–R stars can be approximately written as $$\log Q = -2.6 + 0.1 M_\odot .$$
Properties of some hydrodynamic models are listed in Table 2. The mass of outer pulsating layers is negligible in comparison with the total mass of the star, so that the abundances of helium $X(\hel)$, carbon $X(\carb)$ and oxygen $X(\ox)$ are constant through the envelope. The mean pulsation period $\Pi$ was evaluated using the discrete Fourier transform of the kinetic energy of the oscillating envelope within the time interval $10^2\lesssim t/\Pi\lesssim 10^3$. However, strictly speaking, the definition of the period of radial oscillations can be applied to W–R stars with rather low stellar masses because for $M > 15M_\odot$ the frequency of oscillations at the bottom of the envelope becomes somewhat higher than that of the outer layers. In Fig. 7 are shown the power spectra of the velocity of gas in the outer ($r\approx R$) and the inner ($r\approx 0.79R$) layers of the envelope of the W–R star with mass of $M=16M_\odot$. Contribution of short–period oscillations in the inner layers becomes significant in stars with $M > 20M_\odot$ because they affect the radiative flux emerging from the outer boundary and the period of light variatins becomes shorter than that of hydrodynamic motions in the outer layers of the pulsating envelope.
![The power spectrum of velocity $U(\omega)$ in the outer (upper panel) and the inner (lower panel) layers of the W–R star with mass $M=16M_\odot$. []{data-label="fig7"}](fig7.ps){width="9cm"}
$\mzams/M_\odot$ $\fhe$ $M/M_\odot$ $\log L/L_\odot$ $X(\hel)$ $X(\carb)$ $X(\ox)$ $\Pi$ day $\umax/\vesc$ $\bar{R}/R$
------------------ -------- ------------- ------------------ ----------- ------------ ---------- ----------- --------------- -------------
90 2 15 5.541 0.639 0.311 0.031 0.0525 0.21 2.1
10 5.272 0.467 0.436 0.078 0.0085 0.13 1.1
3 18 5.660 0.786 0.186 0.009 0.124 0.30 3.4
15 5.524 0.745 0.223 0.014 0.0321 0.17 1.5
10 5.221 0.635 0.316 0.030 0.0061 0.09 1.1
5 4.733 0.368 0.502 0.111 0.0021 0.03 1.0
120 2 15 5.553 0.634 0.315 0.033 0.0652 0.25 2.5
10 5.289 0.461 0.439 0.081 0.0107 0.14 1.2
3 22 5.772 0.827 0.148 0.006 0.221 0.34 6.4
20 5.712 0.807 0.166 0.008 0.146 0.30 3.8
15 5.528 0.742 0.225 0.014 0.0336 0.18 1.5
10 5.248 0.632 0.319 0.031 0.0083 0.11 1.1
5 4.743 0.369 0.501 0.111 0.0018 0.06 1.0
: Properties of some hydrodynamic models of W–R stars.
As is seen in Table 2 the pulsational properties of W–R stars depend mostly on the mass loss rate $\dmhe$ whereas the role of the both initial stellar mass $\mzams$ and mass loss rate during hydrogen burning $\dmh$ is significantly weaker. This is due to the fact that variations of $\dmhe$ are accompanied by significant changes of the both stellar luminosity and surface abundances of carbon and oxygen. In particular, increase of the mass loss rate during helum burning leads to the smaller stellar luminosity and therefore to the smaller nonadiabaticity of stellar pulsations.
An important consequence of nonlinear radial stellar oscillations is the increase of the mean radius of pulsating layers of the gas. This effect is illustrated by Table 2 where the last column gives the mean radius of the outer boundary of the hydrodynamic model $\bar R$ expressed in units of the initial equilibrium radius $R$.
Mechanism of pulsational instability {#mechanism-of-pulsational-instability .unnumbered}
====================================
The elementary spherical layer of gas contributing into excitation of pulsational instability performs the positive mechanical work during the pulsation cycle, that is the integral of mechanical work is positive : $W = \displaystyle\oint PdV > 0$, where $P$ is the total pressure and $V$ is the specific volume. Unfortunately, for hydrodynamical models of W–R stars exact calculation of the radial dependence of the mechanic work $W$ is impossible because of strongly nonlinear radial oscillations. The only exception is W–R stars with mass of $M \le 10M_\odot$ where pulsation motions are characterized by a good repetition of pulsation cycles. Formation of W–R stars with so small masses implies rather high mass loss rates during the core helium burning phase ($\fhe\ge 3$). Omitting discussion on possibility of the existence of such W–R stars we consider their pulsational properties because the results obtained can be generalized to more massive W–R stars undergoing substantially smaller mass loss.
The upper panel of Fig. 8 shows radial dependences of the inetgral of mecahnical work $W$ for two models of W–R stars with mass of $M=6M_\odot$ and $M=8M_\odot$. It is clearly seen that excitation of oscillations ($W>0$) occurs in the outer layers of the stars and the maximum of $W$ moves to the surface as the stellar mass $M$ decreases. To clarify the origin of the pulsational instability one should compare the radial dependence of $W$ with that of the amplitude of radiative flux variations. To this end we consider the spectral density of luminosity $$L_r(\omega) = \int\limits_{-\infty}^\infty L_r e^{i\omega t} dt$$ at the angular frequency $\omega = 2\pi/\Pi$. The spectral density $L_r(\omega)$ was computed in all Lagrangeam mass zones using the discrete Fourier transform within time intervals $t/\Pi\lesssim 10^3$. The lower panel of Fig. 8 shows two radial dependences of $L_r(\omega)$ that are normalized to the surface value.
![Radial dependences of the mechanical work over the pulsation cycle $W$ (upper panel) and the normalized spectral density of luminosity $L_r(\omega)$ (lower panel) for models of W–R stars with masses $M=6M_\odot$ and $M=8M_\odot$. []{data-label="fig8"}](fig8.ps){width="8cm"}
Coincidence of radial coordinates of maxima of $W$ and $L_r(\omega)$ (see Fig. 8) allows us to conclude that excitation of oscillations is due to the interaction of radiative flux with gas of the envelope. In particular, for $W > 0$ it is necessary that the gas absorbs radiation at maximum compression and becomes more transparent at maximum expansion. Fig. 9 shows the plots of variations of the gas density $\rho$ and opacity $\kappa$ in the mass zone with maximum mechanical work $W$ of the model of W–R stars with mass $M=8M_\odot$. It is clearly seen that the positive mechanical work $W$ is due to the $\kappa$–mechanism. The average temperature of gas in this zone is $T\sim 2\times 10^5\K$ and the positive temperature derivative of opacity $(\partial\ln\kappa/\partial\ln T)_\rho > 0$ is due to the iron Z-bump.
![Variations of the gas density $\rho$ and opacity $\kappa$ in the vicinity of the maximum of mechanical work $W$ in the model of the W–R star with mass $M=8M_\odot$. The plots of $\rho$ and $\kappa$ are arbitrarily shifted along the vertical axis. []{data-label="fig9"}](fig9.ps){width="9cm"}
Fig. 10 shows the plots of radial dependencies of the gas density $\rho$ and the spectral density of luminosity $L_r(\omega)$ for outer layers of W–R stars with masses $12M_\odot\le M\le 18M_\odot$. The narrow maximum of $L_r(\omega)$ (see the lower panel of Fig. 10) corresponds to layers with agerage temperature $T\sim 2\times 10^5\K$ where the gas density $\rho$ and opacity $\kappa$ reach the maximum simultaneously, that is excitation of oscillations is also due to the $\kappa$–mechanism. In layers above the excitation zone $\partial L_r(\omega)/\partial r \approx 0$, that is there are neither excitation nor damping zones. More massive W–R stars possess more extended envelopes with smaller gradient of the gas density. The excitation zone is at the bottom of the envelope and radial oscillations exist as successive nonlinear waves propagating from the outer boundary of the core to the stellar surface.
![Radial dependencies of gas density $\rho$ (upper panel) and normalized spectral density of luminosity $L_r(\omega)$ (lower panel) in W–R stars with mass $M=12M_\odot$, $15M_\odot$ and $18M_\odot$. []{data-label="fig10"}](fig10.ps){width="8cm"}
Conclusion {#conclusion .unnumbered}
==========
As follows from results presented above the mass and luminosity of Population I massive stars are related by the power dependence during the large part of the helium burning phase. Boundaries of the segment of the evolutionary track within which the coefficients of the power dependence are constant depends on the initial stellar mass $\mzams$ and the mass loss rate $\dot M$ during the preceding evolution. At the initial point of the power dependence the central helium abundance ranges from $Y_c\approx 0.78$ ($\mzams=70M_\odot$, $\fhe=0.5$) to $Y_c\approx 0.96$ ($\mzams=130M_\odot$, $\fhe=3$), whereas the effective temperature is in the range from $\Teff\approx 3\times 10^4\K$ ($\mzams=120M_\odot$) to $\Teff\approx 5\times 10^4\K$ ($\mzams=70M_\odot$). At the end of the power dependence the central helium abundance ranges within $0.01\lesssim Y_c\lesssim 0.15$. Thus, the limits of applicability of the mass–luminosity relation depend on the both initial mass $\mzams$ and mass loss rate $\dot M$, so that relations (\[wrml2\]) should be used with corresponding boundary values of the stellar mass $M$ listed in Table 1.
In the present study we considered pulsational properties of W–R stars as a function of three parameters: the initial stellar mass $\mzams$, the mass loss rate during hydrogen burning $\dmh$ and the mass loss rate during the helium burning phase $\dmhe$. However the most important is $\dmhe$ since its variations lead to significant changes in surface abundances of helium, carbon and oxygen. In particular, at lower mass loss rate $\dmhe$ radial oscillations of W–R stars have larger amplitudes due to higher surface abundances of carbon and oxygen.
As is seen from Fig. 6 and Table 2 the period of radial oscillations is a sensitive indicator of the stellar mass and during evoulution of the W–R star decreases from $\approx 5.5$ hr at $M=22M_\odot$ to $\approx 2.6$ min at $M=5M_\odot$. Thus, observational estimates of pulsation periods could provide with a direct evaluation of the masses of W–R stars. For example, aAccording to Veen et al. (2002a, 2002b, 2002c) the pulsation period of WR46 is $\Pi\approx 0.14$ day and as follows from our hydrodynamical calculations the stellar mass is $M\approx 20M_\odot$.
The present study is confined to hydrodynamical models of W–R stars near their final stage of evolution and such a choice are is to their longer life time. However of great interest are more massive W–R stars with higher luminosity and much stronger instability similar to that of LBV stars. The study of these objects will be presented in the forthcoming paper.
References {#references .unnumbered}
==========
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2. P.M. Veen, A.M. van Genderen, K.A. van der Hucht, et al., Astron.Astrophys., **385**, 585 (2002�).
3. P.M. Veen, A.M. van Genderen, K.A. van der Hucht, et al., Astron.Astrophys., **385**, 600 (2002�).
4. P.M. Veen, A.M. van Genderen, K.A. van der Hucht, et al., Astron.Astrophys., **619**, 585 (2002�).
5. E.A. Dorfi, A. Gautschy, H. Saio, Astron.Astrophys., **453**, L35 (2006).
6. C. Doom, J.P. de Greve, C. de Loore, Astrophys. J., **303**, 136 (1986).
7. W. Glatzel, M. Kiriakidis, S. Chernigovskij, et al., MNRAS, **303**, 116 (1999).
8. N. Langer, Astron. Astrophys., **210**, 93 (1989).
9. A. Maeder, Astron. Astrophys., **120**, 113 (1983).
10. A. Maeder, Astron. Astrophys., **173**, 247 (1987).
11. A. Maeder, G. Meynet, Astron. Astrophys., **182**, 243 (1987).
12. N. Nugis, H. J. G. L. M. Lamers, Astron. Astrophys., **360**, 227 (2000).
13. H. Nieuwenhuijzen and C. de Jager, Astron.Astrophys., **231**, 134 (1990).
14. F.J. Rogers, F.J. Swenson, and C.A. Iglesias, Astrophys.J., **456**, 902 (1996).
15. Yu.A. Fadeyev, M.F. Novikova, Ast.Let., **29**, 522 (2003).
16. Yu.A. Fadeyev, M.F. Novikova, Ast.Let., **30**, 707 (2004).
17. Yu.A. Fadeyev, Ast.Let., **33**, 692 (2007).
18. Yu.A. Fadeyev, Ast.Rep., in press (2008).
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20. D. Schaerer, A. Maeder, Astron. Astrophys., **263**, 192 (1992).
[^1]: e–mail: [email protected]
|
---
abstract: 'The thermal dependence of the electrical conductance of the single-electron transistor (SET) in the zero-bias Kondo regime is discussed. An exact mapping to the universal curve for the symmetric Anderson model is established. Linear, the mapping is parametrized by the Kondo temperature and the charge in the Kondo cloud. Illustrative numerical renormalization-group results, in excellent agreement with the mapping, are presented.'
author:
- 'M. Yoshida'
- 'A. C. Seridonio'
- 'L. N. Oliveira'
title: 'Universality and the zero-bias conductance of the single-electron transistor'
---
Nearly five decades ago, Anderson conceived a Hamiltonian to describe the interaction between a magnetic impurity and otherwise free conduction electrons.[@An61:41] Once a daunting theoretical challenge, the Anderson Hamiltonian yielded to an essentially exact numerical diagonalization,[@KWW80:1003] followed by an exact analytical diagonalization.[@AFL83:331; @TW83:453] From these and alternative approaches, physical properties were extracted, which eased the interpretation of experimental data;[@GGK+98:5225] theoretical results provided unifying views of apparently unrelated phenomena;[@Wil82:1] and quantitative comparisons brought forth novel perceptions.[@LWC+87:1232]
The last ten years were especially fruitful. Parallel advances in scanning tunneling spectroscopy and in the fabrication of nanostructured semiconductor devices enhanced the interest in transport properties.[@MCJ+98.567; @WFF+00:2105; @HKS01:156803; @WDE03.01; @AYR03.81; @KZS+05.18824; @Crommie05.1501; @FFA03:155301; @RWH+06:196601; @SSI+06_096603; @ZB06.035332; @SIS08:155304] In both areas, numerous experimental breakthroughs and theoretical analyses were reported, and the Anderson Hamiltonian proved spectacularly successful in more than one occasion.[@OG03.02; @FJC+07:256601]
Notwithstanding the substantial volume of exact results, certain aspects of the model remain obscure. Consider universality, a concept important in its own right and by virtue of its diverse applications. Universal relations serve as benchmarks checking the accuracy of numerical data; as resources promoting the convergence of theoretical findings; and as instruments bridging the gap between the theorist’s tablet and the laboratory logbook. The conditions under which the Anderson model exhibits universal thermodynamical properties were identified.[@KWW80:1003; @AFL83:331; @TW83:453] Although one expects all properties of the model to be universal in the same domain, few firm results for the dynamical and transport properties can be found in libraries.[@BCP08:395] Costi’s et al’s early effort showed that the transport coefficients for the symmetric Anderson model are universal.[@CHZ94.19] For asymmetric models—even ones that display universal thermodynamical properties—, nonetheless, the universal curves fail to fit the numerical data, the disagreement growing with the (particle-hole) asymmetry.
Puzzled by such contrasts, we have conducted a systematic study of the transport properties for the Anderson Hamiltonian. We combined analytical and numerical-renormalization group (NRG) tools and paid special attention to universality. In a preliminary report,[@SYO2007] we have discussed an Anderson model for a quantum dot [*side-coupled*]{} to a quantum wire, a device comprising two conduction paths whose transport properties are marked by interference.[@KAS+04:035319; @SAK+05:066801; @KSA+06:36; @OAK+07:084706; @Kat07:233201] Notwithstanding the constructive or destructive effects, we have been able to identify universal behavior throughout the [*Kondo regime*]{}, the parametrical domain favoring the formation of a magnetic moment at the quantum dot and its progressive screening by the conduction electrons as the temperature is lowered past the scale set by the [ *Kondo temperature*]{} $T_K$. Specifically, we found the thermal dependence of the conductance to map linearly onto a universal function of the temperature $T$ scaled by the Kondo temperature $T_K$. The mapping is itself universal, [i. e.,]{} it depends on a single physical property, the ground-state phase shift $\delta$, into which the contributions from all model parameters are lumped.
This report examines the alternative experimental set-up in which a quantum dot or molecule, instead of side-coupled to, is [*embedded*]{} in the conduction path.[@GSM+98.156; @GGK+98.5225; @GGH+00:2188; @LSB02:725; @OG03.02; @YKC+04.266802; @YKC+05.256803; @KZS+05.18824] We show that the thermal dependence of the conductance maps onto the same universal function. Although linear, the mapping now depends explicitly on a model parameter—an external potential applied to the conduction electrons—and hence contrasts with the conclusion in our previous report. This dependence accounts for distinctions between the transport properties in the embedded and side-coupled arrangements. At high temperatures, for instance, potentials appropriately applied to the conduction electrons in the side-coupled geometry drive the conductance from low values up to the ballistic limit ${{\mathcal G}_2}=2e^2/h$. If the quantum dot is embedded in the conduction path, by contrast, the high-temperature conductance is pinned at low values and virtually insensitive to potentials applied to the conduction electrons. Our analysis shows that, in the embedded configuration, the screening charge in the Kondo cloud parametrizes the mapping to the universal conductance curve. Since that charge is always close to unity, the mapping is never far from the identity, with maximum relative deviations around 20%.
Our presentation focuses the mapping between the SET and the universal conductances. As illustrations we will present the results of a few Numerical Renormalization Group (NRG) runs. A discussion of the numerics, a comprehensive survey of the Kondo regime, and the comparison with the side-coupled geometry will be deferred to another report.
The text is divided in five Sections, more technical aspects of the analysis having been confined to the three Appendices. Section \[sec:model\] defines the model. Section \[sec:anderson-model\] derives an expression relating the conductance to the spectral density of the quantum dot level. Section \[sec:universality\] is dedicated to universality, and Section \[sec:fixed-points\], to the fixed points of the model Hamiltonian and to an extension of Langreth’s exact expression for the ground-state spectral density. Section \[sec:cross\] then shows that, in the Kondo regime, the thermal dependence of the conduction can be mapped onto the symmetric-SET universal conductance. Finally, Section \[sec:conclusions\] collects our conclusions.
Single-electron transistor\[sec:model\]
=======================================
Figure \[fig:1\] depicts a single-electron transistor (SET), the prototypical example of embedding. The subject of numerous experimental studies, the SET comprises two independent conduction bands coupled by a localized level.
![\[fig:1\]Single-electron transistor. A quantum dot (circle) bridges two quantum wires (rectangles). A gate potential $V_d$ controls the dot energy, while the symmetric potentials $V_w$ shift the the energy of the wire orbitals close to the dot.](fig1){width="\columnwidth"}
Qualitatively, the physics of Fig. \[fig:1\] was understood long before the first device was developed. Figure \[fig:2\] displays the spectrum of the SET Hamiltonian $H$ for zero coupling. The dot levels being then decoupled from the conduction bands, the eigenstates and eigenvalues of $H$ can be labeled by the dot quantum numbers. For simplicity, we will only refer to the dot occupation $n_d$. For fixed $n_d$, the product of the lowest dot state by the conduction-band ground state is shown as a bold dash. The gray levels above it represent the excited states consistent with the same $n_d$ label.
![\[fig:2\]SET energies in the weak-coupling limit. The dot-level occupation $n_d$ labels the energies. For each $n_d$, the bold dash represents the conduction-band ground state, while the thinner lines represent excitations. The coupling between the dot and the two quantum wires mixes each level to the neighboring columns.](fig2){width="\columnwidth"}
A small transition amplitude $V$ between the quantum dot and the wires is sufficient to modify this picture. The amplitude $V$ couples strongly each gray level to the degenerate or nearly degenerate states in the neighboring columns. Exceptions are the lowest levels in the column labeled $n_d={\mathcal{N}}$ in Fig. \[fig:2\], which are energetically distant from their neighbors and thus remain unperturbed to first order in the coupling. At low temperatures, with $k_BT$ small in comparison with the energy $\delta{\mathcal{E}}$ separating the ground state from the closest level in the neighboring columns, the dot occupation is frozen at $n_d={\mathcal{N}}$, a constraint that raises the Coulomb blockade against conduction through the dot.
Adjustment of the gate potential $V_d$ in Fig. \[fig:1\] brings down the blockade. The potential shifts the energies of the dot levels and can be tuned to the condition $\delta{\mathcal{E}}\approx0$, to impose degeneracy between the bold dashes in the $n_d={\mathcal{N}}$ and $n_d={\mathcal{N}}+1$ columns. An infinitesimal bias is then sufficient to induce electronic flow between the wires throught the dot. The conductance peak, we see, at gate potentials $V_d$ such that the ground-state expectation value of $n_d$ is half-integer, [e. g.,]{}${\langle\Omega|\,n_d\,|\Omega\rangle}={\mathcal{N}}+1/2$ as $\delta{\mathcal{E}}\to0$ in Fig. \[fig:2\].
Each peak identifies a resonance at the Fermi level. As the gate voltage is swept past $\delta{\mathcal{E}}=0$, the ground-state occupation changes rapidly from $n_d={\mathcal{N}}$ to $n_d={\mathcal{N}}+1$, and as required by the Friedel sum rule, so does the ground-state phase shift. At moderately low temperatures, for thermal energies smaller than the average spacing between the bold dashes in the figure, the plot of the conductance as a function of the gate voltage is a succession of peaks. Data collected in the laboratory at moderately low temperatures do display a sequence of resonances. At very low temperatures, however, the pattern changes to a sequence of intervals alternating between insulation and conduction.
The conducting plateaus are due to the Kondo effect. For gate voltages corresponding to odd ground-state dot occupations, the magnetic moment of the resulting dot spin interacts antiferromagnetically with the conduction electrons. As the device is cooled past the Kondo temperature, the screening of the moment creates the [*Kondo resonance*]{}, a spiked enhancement of the density of states pinned at the Fermi level. Notwithstanding the Coulomb blockade, the pinned resonance allows conduction.
Anderson model\[sec:anderson-model\]
====================================
A variant of the Anderson Hamiltonian encapsulates the physics of the device in Fig. \[fig:1\]. A spin degenerate level ${c_{d}}$ represents the dot level, and two structureless half-filled conduction bands, labeled $L$ (left) and $R$ (right), represent the two quantum wires. The $L$ ($R$) wire comprises $N$ state $c_{kL}$ ($c_{kR}$) with energies defined by the linear dispersion relation $\epsilon_k =
(k-k_F)v_F$ ($0\le k \le 2k_F$), so that the bandwidth is $2D=2v_Fk_F$. The per-particle, per-spin density of conduction states is $\rho=1/2D$, and we will let $\Delta\equiv D/N$ denote the energy splittings in the conduction bands. The model Hamiltonian is then the sum of three terms, $H=H_w+H_d+H_{wd}$, where the first term describes the wires: $$\label{eq:hw}
H_w =\sum_{k\alpha}\epsilon_k c_{k\alpha}^\dagger c_{k\alpha}
+\frac{{W}}{N}\sum_{kq\alpha}c_{k\alpha}^\dagger c_{q\alpha},$$ with an intra-wire scattering potential ${W}$, fixed by the potential $V_w$ in Fig. \[fig:1\], and $\alpha=L, R$. The Hamiltonian $H_d$ describes the dot: $$\label{eq:dot}
H_{d}={\varepsilon_d}n_d + Un_{d\uparrow}n_{d\downarrow},$$ with a dot energy ${\varepsilon_d}$ controlled by the gate potential $V_d$ in Fig. \[fig:1\]; and the Hamiltonian $H_{wd}$ couples the wires to the dot: $$\label{eq:wd}
H_{wd}= \frac{V}{\sqrt{2N}}\sum_{k\alpha} (c_{k\alpha}^\dagger{c_{d}}+{{\text{H.\ c.}}}).$$
Parity \[sec:parity\]
---------------------
To exploit the inversion symmetry of Fig. \[fig:1\], we define the normalized even ($a_k$) and odd ($b_k$) operators
\[eq:abk\] $$\begin{aligned}
\label{eq:ak}
a_k = \frac1{\sqrt2}(c_{kL}+c_{kR});\\
b_k = \frac1{\sqrt2}(c_{kL}-c_{kR}).\label{eq:bk}
\end{aligned}$$
The projection of the model Hamiltonian on the basis of the $a_k$’s and $b_k$’s splits it in two decoupled pieces, $H={H_A}+H_B$, where $$\label{eq:ha}
{H_A}=\sum_{k}\epsilon_k a_{k}^\dagger a_{k}
+{W}f_0^\dagger f_0 +V(f_0^\dagger {c_{d}}+{{\text{H.\ c.}}})+H_{d},$$ where we have introduced the traditional NRG shorthand $$\label{eq:f0}
f_0\equiv\sum_k a_k/{\sqrt N},$$ and $$\label{eq:hb}
H_B= \sum_{k}\epsilon_k b_{k}^\dagger b_{k} +\frac{{W}}N\sum_{kq}
b_k^\dagger b_q.$$
Conductance\[sec:cond\]
-----------------------
The odd Hamiltonian $H_B$ is decoupled from the quantum dot. It is, moreover, quadratic, and hence easily diagonalizable. Appendix \[sec:expr-cond\] determines its spectrum, analyzes the response of the conduction and dot electrons to the application of an infinitesimal bias and turns the result into the following Linear Response expression for the conductance: $$\label{eq:glin}
G(T) = {{\mathcal G}_2}\,\pi{\Gamma_{W}}\,\int_{-D}^{D}\rho_{d}(\epsilon,T) \left[-\frac{\partial
f(\epsilon)}{\partial\epsilon}\right]\,d\epsilon,$$ where $f(\epsilon)$ is the Fermi function; $$\label{eq:gammaW}
{\Gamma_{W}}= \frac{\Gamma}{1+\pi^2\rho^2W^2}$$ is the width $\Gamma=\pi\rho V^2$ of the ${c_{d}}$ level, here renormalized by the scattering potential $W$; and $$\label{eq:rhod}
\rho_d(\epsilon,T)= \frac1{f(\epsilon)}\sum_{mn}\frac{e^{-\beta
E_m}}{{\mathcal{Z}}}|{\langlen|\,{c_{d}^\dagger}\,|m\rangle}|^2
\delta(\epsilon_{mn}-\epsilon)$$ is the spectral density for the dot level. Here ${|m\rangle}$ and ${|n\rangle}$ are eigenstates of ${H_A}$ with eigenvalues $E_m$ and $E_n$, respectively, $\epsilon_{mn}\equiv E_m-E_n$, and ${\mathcal{Z}}$ is the partition function for the Hamiltonian ${H_A}$.
As one would expect, given that the odd Hamiltonian $H_B$ commutes with ${c_{d}}$, only the eigenvalues and eigenvectors of ${H_A}$ are needed to compute the right-hand side of Eqs. (\[eq:glin\]) and (\[eq:rhod\]). The following discussion will hence focus the even Hamiltonian, Eq. (\[eq:ha\]), which is equivalent to the conventional spin-degenerate Anderson Hamiltonian.[@An61:41]
Characteristic energies\[sec:char-energ\]
-----------------------------------------
Four characteristic energies govern the physical properties of the Anderson Hamiltonian. Two of them are displayed in Fig. \[fig:3\]: the energy $-\epsilon_d$ needed to remove an electron from the dot level; and the energy $\epsilon_d+U$ needed to add an electron to the level. The particle-hole transformation $c_k\to
c_k^\dagger$; $c_d\to -c_d^\dagger$ swaps the two energies, so that, the transformed dot Hamiltonian is given by the right-hand side of Eq. (\[eq:dot\]) with $\epsilon_d\to -(\epsilon_d+U)$.
![Spectrum of the spin-degenerate Anderson model, displayed as in Fig. \[fig:2\]. In the weak-coupling limit, the eigenstates are labeled by the occupation $n_d$ and spin component of the dot configuration displayed at the bottom. For $V\ne0$, each level in the left and right columns hybridizes with nearly degenerate levels in the central columns and acquires the width ${\Gamma_{W}}$ in Eq. (\[eq:gammaW\]). At low energies, the levels in the two central columns combine into a singlet and acquire a width $\Gamma_K\sim k_BT_K$.The vertical arrows near the right border mark the domains of the LM and FL fixed points.\[fig:3\] ](fig3){width="0.99\columnwidth"}
If $2\epsilon_d+U=0$, the dot Hamiltonian remains invariant under the particle-hole transformation. If, in addition, $W=0$, Eq. (\[eq:ha\]) reduces to the symmetric Hamiltonian $$\label{eq:hasym}
H_A^S= \sum_{k}\epsilon_k a_{k}^\dagger a_{k}
+V(f_0^\dagger {c_{d}}+{{\text{H.\ c.}}})-\frac{U}2(n_{d\uparrow}-n_{d\downarrow})^2.$$
With $V\ne0$, two other energies arise: the level width ${\Gamma_{W}}$ \[Eq. (\[eq:gammaW\])\] and the Kondo energy $k_BT_K$, given by $$\label{eq:tk}
T_K\sim \sqrt{\rho J}\exp(-1/\rho J),$$ where $J$ is the antiferromagnetic interaction between the conduction electrons and the dot magnetic moment,[@SW66:491] $$\label{eq:jk}
\rho J=2\frac{{\Gamma_{W}}}{\pi|{\varepsilon_d}|}\frac{U}{{\varepsilon_d}+U}.$$
In the Kondo regime, thermal and excitation energies are much smaller than $\min(|{\varepsilon_d}|, {\varepsilon_d}+U)$. In Fig. \[fig:3\], only the lowest levels in the central columns are energetically accessible. The energy ${\Gamma_{W}}$, associated with transitions from the central to the external columns in the figure ([i. e.,]{} with $c_d^1\to c_d^2$ and $c_d^1\to c_d^0$ transitions) becomes inoperant. Instead, at very low excitation and thermal energies, smaller than the Kondo energy $k_BT_K$, the dot spin binds antiferromagnetically to the conduction spins. In Fig. \[fig:3\], the lowest states in the left and right central columns hybridize to constitute a Kondo singlet.
Universality\[sec:universality\]
================================
The concepts recapitulated in Section \[sec:char-energ\] emerged over three decades ago, with the first accurate computation of the magnetic susceptibility of the Anderson model,[@KWW80:1003] long before the first essentially exact computation of the conductance. A particularly important result in Costi’s, Hewson’s, and Zlatic’s survey of transport properties[@CHZ94.19] is the the thermal dependence of the conductance for the symmetric Hamiltonian $H_A^S$, the universal curve ${G_{}^S}(T/T_K)$, depicted by the solid line in Fig. \[fig:4\]. For $k_BT\ll D$ and any pair $(\Gamma, U)$ satisfying $\Gamma\ll U$ in Eq. (\[eq:hasym\]), proper adjustment of the Kondo temperature $T_K$ gives a conductance curve $G(T/T_K)$ that reproduces ${G_{}^S}(T/T_K)$.
In Fig. \[fig:4\], for instance, the solid line was computed from the eigenvalues and eigenvectors of $H_A^S$ with $\Gamma=0.1\,D$ and $U=3\,D$. The definition $G(T_K)\equiv0.5{{\mathcal G}_2}$ yielded the Kondo temperature $T_K=2.4\times10^{-6}\,D$. When the calculation was repeated for $U=0.6\,D$ and the same $\Gamma$, the Kondo temperature grew four orders of magnitude, to $T_K=2.2\times10^{-2}\,D$. Still, for $k_BT<0.1\,D$, the plot of $G(T/T_K)$ resulted indistinguishable from the solid curve. While $T_K$ is model-parameter dependent, $G(T/T_K)$ is not.
![\[fig:4\]Thermal dependences of the conductance for two sets of model paramters, obtained from Eqs. (\[eq:glin\]) and (\[eq:rhod\]). The solid line depicts the universal conductance curve[@CHZ94.19] for the symmetric Hamiltonian (\[eq:hasym\]) . Here, it was computed with $\Gamma= 0.1\,D$ and $U=3\,D$. The temperatures were scaled by the Kondo temperature $T_K=2.4\times10^{-6}\,D/k_B$, fixed by the requirement $G(T_K)=0.5{{\mathcal G}_2}$. The dashed curve is the conductance for the Hamiltonian (\[eq:ha\]) with $\Gamma=0.1\,D$, $U=3\,D$, ${\varepsilon_d}=-0.3\,D$, and $W=0$, which yielded $T_K=4.0\times10^{-3}\,D$. To keep the data within the temperature range $k_BT<0.1\,D$, the dashed plot stops at $T=25\,T_K$. The horizontal arrows pointing to the vertical axes indicate the corresponding fixed-point conductances, given by Eqs. (\[eq:glm\]) and (\[eq:gfl\]).](fig4){width="0.99\columnwidth"}
Particle-hole asymmetry drives $G$ away from ${G_{}^S}$. For $U+2\epsilon_d\ne0$ or $W\ne0$, the universal curve ${G_{}^S}(T/T_K)$ no longer matches $G(T/T_K)$. An example is the dashed curve in Fig. \[fig:4\], calculated with $\Gamma=0.1\,D$, $U=3\,D$, ${\varepsilon_d}=-0.3\,D$, and $W=0$. The definition $G(T_K)=0.5{{\mathcal G}_2}$, which in this case yields $T_K=4\times10^{-3}\,D$, forces the solid and the dashed lines to agree at $T=T_K$; the conductance for the asymmetric model nonetheless undershoots (overshoots) the universal curve for $T < T_K$ ($T>T_K$). To reconcile this discrepancy with the concept of universality, the following sections rely on renormalization-group concepts.
Fixed points\[sec:fixed-points\]
================================
Renormalization-goup theory probes the spectrum of Hamiltonians in search of characteristic energies and scaling invariances. The wire Hamiltonian (\[eq:hw\]), for instance, exhibits a single, trivial characteristic energy: the conduction bandwidth $2D$. For energies $\epsilon\ll D$, therefore, its spectrum is invariant under the scaling transformation $H_w\to \Lambda H_w$, for arbitrary scaling parameter $\Lambda>1$. Accordingly, for $\epsilon\ll D$, the wire Hamiltonian is a stable [*fixed point*]{} of the renormalization-group transformation in Ref. .
Latent in the Anderson Hamiltonian (\[eq:ha\]), by contrast, are the four nontrivial characteristic energies discussed in Section \[sec:char-energ\]. Part of the spectrum of ${H_A}$ lies close to fixed points; the remainder is in transition ranges. In the vicinity of a fixed point, the spectrum remains approximately invariant under scaling; in the transition intervals, the eigenvalues are comparable to one or more characteristic energies and hence change rapidly under scale transformations. In particular, the portion of the spectrum pertinent to the Kondo regime comprises two lines of fixed points and a crossover region.
For given thermal or excitation energy ${\mathcal{E}}$, the inequality $\max({\mathcal{E}},{\Gamma_{W}}) \ll\min(|{\varepsilon_d}|, {\varepsilon_d}+U, D)$ defines the Kondo regime. As Fig. \[fig:3\] shows, the dot occupation is then nearly unitary. In the energy range $k_BT_K \ll {\mathcal{E}}\ll \min(|{\varepsilon_d}|,{\varepsilon_d}+U,
D)$, which is removed from characteristic energies, the Hamiltonian $H_A$ is near the [*Local Moment*]{} fixed point (LM). At very low energies, ${\mathcal{E}}\ll k_BT_K$, [i. e.,]{} below the energy scale defined by the narrow set of levels at the center of Fig. \[fig:3\], the spectrum becomes asymptotically invariant under scaling as the Hamiltonian approaches the Frozen Level fixed-point (FL). In the intermediate region ${\mathcal{E}}\sim k_BT_K$, the Hamiltonian crosses over from the LM to the FL.
Fixed-point Hamiltonians\[sec:fixed-point-hamilt\]
--------------------------------------------------
As the two central columns in Fig. \[fig:3\] indicate, the LM is an unstable fixed-point consistent of a conduction band and a free spin-1/2 variable. In the FL, a singlet replaces the spin, and the Hamiltonian is equivalent to a conduction band—a stable fixed point. In their most general form, the fixed-point conduction bands mimic the wire Hamiltonian, [i. e.,]{}$$\label{eq:hlm}
H_{LM}^*=\sum_k\epsilon_k a_k^\dagger a_k +W_{LM}f_0^\dagger f_0,$$ and $$\label{eq:hfl}
H_{FL}^*=\sum_k\epsilon_k a_k^\dagger a_k +W_{FL}f_0^\dagger f_0,$$ with scattering potentials $W_{FL}$ and $W_{LM}$ dependent on $V$, $W$, $U$ and $\epsilon_d$. Equations (\[eq:hlm\]) and (\[eq:hfl\]) identify two lines of fixed points, parametrized by $W_{LM}$ and $W_{FL}$, respectively.
The Schrieffer-Wolff transformation offers an approximation for the LM potential: $$\label{eq:wSchW}
\rho W_{LM} = \rho W + 2\frac{{\Gamma_{W}}}{\pi|{\varepsilon_d}|}\frac{2{\varepsilon_d}+U}{{\varepsilon_d}+U}.$$ For most applications, this expression is insufficiently accurate, and an NRG computation is necessary to determine $W_{LM}$ and $W_{FL}$. The exception is the Hamiltonian (\[eq:hasym\]), for which $W_{LM}=0$, as required by particle-hole symmetry.
Fixed-point phase shifts \[sec:fixed-point-phase\]
--------------------------------------------------
Appendix \[sec:diag\] diagonalizes the quadratic Hamiltonians (\[eq:hlm\]) and (\[eq:hfl\]). For the LM, the diagonal form reads $$\label{eq:hlmdiag}
H_{LM}^*=\sum_k\varepsilon_\ell g_\ell^\dagger g_\ell,$$ with phase-shifted energies $$\label{eq:erglm}
\varepsilon_\ell = \epsilon_\ell -\frac{\delta_{LM}}{\pi}\Delta.$$ At the LM, all conduction states are uniformly phase-shifted, with $$\label{eq:deltalm}
\tan\delta_{LM} = -\pi\rho W_{LM}.$$ For ${H_A}=H_A^S$, in particular, $\delta_{LM}=0$, and the low-energy eigenvalues $\varepsilon_k$ coincide with the $\epsilon_k$.
The FL eigenvalues are likewise uniformly phase-shifted, $$\label{eq:hfldiag}
H_{FL}^*=\sum_k\tilde\varepsilon_k \tilde g_k^\dagger \tilde g_k,$$ where $\tilde\varepsilon_k=\epsilon_k-(\delta/\pi)\Delta$. From the Friedel sum rule, it follows that[@La66:516] $$\label{eq:deltafl}
\delta = \delta_{LM}-\frac{\pi}2.$$ For ${H_A}=H_A^S$, in particular, $\delta=\pi/2$.
Conductance at the fixed points\[sec:conductance-at-fixed\]
-----------------------------------------------------------
The LM is the fixed point to which the Anderson Hamiltonian would come if $\Gamma=0$. For $0<\Gamma\ll \min(|\epsilon_d|, \epsilon_d+U, D)$, although the renormalization-group flow never reaches the LM, it brings ${H_A}$ close to the fixed point. The substantial portion of the spectrum of ${H_A}$ marked by the thin double-headed arrow in Fig. \[fig:3\] is approximately described by the many-body eigenvalues of $H_{LM}^*$, and in the pertinent energy range, the physical properties of ${H_A}$ and $H_{LM}^*$ are approximately the same. Likewise, at low temperatures, the properties of ${H_A}$ approach those of $H_{FL}^*$.
The renormalization-group evolution of the Hamiltonian can be traced in the termal dependence of the conductance. As the temperature is reduced from $T\gg T_K$ to $T\ll T_K$, each curve in Fig. \[fig:4\] crosses over from a lower plateau to a higher one. The extension of Langreth’s expression [@La66:516] derived in Appendix \[sec:extens-langr\] determines the plateau conductances:
\[eq:plateaus\] $$\begin{aligned}
\label{eq:glm}
G_{LM}&=& {{\mathcal G}_2}\sin^2(\delta_{LM} - \delta_W)
={{\mathcal G}_2}\cos^2(\delta-\delta_W);\quad\\
G_{FL}&=& {{\mathcal G}_2}\sin^2(\delta - \delta_W),\label{eq:gfl}
\end{aligned}$$
where $\delta_W$ is the ground-state phase shift for $V=0$. According to the analysis in Appendix \[sec:diag\], $$\label{eq:tanDeltaW}
\tan\delta_W = -\pi\rho W.$$
The solid curve in Fig \[fig:4\] was computed for ${H_A}=H_A^S$, so that $\delta_W=0$, while the ground-state ([i. e.,]{} FL) phase shift is $\delta=\pi/2$. According to Eqs. (\[eq:plateaus\]), $G_{LM}=0$ and $G_{FL}={{\mathcal G}_2}$, in agreement with the plot. The ground-state phase shift for the dashed curve, extracted from the low-energy eigenvalues in the NRG run that generated it, is somewhat lower: $\delta=0.43\pi$. Again $\delta_W=0$, and the two horizontal arrows pointing to the vertical axes in Fig. \[fig:4\] indicate the conductances predicted by Eqs. (\[eq:plateaus\]). Given the relatively high Kondo temperature ($k_BT_K=4\times10^{-3}\,D$) in this run, the condition $k_BT \ll D$ restricts the curve to the range $T<25\,T_K$, so that, even at the highest temperature shown, ${H_A}$ is relatively distant from the LM, and Eq. (\[eq:glm\]) cannot be accurately checked. At low temperatures, however, the renormalization-group flow bringing ${H_A}$ asymptotically close to $H_{FL}^*$, the agreement with Eq. (\[eq:gfl\]) is excellent.
Crossover\[sec:cross\]
======================
In the Kondo regime, the Schrieffer-Wolff transformation[@SW66:491] brings the Anderson Hamiltonian ${H_A}$ to the Kondo form $$\label{eq:swkondo}
H_J = \sum_k \epsilon_k a_k^\dagger a_k +W_{LM}f_0^\dagger f_0 + J
\sum_{\mu\nu}f_{0\mu}^\dagger \bm{\sigma}_{\mu\nu}f_{0\nu}\cdot\bm{S},$$ with $J$ defined in Eq. (\[eq:jk\]).
To eliminate the scattering potential on the right-hand side, it is convenient to project $H_J$ upon the basis of the eigenoperators $g_k$ of the LM, which yields[@irrelevant] $$\label{eq:lmkondo}
H_J = \sum_k\varepsilon_\ell g_\ell^\dagger g_\ell +
{J_W}\sum_{\mu\nu}\phi_{0\mu}^\dagger \bm{\sigma}_{\mu\nu}\phi_{0\nu}\cdot\bm{S},$$ where ${J_W}=J\cos^2\delta_{LM}$, and $$\label{eq:phi0}
\phi_0 = \frac1{\sqrt{N}}\sum_\ell g_\ell.$$ In the symmetric case $\delta_{LM}$ vanishes, and the operator $\phi_0$ reduces to $f_0$.
The second term on the right-hand side of Eq. (\[eq:lmkondo\]) drives the Hamiltonian from the LM to the FL. Along the resulting trajectory, the eigenvalues of $H_J$ scale with $T_K$.[@Wi75:773; @KWW80:1044; @TW83:453; @AFL83:331] Let $T_{K}$ and $\bar T_{K}<T_K$ be the Kondo temperatures correspondig to two sets of model parameters in the Kondo regime: $\mathcal{M}\equiv\left\{\Gamma,W, U, {\varepsilon_d}\right\}$ and $\bar{\mathcal{M}}\equiv\left\{\bar\Gamma,\bar W, \bar U,\bar
{\varepsilon_d}\right\}$, to which correspond the antiferromagnetic couplings $J$ and $\bar J$, respectively. If ${|m\rangle}$ is an eigenvector of $H_{J}$ with eigenvalue $E_{m}$, then a corresponding eigenvector ${|\bar m\rangle}$ of $H_{\bar J}$, the [*scaling image of ${|m\rangle}$*]{}, can always be found, with the same quantum numbers and eigenvalue $\bar E_{m}$ such that $E_{m}/T_{K}=\bar E_{m}/\bar T_{K}$.
The matrix elements of any linear combination of the operators $g_k$ are moreover universal. Given two eigenstates ${|m\rangle}$ and ${|n\rangle}$ of $H_{J}$ and their scaling images ${|\bar m\rangle}$ and ${|\bar
n\rangle}$, then the matrix elements of $\phi_0$, for example, are equal: ${\langlem|\,\phi_0\,|n\rangle}={\langle\bar m|\,\phi_0\,|\bar n\rangle}$. Likewise, the matrix elements of the operator $$\label{eq:phi1}
\phi_1 = \sqrt{\frac3N}\sum_\ell\frac{\varepsilon_\ell}{D} g_\ell$$ are universal: ${\langlem|\,\phi_1\,|n\rangle}={\langle\bar m|\,\phi_1\,|\bar n\rangle}$.
Thermal dependence of the conductance\[sec:therm-depend\]
---------------------------------------------------------
By contrast, the matrix elements ${\langlem|\,{c_{d}}\,|n\rangle}$ on the right-hand side of Eq. (\[eq:rhod\]) are non-universal. Even at the lowest energies, as Eq. (\[eq:gamma\_l0\]) shows, they depends explicitly on the model parameters. To discuss universal properties, therefore, we must relate them to universal matrix elements, such as ${\langlem|\,\phi_0\,|n\rangle}$, ${\langlem|\,\phi_1\,|n\rangle}$, or ${\langlem|\,g_\ell\,|n\rangle}$. As a first step towards that goal, we evaluate the commutator $$\label{eq:ha_comm_aq}
[{H_A}, a_q^\dagger] = \epsilon_q a_q^\dagger +
\frac{V}{\sqrt{N}}{c_{d}^\dagger}+ \frac{W}{N}\sum_{p}a_p^\dagger,$$ and sum the result over $q$, to find that $$\label{eq:ha_comm_f0}
[{H_A},f_0^\dagger] = \frac1{\sqrt3}f_1^\dagger + V{c_{d}^\dagger}+ W f_0^\dagger.$$ Here we have defined another shorthand $$\label{eq:f1}
f_1=\sqrt{\frac3N}\sum_q \frac{\epsilon_q}D\, a_q.$$
Equation (\[eq:ha\_comm\_f0\]) relates the matrix elements of ${c_{d}^\dagger}$ between two (low energy) eigenstates ${|m\rangle}$ and ${|n\rangle}$ of ${H_A}$ to those of the operators $f_0$ and $f_1$: $$\label{eq:aq_matel_cd}
V{\langlem|\,{c_{d}^\dagger}\,|n\rangle} = (E_m-E_n-W){\langlem|\,f_0^\dagger\,|n\rangle}-\sqrt3
D{\langlem|\,f_1^\dagger\,|n\rangle}.$$ In the Kondo regime, with $\max(E_m,E_n)\ll D$, the first two terms within the parentheses on the right-hand side can be dropped.
In the symmetric case, since $f_0$ ($f_1$) coincides with $\phi_0$ ($\phi_1$), Eq. (\[eq:aq\_matel\_cd\]) shows that the product $V{\langlem|\,{c_{d}}\,|n\rangle}$ is universal, in line with the firmly established notion that $\Gamma\rho_d(\epsilon/k_BT_K, T/T_K)$, and $G^S(T/T_K)$ are universal functions.[@CHZ94.19; @BCP08:395] To discuss asymmetric Hamiltonians, we have to relate the operators $f_0$ and $f_1$ to $\phi_0$ and $\phi_1$. This is done in Appendix \[sec:higher\], which shows that, in the Kondo regime, a linear transformation with model-parameter dependent coefficients relates the matrix elements of both $f_0$ and $f_1$ to those of $\phi_0$ and $\phi_1$. When Eq. (\[eq:f0matelphi0\]) is substituted for $f_0$ and $f_1$ on the right-side of Eq. (\[eq:aq\_matel\_cd\]), it results that $$\label{eq:cd_phi01}
\sqrt{\pi\rho{\Gamma_{W}}}{\langlem|\,{c_{d}^\dagger}\,|n\rangle} =
\alpha_0{\langlem|\,\phi_0^\dagger\,|n\rangle}+\alpha_1{\langlem|\,\phi_1^\dagger\,|n\rangle}.$$ Here, the constants $\alpha_0$ and $\alpha_1$ are combinations of the (unknown) linear coefficients on the right-hand side of Eq. (\[eq:f0matelphi0\]), the parameter $W$ on the right-hand side of Eq. (\[eq:aq\_matel\_cd\]), and the ratio $\sqrt{\pi\rho{\Gamma_{W}}}/V$, by which we multiplied Eq. (\[eq:aq\_matel\_cd\]) to shorten the following algebra.
Substitution in Eq. (\[eq:rhod\]) yields an expression relating the spectral density $\rho_d$ to universal functions: $$\label{eq:cdUniversal}
\pi\rho{\Gamma_{W}}\rho_d(\epsilon,T) =
\alpha_0^2\rho_0(\epsilon,T)+\alpha_1^2\rho_1(\epsilon,T)
+\alpha_0\alpha_1\rho_{{(01)}}(\epsilon, T),$$ where $$\begin{aligned}
\label{eq:rhos}
\rho_{j}\,(\epsilon,T)&= \displaystyle\sum_{mn}&\frac{e^{-\beta
E_m}}{{\mathcal{Z}}f(\epsilon)}|{\langlen|\,\phi_j\,|m\rangle}|^2\nonumber\\
&&\times\,\delta(E_m-E_n-\epsilon)\qquad(j=0,1),\end{aligned}$$ and $$\begin{aligned}
\rho_{{(01)}}\,(\epsilon,T)&= \displaystyle\sum_{mn}&\frac{e^{-\beta E_m}}{{\mathcal{Z}}f(\epsilon)}\,\left({\langlem|\,\phi_0^\dagger\,|n\rangle}{\langlen|\,\phi_1\,|m\rangle}+\text{c.~c.}
\right)\nonumber\\
&&\times\,\delta(E_m-E_n-\epsilon)\label{eq:rho01}.
\end{aligned}$$
Next, we substitute Eq. (\[eq:cdUniversal\]) on the right-hand side of Eq. (\[eq:glin\]), to split the conduction into three pieces: $$\label{eq:g01bar}
G(T) = \alpha_0^2 G_0(T) +\alpha_1^2G_1(T)
+ \alpha_0\alpha_1 G_{{(01)}}(T),$$ where $$\label{eq:g01}
G_j(T)=\frac{{{\mathcal G}_2}}{\rho}\,\int_{-D}^{D}\rho_{j}(\epsilon,T) \left[-\frac{\partial
f(\epsilon)}{\partial\epsilon}\right]\,d\epsilon\qquad(j=0,1),$$ and $$\label{eq:g_0110}
G_{{(01)}}(T)=\frac{{{\mathcal G}_2}}{\rho}\int_{-D}^{D}
\rho_{{(01)}}(\epsilon,T)
\left[-\frac{\partial f(\epsilon)}{\partial\epsilon}\right]\,d\epsilon.$$
Universal contributions to the conductance\[sec:univ-contrib\]
--------------------------------------------------------------
Given the universality of the energies $E_{m}$ and of the matrix elements ${\langlem|\,\phi_j\,|n\rangle}$ ($j=1,2$) on the right-hand sides of Eqs. (\[eq:rhos\]) and (\[eq:rho01\]), we see that the spectral densities $\rho_j(\epsilon,T)$ ($j=0,1$), and $\rho_{{(01)}}(\epsilon,T)$ are universal. Inspection of the right-hand sides of Eqs. (\[eq:g01\]-\[eq:g\_0110\]) shows that the functions $G_j$ ($j=0,1$) and $G_{{(01)}}$ are likewise universal. To compute them, we are free to consider any convenient Kondo-regime Hamiltonian.
Particle-hole symmetry makes ${H_A}^S$ especially convenient. To show that the cross terms make no contribution to the conductance, [i. e.,]{}that $G_{{(01)}}(T)=0$, we only have to notice that, while leaving ${H_A}^S$ unchanged, the particle-hole transformation ${c_{d}}\to-{c_{d}^\dagger}$, $g_k\to g_k^\dagger$ ([i. e.,]{} $a_k\to a_k^\dagger$) reverses the sign of the product of matrix elements ${\langlem|\,\phi_i^\dagger\,|n\rangle}{\langlen|\,\phi_j\,|m\rangle}+\text{c.~c.}$ on the right-hand side of Eq. (\[eq:rho01\]). We see that $\rho^{{(01)}}(\epsilon, T)$ is an odd function of $\epsilon$, so that the integral on the right-hand side of Eqs (\[eq:g\_0110\]) vanishes.
To evaluate $G_0$ and $G_1$, we start out from the closed form resulting from the diagrammatic expansion (in the coupling $V$) of the conduction-electron retarded Green’s function for the symmetric Hamiltonian: $$\label{eq:gkk}
{\mathbb{G}_{kk'}}^S(\epsilon) = {\mathbb{G}^{(0)}_{k}}(\epsilon)\delta_{kk'} +
\frac{V^2}N{\mathbb{G}^{(0)}_{k}}(\epsilon){\mathbb{G}_d}^S(\epsilon){\mathbb{G}^{(0)}_{k'}},$$ where ${\mathbb{G}_d}^S$ is the retarded dot-level Green’s function for the symmetric Hamiltonian, and $$\label{eq:gzerok}
{\mathbb{G}^{(0)}_{k}}(\epsilon)=\frac{1}{\epsilon-\epsilon_k+i\eta}$$ is the free conduction-electron retarded Green’s function.
From ${\mathbb{G}_{kk'}}^S$, it is a simple matter to obtain the spectral densities on the right-hand side of Eq. (\[eq:cdUniversal\]): $$\label{eq:rho0gk}
\rho_0(\epsilon,T) = -\frac1{\pi N}\Im\sum_{k k'}{\mathbb{G}_{kk'}}^S(\epsilon),$$ and $$\label{eq:rho1gk}
\rho_1(\epsilon,T) = -\frac{3}{\pi ND^2}
\Im\sum_{k k'}\epsilon_k\epsilon_{k'}{\mathbb{G}_{kk'}}^S(\epsilon).$$
To compute the conductances at temperatures $T$ satisfying $k_BT\ll
D$, we only need the spectral densities for $\epsilon \ll D$. It is appropriate, therefore, to expand the right-hand side of Eq. (\[eq:gzerok\]) to linear order in $\epsilon/D$: $$\label{eq:gzeroklinear}
{\mathbb{G}^{(0)}_{k}}(\epsilon)=\frac{2\epsilon}{D}
-i\pi\delta(\epsilon-\epsilon_k)\qquad(\epsilon\ll D).$$
The sums over momenta on the right-hand side of Eqs. (\[eq:rho0gk\]) and (\[eq:rho1gk\]) are then easily computed. Among the resulting terms, only the even powers of $\epsilon$ contribute to the integral on the right-hand side of Eq. (\[eq:g01\]). To compute the conductance to ${\mathcal{O}}[(k_BT/D)^2]$ we hence neglect the terms of ${\mathcal{O}}(\epsilon/D)$. Equation (\[eq:rho0gk\]) then gives $$\label{eq:rho0eps}
\rho_0(\epsilon, T) = \rho -{\pi\rho
\Gamma}\rho_d^S(\epsilon,T),$$ equivalent to an expression obtained in Ref.[@MNU04:3239].
Substitution of this result for $\rho_0$ on the right-hand side of Eq. (\[eq:g01\]) establishes a simple relation between the universal function $G_0$ and the universal conduction for the symmetric Hamiltonian: $$\label{eq:g0}
G_0(T) = {{\mathcal G}_2}- {G_{}^S}(T).$$
Equation (\[eq:g0\]) becomes exact, asymptotically, at low temperatures. The deviations, of ${\mathcal{O}}[(k_BT/D)^2]$, are insifignicant. As an illustration, the open circles in Fig. \[fig:5\] show NRG data for the conductance $G_0(T)$, Eq. (\[eq:g01\]), in excellent agreement with the solid line representing the right-hand side of Eq. (\[eq:g0\]).
![NRG results for the thermal dependence of the auxiliary conductance $G_0(T)$, associated with the spectral density for the operator $\phi_0$. The open circles show Eq. (\[eq:g01\]) for $j=0$, computed for the symmetric Hamiltonian with the displayed model parameters. The solid line is the right-hand side of Eq. (\[eq:g0\]), [i. e.,]{} the universal curve in Fig. \[fig:4\] subtracted from the quantum conductance ${{\mathcal G}_2}$.[]{data-label="fig:5"}](fig5){width="0.99\columnwidth"}
To the same accuracy, we can neglect the ${\mathcal{O}}(\epsilon/D)$ terms resulting from the summation on the right-hand side of Eq. (\[eq:rho1gk\]), which yields $$\label{eq:rho1eps}
\rho_1(\epsilon, T) = \frac{6\rho \Gamma}{\pi}\rho_d^S(\epsilon,T).$$ Equation (\[eq:g01\]) then shows that $G_1$ is also related to the conductance for the symmetric Hamiltonian: $$\label{eq:g1}
G_1(T)= \frac{6}{\pi^2}{G_{}^S}(T),$$
Mapping to the universal conductance\[sec:mapp-univ-cond\]
----------------------------------------------------------
The combination of Eqs. (\[eq:g0\]) and (\[eq:g1\]) with the result $G_{{(01)}}=0$ reduces Eq. (\[eq:g01bar\]) to the equality $$\label{eq:ga0a1set}
G(T) = \alpha_0^2{\bm{\left(}}{{\mathcal G}_2}-{G_{}^S}(T){\bm{\right)}}+\alpha_1^2\,\frac6{\pi}\,\rho\,{G_{}^S}(T)$$
To determine the coefficients $\alpha_0$ and $\alpha_1$, we need only compare the right-hand side with the fixed-point expressions for the conductance. At the LM, ${G_{}^S}=0$, and Eq. (\[eq:glm\]) shows that $\alpha_0^2= \cos^2(\delta-\delta_W)$. At the FL, ${G_{}^S}={{\mathcal G}_2}$, and Eq. (\[eq:gfl\]) shows that $(6/{\pi}^2)\alpha_1^2=\sin^2(\delta-\delta_W)$. These two results turn Eq. (\[eq:ga0a1set\]) into the mapping $$\label{eq:guniversal}
G{\bm{\left(}}\frac{T}{T_K}{\bm{\right)}}-\frac{{{\mathcal G}_2}}2 = -
{\bm{\left(}}{G_{}^S}{\bm{\left(}}\frac{T}{T_K}{\bm{\right)}}-\frac{{{\mathcal G}_2}}2{\bm{\right)}}\cos2(\delta-\delta_W).$$
Illustrative numerical results\[sec:illustration\]
--------------------------------------------------
Equation (\[eq:tk\]) offers an approximation for $T_K$, and Eqs. (\[eq:wSchW\]), (\[eq:deltalm\]) and (\[eq:deltafl\]) provide an approximation for the ground-state phase shift $\delta$. These estimates are far from the accuracy needed to fit numerical or experimental data. In the laboratory, $T_K$ and $\delta-\delta_W$ are adjustable parameters; the former, in particular, is determined by the condition $G(T_K) = {{\mathcal G}_2}/2$.[@GGK+98.5225; @KSA+06:36; @SAK+05:066801] In the computer office, the two unknown parameters on the right-hand side of Eq. (\[eq:guniversal\]) can can be extracted from the conductance itself, or from other properties of the model Hamiltonian. The phase shift $\delta$ is most easily obtained from the ground-state eigenvalues of ${H_A}$. To determine the Kondo temperature $T_K$, it has been traditional to fit the thermal dependence of the magnetic susceptibility $\chi(T)$ with the universal curve for $k_B
T\chi(T/T_K).$[@KWW80:1003; @TW83:453] Here, however, we prefer the laboratory definition, which insures that both sides of Eq. (\[eq:guniversal\]) vanish at $T=T_K$.
Figure \[fig:6\] displays the results of two NRG runs for the same assymetric parameters $U=3\,D$, $\epsilon=-0.3\,D$, $\Gamma=0.1\,D$, with two scattering potentials $W=0$ and $W=-0.6\,D$. The open circles reproduce the dashed curve in Fig. \[fig:4\]. The particle-hole asymmetry, combined with the relatively high ratio between the dot width and excitation energy, $\Gamma/|\epsilon|=1/3$, place the model Hamiltonian close to the border of the Kondo domain. Although $W=0$, the ground-state phase shift deviates significantly from $\pi/2$: from the FL eigenvalues generated by the NRG diagonalization of the model Hamiltonian, we find $\delta=0.43\pi$. The solid curve through the center of the circles is a plot of Eq. (\[eq:guniversal\]) with $\delta-\delta_W=0.43\pi$ and $k_BT_K=4\times10^{-3}\,D$.
![Numerical data for the temperature dependence of the conductance, compared to Eq. (\[eq:guniversal\]). The open circles and triangles show the NRG computed conductances for the indicated model parameters. The solid lines represent the mapping , with ground-phase shifts calculated from the FL eigenvalues of the model Hamiltonian and $T_K$ determined by the condition $G(T_K)={{\mathcal G}_2}/2$. The small disagreement between the triangles and the solid line above $k_BT=10^{-2}\,D$ is due to the relatively large irrelevant operators introduced by the scattering potential $W$, whose contribution to $G$ decays in proportion to $k_BT/D$.[]{data-label="fig:6"}](fig6){width="0.99\columnwidth"}
The scattering potential $W=-0.6\,D$ reduces the Kondo temperature and raises the FL conductance $G(T=0)$. The former shrinks to $k_BT_K=2.2\times10^{-8}\,D$, while the latter rises to nearly ${{\mathcal G}_2}$. Both changes are due to the reduced dot width ${\Gamma_{W}}=
\cos\delta_W \Gamma$. Here Eq. (\[eq:gammaW\]) yields $\rho{\Gamma_{W}}=0.74\rho\Gamma=0.074$, and the resulting smaller antiferromagnetic coupling (\[eq:jk\]) brings the Kondo temperature (\[eq:tk\]) down five orders of magnitude.
The diminished Kondo temperature indicates that the scattering potential has pushed the model Hamiltonian deeper inside the Kondo regime. Other indications are the minute high-temperature conductance; the nearly ballistic low-temperature conductance; and the overall similarity between $G(T/T_K)$ and the solid line in Fig. \[fig:4\].
Discussion\[sec:discussion\]
----------------------------
In the Kondo regime, Eqs. (\[eq:glm\]) and (\[eq:gfl\]) fix the high- and the low-temperature conductances, respectively. Equation (\[eq:guniversal\]) shows that the universal function ${G_{}^S}(T/T_K)$ controls the monotonic transition between the two limits. For $W=0$, in particular, the fixed-point values depend only on the ground-state phase shift $\delta$ and are symmetric with respect to ${{\mathcal G}_2}/2$: $G_{LM}={{\mathcal G}_2}\cos^2\delta$ and $G_{FL}={{\mathcal G}_2}\sin^2\delta$. Thus, depending on $\delta$, the transition from $G_{LM}$ to $G_{FL}$ can be steeper or flatter. Since $\delta$ can never depart much from $\pi/2$ in the Kondo regime, the argument of the trigonometric function on the right-hand side of Eq. (\[eq:guniversal\]) can never depart substantially from $\pi$, and as indicated by the two curves in Fig. \[fig:4\], $G(T/T_K)\approx{G_{}^S}(T/T_K)\pm20\%$. By contrast with this crude estimate, the mapping (\[eq:guniversal\]) gives excellent agreement with the circles in Fig. \[fig:6\].
The wire potential $W$ narrows the dot level and displaces the ground-state phase shift. Depending on the sign and magnitude of $W$, the phase shift can take any value in its domain of definition $-\pi/2\le \delta\le\pi/2$. In the Kondo regime, the Friedel sum rule nonetheless prevents the difference $\delta-\delta_W$ from straying away from $\pi/2$. All effects considered, the scattering potential $W$ displaces the conductance curve towards the symmetric limit $G(T/T_K)={G_{}^S}(T/T_K)$.
These findings are in line with the experimentally established notion that, in the Kondo regime, SET conductances always decay with temperature.[@GGK+98:5225; @GSM+98.156; @WFF+00:2105; @LSB02:725] A brief comparison between this behavior and that of the side-coupled device [@KAS+04:035319; @SAK+05:066801] seems appropriate. As demonstrated in Ref. , a linear mapping analogous to Eq. (\[eq:guniversal\]) can be established between the side-coupled conductance and ${G_{}^S}(T/T_K)$; in that case, however, the coefficient relating the two functions is independent of $\delta_W$ and hence free from the constraint imposed by the Friedel sum rule. Under a sufficiently strong wire potential, its sign can be reversed. Thus, the thermal dependence of $G_{SC}$ is tunable:[@Kat07:233201] a wire potential can turn a monotonically increasing function into a monotonically decreasing one. The embedded geometry of Fig. \[fig:1\] is much less sensitive to $W$.
The parameter $\delta_W$ is ($\pi$ times) the charge induced under the wire electrodes by the potential $W$. According to the Friedel sum rule, [@La66:516] the difference $\delta-\delta_W$ is the charge of the Kondo cloud, the additional charge that piles up at the wire tips surrounding the dot as the temperature is lowered past $T_K$. Neutrality makes the charge of the Kondo cloud equal to the dot occupancy. Since the symmetric condition $n_d=1$ maximizes the the low-temperature conductance, one expects $G(T=0)$ to be ballistic for $2(\delta-\delta_W)=\pi$, a conclusion in agreement with Eq. (\[eq:guniversal\]). Since the screening charge is always nearly unitary, one expects the low-temperature conductance to be close to the conductance quantum, in agreement with the plots in Fig. \[fig:6\].
Conclusions\[sec:conclusions\]
==============================
Our central result, Eq. (\[eq:guniversal\]) maps the conductance in the embedded geometry onto the universal conductance for the symmetric Anderson model (\[eq:hasym\]). Different from the universal result for the side-coupled geometry, the mapping depends explicitly on the potential $W$ applied to the wire. Section \[sec:cross\] showed that, in the Kondo regime, the Friedel sum rule anchors the the argument of the cosine on the right-hand side of Eq. (\[eq:guniversal\]) to the vicinity of $\pi$; it results that $G(T/T_K)$ reproduces semiquantitatively the universal function ${G_{}^S}(T/T_K)$.
At the quantitative level, Eq. (\[eq:guniversal\]) affords comparison with experimental data collected anywhere in the Kondo regime. For that purpose, its linearity is particularly convenient. Once fitted to a set of experimental points, the mapping determines the Kondo temperature $T_K$, as well as the phase shift difference $\delta-\delta_W$. Both are quantities of physical significance. According to the Friedel sum rule, the phase shift difference is ($\pi/2$ times) the screening charge surrounding the dot at low temperatures.
In summary, we have derived an exact expression relating the SET conductance in the Kondo regime to the universal conductance function for the symmetric Anderson Hamiltonian. A subsequent report will exploit this result in an attempt to offer a unified view of an NRG survey of conductance in the Kondo regime.
This work was supported by the CNPq and FAPESP.
Properties of the fixed-point Hamiltonians\[sec:diag\]
======================================================
Diagonalization\[sec:diagonalization\]
--------------------------------------
The LM and FL are described by conduction-band Hamiltonians of the form $$\label{eq:fpHamilt}
H^*= \sum_k \epsilon_k a_k^\dagger a_k + W^*f_0^\dagger f_0.$$
We want to bring $H^*$ to the diagonal form $$\label{eq:diagFPHamilt}
H^* = \sum_\ell \varepsilon_\ell g_\ell^\dagger g_\ell,$$ where $$\label{eq:gks}
g_\ell = \sum_q \alpha_{\ell q}a_q.$$ To this end, we compare the expressions for the commutator $[g_\ell,H^*]$ obtained from Eqs. (\[eq:fpHamilt\]) and (\[eq:diagFPHamilt\]), from which it follows that $$\label{eq:alphaks}
\alpha_{\ell q} = \frac{1}{\varepsilon_\ell -\epsilon_q}\frac{W^*}{N}\sum_k
\alpha_{\ell k}.$$
Summation of both sides over $q$ then leads to the eigenvalue condition: $$\label{eq:eigenval}
1 = \frac{W^*}{N} \sum_q\frac{1}{\varepsilon_\ell -\epsilon_q}.$$ Inspection of this equality shows that, with exception of a split-off energy, which makes $\mathcal{O}(1/N)$ contributions to the low-energy properties, the $\varepsilon_\ell$ are shifted by less than $\Delta$ from the $\epsilon_k$. We therefore refer to the closest conduction energy $\epsilon_\ell$ to label each eigenvalue and define its phase shift $\delta_\ell$ with the expression $$\label{eq:delta}
\varepsilon_\ell \equiv\epsilon_\ell-\frac{\Delta}{\pi}\delta_\ell.$$
This definition substituted in Eq. (\[eq:eigenval\]), a Sommerfeld-Watson transformation[@matthews71:_mathem_method_of_physic] determines the sum on the right-hand side: $$\label{eq:sommer-wat}
\frac1N\sum_q\frac{1}{\varepsilon_\ell -\epsilon_q} = -\pi\rho \cot
\delta_\ell +\rho \,\fint_{-D}^D\frac{1}{\varepsilon_\ell -\epsilon}\,d\epsilon.$$ Substitution in Eq. (\[eq:eigenval\]) results in an expression for the phase shifts: $$\label{eq:cotdelta}
\cot\delta_\ell = -\frac1{\pi\rho W^*}
+\frac1{\pi} \,\fint_{-D}^D\frac{1}{\varepsilon_\ell -\epsilon}\,d\epsilon.$$ At low energies, the contribution of the last term on the right-hand side, of $\mathcal{O}(\epsilon/D)$, can be neglected, which shows that the phase shift becomes uniform: $$\label{eq:tandelta}
\tan\delta= -\pi\rho W^*.$$
To determine the coefficients $\alpha_{\ell q}$, we square both sides of Eq. (\[eq:alphaks\]) and sum the result over $q$: $$\label{eq:alpha2}
\sum_q\alpha^2_{\ell q} = {\bm{\left(}}\sum_k\alpha_{\ell k}\frac{W^*}{N}{\bm{\right)}}^2
\sum_q\frac{1}{(\varepsilon_\ell -\epsilon_q)^2}.$$ To evaluate the sum on the left-hand side, we differentiate Eq. (\[eq:sommer-wat\]) with respect to $\epsilon_\ell$, which yields, with relative error $\mathcal{O}(1/N)$, $$\label{eq:sumeps2}
\frac1{N^2}\sum_q\frac{1}{(\varepsilon_\ell -\epsilon_q)^2} =
{\bm{\left(}}\frac{\pi\rho}{\sin \delta_\ell}{\bm{\right)}}^2.$$
The sum on the left-hand side of Eq. (\[eq:alpha2\]) being unitary, Eq. (\[eq:sumeps2\]) shows that $$\label{eq:eigenvect}
W^*\sum_k\alpha_{\ell k} = -\frac1{\pi\rho} \sin\delta_\ell,$$ the negative phase insuring that $\alpha_{k k}\to1$ for $W^*\to0$. Equation (\[eq:alphaks\]) then gives $$\label{eq:alphasergs}
\alpha_{\ell q} = \frac{\Delta}{\epsilon_q-\varepsilon_\ell}
\frac{\sin\delta_\ell}{\pi}\qquad(\varepsilon_\ell\ll D).$$
Energy moments of the matrix elements of the eigenoperators $g_\ell$ {#sec:higher}
--------------------------------------------------------------------
The Hamiltonian (\[eq:fpHamilt\]) diagonalized, we turn our attention to the following energy moments $$\label{eq:moments}
{M_{mn}^{\,(p)}}\equiv\frac1{\sqrt{N}}\sum_\ell{\bm{\left(}}\frac{\varepsilon_\ell}{D}{\bm{\right)}}^p{\langlem|\,
g_\ell\,|n\rangle}\qquad(p=0,1,\ldots).$$ Chiefly important are ${M_{mn}^{\,(0)}}\equiv{\langlem|\,\phi_0\,|n\rangle}$ and ${M_{mn}^{\,(1)}}\equiv{\langlem|\,\phi_1\,|n\rangle}/\sqrt3$; the other moments, as shown below, are proportional to ${M_{mn}^{\,(1)}}\equiv{\langlem|\,\phi_1\,|n\rangle}$. Since the $M_{mn}^p$ are universal, to evaluate them it is sufficient to consider the symmetric Hamiltonian (\[eq:hasym\]), for which the phase shift $\delta_{LM}=0$, so that $g_k$, $\varepsilon_k$, $\phi_0$, and $\phi_1$ coincide with $a_k$, $\epsilon_k$, $f_0$, and $f_1$, respectively.
From Eq. (\[eq:hasym\]), we then have that $$\label{eq:syscomm}
[g_\ell,{H_A}^S] = \varepsilon_\ell g_{\ell} + \frac V{\sqrt N}{c_{d}}.$$ With the shortand ${\mathcal{E}_{mn}}\equiv(E_m-E_n)/D$, the multiplication of both sides by $(\varepsilon_\ell/D)^{p-1}$ followed by summation over $\ell$ leads to the coupled recursive relations $$\begin{array}{lll}
{M_{mn}^{\,(p)}} =& -{\mathcal{E}_{mn}}{M_{mn}^{\,(p-1)}}-
\displaystyle\frac V{p}{\langlem|\,{c_{d}}\,|n\rangle}
&\qquad(p=1,3,\ldots);\\
{M_{mn}^{\,(p)}}=&-{\mathcal{E}_{mn}}{M_{mn}^{\,(p-1)}}&\qquad(p=2,4,\ldots).
\end{array}$$ Reduced to a matrix equation, this system is easily solved. The result is $${M_{mn}^{\,(p)}} =\left\{\begin{array}{ll}
{M_{mn}^{\,(0)}}{\mathcal{E}_{mn}}^p-\displaystyle\frac Vp{\langlem|\,c_d\,|n\rangle}{\bm{\left(}}1+
{\displaystyle\sum_{r=1}^{p-1}}
\mbox{\raisebox{7pt}{$\prime$}}\frac{{{\mathcal{E}_{mn}}}^r}{r}{\bm{\right)}}&\quad(p=\mathrm{odd})\nonumber\\
{M_{mn}^{\,(0)}}{\mathcal{E}_{mn}}^p-\displaystyle\frac Vp{\langlem|\,c_d\,|n\rangle}
{\displaystyle\sum_{r=1}^{p-1}}
\mbox{\raisebox{7pt}{$\prime$}}\frac{{{\mathcal{E}_{mn}}}^r}{r}&\quad(p=\mathrm{even})\nonumber
\label{eq:matrixmv}
\end{array}\right.,$$ where the primed sum is restricted to odd $r$’s.
The pertinent energies satisfy the condition $k_BT\ll D$, which implies ${\mathcal{E}_{mn}}\ll 1$. It is therefore safe to discard the terms proportional to ${\mathcal{E}_{mn}}$ and its powers. Within this approximation, the only nonzero even moment is ${M_{mn}^{\,(0)}}$, and all odd moments are proportional to ${\langlem|\,{c_{d}}\,|n\rangle}$. It follows that all the odd moments are proportional to ${M_{mn}^{\,(1)}}={\langlem|\,\phi_1\,|n\rangle}$: $$\label{eq:mmnsvsmmn1}
{M_{mn}^{\,(p)}}=\left\{
\begin{array}{ll}
\displaystyle\frac{{\langlem|\,\phi_1\,|n\rangle}}p&\qquad(p=\text{odd})\\
0&\qquad(p=2,4,\ldots)
\end{array}\right..$$
An orthonormal basis describing the conduction band (\[eq:diagFPHamilt\]) can be constructed from the definition $$\label{eq:lagrange}
\phi_p \equiv
\sqrt{\frac{2p+1}N}\sum_{\ell}P_p(\epsilon_\ell)g_{\ell}
\qquad(p=0,1,\ldots),$$ where $P_p(\epsilon)$ denotes a Legendre polynomial.
According to Eq. (\[eq:mmnsvsmmn1\]), ${\langlem|\,\phi_p\,|n\rangle}\sim{\langlem|\,\phi_1\,|n\rangle}$ ($p=3,5,\ldots$), while ${\langlem|\,\phi_p\,|n\rangle}=0$ ($p=2,4,\ldots$). This shows that the matrix element of any conduction operator is a linear combination of ${\langlem|\,\phi_0\,|n\rangle}$ and ${\langlem|\,\phi_1\,|n\rangle}$. In particular $$\label{eq:f0matelphi0}
{\langlem|\,f_i\,|n\rangle} = \sum_{j=0}^1\alpha_{ij}{\langlem|\,\phi_j\,|n\rangle}\qquad(i=0, 1),$$ where $f_0$ and $f_1$ are the operators defined by Eqs. (\[eq:f0\]) and (\[eq:f1\]), respectively, and the $\alpha_{ij}$ ($i,j=0,1$) are constants that depend on the model parameters.
Fixed-point conductances\[sec:extens-langr\]
============================================
This appendix derives an expression for the spectral density $\rho_d(\epsilon,T)$, defined by Eq. (\[eq:rhod\]), at the fixed points. The procedure is analogous to the one in Appendix \[sec:diag\].
From Eq. (\[eq:ha\_comm\_aq\]) we obtain an expression for the matrix element of $a_q$ between two low-energy eigenstates ${|m\rangle}$ and ${|n\rangle}$ of eigenstates of ${H_A}$: $$\begin{aligned}
\label{eq:amatel}
{\langlem|\,a_q^\dagger\,|n\rangle} &=& \frac1{\sqrt{N}}\frac{V}{E_m-E_n-\epsilon_q}
{\langlem|\,{c_{d}^\dagger}\,|n\rangle}\nonumber\\
&&+\frac W{N}\frac{1}{E_m-E_n-\epsilon_q}
{\langlem|\,\sum_p a_p^\dagger\,|n\rangle}.\end{aligned}$$ Summation of both sides over $q$ leads to an expression for the matrix element in the last term on the right-hand side: $$\begin{aligned}
{\langlem|\,\sum_p a_p^\dagger\,|n\rangle} {\bm{\left(}}1-
W{\mathcal{S}_{mn}}{\bm{\right)}}=\sqrt N V{\langlem|\,{c_{d}^\dagger}\,|n\rangle}{\mathcal{S}_{mn}},\end{aligned}$$ where $$\label{eq:smn}
{\mathcal{S}_{mn}}\equiv\frac1N\sum_q\frac{1}{E_m-E_n-\epsilon_q},$$ which brings Eq. (\[eq:amatel\]) to the form $$\label{eq:amatelnosum}
{\langlem|\,a_q^\dagger\,|n\rangle} =
\frac{{\langlem|\,{c_{d}^\dagger}\,|n\rangle}}{\sqrt N(E_m-E_n-\epsilon_q)}
\frac{V}{1-W{\mathcal{S}_{mn}}}.$$
Consider now this equality at one of the two fixed points, LM or FL. The fixed-point Hamiltonian has then the quadratic form (\[eq:diagFPHamilt\]), which defines the complete basis of the operators $g_\ell$. The matrix element ${\langlem|\,g_\ell^\dagger\,|n\rangle}$ vanishes unless ${|m\rangle}=g_\ell^\dagger{|n\rangle}$, which implies $E_m=E_n+\varepsilon_\ell$. At a fixed point, therefore, the sum on the right-hand side of Eq. (\[eq:smn\]) reduces to that in Eq. (\[eq:sommer-wat\]), [i. e.,]{}$$\label{eq:smnFP}
{\mathcal{S}_{mn}}= -\pi\rho \cot \delta_*,$$ where we have disconsidered the last term on the right-hand side of Eq. (\[eq:sommer-wat\]) because at a fixed point the ratio $\varepsilon_\ell/D\to 0$. This result suggests that we introduce the phase shift $\delta_W$, defined by $$\label{eq:deltaw}
{W} \equiv -\frac{\tan\delta_W}{\pi\rho},$$ to simplify Eq. (\[eq:amatelnosum\]): $$\label{eq:amatelnosum_simple}
{\langlem|\,a_q^\dagger\,|n\rangle} =
\frac{V{\langlem|\,{c_{d}^\dagger}\,|n\rangle}}{\sqrt N(E_m-E_n-\epsilon_q)}
\frac{\sin\delta_*\cos\delta_W}{\sin(\delta_*-\delta_W)}.$$ In analogy with Eq. (\[eq:gks\]) we can, moreover, write $$\label{eq:gkAlphasCd}
g_\ell = \gamma_{\ell0} {c_{d}}+ \sum_q\gamma_{\ell q}a_q,$$ with normalized coefficients: $$\label{eq:norm_gks}
\gamma_{\ell 0}^2+\sum_q \gamma_{\ell q}^2 = 1.$$
At each fixed point, therefore, once squared, Eq. (\[eq:amatelnosum\]) reads $$\label{eq:aq_cd_sum}
\gamma_{\ell q}^2=
\frac{V^2\gamma_{\ell 0}^2}{N(\varepsilon_\ell-\epsilon_q)^2}
{\bm{\left(}}\frac{\sin\delta_*\cos\delta_W}{\sin(\delta_*-\delta_W)} {\bm{\right)}}^2.$$ We divide both sides by $N$, sum them over $q$, and substitute Eq. (\[eq:sumeps2\]) for the resulting sum on the right-hand side, to find that $$\label{eq:aq_cd_delta}
\sum_q\gamma_{\ell q}^2= {NV^2}{\gamma_{\ell 0}^2}
{\bm{\left(}}\frac{\pi\rho\cos\delta_W}{\sin(\delta_*-\delta_W)} {\bm{\right)}}^2.$$
Substitution in the second term on the left-hand side of Eq. (\[eq:norm\_gks\]) now shows that, with error $\mathcal{O}(1/N)$: $$\label{eq:gamma_l0}
|{\langlem|\,{c_{d}^\dagger}\,|n\rangle}|^2= \frac1{NV^2}
\frac{\sin^2(\delta_*-\delta_W)}{\pi^2\rho^2\cos^2\delta_W}.$$
A the fixed points, the matrix elements are constants, dependent only on the phase shift and scattering potential. The spectral density $\rho_d$, as one would expect, becomes independent of the temperature: $$\rho_{d}(\epsilon,T)=\frac1{NV^2{{\mathcal{Z}}}}
\sum_{m,n} e^{-\beta E_m}
\frac{\sin^2(\delta_*-\delta_W)}
{\pi^2\rho^2\cos^2\delta_W}\delta(\epsilon_\ell-\epsilon),$$ equivalent to $$\label{eq:rho_d}
\rho_{d}(\epsilon)=\frac{\sin^2(\delta_*-\delta_W)}
{\pi\Gamma\cos^2\delta_W}.$$
$W=0$ recovers the celebrated expression[@La66:516] $$\label{eq:langreth}
\rho_{d}(\epsilon)=\frac{\sin^2\delta_*}
{\pi\Gamma}.$$ More generally, however, to obtain the fixed-point spectral densities, we set $\delta^*=\delta$ at the FL, and $\delta^*=\delta-\pi/2$ at the LM, from which it results that
\[eq:rho\_FPs\] $$\begin{aligned}
\label{eq:rho_LM}
\rho_d^{LM}&=&
\frac{\cos^2(\delta-\delta_W)}{\pi\Gamma\cos^2\delta_W};\\
\rho_d^{FL}&=&
\frac{\sin^2(\delta-\delta_W)}{\pi\Gamma\cos^2\delta_W}.
\label{eq:rho_FL}
\end{aligned}$$
Substitution of Eqs. (\[eq:rho\_LM\]) and (\[eq:rho\_FL\]) for $\rho_d$ on the right-hand side of Eq. (\[eq:glin\]) leads to Eqs. (\[eq:glm\]) and (\[eq:gfl\]), respectively.
Zero-bias conductance\[sec:expr-cond\]
======================================
By contrast with the coupling to the impurity, which is independent of the odd operators $b_k$ defined by Eq. (\[eq:bk\]), the Hamiltonian describing a bias voltage couples to the $b_k$’s. Preliminary to the discussion of the conductance, it is therefore convenient to derive results for the $b_k$’s analogous to those in Appendix \[sec:diag\]. Specifically, given the formal equivalence between Eqs. (\[eq:hb\]) and (\[eq:fpHamilt\]), we can follow the steps in that Appendix to write $H_B$ in the diagonal form $$\label{eq:hbdiag}
H_B=\sum \tilde\varepsilon_\ell {\tilde g}^\dagger_\ell\tilde g_\ell,$$ with $$\label{eq:glbetakl}
\tilde g_\ell = \sum_k \tilde \alpha_{\ell,k} b_k,$$ and to derive a result analogous to Eq (\[eq:eigenvect\]). At low energies, in particular, [i. e.,]{} for $|\tilde\varepsilon_\ell|\ll D$, the eigenvalues $\tilde\varepsilon_\ell$ are uniformly phase shifted by $\delta_W$ \[see Eq. (\[eq:deltaw\])\], and $$\sum_k \tilde\alpha_{\ell,k} = \cos\delta_W\label{eq:sumtildalphas}.$$ Multiplication of both sides by $\tilde g_\ell$ and summation over $\ell$ then shows that $$\label{eq:gl_matell}
\sum_k{\langle\tilde m|\,b_k\,|\tilde n\rangle} =\cos\delta_w \sum_\ell{\langle\tilde
m|\,\tilde g_\ell\,|\tilde n\rangle},$$ for any pair ${|\tilde m\rangle},{|\tilde n\rangle}$ of low-energy eigenstates of $H_B$.
It is likewise convenient to compute the following commutator: $$\label{eq:abmatel_raw}
[{H_A},a_k^\dagger b_k] = \frac{V}{\sqrt{N}} {c_{d}^\dagger}\, b_k
+\frac{W}{N}\sum_q(a_q^\dagger b_k- a_k^\dagger b_q),$$ from which we see that, given two eigenstates ${{|\Psi_m\rangle}}$ and ${|n\rangle}$ of ${H_A}$ with eigenvalues $E_m$ and $E_n$, respectively, $$\begin{aligned}
\label{eq:abmatel}
{\langle{\Psi_m}|a_k^\dagger b_k|{\Psi_{n}}\rangle} &=&
\frac{V}{\sqrt{N}}\frac{{\langle{\Psi_m}|{c_{d}^\dagger}\,b_k|{\Psi_{n}}\rangle}}{E_m-E_n}\nonumber\\
&&+\frac{W}{N}\sum_q\frac{{\langle{\Psi_m}|a_q^\dagger b_k- a_k^\dagger b_q|{\Psi_{n}}\rangle}}{E_m-E_n}.\end{aligned}$$
Current\[sec:current\]
----------------------
To calculate the conductance, we can, for instance, examine the current flowing into the $R$ wire: $$\hat I = \frac{dq_R}{dt}=-\frac{i{\mathrm{e}}}{\hbar}
[{H_A},\sum_{k} c_{kR}^\dagger c_{kR}],$$ [i. e.,]{} $$\label{eq:current}
\hat I =-\frac{i{\mathrm{e}}}{2\hbar}
[{H_A},\sum_{k}{\bm{\left(}}a_{k}^\dagger a_{k}+ b_{k}^\dagger b_{k}
-(a_{k}^\dagger b_{k}+{{\text{H.\ c.}}}){\bm{\right)}}],$$ which is equivalent to $$\label{eq:currentNoW}
\hat I = \frac{i{\mathrm{e}}}{2\hbar}\frac{V}{\sqrt N}\,{c_{d}^\dagger}\sum_{k}(a_k+b_k)+{{\text{H.\ c.}}},$$ because summed over $k$, the last term on the right-hand side of Eq. (\[eq:abmatel\_raw\]) vanishes.
Conductance\[sec:conductance\]
------------------------------
To induce a current, we add to the model Hamiltonian an infinitesimal, slowly growing perturbation that lowers the chemical potential of the $R$ wire relative to that of the $L$ wire: $$H_\mu \equiv \Delta\mu\,{h_{\mu}}(t) = -{\mathrm{e}}\frac{\Delta\mu}2 \sum_{k}{\bm{\left(}}c_{kR}^\dagger
c_{kR}-c_{kL}^\dagger c_{kL}{\bm{\right)}}e^{\eta t/\hbar},$$ with an infinitesimal shift $\Delta\mu$.
Projected on the basis of the $a_k$’s and $b_k$’s, ${h_{\mu}}$ reads $$\label{eq:hnu}
{h_{\mu}}(t) = -\frac{{\mathrm{e}}}2\sum_{k}(a_{k}^\dagger b_{k}+{{\text{H.\ c.}}})e^{\eta t/\hbar},$$ and Eq. (\[eq:abmatel\]) shows that $$\label{eq:hnumatel}
{\langle{\Psi_m}|{h_{\mu}}(t)|{\Psi_{n}}\rangle} =-\frac{{\mathrm{e}}V}{2\sqrt{N}}e^{\eta t/\hbar}\sum_{k>0}
\frac{{\langle{\Psi_m}|{c_{d}^\dagger}b_k- b_k^\dagger{c_{d}}|{\Psi_{n}}\rangle}}{E_m-E_n}.$$
Standard Linear Response Theory relates ${h_{\mu}}$ to the conductance: $$\label{eq:gih}
G(T) = -\frac{i}{{\mathcal{Z}}\hbar}
\int_{-\infty}^{0}\sum_m e^{-\beta E_m}{\langle{\Psi_m}|[\hat I,{h_{\mu}}(t)]|{\Psi_{m}}\rangle}\,dt,$$ where ${\mathcal{Z}}$ is the partition function at the temperature $T$.
Comparison with Eq. (\[eq:hnu\]) shows that the operators $a_k$ within the parentheses on the right-hand side of that equality make no contribution to the conductance. We therefore define $$\label{eq:ib}
\hat I_b \equiv \frac{i{\mathrm{e}}}{2\hbar}\frac{V}{\sqrt N}\,{c_{d}^\dagger}\sum_k b_k +{{\text{H.\ c.}}},$$ and rewrite Eq. (\[eq:gih\]): $$\label{eq:gcom}
G(T) = -\frac{i}{{\mathcal{Z}}\hbar}
\int_{-\infty}^{0}\sum_m e^{-\beta E_m}{\langle{\Psi_m}|[\hat I_b,{h_{\mu}}(t)]|{\Psi_{m}}\rangle}\,dt.$$
Following the insertion of a completeness sum $\sum_n{|n\rangle}{\langlen|}$ on the right-hand side of Eq. (\[eq:gcom\]), straightforward manipulations lead to the familiar expression $$G(T) = \frac{1}{{\mathcal{Z}}}
\sum_{m,n} {\bm{\left(}}e^{-\beta E_m}-e^{-\beta E_n}{\bm{\right)}}\frac{{\langle{\Psi_m}|\hat I_b|{\Psi_{n}}\rangle}{\langle{\Psi_n}|{h_{\mu}}(0)|{\Psi_{m}}\rangle}}{E_m-E_n+i\eta}.$$
On the right-hand side, we now substitute Eq. (\[eq:hnumatel\]) \[Eq. (\[eq:ib\])\] for ${h_{\mu}}$ ($\hat I_b$). This yields $$\begin{aligned}
\displaystyle
G(T) = -i\frac{{\mathrm{e}}^2}{4\hbar}\frac{V^2}{N{\mathcal{Z}}} \sum_{m,n,k,q}
&\displaystyle{\bm{\left(}}\frac{{\langle{\Psi_m}|b_{q}^\dagger
{c_{d}}|{\Psi_{n}}\rangle}{\langle{\Psi_n}|{c_{d}^\dagger}b_{k}|{\Psi_{m}}\rangle}}{E_m-E_n+i\eta}\right.\nonumber\\
&\left.\displaystyle+\frac{{\langle{\Psi_m}|{c_{d}^\dagger}b_k|{\Psi_{n}}\rangle}
{\langle{\Psi_n}|b_{q}^\dagger {c_{d}}|{\Psi_{m}}\rangle}}
{E_m-E_n+i\eta}{\bm{\right)}}\nonumber\\
&\times\displaystyle\frac{e^{-\beta E_m}-e^{-\beta E_n}}{E_m-E_n}.
\end{aligned}$$ Aided by Eq. (\[eq:gl\_matell\]), we can now trade the sum over the conduction operators $b_k$ for a sum over the eigenoperators $\tilde g_{\ell}$:
$$\begin{aligned}
\displaystyle
G(T) = -i\frac{{\mathrm{e}}^2}{2h}\frac{{\Gamma_{W}}}{N\rho{\mathcal{Z}}}
\sum_{m,n,\ell,\ell'}&&
\displaystyle{\bm{\left(}}\frac{{\langle{\Psi_m}|\tilde g_{\ell'}^\dagger
{c_{d}}|{\Psi_{n}}\rangle}{\langle{\Psi_n}|{c_{d}^\dagger}\tilde g_{\ell}|{\Psi_{m}}\rangle}}{E_m-E_n+i\eta}\right.\nonumber\\
&&\left.\displaystyle+\frac{{\langle{\Psi_m}|{c_{d}^\dagger}\tilde g_\ell|{\Psi_{n}}\rangle}{\langle{\Psi_n}|\tilde g_{\ell'}^\dagger {c_{d}}|{\Psi_{m}}\rangle}}
{E_m-E_n+i\eta}{\bm{\right)}}\nonumber\\
&\times&\frac{e^{-\beta E_m}-e^{-\beta E_n}}{E_m-E_n}.
\end{aligned}$$
Since the $\tilde g_\ell$ diagonalize $H_B$, only the terms with $\ell=\ell'$ contribute to the sum on the right-hand side. We interchange the indices $m$ and $n$ in the second term within the parentheses on the right-hand side to show that $$\label{eq:gdelta}
G(T) = \frac{\pi{\mathrm{e}}^2}{h}\frac{\beta\,\Gamma_w}{\rho N{\mathcal{Z}}} \sum_{m,n,\ell}e^{-\beta E_m}
|{\langle{\Psi_m}|{c_{d}^\dagger}g_{\ell}|{\Psi_{n}}\rangle}|^2\delta(E_m-E_n).$$
Since ${{|\Psi_m\rangle}}={|m\rangle}{|\tilde m\rangle}$, where ${|m\rangle}$ (${|\tilde
m\rangle}$) is an eigenstate of $H_A$ (of the quadratic Hamiltonian $H_B$), the right-hand side splits into two coupled sums: $$\begin{aligned}
\label{eq:oneboltzmann}
G(T) =\frac{\pi{\mathrm{e}}^2}{h}\frac{\beta\,{\Gamma_{W}}}{N\rho{\mathcal{Z}}}
&\displaystyle\sum_{m,n,\ell}e^{-\beta E_{m}}
|{\langlem|\,V{c_{d}^\dagger}\,|n\rangle}|^2\delta(E_{m}-E_{n}-\tilde\epsilon_\ell)\nonumber\\
&\displaystyle\times\sum_{\tilde m,\tilde n}e^{-\beta E_{\tilde m}}{\langle\tilde
m|\,g_{\ell}\,|\tilde n\rangle}{\langle\tilde n|\,g_\ell^\dagger\,|\tilde m\rangle}.\end{aligned}$$
The second sum is equal to ${\mathcal{Z}}_b[1-f(\tilde\epsilon_p)]$, where $f(\epsilon) $ is the Fermi function, and ${\mathcal{Z}}_b$, the partition function for the Hamiltonian $H_b$. The identity $$-\frac1{f(\epsilon)}\frac{\partial f}{\partial \epsilon}= \beta {\bm{\left(}}1-f(\epsilon){\bm{\right)}}$$ then turns Eq. (\[eq:oneboltzmann\]) into $$\begin{aligned}
\label{eq:gforha}
G(T) = \frac{{\mathrm{e}}^2}{h{\mathcal{Z}}_a}\frac{\pi{\Gamma_{W}}}{\rho N}
\sum_{m,n,\ell}&\displaystyle\frac{e^{-\beta E_{m}}}{f(\tilde\epsilon_\ell)}
{\bm{\left(}}-\frac{\partial f}{\partial \epsilon}{\bm{\right)}}_{\tilde\varepsilon_\ell}
|{\langlem|\,{c_{d}^\dagger}\,|n\rangle}|^2\nonumber\\&\times\,\delta(E_{m}-E_{n}-\tilde\epsilon_\ell),\end{aligned}$$ where ${\mathcal{Z}}_a$ is the partition function for the Hamiltonian ${H_A}$.
The definition (\[eq:rhod\]) of the spectral density $\rho_d(\epsilon, T)$ allows us to rewrite Eq. (\[eq:gforha\]) as $$G(T) = \frac{{\mathrm{e}}^2}{h}\frac{\pi{\Gamma_{W}}}{\rho N}
\sum_{\ell} {\bm{\left(}}-\frac{\partial f}{\partial
\epsilon}{\bm{\right)}}_{\tilde\varepsilon_\ell}\rho_{d}(\tilde\epsilon_\ell),$$ from which Eq. (\[eq:glin\]) follows.
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---
abstract: 'The collapse and revival of quantum states appear in diverse areas of physics. In quantum optics the occurrence of such a phenomena in the evolution of an atomic state, interacting with a light field initially in a coherent state, was predicted by using the Jaynes-Cummings model (JCM), and subsequently demonstrated experimentally. In this paper we revisit the JCM with the Monte-Carlo wave function approach and investigate the time evolution of the photon emission rate of the atom in a cavity. Analytical and numerical quantum trajectory calculations show that the cavity and the initial field statistics strongly influence the photon emission dynamics. A coherent field indeed gives rise to a collapse and revival behavior that mirrors atomic state evolution. However, there are differences between the two. The emission rate for a field in a Fock number state exhibits a sinusoidal oscillation, and there exists a quiescent period for a thermal field. These properties are quite different from those in free space. It is also shown that the fluctuation in photon emission is much less than that of the atomic population.'
author:
- Chang Jae
title: 'Photon Emission Dynamics of a Two-Level Atom in a Cavity'
---
[^1]
INTRODUCTION
============
The collapse and revival of quantum states by either forced or self-regulated processes is a fascinating phenomena with a long history. Hahn discovered spin echo [@spin_echo] in 1950, and photon echoes were detected at optical frequencies in mid 1960s [@photon_echo]. Analogous echo phenomena can be found in vibrational spectroscopy [@vib_echo] and in the dynamics of an atomic matter wave in a light field [@atom_echo; @tppi]. Echo techniques are of great interest in quantum computing and quantum state engineering applications where maintaining coherence is a critical requirement [@decoherence]. The time evolution of the population inversion of a two-level atom (TLA) interacting with a single-mode light field, as modeled by the JC Hamiltonian [@jcm], also shows the collapse [@cum] and revival behavior when the atom interacts with a field that is initially in a coherent state [@eberly; @rempe]. The JCM, despite simplicity and age, still provides a fertile ground for testing foundations of quantum theory and prototyping practical applications [@jopb_jcm].
Previously, we approached the JCM with a viewpoint based on the Monte-Carlo Wave Function (MCWF) method [@mcwf], and found that a TLA interacting with a field initially in a number state emits (and absorbs) photons with a unique counting statistics [@jkps]. In this paper, we revisit the JCM with fields in coherent and thermal states in addition to the number state. The goal is to examine what effects the field statistics has on emitted photons, rather than the usual atomic state dynamics. In the next section, we briefly describe our adaptation of the MCWF method, and after that, results of MCWF simulations are given along with a quantum trajectory analysis on the results. Finally, the main discoveries and further discussions are given in Conclusions section.
MCWF Method for Photon Emission
===============================
MCWF approach for the JCM dynamics
----------------------------------
We consider a fully-quantized Hamiltonian for a system of a TLA interacting with a single mode field in a lossless cavity. The Hamiltonian considers only the internal energy, neglecting the center-of-mass motion of the atom. With the use of the rotating-wave approximation for the interaction between the TLA and the quantized-field, the total Hamiltonian is given by: $$H = \frac{1}{2} \hbar\omega_{0}\sigma_{z} + \hbar\omega a^{\dagger}a + \frac{\hbar\Omega_{0}}{2} (a \sigma_{+} + a^{\dagger} \sigma_{-}).
\label{eq:jcm}$$ In the above, $\omega_{0} = (E_{e} - E_{g})/\hbar$ is the frequency between the upper and the lower states of the atom, $|e \rangle$ and $|g \rangle$. The frequency of the field is $\omega$, and $a$ and $a^{\dagger}$ are the field annihilation and creation operators, respectively, with the Fock number state being the eigenstate of the photon number operator, $a^{\dagger} a | n \rangle = n | n \rangle$. The Pauli matrices $\sigma_{z}, \sigma_{\pm}$ are operators for the atomic population and transitions $$\sigma_{z} = \begin{pmatrix}
1 & 0 \\
0 & -1
\end{pmatrix} ,
\sigma_{+} = \begin{pmatrix}
0 & 1\\
0 & 0
\end{pmatrix},
\sigma_{-} = \begin{pmatrix}
0 & 0\\
1 & 0
\end{pmatrix},$$ and $\Omega_{0}$ is the vacuum Rabi frequency for the atom-field interaction.
The evolution of the TLA-field system is governed by $$| \psi (t) \rangle = \exp \left( - \frac{i }{\hbar} H t \right) | \psi (0) \rangle .
\label{eq:evol}$$ In case that the atom is initially in the upper state and the field in the number state $|n \rangle $, namely $ | \psi (0) \rangle = |n , e \rangle $, the transition probability of the system to $| n+1 , g \rangle$ is given by [@louisell] $$P_{g}(t) = {\left| \langle n+1 , g | \psi(t) \rangle \right|}^{2}=\frac{{\Omega_{n}}^{2}}{ {\Omega_{\rm eff}}^{2}}\ \sin^{2}\, \frac{\Omega_{\rm eff} }{2}t,
\label{eq:transProb}$$ where $\Omega_{n} = \Omega_{0}\ \sqrt{n+1}$ is the $n-$photon Rabi frequency, $\Omega_{\rm eff}={\sqrt{(\Delta\omega )^{2} + {\Omega_{n}}^{2} }}$, with $\Delta \omega = \omega_{0} - \omega$ being the detuning. We are not interested in the effects of detuning in this paper, so we will set it to zero: $\Delta \omega =0$. In that case Eq. (\[eq:transProb\]) simplifies to $$P_{g}(t) = \sin^{2}\, \frac{ \Omega_{n}}{2} t,
\label{eq:tpsimple}$$ and the probability of transition to the upper state is $P_{e}(t) = 1-P_{g}(t)$.
For a general initial field state with a photon number distribution $p_{n}(0)$, the probability of transition to the upper state at resonance becomes $$P_{e}(t) = \frac{1}{2} \left[ 1 + \sum_{n=0}^{\infty} p_{n}(0)\ \cos \, \Omega_{n} t \right].
\label{eq:pe}$$
Our MCWF approach is described in detail in Refs. [@bkcs; @jkps], so we will give only an outline here. In the MCWF approach, the system is in the superposition state, Eq., before a measurement is made, and the measurement causes the state jump to either $ |n, e \rangle$ or $|n+1, g \rangle$. The algorithm to simulate the reduction of the state may be summarized as follows:
1. Divide the interaction time (the Rabi cycle) into $N$ segments: $N\Delta t = 2 \pi$.
2. At time $t_{m} = t_{0} + m \Delta t, \ (t_{0}=0; m = 1, 2, \ldots , N)$ generate a random number $r_{m}$ in the range $[0,1]$.
3. Compute $P_{e}(t_{m})$ given by Eq. (\[eq:pe\]) and compare it with $r_{m}$.
1. If $P_{e}(t_{m}) > r_{m}$, the quantum jump $| \psi (t) \rightarrow |n, e \rangle$ occurs.
2. If $P_{e}(t_{m}) < r_{m}$, the quantum jump $| \psi (t) \rightarrow |n+1, g \rangle$ occurs.
The procedure is to be repeated many times, then the simulated population should approach the analytical value obtained by Eq. (\[eq:pe\]).
An important benefit of the MCWF approach, unlike conventional treatments [@trprob], is that it provides a means to connect the atomic evolution with photon absorption and emission [*dynamics*]{}, by unraveling macroscopic observations in terms of the dynamics of individual quantum trajectories. For example, the photon emission probability in the time interval $(t_{m-1}, t_{m}]$ for a quantum trajectory is given by the joint probability $${\cal P}_{\rm emission}(t_{m}) \Delta t = P_{e}(t_{m-1}) \cdot P_{g}(t_{m}) \Delta t .
\label{eq:probem}$$ A typical quantum trajectory that gives rise to a photon emission in the time interval $(t_{m-1}, t_{m}]$ is depicted schematically in Fig. \[fig:onetraj\].
![Schematic diagram for a quantum trajectory traced by the TLA-field system. Photon emission occurs in the time interval $(t_{m-1}, t_{m}] = [m - (m-1)]\ \Delta t$.[]{data-label="fig:onetraj"}](fig1.png){width="8.5cm"}
MCWF Simulation Results
-----------------------
We consider three initial field states to study the dependence of the atom-field dynamics on the field statistics–number, coherent, and thermal states. The latter two fields have the following photon number distributions: $$\begin{aligned}
&p_{\rm coherent} = p_{n}(0) = \frac{{\bar n}^{n}}{n!}e^{-{\bar n} } , \\
&p_{\rm thermal} = p_{n}(0) = \frac{1}{{\bar n} + 1} \left( \frac{{\bar n}}{{\bar n} + 1} \right)^{n}. \nonumber
\label{eq:fields}\end{aligned}$$
In these simulations we chose a moderate value for the photon number $n=15$ for the number state and the average photon number ${\bar n}=15$ for other field states. The Rabi cycle (for the coherent and the thermal fields, the average Rabi cycle, $\Omega_{\bar n} t = 2 \pi$) was divided into $N= 10^{4}$ segments, and the vacuum Rabi frequency $\Omega_{0}$ was adjusted accordingly. The infinite sum appearing in Eq. (\[eq:pe\]) was truncated at the highest photon number $n_{\rm max}$ for which $p_{n_{\rm max}}(0)$ is less than $10^{-3}$ of the peak value of either of the distributions. Thus, for ${\bar n}=15$ the sum was truncated with 32 and 108 terms in the summation for the coherent and the thermal states, respectively.
The simulated emission rates from Eq. (\[eq:probem\]) are given in Fig. \[fig:emissions\]. For the simulation $10^{5}$ trajectory calculations are performed for the duration of 10 Rabi cycles for each field state. We can immediately see the effects of the cavity on the emission dynamics as compared to free space. The number state shows two sinusoidal oscillations per Rabi cycle with an amplitude corresponding to 1/4 of the total number of trajectories used. This oscillatory behavior is understandable, because at about $t=0$ the atom does not have enough interaction time to cause emission, despite all the atoms are initially prepared to occupy the upper state. Also, the emission rate is the lowest at about $1/2$ Rabi cycle, because at this time almost all the atoms are at the lower state. At $1/4$ and $3/4$ Rabi cycle, the atom has a $50\%$ probability to be in either the upper or the lower state, so the transition rate is the maximum there [@jkps]. In the case of the coherent field the oscillation is quenched at about three Rabi cycles, and after that we see a collapse of the photon emission rate similar to the case of atomic population. The collapse time arrives much sooner for the thermal photons. A remarkable fact is that the emission rates collapse to 1/4, the ceiling value, rather than to the middle region of the ‘oscillations’ as in the atomic population dynamics.
In order to understand this behavior we note that $t_{m-1} \rightarrow t_{m}$ as $\Delta t \rightarrow 0$, so we arrive at an asymptotic expression for the emission rate: $$\begin{aligned}
\label{eq:analem}
{\cal P}_{\rm emission} (t) &= \frac{1}{2} \left[ 1 + \sum_{n=0}^{\infty} p_{n}(0)\, \cos \, \Omega_{n} t \right] \nonumber \\
& \times \frac{1}{2} \left[ 1 - \sum_{n=0}^{\infty} p_{n}(0)\, \cos, \ \Omega_{n} t \right].\end{aligned}$$
![Simulated photoemission rates as a function of time in units of Rabi cycle. From top to bottom: number, coherent, and thermal initial states. $10^{5}$ quantum trajectories are calculated for each simulation.[]{data-label="fig:emissions"}](fig2.png){width="8.5cm"}
With the analytic expression, Eq. , we can afford to investigate the long-time behavior of photon emission. Equation (\[eq:analem\]) was numerically evaluated with the highest $n$ values in the sum determined above, keeping all other parameters the same. In Figs. \[fig:cslong\] and \[fig:thlong\] the atomic population and the photon emission dynamics for the coherent and the thermal fields are compared up to 100 Rabi cycles. The results agree well with the corresponding simulations given in Fig. \[fig:emissions\].
It is evident that for the coherent state the photon emission dynamics, in addition to the usual atomic population, also has a collapse and revival behavior. We also observe that the collapse and revival periods mirror those of the atomic population. However, the photon emission dynamics has more rapid oscillations and spends in the collapsed state longer as compared to the atomic population dynamics. These observations may be attributed to the fact the oscillation frequency in Eq. is effectively twice bigger than that in Eq. . The ceiling value of the photon emission rate is still 1/4, which is essentially the average emission rate, $\langle \cos^{2} \, \Omega_{n}t/2 \rangle \cdot \langle \sin^{2}\, \Omega_{n}t/2 \rangle $. For the thermal field the atomic population goes into a chaotic dynamics after a brief irregular oscillation, while the photon emission rate quickly reaches the uniform ceiling value till about 10 Rabi cycles and then goes into a chaotic regime without any sign of revival. Nonetheless, we note that the fluctuation of the photon emission rate is much smaller than that of the atomic dynamics.
![Long-time behavior of the atomic population (top) and the photon emission (bottom) dynamics for the field initially in a coherent state, calculated using the analytic expressions, Eqs. (\[eq:pe\]) and (\[eq:analem\]). Note the difference in vertical scales.[]{data-label="fig:cslong"}](fig3.png){width="8.5cm"}
![Same as in Fig. \[fig:cslong\] for the field initially in a thermal state. []{data-label="fig:thlong"}](fig4.png){width="8.5cm"}
CONCLUSIONS
===========
In this paper we considered the dynamics of a coupled TLA-field system in a cavity, using the JCM from the MCWF viewpoint. The effects of the statistics of the initial light field on the dynamics of the atomic population and the photon emission were investigated. The simulation regained the familiar behavior for the atomic dynamics, for example, the collapse and revival behavior for the coherent state. More importantly, this approach enabled us to extract information on the photon emission dynamics as well, by unraveling the ensemble dynamics in terms of individual quantum trajectories. With the quantum trajectory approach, in addition to the simulations, we also obtained an analytical expression for the photon emission process that is useful for investigating long-time behavior, which is very expensive to carry out by means of simulations alone. For the coherent field in a cavity, we found that the photon emission dynamics also exhibits a series of collapses and revivals that mirror the behavior of the atomic dynamics. However, the oscillations and the collapses are not about the halfway between the maximum and minimum as in the case of atomic population, but has a ceiling value of 1/4. Photon emission for the thermal field also collapses to the same ceiling value, but shows no revival behavior. For both fields, the fluctuation in photon emission rate is seen to be much less than that in the atomic population dynamics.
The collapse region is of interest, since it may be useful for obtaining ‘quiet’ light, by opening the cavity during this period. With the quantum trajectory approach, it is also possible to delineate the effects of initial fields on the emitted photon counting statistics in a cavity. Work along these lines are in progress.
This work was supported by Sun Moon University Research Grant of 2014.
E. Hahn, Phys. Rev. [**80**]{}, 580 (1950). N. A. Kurnit, I. D. Abella, S. R. Hartmann, Phys. Rev. Lett. [**13**]{}, 567 (1964); N. A. Kurnit, I. D. Abella, S. R. Hartmann, Phys. Rev. [**141**]{}, 391 (1966). M. D. Fayer, in [*Ultrafast infrared vibrational spectroscopy*]{} edited by M. D. Fayer (CRC Press, Boca Raton, 2013), p. 1. C. J. Lee, Phys. Rev. A [**53**]{}, 4238 (1996). C. J. Lee, Phys. Rev. A [**58**]{}, 3342 (1998). See, for example, M. A. Nielsen and I. L. Chuang. [*Quantum computation and quantum information*]{} (Cambridge University Press, New York, 2011); J.-M. Raimond and S. Haroche, in [*Quantum Decoherence: Poincaré Seminar 2005*]{}, B. Duplantier, J.-M. Raimond, and V. Rivasseau Eds. (Birkhäuser Verlag, Basel, 2007), p. 33.
E. T. Jaynes, F.W. Cummings, Proc. IEEE [**51**]{}, 89 (1963). F. W. Cummings, Phys. Rev. [**140**]{}, A1051 (1965). J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, Phys. Rev. Lett. [**44**]{}, 1323 (1980). G. Rempe, H. Walther, and N. Klein, Phys. Rev. Lett. [**58**]{}, 353 (1987). See, for example, J. Phys. B [**46**]{}, Special issue on Jaynes¡©Cummings physics (2013). R. Dum, P. Zoller, and H. Ritsch, Phys. Rev. A [**45**]{}, 4879 (1992); H. J. Carmichael, [*An Open Systems Approach to Quantum Optics*]{} (Springer-Verlag, New York, 1993); K. M[ø]{}lmer, Y. Castin, and J. Dalibard, J. Opt. Soc. Am. B [**10**]{}, 524 (1993). W. H. Louisell, [*Quantum Statistical Properties of Radiation*]{} (Wiley, New York, 1973). C. J. Lee, Bull. Korean Chem. Soc. [**27**]{}, 1186 (2006). C. J. Lee, J. Korean Phys. Soc. [**60**]{}, 766 (2012). See, for example, L. Mandel and E. Wolf, [*Optical Coherence and Quantum Optics*]{} (Cambridge University Press, New York, 1995), Chap. 15.
[^1]: Tel:+82-041-530-2243
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---
abstract: |
We prove that if $T$ is an $\omega$-categorical supersimple theory with nontrivial dependence (given by forking), then there is a nontrivial regular 1-type over a finite set of reals which is realized by real elements; hence forking induces a nontrivial pregeometry on the solution set of this type and the pregeometry is definable (using only finitely many parameters). The assumption about $\omega$-categoricity is necessary. This result is used to prove the following: If $V$ is a finite relational vocabulary with maximal arity 3 and $T$ is a supersimple $V$-theory with elimination of quantifiers, then $T$ has trivial dependence and finite SU-rank. This immediately gives the following strengthening of [@Kop17a Theorem 4.1]: if ${\mathcal{M}}$ is a ternary simple homogeneous structure with only finitely many constraints, then $Th({\mathcal{M}})$ has trivial dependence and finite SU-rank.
[*Keywords*]{}: model theory, simple theory, regular type, pregeometry, omega-categorical, elimination of quantifiers, homogeneous structure
address: 'Vera Koponen, Department of Mathematics, Uppsala University, Box 480, 75106 Uppsala, Sweden.'
author:
- Vera Koponen
date: '29 June 2018, revised version'
title: 'Supersimple $\omega$-categorical theories and pregeometries'
---
Introduction
============
The idea that “global” properties of a theory can, under some circumstances, be understood to a large part by its “local” properties dates back, at least, to Zilber’s studies of uncountably categorical theories in the 80’ies. (For a monograph in English on this topic see [@Zilber_mono].) Ever since, this idea has been an important guideline in model theory giving rise to many results, in particular in the studies of stable theories and later and more generally in the studies of simple theories.
By a local property we mean a property of (the set of elements realizing) a type (i.e. a consistent set of formulas). Certain types are of particular interest. Zilber considered minimal types and Hrushovski [@Hru85; @Hru90; @Hru92] generalized many of the results to regular types, in both cases in the context of stable theories. Later such results where generalized to the context of simple theories. If $T$ is a simple theory and $p$ is a regular type, then dividing (or equivalently forking) dependence, induces a pregeometry (or matroid) on the set of elements that realize $p$. The interesting thing about this is that the complexity of the pregeometries on regular types tend to reflect the complexity of dividing dependence of the theory. This is, for example, the case when if one considers supersimple 1-based structures with finite SU-rank as shown by Hart, Kim and Pillay [@HKP] and by de Piro and Kim [@DK]. What is more directly relevant for this article is the following result of Goode [@Goode Propositions 2 and 5]: if $T$ is superstable with finite SU-rank (usually called U-rank in the context of stable theories) and $T$ has nontrivial dependence (Definition \[definition of trivial dependence\]), then there is a nontrivial regular type (Definition \[definition of minimal dependent set\]). Palacín [@Pal17] observed that Goode’s proof generalizes to supersimple theories with finite SU-rank. On the other hand the assumption that the SU-rank is finite is necessary (even in the stable context) which is demonstrated by an example in [@Goode] (see Remark \[remark about the use of omega-categoricity\] below).
However, if we assume that the theory is $\omega$-categorical, then the assumption about finite SU-rank can be removed, which is our first main result (Theorem \[nontrivial dependence implies a nontrivial regular type\]). More precisely: if $T$ is supersimple and $\omega$-categorical with nontrivial dependence, then there are ${\mathcal{M}}\models T$, a finite set $C \subseteq M$ and a nontrivial regular 1-type $p$ over $C$ realized by real elements. In our application of this result it matters that $p$ is realized by real elements, as opposed to properly imaginary ones. A result like this can be useful to show that certain theories cannot have complicated behaviour of dependence, by showing that they cannot accomodate a complicated definable pregeometry. In this article we consider theories with elimination of quantifiers in a finite relational language. Such theories are $\omega$-categorical by a well known characterization of $\omega$-categorical theories. To facilitate this approach we prove the following (Theorem \[nontrivial dependence implies nontrivial pregeometry\]): If $T$ is a supersimple $V$-theory with elimination of quantifiers, where $V$ is a finite and relational vocabulary and $T$ has nontrivial dependence, then there is another finite relational vocabulary $V'$ of the same maximal arity as $V$ and a supersimple $V'$-theory $T'$ with elimination of quantifiers such that dependence induces a nontrivial pregeometry on any model of $T'$ and $T'$ has a unique 1-type over the algebraic closure of ${\emptyset}$ with imaginaries. Then we show that no such $T'$ can exist if the maximal arity of $V'$ is 3 (Theorem \[impossibility of a structure satisfying the other result\]). Hence, every supersimple $T$ with elimination of quantifiers in a finite relational language with maximal arity 3 (of the relation symbols) has trivial dependence. It follows from work of Macpherson [@Mac91] and de Piro and Kim [@DK] that if $T$ is simple with elimination of quantifiers in a finite relational langauge and $T$ has [*non*]{}trivial dependence, then $T$ is [*not*]{} 1-based, so if $T$ is supersimple then it must have a [*non*]{}modular regular type. This seems very difficult to achieve (with elimination of quantifiers in a finite relational langauge) so my guess is that dependence is always trivial in such a theory. As a contrast, if we forget about elimination of quantifiers, Hrushovski has constructued, with only a ternary relation symbol, a supersimple $\omega$-categorical structure with SU-rank 1 and nontrivial (even not 1-based) independence (see for instance [@Wag Chapter 6.2] or [@Kim_book Chapter 6]).
Besides the nature of regular types and pregeometries, the SU-rank gives important information about a supersimple theory. Every $\omega$-categorical superstable theory is $\omega$-stable as proved by Lachlan [@Lach74] and has finite Morley-rank as proved by Cherlin, Harrington and Lachlan [@CHL], from which it follows that it has finite SU-rank. There is a conjecture [@Wag p. 205] that every $\omega$-categorical supersimple theory has finite SU-rank, but the cases in which it has been proved are still limited. For every smoothly approximable structure its theory is $\omega$-categorical and supersimple with finite SU-rank [@CH; @HKP]. More generally, Evans and Wagner proved that every theory which is $\omega$-categorical, supersimple and CM-trivial has finite SU-rank [@EW]. If $T$ is simple with elimination of quantifiers in a finite relational language with maximal arity 2, then $T$ is supersimple with finite SU-rank [@Kop16PAM]. We show here, in Corollary \[trivial dependence and finite rank\], that if $T$ is supersimple with elimination of quantifiers in a finite relational language with maximal arity 3, then $T$ has finite SU-rank. This is actually a consequence of the mentioned result that such a theory has trivial dependence and the following result, due to Palacín [@Pal17]: If $T$ is $\omega$-categorical and supersimple with trivial dependence, then $T$ has finite SU-rank. From Corollary \[trivial dependence and finite rank\] we can directly improve the main result in [@Kop17a] to the following: If ${\mathcal{M}}$ is ternary, homogeneous (definition follows below) and simple with only finitely many constraints, then its theory is supersimple with finite SU-rank and trivial dependence. To give some more background, Lachlan [@Lach74] once conjectured that every $\omega$-categorical stable theory is superstable, but this was disproved by Hrushovski [@Hru_pseudoplane].[^1]
In some sense the simplest $\omega$-categorical theories are those with elimination of quantifiers, and yet, via so-called amalgamation constructions, it is possible to construct uncountably many supersimple theories with SU-rank 1 and with elimination of quantifiers by using only one ternary relation symbol [@Kop17a]. Then it is natural to first try to solve hard problems in this special case, which is still challenging. Moreover, theories with elimination of quantifiers, and in particular their countable models, are interesting from other perspectives. Suppose that $T$ is a complete $V$-theory where $V$ is a finite relational vocabulary and let ${\mathcal{M}}$ be a [*countable*]{} model of $T$. Then the following three conditions are equivalent (see for example [@Hod Chapter 7]): (a) $T$ has elimination of quantifiers; (b) ${\mathcal{M}}$ is [*homogeneous*]{}[^2], meaning that every isomorphism between finite substructures of ${\mathcal{M}}$ can be extended to an automorphism of ${\mathcal{M}}$; (c) the class of finite $V$-structures that can be embedded into ${\mathcal{M}}$ has the amalgamation property and consequently ${\mathcal{M}}$ is the so called Fraïssé limit of this class.
Homogeneous structures have turned out to be interesting from a variety of viewpoints. Since they have a rich automorphism group they are interesting in permutation group theory and there are, via structural Ramsey theory, interesting connections to topological dynamics. Homogeneous structures have also become an important object of study in the area of constraint satisfaction problems. See the survey article [@Mac11] by Macpherson for more about the mentioned aspects of homogeneous structures and further references. All stable homogeneous structures are well understood through the work of Lachlan and others [@Lach97], but new challenges are offered in the broader class of simple homogeneous structures. (For example, every stable homogeneous structure is finitely constrained, but given only a ternary relation symbol one can construct uncountably many [*not*]{} finitely constrained simple homogeneous structures [@Kop17a].)
The idea to understand (super)simple theories with elimination of quantifiers by considering a bound on the maximal arity of the relation symbols may seem futile as the bound can always be increased. But I suspect that once the case of maximal arity 4 is understood with respect to general questions such as triviality of dependence and SU-rank, then we also get the answer for any maximal arity. The reason for thinking so has to do with the nature of some proofs in this article and in [@Kop17a]. Moreover, there are related results where the arity 4 is essential, such as [@KopBin Corollaries 5.3–5.4] and the theory of smoothly approximable structures [@CH Theorems 2–6].
The structure of this article is the following: The next section recalls definitions and results which will be used later and clarifies the notation and terminology that will be used. Section \[Main results and their relationships\] states the main results and explains how they are related. Once this has been done it only remains to prove Theorems \[nontrivial dependence implies a nontrivial regular type\], \[nontrivial dependence implies nontrivial pregeometry\] and \[impossibility of a structure satisfying the other result\]. The first two of these are proved in Section \[Section about regular types\] and the third one in Section \[impossibility of a nontrivial pregeometry\].
Preliminaries
=============
Familiarity with model theory including the basics of simplicity theory is assumed. Elementary model theoretic results of relevance here can be found in [@Hod; @TZ]. For basic notions and results about simple theories we refer to any of the books [@Cas; @Kim_book; @Wag]. The terminology and notation used here is relatively standard but nevertheless we make some clarifications. A vocabulary (or signature) $V$ is [*finite and relational*]{} if it is finite and contains only relation symbols. Its [*maximal arity*]{} is the maximal $k$ for which some symbol in it has arity $k$. A finite relational vocabulary is called [*ternary*]{} if its maximal arity is 3. If $V$ is a ternary finite vocabulary then a $V$-structure is called [*ternary*]{}. Structures will be denoted by calligraphic letters such as ${\mathcal{A}}, \ldots, {\mathcal{M}}, {\mathcal{N}}$ and their universes by $A, \ldots, M, N$. Finite sequences/tuples of objects are denoted by $\bar{a}, \bar{b}, \ldots, \bar{x}, \bar{y}, \ldots$. When writing $\bar{a} \in A$ it can mean that $\bar{a}$ belongs to $A$, but it can also mean that each coordinate of the tuple $\bar{a}$ belongs to $A$; hopefully the meaning will be revealed by the context. Occasionally we may write ${\mathrm{rng}}(\bar{a})$ for the set of all elements in the tuple $\bar{a}$ but we often abuse notation and identify (when convenient) $\bar{a}$ with the [*set*]{} of elements in it. Also, for sets $A$ and $B$ we often abbreviate ‘$A \cup B$’ by ‘$AB$’.
Let ${\mathcal{M}}$ be a structure. Its complete theory is denoted by $Th({\mathcal{M}})$. If $A \subseteq M$ then ${\mathcal{M}}{{\negthickspace}\upharpoonright {\negthickspace}}A$ denotes the substructure of ${\mathcal{M}}$ which is generated by $A$. If $p$ is an $n$-type with respect to ${\mathcal{M}}$, then $p({\mathcal{M}})$ denotes the set of all $n$-tuples of elements from $M$ which realize $p$. By $p {{\negthickspace}\upharpoonright {\negthickspace}}A$ we denote the restriction of $p$ to formulas with parameters from $A$. Since the proofs will sometimes involve two structures with different complete theories (even different languages) we will sometimes use the notation ‘${\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }^{\mathcal{M}}$’, ‘${\mathrm{tp}}_{\mathcal{M}}$’, ‘${\mathrm{acl}}_{\mathcal{M}}$’ and ‘${\mathrm{dcl}}_{\mathcal{M}}$’ to clarify that dividing/forking, types, algebraic closure and definable closure, respectively, is with respect to the structure ${\mathcal{M}}$. By ‘${\mathrm{tp}}^{\mathrm{qf}}_{\mathcal{M}}(\bar{a} / B)$’ we denote the type of $\bar{a}$ over $B$ (in ${\mathcal{M}}$) [*restricted to quantifier-free formulas*]{}. The notation ‘$\bar{a} \equiv_{\mathcal{M}}\bar{b}$’ (‘$\bar{a} \equiv^{\mathrm{qf}}_{\mathcal{M}}\bar{b}$’) means the same as ‘${\mathrm{tp}}_{\mathcal{M}}(\bar{a}) = {\mathrm{tp}}_{\mathcal{M}}(\bar{b})$’ (‘${\mathrm{tp}}^{\mathrm{qf}}_{\mathcal{M}}(\bar{a}) = {\mathrm{tp}}^{\mathrm{qf}}_{\mathcal{M}}(\bar{b})$’).
For definitions of the notions [*orthogonality*]{} (of types), [*regular types*]{} and [*weight*]{} see [@Kim_book; @Wag]. A definition of a [*pregeometry*]{} is found in [@Kim_book Definition 4.4.3] and in several other books on model theory. Recall also that in a simple theory [*forking*]{} and [*dividing*]{} are equivalent.
\[definition of minimal dependent set\][Let $T$ be a simple theory, ${\mathcal{M}}\models T$, $\bar{a}_1, \ldots, \bar{a}_n \in M{^{\mathrm{eq}}}$, and $B \subseteq M{^{\mathrm{eq}}}$.\
(i) We call $A = \{\bar{a}_1, \ldots, \bar{a}_n\}$ [*independent over $B$*]{} if for every $\bar{a}_i \in A$, $\bar{a}_i \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} (A \setminus \{\bar{a}_i\})$.[^3] Otherwise we call $A$ [*dependent over $B$*]{}. If $B = {\emptyset}$ we may omit saying ‘over $B$’.\
(ii) If $A = \{\bar{a}_1, \ldots, \bar{a}_n\}$ is dependent over $B$ and every proper subset of $A$ is independent over $B$, then we call $A$ [*minimal dependent over $B$*]{} (and if $B = {\emptyset}$ we may omit ‘over $B$’).\
(iii) A type $p$ over $B \subseteq M{^{\mathrm{eq}}}$ is called [*nontrivial*]{} if $p({\mathcal{M}}{^{\mathrm{eq}}})$ contains a set of cardinality at least $3$ which is minimal dependent over $B$. ]{}
Note that a minimal dependent set (over some set) must always be finite, by the finite character of dividing. I have not been able to find the following useful result in the literature besides as an exercise in [@TZ], and its proof is indeed an elementary exercise in using dividing/forking.
\[useful fact about independence\][([@TZ Exercise 7.2.6])]{} Let $T$ be simple, ${\mathcal{M}}\models T$, $\bar{a}_1, \ldots, \bar{a}_n \in M{^{\mathrm{eq}}}$, $A \subseteq B \subseteq M{^{\mathrm{eq}}}$ and suppose that $\bar{a}_1 \ldots \bar{a}_n \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} B$. Then $\{\bar{a}_1, \ldots, \bar{a}_n\}$ is independent over $A$ if and only if it is independent over $B$.
\[definition of minimal dependent set in a pregeometry\][Let $(X, {\mathrm{cl}})$ be a pregeometry.\
(i) We call $Y \subseteq X$ [*independent*]{} if, for every $y \in Y$, $y \notin {\mathrm{cl}}(Y \setminus \{y\})$.[^4] Otherwise we call $Y$ [*dependent*]{}.\
(ii) If $Y \subseteq X$ is dependent and every proper subset of $Y$ is independent, then we call $Y$ [*minimal dependent*]{}.\
(iii) Let $Y \subseteq X$. An independent set $Z \subseteq Y$ such that $Y \subseteq {\mathrm{cl}}(Z)$ is called a [*basis*]{} of $Y$ and $|Z|$ is called the [*dimension*]{} of $Y$; an elementary result is that all bases of $Y$ have the same cardinality so the dimension of $Y$ is well defined.\
(iv) $(X, {\mathrm{cl}})$ is [*nontrivial*]{} if there is $Y \subseteq X$ with cardinality at least $3$ such that $Y$ is minimal dependent. ]{}
\[existence of regular types\] [([@Kim_book Proposition 4.4.5], [@Wag Proposition 5.1.11])]{} Suppose that $T$ is supersimple, ${\mathcal{M}}\models T$ and $A \subseteq M{^{\mathrm{eq}}}$. Let $n < \omega$ and let $p$ be a nonalgebraic $n$-type over $A$ which is realized in $M$. Then, assuming that ${\mathcal{M}}$ is sufficiently saturated, there are $B \subseteq M{^{\mathrm{eq}}}$ and a regular 1-type $q$ over $B$ such that $q$ is realized by a real element in $M$, $q$ is nonorthogonal to $p$ and ${\mathrm{SU}}(q) \leq {\mathrm{SU}}(p)$.
Some comments about the above fact may be in order. Given $p$ as in the fact and a tuple $\bar{a} = (a_1, \ldots, a_n) \in M^n$ which realizes $p$, $p$ will be nonorthogonal to $q' = {\mathrm{tp}}(a_1 /A)$ where (by the Lascar inequalities [@Kim_book; @Wag]) ${\mathrm{SU}}(q') \leq {\mathrm{SU}}(q)$. Now we can let $q$ be a 1-type of minimal SU-rank among all 1-types that are nonorthogonal to $p$ and realized by a real element, so ${\mathrm{SU}}(q) \leq {\mathrm{SU}}(p)$, and then we can argue as in the proof of [@Kim_book Proposition 4.4.5] or [@Wag Proposition 5.1.11] to show that $q$ is regular.
\[regular types and nondividing extensions\] [([@Kim_book Remarks 4.4.2 (2) and 4.4.4])]{} Let $T$ be simple, ${\mathcal{M}}\models T$, $\bar{a} \in M{^{\mathrm{eq}}}$ and $A \subseteq B \subseteq M{^{\mathrm{eq}}}$.\
(i) Suppose that $\bar{a} \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} B$. Then ${\mathrm{tp}}(\bar{a} / A)$ is regular if and only if ${\mathrm{tp}}(\bar{a} / B)$ is regular.\
(ii) Suppose that $p$ is a regular type over $A$. Then $(p({\mathcal{M}}), {\mathrm{cl}})$ is a pregeometry if, for all $b \in p({\mathcal{M}})$ and all $C \subseteq p({\mathcal{M}})$, $b \in {\mathrm{cl}}(C)$ if and only if $b \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} C$.
By the finite character of dividing the following conditions are equivalent when $T$ is simple:
- For every ${\mathcal{M}}\models T$, all $A, B \subseteq M{^{\mathrm{eq}}}$ and every finite tuple $\bar{a} \in M{^{\mathrm{eq}}}$, if $\bar{a} \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} B$ then there is $b \in B$ such that $\bar{a} \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} b$.
- For every ${\mathcal{M}}\models T$, all finite tuples $\bar{a}, \bar{b}, \bar{c} \in M{^{\mathrm{eq}}}$ and every $A \subseteq M{^{\mathrm{eq}}}$, if $\bar{a} \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{b}\bar{c}$ then $\bar{a} \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{b}$ or $\bar{a} \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{c}$.
\[definition of trivial dependence\][A simple theory $T$ has [*trivial dependence*]{} if the two equivalent conditions (a) and (b) above hold. (Otherwise it has [*nontrivial dependence*]{}.) ]{}
Goode [@Goode] studied a few variations of the notion of trivial dependence in the context of stable theories. When he says ‘$T$ is totally trivial’ it means the same as when we say ‘$T$ has trivial dependence’. When he says ‘$T$ is trivial’ it means that whenever $\bar{a}, \bar{b}$ and $\bar{c}$ are pairwise independent tuples over some set of elements, then $\{\bar{a}, \bar{b}, \bar{c}\}$ is independent over the same set. The next fact is stated in [@Goode] for stable theories, but the proof uses only basic properties of forking/dividing which hold also for simple structures.
\[trivial dependence for real elements gives trivial dependence\] [(Goode [@Goode Lemma 4])]{} Let $T$ be a simple theory and suppose that ${\mathcal{M}}\models T$, $A \subseteq M{^{\mathrm{eq}}}$, $\bar{a}_1, \bar{a}_2, \bar{a}_3 \in M{^{\mathrm{eq}}}$, $\bar{a}_1 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{a}_2 \bar{a}_3$, $\bar{a}_1 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_2$ and $\bar{a}_1 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_3$. Then there are tuples of real elements $\bar{b}_1, \bar{b}_2, \bar{b}_3 \in M$ such that $\bar{b}_1 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{b}_2 \bar{b}_3$, $\bar{b}_1 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{b}_2$ and $\bar{b}_1 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{b}_3$.
\[definition of constraint\][Let $V$ be a vocabulary and ${\mathcal{M}}$ a $V$-structure. A finite $V$-structure ${\mathcal{A}}$ is called a [*constraint of ${\mathcal{M}}$*]{} if ${\mathcal{A}}$ cannot be embedded into ${\mathcal{M}}$ but every proper substructure of ${\mathcal{A}}$ can be embedded into ${\mathcal{M}}$. We say that ${\mathcal{M}}$ is [*finitely constrained*]{} if ${\mathcal{M}}$ has (up to isomorphism) only finitely many constraints. ]{}
Every supersimple theory has elimination of hyperimaginaries (see [@Kim_book Theorem 5.4.9] or [@Wag Theorem 5.3.1]). Also, every simple and ‘small’ theory has elimination of hyperimaginaries (see for example [@Kim_book Corollary 5.3.5]) and we note that $\omega$-categorical theories are small. It follows that in the context of this article we need not consider hyperimaginary elements and two tuples of imaginaries have the same Lascar strong type (over some set) if and only if they have the same strong type (over the same set). For small, and hence for $\omega$-categorical, simple theories this is perhaps most clearly stated in [@Kim98 Theorem 23]. It follows that when we use the independence theorem of simple theories (also called the type amalgamation theorem) [@Cas; @Kim_book; @Wag] we do not need to consider hyperimaginary elements; it suffices that the types that are to be amalgamated extend the same strong type over the “base set” rather than the same Lascar strong type. When saying that a supersimple theory $T$ has [*finite SU-rank*]{} we mean that the SU-rank of every type with finitely many variables (realized by real elements) is finite. The well known characterization of $\omega$-categorical theories by Engeler, Ryll-Nardzewski and Svenonius (see [@Hod Theorem 7.3.1] or [@TZ Theorem 4.3.1] for example) has the following consequence which will be used: if $T$ is $\omega$-categorical then every model of it is $\omega$-saturated and if $A$ is a [*finite*]{} subset of some model of $T$, then there are only finitely many $n$-types over $A$ and each such type is isolated.
Main results and their relationships {#Main results and their relationships}
====================================
\[nontrivial dependence implies a nontrivial regular type\] Suppose that $T$ is an $\omega$-categorical supersimple theory with nontrivial dependence. Then there are ${\mathcal{M}}\models T$, a finite set $C \subseteq M$ and a nontrivial regular 1-type $p$ over $C$ realized by real elements. (Consequently, $T$ is not trivial in the sense of Goode [@Goode] or Palacín [@Pal17].)
If $p$ is a regular type like in Theorem \[nontrivial dependence implies a nontrivial regular type\], then (by Fact \[regular types and nondividing extensions\]) $(p({\mathcal{M}}), {\mathrm{cl}})$ is a pregeometry if, for all $a \in p({\mathcal{M}})$ and all $A \subseteq p({\mathcal{M}})$, $a \in {\mathrm{cl}}(A)$ if and only if $a \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} A$. By the $\omega$-categoricity of $T$, for every $0 < n < \omega$, the relation $x_n \in {\mathrm{cl}}(x_1, \ldots, x_{n-1})$ on $p({\mathcal{M}})$ is $C$-definable. Theorem \[nontrivial dependence implies a nontrivial regular type\] is proved in Section \[Section about regular types\]. By continuing the proof of Theorem \[nontrivial dependence implies a nontrivial regular type\], in the same section with some extra assumptions added, we get the following:
\[nontrivial dependence implies nontrivial pregeometry\] Suppose that $V$ is a finite relational vocabulary and $T$ a complete $V$-theory such that $T$ is supersimple with elimination of quantifiers. If $T$ has nontrivial dependence, then there is a finite relational vocabulary $V'$ with the same maximal arity as $V$ and a $V'$-structure ${\mathcal{M}}'$ satisfying the following conditions:
- $Th({\mathcal{M}}')$ has elimination of quantifiers and is supersimple.
- All elements of $M'$ have the same type over ${\mathrm{acl}}_{({\mathcal{M}}'){^{\mathrm{eq}}}}({\emptyset})$.
- $(M', {\mathrm{cl}})$ is a pregeometry if, for every $X \subseteq M'$ and every $x \in M'$, $x \in {\mathrm{cl}}(X)$ if and only if $x {\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }^{{\mathcal{M}}'} X$.
- $M'$ has a minimal dependent subset of cardinality at least 3 (where dependence is with respect to $Th({\mathcal{M}}')$, or equivalently, with respect to the pregeometry from condition (iii)).
In Section \[impossibility of a nontrivial pregeometry\] we prove the following:
\[impossibility of a structure satisfying the other result\] Let $V'$ be a ternary finite relational vocabulary. Then there does [*not*]{} exist a $V'$-structure ${\mathcal{M}}'$ such that (i) – (iv) of Theorem \[nontrivial dependence implies nontrivial pregeometry\] hold. (This holds also if we replace ‘supersimple’ with ‘simple’ in (i).)
Recall the definition of [*homogenous*]{} structure from the introduction (which implies that a homogeneous structure is countable and has a finite relational vocabulary). By combining Theorems \[nontrivial dependence implies nontrivial pregeometry\] and \[impossibility of a structure satisfying the other result\] and Proposition \[triviality of dependence implies finite rank\] below, we immediately get:
\[trivial dependence and finite rank\] If $V$ is a ternary finite relational vocabulary and $T$ a complete $V$-theory such that $T$ is supersimple with elimination of quantifiers (so its unique countable model is homogeneous), then $T$ has trivial dependence and finite SU-rank.
Corollary \[trivial dependence and finite rank\] easily gives the following improvement of Theorem 4.1 in [@Kop17a]:
\[ternary homogeneous finitely constrained theories have trivial dependence\] Suppose that ${\mathcal{M}}$ is a ternary homogeneous finitely constrained simple structure. Then $Th({\mathcal{M}})$ has trivial dependence and finite SU-rank.
[**Proof.**]{} Suppose that ${\mathcal{M}}$ is ternary, homogeneous, finitely constrained and simple. By [@Kop17a Theorem 4.1], $Th({\mathcal{M}})$ is supersimple with finite SU-rank. Then Corollary \[trivial dependence and finite rank\] implies that $Th({\mathcal{M}})$ has trivial dependence. $\square$\
The notion of trivial dependence used in this article implies the notion of ‘triviality’ (or ‘1-triviality’) used in Palacín’s article [@Pal17] (which uses the terminology of Goode [@Goode]). Hence the next result was proved by Palacín in that article. Here we offer a different proof which is rather short, straightforward and avoids some technical notions such as Lascar strong types, canonical bases, orthogonality and regular types.
\[triviality of dependence implies finite rank\][(Palacín [@Pal17 Corollary 3.12])]{} Suppose that $T$ is an $\omega$-categorical supersimple theory with trivial dependence. Then $T$ has finite SU-rank.
[**Proof.**]{} Let $T$ be $\omega$-categorical and supersimple with trivial dependence. Suppose that $T$ does not have finite SU-rank, so there is $p \in S_1(T)$ with ${\mathrm{SU}}(p) \geq \omega$. Let ${\mathcal{M}}\models T$ be sufficently saturated so that all elements and sets that we talk about can be assumed to come from ${\mathcal{M}}$. Since ${\mathrm{SU}}(p) \geq \omega$ there are (by for example [@Cas Remark 13.14]) $A \subseteq M$ and an extension $p' \in S_1(A)$ of $p$ such that ${\mathrm{SU}}(p') = \omega$. Since $T$ is supersimple there is finite $A_0 \subseteq A$ such that $p'$ does not divide over $A_0$. Let $p_0 = p' {{\negthickspace}\upharpoonright {\negthickspace}}A_0$, so ${\mathrm{SU}}(p_0) = {\mathrm{SU}}(p') = \omega$. For every $0 < n < \omega$ there are $B_n \subseteq M$ and an extension $p_n \in S_1(A_0B_n)$ of $p_0$ such that ${\mathrm{SU}}(p_n) = n$. Since $T$ has trivial dependence there is, for each $0 < n < \omega$, $b_n \in B_n$ such that $q_n = p_n {{\negthickspace}\upharpoonright {\negthickspace}}A_0b_n$ is a dividing extension of $p_0$. Then $$n = {\mathrm{SU}}(p_n) \leq {\mathrm{SU}}(q_n) < {\mathrm{SU}}(p_0) = \omega.$$ Let $k_1 = {\mathrm{SU}}(q_1)$. Suppose that $0 < k_1 < k_2 < \ldots < k_m < \omega$ have been defined and that for each $i$, $k_i = {\mathrm{SU}}(q_n)$ for some $n$. Then choose $n < \omega$ such that $k_m < n$ and let $k_{m+1} = {\mathrm{SU}}(q_n)$, so $k_{m+1} > k_m$. Thus, by renaming (some of) the types and elements, we now have an infinite sequence $0 < k_1 < k_2 < k_3 < \ldots$ of natural numbers, an infinite sequence of elemens $c_1, c_2, c_3, \ldots$ and, for each $0 < i < \omega$ a type $r_i \in S_1(A_0c_i)$ such that ${\mathrm{SU}}(r_i) = k_i$. Let $\bar{a}$ enumerate $A_0$ and, for every $0 < i < \omega$, let $d_i$ realize $r_i$. Since $${\mathrm{SU}}(d_i / A_0c_i) \neq {\mathrm{SU}}(d_j / A_0c_j) \ \text{ if } \ i \neq j$$ it follows that ${\mathrm{tp}}(d_i, \bar{a}, c_i) \neq {\mathrm{tp}}(d_j, \bar{a}, c_j)$ if $i \neq j$. Thus we have infinitely many $n$-types over ${\emptyset}$ if $n = |\bar{a}| + 2$ and this contradicts that $T$ is $\omega$-categorical. $\square$
Proofs of Theorems \[nontrivial dependence implies a nontrivial regular type\] and \[nontrivial dependence implies nontrivial pregeometry\]: nontrivial dependence gives (in the context) a nontrivial pregeometry {#Section about regular types}
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Let $T$ be an $\omega$-categorical supersimple theory with nontrivial dependence. We also assume that ${\mathcal{M}}$ is a sufficiently saturated model of $T$, so that all elements or sets that we claim exist (in some elementary extension of ${\mathcal{M}}{^{\mathrm{eq}}}$) actually exist in ${\mathcal{M}}{^{\mathrm{eq}}}$. In this section, ${\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }$, ${\mathrm{tp}}$, ${\mathrm{acl}}$ and ${\mathrm{dcl}}$ mean ${\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }^{{\mathcal{M}}{^{\mathrm{eq}}}}$, ${\mathrm{tp}}_{{\mathcal{M}}{^{\mathrm{eq}}}}$, ${\mathrm{acl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}$ and ${\mathrm{dcl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}$, respectively. Since $T$ has nontrivial dependence it follows from Fact \[trivial dependence for real elements gives trivial dependence\] that there are finite tuples $\bar{a}_1, \bar{a}_2, \bar{a}_3 \in M$ and $A \subseteq M{^{\mathrm{eq}}}$ such that, for some permutation $(i, j, k)$ of $(1, 2, 3)$, $\bar{a}_i \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{a}_j \bar{a}_k$, $\bar{a}_i \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_j$ and $\bar{a}_i \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_k$. Since $T$ is supersimple we may assume that
- there do not exist $\bar{a}'_i, \bar{a}'_j, \bar{a}'_k \in M$ and $A' \subseteq M{^{\mathrm{eq}}}$ such that
- $\bar{a}'_i \underset{A'}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{a}'_j \bar{a}'_k$, $\bar{a}'_i \underset{A'}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}'_j$, $\bar{a}'_i \underset{A'}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}'_k$,
- ${\mathrm{SU}}(\bar{a}'_n / A') \leq {\mathrm{SU}}(\bar{a}_n / A)$ for all $n = 1, 2, 3$, and
- for some $1 \leq n \leq 3$, ${\mathrm{SU}}(\bar{a}'_n / A') < {\mathrm{SU}}(\bar{a}_n / A)$.
Theorems \[nontrivial dependence implies a nontrivial regular type\] and \[nontrivial dependence implies nontrivial pregeometry\] will be proved via a sequence of lemmas. The proof of Theorem \[nontrivial dependence implies a nontrivial regular type\] is finished after Lemma \[finding a minimal dependent set in the same type\] and then the argument goes on, with added assumptions about $T$, to prove Theorem \[nontrivial dependence implies nontrivial pregeometry\].
\[getting regular types\] There are real elements $a'_1, a'_2, a'_3 \in M$ and $B \subseteq M{^{\mathrm{eq}}}$ such that
- $a'_i \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} a'_ja'_k$, $a'_i \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a'_j$, $a'_i \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a'_k$,
- ${\mathrm{tp}}(a'_n /B)$ is regular for every $1 \leq n \leq 3$, and
- ${\mathrm{SU}}(a'_n / B) = {\mathrm{SU}}(\bar{a}_n / A)$ for every $1 \leq n \leq 3$.
[**Proof.**]{} Let $p = {\mathrm{tp}}(\bar{a}_1 / A)$. By Fact \[existence of regular types\] there is $A'' \subseteq M{^{\mathrm{eq}}}$ and a regular type $p'' \in S_1^{\mathcal{M}}(A'')$ realized by a real element such that $p''$ and $p$ are nonorthogonal and ${\mathrm{SU}}(p'') \leq {\mathrm{SU}}(p)$. This means that there are $B''$, $a''_1, \bar{a}^*_1$ such that $A'', A \subseteq B''$, $a''_1$ realizes a (complete) nonforking extension of $p''$ to $B''$, $\bar{a}^*_1$ realizes a nonforking extension of $p$ to $B''$ and $a''_1 \underset{B''}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{a}^*_1$. Since ${\mathrm{tp}}(\bar{a}^*_1 /A) = {\mathrm{tp}}(\bar{a}_1 / A)$ there are $B \supseteq A$, $A' \subseteq B$ and $a'_1$ such that $p' = {\mathrm{tp}}(a'_1 / A')$ is regular, ${\mathrm{SU}}(p') = {\mathrm{SU}}(p'')$ ($\leq {\mathrm{SU}}(p)$) and $$\label{properties of a'-1 and bar-a-1}
\bar{a}_1 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} B, \quad a'_1 \underset{A'}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} B, \ \text{ and } \ \bar{a}_1 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} a'_1.$$ Since $a'_1 \underset{A'}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} B$ it follows from Fact \[regular types and nondividing extensions\] that ${\mathrm{tp}}(a'_1 / B)$ is regular and ${\mathrm{SU}}(a'_1 / B) = {\mathrm{SU}}(p') \leq {\mathrm{SU}}(p)$. By the existence of nonforking extensions we may assume that $$\label{a'-1B independent over}
a'_1B \underset{\bar{a}_1 A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_2 \bar{a}_3.$$
From (\[properties of a’-1 and bar-a-1\]) we get $\bar{a}_1 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} B$, so (\[a’-1B independent over\]) and transitivity gives $$\label{B independent from ... over A}
\bar{a}_1 \bar{a}_2 \bar{a}_3 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} B.$$ This together with Fact \[useful fact about independence\] implies that $$\label{every proper subset is independent over B}
\text{for all $n, m \in \{1, 2, 3\}$, \ $\bar{a}_n \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_m$ if and only if $\bar{a}_n \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_m$.}$$ Note that from (\[B independent from ... over A\]) it follows that ${\mathrm{SU}}(\bar{a}_n / B) = {\mathrm{SU}}(\bar{a}_n / A)$ for all $1 \leq n \leq 3$. We now prove two claims and then explain how the lemma follows from these claims.
[**Claim 1.**]{} Suppose that $\bar{a}_1 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{a}_2 \bar{a}_3$, $\bar{a}_1 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_2$ and $\bar{a}_1 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_3$. Then $a'_1 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{a}_2 \bar{a}_3$, $a'_1 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_2$ and $a'_1 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_2$.
[**Proof of Claim 1.**]{} By assumption, $\bar{a}_1 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_n$, for $n = 2, 3$, so (\[a’-1B independent over\]) and transitivity implies that $a'_1\bar{a}_1B \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_n$ and hence $$\label{a'-1 ind from a-i}
a'_1 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_n \ \ \text{ and } \ \ \bar{a}_1 \underset{a'_1 B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_n \ \ \text{ for } n = 2, 3.$$ Towards a contradiction, suppose that $a'_1 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_2 \bar{a}_3$. By assumption, $\bar{a}_1 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{a}_2 \bar{a}_3$, so $a'_1 \bar{a}_1 B \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{a}_2 \bar{a}_3$. Transitivity and (\[B independent from ... over A\]) now implies that $a'_1 \bar{a}_1 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{a}_2 \bar{a}_3$. By assumption, $a'_1 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_2 \bar{a}_3$, so transitivity gives $\bar{a}_1 \underset{a'_1 B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{a}_2 \bar{a}_3$. From (\[properties of a’-1 and bar-a-1\]) it follows that ${\mathrm{SU}}(\bar{a}_1 / a'_1 B) < {\mathrm{SU}}(\bar{a}_1 / B) = {\mathrm{SU}}(\bar{a}_1 / A)$. But now, taking $A' = a'_1B$, we have a situation which contradicts the assumption (A). Hence $a'_1 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{a}_2 \bar{a}_3$, so the claim is proved. $\square$
[**Claim 2.**]{} Suppose that $\bar{a}_1 \bar{a}_2 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{a}_3$, $\bar{a}_1 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_3$ and $\bar{a}_2 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_3$. Then $a'_1 \bar{a}_2 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{a}_3$, $a'_1 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_3$ and $\bar{a}_2 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_3$. The claim also holds if ‘$\bar{a}_2$’ and ‘$\bar{a}_3$’ switch places by letting ‘$\bar{a}_2$’ and ‘$\bar{a}_3$’ switch roles in the proof.
[**Proof of Claim 2.**]{} By assumption, $\bar{a}_2 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_3$ so (\[every proper subset is independent over B\]) gives $\bar{a}_2 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_3$. The assumption that $\bar{a}_1 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_3$ together with (\[a’-1B independent over\]) and transitivity gives $a'_1 \bar{a}_1 B \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_3$ and hence $a'_1 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_3$.
Towards a contradiction, suppose that $a'_1 \bar{a}_2 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_3$. By assumption, $\bar{a}_1 \bar{a}_2 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{a}_3$ so $$a'_1 \bar{a}_1 \bar{a}_2 B \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{a}_3.$$ By assumption, $\bar{a}_1 \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_3$ and by (\[a’-1B independent over\]) we have $a'_1 \bar{a}_1 B \underset{A}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_3$, so $\bar{a}_1 \underset{a'_1 B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_3$. The assumption that $a'_1 \bar{a}_2 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_3$ gives $\bar{a}_2 \underset{a'_1 B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_3$. From (\[properties of a’-1 and bar-a-1\]) we get ${\mathrm{SU}}(\bar{a}_1 / a'_1 B) < {\mathrm{SU}}(\bar{a}_1 / B) = {\mathrm{SU}}(\bar{a}_1 / A)$. Thus we have, with $A' = a'_1 B$, a situation that contradicts the assumption (A). Hence $a'_1 \bar{a}_2 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} \bar{a}_3$, so the claim is proved. $\square$\
Let $a^*_1 = a'_1$, $a^*_2 = \bar{a}_2$ and $a^*_3 = \bar{a}_3$. If $i = 1$ then, by Claim 1, $a^*_i \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} a^*_j a^*_k$, $a^*_i \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a^*_j$ and $a^*_i \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a^*_k$. If $i = 2$ or $i = 3$ then we use Claim 2 to get the same conclusion. Next, starting with $a'_1, \bar{a}_2, \bar{a}_3$ and $B$ instead of $\bar{a}_1, \bar{a}_2, \bar{a}_3$ and $A$, we argue in the same way as before Claim 1 to get $B' \supseteq B$ and $a'_2$ such that ${\mathrm{tp}}(a'_2 / B')$ is regular, ${\mathrm{SU}}(a'_2 / B') \leq {\mathrm{SU}}(\bar{a}_2 / B)$, (\[properties of a’-1 and bar-a-1\]) holds if $\bar{a}_1$, $a'_1$, $A$ and $B$ are replaced by $\bar{a}_2$, $a'_2$, $B$ and $B'$, respectively, $a'_2 B' \underset{\bar{a}_2 B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a'_1\bar{a}_3$ (which corresponds to (\[a’-1B independent over\])) and (\[B independent from ... over A\])–(\[every proper subset is independent over B\]) hold if $\bar{a}_1$, $A$ and $B$ are replaced by $a'_1$, $B$ and $B'$, respectively. Then Claims 1 and 2 hold if $\bar{a}_1, \bar{a}_2, \bar{a}_3$, $a'_1$, $A$ and $B$ are replaced by $\bar{a}_2, a'_1, \bar{a}_3$, $a'_2$, $B$ and $B'$, respectively (so $\bar{a}_2$ and $a'_2$ take the “active” role now). Now let $a^*_1 = a'_1$, $a^*_2 = a'_2$ and $a^*_3 = \bar{a}_3$. If $i = 2$ then, by Claim 1, $a^*_i \underset{B'}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} a^*_j a^*_k$, $a^*_i \underset{B'}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a^*_j$ and $a^*_i \underset{B'}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a^*_k$. Otherwise we get the same conclusion by Claim 2. Now we repeat the same kind of argument a final round to get $a'_3$ and $B'' \supseteq B$ such that ${\mathrm{tp}}(a'_3 / B'')$ is regular, ${\mathrm{SU}}(a'_3 / B'') \leq {\mathrm{SU}}(\bar{a}_3 / B')$, $a'_i \underset{B''}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} a'_j a'_k$, $a'_i \underset{B''}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a'_j$ and $a'_i \underset{B''}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a'_k$. For every $1 \leq n \leq 3$ we have ${\mathrm{SU}}(a'_n / B'') \leq {\mathrm{SU}}(\bar{a}_n / A)$ (this follows from the three versions of (\[B independent from ... over A\]) that have been obtained during the three “rounds”). Hence the assumption (A) implies that ${\mathrm{SU}}(a'_n / B'') = {\mathrm{SU}}(\bar{a}_n / A)$ for every $1 \leq n \leq 3$. So if $B''$ is renamed as $B$, then we have proved the statement of the lemma. $\square$\
Let $a'_1, a'_2, a'_3$ and $B$ be as in Lemma \[getting regular types\] and to simplify notation we assume that $(i, j, k) = (1, 2, 3)$.
\[a finite base set of reals\] There is a finite set (of real elements) $C \subseteq M$ such that
- ${\mathrm{tp}}(a'_n / C)$ is regular for every $1 \leq n \leq 3$,
- ${\mathrm{SU}}(a'_n / C) = {\mathrm{SU}}(a'_n / B)$ for every $1 \leq n \leq 3$, and
- $a'_1 \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} a'_2 a'_3$ and $a'_1 \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a'_n$ for every $2 \leq n \leq 3$.
[**Proof.**]{} Let $B' \subseteq M$ be such that $B \subseteq {\mathrm{dcl}}(B')$. We may assume that $B' \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a'_1 a'_2 a'_3$, so in particular $a'_n \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} B'$ for each $n$. Since $B \subseteq {\mathrm{acl}}(B')$ it follows from Fact \[useful fact about independence\] that $a'_1 \underset{B'}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} a'_2 a'_3$ and $a'_1 \underset{B'}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a'_n$ for $n = 2, 3$. By Fact \[regular types and nondividing extensions\], ${\mathrm{tp}}(a'_n / B')$ is regular for each $n = 1, 2, 3$.
Since $T$ is supersimple there is finite $C \subseteq B'$ such that $a'_1 a'_2 a'_3 \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} B'$. Again it follows from Facts \[useful fact about independence\] and \[regular types and nondividing extensions\] that $a'_1 \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} a'_2 a'_3$, $a'_1 \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a'_n$ for $n = 2, 3$ and ${\mathrm{tp}}(a'_n / C)$ is regular for each $n = 1, 2, 3$. Moreover, we have that ${\mathrm{SU}}(a'_n / C) = {\mathrm{SU}}(a'_n / B') = {\mathrm{SU}}(a'_n / B)$. $\square$\
Let $C$ be as in Lemma \[a finite base set of reals\] and, for $n = 1, 2, 3$, let $p_n = {\mathrm{tp}}(a'_n / C)$.
\[getting a minimal dependent set\] We have $a'_2 \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a'_3$, so $\{a'_1, a'_2, a'_3\}$ is minimal dependent over $C$.
[**Proof.**]{} For every $1 \leq n \leq 3$, $p_n$ is regular so it has weight 1 (see [@Kim_book Proposition 4.4.9] or [@Wag Lemma 5.2.11]) from which it follows that the relation $x \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} y$ is an equivalence relation on $X = p_1({\mathcal{M}}) \cup p_2({\mathcal{M}}) \cup p_3({\mathcal{M}})$. Denote the relation $x \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} y$ on $X$ by $E$. Since $T$ is $\omega$-categorical and $C$ is a finite set of reals, $E$ is $C$-definable. Let $\bar{c}$ enumerate $C$. For $\bar{x}, \bar{y}$ of the same length as $\bar{c}$, define $$\text{$F(\bar{x}x, \bar{y}y)$ if and only if
$\bar{x} = \bar{y}$, ${\mathrm{tp}}(\bar{x}) = {\mathrm{tp}}(\bar{c})$ and $x \underset{\bar{x}}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} y$.}$$ Then $F$ is an equivalence relation which is ${\emptyset}$-definable, so every $F$-class is represented by an imaginary element, and hence every $E$-class is also represented by an imaginary element. Let $b$ be the imaginary element representing the $E$-class of $a'_2$ (or strictly speaking the $F$-class of $\bar{c}a'_2$), so $b \in {\mathrm{dcl}}(a'_2C)$.
Suppose for a contradiction that $a'_2 \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} a'_3$. If there would only be finitely many $E$-classes, then $b \in {\mathrm{acl}}(C)$ and hence $a'_2 \underset{Cb}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} a'_3$. By the existence of nonforking extensions we could find $a$ in the same $E$-class as $a'_2$ such that $a'_2 \underset{Cb}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a$, which (as $b \in {\mathrm{acl}}(C)$) implies that $a'_2 \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a$, but this contradicts that (by definition) $E(x, y)$ implies $x \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} y$. Hence there are infinitely many $E$-classes and thus $b \notin {\mathrm{acl}}(C)$. From the assumption that $a'_2 \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} a'_3$ we get $E(a'_2, a'_3)$ and hence $b \in {\mathrm{dcl}}(a'_3C)$. Consequently $a'_n \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} b$ for $n = 2, 3$ and hence ${\mathrm{SU}}(a'_n / bC) < {\mathrm{SU}}(a'_n / C)$ for $n = 2, 3$. By the choice of $a'_1, a'_2$ and $a'_3$, we have $a'_1 \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} a'_2 a'_3$ and $a'_1 \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a'_n$ for $n = 2, 3$. As $b \in {\mathrm{acl}}(a'_n C)$ for $n = 2, 3$ we get $a'_1 \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a'_nb$ and hence $a'_1 \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} b$ and $a'_1 \underset{bC}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a'_n$ for $n = 2, 3$. Using that $a'_1 \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} a'_2 a'_3 b$ and transitivity it also follows that $a'_1 \underset{bC}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} a'_2 a'_3$. But since ${\mathrm{SU}}(a'_n / C) = {\mathrm{SU}}(\bar{a}_n / A)$ for all $n = 1, 2, 3$, this situation contradicts the assumption (A). $\square$
\[finding a minimal dependent set in the same type\] For some $1 \leq n \leq 3$, $p_n({\mathcal{M}})$ contains a set of cardinality at least 3 which is minimal dependent over $C$.
[**Proof.**]{} If all of $a'_1, a'_2$ and $a'_3$ realize the same $p_n$ then we are done. So suppose that this is not the case. Without loss of generality we can assume that $a'_2, a'_3 \notin p_1({\mathcal{M}})$. By the existence of nonforking extensions there are $a''_2, a''_3 \in p_2({\mathcal{M}}) \cup p_3({\mathcal{M}})$ so that ${\mathrm{tp}}(a''_2, a''_3 / a'_1 C) = {\mathrm{tp}}(a'_2, a'_3 / a'_1 C)$ and $a'_2 a'_3 \underset{a'_1 C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a''_2 a''_3$. By Lemma \[getting a minimal dependent set\], $\{a'_1, a'_2, a'_3\}$ is minimal dependent over $C$ and hence $a'_1 \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} a'_2 a'_3$ and consequently $a'_1 \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} a''_2 a''_3$. Since $p_1$ is regular and therefore has weight 1 it follows that $a'_2 a'_3 \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} a''_2 a''_3$ and consequently the set $\{a'_2, a'_3, a''_2, a''_3\}$ is dependent over $C$. The fact that $\{a'_1, a'_2, a'_3\}$ is minimal dependent over $C$ together with $a'_2 a'_3 \underset{a'_1 C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a''_2 a''_3$ and transitivity implies that every proper subset of $Y = \{a'_2, a'_3, a''_2, a''_3\}$ is independent over $C$. Note also that $p_1$ is not realized by any element in $Y$.
If only one of $p_2$ and $p_3$ is realized in $Y$ then we are done. So suppose that this is not the case. Then, by the choice of the elements, two elements in $Y$ realize $p_2$ and two elements realize $p_3$. So by renaming the elements we have $Y = \{b_1, b_2, b_3, b_4\}$ where $b_3$ and $b_4$ realize $p_2$. Choose $b'_1, b'_2 \in p_3({\mathcal{M}})$ so that $$\label{the copy of b-1 b-2 over b-3 and b-4}
\text{${\mathrm{tp}}(b'_1, b'_2, b_3 / b_4 C) = {\mathrm{tp}}(b_1, b_2, b_3 / b_4 C)$ \ and \
$b_1 b_2 \underset{C b_3 b_4}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} b'_1 b'_2$.}$$ Let $n \in \{1, 2\}$. Since $\{b_n, b_3, b_4\}$ is independent over $C$ it follows from (\[the copy of b-1 b-2 over b-3 and b-4\]) and transitivity that $b_n \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} b'_1 b'_2 b_3 b_4$. From (\[the copy of b-1 b-2 over b-3 and b-4\]) it follows that $\{b'_1, b'_2, b_3\}$ is independent over $C$ and hence $\{b_n, b'_1, b'_2, b_3\}$ is independent over $C$. In the same way it follows that $\{b_1, b_2, b'_n, b_3\}$ is independent over $C$.
Since $\{b_3, b_4\}$ is independent over $C$ it follows that $b_4$ realizes a nonforking extension of $p_2$ to $Cb_3$. As $Y$ is minimal dependent over $C$ we have $b_1 b_2 \underset{C b_3}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} b_4$ and (using (\[the copy of b-1 b-2 over b-3 and b-4\])) $b'_1 b'_2 \underset{C b_3}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} b_4$. Since $p_2$ has weight 1 it follows that $b_1 b_2 \underset{C b_3}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} b'_1 b'_2$ and hence $\{b_1, b_2, b'_1, b'_2, b_3\}$ is dependent over $C$. If already $\{b_1, b_2, b'_1, b'_2\}$ is dependent over $C$, then it is minimal dependent and only $p_3$ is realized in it, so we are done. If it is independent over $C$, then $Y' = \{b_1, b_2, b'_1, b'_2, b_3\}$ is minimal dependent over $C$ and all elements in $Y'$ except $b_3$ realizes $p_3$. This means that we can argue similarly as we did in the beginning of the proof of this lemma to get a set of cardinality $8$ which is minimal dependent and in which all elements realize $p_3$. $\square$\
By Lemma \[finding a minimal dependent set in the same type\], for some $1 \leq n \leq 3$, $p_n({\mathcal{M}})$ contains a set of cardinality at least 3 which is minimal dependent over $C$. This proves Theorem \[nontrivial dependence implies a nontrivial regular type\].
\[remark about the use of omega-categoricity\][ In proving Theorem \[nontrivial dependence implies a nontrivial regular type\] the assumption that $T$ is $\omega$-categorical was used only once and it was in the proof of Lemma \[getting a minimal dependent set\]. There $\omega$-categoricity was used to conclude that the relation $x \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} y$ on $p_1({\mathcal{M}}) \cup p_2({\mathcal{M}}) \cup p_3({\mathcal{M}})$ is $C$-definable. In fact, in every simple theory, if $p \in S(C)$ then the set $\{(a, b) : \text{ $a$ realizes $p$ and } a \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} b\}$ is type-definable over $C$. As a finite union of type-definable sets is type-definable, the relation $x \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} y$ is type-definable on $p_1({\mathcal{M}}) \cup p_2({\mathcal{M}}) \cup p_3({\mathcal{M}})$. So if the relation $x \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} y$ is type-definable on $p_1({\mathcal{M}}) \cup p_2({\mathcal{M}}) \cup p_3({\mathcal{M}})$, then, by compactness, it is definable. However, if $T$ is not $\omega$-categorical then the relation $x \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} y$ need not be type-definable. The example called $T_2$ which follows after the proof of Proposition 5 in [@Goode] is superstable (but not $\omega$-categorical) and every regular type is trivial, but $T_2$ is not ‘totally trivial’ in the sense of [@Goode] which means that it has nontrivial dependence in the sense of this article. Moreover, there is a regular 1-type $q$ of $T_2$ over ${\emptyset}$ (with SU-rank $\omega$) such that the relation $x {\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }y$ on $q$ is not type-definable. ]{}
By Lemma \[finding a minimal dependent set in the same type\] we may, without loss of generality, assume that $p_1({\mathcal{M}})$ contains a set of cardinality at least $3$ which is minimal dependent over $C$. By Fact \[regular types and nondividing extensions\], $(p_1({\mathcal{M}}), {\mathrm{cl}})$ is a pregeometry if ${\mathrm{cl}}$ is defined as follows for $x \in p({\mathcal{M}})$ and $X \subseteq p_1({\mathcal{M}})$: $x \in {\mathrm{cl}}(X)$ if and only if $x \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }} X$.
We now continue with the proof of Theorem \[nontrivial dependence implies nontrivial pregeometry\]. Thus, [***we now add the assumption that $T$ is a $V$-theory with elimination of quantifiers where $V$ is a finite relational vocabulary.***]{} Let $P = \{p_1\}$. (The choice of notation may seem awkward in the present context, but makes the correspondence to results in [@Kop17a Section 3] clear.) We now define a new vocabulary $V_C$, which will be finite and relational with the same maximal arity as $V$, and then we define a $V_C$-structure called ${\mathcal{M}}_P$. (This corresponds to Definition 3.1 in [@Kop17a].)
\[definition of structure M-p generated by p\]
\(i) Let $V_C$ be a finite relational vocabulary such that $V \subseteq V_C$ and for every $R \in V$ of arity $r > 1$, every $0 < k < r$, every permutation $\pi$ of $\{1, \ldots, r\}$, and every $\bar{a} \in C^k$, $V_C$ has a relation symbol $Q_{R, \bar{a}, \pi}$ of arity $r-k$. We also assume that $V_C$ has no other symbols than those described.
\(ii) Let ${\mathcal{M}}_P$ be the (infinite) $V_C$-structure with universe $M_P = p_1({\mathcal{M}})$ and where the symbols in $V_C$ are interpreted as follows:
- If $R \in V$ has arity $r$, then $R^{{\mathcal{M}}_P} = R^{\mathcal{M}}\cap (M_P)^r$.
- If $Q_{R, \bar{a}, \pi} \in V_C \setminus V$ where $R \in V$ has arity $r$ and $|\bar{a}| = k$, then for every $\bar{b} \in (M_P)^{r-k}$, $\bar{b} \in (Q_{R, \bar{a}, \pi})^{{\mathcal{M}}_P}$ if and only if $\pi(\bar{b}\bar{a}) \in R^{\mathcal{M}}$ (where, if $\bar{b}\bar{a} = (d_1, \ldots, d_r)$ then $\pi(\bar{b}\bar{a}) = (d_{\pi(1)}, \ldots, d_{\pi(r)})$).
In [@Kop17a Lemmas 3.2–3.4] the following was proved:
\[properties of M-P\] (i) Let $\bar{c}$ be an enumeration of $C$. For all $\bar{a}, \bar{b} \in M_P$, $$\bar{a} \equiv_{{\mathcal{M}}_P}^{\mathrm{qf}}\bar{b} \quad \text{ if and only if } \quad \bar{a}\bar{c} \equiv_{\mathcal{M}}^{\mathrm{qf}}\bar{b}\bar{c}.$$ (ii) $Th({\mathcal{M}}_P)$ has elimination of quantifiers and is simple.[^5]\
(iii) For all $\bar{a}, \bar{b} \in M_P$, $\bar{a} {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }^{{\mathcal{M}}_P} \bar{b}$ if and only in $\bar{a} \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }}^{\mathcal{M}}\bar{b}$.
\[interpretability and supersimplicity\][(i) One characterization of supersimplicity is that for all $\bar{a}$ and $B$ there is a finite $B' \subseteq B$ such that the type of $\bar{a}$ over $B$ does not fork over $B'$. Using this characterization it is easy to prove that if ${\mathcal{M}}$ is supersimple and $d_1, \ldots, d_n \in M$, then the expansion of ${\mathcal{M}}$ with constants for the elements $d_1, \ldots, d_n$ is also supersimple.\
(ii) By [@Wag Remark 2.8.14], every theory which is interpretable in a supersimple theory is supersimple. It follows that ${\mathcal{M}}_P$ is supersimple. From these observations it also follows that if $d_1, \ldots, d_n \in M{^{\mathrm{eq}}}$, then the expansion of ${\mathcal{M}}{^{\mathrm{eq}}}$ by constants for the elements $d_1, \ldots, d_n$ is also supersimple and every structure which is interpretable in this expansion is supersimple. ]{}
\[definition of the equivalence relation\][(i) For $x \in M_P$ and $X \subseteq M_P$, define $x \in {\mathrm{cl}}(X)$ if and only if $x \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }^{\mathcal{M}}} X$.\
(ii) For $x, y \in M_P$, define $x \sim y$ if and only if ${\mathrm{tp}}(x / {\mathrm{acl}}(C)) = {\mathrm{tp}}(y / {\mathrm{acl}}(C))$, where types and algebraic closure are taken in ${\mathcal{M}}{^{\mathrm{eq}}}$.\
(iii) If $X$ is an equivalence class of ‘$\sim$’, then define for all $x \in X$ and all $Y \subseteq X$, ${\mathrm{cl}}_X(Y) = {\mathrm{cl}}(Y) \cap X$. ]{}
Note that ‘$\sim$’ has only finitely many equivalence classes (as $C$ is finite and $T$ is $\omega$-categorical). Moreover, since $p_1$ is nonalgebraic and regular every $\sim$-class is infinite. By Fact \[regular types and nondividing extensions\], $(M_P, {\mathrm{cl}})$ is a pregeometry. Since ${\mathcal{M}}$ is $\omega$-categorical, $M_P$ is a $C$-definable set in ${\mathcal{M}}$. Note also that, by part (i) of Lemma \[properties of M-P\], for every relation $R \subseteq (M_P)^n$, $R$ is ${\emptyset}$-definable in ${\mathcal{M}}_P$ if and only if it is $C$-definable in ${\mathcal{M}}$. By $\omega$-categoricity, $\sim$ is $C$-definable in ${\mathcal{M}}$ and hence (by Lemma \[properties of M-P\] (i)) it is ${\emptyset}$-definable in ${\mathcal{M}}_P$.
\[the structure on a single equivalence class\] Let $X$ be any equivalence class of $\sim$ and let ${\mathcal{N}}= {\mathcal{M}}_P {{\negthickspace}\upharpoonright {\negthickspace}}X$. Then:\
(i) $Th({\mathcal{N}})$ has elimination of quantifiers and is supersimple.\
(ii) For all $\bar{a}, \bar{b} \in X$, $\bar{a} {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }^{{\mathcal{N}}} \bar{b}$ if and only if $\bar{a} {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }^{{\mathcal{M}}_P} \bar{b}$.\
(iii) All elements of $X$ have the same type, in ${\mathcal{N}}{^{\mathrm{eq}}}$, over ${\mathrm{acl}}_{{\mathcal{N}}{^{\mathrm{eq}}}}({\emptyset})$.\
(iv) $(X, {\mathrm{cl}}_X)$ is a pregeometry and for every $x \in X$ and every $Y \subseteq X$, $x \in {\mathrm{cl}}_X(Y)$ if and only if $x {\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }^{\mathcal{N}}Y$.
[**Proof.**]{} (i) To show that $Th({\mathcal{N}})$ has elimination of quantifiers it suffices (by the use of back and forth arguments or Ehrenfeucht-Fraïssé games, see for example [@Hod Section 3.3]) to show that if $\bar{a}, \bar{b} \in X$, $\bar{a} \equiv_{{\mathcal{M}}_P}^{\mathrm{qf}}\bar{b}$ and $d \in X$, then there is $e \in X$ such that $\bar{a}d \equiv_{{\mathcal{M}}_P}^{\mathrm{qf}}\bar{b}e$. For this it suffices to show that if $\bar{a}, \bar{b} \in X$, $\bar{a}\bar{c} \equiv_{\mathcal{M}}^{\mathrm{qf}}\bar{b}\bar{c}$ and $d \in X$, then there is $e \in X$ such that $\bar{a}\bar{c}d \equiv_{\mathcal{M}}^{\mathrm{qf}}\bar{b}\bar{c}e$. But the later implication follows directly since ${\mathcal{M}}$ has elimination of quantifiers and $\sim$ is definable in ${\mathcal{M}}$ with parameters from $\bar{c}$.
That $Th({\mathcal{N}})$ is supersimple follows from Remark \[interpretability and supersimplicity\] since ${\mathcal{N}}$ is interpretable in ${\mathcal{M}}{^{\mathrm{eq}}}$ using the parameters in $C$ and the imaginary element which corresponds to the equivalence class $X$.
\(ii) Let $\bar{a}, \bar{b} \in X$ and let $\varphi(\bar{x}, \bar{y})$ be a quantifier free $V_C$-formula that isolates ${\mathrm{tp}}_{\mathcal{N}}(\bar{a}, \bar{b})$. Also let $E(x, y)$ be a quantifier free $V_C$-formula which, in ${\mathcal{M}}_P$, defines ‘$\sim$’. Then, whenever $x_i, x_j \in {\mathrm{rng}}(\bar{x})$ and $y_k, y_l \in {\mathrm{rng}}(\bar{y})$, we have $$\label{x-i and y-j in the same sim-class}
\models \forall \bar{x}, \bar{y} \Big(\varphi(\bar{x}, \bar{y}) \ \rightarrow \
\big(E(x_i, x_j) \wedge E(y_k, y_l) \wedge E(x_i, y_k) \big) \Big).$$
First suppose that $\bar{a} {\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }^{\mathcal{N}}\bar{b}$, so there are $\bar{b}_i \in X$, $i < \omega$, such that $(\bar{b}_i : i < \omega)$ is an ${\emptyset}$-indiscernible sequence in ${\mathcal{N}}$, $\bar{b}_0 = \bar{b}$ and $\{\varphi(\bar{x}, \bar{b}_i) : i < \omega\}$ is $k$-inconsistent (with respect to ${\mathcal{N}}$) for some $k < \omega$. As ${\mathcal{N}}$ is a substrructure of ${\mathcal{M}}_P$ and both have elimination of quantifiers, $(\bar{b}_i : i < \omega)$ is ${\emptyset}$-indiscernible in ${\mathcal{M}}_P$ as well. If there would be $\bar{a}' \in M_P$ such that ${\mathcal{M}}\models \bigwedge_{i=0}^{k-1} \varphi(\bar{a}', \bar{b}_i)$, then, since some $a' \in {\mathrm{rng}}(\bar{a}')$ must not belong to $X$, we get, for any $b \in {\mathrm{rng}}(\bar{b}_0)$, $${\mathcal{M}}_P \models \varphi(\bar{a}', \bar{b}_0) \wedge \neg E(a', b)$$ and this contradicts (\[x-i and y-j in the same sim-class\]). Hence $\{\varphi(\bar{x}, \bar{b}_i) : i < \omega\}$ is $k$-inconsistent with respect to ${\mathcal{M}}_P$ so $\bar{a} {\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }^{{\mathcal{M}}_P} \bar{b}$.
Now suppose that $\bar{a} {\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }^{{\mathcal{M}}_P} \bar{b}$. Since ${\mathcal{N}}$ is a substructure of ${\mathcal{M}}_P$ and both have elimination of quantifiers it follows that $\varphi(\bar{x}, \bar{y})$ also isolates ${\mathrm{tp}}_{{\mathcal{M}}_P}(\bar{a}, \bar{b})$. Hence there are $\bar{b}_i \in M_P$ such that $(\bar{b}_i : i < \omega)$ is ${\emptyset}$-indiscernible, $\bar{b}_0 = \bar{b}$ and $\{\varphi(\bar{x}, \bar{b}_i) : i < \omega\}$ is $k$-inconsistent (with respect to ${\mathcal{M}}_P$) for some $k < \omega$. From (\[x-i and y-j in the same sim-class\]) it follows that, for all $i < \omega$, all elements in ${\mathrm{rng}}(\bar{b}_i)$ belong to the same $\sim$-class. Since $(\bar{b}_i : i < \omega)$ is ${\emptyset}$-indiscernible we have either
- for all $i < j$, all elements in ${\mathrm{rng}}(\bar{b}_i) \cup {\mathrm{rng}}(\bar{b}_j)$ belong to the same $\sim$-class, or
- for all $i < j$, every $b \in {\mathrm{rng}}(\bar{b}_i)$ belongs to a different $\sim$-class than any $b' \in {\mathrm{rng}}(\bar{b}_j)$.
However, as there are only finitely many $\sim$-classes it follows that we are in the first case. Since $\bar{b}_0 = \bar{b} \in X$ it follows that $\bar{b}_i \in X$ for all $i < \omega$. If $\{\varphi(\bar{x}, \bar{b}_i) : i < \omega\}$ would be $k$-consistent with respect to ${\mathcal{N}}$, then, as ${\mathcal{N}}\subseteq {\mathcal{M}}_P$ and $\varphi$ is quantifier free, the same set would be $k$-consistent with respect to ${\mathcal{M}}_P$, which contradicts the assumption. Hence we conclude that $\bar{a} {\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }^{\mathcal{N}}\bar{b}$.
\(iii) Suppose, for a contradiction, that there are elements $a, b \in X$ such that $${\mathrm{tp}}_{{\mathcal{N}}{^{\mathrm{eq}}}}(a / {\mathrm{acl}}_{{\mathcal{N}}{^{\mathrm{eq}}}}({\emptyset})) \neq
{\mathrm{tp}}_{{\mathcal{N}}{^{\mathrm{eq}}}}(b / {\mathrm{acl}}_{{\mathcal{N}}{^{\mathrm{eq}}}}({\emptyset})).$$ Then there is a nontrivial equivalence relation ‘$\approx$’ on $X$ which is ${\emptyset}$-definable in ${\mathcal{N}}$ and such that $a$ and $b$ belong to different $\approx$-classes. Define an equivalence relation on $M$ as follows: $$F(x, y) \ \Longleftrightarrow \ p_1(x) \ \wedge \ p_1(y) \ \wedge \ x \sim y \ \wedge \ x \approx y.$$ Then $F$ is $C$-definable in ${\mathcal{M}}$. Since $a, b \in X$ and since they belong to different $F$-classes it follows that $a$ and $b$ have different types, in ${\mathcal{M}}{^{\mathrm{eq}}}$, over ${\mathrm{acl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}(C)$ so they belong to different $\sim$-classes which contradicts that $a, b \in X$.
\(iv) We already noted (after Definition \[definition of the equivalence relation\]) that $(M_P, {\mathrm{cl}})$ is a pregeometry and from this it is a straightforward exercise to show that $(X, {\mathrm{cl}}_X)$ is a pregeometry. Let $Y \subseteq X$ and $x \in X$. Suppose that $x \in {\mathrm{cl}}_X(Y)$. Then $x \in {\mathrm{cl}}(Y) \cap X$, so $x \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }}^{\mathcal{M}}Y$ and hence $x \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }}^{\mathcal{M}}\bar{y}$ for some finite tuple $\bar{y} \in Y$. By Lemma \[properties of M-P\] (iii), $x {\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }^{{\mathcal{M}}_P} \bar{y}$ and by part (ii) of this lemma, $x {\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }^{\mathcal{N}}\bar{y}$ so $x {\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }^{\mathcal{N}}Y$.
Now suppose that $x {\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }^{\mathcal{N}}Y$, so $x {\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }^{\mathcal{N}}\bar{y}$ for some finite tuple $\bar{y} \in Y$. By part (ii) of this lemma and part (iii) of Lemma \[properties of M-P\], $x \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }}^{\mathcal{M}}\bar{y}$, so $x \underset{C}{{\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }}^{\mathcal{M}}Y$ and hence $x \in {\mathrm{cl}}(Y)$ (and by assumption $x \in X$). $\square$
\[all elements have the same type\] There is an equivalence class $X$ of ‘$\sim$’ such that $X$ contains a set of cardinality at least 3 which is minimal dependent over ${\emptyset}$ with respect to ${\mathcal{M}}_P$.
[**Proof.**]{} We know that $M_P$ ($= p_1({\mathcal{M}})$) contains a set of cardinality at least 3 which is minimal dependent over $C$ when dividing is considered with respect to ${\mathcal{M}}$. By Lemma \[properties of M-P\] (iii), this set is minimal dependent over ${\emptyset}$ when dividing is considered with respect to ${\mathcal{M}}_P$. Now Lemma \[all elements have the same type\] follows by using Lemma \[reducing the number of types\] below as many times as needed. $\square$
\[reducing the number of types\] Suppose that ${\mathcal{N}}$ is a simple $\omega$-saturated structure and $(N, {\mathrm{cl}}_{\mathcal{N}})$ a pregeometry such that for every $x \in N$ and every $X \subseteq N$, $x \in {\mathrm{cl}}_{\mathcal{N}}(X)$ if and only if $x {\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }^{\mathcal{N}}X$. Also let $E$ be a ${\emptyset}$-definable equivalence relation on $N$ with finitely many equivalence classes. Suppose that $Y$ is a minimal dependent set and $X$ an $E$-class such that $|Y| \geq 3$, $Y \setminus X \neq {\emptyset}$, $Y \cap X \neq {\emptyset}$ and $|Y \cap X|$ is minimal as $X$ ranges over the $E$-classes with which $Y$ has nonempty intersection. Then there is a minimal dependent set $Z$ such that $|Z| \geq |Y|$, $|Z \cap X| = |Y \cap X| - 1$ and the number of $E$-classes with which $Z \setminus X$ has nonempty intersection is equal to the number of $E$-classes with which $Y \setminus X$ has nonempty intersection.
[**Proof.**]{} Suppose that $Y$ is a minimal dependent set of cardinality at least 3 which has nonempty intersection with at least two $E$-classes. Let $X$ be an $E$-class such that $|Y \cap X|$ is minimal as $X$ ranges over the $E$-classes with which $Y$ has nonempty intersection. Let $1 \leq k < l$ and let $$Y = \{y_1, \ldots, y_l\} \ \text{ where } \ Y \cap X = \{y_1, \ldots, y_k\}.$$ Let $\bar{y}' = (y_1, \ldots, y_k)$, $y_i^0 = y_i$ for $i = 1, \ldots, l$ and $\bar{y}^0 = (y_{k+1}^0, \ldots, y_l^0)$. By the existence of nonforking extensions (and $\omega$-saturation) there are, for $0 < n < \omega$, $\bar{y}^n = (y_{k+1}^n, \ldots, y_l^n)$ such that for all $n < \omega$ $$\bar{y}'\bar{y}^n \equiv_{\mathcal{N}}\bar{y}'\bar{y}^0 \ \text{ and } \
\bar{y}^{n+1} \underset{\bar{y}'}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{y}^0 \ldots \bar{y}^n.$$ It follows that, for every $n < \omega$, $\bar{y}'\bar{y}^n$ is minimal dependent. Since (by assumption) there are only finitely many $E$-classes, there are $s < t < \omega$ such that for every $k < i \leq l$, $y_i^s$ and $y_i^t$ belong to the same $E$-class. The lemma now follows if we can show that, for some proper subset $Y^* \subset \{y_1, \ldots, y_k\}$, $Y^* \cup {\mathrm{rng}}(\bar{y}^s) \cup {\mathrm{rng}}(\bar{y}^t)$ is minimal dependent. This follows from the following:
[**Claim.**]{} Suppose that $U$, $V$ and $W$ are mutually disjoint nonempty sets and that $U \cup V$ and $W \cup V$ are minimal dependent sets such that $U \underset{V}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} W$. Then there is a proper subset $V^* \subset V$ such that $U \cup V^* \cup W$ is minimal dependent.
[**Proof of the claim.**]{} Let $v \in V$ and let $V' = V \setminus \{v\}$. Also let $u \in U$ and let $U' = U \setminus \{u\}$. By assumption, $U \underset{V}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} W$ so $U' \underset{V}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} W$. Since $UV$ is minimal dependent we have $U' {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }V$, so transitivity gives $U' {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }VW$ and hence $U' {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }V'W$. As $V' W$ is independent (since by assumption $VW$ is minimal dependent) it follows that $U'V'W$ is independent. From the minimal dependence of $VW$ it follows that $v \in {\mathrm{cl}}_{\mathcal{N}}(V'W)$ and as $UV$ is minimal dependent we get $u \in {\mathrm{cl}}_{\mathcal{N}}(U'V'W)$, so $UV'W$ is dependent and has dimension $|U'V'W| = |UVW|-2$.
By arguing in the same way as above it follows that if $w \in W$ and $W' = W \setminus \{w\}$, then $UV'W'$ is independent and $UV'W \subseteq {\mathrm{cl}}_{\mathcal{N}}(UV'W')$. If $V' = {\emptyset}$ then $UW$ is minimal dependent and we are done, so now suppose that $V' \neq {\emptyset}$. If for any $v' \in V'$, and letting $V'' = V' \setminus \{v'\}$, the set $UV''W$ is independent then $UV'W$ is minimal dependent and we are done.
Now suppose that $UV''W$ is dependent, where $V''$ is as above for some $v' \in V'$. Then there is $S \subseteq UV''W$ such that $S$ is minimal dependent. Since (as we have shown above) $U'V'W$ and $UV'W'$ are independent sets whenever $U' = U \setminus \{u\}$, $W' = W \setminus \{w\}$ for $u \in U$ and $w \in W$ it follows that $UW \subseteq S$. By letting $V^* = S \cap V''$ it follows that $UV^*W = S$ is minimal dependent. This ends the proof of the claim and also of Lemma \[reducing the number of types\]. $\square$\
Now we can finish the proof of Theorem \[nontrivial dependence implies nontrivial pregeometry\]. By Lemma \[all elements have the same type\], there is an equivalence class $X$ of ‘$\sim$’ such that $X$ contains a set of cardinality at least 3 which is minimal dependent (over ${\emptyset}$) with respect to ${\mathcal{M}}_P$. By Lemma \[the structure on a single equivalence class\] (ii), this set is also minimal dependent with respect to ${\mathcal{M}}_P {{\negthickspace}\upharpoonright {\negthickspace}}X$. This together with the other parts of the same lemma implies that if ${\mathcal{M}}' = {\mathcal{M}}_P {{\negthickspace}\upharpoonright {\negthickspace}}X$, then conditions (i) – (iv) of Theorem \[nontrivial dependence implies nontrivial pregeometry\] hold and hence it is proved.
Proof of Theorem \[impossibility of a structure satisfying the other result\]: impossibility of a nontrivial pregeometry (in the given context) {#impossibility of a nontrivial pregeometry}
===============================================================================================================================================
In order to prove Theorem \[impossibility of a structure satisfying the other result\] it suffices to derive a contradiction from the following assumptions which we now make. Let $V$ be a ternary finite relational vocabulary and ${\mathcal{M}}$ a $V$-structure such that the following conditions hold:
- $Th({\mathcal{M}})$ has elimination of quantifiers and is simple.
- All elements of $M$ have the same type over ${\mathrm{acl}}_{({\mathcal{M}}){^{\mathrm{eq}}}}({\emptyset})$.
- $(M, {\mathrm{cl}})$ is a pregeometry if, for every $X \subseteq M$ and every $x \in M$, $x \in {\mathrm{cl}}(X)$ if and only if $x {\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }X$, where ${\raisebox{-2pt}[5pt][0pt]{$\smile$}
\hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$}\hspace*{-6.8pt}
\raisebox{3pt}[5pt][0pt]{$\diagup$} }$ is with respect to $Th({\mathcal{M}})$.
- $M$ has a minimal dependent subset of cardinality at least $3$.
We first prove two lemmas and then derive a contradiction with the help of these. For these lemmas the full assumption (i) is not necessary, but it suffices to assume (besides ((ii)–(iv)) that $Th({\mathcal{M}})$ is simple and that ${\mathcal{M}}$ is $\omega$-saturated.
\[every set is contained in a minimal dependent set\] If $A \subseteq M$ is finite and independent then there is a minimal dependent $A' \subseteq M$ such that $A \subset A'$ and $|A'| \geq |A|+2$.
[**Proof.**]{} We prove the lemma by induction on $n = |A|$. If $n = 1$ then the conclusion follows from assumptions (ii) and (iv). So now suppose that $n \geq 1$ and $\{a_1, \ldots, a_n, b_1\} \subseteq M$ is independent. By the induction hypothesis there is $a_{n+1} \in M$ such that $\{a_1, \ldots, a_n, a_{n+1}\}$ is independent and included in a minimal dependent set. By the existence of nonforking extensions we may assume that $a_{n+1} \underset{a_1 \ldots a_n}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} b_1$. Since $\{a_1, \ldots, a_n, b_1\}$ is independent it follows from transitivity that $\{a_1, \ldots, a_{n+1}, b_1\}$ is independent. Choose $b_{n+1} \in M$ such that $a_1a_{n+1} \equiv_{\mathcal{M}}b_1b_{n+1}$. Note that we now have $a_1\ldots a_n {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }b_1$, $a_1\ldots a_n {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }a_{n+1}$ and $b_1 {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }b_{n+1}$. By assumption (ii), all elements have the same type over ${\mathrm{acl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}({\emptyset})$, so (as ${\mathcal{M}}$ is $\omega$-categorical) the independence theorem of simple theories implies that there is $c_{n+1} \in M$ such that $$\begin{aligned}
\label{c-n+1 has the right types}
&a_1\ldots a_n a_{n+1} \equiv_{\mathcal{M}}a_1\ldots a_n c_{n+1}, \\
&b_1 b_{n+1} \equiv_{\mathcal{M}}b_1 c_{n+1} \ \ \text{ and} \nonumber \\
&c_{n+1} {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }a_1\ldots a_n b_1. \nonumber\end{aligned}$$ As $\{a_1, \ldots, a_n, b_1\}$ is independent (by assumption) it follows that $$\label{a-1 to a-n, b-1 and c-n+1}
\{a_1, \ldots, a_n, c_{n+1}, b_1\} \text{ is independent, so in particular }
a_1 \ldots a_n c_{n+1} {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }b_1.$$ By the choice of $a_{n+1}$ and (\[c-n+1 has the right types\]), there are $m \geq n+2$ and $a_{n+2}, \ldots, a_m \in M$ such that $$\{a_1, \ldots, a_n, c_{n+1}, a_{n+2}, \ldots, a_m\} \text{ is minimal dependent.}$$ By the existence of nonforking extensions we may assume that $$a_{n+2} \ldots a_m \underset{a_1 \ldots a_n c_{n+1}}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} b_1.$$ This together with (\[a-1 to a-n, b-1 and c-n+1\]) and transitivity implies that $$\label{b-1 is independent from c-n+1 and the a}
a_1 \ldots a_n c_{n+1} a_{n+2} \ldots a_m {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }b_1.$$ As $b_1 c_{n+1} \equiv_{\mathcal{M}}b_1 b_{n+1} \equiv_{\mathcal{M}}a_1 a_{n+1}$ there are $b_2, \ldots,$ $b_n \in M$ and $b_{n+2}, \ldots, b_m \in M$ such that $$\{b_1, \ldots, b_n, c_{n+1}, b_{n+2}, \ldots, b_m\} \text{ is minimal dependent.}$$ By the existence of nonforking extensions we may assume that $$\label{independence of all b from all a over}
b_2 \ldots b_n b_{n+2} \ldots b_m \underset{b_1 c_{n+1}}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a_1 \ldots a_n a_{n+2} \ldots a_m.$$ Let $$A = \{a_1, \ldots, a_n, a_{n+2}, \ldots, a_m\} \ \text{ and } \ B = \{b_1, \ldots, b_n, b_{n+2}, \ldots, b_m\}.$$
It now suffices to prove that $AB$ is minimal dependent, because $a_1,$ $\ldots,$ $a_n, b_1 \in AB$ and $|AB| = 2m-2 \geq |\{a_1, \ldots, a_n, b_1\}| + 2$, since $m \geq n+2$. Since $Ac_{n+1}$ and $Bc_{n+1}$ are minimal dependent it follows that $c_{n+1} \in {\mathrm{cl}}(A)$ and $b_m \in {\mathrm{cl}}(Bc_{n+1} \setminus \{b_m\})$. Consequently (as $(M, {\mathrm{cl}})$ is a pregeometry) $b_m \in {\mathrm{cl}}(AB \setminus \{b_m\})$, so $AB$ is dependent. It remains to prove that every proper subset of $AB$ is independent. We will divide the argument into a few cases.
Let $A' \subset A$ be a proper subset (and we will show that $A'B$ is independent). Then $A' {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }c_{n+1}$. From (\[b-1 is independent from c-n+1 and the a\]) we get (by monotonicity) $A' \underset{c_{n+1}}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} b_1$, so transitivity gives $A' {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }b_1 c_{n+1}$. This together with (\[independence of all b from all a over\]) and transitivity gives $$A' {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }b_1 \ldots b_n c_{n+1} b_{n+2} \ldots b_n$$ and hence $A' {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }B$ from which it follows (by repeatedly using monotonicity and transitivity) that $A'B$ is independent. Since we can choose $A' \subset A$ so that $|A'| = |A|-1$ it follows that $AB$ has dimension $|AB|-1$ (where dimension is with respect to ‘${\mathrm{cl}}$’).
Now let $B' \subset B$ with $|B'| = |B|-1$. To complete the proof it suffices to prove that $AB'$ is independent. There is $b \in B$ such that $B' = B \setminus \{b\}$. First suppose that $b \neq b_1$, so $b_1 \in B'$. As $Bc_{n+1}$ is minimal dependent, $B'c_{n+1}$ is independent, so $b_1c_{n+1} {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }(B' \setminus \{b_1\})$. This together with (\[independence of all b from all a over\]) and transitivity (and monotonicity) implies that $$(B' \setminus \{b_1\}) {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }b_1 a_1 \ldots a_n a_{n+2} \ldots a_m$$ and hence $$(B' \setminus \{b_1\}) \underset{b_1}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a_1 \ldots a_n a_{n+2} \ldots a_m.$$ This together with (\[b-1 is independent from c-n+1 and the a\]) and transitivity gives $B' {\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }A$, so $AB'$ is independent.
Now suppose that $b = b_1$, so $B' = B \setminus \{b_1\}$. Towards a contradiction suppose that $AB'$ is dependent. Then there is a proper subset $C \subset AB'$ such that $AB' \subseteq {\mathrm{cl}}(C)$. As $Ac_{n+1}$ is minimal dependent we have $c_{n+1} \in {\mathrm{cl}}(A)$. As $Bc_{n+1}$ is minimal dependent we have $b_1 \in {\mathrm{cl}}(B'c_{n+1})$ and hence $b_1 \in {\mathrm{cl}}(AB')$. Since $AB' \subseteq {\mathrm{cl}}(C)$ we get $AB = AB'b_1 \subseteq {\mathrm{cl}}(C)$ where $|C| \leq |AB| - 2$. Thus the dimension of $AB$ is at most $|AB| - 2$. But this contradicts our earlier conclusion that the dimension of $AB$ is $|AB| - 1$. $\square$
\[turning a minimal dependent set into an independent set\] Let $A \subset M$ be a minimal dependent set of cardinality at least $3$ and let $a_1, a_2, a_3 \in A$ be distinct. Then there is $a'_3 \in M$ such that $(A \setminus \{a_3\}) \cup \{a'_3\}$ is independent and for all $b, c \in A \setminus \{a_3\}$, if $\{b, c\} \neq \{a_1, a_2\}$, then $bca'_3 \equiv_{\mathcal{M}}bca_3$.
[**Proof.**]{} Let $A = \{a_1, \ldots, a_n\} \subset M$ be a minimal dependent where $n \geq 3$. Let $B = \{a_4, \ldots, a_n\}$ (or $B = {\emptyset}$ if $n = 3$). Then $a_1 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a_2$, $a_3 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a_1$ and $a_3 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a_2$. By the independence theorem of simple theories, there is a type $q$ over ${\mathrm{acl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}(B) \cup \{a_1, a_2\}$ which extends the type of $a_3$ over ${\mathrm{acl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}(B) \cup \{a_1\}$ and the type of $a_3$ over ${\mathrm{acl}}_{{\mathcal{M}}{^{\mathrm{eq}}}}(B) \cup \{a_2\}$, and $q$ does not divide over $B$. Since $Th({\mathcal{M}})$ is $\omega$-categorical, and hence ${\mathcal{M}}{^{\mathrm{eq}}}$ is $\omega$-saturated, there is $a'_3 \in M$ which realizes $q$, so ${\mathrm{tp}}_{\mathcal{M}}(a'_3 / a_1 B) = {\mathrm{tp}}{\mathcal{M}}(a_3 / a_1 B)$, ${\mathrm{tp}}_{\mathcal{M}}(a'_3 / a_2 B) = {\mathrm{tp}}_{\mathcal{M}}(a_3 / a_2 B)$ and $a'_3 \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} a_1 a_2$. It follows, since $A$ is minimal dependent, that $\{a_1, a_2, a'_3, a_4, \ldots, a_n\} = \{a'_3, a_1, a_2\} \cup B$ is independent. From the choice of $a'_3$ it also follows that if $b, c \in A \setminus \{a_3\}$ and $\{b, c\} \neq \{a_1, a_2\}$, then $bca'_3 \equiv_{\mathcal{M}}bca_3$. $\square$\
Now we are ready to make a construction which will lead to a contradiction. By the existence of nonforking extensions we find $a_1, a_2, a_3 \in M$ such that $A_0 = \{a_1, a_2, a_3\}$ is independent. Let $a_3^0 = a_3$. We now find elements $a_3^n \in M$, for $0 < n < \omega$, and finite sets $A_n, B_n \subseteq M$, for $n < \omega$, such that for every $n < \omega$ the following conditions hold:
- $A_n = \{a_1, a_2, a_3^n\}$, $A_{n+1} = \{a_1, a_2, a_3^{n+1}\}$ and both $A_n$ and $A_{n+1}$ are independent,
- $A_n B_0 \ldots B_n$ is minimal dependent,
- $A_{n+1}B_0 \ldots B_n$ is independent, and
- if $b, c \in A_{n+1}B_0 \ldots B_n \setminus \{a_3^{n+1}\}$ and $\{b, c\} \neq \{a_1, a_2\}$, then $bca_3^{n+1} \equiv_{\mathcal{M}}bca_3^n$.
Suppose that (a)–(d) hold. By Lemma \[every set is contained in a minimal dependent set\] there is $B_{n+1}$ such that $A = A_{n+1}B_0 \ldots B_{n+1}$ is minimal dependent. By (a) and Lemma \[turning a minimal dependent set into an independent set\] there is $a_3^{n+2} \in M$ such that
- $(A_{n+1}B_0 \ldots B_{n+1} \setminus \{a_3^{n+1}\}) \cup \{a_3^{n+2}\}$ is independent and
- if $b, c \in A_{n+1}B_0 \ldots B_{n+1} \setminus \{a_3^{n+1}\}$ and $\{b, c\} \neq \{a_1, a_2\}$, then\
$bca_3^{n+2} \equiv_{\mathcal{M}}bca_3^{n+1}$.
Let $A_{n+2} = \{a_1, a_2, a_3^{n+2}\}$. Now (a)–(d) holds when ‘$n$’ is replaced by ‘$n+1$’. Note that if $n=0$ then (a) holds by the choices of $A_0$ and $a_3^0$ and then we find $B_0$, $A_1$ and $a_3^1$ such that (b)–(d) hold for $n=0$ in the same way as we did for the general case $n$.
Since $Th({\mathcal{M}})$ is $\omega$-categorical (because ${\mathcal{M}}$ has elimination of quantifiers) there are only finitely many 3-types over ${\emptyset}$. Thus there are $i < j$ such that $a_1a_2a_3^{i+1} \equiv_{\mathcal{M}}a_1a_2a_3^{j+1}$. Since the vocabulary is ternary it follows from elimination of quantifiers and (d) applied to $n = i+1, \ldots, j$ that $$\label{same type for i and j}
{\mathrm{tp}}_{\mathcal{M}}(a_1, a_2, a_3^{i+1} / B_0 \ldots B_{i+1}) = {\mathrm{tp}}_{\mathcal{M}}(a_1, a_2, a_3^{j+1} / B_0 \ldots B_{i+1}).$$ By (b) for $n = i$, $A_{i+1}B_0 \ldots B_{i+1}$ is dependent. By (c) for $n = j$,\
$A_{j+1}B_0 \ldots B_j$ is independent and as $i < j$ it follows that $A_{j+1}B_0 \ldots B_{i+1}$ is independent. Since (by (a) for $n = i+1$ and $n = j+1$) $A_{i+1} = \{a_1, a_2, a_3^{i+1}\}$ and $A_{j+1} = \{a_1, a_2, a_3^{j+1}\}$ we must have $${\mathrm{tp}}_{\mathcal{M}}(a_1, a_2, a_3^{i+1} / B_0 \ldots B_{i+1}) \neq {\mathrm{tp}}_{\mathcal{M}}(a_1, a_2, a_3^{j+1} / B_0 \ldots B_{i+1})$$ which contradicts (\[same type for i and j\]). Thus the proof of Theorem \[impossibility of a structure satisfying the other result\] is finished.
\[remark about higher dimensional amalgamation\][The notion of “$n$-amalgamation property” has been defined in slightly different ways and with slightly different names in various articles. Let us use the definition of [*$n$-complete amalgamation (property)*]{} used in [@KKT; @Pal17]. Then the independence theorem is equivalent to the $3$-complete amalgamation property. Now one can modify Lemma \[turning a minimal dependent set into an independent set\], its proof and the argument after it so that one gets the following result: [*Suppose that $V$ is a finite relational vocabulary with maximal arity $k$ and $T$ a simple $V$-theory with elimination of quantifiers and $k$-complete amalgamation. Then $T$ has no model ${\mathcal{M}}$ such that (ii) – (iv) in the beginning of this section hold.*]{} Since $n$-complete amalgamation, when restricted to (finite tuples of) real elements, is preserved when passing from $T$ to $Th({\mathcal{M}}')$ in Theorem \[nontrivial dependence implies nontrivial pregeometry\] it follows that if $T$ is as above then $T$ has trivial dependence and finite SU-rank. ]{}
[**Acknowledgement.**]{} I thank the anonymous referee for a detailed examination of the article, including finding some minor mistakes (now corrected) and giving suggestions that improved the clarity of the arguments.
[99]{}\[References\]
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T. de Piro, B. Kim, The geometry of 1-based minimal types, [*Transactions of the American Mathematical Society*]{}, Vol. 355 (2003) 4241–4263.
D. Evans, F. O. Wagner, Supersimple $\omega$-categorical groups and theories, [*The Journal of Symbolic Logic*]{}, Vol. 65 (2000) 767–776.
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E. Hrushovski, Unidimensional theories are superstable, [*Annals of Pure and Applied Logic*]{}, Vol. 50 (1990) 117–138.
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V. Koponen, Binary simple homogeneous structures, [*Annals of Pure and Applied Logic*]{}, to appear, online\
<https://arxiv.org/abs/1609.02433>
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<https://arxiv.org/abs/1707.05954>
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[^1]: His counterexample of an $\omega$-categorical stable “pseudoplane” which is not superstable has never been published as far as I know, but it is mentioned in various places in the literature.
[^2]: The terminology [*ultrahomogeneous*]{} and [*finitely homogeneous*]{} is also used in the literature.
[^3]: A basic result is that $\{\bar{a}_1, \ldots, \bar{a}_n\}$ is independent over $B$ if and only if, for every $0 < i < n$, $\bar{a}_{i+1} \underset{B}{{\raisebox{-2pt}[5pt][0pt]{$\smile$} \hspace*{-6.8pt}\raisebox{3pt}[5pt][0pt]{$|$} \; \: }} \bar{a}_1, \ldots, \bar{a}_i$.
[^4]: A basic result is that $Y = \{y_1, \ldots, y_n\}$ is independent if and only if, for every $0 < i < n$, $y_{i+1} \notin {\mathrm{cl}}(y_1, \ldots, y_i)$.
[^5]: Lemma 3.2 in [@Kop17a] uses the terminology ‘homogeneous’ instead of ‘elimination of quantifiers’. But ${\mathcal{M}}$ has elimination of quantifiers if and only if its unique, up to isomorphism, countable elementary substructure has elimination of quantifiers, and the same holds for ${\mathcal{M}}_P$. Therefore we can formulate part (ii) as done here.
|
---
abstract: |
We describe a diffraction microscopy technique based on refractive optics to study structural variations in crystals. The X-ray beam diffracted by a crystal was magnified by Beryllium parabolic refractive lenses on a 2D X-ray camera. The microscopy setup was integrated into the 6-circle Huber diffractometer at the ESRF beamline ID06. Our setup allowed us to visualize structural imperfections with a resolution of $\approx
1{\ensuremath{\,\mathrm{\mu m}}}$. The configuration, however, can easily be adapted for sub-$\mathrm{\mu m}$ resolution.
author:
- |
Thomas Roth[^1], Carsten Detlefs, Irina Snigireva, Anatoly Snigirev\
\
European Synchrotron Radiation Facility,\
B.P. 220, 6 rue Jules Horowitz, 38043 Grenoble Cedex 9, France
bibliography:
- 'DiffMicro\_Arxiv.bib'
date: 'September 21, 2014'
title: 'X-ray diffraction microscopy based on refractive optics'
---
Keywords: Diffraction imaging, X-ray microscopy, Compound refractive lenses, X-ray topography
Introduction
============
X-ray diffraction imaging, known later as X-ray topography, originated in the late 1920ies and early 1930ies, when researchers revealed the internal structure of individual Laue spots in diffraction patterns [@Berg31; @Barrett31]. To improve the resolution, fine grain photographic emulsions were exposed and examined under optical microscopes – for this reason the technique was sometimes called X-ray microscopy [@Barrett45]. This method was applied for both mapping of strains in heavily deformed materials such as cold-worked metals and alloys [@Barrett45] and studies of individual defects in near-perfect single crystals [@Ramachandran44].
It was assumed that each point on the film or detector corresponds to a small volume in the reflecting crystal. Simple geometrical optics then requires the incoming X-ray beam to be tightly collimated, and the film to be placed as closely as possible to the sample. The achievable resolution is then limited by the detector resolution, at best $500{\ensuremath{\,\mathrm{nm}}}$ [@Martin06] but more typically $1{\ensuremath{\,\mathrm{\mu m}}}$.
However, in the absence of X-ray optics between the sample and the film diffraction effects progressively blur the image with increasing sample-to-detector distance. For a typical experimental setup (wavelength $\lambda=1\,\textrm{\AA}$, sample-to-detector distance $s=1{\ensuremath{\,\mathrm{cm}}}$, and sample feature size $d=1{\ensuremath{\,\mathrm{\mu m}}}$) the diffraction limited resolution due to propagation of the perturbed wavefront from the exit surface of the crystal to the detector can be approximated as $$\delta
\approx \frac{\lambda}{d}\cdot s
= \frac{10^{-10}{\ensuremath{\,\mathrm{m}}}}{10^{-6}{\ensuremath{\,\mathrm{m}}}}\cdot 10^{-2}{\ensuremath{\,\mathrm{m}}}
= 1{\ensuremath{\,\mathrm{\mu m}}}.$$ Note that to image $10\times$ smaller features on the sample ($d=100{\ensuremath{\,\mathrm{nm}}}$ and therefore $\delta=100{\ensuremath{\,\mathrm{nm}}}$) the sample-to-detector distance would have to be decreased by a factor of 100, $s=100{\ensuremath{\,\mathrm{\mu m}}}$. In most cases this is technically not feasible. Furthermore, to the best of our knowledge, 2D imaging detectors with a spatial resolution of $100{\ensuremath{\,\mathrm{nm}}}$ are not yet available.
On the other hand, conventional X-ray microscopy techniques as proposed by Kirkpatrick and Baez [@kirkpatrick48; @baez52] have been implemented in the hard X-ray domain rather late. Here, an in-line scheme is used where the beam transmitted through the sample is magnified by X-ray optics such as mirrors [@Underwood86], Fresnel zone plates [@lai95], Bragg-Fresnel lenses [@Snigirev97], or refractive lenses [@lengeler99]. Such forward scattering techniques are primarily sensitive to spatial variations of the X-ray index of refraction which depends mostly on the local density of the sample.
In this paper we propose a compact scheme for diffraction microscopy using X-ray refractive lenses between the sample and the detector. The insertion of refractive optics into the diffracted beam allows significant improvements of the resolution, potentially down to below 100nm (a resolution of 300nm has been demonstrated using a similar lens in transmission X-ray microscopy[@Bosak10]). Furthermore, the progressive blurring due to the wavefront propagating from the sample to the detector can be overcome by a lens, thus reestablishing the direct mapping of intensity variations on the detector to the reflectivity variations on the sample. In this case the image resolution can, in principle, reach the limit imposed by dynamical diffraction effects within the crystal.
Recently, Fresnel zone plates have been used in X-ray reflection microscopy to image monomolecular steps at a solid surface [@Fenter06] and for scanning X-ray topography of strained silicon oxide structures [@Tanuma06]. CRLs have the advantage that efficient focusing can be achieved at higher photon energies, $E \gg
10{\ensuremath{\,\mathrm{keV}}}$. Please note that standard KB mirrors are not suited for imaging setups, as they do not fulfill the Abbe-sine condition. More complicated multi-mirror setups are however being developed to overcome this limitation in transmission geometry [@matsuyamaHard2012].
Experimental details
====================
![ \[fig:setup\]Experimental setup for Bragg diffraction microscopy. 11[$\,\mathrm{keV}$]{} X-rays impinge on the sample. The diffracted intensity is imaged onto a Sensicam camera via a set of 66 Beryllium compound refractive lenses (CRLs) with an apex-radius of curvature of $50{\ensuremath{\,\mathrm{\mu m}}}$. The scattering plane is horizontal, and the imaged features on the sample were aligned parallel to the scattering plane. ](setup_rev.pdf){width="0.7\columnwidth"}
Our experiment was carried out at the undulator beamline ID06 of the European Synchrotron Radiation Facility. A cryogenically cooled permanent magnet in-vacuum undulator [@Chavanne09] with a period of 18[$\,\mathrm{mm}$]{} and a conventional in-air undulator with a period of 32[$\,\mathrm{mm}$]{}, combined with a liquid nitrogen cooled Si (111) monochromator, delivered photons at an energy of 11[$\,\mathrm{keV}$]{}. A transfocator located at 38.7[$\,\mathrm{m}$]{} from the source point (electron beam waist position in the middle between the two undulators) acted as a condenser, i.e. it focused the photons onto the sample at 67.9[$\,\mathrm{m}$]{} distance from the source, using a combination of paraboloid (2D) compound refractive lenses, CRLs, [@lengeler99]: one lens with radius of curvature at the apex $R$= 1.5[$\,\mathrm{mm}$]{} and two lenses with $R$= 0.2[$\,\mathrm{mm}$]{}, all made out of high-purity Beryllium (Be). The use of the condenser-CRLs improved the optical efficiency of the system (absorption of X-rays in the condenser-CRLs was only about 6%.), as it increased the flux on the imaged sample area. The divergence of the photon beam is not altered significantly, as the condenser CRL works almost in a 1:1 magnification geometry. A flux of approximately $2 \cdot 10^{12} {\ensuremath{\,\mathrm{photons/s}}}$ was incident to the sample. The sample was mounted on a six circle diffractometer. The scattering plane coincided with the horizontal plane.
The detector consisted of a scintillator screen, magnifying optics, and a high resolution CCD-camera. The 9.9[$\,\mathrm{\mu m}$]{} thick LAG:Eu scintillator on a 170[$\,\mathrm{\mu m}$]{} YAG substrate converted X-rays into visible light, which was projected onto the CCD by the objective lens (Olympus UPLAPO $\times 10$, numerical aperture 0.4). The CCD camera (pco SensicamQE) had $1376 \times 1040$ pixels (px) of size $6.45{\ensuremath{\,\mathrm{\mu m/px}}} \times 6.45{\ensuremath{\,\mathrm{\mu m/px}}}$ and 12 bit depth, yielding a field of view on the scintillator of $887 \times
670{\ensuremath{\,\mathrm{\mu m^2}}}$ with an effective resolution of 1.3[$\,\mathrm{\mu m}$]{}. Each CCD pixel imaged an area of $0.645\times 0.645{\ensuremath{\,\mathrm{\mu m^2}}}$ on the scintillator.
In front of the detector, on the same diffractometer arm, a second set of paraboloid Be CRLs (66 lenses with apex-radius of curvature $R$=$50{\ensuremath{\,\mathrm{\mu m}}}$) was mounted as X-ray objective lens, i.e. to image the diffracted intensity pattern at the sample exit surface onto the detector. These lenses were mounted on translation and rotation stages to align the lens stack, in particular to tune the sample-to-lens distance to achieve best focusing onto the detector.
The focal length of this lens stack at 11[$\,\mathrm{keV}$]{} was about 14[$\,\mathrm{cm}$]{}, so that a $\approx 4$-fold magnified image was achieved with the lens center placed about 18[$\,\mathrm{cm}$]{} downstream of the sample. The effective aperture was about 240[$\,\mathrm{\mu m}$]{}, giving a corresponding diffraction limit of 130[$\,\mathrm{nm}$]{}. The transmission through the lens stack is reduced by the absorption from the thinnest lens part, plus the increased absorption for rays travelling further away from the lens center, resulting in an effective aperture with Gaussian profile [@Snigirev97; @lengeler99]. The first contribution is easy to calculate and gives an absorption of 18%. Considering the size of the illuminated sample ($\approx$ 200nm, see below) and approximating the reflected beam as a parallel beam, the total absorption is closer to 50%.
Scaling the effective detector resolution by the magnification factor 4 to $1.3{\ensuremath{\,\mathrm{\mu m}}}/4 = 0.33{\ensuremath{\,\mathrm{\mu m}}}$, we expected a resolution limit of $\sqrt{(0.33{\ensuremath{\,\mathrm{\mu m}}})^2+(130{\ensuremath{\,\mathrm{nm}}})^2} \approx 350{\ensuremath{\,\mathrm{nm}}}$ with this set-up.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----
(a) (b)
![\[fig:stripe\_results\] Diffraction microscopy images (exposure time 2.5[$\,\mathrm{s}$]{}) of the [$\mathrm{SiO_{2}}$]{} stripe structure at different Bragg angles: a) at $0.006^\circ$ below the maximum diffracted intensity; b) almost at the maximum; c) $0.005^\circ$ above the maximum; d) $0.009^\circ$ above the maximum. The beam travels from left to right. e) Rocking curve as measured by a photo diode, indicating also the angle positions corresponding to Figs. \[fig:stripe\_results\]a) to \[fig:stripe\_results\]d). f) Scanning electron microscope image of the same [$\mathrm{SiO_{2}}$]{} stripe system. Note that b) shows stripe like intensity in a region where e) shows a homogeneous [$\mathrm{SiO_{2}}$]{} surface. This indicates a strain propagation in the Si beyond the etched areas.](SiO2_0007scale.pdf "fig:"){width="\mywidth"}
(c) (d)
(e) (f)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----
Two samples were imaged in Bragg geometry. The first sample was a Si (111) wafer upon which a regular stripe pattern of amorphous $\mathrm{SiO_2}$ has been fabricated by thermal oxidation followed by standard photo-resist etching. The $\mathrm{SiO_2}$ layer was $z=1.15{\ensuremath{\,\mathrm{\mu m}}}$ thick, and was etched to fabricate 2[$\,\mathrm{\mu m}$]{} wide windows with a period of 4[$\,\mathrm{\mu m}$]{}. The substrate was aligned in the diffractometer to set the oxide stripes parallel to the diffraction plane. In order to record magnified diffraction images of the sample the Bragg (333) reflection of the Si substrate (Bragg angle $\theta_B=32.63^\circ$) was used.
The second sample was a linear Bragg-Fresnel lens (BFL) fabricated on a Si (111) substrate (for the fabrication process see [@aristov87]). The basic geometrical parameters were: an outermost zone width of 0.5[$\,\mathrm{\mu m}$]{}, a height of the structure of 4.4[$\,\mathrm{\mu m}$]{}, and an aperture of 200[$\,\mathrm{\mu m}$]{}. Again, the sample was aligned with the structures parallel to the horizontal scattering plane. Again, the Si (333) ($\theta_B= 32.63^\circ$) reflection was studied.
Resolution
==========
The homogeneous periodicity of the $\mathrm{SiO_2}$ line pattern (Fig. \[fig:stripe\_results\]) was used to calibrate the effective magnification of our configuration. The mask used to produce the pattern had a period of 4[$\,\mathrm{\mu m}$]{}, in good agreement with the value, 4.1(1)[$\,\mathrm{\mu m}$]{}, obtained by scanning electron microscopy (SEM), see Fig. \[fig:stripe\_results\]f. Our X-ray image of this structure shows 15 periods over 395(5)[$\,\mathrm{px}$]{}. (Fig. \[fig:stripe\_results\]b) The line spacing on the fluorescence screen of the detector was therefore $0.645{\ensuremath{\,\mathrm{\mu m/px}}}\cdot 395{\ensuremath{\,\mathrm{px}}}/15 = 17.0(2){\ensuremath{\,\mathrm{\mu m}}}$, yielding a magnification factor of $17.0{\ensuremath{\,\mathrm{\mu m}}}/4.1{\ensuremath{\,\mathrm{\mu
m}}}=4.2(1)$ for the CRL stack and $4.1{\ensuremath{\,\mathrm{\mu m}}}\cdot 15/395{\ensuremath{\,\mathrm{px}}} =
0.156(4){\ensuremath{\,\mathrm{\mu m/px}}}$ for the overall experiment. The resulting field of view on the sample was $\approx 162{\ensuremath{\,\mathrm{\mu m}}}/\sin(\theta_B)$ in the horizontal (within the scattering plane) and $215{\ensuremath{\,\mathrm{\mu m}}}$ in the vertical direction (perpendicular to the scattering plane).
An upper limit for the effective resolution of our imaging system can be estimated from the Fourier transform (FT) of the image (see Fig. \[fig\_ft\_sio2\]). Peaks corresponding to the fundamental, second and third harmonics of the structure are clearly visible above a two-component background. For quantitative analysis, the FT was fitted to a model function $$\tilde{I}(f) = \sum\limits_{n=1}^{3} \left( A_n \cdot g(f-n f_1) \right)
+ a e^{-f/f_\mathrm{BG}} + b,
\label{eq_model_sio2}$$ where $\tilde{I}(f)$ is the Fourier transform of the image at spatial frequency $f$. The background is composed of a constant and an exponentially decaying term with characteristic frequency $f_{\mathrm{BG}}$. The harmonics are modelled by Gaussians $g(f) =
(2\pi \tilde{\sigma}^2)^{-1/2} \exp(-f^2/2\tilde{\sigma}^2)$ with amplitude factors $A_n$ for the $n$-the harmonic. The magnification factor determined above (0.156[$\,\mathrm{\mu m/px}$]{}) was used to scale the frequency axis. The resulting parameters are shown in Fig. \[fig\_ft\_sio2\]. The ratio $A_1/A_3$ can be used to estimate the modulation transfer function (MTF), $\tilde{I}(f)=\tilde{c}(f)\cdot \mathrm{MTF}(f)$, where $\tilde{c}(f)$ is the FT of the scattering amplitude of the sample. For an ideal square wave, $\tilde{c}(f_1)/\tilde{c}(3 f_1) = 3$. For a more smooth modulation, e.g. resulting from continuous buckling of the Bragg planes due to strain [@Kuznetsov04], the higher harmonics will be suppressed, $\tilde{c}(f_1)/\tilde{c}(3 f_1) > 3$ so that $$\frac{
\tilde{I}(f_1)
}{
\tilde{I}(3 f_1)
}
=
\frac{
\tilde{c}(f_1)
}{
\tilde{c}(3 f_1)
}
\cdot
\frac{
\mathrm{MTF}(f_1)
}{
\mathrm{MTF}(3 f_1)
}
\geq
3
\cdot
\frac{
\mathrm{MTF}(f_1)
}{
\mathrm{MTF}(3 f_1)
}$$ The presence of a second harmonic at $2 f_1$ indicates that the contrast does not follow an ideal square modulation.
In the absence of any further information, we assume the MTF to be a Gaussian with standard deviation $\tilde{\sigma}_{\mathrm{MTF}}$, so that $\tilde{\sigma}_{\mathrm{MTF}} = 2 f_1
\left(\log[\mathrm{MTF}(f_1)/\mathrm{MTF}(3f_1)]\right)^{-1/2}$. Using the values obtained from the fit we find $\tilde{\sigma}_{\mathrm{MTF}} \geq 0.254 {\ensuremath{\,\mathrm{\mu m^{-1}}}}$, corresponding to a Gaussian point spread function (PSF) with standard deviation $\tilde{\sigma}_{\mathrm{PSF}} = {1}/(2\pi
\tilde{\sigma}_{\mathrm{MTF}}) \leq 0.625{\ensuremath{\,\mathrm{\mu m}}}$ and full width at half maximum (FWHM) $\leq 1.47{\ensuremath{\,\mathrm{\mu m}}}$ on the sample (9.4[$\,\mathrm{px}$]{} on the CCD).
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
(a) (b)
![\[fig\_zone\_plate\] a), b) Diffraction microscopy images of a Bragg-Fresnel lens (Exposure times a) 1[$\,\mathrm{s}$]{} and b) 2.5[$\,\mathrm{s}$]{}). c) Scanning electron microscope image of the same Bragg-Fresnel lens. d) Fourier analysis of the stripe structures. The region of interest (ROI) shown in (d) was divided into 10 vertical slices. Each slice was Fourier transformed. The average of the resulting magnitudes were fit to eq. \[eq\_model\_bfl\]. The modulation transfer function intersects the background at $f_{\mathrm{max}}=1.10{\ensuremath{\,\mathrm{\mu m^{-1}}}}$, indicating a minimum observable peak-to-valley distance of $1/(2f_{\mathrm{max}})=0.46{\ensuremath{\,\mathrm{\mu m}}}$. ](BFL_1597scale.pdf "fig:"){width="\mywidth"} ![\[fig\_zone\_plate\] a), b) Diffraction microscopy images of a Bragg-Fresnel lens (Exposure times a) 1[$\,\mathrm{s}$]{} and b) 2.5[$\,\mathrm{s}$]{}). c) Scanning electron microscope image of the same Bragg-Fresnel lens. d) Fourier analysis of the stripe structures. The region of interest (ROI) shown in (d) was divided into 10 vertical slices. Each slice was Fourier transformed. The average of the resulting magnitudes were fit to eq. \[eq\_model\_bfl\]. The modulation transfer function intersects the background at $f_{\mathrm{max}}=1.10{\ensuremath{\,\mathrm{\mu m^{-1}}}}$, indicating a minimum observable peak-to-valley distance of $1/(2f_{\mathrm{max}})=0.46{\ensuremath{\,\mathrm{\mu m}}}$. ](BFL_1601roi.pdf "fig:"){width="\mywidth"}
(c) (d)
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Further experiments were performed on a Bragg-Fresnel lens. In this sample, the zone width decreases away from the center, thus yielding a richer Fourier spectrum that should provide more detailed information on the MTF and the effective resolution of our system.
Fig. \[fig\_zone\_plate\] shows the Bragg microscopy images of the Bragg-Fresnel lens (panels a and b), a Scanning electron microscopy image of the same structure (panel c), and a Fourier analysis (panel d) of the region of interest shown in panel b.
As above, Fourier analysis was performed to reveal the resolving power of our experiment. The region of interest shown in panel b) was divided into 10 vertical slices. Each slice was Fourier transformed, The average magnitude of the FT is shown in panel d). A Gaussian MTF with an exponential background was fitted to the average FT, again using the magnification factor of 0.156[$\,\mathrm{\mu m/px}$]{} to scale the frequency axis. $$\tilde{I}(f) = A \cdot g(f) + a \cdot e^{-f/f_{\mathrm{BG}}}
\label{eq_model_bfl}$$ From the resulting parameters (listed in Fig. \[fig\_zone\_plate\]d) two estimates of the resolution were derived: The standard deviation of the MTF, and the frequency $f_{\mathrm{max}}$ where the MTF falls below the background.
The fit yielded $\tilde{\sigma}_{\mathrm{MTF}}=0.34{\ensuremath{\,\mathrm{\mu m^{-1}}}}$ and $f_{\mathrm{max}}=1.10{\ensuremath{\,\mathrm{\mu m^{-1}}}}$. The period of a structure at this frequency is $\lambda_{\mathrm{min}}=1/f_{\mathrm{max}}=0.92{\ensuremath{\,\mathrm{\mu m}}}$, so that the smallest observable peak-to-valley valley distance on the sample is $\lambda_{\mathrm{min}}/2 = 0.46{\ensuremath{\,\mathrm{\mu m}}}$. The standard deviation of the MTF corresponds to a PSF with standard deviation $\sigma_{\mathrm{PSF}}=1/(2\pi
\tilde{\sigma}_{\mathrm{MTF}})=0.47{\ensuremath{\,\mathrm{\mu m}}}$ and FWHM 1.1[$\,\mathrm{\mu m}$]{} on the sample (7.1[$\,\mathrm{px}$]{} on the CCD), slightly better than the estimate obtained above for the [$\mathrm{SiO_2}$]{} stripe pattern.
We have thus shown that our setup reaches sub-micrometer resolution in the vertical direction (perpendicular to the scattering plane). In the horizontal direction (parallel to the scattering plane) the sample does not lie perpendicular to the camera plane. Viewing the sample at the Bragg angle $\theta_B$ yields a projected in-plane images size that is smaller by a factor $\sin(\theta_B)$, here $\sin(32.63^\circ)=
0.539$. Assuming that the resolution limit is given by the detector and the diffraction limit of the imaging lenses, the in-plane resolution at the sample surface is then degraded by the factor $1/\sin(\theta_B) = 1.85$ compared to the vertical out-of-plane direction. Furthermore, within the scattering plane, the beam transverses the scatterer partly. The beam path length inside the sample (limited by absorption or extinction) is comparable to the resolution, so that additional blurring is to be expected. This could be avoided in backscattering geometry ($\theta_B \approx 90^\circ$), or when imaging high-Z materials with low penetration depth.
Discussion
==========
Using the example of the [$\mathrm{SiO_2}$]{} stripe sample, we now discuss the information obtainable from diffraction images taken at different Bragg angles. We recall that the reflected intensity stems form the underlying single crystalline Si waver and not from the amorphous top structure of [$\mathrm{SiO_2}$]{} stripes.
Contrast in the diffraction image may arise from several effects: (a) absorption leading to different amplitudes of rays that do or do not travel through the thin [$\mathrm{SiO_2}$]{} layer (b) phase shift between these beams, as the index of refraction of [$\mathrm{SiO_2}$]{} is different from unity, and (c) local variations of the Si(111) reflectivity due to strain in the Si substrate induced by the overlying [$\mathrm{SiO_2}$]{} layer.
For (a) and (b) we can calculate the expected contrast. The index of refraction of [$\mathrm{SiO_2}$]{} at $E=11{\ensuremath{\,\mathrm{keV}}}$ is $n=1-\delta+i \beta$ with $\delta=3.8\cdot 10^{-6}$ and $\beta=
2.7\cdot 10^{-8}$ [@CXROweb], assuming a density of 2.2[$\,\mathrm{g/cm^3}$]{}. The path length of the X-rays through [$\mathrm{SiO_2}$]{} is $L=2z/\sin(\theta_B)=4.3{\ensuremath{\,\mathrm{\mu m}}}$. The E-field amplitude of the beam travelling through the [$\mathrm{SiO_2}$]{} layer is therefore reduced to $\exp(-2\pi L \beta/\lambda) = 0.9935$, whereas its phase is shifted by $L\delta/\lambda\cdot 360^\circ=52^\circ$ as compared to the beam travelling through an adjacent groove via the bare Si surface. Consequently, the absorption contrast (a) is expected to be 1-$\frac{0.9935^2}{1}$ = 1.3%. The phase contrast (b) occurs through interference at the edges. It can be estimated in calculating the intensity resulting from the superposition of two beam parts that travel (i) through the SiO$_2$ and acquiring the 52$^\circ$ phase shift and a part (ii) that does not travel through the SiO$_2$, but through the groove. This intensity is to be compared with the signal from two beams that did not experience a relative phase shift. We obtain a phase contrast of 1-$\frac{|1+\exp(i\cdot 52^\circ)|^2}{|1+1|^2}$ = 20%. For the regular [$\mathrm{SiO_2}$]{} pattern of Fig \[fig:stripe\_results\]b), the measured contrast was 35%. So a part of the contrast must come from strain.
The magnitude of the strain contrast (c) is difficult to estimate. However, for (a) and (b) the contrast at each point of the rocking curve should be identical, whereas strain might shift and broaden the rocking curve, so that for (c) the contrast in the diffraction micrographs at different points of the rocking curve might differ [@Tanuma06].
This can indeed be seen when comparing images recorded at different positions on the rocking curve (Fig. \[fig:stripe\_results\]a–d). Fig. \[fig:stripe\_results\]a and \[fig:stripe\_results\]b show intensity on the right-hand side of the [$\mathrm{SiO_2}$]{} line pattern, caused by strain propagation beyond the etched areas. Fig. \[fig:stripe\_results\]c and d show that control structures etched into the [$\mathrm{SiO_2}$]{} (shown in the right part of the SEM image, Fig. \[fig:stripe\_results\]f) cause strong strain in the [$\mathrm{Si}$]{} substrate.
Such shifts of the rocking curve can occur via two routes, local tilting of the lattice planes, or local modifications of the lattice parameter [@Aristov92; @Aristov92b]. In the former, a positive tilt on one side of a straining feature should be accompanied by a negative tilt on the opposite side. The corresponding areas should be visible at angles symmetric to the center of the rocking curve. A local modification of the lattice parameter, on the other hand, would lead to a unidirectional shift with respect to the unstrained rocking curve [@Tanuma06].
The sharp features visible in Fig. \[fig:stripe\_results\]d appear only $\approx
0.009^\circ$ above the rocking curve, indicating that the lattice parameter is compressed by $$\frac{\Delta d}{d}
\approx \mathrm{cot}(\theta_B) \cdot \Delta\theta
\approx 2.5\cdot 10^{-4}.$$
As shown in Fig. \[fig:stripe\_results\]e, this strain level is clearly resolved in our experiment. The sensitivity to lattice strain could be further improved by selecting higher order Bragg reflections with narrower rocking curves.
Conclusion
==========
X-ray diffraction microscopy combines the advantages of X-ray microscopy in forward scattering geometry and conventional diffraction topography without image magnification:
- As in transmission X-ray microscopy, the effective resolution is greater than that achievable with conventional diffraction topography, which is limited by detector resolution and sample-to-detector distance (diffraction effects).
- As in diffraction topography, the technique is sensitive to microscopic crystallographic imperfections such as strain, dislocations, twinning, etc.
As we have shown here, data acquisition is fast: A single exposure is sufficient and contains all the maximum resolution information, thus the technique is robust with respect to instabilities of the experimental setup, and it has the potential to study transient, non-equilibrium phenomena where it is impossible to acquire several images of the same state.
It should be underlined that the proposed diffraction microscopy technique has great potential for non-destructive studies of highly deformed metals and alloys. By comparing the images taken at different angular settings the contrast due to strain or orientation are easily distinguished [@Afanasev71]. Adding a tomography option ($180^\circ$ rotation) will provide 3D mapping of the orientation and strain of individual grains in polycrystalline materials.
The use of CRLs is of particular interest in diffraction topography since these optics are well adapted to focusing hard X-rays, and are relatively straight-forward to implement on existing diffractometer setups. The technique, however, can also be used with other imaging systems such as Fresnel zone plates [@Tanuma06; @Fenter06] or mirrors such as Wolter optics [@Wolter52; @Takano02]. This flexibility enables the use of X-ray diffraction microscopy over a very wide range of photon energies from sub-keV soft X-rays, e.g. for the study of multilayers, to very hard X-rays with several tens of keV. Furthermore, the technique can be combined with other standard X-ray techniques to access information unobtainable in transmission geometry. Examples include grazing-incidence diffraction to image micro- and nanostructures grown on a surface, magnetic scattering to image ferromagnetic [@Kreyssig09] or antiferromagnetic [@Lang04] magnetic domain patterns and the imaging of ferroelectric domains [@Fogarty96].
Finally, the field of view and the magnification can be adjusted in-situ simply by changing the number of lenses (e.g. by using a Transfocator) and the sample-to-lens or lens-to-detector distance.
[**\
Acknowledgments**]{}
We acknowledge the European Synchrotron Radiation Facility (ESRF) for the provision of beam time on ID06. C. D. thanks R. Barrett for stimulating discussions and critical reading of the manuscript.
[^1]: now at: European XFEL GmbH, Hamburg, Germany. e-mail: [email protected]
|
---
abstract: 'This paper proposes [*Group-In*]{}, a wireless scanning system to detect static or mobile people groups in indoor or outdoor environments. Group-In collects [*only*]{} wireless traces from the Bluetooth-enabled mobile devices for group inference. The key problem addressed in this work is to detect not only static groups but also moving groups with a multi-phased approach based only noisy wireless Received Signal Strength Indicator (RSSIs) observed by multiple wireless scanners without localization support. We propose new centralized and decentralized schemes to process the sparse and noisy wireless data, and leverage graph-based clustering techniques for group detection from short-term and long-term aspects. Group-In provides two outcomes: 1) group detection in short time intervals such as two minutes and 2) long-term linkages such as a month. To verify the performance, we conduct two experimental studies. One consists of 27 controlled scenarios in the lab environments. The other is a real-world scenario where we place Bluetooth scanners in an office environment, and employees carry beacons for more than one month. Both the controlled and real-world experiments result in high accuracy group detection in short time intervals and sampling liberties in terms of the Jaccard index and pairwise similarity coefficient.'
author:
- 'G[ü]{}rkan Solmaz'
- 'Jonathan F[ü]{}rst'
- Samet Aytaç
- 'Fang-Jing Wu'
bibliography:
- 'Main.bib'
title: 'Group-In: Group Inference from Wireless Traces of Mobile Devices'
---
©2020 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.
|
---
abstract: 'The homogeneous Bethe-Salpeter equation is solved in the quenched ladder approximation for the vector positronium states of 4-component quantum electrodynamics in 2 space and 1 time dimensions. Fermion propagator input is from a Rainbow approximation Dyson-Schwinger solution, with a broad range of fermion masses considered. This work is an extension of earlier work on the scalar spectrum of the same model. The non-relativistic limit is also considered via the large fermion mass limit. Classification of states via their transformation properties under discrete parity transformations allows analogies to be drawn with the meson spectrum of QCD.'
address: ' Department of Theoretical Physics, Research School of Physical Sciences and Engineering, Australian National University, Canberra, ACT 0200, Australia\'
author:
- 'T. W. Allen and C. J. Burden'
title: 'Vector Positronium States in three-dimensional QED'
---
Introduction
============
This paper extends our previous nonperturbative studies of Quantum Electrodynamics in three spacetime dimensions (QED$_3$) [@Bu92; @AB96] from scalar to vector positronium states.
Being a confining theory, the low energy behaviour of QED$_3$ must be dealt with nonperturbatively. We consider QED$_3$ to be a simple but effective testing ground for non-perturbative methods commonly applied to Quantum Chromodynamics (QCD$_4$). Again our approach is via a solution to the homogeneous Bethe-Salpeter equation (BSE) with fermion propagator input from the Dyson-Schwinger equation (DSE). We consider a numerically solvable system of integral equations within the quenched, ladder approximation for the BSE and the quenched, rainbow approximation for the DSE. It is well known that the combination of ladder BSE and rainbow DSE respects Goldstone’s theorem, i.e, in the chiral limit of zero current fermion mass, the spectrum admits a massless “pion” [@DS79]. Our four-component fermion version of QED$_3$ admits a spontaneously broken chiral-like symmetry leading to a doublet of Goldstone bosons. It is reasonable to expect therefore that the important effect of chiral symmetry breaking on the light bound state spectrum will be well modelled by this approximation.
We note, however, that our truncation does break local gauge covariance. Of particular concern is the effect of the rainbow approximation on the analytic structure of the fermion propagator. This approximation is known to generate ghost poles in the complex momentum plane in both QED$_4$ [@M94] and QED$_3$ [@M95]. Such poles can be an impediment to the BSE formulated in Euclidean space, which samples the fermion propagator over a region of the complex momentum plane [@SC92]. Herein we assume the position that, provided the ghost poles do not impinge upon this region of the complex plane, the rainbow-ladder approximation is a reasonable one. We acknowledge, however, that this point deserves further investigation. Several QED$_3$ studies [@Sev] have examined the effect on the spacelike behaviour of the fermion propagator if the fermion-photon vertex is replaced by a more sophisticated ansatz satisfying the Ward-Takahashi identity. These studies find the spacelike behaviour to be qualitatively similar to that of the rainbow approximation. Extension of these results to the remainder of the complex plane is beyond the scope of this current paper. We also note that any improvement in the fermion-photon vertex should be matched by a corresponding improvement in the Bethe-Salpeter (BS) kernel in such a way as to preserve Goldstone’s theorem. That this can in principle be achieved by consistently matching a loop expansion of the vertex with crossed diagrams in the BS kernel has recently been demonstrated [@BRS96].
The extension from scalar to vector states enriches considerably the spectrum of bound states. This facilitates an understanding of the importance of symmetry principles in determining the dynamics of the positronium spectrum, and allows analogies to be drawn with the meson spectrum of QCD$_4$. We find that there is a one to one correspondence between the classification of states in 4-component QED$_3$ in terms of “axial parity” and charge conjugation on the one hand, and the conventional $J^{PC}$ classification of charge neutral mesons in QCD$_4$ on the other.
We consider bare fermion mass ranging from zero to large values. For large fermion masses we are able to make contact with the non-relativistic limit. In this limit we extend our previous derivation of the Schrödinger equation as a limit of the BSE formalism to higher spin states, and observe spectrum degeneracies analogous to those of heavy quark effective theory.
The paper is organised as follows. In section II we look at the Bethe-Salpeter and Dyson-Schwinger approximations used and set out the method we employ to find the vector bound state masses. Transformation properties in QED$_3$, with special attention given to the newly considered vector states, are discussed in appendix A. These transformation properties are necessary for understanding the structure of Bethe-Salpeter amplitudes, and classification of the vector bound states. The Bethe-Salpeter coupled integral equations for the vector states are given in appendix B. Section III describes the nonrelativistic limit for the vector states. In section IV numerical vector Bethe-Salpeter solutions are reported and comparisons are made with existing nonrelativistic limit calculations. The results are discussed and conclusions drawn in section V.
Solving the Bethe-Salpeter Equation for Vector States
=====================================================
As in previous work [@AB96], the BS kernel is the quenched ladder approximation kernel (bare one-photon exchange). Again, for convenience, we use Feynman gauge and work with the Euclidean metric. The BSE can then be written as $$\Gamma_{\nu}(p,P) = -e^2 \int \mbox{$ \, \frac{d^3q}{(2\pi)^3} \,$}
D(p-q) \gamma_\mu S(\mbox{$ \, \frac{1}{2} \,$} P+q)
\Gamma_{\nu}(q,P)S(-\mbox{$ \, \frac{1}{2} \,$} P+q)
\gamma_\mu, \label{eq:BS}$$ where $\Gamma_{\nu}(p,P)$ is the one fermion irreducible positronium-fermion-antifermion vertex with external legs amputated. The photon propagator $D(p-q)$ in Feynman gauge is $1/(p-q)^2$. The fermion propagator $S$ is the solution to a truncated DSE. For bare fermion mass $m$ this truncated DSE (the quenched Rainbow Approximation) is $$\Sigma(p)= S(p)^{-1} - (i\!\not \! p + m) =
e^2 \int \mbox{$ \, \frac{d^3q}{(2\pi)^3} \,$} D(p-q) \gamma_\mu
S(q) \gamma_\mu. \label{eq:DS}$$ From here on we use the units $e^2 = 1$ allowable due to the fact that in the quenched approximation the BSE and DSE can be recast in terms of dimensionless momentum $p/e^2$ and mass $m/e^2$.
We write the fermion propagator in the following general form $$S(p)= -i \not\! p \sigma_V(p^2)+\sigma_S(p^2) \;\;\;\;\; \mbox{or} \;\;\;\;\;
S(p)= \frac{1}{ i \not\! p A(p^2)+B(p^2) },
\label{eq:DSEF}$$ where the vector and scalar parts of the propagator are given by $$\sigma_V(p^2)= \frac{A(p^2)}{p^2 A^2(p^2) + B^2(p^2)}, \;\;\;\;\; \mbox{and} \;\;\;\;
\sigma_S(p^2)= \frac{B(p^2)}{p^2 A^2(p^2) + B^2(p^2)}.
\label{eq:SIGDEF}$$ Non-zero $B$ signals dynamical fermion mass generation in the massless limit. Suitable analytic fits to $A$ and $B$ were found in earlier work [@AB96], namely $$A_{\rm fit}(p^2)= \frac{a_1}{(a_2^{\,2} +p^2)^{\frac{1}{2}}}
+a_3 e^{-a_4p^2} +1,$$ $$B_{\rm fit}(p^2)= \frac{b_1}{b_2 +p^2}+b_3 e^{-b_4p^2} + m. \label{eq:FIT}$$ The parameters $a_n$,$b_n$ (functions of fermion mass) were the result of fits to iterative solutions to Eq. (\[eq:DS\]) and can be found in the aforementioned paper. The analytic properties of the propagator are summarised at the end of this section.
In order to solve the BSE we must write it as a set of numerically tractable coupled integral equations. To do this, we write the bound state amplitude $\Gamma_{\nu}$ in its most general form consistent with the parity and charge conjugation of the required bound state, substitute it into the BSE and project out the coefficient functions for the individual Dirac components. The general vector vertex can be found in appendix A and the integral equations in appendix B.
The solution to the BSE involves iteration of the eight coupled integral equations Eq. (\[eq:IE\]). These equations can be written in the form of an eigenvalue problem. Let ${\bf f}=(a,b,c,d,e,f,g,h)^{\rm T}$ then we have $$\int dq_3
\int d\mbox{$\left| {\bf q} \right|$} \, K(\mbox{$\left| {\bf p}
\right|$},p_3;\mbox{$\left| {\bf q} \right|$},q_3;M) {\bf f}
(\mbox{$\left| {\bf q} \right|$},q_3;M)
= \Lambda(M) \,\,\, {\bf f}(\mbox{$\left| {\bf p} \right|$},p_3;M), \label{eq:EV}$$ for a given test mass $M$. This equation is solved for different test bound state masses until an eigenvalue $\Lambda(M)=1$ is obtained. Such an equation exists for each symmetry case (vector ${\cal C}= -1$, vector ${\cal C}= +1$, axivector ${\cal C}=-1$ and axivector ${\cal C}=+1$) and for each fermion mass $m$.
The BSE described in this section requires a fermion propagator input in the form of Eq. (\[eq:DSEF\]) and this needs to be available over a region in the complex momentum plane defined by $Q^2$ from Eq. (\[eq:QDEF\]) with $q_3$ and $\mbox{$\left| {\bf q} \right|$}$ real. This is the region [@SC92] $$\Omega = \left\{ Q^2 = X + iY \left| X > \frac{Y^2}{M^2}
- \frac{1}{4} M^2 \right. \right\}.
\label{eq:REGION}$$ The DSE solution should be well behaved over $\Omega$. In previous work [@AB96] we have studied the analytical properties of the propagator and here we summarise briefly.
Conjugate poles exist where the factor $p^2 A^2 + B^2$ appearing in the denominator of the fermion propagator is zero. Ref. [@AB96] lists the conjugate poles arising from the fits for each fermion mass and the corresponding maximum bound state masses allowed. The maximum $M$ allowed is the value for which the boundary of $\Omega$ in Eq. (\[eq:REGION\]) coincides with the conjugate poles as we should not allow the singularities to enter into the BSE sampling region. If the singularities were to enter that region it would be necessary for compensating zeros to exist in the Dirac coefficients (ie: nodes in the wavefunction). Note that the region boundary (\[eq:REGION\]) is bound state mass $M$ dependent and thus one must take care as $M$ increases during the bound state search.
We have shown [@AB96] that both the DSE solution and our spacelike fits Eq. (\[eq:FIT\]) have conjugate singularities. For small bare fermion mass $m$ we find that the fits reproduce well the position of singularities. However, as $m$ is increased beyond the scale $e^2 = 1$ of the model, the spacelike propagator functions $A$ and $B$ become quite level, and the propagator poles recede further into the timelike half of the complex plane. In this limit, numerical propagator fits are unable to model accurately the important analytic pole structure in the timelike half of the $p^2$-plane. We therefore find our numerical bound state mass solutions contaminated by noise for $m>>1$. For this reason, the non-relativistic limit $m\rightarrow \infty$ of the theory must be treated separately.
Nonrelativistic Limit
=====================
In previous work [@AB96] a nonrelativistic limit ($m\rightarrow \infty$) of the BSE was derived for the scalar states of the positronium system. In this work we follow the same procedure for the vector states which will allow a comparison to be made with the large fermion mass limit of the full BSE calculation (solution to Eq. (\[eq:IE\])).
Consider first the BSE, in which we set the bound state momentum in Eq. (\[eq:BS\]) equal to $P_{\mu}=(2m+\delta)iw_{\mu}$, where $w_{\mu}=(0,0,1)$ and $-\delta$ is a “binding energy”. This gives (according to the momentum distribution in Eq. (\[eq:BS\])) $$\Gamma_{\nu}(p) = - \int \frac{d^3q}{(2\pi)^3} D(p-q) \, \gamma_\mu \,
S\left[-\left(m+\frac{\delta}{2}\right)iw+q\right]
\Gamma_{\nu}(q) \, S\left[\left(m+\frac{\delta}{2}\right)iw+q\right] \gamma_\mu.
\label{eq:NRBSE}$$ An expansion in orders of $1/m$ [@AB96] leads to the propagators $$S\left[\left(m+\frac{\delta}{2}\right)iw+q \right]
= \frac{(1+\gamma_3)/2}{ [ -(1/2) \delta+iq_3+
\left|{\bf q}\right|^2 / 2m ] + \Sigma_{+}(q_3,\left|{\bf q}\right|)} \,
+ \, O\left(\frac{\ln m}{m}\right)
\label{eq:SPLUS}$$ and $$S\left[-\left(m+\frac{\delta}{2}\right)iw+q \right]
= \frac{(1-\gamma_3)/2}{ [ -(1/2) \delta-iq_3+
\left|{\bf q}\right|^2 / 2m ]
+ \Sigma_{-}(q_3,\left|{\bf q}\right|) } \,
+ \, O\left(\frac{\ln m}{m}\right).
\label{eq:SMINUS}$$ where, to one loop order in the self energy, $$\Sigma_{\pm}(q_3,\left|{\bf q}\right|)
= - \frac{1}{4\pi} \ln \left[ \frac{1}{2m} \left( -\frac{1}{2}\delta \pm iq_3
+ \frac{\left|{\bf q}\right|^2}{2m} \right) \right] \,
+ \, O\left(\frac{1}{m^2}\right).
\label{eq:SG1LP}$$ In Ref. [@AB96] we assumed that, if the fermion self energy is calculated to all orders in rainbow approximation, the fermion self energy feeds back into the loop integral via the propagator to replace Eq. (\[eq:SG1LP\]) by $$\Sigma_{\pm}(q_3,\left|{\bf q}\right|)
= - \frac{1}{4\pi} \ln \left[ \frac{1}{2m} \left( -\frac{1}{2}\delta \pm iq_3
+ \frac{\left|{\bf q}\right|^2}{2m} + \Sigma_{\pm}(q_3,\left|{\bf q}\right|) \right)
\right].
\label{eq:SGRAIN}$$ As before we assume that this equation provides an approximation to the rainbow DSE in the nonrelativistic limit.
Since the vertex $\Gamma_{\nu}$ is defined with the fermion legs truncated, and $S \propto \frac{1}{2}(1 \pm \gamma_3)$, we find that the only relevant part of $\Gamma_{\nu}$ is the projection $\mbox{$ \, \frac{1}{2} \,$}(1-\gamma_3) \, \Gamma_{\nu} \,
\mbox{$ \, \frac{1}{2} \,$}(1+\gamma_3)$ and with this in mind, the general vector form in Eq. (\[eq:VERTEX\]) becomes $$\begin{aligned}
\mbox{$ \, \frac{1}{2} \,$}(1-\gamma_3) \, \Gamma_{\nu}^V(q,P) \,
\mbox{$ \, \frac{1}{2} \,$}(1+\gamma_3) & = &
\mbox{$ \, \frac{1}{2} \,$}(1-\gamma_3) \left[ \,\,\,
H_1(q_3,\mbox{$\left| {\bf q} \right|$}) \,\, u_{\nu}(q)\not \! u(q)
\right. \nonumber \\ & & \,\,\, \left.
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, +
H_2(q_3,\mbox{$\left| {\bf q} \right|$}) \,\, v_{\nu}(q)\not \! v(q)
\,\,\, \right] \label{eq:VTMP}\end{aligned}$$ where we have introduced the linear combinations $H_1=f_3+f_4/M$ and $H_2=f_8 + f_7 M$. In the axivector case the general form in Eq. (\[eq:AVS\]) becomes $$\begin{aligned}
\mbox{$ \, \frac{1}{2} \,$}(1-\gamma_3) \, \Gamma_{\nu}^{AV} \,
\mbox{$ \, \frac{1}{2} \,$}(1+\gamma_3) & = &
\mbox{$ \, \frac{1}{2} \,$}(1-\gamma_3)
\left(\begin{array}{c} \gamma_4 \\ \gamma_5\end{array} \right) \left[ \,\,\,
H_1(q_3,\mbox{$\left| {\bf q} \right|$}) \,\, u_{\nu}(q)
\right. \nonumber \\ & & \,\,\, \left. \,\,\,\,\,\,\,\,\,\,
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
+ H_2(q_3,\mbox{$\left| {\bf q} \right|$}) \,\, v_{\nu}(q) \gamma_{45}
\,\,\, \right]
\label{eq:AVTMP}\end{aligned}$$ where $H_1=f_1+if_2M$ and $H_2=f_5+if_6M$.
Substituting Eqs. (\[eq:SPLUS\]), (\[eq:SMINUS\]) and (\[eq:VTMP\]) into Eq. (\[eq:NRBSE\]) one obtains for the vector states two integral equations (with $H_1^{'}=H_1/\mbox{$\left| {\bf q} \right|$}^2$ and $H_2^{'}=-H_2/M^2\mbox{$\left| {\bf q} \right|$}^2$) $$H_1^{'}(p) = \int \frac{d^3q}{(2\pi)^3} \, \frac{1}{(p - q)^2}
\frac{\left[ H_1^{'}(q) \cos^2\theta+H_2^{'}(q)\sin^2\theta \right]}
{ \left|-\frac{1}{2}\delta+iq_3 + \frac{\left|{\bf q}\right|^2}{2m}
+ \Sigma_{\pm}(q_3,\left|{\bf q}\right|)
\right|^2 },$$ $$H_2^{'}(p) = \int \frac{d^3q}{(2\pi)^3} \, \frac{1}{(p - q)^2}
\frac{\left[ H_1^{'}(q) \sin^2\theta+H_2^{'}(q)\cos^2\theta \right]}
{ \left|-\frac{1}{2}\delta+iq_3 + \frac{\left|{\bf q}\right|^2}{2m}
+ \Sigma_{\pm}(q_3,\left|{\bf q}\right|) \right|^2 }.
\label{eq:NRVTMP1}$$ In these equations $\theta$ is the angle separating the 2-vectors $\bf{p}$ and $\bf{q}$, and $H_{1,2}^{'}(q)=H_{1,2}^{'}(q_3,\mbox{$\left| {\bf q} \right|$})$. It is clear that two independent equations can be found by applying simple linear combinations. Let $F=H_1^{'}+H_2^{'}$ and $G=H_1^{'}-H_2^{'}$, then $$F(p) =\int \frac{d^3q}{(2\pi)^3} \, \frac{1}{(p - q)^2} \frac{F(q)}
{ \left|-\frac{1}{2}\delta+iq_3 + \frac{\left|{\bf q}\right|^2}{2m}
+ \Sigma_{\pm}(q_3,\left|{\bf q}\right|) \right|^2 }.
\label{eq:NRV1}$$ This is identical to the non-relativistic limit of the axiscalar equation which has been solved numerically in Ref. [@AB96], and where only a ${\cal C}=+1$ solution could be found (where $F$ is even in $q.P$). As we have seen, the function $F$ in the vector case involves a linear combination of functions $f_3, f_4, f_7$, and $f_8$. In appendix A we see that these are all odd in $q.P$ for the vector ${\cal C}=+1$ case and even for the vector ${\cal C}=-1$ case. Thus it is clear that the axiscalar ${\cal C}=+1$ solution is degenerate, in the nonrelativistic limit, with a vector ${\cal C}=-1$ state.
We also obtain a new equation, namely $$G(p) = \int \frac{d^3q}{(2\pi)^3} \, \frac{1}{(p - q)^2} \frac{\left[
2\left(\frac{\bf{p.q}}{\left|{\bf p}\right|\left|{\bf q}\right|} \right)^2
-1 \right] \,\, G(q)}
{ \left|-\frac{1}{2}\delta+iq_3 + \frac{\left|{\bf q}\right|^2}{2m}
+ \Sigma_{\pm}(q_3,\left|{\bf q}\right|) \right|^2 } \, .
\label{eq:NRV2}$$ Note that the constituents of $G$ used in the linear combination, namely $f_3$, $f_4$, $f_7$ and $f_8$, are all odd in $q.P$ for the ${\cal C} = +1$ case, and even in $q.P$ for the ${\cal C} = -1$ case, and so is the function $G$.
In the axivector case we find that the surviving terms in $\mbox{$ \, \frac{1}{2} \,$}(1-\gamma_3) \, \Gamma^{AV}_{\nu} \,
\mbox{$ \, \frac{1}{2} \,$}(1+\gamma_3)$ can be collected into two linear combinations ($H_1(p)$ and $H_2(p)$ defined below Eq. (\[eq:AVTMP\])) and rescaled by dividing each by $\left|{\bf p}\right|$ to find two identical decoupled equations which are also identical to the scalar equation of previous work. If $F(p)=H_1(p)/\left|{\bf p}\right|$ or $F(p)=H_2(p)/\left|{\bf p}\right|$, we have $$F(p) = \int \frac{d^3q}{(2\pi)^3} \,
\frac{1}{(p - q)^2}
\frac{\bf{p.q}}{\left|{\bf p}\right|\left|{\bf q}\right|}
\frac{F(q)}{ \left|-\frac{1}{2}\delta+
iq_3 + \frac{\left|{\bf q}\right|^2}{2m}
+ \Sigma_{\pm}(q_3,\left|{\bf q}\right|)
\right|^2 }
\label{eq:NRS}$$ In one case $F$ is an even function of $q.P$ and in the other it is an odd function. The same equation (\[eq:NRS\]) could only be solved with $F$ an even function of $q.P$ in previous work [@AB96] where the scalar ${\cal C}=+1$ solution was found. In the current case, solutions to Eq. (\[eq:NRS\]) can be identified with the axivector ${\cal C} = +1$ state by making the identification $F \propto H_1$, or with the axivector ${\cal C} = -1$ state by making the identification $F \propto H_2$. We conclude that the axivector ${\cal C} = \pm 1$ states are mass degenerate in the $m \rightarrow \infty$ limit.
We should therefore see, in the large fermion mass limit, a vector state with ${\cal C}=-1$ degenerate with the axiscalar state with ${\cal C}=+1$, new vector states (possibly ${\cal C}=-1$ and ${\cal C}=+1$) which are solutions to Eq. (\[eq:NRV2\]), and axivector states with ${\cal C}=\pm 1$ which are degenerate with the scalar ${\cal C}=+1$ state. The degeneracies between scalar and vector states are analogous to those of heavy quark effective theory [@N94], in which the hadron is insensitive to the heavy quark spin to leading order in the inverse quark mass.
Eqs. (\[eq:NRV1\]), (\[eq:NRV2\]) and (\[eq:NRS\]) correspond to states with orbital angular momentum $\ell = 0$, 2 and 1 respectively. To prove this, we show that these integral equations are equivalent to the Schrödinger equation of Koures in Ref. [@Ko96]. Following the working of Ref. [@AB96], we see that Eqs. (\[eq:NRV1\]), (\[eq:NRV2\]) and (\[eq:NRS\]) may be rewritten as in Eq. (3.27) of Ref. [@AB96], namely $$\left\{-\delta + \frac{\left|{\bf p}\right|^2}{m} +
2\mbox{Re}\Sigma_{-}(p_3,\left|{\bf p}\right|)\right\}
\Phi(p_3,\left|{\bf p}\right|) = \int \frac{d^2 {\bf q}}{(2\pi)^2}
V(\left|{\bf p - q}\right|)
\Phi(q_3,\left|{\bf q}\right|) \chi({\bf p},{\bf q}) \label{eq:OUREQS}$$ where $\chi({\bf p},{\bf q})$ is one of $1$, $\frac{{\bf p.q}}{\left|{\bf p}\right| \left|{\bf q}\right|}$ or $\left\{ 2\left(\frac{{\bf p.q} }
{ \left|{\bf p}\right| \left|{\bf q}\right|} \right)^2 - 1 \right\}$. In the final analysis the self energy $\Sigma_{-}$ serves the purpose of cancelling infrared divergence arising from the logarithmic potential $V$.
On the other hand, consider the Schrödinger equation for a particle of orbital angular momentum $\ell$ in $(2 + 1)$ dimensions [@Ko96]: $$\left\{-\delta + \frac{\nabla^2}{m} + 2\mbox{Re}\Sigma_{-}\right\}
\left[ \tilde{\phi}(\left|{\bf r}\right|) e^{\pm i \ell \theta} \right] =
V(r) \left[ \tilde{\phi}(\left|{\bf r}\right|)
e^{\pm i \ell \theta} \right]. \label{eq:SCHROL1}$$ One easily shows that $$\mbox{F.T. of }
\left[ \tilde{\phi}(\left|{\bf r}\right|) e^{\pm i \ell \theta} \right] =
\int d^2 {\bf r} \tilde{\phi}(\left|{\bf r}\right|)
e^{i{\bf r}.{\bf p} \pm i\ell\theta} =
\phi(\left|{\bf p}\right|) e^{\pm i \ell \theta_p},
\label{eq:FT1}$$ where we have defined ($J_\ell$ is Bessel function of order $\ell$) $$\phi(\left|{\bf p}\right|) = 2 \pi i^{\ell} \int_{0}^{\infty}
J_\ell(\left|{\bf p}\right| \left|{\bf r}\right|)
\tilde{\phi}(\left|{\bf r}\right|)
\left|{\bf r}\right| d\left|{\bf r}\right|,
\label{eq:PHIP}$$ and $\theta$ is the angle the vector ${\bf r}$ makes with the $r_1$-axis, $\theta_p$ is the angle the vector ${\bf p}$ makes with the $p_1$-axis. From this we find that the F.T of the $\mbox{r.h.s.}$ of Eq. (\[eq:SCHROL1\]) is the convolution integral $$\int \frac{d^2 {\bf q}}{(2\pi)^2} V(\left|{\bf p - q}\right|)
\phi(\left|{\bf q}\right|) e^{\pm i \ell \theta_q} ,
\label{eq:S1RHS}$$ while the F.T of the $\mbox{l.h.s.}$ of Eq. (\[eq:SCHROL1\]) is $$\left\{-\delta + \frac{\left|{\bf p}\right|^2}{m} + 2\mbox{Re}\Sigma_{-}
\right\} \phi(\left|{\bf p}\right|) e^{\pm i \ell \theta_p}
\label{eq:S1LHS}$$ Equating Eqs. (\[eq:S1LHS\]) and (\[eq:S1RHS\]) we obtain $$\begin{aligned}
\left\{-\delta + \frac{\left|{\bf p}\right|^2}{m} +
2\mbox{Re}\Sigma_{-} \right\}
\phi(\left|{\bf p}\right|) & = &
\int \frac{d^2 {\bf q}}{(2\pi)^2}
V(\left|{\bf p - q}\right|)
\phi(\left|{\bf q}\right|) e^{\pm i \ell (\theta_q - \theta_p)}
\nonumber \\
& = & \int \frac{d^2 {\bf q}}{(2\pi)^2}
V(\left|{\bf p - q}\right|)
\phi(\left|{\bf q}\right|) \cos \ell(\theta_q - \theta_p) \nonumber \\
& = & \int \frac{d^2 {\bf q}}{(2\pi)^2}
V(\left|{\bf p - q}\right|)
\phi(\left|{\bf q}\right|) \chi({\bf p},{\bf q}).
\label{eq:EQUATE1}\end{aligned}$$ which is equivalent to Eq. (\[eq:OUREQS\]).
To summarise, we have obtained equations for degenerate $\ell = 0$ vector and axiscalar states (Eq. (\[eq:NRV1\])), a new $\ell = 2$ vector state (Eq. (\[eq:NRV2\])), and degenerate axivector and scalar states with $\ell = 1$ (Eq. (\[eq:NRS\])). In order to explain the occurrence of this particular set of angular momentum and parity combinations we need to study the symmetry properties of bound states in (2 + 1) dimensions.
To classify the bound states in terms of the spin and orbital angular momentum parts, we consider briefly the finite dimensional representations of the (2 + 1) dimensional Lorentz group $SO(2,1)$ [@KN86]. Its Lie algebra is defined by the commutation relations $$\left[\Sigma_{\mu},\Sigma_{\nu}\right] =
i{\epsilon_{\mu \nu}}^\rho \Sigma_\rho,$$ where $\Sigma_0$ is the generator of rotations in the $x_1$-$x_2$ plane and $\Sigma_{1,2}$ generate translations in the $x_{1,2}$ directions. The finite dimensional representations are classified by the Casimir operator $$\Sigma^2 = (\Sigma_0)^2 - (\Sigma_1)^2 - (\Sigma_2)^2 ,$$ which takes eigenvalues $L(L+1)$, while $\Sigma_0$ takes eigenvalues $\ell = -L, -L + 1, \ldots, L - 1, L$. Identifying $\ell$ with the angular dependence of the Schrödinger wavefunction discussed above, we shall refer to $\ell$ as the total orbital angular momentum. To the spin part of the bound state angular momentum we also attach a quantum number $S$, which we call the total spin. For the symmetric spin combinations of two Dirac particles we have $S = 1$, and for the antisymmetric combination, $S = 0$. The spin angular momentum in the $x$-$y$ plane takes values $m = -S, -S + 1, \ldots, S - 1, S$.
The bound states are also classified by their eigenstates under the action of the “axial parity” operator $A$: $$A: \,\, x^\mu = (x^0,x^1,x^2) \rightarrow x'^\mu = (x^0,-x^1,-x^2),
\label{eq:AXP}$$ which is distinct from the parity operator defining the operation Eq. (\[eq:PAR\]) in appendix A. Eq. (\[eq:AXP\]) is a rotation through $\pi$, and is the operator relevant to interchange of constituent fermions. The transformation properties of Dirac spinors and bilinear currents under axial parity are given by Eqs. (\[eq:APAR\]) and (\[eq:APTY\]). The axial parity $A$ of each of the bound states considered herein is listed in Table \[tab1\].
In the non-relativistic limit, the generalised Pauli exclusion principle [@FS82] adapts from (3+1) to (2+1) dimensions to give the following relations between the total orbital angular momentum $L$, the total spin $S$, axial parity $A$ and charge parity ${\cal C}$ : $$A = (-1)^{L + 1}, \label{al1}$$ $${\cal C} = (-1)^{L + S}.$$ Two fermions can form a system with a total spin of $S=0$ or $S=1$. For a specific total angular momentum $J$, vector addition of angular momenta $m + \ell$ determines the possible orbital angular momenta $\ell$. Table \[tab1\] lists all possible scalar and vector states.
Consider the scalar states where $J=0$. For an orbital angular momentum of zero ($L=0$) the total spin must also be zero ($S=0$). In this case the axial parity must be negative ($A=-1$) and charge parity positive (${\cal C}=+1$). We know from our integral equations that this state is the axiscalar state. It is also known from the transformation properties outlined in appendix A that the axiscalar does indeed have $A=-1$. The other possibility with $J=0$ is the $L=1$, $S=1$ state where the axial parity is positive and the charge parity is positive. This is the scalar state which we know from its transformation properties has $A=+1$. So we have seen that it is only possible for scalar states to have $L=0$ or $L=1$ and they must have positive charge parity. Negative charge parity scalars are forbidden by the generalised Pauli Exclusion principle. States such as the $J^{AC}=0^{--}$ state, for example, we refer to as having unnatural parity.
Now consider the vector case ($J=1$). If $L=0$ then $S$ must equal $1$. In this case both the axial parity and charge parity are negative. We know that this is the $L=0$ vector state. With no spin-orbit coupling contribution in our $e^{-}$–$e^{+}$ potential this state is degenerate with the $L=0$ axiscalar state.
If $L=1$ then there are two possibilities. A $J=1$ is possible with either $S=0$ or $S=1$. Both of these cases correspond to the $L=1$ axivector states, both with positive axial parity, one with negative charge parity and the other with positive charge parity. Again, with no spin-orbit coupling the two are degenerate. They are also degenerate with the $L=1$ scalar state.
The only other possible state is the $L=2$, $S=1$ state which has negative axial parity and negative charge parity. This is the $L=2$ vector state found earlier in this section. There appears to be no possibility of an $L=2$ vector state with positive charge parity and so we would hope that there is no such solution in our nonrelativistic BSE calculations.
In general, we find that the axial parities determined from the angular momenta and Eq. (\[al1\]) match up with those found by analysing the transformation properties of bilinear currents in Appendix A. We also note that the “axi” transformation in $(2 + 1)$ dimensions parallels the conventional parity transformation in $(3 + 1)$ dimensions, and that a one to one correspondence between the allowed $J^{AC}$ states in $(2 + 1)$ dimensions and $J^{PC}$ states in $(3 + 1)$ dimensions follows from the generalised Pauli exclusion principle.
Numerical Results
=================
In this section we report our numerical solutions to the Bethe-Salpeter coupled integral equations of Eq. (\[eq:IE\]). Comparisons with the nonrelativistic calculations of Refs. [@THY95], [@Ko96] will be made for large fermion mass.
As in the scalar calculations of Ref. [@AB96], a grid of $25 \times 25$ ($\mbox{$\left| {\bf q} \right|$}$,$q_3$) tiles was used for the iterative solution to the vector equations Eq. (\[eq:IE\]) with the use of linear interpolation on each of those tiles for the sums ($\tau_{r1}$, $\tau_{r2}$) which are supplied at the corners of the tiles from the previous iteration. Again the tiles were non-uniform in size and an upper limit to the momentum components ($\mbox{$\left| {\bf q} \right|$}$ and $q_3$) were made large enough so that results were independent of their values. For each symmetry and a range of fermion masses ($0-5$) the equations were iterated leaving a set of bound state masses.
Table \[tab3\] shows the bound state masses for each of the four nondegenerate states which are the vector ${\cal C}=+1$, vector ${\cal C}=-1$, axivector ${\cal C}=+1$ and the axivector ${\cal C}=-1$ states. We employ the notation $J^{AC}$ introduced in the last section to classify these states. The degeneracy under space reflection parity in 3-dimensions is assumed from this point on and, for example, the vector state refers to the doublet vector / pseudovector. Fig. 1 displays the solutions $M$ for fermion mass 0–0.1. Fig. 2 shows $M-2m$ over the greater range of 0–1. We note that the poles in the fermion propagator fits lie outside the BS integration region $\Omega$ for all solutions obtained. This has been the case in both scalar and vector calculations.
For small $m$ the bound state masses rise rapidly with increasing fermion mass. The $J^{AC} = 1^{--}$ state is the lowest energy state followed by the $1^{-+}$ and then the $1^{++}$ and $1^{+-}$ states respectively. At some point between $m=0.049$ and $0.064$ the $1^{-+}$ and the $1^{++}$ states cross. The curves then level off for masses approaching the nonrelativistic limit $m>1$.
We found that the two axivector states were near degenerate for small $m$. However, numerical problems prevented negative charge parity solutions to the axivector equations for $m=0.036$ and higher. The eigenvalue $\Lambda(M)$ in Eq. (\[eq:EV\]) splits into complex conjugate pairs past this value of $m$. We see no theoretical reason for the disappearance of the negative charge parity axivector state and we attribute it to deficiencies of the bare vertex, ladder approximation.
As expected, for larger fermion masses the bound state mass rises predominantly as twice the fermion mass plus possible logarithmic corrections. However, there appears to be a good deal of noise in the large $m$ solutions, reflecting the difficulty in accurately modelling the fermion propagator deep into the timelike region from spacelike fits. It is for this reason that Figure 2 does not display the larger fermion masses. Despite this noise, the ordering of the states remains unchanged as $m$ becomes large.
It is clear that this approach to the large fermion mass end of the spectrum is not suitable and that a special nonrelativistic treatment is required. What is required is solution to Eqs. (\[eq:NRV1\]), (\[eq:NRV2\]) and (\[eq:NRS\]). However, this involves the solution for the rainbow self energy $\Sigma_{+}$ from a DSE formulated in the large fermion mass limit. This is not within the scope of this paper and is the subject of future work. In previous work for the scalar positronium solutions [@AB96] a 1-loop approximation to this self energy was employed. This appeared warranted because the approximation gave a good fit to the rainbow propagator for spacelike momenta. However, it was found that the 1-loop approximation resulted in bound state masses that were out by a logarithmic correction (because of the noncancelling infrared divergences) and as a result the masses could not be compared to other large fermion mass solutions. In this work, we do not attempt to match 1-loop nonrelativistic solutions with our large $m$ relativistic solutions. Despite this, we believe that the existence or nonexistence of a solution to the 1-loop nonrelativistic BSE equations is evidence of the existence or nonexistence of those positronium bound states. Therefore, Eqs. (\[eq:NRV1\]), (\[eq:NRV2\]) and (\[eq:NRS\]) were solved with $\Sigma_{+}$ replaced by the 1-loop result in Eq. (\[eq:SG1LP\]) neglecting terms of order $\frac{1}{m^2}$.
Because of the degeneracies pointed out in section III, solutions exist to Eqs. (\[eq:NRV1\]) and (\[eq:NRS\]) and from this we could conclude that the vector $\ell=0$ ($1^{--}$) and the axivector $\ell=1$ ($1^{-\pm}$) bound states do exist. We were left to solve Eq. (\[eq:NRV2\]) for possible positive and negative charge parity $\ell=2$ vector solutions. It was found that the $1^{--}$ solution did exist and had a mass increasing linearly in $\ln m$ as did the smaller $\ell$ solutions. However, the $1^{-+}$ solution mass did not level off to the same slope and appeared to have a strongly divergent mass in the nonrelativistic limit. We interpret this by saying that there is no $1^{-+}$ bound state, in agreement with the generalised Pauli exclusion principle.
Although the 1-loop nonrelativistic masses cannot be quantitatively compared with masses found with the rainbow propagator, we again point out that the results do contain useful qualitative information. We find that the ordering of the states in the nonrelativistic limit is $1^{--}$ ($\ell=0$) followed by $1^{+\pm}$ ($\ell=1$) and then $1^{--}$ ($\ell=2$). Based on both the relativistic calculations and the nonrelativistic 1-loop exercise, it appears that it is energetically favourable for the scalars to have positive charge parity, and for the vectors to have negative charge parity, in agreement with the QCD$_4$ meson analogy.
We compare our relativistic integral equation results with those of existing QED$_3$ Schrödinger equation results. Tam, Hamer and Yung [@THY95] perform an analysis of QED$_3$ from the point of view of discrete light cone quantisation. In the non-relativistic limit, their bound state masses $M$ are solution to the Schrödinger equation $$\left\{ -\frac{1}{m} \nabla^2 + \frac{1}{2\pi} \left(C + \ln(mr)
\right)\right\} \phi({\bf r}) = (M - 2m) \phi({\bf r}), \label{eq:THYDE}$$ where $C$ is Euler’s constant and $m$ is the bare fermion mass. This differential equation is solved for the completely symmetric states and the masses are given by the expression $$M = 2m + \frac{1}{4\pi} \ln m + \frac{1}{2\pi} \left(\lambda^{'} -
\frac{1}{2} \ln\frac{2}{\pi} \right).
\label{eq:THY1}$$ Koures [@Ko96] solves the same equation but also for nonzero angular momentum $\ell$. For $0 \le \ell \le 4$ this reference provides the lowest five eigenvalues ($\lambda$) which, after the transformation $$\lambda^{'}=\lambda + \ln(2) + C
\label{eq:LPRIME}$$ can be used in Eq. (\[eq:THY1\]) above. The eigenvalues to be used here are $\lambda' = 1.7969$ ($\ell = 0$), 2.6566 ($\ell = 1$), 2.9316 ($\ell = 0$) and 3.1148 ($\ell = 2$).
The lowest $\ell=$ 0, 1 and 2 and the first excited $\ell=$ 0 Schrödinger equation results of Refs. [@THY95], [@Ko96] are also plotted in Fig. 2. One could associate the $\ell=$ 0 curve with the vector $1^{--}$ state. The $\ell=$ 1 curve can be seen to match up with one or more of $1^{-+}$, $1^{++}$ or $1^{+-}$. We know from our nonrelativistic analysis that the $1^{+\pm}$ (axivector) states are the only $\ell=$ 1 states and become degenerate in the large $m$ limit. It therefore seems as though there is a surprisingly good match up between relativistic and nonrelativistic $\ell=$ 1 states. The lowest $\ell=$ 2 and the first excited $\ell=0$ curves should coincide with higher vector $1^{--}$ states but no such states were solved for in our relativistic BSE treatment. Also there is no nonrelativistic $1^{-+}$ state to match up with the relativistic vector state with positive charge parity as shown in our nonrelativistic BSE exercise and as predicted by the generalised Pauli exclusion principle. We emphasise that the Schr[" o]{}dinger equation results are of course formulated in the nonrelativistic limit and we cannot expect close numerical agreement with the small mass relativistic results. In fact, there is quite a broad region of intermediate masses where neither the relativistic BSE nor the Schr[" o]{}dinger results are suitable.
Finally, we may make qualitative comparisons between the calculated spectrum of $J^{AC}$ states in QED$_3$ and the spectrum of observed $J^{PC}$ meson states in QCD$_4$. Since the observed meson spectrum is best established for light mesons, the comparison is made in the chiral limit $m=0$. In Table \[tabmes\] we list the $m=0$ spectrum of scalar and axiscalar states from Ref. [@AB96], and of vector and axivector states from this work, together with the corresponding observed light meson states [@PRG96]. (Note that, as pointed out in the previous section, the axial parity quantum number $A$ in $(2 + 1)$ dimensions plays the role of conventional parity $P$ in $(3 + 1)$ dimensions.) Firstly we see that the gaps in the observed meson spectrum occur for parity combinations disallowed by the generalised Pauli exclusion principle. In our fully relativistic QED$_3$ calculations, there is nothing in principle to prevent such states from occurring away from the nonrelativistic limit. We are unable to say whether the unnatural parity solutions are a genuine property of QED$_3$, or an artefact of the rainbow, ladder approximation. If these states are ignored, we observe surprising agreement between the ordering and relative magnitudes of bound state masses in QED$_3$ and the observed meson spectrum. In both cases the dynamics of the bound state spectrum is driven mainly by confinement and chiral symmetry breaking, these being the common features of QED$_3$ and QCD$_4$.
Conclusions
===========
We have extended our earlier study of the positronium states of 4-component QED$_3$ from scalar to vector states. QED$_3$ is a confining theory, and the positronium states are in some sense the analogues of mesons in QCD$_4$. In $(2 + 1)$ dimensions the bound states are classified as eigenstates of reflection parity $P$, axial parity $A$ (i.e. a spatial rotation through $180^\circ$), and charge conjugation $C$. In the four component version of massless QED$_3$, a $U(2)$ symmetry analogous to chiral symmetry is spontaneously broken to $U(1)\times U(1)$. The resulting spectrum consists of reflection parity doublets for which $P$ is an exact symmetry. The states are therefore classified by the quantum numbers $J^{AC}$. We had previously studied in detail axiscalar ($0^-$) and scalar ($0^+$) positronium. Herein our focus is mainly on the extension of this work to vector ($1^-$) and axivector ($1^+$) states.
We calculate bound state masses using the combination of rainbow Dyson-Schwinger and ladder Bethe-Salpeter equations. Fermion propagators calculated for spacelike momenta from the Dyson-Schwinger equation are extended into the required part of the complex momentum plane by making analytic fits to the spacelike part of the propagator. There are two important issues raised by this extrapolation procedure, both of which we addressed for the scalar case in our previous work.
Firstly, the solution to the rainbow approximation Dyson-Schwinger equation, and the analytic fits, contain ghost poles in the timelike half of the complex momentum plane. It is necessary for the success of the approximation that these poles do not impinge on the set of complex momentum values sampled by the BSE. As for the scalar case considered previously, we find that the vector states do not pose a problem in this regard. This is because the scalar and vector bound state masses are typically of comparable size over the range of bare fermion masses considered.
Secondly, in both the scalar and vector cases, the extrapolation procedure is inadequate for intermediate or large bare fermion masses (viz. $m/e^2 > 0.5$). This is because the important contributions to the BSE in the heavy fermion limit come from deep into the timelike part of the complex plane. For this reason the heavy fermion limit must be treated separately.
We have obtained numerical solutions to the combination of Dyson-Schwinger and Bethe-Salpeter equations over a range of dimensionless bare fermion masses $m/e^2$ for each of the axial and charge parity combinations $1^{--}$, $1^{-+}$, $1^{+-}$ and $1^{++}$. Solutions exist over the broad range of bare fermion masses considered except for the $1^{+-}$, for which the eigenvalue of the integral operator in the BSE becomes complex for $m/e^2 > 0.036$. We interpret this as a shortcoming of the rainbow-ladder approximation.
In our previous work we obtained the nonrelativistic (i.e. $m\rightarrow \infty$) limit of the BSE for scalar and axiscalar states in the form of a Schrödinger equation. Here we have extended the proof to vector and axivector states in a way which enables us to identify the orbital angular momentum of each state. We find that the relation $A = (-1)^{L+1}$ is automatically satisfied, and are furthermore able to restrict the allowed charge parities by using the generalised Pauli exclusion principle. In this limit it becomes clear that it is axial parity, and not space reflection parity, which is the QED$_3$ counterpart of conventional parity in QCD$_4$.
We also find that in the nonrelativistic limit, the bound state spectrum becomes insensitive to the spin of constituent fermions. This is manifested as a degeneracy between $0^{-+}$ and $1^{--}$ states and between the $0^{++}$ and $1^{+\pm}$ states. Precisely the same phenomenon occurs in the observed heavy meson spectrum to within order of the inverse heavy quark mass, and can be explained in terms of heavy quark effective theory [@N94].
While certain $J^{AC}$ combinations are disallowed by the generalised Pauli exclusion principle in the nonrelativistic limit, there is nothing in principle to prevent their occurrence in the relativistic regime. We have in fact obtained unnatural parity solutions in both our earlier scalar and current vector positronium calculations. This is not in agreement with the observed meson spectrum, even for light quark mesons, where for instance negative charge parity scalar or pseudoscalar mesons are never observed. One is led to question whether the unnatural parity solutions in QED$_3$ are an artefact of the rainbow-ladder approximation. More importantly we find that, if the unnatural parity solutions are ignored, there is surprising agreement between the relative mass scales of calculated $J^{AC}$ states in QED$_3$ and observed $J^{PC}$ mesons in QCD$_4$, as evidenced in Table \[tabmes\].
The original aim of studying 4-component QED$_3$ was to explore nonperturbatively a theory which has properties in common with QCD$_4$, namely confinement and chiral symmetry breaking, but without the complications of being nonabelian. We have demonstrated that the Bethe-Salpeter formalism generates a bound state spectrum with qualitative characteristics in common with the observed meson spectrum. One shortcoming of the rainbow-ladder formalism is that gauge covariance is broken by using a bare fermion-photon vertex. In any practical truncation it is likely that some symmetry of the original problem will be lost. In this paper we have adopted the position that it is more important for bound state mass calculations to maintain the chiral symmetry breaking mechanism than to maintain gauge symmetry. An alternative approach is that of lattice gauge theory, in which gauge symmetry is maintained, but the original chiral symmetry of QED$_3$ survives only as a small remnant subgroup [@BB87]. Specifically, lattice gauge theory simulations of quenched, non-compact QED3 with 4-component fermions can be carried out with comparative ease [@DKK90], and the chiral condensate checked against the predictions of model Dyson-Schwinger calculations [@B92]. We are unaware of any existing bound state spectrum calculation of this simple lattice model. Such a calculation would provide a much needed cross check between two vastly differing non-perturbative field theory methods.
Appendix A - General Vector Amplitude in QED$_3$ {#appendix-a---general-vector-amplitude-in-qed_3 .unnumbered}
================================================
Solution to the BSE in Eq. (\[eq:BS\]) for the vector states requires a general vertex function which will allow individual Dirac components to be projected out leaving a system of coupled integral equations. We assume a knowledge of the transformation properties of QED$_3$ [@Bu92] and of the Scalar BS amplitudes [@AB96].
We work in the four-component version of QED$_3$ where there exists a complete set of 16 matrices (where $\mu$ = 0, 1 and 2) $\{\gamma_A\}=\{I,\gamma_{4},\gamma_{5},\gamma_{45},\gamma_{\mu},
\gamma_{\mu 4},\gamma_{\mu 5},\gamma_{\mu 45} \}$ satisfying $\frac{1}{4} {\rm tr}(\gamma_A \gamma^B) = \delta^B_A$. These matrices can be found in Ref. [@Bu92]. The QED$_3$ action is invariant with respect to discrete parity and charge conjugation symmetries, which for the fermion fields are given by $$\mbox{$\psi(x)$} \rightarrow \psi^\prime(x^\prime) =
\Pi \mbox{$\psi(x)$}, \;\;\;
\mbox{$\overline{\psi}(x)$} \rightarrow \overline{\psi}^\prime
(x^\prime) = \mbox{$\overline{\psi}(x)$} \Pi^{-1},
\label{eq:PAR}$$ $$\mbox{$\psi(x)$} \rightarrow \psi^\prime(x) =
C \overline{\psi}(x)^{\rm T}, \;\;\;
\mbox{$\overline{\psi}(x)$} \rightarrow \overline{\psi}^\prime(x)
= -\mbox{$\psi(x)$}^{\rm T} C^{\dagger},
\label{eq:CH}$$ where $x^{\prime}=(x^0,-x^1,x^2)$. The parity and charge conjugation matrices ($\Pi$ and $C$ respectively) are each determined up to an arbitrary phase by the condition that the action be invariant [@Bu92]: $$\Pi=\gamma_{14} e^{i\phi_P \gamma_{45}}, \;\;\;
C=\gamma_{2} e^{i\phi_C \gamma_{45}}, (0\leq \phi_P,\phi_C < 2\pi)
\label{eq:pic}$$
Vector, pseudovector, axivector and axipseudovector bound states are defined by the following transformation properties under parity transformations $$\begin{aligned}
\Phi_{\mu}^{V}(x) & \rightarrow &
\Phi_{\mu}^{V\prime}(x^{\prime}) =
\Lambda^{\nu}_{\mu} \Phi_{\nu}^{V}(x),\nonumber \\
\Phi_{\mu}^{PV}(x) & \rightarrow & \Phi_{\mu}^{PV\prime}(x^{\prime}) =
-\Lambda^{\nu}_{\mu} \Phi_{\nu}^{PV}(x), \nonumber \\
\Phi_{\mu}^{AV}(x) & \rightarrow & \Phi_{\mu}^{AV\prime}(x^{\prime}) =
R_P \Lambda^{\nu}_{\mu} \Phi_{\nu}^{AV}(x),\nonumber \\
\Phi_{\mu}^{APV}(x) & \rightarrow & \Phi_{\mu}^{APV\prime}(x^{\prime}) =
-R_P \Lambda^{\nu}_{\mu} \Phi_{\nu}^{APV}(x), \label{eq:PTY} \end{aligned}$$ Where $$\begin{aligned}
R_P & = & \left(\begin{array}{cc} -\cos 2\phi_P & -\sin 2\phi_P \\
-\sin 2\phi_P & \cos 2\phi_P \end{array} \right)
\label{eq:RL}\end{aligned}$$ and in Minkowski space $\Lambda_{\mu}^{\nu}={\rm diag}(1,-1,1)$. Similar transformation properties exist for charge conjugation.
We also find the need to classify our nonrelativistic states in terms of the transformation properties under what we call “axial parity” which is the (2+1)d analogue of the (3+1)d parity where $x^{\prime}=(x^0,-x^1,-x^2)$. The fermion fields transform like $$\mbox{$\psi(x)$} \rightarrow \psi^\prime(x^\prime) =
S_{\pi} \mbox{$\psi(x)$}, \;\;\;\;\;\;
\mbox{$\overline{\psi}(x)$} \rightarrow \overline{\psi}^\prime
(x^\prime) = \mbox{$\overline{\psi}(x)$} S_{\pi}^{-1},
\label{eq:APAR}$$ where $S_{\pi}$ is the operator which corresponds to a rotation through angle $\pi$ in the $x^1x^2$ plane. We find that a suitable operator $S_{\pi}$ which performs such a rotation and leaves the action invariant is the matrix $i \gamma_0$. This is the same operator as used in the (3+1)d case. The phase $i$ is responsible for making $S_{\pi}^{2}=-1$ and has no effect on the transformation of bilinear currents of concern here. Given that the bound state wavefunctions transform like the bilinear currents $J_A=\overline{\psi}(x) \gamma_A \psi(x)$ we find the following transformation properties for scalars and vectors under axial parity. $$\begin{aligned}
\Phi^{S}(x) \rightarrow& \,\,\,\,\Phi^{S}(x), \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\Phi_{\mu}^{V}(x) &\rightarrow \,\,\,\, \eta^{\nu}_{\mu} \Phi_{\nu}^{V}(x), \nonumber \\
\Phi^{PS}(x) \rightarrow& \,\,\,\, \Phi^{PS}(x), \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\Phi_{\mu}^{PV}(x) &\rightarrow \,\,\,\, \eta^{\nu}_{\mu} \Phi_{\nu}^{PV}(x), \nonumber \\
\Phi^{AS}(x) \rightarrow& -\Phi^{AS}(x), \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\Phi_{\mu}^{AV}(x) &\rightarrow - \eta^{\nu}_{\mu} \Phi_{\nu}^{AV}(x), \nonumber \\
\Phi^{APS}(x) \rightarrow& -\Phi^{APS}(x), \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\Phi_{\mu}^{APV}(x) &\rightarrow - \eta^{\nu}_{\mu} \Phi_{\nu}^{APV}(x). \label{eq:APTY} \end{aligned}$$
The matrix $\eta_{\mu}^{\nu}$ is defined as ${\rm diag}(1,-1,-1)$. The corresponding axial parities ($A=\pm 1$) for each state are listed in Table \[tab1\]. This definition parallels the (3+1)d case where a vector state transforms like $\eta^{\nu}_{\mu} \Phi_{\nu}$ and is assigned negative parity.
Let us now derive a general vector vertex. Consider the momentum $p_{\mu}$ which transforms like $p_{\mu} \rightarrow \Lambda_{\mu}^{\nu} p_{\nu}$ under parity. The bound state momentum $P_{\mu}$ seen in Eq. (\[eq:BS\]) is another such vector. Three other vectors can be found which transform in this way which are $\gamma_{\mu}$, $u_{\mu}(p)$ and $v_{\mu}(p)\gamma_{45}$. The vectors $u(p)$ and $v(p)$ used in previous work [@Bu92] are mutually orthogonal with $P$ and are defined as $$u(p)=\frac{P^2p-(P.p)P}{P^2p^2-(P.p)^2} \;\;\;\;\; , \;\;\;\;\;
v(p)=\frac{P \times p}{P^2p^2-(P.p)^2}.$$ The proof that these vectors transform like $p$ is simple using the knowledge that a vector $V_{\mu}(p,P)$ transforms under parity like $V_{\mu}(p,P) \rightarrow \Pi V_{\mu}(\Lambda p, \Lambda P) \Pi^{-1} $ and that the five matrices $\gamma_{\mu}$, $\gamma_{4}$ and $\gamma_{5}$ form an anticommuting set and $\gamma_{45}=-i \gamma_{4}\gamma_{5}$.
So we have a set of five vectors with even parity. A linear combination of $P$ and $u$ can be found which gives $p$ and so it is not required. The four vectors we work with are thus $P_{\mu}$, $\gamma_{\mu}$, $u_{\mu}(p)$ and $v_{\mu}(p)\gamma_{45}$. Any other available vectors even in parity would be linear combinations of these.
A massive boson of spin 1 will have a purely transverse wavefunction [@LS69]. Thus the vector vertex will be purely transverse. This zero divergence criterion in momentum space means that $P_{\mu} \Gamma_{\mu} = 0$. We must find linear combinations of the four vectors above which are all purely transverse. Only three can be found which are $T_{\mu}(P)=\gamma_{\mu}-P_{\mu} \not \! P / P^2$, $u_{\mu}(p)$ and $v_{\mu}(p)\gamma_{45}$. It therefore appears as though the most general vector $\Gamma_{\mu}(p,P)$ we can write which is even under parity and purely transverse is a sum of these quantities, each multiplied by the most general scalar vertex [@Bu92]. The result is an expression with twelve terms and with twelve parameters. However, some terms have been included more than once. It can be shown that four of the twelve terms can be eliminated leaving the general vector vertex $$\begin{aligned}
\Gamma^V_{\mu}(p,P) & = & u_{\mu}(p) \left( f_1 + f_2 \not \! P +
f_3 \not \! u(p) + f_4 \not \! v(p)\gamma_{45} \right) \nonumber \\
& + & v_{\mu}(p)\gamma_{45} \left( f_5 + f_6 \not \! P +
f_7 \not \! u(p) + f_8 \not \! v(p)\gamma_{45} \right). \label{eq:VERTEX}\end{aligned}$$ where $f_n$ ($n=$1–8) are functions of $q^2,P^2$ and $q\cdot P$ only. The pseudovector (PV), axivector (AV) and axipseudovector (APV) vertices are given by $$\begin{aligned}
\Gamma^{PV}_{\mu}(p,P) & = & \gamma_{45} \Gamma^V_{\mu}(p,P), \nonumber \\
\Gamma^{AV}_{\mu}(p,P) & = & \left(\begin{array}{c} \gamma_4 \\
\gamma_5 \end{array} \right) \Gamma^V_{\mu}(p,P), \nonumber \\
\Gamma^{APV}_{\mu}(p,P) & = & \gamma_{45} \Gamma^{AV}_{\mu}(p,P) \,\,\, =
\,\,\, i \left(\begin{array}{c} -\gamma_5 \\
\gamma_4 \end{array} \right) \Gamma^V_{\mu}(p,P). \label{eq:AVS}\end{aligned}$$
For a specified charge parity ${\cal C}=\pm 1$ of a bound state, the parity of the Dirac coefficients ($f_n$) under the transformation $q\cdot P \rightarrow -q\cdot P$ can be determined. The quantity $q\cdot P$ is the only Lorentz invariant which changes sign under charge conjugation and thus determines the charge parity of those functions. Table \[tab5\] lists the charge parities of each function for each case.
We can see from this table that multiplying the vector vertex by $\gamma_{45}$ (and thus producing the pseudovector vertex) leaves the function charge parities unchanged but multiplying the axivector vertex by $\gamma_{45}$ (producing the axipseudovector vertex) reverses the function charge parities. We know that the same BSE results when we multiply the vertex $\Gamma_\mu$ by $\gamma_{45}$. This means that the vector ${\cal C}=+1$ and pseudovector ${\cal C}=+1$ states are degenerate as are the vector ${\cal C}=-1$ and pseudovector ${\cal C}=-1$ states. We must also find the degenerate pairs axivector ${\cal C}=+1$ / axipseudovector ${\cal C}=-1$ and axivector ${\cal C}=-1$ / axipseudovector ${\cal C}=+1$.
Our conventions for Euclidean space quantities are summarised in Appendix A of Ref. [@Bu92]. In particular Euclidean momenta and Dirac matrices are defined by $$P_3^{({\rm E})} = -iP_0^{({\rm M})},\hspace{5 mm}
P_{1,2}^{({\rm E})} = P_{1,2}^{({\rm M})},\hspace{5 mm}
\gamma_3^{({\rm E})} = \gamma_0^{({\rm M})}, \hspace{5 mm}
\gamma_{1,2}^{({\rm E})} = i\gamma_{1,2}^{({\rm M})}.$$
Appendix B - Coupled Integral Equations {#appendix-b---coupled-integral-equations .unnumbered}
=======================================
Working in the rest frame of the bound state where $P_{\mu}=(0,0,iM)$, the vector and axivector vertices given in appendix A are used to derive coupled BS equations. After considerable work the eight coupled equations were found to be (after rescaling and angular integration) $$\begin{aligned}
a(p) & = & \frac{3}{(2\pi)^2} \int^{\infty}_{-\infty}dq_3 \,
\int^{\infty}_0 \mbox{$\left| {\bf q} \right|$} d\mbox{$\left|
{\bf q} \right|$} \, X(\alpha,\beta) \,\, \tau_{a1}(q) \nonumber \\
b(p) & = & \frac{1}{(2\pi)^2} \int^{\infty}_{-\infty}dq_3 \,
\int^{\infty}_0 \mbox{$\left| {\bf q} \right|$} d\mbox{$\left|
{\bf q} \right|$} \, X(\alpha,\beta) \,\, \tau_{b1}(q) \nonumber \\
c(p) & = & \frac{1}{(2\pi)^2} \int^{\infty}_{-\infty}dq_3 \, \int^{\infty}_0
\mbox{$\left| {\bf q} \right|$} d\mbox{$\left| {\bf q} \right|$} \,
\left[ Z(\alpha,\beta) \,\, \tau_{c1}(q) + Y(\alpha,\beta) \,\, \tau_{c2}(q)
\right] \nonumber \\
d(p) & = & \frac{1}{(2\pi)^2} \int^{\infty}_{-\infty}dq_3 \, \int^{\infty}_0
\mbox{$\left| {\bf q} \right|$} d\mbox{$\left| {\bf q} \right|$} \,
\left[ Z(\alpha,\beta) \,\, \tau_{d1}(q) + Y(\alpha,\beta) \,\, \tau_{d2}(q)
\right] \nonumber \\
e(p) & = & \frac{3}{(2\pi)^2} \int^{\infty}_{-\infty}dq_3 \, \int^{\infty}_0
\mbox{$\left| {\bf q} \right|$} d\mbox{$\left| {\bf q} \right|$} \,
X(\alpha,\beta) \,\, \tau_{e2}(q) \nonumber \\
f(p) & = & \frac{1}{(2\pi)^2} \int^{\infty}_{-\infty}dq_3 \, \int^{\infty}_0
\mbox{$\left| {\bf q} \right|$} d\mbox{$\left| {\bf q} \right|$} \,
X(\alpha,\beta) \,\, \tau_{f2}(q) \nonumber \\
g(p) & = & \frac{1}{(2\pi)^2} \int^{\infty}_{-\infty}dq_3 \, \int^{\infty}_0
\mbox{$\left| {\bf q} \right|$} d\mbox{$\left| {\bf q} \right|$} \,
\left[ Y(\alpha,\beta) \,\, \tau_{g1}(q) + Z(\alpha,\beta) \,\, \tau_{g2}(q)
\right] \nonumber \\
h(p) & = & \frac{1}{(2\pi)^2} \int^{\infty}_{-\infty}dq_3 \, \int^{\infty}_0
\mbox{$\left| {\bf q} \right|$} d\mbox{$\left| {\bf q} \right|$} \,
\left[ Y(\alpha,\beta) \,\, \tau_{h1}(q) + Z(\alpha,\beta) \,\, \tau_{h2}(q)
\right] \label{eq:IE} \end{aligned}$$ where we have rescaled the functions $f_1,f_2,f_3,f_4,f_5,f_6,f_7$ and $f_8$ seen in appendix A to the new functions $a,b,c,d,e,f,g$ and $h$ to ensure all quantities are real. The rescalings for the vector case were $$\begin{aligned}
\vspace{2mm}
a(p_3,\mbox{$\left| {\bf p} \right|$})&=
\frac{1}{\mbox{$\left| {\bf p} \right|$}} \,
f_1(p_3,\mbox{$\left| {\bf p} \right|$}), \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
e(p_3,\mbox{$\left| {\bf p} \right|$})&=
\frac{1}{M \mbox{$\left| {\bf p} \right|$}} \,
f_5(p_3,\mbox{$\left| {\bf p} \right|$}), \nonumber \\ \vspace{2mm}
b(p_3,\mbox{$\left| {\bf p} \right|$})&=
\frac{M}{\mbox{$\left| {\bf p} \right|$}} \,
f_2(p_3,\mbox{$\left| {\bf p} \right|$}), \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
f(p_3,\mbox{$\left| {\bf p} \right|$})&=
\frac{1}{\mbox{$\left| {\bf p} \right|$}} \,
f_6(p_3,\mbox{$\left| {\bf p} \right|$}),\nonumber \\ \vspace{2mm}
c(p_3,\mbox{$\left| {\bf p} \right|$})&=
\frac{i}{\mbox{$\left| {\bf p} \right|$}^2} \,
f_3(p_3,\mbox{$\left| {\bf p} \right|$}), \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
g(p_3,\mbox{$\left| {\bf p} \right|$})&=
\frac{-i}{M \mbox{$\left| {\bf p} \right|$}^2} \,
f_7(p_3,\mbox{$\left| {\bf p} \right|$}), \nonumber \\ \vspace{2mm}
d(p_3,\mbox{$\left| {\bf p} \right|$})&=
\frac{-i}{M \mbox{$\left| {\bf p} \right|$}^2} \,
f_4(p_3,\mbox{$\left| {\bf p} \right|$}), \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
h(p_3,\mbox{$\left| {\bf p} \right|$})&=
\frac{i}{M^2 \mbox{$\left| {\bf p} \right|$}^2} \,
f_8(p_3,\mbox{$\left| {\bf p} \right|$}),
\label{eq:RESV}\end{aligned}$$ and for the axivector case $$\begin{aligned}
\vspace{2mm}
a(p_3,\mbox{$\left| {\bf p} \right|$})&=
\frac{1}{\mbox{$\left| {\bf p} \right|$}} \,
f_1(p_3,\mbox{$\left| {\bf p} \right|$}), \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
e(p_3,\mbox{$\left| {\bf p} \right|$})&=
\frac{i}{M \mbox{$\left| {\bf p} \right|$}} \,
f_5(p_3,\mbox{$\left| {\bf p} \right|$}), \nonumber \\ \vspace{2mm}
b(p_3,\mbox{$\left| {\bf p} \right|$})&=
\frac{-i M}{\mbox{$\left| {\bf p} \right|$}} \,
f_2(p_3,\mbox{$\left| {\bf p} \right|$}), \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
f(p_3,\mbox{$\left| {\bf p} \right|$})&=
\frac{1}{\mbox{$\left| {\bf p} \right|$}} \,
f_6(p_3,\mbox{$\left| {\bf p} \right|$}), \nonumber \\ \vspace{2mm}
c(p_3,\mbox{$\left| {\bf p} \right|$})&=
\frac{1}{\mbox{$\left| {\bf p} \right|$}^2} \,
f_3(p_3,\mbox{$\left| {\bf p} \right|$}), \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
g(p_3,\mbox{$\left| {\bf p} \right|$})&=
\frac{-i}{M \mbox{$\left| {\bf p} \right|$}^2} \,
f_7(p_3,\mbox{$\left| {\bf p} \right|$}), \nonumber \\ \vspace{2mm}
d(p_3,\mbox{$\left| {\bf p} \right|$})&=
\frac{-i}{M \mbox{$\left| {\bf p} \right|$}^2} \,
f_4(p_3,\mbox{$\left| {\bf p} \right|$}), \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
h(p_3,\mbox{$\left| {\bf p} \right|$})&=
\frac{1}{M^2 \mbox{$\left| {\bf p} \right|$}^2} \,
f_8(p_3,\mbox{$\left| {\bf p} \right|$}).
\label{eq:RESAV}\end{aligned}$$ These functions also occur within the sums ($\tau$) seen in Eq. (\[eq:IE\]) which are defined as (for a function $r=a$–$h$) $$\begin{aligned}
\tau_{r1}(q)&=&T_{ra}\,a(q)+T_{rb}\,b(q)+T_{rc}\,c(q)+T_{rd}\,d(q)
\,\,\,\,\,\, \mbox{and} \nonumber \\
\tau_{r2}(q)&=&T_{re}\,e(q)+T_{rf}\,f(q)+T_{rg}\,g(q)+T_{rh}\,h(q).\end{aligned}$$ Also $$X(\alpha,\beta)=\frac{(\alpha^2-\beta^2)^{\frac{1}{2}}-\alpha}{\beta
(\alpha^2-\beta^2)^{\frac{1}{2}}}, \,\,\,\,\,\,\,\,\,\,\,\,\,
Y(\alpha,\beta)=\frac{1}{\alpha+(\alpha^2-\beta^2)^{\frac{1}{2}}},$$ $$Z(\alpha,\beta)=\frac{\alpha}{\alpha(\alpha^2-\beta^2)^{\frac{1}{2}}+
(\alpha^2-\beta^2)},
\label{eq:XYZdef}$$ where $$\alpha=(p_3-q_3)^2 + \mbox{$\left| {\bf p} \right|$}^2 + \mbox{
$\left| {\bf q} \right|$}^2, \;\;\;\;\;\;
\beta=-2 \mbox{$\left| {\bf p} \right|$} \mbox{$\left| {\bf q} \right|$},$$ The momentum Q is defined by $$Q^2=q_3^2 + \mbox{$\left| {\bf q} \right|$}^2 -\frac{1}{4}M^2
+ iMq_3 , \label{eq:QDEF}$$ and we use the abbreviations $\sigma_V=\sigma_V(Q^2)$ and $\sigma_S=\sigma_S(Q^2)$ for use in the definition of the functions $T_{aa},T_{ab},\ldots$ which are analytic functions of $q_3, \mbox{$\left| {\bf q} \right|$}$, and $M$. The diagonal $T$’s are given by $$\begin{aligned}
T_{aa}=\,\,\,\,\,(\frac{1}{4}M^2 + q_3^{\,2} + \mbox{$\left| {\bf q}
\right|$}^2) |\sigma_V|^2 \mp |\sigma_S|^2, \,\,\,\,\,
&T_{ee}=\,\,\,\,\,(\frac{1}{4}M^2 + q_3^{\,2} + \mbox{$\left| {\bf q}
\right|$}^2) |\sigma_V|^2 \pm |\sigma_S|^2 \nonumber \\
T_{bb}=-(\frac{1}{4}M^2 + q_3^{\,2} - \mbox{$\left| {\bf q}
\right|$}^2) |\sigma_V|^2 \pm |\sigma_S|^2, \,\,\,\,\,
&T_{ff}=-(\frac{1}{4}M^2 + q_3^{\,2} - \mbox{$\left| {\bf q}
\right|$}^2) |\sigma_V|^2 \pm |\sigma_S|^2 \nonumber \\
T_{cc}=\,\,\,\,\,(\frac{1}{4}M^2 + q_3^{\,2} - \mbox{$\left| {\bf q}
\right|$}^2) |\sigma_V|^2 \pm |\sigma_S|^2, \,\,\,\,\,
&T_{gg}=\,\,\,\,\,(\frac{1}{4}M^2 + q_3^{\,2} - \mbox{$\left| {\bf q}
\right|$}^2) |\sigma_V|^2 \pm |\sigma_S|^2 \nonumber \\
T_{dd}=\,\,\,\,\,(\frac{1}{4}M^2 + q_3^{\,2} + \mbox{$\left| {\bf q}
\right|$}^2) |\sigma_V|^2 \pm |\sigma_S|^2, \,\,\,\,\,
&T_{hh}=\,\,\,\,\,(\frac{1}{4}M^2 + q_3^{\,2} + \mbox{$\left| {\bf q}
\right|$}^2) |\sigma_V|^2 \pm |\sigma_S|^2
\label{eq:TDIAGS}\end{aligned}$$ where the upper sign applies to the vector equations and the lower sign to the axivector equations. The nonzero off diagonal $T$’s for the vector case are $$\begin{aligned}
T_{ab}&=T_{ba}&= \frac{i}{2}(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}
\sigma_V)M - (\sigma_V^{\ast}\sigma_S+\sigma_S^{\ast}\sigma_V)q_3 \nonumber \\
T_{ac}&=T_{ca}&=\mbox{$\left| {\bf q} \right|$}(\sigma_V^{\ast}\sigma_S+
\sigma_S^{\ast}\sigma_V) \nonumber \\
T_{ad}&=-T_{da}&=M\mbox{$\left| {\bf q} \right|$} |\sigma_V|^2 \nonumber \\
T_{bc}&=T_{cb}&=2 q_3 \mbox{$\left| {\bf q} \right|$} |\sigma_V|^2 \nonumber \\
T_{bd}&=T_{db}&=i\mbox{$\left| {\bf q} \right|$} (\sigma_V^{\ast}\sigma_S
-\sigma_S^{\ast}\sigma_V) \nonumber \\
T_{cd}&=T_{dc}&= \frac{1}{2}(\sigma_V^{\ast}\sigma_S+\sigma_S^{\ast}
\sigma_V)M + i(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}\sigma_V)q_3 \nonumber \\
T_{ce}& &= M\mbox{$\left| {\bf q} \right|$} |\sigma_V|^2 \nonumber \\
T_{cf}& &= i\mbox{$\left| {\bf q} \right|$} (\sigma_V^{\ast}\sigma_S
-\sigma_S^{\ast}\sigma_V) \nonumber \\
T_{cg}&=T_{gc}&= -\frac{1}{2}(\sigma_V^{\ast}\sigma_S+\sigma_S^{\ast}
\sigma_V)M - i(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}\sigma_V)q_3 \nonumber \\
T_{ch}&=T_{hc}-2\mbox{$\left| {\bf q} \right|$}^2 |\sigma_V|^2&=
-(\frac{1}{4}M^2 + q_3^{\,2} - \mbox{$\left| {\bf q} \right|$}^2)
|\sigma_V|^2 - |\sigma_S|^2 \nonumber \\
T_{de}& &= \mbox{$\left| {\bf q} \right|$}(\sigma_V^{\ast}\sigma_S
+\sigma_S^{\ast}\sigma_V) \nonumber \\
T_{df}& &= 2\mbox{$\left| {\bf q} \right|$}q_3 |\sigma_V|^2 \nonumber \\
T_{dg}&=T_{gd}+2\mbox{$\left| {\bf q} \right|$}^2 |\sigma_V|^2&=
-(\frac{1}{4}M^2 + q_3^{\,2} - \mbox{$\left| {\bf q} \right|$}^2)
|\sigma_V|^2 - |\sigma_S|^2 \nonumber \\
T_{dh}&=T_{hd}&= -\frac{1}{2}(\sigma_V^{\ast}\sigma_S+\sigma_S^{\ast}
\sigma_V)M - i(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}\sigma_V)q_3 \nonumber \\
T_{ef}&=T_{fe}&=\frac{i}{2}(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}
\sigma_V)M - (\sigma_V^{\ast}\sigma_S+\sigma_S^{\ast}\sigma_V)q_3 \nonumber \\
T_{eg}&=T_{ge}&=-\mbox{$\left| {\bf q} \right|$}(\sigma_V^{\ast}\sigma_S
+\sigma_S^{\ast}\sigma_V) \nonumber \\
T_{eh}&=T_{he}&= -M\mbox{$\left| {\bf q} \right|$} |\sigma_V|^2 \nonumber \\
T_{fg}&=T_{gf}&= -2\mbox{$\left| {\bf q} \right|$}q_3 |\sigma_V|^2 \nonumber \\
T_{fh}&=T_{hf}&=-i\mbox{$\left| {\bf q} \right|$}(\sigma_V^{\ast}\sigma_S
-\sigma_S^{\ast}\sigma_V) \nonumber \\
T_{ga}& &= -M\mbox{$\left| {\bf q} \right|$} |\sigma_V|^2 \nonumber \\
T_{gb}& &=-i\mbox{$\left| {\bf q} \right|$}(\sigma_V^{\ast}\sigma_S
-\sigma_S^{\ast}\sigma_V) \nonumber \\
T_{gh}&=T_{hg}&= \frac{1}{2}(\sigma_V^{\ast}\sigma_S+\sigma_S^{\ast}
\sigma_V)M + i(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}\sigma_V)q_3 \nonumber \\
T_{ha}& &=-\mbox{$\left| {\bf q} \right|$}(\sigma_V^{\ast}\sigma_S
+\sigma_S^{\ast}\sigma_V) \nonumber \\
T_{hb}& &= -2\mbox{$\left| {\bf q} \right|$}q_3 |\sigma_V|^2 \end{aligned}$$ and for the axivector case the nonzero elements are $$\begin{aligned}
T_{ab}&=-T_{ba}&= \frac{1}{2}(\sigma_V^{\ast}\sigma_S+\sigma_S^{\ast}
\sigma_V)M + i(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}\sigma_V)q_3 \nonumber \\
T_{ac}&=-T_{ca}&=-i\mbox{$\left| {\bf q} \right|$}(\sigma_V^{\ast}\sigma_S-
\sigma_S^{\ast}\sigma_V) \nonumber \\
T_{ad}&=-T_{da}&=M\mbox{$\left| {\bf q} \right|$} |\sigma_V|^2 \nonumber \\
T_{bc}&=T_{cb}&=2 q_3 \mbox{$\left| {\bf q} \right|$} |\sigma_V|^2 \nonumber \\
T_{bd}&=T_{db}&=-\mbox{$\left| {\bf q} \right|$} (\sigma_V^{\ast}\sigma_S
+\sigma_S^{\ast}\sigma_V) \nonumber \\
T_{cd}&=-T_{dc}&= \frac{i}{2}(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}
\sigma_V)M - (\sigma_V^{\ast}\sigma_S+\sigma_S^{\ast}\sigma_V)q_3 \nonumber \\
T_{ce}& &= -M\mbox{$\left| {\bf q} \right|$} |\sigma_V|^2 \nonumber \\
T_{cf}& &= -\mbox{$\left| {\bf q} \right|$} (\sigma_V^{\ast}\sigma_S
+\sigma_S^{\ast}\sigma_V) \nonumber \\
T_{cg}&=-T_{gc}&= -\frac{i}{2}(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}
\sigma_V)M + (\sigma_V^{\ast}\sigma_S+\sigma_S^{\ast}\sigma_V)q_3 \nonumber \\
T_{ch}&=T_{hc}-2\mbox{$\left| {\bf q} \right|$}^2 |\sigma_V|^2& =
-(\frac{1}{4}M^2 + q_3^{\,2} + \mbox{$\left| {\bf q} \right|$}^2)
|\sigma_V|^2 + |\sigma_S|^2 \nonumber \\
T_{de}& &= i\mbox{$\left| {\bf q} \right|$}(\sigma_V^{\ast}\sigma_S
-\sigma_S^{\ast}\sigma_V) \nonumber \\
T_{df}& &= 2\mbox{$\left| {\bf q} \right|$}q_3 |\sigma_V|^2 \nonumber \\
T_{dg}&=T_{gd}+2\mbox{$\left| {\bf q} \right|$}^2 |\sigma_V|^2&=
-(\frac{1}{4}M^2 + q_3^{\,2} - \mbox{$\left| {\bf q} \right|$}^2)
|\sigma_V|^2 + |\sigma_S|^2 \nonumber \\
T_{dh}&=-T_{hd}&= \frac{i}{2}(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}
\sigma_V)M - (\sigma_V^{\ast}\sigma_S+\sigma_S^{\ast}\sigma_V)q_3 \nonumber \\
T_{ef}&=-T_{fe}&=\frac{1}{2}(\sigma_V^{\ast}\sigma_S+\sigma_S^{\ast}
\sigma_V)M + i(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}\sigma_V)q_3 \nonumber \\
T_{eg}&=-T_{ge}&=i\mbox{$\left| {\bf q} \right|$}(\sigma_V^{\ast}\sigma_S
-\sigma_S^{\ast}\sigma_V) \nonumber \\
T_{eh}&=T_{he}&=M\mbox{$\left| {\bf q} \right|$} |\sigma_V|^2 \nonumber \\
T_{fg}&=T_{gf}&=-2\mbox{$\left| {\bf q} \right|$}q_3 |\sigma_V|^2 \nonumber \\
T_{fh}&=-T_{hf}&=-\mbox{$\left| {\bf q} \right|$}(\sigma_V^{\ast}\sigma_S
+\sigma_S^{\ast}\sigma_V) \nonumber \\
T_{ga}& &=-M\mbox{$\left| {\bf q} \right|$} |\sigma_V|^2 \nonumber \\
T_{gb}& &=-\mbox{$\left| {\bf q} \right|$}(\sigma_V^{\ast}\sigma_S
+\sigma_S^{\ast}\sigma_V) \nonumber \\
T_{gh}&=-T_{hg}&=-\frac{i}{2}(\sigma_V^{\ast}\sigma_S-\sigma_S^{\ast}
\sigma_V)M + (\sigma_V^{\ast}\sigma_S+\sigma_S^{\ast}\sigma_V)q_3 \nonumber \\
T_{ha}& &=-i\mbox{$\left| {\bf q} \right|$}(\sigma_V^{\ast}\sigma_S
-\sigma_S^{\ast}\sigma_V)i \nonumber \\
T_{hb}& &=-2\mbox{$\left| {\bf q} \right|$}q_3 |\sigma_V|^2. \end{aligned}$$
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to the National Centre for Theoretical Physics at the Australian National University where part of this work was undertaken.
C. J. Burden, Nucl. Phys. [**B387**]{} 419 (1992). T. W. Allen, and. C. J. Burden, Phys. Rev. [**D53**]{} 5842 (1996). R. Delbourgo, and. M. Scadron, J. Phys. [**G5**]{} 1621 (1979). P. Maris, Phys. Rev. [**D50**]{} 4189 (1994). P. Maris, Phys. Rev and references therein. [**D52**]{} 6087 (1995). S. J. Stainsby, and. R. T. Cahill, Int. J. Mod. Phys. [**A7**]{} 7541 (1992); S. J. Stainsby, and. R. T. Cahill, Mod. Phys. Lett. [**A9**]{} 3551 (1994). C. J. Burden, and. C. D. Roberts, Phys. Rev. [**D44**]{} 540 (1991); M. R. Pennington and D. Walsh, Phys. Lett. [**B253**]{} 246 (1991); D. C. Curtis, M. R. Pennington and. D. Walsh, Phys. Lett. [**B295**]{} 313 (1992). A. Bender, C. D. Roberts and. L. von Smekal, Phys. Lett. [**B380**]{} 7 (1996). M. Neubert, Phys. Rep. [**245**]{} 259 (1994). V. G. Koures, J. Comp. Phys. [**128**]{} 1 (1996). Y. S. Kim and M. E. Noz, [*Theory and Applications of the Poincare Group*]{}, Dordrecht Holland, (1986). D. Flamm and F. Schöberl, [*Introduction to the Quark Model of Elementary Particles*]{}, Gordon and Breach, (1982). A. Tam, C. J. Hamer and C. M. Yung, J. Phys. [**G21**]{} 1463 (1995). R. M. Barnett et al. (Particle Physics Group) Phys. Rev. [**D54**]{} 1 (1996). C. J. Burden and A. N. Burkitt, Europhys. Lett. [**3**]{} 545 (1987). E. Dagotto et al., Nucl. Phys. [**B334**]{} 279 (1990). C. Burden, How can QED$_3$ help us understand QCD$_4$, in [*QCD Vacuum Structure*]{}, edited by H. M. Fried and B. Müller, World Scientific, Singapore, 1993. C. H. Llewelyn-Smith, Ann. Phys. [**53**]{} 521 (1969).
[|c|c|c|c|c|]{}
------------------------------------------------------------------------
[$J$]{} & [$L$]{} & [$S$]{} & [$J^{AC}$]{} & State\
------------------------------------------------------------------------
$0$ & $0$ & $0$ & $0^{-+}$ & Axiscalar\
------------------------------------------------------------------------
& $1$ & $1$ & $0^{++}$ & Scalar\
------------------------------------------------------------------------
$1$ & $0$ & $1$ & $1^{--}$ & Vector\
------------------------------------------------------------------------
& $1$ & $0$ & $1^{+-}$ & Axivector\
------------------------------------------------------------------------
& & $1$ & $1^{++}$ & Axivector\
------------------------------------------------------------------------
& $2$ & $1$ & $1^{--}$ & Vector\
[|c|c|c|c|c|]{}
------------------------------------------------------------------------
& [Vector ${\cal C}=+1$]{} & [Vector ${\cal C}=-1$]{} & [Axivector ${\cal C}=+1$]{} & [Axivector ${\cal C}=-1$]{}\
------------------------------------------------------------------------
$m$ & $J^{AC} = 1^{-+}$ & $1^{--}$ & $1^{++}$ & $1^{+-}$\
------------------------------------------------------------------------
0 & 0.104 & 0.074 & 0.119 & 0.121\
------------------------------------------------------------------------
0.001 & 0.112 & 0.081 & 0.128 & 0.129\
------------------------------------------------------------------------
0.004 & 0.134 & 0.099 & 0.152 & 0.155\
------------------------------------------------------------------------
0.009 & 0.160 & 0.122 & 0.179 & 0.182\
------------------------------------------------------------------------
0.016 & 0.194 & 0.151 & 0.217 & 0.223\
------------------------------------------------------------------------
0.025 & 0.239 & 0.184 & 0.260 & 0.282\
------------------------------------------------------------------------
0.036 & 0.281 & 0.219 & 0.295 &\
------------------------------------------------------------------------
0.049 & 0.326 & 0.258 & 0.331 &\
------------------------------------------------------------------------
0.064 & 0.372 & 0.299 & 0.369 &\
------------------------------------------------------------------------
0.081 & 0.418 & 0.343 & 0.410 &\
------------------------------------------------------------------------
0.1 & 0.465 & 0.389 & 0.454 &\
------------------------------------------------------------------------
0.5 & 1.341 & 1.252 & 1.316 &\
------------------------------------------------------------------------
1 & 2.325 & 2.236 & 2.294 &\
------------------------------------------------------------------------
2 & 4.344 & 4.236 & 4.301 &\
------------------------------------------------------------------------
3 & 6.357 & 6.235 & 6.305 &\
------------------------------------------------------------------------
4 & 8.385 & 8.253 & 8.326 &\
------------------------------------------------------------------------
5 & 10.370 & 10.232 & 10.306 &\
[|c|c|c|]{}
------------------------------------------------------------------------
$J^{AC}$ or $J^{PC}$ & Positronium mass $M$ & Meson\
------------------------------------------------------------------------
$0^{-+}$ & 0 & $\pi^0$(135)\
------------------------------------------------------------------------
$0^{++}$ & 0.077 & $f_0$(400-1200)\
------------------------------------------------------------------------
$0^{--}$ & 0.111 &\
------------------------------------------------------------------------
$0^{--}$ & 0.123 &\
------------------------------------------------------------------------
$1^{--}$ & 0.074 & $\rho$(770), $\omega$(782)\
------------------------------------------------------------------------
$1^{-+}$ & 0.104 &\
------------------------------------------------------------------------
$1^{++}$ & 0.119 & $a_1$(1260), $f_1$(1285)\
------------------------------------------------------------------------
$1^{+-}$ & 0.121 & $h_1$(1170), $b_1$(1235)\
[|c|c|c|c|]{}
------------------------------------------------------------------------
& [${\cal C}$]{} & [Even functions of $q.P$]{} & [Odd functions of $q.P$]{}\
------------------------------------------------------------------------
Vector & $+1$ & $f_2,f_6$ & $f_1,f_3,f_4,f_5,f_7,f_8$\
------------------------------------------------------------------------
& $-1$ & $f_1,f_3,f_4,f_5,f_7,f_8$ & $f_2,f_6$\
------------------------------------------------------------------------
Pseudovector & $+1$ & $f_2,f_6$ & $f_1,f_3,f_4,f_5,f_7,f_8$\
------------------------------------------------------------------------
& $-1$ & $f_1,f_3,f_4,f_5,f_7,f_8$ & $f_2,f_6$\
------------------------------------------------------------------------
Axivector & $+1$ & $f_1,f_2,f_4,f_7$ & $f_3,f_5,f_6,f_8$\
------------------------------------------------------------------------
& $-1$ & $f_3,f_5,f_6,f_8$ & $f_1,f_2,f_4,f_7$\
------------------------------------------------------------------------
Axipseudovector & $+1$ & $f_3,f_5,f_6,f_8$ & $f_1,f_2,f_4,f_7$\
------------------------------------------------------------------------
& $-1$ & $f_1,f_2,f_4,f_7$ & $f_3,f_5,f_6,f_8$\
Figure captions. {#figure-captions. .unnumbered}
================
Figure 1:
: Bound state masses $M$ against fermion mass $m=$ 0 to 0.1. Full BSE solutions from Table \[tab1\]: $J^{AC} = 1^{-+}$ ($\Diamond$), $1^{--}$ ($\Box$), $1^{+-}$ ($+$) and $1^{++}$ ($\times$) states.
Figure 2:
: Bound state masses $M-2m$ for fermion mass $m=$ 0 to 1. Full BSE solutions from Table \[tab1\]: $1^{-+}$ ($\Diamond$ solid), $1^{--}$ ($\Box$ solid), $1^{+-}$ ($+$ solid) and $1^{++}$ ($\times$ solid). Nonrelativistic predictions of Eq. (\[eq:THY1\]) using Eq. (\[eq:LPRIME\]): (Dashed curves from bottom to top) $\lambda'=$ 1.7969 ($\ell = 0$), 2.5666 ($\ell = 1$), 2.9316 ($\ell = 0$) and 3.1148 ($\ell = 2$) respectively.
|
---
abstract: 'We present an approach for implementing a specific form of collaborative industrial practices—called Industrial Symbiotic Networks (${\mathsf{ISN}}$s)—as MC-Net cooperative games and address the so called ${\mathsf{ISN}}$ implementation problem. This is, the characteristics of ${\mathsf{ISN}}$s may lead to inapplicability of *fair* and *stable* benefit allocation methods even if the collaboration is a collectively desired one. Inspired by realistic ${\mathsf{ISN}}$ scenarios and the literature on normative multi-agent systems, we consider *regulations* and normative socioeconomic *policies* as two elements that in combination with ${\mathsf{ISN}}$ games resolve the situation and result in the concept of *coordinated* ${\mathsf{ISN}}$s.'
author:
- Vahid Yazdanpanah
- Devrim Murat Yazan
- Henk Zijm
bibliography:
- 'References.bib'
subtitle: Extended Abstract
title: Industrial Symbiotic Networks as Coordinated Games
---
Introduction
============
Industrial Symbiotic Networks (${\mathsf{ISN}}$s) are collaborative networks of industries with the aim to reduce their materials and energy footprint by circulating reusable resources (e.g, physical waste material) among the network members [@chertow2000industrial; @lombardi2012redefining; @yazan2016design]. Such a symbiosis leads to socioeconomic and environmental benefits for involved firms and the society. One barrier against stable ${\mathsf{ISN}}$ implementations is the lack of frameworks able to secure such networks against unfair and unstable allocation of obtainable benefits among the involved firms. In other words, even if economic benefits are foreseeable, lack of stability and/or fairness may lead to non-cooperative decisions and hence unimplementability of ${\mathsf{ISN}}$s (${\mathsf{ISN}}$ *implementation* problem). Reviewing recent contributions in the field of industrial symbiosis research, we encounter studies focusing on the interrelations between industrial enterprises [@yazan2016design] and the role of contracts in the process of ${\mathsf{ISN}}$ implementation [@albino2016exploring]. We believe a missed element for shifting from *theoretical* ${\mathsf{ISN}}$ design to *practical* ${\mathsf{ISN}}$ implementation is to model, reason about, and support ${\mathsf{ISN}}$ decisions in a *dynamic* way—and not by using snapshot-based modeling frameworks.
This abstract reports on extending the game-theoretic approach of [@iesm2017] with *regulative* rules and normative socioeconomic *policies*—following the successful line of work on normative multi-agent systems [@shoham1995social; @grossi2013norms; @andrighetto2013normative]. The extension provides a scalable solution to the ${\mathsf{ISN}}$ implementation problem and enables enforcing desired industrial collaborations in a fair and stable manner.
Research Questions
------------------
The following questions guide the design of a game-theoretic framework and its normative coordination mechanism that jointly facilitate the implementation of ${\mathsf{ISN}}$s:
1. ${\mathsf{ISN}}$ *Games*: How to define a game-theoretic basis for ${\mathsf{ISN}}$s that both reflects their operational cost dynamics and allows the integration of normative rules?
2. ${\mathsf{ISN}}$ *Coordination*: How to uniformly represent the regulatory dimension of ${\mathsf{ISN}}$s using incentive rules and normative policies?
3. *Coordinated* ${\mathsf{ISN}}$ *Games*: How to develop a framework that integrates normative coordination methods into ${\mathsf{ISN}}$ games to enable the fair and stable implementation of desirable ${\mathsf{ISN}}$s—with respect to an established policy?
Dealing with ${\mathsf{ISN}}$s’ complex industrial context [@DBLP:conf/eumas/YazdanpanahYZ16], an ideal ${\mathsf{ISN}}$ implementation platform would be tunable to specific industrial settings, scalable for implementing various ${\mathsf{ISN}}$ topologies, and would not require industries to sacrifice financially nor restrict their freedom in the market. Below, we present the overview of an approach for developing an ${\mathsf{ISN}}$ implementation framework with properties close to the ideal one.
Overview of The Approach
========================
As discussed in [@albino2016exploring; @iesm2017], the total obtainable cost reduction (as an economic benefit) and its allocation among involved firms are key drivers behind the stability of ${\mathsf{ISN}}$s. For any set of agents involved in an ${\mathsf{ISN}}$, this value—i.e., the obtainable cost reduction—characterizes the value of the set and hence can be seen as a basis for formulating ${\mathsf{ISN}}$s as cooperative games. On the other hand, in realistic ${\mathsf{ISN}}$s, the symbiotic practice takes place in presence of economic, social, and environmental *policies* and under *regulations* that aim to enforce the policies by nudging the behavior of agents towards desired ones. This is, while policies generally indicate whether an ${\mathsf{ISN}}$ is “good (bad, or neutral)", the regulations are a set of norms that—in case of agents’ compliance—result in an acceptable spectrum of collective behaviors. We follow this normative perspective and aim to use normative coordination to guarantee the implementability of desirable ${\mathsf{ISN}}$s—modeled as games—in a stable and fair manner. In the following subsections, we indicate how ${\mathsf{ISN}}$ games can be modeled and coordinated using regulatory incentive rules and normative socioeconomic policies.
ISNs as Cooperative Games
-------------------------
In the game-theoretic representation of ${\mathsf{ISN}}$s, the value of any set of agents $S$ is defined [@iesm2017] using the difference between the total cost that firms have to pay in case the ${\mathsf{ISN}}$ does not occur, i.e. costs to discharge wastes and to purchase traditional primary inputs (denoted by $T(S)$), and the total cost that firms have to pay collectively in case the ${\mathsf{ISN}}$ is realized, i.e. costs for recycling and treatment, for transporting resources among firms, and transaction costs (denoted by $O(S)$). Formally, the ${\mathsf{ISN}}$ among agents in a non-empty finite set of agents $N$ is a normalized superadditive cooperative game $(N,v)$ where for $S\subseteq N$, $v(S)$ is equal to $T(S)- O(S)$ if $|S|>1$, and $0$ otherwise.
Benefit sharing is crucial in the process of ${\mathsf{ISN}}$ implementation, mainly because of stability and fairness concerns. Roughly speaking, firms are rational agents that defect unbeneficial collaborations (instability) and mostly tend to reject relations in which benefits are not shared according to contributions (unfairness). Focusing on the Core and Shapley allocations [@osborne1994course; @mas1995microeconomic]—as standard methods that characterize stability and fairness—these solution concepts appear to be applicable in a specific class of ${\mathsf{ISN}}$s but are not generally scalable for value allocation in the implementation phase of ${\mathsf{ISN}}$s. In particular, relying on the balancedness of two-person ${\mathsf{ISN}}$ games, denoted by ${\mathsf{ISN}}_{\Lambda}$, we can show that any ${\mathsf{ISN}}_{\Lambda}$ is implementable in a fair and stable manner. However, in larger games—as balancedness does not hold necessarily—the core of the game may be empty which in turn avoids an ${\mathsf{ISN}}$ implementation that is reasonable for all the involved firms. So, even if a symbiosis could result in collective benefits, it may not last due to instable or unfair implementations. A natural response which is in-line with realistic practices is to employ monetary incentives as a means of normative coordination—to guarantee the implementability of “desired” ${\mathsf{ISN}}$s. To allow a smooth integration with normative rules, we transform ${\mathsf{ISN}}$ games into *basic MC-Nets*[^1] through the following steps: let $(N,v)$ be an arbitrary ${\mathsf{ISN}}$ game, $S_{\geq 2}= \{S \subseteq N : |S|\geq 2 \}$ be the set of all groups with two or more members where $K=|S_{\geq 2}|$ denotes its cardinality. We start with an empty set of MC-Net rules. Then for all groups $S_{i} \in S_{\geq 2}$, for $i= 1$ to $K$, we add a rule $\{\rho_i : (S_i,N \setminus S_i) \mapsto v_i=T(S_i)-O(S_i) \}$ to the MC-Net.
Normative Coordination of ISNs
------------------------------
Following [@shoham1995social; @grossi2013norms], we see that norms can be employed as game transformations to bring about more desirable outcomes in ${\mathsf{ISN}}$ games. For this account, given the economic, environmental, and social dimensions and with respect to potential socioeconomic consequences, ${\mathsf{ISN}}$s can be partitioned in three classes by a normative socioeconomic policy function $\wp : 2^N \mapsto {\{p^+,p^\circ , p^-\}}$, where $N$ is a finite set of firms. Moreover, $p^+$, $p^\circ$, and $p^-$ are labels—assigned by a third-party authority—indicating whether an ${\mathsf{ISN}}$ is promoted, permitted, or prohibited, respectively.
The rationale behind introducing policies is mainly to make sure that the set of promoted ${\mathsf{ISN}}$s are implementable in a fair and stable manner while prohibited ones are instable. To ensure this, in real ${\mathsf{ISN}}$ practices, the regulatory agent introduces monetary incentives, i.e., ascribes subsidies to promoted and taxes to prohibited collaborations. We follow this practice and employ a set of rules to ensure/avoid the implementability of desired/undesired ${\mathsf{ISN}}$s by allocating incentives[^2]. Such a set of incentive rules can be represented by an MC-Net $\Re = \{\rho_i : (\mathcal{P}_i,\mathcal{N}_i) \mapsto \iota_i \}_{i \in K}$ in which $K$ is the set of rule indices. Then, the incentive value for $S \subseteq N$, is defined as $\iota(S)\coloneqq\sum_{i \in \Im(S)} \iota_i$ where $\Im(S)$ denotes the set of rule indices that are applicable to $S$. It is provable that for any ${\mathsf{ISN}}$ game there exists a set of incentive rules to guarantee its implementability.
Coordinated ISN Games
---------------------
Having policies and regulations, we integrate them into ${\mathsf{ISN}}$ games and introduce the concept of *Coordinated ${\mathsf{ISN}}$s* (${\mathcal{C-}\mathsf{ISN}}$s). Formally, let $G$ be an ${\mathsf{ISN}}$ and $\Re$ be a set of regulatory incentive rules, both as MC-Nets among agents in $N$. Moreover, for each group $S\subseteq N$, let $v(S)$ and $\iota(S)$ denote the value of $S$ in $G$ and the incentive value of $S$ in $\Re$, respectively. We say the Coordinated ${\mathsf{ISN}}$ Game (${\mathcal{C-}\mathsf{ISN}}$) among agents in $N$ is a cooperative game $(N,c)$ where for each group $S$, we have that $c(S)=v(S) + \iota(S)$.
It can be observed that employing such incentive rules is effective for enforcing socioeconomic policies. In particular, we have that for any promoted ${\mathsf{ISN}}$ game, under a policy $\wp$, there exist an implementable ${\mathcal{C-}\mathsf{ISN}}$ game. Analogously, similar properties hold while *avoiding* prohibited ${\mathsf{ISN}}$s or *allowing* permitted ones. The presented approach for incentivizing ${\mathsf{ISN}}$s is advisable when the policy-maker is aiming to ensure the implementability of a promoted ${\mathsf{ISN}}$ in an ad-hoc way. In other words, an $\Re$ that ensures the implementability of a promoted ${\mathsf{ISN}}$ $G_1$ may ruin the implementability of another promoted ${\mathsf{ISN}}$ $G_2$. To avoid this, the set of collaborations that a policy $\wp$ marks as promoted should be mutually exclusive. Accordingly, we have the desired result that the mutual exclusivity condition is sufficient for ensuring the implementability of *all* the ${\mathsf{ISN}}$s among $\wp$-promoted groups in a fair and stable manner.
Concluding Remarks
==================
The details of the components for developing the ${\mathsf{ISN}}$ implementation framework—rooted in cooperative games and coordinated with normative rules—consist of algorithms for generating incentive rules and policy properties to ensure the implementability of promoted ${\mathsf{ISN}}$s. We plan to explore the possibility of having multiple policies and tools for policy option analysis [@sara2017structured] in ${\mathsf{ISN}}$s. Then, possible regulation conflicts can be resolved using prioritized rule sets (inspired by formal argumentation theory [@modgil2013general; @kaci2008preference]). We also aim to focus on administration of ${\mathsf{ISN}}$s by modeling them as normative multi-agent organizations [@boissier2013organisational; @DBLP:conf/eumas/YazdanpanahYZ16] and relying on norm-aware frameworks [@dastani2016commitments; @aldewereld2007operationalisation] that enable monitoring organizational behaviors.
The project leading to this work has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 680843.
[^1]: A basic MC-Net represents a game in $N$ as a set of rules $\{\rho_i : (\mathcal{P}_i,\mathcal{N}_i) \mapsto v_i \}_{i \in K}$, where $\mathcal{P}_i \subseteq N$, $\mathcal{N}_i \subset N$, $\mathcal{P}_i \cap \mathcal{N}_i = \emptyset$, $v_i \in \mathbb{R} \setminus \{0\}$, and $K$ is the set of rule indices. For a group $S\subseteq N$, a rule $\rho_i$ is applicable if $\mathcal{P}_i \subseteq S $ and $\mathcal{N}_i \cap S=\emptyset $. Then $v(S)$ will be equal to $\sum_{i \in \Pi(S)} v_i$ where $\Pi(S)$ denote the set of rule indices that are applicable to $S$. This rule-based representation allows natural integration with rule-based coordination methods and results in relatively low complexity for computing allocation methods such as the Shapley value [@lesca2017coalition; @ieong2005marginal].
[^2]: See [@DBLP:conf/ijcai/MeirRM11; @DBLP:conf/atal/ZickPJ13] for similar approaches on incentivizing cooperative games.
|
---
abstract: 'Employing a notion of curvature for arbitrary closed sets we prove an ABP-type estimate for a class of singular submanifolds of arbitrary codimension and bounded mean curvature recently introduced by B. White. A weak-Harnack-type estimate is then derived using the ABP estimate. These results generalize analogous results by O. Savin for viscosity solutions of the minimal surface equation.'
author:
- Mario Santilli
title: ABP inequalities for singular submanifolds of bounded mean curvature
---
#### Keywords.
[Singular submanifolds, curvature, ABP estimate, Harnack estimate.]{}
Introduction
============
We prove ABP (Alexandroff-Bakelmann-Pucci) inequalities for a class *singular submanifolds of arbitrary codimension* in the Euclidean space which includes the class of varifolds without boundary and with bounded mean curvature.
More specifically, we consider the following class.
\[definition of (m,h)\] Suppose $ 1 \leq m \leq n $ are integers, $ \Omega $ is an open subset of $ { \mathbf{R}^{n+1}} $, $ \Gamma $ is relatively closed in $ \Omega $ and $ h \geq 0 $. We say that $ \Gamma $ is a $ (m,h) $ subset of $ \Omega $ provided it has the following property: if $ x \in \Gamma $ and $ f $ is a $ \mathscr{C}^{2} $ function in a neighbourhood of $ x $ such that $ f|\Gamma $ has a local maximum at $ x $ and $ \nabla f(x) \neq 0 $, then $$\operatorname{trace}_{m}\operatorname{D}^{2}f(x) \leq h |\nabla f(x)|,$$ where $ \operatorname{trace}_{m} \operatorname{D}^{2}f(x) $ is the sum of the lowest $ m $ eigenvalues of $ \operatorname{D}^{2}f(x) $.
This class has been recently introduced[^1] in [@MR3466806 2.1] in the analysis of the area blow-up of sequences of smooth submanifolds (or varifolds) with a uniform bound on the mean curvature. In fact, as proved in [@MR3466806 2.6], if $ M_{i} $ is a sequence of $ m $ dimensional varifolds in an open subset $ \Omega $ of $ { \mathbf{R}^{n+1}} $ with mean curvatures uniformly bounded in $ {\mathbf{L}^{\infty}} $ by a non-negative number $ h $ and such that $$\limsup_{i \to \infty}|\partial M_{i}|(U) < \infty \quad \textrm{whenever $ U \subset \subset \Omega $},$$ where $ |\partial M_{i}| $ is the measure associated to the generalized boundary of $ M_{i} $, then the smallest closed subset $ Z $ of $ \Omega $ such that the areas of $ M_{i} $ are uniformly bounded as $ i \to \infty $ on compact subsets of $ \Omega \sim Z $ is an $ (m,h) $ subset of $ \Omega $. As a simple consequence of this general fact, it follows that the support of every $ m $ dimensional varifold in $ \Omega $ with no boundary and mean curvature bounded in $ {\mathbf{L}^{\infty}} $ by $ h $ is an $(m,h)$ subset of $ \Omega $, see [@MR3466806 2.8].
Special cases of \[definition of (m,h)\], under the name of viscosity minimal sets, have been studied in [@2017arXiv170507948S] and [@2017arXiv170801549S].
We describe now the results of the present paper.
Firstly we introduce some notation[^2] that we use through the rest of this work. Suppose $ \mathscr{C}_{1}(0)=\mathbf{U}^{n}(0,1) \times \mathbf{R} \subseteq \mathbf{R}^{n+1} $, $ \Gamma \subseteq \mathscr{C}_{1}(0) $ is relatively closed and $ a > 0 $. For every $ x \in \mathbf{R}^{n} $ we define $ P_{a,x} $ to be the paraboloid of opening $ a $ and center $ x $ touching $ \overline{\Gamma} $ from above. For every closed set $ C \subseteq \mathbf{B}^{n}(0,1) $ we define the touching set $$A_{a}(\Gamma; C)$$ as the set of $ (z,\eta) $ such that $z \in \Sigma(P_{a,x}) \cap \overline{\Gamma}$ for some $ x \in C $ and $ \eta \in \operatorname{Nor}(\Sigma(P_{a,x}), z) $ with $ |\eta |=1 $ and $ \eta_{n+1} > 0 $. Moreover, we let $$A'_{a}(\Gamma; C) = \{ z' : \textrm{$(z, \eta) \in A_{a}(\Gamma; C)$ for some $ \eta \in \mathbf{S}^{n} $}\}.$$ When $ C = \mathbf{B}^{n}(0,1) $ we let $ A_{a}(\Gamma; C) = A_{a}(\Gamma) $ and $ A'_{a}(\Gamma;C) = A'_{a}(\Gamma) $.
For submanifolds of arbitrary codimension we obtain the following ABP inequality:
\[ABP arbitrary cod\] Suppose $ 1 \leq m \leq n $ are integers, $ h \geq 0 $, $ a > 0 $ and $ \Gamma $ is an $ (m,h) $ subset of $ \mathscr{C}_{1}(0) $ such that $ { \mathscr{H}^{m} }(\Gamma) < \infty $. For every closed set $ C $ in $ \mathbf{B}^{n}(0,1) $, if $ \varnothing \neq A_{a}(\Gamma; C) \subseteq \Gamma \times \mathbf{S}^{n} $, then $${ \mathscr{H}^{n} }(C) \leq \gamma \Big(1+a+\frac{1}{a}\Big)^{n-m}\Big(1 + a + \frac{h}{a}\Big)^{m} \int_{\Gamma^{(m)}}{ \mathscr{H}^{n-m} }\{ \eta : (z,\eta) \in A_{a}(\Gamma; C) \} d{ \mathscr{H}^{m} }z,$$ where $ \Gamma^{(m)} $ is the set of $ z \in \Gamma $ where $ \Gamma $ can be touched from $ n+1-m $ linearly independent directions by an open ball of $ { \mathbf{R}^{n+1}} $ and $ \gamma = 4^{n-m}(2m + 4h)^{m} $.
In case of hypersurfaces, the previous result can be refined as follows:
\[ABP\] Suppose $ m = n $ in \[ABP arbitrary cod\]. For every closed set $ C $ in $ \mathbf{B}^{n}(0,1) $, if $ \varnothing \neq A_{a}(\Gamma; C) \subseteq \Gamma \times \mathbf{S}^{n} $, then $$\label{ABP:2}
{ \mathscr{H}^{n} }(C) \leq \gamma (1+a+ha^{-1})^{n}(1+4a^{2})^{1/2}{ \mathscr{H}^{n} }(A'_{a}(\Gamma; C)),$$ where $ \gamma = (2n + 4h)^{n} $.
As a special case of \[ABP\] we obtain the following ABP estimate proved in [@MR2334822 2.1] for viscosity subsolutions of the minimal surface equation[^3] .
\[Savin ABP\] Let $ u $ be a continuous viscosity subsolution in $ \mathbf{U}^{n}(0,1) $ of the minimal surface equation, let $ \Gamma \subseteq { \mathbf{R}^{n+1}} $ be the graph of $ u $ and let $ 0 < a \leq 1 $.
Then there exists a small universal constant $ \gamma $ (depending only on $ n $) such that for every closed set $ C \subseteq \mathbf{B}^{n}(0,1) $ such that $ A'_{a}(\Gamma; C) \subseteq \mathbf{U}^{n}(0,1) $, $${ \mathscr{H}^{n} }(A'_{a}(\Gamma;C)) \geq \gamma { \mathscr{H}^{n} }(C).$$
Similar estimates have been proved in several other works since then; see, for example, [@MR3500837 3.5] [@MR3695374 6.5], [@MR3265174 3.5] and, in a Riemannian setting, [@MR2989991 Theorem 1.2].
It is well known, starting with the prioneering work of Krilov and Safanov, that the ABP estimate plays a fundamental role in the derivation of Harnack-type estimates. In this regard \[ABP\] can be employed to extend to $ (n,h) $ sets in $ \mathbf{R}^{n+1} $ the Measure-Estimate in [@MR2757359 6.1] (the same estimate is implicitly contained in the first part of the proof of the Harnack inequality in [@MR2334822 1.1]). This type of estimate is also known as weak Harnack inequality, see [@MR3695374 6.2]. Having the ABP estimate at our disposal the proof goes along the line of [@MR2334822 2.2].
\[Extrinsic weak Harnack inequality\] For every $ n \geq 1 $ there exist $ \alpha > 1 $ and $ 0 < \beta < 1 $ such that the following statement holds. For every $ \mu > 0 $ there exists an integer $ k \geq 1 $ so that if $ \Gamma $ is an $(n,h)$ subset of $ \mathcal{C}_{1}(0) $ with $ { \mathscr{H}^{n} }(\Gamma) < \infty $, $ 0 \leq h < \alpha^{-k-1} $, $$\Gamma \subseteq \{ x : x_{n+1} \leq 0 \} \quad \textrm{and} \quad \Gamma \cap \{ x : |x'|\leq 1/4, \; -\alpha^{-k-1}/48 \leq x_{n+1} \leq 0 \} \neq \varnothing,$$ then $${ \mathscr{L}^{n} }(\mathbf{B}^{n}(0, 1/3) \sim A'_{\alpha^{-1}}(\Gamma)) \leq \mu { \mathscr{L}^{n} }(\mathbf{B}^{n}(0, 1/3)).$$
We finally remark that the proof of \[ABP arbitrary cod\] and \[ABP\], though based on the same general idea of [@MR2334822 2.1], use a substantially different method to treat the arbitrary-codimension setting of \[ABP arbitrary cod\]. This method is based on the theory of curvature for arbitrary closed sets recently developed in [@2017arXiv170801549S] (see also [@MR534512], [@MR2031455] for earlier contributions in this direction). The proof of [@MR2334822 2.1] goes in the following way: firstly the problem is reduced to semi-concave functions by inf-convolution approximation; since such functions are differentiable on the contact set, it is possible to introduce an auxiliary function (*vertex map*), mapping each contact point into the vertex of the corresponding touching paraboloid; the vertex map is Lipschitz and its derivative can be suitably estimated using the assumptions on the elliptic operator $ F $; henceforth, an application of the classical area formula for Lipschitz maps gives the conclusion. In the present paper we also use a vertex map, but now its domain is a subset of the normal bundle of the $ (m,h) $ set $ \Gamma $, namely $ A_{a}(\Gamma; C) $. Then, in order to complete the proof following the idea of [@MR2334822 2.1], we need a suitable area formula for functions whose domain is a subset of the normal bundle, see \[general facts\], and suitable estimates for the differential of the vertex map, that can be deduced from the upper bound on the trace of the second fundamental form in \[trace and viscosity minimal sets\].
#### Notation. {#notation. .unnumbered}
The distance function from a closed set $ \Gamma $ in $ { \mathbf{R}^{n+1}} $ is $ \bm{\delta}_{\Gamma} $ and the nearest point projection onto $ \Gamma $ is $ \bm{\xi}_{\Gamma} $. If $T$ is a linear subspace of $ { \mathbf{R}^{n+1}} $ then $ T_{\natural} $ is the orthogonal projection onto $ T $.
If $ z \in { \mathbf{R}^{n}} \times { \mathbf{R}^{}} \cong { \mathbf{R}^{n+1}} $ we define $ z' \in { \mathbf{R}^{n}} $ so that If $ \rho > 0 $ and $ x \in { \mathbf{R}^{n}} $ we define $$\mathcal{C}_{\rho}(x) = { \mathbf{R}^{n+1}} \cap \{ z : |z'-x| < \rho \},$$ $$\mathbf{B}^{n}(x,\rho) = { \mathbf{R}^{n}} \cap \{y : | y-x| \leq \rho \}, \quad \mathbf{U}^{n}(x,\rho) = { \mathbf{R}^{n}} \cap \{y : | y-x| < \rho \}.$$
For every function $ f : D \rightarrow { \mathbf{R}^{}} $, where $ D \subseteq { \mathbf{R}^{n}} $, we let $$\Sigma(f) = \{ (x,f(x)) : x \in D \} \quad \textrm{and} \quad \Sigma^{+}(f) = \{ (x,y) : y > f(x) \}.$$
If $ a > 0 $ and $ x \in { \mathbf{R}^{n}} $ then a function $ P : { \mathbf{R}^{n}} \rightarrow { \mathbf{R}^{}} $ is a *paraboloid of center $ x $ and opening $ a $* if there exists $ t \in { \mathbf{R}^{}} $ such that $$P(y) = (a/2)|y-x|^{2} + t \quad \textrm{whenever $ y \in { \mathbf{R}^{n}} $}.$$
If $ \Gamma \subseteq { \mathbf{R}^{n+1}} $ is a closed set and $ f : { \mathbf{R}^{n}} \rightarrow { \mathbf{R}^{}} $ is continuous, we say that *$ f $ touches $ \Gamma $ from above* if and only if $$\Sigma(f) \cap \Gamma \neq \varnothing \quad \textrm{and} \quad \Sigma^{+}(f)\cap \Gamma = \varnothing.$$
Curvature of closed sets {#Curvature of minimal varieties}
========================
Suppose $ n \geq 1 $ is an integer and $ \Gamma \subseteq { \mathbf{R}^{n+1}} $ is closed.
For every $ r > 0 $ we define $$N_{r}(\Gamma)= (\Gamma \times \mathbf{S}^{n}) \cap \{ (z,\eta): \bm{\delta}_{\Gamma}(z+r\eta) = r \};$$ the *generalized normal bundle of $ \Gamma $* is then given by $$N(\Gamma) = \bigcup_{r > 0}N_{r}(\Gamma).$$
If $ 0 \leq m \leq n+1 $ the *$ m $-th stratum of $ \Gamma $* is $$\Gamma^{(m)} = \Gamma \cap \{ z : \dim \bm{\xi}_{\Gamma}^{-1}\{z\} = n+1-m \}.$$ It is not difficult to see that $ \Gamma^{(m)} $ is the set of $ z \in \Gamma $ where $ \Gamma $ can be touched from $ n+1-m $ linearly independent directions by an open ball of $ { \mathbf{R}^{n+1}} $; see [@2017arXiv170801549S 5.1-5.3].
We prove in [@2017arXiv170801549S 3.10(1)] that $ N_{r}(\Gamma) $ is a locally $ {(\mathscr{H}^{n},n)} $ rectifiable closed subset of $ { \mathbf{R}^{n+1}} \times \mathbf{S}^{n} $ and we deduce that $ N(\Gamma) $ is a countably $ {(\mathscr{H}^{n},n)} $ rectifiable Borel subset of $ { \mathbf{R}^{n+1}} \times \mathbf{S}^{n} $ since $ N_{r}(\Gamma) \subseteq N_{s}(\Gamma) $ if $ s < r $. We remark that $ N(\Gamma) $ coincides with the normal bundle introduced in [@MR2031455 §2.1] and it is well known that if $ \Gamma $ is a set of positive reach then $ N(\Gamma) $ coincides with the classical unit normal bundle, as defined in [@MR0257325 3.1.21], and it has locally finite $ { \mathscr{H}^{n} } $ measure (see [@2017arXiv170801549S 4.4]).
In [@2017arXiv170801549S 4.5-4.12] we prove that if $ \Gamma $ is an arbitrary closed subset of $ { \mathbf{R}^{n+1}} $ then for $ { \mathscr{H}^{n} } $ a.e. $(z,\eta) \in N(\Gamma) $ there exist a linear subspace $ T_{\Gamma}(z,\eta) $ of $ { \mathbf{R}^{n+1}} $ with $ \dim T_{\Gamma}(z,\eta) \geq 0 $ and a bilinear form $ Q_{\Gamma}(z,\eta) :T_{\Gamma}(z,\eta) \times T_{\Gamma}(z,\eta) \rightarrow { \mathbf{R}^{}} $ that coincides with the second fundamental form introduced in [@MR1021369 4.5] when $ \Gamma $ is a set of positive reach. Henceforth we refer to this bilinear form as *the second fundamental form of $ \Gamma $.* The principal curvatures $ - \infty < \kappa_{1}(z,\eta) \leq \ldots \leq \kappa_{n-1}(z,\eta) \leq \infty $ of $ \Gamma $ at $ (z,\eta) $ can be defined as $$\kappa_{m+1}(z,\eta) = \infty, \quad \textrm{$ \kappa_{1}(z,\eta), \ldots , \kappa_{m}(z,\eta) $ are the eigenvalues of $ Q_{\Gamma}(z,\eta) $}$$ where $ m = \dim T_{\Gamma}(z,\eta) $. They coincide $ { \mathscr{H}^{n} } $ almost everywhere in $ N(\Gamma) $ with the principal curvatures introduced in [@MR2031455 p. 244] by different methods, see [@2017arXiv170801549S 4.16]. We remark that the functions mapping $ { \mathscr{H}^{n} } $ almost all $ (z,\eta) \in N(\Gamma) $ into $ T_{\Gamma}(z,\eta)_{\natural} \in \operatorname{Hom}({ \mathbf{R}^{n+1}}, { \mathbf{R}^{n+1}}) $ and $ Q_{\Gamma}(z, \eta) \circ \bigodot_{2}T_{\Gamma}(z,\eta)_{\natural} \in \operatorname{Hom}(\bigodot_{2}{ \mathbf{R}^{n+1}}) $ are $ { \mathscr{H}^{n} } $ measurable.
For reader’s convenience in the next theorem we summarize two facts from the general theory developed in [@2017arXiv170801549S] that we employ in the proof of the ABP estimate.
\[general facts\] Let $ \Gamma $ be a closed subset of $ { \mathbf{R}^{n+1}} $. Then the following statements hold.
1. \[general facts:basis\] For every $ r > 0 $ and for $ { \mathscr{H}^{n} } $ a.e. $ (z,\eta) \in N_{r}(\Gamma) $ the approximate tangent cone $ \operatorname{Tan}^{n}({ \mathscr{H}^{n} }\operatorname{\llcorner}N_{r}(\Gamma), (z,\eta)) $ is an $ n $-dimensional plane and there exist $ u_{1}, \ldots , u_{n} \in { \mathbf{R}^{n+1}} $ such that $ \{ u_{1}, \ldots , u_{n}, \eta \} $ is an orthonormal basis of ${ \mathbf{R}^{n+1}}$ and the orthonormal vectors $$\bigg(\frac{1}{(1+\kappa_{i}(z,\eta)^{2})^{1/2}} u_{i}, \frac{\kappa_{i}(z,\eta)}{(1+\kappa_{i}(z,\eta)^{2})^{1/2}} u_{i} \bigg)$$ form a basis of $ \operatorname{Tan}^{n}({ \mathscr{H}^{n} }\operatorname{\llcorner}N_{r}(\Gamma), (z,\eta)) $ (if $ \kappa_{i}(z, \eta) = \infty $ then the corresponding vector equals $ (0, u_{i}) $).
2. \[general facts: Coarea\] If $f$ is a $ ({ \mathscr{H}^{n} }\operatorname{\llcorner}N(\Gamma)) $ integrable $ \overline{{ \mathbf{R}^{}}} $ valued function, then
& \_[N()|\^[(m)]{}]{} f(a,u) \_[i=1]{}\^[m]{} d[ \^[n]{} ]{}(a,u)\
& = \_[\^[(m)]{}]{}\_[{z} N(,z)]{}f d[ \^[n-m]{} ]{} d[ \^[m]{} ]{}z.
See [@2017arXiv170801549S 4.14, 5.6].
In order to combine this measure-theoretic theory with techniques developed in PDE’s theory, we have to analize the behaviour of the principal curvatures introduced above in order to produce useful estimates.
At this point it is worth to recall that if $ \kappa_{1} $ is the principal curvature of the graph $ \Gamma $ of the primitive of the (ternary) Cantor function, which is a convex $ \mathscr{C}^{1} $ function from $ { \mathbf{R}^{}} $ to $ { \mathbf{R}^{}} $, then, letting $ L = \bigcap_{r > 0}N_{r}(\Gamma) $, there exists $ M \subseteq L $ such that $ { \mathscr{H}^{1} }(M)> 0 $ and $$\kappa_{1}(z,\eta) = \infty \quad \textrm{for every $(z,\eta) \in M $.}$$ The reader may consult [@2017arXiv170801549S 5.11] for details on this example.
In order to prove the main result of this section, firstly we need a generalization of the barrier principle in [@MR3466806 7.1].
\[weak maximum principle\] Suppose $ 1 \leq m \leq n $ are integers, $ f : { \mathbf{R}^{n}} \rightarrow { \mathbf{R}^{}} $ is function pointwise differentiable of order $ 2 $ at $ 0 $ such that $ f(0) =0 $ and $ \operatorname{D}f(0) =0 $, $ h \geq 0 $, $ \Omega $ is an open subset of $ { \mathbf{R}^{n+1}} $, $ \Gamma $ is an $ (m,h) $ subset of $ \Omega $ such that $ 0 \in \Gamma $ and $$\Gamma \cap V \subseteq \{ z : z_{n+1}\leq f(z') \}$$ for some open neighbourhood $ V $ of $ 0 $. Then, denoting by $ \chi_{1} \geq \ldots \geq \chi_{n} $ the eigenvalues of $ \operatorname{D}^{2}f(0) $, it follows that $$\chi_{1} + \ldots + \chi_{m} \geq -h.$$
Let $ \epsilon > 0 $ and $ \psi(x) = \frac{1}{2}\operatorname{D}^{2}f(0)(x,x) + \epsilon |x|^{2} $ for $ x \in \mathbf{R}^{n} $. There exists $ r > 0 $ such that $ f(x)\leq \psi(x) $ for every $ x \in \mathbf{U}^{n}(0,r) $. By [@MR3466806 7.1], if $ \kappa_{1} \leq \ldots \leq \kappa_{n} $ are the principal curvatures of $ M = \{(x, \psi(x)) : x \in \mathbf{R}^{n} \} $ at $ (0,0) $ with respect to the unit normal that points into $ \{ z : z_{n+1} \leq \psi(z') \} $, then $$\kappa_{1} + \ldots + \kappa_{m} \leq h.$$ Since a standard and straightforward computation shows that $ \kappa_{i} = -\chi_{i} -\epsilon $ for $ i = 1, \ldots , n $, we obtain the conclusion letting $ \epsilon \to 0 $.
The main result of this section provides the crucial estimate for the trace of the second fundamental form of an $ (m,h) $ set. As the example of the primitive of the Cantor function clearly shows, this estimate cannot be expected to hold without a specific geometric hypothesis (as the weak maximum principle for $ (m,h) $ sets). The proof of the following theorem is adapted from [@2017arXiv170801549S 7.5], where certain stationary submanifolds (viscosity minimal sets) are treated.
\[trace and viscosity minimal sets\] If $ \Omega $ is an open subset in $ { \mathbf{R}^{n+1}} $, $\Gamma$ is an $ (m,h) $ subset of $ \Omega $ that is a countable union of sets with finite $ { \mathscr{H}^{m} } $ measure and if $ \kappa_{1} \leq \ldots \leq \kappa_{n} $ are the principal curvatures then $ \kappa_{m+1}(z, \eta) = \infty $ and $$\textstyle \sum_{i=1}^{m}\kappa_{i}(z,\eta) = \operatorname{trace}Q_{\overline{\Gamma}}(z, \eta) \leq h$$ for $ { \mathscr{H}^{n} } $ a.e. $ (z, \eta) \in N(\overline{\Gamma}) \cap \{ (z, \nu) : z \in \Gamma \} $. In particular, if $ \Gamma $ is the support of an $ m $-dimensional integral varifold in $ \Omega $ without boundary and with bounded variational mean curvature $ H $, then $$\textstyle \sum_{i=1}^{m}\kappa_{i}(z,\eta) = \operatorname{trace}Q_{\overline{\Gamma}}(z, \eta) = -H(z) \bullet \eta$$ for $ { \mathscr{H}^{n} } $ a.e. $ (z, \eta) \in N(\overline{\Gamma}) \cap \{ (z, \nu) : z \in \Gamma \} $.
Suppose $ A = \overline{\Gamma} $, $ \lambda > (2m-1) + 1 $ and the functions $ \bm{\nu}_{A} $ and $ \bm{\psi}_{A} $ are defined as in [@2017arXiv170801549S 3.1]. Given such $ \lambda $ we define for $ r > 0 $ the set $ N_{r} $ as in [@2017arXiv170801549S 3.10]. We recall from the proof of [@2017arXiv170801549S 4.14] that for $ { \mathscr{L}^{1} } $ a.e. $ r > 0 $ and for $ { \mathscr{H}^{n} } $ a.e. $ x \in N_{r} $, if $ \chi_{1} \leq \ldots \leq \chi_{n} $ are the eigenvalues of $ \operatorname{ap}\operatorname{D}\bm{\nu}_{A}(x)| \operatorname{Tan}^{n}({ \mathscr{H}^{n} }\operatorname{\llcorner}N_{r}, x) $ then $$\label{trace and viscosity minimal sets:1}
\kappa_{i}(\bm{\psi}_{A}(x)) = \chi_{i}(1-r\chi_{i})^{-1} \quad \textrm{for $ i = 1, \ldots , \dim T_{A}(\bm{\psi}_{A}(x)) $.}$$
We select $ 0 < r < (2h)^{-1} $ so that [@2017arXiv170801549S 3.10(3),(4)] hold for $ { \mathscr{H}^{n} } $ a.e. $ x \in N_{r} $; then we fix $ x \in N_{r} \cap \bm{\xi}_{A}^{-1}(\Gamma) $ satisfying the conclusions of [@2017arXiv170801549S 3.10(2),(3),(4)]. We assume $ \bm{\xi}_{A}(x) =0 $ and let $ T = \{ v : v \bullet \bm{\nu}_{A} (x) =0 \} $. It follows that there exists a Lipschitzian function $ f : T \rightarrow T^{\perp} $ such that $ f $ is pointwise differentiable of order $ 2 $ at $ 0 $, $ \operatorname{D}f(0) =0 $, $$\operatorname{D}^{2}f(0)(u,v) \bullet \bm{\nu}_{A}(x) = -\operatorname{ap}\operatorname{D}\bm{\nu}_{A}(x)(u) \bullet v \quad \textrm{whenever $ u , v \in T $,}$$ $$W \cap \bm{\delta}_{A}^{-1}\{r\} = W \cap \{ \chi + f(\chi) : \chi \in T \} \quad \textrm{for some neighbourhood $ W $ of $ x $.}$$ Choose $ 0 < s < r/2 $ such that $ \mathbf{U}(x,s) \subseteq W $ and let $ g(\zeta) = f(\zeta) - x $ for $ \zeta \in T $, $$U= T_{\natural}\big(\mathbf{U}(x,s) \cap \{ \chi + f(\chi) : \chi \in T \}\big), \quad V = \{ y-x: y \in T_{\natural}^{-1}(U) \cap \mathbf{U}(x,s) \}.$$ Then $ V $ is an open neighbourhood of $ 0 $ and $$V \cap A \subseteq \{ z : z \bullet \bm{\nu}_{A}(z) \leq g(T_{\natural}(z)) \bullet \bm{\nu}_{A}(z) \}.$$ This inclusion can be proved as follows: if there was $y \in \mathbf{U}(x,s) \cap T_{\natural}^{-1}[U]$ such that $y-x \in A $ and $ y \bullet \bm{\nu}_{A}(x) > f(T_{\natural}(y))\bullet \bm{\nu}_{A}(x) $ then, noting that $ \bm{\nu}_{A}(x) = r^{-1}x $, $$T_{\natural}(y) + f(T_{\natural}(y)) \in \mathbf{U}(x,s) \cap \bm{\delta}_{A}^{-1}\{r\}, \quad |T_{\natural}(y) + f(T_{\natural}(y))-y| < r,$$ we would conclude $$|T_{\natural}(y) + f(T_{\natural}(y))-(y-x)| = r - (y-f(T_{\natural}(y)))\bullet \bm{\nu}_{A}(x)< r$$ which is a contradiction. Let $ \chi_{1} \leq \ldots \leq \chi_{n} $ be the eigenvalues of $\operatorname{ap}\operatorname{D}\bm{\nu}_{A}(x)|T$. Then $ 1 - \chi_{1}r, \ldots , 1 - \chi_{n}r $ are the eigenvalues of $ \operatorname{ap}\operatorname{D}\bm{\xi}_{A}(x)|T$ by [@2017arXiv170801549S 3.5] and $ -\chi_{1}, \ldots , - \chi_{n} $ are the eigenvalues of $ \operatorname{pt}\operatorname{D}^{2} g(0) \bullet \bm{\nu}_{A}(x) $. Therefore we apply a rotated version of \[weak maximum principle\] to conclude that $$\label{trace and viscosity minimal sets:2}
\chi_{1} + \ldots + \chi_{m} \leq h.$$ Since $ \chi_{j} \geq -(\lambda -1)^{-1}r^{-1}$ whenever $ j = 1 , \ldots , n $ by [@2017arXiv170801549S 3.10(2)], we get $$\chi_{j} - (m-1)(\lambda-1)^{-1}r^{-1} \leq \sum_{i=1}^{m} \chi_{i} \leq h,$$ whence we conclude $ \chi_{j} < r^{-1} $ whenever $ j = 1 , \ldots , m $. Therefore, $$\| \textstyle \bigwedge_{m}\big(({ \mathscr{H}^{n} }\operatorname{\llcorner}N_{r},n)\operatorname{ap}\operatorname{D}\bm{\xi}_{A}(x)\big) \| > 0.$$ We remind that the assertions proved so far hold for ${ \mathscr{H}^{n} }$ a.e. $ x \in N_{r} \cap \bm{\xi}_{A}^{-1}(\Gamma) $ and for $ { \mathscr{L}^{1} } $ a.e. $ 0 < r < (2h)^{-1} $.
We prove now the following *Lusin $(N)$ condition*: if $ S \subseteq \Gamma $ and $ { \mathscr{H}^{m} }(S \cap A^{(m)}) =0 $ then $ { \mathscr{H}^{n} }(N(A) \cap \{ (x,\nu) : x \in S\} ) =0 $. Using [@2017arXiv170801549S 3.10(1), 4.3, 5.3] we get that $$A \cap \{ x : { \mathscr{H}^{n-m} }(\bm{\xi}_{A}^{-1}\{x\} \cap N_{r} ) >0 \} \subseteq \bigcup_{i=0}^{m}A^{(i)}$$ for every $ r > 0 $ and, noting that $ { \mathscr{H}^{m} }(A^{(i)}) =0 $ for $ i = 0, \ldots , m-1 $ by [@2017arXiv170801549S 5.2], we deduce $${ \mathscr{H}^{m} }\big( S \cap \{ x : { \mathscr{H}^{n-m} }(\bm{\xi}_{A}^{-1}\{x\} \cap N_{r} ) >0 \} \big) =0.$$ Noting [@2017arXiv170801549S 3.10(1)(2)], we apply [@2017arXiv170801549S 7.4] with $ W $ and $ f $ replaced by $ N_{r} \cap \bm{\xi}_{A}^{-1}(\Gamma) $ and $ \bm{\xi}_{A} $ to get $${ \mathscr{H}^{n} }(\bm{\xi}_{A}^{-1}(S) \cap N_{r} ) =0 \quad \textrm{for $ { \mathscr{L}^{1} } $ a.e.\ $ 0 < r < (2h)^{-1} $,}$$ whence the desired Lusin condition follows from [@2017arXiv170801549S 4.3].
By [@2017arXiv170801549S 5.2, 5.9] it follows that $ \dim T_{A}(z,\eta) = m $ and $ \kappa_{m+1}(z,\eta) = \infty $ for $ { \mathscr{H}^{n} } $ a.e. $ (z, \eta) \in N(A) \cap \{ (z, \nu) : z \in \Gamma \} $. Moreover we combine and to infer that $$\label{trace and viscosity minimal sets:3}
\sum_{i=1}^{m}\frac{\kappa_{i}(\bm{\psi}_{A}(x))}{1 + r \kappa_{i}(\bm{\psi}_{A}(x))} \leq h$$ for $ { \mathscr{H}^{n} } $ a.e. $ x \in N_{r} \cap \bm{\xi}_{A}^{-1}(\Gamma) $ and for $ { \mathscr{L}^{1} } $ a.e. $ 0 < r < (2h)^{-1} $. We choose a sequence $ r_{i} > 0 $ converging to $ 0 $ such that if $ M_{i} $ is the set of $ x \in N_{r_{i}} \cap \bm{\xi}_{A}^{-1}(\Gamma) $ such that is satisfied, then $ { \mathscr{H}^{n} }(N_{r} \cap \bm{\xi}_{A}^{-1}(\Gamma) \sim M_{i}) =0 $ for every $ i \geq 1 $. Then we readily infer that $$\operatorname{trace}Q_{A}(z,\eta) \leq h$$ for every $ (z,\eta) \in \bigcap_{i=1}^{\infty}\bigcup_{j=i}^{\infty}\bm{\psi}_{A}(M_{i}) $ and, noting that $ \bm{\psi}_{A}(N_{r}) \subseteq \bm{\psi}_{A}(N_{s}) $ if $ s < r $, we additionally get that $$\textstyle { \mathscr{H}^{n} }\Big( N(A) \cap \{ (z, \nu) : z \in \Gamma \} \sim \bigcap_{i=1}^{\infty}\bigcup_{j=i}^{\infty}\bm{\psi}_{A}(M_{i}) \Big) =0.$$
If $ \Gamma $ is the support of an $ m $ dimensional integral varifold in $ \Omega $ without boundary and with mean curvature bounded in $ {\mathbf{L}^{\infty}} $ by $ h $, then, noting that $ \Gamma $ is an $ (m,h) $ subset of $ \Omega $ by [@MR3466806 2.8], we may achieve the conclusion using the Lusin $(N)$ condition above in combination with [@2017arXiv170801549S 5.2, 5.9] and [@MR2472179 4.2].
We may conjecture that \[trace and viscosity minimal sets\] could also be proved for classes of *singular submanifolds of possibly unbounded mean curvature*. For example, if $ \Gamma $ is the support of an $ m $ dimensional integral varifold in $ { \mathbf{R}^{n+1}} $ without boundary and with mean curvature $ H $ in $ {\mathbf{L}^{m}} $, is it true that $ \kappa_{m+1}(z,\eta) = \infty $ and $$\textstyle \sum_{i=1}^{m}\kappa_{i}(z,\eta) = - H(z) \bullet \eta$$ for $ { \mathscr{H}^{n} } $ a.e. $ (z,\eta) \in N(\Gamma) $? More generally, what can be concluded in case of rectifiable varifolds with a uniform lower bound on the density? To treat these cases the method of \[trace and viscosity minimal sets\] is not powerful enough and new refined tools seem to be necessary.
Proof of \[ABP arbitrary cod\] and \[ABP\]
==========================================
We notice that $ A_{a}(\Gamma; C) $, $ A'_{a}(\Gamma; C) $ and $$\{ z : \textrm{$(z, \eta) \in A_{a}(\Gamma; C)$ for some $ \eta \in \mathbf{S}^{n} $} \}$$ are closed subset of $ \overline{\Gamma} \times \mathbf{S}^{n} $, $ \mathbf{B}^{n}(0,1) $ and $ \overline{\Gamma} $ respectively.
Clearly, for every $ (w,\eta) \in A_{a}(\Gamma; C) $ there exists a unique $ x \in C $ such that $ w \in \Sigma(P_{a,x}) $ and $ \eta \in \operatorname{Nor}(\Sigma(P_{a,x}),w) $; in fact $$x = w' + a^{-1}\eta_{n+1}^{-1}\eta'.$$ Henceforth, we define the map $ \Psi : A_{a}(\Gamma; C) \rightarrow C $ so that $ \Psi(w,\eta)= w' + \frac{1}{a\eta_{n+1}}\eta' $ and we notice that $ \Psi[A_{a}(\Gamma; C)]= C $.
It is not difficult to check that[^4] $$\bm{\delta}_{\overline{\Gamma}}\big(w + (a\eta_{n+1})^{-1}\eta\big) = (a\eta_{n+1})^{-1}$$ whenever $ (w, \eta) \in A_{a}(\Gamma; C) $. Henceforth $ A_{a}(\Gamma; C) \subseteq N_{a^{-1}}(\overline{\Gamma}) $ and, if $ \kappa_{1}, \ldots , \kappa_{n} $ are the principal curvatures of $ \overline{\Gamma} $, it follows from [@2017arXiv170801549S 4.12] that $$\label{ABP arbitrary cod:1}
\kappa_{i}(w, \eta) \geq - a\eta_{n+1}$$ for $ { \mathscr{H}^{n} } $ a.e. $ (w, \eta) \in A_{a}(\Gamma; C) $ and $ i = 1, \ldots , n $. Employing \[trace and viscosity minimal sets\] we deduce that $$\label{ABP arbitrary cod:2}
\kappa_{i}(w, \eta) \leq (m-1) a \eta_{n+1} + h, \quad \kappa_{m+1}(w,\eta) = \infty,$$ for $ { \mathscr{H}^{n} } $ a.e. $ (w,\eta) \in A_{a}(\Gamma; C) $.
We compute now the jacobian of $ \Psi $ using the orthonormal basis provided by \[general facts\]. For $ { \mathscr{H}^{n} } $ a.e. $ (w,\eta) \in A_{a}(\Gamma; C) $, if $ u_{1}, \ldots , u_{n} $ are the orthonormal vectors provided by \[general facts\], noting that $ \eta_{n+1}^{-1} \leq (1+4a^{2})^{1/2}\leq 1+ 2a $ and using -, we can easily estimate $$| \operatorname{D}\Psi(w,\eta)(0,u_{i})| \leq \frac{1}{a}\Big( \frac{1}{\eta_{n+1}} + \frac{|\eta'|}{\eta_{n+1}^{2}}\Big) \leq \frac{1}{a\eta_{n+1}} + \frac{2}{\eta_{n+1}}\leq 4 + 4a + \frac{1}{a}$$ for $ i = m+1, \ldots , n $, and
& |(w,)(u\_[i]{}, \_[i]{}(w,) u\_[i]{})| 1 + +\
& 1 + + 2 (1+a+),
for $ i = 1, \ldots m $, where $ \gamma = 2m + 4h $. Therefore for $ { \mathscr{H}^{n} } $ a.e. $ (w,\eta) \in A_{a}(\Gamma; C) $, we get
& \[(w,)|\^[n]{}([ \^[n]{} ]{}A\_[a]{}(; C) , (w,))\]\
& 4\^[n-m]{}\^[m]{} (1+a+)\^[n-m]{}(1 + a + )\^[m]{} \_[i=1]{}\^[m]{}.
Now we employ [@MR0257325 3.2.22(3)], \[general facts\] and the Lusin $(N)$ condition obtained in the proof of \[trace and viscosity minimal sets\] to conclude
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where $ \gamma_{1} = 4^{n-m}\gamma^{m} \Big(1+a+\frac{1}{a}\Big)^{n-m}\Big(1 + a + \frac{h}{a}\Big)^{m} $.
\[ABP : aux lemma\] Suppose $ \Gamma \subseteq \mathscr{C}_{1}(0) $ is relatively closed such that $ { \mathscr{H}^{n} }(\Gamma) < \infty $, $ a > 0 $, $ C \subseteq \mathbf{B}^{n}(0,1) $ is closed and $$B= \{ z : \textrm{$(z, \eta) \in A_{a}(\Gamma; C)$ for some $ \eta \in \mathbf{S}^{n} $} \}.$$
Then $${ \mathscr{H}^{n} }(B) \leq (1+4a^{2})^{1/2}{ \mathscr{H}^{n} }(A'_{a}(\Gamma;C)).$$
If $ z \in B $, $ x \in C $ and $ \nu \in { \mathbf{R}^{n+1}} $ such that $ z \in \Sigma(P_{a,x}) $ and $ \nu \in \operatorname{Nor}(\Sigma(P_{a,x}),z) $ with $ \nu_{n+1}> 0 $, then elementary considerations imply that for every $ v \in { \mathbf{R}^{n+1}} $ with $ v \bullet \nu > 0 $ there are $ \epsilon > 0 $ and $ r > 0 $ such that $$\{ y : | s(y-z)-v|< \epsilon \; \textrm{for some $ s > 0 $} \} \cap \mathbf{U}(z,r) \subseteq \Sigma^{+}(P_{a,x}),$$ whence we deduce that $ \operatorname{Tan}^{n}({ \mathscr{H}^{n} }\operatorname{\llcorner}B,z) \subseteq { \mathbf{R}^{n+1}} \cap \{ w : w \bullet \nu \leq 0 \} $ by [@MR0257325 3.2.16]. Since $ B $ is $ n $ rectifiable by [@2017arXiv170309561M 4.12], we deduce from [@MR0257325 3.2.19] that for $ { \mathscr{H}^{n} } $ a.e. $ z \in B $ there exists $ x \in C $ such that $ z \in \Sigma(P_{a,x}) $ and $$\operatorname{Tan}^{n}({ \mathscr{H}^{n} }\operatorname{\llcorner}B,z) = \operatorname{Tan}(\Sigma(P_{a,x}),z).$$ Therefore for $ { \mathscr{H}^{n} } $ a.e. $ z \in B $ there exists $ x \in C $ and an orthonormal basis $ e_{1}, \ldots , e_{n} $ of $ { \mathbf{R}^{n}} $ such that $$e_{n} = \nabla P_{a,x}(z')/|\nabla P_{a,x}(z')| \quad \textrm{if $ \nabla P_{a,x}(z') \neq 0 $},$$ and $$\Big\{(e_{1}, 0),\; \ldots , \; (e_{n-1},0),\; \frac{(e_{n}, |\nabla P_{a,x}(z')|)}{(1 + |\nabla P_{a,x}(z')|^{2})^{1/2}}\Big\}$$ is an orthonormal basis of $ \operatorname{Tan}^{n}({ \mathscr{H}^{n} }\operatorname{\llcorner}B,z) $. If $ p : B \rightarrow { \mathbf{R}^{n}} $ is given by $ p(z)= z' $ for $ z \in B $, noting that $ p $ is injective, the conclusion can be easily obtained applying [@MR0257325 3.2.20] with $ W $ and $ f $ replaced by $ B $ and $ p $.
Combine \[ABP arbitrary cod\] and \[ABP : aux lemma\].
Proof of \[Extrinsic weak Harnack inequality\] {#section: A measure estimate}
==============================================
\[measure to point estimate\] For every $ n \geq 1 $ there exist $ \alpha > 1 $ and $ 0 < \beta < 1 $ such that if $ 0 \leq h < a \leq \alpha^{-1} $, $ \Gamma $ is an $ (n,h) $ subset of $ \mathcal{C}_{1}(0) $ with $ { \mathscr{H}^{n} }(\Gamma) < \infty $, $ x_{0} \in \mathbf{U}^{n}(0,1) $, $ r > 0 $ such that $$\mathbf{B}^{n}(x_{0},r) \subseteq \mathbf{U}^{n}(0,1) \quad \textrm{and} \quad A'_{a}(\Gamma) \cap \mathbf{U}^{n}(x_{0},r) \neq \varnothing,$$ then $${ \mathscr{L}^{n} }(A'_{\alpha a}(\Gamma) \cap \mathbf{U}^{n}(x_{0}, r/8)) \geq \beta r^{n}.$$
Let $ x_{1} \in \mathbf{B}^{n}(0,1) $ such that $ \Sigma(P_{a,x_{1}}) \cap \Gamma \cap \mathcal{C}_{r} (x_{0}) \neq \varnothing $. For every $ \gamma > 1 $ we let $$\phi(t) =
\begin{cases}
- \gamma^{-1}(16^{\gamma}-1) & \textrm{if $ 0 \leq t \leq 1/16 $}\\
-\gamma^{-1}(t^{-\gamma} - 1) & \textrm{if $ 1/16 \leq t \leq 1 $}\\
0 & \textrm{if $ t \geq 1 $} \\
\end{cases}$$ and $ \psi(x) = P_{a,x_{1}}(x) + a r^{2} \phi(|x-x_{0}|/r) $ for every $ x \in { \mathbf{R}^{n}} $.
Firstly, we prove the existence of a number $ 0 < \gamma < \infty $ depending only on $ n $ such that if $ h < a \leq (16^{\gamma + 1} + 2)^{-1} $ then[^5] $$\label{measure to point estimate:1}
\operatorname{trace}Q_{\Sigma(\psi)}(z) \bullet (-\nabla \psi(z'),1) < -h(1 + |\nabla \psi(z')|^{2})^{1/2}$$ for every $ z \in \Sigma(\psi) $ with $ r/16 < | z'-x_{0}| < r $. We fix $ x \in { \mathbf{R}^{n}} $ with $ r/16 < | x-x_{0}| < r $ and let $ z = (x, \psi(x)) $ and $ t = |x-x_{0}|/r $. We notice that $$|\nabla \psi(x)| \leq a(2+16^{\gamma+ 1}) \leq 1$$ and if $ e_{1}, \ldots , e_{n} $ is an orthonormal basis of $ { \mathbf{R}^{n}} $ such that $ e_{n} = \nabla \psi(x)/ |\nabla \psi(x) | $ if $\nabla \psi(x) \neq 0 $, then $$\Big\{(e_{1}, 0), \ldots , (e_{n-1}, 0), \frac{(e_{n}, |\nabla \psi(x)|)}{(1+|\nabla \psi(x)|^{2})^{1/2}} \Big\}$$ is an orthonormal basis of $ \operatorname{Tan}(\Sigma(\psi),z) $. Henceforth, letting $ \zeta = (1+|\nabla \psi(x)|^{2})^{-1/2} $ and $ \nu = \zeta (-\nabla \psi(z'),1) $ and noting that $ 1/2 \leq \zeta \leq 1 $, we compute
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Since $ 0 \leq h < a $, we can evidently choose $ \gamma> 0 $ depending only on $ n $ so that the last line in the previous estimate is less than $ -h $ and is proved.
Next we prove that if $ h <a \leq (16^{\gamma + 1}+2)^{-1} $ then there exists $ z \in \Gamma \cap \overline{\mathcal{C}_{r/16}(x_{0})} $ such that $$\label{measure to point estimate:2}
0 \leq P_{a,x_{1}}(z') - z_{n+1} \leq \gamma^{-1}(16^{\gamma} - 1) a r^{2}.$$ In fact, since $ \Sigma^{+}(\psi) \cap \Gamma \cap \mathcal{C}_{r}(x_{0}) \neq \varnothing $ and $ \Sigma^{+}(\psi + ar^{2}\gamma^{-1}(16^{\gamma} - 1)) \cap \overline{\Gamma} = \varnothing $, we infer that there exists $ 0 < t \leq ar^{2}\gamma^{-1}(16^{\gamma} - 1) $ such that $ \psi + t $ touches $ \overline{\Gamma} $ from above. Since $ \psi(x) + t > P_{a,x_{1}}(x) $ if $ |x-x_{0}| \geq r $, we infer that $$\varnothing \neq \Sigma(\psi + t) \cap \Gamma \subseteq \mathcal{C}_{r}(x_{0});$$ moreover, it follows from \[weak maximum principle\] and that $$\Sigma(\psi + t) \cap \Gamma \cap \big(\mathcal{C}_{r}(x_{0}) \sim \overline{\mathcal{C}_{r/16}(x_{0})} \big) = \varnothing.$$ If $ z \in \Sigma(\psi + t) \cap \Gamma \cap \overline{\mathcal{C}_{r/16}(x_{0})} $ then $$z_{n+1}=\psi(z')+t=P_{a,x_{1}}(z') - ar^{2}\gamma^{-1}(16^{\gamma}-1) + t \leq P_{a,x_{1}}(z')$$ and follows.
We fix now $ a $, $ h $ and $ z $ as in and let $ \theta > 0 $ to be chosen later depending only on $ \gamma $. For every $ y \in \mathbf{B}^{n}(z', r/64) $ we select $ c_{y} \in { \mathbf{R}^{}} $ such that the paraboloid $$Q_{y}(x) = P_{a,x_{1}}(x) + \theta\frac{a}{2}|x-y|^{2}+c_{y}$$ touches $ \overline{\Gamma} $ from above. Noting that $ Q_{y}(z') \geq z_{n+1} $ and using , we choose $ \theta $ large enough depending on $ \gamma $ so that $ Q_{y}(x) > P_{a,x_{1}}(x) $ whenever $ | x - z'| \geq r/16 $, which implies that $$\varnothing \neq \Sigma(Q_{y}) \cap \Gamma \subseteq \mathcal{C}_{r/16}(z') \subseteq \mathcal{C}_{r/8}(x_{0}).$$ Since $ Q_{y} $ is a paraboloid of center $ (\theta/(1+\theta))y + (1/(1+\theta))x_{1} $ and opening $ (\theta+1)a $ we can apply \[ABP\] with $ C $ replaced by $$\bigg\{ \frac{\theta}{1 + \theta}y + \frac{1}{\theta + 1}x_{1} : y \in \mathbf{B}^{n}(z', r/64) \bigg\} \subseteq \mathbf{B}^{n}(0,1)$$ to obtain the conclusion.
The proof of \[measure to point estimate\] closely follows [@MR2334822 2.2], where continuous viscosity super-solutions of certain elliptic equations (including the minimal surface equation) are treated.
This is a standard argument that combines the Vitali-type covering lemma in [@MR2334822 2.3] with the measure-estimate \[measure to point estimate\].
Let $ \alpha $ and $ \beta $ be as in \[measure to point estimate\], $ \epsilon = (48\alpha^{k+1})^{-1} $, $ \mu > 0 $ and let $ k $ be a positive integer to be chosen later depending only on $ n $ and $ \mu $. If $ P $ is the paraboloid of center $ 0 $ and opening $ 48\epsilon $ touching $ \overline{\Gamma} $ from above, we observe that the set of touching points of $ P $ with $\overline{\Gamma}$ is contained in $ \mathcal{C}_{1/3}(0) $. For every integer $ j \geq 0 $ we let $ F_{j} = \mathbf{B}^{n}(0,1/3) \cap A'_{48\epsilon\alpha^{j}}(\Gamma) $ and, noting that $ F_{0} \neq \varnothing $, we combine [@MR2334822 2.3] (or [@MR2757359 6.4]) and \[measure to point estimate\] to conclude $${ \mathscr{L}^{n} }(\mathbf{B}^{n}(0,1/3) \sim A'_{\alpha^{-1}}(\Gamma)) \leq (1-\beta_{1})^{k}{ \mathscr{L}^{n} }(\mathbf{B}^{n}(0,1/3)),$$ where $ \beta_{1} $ depends only on $ \beta $. Now we choose $ k $ so that $ (1-\beta_{1})^{k} \leq \mu $.
[[Sav]{}17b]{}
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Institut für Mathematik, Universität Augsburg, Universitätsstr. 14, 86159, Augsburg, Germany, [email protected]
[^1]: This definition is equivalent to [@MR3466806 2.1], by [@MR3466806 8.1].
[^2]: For basic notation, see the end of the introduction.
[^3]: The results in [@MR2334822] are actually proved for a class of elliptic operators $ F $ that are uniformly elliptic in a neighbourhood of the origin and $ F(0,p,z,x) =0 $ whenever $ p $, $ z $ and $ x $ are close to $ 0 $.
[^4]: In fact, a straightforward computation shows that *if $ n \geq 1 $ is an integer, $ P $ is a paraboloid of center $ 0 $ and opening $ a > 0 $, $ x \in { \mathbf{R}^{n}} $ and $ \eta \in \operatorname{Nor}(\Sigma(P), (x, P(x)) ) $ with $ |\eta |= 1 $ and $ \eta_{n+1} > 0 $, then $$\sup\{s : \bm{\delta}_{\Sigma(P)}((x,P(x)) + s \eta ) = s \} = (a\eta_{n+1})^{-1}.$$*
[^5]: $Q_{\Sigma(\psi)}$ is the second fundamental of the $ \Sigma(\psi)$.
|
---
author:
- 'Mridul Agarwal, Vaneet Aggarwal, Arnob Ghosh, and Nilay Tiwari [^1]'
bibliography:
- 'refs.bib'
title: Reinforcement Learning for Mean Field Game
---
[^1]: The author names are written in an alphabetical ordering.
M. Agarwal and V. Aggarwal are with Purdue University. The work of A. Ghosh and N. Tiwari was performed when they were at Purdue University.
|
---
abstract: 'We investigate weak recognizability of deterministic languages of infinite trees. We prove that for deterministic languages the Borel hierarchy and the weak index hierarchy coincide. Furthermore, we propose a procedure computing for a deterministic automaton an equivalent minimal index weak automaton with a quadratic number of states. The algorithm works within the time of solving the emptiness problem.'
address: Warsaw University
author:
- Filip Murlak
title: Weak index versus Borel rank
---
[Filip Murlak]{}
[^1]
Introduction
============
Finite automata on infinite trees are one of the basic tools in the verification of non-terminating programs. Practical applicability of this approach relies on the simplicity of the automata used to express the specifications. On the other hand it is convenient to write the specifications in an expressive language, e. g. $\mu$-calculus. This motivates the search for automatic simplifications of automata. An efficient, yet reasonably expressive, model is offered by weak alternating automata. It was essentially showed by Rabin [@rabin1] that a language $L$ can be recognized by a weak automaton if and only if both $L$ and $L^\complement$ can be recognized by nondeterministic Büchi automata. Arnold and Niwiński [@an] proposed an algorithm that, given two Büchi automata recognizing a language and its complement, constructs a doubly exponential alternation free $\mu$-calculus formula defining $L$, which essentially provides an equally effective translation to a weak automaton. Kupferman and Vardi [@kv] gave an immensely improved construction that involves only quadratic blow-up.
A more refined construction could also simplify an automaton in terms of different complexity measures. A measure that is particularly important for theoretical and practical reasons is the Mostowski–Rabin index. This measure reflects the alternation depth of positive and negative events in the behaviour of a verified system. The index orders automata into a hierarchy that was proved strict for deterministic [@wagner], nondeterministic [@klony], alternating [@brad; @lenzi], and weak alternating automata [@most]. Computing the least possible index for a given automaton is called the index problem. Unlike for $\omega$-words, where the solution was essentially given already by Wagner [@wagner], for trees this problem in its general form remains unsolved. For deterministic languages, Niwiński and Walukiewicz gave algorithms to compute the deterministic and nondeterministic indices [@kwiatek; @hie].
The theoretical significance of the weak index is best reflected by its coincidence with the quantifier alternation depth in the weak monadic second order logic [@most]. Further interesting facts are revealed by the comparison with the Borel rank. In 1993 Skurczyński gave examples of $\Pi^0_n$ and $\Sigma^0_n$-complete languages recognized by weak alternating automata with index $(0,n)$ and $(1,n+1)$ accordingly [@skurcz]. In [@wata] it was shown that weak $(0,n)$-automata can only recognize $\Pi^0_n$ languages (and dually, $(1,n+1)$-automata can only recognize $\Sigma^0_n$ languages), and it was conjectured that the weak index and the Borel hierarchies actually coincide. Here we prove that the conjecture holds for deterministic languages. Consequently, the algorithm calculating the Borel rank for deterministic languages [@split] can be also used to compute the weak index. Since all deterministic languages are at the first level of the alternating hierarchy, this completes the picture for the deterministic case. We also provide an effective translation to a weak automaton with a quadratic number of states and the minimal index.
Automata
========
We will be working with deterministic and weak automata, but to have a uniform framework, we first define automata in their most general alternating form.
A [*parity game*]{} is a perfect information game of possibly infinite duration played by two players, Adam and Eve. We present it as a tuple $(V_\exists, V_\forall, E, v_0,\mathrm{rank})$, where $V_\exists $ and $V_\forall$ are (disjoint) sets of positions of Eve and Adam, respectively, $E \subseteq V \times V $ is the relation of possible moves, with $V = V_\exists \cup V_\forall $, $p_0 \in V$ is a designated initial position, and $ \mathrm{rank} : V \to \{0,1, \ldots, n\}$ is the ranking function.
The players start a play in the position $v_0$ and then move a token according to relation $E$ (always to a successor of the current position), thus forming a path in the graph $(V, E)$. The move is selected by Eve or Adam, depending on who is the owner of the current position. If a player cannot move, she/he looses. Otherwise, the result of the play is an infinite path in the graph, $v_0,v_1,v_2,\ldots $. Eve wins the play if the highest rank visited infinitely often is even, otherwise Adam wins.
An [*alternating automaton*]{} $A=\langle \Sigma, Q_\exists, Q_\forall, q_0, \delta, \mathrm{rank}\rangle$, consists of a finite input alphabet $\Sigma$, a finite set of states $Q$ partitioned into existential states $Q_\exists$ and universal states $Q_\forall$ with a fixed initial state $q_0$, a transition relation $\delta \subseteq Q \times \Sigma \times \{0,1, \varepsilon\}\times Q$, and a ranking function $\mathrm{rank}:Q \to \omega$. Instead of $(p,\sigma, d, q)\in \delta$, one usually writes $p\stackrel{\sigma,d}{\longrightarrow}q$.
An input tree $t$ is accepted by $A$ iff Eve has a winning strategy in the parity game $\langle Q_\exists \times \{0,1\}^*, Q_\forall \times \{0,1\}^*, (q_0,\varepsilon), E, \mathrm{rank'}\rangle$, where $E= \{((p,v),(q,vd))\colon v\in \mathrm{dom}(t),\; (p, t(v), d, q)\in \delta\}$ and $\mathrm{rank'} (q,v) = \mathrm{rank}(q)$. The computation tree of $A$ on $t$ is obtained by unravelling the graph above from the vertex $(q_0, \varepsilon)$ and labelling the node $(q_0,\varepsilon), (q_1, d_1), (q_2, d_2), \ldots, (q_n,d_n)$ with $q_n$. The result of the parity game above only depends on the computation tree.
An automaton is called [*deterministic*]{} iff Eve has no choice at all, and Adam can only choose the direction: left or right (no $\varepsilon$-moves). Formally, it means that $Q_\exists=\emptyset$, and $\delta: Q \times \Sigma \times\{0,1\} \to Q$. For deterministic automata, the computation tree is a full binary tree. The transitions are often written as $p\stackrel{\sigma}{\longrightarrow}q_0,q_1$, meaning $p\stackrel{\sigma, d}{\longrightarrow}q_d$ for $d=0,1$.
A [*weak automaton*]{} is an alternating automaton satisfying the condition $$p \stackrel{\sigma,d}{\longrightarrow} q \quad \implies \quad \mathrm{rank}\, p \leq \mathrm{rank}\, q\,.$$ A more elegant definition of the class of weakly recognizable languages is obtained by using [*weak parity games*]{} in the definition of acceptance by alternating automata. In those games Eve wins a play if the highest rank used at least once is even. For the purpose of the following lemma, let us call the first version [*restricted alternating automata*]{}. Later, we will stick to the second definition.
For every $L$ it holds that $L$ is recognized by a restricted alternating $(\iota,\kappa)$-automaton iff it is recognized by a weak alternating $(\iota,\kappa)$-automaton.
Every restricted automaton can be transformed into an equivalent weak automaton by simply changing the acceptance condition to weak. Let us, then, concentrate on the converse implication.
Fix a weak automaton $A$ using ranks $(\iota, \kappa)$. To construct a restricted automaton we will take one copy of $A$ for each rank: $A^{(\iota)}, A^{(\iota+1)}, \ldots, A^{(\kappa)}$. By $q^{(i)}$ we will denote the counterpart of $A$’s state $q$ in $A^{(i)}$. We set $\mathrm{rank}\, q^{(i)} = i$. We want the number of the copy the computation is in to reflect the highest rank seen so far. To obtain that, we set the initial state of the new automaton to $q_0^{(\mathrm{rank}\, q_0)}$, and for each $i$ and each transition $p\stackrel{\sigma, d}{\longrightarrow}q$ in $A$ we add a transition $p^{(i)}\stackrel{\sigma, d}{\longrightarrow}q^{(\max(i,\, \mathrm{rank}\, q))}$. For each $i$ and $q$, $q^{(i)}$ is universal iff $q$ is universal. Checking the equivalence is straightforward.
For deterministic automata we will assume that all states are productive, i. e., are used in some accepting run, save for one all-rejecting state $\bot$, and that all transitions are productive or go to $\bot$, i. e., whenever $q\stackrel{\sigma}{\longrightarrow} q_1, q_2$, then either $q_1$ and $q_2$ are productive, or $q_1=q_2=\bot$. The assumption of productivity is vital for our proofs. Thanks to this assumption, in each node of an automaton’s run we can plug in an accepting sub-run.
Transforming a given automaton into such a form of course needs calculating the productive states, which is equivalent to deciding a language’s emptiness. The latter problem is known to be in $\textrm{NP} \cap \textrm{co-NP}$, but it has no polynomial solutions yet. Therefore we can only claim that our algorithms are polynomial for the automata that underwent the above preprocessing. We will try to mention it whenever particularly important.
Two Hierarchies
===============
The [*index of an automaton*]{} $A$ is a pair $(\min {\rm rank}\, Q,\max {\rm rank}\, Q )$. Scaling down the rank function if necessary, one may assume that $\min {\rm rank}\, Q$ is either 0 or 1. Thus, the indices are elements of $\{0,1\}\times\omega \setminus \{(1,0)\}$. For an index $(\iota,\kappa)$ we shall denote by $\overline {(\iota,\kappa)}$ [*the dual index*]{}, i. e., $\overline{(0,\kappa)} = (1,\kappa+1)$, $\overline{(1,\kappa)} = (0,\kappa-1)$. Let us define an ordering of indices with the following formula: $$(\iota,\kappa) < (\iota',\kappa') \textrm{ if and only if } \kappa - \iota < \kappa' < \iota' \,.$$ In other words, one index is greater than another if and only if it “uses” more ranks. This means that dual indices are incomparable. [*The Mostowski–Rabin index hierarchy*]{} for a certain class of automata consists of ascending sets (levels) of languages recognized by $(\iota,\kappa)$-automata.
Here, we are mainly interested in the [*weak index hierarchy*]{}, i. e., the hierarchy of languages recognized by weak $(\iota,\kappa)$-automata. The strictness of this hierarchy was established by Mostowski [@most] via equivalence with the quantifier-alternation hierarchy for the weak monadic second order logic, whose strictness was proved by Thomas [@weakthomas]. The [*weak index problem*]{}, i. e., computing the minimal weak index needed to recognize a given weak language, for the time being remains unsolved just like other versions of the index problem.
The weak index hierarchy is closely related to the Borel hierarchy. We will work with the standard Cantor-like topology on $T_{\Sigma}$ induced by the metric $$d(s,t) = \left \{
\begin{array} {l l}
2 ^{-\min \{|x|\;: \;\; x \in \{0,1\}^*, \; s(x) \neq t(x)\}} & \textrm{iff } s \neq t \\
0 & \textrm{iff } s=t
\end{array}
\right . .$$ The class of Borel sets of a topological space $X$ is the closure of the class of open sets of $X$ by countable sums and complementation.
For a topological space $X$, the initial (finite) levels of the [*Borel hierarchy*]{} are defined as follows:
- $\Sigma^0_1(X)$ – open subsets of $X$,
- $\Pi^0_k(X)$ – complements of the sets from $\Sigma^0_k(X)$,
- $\Sigma^0_{k+1}(X)$ – countable unions of sets from $\Pi^0_k(X)$.
For instance, $\Pi^0_1(X)$ are the closed sets, $\Sigma^0_2(X)$ are $F_\sigma$ sets and $\Pi^0_2(X)$ are $G_\delta$ sets. By convention $\Sigma^0_0(X) = \{\emptyset\}$ and $\Pi^0_0(X) = \{X\}$.
A straightforward inductive argument shows that the classes defined above are closed under inverse images of continuous functions. Let ${\mathcal C}$ be one of those classes. A set $A$ is called [*${\mathcal C}$-hard*]{}, if each set in ${\mathcal C}$ is an inverse image of $A$ under some continuous function. If additionally $A\in{\mathcal C}$, $A$ is [*${\mathcal C}$-complete*]{}.
We start the discussion of the relations between the index of a weak automaton and the Borel rank of the language it recognises by recalling Skurczyński’s results. For a tree $t:\{0,1\}^* \to \Sigma$ and a node $v\in \{0,1\}^*$ let $t.v$ denote the tree rooted in $v$, i. e., $t.v(w) = t(vw)$. Let us define a sequence of languages:
- $L_{(0,1)} = \{t\}$, where $t \in T_{\{a,b\}}$ is the tree with no $b$’s,
- $L_{(1,n+1)} = L_{(0,n)} ^\complement$ for $n \geq 1$,
- $L_{(0,n+1)} = \{t \in T_{\{a,b\}}: \;\; \forall_{k} \; t.0^k1 \in L_{(1,n+1)}\}$ for $n \geq 1$.
For each $n \geq 1$,
- $L_{(0,n)}$ is a $\Pi^0_n$-complete language recognized by a weak $(0,n)$-automaton,
- $L_{(1,n+1)}$ is a $\Sigma^0_n$-complete language recognized by a weak $(1,n+1)$-automaton.
We will now show that this construction is as efficient as it can be: ranks $(0,n)$ are necessary to recognize any $\Pi^0_n$-hard language (if it can be weakly recognized at all).
We will actually prove a bit stronger result. We will consider [*weak game languages*]{} $W_{[\iota, \kappa]}$, to which all languages recognized by weak $[\iota, \kappa]$-automata can be reduced, and show that $W_{[0,n]} \in \Pi^0_n$ and $W_{[1,n+1]} \in \Sigma^0_n$ (by Skurczyński’s results, they are hard for these classes). The languages $W_{[\iota, \kappa]}$ are natural weak counterparts of strong game languages that prove the strictness of the strong alternating index hierarchy. Lately Arnold and Niwiński proved that the strong game languages also form a strict hierarchy with respect to continuous reductions, but they are all non-Borel [@gamelang].
Fix a natural number $N$. For $\iota=0,1$ and $\kappa \geq \iota$, let ${\mathcal T}_{(\iota, \kappa)}$ denote the set of full $N$-ary trees over the alphabet $\{\exists,\forall\} \times \{\iota, \iota+1, \ldots, \kappa\}$. Let $W_{(\iota, \kappa)} \subseteq {\mathcal T}_{(\iota, \kappa)}$ be the set of all trees $t$ for which Eve has a winning strategy in the [*weak*]{} parity game $G_t=\langle V_\exists, V_\forall, E, v_0, \mathrm{rank} \rangle$, where $V_\theta = \{v \in \mathrm{dom}\, t \colon t(v)=(\theta, j) \textrm{ for some } j\}$, $E= \{ (v,vk)\colon v\in \mathrm{dom}\, t \,,\; k < N\}$, $v_0 = \varepsilon$, $\mathrm{rank}(v)=j$ iff $t(v) = (\theta,j)$ for some $\theta$.
\[weakgamelang\] For each $n$, $W_{(0,n)}\in \Pi^0_n(\mathcal T_{(0,n)})$ and $W_{(1,n+1)} \in \Sigma^0_n(\mathcal T_{(1,n+1)})$.
We will proceed by induction on $n$. For $n=0$ the claim is obvious: $W_{(0,0)} = {\mathcal T}_{(0, 0)} \in \Pi^0_0(\mathcal T_{(0,0)})$, $W_{(1,1)} = \emptyset \in \Sigma^0_0(\mathcal T_{(1,1)})$.
Take $n>0$. For each $t\in W_{(1,n+1)}$ there exists a strategy $\sigma$ for Eve, such that it guarantees that the play reaches a node with the rank greater or equal to $2$. By König lemma, this must happen in a bounded number of moves. Basing on this observation we will provide a $\Sigma^0_n$ presentation of $W_{(1,n+1)}$.
Let [*$k$-antichain*]{} be a subset of the nodes on the level $k$. Let ${\mathcal A}$ denote the set of all possible $k$-antichains for all $k<\omega$. Obviously this set is countable. For a $k$-antichain $A$ let $W_A$ denote the set of trees such that there exists a strategy for Eve that guarantees visiting a node with the rank $\geq 2$ during the initial $k$ moves and reaching a node from $A$. This set is a clopen. We have a presentation $$W_{(1,n+1)} = \bigcup_{A \in {\mathcal A}} \left ( W_A \cap \bigcap_{v \in A} \left \{t: t'.v \in W_{(0,n-1)} \right \} \right ) \,,$$ where $t'$ is obtained from $t$ by decreasing all the ranks by $2$ (if the result is $-1$, take $0$). The claim follows by induction hypothesis and the continuity of $t \mapsto t'$ and $t \mapsto t.v$.
Now, it remains to see that $W_{(0,n)}\in\Pi^0_n(\mathcal T_{(0,n)})$. For this, note that $$W_{(0,n)} = \left \{t: t'' \in (W_{(1,n+1)}) ^\complement \right \}\,,$$ where $t''$ is obtained from $t$ by swapping $\exists$ and $\forall$, and increasing ranks by 1. The claim follows by the continuity of $t\mapsto t''$.
As a corollary we get the promised improvement of Skurczyński’s result.
\[weakborel\] For every weak alternating automaton $A$ with index $(0,n)$ (resp. $(1,n+1)$) it holds that $L(A)\in \Pi^0_n$ (resp. $L(A)\in \Sigma^0_n$).
Let $A$ be an automaton with priorities inside $[\iota, \kappa]$. For sufficiently large $N$ we may assume without loss of generality that the computation trees of the automaton are $N$-ary trees. By assigning to an input tree the run of $A$, one obtains a continuous function reducing $L(A)$ to $W_{(\iota, \kappa)}$. Hence, the claim follows from the theorem above.
In fact the corollary follows also from Mostowski’s theorem on equivalence of weak automata and weak monadic second order logic on trees [@most]. The present proof of Theorem \[weakgamelang\] is actually just a repetition of Mostowski’s proof in the setting of the Borel hierarchy. An entirely different proof can be found in [@wata].
We believe that the converse implication is also true: a weakly recognizable $\Pi^0_n$-language can be recognized by a weak $(0,n)$-automaton (and dually for $\Sigma^0_n$).
For weakly recognizable languages the weak index hierarchy and the Borel hierarchy coincide.
In this paper we show that the conjecture holds true when restricted to deterministic languages.
The Deterministic Case
======================
In 2002 Niwiński and Walukiewicz discovered a surprising dichotomy in the family of deterministic languages: a deterministic language is either very simple or very sophisticated.
\[topgap\] For a deterministic automaton $A$ with $n$ states, $L(A)$ is either recognizable with a weak alternating $(0,3)$-automaton with ${\mathcal O}(n^2)$ states (and so $\Pi^0_3$) or is non-Borel (and so not weakly recognizable). The equivalent weak automaton can be constructed within the time of solving the emptiness problem.
[ $\begin{array}{cccccccccc|c}
(1,1) &&& (1,2) &&& (1,3) \\
& \diagdown & \diagup && \diagdown & \diagup && \diagdown &\\
& \diagup & \diagdown && \diagup & \diagdown && \diagup & \diagdown \\
(0,0) &&& (0,1) &&& (0,2) &&& (0,3) \quad & \quad \textrm{non-weak}
\end{array}$]{}
An important tool used in the proof of the Gap Theorem (Theorem \[topgap\]) is the technique of difficult patterns. In the topological setting the general recipe goes like this: for a given class identify a pattern that can be unravelled to a language complete for this class; if an automaton does not contain the pattern, then $L(A)$ should be in the dual class. The same technique was later applied to obtain effective characterisations of the remaining Borel classes of deterministic languages [@split].
Let us define the patterns used in these characterisations. A [*loop*]{} in an automaton is a sequence of states and transitions: $$p_0 \stackrel{\sigma_1,d_1}{\longrightarrow} p_1 \stackrel{\sigma_2,d_2}{\longrightarrow} \ldots \stackrel{\sigma_{n},d_n}{\longrightarrow} p_0\,.$$ A loop is called [*accepting*]{} if $\max_i \mathrm{rank}\,(p_i)$ is even. Otherwise it is [*rejecting*]{}.
A [*$(\iota,\kappa)$-flower*]{} is a sequence of loops $\lambda_\iota, \lambda_{\iota+1}, \ldots , \lambda_\kappa$ starting in the same state $p$, such that the highest rank appearing on $\lambda_i$ has the same parity as $i$ and it is higher than the highest rank on $\lambda_{i-1}$ for $i=\iota, \iota+1, \ldots, \kappa$.
A [*weak $(\iota,\kappa)$-flower*]{} is a sequence of loops $\lambda_\iota, \lambda_{\iota+1} \ldots , \lambda_\kappa$ such that $\lambda_{i+1}$ is reachable from $\lambda_i$, and $\lambda_i$ is accepting iff $i$ is even.
A [*split*]{} is a pair of loops $p \stackrel{\sigma, 0}{\longrightarrow} p_0 \longrightarrow \ldots \longrightarrow p$ and $p \stackrel{\sigma,1}{\longrightarrow} p_1 \longrightarrow \ldots \longrightarrow p$ such that the highest ranks occurring on them are of different parity and the highest one is odd.
A state $q$ is [*replicated*]{} by a loop $p \stackrel {\sigma, d_0}\longrightarrow p_0 \longrightarrow \ldots \longrightarrow p $ if there exists a path $p \stackrel{\sigma, d_1} \longrightarrow p_1 \longrightarrow \ldots \longrightarrow q$ such that $d_0 \neq d_1$. We will say that a loop or a flower is replicated by a loop $\lambda$ if it contains a state replicated by $\lambda$.
\[hardborel\] Let $A$ be a deterministic automaton.
1. $L(A) \in \Pi^0_1$ iff $A$ contains no weak $(1,2)$-flower.
2. $L(A) \in \Sigma^0_1$ iff $A$ contains no weak $(0,1)$-flower.
3. $L(A)\in\Pi^0_2$ iff $A$ contains no $(0,1)$-flower.
4. $L(A) \in \Sigma^0_2$ iff $A$ contains neither $(1,2)$-flower nor a weak $(1,2)$-flower replicated by an accepting loop.
5. $L(A)\in \Sigma^0_3$ iff $A$ contains no $(0,1)$-flower replicated by an accepting loop.
6. $L(A)\in\Pi^0_3$ iff $A$ contains no split.
In particular, the Borel rank of $L(A)$ is computable within the time of finding the productive states of $A$.
The patterns defined above were originally introduced to capture the index complexity of recognizable languages. Niwiński and Walukiewicz used flowers to solve the deterministic index problem for word languages [@kwiatek]. Their result may easily be adapted to trees (see [@split] for details).
\[indch\] For a deterministic tree automaton $A$ the language $L(A)$ is recognized by a deterministic $(\iota,\kappa)$-automaton iff $A$ does not contain a $\overline{(\iota,\kappa)}$-flower. An equivalent minimal index automaton with the same number of states can be constructed within the time of solving the emptiness problem.
The weak flowers provide an analogous characterisation of the weak deterministic index.
\[windch\] A deterministic automaton $A$ is equivalent to a weak deterministic $(\iota,\kappa)$-automaton iff it does not contain a weak $\overline{(\iota,\kappa)}$-flower. An equivalent minimal index automaton with the same number of states can be constructed within the time of solving the emptiness problem.
If the automaton contains a weak $(\iota,\kappa)$-flower, for each weak $\overline{(\iota,\kappa)}$-automaton one can build a cheating tree (see [@split] for details). For the converse implication, construct a weak deterministic $(\iota,\kappa)$-automaton by modifying the ranks of the given deterministic automaton. Set ${\rm rank}\,q$ to the lowest number $m$ such that there exists a weak $(m,\kappa)$-flower with a path from $q$ to $\lambda_m$.
The Power of the Weak {#sect_weakindex}
=====================
In this section we finally turn to the weak recognizability of deterministic languages. First we give sufficient conditions for a deterministic automaton to be equivalent to a weak alternating automaton of index $(0,2)$, $(1,3)$, and $(1,4)$. This is the first step to the solution of the weak index problem for deterministic automata.
\[weak02\] For each deterministic $(1,2)$-automaton with $n$ states one can construct an equivalent weak $(0,2)$-automaton with $2n+1$ states.
Fix a deterministic $(1,2)$-automaton $A$. We will construct a weak $(0,2)$-automaton $B$ such that $L(A) = L(B)$. Basically, for each node $v$ the automaton $B$ should check whether on each path in the subtree rooted in $v$ the automaton $A$ will reach a state with rank 2. This can be done as follows. Take two copies of $A$. In the first copy, all states are universal and have rank 0. The transitions are like in $A$ plus for each state $q^{(1)}$ there is an $\varepsilon$-transition to $q^{(2)}$, the counterpart of $q^{(1)}$ in the second copy. In the second copy all states are universal and have rank 1. For the states with rank 1 in $A$, the transitions are like in $A$. For the states with rank 2 in $A$, there is just one transition to an all-accepting state $\top$ (rank 2 in $B$).
Before we proceed with the conditions, let us show a useful property of the replication.
\[replicationlemma\] A state occurs in infinitely many incomparable nodes of an accepting run iff it is productive and is replicated by an accepting loop.
If a state $p$ is replicated by an accepting loop, then by productivity one may easily construct an accepting run with infinitely many incomparable occurrences of $p$. Let us concentrate on the converse implication.
Let $p$ occur in an infinite number of incomparable nodes $v_0, v_1, \ldots$ of an accepting run $\rho$. Let $\pi_i$ be a path of $\rho$ going through the node $v_i$. Since $2^\omega$ is compact, we may assume, passing to a subsequence, that the sequence $\pi_i$ converges to a path $\pi$. Since $v_i$ are incomparable, $v_i$ is not on $\pi$. Let the word $\alpha_i$ be the sequence of states labeling the path from the last common node of $\pi$ and $\pi_i$ to $v_i$. Cutting the loops off if needed, we may assume that $|\alpha_i| \leq |Q|$ for all $i\in \omega$. Consequently, there exist a word $\alpha$ repeating infinitely often in the sequence $\alpha_0, \alpha_1, \ldots$. Moreover, the path $\pi$ is accepting, so the starting state of $\alpha$ must lay on an accepting productive loop. This loop replicates $p$.
\[weak13\] For each deterministic $(0,1)$-automaton with $n$ states which contains no weak $(1,2)$-flower replicated by an accepting loop one can construct effectively an equivalent weak $(1,3)$-automaton with $3n+1$ states.
Let $A$ be a deterministic $(0,1)$-automaton which contains no weak $(1,2)$-flower replicated by an accepting loop. Let us call a state of $A$ [*relevant*]{} if it has the highest rank on some loop. We may change the ranks of productive irrelevant states to $0$, and assume from now on that all odd states are relevant. We claim that the odd states occur only finitely many times on accepting runs of $A$. Suppose that an odd state $p$ occurs infinitely many times in an accepting run $\rho$. Then it must occur in infinitely many incomparable nodes (otherwise we would get a rejecting path). By the Replication Lemma $p$ is replicated by an accepting loop. As $p$ is odd and relevant, it lies on some nontrivial rejecting loop. Since $p$ is also productive, some accepting loop can be reached from $p$. Hence, $A$ contains a weak $(1,2)$-flower replicated by an accepting loop - a contradiction
Now, we can easily construct a weak $(1,3)$-automaton recognising $L(A)$. Intuitively, we will simulate $A$ and check if $A$’s odd states occur finitely many times. This can be done as follows. Take three copies of $A$. In the first copy all the states are universal and have rank 1. The transitions are just like in $A$, only they go to the second copy of $A$. In the second copy of $A$, all the states are existential and have rank 1. From each state $q^{(2)}$ there are two $\varepsilon$-transitions to $q^{(1)}$ in the first copy and to $q^{(3)}$ in the third copy. Finally, in the third copy of $A$ all the states are universal and have rank 2. The transitions from the states ranked 0 in $A$ are just like in $A$, and from the states ranked $1$ in $A$ they go to an all-rejecting state $\bot$ (rank 3 in $B$). It is easy to see that $B$ recognizes $L(A)$.
\[weak14\] For each automaton with $n$ states containing no $(0,1)$-flower replicated by an accepting loop one can construct an equivalent weak alternating $(1,4)$-automaton with ${\mathcal O}(n^2)$ states.
Let $A$ be an automaton without $(0,1)$-flower replicated by an accepting loop. Consider the DAG of strongly connected components of $A$.For each SCC $X$ containing at least one loop we will construct a weak automaton $B_X$ recognising the languages of trees $t$ such that each path of $A$’s run on $t$ that enters $X$ either leaves $X$ or is accepting. Obviously, the conjunction of such automata recognizes exactly $L(A)$. Let us first consider components replicated by an accepting loop. By the hypothesis, such a component must not contain a $(0,1)$-flower. Therefore we may assume that $X$ only uses ranks 1 and 2. To obtain $B_X$ take a copy of $A$. The states outside $X$ can be divided into three disjoint groups: those that can be reached from $X$, those from which $X$ can be reached, and the rest. Give the states from the first group the rank 4, and the states from the second and third group the rank 2. Finally, following the method from Proposition \[weak02\], replace X with an equivalent weak alternating subautomaton using ranks 2,3, and 4. The constructed automaton has ${\mathcal O}(n)$ states.
The case of $X$ not replicated by an accepting loop is more tricky. The key property follows from the Replication Lemma. Let $\rho_X$ denote the restriction of the run $\rho$ to the nodes labeled with a state from $X$ or having a descendant labeled with a state from $X$. By the Replication Lemma, this tree has only finitely many branches (some of them may be infinite). What $B_X$ should do is to guess a node $v$ on each path such that in the subtree rooted in $v$, $\rho_X$ is either empty or consists of one infinite accepting branch. In the latter case we may additionally demand that on this infinite path the highest rank that ever occurs, occurs infinitely many times.
$B_X$ consists of the component $C_{\textrm{guess}}$ realising the guessing, the component $C_{A \setminus X}$ checking that no path of the computation enters $X$, and components $C_{X,r}$ for all ranks $r$ used in $X$, which check that in a given subtree of the run $\rho$ there is exactly one branch of $\rho_X$ and that on this branch $r$ occurs infinitely often and no higher rank is used.
To construct $C_{\textrm{guess}}$, take a copy of $A$ and declare all the states universal and set their ranks to $1$. For each $q$ add a fresh existential state $q'$ of rank $1$ with an $\varepsilon$-transition to $q$ and either to $q^{A\setminus X}\in C_{A \setminus X}$ if $q\notin X$ ($\rho_X$ is empty) or to $q^{X,r} \in C_{X,r}$ for all $r$ if $q\in X$ ($\rho_X$ is one infinite accepting path). Finally replace each transition $p \stackrel{\sigma}{\longrightarrow}p_0,p_1$ with $\stackrel{\sigma}{\longrightarrow}p'_0,p'_1$.
The component $C_{A \setminus X}$ is a copy of $A$ with all ranks equal $2$, and the SCC $X$ replaced with one all-rejecting state $\bot$ with rank $3$.
Finally, let us now describe the automaton $C_{X,r}$. The automaton, staying in rank 2, works its way down the input tree just like $A$ would, with the following modifications:
- if $A$ enters a state in $X$ with rank greater than $r$, $C_{X,r}$ moves to an all rejecting state $\bot$ (rank 3),
- if $A$ takes a transition exiting $X$ on both branches or staying in $X$ on both branches, $C_{X,r}$ moves to $\bot$,
- if $A$ takes a transition whose left branch leaves $X$ and the right branch stays inside, $C_{X,r}$ sends to the right a $(3,4)$-component looking for a state from $X$ with the rank $r$, and moves on to the right subtree (and symmetrically).
In order two see that $C_{X,r}$ does the job, it is enough to observe that if the $(3,4)$ component always succeeds to find a state from $X$ with the rank $r$, then on the unique path that stays forever in $X$ the rank $r$ repeats infinitely often.
The $(3,4)$-component of $C_{X,r}$ can be constructed in such a way that it has $|X|+2$ states, and so in this case $B_X$ has at most $2|X|(|X|+2) + 3n \leq 2|X|^2 + 7n$ states.
In both cases, the number of states of $B_X$ can be bounded by $c_1|X|^2 + c_2n$ for fixed constants $c_1$ and $c_2$, independent of $X$. Since the SCCs are disjoint, the number of states of the conjunction of $B_X$’s is at most $$1 + \sum_{X\in A} (c_1|X|^2 + c_2n) \leq 1+ c_1 \left ( \sum_{X \in A} |X|\right )^2 + c_2n^2 \leq (c_1+c_2)n^2 + 1\,.$$
We have now collected all the ingredients for the solution of the weak index problem for deterministic languages. What is left to be done is to glue together the sufficient conditions for index easiness and Borel hardness using Corollary \[weakborel\].
For deterministic languages the Borel hierarchy and the weak index hierarchy coincide (Fig. \[fig:dethierarchy\]) and are decidable within the time of solving emptiness problem. For a deterministic automaton with $n$ states, an equivalent minimal index automaton with ${\mathcal O}(n^2)$ states can be constructed effectively within the time of solving the emptiness problem.
We will abuse the notation and write $(\iota, \kappa)$ to denote the class of languages recognized by weak $(\iota, \kappa)$-automata. All the classes considered here are relativised to the deterministic languages.
By the two versions of the Gap Theorem we have the equality and decidability of the classes of the classes $\Pi^0_3$ and $(0,3)$.
Let us continue with the third level. Let us see that $\Sigma^0_3 = (1,4)$. We will show that both these classes are equal to the class of languages recognized by deterministic automata without a $(0,1)$-flower replicated by an accessible loop. If a deterministic automaton $A$ does not contain this pattern, then it is equivalent to a weak $(1,4)$-automaton and by Corollary \[weakborel\] recognizes a $\Sigma^0_3$ language. If $A$ does contain this pattern, then by Proposition \[hardborel\] it is not $\Sigma^0_3$ and so is not equivalent to a weak $(1,4)$-automaton. The decidability follows easily, since checking for the pattern above can be done effectively (in polynomial time).
For the equality $\Pi^0_2 = (0,2)$, prove that both classes are equal to the class of languages recognized by deterministic automata without a $(0,1)$-flower. Proceed just like before, only use Proposition \[weak02\] instead of Proposition \[weak14\]. Analogously, using Proposition \[weak13\], show that both $\Sigma^0_2$ and $(1,3)$ are equal to the class of languages recognized by deterministic automata admitting neither a $(1,2)$-flower nor a weak $(1,0)$-flower replicated by an accepting loop.
For the first level use the characterisation given by Proposition \[windch\]. The level zero is trivial.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author thanks Damian Niwiński for reading carefully a preliminary version of this paper and the anonymous referees for their helpful comments.
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[^1]: Supported by the Polish government grant no. N206 008 32/0810.
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---
abstract: |
We generate theoretical albedo and reflection spectra for a full range of extrasolar giant planet (EGP) models, from Jovian to 51-Pegasi class objects. Our albedo modeling utilizes the latest atomic and molecular cross sections, Mie theory treatment of scattering and absorption by condensates, a variety of particle size distributions, and an extension of the Feautrier technique which allows for a general treatment of the scattering phase function.
We find that due to qualitative similarities in the compositions and spectra of objects within each of five broad effective temperature ranges, it is natural to establish five representative EGP albedo classes. At low effective temperatures (T$_{\textrm{eff}} \lesssim$ 150 K) is a class of “Jovian” objects (Class I) with tropospheric ammonia clouds. Somewhat warmer Class II, or “water cloud,” EGPs are primarily affected by condensed H$_2$O. Gaseous methane absorption features are prevalent in both classes. In the absence of non-equilibrium condensates in the upper atmosphere, and with sufficient H$_2$O condensation, Class II objects are expected to have the highest visible albedos of any class.
When the upper atmosphere of an EGP is too hot for H$_2$O to condense, radiation generally penetrates more deeply. In these objects, designated Class III or “clear” due to a lack of condensation in the upper atmosphere, absorption lines of the alkali metals, sodium and potassium, lower the albedo significantly throughout the visible. Furthermore, the near-infrared albedo is negligible, primarily due to strong CH$_4$ and H$_2$O molecular absorption, and collision-induced absorption (CIA) by H$_2$ molecules. In those EGPs with exceedingly small orbital distance (“roasters”) and 900 K $\lesssim$ T$_{\textrm{eff}} \lesssim$ 1500 K (Class IV), a tropospheric silicate layer is expected to exist. In all but the hottest (T$_{\textrm{eff}} \gtrsim$ 1500 K) or lowest gravity roasters, the effect of this silicate layer is insignificant due to the very strong absorption by sodium and potassium atoms above the layer. The resonance lines of sodium and potassium are expected to be salient features in the reflection spectra of these EGPs. In the absence of non-equilibrium condensates, we find, in contrast to previous studies, that these Class IV roasters likely have the lowest visible and Bond albedos of any class, rivaling the lowest albedos of our solar system. For the small fraction of roasters with T$_{\textrm{eff}} \gtrsim$ 1500 K and/or low surface gravity ($\lesssim 10^3$ cm s$^{-2}$; Class V), the silicate layer is located very high in the atmosphere, reflecting much of the incident radiation before it can reach the absorbing alkali metals and molecular species. Hence, the Class V roasters have much higher albedos than those of Class IV.
We derive Bond albedos ($A_B$) and T$_{\textrm{eff}}$ estimates for the full set of known EGPs. A broad range in both values is found, with T$_{\textrm{eff}}$ ranging from $\sim$ 150 K to nearly 1600 K, and $A_B$ from $\sim$ 0.02 to 0.8.
We find that variations in particle size distributions and condensation fraction can have large quantitative, or even qualitative, effects on albedo spectra. In general, less condensation, larger particle sizes, and wider size distributions result in lower albedos. We explore the effects of non-equilibrium condensed products of photolysis above or within principal cloud decks. As in Jupiter, such species can lower the UV/blue albedo substantially, even if present in relatively small mixing ratios.
author:
- 'David Sudarsky, Adam Burrows, & Philip Pinto'
title: Albedo and Reflection Spectra of Extrasolar Giant Planets
---
\#1 \#1[$\underline{\smash{\hbox{#1}}}$]{} \#1 å[Astron. Astrophys. ]{}
Introduction
============
Since the discovery of the extrasolar giant planet (EGP), 51 Pegasi b, in 1995 (Mayor & Queloz 1995), an explosion of similar discoveries has followed. To date, there are $\sim$ 30 known planets orbiting nearby stars, which have collectively initiated the new field of extrasolar giant planet research.
While to date most detections have been via Doppler spectroscopy, other promising methods, both ground-based and space-based, are in development. These include (but are not limited to) astrometric techniques (Horner et al. 1998), nulling interferometry (Hinz et al. 1998), coronographic imaging (Nakajima 1994), and spectral deconvolution (Charbonneau et al. 1998). Furthermore, planned space instrumentation such as the NGST (Next Generation Space Telescope) and SIM (Space Interferometry Mission) may prove to be useful for the detection and characterization of EGP systems.
With the current push for new instruments and techniques, we expect that some of these new EGPs will soon be directly detected. One group (Cameron et al. 1999) has claimed a detection in reflected light of the “roaster,” $\tau$ Boo b, while another group (Charbonneau et al. 1999) has not claimed a detection, but has quoted an upper limit to the albedo which is in conflict with Cameron et al. Our theoretical models of EGP albedos are motivated by and can help guide attempts to directly detect EGPs in reflection by identifying their characteristic spectral features and by illuminating the systematics.
The theoretical study of EGP albedos and reflection spectra is still largely in its infancy. Marley et al. (1999) have explored a range of stratosphere-free EGP geometric and Bond albedos using temperature-pressure profiles of EGPs in isolation (i.e. no stellar insolation), while Goukenleuque et al. (1999) modeled 51 Peg in radiative equilibrium, and Seager & Sasselov (1998) explored radiative-convective models of EGPs under strong stellar insolation. In the present study, our purpose is to provide a broader set of models than previous work, and to establish a general understanding of the albedo and reflection spectra of EGPs over the full range of effective temperatures (T$_{\textrm{eff}}$). Rather than attempting to model these spectra in a fully consistent way for the almost endless combinations of EGP masses, ages, orbital distances, elemental abundances, and stellar spectral types, we concentrate on representative composition classes based loosely on T$_{\textrm{eff}}$. The “Jovian” Class I objects (T$_{\textrm{eff}} \lesssim$ 150 K) are characterized by the presence of ammonia clouds. (Note that the term, “Jovian”, is used here for convenience, not to imply that this entire class of objects will be identical to Jupiter.) In somewhat warmer objects (T$_{\textrm{eff}}
\sim$ 250 K), ammonia is in its gaseous state, but the upper troposphere contains condensed H$_2$O. These objects are designated Class II, or “water cloud” EGPs. Class III, or “clear” EGPs, are so named because they are too hot (T$_{\textrm{eff}} \gtrsim$ 350 K) for significant H$_2$O condensation and so are not expected to contain any principal condensates, though they are not necessarily completely cloud-free. The hotter EGPs (900 K $\lesssim$ T$_{\textrm{eff}}$ $\lesssim$ 1500 K; Class IV) include those objects with very small orbital distances (“roasters”) or those at large distances which are massive and young enough to have similar effective temperatures. In either case, the troposphere of such an EGP is expected to contain significant abundances of neutral sodium and potassium gases, as well as a silicate cloud layer. The hottest (T$_{\textrm{eff}} \gtrsim$ 1500) and/or least massive (g $\lesssim 10^3$ cm s$^{-2}$) have a silicate layer located so high in the atmosphere that much of the incoming radiation is shielded from alkali metal and molecular absorption.
We use a planar asymmetric Feautrier method in conjunction with temperature-pressure (T-P) profiles, equilibrium gas abundances (assuming Anders & Grevesse (1989) elemental abundances), and simple cloud models to account for condensed species. The T-P profiles of isolated EGPs, as well as profiles which are nearly isothermal in the outer atmosphere, are utilized. This allows us to bracket the effects of various T-P profiles on the resulting EGP albedo spectra. Like Marley et al. (1999), we generate model albedo and reflection spectra and Bond albedos, assuming a variety of central star spectral types. Similarly, Rayleigh scattering, Raman scattering, Mie extinction due to condensates, and molecular absorption by a host of species are treated. In addition to our broader range of compositions and T$_{\textrm{eff}}$ than in Marley et al., we treat the important absorption effects of the alkali metals, include a larger number of relevant condensates (including some non-equilibrium products typical of photolysis), and produce a synthetic albedo spectrum of Jupiter which is in reasonable agreement with Jupiter’s actual albedo spectrum (Karkoschka 1994) from the soft UV to the near infrared.
Doppler spectroscopy favors the detection of massive companions at small orbital distances and indeed EGPs with very small orbital radii have been found. $\tau$ Boo b (Butler et al. 1997), 51 Peg b (Mayor & Queloz 1995), $\upsilon$ And b (Butler et al. 1997), HD 75289b (Mayor et al. 1999), HD 187123b (Butler et al. 1998), HD 217107b (Fischer et al. 1999), and HD 209458b (Charbonneau et al. 1999) all have orbital distances of less than 0.1 AU and masses (actually M$_p\sin i$) ranging from $\sim$0.4 to 3.4 Jupiter masses. Under stellar insolation, the elevated temperatures of EGPs depend mostly on the level of stellar insolation, rather than on their masses and ages, which would largely determine their T$_{\textrm{eff}}$ in isolation. Using simple radiative equilibrium arguments (T$_{\textrm{eff}} \propto F_{inc}^{1/4}$, where $F_{inc}$ is the incident stellar flux), most of the EGPs within 0.1 AU are likely to have very high T$_{\textrm{eff}}$ ($\sim$ 800 K to over 1600 K). T$_{\textrm{eff}}$ is only weakly dependent on the Bond albedo for a large range of low-to-moderate albedos, varying only $\sim$ 20% as the Bond albedo varies from 0.01 to 0.6.
At the other end of the scale, several objects with more traditional orbital distances of $\gtrsim$ 1 AU have been discovered. These EGPs include 16 Cyg Bb (Cochran et al. 1997), 47 UMa b (Butler et al. 1996), $\upsilon$ And d (Marcy et al. 1999), Gl 614b (Mayor et al. 1998), HR 5568b, HR 810b, and HD 210277b (Marcy et al. 1998), and have M$_p\sin i$ ranging from $\sim$ 0.75 to 5 M$_J$. At these larger orbital distances, EGPs receive much less stellar radiation and, therefore, have a lower T$_{\textrm{eff}}$ ($\lesssim$ 300K). Still, many other EGPs, such as 70 Vir b (Butler & Marcy 1996), Gl 86 Ab (Queloz et al. 1999), and HD 114762b (Latham et al 1989), have orbital distances between 0.1 and 1 AU and M$_p\sin i$ between 0.7 and 10 M$_J$. Over the full set of currently known EGPs, spectral classes of the central stars range from F7V to M4V.
The albedo of an object is simply the fraction of light that the object reflects. However, there are several different types of albedos. The [*geometric*]{} albedo refers to the reflectivity of the object at full phase ($\Phi = 0$, where $\Phi$ represents the object’s phase angle) relative to that by a perfect Lambert disk of the same radius under the same incident flux. Since planets are essentially spheres, the factor projecting a unit surface onto a disk orthogonal to the line of sight is given by $\cos\phi\sin\theta$, where $\phi$ is the object’s longitude (defined to be in the observer-planet-star plane) and $\theta$ is its polar angle ($\pi\over 2$ - latitude). The geometric albedo is given by integrating over the illuminated hemisphere: $$A_g = {1\over {\pi I_{inc}}}\int_{\phi = {-{\pi\over 2}}}^{\pi\over 2}
\int_{\theta = 0}^\pi I(\phi,\theta,\Phi=0)\cos\phi\sin\theta d\Omega\, ,
\label{geometric}$$ where $I_{inc}$ is the incident specific intensity, $\pi I_{inc}$ is the incident flux, and $I(\phi,\theta,\Phi = 0)$ is the emergent intensity. More generally, $I = I(\phi,\theta,\Phi;\phi_0,\theta_0)$, but at full phase all incident angles ($\phi_0$, $\theta_0$) are equal to the emergent ones. The geometric albedo is usually given as a function of wavelength, although it is sometimes averaged over a wavelength interval and stated as a single number.
The [*spherical*]{} albedo, $A_s$, refers to the fraction of incident light reflected by a sphere at all angles. Usually stated as a function of wavelength, it is obtained by integrating the reflected flux over all phase angles. The flux ($F(\Phi)$) as a function of phase angle ($\Phi$) is given by the more general form of eq. (\[geometric\]). Assuming unit radius (Chamberlain & Hunten 1987), $$F(\Phi) = \int_{\phi = \Phi-{\pi\over 2}}^{\pi\over 2}\int_{\theta = 0}^\pi
I(\phi,\theta,\Phi;\phi_0,\theta_0)\cos\phi\sin\theta d\Omega.$$
The spherical albedo is obtained by integrating $F(\Phi)$ over all solid angles: $$A_s = {1\over {\pi I_{inc}}}\int_{4\pi} F(\Phi) d\Omega =
{2\over I_{inc}} \int_0^\pi F(\Phi)\sin\Phi d\Phi.$$
Note that the spherical and geometric albedos are related by $A_s = A_gq$, where $$q = {2\over {F(\Phi=0)}}\int_0^\pi F(\Phi)\sin\Phi d\Phi$$ is known as the phase integral.
The [*Bond*]{} albedo, $A_B$, is the ratio of the total reflected and total incident powers. It is obtained by weighting the spherical albedo by the spectrum of the illuminating source and integrating over all wavelengths: $$A_B = {\int_0^\infty A_{s,\lambda} I_{inc,\lambda} d\lambda
\over {\int_0^\infty I_{inc,\lambda} d\lambda}}\, ,
\label{bondeq}$$ where the $\lambda$ subscript signifies that the incident intensity varies with wavelength.
Spherical, geometric, and Bond albedos of objects are strong functions of their compositions. Within the solar system, they vary substantially with wavelength, and from object to object. At short wavelengths, gaseous atmospheres can have high albedos due to Rayleigh scattering, and low albedos at longer wavelengths due to molecular ro-vibrational absorption. Icy condensates, whether they reside on a surface or are present in an upper atmosphere, are highly reflective and increase the albedo. Other condensates, such as silicates or non-equilibrium products of photolysis, can lower the albedo substantially over a broad wavelength region.
Some of the lowest albedos seen in the Solar System are those of asteroids containing large amounts of carbonaceous material. Many have Bond albedos of less than 0.03 (Lebofsky et al 1989). The Bond albedo of the Earth is 0.30 (Stephens et al. 1981) and that of the Moon is 0.11 (Buratti 1996). Jupiter and Saturn have somewhat higher Bond albedos, both near 0.35 (Conrath et al. 1989).
In §\[modelsec\], we describe our approach to modeling EGPs. Section \[transfersec\] describes our radiative transfer method, §\[moleculesec\] contains a discussion of molecular absorption and scattering, and §\[condensatesec\] describes the properties of and our treatment of the relevant condensates in EGP atmospheres. In §\[Jupitersec\], we discuss the application of our methods to Jupiter, §\[resultssec\] contains our EGP model albedo and reflection spectra results, as well as T$_{\textrm{eff}}$ and Bond albedo estimates for currently known EGPs, and §\[parametersec\] describes the effects of varying key parameters of the models. We summarize our results in §\[conclusionsec\].
Extrasolar Giant Planet Models \[modelsec\]
===========================================
Depending upon their proximity to their central stars as well as their masses and ages, EGP effective temperatures likely span a large range, from below 100 K to well over 1600 K, with highly varying temperature-pressure-composition profiles. However, an EGP’s outer atmospheric composition, rather than its specific temperature-pressure profile, is of primary importance in modeling albedos and reflection spectra. With our composition classes, we encompass the range of behaviors of EGP albedos and reflection spectra. We do not model emission spectra, nor do our models account for object-specific details, such as elemental abundance differences or cloud patchiness. EGPs are surely at least as rich and varied as the planets of our solar system, but simple modeling reveals many interesting systematics.
Temperature-Pressure Profiles
-----------------------------
Ideally, temperature (T)-pressure (P) profiles are computed directly via radiative equilibrium models of EGPs under stellar insolation. A move toward such models for very strong stellar insolation has been made by Seager & Sasselov (1998) and Goukenleuque et al. (1999), while for lower temperature objects, Marley et al. (1999) utilize T-P profiles of isolated EGPs. The main effect of stellar insolation on the T-P profile of an EGP is to make the outer atmosphere more nearly isothermal. Studies of strong stellar insolation conclude that a stratosphere does not exist in the high-temperature roasters (Seager & Sasselov 1998; Goukenleuque et al. 1999). However, it is not completely clear what might occur in the upper atmosphere if ultraviolet photochemical processes are fully modeled. Under solar insolation, Jupiter and Saturn do exhibit stratospheres, and we suspect that the Class I EGPs are likely to have stratospheres as well. In an albedo spectrum, the existence of a stratosphere is made manifest mainly by the scattering and absorption effects of non-equilibrium aerosols which reside there. Additionally, photochemical processes in the stratosphere may be the origin of “chromophores,” non-equilibrium solids which settle near or in the upper cloud layers and are largely responsible for the coloration of Jupiter.
To bracket the range of albedos under stellar insolation, we use two sets of pressure-temperature profiles. The first is a subset of profiles for theoretical isolated objects (Marley et al. 1999; Marley 1998; Burrows et al. 1997) with T$_{\textrm{eff}} \approx$ 130 K (representing an isolated Class I EGP), 250 K (Class II), 600 K (Class III), and 1200 K (Class IV). We estimate that these representative isolated T-P profiles are valid for surface gravities between $\sim 3 \times 10^3$ to $3 \times 10^4$ cm s$^{-2}$. A set of modified profiles is obtained by altering these isolated profiles to simulate a stellar insolated T-P profile by using the model results of Seager & Sasselov (1998) as a guide. To approximate the T-P profiles of the very hottest close-in objects (Class V), we scale the 1200 K profile up to 1700 K. We stress that these modified profiles are very approximate, but along with the isolated T-P profiles, they bracket a broad range of possible EGP T-P profiles.
Figure \[tpprofiles\] shows both the isolated and modified T-P profiles for Classes I through IV, as well as our modified Class V profile. Also shown are condensation curves, which indicate the highest temperatures and pressures at which species can condense. Cloud bases are located approximately where the profiles intersect the condensation curves (dotted curves). Class I (“Jovian”) objects contain both ammonia and deeper water cloud layers, while water is likely the only principal condensate present in the tropospheres of Class II objects. (As shown in Figure \[tpprofiles\], a thin ammonia haze layer might appear very high in the atmosphere for an isolated Class II T-P profile.) The Class III T-P profile doesn’t cross any principal condensation curves in the upper atmosphere, regardless of the level of stellar insolation. Finally, the Class IV and V roasters contain a silicate cloud deck and a deeper iron cloud deck throughout the full range of possible T-P profiles, though their cloud depths differ considerably.
Determination of Gaseous Abundances \[gasabunsec\]
--------------------------------------------------
Using the analytic formulae in Burrows & Sharp (1999), we calculate gaseous mixing ratios of the main compounds of carbon, oxygen, and nitrogen (CH$_4$, CO, H$_2$O, NH$_3$, N$_2$) over the full range of temperatures and pressures in the model EGP atmospheres. H$_2$ and He mixing ratios are set according to Anders & Grevesse (1989) solar abundances, and the H$_2$S mixing ratio is set in accordance with the Anders & Grevesse abundance of sulfur ($\sim 3 \times 10^{-5}$). The abundances of the alkali metals (Na, K, Rb, Cs), important in the Class III through Class V EGPs, are calculated numerically using the formalism of Burrows & Sharp (1999).
Overall, the effect of differences in the T-P profile on gaseous mixing ratios tends to be greatest for the Class IV objects due to the temperature and pressure dependences of neutral alkali metal mixing ratios and the fact that the T-P profiles are in the vicinity of the CH$_4$/CO and NH$_3$/N$_2$ equilibrium curves. From the standpoint of gaseous abundances, the T-P profiles have little effect on the albedos of cooler EGPs.
Cloud Modeling \[cloudsec\]
---------------------------
Our treatment of clouds in our fiducial EGP models assumes that the gaseous form of a condensable species is completely depleted above the cloud deck and that the species settles within the cloud layer in its condensed form. The base of the cloud resides where the T-P profile of the EGP meets the condensation curve of the given species, and the cloud top is simply set at one pressure scale height above the base. Not all of the given condensable within the cloud is in condensed form. Rather, at the base of the cloud, the gaseous form is assumed to be at the saturation vapor pressure. For a given abundance of a condensable, if we assume that the portion of the condensable which exceeds the saturation vapor pressure is entirely in condensed form, we will refer to this as “full condensation.” However, as in Jupiter’s outer atmosphere (see §\[Jupitersec\]), it is possible that the condensation fraction will be smaller. Hence, we retain the condensation fraction as a parameter. Furthermore, the particle size distributions in EGP atmospheres are impossible to ascertain at this point, so particle size remains a free parameter as well.
The standard model for Jupiter lends some support to our prescription for clouds. The base of Jupiter’s ammonia cloud deck resides approximately where its T-P profile meets the NH$_3$ condensation curve ($\sim$ 0.7 bar) and the cloud tops extend roughly one pressure scale height, to $\sim$ 0.3 bar (West et al. 1986; Griffith et al. 1992). Although present, NH$_3$ gas is largely depleted above the cloud layer.
In the case of silicate condensation, where the condensate and gas molecules are not identical (unlike NH$_3$ and H$_2$O), the condensate abundance is estimated by the Anders & Grevesse abundance of the limiting species. We use enstatite (MgSiO$_3$; though a number of other silicates with differing optical properties are certainly present), for which the limiting element is silicon. For the full condensation limit, it is assumed that the entire mass of silicon above the pressure of the cloud base settles into MgSiO$_3$ within the cloud.
Radiative Transfer Method \[transfersec\]
=========================================
Due to the forward scattering from condensates in EGP atmospheres, an appropriate radiative transfer method must allow for a forward-backward asymmetric scattering phase function. Although the conventional Feautrier method (e.g. Mihalas 1978) does not allow for such an asymmetry, a straightforward extension of this technique can be derived by separating the source function into upward- and downward-propagating rays (Mihalas 1980; Milkey et al. 1975).
At first thought, it may seem inappropriate to use a planar transfer code in the modeling of albedos and reflection spectra from spherical objects. However, it is fairly straightforward to derive the equivalence between uniform radiation from one direction onto a unit sphere and uniform radiation from 2$\pi$ steradians onto a plane with unit area. Hence, provided that we set the incident intensity to be uniform in angle, the spherical albedo is the ratio of the outward and incident fluxes.
The fundamental transfer equation is $$\mu {\partial I(\mu) \over \partial\tau} = I(\mu) - S(\mu),$$ where the source function is given by $$S(\mu) = {1\over 2}\sigma\int_{-1}^{1}R(\mu,\mu^\prime)I(\mu^\prime)d\mu^\prime$$ and the thermal term is neglected in this albedo study. $R(\mu,\mu^\prime)$ is the azimuth-independent angular redistribution function (azimuthal symmetry is assumed) and $\sigma$ is the single-scattering albedo, $\sigma = \sigma_{scat}/\sigma_{ext}$, where $\sigma_{scat}$ is the scattering cross section and $\sigma_{ext}$ is the extinction cross section. Separated into upward ($I^+$) and downward ($I^-$) components, the transfer equation becomes $$\mu{\partial I^+(\mu)\over\partial\tau} = I^+(\mu) - S^+(\mu)
\label{tplus}$$ and $$-\mu{\partial I^-(\mu)\over\partial\tau} = I^-(\mu) - S^-(\mu),
\label{tminus}$$ where the source functions are given by $$S^+(\mu) = {1\over 2}\sigma\int_0^1 \left[R(\mu,\mu^\prime) I^+(\mu^\prime) +
R(\mu,-\mu^\prime)I^-(\mu^\prime)\right]d\mu^\prime$$ and $$S^-(\mu) = {1\over 2}\sigma\int_0^1 \left[R(-\mu,\mu^\prime) I^+(\mu^\prime) +
R(-\mu,-\mu^\prime)I^-(\mu^\prime)\right]d\mu^\prime$$ for the $I^+$ and $I^-$ equations, respectively.
Forming symmetric and antisymmetric averages, and using the Feautrier variables, $u = {1\over 2}(I^+ + I^-)$ and $v = {1\over 2}(I^+ - I^-)$, eqs. (\[tplus\]) and (\[tminus\]) are rewritten as $$\mu{\partial v(\mu)\over\partial\tau} = u(\mu) - {1\over 2}\left[S^+(\mu)
+ S^-(\mu)\right]
\label{mudvdt}$$ and $$\mu{\partial u(\mu)\over\partial\tau} = v(\mu) - {1\over 2}\left[S^+(\mu)
- S^-(\mu)\right].
\label{mududt}$$
Since $R(\mu,\mu^\prime)$ depends only upon the angle between $\mu$ and $\mu^\prime$, the following symmetries exist: $$\begin{aligned}
R(\mu,\mu^\prime) = R(-\mu,-\mu^\prime) \\
R(-\mu,\mu^\prime) = R(\mu,-\mu^\prime).\end{aligned}$$
With the definitions, $R^+(\mu,\mu^\prime) = R(\mu,\mu^\prime) +
R(-\mu,\mu^\prime)$ and $R^-(\mu,\mu^\prime) = R(\mu,\mu^\prime) -
R(-\mu,\mu^\prime)$, eqs. (\[mudvdt\]) and (\[mududt\]) become $$\mu{\partial v\over\partial\tau} = u - {1\over 2}\sigma\int_0^1 R^+
(\mu,\mu^\prime)u(\mu^\prime)d\mu^\prime$$ and $$\mu{\partial u\over\partial\tau} = v - {1\over 2}\sigma\int_0^1 R^-(\mu,\mu^\prime)
v(\mu^\prime)d\mu^\prime.$$
This system of first-order equations is discretized for numerical computation by replacing the derivatives with difference quotients, and by substituting Gaussian quadrature sums for the integrals. The principal equations then become $$\mu_i{{v_{d,i}-v_{d-1,i}}\over \Delta\tau_d} = u_{d,i} - {1\over 2}\sigma
\sum_j\omega_j R^+(\mu_i,\mu_j) u_{d,j}
\label{disc1}$$ and $$\mu_i{{u_{d+1,i}-u_{d,i}}\over \Delta\tau_{d+{1\over 2}}} = v_{d,i}
-{1\over 2}\sigma\sum_j\omega_j R^-(\mu_i,\mu_j) v_{d,j}\, ,
\label{disc2}$$ where $d$ signifies a given depth zone ($d = 1,...,D$), and $i$ and $j$ signify angular bins ($i,j = 1,...,N$). $\Delta\tau_{d+{1\over 2}}$ equals $\tau_{d+1} - \tau_d$ and $\Delta\tau_d$ equals ${1\over 2}
(\Delta\tau_{d+{1\over 2}} + \Delta\tau_{d-{1\over 2}})$. To achieve numerical stability, $\Delta\tau$ is staggered by half a zone in eq. (\[disc2\]) relative to eq. (\[disc1\]). The $\omega_j$ are the Gaussian weights.
The upper boundary conditions are given by the relations, $u_{1,i} - v_{1,i} =
I^-_i$ and $$\mu_i{{u_{2,i}-u_{1,i}}\over\Delta\tau_{1\over 2}} = u_{1,i} - I^-_i
- {1\over 2}\sigma\sum_j\omega_j \left[u_{1,j}-I^-_j\right] R^-(\mu_i,\mu_j)\, ,$$ where $I^-_i$ and $I^-_j$ signify the incident intensity as a function of angle at the surface. We set $I^-$ to unity at all angles since only the [*ratio*]{} of the outward and inward fluxes determines the spherical albedo. The lower boundary conditions are given by $u_{D,i} + v_{D,i} = I^+_i$ and $$\mu_i{{u_{D,i}-u_{D-1,i}}\over\Delta\tau_{D-{1\over 2}}} = I^+_i - u_{D,i}
- {1\over 2}\sigma\sum_j\omega_j \left[I^+_j - u_{D,j}\right] R^-(\mu_i,\mu_j)\, ,$$ where $I^+_i$ and $I^+_j$ signify the outward-traveling intensity at the base of the atmosphere (set to zero in this study).
The system of equations can be represented by angle matrices (${\bf A}_d, {\bf B}_d, {\bf C}_d,...$) and column vectors (${\bf u}_d$ and ${\bf v}_d$) such that equations (\[disc1\]) and (\[disc2\]) can be written as $${\bf A}_d{\bf v}_{d-1} + {\bf B}_d{\bf u}_d + {\bf C}_d{\bf v}_d = 0$$ and $${\bf D}_d{\bf u}_d + {\bf E}_d{\bf v}_d + {\bf F}_d{\bf v}_{d+1} = 0.$$
Given D depth zones and N angles, the system results in a block matrix containing $[2\times D]^2$ submatrices, each of order N. Implementing the boundary conditions described above, this system is solved directly via LU decomposition and substitution.
Our atmosphere models utilize 100 optical depth zones with logarithmic sizing near the surface, where higher resolution is essential, and a continuous transition to linear zoning at depth. Sixteen polar angular bins per hemisphere are used.
Quantitative Comparison for Uniform Atmospheres
-----------------------------------------------
In order to test our asymmetric Feautrier code, we compare our resulting spherical albedos for uniform atmosphere models with those derived employing both Monte Carlo and analytic techniques. Van de Hulst (1974) derived a solution for the spherical albedo of a planet covered with a semi-infinite homogeneous cloud layer. Given a single-scattering albedo of $\sigma$ (= $\sigma_{scat}/\sigma_{ext}$) and a scattering asymmetry factor of $g
= <\cos\theta>$ (the average cosine of the scattering angle), van de Hulst’s expression for the spherical albedo is $$A_s \approx {(1 - 0.139s)(1 - s)\over {1 + 1.170s}}\, ,$$ where $$s = \left[{1 - \sigma}\over {1 - \sigma g}\right]^{1/2}.$$
Figure \[vancompare\] shows the spherical albedo of a homogeneous, semi-infinite atmosphere as a function of scattering asymmetry factor and single scattering albedo. Along with van de Hulst’s semi-analytic curves are our asymmetric Feautrier and Monte Carlo model results using a Henyey-Greenstein scattering phase function, $$p(\theta) = {{1-g^2}\over {(1+g^2-2g\cos\theta)^{3/2}}}.$$
For nearly all values of $g$ and $\sigma$, the agreement is very good, differing by under 1%. There are slightly larger variations when both $g$ and $\sigma$ approach unity due to the finite number of angles and depth zones used in our numerical models, but in actual planetary or EGP atmospheres, this corner of parameter space is rarely realized.
Real planetary atmospheres are usually highly stratified and the optical depth is a strong function of wavelength. Given the atmospheric temperature-pressure-composition profile, an appropriate conversion to optical depth is required. Assuming hydrostatic equilibrium and using an ideal gas equation of state, this conversion is $$d\tau = {\sigma_{ext}(P)\over{\textrm{g}\mu(P)}}dP,$$ where $\sigma_{ext}$ is the effective extinction cross section per particle at depth $P$, $\mu$ is the mean molecular weight, and g is the surface gravity (assumed constant because the depth of the effective atmosphere is a very small fraction of the planet’s radius).
Atomic and Molecular Scattering and Absorption \[moleculesec\]
==============================================================
The gases present in EGP atmospheres are many (Burrows and Sharp 1999). However, only some of them have the requisite abundances and cross sections at the temperatures and pressures of upper EGP atmospheres to have significant spectral effects in the visible and near-infrared. These species include H$_2$, CH$_4$, H$_2$O, NH$_3$, CO, and H$_2$S. Additionally, Na and K are important absorbers in Class III, IV, and V EGPs.
Of course, H$_2$ is the most abundant species, followed by helium. The dominant carbon-bearing molecule is a function of both temperature and pressure. Chemical equilibrium modeling (Burrows and Sharp 1999; Fegley & Lodders 1996) shows that, at solar metallicity, CH$_4$ will dominate over CO in most EGP atmospheres. At high temperatures, the CO abundance overtakes that of CH$_4$ ($\sim$ 1100 K at 1 bar; $\sim$ 1400 K at 10 bars). There is a similar transition for the nitrogen-bearing molecules: NH$_3$ dominates at low temperatures, but it is overtaken by N$_2$ at higher temperatures ($\sim$ 700 K at 1 bar; $\sim$ 900 K at 10 bars). Some species condense into solids at low temperatures, thereby depleting the gaseous phase. In particular, NH$_3$ condenses below 150–200 K (depending upon pressure), as does H$_2$O below 250–300 K.
At visible and near-infrared wavelengths, molecular absorption is due to ro-vibrational transitions, so molecular opacity is a very strong function of wavelength. Even when no permanent dipole moment exists, such as with the H$_2$ molecule, the high gas pressures in EGP atmospheres can induce temporary dipole moments via collisions. This Collision Induced Absorption (CIA) is responsible for broad H$_2$-H$_2$ (and H$_2$-He) absorption bands in Jupiter and Saturn (Zheng & Borysow 1995; Trafton 1967).
The temperature- and pressure-dependent gaseous opacities are obtained from a variety of sources—a combination of theoretical and experimental data as referenced in Burrows et al. (1997). Additionally, for this study the CH$_4$ opacity was extended continuously into the visible wavelength region using the data of Strong et al. (1993) and a methane absorption spectrum from Karkoschka (1994). These two data sets were then extrapolated in temperature and pressure by scaling with existing temperature and pressure-dependent near-infrared CH$_4$ data (Burrows et al. 1997 and references therein).
Many prominent molecular absorption features may be seen in EGP albedo and reflection spectra. At relatively low temperatures, broad H$_2$-H$_2$ and H$_2$-He CIA bands peak at $\sim$ 0.8 $\mu$m, 1.2 $\mu$m, and 2.4 $\mu$m. At higher temperatures and pressures, the CIA cross sections become larger at all wavelengths. CIA is especially important in cloud-free gaseous objects, where incident radiation is absorbed deeper in the atmosphere. NH$_3$ absorption bands shortward of 2.5 $\mu$m occur at $\sim$ 1.5 $\mu$m, 2.0 $\mu$m, and 2.3 $\mu$m. (Note that our database does not contain the visible bands of ammonia.) H$_2$O absorption occurs at $\sim$ 0.6 $\mu$m, 0.65 $\mu$m, 0.7 $\mu$m, 0.73 $\mu$m, 0.82 $\mu$m, 0.91 $\mu$m, 0.94 $\mu$m, 1.13 $\mu$m, 1.4 $\mu$m, 1.86 $\mu$m, and 2.6 $\mu$m. A large number of CH$_4$ features appear in the visible and near-infrared. Some of the more prominent ones occur at $\sim$ 0.54 $\mu$m, 0.62 $\mu$m, 0.67 $\mu$m, 0.7 $\mu$m, 0.73 $\mu$m, 0.79 $\mu$m, 0.84 $\mu$m, 0.86 $\mu$m, 0.89 $\mu$m, 0.99 $\mu$m, 1.15 $\mu$m, 1.4 $\mu$m, 1.7 $\mu$m, and 2.3 $\mu$m. CO absorption bands occur at $\sim$ 1.2 $\mu$m, 1.6 $\mu$m, and 2.3 $\mu$m and H$_2$S features may be found at $\sim$ 0.55 $\mu$m, 0.58 $\mu$m, 0.63 $\mu$m, 0.67 $\mu$m, 0.73 $\mu$m, 0.88 $\mu$m, 1.12 $\mu$m, 1.6 $\mu$m, and 1.95 $\mu$m. Of course, depending upon mixing ratios and cross sections, only some of these features will appear in a given EGP albedo spectrum.
Strong pressure-broadened lines of neutral sodium and potassium are expected to dominate the visible albedos of Class III and Class IV EGPs. The most prominent absorption lines of sodium occur at 3303 Å, 5890 Å, and 5896 Å, while those of potassium occur at 4044 Å, 7665 Å, and 7699 Å.
Atomic and molecular scattering includes conservative Rayleigh scattering as well as non-conservative Raman scattering. In the case of Rayleigh scattering, cross sections are derived from polarizabilities, which are in turn derived from refractive indices. Since the refractive indices are readily available at 5893 Å (Weast 1983), the Rayleigh cross sections are derived at this wavelength via, $$\sigma_{Ray} = {8\over 3}\pi k^4 \left({{n-1}\over{2\pi L_0}}
\right)^2\, ,$$ where $k$ is the wavenumber at this wavelength (2$\pi/\lambda \simeq 106621$ cm$^{-1}$) and $L_0$ is Loschmidt’s number. Assuming that the refractive indices are not strong functions of wavelength, we simply extrapolate these cross sections as $\lambda^{-4}$.
Raman scattering by H$_2$ involves the shift of continuum photons to longer or shorter wavelengths as they scatter off H$_2$, exciting or de-exciting rotational and vibrational transitions. Raman scattering is not coherent in frequency, so a rigorous treatment is not possible with our transfer code. Instead, we adopt the approximate method introduced by Pollack et al. (1986) and used by Marley et al. (1999) in their albedo study. At a given wavelength, the single scattering albedo within a particular depth zone is approximated by $$\sigma = {{\sigma_{Ray}+\sigma^{\prime}_{scat}+
(f_{{\lambda}^*}/f_{\lambda})\sigma_{Ram}}\over {\sigma_{Ray}+\sigma^{\prime}_{ext}
+\sigma_{Ram}}},$$ where $f_\lambda$ denotes the spectrum of incident radiation (the spectrum of an EGP’s central star), $\lambda^{* -1} = \lambda^{-1} + \Delta
\lambda^{-1}$, where $\Delta\lambda$ is the wavelength of the H$_2$ vibrational fundamental ($\Delta\lambda^{-1}$ = 4161 cm$^{-1}$), $\sigma_{Ram}$ is the Raman cross section, and $\sigma^{\prime}_{scat}$ and $\sigma^{\prime}_{ext}$ are the effective condensate scattering and extinction cross sections, respectively. Raman scattering may be significant in deep gaseous planetary atmospheres, where it can lower the UV/blue albedo by up to $\sim$ 15% (Cochran & Trafton 1978). However, our models show that in higher temperature EGP atmospheres, alkali metal absorption can dominate over this wavelength region, while in cooler EGP atmospheres, condensates largely dominate. Over our full set of EGP models, we find that Raman scattering is relatively insignificant.
Mie Theory and Optical Properties of Condensates \[condensatesec\]
==================================================================
Condensed species in EGP atmospheres range from ammonia ice in low temperature objects to silicate grains at high temperatures. Some of the condensates relevant to EGP atmospheres include NH$_3$ ($\lesssim$ 150–200K), NH$_4$SH ($\lesssim$ 200K), H$_2$O ($\lesssim$ 250–300K), low-abundance sulfides and chlorides ($\lesssim$ 700–1100K), silicates such as MgSiO$_3$ ($\lesssim$ 1600–1800K), and iron or iron-rich compounds ($\lesssim$ 1900–2300K). Additionally, photochemical processes in the upper atmosphere can produce non-equilibrium condensates. Stratospheric hazes may be composed of polyacetylene (Bar-Nun et al. 1988) and other aerosols. Chromophores, those non-equilibrium species which cause the coloration of Jupiter and Saturn, might include P$_4$ (Noy et al. 1981) or organic species similar to Titan tholin (Khare & Sagan 1984).
Condensates can have drastic effects on EGP reflection spectra, increasing the albedo at most wavelengths, but sometimes depressing the albedo in the UV/blue. Of course, those condensates which are higher in the atmosphere will generally have a greater effect than those which reside more deeply. The presence and location of a particular condensed species is determined largely by an object’s T-P profile, and by the tendency of the condensate to settle (due to rain) at a depth in the atmosphere near the region where the T-P profile crosses the condensation curve. Hence, a given low-temperature (T$_{\textrm{eff}} \lesssim$ 150K) atmosphere might consist of an ammonia cloud deck high in the troposphere and a water cloud deck somewhat deeper, with purely gaseous regions above, beneath, and between the clouds. Similarly, a high-temperature (T$_{\textrm{eff}} \sim$ 1200K) atmosphere might consist of a tropospheric silicate cloud deck above a deeper iron cloud deck. Depending upon the amount of condensate in the upper cloud and the wavelength region, the presence of deeper clouds may or may not have any effect on the albedo and reflection spectrum.
The scattering and absorption of electromagnetic radiation by condensed species in planetary atmospheres is a very complex problem. The extinction properties of ices, grains, and droplets of various sizes, shapes, and compositions cannot be described accurately by simple means. Most often, these properties are approximated by Mie Theory, which describes the solution of Maxwell’s equations inside and outside a homogeneous sphere with a given complex refractive index.
We use a full Mie Theory approach which utilizes the formalism of van de Hulst (1957) and Deirmendjian (1969), and results in scattering and extinction cross sections as well as a scattering asymmetry factor, $g = <\cos\theta>$, given the complex index of refraction and particle radius ($a$). Larger particles require an increasing number of terms in an infinite series to describe these parameters accurately, and so they require more computing time. But while the cross sections and scattering asymmetry factors of small- to moderately-sized particles ($2\pi a/\lambda \lesssim 75$) vary substantially with wavelength, these variations are greatly reduced for larger spheres. For these larger particles, we use an asymptotic form of the Mie equations outlined fully by Irvine (1965). Interpolation between the full Mie theory results and these asymptotic limits yields the parameters for large particles. However, inherent assumptions in the asymptotic form of the Mie equations render them inadequate for the computation of the scattering cross sections in the weak-absorption limit ($n_{imag} \lesssim 10^{-3}$), in which case we use the geometric optics approximation (Bohren & Huffman 1983), $$Q_{sca} = 2 - {8\over 3}{n_{imag}\over n_{real}}\left[n_{real}^3 - \left(
n_{real}^2 - 1 \right)^{3/2}\right]x \ ,$$ where $Q_{sca}$ is the usual scattering coefficient (the ratio of the scattering cross section to the geometric cross section), $x$ is the size parameter ($=2\pi a/\lambda$), $n_{real}$ is the real index of refraction, and $n_{imag}$ is the imaginary component of the refractive index.
The principal condensates to which we have applied Mie theory include NH$_3$ ice, H$_2$O ice, and MgSiO$_3$ (enstatite), where the optical properties, namely the complex indices of refraction, were obtained from Martonchik et al. (1984), Warren (1984), and Dorschner et al. (1995), respectively. The complex refractive indices of NH$_3$ were interpolated in the 0.7 to 1.4 $\mu$m wavelength region, due to the lack of data there.
Each of these condensates has absorption features, as is made evident by the behavior of the imaginary index of refraction (Figure \[indices1\]). Shortward of 2.5 $\mu$m, NH$_3$ ice absorption occurs at $\sim$ 1.55 $\mu$m, 1.65 $\mu$m, 2.0 $\mu$m, and 2.25 $\mu$m. H$_2$O ice produces broader features at $\sim$ 1.5 $\mu$m and 2.0 $\mu$m. Enstatite is mostly featureless below 2.5 $\mu$m, except shortward of $\sim$ 0.35 $\mu$m.
The non-equilibrium condensates to which we have applied Mie theory include phosphorus (Noy et al. 1981), tholin (Khare & Sagan 1984), and polyacetylene (Bar-Nun et al. 1988). P$_4$ and tholin are chromophore candidates, particularly for the coloration of Jupiter and Saturn, due to their large imaginary indices of refraction in the UV/blue (Figure \[indices2\]) and plausibility of production. A somewhat yellowish allotrope of phosphorus, P$_4$ was produced in the laboratory by Noy et al. (1981) by UV irradiation of an H$_2$/PH$_3$ gaseous mixture. It is believed that this same process may be responsible for its production in Jupiter. Tholin is a dark-reddish organic solid (composed of over 75 compounds) synthesized by Khare and Sagan (1984) by irradiation of gases in a simulated Titan atmosphere. It is believed that a tholin-like solid may be produced similarly in giant planet atmospheres. Polyacetylenes, polymers of C$_2$H$_2$, were investigated by Bar-Nun et al. (1988) and likely are an optically dominant species in the photochemical stratospheric hazes of giant planets, where hydrocarbons are abundant (Edgington et al. 1996; Noll et al. 1986).
Cloud particle sizes are not easily modeled and are a strong function of the unknown meteorology in EGP atmospheres. Inferred particle sizes in solar system giant planet atmospheres can guide EGP models, though they range widely from fractions of a micron to tens of microns.
We have investigated various particle size distributions. A commonly used distribution, and the one that we use in our fiducial models, is $$n(a) \propto \left({a\over a_0}\right)^6 \exp\left[-6\left({a\over a_0}\right)
\right],
\label{cloud}$$ which reproduces the distributions in cumulus water clouds in Earth’s atmosphere fairly well if the peak of the distribution is a$_0 \sim 4 \mu$m (Deirmendjian 1964). Stratospheric aerosols—at least those in Earth’s stratosphere—can be represented by the “haze” distribution (Deirmendjian 1964), $$n(a) \propto {a\over a_0}\exp\left[-2\left({a\over a_0}\right)^{1/2}\right].$$
The Albedo of Jupiter \[Jupitersec\]
====================================
Jupiter is an important testbed for the theory of albedos, since full-disk geometric albedo spectra have been obtained (Karkoschka 1994, 1998), and because space-based and ground-based studies have provided a fair amount of information concerning Jupiter’s atmosphere. At visible and near-infrared wavelengths, Jupiter’s upper troposphere and stratosphere shape its albedo spectrum. According to the standard model, a somewhat heterogeneous cloud deck extends from $\sim$ 0.3 to 0.7 bars in the troposphere (West et al. 1986; Griffith et al. 1992). Although the bulk of the cloud deck consists primarily of particles at least 10 $\mu$m in size, a layer of smaller particles ($\sim$ 0.5–1.0 $\mu$m) resides near the cloud tops (West et al. 1986; Pope et al. 1992). Beneath this upper cloud deck is a NH$_4$SH and NH$_3$ cloud layer at $\sim$ 2–4 bars and an H$_2$O cloud condenses somewhat deeper. Above the NH$_3$ cloud deck, a stratospheric haze resides at pressures near $\sim$ 0.1 bar. It is worth mentioning that the Galileo probe results deviate from this standard model. One difference relevant to the albedo and reflection spectra is a tropospheric haze inferred from the probe data, likely composed primarily of NH$_3$, above a somewhat deeper NH$_3$ cloud deck (Banfield et al. 1998), but it is not known whether the probe entry location is characteristic of the planet as a whole.
In addition to H$_2$, abundant gaseous species in the upper troposphere include He and CH$_4$, with mixing ratios relative to H$_2$ of 0.156 and $\sim 2.1 \times 10^{-3}$, respectively (Niemann et al. 1996). Gaseous NH$_3$, H$_2$O, H$_2$S, and PH$_3$ are present in small mixing ratios.
It is suggested that the color differences of Jupiter’s belts and zones are largely due to the visibility of chromophores residing within the NH$_3$ cloud deck (West et al. 1986). No appreciable altitude differences between the belts and zones are found, although the zones likely contain thicker upper cloud and/or haze layers than the belts (Chanover et al. 1997; Smith 1986). Jupiter’s UV/blue albedo is depressed substantially from what one would expect from the increase with frequency of the Rayleigh scattering cross sections, and Raman scattering cannot account for the albedo in this wavelength region. This depressed UV/blue albedo likely is produced by the large imaginary refractive indices of the tropospheric chromophores and, to a lesser degree, stratospheric aerosols (West et al. 1986).
Due to the large optical depth of Jupiter’s upper ammonia cloud deck at visible and near-infrared wavelengths, a two-cloud model of the atmosphere suffices (West 1979; Kuehn & Beebe 1993). We model the top of the upper cloud deck ($\sim$ 0.35 bar) with a “cloud” distribution (see §\[condensatesec\]) peaked at 0.5 $\mu$m. Deeper in the cloud, from 0.45 to 0.7 bar, a particle distribution peaked at 30 $\mu$m is used. This size distribution is also utilized in the lower cloud, spanning 2 to 4 bars.
In addition to NH$_3$ condensation, a small mixing ratio of a chromophore, either tholin ($2 \times 10^{-8}$) or P$_4$ ($5 \times 10^{-9}$), is added to the upper cloud. As inferred from limb darkening observations, the condensed chromophore becomes well mixed in the upper ammonia cloud, and perhaps deeper as well (West et al. 1986; Pope et al. 1992). As per Noy et al. (1981), the peak of the chromophore particle size distribution is set to 0.05 $\mu$m. However, the nature of the size distribution and whether the chromophore adheres to the ammonia ice particles are as yet unclear.
Gaseous abundances are modeled using the Galileo Probe Mass Spectrometer values (Niemann et al. 1996) as a guide. The H$_2$, He, and CH$_4$ abundances are taken directly from the Probe results. However, the tropospheric NH$_3$ abundance varies considerably with depth. At $\sim$ 0.4 bar, its mixing ratio has been found to be $\sim 5 \times 10^{-6}$ (Griffith et al. 1992; Kunde et al. 1982), while at $\sim$ 0.7 bar, its mixing ratio is $\sim 5 \times 10^{-5}$ to $10^{-4}$. In an infrared study using Voyager IRIS data (Gierasch et al. 1986), it was found that only a small fraction ($\sim$ 1%) of the ammonia is in condensed form. Based on our visible albedo modeling, where the smaller particle size distribution dominates, and using the gaseous NH$_3$ mixing ratios above, we find that a condensation fraction of $\sim$ 5% in the upper cloud is required to provide the necessary reflectance.
We model Jupiter’s stratospheric haze using Deirmendjian’s “haze” particle size distribution (see §\[condensatesec\]) of polyacetylene peaked at 0.1 $\mu$m—a particle size justified by limb darkening studies (Rages et al. 1997; West 1988; Tomasko et al. 1986). The abundance of C$_2$H$_2$ in Jupiter’s stratosphere is $\sim 10^{-8}$ to $10^{-7}$ (Edgington 1998; Noll et. al. 1986), though the polymerized abundance is not known. In this study, the polyacetylene mixing ratio is set to $5 \times 10^{-8}$, in a haze layer from 0.03 to 0.1 bar.
Figure \[Jupiter\] shows two model geometric albedo spectra along with the observed full-disk albedo spectrum of Jupiter (Karkoschka 1994). We convert our model spherical albedo to a geometric albedo using an averaged phase integral of $q$ = 1.25 (Hanel et al. 1981). The upper model utilizes tholin as the chromophore throughout the upper ammonia cloud deck, while the lower model utilizes P$_4$.
Although the general character of Jupiter’s geometric albedo is reproduced fairly well, many of the methane absorption features are modeled too deeply. Furthermore, the gaseous ammonia features at $\sim$ 0.65 $\mu$m and 0.79 $\mu$m do not appear in the models because our database does not include ammonia absorption shortward of $\sim$ 1.4 $\mu$m. Karkoschka (1998) indicates that the absorption feature centered at $\sim$ 0.93 $\mu$m may be due to ammonia as well. Relying upon Mie scattering theory and our choices for chromophore particle size distributions, tholin appears to reproduce the UV/blue region of the albedo better than P$_4$. However, the actual chromophore(s) in Jupiter’s atmosphere remains a mystery.
The published Bond albedo of Jupiter is 0.343 (Hanel et al. 1981). Using our models and limited wavelength coverage (0.3 $\mu$m to 2.5 $\mu$m), we estimate a Bond albedo (see §\[resultssec\]) in the 0.42 to 0.44 range—a fair approximation—depending upon whether P$_4$ or tholin is used as the chromophore.
Uncertainties in the vertical structure of Jupiter’s atmosphere, heterogeneities in Jupiter’s cloud layers, and our use of an averaged phase integral all likely play a role in explaining the differences between observational and modeled albedo spectra. These details aside, Jupiter’s atmosphere remains a useful benchmark for our models of EGP albedo and reflection spectra.
Results for EGPs \[resultssec\]
===============================
We produce fiducial albedo models for each EGP class using both isolated and modified temperature-pressure profiles. We adopt Deirmendjian’s “cloud” particle size distribution with a peak at the moderate size of 5 $\mu$m. Our model EGP spherical albedos for the full range of effective temperatures are shown in Figures \[spherical1\] through \[spherical3\]. For these fiducial models, “full condensation” is assumed (as described in §\[cloudsec\]).
The “Jovian” Class I albedo spectra are determined mainly by the reflectivity of condensed NH$_3$ and the molecular absorption bands of gaseous CH$_4$. Stratospheric and tropospheric non-equilibrium species are not included in these fiducial models. Their effects are explored in §\[parametersec\]. Because both isolated and nearly “isothermal” T-P profiles of EGPs with T$_{\textrm{eff}} \lesssim$ 150 K cross the NH$_3$ condensation curve, the details of the T-P profiles do not have a large impact on the resulting albedos of Class I objects. As shown in Figure \[spherical1\]a, the reflective NH$_3$ clouds keep the albedo fairly high throughout most of the visible spectral region. Toward the infrared, the gaseous absorption cross sections tend to become larger, so photons are more likely to be absorbed above the cloud deck. Hence, at most infrared wavelengths, the albedo is below that in the visible region.
The isolated and modified profile Class II albedos are shown in Figure \[spherical1\]b. Relative to a Class I EGP, a Class II albedo is even higher in the visible due to very strongly reflective H$_2$O clouds in the upper atmosphere. Gaseous absorption features tend to be shallower because these H$_2$O clouds form higher in the atmosphere than the NH$_3$ clouds of most Class I objects. The intersection of the isolated profile and the NH$_3$ condensation curve near 0.01 bars may result in a thin NH$_3$ condensation layer high in the atmosphere, but NH$_3$ condensation is assumed to be negligible for this model.
The “clear” Class III does not contain any principal condensates in the upper atmosphere (irrespective of the T-P profile), although a silicate cloud deck exists deeper, at $\sim$ 50 bars. The presence of alkali metals in the troposphere has a substantial lowering effect on the albedo. As per Figure \[spherical2\]a, sodium and potassium absorption lowers the albedo at short wavelengths, resulting in a spherical albedo below $\sim$ 0.6 throughout most of the UV/blue spectral region. Into the red region, lower Rayleigh scattering cross sections and strong alkali metal absorption result in spherical albedos which drop below 0.1. In contrast, in the absence of the alkali metals, the spherical albedo would remain high ($\gtrsim$ 0.75) throughout most of the visible. In both cases, the near-infrared albedo is essentially negligible, largely due to absorption by CH$_4$, H$_2$O, and H$_2$-H$_2$ CIA. Our models show that, in Class III objects, the details of the T-P profile will have only minor effects on the albedo. If low-abundance sulfide or chloride condensates were to exist in the troposphere, they could appear at pressures as low as a few bars. Based on theoretical abundances (Burrows & Sharp 1999), thick clouds are very unlikely, but it is worth mentioning that even cirrus-like condensation could raise the albedo in the visible and near-infrared.
In the higher-temperature (900 K $\lesssim$ T$_{\textrm{eff}} \lesssim$ 1500 K) Class IV roasters, the effect of the alkali metals is most dramatic. Unlike the Class III EGPs, a silicate cloud deck exists at moderate pressures of $\sim$ 5–10 bars, depending on the details of the T-P profile. An iron or iron-rich condensate likely exists below the silicate deck, but it is sufficiently below the opaque silicate cloud that it does not have any effect on the visible and near-infrared albedos. Figure \[spherical2\]b shows the spherical albedo of a Class IV EGP. Assuming a fairly “isothermal” T-P profile (the modified profile) in the upper atmosphere, absorption by sodium and potassium atoms, coupled with ro-vibrational molecular absorption, results in a surprisingly low albedo throughout virtually the entire visible and near-infrared wavelength region explored in this study ($\leq 2.5 \mu$m). The silicate cloud is deep enough that its effects are rendered negligible by the absorptive gases above it. Although a Class IV model with a modified T-P profile results in an albedo which is significantly lower than that of even a Class III model, the albedo of a Class IV model with an isolated T-P profile is a different story: Because the upper atmosphere in such a model is significantly cooler than in the modified T-P profile case, the equilibrium abundances of the alkali metals are lower. Furthermore, the silicate cloud deck is expected to be somewhat higher in the atmosphere (Figure \[tpprofiles\]) and to have a non-negligible effect on the albedo in both the visible and near-infrared regions (Figure \[spherical2\]b).
Due to the low ionization potentials of sodium (5.139 eV) and potassium (4.341 eV), it is likely that significant Na II and K II layers exist in the outer atmospheres of Class IV EGPs (and perhaps Class III EGPs). Nevertheless, assuming a silicate cloud layer at $\sim$ 5-10 bars, simple ionization equilibrium estimates indicate that these layers should not reach the depths of the silicate layer in Class IV EGPs, and so substantial column depths of Na I and K I should remain to absorb visible radiation. The full absorption and emission features of such ionization layers will be explored in future EGP studies.
The very hot (T$_{\textrm{eff}} \gtrsim$ 1500 K) Class V roasters have a silicate cloud layer which is located much higher in the atmosphere relative to the Class IV roasters, so alkali metal and molecular absorption is reduced. Figure \[spherical2\]b illustrates the much higher albedo expected of the Class V objects, assuming the silicate layer is composed predominantly of enstatite grains. If the sodium and potassium ionization layers are substantial in these objects, then the absorption due to their neutral lines will be reduced even further. We also note that a roaster of particularly low mass (e.g. HD 209458b) is expected to exhibit a significantly larger radius than such an object in isolation (Burrows et al. 2000). For such a low surface gravity ($\lesssim 10^3$ cm s$^{-2}$) object, the silicate layer will form high in the atmosphere even for T$_{\textrm{eff}}$ $< 1500$ K. Hence, the lower limit to T$_{\textrm{eff}}$ required to render a roaster a Class V EGP is reduced in the case of low surface gravity.
Via spectral deconvolution, Charbonneau et al. (1999) have constrained the geometric albedo of the roaster, $\tau$ Boo b, to be below 0.3 at 0.48 $\mu$m. This limit, which was obtained with an assumed phase function and orbital inclination near 90 degrees, contrasts with the findings of Cameron et al. (1999), who infer that the albedo is high in this region. Using our Class IV T-P profile model (T$_{\textrm{eff}} \lesssim$ 1500 K), we find that the geometric albedo at 0.48 $\mu$m is only 0.03. However, if in fact $\tau$ Boo b is a Class V EGP (T$_{\textrm{eff}} \gtrsim$ 1500 K), we derive a geometric albedo of 0.39 at 0.48 $\mu$m, still smaller than the assumed Cameron et al. value of 0.55, from which they derive a planetary radius as high as 1.8 Jupiter radii. The widely varying albedos of Classes IV and V coupled with the fact that $\tau$ Boo b appears to have an effective temperature near the transition region between these classes indicates that the detailed modeling of this EGP will be necessary in order to ascertain its nature. It is instructive to examine the temperatures and pressures to which incident radiation penetrates an EGP’s atmosphere as a function of wavelength. For each class, Figures \[tauoneP\] and \[tauoneT\] show the pressures and temperatures, respectively, corresponding to one mean free path of an incident photon. In clear atmospheres, these temperatures and pressures are very strong functions of wavelength, largely mirroring molecular absorption bands and/or atomic absorption lines. Conversely, when thick cloud layers are present, the wavelength dependence is much weaker, due to the efficient extinction of radiation by a size distribution of condensed particles.
Due to the azimuthal symmetry of our Feautrier technique, we do not compute the phase integrals of EGPs. In their absence, the characteristics of the atmosphere at $\tau_{\lambda}$ $\sim$ 1 are useful for the approximate conversion from spherical albedos to geometric albedos. Using the asymmetry factor and single scattering albedo values, the phase integral, $q_{\lambda}$, is estimated by interpolating within the tables of Dlugach & Yanovitskij (1974) (Marley et al. 1999). Geometric albedos are then obtained using the relation, $A_{g,\lambda} = A_{s,\lambda}/q_{\lambda}$. Estimated geometric albedo spectra are shown in Figure \[geoplot\]. Our Class II geometric albedo compares qualitatively with that of the “quiescent” water cloud model of Marley et al. (1999). Given the differences in particle size distributions, the Marley et al. albedo tends to fall off a bit more sharply with increasing wavelength, while having shallower gaseous absorption features in the visible. Our Class IV models may be compared with the Marley et al. “brown dwarf” model with silicate (enstatite) clouds, as well as with the 51-Peg b model of Goukenleuque et al. (1999). Our inclusion of the alkali metals results in a qualitatively very different, and much lower, albedo spectrum than in these previous studies. Our Class V model may be compared with the high-temperature model (T$_{\textrm{eff}}$ = 1580 K) of Seager & Sasselov (1998). We find that, similar to Seager & Sasselov, the presence of silicate (enstatite) grains results in significant reflection, but our inclusion of the alkali metals results in prominent absorption lines as well.
We combine a geometric albedo spectrum from each EGP class with appropriately calibrated stellar spectra (Silva & Cornell 1992) to produce representative EGP reflection spectra. Figure \[refplot\]a shows theoretical full-phase reflection spectra of EGPs from 0.35 $\mu$m to 1.0 $\mu$m, assuming a G2V central star, orbital distances of 0.05 AU (Class IV), 0.2 AU (Class III), 1.0 AU (Class II), and 5.0 AU (Class I), and a planetary radius of 1 Jupiter radius (R$_J$). For a Class IV roaster, at 0.45 $\mu$m, the ratio of reflected and stellar fluxes is $\sim$ $5 \times 10^{-6}$, while for Class I, II, and III EGPs, it is $\sim$ $5 \times 10^{-9}$, $10^{-7}$, and $10^{-6}$, respectively. This ratio at 0.65 $\mu$m drops to $\sim$ $5 \times 10^{-7}$ for a Class IV object, and is $\sim$ $5 \times 10^{-9}$, $10^{-7}$, and $3 \times 10^{-7}$ for Class I, II, and III EGPs, respectively. Figure \[refplot\]b shows theoretical full-phase reflection spectra of Class IV and V roasters, assuming an F7V central star, orbital distances of 0.1 AU (Class IV) and 0.04 AU (Class V), and 1 R$_J$. At 0.45 $\mu$m, the reflected to stellar flux ratios are $\sim$ $10^{-6}$ (Class IV) and $5 \times 10^{-5}$ (Class V). At 0.65 $\mu$m, these ratios are $\sim$ $10^{-7}$ (Class IV) and $5 \times 10^{-5}$ (Class V). For larger planetary radii and different orbital distances, these ratios should be scaled accordingly.
In the reflection spectrum of a Class IV object (Figure \[refplot\]), absorption by the resonance lines of sodium (5890Å/5896Å) and potassium (7665Å/7699Å) is extreme. These lines are also very significant, though substantially weaker, in Class III objects. Methane absorption bands shortward of 1 $\mu$m, especially those at $\sim$ 0.73 $\mu$m, 0.86 $\mu$m, and 0.89 $\mu$m, are quite prominent in Class I and III objects. These bands are also clearly present in Class II objects, but with sufficient water condensation high in the troposphere, the bands are not as prominent as in Class I or Class III EGPs. Although present, methane absorption is even weaker in Class IV EGPs, where CO is the dominant carbon-bearing molecule. At the high effective temperature of a Class V object (T$_{\textrm{eff}} \gtrsim$ 1500 K), the methane abundance is completely overwhelmed by that of CO, and we expect that no strong methane bands will be seen in reflection.
Bond albedos for EGPs are obtained using eq. (\[bondeq\]). Our lower and upper wavelength limits of integration are 0.3 $\mu$m and 2.5 $\mu$m, respectively, rather than formally from 0 to infinity. Hence, our derivations are estimates of actual Bond albedos, accurate to $\sim$ 10-15%, depending on the central stellar spectral type and the uncertainties in the EGP spherical albedos shortward of 0.3 $\mu$m and longward of 2.5 $\mu$m. The Bond albedos for our fiducial modified T-P profile models and for isolated T-P profile models are shown in Tables \[Bondtaba\] and \[Bondtabb\]. Assuming full condensation of principal condensates and no non-equilibrium species, the Bond albedos of Class I and II objects are high. Over the spectral range, M4V to A8V, the peak of the stellar energy flux ranges from $\sim$ 0.9 $\mu$m to 0.4 $\mu$m. Class I EGP Bond albedos range from $\sim$ 0.4 to 0.65, while those of Class II EGPs reach nearly 0.9. These albedos tend to be significantly lower when smaller condensation fractions and non-equilibrium condensates are considered. For example, the Bond albedo of our Jupiter model about a G2V central star is in the 0.42 to 0.44 range, depending upon whether P$_4$ or tholin is used as the chromophore—somewhat higher than Jupiter’s actual Bond albedo of 0.343 (Hanel et al. 1981).
In contrast, Bond albedos of Class III and IV EGPs are very low. Those of Class III objects vary from $\sim$ 0.01 to 0.2 over the spectral range, M4V to A8V. Class IV EGPs reflect the smallest fraction of incident radiation, with Bond albedos ranging from below 0.01 up to only 0.04, assuming our modified T-P profile model and no non-equilibrium condensates. These Bond albedos are significantly lower than those of Marley et al. (1999) because we include the effects of the alkali metals. For example, assuming a G2V central star, our fiducial Class III model yields a Bond albedo of 0.12, while those of Marley et al. are in the 0.31 to 0.33 range (cloud-free 500 K models), and our Bond albedo for a Class IV EGP is only 0.03, while those of Marley et al. are in the 0.30 to 0.44 range (cloud-free and cloudy 1000K models). The Bond albedos of the very hot Class V objects are much higher than those of Class III or IV, ranging from $\sim$ 0.51 to 0.57 over the the spectral range, M4V to A8V.
Estimated Bond albedos and effective temperatures of known EGPs are shown in Tables 2 through 4. The equilibrium temperature of an irradiated object is $$T_{\textrm{eq}} = \left[{(1-A_B)L_{*}\over {16\pi\sigma a^{2}}}\right]^{1/4}$$ (Saumon et al. 1996), where $L_{*}$ is the stellar luminosity, $\sigma$ is the Stefan-Boltzmann constant, and $a$ is the orbital distance of the planet. For massive and young EGPs with sufficiently large orbital distances, T$_{\textrm{eff}}$ $>$ T$_{\textrm{eq}}$ due to their significant internal energies. We estimate the effective temperatures of such objects simply by adding the stellar-insolated and internal contributions to the luminosity, and noting that $L = 4\pi R_p^2\sigma T_{\textrm{eff}}^4$. The internal contribution is defined to be the luminosity of an isolated object of the given mass and age, and it is found using the evolutionary models of Burrows et al. (1997).
Given the list of over two dozen known EGPs, it is possible that none is cold enough to be a Class I (“Jovian”) object (HR 5568b is an ambiguous case). Classes II, III, and IV are well-represented (Tables 2 through 4), while Class V likely includes HD 209458b, and perhaps $\tau$ Boo b and/or HD 75289b.
Parameter Studies \[parametersec\]
==================================
In EGP atmospheres, variations in condensation fractions and particle size distributions, as well as the possible presence of stratospheric and tropospheric non-equilibrium species, can have large effects on the spherical and Bond albedos. First, we consider the effects of lowering the condensation fraction to 10% and 1%. Figures \[condfrac\]a and \[condfrac\]b show the substantial changes in Class I (“Jovian”) and Class II (“water cloud”) EGPs. The Class II case best illustrates the systematic effects, since only an H$_2$O cloud deck exists. (Recall that the Class I model contains an ammonia cloud deck above a water cloud deck.) The condensation fraction has a substantial effect on the spherical and geometric albedos. Less condensation clearly results in lower albedos, especially in the red/near-infrared, where gaseous opacities are strong (Marley et al. 1999). Note that the effects of the alkali metals, deep in the atmosphere, are apparent in the UV/blue albedo of the Class II, 1% condensation model. As shown in Tables \[Bondtaba\] and \[Bondtabb\], the Bond albedos of these 1% condensation models are significantly lower than those of their “full condensation” counterparts, particularly for the Class II EGPs.
Cloud particle size distributions in EGPs are not known. As alluded to in §\[condensatesec\], for a given condensate abundance, the net extinction by condensates (almost pure scattering for H$_2$O ice) is smaller when particle sizes are larger. This is shown explicitly in Figure \[paramfig\]a, comparing spherical albedos for Deirmendjian H$_2$O ice “cloud” distributions with size peaks of 0.5 $\mu$m, 5 $\mu$m (fiducial), and 50 $\mu$m. The qualitative effect of increasing the peak size is similar to the effect of reducing the condensation fraction. Widening the distribution has similar consequences because the largest particles squander the condensate, reducing the number density of smaller scattering particles.
Non-equilibrium species in the upper atmospheres of EGPs may be produced by UV-induced processes. While both gaseous and condensed species are likely to be produced, the condensates will generally have greater effects on the albedos and reflection spectra. As in the atmosphere of Jupiter, stratospheric hazes and tropospheric chromophores, or impurities within or above the principal cloud layers, can lower the albedo spectra in the UV/blue range and can also modify their character at other wavelengths. In addition to their compositions, the size distributions of these non-equilibrium species play a role. Figure \[paramfig\]b shows the effect of including a representative upper tropospheric “haze” of tholin (with mixing ratio of 10$^{-8}$) on the spherical albedo of a Class I EGP. In analogy with our Jupiter model, the size distribution is peaked at 0.05 $\mu$m. Although the abundances and size distributions of such particles in EGPs are unknown, we present this model as an indication of the qualitative effect that this type of haze would have on the albedo. The associated Class I Bond albedo, assuming a G2V central star, decreases from 0.57 to 0.48.
We represent the optically dominant aerosol within stratospheric hazes by polyacetylene, although other possibilities certainly exist. Our models show that the effect of polyacetylene on the albedo is minor, lowering the UV/blue albedo no more than a few percent, assuming a mixing ratio as large as $10^{-7}$. We stress that the actual compositions, abundances, and the size distributions of non-equilibrium species in EGPs are unknown, and that the quantitative effects on EGP albedos may or may not be significant.
Conclusions \[conclusionsec\]
=============================
The classification of EGPs into five composition classes, related to T$_{\textrm{eff}}$, is instructive, since the albedos of objects within each of these classes exhibit similar features and values. The principal condensate in Class I “Jovian” EGPs (T$_{\textrm{eff}} \lesssim$ 150 K) is NH$_3$, while in Class II “water cloud” EGPs it is H$_2$O ice. Gaseous molecular absorption features, especially those of methane, are exhibited throughout Class I and II albedo spectra. Assuming adequate levels of condensation, Class II EGPs are the most highly reflective of any class. For lower condensation fractions, the albedos of both classes fall off more quickly with increasing wavelength relative to “full condensation” models—especially the Class II objects. Even a small mixing ratio of a non-equilibrium tropospheric condensate within or above a cloud deck can depress the UV/blue albedo and reflection spectrum significantly.
In Class III “clear” EGPs (T$_{\textrm{eff}} \gtrsim$ 350 K), little condensation is likely, and so albedos are determined almost entirely by atomic and molecular absorption and Rayleigh scattering. Radiation generally penetrates more deeply into these atmospheres, to pressures and temperatures where sodium and potassium absorption and H$_2$-H$_2$ collision-induced absorption (CIA) become substantial. Throughout most of the visible spectral region, the albedo decreases with increasing wavelength. In the near-infrared, CIA, H$_2$O, and CH$_4$ conspire to keep the albedo very low.
In the upper atmospheres of the high-temperature (900 K $\lesssim$ T$_{\textrm{eff}} \lesssim$ 1500 K) Class IV roasters, the equilibrium abundances of the alkali metals are higher than in the Class III EGPs, so the absorption lines of sodium and potassium are expected to lower the albedo more dramatically. A silicate cloud exists at moderate depths ($\sim$ 5–10 bars), but the large absorption cross sections of the sodium and potassium gases above it preclude the cloud from having a significant effect on the albedo. Like Class III EGPs, the near-infrared albedo is expected to remain close to zero in the absence of non-equilibrium condensates.
The hottest (T$_{\textrm{eff}} \gtrsim$ 1500 K) and/or lowest gravity (g $\lesssim 10^3$ cm s$^{-2}$) roasters (Class V) have a silicate layer located much higher in the atmosphere relative to the Class IV roasters. This layer is expected to reflect much of the incident radiation before it is absorbed by neutral sodium and potassium and molecular species. Hence, Class V EGPs have much higher albedos than those of Class IV.
While stratospheres generally are not anticipated in high temperature EGPs (Seager & Sasselov 1998; Goukenleuque et al. 1999), it is possible that more detailed modeling will show that they do exist. The presence of a stratosphere would give rise to visible and infrared emission features not otherwise seen. Furthermore, the presence of non-equilibrium solids due to photochemistry may decrease the albedo in the UV/blue, but increase it somewhat in the red/near-infrared because even largely absorbing condensates are more reflective than gaseous molecular species in this spectral region.
Differences in particle size distributions of the principal condensates can have large quantitative, or even qualitative effects on the resulting albedo spectra. In general, less condensation, larger particle sizes, and wider size distributions result in lower albedos.
Despite many uncertainties in the atmospheric details of EGPs, our set of model albedo spectra serves as a useful guide to the prominent features and systematics over a full range of EGP effective temperatures, from $\sim$ 100 K to 1700 K. Full radiative equilibrium modeling of a given EGP at a specific orbital distance from its central star (of given spectral type), and of specific mass, age, and composition is necessary for a detailed understanding of an object. However, as observational EGP spectra become available, our set of model albedo spectra offers a means by which a quick understanding of their general character is possible, and by which some major atmospheric constituents, both gaseous and condensed, may be inferred.
We thank Mark Marley, Sara Seager, Bill Hubbard, Jonathan Lunine, Christopher Sharp, and Don Huffman for a variety of useful contributions. This work was supported under NASA grants NAG5-7073 and NAG5-7499.
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[cccccc]{}
A8V & I & 0.63 & 0.64 & 0.62 & 0.59 & II & 0.88 & 0.88 & 0.79 & 0.47 & III & 0.17 & 0.13 & & & IV & 0.04 & 0.21 & & & V & 0.57 & & & F7V & I & 0.59 & 0.61 & 0.57 & 0.51 & II & 0.84 & 0.83 & 0.74 & 0.40 & III & 0.14 & 0.10 & & & IV & 0.03 & 0.18 & & & V & 0.56 & & & G2V & I & 0.57 & 0.59 & 0.55 & 0.47 & II & 0.81 & 0.81 & 0.71 & 0.37 & III & 0.12 & 0.09 & & & IV & 0.03 & 0.16 & & & V & 0.55 & & &
[cccccc]{}
G7V & I & 0.55 & 0.58 & 0.52 & 0.44 & II & 0.79 & 0.79 & 0.69 & 0.34 & III & 0.10 & 0.07 & & & IV & 0.02 & 0.15 & & & V & 0.55 & & & K4V & I & 0.48 & 0.52 & 0.44 & 0.33 & II & 0.70 & 0.70 & 0.60 & 0.25 & III & 0.05 & 0.04 & & & IV & $<0.01$ & 0.11 & & & V & 0.53 & & & M4V & I & 0.38 & 0.43 & 0.33 & 0.16 & II & 0.56 & 0.55 & 0.47 & 0.16 & III & 0.01 & $<0.01$ & & & IV & $<0.01$ & 0.08 & & & V & 0.51 & & &
[ccccccc]{}
Gl 876b & M4V & 1.9 & $\sim$0.2 & full & 0.56 & 180 &&&& 10% & 0.47 & 182 &&&& 1% & 0.16 & 199 HR 5568b & K4V & 0.75 & $\sim$1.0 & full & - & - &&&& 10% & - & - &&&& 1% & 0.25 & 160 HD 210277b & G7V & 1.28 & $\sim$1.15 & full & 0.79 & 177 &&&& 10% & 0.69 & 194 &&&& 1% & 0.34 & 232 HR 810b & G0V & 2.0 & $\sim$1.2 & full & 0.82 & 192 &&&& 10% & 0.72 & 213 &&&& 1% & 0.38 & 254
[ccccccc]{}
16 Cyg Bb & G2.5V & 1.66 & 1.7 & full & 0.81 & 158 &&&& 10% & 0.71 & 170 &&&& 1% & 0.37 & 198 47 UMa b & G0V & 2.4 & 2.1 & full & 0.82 & 160 &&&& 10% & 0.72 & 172 &&&& 1% & 0.38 & 199 $\upsilon$ And d & F7V & 4.61 & 2.50 & full & 0.84 & 228 &&&& 10% & 0.74 & 233 &&&& 1% & 0.40 & 247 Gl 614b & K0V & 3.3 & 2.5 & full & 0.75 & 168 &&&& 10% & 0.65 & 170 &&&& 1% & 0.30 & 177 55 Cnc c & G8V & $\sim$5 & 3.8 & full & 0.78 & 198 &&&& 10% & 0.68 & 199 &&&& 1% & 0.33 & 201
[cccccc]{}
HD 130322b & K0V & 1.08 & 0.08 & 0.07 & 810 55 Cnc b & G8V & 0.84 & 0.11 & 0.10 & 690 Gl 86 Ab & K1V & 4.9 & 0.11 & 0.07 & 660 HD 195019b & G3V & 3.4 & 0.14 & 0.12 & 720 HD 199263b & K2V & 0.76 & 0.15 & 0.07 & 540 $\rho$ Cr Bb& G0V & 1.13 & 0.23 & 0.13 & 670 HR 7875b & F8V & 0.69 & $\sim$0.25& 0.14 & 650 HD 168443b & G8IV & 5.04 & 0.277 & 0.10 & 620 HD 114762b & F9V & $\sim$10 & 0.38 & 0.13 & 510 70 Vir b & G4V & 6.9 & 0.45 & 0.11 & 380 $\upsilon$ And c&F7V& 2.11 & 0.83 & 0.14 & 370
[cccccc]{}
HD 187123b & G3V & 0.52 & 0.0415 & 0.03 & 1460 51 Peg b & G2.5V& 0.45 & 0.05 & 0.03 & 1240 $\upsilon$ And b&F7V& 0.71 & 0.059 & 0.03 & 1430 HD 217107b& G7V & 1.28 & 0.07 & 0.02 & 1030
|
---
abstract: |
Gabidulin codes over fields of characteristic zero were recently constructed by Augot *et al.*, whenever the Galois group of the underlying field extension is cyclic. In parallel, the interest in sparse generator matrices of Reed–Solomon and Gabidulin codes has increased lately, due to applications in distributed computations. In particular, a certain condition pertaining to the intersection of zero entries at different rows, was shown to be necessary and sufficient for the existence of the sparsest possible generator matrix of Gabidulin codes over finite fields. In this paper we complete the picture by showing that the same condition is also necessary and sufficient for Gabidulin codes over fields of characteristic zero.
Our proof builds upon and extends tools from the finite-field case, combines them with a variant of the Schwartz–Zippel lemma over automorphisms, and provides a simple randomized construction algorithm whose probability of success can be arbitrarily close to one. In addition, potential applications for low-rank matrix recovery are discussed.
author:
- |
**Hikmet Yildiz**^$*$^, **Netanel Raviv**$^\dagger$, and **Babak Hassibi**^\*^\
^\*^Department of Electrical Engineering, California Institute of Technology, Pasadena CA 91125\
$^\dagger$Department of Computer Science and Engineering, McKelvey School of Engineering,\
Washington University in Saint Louis, St. Louis, MO, 63130\
bibliography:
- 'refs.bib'
title: |
Support Constrained Generator Matrices of\
Gabidulin Codes in Characteristic Zero
---
Introduction
============
Over finite fields, Gabidulin codes [@delsarte1978bilinear; @gabidulin1985theory] can be seen as a rank-metric equivalent of Reed–Solomon codes, where instead of evaluating ordinary polynomials, one uses *linearized polynomials* (i.e., whose only nonzero coefficients are for monomials whose degree is a nonnegative integer power of the field characteristic). To properly generalize this definition to fields of characteristic zero, it was recently suggested in [@augot2018generalized] to employ $\theta$–polynomials, which are linear combinations of compositions of a generator $\theta$ of the underlying Galois group of the field extension (that must be cyclic).
Independently, there has been a surge of interest lately in constructing sparsest generator matrices for Reed–Solomon and Gabidulin codes [@yildiz2019gabidulin; @dau2014existence; @halbawi2014distributed; @yildiz2019optimum; @lovett2018mds], for several applications in distributed computing. Since the rows of a generator matrix are codewords, each row cannot contain more than $k-1$ zeros according to the Singleton bound, where $k$ is the dimension of the code. The so called GM–MDS conjecture, posed by [@dau2014existence] and solved by [@yildiz2019optimum] and [@lovett2018mds], asserts that this maximum number of zeros at every row is attainable, as long as a certain condition regarding the position of zeros is satisfied. Specifically, this condition requires the zero-entries at every set of rows to intersect in at most $k$ minus the number of rows in the intersection. In this paper we complete the picture by showing that the same condition is necessary and sufficient for the existence of sparse generator matrices for Gabidulin codes over fields of characteristic zero. We note that while the proof of the equivalent condition for Reed–Solomon codes is identical for finite fields and fields of characteristic zero, for Gabidulin codes this is *not* the case, and the proof from [@yildiz2019gabidulin] fails over the latter fields. However, by adopting notions from the Reed–Solomon equivalent (the “Simplified GM–MDS conjecture” [@yildiz2019optimum Thm. 3]), and combining with a variant of the well-known Schwartz–Zippel lemma, we are able to resolve the problem over fields of characteristic zero. Moreover, our proof also provides a randomized construction algorithm whose probability of success can be arbitrarily high; similar randomized construction algorithms exist for the finite variants of the problem, but their probability of success is lower.
Beyond their application in network coding [@silva2008rank], space-time codes [@lusina2003maximum], and cryptography [@gabidulin1991ideals], Gabidulin codes have applications in *low rank matrix recovery* [@muelich2017low] (LRMR), which is normally performed over fields of characteristic zero. In this problem, one reconstructs a low-rank matrix from a given set of linear measurements. If these linear measurements are given by multiplication of the unknown matrix by a parity-check matrix of a Gabidulin code, this problem reduces to syndrome decoding of the respective zero codeword. Since the parity-check matrix of a Gabidulin code has a similar structure to that of the generator matrix [@augot2018generalized Prop. 8], our results imply that when performing LRMR with Gabidulin codes, one may employ linear measurements that depend on a small number of entries of the unknown matrix.
The problem is formally stated in Section \[section:ProblemSetup\], alongside necessary mathematical background. Our main results are summarized in Section \[section:mainResults\], and proved in Section \[sec:proofs\] by using auxiliary claims given in Section \[section:MoreOn\].
Notations
---------
Let $[n]=\{1,2,\dots,n\}$. Denote the dimension of a subspace $V$ over a field $\mathsf F$ by $\dim_{\mathsf F} V$ and the span of the elements in a set $S$ over the field $\mathsf F$ by $\operatorname{span}_{\mathsf F} S$. The (total) degree of a (multivariate) polynomial $f$ is denoted by $\deg f$ (e.g. $\deg(x^2y^2+x^3)=4$). For an $m\times n$ matrix $\mathbf X$ and $I\subseteq[m],J\subseteq[n]$, $\mathbf X_{I,J}$ is the submatrix with the rows and columns indexed in $I$ and $J$ respectively. Let $\mathbf X_{I,:}=\mathbf X_{I,[n]}$ and $\mathbf X_{:,J}=\mathbf X_{[m],J}$ and when $I$ or $J$ has a single element, we sometimes write the element only instead of the set.
Problem Setup {#section:ProblemSetup}
=============
In this section we will first provide a brief background on cyclic Galois extensions. Then, we will define rank metric codes and Gabidulin codes. Finally, we will define our problem, namely, finding Gabidulin codes with support constrained generator matrices over a field of characteristic zero.
Field extensions
----------------
Let $\mathsf E/\mathsf F$ be a field extension of finite degree, i.e. the dimension of $\mathsf E$ as a vector space over $\mathsf F$ is finite, and let $\dim_{\mathsf F}\mathsf E=m$. The automorphism group of $\mathsf E/\mathsf F$, $\operatorname{Aut}(\mathsf E/\mathsf F)$, is the set of automorphisms of $\mathsf E$ that fix $\mathsf F$, i.e. $$\begin{aligned}
\operatorname{Aut}(\mathsf E/\mathsf F)=\{
&\theta : \mathsf E\to\mathsf E \text{ automorphism}\mid
\forall x\in\mathsf F, \theta(x)=x\},\end{aligned}$$ with the group operation of function composition $\circ$. If $\left|\operatorname{Aut}(\mathsf E/\mathsf F)\right|=m$, $\mathsf E/\mathsf F$ is called a Galois extension, in which case, $\operatorname{Aut}(\mathsf E/\mathsf F)$ is also denoted by $\operatorname{Gal}(\mathsf E/\mathsf F)$ and is called the Galois group of $\mathsf E/\mathsf F$.
In this paper, we will focus on cyclic Galois extensions, whose Galois group is a cyclic group of order $m$: $$\operatorname{Gal}(\mathsf E/\mathsf F)= \{\theta^0,\theta^1,\dots,\theta^{m-1}\}$$ where the automorphism $\theta$ is the generator and $\theta^{i+1}=\theta\circ\theta^i$ for every $i\geq 0$. Notice that $\theta^m=\theta^0$ is the identity automorphism.
For example, for finite fields, when $\mathsf F=\mathbb F_q$ and $\mathsf E=\mathbb F_{q^m}$, the Galois group is cyclic of order $m$ with the generator automorphism $\theta(x)=x^q$: $$\begin{aligned}
\operatorname{Gal}(\mathbb F_{q^m}/\mathbb F_q)
=\left\{x, x^q, x^{q^2},\dots, x^{q^{m-1}}\right\}.\end{aligned}$$ For infinite fields, when $\mathsf F=\mathbb Q$ is the set of rational numbers and $\mathsf E=\mathbb Q(\zeta_n)$, where $\zeta_n$ is the $n$’th root of unity, $\mathbb Q(\zeta_n)/\mathbb Q$ is a Galois extension of degree $\varphi(n)$, where $\varphi(n)$ is the Euler’s phi function ($\mathbb Q(\zeta_n)$ is called the $n$’th cyclotomic field and an interested reader can refer to [@marcus1977number]). Its Galois group is isomorphic to the multiplicative group $\mathbb Z^*_n$ of integers modulo $n$. Since $\mathbb Z_n^*$ is cyclic for $n=p^a,2p^a$ [@Mathworld], where $p$ is any odd prime and $a$ is any positive integer, it follows that for these values of $n$ we have that $\mathbb{Q}(\zeta_n)$ is a cyclic Galois extension of degree $m=\varphi(n)=p^{a-1}(p-1)$. It is also possible to define cyclic extensions of $\mathbb Q$ for any degree $m$ by considering subfields of $\mathbb Q(\zeta_p)$ for an odd prime $p$ such that $p-1$ is divisible by $m$.
Rank metric codes
-----------------
A linear rank metric code, $[n,k,d]_{\mathsf E/\mathsf F}$, over a field extension $\mathsf E/\mathsf F$ is an $\mathsf E$–subspace $\mathcal C$ of $\mathsf E^n$ of dimension $k$ with the rank distance $$\begin{aligned}
\label{eq:rankdistance}
d=d_R(\mathcal C)\triangleq \min_{0\neq \mathbf c\in\mathcal C}\dim_{\mathsf F}(\operatorname{span}_{\mathsf F}\{c_1,\dots,c_n\})\end{aligned}$$ where $c_1,\dots,c_n\in\mathsf E$ represent the entries of $\mathbf c\in\mathsf E^n$. By fixing an ordered basis of $\mathsf E$ over $\mathsf F$, the elements of $\mathsf E$ can be considered as vectors in $\mathsf F^m$, and then the codewords (i.e. the elements of $\mathcal C\subset\mathsf E^n$) can be viewed as $m\times n$ matrices over $\mathsf F$. Then, this definition of the rank distance in (\[eq:rankdistance\]) is equivalent to the minimum of the rank of the matrix representation of a nonzero codeword.
Notice that by definition in (\[eq:rankdistance\]), the rank distance of $\mathcal C$ can be upper bounded by the Hamming distance, $d_H(\mathcal C)\triangleq \min_{0\neq c\in\mathcal C}\|c\|_0$, where $\|c\|_0$ is the number of nonzero entries of $c$. Therefore, the Singleton bound can be written for the rank distance as well: $$\begin{aligned}
\label{equation:dRledH}
d_R(\mathcal C)\leq d_H(\mathcal C)\leq n-k+1.\end{aligned}$$ The codes with $d_R(\mathcal C)=n-k+1$ are called maximum rank distance (MRD), for which we write $[n,k]_{\mathsf E/\mathsf F}$ by omitting $d$. A generator matrix for an $[n,k,d]_{\mathsf E/\mathsf F}$ code $\mathcal C$ is a $k\times n$ matrix over $\mathsf E$ whose rows form a basis for $\mathcal C$.
Gabidulin codes
---------------
Gabidulin codes are defined as the row space of the $k\times n$ matrix $$\begin{aligned}
\label{eq:gabidulin}
\begin{bmatrix}
\theta^0(x_1) & \theta^0(x_2) & \cdots & \theta^0(x_n)\\
\theta^1(x_1) & \theta^1(x_2) & \cdots & \theta^1(x_n)\\
\vdots & \vdots & & \vdots\\
\theta^{k-1}(x_1) & \theta^{k-1}(x_2) & \cdots & \theta^{k-1}(x_n)
\end{bmatrix}\in\mathsf E^{k\times n}\end{aligned}$$ where $\theta\in\operatorname{Aut}(\mathsf E/\mathsf F)$ and $x_1,\dots,x_n\in\mathsf E$ are $\mathsf F$–linearly independent (notice that this requires $n\leq m=\dim_{\mathsf F}\mathsf E$). Note that Gabidulin codes can be seen as evaluation codes of the so-called $\theta$–polynomials; a $\theta$–polynomial is a function $f:\mathsf E\to\mathsf E$ of the form $f(x)=\sum_i f_i \theta^i(x)$ for $f_i\in\mathsf{E}$, and every codeword in a Gabidulin code is the evaluations of some $\theta$–polynomial of $\theta$–degree at most $k-1$. Note also that the generator matrix can be chosen as the product of any $k\times k$ invertible matrix over $\mathsf E$ and the matrix in (\[eq:gabidulin\]).
Originally, this was defined by Delsarte [@delsarte1978bilinear] and Gabidulin [@gabidulin1985theory] for the finite fields, when $\mathsf F=\mathbb F_q$, $\mathsf E=\mathbb F_{q^m}$, and $\theta(x)=x^q$, as the first general constructions of MRD codes over finite fields. Later [@augot2018generalized], it was extended to fields of characteristic zero and it was shown that when $\mathsf E/\mathsf F$ is a cyclic Galois extension and $\theta$ is the generator of $\operatorname{Gal}(\mathsf E/\mathsf F)$, this extension of Gabidulin codes also gives an $[n,k]_{\mathsf E/\mathsf F}$ MRD code [@augot2018generalized]. In the rest of the paper, we will assume that $\mathsf E/\mathsf F$ is a cyclic Galois extension of order $m$ and $\mathsf{F}$ is of characteristic zero.
Problem definition
------------------
We consider the problem of finding an $[n,k]_{\mathsf E/\mathsf F}$ MRD code whose generator matrix $\mathbf G\in\mathsf E^{k\times n}$ has support constraints. We describe the support constraints through the subsets $\mathcal Z_1,\mathcal Z_2,\dots,\mathcal Z_k\subset[n]$ as $$\begin{aligned}
\label{eq:zeroconstraints}
\mathbf G_{ij} = 0,\qquad \forall j\in\mathcal Z_i, i=1,2,\dots,k.\end{aligned}$$
Over finite fields, this problem was studied in [@yildiz2019gabidulin] and it was shown that a necessary and sufficient condition for the existence of MRD codes under support constraints described by the $\mathcal Z_i$ is $$\begin{aligned}
\label{eq:ineqcond}\textstyle
\left|\bigcap_{i\in\Omega}\mathcal Z_i\right|+|\Omega|\leq k,
\quad\forall\varnothing\neq\Omega\subseteq[k].\end{aligned}$$ The same condition also appears in the GM–MDS conjecture for MDS codes (i.e. $d_H=n-k+1$, see [@dau2014existence], and also [@yan2013algorithms; @halbawi2014distributed]) which was proven in [@yildiz2019optimum] and [@lovett2018mds].
Over infinite fields, the fact that is necessary can be shown similar to [@yildiz2019optimum], since MRD codes are also MDS , and since the proof in [@yildiz2019optimum] applies to both finite and infinite fields. However, a similar proof to [@yildiz2019gabidulin] cannot be applied to show that (\[eq:ineqcond\]) is sufficient when $\mathsf F$ has characteristic zero. The reason is that in finite fields, since the generator matrix in consists of entries in the form of polynomials in the $x_i$’s, which, in one step of the proof, allows to reduce the problem to a similar one with a smaller parameter, whereas in the characteristic zero, the entries are in the form of $\theta$–polynomials (defined in [@augot2018generalized]) and applying the same step turns the problem into one of a different kind. Hence, in this paper, we will show that (\[eq:ineqcond\]) is sufficient for the existence of $[n,k]_{\mathsf E/\mathsf F}$ MRD codes under the support constraints on the generator matrix given in (\[eq:zeroconstraints\]) when $\mathsf F$ has characteristic zero.
Main Results {#section:mainResults}
============
In this section, we present our main results on the existence of MRD codes in characteristic zero (see Theorem \[thm:main\]) and the best achievable rank distance for the cases where there does not exist any (see Corollary \[corollary\]). Also, we will give a randomized algorithm for the code construction. The proofs of the theorems will be given in Section \[sec:proofs\].
\[thm:main\] Let $\mathsf E/\mathsf F$ be a cyclic Galois extension of degree $m$ such that $\mathsf F$ has characteristic zero. For some $k\leq n\leq m$, let $\mathcal Z_1,\dots,\mathcal Z_k\subset[n]$ satisfy (\[eq:ineqcond\]). Then, there exists an $[n,k]_{\mathsf E/\mathsf F}$ Gabidulin code with a generator matrix satisfying the constraints in (\[eq:zeroconstraints\]).
If the $\mathcal Z_i$ do not satisfy (\[eq:ineqcond\]), then as given in [@yildiz2019gabidulin] and [@yildiz2019optimum], $d_R\leq d_H\leq n+1-\max\limits_{\varnothing\neq\Omega\subseteq[k]}\left(\left|\bigcap_{i\in\Omega}\mathcal Z_i\right|+|\Omega|\right) < n-k+1$ and hence, an MRD code does not exist. For this case, Corollary \[corollary\] below (which is the analog of [@yildiz2019gabidulin Thm. 2]) shows that this upper bound is achievable by the subcodes (i.e., the subspaces) of Gabidulin codes.
\[corollary\] In Theorem \[thm:main\], if the $\mathcal Z_i$ do not satisfy (\[eq:ineqcond\]), then there exists an $[n,k,n-\ell+1]_{\mathsf E/\mathsf F}$ subcode of an $[n,\ell]_{\mathsf E/\mathsf F}$ Gabidulin code, which satisfies (\[eq:zeroconstraints\]), where $$\begin{aligned}
\textstyle
\ell = \max\limits_{\varnothing\neq\Omega\subseteq[k]}
\left(\left|\bigcap_{i\in\Omega}\mathcal Z_i\right|+|\Omega|\right)
\end{aligned}$$
Define $\mathcal Z_{k+1}=\cdots=\mathcal Z_{\ell}=\varnothing$. Then, for any nonempty $\Omega\subseteq[\ell]$, we have that $\left|\bigcap_{i\in\Omega}\mathcal Z_i\right| + |\Omega|\leq\ell
$. Hence, by Theorem \[thm:main\], there exists an $[n,\ell,n-\ell+1]_{\mathsf E/\mathsf F}$ Gabidulin code with an $\ell\times n$ generator matrix $\mathbf G$ having zeros dictated by $\mathcal Z_1,\dots, \mathcal Z_{\ell}$. The first $k$ rows of $\mathbf G$ will generate a subcode whose rank distance $d_R$ is as good as the Gabidulin code: $d_R\geq n-\ell+1$. Furthermore, $n-\ell+1$ is an upper bound on $d_H$ [@yildiz2019optimum]. Therefore, $n-\ell+1\leq d_R\leq d_H\leq n-\ell+1$. Hence, $d_R=n-\ell+1$.
Code Construction
-----------------
Fix an $\mathsf F$–basis $\{b_1,\dots,b_m\}$ for $\mathsf E$ and assume that the conditions for the $\mathcal Z_i$ in Theorem \[thm:main\] are satisfied, i.e. $\mathcal Z_1,\dots,\mathcal Z_k\subset[n]$ satisfy (\[eq:ineqcond\]). Then, each $\mathcal Z_i$ has at most $k-1$ elements by applying (\[eq:ineqcond\]) with $|\Omega|=1$. In [@dau2014existence Thm. 2] and [@yildiz2019gabidulin Corollary 3], it is shown that one can keep adding elements to these sets from $[n]$ without violating any of the inequalities in (\[eq:ineqcond\]) until each $\mathcal Z_i$ has exactly $k-1$ elements. Note that adding elements to these sets will only put more zero constraints on the generator matrix. Therefore, without loss of generality, we can assume that $|\mathcal Z_i|=k-1$ for all $i$ along with (\[eq:ineqcond\]). Then, we construct a generator matrix for a rank metric code in a randomized manner as described below:
**Inputs:** A finite nonempty set $S\subset\mathsf F$ and subsets $\mathcal Z_1,\dots,\mathcal Z_k\subset[n]$ satisfying (\[eq:ineqcond\]).
**Steps:**
- Add elements to the $\mathcal Z_i$’s from $[n]$ (if necessary) by following the algorithm given in [@dau2014existence Thm. 2] so that they all have *exactly* $k-1$ elements and *still* satisfy (\[eq:ineqcond\]).
- Choose $(\gamma_{ij})_{i\in[n],j\in[m]}$ uniformly at random from $S$.
- Let $x_i=\sum_{j=1}^m\gamma_{ij}b_j$ for $i\in[n]$.
- Construct $\mathbf A\in\mathsf E^{k\times n}$ as in (\[eq:gabidulin\]) in terms of $x_1,\dots,x_n$.
- Define $\mathbf T\in\mathsf E^{k\times k}$ as $$\begin{aligned}
\label{eq:matrixt}
\mathbf T_{ij} =
\det\begin{bmatrix}\mathbf e_j&\mathbf A_{:,\mathcal Z_i}\end{bmatrix},\quad i,j\in[k]
\end{aligned}$$ where $\mathbf e_j$ is the column vector with $1$ at the $j$th entry and $0$’s elsewhere (Note that $|\mathcal Z_i|=k-1$).
**Output:** The generator matrix $\mathbf G=\mathbf T\cdot\mathbf A\in\mathsf E^{k\times n}$.
By Lemma \[lemma:general\] below, $\mathbf G$ in the above construction is guaranteed to satisfy (\[eq:zeroconstraints\]) for any inputs.
\[lemma:general\] Let $\mathcal Z_1,\dots,\mathcal Z_k\subset[n]$ be subsets of size $k-1$. For a given $k\times n$ matrix $\mathbf A$, a $k\times k$ matrix $\mathbf T$ (over the same field as $\mathbf A$) satisfying $(\mathbf T\cdot\mathbf A)_{ij}=0$ for every $j\in\mathcal Z_i$ and $i\in[k]$ can be given as in (\[eq:matrixt\]).
For a fixed $i\in[k]$, the statement $(\mathbf T\cdot \mathbf A)_{ij}=0$ for every $j\in\mathcal Z_i$ is equivalent to the equation $\mathbf T_{i,:}\cdot\mathbf A_{:,\mathcal Z_i}=0$. A solution $\mathbf T_{i,:}$ to this equation can be described in terms of the adjugate of the $k\times k$ square matrix $\mathbf P=\begin{bmatrix} 0_{k\times 1}& \mathbf A_{:,\mathcal Z_i}\end{bmatrix}$. Recall that $\operatorname{adj}\mathbf P$ is the transpose of the cofactor matrix $\left[(-1)^{i+j}\det (\mathbf P_{[k]\backslash\{i\},[k]\backslash\{j\}})\right]_{i,j\in[k]}$ and satisfies $\operatorname{adj}(\mathbf P)\mathbf P=\det(\mathbf P)\mathbf I_{k\times k}$. Since $\mathbf P$ has an all zero column, we have $\det\mathbf P=0$, which implies $\operatorname{adj}(\mathbf P)\mathbf P=0$. Furthermore, due to the zero column in $\mathbf P$, the entries of $\operatorname{adj}\mathbf P$ are zero except the first row, whose entries are for $j\in[k]$, $$\begin{aligned}
(\operatorname{adj}\mathbf P)_{1,j} &= (-1)^{j+1}\det (\mathbf P_{[k]\backslash\{j\},[k]\backslash\{1\}})\\
&= (-1)^{j+1}\det (\mathbf A_{[k]\backslash\{j\},\mathcal Z_i})\\
&= \det\begin{bmatrix}\mathbf e_j & \mathbf A_{:,\mathcal Z_i}\end{bmatrix}=\mathbf{T}_{i,j}.
\end{aligned}$$ Since $(\operatorname{adj}\mathbf P)_{1,:}\cdot\mathbf P=0$ and $(\operatorname{adj}\mathbf P)_{1,:}\cdot\mathbf A_{:,\mathcal Z_i}=0$, the row vector $\mathbf T_{i,:}=(\operatorname{adj}\mathbf P)_{1,:}$ satisfies $\mathbf T_{i,:}\cdot \mathbf A_{:,\mathcal Z_i}=0$.
Furthermore, if $x_1,\dots,x_n$ are $\mathsf F$–linearly independent and the matrix $\mathbf T$ is invertible (i.e. $\det\mathbf T \neq 0)$, then the code generated by $\mathbf G$ is an $[n,k]_{\mathsf E/\mathsf F}$ Gabidulin code since the row spaces of $\mathbf A$ and $\mathbf G=\mathbf T\cdot\mathbf A$ are identical. In Theorem \[thm:construction\], we give a lower bound on the probability of this construction giving an MRD code.
\[thm:construction\] If the conditions in Theorem \[thm:main\] are satisfied, then, the generator matrix $\mathbf G$ randomly constructed as described above will satisfy (\[eq:zeroconstraints\]) and generate an $[n,k]_{\mathsf E/\mathsf F}$ Gabidulin code with probability at least $1-\frac{n+k(k-1)}{|S|}$.
Since $\mathsf F$ is infinite, $S$ can be arbitrarily large. Therefore, the probability of constructing an MRD code can be arbitrarily close to $1$.
Furthermore, if the $\mathcal Z_i$ do not satisfy (\[eq:ineqcond\]), then by following the proof of Corollary \[corollary\], we can construct a rank metric code achieving the largest possible rank distance for the given support constraints.
More on Cyclic Galois Extensions {#section:MoreOn}
================================
Before moving to the proofs of the theorems, in this section, we will give some useful properties of the automorphisms in $\operatorname{Gal}(\mathsf E/\mathsf F)=\{\theta^0,\theta^1,\dots,\theta^{m-1}\}$.
Linear independence of the elements in $\mathsf E$
--------------------------------------------------
Lemma \[lemma:lindep\] below lists some equivalent conditions to the $\mathsf F$–linear dependence of the elements of $\mathsf E$ in terms of the automorphisms in $\operatorname{Gal}(\mathsf E/\mathsf F)$. The first two of these conditions can be also seen as a special case of [@augot2018generalized Prop. 5], where the authors give equivalent rank metrics for the elements of $\mathsf E^n$, whereas Lemma \[lemma:lindep\] only claims these rank metrics simultaneously declare rank deficiency (i.e. returns a rank less than $n$) for a given element of $\mathsf E^n$. It is worth noting, as shown by Augot *et al.* [@augot2018generalized], that the assumption that the extension $\mathsf E/\mathsf F$ is cyclic plays an important role in Lemma \[lemma:lindep\]. This is since its proof relies on the fact that $\theta$ fixes *only* the elements of $\mathsf F$ (i.e. for any $x\in\mathsf E$, $\theta(x)=x$ if and only if $x\in\mathsf F$), which is the case for the cyclic extensions.
\[lemma:lindep\] Let $n\leq m=\dim_{\mathsf F}\mathsf E$, $x_1,\dots,x_n\in\mathsf E$, and $$\begin{aligned}
\label{eq:matrixm}
\mathbf M=\begin{bmatrix}
\theta^0(x_1) & \theta^0(x_2) & \cdots & \theta^0(x_n)\\
\theta^1(x_1) & \theta^1(x_2) & \cdots & \theta^1(x_n)\\
\vdots & \vdots & & \vdots\\
\theta^{m-1}(x_1) & \theta^{m-1}(x_2) & \cdots & \theta^{m-1}(x_n)
\end{bmatrix}\in\mathsf E^{m\times n}\end{aligned}$$ Then, the following are equivalent:
(i) $x_1,\dots,x_n$ are $\mathsf F$–linearly dependent.
(ii) The columns of $\mathbf M$ are $\mathsf E$–linearly dependent.
(iii) The top $n\times n$ minor of $\mathbf M$ is zero, i.e. $\det\mathbf M_{[n],[n]}=0$.
If $x_i=0$ for some $i$ then the claim is trivial, and hence assume that $x_i\neq 0$ for every $i$.
$(ii)\implies(i)$: Let $\ell$ be the minimum number of columns of $\mathbf{M} $ that are $\mathsf E$–linearly dependent and w.l.o.g. assume that $$\begin{aligned}
\mathbf M_{:,\ell}=\sum_{i=1}^{\ell-1}\beta_i\mathbf M_{:,i}
\end{aligned}$$ for some unique $\beta_1,\dots,\beta_{\ell-1}\in\mathsf E$, which implies that $\theta^{j-1}(x_\ell)=\sum_{i=1}^{\ell-1}\beta_i\theta^{j-1}(x_i)$ for every $j\in[m]$. Then, applying $\theta$ to both sides gives $\theta^{j}(x_{\ell})=\sum_{i=1}^{\ell}\theta(\beta_i)\theta^{j}(x_i)$, which implies $\mathbf M_{:,\ell}=\sum_{i=1}^{\ell-1}\theta(\beta_i)\mathbf M_{:,i}$ as $\theta^m=\theta^0$. Since the $\beta_i$’s are unique it follows that $\theta(\beta_i)=\beta_i$, which implies $\beta_i\in\mathsf F$. Since $\theta^0(x)=x$, we have $x_{\ell}=\sum_{i=1}^{\ell}\beta_ix_i$ for $\beta_i\in\mathsf F$.
$(iii)\implies (ii)$: If the top $n\times n$ minor of $\mathbf M$ is zero, then there exists $\ell\leq n$ such that the $\ell$’th row of $\mathbf M$ is in the $\mathsf E$–span of the first $\ell-1$ rows. By induction, it can be shown that for any $i\geq\ell$, the $i$’th row is in the span of the first $\ell-1$ rows. To see how, assume for some $\beta_1,\dots,\beta_{\ell-1}\in\mathsf E$, $\theta^{i-1}(x_j)=\sum_{t=1}^{\ell-1}\beta_t\theta^{t-1}(x_j)$ for all $j$. Then, by applying $\theta$ to both sides, it follows that the $(i+1)$’th row is a linear combination of the first $\ell$ rows; hence it is also in the span of the first $\ell-1$ rows. As a result, $\operatorname{rank}\mathbf M\leq\ell-1< n$, which implies $(ii)$.
$(i)\implies (iii)$: Assume that $\sum_{i=1}^n\beta_ix_i=0$ for some $\beta_i\in\mathsf F$. Then, for any $j$, applying $\theta^j$ to both sides yields $\sum_{i=1}^n\beta_i\theta^j(x_i)=0$ since $\theta^j(\beta_i)=\beta_i$, which implies $(iii)$.
Schwartz–Zippel Lemma for automorphisms
---------------------------------------
Recall the Schwartz–Zippel Lemma, which states that for a nonzero multivariate polynomial $f$ in $n$ variables over a field, a point uniformly chosen at random from $S^n$, where $S$ is a nonempty finite subset of this field, will be a root of $f$ with probability at most $\frac{\deg f}{|S|}$. In this section, we will give an extension of Schwartz–Zippel Lemma for a special type of functions from $\mathsf E^n$ to $\mathsf E$. More precisely, for a given multivariate polynomial $f$ over $\mathsf E$ in $mn$ variables (seen as an $m\times n$ matrix), we will consider the function $g(x_1,\dots,x_n)=f([\theta^{i-1}(x_j)]_{i\in[m],j\in[n]})$ and give a bound on the probability of a randomly chosen point being a zero of $g$. Later, this will help us to derive the bound on the probability given in Theorem \[thm:construction\].
\[lemma:schwartz\_zippel\] Let $\{b_1,\dots,b_m\}$ be an $\mathsf F$–basis for $\mathsf E$. Let $f$ be a nonzero multivariate polynomial over $\mathsf E$ in $mn$ variables. Let $\mathbf M\in\mathsf E^{m\times n}$ be defined as in (\[eq:matrixm\]) for $x_j=\sum_{i=1}^m\mathbf\Gamma_{ij}b_i$, where the $\mathbf\Gamma_{ij}$ are independently uniformly chosen at random from a finite nonempty subset $S\subset\mathsf F$. Then, $$\begin{aligned}
\mathbb P(f(\mathbf M)= 0) \leq \frac{\deg f}{|S|}.
\end{aligned}$$
Define another polynomial $f'$ as $
f'(\mathbf X) = f(\mathbf B\mathbf X)
$ in the variables $\mathbf X_{ij}$, $i\in[m],j\in[n]$, where $\mathbf B=[\theta^{i-1}(b_j)]_{i,j\in[m]}$ is an ${m\times m}$ matrix defined as in (\[eq:matrixm\]) for $b_1,\dots,b_m$. Since $\{b_1,\dots,b_m\}$ is an $\mathsf F$–basis, the $b_i$ are $\mathsf F$–linearly independent and by Lemma \[lemma:lindep\], $\mathbf B$ is invertible. Then, $f$ can be also written as $
f(\mathbf X) = f'(\mathbf B^{-1}\mathbf X)
$. Hence, $f'$ is also nonzero and $\deg f=\deg f'$. Furthermore, $
f'(\mathbf\Gamma)
=f(\mathbf B\mathbf\Gamma)
=f(\mathbf M)
$ since $$\begin{aligned}
\mathbf M_{ij}
&= \theta^{i-1}(x_j)\\
&= \theta^{i-1}\left(\textstyle\sum_{t=1}^mb_t\mathbf\Gamma_{tj}\right)\\
&= \textstyle\sum_{t=1}^m\theta^{i-1}(b_t)\mathbf\Gamma_{tj}\\
&= (\mathbf B\mathbf\Gamma)_{ij}
\end{aligned}$$ where we use $\theta^{i-1}(\mathbf\Gamma_{tj})=\mathbf\Gamma_{tj}$ since $\mathbf\Gamma_{tj}\in\mathsf F$. Now, applying the Schwartz–Zippel Lemma to the polynomial $f'$ gives $\mathbb P(f'(\mathbf\Gamma)=0)\leq\frac{\deg f'}{|S|}$. Hence, $\mathbb P(f(\mathbf M)=0)\leq\frac{\deg f}{|S|}$.
Proofs of Theorem \[thm:main\] and Theorem \[thm:construction\] {#sec:proofs}
===============================================================
First of all, notice that it is sufficient to prove Theorem \[thm:construction\] since it implies Theorem \[thm:main\] when $S$ is chosen sufficiently large. Assume $x_1,\dots,x_n$ are chosen as described in Theorem \[thm:construction\]. We know that the code with the generator matrix $\mathbf T\cdot\mathbf A$, which satisfies (\[eq:zeroconstraints\]) by Lemma \[lemma:general\], is an $[n,k]_{\mathsf E/\mathsf F}$ Gabidulin code if the $x_i$’s are $\mathsf F$–linearly independent and $\mathbf T$ is invertible. Define $\mathbf M\in\mathsf E^{m\times n}$ as in Lemma \[lemma:lindep\], by which the $x_i$’s are $\mathsf F$–linearly independent iff $\det\mathbf M_{[n],:}\neq 0$. Furthermore, since $\mathbf A = \mathbf M_{[k],:}$, we have that $$\begin{aligned}
\mathbf T &= \left[\det\begin{bmatrix}\mathbf e_j&\mathbf A_{:,\mathcal Z_i}\end{bmatrix}\right]_{i,j\in[k]}
= \left[\det\begin{bmatrix}\mathbf e_j&\mathbf M_{[k],\mathcal Z_i}\end{bmatrix}\right]_{i,j\in[k]}.\end{aligned}$$ Therefore, it is sufficient to show that $\mathbb P(\det\mathbf T\cdot\det\mathbf M_{[n],:}\neq 0)\geq 1-\frac{n+k(k-1)}{|S|}$ or that $\mathbb P(\det\mathbf T\cdot\det\mathbf M_{[n],:}= 0)\leq \frac{n+k(k-1)}{|S|}$.
In order to show this, we will appeal to Lemma \[lemma:schwartz\_zippel\]. Define the multivariate polynomial $$\begin{aligned}
\label{eq:polyf}
f(\mathbf X) = \det\left(\left[\det\begin{bmatrix}\mathbf e_j&\mathbf X_{[k],\mathcal Z_i}\end{bmatrix}\right]_{i,j\in[k]}\right)\cdot \det\mathbf X_{[n],:}\end{aligned}$$ for the variables $\mathbf X_{ij}$, $i\in[m],j\in[n]$ seen as an $m\times n$ matrix $\mathbf X$. Then, it suffices to show that $\mathbb P(f(\mathbf M)=0)\leq\frac{n+k(k-1)}{|S|}$. Hence, by Lemma \[lemma:schwartz\_zippel\], all we need to show is that $f$ is a nonzero polynomial with total degree at most $n+k(k-1)$.
To show the bound on the degree of $f$, recall the Leibniz formula for the determinant of an $n\times n$ square matrix $\mathbf Z$, which is $\det\mathbf Z = \sum_{\pi\in S_n}\operatorname{sgn}(\pi)\prod_{i=1}^n\mathbf Z_{\pi(i),i}$, where $S_n$ is the permutation group of size $n$ and $\operatorname{sgn}(\pi)$ is the sign of the permutation $\pi$. Thus, when the entries of $\mathbf Z$ are polynomials, we can write $$\begin{aligned}
\deg\det\mathbf Z\leq \sum_{j\in[n]}\max_{i\in[n]}\deg\mathbf Z_{i,j}.\end{aligned}$$ Hence, $\deg\det\mathbf X_{[n],:}\leq n$ since each entry of $\mathbf X$ has degree one. Furthermore, $\deg\det\begin{bmatrix}\mathbf e_j&\mathbf X_{[k],\mathcal Z_i}\end{bmatrix}\leq k-1$; hence, $\deg\det\left(\left[\det\begin{bmatrix}\mathbf e_j&\mathbf X_{[k],\mathcal Z_i}\end{bmatrix}\right]_{i,j\in[k]}\right)\leq k(k-1)$. As a result, $\deg f\leq n+k(k-1)$.
To show that $f$ is a nonzero polynomial, we will use the simplified GM–MDS conjecture of Dau *et al.* [@dau2014existence], which was proved in [@yildiz2019optimum] and [@lovett2018mds].
\[lemma:gmmds\] Let $\mathcal Z_1,\dots,\mathcal Z_k\subset[n]$ be subsets of size $k-1$. Then, they satisfy (\[eq:ineqcond\]) if and only if the determinant of the $k\times k$ matrix $$\begin{aligned}
\label{eq:matrixp}
\mathbf P &=
\begin{bmatrix}
\prod_{t\in\mathcal Z_1}(-\alpha_t)& \cdots & \sum_{t\in\mathcal Z_1}(-\alpha_t) & 1\\
\prod_{t\in\mathcal Z_2}(-\alpha_t) & \cdots & \sum_{t\in\mathcal Z_2}(-\alpha_t) & 1\\
\vdots & &\vdots & \vdots\\
\prod_{t\in\mathcal Z_k}(-\alpha_t) & \cdots & \sum_{t\in\mathcal Z_k}(-\alpha_t) & 1
\end{bmatrix}
\end{aligned}$$ with entries $\mathbf P_{ij} = \sum_{\mathcal S\subseteq \mathcal Z_i, |\mathcal S|=k-j}\prod_{t\in \mathcal S}(-\alpha_t)$ is not the zero polynomial in the variables $\alpha_1,\dots,\alpha_n$.
Notice that the $i$’th row of $\mathbf P$ in (\[eq:matrixp\]) consists of the coefficients of the polynomial $$\begin{aligned}
\label{equation:PisPoly}
\prod_{j\in\mathcal Z_i}(X-\alpha_j)=\sum_{j=1}^k\mathbf \mathbf \mathbf P_{ij}X^{j-1}\end{aligned}$$ in the variable $X$. We will also show that $\mathbf P$ can be written in the form of (\[eq:matrixt\]). To see how, define the $m\times n$ Vandermonde matrix $\mathbf V=\left[\alpha_j^{i-1}\right]_{i\in[m],j\in[n]}$. Fix $i\in[k]$ and consider the determinant of the $k\times k$ Vandermonde matrix $\mathbf W=\begin{bmatrix}\mathbf v & \mathbf V_{[k],\mathcal Z_i}\end{bmatrix}$, where $\mathbf v$ is a column vector whose $j$’th entry is $X^{j-1}$ for $j\in[k]$: $$\begin{aligned}
\det\mathbf W = c_i\prod_{j\in\mathcal Z_i}(X-\alpha_j) \overset{\eqref{equation:PisPoly}}{=} c_i\sum_{j\in[k]}\mathbf P_{ij}X^{j-1}\end{aligned}$$ where $c_i=\prod_{j_1<j_2\in\mathcal Z_i}(\alpha_{j_1}-\alpha_{j_2})\neq 0$. On the other hand, by the linearity of the determinant in the first column, we can write $$\begin{aligned}
\det\mathbf W = \sum_{j\in[k]}\det\begin{bmatrix}\mathbf e_j & \mathbf V_{[k],\mathcal Z_i}\end{bmatrix} X^{j-1},\end{aligned}$$ since $\mathbf v=\sum_{j\in[k]}\mathbf e_jX^{j-1}$. As a result, the entries of $\mathbf P$ satisfy $$\begin{aligned}
\label{eq:entriesP}
c_i\mathbf P_{ij} = \det\begin{bmatrix}\mathbf e_j & \mathbf V_{[k],\mathcal Z_i}\end{bmatrix}\end{aligned}$$
Now, let us evaluate $f$ in (\[eq:polyf\]) at $\mathbf V$, which will give a multivariate polynomial in the variables $\alpha_j$: $$\begin{aligned}
f(\mathbf V)
&= \det\left(\left[\det\begin{bmatrix}\mathbf e_j&\mathbf V_{[k],\mathcal Z_i}\end{bmatrix}\right]_{i,j\in[k]}\right)\cdot \det\mathbf V_{[n],:}\\
&\overset{\eqref{eq:entriesP}}{=} \det\left(\left[c_i\mathbf P_{ij}\right]_{i,j\in[k]}\right)\cdot \det\mathbf V_{[n],:}\\
&= \det\mathbf P\cdot\left(\prod_{i\in[k]}c_i\right)\cdot\det\mathbf V_{[n],:}.\end{aligned}$$ By Lemma \[lemma:gmmds\], $\det \mathbf P$ is a nonzero polynomial. Furthermore, we have that $c_i\neq 0$ and $\det\mathbf V_{[n],:}=\prod_{j_1<j_2\in[n]}(\alpha_{j_1}-\alpha_{j_2})\neq 0$. Hence, $f(\mathbf V)$ is not the zero polynomial in the variables $\alpha_j$. Therefore, $f(\mathbf X)$ itself cannot be the zero polynomial in the variables $\mathbf X_{ij}$.
|
---
abstract: 'A discrete completeness relation and a continuous one with a positive measure are found for the photon-added squeezed vacuum states. Extension to the photon-added squeezed one-photon states is considered. Photon-added coherent states on a circle are introduced. Their normalization and unity resolution relation are given.'
author:
- |
C. Quesne [^1]\
[*Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles,*]{}\
[*Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium*]{}
date:
title: |
\
Completeness of photon-added squeezed vacuum and one-photon states and of photon-added coherent states on a circle
---
plus 1pt minus 1pt
PACS: 03.65.Fd, 42.50.Dv
Keywords: squeezed states, photon-added states, completeness relations
Corresponding author: C. Quesne, Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus de la Plaine CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium
Telephone: 32-2-6505559
Fax: 32-2-6505045
E-mail: [email protected]
Introduction
============
Coherent states of the harmonic oscillator are known to have properties similar to those of the classical radiation field [@glauber; @klauder; @sudarshan]. There also exist states of the electromagnetic field whose properties, such as squeezing [@slusher], higher-order squeezing [@hong], antibunching [@kimble], and sub-Poissonian statistics [@short], are strictly quantum mechanical in nature. These states are called nonclassical. Since the introduction of squeezed coherent states in the early seventies [@stoler], many nonclassical states of the radiation field have been constructed.
Among the latter, a class of states has attracted an ever increasing attention: the so-called photon-added (or excited) states, which are obtained by repeated application of photon creation operators on a given state and are distinct from displaced or squeezed number states (for a review of the latter see e.g. [@nieto]). The earliest example was the photon-added coherent states (PACS), introduced by Agarwal and Tara [@agarwal91]. They were soon followed by the photon-added squeezed vacuum state (PASVS), constructed by Zhang and Fan [@zhang]. Since then photon-added squeezed coherent states [@xin], photon-added thermal states [@agarwal92], photon-added even (PAECS) and odd (PAOCS) coherent states [@dodonov], for instance, have been considered. Various methods of generating such states have also been proposed [@agarwal91; @welsch].
Some photon-added states have been interpreted in the context of nonlinear coherent states related to a deformed oscillator [@matos] or their generalizations [@mancini]. Such is the case for the PACS and the PASVS, which were shown to be nonlinear coherent states [@sivakumar] and even nonlinear coherent states [@liu], respectively. Similarly, the photon-added squeezed one-photon states (PASOPS) may be considered as odd nonlinear coherent states [@liu].
Most of the theoretical studies of photon-added states have concentrated so far on displaying their nonclassical properties, while leaving aside more fundamental questions, such as their completeness. It should be stressed however that proving their completeness is important both from theoretical and applied viewpoints. This property, together with normalizability and continuity in the label, indeed makes them qualify as generalized coherent states according to Klauder’s prescription [@klauder] on one hand, and allows one to use them as (nonorthogonal) bases in many applications, on the other hand.
The completeness of PACS has recently been proved by Sixdeniers and Penson [@sixdeniers01]. In the present letter, we consider the case of the PASVS and PASOPS, as well as that of the photon-added coherent states on a circle (PACSC), generalizing the PAECS and PAOCS of Ref. [@dodonov].
Definition and properties of photon-added squeezed vacuum states
================================================================
The PASVS are defined by [@zhang] $$|\zeta, m\rangle = \left[N_m(|\zeta|)\right]^{-1/2} (\ap)^m |\zeta\rangle,
\label{eq:PASVS}$$ where $m = 0$, 1, 2, …, $\ap$, $a$ are photon creation and annihilation operators satisfying the relation $[a, \ap] = I$, $N_m(|\zeta|)$ is some normalization coefficient, and $$|\zeta\rangle = S(z) |0\rangle, \label{eq:SVS}$$ with $|0\rangle$ the vacuum state (i.e., $a |0\rangle = 0$). In (\[eq:SVS\]), $S(z)$ is the squeezing operator $$S(z) = e^{\frac{1}{2} \left[z (\ap)^2 - \zz a^2\right]} = e^{\frac{1}{2} \zeta
(\ap)^2} \left(1 - |\zeta|^2\right)^{\frac{1}{2} \left(N + \frac{1}{2}\right)} e^{-
\frac{1}{2} \zzeta a^2},$$ where $N = \ap a$ is the number operator and $\zeta$ is related to $z$ through the relations $$z = r e^{{\rm i} \phi}, \qquad \zeta = \tanh r e^{{\rm i} \phi}. \label{eq:polar}$$ Hence, $\zeta$ is restricted to the unit disc ($|\zeta| < 1$) when $z$ runs over the complex plane. In explicit form, the squeezed vacuum state (\[eq:SVS\]) can be rewritten as $$|\zeta\rangle = \left(1 - |\zeta|^2\right)^{1/4} e^{\frac{1}{2} \zeta (\ap)^2}
|0\rangle. \label{eq:SVSbis}$$ In the limit $\zeta \to 0$ (resp. $m \to 0$), the state $|\zeta, m\rangle$ reduces to the number state $|m\rangle = (m!)^{-1/2} (\ap)^m |0\rangle$ (resp. the squeezed vacuum state $|\zeta\rangle$).
From (\[eq:SVSbis\]), it follows that the expansion of the states (\[eq:PASVS\]) in the number-state basis $|n\rangle$, $n=0$, 1, 2, …, is given by $$|\zeta, m\rangle = \left[N_m(|\zeta|)\right]^{-1/2} \left(1 - |\zeta|^2\right)^{1/4}
\sum_{k=0}^{\infty} \frac{\sqrt{(2k+m)!}}{k!}\, \left(\case{1}{2} \zeta\right)^k
|2k + m\rangle. \label{eq:PASVS-exp}$$ Hence, for a given $m$ value, the states $|\zeta, m\rangle$ belong to the subspace ${\cal F}^{(m)}_{\mu}$ of Fock space $\cal F$, spanned by the states $|2k +
m\rangle$, $k=0$, 1, 2,…, with a photon number not less than $m$ and of the same parity as $m = \mu\, {\rm mod} 2$.
The overlap $\langle \xi, n | \zeta, m \rangle$ of two PASVS vanishes except if $|n-m|$ is an even integer. If $n-m$ is a nonnegative even integer, the overlap can be written in any one of the three following equivalent forms, $$\begin{aligned}
\langle \xi, n | \zeta, m \rangle & = & \left[N_m(|\zeta|) N_n(|\xi|)\right]^{-1/2}
\left[\left(1 - |\zeta|^2\right) \left(1 - |\xi|^2\right)\right]^{1/4}
\frac{n!}{\left(\frac{n-m}{2}\right)!}\, (\case{1}{2} \zeta)^{(n-m)/2}
\nonumber \\
&& \mbox{} \times {}_2F_1 \left(\frac{n+1}{2}, \frac{n+2}{2}; \frac{n-m}{2} + 1;
\xxi \zeta\right) \nonumber \\
& = & \left[N_m(|\zeta|) N_n(|\xi|)\right]^{-1/2} \langle \xi | \zeta \rangle
\frac{n!}{\left(\frac{n-m}{2}\right)!}\, (\case{1}{2} \zeta)^{(n-m)/2}
(1 - \xxi \zeta)^{-(n+m)/2}\nonumber \\
&& \mbox{} \times {}_2F_1 \left(- \frac{m-1}{2}, - \frac{m}{2}; \frac{n-m}{2} + 1;
\xxi \zeta\right) \nonumber \\
& = & \left[N_m(|\zeta|) N_n(|\xi|)\right]^{-1/2} \langle \xi | \zeta \rangle\, n!\,
\xxi^{(m-n)/4} \zeta^{(n-m)/4} (1 - \xxi \zeta)^{-(m+n)/4}\nonumber \\
&& \mbox{} \times P^{(m-n)/2}_{(m+n)/2} \left((1 - \xxi \zeta)^{-1/2}\right),
\label{eq:PASVS-overlap} \end{aligned}$$ where $\langle \xi | \zeta \rangle$ is the overlap of two squeezed vacuum states, $$\langle \xi | \zeta \rangle = \left[\left(1 - |\zeta|^2\right) \left(1 - |\xi|^2\right)
\right]^{1/4} (1 - \xxi \zeta)^{-1/2}.$$ The first equality in (\[eq:PASVS-overlap\]) directly follows from (\[eq:PASVS-exp\]) and the definition of the hypergeometric function ${}_2F_1(a, b; c; z)$, while the other two equalities result from well-known properties of the latter and of Legendre functions of the first kind $P^{\mu}_{\nu}(z)$ [@erdelyi; @prudnikov]. If $n-m$ is a negative even integer, the corresponding overlap can be deduced from (\[eq:PASVS-overlap\]) by using the Hermiticity property $\langle \xi, n |
\zeta, m \rangle = \overline{\langle \zeta, m | \xi, n \rangle}$.
As a special case of (\[eq:PASVS-overlap\]), we get back the overlap $\langle \zeta, n
| \zeta, m \rangle$ determined in [@zhang] by a different method. We also obtain the normalization coefficient of the PASVS, $$N_m(|\zeta|) = m!\, \left(1 - |\zeta|^2\right)^{-m/2} P_m \left((1 - |\zeta|^2)^{-1/2}
\right),$$ in terms of Legendre polynomials.
The states $|\zeta, m\rangle$ are distinct from the squeezed number states [@nieto] $$|m, \zeta\rangle = S(z) |m\rangle, \qquad m = 0, 1, 2, \ldots.$$ Contrary to the former, the latter are defined in a subspace of Fock space $\cal F$ including photon numbers less than $m$, namely the subspace ${\cal F}_{\mu}$ of even or odd number states according to whether $m$ is even ($\mu = 0$) or $m$ is odd ($\mu = 1$). Since $S(z)$ is a unitary operator, the set of squeezed number states, corresponding to a given $z$ or $\zeta$ value and $m = 0$, 1, 2, …, is an orthogonal basis of $\cal F$: $$\begin{aligned}
\langle n, \zeta | m, \zeta \rangle & = & \delta_{n,m}, \\
\sum_{m=0}^{\infty} |m, \zeta \rangle \langle m, \zeta | & = & I. \label{eq:SNS-RU}\end{aligned}$$
Any PASVS can be expressed as a linear combination of squeezed number states $$\begin{aligned}
|\zeta, m\rangle & = & \left[(1- |\zeta|^2)^{m/2} P_m \left((1 -
|\zeta|^2)^{-1/2}\right) \right]^{-1/2} \sqrt{m!} \nonumber\\
&& \mbox{} \times \sum_{k=0}^m \case{1}{2} \left[1 + (-1)^{m-k}\right]
\frac{\zzeta^{(m-k)/2}}{(m-k)!! \sqrt{k!}}\, |k, \zeta \rangle,
\label{eq:PASVS-SNS}\end{aligned}$$ and conversely $$\begin{aligned}
|m, \zeta \rangle & = & \sqrt{m!}\, \sum_{k=0}^m \case{1}{2} \left[1 +
(-1)^{m-k}\right] \left[(1- |\zeta|^2)^{k/2} P_k \left((1 - |\zeta|^2)^{-1/2} \right)
\right]^{-1/2} \nonumber \\
&& \mbox{} \times \frac{(- \zzeta)^{(m-k)/2}}{(m-k)!! \sqrt{k!}}\, |\zeta, k \rangle.
\label{eq:SNS-PASVS}\end{aligned}$$ In proving (\[eq:PASVS-SNS\]), we used the property $S^{-1}(z) \ap S(z) = \left(\ap +
\zzeta a\right)/\sqrt{1 - |\zeta|^2}$, resulting from Baker-Campbell-Hausdorff formula, and equation (2.1) of [@zhang]. We conclude that the set of PASVS corresponding to a given $\zeta$ value and $m=0$, 1, 2, …forms a nonorthogonal basis of $\cal
F$.
Completeness of photon-added squeezed vacuum states
===================================================
We may consider two different types of completeness or resolution of unity for the PASVS: one in $\cal F$, obtained for a given $\zeta$ by summing over the discrete label $m$, and the other in ${\cal F}^{(m)}_{\mu}$, obtained for a given $m$ by integrating over the continuous label $\zeta$.
The former directly follows from the unity resolution relation (\[eq:SNS-RU\]) for the squeezed number states and the relation (\[eq:SNS-PASVS\]) between the latter and the PASVS: $$\begin{aligned}
&& (1 - |\zeta|^2)^{-1/2} \sum_{m=0}^{\infty} \sum_{n=m}^{\infty} (1 +
\delta_{n,m})^{-1} \left[(n!/m!) P_m \left((1 - |\zeta|^2)^{-1/2}\right)
P_n \left((1 - |\zeta|^2)^{-1/2}\right)\right]^{1/2} \nonumber \\
&& \times P^{(m-n)/2}_{(m+n)/2} \left((1 - |\zeta|^2)^{-1/2}\right)
\Bigl[\left(- e^{-{\rm i} \phi}\right)^{(n-m)/2} |\zeta, m\rangle \langle \zeta, n|
+ \left(- e^{{\rm i} \phi}\right)^{(n-m)/2} |\zeta, n\rangle \langle \zeta,
m|\Bigr] \nonumber \\
&& = I. \end{aligned}$$ It has a nondiagonal form characteristic of a nonorthogonal basis, with coefficients given by the elements of the overlap matrix inverse.
The derivation of the latter is more involved. The problem amounts to determining a positive measure $d\rho_m(\zeta, \zzeta)$ such that $$\int d\rho_m(\zeta, \zzeta) |\zeta, m\rangle \langle \zeta, m| = I^{(m)}_{\mu},
\label{eq:PASVS-RU}$$ where the integration is carried out over the unit disc and $I^{(m)}_{\mu} \equiv
\sum_{k=0}^{\infty} |2k + m\rangle \langle 2k + m|$ denotes the unit operator in ${\cal F}^{(m)}_{\mu}$.
Making the polar decomposition $\zeta = |\zeta| e^{{\rm i} \phi}$, given in (\[eq:polar\]), and the ansatz $$\begin{aligned}
d\rho_m(\zeta, \zzeta) & = & m!\, (1 - y)^{-(m+1)/2} P_m \left((1 - y)^{-1/2}\right)
h_m(y) d^2\zeta, \nonumber \\
y & \equiv & |\zeta|^2, \qquad d^2\zeta \equiv |\zeta| d|\zeta| d\phi,\end{aligned}$$ and using the expansion (\[eq:PASVS-exp\]), we find after integrating over $\phi$ that equation (\[eq:PASVS-RU\]) is equivalent to the set of conditions $$\int_0^1 dy\, y^k h_m(y) = \frac{[(2k)!!]^2}{\pi (m+k)!}, \qquad k = 0, 1, 2, \ldots.
\label{eq:moment}$$ Consequently, the requirement that for a given $|m\rangle$, $|\zeta, m\rangle$ form a complete (in fact, overcomplete) set in ${\cal F}^{(m)}_{\mu}$ is equivalent to the resolution of a power-moment problem [@akhiezer].
As is usual in such a problem (see e.g. [@sixdeniers01; @sixdeniers99]), it is convenient to set for complex $s$ and ${\rm Re}\, s > 0$, $k \to s-1$, to define $$g_m(y) = \left\{\begin{array}{ll}
h_m(y) & {\rm if\ }0 < y < 1 \\[0.2cm]
0 & {\rm if\ }1 < y < \infty
\end{array}\right.,$$ and to interpret (\[eq:moment\]) as the Mellin transform $g^*_m(s)$ of $g_m(y)$ [@sneddon], $$\int_0^\infty dy\, y^{s-1} g_m(y) = g^*_m(s) \equiv \left\{\begin{array}{ll}
\frac{1}{2\pi} B\left(s, \frac{1}{2}\right) & {\rm if\ } m=1 \\[0.2cm]
\frac{1}{4 \pi (m-2)!} B\left(s, \frac{m}{2}\right) B\left(s, \frac{m-1}{2}\right)
& {\rm if\ } m = 2, 3, \ldots
\end{array}\right.. \label{eq:mellin}$$ In (\[eq:mellin\]), $B(z,w)$ denotes the beta function, i.e., $B(z,w) = \Gamma(z)
\Gamma(w)/\Gamma(z+w)$. To find $g_m(y)$, we must perform the inverse Mellin transform on $g^*_m(s)$.
For $m=1$, $g_1(y)$ is given in tables of Mellin transforms [@prudnikov], leading to the following result for $h_1(y)$: $$h_1(y) = \frac{1}{2\pi} (1-y)^{-1/2}. \label{eq:h1}$$ This is a positive function on (0, 1), increasing from $1/(2\pi)$ to $+\infty$ when y goes from 0 to 1.
For higher $m$ values, it is convenient to use the Mellin convolution property of inverse Mellin transforms, which states that if $g^*_m(s) = g^*_{m1}(s) g^*_{m2}(s)$ and $g_{m1}(y)$, $g_{m2}(y)$ exist, then the inverse Mellin transform of $g^*_m(s)$ is $$g_m(y) = \int_0^{\infty} \frac{dt}{t}\, g_{m1}\left(\frac{y}{t}\right) g_{m2}(t).
\label{eq:convolution}$$ In applying (\[eq:convolution\]) to (\[eq:mellin\]) for $m \ge 2$, we choose $g^*_{m1}(s) = [4\pi (m-2)!]^{-1} B (s, \frac{m}{2})$ and $g^*_{m2}(s) = B (s,
\frac{m-1}{2})$, which identifies $g_{m1}(y)$ and $g_{m2}(y)$ as $$g_{m1}(y) = \left\{\begin{array}{ll}
[4\pi (m-2)!]^{-1} (1-y)^{(m-2)/2} & {\rm if\ }0 < y < 1 \\[0.2cm]
0 & {\rm if\ }1 < y < \infty
\end{array}\right.,$$ and $$g_{m2}(y) = \left\{\begin{array}{ll}
(1-y)^{(m-3)/2} & {\rm if\ }0 < y < 1 \\[0.2cm]
0 & {\rm if\ }1 < y < \infty
\end{array}\right.,$$ respectively [@prudnikov]. Hence we obtain $$g_m(y) = [4\pi (m-2)!]^{-1} \int_y^1 dt\, t^{-m/2} (t-y)^{(m-2)/2} (1-t)^{(m-3)/2}
\label{eq:integral}$$ for $0 < y < 1$, and $g_m(y) = 0$ for $1 < y < \infty$. It is obvious that the right-hand side of (\[eq:integral\]) is a positive function, thus providing a solution for $h_m(y)$ for $m \ge 2$.
To express the latter in terms of known functions, we introduce a new variable $u =
(1-t)/(1-y)$, thereby obtaining $$\begin{aligned}
h_m(y) & = & [4\pi (m-2)!]^{-1} (1-y)^{m - \frac{3}{2}} \int_0^1 du\, u^{(m-3)/2}
(1-u)^{(m-2)/2} [1 - (1-y)u]^{-m/2}, \nonumber\\
&& m = 2, 3, \ldots. \end{aligned}$$ Formula 3.197.3 of [@gradshteyn] then leads to $$h_m(y) = \frac{1}{2\pi (2m-3)!!}\, (1-y)^{m - \frac{3}{2}}\, {}_2F_1\left(\frac{m}{2},
\frac{m-1}{2}; m - \frac{1}{2}; 1-y\right), \qquad m = 2, 3, \ldots.$$ By using formula 3.2.5 in volume 1 of [@erdelyi], we can rewrite $h_m(y)$, $m \ge
2$, in terms of a Legendre function of the second kind $Q^{\mu}_{\nu}(z)$, for which $\mu = 0$ and $\nu$ is a nonnegative integer, $$h_m(y) = \frac{1}{2\pi (m-2)!}\, (1-y)^{(m - 2)/2} Q_{m-2} \left((1-y)^{-1/2}\right),
\qquad m = 2, 3, \ldots. \label{eq:hm}$$ Such a function can be expressed in terms of Legendre polynomials combined with a logarithmic function [@erdelyi]: $$\begin{aligned}
Q_0 \left((1-y)^{-1/2}\right) & = & \frac{1}{2} \ln \frac{1 + \sqrt{1-y}}{1 -
\sqrt{1-y}}, \label{eq:Q0}\\
Q_{m-2} \left((1-y)^{-1/2}\right) & = & \frac{1}{2} P_{m-2} \left((1-y)^{-1/2}\right)
\ln \frac{1 + \sqrt{1-y}}{1 - \sqrt{1-y}} \nonumber \\
&& \mbox{} - \sum_{k=0}^{[(m-3)/2]} \frac{2m-4k-5}{(m-k-2) (2k+1)} P_{m-2k-3}
\left((1-y)^{-1/2}\right), \nonumber \\
&& \mbox{} m = 3, 4, \ldots. \label{eq:Qm-2} \end{aligned}$$ Here $[x]$ denotes the largest integer contained in $x$.
For the first few values of $m$, we find $$\begin{aligned}
h_2(y) & = & \frac{1}{4\pi} \ln \frac{1 + \sqrt{1-y}}{1 - \sqrt{1-y}}, \\
h_3(y) & = & \frac{1}{4\pi} \left( \ln \frac{1 + \sqrt{1-y}}{1 - \sqrt{1-y}} - 2
\sqrt{1-y}\right), \\
h_4(y) & = & \frac{1}{16\pi} \left[(2+y) \ln \frac{1 + \sqrt{1-y}}{1 - \sqrt{1-y}} - 6
\sqrt{1-y}\right], \\
h_5(y) & = & \frac{1}{144\pi} \left[3(2+3y) \ln \frac{1 + \sqrt{1-y}}{1 - \sqrt{1-y}} -
2 (11+4y) \sqrt{1-y}\right].\end{aligned}$$
From (\[eq:hm\]), (\[eq:Q0\]), and (\[eq:Qm-2\]), it can be shown that $h_m(y)
\to +\infty$ or 0 according to whether $y\to 0$ or 1. This is confirmed by Fig. 1, which displays $h_m(y)$ for several $m$ values.
Having found a solution for the problem stated in (\[eq:moment\]), we may now ask whether this solution is unique. An answer is provided by the (sufficient) condition of Carleman [@akhiezer]: if a solution exists and $$S \equiv \sum_{k=1}^{\infty} a_k, \qquad a_k \equiv \left(\frac{[(2k)!!]^2}{\pi
(m+2k)!}\right)^{-1/(2k)},$$ diverges, then the solution is unique. The convergence of $S$ can be tested by applying the logarithmic test [@prudnikov]: if $\lim_{k \to \infty} [\ln(a_k) / \ln(k)] > -1$, then $S$ diverges. By using Stirling formula for the asymptotic form of $\Gamma(z)$ [@erdelyi], we obtain $\lim_{k \to \infty} [\ln(a_k) / \ln(k)] = 0$. We conclude that $h_m(y)$ given in (\[eq:h1\]) and (\[eq:hm\]) is the unique solution to the problem.
Extension to the photon-added squeezed one-photon states
========================================================
The PASOPS are defined by [@liu] $$|1, \zeta, m\rangle = [N_{1m}(|\zeta|)]^{-1/2} (\ap)^m |1, \zeta\rangle,$$ where $m = 0$, 1, 2, …, $$|1, \zeta\rangle = S(z) |1\rangle = \left(1 - |\zeta|^2\right)^{3/4} e^{\frac{1}{2} \zeta
(\ap)^2} |1\rangle,$$ and $|1\rangle = \ap |0\rangle$. In the limit $\zeta \to 0$ (resp. $m\to 0$), they reduce to the number state $|m+1\rangle$ (resp. the squeezed one-photon state $|1,
\zeta\rangle$).
Their expansion in the number-state basis is given by $$|1,\zeta, m\rangle = \left[N_{1m}(|\zeta|)\right]^{-1/2} \left(1 -
|\zeta|^2\right)^{3/4}
\sum_{k=0}^{\infty} \frac{\sqrt{(2k+m+1)!}}{k!}\, \left(\case{1}{2} \zeta\right)^k
|2k + m + 1\rangle,$$ showing that for a given $m$ value, they belong to the same subspace ${\cal F}^{(m+1)}_{\mu'}$ ($\mu' \equiv (m+1)\, {\rm mod} 2$) of $\cal F$ as the PASVS $|\zeta, m+1\rangle$. We actually obtain $$[N_{1m}(|\zeta|)]^{1/2} \left(1 - |\zeta|^2\right)^{-1/2} |1,\zeta, m\rangle =
[N_m(|\zeta|)]^{1/2} |\zeta, m+1\rangle,$$ which enables us to easily extend some of the results of the two previous sections to the PASOPS.
For instance, their overlap for $n-m$ an even nonnegative integer and their normalization coefficient are given by $$\begin{aligned}
\langle 1, \xi, n | 1, \zeta, m \rangle & = & \left[N_{1m}(|\zeta|)
N_{1n}(|\xi|)\right]^{-1/2} \langle 1, \xi | 1, \zeta \rangle\, (n+1)!\,
\xxi^{(m-n)/4} \zeta^{(n-m)/4} \nonumber \\
&& \mbox{} \times (1 - \xxi \zeta)^{-(m+n-2)/4} P^{(m-n)/2}_{(m+n+2)/2} \left((1 -
\xxi \zeta)^{-1/2}\right), \label{eq:PASOPS-overlap} \end{aligned}$$ and $$N_{1m}(|\zeta|) = (m+1)! \left(1 - |\zeta|^2\right)^{-(m-1)/2} P_{m+1} \left((1 -
|\zeta|^2)^{-1/2} \right),$$ respectively. In (\[eq:PASOPS-overlap\]), $\langle 1, \xi | 1, \zeta \rangle$ is the overlap of two squeezed one-photon states, $$\langle 1, \xi | 1, \zeta \rangle = \left[(1 - |\zeta|^2) (1 - |\xi|^2)
\right]^{3/4} (1 - \xxi \zeta)^{-3/2}.$$
Similarly, it can be shown that they form a nonorthogonal basis of ${\cal F}^{(1)}$ (i.e., the Fock space from which the one-dimensional subspace spanned by the vacuum state has been removed) and an (over)complete set in ${\cal F}^{(m+1)}_{\mu'}$ with a positive measure given by $$\begin{aligned}
d\rho_{1m}(\zeta, \zzeta) & = & (m+1)!\, (1 - y)^{-(m+2)/2} P_{m+1} \left((1 -
y)^{-1/2}\right) h_{1m}(y) d^2\zeta, \qquad y \equiv |\zeta|^2, \nonumber \\
h_{1m}(y) & = &\frac{1}{2\pi (m-1)!}\, (1-y)^{(m - 1)/2} Q_{m-1}
\left((1-y)^{-1/2}\right),
\qquad m = 1, 2, \ldots. \end{aligned}$$
Definition and completeness of photon-added coherent states on a circle
=======================================================================
A special class of multiphoton coherent states is provided by the eigenstates of a power $a^{\lambda}$ ($\lambda = 2$, 3, 4, …) of the photon annihilation operator [@buzek; @sun; @cq], satisfying the relation $$a^{\lambda} |z, \mu\rangle = z |z, \mu\rangle, \qquad \mu = 0, 1, \ldots, \lambda-1.
\label{eq:CSC-def}$$ Here $\mu$ distinguishes between the $\lambda$ orthogonal solutions of (\[eq:CSC-def\]), belonging to the subspaces ${\cal F}_{\mu}$ of Fock space $\cal
F$ spanned by the number states $|k \lambda + \mu\rangle$, $k=0$, 1, 2, …: $$\begin{aligned}
|z, \mu\rangle & = & [N_{\mu}(|z|)]^{-1/2} \sum_{k=0}^{\infty} \left(\frac{\mu!}
{(k\lambda + \mu)!}\right)^{1/2} z^k |k\lambda + \mu\rangle, \label{eq:CSC-exp1}\\
N_{\mu}(|z|) & = & {}_0F_{\lambda-1} \left(\frac{1}{\lambda}+1, \frac{2}{\lambda}+1,
\ldots, \frac{\mu}{\lambda}+1, \frac{\mu+1}{\lambda}, \frac{\mu+2}{\lambda},
\ldots, \frac{\lambda-1}{\lambda}; y \right), \nonumber \\
y & \equiv & |z|^2/\lambda^{\lambda}, \end{aligned}$$ where ${}_pF_q(a_1, \ldots, a_p; b_1, \ldots, b_q; z)$ denotes a generalized hypergeometric function [@erdelyi].
The states (\[eq:CSC-exp1\]) may also be written as linear combinations of $\lambda$ (standard) coherent states equidistantly separated from each other along a circle of radius $|t| = |z|^{1/\lambda}$ [@sun], $$\begin{aligned}
|z, \mu\rangle & = & [N_{\mu}([z])]^{-1/2} \frac{\sqrt{\mu!}}{\lambda} e^{\frac{1}{2} |t|^2}
t^{-\mu} \sum_{\nu=0}^{\lambda-1} \epsilon^{-\mu\nu} |t \epsilon^{\nu}\rangle,
\qquad t^{\lambda} = z, \label{eq:CSC-exp2} \\
|t \epsilon^{\nu}\rangle & = & e^{- \frac{1}{2} |t|^2 + t \epsilon^{\nu} \ap} |0\rangle,
\qquad \epsilon \equiv e^{2 \pi {\rm i}/\lambda}, \\
N_{\mu}(|z|) & = & \mu!\, |t|^{-2\mu} h_{\mu+1}(|t|^2, \lambda), \label{eq:CSC-norm} \end{aligned}$$ hence the name of coherent states on a circle that is often used for them [@janszky]. In (\[eq:CSC-norm\]), $h_i(x,n)$ denotes a hyperbolic function of order $n$, i.e., a generalization of the hyperbolic cosine and sine functions to which it reduces for $n=2$ and $i=1$ or 2, respectively [@erdelyi].
Let us define PACSC by the relation $$|z, \mu, m\rangle = [N_{\mu m}(|z|)]^{-1/2} (\ap)^m |z, \mu\rangle,$$ where $m=0$, 1, 2, …, $N_{\mu m}(|z|)$ is some normalization coefficient, and $|z,
\mu\rangle$ is given by (\[eq:CSC-exp1\]) or (\[eq:CSC-exp2\]). For $\lambda = 2$ and $\mu = 0$ or 1, they reduce to the PAECS or PAOCS, respectively [@dodonov].
According to whether we use the expansions (\[eq:CSC-exp1\]) or (\[eq:CSC-exp2\]), we can express the PACSC either in the number-state basis, $$|z, \mu, m\rangle = [N_{\mu m}(|z|) N_{\mu}(|z|)]^{-1/2} \sum_{k=0}^{\infty} \frac{[\mu!
(k\lambda+m+\mu)!]^{1/2}}{(k\lambda+\mu)!} z^k |k\lambda + m + \mu\rangle,
\label{eq:PACSC-def}$$ or in terms of the PACS $|t \epsilon^{\nu}, m\rangle$ of [@sixdeniers01], $$|z, \mu, m\rangle = [N_{\mu m}(|z|) N_{\mu}(|z|)]^{-1/2} [N_m(|t|)]^{1/2} \frac{\sqrt{\mu!}}
{\lambda} e^{\frac{1}{2} |t|^2} t^{-\mu} \sum_{\nu=0}^{\lambda-1} \epsilon^{-\mu\nu} |t
\epsilon^{\nu}, m\rangle,$$ where $$|t\epsilon^{\nu}, m\rangle = [N_m(|t|)]^{-1/2} (\ap)^m |t \epsilon^{\nu}\rangle, \qquad
N_m(|t|) = m!\, L_m(- |t|^2),$$ and $L_m(x)$ denotes a Laguerre polynomial [@erdelyi]. From (\[eq:PACSC-def\]), it is clear that for given $\mu$ and $m$ values, the states $|z, \mu, m\rangle$ belong to the subspace ${\cal F}^{(m+\mu)}_{\mu_m}$ of Fock space $\cal F$ spanned by the states with photon number $n$ not less than $m+\mu$ and congruent with $\mu_m$, defined by $m + \mu = \mu_m {\rm mod} \lambda$.
By using methods similar to those employed in Secs. 2 and 3, it is straightforward to obtain the overlap of two PACSC and their normalization coefficient, as well as two different kinds of completeness relations. Here we only mention two of these results.
The normalization coefficient can be written either in closed form as $$\begin{aligned}
N_{\mu m}(|z|) & = & [N_{\mu}(|z|) \mu!]^{-1} (m+\mu)!\, {}_{\lambda}F_{2\lambda-1}
\left(\frac{m+\mu+1}{\lambda}, \frac{m+\mu+2}{\lambda}, \ldots,
\frac{m+\mu+\lambda}{\lambda}; \right. \nonumber \\
&& 1, \frac{1}{\lambda}+1, \frac{2}{\lambda}+1, \ldots, \frac{\mu}{\lambda}+1,
\frac{\mu+1}{\lambda}, \frac{\mu+2}{\lambda}, \ldots, \frac{\lambda-1}{\lambda},
\frac{1}{\lambda}+1, \frac{2}{\lambda}+1, \ldots, \nonumber \\
&& \frac{\mu}{\lambda}+1, \frac{\mu+1}{\lambda}, \frac{\mu+2}{\lambda}, \ldots,
\left.\frac{\lambda-1}{\lambda}; y\right), \qquad y \equiv
|z|^2/\lambda^{\lambda}, \end{aligned}$$ or as a linear combination of Laguerre polynomials, $$N_{\mu m}(|z|) = [\lambda N_{\mu}(|z|)]^{-1} \mu!\, m!\, |t|^{-2\mu}
\sum_{\nu=0}^{\lambda-1} \epsilon^{-\mu\nu} e^{|t|^2 \epsilon^{\nu}} L_m(- |t|^2
\epsilon^{\nu}).$$
The PACSC satisfy a unity resolution relation in ${\cal F}^{(m+\mu)}_{\mu_m}$, $$\int d\rho_{\mu m}(z, \zz) |z, \mu, m\rangle \langle z, \mu, m| = I^{(m+\mu)}_{\mu_m},$$ with a positive measure given by $$\begin{aligned}
d\rho_{\mu m}(z, \zz) & = & N_{\mu m}(|z|) N_{\mu}(|z|) (\mu!)^{-1} h_{\mu m}(y) d^2z,
\qquad y \equiv |z|^2/\lambda^{\lambda}, \nonumber \\
h_{\mu m}(y) & = & \frac{1}{\pi \lambda^{\lambda-\mu}} y^{(\mu+1-\lambda)/\lambda}
e^{- \lambda y^{1/\lambda}} U\left(m, 1, \lambda y^{1/\lambda}\right), \end{aligned}$$ where $U(a, b, z) = \Psi(a, b; z)$ is Kummer’s confluent hypergeometric function [@erdelyi].
Conclusion
==========
In the present letter, we demonstrated that the PASVS satisfy two different types of unity resolution relations, a discrete one in $\cal F$ and a continuous one in ${\cal
F}^{(m)}_{\mu}$, and we extended such results to the PASOPS. In addition, we introduced the PACSC and obtained both their normalization and their continuous unity resolution relation in ${\cal F}^{(m+\mu)}_{\mu_m}$.
Proving the completeness of photon-added squeezed coherent states along similar lines is a much more difficult problem since such states depend upon two continuous variables instead of one [@xin]. We hope however to solve it in a near future.
Another interesting open question is whether completeness relations of the second type exist for photon-subtracted squeezed states. It is already clear that this is true neither for photon-subtracted squeezed vacuum states, nor for the even nonlinear coherent states proposed in [@liu] by extending the results for positive integer $m$ values to negative integer ones. In both cases, the states are indeed nonnormalizable in the limit $\zeta \to 0$. Photon-subtracted squeezed excited states might, on the contrary, be good candidates for the existence of completeness relations provided $m$ remains low enough.
As a final point, it is worth stressing that a central requirement of this work has been to find a continuous resolution of unity of the usual type with a positive measure. Relaxing this demand may lead to generalized unity resolution relations of the type considered in Ref. [@manko] for the nonclassical states studied in the present work, as well as for their extensions.
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Figure captions {#figure-captions .unnumbered}
===============
Fig. 1. The weight function $h_m(y)$ as a function of $y$ for various $m$ values: (a) $m=1$ (solid line), $m=2$ (dashed line); (b) $m=3$ (solid line), $m=4$ (dashed line), $m=5$ (dotted line).
(160,100) (35,0)
Figure 1
[^1]: Directeur de recherches FNRS; E-mail: [email protected]
|
---
abstract: 'We report the results from a systematic study of the quasi-elastic (e,e’p) reaction on $^{12}$C, $^{56}$Fe and $^{197}$Au performed at Jefferson Lab. We have measured nuclear transparency and extracted spectral functions (corrected for radiation) over a Q$^2$ range of 0.64 - 3.25 (GeV/c)$^2$ for all three nuclei. In addition we have extracted separated longitudinal and transverse spectral functions at Q$^2$ of 0.64 and 1.8 (GeV/c)$^2$ for these three nuclei (except for $^{197}$Au at the higher Q$^2$). The spectral functions are compared to a number of theoretical calculations. The measured spectral functions differ in detail but not in overall shape from most of the theoretical models. In all three targets the measured spectral functions show considerable excess transverse strength at Q$^2$ = 0.64 (GeV/c)$^2$, which is much reduced at 1.8 (GeV/c)$^2$.'
author:
- 'D. Dutta$^{15,10,a}$, D. van Westrum$^{4,b}$, D. Abbott$^{22}$, A. Ahmidouch$^{7}$, Ts. A. Amatuoni$^{25}$, C. Armstrong$^{24,c}$, J. Arrington$^{2,d}$, K. A. Assamagan$^{6,e}$, K. Bailey$^1$, O. K. Baker$^{22,6}$, S. Barrow$^{18}$, K. Beard$^6$, D. Beatty$^{18,c}$, S. Beedoe$^{14}$, E. Beise$^{11}$, E. Belz$^{4}$, C. Bochna$^{9}$, P. E. Bosted$^{12}$, H. Breuer$^{11}$, E. E. W. Bruins$^{10}$, R. Carlini$^{22}$, J. Cha$^{6,f}$, N. Chant$^{11}$, C. Cothran$^{23}$, W. J. Cummings$^{1}$, S. Danagoulian$^{14}$, D. Day$^{23}$, D. DeSchepper$^{10}$, J.-E. Ducret$^{21}$, F. Duncan$^{11,g}$, J. Dunne$^{22,f}$, T. Eden$^6$, R. Ent$^{22}$, H.T. Fortune$^{18}$, V. Frolov$^{19,h}$, D. F. Geesaman$^1$, H. Gao$^{9,10,a}$, R. Gilman$^{22,20}$, P. Guèye$^6$, J. O. Hansen$^{1,c}$, W. Hinton$^{6,c}$, R. J. Holt$^{9}$, C. Jackson$^{14}$, H. E. Jackson$^1$, C. Jones$^{1}$, S. Kaufman$^1$, J. J. Kelly$^{11}$, C. Keppel$^{22,6}$, M. Khandaker$^{13}$, W. Kim$^8$, E. Kinney$^4$, A. Klein$^{17}$, D. Koltenuk$^{18,i}$, L. Kramer$^{10,j}$, W. Lorenzon$^{18,k}$, A. Lung$^{2,c}$, K. McFarlane$^{13,l}$, D. J. Mack$^{22}$, R. Madey$^{6,7}$, P. Markowitz$^5$, J. Martin$^{10}$, A. Mateos$^{10}$, D. Meekins$^{22,c}$, M. A. Miller$^{9}$, R. Milner$^{10}$, J. Mitchell$^{22}$, R. Mohring$^{11}$, H. Mkrtchyan$^{25}$, A. M. Nathan$^{9}$, G. Niculescu$^{6,m}$, I. Niculescu$^{6,n}$, T. G. O’Neill$^1$, D. Potterveld$^1$, J. W. Price$^{19,o}$, J. Reinhold$^{1,j}$, C. Salgado$^{13}$, J. P. Schiffer$^1$, R. E. Segel$^{15}$, P. Stoler$^{19}$, R. Suleiman$^{7,p}$, V. Tadevosyan$^{25}$, L. Tang$^{22,6}$, B. Terburg$^{9,q}$, Pat Welch$^{16}$, C. Williamson$^{10}$, S. Wood$^{22}$, C. Yan$^{22}$, Jae-Choon Yang$^{22}$ J. Yu$^{18}$, B. Zeidman$^1$, W. Zhao$^{10}$, and B. Zihlmann$^{23}$.'
title: '**A Study of the Quasi-elastic (e,e’p) Reaction on $^{12}$C, $^{56}$Fe and $^{97}$Au.**'
---
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
INTRODUCTION {#introduction .unnumbered}
============
The value of studying electronuclear reactions has long been recognized. In such studies the entire nucleus is accessed via a well-understood interaction. A new avenue of investigations has been opened up with the completion of the continuous beam, multi-GeV electron accelerator at the Thomas Jefferson National Accelerator Facility, also known as Jefferson Lab (JLab). The present paper reports results from the first experiment done at this facility, which is a study of (e,e’p) reactions in the quasi-elastic region. This experiment utilized one of the advantages of electron scattering, namely, the transferred energy and momentum can be varied separately, and one of the main features of JLab, namely, the high intensity continuous electron beam of CEBAF which makes it possible to do coincidence measurements orders of magnitude more extensive than could be done previously.
The simplest model of a nucleus is one of independent nucleons populating the lowest available shell-model orbits. In a simple picture of $e-p$ scattering within a nucleus, the electron scatters from a single protons which is moving due to its Fermi momentum. The struck proton may then interact with the residual A-1 nucleons before leaving the nucleus. Of course, neither the nucleus nor the scattering process are this simple and the deviations from these simple pictures reveal much about nuclei and their constituents, both real and virtual. The present experiment consisted of measuring proton spectra in coincidence with inelastically scattered electrons with the energy of the electrons chosen such as to be in the “quasi-elastic” region, i.e. at energies corresponding to scattering from single off-mass-shell nucleons. The spectra were taken in an angular region about the “conjugate” angle, i.e. the angle for scattering from stationary nucleons, over an angular range sufficient to cover the smearing of the two-body kinematics caused by the Fermi momentum of the confined protons. Data were taken over the range 0.64 $<$ Q$^2~<$ 3.25 (GeV/c)$^2$ where Q$^2$ is the square of the four-momentum transferred to the struck proton.
For an electron knocking a proton, $p$, out of a nucleus $A$ with energy transfer $\omega$ and (three) momentum transfer $\vec{q}$ leaving a scattered proton, $p'$, and a residual nucleus, $A - 1$, two important kinematic quantities are the missing energy: $$E_m = \omega - T_{p'} - T_{A-1}$$ and missing momentum: $$\vec{p}_m = \vec{p}_{p'} - \vec{q}$$ where $T_{p'}$ and $T_{A-1}$ are the kinetic energy of the knocked out proton and recoiling nucleus, respectively. The spectral functions were extracted from the $E_m$ and $\vec{p}_m$ spectra and compared to a variety of theoretical calculations. The total (e,e’p) yields are obtained by integrating over the spectral functions and the transparencies then determined by comparing these yields with those predicted by Plane Wave Impulse Approximation (PWIA) calculations. Because the PWIA does not allow for final-state interactions the ratio of measured to calculated yield should just be the fraction of outgoing protons which do not suffer a final-state interaction and this is what is defined to be the transparency. Determinations of nuclear transparencies using the (e,e’p) reaction have been reported for a range of targets covering the periodic table, at Bates for Q$^2$ = 0.34 (GeV/c)$^2$ [@gerry92], at SLAC for Q$^2$ between 1 and 7 (GeV/c)$^2$ [@ton95; @mak94], and more recently at JLAB between 3 and 8.1 (GeV/c)$^2$ [@garrow01]. The present work maps out regions not previously covered and is of greater statistical accuracy. Longitudinal - Transverse (L - T) separations were performed at two values of Q$^2$ from which the first reported extensive separated spectral functions are obtained. Some transparency results from the present experiment have been previously published [@prl1], as have the separated spectral functions for carbon [@prc2].
The differential cross section for elastic electron-proton scattering is given by the well-known Rosenbluth formula: $$\label{cc1}
\frac{d\sigma}{d\Omega} = \left(\frac{d\sigma}{d\Omega}\right)_
{\mbox{\small Mott}}\frac{Q^2}{|\vec{q}|^2}[G_{E}^{2}(Q^2) +
\tau\epsilon^{-1}G_{M}^{2}(Q^2)]$$ where $\left(\frac{d\sigma}{d\Omega}\right)_{\mbox{\tiny Mott}}$ is the differential cross section for the scattering of an electron off a unit point charge, $\epsilon =\frac{1}{1+2(1+\tau)\tan^2(\frac{\theta}{2})}$ is the virtual polarization parameter, ${\tau = {{|\vec{q}|^2}\over{Q^2}} - 1}$, G$_E$ is the proton electric form factor and G$_M$ is the proton magnetic form factor in units of the nuclear magneton, $\frac{e\hbar}{2M_{p}c}$ where $M_p$ is the proton mass.
The L - T separation is performed by measuring the cross section at different values of $\epsilon$ while keeping Q$^2$ constant, thus permitting the extraction of G$_E$ and G$_M$.
In scattering from a nucleus the cross section is expressed in terms of four response functions and in the PWIA the coincidence (e,e’p) cross sections can be written: $$\frac{d^6\sigma}{dE_{e'}d\Omega_{e'}dE_{p'}d\Omega_{p'}}= p'E_{p'}\sigma_
{\mbox{\tiny Mott}}$$ $$[\lambda^{2}W_{L}(q,\omega) + [\frac{\lambda}{2}+ \tan^{2}(\frac{\theta}{2})]W_{
T}(q,\omega)+$$ $$\label{cc2}
\lambda[\lambda + \tan^{2}(\frac{\theta}{2})]^{1/2}W_{LT}(q,\omega)\cos(\phi) +
\frac{\lambda}{2}W_{TT}(q,\omega)\cos(2\phi)].$$
where $\lambda =Q^2/|\vec{q}|^2$, $\theta$ is the scattering angle and $\phi$ is the azimuthal angle between the scattering plane and the plane containing $\vec{q}$ and $\vec{p'}$.
The physics of interest is contained in the 4 response functions W$_L$, W$_T$, W$_{LT}$ and W$_{TT}$. Both of the interference terms, W$_{LT}$ and W$_{TT}$ are proportional to sin$\gamma$, where $\gamma$ is the angle between the scattered proton and the transferred momentum $\vec{q}$. Therefore, when measurements are made along $\vec{q}$, i.e. in “parallel kinematics”, the interference terms are absent. Varying the incident energy makes it possible to vary $\theta$ at constant $q$ and $\omega$ and thus disentangle W$_L$ and W$_T$, that is, perform an L - T separation. Although, the position of the spectrometers allowed measurements only in the scattering plane, the interference term W$_{LT}$ could be investigated by varying the proton angle about the direction of $\vec{q}$. Measurements were taken by varying both $\theta$ and $\gamma$. This is the first L - T separation measured for quasi-elastic (e,e’p) scattering that covers a large range in both A and Q$^2$.
EXPERIMENT {#experiment .unnumbered}
==========
Electron Beam {#electron-beam .unnumbered}
-------------
The experiment was performed in 1995 - 1996 in Hall C at JLab and, was the first experiment performed at the Laboratory. Data were taken at (nominal) electron energies E$_e$ = (0.8N + 0.045) GeV with N = 1 - 4 representing the number of “passes” the electrons made around the accelerating track. The absolute beam energy was determined at one-pass by two independent methods. One method is to use the inelastic scattering to an excited state whose energy is accurately known to calibrate the dispersion of a spectrometer and then use the calibrated spectrometer to measure the energy of the scattered electron as a function of nuclear target mass. For these measurements a carbon target was used and the dispersion determined by measuring the difference in position of the electrons scattered to the ground and the 4.43891 $\pm$ 0.00031 MeV [@offerman91] first excited state. A BeO target was then substituted and the energy of the beam, E, determined using the formula:
$$\Delta E_{\mbox{\small recoil}} = 2E^2\sin{\frac{\theta}{2}}^2(\frac{1}{M_1}-
\frac{1}{M_2})
\label{disp}$$
One can accurately determine $\Delta E_{\mbox{\small recoil}}$ because once the dispersion has been accurately measured the only unknown in Eq. \[disp\] is the beam energy $E$. This procedure was repeated for several values of the spectrometer magnetic field. With both targets a small correction was made for the energy loss of the electrons in the target.
The other method is to determine the angle of the diffraction minimum for scattering to a state where the position can be accurately calculated. The minimum for scattering to the $^{12}$C first excited state has been determined to be at $Q^2$ = 0.129$\pm$0.0006 (GeV/c)$^2$ [@fas85]. The (four) momentum transfer can be written:
$$Q^2=4EE'\sin^2{\frac{\theta}{2}}, E'= \frac{E}{1 +
\frac{2E\sin^2{\frac{\theta}{2}}}{M}}
\label{qande}$$
where $M$ is the mass of the scattering nucleus and $\theta$ is the electron scattering angle. An improvement in accuracy in the measurement of $Q$ is obtained by using the ratio of elastic scattering to inelastic scattering. Again, then, the only unknown is the incident electron energy $E$. The two methods agreed to 1 part in 2000 and the absolute energy determination using these methods is believed to be accurate to 10$^{-3}$. These methods become less feasible as the energy is increased. The beam energy can also be determined by measuring the energy and angle of the scattered particles in electron-proton elastic scattering. Because of the uncertainties in the angle and momentum measurements this method is less accurate than the other two but has the advantage that it can be used over the entire range of incident electron energies. Elastic $e-p$ scattering was used to measure the energy of the three-pass beam with an uncertainty of 1 part in 500. Beam energy was also determined by measuring the magnetic field needed to bend the beam around the Hall C arc. The energy calibration as well as other aspects of the experiment are discussed more completely elsewhere [@thesis].
Beam currents of 10 to 60 $\mu$Amps were used. The currents were monitored by 3 microwave cavities that were installed for this purpose in the Hall C beam line[@jlab1]. The absolute calibration was performed by comparison with an Unser cavity, which is a parametric DC current transformer with very stable gain but a drifting offset which was determined as part of our daily calibration procedure. The overall accuracy in the beam current measurement was $\pm$1%.
Targets {#targets .unnumbered}
-------
Data were taken with $\approx$ 200 mg/cm$^2$ C, Fe and Au targets mounted on a steel ladder in an aluminum scattering chamber. The target thicknesses were determined to about 0.1%. The $e-p$ elastic scattering data used for calibration were taken using the 4.0 cm cell of the Hall C cryogenic target [@jlab2]. During the early part of the experiment, before the cryogenic target was available, some data were taken with a solid CH$_2$ target but these data were used to check some kinematic offsets only. The compositions of hydrocarbon targets are subject to change under beam irradiation and therefore all the calibration data were taken with the liquid hydrogen target. The cryogenic targets are also mounted on a ladder with both ladders contained in the aluminum scattering chamber. The 123.0 cm diameter scattering chamber has entrance and exit snouts for the beam and several pumping and viewing ports. The particles that went to the High Momentum Spectrometer (HMS) spectrometer exited through a 0.4 mm aluminum window and those to the Short Orbit Spectrometer (SOS) through a 0.2 mm aluminum window. For both spectrometers the particles had to pass through about 15 cm of air before entering the spectrometer.
Spectrometers {#spectrometers .unnumbered}
-------------
Data were taken with the HMS and the SOS in coincidence. This experiment served as the commissioning experiment for these spectrometers. The HMS detected the electrons and the SOS the protons, except at the highest Q$^2$ where the roles of the spectrometers were reversed.
### High Momentum Spectrometer {#high-momentum-spectrometer .unnumbered}
The HMS is a 25$^o$ vertical bend spectrometer made up of superconducting magnets in a QQQD configuration. The dipole field is monitored and regulated with an NMR probe and kept constant at the 10$^{-4}$ level. The spectrometer rotates on a pair of rails between 12.5$^o$ and 90$^o$ with respect to the beam line. The HMS maximum central momentum is 7.3 GeV/c and in preparing for the present experiment the spectrometer was tested up to 4.4 GeV/c although the highest setting at which data were taken was 2.6 GeV/c. The usable momentum bite is of the spectrometer is $\approx$20%. A momentum resolution ($\sigma$) of $<$1.4 10$^{-3}$, and an in-plane (out-of-plane) angular resolution of 0.8 (1.0) mrad was achieved for the HMS. With no collimator in place the solid angle subtended for a point target is 8.1 msr. A 6.35 cm thick HEAVYMET (machinable Tungsten alloy,10% Cu Ni; density = 17 g/cm$^3$) collimator with a flared octagonal aperture limited the solid angle to 6.8 msr. The higher momentum particles were usually detected in the HMS and except at the backward (electron) angles these were the electrons. Detailed information about the HMS can be found in [@jlab3].
### Short Orbit Spectrometer {#short-orbit-spectrometer .unnumbered}
The SOS consists of 3 (normal conducting) magnets in a QD$\overline{\mbox{D}}$ configuration. The deflection is vertical with the net bend of 18$^o$ at the central momentum. The magnetic fields are monitored with Hall probes. With its short path length of 11 m this spectrometer is particularly well suited for detecting short-lived particles though obviously this attribute was not used in the present experiment. The spectrometer can be moved between 13.1$^o$ and 168.4$^o$ with respect to the beam line (during this experiment the minimum angle was 14.5 $^o$) and can be moved up to 20$^o$ out of the horizontal plane, though this was not done in this experiment. The spectrometer maximum central momentum is 1.8 GeV/c with a nominal momentum bite of 40%. A momentum resolution ($\sigma$) of $<$ 1.0 10$^{-3}$, an inplane (out-of-plane) angular resolution of 4.5(0.5) msr was achieved for the SOS. The solid angle subtended is $\approx$ 9 msr for a point target, although a collimator similar to that used with the HMS limited the solid angle to 7.5 msr. As with the HMS, further details about the SOS can be found in the spectrometer documentation [@jlab3; @moh3].
### Detector Stacks {#detector-stacks .unnumbered}
The detector stacks in the two spectrometers are virtually identical. The particles pass through, in order, a set of drift chambers, a pair of hodoscopes, a gas Čerenkov detector, another pair of hodoscopes and then a lead-glass calorimeter. The particle velocity is inferred from the time-of-flight between the two pairs of hodoscopes though the spectra proved to be so clean that it was not necessary to use time-of-flight for particle identification. Signals from the hodoscope planes provide the trigger and in the electron arm particle identification can be incorporated into the trigger by requiring a signal from the Čerenkov counter and/or a sufficiently large pulse from the calorimeter. Coincidences between the triggers selected out the (e,e’p) events that make up the physics data.
The drift chambers serve to determine the particles’ position, x (y), and direction, x’ (y’), in the bend (nonbend) plane of the spectrometer and it is these quantities that are used to reconstruct the events. Each spectrometer has two chambers and each chamber contains six planes of wires. In each HMS chamber one pair measures x, one pair measures y and the remaining two planes are rotated $\pm$15$^o$ with respect to the x plane. The purpose of the third pair of planes is to correlate the xy information when more than one particle traverses a chamber during the readout interval. In the SOS chambers one pair is in the x plane and the other two pairs of planes are at $\pm$60$^o$ with respect to the x plane. Position resolution per plane is $<$ 250 $\mu$m in the HMS chambers and $<$ 200 $\mu$m in the SOS. The wire chamber data was used to reconstruct the trajectory of the particles and determine the particles momentum fraction relative to the central momentum, $\delta$p/p.
Wire chamber tracking efficiency is an important element in the overall system efficiency and, as such, must be accurately measured. This was done by using the position information in the hodoscopes to tag particles passing through a small central region of the chambers and then see what fraction of such events was reconstructed from the wire chamber signals. In both spectrometers typical tracking efficiency was greater than 97% which was determined to better than 1%. The main sources of wire chamber tracking inefficiency are inefficiencies in the chambers themselves (we require 5 of the 6 planes have good hits) and inefficiency in the reconstruction algorithm. The measured inefficiency was the sum of these inefficiencies and no attempt was made to disentangle the two.
Calibrations {#calibrations .unnumbered}
------------
### Spectrometer optimizations {#spectrometer-optimizations .unnumbered}
Because this was the first experiment performed in Hall C, considerable effort went into first optimizing the performance of the spectrometers and then optimizing the data analysis so as to achieve the highest possible accuracy. The magnetic field of the HMS quadrupoles was mapped to determine its optical axis and its effective field length versus current, with effective field length defined as the line integral of the field divided by the average field. However, the HMS dipole was not mapped and its magnetic field to current (B-to-I) relation was calculated using the TOSCA program [@tosca]. The measured field map of the quadrupole and the TOSCA generated map of the dipole were used to build an optics model of the spectrometer with the COSY program [@cosy1]. For a desired magnetic field of the dipole (i.e. a desired central momentum) the dipole current was set according to the B-to-I relation predicted by the TOSCA program, while the COSY model was used to get the starting value of the quadrupole to dipole ratio (Q/D). The Q/D ratio was then varied to get the best focus in the spectrometer and these optimized ratios were used to determine the current settings of the quadrupole for a desired central momentum of the spectrometer. From elastic $e-p$ scattering data it was later determined that the B-to-I relation of the dipole predicted by TOSCA was wrong by about 0.9%. The dipole currents were adjusted accordingly to correct for this difference. A similar procedure was followed for the SOS except that the quadrupole was not mapped and the optics model was formulated using the COSY program assuming the field of the quadrupole magnet to be an ideal quadrupole. The SOS dipole B-to-I relation was also found to be slightly wrong (0.55%) and suitable corrections were made to the setting procedure.
The basic strategy in determining the momentum and direction of the scattered particles is to use the wire chamber data to determine the position, (x,y), and the angles, (x’,y’), of the particles at the focal plane which, in turn, specifies the trajectory of the particle through the spectrometer. This of course requires knowing the fields of the spectrometer, which are represented by a set of matrix elements that relate the position and direction of the particles as they cross the focal plane, to the particle’s momentum, angles of emission, and starting position along the beam direction. The accuracy of the final results then depends on how well the matrix elements simulate the spectrometers and hence a great deal of effort went into optimizing these matrix elements.
The COSY program was used to calculate an initial set of reconstruction matrix elements using the mapped fields for the HMS magnets and the SOS dipoles and an assumed pure quadrupole field for the SOS quadrupole. The Hall C Matrix Element Optimization Package CMOP [@cosy2] was used to optimize the reconstruction matrix elements. In this package the dispersion matrix elements are optimized using momentum scans, i.e. varying the central momentum by varying the magnetic fields. For each spectrometer these momentum scans were performed for both elastic p(e,e’) and elastic $^{12}$C(e,e’) scattering. In order to obtain the angular matrix elements sieve slits, which are collimators containing accurately positioned holes, were placed in front of each of the spectrometers so that rays of known initial position and direction could be traced. The angular matrix elements were then fit by the CMOP package (using singular value decomposition method) to accurately reproduce the known positions of the sieve slit holes. Similarly the target $y$ position (projection of the target length along the beam) reconstruction was optimized by utilizing the CMOP package with data from scans along the beam direction. These scans were performed by raising and lowering a slanted carbon target and the continuum portion of the carbon spectrum was used. Most of these calibration data were taken at one-pass, 845 MeV, with a check for reproducibility made with two-pass, 1645 MeV, electrons.
### Acceptances {#acceptances .unnumbered}
The spectrometer’s acceptances were studied with the aid of the simulation code SIMC, which is an adoption to the JLab Hall C spectrometers of the (e,e’p) simulation code written for SLAC experiment NE18 [@makth]. This simulation package employs models for each of the spectrometers (HMS and SOS). The same models were also used to study the optical properties of the spectrometers. These models use COSY generated sets of matrices to simulate the transport of charged particles through the magnetic field of the spectrometer to each major aperture of the spectrometer. Energy loss and multiple scattering in the intervening material were also included. The events that passed through all apertures were then reconstructed back to the target using another set of matrices generated by COSY. Surviving events were assigned a weight based on the PWIA cross-section, radiative corrections and coulomb corrections. The PWIA cross-section was calculated using the deForest [@defor83] prescription $\sigma_{cc1}$ for the off-shell $e-p$ cross-section and an Independent Particle Shell Model (IPSM) spectral function for the target nucleus involved. The PWIA calculations and the IPSM spectral functions are elaborated in the next two sections. The radiative corrections in SIMC were performed according to the Mo and Tsai [@motsai69] formulation adapted for the coincidence (e,e’p) reaction as described in Ref. [@ent02]. Further, a normalization factor was calculated from the experimental luminosity, phase space volume and the total number of events generated, so that the simulation provided a prediction of the absolute yield.
The reconstructed momentum, scattering angle, out-of-plane angle and target length distributions generated by the model were compared with the distributions obtained from the $e-p$ elastic scattering data as shown in Fig. 1. These results are an indicator of how well the model acceptance simulated the true acceptance of the spectrometer. This was the status of the model during the experiment, there has been significant improvement in the model since then.
![Comparison of calculated (dark line) and measured (light line) distributions. Top row is momentum, angle, and out of plane angle for electrons and the middle row the same for protons. Last picture is the projection of the distribution along the target for electrons.[]{data-label="fig1"}](fig1.ps){width="9.5cm" height="10.0cm"}
Corrections {#corrections .unnumbered}
-----------
### Radiative corrections {#radiative-corrections .unnumbered}
A major issue in electron scattering experiments is radiative corrections. The incoming and outgoing electrons can interact with the Coulomb field of the nucleus involved in the scattering which results in the emission and absorption of virtual photons and emission of real, primarily soft, photons. Formulas for correcting for these radiative losses have been worked out by Mo and Tsai [@motsai69]. Correcting spectral functions deduced from (e,e’p) coincidence spectra is considerably more complicated because in this case the radiated momentum as well as the lost energy must be allowed for. Although these are real physical processes they are experiment specific and so most theoretical calculations do not take them into account. The prescription for doing this for coincidence (e,e’p) reactions developed by Ent et al. [@ent02] was used in the present work. Using this prescription, radiated spectra are generated which can be directly compared with the experimentally measured spectra. This point is discussed further in the section on spectral functions.
### Nuclear reactions {#nuclear-reactions .unnumbered}
Protons, being hadrons, will undergo strong interactions in traversing the detector stack and valid coincidences will be lost. This loss was measured directly using $e-p$ elastic scattering. Each scattered electron must have an accompanying proton and electrons were selected from a small region at the center of the acceptance thus insuring that protons could only be lost through nuclear interactions and other spectrometer inefficiencies. Transmissions of close to 95% were measured for both spectrometers and are believed to be known to 1%. The absorption is virtually constant over the range of proton energies encountered in this experiment and therefore the small uncertainty in the absorption has little effect on any of the results.
### Deadtimes {#deadtimes .unnumbered}
There were two data acquisition deadtimes of possible concern: electronic deadtime and computer deadtime. Electronic deadtime occurs when triggers are not counted because the electronics hardware is busy processing previous triggers. Electronic deadtime is dependent on the width of the logic signals, which for nearly all of the gates was 30 ns. This deadtime was measured by recording the rates of multiple copies of the trigger with varying widths and then extrapolating to the rate at zero width. For both spectrometers the electronic deadtime was found to be $<$ 0.1%. Computer deadtime is a more serious matter. Most of the earlier data were taken in non-buffered mode where the processing time was about 400 $\mu$s. Later data were taken in the buffered mode with processing times of about 75 $\mu$s. Over 80% of the data were taken with deadtimes of $<$10% but there were a few runs where deadtimes were as great as 60%. Even in these extreme cases the loss of event is known to better than 0.5% from the ratio of the number of triggers generated to the number of triggers recorded by the data acquisition. This method was checked by measuring a large rate run and then varying the fraction of triggers recorded by the data acquisition.
RESULTS {#results .unnumbered}
=======
Kinematics {#kinematics .unnumbered}
----------
Table I shows the kinematics settings where data were taken. The protons in the nucleus have finite momentum and therefore the struck protons from quasi-elastic scattering will emerge in a cone about the three-momentum transfer $\vec{q}$ and measurements must be taken across this cone. The lower the magnitude of $\vec{q}$ the broader the cone but, fortunately the cross section increases with decreasing Q$^2$. While it is desirable to take data over as large a range of Q$^2$ as possible the cross section falls off so rapidly with increasing Q$^2$ that at the highest Q$^2$ point, 3.25 (GeV/c)$^2$, the cross section is so small that data could only be taken on one side of the conjugate angle. L - T separations were performed at Q$^2$ of 0.64(GeV/c)$^2$ and 1.8(GeV/c)$^2$. In order to get a good separation, data should be taken at as divergent values of $\epsilon$ (Eq 1) as possible, which translates into a large $\epsilon$ point at small (electron) angle and large incident energy and a low $\epsilon$ point at large angle and small energy (Table I). The cross section decreases rapidly with increasing angle and so it was only possible to cover one side of the proton cone at $\epsilon$ = 0.31, Q$^2$ = 1.8(GeV/c)$^2$ and even at Q$^2$ = 0.64(GeV/c)$^2$ there was time for only one point on the low-angle side of the cone. Furthermore, no gold data were taken at the larger angle and higher Q$^2$ (1.8 GeV/c$^2$).
-------- ---------- ---------- --------- -------------------------------------------------------- --------------------- ------------
Central Central Central Central
Beam electron electron proton proton Q$^2$
Energy Energy Angle Energy Angle $\epsilon$
(GeV) (GeV) (deg) (MeV) (deg) $\frac{GeV^2}{c^2}$
36.4,39.4
43.4,47.4
2.445 2.075 20.5 350 51.4,**[55.4]{}& 0.64 & 0.93\
& & & & 59.4,63.4 & &\
& & & & 67.4,71.4 & &\
& & & & 75.4 & &\
& & & & & &\
& & & & 27.8 & &\
& & & & **[31.8]{} & 0.64 & 0.38\
0.845 & 0.475 & 78.5 & 350 & 35.8,39.8,& &\
& & & & 43.8,47.8 & &\
& & & & & &\
& & & & 32.6.36.6,& &\
3.245 & 2.255 & 28.6 & 970 & **[40.6]{},& 1.80 & 0.83\
& & & & 44.6,48.6,& &\
& & & & 52.6 & &\
& & & & & &\
& & & & **[22.8]{},& &\
1.645 & 0.675 & 80.0 & 970 & 26.8,30.8 & 1.83 & 0.31\
& & & & 34.8 & &\
& & & & & &\
2.445 & 1.725 & 32.0 & 700 & 31.5,35.5 & 1.28 &\
& & & & 39.5,**[43.5]{}& & 0.81\
& & & & 47.5,51.4 & &\
& & & & 55.4 & &\
& & & & & &\
3.245 & 1.40 & 50.0 & 1800 & **[25.5]{} & 3.25 & 0.54\
& & & & 28.0,30.5 & &\
************
-------- ---------- ---------- --------- -------------------------------------------------------- --------------------- ------------
: Table of kinematics for Experiment E91-013, the central proton angles in bold represents the conjugate angle.[]{data-label="kmat"}
Spectral Functions {#spectral-functions .unnumbered}
------------------
The spectral function for protons in a nucleus S$(E_s, \bf{p}_m)$ is defined as the probability of finding a proton with separation energy $E_s$ and momentum $\bf{p}_m$ inside that nucleus. Obtaining spectral functions was a major objective of the present work and this section details how the spectral functions were deduced from the measured missing energy and missing momentum spectra.
### Hydrogen Data {#hydrogen-data .unnumbered}
A missing energy and a missing momentum spectrum was obtained at each data point. For the hydrogen target this served as a measure of the response of the system while for the other targets these are the spectra from which the spectral functions are determined. Hydrogen missing energy spectra along with the Monte Carlo calculated spectra at the various kinematics are shown in Fig. \[radtest\]. The fact that the low energy tail is well reproduced out to the highest missing energy accepted (80 MeV), shows that the radiative corrections are being handled correctly. Energy resolution, which is not of primary importance in the present work, is clearly not well incorporated into the code in that the calculated zero missing energy peak is always narrower than that observed. The peaks get broader with increasing energy of the scattered particle (see Table I), as could be expected, and this effect is not adequately accounted for. The effect is most dramatic at the two values of Q$^2$ where data was taken at two different electron angles, and the peak is much broader at the forward angle where the electron energy is higher, while the proton energy remains the same.
![Measured missing energy spectra for hydrogen (dark line) compared to spectra calculated using the Monte Carlo code SIMC(light line). The spectra with the same Q$^2$ refer to the forward and backward electron angle kinematics respectively for the L/T separation kinematics.[]{data-label="radtest"}](fig2.eps){width="9.5cm" height="13.0cm"}
The ratio of the observed to predicted $e-p$ elastic scattering yield is shown in Table II. In calculating the predicted yield the electric form factor G$_E$ is taken to have the dipole form: $$G_{E} = \left( 1 + \frac{Q^2}{0.71} \right)^{-2}$$ and $G_M$ is taken from the Gari-Krümpelmann [@gk73] parameterization which, to a good approximation, yields $G_M = \mu_{p}G_E$. Rosenbluth separation measurements of $e-p$ scattering [@walker94] support the validity of this relationship.
Q$^2$ $\epsilon$
------------- ------------ ----------------------------- -------------------
(GeV/c)$^2$ H(e,e’p) H(e,e’)
0.64 0.93 1.006 $\pm$ 0.005 1.015 $\pm$ 0.005
0.64 0.38 0.986 $\pm$ 0.005 0.997 $\pm$ 0.005
1.28 0.81 1.007 $\pm$ 0.005 1.009 $\pm$ 0.005
1.80 0.83 0.991 $\pm$ 0.005 1.003 $\pm$ 0.005
1.83 0.31 0.987 $\pm$ 0.005 0.989 $\pm$ 0.005
3.25 0.54 0.94 $\pm$ 0.012 $\pm$ 0.06 0.991 $\pm$0.007
: Ratio of observed to predicted yield for $e-p$ elastic scattering. Uncertainties are statistical only, except for the (e,e’p) point at 3.25 (GeV/c)$^2$ where there is an additional systematic uncertainty that is discussed in the text.[]{data-label="hydrorat"}
The typical systematic uncertainty for these measurements was 2.3%. However, the large uncertainty in the (e,e’p) yield at Q$^2$ = 3.25 (GeV/c)$^2$ is due to an uncertainty in the proton efficiency due to malfunctioning wire chambers in the HMS. For all of the other points, including the single-arm electrons at 3.25 (GeV/c)$^2$, calculated and measured yield agree to within about 1%. The setting for Q$^2$ = 3.25 (GeV/c)$^2$ was the only one at which the protons were detected in the HMS and this efficiency problem was corrected before the data on the complex nuclei was taken.
As an alternative to performing a Rosenbluth separation, a polarization transfer method has been developed [@acg81] for measuring the ratio of the electric to the magnetic form factor and a recent experiment using this method reports that for the free proton $\mu_{p}G_E/G_M$ decreases with increasing Q$^2$ declining to a value of 0.61 at Q$^2$ = 3.47 (GeV/c)$^2$ [@mjones00]. A value of 0.79 is found at Q$^2$ = 1.8 (GeV/c)$^2$ while at Q$^2$ = 0.64 (GeV/c)$^2$ it is only 5% less than the Q$^2$ = 0 value of unity. In calculating the simulation cross sections for Table I the dipole (Eqn. 7) and Gari-Krümpelmann [@gk73] values for G$_E$ and G$_M$, respectively, are used. The implications of the results of Jones et. al. [@mjones00] for the present work are discussed in the section on L-T separations.
### Missing Energy Spectra for the Nuclear Targets {#missing-energy-spectra-for-the-nuclear-targets .unnumbered}
A missing energy and missing momentum spectrum was obtained at each data point for all three nuclear targets. These are the raw spectra from which the spectral functions were extracted after unfolding the radiative effects, the phase space weight and the $e-p$ cross-section weight. The raw missing energy spectra are shown in Figs. \[carbonem\], \[ironem\], and \[goldem\].
![Measured missing energy spectra for carbon at the different Q$^2$, panels with the same Q$^2$ refer to the forward and backward electron angle kinematics respectively for the L/T separation kinematics. []{data-label="carbonem"}](fig3.eps){width="9.5cm" height="13.0cm"}
![Measured missing energy spectra for iron at the different Q$^2$, panels with the same Q$^2$ refer to the forward and backward electron angle kinematics respectively for the L/T separation kinematics.[]{data-label="ironem"}](fig4.eps){width="9.5cm" height="13.0cm"}
![Measured missing energy spectra for gold at the different Q$^2$, panels with the same Q$^2$ refer to the forward and backward electron angle kinematics respectively for the L/T separation kinematics.[]{data-label="goldem"}](fig5.eps){width="9.5cm" height="13.0cm"}
Fig. \[carbonem\] shows the missing energy spectra for carbon. At Q$^2$ = 0.64 (GeV/c)$^2$ the spectra show a rather sharp peak corresponding to populating low-lying levels in $^{11}$B which can be attributed to removing p - shell protons from $^{12}$C and a broader peaking at higher missing energies which is primarily due to removing s - shell protons. The valley between the two groups is increasingly filled in as Q$^2$ increases, because the (absolute) energy resolution broadens as the energy of the particles increases, as noted above in discussing the hydrogen spectra of Fig. 2. At the two values of Q$^2$ at which L - T separations were performed the valley between the s - shell and p - shell region is less distinct at the forward electron angle, again reflecting the poorer energy resolution that was also observed in the hydrogen spectra. The missing energy spectra for iron are shown in Fig. \[ironem\]. The ground-state region peak is more prominent at low Q$^2$ and backward angles. The missing energy spectra for gold are shown in Fig. \[goldem\]. The statistical uncertainties are much poorer for gold than for the other targets and no trends are apparent.
### Radiative and Acceptance Corrections {#radiative-and-acceptance-corrections .unnumbered}
As previously noted, energy and momentum are lost by the electrons radiating photons in the Coulomb field of the target nucleus both before and after the scattering. The electrons can also emit bremsstrahlung radiation in passing through material in the spectrometers. The net result is a distortion of the spectra and the corrections to this distortion are model dependent. The code SIMC was used to generate correction factors for “deradiating” the observed spectral functions. Model spectral functions were used to populate bins in $p_m$ and $E_m$ space with both the radiative corrections turned on and turned off and the ratio was applied as a correction factor, bin by bin, to the spectral functions derived from the experimental data. The Monte Carlo was also used to calculate the experimental phase space for each ($E_m$,$p_m$) bin. The experimental counts in each ($E_m$,$p_m$) bin corrected for radiation and divided by the phase space for that bin was used to obtain the “experimental” spectral function: $$S^{\mbox{derad}}(E_{m},p_{m})~~~=~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$ $$\frac{1}{{\cal L}~H(E_m,p_m)}\sum_{\mbox{counts}}{\frac{1}{\sigma_{ep}E_{e'}p_{p'}(E_m,p_m)}}C^{{\mbox{rad}}}(E_m,p_m)$$
where ${\cal L}$ is the luminosity, $H(E_m,p_m)$ the phase space for the given $E_m$, $p_m$ bin, $C^{{\mbox{rad}}}(E_m,p_m)$ the correction factor for the same bin and $\sigma_{ep}E_{e'}p_{p'}(E_m,p_m)$ the off-shell $e-p$ cross- section and kinematic factors averaged over the $E_m$ and $p_m$ bin. This “experimental” spectral function is then compared to the input model spectral function and if the two differ by more than a specified amount the experimental spectral functions become the new model spectral functions and the whole process is iterated until a satisfactory convergence is achieved. In order to test the validity of this procedure non-physical spectral functions were input as the model spectral functions and it was demonstrated that after several iterations the extracted spectral functions are virtually independent of the initial model function. The consistency of this de-radiation procedure was also checked using Monte Carlo generated data. It should be noted that these corrected spectral functions still include distortions due the effects of final state nuclear interactions, including absorption.
### Experimental Spectral Functions {#experimental-spectral-functions .unnumbered}
At each electron angle the above procedure was used for each proton angle to obtain experimental (distorted, as defined above) spectral functions and these were integrated over the proton angles to obtain the experimental spectral functions for that target, electron angle and Q$^2$. These summed spectral functions are functions of both missing momentum and missing energy and therefore the missing momentum was integrated over in order to obtain the energy spectral functions and the missing energy was integrated over to obtain momentum distributions. The momentum distributions are shown in Figs. \[carbonpm\], \[ironpm\] and \[goldpm\].
![Momentum distributions for carbon p - shell (top panel, 10$<$$E_m$$<$25 MeV) and s - shell (bottom panel, 30$<$$E_m$$<$50 MeV). They have been normalized so that the integral of the measured spectral functions over $|p_{m}|<$ 300 MeV/c is equal to the integral of the spectral function at Q$^2$ of 1.8 (GeV/c)$^2$.[]{data-label="carbonpm"}](fig6.eps){width="9.5cm" height="12.0cm"}
![Momentum distributions for iron integrated over an $E_m$ range 0$<$$E_m$$<$80 MeV. They have been normalized so that the integral of the measured spectral functions over $|p_{m}|<$ 300 MeV/c is equal to the integral of the spectral function at Q$^2$ of 1.8 (GeV/c)$^2$.[]{data-label="ironpm"}](fig7.eps){width="9.0cm" height="9.0cm"}
![Momentum distributions for gold integrated over an $E_m$ range 0$<$$E_m$$<$80 MeV. They have been normalized so that the integral of the measured spectral functions over $|p_{m}|<$ 300 MeV/c is equal to the integral of the spectral function at Q$^2$ of 1.8 (GeV/c)$^2$.[]{data-label="goldpm"}](fig8.eps){width="9.0cm" height="9.0cm"}
The carbon momentum distributions are shown in Fig. \[carbonpm\]. They have been normalized to the spectral functions at Q$^2$ of 1.8 (GeV/c)$^2$ to remove the effect of variation in final state interactions between the different Q$^2$ points. These spectra show little variation with Q$^2$. The dip at zero missing momentum for missing energy between 10 and 25 MeV is attributable to the fact that the protons in this energy region are primarily l = 1 while only l = 0 protons can have zero missing momentum. There is a left-right (or $\pm$) asymmetry in the momentum distributions that is discussed below. As with carbon the iron momentum distributions (Fig. \[ironpm\]) and gold momentum distributions (Fig. \[goldpm\]) show little change with Q$^2$.
### Independent Particle Shell Model {#independent-particle-shell-model .unnumbered}
Model spectral functions were calculated in the Independent Particle Shell Model (IPSM) approximation, in which the nucleus is considered a sum of nucleons occupying distinct shells with each proton in the lowest possible shell. The parameters of the spectral function were adjusted to reproduce data from low-Q$^2$ A$(e,e'p)$ and A$(p,2p)$ experiments. For $^{12}$C the removal energy and energy width of the two shells, s$_{1/2}$ and p$_{3/2}$ is based on the Saclay $^{12}$C$(e,e'p)$ data [@moug76]. The removal energy and energy width for the $^{56}$Fe shells were based on the $^{58}$Ni$(e,e'p)$ data from Saclay [@moug76; @fesaclay], with the removal energy corrected for the 2 MeV difference between $^{56}$Fe and $^{58}$Ni. The removal energy for the shells not resolved in the Saclay experiment were obtained from Hartree-Fock calculations [@quintth] and the widths for these shells were calculated according to the Brown and Rho [@rho81] parametrization of data for A $<$ 58. Similarly for $^{197}$Au the removal energies and widths are based on those measured for nearby nucleus $^{208}$Pb in A$(e,e'p)$ experiments at NIKHEF [@quintth], with removal energies corrected for the 2.2 MeV difference between $^{208}$Pb and $^{197}$Au. The parameters for the unmeasured shells were obtained from Hartree-Fock calculations [@quintth] and the Brown and Rho parametrization as mentioned above. Further details are given elsewhere [@thesis].
Momentum distributions were obtained for each shell by solving the Schröedinger equation in a Woods-Saxon potential using the code DWEEPY [@guisti87]. For $^{12}$C the parameters used in the potential were based on the Saclay $^{12}$C$(e,e'p)$ data [@moug76]. The $^{56}$Fe and $^{197}$Au momentum distributions were based on those measured for the nearby nucleus $^{58}$Ni and $^{208}$Pb, modified to agree with the $^{56}$Fe$(e,e'p)$ and $^{197}$Au$(e,e'p)$ data from SLAC experiment NE-18 [@tonth], respectively. For $^{56}$Fe and $^{197}$Au a Perey factor (with $\beta$ = 0.85) [@perey63] was used to correct for the non-locality or energy dependence of the potential.
The experimental missing energy spectral function for carbon at Q$^2$ = 1.28 (GeV/c)$^2$ is compared to the IPSM spectral function in Fig. \[carbonsem\]. The model predicts slightly too much yield in the dip region between the s$_{1/2}$ and the p$_{3/2}$ shells possibly implying that the s - shell is more tightly bound than generally thought. The momentum distribution (Fig. \[carbonspm\]) in the region of the low missing energy peak, considered to be the p - shell region, shows a much shallower minimum at $p_m$ = 0 than the IPSM prediction, while for protons from the s - shell region the $p_m$ = 0 yield is smaller than predicted. Agreement is much better if an 8% p - s mixing is included (the $E_m$ cut allows some s-shell strength into the p-shell region and vice-a-versa). The spectroscopic factors found in a high-resolution (e,e’p) experiment done at NIKHEF [@gerard88] support the amount of s - p “mixing” invoked to explain the carbon missing momentum distributions.
![Measured missing energy spectral function for carbon at Q$^2$ = 1.28 (GeV/c)$^2$ compared to Independent Particle Shell Model (IPSM).[]{data-label="carbonsem"}](fig9.eps){width="9.0cm" height="10.0cm"}
![Measured momentum distribution for carbon at Q$^2$ = 1.28 (GeV/c)$^2$ in the s-removal energy region (top panel, 10$<$$E_m$$<$25 MeV) and p-removal energy region (bottom shell, 30$<$$E_m$$<$50 MeV) compared to theoretical predictions. The solid line is the IPSM model; dashed line is IPSM with an 8% s-p mixing. Dotted line is a DWIA calculation from Zhalov [*et al.*]{} [@zalsf] and the dot-dashed line is the same DWIA calculation with color transparency included.[]{data-label="carbonspm"}](fig10.eps){width="9.0cm" height="10.0cm"}
The IPSM predicts sharper structure in the iron missing energy spectral functions (Fig. \[ironsem\]) than is observed indicating that the shell widths are underestimated. This model also predicts too few of the most loosely bound nucleons. Similar differences between calculation and experiment are seen in the gold data (Fig. \[goldsem\]). For both iron and gold the momentum spectral functions are fairly well predicted although in both cases the yield for $|p_m|>$ 250 MeV/c is under-predicted, which is probably because the calculations under-estimate the contribution from short-range correlations. It must be emphasized that in obtaining the transparencies, discussed in the next section, the data were integrated out to a missing energy of 80 MeV and therefore differences in spectral function structure between model and experiment are pretty well averaged out.
![Measured missing energy spectral function for iron at Q$^2$ = 1.28 (GeV/c)$^2$ compared to theoretical models. The solid line is using the IPSM model. The dashed line is a calculation from Benhar [*et al.*]{} [@benharsf] and the dot-dashed line is from calculations using the TIMORA code described in Ref. [@hor81] with spreading widths taken from the IPSM.[]{data-label="ironsem"}](fig11.eps){width="9.0cm" height="10.0cm"}
![Measured momentum distribution for iron integrated over an $E_m$ range 0$<$$E_m$$<$80 MeV at Q$^2$ = 1.28 (GeV/c)$^2$, compared to theoretical predictions. Solid line is using the IPSM model. Dotted line is DWIA calculation from Zhalov [*et al.*]{} [@zalsf] without including color transparency and dot-dashed is the same with color transparency included. Dot-dot-dash line is a calculation from Benhar [*et al.*]{} [@benharsf] and dash-dot-dash line is from calculations using the TIMORA code described in Ref. [@hor81].[]{data-label="ironspm"}](fig12.eps){width="9.0cm" height="10.0cm"}
![Measured missing energy spectral function for gold at Q$^2$ = 1.28 (GeV/c)$^2$ compared to the IPSM model.[]{data-label="goldsem"}](fig13.eps){width="9.0cm" height="10.0cm"}
![Measured momentum distribution for gold integrated over an $E_m$ range 0$<$$E_m$$<$80 MeV at Q$^2$ = 1.28 (GeV/c)$^2$ compared to theoretical predictions. Solid line is using the IPSM model and dashed line is a calculation from Benhar [*et al.*]{} [@benharsf][]{data-label="goldspm"}](fig14.eps){width="9.0cm" height="10.0cm"}
### Other Calculated Spectral Functions {#other-calculated-spectral-functions .unnumbered}
Distorted Wave Impulse Approximation (DWIA) calculations of the (distorted) spectral functions using the Hartree-Fock model with Skyrme’s interaction to describe the single particle aspects of the nuclear structure [@llf94] have been performed by Zhalov [@zalsf]. These calculations include an estimate of the effects of color transparency, which are negligible for carbon (Fig. \[carbonspm\]) and barely discernible in iron (Fig. \[ironspm\]). These calculations overestimate the yield at small missing momentum and fall off too rapidly at large $|p_m|$. Spectral functions have also been calculated by Benhar [@benharsf]. Here single-particle spectral functions are modified by adding terms dependent on the nuclear density. Results are shown in Fig. \[ironspm\] (iron) and \[goldspm\] (gold). Including the density dependence does increase the large $p_m$ tail, though not by enough to reproduce the data. These calculations also underestimate the $p_m$ = 0 region (it must be remembered that the momentum distribution is weighted by $p_m^2$ in normalizing calculation to experiment). The calculated energy spectral function for iron shows more structure than is observed, reflecting the fact that the IPSM spreading width was also used in the Benhar calculation (Fig. \[ironsem\]).
Energy and momentum distributions for iron have been calculated using the TIMORA code written by Horowitz [@hor81] and based on the $\sigma - \omega$ mean field theory of Walecka [@wel74]. Details of this calculation are given elsewhere [@derekth]. As can be seen in Fig. \[ironsem\] this calculation gives a better fit to the observed structure, or lack thereof, than does either the IPSM or the Benhar [@benharsf] calculations.
Transparencies {#transparencies .unnumbered}
--------------
As noted in the Introduction the basic strategy used to obtain nuclear transparencies was to compare the measured yield to that calculated under the assumption that the struck proton escapes the nucleus without further interaction, i.e. the transparency is defined as the ratio of the measured yield to that calculated using the Plane Wave Impulse Approximation, or PWIA.
### PWIA {#pwia .unnumbered}
For each target, incident electron energy, outgoing electron angle and outgoing proton angle, the transparency was determined as the ratio of the observed $e-p$ coincidence yield, integrated over missing momentum ($\pm$ 300 MeV/c) and missing energy (up to 80 MeV), to that calculated using the PWIA. However, before the expected coincidence $e-p$ spectra in the absence of final state interactions can be calculated, a number of complications must be dealt with. As its name implies the PWIA treats the incoming and outgoing particles as plane waves. There are, of course, the radiative corrections that are discussed above. Additionally, the incident and outgoing waves are distorted by the Coulomb field of the target and residual nucleus, respectively. It has been shown [@hor81] that these distortions can be approximated by attaching a phase factor to the plane wave expansion. The acceleration by the Coulomb field increases the electron momentum $k$ by: $$\delta k~=~f~\frac{Z~\alpha}{R}$$ where factor $f$ varies between 1.1 and 1.5 depending on the size of the nucleus and $R$ is the coulomb radius of the nucleus. This can be used to estimate the effect of coulomb distortion on the cross-section with satisfactory accuracy [@knoll74]. This coulomb acceleration of the electron necessitates using an effective momentum transfer and also alters the missing momentum [@ent02]. All of these effects were incorporated into the PWIA and spectral functions calculations.
The PWIA calculations were done using the “traditional” $e-p$ free cross sections in which $\mu_{p}G_E/G_M$ = 1, with the ramifications of recent polarization transfer results [@mjones00] discussed below in L - T separations section. The fact that the target proton is moving and is bound to a nucleus (i.e. is “off shell”) introduces considerable complications. Off-shell prescriptions for quasi-free $e-p$ cross sections have been given by deForest [@defor83] and the prescription $\sigma_{cc1}$ was used in the present work in calculating the PWIA cross sections. Another complication is the fact that the response function is no longer the incoherent sum of the longitudinal and transverse response functions but there are also the interference terms W$_{LT}$ and W$_{TT}$ (Eq. 2). The response function W$_{LT}$ is anti symmetric about the conjugate, or free $e-p$ scattering, angle and thus vanishes in this direction, known as “parallel kinematics”. Of course parallel kinematics is the only kinematics in free $e-p$ scattering and the cross section is given by the familiar Rosenbluth formula.
While it is a reasonable first approximation to take complex nuclei as a collection of $A$ nucleons moving in an average potential with orbits filled in order of increasing energy this is too simplistic a picture to use in extracting transparencies. Short-range nucleon-nucleon correlations are present and one effect of these is to extend some single particle strength up to hundreds of MeV in E$_{m}$ and well beyond the Fermi momentum in p$_{m}$. The missing energy spectra are indeed above the IPSM predictions at the high energy end but because of the acceptance cutoff of the spectrometers only a small portion of this “pushed-up” strength could be detected. Under the assumption that the correlations produce a uniform suppression of the spectral function below the Fermi momentum and the missing energy limit, correlation factors of 1.11 $\pm$ 0.03, 1.26 $\pm$ 0.08 and 1.32 $\pm$ 0.08 for carbon, iron and gold, respectively, are calculated [@vppw92] and these corrections have been applied to the PWIA cross sections in extracting the transparencies.
### Extracted Transparencies {#extracted-transparencies .unnumbered}
The apparent transparencies (i.e. ratio of measured to PWIA calculated (e,e’p) coincidence yield ) relative to that at the conjugate angle are shown in Fig. \[cral\] for the carbon (top), iron (middle) and gold (bottom) targets, for the various electron kinematic settings. The transparencies are significantly asymmetric. One possible reason could be the presence of interference terms in the response function, i. e. a W$_{LT}$ (Eq. 4) in excess of that included in the de Forest prescription $\sigma_{cc1}$. This is not unexpected because modern relativistic models predict such asymmetries . However, it should be noted that coulomb distortion of the electron waves can alter the effective scattering angle and therefore induce an asymmetry about the free conjugate angle. While much of the coulomb distortion can be allowed for by introducing the momentum increase given by Eq. 10 it could well be that this correction is not adequate. Coulomb distortions are known to increase with Z [@couldistth]. The angular dependence of the quasi-free scattering depends directly on the momentum distribution of the scattering nucleons and the tendency of the transparency to peak at the conjugate angle that is seen in the iron and gold distributions could be due to an underestimate of the number of high-momentum protons in the nucleus. None of these complications appear to be present in the carbon data and so we can conclude that in carbon at least there is evidence of an interference term in the response function that decreases with increasing Q$^2$.
![Normalized transparency as a function of angle relative to the conjugate angle for carbon (top), iron (middle) and gold (bottom). Normalization was done at the conjugate angle.[]{data-label="cral"}](fig15.ps){width="11.0cm" height="12.0cm"}
The outgoing proton cone was integrated over in order to determine the transparency for that electron kinematic setting. The values thus obtained are shown in Table III and are plotted as a function of Q$^2$ for the various targets along with previous measurements in Fig. \[transp\]. There are three types of errors in the transparencies:\
(i) Statistical: These are down in the 0.01 region and are never greater than 0.02.\
(ii) Systematic: These are about 2.5% overall and about 2% from point to point.\
(iii) Model dependence: These include uncertainties in the radiative corrections, the off-shell $e-p$ cross sections and the correlation corrections. The sum in quadrature of the model dependent uncertainties is about 5% for C and 8% for Fe and Au. The relative uncertainties in comparing different points with the same target are less than 5%.
In addition to the obvious trend of decreasing with increasing A, the transparencies also decrease with increasing Q$^2$, at least at the low end of the Q$^2$ range covered here. The A and Q$^2$ dependence of the transparencies has already been described and discussed [@prl1]. At the two values of Q$^2$ where data were taken at 2 different angles the transparency, as defined as the ratio of observed cross section to that predicted by the PWIA, is higher at the backward (i.e., high $\epsilon$) angle. This is a manifestation of the enhancement of the transverse component of the cross section, discussed below in the section on the L - T separated spectral functions. Also shown in Fig. \[transp\] are the transparencies extracted from the longitudinal part of the spectral functions (extrapolated to include all $p_{m}$). These transparencies are lower than the transparencies extracted by comparing to PWIA yields and the difference increases with A.
 and SLAC [@ton95; @mak94](open triangle). The solid circles show the transparencies extracted from the longitudinal spectral functions extrapolated to all $p_m$, these have been slightly displaced in Q$^2$ for clarity.[]{data-label="transp"}](fig16.eps){width="9.0cm" height="11.0cm"}
Q$^2$ (GeV/c)$^2$ carbon iron gold
---------------------------- ------------ ------------ ------------
0.64 ($\theta_e$ forward) 0.61(0.02) 0.47(0.01) 0.38(0.01)
0.64 ($\theta_e$ backward) 0.64(0.02) 0.54(0.01) 0.43(0.01)
1.28 0.60(0.02) 0.44(0.01) 0.32(0.01)
1.80 ($\theta_e$ forward) 0.57(0.01) 0.40(0.01) 0.29(0.01)
1.83 ($\theta_e$ backward) 0.59(0.01) 0.44(0.01)
3.25 0.58(0.02) 0.42(0.01) 0.28(0.01)
: Transparencies found at the various Q$^2$ and $\epsilon$ for the 3 targets. Numbers in parenthesis are statistical errors only.[]{data-label="transptable"}
### L - T Separations {#l-t-separations .unnumbered}
L - T separations were performed at 0.64 and 1.8 (GeV/c)$^2$. While at the low Q$^2$, small angle, point the entire cone of outgoing protons was covered just about as quickly as the spectrometer could be moved, because of the kinematic factors some compromises had to be made at the other settings. Performing L - T separations requires accurate data, partially because the anomalous proton magnetic moment leads to the response function being primarily transverse which, in turn, means that it is necessary to separate out a longitudinal response from a response function that is dominated by the transverse over the entire range. As noted above, except at the large $\epsilon$, small Q$^2$ point it was not possible to cover the entire cone, which would have made it possible to average over the interference terms in the response function. The fact that the differential cross sections are not symmetric about the conjugate angle (Fig. 15) demonstrates that these terms are not necessarily negligible. For the L - T separations it was therefore decided to use only data where these terms must be small, namely, requiring that $|p_m|$ be less than 80 MeV/c.
The spectral functions obtained using the PWIA are the weighted average of what can be called separated spectral functions, S$_L$ and S$_T$, and can be written: $$S(E_m,{\bf p}_m) = \frac{\sigma_L~S_L~(E_m,{\bf p}_m) + \sigma_T~S_T(E_m,{\bf
p}_m)}{\sigma_L + \sigma_T},$$ and the L - T separation then separates out $S_L$ and $S_T$ with the deForest prescription [@defor83] used to modify $\sigma_L$ and $\sigma_T$ from the free nucleon values in order to account for the fact that the nucleons are bound in a nucleus. The separated spectral functions for carbon have already been reported [@prl1]. Separated spectral functions for iron are shown in Fig. 17. Because of the increasing dominance of the magnetic scattering with increasing Q$^2$ (Eq. 1) the errors in $S_L$ increase with increasing Q$^2$ while the errors in $S_T$ decrease somewhat. The transverse strength is clearly smaller at the higher Q$^2$ and, at 0.64 (GeV/c)$^2$, $S_T$ is clearly greater than $S_L$. At Q$^2$ = 1.8 (GeV/c)$^2$, the errors on $S_L$ are too great to allow any conclusions as to whether there are (relative) changes in $S_L$ similar in magnitude to those found in $S_T$. Similar results were found for carbon [@prc2].
An L - T separation for gold was only done at 0.64 (GeV/c)$^2$ and the resultant spectral functions are shown in Fig. 18. As with the other two targets at this momentum transfer there is an excess of transverse strength.
The results described above were obtained using the proton form factors discussed in the [*hydrogen data*]{} section, with $\mu_p$G$_E~\approx$ G$_M$. However, the polarization transfer measurements which now have been extended up to 5.5 (GeV/c)$^2$ show $\mu_p$G$_E$/G$_M$ [@pedrisat] continuing to decrease approximately linearly with Q$^2$. These ratios disagree with the series of L - T separation studies of $e-p$ scattering going back over 30 years which in the aggregate [@walker94; @bosted94] find $\mu_p$G$_E$/G$_M$ consistent with unity in this momentum transfer range (and beyond). Because the spectral functions are close to inversely proportional to the square of the form factors large changes in the form factors lead to large changes in the separated spectral functions.
![Iron separated spectral functions integrated over a $p_m$ range 0$<$$p_m$$<$80 MeV/c. The Q$^2$ = 1.8 (GeV/c)$^2$ points have been displaced slightly for clarity. The lowest $E_m$ point has been averaged over 10 $<E_m<$ 25 MeV. In obtaining these spectral functions the proton electric form factor was assumed to have the dipole form and the proton magnetic from factor was taken from Ref. [@gk73][]{data-label="feslst"}](fig17.eps){width="9.0cm" height="11.0cm"}
![Gold separated spectral functions integrated over a $p_m$ range 0$<$$p_m$$<$80 MeV/c. In obtaining these spectral functions the proton electric form factor was assumed to have the dipole form and the proton magnetic from factor was taken from Ref. [@gk73][]{data-label="auslst"}](fig18.eps){width="9.0cm" height="11.0cm"}
![Comparison of the carbon longitudinal (top panel) and transverse (bottom panel) spectral functions at Q$^2$ = 0.64 (GeV/c)$^2$, integrated over a $p_m$ range 0$<$$p_m$$<$80 MeV/c, using the proton form factors obtained by the Rosenbluth separation [@walker94], [@bosted94] (open symbols) and the polarization transfer [@mjones00] methods (solid symbols). The lowest $E_m$ point has been averaged over 10 $<E_m<$ 25 MeV.[]{data-label="caslst"}](fig19.eps){width="9.0cm" height="10.0cm"}
![Comparison of the carbon longitudinal (top panel) and transverse (bottom panel) spectral functions at Q$^2$ = 1.8 (GeV/c)$^2$ , integrated over a $p_m$ range 0$<$$p_m$$<$80 MeV/c, using the proton form factors obtained by the Rosenbluth separation [@walker94],[@bosted94] (open symbols) and the polarization transfer [@mjones00] methods (solid symbols). The lowest $E_m$ point has been averaged over 10 $<E_m<$ 25 MeV. The polarization transfer form factor points have been displaced slightly for clarity.[]{data-label="ccslst"}](fig20.eps){width="9.0cm" height="10.0cm"}
A comparison of the spectral functions obtained using the L - T separation [@walker94],[@bosted94] and the polarization transfer [@mjones00] form factors is shown in Fig. 19(20) for carbon at Q$^2$ = 0.64(1.8) (GeV/c)$^2$. At 0.64 (GeV/c)$^2$ there is little effect on either spectral function and the decrease in transverse strength at the higher Q$^2$ shows little change. However, the form factors of Ref. [@mjones00] lead to a 60% increase in the longitudinal strength between the two values of momentum transfer. It is hard to imagine a mechanism that would lead to such a Q$^2$ dependency and it is clear that the final interpretation of the present (and a great deal of other) data must await a resolution of the question of the free proton electric form factor.
The extra transverse strength at low Q$^2$, which we attribute to multi-nucleon exchange currents and perhaps other multi-nucleon effects, could lead to an overestimation of the transparency because the PWIA only deals with single nucleon currents. Therefore, we also show in Fig. 16 the transparencies at Q$^2$ = 0.64 (GeV/c)$^2$ deduced from the longitudinal spectral function alone, and these deduced transparencies are substantially lower than the nominal transparencies. However, we must note that the same procedure at Q$^2$ = 1.8 (GeV/c)$^2$ does not have a big effect on the deduced transparency.
The behavior of the transverse spectral function as a function of Q$^2$ is consistent with a recent calculation of the separated cross-sections on $^{16}$O [@jan01]. This calculation includes contribution from two-nucleon photo-absorption and predicts a reduction in the transverse strength with increasing Q$^2$, as observed in this experiment. However, it also predicts a large effect due to the two-nucleon photo-absorption on the longitudinal strength which is inconsistent with the present results. It should be pointed out that the effects due to two-nucleon photo-absorption calculated in Ref. [@jan01] are an upper limit rather than an exact prediction.
CONCLUSIONS {#conclusions .unnumbered}
===========
Taking advantage of the high-quality electron beams and associated detection systems that have become available with JLab coming into operation, (e,e’p) coincidence measurements were made on carbon, iron and gold targets at momentum transfers Q$^2$ of 0.64, 1.28, 1.8 and 3.25(GeV/c)$^2$. Spectral functions were measured for missing momentum out to 300 (MeV/c) and missing energy up to 80 MeV and these differ in detail, but not in overall shape, from Independent Particle Shell Model calculations. Other reported calculations do not give much better fits except perhaps those from a code based on a $\sigma - \omega$ mean field theory. By comparing the experimental yields integrated over missing energy and missing momentum with PWIA calculations nuclear transparencies for 350 - 1800 MeV protons were determined with an accuracy that is considerably greater than previously reported transparency determinations.
Longitudinal - Transverse separations were performed at 0.64 (GeV/c)$^2$ and 1.8 (GeV/c)$^2$ with the iron and gold separations being the first such data on medium and heavy nuclei. Considerable excess transverse strength is found at Q$^2$ = 0.64 (GeV/c)$^2$ which is much reduced at 1.8 (GeV/c)$^2$. This excess strength is attributed to multi-nucleon effects that have less effect on smaller distance probes. Recently reported determinations of G$_E$/G$_M$ for the proton which are in substantial disagreement with previously accepted values will, if they are confirmed, substantially alter the magnitude of the longitudinal spectral function at 1.8 (GeV/c)$^2$. However, because G$_M$ is primarily determined by the absolute cross section the transverse spectral function will be little affected.
We would like to gratefully acknowledge the outstanding efforts of the staff of the Accelerator and Physics Divisions of Jefferson Laboratory to make these experiments possible. This work is supported in part by the U.S. Department of Energy and the National Science Foundation.
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|
---
abstract: 'An explicit formula for a strong connection form in a principal extension by a coseparable coalgebra is given.'
address: ' Department of Mathematics, Swansea University, Singleton Park, Swansea SA2 8PP, U.K.'
author:
- 'E.J. Beggs & Tomasz Brzeziński'
date: June 2006
title: An explicit formula for a strong connection
---
[**1.**]{} In the studies of geometry of non-commutative principal bundles or coalgebra-Galois extensions (cf. [@BrzMaj:coa]) an important role is played by the notion of a [*strong connection*]{} (for the universal differential structure) first introduced in the context of Hopf-Galois extensions in [@Haj:str]. The existence of a strong connection guarantees that a bundle associated to a coalgebra-Galois extension is a (finitely generated and) projective module, hence it is (a module of sections on) a vector bundle in the sense of non-commutative geometry (cf. [@Con:ncg]). Furthermore, a strong connection form gives rise to a [*Chern-Galois character*]{} [@BrzHaj:che], a mapping from the Grothendieck group of isomorphism classes of finite dimensional corepresentations of the structure coalgebra to the cyclic homology of the base algebra (see [@BohBrz:str] for the most general, relative formulation).
The existence of a strong connection in a Hopf-Galois extension is assured by the classical Schneider Theorem I [@Sch:pri]. This states that a free coaction of a Hopf algebra with bijective antipode on its injective comodule algebra determines a Hopf-Galois extension with a strong connection. This theorem has been extended to coalgebra (entwined) extensions with a [*coseparable colagebra*]{} [@Brz:gal Theorem 4.6] [@SchSch:gen Theorem 5.9]. In all these cases the proof of existence is not a constructive proof: the existence follows by general arguments, but no explicit form of connection is given. On the other hand, the knowledge of this form is needed for construction and calculation of Levi-Civitá connections and projectors for associated bundles, and the Chern-Galois characters. Recently, several examples of strong connections have been constructed (cf. [@BrzMaj:geo], [@HajMat:loc], [@Lanvan:pri], [@LanPag:Hop]) or their form conjectured [@BonCic:bij], but no general procedure has been established. The aim of this note is to give a direct proof of a Schneider type theorem for coalgebra extensions in which the connection is explicitly given.
We work over a field $k$, unadorned tensor product is over $k$. For a vector space $V$, the identity map is denoted by the same symbol $V$. All algebras are associative and unital. In an algebra $A$, $1$ denotes the unit both as an element and as a $k$-linear map $k\to A$ and $\mu:A\ot A\to A$ denotes the product. In a coalgebra $C$, the coproduct is denoted by $\Delta$ and counit by $\eps$. We denote coactions of $C$ on a vector space $A$ by $\roA$ (the right coaction) and $\Aro$ (the left coaction). The following Sweedler’s notation is used: $\Delta(c) = c\sw 1\ot c\sw 2$, $\roA(a) = a\sw 0\ot a\sw 1$, $\Aro(a) = a\sw{-1}\ot a\sw 0$ (summation suppressed).
[**2.**]{} In this section a strong connection form is explicitly constructed from a cointegral in a coseparable coalgebra. First we recall the definition of a non-commutative object which captures most of the geometric information carried by (locally trivial) principal bundles.
\[def.principal\] Let $C$ be a coalgebra and $A$ an algebra and a right $C$-comodule via $\varrho^A:A\to A\ot C$. Let $$B=A^{co C}:=\{b\in A~|~\varrho^A(ba)=b\varrho^A(a),\ \forall a\in A\},$$ denote the subalgebra of $C$-coinvariants of $A$. The inclusion of algebras $B\subseteq A$ is called a [*principal $C$-extension*]{} if
$
\can: A\ot_BA{\to} A\ot C,\;a\ot a'\mapsto a\varrho^A(a')
$ is bijective (the Galois condition);
$A$ is $C$-equivariantly projective as a left $B$-module, i.e. there exists a left $B$-module, right $C$-comodule section of the product $B\ot A\to A$;
$\psi:C\ot A{\to} A\ot C$, $c\ot a\mapsto
\can(\can^{-1}(1\ot c)a)$ is bijective;
there is a group-like element $e\in C$ such that $\varrho^A(a)=\psi(e\ot a)$, for all $a\in
A$.
By [@BrzHaj:coa Theorem 3.5], the map $\psi$ defined in Definition \[def.principal\](c) is an example of a [*right-right entwining map*]{}, i.e. it is a map $\re : C \otimes A \to A \otimes C$, which, satisfies the following relations: $$\psi\circ({C}\tens \mu) = (\mu\tens {C})\circ
(A\tens\psi)\circ(\psi\tens
{A}), \qquad
\psi\circ ({C}\tens 1) = 1\tens {C},$$ $$({A}\tens\Delta)\circ\psi = (\psi\tens
C)\circ({C}\tens\psi)\circ(\Delta\tens {A}), \qquad
({A}\tens \eps)\circ\psi =
\eps\tens A.$$ Consequently, the inverse of $\psi$ is a [*left-left entwining map*]{}, i.e. the following relations $$\psi^{-1}\circ(\mu\tens C) = (C\tens \mu)\circ
(\psi^{-1}\tens
{A})\circ (A\tens\psi^{-1}), \qquad
{C}\tens 1 = \psi^{-1}\circ (1\tens {C}),$$ $$\label{le2}
(\Delta\tens)\circ\psi^{-1} = ({C}\tens\psi^{-1})\circ(\psi^{-1}\tens
C)\circ(A\tens \Delta), \qquad
{A}\tens \eps =
(\eps\tens A)\circ\psi^{-1}.$$ are satisfied.
Furthermore, $A$ is a [*right entwined module*]{}, i.e. the map $\psi$ makes the $C$-coaction $\roA$ compatible with the product in the sense that, for all $a,\ta\in A$, $$\roA(a\ta) = a\sw 0\psi (a\sw 1\ot \ta).
\label{rem}$$ Since $\psi$ is bijective, $A$ is also a left $C$-comodule with the coaction $$\label{gen.left.coa}
\forall a\in A, \quad \Aro(a)=\psi^{-1}(a\roA(1)).$$ In view of condition (d) in Definition \[def.principal\], in the case of a principal extension this left coaction comes out explicitly as $\roA (a) = \psi^{-1}(e\ot a)$. With this coaction $A$ is a [*left entwined module*]{}, i.e., for all $a, \ta\in A$, $\Aro(a\ta) = \psi^{-1}(a\ot \ta\sw{-1})\ta\sw 0$. Note that if $C$ is a Hopf algebra with a bijective antipode $S$ and $A$ is a right $C$-comodule algebra, then the map $\psi$ in Definition \[def.principal\](c) and its inverse come out as $$\psi (c\ot a) = a\sw 0 \ot ca\sw 1, \qquad \psi^{-1}(a\ot c) = cS^{-1}a\sw 1\ot a\sw 0.$$ Hence, by setting $e=1_C$ we obtain that $\roA(a) = \psi( 1_C\ot a)$ and $\Aro(a) = S^{-1}a\sw 1\ot a\sw 0$. In general, we make the following
\[def.ee\] Let $C$ be a coalgebra and let $A$ be an algebra and a right $C$-comodule. An inclusion of algebras $B\subseteq A$ is called an [*entwined $C$-extension*]{} if $B$ is a subalgebra of coinvariants $B=A^{coC}$ and there exists a bijective right-right entwining map $\psi: C\ot A\to A\ot C$ such that compatibility condition is satisfied.
Note that the compatibility condition imply that the right coaction in an entwined extension is given by $$\label{gen.right.coa}
\forall a\in A, \quad \roA(a)= 1\sw 0\psi(1\sw 1\ot a).$$
In particular, conditions (a) and (c) in Definition \[def.principal\] imply that a principal extension is an example of an entwined extension. The condition (c) in Definition \[def.principal\] is equivalent to the existence of a [*strong connection form*]{}.
\[str.con.form\] Let $B\subseteq A$ be an entwined $C$-extension and let $\cocan$ be the [*lifted canonical map*]{}, $$\cocan: A\ot A{\to} A\ot C,\;a\ot a'\mapsto a\varrho^A(a').$$ A $k$-linear map $\ell: C\to A\ot A$ satisfying the following properties:
$\cocan\circ\ell = 1_A\otimes C$;
$(\ell\otimes C)\circ\Delta= (A\otimes \roA)\circ\ell$;
$(C\otimes\ell)\circ\Delta = (\Aro\otimes A)\circ\ell$,
is called a [*strong connection form*]{} or a [*strong connection lifting*]{}. Here $\Aro$ is the induced left coaction as in .
Existence of a strong connection form $\ell$ in an entwined $C$-extension $B\subseteq A$ implies that it is a Galois extension, i.e. the canonical map $\can$ is bijective, and that $A$ is $C$-equivariantly projective as a left $B$-module (cf. [@BohBrz:str Theorem 3.7, Corollary 3.8] for a detailed proof in the most general case). Explicitly, the splitting $s: A\to B\ot A$ of the product is given by $s(a) = a\sw 0\ell(a\sw 1)$. If, in addition, there is a group-like element $e\in C$ such that $\roA(1) =1\ot e$, then an entwined extension with a strong connection form is a principal extension. In this case, $\ell$ can always be chosen in such a way that $\ell(e) = 1\ot 1$. The existence of a group-like element $e$ is needed in order to have a bijective correspondence between strong connection forms $\ell$ and strong covariant differentials (and also to make the universal differential calculus on $A$ a [*$C$-covariant*]{} calculus, cf. discussion in [@BrzMaj:geo Sections 4,5]).
Recall that a coalgebra $C$ is said to be [*coseparable*]{} if the coproduct has a retraction in the category of $C$-bicomodules, equivalently, if there exists a $k$-linear map $\delta: C\ot C\to k$ with the following properties, for all $c,c'\in C$, $$\delta(c\sw 1\ot c\sw 2) = \eps(c), \qquad c\sw 1\delta(c\sw 2\ot c') = \delta(c\ot c'\sw 1)c'\sw 2.
\label{coint}$$ Such a map $\delta$ is called a [*cointegral*]{}. For example, if $C$ is a coalgebra spanned by a set of group-like elements $x_i$, then $C$ is a coseparable with a cointegral $
\delta(x_i\ot x_j) = \delta_{ij}.
$ If $C$ is a Hopf algebra, then it is coseparable if and only if there exists a normalised left (or right) integral on $C$, i.e. a linear map $\lambda : C\to k$ such that, for all $c\in C$, $$c\sw 1\lambda(c\sw 2)= \lambda(c), \qquad \lambda(1) =1.$$ The corresponding cointegral is $
\delta (c\ot c') = \lambda \left(c S(c')\right).
$ Conversely, given a cointegral $\delta$ on a Hopf algebra $C$, the left integral is obtained as $\lambda( c )= \delta(c\ot 1)$. Since, by the Woronowicz theorem [@Wor:com Theorem 4.2] every compact quantum group has an integral (or a [*Haar measure*]{}) most of the coalgebras which are of interest in non-commutative differential geometry are coseparable.
The main result of this note is contained in the following theorem, which gives the explicit form of a strong connection.
\[thm.main\] Let $B\subseteq A$ be an entwined $C$-extension. Assume that $C$ is a coseparable coalgebra with a cointegral $\delta: C\ot C\to k$ and that the (lifted) canonical map $$\cocan: A\ot A{\to} A\ot C,\;a\ot a'\mapsto a\varrho^A(a')$$ is surjective. Write $\sigma: C\to A\ot A$ for a $k$-linear map such that $\cocan\circ\sigma = 1\ot C$ and define maps $\gamma : C\ot A\to A$ and $\alpha : A\ot C\to A$ by $$\gamma = (\delta\ot A)\circ (C\ot \Aro), \qquad \alpha = (A\ot \delta)\circ (\roA\ot C).$$ Then $$\label{ell}
\ell = (\gamma\ot\alpha)\circ (C\ot\sigma\ot C)\circ (\Delta\ot C)\circ \Delta,$$ is a strong connection form.
Furthermore, if $\roA(1) = 1\ot e$, then $A$ is a principal $C$-extension.
Using the definition of a cointegral, one easily checks that the map $\gamma$ is left $C$-colinear, where $C\ot A$ as understood as a left $C$-comodule via $\Delta\ot A$, and $\alpha$ is right $C$-colinear, where $A\ot C$ is a right $C$-comodule via $A\ot \Delta$. By the colinearity of $\gamma$ and $\alpha$ the map $\ell$ is $C$-bicolinear.
To prove that $\ell$ is a section of the map $\cocan$ we start with the following simple calculation, for all $a,\ta\in A$, $$\psi^{-1}(a\ta\sw 0\ot \ta\sw 1) =
\psi^{-1}\left(a1\sw 0\psi(1\sw 1\ot\ta)\right)
= \psi^{-1}(a1\sw 0\ot 1\sw 1)\ta = a\sw{-1}\ot a\sw 0\ta.$$ Here the first and last equalities follow from the definitions of the right and left $C$-coactions on $A$ (cf. , ), and the second equality follows by and by the fact that $\psi^{-1}$ is the inverse of $\psi$. Thus we obtain the equality $$\psi^{-1}(a\ta\sw 0\ot \ta\sw 1)\ot \ta\sw 2 = a\sw{-1}\ot a\sw 0\ta\sw 0\ot \ta\sw 1. \label{key}$$ For any $c\in C$, write explicitly $
c\suc 1 \ot c\suc 2 := \sigma(c),
$ so that $
c\suc 1c\suc 2\sw 0\ot c\suc 2\sw 1 = 1\ot c.
$ This leads to the equality $$c\sw 1\ot c\sw 2\suc 1c\sw 2\suc 2\sw 0 \ot c\sw 2\suc 2\sw 1\ot c\sw 3 = c\sw 1\ot 1\ot c\sw 2 \ot c\sw 3.$$ Apply $(C\ot \psi^{-1}\ot C\ot \Delta)\circ (C\ot A\ot\Delta\ot C)$ and then use on the left hand side and on the right hand side to obtain $$c\sw 1\ot c\sw 2\suc 1\sw{-1} \ot c\sw 2\suc 1\sw{0}c\sw 2\suc 2\sw 0
\ot \, c\sw 2\suc 2\sw 1\ot c\sw 3\ot c\sw 4
= c\sw 1\ot c\sw 2\ot 1\ot c\sw 3 \ot c\sw 4\ot c\sw 5.$$ Now apply $\delta\ot A\ot\delta\ot C$ and use the definitions of maps $\gamma$ and $\alpha$ in terms of $\delta$ on the left hand side, and the properties of the cointegral on the right, to conclude that $$\gamma(c\sw 1 \ot c\sw 2\suc 1)\alpha(c\sw 2\suc 2\ot c\sw 3)\ot c\sw 4 = 1\ot c.$$ By the right $C$-colinearity of $\alpha$ this implies that $\cocan \circ \ell = 1\ot C$ as required. In view of the discussion that follows Definition \[str.con.form\], the second assertion is obvious.
Note that if $\roA(1) = 1\ot e$, then $\cocan (1\ot 1) = 1\ot e$, hence the linear map $\sigma$ can always be normalised so that $\sigma(e) = 1\ot 1$ by making the linear change $$\sigma \mapsto \sigma +1\ot 1\eps - \sigma(e)\eps.$$ The strong connection form obtained with such normalised $\sigma$ is also normalised, i.e. $\ell(e) = 1\ot 1$. Furthermore, if $\sigma$ is right (resp. left) $C$-colinear, then the formula reduces to $$\ell = (\gamma \ot A)\circ (C\ot\sigma)\circ\Delta, \qquad \mbox{(resp.\
$\ell = (A\ot \alpha)\circ (\sigma\ot C)\circ\Delta$)}.$$ Thus, if $\sigma$ is $C$-bicolinear, then $\ell =\sigma$. This gives an effective way of testing whether the map $\sigma$ is bicolinear.
[**3.**]{} The main usefulness of formula lies in the fact that usually the map $\sigma$ in Theorem \[thm.main\] is already obtained as the first step of checking whether a given extension is a Galois extension. Furthermore, in geometrically most interesting cases, the coalgebra $C$ is a Hopf algebra (although the coaction is not necessarily an algebra map), for which the explicit form of the left integral (or Haar measure) is known. As an illustration one can consider an entwined extension $\cA(\Sigma_q^4)\subseteq \cA(S_q^7)$ by a coalgebra $\cA(SU_q(2))$ constructed in [@BonCic:ins]. Here $\cA(\Sigma_q^4)$ is the algebra of functions on a quantum four-sphere, $ \cA(S_q^7)$ is the algebra of functions on a quantum seven-sphere (obtained as a quotient of the quantum group $U(4)$), while $\cA(SU_q(2))$ is the algebra of functions on the quantum group $SU(2)$. The $k$-linear splitting $\sigma$ required in Theorem \[thm.main\] is constructed explicitly in [@BonCic:bij Equations (11)] (as a part of a complete proof that this is a coalgebra-Galois extension), while the formula for the left integral on $\cA(SU_q(2))$ is derived in [@Wor:com Appendix A1]. Once these two are combined with each other, finding a strong connection $\ell$ is a matter of straightforward albeit tedious calculations. In view of the discussion at the end of [2]{}, this provides one also with a means of checking if [@BonCic:bij Equations (11)] indeed define a strong connection form as conjectured at the end of [@BonCic:bij].
An example of quantum principal bundles is provided by [*quantum homogeneous spaces*]{}. In this case $A$ is a Hopf algebra and a quantum homogeneous space is defined as a subalgebra $B\subseteq A$ such that $\Delta(B) \subseteq A\ot B$. The coalgebra $C$ is defined as a quotient $C = A/B^+A$, where $B^+ = B\cap\ker\eps$, while the surjection $\pi: A\to C$ induces left and right coactions of $C$ on $A$ via $(\pi\ot A)\circ\Delta$ and $(A\ot\pi)\circ\Delta$. If the antipode in $A$ is bijective, this produces an entwined extension $A^{coC}\subseteq A$ (and $B=A^{coC}$ for example if there is a strong connection). By [@BrzMaj:geo Proposition 4.4], left invariant strong connection forms are in bijective correspondence with $C$-bicolinear maps $\iota: C\to A$ such that $\pi\circ\iota = C$. In view of Theorem \[thm.main\] (or directly using ), if $C$ is a coseparable coalgebra with a cointegral $\delta$, then any $k$-linear section $i$ of $\pi$ gives rise to such a $C$-bicolinear section $\iota$ by the formula, for all $c\in C$, $$\iota(c) = \delta\left( c\sw 1 \ot \pi(i(c\sw 2)\sw 1)\right)i(c\sw 2)\sw 2\delta\left( \pi(i(c\sw 2))\sw 3\ot c\sw 3\right).
\label{iota}$$ For example, [@BrzMaj:geo Proposition 6.1] gives an explicit formula for such a $k$-linear section of $\pi$ in the case of a quantum Hopf fibration over a general quantum two-sphere and hence a $C$-bicolinear map such as in [@BrzMaj:geo Proposition 6.3] can be obtained by the above averaging procedure .
Finally, the explicit formula can be used to calculate Chern-Galois characters for principal extensions with coseparable coalgebras. The components of the Chern-Galois character are defined in [@BrzHaj:che Corollary 3.2] in terms of the strong connection form $\ell$. However, settling the question whether the explicit knowledge of $\ell$ in terms of a cointegral provides one with an effective method of gaining information about a principal extension seems to require a case by case analysis. This is hoped to be attempted elsewhere.
[Bibliography]{} F. Bonechi, N. Ciccoli, L. Dabrowski and M. Tarlini. 51 (2004), 71–81. F. Bonechi, N. Ciccoli and M. Tarlini. Noncommutative instantons and the 4-sphere from quantum groups. [*Comm. Math. Phys.*]{}, 226: 419–432, 2003. G. Böhm and T. Brzeziński, [*Strong connections and the relative Chern-Galois character for corings*]{}, Int. Math. Res. Notices, 2005:42 (2005), 2579–2625. T. Brzeziński, [*Galois comodules,*]{} J. Algebra 290 (2005), 503–537. T. Brzeziński and P.M. Hajac, [*Coalgebra extensions and algebra coextensions of Galois type*]{}, Comm. Alg. 27 (1999), 1347–1367. T. Brzeziński and P.M. Hajac, [*The Chern-Galois character*]{}, C. R. Acad. Sci. Paris, Ser. I 338 (2004), 113–116. T. Brzeziński and S. Majid, [*Coalgebra bundles*]{}, Comm. Math. Phys. 191 (1998), 467–492. T. Brzeziński and S. Majid, [*Quantum geometry of algebra factorisations and coalgebra bundles*]{}, Commun. Math. Phys. 213 (2000), 491–521. A. Connes, [*Noncommutative Geometry.*]{} Academic Press, New York 1994. P. M. Hajac, [ *Strong connections on quantum principal bundles*]{}, Commun. Math. Phys. 182 (1996), 579–617. P.M. Hajac, R. Matthes and W. Szymański, [*A locally trivial quantum Hopf fibration*]{}, Alg. Repr. Theory 9 (2006), 121–146. G. Landi, C. Pagani and C. Reina, [*A Hopf bundle over a quantum four-sphere from the symplectic group*]{}, Commun. Math. Phys. 263 (2006), 65–88 G. Landi and W. van Suijlekom, [*Principal fibrations from noncommutative spheres*]{}, Commun. Math. Phys. 260 (2005), 203–225. P. Schauenburg and H.-J. Schneider, [*On generalised Hopf-Galois extensions*]{}, J. Pure Appl. Alg. 202 (2005), 168–194 H.-J. Schneider. Principal homogeneous spaces for arbitrary Hopf algebras. [*Israel J. Math*]{} 72 (1990), 167–195. S. L. Woronowicz, [*Compact matrix pseudogroups,*]{} Commun. Math. Phys. 111 (1987), 613–665.
|
---
author:
- Menglu WANG and Hao WU
bibliography:
- 'bibliography.bib'
title: |
Level Lines of Gaussian Free Field II:\
Whole-Plane $\operatorname{GFF}$
---
Introduction
============
Sequence of level loops starting from interior {#sec::interior_levelloops}
==============================================
The whole-plane $\operatorname{GFF}$ {#subsec::wholeplane_gff}
------------------------------------
Alternating height-varying sequence of level loops {#subsec::alternate_levelloops_construction}
--------------------------------------------------
Interaction {#subsec::interaction}
-----------
Continuum exploration process starting from interior {#sec::continuum_exploration}
====================================================
Growing process of $\operatorname{CLE}_4$ {#subsec::growing_cle4}
-----------------------------------------
From discrete to continuum exploration of $\operatorname{GFF}$ {#subsec::gff_discrete_continuum}
--------------------------------------------------------------
Alternating continuum exploration process of $\operatorname{GFF}$ {#subsec::gff_alternate_continuum}
-----------------------------------------------------------------
Interaction {#subsec::continuum_interaction}
-----------
Reversibility {#subsec::continuum_reversibility}
-------------
Menglu Wang\
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA\
[email protected]
Hao Wu\
NCCR/SwissMAP, Section de Mathématiques, Université de Genève, Switzerland\
*and* Yau Mathematical Sciences Center, Tsinghua University, China\
[email protected]
|
---
abstract: |
Limits on $\nu_\mu (\overline{\nu}_\mu) \to \nu_e (\overline{\nu}_e)$ oscillations based on a statistical separation of $\nu_e N$ charged current interactions in the CCFR detector at Fermilab are presented. $\nu_e$ interactions are identified by the difference in the longitudinal shower energy deposition pattern of $\nu_e N \rightarrow
eX$ versus $\nu_\mu N \rightarrow \nu_\mu X$ interactions. Neutrino energies range from 30 to 600 GeV with a mean of 140 GeV, and $\nu_\mu$ flight lengths vary from 0.9 km to 1.4 km. The lowest 90% confidence upper limit in $\sin^2 2\alpha$ of $1.1 \times 10^{-3}$ is obtained at $\Delta m^2 \sim 300$ ${\rm eV^2}$. For $\sin^2 2\alpha =
1$, $\Delta m^2 > 1.6$ ${\rm eV^2}$ is excluded, and for $\Delta m^2
\gg 1000$ ${\rm eV^2}$, $\sin^2 2\alpha > 1.8 \times 10^{-3}$ is excluded. This result is the most stringent limit to date for $\Delta
m^2 > 25$ ${\rm eV^2}$ and it excludes the high $\Delta m^2$ oscillation region favoured by the LSND experiment. The $\nu_\mu$-to-$\nu_e$ cross-section ratio was measured as a test of $\nu_\mu (\bar\nu_\mu) \leftrightarrow \nu_e (\bar\nu_e)$ universality to be $1.026 \pm 0.055$.
address: |
$^1$ University of Cincinnati, Cincinnati, OH 45221\
$^2$ Columbia University, New York, NY 10027\
$^3$ Fermi National Accelerator Laboratory, Batavia, IL 60510\
$^4$ Kansas State University, Manhattan, KS 66506\
$^5$ Northwestern University, Evanston, IL 60208\
$^6$ University of Oregon, Eugene, OR 97403\
$^7$ University of Rochester, Rochester, NY 14627\
$^8$ University of Wisconsin, Madison, WI 53706\
author:
- 'A. Romosan,$^2$ C. G. Arroyo,$^2$ L. de Barbaro,$^5$ P. de Barbaro,$^7$ A. O. Bazarko,$^2$ R. H. Bernstein,$^3$ A. Bodek,$^7$ T. Bolton,$^4$ H. Budd,$^7$ J. Conrad,$^2$ R. B. Drucker,$^6$ D. A. Harris,$^7$ R. A. Johnson,$^1$ J. H. Kim,$^2$ B. J. King,$^2$ T. Kinnel,$^8$ M. J. Lamm,$^3$ W. C. Lefmann,$^2$ W. Marsh,$^3$ K. S. McFarland,$^3$ C. McNulty,$^2$ S. R. Mishra,$^2$ D. Naples,$^4$ P. Z. Quintas,$^2$ W. K. Sakumoto,$^7$ H. Schellman,$^5$ F. J. Sciulli,$^2$ W. G. Seligman,$^2$ M. H. Shaevitz,$^2$ W. H. Smith,$^8$ P. Spentzouris,$^2$ E. G. Stern,$^2 $ M. Vakili,$^1$ U. K. Yang,$^7$ and J. Yu$^3$'
title: |
[NEVIS-1529]{}\
A High Statistics Search for $\nu_\mu(\overline\nu_\mu) \rightarrow
\nu_e(\overline\nu_e)$ Oscillations in the Small Mixing Angle Regime
---
The existence of neutrino mass and mixing would have important implications for fundamental problems in both particle physics and cosmology. These include violation of lepton family number conservation, the mass of the universe, and the observed neutrino deficits from the sun and from atmospheric sources. Neutrino oscillations are a necessary consequence of non-zero neutrino mass and mixing since neutrinos are produced and detected in the form of weak-interaction eigenstates whereas their motion as they propagate from the point of production to their detection is dictated by the mass eigenstates [@pcrv]. In the two-generation formalism, the mixing probability is: $$P(\nu_1 \rightarrow \nu_2) = \sin^2 2\alpha \sin^2 \left(\frac{1.27
\Delta m^2 L}{E_\nu}\right)
\label{eq:posc}$$ where $\Delta m^2$ is the mass squared difference of the mass eigenstates in ${\rm eV^2}$, $\alpha$ is the mixing angle, $E_\nu$ is the incoming neutrino energy in GeV, and $L$ is the distance between the point of creation and detection in km.
To date the best limits from accelerator experiments for $\nu_\mu
\rightarrow \nu_e$ oscillations come from fine-grained calorimetric (e.g.: BNL-E734 [@e734], BNL-E776 [@e776]) or fully active detectors (e.g. KARMEN [@karm], LSND [@lsnd]) searching for quasi-elastic charged current production of electrons. The LSND experiment, using a liquid scintillator neutrino target, has reported a signal consistent with $\bar{\nu}_\mu \rightarrow \bar{\nu}_e$ oscillations at a $\sin^2 2\alpha \approx 10^{-2}$ and $\Delta m^2
\stackrel{>}{\scriptstyle\sim} 1$ eV${^2}$ [@lsnd]. The CCFR collaboration has previously reported a limit on $\nu_\mu \rightarrow
\nu_e$ oscillations using the ratio of neutral to charged current neutrino events comparable in sensitivity to the above mentioned limits [@donna].
In this report we present new limits on $\nu_\mu \rightarrow \nu_e$ oscillations based on the statistical separation of $\nu_e N$ charged current interactions.
The CCFR detector [@ws90; @bk91] consists of an 18 m long, 690 ton total absorption target calorimeter with a mean density of ${\rm 4.2
g/cm^3}$, followed by a 10 m long iron toroidal spectrometer. The target consists of 168 steel plates, each ${\rm 3 m \times 3 m \times
5.15 cm}$, instrumented with liquid scintillation counters placed every two steel plates and drift chambers spaced every four plates. The separation between scintillation counters corresponds to 6 radiation lengths, and the ratio of electromagnetic to hadronic response of the calorimeter is $1.05$. The toroid spectrometer is not directly used in this analysis which is based on the shower profiles in the target-calorimeter.
The Fermilab Tevatron Quadrupole Triplet neutrino beam is a high-intensity, non-sign-selected wideband beam with a $\nu$ : $\overline\nu$ flux ratio of about 2.5 : 1 and usable neutrino energies up to 600 GeV. The production target is located 1.4 km upstream of the neutrino detector and is followed by a 0.5 km decay region. The resulting neutrino energy spectra for $\nu_\mu$, $\overline\nu_\mu$, $\nu_e$, and $\overline\nu_e$ induced events are shown in Figure \[fig:enu\]. The beam contains a 2.3% fraction of electron neutrinos, 82% of which are produced from $K^\pm \rightarrow \pi^0 e^\pm
\stackrel{_{(-)}}{\nu_e}$.
The neutrino interactions observed in the detector can be divided into three classes depending on the type of the incoming neutrino and on the interaction type:
1. $\nu_{\mu}N \rightarrow \mu^-X$ ($\nu_{\mu}$ charged current (CC) events).
2. $\nu_{\mu,e}N \rightarrow \nu_{\mu,e}X$ ($\nu_{\mu,e}$ neutral current (NC) events).
3. $\nu_e N \rightarrow eX$ ($\nu_e$ CC events).
All three types of neutrino interactions initiate a cascade of hadrons that is registered by the drift chambers and scintillation counters. The $\nu_\mu$ CC events are characterized by the presence of a muon produced in the final state which penetrates beyond the end of the hadron shower, depositing energy characteristic of a minimum ionizing particle [@ws90] in a large number of consecutive scintillation counters. Conversely, the electron produced in a $\nu_e$ CC event deposits energy in a few counters immediately downstream of the interaction vertex which changes the energy deposition profile of the shower. The electromagnetic shower is typically much shorter than the hadron shower and the two cannot be separated for a $\nu_e$ CC event.
In this analysis four experimental quantities are calculated for each event: the length, the transverse vertex position, the visible energy and the shower energy deposition profile. The event length is determined to be the number of scintillation counters spanned from the event vertex to to the last counter with a minimum-ionizing pulse height. The mean position of the hits in the drift chamber immediately downstream of the interaction vertex determines the transverse vertex position. The visible energy in the calorimeter, $E_{vis}$ is obtained by summing the energy deposited in the scintillation counters from the interaction vertex to five counters beyond the end of the shower. The shower energy deposition profile is characterized by the ratio of the sum of the energy deposited in the first three scintillation counters to the total visible energy. Accordingly, we define $$\eta_3 = 1 - \frac{E_1 + E_2 + E_3}{E_{vis}} \label{eq:eta3}$$ where $E_i$ is the energy deposited in the $i^{th}$ scintillation counter downstream of the interaction place.
The most downstream counter with energy deposited from the products of the neutrino interaction (CEXIT) occurs at the end of the hadron shower for $\nu_\mu$ NC and $\nu_e$ CC events but is determined by the muon track for most $\nu_\mu$ CC events. We isolate the events without a muon track by requiring CEXIT to be no more than 10 counters downstream from the end of the hadron shower. We parametrize the event length which contains 99% of such events as: $$L_{NC} = 4.+ 3.81\times \log(E_{vis})$$
In order to measure the number of $\nu_e$ CC events we divide the neutrino events into two classes: “short” if they deposit energy over an interval shorter than $L_{NC}$, and “long” otherwise. The long events consist almost exclusively of class 1 events, while the short ones are a mixture of class 2, class 3 and class 1 events with a low energy muon which cannot be separated on an event-by-event basis.
Based on Lund studies, we take the hadron showers produced in NC and CC interactions to be the same. Any difference in the shower energy deposition profile of long and short events is attributed to the presence of $\nu_e$ CC interactions in the short sample. To compare directly the long and short events a muon track from the data was added to the short events to compensate for the absence of a muon in NC events. The fraction, [*f*]{}, of $\nu_\mu$ CC events with a low energy muon contained in the short sample which now have two muon tracks was estimated from a detailed Monte Carlo of the experiment in the range of 20%. A simulated sample of such events was obtained by choosing long events with the appropriate energy distribution from the data to which a second short muon track was added in software. The length of the short track and the angular distribution were obtained from a Monte Carlo of $\nu_\mu$ CC events.
To simulate $\nu_e$ interactions in our detector we assume $\nu_\mu -
\nu_e$ universality. The electron neutrino showers were generated by adding a GEANT [@g321] generated electromagnetic shower of the appropriate energy to events in the long data sample. The energy distribution of the electron neutrinos and the fractional energy transfer $y$ were generated using a detailed Monte Carlo simulation of the experiment. Since the hadron showers in the long sample already have a muon track, the $\nu_e$ sample can be compared directly with the short and long events.
The long and short $\eta_3$ distributions were further corrected by subtracting the contamination due to cosmic ray events. The cosmic ray background was estimated from the event sample collected during a beam off gate using an identical analysis procedure as for the data gates. Additionally, the $\eta_3$ distribution of short $\nu_\mu$ CC events, normalized to the predicted fraction [*f*]{}, was subtracted from the short event sample. The $\eta_3$ distributions for short, long, and $\nu_e$ events for various energy bins are shown in Figure \[fig:etas\].
For this oscillation search we measure the absolute flux of $\nu_e$’s at the detector and compare it to the flux predicted by a detailed beamline simulation [@ca94]. Any excess could be interpreted as a signal of $\nu_\mu \rightarrow \nu_e$ oscillations. The $\nu_\mu$ flux was determined directly from the low hadron energy CC event sample, normalized to the total neutrino cross-section [@flux]. The same beamline simulation is used to tag the creation point of each simulated $\nu_\mu$ along the decay pipe, and give the number of predicted $\nu_\mu$’s at the detector normalized to the number observed at the detector divided by $1-P(\nu_\mu \rightarrow \nu_e)$. $P(\nu_\mu \rightarrow \nu_e)= P(\overline\nu_\mu \rightarrow
\overline\nu_e)$ is the oscillation probability determined from eq. (\[eq:posc\]), assuming CP invariance. The predicted electron neutrino flux is normalized to the [*produced*]{} number of $\nu_\mu$’s. The $\nu_e$ flux from neutrino oscillations is calculated by multiplying the [*produced*]{} number of $\nu_\mu$’s by $P(\nu_\mu
\rightarrow \nu_e)$.
The events selected are required to deposit a minimum of 30 GeV in the target calorimeter to ensure complete efficiency of the energy deposition trigger. Additionally, we require the event vertex to be more than 5 counters from the upstream end of the target and five counters plus the separation length from the downstream end and less than $50"$ from the detector centre-line. The resulting data sample consists of 632338 long events and 291354 short ones.
To extract the number of $\nu_e$ CC events in each of 15 $E_{vis}$ bins, we fit the corrected shape of the observed $\eta_3$ distribution for the short sample to a combination of $\nu_\mu$ CC and $\nu_e$ CC distributions with appropriate muon additions: $${\rm \nu_\mu NC (+ \mu) = \alpha \: \nu_\mu CC + \beta \: \nu_e CC (+
\mu)}$$
The $\chi^2$ of the fit in each of the 15 $E_{vis}$ bins ranges from 33.2 to 77.7 for 41 degrees of freedom (DoF) with a mean value of 48.4. Figure \[fig:result\] shows that the measured number of $\nu_e$ CC’s agrees with the Monte Carlo prediction in each energy bin. The $\chi^2$ value with a no-oscillations assumption is $9.97/15$ DoF.
The major sources of uncertainties in the comparison of the electron flux extracted from the data to that predicted by the Monte Carlo are: (i) The statistical error from the fit in the extraction of the $\nu_e$ flux. (ii) The error in the shower shape modeling, estimated by extracting the $\nu_e$ flux using two definitions of $\eta$. Analogous to the definition of $\eta_3$ given in eq. (\[eq:eta3\]), we define $\eta_4$ to be the ratio of the sum of the energy deposited outside the first four scintillation counters to the total visible energy. If the modeling of the showers were correct, the difference in the number of electron neutrinos measured by the two methods should be small, any difference is used to estimate the systematic error. Since this error was shown not to be correlated among energy bins, we add it in quadrature to the statistical error from the fit and take this to be the combined basic error. (iii) The 1% uncertainty in the absolute energy calibration of the detector changes the relative neutrino flux which is extracted using the subset of the data sample with low hadron energy [@flux]. (iv) The uncertainty in the incident flux of $\nu_e$’s on the detector is estimated to be $4.1\%$ [@ca94]. This error is dominated by a 20% production uncertainty in the $K_L$ content of the secondary beam which produces 16% of the $\nu_e$ flux. The majority of the $\nu_e$ flux comes from $K_{e_{3}}^\pm$ decays, which are well-constrained by the observed $\nu_\mu$ spectrum from $K_{\mu_{2}}^\pm$ decays [@ca94]. Other sources of systematic errors were also investigated and found to be small.
[cccccc]{} $\Delta m^2$ (eV$^2$) & Best fit & $\sigma$ & $\Delta m^2$ (eV$^2$) & Best fit & $\sigma$\
-------
1.0
2.0
3.0
4.0
5.0
7.0
9.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0
100.0
125.0
150.0
-------
: The result for $\sin^2 2\alpha$ from the fit at each $\Delta
m^2$ for $\nu_\mu \rightarrow \nu_e$ oscillations. The 90% C.L. upper limit is equal to the best fit $\sin^2 2\alpha +
1.28\sigma$.[]{data-label="tab:bestfit"}
&
---------
-0.1741
-0.0501
-0.0153
-0.0112
-0.0051
-0.0036
-0.0021
-0.0023
-0.0004
-0.0003
-0.0002
-0.0002
-0.0002
-0.0002
-0.0003
-0.0003
-0.0002
0.0004
0.0005
---------
: The result for $\sin^2 2\alpha$ from the fit at each $\Delta
m^2$ for $\nu_\mu \rightarrow \nu_e$ oscillations. The 90% C.L. upper limit is equal to the best fit $\sin^2 2\alpha +
1.28\sigma$.[]{data-label="tab:bestfit"}
&
--------
1.6501
0.4107
0.1852
0.1041
0.0671
0.0345
0.0213
0.0173
0.0048
0.0026
0.0018
0.0015
0.0014
0.0014
0.0014
0.0015
0.0015
0.0018
0.0019
--------
: The result for $\sin^2 2\alpha$ from the fit at each $\Delta
m^2$ for $\nu_\mu \rightarrow \nu_e$ oscillations. The 90% C.L. upper limit is equal to the best fit $\sin^2 2\alpha +
1.28\sigma$.[]{data-label="tab:bestfit"}
&
---------
175.0
200.0
225.0
250.0
275.0
300.0
350.0
400.0
450.0
500.0
600.0
700.0
800.0
1000.0
1500.0
2000.0
5000.0
10000.0
20000.0
---------
: The result for $\sin^2 2\alpha$ from the fit at each $\Delta
m^2$ for $\nu_\mu \rightarrow \nu_e$ oscillations. The 90% C.L. upper limit is equal to the best fit $\sin^2 2\alpha +
1.28\sigma$.[]{data-label="tab:bestfit"}
&
---------
0.0000
-0.0002
-0.0003
-0.0004
-0.0004
-0.0004
-0.0004
-0.0003
-0.0003
-0.0004
-0.0005
-0.0003
-0.0002
-0.0004
-0.0003
-0.0004
-0.0003
-0.0004
-0.0004
---------
: The result for $\sin^2 2\alpha$ from the fit at each $\Delta
m^2$ for $\nu_\mu \rightarrow \nu_e$ oscillations. The 90% C.L. upper limit is equal to the best fit $\sin^2 2\alpha +
1.28\sigma$.[]{data-label="tab:bestfit"}
&
--------
0.0016
0.0014
0.0013
0.0012
0.0012
0.0012
0.0012
0.0013
0.0015
0.0016
0.0019
0.0018
0.0018
0.0017
0.0017
0.0017
0.0018
0.0017
0.0017
--------
: The result for $\sin^2 2\alpha$ from the fit at each $\Delta
m^2$ for $\nu_\mu \rightarrow \nu_e$ oscillations. The 90% C.L. upper limit is equal to the best fit $\sin^2 2\alpha +
1.28\sigma$.[]{data-label="tab:bestfit"}
The data are fit by forming a $\chi^2$ which incorporates the Monte Carlo generated effect of oscillations, the basic error, and terms with coefficients accounting for systematic uncertainties. A best fit $\sin^2 2\alpha$ is determined for each $\Delta m^2$ by minimizing the $\chi^2$ as a function of $\sin^2 2\alpha$ and these systematic coefficients. At all $\Delta m^2$, the data are consistent with no observed $\nu_\mu \rightarrow \nu_e$ oscillations. The statistical significance of the best-fit oscillation at any $\Delta m^2$ is at most $0.3 \sigma$.
The frequentist approach [@pdg] is used to set a 90% confidence upper limit for each $\Delta m^2$. The limit in $\sin^2 2\alpha$ corresponds to a shift of 1.64 units in $\chi^2$ from the minimum $\chi^2$ (at the best fit value in Table \[tab:bestfit\]). The 90% confidence upper limit is plotted in Figure \[fig:osc\] for $\nu_\mu \rightarrow \nu_e$. The best limit of $\sin^2 2\alpha < 1.1
\times 10^{-3}$ is at $\Delta m^2 = 300$ ${\rm eV^2}$. For $\sin^2
2\alpha = 1$, $\Delta m^2 > 1.6$ ${\rm eV^2}$ is excluded, and for $\Delta m^2 \gg 1000$ ${\rm eV^2}$, $\sin^2 2\alpha > 1.8 \times
10^{-3}$.
Under the assumption that there are no oscillations, this data can also be used to test $\nu_\mu (\bar\nu_\mu) \leftrightarrow \nu_e
(\bar\nu_e)$ universality by comparing the observed $\nu_e$ flux to that predicted by the Monte Carlo. From this comparison we determine the ratio of the cross sections averaged over our flux to be $\sigma_{CC}(\nu_\mu)/\sigma_{CC}(\nu_e) = 1.026 \pm 0.055$. This is currently the most stringent test of universality at high space-like momentum transfer.
In conclusion, we have used the difference in the longitudinal shower energy deposition pattern of $\nu_e N$ versus $\nu_\mu N$ interactions to search for $\nu_\mu \rightarrow \nu_e$ oscillations with a coarse-grained calorimetric detector. We see a result consistent with no neutrino oscillations and find 90% confidence level excluded regions in the $\sin^2 2\alpha - \Delta m^2$ phase space. This result is the most stringent limit to date for $\nu_\mu \rightarrow \nu_e$ oscillation for $\Delta m^2 > 25$ ${\rm eV^2}$. We also tested $\nu_\mu (\bar\nu_\mu) \leftrightarrow \nu_e (\bar\nu_e)$ universality and found the ratio of the $\nu_\mu$-to-$\nu_e$ cross-section to be $1.026 \pm 0.055$.
B. Pontecorvo, JETP, [**6**]{}, 429 (1958); Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. [**28**]{}, 870 (1962). L. A. Ahrens [*et al.* ]{}, Phys. Rev. [**D36**]{}, 702 (1987). L. Borodovsky [*et al.* ]{}, Phys. Rev. Lett. [**68**]{}, 274 (1992). B.A. Bodmann [*et al.*]{}, Nucl. Phys. [**A553**]{}, 831c (1993). C. Athanassopolous [*et al.*]{}, Phys. Rev. Lett. [**77**]{}, 3082 (1996). K.S. McFarland, D. Naples [*et al.*]{}, Phys. Rev. Lett., [**75**]{}, 3993 (1995). W.K. Sakumoto [*et al.*]{}, Nucl. Instrum. Methods, [**A294**]{}, 179 (1990). B.J. King [*et al.*]{}, Nucl. Instrum. Methods, [**A302**]{}, 254 (1991). CN/ASD, GEANT, detetector description and simulation tool, CERN (1995). C. Arroyo [*et al.*]{}, Phys. Rev. Lett. [**72**]{}, 3452 (1994); Bruce J. King, PhD Thesis, Columbia University (1994), Nevis preprint 284, unpublished. P.Z. Quintas, PhD Thesis, Columbia University (1992), Nevis Preprint 277, unpublished; W.C. Leung, PhD Thesis, Columbia University (1991), Nevis Preprint 276, unpublished. Particle Data Group, Phys. Rev. [**D54**]{},164 (1996).
|
---
abstract: 'Recently, several discriminative learning approaches have been proposed for effective image restoration, achieving convincing trade-off between image quality and computational efficiency. However, these methods require separate training for each restoration task (e.g., denoising, deblurring, demosaicing) and problem condition (e.g., noise level of input images). This makes it time-consuming and difficult to encompass all tasks and conditions during training. In this paper, we propose a discriminative transfer learning method that incorporates formal proximal optimization and discriminative learning for general image restoration. The method requires a single-pass training and allows for reuse across various problems and conditions while achieving an efficiency comparable to previous discriminative approaches. Furthermore, after being trained, our model can be easily transferred to new likelihood terms to solve untrained tasks, or be combined with existing priors to further improve image restoration quality.'
author:
- |
Lei Xiao\
University of British Columbia\
- |
Felix Heide\
Stanford University\
- |
Wolfgang Heidrich\
KAUST\
- |
Bernhard Schölkopf\
MPI for Intelligent Systems\
- |
Michael Hirsch\
MPI for Intelligent Systems\
bibliography:
- 'egbib.bib'
title: Discriminative Transfer Learning for General Image Restoration
---
|
---
abstract: 'We examine the effect of isospin-violating meson-nucleon coupling constants on low-energy pion-nucleon scattering. We compute the couplings in the context of a nonrelativistic quark model. The difference between the up and down constituent masses induces a coupling of the neutral pion to the proton that is slightly larger than the corresponding one for the neutron. This difference generates a large isospin-violating correction—proportional to the isospin-even contribution arising from the nucleon Born terms—to the charge-exchange ($\pi^{-}p \rightarrow \pi^{0}n$) amplitude. In contrast to the isospin-conserving case, this correction is not cancelled by $\sigma$-meson exchange; in our model there is no isospin-violating $NN\sigma$ coupling at $q^2=0$. As a result, we find a violation of the triangle identity consistent with the one reported by Gibbs, Ai, and Kaufmann from a recent analysis of pion-nucleon data.'
address: |
Supercomputer Computations Research Institute,\
Florida State University, Tallahassee, FL 32306
author:
- 'J. Piekarewicz'
title: 'Isospin Violations in the Pion-Nucleon System'
---
Introduction {#secintro}
============
Low energy pion-nucleon ($\pi N$) scattering is one of the best available tools for testing small violations to approximate symmetries of nature. Such violations are expected to be amplified in low-energy $\pi N$ scattering because of the constraints imposed on the symmetry-conserving amplitudes by chiral symmetry. At low energies (i.e., in the soft-pion limit) the pions couple very weakly to the nucleons as a direct consequence of chiral symmetry. Thus, although the violations to the symmetry might be small, they must be considered relative to intrinsically small symmetry-conserving amplitudes.
An example of such a scenario has been reported recently by Gibbs, Ai, and Kaufmann [@gak95]. They have analyzed low-energy pion-nucleon data in search of isospin violations. From very precise data on elastic ($\pi^{\pm}p$) and charge-exchange ($\pi^{-}p \rightarrow \pi^{0}n$) reactions they have extracted $\pi N$ scattering amplitudes from which they have computed violations to the “triangle identity” $$D \equiv f(\pi^{-}p \rightarrow \pi^{0}n) -
{1 \over \sqrt{2}} \, \Big[f(\pi^{+}p)-f(\pi^{-}p)\Big] \;.
\label{triangle}$$ They observed a large isospin violation—of the order of 7%—even after accounting for Coulomb effects and hadronic mass differences. This is particularly interesting since isospin-breaking mechanisms, having their origin in the up-down quark mass difference and electromagnetic effects, are expected to be present at the $\sim$ 1% level.
Evidence for the loss of isospin symmetry in the nucleon-nucleon ($NN$) system is well documented. The difference in the $pp$ and $nn$ scattering lengths [@slaus89], the Nolen-Schiffer anomaly[@nolsch69; @miller90], and the neutron-proton analyzing-power difference [@knut90; @abegg89; @abegg94] are all well known examples. Most theoretical efforts directed at understanding isospin-violating observables in the $NN$ system proceed from a two-body interaction constrained from fits to two-nucleon data and incorporate isospin-violating corrections from a variety of sources. These can be classified as arising from: (i) isovector-isoscalar mixing in the meson propagator—such as $\rho$-$\omega$ mixing, (ii) isospin-breaking in the nucleon wave function—through the neutron-proton mass difference, and (iii) isospin-breaking in the meson-nucleon and photon-nucleon vertices—as in the case of electromagnetic scattering. It is important to note that all these isospin-breaking mechanisms also operate in the pion-nucleon system. Thus, a clear understanding of their role in $NN$ scattering could be of great value to the analysis of low-energy $\pi N$ data. A particularly important—and timely—example is $\rho$-$\omega$ mixing. Naively, one would expect large violations to the triangle identity (also known as the “triangle discrepancy”) to arise from $\rho$-$\omega$ mixing because of the strong $NN\omega$ and $\pi\pi\rho$ couplings. Note, however, that in computing near-threshold $\pi N$ observables it is the mixing amplitude near $q^2=0$ that is relevant. The traditional mechanism of $\rho$-$\omega$ mixing, with the mixing amplitude fixed at the on-shell point, has been called recently into question [@ght92]. Indeed, a large number of calculations using a variety of models have found a value of the $\rho$-$\omega$ mixing amplitude at $q^2=0$ that is strongly suppressed relative to its on-shell value [@piewil93; @hats93; @krein93; @mitch94; @oconn94]. Moreover, for models in which the vector mesons couple to conserved currents, the $\rho$-$\omega$ mixing amplitude is identically zero at $q^2=0$ [@piewil93; @oconn94]. Thus, we believe that $\rho$-$\omega$ mixing should play a small role in low-energy pion-nucleon scattering.
Removing $\rho$-$\omega$ mixing as a viable source of isospin-breaking has important phenomenological consequences; on-shell $\rho$-$\omega$ mixing accounts for a substantial fraction of the neutron-proton analyzing-power difference at 183 MeV [@knut90; @miller86; @willia87]. Hence, if $\rho$-$\omega$ mixing is no longer important at $q^2$ [to 8pt[-6pt]{}]{} 0, additional sources of isospin violation must be found. In a recent study of hadronic structure Dmitrašinović and Pollock have computed isospin-violating corrections to the electroweak form factors of the nucleon [@dmitra95]. Motivated by their findings we have investigated new sources of charge-symmetry violation in the $NN$ potential which resulted from isospin-violating meson-nucleon coupling constants [@ghp95]. The resulting class IV contribution to the charge-symmetry-breaking $NN$ potential is comparable in magnitude and identical in sign to the one obtained from on-shell $\rho$-$\omega$ mixing. We showed that this new contribution—without on-shell $\rho$-$\omega$ mixing—is consistent with the measured value of $\Delta A$ at 183 MeV [@ghp295]. It is the purpose of this paper to estimate the effect of isospin-violating meson-nucleon coupling constants on low-energy pion-nucleon scattering.
Low-energy pion-nucleon scattering {#secpin}
==================================
We approach the study of low-energy pion-nucleon scattering in a conventional way; we include contributions arising from the (s- and u-channel) nucleon Born terms and from (t-channel) meson exchanges [@camp78]. These contributions—particle-exchange poles—give a good representation of the amplitude when the poles are close to the physical region, such as in low-energy $\pi N$ scattering in the chiral ($m_{\pi} \rightarrow 0$) limit. The linear $\sigma$-model [@gell60] and Quantum Hadrodynamics (QHD-II) [@serwal86] are appropriate theoretical frameworks to generate these tree-level contributions. The models differ, at tree level, in the allowed t-channel exchanges and, hence, in the prediction of low-energy $\pi N$ parameters. However, as we shall see, they generate the same isospin-violating contributions in our model.
The $\pi N$ scattering matrix can be written in terms of two sets (one for each isospin combination) of two Lorentz invariant amplitudes ($A$ and $B$) which contain all dynamical information about the reaction [@camp78] $$\hat{{\cal T}} =
\Big[ A^{(+)}(s,t)+{1 \over 2}({\rlap/k}+{\rlap/k'})B^{(+)}(s,t)\Big] -
\Big[ A^{(-)}(s,t)+{1 \over 2}({\rlap/k}+{\rlap/k'})B^{(-)}(s,t)\Big]
({\bf T}\cdot\mbox{\boldmath$\tau$}) \;.
\label{relt}$$ Note, the Lorentz invariant amplitudes are written in terms of the relevant Mandelstam variables ($t \equiv q^{2}$)
$$\begin{aligned}
s &=& (p+k)^{2}=(p'+k')^{2} \;, \\
t &=& (k-k')^{2}=(p'-p)^{2} \;, \\
u &=& (p-k')^{2}=(p'-k)^{2} \;,
\label{mandel}
\end{aligned}$$
where $k(k')$ and $p(p')$ are the initial(final) four-momenta of the pion and nucleon, respectively. The Mandelstam variables are related by $s+t+u=2m_{\pi}^{2}+2M^{2}$. We have also introduced pion (${\bf T}$) and nucleon () isospin matrices \[note, $(T_{a})_{bc} \equiv -i\epsilon_{abc}$\]. Isospin invariance, which is still assumed unbroken, allows for only two isospin combinations: isospin even \[denoted by $(+)$\] and isospin odd \[denoted by $(-)$\]. The connection to the reaction amplitudes is given through the following relations:
$$\begin{aligned}
{\cal T}(\pi^{+}p \rightarrow \pi^{+}p) &=&
{\cal T}^{(+)} - {\cal T}^{(-)} \;, \\
{\cal T}(\pi^{-}p \rightarrow \pi^{-}p) &=&
{\cal T}^{(+)} + {\cal T}^{(-)} \;, \\
{\cal T}(\pi^{-}p \rightarrow \pi^{0}n) &=&
- \sqrt{2}\,{\cal T}^{(-)} \;.
\label{reacmpl}
\end{aligned}$$
From these, the triangle identity \[see Eq. (\[triangle\])\] follows by inspection.
The partial-wave decomposition of the scattering amplitude is simplest if carried out after the Lorentz-invariant scattering matrix has been evaluated between on-shell spinors in the center-of-mass (CM) frame. Thus, as an operator in the spin space of the nucleon the $\pi N$ scattering amplitude can be written as, $$\hat{f}^{(\pm)} =
f^{(\pm)}_{1}(W,\theta) + f^{(\pm)}_{2}(W,\theta)
{(\mbox{\boldmath$\sigma$}\cdot{\bf k}')
(\mbox{\boldmath$\sigma$}\cdot{\bf k} ) \over k^2} \;,
\label{pinamp}$$ where the connection to the Lorentz-invariant amplitudes is given through the relations
$$\begin{aligned}
f^{(\pm)}_{1}(W,\theta) &=&
\left({E_{k}+M \over 8\pi W}\right)
\Big[ +A^{(\pm)}(s,t) + (W-M)B^{(\pm)}(s,t) \Big] \;, \\
f^{(\pm)}_{2}(W,\theta) &=&
\left({E_{k}-M \over 8\pi W}\right)
\Big[ -A^{(\pm)}(s,t) + (W+M)B^{(\pm)}(s,t) \Big] \;.
\label{f12}
\end{aligned}$$
Here $\theta$ denotes the CM scattering angle and $W=(\epsilon_{k}+E_{k})$ is the total energy of the system in the CM frame; it is written in terms of the individual pion ($\epsilon_{k}$) and nucleon ($E_{k}$) contributions. Finally, by introducing the partial-wave amplitudes $f{\lower 2pt \hbox{$\scriptstyle l^{\pm}$}}$, appropriate for scattering in a total angular-momentum channel $j=l\pm 1/2$, the amplitudes $f_{1}$ and $f_{2}$ can be expanded in a partial-wave series:
$$\begin{aligned}
f^{(\pm)}_{1}(W,\theta) &=& \sum_{l}
\Big[
f^{(\pm)}_{\lower 2pt \hbox{$\scriptstyle l^{+}$}}(W)
P^{'}_{l+1}(\cos\theta) -
f^{(\pm)}_{\lower 2pt \hbox{$\scriptstyle l^{-}$}}(W)
P^{'}_{l-1}(\cos\theta)
\Big] \;, \\
f^{(\pm)}_{2}(W,\theta) &=& \sum_{l}
\Big[
f^{(\pm)}_{\lower 2pt \hbox{$\scriptstyle l^{-}$}}(W) -
f^{(\pm)}_{\lower 2pt \hbox{$\scriptstyle l^{+}$}}(W)
\Big] P^{'}_{l}(\cos\theta) \;.
\label{f12pw}
\end{aligned}$$
We compute the Lorentz invariant amplitudes $A$ and $B$ in the linear sigma model [@camp78]. The connection to other models, specifically to QHD, will be done below. At tree-level, the amplitudes receive contribution from only three Feynman diagrams: the two nucleon Born terms and $\sigma$-meson exchange. That is,
$$\begin{aligned}
A^{(+)}(s,t) &=&
-{g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}^{2}\over M}
{m_{\sigma}^{2}-m_{\pi}^{2} \over t-m_{\sigma}^{2}}
\mathop{\longrightarrow}\limits_{|{\bf k}| \rightarrow 0} \;
{g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}^{2}\over M}
\left(1-{m_{\pi}^{2} \over m_{\sigma}^{2}}\right) \; \\
A^{(-)}(s,t) &=& 0 \;, \\
B^{(+)}(s,t) &=&
-{g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}^{2} \over s-M^{2}}
+{g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}^{2} \over u-M^{2}}
\mathop{\longrightarrow}\limits_{|{\bf k}| \rightarrow 0} \;
-{g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}^{2}\over Mm_{\pi}}
\left(1-{m_{\pi}^{2} \over 4M^{2}}\right)^{-1} \; \\
B^{(-)}(s,t) &=&
-{g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}^{2} \over s-M^{2}}
-{g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}^{2} \over u-M^{2}}
\mathop{\longrightarrow}\limits_{|{\bf k}| \rightarrow 0} \;
\hskip 12pt
{g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}^{2}\over 2M^{2}}
\left(1-{m_{\pi}^{2} \over 4M^{2}}\right)^{-1} \;.
\label{bminus}
\end{aligned}$$
where the limit follows from evaluating the amplitudes at threshold: $t=0$, $s=(M+m_{\pi})^{2}$, and $u=(M-m_{\pi})^{2}$. The extraction of the $\pi N$ scattering lengths, defined by $$a_{0}^{(\pm)}=\lim_{|{\bf k}|\rightarrow 0}f^{(\pm)}_{1}
={1 \over 4\pi(1+m_{\pi}/M)}
\Big[A^{(\pm)} + m_{\pi} B^{(\pm)} \Big] \;,
\label{scattl}$$ is now straightforward. We obtain,
$$\begin{aligned}
a_{0}^{(+)} &=& {1 \over 4\pi(1+m_{\pi}/M)}
{g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}^{2}\over M}
\left[\left(1-{m_{\pi}^{2} \over m_{\sigma}^{2}}\right) -
\left(1-{m_{\pi}^{2} \over 4M^{2}}\right)^{-1}\right]
\mathop{\longrightarrow}\limits_{m_{\pi} \rightarrow 0} 0 \;, \\
a_{0}^{(-)} &=& {1 \over 4\pi(1+m_{\pi}/M)}
{g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}^{2}\over M}
\left({m_{\pi} \over 2M}\right)
\left(1-{m_{\pi}^{2} \over 4M^{2}}\right)^{-1}
\mathop{\longrightarrow}\limits_{m_{\pi} \rightarrow 0} 0 \;.
\label{slengths}
\end{aligned}$$
The $\sigma$-exchange contribution is a direct consequence of the underlying chiral symmetry of the model; it is essential for effecting the sensitive cancellation of the isospin-even scattering length. Indeed, each individual contribution to $a_{0}^{(+)}$ is approximately two orders of magnitude larger than the experimental value. Instead, the isospin-odd scattering length vanishes in the chiral limit without the need for sensitive cancellations; in the linear $\sigma$ model no additional t-channel exchanges are included.
A model that allows for additional t-channel exchanges is QHD-II [@serwal86]. Note, even though QHD-II is not a chiral model, a reasonable description of low-energy $\pi N$ scattering has been achieved through a “fine tuning” of parameters [@serwal86; @matser82]. A potentially important (t-channel) isospin-breaking contribution to $\pi N$ scattering might come via $\rho$-meson exchange. Indeed, recently we have computed a large isospin violation in the $NN\rho$ coupling constant [@ghp95]. This, combined with the large isospin-conserving $\pi\pi\rho$ coupling, could have a substantial impact on the triangle discrepancy. However, as we shall see below, in our model all isospin violations arising from the vector-meson sector must vanish as $q^{2} \rightarrow 0$.
Isospin-violating meson-nucleon coupling constants {#seciv}
==================================================
In this section we concentrate on isospin violations to the triangle identity which arise, exclusively, from isospin-violating meson-nucleon coupling constants. Additional isospin-breaking mechanisms, particularly those associated with Coulomb effects and hadronic mass differences, have been treated elsewhere [@gak95]. Recently, we have estimated the effect of isospin-violating meson-nucleon coupling constants on the $NN$ potential [@ghp95]. We have reported a large contribution from vector-meson exchange to the class IV nucleon-nucleon potential. The isospin-violating couplings that we have computed emerged from evaluating matrix elements of quark currents between nucleon states; the violations are driven by the up-down quark mass difference.
The isospin violations that we have computed arise on rather general grounds; we have assumed that the vector mesons ($\omega$ and $\rho$) couple to appropriate isospin components of the quark electromagnetic current. Moreover, at $q^2=0$ our results are insensitive to the quark-momentum distribution; they depend merely on the spin and flavor structure of the nucleon wave function. As a result, some important constraints emerge at $q^2=0$. In particular, only isospin violations in the tensor (or anomalous) couplings are allowed at $q^2=0$; the vector couplings are “protected” by gauge invariance and remain unchanged. However, since all tensor-driven contributions to $\pi N$ scattering vanish in the soft-pion limit ($q_{\mu} \rightarrow 0$) isospin-violating vector-meson-nucleon coupling constants can not contribute to the triangle discrepancy. Moreover, there is no contribution from $\rho$-$\omega$ mixing at $q^2=0$ [@piewil93; @oconn94]. Note that, contrary to the claim of Ref. [@cohmil95], the momentum-dependence of the $\rho$-$\omega$ mixing amplitude can not be absorbed into the vertex without violating gauge invariance. Thus, in our model, all three sources of isospin breaking in the vector-meson sector must vanish at $q^2=0$. In our model, there is no isospin-violating $NN\sigma$ coupling either; the $NN\sigma$ vertex, which has the same nonrelativistic limit as the timelike component of the vector, is also protected at $q^2=0$.
However, there is no symmetry that protects the $NN\pi$ coupling at $q^2=0$. We are interested in computing the coupling of the neutral pion to the nucleon in a nonrelativistic quark model. At $q^{2}=0$ the coupling is determined from the spin and flavor content of the nucleon wave function. In contrast, the isospin-violating coupling of the nucleon to the charged pions is sensitive to the quark momentum distribution and, therefore, more uncertain [@miller90]. It seems, however, that under reasonable assumptions the quark model is able to generate isospin-violating ($NN\pi^{\pm}$) couplings of comparable strength as those obtained in conventional hadronic treatments based on the neutron-proton mass difference. Presumably, these effects have been included in Ref. [@gak95].
The most general form for the on-shell $NN\pi^0$ vertex function consistent with Lorentz covariance and parity invariance is given by $$g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}
\Lambda^{5}_{\lower 2pt \hbox{$\scriptstyle NN\pi$}} =
g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}
\Big[ g^{\pi}_N \gamma^{5} \Big] \;.
\label{vertexa}$$ Here $g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}$ is the isospin-conserving $NN\pi$ coupling constant known phenomenologically from fits to $NN$ phase shifts and to the properties of the deuteron: $g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}^{2}/4\pi=
14.21$ [@machl87; @machl89]. The isospin-violating component is assumed to emerge from evaluating matrix elements of a flavor odd, pseudoscalar quark current between nucleon states, i.e., $$\langle N(p',s') |
\left[ {1\over 5}\, \bar{u} \gamma^{5} u
- {1\over 5}\, \bar{d} \gamma^{5} d \right]
| N(p,s) \rangle =
\bar{U}(p',s')
\Lambda^{5}_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}
U(p,s) \;.
\label{vertexb}$$ Here $U(p,s)$ denotes an on-shell nucleon spinor of mass $M_{N}$, momentum $p$ and spin $s$. Moreover, the constituent quarks are assumed elementary as no quark form factors are introduced. The coupling constants are computed at $q^2=0$ by examining the nonrelativistic reduction of Eq. (\[vertexb\]); this is the essence of the quark-pion model of Mitra and Ross [@mitra67]. In particular, in this limit the derivation closely resembles that which is used in computing the nucleon magnetic moments [@perkins82]. We obtain,
$$\begin{aligned}
{g^{\pi}_p \over 2M_{p}} &=&
{4 \over 3} \left({+1/5 \over 2m_{u}}\right) -
{1 \over 3} \left({-1/5 \over 2m_{d}}\right) =
+{4 \over 30 m_{u}} + {1 \over 30 m_{d}} \;, \\
{g^{\pi}_n \over 2M_{n}} &=&
{4 \over 3} \left({-1/5 \over 2m_{d}}\right) -
{1 \over 3} \left({+1/5 \over 2m_{u}}\right) =
-{4 \over 30 m_{d}} - {1 \over 30 m_{u}} \;,
\end{aligned}$$
where $m_{u}$ and $m_{d}$ are the up and down constituent quark masses. Alternatively, one can construct nucleon isoscalar and isovector combinations: $${g^{\pi}_p \over 2M_{p}}\;{1 \over 2}(1+\tau_z) +
{g^{\pi}_n \over 2M_{n}}\;{1 \over 2}(1-\tau_z) =
{1 \over 6m} \left(
{3\over 10}{\Delta m \over m} + \tau_z \right) \equiv
{1 \over 2M} \Big(g^{\pi}_0 + g^{\pi}_1\tau_z\Big) \;.$$ Note that we have introduced the following definitions: $$M \equiv {1 \over 2}(M_{n}+M_{p}) \;; \quad
m \equiv {1 \over 2}(m_{d}+m_{u}) \;; \quad
\Delta m \equiv (m_{d}-m_{u}) \;.$$ The above relations are correct to leading order in $\Delta m /m$. Moreover, they reveal an isospin-violating component ($g^{\pi}_0$) in the $NN\pi^{0}$ coupling constant. In particular, by selecting $m=M/3=313$ MeV and $\Delta m=4.1$ MeV [@licht89] we obtain: $$g^{\pi}_0 = {3\over 10}{\Delta m \over m} \approx 0.004 \;.$$ Ultimately, this isospin-violation can be traced back to the up-down quark mass difference; the up quark, which is lighter, generates a stronger coupling of the neutral pion to the proton than to the neutron. Note that the isospin breaking computed in the quark model is substantially larger—by about a factor of six—than in the nucleon model of Ref. [@cheung80] where the scale of the breaking is set by the neutron-proton mass difference. In contrast, for the coupling of the nucleon to charged pions both models seem to generate an isospin violation of comparable strength [@miller90].
Incorporating the isospin-violating correction from $g^{\pi}_0$ into the evaluation of the triangle discrepancy is straightforward. First, the elastic $\pi^{\pm}p$ amplitudes remain unchanged. Second, it modifies the charge-exchange (CEX) amplitude $f(\pi^{-}p \rightarrow \pi^{0}n)$ through a simple renormalization of the nucleon Born terms; the s-channel, which has a neutron in the intermediate state, gets reduced relative to the u-channel, which contains a proton in the intermediate state. Thus, in computing the charge-exchange amplitude one must use an isospin-odd contribution given by \[see Eq. (\[bminus\])\]: $$\widetilde{B}^{(-)}(s,t) \equiv
-{g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}^{2}
(1-g^{\pi}_0) \over s-M^{2}}
-{g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}^{2}
(1+g^{\pi}_0)\over u-M^{2}} =
B^{(-)}(s,t) - g^{\pi}_0 \, B^{(+)}(s,t) \;.
\label{btilde}$$ Note that the “small” isospin-odd contribution $B^{(-)}$ is being corrected by the “large” isospin-even term $B^{(+)}$. Indeed, at threshold $|B^{(+)}/B^{(-)}|=2M/m_{\pi} \approx 14$. Now, however, there is no cancellation due to chiral symmetry; there is no isospin-violating $NN\sigma$ coupling at $q^2=0$. Using the above expression for $\widetilde{B}^{(-)}$ we compute the value of the triangle discrepancy at threshold. We obtain, $$D = - \sqrt{2} \;
{g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}^{2}\over 4\pi}
{g^{\pi}_0 \over M}
{1 \over (1+m_{\pi}/M)(1-m_{\pi}^{2}/4M^{2})}
\mathop{\longrightarrow}\limits_{m_{\pi} \rightarrow 0}
- \sqrt{2} \;
{g_{\lower 2pt \hbox{$\scriptstyle NN\pi$}}^{2}\over 4\pi}
{g^{\pi}_0 \over M} \;.$$ This generates an isospin violation to the triangle identity of $D=-0.0145$ fm. The s-wave contribution to the triangle discrepancy shows a very weak energy dependence. Indeed, its contribution at $T_{\rm lab}=40$ MeV is $D=-0.014$ fm; we obtain a much smaller effect from the p-waves: $1.3\times 10^{-4}$ fm and $-2.0\times 10^{-4}$ fm for the $1^{+}$ and $1^{-}$ partial waves, respectively. This result is in good agreement with the value reported recently by Gibbs, Ai, and Kaufmann of $D=-0.012\pm 0.003$ fm from the s-wave alone or $D=-0.011\pm 0.003$ fm for the sum of s and p waves at 40 MeV [@gak95]
Conclusions {#secconcl}
===========
We have examined violations to the triangle identity that arise from isospin-violating meson-nucleon coupling constants. In our model, gauge invariance precludes the contribution from vector-meson exchanges at $q^2=0$; these include $\rho$-$\omega$ mixing as well as isospin-violating $NN\omega$ and $NN\rho$ coupling constants. There is no symmetry, however, that protects the $NN\pi^{0}$ coupling at threshold. We have computed isospin violations in the $NN\pi^{0}$ coupling using a nonrelativistic quark model. We have obtained a larger coupling of the neutral pion to the proton than to the neutron as a result of the up quark being lighter than the down quark. The observed isospin violation is about a factor of six larger than the one computed in nucleon models where the breaking is generated by the neutron-proton mass difference. These results were used to modify the relative weights of the s- and u-channel contributions to the charge-exchange reaction $\pi^{-}p\rightarrow\pi^{0}n$.
The isospin violation in the CEX amplitude became proportional to the large isospin-even amplitude $B^{(+)}$; this amplitude does not vanish in the chiral limit. In chiral models, such as the linear $\sigma$ model used here, the large contribution from $B^{(+)}$ to the isospin-even scattering length is cancelled by an almost equally large and opposite contribution \[$A^{(+)}$\] arising from $\sigma$-meson exchange. However, in our model all isospin violations in the $NN\sigma$ coupling must vanish at $q^{2}=0$. As a result, we obtained a large violation to the triangle identity: $D=-0.014$ fm. This value is in good agreement to the one reported from a recent analysis of high-quality $\pi N$ data which yielded $D=-0.012\pm 0.003$ fm [@gak95].
A particularly interesting test of this mechanism could be a comparison of the “mirror” reactions $\pi^{-}p\rightarrow\pi^{0}n$ and $\pi^{+}n\rightarrow\pi^{0}p$ [@gak95]. For the first case, namely, the one treated here, it was the s-channel that was suppressed relative to the u-channel. In contrast, it is the s-channel—now with a proton in the intermediate state—that becomes enhanced in the $\pi^{+}n\rightarrow\pi^{0}p$ reaction. One could quantify this isospin violation by measuring the difference of these two amplitudes, i.e., $$\widetilde{D} \equiv
f(\pi^{-}p \rightarrow \pi^{0}n) -
f(\pi^{+}n \rightarrow \pi^{0}p) \;.
\label{triangleb}$$ Note that the difference between the $pp\pi^{0}$ and $nn\pi^{0}$ coupling constants, alone, gives $\widetilde{D}=2D\approx-0.029$ fm. This value should be compared to a charge-exchange scattering length of $a_{0}=-0.19$ fm—it represents an isospin violation of 15%.
Undoubtedly, much work remains to be done before a clear understanding of the underlying mechanism behind the large isospin violation reported in Ref. [@gak95] will emerge. Yet, we believe that isospin violations in the $NN\pi^{0}$ coupling constant are likely to play an important role in the final analysis.
I thank S. Gardner, C.J. Horowitz, and B.D. Serot for many helpful conversations. This work was supported by the DOE under Contracts Nos. DE-FC05-85ER250000 and DE-FG05-92ER40750.
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---
abstract: 'We describe a new method of calculation of generic multi-loop master integrals based on the numerical solution of systems of difference equations in one variable. We show algorithms for the construction of the systems using integration-by-parts identities and methods of solutions by means of expansions in factorial series and Laplace’s transformation. We also describe new algorithms for the identification of master integrals and the reduction of generic Feynman integrals to master integrals, and procedures for generating and solving systems of differential equations in masses and momenta for master integrals. We apply our method to the calculation of the master integrals of massive vacuum and self-energy diagrams up to three loops and of massive vertex and box diagrams up to two loops. Implementation in a computer program of our approach is described. Important features of the implementation are: the ability to deal with hundreds of master integrals and the ability to obtain very high precision results expanded at will in the number of dimensions.'
author:
- |
S. Laporta[^1]\
\
[*Dipartimento di Fisica, Università di Bologna,* ]{}\
[*Via Irnerio 46, I-40126 Bologna, Italy*]{}
title: ' High-precision calculation of multi-loop Feynman integrals by difference equations'
---
Introduction
============
Nowadays the technique probably more popularly used for calculating the contribution of Feynman diagrams is that based on the integration-by-parts in $D$ dimensions [@Tkachov; @Tkachov2]. After some algebraic operations, like contracting Lorentz indices and calculating fermion traces, the contribution of the diagram is expressed as a combination of several Feynman integrals with different powers of numerators and denominators.
This expression composed of many integrals is reduced to a combination of a limited number of ‘master integrals’ using recurrence relations obtained by combining the identities obtained by integration-by-parts. Then, the master integrals are calculated numerically or analytically, with some other method.
As this technique is applied to more and more complicated diagrams, some difficulties appear.
First, a working general algorithm for identifying master integrals and for obtaining such recurrence relations is not known at present[^2]. Up to now, for each diagram laborious handwork was needed for obtaining such recurrence relations[^3].
Second, the number of master integrals grows rapidly with the number of loops and legs of the diagrams. Considering for example the reduction to master integrals of the contribution to the $g$-$2$ of the electron in QED, it is known that the one-, two-, and three-loop contributions are reduced by integration-by-parts to respectively, 1, 3 and 17 master integrals (see [@3-loop; @pol] for the analytical calculation of the three-loop contribution). At the four-loop level, to which integration-by-parts has been still not applied, one expects several hundreds of master integrals.
Third, the calculation of a single multi-loop master integral is a difficult problem for which a general method applicable to any diagram, with any values of masses and momenta, able to provide high-precision values and suitable for automatic calculation, is not known at present. Up to now a variety of different methods has been used, according to the topology and the values of masses and momenta of the particular diagram considered.
Therefore, to face problems like four-loop $g$-$2$, it arises the need of a completely automated approach to the calculation of Feynman integrals, and of finding out suitable new methods, algorithms and techniques of calculations which allow one to solve or avoid the above difficulties.
In this paper we describe the methods and techniques to be used in such automated approach. In particular we present:
1. \[met1\] A new method for determining the master integrals and for reducing generic Feynman integrals to master integrals, applicable to arbitrary Feynman diagrams.
2. \[met2\] A new method for calculating master integrals based on the numerical solution of the recurrence relations provided by the integration-by-parts method, seen as linear difference equations in one index, applicable to arbitrary Feynman diagrams.
3. \[met3\] A new method of generation and solution of systems of differential equations in masses and momenta for master integrals, applicable to arbitrary Feynman diagrams.
The part more innovative and important of this work is the method of calculation of master integrals based on *difference equations*. As this mathematical topic appears (surprisingly) to be practically absent from the literature, in order to improve the intelligibility of the paper we will give an extensive discussion on this argument, including methods of solutions, techniques of calculations and examples of applications. Moreover, study of boundary conditions of difference equations will lead us to a detailed discussion of asymptotic behaviour of Feynman integrals *for large powers of one denominator*, another topic which has received very scarce attention in the literature.
Another important point is the automation of the approach. All methods, algorithms and techniques developed in this work have been implemented in a comprehensive program called $\SYS$, written on purpose, which, among other things, contains a simplified algebraic manipulator. The aim of this program is to calculate the value of a generic Feynman integral in completely automatic way, by reducing it to master integrals and by calculating the master integrals. The program turned out to be able to deal with diagrams with hundreds of master integrals and to obtain very high precision values (for example 100-200 digits) expanded at will in $\e=(4-D)/2$, with all coefficients in numerical form, divergent terms included. Limitations of the current implementation will be described.
A crucial point is the test of the approach. By using $\SYS$ we have analyzed vacuum, self-energy, vertex and box diagrams up to three or two loops and we have calculated the values of master integrals for some values of masses and momenta, comparing them with already known results, when possible. We will show the results obtained. In particular we have calculated all the previously unknown single-scale massive three-loop self-energy master integrals. These numerical calculations, involving the calculation of hundreds of master integrals at the same time, allowed us to prove the reliability of the approach in real cases and to accumulate a considerable experience of calculation.
The plan of the paper is as follows: In section \[idebewin\] we illustrate an algorithm for the construction and solution of systems of integration-by-parts identities which allows one to reduce a generic Feynman integral to a combination of master integrals; this algorithm is the basis for the algorithms developed in the following sections. In section \[difequfey\] we show how to obtain systems of difference equations in one variable satisfied by the master integrals. In section \[secsolfact\] we show how to find solutions of difference equations using expansions in factorial series. In section \[detcon\] we discuss the methods to find the values of the arbitrary constants appearing in the solutions of the difference equations. In section \[sumfact\] we describe the techniques used to sum the factorial series expansions. In section \[example1loop\] we describe in detail the application of our methods to the one-loop self-energy diagram. In section \[Laplacesec\] we illustrate an alternative method of solution of difference equations based on the Laplace’s transformation. In section \[resu0\] we apply our methods to various diagrams from one to three loops. In section \[differentialequ\] we show how to deal with integrals with some particular values of masses and momenta, by solving systems of differential equations in masses and momenta. In section \[calcprog\] we show some technical information on the computer program used in the calculations. In section \[Conclusions\] we give our conclusions.
Systems of identities between Feynman integrals {#idebewin}
===============================================
After the introduction of the notation used for integrals and identities, in sections \[ssystem\]-\[algsolsys\] we describe the new method used for solving the systems of integration-by-parts identities and for obtaining reduction to master integrals. The description is much detailed because the algorithm here shown is an essential part of this work, being the basis of all similar algorithms of solution of systems of identities which will be described in sections \[constsysdif\], \[trasys\] and \[consys\]. In section \[identificamaster\] we show how the algorithm can be used for determining the master integrals. In sections \[secab\]-\[exasecab\] we show how to use the algorithm for reducing integrals, and we describe its more interesting feature: the absence of intermediate integrals with increasing exponents of the denominators in the intermediate steps of the reduction to master integrals.
Generalities {#Generalities}
------------
Let us introduce some notations used throughout the paper. We consider a generic Feynman diagram with $\NK$ loops, $\NE$ external and $\ND$ internal lines. The loop momenta are $k_i$, $i=1,\ldots,\NK$; the independent external momenta are $p_i$, $i=1,\ldots,\NP$, where $\NP=\NE-1$ (if $\NE>0$) for total momentum conservation. The denominators of the propagators are $D_i=q_i^2+m_i^2 $, where $q_i$ is the momentum flowing in the internal line $i$ and $m_i$ is the mass of the internal line; each $q_i$ is a linear combination of the momenta $\{p_j\}$ and $\{k_j\}$. The usual prescription $q_i^2+m_i^2=q_i^2+m_i^2-i0$ is implied if necessary. A generic Feynman integral regularized in $D$-dimensional euclidean momentum space has the form $$\label{genericterm}
\int {\dk1\; \dk2 \dots \dk{\NK}} \ V_{\gamma\delta} \ ,$$ where $\dk{i}=d^D k_i/\pi^{D/2}$ and $V_{\gamma\delta}$ is the generic integrand $$\label{genericv}
V_{\gamma\delta}=\dfrac
{\prod_{i=1}^\NP \prod_{j=1}^\NK (p_i\cdot k_j)^{\delta_{ij1}}
\prod_{i=1}^\NK \prod_{j=i}^\NK (k_i\cdot k_j)^{\delta_{ij2}}}
{\prod_{i=1}^\ND D_i^{\gamma_i}} \ ,
\quad \gamma_i\ge 0,\ \delta_{ijl}\ge 0\ ,\quad$$ $\gamma=\{\gamma_1,\ldots,\gamma_\ND\}$ and $\delta=\{\delta_{ijl}\}$. The numerator of is a product of powers of all the possible scalar products involving the loop momenta $k$; the total number $\NPS$ of such scalar products is $$\label{npsdef}
\NPS= \NP\NK + {\NK(\NK+1)}/{2}
\ .$$
Algebraic and integration-by-parts identities {#introduceide}
---------------------------------------------
Let us consider the generic integrand . For each denominator $D_j$ we write the identity $$\label{simpalg}
\frac{{\genpk}_j}{D_j}=\frac{1}{C_j}
\left(1-\frac{D_j-C_j \;{\genpk}_j}{D_j}\right) \ ,\qquad j=1,\ldots,\ND\ ,$$ where ${\genpk}_j$ indicates one scalar product involving loop momenta which appears in the expression of $D_j$, and $C_j$ is its coefficient in the expression. The scalar products ${\genpk}_j$ must be chosen all different. These algebraic identities are applied in sequence to $V_{\gamma\delta}$ and to the terms subsequently generated, more times if it is necessary. As a result, the original integrand $V_{\gamma\delta}$ is transformed into a sum of new terms, each one not containing ${\genpk}_j$ and $D_j$ simultaneously, with the general form $$\label{genericvind}
V'_{ni\alpha\beta}=\dfrac
{\prod_{j=1}^{\NPS-n} {\indpk}_j^{\beta_{j}} }
{\prod_{j=1}^n D_{i_j}^{\alpha_j}}\ , \quad n\le\ND\ , \quad
\alpha_j,\beta_j \ge 0\ ,$$ where the subscript $ni\alpha\beta$ shows the dependence on the number $n$ of denominators, their particular combination $i=\{i_1,\ldots,i_n\}$, and the exponents $\alpha=\{\alpha_1,\ldots,\alpha_n\}$ and $\beta=\{\beta_1,\ldots,\beta_{\NPS-n}\}$. The symbols ${\indpk}_j$, $j=1,\ldots,\NPS-n$ indicate the $\NPS - n$ ‘irreducible’ scalar products which cannot be simplified further on by with one of the denominators $D_{i_1}$, $\ldots$, $D_{i_n}$ of $V'_{ni\alpha\beta}$; if $n=\NPS$ the numerator of is unity. We stress that one different set of irreducible scalar products corresponds to each different combination of denominators.
Integrating by parts in $D$ dimensions [@Tkachov; @Tkachov2] one can write the identities $$\label{intbyparts}
\begin{split}
&\int {\dk1 \dots \dk{\NK}}
\frac{\partial}{\partial (k_j)_\mu}
\left( (p_l)_\mu
V'_{ni\alpha\beta}
\right) =0 \;, \quad j=1,\dots,\NK, \
l=1,\dots,\NP, \\
&\int {\dk1 \dots \dk{\NK}}
\frac{\partial}{\partial (k_j)_\mu}
\left( (k_l)_\mu
V'_{ni\alpha\beta}
\right) =0 \;, \quad j,l=1,\dots,\NK\ , \\
\end{split}$$ where $V'_{ni\alpha\beta}$ is defined in . For each different $V'_{ni\alpha\beta}$ gives $\NK(\NP+\NK)$ identities. The ratio $V'_{ni\alpha\beta}$ contains only irreducible scalar products (relative to the particular combination of denominators); the calculation of the derivative and the contraction of the index $\mu$ form terms also containing reducible scalar products, which must be transformed into irreducible scalar products using the algebraic identities . As a final result, the identity will contain a linear combination of integrals of two kinds: integrals containing all the $n$ denominators $\{D_{i_1}$, $D_{i_2}$, $\ldots$, $D_{i_n}\}$ appearing in $V'_{ni\alpha\beta}$ and integrals with one denominator missing as effect of algebraic identities.
A new method for solving the system of identities {#ssystem}
-------------------------------------------------
Each identity obtained from is a linear combination of integrals like $$\label{generictermvp}
\int {\dk1 \dots \dk{\NK}} \ V'_{ni\alpha\beta} \ ,$$ with polynomials of degree zero or one in the number of dimensions $D$ as coefficients, and where the integrand has the general form of . A generic set of these identities forms a homogeneous linear system of equations, with the integrals as unknowns. As it is well-known, the system is under-determined, and some integrals exist, the “master integrals”, whose values cannot be determined from the system.
In the literature[^4] it is usual to look for the general solution of the *infinite* system in the form of combination of identities which, lowering and raising exponents, transform integrals of the form or into linear combinations of carefully chosen master integrals. Up to now, the methods used to find these identities have been based on laborious human analysis; as the number of these identities (and the difficulties encountered) grow rapidly with the number of denominators and loops, it becomes very difficult to analyze in this way diagrams beyond a certain limit.
On the contrary, our approach is different, as it consists in the solution of systems made up of a *finite* number of identities. The identities are generated explicitly using suitable $V'_{ni\alpha\beta}$, with parameters $n$, $i$, $\alpha$, $\beta$ taken from a large (but finite) set of values carefully chosen. The set of the generated identities forms a linear system with integrals as unknowns, which is solved using the well-known Gauss elimination method; the solution gives the expressions of the integrals as linear combinations of the master integrals with coefficients rational in $D$. The advantage of this approach is that it is simple, applicable without modifications to diagrams with any topology, and suitable for completely automatic calculations.[^5] It does not consist only in a mere “mechanical” approach because, as we will see in section \[secab\], the solution of the whole system allows us to discover very useful identities of a kind *a priori* not expected; these identities allow one to avoid the appearing of intermediate integrals with increasing exponents of the denominators in the intermediate steps of the reduction to master integrals.
Let us now consider more in detail the solution of the system. The identities are generated and inserted in the system one at a time. Let $\sum_j c_j W_j=0$ be an identity obtained from , where $W_j$ are integrals and $c_j$ the coefficients. The identities already existing in the system, expressing some of the integrals $W_j$ in terms of other integrals are substituted in the new identity, which becomes $\sum_j c_j' W_j'=0$. One particular integral $W_l'$ is chosen between the integrals $\{W_j'\}$, and the identity itself is rewritten as $W_l'=\sum_{j\not =l} c_j'' W_j'$ in order to express the particular integral in terms of the remaining integrals. Then the new identity is added to the system and the chosen integral $W_l'$ is substituted in the rest of the system.
The choice of the integral $W_l'$ is carried out following an ordering[^6] of all the integrands $V'_{ni\alpha\beta}$ as functions of the parameters: the number $n$ of denominators, the combination $i$ of denominators, and the exponents. The ordering defines the priority of extraction between integrals. Priorities are arranged so that integrals with a higher number of denominators are extracted first, and expressed as linear combinations of the integrals with a lower number of denominators. Remaining details of ordering used are shown in the algorithm of the next section. The form of the master integrals depends on the choice of the ordering: see section \[identificamaster\].
Now we consider the choice of the order of the generation and processing of the identities. The final solution of the system is independent of the order of processing, but the computing time is not. Each addition of a new identity to the system implies a substitution of an integral in all identities of the system which contains it; therefore the order must be carefully chosen in order to minimize the number of substitutions required. A bad choice may cause the computing time to blow up.
A good choice of the ordering of the ratios $V'_{ni\alpha\beta}$ appearing in is the inverse of the above considered ordering of integrands used for the extraction of integrals. The identities corresponding to ratios $V'_{ni\alpha\beta}$ with the lowest priority of extraction (as integrands), therefore with the lowest number of denominators, are generated and processed first; then the identities with a higher number of denominators are processed. The last processed identities will have the highest number of denominators. This choice is efficient as long as the number $n$ of denominators of the ratio $V'_{ni\alpha\beta}$ used in is equal to the number $n'$ of denominators of the integral $W_l'$ extracted from the identity generated. In some cases (relatively rare, but of great importance, see section \[secab\]) one finds $n'<n$; the identities obtained with ratios $V'_{n'i'\alpha'\beta'}$, containing integrals with $n'$ denominators, have already been inserted into the system, so that the “late” integral $W_l'$ must be substituted in a large number of identities, and that may require a considerable amount of time.
It is useful to split the system of identities into several *subsystems*. Each subsystem is made up of all the identities obtained by inserting in terms $V'$ containing one particular combination $i$ of denominators $\{D_{i_1}$, $D_{i_2}$, …, $D_{i_n}\}$. The integrals with a number of denominators $n$ less than the number of loops $\NK$ of the diagram are null in the framework of the dimensional regularization, and must be not considered, so that the total number $S$ of subsystems is $$\label{defins}
S=\sum_{n=\NK}^\ND \binom{\ND}{n} < 2^\ND\ .$$ Let us now define the non-negative quantities $\MPRD$, $\MDEN$, $$\label{groupdef}
\sum_{j=1}^{\NPS-n} \beta_j=\MPRD \ , \quad
\sum_{j=1}^{n} \left(\alpha_j-1\right)=\MDEN \ ,$$ which are the total sum of the powers of the scalar products in the numerator and the sum of the powers of the denominators minus the number of denominators of the generic integrand $V'_{ni\alpha\beta}$ of , respectively. Using this definition we can split the infinite set of the integrands $V'_{ni\alpha\beta}$ into finite sets of integrands with equal $n$, $\MPRD$ and $\MDEN$ that we indicate with the symbol $\group{n}{\MPRD}{\MDEN}$. The set of the integrands with $0\le\MPRD\le a $ and $0\le\MDEN\le b$ will be indicated with the symbol $\groupl{n}{a}{b}$. In the following sections we will say in short that an integral or an identity belongs to a specified set, understanding that it is the corresponding $V'_{ni\alpha\beta}$ which belongs to that set.
The algorithm of solution of the system {#algsolsys}
---------------------------------------
The algorithm of solution is the following:
\[algsys\]\[algsys3\]
1. Let $n=\NK$. \[quin1\]
2. Let $i_1=1$, $i_2=2$, …, $i_n=n$\[quin15\].
3. Consider the combinations of $n$ different denominators, chosen in the set $\{D_1$, $\ldots$, $D_\ND\}$; let $D_{i_1},D_{i_2},\ldots,D_{i_n}$ be one of these combinations. \[quiN\]
4. Choose two integer non-negative constants $a_i$ and $b_i$, $ i = \{ i_1, \ldots, i_n \} $. \[quiab\]
5. Let $\MDEN=0$. \[quiMden\]
6. Let $\MPRD=0$. \[quiMprd\]
7. \[cf6a\] Consider the ratios of the kind of , \[quiset\] containing the $n$ denominators
$D_{i_1},D_{i_2},\ldots,D_{i_n}$ raised to some power; let $W$ be one of these ratios:\[quissset\] $$W(n,i,\alpha,\beta)=\dfrac
{\prod_{j=1}^{\NPS-n} {\indpk}_j^{\beta_{j}} }
{D_{i_1}^{\alpha_1} D_{i_2}^{\alpha_2} \cdots D_{i_n}^{\alpha_n}},$$ where the non-negative exponents $\alpha_j$, $\beta_j$ are such that $$\sum_{j=1}^{\NPS-n} \beta_j=\MPRD \ , \quad
\sum_{j=1}^{n} \left(\alpha_j-1\right)=\MDEN \ ,$$ i.e., they are such that $W$ belongs to the set $\group{n}{\MPRD}{\MDEN}$.
8. Set $V_{ni\alpha\beta}'=W$ in and generate all the integration-by-parts identities.\[quiW\]\[quiins1\]
9. \[quiins2\] Let $\int \dk1\ldots\dk\NK \ \sum_j c_j W_j =0 $ be one of the identities of , where $W_j$ is a generic integrand with $n$ or $n-1$ denominators, \[quiIde\] then:
1. Substitute all the integrals already known in the left-hand side of the new identity;
let $\int \dk1\ldots\dk\NK \sum_j c_j' W_j'=0$ be the result. If the new identity is a linear combination of other identities of the system, go to step \[nullo\].
2. If the new identity is linearly independent, choose an integrand $W_l'$ from the identity, $$W_l'(n',i',\alpha',\beta')=\dfrac
{\prod_{j=1}^{\NPS-n'} {\indpk}_j^{\beta'_{j}} }
{D_{i_1'}^{\alpha_1'} D_{i_2'}^{\alpha_2'} \cdots D_{i_n'}^{\alpha_n'}} \ ,$$ belonging to $\group{n'}{\MPRD'}{\MDEN'}$, $\NK \le n' \le n$, following the order of priority:
1. the greatest number of the denominators $n'$;
2. the greatest $\MDEN'$; \[ord2\]
3. the greatest $\MPRD'$; \[ord3\]
4. the greatest $i_1'$, the greatest $i_2'$, …, the greatest $i_n'$;
5. the greatest $\alpha_1'$, the greatest $\alpha_2'$, …, the greatest $\alpha_n'$;
6. the greatest $\beta_1'$, the greatest $\beta_2'$, …, the greatest $\beta_{\NPS-n'}$. \[cf6b\]
3. Substitute and add the following identity to the system: $$\int \dk1\ldots\dk\NK \; W_l'= - \int \dk1\ldots\dk\NK \sum_{j\not=l}
( c_j'/ c_l' ) W_j' \ .$$
10. Generate a new integration-by-parts identity among the $\NK(\NP+\NK)$ possible identities of and go to step \[quiIde\], otherwise continue.\[nullo\]
11. Choose a new integrand $W$ with different exponents $\alpha$ and $\beta$, belonging to the set $\group{n}{\MPRD}{\MDEN}$, among the $\binom{\NPS-n+\MPRD-1}{\MPRD}\binom{n+\MDEN-1}{\MDEN}$ elements of the set and go to step \[quiW\], otherwise continue.
12. $\MPRD=\MPRD+1$; if $\MPRD\le a_i$ go to step \[quiset\].
13. $\MDEN=\MDEN+1$; if $\MDEN\le b_i$ go to step \[quiMprd\].
14. Choose a new combination of indices $i_1<i_2<\ldots<i_n$ among the $\binom{\ND}{n}$ possible combinations of numbers $\{1$, $2$, $\ldots$, $\ND\}$ and go to step \[quiN\], otherwise continue.
15. $n=n+1$; if $n\le \ND$ go to step \[quin15\], otherwise end. \[quin2\]
In this algorithm two arbitrary integer constants, $a_i$ and $b_i$, must be chosen for each different combination of denominators $\{D_{i_1}, \ldots, D_{i_n} \}$. These constants are cutoffs and define which identities must be included in the system and which must be excluded; $a_i$ limits the sum of the powers of scalar products in the numerator and $b_i$ limits the sum of the exponents of the denominators.
With a suitable choice of the parameters $a_i$ and $b_i$ (see section \[secab\]), this algorithm allows one to reduce any given integral $I$ to a sum of master integrals $B_l$ $$\label{sumc3}
I=\sum_{l=1}^L r_l B_l\ ,$$ where $r_l$ are rational functions of $D$, masses and scalar products of external momenta.
Determination of master integrals {#identificamaster}
---------------------------------
Our algorithm of solution of the system of identities provides a general method for determining the master integrals: it suffices, for each different combination of $n$ denominators $\{D_{i_1}$, $D_{i_2}$, …, $D_{i_n}\}$ to build the subsystem with suitable $a_i$ and $b_i$, to solve it using the algorithm, and to find the integrals belonging to $\groupl{n}{a_i}{b_i}$ which are not reduced to terms with $n-1$ denominators. These should be master integrals. Of course, since we are dealing with a system made up with a finite number of identities, there is the possibility that a master integral is erroneously identified or not found because some essential identities have not been included in the system as they exceed the limits of generation. Explicit solutions of subsystems with increasing values of $a_i$ and $b_i$ show that empirical ‘thresholds’ $\bar a_i$ and $\bar b_i$ exist, depending on the combination of denominators, such that all the subsystems built with $a_i\ge \bar a_i$ and $b_i\ge \bar b_i$ give a stable and correct identification of the master integrals; for the diagrams so far examined (see section \[resu0\]) one finds $0\le \bar a_i \le 3$ (it is equal to the maximum number of scalar products appearing in the master integrals) and $\bar b_i=0$. Corresponding number of identities varies from some tens to some hundreds.
The particular order of selection of the integrals used in step \[ordering\] of the algorithm favours for each different combination of denominators the choice of master integrals with all the denominators $D_j$ raised to the first power and with the numerator containing increasing numbers of scalar products: first the scalar integral, with 1 as numerator, then integrals with one scalar product, and so on.
As result of the procedure just described we find the set of $L$ master integrals $B_l$ with the form $$B_{l}=
\int {\dk1 \dots \dk{\NK}} \
\dfrac
{\prod_{j=1}^{\NPS-n} {\indpk}_j^{\beta_{jl}} }
{D_{i_1} D_{i_2} \cdots D_{i_n}} \ ,$$ where the combination of indices $i$ and exponents $\beta$ depends on the index $l$. The ordering $B_{1}$, $B_{2}$, …, $B_L$ follows the ordering of integrals provided by the algorithm.
Choice of constants $a_i$ and $b_i$: the “golden rule” {#secab}
------------------------------------------------------
The minimal values of $a_i$ and $b_i$ necessary to reduce one particular integral strongly depend on the structure of the integral and are not easy to determine. Therefore we limit ourselves to give some general rules, based on the experience acquired by using the algorithm, rather than rigorous results. Actually, these rules turned out to be very effective.
Let us consider the system of identities obtained by inserting in all the terms $V'_{ni\alpha\beta}$ belonging to the set $G_{ab}$ $$\label{G2}
G_{ab} =\bigcup_{n=\NK}^{\ND}\groupl{n}{a}{b} = \groupl{\NK \ldots
\ND}{a}{b} $$ containing all the possible $S$ subsystems of identities (see ), and choosing $a_i=a$ and $b_i=b$ for all the combinations of denominators. The number of elements $V'_{ni\alpha\beta}$ forming the set $G_{ab}$ is given by $$\label{ng2bis}
N_{ele}\left(G_{ab}\right)=
\sum_{n=\NK}^\ND
\binom{\ND}{n}
\binom{\NPS-n+a}{a}
\binom{n+b}{b}
\ ,$$ while the number of identities of the system is $$\label{ng2}
N_{ide}\left(G_{ab}\right)=
\NK\left(\NP+\NK\right) N_{ele}(G_{ab}) \ .$$ Explicit calculations suggest the following empirical “golden rule”: the solution of the system provides identities which reduce any integral belonging to the set $G_{ab}$ to master integrals $$\label{expnabtot2}
\text{Integrals}\
G_{ab}
@>>{\text{Identities }G_{ab}}>
\text{Master Integrals} \ .$$
In order to better understand this rule, we have analyzed in some test cases the reduction of single integrals belonging to $\groupl{\ND}{a}{b}$, by modifying the steps \[quin1\] and \[quin2\] of the algorithm so that the various subsystems of identities $\groupl{n}{a}{b}$ are generated starting from $\ND$ denominators and decreasing $n$ up to $\NK$ denominators (note that the solution of the whole system or parts of it with such ‘inverted’ order is extremely time consuming). For each value of $n$, intermediate expressions contain master integrals with a number of denominators from $n$ to $\ND$, and integrals belonging to the set $\groupl{n-1}{a}{b+1}$, that is, integrals with one exponent of denominators increased by one compared with the original integral *whichever is $n$*. This is very different from what one obtains by performing the reduction one denominator at a time with recurrence relations which lower one index and raise another one, where the exponents of denominators of the integrals are observed to increase whenever the number of denominators decreases, causing the number of intermediate integrals to blow up. Our algorithm avoids this blowing-up.
The quite unexpected behaviour of our algorithm seems to be due to the fact that solving the whole system we consider all the identities generated, not only those which reduce single integrals, but also those many “additional” identities apparently useless which reduce *combinations* of integrals with the same number of denominators. Their effect is to reduce systematically all the generated combinations of integrals with increased exponents of the denominators. The reason for this behaviour is not known; however, its systematic presence suggests that (hopefully) it may have some simple explanation. The presence of these “additional” identities is important, particularly if $\ND$ is large, because without them the number of identities to consider (and the number of intermediate integrals) would be orders of magnitude larger.
The “golden rule” turns out to be valid if $a\ge a_0$ and $b\ge b_0$, where $a_0$ and $b_0$ are some empirical ‘thresholds’ depending on the structure of denominators of the diagram; for the diagrams examined in section \[resu0\] we found $1 \le a_0 \le 3$ (values almost always equal to the maximum number of scalar products appearing in the master integrals) and $b_0=0,1$. Values of the thresholds are unfortunately not known *a priori*; if the chosen values of $a$ and $b$ are below the threshold, the result of the solution of the system is that some integrals belonging to $G_{ab}$ are not completely reduced to master integrals, and some non-master integrals with a few denominators still survive in the final expressions. In this case one may increase $a$ or $b$, or alternatively one may suitably enlarge only the subsystems of identities corresponding to residual integrals.
We found some very rare exceptions to the “golden rule” , when some denominators have zero mass; in these cases, whatever the values of $a$ and $b$ are, there are always few integrals belonging to $\group{\ND}{0}{b}$ (note, changing with the value of $b$) which are not completely reduced to master integrals. In the example of the next section we will show just one of these exceptions.
Now, let us consider an important application: the reduction to master integrals of one combination of many integrals, for example, expressing the contribution of a diagram to some physical quantity. Let us suppose that all the integrals of this combination belong to a set $G_{\bar a \bar b}$ with some (minimal) $\bar a$ and $\bar b$; if the values of $\bar a$ and $\bar b$ are over thresholds, according the solution of the system will provide identities which reduce to master integrals all the integrals of the set.
An example {#exasecab}
----------
Let us consider the self-energy diagram with 5 denominators $D_1=(p-k_1)^2+1$, $D_2=(p-k_1-k_2)^2+1$, $D_3=(p-k_2)^2+1$, $D_4=k_1^2$, $D_5=k_2^2$ and $p^2=-1$. Following the notation of section \[Generalities\], this diagram has $\NP=1$, $\NK=2$, $\ND=5$ and $\NPS=5$. For instance, we want to transform the integral $$J= \int
\frac
{[dk]}
{D_1^2 D_2 D_3 D_4 D_5} \ ,$$ where $[dk]=\dk1 \; \dk2$, into a combination of master integrals. The pairs between scalar products and denominators used in the algebraic identities are $(p\cdot k_1,D_1)$, $(k_1\cdot k_2,D_2)$, $(p\cdot k_2, D_3)$, $(k_1\cdot k_1,D_4)$ and $(k_2\cdot k_2,D_5)$. We must identify the master integrals; therefore, following section \[identificamaster\], we build the system of identities with $a_i=1$ and $b_i=0$ for all the combinations of denominators, and we look for the integrals which are not reduced. The master integrals turn out be: $B_{123}=\int [dk] / D_1 D_2 D_3 $, $B_{345}=\int [dk] / D_3 D_4 D_5 $, $B_{12} =\int [dk] / D_1 D_2 $, $B_{13} =\int [dk] / D_1 D_3 $ and $B_{23} =\int [dk] / D_2 D_3 $.
We consider the whole system made up of the identities of the set $G_{11}$, choosing $a=b=1$; according to it contains 1776 identities. Solving it with the algorithm \[algsys\] (by using the program described in section \[calcprog\]) we find 1122 independent identities. Examining them, we find the identities which reduce to master integrals all the 291 integrals $\groupl{2\ldots 5}{1}{1}$, integral $J$ included, with the exception of the two integrals $\int {[dk] }/D_1 D_2 D_3 D_4^2 D_5$ and $\int {[dk] }/D_1 D_2 D_3 D_4 D_5^2$, whose reduction still contains the integrals $\int {[dk] }/D_2 D_4^2 D_5^2$ and $\int {[dk] }/D_3 D_4^2 D_5^2$; in order to complete the reduction of these two integrals, we may add the two sets of 30 identities $\groupl{3}{1}{2}$ with these combinations of three denominators to the system. The remaining 831 identities reduce to master integrals complicated combinations of integrals $\group{2\ldots 5}{0\ldots 2}{2}$.
We want to illustrate how the mechanism of “additional” identities works. Let us consider the partial reduction of the integral $J$ from 5 to 4 denominators. We consider the subsystem of the identities $\group{5}{0}{0 \ldots 1}$, which is formed by 36 identities; solving it, we find:
1. \[p5001\] 6 identities reducing the integrals $\group{5}{0}{0\ldots 1}$ to 4 denominators;
2. \[p5002\] 9 identities reducing combinations of integrals $\group{5}{0}{2}$ to 4 denominators;
3. \[p4012\]20 identities between integrals $\group{4}{0}{1}$ and $\group{4}{0}{2}$ with different combinations of denominators;
The identities of the groups \[p5002\] and \[p4012\] are the “additional” ones. Among the identities of the group \[p5001\] we find $$\label{J}
J =
\int {[dk] } \left(
\frac{1}{4}W_{1} +\frac{1}{2}W_{2} \right) \ ,$$ $$\begin{split}
W_{1}&=
\frac{3}{D_2^2 D_3 D_4 D_5}
-\frac{3}{D_1 D_2^2 D_3 D_4}
+\frac{2}{D_2 D_3 D_4^2 D_5}
-\frac{1}{D_1 D_2 D_3^2 D_4} +\\
&+\frac{1}{D_1 D_2 D_3^2 D_5}
+\frac{1}{D_2 D_3 D_4 D_5^2}
-\frac{1}{D_1 D_3^2 D_4 D_5}
-\frac{1}{D_1 D_3 D_4 D_5^2}
\ ,\\
W_{2}&=
\frac{1}{D_1 D_3^2 D_4^2 D_5}
-\frac{1}{D_1 D_2 D_4^2 D_5^2}
-\frac{1}{D_1 D_2^2 D_4^2 D_5}
+\frac{1}{D_1 D_3 D_4^2 D_5^2}
-\frac{1}{D_2 D_3 D_4^2 D_5^2}
\ .\\
\end{split}$$ We see that the original integral, which contains a square denominator (set $\group{5}{0}{1}$), is transformed into a combination of integrals with 4 denominators, with one or two square denominators (sets $\group{4}{0}{1}$ and $\group{4}{0}{2}$).
Considering the reduction from 4 denominators to master integrals, if we peruse the solution of the subsystems $\groupl{2\ldots 4}{1}{1}$ we find 8 relations which transform each one of the terms of $W_{1}$ into master integrals; on the contrary we do not find relations which transform *single* terms of $W_{2}$ into master integrals. But if we also consider the previous identities of the group \[p4012\], we find one relation which transforms the *whole* combination $W_{2}$. The presence of this relation makes it unnecessary to search for identities which singly transform the integrals of $W_{2}$.
Difference equations for master integrals {#difequfey}
=========================================
By using the algorithm shown in the above sections, a generic Feynman integral can be reduced to master integrals. Now we consider their calculation.
We consider the generic integral as a function of the exponents $U(\alpha,\beta)$; the integration-by-parts identities form a system of recurrence relations satisfied by the function $U$. The key observation is that such recurrence relations are *linear difference equations*[^7] satisfied by the function $U(\alpha,\beta)$. This observation has two important implications:
1. Theory of linear difference equations[@Milne] is a well established (but not very well-known) mathematical topic for which various useful mathematical tools exist.
2. Difference equations can be solved numerically, and therefore this establishes a new method of calculation of the integrals $U(\alpha,\beta)$.
Another important observation is that the integration-by-parts provides recurrence relations in *several* variables for the functions $U$, that is, a system of partial difference equations. In general, the numerical solution of such a system is, from the point of view of the numerical calculation, a task comparable to the solution of a system of partial derivative equations or to the multidimensional integration, so that high-precision results may be very difficult to obtain if the number of variables is high. Therefore, if we desire to obtain high-precision results, we must consider only difference equations in only *one* variable.
In section \[introx\] we show where to introduce such single parameter; in section \[constsysdif\] we show how to obtain recurrence relations (that is, difference equations) in this parameter from the integration-by-parts identities .
Difference equations in one exponent {#introx}
------------------------------------
Let us consider, for instance, a master integral of the form $$\label{inti}
B=
\int
\dfrac
{\dk1 \dots \dk{\NK}}
{\displaystyle D_1 D_2 \ldots D_\ND} \ .$$ We raise one denominator to power $x$. If we choose $D_1$, we define a function $U_{D_1}$ $$U_{D_1}(x)=
\int \dfrac
{\dk1 \dots \dk{\NK}}
{\displaystyle D_1^x D_2 \ldots D_\ND} \ ,$$ where the subscript shows that the exponent has been introduced in the denominator $D_1$. The value of the integral is recovered as $U_{D_1}(1)=B$.
Using integration-by-parts identities, we look for an identity which relates $U_{D_1}$ with integrals with a smaller number of denominators. As $B$ is a master integral, an identity which directly transforms $U_{D_1}(x)$ into simpler integrals cannot be found; instead, as we will see in the next section, one finds identities of the form $$\label{equsimple}
\sum_{j=0}^\R p_j(x) U_{D_1}(x+j)=F(x)\ ,$$ where $p_j(x)$ are polynomials in $x$, and $F(x)$ is a known function. $U_{D_1}$ appears with arguments shifted by an integer. This identity is a linear difference equation of order $R$ in the variable $x$ satisfied by the function $U_{D_1}(x)$.
The right-hand side function $F(x)$ is in general a linear combination of functions analogous to $U_{D_1}$, derived from master integrals containing $D_1$, but with some of the denominators ${D_2, D_3, \ldots, D_\ND}$ missing.
We note that it is possible to raise to $x$ another denominator, $D_2$, $D_3$, …, etc. (or equivalently to permute the numbering of the lines of the diagram); for each denominator $D_j$ a function $U_{D_j}(x)$ will be defined. In general the functions $U_{D_j}(x)$ are different each one and satisfy different difference equations, but for $x=1$ they all have the same value $U_{D_j}(1)=B$. This fact is particularly useful for checking the consistency of the calculations.
Construction of systems of difference equations {#constsysdif}
-----------------------------------------------
Let us suppose that, by means of the procedure described in section \[identificamaster\], we establish that there are $L$ master integrals, with a number of denominators ranging from $\NK$ to $\ND$.
We note that some master integrals may have the first denominators missing. Then we group the master integrals in $\ND-\NK+1$ sets $S_m$ such that each set includes the integrals which contain the denominators $D_i$ with $i \ge m$. Each set will contain $L_m$ master integrals $B_{ml}$, $$B_{ml}=
\int {\dk1 \dots \dk{\NK}}
\dfrac
{\prod_{j=1}^{\NPS-n} {\indpk}_j^{\beta_{jml}} }
{D_{m} D_{i_2} D_{i_3} \ldots D_{i_n} }\ ,
\quad
m<i_2<\ldots<i_n \ ,$$ where $ m=1,\ldots,\ND-\NK+1$, $ l=1, \ldots,L_m $ and $\sum_{m} L_m =L$; the number of denominators $n$ ranges from $\NK$ to $\ND-m+1$. Note that $n$, the indices $i$ and the exponents $\beta$ of the various integrals depend on $m$ and $l$. The ordering $B_{m1}$, $B_{m2}$, …, $B_{mL_m}$ follows the ordering of master integrals with the same first denominator $D_m$ provided by the algorithm \[algsys\]. Changing the exponent of $D_m$ from 1 to $x$ in each master integral, we define the “master functions” $$\label{definitionu}
U_{ml}(x)=
\int {\dk1 \dots \dk{\NK}} \
\dfrac
{\prod_{j=1}^{\NPS-n} {\indpk}_j^{\beta_{jml}} }
{D_{m}^{x} D_{i_2} D_{i_3} \ldots D_{i_n} } \ ;$$ $U_{ml}(1)=B_{ml}$ recovers the original master integral. Now we must find the difference equations satisfied by the functions $U_{ml}(x)$. We follow a method quite similar to that used for the reduction of a generic integral to master integrals. We build a system of identities obtained from integration-by-parts using , modifying the integrand $V'_{ni\alpha\beta}$ of by adding $x-1$ to the exponent of the first denominator: $$\label{genericvindnn}
V'_{ni\alpha\beta}(x)=\dfrac
{\prod_{j=1}^{\NPS-n} {\indpk}_j^{\beta_{j}} }
{D_{i_1}^{x-1+\alpha_1} D_{i_2}^{\alpha_2} \ldots D_{i_n}^{\alpha_n} },
\quad n\le\ND, \quad
\alpha_j,\beta_j \ge 0\ .$$ In this way all the integrals become functions of $x$ $$\label{generictermvpn}
U_{ni\alpha\beta}(x)=
\int {\dk1 \dots \dk{\NK}} \ V'_{ni\alpha\beta}(x) \ ,$$ and the identities become difference equations between these functions. Integrals which have a different first denominator $D_{m}$ never appear in the same identity, so that it is convenient to build and solve separately the systems of the identities with different values of $m$. We do this with the algorithm
\[algsys4\] Consider the algorithm \[algsys\] with the following modifications:
1. \[cond01\] Set $i_1=m$.
2. Add $x-1$ to the exponent of $D_{m}$.
3. \[cond03\] Ignore the addition of $x-1$ to $\MDEN$ in the definition of set .
4. \[cond04\] Add the following conditions to the order of priority of the extraction of integrals:
1. the generic functions $U_{ni\alpha\beta}$ must be extracted before of the functions $U_{ml}$;
2. the functions $U_{ml}(x-1)$ (generated by the algebraic identities) must be extracted before of the functions $U_{ml}(x+i)$, where $i\ge 0$.
As result of the solution of the system, we find lots of identities analogous to , not interesting, which express generic functions $U_{ni\alpha\beta}$ as combinations of master functions $U_{m l}$ with argument possibly shifted, and rational functions of $x$, $D$, masses and scalar products of external momenta as coefficients; we also find much more interesting identities containing only master functions $U_{m l}$.
Let us suppose, for simplicity, that each master integral $B_{ml}$ contains a different combination of denominators. With a suitable choice of the constants $a_i$ and $b_i$, the solution of the system provides difference equations satisfied by the functions $U_{ml}$, with the structure $$\label{diffu1}
\sum_{i=0}^{\R_l} p_{il}(x) U_{ml}(x+i) =
\sum_{k=1}^{l-1} \sum_{j=0}^{Q_{lk}} q_{jkl}(x) U_{mk}(x+j) \ , \quad
l=1,\ldots,L_m \ ,\quad$$ where $p_{il}$ and $q_{jkl}$ are polynomials in $x$, $D$, masses and scalar products of external momenta. The right-hand side of the $l$th equation contains the functions from $U_{m1}$ to $U_{m,l-1}$. Therefore the set of $L_m$ equations forms a triangular system of difference equations. The triangular structure is particularly useful for simplifying the numerical solution: the equations are solved in ascending order, one at a time, for $l=1, 2, \ldots, L_m$; when the equation for $U_{ml}$ has to be solved, the functions $U_{m1}$,…,$U_{m,l-1}$ in the right-hand side are already known. Note that some functions may be missing from the right-hand side; for example, if the master function $U_{ml}$ has $\NK$ denominators, the right-hand side is zero.
Now we consider the more general case where $G$ different master integrals have the same denominator $(D_{i_1}\cdots D_{i_n})$ and different numerators. These master integrals correspond to master functions with contiguous indices $U_{m,l'+1}$, $U_{m,l'+2}$, …, $U_{m,l'+G}$ as effect of the ordering of master integrals. The solution of the system provides for these functions a subsystem of simultaneous $G$ difference equations of the kind $$\label{diffu2}
\sum_{g=1}^{G} \sum_{i=0}^{\R_{hg}} p_{igl'h}'(x) U_{m,l'+g}(x+i) =
\sum_{k=1}^{l'} \sum_{j=0}^{Q_{hl'k}} q'_{jkl'h}(x) U_{mk}(x+j) \ , \quad h=1,\ldots,G\ ,$$ not in triangular form, where the left-hand side of each equation contains all the master functions $U_{m,l'+1}$, …, $U_{m,l'+G}$. We prefer not deal with the solution of the subsystem of simultaneous difference equations, so that we transform it into triangular form; this is not obligatory, is only convenient to simplify the subsequent numerical solution. We make the replacement $x\to x+c$ in the equations , with $c=1,2,\ldots$, generating new identities which are inserted in the subsystem, and taking care of extracting the terms containing the function $U_{m,l'+j}$ before of the terms containing $U_{m,l'+k}$ if $j>k$. This procedure is repeated until one obtains a set of $G$ equations in triangular form, $$\label{diffu1a}
\sum_{i=0}^{\R'_h} p_{il'h}''(x) U_{m,l'+h}(x+i) =
\sum_{k=1}^{l'+h-1} \sum_{j=0}^{Q'_{hl'k}} q_{jkl'h}''(x) U_{mk}(x+j) \ , \quad h=1,\ldots,G
\ .$$ Another advantage of the transformation into triangular form is that one obtains a difference equation containing the function $U_{m,l'+1}$, but not containing the other functions $U_{m,l'+2}$, $U_{m,l'+3}$,…, so that $U_{m,l'+1}$, which corresponds to the master integral with 1 as numerator, can be found independently. Unfortunately the transformation into triangular form has a price: one obtains for $U_{m,l'+1}$ an equation of order $\R'_1 >\max_h R_{h1}$ which has coefficients $q_{jkl'1}''(x)$ much more complicated and cumbersome than the coefficients $q_{jkl'h}'(x)$ of the equations of the subsystem . If $\R'_1$ is large (say $\R'_1>10$) these coefficients become huge and difficult to obtain so that it may be more convenient in these cases to solve the system of simultaneous equations directly; this will be described in some next paper.
Once all the subsystems of simultaneous equations corresponding to the various groups of master integrals with equal denominators are transformed into triangular form, the whole system takes the form .
Concerning the choice of $a_i$ and $b_i$, all the considerations done in section \[secab\] remain valid; for each combination of $n$ denominators $\{D_{i_1}\ldots D_{i_n}\}$ there is a minimal subsystem $\groupl{n}{\tilde a_i}{\tilde b_i}$ whose solution allows one to obtain the equations . For the diagrams so far examined $\tilde a_i$ turns out to be always equal to the maximum number of scalar products appearing in the master integrals (typically $1\ldots 3$); typical values of $\tilde b_i$ are $0,1,2$.
Solutions of difference equations with [0.8pt]{}factorial series {#secsolfact}
================================================================
By using the algorithms of section \[difequfey\], the triangular system of difference equations satisfied by the master functions is worked out. Now we consider the solution of each difference equation.
Let us suppose, for instance, that the master function $U(x)$ defined as $$\label{init1}
U(x)=
\int {\dk1 \dots \dk{\NK}} \
\dfrac
{\prod_{j=1}^{\NPS-\ND} \indpk_j^{\beta_{j}}}
{\displaystyle D_1^x D_2 \ldots D_\ND} \ ;$$ satisfies a difference equation of order $R$ $$\label{equdifref}
\sum_{i=0}^\R p_i(x) U(x+i)=F(x)\ ,$$ where $p_i(x)$ are polynomials and $F(x)$ is some known function. The solution of this nonhomogeneous equation can be written as $$U(x)=\UOMOG(x)+\UNOMOG(x)\ ,$$ where $\UNOMOG$ is a particular solution of the nonhomogeneous equation and $\UOMOG$ is the general solution of the homogeneous equation $$\label{equr1}
\sum_{i=0}^\R p_i(x) \UOMOG(x+i)=0\ .$$ The general solution of can be written as $$\label{equhom}
\UOMOG(x)=\sum_{j=1}^\R \tilde\omega_j(x) \UOMOG_j(x) \ ,$$ where $\tilde\omega_j(x)$ are periodic functions of period 1, and $\{\UOMOG_1(x),\ldots,\UOMOG_R(x)\}$ is a fundamental system of independent solutions of the homogeneous equation. In the following, recalling from [@Milne] for the ease of the unfamiliar reader the essential matter on the solution of linear difference equations with factorial series, we describe how to obtain factorial series expansions of $\UOMOG$ and $\UNOMOG$.
Factorial series
----------------
The series $$\begin{gathered}
\label{remfac}
\sum_{s=0}^\oo\frac{a_s \Gamma(x+1)}{\Gamma(x-\K+s+1)}
=\frac{\Gamma(x+1)}{\Gamma(x-\K+1)}\left(
a_0+\frac{a_1}{x-\K+1} +\frac{a_2}{(x-\K+1)(x-\K+2)} +\ldots \right)\ \end{gathered}$$ is known as factorial series of the first kind [@Milne12], or series of inverse factorials [@Milne271], or faculty series [@Knopp446]. We refer to it in brief as factorial series. The series converges for every point in a half-plane which is limited on the left by $\Re x =\lambda$ (excluding $x=\K-1, \K-2,\ldots $). The number $\lambda$ is the *abscissa of convergence*. If $\lambda=\oo$ the series is everywhere divergent.
As coefficients of the series encountered in this work behave for large $s$ as $|a_s|\sim s! s^\alpha$, it is useful (especially for numerical applications) to define the reduced coefficients $a'_s=a_s/s!$. For large $s$ the generic term of the sum tends to $ a'_s\Gamma(x+1) s^{\K-x}$, so that the factorial series has the same convergence properties as the Dirichlet series $\sum_{s=0}^\oo a'_s s^{\K-x}$.
Operators $\PI$ and $\RHO$
--------------------------
Given an arbitrary number $m$ one defines the operator $\RHO$ as[^8] $$\RHO^m U(x)=\frac{\Gamma(x+1)}{\Gamma(x-m+1)}U(x-m) \ .$$ This operator has the property $$\RHO^m\RHO^n U(x)= \RHO^{m+n} U(x)\ ;$$ if the operand is the unity, we omit it and write $$\RHO^m 1 = \RHO^m = \frac{\Gamma(x+1)}{\Gamma(x-m+1)} \ .$$ One defines the operator $\PI$ as $$\PI U(x)=x(U(x)-U(x-1)) \ .$$ The following properties can be proven: $$\begin{aligned}
\begin{split}\label{xpiro}
[\PI,\RHO]U(x)&=\RHO U(x)\ ,\\
p(\PI) \RHO^m U(x) &= \RHO^m p(\PI+m) U(x)\ ,
\end{split}\\
\notag\\
\begin{split}\label{xpirorel12}
xU(x)&=(\PI+\RHO)U(x), \\
x^n U(x)&=\sum_{k=0}^n \left(
\sum_{j=k}^n (-1)^{j-k}\binom{n}{j}S_{jk} \PI^{n-j} \right) \RHO^k U(x) \ , \\
p(x) U(x)&=\left[p(\PI)+p_1(\PI)\RHO +p_2(\PI)\RHO^2/2! +
\ldots \right]U(x) \ ,
\end{split}\end{aligned}$$ where $p$ is a polynomial, $p_n$ is $$p_n(\lambda)=\Delta_n p(\lambda)=
\sum_{j=0}^n (-1)^j\binom{n}{j}p(\lambda-j) \ ,$$ and $S_{jk}$ are the Stirling’s numbers of second kind [@Abramowitz]. Using these operators an expansion in factorial series becomes an expansion in powers of $\RHO^{-1}$ so that we will able to obtain solutions in factorial series of difference equations in the same manner as power series solutions of differential equations are obtained.
Solution of the homogeneous difference equation {#solfactser}
-----------------------------------------------
Let us consider the homogeneous difference equation of order $\R$ $$\label{diff01}
p_0(x) \UOMOG(x) +p_1(x) \UOMOG(x+1) + \ldots + p_\R(x) \UOMOG(x+\R) =0 \ ;$$ making the replacement $x\to x-\R$ and defining $q_{\R-i}(x)=p_i(x-\R)$ the equation becomes $$\label{diff1}
q_0(x) \UOMOG(x) +q_1(x) \UOMOG(x-1) + \ldots + q_\R(x) \UOMOG(x-\R) =0 \ .$$ Using the previous definitions of operators now we search for a formal solution in factorial series. We make the change of variable $\UOMOG(x)=\mu^x \VOMOG(x)$ in , where $\mu$ is an unspecified parameter, obtaining $$\label{diff2}
\mu^\R q_0(x) \VOMOG(x) +\mu^{\R-1} q_1(x) \VOMOG(x-1)
+ \ldots + q_\R(x) \VOMOG(x-\R) =0 \ .$$ Now we multiply the equation by $x(x-1)(x-2)\ldots (x-\R+1)$ and we observe that $xV(x-1)=\RHO V(x)$, $x(x-1)V(x-2)=\RHO^2V(x)$, etc.; the equation takes the form $$\label{diff3}
\left[ \phi_0(x,\mu) +\phi_1(x,\mu) \RHO + \ldots +
\phi_\R(x,\mu) \RHO^\R \right] \VOMOG(x) =0 \ ,$$ where $\phi_j$ are polynomials in $x$ and $\mu$. The multiplication by $x$ is equivalent to the multiplication by $\PI+\RHO$, therefore substituting the relations in one obtains the *first canonical form* of the difference equation: $$\label{cano1}
\left[f_0(\PI,\mu) +f_1(\PI,\mu)\RHO +f_2(\PI,\mu)\RHO^2
+ \ldots + f_{m+1}(\PI,\mu)\RHO^{m+1}\right]\VOMOG(x) =0 \ ,$$ where $f_i$ are polynomials in $\PI$ and $\mu$. In the case of the difference equations encountered in this work we find that $f_{m+1}(\PI,\mu)$ is independent of $\PI$, but not independent of $\mu$; therefore $f_{m+1}(\PI,\mu)=f_{m+1}(\mu)$. The algebraic equation in $\mu$ $$\label{equchar}
f_{m+1}(\mu)=0$$ is the *characteristic equation*[^9]. It turns out that the characteristic equation has always $R$ solutions different from zero. Let $\mu_1$, $\mu_2$, …, $\mu_\lambda$ be the $\lambda$ *distinct* solutions of . For each of these values, $\mu=\mu_i$, $i=1,\ldots,\lambda$, the first canonical form takes the form $$\label{cano2}
\left[f_0(\PI) +f_1(\PI)\RHO +f_2(\PI)\RHO^2
+ \ldots + f_{m}(\PI)\RHO^{m}\right]\VOMOG(x) =0 \ .$$ Now we try to satisfy the equation in $\VOMOG$ with the factorial series $$\label{serrho}
\VOMOG(x)=\sum_{s=0}^\oo\frac{a_s \Gamma(x+1)}{\Gamma(x-\K+s+1)}
=\sum_{s=0}^\oo a_s \RHO^{\K-s} = a_0\RHO^\K +a_1\RHO^{\K-1} +\ldots \ ,$$ whose asymptotic behaviour for large $x$ is $\VOMOG(x)\approx a_0 x^\K$. Substituting in one obtains the recurrence relations $$\begin{aligned}
\begin{split}\notag
&a_0 f_m(\K+m)=0 \ ,\\
&a_1 f_m(\K+m-1) +a_0 f_{m-1}(\K+m-1)=0 \ ,
\end{split} \\
\begin{split}\label{systa2}
&\cdots \\
&a_s f_m(\K+m-s) +a_{s-1} f_{m-1}(\K+m-s) +\ldots
+a_{s-m} f_0(\K+m-s)=0 \quad (s\ge m) \ . \qquad
\end{split}\end{aligned}$$ Supposing that $a_0\not=0$, the equation $$\label{indic}
f_m(\K+m)=0$$ is the *indicial equation*. Let $\K_1, \K_2, \ldots, \K_\nu$ the roots of this equation. In the case of the difference equations encountered in this work all the roots turn out to be distinct, and $\nu$ turns out to be the multiplicity of $\mu_i$. For each of these values of $\K$ we solve the system of recurrence relations. If there are no roots differing by a positive integer, $f_m(\K+m-s)\not=0$ for $s\ge1$, and therefore the recurrence relation provides the coefficient $a_s$ of the factorial series for every $s$. If there are roots differing by a positive integer (so-called *congruent* roots) we find $f_m(\K+m-s_0)=0$ for some $s_0$, so that the term $a_{s_0} f_m(\K+m-s_0)$ vanishes from the relation used to obtain $a_{s_0}$. The remaining terms of this relation always vanish[^10] in the case of the difference equations encountered in this work, therefore the value of $a_{s_0}$ remains undetermined and can be chosen at will (usually one puts $a_{s_0}=0$). For details on the convergence of the factorial series so obtained, see section \[sumfact2\].
In order to obtain the general solution of the difference equation, for each distinct solution of the characteristic equation $\mu_i$, $i=1,\ldots,\lambda$, we find an indicial equation, whose solutions are $\K_{ij}$, $j=1,\ldots,\nu_i$. For each solution of the indicial equation we solve the system of recurrence relations between the coefficients $a^{(i,j)}_s$ and we find $\nu_i$ solutions of , $$\VOMOG_{ij}(x)=\sum_{s=0}^\oo\frac{a^{(i,j)}_s
\Gamma(x+1)}{\Gamma(x-\K_{ij}+s+1)} \ , \qquad j=1,\ldots,\nu_i \ .$$ Multiplying by $\mu_i^x$ we find solutions of the difference equation . In total we find $\sum_{i=1}^\lambda \nu_i= \R$ different solutions. The general solution of the homogeneous difference equation will be a linear combination of the these solutions with periodic functions $\tilde\omega_{ij}(x)$ of period one as coefficients, $$\label{resho}
\UOMOG(x) =\sum_{i=1}^\lambda \sum_{j=1}^{\nu_i}
\tilde\omega_{i j}(x) \mu_i^x \VOMOG_{i j}(x) \ .$$
Solution of the nonhomogeneous difference equation
--------------------------------------------------
Let us consider the nonhomogeneous difference equation of order $\R$ $$\label{diff01n}
p_0(x) \UNOMOG(x) +p_1(x) \UNOMOG(x+1) + \ldots + p_\R(x) \UNOMOG(x+\R)
=F(x) \ ;$$ making the same replacements as the equation becomes $$\label{diff1n}
q_0(x) \UNOMOG(x) +q_1(x) \UNOMOG(x-1) + \ldots + q_R(x) \UNOMOG(x-\R)
=F'(x) \ ,$$ where $F'(x)=F(x-R)$. In the difference equations encountered in this work, the right-hand side can be written as a sum of factorial series expansions. We assume for simplicity that $F'(x)=\mu^x T(x)$ and $T(x)$ has the factorial series expansion $$\label{expa1b}
T(x)=c_0\RHO^\K +c_1\RHO^{\K-1} +c_2\RHO^{\K-2} +\ldots \ ,$$ where $\mu$, $K$ and $c_i$ are known.
Making the substitution $\UNOMOG(x)=\mu^x \VNOMOG(x)$ in and using the relations (\[xpiro\]-\[xpirorel12\]) we obtain the canonical form $$\label{cano1b}
\left[f_0(\PI) +f_1(\PI)\RHO +f_2(\PI)\RHO^2
+ \ldots + f_m(\PI)\RHO^m\right]\VNOMOG(x) =T(x) \ .$$ The expansion in factorial series of $\VNOMOG(x)$ has the form $$\label{serroh}
\VNOMOG(x)=a_0\RHO^{\K-m} +a_1\RHO^{\K-m-1} +a_2\RHO^{\K-m-2} +\ldots \ ;$$ inserting this expansion in and equating the coefficients of the powers of $\RHO$ one finds the recurrence relations $$\begin{aligned}
\begin{split}\notag
&a_0 f_m(\K)=c_0 \ ,\\
&a_1 f_m(\K-1) +a_0 f_{m-1}(\K-1)=c_1 \ ,
\end{split}\\
\begin{split}\label{systb2}
&\cdots \\
&a_s f_m(\K-s) +a_{s-1} f_{m-1}(\K-s) +\ldots
+a_{s-m} f_0(\K-s)=c_s \quad (s\ge m) \ . \quad
\end{split}\end{aligned}$$ Solving this system we can find the coefficients $a_s$. If $f_m(\K-s_0)=0$ for some $s_0$, the term $a_{s_0} f_m(\K-s_0)$ vanishes from the relation and, in the difference equations encountered in this work, the remaining terms form the trivial identity $c_s=c_s$, so that the value of $a_{s_0}$ remains undetermined and can be chosen at will. If the factorial series converges, $\UNOMOG(x)=\mu^x \VNOMOG(x)$ is a particular solution of the nonhomogeneous difference equation.
The case $F'(x)=\mu^x p(x) T(x)$, where $p$ is a polynomial, can be reduced to the previous case by transforming $x$ into $\PI+\RHO$ in the polynomial and letting the operators act on the expansion of $T(x)$ following .
Determination of arbitrary constants {#detcon}
====================================
The general solution of the difference equation can be written as $$U(x)=\sum_{j=1}^\R \tilde\omega_j(x) \UOMOG_j(x) +\UNOMOG(x)\ ,$$ where $\tilde\omega_j(x)$ are periodic functions of period 1. If we consider only integer values of $x$, the values of $\tilde\omega_j(x)$ are independent of $x$ so that we can replace them with arbitrary constants $\C_j$: $$\label{equhom2}
U(x)=\sum_{j=1}^\R \C_j \UOMOG_j(x) +\UNOMOG(x)\ .$$
The unambiguous solution of the difference equation requires the determination of the $R$ constants $\C_j$. The value of these constants can be determined
1. \[c1\] from the large-$x$ behaviour of the solution and of the integral , by equating the first coefficients of the expansions in factorial series;
2. \[c2\] by equating the values at $x=0$ of and .
As we will see in the following sections, the method \[c1\] may provide the values of all the constants, but the determination of the coefficients of the factorial series turns out to be a simple task only for euclidean massive integrals (see sections \[euclsec0\], \[neuclsec0\] and \[ondeft\]), where it involves integrals with one-loop less, and gets more complicated for non-euclidean integrals or integrals with zero masses (see section \[othercases\]); on the contrary the method \[c2\] has the limitation that it provides only one relation between the constants, but on the other hand it is not affected by the kind of the external momenta or masses (see section \[valx0\]).
Large-$x$ behaviour of integrals: euclidean massive case {#euclsec0}
--------------------------------------------------------
In this section we work out a relation between the coefficients of the expansion in factorial series of the master integral and simpler integrals with one-loop less. We assume that all the external momenta are euclidean, that is, the matrix of the scalar products $p_i \cdot p_j$ is semidefinite non-negative, and no mass is zero.
Choosing the momentum routing of the integral such that $D_1=k_1^2+m_1^2$, we write it as $$\label{init2}
U(x)=
\int
\dfrac {\dk1}{\displaystyle (k_1^2+m_1^2)^x } {g(k_1)} \ ,$$ where $g$ includes the contribution of the remaining $\NK-1$ loops $$\label{intgi}
g(k_1)=
\int {{\dk2 \; \dots \dk{\NK}} \
\dfrac
{\prod_{j=1}^{\NPS-\ND} \indpk_j^{\beta_{j}}}
{\displaystyle D_2 D_3 \ldots D_\ND} \ } \ .$$ Note that the function $g$ also depends on the external momenta $p_j$ and the masses $m_2$, $m_3$, $\ldots,$ $m_\ND$ of the other denominators. From a graphical point of view, $g(k_1)$ corresponds to the original diagram with the line $D_1$ cut. Introducing hyperspherical polar coordinates for the integration over $k_1$ $$d^D k_1= |k_1|^{D-1} d|k_1| \,d\Omega_D(\hat k_1) \ ,$$ ($\Omega_{D}=2\pi^{D/2}/\Gamma(D/2)$ is the $D$-dimensional solid angle) and separating the angular and radial part, becomes $$\label{intk2}\label{int00}
U(x) = \dfrac{1}{\Gamma(D/2)}
\int_0^\oo\dfrac{d k_1^2 \;(k_1^2)^{D/2-1}} {(k_1^2+m_1^2)^x}
f(k_1^2) \ ,$$ where $f$ is the angular mean over $k_1$ of $g$ $$\label{definf}
f(k_1^2)=
\dfrac{1}{\Omega_D}
\int d\Omega_{D}(\hat k_1) \; g(k_1) \ .$$
For large $x$ the factor $(k_1^2+m_1^2)^{-x}$ of peaks strongly around $k_1^2=0$; because of the assumption on external momenta and masses, the function $f(k_1^2)$ has no singularities for $k_1^2\ge 0$, and therefore we expect that the large-$x$ behaviour of the integral is controlled only by the behaviour of $f(k_1^2)$ near $k_1^2=0$. Making the change of variable $k_1^2=m_1^2 \dfrac{u}{1-u}$ in we obtain $$\label{intuf}
U(x)=\dfrac{(m_1^2)^{D/2-x}}{\Gamma(D/2)}\int^1_0 du \; u^{D/2-1}
(1-u)^{x-1-D/2} \tilde f(u)\ ,$$ where $\tilde f(u)=f(m_1^2 u/(1-u))$. We expand $\tilde f(u)$ in $u$ $$\label{expuf}
\tilde f(u)=u^\alpha(1-u)^\beta \sum_{s=0}^\oo b_s u^s \ ;$$ $\alpha$ is an integer greater than or equal to zero ($f$ is regular in the origin), while the factor $(1-u)^\beta$ has been introduced for convenience. Expressing the integrals over $u$ in terms of Beta function (which takes care of analytical continuation if the integral diverges at the endpoints), and choosing $\beta=D/2+1$, takes the desired form of a factorial series $$\label{resu1}
U(x)=\mu_0^x\sum_{s=0}^\oo a_s \RHO^{\K_0-s}
=\mu_0^x\sum_{s=0}^\oo a_s
\dfrac{\Gamma(x+1)}
{\Gamma(x+1-\K_0+s)}\ ,$$ where $$\label{resu2}
\mu_0=1/m_1^2,\quad \K_0=-D/2-\alpha, \quad \text{and }\
a_s= b_s m_1^D \Gamma(s+D/2+\alpha)/\Gamma(D/2)\ .$$ The coefficients $b_s$ of can be easily expressed in terms of the coefficients $f_s$ of the expansion in $k_1^2$ of $f(k_1^2)$ $$\label{expuf2}
f(k_1^2)=(k_1^2)^\alpha \sum_{s=0}^\oo f_s k_1^{2s}
\ ,$$ ($b_0=m_1^{2\alpha} f_0\ $, $\ldots$), so that the large-$x$ behaviour of $U(x)$ proves to be determined by the behaviour of $f(k_1^2)$ for small $k_1^2$. For large $x$ the leading behaviour is given by the first term of $$\label{expkf0}
U(x)\approx (m_1^2)^{D/2-x+\alpha} x^{-D/2-\alpha} f_0
\dfrac{\Gamma(D/2+\alpha)}{\Gamma(D/2)}\ .$$ In the frequent case of an integral with numerator equal to one, $\alpha=0$ and $f_0=f(0)$, it becomes $$\label{expkf00}
U(x)\approx (m_1^2)^{D/2-x} x^{-D/2} f(0) \ .$$
Now we equate with the expansion in factorial series of the general solution (see and ), obtaining $$\label{gen1}
\left(\frac{1}{m_1^2}\right)^x \sum_{s=0}^\oo a_s \RHO^{-D/2-\alpha-s} =
\sum_{j=1}^R \C_j \, \mu_j^x \sum_{s=0}^\oo \hat a_{js} \RHO^{\K_j-s}
+
\sum_l (\mu^{NH}_l)^x \sum_{s=0}^\oo a_{ls}^{NH} \RHO^{\K^{NH}_l-s}
\ .$$ In this equation only the $R$ constants $\C_j$ are unknowns, each one corresponding to a solution $\UOMOG_j$ with a different pair of values $\mu_j$ and $K_j$, $\mu_j$ being one solution of the characteristic equation of the homogeneous equation and $\K_j$ being the solution of the corresponding indicial equation; we have set $\hat a_{j0}=1$. We have also assumed that the expansion of the nonhomogeneous term contains a sum of expansions with different pairs of values $\mu^{NH}_l$ and $\K^{NH}_l$.
Fortunately, the number of constants $\C_j$ to find can be drastically reduced. If the difference equation is homogeneous, the constants $\C_{j}$ which may be different from zero are only those such that the corresponding $\mu_j$ and $\K_j$ satisfy the condition $$\label{cond1}
\mu_j=1/m_1^2\ , \quad \K_j+D/2+\alpha=\text{integer}\le 0
\qquad \Rightarrow \qquad \C_j\not=0\ ;$$ all the other constants must be zero.
In the case of nonhomogeneous equation, one must recall that it is part of a triangular system of difference equations; the nonhomogeneous term receives contributions from master integrals with a smaller number of denominators, which in their turn satisfy other nonhomogeneous difference equations, up to the simplest master integrals which satisfy homogeneous equations. Clearly $\mu^{NH}_l=1/m_1^2$ and $\K^{NH}_l+D/2$ is always an integer, therefore $\C_{j}$ may be different from zero if $\mu_j$ and $\K_j$ satisfy , or the new condition $$\label{cond1a}
\mu_j=1/m_1^2\ , \quad
0<\K_j+D/2+\alpha=\text{integer}\le \max_l \K^{NH}_{l} +D/2+\alpha
\qquad \Rightarrow \qquad \C_j\not=0$$ which implies cancellations between the first coefficients of the homogeneous and nonhomogeneous expansions. We found that in the equations considered in this work, the condition is never satisfied, and the condition is satisfied by a small number of pairs of $\mu_j$ and $\K_j$, often only one; if no pair satisfies the conditions, $U(x)$ is completely determined by the nonhomogeneous term.
The values of the non-zero constants $\C_j$ are determined by comparing the coefficients of the same powers of $\RHO$ of the two sides of . The required few coefficients $a_s$ are easily calculated from the first coefficients $f_s$ of the expansion in $k_1^2$ of $f(k_1^2)$. \[euclsec\]
The required coefficients $f_s$ are calculated by expanding numerators and denominators of for small $k_1$, and by performing the angular integration over $k_1$ of . Angular integrals are straightforward as they contain exclusively powers of scalar products containing $k_1$. As a result, the coefficients $f_s$ will be expressed by integrals over $k_2$,…$k_\NK$, belonging to diagrams with *one loop less* (for example, if the numerator of is unity then $f_0=f(0)=g(0)$). These integrals can be expressed using algorithm \[algsys\] as combinations of the master integrals of the new diagrams; in their turn, these new master integrals can be calculated by inserting an exponent in a denominator and building and solving new difference equations. The arbitrary constants of their solutions can be found using the large-exponent behaviours, which can be expressed in terms of integrals with another loop less, and so on. In this way we can explicitly calculate the values of all the arbitrary constants $\C_j$. This fact is very important.
Non-euclidean case: deformation of the radial path {#neuclsec0}
--------------------------------------------------
Now we consider the non-euclidean case: the matrix of the scalar products $p_i \cdot p_j$ is definite negative. We write as $$\label{intkh}
U(x,P) =
\dfrac{1}{\Gamma(D/2)}
\int_{l_0} \dfrac{d k_1^2 (k_1^2)^{D/2-1}} {(k_1^2+m_1^2)^x}
f(k_1^2,P)\ ,$$ where we explicitly show the dependence on the external momenta $P=$ $\{p_1,$ $\ldots,p_{\NP}\}$, and where $l_0$ indicates the path of the radial integration. For the moment the path is assumed to be the positive axis in the complex plane.
The integral $U(x,P)$, considered as a function of the external momenta $P$, is defined in the non-euclidean region by an analytical continuation through a generic path in the complex $P$-space. The path begins in some euclidean initial point $P_{in}$ and ends in the desired non-euclidean final point $P_{end}$. It is possible that in some intermediate point of the continuation path in the $P$-space, for a particular value of the external momenta $P_s$, $f(k_1^2,P_s)$ is singular in a point $k_1^2=q^2>0$, just on the radial integration path, breaking off the analytical continuation. If this singularity in $k_1^2$ cannot be avoided by modifying the continuation path in the $P$-space, then we are forced to deform[^11] the integration path in the $k_1^2$-space by turning around the singularity; the singularity in $k_1^2$ itself moves in the complex $k_1^2$-plane to a final point $k_{end}^2$ when we complete the analytical continuation from $P_s$ to $P_{end}$. The final integration path $l_0$ starts in $k_1^2=0$, turns around the singularity $k_{end}^2$ (usually on the negative axis) and comes back to $k_1^2\to +\oo$. In the general case of multiple singularities the path turns around all them. The deformation of the radial path must be performed when the values of the scalar products $p_i \cdot p_j$ get over some “deformation thresholds” in the non-euclidean region, the exact point of these thresholds being depending on the values of masses and the structure of the diagram.
One can show using Feynman parameters that the deformation thresholds are the thresholds (anomalous thresholds included) in $P$ of $g(0,P)$, the diagram obtained by eliminating the line $D_1$ and setting $k_1=0$ everywhere. Possible denominators depending only on $k_1$ give “additional” deformation thresholds of the kind $p^2=-m^2$.
Once the deformation thresholds are determined, we can split the non-euclidean region of the $P$-space into two regions:
1. one region below the deformation thresholds which adjoins the euclidean region, where the integration path is the same as the euclidean region;
2. the remaining region above the deformation thresholds where the integration path turns around some singularities.
### Example: deformation of the path for the one-loop self-energy integral
As an example, now we consider the simplest case, the one-loop integral $$\label{hyp14r}
I=\int
\dfrac {\dk{}}{\displaystyle (k^2+m_1^2) ((p-k)^2+m_2^2)}
=
\dfrac{1}{\Gamma(D/2)}
\int_{l_0} \dfrac{d k^2 (k^2)^{D/2-1}} {k^2+m_1^2}
\dfrac{1}{\Omega_D}\int \dfrac{d\Omega_D(\hat k)}{(p-k)^2+m_2^2}\ .$$ The angular integral is[^12] $$\label{hyp14}
\dfrac{1}{\Omega_D}\int \dfrac{d\Omega_{D}(\hat k)}{(p-k)^2+m_2^2}=
\dfrac{Z}{pk} \Phi(Z^2) \ ,$$ where $\Phi(Z^2)$ is the hypergeometric function $F(1,2-D/2;D/2;Z^2)$, $$Z^{\pm1}=\dfrac{p^2+k^2+m_2^2\mp R(p^2,k^2,-m_2^2)}{2p k}\ ,
\qquad p=\sqrt{p^2}\;, \quad k=\sqrt{k^2}\;,\quad$$ and $R(x,y,z)$ is the usual two-body phase space square root $$\label{defrxyz}
R(x,y,z)=\sqrt{x^2+y^2+z^2-2xy-2xz-2yz}\ .$$ The angular integral has two branching points in the complex $k^2$-plane, $k^2=(p \pm i m_2)^2$. Considering in the euclidean case, $p^2>0$, the path of integration is always the positive real axis. If $p^2 < 0$, must be analytically continued in the complex $p-$plane from a generic initial euclidean point $p=p_{in}>0$ to the non-euclidean final point $p=i\sqrt{-p^2}$ which is on the imaginary axis. The initial point has $\Im p_{in} =0$. If $p^2 \ge -m_2^2$ no radial singularity appears so that $$\label{hypbelow}
I=
\dfrac{1}{\Gamma(D/2)}
\int_0^\oo \dfrac{d k^2 (k^2)^{D/2-1}} {k^2+m_1^2}
\dfrac{Z}{pk}\Phi(Z^2) \ .$$ If $p^2<-m_2^2$, the final point has $\Im p >m_2$ so that any path connecting the initial and final point must cross the line $\Im p =m_2$ in some point. Let $p_c=im_2+c$ be this point; if we consider with momentum $p_c$, one of the branching points falls in $k^2=c^2>0$ which is exactly on the path of integration. Therefore the path of integration $l_0$ must be deformed avoiding the branching point $(p-im_2)^2$ which is on the negative real axis. One chooses as path of integration[@Levinehyp] first the segment $((p-im_2)^2,0)$ above the negative real axis, taking the square root $R$ with the plus sign and using $Z^{-1}$ in the place of $Z$, then the straight line $((p-im_2)^2,\oo)$ below the axis. Therefore, if $p^2<-m_2^2$ becomes $$\label{hypabove}
I=
\dfrac{1}{\Gamma(D/2)}
\int_0^{(p-im_2)^2} \dfrac{d k^2 (k^2)^{D/2-1}} {k^2+m_1^2}
\dfrac{Z^{-1}}{pk}\Phi(Z^{-2})
+
\dfrac{1}{\Gamma(D/2)}
\int_{(p-im_2)^2}^\oo \dfrac{d k^2 (k^2)^{D/2-1}} {k^2+m_1^2}
\dfrac{Z}{pk}\Phi(Z^2) \ .$$ This result will be used in section \[1loopself\].
Large-$x$ behaviour of integrals: non-euclidean massive case below and at the deformation threshold
---------------------------------------------------------------------------------------------------
In the region below the deformation thresholds, in the case of non-euclidean massive integrals, the determination of the large-$x$ behaviour is quite similar to that described in section \[euclsec0\].
\[ondeft\] At the deformation threshold the determination of the large-$x$ behaviour presents some peculiarities, since some external momenta have on-mass-shell values. In the integral denominators of the form $k_1^2-2 \bar p\cdot k_1$ appear, where $\bar p$ is some on-mass-shell momentum. Considering the expansion in $k_1$ of $g(k_1)$, these denominators vanish at $k_1=0$ and cannot be expanded in a way as straightforward as for off-mass-shell denominators; they must be included in the angular integrals, complicating quite a lot the integration.
In order to explain the situation, let us consider here the case of a one-loop diagram with $N+1$ external lines, and the integral $$\int \dfrac {\dk{}}{\displaystyle
(k^2+m_0^2)((p_1-k)^2+m_1^2)\ldots ((p_N-k)^2+m_N^2)} \ .$$ Setting $k=0$ we see that the deformation threshold is $p_i^2=-m_i^2$, $i=1$, …, $N$. Now we set the external momenta to these on-mass-shell values, and we consider the integral $$\label{intkn1}
W(x)=\int \dfrac{\dk{}}{(k^2+m_0^2)^x
(k^2-2 p_1\cdot k)\cdots (k^2-2 p_N\cdot k)} \ ,$$ whose we want to calculate the large-$x$ leading behaviour. Following the notation of section \[euclsec0\] we write $$\label{hyp000}
W(x) =\int_0^\oo\dfrac{d k^2 (k^2)^{D/2-1}} {(k^2+m_0^2)^x}
\dfrac{f(k^2)}{\Gamma(D/2)}\ ,$$ $$\label{hypint1}
f(k^2)=\dfrac{1}{\Omega_D}\int \dfrac{d\Omega_D(\hat k)}
{(k^2-2 p_1\cdot k)\cdots (k^2-2 p_N\cdot k)}\ .$$
Now we extract the leading behaviour of $f(k^2)$ for $k^2 \to 0$. Inserting Feynman parameters $x_i$, $i=1$, $\ldots$, $N-1$ one finds $$\label{hypfe}
f(k^2)=\frac{\Gamma(N)}{\Omega_D}
\int {dx_1 \ldots dx_{N-1}} \int \dfrac{d\Omega_D(\hat k)}
{(k^2-2P(x_i)\cdot k)^N}\ ,$$ where $P(x_i)=\sum_{i=1}^N x_i p_i$ and $\sum_{i=1}^{N}x_i=1$. The angular integral in $D$ dimensions can be expressed using a generalization of the formula $$\label{hyp14f}
\dfrac{1}{\Omega_D}\int \dfrac{d\Omega_{D}(\hat k)}{((p-k)^2+m^2)^x}=
\left(\dfrac{Z}{pk}\right)^x F(x,x+1-D/2;D/2;Z^2) \ ;$$ using properties of hypergeometric function one finds for $k^2\to 0$ $$\label{hypsvil0}
\dfrac{1}{\Omega_D}\int \dfrac{d\Omega_{D}(\hat k)}{(k^2-2P(x_i)\cdot k)^N} \approx
(-P^2(x_i) k^2)^{-N/2}
\dfrac{\Gamma(D/2)\Gamma(N/2)}{2\Gamma(N)\Gamma((D-N)/2)}\ ,\quad$$ then $$\label{fk2a1}
f(k^2)\approx\dfrac{\Gamma(D/2)\Gamma(N/2)}{2\Gamma((D-N)/2)}
(k^2)^{-N/2} \int \dfrac{dx_1 \ldots dx_{N-1}}{\left(-P^2(x_i)\right)^{N/2}}
\ .$$ We see that $f(k^2)$ is singular in $k=0$ and, if $N>D$, the integral over $k^2$ of diverges for small $k^2$; with arbitrary $\alpha$ takes care of the necessary analytical continuation.
Let us now consider a one-loop integral in $E$ dimensions with the same denominators as the angular integral $$\label{fdefle}
L_E(p_i) \equiv \dfrac{1}{2}
\int\dfrac{[d^E q]}{
{(q^2-2 p_1\cdot q)\cdots (q^2-2 p_N\cdot q)}
}\ ;$$ introducing Feynman parameters in the same way as for we obtain $$\label{fk2a2}
L_E(p_i)= \dfrac{\Gamma(N-E/2)}{2} \int \dfrac{dx_1 \ldots
dx_{N-1}}{\left(-P^2(x_i)\right)^{N-E/2}}\ .$$ If we choose $E=N$ the integrals over $x_i$ of and become identical, so that we can rewrite as $$\label{connhyp2}
f(k^2)\approx{(k^2)^{-N/2}}\dfrac{\Gamma(D/2)}{\Gamma((D-N)/2)}
L_N(p_i)\ .$$ This result shows that the angular integral with $N$ on-mass-shell denominators in the limit $k^2\to 0$ is proportional to a one-loop integral with the same denominators (one less than ), calculated in a *number of dimensions equal to the number of on-mass-shell denominators*. Moreover, using and (with $\alpha=-N/2$ according to ) one finds the large-$x$ leading behaviour of : $$\label{expkf11}
W(x)\approx (m_0^2)^{D/2-N/2-x} x^{-D/2+N/2} L_N(p_i)\ .$$ This result is similar to euclidean leading behaviour for a one-loop vacuum integral ( with $f(0)=1$); the differences are the change of the exponents and the multiplication by the factor $L_N(p_i)$ which is *independent* of the number of dimensions $D$ of the integral . As a consequence of the divergence of the radial integral and of the necessary analytical continuation, we observe an apparently paradoxical result: if $N>D$ and $m_0=1$, from we see that the integral $W(x)$ increases as the exponent $x$ increases, while in the euclidean case the integral $U(x)$ decreases as $x$ increases. The integral $L_N(p_i)$ may be calculated analytically, for example, extracting the $k^2\to 0$ behaviour from the known analytical expressions of the angular integral for $D=4$ and $N=1$ [@Levinehyp] and $N=2,3$ [@NChyp], or numerically, by inserting an exponent in one denominator of and building and solving a difference equation, or by using the identities of the section \[valx0\] (for example see ). In section \[resu\] we will need the following values $\tilde L_N$ of $L_N(p_i)$ in the on-mass-shell and equal masses case $m_i=1$, $p_i^2=-1$ and $(p_i-p_j)^2=-1$ for every $i$ and $j$: $$\label{valuesl}
\begin{split}
\tilde L_1=&{\sqrt{\pi}}/{2} \;,
\qquad
\tilde L_2={\pi\sqrt{3}}/{9}\;,\\
\tilde L_3=&\left(\arctan{\sqrt{8}}-\arctan{\sqrt{3}}\right)\sqrt{{9\pi}/{8}} \;,
\quad
\tilde L_4=0.172751462\ldots \;.\\
\end{split}$$ The generalization of to multi-loop integrals is straightforward; considering $$W'(x)=\int \dfrac{\dk{}}{(k^2+m_0^2)^x
(k^2-2 p_1\cdot k) \cdots (k^2-2 p_N\cdot k) }
\,h(k)\ ,$$ $$h(k)=\int
\dfrac
{\dk2 \cdots \dk{\NK}}
{D_{N+2} \cdots D_\ND}\ ,$$ the large-$x$ leading behaviour is $$W'(x) \approx (m_0^2)^{D/2-N/2-x} x^{-D/2+N/2}
L_N(p_i) \, h(0)\ .$$ The condition in this case must be modified to $$\label{cond1mod}
\mu_j=1/m_1^2\ , \quad \K_j+D/2-N/2=\text{integer or half-integer}\le 0
\qquad \Rightarrow \qquad \C_j\not=0\ .$$
Non-euclidean case above the deformation threshold and massless case {#othercases}
---------------------------------------------------------------------
The cases described above, euclidean and non-euclidean below the deformation thresholds, are both united by the fact that the large-$x$ behaviour of $U(x)$ depends only on the behaviour of $f(k_1^2)$ near the origin. In these cases it is easy to express this behaviour in terms of simpler diagrams obtained by putting $k_1=0$ in the main diagram. The situation gets more complicated if $f(k_1^2)$ has other singularities on the integration path: for the $i$th singularity placed in $q_i^2 \not= 0 $, an additional contribution to the large-$x$ behaviour of $U(x)$ appears, of the form $(q_i^2+m_1^2)^{-x} V(x)$, with $V(x)\sim x^\K$ for large $x$. The precise form of $V(x)$ depends on the behaviour of $f$ near the singularity; unfortunately its determination requires the calculation of angular integrals for $k_1^2 \not=0 $ (see ) $$\int_{k_1^2\approx q_i^2} d\Omega_D(\hat k_1) \; g(k_1)\ ,$$ which is difficult and case-dependent. In this work we have performed these calculations only in the one-loop example of section \[1loopself\]. In the case of more complicated diagrams we have preferred to avoid the calculation of these angular integrals and to look at the problem from a different point of view, using a method with broader applicability based on differential equations (see section \[differentialequ\]).
The cases where singularities of $f$ other than the origin appear are:
- The non-euclidean massive case, above the deformation threshold: the path of integration must be deformed, turning around a number of singularities of $f(k_1^2)$. See section \[m1m2nz\] for an example.
- The zero mass case, where $m_1>0$ and some of the masses $m_2$, …, $m_\ND$ are zero. If the external momenta are euclidean, singularities on the positive axis of $k_1^2$ may appear, but this fact does *not* require deformation of the path. See section \[m2zero\] for an example.
The situation worsens if the mass $m_1$ of the denominator raised to $x$ is zero. In this case the behaviour of $U(x)$ for large $x$ depends on the behaviour of $f(k_1^2)$ on the whole real axis, and not in some isolated points. This can be easily understood by noting that in this case $U(x+D/2-1)$ is the Mellin transform of $f(k_1^2)$. See section \[m1zero\] for an example.
Zero exponent condition {#valx0}
-----------------------
A very useful relation between the constants $\C_j$ arises from calculated at $x=0$: $$\label{rel0}
U(0)-\UNOMOG(0)=\sum_{j=1}^\R \C_j\UOMOG_j(0)\ .$$ $U(0)$ is an integral without the denominator $D_1$; if the value of $U(0)$ can be found by some method, as an example, by adding an exponent to $D_2$ and by solving a difference equation, and if at least one $\UOMOG_j(0)$ has a non-zero value, establishes one new relation between the constants $\C_j$. The existence of this relation must be verified in each particular case. The advantage of this relation over the relations found using the asymptotic behaviour of $U(x)$, is that it is valid for every kind of integral, regardless of the value of masses or external momenta.
It is possible to construct identities analogous to , by inserting in $x=-1,-2,\ldots $ instead of $x=0$. Unfortunately in all the analyzed cases the identities so found turned out to be equivalent to .
Evaluation of factorial series {#sumfact}
==============================
Once the constants $\C_j$ of have been determined using the methods described in section \[detcon\], in order to obtain the value of the master integral $U(1)$ we must calculate the values of the homogeneous solutions $\UOMOG_j(1)$ and of the nonhomogeneous solution $\UNOMOG(1)$.
Convergence of factorial series and instabilities of recurrence relations {#sumfact2}
-------------------------------------------------------------------------
Let us unify the notation by considering the solution $U^{(\alpha)}(x)$, where $(\alpha)$ indicates one of the solutions of the homogeneous or nonhomogeneous equation, and expand it in factorial series: $$\label{vaser}
U^{(\alpha)}(x)=\left(\mu^{(\alpha)}\right)^x \sum_{s=0}^\oo a_s^{(\alpha)}
\dfrac{\Gamma(x+1)}{\Gamma(x+1-\K^{(\alpha)}+s)} \ .$$
Convergence of the series depends on the value of the abscissa of convergence $\lambda$. Analyzing the large-$s$ behaviour of the coefficients $a_s^{(\alpha)}$ one finds that the series has $\lambda<\oo$ if none of the solutions $\mu_j$ of the characteristic equation satisfies the condition $$\label{conv}
0<|\mu_j/\mu^{(\alpha)}-1|< 1 \ , \qquad j=1,\ldots,\R \ .$$ If $\lambda=\oo$ the series is everywhere divergent and the expansion in factorial series is only formal; in this case another method must be used to calculate $U^{(\alpha)}(1)$ (see section \[Laplacesec\] on the Laplace’s transformation).
If $\lambda<\oo$ the convergence of the factorial series is logarithmic, that is, if $S_m(x)$ is the sum of the first $m$ terms of , one finds that $|S_m(x)-U^{(\alpha)}(x)|\sim {m^{\lambda-x}}$ for large $m$. The series converges if $x>\lambda$, more and more quickly as $x$ increases. As usually one finds $\lambda\sim 1$, it is not convenient or possible to calculate $U^{(\alpha)}(1)$ directly by summing the factorial series. Therefore, chosen a number $x_{max}$ conveniently large, one calculates $U^{(\alpha)}(x)$ for $R$ contiguous values of $x$, $U^{(\alpha)}(x_{max})$, $U^{(\alpha)}(x_{max}+1)$, …, $U^{(\alpha)}(x_{max}+R-1)$, where the series converges faster, and one uses repeatedly the corresponding recurrence relation, or , in order to obtain the values of $U^{(\alpha)}(x)$ for $x=x_{max}-1,x_{max}-2,\ldots$ up to $x=1$. A drawback of this procedure is that the recurrence relation may be unstable, so that each iteration causes a loss of precision.
Let $A^{-1}=\min_{j}|\mu_j/\mu^{(\alpha)}|$. The recurrence relation is unstable if
1. $A>1$: \[instab1\] in this case each iteration increases the error on $U^{(\alpha)}(x)$ of a factor $A$.
2. $A=1$, and $\mu^{(\alpha)}$ is a root of of multiplicity $m>1$: in this case $n$ iterations of the recurrence relation increase the error on $U^{(\alpha)}(x)$ of a factor $n^{m-1}$; this is a kind of instability weaker than the preceding one.
If the recurrence relation is stable $x_{max}$ can be chosen large at will, with no effect on the precision of $U^{(\alpha)}(1)$. In the case of instability with $A>1$, in order to obtain the result $U^{(\alpha)}(1)$ with a number $E$ of exact digits, the calculations of $U^{(\alpha)}(x_{max}+i)$ must be performed with a greater number of decimal digits $C=E+x_{max}\log_{10}A $. Supposing $a_s^{(\alpha)}\sim s!$, a rough estimate of the number $s_{max}$ of terms of the series needed to obtain the sum with such a precision is $s_{max}\sim A^{\frac{C}{C-E}}{x_{max}}$. Performing the calculations with fixed precision arithmetic with $C$ digits, it is convenient to choose a value of $x_{max}$ as low as possible in order to obtain the greatest $E$, compatibly with the rapid increases of $s_{max}$ and of the computing time (see an example in section \[numexa\]). Performing calculations with multiprecision arithmetic, $C$ can be chosen at will, and $x_{max}$ can be increased; a convenient choice which minimizes the estimate of $s_{max}$ is $C/E\sim 1+\ln A$ (see an example in section \[pair2\]). Fixed $C/E$, by varying $E$ one sees that $x_{max}\propto E$ and $s_{max} \propto E$. Therefore the number of terms of the series (and even the computation time) is proportional to the number of digits of precision of the results; this is true even in the stable case provided that one chooses $x_{max}\propto E$.
Truncated expansion in $\e$ {#truncd}
---------------------------
We are interested in the results in the limit $D \to 4$; therefore, defined $\e=(4-D)/2$, we expand all the quantities in $\e$, truncating the expansions at the first $n_\e$ terms, and we perform all the calculations using truncated series; in this way the coefficients of all the powers of $\e$ are found numerically, including the *negative* powers. This technique is perhaps not the most efficient, but is very versatile. We have implemented in the program (see section \[calcprog\]) the arithmetic of truncated series, so that the use of series becomes as simple as with ordinary numbers.
The time of computation of a factorial series grows approximately as $n_\e^2$, and it is mainly due to multiplications. The division between series is an operation less frequent than the multiplication as it occurs only once in the calculation of each $a_s$ with the recurrence relations or or in the calculation of each $U(x)$ using the recurrence relations or . As effect of cancellations of terms when series are summed, and of divisions by series beginning with a non-zero power of $\e$, first or last terms of the expansions in $\e$ may be lost; in this case the calculations must be necessarily performed with an initial number $n_\e$ of terms of the expansions greater than the desired final number $n'_\e$ of terms of $U(x)$. The number $n_\e$ is chosen empirically. The loss of terms of the expansions in $\e$ is frequent when the recurrence relations or are used to obtain $U(x)$ for $x\le \lambda$; this is due to the fact that for such values of $x$ the expansion in $\e$ of $p_0(x)$ begins with a non-zero power of $\e$. Furthermore, we found that it is critical to recognize the cancellation of the first coefficient of a series used as divisor when, because of the numerical errors, the coefficient is a very small number instead of zero; in this case a numerical cutoff must be carefully used.
Applications to simple one-loop integrals {#example1loop}
=========================================
In this section we discuss in detail (as we assume the reader unfamiliar with these operator techniques) the solution of the difference equations for the one-loop vacuum and self-energy integrals.
One-loop vacuum integral {#1loopv}
------------------------
Defining $$\label{inte0}
J(x)=\int \dfrac{\dk{}}{(k^2+m_1^2)^x}\ ,$$ we want to calculate the master integral $J(1)$. According the single identity of the set $\group{1}{0}{0}$ $J(x)$ satisfies the homogeneous difference equation $$\label{equ1den}
m_1^2 (x-1) J(x)-(x-1-D/2)J(x-1)=0\ .$$ We look for a solution[^13] of this equation in the form of a factorial series (see section \[solfactser\]), $$\label{factserese1}
J(x)=\mu^x V(x) =\mu^x \sum_{s=0}^\oo a_s \RHO^{K-s}\ .$$ Introducing the operators $\PI$ and $\RHO$, multiplying by $x$ and using we find the first canonical form of the difference equation $$\label{fcf1}
\left(
(\mu m_1^2 -1 )\RHO^2
+((2\mu m_1^2-1) (\PI - 1) + D/2 )\RHO
+\mu m_1^2 \PI(\PI-1)
\right) V(x)=0 \ .$$ The characteristic equation , $f_2(\mu)=0$, gives the value $\mu=1/m_1^2$. Substituting this value of $\mu$ in one obtains $$\left(
( \PI - 1 + D/2 ) \RHO
+\PI(\PI-1)
\right) V(x) =0 \ .$$ The indicial equation , $f_1(\PI=1+\K)=0$, gives the value $\K=-D/2$. Using the recurrence relation one finds $$\label{asex1}
a_s=
\dfrac{\Gamma(s+D/2)\Gamma(s+D/2+1)}{\Gamma(D/2)\Gamma(D/2+1)\Gamma(s+1)}
a_0 \ .$$ \[vala0\] From one finds that $ a_0=(m_1^2)^{D/2}$. The final result is $$\label{resfin0}
J(x)=(m_1^2)^{D/2-x} \sum_{s=0}^\oo
\dfrac{\Gamma(s+D/2)\Gamma(s+D/2+1)}{\Gamma(D/2)\Gamma(D/2+1)\Gamma(s+1)}
\dfrac{\Gamma(x+1)}{\Gamma(x+1+D/2+s)}\ .\quad$$ The coefficients $a_s$ behave for large $s$ as $a_s/s! \propto s^{D-1}$, therefore the term of the factorial series behaves as $s^{D/2-1-x}$, so that the series has abscissa of convergence $\lambda=D/2$. This value signals the divergence of the integral . The recurrence relation , being of order one, is numerically stable. Therefore it is possible to obtain $J(x)$ for a large integer $x \gg D/2$ by summing a few terms of the factorial series and to use the recurrence relation to obtain $J(1)$. For a numerical example in the limit $D\to 4$ see section \[numexa\].
One-loop self-energy integral {#1loopself}
-----------------------------
Let us consider the one-loop self-energy integral $$I(1)=\int \dfrac {\dk{}} {D_1 D_2}\ ,$$ where $D_1=k^2+m_1^2$ and $D_2=(p-k)^2+m_2^2$. A simple inspection of the system of identities $\group{1 \ldots 2}{0 \ldots 1}{0}$ shows that there are three master integrals, $\int \dk{}/D_1D_2$, $\int \dk{}/D_1$ and $\int \dk{}/D_2$. We must find three difference equations for $$\label{inte1}
I(x)=\int \dfrac {\dk{}} {D_1^x D_2} \ ,
\qquad
J(x)=\int \dfrac {\dk{}} {D_1^x} \ ,
\qquad
K(x)=\int \dfrac {\dk{}} {D_2^x} \ .$$ The solution of the system made up of the set of identities $\group{1 \ldots 2}{0}{0 \ldots 1}$ gives the system of difference equations $$\begin{gathered}
\label{diffequ1}
(x-D)I(x-2)+
(-p^2+m_2^2-m_1^2)(2 x-D-1)I(x-1)
+R^2(p^2,-m_1^2,-m_2^2) (x-1) I(x) \\
+J(x-1) ((D/2-1) (p^2+m_2^2+m_1^2)-(x-2) (p^2+m_2^2))/m_1^2=0 \ , \end{gathered}$$ $$\label{diffequ1b}
m_1^2 (x-1) J(x)-(x-1-D/2)J(x-1)=0 \ ,$$ $$\label{diffequ1c}
m_2^2 (x-1) K(x)-(x-1-D/2)K(x-1)=0 \ ,$$ where $R$ was defined in . The difference equation for $J(x)$ and $K(x)$ has been solved in the preceding section, so that we consider here only the second-order difference equation for $I(x)$. We discuss separately the massive case and the cases where one of the masses $m_1$ or $m_2$ is zero.
### Case 1: $m_1\not = 0$, $m_2\not = 0$ {#m1m2nz}
Following section \[solfactser\] we substitute $I(x)=\mu^x V(x)$ and $J(x)=\mu^x W(x)$ in , multiply it by $x(x-1)$ and insert the operators $\PI$ and $\RHO$. We obtain the canonical form of the equation $$\label{chan1}
\left( f_3(\mu)\RHO^3 +f_2(\PI,\mu)\RHO^2 +f_1(\PI,\mu)\RHO +f_0(\PI,\mu)
\right)V(x)=
\mu\left(g_3(\PI)\RHO^3+g_2(\PI)\RHO^2+g_1(\PI)\RHO \right)W(x) \ ;$$ $f_i$ and $g_i$ are polynomials in $\PI$, $\mu$, $D$, $p^2$, $m_1^2$ and $m_2^2$, whose explicit expressions are not shown for brevity. The characteristic equation $f_3(\mu)=0$ has the two different roots $$\mu=\mu_\pm=\dfrac{1}{(p\pm i m_2)^2 + m_1^2}$$ (we define $p=\sqrt{p^2}$, and if $p^2<0$ then $p=i\sqrt{-p^2}$). The solution of the difference equation is $$\label{solgeni}
I(x)=\C_{+}\IOMOG_{+}(x) + \C_{-}\IOMOG_{-}(x) +\INOMOG(x)\ ,$$ where $\IOMOG_{\pm}$ are the solutions of the homogeneous equation corresponding to $\mu_{\pm}$ and $\INOMOG$ is one solution of the nonhomogeneous equation.
Let us now consider the homogeneous solution $\IOMOG_{-}$. We write $\IOMOG_{-}(x) = \mum^x \VOMOG_{-}(x) $ and we look for the coefficients of the expansion in factorial series of $\VOMOG_{-}(x)$ $$\label{factserm}
\VOMOG_{-}(x)=\sum_{s=0}^\oo a_s^{-} \RHO^{\K_{-}-s}\ .$$ Substituting $\mu=\mum$, the homogeneous part of the canonical form becomes $$\label{chan2}
\left( f_2^{-}(\PI)\RHO^2 +f_1^{-}(\PI)\RHO +f_0^{-}(\PI) \right)\VOMOG_{-}(x)
= 0\ ,$$ $$\begin{split}
f_2^{-}(\PI)=& 2im_2p (2 \PI+D-5) \ ,\\
f_1^{-}(\PI)=& \left((p^2+m_1^2-m_2^2) (\PI+D-3)+2im_2p(3 \PI-4) \right)
(\PI-1) \ ,\\
f_0^{-}(\PI)=&(p^2+m_1^2-m_2^2+2im_2p) \PI (\PI-1)^2 \ ;\\
\end{split}$$ the indicial equation $f_2^{-}(\K_{-}+2)=0$ gives $\K_{-}=(1-D)/2$. Fixed $a_0^{-}=1$, the other coefficients $a^{-}_s$ can be found using the recurrence relation ; the behaviour of $a_s^{-}$ for large $s$ can be determined by considering the recurrence relation as a difference equation in $s$ for $a^{-}_s$ and solving it; one finds $$\label{asminf}
a_s^{-}/s!\approx C_1 s^{(3D-7)/2} + C_2 B^s/s\ ,$$ where $B=\mum/(\mum-\mu_{+})$, while $C_1$ and $C_2$ are constants. If $|B|>1$ the series never converges (in fact the condition is satisfied); if $|B|\le 1$, the series converges with abscissa of convergence $\lambda=D-2$. The solution $\IOMOG_{+}$ can be obtained in analogous way with the replacement $im_2p \to -im_2 p$.
Let us now consider the nonhomogeneous solution $\INOMOG(x)$. We write $\INOMOG(x)=\mu^x \VNOMOG(x)$. The value of $\mu$ must be taken from the expansion in factorial series of $J(x)$ obtained in section \[1loopv\]: $\mu=1/m_1^2$. Substituting this value of $\mu$ in the canonical form, becomes $$\label{chan3}
\left( \hat f_3(\PI)\RHO^3 +\hat f_2(\PI)\RHO^2
+\hat f_1(\PI)\RHO +\hat f_0(\PI) \right)\VNOMOG(x) =
\left(\hat g_3(\PI)\RHO^3 +\hat g_2(\PI)\RHO^2 +\hat g_1(\PI)\RHO
\right)W(x)\ ,$$ $$\begin{split}
\hat f_3(\PI)=&(p^2+m_2^2)^2\ ,\\
\hat f_2(\PI)=& \left( 3 (p^2+m_2^2)^2 +2 m_1^2 (p^2-m_2^2)\right) \PI
-5 (p^2+m_2^2)^2
+(D-5) m_1^2 (p^2-m_2^2) \ ,\\
\hat f_1(\PI)=& \left[
\left((p^2+m_2^2)^2+m_1^2 (p^2-m_2^2)\right)(3 \PI-4)
+m_1^2 (p^2+m_1^2-m_2^2) (\PI+D-3)
\right](\PI-1)\ ,\\
\hat f_0(\PI)=& R^2(p^2,-m_1^2,-m_2^2) \PI (\PI-1)^2 \ ,\\
\hat g_3(\PI)=& p^2+m_2^2 \ ,\\
\hat g_2(\PI)=& (p^2+m_2^2) (2 \PI-3) -(D/2)(p^2+m_1^2+m_2^2)+m_1^2 \ ,\\
\hat g_1(\PI)=& (p^2+m_2^2) (\PI-1)^2
-\left((D/2)(p^2+m_1^2+m_2^2)-m_1^2\right)(\PI-1) \ ,\\
\end{split}$$ where $J(x)=(m_1^2)^{-x} W(x)$, $W(x)=\sum_{s=0}^\oo b_s \RHO^{\K_b-s}$, $\K_b=-D/2$, and the coefficients $b_s$ are the coefficients $a_s$ defined in .
The right-hand side of can be written as $ \sum_{s=0}^\oo c_s \RHO^{\K_c-s} $, where $\K_c=\K_b+3$, and the expression of $c_s$ is given by $$c_s= b_s \hat g_3(\K_c-s) +b_{s-1} \hat g_2(\K_c-s) +b_{s-2} \hat
g_1(\K_c-s) \ .$$ Then, the coefficients $\hat a_s$ of the expansion of $\VNOMOG(x)=\sum_{s=0}^\oo \hat a_s \RHO^{\K_b-s}$ can be found by using the recurrence relation ; the large-$s$ behaviour is $$\hat a_s/s!\approx C_1 s^{(3D-6)/2} +C_2 s^{D-2} +C_3 B_+^s/\sqrt{s} +C_4
B_-^s/\sqrt{s}\ ,$$ where $B_{\pm}=m_1^{-2}/(m_1^{-2}-\mu_{\pm})$ and $C_i$ are constants. If $|B_+|>1$ or $|B_-|>1$ the expansion of $\VNOMOG$ never converges (the condition is satisfied); if $|B_+|<1$ and $|B_-|<1$, the series converges, and the abscissa of convergence is $\lambda=\max (D-2,D/2-1)$ if $C_1\not = 0$ and $C_2\not=0$.
Now we must determine the constants $\C_{+}$ and $\C_{-}$; we compare the large-$x$ behaviours of the solutions $\IOMOG_{\pm}$ and $\INOMOG$, $$\begin{aligned}
\label{lrn1}
\IOMOG_{\pm}(x) &\approx \mu_{\pm}^x x^{(1-D)/2}\ , \\
\label{lrn1nomog}
\INOMOG(x) &\approx
\begin{cases}
(p^2+m_2^2)^{-1}(m_1^2)^{D/2-x} x^{-D/2}
&
\text{$p^2\not = -m_2^2$}\ ,\\
(2-D)(4m_2^2)^{-1} (m_1^2)^{D/2-x} x^{-D/2}
&
\text{$p^2 = -m_2^2$}\ ,
\end{cases}\end{aligned}$$ obtained from the first term of factorial series, with the large-$x$ behaviour of $I(x)$. We point out that these behaviours and the values of $\C_{\pm}$ later inferred are rigorously valid only if all the factorial series have finite abscissa of convergence; however the deduction may be extended to the cases where the series have $\lambda=\oo$ with the help of the integral representations of section \[2den\].
If $p^2\ge -m_2^2$ we are below the deformation threshold, and we can write $I(x)$ as radial integral (see ) $$\label{Ia}
I(x)=\dfrac{1}{\Gamma(D/2)}\int_0^\oo
\dfrac{dk^2 (k^2)^{D/2-1}}{(k^2+m_1^2)^x} \dfrac{Z}{pk}
\Phi(Z^2) \ ;$$ the large-$x$ behaviour is given by $$\label{ian}
I(x)\approx (m_1^2)^{D/2-x} x^{-D/2} (p^2+m_2^2)^{-1}\ .$$ The comparison of with shows that, as $\mu_{\pm}\not = 1/m_1^2$, $\IOMOG_{\pm}$ do not contribute to $I$, and that only $\INOMOG$ contributes, so that $$\label{cond11eu}
\C_{+}=\C_{-}=0 \qquad (p^2 > -m_2^2,\ m_1 \not=0, \ m_2\not=0) \ .$$ At the deformation threshold $p^2=-m_2^2$ one finds from for large $x$ $$\label{Ixth}
I(x) \approx (m_1^2)^{D/2-1/2-x} x^{(1-D)/2} \dfrac{\sqrt{\pi}}{2m_2} \ ;$$ as $\mu_{-}=1/m_1^2$ and $\K_{-}=-D/2+1/2$, is satisfied, so that $\IOMOG_{-}$ contributes to $I$, and $\C_{-}$ is different from zero. Comparing and one finds $$\label{cond11eu0}
\C_{-}=m_1^{D-1} m_2^{-1} \sqrt{\pi}/2\ ,\quad \C_{+}=0
\qquad (p^2 = -m_2^2,\ m_1 \not=0, \ m_2\not=0) \ .$$
If $p^2< -m_2^2$ we are above the deformation threshold, and the radial path of integration turns around the point $k^2=(p-im_2)^2$. We split $I(x)=I_{a}(x)+I_b(x)$ (see ), where $I_{a}$ is given by and $I_b$ is $$\label{Ib}
I_b(x)=\dfrac{1}{\Gamma(D/2)}
\int_{(p-im_2)^2}^0 \dfrac{dk^2 (k^2)^{D/2-1}}{(k^2+m_1^2)^x}
\left(
\dfrac{Z}{pk} \Phi(Z^{2})
-\dfrac{Z^{-1}}{pk} \Phi(Z^{-2})
\right) \ .$$
Noting that for large $x$ the contribution to this integral comes from the neighbourhood of the singularity, one finds $$I_b(x)\approx
\mum^x x^{(1-D)/2}
\dfrac{\sqrt{\pi}}{m_2} \left(\dfrac{im_2/p}{\mum}\right)^{(D-1)/2}
\ ,$$ so that $$\label{cond11neu}
\C_{-}=\dfrac{\sqrt{\pi}}{m_2} \left(\dfrac{im_2/p}{\mum}\right)^{(D-1)/2} \ ,
\quad
\C_{+}=0 \qquad (p^2 < -m_2^2,\ m_1 \not=0, \ m_2\not=0) \ .$$
### Case 2: $m_1\not= 0$, $m_2=0$ {#m2zero}
In this case we have $\mup=\mum=1/(p^2+m_1^2)$. The second-order difference equation simplifies to the first-order equation $$\label{diffequ10}
(D-x-1)I(x-1) +(x-1)(p^2+m_1^2)I(x) -(x-1)J(x) =0 \ , $$ whose solution is of the kind $$I(x)=\C\IOMOG(x)+\INOMOG(x)\ .$$ In a manner analogous to the previous case, one obtains the expansion in factorial series $$\IOMOG(x)=(p^2+m_1^2)^{-x} \sum_{s=0}^\oo a_s \RHO^{2-D-s}\ ;$$ the large-$s$ behaviour of the coefficients is $a_s/s!\propto s^{2D-5}$, so that the series has abscissa of convergence $\lambda=D-2$. Considering the nonhomogeneous solution, $$\INOMOG(x)=(m_1^2)^{-x} \sum_{s=0}^\oo \hat a_s
\RHO^{-D/2-s}\ ,$$ the coefficients have the large-$s$ behaviour $$\hat a_s/s!\approx C_1 s^{(3D-6)/2} +C_2 s^{D-2} +B_0^s s^{-(D-2)/2}\ ,$$ where $B_0=1+m_1^2/p^2$ and $C_i$ are constants. If $p^2>-m_1^2/2$, $|B_0|>1$ and the series does not converge; if $p^2\le -m_1^2/2$ the series converges with abscissa of convergence $\lambda=\max (D-2,D/2-1)$, if $C_1\not= 0 $ and $C_2\not =0$.
Now we determine the constant $\C$. The large-$x$ behaviour of the solutions is given by the first term of the factorial series $$\begin{aligned}
\label{lrn2}
\IOMOG(x) &\approx (p^2+m_1^2)^{-x} x^{2-D}\ , \\
\label{lrn2nomog}
\INOMOG(x) &\approx
\begin{cases}
p^{-2}(m_1^2)^{D/2-x} x^{-D/2} + ?
&
\text{$p^2 > 0$}\ ,\\
p^{-2}(m_1^2)^{D/2-x} x^{-D/2}
&
\text{$p^2 < 0$}\ .
\end{cases}\end{aligned}$$ In the whole region $p^2>0$ the factorial series expansion of $\INOMOG$ never converges so that the large-$x$ behaviour shown above is valid only in asymptotic sense. Using the integral representations of section \[2den\] one can show that if $p^2>0$ the large-$x$ behaviour of $\INOMOG$ contains an additional contribution, exponentially small in comparison with the main contribution $$\label{I0nbis}
\INOMOG(x)\approx p^{-2} (m_1^2)^{D/2-x} x^{-D/2}
+\Gamma(D/2-1)(-p^2)^{1-D/2}(p^2+m_1^2)^{D-2-x} x^{2-D} .$$
The large-$x$ behaviour of $I(x)$ can be deduced from and , as in the massive case. But if $p^2>0$ there is a difference: the integrand of is discontinuous in the point $k^2=p^2$, in fact $$\label{Zdiscont}
Z = \dfrac{p^2+k^2-|p^2-k^2|}{2pk} =
\begin{cases}
k/p \quad\text{$k<p$}\ ,\\
p/k \quad\text{$k\ge p$}\ .
\end{cases}$$ The presence of this discontinuity gives rise to an exponentially small additional contribution proportional to $(p^2+m_1^2)^{-x}$ in the large-$x$ behaviour of $I(x)$. So we split the integral over $k^2$ into two parts, $$\begin{gathered}
I(x)=
\dfrac{1}{\Gamma(D/2)}
\int^\oo_0
\dfrac{dk^2 (k^2)^{D/2-1}}{(k^2+m_1^2)^x}
\dfrac{1}{p^2}\Phi\left(\dfrac{k^2}{p^2}\right) \\
+
\dfrac{1}{\Gamma(D/2)}
\int^\oo_{p^2}
\dfrac{dk^2 (k^2)^{D/2-1}}{(k^2+m_1^2)^x}
\left(
\dfrac{1}{k^2}\Phi\left(\dfrac{p^2}{k^2}\right)
-\dfrac{1}{p^2}\Phi\left(\dfrac{k^2}{p^2}\right)
\right)\ ,\end{gathered}$$ each one having a different large-$x$ behaviour. One finds that the large-$x$ behaviour of $I(x)$ is identical to that of $\INOMOG(x)$ for $p^2>0$, , so that $$\label{cond11eu1}
\C=0 \qquad (p^2 > 0,\ m_1 \not=0, \ m_2=0) \ .$$
If $p^2<0$ the large-$x$ behaviour of $I(x)$ can obtained from (with $m_2=0$). One finds the same result found for $p^2>0$. Therefore for $p^2<0$ the exponentially small contribution is present in the large-$x$ behaviour of $I(x)$ but not in that of $\INOMOG(x)$: it must come from $\C\IOMOG(x)$. Therefore the constant $\C$ is $$\label{cond11neu2}
\C=\Gamma(D/2-1)\left((p^2+m_1^2)/(-i\;p)\right)^{D-2}
\qquad (p^2 < 0,\ m_1 \not=0, \ m_2=0) \ .$$
### Case 3: $m_1=0$, $m_2\not=0$ {#m1zero}
In this case the denominator raised to $x$ has zero mass, and the term containing $J$ in disappears, so that the difference equation becomes homogeneous; the solution is of the kind $$\label{solgeni3}
I(x)=\C_{+}\IOMOG_{+}(x) + \C_{-}\IOMOG_{-}(x) \ .$$ $\IOMOG_{\pm}(x)$ are the same functions considered in section \[m1m2nz\] with $m_1=0$, and their large-$x$ behaviour was given in . But the large-$x$ behaviour of $I(x)$ cannot be found as in the massive case; in fact if $m_1=0$ the integrals and are strongly divergent in $k^2=0$ for large $x$ and only dimensional regularization can give a finite value to $I(x)$. The large-$x$ behaviour of $I(x)$ must be determined by other methods, for example by writing $I(x)$ as integral over one Feynman parameter and using the saddle-point method; one finds $$\label{cond11neu3}
\C_{\pm}=\dfrac{\sqrt{\pi}}{2m_2\sin(\pi D/2)}
\left(\mp ip\mu_{\pm}/m_2\right)^{(1-D)/2}
\qquad ( m_1 =0, \ m_2\not=0) \ . \quad$$
### Numerical example {#numexa}
Let us describe in some detail the calculation of $I(1)$, $J(1)$ and $K(1)$ in the case $-p^2=m_1^2=m_2^2=1$. As $m_1=m_2$ then $K(x)=J(x)$, and we consider only $J(x)$ and $I(x)$. The root of characteristic equation of the difference equation for $J(x)$ () is $\mu=1$; the roots of the characteristic equation of the homogeneous part of the equation for $I(x)$ () are $\mum=1$ and $\mup=-1/3$. $I(x)$ is written as sum of the homogeneous and nonhomogeneous part, ; from one finds $\C_{+}=0$, so that $\IOMOG_{+}$ does not contribute. Therefore we have to evaluate three functions: $J(x)$, $\IOMOG_{-}(x)$ and $\INOMOG(x)$ for $x=1$ and $D\to 4$. The factorial series expansions of all the three functions have abscissa of convergence $\lambda \to 2$ for $D \to 4$ and do not converge for $x=1$. Following the procedure of section \[sumfact\] we evaluate the series for a large value $x=x_{max}$ and, as the equation for $I(x)$ is of second order, even for $x=x_{max}+1$. The convergence becomes faster by increasing $x_{max}$, but, as $A=|\mum/\mup|=3>1$, the recurrence relation for $I(x)$ , rewritten in order to obtain $I(x-2)$ from $I(x-1)$ and $I(x)$, is unstable. Each application of the recurrence relation increases the error of the value of $I(x)$ of about a factor $3$. Therefore we must choose a value of $x_{max}$ which is a compromise between speed of computation and loss of precision in the result. We choose $x_{max}=8$. Calculations are performed by setting $D=4-2\e$ and expanding in $\e$, truncating the expansions at the first three terms. Doing the calculations with 19 digits, the convergence is attained for all the factorial series for $x=8$ (and consequently for $x=9$) in about 4000 terms. The values for $x<8$ are calculated by using repeatedly the recurrence relations and . The application of the unstable recurrence relation enlarges the error of about a factor $3^7\approx 2000$, corresponding to a loss of about 3 significant digits in $I(1)$. Values of $J(x)$, $\IOMOG_{-}(x)$ and $\INOMOG(x)$ are shown in Table \[tableij\], with only the first two terms of the expansion in $\e$, and the coefficients with only 6 digits to save space. The value of $J(1)$ calculated agrees with the exact result $-\e^{-1} +\gamma-1+O(\e)$ within the precision of the calculation ($\gamma$ is the Euler’s constant). Inserting $\IOMOG_{-}(1)$, $\INOMOG(1)$ and the value $\C_{-}=\sqrt{\pi}/2$ (from ) in one finds $$\label{vali1}
I(1)= (1-2\times 10^{-17})\e^{-1} -0.391 015 029 135 750 3 +O(\e)\ ,$$ with an error of $4 \times10^{-16}$ on the constant term in comparison with the exact result $\e^{-1} +2-\pi/\sqrt{3}-\gamma+O(\e)$.
A numerical value of $\C_{-}$ can be obtained independently, using the identity $$I(0)=\C_{-}\IOMOG_{-}(0)+\INOMOG(0)\ ;$$ inserting the values of $I(0)=K(1)=J(1)$, $\IOMOG_{-}(0)$ and $\INOMOG(0)$ one finds $$\label{L1num}
\C_{-}= 0.886 226 925 452 757 8 +5\times10^{-15} \e +O(\e^2)\ ,$$ with an error of $2 \times10^{-16}$ on the constant term. Using this numerical value in the place of the analytical value one obtains a value of $I(1)$ with the same precision as .
$x$ $J(x)$ $\IOMOG_{-}(x)$ $\INOMOG(x)$
----- ------------------------- ----------------------------------- ------------------------------
9 $0.017857 +0.033442 \e$ $ 0.047713 +0.089160 \e $ $ -0.008928 -0.007323 \e$
8 $0.023809 +0.040621 \e$ $ 0.059006 +0.100337 \e $ $ -0.011904 -0.007663 \e$
7 $0.033333 +0.050203 \e$ $ 0.075609 +0.113249 \e $ $ -0.016666 -0.007159 \e$
6 $0.05 +0.062805 \e$ $ 0.101857 +0.126598 \e $ $ -0.025 -0.003929 \e$
5 $0.083333 +0.076898 \e$ $ 0.148085 +0.133074 \e $ $ -0.041666 +0.009010 \e$
4 $0.166666 +0.070464 \e$ $ 0.245635 +0.090010 \e $ $ -0.083333 +0.067207 \e$
3 $0.5 -0.288607 \e$ $ 0.548843 -0.442122 \e $ $ -0.25 +0.534955 \e$
2 $\e^{-1} -0.577215\fe$ $ 0.282094\e^{-1} +0.519388 \fe$ $ -0.25\e^{-1}+0.144303 \fe$
1 $-\e^{-1} -0.422784\fe$ $ 0.282094\e^{-1} -1.645293 \fe$ $ 0.75\e^{-1}+1.067088 \fe$
0 $0\phantom{.123456\fe}$ $ -0.564189\e^{-1} -0.238530 \fe$ $ -0.5\e^{-1} -0.211392 \fe$
: Values of $J(x)$, $\IOMOG_{-}(x)$ and $\INOMOG(x)$.[]{data-label="tableij"}
$x_{max}$ terms finite part of $I(1)$
----------- ---------- -----------------------
$30$ $ 125$ $-0.3910008887063124$
$25$ $ 154$ $-0.3910149952724784$
$20$ $ 217$ $-0.3910150292106927$
$15$ $ 395$ $-0.3910150291388126$
$10$ $ 1470$ $-0.3910150291357554$
$ 9$ $ 2454$ $-0.3910150291357472$
$ 8$ $ 4439$ $-0.3910150291357503$
$ 7$ $ 13086$ $-0.3910150291357507$
$ 6$ $ 36210$ $-0.3910150291357507$
: Dependence of the finite part of $I(1)$ on $x_{max}$.[]{data-label="tablei1"}
In Table \[tablei1\] we show for different choices of $x_{max}$ the number of terms of series necessary to evaluate $J(8)$, $\IOMOG_{-}(8)$ and $\INOMOG(8)$ with 19 digits of precision and the obtained values of the finite part of $I(1)$; it is evident that by increasing $x_{max}$ the series converges faster, but the precision degrades because of the increasing number of applications of the unstable recurrence relation.
Solutions of difference equations by means of Laplace’s transformation {#Laplacesec}
======================================================================
The expansion in factorial series is certainly the most direct method of solution of difference equations with polynomial coefficients; however, for some values of masses and external momenta of the diagram, as we have seen in the above example, the abscissa of convergence of factorial series may become infinite. In this case the factorial series become divergent for every value of $x$ and therefore useless, so that another method of solution must be used: the Laplace’s transformation method[@Milne15].
This method is described in section \[laplacedisc\], and applied to the systems of difference equations in section \[trasys\], \[consyslaplace\] and \[inicondf\]. Techniques used for integrating the differential equations obtained from the application of the method are described in section \[solvedifequ\]. The application to simple one-loop integrals is shown in section \[example1looplapla\].
Transformation of a difference equation {#laplacedisc}
---------------------------------------
Let us consider the difference equation $$\label{diffequl}
p_0(x) U(x) + p_1(x) U(x+1) + \ldots + p_N(x) U(x+N) = 0\ ,$$ where $p_i(x)$ are polynomials in $x$ of maximum degree $P$. The Laplace’s transformation method consists in the substitution $$\label{subLaplace}
U(x)=\int_l dt\; t^{x-1} v(t) \ ,$$ where $l$ is a line of integration suitably determined and where $v(t)$ is found from a certain differential equation. Writing the coefficients as $$p_k(x)= A_{k0} +\sum_{i=1}^P A_{ki} \prod_{j=0}^{i-1} (x+k+j) \ ,$$ substituting in and integrating by parts one finds $$\sum_{k=0}^N p_k(x) U(x+k) =
\int_l dt \; t^{x-1} \sum_{i=0}^P \Phi_i(t) (-t)^{i} v^{(i)}(t)
+[I(x,t)]_l \ ,$$ where $$\Phi_i(t)=\sum_{k=0}^N A_{ki}t^k\ ,$$ $$I(x,t)=\sum_{i=0}^{P-1} (-1)^i v^{(i)}(t)
\sum_{m=0}^{P-1-i}
\left(\dfrac{d}{dt}\right)^m \left( \Phi_{m+i+1}(t) t^{x+m+i} \right)\ .$$
provides a solution of the difference equation if $v(t)$ is a solution of the differential equation $$\label{diffequv}
\sum_{i=0}^P \Phi_i(t) (-t)^{i} v^{(i)}(t) =0 \ ,$$ provided that the line of integration $l$ be chosen so that $I(x,t)$ has the same value at each endpoint of the line, if the line is open. Note that the difference equation has order $N$ with coefficients of degree $P$, while the differential equation has order $P$ with coefficients of degree $N+P$.
The singular points of this differential equation are $0$, $\oo$ and the zeros $t_i$ (of multiplicity $m_i$) of the *characteristic equation* $$\label{char1v}
\Phi_P(t)=0\ ;$$ these points turn to be always regular singular points in the case of the differential equations encountered in this work.
We choose as lines of integration the lines which begin in the origin and end in one of the singular points $t_i$. This is a convenient choice. One can show that $I(x,t)=0$ at each endpoint of such lines, if the integral over $t$ of is finite. Under these conditions, $U(x)$ is a solution of the difference equation.
We can construct a set of $\sum_i m_i=N$ functions $U_{ij}(x)$ which form a fundamental system of solutions of the difference equation by defining $$U_{ij}(x)=\int_{l_i} dt \; t^{x-1} v_{ij}(t) \qquad j=1,\ldots,m_i \ ,$$ where $v_{ij}(t)$ is one of the $m_i$ solutions of the differential equation singular in $t_i$, and $l_i$ is the line which begins in $t=0$ and ends in $t=t_i$.
It is important to note that the characteristic equation of the differential equation and the characteristic equation of the difference equation turn out to be identical, so that we can readily identify the singular points $t_i$ of the differential equation with the solutions $\mu_i$ of .
Now we consider the nonhomogeneous equation $$\label{diffequl2}
\sum_{k=0}^N p_k(x) U(x+k) = \sum_{k=0}^{N'} q_k(x) T(x+k) \ ,$$ where $q_k(x)$ are polynomials and $T(x)$ is a solution of some difference equation. A particular solution $\UNOMOG(x)$ is found by substituting into the equation $$\label{subLaplacew}
T(x)=\int_{l_T} dt \; t^{x-1} w(t) \ , \qquad
\UNOMOG(x)=\int_{l_T} dt \; t^{x-1} v_{NH}(t) \ ,$$ where $l_T$ is a known line of integration and $w(t)$ is a known function, solution of a differential equation analogous to , obtained from the difference equation satisfied by $T(x)$. Provided that $I(x,t)=0$ at the endpoints of $l_T$, one finds the nonhomogeneous differential equation $$\label{diffequv2}
\sum_{i=0}^P \Phi_i(t) (-t)^{i} v^{(i)}_{NH}(t) =
\sum_{i=0}^{P'} \Psi_i(t) (-t)^{i} w^{(i)}(t) \ ,$$ whose solution gives $v_{NH}$. Then $\UNOMOG$ is found using .
Transformation of the system of difference equations {#trasys}
----------------------------------------------------
In section \[constsysdif\] we described the construction of the system of difference equations. The algorithm used yields a system in triangular form; applying to this system the Laplace’s transformation one obtains a system of differential equations with the same triangular structure and the same ease of solution. There is, however, a complication: the $l$th difference equation $$\label{diffequl2a}
\sum_{k=0}^N p_k(x) U_l(x+k) = \sum_{j=0}^{l-1}\sum_{k=0}^{N'_j} q_{jk}(x)
U_j(x+k) $$ of order $N$ and with polynomial coefficients of degree $P$ is transformed, using the Laplace’s transformation, into a differential equation of order $P$ $$\label{diffequv2a}
\sum_{i=0}^P \Phi_i(t) (-t)^{i} v_l^{(i)}(t) =
\sum_{j=0}^{l-1} \sum_{i=0}^{P'_j} \Psi_{ij}(t) (-t)^{i} v_j^{(i)}(t) $$ and coefficients of degree $N+P$. As effect of the particular algorithm of construction and solution of the system of identities, $N$ is usually small (typically $1\sim 4$) while $P$ may be large ($3\sim 30$). Therefore the differential equation may have a high order. Calculations in some test cases have shown that solution of high order equations slows down the calculations and is source of undesired numerical errors, so that, if possible, it is better to avoid it.
We have discovered that this difficulty can be overcome by applying the Laplace’s transformation to the identities obtained by integration-by-parts *before* the insertion in the system of identities, building a system of “transformed” identities between the “transformed” integrals $v(t)$ instead of the real integrals $U(x)$. The solution of the system of transformed identities will provide a system of differential equations of smaller order; a small price to pay is the (possible) appearance of spurious singular points in the equations so obtained (see section \[mobile\]). More in detail, a generic integral is transformed into $$U_{ni\alpha\beta}(x)=\int
\dk1 \ldots \dk{\ND}
\dfrac
{\prod_{j=1}^{\NPS-n} {\indpk}_j^{\beta_{j}} }
{D_{i_1}^{x+\alpha_1} D_{i_2}^{\alpha_2} \cdots D_{i_n}^{\alpha_n}}
= \int_l dt \; t^{x-1} v_{ni\alpha\beta}(t)\ ,$$ where the line $l$ is unspecified; the values of the functions $I(x,t)$ at the endpoints of the line are always zero because of dimensional regularization. Therefore, a generic integration-by-parts identity, $$\sum_{ni\alpha \beta} (x \; r_{ni\alpha \beta} +s_{ni\alpha \beta})
U_{ni\alpha\beta}(x)=0 \ ,$$ ($r_{ni\alpha \beta}$ and $s_{ni\alpha \beta}$ are independent of $x$) becomes the transformed identity $$\label{idet}
\sum_{ni\alpha \beta} \left( (s_{ni\alpha\beta}
-r_{ni\alpha\beta}\alpha_1)v_{ni\alpha\beta}(t)
-t\;r_{ni\alpha\beta} \dfrac{dv_{ni\alpha\beta}}{dt}(t) \right)
t^{\alpha_1} =0 $$ which is a differential equation between the functions $v_{ni\alpha\beta}(t)$.
Construction of the system of differential equations {#consyslaplace}
----------------------------------------------------
We want to build a triangular system of differential equations $$\label{triform}
\sum_{i=0}^{P_l} p_{il}(t)
v^{(i)}_{ml}(t)
=
\sum_{k=1}^{l-1}\sum_{j=0}^{P_{lk}} q_{jkl}(t)
v^{(j)}_{mk}(t)\ , \quad l=1,\ldots,L_m'\ ,$$ between a set of “master transformed” functions $v_{ml}(t)$ ($m=1,\ldots,\ND-\NK+1$) analogous to the system of difference equations between the master functions $U_{ml}(x)$ discussed in section \[constsysdif\]. For this reason we devise an algorithm of construction and solution of the system of identities similar to the algorithm \[algsys4\]:
\[algsys6\] Consider the algorithm \[algsys4\] with the following modifications:
1. After the step \[quiins1\] of algorithm \[algsys\], transform each integration-by-parts identity using ; in the subsequent steps replace everywhere the integrals $\int \dk1\ldots\dk\NK W_{ni\alpha\beta}$ with the functions $v_{ni\alpha\beta}(t)$.
2. The transformed identity obtained does not depend on the index $\alpha_1$ of $W$ in the step \[quissset\] of algorithm \[algsys\] because it appears as an overall factor $t^{\alpha_1}$; in order to restore the total number of different identities we have decided (somewhat arbitrarily) to differentiate $\alpha_1$ times with respect to $t$ each transformed identity.
3. Ignore step \[cond04\] of algorithm \[algsys4\].
4. Derivatives of master transformed functions have priority of extraction lower than other generic transformed integrals.
5. Add the new entry *the greatest derivative* to the list of priorities after the entry \[cf6b\] of algorithm \[algsys\].
With a suitable choice of the parameters $a_i$ and $b_i$ (see the end of section \[constsysdif\]), by means of this algorithm we can identify the master transformed functions $v_{ml}(t)$ as the functions which satisfy equations of non-zero order, and we can work out a set of differential equations among them. If each function $v_{ml}(t)$ corresponds to an integral with a different combination of denominators, the system is obtained directly in the triangular form ; if, on the contrary, there are different master transformed functions $v_{m,l+1}(t)$,…,$v_{m,l+G}(t)$ corresponding to integrals containing the same combination of denominators, the algorithm provides a set of $G$ simultaneous differential equations containing all these $G$ functions, which are conveniently transformed into triangular form using a procedure quite analogous to that described in section \[constsysdif\]. As final result one obtains a system of differential equations with the triangular form . It is important to note that the functions $$F_{ml}(x)=\int_l dt \; t^{x-1} v_{ml}(t)$$ are not necessarily identical to the master functions $U_{ml}(x)$ defined in . While $v_{ml}(t)$ is a master transformed function and satisfies a differential equation of non-zero order, $F_{ml}(x)$ satisfies a difference equation of different order, which may be zero; if so, $F_{ml}(x)$ is not a *master* function. This fact frequently occurs when many master integrals have the same combination of denominators. For these reasons the number and the structure of the master transformed functions $v_{ml}(t)$ must be found independently of the master functions $U_{ml}(x)$. See section \[pair2\] for an example.
Correspondence with factorial series and initial conditions {#inicondf}
-----------------------------------------------------------
Let $U(x)=\sum_\alpha U_\alpha(x)$ be the solution of a generic difference equation; the initial conditions for the integration of the differential equation can be determined by comparing the integral representation of $U_\alpha(x)$ with the factorial series expansion $$\label{factserua}
U_\alpha (x) = \int_0^{\mu_\alpha} dt \; t^{x-1} v_\alpha(t)
= \mu_\alpha^x
\sum_{s=0}^\oo a_{s\alpha} \RHO^{\K^F_\alpha-s} \ .$$ The behaviour of $v_\alpha(t)$ near the singular point $t=\mu_\alpha$ $$\label{va0k}
v_\alpha(t)\approx A_{0\alpha} (\mu_\alpha-t)^{\K^L_\alpha}, \qquad t\approx
\mu_\alpha\ ,$$ can be deduced from the known behaviour of $U_\alpha(x)$ for large $x$. Substituting in , integrating over $t$ and comparing the result with the large-$x$ leading behaviour of the first term of the series $U_\alpha(x) \approx \mu_\alpha^x a_{0\alpha} x^{\K^F_\alpha}$ one finds $K^L_\alpha$ and $A_{0\alpha}$: $$\label{aA0}
\K^L_\alpha=-\K^F_\alpha-1 \ , \qquad
A_{0\alpha}=
a_{0\alpha}/(\mu_\alpha^{\K^L_\alpha} \Gamma(\K^L_\alpha+1))\ .$$ If necessary, the subsequent coefficients of the series expansion of $v_\alpha(t)$ can be deduced in the same way from the coefficients $a_{s\alpha}$.
Integrating differential equations {#solvedifequ}
----------------------------------
The choice of a effective numerical method of integration of the differential equations obtained by applying Laplace’s transformation is not simple. In fact we must consider that:
- The initial and final point of the path of integration $l$ are singular points of the solutions; the numerical methods usually used to integrate differential equations (for example, the Runge-Kutta method) cannot be used in singular points.
- Master functions, coefficients of differential equations and (sometimes) singular points depend on $D$ and are represented by truncated series in $\e$.
- The method must be able to provide very high-precision values, and the time of computation must grow linearly with the number of exact digits of the result; by using fixed-order methods (like the Runge-Kutta method) the time grows exponentially.
Therefore we have decided to solve the differential equations by using power series, expanding the general solution around a number of selected points near or on the path $l$ and equating the sums of the series in some intermediate points. Each power series will be evaluated inside the respective circle of convergence, so that the number of terms of the series necessary to attain a precision of $E$ digits in the results (and also the computation time) will be proportional to $E$.
### Integration over the path
The line of integration can have any shape, but for convenience it is assumed here to be the segment $[0,\mu]$. The segment is subdivided into $M$ intervals $[t_{M+1},t_{M}]$, $[t_{M},t_{M-1}]$, …, $[t_{2},t_{1}]$, where $t_1=\mu$ and $t_{M+1}=0$; the (possible) singular points placed on the segment are $0= t^{\sng}_P<\ldots<t^{\sng}_2<t^{\sng}_1<\mu$. The choice of a line of integration which passes through singular points of the differential equation, instead of avoiding them, may be convenient: it allows one to check if $v(t)$ is regular or singular in these points and it speeds up the calculations, avoiding the use of complex numbers. Let us consider the first interval $[t_2,t_1]$. The solution is expanded around the point $t_1=\mu$ (which may be a singular point). The first coefficients of the expansions of the $P$ solutions corresponding to the roots of the indicial equation are obtained using . All the subsequent coefficients are obtained using recurrence relations obtained by substituting the expansions into the differential equation. Then the power series are evaluated in the point $t_2=t_1-r_1/2$, where $r_1$ is the radius of convergence of the series and the factor $1/2$ has been chosen in order to minimize the total time of calculations. In the next interval the solution is expanded around the regular point $t_2$, the first $P$ coefficients of the expansion are obtained from the already known values of $v$ and its derivatives in $t_2$, the subsequent coefficients are obtained using the recurrence relations, and the power series is evaluated in a point $t_3=t_2-r_2/2$. This process involving expansions around regular points continues for $m$ steps until a point $t_m$ is reached, placed inside the circle of convergence of the series with center the singular point $t^{\sng}_1$, at a distance less than one half of the radius $r_1^{\sng}$ of the circle. Now we expand the solution around the singular point $t^{\sng}_1$. We write the general solution as $v(t)=\sum_{j=1}^P c_j v_j(t)$, where each $v_j$ corresponds to one of the $P$ solutions of the indicial equation. The values of $c_j$ are found by solving the linear system[^14] $\sum_{j=1}^P c_j v^{(i)}_j(t_m)=v^{(i)}(t_m)$, $i=1,\ldots,P$. The right-hand sides contain the already known values of $v$ and its derivatives in $t_m$; the necessary values of partial solutions $v_j(t_m)$ are worked out by summing the corresponding expansions about $t^{\sng}_1$, whose coefficients are found using recurrence relations. Then the singular point is got over by evaluating $v$ (and its derivatives) in the regular point $t_{m+1}=t^{\sng}_1 -r_1^{\sng}/2$. The process is repeated until the next singular point $t^{\sng}_2$ is reached, then $t^{\sng}_3$, $t^{\sng}_4$, etc., up to the final point $t^{\sng}_P=t_{M+1}=0$.
The integration of $v(t)$ over $t$ needed to obtain $U(x)$ can be easily carried out by integrating the expansions in series in the corresponding intervals: $$\label{usum}
U(x)=\int_0^\mu dt \; t^{x-1} v(t) = \sum_{i=1}^{M} \int_{t_i}^{t_{i+1}}
dt \; t^{x-1}
\sum_{j=1}^P \sum_{s=0}^\oo a_s^{(i,j)} (t-\bar t_i)^{\K_{ij}+s} \ .$$ The integrals $$I(j,s)=
\int_{\bar t+a}^{\bar t+b} dt \; t^j (t-\bar t)^{k+s}
=
\int_a^b dy \; (y+\bar t)^j y^{k+s}$$ which appear in can be expressed in terms of incomplete Beta function (note that $k$ is not an integer). If $j=0$ the integral is immediate; if $j$ is positive integer the value can be efficiently computed using the recurrence relation $$I(j,s)=I(j-1,s+1)+\bar t I(j-1,s) \ .$$
### Singular points depending on $D$ {#mobile}
The coefficients $a^{(i,j)}_s$ and the exponents $\K_{ij}$ in depend on $D$; therefore all quantities are expanded around $D=4$, and truncated series are used in the calculation, as described in section \[truncd\]. A new feature, characteristic of the differential equations obtained by solving the system of transformed identities, is the appearance of spurious apparent singular points $\bar t$, not corresponding to any solution $\mu$ of the characteristic equations of the difference equations obtained by solving the system of ‘original’ identities. These apparent singular points are depending on $D$, and correspond to regular points of the solution of differential equation, in contrast with regular singular points, which are independent of $D$. In general the line of integration can be deformed in order to avoid these spurious singular points; but if for $D\to 4$ one or more of these *mobile* points tends to one of the endpoints of the line, $t=\mu$ or $t=0$, we cannot avoid it, and we encounter difficulty in working out the solution near these coalescing points. Let us explain this fact, by considering a homogeneous differential equation with polynomial coefficients $$\label{equhm0}
\sum_{i=0}^P p_i(t) v^{(i)}(t)=0\ ,
\qquad
p_i(t)= t^i \sum_{j=0}^{g_i} p_{ij} t^{j}\ ,$$ and supposing for simplicity that the equation has only one apparent singular point $t_0(D)$ such that $$\label{critp}
t_0(D)=O(D-4) \quad \text{for}\quad D\to 4 \ .$$ The coefficients of the expansion $v(t)=\sum_{s=0}^\oo a_s t^{s+\K}$ can be found using the recurrence relation $$\label{recas1}
a_s=-\dfrac{\sum_{j=1}^m a_{s-j} f_j(\K+s-j) }{f_0(\K+s)}\ ,$$ where $a_i \equiv 0$ if $i<0$, $m=\max_i g_i$, $
f_j(k) =\sum_{i=0}^P k(k-1)\cdots (k-i+1) p_{ij} \ ,
$ and $\K$ is one of the roots of indicial equation $f_0(\K)=0$ (note the analogy with the solution of a difference equation with expansions in factorial series).
As the point $t_0$ is a regular point, the coefficients $a_s$, which are functions of $D$, have values in $D=4$ finite and, in general, different from zero; unfortunately, there are problems for calculating them. As a consequence of , the function $f_0$ vanishes if $D=4$, and the other $f_j$ do not vanish (but the sum in the numerator of always vanishes for $D=4$), so that the recurrence relation turns out to be very unstable, with a degree of instability proportional to $1/(D-4)$. Performing the calculations of $a_s$ using truncated expansions in $D-4$, each iteration of (with increasing $s$) yields one new coefficient $a_s$, whose expansion in $D-4$ has a number of terms reduced by one in comparison with $a_{s-1}$; after a few iterations, the given number of terms of the expansion in $D-4$ is exhausted. We found that a solution of the problem is to modify the recurrence relation to $$\label{recas2}
a_{s-1}^{(n+1)}=-\dfrac{\sum_{j=2}^m a_{s-j}^{(n)} f_j(\K+s-j) + a_s^{(n)}
f_0(\K+s)}{f_1(\K+s-1)}\ ,$$ so that the denominator is $f_1$, which does not vanish for $D=4$. The new recurrence relation requires as input the value of the $s$th coefficient before that its value is obtained; therefore must be seen as part of an iterative process:
1. set $a_s^{(0)}=0$ for $s=1,2,\ldots$, $s_{{max}}$;
2. \[rpoint\] apply for $s=1,2,\ldots$, $s_{{max}}$ obtaining the coefficients $a_s^{(1)}$;
3. repeat $n$ times the step \[rpoint\] until $|a_s^{(n+1)}-a_s^{(n)}|=O((D-4)^m)$ for every $s$, where $m$ is the desired number of term of the expansion in $D-4$; the convergence is guaranteed in about $m$ steps by the fact that $f_0= O(D-4)$.
Analogous modifications must be made to the recurrence relation in the case of nonhomogeneous equations, or in case of two or more spurious singular points coalescing to the same endpoint. An example of equation with one mobile singular point is the third-order equation[^15] $$\begin{gathered}
\label{equmob3}
-2(t-1)(8t+1)(4(D-1)t-D+4))t^3v_{\ref{figself}\text h}'''
+\bigl( -32(D-1)(5D-7)t^3 \\
+8(D-1)(25D-63)t^2
+(-4D^2 +100D -192)t -(D-4)(9D-22) \bigr)t^2v_{\ref{figself}\text h}'' \\
+\bigl( -32(D-1)(D-2)(3D-5)t^3
+32(D-1)(D-3)(5D-12)t^2 \\
+(18D^3 -70D^2 -44D +288)t
+(D-4)(-13D^2 +72D-100) \bigr)tv_{\ref{figself}\text h}' \\
+2(D-4)(D-3)^2\bigl(16(D-1)t^2 +8Dt -3D+10\bigr) v_{\ref{figself}\text h}=0\ , \end{gathered}$$ which is the homogeneous part of the equation satisfied by the function $v_{\ref{figself}\text h}(t)$, corresponding to the integral of Fig. \[figself\]h of section \[resu\], used in the calculation of ( remains the same for both choices of the line $D_1$). The singular points are $t=1$, $-1/8$, $0$ and $(D-4)/(4D-4) $. The last singular point, which satisfies the condition , is a regular point with exponents $0$, $1$ and $3$. The characteristic equation of the homogeneous part of the corresponding difference equation, $$\begin{gathered}
\label{equmob2}
(x-D+1) (x-2 D+4) (2 x-3 D+6) (3 x-4 D+10) U_{\ref{figself}\text h}(x-1) \\
+ 2(x-D+2)\bigl( 21x^3 +( 136 - 67 D )x^2
+ ( 243 + 64 D^2 - 253 D )x \\ -8(D-1)(D-2)(2D-5) \bigr)
U_{\ref{figself}\text h}(x) \\
- 8x (x-D+3) (2 x-3 D+7) (3 x-4 D+7) U_{\ref{figself}\text h}(x+1) =0 \ ,\end{gathered}$$ has only the roots $1$ and $-1/8$.
Applications to simple one-loop integrals {#example1looplapla}
-----------------------------------------
Now we consider the solution with Laplace’s transformation of the difference equations analyzed in the examples of sections \[1loopv\] and \[1loopself\].
### One-loop vacuum integral {#1loopv_dif}
Considering the integral $J(x)$ of , the endpoints of the line of integration are the origin and the root of the characteristic equation $\mu=1/m_1^2$, so that we write $$\label{inint}
J(x)=\int_0^{1/m_1^2} dt \; t^{x-1} v_J(t) \ ,$$ where $v_J$ satisfies the differential equation $$\label{equde1}
-t (m_1^2 t-1) {v'_J}(t) + (D/2-m_1^2 t) v_J(t) =0 \ .$$ The solution is $$\label{v000}
v_J(t)= C (1/m_1^2-t)^{D/2-1} t^{-D/2} \ ;$$ the constant $C$ can be deduced from the value of $a_0$ (see section \[vala0\]), the value of $\K=-D/2$ and the relation . One finds $$C= (m_1^2)^{D/2-1}/\Gamma(D/2) \ .$$
### One-loop self-energy integral {#2den}
Here we consider $I(x)$ of , and in particular the case of non-zero masses. The homogeneous solution can be written using the Laplace’s transformation as $$\IOMOG_{\pm}(x)=\int_0^{\mu_{\pm}} dt \; t^{x-1} \vOMOG(t) \ .$$ The function $\vOMOG(t)$ satisfies the differential equation $$\label{equde2}
-t \Phi_1(t) \vOMOG'(t) + \Phi_0(t) \vOMOG(t) =0\ ,$$ where $$\label{Phi10}
\begin{split}
\Phi_1(t)
&= R^2(p^2,-m_1^2,-m_2^2) (t-\mup)(t-\mum) \ ,\\
\Phi_0(t)&=-t^2 R^2(p^2,-m_1^2,-m_2^2) +(D-1)(p^2+m_1^2-m_2^2)t +2-D \ .\qquad
\end{split}$$ The solution of this equation is $$\vOMOG(t)= C \left((\mup-t)(\mum-t)\right)^{(D-3)/2} t^{2-D} \ .$$ Values of $C$ such that $\IOMOG_{\pm}$ are the same functions considered in section \[m1m2nz\] are $$C_{\pm}= \mu_{\pm}^{(D-1)/2}
\left(\mu_{\mp}-\mu_{\pm}\right)^{(3-D)/2}/\Gamma((D-1)/2) \ .$$ Considering now the nonhomogeneous solution $$\INOMOG(x)=\int_0^{1/m_1^2} dt \; t^{x-1} \vNOMOG(t) \ ,$$ the function $\vNOMOG$ satisfies the differential equation $$\label{equde3}
-t \Phi_1(t) \vNOMOG'(t) + \Phi_0(t) \vNOMOG(t) = t\phi_1(t) v_J'(t) - \phi_0(t)
v_J(t)\ ,$$ where $\Phi_0$ and $\Phi_1$ are given in , $\phi_0$ and $\phi_1$ are $$\begin{split}
\phi_1(t)&= -t(p^2+m_2^2)/m_1^2 \ , \\
\phi_0(t)&= t(D(p^2+m_1^2+m_2^2)/(2m_1^2)-1) \ , \\
\end{split}$$ and $v_J$ is given in .
### Numerical example {#numerical-example}
We consider the calculation using Laplace’s transformation of $J(1)$, $\IOMOG_{-}(1)$ and $\INOMOG(1)$, numerically calculated in section \[numexa\]. The corresponding differential equations are , and . The singular points of the system are $t=1$, $-1/3$ and $0$. The equations are integrated using the method described in section \[solvedifequ\]. The integral over $t$ is divided into 4 intervals, with endpoints $0$, $1/8$, $1/4$, $1/2$, $1$; a cutoff $\lambda \ll 1$ is conveniently introduced in the first interval $[\lambda,1/8]$ because the solutions are not regular in $t=0$. Doing the calculations with 19 digits, the convergence of the expansions in $t$ is attained in about 80 terms; the finite values of $J(x)$, $\IOMOG_{-}(x)$ and $\INOMOG(x)$ for $x=3$ and $x=4$ are obtained by calculating the integrals with ; the unstable recurrence relations are used only to calculate the values for $x\le 2$, where the integrals are divergent, so that the error on $I(1)$ is reduced by two orders of magnitude with respect to .
Application to multi-loop diagrams {#resu0}
==================================
After the self-energy diagram discussed in section \[1loopself\], now we consider more complicated diagrams: the vacuum and the self-energy diagrams up to three loop, shown in Figs. \[figvac\] and \[figself\], and the vertex and box diagrams up to two loops, shown in Figs. \[figvert\] and \[figbox\] (we have considered all diagrams such that the scalar integral is always a master integral which does not factorize in a product of simpler master integrals). A complete discussion of all these diagrams would be rather long and we postpone it to future papers. However, to give an idea of the kind and complexities of the equations involved in the calculations, we will show some results.
First of all, we distinct in each diagram $g$ the topologically different lines which are indicated in the figures with a number; of course in diagrams without numbers all lines are topologically equivalent. For each topologically different line $l$ we set $D_1$ equal to the denominator of such line and we consider the difference equation satisfied by the scalar master function $U_{gl}(x)$ and the differential equation satisfied by the Laplace-transformed function $v_{gl}(t)$, both corresponding to the scalar master integral, $$U_{gl}(x)= \int_l dt \; t^{x-1} v_{gl}(t) =
\int \dfrac {\dk1 \dots \dk{\NK}}
{\displaystyle D_1^x D_2 \ldots D_\ND} \ .$$
(380,120)(0,0) (060,080) (040,080) (115,080) (095,080)[(1,0)[40]{}]{} (190,080) (170,080)(190,100)(209.5,080) (170,080)(190,060)(209.5,080) (265,080) (265,099.2)[(+3,-5)[17]{}]{} (265,099.2)[(-3,-5)[17]{}]{} (340,080) (340,080)[(0,1)[20]{}]{} (340,080)[(+5,-3)[17]{}]{} (340,080)[(-5,-3)[17]{}]{}
(370,480)(0,0)
(050,460) (030,460)[(-1,0)[10]{}]{} (070,460)[(+1,0)[10]{}]{} (140,460) (120,460)[(1,0)[40]{}]{} (120,460)[(-1,0)[10]{}]{} (160,460)[(+1,0)[10]{}]{} (230,460) (210,460)[(-1,0)[10]{}]{} (250,460)[(+1,0)[10]{}]{} (210,460)(228,462)(230,480) (320,460) (320,440)[(0,+1)[40]{}]{} (300,460)[(-1,0)[10]{}]{} (340,460)[(+1,0)[10]{}]{} (050,380) (030,380)(050,400)(070,380) (030,380)(050,360)(070,380) (030,380)[(-1,0)[10]{}]{} (070,380)[(+1,0)[10]{}]{} (140,380) (120,380)[(-1,0)[10]{}]{} (160,380)[(+1,0)[10]{}]{} (120,380)(130,390)(140,400) (120,380)(138,382)(140,400) (230,380) (210,380)[(-1,0)[10]{}]{} (250,380)[(+1,0)[10]{}]{} (210,380)(227,383)(230,400) (210,380)[(1,0)[40]{}]{} (320,380) (300,380)(318,382)(320,400) (340,380)(322,382)(320,400) (300,380)[(-1,0)[10]{}]{} (340,380)[(+1,0)[10]{}]{} (050,300) (030,300)[(-1,0)[10]{}]{} (070,300)[(+1,0)[10]{}]{} (030,300)[(+1,0)[40]{}]{} (050,300)[(0,+1)[20]{}]{} (140,300) (120,300)[(-1,0)[10]{}]{} (160,300)[(+1,0)[10]{}]{} (140,280)(160,300)(140,320) (140,280)(120,300)(140,320) (230,300) (210,300)[(-1,0)[10]{}]{} (250,300)[(+1,0)[10]{}]{} (210,300)(227,303)(230,320) (210,300)(233,304)(240,317) (320,300) (300,300)[(-1,0)[10]{}]{} (340,300)[(+1,0)[10]{}]{} (300,300)(317,303)(320,320) (320,280)[(0,+1)[40]{}]{} (050,220) (030,220)[(-1,0)[10]{}]{} (070,220)[(+1,0)[10]{}]{} (030,220)(047,223)(050,240) (050,240)(052,224)(064,233) (140,220) (120,220)[(-1,0)[10]{}]{} (160,220)[(+1,0)[10]{}]{} (120,220)(137,223)(140,240) (120,220)(137,217)(140,200) (230,220) (210,220)[(-1,0)[10]{}]{} (250,220)[(+1,0)[10]{}]{} (230,200)[(0,+1)[20]{}]{} (230,230) (320,220) (300,220)[(-1,0)[10]{}]{} (340,220)[(+1,0)[10]{}]{} (320,200)[(0,+1)[40]{}]{} (320,240)(322,224)(334,233) (095,140) (075,140)[(-1,0)[10]{}]{} (115,140)[(+1,0)[10]{}]{} (085,157)(088,150)(092,143) (075,140)(100,140)(105,157) (185,140) (165,140)[(-1,0)[10]{}]{} (205,140)[(+1,0)[10]{}]{} (185,120)[(0,+1)[40]{}]{} (165,140)[(+1,0)[20]{}]{} (275,140) (255,140)[(-1,0)[10]{}]{} (295,140)[(+1,0)[10]{}]{} (275,159.2)[(+3,-5)[17]{}]{} (275,159.2)[(-3,-5)[17]{}]{} (095,060) (075,060)[(-1,0)[10]{}]{} (115,060)[(+1,0)[10]{}]{} (085,043)[(0,+1)[34]{}]{} (105,043)[(0,+1)[34]{}]{} (185,060) (165,060)[(-1,0)[10]{}]{} (205,060)[(+1,0)[10]{}]{} (185,060)[(0,-1)[20]{}]{} (185,060)[(+5,+3)[17]{}]{} (185,060)[(-5,+3)[17]{}]{} (275,060) (255,060)[(-1,0)[10]{}]{} (295,060)[(+1,0)[10]{}]{} (261,074)[(+1,-1)[28]{}]{} (261,046)(267,052)(273,058) (289,074)(283,068)(277,062) (273,058)(271,064)(277,062)
(300,200)(0,0) (030,140)[(+1,0)[40]{}]{} (030,140)[(+3,+5)[20]{}]{} (070,140)[(-3,+5)[20]{}]{} (030,140)[(-5,-3)[10]{}]{} (070,140)[(+5,-3)[10]{}]{} (050,173)[(0,+1)[10]{}]{} (130,140)[(+1,0)[40]{}]{} (130,140)[(+3,+5)[20]{}]{} (170,140)[(-3,+5)[20]{}]{} (130,140)[(-5,-3)[10]{}]{} (170,140)[(+5,-3)[10]{}]{} (150,173)[(0,+1)[10]{}]{} (130,140)(150,155)(169,140) (230,140)[(+1,0)[40]{}]{} (230,140)[(+3,+5)[20]{}]{} (270,140)[(-3,+5)[20]{}]{} (230,140)[(-5,-3)[10]{}]{} (270,140)[(+5,-3)[10]{}]{} (250,173)[(0,+1)[10]{}]{} (230,140)(240,152)(250,140) (030,040)[(+1,0)[40]{}]{} (030,040)[(+3,+5)[20]{}]{} (070,040)[(-3,+5)[20]{}]{} (030,040)[(-5,-3)[10]{}]{} (070,040)[(+5,-3)[10]{}]{} (050,073)[(0,+1)[10]{}]{} (050,073)[(0,-1)[33]{}]{} (130,040)[(+1,0)[40]{}]{} (130,040)[(+3,+5)[20]{}]{} (170,040)[(-3,+5)[20]{}]{} (130,040)[(-5,-3)[10]{}]{} (170,040)[(+5,-3)[10]{}]{} (150,073)[(0,+1)[10]{}]{} (140,057)[(+1,0)[20]{}]{} (230,040)[(+3,+5)[20]{}]{} (270,040)[(-3,+5)[20]{}]{} (230,040)[(-1,-1)[10]{}]{} (270,040)[(+1,-1)[10]{}]{} (250,073)[(0,+1)[10]{}]{} (230,040)[(+5,+3)[29]{}]{} (270,040)[(-5,+3)[17]{}]{} (246,054)(243,056)(240.5,057.5) (246,054)(254,058)(252.5,050.5)
(370,200)(0,0) (030,140)[(+1,0)[40]{}]{} (030,140)[(0,+1)[40]{}]{} (070,140)[(0,+1)[40]{}]{} (030,180)[(+1,0)[40]{}]{} (030,140)[(-1,-1)[10]{}]{} (030,180)[(-1,+1)[10]{}]{} (070,140)[(+1,-1)[10]{}]{} (070,180)[(+1,+1)[10]{}]{} (120,140)[(+1,0)[40]{}]{} (120,140)[(0,+1)[40]{}]{} (160,140)[(0,+1)[40]{}]{} (120,180)[(+1,0)[40]{}]{} (120,140)[(-1,-1)[10]{}]{} (120,180)[(-1,+1)[10]{}]{} (160,140)[(+1,-1)[10]{}]{} (160,180)[(+1,+1)[10]{}]{} (120,140)(140,155)(160,140) (210,140)[(+1,0)[40]{}]{} (210,140)[(0,+1)[40]{}]{} (250,140)[(0,+1)[40]{}]{} (210,180)[(+1,0)[40]{}]{} (210,140)[(-1,-1)[10]{}]{} (210,180)[(-1,+1)[10]{}]{} (250,140)[(+1,-1)[10]{}]{} (250,180)[(+1,+1)[10]{}]{} (210,180)[(+1,-1)[40]{}]{} (300,140)[(+1,0)[40]{}]{} (300,140)[(0,+1)[40]{}]{} (340,140)[(0,+1)[40]{}]{} (300,180)[(+1,0)[40]{}]{} (300,140)[(-1,-1)[10]{}]{} (300,180)[(-1,+1)[10]{}]{} (340,140)[(+1,-1)[10]{}]{} (340,180)[(+1,+1)[10]{}]{} (300,180)[(+2,-1)[40]{}]{} (025,040)[(+1,0)[40]{}]{} (025,040)[(0,+1)[40]{}]{} (065,040)[(0,+1)[40]{}]{} (025,080)[(+1,0)[40]{}]{} (025,040)[(-1,-1)[10]{}]{} (025,080)[(-1,+1)[10]{}]{} (065,040)[(+1,-1)[10]{}]{} (065,080)[(+1,+1)[10]{}]{} (025,040)(035,055)(045,040) (095,040)[(+1,0)[40]{}]{} (095,040)[(0,+1)[40]{}]{} (135,040)[(-1,+2)[20]{}]{} (135,080)[(-1,-1)[10]{}]{} (095,040)[(+1,+1)[23]{}]{} (117.5,063)(115,073)(124,070) (095,080)[(+1,0)[40]{}]{} (095,040)[(-1,-1)[10]{}]{} (095,080)[(-1,+1)[10]{}]{} (135,040)[(+1,-1)[10]{}]{} (135,080)[(+1,+1)[10]{}]{} (165,040)[(+1,0)[40]{}]{} (165,040)[(0,+1)[40]{}]{} (205,040)[(0,+1)[40]{}]{} (165,080)[(+1,0)[40]{}]{} (165,040)[(-1,-1)[10]{}]{} (165,080)[(-1,+1)[10]{}]{} (205,040)[(+1,-1)[10]{}]{} (205,080)[(+1,+1)[10]{}]{} (185,080)[(+1,-1)[20]{}]{} (235,040)[(+1,0)[40]{}]{} (235,040)[(0,+1)[40]{}]{} (275,040)[(0,+1)[40]{}]{} (235,080)[(+1,0)[40]{}]{} (235,040)[(-1,-1)[10]{}]{} (235,080)[(-1,+1)[10]{}]{} (275,040)[(+1,-1)[10]{}]{} (275,080)[(+1,+1)[10]{}]{} (255,080)[(0,-1)[40]{}]{} (305,040)[(+1,0)[40]{}]{} (305,040)[(0,+1)[40]{}]{} (345,040)[(-1,+2)[20]{}]{} (345,080)[(-1,-2)[8]{}]{} (325,040)[(+1,+2)[8]{}]{} (332.5,056)(327,064)(336,064) (305,080)[(+1,0)[40]{}]{} (305,040)[(-1,-1)[10]{}]{} (305,080)[(-1,+1)[10]{}]{} (345,040)[(+1,-1)[10]{}]{} (345,080)[(+1,+1)[10]{}]{}
Arbitrary case {#resuarb}
--------------
In the left half of Table \[tablevacself\], considering arbitrary (non exceptional) values of masses and momenta, we list[^16]
- The diagram considered and, if present, the indication of the possible values of the index of the topologically different lines.
- The number $n_b$ of master integrals containing all the $\ND$ denominators, determined with the procedure of section \[identificamaster\], subdivided according to the number of scalar products in the numerator; for example, 1,4,3 means that we have found 8 master integrals altogether, of which 1 with no scalar product, 4 with 1 scalar product and 3 with a product of 2 scalar products.
- In the $R_C$ column we list the values of the order $R$ of the difference equations in $x$ satisfied by $U_{gl}(x)$, for each possible choice of $D_1$ as one of the topologically distinct lines; the index $C$, where present, indicates (assuming values of external momenta below the deformation threshold) the number of constants $\C_j$ which are different from zero because the corresponding partial solutions of the homogeneous equation satisfy the condition .
- The order $S$ of the differential equations in $t$ satisfied by the function $v_{gl}(t)$, for each possible choice of $D_1$.
The orders shown within parenthesis are estimated from the subsystems of simultaneous equations , avoiding the transformation into triangular form; the index $C$ is not shown in these cases. Analyzing the data of the table we observe that
1. The differential equation has order less than the order of the difference equation; this is expected as $v_{gl}(t)$ is an object simpler than $U_{gl}(x)$.
2. The order of the differential equation in $t$ is equal to the number of master integrals, with the curious exception of the diagrams \[figself\]f3 and \[figself\]g1 where $S<n_b$.
3. The choice of the line heavily affects the order and the complexity of the equations, as in the case of the diagram \[figself\]f, where the difference equation may have order 5,15 or 2.
4. \[simil\] Some similarities appear between vertex diagrams, self-energy diagrams and vacuum diagrams with different number of loops:[^17]
1. between a vertex diagram and the self-energy diagrams obtained by connecting two external vertices of the vertex diagram with a line and inserting an external momentum in this new line;
2. between a self-energy diagram and the vacuum diagram obtained by connecting its two external lines.
The total number of master integrals and the number of master integrals subdivided according to the number of scalar products turn out to be the same, as well as the orders of the equations for the lines present in both diagrams. For example, compare the diagrams (\[figvert\]a, \[figself\]d, \[figvac\]e), (\[figself\]a, \[figvac\]b), (\[figself\]b, \[figvac\]c), (\[figself\]c, \[figvac\]d), (\[figvert\]b, \[figself\]j, \[figself\]l), (\[figvert\]c, \[figself\]o, \[figself\]p), (\[figvert\]d, \[figself\]r, \[figself\]s), (\[figvert\]e, \[figself\]t, \[figself\]u) and (\[figvert\]f, \[figself\]v). The similarities presumably exist even between three-loop self-energy diagrams and four-loop vacuum diagrams (some preliminary results seem to confirm it). Perhaps there is a relation with the heuristic “rule of the mapping” described in [@Tkachov] for massless self-energy diagrams.
5. The number of master integrals grows probably exponentially[^18] with the number of loops and external vertices, and may be large. Consider for example the class of $L$-loop “sunset” self-energy diagrams with $L+1$ denominators, shown for one, two and three loops respectively in Fig. \[figself\]a, \[figself\]b and \[figself\]e. The number $n_b(L)$ of master integrals for $L=1$ to $L=5$ is $1,4,11,26,57$, respectively (the last two values come from preliminary analysis). These values seem to follow the law $n_b(L)=2^{L+1}-L-2=\sum_{i=2}^{L+1}\binom{L+1}{i}$, corresponding to the (alternative) choice of all the master integrals with numerator equal to one and with one or two as exponents of the denominators, with at least two exponents one.
The test case {#resu}
-------------
As first test of our approach, we have considered in particular detail the case where all masses are equal, $m_1=\ldots=m_\ND=1$, and all the external lines are on-mass-shell. Denoting the incoming external momenta $-p_1$, $p_1-p_2$,…, $p_{\NP-1}-p_\NP$, $p_\NP$, we choose $p_i^2=-1$ and $(p_i-p_j)^2=-1$ for every $i$ and $j$. In the case of box diagrams this corresponds to setting the Mandelstam variables $s=t=1$ and $u=2$. These values of masses and momenta have been chosen because they introduce symmetries which allow several consistency checks on equations and results. Similarly to the arbitrary case, in the right half of Table \[tablevacself\], for each diagram we list the number $n_b'$ of master integrals with all the denominators, the order $R'$ of the difference equation satisfied by the function $U_{gl}(x)$, the order $S'$ of the differential equation satisfied by the corresponding function $v_{gl}(t)$ and, if present, the number $C'$ of non-zero constants needed to determine the solution.
For these particular values of masses and momenta the equations turn out to be, as expected, simpler than in the arbitrary case. The homogeneous parts of the equations which presented some similarities in the arbitrary case (see the observation \[simil\] of section \[resuarb\]) here turn out to be *identical*; of course the nonhomogeneous parts of these equations are quite different.
The number of constants $\C_j$ to find turns out to be greater than or equal to that of the arbitrary case (except for diagram \[figself\]e); in this connection the test case is more complicated than the arbitrary case. With the exception of the diagram \[figbox\]d6, the calculation of the scalar master integrals requires no more than two constants, easily determined using the identities of section \[valx0\] (which turn out always to provide one useful relation involving the constants) and the large-$x$ leading behaviours. One can prove that the values of external momenta are always below or at the deformation threshold, with the exception of some subdiagrams of diagrams \[figbox\]f and \[figbox\][i]{}; in the cases at deformation threshold the values of the constants $\tilde L_1, \tilde L_2$ and $\tilde L_3$ given in are needed (for example, for the diagrams \[figself\]a, \[figvert\]a and \[figbox\]a).
The characteristic equations of all difference equations have always the solution $\mu=1$. We list the values of the other roots for some diagrams: (Fig. \[figvac\]b: $\mu=-1/3$), (\[figvac\]c: $-1/8$), (\[figvac\]e: $-1/2$), (\[figself\]e: $-1/3$, $-1/15$), (\[figself\]i1: $-1/3$, $-1/2$), (\[figself\]i2: $-1/3$), (\[figself\]i3, \[figself\]j2, \[figself\]l1 and \[figvert\]b2 : $1\pm\sqrt{4/3}$), (\[figself\]v1-2 and \[figvert\]f: $-1/3$, $1/9$), (\[figvert\]a: $-1/2$), (\[figbox\]a: $-3/5$), (\[figbox\]b1-2: $-1/2$, $-1/4\pm i\sqrt{1/8}$), (\[figbox\]b3: $3/4\pm \sqrt{27/32}$), (\[figbox\]f1: $-1/2$, $-1/3$, $-1$, $(1\pm\sqrt{3})/2$, $-0.098\pm0.050i$, $-0.934$, $3.632$), (\[figbox\]f2: $-1/2$, $-1/3$, $-1$, $(1\pm\sqrt{3})/2$, $0.049\pm0.099i$, $-2.607$, $-0.819$, $1.756$, $12.07$), (\[figbox\]i1: $1/9$, $3/2$, $-1/2$, $-1/3$, $-1$) and (\[figbox\]i2: $1/9$, $3/2$, $-1/3$, $-3$). Most of these values, written in the form $1/(q^2+1)$, are due to presence of singularities at $k^2=q^2$ in the function $f(k^2)$ (see section \[othercases\]). All characteristic equations turn out to have one negative rational solution $-1<\mu<0$, so that, according to section \[sumfact2\], all recurrence relations are unstable; the diagram \[figself\]e shows the greatest instability ($A=15$). One can ask whether the difference equations of all the analyzed diagrams admit as solutions *convergent* factorial series expansions. The answer is negative. In fact the root $\mu=1/9$ of the diagrams \[figself\]v1-2 and \[figvert\]f, the root $(1+\sqrt{3})/2$ and the complex roots of diagrams \[figbox\]f1-2, the roots $1/9$ and $3/2$ of diagrams \[figbox\]i1-2 (note, all the diagrams with lines crossed) and the root $3/4+\sqrt{27/32}$ of the diagram \[figbox\]b3 satisfy the condition (with $\mu^{(\alpha)}=1$) so that the factorial series expansions of the solutions never converge. Therefore we are forced to use Laplace’s transformation for these diagrams and these choices of the line $D_1$, and for all the diagrams which become these diagrams by deleting lines. In all the other cases factorial series expansions can be used.
Numerical results {#resuvalues}
-----------------
For each diagram shown in Figs. \[figvac\], \[figself\], \[figvert\] and \[figbox\], except for the diagrams \[figbox\]f and \[figbox\]i (see section \[boxdiagrams\]), we have calculated the values of the master integrals for $D=4-2\e$, using the values of masses and momenta shown in the previous section.
Calculations were carried out by using the program described in section \[program\]. The master integrals of the diagrams \[figvac\]b, \[figvac\]e, \[figself\]a, \[figself\]d, \[figself\]i, \[figself\]j, \[figself\]l, \[figvert\]a, \[figvert\]e, \[figbox\]a and \[figbox\]c were first calculated using expansions in factorial series; these integrals were also recalculated using Laplace’s transformation in order to provide important checks of the calculations. The master integrals of all the remaining diagrams were calculated using Laplace’s transformation. Excluding the simplest diagrams, calculations with Laplace’s transformation turned out faster than calculations with factorial series; this is due to the instabilities of the recurrence relations, which become deeper increasing the number of loops, and which force calculations with factorial series to be performed with a larger number of digits. To give an idea of the size of calculations, the systems of difference equations between the master integrals of the diagrams \[figvac\]e, \[figself\]d, \[figself\]t, \[figself\]u, \[figself\]v, \[figvert\]f, \[figbox\]g, and \[figbox\]h are formed with 44, 28, 245, 304, 362, 81, 139 and 158 equations, respectively; note that in each system, in the right-hand side of the equation corresponding to the more complicated master integral almost all the other master functions appear. We made no use of the symmetries due to the particular values of masses and momenta in order to simplify or reduce the number of the equations, as the aim of program is to deal with calculations of multi-scale integrals, lacking in such symmetries; we used them only to check the final results. In order to guarantee results with at least 20 digits of precision, calculations with factorial series were performed with precision up to 77 digits (depending on the degree of instability), while calculations with Laplace’s transformation were all performed with 38 digits of precision.
For example, the calculation from scratch of the integral $I(\ref{figself}\text t)$, , with Laplace’s transformation, requested about 128 hours of CPU time on a 133 MHz Pentium PC; 16 hours were used for the determination of the systems of difference and differential equations, obtained by solving systems up to 43000 identities. The solution of the systems (245 equations) yields, as a byproduct, also the values of all simpler master integrals, including , , and . We stress that at this preliminary stage of development we directed our efforts to devise tests and cross checks rather than to speed up the program.
For brevity, for each diagram we list here only the values of the master scalar integrals $$\label{ival}
I(diagram)=
\int \dfrac {\dk1 \dots \dk{\NK}}
{\displaystyle D_1 D_2 \ldots D_\ND}$$ without scalar products and containing all the $\ND$ denominators. As usual, the results have been normalized with the division by $\Gammae\equiv\Gamma(1+\e)$ raised to the number of loops of the diagram. Coefficients are shown with only 13 digits to save space. Values for $\e=0$ of all finite integrals have been checked by comparing them with numerical values obtained by performing Monte-Carlo integrations over Feynman parameters, or by performing low dimensional Gaussian integrations on integrands obtained using dispersion relations and hyperspherical variables. As additional consistency check we have repeated the calculation of some diagrams with different choices of the line $D_1$, and we have checked that the results obtained are the same.
### Vacuum diagrams
$$\begin{gathered}
I(\ref{figvac}\text a)\Gammae^{-1}= -\e^{-1} -1 -\e -\e^2 -\e^3 -\e^4 +O(\e^5)\;,
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \end{gathered}$$
$$\begin{gathered}
\label{2b}
I(\ref{figvac}\text b)\Gammae^{-2}=
-1.5 \e^{-2} -4.5 \e^{-1} -6.984139141966 -18.00878162355 \e \\
-27.99422356368 \e^2 -72.00378659799 \e^3 -111.9974983355 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
\label{2c}
I(\ref{figvac}\text c)\Gammae^{-3}=
2 \e^{-3} +7.666666666667 \e^{-2} +17.5 \e^{-1} +22.91666666667 \\
+21.25179105129 \e -184.2300051053 \e^2 \\
-661.1105861534 \e^3 -3685.054779382 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figvac}\text d)\Gammae^{-3}=
-\e^{-3} -5.666666666667 \e^{-2} -15.30161161726 \e^{-1} \\
-46.07511172933 -148.3508545129 \e -394.1378145809 \e^2 \\
-1375.669435211 \e^3 -3466.9749998996 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
\label{2e}
I(\ref{figvac}\text e)\Gammae^{-3}=
2.404113806319 \e^{-1} -10.03527847977 +35.94478903214 \e \\
-119.1503507802 \e^2 +379.7433345095 \e^3 -1183.320931551 \e^4
+O(\e^5)\;.\end{gathered}$$
, first six terms of and first two terms of agree with the analytical expressions given in [@Davy2b], [@3-loop] and [@2e], respectively; remaining terms and other results are new.
### Self-energy diagrams
$$\begin{gathered}
\label{res3a}
I(\ref{figself}\text a)\Gammae^{-1}= \e^{-1} +0.186200635766 +0.021156303568\e
+0.001726745353 \e^2 \\
+0.000109897792 \e^3 +0.000005730593\e^4 +O(\e^5)\;, \end{gathered}$$
$$\begin{gathered}
\label{res3b}
I(\ref{figself}\text b)\Gammae^{-2}=
-1.5 \e^{-2} -4.25 \e^{-1} -7.375 -17.22197253479 \e \\
-29.55920705372 \e^2 -68.87789517038 \e^3 -118.2464846454 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
\label{res3c}
I(\ref{figself}\text c)\Gammae^{-2}=
0.5 \e^{-2} +0.6862006357658 \e^{-1} -0.6868398873414 \\ +1.486398391913 \e
-2.938796587745 \e^2 +5.871086365958 \e^3 \\ -11.73616571449 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
\label{res3d}
I(\ref{figself}\text d)\Gammae^{-2}=
0.9236318265199 -1.284921671848 \e +2.689507626490 \e^2 \\
-5.338399227511 \e^3 +10.67136736912 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
\label{res3e}
I(\ref{figself}\text e)\Gammae^{-3}=
2 \e^{-3} +7.333333333333 \e^{-2} +16.02777777778 \e^{-1}\\
+21.92956264368 +3.605127475161 \e -184.1413665431 \e^2 \\
-838.2364324178 \e^3 -3647.102197031 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figself}\text f)\Gammae^{-3}=
-\e^{-3} -2.612634286982 \e^{-2} -3.906420490690 \e^{-1} \\
+0.5840769314959 +1.76460041453 \e +107.0072031435 \e^2 \\
+163.2855293783 \e^3 +1372.241466189 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figself}\text g)\Gammae^{-3}=
-\e^{-3} -5.5 \e^{-2} -15.48413914197 \e^{-1} -45.68793012675 \\
-149.1607636537 \e -392.298867227 \e^2 \\
-1380.125833167 \e^3 -3455.548194007 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
\label{treh}\label{res3h}
I(\ref{figself}\text h)\Gammae^{-3}=
-\e^{-3} -5.333333333333 \e^{-2} -16 \e^{-1} -43.91483126325 \\
-154.918028663 \e -374.0941853334 \e^2 \\
-1436.672712535 \e^3 -3281.940436319 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figself}\text i)\Gammae^{-3}=
2.404113806319 \e^{-1} -9.763424447585 +34.99888165588 \e \\
-116.0420477564 \e^2 +370.0407274069 \e^3 -1153.646312515 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
\label{res3j}
I(\ref{figself}\text j)\Gammae^{-3}=
0.3333333333333 \e^{-3} +0.5195339690991 \e^{-2}\\ +0.5753609494269 \e^{-1}
-2.981838135558 +12.56596204108 \e \\-44.85302351538 \e^2
+149.1742811721 \e^3 -477.1440886129 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figself}\text k)\Gammae^{-3}=
0.1666666666667 \e^{-3} +0.5931003178829 \e^{-2}\\ +0.06234894542402 \e^{-1}
-1.364667486582 +7.062482427894 \e \\-26.87419915573 \e^2
+91.91641284417 \e^3 -298.196943613 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figself}\text l)\Gammae^{-3}=
0.1666666666667 \e^{-3} +0.5931003178829 \e^{-2} \\+0.06234894542402 \e^{-1}
-1.158877567105 +6.268660583427 \e \\ -24.1193749759 \e^2
+82.9872059343 \e^3 -270.1688760103 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figself}\text m)\Gammae^{-3}=
0.3333333333333 \e^{-3} +0.8528673024324 \e^{-2}\\ -1.728169411584 \e^{-1}
+6.070141409747 -19.48651365516 \e \\ +61.38828756627 \e^2
-190.3302695306 \e^3 +583.8045381529 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
\label{res3n}
I(\ref{figself}\text n)\Gammae^{-3}=
0.3333333333333 \e^{-3} +0.8528673024324 \e^{-2} \\ -1.728169411584 \e^{-1}
+6.120359708375 -19.67063042467 \e \\ +62.00178253235 \e^2
-192.2586253184 \e^3 +589.7212544716 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figself}\text o)\Gammae^{-3}=
0.9236318265199 \e^{-1} -2.423491634417 +8.381349710069 \e \\
-26.99362121677 \e^2 +85.10096322999 \e^3 -263.903318629 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
\label{res3p}
I(\ref{figself}\text p)\Gammae^{-3}=
0.9236318265199 \e^{-1} -2.116169718457 +6.929544685259 \e \\
-21.50327837738 \e^2 +66.32213380401 \e^3 -202.887025717 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figself}\text q)\Gammae^{-3}=
1.326448208272 -5.196648136965 \e +18.37758387804 \e^2 \\
-60.41191503661 \e^3 +191.5963941 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figself}\text r)\Gammae^{-3}=
1.341399241447 -5.197752955896 \e +18.38704656407 \e^2 \\
-60.4233521301 \e^3 +191.614009625 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
\label{res3s}
I(\ref{figself}\text s)\Gammae^{-3}=
2.002500041105 -8.162562835907 \e +29.46716463085 \e^2 \\
-98.13080591819 \e^3 +313.871881187 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
\label{res3t}
I(\ref{figself}\text t)\Gammae^{-3}=
0.2796089232826 -0.1380294113932 \e +0.3194688268113 \e^2 \\
-0.4399664109267 \e^3 +0.6650515012166 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figself}\text u)\Gammae^{-3}=
0.1826272375392 -0.06746690965803 \e +0.1865462420623 \e^2 \\
-0.2498713405447 \e^3 +0.3796187113121 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figself}\text v)\Gammae^{-3}=
0.1480133039584 -0.009263002043238 \e +0.1053308537397 \e^2 \\
-0.1224292041846 \e^3 +0.1898480457555 \e^4
+O(\e^5)\;.\end{gathered}$$
, first four terms of , first term of and first six terms of agree with the analytical expressions given in [@3abcd1], [@3abcd3], [@3abcd2] and [@3-loop; @pol], respectively; remaining terms and other results are new.
### Vertex diagrams
$$\begin{gathered}
I(\ref{figvert}\text a)\Gammae^{-1}=
0.671253105748 +0.1998957762816 \e +0.03189366853371 \e^2 \\
+0.003532937320333 \e^3 +0.0003018185047825 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figvert}\text b)\Gammae^{-2}=
0.5 \e^{-2} +0.6862006357658 \e^{-1} -0.5916667014024 \\ +1.356196533114 \e
-2.669112118814 \e^2 +5.336651358516 \e^3 \\ -10.66866283741 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figvert}\text c)\Gammae^{-2}=
0.671253105748 \e^{-1} -0.08774519609257 +0.7262375626947 \e \\
-1.32112948587 \e^2 +2.667431469376 \e^3 -5.332675337091 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figvert}\text d)\Gammae^{-2}=
0.937139527315 -1.27184968708 \e +2.69185047506 \e^2 \\
-5.336932134961 \e^3 +10.67100342934 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figvert}\text e)\Gammae^{-2}=
0.2711563494022 +0.1833941077514 \e +0.05375101058769 \e^2 \\
+0.01446103368419 \e^3 +0.000746187372276 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figvert}\text f)\Gammae^{-2}=
0.173896742268 +0.1816664876962 \e +0.04440899181832 \e^2 \\
+0.02231547385785 \e^3 -0.003079810479797 \e^4
+O(\e^5)\;.\end{gathered}$$
### Box diagrams {#boxdiagrams}
$$\begin{gathered}
I(\ref{figbox}\text a)\Gammae^{-1}=
0.3455029252972 +0.4731008318818 \e +0.1519459537543 \e^2 \\
+0.0275179284554 \e^3 +0.00348492177519 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figbox}\text b)\Gammae^{-2}=
0.671253105748 \e^{-1} -0.06425178040942 +0.7393966927045 \e \\
-1.317187699112 \e^2 +2.668107343399 \e^3 -5.332500882459 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figbox}\text c)\Gammae^{-2}=
0.9509235623171 -1.258189955235 \e +2.694588643167 \e^2 \\
-5.335347124508 \e^3 +10.67067120761 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figbox}\text d)\Gammae^{-2}=
0.276209225359 +0.1937422320842 \e +0.06034849310181 \e^2 \\
+0.01640828588681 \e^3 +0.001301642516765 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figbox}\text e)\Gammae^{-2}=
0.3455029252972 \e^{-1} +0.4347670080988 +0.17885718095363 \e \\
+0.05005382113385 \e^2 +0.006936292250698 \e^3 +0.002511559375421 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figbox}\text g)\Gammae^{-2}=
0.1723367907503 +0.2679578491711 \e +0.13552112755141 \e^2 \\
+0.04468531532833 \e^3 +0.008430602827459 \e^4
+O(\e^5)\;,\end{gathered}$$
$$\begin{gathered}
I(\ref{figbox}\text h)\Gammae^{-2}=
0.1036407209893 +0.2142416932987 \e +0.14046068671363 \e^2 \\
+0.04437197236388 \e^3
+O(\e^4)\;.\end{gathered}$$
In the case of the diagrams \[figbox\]f and \[figbox\]i (from which the diagram \[figbox\]f is derived with the deletion of a line) the equations , obtained after the transformation of the subsystems of equations into triangular form, have orders relatively high ($\ge 10$) and expressions of size greater than the limit of the computer used, so that our program failed to work out them (the coefficients of the equations would be polynomials in two variables of degree $\sim 100$); we were able to find only the characteristic and indicial equations. The calculation of these integrals without the transformation into triangular form will be considered in the next paper.
Some examples of equations {#pair1}
--------------------------
Here we discuss the equations satisfied by the functions $I_{g}(x)= U_{gl}(x)$ for some simple diagrams with all lines topologically equivalent, using the values of masses and momenta of section \[resu\]. For the diagrams \[figself\]a and \[figvac\]b the difference equations are $$\begin{aligned}
\label{diffeqxa}
\bW1 I_{\ref{figself}\text a}(x)&=-\frac{1}{2}(D -2)I_{\ref{figvac}\text a}(x) \ , \\
\label{diffeqxb}
\bW1 I_{\ref{figvac}\text b}(x)&=
-(D -2)I_{\ref{figvac}\text a}(1)I_{\ref{figvac}\text a}(x) \ , \end{aligned}$$ where $\bW1$ is the operator $$\label{bw1}
\bW1 I(x)= -3x I(x+1) +(2x-D+1)I(x) +(x-D+1)I(x-1) \ ,$$ and $I_{\ref{figvac}\text a}(x)$ is the integral with $m=1$. The identity of the homogeneous parts of the equations is an example of the similarities between equations described in section \[resuarb\]. The solutions of can be written as $$\begin{aligned}
I_{\ref{figself}\text a}(x)&=\C_{\ref{figself}\text a} \IOMOG_{\ref{figself}\text a}(x)
+\INOMOG_{\ref{figself}\text a}(x)\ , \\
I_{\ref{figvac}\text b}(x) &=\C_{\ref{figvac}\text b} \IOMOG_{\ref{figself}\text a}(x)
+2I_{\ref{figvac}\text a}(1) \INOMOG_{\ref{figself}\text a}(x)\ . \end{aligned}$$ Numerical calculation of $\IOMOG_{\ref{figself}\text a}(1)$ and $\INOMOG_{\ref{figself}\text a}(1)$ was described in section \[numexa\]. The constant $\C_{\ref{figvac}\text b}$ may be found by comparing the large-$x$ behaviours of the solutions $\IOMOG_{\ref{figself}\text a}(x)\approx x^{-D/2+1/2}$, $\INOMOG_{\ref{figself}\text a}(x)\propto x^{-D/2}$ (see ) with $$I_{\ref{figvac}\text b}(x) \approx x^{-D/2} \int \dfrac{\dk2 }{(k_2^2+1)^2}
= x^{-D/2} I_{\ref{figvac}\text a}(2)$$ (see ). One finds $\C_{\ref{figvac}\text b}=0$. Therefore $I_{\ref{figvac}\text b}(1) =2I_{\ref{figvac}\text a}(1)
\INOMOG_{\ref{figself}\text a}(1)$; we see that the two-loop integral $I_{\ref{figvac}\text b}(1)$ factorizes into a product of a vacuum one-loop integral and *a part* of a one-loop self-energy integral.
\[pair2\] Now let us consider the diagrams of Figs. \[figself\]b and \[figvac\]c. Both diagrams have only one master integral with all the denominators, the scalar integral (compared with the four master integrals of the arbitrary case); the corresponding master functions $I_{\ref{figself}\text b}(x)$ and $I_{\ref{figvac}\text c}(x)$ satisfy the difference equations $$\begin{aligned}
\label{twoll}
\bW2 I_{\ref{figself}\text b}(x)&=(D-2)^2 I_{\ref{figvac}\text a}(1) I_{\ref{figvac}\text a}(x)
\ , \\
\label{twoll2}
\bW2 I_{\ref{figvac}\text c}(x)&={\tfrac{3}{2}}(D-2)^2 I_{\ref{figvac}\text a}^2(1) I_{\ref{figvac}\text a}(x)
\ ,\end{aligned}$$ where $\bW2$ is the operator $$\begin{gathered}
\label{bw2}\label{twol}
\bW2 I(x)= -8x(x-D+2)I(x+1)
+\bigl( 7x^2 +( 13 - 10D )x \\
+ (3D-4)(D-1)\bigr) I(x) +(x-D+1)(x-3D/2+2) I(x-1)
\ .\end{gathered}$$ Note again the identity of the homogeneous parts of the equations. The characteristic equation has the solutions $1$ and $-1/8$. The index associated to $\mu=1$ is $\K=-D/2$; therefore the condition is satisfied and the corresponding solution of the homogeneous equation $\IOMOG_{\ref{figself}\text b}(x)$ contributes to $I_{\ref{figself}\text b}(x)$ and $I_{\ref{figvac}\text c}(x)$. The solutions of can be written as $$\begin{aligned}
I_{\ref{figself}\text b}(x)&= \C_{\ref{figself}\text b} \IOMOG_{\ref{figself}\text b}(x)
+\INOMOG_{\ref{figself}\text b}(x)\ , \\
I_{\ref{figvac}\text c}(x)&= \C_{\ref{figvac}\text c} \IOMOG_{\ref{figself}\text b}(x)
+\tfrac{3}{2} I_{\ref{figvac}\text a}(1) \INOMOG_{\ref{figself}\text b}(x)\ .\end{aligned}$$ The constants may be obtained by comparing the large-$x$ behaviours $\IOMOG_{\ref{figself}\text b}(x)$$\approx$ $x^{-D/2}$ and $\INOMOG_{\ref{figself}\text b}(x)\propto x^{-D/2-1}$ with $I_{\ref{figself}\text b}(x)\approx x^{-D/2} I_{\ref{figself}\text a}(1)$ and $I_{\ref{figvac}\text c}(x)\approx x^{-D/2}$ $ I_{\ref{figvac}\text b}(1)$. One finds $\C_{\ref{figself}\text b}=I_{\ref{figself}\text a}(1)$ and $\C_{\ref{figvac}\text c}=I_{\ref{figvac}\text b}(1)$. We consider the calculation of $\IOMOG_{\ref{figself}\text b}(1)$ and $\INOMOG_{\ref{figself}\text b}(1)$ with factorial series. The recurrence relation is unstable with $A=8$. We fix a precision of the coefficients of the powers of $\e$ of the results of $E=13$ digits, with $n'_\e=7$ terms of the expansions in $\e$; following section \[sumfact\] we choose $x_{max}=25$, and we perform the calculations with $C=38$ digits. Expansions in factorial series of $\IOMOG_{\ref{figself}\text b}(x)$ and $\INOMOG_{\ref{figself}\text b}(x)$ converge for $x=25$ in about 1800 terms. Values for $x=1$ of the solutions are calculated using repeatedly the recurrence relations . Two terms of the expansions in $\e$ are lost going beyond the abscissa of convergence $\lambda=3$, so that we must retain the first $n_e=9$ terms of the expansions. One obtains $$\begin{gathered}
\C_{\ref{figself}\text b}\IOMOG_{\ref{figself}\text b}(1)\Gammae^{-2}=
0.09188814923697 \e^{-3} +0.194632539439 \e^{-2} \\
-0.045490472375 \e^{-1} +6.50912255436 +10.43240978278 \e \\
+76.7023111407 \e^2 +118.6149739413 \e^3 +732.0021187015 \e^4 +O(\e^5)\ ,\end{gathered}$$ $$\begin{gathered}
\INOMOG_{\ref{figself}\text b}(1)\Gammae^{-2}=
-0.09188814923697 \e^{-3} -1.694632539439 \e^{-2} \\ -4.204509527625 \e^{-1}
-13.88412255436 -27.65438231756 \e \\ -106.2615181944 \e^2
-187.4928691117 \e^3 -850.2486033469 \e^4 +O(\e^5).\end{gathered}$$ Summing the results one finds ; note the cancellation of the $\e^{-3}$ term, and the partial cancellations of digits in the $\e^{-2}$, $\e^{-1}$ and constant terms. Considering now the solution of with Laplace’s transformation, there are *three* master transformed functions: $v_{\ref{figself}\text b}(t)$, $w_{1}(t)$ and $w_{2}(t)$ defined by $$I_{\ref{figself}\text b}(x)=
\int \dfrac{\dk1 \;\dk2}{(k_1^2+1)^x (k_2^2+1) ((p-k_1-k_2)^2+1)}=
\int_0^1 dt \; t^{x-1} v_{\ref{figself}\text b}(t) \ , \qquad\qquad\qquad$$ $$K_\alpha(x) =\int \dfrac{\dk1 \;\dk2 \ (p\cdot k_\alpha)}
{(k_1^2+1)^x (k_2^2+1) ((p-k_1-k_2)^2+1)}
=\int_0^1 dt \; t^{x-1} w_{\alpha}(t) \ , \qquad \alpha=1,2 \ .$$ Clearly $w_1(t)\not = w_2(t)$ and $K_1(x)\not =K_2(x)$ as the insertion of $x$ breaks the symmetry $k_1 \leftrightarrow k_2$. The function $v_{\ref{figself}\text b}(t)$ satisfies the differential equation $$\begin{gathered}
t^2(1-t)(1+8t)v_{\ref{figself}\text b}''
+t(8(1-D)t^2 +10(D -2)t +(5D/2 -6))v_{\ref{figself}\text b}' \\
+(3D-8)(D-3)(t+1/2) v_{\ref{figself}\text b} =(D-2)^2t
I_{\ref{figself}\text a}(1) v_{\ref{figself}\text a}\ ,\end{gathered}$$ where $v_{\ref{figself}\text a}$ is $v_J$ of with $m=1$. $K_1(1)$ and $K_2(1)$ are not master integrals, as $K_{1}(1)=K_{2}(1)=-\tfrac{1}{3}I_{\ref{figself}\text b}(1)$; therefore $K_1(x)$ and $K_2(x)$ are not master functions and satisfy difference equations of order *zero*, for example $$(x+2-D)(-3K_{1}(x+1)
-I_{\ref{figself}\text b}(x+1)+I_{\ref{figself}\text b}(x))=
(2-D)I_{\ref{figself}\text a}(x+1)I_{\ref{figself}\text a}(1) \ .$$ On the contrary, $w_1(t)$ and $w_2(t)$ satisfy *first* order differential equations, for example $$-6t(tw_{1}'+(D-2)w_{1}) +2t(1-t)v_{\ref{figself}\text b}'
+2((2-D)t +D-3)v_{\ref{figself}\text b}=
2(D-2)I_{\ref{figself}\text a}(1) tv_{\ref{figself}\text a} \ ,$$ and therefore they are master functions.
Differential equations in masses and momenta {#differentialequ}
============================================
As shown in the one-loop example, the calculation of the constants $\C_j$, the factors multiplying the solutions of the homogeneous equation, presents different degrees of difficulty according to the values of masses and momenta. For values below the deformation threshold, these factors (when non-zero) are easily expressed in form of integrals with one loop missing, but, above the deformation threshold or when some masses are zero, as discussed in section \[othercases\], the calculation of the factors becomes more complicated. An alternative to this calculation is to calculate the master integrals for values of masses and momenta such that the calculation of the factors is simple, and subsequently to build and integrate a system of differential equations in the masses and momenta in order to reconstruct the integrals for the requested values of masses and momenta. The approaches to calculation of diagrams based on construction of differential equations in masses or momenta were introduced in [@difmas; @difmom]; up to now they have been used in the analysis of particular diagrams (e.g. massless box diagrams in [@prep1]) with a small number of master integrals by building and solving small systems of differential equations. Here we describe how to build and solve systems of differential equations in masses and momenta in way very similar to that of the differential equations obtained by Laplace’s transformation. In this way even large systems containing hundreds of equations can be built and solved, obtaining results expanded at will in $\e$.
Differentiation of master integrals {#diffint}
-----------------------------------
A generic master integral is a function $B(m_i^2, p_i\cdot p_j)$ of scalar products and masses. We parametrize the trajectory from the initial point to the final point in the space of scalar products and masses with a parameter $\Y$; the scalar products and masses of the points of the trajectory will be in general functions of $\Y$. In analogy with the differential equations obtained by using the Laplace’s transformation, we introduce the parametrization in such a way that the initial point corresponds to $\Y=1$ and the final point corresponds to $\Y=0$. Derivatives with respect to $\Y$ of master integrals will be written as derivatives with respect to scalar products and square masses: $$\label{deriv3}
\dfrac{dB}{d\Y}=
\sum_{i=1}^\NP \dfrac{dp_i^2}{d\Y} \dfrac{\partial B}{\partial p_i^2}
+\sum_{i<j} \dfrac{d(p_i\cdot p_j)}{d\Y}
\dfrac{\partial B}{\partial (p_i \cdot p_j)}
+\sum_{i=1}^\ND \dfrac{dm_i^2}{d\Y} \dfrac{\partial B}{\partial m_i^2} \ .$$
Differentiation with respect to masses of the integral is trivial; differentiation with respect to scalar products is less immediate. Let us consider a diagram with $\NP$ independent external momenta $p_i$; the scalar products $p_\alpha \cdot p_\beta$ are $\NPP=\NP (\NP+1)/2$. The derivative with respect to $p_\alpha \cdot p_\beta$ has the general form $$\label{deriv1}
\dfrac{\partial}{\partial(p_\alpha\cdot p_\beta)}= \sum^\NP_{i=1}\sum^\NP_{j=1}
a_{ij\alpha\beta} \; p_i \cdot
\dfrac{\partial}{\partial p_j}\ ,$$ for $(\alpha,\beta)$ equal to each one of the $\NPP$ possible different pairs of integer. The coefficients $a_{ij\alpha\beta}$ can be determined by imposing the $\NPP^2$ conditions $$\label{condi1}
\dfrac{\partial(p_\gamma \cdot p_\delta) }{\partial(p_\alpha\cdot p_\beta)}=
\begin{cases}
1 \text{ if } p_\gamma\cdot p_\delta = p_\alpha \cdot p_\beta \ ,\\
0 \text{ otherwise } \ .
\end{cases}$$ But the number of the coefficients $a_{ij\alpha\beta}$ is $\NP^2\NPP$ which is greater than $\NPP^2$; therefore some of these coefficients are not constrained by these conditions and may be chosen arbitrarily. We conveniently rewrite as $$\label{deriv2}
\dfrac{\partial}{\partial(p_\alpha\cdot p_\beta)}=
\sum_{i=1}^\NP a_{ii\alpha\beta} \; p_i\cdot \dfrac{\partial}{\partial p_i}
+\sum_{i<j} a_{ij\alpha\beta} \left(
p_i\cdot \dfrac{\partial}{\partial p_j} +M_{ij} \;
p_j\cdot \dfrac{\partial}{\partial p_i}
\right)\ ,$$ where the coefficients $a_{ij\alpha\beta}$ to be found are $\NPP^2$ and there are $\NP(\NP-1)/2$ arbitrary constants $M_{ij}$. In order to calculate the coefficients it is convenient, fixed $\alpha$ and $\beta$, to apply to a generic scalar product $p_k \cdot p_l$, and to construct the tensor $A_{ijkl}$ formed by the factors multiplying $a_{ij\alpha\beta}$ in the obtained expression. Now, considering the various pairs of indices $\{i,j\}$ as one single index $ij$, $A$ becomes a square matrix $A_{ij,kl}$ of dimension $\NPP$. Inverting it we find the desired coefficients $$a_{kl,ij} = \left(A^{-1}\right)_{ij,kl} \ .$$ As an example, for a vertex diagram $\NP=2$, and the matrix $A$ is $$A=
\begin{pmatrix}
2p_1^2 & 2M_{12} p_1\cdot p_2 & 0 \\
p_1\cdot p_2 & p_1^2+M_{12} p_2^2 & p_1\cdot p_2 \\
0 & 2p_1\cdot p_2 & 2p_2^2 \\
\end{pmatrix}
\ ,$$ where the indices 1,2,3 correspond to $ij=11,12,22$.
Construction of the system of differential equations {#consys}
----------------------------------------------------
The algorithm used for the generation and solution of the system of identities is the following:
\[algsys5\] Follow the algorithm \[algsys\] with the following modifications:
1. New identities which transform the derivatives with respect to $\Y$ of master integrals into combinations of integrals obtained using are added to the system.
2. Master integrals and their derivatives have a priority of extraction lower than other integrals.
3. Add the new entry [“the greatest derivative”]{} to the list of priorities after the entry \[cf6b\].
Analogously to sections \[identificamaster\] and \[constsysdif\], the previous algorithm allows one to determine the list of the master integrals $B_n(y)$, and to obtain a system of differential equations between them. Applying a procedure of transformation into triangular form of the subsystems of equations between master integrals with the same denominators, the whole system of differential equations takes the triangular form. The arbitrary constants $M_{ij}$ turn out to be particularly useful as a check, because the expression of derivatives contains $M_{ij}$ (through ), while the differential equations obtained from the system of identities are obviously independent of them.
The integration of the system of differential equations requires the knowledge of the values of the master integrals and of some derivatives in the initial point $\Y=1$. The values of the master integrals are obtained by solving a system of difference equations. The derivatives of the master integrals in the initial point are expressed in terms of the master integrals of the system of difference equations using the identities provided by the algorithm \[algsys\].
Note that the master integrals of the system of difference equations may be different in number and structure from the master integrals of the system of differential equations. The system of differential equation is solved with the same procedure used for the Laplace’s transformation method. We stress that mobile apparent singular points, as described in section \[mobile\], may appear in the system of differential equations.
Expansion in $\e$ and infrared divergences {#infrared}
------------------------------------------
All quantities, coefficients of differential equations, solutions, $\Gamma$ functions, etc. are expanded in $\e=(4-D)/2$. Solution of differential equations using power series in $\Y$ with coefficients expanded in $\e$ does not cause difficulties, unless we deal with IR divergences. If a master integral $B(\Y,\e)$ is IR finite in the initial point $\Y=1$, but is IR divergent in the final point $\Y=0$, it is not possible to obtain the correct dimensionally regularized value, which contains an additional IR pole in $\e$, by integrating the differential equation from the initial point, where the integral is IR finite: the coefficients of the expansion in $\e$ of $B(\Y,\e)$ tend to infinity as $\Y \to 0$, so that $\lim_{\Y \to 0}B(\Y,\e) \not = B(0,\e)$. In this unfortunate case the only remedy is to substitute the master integral IR divergent with another integral (or combination of integrals) IR finite. Our preference for master integrals with scalar products in the numerator and all exponents 1 in the denominator rather than integrals with exponents greater than 1 in the denominator was devised to limit the appearance of master integrals IR divergent.
In the case where $B(0,\e)$ is IR finite, but some derivatives with respect to $\Y$ of $B(\Y,\e)$ are IR divergent in $\Y=0$, the problem is overcome by integrating the differential equation up to a small value $\Y\approx 0$, suitable chosen so that the value of the master integral will be calculated with the requested precision.
Examples of calculation of integrals with zero masses {#resumass0}
-----------------------------------------------------
(370,160)(0,0) (040,080)[(-1,0)[15]{}]{} (120,080)[(+1,0)[15]{}]{} (040,080)(080,080)(120,080) (+120.0,+080.0)(+116.6,+116.6)(+080.0,+120.0) (+080.0,+120.0)(+043.4,+116.6)(+040.0,+080.0) (+040.0,+080.0)(+043.4,+043.4)(+080.0,+040.0) (+080.0,+040.0)(+116.6,+043.4)(+120.0,+080.0) (+280.0,+120.0)(+257.3,+119.3)(+245.4,+100.0) (+245.4,+100.0)(+234.6,+080.0)(+245.4,+060.0) (+245.4,+060.0)(+257.3,+040.7)(+280.0,+040.0) (+280.0,+040.0)(+302.7,+040.7)(+314.6,+060.0) (+314.6,+060.0)(+325.4,+080.0)(+314.6,+100.0) (+314.6,+100.0)(+302.7,+119.3)(+280.0,+120.0) (280,080)(297.3,070)(314.6,060) (280,080)(262.7,070)(245.4,060) (280,080)(280,100)(280,120)
For illustrative purpose, we consider here only two simple diagrams; applications to more complicated diagrams will be shown in future papers. Let us consider the diagram of Fig. \[figder\]a, and the integral $$\bar I(x)=\int \dfrac{[dk]}{(k_1^2+1)^x \; k_2^2\;(p-k_1-k_2)^2}\ ,
\quad p^2=-1\ ,$$ where $[dk]=\dk1 \;\dk2$. It satisfies the homogeneous difference equation $$x(x-2D+5)\bar I(x+1)-(x-3D/2+3)(x-D+2)\bar I(x)=0 \ ;$$ the solution of this equation is $$\bar I(x)=\C
\dfrac{\Gamma(x-3D/2+3)\Gamma(x-D+2)}{\Gamma(x)\Gamma(x-2D+5)} \ .$$ The calculation of the constant $\C$ is not simple; in fact, the value of $-p^2=1$ is above the deformation threshold $p^2=0$, so that the large-$x$ behaviour of $\bar I(x)$ contains an additional contribution due to the turning point of the integration path; moreover, the identity is useless because $\bar I(0)=0$.
We replace the zero mass with a square mass $\Y$ in the denominators $$D_1=k_1^2+1 \ , \ D_2=k_2^2+\Y\ ,\ D_3=(p-k_1-k_2)^2+\Y \ ,$$ then we build the system of identities following the algorithm \[algsys5\]. The master integrals are $I_1(\Y)=\int [dk]/{D_1 D_3}$, $I_2(\Y)=\int [dk]/{D_1 D_2}$, $I_3(\Y)=\int [dk]/{D_2 D_3}$, $I_4(\Y)=\int [dk]/{D_1 D_2 D_3}$ and $I_5(\Y)=\int [dk](p \cdot k_2)/{D_1 D_2 D_3}$, and satisfy the system of differential equations $$\begin{split}
\Y I_1'&=(D-2)I_1 \ ,\qquad
2\Y I_2'=(D-2)I_2 \ ,\qquad
2\Y I_3'=(D-2)I_3 \ ,\\
8\Y (\Y -1)I_4''&=4\left((4D-13)\Y +7-2D\right)I_4' -2 (3D-8)(D-3)I_4
+(D-2)^2(I_1-I_2-I_3) \ , \\
6(D-2)I_5&=-4\Y (\Y -1)I_4'+2\left((D-3)\Y +5-2D\right)I_4
+(D-2)(2I_2-I_1-I_3)\ .\\
\end{split}$$ The values of the integrals in the initial point $\Y=1$ can be expressed using the values of integrals with masses equal to one given in section \[resuvalues\]: $I_1(1)=I_2(1)=I_3(1)=I^2(\ref{figself}\text a)$, $I_4(1)=I(\ref{figself}\text b)$, $I_5(1)=-I(\ref{figself}\text b)/3$. The singular points of the system are $\Y =1$ and $\Y =0$. Considering the equation for $I_4$, the exponents in these points are $$\rho_{1,2}^{(\Y=1)}=0, 2 -2\e \ , \qquad
\rho_{1,2}^{(\Y=0)}=0, 3/2-2\e \ .$$ In the initial point the integral $I_4(1)$ and its derivatives are IR finite, so that the solution must be regular; therefore the singular solution associated with $\rho_2^{(\Y=1)}$ does not contribute, and the second-order equation for $I_4$ needs only $I_4(1)$ as initial condition. In the final point the integral $I_4(0)$ is IR finite but $I_4'(0)$ is IR divergent, so that the singular solution associated with $\rho_2^{(\Y=0)}$ must contribute to $I_4(\Y )$.
The integration of the system is divided into two parts: from $\Y =1$ to $\Y =1/2$ expanding in $\Y =1$, and from $\Y =1/2$ to the value $\Y =\lambda^2 \ll 1$ expanding in $\Y =0$. The presence of the cutoff $\lambda$ is needed because the solution is not regular in the origin. Perusing the numerical results, the effect of the cutoff on the coefficient of $\e^{s}$ of the expansion in $\e$ of the final value turns out to be approximately proportional to $\lambda^2\log^{m+s} \lambda$, where $m$ is some small integer. A value $\lambda= 10^{-15}$ suffices to obtain the first coefficients with 20 exact digits. The normalized result is $$\begin{gathered}
I_4(0)\Gammae^{-2}=
-0.5 \e^{-2} -1.25 \e^{-1} -4.6648681336964528729 \\
-9.595397946879509324 \e
-26.045799878017610383 \e^2 \\
-49.501934187562851546 \e^3
-120.38235865133474218 \e^4 +O(\e^5)\ . \end{gathered}$$
Another more complicated example is the integral of Fig. \[figder\]b $$L=\int \dfrac{\dk1 \;\dk2 \;\dk3 }
{(k_1^2+1) \left((k_1-k_2)^2+1\right) \left((k_2-k_3)^2+1\right) k_2^2\;
k_3^2\; (k_1-k_3)^2}\ .\quad$$ As before, we give a square mass $\Y $ to massless denominators. In this case there are 43 master integrals from 2 to 6 denominators, of which only 1 with 6 denominators. The system of differential equations is too long to be shown here. The singular points are $\Y =-1,0,1/9,1/4,(3\pm\sqrt{5})/2,1,4,9$; the points $1/9$, $1/4$, and $(3-\sqrt{5})/2$ on the interval $[0,1]$ turn out to be regular points of the solutions. The effect of the cutoff is different from the previous case, being proportional to $\lambda\log^{m+s} \lambda$. A value $\lambda=10^{-30}$ suffices to obtain the first coefficients with 20 exact digits. The result is $$\begin{gathered}
\label{resl}
L\Gammae^{-3}=
2.4041138063191885708 \e^{-1} -3.0270094939876520198 \\
+22.804522068631748454 \e -53.102275435449702689 \e^2 \\
+201.84333994219396694 \e^3 -577.74024368094326834 \e^4
+O(\e^5)\ .\end{gathered}$$ The first two terms agree with the results [@r3.027]; subsequent terms are new.
These calculations were carried out by using the program ; the calculation from scratch of the integral $L$ and all the other master integrals, including the calculation of the master integrals in the initial point, required about 1.5 hours on a 133 MHz Pentium PC.
The program {#calcprog}
============
\[program\] Here we report in brief some information concerning the program used to calculate the values of the integrals of the sections \[resu\] and \[resumass0\]. Further details will be given in a separate paper.
- C program, about 23000 lines, 1Mb executable.
- The program allows one to calculate the value of integrals in almost completely automatic way; only needed input is a description of the diagram, the constants $a_i$ and $b_i$ of section \[secab\] which limit the identities generated, the number of dimensions $D_0$ about which to expand the integrals, and the number of terms of the expansions in $D-D_0$.
- The program contains a simplified algebraic manipulator, used for the solution of the systems of identities.
- Coefficients of the integrals in the identities can be unlimited precision integers, rationals, ratios of polynomials in one and two variables (for example $D$ and $x$) with integer coefficients. At present the values of square masses and products of external momenta must be rational numbers.
- Efficient management of systems of identities of size up to the limit of disk space (tested up to half million of identities).
- Numerical solution of systems of difference/differential equations up to 500 equations.
- Numerical variables used in the solution are arbitrary precision floating point complex numbers and truncated series in $\e$ with this kind of coefficients.
- Arithmetic libraries which deal with operations on integers, polynomials, rationals, floating point numbers and truncated series in $\e$ were written on purpose.
Two versions of the program exist, one using factorial series and one using Laplace’s transformation. The Laplace’s transformation version is much more complicated than the factorial series version but it turns to be faster in many cases: both systems of difference and Laplace-transformed differential equations are generated, then the system of differential equations is solved, and the functions $U(x)$ are obtained by integrating over $t$ the solutions $v(t)$, also checking that $U(x)$ are solutions of the system of difference equations. Both versions of program were used for calculating the single-scale integrals of section \[resu\].
Conclusions {#Conclusions}
===========
Most part of this paper has been devoted to the description of a new method of calculation of master integrals, based on numerical solution of systems of difference equations, obtained by solving systems of identities obtained by integration-by-parts. Important features of the method are the applicability to arbitrary diagrams, inherited from integration-by-parts method, and the ability to obtain very high precision results (even 100-200 digits) expanded at will in $\e$, due to fact that the calculation of integrals is reduced to sums of *convergent* factorial or power series in one variable.
We also have described two important complements to this method: an algorithm for the reduction of generic Feynman integrals to master integrals, and a procedure for construction and numerical solution of differential equations is masses and momenta; among other things, at present the latter is needed to calculate generic master integrals with massless denominators or with external momenta ‘deep’ in the non-euclidean region (over the deformation thresholds), at least until a working automated general procedure for the calculation of the arbitrary constants of difference equations in these cases will be found.
The implementation of these methods and algorithms in the program $\SYS$ turned out to be essential to test and prove the validity of the approach.
In this first exploratory work, mostly devoted to test our approach, we have focused our attention to the calculation of single-scale master integrals, in particular vacuum and self-energy diagrams up to three loops and vertex and box diagrams up to two loops. The calculated values of these master integrals are useful, as they may appear expanding Feynman integrals with respect to ratio of different scale parameters.
But the final targets of our approach are the calculation of four-loop $g$-$2$ contribution in QED, and the calculation of multi-scale multi-loop master integrals especially in cases where there are no hierarchies between scales, where asymptotic expansions in large masses or momenta seem to be not useful. No insurmountable difficulty seems to exist for applying our approach to these problems. Clearly, that means to deal with a larger number of master integrals, or with more complicated equations, and that will imply modifications or improvements of various parts of the algorithms implemented in the program $\SYS$ which, at this stage of development, is far to be optimal. The experience gained by performing the calculations of this work has given many suggestions on the changes which should be made and which will be discussed in future papers.
Acknowledgement {#acknowledgement .unnumbered}
===============
The author wants to thank E. Remiddi and M. Caffo for useful discussions and encouragement in the very early stage of this work.
[9]{} K. G. Chetyrkin and F. V. Tkachov, (1981) 159. F. V. Tkachov, (1981) 65. O. V. Tarasov, [*Acta Phys. Polon.* ]{}[**B29**]{} (1998) 2655. S. Laporta and E. Remiddi, (1996) 283. S. Laporta and E. Remiddi, [*Acta Phys. Polon.*]{} [**B28**]{} (1997) 959. R. Harlander and M. Steinhauser, [*Prog.Part.Nucl.Phys.*]{} [**43**]{} (1999) 167; hep-ph/9812357. T. van Ritbergen and R. G. Stuart, (2000) 343; hep-ph/9904240. P. A. Baikov, (1996) 404; [*Nucl. Inst. Meth.*]{} [**A389**]{} (1997) 347; P. A. Baikov and M. Steinhauser, [*Comp. Phys. Commun.*]{} [**115**]{} (1998) 161. T. Gehrmann and E. Remiddi, (2000) 485, hep-ph/9912329. L. M. Milne-Thomson, [*The calculus of finite differences*]{} (Macmillan, London, 1951). S. Moch and J. A. M. Vermaseren, (2000) 853, hep-ph/9912355; [*Nucl. Phys. Proc. Suppl.* ]{} 89 (2000) 131, hep-ph/0004235. Talks presented by J. A. M. Vermaseren and O.V. Tarasov at 5th Zeuthen Workshop on Elementary Particle Theory, “Loops and Legs in Quantum Field Theory”, Bastei/Königstein, Germany, 9-14 april 2000. See [@Milne], chapters XII-XIV. See [@Milne], chapter X, p. 271. K. Knopp, [*Theory and application of infinite series*]{} (Hafner, New York, 1971) p. 446. M. Abramowitz and I. Stegun, [*Handbook of mathematical functions*]{}, (Dover, New York, 1972) p. 824. M. J. Levine and R. Roskies, D[**9**]{} (1974) 421; M. J. Levine, E. Remiddi and R. Roskies, in [*Quantum Electrodynamics*]{}, edited by T. Kinoshita, Advanced series on Directions in High Energy Physics, Vol. 7 (World Scientific, Singapore, 1990) p. 162. See [@Milne], chapter XV. P. A. Baikov and V. A. Smirnov, (2000) 367, hep-ph/0001192. A. I. Davydychev and J. B. Tausk, (1993) 123. D. Broadhurst, [*Eur. Phys. J.*]{} [**C8**]{} (1998) 311. D. Broadhurst, [*Z. Phys.*]{} [**C54**]{} (1992) 599. J. Fleischer, M. Yu. Kalmykov and A. V. Kotikov, (1999) 169. V. Borodulin and G. Jikia, (1997) 434. A. V. Kotikov, (1991) 158; [**B259**]{} (1991) 314; [**B267**]{} (1991) 123 (err. [**B295**]{} (1992) 409). E. Remiddi, [**110A**]{} (1997) 1435; M. Caffo, H. Czyż, S. Laporta and E. Remiddi, [*Acta Phys. Polon.* ]{}[**B29**]{} (1998) 2627; (1998) 365. L. Avdeev, J. Fleischer, S. Mikhailov and O. Tarasov, (1994) 560; (1995) 597(E); J. Fleischer and O. Tarasov, (Proc. Supp.) [**37B**]{} (1994) 115; K. G. Chetyrkin, J. H. Kühn and M. Steinhauser, (1995) 331.
Figure Captions {#figure-captions .unnumbered}
===============
Figure 1: Vacuum diagrams up to three loops.
Figure 2: Self-energy diagrams up to three loops.
Figure 3: Vertex diagrams up to two loops.
Figure 4: Box diagrams up to two loops.
Figure 5: Diagrams with massless lines (dashed).
Condensed paper title {#condensed-paper-title .unnumbered}
=====================
[Calculation of Feynman integrals by difference equations]{}
$x$ $J(x)$ $\IOMOG_{-}(x)$ $\INOMOG(x)$
----- ------------------------- ----------------------------------- ------------------------------
9 $0.017857 +0.033442 \e$ $ 0.047713 +0.089160 \e $ $ -0.008928 -0.007323 \e$
8 $0.023809 +0.040621 \e$ $ 0.059006 +0.100337 \e $ $ -0.011904 -0.007663 \e$
7 $0.033333 +0.050203 \e$ $ 0.075609 +0.113249 \e $ $ -0.016666 -0.007159 \e$
6 $0.05 +0.062805 \e$ $ 0.101857 +0.126598 \e $ $ -0.025 -0.003929 \e$
5 $0.083333 +0.076898 \e$ $ 0.148085 +0.133074 \e $ $ -0.041666 +0.009010 \e$
4 $0.166666 +0.070464 \e$ $ 0.245635 +0.090010 \e $ $ -0.083333 +0.067207 \e$
3 $0.5 -0.288607 \e$ $ 0.548843 -0.442122 \e $ $ -0.25 +0.534955 \e$
2 $\e^{-1} -0.577215\fe$ $ 0.282094\e^{-1} +0.519388 \fe$ $ -0.25\e^{-1}+0.144303 \fe$
1 $-\e^{-1} -0.422784\fe$ $ 0.282094\e^{-1} -1.645293 \fe$ $ 0.75\e^{-1}+1.067088 \fe$
0 $0\phantom{.123456\fe}$ $ -0.564189\e^{-1} -0.238530 \fe$ $ -0.5\e^{-1} -0.211392 \fe$
Table \[tableij\]: Values of $J(x)$, $\IOMOG_{-}(x)$ and $\INOMOG(x)$
$x_{max}$ terms finite part of $I(1)$
----------- ---------- -----------------------
$30$ $ 125$ $-0.3910008887063124$
$25$ $ 154$ $-0.3910149952724784$
$20$ $ 217$ $-0.3910150292106927$
$15$ $ 395$ $-0.3910150291388126$
$10$ $ 1470$ $-0.3910150291357554$
$ 9$ $ 2454$ $-0.3910150291357472$
$ 8$ $ 4439$ $-0.3910150291357503$
$ 7$ $ 13086$ $-0.3910150291357507$
$ 6$ $ 36210$ $-0.3910150291357507$
Table \[tablei1\]: Dependence of the finite part of $I(1)$ on $x_{max}$.
(380,120)(0,0) (060,080) (040,080) (115,080) (095,080)[(1,0)[40]{}]{} (190,080) (170,080)(190,100)(209.5,080) (170,080)(190,060)(209.5,080) (265,080) (265,099.2)[(+3,-5)[17]{}]{} (265,099.2)[(-3,-5)[17]{}]{} (340,080) (340,080)[(0,1)[20]{}]{} (340,080)[(+5,-3)[17]{}]{} (340,080)[(-5,-3)[17]{}]{}
Figure \[figvac\]
(370,480)(0,0)
(050,460) (030,460)[(-1,0)[10]{}]{} (070,460)[(+1,0)[10]{}]{} (140,460) (120,460)[(1,0)[40]{}]{} (120,460)[(-1,0)[10]{}]{} (160,460)[(+1,0)[10]{}]{} (230,460) (210,460)[(-1,0)[10]{}]{} (250,460)[(+1,0)[10]{}]{} (210,460)(228,462)(230,480) (320,460) (320,440)[(0,+1)[40]{}]{} (300,460)[(-1,0)[10]{}]{} (340,460)[(+1,0)[10]{}]{} (050,380) (030,380)(050,400)(070,380) (030,380)(050,360)(070,380) (030,380)[(-1,0)[10]{}]{} (070,380)[(+1,0)[10]{}]{} (140,380) (120,380)[(-1,0)[10]{}]{} (160,380)[(+1,0)[10]{}]{} (120,380)(130,390)(140,400) (120,380)(138,382)(140,400) (230,380) (210,380)[(-1,0)[10]{}]{} (250,380)[(+1,0)[10]{}]{} (210,380)(227,383)(230,400) (210,380)[(1,0)[40]{}]{} (320,380) (300,380)(318,382)(320,400) (340,380)(322,382)(320,400) (300,380)[(-1,0)[10]{}]{} (340,380)[(+1,0)[10]{}]{} (050,300) (030,300)[(-1,0)[10]{}]{} (070,300)[(+1,0)[10]{}]{} (030,300)[(+1,0)[40]{}]{} (050,300)[(0,+1)[20]{}]{} (140,300) (120,300)[(-1,0)[10]{}]{} (160,300)[(+1,0)[10]{}]{} (140,280)(160,300)(140,320) (140,280)(120,300)(140,320) (230,300) (210,300)[(-1,0)[10]{}]{} (250,300)[(+1,0)[10]{}]{} (210,300)(227,303)(230,320) (210,300)(233,304)(240,317) (320,300) (300,300)[(-1,0)[10]{}]{} (340,300)[(+1,0)[10]{}]{} (300,300)(317,303)(320,320) (320,280)[(0,+1)[40]{}]{} (050,220) (030,220)[(-1,0)[10]{}]{} (070,220)[(+1,0)[10]{}]{} (030,220)(047,223)(050,240) (050,240)(052,224)(064,233) (140,220) (120,220)[(-1,0)[10]{}]{} (160,220)[(+1,0)[10]{}]{} (120,220)(137,223)(140,240) (120,220)(137,217)(140,200) (230,220) (210,220)[(-1,0)[10]{}]{} (250,220)[(+1,0)[10]{}]{} (230,200)[(0,+1)[20]{}]{} (230,230) (320,220) (300,220)[(-1,0)[10]{}]{} (340,220)[(+1,0)[10]{}]{} (320,200)[(0,+1)[40]{}]{} (320,240)(322,224)(334,233) (095,140) (075,140)[(-1,0)[10]{}]{} (115,140)[(+1,0)[10]{}]{} (085,157)(088,150)(092,143) (075,140)(100,140)(105,157) (185,140) (165,140)[(-1,0)[10]{}]{} (205,140)[(+1,0)[10]{}]{} (185,120)[(0,+1)[40]{}]{} (165,140)[(+1,0)[20]{}]{} (275,140) (255,140)[(-1,0)[10]{}]{} (295,140)[(+1,0)[10]{}]{} (275,159.2)[(+3,-5)[17]{}]{} (275,159.2)[(-3,-5)[17]{}]{} (095,060) (075,060)[(-1,0)[10]{}]{} (115,060)[(+1,0)[10]{}]{} (085,043)[(0,+1)[34]{}]{} (105,043)[(0,+1)[34]{}]{} (185,060) (165,060)[(-1,0)[10]{}]{} (205,060)[(+1,0)[10]{}]{} (185,060)[(0,-1)[20]{}]{} (185,060)[(+5,+3)[17]{}]{} (185,060)[(-5,+3)[17]{}]{} (275,060) (255,060)[(-1,0)[10]{}]{} (295,060)[(+1,0)[10]{}]{} (261,074)[(+1,-1)[28]{}]{} (261,046)(267,052)(273,058) (289,074)(283,068)(277,062) (273,058)(271,064)(277,062)
Figure \[figself\]
(300,200)(0,0) (030,140)[(+1,0)[40]{}]{} (030,140)[(+3,+5)[20]{}]{} (070,140)[(-3,+5)[20]{}]{} (030,140)[(-5,-3)[10]{}]{} (070,140)[(+5,-3)[10]{}]{} (050,173)[(0,+1)[10]{}]{} (130,140)[(+1,0)[40]{}]{} (130,140)[(+3,+5)[20]{}]{} (170,140)[(-3,+5)[20]{}]{} (130,140)[(-5,-3)[10]{}]{} (170,140)[(+5,-3)[10]{}]{} (150,173)[(0,+1)[10]{}]{} (130,140)(150,155)(169,140) (230,140)[(+1,0)[40]{}]{} (230,140)[(+3,+5)[20]{}]{} (270,140)[(-3,+5)[20]{}]{} (230,140)[(-5,-3)[10]{}]{} (270,140)[(+5,-3)[10]{}]{} (250,173)[(0,+1)[10]{}]{} (230,140)(240,152)(250,140) (030,040)[(+1,0)[40]{}]{} (030,040)[(+3,+5)[20]{}]{} (070,040)[(-3,+5)[20]{}]{} (030,040)[(-5,-3)[10]{}]{} (070,040)[(+5,-3)[10]{}]{} (050,073)[(0,+1)[10]{}]{} (050,073)[(0,-1)[33]{}]{} (130,040)[(+1,0)[40]{}]{} (130,040)[(+3,+5)[20]{}]{} (170,040)[(-3,+5)[20]{}]{} (130,040)[(-5,-3)[10]{}]{} (170,040)[(+5,-3)[10]{}]{} (150,073)[(0,+1)[10]{}]{} (140,057)[(+1,0)[20]{}]{} (230,040)[(+3,+5)[20]{}]{} (270,040)[(-3,+5)[20]{}]{} (230,040)[(-1,-1)[10]{}]{} (270,040)[(+1,-1)[10]{}]{} (250,073)[(0,+1)[10]{}]{} (230,040)[(+5,+3)[29]{}]{} (270,040)[(-5,+3)[17]{}]{} (246,054)(243,056)(240.5,057.5) (246,054)(254,058)(252.5,050.5)
Figure \[figvert\]
(370,200)(0,0) (030,140)[(+1,0)[40]{}]{} (030,140)[(0,+1)[40]{}]{} (070,140)[(0,+1)[40]{}]{} (030,180)[(+1,0)[40]{}]{} (030,140)[(-1,-1)[10]{}]{} (030,180)[(-1,+1)[10]{}]{} (070,140)[(+1,-1)[10]{}]{} (070,180)[(+1,+1)[10]{}]{} (120,140)[(+1,0)[40]{}]{} (120,140)[(0,+1)[40]{}]{} (160,140)[(0,+1)[40]{}]{} (120,180)[(+1,0)[40]{}]{} (120,140)[(-1,-1)[10]{}]{} (120,180)[(-1,+1)[10]{}]{} (160,140)[(+1,-1)[10]{}]{} (160,180)[(+1,+1)[10]{}]{} (120,140)(140,155)(160,140) (210,140)[(+1,0)[40]{}]{} (210,140)[(0,+1)[40]{}]{} (250,140)[(0,+1)[40]{}]{} (210,180)[(+1,0)[40]{}]{} (210,140)[(-1,-1)[10]{}]{} (210,180)[(-1,+1)[10]{}]{} (250,140)[(+1,-1)[10]{}]{} (250,180)[(+1,+1)[10]{}]{} (210,180)[(+1,-1)[40]{}]{} (300,140)[(+1,0)[40]{}]{} (300,140)[(0,+1)[40]{}]{} (340,140)[(0,+1)[40]{}]{} (300,180)[(+1,0)[40]{}]{} (300,140)[(-1,-1)[10]{}]{} (300,180)[(-1,+1)[10]{}]{} (340,140)[(+1,-1)[10]{}]{} (340,180)[(+1,+1)[10]{}]{} (300,180)[(+2,-1)[40]{}]{} (025,040)[(+1,0)[40]{}]{} (025,040)[(0,+1)[40]{}]{} (065,040)[(0,+1)[40]{}]{} (025,080)[(+1,0)[40]{}]{} (025,040)[(-1,-1)[10]{}]{} (025,080)[(-1,+1)[10]{}]{} (065,040)[(+1,-1)[10]{}]{} (065,080)[(+1,+1)[10]{}]{} (025,040)(035,055)(045,040) (095,040)[(+1,0)[40]{}]{} (095,040)[(0,+1)[40]{}]{} (135,040)[(-1,+2)[20]{}]{} (135,080)[(-1,-1)[10]{}]{} (095,040)[(+1,+1)[23]{}]{} (117.5,063)(115,073)(124,070) (095,080)[(+1,0)[40]{}]{} (095,040)[(-1,-1)[10]{}]{} (095,080)[(-1,+1)[10]{}]{} (135,040)[(+1,-1)[10]{}]{} (135,080)[(+1,+1)[10]{}]{} (165,040)[(+1,0)[40]{}]{} (165,040)[(0,+1)[40]{}]{} (205,040)[(0,+1)[40]{}]{} (165,080)[(+1,0)[40]{}]{} (165,040)[(-1,-1)[10]{}]{} (165,080)[(-1,+1)[10]{}]{} (205,040)[(+1,-1)[10]{}]{} (205,080)[(+1,+1)[10]{}]{} (185,080)[(+1,-1)[20]{}]{} (235,040)[(+1,0)[40]{}]{} (235,040)[(0,+1)[40]{}]{} (275,040)[(0,+1)[40]{}]{} (235,080)[(+1,0)[40]{}]{} (235,040)[(-1,-1)[10]{}]{} (235,080)[(-1,+1)[10]{}]{} (275,040)[(+1,-1)[10]{}]{} (275,080)[(+1,+1)[10]{}]{} (255,080)[(0,-1)[40]{}]{} (305,040)[(+1,0)[40]{}]{} (305,040)[(0,+1)[40]{}]{} (345,040)[(-1,+2)[20]{}]{} (345,080)[(-1,-2)[8]{}]{} (325,040)[(+1,+2)[8]{}]{} (332.5,056)(327,064)(336,064) (305,080)[(+1,0)[40]{}]{} (305,040)[(-1,-1)[10]{}]{} (305,080)[(-1,+1)[10]{}]{} (345,040)[(+1,-1)[10]{}]{} (345,080)[(+1,+1)[10]{}]{}
Figure \[figbox\]
(370,160)(0,0) (040,080)[(-1,0)[15]{}]{} (120,080)[(+1,0)[15]{}]{} (040,080)(080,080)(120,080) (+120.0,+080.0)(+116.6,+116.6)(+080.0,+120.0) (+080.0,+120.0)(+043.4,+116.6)(+040.0,+080.0) (+040.0,+080.0)(+043.4,+043.4)(+080.0,+040.0) (+080.0,+040.0)(+116.6,+043.4)(+120.0,+080.0) (+280.0,+120.0)(+257.3,+119.3)(+245.4,+100.0) (+245.4,+100.0)(+234.6,+080.0)(+245.4,+060.0) (+245.4,+060.0)(+257.3,+040.7)(+280.0,+040.0) (+280.0,+040.0)(+302.7,+040.7)(+314.6,+060.0) (+314.6,+060.0)(+325.4,+080.0)(+314.6,+100.0) (+314.6,+100.0)(+302.7,+119.3)(+280.0,+120.0) (280,080)(297.3,070)(314.6,060) (280,080)(262.7,070)(245.4,060) (280,080)(280,100)(280,120)
Figure \[figder\]
[^1]: [E-mail: laporta["40]{}bo.infn.it]{}
[^2]: An algorithm which in principle may solve this problem has been recently proposed in [@Tar99], but up to now no practical application was shown.
[^3]: See section \[ssystem\].
[^4]: For a review see [@Harlander]. In the program [BUBBLES]{}[@Ritbergen] reduction of systems of identities is performed by semi-automatic means. A different approach to solving integration-by-parts identities was developed in [@Baikov].
[^5]: A similar method, in very preliminary form, was developed by the author in [@3-loop] to reduce to master integrals the triple-cross vertex diagrams contribution to the $g-2$ of the electron. In that work a system of about 100000 identities was built and solved, and this required some months of computer time; with our new method the same calculation may be performed in a fraction of hour. Some similar technique has been recently used in [@prep1], to solve small systems of identities in order to obtain differential equations for two-loop massless box diagrams.
[^6]: The importance of a total ordering for the solution of systems of differential equations derived from integration-by-parts identities was remarked in [@Tar99].
[^7]: Other authors are also encountering other kinds of difference equations in the evaluation of diagrams [@Vermaseren; @Bastei] and solving them, by using techniques different from the standard methods of [@Milne] used in this paper.
[^8]: In Ref.[@Milne] the operators $\PI$ and $\RHO$ are defined more generically as $\RHO^m U(x)=(\Gamma(x-r+1)/\Gamma(x-r-m+1))U(x-m)$ and $\PI U(x)=(x-r)(U(x)-U(x-1))$, where $r$ is a fixed number. Here for simplicity we have set $r=0$.
[^9]: If the coefficients of the equation are $ p_i(x)=\sum_{j=0}^g p_{ij} x^{g-j} $, the characteristic equation has the explicit expression $\sum_{i=0}^\R p_{i0} \; \mu^i =0 $.
[^10]: Otherwise $\VOMOG(x)$ would be a derivative with respect to $\K$ of a factorial series, with asymptotic behaviour $\sim x^\K \log^n(x)$.
[^11]: The deformation of the radial integration path was first discussed in [@Levinehyp] for self-energy diagrams in $D=4$.
[^12]: Here we generalize to arbitrary $D$ dimensions the result obtained in [@Levinehyp] in the four-dimensional case.
[^13]: The exact solution $J(x)=a_0\Gamma(x-D/2)/\Gamma(x)$ does not have the form of .
[^14]: It is important to note that if $m+1$ solutions of the indicial equation coincide for $D\to 4$, the Wronskian determinant of the linear system is proportional to $(D-4)^m$; this causes a loss of $m$ terms in the expansions in $D-4$ of $c_j$ and, consequently, in the expansions of $v$ in the rest of integration, and therefore in $U(x)$. This mishap sometimes occurred in the final singular point $t=0$ in the calculations of section \[resu\].
[^15]: The equation was found by solving the system of transformed identities. Applying the Laplace’s transformation directly to the difference equation one gets a higher order differential equation which does not have the mobile singular point. This equation can be also derived from by writing $(3t d\Phi(t)/dt +4(D-4)\Phi(t))/(4(D-1)t-(D-4))=0$ where $\Phi(t)$ is the left-hand side of .
[^16]: Due to the present limitations of the program used for the calculations, the values listed in this part of the table were calculated by giving arbitrary rational values to square masses and scalar products of external momenta. It is possible, but very unlikely, that the chosen values correspond to some particular case so that the results obtained do not correspond to the real arbitrary case.
[^17]: The equivalence of recurrence relations very recently described in [@prep2] may probably throw light on this.
[^18]: We consider here the master integrals containing *one particular* combination of denominators. The total number of combinations itself grows exponentially with $\ND$, clearly along with the total number of master integrals with every possible combination of denominators.
|
---
abstract: 'The excitation spectra in the deformed nucleus $^{228}$Th have been studied by means of the (p,t) reaction, using the Q3D spectrograph facility at the Munich Tandem accelerator. The angular distributions of tritons were measured for about 110 excitations seen in the triton spectra up to 2.5 MeV. Firm $0^+$ assignments are made for 17 excited states by comparison of experimental angular distributions with the calculated ones using the CHUCK3 code. Assignments up to spin $6^+$ are made for other states. Sequences of states are selected which can be treated as rotational bands and as multiplets of excitations. Moments of inertia have been derived from these sequences, whose values may be considered as evidence of the two-phonon nature of most $0^+$ excitations. Experimental data are compared with interacting boson model (IBM) and quasiparticle-phonon model (QPM) calculations and with experimental data for $^{229}$Pa.'
author:
- 'A. I. Levon$^{1}$, G. Graw$^2$, R. Hertenberger$^2$, S. Pascu$^{3}$, P. G. Thirolf$^2$, H.-F. Wirth$^2$, and P. Alexa$^4$'
title: '$0^+$ states and collective bands in $^{228}$Th by the (p,t) reaction'
---
Introduction
============
The nucleus $^{228}$Th is located in a region where strong octupole correlations are important in the properties already of the low-lying excitations. Besides the interplay of collective and single-particle excitations, which takes place in deformed rare earth nuclei, the reflection asymmetry additionally complicates the picture of excitations. Already in an earlier publication [@Mah72], a conclusion was made that the nature of the first excited 0$^+$ states in the actinide nuclei is different from that in the rare earth region, where they are due to the quadrupole vibration. The strong excitations in the (p,t)-reaction suggest that these states represent a collective excitation different from the $\beta$-vibration. Decay modes of the levels of the band on the first excited 0$^+$ state in $^{228}$Th have led to the suggestion that this band might predominantly have an octupole two-phonon structure [@Dal87]. One has to expect a complicated picture at higher excitations: residual interactions could mix the one-phonon and multiphonon vibrations of quadrupole and octupole character with each other and with quasiparticle excitations. Detailed experimental information on the properties of such excitations is needed. On the experimental side, two-neutron transfer reactions are very useful. On the theoretical side, a test of the advanced interacting boson model (IBM) and a microscopic approach, such as the quasiparticle-phonon model (QPM), would be very interesting.
After the first observation of a large number of excitations with the $L=0^+$ transfer in the (p,t) reaction seen in the odd nucleus $^{229}$Pa [@Lev94], it was logical to investigate such excitations in the even-even nucleus $^{228}$Th, since $^{229}$Pa corresponds to $^{228}$Th + p, as well as in other actinide nuclei. Such measurements were carried out for the nuclei $^{228}$Th, $^{230}$Th and $^{232}$U, and the results of a limited analysis have been published in [@Wir04] (besides the earlier preliminary study of $^{228}$Th and $^{232,234,236}$U in [@Bal96]). The paper [@Wir04] concentrated only on the energies of the excited $0^+$ states in these actinide nuclei and the (p,t) transfer strengths to these states. The (p,t) reaction, however, gives much more extensive information on specific excitations in these nuclei, which was not analyzed previously. Such information was obtained for $^{230}$Th in our paper [@Lev09] after detailed analysis of the experimental data from the (p,t) reaction. For the $0^+$ excitation, we were able to derive additional information on the moments of inertia, which can be useful in clarifying the structure of these excitations. In this paper we present the results of a careful and detailed analysis of the experimental data from the high-resolution study of the $^{230}$Th(p,t)$^{228}$Th reaction carried out to obtain deeper insight into all excitations in $^{228}$Th. The total picture for $^{228}$Th has to differ from the one for $^{230}$Th, since the first one is considered as an octupole soft and the latter as a vibration-like nucleus. It would be interesting to compare the $0^+$ excitations in the even nucleus $^{228}$Th and the odd nucleus $^{229}$Pa, the data for which in the low-energy part of excitations are known from the publication [@Lev94].
Information on excited states of $^{228}$Th prior to this study was obtained mainly from the $\alpha$-decay of $^{232}$U, the $\beta$- and EC-decay of $^{228}$Ac and $^{228}$Pa, as well as from the ($\alpha$, xn$\gamma$)-reaction. The most complete information was obtained from the $\beta$-decay of $^{228}$Ac reported by Dalmasso et al. [@Dal87] and from the EC-decay study of $^{228}$Pa by Weber et al. [@Web98]. The lowest collective bands in $^{228}$Th were studied in the ($\alpha$, xn$\gamma$)-reaction [@Schu86]. A total of 58 levels were reported in [@Dal87] and 80 levels were observed in [@Web98] below 2.1 MeV, connected by more than 240 $\gamma$-rays that were established in these studies.
Present results, derived from the $^{230}$Th(p,t)$^{228}$Th reaction, lead to about 163 levels in the energy range up to 3.25 MeV. Unfortunately, during the experiment the radioactive target was destroyed and assignments were made only for 106 levels in the range up to 2.5 MeV. Energies and cross sections for one angle were obtained additionally for 57 levels. Besides 0$^+$ excitations, where the number of reliable assignments could be increased for five states in comparison with the preliminary analysis in publication [@Wir04], information on the spins for many other states was obtained. This information was essentially complementary to what was known from publications [@Dal87; @Web98]. Some levels are grouped into rotational bands, thus allowing to derive the moment of inertia for some $0^+$, $2^+$ and $0^-$, $1^-$, $2^-$, $3^-$ bands. One of the results of Ref. [@Web98] was the establishment of the one-phonon octupole-quadruplet with $K^{\pi} = 0^-, 1^-, 2^-, 3^-$ states. In this paper we suggested the two-phonon octupole-quadruplet with $K^{\pi} = 0^+, 2^+, 4^+, 6^+$ states.
\[sec:ExAnRe\] Experiment, analysis and experimental results
============================================================
\[sec:det\_exp\] Details of the experiment
-------------------------------------------
A radioactive target of 100 $\mu$g/cm$^2$ $^{230}$Th with half-life T$_{1/2}$ = 8$\cdot10^4$ years, evaporated on a 22 $\mu$g/cm$^2$ thick carbon backing, was bombarded with 25 MeV protons at an intensity of 1-2 $\mu$A from the Tandem accelerator of the Maier-Leibnitz-Labor of the Ludwig-Maximilians-Universität and Technische Universität München. The isotopic purity of the target was about 99%. The tritons were analyzed with the Q3D magnetic spectrograph and then detected in a focal plane detector. The focal plane detector is a multiwire proportional chamber with readout of a cathode foil structure for position determination and dE/E particle identification [@Zan91; @Wir01]. The acceptance of the spectrograph was 11 msr, except for the most forward angle of 5$^\circ$ with an acceptance of 6 msr. The resulting triton spectra have a resolution of 4–7 keV (FWHM) and are background-free. The angular distributions of the cross sections were obtained from the triton spectra at ten laboratory angles from 5$^\circ$ to 40$^\circ$ for energies up to 1800 keV, but only at five angles from 7.5$^\circ$ to 30$^\circ$ for energies from 1800 to 2500 keV. The energies and cross sections for the states from 2500 keV to 3250 keV were measured only for 10$^\circ$.
A triton energy spectrum measured at a detection angle of 10$^\circ$ is shown in Fig. \[fig:specTh228\_10deg\]. The analysis of the triton spectra was performed with the program GASPAN [@Rie91]. Measurements were carried out with two magnetic settings: one for excitations up to 1.75 MeV, and another one for the energy region from 1.6 MeV to 3.3 MeV, respectively. For the calibration of the energy scale, the triton spectra from the reactions $^{184}$W(p,t)$^{182}$W, $^{186}$W(p,t)$^{184}$W and $^{234}$U(p,t)$^{232}$U were measured at the same magnetic settings. The levels in $^{230}$Th known from the study [@Lev09] were also included in the calibration.
From 106 levels identified in the spectra, 60 levels were identified for all ten angles and 46 levels only for 5 angles. They are listed in Table \[tab:expEI\]. The energies and spins of the levels as derived from this study are compared to known energies and spins, mainly from the published data [@Dal87; @Web98]. They are given in the first four columns. The ratios of cross sections at angles 7.5$^\circ$ and 26$^{\circ}$ to the one at 16$^{\circ}$, given in the next two columns, help to highlight the $0^+$ excitations (large values). The column labelled $\sigma_{\mbox{integ.}}$ gives the cross section integrated in the region from 7.5$^{\circ}$ to 30$^{\circ}$, where the cross sections are measured for all level energies. The column titled [$\sigma_{\mbox{exp.}}/\sigma_{\mbox{calc.}}$]{} gives the ratio of the integrated cross sections, from experimental values and calculations in the DWBA approximation (see Sec. \[sec:DWBA\]). The last column lists the notations of the schemes used in the DWBA calculations: sw.jj means one-step direct transfer of the $(j)^2$ neutrons in the (p,t) reaction; notations of the multi-step transfers used in the DWBA calculations are displayed in Fig. \[fig:schemes\]. Additionally, energies of 57 levels seen only in the spectrum measured at 10$^\circ$ and corresponding cross sections are listed in Table \[tab:expEsigma\].\
[ll l ccc cc r c c]{}
\
\
&&&& &[Ratio]{}&Way of\
&&\[2,7\] &This work&&[(7.5$^o$/16$^o$)]{} &[(26$^o$/16$^o$)]{}&\[[$\mu$b]{}\]& [$\sigma_{\mbox{expt.}}/\sigma_{\mbox{calc.}}$]{}&fitting\
\
\
\
&&&& &[Ratio]{}&Way of\
&&\[2,7\]& This work&&[(7.5$^o$/16$^o$)]{} &[(26$^o$/16$^o$)]{}&\[[$\mu$b]{}\]& [$\sigma_{\mbox{expt.}}/\sigma_{\mbox{calc.}}$]{}&fitting\
\
0.0 *2 && 0.00 & $0^+$ & $0^+$ && 5.83 & 5.61 & 165.56 &6.20& sw.gg\
57.8 *2 && 57.76 & $2^+$ & $2^+$ && 1.59 &0.68 &37.07 &8.30&m1a.gg\
186.8 *2 && 186.83 & $4^+$ & $4^+$ && 0.74 &0.38 &9.07 &1.90&m1a.gg\
328.0 *2 && 328.00 & $1^-$ & $1^-$ && 0.45 &0.66 &0.82 &0.50&m2a.gg\
378.2 *2 && 378.18 & $6^+$ & $6^+$ && 0.58 &0.71 &4.48 &1.60&m2a.gg\
396.9 *2 && 396.08 & $3^-$ & $3^-$ && 0.54 &0.33 &2.89 &0.56&m3a.gg\
519.2 *3 && 519.20 & $5^-$ & ($5^-$) && 1.23 & 1.33 & 0.43 &0.90&sw.gg\
622.5 *4 && 622.50 & $8^+$ & $(8^+)$ && 0.20 & 0.94 & 0.26 &&\
695.6 *3 && 695.50 & $7^-$ & $(7^-)$ && 0.15 & 0.41 & 0.37 &&\
831.9 *2 && 831.83 & $0^+$ & $0^+$ && 12.06& 7.50 & 39.10 &360 &sw.ii\
874.4 *2 && 874.42 & $2^+$ & $2^+$ && 1.22 & 0.58 & 9.57 &160 &m1a.ii\
911.6 *5 && 911.80 & $10^+$ & && & & & &\
920.6 *5 && 920.80 & $9^-$ & && & & & &\
938.7 *2 && 938.55 & $0^+$ & $0^+$ && 18.38 &7.21 & 6.83 &8.20 &sw.ii\
943.8 *4 && 944.19 & $1^-$ & $1^-$ && 0.12 & 0.67 & 0.37 &1.00 &sw.gg\
&& 968.33 & $2^-$ & && & & & &\
&& 968.43 & $4^+$ & && & & & &\
968.8 *2 && 968.97 & $2^+$ & $2^+$ && 0.67 & 0.47 & 20.0 &132&sw.ig\
979.4 *2 && 979.50 & $2^+$ & $2^+$ && 0.78 & 0.59 & 9.25 &55.6&sw.ig\
1016.4 *2 && 1016.43 & $2^+,3^-$ & $3^-$ && 0.80 & 0.47 & 5.37 &1.10&m3a.gg\
&& 1022.53 & $3^+$ & && & & & &\
&& 1059.93 & $(3^-,4^+)4^-$ &$4^-$ && & & & &\
1074.8 *3 && 1074.8 & $4^+$ & $4^+$ && 0.74 & 1.32 & 1.62 &0.26 &m1a.gg\
1091.0 *3 && 1091.01 & $4^+$ & $4^+$ && 0.74 & 0.44 & 0.42 &0.10 &m1a.gg\
1105.5 *3 && & & $6^+$ && 0.61 & 0.56 & 0.77 &21.0 &sw.ii\
1120.1 *3 && 1120.1 & $0^+$ & $0^+$ && 2.63 & 3.71 & 1.24 &0.04 &sw.gg\
&& 1122.95 & $2^-$ & && & & & &\
1142.8 *3 && 1143.2 & $5^-$ & $5^-$ && 0.80 & 0.98 & 1.10 &26.0 &sw.jj\
1153.3 *3 && 1153.48 & $2^+$ & $2^+$ && 0.65 & 0.49 & 23.89 &140 &sw.ig\
1168.0 *4 && 1168.37 & $3^-$ & $3^-$ && 0.36 & 0.58 & 0.68 &1.00 &w.gg\
&& 1174.52 & $5^+$ & && & & & &\
1175.2 *4 && 1175.40 & $2^+$ & $2^+$ && 1.05 & 0.91 & 2.09 &13.0 &sw.ig\
1201.0 *9 && 1200.54 & $3^+$ & $3^+$ && 0.31 & 1.10 & 0.40 &0.56 &m2a.gg\
1225.7 *6 && & & $4^+$ && 1.00 & 0.64 & 0.25 &1.75 &sw.jj\
&& 1226.55 & $4^-$ & && & & & &\
1261.6 *3 && 1261.5 & $4^+$ & $4^+$ && 1.33 & 1.12 & 3.64 &67.0 &sw.ii\
1270.2 *6 && 1270.0 & & $6^+$ && 0.40 & 0.97 & 0.31 &0.15 &sw.gg\
1290.4 *3 && 1290.2 & $4^+$ & $4^+$ && 1.14 & 0.88 & 3.59 &67.0 &sw.ii\
1296.0 *5 && 1297.34 & $5^-$ & $(5^-)$ && 1.33 & 1.23 & 0.50 &1.00 &sw.gg\
1319.2 *4 && & & $(2^+)$ && 0.74 & 1.08 & 0.24 &1.50 &sw.ig\
1343.9 *5 && 1344.03 & $3^-$ & $3^-$ && 0.77 & 0.33 & 0.31 &0.08 &m3a.gg\
&& 1393.4 & $1^+,2,3^-$ & $(1^+)$ && & & & &\
1415.8 *6 && 1415.92 & $2^+,3^-$ &$(3^-)$ && 1.15 & 1.20 & 0.05 &2.80 &sw.jj\
1423.8 *5 && & & $(2^+)$ && 2.20 & 1.33 & 0.16 &0.03 &m1a.gg\
1432.1 *5 && 1431.98 & $3^+,4^+$ & $4^+$ && 1.61 & 1.17 & 0.21 &6.80 &sw.ii\
&& 1448.80 & $3,4^-$ & && & & & &\
&& 1450.29 & $3^-,4^-$ & && & & & &\
1453.5 *5 && & & $(3^-)$ && 0.61 & 0.63 & 1.34 &1.80 &sw.jj\
1470.0 *5 && & & $(6^+)$ && 0.94 & 1.81 & 0.19 &0.01 &m3a.gg\
1497.4 *4 && 1497.7 & $4^+,5^-$ & $(5^-)$ && 1.07 & 0.91 & 0.37 &0.56 &sw.gg\
1511.2 *3 && & & $0^+$ && 7.96 & 6.96 & 2.13 &1.10 &sw.ig\
1531.7 *3 && & & $0^+$ && 2.21 & 0.83 & 0.47 &2.60 & sw.ii\
plus && 1531.48 & $3^+$ & $3^+$ && & & & 0.02 &m2a.gg\
&& 1539.13 & $(2,3)$ & && & & & &\
1544.4 *3 && & & $2^+$ && 1.27 & 0.65 & 1.61 &1.53 &m1a.gg\
&& 1581.0 & $(2^-)$ & && & & & &\
1586.9 *4 && & & $2^+$ && 0.98 & 0.71 & 0.31 &1.00 &sw.jj\
&& 1588.33 & $4^-$ & && & & & &\
1613.0 *5 && & & $4^+$ && 1.06 & 1.26 & 0.54 &12.0 &sw.ii\
1618.3 *5 && 1617.74 & ($3,4)^+$ & $4^+$ && 0.88 & 0.76 & 1.22 &0.16 &m2a.ii\
1627.9 *3 && & & $0^+$ && 7.44 & 5.21 & 9.66 &10.0 &sw.ig\
1638.4 *4 && 1638.25 & $2^+$ & $2^+$ && 0.59 & 0.37 & 1.45 &23.5 &sw.ii\
&& 1643.18 & $(2,3)^-$ & && & & & &\
1643.8 *3 && 1643.8 & $(2,3,4)^+$ & $4^+$ && 1.58 & 1.08 & 8.54 &160 &sw.ii\
&& 1645.89 & $3^+$ & && & & & &\
1651.4 *3 && & & $(3^-)$ && 0.08 & 0.79 & 0.86 &1.20 &sw.gg\
1667.3 *5 && 1667.3 & & $2^+$ && 0.71 & 0.61 & 3.17 &46.0 &sw.ii\
1672.3 *5 && & & $2^+$ && 0.91 & 0.53 & 1.85 &3.80 &sw.jj\
1678.4 *5 && 1678.4 & $2,3,4^+$ & $2^+$ && 0.88 & 0.75 & 1.48 &19.5 &sw.ii\
&& 1682.70 & $(3,4)^+$ & && & & & &\
&& 1683.74 & $(4^-)$ & && & & & &\
&& 1688.39 & $3^+$ & && & & & &\
1691.3 *4 && & & $0^+$ && 2.66 & 2.06 & 1.26 &0.75 &sw.ig\
&& 1707.2 & $2,3^-$ & && & & & &\
1710.7 *6 && & & $0^+$ && 1.38 & 1.86 & 0.54 &0.02 &sw.gg\
1724.6 *4 && 1724.29 & $2^+$ & $2^+$ && 1.06 & 0.66 & 2.73 &5.50 &sw.ii\
1733.8 *4 && 1735.62 & $4^+$ & $4^+$ && 1.13 & 0.86 & 2.28 &3.50 &sw.ij\
1742.8 *4 && 1743.86 & $3,4^+$ & $4^+$ && 0.81 & 0.56 & 1.36 &0.16 &m2a.gg\
1750.7 *3 && & & $0^+$ && 1.27 & 1.84 & 1.75 &0.70 &sw.jj\
1758.1 *3 && 1757.9 & $1^-,2, 3^-$ & $2^+$ && 0.85 & 0.75 & 4.35 &26.0 &sw.ig\
&& 1758.24 & $(3,4)^+$ & && & & & &\
&& 1760.25 & $(2,3,4)^+$ & && & & & &\
&& 1795.9 & $3^-,4^+$ & && & & & &\
1796.8 *3 && 1796.4 & $3^+,4,5^+$ & $4^+$ && 1.40 & 0.83 & 6.47 &89.0 &sw.ig\
&& 1797.65 & $(2^+,1^-)$ & && & & & &\
1803.0 *4 && 1802.9 & $1^-,2,3^-$ & $2^+$ && 0.65 & 0.49 & 15.34 &90.0 &sw.ig\
&& 1804.60 & $(4^+)$ & && & & & &\
&& 1811.5 & $1^-,2,3^-$ & && & & & &\
1812.7 *4 && & & $(6^+)$ && 1.35 & 1.63 & 0.62 &0.04 &sw.ig\
&& 1817.43 & $4^-$ & && & & & &\
&& 1823.4 & $3^-,4,5$ & && & & & &\
1826.2 *4 && & & $(4^+)$ && 1.16 & 0.83 & 1.91 &7.50 &sw.ij\
1840.3 *8 && 1842.2 & $2^+,3^-$ & && 1.41 & 0.33 & 0.21 & &\
1858.6 *5 && & & $(6^+)$ && 0.65 & 1.19 & 1.28 &0.06 &sw.ig\
1863.9 *5 && 1864.8 & $1^-,2,3^-$ & $(2^+)$ && 0.75 & 0.79 & 1.47 &8.10 &sw.ig\
&& 1876.5 & $3^-,4,5^-$ & && & & & &\
1878.9 *5 && 1879.0 & $3^-,4,5^-$ & $(3^-)$ && 1.05 & 0.91 & 1.93 &110 &sw.ii\
&& 1892.98 & $3^+$ & && & & & &\
1898.2 *4 && 1899.98 & $2^+$ & $(2^+)$ && 0.84 & 0.81 & 2.55 &140 &sw.ii\
&& 1901.90 & $4^+$ & && & & & &\
1903.9 *4 && & & ($6^+$) && 0.69 & 1.58 & 1.54 &0.07 &sw.gg\
&& 1906.78 & $(2^+,1^-)$ & && & & & &\
&& 1908.4 & $3^-$ & && & & & &\
1908.9 *7 && & & $0^+$ && 2.17 & 1.91 & 4.56 &1.30 &sw.jj\
&& 1924.1 & $2^-,3,4$ & && & & & &\
&& 1924.6 & $(4,5^+)$ & && & & & &\
1925.4 *4 && 1925.20 & $4^+$ & $4^+,5^-$ && 0.61 & 1.73 & 0.54 &21.0 &sw.ii\
&& 1928.54 & $3^+$ & && & & & &\
&& 1937.16 & $(3,4)^+$ & && & & & &\
1938.3 *4 && 1938.9 & $4^+$ & ($4^+$) && 1.06 & 0.76 & 1.99 &0.67 &m2a.gg\
&& 1944.85 & $3^+$ & && & & & &\
&& 1945.8 & $4^+,5^-$ & && & & & &\
1947.8 *7 && & & ($2^+$) && 1.02 & 0.75 & 0.77 &3.50 &sw.ig\
&& 1949.7 & $1^+,2,3^+$ & && & & & &\
1959.7 *6 && 1958.5 & $2^+$ & $(2^+)$ && 0.10 & 1.69 & 0.43 &1.50 &sw.ig\
&& 1964.90 & $(2^+)$ & && & & & &\
1971.7 *4 && & & $(2^+,3^-)$ && 0.66 & 0.81 & 0.79 &3.10 &sw.ig\
&& 1974.20 & $4^+$ & && & & & &\
1981.9 *4 && 1981.97 & $3,4^+$ & $(3^-)$ && 1.68 & 0.77 & 1.70 & 2.60 &sw.gg\
&& 1987.46 & $4^+$ & && & & & &\
1993.9 *5 && & & ($3^-$) && 0.97 & 0.72 & 1.80 &2.80 &sw.gg\
2010.4 *6 && 2010.15 & $2^+,3,4^+$ & $(2^+)$ && 0.46 & 0.43 & 0.76 &13.0 &sw.ig\
&& 2013.6 & $(3,4)^+$ & && & & & &\
&& 2016.75 & $4^+,5^-$ & && & & & &\
&& 2022.73 & $2^+$ & && & & & &\
2030.3 *4 && 2029.6 & $(2^+)$ & $2^+$ && 0.54 & 0.23 & 0.84 &16.0 &sw.ig\
&& 2037.0 & $(3,4)^+$ & && & & & &\
2044.7 *5 && & & $0^+$ && 9.22 & 4.56 & 0.57 &3.10 &sw.ii\
2052.1 *4 && & & ($6^+$) && 0.72 & 1.30 & 3.70 &180 &sw.ii\
2069.6 *5 && & & $2^+$ && 0.76 & 0.56 & 1.38 &6.10 &sw.ig\
2079.9 *5 && & & $0^+$ && 17.08 & 13.13 & 4.62 &25.9 &sw.ii\
2091.2 *7 && & & $(6^+)$ && 0.62 & 0.82 & 1.20 &35.0 &sw.ii\
2111.6 *5 && & & $(2^+)$ && 0.70 & 0.71 & 2.57 &11.0 &sw.ig\
&& 2123.1 & $(2^+,3^-)$ & && & & & &\
2131.3 *6 && & & $0^+$ && 6.84 & 4.53 & 24.80 &120 &sw.ii\
2152.8 *4 && & & $(4^+)$ && 1.30 & 0.90 & 4.13 &98.0 &sw.ii\
2159.4 *6 && & & $0^+$ && 3.78 & 1.54 & 1.18 &8.10 &sw.ii\
2170.3 *4 && & & $(2^+)$ && 1.00 & 0.85 & 5.61 &26.0 &sw.ig\
2198,2 *4 && & & $2^+$ && 0.59 & 0.62 & 3.81 &19.5 &sw.ig\
2215.9 *4 && & & ($4^+$) && 1.40 & 1.15 & 6.00 &130 &sw.ii\
2235.2 *7 && & & ($4^+$) && 0.98 & 0.86 & 2.82 &61.0 &sw.ii\
2290.0 *7 && & & $0^+$ && 9.96 & 5.75 & 11.00 &61.0 &sw.ii\
2302.9 *5 && & & $(4^+)$ && 1.09 & 0.84 & 2.75 &62.0 &sw.ii\
2323.2 *5 && & & $2^+$ && 0.41 & 0.62 & 2.24 &16.0 &sw.ig\
2335.9 *5 && & & ($4^+$) && 2.13 & 1.65 & 17.10 &370 &sw.ig\
&& & & ($0^+$) && & & 4.50$^a$ & 25.0 &sw.gg+14\
2344.2 *5 && & & ($3^-$) && 0.77 & 0.58 & 6.65 &10.0 &sw.gg\
2356.2 *5 && & & ($2^+$) && 0.63 & 0.61 & 4.61 &21.5 &sw.ig\
2375.5 *8 && & & ($2^+$) && 0.78 & 0.60 & 4.87 &22.0 &sw.ig\
2398.3 *9 && & & ($3^-$) && 0.76 & 0.75 & 7.36 &11.0 &sw.gg\
2408.8 *9 && & & ($4^+$) && 1.87 & 1.28 & 2.34 &60.0 &sw.ii\
2441.7 *5 && & & ($2^+$) && 0.71 & 0.51 & 10.32 &47.0 &sw.ig\
2456.8 *5 && & & $0^+$ && 16.18 & 1.27 & 0.53 &5.20 &sw.ii\
2476.7 *5 && & & ($2^+$) && 0.62 & 0.52 & 10.38 &48.0 &sw.ig\
2494.1 *5 && & & ($2^+$) && 0.65 & 0.47 & 12.74 &63.5 &sw.ig\
\
**********************************************************************************************************
$^a$ The value after subtracting a constant of 14 $\mu$b (see text in Sec. \[sec:DWBA\] for explanation.\
[lclclc]{}\
E \[keV\] & $d\sigma/d\Omega$ & E \[keV\] & $d\sigma/d\Omega$ & E \[keV\] & $d\sigma/d\Omega$\
\
\
2513.5 *7 & 2.00 & 2742.3 *4 &5.50 & 3035.6 *9 & 0.95\
2531.5 *7 & 6.60 & 2763.7 *4 &8.60 & 3046.4 *6 & 2.10\
2536.8 *9 & 3.20 & 2781.4 *5 &1.75 & 3059.2 *5 & 2.15\
2542.4 *9 & 1.85 & 2798.6 *8 &1.55 & 3075.2 *5 & 2.20\
2554.5 *5 & 6.00 & 2805.6 *7 &2.00 & 3085.2 *8 & 1.25\
2566.3 *6 & 2.20 & 2821.0 *5 &2.90 & 3097.0 *6 & 3.10\
2595.4 *5 & 5.40 & 2839.3 *6 &1.30 & 3104.7 *6 & 3.40\
2606.1 *5 & 23.5 & 2853.7 *5 &2.75 & 3112.7 *11 &1.70\
2615.1 *9 & 0.15 & 2868.1 *5 &3.20 & 3119.9 *9 &2.30\
2634.8 *5 & 1.60 & 2877.5 *8 &1.80 & 3128.2 *10 &1.25\
2644.0 *3 & 9.20 & 2883.7 *9 &1.60 & 3158.8 *8 &1.50\
2657.1 *4 & 5.20 & 2918.8 *6 &1.85 & 3165.7 *6 &2.00\
2660.1 *5 & 6.00 & 2927.4 *5 &3.25 & 3186.0 *6 &2.00\
2667.1 *5 & 3.30 & 2936.8 *9 &1.40 & 3195.2 *6 &2.60\
2676.0 *6 & 67.2 & 2945.3 *9 &1.35 & 3209.6 *12&1.40\
2688.4 *4 & 2.10 & 2955.1 *8 &1.25 & 3214.8 *9 &2.20\
2695.6 *7 & 1.10 & 2993.1 *12 &1.00 &3225.0 *20 &0.50\
2705.5 *5 & 1.35 & 2999.5 *10 &1.50 &3232.9 *13 &1.20\
2718.4 *5 & 2.10 & 3014.3 *11 &0.80 &3239.9 *8 &3.40\
*********************************************************
\[sec:DWBA\] DWBA analysis
--------------------------
The spins of the excited states in the final nucleus $^{228}$Th were assigned via an analysis of the angular distributions of tritons from the (p,t) reaction. The angular distributions for $0^+$ excitations have a steeply rising cross section at very small reaction angles, and a sharp minimum at a detection angle of about 14$^\circ$. This pronounced feature helped to identify most of these states in complicated and dense spectra, even without fitting experimental angular distributions. No complication of the angular distributions was expected, since the excitation of $0^+$ states predominantly proceeds via a one-step process. This is not the case for the excitation of states with other spins, where multi-step processes could play a very important role.
The identification of these states is possible by fitting the experimental angular distributions with those calculated in the distorted-wave Born approximation (DWBA). The potential parameters suggested by Becchetti and Greenlees [@Bec69] for protons and by Flynn [*et al.*]{} [@Fly69] for tritons were used in the calculations. These parameters have been tested via their description of angular distributions for the ground states of $^{228}$Th, $^{230}$Th and $^{232}$U [@Wir04]. Minor changes of the parameters for tritons were needed only for some $3^-$ states, particularly for the states at 396.9 and 1016.4 keV. For these states, the triton potential parameters suggested by Becchetti and Greenlees [@Bec71] were used. For each state the binding energies of the two neutrons are calculated to match the outgoing triton energies. The corrections to the reaction energy are introduced depending on the excitation energy. For more details see [@Lev09].
A problem arising in such calculations is the lack of prior knowledge of the microscopic structure of these states. We can assume, however, that the overall shape of the angular distribution of the cross section is rather independent of the specific structure of the individual states, since the wave function of the outgoing tritons is restricted to the nuclear exterior and therefore to the tails of the triton form factors. To verify this assumption, DWBA calculations of angular distributions for different $(j)^2$ transfer configurations to states with different spins were carried out in our previous paper [@Lev09]. The coupled-channel approximation (CHUCK3 code of Kunz [@Kun]) was used in these calculations. Indeed, the calculated angular distributions are very similar. Nevertheless, they depend to some degree on the transfer configuration, the most pronounced being found for the 0$^+$ states, what is confirmed by the experimental angular distributions. The best reproduction of the angular distribution for the ground state was obtained for the transfer of the $(2g_{9/2})^2$ configuration in the one-step process. This orbital is close to the Fermi surface and was considered as the most probable one in the transfer process. Other transfer configurations that might be of importance are $(1i_{11/2})^2$ and $(1j_{15/2})^2$, also near the Fermi surface. Better reproduction of the angular distribution for some $0^+$ states is obtained just for these configurations. The main features of the angular distribution shapes for 2$^+$ and 4$^+$ states are even more weakly dependent on the transfer configurations. Nevertheless the $(2g_{9/2})^2$, $(1i_{11/2})^2$ and $(1j_{15/2})^2$ configuration, alone or in combination, were used in the calculations for these states too.
Results of fitting the angular distributions for the states assigned as $0^+$ excitations are shown in Fig. \[fig:angl\_distr\_0+\]. The agreement between the fit and the data is excellent for most of the levels. Remarks are needed only for the levels at 1531.7 and 2335.9 keV. The spin $3^+$ was assigned to the level at 1531.47 keV in [@Dal87; @Web98]. A level at the close-lying energy of 1531.7 keV has been observed also in the (p,t) reaction, but the angular distribution of tritons cannot be fitted by calculations for transition to the $3^+$ state. The maximum cross section for forward angles suggests the presence of a $0^+$ excitation, though the angular distribution fitted by a calculation for transfer of the $(1i_{11/2})^2$ configuration to the $0^+$ state is not perfect. A satisfactory fit of the experimental angular distribution was obtained assuming overlapping states with spins $0^+$ and $3^+$ (Fig. \[fig:angl\_distr\_0+\]), thus confirming the assignment for both states. An ambiguous picture is observed for the 2335.9 keV state, where the angular distribution is measured for a limited range of angles. The fitting agreement is perfect for a transition to the 4$^+$ state, but the cross section is surprisingly large for a 4$^+$ state, twice larger than for the $4^+$ member of the g.s. band. Therefore, the possibility of a 0$^+$ excitation can not be excluded, but the experimental angular distribution is fitted for a $0^+$ state only with adding a constant of 14 $\mu$b. This ambiguity can be resolved by measurements in wider angular regions.
Thus we can make firm $0^+$ assignments for 17 states for energies excitations below 2.5 MeV, in comparison with 12 states found in the preliminary analysis of the experimental data [@Wir04]. Of course, some higher lying 0$^+$ levels are lost because of the cutoff of the investigated energy region. But as follows from a similar study for $^{230}$Th, only a few $0^+$ states are observed above 2500 keV, where the density of $0^+$ excitations decreases for higher energies (or else that the cross section of such excitations is very low and they are hidden in very dense and complicated spectra). Therefore, we can compare 24 $0^+$ states in $^{230}$Th with only 17 $0^+$ states in $^{228}$Th in the same energy region.
Similar to $0^+$ excitations, the one-step transfer calculations give a satisfactory fit of angular distributions for about 80% of the states with spins different from $0^+$, but about 20% of these states need the inclusion of multi-step excitations. Multi-step excitations have to be included to fit the angular distributions already for the $2^+$, $4^+$ and $6^+$ states of the g.s. band. Fig. \[fig:schemes\] shows the schemes of the multi-step excitations, tested for every state in those cases, where one-step transfer did not provide a successful fit. Fig. \[fig:angl\_distr\_natur\] demonstrates the quality of the fit of some different-shaped angular distributions for the excitation of states with spin $2^+$ by calculations assuming one-step and one-step plus two-step excitations, respectively. Results of similar fits for the states assigned as $4^+$, $6^+$ and $1^-$, $3^-$, $5^-$ excitations are shown in Fig. \[fig:angl\_distr\_unnatur\].
The assignments of the spins resulting from such fits are presented in Table \[tab:expEI\], together with other experimental data. Special comments are needed for the column displaying the ratio $\sigma_{exp}/\sigma_{cal}$. Calculated cross sections for the specific transfer configurations differ very strongly. Since we have no a priori knowledge of the microscopic structure of the excited states, and thus do not know the relative contributions of the specific $(j)^2$ transfer configurations to each of these states, these ratios cannot be considered as spectroscopic factors. Nevertheless, a very large ratio, such as in the case of the $(1i_{11/2})^2$ transfer configurations used in the calculation for some $0^+$ and even $2^+$ and $4^+$ states, is unexpected. Surprisingly, the shape just for this neutron configuration gives the best agreement with experiment.
Some additional comments on Table \[tab:expEI\] are needed. In all cases, where the firm assignment were known from the previous studies [@Dal87; @Web98], they are confirmed by the (p,t) angular distribution analysis. In those cases, where two or three possible spin assignments were proposed earlier, the (p,t) angular distribution analysis allows to select only one assignment almost in all cases. For the energies above 2030 keV only the assignments from the (p,t) reaction are possible at present. The following remarks are needed in those cases, where the assignments from different publications are in contradiction, which can be removed using the data from the (p,t) reaction.
*938.7 keV.* Spin $2^-$ was assigned for this level in [@Dal87]. Our fit of the angular distribution gives reliably spin $0^+$ in agreement with [@Web98].
*943.8 keV.* Spin $2^+$ was assigned for this level in [@Dal87]. The angular distribution rejects this value and agrees with the assignment of spin $1^-$ accepted in [@Web98].
*968-969 keV.* Three levels around 968 keV with spins $2^+, 4^+, 2^-$ were identified in [@Web98] and two levels with spins $2^+, 3^-$ in [@Dal87]. There is a discrepancy in assignment for the 968.33 keV level as $3^-$ in [@Dal87] and as $2^-$ in [@Web98]. This line is masked in the (p,t) spectrum by a strong line from the transition to the 969 keV level. But since the angular distribution is very well described by a calculation leading to a $2^+$ state and is not disturbed by a transition to spin $3^-$, the assignment $2^-$ is preferable (transition is weak).
*1016.4 keV.* The discrepancy in assignment for the 1016.4 keV level as $2^+$ in [@Dal87] and as $3^-$ in [@Web98] can not be removed by the (p,t) angular distribution, since it can be fitted by a transition sw.ig to $2^+$ and m3a.gg to $3^-$, respectively. However, transitions seen in the decay of $^{228}$Pa [@Web98] from this state to the $5^-$ and $4^+$ states leads to the assignment of $3^-$. We accepted this spin also due to strong arguments in [@Web98], including the assignment of spins $2^-$ and $1^-$ to the levels at 968.3 keV and 943.8 keV as members of the $K^{\pi}=1^-$ band.
*1059.9 keV.* From the tentative assignments $(4^+,3^-)$ in [@Dal87] and the firm assignment $4^-$ in [@Web98] for this level, the latter has to be additionally supported by the fact, that the corresponding line in the (p,t) spectrum was not seen (transition to the state of unnatural parity).
*1225.7 and 1226.56 keV.* Spin $4^-$ was assigned for the level 1226.56 keV in both [@Dal87] and [@Web98]. The level with the close energy 1225.7 keV is seen in the (p,t) reaction, but the angular distribution agrees with the assignment of spin $4^+$. Therefore both levels are present in Table \[tab:expEI\].
*1393.4 keV*. This level was observed in the decay of $^{228}$Pa with restriction of the spin-parity to $1^+$, $2$ and $3^-$ by its population and depopulation [@Web98]. Additional restriction from the $W(90^\circ)/W(180^\circ)$ angular distribution ratio indicates that this level has most likely $I^\pi=1^+$ and that the $2^+$ and $3^-$ assignments are nearly excluded. This level was not observed in the (p,t) reaction, thus supporting such conclusion.
*1415.8 keV.* Spin $2^+$ was assigned to this level in [@Dal87] and spins $2^+$ or $3^-$ were allowed in [@Web98]. The angular distribution of tritons gives preference to spin $3^-$.
*1432.1 keV.* The discrepancy between $3^+$ in [@Dal87] and $4^+$ in [@Web98] for the 1432.1 keV level is removed already by the fact of the excitation of this state in the (p,t) reaction, and additionally by the angular distribution leading to the $4^+$ assignment. Also additional lines observed in the decay of $^{228}$Pa [@Web98], leading to the 6$^+$ level, confirm this assignment.
*1450.29 keV.* Spin $3^-$ was assigned to this level in [@Dal87] and spin $4^-$ in [@Web98]. The fact that this level is not observed in the (p,t) reaction gives preference for an assignment of spin $4^-$ not excluding spin $3^-$.
*1531.7 keV.* Spin $3^+$ was assigned for the level at 1531.47 keV both in [@Dal87] and [@Web98]. However, the angular distribution of tritons for the level with the close-lying energy 1531.7 keV indicates another spin value. It has a steeply rising cross section at small angles as for a $0^+$ excitation, however, the minimum at a detection angle about $14^{\circ}$ is not sharp. Therefore we assumed an overlapping of peaks of two levels, one of which is a $0^+$ level.
*1643.8 keV.* There are two close-lying levels at 1643.18 keV with spin 2$^-$ or 3$^-$ identified both in [@Dal87] and [@Web98] and 1643.8 keV, respectively, as identified in [@Web98] with an assignment of possible spins (2,3,4)$^+$. Only the level at 1643.8 keV is observed in the (p,t) reaction with clear assignment of spin 4$^+$.
*1733.8 keV.* We assumed that the level at 1735.6 keV, identified as a 4$^+$ state in [@Dal87] and as 2$^+$,3,4$^+$ state in [@Web98], and the level at 1733.8 keV seen in the (p,t) reaction with an assignment of spin 4$^+$ are identical, though the energy difference is larger than the energy error.
*1742.8 keV.* Spin 3 was assigned to the level at 1743.86 keV in [@Dal87] and spin 4$^+$ in [@Web98]. The angular distribution from the (p,t) reaction prefers the assignment of spin 4$^+$.
*1758-1760 keV.* Several close-lying levels were identified at 1757.9 keV with spin $1^-,2,3^-$ [@Web98], at 1758.24 keV with spin (3,4)$^+$ [@Dal87], at 1760.25 keV with spin 4$^+$ [@Dal87] and with spin $(2,3)^+$ [@Web98]. Different $\gamma$-lines were used in the identification of the levels at 1757.9 and 1758.24 keV: 741.9, 1361.4, 1430.0 keV for the first one and 1571.52 and 1700.59 keV for the latter. At the same time, in [@Dal87] the line at 1430.0 keV was used for the identification of another level at 1617.74 keV, and the important line at 1758.24 keV was used for the identification of the level at 1944.85 keV. The line at 1758.11 keV can be attributed to the decay of the level at 1758.24 keV, then the spin of this state distinctly has to be $2^+$. The ambiguity cannot be solved with the (p,t) data. Therefore we put the level at 1758.1 keV with an assignment of spin $2^+$ from the (p,t) study in correspondence with the level at 1757.9 keV in [@Web98], but for the level at 1758.24 keV in [@Dal87] we do not exclude the spin $2^+$, too. As far as the level at 1760.25 keV is concerned, a spin $2^+$ can be nearly excluded, since this line is not observed in the (p,t) reaction.
*1796.8 keV.* Two close-lying levels were identified: 1795.9 keV with an assignment $(4^+,3^-)$ in [@Dal87] and 1796.4 keV with an assignment as $3^+,4,5^+$ in [@Web98]. The level at 1796.8 keV with spin $4^+$ is observed in the (p,t) reaction. It is problematic to put this level in correspondence with one of the observed ones in decay, considering the assignments. Therefore only energetic proximity was taken into account.
*1908.9 keV.* The level at 1908.4 keV, $(3^-)$ was identified in [@Web98], however, a level with almost the same energy of 1908.9 keV as observed in the (p,t) reaction was clearly identified as a $0^+$ state, they must be considered as a different levels.
*2010.4 keV.* There is discrepancy in the assignment of spin to the level at 2010.15 keV: $2^+,3$ in [@Web98] and $4^+$ in [@Dal87]. The angular distribution from the (p,t) reaction prefers spin $(2^+)$.\
[cccccccccc]{}\
$0^+$ & $1^+$ & $2^+$ & $3^+$ & $4^+$ & $5^+$ & $6^+$ & $7^+$ & $ 8^+$ &\
\
0.0 && 57.8 && 186.8 && 378.2 && 622.5 &\
831.9 && 874.4 && 968.4 &&1105.5 &&1281 &\
&& 968.8 & 1022.5 & 1091.0 &1174.5 & 1270.2 &1380 &1497 &\
&& 1153.5 & 1200.5 & 1261.5 & & & & &\
938.6&& 979.5 && 1074.8 &&&&&\
1120.1 && 1175.4 && 1290.2 && 1470.0& &&\
&& 1319.2 && 1432.0 && && &\
1511.2 && 1544.4 && 1618.3 && && &\
&& 1638.3 & 1688.4 & 1760.2 &&&&&\
&& && 1643.8 && (1812.7) && &\
(1531.7) && 1586.9 && && &&&\
1627.9 && 1667.3 && 1742.8 && 1858.6 &&&\
&& & & 1796.8 && 1903.9 &&&\
&& 1724.6 & 1760.3 & 1826.2 && &&&\
&& 1803.0 && 1938.3 && &&&\
&& 1898.2 & 1937.2 & 1987.5 &&&&&\
&& 1899.98 & 1944.85 & (2010.17) &&&&&\
1908.9 && 1947.8 && 2037.0 &&&&&\
2079.9 && 2111.6 && && &&&\
2131.3 && 2170.3 && && &&&\
2290.0 && 2323.2 && && &&&\
(2335.9) && 2375.5 && && &&&\
\
$K^\pi$& $1^-$ & $2^-$ & $3^-$ & $4^-$ & $5^-$ & $6^-$ & $7^-$ & $8^-$ &$9^-$\
\
$0^-$& 328.0 && 396.9 && 519.2 && 695.6 && 920.6\
$1^-$& 943.8 & 968.3 & 1016.4 & 1059.2 & 1143.2 &&&&\
$2^-$ && 1122.9 & 1168.4 & 1226.6 & 1297.3 &&&&\
$3^-$ && & 1344.0 & & 1497.7 &&&&\
\[Disc\] Discussion
===================
\[sec:bands\]Collective bands in $^{228}$Th
-------------------------------------------
After the assignment of spins to all excited states, those sequences of states can be identified, which show the characteristics of a rotational band structure. An identification of the states attributed to rotational bands was made on the basis of the following conditions:
a\) the angular distribution for a band member candidate state is fitted by DWBA calculations for the spin necessary to put this state in the band;
b\) the transfer cross section in the (p,t) reaction to states in the potential band has to decrease with increasing spin;
c\) the energies of the states in the band can be fitted approximately by the expression for a rotational band $E = E_0+
AI(I+1)$ with a small and smooth variation of the inertial parameter $A$.
Collective bands identified in such a way are shown in Fig. \[fig:bands\] and are listed in Table \[tab:bands\] (for a calculation of the moments of inertia). The procedure can be justified in that some sequences meeting the above criteria are already known from gamma-ray spectroscopy to be rotational bands [@Dal87; @Web98], so similar sequences are rotational bands, too. The straight lines in Fig. \[fig:bands\] strengthen the argument for these assignments. It is worth mentioning, that the assignments of $0^+$ even for the states at 1531.7 and 2335.9 keV are supported by one $2^+$ state on top of them. The bands built on states of one-phonon octupole-quadruplet (the band $K^{\pi}=1^-$ was not correctly identified in [@Dal87]), the band with $K^{\pi}=0^+$, 831.9 keV [@Dal87; @Web98], $K^{\pi}=0^+$, 938.6, 1120.1 keV [@Web98] and $K^{\pi}=2^+$, 968.8 and 1153.5 keV [@Dal87; @Web98] were identified earlier. Additional levels are added to these bands from the (p,t) study (Table \[tab:bands\]). Two bands with $K^{\pi}=2^+$ are added in Table \[tab:bands\], based only on the analysis of the decay of $^{228}$Ac [@Dal87]: at 1638.23 keV and at 1899.98 keV. There is only contradiction in the spin assignment for the 2010.17 keV level. The (p,t) data do not support spin $4^+$ and prefer a $2^+$ assignment instead.
In Fig. \[fig:moments\] we present moments of inertia (MoI) obtained by fitting the level energies of the bands displayed in Fig. \[fig:bands\] by the expression $E = E_0 + AI(I+1)$ for close-lying levels, i.e. they were determined for band members using the ratio of $\Delta
E$ and $\Delta [I(I+1)]$, thus saving the spin dependence of the MoI. This procedure is valid for all bands except the 943.8 keV, $1^-$ band. The usual procedure leads to strongly staggering values. In the case of the $K^{\pi} = 0^-$ and $K^{\pi} = 1^-$ bands, the Coriolis interaction mixes the band members only for $I$ odd. The $I$ even members of the $K^{\pi} = 1^-$ band remain unperturbed. In a simple two level model ($K^{\pi} = 0^-$ and $K^{\pi} = 1^-$ bands) the following expression can be obtained for the band energies $$\begin{aligned}
E(I,K^{\pi}=1^-) \sim E_1 +(A_1+B)I(I+1)\enspace \makebox{for $I$ odd} \nonumber \\
E(I,K^{\pi}=1^-) \sim E_1+A_1I(I+1) \enspace \makebox{for $I$ even}\end{aligned}$$ where $E_1=E_1^{'}-A_1$ and $B={C^2}/({E_1^{'}-A_1-E_0^{'}})$, $E_0^{'}$ and $E_1^{'}$ are the intrinsic bandhead energies, $A_1$ is the inertial parameter and $C$ is the strength of the Coriolis interaction, which is believed to be small. An effective parameters of inertia behaves then as $$\begin{aligned}
A_1(eff) = A_1 + \frac{1}{2}B(I+1)\enspace \makebox{for $I$ odd} \nonumber \\
A_1(eff) = A_1 - \frac{1}{2}B(I-1)\enspace \makebox{for $I$ even}\end{aligned}$$\
Fitting these expressions to the experimental data gives the smoothly changing values of the moment of inertia between 76.9 and 78.6 as shown in Fig.\[fig:moments\] with parameter $B=0.75 \div 0.68$ (thus staggering is removed).
In most cases MoI slightly increase with the increasing spin. There are few cases of $0^+$ bands where MoI are sloping down. This can be explained as an effect of the Coriolis interaction of a 2qp $K^{\pi}=0^ +$ band with a nearby lying 2qp $K^{\pi}=1^+$ band (not seen in the (p,t) reaction). A similar effect was observed e.g. in $^{168}$Er for the $3^-$ and $4^-$ 2qp bands [@Bur85]. In $^{228}$Th an additional MoI even-odd spin staggering is expected for the $K^{\pi}=1^+$ band similar to that for the octupole $K^{\pi}=1^-$ band since the odd spin states of the $K^{\pi}=1^+$ band have no counterparts in the $K^{\pi}=0^+$ band.
The obtained MoI cover a broad range, from $\sim$50 MeV$^{-1}$ to $\sim$100 MeV$^{-1}$. The negative parity bands based on the states with spin 1$^-$, interpreted as the octupole-vibrational bands [@Dal87; @Web98], have high MoI. The $0^+$ band at 1120.1 keV, considered as $\beta$ - vibrational band, has the smallest MoI, close to the one of the ground-state band. At this stage, it is difficult to state a complete correlation between the intrinsic structure of the bands and the magnitude of their MoI. Nevertheless, one can assume for the $0^+$ bands that some of the larger MoI could be related to the two-phonon octupole structure and the smallest MoI could be related to the one-phonon quadrupole structure. The bands with intermediate values of the MoI could be based on the two-phonon quadrupole excitations. If the moments of inertia do indeed carry information on the inner structure of the bands, then the number of excitations with a structure as in the g.s. or $\beta$-vibrational states in $^{228}$Th is small.
\[sec:quadr\] Quadruplets of octupole excitations
-------------------------------------------------
The lowest negative-parity excitations with $K^{\pi} = 0^-, 1^-, 2^-, 3^-$ are generally interpreted as octupole vibrational. They are one-phonon octupole excitations. The corresponding energies in $^{228}$Th as 328.0, 944.2, 1123.0, and 1344.0 keV were established already in [@Web98]. Here we confirmed these assignments after the removal of some ambiguities. They are the bandheads of the rotational bands which are displayed in Fig. \[fig:quadruplet\] together with the (p,t) cross sections and their parameters $A$.
In the case of $^{230}$Th, the assumption was made that the strongly excited first 0$^+$ state, together with also strongly excited states with spins 2$^+$ and 4$^+$, accompanied by somewhat weaker excited state with spin 6$^+$, belong to the two-phonon octupole quadruplet [@Lev09]. Strong excitations and close rotational parameters were the arguments for assigning the same structure to these states. As one can see in Fig. \[fig:strength\], the first 0$^+$ state in $^{228}$Th at 831.9 keV is also strongly excited. Taking into account the decay properties of the band on this 0$^+$ state, the suggestion was made that this band has an octupole two-phonon structure [@Dal87]. The picture for other states is not so transparent. There is no prominent excitation strength of the $2^+$ and $4^+$ states just above this 0$^+$ state. The first excited 2$^+$ state, which is not a member of the 0$^+$ band, is the state at 968.8 keV. But the de-excitation of the band built on this state demonstrates the properties expected for a $\gamma$-vibrational band. Moreover, its moment of inertia is much smaller than the one derived for the 0$^+$ band at 831.9 keV. For the band built on the $2^+$ state at 1153.3 keV, the moment of inertia is close to the one of the band built on the $0^+$ state at 831.9 keV and the state at 1153.3 keV is relatively strong excited in the (p,t) reaction. The $4^+$ and $6^+$ states, which do not belong to rotational bands, which are strongly excited in the (p,t) reaction and could be members of the two-phonon octupole quadruplet, are the states at 1643.8 and 1905.8 keV. The level at 1812.7 keV, tentatively assigned as a $6^+$ state, can be attributed to the band based on the $4^+$ state, the corresponding inertial parameter again is very close to the one for $0^+$ and $2^+$ bands. No members of a rotational band can be related to the $6^+$ band head at 1905.8 keV.
\[sec:IBM\] IBM calculations
----------------------------
In the Interacting Boson Model (IBM), the positive-parity states are described by introducing $s$ and $d$ bosons, while for the negative parity states one has to introduce additional bosons with odd values of angular momentum (at least one $f$ boson). In the region of transitional actinides, where octupole deformation might develop, the IBM-$spdf$ (which uses $p$ and $f$ bosons) was applied with success in Refs. [@Eng87; @Zam01; @Zam03].
In the present paper, we adopt the IBM-$spdf$ framework for calculating the low-lying positive and negative parity states in $^{228}$Th. In Ref. [@Zam01], the IBM calculations for this nucleus have been already performed. However, these calculations used only a simplified Hamiltonian to describe the existing (up to that date) electromagnetic data. More recent calculations (which also used a simplified Hamiltonian) [@Wir04] indicated that IBM fails completely to reproduce the (p,t) spectroscopic factors. The calculated first excited states were found with a transfer strength of $\simeq$1$\%$ of that of the ground state and the higher states were even weaker, whereas experimentally the first excited state is seen with $\simeq$30$\%$ of the ground-state intensity. In order to treat these spectroscopic observables in a reasonable approach, we used the method suggested in Ref. [@Pas10], where it was shown that the addition of the second-order O(5) Casimir operator in the Hamiltonian can account for the observed (p,t) spectroscopic factors.
The Hamiltonian employed in the present paper is similar to the one used in Refs. [@Zam01; @Zam03] and is able to describe simultaneously the positive and negative parity states: $$\begin{aligned}
\hat{H}_{spdf}=\mathrm{\epsilon}_{d} \hat{n}_{d}+\mathrm{\epsilon}_{p}
\hat{n}_{p}+\mathrm{\epsilon}_{f} \hat{n}_{f} + \mathrm{\kappa}(\hat{Q}_{spdf}\cdot
\hat{Q}_{spdf})^{(0)}\nonumber\\
+ \mathrm{\mathit a_{3}}
[(\hat{d}^{\dagger}\tilde{d})^{(3)} \times (\hat{d}^{\dagger}\tilde{d})^{(3)}]^{(0)}\label{eq1},\end{aligned}$$ where $\epsilon_{d}$, $\epsilon_{p}$, and $\epsilon_{f}$ are the boson energies and $\hat{n}_{p}$, $\hat{n}_{d}$, and $\hat{n}_{f}$ are the boson number operators, $\kappa$ is the quadrupole-quadrupole interaction strength and $a_3$ is the strength of the O(5) second order Casimir operator. In the $spdf$ model, the quadrupole operator is considered as being [@Kuz90]: $$\begin{aligned}
\hat{Q}_{spdf}=\hat{Q}_{sd}+\hat{Q}_{pf}=\nonumber\\
(\hat{s}^{\dagger}\tilde{d}+\hat{d}^{\dagger}\hat{s})^{(2)}+\chi^{(2)}_{sd}(\hat{d}^{\dagger}\tilde{d})^{(2)}
+\frac{3\sqrt{7}}{5}[(p^{\dagger}\tilde{f}+f^{\dagger}\tilde{p})]^{(2)}\nonumber\\
+\chi^{(2)}_{pf} \left\{
\frac{9\sqrt{3}}{10}(p^{\dagger}\tilde{p})^{(2)}+\frac{3\sqrt{42}}{10}(f^{\dagger}\tilde{f})^{(2)}\right\}\label{eq2}\end{aligned}$$
The quadrupole electromagnetic transition operator is: $$\begin{aligned}
\hat{T}(E2)=e_{2} \hat{Q}_{spdf}\label{eq3},\end{aligned}$$ where $e_{2}$ represents the boson effective charge. To ensure no-vanishing E2 transitions between the states containing no $pf$ bosons and those having $(pf)^{2}$ components we follow the approach described in Refs. [@Zam01; @Zam03], where the mixing of different positive parity-states with different $pf$ components is achieved by introducing in the Hamiltonian a dipole-dipole interaction term of the form: $$\begin{aligned}
\hat{H}_{int}=\alpha \hat{D}^{\dagger}_{spdf}\cdot \hat{D}_{spdf}+ H.c.\label{eq6}\end{aligned}$$ where $$\begin{aligned}
\hat{D}_{spdf}=-2\sqrt{2}[{p}^{\dagger}\tilde{d}+{d}^{\dagger}\tilde{p}]^{(1)}
+\sqrt{5}[{s}^{\dagger}\tilde{p}+{p}^{\dagger}\tilde{s}]^{(1)}\\\nonumber
+\sqrt{7}[{d}^{\dagger}\tilde{f}+{f}^{\dagger}\tilde{d}]^{(1)}\label{eq7}\end{aligned}$$ is the dipole operator arising from the $O$(4) dynamical symmetry limit, which does not conserve separately the number of positive and negative parity bosons [@Kuz90; @Kuz89]. This term will also be important later in the calculations of the two-neutron transfer intensities. The interaction strength is given by the $\alpha$ parameter and is chosen to have a very small value, $\alpha$=0.0005 MeV, similar to Refs. [@Zam01; @Zam03].
For the $E1$ transitions, a linear combination of the three allowed one-body interactions was taken: $$\begin{aligned}
\hat{T}(E1)=e_{1}[\chi_{sp}^{(1)}({s}^{\dagger}\tilde{p}+{p}^{\dagger}\tilde{s})^{(1)}
+({p}^{\dagger}\tilde{d}+{d}^{\dagger}\tilde{p})^{(1)}\nonumber\\
+\chi_{df}^{(1)}({d}^{\dagger}\tilde{f}+{f}^{\dagger}\tilde{d})^{(1)}]\label{eq4},\end{aligned}$$ where $e_{1}$ is the effective charge for the $E1$ transitions and $\chi_{sp}^{(1)}$ and $\chi_{df}^{(1)}$ are model parameters.
{width="14cm"}
The goal of the present paper is to describe simultaneously both the existing electromagnetic and the hadronic (transfer strength) data. To achieve this goal, two-neutron transfer intensities between the ground state of the target nucleus and the excited states of the residual nucleus were also calculated. The L=0 transfer operator may contain various terms, but we shall restrict our operator to the following form:
$$\begin{aligned}
\hat{P}^{(0)}_{\nu}=(\alpha_{p}\hat{n}_{p}+\alpha_{f}\hat{n}_{f})\hat{s}+\nonumber\\
+\alpha_{\nu} \left (\Omega_{\nu}-N_{\nu}-\frac{N_{\nu}}{N} \hat{n}_{d}\right)^{\frac{1}{2}}
\left(\frac{N_{\nu}+1}{N+1}\right)^{\frac{1}{2}} \hat{s}\label{eq5},\end{aligned}$$
where $\Omega_{\nu}$ is the pair degeneracy of neutron shell, $N_{\nu}$ is the number of neutron pairs, $N$ is the total number of bosons, and $\alpha_{p}$, $\alpha_{f}$, and $\alpha_{\nu}$ are constant parameters. The L=0 transfer operator contains two additional terms beside the leading order term, proportional to the bosonic $\hat{s}$ operator [@Iach87]. Details about the contributions of different terms in calculating the (p,t) spectroscopic factors will be given in a forthcoming paper [@Pas_unp].
The calculations were performed using the computer code OCTUPOLE [@Kuz_comp]. The Hamiltonian is diagonalized in a Hilbert space with a total number of bosons $N_{B}=n_{s}+n_{d}+n_{p}+n_{f}$. For the present calculations we used an extended basis allowing up to three negative parity bosons ($n_{p}+n_{f}$=3). The vibrational strengths used in the calculations are $\epsilon_{d}$=0.2 MeV, $\epsilon_{p}$=1.0 MeV, and $\epsilon_{f}$=1.1 MeV, while the quadrupole-quadrupole interaction strength has a value of $\kappa$=-21 keV. The strength of the O(5) second order Casimir operator is set to $a_{3}$=0.053 MeV, while the quadrupole operator parameters are ($\chi^{(2)}_{sd}$=-1.09, $\chi^{(2)}_{pf}$=-1).
The full spectrum of excited 0$^{+}$ states obtained in the present experiment is displayed in Fig. \[fig10\] in comparison to the corresponding calculated values. In the energy range covered experimentally (up to 2.5 MeV), the IBM-$spdf$ calculations predict 10 excited 0$^{+}$ states in comparison to the 17 experimentally observed 0$^{+}$ excitations (firm spin assignment). Given that there was no attempt to fine tune the calculations to the empirical 0$^{+}$ states, there is no point in invoking a precise energy cutoff for the IBM calculations. Therefore, it is appropriate to look also above 2.5 MeV, where there is a continuous spectrum of 0$^{+}$ states consisting of 20 states up to 3.3 MeV and as many as 30 up to 4 MeV. The IBM predicts that some of these states are having 2$pf$ bosons in their structure and are related (according to Ref. [@Zam01]) to the presence of double dipole/octupole excitations. For example, the boson admixtures for the first excited $0^+$ state are $n_d$=4.2, $n_p$=1.4, $n_f$=0.6 in comparison with those for the $\beta$-vibrational state as $n_d$=4.5, $n_p$=0.006, $n_f$=0.001. However, the present data cannot allow for a final decision on the nature of the 0$^{+}$ states. Additional experimental information is needed to measure the branching ratios and the absolute transition probabilities stemming from these states. In Fig. \[fig10\], the 2$^{+}$ and 4$^{+}$ levels revealed in the present experiment are also compared to the predictions of the IBM. The experiment revealed 33 excited 2$^{+}$ and 25 excited 4$^{+}$ states up to 2.5 MeV. In the same energy range, the calculations produced only 16 excited 2$^{+}$ states and 15 4$^{+}$ excitations. If one looks above this limit, the IBM predicts 32 excited 2$^{+}$ states and 33 excited 4$^{+}$ states up to 3.3 MeV.
K$^{\pi}$ E$_{i}$ (keV) J$_{i}$ J$_{f1}$ J$_{f2}$ Exp. (10$^{-4}$ b$^{-1}$) IBM (10$^{-4}$ b$^{-1}$)
------------- --------------- --------- ------------- ------------- --------------------------- --------------------------
0$^{+}_{1}$ 832 0$^{+}$ 1$^{-}_{1}$ 2$^{+}_{1}$ 5.1(4) 6.1
874 2$^{+}$ 3$^{-}_{1}$ 4$^{+}_{1}$ 7.1(15) 7.6
2$^{+}$ 3$^{-}_{1}$ 2$^{+}_{1}$ 24.5(31) 15.2
2$^{+}$ 3$^{-}_{1}$ 0$^{+}_{1}$ 14.7(24) 23.6
2$^{+}$ 1$^{-}_{1}$ 4$^{+}_{1}$ 4.2(9) 4.4
2$^{+}$ 1$^{-}_{1}$ 2$^{+}_{1}$ 14.5(19) 8.9
2$^{+}$ 1$^{-}_{1}$ 0$^{+}_{1}$ 8.7(14) 13.7
968 4$^{+}$ 5$^{-}_{1}$ 6$^{+}_{1}$ 22.8(80) 9.2
4$^{+}$ 5$^{-}_{1}$ 4$^{+}_{1}$ 10.8(27) 18.2
4$^{+}$ 5$^{-}_{1}$ 2$^{+}_{1}$ 6.7(13) 20.7
4$^{+}$ 3$^{-}_{1}$ 6$^{+}_{1}$ 19.1(67) 5.9
4$^{+}$ 3$^{-}_{1}$ 4$^{+}_{1}$ 9.0(23) 11.8
4$^{+}$ 3$^{-}_{1}$ 2$^{+}_{1}$ 5.6(11) 13.4
0$^{+}_{3}$ 1176 2$^{+}$ 1$^{-}_{1}$ 4$^{+}_{1}$ 0.060(25) 0.08
2$^{+}$ 1$^{-}_{1}$ 2$^{+}_{1}$ 0.25(10) 0.27
2$^{+}$ 1$^{-}_{1}$ 0$^{+}_{1}$ 0.62(28) 0.38
2$^{+}$ 3$^{-}_{1}$ 4$^{+}_{1}$ 0.09(5) 0.16
2$^{+}$ 3$^{-}_{1}$ 2$^{+}_{1}$ 0.39(20) 0.52
2$^{+}$ 3$^{-}_{1}$ 0$^{+}_{1}$ 0.95(51) 0.72
: \[BE\_IBM\] Experimental and calculated B(E1)/B(E2) transition ratios in $^{228}$Th. The parameters of the $E1$ operator are fitted to the experimental data available.
In $^{228}$Th there are no lifetimes measured for the negative-parity states, hence no absolute transition probabilities could be extracted. Therefore we would restrict the present discussion to reproducing the B(E1)/B(E2) ratios. A detailed comparison between the experimental data and the present calculations is presented in Table \[BE\_IBM\]. The agreement is obtained by using $e_{1}$=0.005 $e$fm and $e_{2}$=0.19 $e$b as the effective charges in Eq.(\[eq4\]) and (\[eq3\]), respectively. The remaining $E1$ parameters are $\chi_{sp}$=0.4 and $\chi_{df}$=-1.4.
The B(E1)/B(E2) ratios discussed in Table \[BE\_IBM\] belong to the $K^{\pi}$=0$_{1}^{+}$ (the predicted double-octupole phonon band) and $K^{\pi}$=0$_{3}^{+}$ ($\beta$-vibrational) bands. The comparison between them is important, because it can be used as a tool for providing additional information about the nature of the $K^{\pi}$=0$_{1}^{+}$ band. All the states belonging to this band are having 2 $pf$ bosons in their structure in the IBM calculations and are supposed to have a double-octupole phonon character. Further information confirming this hypothesis comes from the analysis of the $E1$ and $E2$ branching ratios. If the picture proposed by the IBM is correct, the states belonging to the $K^{\pi}$=0$_{1}^{+}$ band will show strong transitions into the negative-parity states (if they have a double-octupole character), while the levels stemming from the $K^{\pi}$=0$_{3}^{+}$ ($\beta$-band) will show very weak $E1$ transitions to these states. The experimental values in Table \[BE\_IBM\] fully confirm this hypothesis, showing that the B(E1)/B(E2) ratios are at least one order of magnitude larger for the $K^{\pi}$=0$_{1}^{+}$ band.
In Fig. \[fig11\], we display the calculated two-neutron intensities for $^{228}$Th in comparison to the integrated experimental cross sections normalized to that of the ground state. The calculations reproduce the strong excitation of the first 0$^+$ state at 832 keV in good agreement with the experiment. The experimental spectrum of 0$^{+}$ states is dominated also by a single state located at an energy of 2.1 MeV, showing a high cross section of about 15$\%$ of that of the ground state. In the IBM, there is predicted a state located at 2.1 MeV, which have the transfer intensities of about 18$\%$. This state has a double-octupole phonon structure. Another state at 2.29 MeV with a relative cross section of about 7$\%$ can be put in correspondence to the predicted in the IBM state at 2.2 MeV also with a double-octupole phonon structure. The running sum in Fig. \[fig11\] is taken up to 3.25 MeV, where another group of states with significant transfer strength is predicted by the IBM. The parameters from Eq.(\[eq5\]) were estimated from the fit of the known two-neutron transfer intensities (integrated cross sections) in Table \[tab:expEI\]. The values employed in the present paper are $\alpha_{p}$=1.3 mb/sr, $\alpha_{f}$=-0.4 mb/sr, and $\alpha_{\nu}$=0.03 mb/sr. The location and transfer intensity of the strongest states is very well reproduced by the calculations. Because the calculated energy distribution of the 0$^{+}$ states is underestimating the experimental data, this also affects the fragmentation of the transferred strength. However, the main characteristics are well reproduced by the present calculations.
![\[fig11\] (Color online) Comparison between the experimental two-neutron transfer intensities (panel (a)) for the 0$^{+}$ states and the IBM predictions (panel (b)). In panel (c) the experimental versus computed running sum of the (p,t) strengths is given.](fig11){width="8"}
\[sec:QPM\] QPM calculations
----------------------------
The IBM is a phenomenological approach. To gain a detailed information on the properties of the states excited in the (p,t) reaction, a microscopic approach is necessary. The ability of the QPM to describe multiple $0^+$ states (energies, $E2$ and $E0$ strengths, two-nucleon spectroscopic factors) was demonstrated for $^{158}$Gd [@LoI04]. An extension of the QPM to describe the $0^+$ states in the actinides [@LoI05] was made after our publication on the results of a preliminary analysis of the experimental data [@Wir04]. These calculations are used to compare to the present detailed analysis of the experimental data for $^{228}$Th. As for the theoretical basis of the calculations, we refer to the publications [@LoI05; @Sol92].
The experimental spectrum of the $0^{+}$ relative level reaction strength for the (p,t) transfer (the ratios of the (p,t) strength for every state to those for the ground state) is compared to the results of the QPM calculations in Fig. \[fig:compil-zeroQPM\]. The (p,t) normalized transfer spectroscopic strengths in the QPM are expressed also as ratios $$\label{spec-fact}
S_n(p,t) = \left[\frac{\Gamma_n(p,t)}{\Gamma_0(p,t)}\right]^2.$$ The amplitude $\Gamma_0(p,t)$ refers to the transitions to the $I$ members of the ground-state rotational band, i.e. to the ground state at an analysis of the $0^+$ excitations. The amplitude $\Gamma_n(p,t)$ includes the transitions between the ground state and the one- and two-phonon components of the wave function. The numerical results of the calculations obtained according to the QPM investigation in the publication [@LoI05] are provided to us by A. V. Sushkov [@Sush]. The QPM generates 15 $0^+$ states below 2.5 MeV, in fair agreement with the 17 firmly assigned states. The calculations reproduce the strong excitation of the first 0$^+$ state in accordance with the experiment. In Fig. \[fig:compil-zeroQPM\], we present also the increments of the (p,t) strength ratios in comparison to those of the QPM calculated normalized spectroscopic strengths. As one can see, the calculations are in fair agreement with the experiment. The (p,t) strength for the questionable 0$^+$ state at 2335.9 keV does not influence considerably the results of comparison (hence it is not included in the comparison).
Visible deviation of calculated strength from experiment is seen above 2 MeV. Theory predicts many $0^+$ excitations at higher energies (more than 80 states in the energy range below 4 MeV), but with small strength. At the same time, two strong excitations are observed in the experiment at 2.13 and 2.29 MeV (see Table \[QPM\_structure\]), respectively. It is interesting to note that both the IBM and the QPM predict two strongly excited states and therefore a jump in the increments of the (p,t) strength in the vicinity of 2 MeV, thus reproducing partly the sharp increase in the experimental increment.
In these QPM calculations, the dominant phonon structure of the 0$^+$ states in low part of energies is the one-phonon quadrupole nature. For higher energies admixtures of two quadrupole and two octupole phonons are present in the structure of these states, and for some of the states they became dominant. The relatively modest role of the octupole phonons in the structure of the low energy $0^+$ states is explained in [@LoI05] by the enforcement of the Pauli principle, leading to a spreading of the lowest two octupole phonon components among several QPM $0^+$ states and pushing them to higher energies.
[ccccccl]{} $K^{\pi}_n$ & $E_{n}(exp)$ & $E_{n}(calc)^*$ & $E_{n}(calc)^{**}$ & $S(p,t)_{exp}$ & $S(p,t)_{calc}^{**}$ & Structure from [@LoI05] Structure from[@Web98]\
\
0$_1^{+}$ & 0.832 & 0.8 & 0.724 & 0.236 & 0.281 & $(20)_196$ $(20)_197;[(30)_1(30)_2]0.3$\
0$_2^{+}$ & 0.939 & 1.0 & 1.496 & 0.041 & 0.002 & $(20)_294;(20)_34$ $(20)_295;(20)_10.8;[(30)_1(30)_1]0.7$\
0$_3^{+}$ & 1.120 & 1.2 & 1.570 & 0.008 & 0.001 & $ (20)_24;(20)_393$ $(20)_382;(20)_414;(20)_21$\
0$_4^{+}$ & 1.511 & 1.4 & 1.831 & 0.013 & 0.021 & $(20)_455;(20)_58;(20)_612;[(30)_1(30)_1]21$ $(20)_478;(20)_317$\
0$_5^{+}$ & 1.532 & & & 0.003 & &\
0$_6^{+}$ & 1.628 & 1.6 & 1.950 & 0.058 & 0.089 & $(20)_420;(20)_573;(20)_63$$(20)_597;(20)_41;[(30)_1(30)_1]0.5$\
0$_7^{+}$ & 1.691 & 1.7 & 1.962 & 0.008 & 0.046 & $(20)_415;(20)_512;(20)_662;[(30)_1(30)_1]6$$(20)_692;[(30)_1(30)_1]4$\
0$_8^{+}$ & 1.710 & 1.8 & 2.138 & 0.003 & 0.001 & $(20)_789;[(22)_1(22)_1]4$$(20)_797;[(30)_1(30)_2]0.2$\
0$_9^{+}$ & 1.750 & & & 0.011 & &\
0$_{10}^{+}$ & 1.909 & 1.9 & 2.162 & 0.028 & 0.016 & $(20)_69;(20)_853;[(22)_1(22)_1]4;[(30)_1(30)_1]23$$(20)_874;(20)_94;(20)_64$\
0$_{11}^{+}$ & 2.045 & & 2.190 & 0.003 & 0.008 & $(20)_76;[(22)_1(22)_1]87$\
0$_{12}^{+}$ & 2.080 & & 2.270 & 0.028 & 0.013 & $(20)_43;(20)_65;(20)_837;(20)_914;(20)_{10}3;[(30)_1(30)_2]32$\
0$_{13}^{+}$ & 2.131 & & 2.290 & 0.150 & 0.008 & $(20)_85;(20)_980;[(30)_1(30)_1]6;[(31)_1(31)_1]7$\
0$_{14}^{+}$ & 2.159 & & 2.334 & 0.007 & 0.001 & $(20)_94;[(30)_1(30)_2]87$\
0$_{15}^{+}$ & 2.290 & & 2.350 & 0.067 & 0.003 & $(20)_{10}2;(20)_{14}30;[(22)_1(22)_2]65$\
0$_{17}^{+}$ & 2.456 & & 2.359 & 0.003 & 0.001 & $(20)_{10}84;[(30)_1(30)_1]3;[(30)_1(30)_3]3$\
0$_1^{-}$ & 0.328 & 0.5 & & 0.005 & & $(30)_199 $\
1$_1^{-}$ & 0.944 & 1.0 & & 0.002 & & $(31)_198 $\
3$_1^{-}$ & 1.344 & 1.4 & & 0.002 & & $(33)_195;~[(20)_1(33)_1]3 $\
2$_1^{+}$ & 0.969 & 1.0 & & 0.121 & & $(22)_198 $\
2$_2^{+}$ & 1.153 & 1.3 & & 0.145 & & $(22)_299 $\
4$_1^{+}$ & 1.432 & 1.5 & & 0.001 & & $(44)_199 $\
$^*$ Data are taken from [@Web98]. $^{**}$ Data are taken from [@LoI05]\
[crccrccrcccc]{} K$^{\pi}_i$ & E$_{i}$ & I$_{i}$ && E$_{f1}$ & I$_{f1}$ && E$_{f2}$ & I$_{f2}$ && Exp. (10$^{-4}$ b$^{-1}$) & QPM (10$^{-4}$ b$^{-1}$)\
\
0$^{+}$ & 831.8 & 0$^{+}$ &&328.0 & 1$^{-}$ &&57.8 & 2$^{+}$ && 5.1(4) & 2.25\
0$^{+}$ & 938.6 & 0$^{+}$ &&328.0 & 1$^{-}$ &&57.8 & 2$^{+}$ && 6.7(6) & 54.8\
0$^{+}$ & 1175.5 & 2$^{+}$ &&328.0 & 1$^{-}$ &&186.8 & 4$^{+}$ && 0.06(3) & 1.45\
Besides the publication [@LoI05], other calculations of the same Dubna group were carried out for a microscopic description of the level structure, and transition rates between excited states in $^{228}$Th, observed in the decay of $^{228}$Pa [@Web98]. The wave functions, the level energies from two publication in correspondence to the experimental ones and to the transfer factors are given in Table \[QPM\_structure\]. The transfer factors and also the moments of inertia are taken into account to put in correspondence the experimental and calculated levels. The large difference in the transfer factors is seen only for two levels at 2131 and 2290 keV. There is an essential difference in the energies of the lowest $0^+$ states in two publications, as they are more close to the experimental ones in [@Web98]. This is caused by the choice of the isoscalar quadrupole-quadrupole interaction strength stronger than the critical value in Ref. [@Web98]. As a consequence the energy of the lowest collective $0^+$ state becomes imaginary and its properties such as the structure, E2 reduced transition probabilities and the transfer factor are partially transferred to the next $0^+$ collective state. Among the transition properties, the calculated B(E1)/B(E2) ratios at the decay of some 0$^+$ states are of special interest in this publication. We present here some data on the B(E1)/B(E2) ratios, in order to note the difference in the explanation of the experimental data by the QPM and the IBM. The calculated ratios for transitions from the $0^+_1$, $0^+_2$, and $0^+_3$ states to the octupole $0^-_1$ state and the ground state band are compared to experimental ratios in Table \[BE\_QPM\]. As one can see from Table \[QPM\_structure\] and Table \[BE\_QPM\], the small admixtures of the octupole two-phonon components to the wave functions of the $0^+_1$ and $0^+_2$ states are responsible for the fast $E1$ transitions. In the IBM, similar results are obtained for the $0^+_1$ state, having 2 $pf$ bosons in their structure and which are supposed to have mainly a double-octupole phonon character (see Table \[BE\_IBM\] and corresponding discussion). At the same time, the B(E1)/B(E2) ratio is considerably smaller for the decay of the $0^+_3$ state, which is a $\beta$-vibrational state, again in agreement with the experiment. The same result is obtained in the IBM.
Generally, the QPM is quite accurate in nuclei with small ground-state correlations. These correlations increase with the collectivity of the first one-phonon states, which is exactly the case of the $K=0^-$ octupole phonon state in $^{228}$Th. To decrease the correlations the value of the octupole-octupole isoscalar interaction strength was diminished so that the calculated energy of the $K=0^-$ state was almost 200 keV higher than the experimental value [@Web98]. In addition, the effect of multi-phonon admixtures (three and more phonons) that pushes two-octupole phonon poles and consequently two-octupole phonon energies to lower values was then underestimated. In summary, the accuracy of the calculations of $^{228}$Th as stated in Ref. [@Web98] is worse due to the increased ground-state correlations and the shift of two-phonon poles towards smaller energies. In future QPM studies one also has to take into account the spin-quadrupole interaction that is known to increase the density of low-lying $0^+$ states [@Pya67; @Sol76].
\[sec:0\_concl\] To the nature of $0^+$ excitations
---------------------------------------------------
At a microscopic approach there can be a few situations of structure of the $0^{+}$ states. A $\beta$-vibrational mode can be characterized by the relatively small two-nucleon transfer strength and a relatively large B(E2) value with a moment of inertia close to the one of the ground state. The large ratio B(E1)/B(E2) and the increase of the moment of inertia indicate the presence of the octupole two-phonon component. If a state has a relatively weak B(E2) value and also a weak two-nucleon transfer strength, but exhibits an increase of the moment of inertia, it should be a state with one dominant 2qp configuration. The pairing vibrational excitations can be characterized by their large two-nucleon transfer strengths and relatively small B(E2) values.
It is clear that the first and second $0^+$ excited states cannot be the $\beta$-vibrational states as usually is observed in deformed rare-earth nuclei since their moments of inertia are much larger than those of the ground state and also their (p,t) strengths are large. The actual $\beta$-vibrational state is observed at 1120 keV and it is excited very weakly in the (p,t) reaction. As we have seen, both the IBM and the QPM reproduce the $0^{+}$ relative level reaction strength for the (p,t) reaction and the B(E1)/B(E2) ratios of the decay of the lowest $0^+$ states reasonably well. At the same time, the nature of $0^+$ excitations in the QPM differs significantly from the one in the [*spdf*]{}-IBM. In all low-lying states of the QPM calculations, quadrupole phonons are dominant and the octupole phonons are predicted to play a relatively modest role, whereas the IBM predicts the lowest $0^+$ state as having mainly 2$pf$ bosons in their structure [@Zam01]. The analysis of the lowest quadrupole phonon wave function in the QPM reveals that the backward RPA amplitudes $\varphi$ contribute considerably to the relative (p,t) reaction strength thus indicating that the lowest excited $0^+$ state describe pairing vibrations arising from ground state fluctuations [@LoI05].
For an additional hint to the nature of the lowest $0^+$ states, we include Fig. \[fig:B(E1)-B(E2)\] with the B(E1)/B(E2) ratio stemming from the lowest excited states in $^{228}$Th and $^{230}$Th. Intuitively, one would expect that a large B(E1)/B(E2) ratio might be characteristic for a two-octupole-phonon excitation, whereas a small ratio might indicate a shape oscillation. Such a picture is observed for $^{228}$Th: large B(E1)/B(E2) ratios for the $0^+_1$ and $0^+_2$ states, and vanishing values for $\beta$- and $\gamma$-vibrational states. The assumption in [@Gra03] that just the two-phonon structure is a reason of the very strong excitation of the $0^+_1$ state in the (p,t) reaction has to be rejected, since the B(E1)/B(E2) ratio for the $0^+_1$ state in $^{230}$Th is small at strong excitation in the (p,t) reaction (or the B(E1)/B(E2) ratio is no reliable manifestation of the two-phonon nature of a state).
.
In this aspect a comparison of the relative transfer strengths in the (p,t) reaction leading to the $0^+$ states in $^{228}$Th and the $3/2^-$ states in $^{229}$Pa (data are taken from [@Lev94]) has to be considered additionally (see Fig. \[fig:B(E1)-B(E2)\]). Since the ground-state spin in the target nucleus $^{231}$Pa is $3/2^-$, just these spins are excited in a two-neutron $L=0$ transfer. $^{229}$Pa can be regarded as $^{228}$Th plus a strongly coupled proton. The rotational band built on the first $3/2^-$ excited state in $^{229}$Pa at 11.6 keV [@Lev98] corresponds to the g.s. band in $^{228}$Th and is excited very strongly. The main component in the structure of the ground state in the target nucleus $^{231}$Pa is 1/2\[530\] [@Lev94]. Therefore the levels excited in the $L=0$ transfer are $I^{\pi},K = 3/2^-,1/2$ states and they are members of collective bands based on the state originating from coupling the $K^{\pi} = 1/2^-$ state to the $^{228}$Th core-excited states. Such bands, as identified in [@Lev94], can be used to derive the moments of inertia for at least three $3/2^-$ states. Values of $J/\hbar^2$ in MeV$^{-1}$ are given below (energies in $^{229}$Pa are relative to the lowest $3/2^-$ state)\
$^{229}$Pa: [**79.2**]{}(0.0 keV), [**78.4**]{}(703 keV), [**127.5**]{}(830 keV)\
(1524 keV).\
$^{228}$Th: [**51.9**]{}(0.0 keV), **70.3**(832 keV), **73.2**(939 keV).\
The moments of inertia in $^{229}$Pa are larger than those in $^{228}$Th which is a manifestation of the contribution of the odd proton. The large moment of inertia for the 830.5 keV state can probably be explained (at least partly) by neglecting the Coriolis coupling when fitting the energies of the corresponding band. Nevertheless the increment of the moment of inertia for the state at 830.5 keV relative to other states in $^{229}$Pa can be put in correspondence to similar increments for the states at 831.8 and 938.6 keV relative to the g.s. in $^{228}$Th. But this state, as well as other low-lying states in $^{229}$Pa, are only weakly excited in contrast to the strong excitation of the state 831.8 keV in $^{228}$Th. At the same time, the $3/2^-$ state at 1523.7 keV is excited strongly, but it cannot be put in correspondence to the first excited state in $^{228}$Th: there is practically no increment of its moment of inertia relative to the lowest 3/2$^-$ state. Besides that, its energy is almost twice larger than the first excited $0^+$ state in $^{228}$Th.
From the insufficient information we can only conclude that the 831.8 keV state in $^{228}$Th has the largest pairing vibrational component and in $^{229}$Pa the additional proton has the effect that the largest pairing vibrational component is moved to the 1523.7 keV state. No theoretical explanation was undertaken since the publication of the experimental results on $^{229}$Pa [@Lev94]. It would be interesting to undertake the theoretical analysis of the excitations with the $L=0^+$ transfer in odd nuclei and first of all in the $^{229}$Pa nucleus. Experimental study of excitations in other odd nuclei, as we have seen, may promise unexpected phenomena.
As for the experimental evidence of the nature of other $0^+$ states, we have only the moments of inertia derived from the sequences of states treated as rotational bands and thus only tentative conclusions can be drawn about their structure. In contrast to $^{230}$Th [@Lev94], for which they are distributed almost uniformly over the region from 47 to 98 MeV$^{-1}$, the moments of inertia in $^{228}$Th have values close to 50 MeV$^{-1}$ only for the g.s., $\beta$-vibrational $0^+$ states and for the state at 1531.7 keV, all other $0^+$ states have larger values from 70 to 95 MeV$^{-1}$. This fact can indicate that corresponding states are possibly of two-phonon nature too, or two quasi-particle states with an admixture of the pairing vibrations.
Conclusion
==========
Excited states in $^{228}$Th have been studied in (p,t) transfer reactions. 106 levels were assigned, using a DWBA fit procedure, additionally only the energies are determined for 57 states. Among them, 17 excited $0^+$ states have been found in this nucleus up to an energy 2.5 MeV, most of them have not been experimentally observed before. Their accumulated strength makes up for more than 70% of the ground-state strength. Firm assignments have been made for most of the 2$^+$ and 4$^+$ states and for some of the 6$^+$ states. These assignments allowed to suggest multiplets of states, which can be treated as one- and two-phonon octupole quadruplets, and to identify the sequences of states, which have the features of rotational bands with definite inertial parameters. Moments of inertia are derived from these sequences. Only for the g.s. and $\beta$-vibrational states and additionally for the state at 1531.7 keV (for which the shape of the angular distribution is different from most other ones), the moments of inertia are about 50 MeV$^{-1}$. For all other states they are larger than 70 MeV$^{-1}$, i.e. the value for the first excited $0^+$ state. This information, together with the spectroscopic information on some ${\gamma}$-transitions, were used for conclusions on the nature of the $0^+$ states. The experimental data have been compared to [*spdf*]{}-IBM and QPM calculations. Spectroscopic factors from the (p,t) reaction, and the trend in their change with excitation energy, are approximately reproduced by both the IBM and QPM for the 0$^+$ states. A remarkable feature of the IBM and QPM is the prediction of strong first vibrational excitations, close in magnitude and position to the experimental ones. Giving also an approximately correct number of $0^+$ states, these models provide different predictions for the structure of these states. The lack of additional information does not allow for final conclusions on the validity of the theoretical approaches. Challenging experiments on gamma spectroscopy following (p,t) reactions would give much needed information.
Acknowledgements
================
The work was supported by the DFG (C4-Gr894/2-3, Gu179/3, Jo391/2-3, Zl510/4-2), MLL, and US-DOE, contract number DE-FG02-91ER-40609, and CZ.1.05/2.1.00/03.0082.
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|
---
abstract: 'We consider how to accelerate fermionic molecular dynamics algorithms by introducing $n$ pseudofermion fields coupled with the $n$th root of the fermionic kernel. This reduces the maximum pseudofermionic force, and thus allows a larger molecular dynamics integration step size without hitting an instability in the integrator.'
address: |
School of Physics, The University of Edinburgh,\
Edinburgh EH9 3JZ, United Kingdom
author:
- 'M. A. Clark and A. D. Kennedy'
title: Accelerating Fermionic Molecular Dynamics
---
Introduction
============
For over fifteen years the algorithm of choice for generating lattice field theory configurations including the dynamical effect of fermions has been Hybrid Monte Carlo (HMC) [@duane87a]. Unfortunately the cost of this algorithm increases rapidly as the fermion mass $\mq$ decreases; in order to keep the acceptance rate $\pacc$ constant the integration step size $\dt$ has to be reduced, and for trajectories of length $\trjlen=1$ this corresponds directly to an increased number of integration steps and hence larger cost.
This required decrease in step size is because of the breakdown of symmetric symplectic integrators. For light dynamical fermions there is an instability for a few isolated light fermion modes, whose frequency is well separated from the bulk of the modes. This instability is seen to be directly responsible for the exponential decrease of acceptance with integration step size above some critical value.
We introduce a method for reducing the severity of this problem by reducing the highest “effective frequency” of the fermionic modes, or equivalently of decreasing the magnitude of the fermionic contribution to the force acting on the gauge fields. The basic idea follows the suggestion of Hasenbusch [@Hasenbusch:2001ne] to split the fermionic action into two parts, and to introduce separate pseudofermion fields for each part. Our approach can be easily generalised to an arbitrary number of pseudofermion fields.[^1]
Non-linearity of CG {#sec:CG}
===================
We observe that the force due to the fermion kernel $\M^{-1}$ is dominated by the smallest eigenvalues of $\M$. The condition number $\k(\M)$ is the ratio of the largest eigenvalue to the smallest eigenvalue, and to a first approximation controls the rate of convergence of iterative Krylov space solvers. The largest eigenvalue remains approximately constant as the fermion mass $\mq$ is decreased, and the smallest eigenvalue is typically of the order $\mq^\alpha$ where $\alpha$ is $1$ or $2$, so we expect $\k(\M)\propto \mq^{-\alpha}$.
Consider the numerical solution of the linear system $\M\chi=\phi$, where $\M>0$ and has condition number $\k(\M)$. The cost of solving these linear equations is proportional to $m^{-\alpha}$. On the other hand we could equivalently solve the set of coupled linear equations $\sqrt{\M}\chi=\psi$ and $\sqrt{\M}\psi=\phi$, each of which has condition number $\k(\sqrt
M)=\sqrt{\k(\M)}$,[^2] leading to a cost of order $2\mq^{-\alpha/2}$ in this case, which is cheaper for sufficiently small $\mq$. This reflects the essential non-linearity of Krylov space solvers. Indeed, we may even be more adventurous and solve the set of $n$ coupled systems $\root n\of\M\psi_j =\psi_{j+1}$, where $\psi_0=\chi$ and $\psi_n=\phi$, for which we have to perform $n$ solves each with condition number $\k(\M^{1/n}) =\k(\M)^{1/n}$, leading to a total cost of order $n\k(\M)^{1/n}$.
Unfortunately we cannot take advantage of this non-linearity, the problem being that it is not straightforward to apply $\sqrt\M$ to a vector when $\M$ is not serendipitously a manifest square. There are efficient techniques for evaluating matrix functions, such as computing the optimal polynomial or rational Chebyshev approximation to the function over the spectrum of the matrix [@kennedy:2003a]. For the rational case the approximations may be found using the Remez algorithm, and they usually converge exponentially in the degree of the rational function. In practice only a relatively low degree rational function is needed to achieve machine floating-point precision [@Clark:2003na]. Furthermore, if we take a rational approximation then we can express it as a partial fraction, and apply all the terms simultaneously (in the same Krylov space) using a multi-shift solver. This reduces the cost of solving $\M^{1/n}x=b$ to about the same cost as solving $\M x=b$. This in turn is expected to be proportional to the condition number $\k(\M)$. Sadly, this means that the cost of the proposed method is of order $n\k(\M)$ rather than $n\k(\M)^{1/n}$, and we are worse off than when we started.
Although this method is clearly useless to accelerate the convergence of Krylov space solvers, it still does significantly reduce the condition number of each of the $n$ solves, and this is what we shall make use of to decrease the pseudofermionic force. Indeed, we expect the force to be of order $n\k(\M)^{1/n}\dt$, which is small compared to $\k(\M)\dt$ for large $\k(\M)$. A na[ï]{}ve calculation minimising $n\k(\M)^{1/n}/\k(\M)$ leads to the conclusion that the optimal number of pseudofermion fields should be[^3] $\opt{n}= \ln\k(\M)$.
Pseudofermion Sampling
======================
Recall that we represent the fermion determinant as a pseudofermion Gaussian functional integral, $\det\M \propto \int d\phi\,d\phi^\dagger\, \exp{\left(-
\phi^\dagger\M^{-1}\phi\right)}$, and then select a single equilibrium pseudofermion configuration using a Gaussian heatbath. We expect, therefore, that the variance of this stochastic estimate of the fermion determinant will lead to statistical fluctuations in the fermionic force: in other words the pseudofermionic force may be larger than the exact fermionic force, which is the functional derivative $\partial\tr\ln\M(U)/\partial U$ with respect to the gauge field $U$. This means that the pseudofermionic force may trigger the instability in the symplectic integrator even though the exact fermionic force would not.
An obvious way of ameliorating this effect is to use $n>1$ pseudofermion fields (which we shall call *multipseudofermions*) to sample the functional integral representing the fermion determinant, and this is achieved simply by writing $$\begin{aligned}
\det\M & = &[\det\M^{1/n}]^n \nonumber \\
& \propto & \prod_{j=1}^n d\phi_j\, d\phi^\dagger_j\,
\exp{\left(-\phi^\dagger_j\M^{-1/n}\phi_j \right)};
\label{eq:multipseudo}\end{aligned}$$ that is, introducing $n$ pseudofermion fields $\phi_j$ each with kernel $\M^{-1/n}$.
We may now follow a similar argument to that in §\[sec:CG\] to estimate the optimal value for $n$. We must keep the maximum force fixed so as to avoid the instability in the integrator, so we may increase the integration step size to $\dt'$ such that $n\k(\M)^{1/n}\dt' = \k(\M)\dt$. At constant trajectory length, and hence constant autocorrelation time, the cost of an trajectory is proportional to the step size, and thus is minimised by choosing $n$ so as to minimise $\dt'/\dt = \k(\M)^{1-1/n}/n$, which leads to the condition $\opt{n} \approx \ln\k(\M)$, corresponding to cost reduction by a factor of $\dt/\dt' \approx e\ln\k(\M)/\k(\M)$.
Acceleration {#sec:RHMC}
=============
Our method is to apply the algorithm [@Clark:2003na] to generate gauge field and pseudofermion configurations distributed according to the probability density $$P(U,\phi_1,\ldots,\phi_n) = \frac1Z \exp{\bigl[-S_B(U) -
S_F(\M)\bigr]},$$ where $S_B$ is the bosonic (pure gauge) action and $S_F(\M) = \sum_{j=1}^n \phi^\dagger_j \M^{-1/n}\phi_j$. Optimal rational approximations are used to evaluate these matrix functions, and we proceed as we would for conventional [@duane87a]. Using a multi-shift solver the computational cost is very similar to , with the additional overhead of having to perform a matrix inversion to evaluate the heatbath.
We have performed tests of the algorithm using small lattices with four flavours of naïve staggered fermions. As well as being the least computationally demanding simulations to perform, the improvement for such systems should be a lower bound on the improvement for large volume and/or Wilson type fermion simulations, where the fermion matrix is less well-conditioned.
The $n=1$ case runs were performed using conventional , and the $n>1$ runs using . We define an efficiency measure $E\defn \langle\pacc
\rangle/\ninv$, where $\ninv$ is number of Krylov solves performed per trajectory. For this is given by $1+\trjlen/\dt$ and for by $(2+\trjlen/\dt)n$, where $\trjlen$ is the trajectory length and $\dt$ the step size. The additional inversion per field for is for the pseudofermion heatbath.
=0.45
-6ex
Figure \[fig:eff-dt-1\] is a plot of the efficiency against step size for various value of $n$. The conjecture that there is some optimal value of $n$ is confirmed, and it can be seen that $n=2$ is the value for this particular lattice. It represents an increase in efficiency of 33% ($n=2$ peak divided by $n=1$ peak). We have conducted the same test with a mass parameter of $\mq=0.01$, which shows the same optimal value of $n$, but with a substantial efficiency increase of $60\%$. This confirms that the improvement factor increases as less well-conditioned systems are studied.
Conclusions {#sec:conclusions}
===========
We have demonstrated that our method for accelerating the Monte Carlo acceptance test improves the efficiency over conventional . This is clearly still a very preliminary study, and shall be extended to large scale systems for both and domain wall simulations using . On such systems, we would expect the improvement in efficiency to increase, and it will be interesting to see how much improvement can be gained. It would also be interesting to compare our method with that proposed by Hasenbusch in [@Hasenbusch:2001ne].
Acknowledgements {#acknowledgements .unnumbered}
================
ADK would like to thank Brian Pendleton and Artan Boriçi for useful discussions and encouragement.
[1]{}
S. Duane, A. D. Kennedy, B. J. Pendleton, and D. Roweth, Phys. Lett. [**195B**]{}, 216 (1987).
M. Hasenbusch, Phys. Lett. [**B519**]{}, 177 (2001), [hep-lat/0107019](http://xxx.soton.ac.uk/abs/hep-lat/0107019).
A. D. Kennedy, Nuclear Physics (Proceedings Supplements), [**B128**]{}, 107–116 (2004), [hep-lat/0402037](http://xxx.soton.ac.uk/abs/hep-lat/0402037).
M. A. Clark and A. D. Kennedy, Nuclear Physics (Proceedings Supplements), [**B129**]{}, 850–852 (2004), [hep-lat/0309084](http://xxx.soton.ac.uk/abs/hep-lat/0309084).
[^1]: Hasenbusch’s method also allows an arbitrary number of pseudofermion fields to be used, but the parameters in the action have to be tuned to ensure that the contributions to the force are divided up equally.
[^2]: For $\M>0$ the positive square root is uniquely defined.
[^3]: Strictly speaking $\opt{n}\in\Z$, so it must be either $\lfloor
\ln\k(\M)\rfloor$ or $\lceil\ln\k(\M)\rceil$ .
|
---
abstract: 'We construct a rank one infinite measure preserving transformation $T$ such that for all sequences of nonzero integers $\{k_{1},\ldots, k_{r}\}$, $T^{k_{1}}\times\ldots\times T^{k_{r}}$ is ergodic.'
author:
- 'Sarah L. Day [^1]'
- 'Brian R. Grivna [^2]'
- 'Earle P. McCartney [^3]'
- 'Cesar E. Silva [^4]'
date: 'March 1, 1998'
title: Power Weakly Mixing Infinite Transformations
---
1.5in
AMS 1991 subject classification: 28D.\
Key words: ergodic index, weak mixing, infinite measure transformation
Introduction
============
It is well known that for the case of finite measure preserving transformations, if $T$ is weakly mixing then $T^{k_{1}}\times
\ldots\times T^{k_{r}}$ is ergodic for any sequence of nonzero integers $\{k_{1},\ldots, k_{r}\}$. Kakutani and Parry proved in \[KP\] that there exist infinite (measure preserving) transformations such that $T\times \cdots\times T$ ($r$ terms) is ergodic but $T\times \cdots\times T$ ($r+1$ terms) is not; in this case the transformation is said to have [**ergodic index $r$**]{}. $T$ is said to have [**infinite ergodic index**]{} if it has ergodic index $r$ for all $r > 0$. In \[KP\], they also constructed infinite Markov shifts of infinite ergodic index. Furthermore, for the case of infinite transformations, it was shown in [@alw] that ergodicity of $T\times T$ implies weak mixing but that there exist infinite weak mixing transformation with $T\times T$ not conservative, hence not ergodic. Later it was shown that $T$ may be weakly mixing with $T\times T$ conservative but still not ergodic, and that there exist rank one infinite transformations of infinite ergodic index [@afs].
In this paper we introduce a condition stronger than infinite ergodic index. Define a transformation $T$ to be [**power weakly mixing**]{} if for all finite sequences of nonzero integers $\{k_{1},\ldots,k_{r}\}$, $$T^{k_{1}}\times \ldots\times
T^{k_{r}}$$ is ergodic. Clearly, any power weakly mixing transformation has infinite ergodic index. As $T$ is weakly mixing, it follows that for all ergodic finite measure preserving transformations $S$, $T\times S$ is ergodic; however, there always exists a conservative ergodic infinite measure preserving transformation $R$ such that $T\times R$ is not conservative, hence not ergodic [@alw]. Recently, it has been shown that infinite ergodic index does not imply power weakly mixing [@afs2].
In section 2 we prove some preliminaries on approximation and in section 3 we construct a rank one infinite measure preserving transformation which is power weakly mixing. We refer to [@afs] for terms not defined here.
[**Acknowledgments.**]{} This paper is based on research in the Dynamical Systems group of the 1997 SMALL Undergraduate Summer Research Project at Williams College with Prof. C. Silva as faculty advisor. Support for the project was provided by a National Science Foundation REU Grant and the Bronfman Science Center of Williams College.
Approximation Properties
==========================
In this section we prove an approximation lemma for transformations defined by cutting and stacking \[F\]. This idea has been used earlier in e.g. [@afs] to show that a specific transformation has infinite ergodic index. However, here we present it in greater generality that permits other applications such as in [@afs2]. Thus we first describe cutting and stacking constructions [@f70].
Let $X$ be a finite or infinite interval of real numbers and $\mu$ be Lebesgue measure. A [**column**]{} consists of a collection of pairwise disjoint intervals in $X$ of the form $B^{0}, B^{1},\ldots, B^{h-1}$, where $\mu(B)>0$ and $h>0$. The elements of $\cal C$ are called [**levels**]{} and $h$ is the [**height**]{} of $\cal C$. The column ${\cal C}$ partially defines a transformation $T$ on levels $B^{i}$, $i=0,\dots, h-2$, by the translation that takes interval $B^{i}$ to interval $B^{i+1}$. Thus sometimes we write $B^{i}$ as $T^{i}B$.
A [**cutting and stacking** ]{} construction for a measure preserving transformation $T:X\to X$ consists of a sequence of columns $${{\cal C}_{n}=\{{B_{n}^{0}, B_{n}^{1},\ldots,
B_{n}^{h_{n}-1}}}\}$$ of height $h_{n}$ such that:
i\) ${\cal C}_{n+1}$ is obtained by cutting ${\cal C}_{n}$ into $c_{n}$ equal-measure subcolumns or [**copies**]{}, putting a number of [**spacers**]{} (new levels of the same measure as any of the levels in the $c_{n}$ subcolumns) above each subcolumn, and stacking from left to right (i.e., the top (or top spacer if it exists) of the left subcolumn is sent by translation to the bottom of its right subcolumn). We assume $c_n \geq 2$. In this way ${\cal C}_{n+1}$ consists of $c_n$ copies of ${\cal C}_n$, possibly separated by spacers.
ii\) $B_{n}$ is a union of elements from $\{B_{n+1}, TB_{n+1},\ldots,
T^{h_{n+1}-h_{n}}B_{n+1}\}$.
iii\) $\bigcup_{n}{\cal C}_{n}$ generates the Borel sets, i.e., for all subsets $A$ in $X$, $\mu(A)>0$, and for all $\epsilon > 0$, there exists $C$, a union of elements from ${\cal C}_{n}$, for some $n$, such that $\mu (A
\bigtriangleup C) < \epsilon$.
Suppose $I = T^{j}B_{k}$ is in ${\cal C}_{k}$. For any $n > k$, $I$ is the union of some elements in ${\cal C}_{n}=\{B_{n}, TB_{n},\ldots,
T^{h_{n}-1}B_{n}\}$. We call the elements in this union [**sublevels**]{} of $I$.
Given a real number $0 < \epsilon < 1$, and a subset $A$ of $X$ with $\mu(A) > 0$, we say that a subset $I$ of $X$ is [**$(1-\epsilon)$-full**]{} of A provided $$\mu(I\cap A) > (1 - \epsilon)\mu (I).$$
A set $I$ in the product space $\Pi _{i=1}^{r}X$ is a [**rectangle**]{} if $I$ can be written as the Cartesian product of levels in some column ${\cal
C}_{k}$. We let $\nu$ be the product measure $\mu^{r}$. Rectangles $I$ are defined to be $(1-\epsilon)$-full of a set $A$ in a similar way as before.
\[iabovej\] Given subsets $A$ and $B$ of $\Pi _{i=1}^{r}X$, and $\epsilon>0$, there exist rectangles $I=I_{1}\times \ldots
\times I_{r}$ and $J=J_{1}\times
\ldots
\times J_{r}$ with $I_{1}, \ldots, I_{r},
J_{1},\ldots, J_{r}$ in a column ${\cal C}_{k}$ such that for all $m=1,\ldots, r$, $I_{m}$ may be chosen to be either above or below $J_{m}$ and with $I$ and $J$ $(1-\epsilon)$-full of sets $A$ and $B$ respectively.
Choose rectangles $I'=I_{1}'\times
\ldots
\times I_{r}'$ and $J'=J_{1}'\times
\ldots
\times J_{r}'$, with $I_{m}'$ and $ J_{m}'$ in ${\cal C}_{k-1}$, such that $I'$ and $J'$ are $(1-\frac{\epsilon}{c_{k-1}^r})$-full of $A$ and $B$ respectively. Now look at any two copies of ${\cal
C}_{k-1}$ in ${\cal C}_{k}$. To have $I_{m}$ above $J_{m}$, let $I_{m} $ be the top copy of $I_{m}'$ in ${\cal
C}_{k}$ and let $J_{m} $ be the bottom copy of $J_{m}'$ in ${\cal C}_{k}$. To have $I_{m}$ below $J_{m}$ make an analogous choice. Let $I =
I_{1}\times
\ldots \times I_{r}$ and $J = J_{1}\times \ldots
\times J_{r}$. One verifies that $I$ and $J$ are $(1-\epsilon)$-full of $A$ and $B$.
\[doubleapprox\][**(Double Approximation Lemma)**]{} Suppose $A$ is a subset of the product space $\Pi_{i=1}^{{r}}X$ with $\nu (A) > 0$. Let $I=I_{1}\times
\ldots \times I_{r}$ be a rectangle in ${\cal C}_k$ that is $(1- \epsilon)$-full of $A$. For $n>k$, let $P_n= c_k\cdots c_{n-1}$, let $V_n$ index the $P_n$ copies of $C_k$ in $C_n$, and let $V= V(n)=V_n\times\dots\times V_n$ ($r$ times). Then for any $\delta$, $0< \delta < 1$, and for any $\tau$, $0 < \tau < 100(1-\epsilon)$, there exists an integer $N$ such that for all $n >
N$, there is a set $V''$ of size at least $\tau$ percent of $V$ such that for all $v=(v_{1},\dots,v_{r})\in V''$, $I_{v}$ is $(1- \delta)$-full of $A$ and each $I_{v}$ is of the form $I_{v}=I_{1}''\times \ldots
\times I_{r}''$ where $I_{m}''$ is a sublevel of $I_{m}$ in the $v_{m}$-copy of ${\cal C}_{k}$.
For convenience, let $A$ denote $I\cap A$ and let $t$ denote $\frac{\tau}{100}$. Then $\nu(I\bigtriangleup A) < \epsilon\nu (I)$. We have that $V_{n} = \{1, \ldots, P_{n}\}$ and $V =
\{(v_{1}, \ldots, v_{r})| v_{i}\in
V_{n}\}$. Then $I = \cup_{v\in V}I_{v}$.
Choose $c > \frac {\delta + 1}{1 - t - \epsilon} >
0.$ Next pick $N>k$ sufficiently large so that for any $n\geq N$ there exists $V'$ a subset of $V$ such that $I' =
\cup_{v\in V'}I_{v}$ satisfies $$\nu(I' \bigtriangleup A) < \frac {\delta}{c}\nu
(I).$$
Thus,
$$\begin{aligned}
\nu(I' \bigtriangleup I) &< \frac {\delta}{c}\nu (I)
+ \epsilon\nu (I) \\ &= (\frac {\delta}{c} +
\epsilon)\nu (I).\end{aligned}$$
Now let $V''=\{{v\in V'|\nu
(I_{v}\bigtriangleup A) <\delta\nu (I_{v})}\}$ and set $I''=\cup_{v\in V''} I_{v}$, the union of the $(1-\delta)$-full $I_{v}$ subintervals.
Then, $$\begin{aligned}
\delta\nu (I'\bigtriangleup I'') &= \sum_{v\in
V'\bigtriangleup V''}\delta\nu (I_{v})\\ &\leq
\sum_{v\in V'\bigtriangleup V''}\nu
(I_{v}\bigtriangleup A)\\ &\leq \nu
(I'\bigtriangleup A).\end{aligned}$$
So, $$\begin{aligned}
\nu (I''\bigtriangleup I) &\leq
\frac {1}{\delta}\nu (I'\bigtriangleup A) + \nu
(I'\bigtriangleup I)\\
&< \frac{1}{c}\nu (I) + (\frac{\delta}{c} +
\epsilon)\nu (I)\\ &< (1-t)\nu (I).\end{aligned}$$
Therefore, more than $\tau$ percent of the subrectangles contained in $I$ are in $I''$ and are thus $(1- \delta)$-full of $A$.
A Power Weakly Mixing $T$
==========================
In this section we construct a rank one infinite measure preserving transformation $T$ that is power weakly mixing; then we mention how the proof gives a family of such transformations. We start by defining inductively a sequence of columns $\{{\cal C}_{n}\}$. Let ${\cal C}_{1}$ have base $B_{1}=[0,1)$ and height $h_{1}=1$. Given a column ${\cal C}_{k}$ with base $B_{k}=[0,\frac{1}{4^{k-1}})$ and height $h_{k}$, ${\cal C}_{k+1}$ is formed by cutting ${\cal C}_{k}$ vertically three times so that $B_{k}$ is cut into the intervals $B_{k,1}=[0,\frac{1}{4^{k}})$, $B_{k,2}=[\frac{1}{4^{k}},\frac{1}{2}(\frac{1}{4^{k-1}}))$, $B_{k,3}=[\frac{1}{2}(\frac{1}{4^{k-1}}),\frac{3}{4}(\frac{1}{4^{k-1}}))$, $B_{k,4}=[\frac{3}{4}(\frac{1}{4^{k-1}}),\frac{1}{4^{k-1}})$. We then add a column of spacers $h_{k}$ high to the top of the subcolumn whose base is $B_{k,2}$. Next we add one spacer to the top of the subcolumn whose base is $B_{k,4}$; this is called the [**staircase spacer**]{} of ${\cal
C}_k$. Then stack from left to right, i.e., the top level on the left is sent to the bottom level on the right by the translation map. The resulting column ${\cal C}_{k+1}$ now has base $B_{k+1}=[0,\frac{1}{4^{k}})$ and height $h_{k+1}=5h_{k}+1$. The union of the columns is $X=[0,\infty)$. This defines a conservative ergodic rank one infinite measure preserving transformation $T$.
Any column ${\cal C}_{n} = \{B_{n}, \ldots,
T^{h_{n}-1}B_{n}\}$ has four subcolumns ${\cal
C}_{n,i}= \{B_{n,i}, \ldots, T^{h_{n}-1}B_{{n,i}}\}$ for $i=1,\ldots,4$. Given a level $L$ in ${\cal C}_{n}$ and an integer $\ell>0$, we will be interested in studying $T^{\ell h_{n}}L$ (a translation of $L$ through ${\cal C}_{n}$ $\ell$ times). To simplify our estimates, we will only be concerned with the part of $T^{\ell h_{n}}L$ that is in ${\cal C}_{n,1}$; this will consist of a sequence of sublevels that we call an $L$[**-crescent**]{} (refer to figure 1).
=6truein
For any sequence of of nonzero integers $\{k_{1},
\ldots, k_{r}\}$, the transformation $T^{k_{1}}\times
\dots \times T^{k_{r}}$ is ergodic.
Let $A$ and $B$ be in $\Pi_{i=1}^{r}X$ with $\nu (A) > 0$ and $\nu (B) >
0$. Find rectangles $I = I_{1}\times
\ldots\times I_{r}$ and $J = J_{1}\times
\ldots
\times J_{r}$ such that $$\nu(A \cap I) > \frac {3}{4} \nu (I) ,$$ $$\nu(B \cap J) > \frac {3}{4} \nu (J) ,$$ and $I_{m}, J_{m}, m = 1,\ldots, r$ are all in the same column ${\cal C}_{k}$, and $I_{m}$ is above $J_{m}$ if $k_{m}$ is positive, and $I_{m}$ is below $J_{m}$ if $k_{m}$ is negative.
Suppose $L$ is a level in ${\cal
C}_{n}$ for any $n\geq k$. Translating $L$ by some multiple $\ell h_n$ of the height of the column results in crescents. It suffices to consider a worst case lower bound to various intersections, thus we will only consider crecents in the leftmost subcolumn ${\cal C}_{n,1}$. The minimum size of the top of any such crescent is at least $\frac {1}{8^{\ell}}\mu(L)$ (given the nature of our construction, the size of the crescent may not decrease after each step, but to simplify our calculations we use a conservative estimate). In addition, each crescent has moved through the staircase spacers $\ell$ times. The maximum total number of staircase spacers that any given crescent has moved through is $s_{\ell} =
\sum_{i=1}^{\ell}i$. Therefore, any level $J$ more than $s_{\ell}$ below $I$ contains some pieces of the crescent from $I$. Furthermore, the minimum amount of the crescent from $I$ that intersects $J$ has measure at least $\frac{1}{8^{\ell+d}}\mu(J)$, where $d$ is the distance $J$ is below $I$.
To account for the fact that after each pass through a staircase spacer the crescent is moved down by one, we note that translating $I$ by $\ell h_{n} +
c$, with $ c > s_{\ell}$, ensures that any level $J$ below (or at) $I$ contains pieces of the $I$-crescent having total measure at least $\frac {1}{8^{\ell+d +
c}}\mu(J)$, where $d$ is as above.
For each $k_{i}$, let $s_{k_{i}} =
\sum_{j=1}^{k_{i}}j$, let $d_i$ be the distance between $I_{i}$ and $J_{i}$ for all $i$, and put $K =
\max\{k_{i}\}$, $S =
\max\{s_{k_{i}} \}$, and $d =
\max\{d_{i} \}$. Choose $\delta$ so that $$0 <
\delta < (\frac {1}{8^{K + d + KS}})^{r}.$$
By the Double Approximation Lemma find $I' =
I_{1}'\times \ldots \times I_{r}'$ and $J' = J_{1}'\times \ldots \times J_{r}'$ such that $I'$ and $J'$ are $(1-\delta)$-full of $A$ and $B$ respectively, $I'_{1},\ldots, I'_{r}, J'_{1},\ldots, J'_{r}$ are all in some column ${\cal C}_{n}$, and for each $i$, $I'_{i}$ and $J'_{i}$ are in the same ${\cal C}_{k}$-copy in ${\cal C}_{n}$, and $I_{i}$ is more than $SK$ levels below the top of ${\cal C}_{n}$.
Let $H = h_{n} + S$. Then for all positive $k_{i}$, $$\mu (T^{k_{i}H}I'_{i}\cap J'_{i}) \geq \frac
{1}{8^{k_i+ d_i + k_{i}s_i}}\mu (I'_{i}) \geq
\frac {1}{8^{K+d+KS }}\mu (I'_{i}).$$
For all negative $k_{i}$, $$\mu (T^{k_{i}H}I'_{i}\cap J'_{i}) = \mu (I'_{i}\cap
T^{|k_{i}|H}J'_{i}) \geq \frac {1}{8^{k_i+ d_i +
k_{i}s_i}}\mu (J'_{i}) \geq \frac {1}{8^{K+d+KS}}\mu
(I'_{i}).$$
Therefore, $$\nu [(T^{k_{1}}\times\ldots\times
T^{k_{r}})^{H}I'\cap J'] \geq (\frac
{1}{8^{K+d+KS}})^{r}\nu (I').$$
Thus, $$\begin{aligned}
\nu [(T^{k_{1}}\times\ldots\times
T^{k_{r}})^{H}A\cap B] & \geq \nu [(T^{k_{1}}\times
\ldots\times T^{k_{r}})^{H}I'\cap J']
- \delta \nu (I') \\ & \geq (\frac
{k}{8^{K+d+KS}})^{r}\nu (I') - \delta \nu (I') > 0.\end{aligned}$$
Therefore $T^{k_{1}}\times
\dots \times T^{k_{r}}$ is ergodic.
[**Remark.**]{} 1. The same proof will apply to a transformation where at the $k^{\rm th}$ stage column $C_k$ is cut into $c>1$ equally-spaced subcolumns ${\cal C}_{k,1},
\ldots,{\cal C}_{k,c}$, a single (staircase) spacer is put on top of column ${\cal C}_{k,c}$ and a stack of $h_k$ spacers is put on top of any of the middle subcolumns.
2\. There exists a rank one infinite measure preserving transformation $S$ such that $S$ has infinite ergodic index but $S\times S^2$ is not conservative, hence $S$ is not power weakly mixing [@afs2].
[AFS2]{}
J. Aaronson, M. Lin, and B. Weiss, [*Mixing properties of Markov operators and ergodic transformations, and ergodicity of Cartesian products,*]{} Israel J. Math. 33, 1979, 198-224.
T. Adams, N. Friedman, and C.E. Silva, [*Rank-one weak mixing for nonsingular transformations,*]{} Israel J. Math. 102 (1997), 269-281.
T. Adams, N. Friedman, and C.E. Silva, [*Rank-one weak mixing for nonsingular transformations II,*]{} preprint.
N.A. Friedman, [Introduction to Ergodic Theory,]{} Van Nostrand, 1970.
S. Kakutani and W. Parry, [Infinite measure preserving transformations with “mixing”,]{} Bull. Amer. Math. Soc. 69, 1963, 752-756.
[^1]: Emory University, Atlanta, GA 30332
[^2]: St. Olaf College, Northfield, MN 55057
[^3]: Williams College, Williamstown, MA 01267
[^4]: Williams College, Williamstown, MA 01267, [email protected]
|
---
abstract: |
We consider the (noisy) Kuramoto model, that is a population of $N$ oscillators, or rotators, with mean-field interaction. Each oscillator has its own randomly chosen natural frequency (quenched disorder) and it is stirred by Brownian motion. In the limit $N \to \infty$ this model is accurately described by a (deterministic) Fokker-Planck equation. We study this equation and obtain quantitatively sharp results in the limit of weak disorder. We show that, in general, even when the natural frequencies have zero mean the oscillators synchronize (for sufficiently strong interaction) around a common rotating phase, whose frequency is sharply estimated. We also establish the stability properties of these solutions (in fact, limit cycles). These results are obtained by identifying the stable hyperbolic manifold of stationary solutions of an associated non disordered model and by exploiting the robustness of hyperbolic structures under suitable perturbations. When the disorder distribution is symmetric the speed vanishes and there is a one parameter family of stationary solutions, as pointed out by H. Sakaguchi [@cf:Sakaguchi]: in this case we provide more precise stability estimates. The methods we use apply beyond the Kuramoto model and we develop here the case of active rotator models, that is the case in which the dynamics of each rotator in absence of interaction and noise is not simply a rotation.\
2010 *Mathematics Subject Classification: 37N25, 82C26, 82C31, 92B20*\
*Keywords: Coupled oscillator systems, Kuramoto model, Fokker-Planck PDE, Normally hyperbolic manifolds, coherence stability, rotating waves*
address:
- ' Universit[é]{} Paris Diderot (Paris 7) and Laboratoire de Probabilit[é]{}s et Modèles Aléatoires (CNRS), U.F.R. Mathématiques, Case 7012 (site Chevaleret) 75205 Paris Cedex 13, France '
- 'Universit[é]{} Paris 6 – Pierre et Marie Curie and Laboratoire de Probabilit[é]{}s et Modèles Aléatoires (CNRS U.M.R. 7599), U.F.R. Mathematiques, Case 188, 4 place Jussieu, 75252 Paris cedex 05, France '
- ' Universit[é]{} Paris Diderot (Paris 7) and Laboratoire de Probabilit[é]{}s et Modèles Aléatoires (CNRS), U.F.R. Mathématiques, Case 7012 (site Chevaleret) 75205 Paris Cedex 13, France '
author:
- Giambattista Giacomin
- Eric Luçon
- Christophe Poquet
title: Coherence stability and effect of random natural frequencies in populations of coupled oscillators
---
Introduction {#sec:intro}
============
Collective phenomena in noisy coupled oscillators
-------------------------------------------------
Coupled oscillator models are omnipresent in the scientific literature because the emergence of coherent behavior in large families of interacting units that have a periodic behavior, that we generically call [*oscillators*]{}, is an extremely common phenomenon (crickets chirping, fireflies flashing, planets orbiting, neurons firing,...). It is impossible to properly account for the literature and the various models proposed for this kind of phenomena, but while a precise description of each of the different instances in which synchronization emerges demands specific, possibly very complex, models, the [*Kuramoto model*]{} has emerged as capturing some of the fundamental aspects of synchronization [@cf:acebron]. It can be introduced via the system of $N$ stochastic differential equations $$\label{eq:Kmod}
{\,\text{\rm d}}{\varphi}^{\omega}_j(t)\, =\,
{\omega}_j {\,\text{\rm d}}t - \frac KN \sum_{i=1}^N \sin({\varphi}^{\omega}_j(t)-{\varphi}^{\omega}_i(t)) {\,\text{\rm d}}t + {\sigma}{\,\text{\rm d}}B_j(t)\, ,$$ for $j=1, \ldots, N$, where
1. $\{B_j\}_{j=1, \ldots, N}$ is a family of standard independent Brownian motions: in physical terms, this is a [*thermal noise*]{};
2. $\{{\omega}_j\}_{j=1\cdots N}$ is a family of independent identically distributed random variables of law $\mu$: they are are the [*natural frequencies*]{} of the oscillators and, in physical terms, they can be viewed as a [*quenched disorder*]{};
3. $K$ and ${\sigma}$ are non-negative parameters, but one should think of them as positive parameters since the cases in which they vanish have only a marginal role in the what follows.
The variables ${\varphi}^{\omega}_j$ are meant to be angles (describing the position of rotators on the circle ${{\ensuremath{\mathbb S}} }$), so we focus on ${\varphi}^{\omega}_j\;\text{mod}\;2\pi$ and defines, once an initial condition is supplied, a diffusion process on ${{\ensuremath{\mathbb S}} }^N$. Note that if $\{ {\varphi}^{\omega}_j (\cdot)\}_{j =1, \ldots, N}$ solves , also $\{ {\varphi}^{\omega}_j (\cdot)+ {\varphi}\}_{j =1, \ldots, N}$, with ${\varphi}\in {{\ensuremath{\mathbb S}} }$, is a solution: this is the rotation symmetry of the system that will repeatedly make surface in the remainder of the paper.
Some of the main features are easily grasped: each oscillator rotates at its own speed, it is perturbed by independent noise and it interacts with all the other oscillators: the interaction tends to align the rotators. It may be helpful at this stage to point out that if $\mu = {\delta}_0$, that is the natural frequencies are just zero, then the dynamics is reversible with invariant probability measure that, up to normalization, is $$\label{eq:mfr}
\exp\left(
\frac{K}{{\sigma}^2}
\sum_{i,j=1}^N \cos \left( {\varphi}_i -{\varphi}_j\right)
\right) {\lambda}_N ({\,\text{\rm d}}{\varphi})\, ,$$ where ${\lambda}_N$ is the uniform measure on ${{\ensuremath{\mathbb S}} }^N$. The Gibbs measure in is a well known statistical mechanics model – it is the classical XY spin mean field model or rotator mean field model – treated analytically in [@cf:SFN; @cf:Pearce] in the $N \to \infty$ limit. In particular, the model exhibits a phase transition at $K=K_c:= 1/{\sigma}^2$, that is effectively a [*synchronization transition*]{}: in the $N \to \infty$ limit we have that for $K\le K_c$ the rotators become independent and uniformly distributed over ${{\ensuremath{\mathbb S}} }$, while for $K>K_c$ the limit measure is obtained by choosing a phase $\theta$ uniformly in ${{\ensuremath{\mathbb S}} }$ and by choosing the values of the phase of each oscillator by drawing it at random following a suitable distribution that concentrates around $\theta$. However, in [@cf:BGP Prop. 1.2], it is shown that, unless $\mu={\delta}_0$, the model is not reversible (for $\mu$ almost surely all the realization of ${\omega}$) and one effectively steps into the domain of non-equilibrium statistical mechanics.
Our approach actually relies on a sharp control of the reversible case and works when the system is not too far from reversibility, that is for weak disorder. Our approach actually applies well beyond : here we will treat explicitly the case ${\omega}_j$ is replaced by $U({\varphi}_j^{\omega}, {\omega}_j)$, that is the natural frequency ${\omega}_j$ is replaced by a [*natural dynamics*]{} that can be substantially different from one oscillator to another. This model is a disordered version of the active rotator model considered for example in [@cf:shinomoto1986a].
Since we will focus on ${\sigma}>0$, from now on, for ease of exposition, we set ${\sigma}:=1$.
The Fokker-Planck or McKean-Vlasov limit
----------------------------------------
An efficient way to tackle is to consider the empirical probability on ${{\ensuremath{\mathbb S}} }\times {{\ensuremath{\mathbb R}} }$ $$\label{eq:empP}
\nu_{N, t}^{\omega}({\,\text{\rm d}}\theta, {\,\text{\rm d}}{\omega})\, :=\, \frac 1N \sum_{j=1}^N
{\delta}_{({\varphi}_j^{\omega}(t), {\omega}_j)} ({\,\text{\rm d}}\theta, {\,\text{\rm d}}{\omega})\, .$$ In fact, in the $N\to \infty$ limit, the sequence of measures $\{ \nu_{N, t}^{\omega}\}_{N=1,2 , \ldots}$ converges to a limit measure whose density (with respect to ${\lambda}_1 \otimes \mu$) solves the nonlinear Fokker-Planck equation $$\label{FKP kuramoto disorder}
\partial_t p_t({\theta},{\omega})\, =\, \frac{1}{2} {\Delta}p_t({\theta},{\omega}) -\partial_{\theta}\Big(p_t({\theta},{\omega})(\langle
J*p_t\rangle_\mu({\theta}) +{\omega})\Big),$$ where $J({\theta})=-K\sin({\theta})$, $\ast$ denotes the convolution and $\langle \cdot\rangle_\mu$ is a notation for the integration with respect to $\mu$, so $\langle J\ast u\rangle_\mu({\theta}) = {\int_{{{\ensuremath{\mathbb R}} }}}{\int_{{{\ensuremath{\mathbb S}} }}}{J({\varphi})
u({\theta}-\varphi, {\omega}){\,\text{\rm d}}{\varphi}\mu({\,\text{\rm d}}{\omega})}$ is the convolution of $J$ and $u$, averaged with respect to the disorder. Here and throughout the whole paper ${\Delta}$ means $\partial^2 _{\vartheta}$. The Fokker-Planck PDE appears repeatedly in the physics and biology literature, see e.g. [@cf:acebron; @cf:Sakaguchi; @cf:StrogatzMirollo], and a mathematical proof (and precise statement) of the result we just stated can be found in [@cf:dPdH; @cf:eric]. Notably, in [@cf:eric] the result is established under the assumption that $\int \vert {\omega}\vert \mu( {\,\text{\rm d}}{\omega})< \infty$ and emphasis is put on the fact that the result holds for almost every realization of the disorder sequence $\{ {\omega}_j\}_{j=1,2, \ldots}$. Let us point out that in ${\omega}$ is a one dimensional real variable, while in the superscript ${\omega}$ is a short for the whole sequence of natural frequencies. Since what follows is really about this abuse of notation will be of limited impact.
In Appendix \[sec:appendix regularity pt with disorder\], we detail the fact that generates an evolution semigroup in suitable spaces. Here we want to stress that can be viewed as a family of coupled PDEs, one for each value of ${\omega}$ in the support of $\mu$: $p_t(\cdot, {\omega})$ is the distribution of phases in the population of oscillators with natural frequency ${\omega}$.
About stationary solutions to
------------------------------
Remarkably ([@cf:Sakaguchi], see also [@cf:dH]), if $\mu$ is symmetric all the stationary solutions to can be written in a semi-explicit way as ${q}({\theta}+{\theta}_0, {\omega})$ (${\theta}_0$ is an arbitrary constant that reflects the rotation symmetry) where $$\label{eq:qhatom}
{q}({\theta}, {\omega}) \,:=\, \frac{S({\theta}, {\omega}, 2Kr)}{Z({\omega},2Kr)}\, ,$$ with $$\label{eq:Sq}
S({\theta}, {\omega}, x) \,=\, e^{G({\theta}, {\omega}, x)}\left[ (1-e^{4\pi{\omega}})
\int_{0}^{{\theta}}{e^{-G(u,
{\omega}, x)}{\,\text{\rm d}}u} + e^{4\pi{\omega}}\int_{0}^{2\pi}{e^{-G(u, {\omega}, x)} {\,\text{\rm d}}u} \right],$$ and $G(u, y, x)= x\cos(u) + 2y u$, $Z({\omega},x)= {\int_{{{\ensuremath{\mathbb S}} }}}S({\theta}, {\omega}, x) {\,\text{\rm d}}{\theta}$ is the normalization constant and $r\in[0,1]$ satisfies the fixed-point relation $$\label{eq:fixedpointom}
r\,=\, \Psi^\mu(2Kr), \ \ \ \text{where} \ \ \ \Psi^\mu(x)\,:=\, {\int_{{{\ensuremath{\mathbb R}} }}}\frac{{\int_{{{\ensuremath{\mathbb S}} }}}\cos({\theta})S({\theta}, {\omega},
x){\,\text{\rm d}}{\theta}}{Z({\omega},x)}\mu({\,\text{\rm d}}{\omega})\, .$$ A series of remarks are in order:
1. $r=0$ solves and this corresponds to the fact that $q(\cdot)\equiv\frac1{2\pi}$ is a stationary solution. It is the only one as long as $K$ does not exceed critical value $K_c$ which is in any case not larger than $$\label{eq:Ktilde}
\widetilde{K} \, :=\,
\left( \int_{{\ensuremath{\mathbb R}} }\frac{\mu({\,\text{\rm d}}{\omega})}{1+4{\omega}^2} \right)^{-1}\, ,$$ as one can easily see by computing (see e.g. [@cf:dH]) the derivative of $\Psi^\mu(2K\cdot)$ at the origin and noticing that is larger than one if and only if $K >{\widetilde}K$ and that $\Psi^\mu(\cdot) <1$, see Figure \[fig:fixed point Psi\].
2. When admits a fixed point $r>0$, and this is certainly the case if $K>\widetilde K$, a nontrivial stationary solution is present and in fact, by rotation symmetry, a circle of non-trivial stationary solutions. Such solutions correspond to a synchronization phenomenum, since the distribution of the phases is no longer trivial.
3. As explained in Figure \[fig:fixed point Psi\] and its caption, in general there can be more than one fixed point $r>0$: in absence of disorder there is only one positive fixed point (when it exists, that is for $K>1$), but this fact is non-trivial even in this case (see below). Uniqueness is expected for $\mu$ which is unimodal, but this has not been established.
4. While the local stability of $\frac 1{2\pi}$ is understood [@cf:StrogatzMirollo] and it holds only if $K \le \widetilde{K}$, the stability properties of the non-trivial solutions are a more delicate issue.
An overview of the results we present
-------------------------------------
Here are two natural questions:
- What are the stability properties of the non-trivial stationary solutions?
- What happens if $\mu$ is not symmetric?
Our work addresses these two questions and provides complete answers for weak disorder. The precise set-up of our work is better understood if we remark from now that we can assume $m_{\omega}:=\int {\omega}\mu ({\,\text{\rm d}}{\omega})=0$. In fact, if this is not the case we can map the model to a model with $m_{\omega}=0$ by putting ourselves on the frame that rotates with speed $m_{\omega}$, that is if we consider the diffusion $\{{\varphi}^{\omega}_{j}(t)- m_{\omega}t\}_{j=1, \ldots, N}$. So, we assume henceforth $m_{\omega}=0$ and we rewrite the natural frequencies as ${\delta}{\omega}$, with ${\delta}$ a non-negative parameter. We assume moreover that $$\label{eq:suppHyp}
\operatorname*{Supp}(\mu)\subseteq [-1, 1]\, .$$ In this set-up, becomes $$\label{FKP kuramoto disorder delta}
\partial_t p_t^{\delta}({\theta},{\omega})\, =\, \frac{1}{2} {\Delta}p_t^{\delta}({\theta},{\omega}) -\partial_{\theta}\Big(p_t^{\delta}({\theta},{\omega})(\langle
J*p_t^{\delta}\rangle_\mu({\theta}) +{\delta}{\omega})\Big)\, .$$ Note that this leads to (obvious) changes to -. We have introduced this parameterization because the results that we present are for small values of ${\delta}$. In particular we are going to show that for any $K>1$, there exists ${\delta}_0>0$ such that for ${\delta}\in [0, {\delta}_0]$
- there exists a solution $p_t^{\delta}({\theta},{\omega})$ to of the form $q( {\theta}- c_\mu ({\delta}) t)$, we show that $c_\mu ({\delta}) =O({\delta}^3)$ and we actually give an expression for $\lim_{{\delta}\searrow 0}c_\mu ({\delta}) / {\delta}^3$: this is a rotating wave (or limit-cycle) for the dynamical system and we establish its stability under perturbations;
- when $\mu$ is symmetric and $K> \widetilde K$ we show that there is, up to rotation symmetry, only one non-trivial solution and that it is (linearly and non-linearly) stable.
The results we obtain are based on the rather good understanding that we have of the case ${\delta}=0$ that, as we have already explained, is reversible and the corresponding Fokker-Planck PDE is of gradient flow type (e.g. [@cf:Otto] and references therein). These properties have been exploited in [@cf:BGP] in order to extract a number of properties of the Fokker-Planck PDE (denoted from now on: reversible PDE) $$\label{eq:revK}
\partial_t p_t({\theta})\, =\, \frac{1}{2} {\Delta}p_t({\theta}) -\partial_{\theta}\Big(p_t({\theta})( J*p_t)({\theta}) \Big) \, ,$$ and notably the linear stability of the non-trivial stationary solutions. In fact one can find in [@cf:BGP] an analysis of the evolution operator linearized around the non-trivial stationary solutions. Some of the results in [@cf:BGP] are recalled in the next section, but they are not directly applicable because the ${\delta}=0$ case that corresponds to what interests us is rather $$\label{FKP kuramoto disorder without drift}
\partial_t p_t({\theta}, {\omega})\, =\, \frac{1}{2} {\Delta}p_t({\theta}, {\omega}) -\partial_{\theta}\Big(p_t({\theta}, {\omega})(\langle J*p_t\rangle_\mu({\theta}) \Big) \, ,$$ which we call [*non-disordered PDE*]{}. So the [*natural frequencies*]{} have no effective role beyond separating the various rotators into populations with given natural (ineffective) frequency that now are just labels. But in order to set-up a proper perturbation procedure we need to control and, in particular, we need (and establish) a spectral gap inequality for the evolution linearized around the non-trivial solutions. This spectral analysis is going to be central both for the general and for the symmetric disorder case. In the general set-up we are going to exploit the [*normally hyperbolic structure*]{} [@cf:HPS; @cf:SellYou] of the manifold of stationary solutions of and the robustness of such structures (like in [@cf:GPPP]). In the case of symmetric $\mu$ we can get more precise results by ad hoc estimates, made possible by the explicit expressions -, and use results in the general theory of operators [@Pazy1983] and perturbation theory of self-adjoint operators [@Kato1995].
The normal hyperbolic manifold approach allows to treat cases that are substantially more general and notably the case of $$\label{eq:AR}
\partial_t p_t({\theta},{\omega})\, =\, \frac{1}{2} {\Delta}p_t({\theta},{\omega}) -\partial_{\theta}\Big(p_t({\theta},{\omega})(\langle
J*p_t\rangle_\mu({\theta}) +{\delta}U(\theta, {\omega}))\Big)\, ,$$ which is the large $N$ limit of with the term ${\omega}_j {\,\text{\rm d}}t$ replaced by $U({\varphi}^{\omega}_j (t), {\omega}_j) {\,\text{\rm d}}t$, with $U \in C^1( {{\ensuremath{\mathbb S}} }\times {{\ensuremath{\mathbb R}} }; \, {{\ensuremath{\mathbb R}} })$. In this case each oscillator has its own non-trivial dynamics which may be very different from the dynamics of other oscillators: consider for example $$U({\varphi}, {\omega})\, =\,b+ {\omega}+ a\sin ({\varphi})\, , \ \ \ a, b \in {{\ensuremath{\mathbb R}} }\, ,$$ and $\mu$ uniform over $[-1,1]$. For $a \in (-1,1)$ there are some [*active rotators*]{} [@cf:shinomoto1986a; @cf:GPPP] that in absence of noise and interaction (${\sigma}=K=0$) rotate (this happens if $\vert b+{\omega}\vert >\vert a\vert$ and of course the direction of rotation depends on the sign of $b+{\omega}$) and others that instead are stuck at a fixed point (this happens if $\vert b+{\omega}\vert \le \vert a\vert$). Our approach allows us to establish that there is a synchronization regime for $K>1$ and ${\delta}$ small and to describe the dynamics of the system in this regime. This is going to be detailed in Section \[sec:AR\].
The two questions raised at the beginning of this section have been already repeatedly approached but looking at synchronized solutions as bifurcation from incoherence. The results are hence for $K$ close to the critical value corresponding to the breakdown of linear stability of $1/2\pi$: one can find a detailed review of the vast literature on this issue in [@cf:acebron Sec. III]. Our results are instead for arbitrary $K>1$, but ${\delta}$ smaller than ${\delta}_0(K)$ and of course ${\delta}_0(K)$ vanishes as $K$ approaches $1$.
Mathematical set-up and main results {#sec:mainresults}
====================================
The reversible and the non-disordered PDE {#subsec:organizednondisorderedcase}
-----------------------------------------
We first recall some results about the reversible PDE . The stationary solutions $q_0(\theta)=q(\theta,0)$ are, up to rotation invariance, given by -, but formulas get simpler, namely $$\label{eq:defstationarysolution nodisorder}
q_0({\theta})\, =\, \frac 1{Z_0(2Kr_0)} \exp(2Kr_0 \cos({\theta}))\, ,$$ where $Z_0(x):= Z(0, x)^\frac 12$ and this time we have the more explicit expression $Z_0(x)\, =\, {\int_{{{\ensuremath{\mathbb S}} }}}e^{x\cos({\theta})} {\,\text{\rm d}}{\theta}= 2\pi I_0(x)$ is the normalization constant and $r_0$ is a solution of the fixed-point problem $$\label{eq:deffixedpoint nodisorder}
r_0\, =\, \Psi_0 (2Kr_0) \qquad \text{where} \qquad \Psi_0(x)\, :=\, \frac{I_1(x)}{I_0(x)}\, ,$$ where we used standard notations for the modified Bessel functions $$\label{def bessel}
I_i(x)\, =\, \frac 1{2\pi}\int_{{{\ensuremath{\mathbb S}} }} (\cos({\theta}))^i\exp(x\cos({\theta})){\,\text{\rm d}}{\theta}\, \qquad i=0,1\, .$$ The mapping $\Psi_0$ is increasing, concave (see [@cf:Pearce]) and with derivative at $0$ equal to $\frac 12$. Consequently if $K{\;\leqslant\;}1$, $r_0=0$ is the unique solution of the fixed-point problem, and $q(\cdot)\equiv\frac{1}{2\pi}$ is the only stationary solution of . If $K>1$, we get in addition a circle (because of the rotation invariance) of nontrivial stationary solutions $$M_{\text{rev}}\, :=\, \{q_{\psi, 0}(\cdot):=q_0(\cdot-\psi):\, \psi\in{{\ensuremath{\mathbb S}} }\} \qquad \text{with} \qquad q_0({\theta})\,
:=\,\frac{\exp(2Kr_0\cos({\theta}))}{\int_{{{\ensuremath{\mathbb S}} }}\exp(2Kr_0\cos({\theta}))}$$ where $r_0=r_0(K)$ is the unique non trivial fixed-point .
Let us now focus on the non-disordered PDE and let us insist on the fact that we are interested in solutions such that ${\varphi}^{\delta}_t(\cdot, {\omega})$ is a probability density. Observe then that if $q({\theta},{\omega})$ is a stationary solution of , we see (Appendix \[sec:appendix regularity pt with disorder\]) that $q$ is $C^\infty$ with respect to ${\theta}$ and that $\langle q \rangle_\mu$ is a stationary solution for . So there exists $\psi\in{{\ensuremath{\mathbb S}} }$ such that $\langle q\rangle_\mu = q_\psi$ and a short computation leads to $$\langle J*q \rangle_\mu({\theta})\, =\, -K\sin({\theta}-\psi)\, ,$$ and, since ${\int_{{{\ensuremath{\mathbb S}} }}}q({\theta},{\omega}) {\,\text{\rm d}}{\theta}=1$ for almost all ${\omega}$, we obtain that $q(\cdot,{\omega})=q_{ \psi}(\cdot)$ for almost all ${\omega}$. In conclusion, with some abuse of notation, we can say the stationary solutions of and are the same: of course in the second case the function space includes the dependence on ${\omega}$, so we choose a different notation, that is $M_0$, for the corresponding circle of non-trivial stationary solutions.
An important issue for us is the stability of $M_0$ (for its existence we are assuming $K>1$) and for this we denote by $A$ the linearized evolution operator of around $q_0$ $$\label{def A}
Au({\theta},{\omega})\, :=\, \frac 12 {\Delta}u({\theta},{\omega}) - \partial_{\theta}\Big(q_0({\theta}) \langle J*u \rangle_\mu({\theta}) + u({\theta},{\omega})
J*q_0({\theta})\Big)$$ with domain $$\label{eq:defdomain A}
{{\ensuremath{\mathcal D}} }(A)\, :=\, \left\{u\in C^2({{\ensuremath{\mathbb S}} }\times{{\ensuremath{\mathbb R}} },{{\ensuremath{\mathbb R}} }):\, {\int_{{{\ensuremath{\mathbb S}} }}}u({\theta},{\omega}){\,\text{\rm d}}{\theta}=0\, \text{ for all }{\omega}\right\}\, .$$ For any smooth positive function $k:{{\ensuremath{\mathbb S}} }\mapsto {{\ensuremath{\mathbb R}} }$, we introduce the Hilbert space $H^{-1}_{k,\mu}$ defined by the closure of ${{\ensuremath{\mathcal D}} }(A)$ for the norm $\Vert \cdot \Vert_{-1,k,\mu}$ associated with the scalar product $$\label{def scalar product sobolev with wage and mu}
\langle u,\, v \,\rangle_{-1,k,\mu}\, :=\, {\int_{{{\ensuremath{\mathbb R}} }\times{{{\ensuremath{\mathbb S}} }}}}\frac{{{\ensuremath{\mathcal U}} }({\theta},{\omega}){{\ensuremath{\mathcal V}} }({\theta},{\omega})}{k({\theta})}{\,\text{\rm d}}{\theta}\mu({\,\text{\rm d}}{\omega}),$$ where ${\omega}$ a.s., ${{\ensuremath{\mathcal U}} }(\cdot,{\omega})$ is the primitive of $u(\cdot,{\omega})$ such that $\int_{{{\ensuremath{\mathbb S}} }}\frac{{{\ensuremath{\mathcal U}} }({\theta},{\omega})}{k({\theta})}{\,\text{\rm d}}{\theta}=0$, and ${{\ensuremath{\mathcal V}} }(\cdot,{\omega})$ is defined in the analogous fashion. Let us remark (see [@cf:GPPP Sec. 2]) immediately that $$\Vert u \Vert_{-1,k_1,\mu}^2\, {\;\leqslant\;}\, \frac{\Vert k_2 \Vert_\infty}{ \Vert k_1 \Vert_\infty}\Vert u \Vert_{-1,k_2,\mu}^2\, ,$$ so that all the norms we have introduced are equivalent. For the case $k(\cdot) \equiv
1$ we use the notations $H^{-1}_\mu$ and $\Vert\cdot\Vert_{-1,\mu}$. We will prove the following result, which is just technical, but it will be of help to understand our main results:
\[th:spectral gap A\] A is essentially self-adjoint in $H^{-1}_{{q_0},\mu}$. Moreover the spectrum lies in $(-\infty,0]$, $0$ is a simple eigenvalue, with eigenspace spanned by $\partial_{\theta}q_0$, and there is a spectral gap, that is the distance ${{\lambda}_{K}}$ between $0$ and the rest of the spectrum is positive.
The proof of this result builds on [@cf:BGP Th. 1.8] that deals with the reversible case and the (lower) bound on the spectral gap ${{\lambda}_{K}}$ that we obtain coincides with the quantity ${\lambda}(K)$ in [@cf:BGP Th. 1.8] (this bound can be improved as explained in in [@cf:BGP Sec. 2.5] and sharp estimates on the spectral gap can be obtained in the limit $K \searrow 1$ and $K \nearrow \infty$). For the reversible evolution, the linear operator $L_{q_0}$ is defined by $$\label{def Lq0}
L_{q_0}u({\theta})\, :=\, \frac 12 {\Delta}u({\theta}) -\partial_{\theta}\Big( q_0({\theta})J*u({\theta})+u({\theta})J*q_0({\theta})\Big),$$ with domain $D(L_{q_0})$ given by the $C^2({{\ensuremath{\mathbb S}} },{{\ensuremath{\mathbb R}} })$ functions with zero integral.
Synchronization: the main result without symmetry assumption
------------------------------------------------------------
Proposition \[th:spectral gap A\] is a key ingredient for our main results and the functional space $H^{-1}$ appears in it, but an important role is played also by $L^2({\lambda}\otimes \mu)$, ${\lambda}$ is the Haar measure on ${{\ensuremath{\mathbb S}} }$, whose norm is denoted by $\Vert \, \cdot \, \Vert_{2, \mu}$. For $C>0$ and $M \subset L^2({\lambda}\otimes \mu)$ we set ${{\ensuremath{\mathcal N}} }_{2, \mu}(M, C):= \{u: \,$there exists $v\in M$ such that $\Vert u-v \Vert_{2, \mu} \le C\}$. In the statement below $q\in M_0$ is the element of the manifold such that $q(\cdot,{\omega})=
q_0(\cdot)$, cf. , with $r_0(K)>0$ (hence $K>1$).
\[th:expansion speed\] For every $K>1$ there exists ${\delta}_0={\delta}_0(K)>0$ such that for $\vert {\delta}\vert \le {\delta}_0$ there exists $\widetilde q _{\delta}\in L^2({\lambda}\otimes \mu)$, satisfying $\Vert \widetilde q _{\delta}-q \Vert_{2, \mu}=O({\delta})$ and a value $c_\mu ({\delta}) \in {{\ensuremath{\mathbb R}} }$ such that if we set $$\label{eq:main1.1}
q^{(\psi)}_t (\theta, {\omega})\, :=\, \widetilde q _{\delta}(\theta -c_\mu ({\delta}) t-\psi)\, ,$$ then $q^{(0)}_t$ solves . Moreover
1. the family of solutions $\{q^{(\psi)}_\cdot\}_\psi$ is stable in the sense that there exist two positive constants ${\beta}={\beta}(K)$ and $C=C(K)$ such that if $p_0^{\delta}\in {{\ensuremath{\mathcal N}} }_{2, \mu}(M_0,{\delta})$, and $\int_{{\ensuremath{\mathbb S}} }p_0^{\delta}(\theta, {\omega}) {\,\text{\rm d}}\theta=0$ $\mu({\,\text{\rm d}}{\omega})$-a.s., then there exists $\psi_0\in {{\ensuremath{\mathbb S}} }$ such that for all $t\ge 0$ $$\Vert q_t^{(\psi_0)}-p_t^{\delta}\Vert_{2, \mu}\, \le \, 2 C\exp(-{\beta}t)\, .$$
2. we have $$\label{eq:main1.2}
c_\mu({\delta})\, =\, {\delta}^3
\frac{
\left\langle {\omega}\partial_{\theta}n^{(2)} , \partial_\theta q_0 \right \rangle_{-1, {q_0},\mu} }
{ \left\langle \partial_\theta q_0 ,\partial_\theta q_0
\right \rangle_{-1, {q_0}} } + O({\delta}^5)\, ,$$ where $n^{(2)}$ is the unique solution of $$An^{(2)}\, =\, {\omega}\partial_{\theta}n^{(1)} \quad\text{ and } \quad \left\langle n^{(2)}, \partial_\theta q_0 \right \rangle_{-1, {q_0},\mu}
\, =\, 0\, ,$$ and $n^{(1)}$ is the unique solution of $$A n^{(1)}\, =\,
{\omega}\, \partial_\theta q_0 \quad\text{ and } \quad {\ensuremath{\left\langlen^{(1)}\, ,\, \partial_\theta q_0 \right\rangle_{ -1, {q_0},\mu}}}\, =\, 0\, .$$
In the proof of Theorem \[th:expansion speed\] one finds also further estimates, in particular (see ) that one has $$\label{eq:qgddevel}
{\widetilde}q_{\delta}\, =\, q_0 + {\delta}n^{(1)} + {\delta}^2 n^{(2)} + O_{L^2}({\delta}^3)\, .$$ Actually, see Remark \[rem:pa\], the argument of proof can be pushed farther to obtain arbitrarily many terms in development , as well as in $$c_\mu ({\delta})\, =\, c_3 {\delta}^3+ c_5 {\delta}^5 +\ldots \, .$$ In Table \[tab:1\] we report a comparison between the $c_\mu({\delta})$ obtained by solving numerically and by evaluating the leading order $c_3$, i.e. by using .
[ | c | c | c | c | ]{}
${\delta}$ & $K=2$ & $ K=1.5 $ & $ K=1.1 $\
$0.5$ & $-1.56300 \cdot 10^{-2}$ & $-8.59626 \cdot 10^{-2}$ & $-3.01064 \cdot 10^{-1}$\
$0.1$ & $-1.23998 \cdot 10^{-2}$ & $-6.84835 \cdot 10^{-2}$ & $-2.72117 \cdot 10^{-1}$\
$0.05$ & $-1.23072 \cdot 10^{-2}$ & $-6.79553 \cdot 10^{-2}$ & $-2.69460 \cdot 10^{-1}$\
$0.01$ & $-1.22776 \cdot 10^{-2}$ & $-6.77921 \cdot 10^{-2}$ & $-2.68603 \cdot 10^{-1}$\
$0.005$& $-1.22767 \cdot 10^{-2}$ & $-6.77869 \cdot 10^{-2}$ & $-2.68576 \cdot 10^{-1}$\
\
$c_3$ & $-1.22764 \cdot 10^{-2}$ & $-6.77851 \cdot 10^{-2}$ & $-2.68567 \cdot 10^{-1}$\
Symmetric disorder case
-----------------------
Let us focus on the case in which the distribution of the disorder $\mu$ is symmetric. In this case, at least for small disorder, Theorem \[th:expansion speed\] is just telling us that the leading order in the development for the speed $c_\mu({\delta})$ is zero: one can actually work harder and show that such a development yields zero terms to all orders. In reality in this case we already know, see -, that for $K$ sufficiently large there is at least a non-trivial stationary profile, hence, by rotation symmetry, at least one whole circle of stationary solutions. Actually, we can show that for ${\delta}$ small there is just one circle, that we call $M_{\delta}$, of non-trivial stationary solutions and this circle converges to $M_0$ as ${\delta}\searrow 0$ (in $C^j$, for every $j$) so the rotating solutions found in Theorem \[th:expansion speed\] must be the stationary solutions in $M_{\delta}$.
In order to be precise about this issue, we point out that - are written for while we work rather with . The changes are obvious, but we introduce a notation for the analog of : $$\label{eq:deffixedpoint disorder}
r_{\delta}\,=\, \Psi_{\delta}^\mu(2Kr_{\delta}), \qquad \text{where,}\, \Psi_{\delta}^\mu(x)\,:=\, {\int_{{{\ensuremath{\mathbb R}} }}}\frac{{\int_{{{\ensuremath{\mathbb S}} }}}\cos({\theta})S({\theta}, {\delta}{\omega},
x){\,\text{\rm d}}{\theta}}{Z({\delta}{\omega}, x)}\mu({\,\text{\rm d}}{\omega})\, .$$
\[th: Psimu concave\] For all $K_{\min}<K_{\max}$, there exists ${\delta}_1={\delta}_1(K_{\min}, K_{\max})>0$ such that, for all $0<K_{\min}<K<K_{\max}$ and all ${\delta}{\;\leqslant\;}{\delta}_1$ the function $\Psi_{\delta}^\mu$ is strictly concave on $[0,1]$. Therefore for has only a positive solution $r_{\delta}=r_{\delta}(K, \mu)$. Moreover $
\lim_{{\delta}\searrow 0} r_{\delta}= r_0$.
We point out that in spite of the fact that $\Psi^\mu$ is explicit (cf. ), it is not so straightforward to show that it is concave. We show that $\Psi_{\delta}^\mu$ remains strictly concave for a small ${\delta}$ via a perturbation argument. But the conjecture (see [@cf:dH] and [@cf:dPdH]) that $\Psi^\mu$ is strictly concave for unimodal distributions $\mu$ is still an open issue.
A direct computation shows that the derivative of $\Psi^\mu_{\delta}$ at the origin is $1/(2\widetilde K _{\delta})$, for $\widetilde K _{\delta}\,:=\,
\left({\int_{{{\ensuremath{\mathbb R}} }}}\frac{\mu({\,\text{\rm d}}{\omega})}{1+4{\delta}^2{\omega}^2}\right)^{-1}$ (of course $\widetilde K_1$ coincides with $\widetilde K$, introduced in ). Under the hypothesis of Lemma \[th: Psimu concave\], one therefore sees that there is a synchronization transition at $K=\widetilde K _{\delta}$ in the sense that for $K\le \widetilde K _{\delta}$ the only stationary solution is $\frac 1{2\pi}$ while for $K>\widetilde K _{\delta}$ also the manifold of non-trivial stationary solutions appears (and there is no other stationary solution).
Theorem \[th:expansion speed\] provides a stability statement for $M_{\delta}$. This result can be sharpened and for this let us introduce the linear operator $$\label{eq:defLqmu}
L^{\omega}_{{q}}u({\theta}, {\omega})\, :=\, \frac 12 \Delta u({\theta}, {\omega}) - \partial_{{\theta}} \left(
u({\theta}, {\omega})\left(
\langle J \ast {q}\rangle_\mu({\theta}) + {\delta}{\omega}\right) + {q}({\theta}, {\delta}{\omega}) \langle J\ast u\rangle_\mu({\theta})
\right),$$ The domain ${{\ensuremath{\mathcal D}} }(L^{\omega}_{{q}})$ of the operator $L^{\omega}_{{q}}$ is chosen to be the same as for $A$, cf. .
We place ourselves within the framework of Lemma \[th: Psimu concave\], in the sense that $\delta$ is small enough to ensure the uniqueness of a non-trivial stationary solution (of course existence requires $K> {\widetilde}K_{\delta}$ and this is implied by $K>1$ if ${\delta}$ is sufficiently small). We prove a number of properties of the linear operator , saying notably that it has a simple eigenvalue at zero and the rest of spectrum is at a positive distance from zero and it is in a cone in that lies in the negative complex half plane. We summarize in the next statement the qualitative features of our results on $L_q^{\omega}$, but what we really prove are quantitative explicit estimates: the interested reader finds them in Section \[sec:sym\].
\[th:spectral prop L disorder\] The operator $L^{\omega}_{{q}}$ has the following spectral properties: $0$ is a simple eigenvalue for $L^{\omega}_{{q}}$, with eigenspace spanned by $({\theta}, {\omega})\mapsto q'({\theta}, {\omega})$. Moreover, for all $K>1$, $\rho\in(0,1)$, $\alpha\in(0,\pi/2)$, there exists ${\delta}_2=\delta_2(K, \rho, \alpha)$ such that for all $0{\;\leqslant\;}\delta{\;\leqslant\;}\delta_2$, the following is true:
- $L^{\omega}_{{q}}$ is closable and its closure has the same domain as the domain of the self-adjoint extension of $A$;
- The spectrum of $L^{\omega}_{{q}}$ lies in a cone $C_{\alpha}$ with vertex $0$ and angle $\alpha$ $$C_{\alpha}\, :=\, {\left\{\lambda\in{{\ensuremath{\mathbb C}} }\,;\,\frac{\pi}{2} + \alpha{\;\leqslant\;}\arg(\lambda){\;\leqslant\;}\frac{3\pi}{2}-\alpha\right\}}\subseteq {\left\{z\in{{\ensuremath{\mathbf C}} }\,;\,\Re(z){\;\leqslant\;}0\right\}}\, ;$$
- There exists $\alpha'\in(0, \frac\pi2)$ such that $L^{\omega}_{{q}}$ is the infinitesimal generator of an analytic semi-group defined on a sector $\{\lambda\in{{\ensuremath{\mathbb C}} },\, |\arg(\lambda)|< \alpha'\}$;
- The distance between $0$ and the rest of the spectrum is strictly positive and is at least equal to $\rho{{\lambda}_{K}}$, where ${{\lambda}_{K}}$ is the spectral gap of the operator $A$ introduced in Proposition \[th:spectral gap A\].
Organization of remainder of the paper
--------------------------------------
In Section \[sec:hyperbolic structure\] we introduce the notion of stable normally hyperbolic manifold, we recall its robustness properties, and show that $M_0$ is in this class of manifolds. The essential ingredient is Proposition \[th:spectral gap A\] that, directly or indirectly, plays a role in each subsequent section. Section \[sec:hyperbolic structure\] is also devoted to the proof of Proposition \[th:spectral gap A\]. The proof of Theorem \[th:expansion speed\] is then completed in Section \[sec:pa\], that is mainly devoted to perturbation arguments. The case of the active rotators is treated in Section \[sec:AR\], while Section \[sec:sym\] deals with the case symmetric disorder distribution and, notably, with the proof of Theorem \[th:spectral prop L disorder\] and of a number of related quantitative estimates.
Hyperbolic structures and periodic solutions {#sec:hyperbolic structure}
============================================
In this section we present the arguments proving the existence of the periodic solution of Theorem \[th:expansion speed\]. We rely on the fact that the circle of stationary solutions $M_0$ is a stable normally hyperbolic manifold, and on the robustness of this kind of structure : adding the perturbation term $-{\delta}\partial_{\theta}(p_t({\theta},{\omega}){\omega})$ in , this manifold $M_0$ is deformed into another manifold $M_{\delta}$, and thanks to the rotation invariance of the problem, $M_{\delta}$ is a circle too. The spectral gap of operator $A$ (Property \[th:spectral gap A\]) which induces the hyperbolic property of $M_0$ is proved at the end of this section.
Stable normally hyperbolic manifolds
------------------------------------
We start by quickly reviewing the notion of of stable normally hyperbolic manifold (SNHM). The evolution of will be studied in the space $X^1_\mu$ defined by $$X^1_\mu\, :=\, \left\{u\in L^2({\lambda}\otimes\mu),\, \int_{{{\ensuremath{\mathbb S}} }}u({\theta},{\omega}){\,\text{\rm d}}{\theta}=1\quad {\omega}\text{ a.s.}\right\}$$ where ${\lambda}$ denotes the Lebesgue measure on ${{\ensuremath{\mathbb S}} }$. This is made possible by the conservative character of the dynamics. The $L^2$-norm with respect to the measure ${\lambda}\otimes\mu$ will be denoted by $\Vert \cdot\Vert_{2,\mu}$. We will also use the space $X^0_\mu$ defined by $$X^0_\mu\, :=\, \left\{u\in L^2({\lambda}\otimes\mu),\, \int_{{{\ensuremath{\mathbb S}} }}u({\theta},{\omega}){\,\text{\rm d}}{\theta}=0\quad {\omega}\text{ a.s.}\right\}\, .$$
To define a SNHM, we need a dynamics: we have in mind but for the moment let us just think of an evolution semigroup in $X^1_\mu$ that gives rise to $\{ u_t\}_{t\ge 0}$, with $u_0=u$, to which we can associate a linear evolution semigroup $\{\Phi(u, t)\}_{t \ge 0}$ in $X^0_\mu$, satisfying $\partial_t \Phi(u, t)v =L(t) \Phi(u, t)v$ and $\Phi(u, 0)v=v$, where $L(t)$ is the operator obtained by linearizing the evolution around $u_t$.
For us a SNHM $M\subset X^1_\mu$ (in reality we are interested only in $1$-dimensional manifolds, that is curves, but at this stage this does not really play a role) of characteristics ${\lambda}_1$, ${\lambda}_2$ ($0\le {\lambda}_1< {\lambda}_2$) and $C>0$ is a $C^1$ compact connected manifold which is invariant under the dynamics and for every $u \in M$ there exists a projection $P^o(u)$ on the tangent space of $M$ at $u$, that is ${{\ensuremath{\mathcal R}} }(P^o(u))=:T_uM$, which, for $v \in L^2_0$, satisfies the following properties:
1. for every $t\ge 0$ we have $$\Phi(u,t) P^o (u_0)v\, =\, P^o (u_t)\Phi(u,t) v\, ,$$
2. we have $$\label{eq:shyp1}
\Vert \Phi(u,t) P^o(u_0) v\Vert_{2,\mu} \, \le \, C \exp( {\lambda}_1 t)
\Vert v \Vert_{2,\mu}\, ,$$ and, for $P^s\, :=\, 1-P^o$, we have $$\label{eq:shyp2}
\Vert \Phi(u,t) P^s(u_0) v\Vert_{2,\mu} \, \le \, C \exp( -{\lambda}_2 t)
\Vert v \Vert_{2,\mu}\, ,$$ for every $t\ge 0$;
3. there exists a negative continuation of the dynamics $\{ u_t\}_{t \le 0}$ and of the linearized semigroup $\{ \Phi(u, t) P^o(u_0) v \}_{t \le 0}$ and for any such continuation we have $$\label{eq:shyp3}
\Vert \Phi(u,t) P^o(u_0) v\Vert_{2,\mu} \, \le \, C \exp( -{\lambda}_1 t)
\Vert v \Vert_{2,\mu}\, ,$$ for $t \le 0$.
$M_0$ is a SNHM
---------------
First of all: the dynamics on $M_0$ is trivial. For $q_\psi\in M_0$, the projection $P^o_{q_\psi}$ on the tangent space is the projection on the subspace spanned by $q^\prime_\psi$: $$\label{def P^o}
P^o_{q_\psi} u\, =\, \frac{\left\langle u,q^\prime_\psi\right\rangle_{-1,q_\psi,\mu}}{\left\langle q^\prime_\psi,q^\prime_\psi\right\rangle_{-1,q_\psi}}q^\prime_\psi$$ and since the dynamic on the manifold is trivial, we are allowed to choose for the parameters ${\lambda}_1=0$ and ${\lambda}_2={{\lambda}_{K}}$ (where we recall that ${{\lambda}_{K}}$ is given by Proposition \[th:spectral gap A\]).
We are in the same situation as in [@cf:GPPP]. For a suitable perturbation and if ${\delta}$ is small enough, the circle $M_0$ is smoothly transformed into another SNHM $M_{\delta}$, which is close to $M_0$. The proof is the same as in [@cf:GPPP Sec. 5], which, in turn builds on results in [@cf:SellYou]): the spaces we are working in are more general since we have to deal with the disorder. Here suitable perturbation means being an element of $C^1(X^0_\mu,H^{-1}_{\mu})$, but it is clearly the case for the perturbation $u\mapsto
-{\delta}\, {\omega}\, \partial_{\theta}u$ when $\mu$ is of compact support. The following theorem works for all $C^1(X^0_\mu,H^{-1}_{\mu})$ perturbations:
\[th:M\] [@cf:GPPP Sec. 5] For every $K>1$ there exists ${\delta}_0>0$ such that if ${\delta}\in [0, {\delta}_0]$ there exists a stable normally hyperbolic manifold $M_{\delta}$ in $X^1_\mu$ for the perturbed equation . Moreover we can write $$\label{eq:M}
M_{\delta}\, =\, \left\{q_\psi + \phi_{\delta}\left(q_\psi\right):\, \psi \in {{\ensuremath{\mathbb S}} }\right\}\, ,$$ for a suitable function $\phi_{\delta}\in C^1(M_0,X^0_\mu)$ with the properties that
- $\phi_{\delta}(q) \in {{\ensuremath{\mathcal R}} }(A)$;
- there exists $C>0$ such that $\sup_{\psi }(\Vert \phi _{\delta}\left(q_\psi\right)\Vert_{2,\mu}+
\Vert \partial_\psi\phi_{\delta}(q_\psi)\Vert _{2,\mu}) \le C {\delta}$.
\[rem:un\] A byproduct of the proof in [@cf:GPPP Sec. 5] is also that $M_{\delta}$ is the unique invariant manifold in a $L^2({\lambda},\mu)$-neighborhood of $M_0$. So in the case of , thanks to the symmetry of the problem that tells us that any rotation of $M_{\delta}$ is still a invariant manifold, $M_{\delta}$ is in fact a circle, and that the dynamics on this circle is a traveling wave of constant (possibly zero) speed $c_\mu({\delta})$. So the invariant manifold we get for is even $C^\infty$. In this sense, when dealing with , we are using only part of the strength of Theorem \[th:M\]. Of course this symmetry argument does not apply when dealing with .
\[rem:stability\] Theorem \[th:M\] addresses the existence and the linear stability of the manifold $M_{\delta}$. The non-linear stability statement in Theorem \[th:expansion speed\](1) follows from Theorem \[th:M\] combined with [@cf:Henry Theorem 8.1.1], when the dynamics is periodic with non zero speed on $M_{\delta}$. If $M_{\delta}$ is a manifold of stationary points, the argument for the non-linear stability follows by repeating the argument in [@cf:GPP Th. 4.8], where the non-disordered case is treated.
We now prove Proposition \[th:spectral gap A\] and thus that $M_0$ is a SNHM.
The spectral gap estimate (proof of Proposition \[th:spectral gap A\])
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We start by remarking that $A$ is symmetric for the scalar product ${\ensuremath{\left\langle\cdot\, ,\, \cdot\right\rangle_{ -1, {q_0},\mu}}}$ (recall ). In fact, for $u$ and $v$ in ${{\ensuremath{\mathcal D}} }(A)$, a short computation gives (in the following we use the notation $u^\prime({\theta},{\omega})=\partial_{\theta}u({\theta},{\omega})$) $$\begin{aligned}
{\ensuremath{\left\langlev\, ,\, Au\right\rangle_{ -1, {q_0},\mu}}} & \, =\, {\int_{{{\ensuremath{\mathbb R}} }\times{{{\ensuremath{\mathbb S}} }}}}\left[\frac{{{\ensuremath{\mathcal V}} }(\theta,{\omega})}{q_0(\theta)}\left(\frac{ u^\prime(\theta,{\omega})}{2}-u(\theta,{\omega})J*q_0(\theta)-q_0(\theta)\langle J*u
\rangle_\mu({\theta}) \right) \right]{\,\text{\rm d}}\theta{\,\text{\rm d}}\mu\\ \notag
& \, =\, -\frac 12 {\int_{{{\ensuremath{\mathbb R}} }\times{{{\ensuremath{\mathbb S}} }}}}\frac{u(\theta,{\omega})v(\theta,{\omega})}{q_0(\theta)}{\,\text{\rm d}}\theta{\,\text{\rm d}}\mu+\int_{{{\ensuremath{\mathbb R}} }}\int_{({{\ensuremath{\mathbb S}} })^2}
v(\theta,{\omega}){\widetilde}J*u(\theta,{\omega}'){\,\text{\rm d}}\theta {\,\text{\rm d}}\mu \otimes\mu\, ,\end{aligned}$$ where ${\widetilde}J({\theta})=K\cos({\theta})$. We now first prove an inequality for $A$ that is stronger than the spectral gap inequality and then deduce that $A$ is (essentially) self-adjoint. We define the two following scalar products, which were used for the non-disordered case in [@cf:BGP]: $$\label{eq:defNorm-1 non-des}
{\ensuremath{\left\langleu\, ,\, v\right\rangle_{ -1, {q_0}}}}\, :=\, {\int_{{{\ensuremath{\mathbb S}} }}}\frac{{{\ensuremath{\mathcal U}} }({\theta}){{\ensuremath{\mathcal V}} }({\theta})}{q_{0}({\theta})}{\,\text{\rm d}}{\theta}\, ,$$ where ${{\ensuremath{\mathcal U}} }(\cdot)$ is the primitive of $u(\cdot)$ such that ${\int_{{{\ensuremath{\mathbb S}} }}}\frac{{{\ensuremath{\mathcal U}} }({\theta})}{q_{0}({\theta})}{\,\text{\rm d}}{\theta}=0$ and $$\label{eq:defNorm2q non-des}
{\ensuremath{\left\langleu\, ,\, v\right\rangle}}_{2,q_0}\, :=\, {\int_{{{\ensuremath{\mathbb S}} }}}\frac{u({\theta})v({\theta})}{q_0({\theta})}{\,\text{\rm d}}{\theta}\, .$$ We denote the closures of ${{\ensuremath{\mathcal D}} }(L_0)$ for these scalar products respectively by $H^{-1}_{{q_0}}$ and $L^2_{{q_0}}$. In the disordered case, $L^2_{{q_0}}$ corresponds to the space $L^2_{{q_0},\mu}$, which we define by the closure of ${{\ensuremath{\mathcal D}} }(A)$ with respect to the norm $\Vert \cdot\Vert_{2,{q_0},\mu}$ associated with the scalar product $$\label{def scalar product L2 with wage and mu}
{\ensuremath{\left\langleu\, ,\, v\right\rangle_{ 2, {q_0},\mu}}}\, :=\, {\int_{{{\ensuremath{\mathbb R}} }}}{\int_{{{\ensuremath{\mathbb S}} }}}\frac{u({\theta},{\omega})v({\theta},{\omega})}{q_0({\theta})} {\,\text{\rm d}}{\theta}{\,\text{\rm d}}\mu \, .$$ The two Dirichlet forms for the disordered and non-disordered case are respectively $$\label{dirichlet disorder}
{{\ensuremath{\mathcal E}} }_\mu(u) \, =\, -{\ensuremath{\left\langleAu\, ,\, u\right\rangle_{ -1, {q_0},\mu}}}\, ,$$ and $$\label{dirichlet non-disorder}
{{\ensuremath{\mathcal E}} }(u)\, =\, -{\ensuremath{\left\langle L_{q_0}u\, ,\, u\right\rangle_{ -1, {q_0}}}} \, .$$ As in [@cf:BGP], we first prove a spectral gap type inequality that involves the scalar product ${\ensuremath{\left\langle\cdot\, ,\, \cdot\right\rangle}}_{2,q_0}$. For this we introduce the projections on the line spanned by $q^\prime_0$ in the spaces $L^2_{{q_0},\mu}$ and $L^2_{{q_0}}$ $$P_{2,{q_0},\mu} u=\frac{{\ensuremath{\left\langleu\, ,\, q^\prime_0\right\rangle_{ 2, {q_0},\mu}}}}{{\ensuremath{\left\langleq^\prime_0\, ,\, q^\prime_0\right\rangle_{ 2, {q_0}}}}}q^\prime_0 \qquad \text{for all}
\,u=u({\theta},{\omega})\in L^2_{{q_0},\mu} \, ,$$ and $$P_{2,{q_0}}u\, =\, \frac{{\ensuremath{\left\langleu\, ,\, q^\prime_0\right\rangle_{ 2, {q_0}}}}}{{\ensuremath{\left\langleq^\prime_0\, ,\, q^\prime_0\right\rangle_{ 2, {q_0}}}}}q^\prime_0 \qquad \text{for all}\ u \in
L^2_{{q_0}} \, .$$ Remark that since $q^\prime_0$ does not depend on ${\omega}$, $${\ensuremath{\left\langleq^\prime_0\, ,\, q^\prime_0\right\rangle_{ 2, {q_0},\mu}}}\, =\, {\ensuremath{\left\langleq^\prime_0\, ,\, q^\prime_0\right\rangle_{ 2, {q_0}}}} \quad
\text{ and } \quad {\ensuremath{\left\langleq^\prime_0\, ,\, q^\prime_0\right\rangle_{ -1, {q_0},\mu}}}\, =\, {\ensuremath{\left\langleq^\prime_0\, ,\, q^\prime_0\right\rangle_{ -1, {q_0}}}}\, ,$$ and that for all $u\in L^2_{{q_0},\mu}$ $$\label{link projections}
P_{2,{q_0},\mu}u\, =\, \langle P_{2,{q_0}}u \rangle_\mu\, =\, P_{2,{q_0}}\langle u\rangle_\mu\, .$$
\[prop:min dirichlet des\] For all $u\in L^2_{{q_0},\mu}$ such that for almost every ${\omega}$, ${\int_{{{\ensuremath{\mathbb S}} }}}u(\cdot, {\omega}) =0$ $${{\ensuremath{\mathcal E}} }_{\mu}(u)\, {\;\geqslant\;}\, c_K {\ensuremath{\left\langleu-P_{2,{q_0},\mu}u\, ,\, u-P_{2,{q_0},\mu}u\right\rangle_{ 2, {q_0},\mu}}}\, ,$$ with $$c_K\, =\, 1-K(1-r_0^2)\in(0,1/2)\, .$$
The proof of this proposition relies on the corresponding result for the non-disordered case.:
(see [@cf:BGP Prop. 2.3]) \[prop:min dirichlet non-des\] For all $u\in L^2_{{q_0}}$ such that for almost every ${\omega}$, ${\int_{{{\ensuremath{\mathbb S}} }}}u(\cdot, {\omega}) =0$ $${{\ensuremath{\mathcal E}} }(v)\, {\;\geqslant\;}\, c_K{\ensuremath{\left\langle u-P_{2,{q_0}}u\, ,\, u-P_{2,{q_0}}u\right\rangle_{ 2, {q_0}}}}\, .$$
*Proof of Proposition .* The first step of the proof is to make the Dirichlet form of the non-disordered case appear in the the disordered case one, that is $$\begin{aligned}
{{\ensuremath{\mathcal E}} }_\mu(u) & \, =\, \langle {{\ensuremath{\mathcal E}} }(u) \rangle_\mu +\int_{{{\ensuremath{\mathbb R}} }}\int_{({{\ensuremath{\mathbb S}} })^2} u(\theta,{\omega}){\widetilde}J*[u(\theta,{\omega})-u(\theta,{\omega}')]{\,\text{\rm d}}\theta
{\,\text{\rm d}}\mu\otimes \mu\\ \label{link Dirichlet}
& \, =\, \langle {{\ensuremath{\mathcal E}} }(u) \rangle_\mu +\frac 12 \int_{{{\ensuremath{\mathbb R}} }}\int_{({{\ensuremath{\mathbb S}} })^2} [u(\theta,{\omega})-u(\theta,{\omega}')]{\widetilde}J*[u(\theta,{\omega})-u(\theta,{\omega}')]{\,\text{\rm d}}\theta {\,\text{\rm d}}\mu\otimes \mu\, ,\end{aligned}$$ and from Proposition we see that $$\langle {{\ensuremath{\mathcal E}} }(u)\rangle_\mu \, {\;\geqslant\;}\, c_K {\ensuremath{\left\langle u-P_{2,{q_0}}u\, ,\, u-P_{2,{q_0}}u\right\rangle_{ 2, {q_0}}}}\, .$$ Now remark that if we define $$v\, =\, u-P_{2,{q_0},\mu}u\, ,$$ using we get $$v-P_{2,{q_0}}v\, =\, u-P_{2,{q_0}}u\, ,$$ and so $$\langle {{\ensuremath{\mathcal E}} }(u)\rangle_\mu \, {\;\geqslant\;}\, c_K {\ensuremath{\left\langlev-P_{2,{q_0}}v\, ,\, v-P_{2,{q_0}}v\right\rangle_{ 2, {q_0},\mu}}}\, .$$
We now introduce an orthogonal decomposition of the space $L^2_{{q_0}}$ which is well adapted to the convolution with ${\widetilde}J$.
(See [@cf:BGP Lemma 2.1].) \[lem:decomposition L2q\] We have the following decomposition $$L^2_{{q_0}}\, =\, F_0\oplus^\bot F_{1/2}\oplus^\bot F_{K-1/2}$$ where $$F_0\, :=\, \left\{\theta\mapsto a_0+\sum_{j{\;\geqslant\;}2}a_j\cos(j\theta)+b_j\sin(j\theta)\, ;\, \sum_j a_j^2+b^2_j<\infty \right\}$$ and both $F_{1/2}$ and $F_{K-1/2}$ are one dimensional subspaces generated respectively by $\theta\mapsto \sin(\theta)q(\theta)\,
(=-q^\prime_0(\theta)/2Kr_0)$ and by $\theta \mapsto \cos(\theta) q_0 (\theta)$. Moreover, when $u\in F_\lambda$, then $${\widetilde}J *u\, =\, \frac{\lambda}{q_0} u\, .$$
With the help of Lemma \[lem:decomposition L2q\] we can find a lower bound for the last term in : choose ${\alpha}$ such that $P_{2,{q_0}}u=\alpha q^\prime_0$, so that we can write $$\label{min Dirchlet}
{{\ensuremath{\mathcal E}} }_\mu(u) \, {\;\geqslant\;}\,c_K {\ensuremath{\left\langle v-P_{2,{q_0}}v\, ,\, v-P_{2,{q_0}}v\right\rangle_{ 2, {q_0},\mu}}} + \frac{{\ensuremath{\left\langleq^\prime_0\, ,\, q^\prime_0\right\rangle_{ 2, {q_0}}}}}{4} \int_{({{\ensuremath{\mathbb S}} })^2}
(\alpha({\omega})-\alpha({\omega}'))^2 {\,\text{\rm d}}\mu\otimes \mu\, .$$ But if $P_{2,{q_0}}v=\beta q^\prime_0$ (recall that $v=u-P_{2,{q_0},\mu}u$), then since $P_{2,{q_0},\mu}u$ is colinear to $q^\prime_0$, for almost all ${\omega}$, ${\omega}'$ $$\beta({\omega})-\beta({\omega}')\, =\, \alpha({\omega})-\alpha({\omega}')$$ and since $v$ is orthogonal to $q^\prime_0$ (with respect to $\langle \cdot,\cdot\rangle_{2,q_0,\mu}$) we get $$\int_{{{\ensuremath{\mathbb R}} }} \beta({\omega}){\,\text{\rm d}}\mu\, =\, 0 \, .$$ So becomes $${{\ensuremath{\mathcal E}} }_\mu(u) \, {\;\geqslant\;}\,c_K {\ensuremath{\left\langlev-P_{2,{q_0}}v\, ,\, v-P_{2,{q_0}}v\right\rangle_{ 2, {q_0},\mu}}} + \frac{{\ensuremath{\left\langleq^\prime_0\, ,\, q^\prime_0\right\rangle_{ 2, {q_0}}}}}{2} \int_{{{\ensuremath{\mathbb S}} }} \beta^2({\omega}) {\,\text{\rm d}}\mu\,
.$$ It is sufficient to compare this last minoration with the norm ${\ensuremath{\left\langlev\, ,\, v\right\rangle_{ 2, {q_0},\mu}}}$, and from Lemma \[lem:decomposition L2q\] it comes $${\ensuremath{\left\langlev\, ,\, v\right\rangle_{ 2, {q_0},\mu}}}\, =\, {\ensuremath{\left\langle v-P_{2,{q_0}}v\, ,\, v-P_{2,{q_0}}v\right\rangle_{ 2, {q_0},\mu}}} + {\ensuremath{\left\langle q^\prime_0\, ,\, q^\prime_0\right\rangle_{ 2, {q_0}}}} \int_{{{\ensuremath{\mathbb S}} }} \beta^2({\omega}) {\,\text{\rm d}}\mu\, .$$ This completes the proof of Proposition \[prop:min dirichlet des\].
We now need two lemmas comparing the scalar products ${\ensuremath{\left\langle\cdot\, ,\, \cdot\right\rangle_{ 2, {q_0},\mu}}} $ and ${\ensuremath{\left\langle\cdot\, ,\, \cdot\right\rangle_{ -1, {q_0},\mu}}} $. They correspond to Lemmas 2.4 and 2.5 in [@cf:BGP]. Their proofs are very similar to the proofs of the results corresponding results in [@cf:BGP] (to which we refer also for the explicit values of the constants $C$ and $c$ appearing below) and they use in particular the rigged Hilbert space representation of $H^{-1}_{{q_0},\mu}$ (see [@MR697382 p.82]): namely, one can identify $H^{-1}_{{q_0},\mu}$ as the dual space $V'$ of the space $V$ closure of ${{\ensuremath{\mathcal D}} }(A)$ with respect to the norm $\Vert{u}\Vert_V:= \left( \int_{{{\ensuremath{\mathbb R}} }\times{{\ensuremath{\mathbb S}} }} v'({\theta},
{\omega})^2
{\,\text{\rm d}}{\theta}\mu({\,\text{\rm d}}{\omega}) \right)^\frac12$. The pivot space $H$ is the usual $L^2({\lambda}\otimes\mu)$ (endowed with the Hilbert norm $\Vert u\Vert_{2,\mu}:=\left( {\int_{{{\ensuremath{\mathbb R}} }\times{{{\ensuremath{\mathbb S}} }}}}u({\theta},
{\omega})^2{\,\text{\rm d}}{\theta}\mu({\,\text{\rm d}}{\omega}) \right)^\frac12$). In particular, one easily sees that the inclusion $V\subseteq H$ is dense. Consequently, one can define $T:H\rightarrow V'$ by setting $Tu(v)= {\int_{{{\ensuremath{\mathbb R}} }\times{{{\ensuremath{\mathbb S}} }}}}u({\theta}, {\omega}) v({\theta}, {\omega}){\,\text{\rm d}}{\theta}\mu({\,\text{\rm d}}{\omega})$. One can prove that $T$ continuously injects $H$ into $V'$ and that $T(H)$ is dense into $V'$ so that one can identify $u\in H$ with $Tu\in V'$. Then for $u\in H$, $$\Vert u\Vert_{V'} = \Vert Tu\Vert_{V'} = \sup_{v\in V} \frac{\int {{\ensuremath{\mathcal U}} }v'}{\Vert v\Vert_{V}}= \sqrt{\int \frac{{{\ensuremath{\mathcal U}} }^2}{q_0}}\, ,$$ which enables us to identify $H^{-1}_{{q_0},\mu}$ with $V'$.
We define the projection in $H^{-1}_{{q_0},\mu}$: $$P_{-1,{q_0},\mu}u\, =\, \frac{{\ensuremath{\left\langleu\, ,\, q^\prime_0\right\rangle_{ -1, {q_0},\mu}}}}{{\ensuremath{\left\langleq^\prime_0\, ,\, q^\prime_0\right\rangle_{ -1, {q_0}}}}}q^\prime_0\, .$$
\[lem:comparaison norm projections\] For every $K>1$ there exists a constant $C=C(K)>0$ such that for $u\in L^2_\mu$ such that ${\int_{{{\ensuremath{\mathbb S}} }}}u =0$ for almost every ${\omega}$ $$\begin{split}
{\ensuremath{\left\langleu-P_{2,{q_0},\mu}u\, ,\, u-P_{2,{q_0},\mu}u\right\rangle_{ 2, {q_0},\mu}}}\, &{\;\geqslant\;}\, e^{4Kr_0}C
{\ensuremath{\left\langleu-P_{-1,{q_0},\mu}u\, ,\, u-P_{-1,{q_0},\mu}u\right\rangle_{ 2, {q_0},\mu}}}\\
&{\;\geqslant\;}\, C {\ensuremath{\left\langleu-P_{-1,{q_0},\mu}u\, ,\, u-p_{-1,{q_0},\mu}u\right\rangle_{ -1, {q_0},\mu}}}\, .
\end{split}$$
\[lem:comparaison norm\] For every $K>1$ there exists $c=c(K)>0$ such that for $u\in L^2_\mu$ such that ${\int_{{{\ensuremath{\mathbb S}} }}}u =0$ for almost every ${\omega}$ and $${\ensuremath{\left\langleu\, ,\, u\right\rangle_{ -1, {q_0},\mu}}}\, {\;\geqslant\;}\, c {\ensuremath{\left\langleP_{2,{q_0},\mu}u\, ,\, P_{2,{q_0},\mu}u\right\rangle_{ 2, {q_0},\mu}}}\, .$$
[*Proof of Proposition \[th:spectral gap A\].*]{} Of course Proposition \[prop:min dirichlet des\] and Lemma \[lem:comparaison norm projections\] imply directly the spectral gap inequality for the Dirichlet form: $$\label{low bound Dirichlet}
{{\ensuremath{\mathcal E}} }(u) \, {\;\geqslant\;}\, c_KC {\ensuremath{\left\langleu-P_{-1,{q_0},\mu}u\, ,\, u-P_{-1,{q_0},\mu}u\right\rangle_{ -1, {q_0},\mu}}}\qquad \text{for all}\, u\in H^{-1}_{{q_0},\mu}\, .$$ We now prove the self-adjoint property of $A$. It is sufficient to prove that the range of $1-A$ is dense in $H^{-1}_{\mu}$ (see [@MR697382 p.113]). For $u, v \in D(A)$, we have $$\begin{gathered}
\label{expr scal prod 1-A}
{\ensuremath{\left\langlev\, ,\, (1-A)u\right\rangle_{ -1, {q_0},\mu}}}\, =\, -\int_{{\ensuremath{\mathbb R}} }{\int_{{{\ensuremath{\mathbb S}} }}}v({\theta},{\omega})\left(\int_0^{\theta}\frac{{{\ensuremath{\mathcal U}} }}{q_0} \right){\,\text{\rm d}}{\theta}{\,\text{\rm d}}\mu+\frac12 \int_{{\ensuremath{\mathbb R}} }{\int_{{{\ensuremath{\mathbb S}} }}}\frac{vu}{q_0}{\,\text{\rm d}}{\theta}{\,\text{\rm d}}\mu \\
-\int_{{{\ensuremath{\mathbb R}} }}\int_{({{\ensuremath{\mathbb S}} })^2} v(\theta,{\omega}){\widetilde}J*u(\theta,{\omega}'){\,\text{\rm d}}\theta {\,\text{\rm d}}\mu \otimes\mu \, .\end{gathered}$$ The right side of this expression is still defined for $u,v\in L^2({\lambda}\otimes\mu)$ (recall that ${\lambda}$ denotes the Lebesgue measure on ${{\ensuremath{\mathbb S}} }$, and that we denote the usual scalar product on $L^2({\lambda}\otimes\mu)$ by $\Vert\cdot\Vert_{2,\mu}$) and there exists $c>0$ such that $$\label{ineq:1-A norm vs L2}
{\ensuremath{\left\langlev\, ,\, (1-A)u\right\rangle_{ -1, {q_0},\mu}}}\, {\;\leqslant\;}\, c \Vert u\Vert_{2,\mu} \Vert v\Vert_{2,\mu}\, ,$$ Furthermore from and Lemma \[lem:comparaison norm\] we have $$\label{ineq:1-A norm vs L2 bis}
{\ensuremath{\left\langleu\, ,\, (1-A)u\right\rangle_{ -1, {q_0},\mu}}}\, {\;\geqslant\;}\, \frac1c \Vert u\Vert_{2,\mu}^2\, .$$ So the bilinear form $(u,v)\mapsto {\ensuremath{\left\langlev\, ,\, (1-A)u\right\rangle_{ -1, {q_0},\mu}}}$ is continuous and coercive on $H^{-1}_{\mu}\times H^{-1}_{\mu}$. If $f\in H^{-1}_{\mu}$, the linear form $v\mapsto {\ensuremath{\left\langlev\, ,\, f\right\rangle_{ -1, {q_0},\mu}}} $ is continuous on $L^2({\lambda}\otimes\mu)$, therefore from Lax-Milgram Theorem we get that there exists a unique $u\in L^2({\lambda}\otimes\mu)$ such that for all $v\in L^2({\lambda}\otimes\mu)$ $${\ensuremath{\left\langlev\, ,\, (1-A)u\right\rangle_{ -1, {q_0},\mu}}} \, =\, {\ensuremath{\left\langlev\, ,\, f\right\rangle_{ -1, {q_0},\mu}}} \, .$$ Since $${\ensuremath{\left\langlev\, ,\, f\right\rangle_{ -1, {q_0},\mu}}} \, =\, -\int_{{\ensuremath{\mathbb R}} }{\int_{{{\ensuremath{\mathbb S}} }}}v({\theta},{\omega})\left(\int_0^{\theta}\frac{{{\ensuremath{\mathcal F}} }}{q_0} \right){\,\text{\rm d}}{\theta}{\,\text{\rm d}}\mu\, ,$$ from we obtain that for almost ${\theta}$ and ${\omega}$ $$-\int_0^{\theta}\frac{{{\ensuremath{\mathcal U}} }({\theta}',{\omega})}{q_0({\theta}')}{\,\text{\rm d}}{\theta}' +\frac{u({\theta},{\omega})}{2q_0({\theta})} -\int_{{\ensuremath{\mathbb R}} }\left({\widetilde}J*u\right)({\theta},{\omega}){\,\text{\rm d}}\mu\, =\, -\int_0^{\theta}\frac{{{\ensuremath{\mathcal F}} }({\theta}',{\omega})}{q_0({\theta}')}{\,\text{\rm d}}{\theta}'\, .$$ So it is clear that if $f$ is continuous with respect to ${\theta}$, then $u$ has a version $C^2$ with respect to ${\theta}$. Thus $u\in
D(A)$ and applying $\partial_{\theta}(q_0({\theta})\partial_{\theta}\cdot)$ to the both sides of this last expression, we get $(1-A)u=f$. Since this kind of functions $f$ is dense in $H^{-1}_{\mu}$, we can conclude that the range of $1-A$ is dense, and that $A$ is essentially self-adjoint. This completes the proof of Proposition \[th:spectral gap A\].
Perturbation arguments (completion of the proof of Theorem \[th:expansion speed\]) {#sec:pa}
==================================================================================
In this section we complete the proof of Theorem \[th:expansion speed\]. Essentially, this section is devoted to computing the expansion of the speed $c_\mu({\delta})$ in. We first recall a lemma that gives a useful parametrization in the neighborhood of $M_0$. The proof of this lemma is given in [@cf:SellYou] , and it is used in the proof of Theorem \[th:M\] (see [@cf:GPPP; @cf:SellYou]).
\[lem:parametrisation\] There exists a ${\sigma}>0$ such that for all $p$ in the neighborhood $$\label{eq:Nsigma}
N_{\sigma}\, :=\,
\cup_{q\in M_0} B_{L^2({\lambda}\otimes\mu)}(q,\sigma)\, ,$$ of $M_0$ there is one and only one $q=v(p) \in M_0$ such that ${\ensuremath{\left\langlep-q\, ,\, \partial_{\theta}q\right\rangle_{ -1, {q_0},\mu}}}=0$. Furthermore the mapping $p \mapsto v(p)$ is in $C^\infty(X^1_\mu,X^1_\mu)$, and $$Dv(p)\, =\, P^o_{v(p)}\, .$$
[*Proof of Theorem \[th:expansion speed\].*]{} The existence and stability of a [*rotating*]{} solution ${\widetilde}q_{\delta}({\theta}-\psi-c_\mu({\delta})t)$ of ($\psi$ is arbitrary) has been established in Section \[sec:hyperbolic structure\] for ${\delta}{\;\leqslant\;}{\delta}_0$, see Theorem \[th:M\] and the two remarks that follow it. We are left with proving Theorem \[th:expansion speed\](2).
Thanks to the invariance by rotation, we can define ${\widetilde}q_{\delta}$ such that $v({\widetilde}q_{\delta})=q_0 $. Now if we denote $$n_{\delta}\, :=\, {\widetilde}q_{\delta}-v\left({\widetilde}q_{\delta}\right)\, ,$$ then $n_{\delta}$ verifies $n_{\delta}=\phi_{\delta}(q_0)$ and (see Lemma \[lem:parametrisation\]) $$\label{eq:orth n q}
{\ensuremath{\left\langlen_{\delta}\, ,\, q^\prime_0\right\rangle_{ -1, {q_0},\mu}}}=0$$ $$\label{eq:orth An q}
{\ensuremath{\left\langleAn_{\delta}\, ,\, q^\prime_0\right\rangle_{ -1, {q_0},\mu}}}=0\, .$$ Moreover the estimates we have on the mapping $\phi_{\delta}$ in Theorem \[th:M\] give $$\label{ineq:bound n}
\Vert n_{\delta}\Vert_{2,\mu}\, {\;\leqslant\;}\, C{\delta}\, ,$$ $$\label{ineq:bound dn}
\Vert \partial_{\theta}n_{\delta}\Vert_{2,\mu}\, {\;\leqslant\;}C {\delta}\, .$$ Taking the derivative with respect to $t$, at time $t=0$, we get (we recall the notation $p^{(\psi)}_t({\theta},{\omega})={\widetilde}q_{\delta}({\theta}-\psi-c_\mu({\delta})t)$) : $$\label{eq:equality derivate}
-c_\mu({\delta})\left(q^\prime_0+\partial_{\theta}n_{\delta}\right)\, =\, \partial_t p^{(0)}_0\, .$$ So at time $t=0$ becomes (recall that $q_0$ is a stationary solution of ) : $$\label{eq:FKP delta projected}
-c_\mu({\delta})\left( q^\prime_0+\partial_{\theta}n_{\delta}\right)\, =\, A n_{\delta}-\partial_{\theta}\left[n_{\delta}\langle J*n_{\delta}\rangle_\mu\right]-{\delta}{\omega}q^\prime_0-{\delta}{\omega}\partial_{\theta}n_{\delta}\, .$$ From we deduce the bound $$\label{ineq:bound nJn}
\left\Vert \partial_{\theta}\left[n_{\delta}\langle J*n_{\delta}\rangle_\mu\right]\right\Vert_{-1,\mu}\, {\;\leqslant\;}\, \Vert J\Vert_2 C^2
{\delta}^2\, ,$$ so by taking the $H^{-1}_{{q_0},\mu}$ scalar product of $q^\prime$ in , using , , and the fact that $\int_{{\ensuremath{\mathbb R}} }w{\,\text{\rm d}}\mu=0$, we get that $c_\mu({\delta})$ is of order ${\delta}^2$. This implies, using the same arguments, that $$\left\Vert A n_{\delta}-{\delta}{\omega}q^\prime_0 \right\Vert_{-1,\mu}\, =\, O({\delta}^2)\, .$$ So $$\label{comparaison n0 n1}
\Vert A(n_{\delta}-{\delta}n^{(1)}) \Vert_{-1,\mu}\, =\, O({\delta}^2)\, ,$$ and since $\Vert (1-A)^{(1/2)}u\Vert_{-1,\mu} \sim \Vert u \Vert_{2,\mu}$ (see and ), we have in particular $$\Vert n_{\delta}-{\delta}n^{(1)} \Vert_{2,\mu}\, =\, O({\delta}^2)\, .$$ It allows us to make a second order expansion for $c_\mu({\delta})$ : taking again the $H^{-1}_{{q_0},\mu}$ scalar product of $q^\prime_0$ in , using the same bounds as for the first order expansion and , we get : $$\label{eq:second order expansion psi prime}
c_\mu({\delta})\, =\, {\delta}^2 \frac{{\ensuremath{\left\langle{\omega}\partial_{\theta}n^{(1)}+n^{(1)}\langle J*n^{(1)}\rangle_\mu\, ,\, q^\prime_0\right\rangle_{ -1, {q_0},\mu}}}}{{\ensuremath{\left\langleq^\prime_0\, ,\, q^\prime_0\right\rangle_{ -1, {q_0},\mu}}}} +O({\delta}^3)\, .$$ Indeed, from , $\Vert {\omega}\partial_{\theta}(n_{\delta}-{\delta}n^{(1)}) \Vert_{-1,\mu}$, $\Vert
\partial_{\theta}[(n_{\delta}-{\delta}n^{(1)})\langle J*n^{(1)}\rangle_\mu]\Vert_{-1,\mu}$, $ \Vert \partial_{\theta}[n^{(1)}\langle J*(n_{\delta}-{\delta}n^{(1)})\rangle_\mu]\Vert_{-1,\mu}$ are of order ${\delta}^2$ and $ \Vert \partial_{\theta}[(n_{\delta}-{\delta}n^{(1)})\langle J*(n_{\delta}-{\delta}n^{(1)})\rangle_\mu]\Vert_{-1,\mu}$ of order ${\delta}^4$. Since $c_\mu({\delta})$ is odd with respect to ${\delta}$, the second order term in is equal to $0$. It is possible to get this fact directly : we remark that $n^{(1)}$ satisfies : $$\label{eq:kdge3}
L_{q_0}\int_{{\ensuremath{\mathbb R}} }n^{(1)}{\,\text{\rm d}}\mu \, =\, \int_{{\ensuremath{\mathbb R}} }A n^{(1)} {\,\text{\rm d}}\mu\, =\, \left(\int_{{\ensuremath{\mathbb R}} }{\omega}{\,\text{\rm d}}\mu\right)q^\prime_0\, =\, 0\, ,$$ $${\ensuremath{\left\langle\int_{{\ensuremath{\mathbb R}} }n^{(1)}{\,\text{\rm d}}\mu\, ,\, q^\prime_0\right\rangle_{ -1, {q_0}}}}\, =\, {\ensuremath{\left\langlen^{(1)}\, ,\, q^\prime_0\right\rangle_{ -1, {q_0},\mu}}}\, =\, 0\, .$$ So since $L_{q_0}$ is bijective on the orthogonal of $q^\prime_0$ in $H^{-1}_{1/q}$ (see [@cf:BGP]), we have $\int_{{\ensuremath{\mathbb R}} }n^{(1)} {\,\text{\rm d}}\mu=0$ and $\langle J*n^{(1)}\rangle\mu=0$. On the other hand, since the operator $A$ conserves the parity with respect to ${\theta}$, $n^{(1)}$ is odd with respect to ${\theta}$ and thus $${\ensuremath{\left\langle{\omega}\partial_{\theta}n^{(1)}\, ,\, q^\prime_0\right\rangle_{ -1, {q_0},\mu}}}\, =\, \int_{{\ensuremath{\mathbb S}} }\int_{{\ensuremath{\mathbb R}} }\frac{{\omega}n^{(1)}}{q_0}\left(q_0-\frac{1}{2\pi I_0^2(2Kr_0)}\right){\,\text{\rm d}}{\theta}{\,\text{\rm d}}\mu\, =\, 0\, .$$ Now back to : since $c_\mu({\delta})$ is of order ${\delta}^3$ and using $\int_{{\ensuremath{\mathbb S}} }n^{(1)}{\,\text{\rm d}}\mu=0$, we get $$\left\Vert A \left(n_{\delta}-{\delta}n^{(1)}-{\delta}^2{\omega}\partial_{\theta}n^{(1)} \right)\right\Vert_{-1,\mu}\, =\, O({\delta}^3)\, ,$$ and thus $$\label{eq:n_2}
\Vert n_{\delta}-{\delta}n^{(1)}-{\delta}^2 n^{(2)}\Vert_{2,\mu} \, =\, O({\delta}^3)\, .$$ This allows us this time to do a third order expansion in : $$\label{eq:rt5p}
c_\mu({\delta})\, =\, {\delta}^3\frac{{\ensuremath{\left\langle{\omega}\partial_{\theta}n^{(2)}\, ,\, q^\prime_0\right\rangle_{ -1, {q_0},\mu}}}}{{\ensuremath{\left\langleq^\prime_0\, ,\, q^\prime_0\right\rangle_{ -1, {q_0},\mu}}}} + O({\delta}^4)\, .$$ This procedure may be repeated recursively at any order: we do not go through the details again, but we do report the result below (Remark \[rem:pa\]) and we point out that the $O({\delta}^4)$ turns out to be $O({\delta}^5)$, in agreement with the fact that $c_\mu({\delta})$ is odd in ${\delta}$.
\[rem:pa\] As anticipated above, one can get arbitrarily many terms in the formal series $c_\mu({\delta})= \sum_{i=1,2, \ldots} c_{2i+1} {\delta}^{2i+1}$ and the remainder, when the series is stopped at $i=n$, is $O({\delta}^{2i+3})$. In fact, by arguing like above, we have $$c_5\, =\, \frac{\left\langle \partial_{\theta}[n^{(2)}\langle J*n^{(3)}\rangle_\mu]+ \partial_{\theta}[n^{(3)}\langle J*n^{(2)}\rangle_\mu]+ w\partial_{\theta}n^{(4)} ,q'_0\right\rangle_{-1,q_0,\mu}}{\langle q'_0,q'_0 \rangle_{-1,q_0}}\, ,$$ where $$An^{(3)}\, =\, \partial_{\theta}[n^{(1)}\langle J*n^{(2)}\rangle_\mu]+ w\partial_{\theta}n^{(2)}-\frac{\left\langle w\partial_{\theta}n^{(2)} ,q'_0\right\rangle_{-1,q_0,\mu}}{\langle q'_0,q'_0 \rangle_{-1,q_0}}q'_0 \, ,$$ and $$An^{(4)}\, =\, \partial_{\theta}[n^{(2)}\langle J*n^{(2)}\rangle_\mu]+ \partial_{\theta}[n^{(1)}\langle J*n^{(3)}\rangle_\mu]+ w\partial_{\theta}n^{(3)}\, .$$ Actually, by induction we obtain $$c_{2i+1}\, =\, \frac{\left\langle \sum_{k+l=2i+1,k>0,l>0}\partial_{\theta}[n^{(l)}\langle J*n^{(k)}\rangle_\mu]+ w\partial_{\theta}n^{(2i)} ,q'_0\right\rangle_{-1,q_0,\mu}}{\langle q'_0,q'_0 \rangle_{-1,q_0}}\, ,$$ and $$n^{(2i)}\, =\, \sum_{k+l=2i,k>0,l>0}\partial_{\theta}[n^{(l)}\langle J*n^{(k)}\rangle_\mu] + w\partial_{\theta}n^{(2i-1)}\, ,$$ $$n^{(2i+1)}\, =\, \sum_{k+l=2i+1,k>0,l>0}\partial_{\theta}[n^{(l)}\langle J*n^{(k)}\rangle_\mu] + w\partial_{\theta}n^{(2i)}- c_{2i+1}q'_0\, .$$ Since this procedure yields also $n^{(j)}$ for arbitrary $j$, one can generalizes also and, hence, .
Active rotators {#sec:AR}
===============
In this section we deal with the equation and we do it in a rather informal way, because on one hand a formal statement would be very close to Theorem \[th:expansion speed\] and, on the the other hand, the large scale behavior of disordered active rotators is qualitatively and quantitatively close to the non disordered case, treated in [@cf:GPPP], in a way that we explain below.
First of all, from a technical viewpoint the main difference between and is that is (in general) not rotation invariant, so the manifold $M_{\delta}=\{q_\psi+\phi(q_\psi)\}$ we get after perturbation is not necessarily a circle. Unlike Theorem \[th:expansion speed\], the motion on $M_{\delta}$ is not uniform, and we describe the behaviour on $M_{\delta}$ by the phase derivate $\dot{\psi}$. We follow the same procedure as in the previous section : if $p^{\delta}_t$ is a solution belonging to $M_{\delta}$, we define (see Lemma \[lem:parametrisation\]) $$q_{\psi^{\delta}_t}\, =\, v(p^{\delta}_t)\ , \ \ \text{ and } \ \ \
n^{\delta}_t\,=\, p^{\delta}_t- v(p^{\delta}_t)\, .$$ In this context, becomes $$\label{eq:FKP delta projected AR}
-\dot{\psi}^{\delta}_t q'_{\psi^{\delta}_t} + \partial_t n^{\delta}_t\, =\, A^{\psi^{\delta}_t} n^{\delta}_t -\partial_\theta [n^{\delta}_t \langle J*n^{\delta}_t \rangle_\mu ]-{\delta}U q'_\psi -{\delta}U\partial_\theta n^{\delta}_t \, ,$$ where $A^\psi$ is the rotation of the operator $A$ $$A^\psi u({\theta},{\omega}) \, :=\, \frac12 \Delta u({\theta},{\omega})-\partial_{\theta}\Big(q_0({\theta}-\psi)\langle J*u\rangle_\mu({\theta})+u({\theta},{\omega})J*q_0({\theta}-\psi)\Big)\, .$$ Note that we can reformulate the second term of the left hand side in : $$\partial_t n^{\delta}_t\, =\, \dot{\psi}^{\delta}_t\partial_\psi \phi(q_\psi) |_{\psi=\psi^{\delta}_t}\, .$$ So, as in the previous section, using the estimates on the mapping $\phi$ given in Theorem \[th:M\], we get the bounds $$\Vert n^{\delta}_t \Vert_{2,\mu}\, {\;\leqslant\;}\, C{\delta}\, , \ \
\Vert \partial_t n^{\delta}_t \Vert_{2,\mu}\, {\;\leqslant\;}\, C{\delta}|\dot{\psi}^{\delta}_t|
\ \text{ and } \
\Vert \partial_\theta [n^{\delta}_t \langle J*n^{\delta}_t \rangle_\mu ] \Vert_{2,\mu} \, {\;\leqslant\;}\, \Vert J\Vert_2 C^2{\delta}\, ,$$ and we deduce the first order expansion $$\label{eq:AReq}
{\dot{\psi}}^{\delta}_t\,
=\, {\delta}\frac{ \langle (U q_{\psi^{\delta}_t})', q'_{\psi^{\delta}_t} \rangle_{-1,q_{\psi^{\delta}_t},\mu}}{ \langle q'_0,q'_0 \rangle_{-1,q_0}}+O({\delta}^2)\, .$$ Since $\dot{\psi}$ is odd in ${\delta}$ and the expansion can be pushed further in ${\delta}$, this $O({\delta}^2)$ is in reality a $O({\delta}^3)$ and one can actually improve this result both in the direction of obtaining a regularity estimate on the $O({\delta}^2)$ rest in (like in [@cf:GPPP Th. 2.3]) and of going to higher orders (like in Remark \[rem:pa\]).
However the evolution for small ${\delta}$ is dominated by the leading order and from we can directly read that, to first order, the effect of the disorder is rather simple: in fact $$\langle (U q_{\psi})', q'_{\psi} \rangle_{-1,q_{\psi},\mu}\, =\,
\int_{{\ensuremath{\mathbb R}} }\int_{{\ensuremath{\mathbb S}} }U(\theta, {\omega}) q_{\psi} (\theta)\left( q_{\psi}(\theta)- c\right) {\,\text{\rm d}}\theta \mu ({\,\text{\rm d}}{\omega})\, ,$$ where $c$ is such that $\int_{{\ensuremath{\mathbb S}} }(q_\psi -c)=0$, that is $1/c={2\pi (I_0(2Kr_0))^2}$ (recall -: this computation is analogous to ). Since the integrand depends on ${\omega}$ only via $U$, this integration can be performed first and the system behaves to leading order in ${\delta}$ as the non-disordered model with active rotator dynamics led by the deterministic force $\int_{{\ensuremath{\mathbb R}} }U(\cdot, {\omega}) \mu({\,\text{\rm d}}{\omega})$. The rich phenomenology connected to these models is worked out in [@cf:GPPP Sec. 3].
Symmetric case: stability of the stationary solutions {#sec:sym}
=====================================================
On the non-trivial stationary solutions (proof of Lemma \[th: Psimu concave\])
------------------------------------------------------------------------------
We start by observing that in the case with no disorder the strict concavity of the fixed-point function $\Psi_0$ has been proven in [@cf:Pearce Lemma 4, p.315], in the apparently different context of classical XY-spin model (for a detailed discussion on the link with these models see [@cf:BGP]). We are going to obtain the concavity of $\Psi^\mu_{\delta}$ for small ${\delta}$ via a perturbation argument, by relying on the result in [@cf:Pearce].
Since $\Psi^\mu_{\delta}$ is a smooth perturbation of $\Psi_0$, one expects that the strict concavity of $\Psi_0$ will be preserved to $\Psi^\mu_{\delta}$ for small ${\delta}>0$, namely $\sup_{x} (\Psi_{\delta}^\mu)''(x)<0$. Nevertheless, an easy calculation shows that $\Psi''_0(0)=0$; in that sense one has to treat the concavity in a neighborhood of $0$ as a special case.
In what follows, we suppose that the coupling strength $K$ is bounded above and below by fixed constants $K_{\min}$ and $K_{\max}$: $$0\, <\, K_{\min} \, {\;\leqslant\;}\, K\, {\;\leqslant\;}\, K_{\max}\, <\, \infty\, .$$
We first prove the statement on the concavity in a neighborhood of $0$: there exist $\eta_0>0$, ${\delta}>0$ such that for all $K\in[K_{\min}, K_{\max}]$, for all $\mu$ such that $\operatorname*{Supp}(\mu)\subseteq[-1,1]$, $\Psi_{\delta}^\mu$ is strictly concave on $[0,
\eta_0]$.
Indeed, one easily shows (using that the function $x\mapsto \Psi_\mu^{\delta}(x)$ is odd) that we have the following Taylor’s expansion: $$(\Psi_{\delta}^\mu)''(x) = -6 D^{\delta}(\mu)K^3 x + \epsilon(x)\, ,$$where $\epsilon(x)=o(x)$ as $x\rightarrow 0$ and where for fixed $\mu$, we write $$D^{\delta}(\mu) := \int_{{\mathbb{R}}{}}{h({\delta}{\omega})\mu({\,\text{\rm d}}{\omega})},$$where $$h({\omega})
:= \frac{1}{2(1+{\omega}^{2})}-\frac{8{\omega}^{2}}{(1+4{\omega}^{2})^2}.$$ Note that the $o(x)$ only depends on $K_{\max}$ (in particular it can be chosen independently of $\mu$). A closer look at the function $h$ shows that there exists ${\delta}>0$ such that for all $\mu$ with $\operatorname*{Supp}(\mu)\subseteq[-1, 1]$, $D^{\delta}(\mu)>\frac
14$. If we choose $\eta_{0}>0$ such that $\frac{1}{\eta_0}\sup\limits_{0{\;\leqslant\;}x< \eta_0}|\epsilon(x)|< \frac 32 K_{\min}^3$ then $(\Psi_{\delta}^\mu)''(x)<0$ for all $0<x<\eta_0$, which is the desired result.
We are now left with proving concavity away from $0$: namely, we prove that for all $\eta>0$, all $K_{\max}$, there exists ${\delta}_0>0$ such that for all $K{\;\leqslant\;}K_{\max}$, for all $0<{\delta}<{\delta}_0$, for any measure $\mu$ such that $\operatorname*{Supp}(\mu)\subseteq[-1, 1]$, $\Psi^\mu_{\delta}$ is strictly concave on $[\eta, 2K_{\max}]$.
Indeed, using the strict concavity of $\Psi_0$ proved in [@cf:Pearce], there exists a constant $\alpha>0$ such that for all $x\in[\eta, 2K_{\max}]$, $\Psi_{0}''(x)<-\alpha<0$. But then, it easy to see that $$\sup_{0<{\delta}<{\delta}_0}\sup_{\mu,\
\operatorname*{Supp}(\mu)\subseteq [-1,
1]}\sup_{x\in[0, 2K_{\max}]} \left|(\Psi^\mu_{\delta})''(x) - \Psi_{0}''(x)\right|
\stackrel{{\delta}_0\searrow 0}{\rightarrow}0.$$If one chooses ${\delta}_0$ such that the latter quantity is smaller than or equal to $\frac{\alpha}{2}$, the result follows. The proof of Lemma \[th: Psimu concave\] is therefore complete.
On the linear stability of non-trivial stationary solutions
-----------------------------------------------------------
We now prove Theorem \[th:spectral prop L disorder\] along with a number of explicit estimates.
Note that, since the whole operator $L^{\omega}_{{q}}$ is no longer self-adjoint nor symmetric, its spectrum need not be real. In that extent, one has to deal in this section with the complexified versions of the scalar products defined in Section \[sec:mainresults\], and in Section \[sec:hyperbolic structure\], . Thus, we will assume for the rest of this section that we work with complex versions of these scalar products. The results concerning the operator $A$ are obviously still valid, since $A$ is symmetric and real.
We will also use the following standard notations: for an operator $F$, we will denote by $\rho(F)$ the set of all complex numbers $\lambda$ for which $\lambda-F$ is invertible, and by $R(\lambda, F):= \left( \lambda - F \right)^{-1}$, $\lambda\in\rho(F)$ the resolvent of $F$. The spectrum of $F$ will be denoted as $\sigma(F)$.
### Decomposition of $L^{\omega}_{{q}}$
In what follows, $K>1$ and $r_0= \Psi_{0}(2Kr_0)>0$ are fixed.
In order to study the spectral properties of the operator $L^{\omega}_{{q}}$ for general distribution of disorder, we decompose $L^{\omega}_{{q}}$ in into the sum of the self-adjoint operator $A$ defined in and a perturbation $B$ which will be considered to be small w.r.t. $A$, namely: $$\label{eq:defBmu}
Bu({\theta}, {\omega})\, :=\, - \partial_{{\theta}} \left( u({\theta}, {\omega}) \langle J \ast {\varepsilon}(q) \rangle_\mu + {\varepsilon}(q)({\theta}, {\omega}, {\delta})
\langle J \ast
u\rangle_\mu({\theta}) + {\delta}{\omega}u({\theta}, {\omega})\right),$$ where $$\label{eq:defdeltaqmu}
{\varepsilon}(q) \, :=\, ({\theta}, {\omega}, {\delta})\mapsto q({\theta}, {\delta}{\omega}) - q_0({\theta}),$$ is the difference between the stationary solution with disorder and the one without disorder.
\[prop:Apositivemu\] The (extension of the) operator $A$ is the infinitesimal generator of a strongly continuous semi-group of contractions $T_A(t)$ on $H^{-1}_{{q_0},\mu}$.
Moreover, for every $0<\alpha<\frac{\pi}{2}$ this semigroup can be extended to an analytic semigroup $T_A(z)$ defined on $\Delta_\alpha\, :=\, {\left\{z\in{{\ensuremath{\mathbb C}} }\,;\,|\arg(z)|<\alpha\right\}}$.
We recall here the result we use concerning analytic extensions of strongly continuous semigroups. Its proof can be found in [@Pazy1983 Th 5.2, p.61].
\[prop:pazysemgps\] Let $T(t)$ a uniformly bounded strongly continuous semigroup, whose infinitesimal generator $F$ is such that $0\in\rho(F)$ and let $\alpha\in(0, \frac\pi2)$. The following statements are equivalent:
1. \[it:prop:pazysemgps1\]$T(t)$ can be extended to an analytic semigroup in the sector $\Delta_\alpha\, =\, {\left\{\lambda\in{{\ensuremath{\mathbf C}} }\,;\,|\arg(\lambda)|<\alpha\right\}}$ and $\Vert{T(z)}\Vert$ is uniformly bounded in every closed sub-sector $\bar{\Delta}_\alpha'$, $\alpha'<\alpha$, of $\Delta_\alpha$,
2. \[it:prop:pazysemgps2\] There exists $M>0$ such that$$\rho(F) \supset \Sigma\, =\,
{\left\{\lambda
\in{{\ensuremath{\mathbf C}} }\,;\,|\arg(\lambda)|<\frac{\pi}{2}+\alpha\right\}} \cup \{0\},$$and$$\Vert{R(\lambda, F)}\Vert \, {\;\leqslant\;}\, \frac{M}{|\lambda|}, \quad
\lambda\in\Sigma, \lambda\neq0\, .$$
The proof in Section \[sec:hyperbolic structure\], Theorem \[th:spectral gap A\] of the self-adjointness of $A$ shows that $A$ satisfies the hypothesis of Lumer-Phillips Theorem (see [@Pazy1983 Th 4.3, p.14]): $A$ is the infinitesimal generator of a $C_0$ semi-group of contractions denoted by $T_A(t)$.
The rest of the proof is devoted to show the existence of an analytic extension of this semigroup in a proper sector. We follow here the lines of the proof of Th 5.2, p. 61-62, in [@Pazy1983], but with explicit estimates on the resolvent, in order to quantify properly the appropriate size of the perturbation.
Let us first replace the operator $A$ by a small perturbation: for all ${\varepsilon}>0$, let $A_{{\varepsilon}}\, :=\, A-{\varepsilon}$, so that $0$ belongs to $\rho(A_{\varepsilon})$. The operator $A_{{\varepsilon}}$ has the following properties: as $A$, it generates a strongly continuous semigroup of operators (which is $T_{A, {\varepsilon}}(t)=T_A(t)e^{-{\varepsilon}t}$).
Since $A$ is self-adjoint, it is easy to see that $$\label{eq:estimRAeps1}
\forall
\lambda\in{{\ensuremath{\mathbf C}} }\smallsetminus{{\ensuremath{\mathbb R}} }, {\ensuremath{\left\|\,R(\lambda, A_{\varepsilon})\,\right\|_{ -1, {q_0}, \mu}}}\, {\;\leqslant\;}\,
\frac{1}{|\Im(\lambda)|}\, ,$$ and since the spectrum of $A$ is negative, for every $\lambda\in{{\ensuremath{\mathbf C}} }$ such that $\Re(\lambda)>0$, $$\label{eq:estimRAepsdef1}
{\ensuremath{\left\|\,R(\lambda, A_{\varepsilon})\,\right\|_{ -1, {q_0}, \mu}}}\, {\;\leqslant\;}\, \frac{1}{|\lambda|}\, .$$ For any $\alpha \in(0, \frac{\pi}{2})$, let $$\Sigma_\alpha \,:=\, {\left\{\lambda\in{{\ensuremath{\mathbb C}} }\,;\,|\arg(\lambda)|<\frac{\pi}{2}+\alpha\right\}}\, .$$ Let us prove that for $\lambda\in\Sigma_\alpha$, $$\label{eq:estim R lambda alpha}
{\ensuremath{\left\|\,R(\lambda, A_{\varepsilon})\,\right\|_{ -1, {q_0}, \mu}}}\, {\;\leqslant\;}\, \frac{1}{1-\sin(\alpha)}\cdot \frac{1}{|\lambda|}\, .$$ Note that is clear from and when $\lambda$ is such that $\Re(\lambda){\;\geqslant\;}0$.
Let us consider $\sigma>0, \tau\in{{\ensuremath{\mathbb R}} }$ to be chosen appropriately later.
Let us write the following Taylor expansion for $R(\lambda, A_{\varepsilon})$ around $\sigma+i\tau$ (at least well defined in a neighborhood of $\sigma+i\tau$ since $\sigma>0$): $$\label{eq:taylor resolvent}
R(\lambda, A_{\varepsilon}) \, =\, \sum_{n=0}^{\infty}{R(\sigma+i\tau, A_{\varepsilon})^{n+1}((\sigma + i\tau)-\lambda)^n}\, .$$
From now, we fix $\lambda\in\Sigma_\alpha$ with $\Re(\lambda)<0$. This series $R(\lambda, A_{\varepsilon})$ is well defined in $\lambda$ if one can choose $\sigma$, $\tau$ and $k\in(0,1)$ such that ${\ensuremath{\left\|\,R(\sigma
+i\tau, A_{\varepsilon})\,\right\|_{ -1, {q_0}, \mu}}}|\lambda-(\sigma+i\tau)|{\;\leqslant\;}k<1$. In particular, using , it suffices to have $|\lambda-(\sigma+i\tau)|{\;\leqslant\;}k|\tau|$ and since $\sigma>0$ is arbitrary, it suffices to find $k\in(0,1)$ and $\tau$ with $|\lambda-i\tau|{\;\leqslant\;}k|\tau|$ to obtain the convergence of . For this $\lambda\in\Sigma_\alpha$ with $\Re(\lambda)<0$, let us define $\lambda'$ and $\tau$ as in Figure \[fig:angle alpha\]. Then, $|\lambda-i\tau|{\;\leqslant\;}|\lambda'-i\tau|= \sin(\alpha)|\tau|$ with $\sin(\alpha)\in(0,1)$. So the series converges for $\lambda\in\Sigma_\alpha$ and one has, using again , $$\label{eq:estimRAepsdef}
{\ensuremath{\left\|\,R(\lambda, A_{\varepsilon})\,\right\|_{ -1, {q_0}, \mu}}} \, {\;\leqslant\;}\, \frac{1}{(1-\sin(\alpha))|\tau|} \, {\;\leqslant\;}\, \frac{1}{1-\sin(\alpha)}
\cdot\frac{1}{|\lambda|}\, .$$
![The set $\Sigma_\alpha$.[]{data-label="fig:angle alpha"}](resolvent.eps){width="50.00000%"}
The fact that $T_{A, {\varepsilon}}(t)$ can be extended to an analytic semigroup $T_{A, {\varepsilon}}(z)$ on the domain $\Delta_\alpha$ is a simple application of and Proposition \[prop:pazysemgps\], with $M:=\frac{1}{1-\sin(\alpha)}$.
Let us then define ${\widetilde}{T_A}(z):= e^{{\varepsilon}z} T_{A,{\varepsilon}}(z)$, for $z\in\Delta_\alpha$ so that ${\widetilde}{T_A}$ is an analytic extension of $T_A$ (an argument of analyticity shows that ${\widetilde}{T_A}$ does not depend on ${\varepsilon}$).
Note that estimate is also valid in the limit as ${\varepsilon}\to 0$: for all $\alpha\in(0, \frac\pi2)$, $\lambda\in\Sigma_\alpha$, $$\label{eq:estim R lambda alpha without gep}
{\ensuremath{\left\|\,R(\lambda, A)\,\right\|_{ -1, {q_0}, \mu}}}\, {\;\leqslant\;}\, \frac{1}{1-\sin(\alpha)}\cdot \frac{1}{|\lambda|}\, .$$
### Spectral properties of $L^{\omega}_{{q}}=A+B$
In this part, we show that if the perturbation $B$ is small enough with respect to $A$, one has the same spectral properties for $L^{\omega}_{{q}}= A+B$ as for $A$. In this extent, we recall that $\mu$ is of compact support in $[-1, 1]$, and the disorder is rescaled by ${\delta}>0$.
\[prop:BAboundedmu\]
The operator $B$ is $A$-bounded, in the sense that there exist explicit constants $a_{K, {\delta}}$ and $b_{K,{\delta}}$, depending on $K$ and ${\delta}$ such that for all $u$ in the domain of (the closure of) $A$ $$\label{eq:BAboundedmu}
{\ensuremath{\left\|\,Bu\,\right\|_{ -1, {q_0}, \mu}}}\, {\;\leqslant\;}\, a_{K, {\delta}}{\ensuremath{\left\|\,u\,\right\|_{ -1, {q_0}, \mu}}} + b_{K, {\delta}}{\ensuremath{\left\|\,Au\,\right\|_{ -1, {q_0}, \mu}}}\, .$$ Moreover, for fixed $K>1$, $a_{K,{\delta}}=O({\delta})$ and $b_{K,{\delta}}=O({\delta})$, as ${\delta}\rightarrow0$.
The latter proposition is based on the fact that the difference ${\varepsilon}(q)({\theta}, {\omega}, {\delta}) = q({\theta}, {\delta}{\omega})- q_0({\theta})$ in is small if the scale parameter ${\delta}$ tend to $0$:
\[lem:estimdeltaq\] For ${\delta}>0$, let us define
$$\label{eq:def Ninf eps(q)}
{\left\|\,{\varepsilon}(q)\,\right\|_{\infty}}\, :=\, {\sup_{\substack{{\theta}\in{{\ensuremath{\mathbb S}} }, |{\omega}|{\;\leqslant\;}1 \\ 0<u<{\delta}}}} |{\varepsilon}(q)({\theta}, {\omega}, u)|\, .$$
Then for all $K>1$, ${\left\|\,{\varepsilon}(q)\,\right\|_{\infty}} = O({\delta})$, as ${\delta}\rightarrow 0$. More precisely, for $K>1$, ${\delta}>0$, the following inequality holds: $$\label{eq:estimdeltaq}
{\left\|\,{\varepsilon}(q)\,\right\|_{\infty}}\, {\;\leqslant\;}\, {\varepsilon}_{K,{\delta}}\, ,$$ where the constant ${\varepsilon}_{K, {\delta}}$ can be chosen explicitly in terms of $K$ and ${\delta}$: $$\label{eq:estimdeltaK}
{\varepsilon}_{K, {\delta}}\, :=\, \frac{{\delta}}{\pi} e^{8\pi{\delta}}\left( 2+ 3e^{4\pi{\delta}} \right) e^{14K\bar{r}_{\delta}}\left( 1+ 2\pi
e^{2K\bar{r}_{\delta}}\right)\, ,$$ where we recall that $\bar{r}_{\delta}= \max\left( r_0, r_{\delta}\right)$.
Recall that the disordered stationary solution $q$ is given by $$q({\theta}, {\delta}{\omega}) \,:=\, \frac{S({\theta}, {\delta}{\omega}, 2Kr_{\delta})}{Z({\delta}{\omega}, 2Kr_{\delta})},$$ where $S({\theta}, {\omega}, x)$ is defined in and that the non-disordered one is given by $q_0({\theta})=\frac{S({\theta}, 0, 2Kr_0)}{Z(0, 2Kr_0)}= \frac{e^{2Kr_0\cos({\theta})}}{{\int_{{{\ensuremath{\mathbb S}} }}}e^{2Kr_0\cos({\theta})}{\,\text{\rm d}}{\theta}}$. Since $q({\theta}, {\delta}{\omega})=q(-{\theta}, -{\delta}{\omega})$, it suffices to consider the case ${\delta}{\omega}>0$. A simple computation shows that $$\label{eq:lower bound Zomega}
Z({\delta}{\omega}, 2Kr_{\delta})\, {\;\geqslant\;}\, 4\pi^2 e^{-4K r_{\delta}}e^{-4\pi{\delta}}\, ,$$ and that $$\label{eq:upper bound S0}
|S({\theta}, 0)|\, {\;\leqslant\;}\, 2\pi e^{4Kr_0}\, .$$
Using $|q({\theta}, {\delta}{\omega})- q_0({\theta})|{\;\leqslant\;}\frac{1}{Z({\delta}{\omega})Z(0)}\left( Z(0)|S({\theta}, {\delta}{\omega}) - S({\theta}, 0)| +
|S({\theta}, 0)||Z(0)-Z({\delta}{\omega})|\right)$, one has to deal with, successively:
- for fixed ${\theta}\in{{\ensuremath{\mathbb S}} }$, $|S({\theta}, {\delta}{\omega}) - S({\theta}, 0)|{\;\leqslant\;}{\delta}\cdot \sup_{|{\omega}|{\;\leqslant\;}1}|\frac{{\,\text{\rm d}}}{{\,\text{\rm d}}{\omega}} S({\theta},
{\delta}{\omega})|$. A long calculation shows that the latter expression $|\frac{{\,\text{\rm d}}}{{\,\text{\rm d}}{\omega}} S({\theta}, {\delta}{\omega})|$ can be bounded above by $8\pi^2 e^{4Kr_{\delta}}e^{4\pi{\delta}}\left( 2+ 3e^{4\pi{\delta}}\right)$, that is, $$\label{eq:upper bound diff S}
|S({\theta}, {\delta}{\omega}) - S({\theta}, 0)|\, {\;\leqslant\;}\, {\delta}8\pi^2 e^{4Kr_{\delta}}e^{4\pi{\delta}}\left( 2+ 3e^{4\pi{\delta}}\right)\, .$$
- Using $|Z({\delta}{\omega}) - Z(0)|= \left|{\int_{{{\ensuremath{\mathbb S}} }}}(S({\theta}, {\delta}{\omega}) - S({\theta}, 0)) {\,\text{\rm d}}{\theta}\right|$ and , one has directly: $$\label{eq:upper bound diff Z}
|Z({\delta}{\omega}) - Z(0)|\, {\;\leqslant\;}\, {\delta}16\pi^3 e^{4Kr_{\delta}}e^{4\pi{\delta}}\left( 2+ 3e^{4\pi{\delta}}\right)\, .$$
Putting together , , and , one obtains the result.
We are now in position to prove the $A$-boundedness of $B$:
$B$ is $A$-bounded: let us fix a $u$ in the domain of the closure of $A$. Then we have ${\ensuremath{\left\|\,Bu\,\right\|_{ -1, {q_0}, \mu}}}= {\left\|\,{{\ensuremath{\mathcal B}} }u\,\right\|_{2, {q_0}, \mu}}$, where ${{\ensuremath{\mathcal B}} }u$ is the appropriate primitive of $Bu$, namely: $$\begin{aligned}
{{\ensuremath{\mathcal B}} }u({\theta}, {\omega})&\, :=\, {} -\left( u({\theta}, {\omega}) \langle J \ast {\varepsilon}(q) \rangle_\mu + {\varepsilon}(q)({\theta}, {\omega}, {\delta}) \langle J \ast
u\rangle_\mu({\theta}) + {\delta}{\omega}u({\theta}, {\omega})\right)\nonumber\\
&+{} \left( {\int_{{{\ensuremath{\mathbb S}} }}}\frac{1}{q_0} \right)^{-1}\left( {\int_{{{\ensuremath{\mathbb S}} }}}\frac{u({\theta}, {\omega}) \langle J \ast {\varepsilon}(q) \rangle_\mu + {\varepsilon}(q)({\theta},
{\omega}, {\delta}) \langle J \ast
u\rangle_\mu({\theta}) + {\delta}{\omega}u({\theta}, {\omega})}{q_0({\theta})} {\,\text{\rm d}}{\theta}\right)\, .\label{eq:primitiveBu}\end{aligned}$$
One can easily shows that there exists a constant $c^{(1)}_{K, {\delta}}$, depending only on $K>1$ and ${\delta}>0$ such that: $$\label{eq:estim1Bu}
{\ensuremath{\left\|\,Bu\,\right\|_{ -1, {q_0}, \mu}}} \, {\;\leqslant\;}\, c^{(1)}_{K, {\delta}} {\left\|\,u\,\right\|_{2, {q_0}, \mu}}\, .$$ Indeed, an easy calculation shows that $|\langle J\ast {\varepsilon}(q)\rangle_\mu|{\;\leqslant\;}4K{\left\|\,{\varepsilon}(q)\,\right\|_{\infty}}$ and that $$\begin{split}
|\langle J\ast
u\rangle_\mu(\cdot)|&\, {\;\leqslant\;}\, K\left( {\int_{{{\ensuremath{\mathbb S}} }}}\sin(\cdot-\varphi)^2 q_0(\varphi){\,\text{\rm d}}\varphi \right)^{\frac12}{\left\|\,u\,\right\|_{2, {q_0}, \mu}}\\
&\,{\;\leqslant\;}K\left( {\int_{{{\ensuremath{\mathbb S}} }}}q_0(\varphi){\,\text{\rm d}}\varphi \right)^{\frac12}{\left\|\,u\,\right\|_{2, {q_0}, \mu}} = K {\left\|\,u\,\right\|_{2, {q_0}, \mu}}\, .
\end{split}$$ So we have for all ${\theta}, {\omega}$ (recall that $Z_0$ is the normalization constant in ): $$\begin{split}
|{{\ensuremath{\mathcal B}} }u({\theta}, {\omega})|\, {\;\leqslant\;}\, & \left(4 K{\left\|\,{\varepsilon}(q)\,\right\|_{\infty}}+{\delta}|{\omega}|\right)|u|+ 2K {\left\|\,{\varepsilon}(q)\,\right\|_{\infty}}{\left\|\,u\,\right\|_{2, {q_0}, \mu}}\\ &+
Z_{0}^{-1}\left(4 K
{\left\|\,{\varepsilon}(q)\,\right\|_{\infty}} + {\delta}|{\omega}|\right) \left( {\int_{{{\ensuremath{\mathbb S}} }}}\frac{|u|^2}{q_0} \right)^{\frac 12}\, .
\end{split}$$ Hence, inequality is true for the following choice of $c^{(1)}_{K, {\delta}}$ (recall that ${\varepsilon}_{K, {\delta}}$ is defined in ): $$\label{eq:defc1}
c^{(1)}_{K, {\delta}}\, :=\, \left( 6\left(4 K{\varepsilon}_{K, {\delta}}+{\delta}\right)^2 +12 K^2 Z_{0}^2 {\varepsilon}_{K, {\delta}}^2 \right)^{\frac
12}\, .$$
Note that, thanks to Lemma \[lem:estimdeltaq\], one has that $c^{(1)}_{K, {\delta}}=O({\delta})$ as ${\delta}\rightarrow0$.
In order to complete the proof of the inequality , it suffices to prove that there exist constants $c^{(2)}_K$ and $c^{(3)}_K$, only depending on $K$ such that, for all $u$:
$$\label{eq:estim2Bu}
{\left\|\,u\,\right\|_{2, {q_0}, \mu}} \, {\;\leqslant\;}\, c^{(2)}_K {\ensuremath{\left\|\,Au\,\right\|_{ -1, {q_0}, \mu}}} + c^{(3)}_K {\ensuremath{\left\|\,u\,\right\|_{ -1, {q_0}, \mu}}}\, .$$
The rest of this first of the proof is devoted to find explicit expressions of $c^{(2)}_K$ and $c^{(3)}_K$, and is based on an interpolation argument.
For all integer $n>1$, one can compute the linear operator $f\mapsto f'$ in terms of a sum of two integral operators, namely: $$\label{eq:identuprimemu}
f' \, =\, {\ensuremath{I_n}}(f'') + {\ensuremath{J_n}}(f)\, ,$$ where ${\ensuremath{I_n}}:f\mapsto \int_{0}^{2\pi} i_n({\theta}, {\varphi})f({\varphi}){\,\text{\rm d}}{\varphi}$ (resp. ${\ensuremath{J_n}}:f\mapsto \int_{0}^{2\pi} j_n({\theta}, {\varphi})f({\varphi}){\,\text{\rm d}}{\varphi}$) is the integral operator whose kernel $i_n({\theta},
{\varphi})$ (resp. $j_n({\theta}, {\varphi})$) is defined by: $$\left\{\begin{array}{lll}
i_n({\theta}, {\varphi})\, :=\, \frac{{\varphi}^{n+1}}{2\pi{\theta}^{n}}\, ,& j_n({\theta}, {\varphi})\, :=\, -\frac{n(n+1){\varphi}^{n-1}}{2\pi{\theta}^n}\, ,&
0\, {\;\leqslant\;}\, {\varphi}\, <\, {\theta}\, {\;\leqslant\;}\, 2\pi\, ,\\[10pt]
i_n({\theta}, {\varphi})\, :=\, \frac{-(2\pi-{\varphi})^{n+1}}{2\pi(2\pi-{\theta})^n}\ , & j_n({\theta}, {\varphi})\, :=\, \frac{n(n+1)
(2\pi-{\varphi})^{n-1}}{2\pi (2\pi-{\theta})^n}\ , & 0\, {\;\leqslant\;}\, {\theta}\, <\, {\varphi}\, {\;\leqslant\;}\, 2\pi\, .
\end{array}\right.$$ Equality can be easily verified by integrations by parts. Since, $$\left\{\begin{array}{ll}
\int_{0}^{2\pi} \left|i_n({\theta}, {\varphi})\right|{\,\text{\rm d}}{\varphi}\, {\;\leqslant\;}\, \frac{2\pi}{n+2},& \int_{0}^{2\pi} \left|i_n({\theta},
{\varphi})\right|{\,\text{\rm d}}{\theta}\, {\;\leqslant\;}\, \frac{2\pi}{n-1}\, ,\\[10pt]
\int_{0}^{2\pi} \left|j_n({\theta}, {\varphi})\right|{\,\text{\rm d}}{\varphi}\, {\;\leqslant\;}\, \frac{n+1}{\pi}\, ,& \int_{0}^{2\pi} \left|j_n({\theta},
{\varphi})\right|{\,\text{\rm d}}{\theta}\, {\;\leqslant\;}\, \frac{n(n+1)}{\pi(n-1)}\, ,
\end{array}\right.$$ we see (cf. [@Kato1995 p.143-144]) that ${\ensuremath{I_n}}$ and ${\ensuremath{J_n}}$ are bounded operators on $L^2({{{\ensuremath{\mathbb S}} }})$, namely: $$\Vert{{\ensuremath{I_n}}}\Vert\, {\;\leqslant\;}\, \frac{2\pi}{n-1},\quad \Vert{{\ensuremath{J_n}}}\Vert\, {\;\leqslant\;}\, \frac{n(n+1)}{\pi(n-1)}\, .$$ So, applying relation for $f={{\ensuremath{\mathcal U}} }$ we get, for $\mu$-almost every ${\omega}$: $$\left( {\int_{{{\ensuremath{\mathbb S}} }}}|u({\theta}, {\omega})|^2 {\,\text{\rm d}}{\theta}\right)^{\frac 12} {\;\leqslant\;}\frac{2\pi}{n-1}\left( {\int_{{{\ensuremath{\mathbb S}} }}}|u'({\theta}, {\omega})|^2 {\,\text{\rm d}}{\theta}\right)^{\frac 12} +
\frac{n(n+1)}{\pi(n-1)}\left({\int_{{{\ensuremath{\mathbb S}} }}}|{{\ensuremath{\mathcal U}} }({\theta}, {\omega})|^2 {\,\text{\rm d}}{\theta}\right)^{\frac 12}.$$
This gives $$\label{eq:estuUmu}
{\left\|\,u\,\right\|_{2, \mu}}\, {\;\leqslant\;}\,\frac{2\pi}{n-1}{\left\|\,u'\,\right\|_{2, \mu}} +\frac{n(n+1)}{\pi(n-1)}{\left\|\,{{\ensuremath{\mathcal U}} }\,\right\|_{2, \mu}}\, .$$ Since ${\left\|\,{{\ensuremath{\mathcal U}} }\,\right\|_{2, {q_0}, \mu}}= {\ensuremath{\left\|\,u\,\right\|_{ -1, {q_0}, \mu}}}$, it only remains to control ${\left\|\,u'\,\right\|_{2, {q_0}, \mu}}$ with ${\ensuremath{\left\|\,Au\,\right\|_{ -1, {q_0}, \mu}}}$: like for the beginning of this proof for the operator $B$, we have ${\ensuremath{\left\|\,Au\,\right\|_{ -1, {q_0}, \mu}}}= {\left\|\,{{\ensuremath{\mathcal A}} }u\,\right\|_{2, {q_0}, \mu}}$, where ${{\ensuremath{\mathcal A}} }u$ is the appropriate primitive of $Au$: $$\begin{aligned}
{{\ensuremath{\mathcal A}} }u({\theta}, {\omega})&\, :=\, {} {\frac 12} u'({\theta}, {\omega}) -\left( u({\theta}, {\omega}) (J \ast q_0) + q_0({\theta}) \langle J
\ast u\rangle_\mu({\theta})\right)\nonumber\\
&+{} \left( {\int_{{{\ensuremath{\mathbb S}} }}}\frac{1}{q_0} \right)^{-1}\left( {\int_{{{\ensuremath{\mathbb S}} }}}\left\{\frac{u({\theta}, {\omega}) (J \ast
q_0)}{q_0({\theta})} + {\frac 12} u({\theta}, {\omega}) \partial_{\theta}\left( \frac{1}{q_0({\theta})}
\right)\right\}{\,\text{\rm d}}{\theta}\right)\, .\label{eq:primitiveAu}\end{aligned}$$ Using inequalities $|\langle J\ast u\rangle|_{\mu}(\cdot){\;\leqslant\;}K\sqrt{\pi}{\left\|\,u\,\right\|_{2, \mu}}$, and ${\int_{{{\ensuremath{\mathbb S}} }}}\frac{|u(\cdot, {\omega})|}{q_0}{\;\leqslant\;}Z_0^{\frac12}e^{Kr_0}\left( {\int_{{{\ensuremath{\mathbb S}} }}}|u(\cdot, {\omega})^2|\right)^{\frac12}$, an easy calculation shows that: $$\label{eq:estimuprimeAu1}
|u'(\cdot, {\omega})|\, {\;\leqslant\;}\, 2|{{\ensuremath{\mathcal A}} }u(\cdot, {\omega})| + 2Kr_0|u(\cdot, {\omega})|+2\sqrt{\pi}K q_0(\cdot){\left\|\,u\,\right\|_{2, \mu}} +
\frac{4Kr_0}{Z_0^\frac12}e^{Kr_0}\left( {\int_{{{\ensuremath{\mathbb S}} }}}|u(\cdot, {\omega})^2|\right)^{\frac12}\, ,$$ and thus, $$\label{eq:estimuprimeAu}
{\left\|\,u'\,\right\|_{2, \mu}}\, {\;\leqslant\;}\, 4 {\left\|\,{{\ensuremath{\mathcal A}} }u\,\right\|_{2, \mu}} + 4K \left( r_0^2+ \pi Z_0^{-1}e^{2Kr_0}(1+8r_0^2) \right)^{\frac 12}{\left\|\,u\,\right\|_{2, \mu}}\, ,$$ and by putting and together we obtain $$\begin{split}
{\left\|\,u\,\right\|_{2, \mu}} \, {\;\leqslant\;}\,& \frac{8\pi}{n-1}{\left\|\,{{\ensuremath{\mathcal A}} }u\,\right\|_{2, \mu}} + \frac{2\pi}{n-1}4K \left( r_0^2+ \pi Z_0^{-1}e^{2Kr_0}(1+8r_0^2) \right)^{\frac 12}{\left\|\,u\,\right\|_{2, \mu}}\\
&+\frac{n(n+1)}{\pi(n-1)}{\ensuremath{\left\|\,u\,\right\|_{ -1, {q_0}, \mu}}}\, .
\end{split}$$ Let us choose the integer $n=\left\lfloor 16\pi K\left( r_0^2+ \pi Z_0^{-1}e^{2Kr_0}(1+8r_0^2) \right)^{\frac 12} +1\right\rfloor$ so that $$\frac{2\pi}{n-1}4K \left( r_0^2+ \pi Z_0^{-1}e^{2Kr_0}(1+8r_0^2) \right)^{\frac 12}\, {\;\leqslant\;}\, {\frac 12}\, .$$ In this case, we obtain: $$\begin{aligned}
{\left\|\,u\,\right\|_{2, {q_0}, \mu}} &\, {\;\leqslant\;}\, \frac{e^{2Kr_0}}{4K\left( r_0^2+ \pi Z_0^{-1}e^{2Kr_0}(1+8r_0^2) \right)^{\frac 12}}{\ensuremath{\left\|\,Au\,\right\|_{ -1, {q_0}, \mu}}}\nonumber\\&+
\frac{e^{2Kr_0}\left(16 K\left(r_0^2+ \pi Z_0^{-1}e^{2Kr_0}(1+8r_0^2)\right)^{\frac 12}
+3\right)^2}{16\pi^2 K\left(r_0^2+ \pi Z_0^{-1}e^{2Kr_0}(1+8 r_0^2)\right)^{\frac 12}}{\ensuremath{\left\|\,u\,\right\|_{ -1, {q_0}, \mu}}}\, ,\end{aligned}$$ which is precisely the inequality we wanted to prove. Inequalities and give the result, for $a_{K, {\delta}}:= c^{(1)}_{K, {\delta}} \cdot c_K^{(3)}$ and $b_{K, {\delta}}:= c^{(1)}_{K, {\delta}} \cdot
c_K^{(2)}$.
\[prop:Lcompactresolvent\] For all $K>1$, there exists ${\delta}_{3}(K)>0$ such that for all $0<{\delta}{\;\leqslant\;}{\delta}_{3}(K)$, the operator $L^{\omega}_{{q}}$ is closable. In that case, its closure has the same domain as the closure of $A$.
Let us choose ${\delta}_3(K)>0$ so that $$\label{eq:condomega37}
b_{K, {\delta}_3(K)}<1$$ where $b_{K, {\delta}}$ is the constant introduced in , then, for all $0<{\delta}{\;\leqslant\;}{\delta}_3(K)$, the operator $B$ is $A$-bounded with $A$-bound strictly lower than $1$. The result is then a consequence of Th. IV-1.1, p.190 in [@Kato1995].
### The spectrum of $L^{\omega}_{{q}}$
We divide our study into two parts: the determination of the position of the spectrum within a sector and its position near $0$.
### Position of the spectrum away from $0$
We prove mainly that the perturbed operator $L^{\omega}_{{q}}$ still generates an analytic semigroup of operators on an appropriate sector. An immediate corollary is the fact that the spectrum lies in a cone whose vertex is zero.
We know (Proposition \[prop:Apositivemu\]) that for all $0<\alpha<\frac{\pi}{2}$, $A$ generates an analytic semigroup of operators on $\Delta_\alpha:={\left\{\lambda\in{{\ensuremath{\mathbf C}} }\,;\,|\arg(\lambda)|<\alpha\right\}}$.
\[prop:semigroupAB\] For all $K>1$, $0<\alpha<\frac{\pi}{2}$ and ${\varepsilon}>0$, there exists ${\delta}_4>0$ (depending on $\alpha$, $K$ and ${\varepsilon}$) such that for all $0<{\delta}<{\delta}_{4}$, the spectrum of $L^{\omega}_{{q}}= A+B$ lies within $\Theta_{{\varepsilon},
\alpha}:= {\left\{\lambda\in{{\ensuremath{\mathbf C}} }\,;\,\frac{\pi}{2} + \alpha{\;\leqslant\;}\arg(\lambda){\;\leqslant\;}\frac{3\pi}{2}-\alpha\right\}}\cup {\left\{\lambda\in{{\ensuremath{\mathbf C}} }\,;\,|\lambda|{\;\leqslant\;}{\varepsilon}\right\}}$. Moreover, there exists $\alpha'\in(0, \frac\pi2)$ such that the operator $L^{\omega}_{{q}}$ still generates an analytic semigroup on $\Delta_{\alpha'}$.
Let $0<\alpha<\frac{\pi}{2}$ be fixed. Following and using , one can easily deduce an estimate on the bounded operator $BR(\lambda, A)$, for $\lambda\in\Sigma_\alpha$: $$\begin{split}
{\ensuremath{\left\|\,BR(\lambda, A)u\,\right\|_{ -1, {q_0}, \mu}}} &\,{\;\leqslant\;}\, a_{K,{\delta}}{\ensuremath{\left\|\,R(\lambda, A)u\,\right\|_{ -1, {q_0}, \mu}}} + b_{K,
{\delta}}{\ensuremath{\left\|\,A R(\lambda,A)u\,\right\|_{ -1, {q_0}, \mu}}} \\
&\, {\;\leqslant\;}\, a_{K, {\delta}}\frac{1}{(1-\sin(\alpha))|\lambda|}{\ensuremath{\left\|\,u\,\right\|_{ -1, {q_0}, \mu}}}\\ &
\ \ \ \ \ \
+ b_{K, {\delta}}\left(1+
\frac{1}{1-\sin(\alpha)}\right) {\ensuremath{\left\|\,u\,\right\|_{ -1, {q_0}, \mu}}}\, .
\end{split}$$ Let us fix ${\varepsilon}>0$ and choose ${\delta}$ so that: $$\label{eq:condomega00}
\max\left(4b_{K, {\delta}}\left(\frac{1}{1-\sin(\alpha)}+1\right) , \frac{4 a_{K, {\delta}}}{(1-\sin(\alpha)){\varepsilon}} \right) \,
{\;\leqslant\;}\, 1\, .$$
Then, for $\lambda\in\Sigma_\alpha$ such that $|\lambda|>{\varepsilon}{\;\geqslant\;}\frac{4a_{K, {\delta}}}{1-\sin(\alpha)}$, we have $${\ensuremath{\left\|\,BR(\lambda, A)u\,\right\|_{ -1, {q_0}, \mu}}} \, {\;\leqslant\;}\, \frac{1}{2}{\ensuremath{\left\|\,u\,\right\|_{ -1, {q_0}, \mu}}}\, .$$ In particular, $1 - BR(\lambda, A)$ is invertible with ${\ensuremath{\left\|\,\left( 1 - BR(\lambda, A) \right)^{-1}\,\right\|_{ -1, {q_0}, \mu}}}{\;\leqslant\;}2$. A direct calculation shows that $$\left( \lambda - (A+B) \right)^{-1} \, =\, R(\lambda, A) \left( 1 -
BR(\lambda, A)\right)^{-1}\, .$$ One deduces the following estimates on the resolvent: for $\lambda\in\Sigma_\alpha$, $|\lambda|>{\varepsilon}$, $$\label{eq:estim resolvent Lq gep prime}{\ensuremath{\left\|\,R(\lambda, L^{\omega}_{{q}})\,\right\|_{ -1, {q_0}, \mu}}}\, {\;\leqslant\;}\, \frac{2}{(1-\sin(\alpha))|\lambda|}\,
.$$ Estimate has two consequences: firstly, one deduces immediately that the spectrum $\sigma(L^{\omega}_{q})$ of $L^{\omega}_{q}$ is contained in $\Theta_{{\varepsilon},\alpha}$: $$\label{eq:subset rho Lq}
\sigma(L_q^{{\omega}})\subseteq{\left\{\lambda\in{{\ensuremath{\mathbf C}} }\,;\,\frac{\pi}{2} + \alpha{\;\leqslant\;}\arg(\lambda){\;\leqslant\;}\frac{3\pi}{2}-\alpha\right\}}\cup
{\left\{\lambda\in{{\ensuremath{\mathbf C}} }\,;\,|\lambda|{\;\leqslant\;}{\varepsilon}\right\}}.$$ Secondly, entails that $L^{\omega}_{q}$ generates an analytic semigroup of operators on an appropriate sector. Indeed, if one denotes by $L^{{\omega}}_{q, {\varepsilon}}:= L^{{\omega}}_{q} -{\varepsilon}$, one deduces from that $0\in\rho(L^{{\omega}}_{q, 2{\varepsilon}})$ and that for all $\lambda\in{{\ensuremath{\mathbf C}} }$ with $\Re(\lambda)>0$ (in particular, $|\lambda|<|\lambda+2{\varepsilon}|$) $$\begin{aligned}
{\ensuremath{\left\|\,R(\lambda, L^{\omega}_{{q,2{\varepsilon}}})\,\right\|_{ -1, {q_0}, \mu}}}\, &=\, {\ensuremath{\left\|\,R(\lambda+2{\varepsilon}, L^{\omega}_{{q}})\,\right\|_{ -1, {q_0}, \mu}}}\, {\;\leqslant\;}\, \frac{2}{(1-\sin(\alpha))|\lambda+2{\varepsilon}|}\,
,\nonumber\\
&\, {\;\leqslant\;}\, \frac{2}{(1-\sin(\alpha))|\lambda|}\, .\end{aligned}$$ Hence, using the same arguments of Taylor expansion as in the proof of Proposition \[prop:Apositivemu\] and applying Proposition \[prop:pazysemgps\], one easily sees that $L^{\omega}_{{q,2{\varepsilon}}}$ generates an analytic semigroup in a (a priori) smaller sector $\Delta_{\alpha'}$, where $\alpha'\in(0, \frac\pi2)$ can be chosen as $\alpha':= \frac12 \arctan\left( \frac{1-\sin(\alpha)}{2} \right)$. But if $L^{\omega}_{{q,2{\varepsilon}}}$ generates an analytic semigroup, so does $L^{\omega}_{q}$.
### Position of the spectrum near $0$ {#subsubsec:loczero}
Let us apply Proposition \[prop:semigroupAB\] for fixed $K>1$, $\alpha\in(0, \frac\pi2)$, $\rho\in(0,1)$ and ${\varepsilon}:=\rho{{\lambda}_{K}}$, where we recall that ${{\lambda}_{K}}$ is the spectral gap between the eigenvalue $0$ for the non perturbed operator $A$ and the rest of the spectrum $\sigma(A)\smallsetminus\{0\}$. Let $\Theta_{{\varepsilon}, \alpha}^{+}:= {\left\{\lambda\in\Theta_{{\varepsilon}, \alpha}\,;\,\Re(\lambda){\;\geqslant\;}0\right\}}$ be the subset of $\Theta_{{\varepsilon}, \alpha}$ which lies in the positive part of the complex plane (see Fig. \[fig:Thetaeps\]). In order to show the linear stability, one has to make sure that one can choose a perturbation $B$ small enough so that no eigenvalue of $A+B$ remains in the small set $\Theta_{{\varepsilon}, \alpha}^{+}$.
![The set $\Theta_{{\varepsilon},
\alpha}$.[]{data-label="fig:Thetaeps"}](coin.eps){width="50.00000%"}
Since ${{\lambda}_{K}}>0$, one can separate $0$ from the rest of the spectrum of $A$ by a circle $\mathscr{C}$ centered in $0$ with radius $(\frac{\rho+1}{2}){{\lambda}_{K}}$. The appropriate choice of ${\varepsilon}$ ensures that the interior of the disk delimited by $\mathscr{C}$ contains $\Theta_{{\varepsilon}, \alpha}^{+}$ (see Figure \[fig:Thetaeps\]).
The main argument is the following: by construction of $\mathscr{C}$, $0$ is the only eigenvalue (with multiplicity $1$) of the non-perturbed operator $A$ lying in the interior of $\mathscr{C}$. A principle of local continuity of eigenvalues shows that, while adding a sufficiently small perturbation $B$ to $A$, the interior of $\mathscr{C}$ still contains exactly one eigenvalue (which is *a priori* close but not equal to $0$) with the same multiplicity.
But we already know that for the perturbed operator $L^{\omega}_{{q}}=A+B$, $0$ is always an eigenvalue (since $L^{\omega}_{{q}}q'=0$). One can therefore conclude that, by uniqueness, $0$ is the only element of the spectrum of $L^{\omega}_{{q}}$ within $\mathscr{C}$, and is an eigenvalue with multiplicity $1$. In particular, there is no element of the spectrum in the positive part of the complex plane.
In order to quantify the appropriate size of the perturbation $B$, one has to have explicit estimates on the resolvent $R(\lambda,
A)$ on the circle $\mathscr{C}$.
\[lem:estimRGamma\] There exists some explicit constant $c_{\mathscr{C}}=c_{\mathscr{C}}(K, \rho)$ such that for all $\lambda\in\mathscr{C}$, $$\begin{aligned}
{\ensuremath{\left\|\,R(\lambda, A)\,\right\|_{ -1, {q_0}, \mu}}} &\, {\;\leqslant\;}\, c_{\mathscr{C}}\, ,\label{eq:estimRcercle}\\
{\ensuremath{\left\|\,AR(\lambda, A)\,\right\|_{ -1, {q_0}, \mu}}} &\, {\;\leqslant\;}\, 1+ \left(\frac{1+\rho}{2}\right){{\lambda}_{K}}\cdot c_{\mathscr{C}}\, .\label{eq:estimARcercle}\end{aligned}$$ One can choose $c_{\mathscr{C}}$ as $\frac{1}{{{\lambda}_{K}}} \max\left( \frac{2}{\rho+1},\, \frac{2}{1-\rho}\right):= \frac{\ell(\rho)}{{{\lambda}_{K}}}$.
Applying the spectral theorem (see [@cf:Dunford Th. 3, p.1192]) to the essentially self-adjoint operator $A$, there exists a spectral measure $E$ vanishing on the complementary of the spectrum of $A$ such that $A={\int_{{{\ensuremath{\mathbb R}} }}}\lambda {\,\text{\rm d}}E(\lambda)$. In that extent, one has for any $\zeta\in\mathscr{C}$ $$R(\zeta, A)\, =\, {\int_{{{\ensuremath{\mathbb R}} }}}\frac{{\,\text{\rm d}}E(\lambda)}{\lambda -\zeta}\, .$$ In particular, for $\zeta\in\mathscr{C}$ $${\ensuremath{\left\|\,R(\zeta, A)\,\right\|_{ -1, {q_0}, \mu}}} {\;\leqslant\;}\sup_{\lambda\in \sigma(A)} \frac{1}{|\lambda-\zeta|}{\;\leqslant\;}\frac{\ell(\rho)}{{{\lambda}_{K}}}\, .$$
The estimation is straightforward.
We are now in position to apply our argument of local continuity of eigenvalues: Following [@Kato1995 Th III-6.17, p.178], there exists a decomposition of the operator $A$ according to $H^{-1}_{{q_0},\mu}= H_0 \oplus H'$ (in the sense that $AH_0\subset H_0$, $A H'\subset H'$ and $P {{\ensuremath{\mathcal D}} }(A) \subset {{\ensuremath{\mathcal D}} }(A)$, where $P$ is the projection on $H_0$ along $H'$) in such a way that $A$ restricted to $H_0$ has spectrum $\{0\}$ and $A$ restricted to $H'$ has spectrum $\sigma(A)\smallsetminus\{0\}$.
Let us note that the dimension of $H_0$ is $1$, since the characteristic space of $A$ in the eigenvalue $0$ is reduced to its kernel which is of dimension $1$.
Then, applying [@Kato1995 Th. IV-3.18, p.214], and using Proposition \[prop:BAboundedmu\], we find that if one chooses ${\delta}>0$, such that $$\label{eq:condperturbGamma}
\sup_{\lambda\in\mathscr{C}} \left(a_{K, {\delta}} {\ensuremath{\left\|\,R(\lambda, A)\,\right\|_{ -1, {q_0}, \mu}}} + b_{K, {\delta}}{\ensuremath{\left\|\,AR(\lambda,
A)\,\right\|_{ -1, {q_0}, \mu}}}\right)<1,$$ then the perturbed operator $L^{\omega}_{{q}}$ is likewise decomposed according to $H^{-1}_{{q_0},\mu}= {\widetilde}{H}_0\oplus {\widetilde}{H}'$, in such a way that $\dim(H_0)= \dim({\widetilde}{H}_0)= 1$, and that the spectrum of $L^{\omega}_{{q}}$ is again separated in two parts by $\mathscr{C}$ . But we already know that the characteristic space of the perturbed operator $L^{\omega}_{{q}}$ according to the eigenvalue $0$ is, at least, of dimension $1$ (since $L^{\omega}_{{q}}q'=0$).
We can conclude, that for such an ${\delta}>0$, $0$ is the only eigenvalue in $\mathscr{C}$ and that $\dim({\widetilde}{H}_0)=1$.
Applying Lemma \[lem:estimRGamma\], we see that condition is satisfied if we choose ${\delta}>0$ so that: $$\label{eq:condomega99}
a_{K, {\delta}} c_{\mathscr{C}} + b_{K, {\delta}}\left( 1 + \left( \frac{1+\rho}{2} \right){{\lambda}_{K}}c_{\mathscr{C}}\right)<1.$$
In particular, in that case, the spectrum of $L^{\omega}_{{q}}$ is contained in $${\left\{\lambda\in{{\ensuremath{\mathbb C}} }\,;\,\frac{\pi}{2} + \alpha{\;\leqslant\;}\arg(\lambda){\;\leqslant\;}\frac{3\pi}{2}-\alpha\right\}}\subseteq
{\left\{z\in{{\ensuremath{\mathbf C}} }\,;\,\Re(z){\;\leqslant\;}0\right\}}\, .$$ Finally, the following proposition sums-up the sufficient conditions on ${\delta}$ for the conclusions of Theorem \[th:spectral prop L disorder\] to be satisfied:
\[prop:resume conditions delta\_2\]
Recall the definitions of $a_{K, {\delta}}$ and $b_{K, {\delta}}$ in Proposition \[prop:BAboundedmu\]. If ${\delta}>0$ satisfies the following conditions $$\label{eq:conds4}
\begin{split}
b_{K, {\delta}} &\, {\;\leqslant\;}\, 1\, ,\\
4b_{K, {\delta}}\left( \frac{1}{1-\sin(\alpha)} +1\right) &\, {\;\leqslant\;}\, 1\, ,\\
\frac{4 a_{K, {\delta}} }{\rho{{\lambda}_{K}}\left( 1-\sin(\alpha) \right)}&\, {\;\leqslant\;}\, 1\, ,\\
a_{K, {\delta}}\frac{\ell(\rho)}{{{\lambda}_{K}}} + b_{K, {\delta}} \left( 1+ \left(\frac{1+\rho}{2}\right)\ell(\rho)\right)&\, <\, 1\, .
\end{split}$$ the conclusions of Theorem \[th:spectral prop L disorder\] are true.
One has simply to sum-up conditions , with ${\varepsilon}=\rho{{\lambda}_{K}}$ and . can be obtained by (long) estimations on the coefficients $a_{K, {\delta}}$ and $b_{K, {\delta}}$.
The conditions in Proposition \[prop:resume conditions delta\_2\] can be simplified. For example one can exhibit an explicit constant $c$ such that if ${\delta}$ satisfies $$\label{eq:estim gd K}
\begin{split}
{\delta}e^{12\pi{\delta}}\,{\;\leqslant\;}\, c e^{-20K\bar{r_{\delta}}}\max&\left(1, \left( \frac{1-\sin(\alpha)}{2-\sin(\alpha)} \right),
\frac{\rho{{\lambda}_{K}}(1-\sin(\alpha))e^{-4K\bar{r_{\delta}}}}{K^2},\right.\\
&\left.\ \ \ \ \frac{{{\lambda}_{K}}}{K^2e^{4K\bar{r_{\delta}}}\ell(\rho) + {{\lambda}_{K}}\left( 1+ \left( \frac{1+\rho}{2} \right)\ell(\rho) \right)}\right)
\end{split}$$ the conditions in are fulfilled. Explicit estimates on the spectral gap ${\lambda}_K$ can be found in [@cf:BGP Sec. 2.5].
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to K. Pakdaman and G. Wainrib for helpful discussions. G. G. acknowledges the support of the ANR (projects Mandy and SHEPI) and the support of the Petronio Fellowship Fund at the Institute for Advanced Study (Princeton, NJ) where this work has been completed.
Regularity in the non-linear Fokker-Planck equation {#sec:appendix regularity pt with disorder}
===================================================
The purpose of this section is to establish regularity properties of the solution of the non-linear equation (where we fix ${\delta}=1$ for simplicity). Note that this case also captures the situation where $U(\cdot, {\omega})\equiv
{\omega}$ (evolution ), as well as the situation where $U(\cdot, \cdot)\equiv 0$ (evolution ). In what follows we make the assumption that $U$ is bounded and that for all ${\omega}\in\operatorname*{Supp}(\mu)$, ${\theta}\mapsto
U({\theta}, {\omega})\in C^\infty({{\ensuremath{\mathbb S}} }; \, {{\ensuremath{\mathbb R}} })$ with bounded derivatives.
The existence and uniqueness in $L^2({\lambda}\otimes{\omega})$ of a solution to can be tackled using Banach fixed point arguments (see [@cf:SellYou Section 4.7]), but one can obtain more regularity from the theory of fundamental solutions of parabolic equations.
More precisely, it is usual to interpret Equation as the strong formulation of the weak equation (where $\nu \in{{\ensuremath{\mathcal C}} }([0, T],
{{\ensuremath{\mathcal M}} }_{1}({{\ensuremath{\mathbb S}} }\times{{\ensuremath{\mathbb R}} }))$ and $F$ is any bounded function on ${{\ensuremath{\mathbb S}} }\times{{\ensuremath{\mathbb R}} }$ with twice bounded derivatives w.r.t. ${\theta}$): $$\begin{aligned}
{\int_{{{\ensuremath{\mathbb R}} }\times{{{\ensuremath{\mathbb S}} }}}}F({\theta}, {\omega})\nu_t({\,\text{\rm d}}{\theta}, {\,\text{\rm d}}{\omega})&= {\int_{{{\ensuremath{\mathbb R}} }\times{{{\ensuremath{\mathbb S}} }}}}F({\theta}, {\omega})\nu_0({\,\text{\rm d}}{\theta}, {\,\text{\rm d}}{\omega}) +\frac12\int_0^t {\int_{{{\ensuremath{\mathbb R}} }\times{{{\ensuremath{\mathbb S}} }}}}F''({\theta}, {\omega})\nu_s({\,\text{\rm d}}{\theta},
{\,\text{\rm d}}{\omega}){\,\text{\rm d}}s\nonumber\\
&+ \int_0^t {\int_{{{\ensuremath{\mathbb R}} }\times{{{\ensuremath{\mathbb S}} }}}}F'({\theta}, {\omega}) \left({\int_{{{\ensuremath{\mathbb R}} }\times{{{\ensuremath{\mathbb S}} }}}}J({\theta}-\cdot){\,\text{\rm d}}\nu_s+ U({\theta}, {\omega})\right)\nu_s({\,\text{\rm d}}{\theta},{\,\text{\rm d}}{\omega}){\,\text{\rm d}}s,
\label{eq:weak formulation qt with U}\end{aligned}$$ where the second marginal (w.r.t. to the disorder ${\omega}$) of the initial condition $\nu_0({\,\text{\rm d}}{\theta}, {\,\text{\rm d}}{\omega})$ is $\mu({\,\text{\rm d}}{\omega})$ so that one can write $$\label{eq:initial condition nu zero}
\nu_0({\,\text{\rm d}}{\theta}, {\,\text{\rm d}}{\omega}) = \nu_0^{\omega}({\,\text{\rm d}}{\theta}) \mu({\,\text{\rm d}}{\omega})\, ,$$ where $\nu_0^{\omega}$ is a probability measure on ${{\ensuremath{\mathbb S}} }$, for $\mu$-a.e. ${\omega}$.
As already mentioned, a proof of the existence of a solution on $[0, T]$ of can be obtained from the almost-sure convergence of the empirical measure of the microscopic system [@cf:eric]. One can also find a proof of uniqueness of such a solution relying on arguments introduced in [@cf:Oelschlager].
The regularity result can be stated as follows:
\[prop:regularity solution qt\] For all probability measure $\nu_0({\,\text{\rm d}}{\theta}, {\,\text{\rm d}}{\omega})=\nu_0^{\omega}({\,\text{\rm d}}{\theta})\mu({\,\text{\rm d}}{\omega})$ on ${{\ensuremath{\mathbb S}} }\times{{\ensuremath{\mathbb R}} }$, for all $T>0$, there exists a unique solution $\nu$ to in ${{\ensuremath{\mathcal C}} }([0, T], {{\ensuremath{\mathcal M}} }_1({{\ensuremath{\mathbb S}} }\times{{\ensuremath{\mathbb R}} }))$ such that for all $F\in{{\ensuremath{\mathcal C}} }({{\ensuremath{\mathbb S}} }\times{{\ensuremath{\mathbb R}} })$, $$\lim_{t\searrow0}{\int_{{{\ensuremath{\mathbb R}} }\times{{{\ensuremath{\mathbb S}} }}}}F({\theta}, {\omega})\nu_t({\,\text{\rm d}}{\theta}, {\,\text{\rm d}}{\omega})= {\int_{{{\ensuremath{\mathbb R}} }\times{{{\ensuremath{\mathbb S}} }}}}F({\theta}, {\omega}) \nu_0^{\omega}({\,\text{\rm d}}{\theta}) \mu({\,\text{\rm d}}{\omega}).$$
Moreover, for all $t>0$, $\nu_t$ is absolutely continuous with respect to $\lambda_1\otimes\mu$ and for $\mu$-a.e. ${\omega}\in Supp(\mu)$, its density $(t,{\theta}, {\omega})\mapsto p_t({\theta}, {\omega})$ is strictly positive on $(0, T]\times {{\ensuremath{\mathbb S}} }$, is ${{\ensuremath{\mathcal C}} }^\infty$ in $(t,
{\theta})$ and solves the Fokker-Planck equation .
Let us fix $T>0$, ${\omega}\in\operatorname*{Supp}(\mu)$ and $t\mapsto \nu_t$ the unique solution in ${{\ensuremath{\mathcal C}} }([0, T], {{\ensuremath{\mathcal M}} }_1({{\ensuremath{\mathbb S}} }\times{{\ensuremath{\mathbb R}} }))$ to . Let us define $R(t, {\theta}, {\omega}):= {\int_{{{\ensuremath{\mathbb R}} }\times{{{\ensuremath{\mathbb S}} }}}}J({\theta}-\cdot){\,\text{\rm d}}\nu_t+ U({\theta}, {\omega})$ and consider the linear equation $$\label{eq:linear equation qt with Rt}
\partial_t p_t({\theta},{\omega})\, =\, \frac{1}{2} {\Delta}p_t({\theta},{\omega}) -\partial_{\theta}\Big(p_t({\theta},{\omega})R(t, {\theta}, {\omega})\Big)\, ,$$ such that for $\mu$-a.e. ${\omega}$, for all $F\in{{\ensuremath{\mathcal C}} }({{\ensuremath{\mathbb S}} })$, $$\label{eq:linear equation qt with Rt cond limit}
{\int_{{{\ensuremath{\mathbb S}} }}}F({\theta})p_t({\theta}, {\omega}) {\,\text{\rm d}}{\theta}\,
\stackrel{t\searrow 0}{\longrightarrow}
{\int_{{{\ensuremath{\mathbb S}} }}}F({\theta}) \nu_0^{\omega}({\,\text{\rm d}}{\theta})\, .$$
For fixed ${\omega}\in\operatorname*{Supp}(\mu)$, $R(\cdot, \cdot, {\omega})$ is continuous in time and ${{\ensuremath{\mathcal C}} }^\infty$ in ${\theta}$.
Suppose for a moment that we have found a weak solution $p_t({\theta},{\omega})$ to - such that for $\mu$-a.e. ${\omega}$, $p_t(\cdot, {\omega})$ is strictly positive on $(0,T]\times{{\ensuremath{\mathbb S}} }$. In particular for such a solution $p$, the quantity ${\int_{{{\ensuremath{\mathbb S}} }}}p_t({\theta}, {\omega}){\,\text{\rm d}}{\theta}$ is conserved for $t>0$, so that $p_t(\cdot, {\omega})$ is indeed a probability density for all $t>0$. Then both probability measures $\nu_t({\,\text{\rm d}}{\theta}, {\,\text{\rm d}}{\omega})$ and $p_t({\theta},{\omega}){\,\text{\rm d}}{\theta}\mu({\,\text{\rm d}}{\omega})$ solve
$$\begin{aligned}
{\int_{{{\ensuremath{\mathbb R}} }\times{{{\ensuremath{\mathbb S}} }}}}F({\theta}, {\omega})\nu_t({\,\text{\rm d}}{\theta}, {\,\text{\rm d}}{\omega})&= {\int_{{{\ensuremath{\mathbb R}} }\times{{{\ensuremath{\mathbb S}} }}}}F({\theta}, {\omega})\nu_0({\,\text{\rm d}}{\theta}, {\,\text{\rm d}}{\omega}) +\frac12\int_0^t {\int_{{{\ensuremath{\mathbb R}} }\times{{{\ensuremath{\mathbb S}} }}}}F''({\theta}, {\omega})\nu_s({\,\text{\rm d}}{\theta},
{\,\text{\rm d}}{\omega}){\,\text{\rm d}}s\nonumber\\
&+ \int_0^t {\int_{{{\ensuremath{\mathbb R}} }\times{{{\ensuremath{\mathbb S}} }}}}F'({\theta}, {\omega}) R(t, {\theta}, {\omega})\nu_s({\,\text{\rm d}}{\theta},{\,\text{\rm d}}{\omega}){\,\text{\rm d}}s.
\label{eq:weak formulation qt with Rt}\end{aligned}$$
By [@cf:eric] or [@cf:Oelschlager Lemma 10], uniqueness in is precisely a consequence of uniqueness in . Hence, by uniqueness in , $\nu_t({\,\text{\rm d}}{\theta}, {\,\text{\rm d}}{\omega})=p_t({\theta},{\omega}){\,\text{\rm d}}{\theta}\mu({\,\text{\rm d}}{\omega})$, which is the result. So it suffices to exhibit a weak solution $p_t({\theta},{\omega})$ to such that is satisfied.
This fact can be deduced from standard results for uniform parabolic PDEs (see [@cf:Aronson] and [@cf:Friedman] for precise definitions). In particular, a usual result, which can be found in [@cf:Aronson §7 p.658], states that admits a fundamental solution $\Gamma({\theta}, t; {\theta}', s, {\omega})$ ($t>s$), which is bounded above and below (see [@cf:Aronson Th.7, p.661]): $$\label{eq:control Gamma}
\frac{1}{C\sqrt{t-s}} \exp\left( \frac{-C({\theta}- {\theta}')^2}{\sqrt{t-s}}\right){\;\leqslant\;}\Gamma({\theta}, t; {\theta}', s, {\omega}) {\;\leqslant\;}\frac{C}{\sqrt{t-s}}
\exp\left( \frac{-({\theta}- {\theta}')^2}{C\sqrt{t-s}}\right)\, .$$ Note that the constant $C>0$ only depends on $T$ and the *structure* of the linear operator in (see [@cf:Aronson Th.7, p.661] and [@cf:Aronson §1, p.615]). In particular, since $({\theta}, {\omega})\mapsto U({\theta}, {\omega})$ is bounded, this constant does not depend on ${\omega}$.
Note that the proof given in [@cf:Aronson] is done for ${\theta}\in{{\ensuremath{\mathbb R}} }$ but can be readily adapted to our case (${\theta}\in{{\ensuremath{\mathbb S}} }$).
Moreover, thanks to Corollary 12.1, p.690 in [@cf:Aronson], the following expression of $p_t({\theta}, {\omega})$ $$\label{eq:pt Gamma}
p_t({\theta}, {\omega})={\int_{{{\ensuremath{\mathbb S}} }}}\Gamma({\theta}, t; {\theta}', 0, {\omega})\nu_0^{\omega}({\,\text{\rm d}}{\theta}')$$ defines a weak solution of on $(0,T]\times{{\ensuremath{\mathbb S}} }$ (namely a weak solution on $(\tau, T]\times {{\ensuremath{\mathbb S}} }$, for all $0<\tau<T$) such that is satisfied. The positivity and boundedness of $p_t(\cdot, {\omega})$ for $t>0$ is an easy consequence of . The smoothness of $p_\cdot(\cdot, {\omega})$ on $(0, T]\times {{\ensuremath{\mathbb S}} }$ can be derived by standard bootstrap methods.
We focus now on the regularity of the solution $p_t({\theta}, {\omega})$ of with respect to the disorder ${\omega}$. We assume here that the initial condition $\nu_0$ is such that for all ${\omega}\in\operatorname*{Supp}(\mu)$, $\nu_0^{\omega}({\,\text{\rm d}}{\theta})$ is absolutely continuous with respect to the Lebesgue measure $\lambda_1$ on ${{\ensuremath{\mathbb S}} }$: there exists a positive integrable function $\gamma(\cdot, {\omega})$ of integral $1$ on ${{\ensuremath{\mathbb S}} }$ such that $\nu_0^{\omega}({\,\text{\rm d}}{\theta})=\gamma({\theta}, {\omega}){\,\text{\rm d}}{\theta}$. Then we have
\[lem:regularity density disorder\] For every $(t_0, {\theta}_0)\in(0, \infty)\times{{\ensuremath{\mathbb S}} }$, for every ${\omega}_0$ which is an accumulation point in $\operatorname*{Supp}(\mu)$ such that the following holds $$\label{eq:condition continuity gamma}
{\int_{{{\ensuremath{\mathbb S}} }}}|\gamma({\theta}, {\omega})-\gamma({\theta}, {\omega}_0)|{\,\text{\rm d}}{\theta}\to 0, \quad\text{as ${\omega}\to{\omega}_0$}\, ,$$ then the solution $p$ of defined on $(0, \infty)\times{{\ensuremath{\mathbb S}} }\times \operatorname*{Supp}(\mu)$ is continuous at the point $(t_0, {\theta}_0, {\omega}_0)$.
For any ${\omega}$ in the support of $\mu$, let for all $t>0$, ${\theta}\in{{\ensuremath{\mathbb S}} }$ $$\label{eq:def u diff q1 q2}
u(t, {\theta}, {\omega}) := p_t({\theta}, {\omega}) - p_t({\theta}, {\omega}_0),$$ where $(p_t(\cdot, \cdot))_{t{\;\geqslant\;}0}$ is the unique solution of . It is easy to see that $u$ is a strong solution to the following PDE $$\label{eq:pde verified by u}
\partial_t u(t, {\theta}, {\omega}) - \left[\frac12 {\Delta}u(t, {\theta}) - \partial_{\theta}\left( u(t, {\theta}) R(t, {\theta}, {\omega}_0)\right)\right] =
\mathcal{R}(t, {\theta}, {\omega}),$$ where $\mathcal{R}(t, {\theta}, {\omega}):= \partial_{\theta}\left[ p_t({\theta}, {\omega})\left( R(t, {\theta}, {\omega}) -R(t, {\theta}, {\omega}_0) \right)\right]$ and with initial condition (since $\nu_0^{\omega}({\,\text{\rm d}}{\theta})=\gamma({\theta}, {\omega}){\,\text{\rm d}}{\theta}$ for all ${\omega}$) $$\label{eq:initial condition u}
u(t, {\theta}, {\omega})|_{t\searrow0} =\gamma({\theta}, {\omega}) - \gamma({\theta}, {\omega}_0).$$ Then applying [@cf:Friedman Th. 12 p.25], $u(t, {\theta}, {\omega})$ can be expressed as $$\label{eq:u Gamma}
u(t, {\theta}, {\omega})={\int_{{{\ensuremath{\mathbb S}} }}}\Gamma({\theta}, t; {\theta}', 0, {\omega}_0)(\gamma({\theta}, {\omega}) - \gamma({\theta}, {\omega}_0)){\,\text{\rm d}}{\theta}' - \int_0^t {\int_{{{\ensuremath{\mathbb S}} }}}\Gamma({\theta},
t;{\theta}', s, {\omega}_0)\mathcal{R}(s, {\theta}', {\omega}){\,\text{\rm d}}{\theta}'{\,\text{\rm d}}s.$$ For the first term of the RHS of , we have $$\left|{\int_{{{\ensuremath{\mathbb S}} }}}\Gamma({\theta}, t; {\theta}', 0, {\omega}_0)(\gamma({\theta}, {\omega}) - \gamma({\theta}, {\omega}_0)){\,\text{\rm d}}{\theta}'\right|{\;\leqslant\;}\frac{C}{\sqrt{t}}{\int_{{{\ensuremath{\mathbb S}} }}}|\gamma({\theta}, {\omega}) - \gamma({\theta}, {\omega}_0)|{\,\text{\rm d}}{\theta}'\, ,$$ which converges to $0$, for fixed $t>0$, by hypothesis .
Secondly, it is easy to see from the definition of the density $p$ and the estimates and [@cf:Friedman Th.9 p.263] concerning the fundamental solution $\Gamma$ that both $p_t({\theta}, {\omega})$ and $\partial_{\theta}p_t({\theta}, {\omega})$ are bounded uniformly on $(t, {\theta}, {\omega})\in [0, T]\times {{\ensuremath{\mathbb S}} }\times \operatorname*{Supp}(\mu)$. In particular, a standard result shows that for fixed $(t, {\theta})$, the second term of the RHS of goes to $0$ as ${\omega}\to{\omega}_0$. But then the joint continuity of $p$ at $(t_0,{\theta}_0, {\omega}_0)$ follows from and uniform estimates on $\Gamma$ (see [@cf:Friedman Th.9 p.263]).
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|
---
address: 'Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131.'
author:
- 'Charles P. Boyer and Krzysztof Galicki'
title: 'Sasakian Geometry, Holonomy, and Supersymmetry'
---
[^1]
Introduction
============
Supersymmetry has emerged in physics as an attempt to unify the way physical theories deal with bosonic and fermionic particles. Since its birth around the early 70ties it has come to dominate theoretical high energy physics (for a historical perspective see [@KaSh00] with the introduction by Kane and Shifman, and for a mathematical treatment see [@Var04]). This dominance is still ongoing in spite of the fact that after almost 40 years there is no single experimental evidence that would directly and convincingly “prove" or “discover" the existence of supersymmetry in nature. On the other hand, especially in the last 20 years, supersymmetry has given birth to many beautiful mathematical theories. Gromov-Witten Theory, Seiberg-Witten Theory, Rozansky-Witten Theory as well as the Mirror Duality Conjecture are just a few of the more famous examples of important and deep mathematics having its origins in the physics of various supersymmetric theories.
Various supersymmetric field theories naturally include both Riemannian and pseudo-Riemannian manifolds. The latter is necessary in order to incorporate the physical space-time into the picture while the former typically describes the geometry associated with ‘invisible’ extra dimensions. It is mainly in such a context that Sasaki-Einstein manifolds appear in physics: they are compact Einstein manifolds of positive scalar curvature that occur in abundance in the physically interesting dimensions five and seven. Moreover, when they are simply connected they admit real Killing spinors. It is this last property that vitally connects them to Supergravity, Superstring, and $M$-Theory.
The main purpose of this review is to describe geometric properties of Sasaki-Einstein manifolds which make them interesting in modern theoretical physics. In spite of the fact that it is [*supersymmetry*]{} that connects Sasaki-Einstein spaces to physics, it is not the purpose of this article to describe what this concept really means to either physicists or mathematicians. There have been many recent attempts to frame these important notions of theoretical physics in precise mathematical terms. This enormous task is far beyond the scope of this article, so we refer the reader to recent monographs and references therein [@QFTS99; @Var04; @Jos01; @AJPS97]. Here we content ourselves with providing the main theorems and results concerning Killing spinors.
It is most remarkable that, even though Sasaki-Einstein manifolds always have holonomy $SO(TM),$ [*i.e.*]{}, the holonomy of any generic Riemannian metric, they are far from being generic. In fact, the most interesting thing about this geometry is that it naturally relates to several different Riemannian geometries with reduced holonomies. It is this point that we will try to stress throughout this article. For more detailed exposition we refer the interested reader to our recent monograph on [*Sasakian Geometry*]{} [@BG05].
The key to understanding the importance of Sasakian geometry is through its relation to Kählerian geometry. Before we define Sasakian manifolds and describe some of their elementary properties in Section \[sake.snd\] let us motivate things in the more familiar context of contact and symplectic manifolds. These two provide the mathematical foundations of Lagrangian and Hamiltonian Mechanics. Let $(M,\eta,\xi)$ be a contact manifold where $\eta$ is a contact form on $M$ and $\xi$ is its Reeb vector field. It is easy to see that the cone $(C(M)=\bbr_+\times M,
\omega=d(t\eta))$ is symplectic. Likewise, the Reeb field defines a foliation of $M$ and the transverse space $\calz$ is also symplectic. When the foliation is regular the transverse space is a smooth symplectic manifold giving a projection $\pi$ called Boothby-Wang fibration, and $\pi^*\Omega=d\eta$ relates the contact and the symplectic structures as indicated by
\#1\#2[ ]{}
$$\rahmen{.5cm}{
\hbox{$\begin{matrix}(\calc(M),\omega)&\hookleftarrow&(M,\eta,\xi)\\
& {}&\decdnar\pi\cr {}&{}&(\calz,\Omega).\\\end{matrix}$} }$$
We do not have any Riemannian structure yet. It is quite reasonable to ask if there is a Riemannian metric $g$ on $M$ which “best fits" into the above diagram. As the preferred metrics adapted to symplectic forms are Kähler metrics one could ask for the Riemannian structure which would make the cone with the metric $\bar{g}=dt^2+t^2g$ together with the symplectic form $\omega$ into a Kähler manifold. Then $\bar{g}$ and $\omega$ define a complex structure $\bar{\Phi}$. Alternatively, one could ask for a Riemannian metric $g$ on $M$ which would define a Kähler metric $h$ on $\calz$ via a Riemannian submersion. Surprisingly, in both cases the answer to these questions leads naturally and uniquely to [**Sasakian Geometry**]{}. Our diagram becomes
\#1\#2[ ]{}
$$\rahmen{.5cm}{
\hbox{$\begin{matrix}(\calc(M),\omega,\bar{g},\bar{\Phi})&\hookleftarrow&(M,\xi,\eta,g,\Phi)\\
& {}&\decdnar\pi\cr {}&{}&(\calz,\Omega,h,J).\end{matrix}$} }$$
From this point of view it is quite clear that Kählerian and Sasakian geometries are inseparable, Sasakian Geometry being naturally [*sandwiched*]{} between two different types of Kählerian Geometry.
Cones, Holonomy, and Sasakian Geometry {#holonomy}
======================================
As we have just described Sasakian manifolds can and will be (cf. Theorem-Definition \[sas.def\]) defined as bases of metric cones which are Kähler. Let us begin with the following more general
\[conemetric\] For any Riemannian metric $g_M$ on $M,$ [**the warped product metric**]{} on $C(M)=\bbr^+\times M$ is the Riemannian metric defined by $$g=dr^2+\phi^2(r)g_M\,,$$ where $r\in\bbr^+$ and $\phi=\phi(r)$ is a smooth function, called the [**warping function**]{}. If $\phi(r)=r$ then $(C(M),g)$ is simply called the [**Riemannian cone**]{} or [**metric cone**]{} on $M.$ If $\phi(r)=\sin r$ then $(C(M),g)$ is called the [**sine-cone**]{} on $M.$
The relevance of sine-cones will become clear later while the importance of metric cones in relation to the Einstein metrics can be summarized by the following fundamental
\[einlemma\] Let $(M,g)$ be a Riemannian manifold of dimension $n,$ and consider $(C(M)=M\times \bbr^+,\bar{g})$ the cone on $M$ with metric $\bar{g}=dr^2+r^2g.$ Then if $\bar{g}$ is Einstein, it is Ricci-flat, and $\bar{g}$ is Ricci-flat if and only if $g$ is Einstein with Einstein constant $n-1.$
Interestingly, there is a similar lemma about sine-cone metrics.
\[sine-coneEinstein\] Let $(M^n,g)$ be an Einstein manifold with Einstein constant $n-1$ and consider $(C_s(M)=M\times (0,\pi),\bar{g}_s)$ the sine-cone on $M$ with metric $\bar{g}_s=dr^2+(\sin^2 r)g.$ Then $\bar{g}_s$ is Einstein with Einstein constant $n.$
It is well-known that one characterization of Kählerian geometry is via the holonomy reduction. We now recall some basic facts about the holonomy groups of irreducible Riemannian manifolds. Let $(M,g)$ be a Riemannian manifold and consider parallel translation defined by the Levi-Civita connection and its associated holonomy group which is a subgroup of the structure group $O(n,\bbr)$ ($SO(n,\bbr)$ in the oriented case). Since this connection $\nabla^g$ is uniquely associated to the metric $g$, we denote it by ${\rm Hol}(g)$, and refer to it as the [*Riemannian holonomy*]{} group or just the [*holonomy group*]{} when the context is clear. Indeed, it is precisely this Riemannian holonomy that plays an important role here. Now on a Riemannian manifold $(M,g)$ there is a canonical epimorphism $\pi_1(M)\ra{1.8} {\rm
Hol}(g)/{\rm Hol}^0(g),$ in particular, if $\pi_1(M)=0$ then ${\rm
Hol}(g)={\rm Hol}^0(g).$ In 1955 Berger proved the following theorem [@Ber55] concerning Riemannian holonomy:
\[Berger.main.thm\] Let $(M,g)$ be an oriented Riemannian manifold which is neither locally a Riemannian product nor locally symmetric. Then the restricted holonomy group ${\rm
Hol}^0(g)$ is one of the following groups listed in Table 1.1.
\[bergertable\]
[|l|l|l|l|]{}\
${\rm Hol}^0(g)$&dim$(M)$ & Geometry of $M$ & Comments\
$SO(n)$ & $n$ &orientable Riemannian & generic Riemannian\
$U(n)$ & $2n$ &Kähler & generic Kähler\
$SU(n)$&$2n$ & Calabi-Yau & Ricci-flat Kähler\
$Sp(n)\cdot Sp(1)$ &$4n$ & quaternionic Kähler & Einstein\
$Sp(n)$ &$4n$ & hyperkähler & Ricci-flat\
$G_2$ &$7$ & $G_2$-manifold & Ricci-flat\
$Spin(7)$ &$8$& $Spin(7)$-manifold & Ricci-flat\
Originally Berger’s list included $Spin(9)$, but Alekseevsky proved that any manifold with such holonomy must be symmetric [@Al1]. In the same paper Berger also claimed a classification of all holonomy groups of torsion-free affine (linear) connections that act irreducibly. He produced a list of possible holonomy representations up to what he claimed was a finite number of exceptions. But his classification had some gaps discovered 35 years later by Bryant [@Bry91]. An infinite series of exotic holonomies was found in [@CMS96] and finally the classification in the non-Riemannian affine case was completed by Merkulov and Schwachhöfer [@MeSc99]. We refer the reader to [@MeSc99] for the proof, references and the history of the general affine case. In the Riemannian case a new geometric proof of Berger’s Theorem is now available [@Olm05]. An excellent review of the subject just prior to the Merkulov and Schwachhöfer’s classification can be found in [@Bry96]. We should add that one of the first non-trivial results concerning manifolds with the exceptional holonomy groups of the last two rows of Table 1.1 is due to Bonan [@Bon66] who established Ricci-flatness of manifolds with parallel spinors.
Manifolds with reduced holonomy have always been very important in physics. Partly because Calabi-Yau, hyperkähler, quaternionic Kähler, $G_2$ and $Spin(7)$ manifolds are automatically Einstein. In addition, all of these spaces appear as $\sigma$-model geometries in various supersymmetric models. What is perhaps less known is that all of these geometries are also related, often in more than one way, to Sasakian structures of various flavors. Let us list all such known relations.
- [**$SO(n)$-holonomy.**]{} As remarked this is holonomy group of a generic metric on an oriented Riemannian manifold $(M^n,g).$ As we shall see Sasaki-Einstein metrics necessarily have maximal holonomy.
- [**$U(n)$-holonomy and Kähler geometry.**]{}
1. Metric cone on a [*Sasakian*]{} manifold is [*Kähler*]{}.
2. Transverse geometry of a [*Sasakian*]{} manifold is [*Kähler*]{}.
3. Transverse geometry of a [*positive Sasakian*]{} manifold is [*Fano*]{}.
4. Transverse geometry of a [*Sasaki-Einstein*]{} manifold is [*Fano*]{} and [*Kähler-Einstein*]{} of [*positive*]{} scalar curvature.
5. Transverse geometry of a [*negative Sasakian*]{} manifold is [*canonical*]{} in the sense that the transverse canonical bundle is ample.
6. Transverse geometry of a [*$3$-Sasakian*]{} manifold is a [*Kähler-Einstein*]{} with a [*complex contact structure*]{}, [*i.e.,*]{} [*twistor geometry*]{}.
- [**$SU(n)$-holonomy and Calabi-Yau geometry.**]{}
1. Metric cone on a [*Sasaki-Einstein*]{} manifold is [*Calabi-Yau*]{}.
2. Transverse geometry of a [*null Sasakian*]{} manifold is [*Calabi-Yau*]{}.
- [**$Sp(n)Sp(1)$-holonomy and Quaternionic Kähler geometry.**]{}
1. Transverse geometry of the 3-dimensional foliation of a [*$3$-Sasakian*]{} manifold is [*quaternionic-Kähler*]{} of [*positive*]{} scalar curvature.
2. [*$3$-Sasakian*]{} manifolds occur as conformal infinities of complete [*quaternionic Kähler*]{} manifolds of [*negative*]{} scalar curvature.
- [**$Sp(n)$-holonomy and hyperähler geometry.**]{}
1. Metric cone on a [*$3$-Sasakian*]{} manifold is [*hyperkähler*]{}.
2. Transverse geometry of a [*null Sasakian*]{} manifold with some additional structure is [*hyperkähler*]{}.
- [**$G_2$-holonomy.**]{}
1. The [*‘squashed’ twistor space of a $3$-Sasakian*]{} $7$-manifold is [*nearly Kähler*]{}; hence, the metric cone on it has holonomy inside $G_2$.
2. [*Sine-cone*]{} on a Sasaki-Einstein $5$-manifold is [*nearly Kähler*]{}; hence, its [*metric cone*]{} has holonomy inside $G_2$.
- [**$Spin(7)$-holonomy.**]{}
1. The [*‘squashed’ $3$-Sasakian*]{} $7$-manifold has a nearly parallel $G_2$-structure; hence, its metric cone has holonomy in $Spin(7)$.
2. [*Sine-cone*]{} on a [*‘squashed’ twistor space of a $3$-Sasakian*]{} $7$-manifold has a [*nearly parallel $G_2$ structure*]{}; hence, its [*metric cone*]{} has holonomy inside $Spin(7)$.
3. [*Sine cone on a sine cone*]{} on a $5$-dimensional Sasaki-Einstein base has a [*nearly parallel $G_2$-structure*]{}; hence, its [*metric cone*]{} has holonomy inside $Spin(7).$
Note that Sasakian manifolds are related to various other geometries in two very distinct ways. On one hand we can take a Sasakian (Sasaki-Einstein, $3$-Sasakian, etc.) manifold and consider its metric or sine-cone. These cones frequently have interesting geometric properties and reduced holonomy. On the other hand, a Sasakian manifold is always naturally foliated by one-dimensional leaves (three-dimensional leaves in addition to the one-dimensional canonical foliation when the manifold is $3$-Sasakian) and we can equally well consider the transverse geometries associated to such fundamental foliations. These too have remarkable geometric properties including reduced holonomy. In particular, Sasakian manifolds are not just related to all of the geometries on Berger’s holonomy list, but more importantly, they provide a [*bridge*]{} between the different geometries listed there. We will investigate some of these bridges in the next two sections.
Sasakian and Kählerian geometry {#sake.snd}
===============================
\[contactdef\] A $(2n+1)$-dimensional manifold $M$ is a [**contact manifold**]{} if there exists a $1$-form $\eta$, called a [**contact $1$-form**]{}, on $M$ such that $$\eta\wedge (d\eta)^n \neq 0$$ everywhere on $M.$ A [**contact structure**]{} on $M$ is an equivalence class of such $1$-forms, where $\eta'\sim \eta$ if there is a nowhere vanishing function $f$ on $M$ such that $\eta'=f\eta.$
On a contact manifold $(M,\eta)$ there is a unique vector field $\xi$, called the [**Reeb vector field**]{}, satisfying the two conditions $$\xi \hook \eta=1, \qquad \xi \hook d\eta =0.$$
\[almostcontactdef\] An [**almost contact structure**]{} on a differentiable manifolds $M$ is a triple $(\xi,\eta,\Phi),$ where $\Phi$ is a tensor field of type $(1,1)$ ([*i.e.*]{}, an endomorphism of $TM$), $\xi$ is a vector field, and $\eta$ is a $1$-form which satisfy $$\eta(\xi)=1 ~~~\hbox{and}~~~ \Phi\circ \Phi= -\BOne + \xi\otimes\eta,$$ where $\BOne$ is the identity endomorphism on $TM.$ A smooth manifold with such a structure is called an [**almost contact manifold**]{}.
\[contactalmostconremark\] The reader will notice from Definitions \[contactdef\] and \[almostcontactdef\] that an almost contact structure actually has more structure than a contact structure! This is in stark contrast to the usual relationship between a structure and its ‘almost structure’; however, we feel that the terminology is too well ensconced in the literature to be changed at this late stage.
Let $(M,\eta)$ be a contact manifold with a contact $1$-form $\eta$ and consider $\cald = \ker~\eta\subset TM.$ The subbundle $\cald$ is maximally non-integrable and it is called the [*contact distribution*]{}. The pair $(\cald,\gro)$, where $\gro$ is the restriction of $d\eta$ to $\cald$ gives $\cald$ the structure of a symplectic vector bundle. We denote by $\calj(\cald)$ the space of all almost complex structures $J$ on $\cald$ that are compatible with $\gro,$ that is the subspace of smooth sections $J$ of the endomorphism bundle ${\rm End}(\cald)$ that satisfy $$J^2= -\BOne, \ \ d\eta(JX,JY)=d\eta(X,Y),\ \ d\eta(JX,X)>0$$ for any smooth sections $X,Y$ of $\cald.$ Notice that each $J\in
\calj(\cald)$ defines a Riemannian metric $g_\cald$ on $\cald$ by setting $$\label{tranmetric}
g_\cald(X,Y) =d\eta(JX,Y).$$ One easily checks that $g_\cald$ satisfies the compatibility condition $g_\cald(JX,JY)=g_\cald(X,Y).$ Furthermore, the map $J\mapsto g_\cald$ is one-to-one, and the space $\calj(\cald)$ is contractible. A choice of $J$ gives $M$ an almost CR structure. Moreover, by extending $J$ to all of $TM$ one obtains an almost contact structure. There are some choices of conventions to make here. We define the section $\Phi$ of ${\rm End}(TM)$ by $\Phi =J$ on $\cald$ and $\Phi\xi=0$, where $\xi$ is the Reeb vector field associated to $\eta.$ We can also extend the transverse metric $g_\cald$ to a metric $g$ on all of $M$ by $$\label{sametric}
g(X,Y)= g_\cald +\eta(X)\eta(Y)= d\eta(\Phi X,Y)+ \eta(X)\eta(Y)$$ for all vector fields $X,Y$ on $M.$ One easily sees that $g$ satisfies the compatibility condition $g(\Phi X,\Phi
Y)=g(X,Y)-\eta(X)\eta(Y).$
A contact manifold $M$ with a contact form $\eta$, a vector field $\xi,$ a section $\Phi$ of ${\rm End}(TM),$ and a Riemannian metric $g$ which satisfy the conditions $$\eta(\xi)=1,\qquad \Phi^2=-\BOne +\xi\otimes \eta,$$ $$g(\Phi X,\Phi Y)
=g(X,Y)-\eta(X)\eta(Y)$$ is known as a [**metric contact structure**]{} on $M.$
\[sas.def\] A Riemannian manifold $(M,g)$ is called a [**Sasakian manifold**]{} if any one, hence all, of the following equivalent conditions hold:
1. There exists a Killing vector field $\xi$ of unit length on $M$ so that the tensor field $\Phi$ of type $(1,1)$, defined by $\Phi(X) ~=~ -\nabla_X \xi$, satisfies the condition $$(\nabla_X \Phi)(Y) ~=~ g(X,Y)\xi -g(\xi,Y)X$$ for any pair of vector fields $X$ and $Y$ on $M.$
2. There exists a Killing vector field $\xi$ of unit length on $M$ so that the Riemann curvature satisfies the condition $$R(X,\xi)Y ~=~ g(\xi,Y)X-g(X,Y)\xi,$$ for any pair of vector fields $X$ and $Y$ on $M.$
3. The metric cone $(\calc(M), \bar{g}) ~=~ (\bbr_+\times M, \
dr^2+r^2g)$ is Kähler.
We refer to the quadruple $\cals=(\xi,\eta,\Phi,g)$ as a [*Sasakian structure*]{} on $M$, where $\eta$ is the 1-form dual vector field $\xi.$ It is easy to see that $\eta$ is a contact form whose Reeb vector field is $\xi$. In particular $\cals=(\xi,\eta,\Phi,g)$ is a special type of [*metric contact structure*]{}.
The vector field $\xi$ is nowhere vanishing, so there is a 1-dimensional foliation $\calf_\xi$ associated with every Sasakian structure, called the [*characteristic foliation*]{}. We will denote the space of leaves of this foliation by $\calz$. Each leaf of $\calf_\xi$ has a holonomy group associated to it. The dimension of the closure of the leaves is called the [*rank*]{} of $\cals$. We shall be interested in the case $\hbox{rk}(\cals)=1$. We have
\[quasi-regular\] The characteristic foliation $\calf_\xi$ is said to be [**quasi-regular**]{} if there is a positive integer $k$ such that each point has a foliated coordinate chart $(U,x)$ such that each leaf of $\calf_\xi$ passes through $U$ at most $k$ times. Otherwise $\calf_\xi$ is called [**irregular**]{}. If $k=1$ then the foliation is called [**regular**]{}, and we use the terminology [**non-regular**]{} to mean quasi-regular, but not regular.
Let $(M,\cals)$ be a Sasakian manifold, and consider the contact subbundle $\cald=\ker~\eta.$ There is an orthogonal splitting of the tangent bundle as $$TM=\cald \oplus L_\xi,$$ where $L_\xi$ is the trivial line bundle generated by the Reeb vector field $\xi.$ The contact subbundle $\cald$ is just the normal bundle to the characteristic foliation $\calf_\xi$ generated by $\xi.$ It is naturally endowed with both a complex structure $J=\Phi|_\cald$ and a symplectic structure $d\eta.$ Hence, $(\cald,J,d\eta)$ gives $M$ a [*transverse Kähler*]{} structure with Kähler form $d\eta$ and metric $g_\cald$ defined as in (\[tranmetric\]) which is related to the Sasakian metric $g$ by $g=g_\cald \oplus \eta\otimes \eta$ as in (\[sametric\]). We have [@BG00a] the following fundamental structure theorem:
Let $(M,\xi,\eta,\Phi,g)$ be a compact quasi-regular Sasakian manifold of dimension $2n+1$, and let $\calz$ denote the space of leaves of the characteristic foliation. Then the leaf space $\calz$ is a Hodge orbifold with Kähler metric $h$ and Kähler form $\gro$ which defines an integral class $[\gro]$ in $H^2_{orb}(\calz,\bbz)$ so that $\pi:(M,g) \ra{1.3} (\calz,h)$ is an orbifold Riemannian submersion. The fibers of $\pi$ are totally geodesic submanifolds of $M$ diffeomorphic to $S^1.$
and its converse:
Let $(\calz,h)$ be a Hodge orbifold. Let $\pi:M\ra{1.3} \calz$ be the $S^1$ V-bundle whose first Chern class is $[\gro],$ and let $\eta$ be a connection $1$-form in $M$ whose curvature is $2\pi^*\gro,$ then $M$ with the metric $\pi^*h+\eta\otimes\eta$ is a Sasakian orbifold. Furthermore, if all the local uniformizing groups inject into the group of the bundle $S^1,$ the total space $M$ is a smooth Sasakian manifold.
Irregular structures can be understood by the following result of Rukimbira [@Ruk95a]:
\[Rukapproxthm\] Let $(\xi,\eta,\Phi,g)$ be a compact irregular Sasakian structure on a manifold $M.$ Then the group $\gA\gu\gt(\xi,\eta,\Phi,g)$ of Sasakian automorphisms contains a torus $T^k$ of dimension $k\geq
2.$ Furthermore, there exists a sequence $(\xi_i,\eta_i,\Phi_i,g_i)$ of quasi-regular Sasakian structures that converge to $(\xi,\eta,\Phi,g)$ in the $C^\infty$ compact-open topology.
The orbifold cohomology groups $H_{orb}^p(\calz,\bbz)$ were defined by Haefliger [@Hae84]. In analogy with the smooth case a [*Hodge orbifold*]{} is then defined to be a compact Kähler orbifold whose Kähler class lies in $H_{orb}^2(\calz,\bbz)$. Alternatively, we can develop the concept of basic cohomology which works equally well in the irregular case, but only has coefficients in $\bbr.$ It is nevertheless quite useful in trying to extend the notion of $\calz$ being Fano to both the quasi-regular and the irregular situation. This can be done in several ways. Here we will use the notion of basic Chern classes. Recall [@Ton] that a smooth p-form $\gra$ on $M$ is called [*basic*]{} if $$\xi\hook \gra=0, \qquad \call_\xi\gra=0\,,$$ and we let $\grL^p_B$ denote the sheaf of germs of basic $p$-forms on $M,$ and by $\grO_B^p$ the set of global sections of $\grL^p_B$ on $M.$ The sheaf $\grL^p_B$ is a module over the ring, $\grL^0_B,$ of germs of smooth basic functions on $M.$ We let $C^\infty_B(M)=\grO^0_B$ denote global sections of $\grL^0_B,$ i.e. the ring of smooth basic functions on $M.$ Since exterior differentiation preserves basic forms we get a de Rham complex $$\cdots\ra{2.5}\grO_B^p\fract{d}{\ra{2.5}}\grO_B^{p+1}\ra{2.5}\cdots$$ whose cohomology $H^*_B(\calf_\xi)$ is called the [*basic cohomology*]{} of $(M,\calf_\xi).$ The basic cohomology ring $H^*_B(\calf_\xi)$ is an invariant of the foliation $\calf_\xi$ and hence, of the Sasakian structure on $M.$ It is related to the ordinary de Rham cohomology $H^*(M,\bbr)$ by the long exact sequence [@Ton] $$\label{exact}
\cdots\ra{2.5}H_B^p(\calf_\xi)\ra{2.5}H^p(M,\bbr)\fract{j_p}{\ra{2.5}}
H_B^{p-1}(\calf_\xi) \fract{\grd}{\ra{2.5}}
H^{p+1}_B(\calf_\xi)\ra{2.5}\cdots$$ where $\grd$ is the connecting homomorphism given by $\grd[\gra]=[d\eta\wedge \gra]=[d\eta]\cup[\gra],$ and $j_p$ is the composition of the map induced by $\xi\hook$ with the well-known isomorphism $H^r(M,\bbr)\approx H^r(M,\bbr)^{S^1}$ where $H^r(M,\bbr)^{S^1}$ is the $S^1$-invariant cohomology defined from the $S^1$-invariant $r$-forms $\grO^r(M)^{S^1}.$ We also note that $d\eta$ is basic even though $\eta$ is not. Next we exploit the fact that the transverse geometry is Kähler. Let $\cald_\bbc$ denote the complexification of $\cald,$ and decompose it into its eigenspaces with respect to $J,$ that is, $\cald_\bbc=
\cald^{1,0}\oplus \cald^{0,1}.$ Similarly, we get a splitting of the complexification of the sheaf $\grL^1_B$ of basic one forms on $M,$ namely $$\grL^1_B\otimes \bbc = \grL^{1,0}_B\oplus \grL^{0,1}_B\,.$$ We let $\cale^{p,q}_B$ denote the sheaf of germs of basic forms of type $(p,q),$ and we obtain a splitting $$\grL^r_B\otimes \bbc = \bigoplus_{p+q=r}\cale^{p,q}_B\,.$$
The basic cohomology groups $H^{p,q}_B(\calf_\xi)$ are fundamental invariants of a Sasakian structure which enjoy many of the same properties as the ordinary Dolbeault cohomology of a Kähler structure.
Consider the complex vector bundle $\cald$ on a Sasakian manifold $(M,\xi,\eta,\Phi,g).$ As such $\cald$ has Chern classes $c_1(\cald),\ldots,c_n(\cald)$ which can be computed by choosing a connection $\nabla^\cald$ in $\cald$ [@Kobbook]. Let us choose a local foliate unitary transverse frame $(X_1,\ldots,X_n),$ and denote by $\grO^T$ the transverse curvature $2$-form with respect to this frame. A simple calculation shows that $\grO^T$ is a basic $(1,1)$-form. Since the curvature $2$-form $\grO^T$ has type $(1,1)$ it follows as in ordinary Chern-Weil theory that
\[transChernWeil\] The $k^{\rm th}$ Chern class $c_k(\cald)$ of the complex vector bundle $\cald$ is represented by the basic $(k,k)$-form $\grg_k$ determined by the formula $$\det\Bigl(\BOne_n-\frac{1}{2\pi i}\grO^T\Bigr)=1+\grg_1+\cdots +\grg_k.$$ Since $\grg_k$ is a closed basic $(k,k)$-form it represents an element in $H_B^{k,k}(\calf_\xi)\subset H_B^{2k}(\calf_\xi)$ that is called the [*basic $k^{\rm th}$ Chern class*]{} and denoted by $c_k(\calf_\xi).$
We now concentrate on the first Chern classes $c_1(\cald)$ and $c_1(\calf_\xi).$ We have
\[c1def\] A Sasakian structure $\cals=(\xi,\eta,\Phi,g)$ is said to be [**positive (negative)**]{} if $c_1(\calf_\xi)$ is represented by a positive (negative) definite $(1,1)$-form. If either of these two conditions is satisfied $\cals$ is said to be [**definite**]{}, and otherwise $\cals$ is called [**indefinite**]{}. $\cals$ is said to be [**null**]{} if $c_1(\calf_\xi)=0.$
Notice that irregular structures cannot occur for negative or null Sasakian structures, since the dimension of $\gA\gu\gt(\xi,\eta,\Phi,g)$ is greater than one. In analogy with common terminology of smooth algebraic varieties we see that a positive Sasakian structure is a [*transverse Fano structure*]{}[^2], while a null Sasakian structure is a [*transverse Calabi-Yau structure*]{}. The negative Sasakian case corresponds to the canonical bundle being ample.
Sasaki-Einstein and 3-Sasakian Geometry {#se.3se}
=======================================
A Sasakian manifold $(M,\cals)$ is [**Sasaki-Einstein**]{} if the metric $g$ is also Einstein.
For any 2n+1-dimensional Sasakian manifold ${\rm Ric}(X,\xi)=2n\eta(X)$ implying that any Sasaki-Einstein metric must have positive scalar curvature. Thus any complete Sasaki-Einstein manifold must have a finite fundamental group. Furthermore the metric cone $(\calc(M),\bar{g})= (\bbr_+\times M, \ dr^2+r^2g)$ on $M$ is Kähler Ricci-flat (Calabi-Yau).
The following theorem [@BG00a] is an orbifold version of the famous Kobayashi bundle construction of Einstein metrics on bundles over positive Kähler-Einstein manifolds [@Bes; @Kob56].
\[kob\] Let $(\calz, h)$ be a compact Fano orbifold with $\pi_1^{orb}(\calz)=0$ and Kähler-Einstein metric $h$. Let $\pi:M\ra{1.3} \calz$ be the $S^1$ V-bundle whose first Chern class is $\frac{c_1(\calz)}{\hbox{Ind}(\calz)}.$ Suppose further that the local uniformizing groups of $\calz$ inject into $S^1.$ Then with the metric $g=\pi^*h+\eta\otimes\eta$, $M$ is a compact simply connected Sasaki-Einstein manifold.
Here ${\hbox{Ind}(\calz)}$ is the [*orbifold Fano index*]{} [@BG00a] defined to be the largest positive integer such that $\frac{c_1(\calz)}{\hbox{Ind}(\calz)}$ defines a class in the orbifold cohomology group $H^2_{orb}(\calz,\bbz).$ A very special class of Sasaki-Einstein spaces is naturally related to several quaternionic geometries.
\[3s.def\] Let $(M,g)$ be a Riemannian manifold of dimension $m$. We say that $(M,g)$ is [**$3$-Sasakian**]{} if the metric cone $(\calc(M),\bar{g})=
(\bbr_+\times M, \ dr^2+r^2g)$ on $M$ is hyperkähler.
We emphasize the important observation of Kashiwada [@Kas71] that a 3-Sasakian manifold is automatically Einstein. We denote a Sasakian manifold with a $3$-Sasakian structure by $(M,\boldsymbol{\cals}),$ where $\boldsymbol{\cals}=(\cals_1,\cals_2,\cals_3)$ is a triple or a $2$-sphere of Sasakian structures $\cals_i=(\eta_i,\xi_i,\Phi_i,g).$
In the $3$-Sasakian case there is an extra structure, [*i.e.*]{}, the transverse geometry $\calo$ of the $3$-dimensional foliation which is quaternionic-Kähler. In this case, the transverse space $\calz$ is the twistor space of $\calo$ and the natural map $\calz\ \ra{1.3}\ \calo$ is the orbifold twistor fibration [@Sal82]. We get the following diagram which we denote by $\boldsymbol{\diamondsuit}(M,\boldsymbol{\cals})$ [@BGM93a; @BGM94a]:
---------------------
[**Hyperkähler**]{}
[**Geometry**]{}
---------------------
$$\begin{tabular}{|c|}\hline {\bf Twistor}\\{\bf Geometry}\\ \hline
\end{tabular}
\begin{array}{ccccc}
&&\calc(M)&&\\
&\swarrow&&\searrow&\\
\calz &&\hskip -15pt\la{4}\hskip -30pt\decdnar{}&& M\\
&\searrow& &\swarrow&\\
&&\calo&&
\end{array}
\begin{tabular}{|c|}\hline {\bf 3-Sasakian}\\{\bf Geometry}\\ \hline
\end{tabular}$$
---------------------------
[**Quaternion Kähler**]{}
[**Geometry**]{}
---------------------------
The table below summarizes properties of the cone and transverse geometries associated to various metric contact structures.
Cone Geometry of $\calc(M)$ $M$ Transverse Geometry of $\calf_\xi$
----------------------------- ------------------- ----------------------------------------
Symplectic Contact Symplectic
Kähler Sasakian Kähler
Kähler positive Sasakian Fano, $c_1(\calz)>0$
Kähler null Sasakian Calabi-Yau, $c_1(\calz)=0$
Kähler negative Sasakian ample canonical bundle, $c_1(\calz)<0$
Calabi-Yau Sasaki-Einstein Fano, Kähler-Einstein
Hyperkähler 3-Sasakian $\bbc$-contact, Fano, Kähler-Einstein
For numerous examples and constructions of and $3$-Sasakian manifolds see [@BG05]. We finish this section with a remark that both the $3$-Sasakian metric on $M$ and the twistor space metric on $\calz$ admit ‘squashings’ which are again Einstein. More generally, let $\pi:M\longrightarrow B$ be an orbifold Riemannian submersion with $g$ the Riemannian metric on $M.$ Let $\calv$ and $\calh$ denote the vertical and horizontal subbundles of the tangent bundle $TM.$ For each real number $t>0$ we construct a one parameter family $g_t$ of Riemannian metrics on $M$ by defining $$g_t|_{\calv} =tg|_{\calv}\,, \qquad g_t|_\calh =g|_\calh\,, \qquad
g_t(\calv,\calh)=0\,.$$ So for each $t>0$ we have an orbifold Riemannian submersion with the same base space. Furthermore, if the fibers of $g$ are totally geodesic, so are the fibers of $g_t\,.$ We apply the canonical variation to the orbifold Riemannian submersion $\pi:M\longrightarrow \calo$ and $\pi:\calz\longrightarrow \calo$
\[secondE.thm\] Every $3$-Sasakian manifold $M$ admits a second Einstein metric of positive scalar curvature. Furthermore, the twistor space $\calz$ also admits a second orbifold Einstein metric which is Hermitian-Einstein, but not Kähler-Einstein.
Toric Sasaki-Einstein $5$-Manifolds {#toric.SE.mfds}
===================================
Examples of manifolds are plentiful and we refer the interested reader to our monograph for a detailed exposition [@BG05]. Here we would like to consider the toric structures in dimension $5$ again referring to [@BG05] for all necessary details. Toric $5$-manifolds recently emerged from physics in the context of [*supersymmetry*]{} and the so-called AdS/CFT duality conjecture which we will discuss in the last section. It is known that, in dimension $5$, toric structures can only occur on the $k$-fold connected sums $k(S^2\times S^3)$ [@BG05]. The first inhomogeneous toric structures on $S^2\times S^3$ were constructed by Gauntlett, Martelli, Sparks, and Waldram. It follows that $S^2\times S^3$ admits infinitely many distinct quasi-regular and irregular toric structures [@GMSW04a]. Toric geometry of these examples was further explored in [@MaSp05b; @MaSpYau05; @MaSpYau06]. We will now describe a slightly different approach to a more general problem.
Consider the symplectic reduction of $\bbc^n$ (or equivalently the Sasakian reduction of $S^{2n-1}$) by a $k$-dimensional torus $T^k.$ Every complex representation of a $T^k$ on $\bbc^{n}$ can be described by an exact sequence $$0\ra{1.3}T^k\fract{f_\Omega}{\ra{1.3}} T^{n}\ra{1.3} T^{n-k}\ra{1.3}0\, .$$ The monomorphism $f_\grO$ can be represented by the diagonal matrix $$f_\Omega(\tau_1,\ldots,\tau_k)={\rm diag}\Biggl(
\displaystyle{\prod_{i=1}^k\tau_i^{a^i_1}},\ldots,
\displaystyle{\prod_{i=1}^k\tau_i^{a^i_{n}}}\Biggr)\, ,$$ where $(\tau_1,..,\tau_k)\in S^1\times\cdots\times S^1=T^k$ are the complex coordinates on $T^k,$ and $a^i_\gra\in \bbz$ are the coefficients of a $k\times n$ integral [*weight*]{} matrix $\Omega\in\calm_{k,n}(\bbz).$ We have [@BG05]
\[toric.Sasakian.quotients\] Let $X(\Omega)=(\bbc^n\setminus{\bf0})/\!\!/T^k(\Omega)$ denote the Kähler quotient of the standard flat Kähler structure on $(\bbc^n\setminus{\bf0})$ by the weighted Hamiltonian $T^k$-action with an integer weight matrix $\Omega.$ Consider the Kähler moment map $$\mu^i_\Omega(\bfz)=
\sum_{\alpha=1}^{n}a^i_{\alpha}|z_\alpha|^2,\qquad i=1,\ldots,k\,.$$ If all minor $k\times k$ determinants of $\Omega$ are non-zero then $X(\Omega)=C(Y(\Omega))$ is a cone on a compact Sasakian orbifold $Y(\Omega)$ of dimension $2(n-k)-1$ which is the Sasakian reduction of the standard Sasakian structure on $S^{2n-1}.$ In addition, the projectivization of $X(\Omega)$ defined by $\calz(\Omega)=X(\Omega)/\bbc^*$ is a Kähler reduction of the complex projective space $\bbc\bbp^{n-1}$ by a Hamiltonian $T^k$-action defined by $\Omega$ and it is the transverse space of the Sasakian structure on $Y(\Omega)$ induced by the quotient. If $$\label{CY.condition}
\sum_\alpha a^i_\alpha=0,\qquad \forall~ i=1,\ldots,k$$ then $c_1(X(\Omega))=c_1(\cald)=0.$ In particular, the orbibundle $Y(\Omega)\ra{1.2}\calz(\Omega)$ is anticanonical. Moreover, the cone $C(Y(\Omega))$, its Sasakian base $Y(\Omega)$, and the transverse space $\calz(\Omega)$ are all toric orbifolds.
\[reductproprem\] The conditions on the matrix $\grO$ that assure that $Y(\grO)$ is a smooth manifold are straightforward to work out. They involve gcd conditions on certain minor determinants of $\grO.$
This proposition is nicely summarized by the ‘reduction’ diagram $$\label{commquot4}
\begin{matrix} \bbc\bbp^{n-1} &\longleftarrow &S^{2n-1} &\longleftarrow & \bbc^n \setminus (\bf0) \\
\Downarrow & &\Downarrow & &\Downarrow \\
\calz(\grO) &\longleftarrow &Y(\grO) &\longleftarrow
&C(Y(\grO)).
\end{matrix}$$
Both the toric geometry and the topology of $Y(\Omega)$ depend on $\Omega$. Furthermore, $Y(\Omega)$ comes equipped with a family of Sasakian structures. When $n-k=3,$ assuming that $Y(\Omega)$ is simply connected (which is an additional condition on $\Omega$), we must have $m(S^2\times S^3)$ for some $m\leq k.$ We will be mostly interested in the case when $m=k.$
Gauntlett, Martelli, Sparks, and Waldram [@GMSW04a] gave an explicit construction of a metric for $\grO=(p,p,-p+q,-p-q),$ where $p$ and $q$ are relatively prime nonnegative integers with $p>q.$ (The general case for $k=1$ was treated later in [@CLPP05; @MaSp05b], see Remark \[CLPP.metrics\] below). To connect with the original notation we write $Y(\Omega)=Y_{p,q}.$ Then we get:
One can check that $Y_{1,0}$ is just the homogeneous metric on $S^2\times S^3$ which is both toric and regular. The next simplest example is $Y_{2,1}$ which, as a toric contact (Sasakian) manifold, is a circle bundle over the blow up of $\bbc\bbp^2$ at one point $F_1=\bbc\bbp^2\#\overline{\bbc\bbp^2}$ [@MaSp06]. As $F_1$ cannot admit any Kähler-Einstein metric, Kobayashi’s bundle construction cannot give a compatible Sasaki-Einstein structure. But there is a choice of a Reeb vector field in the torus which makes it possible to give $Y_{2,1}$ a metric. The structure on $Y_{2,1}$ is not quasi-regular and this was the first such example in the literature. Hence, $S^2\times S^3$ admits infinitely many toric quasi-regular structures and infinitely many toric irregular structures of rank $2.$ We have the following generalization of the $Y_{p,q}$ metrics due to [@FOW06; @CFO07]:
\[toric.SE.exist.unique\] Let $Y(\Omega)$ be as in Proposition \[toric.Sasakian.quotients\]. Then $Y(\Omega)$ admits a toric structure which is unique up to a transverse biholomorphism.
This existence of a metric is proved in [@FOW06] although the authors do not draw all the conclusions regarding possible toric manifolds that can be obtained. They give one interesting example of an irregular structure which generalizes the $Y_{2,1}$ example of [@MaSp05b] in the following sense: One considers a regular positive Sasakian structure on the anticanonical circle bundle over the del Pezzo surface $\bbc\bbp^2\#2\overline{\bbc\bbp^2}$ which gives a toric Sasakian structure on $2(S^2\times S^3).$ The regular Sasakian structure on $2(S^2\times S^3)$ cannot have any metric. However, as it is with $Y_{2,1}$ Futaki, Ono and Wang [@FOW06] show that one can deform the regular structure to a unique irregular structure. A slightly different version of the Theorem \[toric.SE.exist.unique\] is proved in [@CFO07] where uniqueness is also established. Cho, Futaki and Ono work with toric diagrams rather than with Kähler (Sasakian) quotients which amounts to the same thing by Delzant’s construction. We should add that the results of [@CFO07] apply to the toric manifolds in general dimension and not just in dimension $5$.
\[toric.SE.5man\] The manifolds $k(S^2\times S^3)$ admit infinite families of toric structures for each $k\geq1.$
As in the $k=1$ case one would expect infinitely many quasi-regular and infinitely many irregular such structures for each $\Omega$ satisfying all the condition.
\[CLPP.metrics\] The general anticanonical circle reduction was considered independently in two recent papers, [@CLPP05; @MaSp05b]. There it was shown that for $\Omega=\bfp=(p_1,p_2,-q_1,-q_2)$, with $p_i,q_i\in\bbz^+$, $p_1+p_2=q_1+q_2$, and ${\rm gcd}(p_i,q_j)=1$ for all $i,j=1,2,$ the $5$-manifold $Y(\Omega)\approx S^2\times
S^3$ admits a structure which coincides with that on $Y_{p,q}$ when $p_1=p_2=p$ and $q_1=p-q,q_2=p+q.$ In [@CLPP05] this family is denoted by $L^5(a,b,c),$ where $\bfp=(a,b,-c,-a-b+c)$ and they write the metric explicitly. However, in this case it appears to be harder (though, in principle, possible) to write down the condition under which the Reeb vector field $\xi=\xi(a,b,c)$ is quasi-regular. A priori, it is not even clear whether the quasi-regularity condition has any additional solutions beyond those obtained for the subfamily $Y_{p,q}.$ Moreover, it follows from [@CFO07] that the metrics of [@CLPP05; @MaSp05b] describe all possible toric structures on $S^2\times S^3.$
There have been similar constructions of a two-parameter family $X_{p,q}$ of toric metrics on $2(S^2\times S^3)$ [@HKW05], and another two-parameter family, called $Z_{p,q},$ on $3(S^2\times S^3)$ [@OoYa06]. All these examples, and many more, can be obtained as special cases of Theorem \[toric.SE.exist.unique\] as they are all $Y(\Omega)$ for some choice of $\Omega$. The $Y_{p,q},$ $L^5(a,b,c),$ $X_{p,q}$ and $Z_{p,q}$ metrics have received a lot of attention because of the role such manifolds play in the AdS/CFT Duality Conjecture. They created an avalanche of papers studying the properties of these metrics from the physics perspective [@ABCC06; @OoYa06; @OoYa06c; @OoYa06b; @KSY05; @HEK05; @BuZa05; @BBC05; @BFZ05; @SfZo05; @HKW05; @BLMP05; @BHK05; @BFHMS05; @Pal05; @HSY04]. The AdS/CFT duality will be discussed in the last section.
The Dirac Operator and Killing Spinors {#Diracsect}
======================================
We begin with a definition of spinor bundles and the bundle of Clifford algebras of a vector bundle [@LaMi89; @Fri00]. Recall that the [*Clifford algebra*]{} $Cl(\bbr^n)$ over $\bbr^n$ can be defined as the quotient algebra of the tensor algebra $\calt(\bbr^n)$ by the two-sided ideal $\cali$ generated by elements of the form $v\otimes v +q(v)$ where $q$ is a quadratic form on $\bbr^n.$
\[Clifford\] Let $E$ be a vector bundle with inner product $\langle\cdot,\cdot
\rangle$ on a smooth manifold $M$, and let $\calt(E)$ denote the tensor bundle over $E.$ The [**Clifford bundle**]{} of $E$ is the quotient bundle $Cl(E)=\calt(E)/\cali(E)$ where $\cali$ is the bundle of ideals (two-sided) generated pointwise by elements of the form $v\otimes v+\langle v,v\rangle$ with $v\in E_x\,.$ A [**real spinor bundle**]{} $S(E)$ of $E$ is a bundle of modules over the Clifford bundle $Cl(E).$ Similarly, a [**complex spinor bundle**]{} is a bundle of complex modules over the complexification $Cl(E)\otimes \bbc.$
As vector bundles $Cl(E)$ is isomorphic to the exterior bundle $\grL(E),$ but their algebraic structures are different. The importance of $Cl(E)$ is that it contains the spin group $Spin(n)$, the universal (double) covering group of the orthogonal group $SO(n),$ so one obtains all the representations of $Spin(n)$ by studying representations of $Cl(E).$ We assume that the vector bundle $E$ admits a spin structure, so $w_2(E)=0.$ We are interested mainly in the case when $(M,g)$ is a Riemannian spin manifold and $E=TM$ in which case we write $S(M)$ instead of $S(TM).$ The Levi-Civita connection $\nabla$ on $TM$ induces a connection, also denoted $\nabla,$ on any of the spinor bundles $S(M),$ or more appropriately on the sections $\grG(S(M)).$
\[Dirac\] Let $(M^n,g)$ be a Riemannian spin manifold and let $S(M)$ be any spinor bundle. The [**Dirac operator**]{} is the first order differential operator $D:\grG(S(M))\ra{1.8} \grG(S(M))$ defined by $$D\psi=\sum_{j=1}^n E_j\cdot \nabla_{E_j}\psi\, ,$$ where $\{E_j\}$ is a local orthonormal frame and $\cdot$ denotes Clifford multiplication.
The Dirac operator, of course originating with the famous Dirac equation describing fermions in theoretical physics, was brought into mathematics by Atiyah and Singer in [@AtSi63]. Then Lichnerowicz [@Lic63b] proved his famous result that a Riemannian spin manifold with positive scalar curvature must have vanishing $\hat{A}$-genus. An interesting question on any spin manifold is: what are the eigenvectors of the Dirac operator. In this regard the main objects of interest consists of special sections of certain spinor bundles called [*Killing spinor fields*]{} or just [*Killing spinors*]{} for short. Specifically, (cf. [@BFGK91; @Fri00])
\[real.Killing.spinors\] Let $(M,g)$ be a complete $n$-dimensional Riemannian spin manifold, and let $S(M)$ be a spin bundle (real or complex) on $M$ and $\psi$ a smooth section of $S(M).$ We say that $\psi$ is a [**Killing spinor**]{} if for every vector field $X$ there is $\gra\in \bbc,$ called [**Killing number**]{}, such that $$\nabla_X\psi=\alpha X\!\cdot\!\psi\, .$$ Here $X\!\cdot\!\psi$ denotes the Clifford product of $X$ and $\psi.$ We say that $\psi$ is [**imaginary**]{} when $\alpha\in {\rm
Im}(\bbc^*),$ $\psi$ is [**parallel**]{} if $\alpha=0$ and $\psi$ is [**real**]{}[^3] if $\alpha\in {\rm Re}(\bbc^*).$
We shall see shortly that the three possibilities for the Killing number $\gra:$ real, imaginary, or $0,$ are the only possibilities. The name Killing spinor derives from the fact that if $\psi$ is a non-trivial Killing spinor and $\gra$ is real, the vector field $$\label{KspinKvector}
X_\psi =\sum_{j=1}^n g(\psi,E_j\cdot \psi)E_j$$ is a Killing vector field for the metric $g$ (which, of course, can be zero). If $\psi$ is a Killing spinor on an $n$-dimensional spin manifold, then $$\label{KspineigenD}
D\psi= \sum_{j=1}^n E_j\cdot \nabla_{E_j}\psi =\sum_{j=1}^n\alpha
E_j\cdot E_j\cdot\!\psi= -n\gra \psi\, .$$ So Killing spinors are eigenvectors of the Dirac operator with eigenvalue $-n\gra.$ In 1980 Friedrich [@Fr80] proved the following remarkable theorem:
\[thm.fried\] Let $(M^n,g)$ be a Riemannian spin manifold which admits a non-trivial Killing spinor $\psi$ with Killing number $\gra.$ Then $(M^n,g)$ is Einstein with scalar curvature $s=4n(n-1)\gra^2.$
A proof of this is a straightforward curvature computation which can be found in either of the books [@BFGK91; @Fri00]. It also uses the fact that a non-trivial Killing spinor vanishes nowhere. It follows immediately from Theorem \[thm.fried\] that $\gra$ must be one of the three types mentioned in Definition \[real.Killing.spinors\]. So if the Killing number is real then $(M,g)$ must be a positive Einstein manifold. In particular, if $M$ is complete, then it is compact. On the other hand if the Killing number is pure imaginary, Friedrich shows that $M$ must be non-compact.
The existence of Killing spinors not only puts restrictions on the Ricci curvature, but also on both the Riemannian and the Weyl curvature operators [@BFGK91].
\[K.S.curvature\] Let $(M^n,g)$ be a Riemannian spin manifold. Let $\psi$ be a Killing spinor on $M$ with Killing number $\alpha$ and let $\calr,
\calw:\Lambda^2M\ra{1.2}\Lambda^2M$ be the Riemann and Weyl curvature operators, respectively. Then for any vector field $X$ and any $2$-form $\beta$ we have $$\begin{aligned}
&\calw(\beta)\cdot\psi=0\, ; \\
&(\nabla_X\calw)(\beta)\cdot\psi=-2\alpha\bigl(X\hook\calw(\beta)\bigr)\cdot\psi\, ; \\
&(\calr(\beta)+4\alpha^2\beta)\cdot\psi=0\, ;\\
&(\nabla_X\calr)(\beta)\cdot\psi=-2\alpha\bigl(X\hook\calr(\beta)+4\alpha^2\beta(X)\bigr)
\cdot\psi\, .\end{aligned}$$
These curvature equations can be used to prove (see [@BFGK91] or [@Fri00])
\[K.S.irreducibility\] Let $(M^n,g)$ be a connected Riemannian spin manifold admitting a non-trivial Killing spinor with $\alpha\not=0.$ Then $(M,g)$ is locally irreducible. Furthermore, if $M$ is locally symmetric, or $n\leq4,$ then $M$ is a space of constant sectional curvature equal to $4\alpha^2.$
Friedrich’s main objective in [@Fr80] was an improvement of Lichnerowicz’s estimate in [@Lic63b] for the eigenvalues of the Dirac operator. Indeed, Friedrich proves that the eigenvalues $\grl$ of the Dirac operator on any compact manifold satisfy the estimate $$\label{fried.bound}
\lambda^2\geq \frac{1}{4}\frac{ns_0}{n-1}\, ,$$ where $s_0$ is the minimum of the scalar curvature on $M.$ Thus, Killing spinors $\psi$ are eigenvectors that realize equality in equation . Friedrich also proves the converse that any eigenvector of $D$ realizing the equality must be a Killing spinor with $$\label{mineigen}
\gra =\pm \frac{1}{2}\sqrt{\frac{s_0}{n(n-1)}}\, .$$
\[Kspinsphere\][**\[Spheres\]**]{} In the case of the round sphere $(S^n,g_0)$ equality in equation is always attained. So normalizing such that $s_0=n(n-1),$ and using Bär’s Correspondence Theorem \[Barcorresthm\] below the number of corresponding real Killing spinors equals the number of constant spinors on $\bbr^{n+1}$ with the flat metric. The latter is well known (see the appendix of [@PeRi88]) to be $2^{\lfloor n/2\rfloor}$ for each of the values $\gra=\pm\frac{1}{2},$ where $\lfloor n/2\rfloor$ is the largest integer less than or equal to $n/2.$
Actually (without making the connection to Sasakian geometry) already in [@Fr80] Friedrich gives a non-spherical example of a compact $5$-manifold with a real Killing spinor: $M=SO(4)/SO(2)$ with its homogeneous Kobayashi-Tanno structure.
We now wish to relate Killing spinors to the main theme of this article, Sasakian geometry. First notice that if a Sasakian manifold $M^{2n+1}$ admits a Killing spinor, Theorem \[thm.fried\] says it must be Sasaki-Einstein, so the scalar curvature $s_0=2n(2n+1)$, and equation implies that $\gra=\pm \frac{1}{2}.$ We have the following result of Friedrich and Kath [@FrKat2]
\[FrKat.thm\] Every simply connected manifold admits non-trivial real Killing spinors. Furthermore,
1. if $M$ has dimension $4m+1$ then $(M,g)$ admits exactly one Killing spinor for each of the values $\gra=\pm \frac{1}{2},$
2. if $M$ has dimension $4m+3$ then $(M,g)$ admits at least two Killing spinors for one of the values $\gra=\pm
\frac{1}{2}.$
(Details can be found in [@FrKat2] or the book [@BFGK91].) Every simply connected manifold is known to be spin, so $M$ has a spin bundle $S(M).$ Given a fixed Sasakian structure $\cals=(\xi,\eta,\Phi,g)$ we consider two subbundles $\cale_\pm(\cals)$ of $S(M)$ defined by $$\label{Rebb.Killing}
\cale_\pm(\cals)=\{\psi\in S(M)\ \ |\ \ (\pm2\Phi X+\pounds_\xi
X)\cdot\psi=0,\ \ \ \forall X\in\Gamma(TM)\}\, .$$ Set $\nabla^\pm_X= \nabla_X \pm \frac{1}{2}X\cdot.$ A straightforward computation shows that $\nabla^\pm$ preserves the subbundles $\cale_\pm$ and defines a connection there. Moreover, by standard curvature computations it can be shown that the connection $\nabla^\pm$ is flat in $\cale_\pm(\cals).$ So it has covariantly constant sections which are precisely the Killing spinors. One then uses some representation theory of ${\rm
Spin}(2n+1)$ to compute the dimensions of $\cale_+(\cals)$ and $\cale_-(\cals)$ proving the result.
We have the following:
\[SE.locally.irr\] Let $(M,g)$ be a manifold of dimension $2m+1.$ Then $(M,g)$ is locally symmetric if and only if $(M,g)$ is of constant curvature. Moreover, ${\rm Hol}(g)=SO(2m+1)$ and $(M,g)$ is locally irreducible as a Riemannian manifold.
If necessary, go to the universal cover $\tilde M$. This is a compact simply connected manifold; hence, it admits a non-trivial Killing spinor by Theorem \[FrKat.thm\]. The first statement then follows from the Theorem \[K.S.irreducibility\]. The second statement follows from the Berger Theorem \[Berger.main.thm\]. Since $M$ has dimension $2m+1$ the only possibilities for ${\rm Hol}(g)$ are $SO(2m+1)$ and $G_2$. But the latter is Ricci flat, so it cannot be Sasaki-Einstein.
Friedrich and Kath began their investigation in dimension $5$ [@FrKa89] where they showed that a simply-connected compact $5$-manifold which admits a Killing spinor must be Sasaki-Einstein. In dimension $7$ they showed that there are exactly three possibilities: weak $G_2$-manifolds, manifolds which are not $3$-Sasakian, and $3$-Sasakian manifolds [@FrKat2]. Later Grunewald gave a description of $6$-manifolds admitting Killing spinors [@Gru90]. We should add an earlier result of Hijazi who showed that the only $8$-dimensional manifold with Killing spinors must be the round sphere [@Hij86]. By 1990 a decade of research by many people slowly identified all the ingredients of a classification of such manifolds in terms of their underlying geometric structures. The pieces of the puzzle consisting of round spheres in any dimension, manifolds in odd dimensions, nearly Kähler manifolds in dimension $6$, and weak $G_2$-holonomy manifolds in dimension $7$ were all in place with plenty of interesting examples to go around [@BFGK91]. What remained at that stage was to show that in even dimensions greater than $8$ there is nothing else but the round spheres, while in odd dimensions greater than $7$ the only such examples must be Sasaki-Einstein. The missing piece of the puzzle was finally uncovered by Bär: real Killing spinors on $M$ correspond to parallel spinors on the cone $C(M)$ [@Bar93]. A bit earlier Wang [@Wan89] had shown that on a simply connected complete Riemannian spin manifold the existence of parallel spinors corresponds to reduced holonomy. This led Bär to an elegant description of the geometry of manifolds admitting real Killing spinors (in any dimension) in terms of special holonomies of the associated cones. We refer to the correspondence between real Killing spinors on $M$ and parallel spinors on the cone $C(M)$ (equivalently reduced holonomy) as [*Bär’s correspondence*]{}. In particular, this correspondence not only answered the last remaining open questions, but also allowed for simple unified proofs of most of the theorems obtained earlier.
Real Killing Spinors, Holonomy and Bär’s Correspondence
=======================================================
As mentioned the Bär correspondence relates real Killing spinors on a compact Riemannian spin manifold $(M,g)$ to parallel spinors on the Riemannian cone $(C(M),\bar{g}).$ We now make this statement precise.
\[Barcorresthm\] Let $(M^n,g)$ be a complete Riemannian spin manifold and $(C(M^n),\bar{g})$ be its Riemannian cone. Then there is a one to one correspondence between real Killing spinors on $(M^n,g)$ with $\gra=\pm\frac{1}{2}$ and parallel spinors on $(C(M^n),\bar{g}).$
The existence of a parallel spinor on $(C(M^n),\bar{g})$ implies that $\bar{g}$ is Ricci flat by Theorem \[thm.fried\]. Then by Lemma \[einlemma\] $(M^n,g)$ is Einstein with scalar curvature $s=n(n-1).$ So any Killing spinors must have $\gra=\pm\frac{1}{2}$ by equation . As in the proof of Theorem \[FrKat.thm\], $\nabla^\pm_X= \nabla_X \pm \frac{1}{2}X\cdot$ defines a connection in the spin bundle $S(M).$ The connection $1$-forms $\gro^\pm$ of $\nabla^\pm$ are related to the connection $1$-form $\gro$ of the Levi-Civita connection by $\gro^\pm =\gro
\pm \frac{1}{2}\grb,$ where $\grb$ is a $1$-form called the [*soldering form*]{}. This can be interpreted as a connection with values in the Lie algebra $\gs\gp\gi\gn(n+1)=
\gs\gp\gi\gn(n)\oplus \bbr^n,$ and pulls back to the Levi-Civita connection in the spin bundles on the cone $(C(M^n),\bar{g})$. So parallel spinors on the cone correspond to parallel spinors on $(M,g)$ with respect to the connection $\nabla^\pm$ which correspond precisely to real Killing spinors with respect to the Levi-Civita connection.
Now we have the following definition:
\[p.q.type\] We say that a Riemannian spin manifold $(M,g)$ is of [**type $(p,q)$**]{} if it carries exactly $p$ linearly independent real Killing spinors with $\alpha>0$ and exactly $q$ linearly independent real Killing spinors with $\alpha<0$.
The following theorem has an interesting history. As mentioned above it was Bär [@Bar93] who recognized the correspondence between real Killing spinors on $(M,g)$ and parallel spinors on the Riemannian cone $(C(M),\bar{g})$. The relation between parallel spinors and reduced holonomy was anticipated in the work of Hitchin [@Hit74a] and Bonan [@Bon66], but was formalized in the 1989 paper of Wang [@Wan89]. It has also been generalized to the non-simply connected case in [@Wan95; @MoSe00].
\[Bar.main.thm1\] Let $(M^n,g)$ be a complete simply connected Riemannian spin manifold, and let ${\rm Hol}(\bar{g})$ be the holonomy group of the Riemannian cone $(C(M),\bar{g})$. Then $(M^n,g)$ admits a non-trivial real Killing spinor with $(M^n,g)$ of type $(p,q)$ if and only if $\bigl(\dim M, {\rm Hol}(\bar{g}), (p,q)\bigr)$ is one of the $6$ possible triples listed in the table below:
Here $m\geq 1,$ and $n>1.$
Since $(M,g)$ is complete and has a non-trivial real Killing spinor, it is compact by Theorem \[thm.fried\]. It then follows from a theorem of Gallot [@Gal79] that if the Riemannian cone $(C(M),\bar{g})$ has reducible holonomy it must be flat. So we can apply Berger’s Theorem \[Berger.main.thm\]. Now Wang [@Wan89] used the spinor representations of the possible irreducible holonomy groups on Berger’s list to give the correspondence between these holonomy groups and the existence of parallel spinors. First he showed that the groups listed in Table \[bergertable\] that are not on the above table do not admit parallel spinors. Then upon decomposing the spin representation of the group in question into irreducible pieces, the number of parallel spinors corresponds to the multiplicity of the trivial representation. Wang computes this in all but the first line of the table when $(C(M),\bar{g})$ is flat. In this case $(M,g)$ is a round sphere as discussed in Example \[Kspinsphere\], so the number of linearly independent constant spinors is $(2^{\lfloor
n/2\rfloor},2^{\lfloor n/2\rfloor}).$ By Bär’s Correspondence Theorem \[Barcorresthm\] real Killing spinors on $(M,g)$ correspond precisely to parallel spinors on $(C(M),\bar{g}).$ Note that the hypothesis of completeness in Wang’s theorem [@Wan89] is not necessary, so that the correspondence between the holonomy groups and parallel spinors holds equally well on Riemannian cones. However, the completeness assumption on $(M,g)$ guarantees the irreducibility of the cone $(C(M),\bar{g})$ as mentioned above.
Let us briefly discuss the types of geometry involved in each case of this theorem. As mentioned in the above proof the first line of the table corresponds to the round spheres. The next three lines correspond to geometry, so Theorem \[Bar.main.thm1\] generalizes the Friedrich-Kath Theorem \[FrKat.thm\] in this case. The last of these three lines corresponds precisely to 3-Sasakian geometry by Definition \[3s.def\]. Finally the two cases whose cones have exceptional holonomy will be discussed in more detail in Section \[weakG2sect\] below. Suffice it here to mention that it was observed by Bryant and Salamon [@BrSa89] that a cone on a nearly parallel $G_2$ manifold has its own holonomy in $Spin(7).$ It is interesting to note that Theorem \[Bar.main.thm1\] generalizes the result of Hijazi in dimension eight mentioned earlier as well as part of the last statement in Theorem \[K.S.irreducibility\], namely
\[Bar.main.thm0\] Let $(M^{2n},g)$ be a complete simply connected Riemannian spin manifold of dimension $2n$ with $n\neq 3$ admitting a non-trivial real Killing spinor. Then $M$ is isometric to the round sphere.
We end this section with a brief discussion of the non-simply connected case. Here we consider two additional cases for ${\rm
Hol}(\bar{g})$, namely $SU(2m+2)\rtimes\bbz_2$ and $Sp(2)\times
\bbz_d.$ See [@Wan95; @MoSe00] for the list of possibilities.
\[Zmod2Stiefel\] ${\rm Hol}(\bar{g})=SU(2m)\rtimes \bbz_2$. Consider the $(4m-1)$-dimensional Stiefel manifold $V_2(\bbr^{2m+1})$ with its homogeneous metric. The quotient manifold $M^{4m-1}_\grs$ of $V_2(\bbr^{2m+1})$ by the free involution $\grs$ induced from complex conjugation has an Einstein metric which is “locally Sasakian”. The cone $C(M^{4m-1}_\grs)$ is not Kähler and its holonomy is ${\rm Hol}(\bar{g})=SU(2m+2)\rtimes\bbz_2$. According to Wang [@Wan95] $C(M^{4m-1}_\grs)$ admits a spin structure with precisely one parallel spinor if and only if $m$ is even, and according to Moroianu and Semmelmann [@MoSe00] $C(M^{4m-1}_\grs)$ admits exactly two spin structures each with precisely one parallel spinor if $m$ is even. Thus, by Theorem \[Barcorresthm\] $M^{4m-1}_\grs$ admits exactly two spin structures each with exactly one Killing spinor if and only if $m$ is even.
\[3Sas->SE\] Consider a $3$-Sasakian manifold $(M^{4n-1},\boldsymbol{\cals})$ and choose a Reeb vector field $\xi(\boldsymbol{\grt}).$ Let $C_m$ be the cyclic subgroup of order $m>2$ of the circle group generated by $\xi(\boldsymbol{\grt}).$ Assume that $m$ is relatively prime to the order $\upsilon(\boldsymbol{\cals})$ of $\boldsymbol{\cals}$ and that the generic fibre of the fundamental $3$-dimensional foliation $\calf_Q$ is $SO(3),$ so that $C_m$ acts freely on $M^{4n-1}.$ This last condition on the generic fibre is easy to satisfy; for example, it holds for any of the $3$-Sasakian homogeneous spaces other than the standard round sphere, as well as the bi-quotients described in [@BGM94a]. (To handle the case when the generic fibre is $Sp(1)$ we simply need to divide $m$ by two when it is even). Since $C_m$ is not in the center of $SO(3),$ the quotient $M^{4n-1}/C_m$ is not $3$-Sasakian. However, $C_m$ does preserve the Sasakian structure determined by $\xi(\boldsymbol{\grt}),$ so $M^{4n-1}/C_m$ is Sasaki-Einstein. The cone $C(M^{4n-1}/C_m)$ has holonomy $Sp(n)\times \bbz_m$, and admits precisely $\frac{n+1}{m}$ parallel spinors if and only if $m$ divides $n+1$ [@Wan95; @MoSe00]. Thus, by Theorem \[Barcorresthm\] $M^{4n-1}/C_m$ admits precisely $\frac{n+1}{m}$ Killing spinors when $m$ divides $n+1.$
Geometries Associated with 3-Sasakian 7-manifolds {#lbspin3}
=================================================
It is most remarkable that to each $4n$-dimensional positive QK metric $(\calo,g_\calo)$ (even just locally) one can associate [*nine*]{} other Einstein metrics in dimensions $4n+k$, $k=1,2,3,4$. Alternatively, one could say that each $3$-Sasakian metric $(M,g)$ canonically defines an additional nine Einstein metrics in various dimensions. We have already encountered all of these metrics. First there are the four geometries of the diamond diagram $\boldsymbol{\diamondsuit}(M,\boldsymbol{\cals}).$ Then $M$ and $\calz$ admit additional “squashed" Einstein metrics discussed in Theorem \[secondE.thm\]. Thus we get five Einstein metrics with positive Einstein constants: $(\calo,g_\calo), (M,g),
(M',g'), (\calz, h), (\calz', h')$. Of course $M\simeq M'$ and $\calz\simeq \calz'$ as smooth manifolds (orbifolds) but they are different as Riemannian manifolds (orbifolds), hence, the notation. Let us scale all these metrics so that the Einstein constant equals the dimension of the total space minus 1. Note that any $3$-Sasakian metric already has this property. In the other four cases this is a choice of scale which is quite natural due to Lemma \[einlemma\]. However, note that this is not the scale one gets for $(\calz,h),$ and $(\calo,g_\calo)$ via the Riemannian submersion from $(M,g).$ Now, in each case one can consider its Riemannian cone which will be Ricci-flat by Lemma \[einlemma\]. We thus obtain five Ricci-flat metrics on the corresponding Riemannian cones. In addition, one can also take (iterated) sine-cone metrics defined in on the same five bases. These metrics are all Einstein of positive scalar curvature (cf. Lemma \[sine-coneEinstein\]). Let us summarize all this with the following extension of $\boldsymbol{\diamondsuit}(M,\boldsymbol{\cals})$:
$$\label{10.Einstein.metrics}
\xymatrix{&C(\calz')\ &\ar@{_{(}->}@<.2ex>[l]\ \calz'\ar[rd]&&M'\ar[ld]\ \ar@{^{(}->}@<-.2ex>[r]&\
C(M')\\
&&&\calo\ \ar@{^{(}->}@<-.2ex>[r]&\ \ C(\calo)\\
&C(\calz)\ &\ar@{_{(}->}@<.2ex>[l]\ \calz\ar[ru]&&\ar[ll]M\ar[lu]\
\ar@{^{(}->}@<-.2ex>[r]&\ C(M)}$$
There would perhaps be nothing special about all these $10$ (and many more by iterating sine-cone construction) geometries beyond what has already been discussed in the previous sections. This is indeed true when ${\rm dim}(M)>7.$ However, when ${\rm dim}(M)=7$, or, alternatively, when $\calo$ is a positive self-dual Einstein orbifold metric (more generally, just a local metric of this type) some of the metrics occurring in diagram have additional properties. We shall list all of them first. For the moment, let us assume that $(M,g)$ is a compact $3$-Sasakian $7$-manifold, then the following hold:
1. $(\calo,g_\calo)$ is a positive self-dual Einstein manifold (orbifold). We will think of it as the [*source*]{} of all the other geometries.
2. $(C(\calo),dt^2+t^2g_\calo)$ is a $5$-dimensional Ricci-flat cone with base $\calo.$
3. $(\calz,
h)$ is the orbifold twistor space of $\calo.$
4. $(\calz', h')$ is a nearly-Kähler manifold (orbifold).
5. $(M,g)$ is the $3$-Sasakian manifold.
6. $(M',g')$ is a $7$-manifold with weak $G_2$ structure.
7. $(C(\calz'),dt^2+t^2h')$ is a $7$-manifold with holonomy inside $G_2.$
8. $(C_s(\calz'),dt^2+(\sin
^2t)h')$ is a $7$-manifold with weak $G_2$ structure.
9. $(C(\calz),dt^2+t^2h)$ is a $7$-dimensional Ricci-flat cone with base $\calz.$
10. $(C(M),dt^2+t^2g)$ is hyperkähler with holonomy contained in $Sp(2).$
11. $(C(M'),dt^2+t^2g')$ has holonomy contained in $Spin(7).$
The cases (2) and (8) do not appear to have any special properties other than Ricci-flatness. The cases (1), (3), (5), and (10) are the four geometries of $\boldsymbol{\diamondsuit}(M,\boldsymbol{\cals}).$ The five remaining cases are all very interesting from the point of view of the classification of Theorem \[Bar.main.thm1\]. Indeed $\calz'$ and $C(\calz')$ are examples of the structures listed in the last row of the table while $C_2(\calz')$, $M'$ and $C(M')$ give examples of the structures listed in the fifth row. In particular, our diagram provides for a cornucopia of the orbifold examples in the first case and smooth manifolds in the latter.
Nearly Parallel $G_2$-Structures and $Spin(7)$ Holonomy Cones {#weakG2sect}
-------------------------------------------------------------
Recall, that geometrically $G_2$ is defined to be the Lie group acting on the imaginary octonions $\bbr^7$ and preserving the $3$-form $$\begin{split}
\varphi&=\alpha_1\wedge\alpha_2\wedge\alpha_3+
\alpha_1\wedge(\alpha_4\wedge\alpha_5- \alpha_6\wedge\alpha_7) \\
&+ \alpha_2\wedge(\alpha_4\wedge\alpha_6- \alpha_7\wedge\alpha_5)+
\alpha_3\wedge(\alpha_4\wedge\alpha_7-
\alpha_5\wedge\alpha_6),\label{bg2.2.1}
\end{split}$$ where $\{\alpha_i\}_{i=1}^7$ is a fixed orthonormal basis of the dual of $\bbr^7$. A $G_2$ structure on a $7$-manifold $M$ is, by definition, a reduction of the structure group of the tangent bundle to $G_2$. This is equivalent to the existence of a global 3-form $\varphi\in\Omega^3(M)$ which may be written locally as \[bg2.2.1\]. Such a 3-form defines an associated Riemannian metric, an orientation class, and a spinor field of constant length.
\[weak.G2\] Let $(M,g)$ be a complete $7$-dimensional Riemannian manifold. We say that $(M,g)$ is a [**nearly parallel**]{}[^4] $G_2$ structure if there exist a global $3$-form $\varphi\in\Omega^3(M)$ which locally can be written in terms of a local orthonormal basis as in \[bg2.2.1\], and $d\varphi=c\star \varphi$, where $\star$ is the Hodge star operator associated to $g$ and $c\neq 0$ is a constant whose sign is fixed by an orientation convention.
The case $c=0$ in Definition \[weak.G2\] is somewhat special. In particular, it is known [@Sal89] that the condition $d\varphi =0=d\star\varphi$ is equivalent to the condition that $\varphi$ be parallel, [*i.e.*]{}, $\nabla\varphi=0$ which is equivalent to the condition that the metric $g$ has holonomy group contained in $G_2.$ The following theorem provides the connection with the previous discussion on Killing spinors [@Bar93]
\[weak.G2.Spin7\] Let $(M,g)$ be a complete $7$-dimensional Riemannian manifold with a nearly parallel $G_2$ structure. Then the holonomy ${\rm Hol}(\bar{g})$ of the metric cone $(C(M),\bar{g})$ is contained in $Spin(7).$ In particular, $C(M)$ is Ricci-flat and $M$ is Einstein with positive Einstein constant $\lambda=6.$
The sphere $S^7$ with its constant curvature metric is isometric to the isotropy irreducible space $Spin(7)/G_2.$ The fact that $G_2$ leaves invariant (up to constants) a unique $3$-form and a unique $4$-form on $\bbr^7$ implies immediately that this space has a nearly parallel $G_2$ structure.
\[proper.weak.G2\] Let $(M,g)$ be a complete $7$-dimensional Riemannian manifold. We say that $g$ is a [**proper $G_2$-metric**]{} if ${\rm
Hol}(\bar{g})={\rm Spin}(7).$
We emphasize here that $G_2$ is the structure group of $M,$ not the Riemannian holonomy group. Specializing Theorem \[Bar.main.thm1\] to dimension $7$ gives the following theorem due to Friedrich and Kath [@FrKat2].
\[killing.spinor.7D\] Let $(M^7,g)$ be a complete simply-connected Riemannian spin manifold of dimension $7$ admitting a non-trivial real Killing spinor with $\alpha>0$ or $\alpha<0$. Then there are four possibilities:
1. $(M^7,g)$ is of type $(1,0)$ and it is a proper $G_2$-manifold,
2. $(M^7,g)$ is of type $(2,0)$ and it is a Sasaki-Einstein manifold, but $(M^7,g)$ is not $3$-Sasakian,
3. $(M^7,g)$ is of type $(3,0)$ and it is $3$-Sasakian,
4. $(M^7,g)=(S^7,g_{can})$ and is of type $(8,8).$
Conversely, if $(M^7,g)$ is a compact simply-connected proper $G_2$-manifold then it carries precisely one Killing spinor with $\alpha>0.$ If $(M^7,g)$ is a compact simply-connected $7$-manifold which is not $3$-Sasakian then $M$ carries precisely $2$ linearly independent Killing spinors with $\alpha>0$. Finally, if $(M^7,g)$ is a $3$-Sasakian $7$-manifold, which is not of constant curvature, then $M$ carries precisely $3$ linearly independent Killing spinors with $\alpha>0.$
\[complete.G2\] The four possibilities of the Theorem \[killing.spinor.7D\] correspond to the sequence of inclusions $${\rm Spin}(7) \supset SU(4)\supset Sp(2)\supset \BOne\, .$$ All of the corresponding cases are examples of nearly parallel $G_2$ metrics. If we exclude the trivial case when the associated cone is flat, we have three types of nearly parallel $G_2$ geometries. Following [@FKMS97] we use the number of linearly independent Killing spinors to classify these geometries, and call them type I, II, and III corresponding to cases (i), (ii), and (iii) of Theorem \[killing.spinor.7D\], respectively.
We are now ready to describe the $G_2$ geometry of the $M'\hookrightarrow C(M')$ part of Diagram \[10.Einstein.metrics\] [@GaSal96; @FKMS97]:
\[Strick.G2.from.3Sas\] Let $(M,\boldsymbol{\cals})$ be a $7$-dimensional $3$-Sasakian manifold. Then the $3$-Sasakian metric $g$ is a nearly parallel $G_2$ metric. Moreover, the second Einstein metric $g'$ given by Theorem \[secondE.thm\] and scaled so that the Einstein constant $\lambda=6$ is a nearly parallel $G_2$ metric; in fact, it is a proper $G_2$ metric.
For the second Einstein metric $g'$ we have three mutually orthonormal $1$-forms $\alpha^1=\sqrt{t}\eta^1,\quad\alpha^2=\sqrt{t}\eta^2,\quad
\alpha^3=\sqrt{t}\eta^3,$ where $t$ is the parameter of the canonical variation. Let $\{\alpha^4,\alpha^5,\alpha^6,\alpha^7\}$ be local $1$-forms spanning the annihilator of the vertical subbundle $\calv_3$ in $T^*\cals$ such that $$\hi^1= 2(\alpha^4\wedge\alpha^5-\alpha^6\wedge\alpha^7)\, ,$$ $$\hi^2= 2(\alpha^4\wedge\alpha^6-\alpha^7\wedge\alpha^5)\, ,$$ $$\hi^3= 2(\alpha^4\wedge\alpha^7-\alpha^5\wedge\alpha^6)\, .$$ Then the set $\{\gra^1,\ldots, \gra^7\}$ forms a local orthonormal coframe for the metric $g'.$ Let $$\Eta = \eta_1\wedge\eta_2\wedge \eta_3\,,\qquad
\Theta=\sum_a\eta_a\wedge \hi_a=\sum_a\eta_a\wedge d\eta_a+
6\Eta\label{btop.2.2}$$ In terms of the $3$-forms $\Eta$ and $\Theta$ we have $\varphi=\frac{1}{2}\sqrt{t}\Theta+\sqrt{t}^3\Eta.$ One easily sees that this is of the type of equation and, therefore, defines a compatible $G_2$-structure. Moreover, a straightforward computation gives $$d\varphi =\frac{1}{2}\sqrt{t}\Omega + \sqrt{t}(t+1)d\Eta,\qquad \star\varphi =
-\frac{1}{2}td\Eta - \frac{1}{24}\Omega\, .$$ Thus, $d\varphi=c\star\varphi$ is solved with $\sqrt{t}=1/\sqrt5$, and $c=-12/\sqrt5$. So $g'$ is nearly parallel. That $g'$ is a proper $G_2$ metric is due to [@FKMS97]. The idea is to use Theorem \[killing.spinor.7D\]. Looking at the four possibilities given in that theorem, we see that it suffices to show that $g'$ is not Sasaki-Einstein. The details are in [@FKMS97].
$3$-Sasakian $7$-manifolds are plentiful [@BG05]. All of them give, by Theorem \[Strick.G2.from.3Sas\], examples of type I and type III geometries. Examples of simply connected type I geometries that do not arise via Theorem \[Strick.G2.from.3Sas\] are the homogeneous Aloff-Wallach spaces $M^7_{m,n},$ $(m,n)\not=(1,1)$ which, as special cases of Eschenburg bi-quotients [@CMS96; @BFGK91], are together with an isotropy irreducible homogeneous space defined as follows: Consider the space $\calh_2$ of homogeneous polynomials of degree $2$ in three real variables $(x_1,x_2,x_3).$ As ${\rm
dim}(\calh_2)=5$ it gives rise to the embedding $SO(3)\subset
SO(5).$ We take $M=SO(5)/SO(3).$ This example was used by Bryant to get the first $8$-dimensional metric with holonomy $Spin(7)$ [@Bry87]. Examples of type II geometries (Sasaki-Einstein) are equally rich [@BG05]. In particular, there are hundreds of examples of type II nearly parallel $G_2$ metrics on each of the 28 homotopy spheres in dimension $7.$
According to [@CMS96] the Aloff-Wallach manifold $M^7_{1,1}$ has three Einstein metrics. One is the homogeneous $3$-Sasakian metric. The second is the proper $G_2$ metric of Theorem \[Strick.G2.from.3Sas\]. The third Einstein metric is also nearly parallel most likely being of type I, but we could not positively exclude type II as a possibility.
Classify all compact $7$-manifolds with nearly parallel $G_2$ structures of type I, II, or III, respectively.
The classification of type III consists of the classification of all compact $3$-Sasakian $7$-manifolds. This is probably very hard. The case of $3$-Sasakian $7$-manifolds with vanishing $\ga\gu\gt(M,\boldsymbol{\cals})$ appears quite difficult. The type II classification ($7$-dimensional manifolds which are not $3$-Sasakian) is clearly completely out of reach at the moment. A classification of proper nearly parallel $G_2$ structures on a compact manifold that do not arise via Theorem \[Strick.G2.from.3Sas\] would be very interesting and it is not clear how hard this problem really is.
\[complete.hol.Spin7\] The holonomy $Spin(7)$ cone metrics are plentiful but never complete. However, some of these metrics can be deformed to complete holonomy $Spin(7)$ ones on non compact manifolds. The first example was obtained by Bryant and Salamon who observed that the spin bundle over $S^4$ with its canonical metric carries a complete metric with holonomy $Spin(7)$ [@BrSa89]. Locally the metric was later considered also in [@GPP90]. More generally, spin orbibundles over positive QK orbifolds also carry such complete orbifold metrics as observed by Bryant and Salamon in [@BrSa89]. Other complete examples were constructed later by physicists [@CGLP02; @CGLP04; @KaYa02a; @KaYa02b]. Finally, the first compact examples were obtained in 1996 by Joyce [@Joy96b; @Joy99]. See Joyce’s book [@Joy00] for an excellent detailed exposition of the methods and the discussion of examples.
\[3S.complete.Spin7\] [**\[Complete metrics on cones\]**]{} Let $(M^7,\boldsymbol{\cals})$ be any $3$-Sasakian $7$-manifold and let $(M^7,g')$ be the associated proper nearly parallel $G_2$ squashed metric. Consider the two Riemannian cones for these metrics.
1. When does the metric cone $(C(M),dt^2+t^2g')$ admit complete holonomy $Spin(7)$ deformations?
2. When does the metric cone $(C(M),dt^2+t^2g)$ admit complete holonomy $Sp(2)$ (hyperkähler) deformations?
In other dimensions one also could ask the following more general questions:
1. Let $(M^{4n+3},\boldsymbol{\cals})$ be a compact $3$-Sasakian manifold. When does the metric cone $(C(M),dt^2+t^2g)$ admit complete hyperkähler (or just Calabi-Yau) deformations?
2. Let $(M^{2n+1},\cals)$ be a compact Sasaki-Einstein manifold. When does the metric cone $(C(M),dt^2+t^2g)$ admit complete Calabi-Yau deformations?
3. Let $(M^7,g)$ be a compact nearly parallel $G_2$-manifold. When does the metric cone $(C(M),dt^2+t^2g)$ admit complete holonomy $Spin(7)$ deformations?
4. Let $(M^6,g)$ be a compact strict nearly Kähler manifold. When does the metric cone $(C(M),dt^2+t^2g)$ admit complete holonomy $G_2$ deformations?
The metric on the spin bundle $S(S^4)$ by Bryant and Salamon is a deformation of the $Spin(7)$ holonomy metric on the cone over the squashed metric on $S^7$ [@CGLP02; @CGLP04], so there are examples of such deformations regarding question (i). Regarding (ii), we recall that every compact $3$-Sasakian $3$-manifold is isometric to $S^3/\Gamma$ and the metric cone is the flat cone $\bbc^2/\Gamma$. Hence, one could think of (ii) as a $7$-dimensional analogue of a similar problem whose complete solution was given by Kronheimer [@Kro89a]. There are non-trivial examples also in the higher dimensional cases. The metric cone on the homogeneous $3$-Sasakian manifold $\cals(1,1,1)$ of [@BGM94a] admits complete hyperkähler deformations, namely the Calabi metric on $T^*\bbc\bbp^2.$ We do not know of any other examples at the moment. In case (iv) of the Calabi-Yau cones on manifolds, however, there are many such examples. Futaki very recently proved that such a complete Calabi-Yau metric exists for all the regular toric manifolds of Section \[toric.SE.mfds\] [@Fut07]. In such cases the metric can be thought of as a complete Ricci-flat Kähler metric on the canonical bundle over a toric Fano manifold. Futaki’s result should generalize to the case of toric log Fano orbifolds.
Nearly Kähler $6$-Manifolds and $G_2$ Holonomy Cones. {#nearlyKsect}
-----------------------------------------------------
In this section we explain the geometry of the $\calz'\hookrightarrow
C(\calz')$ part of the diagram \[10.Einstein.metrics\]. Before we specialize to dimension 6 we begin with a more general introduction. Nearly Kähler manifolds were first studied by Tachibana in [@Tach59] and they appear under the name of almost Tachibana spaces in Chapter VIII of the book [@Yan65]. They were then rediscovered by Gray [@Gra70] and given the name nearly Kähler manifolds which by now is the accepted name.
\[nearly.K\] A [**nearly Kähler manifold**]{} is an almost Hermitian manifold $(M,g,J,\gro)$ such that $(\nabla_XJ)X=0$ for all tangent vectors $X$, where $\nabla$ is the Levi-Civita connection and $J$ is the almost complex structure. One says that a nearly Kähler manifold is [**strict**]{} if it is not Kähler.
This definition is equivalent to the condition $$\label{JKtensor}
(\nabla_XJ)Y+ (\nabla_YJ)X =0$$ for all vector fields $X,Y,$ which is to say that $J$ is a [*Killing tensor*]{} field. An alternative characterization of nearly Kähler manifolds is given by
\[NKcovd\] An almost Hermitian manifold $(M,g,J,\gro)$ is nearly Kähler if and only if $$\nabla \gro= \frac{1}{3}d\gro\, .$$ In particular, a strict nearly Kähler structure is never integrable.
Any nearly Kähler manifold can be locally decomposed as the product of a Kähler manifold and a strict nearly Kähler manifold. Such a decomposition is global in the simply connected case [@Nagy02a]. Hence, the study of nearly Kähler manifolds reduces to the case of strict ones. In addition every nearly Kähler manifold in dimension $4$ must be Kähler so that the first interesting dimension is six.
The following theorem establishes relationship between the twistor space $\calz\ra{1.2}\calo$ of a quaternionic Kähler manifold (orbifold) and nearly Kähler geometry.
\[NK.structure.twistor\] Let $\pi:(\calz,h)\ra{1.2}(\calo,g_\calo)$ be the twistor space of a positive QK manifold with its Kähler structure $(J,h,\omega_h).$ Then $\calz$ admits a strict nearly Kähler structure $(J_1,h_1,\omega_{h_1})$. If $TM=\calv\oplus\calh$ is the natural splitting induced by $\pi$ then $$h\!\!\mid_\calv=2h_1\!\!\mid_\calv\, ,\ \ \
h\!\!\mid_\calh=h_1\!\!\mid_\calh=\pi^*(g_\calo)\, ,\ \ $$ $$J\!\!\mid_\calv=-J_1\!\!\mid_\calv\, ,\ \ \
J\!\!\mid_\calh=J_1\!\!\mid_\calh\, .\ \ $$
Theorem \[NK.structure.twistor\] is due to Eells and Salamon [@EeSa83] when $\calo$ is $4$-dimensional. The higher dimensional analogue was established in [@AGI98] (see also [@Nagy02a]).
Observe that the metric of the nearly Kähler structure of Theorem \[NK.structure.twistor\], in general, is [*not*]{} Einstein. In particular, $h_1$ is not the squashed metric $h'$ introduced in the diagram \[10.Einstein.metrics\], unless ${\rm dim}(\calz)=6$. In six dimensions, we can scale $h_1$ so that it has scalar curvature $s=30$ and then indeed $h_1=h'$ as one can easily check.
\[3symmetric.space\] Let $M=G/H$ be a homogeneous space. We say that $M$ is [**$3$-symmetric**]{} if $G$ has an automorphism $\sigma$ of order $3$ such that $G_0^\sigma\subset H\subset G^\sigma$, where $G^\sigma$ is the fixed point set of $\sigma$ and $G_0^\sigma$ is the identity component in $G_0^\sigma$.
We have the following two theorems concerning nearly Kähler homogeneous Riemannian manifolds. The first is due to Wolf and Gray in all dimensions but six [@WoGr68a; @WoGr68b]. They also conjectured that the result is true for strict nearly Kähler $6$-manifolds. The Wolf-Gray conjecture was proved quite recently by Butruille [@But05; @But06] which is the second theorem below.
\[Gray.Wolf\] Every compact homogeneous strict nearly Kähler manifold $M$ of dimension different than $6$ is $3$-symmetric.
\[But.thm\] Let $(M,g)$ be a strict nearly Kähler $6$-dimensional Riemannian homogeneous manifold. Then $M$ is isomorphic as a homogeneous space to a finite quotient of $G/H,$ where $G$ and $H$ are one of the following:
1. $G=SU(2)\times SU(2)$ and $H=\{{\rm id}\};$
2. $G=G_2$ and $H=SU(3),$ where metrically $G/H=S^6$ the round sphere;
3. $G=Sp(2)$ and $H=SU(2)U(1),$ where $G/H=\bbc\bbp^3$ with its nearly Kähler metric determined by Theorem \[NK.structure.twistor\];
4. $G=SU(3)$ and $H=T^2$, where $G/H$ is the flag manifold with its nearly Kähler metric determined by Theorem \[NK.structure.twistor\].
Each of these manifolds carries a unique invariant nearly Kähler structure, up to homothety.
In every dimension, the only known compact examples of nearly Kähler manifolds are $3$-symmetric. On the other hand, Theorem \[NK.structure.twistor\] can be easily generalized to the case of orbifolds so that there are plenty examples of compact inhomogeneous strict nearly Kähler orbifolds in every dimension.
\[nagy.main.class\] Let $M$ be a compact simply-connected strict nearly Kähler manifold. Then, in all dimensions, as a Riemannian manifold $M$ decomposes as a product of
1. $3$-symmetric spaces,
2. twistor spaces of positive QK manifolds $\calq$ such that $\calq$ is not symmetric,
3. $6$-dimensional strict nearly Kähler manifold other than the ones listed in Theorem \[But.thm\].
This theorem is due to Nagy [@Nagy02], but our formulation uses the result of Butruille together with the fact that the twistor spaces of all symmetric positive QK manifolds are $3$-symmetric. The LeBrun-Salamon conjecture can now be phrased as follows
\[L.S.NK.conj\] Any compact simply connected strict irreducible nearly Kähler manifold $(M,g)$ of dimension greater than $6$ must be a $3$-symmetric space.
In particular, the Conjecture \[L.S.NK.conj\] is automatically true in dimensions $4n$ because of Nagy’s classification theorem and also true in dimensions $10$ and $14$ because all positive QK manifolds in dimension $8$ and $12$ are known. The third case leads to an important
\[NK.6.mfds\] Classify all compact strict nearly Kähler manifolds in dimension $6$.
Dimension six is special not just because of the rôle it plays in Theorem \[nagy.main.class\]. They have several remarkable properties which we summarize in the following theorem.
\[NK6.properties\] Let $(M,J,g,\omega_g)$ be a compact strict nearly Kähler $6$-manifold. Then
1. The metric $g$ is Einstein of positive scalar curvature.
2. $c_1(M)=0$ and $w_2(M)=0$.
3. If $g$ is scaled so it has Einstein constant $\lambda=5$ then the metric cone $(C(M), dt^2t+t^2g)$ has holonomy contained in $G_2$. In particular, $C(M)$ is Ricci-flat.
The first property is due to Matsumoto [@Mat72t] while the second is due to Gray [@Gra76]. The last part is due to Bär [@Bar93]. In fact, nearly Kähler $6$-manifolds is the geometry of the last row of the table of Theorem \[Bar.main.thm1\]. More precisely we have the following theorem proved by Grunewald [@Gru90]:
\[NK.Gru.thm\] Let $(M^6,g)$ be a complete simply connected Riemannian spin manifold of dimension $6$ admitting a non-trivial Killing spinor with $\alpha>0$ or $\alpha<0.$ Then there are two possibilities:
1. $(M,g)$ is of type $(1,1)$ and it is a strict nearly Kähler manifold,
2. $(M,g)=(S^6,g_{can})$ and is of type $(8,8).$
Conversely, if $(M,g)$ is a compact simply-connected strict nearly Kähler $6$-manifold of non-constant curvature then $M$ is of type $(1,1).$
Compact strict nearly Kähler manifolds with isometries were investigated in [@MNS05] where it was shown that
Let $(M,J,g,\omega_g)$ be a compact strict nearly Kähler $6$-manifold. If $M$ admits a unit Killing vector field, then up to finite cover $M$ is isometric to $S^3\times S^3$ with its standard nearly Kähler structure.
\[holonomy.G2.metric\] The first example of a non-trivial $G_2$ holonomy metric was found by Bryant [@Bry87], who observed that a cone on the complex flag manifold $U(3)/T^3$ carries an incomplete metric with $G_2$-holonomy. The flag ${\rm U}(3)/T^3$ is the twistor space of the complex projective plane $\bbc\bbp^2$ and as such it also has a strict nearly Kähler structure. As explained in this section, this therefore is just one possible example. One gets such non-trivial metrics also for the cones with bases $\bbc\bbp^3$ and $S^3\times S^3$ with their homogeneous strict nearly Kähler structures. Interestingly, in some cases there exist complete metrics with $G_2$ holonomy which are smooth deformations of the asymptotically conical ones. This fact was noticed by Bryant and Salamon [@BrSa89] who constructed complete examples of $G_2$ holonomy metrics on bundles of self-dual $2$-forms over $\bbc\bbp^2$ and $S^4.$ Replacing the base with any positive QK orbifold $\calo$ gives complete (in the orbifold sense) metrics on orbibundles of self-dual $2$-forms over $\calo$. Locally some of these metrics were considered in [@San03]. More complete examples of explicit $G_2$ holonomy metrics on non-compact manifolds were obtained by Salamon [@Sal04]. $G_2$ holonomy manifolds with isometric circle actions were investigated by Apostolov and Salamon [@ApSa04]. The first compact examples are due to the ground breaking work of Joyce [@Joy96a].
Geometries Associated with Sasaki-Einstein $5$-manifolds
========================================================
Like $3$-Sasakian manifolds $5$-manifolds are naturally associated to other geometries introduced in the previous section. Of course, each such space $(M^5,\cals)$ comes with its Calabi-Yau cone $(C(M),\bar{g})$ and, if the structure $\cals$ is quasi-regular, with its quotient log del Pezzo surface $(\calz,h).$ But as it turns out, there are [*two*]{} more Einstein metrics associated to $g$. The examples of this section also illustrate how the Theorem \[Bar.main.thm1\] and Bär’s correspondence break down when $(M,g)$ is a manifold with Killing spinors which is, however, [*not complete*]{}.
We begin by describing a relation between $5$-dimensional structures and six-dimensional nearly Kähler structures which was uncovered recently in [@FIMU06]. This relation involves the sine-cones of Definition \[conemetric\]. We use the notation $\bar{g}_{s}$ to distinguish the sine-cone metric from the usual Riemannian cone metric $\bar{g}$. Of course this metric is not complete, but one can compactify $M$ obtaining a very tractable stratified space $\bar{M}=N\times [0,\pi]$ with conical singularities at $t=0$ and $t=\pi.$ Observe the following simple fact which shows that the Riemannian cone on a sine cone is always a Riemannian product.
\[CsCm.lemma\] Let $(M,g)$ be a Riemannian manifold. Then the product metric $ds^2=dx^2+dy^2+y^2g$ on $\bbr\times C(M)$ can be identified with the iterated cone metric on $C(C_s(M))$.
Consider the map $\bbr^+\times(0,\pi)\ra{1.2}\bbr\times\bbr^+$ given by polar coordinate change $(r,t)\mapsto(x,y)=(r\cos t,r\sin
t),$ where $r>0$ and $t\in(0,\pi).$ We get $$ds^2=dx^2+dy^2+y^2g=dr^2+r^2dt^2+r^2\sin^2tg=dr^2+r^2(dt^2+\sin^2
tg)\, . \qedhere$$
So the iterated Riemannian cone $(C(C_s(M)),ds^2)$ has reducible holonomy $1\times {\rm Hol}(C(M)).$ This leads to
\[sine-coneEinsteinSE\] Let $(N,g)$ be a manifold of dimension $2n+1.$ Then the sine-cone $C_s(N)$ with the metric $\bar{g}_s=dr^2+(\sin^2 r)g$ is Einstein with Einstein constant $2n+1.$
We are particularly interested in the case $n=2.$ Compare Lemma \[CsCm.lemma\] with the following result in [@Joy00], Propositions 11.1.1-2:
\[Joy.G2\] Let $(M^4,g_4)$ and $(M^6,g_6)$ be Calabi-Yau manifolds. Let $(\bbr^3, ds^2=dx^2+dy^2+dz^2)$ and $(\bbr,
ds^2=dx^2)$ be the Euclidean spaces. Then
1. $(\bbr^3\times M^4,g=ds^2+g_4)$ has a natural $G_2$ structure and $g$ has holonomy ${\rm Hol}(g)\subset \BOne_3\times
SU(2)\subset G_2,$
2. $(\bbr\times M^6,g=ds^2+g_6)$ has a natural $G_2$ structure and $g$ has holonomy ${\rm Hol}(g)\subset
1\times SU(3)\subset G_2.$
As long as $(M^4,g_4)$ and $(M^6,g_6)$ are simply connected then the products $\bbr^3\times M^4$ and $\bbr\times M^6$ are simply connected $G_2$-holonomy manifolds with reducible holonomy groups and parallel Killing spinors. Note that this does not violate Theorem \[Bar.main.thm1\] as these spaces are not Riemannian cones over complete Riemannian manifolds. Using (ii) of Proposition \[Joy.G2\] we obtain the following corollary of Theorem \[sine-coneEinstein\] first obtained in [@FIMU06]
\[Se-nearlyK\] Let $(N^5,g)$ be a Sasaki-Einstein manifold. Then the sine cone $C_s(N^5)=N^5\times (0,\pi)$ with metric $\bar{g}_{s}$ is nearly Kähler of Einstein constant $\lambda=5.$ Furthermore $\bar{g}_s$ approximates pure $SU(3)$ holonomy metric near the cone points.
Using Corollary \[Se-nearlyK\] we obtain a host of examples of nearly Kähler $6$-manifolds with conical singularities by choosing $N^5$ to be any of the manifolds constructed in [@BGN03c; @BGN02b; @BG03; @Kol04; @Kol05b; @GMSW04a; @GMSW04b; @CLPP05; @FOW06; @CFO07]. For example, in this way we obtain nearly-Kähler metrics on $N\times (0,\pi)$ where $N$ is any Smale manifold with a metric such as $S^5$ or $k(S^2\times S^3),$ etc. Note that every simply connected strict nearly Kähler manifold has exactly two real Killing spinors. So as long as $N^5$ is simply connected $C_s(N^5)$ will have two real Killing spinors. Using Theorem \[sine-coneEinstein\] the metrics constructed in [@BG00a; @BGK05; @BGKT05; @GhKo05; @BG06b] in all odd dimensions also give new Einstein metrics on $C_s(N^{2n+1}).$ For example, one obtains many positive Einstein metrics on $\grS^{2n+1}\times
(0,\pi)$ where $\grS^{2n+1}$ is any odd dimensional homotopy sphere bounding a parallelizable manifold. Of course, there are no Killing spinors unless $n=2.$ Returning to the case of dimension 6, a somewhat more general converse has been obtained in [@FIMU06], namely
\[nK-Se\] Any totally geodesic hypersurface $N^5$ of a nearly Kähler $6$-manifold $M^6$ admits a structure.
The method in [@FIMU06] uses the recently developed notion of hypo $SU(2)$ structure due to Conti and Salamon [@CoSa06]. The study of sine cones appears to have originated in the physics literature [@BiMe03; @ADHL03], but in one dimension higher. Now recall the following result of Joyce (cf. [@Joy00], Propositions 13.1.2-3)
\[Joy.Spin7\] Let $(M^6,g_6)$ and $(M^7,g_7)$ be Calabi-Yau and $G_2$-holonomy manifolds, respectively. Let $(\bbr^2, ds^2=dx^2+dy^2)$ and $(\bbr, ds^2=dx^2)$ be Euclidean spaces. Then
1. $(\bbr^2\times M^6,g=ds^2+g_6)$ has a natural $Spin(7)$ structure and $g$ has holonomy ${\rm Hol}(g)\subset \BOne_2\times
SU(3)\subset Spin(7),$
2. $(\bbr\times M^7,g=ds^2+g_7)$ has a natural $Spin(7)$ structure and $g$ has holonomy ${\rm
Hol}(g)\subset 1\times G_2\subset Spin(7).$
Again, if $(M^6,g_6)$ and $(M^7,g_7)$ are simply connected so are the $Spin(7)$-manifolds $\bbr^2\times M^6$ and $\bbr\times M^7$ so that they have parallel spinors. Not surprisingly, in view of Lemma \[CsCm.lemma\] and Proposition \[Joy.Spin7\], the sine cone construction now relates strict nearly Kähler geometry in dimension $6$ to nearly parallel $G_2$ geometry in dimension $7$. More precisely [@BiMe03]
\[NK.to.G2\] Let $(N^6,g)$ be a strict nearly Kähler $6$-manifold such that $g$ has Einstein constant $\lambda_6=5$. Then the manifold $C_s(N)=N^6\times (0,\pi)$ with its sine cone metric $\bar{g}_{s}$ has a nearly parallel $G_2$ structure with Einstein constant $\lambda_7=6$ and it approximates pure $G_2$ holonomy metric near the cone points.
Just as before, starting with $(N^6,g_6)$ we consider its metric cone $C(N^6)$ with the metric $\bar{g}=dy^2+y^2g_6$ and the product metric $g_8$ on $\bbr\times C(N^6).$ With the above choice of the Einstein constant we see that $g_8=dx^2+dy^2+y^2g_6$ must have holonomy ${\rm Hol}(g_8)\subset1\times G_2\subset
Spin(7).$ By Lemma \[CsCm.lemma\] $g_8$ is a metric cone on the metric $g_7=dt^2+\sin^2 t g_6,$ which must, therefore, have weak $G_2$ holonomy and the Einstein constant $\lambda_7=6.$
Again, any simply connected weak $G_2$-manifold has at least one Killing spinor. That real Killing spinor on $C_s(N^6)$ will lift to a parallel spinor on $C(C_s(N^6))=\bbr\times C(N^6)$ which is a non-complete $Spin(7)$-manifold of holonomy inside $1\times G_2.$ One can iterate the two cases by starting with a compact $5$-manifold $N^5$ and construct either the cone on the sine cone of $N^5$ or the sine cone on the sine cone of $N^5$ to obtain a nearly parallel $G_2$ manifold. We list the Riemannian manifolds coming from this construction that are irreducible.
\[Se-NK-G2-Spin7\] Let $(N^5,g_5)$ be a compact manifold which is not of constant curvature. Then the following have irreducible holonomy groups:
1. the manifold $C(N^5)$ with the metric $g_6=dt^2+t^2g_5$ has holonomy $SU(3);$
2. the manifold $C_s(N^5)=N^5\times (0,\pi)$ with metric $g_6=dt^2 +\sin^2t~g_5$ is strict nearly Kähler;
3. the manifold $C_s(C_s(N^5))=N^5\times (0,\pi)\times (0,\pi)$ with the metric $g_7=d\gra^2+\sin^2\gra(dt^2 +\sin^2t~g_5)$ has a nearly parallel $G_2$ structure.
In addition we have the reducible cone metrics: $C(C_s(N^5))=\bbr\times C(N^5)$ has holonomy in $1\times
SU(3)\subset G_2$ and $C(C_s(C_s(N^5)))=\bbr\times
C(C_s(N^5))=\bbr\times\bbr\times C(N^5)$ has holonomy $\BOne_2\times SU(3)\subset 1\times G_2\subset Spin(7).$ If $N^5$ is simply connected then $G_5,g_6$ and $g_7$ admit two Killing spinors. For a generalization involving conformal factors see [@MoOr07].
\[incomplete.G2\] Recall Remark \[complete.G2\]. Note that when a nearly parallel $G_2$ metric is not complete then the type I-III classification is no longer valid. The group $Spin(7)$ has other subgroups than the ones listed there and we can consider the following inclusions of (reducible) holonomies $${\rm Spin}(7)
\supset G_2\times1\supset SU(3)\times\BOne_2\times \supset
SU(2)\times\BOne_3 \supset\BOne_8 \, .$$ According to the Friedrich-Kath Theorem \[killing.spinor.7D\] the middle three cannot occur as holonomies of Riemannian cones of complete $7$-manifolds with Killing spinors. But as the discussion of this section shows, they most certainly can occur as holonomy groups of Riemannian cones of incomplete nearly parallel $G_2$ metrics. These metrics can be still separated into three types depending on the holonomy reduction: say the ones that come from strict nearly Kähler manifolds are generically of type $I_s$ while the ones that come from $5$-manifolds via the iterated sine cone construction are of type $II_s$ and of type $III_s$ when $H\subset
SU(3)$ is some proper non-trivial subgroup. On the other hand, it is not clear what is the relation between the holonomy reduction and the actual number of Killing spinors one gets in each case.
Geometric Structures on Manifolds and Supersymmetry
===================================================
The intricate relationship between supersymmetry and geometric structures on manifolds was recognized along the way the physics of supersymmetry slowly evolved from its origins: first globally supersymmetric field theories (70ties) arose, later came supergravity theory (80ties), which evolved into superstring theory and conformal field theory (late 80ties and 90ties), and finally into M-Theory and the supersymmetric branes of today. At every step the “first" theory would quickly led to various generalizations creating many different new ones: so it is as if after discovering plain vanilla ice cream one would quickly find oneself in an Italian ice cream parlor confused and unable to decide which flavor was the right choice for the hot afternoon. This is a confusion that is possibly good for one’s sense of taste, but many physicists believe that there should be just one theory, the Grand Unified Theory which describes our world at any level.[^5] An interesting way out of this conundrum is to suggest that even if two theories appear to be completely different, if both are consistent and admissible, they actually [*do*]{} describe the same physical world and, therefore, [*they should be dual*]{} to one another in a certain sense. This gave rise to various duality conjectures such as the Mirror Symmetry Conjecture or the AdS/CFT Duality Conjecture.
The first observation of how supersymmetry can restrict the underlying geometry was due to Zumino [@Zum79] who discovered that globally $N=1$ supersymmetric $\sigma$-models in $d=4$ dimensions require that the bosonic fields (particles) of the theory are local coordinates on a Kähler manifold. Later Alvarez-Gaumé and Friedman observed that $N=2$ supersymmetry requires that the $\sigma$-model manifold be not just Kähler but hyperkähler [@AlFr81]. This relation between globally supersymmetric $\sigma$-models and complex manifolds was used by Lindström and Roček to discover the hyperkähler quotient construction in [@LiRo83; @HKLR].
The late seventies witnessed a series of attempts to incorporate gravity into the picture which quickly led to the discovery of various supergravity theories. Again the $N=1$ supergravity-matter couplings in $d=4$ dimensions require bosonic matter fields to be coordinates on a Kähler manifold with some special properties [@WiBa82] while $N=2$ supergravity demands that the $\sigma$-model manifold be quaternionic Kähler [@BaWi83]. The quaternionic underpinnings of the matter couplings in supergravity theories lead to the discovery of quaternionic Kähler reduction in [@Gal87a; @GaLa88].
At the same time manifolds with Killing spinors emerged as important players in the physics of the supergravity theory which in $D=11$ dimensions was first predicted by Nahm [@Nah78] and later constructed by Cremmer, Julia and Scherk [@CJS78]. The well-known Kaluza-Klein trick applied to a $D=11$ supergravity model is a way of constructing various limiting [*compactifications*]{} which would better describe the apparently four-dimensional physical world we observe. The geometry of such a compactification is simply a Cartesian product $\bbr^{3,1}\times
M^7$, where $\bbr^{3,1}$ is the Minkowski space-time (or some other Lorentzian $4$-manifold) and $M^7$ is a compact manifold with so small a radius that its presence can only be felt and observed at the quantum level. Many various models for $M^7$ were studied in the late seventies which by the eighties had already accrued into a vast physics literature (cf. the extensive three-volume monograph by Castellani, D’Auria and Fré [@CDF91]). Most of the models assumed a homogeneous space structure on $M^7=G/H$ (see Chapter V.6 in [@CDF91], for examples). Two things were of key importance in terms of the required physical properties of the compactified theory. First, the compact space $M^7$, as a Riemannian manifold, had to be Einstein of positive scalar curvature. Second, although one could consider any compact Einstein space for the compactification, the new theory would no longer be supersymmetric unless $(M^7,g)$ admitted Killing spinor fields, and the number of them would be exactly the number of [*residual*]{} supersymmetries of the compactified theory. For that reason compactification models involving $(S^7,g_0)$ were quite special as they gave the maximally supersymmetric model. However, early on it was realized that there are other, even homogeneous, $7$-manifolds of interest. The $Sp(2)$-invariant Jensen metric on $S^7$, or as physicists correctly nicknamed it, the [*squashed $7$-sphere*]{} is one of the examples. Indeed, Jensen’s metric admits exactly one Killing spinor field since it has a nearly parallel $G_2$ structure. Of course, any of the Einstein geometries in the table of Theorem \[Bar.main.thm1\] can be used to obtain such supersymmetric models.
The $D=11$ supergravity theory only briefly looked liked it was the Grand Theory of Einstein’s dream. It was soon realized that there are difficulties with getting from $D=11$ supergravity to the standard model. The theory which was to solve these and other problems was Superstring Theory and later M-Theory (which is yet to be constructed). With the arrival of superstring theory and M-theory, supersymmetry continues its truly remarkable influence on many different areas of mathematics and physics: from geometry to analysis and number theory. For instance, once again five, six, and seven-dimensional manifolds admitting real Killing spinors have become of interest because of the so called AdS/CFT Duality. Such manifolds have emerged naturally in the context of $p$-brane solutions in superstring theory. These so-called $p$-branes, “near the horizon" are modelled by the pseudo-Riemannian geometry of the product ${\rm AdS}_{p+2}\times M$, where ${\rm AdS}_{p+2}$ is the $(p+2)$-dimensional anti-de-Sitter space (a Lorentzian version of a space of constant sectional curvature) and $(M, g)$ is a Riemannian manifold of dimension $d=D-p-2$. Here $D$ is the dimension of the original supersymmetric theory. In the most interesting cases of M2-branes, M5-branes, and D3-branes $D$ equals either 11 (M$p$-branes of M-theory) or 10 (D$p$-branes in type IIA or type IIB string theory). String theorists are particularly interested in those vacua of the form ${\rm
AdS}_{p+2}\times M$ that preserve some residual supersymmetry. It turns out that this requirement imposes constraints on the geometry of the Einstein manifold $M$ which is forced to admit real Killing spinors. Depending on the dimension $d$, the possible geometries of $M$ are as follows:
d Geometry of M $(\mu, \Bar\mu)$
----- ----------------------- ----------------------
any round sphere $(1,1)$
$7$ nearly parallel $G_2$ $(\frac18,0)$
Sasaki–Einstein $(\frac14,0)$
3-Sasakian $(\frac38, 0)$
$6$ nearly Kähler $(\frac18,\frac18)$
$5$ Sasaki–Einstein $(\frac14,\tfrac14)$
where the notation $(\mu,\bar{\mu}),$ which is common in the physics literature, represents the ratio of the number of real Killing spinors of type $(p,q)$ to the maximal number of real Killing spinors that can occur in the given dimension. This maximum is, of course, realized by the round sphere of that dimension. So this table is just a translation of the table of Theorem \[Bar.main.thm1\] for the special dimensions that occur in the models used by the physicists.
Furthermore, given a $p$-brane solution of the above type, the interpolation between ${\rm AdS}_{p+2}\times M$ and $\bbr^{p,1}\times \calc(M)$ leads to a conjectured duality between the supersymmetric background of the form ${\rm AdS}_{p+2}\times M$ and a $(p+1)$-dimensional superconformal field theory of $n$ coincident $p$-branes located at the conical singularity of the $\bbr^{p,1}\times \calc(M)$ vacuum. This is a generalized version of the Maldacena or AdS/CFT Conjecture [@Mal99]. In the case of D3-branes of string theory the relevant near horizon geometry is that of ${\rm AdS}_{5}\times M$, where $M$ is a 5-manifold. The D3-brane solution interpolates between ${\rm AdS}_{5}\times M$ and $\bbr^{3,1}\times \calc(M)$, where the cone $\calc(M)$ is a Calabi-Yau threefold. In its original version the Maldacena conjecture (also known as AdS/CFT duality) states that the ’t Hooft large $n$ limit of $N=4$ supersymmetric Yang-Mills theory with gauge group $SU(n)$ is dual to type IIB superstring theory on ${\rm AdS}_{5}\times S^5$ [@Mal99]. This conjecture was further examined by Klebanov and Witten [@KlWi99] for the type IIB theory on ${\rm AdS}_{5}\times T^{1,1}$, where $T^{1,1}$ is the other homogeneous $5$-manifold $T^{1,1}=S^2\times S^3$ and the Calabi-Yau 3-fold $\calc(T^{1,1})$ is simply the quadric cone in $\bbc^4$. Using the well-known fact that $\calc(T^{1,1})$ is a Kähler quotient of $\bbc^4$ (or, equivalently, that $S^2\times
S^3$ is a Sasaki-Einstein quotient of $S^7$), a dual super Yang-Mills theory was proposed, representing D3-branes at the conical singularities. In the framework of D3-branes and the AdS/CFT duality the question of what are all the possible near horizon geometries $M$ and $\calc(M)$ might be of importance. Much of the interest in manifolds is precisely due to the fact that each such explicit metric, among other things, provides a useful model to test the AdS/CFT duality. We refer the reader interested in the mathematics and physics of the AdS/CFT duality to the recent book in the same series [@Biq05]. In particular, in this context, Sasaki-Einstein geometry is discussed in one of the articles there [@GMSW05pc].
[**\[$G_2$ holonomy manifolds unification scale and proton decay\]**]{} Until quite recently the interest in $7$-manifolds with $G_2$ holonomy as a source of possible physical models was tempered by the fact the Kaluza-Klein compactifications on smooth and complete manifolds of this type led to models with no charged particles. All this has dramatically changed in the last few years largely because of some new developments in M-theory. Perhaps the most compelling reasons for reconsidering such $7$-manifolds was offered by Atiyah and Witten who considered the dynamics on manifolds with $G_2$ holonomy which are asymptotically conical [@AtWi02]. The three models of cones on the homogeneous nearly Kähler manifolds mentioned earlier are of particular interest, but Atiyah and Witten consider other cases which include orbifold (quotient) singularities. Among other things they point to a very interesting connection between Kronheimer’s quotient construction of the ALE metrics [@Kro89a; @Kro89b] and asymptotically conical manifolds with $G_2$-holonomy. To explain the connection, consider Kronheimer’s construction for $\Gamma=\bbz_{n+1}.$ Suppose one chooses a circle $S^1_{k,l}\simeq{\rm U}(1)\in
K(\bbz_{n+1})={\rm U}(1)^n$ and then one considers a $7$-manifold obtained by performing Kronheimer’s HK quotient construction with zero momentum level $(\boldsymbol{\xi}=0)$ while “forgetting” the three moment map equations corresponding to this particular circle. An equivalent way of looking at this situation is to take the Kronheimer quotient with nonzero momentum $\boldsymbol{\xi}=a\in \gs\gp(1)$ but only for the moment map of the chosen circle $S^1_{k,l}$ (such $\boldsymbol{\xi}$ is never in the “good set”) and then consider the fibration of singular Kronheimer quotients over a $3$-dimensional base parameter space. Algebraically this corresponds to a partial resolution of the quotient singularity and this resolution depends on the choice of $S^1_{k,l}$, hence $\boldsymbol{\xi}.$ This example was first introduced in [@AtWi02]. It can be shown that the $7$-manifold is actually a cone on the complex weighted projective $3$-space with weights $(k,k,l,l),$ where $k+l=n+1.$ It then follows from the physical model considered that such a cone should admit a metric with $G_2$ holonomy. However, unlike the homogeneous cones over the four homogeneous strict nearly Kähler manifolds of Theorem \[But.thm\], the metric in this case is not known explicitly. This construction appears to differ from all previous geometric constructions of metrics with $G_2$ holonomy. One can consider similar constructions for other choices of $S^1\subset
K(\Gamma)$ [@BeBr02].
In [@FrWi03] using a specific models of M-theory compactifications on manifolds with $G_2$ holonomy, Friedman and Witten address the fundamental questions concerning the unification scale ([*i.e.*]{}, the scale at which the Standard Model of $SU(3)\times SU(2)\times U(1)$ unifies in a single gauge group) and proton decay. The authors point out that the results obtained are model dependent, but some of the calculations and conclusions apply to a variety of different models.
\[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{}
[GMSW04b]{}
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[^1]: During the preparation of this work the authors were partially supported by NSF grant DMS-0504367.
[^2]: For a more algebro-geometric approach to positivity and fundamentals on log Fano orbifolds see [@BG05].
[^3]: Here the standard terminology real and imaginary Killing spinors can be somewhat misleading. The Killing spinor $\psi$ is usually a section of a complex spinor bundle. So a real Killing spinor just means that $\gra$ is real.
[^4]: It had become customary to refer to this notion as ‘weak holonomy $G_2$’, a terminology introduced by Gray [@Gra71]. However, it was pointed out to us by the anonomous referee that this terminology is misleading due to the fact that Gray’s paper contains errors rendering the concept of weak holonomy useless as discovered by Alexandrov [@Ale05]. Hence, the term ‘nearly parallel’ used in [@FKMS97] is preferred.
[^5]: Actually, string theory of today appears to offer a rather vast range of vacua (or possible universes). Such possible predictions have been nicknamed the [*string landscape*]{} [@Sus03]. This fact has been seen as a drawback by some, but not all, physicists (see more recent discussion on [*landscape*]{} and [*swampland*]{} in [@Vaf05; @OgVa06]). The insistence that the universe we experience, and this on such a limited scale at best, [*is the only Universe*]{}, is largely a matter of ‘philosophical attitude’ towards science. See the recent book of Leonard Susskind on the anthropic principle, string theory and the cosmic landscape [@Sus05].
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