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abstract: 'We study in this paper a smoothness regularization method for functional linear regression and provide a unified treatment for both the prediction and estimation problems. By developing a tool on simultaneous diagonalization of two positive definite kernels, we obtain shaper results on the minimax rates of convergence and show that smoothness regularized estimators achieve the optimal rates of convergence for both prediction and estimation under conditions weaker than those for the functional principal components based methods developed in the literature. Despite the generality of the method of regularization, we show that the procedure is easily implementable. Numerical results are obtained to illustrate the merits of the method and to demonstrate the theoretical developments.'
address:
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Milton Stewart School of Industrial and\
Systems Engineering\
Georgia Institute of Technology\
Atlanta, Georgia 30332\
USA\
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Department of Statistics\
The Wharton School\
University of Pennsylvania\
Philadelphia, Pennsylvania 19104\
USA\
author:
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-
title: A reproducing kernel Hilbert space approach to functional linear regression
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Introduction
============
Consider the following functional linear regression model where the response $Y$ is related to a square integrable random function $X(\cdot
)$ through $$Y=\alpha_0+\int_\mathcal{T}X(t)\beta_0(t)\,dt+\varepsilon.$$ Here $\alpha_0$ is the intercept, $\mathcal{T}$ is the domain of $X(\cdot)$, $\beta_0(\cdot)$ is an unknown slope function and $\varepsilon$ is a centered noise random variable. The domain $\mathcal{T}$ is assumed to be a compact subset of an Euclidean space. Our goal is to estimate $\alpha_0$ and $\beta_0(\cdot)$ as well as to retrieve $$\label{eq:lr}
\eta_0(X):=\alpha_0+\int_\mathcal{T}X(t)\beta_0(t)\,dt$$ based on a set of training data $(x_1,y_1),\ldots,(x_n,y_n)$ consisting of $n$ independent copies of $(X,Y)$. We shall assume that the slope function $\beta_0$ resides in a reproducing kernel Hilbert space (RKHS) $\mathcal{H}$, a subspace of the collection of square integrable functions on $\mathcal{T}$.
In this paper, we investigate the method of regularization for estimating $\eta_0$, as well as $\alpha_0$ and $\beta_0$. Let $\ell_n$ be a data fit functional that measures how well $\eta$ fits the data and $J$ be a penalty functional that assesses the “plausibility” of $\eta$. The method of regularization estimates $\eta_0$ by $$\label{eq:mor}
\hat{\eta}_{n\lambda}=\mathop{\arg\min}_{\eta} [\ell_n (\eta
|{\rm
data} )+\lambda J(\eta) ],$$ where the minimization is taken over $$\biggl\{\eta\dvtx\mathcal{L}_2(\mathcal{T}) \to\mathbb{R}\big| \eta
(X)=\alpha+\int
_\mathcal{T}X\beta\dvtx
\alpha\in\mathbb{R}, \beta\in\mathcal{H}\biggr\},$$ and $\lambda\ge0$ is a tuning parameter that balances the fidelity to the data and the plausibility. Equivalently, the minimization can be taken over $(\alpha,\beta)$ instead of $\eta$ to obtain estimates for both the intercept and slope, denoted by $\hat{\alpha}_{n\lambda}$ and $\hat{\beta}_{n\lambda}$ hereafter. The most common choice of the data fit functional is the squared error $$\label{eq:ls}
\ell_n(\eta)={1\over n}\sum_{i=1}^n [y_i-\eta(x_i) ]^2.$$ In general, $\ell_n$ is chosen such that it is convex in $\eta$ and $E\ell_n(\eta)$ in uniquely minimized by $\eta_0$.
In the context of functional linear regression, the penalty functional can be conveniently defined through the slope function $\beta$ as a squared norm or semi-norm associated with $\mathcal{H}$. The canonical example of $\mathcal{H}$ is the Sobolev spaces. Without loss of generality, assume that $\mathcal{T}=[0,1]$, the Sobolev space of order $m$ is then defined as $$\begin{aligned}
\mathcal{W}_2^m([0,1])&=& \bigl\{\beta\dvtx[0,1]\to\mathbb{R}|
\beta,
\beta^{(1)},\ldots,\beta^{(m-1)} \mbox{ are absolutely}
\\
&&\hspace*{104pt}{}\mbox{continuous and } \beta^{(m)}\in\mathcal{L}_2
\bigr\}.\end{aligned}$$ There are many possible norms that can be equipped with $\mathcal
{W}_2^m$ to make it a reproducing kernel Hilbert space. For example, it can be endowed with the norm $$\label{eq:sobnorm}
\Vert\beta\Vert^2_{\mathcal{W}_2^m}=\sum_{q=0}^{m-1} \biggl(\int
\beta
^{(q)} \biggr)^2+\int\bigl(\beta^{(m)} \bigr)^2.$$ The readers are referred to Adams ([-@Adams1975]) for a thorough treatment of this subject. In this case, a possible choice of the penalty functional is given by=1 $$\label{eq:sobpen}
J(\beta)=\int_0^1 \bigl[\beta^{(m)}(t) \bigr]^2\,dt.$$=0 Another setting of particular interest is $\mathcal{T}=[0,1]^2$ which naturally occurs when $X$ represents an image. A popular choice in this setting is the thin plate spline where $J$ is given by $$J(\beta)=\int_0^1\!\!\int_0^1
\biggl[ \biggl({\partial^2 \beta\over\partial x_1^2} \biggr)^2
+2 \biggl({\partial^2 \beta\over\partial x_1\,\partial x_2} \biggr)^2
+ \biggl({\partial^2 \beta\over\partial x_2^2} \biggr)^2 \biggr]\,
dx_1\,dx_2,$$ and $(x_1,x_2)$ are the arguments of bivariate function $\beta$. Other examples of $\mathcal{T}$ include $\mathcal{T}=\{1,2,\ldots, p\}$ for some positive integer $p$, and unit sphere in an Euclidean space among others. The readers are referred to Wahba ([-@Wahba1990]) for common choices of $\mathcal{H}$ and $J$ in these as well as other contexts.
Other than the methods of regularization, a number of alternative estimators have been introduced in recent years for the functional linear regression \[James ([-@James2002]); Cardot, Ferraty and Sarda ([-@CardotFerratySarda2003]); Ramsay and Silverman ([-@RamsaySilverman2005]); Yao, Müller and Wang ([-@YaoMullerWang2005]); Ferraty and Vieu ([-@FerratyVieu2006]); Cai and Hall ([-@CaiHall2006]); Li and Hsing ([-@LiHsing2007]); Hall and Horowitz ([-@HallHorowitz2007]); Crambes, Kneip and Sarda ([-@CrambesKneipSarda2009]); Johannes ([-@Johanness2009])\]. Most of the existing methods are based upon the functional principal component analysis (FPCA). The success of these approaches hinges on the availability of a good estimate of the functional principal components for $X(\cdot)$. In contrast, the aforementioned smoothness regularized estimator avoids this task and therefore circumvents assumptions on the spacing of the eigenvalues of the covariance operator for $X(\cdot)$ as well as Fourier coefficients of $\beta_0$ with respect to the eigenfunctions, which are required by the FPCA-based approaches. Furthermore, as we shall see in the subsequent theoretical analysis, because the regularized estimator does not rely on estimating the functional principle components, stronger results on the convergence rates can be obtained.
Despite the generality of the method of regularization, we show that the estimators can be computed rather efficiently. We first derive a representer theorem in Section \[representer.sec\] which demonstrates that although the minimization with respect to $\eta$ in (\[eq:mor\]) is taken over an infinite-dimensional space, the solution can actually be found in a finite-dimensional subspace. This result makes our procedure easily implementable and enables us to take advantage of the existing techniques and algorithms for smoothing splines to compute $\hat{\eta}_{n\lambda}$, $\hat{\beta}_{n\lambda}$ and $\hat\alpha_{n\lambda}$.
We then consider in Section \[diagonal.sec\] the relationship between the eigen structures of the covariance operator for $X(\cdot)$ and the reproducing kernel of the RKHS $\mathcal{H}$. These eigen structures play prominent roles in determining the difficulty of the prediction and estimation problems in functional linear regression. We prove in Section \[diagonal.sec\] a result on simultaneous diagonalization of the reproducing kernel of the RKHS $\mathcal{H}$ and the covariance operator of $X(\cdot)$ which provides a powerful machinery for studying the minimax rates of convergence.
Section \[rate.sec\] investigates the rates of convergence of the smoothness regularized estimators. Both the minimax upper and lower bounds are established. The optimal convergence rates are derived in terms of a class of intermediate norms which provide a wide range of measures for the estimation accuracy. In particular, this approach gives a unified treatment for both the prediction of $\eta_0(X)$ and the estimation of $\beta_0$. The results show that the smoothness regularized estimators achieve the optimal rate of convergence for both prediction and estimation under conditions weaker than those for the functional principal components based methods developed in the literature.
The representer theorem makes the regularized estimators easy to implement. Several efficient algorithms are available in the literature that can be used for the numerical implementation of our procedure. Section \[numerical.sec\] presents numerical studies to illustrate the merits of the method as well as demonstrate the theoretical developments. All proofs are relegated to Section \[proof.sec\].
Representer theorem {#representer.sec}
===================
The smoothness regularized estimators $\hat{\eta}_{n\lambda}$ and $\hat\beta_{n\lambda}$ are defined as the solution to a minimization problem over an infinite-dimensional space. Before studying the properties of the estimators, we first show that the minimization is indeed well defined and easily computable thanks to a version of the so-called representer theorem.
Let the penalty functional $J$ be a squared semi-norm on $\mathcal{H}$ such that the null space $$\mathcal{H}_0:= \{\beta\in\mathcal{H}\dvtx J(\beta)=0 \}$$ is a finite-dimensional linear subspace of $\mathcal{H}$ with orthonormal basis $\{\xi_1,\ldots,\break \xi_N\}$ where $N:=\operatorname
{dim}(\mathcal{H}_0)$. Denote by $\mathcal{H}_1$ its orthogonal complement in $\mathcal{H}$ such that $\mathcal{H}=\mathcal{H}_0\oplus\mathcal{H}_1$. Similarly, for any function $f\in\mathcal{H}$, there exists a unique decomposition $f=f_0+f_1$ such that $f_0\in\mathcal{H}_0$ and $f_1\in\mathcal{H}_1$. Note $\mathcal
{H}_1$ forms a reproducing kernel Hilbert space with the inner product of $\mathcal{H}$ restricted to $\mathcal{H}_1$. Let $K(\cdot,\cdot)$ be the corresponding reproducing kernel of $\mathcal{H}_1$ such that $J(f_1)=\|f_1\|
_{K}^2=\|f_1\|
_\mathcal{H}^2$ for any $f_1\in\mathcal{H}_1$. Hereafter we use the subscript $K$ to emphasize the correspondence between the inner product and its reproducing kernel.
In what follows, we shall assume that $K$ is continuous and square integrable. Note that $K$ is also a nonnegative definite operator on $\mathcal{L}_2$. With slight abuse of notation, write $$(Kf)(\cdot)=\int_\mathcal{T}K(\cdot,s)f(s)\,ds.$$ It is known \[see, e.g., Cucker and Smale ([-@CuckerSmale2001])\] that $Kf\in\mathcal{H}_1$ for any $f\in\mathcal{L}_2$. Furthermore, for any $f\in\mathcal{H}_1$ $$\int_\mathcal{T}f(t)\beta(t)\,dt=\langle Kf, \beta\rangle
_{\mathcal{H}}.\vadjust{\goodbreak}$$ This observation allows us to prove the following result which is important to both numerical implementation of the procedure and our theoretical analysis.
\[th:rep\] Assume that $\ell_n$ depends on $\eta$ only through $\eta(x_1),\eta
(x_2),\ldots,\break\eta(x_n)$; then there exist $\mathbf
{d}=(d_1,\ldots,
d_{N})'\in
\mathbb{R}^N$ and $\mathbf{c}=(c_1,\ldots, c_n)'\in\mathbb{R}^n$ such that $$\label{eq:rep}
\hat{\beta}_{n\lambda}(t)=\sum_{k=1}^{N} d_k\xi_k(t) +\sum
_{i=1}^n c_i
(Kx_i)(t).$$
Theorem \[th:rep\] is a generalization of the well-known representer lemma for smoothing splines (Wahba, [-@Wahba1990]). It demonstrates that although the minimization with respect to $\eta$ is taken over an infinite-dimensional space, the solution can actually be found in a finite-dimensional subspace, and it suffices to evaluate the coefficients $\mathbf{c}$ and $\mathbf{d}$ in (\[eq:rep\]). Its proof follows a similar argument as that of Theorem 1.3.1 in Wahba ([-@Wahba1990]) where $\ell_n$ is assumed to be squared error, and is therefore omitted here for brevity.
Consider, for example, the squared error loss. The regularized estimator is given by $$\label{eq:pls0}
(\hat{\alpha}_{n\lambda},\hat{\beta}_{n\lambda} )
=\mathop{\arg\min}_{\alpha\in\mathbb{R}, \beta\in\mathcal{H}}
\Biggl\{{1\over n}\sum_{i=1}^n\biggl[y_i- \biggl(\alpha+\int
_\mathcal{T}x_i(t)\beta(t)\,dt \biggr) \biggr]^2+\lambda J(\beta)
\Biggr
\}.\hspace*{-30pt}$$ It is not hard to see that $$\hat{\alpha}_{n\lambda}= \bar y - \int_\mathcal{T}\bar x(t) \hat
{\beta
}_{n\lambda}(t)\,dt,$$ where $\bar{x}(t)={1\over n} \sum_{i=1}^n x_i(t)$ and $\bar
{y}={1\over
n} \sum_{i=1}^n y_i$ are the sample average of $x$ and $y$, respectively. Consequently, (\[eq:pls0\]) yields $$\hspace*{30pt}\hat{\beta}_{n\lambda}=\mathop{\arg\min}_{\beta
\in\mathcal{H}}
\Biggl\{{1\over n}\sum_{i=1}^n
\biggl[(y_i-\bar{y})-\int_\mathcal{T}\bigl(x_i(t)-\bar{x}(t)\bigr
)\beta
(t)\,dt \biggr]^2+\lambda J(\beta) \Biggr\}.$$ For the purpose of illustration, assume that $\mathcal{H}=\mathcal
{W}_2^2$ and $J(\beta)=\int(\beta'')^2$. Then $\mathcal{H}_0$ is the linear space spanned by $\xi_1(t)=1$ and $\xi_2(t)=t$. A popular reproducing kernel associated with $\mathcal{H}_1$ is $$K(s,t)={1\over(2!)^2}B_2(s)B_2(t)-{1\over4!}B_4(|s-t|),$$ where $B_m(\cdot)$ is the $m$th Bernoulli polynomial. The readers are referred to Wahba ([-@Wahba1990]) for further details. Following Theorem \[th:rep\], it suffices to consider $\beta$ of the following form: $${\beta}(t)=d_1 + d_2 t + \sum_{i=1}^n c_i \int_{\mathcal{T}}
[x_i(s)-\bar{x}(s)]K(t,s)\,ds
\label{beta.w2}$$ for some $\mathbf{d}\in\mathbb{R}^2$ and $\mathbf{c}\in\mathbb
{R}^n$. Correspondingly, $$\begin{aligned}
&&\int_\mathcal{T}[X(t)-\bar{x}(t)]\beta(t)\,dt
\\
&&\qquad=d_1 \int_\mathcal{T}[X(t)-\bar{x}(t)]\,dt + d_2\int
_\mathcal{T}
[X(t)-\bar{x}(t)]t\,dt \\
&&\qquad\quad{}+\sum_{i=1}^n c_i \int_\mathcal{T}\int_{\mathcal{T}}
[x_i(s)-\bar
{x}(s)]K(t,s)[X(t)-\bar{x}(t)]\,ds\,dt.\end{aligned}$$ Note also that for $\beta$ given in (\[beta.w2\]) $$J({\beta})=\mathbf{c}'\Sigma\mathbf{c},$$ where $\Sigma=(\Sigma_{ij})$ is a $n\times n$ matrix with $$\Sigma_{ij}=\int_\mathcal{T}\int_{\mathcal{T}} [x_i(s)-\bar
{x}(s)]K(t,s)[x_j(t)-\bar
{x}(t)]\,ds\,dt.$$ Denote by $T=(T_{ij})$ an $n\times2$ matrix whose $(i,j)$ entry is $$T_{ij}=\int[x_i(t)-\bar{x}(t)]t^{j-1}\,dt$$ for $j=1,\; 2$. Set $\mathbf{y}=(y_1,\ldots,y_n)'$. Then $$\label{eq:lsvec}
\ell_n(\eta)+\lambda J(\beta)={1\over n} \Vert\mathbf{y}-
(T\mathbf{d}+
\Sigma\mathbf{c}) \Vert^2_{\ell_2}+\lambda\mathbf{c}'\Sigma
\mathbf{c},$$ which is quadratic in $\mathbf{c}$ and $\mathbf{d}$, and the explicit form of the solution can be easily obtained for such a problem. This computational problem is similar to that behind the smoothing splines. Write $W=\Sigma+n\lambda I$; then the minimizer of (\[eq:lsvec\]) is given by $$\begin{aligned}
\mathbf{d}&=& (T'W^{-1}T )^{-1}T'W^{-1}\mathbf{y},\\
\mathbf{c}&=&W^{-1} [I-T (T'W^{-1}T )^{-1}T'W^{-1} ]\mathbf{y}.\end{aligned}$$
Simultaneous diagonalization {#diagonal.sec}
============================
Before studying the asymptotic properties of the regularized estimators $\hat{\eta}_{n\lambda}$ and $\hat\beta_{n\lambda}$, we first investigate the relationship between the eigen structures of the covariance operator for $X(\cdot)$ and the reproducing kernel of the functional space $\mathcal{H}$. As observed in earlier studies \[e.g., Cai and Hall ([-@CaiHall2006]); Hall and Horowitz ([-@HallHorowitz2007])\], eigen structures play prominent roles in determining the nature of the estimation problem in functional linear regression.
Recall that $K$ is the reproducing kernel of $\mathcal{H}_1$. Because $K$ is continuous and square integrable, it follows from Mercer’s theorem \[Riesz and Sz-Nagy ([-@RieszSznagy1955])\] that $K$ admits the following spectral decomposition: $$K(s,t)=\sum_{k=1}^\infty\rho_k \psi_k(s)\psi_k(t).$$ Here $\rho_1\ge\rho_2\ge\cdots$ are the eigenvalues of $K$, and $\{\psi_1, \psi_2, \ldots\}$ are the corresponding eigenfunctions, that is, $$K\psi_k=\rho_k \psi_k, \qquad k=1,2,\ldots.$$ Moreover, $$\langle\psi_i, \psi_j \rangle_{\mathcal{L}_2}=\delta_{ij}
\quad\mbox{and}\quad\langle\psi_i, \psi_j \rangle_{K}=\delta
_{ij}/\rho_j,$$ where $\delta_{ij}$ is the Kronecker’s delta.
Consider, for example, the univariate Sobolev space $\mathcal{W}_2^m([0,1])$ with norm (\[eq:sobnorm\]) and penalty (\[eq:sobpen\]). Observe that $$\mathcal{H}_1= \biggl\{f\in\mathcal{H}\dvtx\int f^{(k)}=0,
k=0,1,\ldots
,m-1\biggr\}.$$ It is known that \[see, e.g., Wahba ([-@Wahba1990])\] $$\label{eq:sobrk}
K(s,t)={1\over(m!)^2}B_m(s)B_m(t)+{(-1)^{m-1}\over(2m)!}B_{2m}(|s-t|).$$ Recall that $B_m$ is the $m$th Bernoulli polynomial. It is known \[see, e.g., Micchelli and Wahba ([-@MicchelliWahba1981])\] that in this case, $\rho_k\asymp
k^{-2m}$, where for two positive sequences $a_k$ and $b_k$, $a_k\asymp b_k$ means that $a_k/b_k$ is bounded away from $0$ and $\infty$ as $k \to\infty$.
Denote by $C$ the covariance operator for $X$, that is, $$C(s,t)=E \{[X(s)-E(X(s))][X(t)-E(X(t))] \}.$$ There is a duality between reproducing kernel Hilbert spaces and covariance operators \[Stein ([-@Stein1999])\]. Similarly to the reproducing kernel $K$, assuming that the covariance operator $C$ is continuous and square integrable, we also have the following spectral decomposition $$C(s,t)=\sum_{k=1}^\infty\mu_k \phi_k(s)\phi_k(t),$$ where $\mu_1\ge\mu_2\ge\cdots$ are the eigenvalues and $\{\phi
_1,\phi
_2,\ldots\}$ are the eigenfunctions such that $$C\phi_k:=\int_\mathcal{T}C(\cdot, t)\phi_k(t)\,dt=\mu_k\phi
_k,\qquad
k=1,2,\ldots.$$
The decay rate of the eigenvalues $\{\mu_k\dvtx k\ge1\}$ can be determined by the smoothness of the covariance operator $C$. More specifically, when $C$ satisfies the so-called Sacks–Ylvisaker conditions of order $s$ where $s$ is a nonnegative integer \[Sacks and Ylvisaker ([-@SacksYlvisaker1966; -@SacksYlvisaker1968; -@SacksYlvisaker1970])\], then $\mu_k\asymp k^{-2(s+1)}$. The readers are referred to the original papers by Sacks and Ylvisaker or a more recent paper by Ritter, Wasilkowski and Woźniakwski ([-@RitterWasilkowskiWozniakowski1995]) for detailed discussions of the Sacks–Ylvisaker conditions. The conditions are also stated in the for . Roughly speaking, a covariance operator $C$ is said to satisfy the Sacks–Ylvisaker conditions of order $0$ if it is twice differentiable when $s\neq t$ but not differentiable when $s=t$. A covariance operator $C$ satisfies the Sacks–Ylvisaker conditions of order $r$ for an integer $r>0$ if $\partial^{2r} C(s,t)/(\partial s^r\,\partial t^r)$ satisfies the Sacks–Ylvisaker conditions of order $0$. In this paper, we say a covariance operator $C$ satisfies the Sacks–Ylvisaker conditions if $C$ satisfies the Sacks–Ylvisaker conditions of order $r$ for some $r\ge0$. Various examples of covariance functions are known to satisfy Sacks–Ylvisaker conditions. For example, the Ornstein–Uhlenbeck covariance function $C(s,t)=\exp(-|s-t|)$ satisfies the Sacks–Ylvisaker conditions of order $0$. Ritter, Wasilkowski and Wo' zniakowski ([-@RitterWasilkowskiWozniakowski1995]) recently showed that covariance functions satisfying the Sacks–Ylvisaker conditions are also intimately related to Sobolev spaces, a fact that is useful for the purpose of simultaneously diagonalizing $K$ and $C$ as we shall see later.
Note that the two sets of eigenfunctions $\{\psi_1, \psi_2, \ldots\}$ and $\{\phi_1,\phi_2,\ldots\}$ may differ from each other. The two kernels $K$ and $C$ can, however, be simultaneously diagonalized. To avoid ambiguity, we shall assume in what follows that $Cf\neq0$ for any $f\in\mathcal
{H}_0$ and $f\neq0$. When using the squared error loss, this is also a necessary condition to ensure that $E\ell_n(\eta)$ is uniquely minimized even if $\beta$ is known to come from the finite-dimensional space $\mathcal{H}_0$. Under this assumption, we can define a norm $\|
\cdot
\|
_R$ in $\mathcal{H}$ by $$\label{eq:Rrk}
\|f\|_R^2=\langle Cf, f\rangle_{\mathcal{L}_2}+J(f)=\int_{\mathcal
{T}\times
\mathcal{T}
}f(s)C(s,t)f(t)\,ds\,dt+J(f).$$ Note that $\|\cdot\|_R$ is a norm because $\|f\|_R^2$ defined above is a quadratic form and is zero if and only if $f=0$.
The following proposition shows that when this condition holds, $\|\cdot\|_R$ is well defined on $\mathcal{H}$ and equivalent to its original norm, $\|\cdot\|_{\mathcal{H}}$, in that there exist constants $0<c_1<c_2<\infty$ such that $c_1\|f\|_R\le\|f\|_\mathcal{H}\le
c_2\|f\|_R$ for all $f\in\mathcal{H}$. In particular, $\|f\|_R<\infty$ if and only if $\|f\|_\mathcal{H}<\infty$.
\[prop:Rnorm\] If $Cf\neq0$ for any $f\in\mathcal{H}_0$ and $f\neq0$, then $\|
\cdot\|_R$ and $\|\cdot\|_{\mathcal{H}}$ are equivalent.
Let $R$ be the reproducing kernel associated with $\|\cdot\|_R$. Recall that $R$ can also be viewed as a positive operator. Denote by $\{(\rho
'_k,\psi'_k)\dvtx k\ge1\}$ the eigenvalues and eigenfunctions of $R$. Then $R$ is a linear map from $\mathcal{L}_2$ to $\mathcal{L}_2$ such that $$R\psi'_k=\int_\mathcal{T}R(\cdot, t)\psi'_k(t)\,dt=\rho'_k\psi
_k',\qquad
k=1,2,\ldots.$$ The square root of the positive definite operator can therefore be given as the linear map from $\mathcal{L}_2$ to $\mathcal{L}_2$ such that $$R^{1/2}\psi'_k=(\rho'_k)^{1/2}\psi_k',\qquad k=1,2,\ldots.\vadjust{\goodbreak}$$ Let $\nu_1\ge
\nu_2\ge\cdots$ be the eigenvalues of the bounded linear operator $R^{1/2}CR^{1/2}$ and $\{\zeta_k\dvtx k=1,2, \ldots\}$ be the corresponding orthogonal eigenfunctions in $\mathcal{L}_2$. Write $\omega_k=\nu
^{-1/2}_kR^{1/2}\zeta_k$, $k=1,2,\ldots.$ Also let $\langle\cdot
,\cdot
\rangle_R$ be the inner product associated with $\|\cdot\|_R$, that is, for any $f,g\in\mathcal{H}$, $$\langle f, g\rangle_R =\tfrac14 (\|f+g\|_{R}^2-\|f-g\|_{R}^2 ).$$ It is not hard to see that $$\hspace*{30pt}\langle\omega_j,\omega_k\rangle_R =\nu_j^{-1/2}\nu
_k^{-1/2}\langle
R^{1/2}\zeta_j,R^{1/2}\zeta_k\rangle_R=\nu_k^{-1}\langle\zeta
_j,\zeta
_k\rangle_{\mathcal{L}_2}=\nu_k^{-1}\delta_{jk},$$ and $$\begin{aligned}
\langle C^{1/2}\omega_j,C^{1/2}\omega_k\rangle_{\mathcal
{L}_2}&=&\nu
_j^{-1/2}\nu_k^{-1/2}\langle C^{1/2}R^{1/2}\zeta
_j,C^{1/2}R^{1/2}\zeta
_k\rangle_{\mathcal{L}_2}\\
&=&\nu_j^{-1/2}\nu_k^{-1/2}\langle R^{1/2}CR^{1/2}\zeta_j,\zeta
_k\rangle
_{\mathcal{L}_2}\\
&=&\delta_{jk}.\end{aligned}$$ The following theorem shows that quadratic forms $\|f\|_R^2=\langle
f,f\rangle_R$ and $\langle Cf,\break f\rangle_{\mathcal{L}_2}$ can be simultaneously diagonalized on the basis of $\{\omega_k: k\ge1\}$.
\[th:simdiag\] For any $f\in\mathcal{H}$, $$f=\sum_{k=1}^\infty f_k\omega_k,$$ in the absolute sense where $f_k=\nu_k\langle f, \omega_k\rangle_R$. Furthermore, if $\gamma_k=(\nu_k^{-1}-1)^{-1}$, then $$\langle f,f\rangle_R = \sum_{k=1}^\infty(1+\gamma_k^{-1}
)f_k^2\quad\mbox{and}\quad\langle Cf,f\rangle_{\mathcal{L}_2} =
\sum
_{k=1}^\infty f_k^2.$$ Consequently, $$J(f)=\langle f,f\rangle_R-\langle Cf,f\rangle_{\mathcal{L}_2}=\sum
_{k=1}^\infty\gamma_k^{-1}f_k^2.$$
Note that $\{(\gamma_k,\omega_k)\dvtx k\ge1\}$ can be determined jointly by $\{(\rho_k,\psi_k)\dvtx k\ge1\}$ and $\{(\mu_k,\phi_k)\dvtx
k\ge1\}$. However, in general, neither $\gamma_k$ nor $\omega_k$ can be given in explicit form of $\{(\rho_k,\psi_k)\dvtx k\ge1\}$ and $\{(\mu_k,\phi_k)\dvtx
k\ge1\}$. One notable exception is the case when the operators $C$ and $K$ are commutable. In particular, the setting $\psi_k=\phi_k$, $k=1,2,\ldots,$ is commonly adopted when studying FPCA-based approaches \[see, e.g., Cai and Hall ([-@CaiHall2006]); Hall and Horowitz ([-@HallHorowitz2007])\].
\[prop:simdiag\] Assume that $\psi_k=\phi_k$, $k=1,2,\ldots,$ then $\gamma_k=\rho
_k\mu
_k$ and $\omega_k=\mu_k^{-1/2}\psi_k$.
In general, when $\psi_k$ and $\phi_k$ differ, such a relationship no longer holds. The following theorem reveals that similar asymptotic behavior of $\gamma_k$ can still be expected in many practical settings.
\[th:eigen\] Consider the one-dimensional case when $\mathcal{T}=[0,1]$. If $\mathcal{H}$ is the Sobolev space $\mathcal{W}^m_2([0,1])$ endowed with norm (\[eq:sobnorm\]), and $C$ satisfies the Sacks–Ylvisaker conditions, then $\gamma
_k\asymp
\mu_k\rho_k$.
Theorem \[th:eigen\] shows that under fairly general conditions $\gamma_k\asymp\mu_k\rho_k$. In this case, there is little difference between the general situation and the special case when $K$ and $C$ share a common set of eigenfunctions when working with the system $\{
(\gamma_k,\omega_k), k=1,2,\ldots\}$. This observation is crucial for our theoretical development in the next section.
Convergence rates {#rate.sec}
=================
We now turn to the asymptotic properties of the smoothness regularized estimators. To fix ideas, in what follows, we shall focus on the squared error loss. Recall that in this case $$\hspace*{25pt} (\hat{\alpha}_{n\lambda},\hat{\beta}_{n\lambda}
)=\mathop{\arg\min}
_{\alpha\in\mathbb{R}, \beta\in\mathcal{H}}
\Biggl\{{1\over n}\sum_{i=1}^n
\biggl[y_i- \biggl(\alpha+\int_\mathcal{T}x_i(t)\beta(t)\,dt
\biggr)
\biggr]^2+\lambda
J(\beta) \Biggr\}.$$ As shown before, the slope function can be equivalently defined as $$\hspace*{25pt}\hat{\beta}_{n\lambda}=\mathop{\arg\min}_{\beta
\in\mathcal{H}}
\Biggl\{{1\over n}\sum
_{i=1}^n \biggl[(y_i-\bar{y})-\int_\mathcal{T}\bigl(x_i(t)-\bar
{x}(t)\bigr)\beta
(t)\,dt \biggr]^2+\lambda J(\beta) \Biggr\},$$ and once $\hat{\beta}_{n\lambda}$ is computed, $\hat{\alpha
}_{n\lambda
}$ is given by $$\hat{\alpha}_{n\lambda}=\bar{y} - \int_\mathcal{T}\bar{x}(t)\hat
{\beta
}_{n\lambda}(t)\,dt.$$ In light of this fact, we shall focus our attention on $\hat{\beta
}_{n\lambda}$ in the following discussion for brevity. We shall also assume that the eigenvalues of the reproducing kernel $K$ satisfies $\rho_k\asymp k^{-2r}$ for some $r>1/2$. Let $\mathcal{F}(s,M,K)$ be the collection of the distributions $F$ of the process $X$ that satisfy the following conditions:
(a) The eigenvalues $\mu_k$ of its covariance operator $C(\cdot,\cdot)$ satisfy $\mu_k\asymp k^{-2s}$ for some $s>1/2$.
(b) For any function $f\in\mathcal{L}_2(\mathcal{T})$, $$\begin{aligned}
&&E \biggl(\int_\mathcal{T}f(t)[X(t)-E(X)(t)]\,dt \biggr
)^4\nonumber
\\[-8pt]\\[-8pt]
&&\qquad\le M \biggl[E \biggl(\int_\mathcal{T}f(t)[X(t)-E(X)(t)]\,dt
\biggr)^2 \biggr]^2.\nonumber\end{aligned}$$
(c) When simultaneously diagonalizing $K$ and $C$, $\gamma
_k\asymp\rho_k\mu_k$, where $\nu_k=(1+\gamma_k^{-1})^{-1}$ is the $k$th largest eigenvalue of $R^{1/2}CR^{1/2}$ where $R$ is the reproducing kernel associated with $\|\cdot\|_R$ defined by (\[eq:Rrk\]).
The first condition specifies the smoothness of the sample path of $X(\cdot)$. The second condition concerns the fourth moment of a linear functional of $X(\cdot)$. This condition is satisfied with $M=3$ for a Gaussian process because $\int f(t)X(t)\,dt$ is normally distributed. In the light of Theorem \[th:eigen\], the last condition is satisfied by any covariance function that satisfies the Sacks–Ylvisaker conditions if $\mathcal{H}$ is taken to be $\mathcal{W}_2^m$ with norm (\[eq:sobnorm\]). It is also trivially satisfied if the eigenfunctions of the covariance operator $C$ coincide with those of $K$.
Optimal rates of convergence
----------------------------
We are now ready to state our main results on the optimal rates of convergence, which are given in terms of a class of intermediate norms between $\|f\|_K$ and $$\biggl(\int\!\!\int f(s)C(s,t)f(t)\,ds\,dt \biggr)^{1/2},$$ which enables a unified treatment of both the prediction and estimation problems. For $0\le a\le1$ define the norm $\|\cdot\|_a$ by $$\Vert f\Vert_a^2=\sum_{k=1}^\infty(1+\gamma_k^{-a})f_k^2,$$ where $f_k=\nu_k \langle f, \omega_k\rangle_R$ as shown in Theorem \[th:simdiag\]. Clearly $\|f\|_0$ reduces to $\langle Cf, f\rangle_{\mathcal{L}_2}$ whereas $\|f\|_1=\|f\|_R$. The convergence rate results given below are valid for all $0\le a \le1$. They cover a range of interesting cases including the prediction error and estimation error.
The following result gives the optimal rate of convergence for the regularized estimator $\hat{\beta}_{n\lambda}$ with an appropriately chosen tuning parameter $\lambda$ under the loss $\|\cdot\|_a$.
\[th:main\] Assume that $E(\varepsilon_i)=0$ and $\operatorname{Var}(\varepsilon
_i)\le M_2$. Suppose the eigenvalues $\rho_k$ of the reproducing kernel $K$ of the RKHS $\mathcal{H}$ satisfy $\rho_k \asymp k^{-2r}$ for some $r > 1/2$. Then the regularized estimator $\hat\beta_{n\lambda}$ with $$\label{eq:lamopt}
\lambda\asymp n^{-2(r+s)/(2(r+s)+1)}$$ satisfies $$\begin{aligned}
\label{ubd.eq}
\hspace*{30pt}&&\lim_{D\to\infty} \mathop{\overline{\lim}}_{n\to\infty} \sup
_{F\in\mathcal{F}
(s,M,K),\beta_0\in
\mathcal{H}} P \bigl(\|\hat{\beta}_{n\lambda}-\beta_0\|
_a^2>Dn^{-\afrac{2(1-a)(r+s)}{2(r+s)+1}} \bigr)\nonumber
\\[-8pt]\\[-8pt]
&&\qquad=0.\nonumber\end{aligned}$$
Note that the rate of the optimal choice of $\lambda$ does not depend on $a$. Theorem \[th:main\] shows that the optimal rate of convergence for the regularized estimator $\hat{\beta}_{n\lambda}$ is $n^{-\afrac{2(1-a)(r+s)}{2(r+s)+1}}$. The following lower bound result demonstrates that this rate of convergence is indeed optimal among all estimators, and consequently the upper bound in equation (\[ubd.eq\]) cannot be improved. Denote by $\mathcal{B}$ the collection of all measurable functions of the observations $(X_1,Y_1),\ldots, (X_n,Y_n)$.
\[th:main1\] Under the assumptions of Theorem \[th:main\], there exists a constant $d>0$ such that $$\hspace*{25pt}\mathop{\underline{\lim}}_{n\to\infty}\inf_{\tilde{\beta}\in
\mathcal{B}}\sup_{F\in
\mathcal{F}
(s,M,K),\beta_0\in\mathcal{H}}
P \bigl(\|\tilde{\beta}-\beta_0\|_a^2>d n^{-\afrac{2(1-a)(r+s)}{
2(r+s)+1}} \bigr)>0.$$ Consequently, the regularized estimator $\hat{\beta}_{n\lambda}$ with $\lambda\asymp n^{-2(r+s)/(2(r+s)+1)}$ is rate optimal.
The results, given in terms of $\|\cdot\|_a$, provide a wide range of measures of the quality of an estimate for $\beta_0$. Observe that $$\Vert\tilde{\beta}-\beta_0 \Vert_0^2=E_{X^\ast} \biggl(\int
\tilde{\beta}(t)X^\ast(t)\,dt-\int\beta_0(t)X^\ast(t)\,dt \biggr)^2,$$ where $X^\ast$ is an independent copy of $X$, and the expectation on the right-hand side is taken over $X^\ast$. The right-hand side is often referred to as the prediction error in regression. It measures the mean squared prediction error for a random future observation on $X$. From Theorems \[th:main\] and \[th:main1\], we have the following corollary.
\[co:me\] Under the assumptions of Theorem \[th:main\], the mean squared optimal prediction error of a slope function estimator over $F\in\mathcal{F}(s,M,K)$ and $\beta_0\in\mathcal{H}$ is of the order $n^{-{2(r+s)\over
2(r+s)+1}}$ and it can be achieved by the regularized estimator $\hat{\beta
}_{n\lambda}$ with $\lambda$ satisfying (\[eq:lamopt\]).
The result shows that the faster the eigenvalues of the covariance operator $C$ for $X(\cdot)$ decay, the smaller the prediction error.
When $\psi_k=\phi_k$, the prediction error of a slope function estimator $\tilde{\beta}$ can also be understood as the squared prediction error for a fixed predictor $x^\ast(\cdot)$ such that $|\langle x^\ast,
\phi_k\rangle_{\mathcal{L}_2}|\asymp k^{-s}$ following the discussed from the last section. A similar prediction problem has also been considered by Cai and Hall ([-@CaiHall2006]) for FPCA-based approaches. In particular, they established a similar minimax lower bound and showed that the lower bound can be achieved by the FPCA-based approach, but with additional assumptions that $\mu_k-\mu_{k+1}\ge
C_0^{-1}k^{-2s-1}$, and $2r>4s+3$. Our results here indicate that both restrictions are unnecessary for establishing the minimax rate for the prediction error. Moreover, in contrast to the FPCA-based approach, the regularized estimator $\hat{\beta}_{n\lambda}$ can achieve the optimal rate without the extra requirements.
To illustrate the generality of our results, we consider an example where $\mathcal{T}=[0,1]$, $\mathcal{H}=\mathcal
{W}_2^{m}([0,1])$ and the stochastic process $X(\cdot)$ is a Wiener process. It is not hard to see that the covariance operator of $X$, $C(s,t)=\min\{s,t\}$, satisfies the Sacks–Ylvisaker conditions of order $0$ and therefore $\mu_k\asymp k^{-2}$. By Corollary \[co:me\], the minimax rate of the prediction error in estimating $\beta_0$ is $n^{-\vafrac{2m+2}{r2m+3}}$. Note that the condition $2r>4s+3$ required by Cai and Hall ([-@CaiHall2006]) does not hold here for $m\le7/2$.
The special case of phi k = psi k
---------------------------------
It is of interest to further look into the case when the operators $C$ and $K$ share a common set of eigenfunctions. As discussed in the last section, we have in this case $\phi_k=\psi_k$ and $\gamma_k\asymp
k^{-2(r+s)}$ for all $k\ge1$. In this context, Theorems \[th:main\] and \[th:main1\] provide bounds for more general prediction problems. Consider estimating $\int x^\ast\beta_0$ where $x^\ast$ satisfies $
|\langle x^\ast, \phi_k\rangle_{\mathcal{L}_2} |\asymp k^{-s+q}$ for some $0<q< s-1/2$. Note that $q<s-1/2$ is needed to ensure that $x^\ast
$ is square integrable. The squared prediction error $$\label{eq:prederr}
\biggl(\int\tilde{\beta}(t)x^\ast(t)\,dt-\int\beta_0(t)x^\ast(t)\,dt \biggr)^2$$ is therefore equivalent to $\Vert\tilde{\beta}-\beta_0\Vert_{(s-q)/(r+s)}$. The following result is a direct consequence of Theorems \[th:main\] and \[th:main1\].
Suppose $x^\ast$ is a function satisfying $ |\langle x^\ast, \phi
_k\rangle_{\mathcal{L}_2} |\asymp k^{-s+q}$ for some $0<q< s-1/2$. Then under the assumptions of Theorem \[th:main\], $$\begin{aligned}
\hspace*{30pt}&&\mathop{\underline{\lim}}_{n\to\infty}\inf_{\tilde{\beta}\in
\mathcal{B}}
\sup_{F\in\mathcal{F}(s,M,K),\beta_0\in\mathcal{H}} P
\biggl\{ \biggl(\int\tilde{\beta}(t)x^\ast(t)\,dt-\int\beta_0(t)x^\ast(t)\,dt \biggr)^2\nonumber
\\[-9pt]\\[-9pt]
&&\hspace*{195pt}>dn^{-\afrac{2(r+q)}{2(r+s)+1}} \biggr\}>0\nonumber\end{aligned}$$ for some constant $d>0$, and the regularized estimator $\hat{\beta
}_{n\lambda}$ with $\lambda$ satisfying (\[eq:lamopt\]) achieves the optimal rate of convergence under the prediction error (\[eq:prederr\]).
It is also evident that when $\psi_k=\phi_k$, $\|\cdot\|_{s/(r+s)}$ is equivalent to $\|\cdot\|_{\mathcal{L}_2}$. Therefore, Theorems \[th:main\] and \[th:main1\] imply the following result.
If $\phi_k=\psi_k$ for all $k\ge1$, then under the assumptions of Theorem \[th:main\] $$\mathop{\underline{\lim}}_{n\to\infty}\inf_{\tilde{\beta}\in
\mathcal{B}}
\sup_{F\in\mathcal{F}(s,M,K),\beta_0\in\mathcal{H}} P \bigl( \Vert
\tilde{\beta}-\beta_0 \Vert_{\mathcal{L}_2}^2>d n^{-\afrac{2r}{2(r+s)+1}} \bigr)>0$$ for some constant $d>0$, and the regularized estimate $\hat{\beta
}_{n\lambda}$ with $\lambda$ satisfying (\[eq:lamopt\]) achieves the optimal rate.
This result demonstrates that the faster the eigenvalues of the covariance operator for $X(\cdot)$ decay, the larger the estimation error. The behavior of the estimation error thus differs significantly from that of prediction error.
Similar results on the lower bound have recently been obtained by Hall and Horowitz ([-@HallHorowitz2007]) who considered estimating $\beta_0$ under the assumption that $ |\langle\beta_0,\phi_k\rangle_{\mathcal{L}_2} |$ decays in a polynomial order. Note that this slightly differs from our setting where $\beta_0\in\mathcal{H}$ means that $$\label{eq:hilcond}
\sum_{k=1}^\infty\rho_k^{-1}\langle\beta_0,\psi_k\rangle
_{\mathcal{L}
_2}^2=\sum_{k=1}^\infty\rho_k^{-1}\langle\beta_0,\phi_k\rangle
_{\mathcal{L}
_2}^2<\infty.$$ Recall that $\rho_k\asymp k^{-2r}$. Condition (\[eq:hilcond\]) is comparable to, and slightly stronger than, $$\label{eq:hhcond}
|\langle\beta_0,\phi_k\rangle_{\mathcal{L}_2} |\le M_0k^{-r-1/2}$$ for some constant $M_0>0$. When further assuming that $2s+1<2r$, and $\mu_k-\mu_{k+1}\ge M_0^{-1}k^{-2s-1}$ for all $k\ge1$, Hall and Horowitz ([-@HallHorowitz2007]) obtain the same lower bound as ours. However, we do not require that $2s+1<2r$ which in essence states that $\beta_0$ is smoother than the sample path of $X$. Perhaps, more importantly, we do not require the spacing condition $\mu_k-\mu_{k+1}\ge M_0^{-1}k^{-2s-1}$ on the eigenvalues because we do not need to estimate the corresponding eigenfunctions. Such a condition is impossible to verify even for a standard RKHS.
Estimating derivatives
----------------------
Theorems \[th:main\] and \[th:main1\] can also be used for estimating the derivatives of $\beta_0$. A natural estimator of the $q$th derivative of $\beta_0$, $\beta_0^{(q)}$, is $\hat{\beta
}_{n\lambda}^{(q)}$, the $q$th derivative of $\hat{\beta}_{n\lambda}$. In addition to $\phi_k=\psi_k$, assume that $ \Vert\psi
_k^{(q)}/\psi
_k \Vert_\infty\asymp k^{q}$. This clearly holds when $\mathcal
{H}=\mathcal{W}
_2^m$. In this case $$\big\|\tilde{\beta}^{(q)}-\beta_0^{(q)} \big\|_{\mathcal{L}_2}\le C_0\|
\tilde{\beta}-\beta_0\|_{(s+q)/(r+s)}.$$ The following is then a direct consequence of Theorems \[th:main\] and \[th:main1\].
Assume that $\phi_k=\psi_k$ and $ \Vert\psi_k^{(q)}/\psi_k
\Vert_\infty\asymp k^{q}$ for all $k\ge1$. Then under the assumptions of Theorem \[th:main\], for some constant $d>0$, $$\begin{aligned}
\hspace*{25pt}&&\mathop{\underline{\lim}}_{n\to\infty} \inf_{\tilde{\beta
}^{(q)}\in\mathcal{B}}\sup
_{F\in\mathcal{F}
(s,M,K),\beta_0\in\mathcal{H}} P \bigl( \big\|\tilde{\beta}^{(q)}-\beta_0^{(q)}\big \|_{\mathcal{L}_2}^2
>dn^{-\afrac{2(r-q)}{2(r+s)+1}} \bigr)\nonumber
\\[-8pt]\\[-8pt]
&&\qquad>0,\nonumber\end{aligned}$$ and the regularized estimate $\hat{\beta}_{n\lambda}$ with $\lambda$ satisfying (\[eq:lamopt\]) achieves the optimal rate.
Finally, we note that although we have focused on the squared error loss here, the method of regularization can be easily extended to handle other goodness of fit measures as well as the generalized functional linear regression \[Cardot and Sarda ([-@CardotSarda2005]) and Müller and Stadtmüller ([-@MullerStadtmuller2005])\]. We shall leave these extensions for future studies.
Numerical results {#numerical.sec}
=================
The Representer Theorem given in Section \[representer.sec\] makes the regularized estimators easy to implement. Similarly to smoothness regularized estimators in other contexts \[see, e.g., Wahba ([-@Wahba1990])\], $\hat{\eta}_{n\lambda}$ and $\hat{\beta}_{n\lambda}$ can be expressed as a linear combination of a finite number of known basis functions although the minimization in (\[eq:mor\]) is taken over an infinitely-dimensional space. Existing algorithms for smoothing splines can thus be used to compute our regularized estimators $\hat{\eta}_{n\lambda}$, $\hat{\beta
}_{n\lambda
}$ and $\hat\alpha_{n\lambda}$.
To demonstrate the merits of the proposed estimators in finite sample settings, we carried out a set of simulation studies. We adopt the simulation setting of Hall and Horowitz ([-@HallHorowitz2007]) where $\mathcal{T}=[0,1]$. The true slope function $\beta_0$ is given by $$\beta_0=\sum_{k=1}^{50} 4(-1)^{k+1}k^{-2} \phi_k,$$ where $\phi_1(t)=1$ and $\phi_{k+1}(t)=\sqrt{2}\cos(k\pi t)$ for $k\ge
1$. The random function $X$ was generated as $$X=\sum_{k=1}^{50} \zeta_kZ_k \phi_k,$$ where $Z_k$ are independently sampled from the uniform distribution on$[-\sqrt{3},\sqrt{3}]$ and $\zeta_k$ are deterministic. It is not hard to see that $\zeta_k^2$ are the eigenvalues of the covariance function of $X$. Following Hall and Horowitz ([-@HallHorowitz2007]), two sets of $\zeta_k$ were used. In the first set, the eigenvalues are well spaced: $\zeta
_k=(-1)^{k+1}k^{-\nu/2}$ with $\nu=1.1, 1.5, 2$ or $4$. In the second set, $$\hspace*{20pt}\zeta_k= \cases{
%
1,&\quad$k=1$,\cr
0.2(-1)^{k+1}(1-0.0001k),&\quad$2\le k\le4$,\cr
0.2(-1)^{k+1} [(5\lfloor k/5\rfloor)^{-\nu/2}-0.0001(k \operatorname{mod}\ 5) ],&\quad$k\ge5$.
}
%$$
As in Hall and Horowitz ([-@HallHorowitz2007]), regression models with $\varepsilon\sim
N(0,\sigma^2)$ where $\sigma=0.5$ and $1$ were considered. To comprehend the effect of sample size, we consider $n=50, 100, 200$ and $500$. We apply the regularization method to each simulated dataset and examine its estimation accuracy as measured by integrated squared error $\|\hat
{\beta}_{n\lambda}-\beta_0\|_{\mathcal{L}_2}^2$ and prediction error $\|
\hat
{\beta}_{n\lambda}-\beta_0\|_{0}^2$. For the purpose of illustration, we take $\mathcal{H}=\mathcal{W}_2^2$ and $J(\beta)=\int(\beta'')^2$, for which the detailed estimation procedure is given in Section \[representer.sec\]. For each setting, the experiment was repeated 1000 times.
As is common in most smoothing methods, the choice of the tuning parameter plays an important role in the performance of the regularized estimators. Data-driven choice of the tuning parameter is a difficult problem. Here we apply the commonly used practical strategy of empirically choosing the value of $\lambda$ through the generalized cross validation. Note that the regularized estimator is a linear estimator in that $\hat{\mathbf{y}}=H(\lambda)\mathbf{y}$ where $\hat{\mathbf{y}}=(\hat{\eta}_{n\lambda}(x_1),\ldots, \hat{\eta
}_{n\lambda}(x_n))'$ and $H(\lambda)$ is the so-called hat matrix depending on $\lambda$. We then select the tuning parameter $\lambda$ that minimizes $$\operatorname{GCV}(\lambda)={(1/ n) \Vert\hat{\mathbf{y}}-\mathbf
{y}\Vert_{\ell
_2}^2\over(1-\operatorname{tr}(H(\lambda))/n )^2}.$$ Denote by $\hat{\lambda}^{\mathrm{GCV}}$ the resulting choice of the tuning parameter.
![Prediction errors of the regularized estimator ($\sigma=0.5$): $X$ was simulated with a covariance function with well-spaced eigenvalues. The results are averaged over 1000 runs. Black solid lines, red dashed lines, green dotted lines and blue dash-dotted lines correspond to $\nu= 1.1,\; 1.5,\; 2$ and $4$, respectively. Both axes are in log scale.[]{data-label="fig:sim-well-pred-small"}](772f01.eps)
We begin with the setting of well-spaced eigenvalues. The left panel of Figure \[fig:sim-well-pred-small\] shows the prediction error, $\|\hat{\beta}_{n\lambda}-\beta_0\|_{0}^2$, for each combination of $\nu$ value and sample size when $\sigma=0.5$. The results were averaged over 1000 simulation runs in each setting. Both axes are given in the log scale. The plot suggests that the estimation error converges at a polynomial rate as sample size $n$ increases, which agrees with our theoretical results from the previous section. Furthermore, one can observe that with the same sample size, the prediction error tends to be smaller for larger $\nu$. This also confirms our theoretical development which indicates that the faster the eigenvalues of the covariance operator for $X(\cdot)$ decay, the smaller the prediction error.=1
![Estimation errors of the regularized estimator ($\sigma=0.5$): $X$ was simulated with a covariance function with well-spaced eigenvalues. The results are averaged over 1000 runs. Black solid lines, red dashed lines, green dotted lines and blue dash-dotted lines correspond to $\nu= 1.1, 1.5, 2$ and $4$, respectively. Both axes are in log scale.[]{data-label="fig:sim-well-est-small"}](772f02.eps)
To better understand the performance of the smoothness regularized estimator and the GCV choice of the tuning parameter, we also recorded the performance of an oracle estimator whose tuning parameter is chosen to minimize the prediction error. This choice of the tuning parameter ensures the optimal performance of the regularized estimator. It is, however, noteworthy that this is not a legitimate statistical estimator since it depends on the knowledge of unknown slope function $\beta_0$. The right panel of Figure \[fig:sim-well-pred-small\] shows the prediction error associated with this choice of tuning parameter. It behaves similarly to the estimate with $\lambda$ chosen by GCV. Note that the comparison between the two panels suggest that GCV generally leads to near optimal performance.
We now turn to the estimation error. Figure \[fig:sim-well-est-small\] shows the estimation errors, averaged over 1000 simulation runs, with $\lambda$ chosen by GCV or minimizing the estimation error for each combination of sample size and $\nu$ value. Similarly to the prediction error, the plots suggest a polynomial rate of convergence of the estimation error when the sample size increases, and GCV again leads to near-optimal choice of the tuning parameter.
![Estimation and prediction errors of the regularized estimator ($\sigma^2=1^2$): $X$ was simulated with a covariance function with well-spaced eigenvalues. The results are averaged over 1000 runs. Black solid lines, red dashed lines, green dotted lines and blue dash-dotted lines correspond to $\nu= 1.1, 1.5, 2$ and $4$, respectively. Both axes are in log scale.[]{data-label="fig:sim-well-large"}](772f03.eps)
A comparison between Figures \[fig:sim-well-pred-small\] and \[fig:sim-well-est-small\] suggests that when $X$ is smoother (larger $\nu$), prediction (as measured by the prediction error) is easier, but estimation (as measured by the estimation error) tends to be harder, which highlights the difference between prediction and estimation in functional linear regression. We also note that this observation is in agreement with our theoretical results from the previous section where it is shown that the estimation error decreases at the rate of $n^{-2r/(2(r+s)+1)}$ which decelerates as $s$ increases; whereas the prediction error decreases at the rate of $n^{-2(r+s)/(2(r+s)+1)}$ which accelerates as $s$ increases.
Figure \[fig:sim-well-large\] reports the prediction and estimation error when tuned with GCV for the large noise ($\sigma=1$) setting. Observations similar to those for the small noise setting can also be made. Furthermore, notice that the prediction errors are much smaller than the estimation error, which confirms our finding from the previous section that prediction is an easier problem in the context of functional linear regression.
The numerical results in the setting with closely spaced eigenvalues are qualitatively similar to those in the setting with well-spaced eigenvalues. Figure \[fig:close\] summarizes the results obtained for the setting with closely spaced eigenvalues.
![Estimation and prediction errors of the regularized estimator: $X$ was simulated with a covariance function with closely-spaced eigenvalues. The results are averaged over 1000 runs. Both axes are in log scale. Note that $y$-axes are of different scales across panels.[]{data-label="fig:close"}](772f04.eps)
We also note that the performance of the regularization estimate with $\lambda$ tuned with GCV compares favorably with those from Hall and Horowitz ([-@HallHorowitz2007]) using FPCA-based methods even though their results are obtained with optimal rather than data-driven choice of the tuning parameters.
Proofs {#proof.sec}
======
Proof of Proposition 2
----------------------
Observe that $$\int_{\mathcal{T}\times\mathcal{T}}f(s)C(s,t)f(t)\,ds\,dt\le\mu_1\|f\|_{\mathcal{L}_2}^2\le c_1\|f\|_\mathcal{H}^2$$ for some constant $c_1>0$. Together with the fact that $J(f)\le\|f\|
_\mathcal{H}^2$, we conclude that $$\|f\|_R^2= \int_{\mathcal{T}\times\mathcal
{T}}f(s)C(s,t)f(t)\,ds\,dt+J(f)\le
(c_1+1)\|
f\|_\mathcal{H}^2.$$
Recall that $\xi_k$, $k=1,\ldots, N,$ are the orthonormal basis of $\mathcal{H}
_0$. Under the assumption of the proposition, the matrix $\Sigma
=(\langle C\xi_j,\xi_k\rangle_\mathcal{H})_{1\le j,k\le N}$ is a positive definite matrix. Denote by $\mu_1'\ge\mu_2'\ge\cdots\ge\mu_N'>0$ its eigenvalues. It is clear that for any $f_0\in\mathcal{H}_0$ $$\|f_0\|_R^2\ge\mu_N'\|f_0\|_\mathcal{H}^2.$$ Note also that for any $f_1\in\mathcal{H}_1$, $$\|f_1\|_\mathcal{H}^2=J(f_1)\le\|f_1\|_R^2.$$
For any $f\in\mathcal{H}$, we can write $f:=f_0+f_1$ where $f_0\in
\mathcal{H}_0$ and $f_1\in\mathcal{H}_1$. Then $$\|f\|_R^2=\int_{\mathcal{T}\times\mathcal{T}}f(s)C(s,t)f(t)\,ds\,dt+\|
f_1\|_{\mathcal{H}}^2.$$ Recall that $$\|f_1\|_{\mathcal{H}}^2\ge\rho_1^{-1}\|f_1\|_{\mathcal{L}_2}^2\ge
\rho
_1^{-1}\mu
_1^{-1}\int_{\mathcal{T}\times\mathcal{T}}f_0(s)C(s,t)f_0(t)\,ds\,dt.$$ For brevity, assume that $\rho_1=\mu_1=1$ without loss of generality. By the Cauchy–Schwarz inequality, $$\begin{aligned}
\|f\|_R^2&\ge&{1\over2}\int_{\mathcal{T}\times\mathcal{T}
}f(s)C(s,t)f(t)\,ds\,dt+\|
f_1\|_{\mathcal{H}}^2\\
&\ge&{1\over2}\int_{\mathcal{T}\times\mathcal{T}
}f_0(s)C(s,t)f_0(t)\,ds\,dt+{3\over
2}\int_{\mathcal{T}\times\mathcal{T}}f_1(s)C(s,t)f_1(t)\,ds\,dt\\
&&{}- \biggl(\int_{\mathcal{T}\times\mathcal{T}}f_0(s)C(s,t)f_0(t)\,ds\,dt \biggr)^{1/2}
\\
&&\hspace*{12pt}{}\times\biggl(\int_{\mathcal{T}\times\mathcal{T}}f_1(s)C(s,t)f_1(t)\,ds\,dt \biggr)^{1/2}\\
&\ge&{1\over3}\int_{\mathcal{T}\times\mathcal{T}}f_0(s)C(s,t)f_0(t)\,ds\,dt,\end{aligned}$$ where we used the fact that $3a^2/2-ab\ge-b^2/6$ in deriving the last inequality. Therefore, $${3\over\mu_N'}\|f\|_R^2\ge\|f_0\|_\mathcal{H}^2.$$ Together with the facts that $\|f\|_\mathcal{H}^2=\|f_0\|_\mathcal
{H}^2+\|f_1\|
_\mathcal{H}^2$ and $$\|f\|_R^2\ge J(f_1)\ge\|f_1\|_\mathcal{H}^2,$$ we conclude that $$\|f\|_R^2\ge(1+3/\mu_N')^{-1}\|f\|_\mathcal{H}^2.$$ The proof is now complete.
Proof of Theorem 3
------------------
First note that $$\begin{aligned}
R^{-1/2}f&=&\sum_{k=1}^\infty\langle R^{-1/2}f, \zeta_k\rangle
_{\mathcal{L}
_2} \zeta_k
=\sum_{k=1}^\infty\langle R^{-1/2}f, \nu_k^{1/2}R^{-1/2}\omega
_k\rangle
_{\mathcal{L}_2} \nu_k^{1/2}R^{-1/2}\omega_k\\
&=&R^{-1/2} \Biggl(\sum_{k=1}^\infty\nu_k\langle R^{-1/2}f,
R^{-1/2}\omega_k\rangle_{\mathcal{L}_2} \omega_k \Biggr)
=R^{-1/2} \Biggl(\sum_{k=1}^\infty\nu_k\langle f, \omega_k\rangle_R
\omega_k \Biggr).\end{aligned}$$ Applying bounded positive definite operator $R^{1/2}$ to both sides leads to $$f=\sum_{k=1}^\infty\nu_k\langle f, \omega_k\rangle_R \omega_k.$$
Recall that $\langle\omega_k, \omega_j\rangle_R=\nu_k^{-1}\delta
_{kj}$. Therefore, $$\begin{aligned}
\|f\|_R^2&=& \Bigg\langle\sum_{k=1}^\infty\nu_k\langle f, \omega
_k\rangle_R \omega_k, \sum_{j=1}^\infty\nu_j\langle f, \omega
_j\rangle
_R \omega_j \Bigg\rangle_R
\\
&=&\sum_{k,j=1} \nu_k\nu_j\langle f, \omega_k\rangle_R\langle f,
\omega
_j\rangle_R\langle\omega_k,\omega_j\rangle_R\\
&=&\sum_{k=1} \nu_k \langle f, \omega_k\rangle_R^2.\end{aligned}$$
Similarly, because $\langle C\omega_k, \omega_j\rangle_{\mathcal{L}
_2}=\delta_{kj}$, $$\begin{aligned}
\langle Cf, f\rangle_{\mathcal{L}_2}
&=& \Bigg\langle C \Biggl(\sum_{k=1}^\infty
\nu_k\langle f, \omega_k\rangle_R \omega_k \Biggr), \sum_{j=1}^\infty
\nu
_j\langle f, \omega_j\rangle_R \omega_j \Bigg\rangle_{\mathcal{L}_2}\\
&=& \Bigg\langle\sum_{k=1}^\infty\nu_k\langle f, \omega_k\rangle_R
C\omega_k, \sum_{j=1}^\infty\nu_j\langle f, \omega_j\rangle_R
\omega
_j \Bigg\rangle_{\mathcal{L}_2}\\
&=&\sum_{k,j=1} \nu_k\nu_j\langle f, \omega_k\rangle_R\langle f,
\omega
_j\rangle_R\langle C\omega_k,\omega_j\rangle_{\mathcal{L}_2}\\
&=&\sum_{k=1} \nu_k^2 \langle f, \omega_k\rangle_R^2.\end{aligned}$$
Proof of Proposition 4
----------------------
Recall that for any $f\in\mathcal{H}_0$, $Cf\neq0$ if and only if $f=0$, which implies that $\mathcal{H}_0\cap{\rm l.s.}\{\phi_k\dvtx k\ge1\}
^{\perp
}=\{
0\}$. Together with the fact that $\mathcal{H}_0\cap\mathcal{H}_1=\{
0\}$, we conclude that $\mathcal{H}=\mathcal{H}_1={\rm l.s.}\{\phi_k\dvtx k\ge1\}
$. It is not hard to see that for any $f,g\in\mathcal{H}$, $$\langle f,g\rangle_R=\int_{\mathcal{T}\times\mathcal{T}
}f(s)C(s,t)g(t)\,ds\,dt+\langle
f, g\rangle_K.$$ In particular, $$\langle\psi_j,\psi_k\rangle_R= (\mu_k+\rho_k^{-1} )\delta_{jk},$$ which implies that $\{((\mu_k+\rho_k^{-1})^{-1},\psi_k)\dvtx k\ge1\}$ is also the eigen system of $R$, that is, $$R(s,t)=\sum_{k=1}^\infty(\mu_k+\rho_k^{-1})^{-1}\psi_k(s)\psi_k(t).$$ Then $$R\psi_k:=\int_\mathcal{T}R(\cdot, t)\psi_k(t)\,dt=(\mu_k+\rho
_k^{-1})^{-1}\psi
_k,\qquad k=1,2,\ldots.$$ Therefore, $$\begin{aligned}
R^{1/2}CR^{1/2}\psi_k&=&R^{1/2}C \bigl((\mu_k+\rho_k^{-1})^{-1/2}\psi
_k \bigr)\\
&=&R^{1/2} \bigl(\mu_k(\mu_k+\rho_k^{-1})^{-1/2}\psi_k \bigr)
= (1+\rho_k^{-1}\mu_k^{-1} )^{-1} \psi_k,\end{aligned}$$ which implies that $\zeta_k=\psi_k=\phi_k$, $\nu_k= (1+\rho
_k^{-1}\mu_k^{-1} )^{-1}$ and $\gamma_k=\rho_k\mu_k$. Consequently, $$\omega_k=\nu_k^{-1/2}R^{1/2}\psi_k=\nu_k^{-1/2}(\mu_k+\rho
_k^{-1})^{-1/2}\psi_k=\mu_k^{-1/2}\psi_k.$$
Proof of Theorem 5
------------------
Recall that $\mathcal{H}=\mathcal{W}_2^m$, which implies that $\rho
_k\asymp
k^{-2m}$. By Corollary 2 of Ritter, Wasilkowski and Woźniakowski ([-@RitterWasilkowskiWozniakowski1995]), $\mu_k\asymp
k^{-2(s+1)}$. It therefore suffices to show $\gamma_k\asymp
k^{-2(s+1+m)}$. The key idea of the proof is a result from Ritter, Wasilkowski and Woźniakowski ([-@RitterWasilkowskiWozniakowski1995]) indicating that the reproducing kernel Hilbert space associated with $C$ differs from $\mathcal{W}^{s+1}_2([0,1])$ only by a finite-dimensional linear space of polynomials.
Denote by $Q_r$ the reproducing kernel for $\mathcal{W}^{r}_2([0,1])$. Observe that $Q_r^{1/2}(\mathcal{L}_2)=\mathcal{W}^r_2$ \[e.g., Cucker and Smale ([-@CuckerSmale2001])\]. We begin by quantifying the decay rate of $\lambda_{k}
(Q_{m}^{1/2}Q_{s+1}Q_m^{1/2} )$. By Sobolev’s embedding theorem, $(Q_{s+1}^{1/2}Q_{m}^{1/2})(\mathcal{L}_2)=Q_{s+1}^{1/2}(\mathcal
{W}_2^{m})=\mathcal{W}
_2^{m+s+1}$. Therefore, $Q_{m}^{1/2}Q_{s+1}Q_m^{1/2}$ is equivalent to $Q_{m+s+1}$. Denote by $\lambda_k(Q)$ be the $k$th largest eigenvalue of a positive definite operator $Q$. Let $\{h_k\dvtx k\ge1\}$ be the eigenfunctions of $Q_{m+s+1}$, that is, $Q_{m+s+1}h_k=\lambda
_k(Q_{m+s+1})h_k$, $k=1,2,\ldots.$ Denote by $\mathcal{F}_k$ and $\mathcal{F}
_k^{\perp}$ the linear space spanned by $\{h_j\dvtx 1\le j\le k\}$ and $\{
h_j\dvtx j\ge k+1\}$, respectively. By the Courant–Fischer–Weyl min–max principle, $$\begin{aligned}
\lambda_k (Q_{m}^{1/2}Q_{s+1}Q_m^{1/2} )&\ge& \min_{f\in\mathcal{F}
_k} \Vert Q_{s+1}^{1/2}Q_m^{1/2}f \Vert_{\mathcal{L}_2}^2/\|f\|
_{\mathcal{L}_2}^2\\
&\ge& C_1 \min_{f\in\mathcal{F}_k} \Vert Q_{m+s+1}^{1/2}f \Vert
_{\mathcal{L}_2}^2/\|f\|_{\mathcal{L}_2}^2\\
&\ge& C_1\lambda_k(Q_{m+s+1})\end{aligned}$$ for some constant $C_1>0$. On the other hand, $$\begin{aligned}
\lambda_{k} (Q_{m}^{1/2}Q_{s+1}Q_m^{1/2} )&\le& \max_{f\in\mathcal{F}
_{k-1}^{\perp}} \Vert Q_{s+1}^{1/2}Q_m^{1/2}f \Vert_{\mathcal{L}
_2}^2/\|f\|_{\mathcal{L}_2}^2\\
&\le& C_2 \min_{f\in\mathcal{F}_{k-1}^{\perp}} \Vert
Q_{m+s+1}^{1/2}f \Vert_{\mathcal{L}_2}^2/\|f\|_{\mathcal{L}_2}^2\\
&\le& C_2\lambda_k(Q_{m+s+1})\end{aligned}$$ for some constant $C_2>0$. In summary, we have $\lambda_{k}
(Q_{m}^{1/2}Q_{s+1}Q_m^{1/2} )\asymp\break\times k^{-2(m+s+1)}$.
As shown by Ritter, Wasilkowski and Woźniakowski \[([-@RitterWasilkowskiWozniakowski1995]), Theorem 1, page 525\], there exist $D$ and $U$ such that $Q_{s+1}=D+U$, $D$ has at most $2(s+1)$ nonzero eigenvalues and $\|U^{1/2}f\|_{\mathcal{L}_2}$ is equivalent to $\|
C^{1/2}f\|
_{\mathcal{L}_2}$. Moreover, the eigenfunctions of $D$, denoted by $\{
g_1,\ldots, g_d\}$ ($d\le2(s+1)$) are polynomials of order no greater than $2s+1$. Denote $\mathcal{G}$ the space spanned by $\{g_1,\ldots
,g_d\}$. Clearly $\mathcal{G}\subset\mathcal{W}_2^{m}=Q_m^{1/2}(\mathcal
{L}_2)$. Denote $\{
\tilde
{h}_j \dvtx j\ge1\}$ the eigenfunctions of $Q_{m}^{1/2}Q_{s+1}Q_m^{1/2}$. Let $\tilde{\mathcal{F}}_k$ and $\tilde{\mathcal{F}}_k^\perp$ be defined similarly as $\mathcal{F}_k$ and $\mathcal{F}_k^{\perp}$. Then by the Courant–Fischer–Weyl min–max principle, $$\begin{aligned}
\lambda_{k-d} (Q_{m}^{1/2}UQ_m^{1/2} )&\ge& \min_{f\in\tilde
{\mathcal{F}
}_k\cap Q_m^{-1/2}(\mathcal{G})^{\perp}} \Vert U^{1/2}Q_m^{1/2}f
\Vert
_{\mathcal{L}
_2}^2/\|f\|_{\mathcal{L}_2}^2\\
&=& \min_{f\in\tilde{\mathcal{F}}_k\cap Q_m^{-1/2}(\mathcal
{G})^{\perp}}
\Vert
Q_{s+1}^{1/2}Q_m^{1/2}f \Vert_{\mathcal{L}_2}^2/\|f\|_{\mathcal
{L}_2}^2\\
&=& \min_{f\in\tilde{\mathcal{F}}_k} \Vert Q_{s+1}^{1/2}Q_m^{1/2}f
\Vert
_{\mathcal{L}_2}^2/\|f\|_{\mathcal{L}_2}^2\\
&\ge& C_1\lambda_k(Q_{m+s+1})\end{aligned}$$ for some constant $C_1>0$. On the other hand, $$\begin{aligned}
\lambda_{k+d} (Q_{m}^{1/2}Q_{s+1}Q_m^{1/2} )&\le& \max_{f\in\tilde
{\mathcal{F}
}_{k-1}^{\perp}\cap Q_m^{-1/2}(\mathcal{G})^{\perp}} \Vert U^{1/2}Q_m^{1/2}f
\Vert_{\mathcal{L}_2}^2/\|f\|_{\mathcal{L}_2}^2\\
&=& \max_{f\in\tilde{\mathcal{F}}_{k-1}^{\perp}\cap
Q_m^{-1/2}(\mathcal{G}
)^{\perp
}} \Vert Q_{s+1}^{1/2}Q_m^{1/2}f \Vert_{\mathcal{L}_2}^2/\|f\|
_{\mathcal{L}
_2}^2\\
&=& \min_{f\in\tilde{\mathcal{F}}_{k-1}^{\perp}} \Vert
Q_{s+1}^{1/2}Q_m^{1/2}f \Vert_{\mathcal{L}_2}^2/\|f\|_{\mathcal
{L}_2}^2\\
&\le& C_2\lambda_k(Q_{m+s+1})\end{aligned}$$ for some constant $C_2>0$. Hence $\lambda_{k} (Q_{m}^{1/2}UQ_m^{1/2}
)\asymp k^{-2(m+s+1)}$.
Because $Q_{m}^{1/2}UQ_m^{1/2}$ is equivalent to $R^{1/2}CR^{1/2}$, following a similar argument as before, by the Courant–Fischer–Weyl min–max principle, we complete the the proof.
Proof of Theorem 6
------------------
We now proceed to prove Theorem \[th:main\]. The analysis follows a similar spirit as the technique commonly used in the study of the rate of convergence of smoothing splines \[see, e.g., Silverman ([-@Silverman1982]); Cox and O’Sullivan ([-@CoxOSullivan1990])\]. For brevity, we shall assume that $EX(\cdot)=0$ in the rest of the proof. In this case, $\alpha_0$ can be estimated by $\bar{y}$ and $\beta_0$ by $$\hat{\beta}_{n\lambda}=\mathop{\arg\min}_{\beta\in\mathcal{H}}
\Biggl[{1\over n}\sum_{i=1}^n
\biggl(y_i-\int_\mathcal{T}x_i(t)\beta(t)\,dt \biggr)^2+\lambda J(\beta) \Biggr].$$ The proof below also applies to the more general setting when $EX(\cdot
)\neq0$ but with considerable technical obscurity.
Recall that $$\ell_n(\beta)={1\over n}\sum_{i=1}^n \biggl(y_i-\int_\mathcal
{T}x_i(t)\beta
(t)\,dt \biggr)^2.$$ Observe that $$\begin{aligned}
\ell_\infty(\beta)&:=& E\ell_n(\beta)
=E \biggl[Y-\int_\mathcal{T}X(t)\beta(t)\,dt\biggr]^2\\
&=&\sigma^2+\int_\mathcal{T}\int_\mathcal{T}[\beta(s)-\beta_0(s) ]C(s,t)[\beta(t)-\beta_0(t) ]\,ds\,dt\\
&=&\sigma^2+ \Vert\beta-\beta_0 \Vert^2_0.\end{aligned}$$ Write $$\bar{\beta}_{\infty\lambda}=\mathop{\arg\min}_{\beta\in
\mathcal{H}} \{\ell
_\infty(\beta
)+\lambda J(\beta) \}.$$ Clearly $$\hat{\beta}_{n\lambda}-\beta_0= (\hat{\beta}_{n\lambda}-\bar
{\beta
}_{\infty\lambda} )+ (\bar{\beta}_{\infty\lambda}-\beta_0 ).$$ We refer to the two terms on the right-hand side stochastic error and deterministic error, respectively.
### Deterministic error
Write $\beta_0(\cdot)=\sum_{k=1}^\infty a_k\omega_k(\cdot)$ and ${\beta}(\cdot)=\break\sum_{k=1}^\infty b_k\omega_k(\cdot)$. Then Theorem \[th:simdiag\] implies that $$\begin{aligned}
\ell_\infty({\beta})=\sigma^2+\sum_{k=1}^\infty(b_{k}-{a}_{k})^2,\qquad
J({\beta})=\sum_{k=1}^\infty\gamma_k^{-1} b_k^2.\end{aligned}$$ Therefore, $$\bar{\beta}_{\infty\lambda}(\cdot)=\sum_{k=1}^\infty{ a_k\over
1+\lambda\gamma_k^{-1}} \omega_k(\cdot)=: \sum_{k=1}^\infty\bar
{b}_k\omega_k(\cdot).$$
It can then be computed that for any $a<1$, $$\begin{aligned}
\Vert\bar{\beta}_{\infty\lambda}-\beta_0 \Vert_a^2&=&\sum
_{k=1}^\infty
(1+\gamma_k^{-a})(\bar{b}_k-a_k)^2\\[-2pt]
&=&\sum_{k=1}^\infty(1+\gamma_k^{-a}) \biggl({ \lambda\gamma_k^{-1}\over
1+\lambda\gamma_k^{-1}} \biggr)^2a_k^2\\[-2pt]
&\le&\lambda^2 \sup_k {(1+\gamma^{-a})\gamma_k^{-1}\over
(1+\lambda
\gamma_k^{-1} )^2}\sum_{k=1}^\infty\gamma_k^{-1}a_k^2\\[-2pt]
&=&\lambda^2J(\beta_0) \sup_k {(1+\gamma^{-a})\gamma_k^{-1}\over
(1+\lambda\gamma_k^{-1} )^2}.\end{aligned}$$ Now note that $$\begin{aligned}
\sup_k {(1+\gamma^{-a})\gamma_k^{-1}\over(1+\lambda\gamma_k^{-1}
)^2}&\le& \sup_{x> 0} {(1+x^{-a})x^{-1}\over(1+\lambda x^{-1} )^2}\\[-2pt]
&\le&\sup_{x> 0} {x^{-1}\over(1+\lambda x^{-1} )^2}+\sup_{x> 0}
{x^{-(a+1)}\over(1+\lambda x^{-1} )^2}\\[-2pt]
&=& {1\over\inf_{x> 0} (x^{1/2}+\lambda x^{-1/2} )^2}+ {1\over\inf
_{x> 0} (x^{(a+1)/2}+\lambda x^{-(1-a)/2} )^2}\\[-2pt]
&=&{1\over4\lambda}+C_0\lambda^{-(a+1)}.\end{aligned}$$ Hereafter, we use $C_0$ to denote a generic positive constant. In summary, we have
\[le:detererr\] If $\lambda$ is bounded from above, then $$\Vert\bar{\beta}_{\infty\lambda}-\beta_0 \Vert^2_{a}=O ({\lambda^{1-a}}
J(\beta_0) ).$$
### Stochastic error
Next, we consider the stochastic error $\hat{\beta}_{n\lambda}-\bar
{\beta}_{\infty\lambda}$. Denote $$\begin{aligned}
D\ell_{n} (\beta)f
&=&-{2\over n}\sum_{i=1}^n \biggl[ \biggl(y_i-\int_\mathcal{T}x_i(t)\beta(t)\,dt \biggr)\int_\mathcal{T}x_i(t)f(t)\,dt \biggr],\\[-2pt]
D\ell_{\infty} (\beta)f&=&-2E_X \biggl(\int_\mathcal{T}X(t) [\beta_0(t)-\beta(t)]\,dt\int_\mathcal{T}X(t)f(t)\,dt \biggr)\\[-2pt]
&=&-2\int_\mathcal{T}\int_\mathcal{T}[\beta_0(s)-\beta(s)]C(s,t)f(t)\,ds\,dt,\\[-2pt]
D^2\ell_{n} (\beta)fg&=&{2\over n}\sum_{i=1}^n \biggl[\int_\mathcal{T}x_i(t)f(t)\,dt\int_\mathcal{T}x_i(t)g(t)\,dt \biggr],\\[-2pt]
D^2\ell_{\infty} (\beta)fg&=&2\int_\mathcal{T}\int_\mathcal{T}f(s)C(s,t)g(t)\,ds\,dt.\end{aligned}$$ Also write $\ell_{n\lambda}(\beta)=\ell_n(\beta)+\lambda J(\beta
)$ and $\ell_{\infty\lambda}=\ell_\infty(\beta)+\lambda J(\beta)$. Denote $G_\lambda=(1/2)D^2\ell_{\infty\lambda}(\bar{\beta}_{\infty
\lambda})$ and $$\tilde{\beta}=\bar{\beta}_{\infty\lambda}-\tfrac12 G_\lambda
^{-1}D\ell
_{n\lambda}(\bar{\beta}_{\infty\lambda}).$$ It is clear that $$\hat{\beta}_{n\lambda}-\bar{\beta}_{\infty\lambda}= (\hat{\beta
}_{n\lambda}-\tilde{\beta} )+ (\tilde{\beta}-\bar{\beta}_{\infty
\lambda
} ).$$ We now study the two terms on the right-hand side separately. For brevity, we shall abbreviate the subscripts of $\hat{\beta}$ and $\bar
{\beta}$ in what follows. We begin with $\tilde{\beta}-\bar{\beta}$. Hereafter we shall omit the subscript for brevity if no confusion occurs.
\[le:stocherr1\] For any $0\le a\le1$, $$E \Vert\tilde{\beta}-\bar{\beta} \Vert_a^2\asymp n^{-1}\lambda^{-
(a+\afrac{1}{2(r+s)} )}.$$
Notice that $D\ell_{n\lambda}(\bar{\beta
})=D\ell_{n\lambda}(\bar{\beta})-D\ell_{\infty\lambda}(\bar
{\beta
})=D\ell_{n}(\bar{\beta})-D\ell_{\infty}(\bar{\beta})$. Therefore $$\begin{aligned}
E [D\ell_{n\lambda}(\bar{\beta})f ]^2
&=&E [D\ell_{n}(\bar{\beta})f-D\ell_{\infty}(\bar{\beta})f ]^2\\[-2pt]
&=&{4\over n} \operatorname{Var}\biggl[ \biggl(Y-\int_\mathcal{T}X(t)\bar{\beta}(t)\,dt \biggr)\int_\mathcal{T}X(t)f(t)\,dt \biggr]\\[-2pt]
&\le&{4\over n} E \biggl[ \biggl(Y-\int_\mathcal{T}X(t)\bar{\beta}(t)\,dt \biggr)\int_\mathcal{T}X(t)f(t)\,dt \biggr]^2\\[-2pt]
&=&{4\over n} E \biggl(\int_\mathcal{T}X(t) [\beta_0(t)-\bar{\beta}(t)]\,dt\int_\mathcal{T}X(t)f(t)\,dt \biggr)^2\\[-2pt]
&&{}+{4\sigma^2\over n}E \biggl(\int_\mathcal{T}X(t)f(t)\,dt \biggr)^2,\end{aligned}$$ where we used the fact that $\varepsilon=Y-\int X\beta_0$ is uncorrelated with $X$. To bound the first term, an application of the Cauchy–Schwarz inequality yields $$\begin{aligned}
&&E \biggl(\int_\mathcal{T}X(t) [\beta_0(t)-\bar{\beta}(t) ]\,dt\int_\mathcal{T}X(t)f(t)\,dt \biggr)^2\\[-2pt]
&&\qquad\le \biggl\{E \biggl(\int_\mathcal{T}X(t) [\beta_0(t)-\bar{\beta}(t) ]\,dt \biggr)^4E\biggl(\int_\mathcal{T}X(t)f(t)\,dt \biggr)^4 \biggr\}^{1/2}\\[-2pt]
&&\qquad\le M \|\beta_0-\bar{\beta}\|^2_0\|f\|_0^2,\end{aligned}$$ where the second inequality holds by the second condition of $\mathcal{F}
(s,M,K)$. Therefore, $$E [D\ell_{n\lambda}(\bar{\beta})f ]^2\le{4M\over n} \|\beta
_0-\bar{\beta
}\|^2_0\|f\|_0^2+{4\sigma^2\over n}\|f\|_0^2,\vadjust{\goodbreak}$$ which by Lemma \[le:detererr\] is further bounded by $(C_0\sigma^2/
n) \|f\|_0^2$ for some positive constant $C_0$. Recall that $\|\omega
_k\|_0=1$. We have $$E [D\ell_{n\lambda}(\bar{\beta})\omega_k ]^2\le C_0\sigma^2/ n.$$ Thus, by the definition of $\tilde{\beta}$, $$\begin{aligned}
E \Vert\tilde{\beta}-\bar{\beta} \Vert_a^2
&=&E \bigg\Vert{1\over 2}G_\lambda^{-1}D\ell_{n\lambda}(\bar{\beta}) \bigg\Vert_a^2\\
&=& {1\over4}E \Biggl[\sum_{k=1}^\infty(1+\gamma_k^{-a})(1+\lambda\gamma_k^{-1})^{-2} (D\ell_{n\lambda}(\bar{\beta})\omega_k )^2 \Biggr]\\
&\le&{C_0\sigma^2\over4n}\sum_{k=1}^\infty(1+\gamma_k^{-a})(1+\lambda\gamma_k^{-1})^{-2}\\
&\le&{C_0\sigma^2\over4n} \sum_{k=1}^\infty\bigl(1+k^{2a(r+s)}\bigr)\bigl(1+\lambda k^{2(r+s)}\bigr)^{-2}\\
&\asymp&{C_0\sigma^2\over4n}\int_1^\infty x^{2a(r+s)}\bigl(1+\lambda x^{2(r+s)}\bigr)^{-2}\,dx\\
&\asymp&{C_0\sigma^2\over4n}\int_1^\infty\bigl(1+\lambda x^{2(r+s)/(2a(r+s)+1)} \bigr)^{-2}\,dx\\
&=&{C_0\sigma^2\over4n}\lambda^{- (a+\afrac{1}{2(r+s)} )} \int_{\lambda^{a+\afrac{1}{2(r+s)}}}^\infty\bigl(1+x^{2(r+s)/(2a(r+s)+1)} \bigr)^{-2}\,dx\\
&\asymp&n^{-1}\lambda^{- (a+\afrac{1}{2(r+s)} )}.\end{aligned}$$ The proof is now complete.
Now we are in position to bound $E\|\hat{\beta}-\tilde{\beta}\|_a^2$. By definition, $$G_\lambda(\hat{\beta}-\tilde{\beta})=\tfrac12D^2\ell_{\infty
\lambda
}(\bar{\beta})(\hat{\beta}-\tilde{\beta}).$$ First-order condition implies that $$D\ell_{n\lambda}(\hat{\beta})=D\ell_{n\lambda}(\bar{\beta
})+D^2\ell
_{n\lambda}(\bar{\beta})(\hat{\beta}-\bar{\beta})=0,$$ where we used the fact that $\ell_{n,\lambda}$ is quadratic. Together with the fact that $$D\ell_{n\lambda}(\bar{\beta})+D^2\ell_{\infty\lambda}(\bar{\beta})(\tilde{\beta}-\bar{\beta})=0,$$ we have $$\begin{aligned}
D^2\ell_{\infty\lambda}(\bar{\beta})(\hat{\beta}-\tilde{\beta})
&=&D^2\ell_{\infty\lambda}(\bar{\beta})(\hat{\beta}-\bar{\beta})+D^2\ell_{\infty\lambda}(\bar{\beta})(\bar{\beta}-\tilde{\beta})\\
&=&D^2\ell_{\infty\lambda}(\bar{\beta})(\hat{\beta}-\bar{\beta})-D^2\ell_{n\lambda}(\bar{\beta})(\hat{\beta}-\bar{\beta})\\
&=&D^2\ell_{\infty}(\bar{\beta})(\hat{\beta}-\bar{\beta})-D^2\ell_{n}(\bar{\beta})(\hat{\beta}-\bar{\beta}).\end{aligned}$$ Therefore, $$(\hat{\beta}-\tilde{\beta})=\tfrac12G_\lambda^{-1} [D^2\ell_{\infty}(\bar{\beta})(\hat{\beta}-\bar{\beta})-D^2\ell_{n}(\bar{\beta})(\hat{\beta}-\bar{\beta}) ].$$ Write $$\hat{\beta}=\sum_{k=1}^\infty\hat{b}_k\omega_k\quad \mbox{and}\quad
\bar{\beta}=\sum_{k=0}^\infty\bar{b}_k\omega_k.$$ Then $$\begin{aligned}
&&\Vert\hat{\beta}-\tilde{\beta} \Vert^2_a
\\
&&\qquad={1\over4}\sum_{k=1}^\infty(1+\lambda\gamma_k^{-1})^{-2}(1+\gamma_k^{-a})
\\
&&\qquad\quad\hspace*{23pt}{} \times\Biggl[\sum_{j=1}^\infty(\hat{b}_j-\bar{b}_j)\int_\mathcal{T}\int_\mathcal{T}
\omega_j(s) \Biggl({1\over n}\sum_{i=1}^nx_i(t)x_i(s)-C(s,t) \Biggr)
\\
&&\qquad\quad\hspace*{217pt}{}\times\omega_k(t)\,ds\,dt\Biggr]^2\\
&&\qquad\le{1\over4}\sum_{k=1}^\infty(1+\lambda\gamma_k^{-1})^{-2}(1+\gamma_k^{-a}) \Biggl[ \sum_{j=1}^\infty(\hat{b}_j-\bar{b}_j)^2(1+\gamma_j^{-c})\Biggr]\\
&&\qquad\quad\hspace*{23pt}{} \times\Biggl(\sum_{j=1}^\infty(1+\gamma_j^{-c})^{-1}
\Biggl[\int_\mathcal{T}\int_\mathcal{T}\omega_j(s) \Biggl({1\over n}\sum_{i=1}^nx_i(t)x_i(s)-C(s,t) \Biggr)
\\
&&\qquad\quad\hspace*{236pt}{}\times\omega_k(t)\,ds\,dt\Biggr]^2 \Biggr),\end{aligned}$$ where the inequality is due to the Cauchy–Schwarz inequality.
Note that $$\begin{aligned}
\hspace*{-5pt}&&E \Biggl(\sum_{j=1}^\infty(1+\gamma_j^{-c})^{-1} \Biggl[\int_\mathcal{T}\omega_j(s)\Biggl({1\over n}\sum_{i=1}^nx_i(t)x_i(s)-C(s,t) \Biggr)\omega_k(t)\,ds\,dt \Biggr]^2 \Biggr)\\
\hspace*{-5pt}&&\qquad={1\over n}\sum_{j=1}^\infty(1+\gamma_j^{-c})^{-1}\operatorname{Var}\biggl(\int_\mathcal{T}\omega_j(t)X(t)\,dt\,\int_\mathcal{T}\omega_k(t)X(t)\,dt \biggr)\\
\hspace*{-5pt}&&\qquad\le{1\over n}\sum_{j=1}^\infty(1+\gamma_j^{-c})^{-1} E \biggl[ \biggl(\int_\mathcal{T}\omega_j(t)X(t)\,dt \biggr)^2 \biggl(\int_\mathcal{T}\omega_k(t)X(t)\,dt \biggr)^2 \biggr]\\
\hspace*{-5pt}&&\qquad\le{1\over n}\sum_{j=1}^\infty(1+\gamma_j^{-c})^{-1} E \biggl[ \biggl(\int_\mathcal{T}\omega_j(t)X(t)\,dt \biggr)^4\biggr]^{1/2}
%&&\qquad\quad\hspace*{20pt}{}\times
E \biggl[ \biggl(\int_\mathcal{T}\omega_k(t)X(t)\,dt \biggr)^4\biggr]^{1/2}\\
\hspace*{-5pt}&&\qquad\le{M\over n}\sum_{j=1}^\infty(1+\gamma_j^{-c})^{-1} E \biggl[ \biggl(\int_\mathcal{T}\omega_j(t)X(t)\,dt \biggr)^2 \biggr]E \biggl[ \biggl(\int_\mathcal{T}\omega_k(t)X(t)\,dt \biggr)^2\biggr]\\
\hspace*{-5pt}&&\qquad={M\over n}\sum_{j=1}^\infty(1+\gamma_j^{-c})^{-1} \asymp n^{-1},\end{aligned}$$ provided that $c>1/2(r+s)$. On the other hand, $$\begin{aligned}
&&\sum_{k=1}^\infty(1+\lambda\gamma_k^{-1})^{-2}(1+\gamma_k^{-a})
\\
&&\qquad\le C_0\sum_{k=1}^\infty\bigl(1+\lambda k^{2(r+s)}\bigr)^{-2}\bigl(1+k^{2a(r+s)}\bigr)\\
&&\qquad\asymp\int_1^\infty\bigl(1+\lambda x^{2(r+s)} \bigr)^{-2}x^{2a(r+s)}\,dx\\
&&\qquad\asymp\int_1^\infty\bigl(1+\lambda x^{2(r+s)/(2a(r+s)+1)} \bigr)^{-2}\,dx\\
&&\qquad=\lambda^{- (a+1/(2(r+s)) )}\int_{\lambda^{a+1/(2(r+s))}}^\infty\bigl(1+x^{2(r+s)/(2a(r+s)+1)} \bigr)^{-2}\,dx\\
&&\qquad\asymp\lambda^{- (a+1/(2(r+s)) )}.\end{aligned}$$
To sum up, $$\label{eq:hat-til}
\Vert\hat{\beta}-\tilde{\beta} \Vert^2_a=O_p \bigl(n^{-1}\lambda^{-
(a+1/(2(r+s)) )} \Vert\hat{\beta}-\bar{\beta} \Vert^2_c \bigr).$$ In particular, taking $a=c$ yields $$\Vert\hat{\beta}-\tilde{\beta} \Vert^2_c=O_p \bigl(n^{-1}\lambda^{-
(c+1/(2(r+s)) )} \Vert\hat{\beta}-\bar{\beta} \Vert^2_c \bigr).$$ If $$n^{-1}\lambda^{- (c+1/(2(r+s)) )} \to0,$$ then $$\Vert\hat{\beta}-\tilde{\beta} \Vert_c=o_p ( \Vert\hat{\beta
}-\bar
{\beta} \Vert_c ).$$ Together with the triangular inequality $$\Vert\tilde{\beta}-\bar{\beta} \Vert_c \ge\Vert\hat{\beta
}-\bar
{\beta} \Vert_c- \Vert\hat{\beta}-\tilde{\beta} \Vert_c=\bigl(1-o_p(1)\bigr)
\Vert\hat{\beta}-\bar{\beta} \Vert_c.$$ Therefore, $$\Vert\hat{\beta}-\bar{\beta} \Vert_c=O_p ( \Vert\tilde{\beta
}-\bar
{\beta} \Vert_c )$$ Together with Lemma \[le:stocherr1\], we have $$\Vert\hat{\beta}-\bar{\beta} \Vert^2_c=O_p \bigl(n^{-1}\lambda^{-
(c+\afrac{1}{2(r+s)} )} \bigr)=o_p(1).$$
Putting it back to (\[eq:hat-til\]), we now have:
\[le:stocherr2\] If there also exists some ${1/2(r+s)}<c\le1$ such that $n^{-1}\times\lambda
^{- (c+{1/2(r+s)} )}\to0$, then $$\Vert\hat{\beta}-\tilde{\beta} \Vert^2_a=o_p \bigl(n^{-1}\lambda^{-
(a+{1/2(r+s)} )} \bigr).$$
Combining Lemmas \[le:detererr\]–\[le:stocherr2\], we have $$\begin{aligned}
\hspace*{30pt}&&\lim_{D\to\infty} \mathop{\overline{\lim}}_{n\to\infty} \sup
_{F\in\mathcal{F}
(s,M,K),\beta_0\in
\mathcal{H}} P \bigl(\|\hat{\beta}_{n\lambda}-\beta_0\|
_a^2>Dn^{-\afrac{2(1-a)(r+s)}{
2(r+s)+1}} \bigr)\nonumber
\\[-8pt]\\[-8pt]
&&\qquad=0\nonumber\end{aligned}$$ by taking $\lambda\asymp n^{-2(r+s)/(2(r+s)+1)}$.
Proof of Theorem 7
------------------
We now set out to show that $n^{-2(1-a)(r+s)/(2(r+s)+1)}$ is the optimal rate. It follows from a similar argument as that of Hall and Horowitz ([-@HallHorowitz2007]). Consider a setting where $\psi
_k=\phi_k$, $k=1,2,\ldots
.$ Clearly in this case we also have $\omega_k=\mu_k^{-1/2}\phi_k$. It suffices to show that the rate is optimal in this special case. Recall that $\beta_0=\sum a_k\phi_k$. Set $$a_k= \cases{L_n^{-1/2} k^{-r}\theta_k, &\quad $L_{n}+1\le k\le2L_n$,\cr
0,&\quad \mbox{otherwise},
}$$ where $L_n$ is the integer part of $n^{1/(2(r+s)+1)}$, and $\theta_k$ is either $0$ or $1$. It is clear that $$\|\beta_0\|_K^2\le\sum_{k=L_n+1}^{2L_n} L_n^{-1} =1.$$ Therefore $\beta_0\in\mathcal{H}$. Now let $X$ admit the following expansion: $X=\sum_k \xi_k k^{-s}\phi_k$ where $\xi_k$s are independent random variables drawn from a uniform distribution on $[-\sqrt
{3},\sqrt
{3}]$. Simple algebraic manipulation shows that the distribution of $X$ belongs to $\mathcal{F}(s, 3)$. The observed data are $$y_i=\sum_{k=L_n+1}^{2L_n} L_n^{-1/2}k^{-(r+s)}\xi_{ik}\theta
_k+\varepsilon_i,\qquad i=1,\ldots, n,$$ where the noise $\varepsilon_i$ is assumed to be independently sampled from $N(0,M_2)$. As shown in Hall and Horowitz ([-@HallHorowitz2007]), $$\lim_{n\to\infty} \inf_{L_n<j\le2L_n}\inf_{\tilde{\theta}_j}\sup^\ast E(\tilde{\theta}_j-\theta_j)^2>0,$$ where $\mathop{\sup^{}}\limits^{*}$ denotes the supremum over all $2^{L_n}$ choices of $(\theta_{L_n+1},\ldots, \theta_{2L_n})$, and $\inf
_{\tilde
{\theta}}$ is taken over all measurable functions $\tilde{\theta}_j$ of the data. Therefore, for any estimate $\tilde{\beta}$, $$\begin{aligned}
\label{eq:mseldb}
\sup^\ast\|\tilde{\beta}-\beta_0\|_a^2
&=&\sup^\ast\sum_{k=L_n+1}^{2L_n}
L_n^{-1}k^{-2(1-a)(r+s)}E(\tilde{\theta}_j-\theta_j)^2\nonumber
\\[-8pt]\\[-8pt]
&\ge& Mn^{-\afrac{2(1-a)(r+s)}{2(r+s)+1}}\nonumber\end{aligned}$$ for some constant $M>0$.
Denote $$\tilde{\hspace*{-2pt}\tilde{\theta}}_k=
\cases{1, &\quad $\tilde{\theta}_k>1$,\cr
\tilde{\theta}_k,&\quad $0\le\tilde{\theta}_k\le1$,\cr
0, &\quad $\tilde{\theta}_k<0$.
}$$ It is easy to see that $$\begin{aligned}
&&\sum_{k=L_n+1}^{2L_n} L_n^{-1}k^{-2(1-a)(r+s)}(\tilde{\theta}_j-\theta_j)^2\nonumber
\\[-8pt]\\[-8pt]
&&\qquad\ge\sum_{k=L_n+1}^{2L_n} L_n^{-1}k^{-2(1-a)(r+s)}(\hspace*{2pt}\tilde{\hspace*{-2pt}\tilde{\theta}}_j-\theta_j)^2.\nonumber\end{aligned}$$ Hence, we can assume that $0\le\tilde{\theta}_j\le1$ without loss of generality in establishing the lower bound. Subsequently, $$\begin{aligned}
\sum_{k=L_n+1}^{2L_n} L_n^{-1}k^{-2(1-a)(r+s)}(\tilde{\theta}_j-\theta_j)^2
&\le&\sum_{k=L_n+1}^{2L_n} L_n^{-1}k^{-2(1-a)(r+s)}\nonumber
\\[-8pt]\\[-8pt]
&\le& L_n^{-2(1-a)(r+s)}.\nonumber\end{aligned}$$ Together with (\[eq:mseldb\]), this implies that $$\lim_{n\to\infty}\inf_{\tilde{\beta}}\sup^\ast
P \bigl(\Vert\tilde{\beta}-\beta\Vert_a^2>dn^{-\afrac{2(1-a)(r+s)}{2(r+s)+1}} \bigr)>0$$ for some constant $d>0$.
Appendix: Sacks–Ylvisaker conditions {#Appendix .unnumbered}
====================================
In Section \[diagonal.sec\], we discussed the relationship between the smoothness of $C$ and the decay of its eigenvalues. More precisely, the smoothness can be quantified by the so-called Sacks–Ylvisaker conditions. Following Ritter, Wasilkowski and Woźniakowski ([-@RitterWasilkowskiWozniakowski1995]), denote $$\begin{aligned}
\Omega_+&=&\{(s,t)\in(0,1)^2\dvtx s>t\}\quad \mbox{and}\nonumber
\\[-8pt]\\[-8pt]
\Omega_-&=&\{(s,t)\in(0,1)^2\dvtx s<t\}.\nonumber\end{aligned}$$ Let $\operatorname{cl}(A)$ be the closure of a set $A$. Suppose that $L$ is a continuous function on $\Omega_+\cup\Omega_-$ such that $L|_{\Omega_j}$ is continuously extendable to $\operatorname{cl}(\Omega_j)$ for $j\in\{+,-\}$. By $L_j$ we denote the extension of $L$ to $[0,1]^2$, which is continuous on $\operatorname{cl}(\Omega_j)$, and on $[0,1]^2 \setminus\operatorname{cl}(\Omega_j)$. Furthermore write $M^{(k,l)}(s,t)=(\partial
^{k+l}/(\partial s^k \,\partial t^l))M(s,t)$. We say that a covariance function $M$ on $[0,1]^2$ satisfies the Sacks–Ylvisaker conditions of order $r$ if the following three conditions hold:
(A) $L=M^{(r,r)}$ is continuous on $[0,1]^2$, and its partial derivatives up to order 2 are continuous on $\Omega_+\cup\Omega_-$, and they are continuously extendable to $\operatorname{cl}(\Omega_+)$ and $\operatorname{cl}(\Omega_-)$.
(B) $$\min_{0\le s\le1} \bigl(L_-^{(1,0)}(s,s)-L_+^{(1,0)}(s,s) \bigr)>0.$$
(C) $L_+^{(2,0)}(s,\cdot)$ belongs to the reproducing kernel Hilbert space spanned by $L$ and furthermore $$\sup_{0\le s\le1} \big\Vert L_+^{(2,0)}(s,\cdot) \big\Vert_L<\infty.$$
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|
---
author:
- |
Wei Min Jin\
[Department of Physics and Astronomy, State University of New York at Buffalo,]{}\
[Buffalo, NY 14260-1500, U.S.A.]{}\
[Electronic mail: [email protected]]{}
title: '**Quantization of Dirac fields in static spacetime**'
---
0.5in
[**Abstract**]{}. On a static spacetime, the solutions of Dirac equation are generated by a time-independent Hamiltonian. We study this Hamiltonian and characterize the split into positive and negative energy. We use it to find explicit expressions for advanced and retarded fundamental solutions and for the propagator. Finally we use a fermion Fock space based on the positive/negative energy split to define a Dirac quantum field operator whose commutator is the propagator.
1. Introduction {#introduction .unnumbered}
===============
In the theoretical framework of Dirac fields in curved space-time, many fundamental results have been obtained by Lichnorowicz \[1\]. There are also quite a few standard references with detailed discussions, for example, see \[2\]\[3\]\[4\]. A very general theory has been established by Dimock \[5\], for Dirac quantum fields on hyperbolic Lorentzian manifold. For most up-to-date research along this line, one can find recent papers by Hollands \[6\] and Kratzert \[7\], and references therein.
Here in this paper we focus on solving a very specific and well-posed problem: the quantization of Dirac fields in static space-time. Technically speaking, the separation of time and space parts of Dirac fields plays a pivotal role in doing field quantization. Since only static space-time is considered here, the following results about other type of fields are also of relevance: the quantization of Klein-Gordon scalar fields on stationary manifold by Kay \[8\], and the quantization of electromagnetic fields and massive vector fields on static space-time by Furlani \[9\]\[10\].
This paper proceeds as follows. First we present some preliminary results for classical Dirac fields. In static space-time, the time component of the spin affine connection of Dirac fields vanishes and the other components are all independent of time. Hence one can separate space and time. The dynamics can be expressed in terms of a time-independent Hamiltonian. We prove that the Hamiltonian is essentially self-adjoint. We also characterize the positive and negative energy subspaces. This leads to explicit expressions for various fundamental solutions and for the propagator function.
For the quantum problem, we first define an appropriate fermion Fock space based on the positive/negative energy split. Then Dirac quantum field operators are defined using the creation and annihilation operators on this Fock space and projections onto the positive and negative energy subspaces. The field operator is shown to have a commutator which is the propagator function. Finally we study the unitary implementability of the time evolution.
2. Dirac fields in static space-time {#dirac-fields-in-static-space-time .unnumbered}
====================================
In this section, we present some preliminary results that are relevant to our work. On Lorentzian manifold $L$, the first-order equation of free Dirac fields can be written as $$(\hbox{\rlap/P}-m)\psi=0,\eqno(2.1a)$$ $$\psi^*(\hbox{\rlap/P}^*-m)=0,\eqno(2.1b)$$ where, by the notation in \[11\]\[12\], $\psi^*=\psi^\dagger\gamma^0$ and $\hbox{\rlap/P}^*=\gamma^0\hbox{\rlap/P}^\dagger\gamma^0$ and $$\hbox{\rlap/P}=i\gamma^\mu\nabla_\mu=i\gamma^\mu(\partial_\mu-\Gamma_\mu).\eqno(2.2)$$ Here $\gamma^\mu=V^\mu_a(x)\gamma^a$ are spinor tensors, with the introduction of vierbein fields $V^\mu_a(x)$ and Dirac matrices $\gamma^a(a=0,1,2,3)$ by the convention in \[13\]. The components of spin affine connection are $$\Gamma_\mu={1\over2}G^{[a,b]}(\hbox{\tensl D}_\mu V^\nu_a)V_{\nu
b},\eqno(2.3)$$ where $G^{[a,b]}={1\over4}[\gamma^a,\gamma^b]$ are the generators of Lorentz group, and $\hbox{\tensl D}_\mu V^\nu_a=\partial_\mu
V^\nu_a+\Gamma^\nu_{\mu\lambda}V^\lambda_a$ are the covariant derivatives of vierbein fields on space.
The second-order equation of Dirac fields can be written as $$(\square -m^2)\psi=0,\eqno(2.4)$$ with an operator $$\square =\hbox{\rlap/P}^2=-\nabla^\mu\nabla_\mu+{1\over4}R,\eqno(2.5)$$ where $R$ is the Riemann scalar.
To preserve manifest covariance on Lorentzian Manifold $L$, we may introduce an indefinite inner product \[1\]: $$<u,v>=\int u^*(x)v(x)d_g^4x,\hskip0.1in u,v\in C^\infty(L;C^4),\eqno(2.6)$$ where $d_g^4x=\sqrt{g}d^4x$ is the invariant density with $g=-\det(g^{\mu\nu})$ and $u^*=u^\dagger\gamma^0$ is the adjoint of $u$. The inner product is invariant under both global coordinate and local Lorentz transformations.
The adjoint operator $A^*$ of $A$ is defined by $$<u,A^*v>=<Au,v>,\eqno(2.7)$$ namely $$A^*=\gamma^0A^\dagger\gamma^0.\eqno(2.8)$$ A symmetric operator $O$ satisfies $$<u,Ov>=<Ou,v>,\eqno(2.9)$$ namely $$O^*=\gamma^0O^\dagger\gamma^0=O.\eqno(2.10)$$ Symmetric operators play essential roles in functional analysis.
Let us consider a static manifold $R\times M$ where $M$ is compact, with metric elements $g^{\mu\nu}$ of signature $(1,-1,-1,-1)$, $$[g^{\mu\nu}]=\left[\begin{array}{cc}1&0\\
0&g^{ij}({\bf x})\end{array}\right],\hskip0.1in(i,j=1,2,3)\eqno(2.11)$$ where Greek indices apply to 4-d static space-time, and Latin indices apply to 3-d static space. The 3-d static space $M$ is a Cauchy surface of the 4-d static space-time $R\times M$.
Let $u$ and $v$ have compact support. Then check Green’s identity in static space-time $L=R\times M$ where $M$ is compact without boundary $$<\hbox{\rlap/P}u,v>-<u,\hbox{\rlap/P}v>=\int_Ld_g^4x[(\hbox{\rlap/P}u)^*v-u^*\hbox{\rlap/P}v]$$ $$=-i\int_Ld_g^4x[(\nabla_\nu u)^*\gamma^\mu v+u^*\gamma^\mu\nabla_\mu v]$$ $$=-i\int_Ld_g^4x\nabla_\mu(u^*\gamma^\mu v),\eqno(2.12)$$ where $\nabla_\mu\gamma^\nu=0$ has been used. This integral is the same as the integral over $[-T,T]\times M$ for $T$ sufficiently large depending on the test functions. The integral of the divergence is an integral over the boundary of this region which is $-T\times M$ and $T\times M$. The surface integrals are zero. We therefore have $$<\hbox{\rlap/P}u,v>=<u,\hbox{\rlap/P}v>,\eqno(2.13)$$ namely the operator [/P]{} is symmetric $$\hbox{\rlap/P}^*=\hbox{\rlap/P}.\eqno(2.14)$$ Obviously $\Delta=\hbox{\rlap/P}^2$ is also symmetric by the same arguments as for [/P]{}.
The vierbein fields satisfy $g_{\mu\nu}(x)V^\mu_a(x)V^\nu_b(x)=\eta_{ab}$ with a Minkowski metric $\left\{\eta_{ab}\right\}=diag(1,-1,-1,-1)$. In static space-time, the vierbein fields $V^0_0=1,\hskip0.05in V^0_i=0=V^i_0$ and $V^i_a({\bf
x})\hskip0.05in(i,a=1,2,3)$ are all independent of time. Then by (2.3) it is easy to show that the time component of the spin affine connection vanishes: $$\Gamma^\nu_{0\lambda}={1\over2}g^{\nu\sigma}(\partial_\lambda
g_{\sigma0}+\partial_0g_{\sigma\lambda}-\partial_\sigma g_{0\lambda})=0,$$ $$\hbox{\tensl
D}_0V^\nu_a=\partial_0V^\nu_a+\Gamma^\nu_{0\lambda}V^\lambda_a=0,$$ $$\Gamma_0={1\over2}G^{[a,b]}(\hbox{\tensl D}_0V^\nu_a)V_{\nu b}=0,\eqno(2.15)$$ and also the other components $\Gamma_i({\bf
x})\hskip0.05in(i=1,2,3)$ are all independent of time.
Separating time from space, we assume there exists a global dreibein field on $M$. If there is not a dreibein field, our analysis should still hold true, however the spinor fields will be sections of a vector bundle \[5\]. The simplest example of a compact manifold $M$ with a global dreibein field is $M=T^3$, the three torus or periodic box.
3. Separation of energy spectrum {#separation-of-energy-spectrum .unnumbered}
================================
With the above results in static space-time, the Dirac equation (2.1a) turns out to be $$i\partial_t\psi(t,{\bf x})=H\psi(t,{\bf x}),\eqno(3.1)$$ where a time-independent Hamiltonian is $$H=-i\gamma^0\gamma^i\nabla_i+\gamma^0m.\eqno(3.2)$$ And the Hamiltonian squared is $$H^2=\nabla^i\nabla_i-{1\over4}R+m^2.\eqno(3.3)$$
Since we know $\gamma^i({\bf x})=V^i_a({\bf x})\gamma^a$ and $-\gamma^0\gamma^a=\sigma^1\otimes\sigma^a$ where $\sigma^a$ are Pauli matrices. The Hamiltonian can then be written in a matrix form $$H=\left[\begin{array}{cc}m&Q\\
Q&-m\end{array}\right],\eqno(3.4)$$ where we denote $Q=i\sigma^i\nabla_i$ and $\sigma^i=V^i_a({\bf
x})\sigma^a\hskip0.05in(i,a=1,2,3)$. The Hamiltonian squared becomes $$H^2=(m^2+Q^2)\left[\begin{array}{cc}I&0\\
0&I\end{array}\right],\eqno(3.5)$$ where $Q^2=\nabla^i\nabla_i-{1\over4}R$.
Define a positive-definite inner product on $M$ $$(\chi,\varphi)=\int d_g^3{\bf x}(\chi^\dagger\varphi),\eqno(3.6)$$ where $d_g^3{\bf x}=\sqrt{g}d^3{\bf x}$. Then define a Hilbert space $$\hbox{\tensl H}=L^2(M,C^4,d_g^3{\bf x}).\eqno(3.7)$$
Since the vierbein fields $V^i_a({\bf x})$ are real functions of space on $M$ and Pauli matrices $\sigma^a$ are symmetric on $C^2$, $\sigma^i=V^i_a({\bf x})\sigma^a$ are thus symmetric in $L^2(M,C^2)$. It can be easily checked that ${G^{[a,b]}}^\dagger=-G^{[a,b]}$. By (2.3), we know $(i\Gamma_i)^\dagger=i\Gamma_i$ and $i\Gamma_i$ are some real functions of space on $M$ after summation of indices. So $i\nabla_i=i(\partial_i-\Gamma_i)$ is symmetric in $L^2(M)$, and $i\sigma^i\nabla_i$ is symmetric in $L^2(M,C^2)$. Then we know the Hamiltonian $H$ is symmetric in $L^2(M,C^4)$.
Let $A$ be the closure of $H$ on $C^\infty(M,C^4)$ in $L^2(M,C^4)$. Define the domain of $A$ as follows $$D(A)=\Bigl\{\psi\in\hbox{\tensl H}:\hskip0.05in\exists\psi_n\in C^\infty M,\hskip0.05in\lim_{n\rightarrow\infty}\psi_n\rightarrow\psi,\hskip0.05in
\lim_{n\rightarrow\infty}A\psi_n\hskip0.05in exists\Bigr\}.\eqno(3.8)$$ Then $A: D(A)\rightarrow\hbox{\tensl H}$ is given by defining $$A\psi=\lim_{n\rightarrow\infty}A\psi_n.\eqno(3.9)$$ Let $B=A^2$ be the closure of $H^2$ on $C^\infty(M,C^4)$.
0.1in [*Lemma 1*]{}. $B$ is self-adjoint in [H]{}, i.e. $H^2$ is essentially self-adjoint on $C^\infty(M,C^4)$.
0.1in [*Proof*]{}: It has been proven that the Laplacian operator $-\nabla^i\nabla_i$ is self-adjoint in [H]{} \[14\]\[15\]. Since $-{1\over4}R+m^2$ is a continuous function on a compact manifold, it is a bounded function and hence a bounded operator in [H]{} \[16\]\[17\]. By Kato-Rellich theorem \[18\], $B$, a closure of $H^2=\nabla^i\nabla_i-{1\over4}R+m^2$, is self-adjoint in [H]{}. This is equivalent to say $H^2$ is essentially self-adjoint on $C^\infty(M,C^4)$. $\square$
0.1in [*Lemma 2*]{}. $A$ is self-adjoint in [H]{}, i.e. $H$ is essentially self-adjoint on $C^\infty(M,C^4)$. 0.1in [*Proof*]{}: This is equivalent to show \[16\] $$Ran(A\pm i)=\hbox{\tensl H}. \eqno(3.10)$$ To prove it, we need to find $\psi$ so that $$(A\pm i)\psi=\chi,\hskip0.1in for\hskip0.1in \chi\in\hbox{\tensl
H}.\eqno(3.11)$$ By observation, the answer should be $$\psi=(A\mp i)(B+1)^{-1}\chi.\eqno(3.12)$$ It then suffices to show $$\varphi=(B+1)^{-1}\chi\in D(A),\eqno(3.13a)$$ $$\psi=(A\mp i)\varphi\in D(A).\eqno(3.13b)$$
By [*Lemma 1*]{}, for any $\varphi\in D(B)$ we can find smooth $\varphi_n$ so that $$\lim_{n\rightarrow\infty}\varphi_n\rightarrow\varphi,\hskip0.1in\lim_{n\rightarrow\infty}B\varphi_n\rightarrow
B\varphi.\eqno(3.14)$$ Now we derive $$\parallel A\varphi_n-A\varphi_m\parallel^2=(\varphi_n-\varphi_m,A^2(\varphi_n-\varphi_m))$$ $$\leq\parallel\varphi_n-\varphi_m\parallel\times\parallel
B(\varphi_n-\varphi_m)\parallel\rightarrow0.\eqno(3.15)$$ So we know $\varphi\in D(A)$ by (3.8). Then we may define smooth $\psi_n=(A\mp i)\varphi_n$ so that $\psi_n \rightarrow\psi$ and derive $$\parallel A\psi_n-A\psi_m\parallel^2=(A^2(\varphi_n-\varphi_m),(A^2+1)(\varphi_n-\varphi_m))$$ $$\leq\parallel B(\varphi_n-\varphi_m)\parallel\times\parallel
(B+1)(\varphi_n-\varphi_m)\parallel\rightarrow0.\eqno(3.16)$$ Thus we know $\psi\in D(A)$. By the above analysis, $A$ is self-adjoint in [H]{}, and equivalently $H$ is essentially self-adjoint on $C^\infty(M,C^4)$. $\square$
0.1in By [*Lemma 1*]{}, $H^2$ is essentially self-adjoint and positive, and hence has a square root. We may define a positive scalar energy operator $$\omega=(m^2+Q^2)^{1/2}=(\nabla^i\nabla_i-{1\over4}R+m^2)^{1/2},\eqno(3.17)$$ where $Q=i\sigma^i\nabla_i$ for $\sigma^i=V^i_a({\bf x})\sigma^a$. By [*Lemma 2*]{}, the closure of $H$ is self-adjoint, and similarly the closure of $Q$ is self-adjoint, then $Q^2=\nabla^i\nabla_i-R/4\geq0$ and $\omega\geq m$. It is now straightforward to prove the following Theorem.
0.1in [*Theorem 1*]{}: The Hilbert space [H]{} splits into the positive and negative subspaces: $\hbox{\tensl H}=\hbox{\tensl H}^+\oplus\hbox{\tensl H}^-$. Then for $\psi_\pm\in\hbox{\tensl H}^\pm\cap D(H)$, we have $$H\psi_\pm=\pm\omega\psi_\pm,\eqno(3.18)$$ where $\omega$ is expressed by (3.17). And also $\psi_\pm$ are of the following form: $$\psi_+=T\left[\begin{array}{c}f\\
0\end{array}\right],\hskip0.1in\psi_-=T\left[\begin{array}{c}0\\
h\end{array}\right],\hskip0.1in f,h\in L^2(M;C^2),\eqno(3.19)$$ where $T$ is the unitary operator $$T=N\left[\begin{array}{cc}{\omega+m}&-Q\\
Q&{\omega+m}\end{array}\right],\eqno(3.20)$$ with $N=[2\omega(\omega+m)]^{-1/2}$.
0.1in [*Proof*]{}: We diagonalize the Hamiltonian (3.4) to $$H'=T^{-1}HT=\left[\begin{array}{cc}\omega&0\\
0&-\omega\end{array}\right],\eqno(3.21)$$ by using a transformation operator (3.20). The inverse of $T$ is $$T^{-1}=N\left[\begin{array}{cc}{\omega+m}&Q\\
-Q&{\omega+m}\end{array}\right].\eqno(3.22)$$ Both $T$ and $T^{-1}$ are norm-preserving: $$\int(T\varphi)^\dagger T\varphi d\mu=\int\varphi^\dagger\varphi d\mu=\int(T^{-1}\varphi)^\dagger T^{-1}\varphi d\mu, \eqno(3.23)$$ so $T$ is unitary.
Any $\psi$ in [H]{} can be written as $$\psi=T\left[\begin{array}{c}f\\
h\end{array}\right],\hskip0.1in f,h\in L^2(M;C^2)\eqno(3.24)$$ and hence $\psi=\psi_+ +\psi_-$ where $$\psi_+=T\left[\begin{array}{c}f\\
0\end{array}\right],\hskip0.1in \psi_-=T\left[\begin{array}{c}0\\
h\end{array}\right],\hskip0.1in f,h\in L^2(M;C^2).\eqno(3.25)$$ If also $\psi_\pm \in D(H)$, then $$H\psi_\pm=\pm\omega\psi_\pm.\eqno(3.26)$$ Thus we have exhibited the split into the positive and negative energy spectra. (Note that 0 is not in the spectra since $\omega\geq m>0$.) $\square$
0.1in By the above [*Theorem 1*]{}, we can now express Dirac fields by summing up both positive and negative energy parts $$\psi(t,{\bf x})=U(t)\psi_+({\bf x})+U(-t)\psi_-({\bf x}),\eqno(3.27)$$ which is a solution of Dirac equation with data $\psi$. Here $\psi_\pm$ satisfy (3.25), and a unitary operator $U(t)=\exp(-i\omega t)$ satisfies $U^\dagger(t)=U(-t)=U^{-1}(t)$ with $\omega$ given by (3.17).
4. Propagator of Dirac equation {#propagator-of-dirac-equation .unnumbered}
===============================
To obtain the propagator of Dirac equation, we start from the second-order inhomogeneous equation $$(\square-m^2)\psi=\rho,\eqno(4.1)$$ where $\square$ is expressed by (2.5). The fundamental solutions of this inhomogeneous equation are defined by $$(\square_x-m^2)E_F(x,y)=\delta(x,y).\eqno(4.2)$$ Here the $\delta$-function is defined by a bispinor \[1\] $$\delta(x,y)=\delta^\alpha_\beta\delta^4(x,y).\eqno(4.3)$$
The general discussions about the existence and uniqueness of fundamental solutions of hyperbolic differential equations such as (4.1) can be found in \[3\]\[4\]. Now we are going to do a formal calculation to find a representation for the advanced and retarded fundamental solutions $E_A, E_R$.
On static metric $R\times M$, we separate the time and space parts of the solutions. It is translation-invariant along time direction but not necessarily along space direction. We make Fourier transform along time axis but leave the space part alone. The fundamental solutions can be written as $$E_F(x,y)={1\over2\pi}\int e^{-ikt}E_k({\bf x},{\bf y})dk,\eqno(4.4)$$ where $$t=x_0-y_0. \eqno(4.5)$$ Inserting (4.4) into (4.2) leads to $$(k^2-\nabla^i\nabla_i+{1\over4}R-m^2)E_k({\bf x},{\bf
y})=\delta({\bf x},{\bf y}).\eqno(4.6)$$ We formally write k-component solution $$E_k({\bf x},{\bf y})=(k^2-\omega^2)^{-1}\delta({\bf x},{\bf y}),\eqno(4.7)$$ with $\omega^2=\nabla^i\nabla_i-R/4+m^2$. So (4.4) formally becomes $$E_F(x,y)={1\over2\pi}\int{e^{-ikt}\over{k^2-\omega^2}}\delta({\bf
x},{\bf y})dk,\eqno(4.8)$$ which is singular with a delta function. Let us smear it by two test functions $\chi({\bf x})$ and $\varphi({\bf y})$ on $M$ $$E_F(t;\chi,\varphi)={1\over2\pi}\int
e^{-ikt}(\chi,{1\over{k^2-\omega^2}}\varphi)dk.\eqno(4.9)$$ Let $P_\lambda=P_{(-\infty,\lambda]}$ be a projection-valued measure of self-adjoint operator $\omega$. Its family $\left\{P_\lambda\right\}$ exists by the Spectral Theorem \[16\]. Thus (4.9) becomes[^1] $$E_F(t;\chi,\varphi)={1\over2\pi}\int\int{e^{-ikt}\over{k^2-\lambda^2}}dk(\chi,dP_\lambda
\varphi),\hskip0.1in \forall\chi,\varphi\in\hbox{\tensl H}.\eqno(4.10)$$ We may now prove the following proposition.
0.1in [*Proposition 1*]{}: The advanced and retarded fundamental solutions in integral representation, defined by $$E_A(t;\chi,\varphi)={1\over2\pi}\int\int_{\Gamma_A}{e^{-ikt}\over{k^2-\lambda^2}}dk(\chi,dP_\lambda
\varphi),\hskip0.1in \forall\chi,\varphi\in\hbox{\tensl H},\eqno(4.11a)$$ $$E_R(t;\chi,\varphi)={1\over2\pi}\int\int_{\Gamma_R}{e^{-ikt}\over{k^2-\lambda^2}}dk(\chi,dP_\lambda
\varphi),\hskip0.1in \forall\chi,\varphi\in\hbox{\tensl H},\eqno(4.11b)$$ vanish in the future and past respectively. Here $\Gamma_A$ ($\Gamma_R$) is a straight line slightly below (above) the real k-axis.
0.1in [*Proof*]{}: Obviously (4.10) has two poles $k=\pm\lambda$ for each $\lambda$ in the integral over $k$. Let us avoid two poles in (4.11a) by taking the integral along a straight line $\Gamma_A$ which passes slightly below the real k-axis. This is equivalent to moving the poles to slightly above the real k-axis. When $t>0$, we calculate the integral by closing the contour in the lower half of the complex k-plane. Since $Re(-ikt)<0$, the infinite semicircle boundary in the lower half plane does not contribute. And there is no pole inside the closed contour, $E_A$ vanishes in the future when $t>0$. Similarly $E_R$ defined by (4.11b) vanishes in the past when $t<0$, by the integral along a straight line $\Gamma_R$ which passes slightly above the real k-axis. It is also straightforward to check that the expressions are actually fundamental solutions. From \[3\]\[4\], the fundamental solutions of hyperbolic differential equations have support only in light cone. So the advanced and retarded fundamental solutions $E_A$ and $E_R$ have support only in the past and future light cones respectively. $\square$
0.1in Let us take the difference of (4.11a) and (4.11b) by closing contour $C$ around both poles $$E(t;\chi,\varphi)=E_A(t;\chi,\varphi)-E_R(t;\chi,\varphi)$$ $$={1\over2\pi}\int\oint_C{e^{-ikt}\over{k^2-\lambda^2}}dk(\chi,dP_\lambda
\varphi).\eqno(4.12)$$ It can also be split into the positive and negative parts by choosing the contours around both poles: $$E(t;\chi,\varphi)=E_+(t;\chi,\varphi)+E_-(t;\chi,\varphi), \eqno(4.13)$$ where $$E_\pm(t;\chi,\varphi)={1\over2\pi}\int\oint_{C_\pm}{e^{-ikt}\over{k^2-\lambda^2}}dk(\chi,dP_\lambda
\varphi)$$ $$=\pm i\int{e^{\mp i\lambda t}\over{2\lambda}}(\chi,dP_\lambda
\varphi)$$ $$=\pm i(\chi, {e^{\mp i\omega
t}\over{2\omega}}\varphi).\eqno(4.14)$$
The fundamental solutions of the inhomogeneous Dirac equation $$(\hbox{\rlap/P}-m)\psi=\rho,\eqno(4.15)$$ are defined by $$(\hbox{\rlap/P}-m)S_F(x,y)=\delta(x,y).\eqno(4.16)$$ From $(\hbox{\rlap/P}-m)(\hbox{\rlap/P}+m)=\square-m^2$, we see $$S_F(x,y)=(\hbox{\rlap/P}+m)E_F(x,y),\eqno(4.17)$$ which can be smeared by two test functions $\chi({\bf x})$ and $\varphi({\bf y})$ on $M$: $$S_F(t;\chi,\varphi)={1\over2\pi}\int
e^{-ikt}dk(\chi,{{\hbox{\rlap/P}(k)+m}\over{k^2-\omega^2}}\varphi),\eqno(4.18)$$ where the operator [/P]{}$(k)$ is also a function of $k$: $$\hbox{\rlap/P}(k)=\gamma^0k+i\gamma^i\nabla_i.\eqno(4.19)$$
Following the discussions about $E_\pm$ (4.14), we obtain $$S_\pm(t;\chi,\varphi)={1\over2\pi}\int\oint_{C_\pm}e^{-ikt}dk(\chi,{{\hbox{\rlap/P}(k)+m}\over{k^2-\lambda^2}}dP_\lambda
\varphi)$$ $$=\pm i\int e^{\mp i\lambda t}(\chi,{{\hbox{\rlap/P}(\pm\lambda)+m}\over{2\lambda}}dP_\lambda
\varphi)$$ $$=(\chi,ie^{\mp i\omega t}\pi_\pm\gamma^0\varphi).\eqno(4.20)$$ Here $\pi_\pm$ are two orthogonal projection operators onto the positive and negative energy parts in Hilbert space respectively, $$\pi_\pm={{\omega\pm(-i\gamma^0\gamma^i\nabla_i+\gamma^0m)}\over2\omega}.\eqno(4.21)$$ We can check they are the correct projection operators by applying them on the positive and negative energy solutions (3.19): $$\pi_\pm\psi_\pm=\psi_\pm,\hskip0.1in\pi_\pm\psi_\mp=0.\eqno(4.22)$$ Generally, $\pi_\pm$ have the following relations: $$\pi^\dagger_\pm=\pi_\pm=\pi^2_\pm,\eqno(4.23a)$$ $$\pi_\pm\pi_\mp=0,\eqno(4.23b)$$ $$\pi_++\pi_-=1.\eqno(4.23c)$$ And $\pi_\pm$ commute with $\omega$, since $\omega^2$ commutes with $\gamma^i\nabla_i$ and $\omega$ is a scalar.
By (4.20), the positive and negative energy parts of $S$-function in space-time representation can be formally expressed by $$S_\pm(x,y)=ie^{\mp i\omega t}\pi_\pm({\bf x})\gamma^0\delta({\bf x},{\bf
y}),\eqno(4.24)$$ which make sense mathematically only if they are smeared as in (4.20).
Similar to $E(x,y)$, the propagator of Dirac equation is defined as the difference between the advanced and retarded fundamental solutions: $$S(x,y)=S_A(x,y)-S_R(x,y).\eqno(4.25)$$ where $S_A, S_R$ are related to $E_A, E_R$ by (4.17), or the summation of the positive and negative energy fundamental solutions: $$S(x,y)=S_+(x,y)+S_-(x,y).\eqno(4.26)$$ Since $supp(S_Ru)\cap
supp(S_Av)$ is compact, by using the same procedure of deriving equation (2.13) we compute $$<S_Ru,v>=<S_Ru,(\hbox{\rlap/P}-m)S_Av>$$ $$=<(\hbox{\rlap/P}-m)S_Ru,S_Av>=<u,S_Av>.\eqno(4.27)$$ Here we can see the advanced and retarded fundamental solutions are the adjoints of each other in spin space $$[S_R(x,y)]^*=S_A(y,x),\eqno(4.28)$$ and similarly $$[S_\pm(x,y)]^*=-S_\pm(y,x).\eqno(4.29)$$ Therefore the propagator satisfies $$[S(x,y)]^*=-S(y,x),\eqno(4.30)$$ and there is an obvious result $$[-iS(x,y)]^*=-iS(y,x).\eqno(4.31)$$ This completes our discussions on the propagator of Dirac equation.
5. Quantization of Dirac fields {#quantization-of-dirac-fields .unnumbered}
===============================
To construct field theory one needs to define a Fock space with one particle space as its base space \[19\]. A general Fock space is defined by \[20\]\[21\] $$\hbox{\tensl F}(\hbox{\tensl H})=\oplus_{n=0}^\infty\hbox{\tensl F}^{(n)}(\hbox{\tensl
H})=\hbox{\tensl F}^{(0)}(\hbox{\tensl H})\oplus\dots\oplus\hbox{\tensl
F}^{(n)}(\hbox{\tensl H})\oplus\dots,\eqno(5.1)$$ where n-fold tensor subspace is $$\hbox{\tensl F}^{(n)}(\hbox{\tensl H})=\otimes\hbox{\tensl
H}^{(n)}=\hbox{\tensl H}\otimes\dots\otimes\hbox{\tensl
H},\eqno(5.2)$$ and $\hbox{\tensl F}^{(0)}(\hbox{\tensl H})=C$ is the vacuum space with complex constants as its elements. [H]{} is any complex Hilbert space with a positive-definite inner product. In Fock space $\psi\in\hbox{\tensl
F}(\hbox{\tensl H})$ $$\psi=(\psi^{(0)},\psi^{(1)},\dots\psi^{(n)},\dots),\eqno(5.3)$$ where $\psi^{(n)}\in\hbox{\tensl F}^{(n)}(\hbox{\tensl H})$. A dense set in $\hbox{\tensl F}^{(n)}(\hbox{\tensl H})$ is linear combinations of vectors of the form $$\psi^{(n)}=\psi_1\otimes\dots\otimes\psi_n.\eqno(5.4)$$ The inner product on $\hbox{\tensl F}(\hbox{\tensl H})$ is induced by the inner product on $\hbox{\tensl H}$: $$(\psi,\psi)_{\hbox{\tensl
F}}=|\psi^{(0)}|^2+(\psi^{(1)},\psi^{(1)})_{\hbox{\tensl
H}^{(1)}}+\dots+(\psi^{(n)},\psi^{(n)})_{\hbox{\tensl H}^{(n)}}+\dots.\eqno(5.5)$$ If $\psi^{(n)}$ has the form (5.4), then $$(\psi^{(n)},\psi^{(n)})_{\hbox{\tensl
H}^{(n)}}=\prod^n_{i=1}(\psi_i,\psi_i)_{\hbox{\tensl H}}.\eqno(5.6)$$
The construction of Fock space can be put in the above form for both fermion and boson fields. Here we are only interested in Dirac fermion fields which obey antisymmetric rule. Define linear permutation operators on [F]{}([H]{}) by $$\Pi(\psi_1\otimes\dots\otimes\psi_n)={1\over n!}\sum_\pi(-1)^\pi\psi_{\pi(1)}\otimes\dots\otimes\psi_{\pi(n)},\eqno(5.7)$$ which induces orthogonal projections onto $\hbox{\tensl F}^{(n)}$([H]{}). For example, $$\Pi(\psi_1\otimes\psi_2)={1\over2}(\psi_1\otimes\psi_2-\psi_2\otimes\psi_1).\eqno(5.8)$$ Applying this operator to [F]{}([H]{}), we obtain antisymmetric fermionic Fock space $$\hbox{\tensl F}_a(\hbox{\tensl H})=\oplus_{n=0}^\infty\hbox{\tensl F}^{(n)}_a(\hbox{\tensl H}),\eqno(5.9)$$ in which n-fold subspaces are defined by $$\hbox{\tensl F}^{(n)}_a(\hbox{\tensl H})=\Pi\hbox{\tensl
F}^{(n)}(\hbox{\tensl H}).\eqno(5.10)$$
To have a complete description of Fock space, one should define creation and annihilation operators. On the vacuum space $\hbox{\tensl
F}^{(0)}(\hbox{\tensl H})$, one defines $$a_0(\chi)\psi_0=0,\eqno(5.11a)$$ $$a_0^\dagger(\chi)\psi_0=\chi.\eqno(5.11b)$$ Generally, one can define the creation and annihilation operators on $\hbox{\tensl F}(\hbox{\tensl H})$ by $$a_0^\dagger(\chi)(\psi_1\otimes\dots\otimes\psi_n)=\sqrt{n+1}(\chi\otimes\psi_1\otimes\dots\otimes\psi_n),\eqno(5.12a)$$ $$a_0(\chi)(\psi_1\otimes\dots\otimes\psi_n)=\sqrt{n}(\chi,\psi_1)_{\hbox{\tensl
H}}(\psi_2\otimes\dots\otimes\psi_n).\eqno(5.12b)$$ Then on Fermi-Fock space $\hbox{\tensl F}_a(\hbox{\tensl H})$, according to Bratteli and Robinson \[22\], the creation operator $a^\dagger(\chi)$ and annihilation operator $a(\chi)$ can be defined as $$a^\dagger(\chi)=\Pi a_0^\dagger(\chi)\Pi,\eqno(5.13a)$$ $$a(\chi)=\Pi a_0(\chi)\Pi.\eqno(5.13b)$$ It is straightforward to show the creation and annihilation operators satisfy the following canonical anticommutation relations (CAR) $$\Bigl\{a(\chi),a(\varphi)\Bigr\}=0,\eqno(5.14a)$$ $$\Bigl\{a^\dagger(\chi),a^\dagger(\varphi)\Bigr\}=0,\eqno(5.14b)$$ $$\Bigl\{a(\chi),a^\dagger(\varphi)\Bigr\}=(\chi,\varphi)_{\hbox{\tensl
H}},\eqno(5.14c)$$ where $(\chi,\varphi)_{\hbox{\tensl
H}}$ represents an inner product in Hilbert space.
Let us define a Fermi-Fock space in static space-time $$\hbox{\tensl F}_a(\hbox{\tensl H})=\hbox{\tensl F}_a(\hbox{\tensl H}_+)\otimes\hbox{\tensl F}_a(\hbox{\tensl H}_-),\eqno(5.15)$$ where $\hbox{\tensl F}_a(\hbox{\tensl H}_\pm)$ are the positive and negative subspaces respectively. Also define $$a_+(\chi)=a(\chi)\otimes I, \hskip0.1in a^\dagger_+(\chi)=a^\dagger(\chi)\otimes
I;\eqno(5.16a)$$ $$a_-(\chi)=(-1)^N\otimes a(\chi), \hskip0.1in a^\dagger_-(\chi)=(-1)^N\otimes
a^\dagger(\chi), \eqno(5.16b)$$ where $(-1)^N$ is necessary so that $a_+$ and $a_-$ anti-commute \[20\], and $a(\chi), a^\dagger(\chi)$ are defined in (5.13) with inner product $(\chi,\varphi)_{\hbox{\tensl H}}=(\chi,\varphi)$. It is straightforward to check that the so-defined annihilation and creation operators satisfy the following CAR $$\Bigl\{a_\pm(f_\pm),a_\pm(h_\pm)\Bigr\}=0,\eqno(5.17a)$$ $$\Bigl\{a_\pm^\dagger(f_\pm),a_\pm^\dagger(h_\pm)\Bigr\}=0,\eqno(5.17b)$$ $$\Bigl\{a_\pm(f_\pm),a_\pm^\dagger(h_\pm)\Bigr\}=(f_\pm,h_\pm),\eqno(5.17c)$$ where $f_\pm$ are any vectors in the positive and negative energy subspaces respectively.
The classical solutions of the Dirac equation are given by (3.27). Correspondingly we define a quantum field operator to be the solution with data which are the positive and negative energy annihilation operators $a_\pm({\bf x}), a_\pm^\dagger({\bf x})$ given formally by $$a_\pm(\chi)=\int \chi^\dagger({\bf x})a_\pm({\bf x})d_g^3{\bf
x},\eqno(5.18a)$$ $$a_\pm^\dagger(\chi)=\int a_\pm^\dagger({\bf x})\chi({\bf x})d_g^3{\bf
x},\eqno(5.18b)$$ where $d_g^3{\bf x}=\sqrt{g}d^3{\bf x}$. Thus we put $$\psi(t,{\bf x})=U(t)\pi_+a_+({\bf x})+U(-t)\pi_-a_-({\bf
x}),\eqno(5.19)$$ where $U(\mp t)=\exp(\pm i\omega t)$. Smearing $\psi$ by a $C^\infty_0$ test function $f(t,{\bf x})$ on $R\times M$, we get the field $$\psi(f)=\int f^*(x)\psi(x)d_g^4x,\eqno(5.20a)$$ and also by our convention $\psi^*=\psi^\dagger\gamma^0$, $$\psi^*(f)=\int\psi^*(x)f(x)d_g^4x,\eqno(5.20b)$$ with a relation $$\psi^*(f)= \psi(f)^\dagger.\eqno(5.21)$$ Then by (5.19), it is easy to see (5.20) leads to $$\psi(f)=a_+(f_+)+a_-(f_-),\eqno(5.22a)$$ $$\psi^*(f)=a^\dagger_+(f_+)+a^\dagger_-(f_-),\eqno(5.22b)$$ where $f_\pm$ on $M$ are the positive and negative energy components defined by $$f_\pm({\bf x})=\int e^{\pm i\omega x_0}\pi_\pm\gamma^0f(x_0,{\bf
x})dx_0.\eqno(5.23)$$ We take (5.22) as the precise definitions of $\psi(f)$ and $\psi^*(f)$. With the above preparations, we are now ready to prove the following theorem.
0.1in [*Theorem 2*]{}: In static space-time, let Dirac field operators be expressed in terms of the creation and annihilation operators on Fermi-Fock space as in (5.22). Then the quantized Dirac field operators satisfy the equation $$\psi((\hbox{\rlap/P}-m)f)=0,\eqno(5.24a)$$ $$\psi^*((\hbox{\rlap/P}-m)h)=0,\eqno(5.24b)$$ and the following CAR $$\Bigl\{\psi(f),\psi(h)\Bigr\}=0,\eqno(5.25a)$$ $$\Bigl\{\psi^*(f),\psi^*(h)\Bigr\}=0,\eqno(5.25b)$$ $$\Bigl\{\psi(f),\psi^*(h)\Bigr\}=-i<f,Sh>,\eqno(5.25c)$$ together with an integral $$<f,Sh>=\int\int f^*(x)S(x,y)h(y)d_g^4xd_g^4y,\eqno(5.26)$$ in terms of a propagator $S(x,y)$ obtained in §4.
0.1in [*Proof*]{}: Let $D=\hbox{\rlap/P}-m$. We can write $$\psi(Df)=a_+((Df)_+)+ a_-((Df)_-).\eqno(5.27)$$ By (5.23), it follows by integration by parts $$(Df)_\pm({\bf x})=\int e^{\pm i\omega x_0}\pi_\pm\gamma^0D_\pm f(x_0,{\bf
x})dx_0,\eqno(5.28)$$ where $\pi_\pm$ are expressed in (4.21) and $$D_\pm=\pm\gamma^0\omega+i\gamma^i\nabla_i-m=\pm2\omega\gamma^0\pi_\mp.\eqno(5.29)$$ It is then clear to see $$\pi_\pm\gamma^0D_\pm=\pm2\omega\pi_\pm\pi_\mp=0,\eqno(5.30)$$ So we end up with (5.24).
By (5.22) and (5.17a) and (5.17b), it is easy to show (5.25a) and (5.25b). To show the nonvanishing CAR (5.25c), we first compute by using (4.20), (4.23) and (5.23) $$-i<f,S_\pm h>=-i\int\int S_\pm(x_0-y_0; \gamma^0f(x_0,\cdot),
h(y_0,\cdot))dx_0dy_0$$ $$=\int\int(\gamma^0f(x_0,\cdot),e^{\mp
i\omega(x_0-y_0)}\pi_\pm\gamma^0h(y_0,\cdot))dx_0dy_0$$ $$=\int\int(e^{\pm i\omega x_0}\pi_\pm\gamma^0f(x_0,\cdot),e^{\pm i\omega y_0}\pi_\pm\gamma^0h(y_0,\cdot))dx_0dy_0$$ $$=(f_\pm,h_\pm).\eqno(5.31)$$ Then using (5.22), (5.17), (5.31) and (4.26), we get $$\Bigl\{\psi(f),\psi^*(h)\Bigr\}=\Bigl\{a_+(f_+),a_+^\dagger(h_+)\Bigr\}+\Bigl\{
a_-(f_-),a_-^\dagger(h_-)\Bigr\}$$ $$=(f_+,h_+)+(f_-,h_-)$$ $$=-i<f,S_+h>-i<f,S_-h>$$ $$=-i<f,Sh>,\eqno(5.32)$$ which is just (5.25c). $\square$
0.1in If we take the sum $$f_s=f_++f_-,\eqno(5.33)$$ then we obtain $$-i<f,Sh>=(f_+,h_+)+(f_-,h_-)=(f_s,h_s).\eqno(5.34)$$ We may just take $\psi(f)$ and $\psi^*(h)$ as the annihilation and creation operators on Fermi-Fock space $\hbox{\tensl F}_a(\hbox{\tensl H})$ and define $$\Bigl\{\psi(f),\psi^*(h)\Bigr\}=(f_s,h_s).\eqno(5.35)$$ This becomes a special case of Dimock’s general theorem \[5\].
Considering Dirac fields as operators, under time evolution we would have by Heisenberg picture $$\psi(x_0+t,{\bf x})=e^{iKt}\psi(x_0,{\bf x})e^{-iKt},\eqno(5.36a)$$ $$\psi^*(x_0+t,{\bf x})=e^{iKt}\psi^*(x_0,{\bf x})e^{-iKt},\eqno(5.36b)$$ where $K$ is the energy operator. Smearing them by test function $f(x_0,{\bf x})$ it is easy to see the left sides of (5.36) become $$\int f^*(x_0-t,{\bf x})\psi(x_0,{\bf x})d_g^4x=\psi(f(\cdot -t)),\eqno(5.37a)$$ $$\int \psi^*(x_0,{\bf x})f(x_0-t,{\bf x})d_g^4x=\psi^*(f(\cdot -t)),\eqno(5.37b)$$ and the right sides of (5.36) become $$e^{iKt}[\int f^*(x_0,{\bf x})\psi(x_0,{\bf
x})d_g^4x]e^{-iKt}=e^{iKt}\psi(f)e^{-iKt},\eqno(5.38a)$$ $$e^{iKt}[\int \psi^*(x_0,{\bf x})f(x_0,{\bf
x})d_g^4x]e^{-iKt}=e^{iKt}\psi^*(f)e^{-iKt}.\eqno(5.38b)$$ So we should get $$\psi_t(f)=\psi(f(\cdot -t))=e^{iKt}\psi(f)e^{-iKt},\eqno(5.39a)$$ $$\psi_t^*(f)=\psi^*(f(\cdot -t))=e^{iKt}\psi^*(f)e^{-iKt}.\eqno(5.39b)$$ Now we show how this works out.
Define unitary operators for time evolution by $$U(t)=U^+(t)\otimes U^-(t),\eqno(5.40)$$ and $$U^\pm(t)=\oplus_{n=0}^\infty U_n^\pm(t),\eqno(5.41)$$ and $$U_n^\pm(t)=e^{\mp i\omega t}\otimes\dots\otimes e^{\mp i\omega t}.\eqno(5.42)$$ These are all unitary groups with generators $K$, $K^\pm$ and $K_n^\pm$ respectively. Here $K$ is given by $$K=K^+\otimes I+ I\otimes K^-,\eqno(5.43)$$ which is not positive-definite, and $K^\pm$ is given by $$K^\pm=\oplus_{n=0}^\infty K^\pm_n,\eqno(5.44)$$ and $K^\pm_n$ is given by $$K^\pm_n=\omega\otimes I\otimes\dots\otimes
I+I\otimes\omega\otimes I\otimes\dots\otimes I$$ $$+\dots+I\otimes\dots\otimes I\otimes\omega\hskip0.2in(\hbox{n}\hskip0.05in \hbox{terms}).\eqno(5.45)$$ We now want to prove (5.39) in terms of the time evolution of creation and annihilation operators by a proposition.
0.1in [*Proposition 2*]{}: With the above definitions of the Dirac field operators $\psi(f),\psi^*(f)$ and the energy operators $K$, $K^\pm$ and $K^\pm_n$, the time evolution of Dirac field operators takes the form (5.39).
0.1in [*Proof*]{}: By (5.22), this is equivalent to show $$a(f_\pm(\cdot-t))=e^{iKt}a(f_\pm)e^{-iKt},\eqno(5.46a)$$ $$a^\dagger(f_\pm(\cdot-t))=e^{iKt}a^\dagger(f_\pm)e^{-iKt}.\eqno(5.46b)$$ By (5.23), we have $$e^{\pm i\omega t}f_\pm=\int e^{\pm i\omega x'_0}\pi_\pm\gamma^0f(x'_0-t,{\bf x})dx'_0=f_\pm(\cdot-t).\eqno(5.47)$$ To show (5.46), it suffices to prove $$a(e^{\pm i\omega t}f_\pm)=e^{iKt}a(f_\pm)e^{-iKt},\eqno(5.48a)$$ $$a^\dagger(e^{\pm i\omega
t}f_\pm)=e^{iKt}a^\dagger(f_\pm)e^{-iKt}.\eqno(5.48b)$$ The proof of (5.48) is standard \[22\]. $\square$
0.1in If one replaces the negative energy annihilation operator by a particle conjugated creation operator, then time evolution is implemented with positive energy (refer to \[11\] for details). This completes our work.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author would like to thank professor Jonathan Dimock for his enlightening comments during numerous discussions. The referee’s comments are also appreciated. The author was partially supported by NSF Grant PHY9722045.
References {#references .unnumbered}
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[^1]: Since we are considering compact manifolds, the spectral integral is actually a sum over discrete eigenvalues. Nevertheless we stick with the more general notation.
|
---
abstract: |
In this work we study numerical construction of optimal clinical diagnostic tests for detecting sporadic Creutzfeldt-Jakob disease (sCJD). A cerebrospinal fluid sample (CSF) from a suspected sCJD patient is subjected to a process which initiates the aggregation of a protein present only in cases of sCJD. This aggregation is indirectly observed in real-time at regular intervals, so that a longitudinal set of data is constructed that is then analysed for evidence of this aggregation. The best existing test [@McGuire11; @McGuire11b] is based solely on the final value of this set of data, which is compared against a threshold to conclude whether or not aggregation, and thus sCJD, is present. This test criterion was decided upon by analysing data from a total of 108 sCJD and non-sCJD samples, but this was done subjectively and there is no supporting mathematical analysis declaring this criterion to be exploiting the available data optimally. This paper addresses this deficiency, seeking to validate or improve the test primarily via support vector machine (SVM) classification. Besides this, we address a number of additional issues such as i) early stopping of the measurement process, ii) the possibility of detecting the particular type of sCJD and iii) the incorporation of additional patient data such as age, sex, disease duration and timing of CSF sampling into the construction of the test.
**Keywords:** Creutzfeld-Jakob disease, support vector machines, real-time quaking-induced conversion, RT-QuIC, coordinate descent method
author:
- 'William Hulme$^*$'
- 'Peter Richtárik[^1]'
- 'Lynne McGuire[^2]'
- 'Alison Green$^\dagger$'
bibliography:
- 'biblio.bib'
date: 'December 10, 2012'
title: '**Optimal diagnostic tests for sporadic Creutzfeldt-Jakob disease based on support vector machine classification of RT-QuIC data**'
---
Introduction
============
**Background.** Current CSF tests for sCJD rely on the detection of surrogate markers of neuronal damage such as CSF 14-3-3 or tau protein. However, these markers are not specific for sCJD, thus reducing the specificity (defined below) of the test. The hallmark of sCJD is the post-translational conformational change of a normal cellular protein, PrP^C^, into a disease-associated form, termed PrP^Sc^, which aggregates together to form plaques within the brain.
Real-time quaking-induced-conversion (RT-QuIC) is a recently developed technique [@wilham10; @atarashi11] that exploits the ability of PrP^Sc^ in brain tissue or CSF from patients with sCJD to induce a hamster recombinant PrP to change shape and aggregate over time. This aggregation is observed by adding Thioflavin T (ThT) to the reaction mixture as it binds to the aggregated PrP causing a change in the ThT emission spectrum, which may be monitored over time using fluorescence spectroscopy. Recording these fluorescence levels during the course of the RT-QuIC thus creates a longitudinal data set, interpreted as a collection of “curves”, and this can be used to detect the occurrence of aggregation, which manifests as an increase in fluorescence over time.
A study by scientists and clinicians at the National Creutzfeldt-Jakob Disease Research & Surveillance Unit found that data obtained using RT-QuIC analysis could provide a test with a sensitivity and specificity of 91% and 98% [@McGuire11; @McGuire11b] which compares favourably against alternative sCJD tests using surrogate marker proteins; CSF 14-3-3 (93% and 56%) and CSF tau protein (93% and 79%).
**Methods and data.** CSF samples were taken from 55 neuropathologically confirmed sCJD patients (30F:23M; aged 31-84 years; mean $\pm$ SD 62.1 $\pm$ 13.5 years) and from 53 patients (26F:27M; aged 43-84 years; mean $\pm$ SD 67.8 $\pm$ 10.4 years) who were initially suspected of having sCJD but were shown to have an alternative diagnosis. The 55 positive samples can be further categorised into three sCJD subtypes; 30 were classified as *CJD-MM*, 17 *CJD-MV* and 8 *CJD-VV*. The clinical details of these sCJD types are detailed in [@McGuire11].
The RT-QuIC analysis was performed in quadruplicate, with fluorescent unit readings (rfu) for each of the replicates being taken every 30 minutes for 90 hours. This gives a total of $4 \times 181= 724$ readings for each sample. RFU readings are capped at 65,000 rfu.
Together with the longitudinal fluorescence data obtained, the following patient data are also available:
- Sex (M/F);
- Age at first symptoms, in years;
- LP Date (this is the date on which the CSF samples were obtained, so called due to the Lumbar Puncture procedure used to obtain the CSF);
- Duration of disease, in months (this is the time to death from first symptoms);
- Time to LP, in months (this is the time of first symptoms until the time the CSF sample was collected).
Information on the final two factors is unavailable for non-sCJD cases.
**The test.** By subjectively examining the plotted rfu data over time, a novel sCJD specific test using the RT-QuIC technique was proposed in [@McGuire11; @McGuire11b], where a positive result was defined as the mean of the two highest rfu readings at 90 hours being over 10,000 rfu. It is this test with this single criterion which provides the high sensitivity and specificity values above. This test criterion essentially discards all but the final rfu reading, and it does not use any of the additional data listed above.
**Problem statement.** There are no rigorous mathematical analyses that suggest this test exploits the rfu data optimally, nor that it will reliably generalise to new CSF samples whose sCJD status is unknown, as is its purpose. It is demonstrably superior to CSF 14-3-3 and CSF tau protein tests, but it may be possible to increase the sensitivity and specificity further by using more and possibly all of the available data. Further, [@McGuire11; @McGuire11b] does not consider the potential of the rfu data to detect differences *between* sCJD types. These issues are addressed in this paper.
Preliminary analysis
--------------------
![Plots of quadruplicates for samples classified as being sCJD positive. The horizontal axis runs from 0 to 90 hours, and the vertical axis runs from 0 to 65,000 rfu.[]{data-label="quad1"}](quadruplicatePos2.eps){width="0.9\linewidth"}
![Plots of quadruplicates for samples classified as being sCJD negative. The horizontal axis runs from 0 to 90 hours, and the vertical axis runs from 4500 to 8000 rfu.[]{data-label="quad-1"}](quadruplicateNeg2.eps){width="0.9\linewidth"}
The graphs in Figures \[quad1\] and \[quad-1\], which plot the fluorescence readings over time for all samples and replicates, show obvious differences between sCJD positive and sCJD negative cases, henceforth simply *positive* and *negative*. Note that the scale for positive cases is much larger than the scale for negative cases. To get an idea of the relative difference between these two classes, see Figure \[rawdataplot\]. As expected, the *curves* for the positive cases in general exhibit an increase in rfu over time whereas the negative curves tend to remain roughly constant. There are some exceptions, most notably sample \#987, which despite being classified as negative, shows an increase in rfu beyond doubt in 3 of its replicates. The maximum rfu reached for each of these three curves ranges from 34,212 to 42,145 rfu, but this is hidden by the range of the vertical axis. [@McGuire11; @McGuire11b] explains that although this case was classified as negative, sCJD could not be ruled out entirely.
Noise levels, that is, the random discrepancies between the idealised rfu level over time and that which is observed, can vary greatly between and within samples. Some curves display high levels of noise whereas others appear much smoother. Discussions with the authors of [@McGuire11; @McGuire11b] explain that there are mechanical explanations for noise that exceeds the usual levels, relating to equipment faults. The noise level is not thought to relate to the sCJD status of the patient.
The general profile of each of the four curves from a given sample can differ greatly. Most positive cases are characterised by three distinct states over the 90 hours:
- an initial base-line period of a constant low-level rfu (5,000-6,000 rfu);
- an increase in rfu which may be anywhere between extremely rapid (up to the maximum fluorescence within hours) to very slow (appears to still be increasing at 90 hours by which time rfu has only doubled from the base-line value);
- a final period where the rfu is constant or slowly increasing.
The times at which the curve passes from one state to the next varies greatly, as does the final rfu reached, even within samples. Some positive samples only have one curve displaying any clear evidence of aggregation while the remaining curves have a constant, low-level rfu over time.
Clearly, there is a great deal of variation between and within samples and the most meaningful way to use the data from the four replicates to diagnose a sample is not immediate. It is unknown whether any of this variation is due to factors such as age, sex or disease duration. The only indisputable source of variation is the presence or not of sCJD in the sample and this is enough to provide a high-performance test for sCJD as detailed in [@McGuire11; @McGuire11b], henceforth the *null test*. However, it is conceivable that within all the data available, enough information is present to make a better or possibly perfect distinction between the two classes, that generalises well to new observations.
Many alternate and potentially better tests might be proposed using educated guesses and trial and error, but a systematic approach—that can automatically find the important features of the data, and use it optimally in some sense—would be useful here, along with well-defined criteria to adequately compare tests.
There is a machine learning / mathematical programming technique known as *support vector machine* classification [@Bishop06] which can be employed to approach the problem in this way, and it is this technique that will be used throughout this paper.
Support vector machine classification {#chap:SVMs}
=====================================
Primarily attributed to the work of Vladimir Vapnik [@Vapnik82; @Vapnik96; @Vapnik98], support vector machines (SVMs) are a powerful and widely-applicable machine learning technique used for binary classification of data. Applications include handwritten digit recognition [@Vapnik95], text categorisation [@Joachim] and bio-informatics [@Ivaniuc07]. SVMs operate by representing the data as points in a multi-dimensional space and finding a hyperplane which separates the two classes, if separation is possible. Certain separating hyperplanes are preferred to others. Ideally, we would wish the nearest point in either class from the hyperplane to be as far a possible; this concept is referred to as *margin maximisation*. The classification of a new observation is performed according to which side of the gap it falls.
The arguments that motivate the SVM formulation and present its properties draw from a range of mathematical ideas, and are an accumulation of decades of progress in machine learning theory. Presented here is a succinct summary of these arguments (which draws upon the expositions in [@Bishop06; @Hastie09; @Cristianini00]), and a discussion of two classification-improving SVM methods that are heavily used throughout the paper.
Training set and separating hyperplane
--------------------------------------
Consider a set of $n$ multivariate observations/examples, $x_i\in\mathbb{R}^m$, $i=1,\dots,n$, where each observation is one of two types/classes. The class of observation $x_i$ is denoted by $y_i$; we assume it is *known* for all $i$. We thus have $n$ pairs $(x_i,y_i)$ forming the *training set* $$T {\overset{\text{def}}{=}}\{(x_1,y_1),(x_2,y_2),\dots,(x_n,y_n) \}.$$ The training set $T$ is used to construct/train a model which is then used to classify new observations the class of which is *not known*.
A typical application is that of spam filtering, where the vectors $x_i$ correspond to emails and the label $y_i$ identifies each email as spam or not spam. In this paper the observation $x_i$ corresponds to longitudinal data corresponding to CSF sample of patient $i$, and $y_i$ identifies each patient as having CJD or not.
An approach to classification, common in the machine learning and mathematical optimization literature, is to seek a hyperplane in $\mathbb{R}^m$ that separates the two classes of observations contained in the training set, if possible. Formally, given a training set $T$, we wish to compute a *weight vector* $w\in \mathbb{R}^m$ and *bias* $b\in \mathbb{R}$, so that $$\label{constraint}
y_i(w^t x_i - b) \geq 0, \qquad i=1,\dots,n,$$ where $u^t v$ denotes the standard inner product between vectors $u,v \in {\mathbb{R}}^m$: $u^t v = \sum_{i} u_i v_i$. Note that, given $(w,b)$ satisfying , the positive and negative observations lie in different half-spaces generated by the hyperplane $$(w,b) {\overset{\text{def}}{=}}\{x \in \mathbb{R}^m \;:\; w^t x - b = 0\}.$$ Indeed, observations $x_i$ with $y_i=+1$ lie in the half-space $\{x\in {\mathbb{R}}^m \;:\; w^t x-b \geq 0\}$, and observations $x_i$ with $y_i=-1$ lie in the half-space $\{x\in {\mathbb{R}}^m \;:\; w^tx-b \leq 0\}$.
The class $y$ of a *new observation* $x$ is estimated/determined by finding which “side” of the hyperplane this point lies on, i.e., by setting $$y {\overset{\text{def}}{=}}\text{sign} \{w^t x - b\}.$$ By convention, we may set $\text{sign}\{0\} = 1$ (or break the tie arbitrarily).
The maximum margin hyperplane
-----------------------------
If a hyperplane $(w,b)$ satisfying (\[constraint\]) exists, we say that the training set $T$ is *linearly separable*. In such a case, usually there is an infinity of separating hyperplanes. There are several algorithms which can be used to identify/compute one of these hyperplanes (see [@Hastie09], Chapter 4). The SVM approach to this problem is to choose a hyperplane maximising the distance (so called *margin*) to the nearest training point $x_i$. The distance from $x_i$ to the hyperplane $(w,b)$ is given by $$\begin{aligned}
\delta_i &{\overset{\text{def}}{=}}& \frac{|w^t x_i + b|}{\|w\|_2},\end{aligned}$$ where $\|w\|_2 {\overset{\text{def}}{=}}(\sum_i w_i^2)^{1/2}$ is the $L_2$ norm of $w$. If $(w,b)$ separates the two classes, then $|w^t x_i + b| = y_i(w^t x_i +b)$ for all $i$, and we can find the maximum-margin hyperplane via solving the following optimisation problem: $$\label{SVM:margin}
\underset{w, b}{\text{maximise}} \min_{i=1,2,\dots,n} \left\{\frac{y_i(w^t x_i + b)}{\|w\|_2} \right\}.$$ In this form, the optimization problem is not easily solvable. However, noting that the objective function in is positively homogeneous of degree one (i.e., $\delta_i$ does not change if we replace $(w,b)$ by $(tw,tb)$ for $t>0$), we may further reformulate as $$\label{SVMcanonical}
\begin{aligned}
& \underset{w, b}{\text{minimise}} & & \tfrac{1}{2}\|w\|_2^2 \\
& \text{subject to } & & y_i(w^t x_i - b) \geq 1, & i=1,2,\dots,n,
\end{aligned}$$ which is a quadratic minimisation problem with $n$ linear constraints. This problem can be solved efficiently using standard convex optimization algorithms [@Ben-Tal_Nemirovskii:2001:book] such as interior point methods [@Nesterov-Nemirovski:1994:IPMbible; @Renegar:1991:book].
Let $(w,b)$ be the optimal solution of . Points $x_i$ for which the constraint $y_i(w^t x_i - b) \geq 1$ is active, i.e., for which it holds as an equality, are referred to as *support vectors*. Note that it is precisely the support vectors which define the hyperplane. Indeed, any non-support vectors can be removed from the training set and the solution $(w,b)$ will not change. The act of solving the optimisation problem is usually called *training a support vector machine.*
Soft margins {#subs:soft}
------------
In general, linear separability of the training set may not be possible in which case the optimization problem is *infeasible* (i.e., there is no pair $(w,b)$ satisfying the $n$ constraints) and hence cannot be used to find $(w,b)$.
One approach to this is to map the data into a space in which separation is possible (so called *kernel trick*); this is not considered in this paper. An alternative approach, pioneered by Vapnik [@Vapnik95], is to allow misclassifications but to discourage this by penalization. We will now describe this approach.
For each observation $(x_i,y_i)$ in the training set $T$ we introduce a slack variable $\xi_i\geq 0$ measuring the degree to which the $i$-th inequality/constraint in is not met. We then relax the $i$-th inequality to $y_i(w^T x_i + b) + \xi_i \geq 1$, and add the term $C\sum_i \xi_i$ to the objective function, where $C\geq 0$, which has the purpose of pushing the slack variables to zero. This leads to the following optimisation problem: $$\label{SVMsoftmargin}
\begin{aligned}
& \underset{w, b, \xi}{\text{minimise}} & & \tfrac{1}{2}\|w\|_2^2 + C\sum_{i=1}^n \xi_i \\
& \text{subject to } & & y_i(w^t x_i - b) + \xi_i \geq 1, & i=1,2,\dots,n,\\
& & & \xi_i \geq 0, & i=1,2,\dots,n.
\end{aligned}$$ Note that if $0\leq \xi_i < 1$, $x_i$ is classified correctly. If $\xi_i=1$, $x_i$ may or may not be classified correctly. If $\xi_i>1$, $x_i$ is misclassified. Whenever $\xi_i > 0 $, penalty $C\xi_i$ is incurred. As $C$ increases, the optimal hyperplane will misclassify fewer observations, and for large enough $C$ the hyperplane will fit as tightly as possible to minimise the sum of the slacks. Once this state has been reached, increasing $C$ further has no effect on the solution, other than to increase optimal objective function value.
The hard-margin case in (\[SVMcanonical\]) corresponds to $C=\infty$. In this case, if the training set is not linearly separable, then the objective function value of (\[SVMsoftmargin\]) is infinite because at least one $\xi_i$ must be non-zero, rendering the solution infeasible, and this agrees with the infeasibility of (\[SVMcanonical\]) for non-separable problems. If the problem is separable, then the $\xi_i$s will be forced to zero by the minimisation, thus removing the slack variables and recovering the original problem.
Feature Selection {#subs:sparsity}
-----------------
Maximisation of the margin between the two classes may not be the only criterion for obtaining a good classifier. Sometimes it is desirable that the dimension $m$ of the space in which the observations are expressed is reduced so that only those variables, or *features*, which contribute the most to the separability of the training set are used, so that the unnecessary features do not over-complicate the model. Reducing the number of non-critical features expressed in $w$ may be desirable for easier interpretation of the classifier. This process is known as *feature selection*. A widely used technique for encouraging $w$ to be sparse is to introduce the sparsity-inducing term term $D\|w\|_1$ into the objective function of the minimisation problem [@Tibshirani96; @Zou05], where $D\geq 0$ is a constant and $\|w\|_{1}=\sum_i |w_i|$ is the $L_1$ norm of $w$. Larger values of $D$ encourage more sparsity in $w$. Thus, instead of we are interested in solving the optimization problem $$\label{SVMcomplete}
\begin{aligned}
& \underset{w, b, \xi}{\text{minimise}} & & \tfrac{1}{2}\|w\|_2^2 + C\sum_{i=1}^n \xi_i + D\|w\|_{1} \\
& \text{subject to } & & y_i(w^t x_i - b) + \xi_i \geq 1, & i=1,2,\dots,n,\\
& & & \xi_i \geq 0, & i=1,2,\dots,n.
\end{aligned}$$ Clearly, is recovered for $D=0$ and is recovered for $D=C=0$. By increasing $D$, the number of zeros in the optimal vector $w$ will grow. More interestingly, by proper choice of $C$ and $D$ a balance can be struck between the conflicting goals of finding $w$ supported at a few of the most important features only (controlled by $D$) and seeking an acceptably large enough margin (controlled by $C$).
Huge-scale $L_1$-regularized optimization problems can be efficiently solved by coordinate descent methods [@RT2012; @RT2011; @RT-TTD-2012]. Greedy coordinate descent methods for $L_1$-regularized problems were first analyzed in [@RT-TTD-2012], a general theory of serial and parallel randomized methods is developed in [@RT2011; @RT2012].
Performance measures
====================
Before applying a SVM classifier to new data, performance measures need to be devised for easy comparison between alternate tests/approaches. Sensitivity (true positives / total positives) and specificity (true negatives / total negatives) have already been mentioned, and these give a good, quantitative account of how many observations in each class are correctly separated by the hyperplane of any given classifier. These measures together will be referred to as the *fit* of the test. Optimising the fit results in a test which minimises the number of classification errors on observations from the training set.
An overly good fit on the training data may not necessarily lead to reliable classifications on *new data*. Indeed, the SVM may be *overfitting* to the training data, increasing generalization error (which we refer in this paper by the term robustness). *Robustness* may be difficult to quantify if there is no new data with which to assess the test. A work-around is to train a SVM on a portion of the data only and then assess the performance of the resulting test on the remaining data. This is called *cross-validation* (CV) [@Hastie09] and can provide useful feedback regarding the amount of overfitting afflicting a given test.
The particular CV technique proposed here is *leave-one-out*, which for a given training set $T$ proceeds as follows: for each observation $x_i$, train a SVM on the training set from which $(x_i,y_i)$ has been removed and classify $x_i$ according to this new SVM. That is, for all $i$, compute $$y_i^\prime {\overset{\text{def}}{=}}\text{sign} \{(w^i)^{T} x_i - b^i\},$$ where $(w^i,b^i)$ is the hyperplane obtained when training an SVM on $T\backslash \{(x_i,y_i)\}$. If $y_i^\prime = y_i$ then the classification is correct, otherwise the classifier has failed for this observation.
The benefit of removing just a single observation from the training set each time is that information analogous to the sensitivity and specificity of a test can be gained by identifying the number of false positives and false negatives produced by the cross-validation. As the classifier may change for each $x_i$ we do not have performance information for a single classifier, but rather a family of classifiers created from the choice of training set and the choice of parameters $C$ and $D$. The performance information gathered in this way shall be referred to as *pseudo-sensitivity* and *pseudo-specificity*, and these will be the measures of robustness. A few remarks:
- If a training set is linearly separable for parameters $C$, $D$, then sensitivity = specificity = 100%. However, the classifier may overfit to the data, and so the test may not be very robust.
- If $x_i$ is not a support vector, then the classification of this vector by the cross-validation method above will be correct, because removing this vector will not alter the hyperplane. The number of support vectors in a training set is therefore an upper-bound on the number of misclassifications possible by cross-validation. Further, the number of positive support vectors bounds the number of false negatives, and the number of negative support vectors bound the number of false positives. This provides an intuitive perspective for understanding how the number of support vectors acts as a proxy for the robustness of the classifier.
- If a test has 100% pseudo-sensitivity and 100% pseudo-specificity, this is no guarantee that new observations will classified correctly.
- The sensitivity is an upper bound on the pseudo-sensitivity, and the specificity is an upper bound on the pseudo-sensitivity. Consequently, if the sensitivity and specificity are equal to the pseudo-sensitivity and pseudo-specificity respectively, then test can be considered to be optimally robust.
For conciseness and easy comparison, the fit of a test will be reported as the non-simplified fraction of the sensitivity and specificity values in the format $$\left(\frac{true\;positives}{total\;positives},\frac{true\;negatives}{total\;negatives}\right)_{F},$$ as all else is already known and easily recovered. The robustness will be referred to in a similar format, only now the numerators are the number of positive or negative observations that, when removed from a training set, are correctly classified by the SVM trained on the reduced training set. The subscript will be $R$.
SVM analysis on untransformed fluorescence data
===============================================
The first application of support vector machines in this paper is an application directly to all or a subset of the raw fluorescence data in an attempt to find superior tests to the null test. The most conspicuous way to do this is by finding a SVM-optimal alternative to the 10,000 threshold, by training a support vector machine on the training set employed by the null test. Also considered is using the single highest fluorescence reading at 90 hours from each sample as a training set.
As readings are taken every half hour it is possible to observe whether an equally high performance test is achievable *before* 90 hours, so that the result of the test may be known sooner. This is investigated here by training a SVM on each of the 181 readings and observing their performance. This is first done by taking the maximum of the four readings at each time, then by training on the average of the two highest readings.
Tests on the fluorescence reading at 90 hours {#subs:adjust}
---------------------------------------------
Recall that the null test misclassifies only 6 from 108 observations, with a fit of $(\frac{50}{55},\frac{52}{53})_{F}$. The threshold of this test, 10,000 fluorescent units, was decided subjectively and it can therefore be adjusted via SVMs so as to maximise the margin between the two classes. Training a soft margin SVM on this one-dimensional training set (separability is clearly not possible), an optimal bias and weight vector of $b^*=2.3089$ and $w^*=2.1580e^{-4}$ respectively are obtained, along with a maximum margin of $\frac{1}{\|w^*\|_2}=\frac{1}{w^*}=4,634$. Recall that for a given test defined by the hyperplane $(w,b)$, the test-classification (as opposed to the true classification) of an observation $x_{i}$ is calculated by $$y_{i}=\text{sign}\{w x_i - b \},$$ when $m=1$, and from this $$\begin{aligned}
x_i \geq \frac{b}{w} & \quad \Longrightarrow \quad & y_i = +1,\\
x_i < \frac{b}{w} & \quad \Longrightarrow \quad & y_i = -1 &,
\end{aligned}$$ can be recovered. Thus, $\frac{b}{w}$ is the threshold above which a sample is defined as positive and for the above test, the threshold is $\frac{b^*}{w^*}= 10,699$, which shows that the 10,000 threshold from the original test was not too far from the SVM-optimal threshold. No values in the training set lie between 10,000 and 10,699 so the sensitivity and specificity remain the same.
Rather than the two highest readings at 90 hours, an alternate test could be to take just the single highest reading at 90 hours. Training a soft margin SVM on this training set delivers an optimal bias of $b^*=1.8878$ and a maximum margin of $\frac{1}{w^*}=7,286$, which together define the threshold as 13,755 fluorescent units. The performance is the same as the null test.
As these adjusted tests are both calibrated by training a SVM, optimality with respect to the training set is guaranteed. Both tests have the same performance, but as the latter of these has a greater margin, the training set it operates on can be considered to exhibit a greater distinction between the two classes, which suggests greater classification accuracy in the face of new samples. Taking the single highest reading at 90 hours as opposed to taking the average of the two highest readings thus may be a more reliable summariser of the data.
Cross-validation shows that the robustness of both these tests matches the fit; that is, no observation is classified differently when training a SVM on a training set with any single observation removed. As such, the tests are optimally robust, strongly indicating the test will perform well when classifying new observations.
Tests on single fluorescence readings before 90 hours {#90toolong}
-----------------------------------------------------
The approach in the preceding section is applied to every single fluorescence reading. For the single highest fluorescence reading, soft margin SVMs are used (separability is not possible) and the number of classification errors and pseudo-classification errors are recorded in Figure \[singletime\](a), together with the value of the threshold and the size of margin. Measurements before 50 hours are discarded as the specificity and sensitivity is extremely poor at these times.
It can be seen that the performance steadily increases for later readings and then stabilises beyond about 83 hours. The threshold stabilises at the same time, settling at just under 14,000 fluorescent units, with a final margin of 7,286 (these are the exact same measures from the SVM taking the maximum fluorescence at 90 hours in Section \[subs:adjust\], because the data is identical). Perhaps more interestingly, there is always only 1 false positive, and this corresponds to sample \#987. As discussed, sCJD could not be entirely ruled out for this sample, and given the results of the RT-QuIC analysis it is highly plausible that this sample is in fact sCJD positive. It may be interesting then to observe the performance of the SVMs trained on single readings when this sample is excluded and these results are presented in Figure \[singletime\](b).
A similar pattern emerges, only this time the final threshold stabilises just above 10,000 fluorescent units, and the final margin is a much smaller 3,760. The number of false positives has actually increased overall, due to the lowering of the threshold towards the maximum readings of the negative cases. Clearly, sample \#987 is having a significant impact on the threshold, raising it by almost 4,000 fluorescent units and increasing the margin in an attempt to reduce the size of $\xi_{987}$, which contributes to the objective function.
Also interesting is that the number of pseudo-errors increases when \#987 is removed. This is likely to be caused by the stability that its inclusion causes, due to it dominating the objective function and therefore largely dictating the location of the threshold. During cross-validation, when individual samples are removed and the SVM re-trained, \#987 still exerts a big influence on the threshold, so each sub-threshold within each cross-validation loop will not change very much, and the classification is more likely to be the same as it was for the parent SVM. Remove \#987 and perform cross validation, and there is now no single sample which dominates the objective function, and the sub-SVMs have more freedom in finding the optimal threshold. The threshold will then vary more greatly and so classification of the single removed sample is less likely to match with the parent SVM. Of course, the threshold will only be altered when support vectors are removed, which is why the difference in the number of pseudo-errors between the two training sets is small. So, the inclusion of sample \#987 improves robustness, but this is due to the way in which it restricts the threshold under cross-validation, distorting the results to accommodate it.
The same analysis is now applied to the average of the two highest readings. The performance measures are presented in Figure \[singletimeavg\], and the results are similar to those in the last section, with performances stabilising around the 83 hour mark. Again, including sample \#987 causes a much larger margin (4,634 fluorescent units at 90 hours) as the SVMs try to reduce $\xi_{987}$ to minimise the objective function, and its inclusion also increases robustness. This time, thresholds are lower (10,699 at 90 hours) due to the readings being reduced by the averaging. It could be argued that when excluding \#987, the thresholds and margins are still increasing slightly during the final few hours, and this is likely due to the second highest readings often being from replicates whose curves are still increasing. The threshold and margin at 90 hours are 10,390 and 3,931 fluorescent units respectively.
It has been argued that the inclusion of sample \#987 distorts the threshold so that it does not accurately reflect the data. This, coupled with the fact that the classification of this sample cannot be guaranteed, suggests that removing the sample from subsequent analyses might provide more accurate results. It remains to compare the remaining two sets of SVMs in \[singletime\](b) and \[singletimeavg\](b) then, but there is not a lot to choose between them. They both result in six classification errors towards the 90^th^ hour, although there are two false positives and four false negatives for the highest fluorescence reading SVMs compared with one false positive and five false negatives for the averages of the two highest readings SVMs. The performance of these tests will deteriorate if final readings are taken before around 83 hours so to be confident of good results, it would probably be unwise to reduce the RT-QuIC time from a 90 hour total.
One notable difference is that for the maximum reading SVMs, the threshold appears to have plateaued by the 90^th^ hour, but this seems not be the case for the average of the two highest reading SVMs. It could be argued therefore that 90 hours is only sufficiently long to get optimal results for the maximum reading SVMs, whereas the averaging SVMs may need longer. However, as the robustness at these times are similar for both sets of SVMs, a clear “winner” cannot be identified.
Note that if the thresholds in Figures \[singletime\](a) and \[singletimeavg\](a) are used for tests on the corresponding readings but sample \#987 is omitted, then the tests will be 100% specific, (with the exception of readings at 74.5 hours in \[singletimeavg\](a)) but the SVM trained without \#987 chooses a threshold much lower and consequently false positives occur. This is due to the fact that a higher threshold would result in heavy penalties for the positive cases whose fluorescence does not increase. It may be worth considering increasing the threshold dictated by the SVM to obtain a more specific test. The thresholds are optimal in the sense that the sum of the distances of the misclassified sample readings from the threshold is minimised, but other considerations such as a preference for greater specificity cannot be recognised by SVMs and so deviating from this optimal threshold can be justified.
Sparsity induction into the training sets is meaningless here, as the training set has only one dimension and so cannot be reduced further.
Tests on alternate feature spaces
=================================
A test which improves upon the sensitivity and specificity of the null test has not been found, although it can arguably be bettered by adjusting the threshold. It is clear that the tests considered so far discard a huge proportion of the data and so the potential of using the whole data to provide a stronger test is not yet known. A sensible progression then is to train a support vector machine on the whole of the fluorescence data, to observe if multiple readings can help distinguish further between the two classes, this is the focus of the following section.
This is first done one the entirety of the fluorescence data, without modification, but due to overfitting this approach fails. Instead, a variety of methods are explored that attempt to condense the data into just a few dimensions that summarise the nature of curve, namely parametrisation, and extracting derivative information using finite differences.
Choosing data within samples {#subs:condense}
----------------------------
Firstly, a problem arising from the arbitrary order of the replicates must be resolved. For a given sample, it would be wrong to use all four replicates to construct one single observation of $4\times 181 = 724$ dimensions because there is no meaningful order in which to put the replicates, as they are simply four instances of an identical experiment. Arbitrarily ordering them would created a meaningless connection between the $j$^th^ replicate in each sample and so should be avoided. A way must be devised to summarise the replicates within each sample.
Selecting a replicate randomly is clearly flawed as it could result in positive samples being represented by a “flat” replicate, despite other replicates from the same sample showing strong evidence of aggregation, and thus the major distinction between the two classes will be lost. Taking averages over the replicates does not seem sensible either. Suppose that for a positive sample, three replicates are flat and one shows a strong increase (for example \#3385), then averaging would dilute the significance of this single increasing curve because the flat readings would lower the average, bringing them closer to the negative samples. The overall effect is that observable differences between classes would diminish.
A sensible heuristic is to only use data from the replicate which has the greatest fluorescence reading at 90 hours. That is, the chosen replicate for sample $i$ is $$\arg\max_{j=1,2,3,4}\{f_j^i(90)\},$$ where $f_j^i(t)$ is the fluorescence at time $t$ for replicate $j$. This ensures that if any significant increase in the curve occurs it is strongly represented and if no increase occurs across replicates then the maximum difference will not be significant and it will not warp the representation of the sample. For reference, this summary method shall be called the *maximum total increase* heuristic. See Figure \[rawdataplot\] for plots of the curves selected by this heuristic for each class.
Training on the whole length of the data {#robustfluoread}
----------------------------------------
Using the maximum fluorescence increase heuristic, a SVM is trained on the full length of the data. Encouragingly, the test is 100% sensitive and 100% specific. However, performing cross-validation on the SVM defined by the maximum total increase replicates with $C=\infty$ and $D=0$, shows that the associated test has a pseudo-sensitivity of $42/55$ and a pseudo-specificity of $49/53$, or simply $(\frac{42}{55},\frac{49}{53})_{R}$. Given that the test has a fit of $(\frac{55}{55},\frac{53}{53})_{F}$ and that the fit is an upper bound on robustness, the test is not optimally robust, and indicates that some level of overfitting is present in the test trained on the full training set.
So the test formed from training a SVM on the whole fluorescence data (with a suitable choice of replicate) has a weaker robustness—both absolutely and relative to the fit—than those tests trained only on the final reading (above), despite having 180 more dimensions and consequently this test can not be considered an improvement.
This result should in fact not be surprising as there are more dimensions than obvservations in the training set. This practically guarantees separability because there is so much space in which to seek separability, compared with the number of observations that restrict it.
A plausible remedy to this overfitting is to induce sparsity into the weight vector $w$, so that the SVM only uses features (fluorescence readings) which are necessary for separation. However, it is clear that the curves are subject to random variation, not only with regards to the noise at each measurement in time, but also the general progression of the curve over time. This becomes obvious when viewing Figures \[quad1\] and \[quad-1\] given the variation present within each sample. From this, it becomes easy to convince oneself that this approach is futile, as the noise is exploitable by the SVM in its attempt to find a separating hyperplane. Analysis confirms this - separability requires a minimum of 25 dimensions, and the robustness of the feature space is poor. However, there may still be information contained in the earlier readings which cannot be extracted by simply stacking the readings, such as the rate of change of the fluorescence levels.
The SVM analysis above proceeds by simply stacking the measurements from each observation into one long vector. Consequently, re-ordering these dimensions in the vector will have zero effect on any aspects of the analysis, because the temporal structure of these vectors is not recognised by the SVM. All dependencies of the readings on adjacent and other local readings are lost. This next section attempts to deal with this by identifying attributes of the curve that in some way measure the relationship *between* the elements within each vector, rather than treating them independently. We therefore seek low-dimensional feature spaces which, unlike sparsity induction, do not discard large portions of the data, but tries to summarise important attributes of it. The idea is to eliminate noise by retaining only the essential information, that which can be used to discriminate between the two classes.
Recall that sample \#987 has been removed, so for comparison with the null test, take $(\frac{50}{55},\frac{52}{52})_F$ as the benchmark fit.
Parametrisation {#sec:params}
---------------
In this section we explore the possibility of a parametrisation of the curves. This is done by fitting some function to the curves by optimising the parameters of that function by a least-squares approach. The space of these parameters can then be used to train an SVM. Each parameter in a given model will describe an attribute of the profile of the curve with a single value, vastly reducing the dimension of the problem, thus providing less opportunity for the SVM to exploit noise.
The most basic way to parametrise the curves would be to perform a simple linear regression on each curve, which would provide a two-parameter feature-space with which to train an SVM. However, this is unlikely to be very helpful; it is clear that most of the positively classed curves are not linear and the only important aspect of a curve it would capture is whether a significant increase has occurred. As this can be observed using the final fluorescence measurement, linear regression is unlikely to reveal anything new.
A more flexible approach is to approximate the curves piecewise-linearly. As discussed, we can roughly identify three major states in the curve profile for those which aggregate. By approximating these stages with three connected lines using a least-squares approach, a simplification of the curve can be achieved which has very clearly defined properties, namely the gradient and intercept for each line.
The location of the two *break-points*, where the lines are allowed to change gradient, can be chosen in different ways: they can be fixed, for example at 30 hours and 60 hours, but this would be highly restrictive, and it is clear from viewing the curves in Figure \[rawdataplot\] that this would not accurately reflect many of the curves; they can be decided heuristically, for example by using the two times where the gradient is thought to have changed the most (using second derivative approximations, see Section \[sec:approxs\]); or most flexibly, they can be decided by absorbing their freedom using the least-squares minimisation [@Hastie09; @Kim09]. Here the latter approach is chosen, and the resulting approximations are shown in for a selection of samples in Figure \[linplot\].
It is easy to verify that there are 6 degrees of freedom in this approximation of the curves, and these can be parametrised in different ways, with an appropriate combination of intercepts, slopes and the locations of the breakpoints of the ‘curve’. Various such parametrisations were tried, but none improved the fit of the original test. Further, extracting the most useful parameters via sparsity induction invariably results in a single parameter being kept, and this is invariably a proxy for the final fluorescence value, such as the intercept at 90 hours, or the slope of the middle line. From this it is reasonable to conclude that additional information provided by parametrising piecewise-linearly does not help find a test to rival the single dimension tests.
Attempts were also made to fit various non-linear functions to the curves via least squares regression, with the intention of using the space of the parameters of these functions to train SVMs.
Polynomial approximations were tried, but these were a poor fit for lower orders, and convergence of polynomials of higher orders was frequently unstable, so the polynomial approach was abandoned. Figure \[polyplot\] displays the fits for polynomials of order 4, and it is clear that for sharply rising curves, the regression fails to capture the trend.
Another family of non-linear functions, called Sigmoid functions, were tried. These are used to describe a monotonic curve with upper and lower asymptotes and intermediate growth - most of the positive cases match this description and it seems plausible that the flat curves which do not follow this trend could be modelled by degenerate or distorted cases of the sigmoid function (linearly, for example, with the lower and upper asymptotes equal). Unfortunately, it was not possible to find a sigmoid function with enough flexibility to accommodate the many idiosyncrasies of each curve adequately. Given that the support vector approach requires the space in which the observations are expressed to be identical for *all* observations, using different functions (and therefore a different parameter-space) is not feasible, and so this approach was also abandoned.
Local approximations to the curve using finite differences {#sec:approxs}
----------------------------------------------------------
One aspect of the data which cannot be easily interpreted by stacking the dimensions is derivative information along the curve, in particular the gradient and the curvature of the curves at different times. If this information is obtained locally at regular intervals it can form a training set to be used for SVM analysis. However, this approach would also involve stacking the dimensions and so the problems associated with stacking, such as wrongly assuming independence of adjacent measurements, will still be present. Instead, by extracting certain potentially significant values—namely, the maximum and minimum fluorescent values, the maximum and minimum first- and second-derivative estimates, and also the times that these extrema occur—a feature space of just 12 dimensions can be formed (potentially reduced via sparsity induction) which may be of use for SVM training.
As no functional representation for each curve could be found, gradient information will be approximated by other means—using *finite differences*—the differences between two points on a curve a distance $h$ apart [@Hildebrand68]. There are a variety of finite difference methods available. This paper uses the *central* difference, defined by $$\delta_h f(t)=f(t+\tfrac{1}{2}h)-f(t-\tfrac{1}{2}h),$$ where $f(t)$ is the fluorescence at time $t$. The central difference is preferred as it does not rely solely on measurements that follow it, so for large $h$, the approximation is less local, and more stable. The gradient, or rate of change, approximation may be calculated by dividing the difference by distance $h$, so we have $$f^\prime(t)\approx\frac{\delta_h f(t)}{h}=\frac{f(t+\frac{1}{2}h)-f(t-\frac{1}{2}h)}{h}.$$ Second derivatives, which measure curvature, can be approximated by $$f^{\prime\prime}(t)\approx\frac{\delta_h^2 f(t)}{h^2} = \frac{f(t+\frac{1}{2}h)-2f(t)+f(t-\frac{1}{2}h)}{h^2}.$$ With respect to the fluorescent data, first derivative approximations are measured using *rfu per hour* and second derivative approximations are measured using *rfu per hour-squared*. Measurements are taken every half-hour, so $h$ must be a multiple of a 1-hour interval in order that there are measurements available for calculations.
The maximum (minimum) first derivative approximation should correspond to the time on the curve where the steepest positive (negative) gradient occurs. The maximum (minimum) second derivative approximation should correspond to the time on the curve where the sharpest upturn (downturn) occurs.
Unfortunately, finite differences are very sensitive to noise and this can severely distort the approximations. This is because each approximation is calculated using just two observations, so significant changes between adjacent observations due to noise can cause marked differences between adjacent approximations, even if the trend suggests they should be equal or close. This issue can be partly resolved if $h$ is large enough, as the noise between $t-\frac{1}{2}h$ and $t+\frac{1}{2}h$ is simply bypassed. However, this will work only where the change dictated by the trend is greater than the noise, which is certainly not the case wherever a curve appears to be flat. Also, if $h$ is too large then the approximation cannot be guaranteed to be local enough for sufficient accuracy.
For the fluorescence data, preliminary analysis shows that only using large $h$ is not enough to bypass noise and gain sufficient accuracy to capture the trend. An additional method to deal with noise, which can be employed in conjunction with different $h$ sizes, is to *smooth* the curves, a process which attempts to identify the underlying temporal trend of longitudinal data series [@Chatfield04]. The smoothed data should approximate the progression of this trend, as if no noise is present. A popular and easy to implement smoothing method is the *moving average filter*, which averages a succession of contiguous, fixed-size subsets of longitudinal data, to provide an estimate for the central point in the subset. The set-size must therefore be odd, so that a central point exists. If the set size is $k$, then the *smoothed* estimate for observation $f(t)$ is $$g(t)= \frac{1}{k}\sum_{r=\frac{k+1}{2}}^{m-\frac{k-1}{2}} f(r),$$ where $m$ is the length of the series. For the first and last $\frac{k-1}{2}$ points where there is not enough data to average over $k$ values, the size of $k$ is reduced accordingly. The choice of $k$ is important; too small and the curve will not be sufficiently smoothed, yet too large and it will not adequately capture the trend, so a compromise must be made. The “best” $k$ is subjective, and can be gauged by visually comparing the smoothed curve against the raw.
The smoothed data for a selection of samples (chosen as representative examples to avoid visual clutter), are plotted in Figure \[smoothed\] for different values of $k$. Also plotted are the first- and second-derivative approximations of this smoothed data via finite differences, $\frac{\delta_hg(t)}{h}$ and $\frac{\delta^2_h g(t)}{h^2}$, for various values of $h$.
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A brief analysis of Figure \[smoothed\] highlights the value in smoothing and justifies its use. Firstly, the derivative information gained from $k=1$ and $h=1$ does not reflect the trend well because, for example, after the steep rise for the green sample the trend suggests a steady linear increase which would dictate that the first derivative is positive constant and the second derivative is zero, but instead the first- and second-derivatives oscillate wildly about zero. As $k$ and $h$ increase these oscillations are greatly reduced, and the locations of the extrema become more consistent with the trend, so that, for example, the maximum second-derivative is where the curve starts to rise, rather than afterwards where the increase appears to have stopped. However, the curves tend to be “flatter” for large $k$, so the first and second derivative approximations may be closer to zero than the trend would suggest.
This assumes, of course, that the non-smooth nature of the curves is in fact due to noise, and not perhaps some oscillatory behaviour which has some relevant meaning. This possibility is not examined in this paper.
As expected, training SVMs on the vectors created by stacking the first- or second-derivative approximations results in overfitting. Table \[stacksmoothfindiff\] presents the robustness measures for a selection of such SVMs and shows that the robustness is consistently poor.
-------------------- ----------- ----------- ----------- ----------- ----------- -----------
$k=1$ $k=7$ $k=21$ $k=1$ $k=7$ $k=21$
$h={\phantom 0}1 $ $(46,48)$ $(41,47)$ $(42,47)$ $(25,45)$ $(26,45)$ $(34,46)$
$h={\phantom 0}5$ $(42,48)$ $(39,46)$ $(40,48)$ $(26,50)$ $(30,46)$ $(38,46)$
$h=15$ $(44,51)$ $(42,47)$ $(40,48)$ $(40,48)$ $(38,46)$ $(44,48)$
-------------------- ----------- ----------- ----------- ----------- ----------- -----------
: Robustness of SVMs trained on first- and second-derivative approximations with smoothing parameter $k$ and finite difference parameter $h$, interpreted as (*number of true pseudo-positives, number of true pseudo-negatives*). The denominators (55 total positive cases and 52 total negative cases) are omitted for clarity.[]{data-label="stacksmoothfindiff"}
Instead, a summarising set of features is used, and this is taken to be the 12-dimensional feature space created by identifying the extrema of the fluorescence values and the first and second derivatives, together with their times. The set is tested for its use in SVM classification, and this can be done for differing values of $k$ and $h$ to assess if any particular combination works better than others. The appropriate optimisation problem for $C=\infty$ and $D=0$ is infeasible (implying non-separability) for many, but not all, $k$ and $h$ combinations. The very fact that separability can occur in just 12 dimensions, rather than the minimum of 25 dimensions needed for separability in the stacked training set, indicates that summarising the curves in this way allows for a more meaningful interpretation of the data by the SVM.
For the remaining non-separable combinations, an appropriately sized $0<C<\infty$ is used ($C\approx 10$ is enough large enough to ensure that only the vectors preventing separability have $\xi_i>0$). The classification performance of these SVMs for varying $k$ and $h$ are presented in Figure \[smoothedcolour\], where the number of misclassifications determines the shade of the colour.
Analysing these graphs, a slight tendency for low $k$ to yield less classification errors can be seen, and for high $k$ to yield less pseudo-classification errors, and consequently, where there are very few classification errors (light shades), the robustness tends to be weaker, indicating that near-separability is achieved only by overfitting to the data. This suggests that there is not an optimal pairing for $k$ and $h$, because improving the fit reduces the robustness.
One potential explanation for this is that the efforts to reduce and bypass noise are not particularly effective, but this goes against the apparent value of these methods that is demonstrated under a graphical interpretation from Figure \[smoothed\]. Another explanation is that the smoothing does improve the accuracy of the first- and second-derivative approximations, but the SVMs are not using these for separations. This latter explanation can be investigated by inducing sparsity into the weight vector and observing which features the SVM selects. For this feature space, sparsity induction is actually more valid than in the stacked case as the SVMs will no longer be dismissing single frames in time, but instead more general aspects of the whole curve which may be irrelevant to the differences between the classes.
The results are enlightening: when $D$ is large enough to perturb the solution, the SVM immediately rejects the dimensions corresponding to the times of the extrema, suggesting that these dimensions have no real value for discriminating between cases. The number of classification errors remains roughly the same, although the balance shifts towards one or two more false positives and one or two less false negatives. The robustness in general is improved, which again points toward overfitting for $D=0$.
If $D$ is completely dominant, the SVM always selects the feature corresponding to the maximum fluorescent level, and it always has a fit of $(\frac{50}{55},\frac{52}{52})$ and a robustness of $(\frac{50}{55},\frac{52}{52})$, regardless of the choice of $k$ and $h$. In fact, as $h$ is a parameter of the finite differences, it has no effect at all on the maximum fluorescence dimension, so $h$ need not be considered when examining only this dimension.
If a SVM is trained on either the maximum first derivative feature, the maximum second derivative feature, or the minimum second derivative feature, then the performances are quite good, but inferior to using the maximum fluorescence. This should be unsurprising, as these measures are all be correlated with maximum fluorescence; if the maximum fluorescence of a curve is high, then the gradient must increase at some point. Similarly, this increase in slope must cause the second derivative to increase. As the curves level, then the second derivative goes negative, and the minimum second derivative will correspond to this. Unfortunately, the SVM does not detect anything extra in these features or the times that they occur that eradicates false positives and false negatives and improves robustness. Consequently, they can only be considered proxies for the maximum fluorescence and so their inclusion is not warranted.
Choosing the maximum fluorescence as the single feature results in the maximum achievable robustness via finite differences and smoothing and consequently, using information from finite differences does not result in a test that supersedes the null test.
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Effects of additional data {#sec:classadditional}
==========================
All the previous analyses used only fluorescence data obtained from RT-QuIC analysis on the CSF samples from each patient. In this section, the possibility of using the additional data (sex, age, date of LP procedure) to strengthen the test is considered. Disease duration and time to LP procedure is not considered here, as this data is unavailable for non-sCJD cases.
This can be investigated preliminarily by plotting summary measures of the curve from Section \[sec:approxs\] against these additional *predictors*, and observing if any correlations or patterns present themselves and these plots are displayed in Figure \[classpatterns\]. Sample \#987 has been reintroduced to the data for plotting.
Firstly, the plots clearly shows that age and sex alone are completely unreliable as predictors of sCJD. There is no clear segregation of these factors by sCJD subtype at all, and this conclusion can be drawn by simply ignoring the dimensions on the vertical axes and observing the distribution of positive and negative cases along the horizontal axes. Unsurprisingly, this is also true for the time to LP procedure. Consequently, if these additional predictors have no impact on the profile of the curves (vertical axes), then they will have no value as additional features in an SVM.
It is clear that no correlations are apparent between the curve profile measures and any of the predictors. A clear difference exists between positive and negative cases, and this is true for all four curve profile measures. By ignoring the data on the horizontal axes, it can be seen that the maximum fluorescence is the best indicator of these differences, proven by the fact that this feature is selected via sparsity induction in Section \[sec:approxs\]. The remaining measures plotted here are merely proxies for the maximum fluorescence, and it is easy to see that there is a greater overlap of cases by these measures.
One small characteristic of the data to note is that there is a more distinct difference between the two cases when the patients are younger. This can most clearly be seen in the panel corresponding to maximum rfu and age, where all of the positive sCJD cases that are misclassified by the null test (those with low maximum rfu values) are at least 70 years old, and hence significantly older than the positive population average (62.1 years). There are also many CSF samples from older sCJD patients with high maximum rfu values, so this characteristic is not indicative of a correlation with age, but an increase in the variance of maximum rfu with age. As this does not constitute a separation of the data, SVM cannot exploit this characteristic to improve the reliability of the test. This would be exploitable by probabilistic classifiers which can report the probability of a case being positive—in this case the probability of an older patient with no suggestion of an increase in fluorescence would be higher than for a younger patient with a similar curve—but such classifiers are not considered in this paper.
Similar graphs plotting the times of the extrema of rfu values and of the first- and second-derivatives against the predictors show no significant patterns either. SVMs were trained with these predictors and curve summarisers, but no tests were found that improved separability without reducing the robustness. The details of these SVMs is omitted.
Seeking differences between CJD types {#chap:type}
=====================================
The null test makes use of the differences between sCJD positive and sCJD negative cases that present themselves in the RT-QuIC data. Further clinical distinctions can be made between positive cases, but whether or not these distinctions are manifest in the RT-QuIC data has not been explored. For the purposes of developing a more detailed diagnosis, this issue is addressed in the following section, using SVM analysis.
There are 55 positive cases, of which 30 are type CJD-MM, 17 are type CJD-MV and 8 are type CJD-VV. Consequently, a brief discussion on extending the binary classification techniques described in Section \[chap:SVMs\] to $p>2$ classes is needed so that the SVM approach can be applied to this 3-class problem.
Two common approaches to multi-class classification using SVMs are *one-versus-all* and *one-versus-one*; see [@Hsu02]. One-vs-all constructs a separate SVM for each of the $p$ classes, where the training set for each SVM is taken as one of the classes versus the remaining classes. Classification of a new observation is achieved by assigning it to the class whose SVM gives it the greatest positive distance from the hyperplane. This is the approach used here, but it soon becomes apparent that neither approach would improve the reliability of a test because no differences between types can be detected in the curve.
Analysis on fluorescence data
-----------------------------
**Stacked data.**\[stackedmulti\] The three SVMs constructed using the maximum yield training set are all linearly separable, which is highly unsurprising given the freedom in the training sets to find separability. Cross-validation confirms that this is purely noise-exploitation (results omitted), and so analysis of this stacked training set ends here. The next step is to try derivative information obtained from finite differences.
**Analysis on curve profile measures.**\[curveprof\] Constructing the 12 dimensional finite-differences training set in the usual way (as described in Section \[sec:approxs\]) for a given $k$ and $h$, we can observe if there are any discernible differences between the types present in features which summarise the profile of the curve. The results show that no meaningful differences can be found. Separability is not possible in any case, the robustness measures are very often below 50%, and the selection of features via sparsity induction is highly dependent on noise. There can be absolutely no expectation that derivative information in the curve can be reliably used to diagnose the specific sCJD subtype.
Effects of additional patient data
----------------------------------
As in Section \[sec:classadditional\], this section deals with the possibility of using additional patient data to help discriminate cases. Together with the sex, age and the date of the LP procedure, further data detailing the duration of disease and the time of the LP since first symptoms is available. An additional predictor may be constructed by dividing the LP time by the duration of disease to observe directly how far into the disease the patient was when the sample was taken, as a proportion of the total disease duration.
Rather predictably, the LP time alone does not influence the sCJD subtype, but it is conceivable that LP time and the other *predictors* can be used to partially explain the profile of the curve. It is worth noting that for a pre-mortem diagnosis, disease duration cannot be used as a predictor as it will not be known. As such, it may be useful to incorporate these predictors into a training set so as to augment the space of features in which to seek differences between the three sCJD subtypes, even though the profile of the curve alone does not provide enough information to be used as a diagnostic tool.
Via Section \[sec:approxs\], derivative information for the curves is obtained for various $k$ and $h$, and are plotted against sex, age and the date of LP procedure to examine if these predictors do influence the curve profile. These are presented in Figure \[typepatterns\] for $k=11$ and $h=11$ and it is clear that no correlations or patterns are present. This is also true for all combinations of $k$ and $h$.
The patient data available only for positive cases is also plotted in a similar way in Figure \[typepatternsdur\] and again, no clear correlations or patterns emerge, which holds for all $k$ and $h$.
Of course, this pair-wise analysis of different measures is not enough to confirm that these additional predictors do not aid in detecting differences between types, but SVM analysis does confirm this. Separability is not possible and sensitivity and specificity measures in the soft margin case are poor. The overall conclusion then is that sCJD subtypes are not identifiable from the data.
Conclusions
===========
Distinguishing between sCJD and non-sCJD cases
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The most reliable tests for sCJD can be formed simply using the rfu readings at 90 hours. Training SVMs on any additional data, despite increasing the fit, does not improve robustness and is therefore less reliable. However, with more data, better cross validation techniques can be used; and it is conceivable that a different test might have both a better fit and better robustness.
Measures which summarise the profile of the curve, such as the maximum first derivative approximation, are strongly correlated with the readings at 90 hours and so provide similar, but inferior, measures of the total rfu increase. Using these measures in addition to the maximum rfu increase would not improve performance. There are no useful patterns to exploit in additional patient data such as age and sex.
The rfu readings at 90 hours can be used in different ways, but the most effective of these is not obvious. The threshold obtained when training all 108 samples on the highest rfu readings at 90 hours has the same performance as the null test, but it was shown that the dubiously classified sample \#987 was distorting this threshold and as such might not reflect the data well. Removing this sample reduced the threshold considerably, and changed the balance of classification errors.
The null test is not significantly improved upon, but one may wish to consider using only the maximum rfu reading at 90 hours for its simplicity. In this case, the threshold suggested by SVM classification (without \#987) is roughly 10,000 rfu but this gives a less specific (but more sensitive) test. If desired, the performance of the null test can be recovered by shifting the threshold to around 14,000 rfu.
The major finding of this paper is that using the 90^th^ hour rfu readings is the most robust way to discriminate between classes—the calibration of the threshold is secondary. Also, it is likely that the result of a test to a new sample whose maximum reading at 90 hours is around 10,000 rfu will be considered inconclusive, as there is a degree of overlap of the classes around this level. The classification of such a sample may be postponed until further tests are carried out, and so the exact value of the threshold is not a crucial factor in the construction of an appropriate test.
Distinguishing between different sCJD types
-------------------------------------------
Section \[chap:type\] confirms that the rfu values cannot be used to distinguish between different subtypes of sCJD, neither on their own or in conjunction with other predictors such as sex and age. Consequently, a test for classifying samples by sCJD subtype is not possible from these data.
Limitations and extensions
--------------------------
Listed here are some of the limitations of this manuscript, and suggestions for possible extensions.
- This manuscript does not describe the training of SVMs using non-linear kernels, but such SVMs were in fact investigated. Both the Gaussian radial basis function and polynomial kernels [@Cristianini00] were tried, but because these kernels cannot be explicitly expressed using a feature mapping, solving the primal solution directly is not possible. Instead the dual must be solved, but due to the inaccuracy of the recovery of the primal solution $(w,b)$ from the dual solution $\alpha$, a reliable cross-validation analysis—which requires $(w,b)$—could not be conducted. However, the unreliable robustness measures on the full set of readings from 0 to 90 hours (with the maximum total fluorescence heuristic invoked) indicated that the Gaussian kernel performed no better than the linear case, with the optimal Gaussian parameter being $\sigma \approx 0.01$, and that the polynomial kernel had a far inferior robustness for polynomial parameter $d>2$. Consequently, linear kernels were deemed sufficient, but this conclusion was not based on any rigorous investigation.
- Alternative classifying algorithms were not considered. For example logistic regression, a probabilistic binary classifier, has the ability to predict the probability of a sample being sCJD positive given a collection of features, rather than simply assigning it to a class. This essentially interprets the signed distance of a vector to the margin as a probability—zero distance corresponds to an equal probability of being in either class—and so would be able to take advantage of the relationship between maximum fluorescence and age (see Section \[sec:classadditional\]), allowing the uncertainty of the classification of older patients with low maximum fluorescence to be expressed, rather than immediately classifying these cases as sCJD negative, as the SVM approach has done.
- The moving-average smoothing technique used throughout this text is crude. More sophisticated smoothing methods exist, such as the Savitsky-Golay filter [@Golay64], which performs local polynomial regression on a contiguous set of points to find the smoothed point. Consequently, derivative information can be extracted directly from the local polynomials defining each smoothed point by analytic differentiation, and finite differences become unnecessary. Further, “flattening” can occur with smoothed curves resulting from a moving-average filters (see Figure \[smoothed\] and accompanying discussion) and so first and second derivative approximations will tend to be closer to zero than the actual trend suggests. Savitsky-Golay filters tend not to suffer from this, and so derivative information is more reflective of the trend. However, SVM analysis found no significant meaning in the first and second derivative information, other than their correlation with the maximum fluorescence, and so re-examining this using more sophisticated filters may be fruitless.
- The noise levels are assumed to be due to accuracy and quality of the equipment and consequently there is no exploration of the possibility of this noise containing information that would aid discrimination between classes. For example, the noise may in fact be oscillatory fluctuations that can reveal something important about the sample, but this is not investigated, and nor are any other possible hypotheses concerning the nature of the noise.
- By employing the maximum total fluorescence heuristic throughout large parts of this paper, three-quarters of the fluorescence data was ignored (apart from its use in the selection of the replicate itself). It is possible that the relationship between replicates from the same sample could contain useful discriminatory information, but this was not investigated.
[^1]: School of Mathematics, James Clerk Maxwell Building, King’s Buildings, Edinburgh, EH9 3JZ
[^2]: National CJD Surveillance Unit, Western General Hospital, Edinburgh, EH4 2XU
|
---
abstract: |
-6.4cm
[****]{}\
5.4cm The advent of high-intensity pulsed laser technology enables the generation of extreme states of matter under conditions that are far from thermal equilibrium. This in turn could enable different approaches to generating energy from nuclear fusion. Relaxing the equilibrium requirement could widen the range of isotopes used in fusion fuels permitting cleaner and less hazardous reactions that do not produce high energy neutrons. Here we propose and implement a means to drive fusion reactions between protons and boron-11 nuclei, by colliding a laser-accelerated proton beam with a laser-generated boron plasma. We report proton-boron reaction rates that are orders of magnitude higher than those reported previously. Beyond fusion, our approach demonstrates a new means for exploring low-energy nuclear reactions such as those that occur in astrophysical plasmas and related environments.
author:
- 'C. Labaune$^1$'
- 'C. Baccou$^1$'
- 'S. Depierreux$^2$'
- 'C. Goyon$^2$'
- 'G. Loisel$^1$'
- 'V. Yahia$^1$'
- 'J. Rafelski$^3$'
date: 'Received 24 Jan 2013, Accepted 27 Aug 2013'
title: |
Fusion reactions initiated by laser-accelerated particle beams\
in a laser-produced plasma
---
-1.0cm
\[sec:intro\]Introduction
=========================
Inertial confinement fusion research over the past 40 year has been focused on the laser-driven reaction of deuterium ($d$) and tritium ($t$) nuclei under near thermal equilibrium conditions [@Nuckolls72; @Lindl95]. The $dt$ fusion reaction was chosen because of its higher thermal reaction rate compared to that of other light isotopes and a sustained burn can be achieved at relatively low temperatures ($\simeq 20$keV; Ref. [@Atzeni04]). However, this reaction produces an intense flux of high energy neutrons ($n)$, which represents a significant radiation hazard and generates nuclear waste. Recent advances in laser technology [@StrickMourou85], laser-plasma interaction physics [@Pesme93], and laser-accelerated particle beams [@Fews94; @Umstadter00; @Snavely00; @Fuchs06] could enable the development of fuels based on aneutronic nuclear reactions that produce substantially less radiation [@FuelReview00; @Nevins98]. In the case of the fusion reaction of protons ($p$) and boron-11 ($^{11\!}$B) nuclei, fusion energy is released predominantly in the form of charged alpha ($\alpha$) particles [@Becker87] rather than neutrons. Moreover, boron is both, more plentiful than tritium, and easy to handle. At high temperature, the equilibrium thermal fusion rate of $p^{11\!}$B is comparable to the $dt$-fusion rate [@Nevins00]. However, the use of $p^{11\!}$B with the spherical laser compression scheme would require excessively high laser energies to reach the high temperature and density necessary to achieve in thermal equilibrium fusion burn. Moreover, energy losses to Bremsstrahlung radiation under such conditions would prevent this reaction from being self-sustaining [@Moreau77; @pBEliezer96].
Such problems could be overcome by driving the $p^{11\!}$B reaction under conditions far from equilibrium, over shorter timescales than those involved in conventional inertial confinement fusion schemes, using short-pulsed high-intensity lasers. Laser-driven nuclear reactions are a new domain of physics [@Ledingham10], which aside from energy generation, are of interest to furthering the understanding of stellar nuclear processes [@Adelberger11; @Nomoto06] and of Big Bang nucleosynthesis [@Coc12]. The first demonstration of a laser-driven $p^{11\!}$B reaction [@Belyaev05] used a picosecond laser pulse at an intensity of $2\times 10^{18}\mathrm{W/cm}^2$ focused onto a composite target $^{11}$B+(CH$_2)_n$1 resulting in $\simeq 10^3$ reactions, or more [@Kimura09], in $4\pi$ steradians. The observed reaction yield was interpreted as the result of in-situ high-energy ions accelerated by high-intensity laser pulses [@Gitormer86]. Other theoretical and numerical schemes have been considered in the past 15 years with the aim of realizing a $p$B fusion reactor: a colliding beam fusion device [@Rostocker97], fusion in degenerate plasma [@Son04], plasma block ignition driven by nonlinear ponderomotive forces [@Hora09], and proton pulse from Coulomb explosion hitting a solid B target [@CoulombExpl11]. In all these schemes one seeks to improve the expected ratio of energy gain to loss by various means, but none of them has come close to achieving this goal.
Here we demonstrate an approach that realizes a substantial increase in the rate of a laser-driven $p^{11\!}$B reaction. We achieve this by using two laser beams. The first is a high energy, long pulse duration (nanosecond regime) laser that is focused on a solid target to form an almost completely ionized boron-11 plasma ($T_e\ge 0.5$keV). The second beam is a high intensity ($6\times 10^{18}\mathrm{W/cm}^2$), short pulse duration (picosecond regime) laser capable of accelerating a high energy proton beam (see appendix \[Methods\]). The picosecond timescale of a laser generated high intensity proton beam limits the ensuing radiation losses. Directing this beam into the plasma results in collisions with boron ions [@Patent12] at energies near to the nuclear resonance energies of $E_p=162$ and $675$keV (for $p + ^{11}$B resonances, see table 12.11 in Ref. [@NPB90]). Unlike the other efforts we do not wish to address an immediate potential for realization of an actual practical device, but to demonstrate scientific progress towards aneutronic fusion with short pulse lasers, and to present opportunities for future continuation of this research objective.
Experimental set-up
===================
Two laser beams at LULI2000
---------------------------
Experiments have been carried out on the Pico2000 laser facility at the LULI laboratory. This installation synchronizes two laser beams as described above, for use in the same vacuum chamber. The long high-energy pulse delivers 400J in \[1.5-4\]ns, square pulse, at 0.53$\mu$m of laser light. It was focused with an f/8 lens through a random phase plate producing a focal spot diameter around 100$\mu$m (full width at half maximum) and an average intensity of $5\times 10^{18}\mathrm{W/cm}^2$. It was used to produce a plasma from a natural boron target (20% of $^{10}$B and 80% of $^{11}$B) placed at an incidence of 45$^0$ from the laser pulse propagation axis. The boron plasma expanded in vacuum producing an electron density profile from zero to solid $(\simeq 6\times 10^{23}\mathrm{cm}^{-3})$. The short laser pulse delivered 20J in 1ps with high contrast at 0.53$\mu$m wavelength. It was tightly focused on target reaching intensities $\simeq 6\times 10^{18}\mathrm{W/cm}^2$ to produce a proton beam by the TNSA (Target Normal Sheath Acceleration) mechanism [@Passoni04]. Thin foils of aluminum, plastic and plastic covered by a thin layer of gold were irradiated at normal incidence. Details of the set-up are shown in [Fig.\[Figure1\]]{}. The two beams were set at a relative angle of 112.5$^0$ from each other. The distance between the thin foil and the boron target was 1.5mm. The time delay between the two beams was adjusted between 0.25 and 1.2ns, so that the proton beam interacted with a plasma state in various conditions of ionization and temperature. Shots were done with either the short pulse only so the proton beam interacted with solid boron, or with the two laser beams so the proton beam interacted with boron plasma.
![\[Figure1\]: [**Experimental set-up.**]{} Scheme of the experimental set-up showing the laser beam configuration, the target arrangement and the diagnostics (CR39 track detectors and a magnetic spectrometer). The picosecond pulse arrives from the left and generates a proton beam in the first 20$\mu$m Al foil, which impacts the boron plasma produced by the nanosecond pulse arriving from the bottom. The second 10$\mu$m Al foil protects the first one from irradiation by the nanosecond beam.](Fig_pBNatCom1){width="\columnwidth"}
Diagnostics
-----------
Track detectors CR-39, [@CR3907], covered by aluminum foils of various thickness between 6 and 80$\mu$m were used to collect impacts by both protons and $\alpha$-particles. Six detectors were used for each shot with angles 0, 15, 35, 70, 100 and 170$^0$ from the picosecond beam axis (0$^0$ is the forward direction). A magnetic spectrometer, with a magnetic field of 0.5T, was placed along the normal of the boron target to analyze the $\alpha$-particle spectrum. An aluminum filter with 12$\mu$m thickness was placed in front of the slit of the spectrometer to block low-energy ions, Boron below 11MeV, Carbon below 14MeV, Oxygen below 19MeV and Aluminum below 23MeV. The tracks observed inside the spectrometer are therefore mainly ascribable to protons and $\alpha$-particles which impact at the same position when having the same entrance point and same energy inside the spectrometer. Taking into account the loss of energy in the aluminum filter at the entrance of the spectrometer, $\alpha$-particles with energy between 3.3 and 7.5MeV and protons between 0.9 and 5MeV could be measured. This is also the method by which we characterized the proton beam spectra in preparation for fusion shots.
Plasma characterization
-----------------------
In the main part of the experiment, the objective was to study the number of $p^{11}$B reactions between the proton beam accelerated by the picosecond laser and the boron for different prepared target conditions. The expansion of the boron plasma produced by the nanosecond pulse was characterized by time-integrated X-ray pinhole images in the range $\simeq$3-5keV. Typically the overall extension of the boron plasma was around 200$\mu$m after 1ns. This diagnostic was also very useful to control the alignment and superposition of the two beams as shown in [Fig.\[Figure2\]]{} where three plasmas are observed along the direction of propagation of the picosecond beam: the first one comes from the 20$\mu$m Al foil which is used to produce the proton beam, the second one comes from a second 10$\mu$m Al foil that was inserted to protect the rear part of the first Al foil from the nanosecond scattered photons by the boron target and the third one is the boron plasma. Without the 10$\mu$m Al shield, the proton beam could not be produced in the two-beam irradiation shots because light scattered from the nanosecond pulse modified the rear surface of the Al foil, and as it is believed, cleaning up all the hydrogen rich impurities [@pBeam08]. Finally, an estimate of the electronic temperature of the boron plasma was obtained from the shift of the time-resolved stimulated Brillouin backscattering (SBS) spectra of the nanosecond pulse [@Kruer03]. They were recorded with a high-dispersion spectrometer and a streak camera. A typical example of such time-resolved SBS spectrum is shown in [Fig.\[Figure3\]]{} in the case of a 4ns pulse irradiating the boron target. The spectral shift of the SBS light was analyzed using the ion-acoustic velocity formula, providing an electron temperature of $T_e \simeq(0.7\pm0.15)$keV.
![\[Figure2\]: [**Observation of the multiple plasmas.**]{} Time-integrated X-ray pinhole image of the three plasmas along the direction of propagation of the picosecond beam. From left to right, we observe the heated parts of the first Al foil that produces the proton beam, the second Al foil that protects the first one and the boron plasma.](Fig_pBNatCom2){width="0.8\columnwidth"}
![\[Figure3\]: [**Stimulated Brillouin scattering spectrum.**]{} Time-resolved spectrum of stimulated Brillouin backscattering (SBS) of the nanosecond pulse from the boron plasma. The light is collected in the focusing optics of the nanosecond beam in the backward direction. The laser pulse is at 0.53$\mu$m with an intensity of $5\times 10^{14}\!\mathrm{\,W/cm}^2$.](Fig_pBNatCom3){width="0.75\columnwidth"}
Experimental results
====================
The total number of tracks observed in the magnetic spectrometer, per unit of surface on the CR39, as a function of the $\alpha$-particle energy is shown in [Fig.\[Figure4\]]{} for various shot conditions. No particle can be observed in the hatched part owing to the aluminum filter in front of the spectrometer. Shots with no boron target behind the Al foil (yellow triangles) display almost no track which means that very few protons are accelerated at an angle of 100° from the pico beam axis as expected from the TNSA process. In the case of shots with the picosecond beam alone, in which the proton beam interacts with a solid boron (blue diamonds), the number of tracks is close to the noise level, indicating very weak activity. Shots with the two beams, in which the proton beam interacted with boron plasma, demonstrate a large increase of the number of tracks by a factor of more than a hundred, in the highest case, compared to the previous ones. Three time-delays between the two beams have been tried: 0.25ns (open circles), 1ns (green triangles) and 1.2ns (blue squares) showing that the highest number of tracks was obtained for the longest time-delay which corresponds to the highest temperature and ionization state of the boron plasma.
![\[Figure4\]: [**$\alpha$-particle spectra.**]{} The total number of tracks observed in the magnetic spectrometer (with an entrance slit of 1mm$^2$) per unit of surface of CR39 as a function of the $\alpha$-particle energy for six shot configurations: yellow triangles = shot with no boron; blue diamonds = interaction of the proton beam with solid boron; blue square, green triangles = interaction of the proton beam with plasma boron and time-delay between the two beams of 1 and 1.2ns respectively; red circles = ibid, where the proton beam is produced in a foam rather than aluminum foil; open circles = short delay (0.25ns) between the nano and the pico pulses. The error bars in energy are given by the width of the CR39 on which the number of impacts has been counted; the error bars in the number of impacts are given by the shot to shot fluctuations ($\simeq\pm10\%$). The low energy domain has no counts as the entrance slit is protected by a 12$\mu$m Al foil.](Fig_pBNatCom4){width="0.9\columnwidth"}
These results were complemented by the analysis of the CR39 detectors which were positioned outside the spectrometer, close to the entrance slit. Tracks were observed only behind aluminum filters of thickness smaller than 24$\mu$m. If scattering of the proton beam by the boron plasma had sent protons into the spectrometer, tracks would have been recorded for all the aluminum thicknesses as the proton beam includes a continuous spectrum of energy up to $\simeq 10$MeV (see appendix \[Methods\]) which can cross a thickness of aluminum larger than 80$\mu$m. This is not the case for the produced $\alpha$-particles. Given our proton spectrum, the kinetic energy range of $\alpha$-particles produced in $p^{11}$B reactions [@Becker87] is 0.5-8MeV. Considering the exponential decrease of proton yield with energy there is little if any production of $\alpha$-particles with energy larger than 7.1MeV required to cross 36$\mu$m or more of aluminum. $\alpha$-particles with typical fusion energy between 3.3 and 5.4 MeV can cross 12 and 24$\mu$m of aluminum as observed in the $p^{11}$B shots. To conclude, the absence of high energy proton signature in control detectors placed near to the spectrometer is our evidence that scattered protons are not producing the track signature inside the spectrometer.
A rough estimate of the fusion rate can be obtained from the number of tracks in the spectrometer and the solid angle of observation ($\delta\Omega=1.1\times 10^{-5}$ sr). The highest event rate measured in this scheme was $9\times 10^6$/sr which is much higher than previous observations [@Belyaev05]. However, as pointed out in Ref. [@Kimura09], the choice of the detection energy region of the reaction products can underestimate the total yield as $\alpha$-particles with energy lower than 3.3MeV are not taken into account. In our experimental conditions, the low energy $\alpha$-particles are not expected to escape the plasma and furthermore those having relatively small energy when escaping may not be observed leading to an underestimate of the absolute fusion yield [@Kimura09]. External $\alpha$-particle detectors can only observe fusion products emitted within the plasma in a backward hemisphere at an energy allowing escape from the plasma. This means that $\alpha$-particles propagating in the forward direction, into the thick solid target, cannot be observed directly. These are accounted for by consideration of the solid angle of observation of the spectrometer. Test shots were performed with either the nano or the pico pulse alone on the boron target to measure the possible reactions in the hot plasma and the number of tracks in both cases was below 10. This demonstrates that the observed high number of particles in the two-beam experiments is definitely the consequence of the interaction of the proton beam with the boron plasma.
Discussion
==========
The features of the energy spectrum of $\alpha$-particles presented in [Fig.\[Figure4\]]{} agree well with the $p^{11}$B fusion spectrum. The rise at $E<5$MeV is a well known feature of the spectra arising from the formation of the broad $^8\mathrm{Be}^*$ resonance in first step with the subsequent $^8\mathrm{Be}^*\to \alpha+\alpha$ decay products seen both experimentally [@Stave11] and understood theoretically [@Dimitriev09]. The indication of a drop in the spectrum at the edge of our sensitivity near 3.5MeV could be the result of reaching the spectrometer edge, but is also a $p^{11}$B fusion feature observed in other experiments, and expected theoretically. The small bump near to 6MeV may correspond to the two-body reaction $p + ^{11}\!\mathrm{B}\to ^8\mathrm{Be} + \alpha + (8.59+E^*)$MeV, at the reaction resonance energy $E^*$, Ref. [@NPB90], where 2/3 of the available energy (6–6.5MeV ) is carried away by the $\alpha$-particle. This bump is experimentally observed in thin target experiments [@Stave11] but, considering our measurement error bars, is not a compelling feature in our results. Overall, the $\alpha$-particle spectrum in [Fig.\[Figure4\]]{} can be explained by the main known characteristic features of the $p^{11}$B fusion.
There are several possible plasma state mechanisms modifying the $p^{11}$B fusion yield. Recall that protons entering a solid atomic target use most of their energy to ionize atoms and do not penetrate beyond a thin layer on the front surface. In the case of a preformed plasma, energetic protons ($>0.5$MeV) can be subject to reduced stopping power [@Inject00], and so penetrate deeper inside the plasma. Moreover, since we are employing the TNSA mechanism to produce the proton pulse, we know that the proton beam is Coulomb-pulled by a relativistic electron pre-pulse cloud. This cloud contains around 10-30% of the pico pulse energy and impacts the boron plasma about 100ps ahead for our geometry: the distance between the thin foil and the boron target was 1.5mm, which at the velocity of light, corresponds to 5ps travel time. In comparison, a proton of kinetic energy $E_p=1$MeV and velocity $(v_p/c)^2= 2E_p/(m_p c^2)$ will take 21.7times longer, that is $108$ps to travel this distance. The relativistic electron cloud may condition the boron plasma just in the proton-target area, pushing out the electrons and forming an ionic channel. The consequence is that the proton pulse energy loss caused by interactions with plasma electrons is reduced while number of interactions with boron atomic nuclei is correspondingly enhanced. Note that because of the proton-boron mass asymmetry, protons lose relatively little energy in each deep near-nuclear Coulomb collision. Therefore across the large width $\simeq 250$keV of the $E_p=675$keV (proton energy $E_p$ on rest boron target) resonance [@NPB90], many such interactions can result in a large probability of fusion yield per proton. Furthermore the fusion cross section may be modified by plasma effects e.g. a modified electron screening [@Angulo93; @Barker02; @Kimura04] of the boron nuclei which may not be completely ionized. However this effect so far has been observed to be significant only at lowest reaction energies but is not well-understood in hot plasmas.
The rate of proton initiated fusion in a $^{11}$B target is $\lambda_f = \sigma_f\,\rho_B\, v_r$, where $v_r$ is the relative $p$ – $^{11}$B velocity, $v_r=c/26$ – $c/33$ in $E_p =0.68$ – $0.43$MeV proton energy domain of interest, $\rho_B$ is $^{11}$B target density, which we take as a fraction of the solid natural target $\rho_{0B} = 1.0\times 10^{23}$atoms/cm$^3$, and $\sigma_f$ is the resonant fusion cross section which averages in the interval of energy of the proton $E_p=0.68$ – $0.43$MeV to 1 barn [@Nevins98]. This gives an average fusion reaction rate of $\lambda_f = 1/(100\mathrm{ns})$. In our situation the number of fusions achievable per proton is limited by the active depth of the reduced density plasma target which protons will traverse in $\simeq (1/30)$ns. Allowing for available higher energy protons in laser generated particle beam, in our present experimental conditions, we expect about 1 in 300 – 3000 protons will be able to induce a $p^{11}$B fusion reaction. Assuming that $10^{-3 }$ of the protons produce a $p^{11}$B fusion reaction, the total number of reactions can be estimated very crudely by $N=n_1\,n_2\,\sigma \,v$, which gives $N\simeq 8\times 10^7$ in $4\pi$, (with $n_1=5\times 10^9$, $n_2=4\times 10^{14}$, assuming a reacting volume of $4\times 10^{-8}$cm$^3$, cylinder of 20$\mu$m radius and 30$\mu$m length, and an average density of $10^{22}$cm$^{-3}$). This corresponds to 88 $\alpha$-particles in the solid angle of observation ($\delta \Omega=1.1\times 10^{-5}$sr), in qualitative agreement with the observed numbers.
Concerning the observability of $\alpha$-particles produced in fusion, it is clear that it depends on their energy spectrum and angular diagram of emission, which then depends on the relative cross sections of the different possible reactions [@Becker87; @Kimura09], which are unknown under our conditions. If for some reason there are some differences in fusion reactions in solid compared to plasma medium, either in the effective cross section or in their capability to escape from the target, this could contribute to the modification in observed $\alpha$-count rates. Future experimental work will be dedicated to test the relative importance of these different mechanisms.
Although our results are specific to the $p^{11\!}$B case, a similar approach could be used to study the reaction of other light isotopes. This provides a new approach to exploring aneutronic nuclear fusion reactions in dense plasma environment. Furthermore, our method could enable progress in the development of so-called fast-ignition fusions scheme [@Roth01] by providing a short lived hot spot generated by both the particle beam and the $\alpha$-particles produced in the reactions, initiating and promoting a propagating burn wave. The $p^{11\!}$B case is unique in that secondary $\alpha$B reactions can regenerate the high energy proton, sustaining a fusion chain. Our experimental approach also suggests opportunities to explore nuclear reactions of astrophysical interest [@40] in an environment more similar to the early universe or stellar interiors.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We acknowledge the support of the LULI teams, and discussions with V. Tikhonchuk, W. Rozmus. JR wishes to thank Ecole Polytechnique for the support of his three months sabbatical research visit in 2012, and Christine Labaune for hospitality at the LULI laboratory where this work was carried out.
Contributions {#contributions .unnumbered}
-------------
C.L., G.L., and V.Y. were involved in the experimental project planning and the target design and carried out the experiments. S.D. and C.G. participated in the experiment and were involved in the optimization of the multiple targets configuration. C.B. implemented, developed, and analyzed the CR39. C.L. and J.R. conceived the idea of the scheme, were leaders of the analysis team and wrote jointly all research reports including this one.
\[Methods\]Methods
==================
[[**Set up:**]{} Experiments were conducted on the LULI2000 laser installation at Ecole Polytechnique. Two beams are produced by synchronized oscillators and amplified in similar neodymium glass chains. The first one delivers nanosecond pulses of 1kJ and the second one delivers picosecond pulses of 100J which are stretched before amplification and then compressed before focusing using the CPA (chirped pulse amplification) method. Both beams are initially at wavelength 1.06$\mu$m and then converted to the second harmonic just before focusing, which produces a high contrast for the pico pulse which is important for proton beam acceleration. CR39 detectors were etched after irradiation during 6–12 hours in a solution of NaOH in H$_2$O at 70C. The number and diameters of the tracks were analyzed using a Nikon microscope with a magnification of 20. The calibration of the CR39 for $\alpha$-particles was done using a $^{233}$U source which delivers $\alpha$-particles with energy of 5MeV. By increasing the thickness of the air layer between the source and the CR-39, lower energy $\alpha$-particles could also be observed.]{}
[**Laser-accelerated proton beam:**]{} Protons with the required high kinetic energy are now routinely produced by short laser pulses having intensity on target higher than $2\times 10^{18}\mathrm{W/cm}^2$. Therefore, high-intensity lasers are a new tool in the study of nuclear fusion reactions in a high density regime. Their unique features are the formation of a high intensity proton pulse of time duration similar to the laser pulse, and a tunable spectral distribution.
The first part of the experiment was dedicated to the proton beam optimization. In their interaction with the boron target, protons with energy around 170keV and 700keV would be most capable to take direct advantage of the resonances in the cross section of the $p^{11}$B reactions. Nevertheless, we believed that in our target condition a broad energy spectrum and high intensity proton pulse had a greater advantage to increase the fusion yield. Our boron target was thick and the electron density profile of the boron plasma created by the nano-pulse displayed all densities up to the solid assuring that the incoming particle pulse would be stopped. So, we chose to optimize the number of above 1MeV protons in the particle pulse. We tried out three types of targets: CH with 2$\mu$m thickness covered by 125nm of gold, low-density (3mg/cc) cellulose-triacetate-(C$_{12}$H$_{16}$O$_{8}$) TAC foams [@TACfoam] with 300$\mu$m length and Al foils with thickness 10 and 20$\mu$m. The largest number of high-energy protons on the laser axis in the forward direction was obtained with the 20$\mu$m Al foils,the protons are known to originate from hydrogenated deposit on the back of the Al foil.
![\[Figure5\]: [**Proton spectrum.**]{} Energy spectrum of the proton beam produced by the interaction of an aluminum foil of 20$\mu$m thickness with the LULI2000 picosecond pulse at 0.53$\mu$m, with a pulse duration of 1ps and an intensity of $6\times 10^{18}\!\mathrm{\,W/cm}^2$. The error bars in energy are given by the differences in thickness of the Al filters covering the CR39 on which the number of impacts has been counted; the error bars in the number of impacts are given by the shot to shot fluctuations ($\simeq\pm10\%$).](Fig_pBNatCom5){width="0.9\columnwidth"}
[An estimate of the energy distribution of the proton beam in forward direction was obtained by analyzing the number of impacts on the CR-39 covered by 24, 36, 44, 56, 60, 72 and 80$\mu$m of aluminum. In addition, the absolute number of protons with energy higher than 5MeV was deduced from the boron activation which produces $^{11}$C through the nuclear reaction [@Ledingham04]: $p + ^{11}\!\mathrm{B} \to ^{12}\!\mathrm{C}\to n + ^{11}\!\mathrm{C} -2.9$MeV. The $^{11}$C has a half-time decay of 20.334 minutes and was measured from the residues of the target just after the shot until one hour later. From these measurements, we deduced that per “fusion” shot more than $5\times 10^7$ protons with energy larger than 5 MeV were produced by the pico laser pulse and arrived on the target. We further determined that the angular emission of protons in the pulse was strongly peaked along the laser axis with a typical half-angle of $\simeq 5^0$. An example of the energy spectrum of the proton beam produced by the interaction of an aluminum foil of 20$\mu$m thickness with the picosecond pulse at an intensity of $6\times 10^{18}\,\mathrm{W/cm}^2$ is shown in [Fig.\[Figure5\]]{}. A continuous spectrum of energy was observed up to $\simeq 10$MeV in agreement with other observations for similar conditions.]{}
[[**Two-beam experiments:**]{} The main goal of the experiments we report here was to demonstrate the effect of the preparation of the target boron plasma state on the observed reaction rate. The largest number of CR-39 associated tracks was observed in the case of the best geometric superimposition of the proton beam and the boron plasma, when the proton beam was produced from a 20$\mu$m thick Al foil and arrived close to the end of the nanosecond pulse, so the boron plasma was at maximum temperature and ionization. When using a 3mg/cc 300$\mu$m long foam to produce the protons, the number of tracks was reduced by $\simeq 7$ compared to the case where the proton beam was from a 20$\mu$m thick Al foil, with all the other parameters being the same (red circles in [Fig.\[Figure4\]]{}). This reduction may be directly attributed to the reduction of the total number of MeV energy protons achieved in the case of the foam target. Complementary shots were dedicated to establish the mechanisms leading to our results, and of most interest in the present discussion is the case in which we fired the picosecond pulse on a target composed of boron covered by a 0.9$\mu$m CH foil to produce high energy ions in a plasma mixture of $p$ and B. The number of tracks was close to our noise level. Those conditions are close to the ones used by Belyaev [@Belyaev05] further demonstrating that the production of a particle pulse comprising high-energy protons (above 1 MeV) in a separate target could be the origin of the significant increase of the fusion yield.]{}
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|
---
abstract: |
We use class field theory, specifically Drinfeld modules of rank $1$, to construct a family of asymptotically good algebraic-geometric (AG) codes over fixed alphabets. Over a field of size $\ell^2$, these codes are within $2/(\sqrt{\ell}-1)$ of the Singleton bound. The functions fields underlying these codes are subfields with a cyclic Galois group of the narrow ray class field of certain function fields. The resulting codes are “folded" using a generator of the Galois group. This generalizes earlier work by the first author on folded AG codes based on cyclotomic function fields. Using the Chebotarev density theorem, we argue the abundance of inert places of large degree in our cyclic extension, and use this to devise a linear-algebraic algorithm to list decode these folded codes up to an error fraction approaching $1-R$ where $R$ is the rate. The list decoding can be performed in polynomial time given polynomial amount of pre-processed information about the function field.
Our construction yields algebraic codes over constant-sized alphabets that can be list decoded up to the Singleton bound — specifically, for any desired rate $R \in (0,1)$ and constant ${\varepsilon}> 0$, we get codes over an alphabet size $(1/{\varepsilon})^{O(1/{\varepsilon}^2)}$ that can be list decoded up to error fraction $1-R-{\varepsilon}$ confining close-by messages to a subspace with $N^{O(1/{\varepsilon}^2)}$ elements. Previous results for list decoding up to error-fraction $1-R-{\varepsilon}$ over constant-sized alphabets were either based on concatenation or involved taking a carefully sampled subcode of algebraic-geometric codes. In contrast, our result shows that these folded algebraic-geometric codes [*themselves*]{} have the claimed list decoding property.
address:
- 'Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA.'
- 'Division of Mathematical Sciences, School of Physical & Mathematical Sciences, Nanyang Technological University, Singapore.'
author:
- Venkatesan Guruswami
- Chaoping Xing
title: Optimal rate algebraic list decoding using narrow ray class fields
---
Introduction
============
Reed-Solomon codes are a classical and widely used family of error-correcting codes. They encode messages, which are viewed as polynomials $f \in {\mathbb{F}}_q[X]$ of degree $< k$ over a finite field ${\mathbb{F}}_q$, into [*codewords*]{} consisting of the evaluations of $f$ at a sequence of $n$ distinct elements $\alpha_1,\dots,\alpha_n \in {\mathbb{F}}_q$ (this requires a field size $q {\geqslant}n$). We refer to $n$ as the block lengh of the code. The rate of this code, equal to the ratio of number of message symbols to the number of codeword symbols, equals $R = k/n$. Since two distinct polynomials of degree $<
k$ can agree on at most $k-1$ distinct points, every pair of Reed-Solomon codewords differ on more than $n-k$ positions. In other words, the [*relative distance*]{} of this code, or the minimum fraction of positions two distinct codewords differ on, is bigger than $(1-R)$. This means that even if up to a fraction $(1-R)/2$ of the $n$ codeword symbols, are corrupted in an [*arbitrary*]{} manner, the message polynomial $f$ is still uniquely determined. Moreover, classical algorithms, starting with [@peterson], can recover the message $f$ in such a situation in polynomial time.
For a fraction of errors exceeding $(1-R)/2$, unambiguous decoding of the correct message is not always possible. This holds not just for the Reed-Solomon code but for [*every*]{} code. However, if we allow the decoder to output in the worst-case a small list of messages whose encodings are close to the corrupted codeword, then it turns out that one can correct a much larger error fraction. This model is called [*list decoding*]{}. Using the probabilistic method, for any ${\varepsilon}> 0$, one can prove the abundance of codes of rate $R$ which can be list decoded up to an error fraction $(1-R-{\varepsilon})$ with a maximum output list size bounded by a constant depending only on ${\varepsilon}$. This error fraction is twice the classicial $(1-R)/2$ bound, and further is optimal as the message has $Rn$ symbols of information and recovering it up to some small ambiguity is impossible from fewer than a fraction $R$ of correct codeword symbols.
Recent progress in algebraic coding theory has led to the construction of explicit codes which can be efficiently list decoded up to an error fraction approaching the $1-R$ information-theoretic limit. The first such construction, due to Guruswami and Rudra [@GR-FRS], was [ *folded Reed-Solomon codes*]{}. In the [*$m$-folded*]{} version of this code (where $m$ is a positive integer), the Reed-Solomon (RS) encoding $(f(1),f(\gamma),\cdots,f(\gamma^{n-1}))$ of a low-degree polynomial $f \in {\mathbb{F}}_q[X]$ is viewed as a codeword of length $N = n/m$ over the alphabet ${\mathbb{F}}_q^m$ by blocking together successive sets of $m$ symbols. Here $\gamma$ is a primitive element of the field ${\mathbb{F}}_q$. The alphabet size of the folded RS codes is $q^m > N^m$. To list decode these codes up to an error fraction $1-R-{\varepsilon}$, one has to choose $m
\approx 1/{\varepsilon}^2$ which makes the alphabet size a larger polynomial in the block length. In comparison, the probabilistic method shows the existence of such list decodable codes over an alphabet size $\exp(O(1/{\varepsilon}))$, which is also the best possible asymptotic dependence on ${\varepsilon}$.
It is possible to bring down the alphabet size of folded RS codes by concatenating them with appropriate optimal codes found by a brute-force search, followed by symbol redistribution using an expander [@GR-FRS]. However, the resulting codes have a large construction and decoding complexity due to the brute-force decoding of the inner codes used in concatenation. Furthermore, these codes lose the nice algebraic nature of folded RS codes which endows them with other useful features like list recovery and soft decoding. It is therefore of interest to find explicitly described algebraic codes over [*smaller*]{} alphabets with list decoding properties similar to folded RS codes.
Algebraic-geometric (AG) codes are a generalization of Reed-Solomon codes based on algebraic curves which have $n \gg q$ ${\mathbb{F}}_q$-rational points. These enable construction of RS-like codes with alphabet size smaller than (and possibly even dependent of) the block length. Thus, they provide a possible avenue to construct the analog of folded RS codes over smaller alphabets.
The algebraic crux in list decoding folded RS codes was the identity $f(\gamma X) \equiv f(X)^q \pmod {E(X)}$ for $E(X) = X^{q-1}-\gamma$ which is an irreducible polynomial over ${\mathbb{F}}_q$. Extending this to other algebraic-geometric codes requires finding a similar identity in the function field setting. As noted by the first author [@Gur-cyclo], this can be achieved using Frobenius automorphisms $\sigma$ in cyclic Galois extensions, and considering the residue of $f^{\sigma}$ at a place of high degree in the function field. Using certain subfields of cyclotomic function fields, Guruswami [@Gur-cyclo] was able to extend the folded RS list decoding result of [@GR-FRS] and obtain folded algebraic-geometric codes of rate $R$ list decodable up to error fraction $1-R-{\varepsilon}$ over an alphabet size $(\log N)^{O(1/{\varepsilon}^2)}$. In other words, the alphabet size was reduced to poly-logarithmic in the block length $N$ of the code.
Our result
----------
The main result in this work is a construction of folded algebraic-geometric codes which brings down the alphabet size to a [*constant*]{} depending only on ${\varepsilon}$. This is based on algebraic function fields constructed via class field theory, utilizing Drinfeld modules of rank $1$.
Let $\ell$ be a square prime power and let $q =\ell^2$. For every $R \in (0,1)$, there is an infinite family of ${\mathbb{F}}_q$-linear algebraic-geometric codes of rate at least $R$ which has relative distance at least $1-R-2/(\sqrt{\ell}-1)$.
For every pair of integers $m {\geqslant}s {\geqslant}1$, the $m$-folded version of these codes (which is a code over alphabet ${\mathbb{F}}_q^m$) can be list decoded from an error fraction $$\tau = \frac{s}{s+1} \biggl( 1 - \frac{m}{m-s+1} \Bigl( R + \frac{2}{\sqrt{\ell}-1} \Bigr) \biggr) \ ,$$ outputting a subspace over ${\mathbb{F}}_q$ with at most $O(N^{(\sqrt{\ell}-1)s})$ elements that includes all message functions whose encoding is within Hamming distance $\tau N$ from the input. (Here $N$ denotes the block length of the code.)
Given a polynomial amount of pre-processed information about the code, the algorithm essentially consists of solving two linear systems over ${\mathbb{F}}_q$, and thus runs in deterministic polynomial time.
Picking suitable parameters in the above theorem, specifically $\ell
\approx 1/{\varepsilon}^2$, $s \approx 1/{\varepsilon}$, and $m \approx 1/{\varepsilon}^2$, leads to folded AG codes with alphabet size $(1/{\varepsilon})^{O(1/{\varepsilon}^2)}$ of any desired rate $R \in (0,1)$ that are list decodable up to error fraction $1-R-{\varepsilon}$ with a maximum output list size bounded by $N^{O(1/{\varepsilon}^2)}$. In other words, the polylogarithmic alphabet size of cyclotomic function fields is further improved to a constant depending only on ${\varepsilon}$. We prove the above theorem by employing the recently developed [ *linear-algebraic*]{} approach to list decoding, which was first used to an alternate, simpler proof of the list decodability of folded RS codes up to error fractions approaching $1-R$ (see [@GW-merged]).
One of the simple but key observations that led to this work is the following. In order to apply the linear algebraic list decoder for a folded version of AG codes (such as the cyclotomic function field based codes of [@Gur-cyclo]), one can use the Frobenius automorphism based argument to just combinatorially [*bound*]{} the list size, but such an automorphism is [*not*]{} needed in the actual decoding algorithm. In particular, we [*don’t*]{} need to find high degree places with a specific Galois group element as its Frobenius automorphism (this was one of the several challenges in the cyclotomic function field based construction [@Gur-cyclo]), but only need the [*existence*]{} of such places. This allows us to devise a linear-algebraic list decoder for folded versions of a family of AG codes, once we are able to construct function fields with certain stipulated properties (such as many rational places compared to the genus, and the existence of an automorphism which powers the residue of functions modulo some places). We then construct function fields with these properties over a fixed alphabet using class field theory, which is our main technical contribution.
This gives the first construction of folded AG codes over constant-sized alphabets list decodable up to the optimal $1-R$ bound, although we are not able to efficiently construct the (natural) representation of the code that is utilized by our polynomial time decoding algorithm. This representation consists of the evaluations of regular functions at the rational places used for encoding (by a regular function at a place, we mean a function having no pole at this place).
In our previous works [@GX-stoc12; @GX-gabidulin], we considered list decoding of folded AG codes and a variant where rational points over a subfield are used for encoding. We were able to show that a [*subcode*]{} of these codes can be efficiently list decoded up to the optimal $1-R-{\varepsilon}$ error fraction. The subcode is picked based on variants of subspace-evasive sets (subsets of the message space that have small intersection with low-dimensional subspaces) that are not explicitly constructed. In contrast, in this work we are able to list decode the folded AG codes [*themselves*]{}, and no randomly constructed subcode is needed.
Techniques
----------
Our main techniques can be summarized as follows.
Our principal algebraic construction is that of an infinite family of function fields over a fixed base field ${\mathbb{F}}_q$ with many rational places compared to their genus, together with certain additional properties needed for decoding. Our starting point is a family of function fields $E/{\mathbb{F}}_\ell$ (where $\ell = \sqrt{q}$) such as those from the Garcia-Stichtenoth towers [@GS95; @GS96] which attain the Drinfeld-Vlădut bound (the best possible trade-off between number of rational places and genus). We consider the constant field extension $L = {\mathbb{F}}_q \cdot E$, and take its narrow ray class field of with respect to some high degree place. We descend to a carefully constructed subfield $F$ of this class field in which the ${\mathbb{F}}_q$-rational places in $L$ split completely, and further the extension $F/L$ has a cyclic Galois group.
A generator $\sigma$ of this cyclic group $\Gal(F/L)$, which is an automorphism of $F$ of high order, is used to order the evaluation points in the AG code and then to fold this code. This last part is similar to the earlier cyclotomic construction, but there the full extension $F/{\mathbb{F}}_q(X)$ was cyclic. This was a stringent constraint that in particular ruled out asymptotically good function fields — in fact even abelian extensions must have the ratio of the number of rational places to genus tend to 0 when the genus grows [@FPS]. In our construction, only the portion $F/L$ needs to be cyclic, and this is another insight that we exploit.
Next, using the Chebotarev density theorem, we argue the existence of many large degree places which are inert in the extension $F/L$ and have $\sigma$ as their Frobenius automorphism. This suffices to argue that the list size is small using previous algebraic techniques. Essentially the values of the candidate message functions at the inert places mentioned above can be found by finding the roots of a univariate polynomial over the residue field, and these values can be combined via Chinese remaindering to identify the message function.
Under the linear-algebraic approach, the above list will in fact be a subspace. Thus knowing that this subspace has only polynomially many elements is enough to list all elements in the subspace in polynomial time by solving a linear system! To solve the linear system, we make use of the local power series expansion of a basis of the Riemann-Roch message space at certain rational places of $F$ (namely those lying above a rational place of $L$ that splits completely in $F/L$). To summarize, some of the novel aspects of this work are:
1. The use of class fields based on rank one Drinfeld modules to construct function fields $F/{\mathbb{F}}_q$ with many ${\mathbb{F}}_q$-rational places compared to its genus, [*and*]{} which have a subfield $L$ such that $F/L$ is a cyclic Galois extension of sufficiently high degree.
2. The use of the Chebotarev density theorem to combinatorially bound the list size.
3. Decoupling the [*proof*]{} of the combinatorial bound on list size from the algorithmic task of [*computing*]{} the list. This computational part is tackled by a linear-algebraic decoding algorithm whose efficiency automatically follows from the list size bound.
Organization
------------
In Section 2, we show a construction of folded algebraic-geometric codes over arbitrary function fields with many rational places and an automorphism of relatively large order. Then we present a linear-algebraic list decoding of the folded codes. Under some assumption about the base function fields, we prove in the same section that the folded codes is deterministically list decodable up to the Singleton bound. Section 3 is devoted to the construction of the base function fields needed in Section 2 for constructing our folded codes. Our construction of the base function fields is through class field theory, specifically Drinfeld modules of rank $1$. In Section 4, we discuss the encoding and decoding of our folded codes by some possible approach of finding explicit equations of the base function fields that are constructed in Section 3. The main result of this paper is then stated after discussion of encoding and decoding.
Linear-Algebraic List Decoding of Folded Algebraic-Geometric Codes {#sec:ALD}
==================================================================
In this section, we first present a construction of folded algebraic geometric codes over arbitrary function fields with certain properties and then give a deterministic list decoding of folded algebraic geometric codes over certain function fields satisfying some conditions.
Preliminaries on Function Fields
--------------------------------
For convenience of the reader, we start with some background on global function fields over finite fields.
For a prime power $q$, let ${\mathbb{F}}_q$ be the finite field of $q$ elements. An [ algebraic function field]{} over ${\mathbb{F}}_q$ in one variable is a field extension $F \supset {\mathbb{F}}_q$ such that $F$ is a finite algebraic extension of ${\mathbb{F}}_q(x)$ for some $x\in F$ that is transcendental over ${\mathbb{F}}_q$. The field ${\mathbb{F}}_q$ is called the full constant field of $F$ if the algebraic closure of ${\mathbb{F}}_q$ in $F$ is ${\mathbb{F}}_q$ itself. Such a function field is also called a global function field. From now on, we always denote by $F/{\mathbb{F}}_q$ a function field $F$ with the full constant field ${\mathbb{F}}_q$.
Let $\PP_F$ denote the set of places of $F$. The divisor group, denoted by ${\rm Div}(F)$, is the free abelian group generated by all places in $\PP_F$. An element $G=\sum_{P\in\PP_F}n_PP$ of ${\rm Div}(F)$ is called a divisor of $F$, where $n_P=0$ for almost all $P\in\PP_F$. The support, denoted by $\Supp(G)$, of $G$ is the set $\{P\in\PP_F:\; n_P\neq 0\}$. For a nonzero function $z\in F$, the principal divisor of $z$ is defined to be ${\rm div}(z)=\sum_{P\in\PP_F}\nu_P(z)P$, where $\nu_P$ denotes the normalized discrete valuation at $P$. The zero and pole divisors of $z$ are defined to be ${\rm div}(z)_0=\sum_{\nu_P(z)>0}\nu_P(z)P$ and ${\rm div}(z)_{\infty}=-\sum_{\nu_P(z)<0}\nu_P(z)P$, respectively.
For a divisor $G$ of $F$, we define the Riemann-Roch space associated with $G$ by $$\mL(G):=\{f\in F^*:\; {\rm div}(f)+G{\geqslant}0\}\cup\{0\}.$$ Then $\mL(G)$ is a finite dimensional space over ${\mathbb{F}}_q$ and its dimension $\ell(G)$ is determined by the Riemann-Roch theorem which gives $$\ell(G)=\deg(G)+1-g+\ell(W-G),$$ where $g$ is the genus of $F$ and $W$ is a canonical divisor of degree $2g-2$. Therefore, we always have that $\ell(G){\geqslant}\deg(G)+1-g$ and the equality holds if $\deg(G){\geqslant}2g-1$.
For a function $f$ and a place $P\in\PP_F$ with $\nu_P(f){\geqslant}0$, we denote by $f(P)$ the residue class of $f$ in the residue class field $F_P$ at $P$. For an automorphism $\phi\in \Aut(F/{\mathbb{F}}_q)$ and a place $P$, we denote by $P^{\phi}$ the place $\{\phi(x):\; x\in P\}$. For a function $f\in F$, we denote by $f^{\phi}$ the action of $\phi$ on $f$. If $\nu_P(f){\geqslant}0$ and $\nu_{P^{\phi}}(f){\geqslant}0$, then one has that $\nu_P(f^{\phi^{-1}}){\geqslant}0 $ and $f(P^{\phi})=f^{\phi^{-1}}(P)$. Furthermore, for a divisor $G=\sum_{P\in\PP_F}m_PP$ we denote by $G^{\phi}$ the divisor $\sum_{P\in\PP_F}m_PP^{\phi}$.
Folded Algebraic Geometric Codes
--------------------------------
To construct our folded codes, we assume that there exists a global function field $F$ with the full constant field ${\mathbb{F}}_q$ having the following property:
[**Property (P1)**]{}
- There exists an automorphism ${\sigma}$ in $\Aut(F/{\mathbb{F}}_q)$;
- $F$ has $mN$ distinct rational places $P_1, P_1^{{\sigma}},\dots, P_1^{{\sigma}^{m-1}}, P_2, P_2^{{\sigma}},\dots,$ $ P_2^{{\sigma}^{m-1}}, \dots, $ $P_N, P_N^{{\sigma}},\dots, P_N^{{\sigma}^{m-1}}$;
- $F$ has a divisor $D$ of degree $e$ such that $D$ is fixed under ${\sigma}$, i.e., $D^{{\sigma}}=D$; and $P_i^{{\sigma}^j}\not\in\Supp(D)$ for all $1{\leqslant}i{\leqslant}N$ and $0{\leqslant}j{\leqslant}m-1$.
A folded algebraic geometric code can be defined as follows.
\[def:f-ag-code\][The folded code from $F$ with parameters $N,l,q,e,m$, denoted by ${\FH}(N,l,q,e,m)$, encodes a message function $f \in \cL(lD)$ as $$\label{eq:f-ag-defn} \pi:\quad
f \mapsto
\left(
\left[\begin{array}{c} f(P_1) \\ f(P_1^{{\sigma}}) \\ \vdots \\ f(P_1^{{\sigma}^{m-1}})\end{array}\right],
\left[\begin{array}{c} f(P_2) \\ f(P_2^{{\sigma}}) \\ \vdots \\ f(P_2^{{\sigma}^{m-1}})\end{array}\right],
\ldots,
\left[\begin{array}{c} f(P_N) \\ f(P_N^{{\sigma}}) \\ \vdots \\ f(P_N^{{\sigma}^{m-1}})\end{array}\right]
\right) \in \left( {\mathbb{F}}_{q}^m \right)^{N} \ .$$ ]{}
Note that the folded code ${\FH}(N,l,q,e,m)$ has the alphabet ${\mathbb{F}}_q^m$ and it is ${\mathbb{F}}_q$-linear. Furthermore, ${\FH}(N,l,q,e,m)$ has the following parameters.
\[lem:para\] If $le<mN$, then the above code ${\FH}(N,l,q,e,m)$ is an ${\mathbb{F}}_q$-linear code with alphabet size $q^{m}$, rate at least $\frac{le-g+1}{Nm}$, and minimum distance at least $N - \frac{le}{m}$.
It is clear that the map $\pi$ in (\[eq:f-ag-defn\]) is ${\mathbb{F}}_q$-linear and the kernel of $\pi$ is $$\cL\Bigl( lD-\sum_{i=1}^N\sum_{j=0}^{m-1}P_i^{{\sigma}^j}\Bigr)$$ which is $\{0\}$ under the condition that $le<mN$. Thus, $\pi$ is injective. Hence, the rate is at least $\frac{le-g+1}{Nm}$ by the Riemann-Roch theorem. To see the minimum distance, let $f$ be a nonzero function in $\cL(lD)$ and assume that $I$ is the support of $\pi(f)$. Then the Hamming weight ${\rm wt}_H(\pi(f))$ of $\pi(f)$ is $|I|$ and $f\in \cL\left(lD-\sum_{i\not\in I}\sum_{j=0}^{m-1}P_i^{{\sigma}^j}\right)$. Thus, $0{\leqslant}\deg\left(lD-\sum_{i\not\in I}\sum_{j=0}^{m-1}P_i^{{\sigma}^j}\right)=le-m(N-|I|)$, i.e., ${\rm wt}_H(\pi(f))=|I|{\geqslant}N-\frac{le}{m}$. This completes the proof.
List Decoding of Folded Algebraic Geometric Codes
-------------------------------------------------
Suppose a codeword (\[eq:f-ag-defn\]) encoded from $f\in \cL(lD)$ was transmitted and received as $$\label{eq:recd-word}
\mathbf{y} =
\left(
\begin{array}{ccccc}
y_{1,1} & y_{2,1} & & & y_{N,1}\\
y_{1,2} & y_{2,2} & & & \vdots\\
& & & \ddots & \\
y_{1,m} & \cdots &&& y_{N,m}
\end{array}
\right),$$ where some columns are erroneous. Let $s {\geqslant}1$ be an integer parameter associated with the decoder.
\[lem:herm-interpolation\] Given a received word as in [(\[eq:recd-word\])]{}, we can find a nonzero linear polynomial in $F[Y_1,Y_2,\dots,Y_s]$ of the form $$\label{eq:form-of-Q}
Q(Y_1,Y_2,\dots,Y_s) =
A_0 + A_1 Y_1 + A_2 Y_2 + \cdots + A_s Y_s $$ satisfying $$\label{eq:interpolation-cond}
Q(y_{i,j+1},y_{i,j+2},\cdots,y_{i,j+s}) =A_0(P_i^{{\sigma}^{j}})+A_1(P_i^{{\sigma}^{j}})y_{i,j+1}+\cdots+A_s(P_i^{{\sigma}^{j}})y_{i,j+s}= 0$$ [for ]{} $i=1,2,\dots,N$ [ and ]{} $j =0,1,\dots,m-s$. The coefficients $A_i$ of $Q$ satisfy $A_i \in \cL(\kappa D)$ for $i=1,2,\dots,s$ and $A_0\in \cL(({\kappa}+l)D)$ for a “degree" parameter $d$ chosen as $$\label{eq:choice-of-kappa}
{\kappa}=\left\lfloor \frac {N(m-s+1)-el+(s+1)(g-1)+1}{e(s+1)}\right\rfloor .$$
Let $u$ and $v$ be dimensions of $\cL({\kappa}D)$ and $\cL(({\kappa}+l)D)$, respectively. Let $\{x_1,\dots,x_u\}$ be an ${\mathbb{F}}_q$-basis of $\cL({\kappa}D)$ and extend it to an ${\mathbb{F}}_q$-basis $\{x_1,\dots,x_v\}$ of $\cL((d+l)D)$. Then $A_i$ is an ${\mathbb{F}}_q$-linear combination of $\{x_1,\dots,x_u\}$ for $i=1,2,\dots,s$ and $A_0$ is an ${\mathbb{F}}_q$-linear combination of $\{x_1,\dots,x_v\}$. Determining the functions $A_i$ is equivalent to determining the coefficients in the combinations of $A_i$. Thus, there are totally $su+v$ freedoms to determine $A_0,A_1,\dots,A_s$. By the Riemann-Roch theorem, the number of freedoms is at least $s({\kappa}e-g+1)+({\kappa}+l)e-g+1$.
On the other hand, there are totally $N(m-s+1)$ equations in (\[eq:interpolation-cond\]). Thus, there must be one nonzero solution by the condition (\[eq:choice-of-kappa\]), i.e., $Q(Y_1,Y_2,\dots,Y_s)$ is a nonzero polynomial.
\[lem:Q-is-good\] If $f$ is a function in $\cL(lD)$ whose encoding (\[eq:f-ag-defn\]) agrees with the received word $\mathbf{y}$ in at least $t$ columns with $$t>\frac{({\kappa}+l)e}{m-s+1} \ ,$$ then $Q(f,f^{{\sigma}^{-1}},\dots,f^{{\sigma}^{-(s-1)}})$ is the zero function, i.e., $$\label{eq:alg-eqn}A_0+A_1f+A_2f^{{\sigma}^{-1}}+\cdots+A_sf^{{\sigma}^{-(s-1)}}=0.$$
Since $D=D^{{\sigma}}$, we have $f^{{\sigma}^i}\in \cL(lD)$ for all $i\in \ZZ$. Thus, it is clear that $Q(f,f^{{\sigma}^{-1}},\dots,f^{{\sigma}^{-(s-1)}})$ is a function in $\cL(({\kappa}+l)D)$.
Let us assume that $I\subseteq\{1,2,\dots,N\}$ is the index set such that the $i$th columns of $\by$ and $\pi(f)$ agree if and only if $i\in I$. Then we have $|I|{\geqslant}t$. For every $i\in I$ and $0{\leqslant}j{\leqslant}m-s$, we have by (\[eq:interpolation-cond\]) $$\begin{aligned}
0&=&A_0(P_i^{{\sigma}^{j}})+A_1(P_i^{{\sigma}^{j}})y_{i,j+1}+A_2(P_i^{{\sigma}^{j}})y_{i,j+2}+\cdots+A_s(P_i^{{\sigma}^{j}})y_{i,j+s}\\
&=&A_0(P_i^{{\sigma}^{j}})+A_1(P_i^{{\sigma}^{j}})f(P_i^{{\sigma}^{j}})+A_2(P_i^{{\sigma}^{j}})f(P_i^{{\sigma}^{j+1}}))+\cdots+A_s(P_i^{{\sigma}^{j}})f(P_i^{{\sigma}^{j+s-1}})\\
&=&A_0(P_i^{{\sigma}^{j}})+A_1(P_i^{{\sigma}^{j}})f(P_i^{{\sigma}^{j}})+A_2(P_i^{{\sigma}^{j}})f^{{\sigma}^{-1}}(P_i^{{\sigma}^{j}})+\cdots+A_s(P_i^{{\sigma}^{j}})f^{{\sigma}^{-s+1}}(P_i^{{\sigma}^{j}})\\
&=&\left(A_0+A_1f+A_2f^{{\sigma}^{-1}}+\cdots+A_sf^{{\sigma}^{-s+1}}\right)(P_i^{{\sigma}^{j}}),\end{aligned}$$ i.e., $P_i^{{\sigma}^{j}}$ is a zero of $Q(f,f^{{\sigma}},\dots,f^{{\sigma}^{s-1}})$. Hence, $Q(f,f^{{\sigma}^{-1}},\dots,f^{{\sigma}^{-(s-1)}})$ is a function in $\cL\left(({\kappa}+l)D-\sum_{i\in I}\sum_{j=0}^{m-s}P_i^{{\sigma}^{j}}\right)$. Our desired result follows from the fact that $\deg\left(({\kappa}+l)D-\sum_{i\in I}\sum_{j=0}^{m-s}P_i^{{\sigma}^{j}}\right)<0$.
By Lemma \[lem:Q-is-good\], we know that all candidate functions $f$ in our list must satisfy the equation (\[eq:alg-eqn\]). In other words, we have to study the solution set of the equation (\[eq:alg-eqn\]). In our previous work [@GX-stoc12], to upper bound the list size, we analyzed the solutions of the equation (\[eq:alg-eqn\]) by considering local expansions at a certain point. This local expansion method only guarantees a structured list of exponential size. Through precoding by using the structure in the list, we were able to obtain a Monte Carlo construction of subcodes of these codes with polynomial time list decoding. The other method used in [@GR-FRS] for decoding the Reed-Solomon codes is to construct an irreducible polynomial $h(x)$ of degree $q-1$ such that every polynomial $f$ satisfies $f^{{\sigma}^{-1}}\equiv f^{q} \mod{h} $. Then the solution set of (\[eq:alg-eqn\]) is the same as the solution set of the equation $ A_0+A_1f+A_2f^{q}+\cdots+A_sf^{q^{s-1}} \equiv 0 \mod{h}$ since $\deg(f)<q-1=\deg(h)$. Thus, there are at most $q^{s-1}$ solutions for the equation (\[eq:alg-eqn\]). In order to generalize the latter idea used for the Reed-Solomon code to upper bound our list size of our folded algebraic geometric codes, we require some further property that $F$ must satisfy.
[**Property (P2)**]{}
- There exists a finite set $T$ of places of $F$ such that $\supp(D)\cap T=\emptyset$ and every place in $T$ has the same degree;
- There exists an integer $u> 0$ such that $f^{{\sigma}^{-1}}\equiv f^{q^u} \mod{R} $, i.e., $f^{{\sigma}^{-1}}(R)\equiv f(R)^{q^u}$ for every $R\in T$ and all $f\in F$ with $\nu_R(f){\geqslant}0$.
- $\sum_{R\in T}\deg(R)>le$.
\[lem:first-list-size\] Assume that $F$ satisfies (P1) and (P2), then the solution set of the equation [(\[eq:alg-eqn\])]{} has size at most $q^{u(s-1)|T|}$.
Consider the map $\psi:\; \cL(lD)\rightarrow \prod_{R\in T}F_R$ by sending $z$ to $\psi(z)=(z(R))_{R\in T}$. It is clear that $
\psi$ is ${\mathbb{F}}_q$-linear. Furthermore, $\psi$ is injective. Indeed, if $\psi(y)=\psi(z)$ for some $y,z\in\cL(lD)$, then $\psi(y-z)=0$, i.e., $(y-z)(R)=0$ for all $R\in T$. Hence, $y-z$ belongs to $\cL(lD-\sum_{R\in T}R)$. So, we must have $y-z=0$ since $\deg(lD-\sum_{R\in T}R)=le-\sum_{R\in T}\deg(R)<0$.
Let $W$ be the solution set of (\[eq:alg-eqn\]). Then for every $R\in T$ and $f\in W$, we have $$\begin{aligned}
0&=&A_0(R)+A_1(R)f(R)+A_2(R)f^{{\sigma}^{-1}}(R)+\cdots+A_s(R)f^{{\sigma}^{-(s-1)}}(R)\\
&=&A_0(R)+A_1(R)f(R)+A_2(R)f^{q^u}(R)+\cdots+A_s(R)f^{q^{u(s-1)}}(R)\in F_R.\end{aligned}$$ The above equation has at most $q^{u(s-1)}$ solutions in $F_R$. This implies that the set $W_R:=\{f(R):\; f\in W\}\subseteq F_R$ has size at most $q^{u(s-1)}$. Moreover, it is clear that $\psi(W)\subseteq \prod_{R\in T}W_R$. Thus, our desired result follows from $$|W|=|\psi(W)|{\leqslant}\left|\prod_{R\in T}W_R\right|=\prod_{R\in T}|W_R|{\leqslant}q^{u(s-1)|T|}.$$ This completes the proof.
- From the proof of Lemma \[lem:first-list-size\], we can see that the places in the set $T$ given in (P2)(i) need not all be of the same degree. In fact, as long as the condition in (P2)(iii), i.e., $\sum_{R\in T}\deg(R)>le$ is satisfied, we can guarantee the list size given in Lemma \[lem:first-list-size\].
- In [@Gur-cyclo], the set $T$ given in (P2)(i) has a single place with degree bigger than $le$. Then the list size would be $q^{u(s-1)|T|}=q^{u(s-1)}$. It seems that we could get a smaller list size. However, it is not possible in our case. The reason is that if we choose $|T|=1$, then the degree of the place $R$ is bigger than $le$ and thus the extension degree of $F/L$ is at least $le/u$, where $L$ is the subfield of $F$ fixed by $\langle{\sigma}\rangle$. On the other hand, we will see that the extension degree of $F/L$ is $e=|\langle{\sigma}\rangle|$ from (P3) below. This means that we must have $u> l$. This restriction on $u$ makes our list size even bigger when we choose a set $T$ of single place.
Now we look at the fraction of errors that we can correct from the above list decoding. By taking $t=1+\left\lfloor\frac{({\kappa}+l)e}{m-s+1}\right\rfloor$ and combining Lemmas \[lem:Q-is-good\] and \[lem:herm-interpolation\], we conclude the fraction of errors $\tau = 1-t/N$ satisfies $$\label{eq:herm-error-frac}
\tau \thickapprox \frac{s}{s+1}- \frac{s}{s+1}\times \frac m{m-s+1}\times\frac{k+g}{mN} , $$ where $k$ is the dimension of $\cL(lD)$ which is at least $le-g+1$.
Let $\ell$ be an even power of a prime and $q=\ell^2$. In Section \[FF\], we will show that, for any given family $\{E/{\mathbb{F}}_{\ell}\}$ with $N(E/{\mathbb{F}}_{\ell})/g(E)\rightarrow\sqrt{\ell}-1$ and $g(E)\rightarrow\infty$, where $N(E/{\mathbb{F}}_{\ell})$ denotes the number of ${\mathbb{F}}_{\ell}$-rational places of $E$, there exists a family $\{F/{\mathbb{F}}_q\}$ of function fields satisfying the following
[**Property (P3)**]{}
- $F/L$ is a cyclic Galois extension of degree $e$, where $L$ is the constant field extension $E\cdot{\mathbb{F}}_q$ and $e=(\ell^r+1)/(\ell+1)$ with $r= 2\lceil N(E/{\mathbb{F}}_{\ell})/(\sqrt{\ell}-1)\rceil +1$;
- There exists a subset $S$ of $\PP_F$ such that $|S|{\geqslant}q^r$ and $\deg(R)=3re$ for all $R\in S$. Moreover, for any $R\in S$ and $z\in F$ with $\nu_R(z){\geqslant}0$, one has $z^{{\sigma}^{-1}}\equiv z^{q^{3r}} \mod{R}$, where ${\sigma}$ is the generator of the Galois group $\Gal(F/L)$ (note that $\Gal(F/L)$ is a subgroup of ${\rm Aut}(F/{\mathbb{F}}_q)$).
- Every rational place of $E$ can be regarded as a rational place of $L$ and it splits completely in $F$. Thus, one has $N(F/{\mathbb{F}}_q){\geqslant}eN(E/{\mathbb{F}}_{\ell})$, where $N(F/{\mathbb{F}}_q$ denotes the number of ${\mathbb{F}}_q$-rational places of $F$. Furthermore, $$\liminf N(F/{\mathbb{F}}_q)/g(F){\geqslant}(\sqrt{\ell}-1)/2=(q^{1/4}-1)/2 \ .$$
\[thm:assum-list-size\] Let $\ell$ be a square prime power and let $q =\ell^2$. For every $R \in (0,1)$, there is an infinite family of folded codes given in [(\[eq:f-ag-defn\])]{} of rate at least $R$ which has relative distance at least $1-R-2/(\sqrt{\ell}-1)$.
For every pair of integers $m {\geqslant}s {\geqslant}1$, these codes can be list decoded from an error fraction $$\tau = \frac{s}{s+1} \biggl( 1 - \frac{m}{m-s+1} \Bigl( R + \frac{2}{\sqrt{\ell}-1} \Bigr) \biggr) \ ,$$ outputting a subspace over ${\mathbb{F}}_q$ with at most $O(N^{(\sqrt{\ell}-1)s})$ elements that includes all message functions whose encoding is within Hamming distance $\tau N$ from the input. (Here $N$ denotes the block length of the code.)
Let $\{F/{\mathbb{F}}_q\}$ be a family of function fields satisfying (P3) constructed in Theorem [\[2.3\]]{}. Choose a rational place $\infty$ of $E$ and regard it as a rational place of $L$. Define the divisor $D:=l\sum_{P_{\infty}|\infty, P_{\infty}\in\PP_F}P$. Then it is easy to see that $D^{{\sigma}}=D$. For every rational place $P$ of $E$, there are exactly $e$ rational places of $F$ lying over $P$ and they can be represented as $P, P^{{\sigma}},\dots, P^{{\sigma}^{e-1}}$. By taking away those rational places lying over $\infty$, we have at least $e(n-1)$ rational places of $F$, where $n=N(E/{\mathbb{F}}_{\ell})$. Thus, for an integer $m$ with $1{\leqslant}m < e$, we can label $Nm$ distinct places $P_1, P_1^{{\sigma}},\dots, P_1^{{\sigma}^{ m- 1}},\dots,P_N, P_N^{{\sigma}},\dots, P_N^{{\sigma}^{ m -1}}$ of F such that none of them lies over $\infty$, as long as $N {\leqslant}(n-1) \lfloor\frac{e}m\rfloor =(N(E/{\mathbb{F}}_{\ell})-1)\lfloor \frac{e}m\rfloor$.
It is clear that the property (P1) is satisfied and hence we can define the folded algebraic geometric code ${\FH}(N,l,q,e,m)$ as in Definition \[def:f-ag-code\]. We choose $l$ to satisfy the condition $le<Nm$. Choose a subset $T$ of $S$ with $|T|=\lceil \sqrt{\ell}-1\rceil$. Then we have $$\sum_{R\in T}\deg(R) {\geqslant}3re(\sqrt{\ell}-1){\geqslant}6N(E/{\mathbb{F}}_{\ell})e=6mN(E/{\mathbb{F}}_{\ell})\frac em>Nm>le.$$ This implies that the property (P2) is also satisfied. Hence, the code ${\FH}(N,l,q,e,m)$ is deterministically list decodable with list size at most $q^{3r(s-1)\lceil \sqrt{\ell}-1\rceil}=O(q^{sn})$. Note that the code length is $N$ which is approximately $en=m = O(n\ell^{2n/(\sqrt{\ell}-1)}/\ell m)$. Thus, the list size is $O(N ^{( \sqrt{\ell}-1)s} )$.
The claimed error fraction follows from (\[eq:herm-error-frac\]) and the fact that $g/Nm\rightarrow (\sqrt{\ell}-1)/2$.
Construction of a Family of Function Fields {#FF}
===========================================
In view of Theorem \[thm:assum-list-size\], it is essential to construct a family of function fields satisfying the Property (P3). In this section, we use class field theory and the Chebotarev Density Theorem to show the existence of such a family.
Narrow-ray class fields and Drinfeld module of rank one {#subsec:narrow-ray}
-------------------------------------------------------
Throughout this subsection, we fix a function field $F$ over ${\mathbb{F}}_q$ and a rational place $\infty$. Denote by $A$ the ring $$A:=\{x\in F:\; \nu_P(x){\geqslant}0 \ \mbox{for all $P\not=\infty$}\}.$$ Let $\Fr$ and $\Prin$ denote the fractional ideal group and the principal ideal group of $A$, respectively. Then the fractional idea class group $\Cl(A)=\Fr/\Prin$ of $A$ is actually isomorphic to the zero degree divisor class group of $F$.
Let $D=\sum_P\nu_P(D)P$ be a positive divisor of $F$ with $\infty\not\in {\rm supp(D)}$. For $x\in F^*$, $x\equiv 1 \, ({\rm mod} \ D)$ means that $x$ satisfies the following condition:
> if $P\in\supp(D)$, then $\nu_P(x-1){\geqslant}\nu_P(D)$.
Let $\Fr_{D}$ be the subgroup of $\Fr$ consisting of the fractional ideals of $A$ that are relatively prime to $D$, that is, $$\Fr_{D} = \{\Re\in \Fr: \nu_P(\Re) = 0 \ {\rm for \ all} \ P\in {\rm supp(D)}
\}.$$ Define the subgroup $\Prin_{D}$ of $\Fr_{D}$ by $$\Prin_{D}=\{xA:x\in F^*, \, x\equiv 1 \, ({\rm mod} \ D)\}.$$ The factor group $\Fr_{D}/\Prin_{D}$ is called the [ $\infty$-ray class group]{} modulo $D$. It is a finite group and denoted by $\Cl_D(A)$. If $D=0$, then we obtain the fractional ideal class group $\Cl(A)$.
Choose a local parameter $t\in F$ at $P$, i.e., $\nu_P(t)=1$. Then the $\infty$-adic completion ${\mathbb{F}}_{\infty}$ of $F$ consists of all power series of the form $\sum_{i=v}^{\infty}a_it^i$, where $v\in\ZZ$ and $a_i\in{\mathbb{F}}_q$ for all $i{\geqslant}v$. We can define a sign function $\sgn$ from ${\mathbb{F}}_{\infty}^*$ to ${\mathbb{F}}_q^*$ by sending $\sum_{i=v}^{\infty}a_it^i$ to $a_v$ if $a_v\neq 0$ (see [@NX01 pages 50-51]). Define $$\Prin_{D}^+=\{xA:\; x\in F^*, \ \sgn(x)=1, \, x\equiv 1 \, ({\rm mod} \ D)\}.$$
The factor group $$\Cl^+_D(A) = \Fr_D/\Prin^+_D$$ is called the narrow ray class group of $A$ modulo $D$ [(]{}with respect to the $\sgn$[)]{}. When $D$ is supported on a single place $Q$, i.e., $D = 1 \cdot Q$, we denote $\Cl_D(A)$ (resp. $\Cl^+_D(A)$) as simply $\Cl_Q(A)$ [(]{}resp. $\Cl^+_Q(A)$[)]{}.
We have the following result [@NX01 Proposition 2.6.4] concerning narrow ray and ideal class groups.
\[2.1.1\]
- $\Prin_D^+$ is a subgroup of $\Prin(D)$ and $\Prin_D/\Prin^+_D\simeq {\mathbb{F}}_q^*.$
- We have the isomorphisms $$\Cl_D^+(A)/{\mathbb{F}}_q^*\backsimeq\Cl_D^+(A)/( \Prin_D/\Prin^+_D)\backsimeq \Cl_D(A).$$
- We have $$\Cl^+_D(A)/(A/\mD)^* \backsimeq\Cl(A),$$ where $\mD$ is the ideal of $A$ corresponding to the divisor $D$, i.e., $\mD=\prod\wp^{n_P}$ if $D=\sum n_P P$ with $\wp$ being the prime ideal of $A$ corresponding to the place $P$.
Let $H_A$ denote the Hilbert class field of $F$ with respect to the place $\infty$, i.e, $H_A$ is the maximal abelian extension in a fixed algebraic closure of $F$ such that $\infty$ splits completely. Then we have $\Gal(H_A/F)\backsimeq \Cl(A)$.
We will use the following result from class field theory (see [@NX01 Sections 2.5-2.6]).
\[prop:CFT\] Now let $Q$ be a place of degree $d>1$ in a function field $F/{\mathbb{F}}_q$. Then there exists an abelian extension $F^Q$ of $F$ (called a narrow ray class field) with the following properties:
- $\Gal(F^Q/F)\simeq\Cl_Q^+(A)$ and the extension degree of $F^Q/F$ is $|\Cl_Q^+(A)|=(q^d-1) |\Cl(A)|=(q^d-1)h_F$, where $h_F:=|\Cl(A)|$ is the zero degree divisor class number of $F$.
- The Hilbert class field $H_A$ of $F$ is a subfield of $F^Q$ and the Galois group $\Gal(F^Q/H_A)$ is isomorphic to $(A/\Q)^* \backsimeq{\mathbb{F}}^*_{q^d}$, where $\Q$ is the ideal of $A$ corresponding to the place $Q$.
- $\infty$ and $Q$ are only ramified places in $F^Q/F$. The inertia group of $Q$ in $F^Q/F$ is $(A/{\mQ})^*$ and the inertia group of $\infty$ is ${\mathbb{F}}_q^*$. In particular, the ramification index of $Q$ is $e_Q=q^d-1$ and the ramification index of $\infty$ is $q-1$. Furthermore, $\infty$ splits into rational places in $F^Q$.
- In the Galois extension $F^Q/F$, the Frobenius automorphism of a place $P$ that is different from $\infty$ and $Q$ is $P$ itself when $P$ is viewed as an element in $\Cl_Q^+(A)$.
From the above, we can easily compute the genus of $F^Q$ by the Hurwitz genus formula, i.e., $$2g(F^Q)-2=(2g(F)-2)h_F(q^d-1)+(q-2)h_F\frac{q^d-1}{q-1}+d(q^d-2)h_F \ .$$
Next, we give a more explicit description of the narrow ray class field $F^Q$ in terms of a Drinfeld module of rank one.
Let $p$ be the characteristic of ${\mathbb{F}}_q$ and let $\pi: c\mapsto c^p$ be the Frobenius endomorphism of $H_A$. Consider the left twisted polynomial ring $H_A[\pi]$ whose elements are polynomials in $\pi$ with coefficients from $H_A$ written on the left; but multiplication in $H_A[\pi]$ is twisted by the rule $$\pi u=u^p\pi\qquad \mbox{for all} \ u\in H_A.$$ Let $\tilde{D} : H_A[\pi]\longrightarrow H_A$ be the map which assigns to each polynomial in $H_A[\pi]$ its constant term.
\[2.1.2\]
A [Drinfeld]{} $A$-[module]{} of rank $1$ over $H_A$ is a ring homomorphism $\phi:
A\longrightarrow H_A[\pi]$, $a\mapsto\phi_a$, such that:
\(i) not all elements of $H_A[\pi]$ in the image of $\phi$ are constant polynomials;
\(ii) $\tilde{D} \circ \phi$ is the identity on $A$;
\(iii) There exists a positive integer $\lambda$ such that $\deg(\phi_a)=-\lambda\nu_{\infty}(a)$ for all nonzero $a\in A$, where $\deg(\phi_a)$ is the degree of $\phi_a$ as a polynomial in $\pi$.
\[ex:2.1.2\] [Consider the rational function field $F={\mathbb{F}}_q(T)$ with $\infty$ being the pole place of $T$. Then we have $A={\mathbb{F}}_q[T]$ and $H_A=F={\mathbb{F}}_q(T)$. A Drinfeld $A$-module $\phi$ of rank $1$ over $F$ is uniquely determined by the image $\phi_T$ of $T$. By Definition \[2.1.2\] we must have $$\tilde{D}(\phi_T)=(\tilde{D}\circ\phi)(T)=T,$$ i.e., $\phi_T$ is a nonconstant polynomial in $\pi$ with the constant term $T$. Since $\deg(\phi_T)=-\lambda\nu_{\infty}(T)=\lambda$, we know that $\phi_T$ is of the form $T+f(\pi)\pi+x\pi^{\lambda}$ for an element $x\in F^*$ and $f(\pi)\in F[\pi]$ with $\deg(f(\pi)){\leqslant}\lambda-2$. Taking $x=1$ and $f(\pi)=0$ gives the so-called [ Carlitz module]{}, which yields the construction of cyclotomic function fields.]{}
[We fix a sign function sgn. We say that a Drinfeld $A$-module $\phi$ of rank $1$ over $H_A$ is sgn-[normalized]{} if ${\rm sgn}(a)$ is equal to the leading coefficient of $\phi_a$ for all $a\in A$. In particular, the leading coefficient of $\phi_a$ must belong to ${\mathbb{F}}^*_q$. ]{}
Given a Drinfeld $A$-module $\phi$ of rank $1$ over $H_A$ and a prime ideal $\Q$ of $A$, let $I_{\Q, \phi}$ be the left ideal generated in $H_A[\pi]$ by the twisted polynomials $\phi_a$, $a\in \Q$. As left ideals are principal, $I_{\Q, \phi}=H_A[\pi]\phi_{\Q}$ for a unique monic twisted polynomial $\phi_{\Q}\in H_A[\pi].$
Let $K$ be any $H_A$-algebra. Then for a polynomial $f(\pi)=\sum_{i=0}^k b_i\pi^i\in H_A[\pi]$ the action of $f(\pi)$ on $K$ is defined by $$f(\pi)(t)=\sum_{i=0}^k b_it^{p^i}\quad \mbox{for all } t\in K \ .$$ Let $\overline{H_A}$ denote a fixed algebraic closure of $H_A$ whose additive group $(\overline{H_A},+)$ is equipped with an $A$-module structure under the action of $\phi$.
[Let $\phi$ be a sgn-normalized Drinfeld $A$-module of rank $1$ over $H_A$ and $\Q$ be a nonzero ideal of $A$. The $\Q$-[torsion module]{} $\Lambda_{\phi}(\Q)$ associated with $\phi$ is defined by $$\Lambda_{\phi}(\Q)=\{t\in (\overline{H_A},+): \phi_{\Q}(t)=0\}.$$]{}
The following are a few basic facts about $\Lambda_{\phi} (\Q)$:
\(i) $\Lambda_{\phi} (\Q)$ is a finite set of cardinality $|\Lambda_{\phi} (\Q)|=
p^{{\rm deg}(\phi _{\Q})}$;
\(ii) $\Lambda_{\phi} (\Q)$ is an $A$-submodule of $(\overline{H_A},+)$ and a cyclic $A$-module isomorphic to $A/\Q $;
\(iii) $\Lambda_{\phi} (\Q)$ has $\Phi (\Q):=|(A/\Q)^*|$ generators as a cyclic $A$-module, where $(A/\Q)^*$ is the group of units of the ring $A/\Q $.
The elements of $\Lambda_{\phi}(\Q)$ are also called the $\Q$-[ torsion elements]{} in $(\overline{H_A},+)$. The following gives an explicit description of narrow ray class fields in terms of extension fields obtained by adjoining these torsion elements.
\[prop:explicit-CFT\] The extension field $H_A(\Lambda_{\phi}(\Q))$ obtained by adjoining these $\Q$-torsion elements to $H_A$ is isomorphic to the narrow ray class field $F^Q$ from Proposition [\[prop:CFT\]]{}, where $Q$ is the place corresponding to the ideal $\Q$.
In the case where $F$ is the rational function field and $\phi$ is the Carlitz module in Example \[ex:2.1.2\], the field $F^Q$ is the cyclotomic function field over $F$ with modulus $\Q$.
A family of function fields {#sec:FF family}
---------------------------
In this subsection, we assume that $\ell$ is a prime power and $q=\ell^2$. The following is the key technical component of our construction of the function fields needed for our list-decodable code construction.
\[2.1.3\] Let $E/{\mathbb{F}}_{\ell}$ be a function field with at least one rational point $\infty$ and a place $Q$ of degree $r$, where $r > 1$ is an odd integer. Then there exists a function field $F/{\mathbb{F}}_q$ such that
- $F/({\mathbb{F}}_q\cdot E)$ is a cyclic abelian extension with $[F:{\mathbb{F}}_q\cdot E]=\frac{\ell^r+1}{\ell+1}$.
- $N(F/{\mathbb{F}}_q){\geqslant}\frac{\ell^{r}+1}{\ell+1}N(E/{\mathbb{F}}_{\ell})$.
- $g(F){\leqslant}(g(E)-1)\frac{\ell^{r}+1}{\ell+1}+\frac r2\left(\frac{\ell^{r}+1}{\ell+1}-1\right)+1.$
Let us outline the idea behind the proof. First we choose a place $Q$ of $E$ of odd degree $r$ and consider the constant extension ${\mathbb{F}}_q\cdot E$. Then $Q$ remains a place of degree $r$ in ${\mathbb{F}}_q\cdot E$ since $r$ is odd (see [@NX01 Theorem 1.5.2(iii)(a)]). We take the narrow ray class field of $ {\mathbb{F}}_q \cdot E$ modulo $Q$ and then descend to a subfield $K$ which is the fixed field of a certain subgroup of the Galois group. This is done to ensure that the rational places of $E$ which can be regarded as rational places of ${\mathbb{F}}_q \cdot E$ split completely in $K/{\mathbb{F}}_q \cdot E$ (note that $K$ may not be a cyclic extension over ${\mathbb{F}}_q \cdot E$). To obtain a cyclic extension over ${\mathbb{F}}_q \cdot E$, we need to descend to a further subfield of $K$, which will be our claimed function field $F$. The reason why we use a place $Q$ of odd degree is that, in the case of odd $r$, the narrow ray class group of ${\mathbb{F}}_q \cdot E$ modulo $Q$ is a cyclic Galois extension over its Hilbert class field. In the end, we can construct our desired function field such that it is a cyclic extension over ${\mathbb{F}}_q \cdot E$.
Put $E_1:=E$ and consider the constant field extension $E_2:={\mathbb{F}}_q\cdot E_1$. Then $\infty$ remains a rational place in $E_2$ and $Q$ remains a place of degree $r$ in $E_2$ as well.
Let $A_i$ be the ring in $E_i$ defined by $$A_i:=\{x\in E_i:\; \nu_P(x){\geqslant}0 \ \mbox{ for all $P\not=\infty$}\}$$ and let $H_i$ be the Hilbert class field of $A_i$ of $E_i$ with respect to $\infty$. Consider the narrow-ray class field $E_i^Q=H_i(\Lambda_{\phi}(\Q))$ where $\Q$ is the ideal corresponding to place $Q$. Then we can identify Gal$(E_{i}^Q/E_{i})$ with $\Cl^+_{{Q}}(A_{i})$.
Now let $K$ be the subfield of the extension $E_{2}^Q/E_2$ fixed by the subgroup $G={\mathbb{F}}_q^*\cdot\Cl^+_{{\mQ}}(A_1)$ of $\Cl^+_{{Q}}(A_{2})$. We have $$|G|=\frac{|{\mathbb{F}}_q^*|\cdot|\Cl^+_{{Q}}(A_1)|}{|{\mathbb{F}}_q^*\cap \Cl^+_{{Q}}(A_1)|}=\frac{(\ell^2-1) \cdot (\ell^r-1) h_{E_1}}{\ell-1 } = (\ell+1)(\ell^{r}-1)h_{E_1}$$ and so $$\label{eq:4.13}
[K:E_{2}]=\frac{|\Cl^+_{{D}}(A_{2})|}{|G|}=\frac{(q^r-1) h_{E_2}}{|G|}
=
\frac{\ell^{r}+1}{\ell+1}\times\frac{h_{E_2}}{h_{E_1}}.$$ Let $P_{\infty}$ be a place of $K$ lying over $\infty$. Then the inertia group of $P_{\infty}$ in the extension $E_2^Q/K$ is ${\mathbb{F}}_q^*
\cap G$, and so the ramification index $e(P_{\infty}|\infty)$ of $P_{\infty}$ over $\infty$ is given by $$e(P_{\infty}|\infty)=\frac{|{\mathbb{F}}_q^*|}{|{\mathbb{F}}_q^*\cap G|}=\frac
{|{\mathbb{F}}_q^*\cdot G|}{|G|}=\frac{|{\mathbb{F}}_q^*\cdot \Cl^+_{{Q}}(A)|}{|G|}=1,$$ i.e., $\infty$ is unramified in $K/E_2$.
Let $R$ be a place of $K$ lying over $Q$. Since the inertia group of $Q$ in $E_2^Q/E_2$ is $(A_{2}/{\Q})^*$ by the theory of narrow ray class fields, the inertia group of $R$ in $E_2^Q/K$ is $(A_{2}/{\mQ})^{*}\cap G={\mathbb{F}}_q^*\cdot(A/{\Q})^*$. Thus, the ramification index $e(R|Q)$ of $R$ over $Q$ is given by $$\label{eq:4.14}
e(R|Q)=\frac{|(A_{2}/{\Q})^{*}|}{|{\mathbb{F}}_q^*\cdot(A_1/{\Q})^{*}|}=\frac{|(A_{2}/{\Q})^{*}|
\cdot|{\mathbb{F}}_q^*\cap(A_1/{\Q})^{*}|}{|{\mathbb{F}}_q^*|\cdot|(A_1/{\Q})^{*}|}=
\frac{(q^r-1)(\ell-1)}{(q-1)(\ell^r-1)}=
\frac{\ell^{r}+1}{\ell+1}.$$ Since $\infty,Q$ are the only ramified places in $E_2^Q/E_2$, and $\infty$ is unramified in $K/E_2$, we conclude that the place $Q$ is the only ramified place in $K/E_2$ with ramification index $(\ell^r+1)/(\ell+1)$.
Now, all ${\mathbb{F}}_{\ell}$-rational places of $E_1$ can be viewed as ${\mathbb{F}}_q$-rational places of $E_2$ and furthermore they split completely in $K$. This is because for a rational place $P$ of $E_1$ with $P\neq \infty,Q$, from Proposition \[prop:CFT\] and our construction, it follows that the Frobenius automorphism of $P$ is contained in the subgroup $\Gal(E_2^Q/K)$. Therefore, the Frobenius automorphism of $P$ in the extension $K/E_2$ is the identity, and therefore $P$ must split completely in $K/E_2$.
Since the decomposition group of $Q$ in $E_2^Q/E_2$ is isomorphic to the cyclic group $(A/{\Q})^*$, the decomposition group of $Q$ in $K/E_2$ is cyclic as well. The inertia group of $Q$, which is a subgroup of the decomposition group of $Q$, has order $\frac{\ell^{r}+1}{\ell+1}$ by (\[eq:4.14\]). Thus, the Galois group $\Gal(K/E_2)$ contains a cyclic subgroup of order $\frac{\ell^{r}+1}{\ell+1}$. This implies that there exists a subfield $F$ of $K/E_2$ such that $\Gal(F/E_2)$ is a cyclic group of order $\frac{\ell^{r}+1}{\ell+1}$. It is clear that all ${\mathbb{F}}_{\ell}$-rational places of $E_1$ split completely in $F$ as well. Hence $$N(F/{\mathbb{F}}_q){\geqslant}[F:E_2]N(E_1/{\mathbb{F}}_{\ell})= \frac{\ell^{r}+1}{\ell+1}N(E_1/{\mathbb{F}}_{\ell}) \ .$$ Moreover, the place $Q$ is the only ramified place in $F/E_2$ (since it is the only ramified place in $K/E_2$) and it is tamely ramified with the ramification index at most $[F:E_2]$. Hence, we can apply the Hurwitz genus formula to the extension $F/E_2$ and get $$2g(F)-2 {\leqslant}(2g(E_2)-2)[F:E_2]+r([F:E_2]-1).$$ The desired result follows from the fact that $g(E_2)=g(E_1)$.
The following theorem provides the family of function fields that we required to construct our folded algebraic geometric codes in Theorem \[thm:assum-list-size\].
\[2.1\] Let $\ell$ be a prime power and let $q=\ell^2$. Assume that there is a family $\{E/{\mathbb{F}}_{\ell}\}$ of function fields such that $g(E)\rightarrow\infty$ and $N(E/{\mathbb{F}}_{\ell})/g(E)\rightarrow A$ for a positive real $A$. Then for any odd integer $r$ with $r>\log(2+7g(E))/\log(\ell)$, there exists a function field $F/{\mathbb{F}}_{q}$ such that $F$ is a finite extension of ${\mathbb{F}}_q\cdot E$ of degree $e:=(\ell^r+1)/(\ell+1)$ and
- $g(F)\rightarrow\infty$ and $g(F){\leqslant}(g(E)-1)e+r(e-1)/2+1$.
- $N(F/{\mathbb{F}}_q){\geqslant}eN(E/{\mathbb{F}}_{\ell})$.
- $F/({\mathbb{F}}_q\cdot E)$ is a cyclic Galois extension of degree $e$.
In particular, we have $\liminf_{g(F)\rightarrow\infty}N(F/{\mathbb{F}}_q)/g(F){\geqslant}A/(1+c)$ if $r/g(E)\rightarrow 2c$ for a constant $c{\geqslant}0$.
The result follows directly from Lemma \[2.1.3\] and the fact that there exists a place of degree $r$ in $E$ as long as $r>\log(2+7g(E))/\log(\ell)$ (see [@stich-book Corollary 5.2.10]). This completes the proof.
Chebotarev Density Theorem
--------------------------
Given Theorem \[2.1\], to show that the family of function fields with property (P3) exists, it remains to find a large set $S$ of places of $\PP_F$ satisfying (P3)(ii). In order to accomplish this task, we need the explicit form of the Chebotarev Density Theorem.
Let $F/L$ be a Galois extension of degree $e$ of function fields over ${\mathbb{F}}_q$. Assume that ${\mathbb{F}}_q$ is the full constant field of both $F$ and $L$. Let $t$ be a separating transcendence element over ${\mathbb{F}}_q$. Let $d=[L:{\mathbb{F}}_q(t)]$.
For a place $Q$ of $F$ lying over $P$ of $L$, let $\left[\frac{F/L}{Q}\right]$ be the Frobenius of $Q$. Then for any ${\sigma}\in \Gal(F/L)$, the Frobenius of ${\sigma}(Q)$ is ${\sigma}\left[\frac{F/L}{Q}\right]{\sigma}^{-1}$. Thus, the conjugacy class $\left\{{\sigma}\left[\frac{F/L}{Q}\right]{\sigma}^{-1}:\; {\sigma}\in \Gal(F/L)\right\}$ is determined by $P$. We denote this conjugacy by $\left[\frac{F/L}{P}\right]$.
Fix a conjugacy class $C$ of $\Gal(F/L)$, let $M_h(C)$ denote the number of places $P$ of degree $h$ in $L$ that are unramified in both $F/L$ and $L/{\mathbb{F}}_q(t)$ such that $\left[\frac{F/L}{P}\right]=C$. Then we have the following result [@FJ08 Proposition 6.4.8] and [@MS94].
\[2.2\] One has $$\left|M_h(C)-\frac{|C|}{e h}q^h\right|{\leqslant}\frac{2|C|}{e h}(e+g_F)q^{h/2}+e(2g_L+1)q^{h/4}+g_F+de,$$ where $g_F$ and $g_L$ denote the genera of $F$ and $L$, respectively.
Finally, we are able to show existence of a family of function fields with (P3).
\[2.3\] There exists a family of function fields with [(P3)]{}. More precisely speaking, we have the following result.
Let $\ell$ be a square prime power and let $\{E/{\mathbb{F}}_{\ell}\}$ be the Garcia-Stichtenoth tower given in [[@GS95]]{}. Let $\{F/{\mathbb{F}}_q\}$ be the family of function fields constructed in Theorem [\[2.1\]]{}. Put $L:={\mathbb{F}}_q\cdot E$, $n:=N(E/{\mathbb{F}}_{\ell})$ and denote by ${\sigma}$ a generator of $\Gal(F/L)$. Let $r= 2\lceil n/(\sqrt{\ell}-1)\rceil+1$ and $h=3r$. Then there exists a set $S$ of places of $L$ such that
- $|S|{\geqslant}q^r$ and $\deg(R)=h$ for all places $R$ in $S$.
- For every place $R$ in $S$, there is a unique place $P$ of $F$ of degree $eh$ lying over $R$ and $\left[\frac{F/L}{P}\right]={\sigma}^{-1}$.
- $\liminf N(F/{\mathbb{F}}_q)/g(F){\geqslant}(\sqrt{\ell}-1)/2=(q^{1/4}-1)/2$.
Let $\{E/{\mathbb{F}}_{\ell}\}$ be the well-known Garcia-Stichtenoth tower [@GS95]. Then one has $N(E/{\mathbb{F}}_{\ell})/g(E)\rightarrow\sqrt{\ell}-1$ with $g(E)\rightarrow\infty$ and $[E:{\mathbb{F}}_{\ell}(t)]{\leqslant}g(E)$ for a separating transcendence element over ${\mathbb{F}}_{\ell}$. Thus, $d=[L:{\mathbb{F}}_q(t)]{\leqslant}g(E)=g(L)$. By our choice of parameters $h, r$, we find that $$\frac{1}{e h}q^h-\left(\frac{2}{e h}(e+g_F)q^{h/2}+e(2g_L+1)q^{h/4}+g_F+de\right){\geqslant}q^r.$$ By Theorem \[2.2\], there exists a set $S$ of places of $L$ with $|S|{\geqslant}q^r$ such that $\deg(R)=h$ and $\left[\frac{F/L}{R}\right]={\sigma}^{-1}$ for every place $R$ in $S$. Let $P$ be a place of $F$ lying over $R$. The Frobenius ${\sigma}^{-1}$ of $P$ belongs to $\Gal(F/Z)$, where $Z$ is the decomposition field of $P$ in $F/L$. Since the order of ${\sigma}^{-1}$ is $e=[F:L]$, we must have $Z=L$ and hence the relative degree $f(P|R)$ is $e$. So $P$ is the only place of $F$ lying over $R$.
Since $r/g(E)\rightarrow 2$, we have $\liminf N(F/{\mathbb{F}}_q)/g(F){\geqslant}(\sqrt{\ell}-1)/2$ by Theorem \[2.1\].
Encoding and Decoding {#sec:ED}
=====================
We have not considered encoding and decoding of the folded algebraic geometric codes constructed in Section \[sec:ALD\]. This section is devoted to the computational aspects of encoding and decoding of our folded codes.
Encoding
--------
Let us consider the folded algebraic geometric code given in the proof of Theorem \[thm:assum-list-size\], where the divisor $D$ is $l\sum_{P_{\infty}|\infty, P_{\infty}\in\PP_F}P$ and the Riemann-Roch space is $\cL(lD)$. To encode, we assume that $le>2g-1$ and there is an algorithm to find a basis $\{z_1,z_2,\dots,z_k\}$ of $\cL(lD)$ with $k= le-g+1$.
Furthermore, we assume that, for every point $P_i^{\sigma^j}$ and each function $f$ with $\nu_{P_i^{\sigma^j}}(f){\geqslant}0$, there is an efficient algorithm to evaluate $f$ at $P_i^{\sigma^j}$, i.e., find $f(P_i^{\sigma^j})$. For a function $f$ and a rational place $P$ with $\nu_P(f){\geqslant}0$, the algorithm of evaluating $f$ at $P$ consists of
- Finding a local parameter $t$ at $P$ (recall that a function $t$ is called a local parameter at $P$ if $\nu_P(t)=1$).
- Finding the unique element ${\alpha}\in{\mathbb{F}}_q$ such that $\nu_P\left(\frac{f-{\alpha}}t\right){\geqslant}0$ (note that this unique element ${\alpha}$ is equal to $f(P)$).
Decoding
--------
As we have seen, encoding is easy as long as we have an efficient algorithm to compute a basis of the Riemann-Roch space and evaluation at rational places. However, we need some further work for decoding.
The idea of decoding is to solve the equation (\[eq:alg-eqn\]) through local expansions at a point. Let us briefly introduce local expansions first. The reader may refer to [@NX01 pages 5-6] for the detailed result on local expansions. Let $F/{\mathbb{F}}_q$ be a function field and let $P$ be a rational place. For a nonzero function $f\in F$ with $\nu_P(f){\geqslant}v$, we have $\nu_P\left(\frac f{t^v}\right){\geqslant}0.$ Put $a_v=\left(\frac f{t^v}\right)(P),$ i.e., $a_v$ is the value of the function $f/t^v$ at $P$. Note that the function $f/t^v-a_v$ satisfies $\nu_P\left(\frac f{t^v}-a_v\right){\geqslant}1,$ hence we know that $ \nu_P\left(\frac {f-a_vt^v}{t^{v+1}}\right){\geqslant}0.$ Put $a_{v+1}=\left(\frac{f-a_vt^v}{t^{v+1}}\right)(P).$ Then $\nu_P(f-a_vt^v-a_{v+1}t^{v+1}){\geqslant}v+2$.
Assume that we have obtained a sequence $\{a_r\}_{r=v}^m$ ($m>v$) of elements of ${\mathbb{F}}_q$ such that $\nu_P(f-\sum_{r=v}^ka_rt^r){\geqslant}k+1$ for all $v{\leqslant}k{\leqslant}m$. Put $a_{m+1}=\left(\frac{f-\sum_{r=v}^ma_rt^r}{t^{m+1}}\right)(P).$ Then $\nu_P(f-\sum_{r=v}^{m+1}a_rt^r){\geqslant}m+2$. In this way we continue our construction of the $a_r$. Then we obtain an infinite sequence $\{a_r\}_{r=v}^{\infty}$ of elements of ${\mathbb{F}}_q$ such that $
\nu_P(f-\sum_{r=v}^ma_rt^r){\geqslant}m+1
$ for all $m{\geqslant}v$. We summarize the above construction in the formal expansion $$\label{eqn_1.1}
f=\sum_{r=v}^{\infty}a_rt^r,$$ which is called the [ local expansion]{} of $f$ at $P$.
It is clear that local expansions of a function depend on choice of the local parameters $t$. Note that if a power series $\sum_{i=v}^{\infty}a_it^i$ satisfies $\nu_P(f-\sum_{i=v}^ma_it^i){\geqslant}m+1$ for all $m{\geqslant}v$, then it is a local expansion of $f$. The above procedure shows that finding a local expansion at a rational place is very efficient as long as the computation of evaluations of functions at this place is easy.
The following fact plays an important role in our decoding.
\[lem:conj-expansion\] Let $F/{\mathbb{F}}_q$ be a function field and let ${\sigma}\in {\rm Aut}(F/{\mathbb{F}}_q)$ be an automorphism. Let $P, P^{{\sigma}^{-1}}$ be two distinct rational places. Assume that $t$ is a common local parameter of $P$ and $P^{{\sigma}}$, i.e., $\nu_P(t)=\nu_{P^{{\sigma}}}(t)=1$ such that $t^{{\sigma}}=t$. Suppose that $f\in F$ has a local expansion $\sum_{i=0}^{\infty}a_it^i$ at $P^{{\sigma}}$ for some $a_i\in {\mathbb{F}}_q$, then the local expansion of $f^{{\sigma}^{-1}}$ at $P$ is $\sum_{i=0}^{\infty}a_it^i$.
By the definition of local expansion, we have $\nu_{P^{{\sigma}}}\left(f-\sum_{i=0}^{m}a_it^i\right){\geqslant}m+1$ for all $m{\geqslant}0$. This gives $\nu_{(P^{{\sigma}})^{{\sigma}^{-1}}}\left((f-\sum_{i=0}^{m}a_it^i)^{{\sigma}^{-1}}\right)=\nu_{P}\left(f^{{\sigma}^{-1}}-\sum_{i=0}^{m}a_it^i\right){\geqslant}m+1$ for all $m{\geqslant}0$. The desired result follows.
Now let $F$ be the function field constructed in Corollary \[2.3\]. Then it satisfies the Property (P3). Assume that $U \neq\infty$ is a rational place of $E$ and $t\in E$ is a local parameter at $U$. Then $U$ can be viewed as an ${\mathbb{F}}_q$-rational point of $L={\mathbb{F}}_q\cdot E$. Moreover, $U$ splits completely in $F/L$. We may assume that all rational places of $F$ lying over $U$ are $P,P^{{\sigma}},\dots,P^{{\sigma}^{e-1}}$, where ${\sigma}$ is a generator of ${\rm Gal}(F/L)$. It is clear that $t$ is a common local parameter of $P,P^{{\sigma}},\dots,P^{{\sigma}^{e-1}}$. Furthermore, we have $t^{{\sigma}}=t$ since $t\in E\subset L$.
To solve for the functions $f$ that satisfy the algebraic equation , let us assume that $f=\sum_{i=1}^kf_iz_i$ for some $f_i\in{\mathbb{F}}_q$, where $k=le-g+1$ is the dimension of $\cL(lD)$. Solving for $f$ in (\[eq:alg-eqn\]) is equivalent to finding $\{f_i\}_{i=1}^k$. Assume that the local expansion of $z_i$ at $P^{{\sigma}^j}$ is given by $\sum_{h=0}^{\infty}{\alpha}_{ijh}t^h$. Then by Lemma \[lem:conj-expansion\], $z_i^{{\sigma}^{-j}}$ have the local expansion $\sum_{h=0}^{\infty}{\alpha}_{ijh}t^h$ at $P$. Thus, $f^{{\sigma}^{-j}}$ has the local expansion $\sum_{i=1}^k\sum_{h=0}^{\infty}{\alpha}_{ijh}t^h$ at $P$. Furthermore assume that $A_i$ have local expansions $\sum_{j=0}^{\infty}a_{ij}t^j$ at $P$ for $0{\leqslant}i{\leqslant}s$. Substitute these local expansions in Equation , we obtain an equation $$\label{eq:coeff-power}
c_0(f_1,f_2,\dots,f_k)+c_1(f_1,f_2,\dots,f_k)t+\dots+c_i(f_1,f_2,\dots,f_k)t^i+\dots=0,$$ where $c_i(f_1,f_2,\dots,f_k)$ is a linear combination of $f_1,f_2,\dots,f_k$ for all $i{\geqslant}0$. Thus, each of the coefficients of the above power series (\[eq:coeff-power\]) must be zero. This produces infinitely many linear equations $c_i(f_1,f_2,\dots,f_k)=0$ for $i{\geqslant}0$ in variables $f_1,f_2,\dots,f_k$. This system of infinitely many linear equations is equivalent to the system $$\label{eq:coeff-sol}
c_i(f_1,f_2,\dots,f_k)=0 \qquad \mbox{for $i=0,1,\dots,le$}$$ due to the fact that $A_0+A_1f+\cdots+A_sf^{{\sigma}^{-(s-1)}}\in\cL(lD)$ and the following simple claim.
\[lem:expansion-equal\] If $x$ is an element in $\cL(lD)$ and has a local expansion $\sum_{i=le+1}^{\infty}{\lambda}_it^i$ for some ${\lambda}_i\in{\mathbb{F}}_q$, then $x$ is identical to $0$ if ${\lambda}_i=0$ for all $i{\leqslant}le$.
By the local expansion of $x$, we know that $x$ belongs to $\cL(lD-(le+1)P)$. The desired result follows from the fact that $\deg(lD-(le+1)P)=-1<0$.
The equation system (\[eq:coeff-sol\]) has $le+1$ equations and contains $k=le-g+1$ variables. Theorem \[thm:assum-list-size\] guarantees that this system has at most $O(N^{(\sqrt{\ell}-1)s})$ solutions.
Given the discussion of encoding and decoding, we rewrite Theorem \[thm:assum-list-size\] as the main result of this paper.
\[Main\]\[thm:main\] For any small ${{\varepsilon}}>0$ and a real $0<R<1$, one can construct a folded algebraic geometric code over alphabet size $(1/{{\varepsilon}})^{O(1/{{\varepsilon}}^2)}$ with rate $R$ and decoding radius $\tau=1-R-{{\varepsilon}}$ such that the length of the code tends to $\infty$ and is independent of ${{\varepsilon}}$. Moreover, the code is deterministically list decodable with a list size $O(N ^{1/{\varepsilon}^2} )$.
Given a polynomial amount of pre-processed information about the code, the algorithm essentially consists of solving two linear systems over ${\mathbb{F}}_q$, and thus runs in deterministic polynomial time.
In Theorem \[thm:assum-list-size\], choose $s\approx 1/{{\varepsilon}}$ and $m\approx 1/{{\varepsilon}}^2$ and $q\approx 1/{{\varepsilon}}^4$, the error fraction $\tau $ given in Theorem \[thm:assum-list-size\] is $1-R-{{\varepsilon}}$. The alphabet size of the folded code is $q^m$, which is $(1/{{\varepsilon}})^{O(1/{{\varepsilon}}^2)}$ and the list size is $O(N^{(\sqrt{\ell}-1)s})=O(N ^{1/{\varepsilon}^2} )$.
Computing A Basis of Riemann-Roch Space
---------------------------------------
Both encoding and decoding described earlier depend on an algorithm to find a basis of the Riemann-Roch space $\cL(lD)$. We divide this job into two steps. The first step is to find an explicit equation defining our function field $F$ constructed in Section \[FF\] through class field method. The second step is to find a basis of our Riemann-Roch space based on the equation form Step 1.
In [@Hess02], a polynomial algorithm of finding a basis of a Riemann-Roch space is given based on an explicit equations of the associated function field. If $F$ is of the form ${\mathbb{F}}_q(x,y)$ with a defining equation $$\label{eq:defining equation}
y^h+a_1(x)y^{h-1}+\cdots+a_{h-1}(x)y+a_h(x)=0$$ with $a_i(x)\in{\mathbb{F}}_q[x]$, then [@Hess02] describes an algorithm with polynomial time in $h$, the divisor degree $le$ and $\Delta$, where $\Delta$ is the largest degree of $a_1(x),a_2(x),\dots,a_h(x)$ in (\[eq:defining equation\]). Thus, if we can find an equation defining the field $F$ with $\Delta$ being a polynomial in the code length $N$, then [@Hess02] provides an polynomial algorithm in the code length $N$ to determine a basis of $\cL(lD)$.
Thus, to get a polynomial time encoding and decoding for our folded algebraic code, it is sufficient to obtain polynomial time algorithms for
- finding a defining equation (\[eq:defining equation\]) of $F$ such that $\Delta$ is a polynomial in code length $N$;
- computing evaluations of functions at rational places.
Part (ii) is usually easier. The key part is to find a defining equation of the underlying function field. To see this, we start with the function field $E$ defined by the Garcia-Stichtenoth tower. Then one has $[E:{\mathbb{F}}_{\ell}(x)]{\leqslant}N(E/{\mathbb{F}}_{\ell})$. Moreover, $E/{\mathbb{F}}_{\ell}(x)$ is a separable extension, thus there exists ${\beta}\in E$ such that $E={\mathbb{F}}_{\ell}(x,{\beta})$. Consequently, we have $L={\mathbb{F}}_q\cdot E={\mathbb{F}}_{q}(x,{\beta})$.
The paper [@DF12] describes a method to find an element ${\alpha}$ of $F$ such that $F=L({\alpha})$. Thus, $F={\mathbb{F}}_q(x,{\alpha},{\beta})$. Now the problem is how to find an element $y\in F$ such that $F={\mathbb{F}}_q(x,{\alpha},{\beta})={\mathbb{F}}_q(x,y)$ with the defining equation given in (\[eq:defining equation\]) and the maximum degree $\Delta$ is a polynomial in $N$.
We summarize what we discussed above into an open problem.
Find a polynomial time algorithm to construct an explicit equation [(\[eq:defining equation\])]{} of the function field $F$ given in Theorem [\[2.3\]]{} and compute a basis of the Riemann-Roch space efficiently.
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|
---
author:
- 'Søren Dahlgaard[^1]'
- 'Mathias Bæk Tejs Knudsen [^2]'
- Eva Rotenberg
- Mikkel Thorup
bibliography:
- 'general.bib'
title: 'Hashing for statistics over $k$-partitions'
---
[^1]: Research partly supported by Mikkel Thorup’s Advanced Grant DFF-0602-02499B from the Danish Council for Independent Research under the Sapere Aude research career programme.
[^2]: Research partly supported by the FNU project AlgoDisc - Discrete Mathematics, Algorithms, and Data Structures.
|
---
abstract: 'Name matching is a key component of systems for entity resolution or record linkage. Alternative spellings of the same names are a common occurrence in many applications. We use the largest collection of genealogy person records in the world together with user search query logs to build name matching models. The procedure for building a crowd-sourced training set is outlined together with the presentation of our method. We cast the problem of learning alternative spellings as a machine translation problem at the character level. We use information retrieval evaluation methodology to show that this method substantially outperforms on our data a number of standard well known phonetic and string similarity methods in terms of precision and recall. Additionally, we rigorously compare the performance of standard methods when compared with each other. Our result can lead to a significant practical impact in entity resolution applications.'
author:
- Jeffrey Sukharev
- Leonid Zhukov
- Alexandrin Popescul
bibliography:
- 'paper.bib'
title: ' [**Learning Alternative Name Spellings**]{}\'
---
Introduction
============
A person’s name, especially the family name, is the key field used in identifying person’s records in databases. Software tools and applications designed for entity resolution of a person’s records usually rely on the person’s name as a primary identification field. In particular, genealogy services provide user access to a person’s record databases and facilitate search for a user’s ancestors/relatives and other person/records of interest. A persons’s records indicate some type of a life event including birth, death, marriage or relocation. Typically, records are indexed by some unique identifier and can also be searched by a combination of last/first names, geographical locations and event dates. Searching a record database is complicated by the user not knowing the exact spelling of the name in the record they are searching for. This task becomes even harder since databases often contain alternate spellings referring to the same person. This transpires due to many factors including optical character recognition errors when scanning the records, errors/misspelling of names in the records themselves, and name transliterations. For instance a common last name “Shepard” has been also commonly spelled as “Shepherd”, “Sheppard”, “Shephard”, “Shepperd”, “Sheperd”. Clearly, having methods that would provide users and the search engines with a list of highly credible misspelling of the query name would significantly improve the search results.
Knowing how to misspell names to find records of interest has always been a part of a professional genealogist’s domain expertise. In this paper we try to bridge this gap and bring this power to the average user by employing data-driven methods that rely on our unique data set[^1]. Through Ancestry.com we have access to the world’s largest genealogy data repository. The main function of a genealogy service is to facilitate discovery of relevant person’s records and the construction of family trees. Tree construction involves user attaching relevant scanned/digitized records, found in Ancestry.com databases, to user-generated tree node. Having records attached to individual tree nodes affords us an opportunity to collect misspellings of names. By leveraging user-provided links between individual user family tree nodes and scanned attached records, we generated a “labeled” dataset of name pairs where the left side of each pair comes from user supplied person names and the right part of the pair comes from the attached record name field. All user-identifying information except for last name pairs is discarded from the final dataset. We filter and pre-process this list of name pairs and use it to train a model using standard machine translation methods. We will go into more details on data pre-processing in Section 5. Additionally, we generated another dataset from company search logs. Often, users modify a previous search query in hope of getting better results. These user-driven modifications are called query reformulations. By identifying logged-in user sessions and extracting names from user queries in sequential order from the same session in a specified time interval and using our assumption that users frequently search for variations of the same name we have been able to accumulate a large number of name pairs that could also be used as a training/testing data for our models.
As a result of our experiments we produce ranked candidate variant spellings for each query name. In addition to providing the translation model we also propose a methodology, adapted from the information retrieval community, for evaluating of the final candidate list and for comparing it with other methods.
In the results section we will show that our methods perform significantly better than other state-of-the-art methods in terms of precision and recall in identifying a quality lists of alternative name spellings.
The remainder of the paper is organized as following. In section 2 we discuss the numerous previous works in related fields. A detailed description of our training data is given in Section 3. In Section 4 we outline the machine translation method used in training our model. We then discuss our results and present comparisons with other methods in Section 5 and conclude in Section 6.
Previous work
==============
The classic reference in the field of record linkage is a paper by Fellegi and Sunter [@fellegi1969] published in 1969. In their work the authors have carefully presented the theory of record matching. They defined the terms of positive disposition (link and non-link) and negative disposition (possible link) and showed that the optimal record matching linkage rule would minimize the possibility of failing to make a positive disposition for fixed levels of errors. Since this seminal work there has been proliferation of work in this area. In the interest of brevity we direct the reader to the outstanding 2006 survey paper by Winkler [@Winkler06] and to the comprehensive work by Christen [@data:matching] published just recently.
With the explosive growth of data coming from web applications it is becoming imperative to discover the best methods for record matching in terms of accuracy and speed. Historically, methods focusing on name matching could be separated into two classes: sequential character methods and bag-of-words methods [@MoreauYC08].
Sequential Character methods
----------------------------
Phonetic similarity methods are an important category of sequential character methods. The key concept of phonetic methods is to map n-grams of characters into phonetic equivalents. The output of using these methods on string pairs is a binary decision and not a degree of similarity. The best-known method from this class is Soundex [@Russell18]. Over the years a numerous improvements of this approach have been made. In particular some of them had to do with accommodating non-English names. Popular methods include Soundex [@Russell18], Double Metaphon [@philips90], and NYSIIS [@Taft70]. While these methods proved to be useful in improving performance in data matching applications they do not solve the problem of relevance ranking of alternative spellings, which is of great importance for search engines when considering using alternative name spellings for query expansion.
Another important category of sequential character methods often used in conjunction with phonetic methods is the class of static string similarity measures. Similarity method based on edit distance (the Levenshtein distance, as it is also known [@Levenshtein66]) is the most well-known method of this type. The edit distance between strings $s$ and $t$ is the cost of the optimal shortest sequence of edit operations (substitute, add, delete) that converts $s$ to $t$. For instance, the mapping of $s$ = ”Johnson” to $t$ = ”Johnston” results in one addition of letter “t” and hence, results in a distance of one. Other common similarity measures include the Jaro [@jaro89] method which takes into account the number and order of common characters between two strings and the Jaro-Winkler [@winkler90] method which extends Jaro by accounting for the common prefixes in both strings [@bilenko03], [@data:matching]. The static similarity measures described above, while useful in measuring similarity and ranking alternative spellings, are not capable of generating alternative spellings. This capability is typically absent from all methods that do not take a dataset’s statistical information into account.
In 2013 Bradford [@Bradford13] published a paper dealing with alternative name spelling generation. He used latent semantic indexing that uses Singular Value Decomposition (SVD) method to identify patterns in the relationships between the terms in unstructured collection of texts.
Because of the difficulty associated with obtaining the experimental data many researchers build their own synthetic datasets by utilizing specialized tools for mining the web and extracting words that appear to be last names. The resulting names are used in forming artificial pairs using one or more similarity measures (typically based on edit distance). Another popular alternative is to hire human annotators who create last name pairs based on their knowledge of name misspelling. Both of these methods may introduce bias.
Our data is being produced by millions of users who act as human annotators and who should be experts in their own genealogy and are motivated to build quality content. Due to the nature of our dataset we can extract best pairs using frequency statistics. We will go into more detail about our filtering process later in this paper. Having frequency information allows us to assemble realistic distribution of name pairs and helps in training more accurate models of alternative name spellings.
Bag-of-words methods
--------------------
Bag-of-words methods typically represent strings as a set of words or word n-grams. There were numerous studies published on the topic of applying bags of words to record linkage over the last decade [@MoreauYC08]. Cosine similarity of term frequency inverse document frequency (TFIDF) weighted vectors is one of the most popular methods of this type. Typical vectors consist of individual words or n-grams. The main shortcoming of cosine similarity TFIDF is that this method requires exact matches between fields. To alleviate this issue cosine similarity SoftTFIDF was introduced by Cohen et. al. [@Cohen03acomparison]. In addition to counting identical fields occurring in both vectors SoftTFIDF compares and keeps track of “similar” fields in both vectors. Bilenko et. al. [@bilenko03] showed how machine learning methods could be successfully employed for learning the combined field similarity. They trained an SVM classifier using feature vectors, and then applied the learned classifier’s confidence in the match as a class score. In this paper we do not consider these approaches because we primarily work with single word last names and bag-of-words methods are more suited for finding similarity between multi-field records.
Spelling correction and Machine Translation literature
------------------------------------------------------
In the 1990 Kernighan et. al. [@kernighan1990spelling] in their short paper proposed a method for spelling corrections based on noisy channels. The same formulation would latter be used in machine translation field. The basic idea was to find best possible correction by optimizing the product of language model ( a prior probability of letters/phrases/words in a given language) and correction model (likelihood of one word being spelled as another). For comprehensive survey of spelling correction methods the reader should look at the excellent chapter on this topic at Jurafsky and Martin 2008 Speech and Language Processing book. [@jurafsky2008speech]
In the last several decades machine translation methods have gained significant traction and recently found their way into the problem of name matching. In 2007 Bhagat et. al. [@BhagatH07] implemented a transducer based method for finding alternative name spellings by employing a graphemes-to-phonemes framework. Their method involved running EM (expectation maximization) algorithm, first presented by Dempster [@dempster1977], to align text from the CMU dictionary with their phoneme sequence equivalents. Next, they built a character language model of phoneme trigrams using the same CMU dictionary phonemes. Their training set was mined from the web. Using both-ways translation models and language models, the authors were able to generate alternative phoneme sequences (pronunciations), given a character string name, and then each of these sequences was converted into an alternative character sequence [@Pfeifer96].
In 1996 Ristad and Yianilos [@Ristad96] presented an interesting solution where they learned the cost of edit distance operations, which are normally all set to one in static edit distance algorithms. The authors used expectation maximization algorithm for training. Their model resulted in the form a transducer. Bilenko et. al. [@BilenkoKDD03] improved on Ristad and Yianilos’s learned edit distance model by including affine gaps. They also presented a learned string similarity measure based on unordered bags of words, using SVM for training. McCallum et. al. [@McCallumBP05] in 2005 approached the same problem from the different angle. Instead of using generative models like [@Ristad96] and [@BilenkoKDD03] they have used discriminative method, conditional random fields (CRF), where they have been able to use both positive and negative string pairs for training.
Datasets
========
Ancestry.com has over the years accumulated over 11 billion records and over 40 million personal family trees [@acom12]. Most of the records in the Ancestry.com database originate from the Western European countries, United States, Canada, Australia, and New Zealand. Scanned collections of census data, and personal public and private documents uploaded by company users comprise the bulk of Ancestry.com datasets. One of the key features of the Ancestry.com web site is the facility for building personal family trees with an option for searching and attaching relevant documents from record databases to the relevant parts of family trees. For example if a family tree contains a node for a person with the name John Smith it would be often accompanied by the birth record, relocation record, marriage record and other records discovered by the owner of the family tree. Since most of the nodes in the deep family trees involve persons who are no longer living, death records can often be discovered and attached to the appropriate nodes.
This linkage between user-specified family tree nodes and the official records present us with a unique opportunity to assemble a parallel corpus of pairs of names, hand-labeled by the users themselves. In the past researchers working on name matching problem were forced to assemble their training datasets by employing text mining techniques. Very often a specific method was needed for identifying names in a given text and then edit distance measure was used to find a list of misspelling candidates. Additionally, in some studies, a small number of dedicated human labelers provided additional level of confidence. These methods would inevitably lead to bias. We believe that our user-labeled dataset contains significantly less bias than previously used training datasets.
Due to the availability of the “labeled” dataset in the Ancestry.com we have a more direct way of generating training data. From the begininning we realized that we could not employ standard supervised machine learning methods for finding alternative name spellings since that would require us to collect positive and negative training sets. While it would have been possible to mine positive sets from user-labeled data, defining the process generating realistic negative examples is ambiguous at best. This would require us finding name pairs that would not be alternative spellings of each other with a high degree of confidence. Even through it may seem doable at first glance this a very tricky proposition. First of all how would we choose each pair item? What is the distribution of negative pairs? We only have user labels for positive pairs, but not having user label for a name pair does not necessarily mean that the pair is negative. Not having any other alternatives we would have to bias our negative set to some kind of similarity measure like the Levenshtein method and this would force us to arbitrary select a threshold that would distinguish negative pairs from positive pairs. However, besides introducing bias this method would make us miss numerous negative pairs which would have high similarity values but would not constitute a positive common misspellings. Due to having this obstacle in front of us, we turned toward machine translation methods because only a parallel corpus was needed to train the translation model.
Given the way Ancestry.com users interact with the genealogy service, we isolated two separate ways of collecting parallel corpus data that would later be used for training translation models and for testing. We felt that having two completely different underlying processes for generating our datasets would strengthen our case if we arrived at similar conclussions.
The first process of assembling a parallel corpus consists of collecting all directed pairs of names drawn from anonymized (striped from all user identifying information except last names) user tree nodes and their attached anonymized records. We chose pair direction as following: last names on the left come from tree nodes and last names on the right come from the records. Since last names in records and tree nodes have different distribution taking directionality into account is important when choosing the training set of pairs. A number of filtering steps have been applied in order to de-noise the datasets and will be discussed in more detail in the later sections. The pairs are directed which implies that a pair “Johansson” - “Johanson” would be different from the reverse pair “Johanson”-“Johansson”. This would manifest in separate co-occurrence count for each pair.
The second process for building a parallel corpus involves using recorded user search queries. Since the Ancestry.com search query form asks the user for specific fields when searching for trees or records, we have been able to extract user queries containing names from a search log. By grouping users by their loginname, sorting the queries in chronological order, and fixing the time interval at a generous $30$ minutes, we have been able to extract directed pairs of names that users use in their search queries. Our build-in assumption is that frequently users do not find what they are looking for on their first attempt and if that is the case they try again. The resulting data set is also noisy and requires extensive filtering before being used as a training set. Each pair has a direction from an older name spelling to a newer reformulation. For example if a user A searches for name “Shephard” at time $t_0$ and then searches for name “Shepperd” at time $t_1$ where $t_1 - t_0 < 30$ then the resulting pair will be: “Shephard” - “Shepperd” and not the other way around.
Table \[table:names\] provides an illustration of a sample of “records” dataset grouped by Levenshtein edit distance and sorted by co-occurrence count. The distribution of values of edit distances between names in each pair and types of individual edit operations needed to transform left-hand member of a pair into a right-hand member are shown on Tables \[table:ed\] and \[table:ed\_pct\] for each dataset. We also demonstrate the breakdown of unique last names by their country of origin on Tables 4 and 5 for both datasets. Country of origin information was gathered from person tree nodes. Each person’s node contains person’s place of birth in addition to first and last names. The most common country of birth was selected as a name’s country of origin. Only the “Old World” countries were chosen in order to avoid mixing names from different regions which are present in the “New World” countries.
[ |l|l|l|l|l|l|l|l|l|l| ]{} Levenshtein edit distance & name\#1 & name\#2 & cooccurrence & count\#1 & count\#2 & Jaro-Winkler & Jaro & Jaccard\
\[0.5ex\]
1 & clark & clarke & 139024 & 1168804 & 335902 & 0.922 & 0.889 & 0.102\
& bailey & baily & 89910 & 725361 & 123012 & 0.922 & 0.889 & 0.119\
& parrish & parish & 77529 & 179308 & 138774 & 0.933 & 0.905 & 0.322\
2 & seymour & seymore & 15583 & 90071 & 24127 & 0.907 & 0.810 & 0.158\
& schumacher & schumaker & 6013 & 52769 & 12867 & 0.884 & 0.793 & 0.101\
& bohannon & bohanan & 5902 & 44770 & 16252 & 0.854 & 0.738 & 0.107\
3 & arsenault & arseneau & 1489 & 11455 & 4305 & 0.838 & 0.769 & 0.104\
& blackshear & blackshire & 1269 & 9556 & 3049 & 0.884 & 0.793 & 0.112\
& grimwade & greenwade & 781 & 1886 & 2480 & 0.764 & 0.611 & 0.218\
4 & sumarlidasson & somerledsson & 671 & 674 & 1526 & 0.752 & 0.628 & 0.439\
& riedmueller & reidmiller & 143 & 438 & 556 & 0.736 & 0.664 & 0.168\
& braunberger & bramberg & 131 & 624 & 277 & 0.802 & 0.674 & 0.170\
Methods
=======
The problem of finding best alternative name spellings given a source name can be posed as maximization of conditional probability $P(t_{name}|s_{name})$ where $t_{name}$ is a target name and $s_{name}$ is a source name. Following the traditions of statistical machine translation methods [@Brown1990] this probability can be expressed using Bayes’ rule as $$P(t_{name}|s_{name}) = \frac{P(s_{name}|t_{name}) * P(t_{name})}{P(s_{name})}$$ where $P(t_{name})$ is a ”name model” (corresponds to language model in machine translation literature) and describes frequencies of particular name/language constructs. $P(s_{name})$ is a probability of a source name. $P(s_{name} | t_{name})$ corresponds to alignment model.
”Name model” can be estimated using character n-grams language model representation by finding the probabilities using the chain rule [@shannon48]): $$P (c_1 c_2 ... c_m) = \prod_{i=1}^{m} P (c_i|c_{max(1,i-(n-1))},...,c_{max(1,i-1)})$$ where $c_i$ is an $i^{th}$ character in the sequence of characters that comprise a name of length $m$. $n$-gram model computes a probability of a character sequence where each subsequent character depends on $n-1$ previous characters in the sequence.
An “alignment model” is used in generating translational correspondences between names in our context and it can be best described by an example shown on Figure \[fig:align\]. Here the name “Thorogood” is aligned with the name “Thoroughgood”. Looking at the Figure \[fig:align\] we can clearly see that second occurrence of letter ’o’ in “Thorogood” alignes with two letters (’o’ and ’u’) in “Thoroughgood”, similar situation happens with the last letter ’g’ in “Thorogood” which gets aligned with 2 letters ’g’ in “Thoroughgood”. Other letters in “Thorogood” aligned 1-to-1 with letters in “Thoroughgood”. Letter ’h’ in “Thoroughgood” does not align with anything in “Thorogood”.
Estimating “alignment model” results in generation alignement rules such as the ones that we just presented.
We are not only interested in the best alternative spelling given by\
$\arg \max\limits_{t_{name}} P(t_{name}|s_{name})$, but also in the ranked list of best suggestions, that can be computed from the same distribution by sorting probabilities in decreasing order:
$$\operatorname*{arg\,max}^{K}_{t_{name}} P(t_{name} | s_{name}) = \operatorname*{arg\,max}^{K}_{t_{name}} P(t_{name}) * P(s_{name} | t_{name})$$
where $\arg \max\limits^{K}_{t_{name}}$ represents operator that finds top $K$ $t_{name}$s that maximize $P(t_{name}|s_{name})$. Finding $P(t_{name} | s_{name})$ accurately without using the equation above would be challenging. However, using Baye’s rule and breaking down $P(t_{name} | s_{name})$ into language model $P(t_{name})$ and alignment model $P(s_{name} | t_{name})$ allows us to get a theoretically good translation even if underlying probabilities are not that accurate [@Brown1990]. $P(s_{name})$ is fixed and does not depend on the optimization variable $t_name$ and hence, will not influence the outcome and can be discarded.
To find probability values corresponding “name model” and “alignment model” we will be using tools developed by machine translation community, replacing sentences with names and words with characters.
For training of our language and alignment models we have chosen the Moses software package which is a widely known open-source statistical machine translation software package [@Koehn2007]. Moses is a package that contains various tools needed in translation process. Typically, translation software deals with words in a sentence as primary tokens, since we compare individual last names we had to transform our input to a format recognizable by Moses while also maintaining characters as primary tokens. In our case single words become sentences and characters become words in the sentence.
When using the Moses software package we chose to use Moses’ Baseline System training pipeline. It includes several stages:
1. Preparing the dataset: tokenization, truecasing and cleaning. Tokenization involves including spaces between every character. Truecasing and cleaning deals with lowercasing each string and removing all non-alphabetic characters among other things.
2. Language model training. A language model is a set of statistics generated for an n-gram representation built with the target language. We used IRSTLM [@Federico08], a statistical language model tool for this purpose. As a result we generated 2-gram through 6-gram language models (6 was the maximum possible). This step adds ”sentence” boundary (”word” boundary here) symbols and, also as in the Baseline System, uses improved Kneser-Ney smoothing. We follow a common practice in machine translation where all examples of the target language, and not only forms present in parallel corpus translation pairs, are used to construct a language model. It is, therefore, based on a larger data set, and can lead to an improved translation quality. In our experiments with search logs and tree attachment datasets we used their respective lists of 250,000 most frequent surname forms for language model estimation.
3. Alignment model building: Moses uses the GIZA++ package for statistical character-alignment [@och03:asc] character (Word)-alignment tools typically implement one of Brown’s IBM generative models [@Brown93] that are being used for determining translation rules for source language to the target language (including fertility rules: maximum number of target characters generated from one source character and so on) We created alignment model, for each of the 2-gram through 6-gram language models created in the previous step. As in the Baseline System, the ”-alignment” option was set to ”grow-diag-final-and” and the ”-reordering” option was set to ”msd-bidirectional-fe”
4. Testing. We tested decoding on test folds in a batch mode with an option ”-n-best-list” to give top 1000 distinct translations. This value was chosen large to well represent the high recall area on respective precision-recall curves. It is possible that using different Moses configuration could give even more accurate results.
We basically followed the Baseline System with the exception of tuning the phase and replacing our source and target languages with sequences of characters and instead of sequences of words. The tuning phase consists of steps optimizing the default model weights used in the training phase. We have omitted this phase because based on our initial tests, it didn’t give immediate accuracy improvements on our datasets and it is relatively slow.
[ |l|l|l| ]{} Edit Distance & “Search” \# of pairs & “Records” \# of pairs\
\[0.5ex\]
1 & 10894 & 21819\
2 & 1312 & 2560\
3 & 155 & 258\
4 & 32 & 70\
5 & 15 & 58\
6 & 20 & 66\
7 & 30 & 68\
8-11 & 42 & 99\
[ |l|l|l| ]{} Operations & “Search” ops type % & “Records” ops type %\
\[0.8ex\]
deletes & 32.18 % & 38.18 %\
inserts & 33.91 % & 20.65 %\
replaces & 33.91 % & 41.47 %\
Results
=======
Data Preparation
----------------
Since we dealt with user-generated data we had to devise an algorithm for treating the data and generating a high confidence training set. We outlined the following procedure:
1. Initially, a ”universe” of names was defined. All names in tests and training sequences came from this “universe”. A set of names was selected by taking top 250,000 most frequent names from both datasets (“search” and “records”).
2. For each pair selected using procedure outlined in Section 3 we made sure that each name comes from our set of high-frequency names. This step resulted in $12,855,829$ pairs in the “search” dataset and $51,744,673$ pairs in the “records” dataset.
3. In order to de-noise the name pairs we selected the top 500k/250k pairs by co-occurrence for the “records” and “search” datasets respectively.
4. The remaining pairs were passed through the Jaccard index filter $J$: $$J(A,B) = |(A \cap B)| / |(A \cup B)|$$ where $A$ is a set of users linked with left name from a name pair and $B$ is set of users linked with the right names from a name pair. Users are identified by either their login session (“search” dataset) or by userid (“records” dataset). The reason that users where used in calculating Jaccard instead of just using co-occurrence counts and marginal counts has to do with preventing a few highly active users from skewing the results. This filter was used to remove name pairs that would be likely to co-occur by chance due to high frequency of each individual name involved in a pair. For instance, “Smith”-“Williams” pair would be filtered out. After filtering we were left with 25k “record” and 12.5k “search” name pairs.
5. In the final step we estimated the rate of “obvious” false positives based on manual checks and similarity measures cross checking. Looking at random samples stratified by edit distance we manually evaluated these samples to estimate the false positive percentage. We estimated that the rate of obvious false positives is 1.5% in “search” dataset and 1.4% in “records” dataset. We specifically avoided using string based similarity criteria when defining parallel corpus to prevent introducing bias. In principle, extra filters can be applied to training sets.
![Machine translation: alignment; the source name “Thoroughgood” and the target name “Thorogood”. Arrows and red circles represent phrase alignment rules learned as a result of the training stage. []{data-label="fig:align"}](align.pdf){width="7.2cm"}
Experiments and results
-----------------------
Comparing phonetic methods with similarity measures and with machine translation methods is not straightforward. Phonetic methods only allow for binary responses (match or mismatch) when applied to name pairs. Therefore, it is impossible to rank positive matches without introducing additional ranking criteria. Our machine translation (MT) method produces a score that we use in ranking. Similarity methods produce a similarity value that is also used in ranking. To get a meaningful comparison of these methods with phonetic methods we had to make use of statistics that we gathered while processing datasets. Additionally, we devised a unified methodology that could be applied to all listed method types.
In all our experiments we used 10-fold cross validation for evaluating how the results of predictions generalize to previously unseen data.
We randomly divided each dataset (“search” and “records”) into ten folds. We train on 9 folds, then test on the remaining 1. This process was repeated 10 times for each test fold. Training folds were used to train the MT models. The same test folds were used to test all methods, including MT generated models, phonetic methods and similarity measures.
Generating results involves building a consistent metric that can be plotted and compared between different methods. We adapted a standard information retrieval performance metric: precision and recall. $$Precision = \frac{TP}{TP+FP}$$
$$Recall = \frac{TP}{TP+FN}$$ where $TP$ are true positives, $FN$ are false negatives and $FP$ are false negatives. The methods with larger precision and recall are superior.
Each test fold contains a source name and one or more target names associated with each source name. Each of our methods for each source name would produce its own list of target names. Since the number of suggested target names (or alternative spellings) can be large we needed to find a suitable method for ranking target names. For all target names for position/rank $i$ in the range from $1$ to $N$ corresponding $recall_i$ and $precision_i$ are calculated. So, we had to agree on what precisely we mean by rank for phonetic methods, similarity methods and machine translation methods.
We decided to view ranking as the product of $$rank(s,t) = alignmentScore(s,t) * languageModelScore(t)$$ For the machine translation method (generated using the Moses software library) we used model-applier output scores which already contain the product of language model score and alignment score. For machine translation where character is a *word*: $$rank(s,t) = mosesScore(s, t)$$ For phonetic algorithms $languageModelScore(t)$ is the frequency of a name in the dataset ($freq$). $$rank(s,t) = hasSameCode(s, t) * freq(t)$$ where $hasSameCode(s, t) \to \{0,1\}$ and $freq(t)$ represents the frequency of name $t$ in the dataset.
For similarity measures we also used name frequency, but we had to experimentally find a suitable exponential constant $\gamma$ to avoid over-penalizing low-frequency names. $$rank(s,t) = sim(s, t) * freq(t)^\gamma$$ where $sim(s,t)\to [0,1]$ represents the floating point similarity values and $\gamma$ is the exponential constant used to control the frequency values. We used $\gamma$ value $0.001$.
After saving precomputed sorted (according to ranking) lists of alternative name spellings for each method (phonetic, similarity, MT methods) we computed Precision and Recall values for each position from $1$ to $N$ separately for each test fold.
After producing 10 precision-recall curves for each method we needed to find a suitable way to visualize confidence in our results without actually drawing 10 curves per method.
Inspired by the work of [@Macskassy04] we designed our own methodology for robust statistical comparisons of our precision-recall curves. Using our ten folds we evaluated confidence bands for each method. Assuming test examples are drawn from the same, fixed, multivariate normal distribution, the expectation is that the model’s precision-recall curves will fall within the bands with probability of at least $1-\delta$ where $\delta$ represents the significance level. We need to find the standard deviation of the sample which is the degree to which individual points within the sample differ from the sample mean.
The density contours of multivariate normal distribution of precision and recall pairs are ellipses centered at the mean of the sample. The eigenvectors of the covariance matrix $\Sigma$ are used as directions of the principal axes of the Gaussian ellipses [@Hansen05thecma]. For our collection of $2D$ precision/recall pairs $X = (X_1,X_2)$ $$\Sigma_{ij} = E[(X_i-\mu_i)(X_j-\mu_j)]$$
The average values of ten points $\mu_1$ and $\mu_2$ have given us centroid curve for each method and the center of density contours.
The standard deviation for each vector direction is found by taking the Cholesky decomposition of the covariance matrix and using the resulting matrix for generating elliptical contours of a two dimensional normal distribution. To capture the $95\%$ confidence level in $2D$ we need to multiply each $\sigma$ by the multiplier. We get the squared $\sigma$-multiplier value from the Chi Square Distribution ($\chi^{2}$) table for $2$ degrees of freedom where the Chi Square Distribution is the distribution of the sum of squared independent standard normal variables. $\sigma$ multiplier equals to $2.447$ in this case. See Figure \[fig:bands\] for a visualization and further explanation of confidence bands.
The resulting bands are formed by connecting by line segment endpoints of the longest principle axis of each ellipse with its corresponding neighbor ellipses. The resulting bands give us a visual cue regarding the variance of precision/recall (PR) curves produced for different data test folds. Also, the resulting bands have at least $95\%$ confidence level because data points that may not be captured be ellipses may still end up inside the bands between ellipses and since the ellipses are already at $95\%$ confidence level this implies that the bands will have a higher confidence level.
We ran 70 experiments on phonetic methods. Seven commonly used phonetic methods were selected for testing and these methods were applied on the same ten test folds. 90 experiments were conducted with distance metrics methods (Jaro, Levenshtein, Winkler-Jaro). We experimented with three values when choosing suitable $\gamma$ parameter for distance measurement methods ranking.
Our results indicate general consistency when using test data from both datasets (“search” and “records”). NYSIIS phonetic method, first introduced by Taft in 1970 [@Taft70] significantly outperforms other phonetic methods. Phonex method appears to be the weakest performer of the phonetic methods we have looked at. Other phonetic methods lie in the middle and their confidence bands overlap. Because of the overlapping regions we cannot definitively rank the performance of these methods.
Figure \[fig:moses\] shows how we selected the best of MT methods. Even though that for all of our data test folds MT methods produced overlapping confidence bands we can still see that the centroid curve for 5-gram MT methods slightly outperforms other n-gram methods. Therefore, we have selected it to represent MT methods when comparing with phonetic and similarity methods.
Our main results are shown on Figures \[fig:search1140\] and \[fig:records1140\]. Here we present the comparison of all alternative name generating methods on precision-recall plots. It is clear that for both datasets MT methods perform better than all other methods and that similarity methods generally outperform phonetic methods.
Implementation details
----------------------
We imported our records/tree datasets into CDH4 Cloudera Hadoop and we perform all our filtering using Hive/Python scripts and Java native implementations. The Febrl library [@Christen08febrl] implemented by Christen was used for calculating phonetic codes and string similarity values.
[ |l|l|l| ]{} Country & number of unique names\
\[0.4ex\]
England & 9341\
Germany & 5679\
France & 1233\
Ireland & 981\
Scotland & 647\
Russia & 448\
Italy & 426\
Switzerland & 377\
Norway & 376\
Netherlands & 300\
Others & 3779\
[ |l|l|l| ]{} Country & number of unique names\
\[0.4ex\]
England & 6690\
Germany & 1323\
Ireland & 900\
France & 631\
Scotland & 468\
Russia & 241\
Italy & 157\
Sweden & 109\
Poland & 90\
Switzerland & 83\
Others & 727\
Discussion and Conclusion
=========================
In this paper we presented a novel way of approaching alternative name spelling generation problem. We utilized a well-known methodology for comparing alternative name spelling methods and presented our results as precision-recall plots which clearly indicate not only that machine translation methods appear to be superior for our datasets to other methods but also show the rankings of other well known methods. We demonstrated our results using a unique dataset from Ancestry.com generated by millions of motivated users who are “experts” at labeling the dataset.
The main conclusion of this work is that machine translation methods that we have employed for finding ranked list of alternative last name spellings far-outperformed all other methods we tried. Our results, also, indicated that the NYSIIS phonetic method significantly outperformed other phonetic algorithms and the Phonex phonetic method did not perform as well on our data. Additionally, Jaro-Winkler similarity method together with the Levenshtein edit distance method performed better than the Jaro method, which was in line with our expectations. On the other hand, we were surprised by how well the NYSIIS method performed compared to other phonetic methods. Our finding regarding phonetic methods performance went against findings reported by Christen in his 2006 paper [@christen2006comparison]. However, he was relying on very different dataset and that may explain the differences in our results.
In future work we plan on training our models specifically on training sets composed of name pairs from the same country we plan on testing them against. We also plan on doing more experiments with full names including first names and initials. Additionally, we plan on trying MT methods on geographical locations such as town/village names.
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![Train/Test datasets preparation []{data-label="fig:dataprep"}](dataprep.pdf){width="9.2cm"}
[^1]: Data is available for research purposes from the authors
|
---
abstract: 'In this paper, we address the line spectral estimation problem with multiple measurement corrupted vectors. Such scenarios appear in many practical applications such as radar, optics, and seismic imaging in which the signal of interest can be modeled as the sum of a spectrally sparse and a block-sparse signal known as outlier. Our aim is to demix the two components and for that, we design a convex problem whose objective function promotes both of the structures. Using positive trigonometric polynomials (PTP) theory, we reformulate the dual problem as a semi-definite program (SDP). Our theoretical results states that for a fixed number of measurements $N$ and constant number of outliers, up to $\mathcal{O}(N)$ spectral lines can be recovered using our SDP problem as long as a minimum frequency separation condition is satisfied. Our simulation results also show that increasing the number of samples per measurement vectors, reduces the minimum required frequency separation for successful recovery.'
author:
- 'Hoomaan Hezave, Sajad Daei, Mohammad Hossein Kahaei [^1]'
bibliography:
- 'references.bib'
title: Demixing Sines and Spikes Using Multiple Measurement Vectors
---
Spectral super resolution, Demixing, Multiple measurement vector, Atomic norm, Convex optimization.
Introduction {#section1}
============
super resolution is the problem of estimating the spectrum of a signal composed of sinusoids using finite number of samples. This problem, also known as line spectral estimation, is of great importance in signal processing applications such as radar[@Heckel2018], multi-path channel estimation[@Pejoski2015], seismic imaging [@Borcea_2002] and magnetic resonance imaging[@Koochakzadeh2018].
There exist three main attitudes toward spectral super resolution problem: non-parametric methods, parametric approaches [@stoica2005spectral], and optimization based methods [@bhaskar2013atomic]. Periodogram as a non-parametric method can localize sinusoids up to a limited resolution[@harris1978use] in the noiseless case. Multiple Signal Classification (MUSIC) is a parametric method which can recover sinusoids perfectly [@schmidt1986multiple]. However, the performance of this method degrades in the presence of noise or outliers. Also, MUSIC needs the correlation matrix of the signal and lack of measurements can highly affect the performance of MUSIC. Other examples of parametric approaches are Estimation of Signal Parameters via Rotational Invariance Technique (ESPRIT)[@roy1989esprit] and Matrix Pencil method [@sarkar1995using]. Optimization based approaches minimize the continuous counterpart of ${l_1}$ norm known as Total Variation(TV) norm[@bhaskar2013atomic]. These methods are shown to be robust against Gaussian noise[@bhaskar2013atomic]. However, their performance degrades when outliers are present. Tang et.al. proposed a mathematical formulation for the spectral super resolution problem using Atomic Norm Minimization (ANM)[@tang2013compressed]. For more illustration, consider a time dispersive multipath channel. The problem is to estimate channel delays and the corresponding complex coefficients using limited number of pilots. This problem is studied using spectral super resolution and ANM[@Pejoski2015].
In most applications, an array of sensors is utilized to receive the signal. In real scenarios, the output of some sensors might be corrupted by perturbations and this makes it harder to super resolve the spectrum of the signal. Thus, the received signal can be described as a mixture of the transmitted signal and a spiky noise. This noise can be due to the interference arising from other signals, lightning discharges, and sensor failures. The problem of estimating the transmitted signal from the latter mixture is known as the demixing of sines and spikes. The demixing problem using Single Measurement Vector (SMV) is studied in [@fernandez2017demixing].
In this work we study the demixing problem using Multiple Measurement Vectors (MMVs). It is shown that using MMVs makes it possible to localize the sines with high precision. According to the fact that the measurement vectors share the same spectral characteristic of the signal of interest, it is possible to use this joint spectral sparsity and distinguish the signal of interest from the outliers. According to the applied signal model, a new method for spectral super resolution in the presence of outliers is proposed. Also, with respect to the infinite dimensionality of the TV norm minimization problem, the dual problem is investigated. Using Positive Trigonometric Polynomials(PTP) theory [@dumitrescu2007positive], a tractable Semi Definite Program (SDP) is proposed. A vector dual polynomial is formed using the dual variables of the latter SDP. Also, a sufficient condition for exact recovery of the proposed method is investigated.
The rest of the paper is as follows: In Section\[section2\] the demixing problem for the MMV case is formulated, in Section\[section3\] the TV norm minimization is applied to distinguish the signal of interest from the outliers, in Section\[section4\] the dual problem is investigated and a new SDP is proposed, in Section\[section5\] numerical results are presented, and Section\[section51\] is devoted to conclusion and future work discussions. Also, the proof of the main theorem can be found in Section\[section6\].
**Notation**. Throughout this paper, scalars are denoted by lowercase letters, vectors by lowercase boldface letters, and matrices by uppercase boldface letters. The ${i}$th element of the vector ${\boldsymbol x}$ is given by ${\boldsymbol x_i}$. ${|.|}$ denotes cardinality for sets and absolute value for scalars. ${f^{(i)}(t)}$ denotes the ${i}$th derivative of ${f(t)}$ with respect to ${t}$. Transpose, conjugate, and hermitian of a matrix or vector are given by ${(.)^T}$, ${(.)^{\ast}}$, and ${(.)^H}$ respectively.
Problem Formulation {#section2}
===================
Suppose that the signal of interest is composed of ${K}$ complex exponentials $$\begin{aligned}
\label{eqn1}
s_{jl} & =\sum_{k=1}^{K}a_{kl}e^{i2\pi jf_k}\:\:\:,j\in \mathcal{N},l\in \mathcal{L}\end{aligned}$$ where ${\mathcal{N}=\{0,\ldots,N-1\}}$, ${\mathcal{L}=\{1,\ldots,L\}}$, ${a_{kl}\in \mathbb{C}}$ is the complex amplitude corresponding to the ${k}$th frequency, ${i=\sqrt{-1}}$, ${N}$ is the length of the sinusoids, ${L}$ is the number of measurements or snapshots taken over time, and ${f_k\in\mathbb{T}}$ where ${\mathbb{T}:=\{f_1,\ldots,f_K\}\subset[0,1]}$ is the support set of the signal. In the Fourier domain, (\[eqn1\]) can be expressed as $$\begin{aligned}
\label{eqn2}
G_l(f) & =\sum_{k=1}^{K}a_{kl}\delta(f-f_k)\end{aligned}$$ where ${\delta(f-f_k)}$ is Dirac delta function located at ${f_k}$. The signal can be expressed in a matrix form ${\boldsymbol S}$ whose columns denote the measurements for one snapshot and the rows correspond to the output of each sensor for different snapshots. Note that we can write
$$\begin{aligned}
s_{jl} &= \sum_{k=1}^{K}a_{kl}e^{i2\pi jf_k}=\int_{0}^{1}e^{i2\pi jf}G_l(df)=(\mathcal{F}_NG_l)_j \nonumber\end{aligned}$$
where ${\mathcal{F}_N}$ maps the measure ${G_l}$ to its first N Fourier series coefficients. Here, we study the full measurement case. The results can be extended to the random sampling case[@yang2016exact].
As stated in Section \[section1\], outliers degrade the performance of recent optimization based spectral super resolution methods. In order to overcome this problem, the effect of the outliers should be considered in the initial model used for the received signal. Following the same approach of [@fernandez2017demixing], the outliers are added to the received signal as a matrix ${\boldsymbol Z}$ $$\begin{aligned}
\label{eqn3}
\boldsymbol Y & = \boldsymbol S + \boldsymbol Z = [(\mathcal{F}_NG_1),\ldots,(\mathcal{F}_NG_L)] + \boldsymbol Z\end{aligned}$$ where ${\boldsymbol Y_{jl}}$ and ${\boldsymbol Z_{jl}}$ are the received signal and the outliers at ${j}$th sensor and ${l}$th snapshot respectively. Note that the outliers are considered to be sparse in each snapshot, but their support may change during the overall measurement interval. In this paper, ${\boldsymbol \Omega\subset\{0,\ldots,N-1\}}$ denotes the overall support set of the outliers.
Total Variation Norm Minimization {#section3}
=================================
Without any prior assumption, the demixing problem is ill-posed. Sparse assumption on the signal structure is proved to be helpful in solving linear inverse problems. In compressed sensing theory, *Restricted-Isometry Property (RIP))* guaranteed that a random sampling operator would preserve most of the signal’s energy with high probability. However, in spectral super resolution it is possible that the non-zero spectral information of the signal lie in the null space of the sampling operator. Thus, an additional condition called *minimum separation condition* should be met[@fernandez2016super].
\[definition1\] (Minimum separation)Consider the set ${\mathbb{T}}$ as defined before. The minimum separation is defined as the minimum distance between any elements of ${\mathbb{T}}$,$${\Delta(\mathbb{T})=\underset{(f_1,f_2)\in \mathbb{T}:f_1\neq f_2}{\inf}|f_2-f_1|}$$
In compressed sensing theory, ${l_1/l_2}$ norm was used to promote group sparsity of the received signals sharing same support sets. The continuous counterpart of ${l_1/l_2}$ norm is group Total Variation (gTV) norm $$\begin{aligned}
\|\boldsymbol X\|_{\rm gTV}&:=\underset{\underset{\boldsymbol F:\mathbb{T}\rightarrow\mathbb{C}^L}{\|\boldsymbol F(t)\|_2\leq1,t\in\mathbb{T}}}{\sup}\sum_{l=1}^{L}Re\{\int_{\mathbb{T}}{\boldsymbol F_l^{H}(t)}\boldsymbol X_l(dt)\}.\nonumber\end{aligned}$$
Fernandez proved that a minimum separation of ${\frac{2.52}{N-1}}$ has to be met so that the gTV norm minimization achieves exact recovery[@fernandez2016super]. Following the same insight of [@fernandez2017demixing], we propose the following optimization problem for demixing in the MMV case $$\begin{aligned}
\label{eqn4}
\underset{\tilde{\boldsymbol G},\tilde{\boldsymbol Z}}{\min} & \|\tilde{\boldsymbol G}\|_{gTV}+\lambda\|\tilde{\boldsymbol Z}\|_{1,2}\:\:\: s.t.\:\: \boldsymbol Y = [\mathcal{F}_N\tilde{ G_1},\ldots,\mathcal{F}_N\tilde{ G_L}] + \tilde{\boldsymbol Z}\end{aligned}$$ where ${\lambda>0}$ is a regularization parameter and ${\|.\|_{1,2}}$ denotes the matrix ${l_{1/2}}$ norm. The main contribution of this paper is to show that under certain assumptions, the above problem has a unique solution.
\[theorem1\] Consider ${N}$ measurements for ${L}$ snapshot. Suppose that in each measurement instance a perturbation vector of ${\boldsymbol z_l}$ with ${s}$ uniformly distributed nonzero elements is added to the signal of interest. If the minimum separation condition of ${\Delta_{min}=\frac{2.52}{N-1}}$ is satisfied and the phases of ${\boldsymbol a_{l}}$ and the nonzero entries of ${\boldsymbol z_l}$ are i.i.d uniformly distributed in ${[0,2\pi]}$, then (\[eqn4\]) with ${\lambda=1/\sqrt{N}}$ provides us exact solution with probability ${1-\epsilon}$ for any ${\epsilon>0}$ as long as $${K<C_K(log\frac{\sqrt{L}N}{\epsilon})^{-2}N}$$ $${s<C_s(log\frac{\sqrt{L}N}{\epsilon})^{-2}N}$$ for some constants ${C_K}$, ${C_s}$, and ${N\geq2\times10^{3}}$.
\[remark1\] Using MMVs lead to an increased probability of successful recovery. To see this, consider demixing the corresponding columns of $\bm S$ and $\bm Z$ in the SMV case [@fernandez2017demixing]. With fixed N, each column of $\bm S$ and $\bm Z$ can be recovered from the corresponding column of of $\bm Y$ with probability at least $1-\epsilon$. Thus, in order to recover all the columns of $\bm S$, the probability of successful recovery would be at least $1-L\epsilon$. However, in order to solve the problem with a single optimization, as proposed in Theorem 1, the probability of successful recovery for the same conditions on $N$ and $K$, is at least $1-\sqrt{L}\epsilon$. This explicitly certifies that the proposed method outperforms $L$ individual SMVs in terms of the success probability.
The proof of theorem\[theorem1\] appears in Appendix A. In Section\[section4\] we look at the dual of (\[eqn4\]) and reformulate it as a SDP.
Dual Problem {#section4}
============
According to the infinite dimensionality of gTV norm in (\[eqn4\]), we look at its dual formulation and analyze it. The proposed demixing problem (\[eqn4\]) is closely related to the atomic norm minimization problem introduced in [@bhaskar2013atomic]. Using the fact that our signal of interest is composed of ${K}$ complex exponentials, we can present it sparsely with an atomic set containing ${N}$ dimensional sinusoids. The measurements of each snapshot or time sample form a measurement matrix as in (\[eqn3\]). As a consequence, it is crucial that we use matrix form atoms to build up our signal. Consider the following atomic set $${\mathcal{A}=\{\boldsymbol a(f,\phi) \boldsymbol b^H:f\in[0,1],\phi\in[0,2\pi],\|\boldsymbol b\|_2=1\}}$$ for any ${\boldsymbol b\in \mathbb{C}^{L\times 1}}$ and $${\boldsymbol a(f,\phi)=\frac{1}{\sqrt{N}}e^{i\phi}[1,e^{i2\pi f},\ldots,e^{i2\pi (N-1)f}]^T\in\mathbb{C}^{N}.}$$ Using the above definition of the atomic set we can define the matrix ${\boldsymbol S}$ as
$$\begin{aligned}
\label{eqn5}
\boldsymbol S & =\sqrt{N}\sum_{k=1}^{K}\boldsymbol a(f_k,\phi)\boldsymbol \psi_k^H=\sum_{k=1}^{K}c_k\boldsymbol a(f_k,\phi)\boldsymbol b_k^H\end{aligned}$$
where ${c_k=\sqrt{N}\|\boldsymbol \psi\|_2>0}$ and ${\boldsymbol b_k=c_k^{-1}\boldsymbol \psi\sqrt{N}}$ with ${\|\boldsymbol b_k\|_2=1}$. According to [@tang2013compressed; @chandrasekaran2012convex], spectral super resolution problem can be treated using atomic norm minimization. This attitude arises from the fact that in spectral super resolution problem the spectrum of the signal of interest is sparse. The atomic norm is defined as follow
$$\begin{aligned}
\label{eqn6}
\|\boldsymbol X\|_{\mathcal{A}} & :=\inf\{t>0:\boldsymbol X\in tconv(\mathcal{A})\},\end{aligned}$$
where ${conv(\mathcal{A})}$ denotes the convex hull of the atomic set ${\mathcal{A}}$.
Using the definition of the atomic norm, (\[eqn4\]) can be represented as follow
$$\begin{aligned}
\label{eqn7}
\underset{\tilde{\boldsymbol S},\tilde{\boldsymbol Z}}{\min} & \|\tilde{\boldsymbol S}\|_{\mathcal{A}}+\lambda\|\tilde{\boldsymbol Z}\|_{1,2}\:\:\: s.t.\:\: \boldsymbol Y =\tilde{\boldsymbol S} + \tilde{\boldsymbol Z}.\end{aligned}$$
In order to formulate the dual problem, we need the definition of dual atomic norm. Then,
$$\begin{aligned}
\|\boldsymbol \Gamma \|_{\mathcal{A}}^{\ast} & = \underset{\|\tilde{\boldsymbol S}\|_{\mathcal{A}}\leq1}{\sup} <\boldsymbol \Gamma,\tilde{\boldsymbol S}>_{\mathbb{F}} ,\nonumber\\
& = \underset{\underset{\|\boldsymbol b\|_2=1}{\underset{\phi\in[0,2\pi]}{f\in[0,1]}}}{\sup}<\boldsymbol \Gamma,e^{i\phi}\boldsymbol a(f,0)\boldsymbol b^H>_{\mathbb{F}} ,\nonumber\\
& = \underset{\underset{\|\boldsymbol b\|_2=1}{f\in[0,1]}}{\sup} |<\boldsymbol \Gamma,\boldsymbol a(f,0)\boldsymbol b^H>_{\mathbb{F}}|, \nonumber\\
& = \underset{f\in[0,1]}{\sup} \| \boldsymbol \Gamma^H\boldsymbol a(f)\|_2, \nonumber\end{aligned}$$
where ${<.>_{\mathbb{F}}}$ denotes Frobenius inner product. Using the above definition, the dual of (\[eqn7\]) can be written as
$$\begin{aligned}
\label{eqn9}
\underset{\boldsymbol \Gamma\in\mathbb{C}^{N\times L}}{\max}Re< {\boldsymbol Y}, \boldsymbol \Gamma >_{\mathbb{F}}\: s.t. && \underset{f\in[0,1]}{\sup} \|{\boldsymbol \Gamma}^H \boldsymbol a(f,0)\|_2 \leq1,\nonumber \\
&& \|\boldsymbol \Gamma\|_{\infty,2}\leq\lambda,\end{aligned}$$
where ${Re<.>}$ denotes the real part of the inner product and ${\|.\|_{\infty,2}}$ is the matrix infinity/2 norm defined as $${\|\boldsymbol \Gamma\|_{\infty,2}=\underset{i}{\max}\|\boldsymbol \Gamma_{i,:}\|_2.}$$ By applying the PTP theory[@dumitrescu2007positive], the maximization constraint in (\[eqn9\]) can be reformulated as a Linear Matrix Inequality (LMI). Therefore, (\[eqn9\]) can be represented as
$$\begin{aligned}
\label{eqn10}
\underset{\boldsymbol \Gamma\in\mathbb{C}^{N\times L},\boldsymbol \Lambda \in\mathbb{C}^{N\times N} }{\max}Re< {\boldsymbol Y}, \boldsymbol \Gamma >_{\mathbb{F}}\: s.t. && \left[
\begin{array}{cc}
\boldsymbol \Lambda & \boldsymbol \Gamma \\
\boldsymbol \Gamma^{H} & \boldsymbol I_{L} \\
\end{array}
\right]\succeq0, \nonumber\\
&& \mathcal{T}^{\ast}(\boldsymbol \Lambda)=\left[
\begin{array}{c}
1 \\
\boldsymbol 0 \\
\end{array}
\right],
\nonumber\\
&& \|\boldsymbol \Gamma\|_{\infty,2}\leq\lambda,\end{aligned}$$
where ${\mathcal{T^{\ast}}}$ is defined as $${\mathcal{T}^{\ast}(\boldsymbol \Lambda)_j=\sum_{i=1}^{N-j+1}\boldsymbol \Lambda_{i,i+j-1},}$$ ${\boldsymbol I_L}$ denotes the identity matrix of size ${L\times L}$, ${\boldsymbol 0\in\mathbb{C}^{N-1}}$ is a zero vector, and ${\succeq0}$ denotes positive semi-definiteness.
In order to localize the frequencies of the signal of interest and the noisy spikes, lemma\[lemma1\] is presented.
\[lemma1\] The solution to (\[eqn7\]) is unique if for ${\boldsymbol \Gamma\in\mathbb{C}^{N\times L}}$ and the vector-valued dual polynomial ${\boldsymbol Q=\boldsymbol a(f,0)^H\boldsymbol \Gamma}$ we have
$$\begin{aligned}
\boldsymbol Q(f_k)&=& \frac{c_k}{|c_k|}\boldsymbol b_k^{H}\:\:\: k:f_k\in\mathbb{T}\label{eqn11}\\
\|\boldsymbol Q(f_j)\|_2<1& &\:\:\:\forall f_j\in[0,1]\backslash \mathbb{T}\label{eqn12}
\end{aligned}$$
${d}$${\boldsymbol \Omega}$ ${l \in \boldsymbol \Omega^c}$ $$\begin{aligned}
\boldsymbol \Gamma_{ d,:} &=& \lambda\frac{\boldsymbol Z_{d,:}}{\|\boldsymbol Z_{d,:}\|_2} \label{eqn13}\\
\|\boldsymbol \Gamma_{l,:}\|_{\infty,2} &<& \lambda\label{eqn14}
\end{aligned}$$
If we find a ${\boldsymbol \Gamma}$ satisfying the above conditions, it is dual feasible. Consider ${\hat{\boldsymbol S}}$ and ${\hat{\boldsymbol Z}}$ as the solutions to (\[eqn7\]). Then we would have $$\begin{aligned}
\|\hat{\boldsymbol S}\|_{\mathcal{A}} &\geq& \|\hat{\boldsymbol S}\|_{\mathcal{A}}\|{\boldsymbol \Gamma}\|_{\mathcal{A}}^{\ast}\nonumber \\
&\geq& <\boldsymbol \Gamma,\hat{\boldsymbol S}>_{\mathbb{R}} \nonumber\\
&=& <\boldsymbol \Gamma,\sum_{k=1}^{K}c_k\boldsymbol a(f_k,\phi_k)\boldsymbol b_k^H>_{\mathbb{R}} \nonumber\\
&=& \sum_{k=1}^{K}Re\{c_k^{\ast}<{\boldsymbol \Gamma},\boldsymbol a(f_k,\phi_k)\boldsymbol b_k^H>\} \nonumber\\
&=& \sum_{k=1}^{K}Re\{c_k^{\ast}<\boldsymbol b_k,\boldsymbol Q(f_k)^H>\}\nonumber\\
&=& \sum_{k=1}^{K}Re\{c_k^{\ast}\frac{c_k}{|c_k|}\}\geq \|\hat{\boldsymbol S}\|_{\mathcal{A}}\nonumber\end{aligned}$$ Also, $$\begin{aligned}
Re<\hat{\boldsymbol Y},{\boldsymbol \Gamma}> &=& Re<\hat{\boldsymbol S},{\boldsymbol \Gamma}>+Re<\hat{\boldsymbol Z},{\boldsymbol \Gamma
}> \nonumber\\
&=& \|\hat{\boldsymbol S}\|_{\mathcal{A}}+\sum_{ d\in{\boldsymbol \Omega}}^{ }Re\{\hat{\boldsymbol Z_{d,:}^{\ast}},\boldsymbol \Gamma_{d,:} \} \nonumber\\
&=& \|\hat{\boldsymbol S}\|_{\mathcal{A}}+ \lambda\sum_{d\in\boldsymbol \Omega}^{ }Re\{\frac{\boldsymbol Z_{d,:}^{\ast}\boldsymbol Z_{d,:}}{\|\boldsymbol Z_{d,:}\|_2}\}=\lambda\|\boldsymbol Z\|_{1,2}\nonumber\end{aligned}$$
where the last equality is derived using \[eqn13\]. Therefore, we must have ${<\boldsymbol \Gamma,\hat{\boldsymbol S}>_{\mathbb{R}}= \|\hat{\boldsymbol S}\|_{\mathcal{A}}+\lambda\|\hat{\boldsymbol Z}\|_{1,2}}$. Thus, by strong duality ${\hat{\boldsymbol S}}$ and ${\hat{\boldsymbol Z}}$ are primal optimal and ${\boldsymbol \Gamma}$ is dual optimal. To investigate uniqueness, we consider ${\tilde{\boldsymbol S}}$ and ${\tilde{\boldsymbol Z}}$ as other solutions to (\[eqn7\]). Because of the independency of atoms in ${\mathbb{T}}$, the support sets of ${\tilde{\boldsymbol S}}$ and ${\tilde{\boldsymbol Z}}$ are different from ${\hat{\boldsymbol S}}$ and ${\hat{\boldsymbol Z}}$. Let ${\tilde{\boldsymbol S}=\sum_{k}^{ }\tilde{c}_k\boldsymbol a(\tilde{f}_k,\tilde{\phi}_k)\tilde{\boldsymbol b}_k^H}$ for ${\tilde{c}_k>0}$ and some supports ${\tilde{f}_k\notin\mathbb{T}}$. Then,
$$\begin{aligned}
& & <\tilde{\boldsymbol Y},\boldsymbol \Gamma>_{\mathbb{R}}=<\tilde{\boldsymbol S},\boldsymbol \Gamma>_{\mathbb{R}}+<\tilde{\boldsymbol Z},\boldsymbol \Gamma>_{\mathbb{R}}\nonumber\\
&=&\sum_{\tilde{f}_k\in\mathbb{T}}^{ }Re\{\tilde{c}_k<\tilde{b}_k,\boldsymbol Q(\tilde{f}_k)^H>\} \nonumber\\
&+&\sum_{\tilde{f}_j\notin\mathbb{T}}^{ }Re\{\tilde{c}_j<\tilde{b}_j,\boldsymbol Q(\tilde{f}_j)^H>\}\nonumber\\
&+& \sum_{ d \in {\boldsymbol \Omega}}^{ }Re\{<\tilde{\boldsymbol Z}_{d,:}^{\ast},\boldsymbol \Gamma_{d,:}>\}+\sum_{l\in{\boldsymbol \Omega}^c}^{ }Re\{<\tilde{\boldsymbol Z}_{l,:}^{\ast},\boldsymbol \Gamma_{l,:}>\} \nonumber\\
&\leq& \sum_{\tilde{f}_k\in\mathbb{T}}^{ }Re\{\tilde{c}_k\|\tilde{\boldsymbol b}_k\|_2\|\boldsymbol Q(\tilde{f}_k)\|_2 + \sum_{\tilde{f}_j\notin\mathbb{T}}^{ }Re\{\tilde{c}_j\|\tilde{\boldsymbol b}_j\|_2\|\boldsymbol Q(\tilde{f}_j)\|_2\nonumber\\
&+& \lambda\sum_{ d \in {\boldsymbol \Omega}}^{ }Re\{\|\tilde{\boldsymbol Z}_{d,:}\|_2\} + \|\boldsymbol \Gamma_{\boldsymbol \Omega^c,:}\|_{\infty,2}\sum_{l\in{\boldsymbol \Omega}^c}^{ }Re\{\|\tilde{\boldsymbol Z}_{l,:}\|_2\} \nonumber\\
&<& \sum_{\tilde{f}_k\in\mathbb{T}}^{ }\tilde{c}_k\|\tilde{\boldsymbol b}_k\|_2 +\sum_{\tilde{f}_j\notin\mathbb{T}}^{ }\tilde{c}_j\|\tilde{\boldsymbol b}_j\|_2 +\lambda\sum_{d \in {\boldsymbol \Omega}}^{ }Re\{\|\tilde{\boldsymbol Z}_{d}\|_2\}\nonumber\\
&+& \lambda\sum_{\boldsymbol l\in{\boldsymbol \Omega}^c}^{ }Re\{\|\tilde{\boldsymbol Z}_{l,:}\|_2\} = \|\tilde{\boldsymbol S}\|_{\mathcal{A}}+\lambda\|\tilde{\boldsymbol Z}\|_{1,2}\nonumber\end{aligned}$$
which contradicts strong duality. Therefore, ${\hat{\boldsymbol S}}$ and ${\hat{\boldsymbol Z}}$ are unique optimal solutions of (\[eqn7\]).
Numerical Results {#section5}
=================
In this section, numerical experiments are presented to evaluate the performance of the method proposed in Section\[section4\]. First, we investigate the constraints (\[eqn11\]) and (\[eqn12\]) on the dual polynomial and the constraints (\[eqn13\]) and (\[eqn14\]) on the dual variable. Using these constraints, one can localize the signal frequencies and the outliers’ spikes. Next, the minimum required frequency separation for successful recovery in MMV case is compared with the one needed in SMV case. In all simulations, the number of the sensors or the signal length is ${N=50}$. In the first part of the simulations, the signal of interest ${\boldsymbol S\in\mathbb{C}^{N\times L}}$ has ${K=3}$ frequencies and the coefficients ${a_{kl}}$ are drawn from a standard i.i.d complex Gaussian distribution. The outliers’ spikes are considered to happen in ${s=3}$ different random positions in each snapshot. For better visualization, it is assumed that outliers happen in each sensor only once. Fig.\[fig1\] depicts ${\|\boldsymbol Q(f)\|_2}$ for ${L=5}$ snapshots and ${\mathbb{T}=\{0.1,0.4,0.8\}}$. As it can be seen, the signal frequencies can be estimated by solving ${\|\boldsymbol Q(f)\|_2=1}$ for all ${f\in[0,1]}$. The outliers are localized in each receiving sensor using (\[eqn13\]). We considered ${s=3}$ noisy spikes happening randomly in each measurement without replacement. Thus, with ${L=5}$ we expect to detect ${15}$ outlier in the receiver. Fig.\[fig2\] depicts the result. As it can be seen,Fig.\[fig2\] verifies the conclusion of lemma\[lemma1\].
![${l_2}$ norm of the dual polynomial and the true frequencies.[]{data-label="fig1"}](Fig1.jpg)
![${l_{2,\infty} }$ norm of ${\boldsymbol \Gamma}$ and the noise spikes.[]{data-label="fig2"}](Fig2.jpg)
Next, we investigated the minimum separation condition. To do this, we considered two frequencies slowly taking distance. The first frequency is fixed at ${f_1=0.2}$ and the second one has a distance of ${f_{\delta}=\{0.1/N:0.1/N:1.5/N\}}$ from ${f_1}$. During this experiment, ${s=10}$ outliers in the overall measurement process was considered and ${N=50}$ was fixed. We define ${\boldsymbol f_{est}=[f_1^{est},f_2^{est}]}$ as the estimated frequencies vector. A successful estimation is defined as when $$\begin{aligned}
& \max\{|\boldsymbol f_{est}-\boldsymbol f_{true}|\}\leq 10^{-4}\end{aligned}$$ where ${\boldsymbol f_{true}}$ denotes the true frequencies. With this definition, Fig\[fig3\] illustrates the probability of successful recovery for ${L=\{1,3,5\}}$ and ${100}$ Monte-Carlo simulations.
![Probability of successful recovery for various number of snapshots.[]{data-label="fig3"}](Fig3_2.jpg)
As it can be seen, the minimum required frequency separation is decreased with an increase in the number of snapshots.
Proof of Theorem 1 {#section6}
==================
In order to prove that problem (\[eqn7\]) achieves exact demixing, we construct a trigonometric dual polynomial. Following the same line of [@fernandez2017demixing], we apply the following kernel to build up the dual polynomial
$$\begin{aligned}
\bar{K}(f)&:=& \mathcal{D}_{0.247m}(f)\mathcal{D}_{0.339m}(f)\mathcal{D}_{0.414m}(f)\\
& =&\sum_{l=-m}^{m}c_le^{i2\pi lf}\label{eqn16}
\end{aligned}$$
where ${N=2m+1}$, ${\boldsymbol c\in\mathbb{C}^N}$ is the convolution of the Fourier coefficients of the above kernels, and ${\mathcal{D}_m}$ is the Dirichlet kernel of order ${m>0}$ defined as $${\mathcal{D}_m(f):=\frac{1}{N}\sum_{l=-m}^{m}e^{i2\pi lf}.}$$ According to the presence of outliers, conventional forms of dual polynomial can not be applied since the constraints (\[eqn14\]) and (\[eqn13\]) will not be met. Therefore, we use the randomized vector form of the dual polynomial presented in [@fernandez2017demixing] as $$\begin{aligned}
\boldsymbol Q(f) &=& \boldsymbol Q_{aux}(f)+\boldsymbol R(f) \label{eqn161}\end{aligned}$$ where $$\begin{aligned}
\boldsymbol Q_{aux}(f) &=& \sum_{l\in \boldsymbol \Omega^c }^{ }\boldsymbol \Gamma_{l,:}e^{-i2\pi l f},\label{eqn17} \\
\boldsymbol R(f) &=&\frac{1}{\sqrt{N}}\sum_{d\in \boldsymbol \Omega}^{}\boldsymbol r_ie^{-i2\pi d f},\label{eqn18}\end{aligned}$$ where ${\boldsymbol r_i=\frac{\boldsymbol Z_{d,:}}{\|\boldsymbol Z_{d,:}\|_2}}$ and ${\boldsymbol r\in\mathbb{C}^{K\times L}}$. Note that (\[eqn13\]) is immediately satisfied since ${\lambda=1/\sqrt{N}}$. Now we should build up the dual polynomial so that the other constraints in lemma\[lemma1\] are met. Using the same interpolation technique of [@fernandez2016super], we set the value of the dual polynomial equal to ${\frac{c_k}{|c_k|}\boldsymbol b_k^{H}=h_k\boldsymbol b_k^{H}}$ at ${f_k\in\mathbb{T}}$ and set the derivative of the dual polynomial equal to zero at the same points. Setting the derivative to zero forces the dual polynomial to shape such that ${f_k}$ be a local extremum and bounds the value of the dual polynomial at these points. Thus, the following set of equations is formed for any ${f_k\in\mathbb{T}}$
$$\begin{aligned}
\boldsymbol Q(f_k) &=& h_k\boldsymbol b_k^H, \label{eqn19}\\
\boldsymbol Q_R^{(1)}(f_k)+i\boldsymbol Q_I^{(1)} &=& 0,\label{eqn20}\end{aligned}$$
where ${\boldsymbol Q_R^{(1)}}$ denotes the real part of the first derivative of ${\boldsymbol Q}$ and ${\boldsymbol Q_I}$ is the imaginary part of ${\boldsymbol Q}$. Using (\[eqn161\]) in the above equations yields
$$\begin{aligned}
& \boldsymbol Q_{aux}(f_k)=h_k\boldsymbol b_k^H-\boldsymbol R_k(f_k)\label{eqn21}\\
& (\boldsymbol Q_{aux})_R^{(1)}(f_k)+i(\boldsymbol Q_{aux})_I^{(1)}(f_k)=-\boldsymbol R_R^{(1)}(f_k)-i\boldsymbol R_I^{(1)}(f_k)\label{eqn22}\end{aligned}$$
Now, to interpolate ${\boldsymbol Q(f)}$ with ${\bar{K}(f)}$, we need to confine the kernel to ${\boldsymbol \Omega^c}$, as discussed for the missing data case in [@tang2013compressed]. Thus, $$\begin{aligned}
\label{eqn23}
K(f) &:=& \sum_{l\in\boldsymbol \Omega^c}^{}c_le^{i2\pi lf}=\sum_{l=-m}^{m}\delta_{\boldsymbol \Omega^c}(l)c_le^{i2\pi lf}\end{aligned}$$ where ${\delta_{\boldsymbol \Omega^c}(l)}$ are Bernoulli random variables with parameter ${\frac{N-s}{N}}$. Thus, ${\mathbb{E}K}$ is an scaled version of ${\bar{K}}$ $$\begin{aligned}
\mathbb{E}K(f) &=& \frac{N-s}{N}\sum_{l=-m}^{m}c_le^{i2\pi lf}=\frac{N-s}{N}\bar{K}(f).\label{eqn24}\end{aligned}$$ The asymptotic behaviour of ${K(f)}$, ${\bar{K}(f)}$ and their derivatives is investigated in [@fernandez2016super]. With ${K(f)}$ restricted to ${\boldsymbol \Omega^c}$ we can express ${\boldsymbol Q_{aux}}$ in terms of ${K(f)}$ and its first derivative as $$\begin{aligned}
\boldsymbol Q_{aux} &=& \sum_{k=1}^{K}\boldsymbol \alpha_kK(f-f_k)+\kappa\boldsymbol \beta_kK^{(1)}(f-f_k)\label{eqn25}\end{aligned}$$ where ${\boldsymbol \alpha\in \mathbb{C}^{K\times L}}$ and ${\boldsymbol \beta\in \mathbb{C}^{K\times L}}$ are such that (\[eqn21\]) and (\[eqn22\]) are satisfied and ${\kappa:=1/\sqrt{\bar{K}^{(2)}(0)}}$. This system of equations can be represented as follow $$\begin{aligned}
\label{eqn26}
\left[
\begin{array}{cc}
\boldsymbol D_0 & \boldsymbol D_1 \\
\boldsymbol D_1^T& \boldsymbol D_2 \\
\end{array}
\right]\left[
\begin{array}{c}
\boldsymbol \alpha \\
\boldsymbol \beta \\
\end{array}
\right]
&=& \left[
\begin{array}{c}
\boldsymbol \Phi \\
\boldsymbol 0 \\
\end{array}
\right]-\frac{1}{\sqrt{N}}\boldsymbol B_{\boldsymbol \Omega}\boldsymbol r\end{aligned}$$ where ${\boldsymbol 0\in\mathbb{C}^{K\times L}}$ is a zero matrix,${\boldsymbol \Phi_{k,:}=h_k\boldsymbol b_k^H}$, $$\begin{aligned}
(D_0)_{jl} &=& K(f_j-f_l) ,\nonumber\\
(D_1)_{jl} &=& \kappa K^{(1)}(f_j-f_l) ,\nonumber\\
(D_2)_{jl} &=& -\kappa^2 K^{(2)}(f_j-f_l),\nonumber\end{aligned}$$ $${\frac{1}{\sqrt{N}}\boldsymbol B_{\boldsymbol \Omega}\boldsymbol r=[\boldsymbol R(f_1),\ldots,\boldsymbol R(f_k),\boldsymbol R^{(1)}(f_1),\ldots
,\boldsymbol R^{(1)}(f_k)]^T,}$$ $${\boldsymbol B_{\boldsymbol \Omega}=[\boldsymbol\nu(d_1),\ldots,\boldsymbol\nu(d_s)],}$$ $$\begin{aligned}
\boldsymbol\nu(g):=[e^{-i2\pi gf_1},\ldots &,& e^{-i2\pi gf_k}, \nonumber\\
i2\pi g\kappa e^{-i2\pi gf_1} &,& \ldots,i2\pi g \kappa e^{-i2\pi gf_k}]^T. \nonumber\end{aligned}$$ By solving (\[eqn26\]), one can find ${\boldsymbol \alpha}$ and ${\boldsymbol \beta}$ and define ${\boldsymbol Q(f)}$ as
$$\begin{aligned}
&\boldsymbol Q(f) = \sum_{k=1}^{K}\boldsymbol \alpha_kK(f-f_k)+\kappa\boldsymbol \beta_kK^{(1)}(f-f_k)+\boldsymbol R(f)\\\label{eqn27}
&=\boldsymbol G_0^T(f)\boldsymbol D^{-1}\left(\left[
\begin{array}{c}
\boldsymbol \Phi \\
\boldsymbol 0 \\
\end{array}
\right]-\frac{1}{\sqrt{N}}\boldsymbol B_{\boldsymbol \Omega}\boldsymbol r
\right)+\boldsymbol R(f)\end{aligned}$$
where ${\boldsymbol G_p(f)}$ is defined as $$\begin{aligned}
\boldsymbol G_p(f):=\kappa^p[K^{(p)}(f-f_1),\ldots,K^{(p)}(f-f_k),&\nonumber\\
\kappa K^{(p+1)}(f-f_1),\ldots,\kappa K^{(P+1)}(f-f_k)]^T&\end{aligned}$$ for ${p=0,1,2,\ldots}$ Now we should verify that the polynomial we formed above is guaranteed to be valid with high probability. If one can prove that ${\boldsymbol D^{-1}}$ exists, then (\[eqn26\]) can be solved and (\[eqn11\]) holds. Consider ${\bar{\boldsymbol D}}$ as the deterministic version of ${\boldsymbol D}$. Lemma 3.8 from [@fernandez2017demixing] helps defining a condition under which ${\boldsymbol D^{-1}}$ exists. We consider ${\varepsilon_{\boldsymbol D}^c}$ as the event in which ${\boldsymbol D^{-1}}$ exists with probability ${1-\epsilon/5}$ for ${\epsilon>0}$ under the assumption of Theorem \[theorem1\]. With this lemma, one can conclude that in ${\varepsilon^c_{\boldsymbol D}}$ (\[eqn11\]) holds. Note that (\[eqn13\]) holds according to the definition of ${\boldsymbol Q(f)}$. All that remains is to prove (\[eqn12\]) and (\[eqn14\]). We use the results of lemma 3.5, lemma 3.6, and lemma 3.7 from [@fernandez2017demixing] which bound ${\boldsymbol\nu(d)}$, ${\boldsymbol B_{\boldsymbol \Omega}}$, and ${\boldsymbol G_p(f)}$ respectively. We use ${\varepsilon_{\boldsymbol B}^c}$ and ${\varepsilon_{\boldsymbol \nu}^c}$ as the events in which ${\boldsymbol B_{\boldsymbol \Omega}}$ and ${\boldsymbol \nu(d)}$ are bounded with probability at least ${1-\epsilon/5}$ under the assumption of Theorem \[theorem1\] respectively.
\[prop1\] Under the assumption of Theorem \[theorem1\] and conditioned on ${\varepsilon_{\boldsymbol B}^c\cap\varepsilon^c_{\boldsymbol D}\cap\varepsilon_{\boldsymbol \nu}^c}$, (\[eqn12\]) holds with probability at least ${1-\epsilon/5}$.
\[prop1\_proof\] Consider ${\bar{\boldsymbol Q}(f)}$ as the dual polynomial constructed using ${\bar{\boldsymbol K}(f)}$. We can rewrite (\[eqn27\]) in a more general form for ${\boldsymbol K(f)}$ and ${\bar{\boldsymbol K}(f)}$ as follows $$\begin{aligned}
&\kappa^{\iota}\bar{\boldsymbol Q}^{(\iota)}(f) := \kappa^{\iota}\sum_{j=1}^{K}\bar{\boldsymbol\alpha}_j\bar{\boldsymbol K}^{(\iota)}(f-f_j)+ \nonumber\\
& \kappa^{\iota+1}\sum_{j=1}^{K}\bar{\boldsymbol\beta}_j\bar{\boldsymbol K}^{(\iota+1)}(f-f_j) =\bar{\boldsymbol G}_{\iota}(f)^T\bar{\boldsymbol D}^{-1}\left[
\begin{array}{c}
\boldsymbol \Phi \\
\boldsymbol 0 \\
\end{array}
\right]\label{eqn28}
\\
&\kappa^{\iota}{\boldsymbol Q}^{(\iota)}(f) := \kappa^{\iota}\sum_{j=1}^{K}{\boldsymbol\alpha}_j{\boldsymbol K}^{(\iota)}(f-f_j)+ \nonumber \\ &\kappa^{\iota+1}\sum_{j=1}^{K}{\boldsymbol\beta}_j{\boldsymbol K}^{(\iota+1)}(f-f_j)+\kappa^{\iota}\boldsymbol R^{(\iota)}(f) \nonumber \\
& ={\boldsymbol G}_{\iota}(f)^T{\boldsymbol D}^{-1}\left(\left[
\begin{array}{c}
\boldsymbol \Phi \\
\boldsymbol 0 \\
\end{array}
\right]-\frac{1}{\sqrt{N}}\boldsymbol B_{\boldsymbol \Omega}\boldsymbol r\right)+\kappa^{\iota}\boldsymbol R^{(\iota)}(f).\label{eqn29}
\end{aligned}$$ We express (\[eqn29\]) as $$\begin{aligned}
& \kappa^{\iota}{\boldsymbol Q}^{(\iota)}(f) :=\kappa^{\iota}\bar{\boldsymbol Q}^{(\iota)}(f)+\kappa^{\iota}{\boldsymbol R}^{(\iota)}(f)-\frac{1}{\sqrt{N}}\boldsymbol G_{\iota}(f)^T\boldsymbol D^{-1}\boldsymbol B_{\boldsymbol \Omega}\boldsymbol r \nonumber\\
& +(\boldsymbol G_{\iota}(f)-\frac{N-s}{N}\bar{\boldsymbol G}_{\iota}(f))^T\boldsymbol D^{-1}\left[
\begin{array}{c}
\boldsymbol \Phi \\
\boldsymbol 0 \\
\end{array}
\right] \nonumber\\
& +\frac{N-s}{N}\bar{\boldsymbol G}_{\iota}(f)^T(\boldsymbol D^{-1}-\frac{N}{N-s}\bar{\boldsymbol D}^{-1})\left[
\begin{array}{c}
\boldsymbol \Phi \\
\boldsymbol 0 \\
\end{array}
\right].\nonumber
\end{aligned}$$ Now note that ${\|\boldsymbol Q(f)\|_2\leq\|\bar{\boldsymbol Q}(f)\|_2+\|\boldsymbol Q(f)-\bar{\boldsymbol Q}(f)\|_2}$. Thus, for (\[eqn12\]) to hold we should have ${\|\bar{\boldsymbol Q}(f)\|_2+\|\boldsymbol Q(f)-\bar{\boldsymbol Q}(f)\|_2\leq1}$. The following lemmas complete the proof.
\[lemma2\] Under the assumptions of Proposition \[prop1\], ${\|\boldsymbol Q(f)-\bar{\boldsymbol Q}(f)\|_2\leq10^{-2}}$.
\[lemma3\] Under the assumptions of Proposition \[prop1\], ${\|\bar{\boldsymbol Q}(f)\|_2<0.99}$. Also $$\begin{aligned}
&\frac{1}{2}\frac{d^2\|\boldsymbol Q(f)\|_2}{df^2} =\|\boldsymbol Q^{\prime}\|_2^2+Re\{\boldsymbol Q^{\prime \prime}\boldsymbol Q^H(d)\}<0\label{eqn30}\\
&\forall f\in A_{near}:=\{f||f-f_j|\leq0.09\:for f_j\in \mathbb{T}\}.\nonumber
\end{aligned}$$
The proof of the above lemmas appear in Appendix.
Now, we prove (\[eqn14\]) as the last step to prove Theorem \[theorem1\].
\[prop2\] Under the assumption of Theorem \[theorem1\] and conditioned on ${\varepsilon_{\boldsymbol B}^c\cap\varepsilon^c_{\boldsymbol D}\cap\varepsilon_{\boldsymbol \nu}^c}$, (\[eqn14\]) holds with probability at least ${1-\epsilon/5}$.
\[prop2\_proof\] We can express ${\boldsymbol \Gamma_{l,:}}$ as $$\begin{aligned}
& \boldsymbol \Gamma_{l,:}=\sum_{j=1}^{K}c_l\boldsymbol \alpha_je^{i2\pi lf_j}+i2\pi l\kappa\sum_{j=1}^{K}\sum_{j=1}^{K}\boldsymbol \beta_je^{i2\pi lf_j} \nonumber\\
& =c_l\boldsymbol\nu(l)^H\left[
\begin{array}{c}
\boldsymbol \alpha \\
\boldsymbol \beta \\
\end{array}
\right]=c_l\boldsymbol\nu(l)^H\boldsymbol D^{-1}\left(\left[
\begin{array}{c}
\boldsymbol \Phi \\
\boldsymbol 0 \\
\end{array}
\right]-\frac{1}{\sqrt{N}}\boldsymbol B_{\boldsymbol \Omega}\boldsymbol r\right)\nonumber\\
&=c_l\left(<P\boldsymbol D^{-1}\boldsymbol \nu(l),\boldsymbol \Phi>+\frac{1}{\sqrt{N}}<\boldsymbol B_{\boldsymbol \Omega}^H\boldsymbol D^{-1}\boldsymbol \nu(l),\boldsymbol r>\right).\label{eqn31}
\end{aligned}$$ We use the results from [@fernandez2017demixing] to bound ${\|P\boldsymbol D^{-1}\boldsymbol \nu(l)\|_2}$ and ${\boldsymbol B_{\boldsymbol \Omega}^H\boldsymbol D^{-1}\boldsymbol \nu(l)}$,
$$\begin{aligned}
& \|P\boldsymbol D^{-1}\boldsymbol \nu(l)\|_2^2\leq640K\leq\frac{0.18^2N}{log40/\epsilon}\:\:\:in\:\: \varepsilon_D^c\label{eqn32}\\
& \|\boldsymbol B_{\boldsymbol \Omega}^H\boldsymbol D^{-1}\boldsymbol \nu(l)\|_2^2\leq640C_B^2KN\leq\frac{0.18^2N^2}{log40/\epsilon}\:\:\:in\:\:\varepsilon_D^c\cap\varepsilon_B^c\label{eqn33}
\end{aligned}$$
Now, applying the vector form Hoeffding’s inequality [@yang2016exact] with ${t=0.18\sqrt{N}}$ for (\[eqn31\]) and ${t=0.18N}$ for (\[eqn32\]), we can conclude that each term in (\[eqn31\]) is greater than its corresponding ${t}$ with probability ${\epsilon/10}$. Thus, $$\begin{aligned}
& \|\boldsymbol \Gamma_{l,:}\|_{\infty,2}\leq\nonumber\\
&\|\boldsymbol c\|_{\infty}\left(\|\boldsymbol\nu(l)^H\boldsymbol D^{-1}P^T\boldsymbol \Phi\|_2+\frac{1}{\sqrt{N}}\|\boldsymbol\nu(l)^H\boldsymbol D^{-1}\boldsymbol B_{\boldsymbol \Omega}\boldsymbol r\|_2\right) \nonumber\\
& \leq\frac{2.6}{N}(0.36\sqrt{N})=\frac{0.936}{\sqrt{N}}<\frac{1}{\sqrt{N}},\nonumber
\end{aligned}$$ with probability at least ${1-\epsilon/5}$.
Conclusion and Future Work {#section51}
==========================
In this paper, the problem of demixing exponential form signals and outliers using MMVs was discussed. A new convex optimization problem was proposed to solve this problem. It was shown that with the minimum frequency separation condition satisfied, there exists a dual polynomial which interpolates the sign pattern of the signal and helps estimating the signal frequencies. Also, the dual variable was utilized to localize the outliers in the receiver.
As an extension to this work, one can investigate the demixing problem using arbitrary sampling scheme. This is the case when integer sampling is not possible. Also, the computational complexity of the available SDPs is high. For practical purposes, it is mandatory to reduce the computational complexity of the proposed method.
Proof of Lemma\[lemma2\]
------------------------
First we bound ${\|\kappa^{\iota}\boldsymbol Q^{(\iota)}(f)-\kappa^{\iota}\bar{\boldsymbol Q}^{(\iota)}(f)\|_2}$ on a grid. Then the result is extended to the continuous domain ${[0,1]}$ and then (\[eqn12\]) is proved. In order to bound ${\|\kappa^{\iota}\boldsymbol Q^{(\iota)}(f)-\kappa^{\iota}\bar{\boldsymbol Q}^{(\iota)}(f)\|_2}$, we can bound each term in $$\begin{aligned}
& \|\kappa^{\iota}{\boldsymbol R}^{(\iota)}(f)\|_2+\|\frac{1}{\sqrt{N}}\boldsymbol G_{\iota}(f)^T\boldsymbol D^{-1}\boldsymbol B_{\boldsymbol \Omega}\boldsymbol r\|_2 \nonumber\\
& +\|(\boldsymbol G_{\iota}(f)-\frac{N-s}{N}\bar{\boldsymbol G}_{\iota}(f))^T\boldsymbol D^{-1}\left[
\begin{array}{c}
\boldsymbol \Phi \\
\boldsymbol 0 \\
\end{array}
\right]\|_2 \nonumber\\
& +\|\frac{N-s}{N}\bar{\boldsymbol G}_{\iota}(f)^T(\boldsymbol D^{-1}-\frac{N}{N-s}\bar{\boldsymbol D}^{-1})\left[
\begin{array}{c}
\boldsymbol \Phi \\
\boldsymbol 0 \\
\end{array}
\right]\|_2.\label{eqn34}
\end{aligned}$$ on a grid ${\mathcal{G}}$ such that ${|\mathcal{G}|=200\sqrt{L}N^3}$ where ${|\mathcal{G}|}$ is the cardinality of ${\mathcal{G}}$. Since, ${\iota\in\{0,1,2,3\}}$ we are dealing with ${|\mathcal{U}|=4|\mathcal{G}|}$ points. To bound each term in (\[eqn34\]), vector form Hoeffding’s inequality [@yang2016exact] is used.
Consider a matrix ${\boldsymbol \Psi\in\mathbb{C}^{K\times L}}$. By sampling the rows of ${\boldsymbol \Psi}$ independently on the zero-mean complex hyper-sphere ${\mathbb{S}^{2L-1}}$ we have $$\begin{aligned}
\mathbb{P}\{\|\boldsymbol \omega^H\boldsymbol\Psi\|_2\geq t\}\leq(L+1)e^{-\frac{t^2}{8\|\boldsymbol \omega\|_2}} & \:\forall\boldsymbol \omega\in\mathbb{C}^K,\boldsymbol\omega\neq0,t>0\label{eqn35}
\end{aligned}$$
Each term in (\[eqn34\]) is associated with an event ${\varepsilon_q}$ and ${q=\{1,2,3,4\}}$. The first term in (\[eqn34\]) can be expressed as $$\begin{aligned}
\kappa^{\iota}{\boldsymbol R}^{(\iota)}(f) & = \frac{\kappa^{\iota}}{\sqrt{N}}\sum_{d\in\boldsymbol \Omega}^{ }\boldsymbol r_{d,:}(i2\pi d)^{(\iota)}e^{-i2\pi df}\:\:\:(\iota)=\{0,1,2,3\}.\nonumber\end{aligned}$$ Therefore, we define $$\begin{aligned}
\varepsilon_1 & :=\{\|\kappa^{\iota}{\boldsymbol R}^{(\iota)}(f)\|_2\geq t\:\:\: for\:\: all\:\: f\in \mathcal{U}\}.\nonumber\end{aligned}$$ By setting ${\boldsymbol\Psi=\boldsymbol r}$ and $$\begin{aligned}
\boldsymbol \omega & =\frac{\kappa^{\iota}}{\sqrt{N}}\left[(i2\pi l_1)^{(\iota)}e^{i2\pi l_1f},\ldots,(i2\pi l_s)^{\iota}e^{i2\pi l_sf}\right]^T,\nonumber
\end{aligned}$$ in (\[eqn35\]) and noting that[@fernandez2017demixing] $$\begin{aligned}
\|\boldsymbol \omega\|_2^2 & \leq\frac{\kappa^{2\iota}}{N}(2\pi m)^{2\iota}s\leq\frac{\pi^6s}{N}\leq C^2_{\mathcal{U}}(log\frac{\sqrt{L}N}{\epsilon})^{-1},\nonumber\end{aligned}$$ and using the union bound, we can conclude that $$\begin{aligned}
\mathbb{P}\{\underset{f\in\mathcal{U}}{sup}\|\kappa^{\iota}\boldsymbol R^{(\iota)}(f)\|_2\geq t\} & \leq 4(L+1)|\mathcal{G}|e^{-\frac{t^2}{8C_{\mathcal{U}}^2(log\frac{\sqrt{L}N}{\epsilon})^{-1}}}\label{eqn351}\end{aligned}$$ After setting ${t=\frac{10^{-2}}{8}}$ and $C_{\mathcal{U}}=10^{-4}$, one can conclude that the event ${\varepsilon_1}$ happens with probability at most ${\epsilon/20}$ under the assumptions of Proposition\[prop1\]. Following the same procedure, one can bound the second term in (\[eqn34\]). Consider ${\boldsymbol \Psi=\boldsymbol r}$ and $$\begin{aligned}
\boldsymbol \omega & =\frac{1}{\sqrt{N}}\boldsymbol G_{\iota}^T(f)\boldsymbol D^{-1}\boldsymbol B_{\boldsymbol \Omega}.\nonumber\end{aligned}$$ Note that we can write $$\begin{aligned}
\|\frac{1}{\sqrt{N}}\boldsymbol G_{\iota}^T(f)\boldsymbol D^{-1}\boldsymbol B_{\boldsymbol \Omega}\|_2 & \leq\|\boldsymbol B_{\boldsymbol \Omega}\|_2\|\boldsymbol D^{-1}\|_2\|\boldsymbol G_{\iota}(f)\|_2.\label{eqn36}\end{aligned}$$ Using modified versions of Lemma 3.6, 3.8, H.8 and Corollary H.9 from [@fernandez2017demixing] according to Theorem\[theorem1\], we can find tight bounds for (\[eqn36\]) as $$\begin{aligned}
& \|\boldsymbol B_{\boldsymbol \Omega}\|_2\|\boldsymbol D^{-1}\|_2\|\boldsymbol G_{\iota}(f)\|_2 \leq\frac{8(C_{\bar{\boldsymbol \nu}}+C_{{\boldsymbol \nu}})\|\boldsymbol B_{\boldsymbol \Omega}\|}{\sqrt{N}} \nonumber\\
& \leq\frac{8(C_{\bar{\boldsymbol \nu}}+C_{{\boldsymbol \nu}})C_{\boldsymbol B}\left(log\frac{\sqrt{L}N}{\epsilon}\right)^{-\frac{1}{2}}\sqrt{N}}{\sqrt{N}}\:\:\:,C_{\boldsymbol B}=\frac{C_{\mathcal{U}}}{8\left(C_{\bar{\boldsymbol \nu}}+C_{{\boldsymbol \nu}}\right)}\nonumber.\end{aligned}$$ Thus, by setting ${t=\frac{10^{-2}}{8}}$ and using the vector form Hoeffding’s inequality and the union bound we have $$\begin{aligned}
\mathbb{P}\{\underset{f\in\mathcal{U}}{sup}\|\frac{1}{\sqrt{N}}\boldsymbol G_{\iota}^T(f)\boldsymbol D^{-1}\boldsymbol B_{\boldsymbol \Omega}\|_2&\geq \frac{10^{-2}}{8}\}\nonumber\\ &\leq 4(L+1)|\mathcal{G}|e^{-\frac{{\frac{10^{-2}}{8}}^2}{8C_{\mathcal{U}}^2(log\frac{\sqrt{L}N}{\epsilon})^{-1}}}.\label{eqn37}\end{aligned}$$ Thus, the event $$\begin{aligned}
\varepsilon_2 & :=\{\|\frac{1}{\sqrt{N}}\boldsymbol G_{\iota}^T(f)\boldsymbol D^{-1}\boldsymbol B_{\boldsymbol \Omega}\|_2\geq t\:\:\: for\:\: all\:\: f\in \mathcal{U}\}.\nonumber\end{aligned}$$ holds with probability at most ${\epsilon/20}$ under the assumptions of Proposition\[prop1\]. For the third term, we can consider ${\boldsymbol \Psi=\boldsymbol \Phi}$ and $$\begin{aligned}
\boldsymbol \omega & =\boldsymbol P\boldsymbol D^{-1}\left(\boldsymbol G_{\iota}(f)-\frac{N-s}{N}\bar{\boldsymbol G}_{\iota}(f)\right)\nonumber\end{aligned}$$ where ${\boldsymbol P\in\mathbb{R}^{K\times 2K}}$ is a projection matrix which selects the first ${K}$ elements in a vector and ${\|\boldsymbol P\|=1}$. According to modified versions of lemmas 3.7 and 3.8 in [@fernandez2017demixing] with respect to Theorem\[theorem1\], we can write $$\begin{aligned}
& \|\boldsymbol P\boldsymbol D^{-1}\left(\boldsymbol G_{\iota}(f)-\frac{N-s}{N}\bar{\boldsymbol G}_{\iota}(f)\right)\|_2 \leq \nonumber\\
&\|\boldsymbol P\|\|\boldsymbol D^{-1}\|\|\boldsymbol G_{\iota}(f)-\frac{N-s}{N}\bar{\boldsymbol G}_{\iota}(f)\|_2\leq \nonumber\\
& 8\|\boldsymbol G_{\iota}(f)-\frac{N-s}{N}\bar{\boldsymbol G}_{\iota}(f)\|_2\leq C_{\mathcal{U}}\left(log\frac{\sqrt{L}N}{\epsilon}\right)^{-\frac{1}{2}}.\nonumber\end{aligned}$$ By setting ${t=\frac{10^{-2}}{8}}$ and applying (\[eqn35\]) and the union bound we have $$\begin{aligned}
\mathbb{P}\{\underset{f\in\mathcal{U}}{sup}\|(\boldsymbol G_{\iota}(f)-\frac{N-s}{N}&\bar{\boldsymbol G}_{\iota}(f))^T\boldsymbol D^{-1}\left[
\begin{array}{c}
\boldsymbol \Phi \\
\boldsymbol 0 \\
\end{array}
\right]\|_2\geq \frac{10^{-2}}{8}\}\nonumber\\ &\leq 4(L+1)|\mathcal{G}|e^{-\frac{{\frac{10^{-2}}{8}}^2}{8C_{\mathcal{U}}^2(log\frac{\sqrt{L}N}{\epsilon})^{-1}}}.\label{eqn38}\end{aligned}$$ Therefore, the event $$\begin{aligned}
\varepsilon_3 :=\{\|(\boldsymbol G_{\iota}(f)-\frac{N-s}{N}\bar{\boldsymbol G}_{\iota}(f))^T\boldsymbol D^{-1}&\left[
\begin{array}{c}
\boldsymbol \Phi \\
\boldsymbol 0 \\
\end{array}
\right]\|_2\geq \frac{10^{-2}}{8}\nonumber\\
&\:\:\: for\:\: all\:\: f\in \mathcal{U}\}\nonumber\end{aligned}$$ holds with probability at most ${\epsilon/20}$ under the assumptions of Proposition\[prop1\]. At last, one can bound the fourth term in (\[eqn34\]) by considering ${\boldsymbol \Psi=\boldsymbol \Phi}$ and $$\begin{aligned}
& \boldsymbol \omega=\frac{N-s}{N}\boldsymbol P\left(\boldsymbol D^{-1}-\frac{N}{N-s}\bar{\boldsymbol D}^{-1}\right)\bar{\boldsymbol G}_{\iota}(f)\nonumber\end{aligned}$$ Using lemmas 3.7 and 3.8 from [@fernandez2017demixing], we have $$\begin{aligned}
& \|\frac{N-s}{N}\boldsymbol P\left(\boldsymbol D^{-1}-\frac{N}{N-s}\bar{\boldsymbol D}^{-1}\right)\bar{\boldsymbol G}_{\iota}(f)\|_2\leq \nonumber\\
& \|\boldsymbol P\|\|\boldsymbol D^{-1}-\frac{N}{N-s}\bar{\boldsymbol D}^{-1}\|\|\bar{\boldsymbol G}_{\iota}(f)\|_2\leq \nonumber\\
& C_{\bar{\boldsymbol \nu}}\|\boldsymbol D^{-1}-\frac{N}{N-s}\bar{\boldsymbol D}^{-1}\|\leq C_{\mathcal{U}}\left(log\frac{\sqrt{L}N}{\epsilon}\right)^{-\frac{1}{2}}.\nonumber\end{aligned}$$ Now, by applying (\[eqn35\]) and the union bound and setting ${t=\frac{10^{-2}}{8}}$ one can write $$\begin{aligned}
\mathbb{P}\{\underset{f\in\mathcal{U}}{sup}\|\frac{N-s}{N}\bar{\boldsymbol G}_{\iota}(f)^T(\boldsymbol D^{-1}&-\frac{N}{N-s}\bar{\boldsymbol D}^{-1})\left[
\begin{array}{c}
\boldsymbol \Phi \\
\boldsymbol 0 \\
\end{array}
\right]\|_2\geq \frac{10^{-2}}{8}\}\nonumber\\ &\leq 4(L+1)|\mathcal{G}|e^{-\frac{({\frac{10^{-2}}{8}})^2}{8C_{\mathcal{U}}^2(log\frac{\sqrt{L}N}{\epsilon})^{-1}}}.\label{eqn39}\end{aligned}$$ Therefore, the event $$\begin{aligned}
\varepsilon_4 :=\{\|\frac{N-s}{N}\bar{\boldsymbol G}_{\iota}(f)^T(\boldsymbol D^{-1}-\frac{N}{N-s}\bar{\boldsymbol D}^{-1})&\left[
\begin{array}{c}
\boldsymbol \Phi \\
\boldsymbol 0 \\
\end{array}
\right]\|_2\geq \frac{10^{-2}}{8}\nonumber\\
&\:\:\: for\:\: all\:\: f\in \mathcal{U}\}\nonumber\end{aligned}$$ holds with probability at most ${\epsilon/20}$ under the assumptions of Proposition\[prop1\]. Thus, using (\[eqn351\]),(\[eqn37\]),(\[eqn38\]),(\[eqn39\]), and the triangle inequality we conclude that $$\begin{aligned}
\label{eqn40}
& \underset{f\in\mathcal{U}}{sup}\|\kappa^{\iota}\boldsymbol Q^{(\iota)}(f)-\kappa^{\iota}\bar{\boldsymbol Q}^{(\iota)}(f)\|_2\leq\frac{10^{-2}}{2}
\end{aligned}$$ holds with probability at least ${1-\epsilon/5}$ under the assumptions of Proposition\[prop1\]. Now, we extend the results to the continuous domain ${[0,1]}$ using Bernstein polynomial inequality[@schaeffer1941inequalities]. Consider ${f\in[0,1]}$ and ${f_g\in\mathcal{G}}$. Then, $$\begin{aligned}
& \|\kappa^{\iota}\boldsymbol Q^{(\iota)}(f)-\kappa^{\iota}\bar{\boldsymbol Q}^{(\iota)}(f)\|_2\leq\|\kappa^{\iota}\boldsymbol Q^{(\iota)}(f_g)-\kappa^{\iota}{\boldsymbol Q}^{(\iota)}(f)\|_2 \nonumber\\
&+\|\kappa^{\iota}\bar{\boldsymbol Q}^{(\iota)}(f_g)-\kappa^{\iota}{\boldsymbol Q}^{(\iota)}(f_g)\|_2+\|\kappa^{\iota}\bar{\boldsymbol Q}^{(\iota)}(f)-\kappa^{\iota}\bar{\boldsymbol Q}^{(\iota)}(f_g)\|_2. \nonumber
\end{aligned}$$ Now, consider the third term in the right side of the above inequality. We had ${\bar{\boldsymbol Q}^{(\iota)}(f)\in\mathbb{C}^{1\times L}}$ and for any ${\boldsymbol v\in\mathbb{C}^{1\times L}}$,${\|\boldsymbol v\|\leq\sqrt{L}\|\boldsymbol v\|_{\infty}}$. The ${j}$th entry of ${\bar{\boldsymbol Q}^{(\iota)}(f)}$ is $$\begin{aligned}
&|\kappa^{\iota}\bar{\boldsymbol Q}_j^{(\iota)}(f)| \leq |<\bar{\boldsymbol D}^{-1}\bar{\boldsymbol G}_{\iota}(f),\boldsymbol \Phi_{:,j}>| \nonumber\\
& \leq8\sqrt{K}\left(256\sqrt{K}\right)=CK\leq CN^2.\nonumber\end{aligned}$$ Next, take ${\kappa^{\iota}\bar{\boldsymbol Q}_j^{(\iota)}(f)}$ as a polynomial of ${z=e^{-i2\pi f}}$ with degree ${m}$ and apply the Bernstein polynomial inequality $$\begin{aligned}
& |\kappa^{\iota}\bar{\boldsymbol Q}_j^{(\iota)}(f)-\kappa^{\iota}\bar{\boldsymbol Q}_j^{(\iota)}(f_g)|\leq \nonumber\\
&|e^{-i2\pi f}-e^{-i2\pi f_g}|\underset{z}{sup}\left|\frac{d\kappa^{\iota}\bar{\boldsymbol Q}_j^{(\iota)}(z)}{dz}\right| \nonumber\\
& \leq |e^{-i\pi(f+f_g)}2sin(\pi(-f+f_g))|m\underset{f}{sup}|\kappa^{\iota}\bar{\boldsymbol Q}_j^{(\iota)}(f)| \nonumber\\
&\leq CN^3|f-f_g|.\nonumber\end{aligned}$$ Thus, $$\begin{aligned}
& \|\kappa^{\iota}\bar{\boldsymbol Q}^{(\iota)}(f)-\kappa^{\iota}\bar{\boldsymbol Q}^{(\iota)}(f_g)\|_2\leq\sqrt{L}\|\kappa^{\iota}\bar{\boldsymbol Q}^{(\iota)}(f)-\kappa^{\iota}\bar{\boldsymbol Q}^{(\iota)}(f_g)\|_{\infty} \nonumber\\
& \leq C\sqrt{L}N^3|f-f_g|.\nonumber\end{aligned}$$ The above calculations reveal that the grid size should be such that ${|f-f_g|\leq\frac{10^{-2}}{4C\sqrt{L}N^3}}$. Using the same arguments, one can obtain the same bound for ${ \|\kappa^{\iota}{\boldsymbol Q}^{(\iota)}(f)-\kappa^{\iota}{\boldsymbol Q}^{(\iota)}(f_g)\|_2}$. Combining the bove results with (\[eqn40\]) proves the lemma.
Proof of Lemma\[lemma3\]
------------------------
Consider ${A_{far}=[0,1]\backslash A_{near}}$ where ${\ A_{near}}$ is defined in lemma\[lemma3\]. We prove that ${\|\bar{\boldsymbol Q}(f)\|_2<0.99}$ in ${A_{far}}$. Next, it is shown that ${\|\boldsymbol Q(f)\|_2<1}$ in ${A_{near}}$. For ${\|\bar{\boldsymbol Q}(f)\|_2}$ we can write $$\begin{aligned}
&\|\bar{\boldsymbol Q}(f)\|_2\nonumber\\
&\leq \sum_{f_k\in\mathbb{T}}^{ }\|\boldsymbol \alpha_k\|_2|\bar{K}(f-f_k)|+\sum_{f_k\in\mathbb{T}}^{}\kappa\|\boldsymbol \beta_k\|_2|\bar{K}^{\prime}(f-f_k)| \nonumber\\
& \leq \|\boldsymbol \alpha\|_{\infty,2}\sum_{f_k\in\mathbb{T}}^{ }|\bar{K}(f-f_k)|+ \|\boldsymbol \beta\|_{\infty,2}\sum_{f_k\in\mathbb{T}}^{}\kappa|\bar{K}^{\prime}(f-f_k)|.\nonumber\end{aligned}$$ Using lemma H.10 from [@fernandez2017demixing], we have $$\begin{aligned}
& \sum_{j=1}^{K}\kappa^{\iota}|\bar{K}^{(\iota)}(f-f_j)|\leq127C_1+2.42C_2\nonumber\end{aligned}$$ for some properly chosen ${C_1}$ and ${C_2}$. Thus, $$\begin{aligned}
& \|\bar{\boldsymbol Q}(f)\|_2\leq(\|\boldsymbol \alpha\|_{\infty,2}+\|\boldsymbol \beta\|_{\infty,2})(127C_1+2.42C_2).\nonumber\end{aligned}$$ In the following we calculate upper bounds for ${\|\boldsymbol \alpha\|_{\infty,2}}$ and ${\|\boldsymbol \beta\|_{\infty,2}}$. Recall (\[eqn26\]) for the deterministic case. Using this equation we have $$\begin{aligned}
& \left[
\begin{array}{c}
\boldsymbol \alpha \\
\boldsymbol \beta \\
\end{array}
\right]=\left[
\begin{array}{c}
\boldsymbol I \\
\bar{\boldsymbol D}_2^{-1}\bar{\boldsymbol D}_1 \\
\end{array}
\right]\bar{\boldsymbol D}_3^{-1}\boldsymbol \Phi\nonumber\end{aligned}$$ where ${\bar{\boldsymbol D}_3\triangleq\bar{\boldsymbol D}_0+\bar{\boldsymbol D}_1\bar{\boldsymbol D}_2^{-1}\bar{\boldsymbol D}_1}$. According to lemma 4.1 from [@fernandez2016super] and the fact that ${\|\boldsymbol \Phi\|_{\infty,2}=1}$ $$\begin{aligned}
& \|\boldsymbol \alpha\|_{\infty,2}=\|\bar{\boldsymbol D}_3^{-1}\boldsymbol \Phi\|_{\infty,2}\leq1+2.37\times 10^{-2}\nonumber\end{aligned}$$ and $$\begin{aligned}
& \|\boldsymbol \beta\|_{\infty,2}\leq\|\bar{\boldsymbol D}_2^{-1}\bar{\boldsymbol D}_1\bar{\boldsymbol D}_3^{-1}\boldsymbol \Phi\|_{\infty,2} \leq \frac{4.247}{m}\times 10^{-2} .\nonumber\end{aligned}$$ Therefore, by proper choices of ${C_1}$ and ${C_2}$ we get $${\|\bar{\boldsymbol Q}(f)\|_2< 0.99\:\:\: for\:f\in A_{far}}$$. Now, to show that ${\|\boldsymbol Q(f)<1\|_2}$ in ${A_{near}}$, it is enough to show that the second derivative of ${\|\boldsymbol Q(f)<1\|_2}$ is negative in ${A_{near}}$. In a mathematical fashion, it is enough to prove the following inequality $$\begin{aligned}
& \frac{1}{2}\frac{d^2\|\boldsymbol Q(f)\|_2}{df^2} =\|\boldsymbol Q^{\prime}\|_2^2+Re\{\boldsymbol Q^{\prime \prime}\boldsymbol Q^H(d)\}<0.\label{eqn41}\end{aligned}$$ Now, we investigate each term in the above inequality. For the first term we can write $$\begin{aligned}
& \|\kappa\boldsymbol Q^{\prime}(f)\|_2^2=\|\kappa\boldsymbol Q^{\prime}(f)-\kappa\bar{\boldsymbol Q}^{\prime}(f)+\kappa\bar{\boldsymbol Q}^{\prime}(f)\|_2^2 \nonumber\\
& \leq 10^{-4}+2\times 10^{-2}\|\kappa \bar{\boldsymbol Q}^{\prime}(f)\|_2+\|\kappa\bar{\boldsymbol Q}^{\prime}(f)\|_2^2\nonumber\end{aligned}$$ Also, applying the kernel bounds of [@fernandez2016super] leads to $$\begin{aligned}
& \|\kappa \bar{\boldsymbol Q}^{\prime}(f)\|_2 \nonumber\\
&\leq\|\boldsymbol \alpha\|_{\infty,2}\sum_{k=1}^{K}\kappa|\bar{K}^{\prime}(f-f_k)|+\|\boldsymbol \beta\|_{\infty,2}\sum_{k=1}^{K}\kappa^2|\bar{K}^{\prime \prime}(f-f_k)| \nonumber\\
&\leq 1.0237\times 2.409\times 10^{-2}+\frac{4.247\times10^{-2}}{m}(0.087)\leq0.0247\nonumber\end{aligned}$$ where the last inequality is achieved using ${m\geq10^3}$. The second term of (\[eqn41\]) can be represented as follow $$\begin{aligned}
&Re\left\{\kappa^2\boldsymbol Q^{\prime \prime}(f)\boldsymbol Q^H(f)\right\}=Re\left\{\kappa^2(\boldsymbol Q^{\prime \prime}(f)-\bar{\boldsymbol Q}^{\prime \prime}(f))\boldsymbol Q^H(f)\right\} \nonumber\\
& +Re\left\{\kappa^2\bar{\boldsymbol Q}^{\prime \prime}(f)(\boldsymbol Q(f)-\bar{\boldsymbol Q}(f))^H\right\}+Re\left\{\kappa^2\bar{\boldsymbol Q}^{\prime \prime}(f)\bar{\boldsymbol Q}^H(f)\right\} \nonumber\\
& \leq 0.0101+ 0.01+Re\left\{\kappa^2\bar{\boldsymbol Q}^{\prime \prime}(f)\bar{\boldsymbol Q}^H(f)\right\}\nonumber\end{aligned}$$ Now, we investigate ${\kappa^2\bar{\boldsymbol Q}^{\prime \prime}(f)\bar{\boldsymbol Q}^H(f)}$. According to (\[eqn28\]), we can write $${\kappa^2\bar{\boldsymbol Q}^{\prime \prime}(f)\bar{\boldsymbol Q}^H(f)=\kappa^2\bar{Q}^{\prime \prime}(f)\boldsymbol b^H\boldsymbol b\bar{ Q}^{\ast}(f)=\kappa^2\bar{Q}^{\prime \prime}(f)\bar{ Q}^{\ast}(f)}$$ which is a scalar value. Also, note that $$\begin{aligned}
& \kappa^2Re \left\{\bar{Q}^{\prime \prime}(f)\bar{ Q}^{\ast}(f)\right\}=\kappa^2\left(\bar{Q}_R^{\prime\prime}(f)\bar{Q}_R(f)+|\bar{Q}_I^{\prime\prime}(f)||\bar{Q}_I|\right) \nonumber\\
& \leq \left(-0.8915\times 2.015+0.0474\times 2.555 \right)\leq-1.6752.\nonumber\end{aligned}$$ Thus, $$\begin{aligned}
& \frac{\kappa^2}{2}\frac{d^2\|\boldsymbol Q(f)\|_2}{df^2}\leq-1.6752+0.0201+12.01\times 10^{-4}<0\nonumber\end{aligned}$$ and the proof is complete.
[^1]: H.Hezave, S. Daei and M.H. Kahaei are with the School of Electrical Engineering, Iran University of Science & Technology.
|
---
author:
- 'Jordan A. Barr$^{1}$, Scott P. Beckman$^{1}$, and Takeshi Nishimatsu$^{2}$[^1]'
bibliography:
- 'ElastoLeadv1.bib'
title: 'Elastocaloric Response of PbTiO$_3$ Predicted from a First-Principles Effective Hamiltonian'
---
=1
Introduction \[introduction\]
=============================
Solid-state caloric effects provide a promising approach to future refrigeration technologies. The electrocaloric [@ECERochelle; @ReviewECEScott; @1956ECE; @1961ECE], magnetocaloric [@RecentMagneto; @FirstMagneto], and barocaloric effects [@BaroNiMnIn; @BarocaloricThermalExpansionPhase] produce a temperature change due to entropic changes induced by the application of an electric field, magnetic field, and pressure, respectively. The application of uniaxial stress to a ferroelectric material affects the spontaneous polarization and produces an adiabatic temperature change. This is called the elastocaloric effect[@ElastocaloricMagnetocaloricShapeMemory; @GiantElastocaloricBaSr; @MulticaloricUSF; @ElastoNiTiWires; @ElastoMartensiticTransition].
Here, a first-principles effective Hamiltonian model implemented within a molecular dynamics (MD) framework is used to predict the elastocaloric response of PbTiO$_{3}$. Following the work of Lisenkov *et al*.[@MulticaloricUSF], the elastocaloric response of PbTiO${_3}$ is examined for tensile uniaxial loads ranging from 0 to $-2.0$ GPa and temperatures ranging from 300 to 1000 K. The results of this study will be compared with those reported in the literature.
Methods \[methods\]
===================
The effective Hamiltonian used is $$\begin{gathered}
\label{eq:Effective:Hamiltonian}
H^{\rm eff}%(\{\bm{u}\},\{\bm{w}\}, \eta_1,\cdots\!,\eta_6)
= \frac{M^*_{\rm dipole}}{2} \sum_{\bm{R},\alpha}\dot{u}_\alpha^2(\bm{R})
+ \frac{M^*_{\rm acoustic}}{2}\sum_{\bm{R},\alpha}\dot{w}_\alpha^2(\bm{R})\\
+ V^{\rm self}(\{\bm{u}\})+V^{\rm dpl}(\{\bm{u}\})+V^{\rm short}(\{\bm{u}\})\\
+ V^{\rm elas,\,homo}(\eta_1,\dots\!,\eta_6)+V^{\rm elas,\,inho}(\{\bm{w}\})\\
+ V^{\rm coup,\,homo}(\{\bm{u}\}, \eta_1,\cdots\!,\eta_6)+V^{\rm coup,\,inho}(\{\bm{u}\}, \{\bm{w}\}).\end{gathered}$$ Here, the collective atomic motion is coarse-grained by the local soft mode vectors $\bm{u}(\bm{R})$ and local acoustic displacement vectors $\bm{w}(\bm{R})$ of each unit cell at $\bm{R}$ in a simulation supercell. $\eta_1,\dots,\eta_6$ are the six components of homogeneous strain in Voigt notation. $\frac{M^*_{\rm dipole}}{2} \sum_{\bm{R},\alpha}\dot{u}_\alpha^2(\bm{R})$ and $\frac{M^*_{\rm acoustic}}{2}\sum_{\bm{R},\alpha}\dot{w}_\alpha^2(\bm{R})$ are the kinetic energies possessed by the local soft modes and local acoustic displacement vectors along with their effective masses of $M^*_{\rm dipole}$ and $M^*_{\rm acoustic}$, $V^{\rm self}(\{\bm{u}\})$ is the local-mode self-energy, $V^{\rm dpl}(\{\bm{u}\})$ is the long-range dipole-dipole interaction, $V^{\rm short}(\{\bm{u}\})$ is the short-range interaction between local soft modes, $V^{\rm elas,\,homo}(\eta_1,\dots,\eta_6)$ is the elastic energy from homogeneous strains, $V^{\rm elas,\,inho}(\{\bm{w}\})$ is the elastic energy from inhomogeneous strains, $V^{\rm coup,\,homo}(\{\bm{u}\}, \eta_1,\dots,\eta_6)$ is the coupling between the local soft modes and the homogeneous strain, and $V^{\rm coup,\,inho}(\{\bm{u}\}, \{\bm{w}\})$ is the coupling between the soft modes and the inhomogeneous strains. Details of this Hamiltonian are explained in Refs. . Additionally, to investigate the effects from stress, we use the enthalpy $\mathcal{H}=H^{\rm eff}+N a_0^3\,\bm{\sigma}\cdot \bm{\eta}$, where $N=L_x\times L_y\times L_z$ is the supercell size and $a_0$ is the unit cell length; therefore, $N a_0^3$ is the supercell volume and $\bm{\sigma}$ is the six components of stress. In this study, we apply uniaxial tensile stress to the system along the $z$-direction. It is implemented in the MD framework, and the MD simulation program is called `feram`. `feram` is distributed as free software under the conditions described in the GNU General Public License from its website[@feram]. Examples of the input files are packaged within the source code under the `feram-0.22.05/src/28example-PbTiO3-elastocaloric-770K/` directory. The model parameters for PbTiO$_{3}$ are determined semi-empirically in a previous work[@Waghmare:R:1997PRB] and adopted for `feram` in Ref. .
Using the above parameters, the results of heating-up and cooling-down test MD simulations for a supercell of $N = 16 \times 16 \times 16$ are shown in Fig. \[heating-cooling\]. From the temperature $T$ dependences of the averaged lattice constants \[shown in Fig. \[heating-cooling\](a)\], a tetragonal-to-cubic ferroelectric-to-paraelectric phase transition is clearly observed upon heating-up to 672 K. During the cooling-down simulation, $90^\circ$ ferroelectric domains are formed at 630 K and are frozen at a low temperature, as described in Ref. . These two transition temperatures are largely dependent on supercell size with a slight dependence on the initial random configurations of $\{\bm{u}\}$. However, their average of 653 K is in good agreement with those obtained in earlier Monte Carlo[@Waghmare:R:1997PRB] and MD[@LeadParameters] simulations, in which the same set of parameters were used. This is lower than the experimental value $T_{\rm C}=763$ K[@SHIRANE:H:S:PHYSICALREVIEW:80:p1105-1106:1950]. This disagreement between simulations and experiments is unavoidable, owing to the errors in the total energy of the first-principles calculations, which is around 10 meV per unit cell. In Fig. \[heating-cooling\](b), the temperature dependence of the total energy per unit cell is plotted for heating-up and cooling-down test simulations. At the transition temperatures, we observe jumps in the total energy, i.e., the latent heat. In Figs. \[heating-cooling\](c) and \[heating-cooling\](d), the relative dielectric constant tensor computed from the fluctuations of the dipoles is plotted. This is defined as $$\epsilon_{\alpha\beta} =
\frac{1}{V \epsilon_0 k_{\rm B} T}
[\langle p_\alpha p_\beta \rangle - \langle p_\alpha \rangle \langle p_\beta \rangle],
\label{eq:epsilon}$$ where $V$ is the volume of the supercell, $\epsilon_0$ is the absolute dielectric constant of vacuum, $k_{\rm B}$ is the Boltzmann constant, $p_\alpha$ is the $\alpha (= x,y,z)$ component of the total electric dipole moment in the supercell, $\bm{p} = Z^*\sum_{\bm{R}}\bm{u}(\bm{R})$, $Z^*$ is the Born effective charge associated with a soft mode vector, and the angle brackets $\langle\rangle$ denote the statistical time average[@PhysRevB.75.014111]. Divergence in the dielectric constants at the transition temperatures can be clearly observed, although it is slightly underestimated compared with the experimentally observed value[@Remeika197037].
Temperature-independent elastic coefficients ($C_{11} = 302$ GPa, $C_{12} = 132$ GPa, $C_{44} = 351$ GPa) determined from the cubic structure are used, although they slightly depend on temperature even with this Hamiltonian of Eq. (\[eq:Effective:Hamiltonian\]) because strains $\eta_1,\dots,\eta_6$ and dipoles $\{\bm{u}\}$ couple through $V^{\rm coup,\,homo}(\{\bm{u}\}, \eta_1,\dots,\eta_6)$.
![(Color online) Heating-up and cooling-down MD simulations of bulk PbTiO$_3$. (a) Averaged lattice constants. (b) Total energy per unit cell. Three components of relative dielectric constant are also plotted for (c) heating-up and (d) cooling-down simulations. Thermal hysteresis can be seen. In (a), experimentally observed lattice constants[@SHIRANE:H:S:PHYSICALREVIEW:80:p1105-1106:1950] are plotted with black solid lines.[]{data-label="heating-cooling"}](a_E_susceptibility){width="0.95\columnwidth"}
As shown in Fig. \[illustrations\], the simulation procedure for determining the elastocaloric response of PbTiO${_3}$ is similar to that used for the “direct” prediction of the electrocaloric effect presented in Ref. . A supercell size of $N = 64 \times 64 \times 64$ is used and is thermalized for 50,000 time steps in a canonical ensemble at the constant initial temperature $T_{\rm initial}$ and constant applied stress. A single-domain $+z$-polarized initial configuration for the first thermalization MD is generated randomly with certain averages and deviations for $\{\bm{u}\}$: $\langle u_x \rangle = \langle u_y \rangle = 0$, $\langle u_z \rangle = 0.33~{\rm \AA}$, $\langle u_x^2 \rangle - \langle u_x \rangle^2 =
\langle u_y^2 \rangle - \langle u_y \rangle^2 = (0.045~{\rm \AA})^2$, and $\langle u_z^2 \rangle - \langle u_z \rangle^2 = (0.021~{\rm \AA})^2$. Once thermalized, the system is switched from being held at a constant temperature to being isolated as a microcanonical ensemble. The mechanical load is removed and the system is allowed to equilibrate for 40,000 time steps. Once equilibrated, the system’s final temperature $T_{\rm final}$ is determined by averaging the acoustic and dipole kinetic energies for 10,000 time steps. The time step for this simulation is 2 fs. One of the advantages of this “direct” prediction method is that the temperature and external-field dependences of heat capacity and latent heat are implicitly and automatically included in the simulations, whereas in the “indirect” method, an experimentally observed heat capacity must be used in the entire temperature and external field ranges, as described in Ref. . The temperature ranges from 300 to 1000 K, incremented with a step size of 1 K, and the applied uniaxial stress ranges from 0 to $-2.0$ GPa, incremented with a step size of $-0.2$ GPa.
![(Color online) (a) Schematic illustration of elastocaloric cooling. (b) Procedure of direct simulation of the elastocaloric effect.[]{data-label="illustrations"}](elastocaloric){width="0.6\columnwidth"}
Results and Discussion\[Results\]
=================================
The elastocaloric response $\Delta T_{\rm raw} = T_{\rm final} - T_{\rm initial}$ of PbTiO$_3$ is presented in Fig. \[ourPTO\](a). Scaling from $\Delta T_{\rm raw}$ to $\Delta T_{\rm corrected} = \frac{2}{5} \Delta T_{\rm raw}$ must be employed to account for the reduced degrees of freedom due to coarse graining, as discussed in Ref. . In Fig. \[ourPTO\](b), polarizations along the $z$-direction before and after the release of the load of $\sigma_3=-1.6$ GPa are compared, i.e., $P_z(T_{\rm initial},\,\sigma_3=-1.6~{\rm GPa})$ and $P_z(T_{\rm final},\,\sigma_3=0)$ are compared, respectively. We redefine the transition temperature under uniaxial stress as $T'_{\rm C}(\sigma_3)$. For $\sigma_3=-1.6$ GPa, $T'_{\rm C}=917$ K. Transformation from $T'_{\rm C}$ is indicated with a dashed blue arrow. Above a certain temperature ($T_{\rm onset}$), there is a temperature range $T_{\rm onset} < T_{\rm initial} \leq T'_{\rm C}$ in which one can obtain a large elastocaloric effect. Transformation from $T_{\rm onset}$ is indicated by a dotted magenta arrow. It can be seen that below $T_{\rm onset}$ ($T_{\rm initial}\leq T_{\rm onset}$), transformation from switching off the uniaxial tensile stress is from an elongated ferroelectric polarized state to a normal ferroelectric polarized state. Between $T_{\rm onset}$ and $T'_{\rm C}$, i.e., $T_{\rm onset} < T_{\rm initial} \leq T'_{\rm C}$, the transformation changes from a stress-enhanced ferroelectric polarized state to a paraelectric nonpolar state, resulting in a large elastocaloric response. Just above $T_{\rm onset}$, a maximum $|\Delta T|$ is obtained and its transformation is indicated by a solid red arrow in Fig. \[ourPTO\](b). Above $T'_{\rm C}$ ($T'_{\rm C}<T_{\rm initial}$), even under the uniaxial tensile stress load, the system remains paraelectric and consequently $|\Delta T|=0$.
In Figs. \[ourPTO\](c)–\[ourPTO\](e), $P_z(T_{\rm initial}, \sigma_3\leq 0)$ and $P_z(T_{\rm final}, \sigma_3=0)$ are plotted also for loads of $\sigma_3=-0.8$, $-0.4$, and $0.0$ GPa. In Fig. \[ourPTO\](c), it is observed that with a load of $-0.8$ GPa, the effective temperature range $T_{\rm onset} < T_{\rm initial} \leq T'_{\rm C}$ becomes narrower than that of $-1.6$ GPa. In Fig. \[ourPTO\](d), it can be seen that the initial uniaxial tensile load of $-0.4$ GPa is not sufficiently large to induce a ferroelectric-to-paraelectric transformation. Therefore, we cannot define $T_{\rm onset}$ for loads of $0.0 < \sigma_3 < -0.4$ GPa. In the case of zero load, in Fig. \[ourPTO\](e), the accuracy of our MD simulations ($\Delta T\equiv 0$) and the simulated and underestimated phase transition temperature of $T_{\rm C}=640$ K under zero pressure are shown. As anticipated, the greater the uniaxial loading, the greater the induced temperature change $|\Delta T|$, and for a loading of $-2.0$ GPa, a temperature change of $-43$ K is predicted.
![(Color online) (a) Simulated elastocaloric effect ($\Delta T$) in PbTiO$_3$ as a function of initial temperature ($T_{\rm initial}$). The applied uniaxial stress ranges from 0 to $-2.0$ GPa. $\Delta T$ is scaled from $\Delta T_{\rm raw}$ to $\Delta T_{\rm corrected}$ by accounting for the reduced degrees of freedom, as discussed in Ref. . (b) Polarization along the $z$-axis both before \[$P_z(T_{\rm initial})$\] (gray solid line) and after \[$P_z(T_{\rm final})$\] (gray dotted line) the release of load of $-1.6$ GPa. (c) $P_z(T_{\rm initial})$ (cyan solid line) and $P_z(T_{\rm final})$ (cyan dotted line) of load of $-0.8$ GPa. $P_z(T_{\rm final})$ after 990,000 MD time steps (thin black chain line). (d) $P_z(T_{\rm initial})$ (green solid line) and $P_z(T_{\rm final})$ (green dashed line) of load of $-0.4$ GPa. (d) $P_z(T_{\rm initial})$ (green solid line) and $P_z(T_{\rm final})$ (green dashed line) of zero load. In (b)–(d), transformations that give $T_{\rm onset}$, ${\rm max}|\Delta T|$, and $T'_{\rm C}$ are indicated by dotted magenta, solid red, and dashed blue arrows, respectively.[]{data-label="ourPTO"}](elastoPolarization){width="1.05\columnwidth"}
In Fig. \[Comparison\], plots show ${\rm max}|\Delta T_{\rm corrected}|$, and $T_{\rm onset}$ and $T'_{\rm C}$ under different applied loads. It can be seen that $T'_{\rm C}$ linearly depends on applied load. $T_{\rm onset}$ depends on applied load nearly linearly in $-0.6<\sigma_3<-2.0$, but less steeply than $T'_{\rm C}$.
![(Color online) Plots of ${\rm max}|\Delta T_{\rm corrected}|$, $T_{\rm onset}$, and $T'_{\rm C}$ vs the different initially applied uniaxial tensile stresses. Data are connected with solid red, dotted magenta, and dashed blue lines, respectively.[]{data-label="Comparison"}](T1){width="0.7\columnwidth"}
$T_{\rm onset}$ is also found to depend on the period of equilibration. Between $T_{\rm C}$ and $T_{\rm onset}$ ($T_{\rm C}<T_{\rm initial}\leq T_{\rm onset}$), when a uniaxial tensile stress is applied and then released, the system stays in a ferroelectric state and does not transform into a paraelectric state. In other words, the system [*remembers*]{} the strength of the stress applied. This is confirmed with a longer equilibration of 990,000 time steps instead of the 40,000 shown in Fig. \[illustrations\]. As indicated by black chain lines in Figs. \[ourPTO\](a) and \[ourPTO\](c), $T_{\rm onset}$ with longer equilibration becomes 736 K, whereas that of 40,000 was 744 K, i.e., the system [*forgets*]{} the strength of the stress applied. Therefore, the stronger load and the shorter period of equilibration result in a higher $T_{\rm onset}$.
In Contrast, in Fig. \[OffOn\], we also perform “heating” simulations with switching-on of uniaxial stress in which the system is firstly thermalized under zero stress and then $\Delta T$ is measured under switched-on uniaxial tensile stresses. Zigzag structures at the final temperatures are observed. A vertical cross section and a horizontal slice of a final state indicated by a “+” mark in Fig. \[OffOn\] are shown in Figs. \[CrossSection\] and \[slice\], respectively. The supercell is divided into the $+z$ and $-z$ domains. It can be understood that the zigzag structures arise from the existence and nonexistence of domain structures. In the elastocaloric effect, domain structures may be formed more easily than in the electrocaloric effect, because there is no significant $+z$- nor $-z$-direction in the uniaxial stress, but there is in the external electric field. It is suggested that the formation of domain structures may cause some degradations in effectiveness in applications of the elastocaloric effect. Note also that when comparing Figs. \[ourPTO\](a) and \[OffOn\], the onset temperature is constant in switching-on “heating” simulations because the simulations are started from zero stress.
![(Color online) Simulated switching-on elastocaloric effect with positive $\Delta T$ in PbTiO$_3$ as functions of initial temperature, $T_{\rm initial}$. The switched-on uniaxial stress ranges from 0 to $-1.6$ GPa. $\Delta T$ is scaled from $\Delta T_{\rm raw}$ to $\Delta T_{\rm corrected}$. Zigzag structures in final temperatures are observed. A vertical cross section and a horizontal slice of a final state indicated with a “+” mark for a $-0.8$ GPa simulation are shown in Figs. \[CrossSection\] and \[slice\], respectively.[]{data-label="OffOn"}](off-on){width="0.99\columnwidth"}
![(Color online) Vertical cross section of a final state indicated with a “+” mark in Fig. \[OffOn\]. Dipole moments of each site are projected onto the $yz$-plane and indicated with arrows. The arrows are colored with red or blue if each dipole has $+z$ or $-z$ component, respectively.[]{data-label="CrossSection"}](CrossSection){width="0.99\columnwidth"}
![(Color online) Horizontal slice of a final state indicated with a “+” mark in Fig. \[OffOn\]. $+z$-polarized and $-z$-polarized sites are denoted by red $\square$ and blue $\blacksquare$, respectively.[]{data-label="slice"}](slice){width="0.86\columnwidth"}
The results presented here can be compared with those presented by Lisenkov *et al*. in their Fig. 1(b) [@MulticaloricUSF]. Note that their $\Delta T$ is positive because they switched on the uniaxial stress from zero stress. For the same reason, their onset initial temperature, which gives ${\rm max}|\Delta T|$, is always $T_{\rm C}$. We have carried out [*switching-off*]{} time-dependent MD simulations and found the applied-stress and equilibration-period dependences of $T_{\rm onset}$ because we consider that hysteretic behavior is important for the cooling application of the elastocaloric effect.
Furthermore, whereas Lisenkov *et al*. reported a continuous linear increase in ${\rm max}|\Delta T|$ as stronger stresses are applied and a maximum of approximately $+35$ K for a tensile load of $-2.0$ GPa, our results of initial stress dependence of ${\rm max}|\Delta T|$ are not continuous at around $-0.5$ GPa. Our MD simulation for a tensile load of $-2.0$ GPa results in ${\rm max}|\Delta T_{\rm corrected}|=|-43|$ K.
Finally, the shapes of the two $\Delta T$ vs $T$ plots differ in the high-temperature regime. Both models have a ${\rm max}|\Delta T|$ that increases with loading, but the results here show a sharper drop in $|\Delta T|$ at $T'_{\rm C}$, although this difference might be due to differences in the implementations, not the underlying physics. In Fig. \[Comparison\], it is observed that $T'_{\rm C}$ increases linearly with increased loading to a temperature of 1000 K, and presumably above, whereas Ref. states that the elastocaloric response disappears for temperatures above 890 K.
Summary\[Summary\]
==================
Note that these simulations are very ideal and unrealistic ones. For example, applying such huge tensile uniaxial stresses is difficult and the phase transition of pure PbTiO$_3$ would cause cracks in a crystal experimentally. However, these ideal simulations suggest that, for example, elastocaloric cooling has the largest effect when a ferroelectric-to-paraelectric phase transition occurs.
In summary, a first-principles effective Hamiltonian is used in a molecular dynamics simulation to study the elastocaloric effect in PbTiO$_3$. The results show that for a modest loading of around $-0.5$ GPa, a thermal response of around $-25$ K can be achieved, but for a large load of around $-2.0$ GPa, the thermal response can be as large as $-44$ K.
The onset temperature $T_{\rm onset}$ and the termination temperature, $T'_{\rm C}$, are identified as the temperatures bracketing the temperature range where the elastocaloric effect is greatest. $T'_{\rm C}$ is found to scale linearly with initial load, whereas $T_{\rm onset}$ has a less steep linearity. Although increasing the initial stress widens the window of temperatures continuously, the initial stress dependence of $\Delta T$ becomes smaller for stresses stronger than $-0.5$ GPa.
The formation of domain structures is observed in switching-on “heating” simulations and it is suggested that the formation of domain structures may cause some malfunctions in applications of the elastocaloric effect.
The results here are in qualitative agreement with those reported in Ref. , which were prepared using an effective Hamiltonian in a Monte Carlo model; however, there are physically significant differences including temperature and applied-stress dependences of $\Delta T$. There is no easy explanation for these differences, and this requires future investigation.
Acknowledgments {#acknowledgments .unnumbered}
===============
The work of JAB and SPB was supported by the US National Science Foundation (NSF) through grant No. DMR-1105641. The NSF acknowledged for sponsoring JAB’s travel to Tohoku University, which was provided through grant No. DMR-1037898. The work of TN was supported in part by JSPS KAKENHI Grant Number 25400314. This work was also supported in part by the Strategic Programs for Innovative Research (SPIRE), MEXT, and the Computational Materials Science Initiative (CMSI), Japan. The computational resources were provided by the Center for Computational Materials Science, Institute for Materials Research (CCMS-IMR), Tohoku University. We thank the staff at CCMS-IMR for their constant effort. This research was also conducted using the Fujitsu PRIMEHPC FX10 System (Oakleaf-FX, Oakbridge-FX) in the Information Technology Center, The University of Tokyo.
[^1]: E-mail: [email protected]
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---
abstract: 'Manevitz and Weinberger proved that the existence of faithful $K$-Lipschitz $\mathbb{Z}/n\mathbb{Z}$-actions implies the existence of faithful $K$-Lipschitz $\mathbb{Q}/\mathbb{Z}$-actions. The $\mathbb{Q}/\mathbb{Z}$-actions were constructed from suitable actions of a sufficiently large hyperfinite cyclic group $\prescript{\ast}{}{\mathbb{Z}}/\gamma\prescript{\ast}{}{\mathbb{Z}}$ in the sense of nonstandard analysis. In this paper, we modify their construction, and prove that the existence of $\varepsilon$-faithful $K$-Lipschitz $G_{\lambda}$-actions implies the existence of $\varepsilon$-faithful $K$-Lipschitz $\varinjlim G_{\lambda}$-actions. In a similar way, we generalise Manevitz and Weinberger’s result to injective direct limits of torsion groups.'
address: |
Research Institute for Mathematical Sciences\
Kyoto University\
Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan
author:
- Takuma Imamura
bibliography:
- '\\string"A nonstandard construction of direct limit group actions\\string".bib'
title: A nonstandard construction of direct limit group actions
---
Introduction
============
Let $M$ be a compact connected manifold with a metric. Using nonstandard analysis, Manevitz and Weinberger [@MW96] proved that if $M$ admits a faithful $K$-Lipschitz action of the finite cyclic group $\mathbb{Z}/n\mathbb{Z}$ for each $n\in\mathbb{Z}_{+}$, then $M$ also admits a faithful $K$-Lipschitz action of the rational circle group $\mathbb{Q}/\mathbb{Z}$. Their proof requires no advanced knowledge of transformation group theory except for Newman’s theorem, although it requires a bit of nonstandard analysis. The basic idea of their proof is as follows: Let $\gamma$ be an infinite hyperinteger that is divisible by any nonzero integer (e.g. the factorial $\omega!$ of an infinite positive hyperinteger $\omega$). Then $\mathbb{Q}/\mathbb{Z}$ can be embedded into the hyperfinite cyclic group $\prescript{\ast}{}{\mathbb{Z}}/\gamma\prescript{\ast}{}{\mathbb{Z}}$ by identifying $\left[k/n\right]\in\mathbb{Q}/\mathbb{Z}$ with $\left[k\left(\gamma/n\right)\right]\in\prescript{\ast}{}{\mathbb{Z}}/\gamma\prescript{\ast}{}{\mathbb{Z}}$. By the transfer principle, the nonstandard extension $\prescript{\ast}{}{M}$ admits an internal faithful $K$-Lipschitz action of $\prescript{\ast}{}{\mathbb{Z}}/\gamma\prescript{\ast}{}{\mathbb{Z}}$. By restricting the domain and by taking its standard part, we obtain the desired $\mathbb{Q}/\mathbb{Z}$-action on $M$. The faithfulness of the resulting action follows from Newman’s theorem in the version of Bredon [@Bre72 9.6 Corollary]. However, their proof contains an error involving the use of the (downward) transfer principle. Fortunately, their proof can be corrected, as we shall see below.
The aim of this paper is to generalise Manevitz and Weinberger’s result. Note here that $\mathbb{Q}/\mathbb{Z}$ is isomorphic to the (injective) direct limit of $\mathbb{Z}/n\mathbb{Z}\ \left(n\in\mathbb{Z}_{+}\right)$, where the index set $\mathbb{Z}_{+}$ is ordered by the divisibility relation. It is natural to attempt to generalise their result to direct limits of groups. In , we prove that if $M$ admits an $\varepsilon$-faithful $K$-Lipschitz $G_{\lambda}$-action for each $\lambda$, then $M$ also admits an $\varepsilon$-faithful $K$-Lipschitz $\varinjlim G_{\lambda}$-action. In we generalise their result to injective direct limits of torsion groups. We obtain a correct proof of their result as a by-product.
We refer to Robinson [@Rob66] for nonstandard analysis. We work within a sufficiently saturated nonstandard universe. Actually we only need the enlargement property.
\[sec:Direct-limit-group-actions\]Direct limit group actions
============================================================
Let $G$ be a group, $M$ a metric space, and $\varepsilon,K>0$. Suppose $G$ acts on $M$. The action is said to be *faithful* if for each $g\in G$ other than the unit, $d_{M}\left(x,gx\right)>0$ for some $x\in M$. The action is said to be *$\varepsilon$-faithful* if for each $g\in G$ other than the unit, $d_{M}\left(x,gx\right)\geq\varepsilon$ for some $x\in M$. The action is said to be *$K$-Lipschitz* if $d_{M}\left(gx,gy\right)\leq Kd_{M}\left(x,y\right)$ for all $g\in G$ and all $x,y\in M$.
\[thm:main-theorem\]Let $M$ be a compact metric space. Let $\left(G_{\lambda},i_{\lambda\mu},\Lambda\right)$ be a direct system of groups. If there exists an $\varepsilon$-faithful $K$-Lipschitz $G_{\lambda}$-action on $M$ for each $\lambda\in\Lambda$, then there exists an $\varepsilon$-faithful $K$-Lipschitz $\varinjlim G_{\lambda}$-action on $M$.
Let us start to prove . Denote $G=\varinjlim G_{\lambda}$. For each $\lambda\in\Lambda$ let $i_{\lambda}\colon G_{\lambda}\to G$ be the canonical homomorphism. By the saturation principle, fix an element $\gamma\in\prescript{\ast}{}{\Lambda}$ which is an upper bound of $\Lambda$ in $\prescript{\ast}{}{\Lambda}$.
\[lem:Embedding-Lemma\]There exists an embedding $j\colon G\to\prescript{\ast}{}{G_{\gamma}}$ such that the following diagram commutes: $$\xymatrix{ & G\ar[dr]^{j}\\
G_{\lambda}\ar[ur]^{i_{\lambda}}\ar[rr]^{\prescript{\ast}{}{i_{\lambda\gamma}}} & & \prescript{\ast}{}{G_{\gamma}}
}$$
Given $g\in G$, find a $g_{\lambda}\in G_{\lambda}$ such that $g=i_{\lambda}\left(g_{\lambda}\right)$, and then define $j\left(g\right)=\prescript{\ast}{}{i_{\lambda\gamma}}\left(g_{\lambda}\right)$. Clearly $j$ commutes the above diagram.
[Well-definedness]{}
: Suppose $g_{\mu}\in G_{\mu}$ is another one such that $g=i_{\mu}\left(g_{\mu}\right)$ holds. Since $i_{\lambda}\left(g_{\lambda}\right)=g=i_{\mu}\left(g_{\mu}\right)$, $i_{\lambda\nu}\left(g_{\lambda}\right)=i_{\mu\nu}\left(g_{\mu}\right)$ holds for some standard $\nu\geq\lambda,\mu$. Since $\nu\leq\gamma$, $\prescript{\ast}{}{i_{\lambda\gamma}}\left(g_{\lambda}\right)=\prescript{\ast}{}{i_{\mu\gamma}}\left(g_{\mu}\right)$ by upward transfer.
[Structure-preservation]{}
: Let $e_{\lambda}\in G_{\lambda}$ be the unit of some $G_{\lambda}$. Since $i_{\lambda}$ is homomorphic, $e=i_{\lambda}\left(e_{\lambda}\right)$ is the unit of $G$. By definition $j\left(e\right)=\prescript{\ast}{}{i_{\lambda\gamma}}\left(e_{\lambda}\right)$. By upward transfer, $\prescript{\ast}{}{i_{\lambda\gamma}}$ is homomorphic, so $e_{\gamma}=\prescript{\ast}{}{i_{\lambda\gamma}}\left(e_{\lambda}\right)$ is the unit of $\prescript{\ast}{}{G_{\gamma}}$.
Next let $g,h\in G$. Choose $g_{\lambda},h_{\lambda}\in G_{\lambda}$ such that $g=i_{\lambda}\left(g_{\lambda}\right)$ and $h=i_{\lambda}\left(h_{\lambda}\right)$. Then $gh=i_{\lambda}\left(g_{\lambda}h_{\lambda}\right)$. By definition and by upward transfer $j\left(gh\right)=\prescript{\ast}{}{i_{\lambda\gamma}}\left(gh\right)=\prescript{\ast}{}{i_{\lambda\gamma}}\left(g\right)\prescript{\ast}{}{i_{\lambda\gamma}}\left(h\right)=j\left(g\right)j\left(h\right)$.
[Injectivity]{}
: For $g,h\in G$ suppose that $j\left(g\right)=j\left(h\right)$. Choose $g_{\lambda},h_{\lambda}\in G_{\lambda}$ such that $g=i_{\lambda}\left(g_{\lambda}\right)$ and $h=i_{\lambda}\left(h_{\lambda}\right)$. Then, $\prescript{\ast}{}{i_{\lambda\gamma}}\left(g_{\lambda}\right)=j\left(g\right)=j\left(h\right)=\prescript{\ast}{}{i_{\lambda\gamma}}\left(h_{\lambda}\right)$ by definition. Hence “there is a $\mu\in\prescript{\ast}{}{\Lambda}$ such that $\prescript{\ast}{}{i_{\lambda\mu}}\left(g_{\lambda}\right)=\prescript{\ast}{}{i_{\lambda\mu}}\left(h_{\lambda}\right)$”. By downward transfer, “there is a $\mu\in\Lambda$ such that $i_{\lambda\mu}\left(g_{\lambda}\right)=i_{\lambda\mu}\left(h_{\lambda}\right)$”. Therefore $g=h$.
There exists an internal $\varepsilon$-faithful $K$-Lipschitz action $\Phi\colon\prescript{\ast}{}{G_{\gamma}}\times\prescript{\ast}{}{M}\to\prescript{\ast}{}{M}$ by upward transfer. Define a map $\Psi\colon G\times M\to M$ by letting $\Psi\left(g,x\right)=\prescript{\circ}{}{\Phi\left(j\left(g\right),x\right)}$, where $\circ\colon\prescript{\ast}{}{M}\to M$ is the standard part map. Let us show that $\Psi$ is the desired action.
\[lem:Action\]$\Psi$ is an action.
$\Psi\left(e\right)=\mathrm{id}_{M}$: $$\begin{aligned}
\Psi\left(e,x\right) & =\prescript{\circ}{}{\Phi\left(j\left(e\right),x\right)}\\
& =\prescript{\circ}{}{\Phi\left(e_{\gamma},x\right)}\\
& =\prescript{\circ}{}{x}\\
& =x.\end{aligned}$$ $\Psi\left(gh\right)=\Psi\left(g\right)\circ\Psi\left(h\right)$: $$\begin{aligned}
\Psi\left(gh,x\right) & =\prescript{\circ}{}{\Phi\left(j\left(gh\right),x\right)}\\
& =\prescript{\circ}{}{\Phi\left(j\left(g\right)j\left(h\right),x\right)}\\
& =\prescript{\circ}{}{\Phi\left(j\left(g\right),\Phi\left(j\left(h\right),x\right)\right)}\\
& =\prescript{\circ}{}{\Phi\left(j\left(g\right),\prescript{\circ}{}{\Phi\left(j\left(h\right),x\right)}\right)}\\
& =\prescript{\circ}{}{\Phi\left(j\left(g\right),\Psi\left(h,x\right)\right)}\\
& =\Psi\left(g,\Psi\left(h,x\right)\right).\end{aligned}$$
\[lem:Lipschitz\]$\Psi$ is $K$-Lipschitz.
Since $\prescript{\ast}{}{d_{M}}\left(\Psi\left(g,x\right),\Phi\left(j\left(g\right),x\right)\right)\approx\prescript{\ast}{}{d_{M}}\left(\Phi\left(j\left(g\right),y\right),\Psi\left(g,y\right)\right)\approx0$ and by the triangle inequality, $$\begin{aligned}
d_{M}\left(\Psi\left(g,x\right),\Psi\left(g,y\right)\right) & \approx\prescript{\ast}{}{d_{M}}\left(\Phi\left(j\left(g\right),x\right),\Phi\left(j\left(g\right),y\right)\right)\\
& \leq Kd_{M}\left(x,y\right).\end{aligned}$$ By taking the standard parts of the both sides, we have that $d_{M}\left(\Psi\left(g,x\right),\Psi\left(g,y\right)\right)\leq Kd_{M}\left(x,y\right)$.
$\Psi$ is $\varepsilon$-faithful.
Let $g\in G$ other than the unit. Clearly $j\left(g\right)$ is not the unit. Since $\Phi$ is (internally) $\varepsilon$-faithful, there is an $x\in\prescript{\ast}{}{M}$ such that $\prescript{\ast}{}{d_{M}}\left(x,\Phi\left(j\left(g\right),x\right)\right)\geq\varepsilon$. Obviously $\prescript{\circ}{}{x}$ is infinitesimally close to $x$. Since $\Phi$ is (internally) $K$-Lipschitz, $\Psi\left(g,\prescript{\circ}{}{x}\right)$ is infinitesimally close to $\Phi\left(j\left(g\right),x\right)$. By the triangle inequality, $d_{M}\left(\prescript{\circ}{}{x},\Psi\left(g,\prescript{\circ}{}{x}\right)\right)$ is infinitesimally close to $\prescript{\ast}{}{d_{M}}\left(x,\Phi\left(j\left(g\right),x\right)\right)$. So $d_{M}\left(\prescript{\circ}{}{x},\Psi\left(g,\prescript{\circ}{}{x}\right)\right)\geq\varepsilon$. Therefore $\Psi$ is $\varepsilon$-faithful.
This completes the proof of .
\[sec:Generalisation-and-correction\]Generalisation and correction of Manevitz and Weinberger’s result
======================================================================================================
\[thm:Injective-limit-of-finite-groups\]Let $M$ be a compact manifold with a metric. Let $\left(G_{\lambda},i_{\lambda\mu},\Lambda\right)$ be a direct system of torsion groups, where $i_{\lambda\mu}$ is injective for all $\lambda\leq\mu$. If there exists a faithful $K$-Lipschitz $G_{\lambda}$-action on $M$ for each $\lambda\in\Lambda$, then there exists a faithful $K$-Lipschitz $\varinjlim G_{\lambda}$-action on $M$.
The proof is almost similar to that of . Fix an infinite element $\lambda\in\prescript{\ast}{}{\Lambda}$ by saturation. There is an embedding $j\colon G\to\prescript{\ast}{}{G_{\gamma}}$ with $j\circ i_{\lambda}=\prescript{\ast}{}{i_{\lambda\gamma}}$ by . The upward transfer principle yields an internal faithful $K$-Lipschitz action $\Phi\colon\prescript{\ast}{}{G_{\gamma}}\times\prescript{\ast}{}{M}\to\prescript{\ast}{}{M}$. Define a map $\Psi\colon G\times M\to M$ by $\Psi\left(g,x\right)=\prescript{\circ}{}{\Phi\left(j\left(g\right),x\right)}$, which is a $K$-Lipschitz action by and .
We only need to prove the faithfulness of $\Psi$. Let $g\in G$ other than the unit. Choose a $g_{\lambda}\in G_{\lambda}$ such that $g=i_{\lambda}\left(g_{\lambda}\right)$. Clearly $g_{\lambda}$ is not the unit. Since $G_{\lambda}$ is torsion, the cyclic group $\braket{g_{\lambda}}$ generated by $g_{\lambda}$ is a *standard finite* subgroup of $\prescript{\ast}{}{G_{\lambda}}$. Consider the *internal* action $\Phi_{\lambda}\colon\braket{g_{\lambda}}\times\prescript{\ast}{}{M}\to\prescript{\ast}{}{M}$ defined by $\Phi_{\lambda}\left(h,x\right)=\Phi\left(\prescript{\ast}{}{i_{\lambda\gamma}}\left(h\right),x\right)$. Since $\prescript{\ast}{}{i_{\lambda\gamma}}$ is injective by upward transfer, $\Phi_{\lambda}$ is faithful. By Dress [@Dre69 Theorem 2] (a version of Newman’s theorem) and by upward transfer, there exists a *standard* constant $\varepsilon>0$ such that “there exist an $h\in\braket{g_{\lambda}}$ and an $x\in\prescript{\ast}{}{M}$ such that $\prescript{\ast}{}{d_{M}}\left(x,\Phi_{\lambda}\left(h,x\right)\right)\geq\varepsilon$”. (The downward transfer principle cannot be applied to the quoted statement, because it contains a *nonstandard* object, namely $\Phi_{\lambda}$. Manevitz and Weinberger [@MW96 p. 152, line 2124] accidentally applied the downward transfer principle to the corresponding statement in the original proof. Such use is illegal.) We can find a *standard* $n\in\mathbb{N}$ such that $h=g_{\lambda}^{n}$. Then $\prescript{\ast}{}{d_{M}}\left(x,\Phi_{\lambda}\left(g_{\lambda}^{n},x\right)\right)\geq\varepsilon$ holds. Obviously $\prescript{\circ}{}{x}$ is infinitesimally close to $x$. Since $\Phi$ is $K$-Lipschitz, $\Psi\left(g^{n},\prescript{\circ}{}{x}\right)$ is infinitesimally close to $\Phi\left(j\left(g^{n}\right),x\right)=\Phi\left(\prescript{\ast}{}{i_{\lambda\gamma}}\left(g_{\lambda}^{n}\right),x\right)=\Phi_{\lambda}\left(g_{\lambda}^{n},x\right)$. By the triangle inequality, $d_{M}\left(\prescript{\circ}{}{x},\Psi\left(g^{n},\prescript{\circ}{}{x}\right)\right)$ is infinitesimally close to $\prescript{\ast}{}{d_{M}}\left(x,\Phi_{\lambda}\left(g_{\lambda}^{n},x\right)\right)$. So $d_{M}\left(\prescript{\circ}{}{x},\Psi\left(g^{n},\prescript{\circ}{}{x}\right)\right)\geq\varepsilon$. Since $\Psi\left(g\right)^{n}=\Psi\left(g^{n}\right)\neq\mathrm{id}_{M}$, it follows that $\Psi\left(g\right)\neq\mathrm{id}_{M}$. Therefore $\Psi$ is faithful.
Let $M$ be a compact manifold with a metric. If there exists a faithful $K$-Lipschitz $\mathbb{Z}/n\mathbb{Z}$-action on $M$ for each $n\in\mathbb{Z}_{+}$, then there exists a faithful $K$-Lipschitz $\mathbb{Q}/\mathbb{Z}$-action on $M$.
In , set $\Lambda=\mathbb{Z}_{+}$, $G_{\lambda}=\mathbb{Z}/\lambda\mathbb{Z}$ and $i_{\lambda\mu}\left(k\text{ mod }\lambda\right)=k\left(\mu/\lambda\right)\text{ mod }\mu$, where the index set $\mathbb{Z}_{+}$ is ordered by the divisibility relation.
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abstract: 'In this paper, we study the endpoint reversed Strichartz estimates along general time-like trajectories for wave equations in $\mathbb{R}^{3}$. We also discuss some applications of the reversed Strichartz estimates and the structure of wave operators to the wave equation with one potential. These techniques are useful to analyze the stability problem of traveling solitons.'
address: 'Department of Mathematics, The University of Chicago, 5734 South University Avenue, Chicago, IL 60615, U.S.A'
author:
- Gong Chen
date: '10/29/16'
title: Wave Equations with Moving Potentials
---
[^1]
Introduction\[sec:Intro\]
=========================
Our starting point is the free wave equation ($H_{0}=-\Delta$) on $\mathbb{R}^{3}$ $$\partial_{tt}u-\Delta u=0$$ with initial data $$u(x,0)=g(x),\,u_{t}(x,0)=f(x).$$ We can write down $u$ explicitly, $$u=\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f+\cos\left(t\sqrt{-\Delta}\right)g.$$ It obeys the energy inequality, $$E_{F}(t)=\int_{\mathbb{R}^{3}}\left|\partial_{t}u(t)\right|^{2}+\left|\nabla u(t)\right|^{2}\,dx\lesssim\int_{\mathbb{R}^{3}}\left|f\right|^{2}+\left|\nabla g\right|^{2}\,dx.$$ We also have the well-known dispersive estimates for the free wave equation ($H_{0}=-\Delta$) on $\mathbb{R}^{3}$: $$\left\Vert \frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f\right\Vert _{L^{\infty}\left(\mathbb{R}^{3}\right)}\lesssim\frac{1}{\left|t\right|}\left\Vert \nabla f\right\Vert _{L^{1}\left(\mathbb{R}^{3}\right)},\label{eq:disper1}$$
$$\left\Vert \cos\left(t\sqrt{-\Delta}\right)g\right\Vert _{L^{\infty}\left(\mathbb{R}^{3}\right)}\lesssim\frac{1}{\left|t\right|}\left\Vert \Delta g\right\Vert _{L^{1}\left(\mathbb{R}^{3}\right)}.\label{eq:disper2}$$
For the sake of completeness, the proofs of estimates and are provided in details in Appendix A. (Notice that the estimate is slightly different from the estimates commonly used in the literature, such as Krieger-Schlag [@KS] where one needs the $L^{1}$ norm of $D^{2}g$ instead of $\Delta g$).
Strichartz estimates can be derived abstractly from these dispersive inequalities and the energy inequality. With some appropriate $\left(p,q,s\right)$, one has $$\|u\|_{L_{t}^{p}L_{x}^{q}}\lesssim\|g\|_{\dot{H}^{s}}+\|f\|_{\dot{H}^{s-1}}\label{eq:IFStri}$$ The non-endpoint estimates for the wave equations can be found in Ginibre-Velo [@GV]. KeelTao [@KT] also obtained sharp Strichartz estimates for the free wave equation in $\mathbb{R}^{n},\,n\geq4$ and everything except the endpoint in $\mathbb{R}^{3}$. See Keel-Tao [@KT] and Tao’s book [@Tao] for more details on the subject’s background and the history.
In $\mathbb{R}^{3}$, there is no hope to obtain such an estimate with the $L_{t}^{2}L_{x}^{\infty}$ norm, the so-called endpoint Strichartz estimate for free wave equations, cf. Klainerman-Machedon [@KM] and Machihara-Nakamura-Nakanishi-Ozawa [@MNNO]. But if we reverse the order of space-time integration, one can obtain a version of reversed Strichartz estimates from the Morawetz estimate, cf. Theorem \[thm:EndRStrichF\]: $$\left\Vert \frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\left\Vert f\right\Vert _{L^{2}\left(\mathbb{R}^{3}\right)},\,\left\Vert \cos\left(t\sqrt{-\Delta}\right)g\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\left\Vert g\right\Vert _{\dot{H}^{1}\left(\mathbb{R}^{3}\right)}.$$ These types of estimates are extended to inhomogeneous cases and perturbed Hamiltonian in Goldberg-Beceanu [@BecGo]. In Section \[sec:revered\], we will study these estimates and their generalizations intensively.
Next, we consider a linear wave equations with a real-valued stationary potential, $$H=-\Delta+V,$$ $$\partial_{tt}u+Hu=\partial_{tt}u-\Delta u+Vu=0,$$ $$u(x,0)=g(x),\,u_{t}(x,0)=f(x).$$ Explicitly, we have $$u=\frac{\sin\left(t\sqrt{H}\right)}{\sqrt{H}}f+\cos\left(t\sqrt{H}\right)g.$$ For the class of short-range potentials we consider in this paper, under our hypotheses $H$ only has pure absolutely continuous spectrum on $[0,\infty)$ and a finite number of negative eigenvalues. It is very crucial to notice that if there is a negative eigenvalue $E<0$, the associated eigenfunction responds to the wave equation propagators with a scalar factor by $\cos\left(t\sqrt{E}\right)$ or $\frac{\sin\left(t\sqrt{E}\right)}{E^{\frac{1}{2}}}$, both of which will grow exponentially since $\sqrt{E}$ is purely imaginary. Thus, Strichartz estimates for $H$ must include a projection $P_{c}$ onto the continuous spectrum in order to get away from this situation.
The problem of dispersive decay and Strichartz estimates for the wave equation with a potential has received much attention in recent years, see the papers by Beceanu-Goldberg [@BecGo], Krieger-Schlag [@KS] and the survey by Schlag [@Sch] for further details and references.
The Strichartz estimates in this case are in the form: $$\|\frac{\sin\left(t\sqrt{H}\right)}{\sqrt{H}}P_{c}f+\cos\left(t\sqrt{H}\right)P_{c}g\|_{L_{t}^{p}L_{x}^{q}}\lesssim\|g\|_{\dot{H}^{1}}+\|f\|_{L^{2}{}^{,}}$$ with $2<p,\,\frac{1}{2}=\frac{1}{p}+\frac{3}{q}.$ One also has the endpoint reversed Strichartz estimates: $$\left\Vert \frac{\sin\left(t\sqrt{H}\right)}{\sqrt{H}}P_{c}f+\cos\left(t\sqrt{H}\right)P_{c}g\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}},$$ see Theorem \[thm:PStriRStrich\].
In Section \[sec:Prelim\] and Section \[sec:revered\], we will systematically pass the estimates for free equations to the perturbed case via the structure formula of wave operators. This strategy also works in many other contexts provided that the free solution operators commute with certain symmetries.
For wave equations in $\mathbb{R}^{3}$, there are several difficulties. For example, the failure of the $L_{t}^{2}L_{x}^{\infty}$ estimate and the weakness of decay power $\frac{1}{t}$ in dispersive estimates. The reversed Strichartz estimates might circumvent these difficulties. Reversed Strichartz estimates along time-like trajectories play an important role in the analysis of wave equations of moving potentials. For example, in [@GC2], we used some preliminary versions of these estimates to show Strichartz estimates for wave equations with charge transfer Hamiltonian.
There are extra difficulties when dealing with time-dependent potentials. For example, given a general time-dependent potential $V(x,t)$, it is not clear how to introduce an analog of bound states and a spectral projection. The evolution might not satisfy group properties any more. It might also result in the growth of certain norms of the solutions, see Bourgain’s book [@Bou].
The second part of this paper, we apply the endpoint reversed Strichartz estimates along trajectories to study the wave equation with one moving potential: $$\partial_{tt}u-\Delta u+V\left(x-\vec{v}(t)\right)=0$$ which appears naturally in the study of stability problems of traveling solitons.
For Schrödinger equations with moving potentials, one can find references and progress, for example in Beceanu-Soffer [@BS], Rodnianski-Schlag-Soffer [@RSS]. Compared with Schrödinger equations, wave equations have some natural difficulties, for example the evolution of bound states of wave equations leads to exponential growth meanwhile the evolution of bound states of Schrödinger equations are merely multiplied by oscillating factors. We also notice that Lorentz transformations are space-time rotations, therefore one can not hope to succeed by the approach used with Schrödinger equations based on Galilei transformations. The geometry becomes much more complicated in the wave equation context. A crucial step to study wave equations with moving potentials is to understand the change of the energy under Lorentz transformations. In Chen [@GC2], we obtained that the energy stays comparable under Lorentz transformations. In this paper, we study this by a different approach based on local energy conservation which requires less decay of the potential. As a byproduct, we also obtain Agmon’s estimates for the decay of eigenfunctions associated to negatives eigenvalues of $H$.
Main results
------------
A trajectory $\vec{v}(t)\in\mathbb{R}^{3}$ is said to be an admissible if $\vec{v}(t)$ is $C^{1}$ and there exist $0\leq\ell<1$ such $\left|\vec{v}(t)\right|<\ell<1$ for $t\in\mathbb{R}$.
Consider the solution to the free wave equation ($H_{0}=-\Delta$), $$u(x,t)=\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f+\cos\left(t\sqrt{-\Delta}\right)g+\int_{0}^{t}\frac{\sin\left(\left(t-s\right)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}F(s)\,ds$$ and let $\vec{v}(t)\in\mathbb{R}^{3}$ be an admissible trajectory. Setting $$u^{S}(x,t):=u\left(x+\vec{v}(t),t\right),$$ we estimate $$\sup_{x\in\mathbb{R}^{3}}\int\left|u^{S}(x,t)\right|^{2}dt$$ in terms of the initial energy and various norms of $F$. The idea behind these estimates is that the fundamental solution of the free wave equation is supported on the light cone. Along a time-like curve, the propagation will only meet the light cone once.
\[thm:reversedF\]Let $\vec{v}(t)$ be an admissible trajectory. Set $$u(x,t)=\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f+\cos\left(t\sqrt{-\Delta}\right)g+\int_{0}^{t}\frac{\sin\left(\left(t-s\right)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}F(s)\,ds$$ and $$u^{S}(x,t):=u\left(x+\vec{v}(t),t\right).$$ Then $$\left\Vert u^{S}(x,t)\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}+\left\Vert F\right\Vert _{\dot{W}_{x}^{1,1}L_{t}^{2}}.$$ If $\vec{v}(t)$ does not change the direction, then $$\left\Vert u^{S}(x,t)\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}+\left\Vert F\right\Vert _{L_{d}^{1}L_{\widehat{d}}^{2,1}L_{t}^{2}},$$ where $d$ is the direction of $\vec{v}(t)$.
Let $\vec{\mu}(t)$ be another admissible trajectory, we have the same estimates above with $F$ replaced by $$F^{S'}(x,t):=F\left(x+\vec{\mu}(t),t\right).$$
We can extend the above estimates to wave equations with perturbed Hamiltonian, $$H=-\Delta+V$$ for $V$ decays with rate $\left\langle x\right\rangle ^{-\alpha}$ for $\alpha>3$, such that $H$ admits neither eigenfunctions nor resonances at $0$. Recall that $\psi$ is a resonance at $0$ if it is a distributional solution of the equation $H\psi=0$ which belongs to the space $L^{2}\left(\left\langle x\right\rangle ^{-\sigma}dx\right):=\left\{ f:\,\left\langle x\right\rangle ^{-\sigma}f\in L^{2}\right\} $ for any $\sigma>\frac{1}{2}$, but not for $\sigma=\frac{1}{2}.$
\[thm:reversedP\]Let $\vec{v}(t)$ be an admissible trajectory. Suppose $$H=-\Delta+V$$ admits neither eigenfunctions nor resonances at $0$. Set $$u(x,t)=\frac{\sin\left(t\sqrt{H}\right)}{\sqrt{H}}P_{c}f+\cos\left(t\sqrt{-H}\right)P_{c}g+\int_{0}^{t}\frac{\sin\left(\left(t-s\right)\sqrt{H}\right)}{\sqrt{H}}P_{c}F(s)\,ds$$ and $$u^{S}(x,t):=u\left(x+\vec{v}(t),t\right),$$ where $P_{c}$ is the projection onto the continuous spectrum of $H$.
Then $$\left\Vert u^{S}(x,t)\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}+\left\Vert F\right\Vert _{\dot{W}_{x}^{1,1}L_{t}^{2}}.$$ If $\vec{v}(t)$ does not change the direction, then $$\left\Vert u^{S}(x,t)\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}+\left\Vert F\right\Vert _{L_{d}^{1}L_{\widehat{d}}^{2,1}L_{t}^{2}},$$ where $d$ is the direction of $\vec{v}(t)$.
Let $\vec{\mu}(t)$ be another admissible trajectory, we have the same estimates above with $F$ replaced by $$F^{S'}(x,t):=F\left(x+\vec{\mu}(t),t\right).$$
We will reply on the structure formula of the wave operators by Beceanu. Although one can obtain similar results without using the structure formula, see [@GC2], the goal of our exposition is the illustrate a general strategy that one can pass the estimates for the free evolution to the perturbed one via the structure formula provided there are some symmetries of the free solution operators.
As applications of the above estimates, we study both regular and reversed Strichartz estimates for scattering states to a wave equation with a moving potential. Suppose $\vec{v}(t)\in\mathbb{R}$ is a trajectory such that there exist $\vec{\mu}\in\mathbb{R}^{3}$ $$\left|\vec{v}(t)-\vec{\mu}t\right|\lesssim\left\langle t\right\rangle ^{-\beta},\,\beta>1.$$ Consider $$\partial_{tt}u-\Delta u+V\left(x-\vec{v}(t)\right)=0\label{eq:meq}$$ with initial data $$u(x,0)=g(x),\,u_{t}(x,0)=f(x).$$
An indispensable tool we need to study wave equations with moving potentials is the Lorentz transformations. Without loss of generality, we assume $\vec{\mu}$ is along $\overrightarrow{e_{1}}$. We apply Lorentz transformation $L$ with respect to a moving frame with speed $\left|\mu\right|<1$ along the $x_{1}$ direction. Writing down the Lorentz transformation explicitly, we have $$\begin{cases}
t'=\gamma\left(t-\mu x_{1}\right)\\
x_{1}'=\gamma\left(x_{1}-\mu t\right)\\
x_{2}'=x_{2}\\
x_{3}'=x_{3}
\end{cases}\label{eq:LorentzT}$$ with $$\gamma=\frac{1}{\sqrt{1-\left|\mu\right|^{2}}}.$$ We can also write down the inverse transformation of the above: $$\begin{cases}
t=\gamma\left(t'+vx_{1}'\right)\\
x_{1}=\gamma\left(x_{1}'+\mu t'\right)\\
x_{2}=x_{2}'\\
x_{3}=x_{3}'
\end{cases}.\label{eq:InvLorentT}$$ Under the Lorentz transformation $L$, if we use the subscript $L$ to denote a function with respect to the new coordinate $\left(x',t'\right)$, we have $$u_{L}\left(x_{1}',x_{2}',x_{3}',t'\right)=u\left(\gamma\left(x_{1}'+\mu t'\right),x_{2}',x_{3}',\gamma\left(t'+\mu x_{1}'\right)\right)\label{eq:Lcoordinate}$$ and $$u(x,t)=u_{L}\left(\gamma\left(x_{1}-\mu t\right),x_{2},x_{3},\gamma\left(t-v\mu x\right)\right).\label{eq:ILcoordinate}$$ In order to study the equation with time-dependent potentials, we need to introduce a suitable projection. Let $$H=-\Delta+V\left(\sqrt{1-\left|\mu\right|^{2}}x_{1},x_{2},x_{3}\right).$$
Let $m_{1},\,\ldots,\,m_{w}$ be the normalized bound states of $H$ associated to the negative eigenvalues $-\lambda_{1}^{2},\,\ldots,\,-\lambda_{w}^{2}$ respectively (notice that by our assumptions, $0$ is not an eigenvalue). In other words, we assume $$Hm_{i}=-\lambda_{i}^{2}m_{i},\,\,\,m_{i}\in L^{2},\,\lambda_{i}>0.$$ We denote by $P_{b}$ the projections on the the bound states of $H$ and let $P_{c}=Id-P_{b}$. To be more explicit, we have $$P_{b}=\sum_{j=1}^{\ell}\left\langle \cdot,m_{j}\right\rangle m_{j}.$$ With Lorentz transformations $L$ associated to the moving frame $\left(x-\vec{\mu}t,t\right)$, we use the subscript $L$ to denote a function under the new frame $\left(x',t'\right)$.
\[AO\]Let $$\partial_{tt}u-\Delta u+V\left(x-\vec{v}(t)\right)u=0\label{eq:eqBSsec-1}$$ with initial data $$u(x,0)=g(x),\,u_{t}(x,0)=f(x).$$ If $u$ also satisfies $$\left\Vert P_{b}u_{L}(t')\right\Vert \rightarrow0\,\,\,t,t'\rightarrow\infty,\label{eq:ao2-1}$$ we call it a scattering state.
\[thm:Stri\]Suppose $u$ is a scattering state in the sense of Definition \[AO\] which solves the equation .Then for $p>2$ and $(p,q)$ satisfying $$\frac{1}{2}=\frac{1}{p}+\frac{3}{q},$$ we have $$\|u\|_{L_{t}^{p}\left([0,\infty),\,L_{x}^{q}\right)}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$
The above theorem can be extended to the inhomogeneous case, see for example [@GC2].
Secondly, one has the energy estimate:
\[thm:Energy\]Suppose $u$ is a scattering state in the sense of Definition \[AO\] which solves the equation .Then we have $$\sup_{t\geq0}\left(\|\nabla u(t)\|_{L^{2}}+\|u_{t}(t)\|_{L^{2}}\right)\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.\label{eq:StriCharWOB-1}$$
We also have the local energy decay:
\[thm:LEnergy\]Suppose $u$ is a scattering state in the sense of Definition \[AO\] which solves the equation .Then for $\forall\epsilon>0,\,\left|\vec{\nu}\right|<1$, we have $$\left\Vert \left(1+\left|x-\vec{\nu}t\right|\right)^{-\frac{1}{2}-\epsilon}\left(\left|\nabla u\right|+\left|u_{t}\right|\right)\right\Vert _{L_{t,x}^{2}}\lesssim_{\mu,\epsilon}\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.\label{eq:LEnergy}$$
We also obtain the endpoint reversed Strichartz estimates for $u$.
\[thm:EndRStri\]Let $\vec{h}(t)$ be an admissible trajectory. Suppose $u$ is a scattering state in the sense of Definition \[AO\] which solves the equation .Then $$\sup_{x\in\mathbb{R}^{3}}\int_{0}^{\infty}\left|u(x,t)\right|^{2}dt\lesssim\left(\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}\right)^{2},$$ and $$\sup_{x\in\mathbb{R}^{3}}\int_{0}^{\infty}\left|u(x+\vec{h}(t),t)\right|^{2}dt\lesssim\left(\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}\right)^{2}.$$
With the endpoint estimate along $\left(x+\vec{v}(t),t\right)$, one can derive the boundedness of the total energy. We denote the total energy of the system as $$E_{V}(t)=\int\left|\nabla_{x}u\right|^{2}+\left|\partial_{t}u\right|^{2}+V\left(x-\vec{v}(t)\right)\left|u\right|^{2}dx.$$
\[cor:ene\] Suppose $u$ is a scattering state in the sense of Definition \[AO\] which solves the equation . Assume $$\left\Vert \nabla V_{2}\right\Vert _{L^{1}}<\infty,$$ then $E_{V}(t)$ is bounded by the initial energy independently of $t$, $$\sup_{t\geq0}E_{V}(t)\lesssim\left\Vert \left(g,f\right)\right\Vert _{\dot{H}^{1}\times L^{2}}^{2}.$$
Notation {#notation .unnumbered}
--------
$A:=B{\ifmmode\begingroup\def\b@ld{bold}
\text{\ifx\math@version\b@ld\bfseries\fi\textquotedblright}\endgroup\else\textquotedblright\fi}$ or ${\ifmmode\begingroup\def\b@ld{bold}
\text{\ifx\math@version\b@ld\bfseries\fi\textquotedblleft}\endgroup\else\textquotedblleft\fi}B=:A{\ifmmode\begingroup\def\b@ld{bold}
\text{\ifx\math@version\b@ld\bfseries\fi\textquotedblright}\endgroup\else\textquotedblright\fi}$ is the definition of $A$ by means of the expression $B$. We use the notation $\langle x\rangle=\left(1+|x|^{2}\right)^{\frac{1}{2}}$. The bracket $\left\langle \cdot,\cdot\right\rangle $ denotes the distributional pairing and the scalar product in the spaces $L^{2}$, $L^{2}\times L^{2}$ . For positive quantities $a$ and $b$, we write $a\lesssim b$ for $a\leq Cb$ where $C$ is some prescribed constant. Also $a\simeq b$ for $a\lesssim b$ and $b\lesssim a$. Throughout, we use $\partial_{tt}u:=\frac{\partial^{2}}{\partial t\partial t}$, $u_{t}:=\frac{\partial}{\partial_{t}}u$, $\Delta:=\sum_{i=1}^{n}\frac{\partial^{2}}{\partial x_{i}\partial x_{i}}$ and occasionally, $\square:=-\partial_{tt}+\Delta$.
Organization {#organization .unnumbered}
------------
The paper is organized as follows: In Section \[sec:Prelim\], we discuss some preliminary results for the free wave equation and the wave equation with a stationary potential. In Section \[sec: Lorentz\], we will analyze the change of the energy under Lorentz transformations. Agmon’s estimates are also presented as a consequence of our comparison results. In Section \[sec:revered\], the endpoint reversed Strichartz estimates of homogeneous and inhomogeneous forms are derived along admissible trajectories. In Section \[sec:one\], we show Strichartz estimates, energy estimates, the local energy decay and the boundedness of the total energy for a scattering state to the wave equation with a moving potential. Finally, in Section \[sec:Scattering\], we confirm that a scattering state indeed scatters to a solution to the free wave equation and also obtain a version of the asymptotic completeness description of the wave equations with one moving potential. In appendices, for the sake of completeness, we show the dispersive estimates for wave equations in $\mathbb{R}^{3}$ based on the idea of reversed Strichartz estimates, the local energy decay of free wave equations and the global existence of solutions to the wave equation with a time-dependent potential. A Fourier analytic proof of the endpoint reversed Strichartz estimates is also presented.
Acknowledgment {#acknowledgment .unnumbered}
--------------
I feel deeply grateful to my advisor Professor Wilhelm Schlag for his kind encouragement, discussions, comments and all the support. I also want to thank Marius Beceanu for many useful discussions.
Preliminaries\[sec:Prelim\]
===========================
Strichartz estimates and the endpoint reversed Strichartz estimates
-------------------------------------------------------------------
We start with Strichartz estimates for free wave equations. Strichartz estimates can be derived abstractly from these dispersive inequalities and the energy inequality. The following theorem is standard. One can find a detailed proof in, for example, Keel-Tao [@KT].
\[thm:StrichF\]Suppose $$\partial_{tt}u-\Delta u=F$$ with initial data $$u(x,0)=g(x),\,u_{t}(x,0)=f(x).$$ Then for $p,\,a>\frac{2}{s}$, $\left(p,q\right),\,\left(a,b\right)$ satisfying $$\frac{3}{2}-s=\frac{1}{p}+\frac{3}{q}$$ $$\frac{3}{2}-s=\frac{1}{a}+\frac{3}{b}$$ we have $$\|u\|_{L_{t}^{p}L_{x}^{q}}\lesssim\|g\|_{\dot{H}^{s}}+\|f\|_{\dot{H}^{s-1}}+\left\Vert F\right\Vert _{L_{t}^{a'}L_{x}^{b'}}\label{eq:StrichF}$$ where $\frac{1}{a}+\frac{1}{a'}=1,\,\frac{1}{b}+\frac{1}{b'}=1.$
The endpoint $\left(p,q\right)=\left(2,\infty\right)$ can be recovered for radial functions in Klainerman-Machedon [@KM] for the homogeneous case and Jia-Liu-Schlag-Xu [@JLSX] for the inhomogeneous case. The endpoint estimate can also be obtained when a small amount of smoothing (either in the Sobolev sense, or in relaxing the integrability) is applied to the angular variable, see Machihara-Nakamura-Nakanishi-Ozawa [@MNNO].
\[thm:inhomAR\]For any $1\leq p<\infty$, suppose $u$ solves the free wave equation $$\partial_{tt}u-\Delta u=0$$ with initial data $$u(x,0)=g(x),\,u_{t}(x,0)=f(x).$$ Then $$\|u\|_{L_{t}^{2}L_{r}^{\infty}L_{\omega}^{p}}\le C(p)\left(\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}\right).\label{eq:inhomoAR}$$
The regular Strichartz estimates fail at the endpoint. But if one switches the order of space-time integration, it is possible to estimate the solution using the fact that the solution decays quickly away from the light cone. Therefore, we introduce reversed Strichartz estimates. Since we will only use the endpoint reversed Stricharz estimate, we will restrict our focus to that case.
\[thm:EndRStrichF\]Suppose $$\partial_{tt}u-\Delta u=F$$ with initial data $$u(x,0)=g(x),\,u_{t}(x,0)=f(x).$$ Then $$\left\Vert u\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}+\left\Vert F\right\Vert _{L_{x}^{\frac{3}{2},1}L_{t}^{2}}.\label{eq:EndRStrichF}$$
See Section \[sec:revered\] for the detailed proof. For the homogeneous case, one can find an alternative proof based on the Fourier transform in Appendix D.
The above results from Theorem \[thm:StrichF\] and Theorem \[thm:EndRStrichF\] can be generalized to the wave equation with a real stationary potentials.
For the perturbed Hamiltonian, $$H=-\Delta+V,$$ with $V\lesssim\left\langle x\right\rangle ^{-\alpha}$ for $\alpha>3$, consider the wave equation with potential in $\mathbb{R}^{3}$:
$$\partial_{tt}u-\Delta u+Vu=0$$
with initial data $$u(x,0)=g(x),\,u_{t}(x,0)=f(x).$$ One can write down the solution to it explicitly: $$u=\frac{\sin\left(t\sqrt{H}\right)}{\sqrt{H}}f+\cos\left(t\sqrt{H}\right)g.$$ Let $P_{b}$ be the projection onto the point spectrum of $H$, $P_{c}=I-P_{b}$ be the projection onto the continuous spectrum of $H$.
With the above setting, we formulate the results from [@BecGo].
\[thm:PStriRStrich\]Consider the perturbed Hamiltonian $H=-\Delta+V$ in $\mathbb{R}^{3}$. Suppose $H$ has neither eigenvalues nor resonance at zero. Then for all $0\leq s\leq1$, $p>\frac{2}{s}$, and $\left(p,q\right)$ satisfying $$\frac{3}{2}-s=\frac{1}{p}+\frac{3}{q}$$ we have $$\left\Vert \frac{\sin\left(t\sqrt{H}\right)}{\sqrt{H}}P_{c}f+\cos\left(t\sqrt{H}\right)P_{c}g\right\Vert _{L_{t}^{p}L_{x}^{q}}\lesssim\|g\|_{\dot{H}^{s}}+\|f\|_{\dot{H}^{s-1}}.\label{PSrich}$$ For the endpoint of reversed Strichartz estimates, we have $$\left\Vert \frac{\sin\left(t\sqrt{H}\right)}{\sqrt{H}}P_{c}f+\cos\left(t\sqrt{H}\right)P_{c}g\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}},\label{eq:PEndRSch}$$ $$\left\Vert \int_{0}^{t}\frac{\sin\left((t-s)\sqrt{H}\right)}{\sqrt{H}}P_{c}F(s)\,ds\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\left\Vert F\right\Vert _{L_{x}^{\frac{3}{2},1}L_{t}^{2}},\label{eq:PEndRSIn}$$
One can find detailed arguments and more estimates in [@BecGo]. We will apply the structure of the wave operators to show the above theorem in Section \[sec:revered\].
Structure of wave operators and its applications
------------------------------------------------
Next, we discuss the structure of wave operators. Again consider $$H=-\Delta+V.$$ For wave operators, we define $$W^{+}=s-\lim_{t\rightarrow\infty}e^{itH}e^{it\Delta}.$$ We know $$W^{+}\left(-\Delta\right)=HW^{+}$$ and $$\left(W^{+}\right)^{*}=s-\lim_{t\rightarrow\infty}e^{itH_{0}}e^{-itH}P_{c}.$$ By Beceanu [@Bec1; @Bec2], we have a structure formula for $W^{+}$ and $\left(W^{+}\right)^{*}$.
\[thm:structure\]Assume $H=-\Delta+V$ admits neither eigenfunction nor resonances at $0$. Then for both $W^{+}$ and $\left(W^{+}\right)^{*},$ we have for $f\in L^{2}$, $$Wf(x)=f(x)+\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}g(x,y,\omega)f\left(S_{\omega}x+y\right)\,dyd\omega,\label{eq:lwaveop}$$ for some $g(x,y,\omega)$ such that $$\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}\left\Vert g(x,y,\omega)\right\Vert _{L_{x}^{\infty}}dyd\omega<\infty$$ and where $$S_{\omega}x=x-(2x\cdot\omega)\omega.$$ is the reflection by the plane orthogonal to $\omega$. Here $W$ is either of $W^{+}$ or $\left(W^{+}\right)^{*}$.
In [@Bec1], for the potential $V$, the author only assumes that $$V\in B^{1}\cap L^{2},$$ where $$B^{\beta}=\left\{ V\,|\sum_{k\in\mathbb{Z}}2^{\beta k}\left\Vert \chi_{\left\{ \left|x\right|\in\left[2^{k},2^{k+1}\right]\right\} }(x)V(x)\right\Vert _{L^{2}}<\infty\right\} .$$
The structure formula in Theorem \[thm:structure\] is very powerful. One can easily pass many estimates from the free case to the perturbed case provided the solution operators of the free problem commute with certain symmetries. Here we illustrate this idea by a concrete computation based on Theorem \[thm:inhomAR\].
\[thm:inhomARH\] Assume $H=-\Delta+V$ admits neither eigenfunction nor resonances at $0$. Setting $$u^{H}=\frac{\sin\left(t\sqrt{H}\right)}{\sqrt{H}}P_{c}f+\cos\left(t\sqrt{H}\right)P_{c}g,$$ then for any $1\leq p<\infty$, one has
$$\left\Vert u^{H}\right\Vert {}_{L_{t}^{2}L_{r}^{\infty}L_{\omega}^{p}}\le C(p)\left(\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}\right)\label{eq:inhomoARH}$$
It suffices to consider $$u^{H}=\frac{\sin\left(t\sqrt{H}\right)}{\sqrt{H}}P_{c}f.$$ By construction, $$\frac{\sin\left(t\sqrt{H}\right)}{\sqrt{H}}P_{c}=W^{+}\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}\left(W^{+}\right)^{*}.$$ $$\frac{\sin\left(t\sqrt{H}\right)}{\sqrt{H}}P_{c}f=W^{+}\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}\left(W^{+}\right)^{*}P_{c}f.$$ Denoting $$h=\left(W^{+}\right)^{*}P_{c}f,$$ we have $$\|P_{c}f\|_{L^{2}}\simeq\|h\|_{L^{2}}.$$ Setting $$G=\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}h,$$ by Theorem \[thm:structure\], it is sufficient to consider the boundedness of $$G+\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}g(x,y,\tau)G\left(S_{\tau}x+y\right)\,dyd\tau.$$ Clearly, by Theorem \[thm:inhomAR\], $$\|G\|_{L_{t}^{2}L_{r}^{\infty}L_{\omega}^{p}}\lesssim\|h\|_{L^{2}}\simeq\|P_{c}f\|_{L^{2}}.$$ Next, by Minkowski’s inequality, $$\begin{aligned}
\left\Vert \int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}g(x,y,\tau)G\left(S_{\tau}x+y\right)\,dyd\tau\right\Vert {}_{L_{t}^{2}L_{r}^{\infty}L_{\omega}^{p}}\\
\lesssim\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}\left\Vert g(x,y,\tau)G\left(S_{\tau}x+y\right)\right\Vert {}_{L_{t}^{2}L_{r}^{\infty}L_{\omega}^{p}}dyd\tau\nonumber \end{aligned}$$ $$\left\Vert g(x,y,\tau)G\left(S_{\tau}x+y\right)\right\Vert {}_{L_{t}^{2}L_{r}^{\infty}L_{\omega}^{p}}\lesssim\left\Vert g(x,y,\tau)\right\Vert _{L_{x}^{\infty}}\left\Vert G\left(S_{\tau}x+y\right)\right\Vert {}_{L_{t}^{2}L_{r}^{\infty}L_{\omega}^{p}}.$$ Since reflections with respect to a fixed plane and translations commute with the solution of a free wave equation, we obtain $$G\left(S_{\tau}x+y\right)=\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}h\left(S_{\tau}x+y\right).$$ Therefore, $$\left\Vert G\left(S_{\tau}x+y\right)\right\Vert {}_{L_{t}^{2}L_{r}^{\infty}L_{\omega}^{p}}\lesssim\|h\left(S_{\tau}x+y\right)\|_{L^{2}}\lesssim\|h\|_{L^{2}}\simeq\|P_{c}f\|_{L^{2}}.$$ It follows $$\begin{aligned}
\left\Vert G+\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}g(x,y,\tau)G\left(S_{\tau}x+y\right)\,dyd\tau\right\Vert {}_{L_{t}^{2}L_{r}^{\infty}L_{\omega}^{p}}\\
\lesssim\left(1+\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}\left\Vert g(x,y,\tau)\right\Vert _{L_{x}^{\infty}}dyd\tau\right)\|P_{c}f\|_{L^{2}}\lesssim\|f\|_{L^{2}}.\nonumber \end{aligned}$$ Then we conclude $$\left\Vert u^{H}\right\Vert {}_{L_{t}^{2}L_{r}^{\infty}L_{\omega}^{p}}\le C(p,V)\left(\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}\right),$$ as claimed.
One can do similar arguments to obtain many other estimates for the perturbed wave equations, for example the local energy decay estimate, the energy estimate and many weighted estimates.
The following Christ-Kiselev Lemma is important in our derivation of Strichartz estimates.
\[lem:Christ-Kiselev\]Let $X$, $Y$ be two Banach spaces and let $T$ be a bounded linear operator from $L^{\beta}\left(\mathbb{R}^{+};X\right)$ to $L^{\gamma}\left(\mathbb{R}^{+};Y\right)$, such that $$Tf(t)=\int_{0}^{\infty}K(t,s)f(s)\,ds.$$ Then the operator $$\widetilde{T}=\int_{0}^{t}K(t,s)f(s)\,ds$$ is bounded from $L^{\beta}\left(\mathbb{R}^{+};X\right)$ to $L^{\gamma}\left(\mathbb{R}^{+};Y\right)$ provided $\beta<\gamma$, and the $$\left\Vert \widetilde{T}\right\Vert \leq C(\beta,\gamma)\left\Vert T\right\Vert$$ with $$C(\beta,\gamma)=\left(1-2^{\frac{1}{\gamma}-\frac{1}{\beta}}\right)^{-1}.$$
Lorentz Transformations and Energy\[sec: Lorentz\]
==================================================
When we consider wave equations with moving potentials, Lorentz transformations will be important for us to reduce some estimates to stationary cases. In order to approach our problem from the viewpoint of Lorentz transformations as in [@GC2], the first natural step is to understand the change of energy under Lorentz transformations.
Indeed, in [@GC2], we shown that under Lorentz transformations, the energy stays comparable to that of the initial data. The method in [@GC2] is based on integration by parts. Here we present an alternative approach based on the local energy conservation which is more natural and requires less decay of the potential. We notice that the method in [@GC2] can be viewed as the differential version of the argument here.
Throughout this section, we perform Lorentz transformations with respect to a moving frame with speed $\left|v\right|<1$, say, along the $x_{1}$ direction, i.e., the velocity is $$\vec{v}=\left(v,0,0\right).\label{eq:l1}$$ Recall that after we apply the Lorentz transformation, for function $u$, under the new coordinates, we denote $$u_{L}\left(x_{1}',x_{2}',x_{3}',t'\right)=u\left(\gamma\left(x_{1}'+vt'\right),x_{2}',x_{3}',\gamma\left(t'+vx_{1}'\right)\right).\label{eq:l6}$$ Now let $u$ be a solution to some wave equation and set $t'=0$. We notice that in order to show under Lorentz transformations, the energy stays comparable to that of the initial data, up to an absolute constant it suffices to prove $$\begin{aligned}
\int\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}dx\nonumber \\
\simeq\int\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}dx.\end{aligned}$$
Throughout this section, we will assume all functions are smooth and decay fast. We will obtain estimates independent of the additional smoothness assumption. It is easy to pass the estimates to general cases with a density argument.
\[rem:dim\]One can observe that all discussions in this section hold for $\mathbb{R}^{n}$.
Energy comparison
-----------------
In this section, a more general situation is analyzed. We consider wave equations with time-dependent potentials $$\partial_{tt}u-\Delta u+V(x,t)u=0$$ with $$\left|V(x,\mu x_{1})\right|\lesssim\frac{1}{\left\langle x\right\rangle ^{2}}$$ uniformly for $0\leq\left|\mu\right|\leq1$. These in particular apply to wave equations with moving potentials with speed strictly less than $1$. For example, if the potential is of the from $$V(x,t)=V\left(x-\nu t\right)$$ with $$\left|V(x)\right|\lesssim\frac{1}{\left\langle x\right\rangle ^{2}}$$ then it is transparent that $$\left|V(x,\mu x_{1})\right|=\left|V(x-\nu\mu x_{1})\right|\lesssim\frac{1}{\left\langle x\right\rangle ^{2}}.$$ Suppose $$\partial_{tt}u-\Delta u+V(x,t)u=0,$$ then it is clear that $$\begin{aligned}
0 & = & u_{t}\left(\square u-V(t)u\right)\nonumber \\
& = & -\partial_{t}\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)+\mathrm{div}\left(\nabla uu_{t}\right)-V(x,t)uu_{t}.\label{eq:vector}\end{aligned}$$
\[lem:upper\]Let $\left|v\right|<1$. Suppose $$\partial_{tt}u-\Delta u+V(x,t)u=0$$ and $$\left|V(x,\mu x_{1})\right|\lesssim\frac{1}{\left\langle x\right\rangle ^{2}}$$ uniformly for $0\leq\left|\mu\right|<1$. Then for arbitrary $R>0$, there exists some constant $M(v)>1$ depending on $v$ such that $$\begin{aligned}
\int_{\left|x\right|>M(v)R}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}dx\nonumber \\
\lesssim\int_{\left|x\right|>R}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}dx,\end{aligned}$$ where the implicit constant depends on $v,\,V$.
Denote $$L_{+}^{U}\left(u,\mu,R\right)=\int_{\left|x\right|>M(\mu)R,\,x_{1}>0}\left|V\left(x,\mu x_{1}\right)\right|\left|u\left(x_{1},x_{2},x_{3},\mu x_{1}\right)\right|^{2}dx.$$ $$T_{+}^{U}\left(u,\mu,R\right)=\int_{\left|x\right|>M(\mu)R,\,x_{1}>0}\left|V\left(x,\mu x_{1}\right)\right|\left|u_{t}\left(x_{1},x_{2},x_{3},\mu x_{1}\right)\right|^{2}dx.$$
$$E_{+}^{U}\left(u,\mu,R\right)=\int_{\left|x\right|>M(\mu)R,\,x_{1}>0}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},\mu x_{1}\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},\mu x_{1}\right)\right|^{2}dx,$$
where $$M(\mu)=\frac{1}{1-\left|\mu\right|}.$$ One observes that if $$\left|V(x,\mu x_{1})\right|\lesssim\frac{1}{\left\langle x\right\rangle ^{2}}$$ uniformly with respect to $0\leq\left|\mu\right|<1$, by Hardy’s inequality, $$L_{+}^{U}\left(u,\mu,R\right)+T_{+}^{U}\left(u,\mu,R\right)\lesssim E_{+}^{U}\left(u,\mu,R\right).$$ With these notations, we need to show $$E_{+}^{U}\left(u,v,R\right)\lesssim E_{+}^{U}\left(u,0,R\right).$$ For fixed $R>0$, we construct two regions as follows:
$L_{R}^{+}$ with equation $$x_{1}^{2}+x_{2}^{2}+x_{3}^{3}\leq\left(R+t\right)^{2},\,\,\,0\leq t<\infty,\,x_{1}\geq0$$ and $L_{R}^{-}$ with equation $$x_{1}^{2}+x_{2}^{2}+x_{3}^{3}\leq\left(R-t\right)^{2},\,\,\,-\infty<t\leq0,\,x_{1}\leq0.$$ Denote the region bounded by $\left([0,\infty)\times\mathbb{R}^{2}\times[0,\infty)\right)\backslash L_{R}^{+}$ and the plane $\left(x_{1},x_{2},x_{3},vx_{1}\right)$ by $Y^{+}$ and use $Y^{-}$ to denote the region bounded by $\left((-\infty,0]\times\mathbb{R}^{2}\times(-\infty,0]\right)\backslash L_{R}^{-}$ and the plane $\left(x_{1},x_{2},x_{3},vx_{1}\right)$.
By symmetry, it suffices to analyze $Y^{+}$. We apply the space-time divergence theorem to $$\left(\nabla uu_{t},-\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)\right)$$ in region $Y^{+}$. We denote the top of $Y^{+}$ by $Y_{T}^{+}$, the bottom as $Y_{B}^{+}$ and the lateral boundary as $Y_{L}^{+}$. We should notice $Y_{B}^{+}$ actually is $\left\{ x_{1}>0\right\} \cap\left(\mbox{\ensuremath{\mathbb{R}}}^{3}\backslash B_{R}(0)\right)$.
The unit outward-pointing normal vector on the plane $\left(x_{1},x_{2},x_{3},vx_{1}\right)$ is $$\frac{1}{\sqrt{v^{2}+1}}\left(-v,0,0,1\right).$$ The outward-pointing normal vector on the bottom of $Y^{+}$ is $$(0,0,0,-1).$$ From , one obtains $$\begin{aligned}
\frac{1}{\sqrt{v^{2}+1}}\int_{Y_{T}^{+}}\left[v\partial_{x_{1}}uu_{t}-\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)\right]\,dx+\int_{Y_{B}^{+}}\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)\,dx\nonumber \\
=\frac{1}{2\sqrt{2}}\int_{Y_{_{L}}^{+}}\left|\nabla u-n_{L}(x)u_{t}\right|^{2}\,d\sigma+\int_{Y^{+}}V(x,t)uu_{t}\,dxdt,\end{aligned}$$ where $n_{L}(x)$ is a vector of norm $1$.
Note that $$\int_{Y^{+}}\left|V(x,t)uu_{t}\right|dxdt\lesssim\int_{Y^{+}}\left|V(x,t)\right|\left(\left|u\right|^{2}+\left|u_{t}\right|^{2}\right)dxdt.$$ Hence we can conclude $$\begin{aligned}
\frac{1}{\sqrt{v^{2}+1}}\int_{Y_{T}^{+}}\left[-v\partial_{x_{1}}uu_{t}+\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)\right]\,dx\nonumber \\
\lesssim\int_{Y_{B}^{+}}\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)\,dx+\int_{Y^{+}}\left|V(x,t)\right|\left(\left|u\right|^{2}+\left|u_{t}\right|^{2}\right)dxdt.\end{aligned}$$ Consider the integral $$\int_{Y^{+}}\left|V(x,t)\right|\left(\left|u\right|^{2}+\left|u_{t}\right|^{2}\right)dxdt,$$ by a change of variable and Fubini’s Theorem, it follows $$\begin{aligned}
\int_{Y^{+}}\left|V(x,t)\right|\left(\left|u\right|^{2}+\left|u_{t}\right|^{2}\right)dxdt & = & \int_{0}^{v}\left(L_{+}^{U}\left(u,\mu,R\right)+T_{+}^{U}\left(u,\mu,R\right)\right)d\mu\nonumber \\
& \lesssim & \int_{0}^{v}E_{+}^{U}\left(u,\mu,R\right)\,d\mu.\end{aligned}$$ Note that with $\left|v\right|<1$ and the AMGM inequality, we obtain $$\left(1-\left|v\right|\right)\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)\leq-v\partial_{x_{1}}uu_{t}+\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right),$$ which implies $$\begin{aligned}
\frac{\left(1-\left|v\right|\right)}{\sqrt{v^{2}+1}}\int_{Y_{T}^{+}}\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)\,dx & \lesssim & \int_{Y_{B}^{+}}\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)\,dx\nonumber \\
& & +\int_{0}^{v}E_{+}^{U}\left(u,\mu,R\right)\,d\mu\end{aligned}$$ With our notations, we have $$E_{+}^{U}\left(u,v,R\right)\lesssim\frac{\sqrt{v^{2}+1}}{\left|1-\left|v\right|\right|}\left(E_{+}^{U}\left(u,0,R\right)+\int_{0}^{v}E_{+}^{U}\left(u,\mu,R\right)\,d\mu\right).$$ By Grönwall’s inequality with respect to $v$, one obtains $$E_{+}^{U}\left(u,v,R\right)\lesssim E_{+}^{U}\left(u,0,R\right)$$ provided $\left|v\right|<1$.
By construction, we have $$\begin{aligned}
\int_{\left|x\right|>M(v)R,\,x_{1}>0}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}dx\nonumber \\
\lesssim\int_{\left|x\right|>R,\,x_{1}>0}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}dx\end{aligned}$$ A similar argument for $Y^{-}$ gives $$\begin{aligned}
\int_{\left|x\right|>M(v)R,\,x_{1}\leq0}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}dx\nonumber \\
\lesssim\int_{\left|x\right|>R,\,x_{1}\leq0}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}dx.\end{aligned}$$ Hence, we get $$\begin{aligned}
\int_{\left|x\right|>M(v)R}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}dx\nonumber \\
\lesssim\int_{\left|x\right|>R}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}dx\end{aligned}$$ as claimed.
\[lem:lower\]Let $\left|v\right|<1$. Suppose $$\partial_{tt}u-\Delta u+V(x,t)u=0$$ and $$\left|V(x,\mu x_{1})\right|\lesssim\frac{1}{\left\langle x\right\rangle ^{2}}$$ uniformly for $0\leq\left|\mu\right|<1$. Then for arbitrary $R>0$, there exists some constant $M_{1}(v)<1$ depending $v$ such that $$\begin{aligned}
\int_{\left|x\right|>R}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}dx\label{eq:lower}\\
\lesssim\int_{\left|x\right|>M_{1}(v)R}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}dx,\nonumber \end{aligned}$$ where the implicit constant depends on $v,\,V$.
Denote $$L_{+}^{L}\left(u,\mu,R\right)=\int_{\left|x\right|>M_{1}(\mu)R,\,x_{1}>0}\left|V\left(x,\mu x_{1}\right)\right|\left|u\left(x_{1},x_{2},x_{3},\mu x_{1}\right)\right|^{2}dx.$$ $$T_{+}^{L}\left(u,\mu,R\right)=\int_{\left|x\right|>M_{1}(\mu)R,\,x_{1}>0}\left|V\left(x,\mu x_{1}\right)\right|\left|u_{t}\left(x_{1},x_{2},x_{3},\mu x_{1}\right)\right|^{2}dx.$$
$$E_{+}^{L}\left(u,\mu,R\right)=\int_{\left|x\right|>M_{1}(\mu)R,\,x_{1}>0}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},\mu x_{1}\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},\mu x_{1}\right)\right|^{2}dx,$$
where $$M_{1}(\mu)=\frac{1}{1+\left|\mu\right|}.$$ One observes that if $$\left|V(x,\mu x_{1})\right|\lesssim\frac{1}{\left\langle x\right\rangle ^{2}}$$ uniformly with respect to $0\leq\left|\mu\right|<1$, by Hardy’s inequality, $$L_{+}^{L}\left(u,\mu,R\right)+T_{+}^{L}\left(u,\mu,R\right)\lesssim E_{+}^{L}\left(u,\mu,R\right).$$ With these notations, we need to show $$E_{+}^{L}\left(u,0,R\right)\lesssim E_{+}^{L}\left(u,v,R\right).$$ For fixed $R>0$, we construct two regions as follows:
$C_{R}^{+}$ with equation $$x_{1}^{2}+x_{2}^{2}+x_{3}^{3}\leq\left(R-t\right)^{2},\,\,\,0\leq t\leq R,\,x_{1}\geq0$$ and $C_{R}^{-}$ with equation $$x_{1}^{2}+x_{2}^{2}+x_{3}^{3}\leq\left(R+t\right)^{2},\,\,\,-R\leq t\leq0,\,x_{1}\leq0.$$ Denote the region bounded by $\left([0,\infty)\times\mathbb{R}^{2}\times[0,\infty)\right)\backslash C_{R}^{+}$ and the plane $\left(x_{1},x_{2},x_{3},vx_{1}\right)$ by $K^{+}$ and use $K^{-}$ to denote the region $\left((-\infty,0]\times\mathbb{R}^{2}\times(-\infty,0]\right)\backslash C_{R}^{-}$ and the plane $\left(x_{1},x_{2},x_{3},vx_{1}\right)$.
By symmetry, it suffices to analyze $K^{+}$. We again apply the space-time divergence theorem to $$\left(\nabla uu_{t},-\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)\right)$$ in region $K^{+}$ as above. We denote the top of $K^{+}$ by $K_{T}^{+}$, the bottom as $K_{B}^{+}$ and the lateral boundary as $K_{L}^{+}$. The unit outward-pointing normal vector on the plane $\left(x_{1},x_{2},x_{3},vx_{1}\right)$ is $$\frac{1}{\sqrt{v^{2}+1}}\left(-v,0,0,1\right).$$ The outward-pointing normal vector on the bottom of $K^{+}$ is $$(0,0,0,-1).$$ One obtains from , $$\begin{aligned}
\frac{1}{\sqrt{v^{2}+1}}\int_{K_{T}^{+}}\left[v\partial_{x_{1}}uu_{t}-\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)\right]\,dx+\int_{K_{B}^{+}}\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)\,dx\nonumber \\
\leq-\frac{1}{2\sqrt{2}}\int_{K_{L}^{+}}\left|\nabla u-n(x)u_{t}\right|^{2}\,d\sigma+\int_{K^{+}}\left|V(x,t)\right|\left(\left|u\right|^{2}+\left|u_{t}\right|^{2}\right)dxdt,\end{aligned}$$ where $n(x)$ is a vector of norm $1$.
Note that $$\int_{K^{+}}\left|V(t)uu_{t}\right|dxdt\lesssim\int_{K^{+}}\left|V(x,t)\right|\left(\left|u\right|^{2}+\left|u_{t}\right|^{2}\right)dxdt.$$ Hence we can conclude $$\begin{aligned}
\int_{K_{B}^{+}}\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)\,dx & \lesssim & \frac{1}{\sqrt{v^{2}+1}}\int_{K_{T}^{+}}\left[-v\partial_{x_{1}}uu_{t}+\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)\right]\,dx\nonumber \\
& & +\int_{K^{+}}\left|V(x,t)\right|\left(\left|u\right|^{2}+\left|u_{t}\right|^{2}\right)\,dxdt.\end{aligned}$$ Again, consider the integral $$\int_{K^{+}}\left|V(x,t)\right|\left(\left|u\right|^{2}+\left|u_{t}\right|^{2}\right)dxdt,$$ by a change of variable and Fubini’s Theorem, it follows $$\begin{aligned}
\int_{K^{+}}\left|V(x,t)\right|\left(\left|u\right|^{2}+\left|u_{t}\right|^{2}\right)dxdt & = & \int_{0}^{v}\left(L_{+}^{L}\left(u,\mu,R\right)+T_{+}^{L}\left(u,\mu,R\right)\right)d\mu\nonumber \\
& \lesssim & \int_{0}^{v}E_{+}^{L}\left(u,\mu,R\right)\,d\mu.\end{aligned}$$ Hence we can conclude $$\begin{aligned}
\int_{K_{B}^{+}}\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)\,dx & \lesssim & \frac{1+\left|v\right|}{\sqrt{v^{2}+1}}\int_{K_{T}^{+}}\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)\,dx\nonumber \\
& & +\int_{0}^{v}E_{+}^{L}\left(u,\mu,R\right)\,d\mu\end{aligned}$$ In other words, $$E_{+}^{L}\left(u,0,R\right)\lesssim\frac{1+\left|v\right|}{\sqrt{v^{2}+1}}\left(E_{+}^{L}\left(u,v,R\right)+\int_{0}^{v}E_{+}^{L}\left(u,\mu,R\right)\,d\mu\right).$$ With Grönwall’s inequality again, it implies $$E_{+}^{L}\left(u,0,R\right)\lesssim E_{+}^{L}\left(u,v,R\right)$$ Therefore, $$\int_{K_{B}^{+}}\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)\,dx\lesssim\int_{K_{T}^{+}}\left(\frac{\left|u_{t}\right|^{2}}{2}+\frac{\left|u_{x}\right|^{2}}{2}\right)\,dx.$$ By construction and a similar argument applied to $K^{-}$, we obtain precisely $$\begin{aligned}
\int_{\left|x\right|>R}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}dx\\
\lesssim\int_{\left|x\right|>M_{1}(v)R}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}dx,\nonumber \end{aligned}$$ as claimed.
\[thm:generalC\]Let $\left|v\right|<1$. Suppose $$\partial_{tt}u-\Delta u+V(x,t)u=0$$ and $$\left|V(x,\mu x_{1})\right|\lesssim\frac{1}{\left\langle x\right\rangle ^{2}}$$ for $0\leq\left|\mu\right|<1$. Then
$$\begin{aligned}
\int\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}dx\nonumber \\
\simeq\int\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}dx,\label{eq:generalC}\end{aligned}$$
where the implicit constant depends on $v$ and $V$.
We first show $$\begin{aligned}
\int\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}dx\nonumber \\
\lesssim\int\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}dx.\end{aligned}$$ This follows from Lemma \[lem:upper\] which implies for $R>0$, $$\begin{aligned}
\int_{\left|x\right|>M(v)R}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}dx\nonumber \\
\lesssim\int_{\left|x\right|>R}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}dx\\
\lesssim\int\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}dx\nonumber \end{aligned}$$ with an implicit constant independent of $R$. By the monotone convergence theorem, letting $R\rightarrow0$, we get $$\begin{aligned}
\int\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}dx\nonumber \\
\lesssim\int\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}dx.\end{aligned}$$ Next, we establish the converse inequality. By Lemma \[lem:lower\], for $R>0$, $$\begin{aligned}
\int_{\left|x\right|>R}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}dx\nonumber \\
\lesssim\int_{\left|x\right|>M_{1}(v)R}\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}dx\\
\lesssim\int\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}dx.\end{aligned}$$ with an implicit constant independent of $R$. Letting $R\rightarrow0$, from the dominated convergence theorem, it follows that $$\begin{aligned}
\int\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}dx\\
\lesssim\int\left|\nabla_{x}u\left(\gamma x_{1},x_{2},x_{3},\gamma vx_{1}\right)\right|^{2}+\left|\partial_{t}u\left(\gamma x_{1},x_{2},x_{3},\gamma vx_{1}\right)\right|^{2}dx.\nonumber \end{aligned}$$ Hence, we conclude $$\begin{aligned}
\int\left|\nabla_{x}u\left(\gamma x_{1},x_{2},x_{3},\gamma vx_{1}\right)\right|^{2}+\left|\partial_{t}u\left(\gamma x_{1},x_{2},x_{3},\gamma vx_{1}\right)\right|^{2}dx\nonumber \\
\simeq\int\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}dx.\end{aligned}$$ The theorem is proved.
Agmon’s estimates via wave equations
------------------------------------
As a by product of Theorem \[thm:generalC\], we show Agmon’s estimates [@Agmon] for the decay of eigenfunctions associated with negative eigenvalues of $$H=-\Delta+V.$$ Again, we restrict our attention to the class of potentials satisfying the assumption $$\left|V(x)\right|\leq C_{V}\left(1+x^{2}\right)^{-1},\,\,\forall x\in\mathbb{R}^{3}.\label{eq:decay}$$ As in Remark \[rem:dim\], all arguments and discussions are valid for $x\in\mathbb{R}^{n}$.
\[thm:Agmon\]Let $V$ satisfy the assumption . Suppose $\phi\in W^{2,2}$ $$-\Delta\phi+V\phi=E\phi,\,\,E<0.\label{eq:eigen}$$ Then $\forall\alpha\in[0,2\sqrt{-E})$ $$\int_{\mathbb{R}^{3}}e^{\alpha\left|x\right|}\left|\phi(x)\right|^{2}\,dx\simeq\int_{\mathbb{R}^{3}}\left|\phi(x)\right|^{2}\,dx,\label{eq:L2decay}$$ with implicit constants depending on $\alpha,\,V$.
Furthermore, if $V\in W^{k}\mbox{\ensuremath{\left(\mathbb{R}^{3}\right)} }$where $k>\frac{3}{2}$ and for $0\leq i\leq k$, $$\left|\nabla^{i}V(x)\right|\leq C_{V,i}\left(1+x^{2}\right)^{-1}$$ then $$\left|\phi(x)\right|\lesssim e^{-\frac{\alpha}{2}\left|x\right|}.\label{eq:pointwise}$$
It suffices to show $\forall\alpha\in[0,2\sqrt{-E})$ $$\int_{\mathbb{R}^{3}}e^{\alpha\left|x_{j}\right|}\left|\phi(x)\right|^{2}\,dx\simeq\int_{\mathbb{R}^{3}}\left|\phi(x)\right|^{2}\,dx,\,\,\forall j=1,2,3.$$ Without loss of generality, we pick $j=1$.
With Theorem \[thm:generalC\], we know if $u_{tt}+Hu=0$, then with $\left|v\right|<1$ $$\begin{aligned}
\int\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}+\left|\partial_{t}u\left(\gamma x_{1},x_{2},x_{3},vx_{1}\right)\right|^{2}dx\nonumber \\
\simeq\int\left|\nabla_{x}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}+\left|\partial_{t}u\left(x_{1},x_{2},x_{3},0\right)\right|^{2}dx.\end{aligned}$$ We can rewrite the above result using half-wave operator $e^{it\sqrt{H}}$, for $f\in L^{2}$ then $$\int\left(\left|e^{ivt\sqrt{H}}f\right|_{t=x_{1}}^{2}\right)\,dx\simeq\int\left|f\right|^{2}\,dx.$$ We pick $f=\phi$ satisfying $$-\Delta\phi+V\phi=E\phi,\,\,E<0,$$ then $$\int e^{-vx_{1}2\sqrt{-E}}\left|\phi\right|^{2}\,dx=\int\left|e^{-vx_{1}\sqrt{-E}}\phi\right|^{2}\,dx\simeq\int\left|\phi\right|^{2}\,dx.$$ With $v$ replaced by $-v$, we obtain $$\int e^{vx_{1}2\sqrt{-E}}\left|\phi\right|^{2}\,dx=\int\left|e^{-vx_{1}\sqrt{-E}}\phi\right|^{2}\,dx\simeq\int\left|\phi\right|^{2}\,dx.$$ Therefore, $$\int e^{\left|2v\sqrt{-E}\right|\left|x_{1}\right|}\left|\phi\right|^{2}\,dx\simeq\int\left|\phi\right|^{2}\,dx.$$ Fixed an $\alpha\in[0,2\sqrt{-E})$, we can find $\left|v\right|\in[0,1)$ such that $\alpha=\left|2v\sqrt{-E}\right|$, then it follows that $$\int e^{\alpha\left|x_{1}\right|}\left|\phi\right|^{2}\,dx\simeq\int\left|\phi\right|^{2}\,dx.$$ Therefore the estimate is proved.
Next we move to . Since $$-\Delta\phi+V\phi=E\phi,\,\,E<0,$$ then $$\int\left|\nabla\phi\right|^{2}dx+\int V\left|\phi\right|^{2}dx=E\int\left|\phi\right|^{2}dx,$$ $$\int\left|\nabla\phi\right|^{2}dx\leq\left\Vert V\right\Vert _{L^{\infty}}\int\left|\phi\right|^{2}dx.$$ Differentiating the equation, for any multi-index $\beta$ $$-\Delta\left(\partial^{\beta}\phi\right)+\partial^{\beta}\left(V\phi\right)=E\partial^{\beta}\phi$$ we can conclude $$\int\left|\nabla\left(\partial^{\beta}\phi\right)\right|^{2}dx\leq\int\partial^{\beta}\left(V\phi\right)\partial^{\beta}\phi\,dx.$$ By induction, we obtain $$\int\left|\nabla\left(\partial^{\beta}\phi\right)\right|^{2}dx\leq\left\Vert V\right\Vert _{W^{\left|\beta\right|,\infty}}\int\left|\phi\right|^{2}dx.$$ Let $\psi$ be a smooth bump-cutoff function such that $\psi=1$ in $B_{1}(0)$ and $\psi=0$ in $\mathbb{R}^{3}\backslash B_{2}(0)$. We localize our estimate, $$\int\left(-\Delta\phi(x)+V\phi(x)\right)\bar{\phi}(x)\psi^{2}(x-y)\,dx=E\int\left|\phi(x)\right|^{2}\psi^{2}(x-y)\,dx.$$ Integrating by parts, we know $$\begin{aligned}
\int\left(-\Delta\phi(x)+V\phi(x)\right)\bar{\phi}(x)\psi^{2}(x-y)\,dx\nonumber \\
=\int V\left|\phi(x)\right|^{2}\psi^{2}(x-y)\,dx\\
+\int\left|\nabla\phi(x)\right|^{2}\psi^{2}(x-y)\,dx\nonumber \\
+2\int\nabla\phi(x)\bar{\phi}(x)\psi(x-y)\nabla\psi(x-y)\,dx.\nonumber \end{aligned}$$ Therefore, by the Cauchy-Schwarz inequality, $$\begin{aligned}
\int\left|\nabla\phi(x)\right|^{2}\psi^{2}(x-y)\,dx & \lesssim & E\int\left|\phi(x)\right|^{2}\psi^{2}(x-y)\,dx\nonumber \\
& & +\int V\left|\phi(x)\right|^{2}\psi^{2}(x-y)\,dx\\
& & +2\int\left|\phi(x)\nabla\psi(x-y)\right|^{2}\,dx.\nonumber \end{aligned}$$ It follows, $$\sup_{y\in\mathbb{R}^{3}}\int_{\left|x-y\right|\leq1}\left|\nabla\phi(x)\right|^{2}\,dx\lesssim\left(\left\Vert V\right\Vert _{L^{\infty}}+1+\left|E\right|\right)\int_{\left|x-y\right|\leq2}\left|\phi(x)\right|^{2}\,dx.$$ Inductively as above, we have $$\sup_{y\in\mathbb{R}^{3}}\int_{\left|x-y\right|\leq1}\left|\nabla\left(\partial^{\beta}\phi\right)\right|^{2}\,dx\lesssim\left(\left\Vert V\right\Vert _{W^{\left|\beta\right|,\infty}}+1+\left|E\right|\right)\int_{\left|x-y\right|\leq2}\left|\phi(x)\right|^{2}\,dx.$$ Finally by Sobolev’s embedding theorem, $$\begin{aligned}
\sup_{y\in\mathbb{R}^{3}}\sup_{\left|x-y\right|\leq1}\left|\phi(x)\right|^{2} & \lesssim & \sum_{\beta\leq k}\sup_{y\in\mathbb{R}^{3}}\int_{\left|x-y\right|\leq1}\left|\left(\partial^{\beta}\phi\right)\right|^{2}\,dx\nonumber \\
& \lesssim & \int_{\left|x-y\right|\leq2}\left|\phi(x)\right|^{2}\,dx\\
& \lesssim & e^{-\alpha\left(\left|y\right|-2\right)}\int_{\left|x-y\right|\leq2}e^{\alpha\left|x\right|}\left|\phi(x)\right|^{2}\,dx\nonumber \\
& \lesssim & e^{-\alpha\left|y\right|}\int\left|\phi(x)\right|^{2}\,dx\nonumber \\
& \lesssim & e^{-\alpha\left|y\right|}.\nonumber \end{aligned}$$ Hence, $$\sup_{y\in\mathbb{R}^{3}}\left|\phi(y)\right|\lesssim e^{-\frac{\alpha}{2}\left|y\right|}$$ as claimed.
Endpoint Strichartz Estimates\[sec:revered\]
============================================
In [@GC2], we analyzed the endpoint reversed Strichartz estimates along slanted lines for both homogeneous and inhomogeneous cases. Intuitively, the reversed Strichartz estimates along slanted lines are based on the fact that the fundamental solutions of the wave equation in $\mathbb{R}^{3}$ is supported on the light cone. For fixed $x$, the propagation will only meet the light cone once. Here we further note that for a general smooth trajectory with velocity strictly less than $1$, it will also only intersect the light cone once. In this section, we will study the reversed Strichartz estimates along general trajectories in several different settings.
Recall that a trajectory $\vec{v}(t)\in\mathbb{R}^{3}$ is called an admissible trajectory if $\vec{v}(t)$ is $C^{1}$ and there exist $0\leq\ell<1$ such $\left|\vec{v}(t)\right|<\ell<1$ for $t\in\mathbb{R}$.
Free wave equations
-------------------
Let $\vec{v}(t)$ be an admissible trajectory. Set $$u(x,t)=\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f+\cos\left(t\sqrt{-\Delta}\right)g+\int_{0}^{t}\frac{\sin\left(\left(t-s\right)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}F(s)\,ds$$ and $$u^{S}(x,t):=u\left(x+\vec{v}(t),t\right).$$ First of all, for the standard case, one has $$\left\Vert u\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}+\left\Vert F\right\Vert _{L_{x}^{\frac{3}{2},1}L_{t}^{2}}.$$ Consider the estimates along the trajectory $\vec{v}(t)$, one has $$\left\Vert u^{S}(x,t)\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}+\left\Vert F\right\Vert _{\dot{W}_{x}^{1,1}L_{t}^{2}}.$$ If $\vec{v}(t)$ does not change the direction, then $$\left\Vert u^{S}(x,t)\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}+\left\Vert F\right\Vert _{L_{d}^{1}L_{\widehat{d}}^{2,1}L_{t}^{2}},$$ where $d$ is the direction of $\vec{v}(t)$.
Let $\vec{\mu}(t)$ be another admissible trajectory, we have the same estimates above with $F$ replaced by $$F^{S'}(x,t):=F\left(x+\vec{\mu}(t),t\right).$$
For the first term, $$u_{1}(x,t)=\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f=\frac{1}{4\pi t}\int_{\left|x-y\right|=t}f(y)\,\sigma\left(dy\right).$$ So in polar coordinates, $$\begin{aligned}
\left\Vert u_{1}^{S}(x,t)\right\Vert _{L_{t}^{2}[0,\infty)}^{2} & \lesssim & \int_{0}^{\infty}\left(\int_{\mathbb{S}}f(x+\vec{v}(r)+r\omega)r\,d\omega\right)^{2}dr\end{aligned}$$
$$\begin{aligned}
\int_{0}^{\infty}\left(\int_{\mathbb{S}}f(x+\vec{v}(r)+r\omega)r\,d\omega\right)^{2}dr & =\int_{0}^{\infty}\left(\int_{\mathbb{S}}f\left[x+\left(\vec{v}(r)/r+\omega\right)r\right]r\,d\omega\right)^{2}dr\nonumber \\
& \lesssim\left(\int_{0}^{\infty}\int_{\mathbb{S}}f(x+r\eta)^{2}r^{2}\,d\eta dr\right)\left(\int_{\mathcal{S}_{v}^{2}}d\eta\right)\end{aligned}$$
Let $$\eta=\vec{v}(r)/r+\omega.$$ Since $\left|v'(t)\right|<\lambda<1$, the Jacobian of this change of variable is bounded from below uniformly. Therefore, $$\begin{aligned}
\int_{0}^{\infty}\left(\int_{\mathbb{S}}f\left[x+\left(\vec{v}(r)/r+\omega\right)r\right]r\,d\omega\right)^{2}dr & \simeq\int_{0}^{\infty}\left(\int_{\mathbb{S}}f\left(x+\eta r\right)r\,d\eta\right)^{2}dr\nonumber \\
& \lesssim\left(\int_{0}^{\infty}\int_{\mathbb{S}}f(x+r\eta)^{2}r^{2}\,d\eta dr\right)\left(\int_{\mathcal{S}_{v}^{2}}d\eta\right)\nonumber \\
& \lesssim\left\Vert f\right\Vert _{L^{2}}^{2}.\end{aligned}$$ A similar argument holds for $$u_{2}(x,t)=\cos\left(t\sqrt{-\Delta}\right)g.$$ Therefore $$\left\Vert u_{1}^{S}\right\Vert _{L_{x}^{\infty}L_{t}^{2}}+\left\Vert u_{2}^{S}\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$ In particular, $$\left\Vert u_{1}\right\Vert _{L_{x}^{\infty}L_{t}^{2}}+\left\Vert u_{2}\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$ as claimed.
Next, we consider the inhomogenous case, $$D(x,t)=\int_{0}^{t}\frac{\sin\left((t-s)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}F(s)\,ds.$$ For the standard case, we consider $$\begin{aligned}
\left\Vert \int_{0}^{t}\frac{\sin\left((t-s)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}F(s)\,ds\right\Vert _{L_{t}^{2}} & = & \left\Vert \int_{0}^{t}\int_{\left|x-y\right|=t-s}\frac{1}{\left|x-y\right|}F(y,s)\,\sigma\left(dy\right)ds\right\Vert _{L_{t}^{2}}\\
& = & \left\Vert \int_{\left|x-y\right|\leq t}\frac{1}{\left|x-y\right|}F\left(y,t-\left|x-y\right|\right)\,dy\right\Vert _{L_{t}^{2}}\\
& \lesssim & \int\frac{1}{\left|x-y\right|}\left\Vert F\left(y,t-\left|x-y\right|\right)\right\Vert _{L_{t}^{2}}dy\\
& \lesssim & \sup_{x\in\mathbb{R}^{3}}\int\frac{1}{\left|x-y\right|}\left\Vert F\left(y,t\right)\right\Vert _{L_{t}^{2}}dy\\
& \lesssim & \left\Vert F\right\Vert _{L_{x}^{\frac{3}{2},1}L_{t}^{2}}.\end{aligned}$$ Therefore, indeed, $$\left\Vert D\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\left\Vert F\right\Vert _{L_{x}^{\frac{3}{2},1}L_{t}^{2}}.$$ Now we consider the estimate along an admissible trajectory $\vec{v}(t)\in\mathbb{R}^{3}$.
We first notice that from the above discussion or the argument in Appendix D, $$T:=\frac{e^{it\sqrt{-\Delta}}}{\sqrt{-\Delta}}$$ is a bounded operator from $L_{x}^{2}$ to $L_{x}^{\infty}L_{t}^{2}$. Also the operator $T^{S}$: $$T^{S}f:=\left(Tf\right)^{S}=\left(\frac{e^{it\sqrt{-\Delta}}}{\sqrt{-\Delta}}f\right)^{S}$$ is a bounded operator from $L_{x}^{2}$ to $L_{x}^{\infty}L_{t}^{2}$.
Writing down the inhomogeneous evolution explicitly, one has $$\begin{aligned}
\left\Vert \int_{0}^{t}\frac{\sin\left((t-s)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}F(s)\,ds\right\Vert _{L_{t}^{2}} & =\left\Vert \int_{0}^{t}\int_{\left|x-y\right|=t-s}\frac{1}{\left|x-y\right|}F(y,s)\,\sigma\left(dy\right)ds\right\Vert _{L_{t}^{2}}\nonumber \\
& \lesssim\left\Vert \int_{\left|x-y\right|\leq t}\frac{1}{\left|x-y\right|}F\left(y,t-\left|x-y\right|\right)\,dy\right\Vert _{L_{t}^{2}}\end{aligned}$$ Therefore, $$\begin{aligned}
\sup_{x\mathbb{\in R}^{3}}\left\Vert \int_{\left|x-y\right|\leq t}\frac{1}{\left|x-y\right|}F\left(y,t-\left|x-y\right|\right)\,dy\right\Vert _{L_{t}^{2}} & \lesssim\sup_{x\mathbb{\in R}^{3}}\left\Vert \int\frac{1}{\left|x-y\right|}\left|F\left(y,t-\left|x-y\right|\right)\,\right|dy\right\Vert _{L_{t}^{2}}\nonumber \\
& \lesssim\sup_{x\mathbb{\in R}^{3}}\left\Vert \int_{0}^{\infty}\frac{\sin\left((t-s)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}\left|F(s)\right|\,ds\right\Vert _{L_{t}^{2}}\nonumber \\
& \lesssim\sup_{x\mathbb{\in R}^{3}}\left\Vert \Re\left(TT^{*}\sqrt{-\Delta}\left|F\right|\right)\right\Vert _{L_{t}^{2}}.\end{aligned}$$ Hence we know $$\begin{aligned}
\sup_{x\mathbb{\in R}^{3}}\left\Vert D^{S}\left(x+v(t),t\right)\right\Vert _{L_{t}^{2}} & \lesssim\sup_{x\mathbb{\in R}^{3}}\left\Vert \Re\left(T^{S}T^{*}\sqrt{-\Delta}\left|F\right|\right)\right\Vert _{L_{t}^{2}}\nonumber \\
& \lesssim\left\Vert \sqrt{-\Delta}\left|F\right|(x,t)\right\Vert _{L_{x}^{1}L_{t}^{2}}\nonumber \\
& \left\Vert F\right\Vert _{\dot{W}_{x}^{1,1}L_{t}^{2}}.\end{aligned}$$ If the trajectory does not change the direction, we can obtain an estimate which does not require $\sqrt{-\Delta}F$ by a similar argument to the estimates along slanted lines in [@GC2]. Without loss of generality, we assume the direction of the trajectory is along $x_{1}$. Then $$D^{S}(x,t)=\int_{0}^{t}\int_{\left|x+\vec{v}(t)-y\right|=t-s}\frac{F(y,s)}{\left|x+\vec{v}(t)-y\right|}\,\sigma\left(dy\right)ds$$ and $$\begin{aligned}
\left\Vert D^{S}(x,\cdot)\right\Vert _{L_{t}^{2}} & = & \left\Vert \int_{0}^{t}\int_{\left|x+\vec{v}(t)-y\right|=t-s}\frac{F(y,s)}{\left|x+\vec{v}(t)-y\right|}\,\sigma\left(dy\right)ds\right\Vert _{L_{t}^{2}}\nonumber \\
& = & \left\Vert \int_{\left|y\right|\leq t}\frac{F(x+\vec{v}(t)-y,t-\left|y\right|)}{\left|y\right|}\,dy\right\Vert _{L_{t}^{2}}\\
& \leq & \left\Vert \int_{\mathbb{R}^{3}}\frac{\left|F(x-y,t-\left|y+\vec{v}(t)\right|)\right|}{\left|y+\vec{v}(t)\right|}\,dy\right\Vert _{L_{t}^{2}}\nonumber \\
& \leq & \left\Vert \int_{\mathbb{R}^{3}}\frac{\left|F(x-y,t-\left|y+\vec{v}(t)\right|)\right|}{\sqrt{y_{2}^{2}+y_{3}^{2}}}\,dy\right\Vert _{L_{t}^{2}},\nonumber \end{aligned}$$ where in the third line, we used a change of variable and for the last inequality and reduce the norm of $y$ to the norm of the component of $y$ orthogonal to the direction of the motion.
Finally, $$\left\Vert \int_{\mathbb{R}^{3}}\frac{F(x-y,t-\left|y+\vec{v}(t)\right|)}{\sqrt{y_{2}^{2}+y_{3}^{2}}}\,dy\right\Vert _{L_{t}^{2}}\leq\int_{\mathbb{R}^{3}}\frac{\left\Vert F(x-y,t-\left|y+\vec{v}(t)\right|)\right\Vert _{L_{t}^{2}}}{\sqrt{y_{2}^{2}+y_{3}^{2}}}\,dy$$ For fixed $y$, if we apply a change of variable of $t$ here, the Jacobian is bounded by $1-|v'|$ and $1+|v'|$, so $$\begin{aligned}
\int_{\mathbb{R}^{3}}\frac{\left\Vert F(x-y,t-\left|y+vt\right|)\right\Vert _{L_{t}^{2}}}{\sqrt{y_{2}^{2}+y_{3}^{2}}}\,dy & \lesssim & \int_{\mathbb{R}^{3}}\frac{\left\Vert F(x-y,\cdot)\right\Vert _{L_{t}^{2}}}{\sqrt{y_{2}^{2}+y_{3}^{2}}}dy\nonumber \\
& \lesssim & \left\Vert F\right\Vert _{L_{x_{1}}^{1}L_{\widehat{x_{1}}}^{2,1}L_{t}^{2}}\end{aligned}$$ where $\widehat{x_{1}}$ denotes the subspace orthogonal to $x_{1}$ (more generally, the subspace orthogonal to the direction of the motion). Here $L^{2,1}$ is the Lorentz norm and the last inequality follows from Hölder’s inequality of Lorentz spaces. Therefore, $$\left\Vert D^{S}\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\left\Vert F\right\Vert _{L_{x_{1}}^{1}L_{\widehat{x_{1}}}^{2,1}L_{t}^{2}}.\label{eq:ersI-1}$$ as claimed.
Finally, we consider the estimate with the source term $F$ along an admissible trajectory. This follows from a duality or the same argument as in [@GC2]. So we conclude that $$\left\Vert D^{S}\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\left\Vert F^{S'}\right\Vert {}_{\dot{W}_{x}^{1,1}L_{t}^{2}},$$ and $$\left\Vert D^{S}\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\left\Vert F^{S'}\right\Vert {}_{L_{x_{1}}^{1}L_{\widehat{x_{1}}}^{2,1}L_{t}^{2}}$$ provided $\vec{v}(t)$ moves along $x_{1}$.
The theorem is proved.
We notice that from Sobolev’s embedding, $$\dot{W}_{x}^{1,1}\hookrightarrow L^{\frac{3}{2},1}.$$ Therefore indeed, the estimates along general curves requires slightly more regularity than the standard cases.
To end this subsection, we consider a truncated version which appears naturally in bootstrap arguments, for example in [@GC2].
Let $\vec{v}(t)$ be an admissible trajectory. For fixed $A>0$, setting $$D_{A}=\int_{0}^{t-A}\frac{\sin\left((t-s)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}F(s)\,ds,$$ we have $$\left\Vert D_{A}^{S}\right\Vert _{L_{x}^{\infty}L_{t}^{2}[A,\infty)}\lesssim\frac{1}{A}\left\Vert F\right\Vert _{L_{x}^{1}L_{t}^{2}},$$ and $$\left\Vert D_{A}^{S}\right\Vert _{L_{t}^{2}}\lesssim\frac{1}{A}\left\Vert F^{S'}\right\Vert _{L_{x}^{1}L_{t}^{2}},$$ where $$F^{S'}:=F\left(x+\vec{h}(t),t\right)$$ with $\vec{h}(t)$ is an admissible trajectory.
By Kirchhoff’s formula, $$\begin{aligned}
\left\Vert D_{A}^{S}\right\Vert _{L_{t}^{2}} & \lesssim\left\Vert \int_{A\leq\left|y\right|}\frac{\left|F(x+\vec{v}(t)-y,t-\left|y\right|)\right|}{\left|y\right|}\,dy\right\Vert _{L_{t}^{2}}\nonumber \\
& \lesssim\frac{1}{A}\left\Vert F\right\Vert _{L_{x}^{1}L_{t}^{2}}.\end{aligned}$$ By duality or the same arguments as in [@GC2], we can also obtain,
$$\,\left\Vert D_{A}^{S}\right\Vert _{L_{t}^{2}}\lesssim\frac{1}{A}\left\Vert F^{S'}\right\Vert _{L_{x}^{1}L_{t}^{2}},$$
as claimed
Perturbed wave equations.
-------------------------
Finally, we extend all of our estimates to the perturbed Hamiltonian. In [@GC2], we relied on Duhamel expansion of the perturbed evolution, the estimates along trajectories for free ones and the standard estimates for the perturbed ones. Here we present an alternative approach based on the structure formula of the wave operators as in Section \[sec:Prelim\].
We only present the standard cases and other estimates can be obtained similarly.
\[thm:PStriRStrich-1\]Let $\vec{v}(t)$ be an admissible trajectory. Suppose $$H=-\Delta+V$$ admits neither eigenfunctions nor resonances at $0$. Set $$u(x,t)=\frac{\sin\left(t\sqrt{H}\right)}{\sqrt{H}}P_{c}f+\cos\left(t\sqrt{-H}\right)P_{c}g+\int_{0}^{t}\frac{\sin\left(\left(t-s\right)\sqrt{H}\right)}{\sqrt{H}}P_{c}F(s)\,ds$$ and $$u^{S}(x,t):=u\left(x+\vec{v}(t),t\right),$$ where $P_{c}$ is the projection onto the continuous spectrum of $H$.
First of all, for the standard endpoint reversed Strichartz estimates, we have
$$\left\Vert \frac{\sin\left(t\sqrt{H}\right)}{\sqrt{H}}P_{c}f+\cos\left(t\sqrt{H}\right)P_{c}g\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}},\label{eq:PEndRSch-1}$$
$$\left\Vert \int_{0}^{t}\frac{\sin\left((t-s)\sqrt{H}\right)}{\sqrt{H}}P_{c}F(s)\,ds\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\left\Vert F\right\Vert _{L_{x}^{\frac{3}{2},1}L_{t}^{2}}.\label{eq:PEndRSIn-1}$$
Consider the estimates along the trajectory $\vec{v}(t)$, one has $$\left\Vert u^{S}(x,t)\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}+\left\Vert F\right\Vert _{\dot{W}_{x}^{1,1}L_{t}^{2}}.$$ If $\vec{v}(t)$ does not change the direction, then $$\left\Vert u^{S}(x,t)\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}+\left\Vert F\right\Vert _{L_{d}^{1}L_{\widehat{d}}^{2,1}L_{t}^{2}},$$ where $d$ is the direction of $\vec{v}(t)$.
Let $\vec{\mu}(t)$ be another admissible trajectory, we have the same estimates above with $F$ replaced by $$F^{S'}(x,t):=F\left(x+\vec{\mu}(t),t\right).$$
It suffices to consider $$\frac{\sin\left(t\sqrt{H}\right)}{\sqrt{H}}P_{c}f.$$ By construction, $$\frac{\sin\left(t\sqrt{H}\right)}{\sqrt{H}}P_{c}=W^{+}\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}\left(W^{+}\right)^{*}.$$ $$\frac{\sin\left(t\sqrt{H}\right)}{\sqrt{H}}P_{c}f=W^{+}\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}\left(W^{+}\right)^{*}P_{c}f.$$ Denoting $$h=\left(W^{+}\right)^{*}P_{c}f,$$ we have $$\|P_{c}f\|_{L^{2}}\simeq\|h\|_{L^{2}}.$$ Setting $$G=\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}h,$$ by Theorem \[thm:structure\], it is sufficient to consider the boundedness of $$G+\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}g(x,y,\tau)G\left(S_{\tau}x+y\right)\,dyd\tau.$$ Clearly, by the endpoint reversed Strichartz estimate for the free case, $$\|G\|_{L_{x}^{\infty}L_{t}^{2}}\lesssim\|h\|_{L^{2}}\simeq\|P_{c}f\|_{L^{2}}.$$ Next, by Minkowski’s inequality, $$\begin{aligned}
\left\Vert \int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}g(x,y,\tau)G\left(S_{\tau}x+y\right)\,dyd\tau\right\Vert {}_{L_{x}^{\infty}L_{t}^{2}}\\
\lesssim\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}\left\Vert g(x,y,\tau)G\left(S_{\tau}x+y\right)\right\Vert {}_{L_{x}^{\infty}L_{t}^{2}}dyd\tau\nonumber \end{aligned}$$ $$\left\Vert g(x,y,\tau)G\left(S_{\tau}x+y\right)\right\Vert {}_{L_{x}^{\infty}L_{t}^{2}}\lesssim\left\Vert g(x,y,\tau)\right\Vert _{L_{x}^{\infty}}\left\Vert G\left(S_{\tau}x+y\right)\right\Vert {}_{L_{x}^{\infty}L_{t}^{2}}.$$ Since reflections with respect to a fixed plane and translations commute with the solution of a free wave equation, we obtain $$G\left(S_{\tau}x+y\right)=\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}h\left(S_{\tau}x+y\right).$$ Therefore, $$\left\Vert G\left(S_{\tau}x+y\right)\right\Vert {}_{L_{x}^{\infty}L_{t}^{2}}\lesssim\|h\left(S_{\tau}x+y\right)\|_{L^{2}}\lesssim\|h\|_{L^{2}}\simeq\|P_{c}f\|_{L^{2}}.$$ It follows $$\begin{aligned}
\left\Vert G+\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}g(x,y,\tau)G\left(S_{\tau}x+y\right)\,dyd\tau\right\Vert {}_{L_{x}^{\infty}L_{t}^{2}}\\
\lesssim\left(1+\int_{\mathbb{S}^{2}}\int_{\mathbb{R}^{3}}\left\Vert g(x,y,\tau)\right\Vert _{L_{x}^{\infty}}dyd\tau\right)\|P_{c}f\|_{L^{2}}\lesssim\|f\|_{L^{2}}.\nonumber \end{aligned}$$ Then we conclude $$\left\Vert \frac{\sin\left(t\sqrt{H}\right)}{\sqrt{H}}P_{c}f\right\Vert _{_{L_{x}^{\infty}L_{t}^{2}}}\lesssim\|f\|_{L^{2}},$$ as claimed.
Wave equations with moving potentials
-------------------------------------
Finally in this section, we consider the wave equation $$\partial_{tt}u-\Delta u+V\left(x-\vec{\mu}t\right)u=0$$ $$u(x,0)=g(x),\,u_{t}(x,0)=f(x).$$ Again without of loss of generality, we assume $\vec{\mu}$ is along $\vec{e}_{1}$ and $\vec{\mu}<1$. Recall that associated to this model, we define $$H=-\Delta+V\left(\sqrt{1-\left|\vec{\mu}\right|^{2}}x_{1},x_{2},x_{3}\right).$$
Let $m_{1},\,\ldots,\,m_{w}$ be the normalized bound states of $H$ associated to the negative eigenvalues $-\lambda_{1}^{2},\,\ldots,\,-\lambda_{w}^{2}$ respectively (notice that by our assumptions, $0$ is not an eigenvalue). We denote by $P_{b}$ the projections on the the bound states of $H$ , respectively, and let $P_{c}=Id-P_{b}$.
Performing a Lorentz transformation $L$ with respect to the moving frame $\left(x-\vec{\mu}t,t\right)$, we have $$\partial_{t't'}u_{L}+Hu_{L}=0,$$ $$u_{L}(x',0)=\tilde{g}(x'),\,\left(u_{L}\right)_{t}(x',0)=\tilde{f}(x')$$ and $$\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}\simeq\|\tilde{f}\|_{L^{2}}+\|\tilde{g}\|_{\dot{H}^{1}}.$$ We can write $$u_{L}\left(x',t'\right)=\sum_{i=1}^{w}a_{i}(t')m_{i}(x')+r_{L}\left(x',t'\right),$$ such that $$P_{c}r_{L}=r_{L}.$$ Return to our original coordinate, we have a decomposition for $u$ that $$u(x,t)=\sum_{i=1}^{w}a_{i}\left(\gamma(t-vx_{1})\right)\left(m_{i}\right)_{\mu}\left(x,t\right)+r\left(x,t\right)\label{eq:decomp}$$ where $$\left(m_{i}\right)_{\mu}(x,t)=m_{i}\left(\gamma\left(x_{1}-\mu t\right),x_{2},x_{3}\right).$$
\[thm:endmove\]Let $\vec{v}(t)\in\mathbb{R}^{3}$ be an admissible trajectory. With the notations from above, we have $$\left\Vert r^{S}\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}},$$ in particular, $$\int_{0}^{\infty}\int_{\mathbb{R}^{3}}\frac{1}{\left\langle x-\vec{h}(t)\right\rangle ^{\alpha}}r^{2}(x,t)\,dxdt\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$
Notice that if $\vec{v}(t)$ is an admissible trajectory in our original frame $\left(x,t\right)$, then if we perform a Lorentz transformation $L(\vec{\mu})$, in the new frame, the trajectory $v(t)$ can be written as $\vec{\nu}(t')$ with $\left|\vec{v}'(t')\right|<\phi\left(\lambda,\vec{\mu}\right)<1.$ In other words, in the new coordinate, the trajectory is still admissible. Then for fixed $x\in\mathbb{R}^{3}$, $$\int\left|r^{S}(x,t)\right|^{2}dt\lesssim\sup_{x'\in\mathbb{R}^{3}}\int\left|r_{L}^{S'}(x',t')\right|^{2}dt',$$ where $$r_{L}^{S'}\left(x',t'\right)=r_{L}\left(x'+\vec{v}'(t'),t'\right).$$ By construction and Theorem \[thm:generalC\], $$\sup_{x'\in\mathbb{R}^{3}}\int\left|r_{L}^{S'}(x',t')\right|^{2}dt'\lesssim\left(\|\tilde{f}\|_{L^{2}}+\|\tilde{g}\|_{\dot{H}^{1}}\right)^{2}\simeq\left(\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}\right)^{2}$$ and hence $$\left\Vert r^{S}\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$ The theorem is proved.
Strichartz Estimates and Energy Estimates\[sec:one\]
====================================================
In this section, we establish Strichartz estimates and energy estimates for scattering states to the wave equation $$\partial_{tt}u-\Delta u+V\left(x-\vec{v}(t)\right)u=0,$$ $$u(x,0)=g(x),\,u_{t}(x,0)=f(x)$$ with $$\left|\vec{v}(t)-\vec{\mu}t\right|\lesssim\left\langle t\right\rangle ^{-\beta},\,\beta>1,\,\left|\vec{\mu}\right|<1.$$ To simplify the problem, we assume $$H=-\Delta+V\left(\sqrt{1-\left|\vec{\mu}\right|^{2}}x_{1},x_{2},x_{3}\right)$$ only has one bound state $m$ such that $$Hm=-\lambda^{2}m,\,\lambda>0.$$ One can observe that our arguments work for the general case.
We start with reversed Strichartz estimates.
\[thm:revemoving\]Let $\vec{h}(t)$ be an admissible trajectory and $u$ be a scattering in the sense of Definition \[AO\]. Then for $$u^{S}(x,t)=u\left(x+\vec{h}(t),t\right)$$ one has $$\left\Vert u^{S}(x,t)\right\Vert _{L_{x}^{\infty}L_{t}^{2}[0,\infty)}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$ In particular, it implies for $\alpha>3$, $$\int_{0}^{\infty}\int_{\mathbb{R}^{3}}\frac{1}{\left\langle x-\vec{h}(t)\right\rangle ^{\alpha}}u^{2}(x,t)\,dxdt\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.\label{eq:moveweighted}$$
First of all, we need to understand the evolution of bound states. Writing the equation as $$\partial_{tt}u-\Delta u+V\left(x-\vec{\mu t}\right)u=\left[V\left(x-\vec{\mu}t\right)-V\left(x+\vec{v}(t)\right)\right]u.\label{eq:Veq}$$ Recall that we assume $\vec{\mu}$ is along $x_{1}$. Suppose $u(x,t)$ is a scattering state. As in , we decompose the evolution as following, $$u(x,t)=a\left(\gamma(t-\mu x_{1})\right)m_{\mu}\left(x,t\right)+r(x,t)\label{eq:evolution}$$ where $$m_{\mu}(x,t)=m\left(\gamma\left(x_{1}-\mu t\right),x_{2},x_{3}\right)$$ and $$P_{c}\left(H\right)r_{L}=r_{L}.$$ Performing the Lorentz transformation $L$ with respect to the moving frame $\left(x-\vec{\mu}t,t\right)$, we have $$u_{L}(x',t')=a\left(t'\right)m\left(x'\right)+r_{L}(x',t'),\label{eq:evolutionL}$$ and $$\partial_{t't'}u_{L}+Hu_{L}=-M(x',t')u_{L}\label{eq:eqL}$$ where $$M(x',t')=-\left[V\left(x-\vec{\mu t}\right)-V\left(x+\vec{v}(t)\right)\right]_{L}.$$ When $u$ is a scattering state in the sense Definition \[AO\], the scattering condition forces $a(t)$ to go $0$.
Plugging the evolution into the equation and taking inner product with $m$, we get $$\ddot{a}(t')-\lambda^{2}a(t')+a(t')\left\langle Mm,m\right\rangle +\left\langle Mr_{L},m\right\rangle =0$$ Notice that $$\left|M(x',t')\right|\lesssim\frac{1}{\left\langle \gamma\left(t'+\mu x_{1}'\right)\right\rangle ^{\beta}}.$$ One can write $$\ddot{a}(t')-\lambda^{2}a(t')+a(t')c(t')+h(t')=0,\label{eq:aode}$$ $$c(t'):=\left\langle Mm,m\right\rangle$$ and $$h(t'):=\left\langle Mr_{L},m\right\rangle .$$ Since $w$ is exponentially localized by Agmon’s estimate, we know $$\left|c(t')\right|\lesssim e^{-b\left|t'\right|},\,b>0.$$ The existence of the solution to the ODE is clear. We study the long-time behavior of the solution. Write the equation as
$$\ddot{a}(t')-\lambda^{2}a(t')=-\left[a(t')c(t')+h(t')\right],$$
and denote $$N(t'):=-\left[a(t')c(t')+h(t')\right].$$ Then $$a(t')=\frac{e^{\lambda t'}}{2}\left[a(0)+\frac{1}{\lambda}\dot{a}(0)+\frac{1}{\lambda}\int_{0}^{t'}e^{-\lambda s}N(s)\,ds\right]+R(t')$$ where $$\left|R(t')\right|\lesssim e^{-ct'},$$ for some positive constant $c>0$. Therefore, the stability condition forces $$a(0)+\frac{1}{\lambda}\dot{a}(0)+\frac{1}{\lambda}\int_{0}^{\infty}e^{-\lambda s}N(s)\,ds=0.\label{eq:stability}$$ Then under the stability condition , $$a(t')=e^{-\lambda t'}\left[a(0)+\frac{1}{2\lambda}\int_{0}^{\infty}e^{-\lambda s}N(s)ds\right]+\frac{1}{2\lambda}\int_{0}^{\infty}e^{-\lambda\left|t-s\right|}N(s)\,ds.$$ By Young’s inequality, to estimate all $L^{p}$ norms of $a(t')$, it suffices to estimate the $L^{1}$ norm of $h(t')$, see [@GC2].
By Cauchy-Schwarz and Theorem \[thm:endmove\] ,
$$\int_{0}^{\infty}\left|\left\langle Mr_{L},m\right\rangle \right|dt\lesssim\left\Vert r_{L}\right\Vert _{L_{x'}^{\infty}L_{t'}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$ Therefore, $$\left\Vert a(t)\right\Vert _{L^{p}[0,\infty)}\lesssim_{p}\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$ Given $\vec{h}(t)$ an admissible trajectory, set $$B(x,t)=a\left(\gamma(t-\mu x_{1})\right)m_{\mu}\left(x,t\right),$$ $$B^{S}(x,t)=B\left(x+\vec{h}(t),t\right).$$ By Agmon’s estimate, see Theorem \[thm:Agmon\], and the $L^{1}$ norm estimate for $h(t')$, we have $$\left\Vert B^{S}(x,t)\right\Vert _{L_{x}^{\infty}L_{t}^{2}[0,\infty)}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$ By Theorem \[thm:endmove\], we also know $$\left\Vert r^{S}(x,t)\right\Vert _{L_{x}^{\infty}L_{t}^{2}[0,\infty)}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}$$ Therefore, one has $$\left\Vert u^{S}(x,t)\right\Vert _{L_{x}^{\infty}L_{t}^{2}[0,\infty)}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$ We notice that this in particular implies $$\int_{0}^{\infty}\int_{\mathbb{R}^{3}}\frac{1}{\left\langle x-\vec{h}(t)\right\rangle ^{\alpha}}u^{2}(x,t)\,dxdt\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$ The theorem is proved.
Next, we show Strichartz estimates following [@RS; @LSch; @GC2]. In the following, we use the short-hand notation $$L_{t}^{p}L_{x}^{q}:=L_{t}^{p}\left([0,\infty),\,L_{x}^{q}\right).$$
\[thm:StriOneM\]Suppose $u$ is a scattering state in the sense of Definition of \[AO\] which solves $$\partial_{tt}u-\Delta u+V\left(x-\vec{v}(t)\right)u=0\label{eq:weq}$$ with initial data $$u(x,0)=g(x),\,u_{t}(x,0)=f(x).$$ Then for $p>2$, and $(p,q)$ satisfying $$\frac{1}{2}=\frac{1}{p}+\frac{3}{q}$$ we have $$\|u\|_{L_{t}^{p}L_{x}^{q}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.\label{eq:StriOneM}$$
Following [@LSch], we set $A=\sqrt{-\Delta}$ and notice that $$\left\Vert Af\right\Vert _{L^{2}}\simeq\left\Vert f\right\Vert _{\dot{H}^{1}},\,\,\forall f\in C^{\infty}\left(\mathbb{R}^{3}\right).\label{eq:nabla}$$ For real-valued $u=\left(u_{1},u_{2}\right)\in\mathcal{H}=\dot{H}^{1}\left(\mathbb{R}^{3}\right)\times L^{2}\left(\mathbb{\mathbb{R}}^{3}\right)$, we write $$U:=Au_{1}+iu_{2}.$$ From , we know $$\left\Vert U\right\Vert _{L^{2}}\simeq\left\Vert \left(u_{1},u_{2}\right)\right\Vert _{\mathcal{H}}.$$ We also notice that $u$ solves if and only if $$U:=Au+i\partial_{t}u$$ satisfies $$i\partial_{t}U=AU+V\left(x-\vec{v}(t)\right)u,$$ $$U(0)=Ag+if\in L^{2}\left(\mathbb{R}^{3}\right).$$ By Duhamel’s formula, $$U(t)=e^{itA}U(0)-i\int_{0}^{t}e^{-i\left(t-s\right)A}V\left(\cdot-\vec{v}(s)\right)u(s)\,ds.$$ Let $P:=A^{-1}\Re$, then from Strichartz estimates for the free evolution, $$\left\Vert Pe^{itA}U(0)\right\Vert _{L_{t}^{p}L_{x}^{q}}\lesssim\left\Vert U(0)\right\Vert _{L^{2}}.\label{eq:Sfirst}$$ Writing $V=V_{1}V_{2}$ and with the Christ-Kiselev lemma, Lemma \[lem:Christ-Kiselev\], it suffices to bound $$\left\Vert P\int_{0}^{\infty}e^{-i\left(t-s\right)A}V_{1}V_{2}\left(\cdot-\vec{v}(s)\right)u(s)\,ds\right\Vert _{L_{t}^{p}L_{x}^{q}}.$$ We only need to analyze $$\left\Vert P\int_{0}^{\infty}e^{-i\left(t-s\right)A}V_{1}V_{2}\left(\cdot-\vec{v}(s)\right)u(s)\,ds\right\Vert _{L_{t}^{p}L_{x}^{q}}\leq\left\Vert \widetilde{K}\right\Vert _{L_{t,x}^{2}\rightarrow L_{t}^{p}L_{x}^{q}}\left\Vert V_{2}\left(x-\vec{v}(s)\right)u\right\Vert _{L_{t,x}^{2}}$$ where $$\left(\widetilde{K}F\right)(t):=P\int_{0}^{\infty}e^{-i\left(t-s\right)A}V_{1}\left(\cdot-\vec{v}(s)\right)F(s)\,ds.$$ To show $\left\Vert \widetilde{K}\right\Vert _{L_{t,x}^{2}\rightarrow L_{t}^{p}L_{x}^{q}}$ is bounded, we test it against $F\in L_{t,x}^{2}$, clearly, $$\left\Vert \widetilde{K}F\right\Vert _{L_{t}^{p}L_{x}^{q}}\leq\left\Vert Pe^{-itA}\right\Vert _{L^{2}\rightarrow L_{t}^{p}L_{x}^{q}}\left\Vert \int_{0}^{\infty}e^{isA}V_{1}\left(\cdot-\vec{v}(s)\right)F(s)\,ds\right\Vert _{L^{2}}.\label{eq:TKF}$$ The first factor on the right-hand side of is bounded by Strichartz estimates for the free evolution. Consider the second factor, by duality, it is sufficient to show $$\left\Vert V_{1}\left(\cdot-\vec{v}(t)\right)e^{-itA}\phi\right\Vert _{L_{t,x}^{2}}\lesssim\left\Vert \phi\right\Vert _{L^{2}},\,\forall\phi\in L^{2}\left(\mathbb{R}^{3}\right).$$ By our assumption, $$\left|\vec{v}(t)-\vec{\mu}t\right|\lesssim\left\langle t\right\rangle ^{-\beta},\,\beta>1,\,\left|\vec{\mu}\right|<1.$$ Therefore, it reduces to show $$\left\Vert \left(1+\left|x-\vec{\mu}t\right|\right)^{-\frac{1}{2}-\epsilon}e^{-itA}\phi\right\Vert _{L_{t,x}^{2}}\lesssim\left\Vert \phi\right\Vert _{L^{2}},\,\forall\phi\in L^{2}\left(\mathbb{R}^{3}\right).\label{eq:DAFT}$$ Notice that this is a consequence of that the energy of the free wave equation stays comparable under Lorentz transformations, Theorem \[thm:generalC\]. To show estimate , one can apply the Lorentz transformation $L$. In the new frame $\left(x',t'\right)$, then we can use the standard local energy decay for free wave equations, estimate in Appendix B. Finally after applying an inverse transformation back to the original frame, we obtain .
From estimate , one does have $$\left\Vert V_{1}\left(\cdot-\vec{v}(t)\right)e^{-itA}\phi\right\Vert _{L_{t,x}^{2}}\lesssim\left\Vert \phi\right\Vert _{L^{2}}.$$ Therefore, indeed, $$\left\Vert \widetilde{K}\right\Vert _{L_{t,x}^{2}\rightarrow L_{t}^{p}L_{x}^{q}}\leq C.$$ Hence $$\left\Vert P\int_{0}^{\infty}e^{-i\left(t-s\right)A}V_{1}V_{2}\left(\cdot-\vec{v}(s)\right)u(s)\,ds\right\Vert _{L_{t}^{p}L_{x}^{q}}\lesssim\left\Vert V_{2}\left(x-\vec{v}(s)\right)u\right\Vert _{L_{t,x}^{2}}.$$ By our estimate , $$\left\Vert V_{2}\left(x-\vec{v}(s)\right)u\right\Vert _{L_{t,x}^{2}}\lesssim\left(\int_{\mathbb{R}^+}\int_{\mathbb{R}^{3}}\frac{1}{\left\langle x-\vec{v}(t)\right\rangle ^{\alpha}}\left|u(x,t)\right|^{2}dxdt\right)^{\frac{1}{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$ Therefore, $$\left\Vert P\int_{0}^{\infty}e^{-i\left(t-s\right)A}V_{1}V_{2}\left(\cdot-\vec{v}(s)\right)u(s)\,ds\right\Vert _{L_{t}^{p}L_{x}^{q}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$ Hence one can conclude $$\left\Vert u\right\Vert _{L_{t}^{p}L_{x}^{q}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}},$$ as we claimed.
The energy estimates can be established in a similar manner.
\[thm:EnergyOne\]Suppose $u$ is a scattering state in the sense of Definition of \[AO\] which solves $$\partial_{tt}u-\Delta u+V\left(x-\vec{v}(t)\right)u=0$$ with initial data $$u(x,0)=g(x),\,u_{t}(x,0)=f(x).$$ Then we have $$\sup_{t\in\mathbb{R}}\left(\|\nabla u(t)\|_{L^{2}}+\|u_{t}(t)\|_{L^{2}}\right)\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$
Again, we set $A=\sqrt{-\Delta}$ and notice that $$\left\Vert Af\right\Vert _{L^{2}}\simeq\left\Vert f\right\Vert _{\dot{H}^{1}},\,\,\forall f\in C^{\infty}\left(\mathbb{R}^{3}\right).$$ For real-valued $u=\left(u_{1},u_{2}\right)\in\mathcal{H}=\dot{H}^{1}\left(\mathbb{R}^{3}\right)\times L^{2}\left(\mathbb{\mathbb{R}}^{3}\right)$, we write $$U:=Au_{1}+iu_{2}.$$ and we know $$\left\Vert U\right\Vert _{L^{2}}\simeq\left\Vert \left(u_{1},u_{2}\right)\right\Vert _{\mathcal{H}}.$$ We also notice that $u$ solves the original equation if and only if $$U:=Au+i\partial_{t}u$$ satisfies $$i\partial_{t}U=AU+V\left(x-\vec{v}(t)\right)u,$$ $$U(0)=Ag+if\in L^{2}\left(\mathbb{R}^{3}\right).$$ By Duhamel’s formula, $$U(t)=e^{itA}U(0)-i\int_{0}^{t}e^{-i\left(t-s\right)A}\left(V\left(\cdot-\vec{v}(s)\right)u(s)\right)\,ds.$$ From the energy estimate for the free evolution, $$\sup_{t\in\mathbb{R}}\left\Vert e^{itA}U(0)\right\Vert _{L_{x}^{2}}\lesssim\left\Vert U(0)\right\Vert _{L^{2}}.\label{eq:Sfirst-1-3}$$ Writing $V=V_{1}V_{2}$, it suffices to bound $$\sup_{t\in\mathbb{R}}\left\Vert \int_{0}^{\infty}e^{-i\left(t-s\right)A}V_{1}V_{2}\left(\cdot-\vec{v}(s)\right)u(s)\,ds\right\Vert _{L_{x}^{2}}.$$ This is can be bounded in a same manner as Theorem \[thm:StriOneM\].
It is clear that $$\left\Vert \int_{0}^{\infty}e^{-i\left(t-s\right)A}V_{1}V_{2}\left(\cdot-\vec{v}(s)\right)u(s)\,ds\right\Vert _{L_{t}^{\infty}L_{x}^{2}}\leq\left\Vert \widetilde{K}\right\Vert _{L_{t}^{2}L_{x}^{2}\rightarrow L_{t}^{\infty}L_{x}^{2}}\left\Vert V_{2}\left(x-\vec{v}(t)\right)u\right\Vert _{L_{t}^{2}L_{x}^{2}},$$ where $$\left(\widetilde{K}F\right)(t):=\int_{0}^{\infty}e^{-i\left(t-s\right)A}V_{1}\left(\cdot-\vec{v}(s)\right)F(s)\,ds.$$ We need to estimate $$\left\Vert \widetilde{K}\right\Vert _{L_{t}^{2}L_{x}^{2}\rightarrow L_{t}^{\infty}L_{x}^{2}}.$$ Testing against $F\in L_{t}^{2}L_{x}^{2}$, clearly, $$\left\Vert \widetilde{K}F\right\Vert _{L_{t}^{\infty}L_{x}^{2}}\leq\left\Vert e^{-itA}\right\Vert _{L^{2}\rightarrow L_{t}^{\infty}L_{x}^{2}}\left\Vert \int_{0}^{\infty}e^{isA}V_{1}\left(\cdot-\vec{v}(s)\right)F(s)\,ds\right\Vert _{L^{2}}.\label{eq:TKF-1-3}$$ The first factors on the right-hand side of is bounded by the energy estimates for the free evolution. Consider the second factor, by duality, it suffices to show $$\left\Vert V_{1}\left(x-\vec{v}(t)\right)e^{-itA}\phi\right\Vert _{L_{t}^{2}L_{x}^{2}}\lesssim\left\Vert \phi\right\Vert _{L^{2}},\,\forall\phi\in L^{2}\left(\mathbb{R}^{3}\right).$$ From our discussions Theorem \[thm:StriOneM\], we know $$\left\Vert V_{1}\left(x-\vec{v}(t)\right)e^{-itA}\phi\right\Vert _{L_{t}^{2}L_{x}^{2}}\lesssim\left\Vert \phi\right\Vert _{L^{2}}.$$ Hence $$\left\Vert \int_{0}^{\infty}e^{isA}V_{1}\left(\cdot-\vec{v}(s)\right)F(s)\,ds\right\Vert _{L^{2}}\lesssim\left\Vert F\right\Vert _{L_{t}^{2}L_{x}^{2}}.$$ Therefore, indeed, we have $$\left\Vert \widetilde{K}\right\Vert _{L_{t}^{2}L_{x}^{2}\rightarrow L_{t}^{\infty}L_{x}^{2}}\leq C$$ and $$\sup_{t\in\mathbb{R}}\left\Vert \int_{0}^{\infty}e^{-i\left(t-s\right)A}V_{1}V_{2}\left(\cdot-\vec{v}(s)\right)u(s)\,ds\right\Vert _{L_{x}^{2}}\lesssim\left\Vert V_{2}\left(\cdot-\vec{v}(s)\right)u\right\Vert _{L_{t,x}^{2}}.$$ By the weighted estimate , $$\left\Vert V_{2}\left(\cdot-\vec{v}(s)\right)u\right\Vert _{L_{t}^{2}L_{x}^{2}}\lesssim\left(\int_{0}^{\infty}\int_{\mathbb{R}^{3}}\frac{1}{\left\langle x-\vec{v}(t)\right\rangle ^{\alpha}}\left|u(x,t)\right|^{2}dxdt\right)^{\frac{1}{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$ It implies $$\sup_{t\in\mathbb{R}}\left\Vert \int_{0}^{\infty}e^{-i\left(t-s\right)A}V_{1}V_{2}\left(\cdot-\vec{v}(s)\right)u(s)\,ds\right\Vert _{L_{x}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.\label{eq:Ssecond-2-2}$$ Therefore, by estimates and , we have $$\sup_{t\geq0}\left(\|\nabla u(t)\|_{L^{2}}+\|u_{t}(t)\|_{L^{2}}\right)\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}$$ as claimed
Similarly, one can also obtain the local energy decay estimate:
\[thm:LEnergyOne\]Suppose $u$ is a scattering state in the sense of Definition of \[AO\] which solves $$\partial_{tt}u-\Delta u+V\left(x-\vec{v}(t)\right)u=0$$ with initial data $$u(x,0)=g(x),\,u_{t}(x,0)=f(x).$$ Then for $\forall\epsilon>0,\,\left|\vec{\nu}\right|<1$, we have $$\left\Vert \left(1+\left|x-\vec{\nu}t\right|\right)^{-\frac{1}{2}-\epsilon}\left(\left|\nabla u\right|+\left|u_{t}\right|\right)\right\Vert _{L_{t,x}^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$
The proof is the same as above with the energy estimate for the free wave equation replaced by the local energy decay estimate of the free wave equation. $$\left\Vert \left(1+\left|x-\vec{\nu}t\right|\right)^{-\frac{1}{2}-\epsilon}e^{it\sqrt{-\Delta}}f\right\Vert _{L_{t,x}^{2}}\lesssim\left\Vert f\right\Vert _{L_{x}^{2}}.$$ The claim follows easily.
To finish this section, we show one important application of Theorem \[thm:revemoving\].
We denote $$E_{V}(t)=\int_{\mathbb{R}^{3}}\left|\nabla_{x}u\right|^{2}+\left|\partial_{t}u\right|^{2}+V\left(x-\vec{v}(t)\right)\left|u\right|^{2}dx.\label{eq:eneOM}$$
\[cor:eneOMB\] Suppose $u$ is a scattering state in the sense of Definition of \[AO\] which solves $$\partial_{tt}u-\Delta u+V(x-\vec{v}(t))u=0$$ with initial data $$u(x,0)=g(x),\,u_{t}(x,0)=f(x).$$ Assume $$\left\Vert \nabla V\right\Vert _{L^{1}}<\infty,$$ then $E_{V}(t)$ is bounded by the initial energy independently of $t$, $$\sup_{t}E_{V}(t)\lesssim\left\Vert \left(g,f\right)\right\Vert _{\dot{H}^{1}\times L^{2}}^{2}.\label{eq:eneOMB}$$
We might assume $u$ is smooth. Taking the time derivative of $E_{V}(t)$ and by the fact that $u$ solves equation, we obtain
$$\partial_{t}E_{V}(t)=\int_{\mathbb{R}^{3}}\partial_{t}V(x-\vec{v}(t))\left|u(x,t)\right|^{2}dx=-\vec{v}'(t)\int_{\mathbb{R}^{3}}\partial_{y}V(y)\left|u^{S}(y,t)\right|^{2}dy.$$
by a simple change of variable.
Note that $$\begin{aligned}
\int_{0}^{\infty}\left|\partial_{t}E_{V}(t)\right|dt & \lesssim & \int_{0}^{\infty}\int_{\mathbb{R}^{3}}\left|\partial_{y}V(y)\right|\left|u^{S}(y)\right|^{2}dydt,\nonumber \\
& \lesssim & \left\Vert \partial_{x}V\right\Vert _{L_{x}^{1}}\left\Vert u^{S}\right\Vert _{L_{x}^{\infty}L_{t}^{2}}^{2}\\
& \lesssim & \left\Vert \left(g,f\right)\right\Vert _{\dot{H}^{1}\times L^{2}}^{2}\nonumber \end{aligned}$$ where in the last inequality, we applied Theorem \[thm:revemoving\].
Therefore, for arbitrary $t\in\mathbb{R}^{+}$, we have $$E_{V}(t)-E_{V}(0)\leq\int_{0}^{\infty}\left|\partial_{t}E_{V}(t)\right|dt\lesssim\left\Vert \left(g,f\right)\right\Vert _{\dot{H}^{1}\times L^{2}}^{2}$$ which implies $$\sup_{t}E_{V}(t)\lesssim\left\Vert \left(g,f\right)\right\Vert _{\dot{H}^{1}\times L^{2}}^{2}.$$ We are done.
With endpoint Strichartz estimates along smooth trajectories, we can also derive inhomogenenous Strichartz estimates. One can find a detailed argument in [@GC2].
Scattering and Asymptotic Completeness\[sec:Scattering\]
========================================================
In this section, we show some applications of the results in this paper. We will study the long-time behaviors for a scattering state in the sense of Definition \[AO\].
Following the notations from above section, we will still use the short-hand notation $$L_{t}^{p}L_{x}^{q}:=L_{t}^{p}\left([0,\infty),\,L_{x}^{q}\right).$$ We reformulate the wave equation as a Hamiltonian system, $$U'=JE'(U)$$ where $J$ is a skew symmetric matrix and $E'(U)$ is the Frechet derivative of the conserved quantity. Setting $$U:=\left(\begin{array}{c}
u\\
\partial_{t}u
\end{array}\right),\label{eq:bigU}$$ $$J:=\left(\begin{array}{cc}
0 & 1\\
-1 & 0
\end{array}\right),\label{eq:Jmatrix}$$ $$H_{F}:=\left(\begin{array}{cc}
-\Delta & 0\\
0 & 1
\end{array}\right),$$ we can rewrite the free wave equation as $$\dot{U}_{0}-JH_{F}U_{0}=0,$$ $$U_{0}[0]=\left(\begin{array}{c}
g_{0}\\
f_{0}
\end{array}\right)$$ The solution of the free wave equation is given by $$U_{0}=e^{tJH_{F}}U_{0}[0].$$
\[thm:scattering\]Suppose $u$ is a scattering state in the sense of Definition of \[AO\] which solves $$\partial_{tt}u-\Delta u+V(x-\vec{v}(t))u=0\label{eq:scateq}$$ with initial data $$u(x,0)=g(x),\,u_{t}(x,0)=f(x).$$ Write $$U=\left(u,u_{t}\right)^{t}\in C^{0}\left([0,\infty);\,\dot{H}^{1}\right)\times C^{0}\left([0,\infty);\,L^{2}\right),$$ with initial data
We will still use the formulation in Theorem . We set $A=\sqrt{-\Delta}$ and notice that $$\left\Vert Af\right\Vert _{L^{2}}\simeq\left\Vert f\right\Vert _{\dot{H}^{1}},\,\,\forall f\in C^{\infty}\left(\mathbb{R}^{3}\right).$$ For real-valued $u=\left(u_{1},u_{2}\right)\in\mathcal{H}=\dot{H}^{1}\left(\mathbb{R}^{3}\right)\times L^{2}\left(\mathbb{\mathbb{R}}^{3}\right)$, we write $$U:=Au_{1}+iu_{2}.$$ We know $$\left\Vert U\right\Vert _{L^{2}}\simeq\left\Vert \left(u_{1},u_{2}\right)\right\Vert _{\mathcal{H}}.$$ We also notice that $u$ solves if and only if $$U:=Au+i\partial_{t}u$$ satisfies $$i\partial_{t}U=AU+V\left(x-\vec{v}(t)\right)u,$$ $$U(0)=Ag+if\in L^{2}\left(\mathbb{R}^{3}\right).$$ By Duhamel’s formula, for fixed $T$ $$U(T)=e^{iTA}U(0)-i\int_{0}^{T}e^{-i\left(T-s\right)A}\left(V\left(\cdot-\vec{v}(s)\right)u(s)\right)\,ds.$$ Applying the free evolution backwards, we obtain $$e^{-iTA}U(T)=U(0)-i\int_{0}^{T}e^{isA}\left(V\left(\cdot-\vec{v}(s)\right)u(s)\right)\,ds.$$ Letting $T$ go to $\infty$, we define $$U_{0}(0):=U(0)-i\int_{0}^{\infty}e^{isA}\left(V\left(\cdot-\vec{v}(s)\right)u(s)\right)\,ds$$ By construction, we just need to show $U_{0}[0]$ is well-defined in $L^{2}$, then automatically, $$\left\Vert U(t)-e^{itA}U_{0}(0)\right\Vert _{L^{2}}\rightarrow0.$$ It suffices to show $$\int_{0}^{\infty}e^{isA}\left(V\left(\cdot-\vec{v}(s)\right)u(s)\right)\,ds\in L^{2}.$$ Then following the argument as in the proof of Theorem \[thm:StriOneM\], we write $V=V_{1}V_{2}$.
We consider $$\left\Vert \int_{0}^{\infty}e^{isA}V_{1}V_{2}\left(\cdot-\vec{v}(s)\right)u(s)\,ds\right\Vert _{L_{x}^{2}}\leq\left\Vert K\right\Vert _{L_{t,x}^{2}\rightarrow L_{x}^{2}}\left\Vert V_{2}\left(\cdot-\vec{v}(s)\right)u\right\Vert _{L_{t,x}^{2}},$$ where $$\left(KF\right)(t):=\int_{0}^{\infty}e^{isA}V_{1}\left(\cdot-\vec{v}(s)\right)F(s)\,ds.$$ By the same argument in the proof of Theorem \[thm:EnergyOne\], one has $$\left\Vert K\right\Vert _{L_{t,x}^{2}\rightarrow L_{x}^{2}}\leq C.$$ Therefore
$$\left\Vert \int_{0}^{\infty}e^{isA}V_{1}V_{2}\left(\cdot-\vec{v}(s)\right)u(s)\,ds\right\Vert _{L_{x}^{2}}\lesssim\left\Vert V_{2}\left(x-\vec{v}(t)\right)u\right\Vert _{L_{t,x}^{2}}.$$
By estimate , $$\left\Vert V_{2}\left(x-\vec{v}(t)\right)u\right\Vert _{L_{t,x}^{2}}\lesssim\left(\int_{\mathbb{R}^{+}}\int_{\mathbb{R}^{3}}\frac{1}{\left\langle x-\vec{v}(t)\right\rangle ^{\alpha}}\left|u(x,t)\right|^{2}dxdt\right)^{\frac{1}{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$ We conclude $$\int_{0}^{\infty}e^{isA}\left(V\left(\cdot-\vec{v}(s)\right)u(s)\right)\,ds\in L^{2}$$ with $$\left\Vert \int_{0}^{\infty}e^{isA}\left(V\left(\cdot-\vec{v}(s)\right)u(s)\right)\,ds\right\Vert _{L^{2}}\lesssim\|f\|_{L^{2}}+\|g\|_{\dot{H}^{1}}.$$ So $$U_{0}(0):=U(0)-i\int_{0}^{\infty}e^{isA}\left(V\left(\cdot-\vec{v}(s)\right)u(s)\right)\,ds$$ is well-defined in $L^{2}$ and $$\left\Vert U(t)-e^{itA}U_{0}(0)\right\Vert _{L^{2}}\rightarrow0.$$ Define $$\left(g_{0},f_{0}\right):=\left(A^{-1}\Re U_{0}(0),\,\Im U_{0}(0)\right).$$ By construction, notice that $$U[t]=\left(A^{-1}\Re U(t),\,\Im U(t)\right)$$ and $$\left\Vert U[t]-e^{tJH_{F}}U_{0}[0]\right\Vert _{\dot{H}^{1}\times L^{2}}\rightarrow0.$$ We are done.
To finish this section, we show the asymptotic completeness for the wave equation with the potential moving along a straight line.
Consider the wave equation $$\partial_{tt}u-\Delta u+V(x-\vec{\mu}t)u=0$$ with initial data $$u(x,0)=g(x),\,u_{t}(x,0)=f(x).$$ Without loss of generality, we still assume that $\vec{u}$ is along $\vec{e}_{1}$.
Let $m_{1},\,\ldots,\,m_{w}$ be the normalized bound states of $$H=-\Delta+V\left(\sqrt{1-\left|\mu\right|}x_{1},x_{2},x_{3}\right)$$ associated with eigenvalues $-\lambda_{1}^{2},\,\ldots,\,-\lambda_{w}^{2}$ respectively with $\lambda_{i}>0,\,i=1,\ldots,w$. Setting $$A_{H}=\left(\begin{array}{cc}
0 & 1\\
-H & 0
\end{array}\right),$$ then the point spectrum of $A_{H}$ is $$\sigma_{p}=\bigcup_{i=1}^{w}\left\{ \pm\lambda_{i}\right\}$$ and the continuous spectrum is $$\sigma_{c}=i\left(-\infty,\infty\right).$$ Setting $$E_{i}^{\pm}=\left(\begin{array}{c}
m_{i}\\
\pm\lambda_{i}m_{i}
\end{array}\right),\,i=1,\ldots,w,$$ we know $E_{i}^{\pm}$ are eigenvectors of $A_{H}$ with with eigenvalues $\pm\lambda_{i}$. One can define the associated Riesz projection $$P_{i,\pm}\left(H\right):=\left\langle \cdot,JE_{i}^{\mp}\right\rangle E_{i}^{\pm}\label{eq:rieszpro}$$ onto $E_{i}^{\pm}$. One can check $$P_{i,\pm}\left(H\right)\left(\begin{array}{c}
u\\
\partial_{t}u
\end{array}\right)=\left\langle \pm\lambda_{i}u(t)+\partial_{t}u(t),\,m_{i}\right\rangle .$$ From the standard asymptotic completeness results, if we write $$\dot{U}=A_{H}U$$ where as , $$U=\left(\begin{array}{c}
u\\
\partial_{t}u
\end{array}\right),$$ and $$U[0]=\left(\begin{array}{c}
g\\
f
\end{array}\right)$$ then one can decompose the evolution as $$U(t)=\sum_{i=1}^{w}\left\langle U[0],JE_{i,\mp}\right\rangle e^{\pm\lambda_{i}t}E_{i}^{\pm}+e^{tH_{F}}U_{0}[0]+R(t)\label{eq:StAC}$$ where $e^{tH_{F}}U_{0}[0]$ is the free evolution with initial data $U_{0}[0]$ and $$\left\Vert R(t)\right\Vert _{\dot{H}^{1}\times L^{2}}\rightarrow0,\,\,t\rightarrow\infty.$$ With notations above, we can obtain a similar decomposition as when the potential is moving.
\[cor:ACOne\]Suppose $H$ admits no eigenfunction nor resonances at zero. Let $u$ solve $$\partial_{tt}u-\Delta u+V(x-\vec{\mu}t)u=0.$$ Write $$U=\left(u,u_{t}\right)^{t}\in C^{0}\left([0,\infty);\,\dot{H}^{1}\right)\times C^{0}\left([0,\infty);\,L^{2}\right),$$ with initial data
Applying a Lorentz transformation such that under the new frame $\left(x',t'\right)$, $V$ is stationary, by the standard asymptotic completeness decomposition, one can write $$U_{L}\left(x',t'\right)=\sum_{i=1}^{w}\left\langle U_{L}[0],JE_{i,\mp}\right\rangle e^{\pm\lambda_{i}t'}E_{i}^{\pm}\left(x'\right)+\mathcal{R}_{L}\left(x',t'\right),\label{eq:Ldecomp}$$ where again, we used subscript $L$ to denote the function under the new frame.
Clearly, by the above decomposition , $$P_{b}\left(H\right)\mathcal{R}_{L}\left(x',t'\right)=0.$$ Then in the original frame, $$U(t)=\sum_{i=1}^{w}a_{i,\pm}e^{\pm\lambda_{i}\gamma\left(t-\mu x_{1}\right)}E_{i,\mu}^{\pm}\left(x,t\right)+\mathcal{R}\left(x,t\right).$$ where $$a_{i,\pm}=\left\langle U_{L}[0],JE_{i,\mp}\right\rangle .$$ By construction, $\mathcal{R}(x,t)$ satisfies the conditions in Theorem \[thm:scattering\]. Hence $$\mathcal{R}(x,t)=e^{tH_{F}}U_{0}[0]+R(t)$$ where $e^{tH_{F}}U_{0}[0]$ is the free evolution with initial data $U_{0}[0]$ and $$\left\Vert R(t)\right\Vert _{\dot{H}^{1}\times L^{2}}\rightarrow0,\,\,t\rightarrow\infty.$$ Therefore, finally, we can write $$U(t)=\sum_{i=1}^{w}a_{i,\pm}e^{\pm\lambda_{i}\gamma\left(t-\mu x_{1}\right)}E_{i,\mu}^{\pm}\left(x,t\right)+e^{tH_{F}}U_{0}[0]+R(t)$$ with $$E_{i,\mu}^{\pm}\left(x,t\right)=E_{i}^{\pm}\left(\gamma\left(x_{1}-\mu t\right),x_{2},x_{3}\right)$$ and $$\left\Vert R(t)\right\Vert _{\dot{H}^{1}\times L^{2}}\rightarrow0,\,\,t\rightarrow\infty.$$ The theorem is proved.
\[rem:asyC\]As a final remark, we point out that there is no hope to establish an elegant asymptotic completeness if the potential is not moving along a straight line. If there is a perturbation from that case, the interaction among bound states becomes complicated. Basically, the mechanism is that if the evolution of one bound state is activated, say the bound state with the highest energy, then it will not only cause exponential growth with highest rate for itself but also make the evolution of other bound states grow exponentially. Meanwhile, if we have a scattering state, the evolution of bound states is controllable.
Appendix A {#appendix-a .unnumbered}
==========
For the sake of completeness, in this appendix, we provide the proof of dispersive estimates for the free wave equation in $\mathbb{R}^{3}$ based on the idea of reversed Strichartz estimates.
Consider $$u_{tt}-\Delta u=0=\square u$$ with initial data $$u(0)=g,\,\,u_{t}(0)=f.$$ One can write down $u$ explicitly, $$u=\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f+\cos\left(t\sqrt{-\Delta}\right)g.$$
\[thm:dispersive\]In $\mathbb{R}^{3}$, suppose $f\in L^{2},\,\nabla f\in L^{1}$ and $g\in L^{2},\,\Delta g\in L^{1}$. Then one has the following estimates:
$$\left\Vert \frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f\right\Vert _{L_{x}^{\infty}}\lesssim\frac{1}{\left|t\right|}\left\Vert \nabla f\right\Vert _{L_{x}^{1}},$$
$$\left\Vert \cos\left(t\sqrt{-\Delta}\right)g\right\Vert _{L_{x}^{\infty}}\lesssim\frac{1}{\left|t\right|}\left\Vert \Delta g\right\Vert _{L_{x}^{1}}.$$
Note that the second estimate is slightly different from the estimates commonly used in the literature. For example, in Krieger-Schlag [@KS] one needs the $L^{1}$ norm of $D^{2}g$ instead of $\Delta g$.
First of all, we consider $$\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f.$$ In $\mathbb{R}^{3}$, one has $$\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f=\frac{1}{4\pi t}\int_{\left|x-y\right|=t}f(y)\,dy.$$ Without loss of generality, we assume $t\geq0$.
Multiplying $t$ and integrating, we obtain $$\begin{aligned}
\int_{0}^{\infty}\left|t\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f\right|dt & \lesssim & \int_{0}^{\infty}\int_{\mathbb{S}^{2}}\left|f(x+r\omega)\right|r^{2}\,d\omega dr\nonumber \\
& \lesssim & \left\Vert f\right\Vert _{L_{x}^{1}}.\end{aligned}$$ Therefore, $$\left\Vert t\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f\right\Vert _{L_{x}^{\infty}L_{t}^{1}}\lesssim\left\Vert f\right\Vert _{L_{x}^{1}}.$$ Notice that, from the above estimate, we also have $$\left\Vert \int_{t}^{\infty}\frac{\sin\left(s\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f\,ds\right\Vert _{L_{x}^{\infty}}\lesssim\frac{1}{\left|t\right|}\left\Vert f\right\Vert _{L_{x}^{1}}.$$ Replacing $f$ with $\Delta f$, it implies that $$\left\Vert \int_{t}^{\infty}\sqrt{-\Delta}\sin\left(s\sqrt{-\Delta}\right)f\,ds\right\Vert _{L_{x}^{\infty}}\lesssim\frac{1}{\left|t\right|}\left\Vert \Delta f\right\Vert _{L_{x}^{1}}.$$ On the other hand, $$\begin{aligned}
\int_{0}^{\infty}\left|t\cos\left(t\sqrt{-\Delta}\right)f\right|dt & \lesssim & \int_{0}^{\infty}\int_{\mathbb{S}^{2}}\left|rf\left(x+r\omega\right)d\omega+r^{2}\partial_{r}f\left(x+r\omega\right)\right|\,d\omega dr\nonumber \\
& \lesssim & \left\Vert \nabla f\right\Vert _{L_{x}^{1}}\end{aligned}$$ where in the last inequality, we applied integration by parts in $r$ in the first term of the RHS of the first line.
Therefore, $$\left\Vert t\cos\left(t\sqrt{-\Delta}\right)f\right\Vert _{L_{x}^{\infty}L_{t}^{1}}\lesssim\left\Vert \nabla f\right\Vert _{L_{x}^{1}}.$$ Hence $$\left\Vert \int_{t}^{\infty}\cos\left(s\sqrt{-\Delta}\right)f\,ds\right\Vert _{L_{x}^{\infty}}\lesssim\frac{1}{\left|t\right|}\left\Vert \nabla f\right\Vert _{L_{x}^{1}}.$$ Finally, we check $$\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f=\int_{t}^{\infty}\cos\left(s\sqrt{-\Delta}\right)f\,ds,$$ and $$\cos\left(t\sqrt{-\Delta}\right)g=\int_{t}^{\infty}\sqrt{-\Delta}\sin\left(s\sqrt{-\Delta}\right)g\,ds.$$ Let $f,\,g,\,h$ be any test functions. Define $$Ag=\cos\left(t\sqrt{-\Delta}\right)g-\int_{t}^{\infty}\sqrt{-\Delta}\sin\left(s\sqrt{-\Delta}\right)g\,ds$$ and $$Bf=\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f-\int_{t}^{\infty}\cos\left(s\sqrt{-\Delta}\right)f\,ds.$$ It is easy to check that $A,\,B$ are independent of $t$.
For $A$, one observes that $$\left\langle \cos\left(t\sqrt{-\Delta}\right)g,\,h\right\rangle \rightarrow0$$ and $$\left\Vert \int_{t}^{\infty}\frac{\sin\left(s\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f\,ds\right\Vert _{L_{x}^{\infty}}\lesssim\frac{1}{\left|t\right|}\left\Vert f\right\Vert _{L_{x}^{1}}.$$ Therefore, $$\left\langle Ag,\,h\right\rangle \rightarrow0,\,t\rightarrow\infty.$$ Since $A$ is independent of $t$, one concludes that $$\left\langle Ag,\,h\right\rangle =0$$ for any pair of test functions and hence $$A=0.$$ Similarly, we get $$B=0.$$ Therefore by our calculations above, we can obtain the dispersive estimates for the free wave equation, $$\begin{aligned}
\left\Vert \frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f\right\Vert _{L_{x}^{\infty}} & = & \left\Vert \int_{t}^{\infty}\cos\left(s\sqrt{-\Delta}\right)f\,ds\right\Vert _{L_{x}^{\infty}}\nonumber \\
& \lesssim & \frac{1}{\left|t\right|}\left\Vert \nabla f\right\Vert _{L_{x}^{1}},\end{aligned}$$ and $$\begin{aligned}
\left\Vert \cos\left(t\sqrt{-\Delta}\right)g\right\Vert _{L_{x}^{\infty}} & = & \left\Vert \int_{t}^{\infty}\sqrt{-\Delta}\sin\left(s\sqrt{-\Delta}\right)g\,ds\right\Vert _{L_{x}^{\infty}}\nonumber \\
& \lesssim & \frac{1}{\left|t\right|}\left\Vert \Delta g\right\Vert _{L_{x}^{1}}.\end{aligned}$$ The theorem is proved.
Appendix B {#appendix-b .unnumbered}
==========
We derive the local energy decay estimate for the free wave equation by the Fourier method.
Recall the coarea formula: for a a real-valued Lipschitz function $u$ and a $L^{1}$ function $g$ then $$\int_{\mathbb{R}^{n}}g(x)\left|\nabla u(x)\right|dx=\int_{\mathbb{R}}\int_{\left\{ u(x)=t\right\} }g(x)\,d\sigma(x)dt,\label{eq:coarea}$$ where $\sigma$ is the surface measure.
\[lem:area\]For $F\in C_{0}^{\infty}$, $\phi$ smooth and non-degenerate,i.e. $\left|\nabla\phi(x)\right|\neq0$, one has $$\int_{\mathbb{R}}\int_{\mathbb{R}^{n}}e^{i\lambda\phi(x)}F(x)\,dxd\lambda=\left(2\pi\right)^{n}\int_{\left\{ \phi=0\right\} }\frac{F(x)}{\left|\nabla\phi(x)\right|}\,d\sigma(x).\label{eq:area}$$
From , $$\int_{\mathbb{R}}\int_{\mathbb{R}^{n}}e^{i\lambda\phi(x)}F(x)\,dxd\lambda=\int_{\mathbb{R}}\int_{\mathbb{R}}e^{i\lambda y}\int_{\left\{ \phi=y\right\} }\frac{F(x)}{\left|\nabla\phi(x)\right|}\,d\sigma(x)dyd\lambda.$$ Denote $\int_{\left\{ \phi=y\right\} }\frac{F(x)}{\left|\nabla\phi(x)\right|}\,d\sigma(x)=g(y)$, then $$\begin{aligned}
\int_{\mathbb{R}}\int_{\mathbb{R}^{n}}e^{i\lambda\phi(x)}F(x)\,dxd\lambda & = & \int_{\mathbb{R}}\int_{\mathbb{R}}e^{i\lambda y}g(y)\,dyd\lambda\nonumber \\
& = & \left(2\pi\right)^{\frac{n}{2}}\int_{\mathbb{R}}\hat{g}(\lambda)\,d\lambda\nonumber \\
& = & \left(2\pi\right)^{n}g(0)\nonumber \\
& = & \int_{\left\{ \phi=0\right\} }\frac{F(x)}{\left|\nabla\phi(x)\right|}\,d\sigma(x).\end{aligned}$$ We are done.
It suffices to consider the half wave evolution, $$e^{it\sqrt{-\Delta}}f.$$
\[thm:local\] Let $\chi\geq0$ be a smooth cut-off function such that $\hat{\chi}$ has compact support. Then $$\left\Vert \chi(x)e^{it\sqrt{-\Delta}}f\right\Vert _{L_{t,x}^{2}}\lesssim\left\Vert f\right\Vert _{L_{x}^{2}}.\label{eq:local}$$
Consider $$\begin{aligned}
\int_{\mathbb{R}}\int_{\mathbb{R}^{n}}\left|e^{it\sqrt{-\Delta}}f\right|^{2}(x)\chi(x)\,dxdt & = & \int_{\mathbb{R}}\left\langle e^{it\sqrt{-\Delta}}f,\chi(x)e^{it\sqrt{-\Delta}}f\right\rangle _{L^{2}}dt\\
& = & \int_{\mathbb{R}}\left\langle e^{it\left|\xi\right|}\hat{f}\left(\xi\right),\left[e^{it\left|\xi\right|}\hat{f}\left(\xi\right)\right]*\hat{\chi}\left(\xi\right)\right\rangle _{L^{2}}dt\nonumber \\
& = & \int_{\mathbb{R}}\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}e^{it\left(\left|\xi\right|-\left|\eta\right|\right)}\hat{\chi}\left(\xi-\eta\right)\hat{f}\left(\xi\right)\hat{f}\left(\eta\right)\,d\eta d\xi dt.\nonumber \end{aligned}$$ Applying Lemma \[lem:area\] with $\phi\left(\xi,\eta\right)=\left|\xi\right|-\left|\eta\right|$, the surface $\left\{ \phi=0\right\} $ becomes $\left\{ \left|\xi\right|=\left|\eta\right|\right\} $ and $\left|\nabla\phi\right|=\sqrt{2}$. It follows that $$\begin{aligned}
\int_{\mathbb{R}}\int_{\mathbb{R}^{n}}\left|e^{it\sqrt{-\Delta}}f\right|^{2}(x)\chi(x)\,dxdt & \simeq & \int_{\left|\xi\right|=\left|\eta\right|}\hat{\chi}\left(\xi-\eta\right)\hat{f}\left(\xi\right)\hat{f}\left(\eta\right)\,d\sigma\\
& \lesssim & \int_{\left|\xi\right|=\left|\eta\right|}\left|\hat{\chi}\left(\xi-\eta\right)\right|\left[\left|\hat{f}\left(\xi\right)\right|^{2}+\left|\hat{f}\left(\eta\right)\right|^{2}\right]\,d\sigma\\
& \lesssim & \int_{\mathbb{R}^{n}}\left|\hat{f}\left(\xi\right)\right|^{2}\int_{\left|\xi\right|=\left|\eta\right|}\left|\hat{\chi}\left(\xi-\eta\right)\right|\,d\sigma d\xi\\
& \lesssim & \sup_{\xi}\left|K\left(\xi\right)\right|\int_{\mathbb{R}^{n}}\left|\hat{f}\left(\xi\right)\right|^{2}d\xi\\
& \lesssim & \int_{\mathbb{R}^{n}}\left|f(x)\right|^{2}dx.\end{aligned}$$ It reduces to show that $$K(\xi)=\int_{\left|\xi\right|=\left|\eta\right|}\left|\hat{\chi}\left(\xi-\eta\right)\right|\,d\sigma$$ is bounded uniformly in $\xi$. Since $\hat{\chi}\left(\xi\right)$ decays fast, we have $$\left|\hat{\chi}(\xi)\right|\lesssim\left\langle \xi\right\rangle ^{-N}$$ and $$\left|\hat{\chi}\left(\xi\right)\right|\lesssim\left|\xi\right|^{1-\epsilon-n},$$ where as usual, $\left\langle \xi\right\rangle =\left(1+\left|\xi\right|^{2}\right)^{\frac{1}{2}}$.
Note that $$\begin{aligned}
K(\xi) & = & \int_{\left|\xi\right|=\left|\eta\right|}\left|\hat{\chi}\left(\xi-\eta\right)\right|\,d\sigma\\
& = & \int_{\left|\zeta-\xi\right|=\left|\xi\right|}\left|\hat{\chi}\left(\zeta\right)\right|\,d\sigma\nonumber \\
& \lesssim & \int_{\left|\zeta-\xi\right|=\left|\xi\right|,\left|\zeta\right|<1}\left|\hat{\chi}\left(\zeta\right)\right|\,d\sigma\nonumber \\
& & +\int_{\left|\zeta-\xi\right|=\left|\xi\right|,\left|\zeta\right|>1}\left|\hat{\chi}\left(\zeta\right)\right|\,d\sigma\nonumber \\
& \lesssim & C(n)\nonumber \end{aligned}$$ which is uniformly bounded in $\xi$ and only depends on $n$.
Therefore, we can conclude $$\left\Vert \chi(x)e^{it\sqrt{-\Delta}}f\right\Vert _{L_{t,x}^{2}}\lesssim\left\Vert f\right\Vert _{L_{x}^{2}}.$$ We are done.
With dyadic decomposition and weights, one has a global version of the above result:
\[cor:fullwave\]$\forall\epsilon>0$, one has $$\left\Vert \left(1+\left|x\right|\right)^{-\frac{1}{2}-\epsilon}e^{it\sqrt{-\Delta}}f\right\Vert _{L_{t,x}^{2}}\lesssim_{\epsilon}\left\Vert f\right\Vert _{L_{x}^{2}}.\label{eq:fullwave}$$
Let $\chi(x)$ from Theorem \[thm:local\] be a smooth version of $1_{B_{1}(0)}$, the indicator function of the unit ball. It follows that $$\begin{aligned}
\left\Vert \chi\left(2^{-j}x\right)e^{it\sqrt{-\Delta}}f\right\Vert _{L_{t,x}^{2}}\nonumber \\
=2^{\frac{jn}{2}}2^{\frac{j}{2}}\left\Vert \chi\left(x\right)\left(e^{it\sqrt{-\Delta}}f\right)\left(2^{j}t,2^{j}x\right)\right\Vert _{L_{t,x}^{2}}\nonumber \\
=2^{\frac{jn}{2}}2^{\frac{j}{2}}\left\Vert \chi\left(x\right)\left(e^{it\sqrt{-\Delta}}f\left(2^{j}\cdot\right)\right)\right\Vert _{L_{t,x}^{2}}\nonumber \\
\lesssim2^{\frac{jn}{2}}2^{\frac{j}{2}}\left\Vert f\left(2^{j}\cdot\right)\right\Vert _{L_{x}^{2}}\nonumber \\
\lesssim2^{\frac{j}{2}}\left\Vert f\right\Vert _{L_{x}^{2}}.\end{aligned}$$ Notice that $$\left(1+\left|x\right|\right)^{-\frac{1}{2}-\epsilon}\simeq\sum_{j\geq0}2^{-j\left(\frac{1}{2}+\epsilon\right)}\chi\left(2^{-j}x\right)$$ then with our computations above, we can conclude that $$\left\Vert \sum_{j\geq0}2^{-j\left(\frac{1}{2}+\epsilon\right)}\chi\left(2^{-j}x\right)e^{it\sqrt{-\Delta}}f\right\Vert _{L_{t,x}^{2}}\lesssim_{\epsilon}\left\Vert f\right\Vert _{L_{x}^{2}},$$ and hence $$\left\Vert \left(1+\left|x\right|\right)^{-\frac{1}{2}-\epsilon}e^{it\sqrt{-\Delta}}f\right\Vert _{L_{t,x}^{2}}\lesssim_{\epsilon}\left\Vert f\right\Vert _{L_{x}^{2}}.$$ The corollary is proved.
Appendix C {#appendix-c .unnumbered}
==========
In this appendix, we discuss the global existence of solutions to the wave equation with time-dependent potentials. Lorentz transformations are important tools in our analysis. Lorentz transformations are rotations of space-time, therefore, a priori, one needs to show the global existence of solutions to wave equations with time-dependent potentials.
\[thm:globalexistence\]Assume $V(x,t)\in L_{t,x}^{\infty}$. Then for each $\left(g,\,f\right)\in H^{1}\left(\mathbb{R}^{3}\right)\times L^{2}\left(\mathbb{R}^{3}\right),$ there is a unique solution $\left(u,\,u_{t}\right)\in C\left(\mathbb{R},\,H^{1}\left(\mathbb{R}^{3}\right)\right)\times C\left(\mathbb{R},\,L^{2}\left(\mathbb{R}^{3}\right)\right)$ to $$\partial_{tt}u-\Delta u+V(x,t)u=0\label{eq:timedepexist}$$ with initial data $$u(x,0)=g,\,\partial_{t}u(x,0)=f.$$
By Duhamel’s formula, we might write the solution as $$u=\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f+\cos\left(t\sqrt{-\Delta}\right)g+\int_{0}^{t}\frac{\sin\left(\left(t-s\right)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}V(\cdot,s)u(s)\,ds.$$ Starting from the local existence, we try to construct the solution in $$X=C\left([0,\,T],\,H^{1}\left(\mathbb{R}^{3}\right)\right)\times C\left([0,\,T),\,L^{2}\left(\mathbb{R}^{3}\right)\right)$$ with $T\leq1$. One can view $u$ as the fixed-point of the map $$S(h)(t)=\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f+\cos\left(t\sqrt{-\Delta}\right)g+\int_{0}^{t}\frac{\sin\left(\left(t-s\right)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}V(\cdot,s)h(s)\,ds.$$ Let $$R=2\left\Vert \frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f+\cos\left(t\sqrt{-\Delta}\right)g\right\Vert _{X}.$$ We will show when $T$ is small enough, $S$ will be a contraction map in $B_{X}(0,R)$.
Clearly,
$$\left\Vert S(h)(t)\right\Vert _{X}\leq\left\Vert \frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f+\cos\left(t\sqrt{-\Delta}\right)g\right\Vert _{X}+\left\Vert \int_{0}^{t}\frac{\sin\left(\left(t-s\right)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}V(\cdot,s)h(s)\,ds\right\Vert _{X}.$$
By direct calculations, $$\left\Vert \int_{0}^{t}\frac{\sin\left(\left(t-s\right)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}V(\cdot,s)h(s)\,ds\right\Vert _{L_{x}^{2}}\leq T^{2}\left\Vert V\left(\cdot,t\right)h(t)\right\Vert _{L^{2}},$$ $$\left\Vert \int_{0}^{t}\frac{\sin\left(\left(t-s\right)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}V(\cdot,s)h(s)\,ds\right\Vert _{\dot{H}_{x}^{1}}\leq T\left\Vert V\left(\cdot,t\right)h(t)\right\Vert _{L^{2}},$$ and $$\left\Vert \partial_{t}\left(\int_{0}^{t}\frac{\sin\left(\left(t-s\right)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}V(\cdot,s)h(s)\,ds\right)\right\Vert _{L_{x}^{2}}\leq T\left\Vert V\left(\cdot,t\right)h(t)\right\Vert _{L^{2}}.$$ Therefore, we can pick $T\left\Vert V\right\Vert _{L_{t,x}^{\infty}}<\frac{1}{10}$, we have $$\left\Vert S(h)(t)\right\Vert _{X}\leq\left\Vert \frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f+\cos\left(t\sqrt{-\Delta}\right)g\right\Vert _{X}+\frac{1}{2}\left\Vert h\right\Vert _{X}.$$ Hence, $S$ maps $B_{X}\left(0,R\right)$ into itself.
Next we show $S$ is a contraction. The calculations are straightforward. $$\left\Vert S(h_{1}-h_{2})(t)\right\Vert _{X}\leq\left\Vert \int_{0}^{t}\frac{\sin\left(\left(t-s\right)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}V(\cdot,s)\left(h_{1}(s)-h_{2}(s)\right)\,ds\right\Vert _{X}.$$ The the same arguments as above give $$\left\Vert S(h_{1}-h_{2})(t)\right\Vert _{X}\leq\frac{1}{2}\left\Vert (h_{1}-h_{2})(t)\right\Vert _{X}.$$ Therefore, by fixed point theorem, there is $u\in X$ such that $$u=S(u),$$ in other words, there exist $u\in C\left([0,\,T],\,H^{1}\left(\mathbb{R}^{3}\right)\right)\times C\left([0,\,T),\,L^{2}\left(\mathbb{R}^{3}\right)\right)$ such that $$u=\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f+\cos\left(t\sqrt{-\Delta}\right)g+\int_{0}^{t}\frac{\sin\left(\left(t-s\right)\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}V(\cdot,s)u(s)\,ds.$$
We notice that the choice of $T$ is independent of the size of the initial data. Then we can repeat the above argument with $\left(u(T),\partial_{t}u(T)\right)$ as initial condition to construct the solution from $T$ to $2T$. Iterating this process, one can easily construct the solution $\left(u,\,u_{t}\right)\in C\left(\mathbb{R},\,H^{1}\left(\mathbb{R}^{3}\right)\right)\times C\left(\mathbb{R},\,L^{2}\left(\mathbb{R}^{3}\right)\right)$.
Finally, we notice the uniqueness of the solution follows from Grönwall’s inequality. Suppose one has two solutions $u_{1}$ and $u_{2}$ to our equation with the same data, then $$\left\Vert u_{1}-u_{2}\right\Vert _{H^{1}\times L^{2}}(t)\leq\int_{0}^{t}\left(t-s\right)\left\Vert u_{1}-u_{2}\right\Vert (s)\,ds.$$ Applying Grönwall’s inequality over $[0,T]$, we obtain $$\left\Vert u_{1}-u_{2}\right\Vert _{X}=0,$$ which means $u_{1}\equiv u_{2}$ on $[0,T]$. Then by the same iteration argument as above, we can conclude that in $C\left(\mathbb{R},\,H^{1}\left(\mathbb{R}^{3}\right)\right)\times C\left(\mathbb{R},\,L^{2}\left(\mathbb{R}^{3}\right)\right)$ $$u_{1}\equiv u_{2}.$$ Therefore, one obtains the uniqueness.
The theorem is proved.
In our setting, $V(x,t)=V\left(x-\vec{v}(t)\right)$ satisfies the assumption of Theorem \[thm:globalexistence\], therefore we have the global existence and uniqueness.
\[cor:GlobalCharge\]For each $\left(g,\,f\right)\in H^{1}\left(\mathbb{R}^{3}\right)\times L^{2}\left(\mathbb{R}^{3}\right),$ there is a unique global solution $\left(u,\,u_{t}\right)\in C\left(\mathbb{R},\,H^{1}\left(\mathbb{R}^{3}\right)\right)\times C\left(\mathbb{R},\,L^{2}\left(\mathbb{R}^{3}\right)\right)$ to the wave equation $$\partial_{tt}u-\Delta u+V\left(x-\vec{v}(t)\right)u=0\label{eq:chargeexist}$$ with initial data $$u(x,0)=g,\,\partial_{t}u(x,0)=f.$$
The above theorem also applies to the charge transfer model in [@GC2]: $$\partial_{tt}u-\Delta u+\sum_{i=1}^{m}\sum_{j=1}^{m}V_{v_{j}}\left(x-\vec{v}_{j}t\right)u=0.$$
Appendix D {#appendix-d .unnumbered}
==========
In this appendix, we present an alternative approach to the homogeneous endpoint reversed Strichartz estimates based on the Fourier transformation.
We only consider $\frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f=\frac{1}{2}\frac{e^{it\sqrt{-\Delta}}}{\sqrt{-\Delta}}f-\frac{1}{2}\frac{e^{-it\sqrt{-\Delta}}}{\sqrt{-\Delta}}f$. We can further reduce to consider $$\frac{e^{it\sqrt{-\Delta}}}{\sqrt{-\Delta}}f$$ With Fourier transform and polar coordinates $\xi=\lambda\omega$, we have $$\begin{aligned}
\frac{e^{it\sqrt{-\Delta}}}{\sqrt{-\Delta}}f & = & \int_{0}^{\infty}\int_{\mathbb{S}^{2}}\frac{e^{2\pi it\lambda}}{\lambda}e^{2\pi i\lambda\left(\omega\cdot x\right)}\lambda^{2}\hat{f}\left(\lambda\omega\right)d\omega d\lambda\nonumber \\
& = & \int_{\mathbb{R}}e^{2\pi it\lambda}\left(\chi_{[0,\infty)}(\lambda)\int_{\mathbb{S}^{2}}e^{2\pi i\lambda\left(\omega\cdot x\right)}\lambda\hat{f}\left(\lambda\omega\right)d\omega\right)d\lambda\nonumber \\
& = & \int_{\mathbb{R}}e^{2\pi it\lambda}G(x,\lambda)d\lambda\end{aligned}$$ where $$G(x,\lambda)=\chi_{[0,\infty)}(\lambda)\int_{\mathbb{S}^{2}}e^{2\pi i\lambda\left(\omega\cdot x\right)}\lambda\hat{f}\left(\lambda\omega\right)d\omega.$$ By Plancherel’s Theorem, we know for fixed $x$, $$\left\Vert \frac{e^{it\sqrt{-\Delta}}}{\sqrt{-\Delta}}f\right\Vert _{L_{t}^{2}}=\left\Vert G(x,\lambda)\right\Vert _{L_{\lambda}^{2}}.$$ $$\begin{aligned}
G^{2}(x,\lambda) & = & \left(\chi_{[0,\infty)}(\lambda)\int_{\mathbb{S}^{2}}e^{2\pi i\lambda\left(\omega\cdot x\right)}\lambda\hat{f}\left(\lambda\omega\right)d\omega\right)^{2}\nonumber \\
& \lesssim & \chi_{[0,\infty)}(\lambda)\int_{\mathbb{S}^{2}}\lambda^{2}\left|\hat{f}\left(\lambda\omega\right)\right|^{2}d\omega\end{aligned}$$ $$\begin{aligned}
\left\Vert \frac{e^{it\sqrt{-\Delta}}}{\sqrt{-\Delta}}f\right\Vert _{L_{t}^{2}}^{2} & \lesssim & \int_{0}^{\infty}\int_{\mathbb{S}^{2}}\lambda^{2}\left|\hat{f}\left(\lambda\omega\right)\right|^{2}d\omega d\lambda\nonumber \\
& \lesssim & \int\left|\hat{f}\left(\xi\right)\right|^{2}d\xi\\
& = & \int\left|f\left(x\right)\right|^{2}dx.\nonumber \end{aligned}$$ Therefore, $$\left\Vert \frac{\sin\left(t\sqrt{-\Delta}\right)}{\sqrt{-\Delta}}f\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\left\Vert f\right\Vert _{L^{2}}$$ as desired.
The two dimension version was obtained in [@Oh] and is mentioned in [@B]: $$\left\Vert e^{it\sqrt{-\Delta}}f\right\Vert _{L_{x}^{\infty}L_{t}^{2}}\lesssim\left\Vert f\right\Vert _{\dot{B}_{2,1}^{1/2}}.$$
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[^1]: This work is part of the authors Ph.D. thesis at the University of Chicago.
|
---
abstract: 'The static estimation of the energy consumed by program executions is an important challenge, which has applications in program optimization and verification, and is instrumental in energy-aware software development. Our objective is to estimate such energy consumption in the form of *functions on the input data sizes of programs*. We have developed a tool for experimentation with static analysis which infers such energy functions at two levels, the instruction set architecture (ISA) and the intermediate code ([LLVM IR]{}) levels, and reflects it upwards to the higher source code level. This required the development of a translation from [LLVM IR]{}to an intermediate representation and its integration with existing components, a translation from ISA to the same representation, a resource analyzer, an ISA-level energy model, and a mapping from this model to [LLVM IR]{}. The approach has been applied to programs written in the XC language running on XCore architectures, but is general enough to be applied to other languages. Experimental results show that our [LLVM IR]{}[level]{}analysis is reasonably accurate (less than $6.4\%$ average error vs. hardware measurements) and more powerful than analysis at the ISA [level]{}. This paper provides insights into the trade-off of precision versus analyzability at these levels.'
author:
- 'U. Liqat'
- 'K. Georgiou'
- 'S. Kerrison'
- 'P. Lopez-Garcia'
- 'John P. Gallagher'
- 'M.V. Hermenegildo'
- 'K. Eder'
title: |
Inferring Parametric Energy Consumption Functions at Different Software Levels:\
ISA vs. [LLVM IR]{}
---
Introduction
============
Energy consumption and the environmental impact of computing technologies have become a major worldwide concern. It is an important issue in high-performance computing, distributed applications, and data centers. There is also increased demand for complex computing systems which have to operate on batteries, such as implantable/portable medical devices or mobile phones. Despite advances in power-efficient hardware, more energy savings can be achieved by improving the way current software technologies make use of such hardware.
The process of developing energy-efficient software can benefit greatly from static analyses that estimate the energy consumed by program executions without actually running them. Such estimations can be used for different software-development tasks, such as performing automatic optimizations, verifying energy-related specifications, and helping system developers to better understand the impact of their designs on energy consumption. These tasks often relate to the source code [level]{}. For example, source-to-source transformations to produce optimized programs are quite common. Specifications included in the source code can be proved or disproved by comparing them with safe information inferred by analysis. Such information, when referred to the procedures in the source code can be useful for example to detect which are the most energy-consuming ones and replace them by more energy-efficient implementations. On the other hand, energy consumption analysis must typically be performed at lower levels in order to take into account the effect of compiler optimizations and to link to an energy model. Thus, the inference of energy consumption information for lower [levels]{}such as the Instruction Set Architecture (ISA) or intermediate compiler representations (such as [LLVM IR]{} [@LattnerLLVM2004]) is fundamental for two reasons: 1) It is an intermediate step that allows propagation of energy consumption information from such lower [levels]{}up to the source code [level]{}; and 2) it enables optimizations or other applications at the ISA and [LLVM IR]{}[levels]{}.
In this paper (an improved version of [@entra-d3.2.4-isa-vs-llvm-short]) we propose a static analysis approach that infers energy consumption information at the ISA and [LLVM IR]{}[levels]{}, and reflects it up to the source code [level]{}. Such information is provided in the form of *functions on input data sizes*, and is expressed by means of *assertions* that are inserted in the program representation at each of these [levels]{}. The user (i.e., the “energy-efficient software developer”) can customize the system by selecting the [level]{}at which the analysis will be performed (ISA or [LLVM IR]{}) and the [level]{}at which energy information will be output (ISA, [LLVM IR]{}or source code). As we will show later, the selection of analysis level has an impact on the analysis accuracy and on the class of programs that can be analyzed.
The main goal of this paper is to study the feasibility and practicability of the proposed analysis approach and perform an initial experimental assessment to shed light on the trade-offs implied by performing the analysis at the ISA or [LLVM]{}[levels]{}. In our experiments we focus on the energy analysis of programs written in XC [@Watt2009] running on the XMOS XS1-L architecture. However, the concepts presented here are neither language nor architecture dependent and thus can be applied to the analysis of other programming languages (and associated lower [level]{}program representations) and architectures as well. XC is a high-level C-based programming language that includes extensions for concurrency, communication, input/output operations, and real-time behavior. In order to potentially support different programming languages and different program representations at different [levels]{}of compilation (e.g., [LLVM IR]{}and ISA) in the same analysis framework we differentiate between the *input language* (which can be XC source, [LLVM IR]{}, or ISA) and the *intermediate semantic program representation* that the resource analysis operates on. The latter is a series of connected code blocks, represented by Horn Clauses, that we will refer to as “[HC IR]{}” from now on. We then propose a transformation from each *input language* into the [HC IR]{}and passing it to a resource analyzer. The [HC IR]{}representation as well as a transformation from [LLVM IR]{}into [HC IR]{}will be explained in Section \[sec:llvm-ciao-translation\]. In our implementation we use an extension of the [CiaoPP]{} [@ciaopp-sas03-journal-scp] resource analyzer. This analyzer always deals with the [HC IR]{}in the same way, independent of its origin, inferring energy consumption functions for all procedures in the [HC IR]{}program. The main reason for choosing Horn Clauses as the intermediate representation is that it offers a good number of features that make it very convenient for the analysis [@decomp-oo-prolog-lopstr07]. For instance, it supports naturally Static Single Assignment (SSA) and recursive forms, as will be explained later. In fact, there is a current trend favoring the use of Horn Clause programs as intermediate representations in analysis and verification tools [@DBLP:conf/tacas/GrebenshchikovGLPR12; @DBLP:conf/fm/HojjatKGIKR12-short; @z3; @hcvs14].
Although our experiments are based on single-threaded XC programs (which do not use pointers, since XC does not support them), our claim about the generality and feasibility of our proposed approach for static resource analysis is supported by existing tools based on the Horn Clause representation that can successfully deal with C source programs that exhibit interesting features such as the use of pointers, arrays, shared-memory, or concurrency in order to analyze and verify a wide range of properties [@DBLP:conf/tacas/GrebenshchikovGLPR12; @DBLP:conf/fm/HojjatKGIKR12-short; @DBLP:conf/tacas/GurfinkelKN15-short]. For example [@DBLP:conf/tacas/GurfinkelKN15-short] is a tool for the verification of safety properties of C programs which can reason about scalars and pointer addresses, as well as memory contents. It represents the bytecode corresponding to a C program by using (constraint) Horn clauses.
Both static analysis and energy models can potentially relate to any language [level]{}(such as XC source, [LLVM IR]{}, or ISA). Performing the analysis at a given [level]{}means that the representation of the program at that [level]{} is transformed into the [HC IR]{}, and the analyzer “mimics” the semantics of instructions at that level. The energy model at a given [level]{}provides basic information on the energy cost of instructions at that level. The analysis results at a given [level]{}can be mapped upwards to a higher level, e.g. from ISA or [LLVM IR]{}to XC. Furthermore, it is possible to perform analysis at a given level with an energy model for a lower level. In this case the energy model must be reflected up to the analysis level.
Our hypothesis is that the choice of [level]{}will affect the accuracy of the energy models and the precision of the analysis in opposite ways: energy models at lower [levels]{}(e.g. at the ISA [level]{}) will be more precise than at higher [levels]{}(e.g. XC source code), since the closer to the hardware, the easier it is to determine the effect of the execution on the hardware. However, at lower [levels]{}more program structure and data type/shape information is lost due to lower-level representations, and we expect a corresponding loss of analysis accuracy. We could devise mechanisms to represent such higher-level information and pass it down to the lower-level [ISA]{}, or to recover it by analysing the [ISA]{}. However, our goal is to compare the analysis at the [LLVM IR]{}and [ISA]{}[levels]{}without introducing such mechanisms, which might be complex or not effective in some cases (e.g., in abstracting memory operations or recovering type information). This hypothesis about the analysis/modelling [level]{}trade-off (and potential choices) is illustrated in Figure \[analysis-model-choices-overview\]. The possible choices are classified into two groups: those that analyze and model at the same [level]{}, and those that operate at different [levels]{}. For the latter, the problem is finding good mappings between software segments from the [level]{}at which the model is defined up to the [level]{}at which the analysis is performed, in a way that does not lose accuracy in the energy information.
![Analysis/modelling [level]{}trade-off and potential choices.[]{data-label="analysis-model-choices-overview"}](\figpath/AnalysisLayersGeneric1)
In this paper we concentrate on two of these choices and their comparison, to see if our hypothesis holds. In particular, the first approach (choice 1) is represented by analysing the generated ISA-[level]{}code using models defined at the ISA [level]{}that express the energy consumed by the execution of individual ISA instructions. This approach was explored in [@isa-energy-lopstr13-final]. It used the precise ISA-[level]{}energy models presented in [@Kerrison13], which when used in the static analysis of [@isa-energy-lopstr13-final] for a number of small numerical programs resulted in the inference of functions that provide reasonably accurate energy consumption estimations for any input data size (3.9% average error vs. hardware measurements). However, when dealing with programs involving structured types such as arrays, it also pointed out that, due to the loss of information related to program structure and types of arguments at the ISA [level]{}(since it is compiled away and no longer relates cleanly to source code), the power of the analysis was limited. In this paper we start by exploring an alternative approach: the analysis of the generated [LLVM IR]{}(which retains much more of such information, enabling more direct analysis as well as mapping of the analysis information back to source [level]{}) together with techniques that map segments of ISA instructions to [LLVM IR]{}blocks [@Georgiou2015arXiv] (choice 2). This mapping is used to propagate the energy model information defined at the ISA [level]{}up to the [level]{}at which the analysis is performed, the [LLVM IR]{}[level]{}. In order to complete the [LLVM IR]{}-[level]{}analysis, we have also developed and implemented a transformation from [LLVM IR]{}into [HC IR]{} and used the [CiaoPP]{}resource analyzer. This results in a parametric analysis that similarly to [@isa-energy-lopstr13-final] infers energy consumption functions, but operating on the [LLVM IR]{}[level]{}rather than the ISA [level]{}.
We have performed an experimental comparison of the two choices for generating energy consumption functions. Our results support our intuitions about the trade-offs involved. They also provide evidence that the [LLVM IR]{}-[level]{}analysis (choice 2) offers a good compromise within the [level]{}hierarchy, since it broadens the class of programs that can be analyzed without the need for developing complex techniques for recovering type information and abstracting memory operations, and without significant loss of accuracy.
In summary, the original contributions of this paper are:
1. A translation from [LLVM IR]{}to [HC IR]{} (Section \[sec:llvm-ciao-translation\]).
2. The integration of all components into an experimental tool architecture, enabling the static inference of energy consumption information in the form of *functions on input data sizes* and the experimentation with the trade-offs described above (Section \[sec:overview\]). The components are: [LLVM IR]{}and [ISA]{} translations, [ISA]{}-[level]{}energy model and mapping technique (Section \[sec:energy-model\] and [@Kerrison13; @Georgiou2015arXiv]), and analysis tools (Section \[sec:ciaopp-analysis\] and [@resource-iclp07; @plai-resources-iclp14]).
3. The experimental results and evidence of trade-off of precision versus analyzability (Section \[sec:experiments\]).
4. A sketch of how the static analysis system can be integrated in a source-level Integrated Development Environment (IDE) (Section \[sec:overview\]).
Finally, some related work is discussed in Section \[sec:related-work\], and Section \[sec:conclusions\] summarises our conclusions and comments on ongoing and future work.
Overview of the Analysis at the [LLVM IR]{}Level {#sec:overview}
================================================
![An overview of the analysis at the [LLVM IR]{}[level]{}using ISA models.](\figpath/CiaoPP_realization_LLVM)
\[fig:ciaopp-framework-llvm\]
An overview of the proposed analysis system at the [LLVM IR]{}[level]{}using models at the ISA [level]{}is depicted in Figure \[fig:ciaopp-framework-llvm\]. The system takes as input an XC source program that can (optionally) contain assertions (used to provide useful hints and information to the analyzer), from which a *Transformation and Mapping* process (dotted red box) generates first its associated [LLVM IR]{}using the xcc compiler. Then, a transformation from [LLVM IR]{}into [HC IR]{}is performed (explained in Section \[sec:llvm-ciao-translation\]) obtaining the intermediate representation (green box) that is supplied to the [CiaoPP]{}analyzer. This representation includes assertions that express the energy consumed by the [LLVM IR]{}blocks, generated from the information produced by the mapper tool (as explained in Section \[sec:energy-model\]). The *[CiaoPP]{}analyzer* (blue box, described in Section \[sec:ciaopp-analysis\]) takes the [HC IR]{}, together with the assertions which express the energy consumed by [LLVM IR]{}blocks, and possibly some additional (trusted) information, and processes them, producing the analysis results, which are expressed also using assertions. Based on the procedural interpretation of these [HC IR]{}programs and the resource-related information contained in the assertions, the resource analysis can infer static bounds on the energy consumption of the [HC IR]{}programs that are applicable to the original [LLVM IR]{}and, hence, to their corresponding XC programs. The analysis results include energy consumption information expressed as functions on data sizes for the whole program and for all the procedures and functions in it. Such results are then processed by the *[CiaoPP]{}printer* (purple box) which presents the information to the program developer in a user-friendly format.
[LLVM IR]{}to [HC IR]{}Transformation {#sec:llvm-ciao-translation}
=====================================
In this section we describe the [LLVM IR]{}to [HC IR]{}transformation that we have developed in order to achieve the complete analysis system at the [LLVM IR]{}[level]{}proposed in the paper (as already mentioned in the overview given in Section \[sec:overview\] and depicted in Figure \[fig:ciaopp-framework-llvm\]).
A Horn clause (HC) is a first-order predicate logic formula of the form $\forall (S_1 \wedge \ldots \wedge S_n \rightarrow S_0)$ where all variables in the clause are universally quantified over the whole formula, and $S_0,S_1,\ldots,S_n$ are atomic formulas, also called literals. It is usually written $S_0 \imp S_1,\ldots,S_n$.
The [HC IR]{}representation consists of a sequence of *blocks* where each block is represented as a *Horn clause*:
Each block has an entry point, that we call the *head* of the block (to the left of the $\imp$ symbol), with a number of parameters $<params>$, and a sequence of steps (the *body*, to the right of the $\imp$ symbol). Each of these $S_i$ steps (or *literals*) is either (the representation of) an [LLVM IR]{}*instruction*, or a *call* to another (or the same) block. The analyzer deals with the [HC IR]{}always in the same way, independent of its origin. The transformation ensures that the program information relevant to resource usage is preserved, so that the energy consumption functions of the [HC IR]{}programs inferred by the resource analysis are applicable to the original [LLVM IR]{}programs.
The transformation also passes energy values for the [LLVM IR]{}[level]{}for different programs based on the ISA/[LLVM IR]{}mapping information that express the energy consumed by the [LLVM IR]{}blocks, as explained in Section \[sec:energy-model\]. Such information is represented by means of *trust* assertions (in the [Ciao]{}assertion language [@hermenegildo11:ciao-design-tplp-short]) that are included in the [HC IR]{}. In general, *trust* assertions can be used to provide information about the program and its constituent parts (e.g., individual instructions or whole procedures or functions) to be trusted by the analysis system, i.e., they provide base information assumed to be true by the inference mechanism of the analysis in order to propagate it throughout the program and obtain information for the rest of its constituent parts.
[LLVM IR]{}programs are expressed using typed assembly-like instructions. Each function is in SSA form, represented as a sequence of basic blocks. Each basic block is a sequence of [LLVM IR]{}instructions that are guaranteed to be executed in the same order. Each block ends in either a branching or a return instruction. In order to transform an [LLVM IR]{}program into the [HC IR]{}, we follow a similar approach as in a previous ISA-[level]{}transformation [@isa-energy-lopstr13-final]. However, the [LLVM IR]{}includes an additional type transformation as well as better memory modelling.
The following subsections describe the main aspects of the transformation.
Inferring Block Arguments
-------------------------
As described before, a *block* in the [HC IR]{}has an entry point (head) with input/output parameters, and a body containing a sequence of steps (here, representations of [LLVM IR]{}instructions). Since the scope of the variables in [LLVM IR]{}blocks is at the function level, the blocks are not required to pass parameters while making jumps to other blocks. Thus, in order to represent [LLVM IR]{}blocks as [HC IR]{}blocks, we need to infer input/output parameters for each block.
For entry blocks, the input and output arguments are the same as the ones to the function. We define the functions $param_{in}$ and $param_{out}$ which infer input and output parameters to a block respectively. These are recomputed according to the following definitions until a fixpoint is reached: $$\begin{aligned}
\mathit{params}_{out}(b) & = \textstyle (kill(b) \ \cup\ \mathit{params}_{in}(b))
\ \cap\ \bigcup_{b'\in \mathit{next}(b)}\mathit{params}_{out}(b') \\
\mathit{params}_{in}(b) & = \textstyle gen(b)\ \cup\ \bigcup_{b'\in \mathit{next}(b)} \mathit{params}_{in}(b')\end{aligned}$$ where $\mathit{next}(b)$ denotes the set of immediate target blocks that can be reached from block $b$ with a jump instruction, while $gen(b)$ and $kill(b)$ are the read and written variables in block $b$ respectively, which are defined as: $$\begin{aligned}
kill(b)&=\textstyle\bigcup\limits_{k=1}^{n}\mathit{def}(k) \\
gen(b)&=\textstyle\bigcup\limits_{k=1}^{n}\{v\mid v\in \mathit{ref}(k) \wedge \forall (j<k). v\notin \mathit{def}(j)\}\end{aligned}$$ where $\mathit{def}(k)$ and $\mathit{ref}(k)$ denote the variables written or referred to at a node (instruction) $k$ in the block, respectively, and $n$ is the number of nodes in the block.
Note that the [LLVM IR]{}is in SSA form at the function level, which means that blocks may have $\phi$ nodes which are created while transforming the program into SSA form. A $\phi$ node is essentially a function defining a new variable by selecting one of the multiple instances of the same variable coming from multiple predecessor blocks:
$x=\phi(x_1, x_2, ..., x_n)$
$\mathit{def}$ and $\mathit{ref}$ for this instruction are $\{x\}$ and $\{x_1, x_2, ... ,
x_n\}$ respectively. An interesting feature of our approach is that $\phi$ nodes are not needed. Once the input/output parameters are inferred for each block as explained above, a post-process gets rid of all $\phi$ nodes by modifying block input arguments in such a way that blocks receive $x$ directly as an input and an appropriate $x_i$ is passed by the call site. This will be illustrated later in Section \[sec:llvm2isa-block-trans\].
Consider the example in Figure \[llvm2hcirblock\] (left), where the [LLVM IR]{}block *looptest* is defined. The body of the block reads from 2 variables without previously defining them in the same block. The fixpoint analysis would yield:
$params_{in}(looptest) = \{Arr, I\}$
which is used to construct the [HC IR]{}representation of the *looptest* block shown in Figure \[llvm2hcirblock\] (right), line 3.
Translating [LLVM IR]{}Types into [HC IR]{}Types
------------------------------------------------
[LLVM IR]{}is a typed representation which allows retaining much more of the (source) program information than the ISA representation (e.g., types defining compound data structures). As already mentioned, this enables a more direct analysis as well as mapping of the analysis information back to source level. Thus, we define a mechanism to translate [LLVM IR]{}types into their counterparts in [HC IR]{}.
The LLVM type system defines primitive and derived types. The primitive types are the fundamental building blocks of the type system. Primitive types include *label, void, integer, character, floating point, x86mmx,* and *metadata*. The *x86mmx* type represents a value held in an MMX register on an x86 machine and the *metadata* type represents embedded metadata. The derived types are created from primitive types or other derived types. They include *array, function, pointer, structure, vector, opaque*. Since the XCore platform supports neither pointers nor floating point data types, the [LLVM IR]{}code generated from XC programs uses only a subset of the LLVM types.
At the [HC IR]{}level we use *regular types*, one of the type systems supported by [CiaoPP]{} [@ciaopp-sas03-journal-scp]. Translating [LLVM IR]{}primitive types into regular types is straightforward. The *integer* and *character* types are abstracted as *num* regular type, whereas the *label*, *void*, and *metadata* types are represented as *atm* (atoms).
For derived types, corresponding non-primitive regular types are constructed during the transformation phase. Supporting non-primitive types is important because it enables the analysis to infer energy consumption functions that depend on the sizes of internal parts of complex data structures. The array, vector, and structure types are represented as follows:
$array\_type\rightarrow (nested) list$\
$vector\_type\rightarrow (nested) list$\
$structure\_type\rightarrow functor\_term$
Both the *array* and *vector* types are represented by the *list* type in [CiaoPP]{}which is a special case of compound term. The type of the elements of such lists can be again a primitive or a derived type. The *structure* type is represented by a compound term which is composed of an atom (called the *functor*, which gives a name to the structure) and a number of *arguments*, which are again either primitive or derived types. LLVM also introduces pointer types in the intermediate representation, even if the front-end language does not support them (as in the case of XC, as mentioned before). Pointers are used in the pass-by-reference mechanism for arguments, in memory allocations in *alloca* blocks, and in memory load and store operations. The types of these pointer variables in the [HC IR]{}are the same as the types of the data these pointers point to.
[ c | c ]{}
``` {language="xc"}
struct mystruct{
int x;
int arr[5];
};
void print(struct mystruct [] Arg, int N)
{
...
}
```
&
``` {language="ciao"}
:- regtype array1/1.
array1:=[] | [~struct|array1].
:- regtype struct/1.
struct:=mystruct(~num,~array2).
:- regtype array2/1.
array2:=[] | [~num|array2].
```
Consider for example the types in the XC program shown in Figure \[xc2regtypes\]. The type of argument $Arg$ of the $print$ function is an array of $mystruct$ elements. $mystruct$ is further composed of an integer and an array of integers. The [LLVM IR]{}code generated by xcc for the function signature $print$ in Figure \[xc2regtypes\] (left) is:
define void @print( $[0 \times \{ i32, [5 \times i32] \}]$\* noalias nocapture)
The function argument type in the [LLVM IR]{}($[0 \times \{ i32, [5
\times i32] \}]$) is the typed representation of the argument $Arg$ to the function in the XC program. It represents an array of arbitrary length with elements of $\{ i32, [5 \times i32] \}$ structure type which is further composed of an $i32$ integer type and a $[5 \times i32]$ array type, i.e., an array of 5 elements of $i32$ integer type.[^1]
This type is represented in the [HC IR]{}using the set of regular types illustrated in Figure \[xc2regtypes\] (right). The regular type $array1$, is a list of $struct$ elements (which can also be simply written as `array1 := list(struct)`). Each $struct$ type element is represented as a functor $mystruct/2$ where the first argument is a $num$ and the second is another list type $array2$. The type $array2$ is defined to be a list of $num$ (which, again, can also be simply written as `array2 := list(num)`).
Transforming [LLVM IR]{}Blocks/Instructions into [HC IR]{} {#sec:llvm2isa-block-trans}
----------------------------------------------------------
In order to represent an [LLVM IR]{}function by an [HC IR]{}function (i.e., a predicate), we need to represent each [LLVM IR]{}block by an [HC IR]{}block (i.e., a Horn clause) and hence each [LLVM IR]{}instruction by an [HC IR]{}literal.
``` {language="llvm"}
alloca:
br label looptest
looptest:
%I=phi i32[%N,%alloca], [%I1,%loopbody]
%Zcmp=icmp ne i32 %I, 0
br i1 %Zcmp, label %loopbody, label %loopend
loopbody:
%Elm=getelementptr [0xi32]*%Arr, i32 0,i32 %I
//process array element `Elm'
%I1=sub i32 %I, 1
br label %looptest
loopend:
ret void
```
``` {language="hcir"}
alloca(N, Arr):-
looptest(N, Arr).
looptest(I, Arr):-
icmp_ne(I, 0, Zcmp),
loopbody_loopend(Zcmp,I,Arr).
icmp_ne(X, Y, 1):- X \= Y.
icmp_ne(X, Y, 0):- X = Y.
loopbody_loopend(Zcmp,I,Arr):-
Zcmp=1,
nth(I, Arr, Elm),
//process list element `Elm'
I1 is I - 1, sub(I,1,I1),
looptest(I1, Arr).
loopbody_loopend(Zcmp,I,Arr):-
Zcmp=0.
```
The [LLVM IR]{} instructions are transformed into equivalent [HC IR]{} literals where the semantics of the execution of the [LLVM IR]{} instructions are either described using trust assertions or by giving definition to [HC IR]{} literals. The *phi assignment* instructions are removed and the semantics of the *phi assignment* are preserved on the call sites. For example, the *phi assignment* is removed from the [HC IR]{}block in Figure \[llvm2hcirblock\] (right) and the semantics of the *phi assignment* is preserved on the call sites of the *looptest* (lines 2 and 14). The call sites $alloca$ (line 2) and $loopbody$ (line 13) pass the corresponding value as an argument to $looptest$, which is received by $looptest$ in its first argument $I$.
Consider the instruction *getelementptr* at line 8 in Figure \[llvm2hcirblock\] (left), which computes the address of an element of an array *%Arr* indexed by *%I* and assigns it to a variable *%Elm*. Such an instruction is represented by a call to an abstract predicate *nth/3*, which extracts a reference to an element from a list, and whose effect of execution on energy consumption as well as the relationship between the sizes of input and output arguments is described using trust assertions. For example, the assertion:
``` {language="ciao" basicstyle="\scriptsize\ttfamily"}
:- trust pred nth(I, L, Elem)
:(num(I), list(L, num), var(Elem))
=> ( num(I), list(L, num), num(Elem),
rsize(I, num(IL, IU)),
rsize(L, list(LL, LU, num(EL, EU))),
rsize(Elem, num(EL, EU)) )
+ (resource(avg, energy, 1215439) ).
```
indicates that if the `nth(I, L, Elem)` predicate (representing the *getelementptr* [LLVM IR]{}instruction) is called with *I* and *L* bound to an integer and a list of numbers respectively, and *Elem* an unbound variable (precondition field “`:`”), then, after the successful completion of the call (postcondition field “`=>`”), *Elem* is an integer number and the lower and upper bounds on its size are equal to the lower and upper bounds on the sizes of the elements of the list *L*. The sizes of the arguments to *nth/3* are expressed using the property *rsize* in the assertion language. The lower and upper bounds on the length of the list *L* are *LL* and *LU* respectively. Similarly, the lower and upper bounds on the elements of the list are *EL* and *EU* respectively, which are also the bounds for *Elem*. The *resource* property (global computational properties field “+”) expresses that the energy consumption for the instruction is an average value (1215439 nano-joules[^2]).
The branching instructions in [LLVM IR]{} are transformed into calls to target blocks in [HC IR]{}. For example, the branching instruction at line 6 in Figure \[llvm2hcirblock\] (left), which jumps to one of the two blocks *loopbody* or *loopend* based on the Boolean variable *Zcmp*, is transformed into a call to a predicate with two clauses (line 5 in Figure \[llvm2hcirblock\] (right)). The name of the predicate is the concatenation of the names of the two [LLVM IR]{}blocks mentioned above. The two clauses of the predicate defined at lines 8-13 and 14-15 in Figure \[llvm2hcirblock\] (right) represent the [LLVM IR]{}blocks *loopbody* and *loopend* respectively. The test on the conditional variable is placed in both clauses to preserve the semantics of the conditional branch.
Obtaining the Energy Consumption of [LLVM IR]{}Blocks {#sec:energy-model}
=====================================================
Our approach requires producing assertions that express the energy consumed by each call to an [LLVM IR]{}block (or parts of it) when it is executed. To achieve this we take as starting point the energy consumption information available from an existing XS1-L ISA Energy Model produced in our previous work of ISA level analysis [@isa-energy-lopstr13-final] using the techniques described in [@Kerrison13]. We refer the reader to [@Kerrison13] for a detailed study of the energy consumption behaviour of the XS1-L architecture, containing a description of the test and measurement process along with the construction and full evaluation of such model. In the experiments performed in this paper a single, constant energy value is assigned to each instruction in the ISA based on this model.
A mechanism is then needed to propagate such ISA-[level]{}energy information up to the [LLVM IR]{}[level]{}and obtain energy values for [LLVM IR]{}blocks. A set of mapping techniques serve this purpose by creating a fine-grained mapping between segments of ISA instructions and [LLVM IR]{}code segments, in order to enable the energy characterization of each [LLVM IR]{}instruction in a program, by aggregating the energy consumption of the ISA instructions mapped to it. Then, the energy value assigned to each [LLVM IR]{}block is obtained by aggregating the energy consumption of all its [LLVM IR]{}instructions. The mapping is done by using the debug mechanism where the debug information, preserved during the lowering phase of the compilation from [LLVM IR]{}to ISA, is used to track ISA instructions against [LLVM IR]{}instructions. A full description and formalization of the mapping techniques is given in [@Georgiou2015arXiv].
Resource Analysis with CiaoPP {#sec:ciaopp-analysis}
=============================
In order to perform the global energy consumption analysis, our approach leverages the [CiaoPP]{}tool [@ciaopp-sas03-journal-scp], the preprocessor of the [Ciao]{} programming environment [@hermenegildo11:ciao-design-tplp-short]. [CiaoPP]{}includes a global static analyzer which is parametric with respect to resources and type of approximation (lower and upper bounds) [@resource-iclp07; @plai-resources-iclp14]. The framework can be instantiated to infer bounds on a very general notion of resources, which we adapt in our case to the inference of energy consumption. As mentioned before, the resource analysis in [CiaoPP]{}works on the intermediate block-based representation language, which we have called [HC IR]{}in this paper. Each block is represented as a Horn Clause, so that, in essence, the [HC IR]{}is a pure Horn clause subset (pure logic programming subset) of the [Ciao]{}programming language. In [CiaoPP]{}, a resource is a user-defined *counter* representing a (numerical) non-functional global property, such as execution time, execution steps, number of bits sent or received by an application over a socket, number of calls to a predicate, number of accesses to a database, etc. The instantiation of the framework for energy consumption (or any other resource) is done by means of an assertion language that allows the user to define resources and other parameters of the analysis by means of assertions. Such assertions are used to assign basic resource usage functions to elementary operations and certain program constructs of the base language, thus expressing how the execution of such operations and constructs affects the usage of a particular resource. The resource consumption provided can be a constant or a function of some input data values or sizes. The same mechanism is used as well to provide resource consumption information for procedures from libraries or external code when code is not available or to increase the precision of the analysis.
For example, in order to instantiate the [CiaoPP]{}general analysis framework for estimating bounds on energy consumption, we start by defining the identifier (“counter”) associated to the energy consumption resource, through the following [Ciao]{}declaration:
``` {language="ciao" basicstyle="\small\ttfamily"}
:- resource energy.
```
We then provide assertions for each [HC IR]{}block expressing the energy consumed by the corresponding [LLVM IR]{}block, determined from the energy model, as explained in Section \[sec:energy-model\]. Based on this information, the global static analysis can then infer bounds on the resource usage of the whole program (as well as procedures and functions in it) as functions of input data sizes. A full description of how this is done can be found in [@plai-resources-iclp14].
Consider the example in Figure \[llvm2hcirblock\] (right). Let $P_e$ denote the energy consumption function for a predicate $P$ in the [HC IR]{}representation (set of blocks with the same name). Let $c_b$ represent the energy cost of an [LLVM IR]{}block $b$. Then, the inferred equations for the [HC IR]{}blocks in Figure \[llvm2hcirblock\] (right) are:
$$\begin{array}{rcl}
alloca_{e}(N, Arr) = c_{alloca} + looptest_{e}(N, Arr) \\
\end{array}$$ $$\begin{array}{rcl}
looptest_{e}(N, Arr) = c_{looptest} + loopbody\_loopend_{e}(0 \neq N, N, Arr)\\
\end{array}$$ $$\begin{array}{rcl}
loopbody\_loopend_{e}(B, N, Arr) =
\left\{
\begin{array}{lr}
looptest_{e}(N-1, Arr) \ \ \text{if } B \text{ is {\tt true}} \\
+ \ c_{loopbody} \\ \\
c_{loopend} \ \quad\quad\quad\quad \text{if } B \text{ is {\tt false}}
\end{array}
\right.
\end{array}$$
If we assume (for simplicity of exposition) that each [LLVM IR]{}block has unitary cost, i.e., $c_b = 1$ for all [LLVM IR]{}blocks $b$, solving the above recurrence equations, we obtain the energy consumed by `alloca` as a function of its input data size ($N$):
$alloca_{e}(N, Arr) = 2 \times N + 3$
Note that using average energy values in the model implies that the energy function for the whole program inferred by the upper-bound resource analysis is an approximation of the actual upper bound (possibly below it). Thus, theoretically, to ensure that the analysis infers an upper bound, we need to use upper bounds as well in the energy models. This is not a trivial task as the worst case energy consumption depends on the data processed, is likely to be different for different instructions, and unlikely to occur frequently in subsequent instructions. A first investigation into the effect of different data on the energy consumption of individual instructions, instruction sequences and full programs is presented in [@pallister2015data]. A refinement of the energy model to capture upper bounds for individual instructions, or a selected subset of instructions, is currently being investigated, extending the first experiments into the impact of data into worst case energy consumption at instruction level as described in Section 5.5 of [@Kerrison13].
Experimental Evaluation {#sec:experiments}
=======================
We have performed an experimental evaluation of our techniques on a number of selected benchmarks. Power measurement data was collected for the XCore platform by using appropriately instrumented power supplies, a power-sense chip, and an embedded system for controlling the measurements and collecting the power data. Details about the power monitoring setup used to run our benchmarks and measure their energy consumption can be found in [@Kerrison13]. The main goal of our experiments was to shed light on the trade-offs implied by performing the analysis at the ISA [level]{}(without using complex mechanisms for propagating type information and representing memory) and at the [LLVM]{}level using models defined at the ISA [level]{}together with a mapping mechanism.
There are two groups of benchmarks that we have used in our experimental study. The first group is composed of four small recursive numerical programs that have a variety of user defined functions, arguments, and calling patterns (first four benchmarks in Table \[tab:isa-llvm-comparison\]). These benchmarks only operate over primitive data types and do not involve any structured types. The second group of benchmarks (the last five benchmarks in Table \[tab:isa-llvm-comparison\]) differs from the first group in the sense that they all involve structured types. These are recursive or iterative.
The second group of benchmarks includes two filter benchmarks namely *Biquad* and *Finite Impulse Response (FIR)*. A filter program attenuates or amplifies one specific frequency range of a given input signal. The `fir(N)` benchmark computes the inner-product of two vectors: a vector of input samples, and a vector of coefficients. The more coefficients, the higher the fidelity, and the lower the frequencies that can be filtered. On the other hand, the Biquad benchmark is an equaliser running Biquad filtering. An equaliser takes a signal and attenuates/amplifies different frequency bands. In the case of an audio signal, such as in a speaker or microphone, this corrects the frequency response. The `biquad(N)` benchmark uses a cascade of Biquad filters where each filter attenuates or amplifies one specific frequency range. The energy consumed depends on the number of banks `N`, typically between 3 and 30 for an audio equaliser. A higher number of banks enables a designer to create more precise frequency response curves. None of the XC benchmarks contain any assertions that provide information to help the analyzer. Table \[tab:isa-llvm-comparison-full\] shows detailed experimental results. Column **SA energy function** shows the energy consumption functions, which depend on input data sizes, inferred for each program by the static analyses performed at the ISA and [LLVM IR]{}[levels]{}(denoted with subscripts $isa$ and $llvm$ respectively). We can see that the analysis is able to infer different kinds of functions (polynomial, exponential, etc.). Column **HW** shows the actual energy consumption in nano-joules measured on the hardware corresponding to the execution of the programs with input data of different sizes (shown in column **Input Data Size**). **Estimated** presents the energy consumption estimated by static analysis. This is obtained by evaluating the functions in column **SA energy function** for the input data sizes in column **Input Data Size**. The value N/A in such column means that the analysis has not been able to infer any useful energy consumption function and, thus, no estimated value is obtained. Column **Err vs. HW** shows the error of the values estimated by the static analysis with respect to the actual energy consumption measured on the hardware, calculated as follows: $\textbf{Err vs.\ HW}= (\frac{\textbf{LLVM} (or \ \textbf{ISA}) - \textbf{HW}}{\textbf{HW}} \times 100 ) \%$. Finally, the last column shows the ratio between the estimations of the analysis at the ISA and [LLVM IR]{}[levels]{}.
[|>p[35mm]{}|>p[13mm]{}|>p[15mm]{}|>p[15mm]{}|>p[15mm]{}|>p[7mm]{}|>p[7mm]{}|p[6mm]{}|]{} **SA energy** & [**Input**]{} & **HW (nJ)** & &
& **isa**/\
**function (nJ)** & **Size** & & **llvm** &
**isa** & **llvm** & **isa** & **llvm**\
$Fact_{isa}(N)$= & N=8 & 227 & 237 & 212 & 4.6 & -6.4 & 0.9\
$ 24.26 \ N + 18.43 $ & N=16 & 426 & 453 & 406 & 6.5 & -4.5 &0.9\
$Fact_{llvm}(N)$= & N=32 & 824 & 886 & 794 & 7.6 & -3.5 & 0.9\
$ 27.03 \ N + 21.28 $ & N=64 & 1690 & 1751 & 1571 & 3.6 & -7.0 & 0.9\
$Fib_{isa}(N)$[^3]=$26.88fib(N)$ & N=2 & 75 & 74 & 65 & -1.16 & -12.5 & 0.89\
$+22.85 \ lucas(N)$[^4]$-30.04$ & N=4 & 219 & 241 & 210 & 10 &-4.1& 0.87\
& N=8 & 1615 & 1853 & 1608 &14.75&-0.4& 0.87\
$Fib_{llvm}(N)^a$=$32.5fib(N)$ & N=15 & $47\times 10^3$ & $54\times 10^3$ & $47\times 10^3$ &16.47&1.2& 0.87\
$+25.6 \ lucas(N)^b-35.65$ & N=26 & $9.30\times10^6$ & $10.9\times 10^6$ & $9.5\times 10^6$ &17.3&1.74& 0.87\
$Sqr_{isa}(N)$= & N=9 & 1242 & 1302 & 1148 & 4.8 &-7.5 & 0.88\
$8.6 \ N^2 + 48.7 \ N + 15.6$ & N=27 & 8135 & 8734 & 7579 & 7.4 & -6.8 & 0.87\
& N=73 & $52\times 10^3$ & $57\times 10^3$ & $49\times 10^3$ & 8.5 & -6.5 & 0.86\
$Sqr_{llvm}(N)$= & N=144 & $19.7\times 10^4$ & $21.4\times 10^4$ & $18.4\times 10^4$ & 8.89 & -6.4 & 0.86\
$10 \ N^2 + 53 \ N + 15.6$ & N=234 & $51\times 10^4$ & $56\times 10^4$ & $48\times 10^4$ & 9.61 & -5.86 & 0.86\
& N=360 & $11.89\times 10^5$ & $13\times 10^5$ & $11.2\times 10^5$ & 10.49 & -5.16 & 0.86\
& N=3 & 326 & 344 & 3.6 & 5.7 & -6.0 & 0.89\
$PowerOfTwo_{isa}(N)$= & N=6 & 2729 & 2965 & 2631 & 8.7 &3.6 & 0.89\
$41.5\times 2^{N}-25.9 $ & N=9 & $21.9\times 10^3$ & $23.9\times 10^3$ & $21.2\times 10^3$ &9 & 3.3 & 0.89\
$PowerOfTwo_{llvm}(N)$ = & N=12 & $17.57\times 10^4$ & $19.1\times 10^4$ & $17\times 10^4$ & 9 & -3.3 & 0.89\
$ 46.8 \times 2^{N} -29.9 $ & N=15 & $13.8\times 10^5$ & $15.3\times 10^5$ & $13.6\times 10^5$ & 11 & -1.5 & 0.89\
& N=57 & 1138 & 1179 & N/A & 3.60 & N/A & N/A\
$reverse_{llvm}(N)$= & N=160 & 3125 & 3185 & N/A & 1.91 & N/A & N/A\
$19.47 \ N + 69.33 $ & N=320 & 6189 & 6301 & N/A &1.82 & N/A & N/A\
& N=720 & 13848 & 14092 & N/A & 1.76 & N/A & N/A\
& N=1280 & 24634 & 24998 & N/A & 1.48 & N/A & N/A\
& N=5 & 7453 & 7569 & N/A & -2 & N/A & N/A\
$matmult_{llvm}(N)$= & N=15 & $15.79\times 10^4$ & $15.9\times 10^4$ & N/A & 1.03 & N/A & N/A\
$42.47 \ N^3 + 68.85 \ N^2+$ & N=20 & $36.29\times 10^4$ & $36.8\times 10^4$ & N/A &1.51 & N/A & N/A\
$49.9 \ N + 24.22 $& N=25 & $69.56\times 10^4$ & $70.8\times 10^4$ & N/A & 1.77 & N/A & N/A\
& N=31 & $13.07\times 10^5$ & $13.3\times 10^5$ & N/A & 1.98 & N/A & N/A\
& N=131; M=69 & $14.5\times 10^3$ & $13.2\times 10^3$ & N/A & 8.65 & N/A & N/A\
$concat_{llvm}(N,M)$= & N=170; M=182 & $25.44\times 10^3$ & $23.3\times 10^3$ & N/A &8.60 & N/A & N/A\
$65.7 \ N + 65.7 \ M + 137 $ & N=188; M=2 & $13.8\times 10^3$ & $12.6\times 10^3$ & N/A & 8.59 & N/A & N/A\
& N=13; M=134 & $10.7\times 10^3$ & $9.79\times 10^3$ & N/A & 8.74 & N/A & N/A\
$biquad_{llvm}(N)$= & N=5 & 871 & 836 & N/A & -4 & N/A & N/A\
$157 \ N + 51.7 $ & N=7 & 1187 & 1151 & N/A & -3.1 & N/A & N/A\
& N=10 & 1660 & 1622 & N/A & -2.31 & N/A & N/A\
& N=14 & 2290 & 2250 & N/A & -1.75 & N/A & N/A\
$fir_{llvm}(N)$= & N=85 & 2999 & 2839 & N/A & -5.3 & N/A & N/A\
$31.8 \ N + 137 $ & N=97 & 3404 & 3221 & N/A & -5.37 & N/A & N/A\
& N=109 & 3812 & 3602 & N/A & -5.5 & N/A & N/A\
& N=121 & 4227 & 3984 & N/A & -5.7 & N/A & N/A\
Table \[tab:isa-llvm-comparison\] shows a summary of results. The first two columns show the name and short description of the benchmarks. The columns under **Err vs. HW** show the average error obtained from the values given in Table \[tab:isa-llvm-comparison-full\] for different input data sizes. The last row of the table shows the average error over the number of benchmarks analyzed at each [level]{}.
------------------- ----------------------------------------- ---------- ---------- ----------
**Program** **Description** **isa/**
**llvm** **isa** **llvm**
`fact(N)` Calculates N! 5.6% 5.3% 0.89
`fibonacci(N)` Nth Fibonacci number 11.9% 4% 0.87
`sqr(N)` Computes $N^2$ performing additions 9.3% 3.1% 0.86
`pow_of_two(N)` Calculates $2^N$ without multiplication 9.4% 3.3% 0.89
`Average` **9%** **3.9%** **0.92**
`reverse(N, M)` Reverses an array 2.18% N/A N/A
`concat(N, M)` Concatenation of arrays 8.71% N/A N/A
`matmult(N, M)` Matrix multiplication 1.47% N/A N/A
`fir(N)` Finite Impulse Response filter 5.47% N/A N/A
`biquad(N)` Biquad equaliser 3.70% N/A N/A
`Average` **3.0%** **N/A** **N/A**
`Overall average` **6.4%** **3.9%** **0.92**
------------------- ----------------------------------------- ---------- ---------- ----------
: [LLVM IR]{}- vs. ISA-[level]{}analysis accuracy.[]{data-label="tab:isa-llvm-comparison"}
The experimental results show that:
- For the benchmarks in the first group, both the ISA- and [LLVM IR]{}-[level]{}analyses are able to infer useful energy consumption functions. On average, the analysis performed at either level is reasonably accurate and the relative error between the two analyses at different [levels]{}is small. ISA-[level]{}estimations are slightly more accurate than the ones at the [LLVM IR]{}[level]{}(3.9% vs. 9% error on average with respect to the actual energy consumption measured on the hardware, respectively). This is because the ISA-[level]{}analysis uses very accurate energy models, obtained from measuring directly at the ISA [level]{}, whereas at the [LLVM IR]{}[level]{}, such ISA-[level]{}model needs to be propagated up to the [LLVM IR]{}[level]{}using (approximated) mapping information. This causes a slight loss of accuracy.
- For the second group of benchmarks, the ISA [level]{}analysis is not able to infer useful energy functions. This is due to the fact that significant program structure and data type/shape information is lost due to lower-level representations, which sometimes makes the analysis at the ISA [level]{}very difficult or impossible. In order to overcome this limitation and improve analysis accuracy, significantly more complex techniques for recovering type information and representing memory in the [HC IR]{}would be needed. In contrast, type/shape information is preserved at the [LLVM IR]{}[level]{}, which allows analyzing programs using data structures (e.g., arrays). In particular, all the benchmarks in the second group are analyzed at the [LLVM IR]{}[level]{}with reasonable accuracy (3% error on average). In this sense, the [LLVM IR]{}-[level]{}analysis is more powerful than the one at the ISA [level]{}. The analysis is also reasonably efficient, with analysis times of about 5 to 6 seconds on average, despite the naive implementation of the interface with external recurrence equation solvers, which can be improved significantly. The scalability of the analysis follows from the fact that it is compositional and can be performed in a modular way, making use of the [Ciao]{}assertion language to store results of previously analyzed modules.
Related Work {#sec:related-work}
============
Few papers can be found in the literature focusing on static analysis of energy consumption. As mentioned before, the approach presented in this paper builds on our previously developed analysis of XC programs [@isa-energy-lopstr13-final] based on transforming the corresponding ISA code into a Horn Clause representation that is supplied, together with an ISA-[level]{}energy model, to the [CiaoPP]{} [@ciaopp-sas03-journal-scp] resource analyzer. In this work we have increased the power of the analysis by transforming and analyzing the corresponding [LLVM IR]{}, and using techniques for reflecting the ISA-[level]{}energy model upwards to the [LLVM IR]{}[level]{}. We also offer novel results supported by our experimental study that shed light on the trade-offs implied by performing the analysis at each of these two [levels]{}. Our approach now enables the analysis of a wider range of benchmarks. We obtained promising results for a good number of benchmarks for which [@isa-energy-lopstr13-final] was not able to produce useful energy functions. A similar approach was proposed for upper-bound energy analysis of Java bytecode programs in [@NMHLFM08], where the Jimple (a typed three-address code) representation of Java bytecode was transformed into Horn Clauses, and a simple energy model at the Java bytecode [level]{} [@LL07] was used. However, this work did not compare the results with actual, measured energy consumption.
In all the approaches mentioned above, instantiations for energy consumption of general resource analyzers are used, namely [@resource-iclp07] in [@NMHLFM08] and [@isa-energy-lopstr13-final], and [@plai-resources-iclp14] in this paper. Such resource analyzers are based on setting up and solving recurrence equations, an approach proposed by Wegbreit [@DBLP:journals/cacm/Wegbreit75] that has been developed significantly in subsequent work [@Rosendahl89; @granularity; @low-bounds-ilps97; @vh-03; @resource-iclp07; @AlbertAGP11a; @plai-resources-iclp14]. Other approaches to static analysis based on the transformation of the analyzed code into another (intermediate) representation have been proposed for analyzing low-level languages [@HenriksenG06] and Java (by means of a transformation into Java bytecode) [@jvm-cost-esop]. In [@jvm-cost-esop], cost relations are inferred directly for these bytecode programs, whereas in [@NMHLFM08] the bytecode is first transformed into Horn Clauses. The general resource analyzer in [@resource-iclp07] was also instantiated in [@estim-exec-time-ppdp08] for the estimation of execution times of logic programs running on a bytecode-based abstract machine. The approach used timing models at the bytecode instruction level, for each particular platform, and program-specific mappings to lift such models up to the Horn Clause level, at which the analysis was performed. The timing model was automatically produced in a one-time, program-independent profiling stage by using a set of synthetic calibration programs and setting up a system of linear equations. By contrast to the generic approach based on [CiaoPP]{}, an approach operating directly on the [LLVM IR]{}representation is explored in [@grech15]. Though relying on similar analysis techniques, the approach can be integrated more directly in the LLVM toolchain and is in principle applicable to any languages targeting this toolchain. The approach uses the same [LLVM IR]{}energy model and mapping technique as the one applied in this paper.
There exist other approaches to cost analysis such as those using dependent types [@DBLP:journals/toplas/0002AH12], SMT solvers [@Alonso2012], or *size change abstraction* [@DBLP:journals/corr/abs-1203-5303] A number of static analyses are also aimed at worst case execution time (WCET), usually for imperative languages in different application domains (see e.g., [@DBLP:journals/tecs/WilhelmEEHTWBFHMMPPSS08] and its references). The worst-case analysis presented in [@DBLP:conf/rtas/JayaseelanML06], which is not based on recurrence equation solving, distinguishes instruction-specific (not proportional to time, but to data) from pipeline-specific (roughly proportional to time) energy consumption. However, in contrast to the work presented here and in [@estim-exec-time-ppdp08], these worst case analysis methods do not infer cost functions on input data sizes but rather absolute maximum values, and they generally require the manual annotation of loops to express an upper-bound on the number of iterations. An alternative approach to WCET was presented in [@Herrmann-WCET-2007]. It is based on the idea of amortisation, which allows to infer more accurate yet safe upper bounds by averaging the worst execution time of operations over time. It was applied to a functional language, but the approach is in principle generally applicable. A timing analysis based on game-theoretic learning was presented in [@DBLP:conf/tacas/SeshiaK11]. The approach combines static analysis to find a set of basic paths which are then tested. In principle, such approach could be adapted to infer energy usage. Its main advantage is that this analysis can infer distributions on time, not only average values.
Conclusions and Future Work {#sec:conclusions}
===========================
We have presented techniques for extending to the [LLVM IR]{}[level]{}our tool chain for estimating energy consumption as functions on program input data sizes. The approach uses a mapping technique that leverages the existing debugging mechanisms in the XMOS XCore compiler tool chain to propagate an ISA-[level]{}energy model to the [LLVM IR]{}[level]{}. A new transformation constructs a block representation that is supplied, together with the propagated energy values, to a parametric resource analyzer that infers the program energy cost as functions on the input data sizes.
Our results suggest that performing the static analysis at the [LLVM IR]{}[level]{}is a reasonable compromise, since 1) [LLVM IR]{}is close enough to the source code [level]{}to preserve most of the program information needed by the static analysis, and 2) the [LLVM IR]{}is close enough to the ISA [level]{}to allow the propagation of the ISA energy model up to the [LLVM IR]{}[level]{}without significant loss of accuracy for the examples studied. Our experiments are based on single-threaded programs. We also have focused on the study of the energy consumption due to computation, so that we have not tested programs where storage and networking is important. However, this could potentially be done in future work, by using the [CiaoPP]{}static analysis, which already infers bounds on data sizes, and combining such information with appropriate energy models of communication and storage. Although the analysis infers sound bound representations in the form of recurrence equations, sometimes the external solvers it uses are not able to find closed form functions for such equations. This is a limitation in applications where such closed forms are needed. Techniques to address such limitation are included in our plans for future work. Our static analysis will also benefit from any improvement of the Computer Algebra Systems used for solving recurrence equations. It remains to be seen whether the results would carry over to other classes of programs, such as multi-threaded programs and programs where timing is more important. In this sense our results are preliminary, yet they are promising enough to continue research into analysis at [LLVM IR]{}[level]{}and into ISA-[LLVM IR]{}energy mapping techniques to enable the analysis of a wider class of programs, especially multi-threaded programs.
Acknowledgements {#acknowledgements .unnumbered}
================
This research has received funding from the European Union 7th Framework Program agreement no 318337, ENTRA, Spanish MINECO TIN’12-39391 *StrongSoft* project, and the Madrid M141047003 *N-GREENS* program.
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[^1]: $[0 \times i32]$ specifies an arbitrary length array of $i32$ integer type elements.
[^2]: nJ, $10^{-9}$ joules
[^3]: \[fibNote\]It uses mathematical functions $fib$ and $lucas$, a function expansion would yield:\
$Fib_{isa}(N)$=$34.87\times 1.62^N+10.8\times(-0.62)^N-30$\
$Fib_{llvm}(N)$=$40.13\times 1.62^N+11.1\times(-0.62)^N-35.65$
[^4]: $Lucas(n)$ satisfy the recurrence relation $L_n=L_{n-1}+L_{n-2}$ with $L_1=1, L_2=3$
|
---
abstract: 'In this paper, we propose a ternary knowing how operator to express that the agent knows how to achieve ${\varphi}$ given $\psi$ while maintaining $\chi$ in-between. It generalizes the logic of goal-directed knowing how proposed by Wang in [@Wang15lori]. We give a sound and complete axiomatization of this logic.'
author:
- Yanjun Li
- Yanjing Wang
bibliography:
- 'kh.bib'
subtitle: A logic of knowing how with intermediate constraints
title: 'Achieving while maintaining:'
---
Introduction
============
Standard epistemic logic proposed by von Wright and Hintikka studies propositional knowledge expressed by “knowing that ${\varphi}$” [@Wright51; @Hintikka:kab]. However, there are very natural knowledge expressions beyond “knowing that”, such as “knowing what your password is”, “knowing why he came late”, “knowing how to go to Beijing”, and so on. In recent years, there have been attempts to capture the logic of such different kinds of knowledge expressions by taking the “knowing X” as a single modality [@WangF13; @WangF14; @FWvD14; @FWvD15; @GW16; @Wang15lori].[^1]
In particular, Wang proposed a logical language of goal-directed knowing how [@Wang15lori], which includes formulas ${\ensuremath{\mathcal{K}h{(\psi, {\varphi})}}}$ to express that the agent knows how to achieve ${\varphi}$ given the precondition $\psi$.[^2] The models are labeled transition systems which represent the agent’s abilities, inspired by [@Wang15]. Borrowing the idea from conformant planning in AI (cf. e.g., [@SW98; @YLW15]), ${\ensuremath{\mathcal{K}h{(\psi, {\varphi})}}}$ holds globally in a labeled transition system, if there is an uniform plan such that from all the $\psi$-states this plan can always be successfully executed to reach some ${\varphi}$-states. As an example, in the following model ${\ensuremath{\mathcal{K}h{(p, q)}}}$ holds, since there is a plan $ru$ which can always work to reach a $q$-state from any $p$-state. $$\xymatrix{
&s_6&{{s_7:q}}&{{s_8: q}} &\\
s_1\ar[r]|r& s_2:p\ar[r]|r\ar[u]|u& s_3:p\ar[r]|r\ar[u]|u&{s_4:q}\ar[r]|r\ar[u]|u&s_5
}$$ In [@Wang15lori], a sound and complete proof system is given, featuring a crucial axiom capturing the compositionality of plans: $${\ensuremath{\mathtt{COMPKh}}}\qquad {\ensuremath{\mathcal{K}h{(p, r)}}}\land{\ensuremath{\mathcal{K}h{(r, q)}}}\to{\ensuremath{\mathcal{K}h{(p, q)}}}$$
However, as observed in [@LauWang], constraints on how we achieve the goal often matter. For example, the ways for me to go to New York are constrained by the money I have; we want to know how to win the game by playing fairly; people want to know how to be rich without breaking the law. Generally speaking, actions have costs, both financially and morally, we need to stay within our “budget” in reaching our goals. Clearly such intermediate constraints cannot be expressed by ${\ensuremath{\mathcal{K}h{(\psi, {\varphi})}}}$ since it only cares about the starting and ending states. This motivates us to introduce a ternary modality ${\ensuremath{\mathcal{K}h{(\psi,\chi,{\varphi})}}}$ where $\chi$ constrains the intermediate states.[^3]
In the rest of the paper, we first introduce the language, semantics, and a proof system of our logic in Section \[Sec.Logic\]. In Section \[Sec.Proof\] we give the highly non-trivial completeness proof of our system, which is much more complicated than the one for the standard knowing how logic. In the last section we conclude with future directions.
The Logic {#Sec.Logic}
=========
Given a set of proposition letters ${\ensuremath{\mathbf{P}}}$, the language ${\mathbf{L_{Khm}}}$ is defined as follows: $${\varphi}:=p\mid \neg{\varphi}\mid ({\varphi}\land{\varphi})\mid {\ensuremath{\mathcal{K}hm{({\varphi},{\varphi},{\varphi})}}}$$ where $p\in{\ensuremath{\mathbf{P}}}$. ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}$ expresses that the agent knows how to guarantee ${\varphi}$ given $\psi$ while maintaining $\chi$ in-between (excluding the start and the end). Note that ${\ensuremath{\mathcal{K}hm{(\psi\land\chi,\chi,{\varphi}\land\chi)}}}$ expresses knowing how with inclusive intermediate constraints. We use the standard abbreviations $\bot, {\varphi}\lor\psi$ and ${\varphi}\to\psi $, and define ${\mathcal{U}}\varphi$ as $ {\ensuremath{\mathcal{K}hm{(\neg{\varphi},\top,\bot)}}}$. ${\mathcal{U}}$ is intended to be an universal modality, and it will become more clear after defining the semantics. Note that the binary know-how operator in [@Wang15] can be defined as ${\ensuremath{\mathcal{K}h{(\psi,{\varphi})}}}:= {\ensuremath{\mathcal{K}hm{(\psi, \top, {\varphi})}}}$.
Given a countable set of proposition letters ${\ensuremath{\mathbf{P}}}$ and a countable non-empty set of action symbols ${\ensuremath{\mathbf{\Sigma}}}.$ A model (also called an ability map) is essentially a labelled transition system $({\mathcal{S}}, {\mathcal{R}},{{\mathcal{V}}})$ where:
- ${\mathcal{S}}$ is a non-empty set of states;
- ${\mathcal{R}}: {\ensuremath{\mathbf{\Sigma}}}\to 2^{{\mathcal{S}}\times {\mathcal{S}}}$ is a collection of transitions labelled by actions in ${\ensuremath{\mathbf{\Sigma}}}$;
- ${{\mathcal{V}}}: S\to 2^{\ensuremath{\mathbf{P}}}$ is a valuation function.
We write $s{\xrightarrow{a}}t$ if $(s, t)\in {\mathcal{R}}(a).$ For a sequence $\sigma=a_1\dots a_n\in{\ensuremath{\mathbf{\Sigma}}}^*$, we write $s{\xrightarrow{\sigma}}t$ if there exist $ s_2\dots s_{n}$ such that $s{\xrightarrow{a_1}}s_2{\xrightarrow{a_2}}\cdots {\xrightarrow{a_{n-1}}}s_n{\xrightarrow{a_n}}t$. Note that $\sigma$ can be the empty sequence $\epsilon$ (when $n=0$), and we set $s{\xrightarrow{\epsilon}}s$ for any $s$. Let $\sigma_k$ be the initial segment of $\sigma$ up to $a_k$ for $k\leq |\sigma|$. In particular let $\sigma_0=\epsilon$. We say $\sigma=a_1\cdots a_n$ is strongly executable at $s'$ if for each $0\leq k <n$: $s'{\xrightarrow{\sigma_k}}t$ implies that $t$ has at least one $a_{k+1}$-successor.
Intuitively, $\sigma$ is *strongly executable* at $s$ if you can always successfully finish the whole $\sigma$ after executing any initial segment of $\sigma$ from $s$. For example, $ab$ is not strongly executable at $s_1$ in the model below, though it is executable at $s_1$. $$\xymatrix@R-25pt{
&{s_2}\ar[r]|b&{s_4: q}\\
{s_1:p}\ar[ur]|a\ar[dr]|a\\
&{s_3}
}
$$
Suppose $s$ is a state in a model ${{\mathcal{M}}}=({\mathcal{S}},{\mathcal{R}},{{\mathcal{V}}})$. Then we inductively define the notion of a formula ${\varphi}$ being satisfied (or true) in ${{\mathcal{M}}}$ at state $s$ as follows:
[|L L L|]{} ,s& & always\
,sp && s(p).\
,s&& ,s.\
,s&& ,s ,s.\
,s[$\mathcal{K}hm{(\psi,\chi,{\varphi})}$]{} &&[$\mathbf{\Sigma}$]{}\^\*\
&&,s’\
&&\
where we say $\sigma=a_1\cdots a_n$ is strongly $\chi$-executable at $s'$ if:
- $\sigma$ is strongly executable at $ s'$, and
- $s'{\xrightarrow{\sigma_k}}t$ implies ${{\mathcal{M}}},t\vDash \chi$ for all $0<k<n$.
It is obvious that $\epsilon$ is strongly $\chi$-executable at each state $s$ for each formula $\chi$. Note that ${\ensuremath{\mathcal{K}hm{(\psi,\bot,{\varphi})}}}$ expresses that there is $\sigma\in{\ensuremath{\mathbf{\Sigma}}}\cup\{\epsilon\}$ such that the agent knows doing $\sigma$ on $\psi$-states can guarantee ${\varphi}$, namely the witness plan $\sigma$ is at most one-step. As an example, ${\ensuremath{\mathcal{K}h{(p,\bot,o)}}}$ and ${\ensuremath{\mathcal{K}h{(p,o,q)}}}$ hold in the following model for the witness plans $a$ and $ab$ respectively. Note that the truth value of ${\ensuremath{\mathcal{K}h{(\psi,\chi,{\varphi})}}}$ does not depend on the designated state. $$\xymatrix@R-25pt{
&{s_2:o}\ar[rd]|b&\\
{s_1:p}\ar[ur]|a\ar[dr]|b&&{s_4: q}\\
&{s_3:\neg o}\ar[ur]|a&
}$$ Now we can also check that the operator ${\mathcal{U}}$ defined by ${\ensuremath{\mathcal{K}hm{(\neg\psi,\top,\bot)}}}$ is indeed an *universal modality*: $$\begin{array}{|rcl|}
\hline
{{\mathcal{M}}},s\vDash {\mathcal{U}}\varphi&\Leftrightarrow& \text{ for all }t\in {\mathcal{S}}, {{\mathcal{M}}}, t\vDash\varphi \\
\hline
\end{array}$$ The following formulas are valid on all models.
\[prop:validEMPKhm\] $\vDash {\mathcal{U}}(p \to q)\to {\ensuremath{\mathcal{K}hm{(p,\bot,q)}}}$
Assuming that ${{\mathcal{M}}},s\vDash{\mathcal{U}}(p\to q)$, it means that ${{\mathcal{M}}},t\vDash p\to q$ for all $t\in{\mathcal{S}}$. Given ${{\mathcal{M}}},t\vDash p$, it follows that ${{\mathcal{M}}},t\vDash q$. Thus, we have $\epsilon$ is strongly $\bot$-executable at $t$. Therefore, we have ${{\mathcal{M}}},s\vDash{\ensuremath{\mathcal{K}hm{(p,\bot,q)}}}$.
$\vDash {\ensuremath{\mathcal{K}hm{(p,o,r)}}}\land{\ensuremath{\mathcal{K}hm{(r,o,q)}}}\land{\mathcal{U}}(r\to o)\to {\ensuremath{\mathcal{K}hm{(p,o,q)}}}$
Assuming ${{\mathcal{M}}},s\vDash {\ensuremath{\mathcal{K}hm{(p,o,r)}}}\land{\ensuremath{\mathcal{K}hm{(r,o,q)}}}\land{\mathcal{U}}(r\to o)$, we will show that ${{\mathcal{M}}},s\vDash {\ensuremath{\mathcal{K}hm{(p,o,q)}}}$. Since ${{\mathcal{M}}},s\vDash {\ensuremath{\mathcal{K}hm{(p,o,r)}}}$, it follows that there exists $\sigma\in{\ensuremath{\mathbf{\Sigma}}}^*$ such that for each ${{\mathcal{M}}},u\vDash p$, $\sigma$ is strongly $o$-executable at $u$ and that ${{\mathcal{M}}},v\vDash r$ for each $v$ with $u{\xrightarrow{\sigma}}v$. Since ${{\mathcal{M}}},s\vDash {\ensuremath{\mathcal{K}hm{(r,o,q)}}}$, it follows that there exists $\sigma'\in{\ensuremath{\mathbf{\Sigma}}}^*$ such that for each ${{\mathcal{M}}},v'\vDash r$, $\sigma'$ is strongly $o$-executable at $v'$ and that ${{\mathcal{M}}},t\vDash q$ for each $t$ with $v'{\xrightarrow{\sigma}}t$. In order to show ${{\mathcal{M}}},s\vDash {\ensuremath{\mathcal{K}hm{(p,o,q)}}}$, we only need to show that $\sigma\sigma'$ is strongly $o$-executable at $u$ and that ${{\mathcal{M}}},t'\vDash q$ for each $t'$ with $u{\xrightarrow{\sigma\sigma'}}t'$, where $u$ is a state with ${{\mathcal{M}}},u\vDash p$.
By assumption, we know that $\sigma$ is strongly $o$-executable at $u$, and for each $v$ with $u{\xrightarrow{\sigma}}v$, it follows by assumption that ${{\mathcal{M}}},v\vDash r$ and $\sigma'$ is strongly $o$-executable at $v$. Moreover, since ${{\mathcal{M}}},s\vDash{\mathcal{U}}(r\to o) $, it follows that ${{\mathcal{M}}},v\vDash o$ for each $v$ with $u{\xrightarrow{\sigma}}v$. Thus, $\sigma\sigma'$ is strongly $o$-executable at $u$. What is more, for each $t'$ with $u{\xrightarrow{\sigma\sigma'}}t'$, there is $v$ such that $u{\xrightarrow{\sigma}}v{\xrightarrow{\sigma'}}t'$ and ${{\mathcal{M}}},v\vDash r$, it follows by assumption that ${{\mathcal{M}}},t'\vDash q$. Therefore, we have ${{\mathcal{M}}},s\vDash {\ensuremath{\mathcal{K}hm{(p,o,q)}}}$.
$\vDash {\ensuremath{\mathcal{K}hm{(p,o,q)}}}\land\neg{\ensuremath{\mathcal{K}hm{(p,\bot,q)}}}\to {\ensuremath{\mathcal{K}hm{(p,\bot,o)}}}$
Assuming ${{\mathcal{M}}},s\vDash {\ensuremath{\mathcal{K}hm{(p,o,q)}}}\land\neg{\ensuremath{\mathcal{K}hm{(p,\bot,q)}}}$, we will show that ${{\mathcal{M}}},s\vDash{\ensuremath{\mathcal{K}hm{(p,\bot,o)}}}$. Since ${{\mathcal{M}}},s\vDash{\ensuremath{\mathcal{K}hm{(p,o,q)}}}$, it follows that there exists $\sigma\in{\ensuremath{\mathbf{\Sigma}}}^*$ such that for each ${{\mathcal{M}}},u\vDash p$, $\sigma$ is strongly $o$-executable at $u$ and ${{\mathcal{M}}},v\vDash q$ for all $v$ with $u{\xrightarrow{\sigma}}v$. If $\sigma\in{\ensuremath{\mathbf{\Sigma}}}\cup\{\epsilon\}$, it follows that ${{\mathcal{M}}},s\vDash{\ensuremath{\mathcal{K}hm{(p,\bot,q)}}}$. Since ${{\mathcal{M}}},s\vDash\neg{\ensuremath{\mathcal{K}hm{(p,\bot,q)}}}$, it follows that $\sigma\not\in{\ensuremath{\mathbf{\Sigma}}}\cup\{\epsilon\}$. Thus, $\sigma=a_1\cdots a_n$ where $n\geq 2$. Let $u$ be a state such that ${{\mathcal{M}}},u\vDash p$. Since $\sigma=a_1\cdots a_n$ is strongly $o$-executable at $u$, it follows that $a_1$ is executable at $u$. Moreover, since $n\geq 2$, we have ${{\mathcal{M}}},v\vDash o$ for each $v$ with $u{\xrightarrow{a_1}}v$. Therefore, we have ${{\mathcal{M}}},s\vDash{\ensuremath{\mathcal{K}hm{(p,\bot,o)}}}$.
\[prop:validREPKhm\] $\vDash {\mathcal{U}}(p'\to p)\land{\mathcal{U}}(o\to o')\land{\mathcal{U}}(q\to q')\land{\ensuremath{\mathcal{K}hm{(p,o,q)}}}\to {\ensuremath{\mathcal{K}hm{(p',o',q')}}}$
Assuming ${{\mathcal{M}}},s\vDash {\mathcal{U}}(p'\to p)\land{\mathcal{U}}(o\to o')\land{\mathcal{U}}(q\to q')\land{\ensuremath{\mathcal{K}hm{(p,o,q)}}}$, we will show ${{\mathcal{M}}},s\vDash{\ensuremath{\mathcal{K}hm{(p',o',q')}}}$. Since ${{\mathcal{M}}},s\vDash{\ensuremath{\mathcal{K}hm{(p,o,q)}}}$, it follows that there exists $\sigma\in{\ensuremath{\mathbf{\Sigma}}}^*$ such that for each ${{\mathcal{M}}},u\vDash p$: $\sigma$ is strongly $o$-executable at $u$ and ${{\mathcal{M}}},v\vDash q$ for each $v$ with $u{\xrightarrow{\sigma}}v$. Let $s'$ be a state with ${{\mathcal{M}}},s'\vDash p'$. Next we will show that $\sigma$ is strongly $o'$-executable at $s'$ and ${{\mathcal{M}}},v'\vDash q'$ for all $v'$ with $s'{\xrightarrow{\sigma}}v'$.
Since ${{\mathcal{M}}},s\vDash{\mathcal{U}}(p'\to p)$, it follows that ${{\mathcal{M}}},s'\vDash p$. Thus, $\sigma$ is strongly $o$-executable at $s'$ and ${{\mathcal{M}}},v'\vDash q$ for each $v'$ with $s'{\xrightarrow{\sigma}} v'$. Since ${{\mathcal{M}}},s\vDash {\mathcal{U}}(o\to o')$, it follows that $\sigma$ is strongly $o'$-executable at $s'$. Since ${{\mathcal{M}}},s\vDash{\mathcal{U}}(q\to q')$, it follows that ${{\mathcal{M}}},v'\vDash q'$ for each $v'$ with $s'{\xrightarrow{\sigma}} v'$.
The axioms and rules shown in Table \[tab:SysSKhm\] constitutes the proof system ${\mathbb{SKHM}}$.
Note that ${\ensuremath{\mathtt{DISTU}}},{\ensuremath{\mathtt{NECU}}}, {\ensuremath{\mathtt{TU}}}$ are standard for the universal modality ${\mathcal{U}}$. ${\ensuremath{\mathtt{4KhmU}}}$ and ${\ensuremath{\mathtt{4KhmU}}}$ are introspection axioms reflecting that ${{\ensuremath{\mathcal{K}hm}}}$ formulas are global. ${\texttt{EMPKhm}}$ captures the interaction between ${\mathcal{U}}$ and ${{\ensuremath{\mathcal{K}hm}}}$ via empty plan. ${\texttt{COMPKhm}}$ is the new composition axiom for ${{\ensuremath{\mathcal{K}hm}}}$. ${\ensuremath{\mathtt{UKhm}}}$ shows how we can weaken the knowing how claims. ${\texttt{ONEKhm}}$ is the characteristic axiom for ${\mathbb{SKHM}}$ compared to the system for binary $\mathcal{K}h$, and it expresses the condition for the necessity of the intermediate steps.
[|lc|]{}\
[$\mathtt{TAUT}$]{}&\
[$\mathtt{DISTU}$]{}& ${\mathcal{U}}p\land{\mathcal{U}}(p\to q)\to {\mathcal{U}}q$\
[$\mathtt{TU}$]{}& ${\mathcal{U}}p\to p $\
[$\mathtt{4KhmU}$]{}& ${\ensuremath{\mathcal{K}hm{(p, o,q)}}}\to{\mathcal{U}}{\ensuremath{\mathcal{K}hm{(p, o,q)}}}$\
[$\mathtt{5KhmU}$]{}& $\neg {\ensuremath{\mathcal{K}hm{(p, o,q)}}}\to{\mathcal{U}}\neg{\ensuremath{\mathcal{K}hm{(p, o,q)}}}$\
[`EMPKhm`]{}&${\mathcal{U}}(p \to q)\to {\ensuremath{\mathcal{K}hm{(p,\bot,q)}}}$\
[`COMPKhm`]{}& ${\ensuremath{\mathcal{K}hm{(p,o,r)}}}\land{\ensuremath{\mathcal{K}hm{(r,o,q)}}}\land{\mathcal{U}}(r\to o)\to {\ensuremath{\mathcal{K}hm{(p,o,q)}}}$\
[`ONEKhm`]{}& ${\ensuremath{\mathcal{K}hm{(p,o,q)}}}\land\neg{\ensuremath{\mathcal{K}hm{(p,\bot,q)}}}\to {\ensuremath{\mathcal{K}hm{(p,\bot,o)}}}$\
[$\mathtt{UKhm}$]{}& ${\mathcal{U}}(p'\to p)\land{\mathcal{U}}(o\to o')\land{\mathcal{U}}(q\to q')\land{\ensuremath{\mathcal{K}hm{(p,o,q)}}}\to {\ensuremath{\mathcal{K}hm{(p',o',q')}}}$\
\
\
Note that the corresponding axioms for ${\texttt{COMPKhm}}$, ${\texttt{EMPKhm}}$ and ${\ensuremath{\mathtt{UKhm}}}$ in the setting of binary $\mathcal{K}h$ are the following:[^4]
----------------------- -----------------------------------------------------------------------------------------------------------------------------------------
[$\mathtt{COMPKh}$]{} ${\ensuremath{\mathcal{K}h{(p,q)}}}\land{\ensuremath{\mathcal{K}h{(q,r)}}}\to {\ensuremath{\mathcal{K}h{(p,r)}}}$
[$\mathtt{EMPKh}$]{} ${\mathcal{U}}(p\to q)\to {\ensuremath{\mathcal{K}h{(p,q)}}}$
[$\mathtt{UKh}$]{} ${\mathcal{U}}(p'\to p) \land {\mathcal{U}}(q\to q')\land {\ensuremath{\mathcal{K}h{(p, q)}}}\to {\ensuremath{\mathcal{K}h{(p', q')}}}$
----------------------- -----------------------------------------------------------------------------------------------------------------------------------------
In the system ${\mathbb{SKH}}$ of [@Wang15lori] [$\mathtt{UKh}$]{} can be derived using ${\ensuremath{\mathtt{COMPKh}}}$ and ${\ensuremath{\mathtt{EMPKh}}}$. However, ${\ensuremath{\mathtt{UKhm}}}$ cannot be derived using ${\texttt{COMPKhm}}$ and ${\texttt{EMPKhm}}$. In particular, ${\ensuremath{\mathcal{K}hm{(p',\bot, p)}}}\land{\ensuremath{\mathcal{K}hm{(p,o,q)}}}\to {\ensuremath{\mathcal{K}hm{(p',o,q)}}}$ is not valid due to the lack of ${\mathcal{U}}(p\to o)$, in contrast with the ${\mathbb{SKH}}$-derivable ${\ensuremath{\mathcal{K}h{(p', p)}}}\land {\ensuremath{\mathcal{K}h{(p, q)}}}\to {\ensuremath{\mathcal{K}h{(p', q)}}}$ which is crucial in the derivation of [$\mathtt{UKh}$]{} in ${\mathbb{SKH}}$.
Since ${\mathcal{U}}$ is an universal modality, [$\mathtt{DISTU}$]{} and [$\mathtt{TU}$]{} are obviously valid. Due to the fact that the modality ${{\ensuremath{\mathcal{K}hm}}}$ is not local, it is easy to show that [$\mathtt{4KhmU}$]{} and [$\mathtt{5KhmU}$]{} are valid. Moreover, by Propositions \[prop:validEMPKhm\]–\[prop:validREPKhm\], we have that all axioms are valid. Due to a standard argument in modal logic, we know that the rules [$\mathtt{MP}$]{}, [$\mathtt{NECU}$]{} and [$\mathtt{SUB}$]{} preserve formula’s validity. The soundness of ${\mathbb{SKHM}}$ follows immediately.
${\mathbb{SKHM}}$ is sound w.r.t. the class of all models.
Below we derive some theorems and rules that are useful in the later proofs.
We can derive the following in ${\mathbb{SKHM}}$:
------------------- ---------------------------------------------------------------------------------------------------------------
[$\mathtt{4U}$]{} ${\mathcal{U}}p\to{\mathcal{U}}{\mathcal{U}}p$
[$\mathtt{5U}$]{} $\neg{\mathcal{U}}p\to {\mathcal{U}}\neg{\mathcal{U}}p$
[`ULKhm`]{} ${\mathcal{U}}(p'\to p)\land{\ensuremath{\mathcal{K}hm{(p,o,q)}}}\to {\ensuremath{\mathcal{K}hm{(p',o,q)}}}$
[`UMKhm`]{} ${\mathcal{U}}(o\to o')\land{\ensuremath{\mathcal{K}hm{(p,o,q)}}}\to {\ensuremath{\mathcal{K}hm{(p,o',q)}}}$
[`URKhm`]{} ${\mathcal{U}}(q\to q')\land{\ensuremath{\mathcal{K}hm{(p,o,q')}}}\to {\ensuremath{\mathcal{K}hm{(p,o,q')}}}$
[`UNIV`]{} ${\mathcal{U}}\neg p\to{\ensuremath{\mathcal{K}hm{(p,\bot,\bot)}}}$
[`REU`]{} from ${\varphi}{\leftrightarrow}\psi$ prove ${\mathcal{U}}{\varphi}{\leftrightarrow}{\mathcal{U}}\psi$
[`RE`]{} from ${\varphi}{\leftrightarrow}\psi$ prove $\chi{\leftrightarrow}\chi'$
where $\chi'$ is obtained by replacing some occurrences of ${\varphi}$ in $\chi$ by $\psi$.
------------------- ---------------------------------------------------------------------------------------------------------------
[`REU`]{} is immediate given ${\ensuremath{\mathtt{DISTU}}}$ and ${\ensuremath{\mathtt{NECU}}}$. ${\ensuremath{\mathtt{4U}}}$ and ${\ensuremath{\mathtt{5U}}}$ are special cases of ${\ensuremath{\mathtt{4KhmU}}}$ and ${\ensuremath{\mathtt{5KhmU}}}$ respectively. ${\texttt{ULKhm}}, {\texttt{UMKhm}}, {\texttt{URKhm}}$ are the special cases of ${\ensuremath{\mathtt{UKhm}}}$. To prove [`UNIV`]{}, first note that ${\mathcal{U}}\neg p{\leftrightarrow}{\mathcal{U}}(p\to\bot)$ due to [`REU`]{}. Then due to ${\texttt{EMPKhm}}$, we have ${\mathcal{U}}\neg p\to {\ensuremath{\mathcal{K}hm{(p, \bot, \bot)}}}$. ${\texttt{RE}}$ can be obtained by using ${\ensuremath{\mathtt{UKhm}}}$ and ${\ensuremath{\mathtt{NECU}}}$.
Completeness {#Sec.Proof}
============
This section will prove that ${\mathbb{SKHM}}$ is complete w.r.t. the class of all models. The key is to build a canonical model based on a fixed maximal consistent set, just as in [@Wang15lori]. However, the canonical model here is much more complicated. Firstly, the state of the canonical model is a pair consisting of a maximal consistent set and a marker which will play an important role in defining the witness plan for ${{\ensuremath{\mathcal{K}hm}}}$-formulas. Secondly, different from the canonical model in [@Wang15lori] where each formula of the form ${\ensuremath{\mathcal{K}h{(\psi,{\varphi})}}}$ is realized by an one-step witness plan, some ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}$ formulas here have to be realized by a two-step witness plan, and the intermediate states need to satisfy $\chi$.
Here are some notions before we prove the completeness. Given a set of ${\mathbf{L_{Khm}}}$ formulas $\Delta$, let $\Delta|_{{{\ensuremath{\mathcal{K}hm}}}}$ and $\Delta|_{\neg{{\ensuremath{\mathcal{K}hm}}}}$ be the collections of its positive and negative ${{\ensuremath{\mathcal{K}hm}}}$ formulas: $$\Delta|_{{{\ensuremath{\mathcal{K}hm}}}}=\{\theta \mid\theta={\ensuremath{\mathcal{K}hm{(\psi,\chi,\varphi)}}}\in\Delta \};$$ $$\Delta|_{\neg{{\ensuremath{\mathcal{K}hm}}}}=\{\theta \mid\theta=\neg{\ensuremath{\mathcal{K}hm{(\psi,\chi,\varphi)}}}\in\Delta \}.$$ In the following, let $\Gamma$ be a maximal consistent set (MCS) of ${\mathbf{L_{Khm}}}$ formulas. We first prepare ourselves with some handy propositions.
Let ${{\ensuremath{\Phi_\Gamma}}}$ be the set of all MCS $\Delta$ such that $\Delta|_{{\ensuremath{\mathcal{K}hm}}}=\Gamma|_{{\ensuremath{\mathcal{K}hm}}}$.
Since every $\Delta\in {{\ensuremath{\Phi_\Gamma}}}$ is maximal consistent it follows immediately that:
\[prop:ShareKHow\] For each $\Delta\in{{\ensuremath{\Phi_\Gamma}}}$, we have ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}\in\Gamma$ if and only if ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}\in\Delta$ for all ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}\in {\mathbf{L_{Khm}}}$.
\[prop:allphiImplyUphi\] If ${\varphi}\in\Delta$ for all $\Delta\in{{\ensuremath{\Phi_\Gamma}}}$ then ${\mathcal{U}}{\varphi}\in\Delta$ for all $\Delta\in{{\ensuremath{\Phi_\Gamma}}}$.
Suppose $\varphi\in \Delta$ for all $\Delta\in \Phi_\Gamma$, then by the definition of $\Phi_\Gamma$, $\neg\varphi$ is not consistent with $\Gamma|_{{{\ensuremath{\mathcal{K}hm}}}}\cup\Gamma|_{\neg{{\ensuremath{\mathcal{K}hm}}}}$, for otherwise $\Gamma|_{{{\ensuremath{\mathcal{K}hm}}}}\cup\Gamma|_{\neg{{\ensuremath{\mathcal{K}hm}}}}\cup\{\neg \varphi\}$ can be extended into a maximal consistent set in $\Phi_\Gamma$ due to a standard Lindenbaum-like argument. Thus there are ${\ensuremath{\mathcal{K}hm{(\psi_1,\chi_1, \varphi_1)}}}$, …, ${\ensuremath{\mathcal{K}hm{(\psi_k,\chi_k, \varphi_k)}}} \in \Gamma|_{{{\ensuremath{\mathcal{K}hm}}}}$ and $\neg {\ensuremath{\mathcal{K}hm{(\psi'_1,\chi'_1, \varphi'_1)}}}$, …, $\neg{\ensuremath{\mathcal{K}hm{(\psi'_l,\chi'_l, \varphi'_l)}}} \in \Gamma|_{\neg {{\ensuremath{\mathcal{K}hm}}}}$ such that $$\vdash \bigwedge_{1\leq i\leq k}{\ensuremath{\mathcal{K}hm{(\psi_i,\chi_i, \varphi_i)}}}\land \bigwedge_{1\leq j\leq l}\neg {\ensuremath{\mathcal{K}hm{(\psi'_j,\chi'_j, \varphi'_j)}}}\to \varphi.$$ By ${\ensuremath{\mathtt{NECU}}}$, $$\vdash {\mathcal{U}}(\bigwedge_{1\leq i\leq k}{\ensuremath{\mathcal{K}hm{(\psi_i,\chi_i, \varphi_i)}}}\land \bigwedge_{1\leq j\leq l}\neg {\ensuremath{\mathcal{K}hm{(\psi'_j,\chi'_j, \varphi'_j)}}}\to \varphi).$$ By ${\ensuremath{\mathtt{DISTU}}}$ we have: $$\vdash{\mathcal{U}}(\bigwedge_{1\leq i\leq k}{\ensuremath{\mathcal{K}hm{(\psi_i,\chi_i, \varphi_i)}}}\land \bigwedge_{1\leq j\leq l}\neg {\ensuremath{\mathcal{K}hm{(\psi'_j,\chi'_j, \varphi'_j)}}}) \to {\mathcal{U}}\varphi.$$ Since ${\ensuremath{\mathcal{K}hm{(\psi_1,\chi_1, \varphi_1)}}}$, …, ${\ensuremath{\mathcal{K}hm{(\psi_k,\chi_k, \varphi_k)}}} \in \Gamma$, we have ${\mathcal{U}}{\ensuremath{\mathcal{K}hm{(\psi_1,\chi_1, \varphi_1)}}}$, …, ${\mathcal{U}}{\ensuremath{\mathcal{K}hm{(\psi_k,\chi_k, \varphi_k)}}}\in \Gamma$ due to ${\ensuremath{\mathtt{4KhmU}}}$ and the fact that $\Gamma$ is a maximal consistent set. Similarly, we have ${\mathcal{U}}\neg {\ensuremath{\mathcal{K}hm{(\psi'_1,\chi'_1, \varphi'_1)}}}$, …, ${\mathcal{U}}\neg{\ensuremath{\mathcal{K}hm{(\psi'_l,\chi'_l, \varphi'_l)}}} \in \Gamma$ due to ${\ensuremath{\mathtt{5KhmU}}}$. By [$\mathtt{DISTU}$]{} and [$\mathtt{NECU}$]{}, it is easy to show that $\vdash {\mathcal{U}}(p\land q){\leftrightarrow}{\mathcal{U}}p\land {\mathcal{U}}q$. Then due to a slight generalization, we have: $${\mathcal{U}}(\bigwedge_{1\leq i\leq k}{\ensuremath{\mathcal{K}hm{(\psi_i,\chi_i, \varphi_i)}}}\land \bigwedge_{1\leq j\leq l}\neg {\ensuremath{\mathcal{K}hm{(\psi'_j,\chi'_j, \varphi'_j)}}}) \in \Gamma.$$ Now it is immediate that ${\mathcal{U}}\varphi\in\Gamma$. Due to Proposition \[prop:ShareKHow\], ${\mathcal{U}}\varphi\in\Delta$ for all $\Delta\in \Phi_\Gamma.$
\[prop:OneStepExistence\] Given ${\ensuremath{\mathcal{K}hm{(\psi,\top,{\varphi})}}}\in \Gamma$ and $\Delta\in\Phi_\Gamma$, if $\psi\in\Delta$ then there exists $\Delta'\in\Phi_\Gamma$ such that ${\varphi}\in\Delta'$.
Assuming ${\ensuremath{\mathcal{K}hm{(\psi,\top,{\varphi})}}}\in \Gamma$ and $ \psi\in\Delta\in \Phi_\Gamma$, if there does not exist $\Delta'\in\Phi_\Gamma$ such that ${\varphi}\in\Delta'$, it means that $\neg{\varphi}\in\Delta'$ for all $\Delta'\in\Phi_\Gamma$. It follows by Proposition \[prop:allphiImplyUphi\] that ${\mathcal{U}}\neg{\varphi}\in\Gamma$, namely ${\ensuremath{\mathcal{K}hm{({\varphi},\top,\bot)}}}\in\Gamma$. Since ${\mathcal{U}}({\varphi}\to \bot)$ and ${\ensuremath{\mathcal{K}hm{(\psi,\top,{\varphi})}}}\in\Gamma$, it follows by [`COMPKhm`]{} that ${\ensuremath{\mathcal{K}hm{(\psi,\top,\bot)}}}\in\Gamma$ namely, ${\mathcal{U}}\neg\psi\in\Gamma$. By Proposition \[prop:ShareKHow\], we have that ${\mathcal{U}}\neg\psi\in\Delta$. It follows by [$\mathtt{TU}$]{} that $\neg\psi\in\Delta$. This is contradictory with $\psi\in\Delta$. Therefore, there exists $\Delta'\in\Phi_\Gamma$ such that ${\varphi}\in\Delta'$.
Let the set of action symbols ${\ensuremath{\mathbf{\Sigma}}}_\Gamma$ be defined as ${\ensuremath{\mathbf{\Sigma}}}_\Gamma=\{{\langle \psi,\bot,{\varphi}\rangle}\mid {\ensuremath{\mathcal{K}hm{(\psi,\bot,{\varphi})}}}\in\Gamma \}\cup\{{\langle \chi^\psi,{\varphi}\rangle}\mid {\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}},\neg{\ensuremath{\mathcal{K}hm{(\psi,\bot,{\varphi})}}}\in\Gamma \}$.
The later part of ${\ensuremath{\mathbf{\Sigma}}}_\Gamma$ is to handle the cases where the intermediate state is indeed necessary: $\neg{\ensuremath{\mathcal{K}hm{(\psi,\bot,{\varphi})}}}$ makes sure that you cannot have a plan to guarantee ${\varphi}$ in less than two steps.
In the following we build a separate canonical model for each MCS $\Gamma$, for it is not possible to satisfy all of ${{\ensuremath{\mathcal{K}hm}}}$ formulas simultaneously in a single model since they are global. Because the later proofs are quite technical, it is very important to first understand the ideas behind the canonical model construction. Note that to satisfy a ${\ensuremath{\mathcal{K}hm{(\psi, \chi, {\varphi})}}}$ formula, there are two cases to be considered:
\(1) ${\ensuremath{\mathcal{K}hm{(\psi, \bot, {\varphi})}}}$ holds and we just need an one-step witness plan, which can be handled similarly using the techniques developed in [@Wang15lori];
\(2) ${\ensuremath{\mathcal{K}hm{(\psi, \bot, {\varphi})}}}$ does not hold, and we need to have a witness plan which at least involves an intermediate $\chi$-stage. By [`ONEKhm`]{}, ${\ensuremath{\mathcal{K}hm{(\psi, \bot, \chi)}}}$ holds. It is then tempting to reduce ${\ensuremath{\mathcal{K}hm{(\psi, \chi, {\varphi})}}}$ to ${\ensuremath{\mathcal{K}hm{(\psi, \bot, \chi)}}}\land {\ensuremath{\mathcal{K}hm{(\chi, \chi, {\varphi})}}}$. However, it is not correct since we may not have a strongly $\chi$-executable plan to make sure ${\varphi}$ from *any* $\chi$-state. Note that ${\ensuremath{\mathcal{K}hm{(\psi, \chi, {\varphi})}}}$ and ${\ensuremath{\mathcal{K}hm{(\psi, \bot, \chi)}}}$ only make sure we can start from *certain* $\chi$-states that result from the witness plan for ${\ensuremath{\mathcal{K}hm{(\psi, \bot, \chi)}}}$. However, we cannot refer to such $\chi$-states in the language of ${\mathbf{L_{Khm}}}$. This is why we include $\chi^\psi$ markers in the building blocks of the canonical model besides maximal consistent set. $\chi^\psi$ roughly tells us where does this state “comes from”. [^5]
The canonical model for $\Gamma$ is a tuple ${\mathcal{M}}^c_{\Gamma}={\langle {\mathcal{S}}^c,{\mathcal{R}}^c,{\mathcal{V}}^c \rangle}$ where:
- ${\mathcal{S}}^c=\{(\Delta,\chi^\psi)\mid \chi\in\Delta\in{{\ensuremath{\Phi_\Gamma}}}$, and ${\langle \chi^\psi,{\varphi}\rangle}\in{\ensuremath{\mathbf{\Sigma}}}_\Gamma$ for some ${\varphi}$ or ${\langle \psi,\bot,\chi \rangle}\in{\ensuremath{\mathbf{\Sigma}}}_\Gamma \}$. We write the pair in ${\mathcal{S}}$ as $w,v,\cdots$, and refer to the first entry of $w\in{\mathcal{S}}$ as ${{\ensuremath{\mathtt{L}(w)}}}$, to the second entry as ${{\ensuremath{\mathtt{R}(w)}}}$;
- $w{\xrightarrow{{\langle \psi,\bot,{\varphi}\rangle}}}_c w'$ iff $\psi\in{{\ensuremath{\mathtt{L}(w)}}}$ and ${{\ensuremath{\mathtt{R}(w')}}}={\varphi}^\psi$;
- $w{\xrightarrow{{\langle \chi^\psi,{\varphi}\rangle}}}_c w'$ iff ${{\ensuremath{\mathtt{R}(w)}}}=\chi^\psi$ and ${\varphi}\in{{\ensuremath{\mathtt{L}(w')}}}$;
- $p\in{\mathcal{V}}^c (w)$ iff $p\in{{\ensuremath{\mathtt{L}(w)}}}$.
For each $w\in{\mathcal{S}}$, we also call $w$ a $\psi$-state if $\psi\in{{\ensuremath{\mathtt{L}(w)}}}$.
In the above definition, ${{\ensuremath{\mathtt{R}(w)}}}$ marks the use of $w$ as an intermediate state. The same maximal consistent set $\Delta$ may have different uses depending on different ${{\ensuremath{\mathtt{R}(w)}}}$. We will make use of the transitions $w{\xrightarrow{{\langle \psi,\bot,\chi \rangle}}}_c v{\xrightarrow{{\langle \chi^\psi,{\varphi}\rangle}}}_c w'$ where ${{\ensuremath{\mathtt{R}(v)}}}=\chi^\psi$. Note that if ${{\ensuremath{\mathtt{R}(w)}}}=\chi^\psi$ then $w{\xrightarrow{{\langle \chi^\psi,{\varphi}\rangle}}}_cv$ for each ${\varphi}$-state $v$. The highly non-trivial part of the later proof of the truth lemma is to show adding such transitions and making them to be composed arbitrarily will not cause some ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}\not\in {{\ensuremath{\mathtt{L}(w)}}}$ to hold at $w$.
We first show that each $\Delta\in \Phi_\Gamma$ appears as ${{\ensuremath{\mathtt{L}(w)}}}$ for some $w\in {\mathcal{S}}^c$.
\[prop:allMCSinS\] For each $\Delta\in{{\ensuremath{\Phi_\Gamma}}}$, there exists $w\in{\mathcal{S}}^c$ such that ${{\ensuremath{\mathtt{L}(w)}}}=\Delta$.
Since $\vdash\top\to\top$, it follows by [$\mathtt{NECU}$]{} that $\vdash{\mathcal{U}}(\top\to\top)$. Thus, we have ${\mathcal{U}}(\top\to\top)\in\Gamma$. It follows by [`EMPKhm`]{} that ${\ensuremath{\mathcal{K}hm{(\top,\bot,\top)}}}\in\Gamma$. It follows that $a={\langle \top,\bot,\top \rangle}\in{\ensuremath{\mathbf{\Sigma}}}_\Gamma$. Since $\top\in\Delta$, it follows that $(\Delta,\top^\top)\in{\mathcal{S}}^c$.
Since $\Gamma\in\Phi_\Gamma$, it follows by Proposition \[prop:allMCSinS\] that ${\mathcal{S}}^c\neq\emptyset$.
Proposition \[prop:allphiImplyUphi\] helps us to prove the following two handy propositions which will play crucial roles in the completeness proof. Note that according to Proposition \[prop:allphiImplyUphi\], to obtain that ${\mathcal{U}}{\varphi}$ in all the $\Delta\in \Phi_\Gamma$, we just need to show that ${\varphi}$ is in all the $\Delta\in \Phi_\Gamma$, not necessarily in all the $w\in {\mathcal{S}}^c$.
\[prop:IMPinHead\] Given $a={\langle \psi',\bot,{\varphi}' \rangle}\in{\ensuremath{\mathbf{\Sigma}}}_\Gamma$, If for each $\psi$-state $w\in{\mathcal{S}}^c$ we have that $a$ is executable at $w$, then ${\mathcal{U}}(\psi\to\psi')\in \Gamma$.
Suppose that every $\psi$-state has an outgoing $a$-transition, then by the definition of ${\mathcal{R}}^c$, $\psi'$ is in all the $\psi$-states. For each $\Delta\in\Phi_\Gamma$, either $\psi\not\in \Delta$, or $\psi\in\Delta$ thus $\psi'\in \Delta$. Now by the fact that $\Delta$ is maximally consistent it is not hard to show $\psi\to\psi'\in\Delta$ in both cases. By Proposition \[prop:allphiImplyUphi\], ${\mathcal{U}}(\psi\to\psi') \in \Delta$ for all $\Delta\in\Phi_\Gamma.$ It follows by $\Gamma\in\Phi_\Gamma$ that ${\mathcal{U}}(\psi\to\psi') \in\Gamma$.
\[prop:IMPinTail\] Given $w\in{\mathcal{S}}^c$ and $a={\langle \psi,\bot,{\varphi}' \rangle}$ or ${\langle \chi^{\psi},{\varphi}' \rangle}\in{\ensuremath{\mathbf{\Sigma}}}_\Gamma$ such that $a$ is executable at $w$, if ${\varphi}\in{{\ensuremath{\mathtt{L}(w')}}}$ for each $w'$ with $w{\xrightarrow{a}}w'$ then ${\mathcal{U}}({\varphi}'\to{\varphi})\in\Gamma$.
Firstly, we focus on the case of $a={\langle \psi,\bot,{\varphi}' \rangle}$. For each $\Delta\in\Phi_\Gamma$ with ${\varphi}'\in \Delta$, we have $v=(\Delta,{\varphi}'^\psi)\in{\mathcal{S}}^c$. Since ${\langle \psi,\bot,{\varphi}' \rangle}$ is executable at $w$, it means that $\psi\in{{\ensuremath{\mathtt{L}(w)}}}$. By the definition, it follows that $w{\xrightarrow{a}}v$. Since ${\varphi}\in{{\ensuremath{\mathtt{L}(w')}}}$ for each $w'$ with $w{\xrightarrow{a}}w'$, it follows that ${\varphi}\in{{\ensuremath{\mathtt{L}(v)}}}$. Therefore, we have ${\varphi}\in\Delta$ for each $\Delta\in\Phi_\Gamma$ with ${\varphi}'\in\Delta$, namely ${\varphi}'\to{\varphi}\in \Delta$ for all $\Delta\in\Phi_\Gamma$. It follows by Proposition \[prop:allphiImplyUphi\] that ${\mathcal{U}}({\varphi}'\to{\varphi})\in\Gamma$.
Secondly, we focus on the case of $a={\langle \chi^\psi,{\varphi}' \rangle}$. For each $\Delta\in\Phi_\Gamma$ with ${\varphi}'\in \Delta$, it follows by Proposition \[prop:allMCSinS\] that there exists $v\in{\mathcal{S}}^c$ such that ${{\ensuremath{\mathtt{L}(v)}}}=\Delta$. Since $a$ is executable at $w$, it follows that $w{\xrightarrow{a}}v$. Since ${\varphi}\in{{\ensuremath{\mathtt{L}(w')}}}$ for each $w'$ with $w{\xrightarrow{a}}w'$, it follows that ${\varphi}\in{{\ensuremath{\mathtt{L}(v)}}}$. Therefore, we have shown that ${\varphi}'\in\Delta$ implies ${\varphi}\in\Delta$ for all $\Delta\in\Phi_\Gamma$. It follows by Proposition \[prop:allphiImplyUphi\] that ${\mathcal{U}}({\varphi}'\to{\varphi})\in\Gamma$.
Before proving the truth lemma, we first need a handy result.
\[prop:phi\_n\] Given a non-empty sequence $\sigma=a_1\cdots a_n\in{\ensuremath{\mathbf{\Sigma}}}^*_\Gamma$ where $a_i={\langle \psi_i,\bot,{\varphi}_i \rangle}$ or $a_i={\langle \chi^{\psi_i}_i,{\varphi}_i \rangle}$ for each $1\leq i\leq n$, we have ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi}_i)}}}\in\Gamma$ for all $1\leq i\leq n $ if for each $\psi$-states $w\in {\mathcal{S}}^c$:
- $\sigma$ is strongly executable at $w$;
- $w{\xrightarrow{\sigma_j}}t'$ implies $\chi\in{{\ensuremath{\mathtt{L}(t')}}}$ for all $1\leq j <n$.
If there is no $\psi$-state in ${\mathcal{S}}^c$, it follows that $\neg\psi\in {{\ensuremath{\mathtt{L}(w')}}}$ for each $w'\in{\mathcal{S}}^c$. It follows by Proposition \[prop:allMCSinS\] that $\neg\psi\in\Delta$ for all $\Delta\in\Phi_\Gamma$. By Proposition \[prop:allphiImplyUphi\], we have ${\mathcal{U}}\neg\psi\in \Gamma$. By [`UNIV`]{}, ${\ensuremath{\mathcal{K}hm{(\psi,\bot,\bot)}}}\in \Gamma$. Since $\vdash \bot\to\chi$ and $\vdash\bot\to {\varphi}$. Then by [$\mathtt{NECU}$]{}, we have $\vdash{\mathcal{U}}(\bot\to\chi)$ and $\vdash{\mathcal{U}}(\bot\to{\varphi})$. By ${\texttt{UMKhm}}$ and ${\texttt{URKhm}}$, it is obvious that ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}\in\Gamma$.
Next, assuming $v\in{\mathcal{S}}^c$ is a $\psi$-state, we will show ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}\in\Gamma$. There are two cases: $n=1$ or $n\geq 2$. For the case of $n=1$, we will prove it directly; for the case of $n\geq 2$, we will prove it by induction on $i$.
- $n=1$. If $a_1$ is in the form of ${\langle \chi_1^{\psi_1},{\varphi}_1 \rangle}$, by the definition of ${\xrightarrow{{\langle \chi_1^{\psi_1},{\varphi}_1 \rangle}}}$ it follows that ${{\ensuremath{\mathtt{R}(w)}}}=\chi_1^{\psi_1}$ for each $\psi$-state $w$. Let $\chi_0$ be a formula satisfying that $\vdash\chi_0{\leftrightarrow}\chi_1$ and $\chi_0\neq \chi_1$. By the rule of Replacement of Equals [`RE`]{}, it follows that ${\langle \chi_0^{\psi_1},{\varphi}_1 \rangle}\in{\ensuremath{\mathbf{\Sigma}}}_\Gamma$. Let $w'=({{\ensuremath{\mathtt{L}(v)}}},\chi_0^{\psi_1})$ then it follows that $w'\in{\mathcal{S}}^c$. Since $\psi\in{{\ensuremath{\mathtt{L}(v)}}}$, then we have $\psi\in{{\ensuremath{\mathtt{L}(w')}}}$. However, since ${{\ensuremath{\mathtt{R}(w')}}}=\chi_1^{\psi_1}\not=\chi_0^{\psi_1}$, $\sigma={\langle \chi_1^{\psi_1},{\varphi}_1 \rangle}$ is not executable at the $\psi$-state $w'$, contradicting the assumption that $\sigma$ is strongly executable at all $\psi$-states. Therefore, we know that $a_1$ cannot be in the form of ${\langle \chi_1^{\psi_1},{\varphi}_1 \rangle}$.
If $a_1={\langle \psi_1,\bot,{\varphi}_1 \rangle}$, it follows that ${\ensuremath{\mathcal{K}hm{(\psi_1,\bot,{\varphi}_1)}}}\in\Gamma$. Since $a_1$ is executable at each $\psi$-state, it follows by Proposition \[prop:IMPinHead\] that ${\mathcal{U}}(\psi\to\psi_1)\in\Gamma$. Since ${\ensuremath{\mathcal{K}hm{(\psi_1,\bot,{\varphi}_1)}}}\in\Gamma$, it follows by [`ULKhm`]{} that ${\ensuremath{\mathcal{K}hm{(\psi,\bot,{\varphi}_1)}}}\in\Gamma$. By [$\mathtt{NECU}$]{} and [`UMKhm`]{}, it is clear that ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi}_1)}}}\in\Gamma$.
- $n\geq 2$. By induction on $i$, next we will show that ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi}_i)}}}\in\Gamma$ for each $1\leq i\leq n$. For the case of $i=1$, with the similar proof as in the case of $n=1$, we can show that $a_1$ can only be ${\langle \psi_1,\bot,{\varphi}_1 \rangle}$ and ${\mathcal{U}}(\psi\to\psi_1)\in\Gamma$. Therefore by ${\ensuremath{\mathtt{UKhm}}}$ we have ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi}_1)}}}\in\Gamma$. Under the induction hypothesis (IH) that ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi}_i)}}}\in\Gamma$ for each $1\leq i\leq k$, we will show that ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi}_{k+1})}}}\in\Gamma$, where $1\leq k\leq n-1$. Because $\sigma$ is strongly executable at $v$, it follows that there are $w',v'\in{\mathcal{S}}^c$ such that $$\xymatrix{
v\ar[r]^{a_1}&\cdots\ar[r]^{a_{k-1}}&w'\ar[r]^{a_k}&v'\ar[r]^{a_{k+1}}&\cdots\ar[r]^{a_n}&t.\\
}$$ Moreover, for each $t'$ with $w'{\xrightarrow{a_{k}}}t'$ we have $\chi\in{{\ensuremath{\mathtt{L}(t')}}}$. It follows by Proposition \[prop:IMPinTail\] that ${\mathcal{U}}({\varphi}_{k}\to \chi)\in\Gamma \ (\blacktriangle)$. Proceeding, there are two cases of $a_{k+1}$:
- $a_{k+1}={\langle \psi_{k+1},\bot,{\varphi}_{k+1} \rangle}$. Since $\sigma$ is strongly executable at $v$, it follows that for each $t'$ with $w'{\xrightarrow{a_{k}}}t'$ we know that $a_{k+1}$ is executable at each $t'$. It follows by the definition of ${\xrightarrow{{\langle \psi_{k+1},\bot,{\varphi}_{k+1} \rangle}}}$ that $\psi_{k+1}\in {{\ensuremath{\mathtt{L}(t')}}}$. Moreover, since $a_k$ is executable at $w'$, it follows by Proposition \[prop:IMPinTail\] that ${\mathcal{U}}({\varphi}_{k}\to\psi_{k+1})\in\Gamma$. Since $a_{k+1}\in{\ensuremath{\mathbf{\Sigma}}}_\Gamma$, it then follows that ${\ensuremath{\mathcal{K}hm{(\psi_{k+1},\bot,{\varphi}_{k+1})}}}\in\Gamma$. It then follows by [`ULKhm`]{} that ${\ensuremath{\mathcal{K}hm{({\varphi}_{k},\bot,{\varphi}_{k+1})}}}\in\Gamma$. Since $\vdash{\mathcal{U}}(\bot\to\chi)$, it follows by [`UMKhm`]{} that ${\ensuremath{\mathcal{K}hm{({\varphi}_{k},\chi,{\varphi}_{k+1})}}}\in\Gamma$. Since by IH we have that ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi}_{k})}}}\in\Gamma$, It follows from $(\blacktriangle)$ and [`COMPKhm`]{} that ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi}_{k+1})}}}\in\Gamma$.
- $a_{k+1}={\langle \chi_{k+1}^{\psi_{k+1}},{\varphi}_{k+1} \rangle}$. Since $\sigma$ is strongly executable at $v$, it follows that for each $t'$ with $w'{\xrightarrow{a_{k}}}t'$ we know that $a_{k+1}$ is executable at $t'$. Then we have that ${{\ensuremath{\mathtt{R}(t')}}}=\chi_{k+1}^{\psi_{k+1}}$ for each $t'$ with $w'{\xrightarrow{a_{k}}}t'$.\
Note that the action $a_{k}$ cannot be in the form of ${\langle \chi_{k}^{\psi_{k}},{\varphi}_{k} \rangle}$. Suppose it can be, let $v''=({{\ensuremath{\mathtt{L}(v')}}},\chi_0^{\psi_{k+1}})$ where $\vdash\chi_0{\leftrightarrow}\chi_{k+1}$ and $\chi_0\neq \chi_{k+1}$. Since $w'{\xrightarrow{a_k}}v'$, it follows that ${\varphi}_k\in{{\ensuremath{\mathtt{L}(v')}}}$. Then it follows by the definition of transitions that $w'{\xrightarrow{a_{k}}}v''$. However, we know that ${{\ensuremath{\mathtt{R}(v'')}}}\neq \chi_{k+1}^{\psi_{k+1}}$ thus $a_{k+1}={\langle \chi_{k+1}^{\psi_{k+1}},{\varphi}_{k+1} \rangle}$ is not executable at $v''$, contradicting the strong executability. Therefore, we know that $a_{k}$ cannot be in the form of ${\langle \chi_{k}^{\psi_{k}},{\varphi}_{k} \rangle}$.
Now $a_{k}={\langle \psi_{k},\bot,{\varphi}_{k} \rangle}$. Since $w'{\xrightarrow{a_{k}}}v'$ and $a_{k+1}={\langle \chi_{k+1}^{\psi_{k+1}},{\varphi}_{k+1} \rangle}$ is executable at $v'$, we have ${{\ensuremath{\mathtt{R}(v')}}}={\varphi}_k^{\psi_k}=\chi_{k+1}^{\psi_{k+1}}$ by definition of transitions. It follows that $\psi_{k}=\psi_{k+1}$ and ${\varphi}_{k}=\chi_{k+1}$. Since $a_{k+1}\in{\ensuremath{\mathbf{\Sigma}}}_\Gamma$, it follows that ${\ensuremath{\mathcal{K}hm{(\psi_{k+1},\chi_{k+1},{\varphi}_{k+1})}}}\in\Gamma$. Thus, we have ${\ensuremath{\mathcal{K}hm{(\psi_{k},{\varphi}_{k},{\varphi}_{k+1})}}}\in\Gamma$. By $(\blacktriangle)$ and [`UMKhm`]{} we then have that ${\ensuremath{\mathcal{K}hm{(\psi_{k},\chi,{\varphi}_{k+1})}}}\in\Gamma \ (\blacktriangledown)$. If $k=1$, by Proposition \[prop:IMPinHead\] it is easy to show that ${\mathcal{U}}(\psi\to\psi_1)\in\Gamma$. Then by [`ULKhm`]{} we have ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi}_{k+1})}}}\in\Gamma$. If $k>1$, there is a state $w''$ such that $$\xymatrix{
v\ar[r]^{a_1}&\cdots\ar[r]^{a_{k-2}}&w''\ar[r]^{a_{k-1}}&w'\ar[r]^{a_k}&v'\ar[r]^{a_{k+1}}&\cdots\ar[r]^{a_n}&t.\\
}$$ Since $\sigma$ is strongly executable at $v$, it follows that for each $t'$ with $w''{\xrightarrow{a_{k-1}}}t'$ we have $a_{k}$ is executable at $t'$. It follows by the definition of ${\xrightarrow{{\langle \psi_k,\bot,{\varphi}_k \rangle}}}$, it follows that $\psi_k\in{{\ensuremath{\mathtt{L}(t')}}}$ for each $t'$ with $w''{\xrightarrow{a_{k-1}}}t'$. Since $a_{k-1}$ is executable at $w''$, it follows by Proposition \[prop:IMPinTail\] that ${\mathcal{U}}({\varphi}_{k-1}\to \psi_{k})\in \Gamma$.Moreover, since $v{\xrightarrow{\sigma_{k-1}}}t'$ for each $t' $ with $w''{\xrightarrow{a_{k-1}}}t'$, it follows that $\chi\in{{\ensuremath{\mathtt{L}(t')}}}$. Thus by Proposition \[prop:IMPinTail\] again, we have ${\mathcal{U}}({\varphi}_{k-1}\to \chi)\in \Gamma$. Since we have proved $(\blacktriangledown)$, it follows by [`ULKhm`]{} that ${\ensuremath{\mathcal{K}hm{({\varphi}_{k-1},\chi,{\varphi}_{k+1})}}}\in\Gamma$. Since by IH we have ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi}_{k-1})}}}\in\Gamma$, it follows by [`COMPKhm`]{} that ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi}_{k+1})}}}\in\Gamma$.
Now we are ready to prove the truth lemma.
For each ${\varphi}$, we have ${\mathcal{M}}^c_\Gamma,w\vDash{\varphi}$ iff ${\varphi}\in {{\ensuremath{\mathtt{L}(w)}}}$.
Boolean cases are trivial, and we only focus on the case of ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}$.
**Left to Right:** If there is no state $w'$ such that ${{\mathcal{M}}}^c_\Gamma,w'\vDash\psi$, it follows by induction that $\neg\psi\in {{\ensuremath{\mathtt{L}(w')}}}$ for each $w'\in{\mathcal{S}}^c$. It follows by Proposition \[prop:allMCSinS\] that $\neg\psi\in\Delta$ for all $\Delta\in\Phi_\Gamma$. By Proposition \[prop:allphiImplyUphi\], we have ${\mathcal{U}}\neg\psi\in {{\ensuremath{\mathtt{L}(w)}}}$. By [`UNIV`]{}, ${\ensuremath{\mathcal{K}hm{(\psi,\bot,\bot)}}}\in {{\ensuremath{\mathtt{L}(w)}}}$. Since $\vdash \bot\to\chi$ and $\vdash\bot\to {\varphi}$. Then by [$\mathtt{NECU}$]{}, we have $\vdash{\mathcal{U}}(\bot\to\chi)$ and $\vdash{\mathcal{U}}(\bot\to{\varphi})$. By ${\texttt{UMKhm}}$ and ${\texttt{URKhm}}$, it is obvious that ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}\in{{\ensuremath{\mathtt{L}(w)}}}$.
Next, assuming ${{\mathcal{M}}}^c_\Gamma,v\vDash\psi$ for some $v\in{\mathcal{S}}^c$, we will show ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}\in{{\ensuremath{\mathtt{L}(w)}}}$. Since ${{\mathcal{M}}}^c_\Gamma,w\vDash{\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}$, it follows that there exists $\sigma\in{\ensuremath{\mathbf{\Sigma}}}^*$ such that for each ${{\mathcal{M}}}^c_\Gamma,w'\vDash\psi$: $\sigma$ is strongly $\chi$-executable at $w'$ and ${{\mathcal{M}}}^c_\Gamma,v'\vDash{\varphi}$ for all $v'$ with $w'{\xrightarrow{\sigma}}v'$. There are two cases: $\sigma$ is empty or not.
- $\sigma=\epsilon$. This means that ${{\mathcal{M}}}^c_\Gamma,w'\vDash{\varphi}$ for each ${{\mathcal{M}}}^c_\Gamma,w'\vDash\psi$. It follows by induction that $\psi\in{{\ensuremath{\mathtt{L}(w')}}}$ implies ${\varphi}\in{{\ensuremath{\mathtt{L}(w')}}}$. Thus, we have $\psi\to{\varphi}\in{{\ensuremath{\mathtt{L}(w')}}}$ for all $w'\in{\mathcal{S}}^c$. By Proposition \[prop:allMCSinS\], we have $\psi\to{\varphi}\in\Delta$ for all $\Delta\in\Phi_\Gamma$. It follows by Proposition \[prop:allphiImplyUphi\] that ${\mathcal{U}}(\psi\to{\varphi})\in {{\ensuremath{\mathtt{L}(w)}}}$. It then follows by [`EMPKhm`]{} that ${\ensuremath{\mathcal{K}hm{(\psi,\bot,{\varphi})}}}\in{{\ensuremath{\mathtt{L}(w)}}}$. By [$\mathtt{NECU}$]{} and [`UMKhm`]{}, it is easy to show that ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}\in{{\ensuremath{\mathtt{L}(w)}}}$.
- $\sigma=a_1\cdots a_n$ where for each $1\leq i\leq n$, $a_i={\langle \psi_i,\bot,{\varphi}_i \rangle}$ or $a_i={\langle \chi^{\psi_i}_i,{\varphi}_i \rangle}$. Since $\sigma$ is strongly $\chi$-executable at each $w'$ with ${\mathcal{M}}^c_\Gamma, w'\vDash \psi$, it follows by IH that for each $\psi$-state $w'$: $\sigma$ is strongly executable at $w'$ and $w'{\xrightarrow{\sigma_j}}t'$ implies $\chi\in{{\ensuremath{\mathtt{L}(t')}}}$ for all $1\leq j <n$. By Proposition \[prop:phi\_n\], we have that ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi}_n)}}}\in{{\ensuremath{\mathtt{L}(v)}}}$. Since ${{\mathcal{M}}}^c_\Gamma,v\vDash\psi$ and $\sigma$ is strongly $\chi$-executable at $v$ and ${{\mathcal{M}}}^c_\Gamma,v''\vDash{\varphi}$ for each $v''$ with $v{\xrightarrow{\sigma}}v''$, it follows that there exists $v'$ such that $a_n$ is executable at $v'$ and ${{\mathcal{M}}}^c_\Gamma,v''\vDash{\varphi}$ for each $v''$ with $v'{\xrightarrow{a_n}}v''$. (Please note that $v'=v$ if $n=1$.) Note that $a_n$ is either ${\langle \psi_n, \bot, {\varphi}_n \rangle}$ or ${\langle \chi_n^{\psi_n}, {\varphi}_n \rangle}$. It follows by Proposition \[prop:IMPinTail\] and IH that ${\mathcal{U}}({\varphi}_n\to{\varphi})\in\Gamma$, then we have ${\mathcal{U}}({\varphi}_n\to{\varphi})\in{{\ensuremath{\mathtt{L}(v)}}}$. It follows by [`URKhm`]{} and Proposition \[prop:ShareKHow\] that ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}\in{{\ensuremath{\mathtt{L}(w)}}}$.
This completes the proof for $w\vDash {\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}$ implies ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}\in{{\ensuremath{\mathtt{L}(w)}}}.$
**Right to Left:** Suppose that ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}\in{{\ensuremath{\mathtt{L}(w)}}}$, we need to show that ${{\mathcal{M}}}^c_\Gamma,w\vDash{\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}$. There are two cases: there is a state $w'\in{\mathcal{S}}^c$ such that ${{\mathcal{M}}}^c_\Gamma,w'\vDash\psi$ or not. If there is no such state, it follows ${{\mathcal{M}}}^c_\Gamma,w\vDash{\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}$.
For the second case, let $w'$ be a state such that ${{\mathcal{M}}}^c_\Gamma,w'\vDash\psi$. It follows by IH that $\psi\in{{\ensuremath{\mathtt{L}(w')}}}$. Since we already have ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}\in{{\ensuremath{\mathtt{L}(w)}}}$, it follows by Proposition \[prop:ShareKHow\] that ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}\in\Gamma$. Since $\vdash{\mathcal{U}}(\chi\to\top)$, it follows by [`UMKhm`]{} that ${\ensuremath{\mathcal{K}hm{(\psi,\top,{\varphi})}}}\in\Gamma$. It follows by Proposition \[prop:OneStepExistence\] that there exists $\Delta'\in\Phi_\Gamma$ such that ${\varphi}\in\Delta'$. There are two cases: ${\ensuremath{\mathcal{K}hm{(\psi,\bot,{\varphi})}}}\in \Gamma$ or not.
- ${\ensuremath{\mathcal{K}hm{(\psi,\bot,{\varphi})}}}\in\Gamma$. It follows that $a={\langle \psi,\bot,{\varphi}\rangle}\in{\ensuremath{\mathbf{\Sigma}}}_\Gamma$. Therefore, we have $v=(\Delta',{\varphi}^\psi)\in{\mathcal{S}}^c$. Since $\psi\in{{\ensuremath{\mathtt{L}(w')}}}$, it follows that $w'{\xrightarrow{a}}v$. Thus, $a$ is strongly $\chi$-executable at $w'$. What is more, ${\varphi}\in{{\ensuremath{\mathtt{L}(v')}}}$ for each $v'$ with $w'{\xrightarrow{a}}v'$ by the definition of the transition. It follows by IH that ${{\mathcal{M}}}^c_\Gamma,v'\vDash{\varphi}$ for all $v'$ with $w'{\xrightarrow{a}}v'$. Therefore, we have ${{\mathcal{M}}}^c_\Gamma,w\vDash{\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}$ witnessed by a single step $\sigma$.
- $\neg{\ensuremath{\mathcal{K}hm{(\psi,\bot,{\varphi})}}}\in\Gamma$. It follows by [`ONEKhm`]{} that ${\ensuremath{\mathcal{K}hm{(\psi,\bot,\chi)}}}\in\Gamma$. We then have $a={\langle \psi,\bot,\chi \rangle}\in{\ensuremath{\mathbf{\Sigma}}}_\Gamma$ and $b={\langle \chi^\psi,{\varphi}\rangle}\in{\ensuremath{\mathbf{\Sigma}}}_\Gamma$. Since ${\ensuremath{\mathcal{K}hm{(\psi,\bot,\chi)}}}\in\Gamma$ and $\vdash{\mathcal{U}}(\bot\to\top)$, it follows by [`UMKhm`]{} that ${\ensuremath{\mathcal{K}hm{(\psi,\top,\chi)}}}\in\Gamma$. It follows by Proposition \[prop:OneStepExistence\] that there exists $\Delta''\in\Phi_\Gamma$ such that $\chi\in\Delta''$. Therefore, we have $t=(\Delta'',\chi^\psi)\in{\mathcal{S}}^c$. Since there exists $\Delta'\in\Phi_\Gamma$ with ${\varphi}\in\Delta'$, it follows by Proposition \[prop:allphiImplyUphi\] that there is $t'\in{\mathcal{S}}^c$ such that ${{\ensuremath{\mathtt{L}(t')}}}=\Delta'$. Now, starting with any $\psi$-state, $a$ is clearly executable and it will lead to a $\chi$-state, and then by a $b$ step we will reach all the ${\varphi}$ states. Therefore, by IH, we have that $ab$ is strongly $\chi$-executable at $w'$, and that for all $v'$ with $w'{\xrightarrow{ab}}v'$ we have ${{\mathcal{M}}}^c_\Gamma,v'\vDash{\varphi}$. Therefore, we have ${{\mathcal{M}}}^c_\Gamma,w\vDash{\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}$. Note that we do need a 2-step $\sigma$ in this case.
Now due to a standard Lindenbaum-like argument, each ${\mathbb{SKHM}}$-consistent set of formulas can be extended to a maximal consistent set $\Gamma$. Due to the truth lemma, ${{\mathcal{M}}}^c_\Gamma, (\Gamma, \top^\top)\vDash\Gamma.$ The completeness of ${\mathbb{SKHM}}$ follows immediately.
${\mathbb{SKHM}}$ is strongly complete w.r.t. the class of all models.
Conclusions
===========
This paper generalizes the knowing how logic presented in [@Wang15lori] and proposes a ternary modal operator ${\ensuremath{\mathcal{K}hm{(\psi,\chi,{\varphi})}}}$ to express that the agent knows how to achieve ${\varphi}$ given $\psi$ while maintaining $\chi$ in-between. This paper also presents a sound and complete axiomatization of this logic. Compared to the completeness proof in [@Wang15lori], the proof here is much more complicated, and the essential difference is that the state of the canonical model here is a pair consisting of a maximal consistent set and a marker of the form $\chi^\psi$ which indicates that this state has a ${\langle \psi,\bot,\chi \rangle}$-predecessor, in order to handle the intermediate constraints.
For future research, besides the obvious questions of decidability and model theory of the logic, we may give some alternative semantics to the same language by relaxing the strong executability. Intuitively, strongly executable plan may be too strong for knowledge-how in some cases. For example, if there is an action sequence $\sigma$ in the agent’s ability map such that doing $\sigma$ at a $\psi$-state will always make the agent *stop* on ${\varphi}$ states, we can probably also say the agent knows how to achieve ${\varphi}$ given $\psi$, e.g., I know how to start the engine in that old car, just turn the key several times until it starts, and three times should suffice at most. Please note that there are two kinds of states on which the agent might stop: either states the agent achieves after doing $\sigma$ successfully, or states on which the agent is unable to continue executing the remaining actions.
Another interesting topic is extending this logic with public announcement operators. Intuitively, $[\theta]{\varphi}$ says that ${\varphi}$ holds after the information $\theta$ is provided. The update of the new information amounts to the change of the background knowledge throughout the model, and this will affect the knowledge-how. For example, a doctor may not know how to treat a patient with the disease $p$ since he is worried that the only available medicine may potentially cause some very bad side-effect $r$, which can be expressed as $\neg{\ensuremath{\mathcal{K}hm{(p,\neg r,\neg p)}}}$. Suppose a new scientific discovery shows that the side-effect is not possible under the relevant circumstance, then the doctor should know how to treat the patient, which can be expresses as $[\neg r]{\ensuremath{\mathcal{K}hm{(p,\neg r,\neg p)}}}$.[^6]
Moreover, we can consider contingent plans which involve conditions based on the knowledge of the agent. A contingent plan is a partial function on the agent’s belief space. Such plans make more sense when the agent has the ability of observations during the execution of the plan. To consider contingent plan, we need to extend the model (ability map) with an epistemic relation. We then can express knowledge-that and knowledge-how at the same time, and discuss their interactions in one unified logical framework.
[^1]: See [@Wang16] for a survey.
[^2]: See [@Gochet13; @KandA15; @Wang15lori] for detailed discussions on related work in AI and Philosophy.
[^3]: This ternary modality is first proposed and discussed briefly in the full version of [@Wang15lori], which is under submission to a journal.
[^4]: We can obtain the corresponding axioms by taking the intermediate constraint as $\top$. Note that in [@Wang15lori], we use the name `WKKh` for ${\ensuremath{\mathtt{UKh}}}$.
[^5]: In [@Wang15lori], the canonical models are much simpler: we just need MCSs and the canonical relations are simply labeled by ${\langle \psi, {\varphi}\rangle}$ for ${\ensuremath{\mathcal{K}h{(\psi,{\varphi})}}}\in \Gamma$.
[^6]: However, the announcement operator $[{\varphi}]$ is not reducible in ${\mathbf{L_{Khm}}}$ as discussed in the full version of [@Wang15lori] which is under submission.
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abstract: 'Every day media generate large amounts of text. An unbiased view on media reports requires an understanding of the political bias of media content. Assistive technology for estimating the political bias of texts can be helpful in this context. This study proposes a simple statistical learning approach to predict political bias from text. Standard text features extracted from speeches and manifestos of political parties are used to predict political bias in terms of political party affiliation and in terms of political views. Results indicate that political bias can be predicted with above chance accuracy. Mistakes of the model can be interpreted with respect to changes of policies of political actors. Two approaches are presented to make the results more interpretable: a) discriminative text features are related to the political orientation of a party and b) sentiment features of texts are correlated with a measure of political power. Political power appears to be strongly correlated with positive sentiment of a text. To highlight some potential use cases a web application shows how the model can be used for texts for which the political bias is not clear such as news articles.'
author:
- 'Felix Biessmann [^1]'
title: Automating Political Bias Prediction
---
Introduction {#sec:intro}
============
Modern media generate a large amount of content at an ever increasing rate. Keeping an unbiased view on what media report on requires to understand the political bias of texts. In many cases it is obvious which political bias an author has. In other cases some expertise is required to judge the political bias of a text. When dealing with large amounts of text however there are simply not enough experts to examine all possible sources and publications. Assistive technology can help in this context to try and obtain a more unbiased sample of information.
Ideally one would choose for each topic a sample of reports from the entire political spectrum in order to form an unbiased opinion. But ordering media content with respect to the political spectrum at scale requires automated prediction of political bias. The aim of this study is to provide empirical evidence indicating that leveraging open data sources of german texts, automated political bias prediction is possible with above chance accuracy. These experimental results confirm and extend previous findings [@Yu2008; @Hirst2014]; a novel contribution of this work is a proof of concept which applies this technology to sort news article recommendations according to their political bias.
When human experts determine political bias of texts they will take responsibility for what they say about a text, and they can explain their decisions. This is a key difference to many statistical learning approaches. Not only is the responsibility question problematic, it can also be difficult to interpret some of the decisions. In order to validate and explain the predictions of the models three strategies that allow for better interpretations of the models are proposed. First the model misclassifications are related to changes in party policies. Second univariate measures of correlation between text features and party affiliation allow to relate the predictions to the kind of information that political experts use for interpreting texts. Third sentiment analysis is used to investigate whether this aspect of language has discriminatory power.
In the following briefly surveys some related work, thereafter gives an overview of the data acquisition and preprocessing methods, presents the model, training and evaluation procedures; in the results are discussed and concludes with some interpretations of the results and future research directions.
Related Work {#sec:related}
============
Throughout the last years automated content analyses for political texts have been conducted on a variety of text data sources (parliament data blogs, tweets, news articles, party manifestos) with a variety of methods, including sentiment analysis, stylistic analyses, standard bag-of-word (BOW) text feature classifiers and more advanced natural language processing tools. While a complete overview is beyond the scope of this work, the following paragraphs list similarities and differences between this study and previous work. For a more complete overview we refer the reader to [@Grimmer2013; @Kaal2014].
A similar approach to the one presented here was taken in [@Yu2008]. The authors extracted BOW feature vectors and applied linear classifiers to predict political party affiliation of US congress speeches. They used data from the two chambers of the US congress, House and Senat, in order to assess generalization performance of a classifier trained on data from one chamber and tested on data from another. They found that accuracies of the model when trained on one domain and tested on another were significantly decreased. Generalization was also affected by the time difference between the political speeches used for training and those used for testing.
Other work has focused on developing dedicated methods for predicting political bias. Two popular methods are WordFish [@Slapin08ascaling] and WordScores [@Laver2003], or improved versions thereof, see e.g. [@Lowe09scalingpolicy]. These approaches have been very valuable for [*a posteriori*]{} analysis of historical data but they do not seem to be used as much for analyses of new data in a predictive analytics setting. Moreover direct comparisons of the results obtained with these so called [*scaling*]{} methods with the results of the present study or those of studies as [@Yu2008] are difficult, due to the different modeling and evaluation approaches: Validations of WordFish/WordScore based analyses often compare parameter estimates of the different models rather than predictions of these models on held-out data with respect to the same type of labels used to train the models.
Finally Hirst et al conducted a large number of experiments on data from the Canadian parliament and the European parliament; these experiments can be directly compared to the present study both in terms of methodology but also with respect to their results [@Hirst2014]. The authors show that a linear classifier trained on parliament speeches uses language elements of defense and attack to classify speeches, rather than ideological vocabulary. The authors also argue that emotional content plays an important role in automatic analysis of political texts. Furthermore their results show a clear dependency between length of a political text and the accuracy with which it can be classified correctly.
Taken together, there is a large body of literature in this expanding field in which scientists from quantitative empirical disciplines as well as political science experts collaborate on the challenging topic of automated analysis of political texts. Except for few exceptions most previous work has focused on binary classification[^2] or on assignment of a one dimensional policy position (mostly left vs right). Yet many applications require to take into account more subtle differences in political policies. This work focuses on more fine grained political view prediction: for one, the case of the german parliament is more diverse than two parliament systems, allowing for a distinction between more policies; second the political view labels considered are more fine grained than in previous studies. While previous studies used such labels only for partitioning training data [@Slapin08ascaling] (which is not possible at test time in real-world applications where these labels are not known) the experiments presented in this study directly predict these labels. Another important contribution of this work is that many existing studies are primarily concerned with [*a posteriori*]{} analysis of historical data. This work aims at prediction of political bias on out-of-domain data with a focus on the practical application of the model on new data, for which a prototypical web application is provided. The experiments on out-of-domain generalization complement the work of [@Yu2008; @Hirst2014] with results from data of the german parliament and novel sentiment analyses.
Data Sets and Feature Extraction {#sec:data}
================================
All experiments were run on publicly available data sets of german political texts and standard libraries for processing the text. The following sections describe the details of data acquisition and feature extraction.
Data
----
Annotated political text data was obtained from two sources: a) the discussions and speeches held in the german parliament ([*Bundestag*]{}) and b) all manifesto texts of parties running for election in the german parliament in the current 18th and the last, 17th, legislation period.
#### Parliament discussion data
Parliament texts are annotated with the respective party label, which we take here as a proxy for political bias. The texts of parliament protocols are available through the website of the german bundestag[^3]; an open source API was used to query the data in a cleaned and structured format[^4]. In total 22784 speeches were extracted for the 17th legislative period and 11317 speeches for the 18th period, queried until March 2016.
#### Party manifesto data
For party manifestos another openly accessible API was used, provided by the Wissenschaftszentrum Berlin (WZB). The API is released as part of the [*Manifestoproject*]{} [@manifesto]. The data released in this project comprises the complete manifestos for each party that ran for election enriched with annotations by political experts. Each sentence (in some cases also parts of sentences) is annotated with one of 56 political labels. Examples of these labels are [*pro/contra protectionism, decentralism, centralism, pro/contra welfare*]{}; for a complete list and detailed explanations on how the annotators were instructed see [@leftright]. The set of labels was developed by political scientists at the WZB and released for public use. All manifestos of parties that were running for election in this and the last legislative period were obtained. In total this resulted in 29451 political statements that had two types of labels: First the party affiliation of each political statement; this label was used to evaluate the party evaluation classifiers trained on the parliament speeches. For this purpose the data acquisition was constrained to only those parties that were elected into the parliament. Next to the party affiliation the political view labels were extracted. For the analyses based on political view labels all parties were considered, also those that did not make it into the parliament.
The length of each annotated statement in the party manifestos was rather short. The longest statement was 522 characters long, the 25%/50%/75% percentiles were 63/95/135 characters. Measured in words the longest data point was 65 words and the 25%/50%/75% percentiles were 8/12/17 words, respectively. This can be considered as a very valuable property of the data set, because it allows a fine grained resolution of party manifestos. However for a classifier (as well as for humans) such short sentences can be rather difficult to classify. In order to obtain less ’noisy’ data points from each party – for the party affiliation task only – all statements were aggregated into political topics using the manifesto code labels. Each political view label is a three digit code, the first digit represents the political domain. In total there were eight political domains (topics): [*External Relations, Freedom and Democracy, Political System, Economy, Welfare and Quality of Life, Fabric of Society, Social Groups*]{} and a topic [*undefined*]{}, for a complete list see also [@leftright]. These 8 topics were used to aggregate all statements in each manifesto into topics. Most party manifestos covered all eight of them, some party manifestos in the 17th Bundestag only covered seven.
Bag-of-Words Vectorization {#sec:bow-vectorization}
--------------------------
First each data set was segmented into semantic units; in the case of parliament discussions this were the speeches, in the case of the party manifesto data semantic units were the sentences or sentence parts associated with one of the 56 political view labels. Parliament speeches were often interrupted; in this case each uninterrupted part of a speech was considered a semantic unit. Strings of each semantic unit were tokenised and transformed into bag-of-word vectors as implemented in scikit-learn [@scikit-learn]. The general idea of bag-of-words vectors is to simply count occurrences of words (or word sequences, also called [*n-grams*]{}) for each data point. A data point is usually a document, here it is the semantic units of parliament speeches and manifesto sentences, respectively. The text of each semantic unit is transformed into a vector ${\mathbf{x}}\in\mathds{R}^d$ where $d$ is the size of the dictionary; the $w$th entry of ${\mathbf{x}}$ contains the (normalized) count of the $w$th word (or sequence of words) in our dictionary. Several options for vectorizing the speeches were tried, including term-frequency-inverse-document-frequency normalisation, n-gram patterns up to size $n=3$ and several cutoffs for discarding too frequent and too infrequent words. All of these hyperparameters were subjected to hyperparameter optimization as explained in .
Classification Model and Training Procedure {#sec:model}
===========================================
Bag-of-words feature vectors were used to train a multinomial logistic regression model. Let $y\in\{1,2,\dots,K\}$ be the true label, where $K$ is the total number of labels and ${\mathbf{W}}=[{\mathbf{w}}_1,\dots,{\mathbf{w}}_K]\in{\mathds{R}}^{d\times K}$ is the concatenation of the weight vectors ${\mathbf{w}}_k$ associated with the $k$th party then $$\begin{aligned}
\label{eq:logreg_multiclass}
p(y=k|{\mathbf{x}},{\mathbf{W}}) = &\frac{e^{z_k}}{\sum_{j=1}^K e^{z_j}} \qquad \textrm{with } z_k=&{\mathbf{w}}_k^{\top}{\mathbf{x}} \\\nonumber\end{aligned}$$ We estimated ${\mathbf{W}}$ using quasi-newton gradient descent. The optimization function was obtained by adding a penalization term to the negative log-likelihood of the multinomial logistic regression objective and the optimization hence found the ${\mathbf{W}}$ that minimized $$\label{eq:objective}
L({\mathbf{W}}, {\mathbf{x}}, \gamma) = - \log{\frac{e^{z_k}}{\sum_{j=1}^K e^{z_j}}}+ \gamma \| {\mathbf{W}} \|_{F}$$ Where $\|~\|_F$ denotes the Frobenius Norm and $\gamma$ is a regularization parameter controlling the complexity of the model. The regularization parameter was optimized on a log-scaled grid from $10^{-4,\dots,4}$. The performance of the model was optimized using the classification accuracy, but we also report all other standard measures, precision ($TP / (FP + TP$), recall ($TP / (TP + FN)$) and f1-score ($2\times (Prec. \times Rec) / (Prec + Rec.)$).\
Three different classification problems were considered:
1. [**Classification of party affiliation**]{} (five class / four class problem)
2. [**Classification of government membership**]{} (binary problem)
3. [**Classification of political views**]{} (56 class problem)
Party affiliation is a five class problem for the 17th legislation period, and a four class problem for the 18th legislation period. Political view classification is based on the labels of the manifesto project, see and [@leftright]. For each of first two problems, party affiliation and government membership prediction, classifiers were trained on the parliament speeches. For the third problem classifiers were trained only on the manifesto data for which political view labels were available.
Optimisation of Model Parameters {#sec:crossvalidation}
--------------------------------
The model pipeline contained a number of hyperparameters that were optimised using cross-validation. We first split the training data into a training data set that was used for optimisation of hyperparameters and an held-out test data set for evaluating how well the model performs on in-domain data; wherever possible the generalisation performance of the models was also evaluated on out-of domain data. Hyperparameters were optimised using grid search and 3-fold cross-validation within the training set only: A cross-validation split was made to obtain train/test data for the grid search and for each setting of hyperparameters the entire pipeline was trained and evaluated – no data from the in-domain evaluation data or the out-of-domain evaluation data were used for hyperparameter optimisation. For the best setting of all hyperparameters the pipeline was trained again on all training data and evaluated on the evaluation data sets. For party affiliation prediction and government membership prediction the training and test set were 90% and 10%, respectively, of all data in a given legislative period. Out-of-domain evaluation data were the texts from party manifestos. For the political view prediction setting there was no out-of-domain evaluation data, so all labeled manifesto sentences in both legislative periods were split into a training and evaluation set of 90% (train) and 10% (evaluation).
Sentiment analysis {#sec:sentiment_analysis_methods}
------------------
A publicly available key word list was used to extract sentiments [@remquahey2010]. A sentiment vector ${\mathbf{s}}\in{\mathds{R}}^d$ was constructed from the sentiment polarity values in the sentiment dictionary. The sentiment index used for attributing positive or negative sentiment to a text was computed as the cosine similarity between BOW vectors ${\mathbf{x}}\in{\mathds{R}}^d$ and ${\mathbf{s}}$
$$\begin{aligned}
\frac{{\mathbf{s}}^\top {\mathbf{x}}}{\|{\mathbf{s}}\|\|{\mathbf{x}}\|}\end{aligned}$$
Analysis of bag-of-words features {#sec:correlations_methods}
---------------------------------
While interpretability of linear models is often propagated as one of their main advantages, doing so naively without modelling the noise covariances can lead to wrong conclusions, see e.g. [@Zien2009; @Haufe2013]; interpreting coefficients of linear models (independent of the regularizer used) implicitly assumes uncorrelated features; this assumption is violated by the text data used in this study. Thus direct interpretation of the model coefficients ${\mathbf{W}}$ is problematic. In order to allow for better interpretation of the predictions and to assess which features are discriminative correlation coefficients between each word and the party affiliation label were computed. The words corresponding to the top positive and negative correlations are shown in .
Results {#sec:results}
=======
The following sections give an overview of the results for all political bias prediction tasks. Some interpretations of the results are highlighted and a web application of the models is presented at the end of the section.
Predicting political party affiliation
--------------------------------------
The results for the political party affiliation prediction on held-out parliament data and on evaluation data are listed in for the 17th Bundestag and in for the 18th Bundestag, respectively. Shown are the evaluation results for in-domain data (held-out parliament speech texts) as well as the out-of-domain data; the party manifesto out-of-domain predictions were made on the sentence level.
When predicting party affiliation on text data from the same domain that was used for training the model, average precision and recall values of above 0.6 are obtained. These results are comparable to those of [@Hirst2014] who report a classification accuracy of 0.61 on a five class problem of prediction party affiliation in the European parliament; the accuracy for the 17th Bundestag is 0.63, results of the 18th Bundestag are difficult to compare as the number of parties is four and the legislation period is not finished yet. For out-of domain data the models yield significantly lower precision and recall values between 0.3 and 0.4. This drop in out of domain prediction accuracy is in line with previous findings [@Yu2008]. A main factor that made the prediction on the out-of-domain prediction task particularly difficult is the short length of the strings to be classified, see also . In order to investigate whether this low out-of-domain prediction performance was due the domain difference (parliament speech vs manifesto data) or due to the short length of the data points, the manifesto data was aggregated based on the topic. The manifesto code political topics labels were used to concatenate texts of each party to one of eight topics, see . The topic level results are shown in and and demonstrate that when the texts to be classified are sufficiently long and the word count statistics are sufficiently dense the classification performance on out of domain data can achieve in the case of some parties reliably precision and recall values close to 1.0. This increase is in line with previous findings on the influence of text length on political bias prediction accuracy [@Hirst2014].
------------- ----------- -------- ---------- ------ ----------- -------- ---------- -------
precision recall f1-score N precision recall f1-score N
cducsu 0.62 0.81 0.70 706 0.26 0.58 0.36 2030
fdp 0.70 0.37 0.49 331 0.38 0.28 0.33 2319
gruene 0.59 0.40 0.48 298 0.47 0.20 0.28 3747
linke 0.71 0.61 0.65 338 0.30 0.47 0.37 1701
spd 0.60 0.69 0.65 606 0.26 0.16 0.20 2278
avg / total 0.64 0.63 0.62 2279 0.35 0.31 0.30 12075
------------- ----------- -------- ---------- ------ ----------- -------- ---------- -------
: \[tab:results\_17\] Classification performance on the party affiliation prediction problem for data from the 17th legislative period on test set and evaluation set, respectively. Predictions on the manifesto data was done on [**sentence level**]{}; $N$ denotes number of data points in the evaluation set.
------------- ----------- -------- ---------- ------ ----------- -------- ---------- -------
precision recall f1-score N precision recall f1-score N
cducsu 0.66 0.82 0.73 456 0.32 0.64 0.43 2983
gruene 0.68 0.54 0.60 173 0.59 0.15 0.23 5674
linke 0.77 0.58 0.66 173 0.36 0.48 0.41 2555
spd 0.60 0.54 0.57 330 0.26 0.31 0.28 2989
avg / total 0.66 0.66 0.65 1132 0.42 0.34 0.32 14201
------------- ----------- -------- ---------- ------ ----------- -------- ---------- -------
: \[tab:results\_18\] Classification performance on the party affiliation prediction problem for data from the 18th legislative period on test set and evaluation set, respectively. Predictions on the manifesto data was done on [**sentence level**]{}.
------------- ----------- -------- ---------- ----
precision recall f1-score N
cducsu 0.64 1.00 0.78 7
fdp 1.00 1.00 1.00 7
gruene 1.00 0.86 0.92 7
linke 1.00 1.00 1.00 7
spd 0.80 0.50 0.62 8
avg / total 0.88 0.86 0.86 36
------------- ----------- -------- ---------- ----
: \[tab:results\_topic\] [**Topic level classification performance**]{} on the party affiliation prediction problem for data from the evaluation set (manifesto texts) of the 17th legislative period. In contrast to single sentence level predictions (see , , for results and for topic definitions) the predictions made on topic level are reliable in many cases. Note that all manifesto topics of the green party in the 18th Bundestag are predicted to be from the parties of the governing coalition, CDU/CSU or SPD.
------------- ----------- -------- ---------- ----
precision recall f1-score N
cducsu 0.50 1.00 0.67 8
gruene 0.00 0.00 0.00 8
linke 1.00 0.88 0.93 8
spd 0.56 0.62 0.59 8
avg / total 0.51 0.62 0.55 32
------------- ----------- -------- ---------- ----
: \[tab:results\_topic\] [**Topic level classification performance**]{} on the party affiliation prediction problem for data from the evaluation set (manifesto texts) of the 17th legislative period. In contrast to single sentence level predictions (see , , for results and for topic definitions) the predictions made on topic level are reliable in many cases. Note that all manifesto topics of the green party in the 18th Bundestag are predicted to be from the parties of the governing coalition, CDU/CSU or SPD.
In order to investigate the errors the models made confusion matrices were extracted for the predictions on the out-of-domain evaluation data for sentence level predictions (see ) as well as topic level predictions (see ). One example illustrates that the mistakes the model makes can be associated with changes in the party policy. The green party has been promoting policies for renewable energy and against nuclear energy in their manifestos prior to both legislative periods. Yet the statements of the green party are more often predicted to be from the government parties than from the party that originally promoted these green ideas, reflecting the trend that these legislative periods governing parties took over policies from the green party. This effect is even more pronounced in the topic level predictions: a model trained on data from the 18th Bundestag predicts all manifesto topics of the green party to be from one of the parties of the governing coalition, CDU/CSU or SPD.\
#### Government membership prediction
Next to the party affiliation labels also government membership labels were used to train models that predict whether or not a text is from a party that belonged to a governing coalition of the Bundestag. In and the results are shown for the 17th and the 18th Bundestag, respectively. While the in-domain evaluation precision and recall values reach values close to 0.9, the out-of-domain evaluation drops again to values between 0.6 and 0.7. This is in line with the results on binary classification of political bias in the Canadian parliament [@Yu2008]. The authors report classification accuracies between 0.8 and 0.87, the accuracy in the 17th Bundestag was 0.85. While topic-level predictions were not performed in this binary setting, the party affiliation results in suggest that a similar increase in out-of-domain prediction accuracy could be achieved when aggregating texts to longer segments.
\[tab:conf\_mat\_four\_class\]
[lccccccc]{}\
\
&&&\
&&& cducsu & fdp& gruene& linke& spd\
& &cducsu &1186 &289& 178& 198& 179\
&&fdp &882& 658& 236& 329& 214\
&&gruene &1174& 404& 764& 941& 464\
&&linke &388& 92& 214& 806& 201\
&&spd &999& 268& 240& 398& 373\
[lcccccc]{}\
\
&&&\
&&& cducsu & gruene& linke& spd\
&&cducsu&1912& 156& 331& 584\
&&gruene&2092& 827& 1311& 1444\
&&linke&596& 186& 1216& 557\
&&spd&1284& 226& 563& 916\
\[tab:conf\_mat\_four\_class\]
[lccccccc]{}\
\
&&&\
&&& cducsu & fdp& gruene& linke& spd\
&&cducsu &7& 0& 0& 0& 0\
&&fdp&0& 7& 0& 0& 0\
&&gruene&0& 0& 6& 0& 1\
&&linke&0& 0& 0& 7& 0\
&&spd&4& 0& 0& 0& 4\
[lcccccc]{}\
\
&&&\
&&& cducsu & gruene& linke& spd\
&&cducsu&8& 0& 0& 0\
&&gruene&4& 0& 0& 4\
&&linke&1& 0& 7& 0\
&&spd&3& 0& 0& 5\
\
------------- ----------- -------- ---------- ------ ----------- -------- ---------- -------
precision recall f1-score N precision recall f1-score N
government 0.83 0.84 0.84 1037 0.49 0.59 0.54 4349
opposition 0.86 0.86 0.86 1242 0.74 0.66 0.70 7726
avg / total 0.85 0.85 0.85 2279 0.65 0.63 0.64 12075
------------- ----------- -------- ---------- ------ ----------- -------- ---------- -------
: \[tab:results\_binary\_17\] Classification performance on the binary prediction problem in the 17th legislative period, categorizing speeches into government (FDP/CDU/CSU) and opposition (Linke, Grüne, SPD).
------------- ----------- -------- ---------- ------ ----------- -------- ---------- -------
precision recall f1-score N precision recall f1-score N
government 0.88 0.95 0.92 786 0.52 0.66 0.58 5972
opposition 0.86 0.71 0.78 346 0.69 0.56 0.62 8229
avg / total 0.88 0.88 0.87 1132 0.62 0.60 0.60 14201
------------- ----------- -------- ---------- ------ ----------- -------- ---------- -------
: \[tab:results\_binary\_18\] Classification performance on the binary prediction problem in the 18th legislative period, categorizing speeches into government (SDP/CDU/CSU) and opposition (Linke, Grüne).
Predicting political views
--------------------------
Parties change their policies and positions in the political spectrum. More reliable categories for political bias are party independent labels for political views, see . A separate suite of experiments was run to train and test the prediction performance of the text classifiers models described in . As there was no out-of-domain evaluation set available in this setting only evaluation error on in-domain data is reported. Note however that also in this experiment the evaluation data was never seen by any model during training time. In results for the best and worst classes, in terms of predictability, are listed along with the average performance metrics on all classes. Precision and recall values of close to 0.5 on average can be considered rather high considering the large number of labels.\
code meaning precision recall f1-score N
------------- ------------------------ ----------- -------- ---------- ------
501 environmentalism + 0.62 0.61 0.61 165
202 democracy + 0.58 0.55 0.57 122
701 labour + 0.57 0.54 0.56 129
201 freedom/human rights + 0.58 0.54 0.56 159
106 peace + 0.52 0.57 0.55 21
…
302 centralism + 0.25 0.20 0.22 10
401 free enterprise + 0.20 0.19 0.20 52
505 welfare - 0.13 0.14 0.14 14
409 keynesian demand + 0.14 0.12 0.13 8
0 undefined 0.09 0.12 0.10 17
avg / total 0.47 0.46 0.46 2946
: \[tab:results\_avg\_political\_view\] Classification performance of 56 political views, see .
Correlations between words and parties {#sec:word_party_correlations}
--------------------------------------
The 10 highest and lowest correlations between individual words and the party affiliation label are shown for each party in . Correlations were computed on the data from the current, 18th, legislative period. Some unspecific stopwords are excluded. The following paragraphs highlight some examples of words that appear to be preferentially used or avoided by each respective party. Even though interpretations of these results are problematic in that they neglect the context in which these words were mentioned some interesting patterns can be found and related to the actual policies the parties are promoting.
#### **Left party (linke)**
The left party mostly criticises measures that affect social welfare negatively, such as the [*Hartz IV*]{} program. Main actors that are blamed for decisions of the conservative governments by the left party are big companies ([*konzerne*]{}). Rarely the party addresses concerns related to security ([*sicherheit*]{}).
#### **Green party (gruene)**
The green party heavily criticised the secret negotiations about the TiSA agreement[^5] and insists in formal inquiries that the representatives of the green party put forward in this matter ([*fragen, anfragen*]{}). They also often ask questions related to army projects ([*Rüstungsprojekte, Wehrbericht*]{}) or the military development in east europe ([*Jalta*]{}[^6]).
#### **Social democratic party (SPD)**
The social democrats often use words related to rights of the working class, as reflected by the heavy use of the [*International Labour Organisation*]{} (ILO) or rights of employes ([*Arbeitnehmerrechte*]{}). They rarely talk about competition ([*Wettbewerb*]{}) or climate change ([*klimapolitik*]{}).
#### **Conservative party (CDU/CSU)**
The conservative christian party often uses words related to a pro-economy attitude, such as competitiveness or (economic) development ([*Wettbewerbsfähigkeit, Entwicklung*]{}) and words related to security ([*Sicherheit*]{}). The latter could be related to the ongoing debates about whether or not the governments should be allowed to collect data and thus restrict fundamental civil rights in order to better secure the population. In contrast to the parties of the opposition, the conservatives rarely mention the word war ([*krieg*]{}) or related words.
![ \[fig:party\_word\_correlations\] Correlations between words and party affiliation label for parliament speeches can help interpreting the features used by a predictive model. Shown are the top 10 positively and negatively correlated text features for the current Bundestag. For interpretations see .](images/party_word_correlations-linke-18.pdf "fig:"){width="2.8cm"} ![ \[fig:party\_word\_correlations\] Correlations between words and party affiliation label for parliament speeches can help interpreting the features used by a predictive model. Shown are the top 10 positively and negatively correlated text features for the current Bundestag. For interpretations see .](images/party_word_correlations-gruene-18.pdf "fig:"){width="2.9cm"} ![ \[fig:party\_word\_correlations\] Correlations between words and party affiliation label for parliament speeches can help interpreting the features used by a predictive model. Shown are the top 10 positively and negatively correlated text features for the current Bundestag. For interpretations see .](images/party_word_correlations-spd-18.pdf "fig:"){width="3cm"} ![ \[fig:party\_word\_correlations\] Correlations between words and party affiliation label for parliament speeches can help interpreting the features used by a predictive model. Shown are the top 10 positively and negatively correlated text features for the current Bundestag. For interpretations see .](images/party_word_correlations-cducsu-18.pdf "fig:"){width="3cm"}
Speech sentiment correlates with political power {#sec:sentiment_result}
------------------------------------------------
In order to investigate the features that give rise to the classifiers’ performance the bag-of-words features were analysed with respect to their sentiment. The average sentiment of each political party is shown in . High values indicate more pronounced usage of positive words, whereas negative values indicate more pronounced usage of words associated with negative emotional content.
The results show an interesting relationship between political power and sentiment. Political power was evaluated in two ways: a) in terms of the number of seats a party has and b) in terms of membership of the government. Correlating either of these two indicators of political power with the mean sentiment of a party shows a strong positive correlation between speech sentiment and political power. This pattern is evident from the data in and in : In the current Bundestag, government membership correlates with positive sentiment with a correlation coefficient of 0.98 and the number of seats correlates with 0.89.
Note that there is one party, the social democrats (SPD), which has many seats and switched from opposition to government with the 18th Bundestag: With its participation in the government the average sentiment of this party switched sign from negative to positive, suggesting that positive sentiment is a strong indicator of government membership.
![ \[fig:party\_sentiments\] Speech sentiments computed for speeches of each party; parties are ordered according to the number of seats in the parliament. There is a trend for more positive speech content with more political power. Note that the SPD (red) switched from opposition to government in the 18th Bundestag: their seats in the parliament increased and the average sentiment of their speeches switched sign from negative to overall positive sentiment. ](images/party-sentiments-17.pdf "fig:"){width="6cm"} ![ \[fig:party\_sentiments\] Speech sentiments computed for speeches of each party; parties are ordered according to the number of seats in the parliament. There is a trend for more positive speech content with more political power. Note that the SPD (red) switched from opposition to government in the 18th Bundestag: their seats in the parliament increased and the average sentiment of their speeches switched sign from negative to overall positive sentiment. ](images/party-sentiments-18.pdf "fig:"){width="5cm"}
Sentiment vs. Gov. Member Seats
---------------- ------------- -------
17th Bundestag 0.84 0.70
18th Bundestag 0.98 0.89
: \[tab:sentiments\] Correlation coefficient between average sentiment of political speeches of a party in the german Bundestag with two indicators of political power, a) membership in the government and b) the number of seats a party occupies in the parliament.
An example web application
--------------------------
To show an example use case of the above models a web application was implemented that downloads regularly all articles from some major german news paper websites[^7] and applies some simple topic modelling to them. For each news article topic, headlines of articles are plotted along with the predictions of the political view of an article and two labels derived deterministically from the 56 class output, a left right index and the political domain of a text, see [@leftright]. Within each topic it is then possible to get an ordered (from left to right) overview of the articles on that topic. An example of one topic that emerged on March 31st is shown in . A preliminary demo is live at [@fipidemo] and the code is available on github[@fipi].
![ \[fig:fipi\] A screen shot of an example web application using the political view prediction combined with topic modelling to provide a heterogeneous overview of a topic. ](images/fipi-screenshot){width="10cm"}
Conclusions, Limitations and Outlook {#sec:conclusion}
====================================
This study presents a simple approach for automated political bias prediction. The results of these experiments show that automated political bias prediction is possible with above chance accuracy in some cases. It is worth noting that even if the accuracies are not perfect, they are above chance and comparable with results of comparable studies [@Yu2008; @Hirst2014]. While these results do not allow for usage in production systems for classification, it is well possible to use such a system as assistive technology for human annotators in an active learning setting.
One of the main limiting factors of an automated political bias prediction system is the availability of training data. Most training data sets that are publicly available have an inherent bias as they are sampled from a different domain. This study tried to quantify the impact of this effect. For the cases in which evaluation data from two domains was available there was a pronounced drop in prediction accuracy between the in domain evaluation set and the out of domain evaluation set. This effect was reported previously for similar data, see e.g. [@Yu2008]. Also the finding that shorter texts are more difficult to classify than longer texts is in line with previous studies [@Hirst2014]. When considering texts of sufficient length (for instance by aggregating all texts of a given political topic) classification performance improved and in some cases reliable predictions could be obtained even beyond the training text domain.
Some aspects of these analyses could be interesting for social science researchers; three of these are highlighted here. First the misclassifications of a model can be related to the changes in policy of a party. Such analyses could be helpful to quantitatively investigate a change in policy. Second analysing the word-party correlations shows that some discriminative words can be related to the political views of a party; this allows for validation of the models by human experts. Third when correlating the sentiment of a speech with measures of political power there is a strong positive correlation between political power and positive sentiment. While such an insight in itself might seem not very surprising this quantifiable link between power and sentiment could be useful nonetheless: Sentiment analysis is a rather domain independent measure, it can be easily automated and scaled up to massive amounts of text data. Combining sentiment features with other measures of political bias could potentially help to alleviate some of the domain-adaptation problems encountered when applying models trained on parliament data to data from other domains.\
All data sets used in this study were publicly available, all code for experiments and the link to a live web application can be found online [@fipi].
Acknowledgements {#acknowledgements .unnumbered}
----------------
I would like to thank Friedrich Lindenberg for factoring out the <https://github.com/bundestag/plpr-scraper> from his bundestag project. Some backend configurations for the web application were taken from an earlier collaboration with Daniel Kirsch. Pola Lehmann and Michael Gaebler provided helpful feedback on an earlier version of the manuscript. Pola Lehman also helped with getting access to and documentation on the Manifestoproject data.
[^1]: [email protected]
[^2]: Many parliaments only have two parties and many studies chose binary classification schemes within manually defined topics and generic schemes such as conservative vs liberal or left vs right.
[^3]: <https://www.bundestag.de/protokolle>
[^4]: <https://github.com/bundestag>
[^5]: <https://en.wikipedia.org/wiki/Trade_in_Services_Agreement>
[^6]: Referring to the <https://en.wikipedia.org/wiki/Yalta_Conference>
[^7]: <http://www.spiegel.de/politik>, <http://www.faz.net/aktuell/politik>, <http://www.welt.de/politik>, <http://www.sueddeutsche.de/politik>, <http://www.zeit.de/politik>
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---
abstract: 'Using two different criteria for continuous variable systems we demonstrated that pump and probe beams became quantum correlated in a situation of Electromagnetically Induced Transparency in a sample of $^{85}$Rb atoms. Our result combines two important features for practical implementations in the field of quantum information processing. Namely, we proved the existence of entanglement between two macroscopic light beams, and this entanglement is intrinsically associated to a strong coherence in an atomic medium.'
author:
- 'C. L. Garrido Alzar, M. França Santos,[@byline] and P. Nussenzveig'
title: '**Entanglement between two macroscopic fields by coherent atom-mediated exchange of photons**'
---
In the last two decades, the philosofical and technological impacts of quantum correlations or entanglement in multipartite physical systems have been accentuated by the theoretical and experimental investigations leading to the development of the area of quantum information processing [@bennett]. The existence of entanglement, which is an evidence of the non-local character of the quantum theory, has been confirmed in quantum optics experiments using dichotomics [@aspect] and continuous variable systems [@kimble1]. These experimental realizations made possible some implementations in the fields of quantum information [@kimble; @gisin] and computation [@haroche1; @ions]. Even so, the unavoidable interaction between the physical system used to process the quantum information and the environment leads to a loss of coherence, and consequently, to a loss of information that introduces important limitations to the practicality of the quantum information processing technology. Since it is impossible to have isolated systems, among the different approaches employed to reduce the influence of the environment we find the use of continuous variable systems, specifically, intense light beams as in the case of quantum teleportation based on the entanglement between twin beams issued from an OPO [@kimble]. In that sense, there are propositions, for exemple, for quantum cryptography [@ralph], quantum computation [@lloyd], dense coding [@samuel], and quantum key distribution [@leuchs]. Another approach is to use coherently-prepared atomic media and, in this case, the Electromagnetically Induced Transparency (EIT) is a good candidate as it has been demonstrated recently by the observation of very slow light pulse propagation [@paq] and light storage [@p]. In this context, a natural question arises: can we produce entanglement between two intense ligth beams in a coherently-prepared atomic medium, combining in that way the two mentioned approaches? As we will see through out this paper, the answer to this question is yes and this result is very important due to the recent huge interest in the applications of coherently-prepared atomic media using EIT [@mastko].
In this paper, we show that three-level atoms can produce entanglement in two intense travelling light fields, in the EIT regime. We study this system from a theoretical point of view, and we show that for some particular parameters, the two fields present quantum correlations after interacting with the atoms, even if they are initially completely uncorrelated. We demonstrate the existence of entanglement between these two propagating fields according to two different criteria.
In our model, we consider three-level atoms in a closed $\Lambda$ configuration (ground states $|1\rangle$ and $|2\rangle$, and excited state $|0\rangle$) interacting with two copropagating fields treated quantum-mechanically. In the Heisenberg picture, the electric field operator for the propagating mode $j$ (pumping laser $j=1$, probe laser $j=2$) is given by the expression $$\widehat{\vec{E}}_{j}(t) = {\cal E}_{0\omega_{Lj}} \bar{\epsilon}_{j}
e^{-i \omega_{Lj} t}\hat{A}_{j}(t) + h.c.\ ,
\label{Eq1}$$ where ${\cal E}_{0\omega_{Lj}}$ , $\bar{\epsilon}_{j}$ and $\omega_{Lj}$ are the amplitude, the polarization direction and the angular frequency of mode (laser) $j$, respectively. $\hat{A}_{j}$ ($\hat{A}^{\dag}_{j}$) is the annihilation (creation) operator and represents the slowly varying amplitude of the laser field. We take the hamiltonian $$\hat{H}_{Lj}=\int_{-\infty}^{+\infty}d\omega\ |G_{j}(\omega)|^{2}
\hbar\omega\ \hat{a}^{\dag}_{j\omega}\hat{a}_{j\omega}\
\label{Eq2}$$ as the energy source of the interacting field where, $\hat{a}_{j\omega}$ is the annihilation operator of the field inside the laser source cavity and, through the non-dimensional function $G_{j}(\omega)$, we take into account the influence of the external vacuum modes. This function is determined by the frequency-dependent reflectivity of the output mirror of the laser cavity and provides the laser linewidth $\gamma_j$, which we assume to be constant here, in accordance with the Markov approximation [@gardiner]. Taking into account the theoretical and experimental studies about the laser sources, we took $G_{j}(\omega)$ as a lorentzian profile centered at the laser frequency $\omega_{Lj}$, allowing the lower integration limit in (\[Eq2\]) be taken equal to $-\infty$ instead of zero.
The coupling between the source and the propagating mode is given by the linear hamiltonian $$\hat{H}_{Lj-Cj}=i \hbar\sqrt{\frac{\gamma_{j}}{2\pi}} \int_{-\infty}^{+\infty}
d\omega\ \hat{a}^{\dag}_{j\omega}\hat{A}_{j}(t)+ h.c.
\label{Eq3}$$ We obtain the interaction hamiltonian for the two light beams and the atoms, within the usual dipole and rotating-wave approximations $$\begin{aligned}
\hat{H}_{int}=\hbar g_{1}\hat{S}^{+}_{1}(t)\hat{A}_{1}(t) +
\hbar g_{2}\hat{S}^{+}_{2}(t)\hat{A}_{2}(t) + h.c. \ ,
\label{Eq4}\end{aligned}$$ where $g_{1}$ ($g_{2}$) is the atom – field 1 (field 2) coupling strength, and $\hat{S}^{+}_{1}$ ($\hat{S}^{+}_{2}$) the slowly varying envelope of the atomic polarization on the transition $|1\rangle \leftrightarrow |0\rangle$ ($|2\rangle \leftrightarrow |0\rangle$). The dynamics of the system is determined by twelve coupled quantum Langevin equations derived from the Heisenberg equations of motion. Since we are dealing with macroscopic systems, the quantum fluctuations of the operators are studied by linearizing them around their steady-state values and the dynamics of these fluctuations is described by a matrix linear stochastic differential equation for the fluctuation operators [@tobe]. Recently, intensity correlations between the pump and probe fields in EIT have been measured [@exper]. These can be understood by inspection of the equations for the fluctuations of one field and for the corresponding atomic polarization: $$\frac{d\delta\hat{A}_{1}(t)}{dt}=-\frac{\gamma_{1}}{2}
\delta\hat{A}_{1}(t) - i g_{1} \delta\hat{S}^{-}_{1}(t)
+\sqrt{\gamma_{1}}\delta\hat{A}_{1in}(t)\ ,
\label{Eq5}$$ $$\begin{aligned}
\frac{d\delta\hat{S}^{-}_{1}(t)}{dt}=-\left(\frac{\Gamma_{1}+\Gamma_{2}}{2}-
i\ \delta_{L1}\right)\delta\hat{S}^{-}_{1}(t) \nonumber\\
+ i g_{1}w_{1} \delta\hat{A}_{1}(t) +i g_{1}\alpha_{1}\delta\hat{W}_{1}(t)
\nonumber\\
-i g_{2}s^{*}_{12}\delta\hat{A}_{2}(t)
-i g_{2}\alpha_{2}\delta\hat{S}^{\dag}_{12}(t)
+ \hat{F}_{S1}(t)
\; .
\label{Eq6}\end{aligned}$$
Here we define $\hat{A}_{1in}$ the annihilation operator of the source or input field 1, $\Gamma_1$ ($\Gamma_2$) the spontaneous emission rate from $|0\rangle \rightarrow |1\rangle$ ($|0\rangle \rightarrow |2\rangle$), $\delta_{L1}$ the detuning between field 1 and the corresponding atomic transition, $w_1$ the steady-state inversion between states $|0\rangle$ and $|1\rangle$, $\alpha_1$ ($\alpha_2$) the steady-state amplitude of field 1 (field 2), $\hat{W}_1$ the inversion (operator) between states $|0\rangle$ and $|1\rangle$, $s^*_{12}$ the steady-state coherence between ground states $|1\rangle$ and $|2\rangle$, $\hat{S}^+_{12}$ the coherence operator, $\hat{F}_{S1}$ the Langevin fluctuation force. The notation $\delta \hat{S}^-_1$ means fluctuations of the corresponding operator. The fluctuations of the input $\delta\hat{A}_{1in}$, the interacting $\delta\hat{A}_{1}$ and the detected $\delta\hat{A}_{1out}$ fields are related by the expression $\delta\hat{A}_{1out}=\delta\hat{A}_{1in}-\sqrt{\gamma_{1}}\delta\hat{A}_{1}$.
From Eq. (\[Eq6\]), we notice that noise correlations between the fields are created, owing to the coherent effect in the atomic medium [@exper]. In Fig. \[cira2\] we show the quadrature correlations of the fields in the frequency domain as a function of probe detuning for a resonant pump. This theoretical prediction (and the following too) corresponds to a system of N=$10^{8}$ atoms of $^{85}$Rb where the states of the $\Lambda$ configuration are designated as follows: $|0 \rangle = |5P_{3/2},F^{'}=3 \rangle$, $|1 \rangle = |5S_{1/2},F=3 \rangle$ and $|2 \rangle = |5S_{1/2},F=2 \rangle$. The pump and probe lasers are taken linearly polarized with equal intensities (2.8 mW/cm$^{2}$) and issued from two independent sources with quantum fluctuations corresponding to a coherent state. We took the analysis frequency $\Omega=\Gamma/6$, where $\Gamma=\Gamma_{1}+\Gamma_{2}=2\pi\ 6$ MHz is the total decay rate of the rubidium excited state. The correlation, taking values in the interval \[-1;1\], is defined as the ratio between the fields covariance and the squared root of the product of the fields’ variances. Outside the EIT window the fields are completely uncorrelated and, in the EIT condition (zero probe detuning), there is the following correlation between the fluctuations of the fields quadratures: $\delta\hat{Y}_{1out,0} \leftrightarrow \delta\hat{Y}_{2out,0}$ and $\delta\hat{Y}_{1out,\pi/2} \leftrightarrow -\delta\hat{Y}_{2out,\pi/2}$. The subscript 0 ($\pi/2$) stands for the field amplitude (phase) quadrature and the general quadrature fluctuation operator is, for the field 2, given by $\delta \hat{Y}_{2out,\phi}(t)= \delta \hat{A}_{2out}(t) e^{-i \phi}+
\delta \hat{A}^{\dag}_{2out}(t) e^{i \phi}$.
![Correlation between amplitude and phase quadratures.[]{data-label="cira2"}](fig1alza)
The observed correlations can be interpreted from the propagation dynamics of the beams. When the beams have the same intensity, the role “pump" and “probe" is interchangeable and for a resonant coupling of both fields the atomic medium presents exactly the same absorptive and dispersive responses for these beams. This regime, that may be called electromagnetically mutual induced transparency (EMIT), is broken down when we introduce a detuning in one of the fields, creating in this way a phase difference between them leading to their decorrelation.
We use two criteria for continuous variable systems to distinguish between quantum and classical correlations. We begin our analysis with the criterion of the inferred variances, described theoretically in [@reid] and experimentally implemented in [@kimble1; @kimble]. Let us suppose that we are interested in the inferrence of the probe field amplitude ($\phi=0$) and phase ($\phi=\pi/2$) quadratures from measurements of the pump field quadratures. In this case, the inferred variances of the probe quadratures are defined by the equations $$\begin{aligned}
\Delta^{2}_{inf}Y_{2,0}(t)\equiv
\langle\Big(\hat{Y}^{'}_{2out,0}(t)-
\eta_{0}\ \hat{Y}^{'}_{1out,0}(t)\Big)^{2} \rangle \ ,
\label{Eq7}\\
\Delta^{2}_{inf}Y_{2,\pi/2}(t)\equiv
\langle\Big(\hat{Y}^{'}_{2out,\pi/2}(t)+
\eta_{\pi/2}\ \hat{Y}^{'}_{1out,\pi/2}(t)\Big)^{2} \rangle \ ,
\label{Eq8}\end{aligned}$$ where $\hat{Y}^{'}_{j out,0}(t)=\hat{Y}_{j out,0}(t)-
\langle \hat{Y}_{j out,0}(t) \rangle $ and $\hat{Y}^{'}_{j out,\pi/2}(t)=
\hat{Y}_{j out,\pi/2}(t)-\langle \hat{Y}_{j out,\pi/2}(t) \rangle $ with $j=1,2$. The parameters $\eta_{0}$ and $\eta_{\pi/2}$ take into account the non-perfect correlation between the fields and the non-ideal efficiency of the measurement procedure. The values of these parameters are taken in order to minimize the inferred variances (\[Eq7\]) and (\[Eq8\]), allowing the following criterion, in the frequency domain, for the entanglement of the pump and probe fields $$[\Delta^{2}_{inf}Y_{2,0}(\Omega)]_{min}
[\Delta^{2}_{inf}Y_{2,\pi/2}(\Omega)]_{min}<1\ .
\label{Eq9}$$
That is to say, if the product of the inferred variances is less than 1, then the correlation between the fields has a quantum nature. In Fig. \[vari1\] we show the product of the inferred variances for the probe field. As expected, outside de EIT region, the inequality (\[Eq9\]) is violated since the fields are uncorrelated (see Fig. \[cira2\]). However, in the EIT condition, the pump and probe fields became quantum correlated.
![Product of the minimal inferred variances of the probe field.[]{data-label="vari1"}](fig2alza)
The other criterion used to establish the nature of the fields correlation is the theorem of Duan [*et al.*]{} [@zoller], which we will abreviate as DGCZ. Taking $a=1$, introducing the equivalence between the fields quadrature operators and the operators defined in [@zoller] as $\hat{x}_{1} \Leftrightarrow \hat{Y}_{1 out,0}$, $\hat{p}_{1} \Leftrightarrow \hat{Y}_{1 out,\pi/2}$, $\hat{x}_{2} \Leftrightarrow \hat{Y}_{2 out,0}$ and $\hat{p}_{2} \Leftrightarrow \hat{Y}_{2 out,\pi/2}$, and using the commutation relation $[\hat{x}_{j},\hat{p}_{j^{'}}]=2 i \delta_{j j^{'}}$ derived from the definition of the quadrature operators, we find the following necessary condition to prove that the join state of the pump and probe fields is separable $$\langle (\Delta\hat{u})^{2} \rangle_{\rho} +
\langle (\Delta\hat{v})^{2} \rangle_{\rho} \ge 4\ .
\label{Eq10}$$
Since this last inequality provides a necessary condition for the separability of the join fields state, then its violation is a sufficient condition for the inseparability or entanglement between the fields. In Fig. \[cira4\] we plot the left hand side of the inequality (\[Eq10\]) and, again, outside the EIT window the equality is satisfied since the fields are uncorrelated. In the EIT condition, the violation of (\[Eq10\]) indicates that the pump and probe fields are entangled. We must point out that both criteria report about 40 % of entanglement of the fields. This amount of quantum correlation is limited by the decay rate of the coherence between the two ground states.
![Pump-probe entanglement according to the DGCZ criterion.[]{data-label="cira4"}](fig3alza)
The predicted quantum correlation between the pump and probe fields in the EIT condition is associated to the existence of phase quadrature squeezing in both fields. This squeezing, produced in the coherent situation, depends on the intensities of the source fields and, as it can be observed from the bistability response of the detected fields, it is produced at the turning point of the bistability curve and is accompanied by an excess noise in the corresponding conjugate quadrature amplitude [@tobe].
These results may be somehow unexpected because it is believed that in the EIT condition the field fluctuations are not altered for field intensities higher or comparable to the saturation intensity of the atomic transition. As we showed theoretically and experimentally [@exper] this is not the case. In the EIT situation there is a coherent atom-mediated exchange of photons between the pump and probe fields that preserves their mean intensities and at the same time modifies their quantum noise properties creating a correlation between them. This modification of the field quantum fluctuations is a direct consequence of the strong coherence induced in the atomic medium by the two beams.
So far, the correlation properties of the pump and probe beams have not been extensively studied in the EIT experiments. Entanglement between two single-photon pulses (quantum fields) has been predicted before in a coherently-prepared medium by two classical beams [@lukin]. In this paper, we show that the *same* intense beams used to prepare the transparent nonlinear medium, in particular circumstances, become entangled even when the investigated system is subjet to the influence of a reservoir and consequently the quantum correlations are predicted in a system that is not “pure". Since we demonstrated the existence of entanglement between the light beams *only*, our result suggests that there exists stronger quantum correlations in the system . Another remarkable point is that the investigation of the correlations and quantum fluctuations of the light beams in the EIT situation provides a precise tool to determine the natural width of the EIT resonance, and this can be a very powerful method to attain the highest sensibility in detecting coherent effects in atomic media.
Given the possible technological applications of such intense entangled fields, it is our understanding that the statistical properties of these fields deserve further experimental investigations in the near future. Not only do they open the possibility to use such systems as a macroscopic resource for different quantum technologies, but they also help understanding the nature of entanglement and how it may arise from non-linear couplings in macroscopic media.
Finally, in Fig. 4 we present an experimental setup that can be employed to measure the entanglement between the pump and probe fields. Considering the light beams have orthogonal polarizations, they can be separated using polarizing cube beam splitters and then the variance of their quadratures and the correlation between them can be determined utilizing homodyne detectors. From practical considerations, the analysis frequency must be chosen as low as possible since in this case we have more sensibility to detect correlated photons.
![Experimental setup to measure the probe field inferred variances. PBS: polarizing cube beam splitter; $\lambda/2$: half-wave plate; D1, D2, D3, and D4: photodetectors; OL1($\theta$) and OL2($\phi$): local oscillators for the pump and probe field, respectively; $\eta_{0}(\eta_{\pi/2})$: controlled-gain amplifier for the amplitude (phase) quadrature.[]{data-label="varinfexpe"}](fig4alza)
M.F.S. would like to thank Prof. S. Salinas for his hospitality at the University of São Paulo. The authors acknowledge the financial support of the Brazilian agencies CAPES, FAPESP and CNPq.
[99]{}
Present address: Blackett Laboratory, Imperial College, London SW7 2BW, United Kingdom.
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abstract: 'Extending our earlier results, we prove that certain tight contact structures on circle bundles over surfaces are not symplectically semi–fillable, thus confirming a conjecture of Ko Honda.'
address:
- |
Dipartimento di Matematica\
Università di Pisa\
I-56127 Pisa, Italy
- |
Rényi Institute of Mathematics\
Hungarian Academy of Sciences\
H-1053 Budapest\
Reáltanoda utca 13–15, Hungary
author:
- Paolo Lisca
- 'András I. Stipsicz'
title: 'Tight, not semi–fillable contact circle bundles'
---
[^1]
Introduction {#s:intro}
============
Let $Y$ be a closed, oriented three–manifold. A *positive, coorientable contact structure* on $Y$ is the kernel $\xi=\ker{\alpha}\subset TY$ of a one–form $\alpha\in{\Omega}^1(Y)$ such that $\alpha\wedge d\alpha$ is a positive volume form on $Y$. The pair $(Y,\xi)$ is a *contact three–manifold*. In this paper we only consider positive, coorientable contact structures, so we call them simply ‘contact structures’. For an introduction to contact structures the reader is referred to [@Ae], Chapter 8 and [@E].
There are two kinds of contact structures $\xi$ on $Y$. If there exists an embedded disc $D\subset Y$ tangent to $\xi$ along its boundary, $\xi$ is called *overtwisted*, otherwise it is said to be *tight*. The isotopy classification of overtwisted contact structures coincides with their homotopy classification as tangent two–plane fields [@El1]. Tight contact structures are much more misterious, and difficult to classify. A contact structure on $Y$ is *virtually overtwisted* if its pull–back to some finite cover of $Y$ becomes overtwisted, while it is called *universally tight* if its pull–back to the universal cover of $Y$ is tight.
A contact three–manifold $(Y,\xi)$ is *symplectically fillable*, or simply *fillable*, if there exists a compact symplectic four–manifold $(W, \omega)$ such that (i) $\partial W=Y$ as oriented manifolds (here $W$ is oriented by ${\omega}\wedge{\omega}$) and (ii) $\omega\vert_{\xi}\not=0$ at every point of $Y$. $(Y,\xi)$ is symplectically *semi*–fillable if there exists a fillable contact manifold $(N, \eta )$ such that $Y\subset N$ and $\eta|_Y=\xi$. Semi–fillable contact structures are tight [@El3; @Gr]. The converse is known to be false by work of Etnyre and Honda, who recently found two examples of tight but not semi–fillable contact three–manifolds [@EH]. Nevertheless, all such examples known at present are virtually overtwisted, so it is natural to wonder whether every universally tight contact structure is symplectically semi–fillable.
In this paper we study certain virtually overtwisted tight contact structures discovered by Ko Honda. Denote by $Y_{g,n}$ the total space of an oriented $S^1$–bundle over ${\Sigma}_g$ with Euler number $n$. Honda gave a complete classification of the tight contact structures on $Y_{g,n}$ [@H2]. The three–manifolds $Y_{g,n}$ carry infinitely many tight contact structures up to diffeomorphism. The hardest part of the classification involves two virtually overtwisted contact structures $\xi_0$ and $\xi_1$, which exist only when $n\geq
2g$. Honda conjectured that $\xi_0$ and $\xi_1$ are not symplectically semi–fillable [@H2]. The main theorem of the present paper extends our earlier results regarding these structures [@LS], establishing Honda’s conjecture:
\[t:main\] For $n\geq 2g>0$, the tight contact structures $\xi_0$ and $\xi_1$ on $Y_{g,n}$ are not symplectically semi–fillable.
The proof of Theorem \[t:main\] consists of two steps. In the first step, we derive a contact surgery presentation for $\xi_0$ and $\xi_1$ in the sense of [@DG2], and we use it to determine the homotopy type of $\xi_0$ and $\xi_1$ considered as oriented two–plane fields. This is done in Sections \[s:surgery\] and \[s:homotopy\].
In the second step, using specific properties of the $Spin ^c$ structures ${\mathbf t}_{\xi _i}$ on $Y_{g,n}$ induced by $\xi_i$ ($i=0,1$) we generalize a result of the first author [@L] so it applies to the situation at hand. Using this generalization together with an analytic computation of Nicolaescu’s [@N], we are able to determine the possible homotopy types of a semi–fillable contact structure inducing either ${\mathbf t}_{\xi_0}$ or ${\mathbf t}_{\xi_1}$. This is done in Section \[s:proof\].
Theorem \[t:main\] follows immediately from the fact that the two sets of homotopy classes determined in the two steps above have empty intersection.
Contact surgery presentations for $\xi _0$ and $\xi _1$ {#s:surgery}
=======================================================
A smooth knot $K$ in a contact three–manifold $(Y,\xi)$ which is everywhere tangent to $\xi$ is called *Legendrian*. The contact structure $\xi$ naturally induces a framing of $K$ called the *contact framing*.
Let ${\Sigma}_g$ be a closed, oriented surface of genus $g\geq 1$, and let $\pi{\colon\thinspace}Y_{g,n}\to{\Sigma}_g$ denote an oriented circle bundle over ${\Sigma}_g$ with Euler number $n$. Let $\xi$ be a contact structure on $Y_{g,n}$ such that a fiber $f=\pi^{-1}(s)\subset Y_{g,n}$ ($s\in{\Sigma}_g$) is Legendrian. We say that $f$ has *twisting number $k$* if the contact framing of $f$ is $k$ with respect to the framing determined by the fibration $\pi$. A contact structure on $Y_{g,n}$ is called *horizontal* if it is isotopic to a contact structure transverse to the fibers of $\pi$.
Let ${\zeta}$ be a horizontal contact structure on $Y_{g,2g-2}$ such that a fiber $f$ of the projection $\pi$ is Legendrian with twisting number $-1$ (the existence of such a contact structure is well–known, cf. [@Gi2], §1.D). Let $n\geq 2g$, and view the bundle $Y_{g,n}\to{\Sigma}_g$ as obtained by performing a $-\frac
1{p+1}$–surgery, where $p=n-2g+1$, along the fiber $f$ of $\pi{\colon\thinspace}Y_{g,2g-2}\to{\Sigma}_g$ with respect to the trivialization induced by the fibration $\pi$. It was observed by Honda ([@H2], §5) that there are two possible ways of extending ${\zeta}$ from the complement of a standard neighborhood of $f$ to a tight contact structure on $Y_{g,n}$. This determines the contact structures $\xi_0$ and $\xi_1$.
The construction of $\xi_0$ and $\xi_1$ can be viewed as a particular case of a more general construction. In fact, given a Legendrian knot $K$ in a contact three–manifold $(Y,\xi)$ and a rational number $r\in{\mathbb Q}$, it is possible to perform a *contact $r$–surgery* along $K$ to obtain a new contact three–manifold $(Y',\xi')$ [@DG1; @DG2]. Here $Y'$ is the three-manifold obtained by a smooth $r$–surgery along $K$ with respect to the contact framing, while $\xi'$ is constructed by extending $\xi$ from the complement of a standard neighborhood of $K$ to a tight contact structure on the glued–up solid torus. Such extension exists once $r\neq 0$. In general there are several ways to extend $\xi$, but up to isotopy there is only one if $r=\frac 1k$, $k\in{\mathbb Z}$, and two if $r=\frac{p}{p+1}$ and $p>1$, as follows from [@DG1], Propositions 3, 4 and 7. When $r=-1$ the corresponding contact surgery coincides with Legendrian surgery [@El2; @Go; @W]. A simple computation using the fact that the fiber $f$ of $Y_{g,2g-2}$ has twisting number $-1$ with respect to the contact structure ${\zeta}$ shows that $\{\xi_0, \xi_1\}$ can be defined as the set of contact structures obtainable by contact $\frac {p}{p+1}$–surgery along $f$.
[From]{} now on, we shall indicate a contact $r$–surgery along a Legendrian knot $K$ by writing the coefficient $r$ next to it. Consider the result of performing contact $(-1)$–surgery on the Legendrian knot in standard form in Figure \[f:figure1\] (here we are using the notation of [@Go], see especially Definition 2.1). Since contact $(-1)$–surgery is equivalent to Legendrian surgery, Figure \[f:figure1\] also represents a Stein four–manifold $W$ with boundary [@Go]. As a smooth four–manifold, $W$ is diffeomorphic to the two–disc bundle $D_{g,2g-2}$ with Euler number $2g-2$ over a surface of genus $g$. This can be checked by converting the contact surgery coefficient into the corresponding smooth surgery coefficient e.g. via the formulas found in [@Go] or [@GS]. Since by construction the boundaries of Stein four–manifolds come equipped with Stein fillable contact structures, we have a Stein fillable contact structure ${\zeta}(g)$ on $Y_{g,2g-2}$, which is tight by [@El3; @Gr].
\[l:horizontal\] The contact structure ${\zeta}(g)$ is horizontal. Moreover, after an isotopy the map $\pi{\colon\thinspace}Y_{g,2g-2}\to{\Sigma}_g$ has a fiber with twisting number equal to $-1$.
The existence of a Legendrian knot isotopic to a fiber with twisting number $-1$ is apparent from Figure \[f:figure1\]. On the other hand, contact $(-1)$–surgery on a Legendrian knot isotopic to a fiber and having twisting number $\geq 0$ would result in a Stein manifold containing a sphere with self-intersection $\geq -1$, contradicting the adjunction inequality for Stein manifolds [@LM]. By the classification of tight contact structures on $Y_{g,2g-2}$ with *negative twisting number* i.e. such that the twisting number of any closed Legendrian curve isotopic to a fiber is $<0$ ([@H2], Theorem 2.11), we conclude that the diagram of Figure \[f:figure1\] represents a horizontal contact structure.
By [@DG2], Proposition 3, any contact $r$–surgery with $r<0$ is equivalent to a Legendrian surgery along a Legendrian link. Moreover, the set of Legendrian links which correspond to some contact $r$–surgery is determined via a simple algorithm by the Legendrian knot and the continued fraction expansion of $r$. For example, let $K$ be a Legendrian unknot in the standard contact three–sphere with Thurston–Bennequin invariant equal to $-1$. Then, a contact $-\frac{p}{p-1}$–surgery ($p>1$) along $K$ is equivalent to Legendrian surgery along one of the Legendrian links in Figure \[f:figure2\].
According to [@DG2], Proposition 7, a contact $\frac{p}{p+1}$–surgery on a Legendrian knot $K$ is equivalent to a contact $\frac{1}{2}$–surgery on $K$ followed by a contact $-\frac{p}{p-1}$–surgery on a Legendrian push–off of $K$. By [@DG1], Proposition 9, a contact $\frac{1}{2}$–surgery on a Legendrian knot $K$ can be replaced by two contact $(+1)$–surgeries, one on $K$ and the other on a Legendrian push–off of $K$.
This implies that if we perform a Legendrian $\frac{p}{p+1}$–surgery on a Legendrian fiber of $(Y_{g,2g-2},{\zeta}(g))$ with twisting number $-1$, the resulting contact structures will have contact surgery presentations obtained by replacing the “dotted ellipse” in Figure \[f:figure1\] with either Figure \[f:figure3\](a) or \[f:figure3\](b). More precisely, we can define $\xi_0$, respectively $\xi_1$, as the contact structure obtained by using Figure \[f:figure3\](a), respectively Figure \[f:figure3\](b).
Homotopy classes of $\xi_0$ and $\xi_1$ {#s:homotopy}
=======================================
Homotopy theory of oriented two–plane fields on three–manifolds {#homotopy-theory-of-oriented-twoplane-fields-on-threemanifolds .unnumbered}
---------------------------------------------------------------
Let $\Xi_Y$ denote the space of oriented two–plane fields on the closed, oriented three–manifold $Y$. Since a $Spin^c$ structure on a three–manifold can be interpreted as an equivalence class of nowhere vanishing vector fields [@Tu], by taking the oriented normal, a two–plane field $\xi\in\Xi_Y$ naturally induces a $Spin^c$ structure ${\mathbf t}_\xi$, which depends only on the homotopy class $[\xi]$. Therefore there is a map $p{\colon\thinspace}\pi_0(\Xi_Y)\to
Spin^c(Y)$ defined as $p([\xi])={\mathbf t}_\xi$. It is not difficult to show that, if $Y$ is connected, there is a non–canonical identification of each fiber $p^{-1}({\mathbf t}_\xi)$ with ${\mathbb Z}/d({\mathbf t}_\xi){\mathbb Z}$, where $d({\mathbf t}_\xi)\in{\mathbb Z}$ is the divisibility of $c_1(\xi)\in H^2(Y; {\mathbb Z})$, and is zero if $c_1(\xi)$ is a torsion element (see, e.g. [@Go], Proposition 4.1).
When $c_1(\xi)$ is torsion the two–plane fields inducing the same $Spin^c$ structure ${\mathbf t}_{\xi}$ can be distinguished by a numerical invariant. Suppose that $X$ is a compact 4-manifold with $\partial X =
Y$, with $X$ carrying an almost–complex structure $J$ whose complex tangents at the boundary form an oriented two–plane field homotopic to $\xi$ on $Y$. Observe that the fact that $c_1(\xi)$ is torsion implies that $c^2_1(X,J)\in{\mathbb Q}$ makes sense.
The rational number $$d_3(\xi) = \frac{1}{4}(c^2_1(X, J) - 3\sigma
(X)-2\chi (X))\in {\mathbb Q}$$ depends only on $[\xi]$, not on the almost–complex four–manifold $(X, J)$. Moreover, two two–plane fields $\xi_1$ and $\xi_2$ inducing the same $Spin^c$ structure with torsion first Chern class are homotopic if and only if $d_3(\xi _1)=d_3(\xi _2)$.
In the following we shall refer to the invariant $d_3$ as the *three–dimensional invariant*.
Attaching two–handles and homotopy invariants {#attaching-twohandles-and-homotopy-invariants .unnumbered}
---------------------------------------------
Recall that contact $(-1)$–surgery, i.e. Legendrian surgery, can be viewed as the result of attaching a symplectic two–handle [@W]. In fact, attaching the two–handle to a contact three–manifold $(Y_1,\xi_1)$ gives rise to a cobordism $W$ between $Y_1$ and the three–manifold underlying the contact three–manifold $(Y_2,\xi_2)$ resulting from the three–dimensional contact surgery. Furthermore, $W$ carries an almost–complex structure whose complex tangent lines at the boundary coincide with $\xi_1$ and $\xi_2$ (see e.g. [@EH2]).
In the case of contact $(+1)$-surgery, there is still a smooth cobordism $W$ between $Y_1$ and $Y_2$. One can easily check the existence of an almost–complex structure $J$ on the complement of a ball $B$ in the interior of $W$, with $J$ inducing $\xi_1$ and $\xi_2$ as tangent complex lines. We define $q$ to be the three–dimensional invariant of the two–plane field induced by $J$ on $\partial
B$. Observe that, although $J$ may not extend to the whole cobordism, $J$ induces a $Spin^c$ structure ${\mathbf s}_J$ which does extend – uniquely – to $W$.
\[l:value\] The value of $q$ is $\frac 12$.
Consider an oriented Legendrian unknot $K$ in the standard contact three–sphere with Thurston–Bennequin invariant equal to $-1$ and vanishing rotation number. We view the standard contact three–sphere as the contact boundary of the unit ball $B_1(0)\subset{\mathbb C}^2$. Attach a smooth two–handle $H_1$ to $B_1(0)$ with framing $+1$ with respect to the contact framing. The result is a smooth four–manifold $X$ diffeomorphic to $S^2{\times}D^2$. The unique $Spin^c$ structure on $B_1(0)$ extends to a $Spin^c$ structure ${\mathbf s}$ on $X$, restricting to $H_1$ as the $Spin^c$ structure defined above. Denote by $k$ the value of $c_1({\mathbf s})$ on a generator of the second homology group of $X$.
Let $K'$ be a Legendrian push–off of $K$, which we may assume disjoint from $H_1$, and attach a symplectic two–handle $H_2$ to $K'$ realizing Legendrian surgery on $K'$. The $Spin^c$ structure ${\mathbf s}$ extends over $H_2$, and the value of its first Chern class on the homology generator corresponding to $K'$ is $0$, because $K'$ has vanishing rotation number (see [@Go], especially the proof of Proposition 2.3). By [@DG1], Proposition 8, the resulting contact three–manifold is just the standard contact three–sphere. Its three–dimensional invariant $d_3$ is $-\frac 12$, but when viewed as the result of the above construction, $d_3$ can also be expressed as $\frac{1}{4}(2k^2-4)+q$.
We can generalize this argument using Legendrian push–offs $K_1$, $K_1', \ldots$, $K_n$, $K_n'$ of $K$ by performing contact $(+1)$–surgeries on $K_1,\ldots , K_n$ and contact $(-1)$–surgeries on $K_1', \ldots , K_n'$. The resulting contact three–manifold is the standard contact three–sphere again. A homological computation as before gives the identity $$\frac{1}{4}(n+1)(k^2n-2)+nq=-\frac 12,$$ which must hold for all $n\in{\mathbb N}$. This implies that $k=0$ and $q=\frac 12$.
$Spin^c$ structures on disc and circle bundles {#spinc-structures-on-disc-and-circle-bundles .unnumbered}
----------------------------------------------
Let $D_{g,n}$ be the oriented disc bundle with Euler number $n$ over a closed oriented surface of genus $g$. By e.g. fixing a metric on $D_{g,n}$ one sees that the tangent bundle of $D_{g,n}$ is isomorphic to the direct sum of the pull–back of $T{\Sigma}_g$ and the vertical tangent bundle, which is isomorphic to the pull–back of the real oriented two–plane bundle $E_{g,n}\to{\Sigma}_g$ with Euler number $n$. In short, we have $$\label{e:split}
TD_{g,n}\cong \pi^*(T{\Sigma}_g\oplus E_{g,n}).$$ This splitting of $TD_{g,n}$ naturally endows $D_{g,n}$ with and almost–complex structure which induces a $Spin^c$ structure ${\mathbf s}_0$ on $D_{g,n}$. The orientation on $D_{g,n}$ determines an isomorphism $H^2(D_{g,n};{\mathbb Z})\cong{\mathbb Z}$, so the set $Spin^c(D_{g,n})={\mathbf s}_0 +
H^2(D_{g,n};{\mathbb Z})$ can be canonically identified with the integers. We denote by ${\mathbf s}_e={\mathbf s}_0+e\in Spin^c(D_{g,n})$ the element corresponding to the integer $e\in{\mathbb Z}\cong H^2(D_{g,n};{\mathbb Z})$.
Consider $Y_{g,n}={\partial}D_{g,n}$. We have $H_1(Y_{g,n};{\mathbb Z})\cong
H^2(Y_{g,n}; {\mathbb Z}) \cong {\mathbb Z}^{2g}\oplus {\mathbb Z}/n{\mathbb Z}$, where the summand ${\mathbb Z}/n{\mathbb Z}$ is generated by the Poincaré dual $F$ of the class of a fiber of the projection $\pi{\colon\thinspace}Y_{g,n}\to{\Sigma}_g$. Each $Spin^c$ structure ${\mathbf s}_e\in Spin^c(D_{g,n})$ determines by restriction a $Spin^c$ structure ${\mathbf t}_e\in Spin^c(Y_{g,n})$ with ${\mathbf t}_e={\mathbf t}_0+eF$, $e\in{\mathbb Z}$. Since $nF=0$, we see that ${\mathbf t}_{e+n}={\mathbf t}_e$ for every $e$. Therefore, ${\mathbf t}_0,\ldots,{\mathbf t}_{n-1}$ is a complete list of *torsion $Spin^c$ structures* on $Y_{g,n}$, i.e. $Spin^c$ structures on $Y_{g,n}$ with torsion first Chern class. In short, the $Spin^c$ structures on $Y_{g,n}$ which extend to the disc bundle are precisely the torsion ones.
Homotopy invariants of the contact structures $\xi _i$ {#homotopy-invariants-of-the-contact-structures-xi-_i .unnumbered}
------------------------------------------------------
Let $W$ be the Stein four–manifold with boundary diffeomorphic to $D_{g, 2g-2}$ as given by Figure \[f:figure1\]. Consider the smooth four–dimensional handlebody $X$ obtained by attaching to $W$ the two–handles realizing the contact surgeries described in Figure \[f:figure3\](a) or \[f:figure3\](b). Converting the contact framing coefficients into the usual ones, we see that a framed link presentation of $X$ is obtained by pasting Figure \[f:figure4\](a) in place of the ‘dotted ellipse’ in Figure \[f:figure1\].
By the discussion above on attaching two–handles we know that, corresponding to each of Figure \[f:figure3\](a) and \[f:figure3\](b), there is an almost–complex structure on $X$ minus two balls lying in the interior of the two–handles realizing the $(+1)$–surgeries. Moreover, the two almost–complex structures determine the two–plane fields $\xi_0$ and $\xi_1$ on ${\partial}X$ and two $Spin^c$ structures ${\mathbf s}_0$ and ${\mathbf s}_1$ on $X$. Observe that, since the rotation number of the Legendrian knot in Figure \[f:figure1\] vanishes, it follows from [@Go], Theorem 4.12, that $c_1(W)=0$. In the same way, it follows that we can choose an orientation of the $n-2g$ linking knots with framing $-3$ in Figure \[f:figure4\](a) so that $c_1({\mathbf s}_i)$ evaluates as $(-1)^i$ on all the corresponding homology classes. Finally, by the argument given in the proof of Lemma \[l:value\], $c_1({\mathbf s}_i)$ evaluates trivially on the generators of $H_2(X;{\mathbb Z})$ determined by the two–handles corresponding to the $(+1)$–surgeries.
The four–manifold $X$ is diffeomorphic to $D_{g,n}\# S^2{\times}S^2\#
(n-2g){\overline{{\mathbb C}{\mathbb P}}}^2$. One can see this by performing a sequence of handleslides on the Kirby diagram as shown in Figure \[f:figure4\]. In fact, start by sliding over the knot $K_1$ in Figure \[f:figure4\](a) the remaining $(n-2g-1)$ $(-3)$–framed circles. Then, slide $K_1$ over $K_2$ and finally $K_2$ over $K_3$, obtaining \[f:figure4\](b). Sliding the long $(2g-2)$–framed arc over the $2$–framed knot and using the $0$–framed normal circle to separate the $2$–framed circle from the rest of the diagram, we get \[f:figure4\](c). Blowing down the $(-1)$–circle results in \[f:figure4\](d), and $(n-2g-1)$ further blow downs give \[f:figure4\](e).
Following the handle slides of Figure \[f:figure4\] on the homological level we see that $c_1({\mathbf s}_i)$ evaluates on the generator of the second homology of $D_{g,n}$ as $(-1)^i(n-2g)$. Moreover, it evaluates as $(-1)^i$ on generators of the ${\overline{{\mathbb C}{\mathbb P}}}^2$ summands, and vanishes when restricted to the $S^2{\times}S^2$ summand. This immediately implies that the $Spin^c$ structure ${\mathbf t}_{\xi_i}$ is equal to the restriction of the unique $Spin^c$ structure ${\mathbf s}_e\in Spin^c(D_{g,n})$ such that $c_1({\mathbf s}_e)$ evaluates on the generator of $H_2(D_{g,n};{\mathbb Z})$ as $(-1)^i(n-2g)$. Since the value of $c_1({\mathbf s}_0)$ on the generator is $2-2g+n$, $e$ satisfies the equation: $$2-2g+n+2e=(-1)^i (n-2g).$$ Therefore we get $e=-1$ or $e=2g-1+n$ respectively for $i=0$ or $i=1$. Since ${\mathbf s}_e|_{Y_{g,n}}={\mathbf t}_e$, we conclude that $${\mathbf t}_{\xi _i }={\mathbf t}_{2ig-1}$$ for $i=0,1$. Observe that this result is consistent with the independent calculation made in [@LS].
\[l:d3ofxi\] The value of the three–dimensional invariant of $\xi_i$ is $$d_3(\xi_i)=\frac{n^2-3n+4g^2}{4n}.$$
We have $\chi(X)=n-4g+4$ and $\sigma (X)=1-n+2g$. [From]{} what we know about $c_1({\mathbf s}_i)$ is easy to deduce that $$c_1^2({\mathbf s}_i)=-2\frac{g(n-2g)}{n}.$$ In order to compute the three–dimensional invariant we need to take into account the correction term $q$ for each of the two contact $(+1)$–surgeries. Using Lemma \[l:value\] we conclude $$d_3(\xi _i)
=\frac{1}{4}(c_1^2({\mathbf s}_i)-2\chi(X)-3{\sigma}(X))+2=
\frac{n^2-3n+4g^2}{4n}.$$
The proof of Theorem \[t:main\] {#s:proof}
===============================
\[t:pani\] Let $n\geq 2g>0$, and let $\xi$ be a two–plane field on $Y_{g,n}$ such that ${\mathbf t}_{\xi}\in\{{\mathbf t}_{\xi_0},{\mathbf t}_{\xi_1}\}$. If $\xi$ is homotopic to a semi–fillable contact structure, then $$d_3(\xi )=\frac{n^2+n+4g^2}{4n}-2g-2.$$
In the proof of Theorem 2.1 of [@L] it is shown that if $Y$ is a closed three–manifold and ${\mathbf t}\in Spin^c(Y)$ is torsion and satisfies:
- all Seiberg-Witten solutions in ${\mathbf t}$ are reducible and
- the moduli space of the Seiberg-Witten solutions in ${\mathbf t}$ is a smooth manifold and the corresponding Dirac operators have trivial kernels,
then the expected dimension $d_1$ of the Seiberg-Witten moduli space of solutions over a potential symplectic semi–filling of $(Y_{g,n}, \xi_i)$ equipped with a cylindrical end metric and fixed asymptotic limit is equal to $-1-b_1(Y_{g,n})$.
The moduli space of Seiberg-Witten solutions on $Y_{g,n}$ has been determined in [@MOY] (see also [@OSz]). These results show that the assumptions listed above hold for the moduli spaces associated to the $Spin ^c$ structures ${\mathbf t}_{\xi_i}$. Therefore, the conclusion $d_1=-1-b_1(Y_{g,n})$ holds. This implies that for each $i=0,1$, $Y_{g,n}$ carries only one homotopy type of two–plane field which contains potentially semi–fillable contact structures inducing ${\mathbf t}_{\xi_i}$, because $d_1$ is equal to the three–dimensional invariant plus an expression involving some topological terms and an $\eta$–invariant ([@L0], Formula 3.1). In fact, such an expression has been explicitely calculated in [@N], in the formula preceding (3.29), so our proof reduces to translating that formula into our notations.
In Nicolaescu’s notations the integer $\kappa $ corresponds to ${\mathbf t}_{g-1+\kappa}$. This is because his “base” $Spin^c$ structure is induced by a $Spin$ structure on $Y_{g,n}$ with associated bundle of spinors ${\mathbb S}=\pi^* K_{{\Sigma}_g}^{-\frac 12}\oplus\pi^*
K_{{\Sigma}_g}^{\frac 12}\to Y_{g,n}$ (see text following Formula (2.6) in [@N]), and $\mathbb S$ is the restriction of $TD_{g,n}\otimes
\pi^* K_{{\Sigma}_g}^{\frac 12}\to D_{g,n}$ to the boundary.
The result we need is obtained by substituting $n$ for $\ell$ and $g$ or $n-g$ in place of $\kappa$ into the formula preceding (3.29) of [@N]. (The formula we are using here differs from Formula (3.29) by the additive term $2g-1$ because (3.29) computes the dimension of the whole moduli space rather than the dimension of the moduli space of solutions with a fixed asymptotic limit, i.e. $d_1$). Explicitely, in our notation we have: $$-1-b_1(Y_{g,n})=d_1=d_3(\xi ) -\frac{1}{2}(2g-1)-\frac{1}{4}(n-1)-
\frac{\kappa ^2}{n} +\kappa$$ where $b_1(Y_{g,n})=2g$ and the value of $\kappa $ to be substituted is either $g$ or $n-g$ according to whether ${\mathbf t}_\xi={\mathbf t}_{\xi_1}$ or ${\mathbf t}_\xi={\mathbf t}_{\xi_0}$, respectively. In both cases we obtain for $d_3(\xi)$ the value given in the statement.
Let $\xi$ be a two–plane field representing a homotopy class inducing ${\mathbf t}_{\xi _i}$ which might be represented by a semi–fillable contact structure. Then, by Theorems \[l:d3ofxi\] and \[t:pani\] we have $d_3(\xi_i) - d_3 (\xi)=2g+1>0$. Therefore, the homotopy classes $[\xi_i]$ cannot be represented by semi–fillable contact structures.
[**Remarks.**]{} (1) For $n<2g$ the circle bundle $Y_{g,n}$ admits no $Spin ^c$ structure for which the Seiberg-Witten moduli space has the properties required by the proof of Theorem \[t:pani\].
\(2) The assumption $g>0$ in Theorem \[t:pani\] is necessary — $Y_{0,n}$ is a lens space on which all tight contact structures are Stein fillable. The proof of Theorem \[t:pani\] breaks down since the formula from [@N] used in the proof holds only for $g\geq 1$.
\(3) Notice that for $n=2g$ the two contact structures $\xi_0$ and $\xi _1$ coincide.
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[^1]: E-mail addresses: [email protected] (P. Lisca), [email protected] (A.I. Stipsicz)The first author was partially supported by MURST, and he is a member of EDGE, Research Training Network HPRN-CT-2000-00101, supported by The European Human Potential Programme. The second author was partially supported by Széchenyi Professzori Ösztönd[í]{}j and OTKA
|
---
abstract: 'We investigate the nature of the broad-band emission associated with the low-power radio hotspots 3C105 South and 3C445 South. Both hotspot regions are resolved in multiple radio/optical components. High-sensitivity radio VLA, NIR/optical VLT and HST, and X-ray [*Chandra*]{} data have been used to construct the multi-band spectra of individual hotspot components. The radio-to-optical spectra of both hotspot regions are well fitted by a synchrotron model with steep spectral indices $\sim 0.8$ and break frequencies between $10^{12}-10^{14}$ Hz. 3C105 South is resolved in two optical components: a primary one, aligned with the jet direction and possibly marking the first jet impact with the surrounding medium, and a secondary, further out from the jet and extended in a direction perpendicular to it. This secondary region is interpreted as a splatter-spot formed by the deflection of relativistic plasma from the primary hotspot. Radio and optical images of 3C445 South show a spectacular 10-kpc arc-shape structure characterized by two main components, and perpendicular to the jet direction. HST images in I and B bands further resolve the brightest components into thin elongated features. In both 3C105 South and 3C445 South the main hotspot components are enshrouded by diffuse optical emission on scale of several kpcs, indicating that very high energy particles, possibly injected at strong shocks, are continuously re-accelerated in situ by additional acceleration mechanisms. We suggest that stochastic processes, linked to turbulence and instabilities, could provide the required additional re-acceleration.'
date: 'Received ; accepted ?'
title: 'Complex particle acceleration processes in the hotspots of 3C105 and 3C445[^1]'
---
\[firstpage\]
radio continuum: galaxies - radiation mechanisms: non-thermal - acceleration of particles
Introduction
============
Radio hotspots are bright and compact regions located at the end of powerful radio galaxies [FRIIs, @fr2] and considered to be the working surfaces of supersonic jets. In these regions, the jet emitted by the active galactic nucleus (AGN) impacts on the surrounding ambient medium producing a shock that may re-accelerate relativistic particles transported by the jet and enhance the radio emission. Electrons responsible for synchrotron emission in the optical band must be very energetic (Lorentz factor $\gamma >
10^{5}$), and therefore with short radiative lifetime. Consequently the detection of optical emission from hotspots supports the scenario where the emitting electrons are accelerated at the hotspots, possibly by strong shocks generated by the impact of the jet with the ambient medium [@meise89; @meise97; @gb03]. The detection of X-ray synchrotron counterparts of radio hotspots would imply the presence of electrons with even higher energies. However the main radiation process responsible for the X-ray emission seems to differ between high and low luminosity hotspots [@hardcastle04]. In bright hotspots, like Cygnus A and 3C295, the X-ray emission is produced by synchrotron-self Compton (SSC) in the presence of a magnetic field that is roughly in equipartition, while in low-luminosity hotspots, like 3C390.3, the emission at such high energies is likely due to synchrotron radiation [@hardcastle07].\
The discovery of optical emission extended to kpc scale questions the standard shock acceleration model, suggesting that other efficient mechanisms must take place across the hotspot region. Although it may seem an uncommon phenomenon due to the difficulty to produce high-energy electrons on large scales, deep optical images showed that diffuse optical emission is present in a handful of hotspots: 3C33, 3C111, 3C303, 3C351 [@valta99], 3C390.3 [@aprieto97], 3C275.1 [@cheung05], Pictor A [@thomson95], and 3C445 [@aprieto02]. A possible mechanism able to keep up the optical emission in the post-shock region on kpc scale is a continuous, relatively efficient, stochastic mechanism[^2].\
The sample of low-power hotspots presented by @mack09 is characterized by low magnetic field strengths between 40 and 130 $\mu$G, a factor 2 to 5 lower than that estimated in hotspots with optical counterparts previously studied in the literature. A surprisingly high optical detection rate ($\geq$ 45%) of the hotspots in this sample was found, and in most cases the optical counterpart extends on kpc scales. This is the case of 3C445 South, 3C 445 North, 3C105 South and 3C227 West [@mack09].\
This paper focuses on a multi-band, from radio to X-rays, high spatial resolution study of the two most interesting cases among the low-luminosity hotspots from @mack09, 3C105 South and 3C445 South, in which the hotspot regions are resolved into multiple components. 3C105 is hosted by a narrow-line radio galaxy (NLRG) at redshift $z=0.089$ [@tadhunter93]. At this redshift 1$^{\prime\prime}$ corresponds to 1.642 kpc. The radio source 3C105 is about 330$^{\prime\prime}$ (542 kpc) in size, and the hotspot complex 3C105 South is located about 168$^{\prime\prime}$ (276 kpc) from the core in the south-east direction. 3C445 is hosted by a broad-line radio galaxy (BLRG) at redshift $z=0.05623$ [@eracleous94]. At this redshift 1$^{\prime\prime}$ corresponds to 1.077 kpc. The radio source 3C445 is about 562$^{\prime\prime}$ (608 kpc) in size, and the hotspot complex 3C445 South is located 270$^{\prime\prime}$ (291 kpc) south of the core.
Throughout this paper, we assume the following cosmology: $H_{0} =
71\; {\rm km/s\, Mpc^{-1}}$, $\Omega_{\rm M} = 0.27$ and $\Omega_{\rm \Lambda} = 0.73$, in a flat Universe. The spectral index is defined as $S {\rm (\nu)} \propto \nu^{- \alpha}$.\
Observations
============
Radio observations
------------------
VLA observations at 1.4, 4.8, and 8.4 GHz of the radio hotspots 3C445 South and 3C105 South were carried out in July 2003 (project code AM772) with the array in A-configuration. Each source was observed for about half an hour at each frequency, spread into a number of scans interspersed with other source/calibrator scans in order to improve the $uv$-coverage. About 4 minutes were spent on the primary calibrator 3C286, while secondary phase calibrators were observed for 1.5 min about every 5 min. Data at 1.4 and 4.8 GHz were previously published by @mack09. The data reduction was carried out following the standard procedures for the VLA implemented in the NRAO AIPS package. Final images were produced after a few phase-only self-calibration iterations. The r.m.s. noise level on the image plane is negligible if compared to the uncertainty of the flux density due to amplitude calibration errors that, in this case, are estimated to be $\sim$3%.\
Besides the [*full-resolution*]{} images, we also produced [ *low-resolution*]{} images at both 4.8 and 8.4 GHz, using the same $uv$-range, image sampling and restoring beam of the 1.4 GHz data. These new images were obtained with natural grid weighting in order to mitigate the differences in the sampling density at short spacing, and to perform a robust spectral analysis.\
Optical observations
--------------------
For both 3C105 South and 3C445 South, VLT high spatial resolution images in standard filters taken with both ISAAC in J-, H-, K-, and FORS in I-, R-, B- and U- bands are used in this work. All the images have excellent spatial resolutions in the range of 0.5$^{\prime\prime}$ $<$ FWHM $<$ 0.7$^{\prime\prime}$. Details on the observations and data reduction are given in @mack09. The pixel scale of the ISAAC images is 0.14 arcsec pixel$^{-1}$. In the case of the FORS images the pixel scale is 0.2 arcsec pixel$^{-1}$, with the exception of the I-band where it is 0.1 arcsec pixel$^{-1}$.\
Further HST observations on 3C445 South only, were obtained with the ACS/HRC camera on 7th July, 2005 in the filters F814W (I-band, exposure time $\sim$ 1.5 hr) and F475W (B-band, exposure time $\sim$ 2.3 hr).\
For science analysis we used the “\*drz” images delivered by the HST ACS pipeline. These final images are calibrated, cosmic-ray cleaned, geometrically corrected, and drizzle-combined, provided in electrons per sec. The final pixel scale of the drizzled images is 0.025$^{\prime\prime}$$\times$0.025$^{\prime\prime}$ per pixel. The flux calibration was done using the standard HST/ACS procedure that relies on the PHOTFLAM keyword in the respective image headers. The quality of the pipeline-delivered images was adequate for the purposes of analyzing the hotspot region.
-------- ------- -------- ----------------------- ------------- ------------ -------------- -------------------- --------------------
Source Comp. z scale S$_{1.4}$ S$_{4.8}$ S$_{8.4}$ $\theta_{\rm maj}$ $\theta_{\rm min}$
kpc/$^{\prime\prime}$ mJy mJy mJy arcsec arcsec
3C105 S1 0.089 1.642 130$\pm$10 67$\pm$5 45$\pm$5 1.0 0.8
S2 1250$\pm$40 620$\pm$20 460$\pm$15 1.30 1.0
S3 1180$\pm$35 510$\pm$15 320$\pm$12 1.5 0.8
Ext 174$\pm$10 75$\pm$5 50$\pm$3
3C445 SE 0.0562 1.077 290$\pm$30 98$\pm$15 65$\pm$10 3.5 1.0
SW 220$\pm$25 51$\pm$10 36$\pm$6 1.5 0.5
Diff 13.0$\pm$1.1
-------- ------- -------- ----------------------- ------------- ------------ -------------- -------------------- --------------------
\[tab\_flux\_rad\]
X-ray observations
------------------
The radio source 3C105 was observed by [*Chandra*]{} on 2007 December 17 (Obs ID 9299) during “The [*Chandra*]{} 3C Snapshot Survey for Sources with z$<$0.3” [@massaro10]. An $\sim$8 ksec exposure was obtained with the ACIS-S camera, operating in VERY FAINT mode. The data analysis was performed following the standard procedures described in the [*Chandra*]{} Interactive Analysis of Observations (CIAO) threads and using the CIAO software package v4.2 (see Massaro et al. 2009 for more details). The [*Chandra*]{} Calibration Database (CALDB) version 4.2.2 was used to process all files. Level 2 event files were generated using the $acis\_process\_events$ task, after removing the hot pixels with $acis\_run\_hotpix$. Events were filtered for grades 0,2,3,4,6, and we removed pixel randomization.\
3C445 South was observed by [*Chandra*]{} on 2007 October 18 [@perlman10], ACIS chip S3, with an exposure time of 45.6 ksec. The data were retrieved from the archive and analysed following the same procedure as for 3C105 South. This re-analysis was necessary in order to achieve a proper alignment with the radio data.\
We created 3 different flux maps in the soft, medium, and hard X-ray bands (0.5 – 1, 1 – 2, and 2 – 7 keV, respectively) by dividing the data with monochromatic exposure maps with nominal energies = 0.8 keV (soft), 1.4 keV (medium), and 4 keV (hard). Both the exposure maps and the flux maps were regridded to a pixel size of 0.25 the size of a native ACIS pixel (native=0.492$^{\prime\prime}\times0.492^{\prime\prime}$). To obtain maps with brightness units of ergs cm$^{-2}$ s$^{-1}$ pixel$^{-1}$, we multiplied each event by the nominal energy of its respective band.\
For 3C445 South, we measured a flux density consistent with what reported by @perlman10. The flux density was extracted from [*Chandra*]{} ACIS-S images in which the hotspot was placed on axis. Both hotspots have been detected also by [*Swift*]{} in the energy range 0.3-10 keV (See Appendix A). This is remarkable given [ *Swift*]{}’s survey operation mode and its poor spatial resolution. The detection level is about 7$\sigma$ and 12$\sigma$ for 3C105 South and 3C445 South, respectively. However, given the large [*Swift*]{} errors in the counts-to-flux conversion and its low angular resolution, we do not provide any further flux estimate.\
Image registration
------------------
The alignment between radio and optical images was done by the superposition of the host galaxies with the nuclear component of the radio source using the AIPS task LGEOM. This results in a shift of 3.5$^{\prime\prime}$. To this purpose, the optical images were previously brought on the same grid, orientation and coordinate system as the radio images by means of the AIPS task CONV and REGR [see also @mack09]. The final overlay of radio and optical images is accurate to 0.1$^{\prime\prime}$.\
For 3C105 South the X-ray image has been aligned with the radio one by comparing the core position. Then, the final overlay of X-ray contours on the VLT image is accurate to 0.1$^{\prime\prime}$. In the case of 3C445 the shape of the nucleus of the galaxy is badly distorted in the [*Chandra*]{} image because of its location far off axis of [*Chandra*]{}. The alignment was then performed using three background sources visible both in X-ray and B band, and located around the hotspot. The achieved accuracy with this registration is better than 0.15 arcsec, allowing us to confirm a shift of about 2$^{\prime\prime}$ in declination between the X-rays and B-band emission centroids, the X-ray one being the closest to the core (Fig. \[fig\_3c445\]).\
-------- ------- -------------- -------------- ------------- ------------- ------------- ------------- ------------- ----------------------- ----------------------- -----------------------
Source Comp. S$_{\rm K}$ S$_{\rm S$_{\rm J}$ S$_{\rm I}$ S$_{\rm R}$ S$_{\rm B}$ S$_{\rm S$_{\rm I}^{\rm HST}$ S$_{\rm B}^{\rm HST}$ S$_{X}$
H}$ U}$
$\mu$Jy $\mu$Jy $\mu$Jy $\mu$Jy $\mu$Jy $\mu$Jy $\mu$Jy $\mu$Jy $\mu$Jy
3C105 S1 4.6$\pm$0.9 4.4$\pm$1.1 $<$2.5 - 0.5$\pm$0.1 0.2$\pm$0.1 - - - 7.5$\pm$2.4
S2 18.4$\pm$1.4 12.3$\pm$1.1 3.4$\pm$1.0 - 0.7$\pm$0.1 0.2$\pm$0.1 - - - $<$2.0
S3 31.9$\pm$2.8 25.7$\pm$2.9 4.4$\pm$1.8 - 0.9$\pm$0.1 0.3$\pm$0.1 - - - 3.2$\pm$1.6
Ext 15.4$\pm$2.0 5.4$\pm$2.0 - - 0.4$\pm$0.1 0.2$\pm$0.1 - - - -
3C445 SE 8.0$\pm$1.0 5.6$\pm$2.0 6.0$\pm$1.5 2.0$\pm$0.2 1.3$\pm$0.2 0.7$\pm$0.1 0.5$\pm$0.3 1.7$\pm$0.2 1.5$\pm$0.3 9.38$\times$10$^{-4}$
SW 4.6$\pm$1.4 3.6$\pm$1.5 3.0$\pm$0.4 1.7$\pm$0.3 1.4$\pm$0.1 0.7$\pm$0.1 0.5$\pm$0.2 1.4$\pm$0.1 0.3$\pm$0.1 -
SC - - - - 0.8$\pm$0.1 0.6$\pm$0.1 0.4$\pm$0.1 - - -
Diff - 2.1$\pm$0.6 3.2$\pm$1.3 1.2$\pm$0.2 1.0$\pm$0.2 0.8$\pm$0.2 -
-------- ------- -------------- -------------- ------------- ------------- ------------- ------------- ------------- ----------------------- ----------------------- -----------------------
\[tab\_flux\_opt\]
Photometry
==========
To construct the spectral energy distribution (SED) of individual hotspot components, the flux density at the various wavelengths must be accurately measured in the same region, avoiding contamination from unrelated features. To this purpose, we produced a cube where each plane consists of radio and optical images regridded to the same size and smoothed to the same resolution. Then the flux density was derived by means of AIPS task BLSUM which performs an aperture integration on a selected polygonal region common to all the images. The values derived in this way were then used to construct the radio-to-optical SED, and they are reported in Tables 1 and 2.\
In addition to the low-resolution approach, we derive the hotspot flux densities and angular sizes on the full resolution images, in order to better describe the source morphology.
On the radio images, we estimate the flux density of each component by means of TVSTAT, which is similar to BLSUM, but instead of working on an image cube it works on a single image. The angular size was derived from the lowest contour on the image plane, and it corresponds to roughly twice the size of the full width half maximum (FWHM) of a conventional Gaussian covering a similar area. In the case of 3C105 South, the hotspot components are unresolved at 1.4 GHz, and we derive the flux density at this frequency by means of AIPS task JMFIT, which performs a Gaussian fit in the image plane. The angular size was measured on the images in which the components were resolved, i.e. in the case of 3C105 South we use the 4.8 and 8.4-GHz images, which provide the same value, while for 3C445 South the components could be reliably resolved in the image at 8.4 GHz only (Table 1).\
Full-resolution infrared and optical flux densities of hotspot sub-components were measured by means of the IDL-based task ATV using a circular aperture centred on each component. Such values were compared to those derived from the analysis of the cube and they were found to be within the expected uncertainties.\
For the X-ray flux we constructed photometric apertures to accommodate the [*Chandra*]{} point spread function and to include the total extent of the radio structures. The background regions, with a total area typically twice that of the source region, have been selected close to the source, and centred on a position where other sources or extended structures are not present. The X-ray flux was measured in any aperture with only a small correction for the ratio of the mean energy of the counts within the aperture to the nominal energy for the band. We note that in 3C105 South, the hotspot components are well separated (2$^{\prime\prime}$), allowing us to accurately isolate the corresponding X-ray emission. In 3C445 South the X-ray emission is not associated with the two main components clearly visible in the radio and optical bands, and flux was derived by using an aperture large enough to include all of the X-ray emission extending over the entire hotspot region. Our estimated value is in agreement with the one reported by @perlman10. All X-ray flux densities have been corrected for the Galactic absorption with the column density N$_H$ = 1.15$\cdot$10$^{21}$cm$^{-2}$ given by @kalberla05. X-ray fluxes are reported in Table \[tab\_flux\_opt\].\
Morphology
==========
3C105 South
-----------
The southern hotspot complex of 3C105 shows a curved structure of about 8$^{\prime\prime}$$\times$4.5$^{\prime\prime}$ ($\sim$13$\times$7 kpc) in size. It is dominated by three bright components, all resolved at radio frequencies, connected by a low surface brightness emission also visible in optical and infrared (Fig. \[fig\_3c105\]). The central component, labeled S2 in Fig. \[fig\_3c105\], is the brightest in radio and, when imaged with high spatial resolution, it is resolved in two different structures separated by about 1.2 kpc. @leahy97 interpreted this as the true jet termination hotspot, while S1, with an elongated structure of (1.6$\times$1.3) kpc and located 5.7 kpc to the north of S2 is considered as jet emission. The southernmost component S3, located about 4.1 kpc from S2, has a resolved structure of (2.4$\times$1.3) kpc in size, and it is elongated in a direction perpendicular to the line leading to S2. Its morphology suggests that S3 is a secondary hotspot similar to 3C20 East [@cox91].\
At 1.4 GHz, an extended tail accounting for $S_{\rm 1.4} =$ 608 mJy and embedding the jet is present to the west of the hotspot complex, in agreement with the structure previously found by @neff95. At higher frequencies the lack of the short spacings prevents the detection of such an extended structure, and only a hint of the jet, accounting for $S_{\rm 4.8} \sim 70$ mJy, is still visible at 4.8 GHz.\
In the optical and NIR the hotspot complex is characterized by the three main components detected in radio. In NIR and optical, the southernmost component S3 is the brightest one, with a radio-to-optical spectral index $\alpha_{r-o}=$0.95$\pm$0.10. It displays an elongated structure rather similar in shape and size to that found in radio. It is resolved in all bands with the only exception of B band, likely due to the lower spatial resolution achieved. Component S1 is resolved in all NIR/optical bands, showing a tail extending towards S2. Its radio-to-optical spectral index is $\alpha_{r-o}=$0.95$\pm$0.10. On the other hand, S2 appears unresolved in all bands, with the exception of K and H bands, i.e. those with the highest resolution achieved. In these NIR bands S2 is extended in the southern direction, resembling what is observed in radio. Its radio-to-optical spectral index is $\alpha_{r-o}=$1.05$\pm$0.10.\
Diffuse emission connecting the main hotspot components and extending to the southwestern part of the hotspot complex is detected in most of the NIR and optical images.\
In the X-ray band, S1 is the brightest component, whereas the emission from S3 is very weak (formally detected at only 2$\sigma$ level). For this reason in the following we will use the nominal X-ray flux of S3 as a conservative upper limit. For component S2 only an upper limit could be set.
3C445 South
-----------
The hotspot 3C445 South displays an extended east-west structure of about 9.3$^{\prime\prime}$ $\times$ 2.8$^{\prime\prime}$ (10$\times$3 kpc) in size in radio (Fig. \[fig\_3c445\]). At 8.4 GHz, the hotspot complex is almost completely resolved out and the two main components, clearly visible in NIR/optical images, are hardly distinguishable. When imaged with enough resolution, these components display an arc-shaped structure both in radio and NIR/optical bands, with sizes of about (3.4$\times$1.5) kpc and (2.1$\times$1.1) kpc for SE and SW respectively. Component SE is elongated in a direction almost perpendicular to the line leading to the source core, while SW forms an angle of about -20$^{\circ}$ with the same line.\
In radio and NIR, the SE component is the brightest one, with a flux density ratio SE/SW $\sim$ 1.6, while in the optical both components have similar flux densities. Both components have a radio-to-optical spectral index $\alpha_{r-o}=$ 0.9$\pm$0.10. In the optical R-, B-, and U-band images a third component (labelled SC in Fig. \[fig\_3c445\]) aligned with the jet direction becomes visible between SE and SW. Despite the good resolution and sensitivity of the radio and NIR images, SC is not present at such wavelengths. When imaged with the high resolution provided by HST, both SE and SW are clearly resolved, and no compact regions can be identified in the hotspot complex. Trace of the SC component is seen in the B-band, in agreement with the VLT images.\
In the VLA and VLT images, the two main components are enshrouded by a diffuse emission, visible in radio and NIR/optical bands. The flux densities of the SE and SW components measured on the HST images are consistent (within the errors) with those derived on the VLT images.\
The optical component W located about 2.8$^{\prime\prime}$ (3 kpc) on the northwestern part of SW does not have a radio counterpart, as it is clearly shown by the superposition of I-band HST and 8.4-GHz VLA images (Fig. \[hst\_vla\]), and thus it is considered an unrelated object, like a background galaxy. Another possibility is that this is a synchrotron emitting region where the impact of the jet produces very efficient particle acceleration. However, its steep optical spectrum ($\alpha \sim 2$ between I and U bands, see Section 5.3, Fig. \[slope\_3c445\]) together with the absence of detected radio emission disfavour this possibility. Future spectroscopic information would further unveil the nature of this optical region.\
[*Chandra*]{} observations of 3C445 South detected X-ray emission from a region that extends over 6$^{\prime\prime}$ in the east-west direction (Fig. \[fig\_3c445\]), and it peaks almost in the middle of the hotspot structure, suggesting a spatial displacement between X-ray and radio/NIR/optical emission [@perlman10].\
Spectral energy distribution
============================
-------- ------- ---------- --------------- -------------- ---------------
Source Comp. $\alpha$ $\nu_{\rm b}$ $B_{\rm eq}$ $t_{\rm rad}$
10$^{13}$ Hz $\mu$G yr
3C105 S1 0.8 0.50 150 12
S2 0.8 0.75 290 4
S3 0.8 1.5 270 3
3C445 SE 0.75 5.2 60 15
SW 0.75 9.4 50 15
-------- ------- ---------- --------------- -------------- ---------------
: Synchrotron parameters. Column 1: Hotspot; Column 2: component; Columns 3, 4: spectral index and break frequency as derived from the fit to the radio-to-optical SED (Section 5.1); Column 5: equipartition magnetic field, computed following the approach presented in Brunetti et al. (2002); Column 6: radiative age computed using Eq. 2.
\[tab\_fit\]
The broad-band energy distribution
----------------------------------
We model the broad band energy distribution, from radio to optical, of the hotspot regions in order to determine the mechanisms at the basis of the emission. The comparison between the model expectation in the X-rays and [*Chandra*]{} data sets additional constraints. In the adopted models, the hotspot components are described by homogeneous spheres with constant magnetic field and constant properties of the relativistic electron populations. The spectral energy distributions of the emitting electrons are modelled assuming the formalism described in @gb02. According to this model a population of seed electrons (with $\gamma \leq
\gamma_{*}$) is accelerated at the shock and is injected in the downstream region with a spectrum dN($\gamma$)/dt $\propto$ $\gamma^{-p}$, for $\gamma_{*} < \gamma < \gamma_{c}$, $\gamma_{c}$ being the maximum energy of the electrons accelerated at the shock. Electrons accelerated at the shock are advected in the downstream region and age due to radiative losses. Based on @gb02, the volume integrated spectrum of the electron population in the downstream region of size $L \sim T v_{\rm adv}$ ($T$ and $v_{\rm adv}$ being the age and the advection velocity of the downstream region) is given by either a steep power-law $N$($\gamma$) $\propto \gamma^{-(p+1)}$ for $\gamma_{b} < \gamma < \gamma_{c}$, where $\gamma_{b}$ is the maximum energy of the “oldest” electrons in the downstream region, or by $N$($\gamma$) $\propto \gamma^{-p}$ for $\gamma_{*} < \gamma < \gamma_{b}$, or by a flatter shape for $\gamma_{\rm low} < \gamma < \gamma_{*}$, where $\gamma_{\rm low}$ is the minimum energy of electrons accelerated at the shock.\
As the first step we fit the SED in the radio-NIR-optical regimes with a synchrotron model, and we derive the relevant parameters of the synchrotron spectrum (injection spectrum $\alpha$, break frequency $\nu_{\rm b}$, cut-off frequency $\nu_{\rm c}$) and the slope of the energy distribution of the electron population as injected at the shock (p $= 2 \alpha +1$). Since hotspots have spectra with injection slope $\alpha$ ranging between 0.5 and 1 (as a reference, the classical value from the diffuse particle acceleration at strong shocks is $\alpha = 0.5$, e.g. Meisenheimer et al. 1997), we decided to consider the injection spectral index as a free parameter. Such constraints allow us to determine the spectrum of the emitting electrons (normalization, break and cut-off energy), once the magnetic field strength has been assumed, and to calculate the emission from either synchrotron-self-Compton (SSC) or inverse-Compton scattering of the cosmic background radiation (IC-CMB) expected from the hotspot (or jet) region [following @gb02]. Models described in @gb02 take also into account the boosting effects arising from a hotspot/jet that is moving at relativistic speeds and oriented at a given angle with respect to our line of sight.\
3C105 South
-----------
In Figures \[fig\_spectra\_105n\] to \[fig\_spectra\_105s\] we show the SED from the radio band to high energy emission measured for the hotspot components of 3C105 South, together with the model fits. Synchrotron models with an injection spectral index $\alpha$=0.8 provide an adequate representation of the SED of the central and southern components of 3C105 South, with break frequencies ranging from 5$\times$10$^{12}$ to 1.5$\times$10$^{13}$ W/Hz, while the cutoff frequencies are between 3$\times$10$^{14}$ and 2$\times$10$^{15}$ W/Hz. In both components, the upper limit to the X-ray emission does not allow us to constrain the validity of the SSC model (Figs. \[fig\_spectra\_105c\] and \[fig\_spectra\_105s\]). On the other hand, the northern component of 3C105 shows a prominent X-ray emission. A synchrotron model (dashed line in Fig. \[fig\_spectra\_105n\]) may fit quite reasonably the radio, NIR and X-ray emission, but it completely fails in reproducing the optical data. An additional contribution of the SSC is not a viable option since it requires a magnetic field much smaller than that obtained assuming equipartition (see Section 5.4) (solid lines), and implying an unreasonably large energy budget. On the other hand, the high energy emission is well modelled by IC-CMB [e.g. @tavecchio00; @celotti01] where the CMB photons are scattered by relativistic electrons with Lorentz factor $\Gamma \sim 6$, and $\theta$=5$^{\circ}$ with a magnetic field of 16 $\mu$G. This model implies that boosting effects play an important role in the X-ray emission of this component, suggesting that S1 is more likely a relativistic knot in the jet, rather than a hotspot feature. The weakness of this interpretation is that 3C105 is a NLRG and its jets are expected to form a large angle with our line of sight. Alternatively, the X-ray emission may be synchrotron from a different population of electrons, as suggested in the case of the jet in 3C273 (Jester et al. 2007).\
3C445 South
-----------
The analysis of the southern hotspot of 3C445 as a single unresolved component was carried out in previous work by @aprieto02 [@mack09; @perlman10]. In this new analysis, the high spatial resolution and multiwavelength VLT and HST data of 3C445 South allow us to study the SED of each component separately in order to investigate in more detail the mechanisms at work across the hotspot region. In Figures \[fig\_spectra\_445w\] to \[fig\_spectra\_445diff\] we show the SED from the radio band to high energy emission measured for the components of 3C445 South, together with the model fits. We must note that at 1.4 GHz the resolution is not sufficient to reliably separate the contribution from the two main components. For this reason, we do not consider the flux density at this frequency in constructing the SED. The X-ray emission (Fig. \[fig\_3c445\]) is misaligned with respect to the radio-NIR-optical position. For this reason, on the SED of both components (Figs. \[fig\_spectra\_445w\] and \[fig\_spectra\_445e\]) we plot the total X-ray flux which must be considered an upper limit. For the components of 3C445 South the synchrotron models with $\alpha$=0.75 reasonably fit the data, providing break frequencies in the range of 10$^{13}$ and $10^{14}$ W/Hz, and cutoff frequencies from 10$^{15}$ Hz and 10$^{18}$ Hz.\
Both the morphology (Fig. \[fig\_3c445\]) and the SED (Figs. \[fig\_spectra\_445w\] and \[fig\_spectra\_445e\]) indicate that the bulk of [*Chandra*]{} X-ray emission detected in 3C445 is not due to synchrotron emission from the two components (Section 6).\
As discussed in Section 4.2, diffuse IR and optical emission surrounds the two components SE and SW of 3C445 South, and a third component, SC, becomes apparent in the optical. We attempt to evaluate the spectral properties of the diffuse emission (including component SC). When possible, depending on statistics, we subtract from the total flux density of the hotspot, the contribution arising from the two main components, obtaining in this way the SED of the diffuse emission (inclusive of SC component) of 3C445 South. In the image we also plot the total X-ray flux. As expected the emission has a hard spectrum ($\alpha
\sim 0.85$) without evidence of a break up to the optical band, 10$^{15}$ Hz $<$ $\nu_{b}$ $\leq$ 8$\times$10$^{16}$ Hz. We also note that this hard component may represent a significant contribution of the observed X-ray emission, although the X-ray peak appears shifted ($\sim$ 1$^{\prime\prime}$) from the SC component. Due to the extended nature of the emission in this hotspot, we created a power-law spectral index map illustrating the change of the spectral index $\alpha$ across the hotspot region (Fig. \[slope\_3c445\]). The spectral energy distributions presented in Figs. \[fig\_spectra\_445w\], \[fig\_spectra\_445e\], and \[fig\_spectra\_445diff\] show the curvature of the integrated spectrum for the main components and the diffuse emission (see Section 5.1). The spectral map in Fig. \[slope\_3c445\] attempts to provide complementary information on the spectral slope for the diffuse inter-knot emission. Extracting these maps using the largest possible frequency range is complicated as it implies combining images from different instruments with different scale sampling, noise pattern, etc. These effects sum up to produce very low contrast maps given the weakness of the hotspot signal. To minimise these effects it was decided to extract the slope maps from the optical and -IR images only.\
The spectral index map between I- and U-band (Fig. \[slope\_3c445\]) shows two sharp edges, at the SW and SE components, with the highest value $\alpha \sim 1.5 $. Between these two main regions there is diffuse emission that is clearly seen in the I-/U-band spectral index map. The slope of this component is flatter than that of the two main regions and rather uniform all over the hotspot, with $\alpha \sim 1$.\
Physical parameters
-------------------
We compute the magnetic field of each hotspot component by assuming minimum energy conditions, corresponding to equipartition of energy between radiating particles and magnetic field, and following the approach by @gb97. We assume for the hotspot components an ellipsoidal volume $V$ with a filling factor $\phi$=1 (i.e. the volume is fully and homogeneously filled by relativistic plasma). The volume $V$ is computed by means:\
$$V = \frac{\pi}{6} d_{\min}^{2} d_{\max}$$
where d$_{\min}$ and d$_{\max}$ are the linear size of the minor and major axis, respectively. We consider $\gamma_{\rm min} =$100, and we assume that the energy densities of protons and electrons are equal. We find equipartition magnetic fields ranging from $\sim$ 50 - 290 $\mu$G (Table \[tab\_fit\]) that is lower than those inferred in high-power radio hotspots which range from $\sim$ 250 to 650 $\mu$G [@meise97; @cheung05]. Remarkably, if we compare these results with those from @mack09, we see that in 3C445 South the value computed considering the entire source volume is similar to those obtained in its individual sub-components, suggesting that compact and well-separated emitting regions are not present in the hotspot volume. On the other hand, the magnetic field averaged over the whole 3C105 South hotspot complex is much smaller than those derived in its sub-components.\
In the presence of such low magnetic fields high-energy electrons may have longer radiative lifetime than in high-power radio hotspots. The radiative age $t_{\rm rad}$ is related to the magnetic field and the break frequency by[^3]:\
$$t_{\rm rad} = 1610 \; B^{-3/2} \nu_{b}^{-1/2} (1+z)^{-1/2}
\label{eq_trad}$$
where B is in $\mu$G, $\nu_{b}$ in GHz and $t_{\rm rad}$ in 10$^{3}$ yr. If in Eq. \[eq\_trad\] we assume the equipartition magnetic field we find that the radiative ages are just a few years (Table 3). As the hotspots extend over kpc distances, it is indicative that a very efficient re-acceleration mechanism is operating in a similar way over the entire hotspot region.\
Discussion
==========
The detection of diffuse optical emission occurring well outside the main shock region and distributed over a large fraction of the whole kpc-scale hotspot structure is somewhat surprising. Deep optical observations pointed out that this is a rather common phenomenon detected in about a dozen hotspots [e.g. @mack09; @cheung05; @thomson95]. First-order Fermi acceleration alone cannot explain optical emission extending on kpc scale and additional efficient mechanisms taking place away from the main shock region should be considered, unless projection effects play an important role in smearing compact regions where acceleration is still occurring.\
Theoretically, we can consider several scenarios that are able to reproduce the observed extended structures. (1) One possibility is that a very wide jet, with a size comparable to the hotspot region, impacts simultaneously into various locations across the hotspot generating a complex shocked region that defines an arc-shaped structure. This, combined with projection effects may explain a wide (projected) emitting region. (2) Another possibility is a narrow jet that impacts into the hotspot in a small region where electrons are accelerated at a strong shock. In this case the accelerated particles are then transported upstream in the hotspot volume where they are continuously re-accelerated by stochastic mechanisms, likely due to turbulence generated by the jet and shock itself. (3) Finally, extended emission may be explained by the “dentist’s drill” scenario, in which the jet impacts into the hotspot region in different locations at different times.\
The peculiar morphology and the rather high NIR/optical luminosity of 3C105 South and 3C445 South, makes these hotspots ideal targets to investigate the nature of extended diffuse emission.\
In 3C105 South, the detection of optical emission in both primary and secondary hotspots implies that in these regions there is a continuous re-acceleration of particles. The secondary hotspot S3 could be interpreted as a splatter-spot from material accelerated in the primary one, S2 [@williams85]. Both the alignment and the distance between these components exclude the jet drilling scenario: the light time between the two components is more than 10$^{4}$ years, i.e. much longer than their radiative time (Table 3), suggesting that acceleration is taking place in both S2 and S3 simultaneously. The secondary hotspot S3 shows some elongation, always in the same direction, in all the radio and optical images with adequate spatial resolution. This elongation is expected in a splatter-spot and it follows the structure of the shock generated by the impact of the outflow from the primary upon the cocoon wall.\
This scenario, able to explain the presence of optical emission from two bright and distant components, fails in reproducing the diffuse optical emission enshrouding the main features, and the extended tail. In this case, an additional contribution from stochastic mechanisms caused by turbulence in the downstream region is necessary. Although this acceleration mechanism is in general less efficient than Fermi-I processes, the (radiative) energy losses of particles are smaller in the presence of low magnetic fields, such as those in between S2 and S3, (potentially) allowing stochastic mechanisms to maintain electrons at high energies.\
In 3C 445 South the observational picture is complex. The optical images of 3C 445 South show a spectacular 10-kpc arc-shape structure. High resolution HST images allow a further step since they resolve this structure in two elongated components enshrouded by diffuse emission. These components may mark the regions where a “dentist’s drill” jet impacts on the ambient medium, representing the most recent episode of shock acceleration due to the jet impact. On the other hand, they could simply trace the locations of higher particle-acceleration efficiency from a wide/complex interaction between the jet and the ambient medium. However, the transverse extension, about 1 kpc, of the two elongated components is much larger than what is derived if the relativistic particles, accelerated at the shock, age in the downstream region (provided that the hotspot advances at typical speeds of 0.05-0.1$c$). Furthermore, the diffuse optical emission on larger scale suggests the presence of additional, complex, acceleration mechanisms, such as stochastic processes, able to keep particle re-acceleration ongoing in the hotspot region. The detection of X-ray emission with [*Chandra*]{} adds a new grade of complexity. This emission and its displacement are interpreted by @perlman10 as due to IC-CMB originating in the fast part of the decelerating flow. Their model requires that the angle between the jet velocity and the observer’s line of sight is small. However, 3C445 is a classical double radio galaxy and the jet should form a large angle with the line of sight (see also Perlman et al. 2010). On the other hand, we suggest that the X-ray/optical offset might be the outcome of ongoing efficient particle acceleration occurring in the hotspot region. An evidence supporting this interpretation may reside on the faint and diffuse blob seen in U- and B-bands (labelled SC in Fig. \[fig\_3c445\]) just about 1$^{\prime\prime}$ downstream the X-ray peak. The surface brightness of this component decreases rapidly as the frequency decreases, as it is shown in Fig. \[fig\_3c445\]: well-detected in U- and B-bands, marginally visible in I-band, and absent at NIR and radio wavelengths. The SED of the diffuse hotspot emission (including SC component and excluding SW and SE) is consistent with synchrotron emission with a break at high frequencies, 10$^{15}$ Hz $<$ $\nu_{b}$ $\leq$ 8$\times$10$^{16}$ Hz, and may significantly contribute to the observed X-ray flux. Such a hard spectrum is in agreement with (i) a very recent episode of particle acceleration (the radiative cooling time of the emitting particles being 10$^{2}$-10$^{3}$ yr); (ii) efficient spatially-distributed acceleration processes, similar to the scenario proposed for the western hotspot of Pictor A (Tingay et al. 2008, see their Fig.5).\
Conclusions
===========
We presented a multi-band, high spatial resolution study of the hotspot regions in two nearby radio galaxies, namely 3C105 South and 3C445 South, on the basis of radio VLA, NIR/optical VLT and HST, and X-ray [*Chandra*]{} observations. At the sub-arcsec resolution achieved at radio and optical wavelengths, both hotspots display multiple resolved components connected by diffuse emission detected also in optical. The hotspot region in 3C105 resolves in three major components: a primary hotspot, unresolved and aligned with the jet direction, and a secondary hotspot, elongated in shape, and interpreted as a splatter-spot arising from continuous outflow of particles from the primary. Such a feature, together with the extremely short radiative ages of the electron populations emitting in the optical, indicates that the jet has been impacting almost in the same position for a long period, making the drilling jet scenario unrealistic. The detection of an excess of X-ray emission from the northern component of 3C105 South suggests that this region is likely a relativistic knot in the jet rather than a genuine hotspot feature. The optical diffuse emission enshrouding the main components and extending towards the tail can be explained possibly assuming additional stochastic mechanisms taking place across the whole hotspot region.\
In the case of 3C445 South the optical observations probe a scenario where the interaction between jet and the ambient medium is very complex. Two optical components pinpointed by HST observations mark either the locations where particle acceleration is most efficient or the remnants of the most recent episodes of acceleration. Although projection effects may play an important role, the morphology and the spatial extension of the diffuse optical emission suggest that particle accelerations, such as stochastic mechanisms, add to the standard shock acceleration in the hotspot region. The X-rays detected by [*Chandra*]{} cannot be the counterpart at higher energies of the two main components. It might be due to IC-CMB from the fast part of a decelerating flow. Alternatively the X-rays could pinpoint synchrotron emission from recent episodes of efficient particle acceleration occurring in the whole hotspot region, similarly to what proposed in other hotspots, that would make the scenario even more complex. A possible evidence supporting this scenario comes from the hard spectrum of the diffuse hotspot emission and from the appearance of a new component (SC) in the optical images.
Acknowledgment {#acknowledgment .unnumbered}
==============
We thank the anonymous referee for the valuable suggestions that improved the manuscript. F.M. acknowledges the Foundation BLANCEFLOR Boncompagni-Ludovisi, n’ee Bildt for the grant awarded him in 2010 to support his research. The VLA is operated by the US National Radio Astronomy Observatory which is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This work has made use of the NASA/IPAC Extragalactic Database NED which is operated by the JPL, Californian Institute of Technology, under contract with the National Aeronautics and Space Administration. This research has made used of SAOImage DS9, developed by the Smithsonian Astrophysical Observatory (SAO). Part of this work is based on archival data, software or on-line services provided by ASI Science Data Center (ASDC). The work at SAO is supported by supported by NASA-GRANT GO8-9114A. We acknowledge the use of public data from the Swift data archive. This research has made use of software provided by the Chandra X-ray Center (CXC) in the application packages CIAO and ChIPS.
[*Swift*]{} images
==================
Both the radio hotspot 3C105 South and 3C445 South have been detected by [*Swift*]{} in the energy range 0.3-10 keV.\
The reduction procedure for [*Swift*]{} data follows that described in @massaro08. In the following we report only the basic details.\
3C105 has been observed by [*Swift*]{} in four occasions (Obs. ID 00035625001-2-3-4) for a total exposure of $\sim$ 22 ks while 3C445 only for $\sim$ 12 ks (Obs. ID 00030944001-2). During all these observations, the [*Swift*]{} satellite was operated with all the instruments in data taking mode. We consider only XRT [@burrows05] data, since our sources were not bright enough to be detected by the BAT high energy experiment. In particular, [*Swift*]{}-XRT observations have been performed in photon-counting mode (PC).\
The XRT data analysis has been performed with the XRT- DAS software, developed at the ASI Science Data Center (ASDC) and distributed within the HEAsoft pack- age (v. 6.9). Event files were calibrated and cleaned with standard filtering criteria using the xrtpipeline task, combined with the latest calibration files available in the [*Swift*]{} CALDB distributed by HEASARC. Events in the energy range 0.3-10 keV with grades 0-12 (PC mode) were used in the analysis (see Hill et al. 2004 for more details). No signatures of pile-up were found in our [*Swift*]{} XRT observations. Events are extracted using a 17 arcsec radius circle centered on the radio position of the southern hotspots in both cases of 3C105 and 3C445 (see Fig. \[appendice\]). we measured 15 counts in the southern hotspot of 3C105 and 12 counts for that of 3C445, while the background estimated from a nearby source-free circular region of the same radius is 1.8 counts and 0.9 respectively.
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[^1]: Based on VLT programs 72B-0360B, 70B-0713B, 267B-5721, and HST program 10434.
[^2]: More recently these stochastic mechanisms have been also proposed for the acceleration of ultra-high energy cosmic-rays in the lobes of radiogalaxies [@hardcastle09].
[^3]: The magnetic field energy density in these hotpots are at least an order of magnitude higher than the energy density of the cosmic microwave background (CMB) radiation. Inverse Compton losses due to scattering of CMB photons are negligible.
|
---
abstract: 'Both in the design and deployment of blockchain solutions many performance-impacting configuration choices need to be made. We introduce BlockSim, a framework and software tool to build and simulate discrete-event dynamic systems models for blockchain systems. BlockSim is designed to support the analysis of a large variety of blockchains and blockchain deployments as well as a wide set of analysis questions. At the core of BlockSim is a Base Model, which contains the main model constructs common across various blockchain systems organized in three abstraction layers (network, consensus and incentives layer). The Base Model is usable for a wide variety of blockchain systems and can be extended easily to include system or deployment particulars. The BlockSim software tool provides a simulator that implements the Base Model in Python. This paper describes the Base Model, the simulator implementation, and the application of BlockSim to Bitcoin, Ethereum and other consensus algorithms. We validate BlockSim simulation results by comparison with performance results from actual systems and from other studies in the literature. We close the paper by a BlockSim simulation study of the impact of uncle blocks rewards on mining decentralization, for a variety of blockchain configurations.'
author:
- 'Maher Alharby$^{1,2}$'
- 'Aad van Moorsel$^{1}$'
bibliography:
- 'mybibliography.bib'
title: 'BlockSim: An Extensible Simulation Tool for Blockchain Systems'
---
Introduction
============
In the design as well as the deployment of blockchain solutions, many architectural, configuration and parameterization questions need to be considered. Since it is usually not feasible or practical to answer these questions using experimentation or trial-and-error, model-based simulation is required as an alternative. In this paper, we propose a discrete-event simulation framework called BlockSim [@alharby2019blocksim] to explore the effects of configuration, parameterization and design decisions on the behavior of blockchain systems.
BlockSim aims to provide simulation constructs that are intuitive, hide unnecessary detail and can be easily manipulated to be applied to a large set of blockchains design and deployment questions (related to performance, reliability, security or other properties of interest). That is, BlockSim has the following objectives:
1. Generality: we want to be able to use BlockSim for a large set of blockchain systems, configurations and design questions.
2. Extensibility: BlockSim should be easy to manipulate by a designer or analyst to study different types and aspects of blockchain systems.
3. Simplicity: the above two objectives should be met while making BlockSim easy to use, both for simulation studies and for extending it.
This paper expands on the short introduction of the BlockSim framework in [@alharby2019blocksim], and discusses all facets of the tool design, implementation and use.
At the core of BlockSim is a Base Model, which contains model constructs at three abstraction layers: the network layer, the consensus layer and the incentives layer [@vanmoorsel:2018]. The network layer captures the blockchain’s nodes and the underlying peer-to-peer protocol to exchange data between nodes. The consensus layer captures the algorithms and rules adopted to reach an agreement about the current state of the blockchain ledger. The incentives layer captures the economic incentive mechanisms adopted by a blockchain to issue and distribute rewards among the participating nodes.
The Base Model contains a number of functional blocks common across blockchains, that can be extended and configured as suited for the system and study of interest. The main functional blocks include Node, Transaction, Block, Consensus and Incentives, as we describe in Section \[s4:BlockSim:model\]. These are then implemented through a number of Python modules, discussed in Section \[s4:BlockSim:imp\], and complemented by modules (event, scheduler, statistics, etc.) that implement the simulation engine.
The public nature of permissionless blockchains provides for particularly powerful opportunities to validate the simulator. We validate the BlockSim simulation results by comparing against theoretical results (invariants such as block rate), against data from the existing public blockchain systems such as Ethereum and Bitcoin and against results from the literature. The BlockSim simulation results are within a statistically acceptable margin of the real-life or published results, as discussed in Section \[ss4:BlockSim:validation\]. We also demonstrate the use of BlockSim for a simulation study that considers stale rate, throughput and mining decentralization, for a range of possible blockchain configurations (not all existing in real-life systems). Using BlockSim we can demonstrate that uncle inclusion (such as in Ethereum) is beneficial for mining decentralization.
The structure of the paper is as follows. Section \[s4:BlockSim:model\] discusses the core Base Model of BlockSim including the design objectives behind it. Section \[s4:BlockSim:imp\] presents the implementation of the Base Model. Section \[S4:BlockSim:cases\] presents the application of BlockSim to Bitcoin, Ethereum and other consensus protocols as case studies. Section \[ss4:BlockSim:validation\] discusses the validation of BlockSim against actual systems and studies from the literature. Sections \[ss4:BlockSim:results\] and \[ss4:eval\] show a BlockSim simulation study as well as the evaluation of BlockSim against the design objectives. Section \[s4:related\] discusses the related work. Section \[s4:conclusion\] concludes the paper.
Background {#s:BC}
==========
Blockchain Overview {#ss:BC:overview}
-------------------
A blockchain is a distributed ledger, with an aim to keep track of all transactions that ever occurred in the blockchain network. This ledger is replicated and distributed among the network’s nodes. Such a ledger has two main purposes, to provide an [*immutable*]{} log of all transactions, and to make the transactions [*transparent*]{} (i.e. visible) to anyone inspecting or using the blockchain.
The technologically most intriguing type of blockchain is the public or permissionless blockchain. The main feature of permissionless blockchains is that the nodes that participate in maintaining the ledger do not need to be trusted or even be known to each other. That is, any user can join and participate in the network. Permissionless blockchains contain a cryptocurrency, to reward nodes for investing resources in maintaining the blockchain. The first and most popular permissionless blockchain system is Bitcoin [@nakamoto:2008], which is a digital payment system that enables non-trusting entities to commit financial transactions. Other blockchains (e.g., Ethereum [@wood:2014]) have introduced the idea of smart contracts to support various distributed applications such as e-voting, health applications, etc.
The term blockchain comes from the fact that data about multiple transactions is grouped into blocks. Each block is uniquely identified by its cryptographic hash and each block is attached and linked to the one that came before it. This results in a chain of blocks. Once a block is generated and attached to the blockchain ledger, the transactions in that block cannot be modified by any node, since it would require the node to rewrite all subsequent blocks. This makes blockchain systems immutable and protected against double-spending attacks [@alharby:2017].
Any participating node in a permissionless blockchain can generate a transaction and broadcast it in the network. Each node has a pool to keep pending incoming transactions (transactions that need to be executed). To generate and attach a new block to the blockchain ledger, a subset of the nodes (called miners) select several pending transactions from their pools, execute them and then create a new block containing those transactions. How and when blocks are generated depends on the consensus protocol adopted by the blockchain system (see Section \[ss:BC:consensus\]).
Once a miner has successfully created a block, it will then broadcast it to other nodes in the network. Upon receiving the block, each node validates the block’s correctness and appends it to its ledger. If the majority of the nodes attach the block to their ledger and start building on top of it, the block will be confirmed and considered as part of the blockchain ledger. The miner of that block can then collect a reward for the block as well as the fees associated with its transactions as compensation for their efforts.
Blockchain in Layers {#ss:BC:layers}
--------------------
Blockchain systems can naturally be divided in three layers, the network, consensus and incentives layer, as depicted in Figure \[Layers\]. We will utilize these layers to structure the BlockSim simulator and therefore here provide a system explanation in layers as well. The network layer captures the network’s nodes and the underlying network protocol to distribute information between nodes. The consensus layer captures the algorithms and rules adopted to reach an agreement about the current state of the blockchain ledger. The incentives layer captures the economic mechanisms adopted by a blockchain to issue and distribute rewards among the participating nodes.
![Blockchain System Layers[]{data-label="Layers"}](Figures/layers.pdf)
### Network Layer {#ss:BC:network}
The network layer in blockchain systems contains the nodes in the network, their geographical and relative locations and the connectivity among them. It defines which information is to be propagated as well as the mechanism to propagate such information.
The main constitute in the network layer is a [*node*]{}. A node can be an ordinary user who wants to create and submit a transaction to be executed and included in the ledger or a special node, known as [*miner*]{}, who maintains and expands the ledger by appending new blocks. A node has a unique identifier and maintains its balance, a local copy of the blockchain ledger and, if the node is a miner, an individual transactions pool. The transactions pool keeps the pending transactions received from other nodes in the network.
Nodes communicate the following information to each other. If a node generates a new transaction, it cryptographically signs it and propagates it to its peers to have it confirmed and recorded in the blockchain ledger. In case the node is a miner, every time it generates a block, it notifies its peers so they can validate it and append it to their copies of the ledger.
As information propagation mechanism for blockchains several protocols have been proposed, including relay networks and advertisement-based protocols [@gervais:2016]. In the advertisement-based protocol used in most blockchains [@gervais:2016], the node sends a notification to its peers about the new data (e.g., a transaction). If the recipient node responds by requesting the data, the node will send it. Otherwise, the node will not send it as the recipient node has already had the data.
### Consensus Layer {#ss:BC:consensus}
The consensus layer in blockchain systems defines the algorithms and rules for reaching an agreement about the blockchain’s state among the network’s nodes. Such rules specify which node is eligible for generating and appending the next block to the blockchain ledger, how often blocks are generated as well as how to resolve potential conflicts that may occur when nodes have multiple, differing copies of the ledger.
There are several consensus algorithms such as Proof of Work (PoW) and Proof of Stake (PoS) that have been proposed for blockchain systems. In PoW, nodes (i.e., miners) invest their computing power to maintain the ledger by attaching new blocks, while in PoS, nodes invest their stake or money. Regardless of what is required to be invested by the nodes, the intuition behind such algorithms is to introduce a cost for maintaining the ledger. The cost introduced should be more than enough to deter nodes from behaving maliciously [@wang2019survey]. At the same time, nodes are only rewarded for their efforts if they follow the rules and maintain the ledger honestly (see Incentives Layer).
To illustrate the consensus layer, we discuss the PoW algorithm here as it is the most common algorithm for permissionless blockchains, used by Bitcoin and Ethereum. In PoW, the computing power invested by a miner determines how frequent that miner will generate and append blocks to the blockchain ledger. To generate a block, the miner has to repeatedly try nonces (random numbers) until the hash of the nonce combined with the block information will be within a certain threshold (referred to as the block difficulty). The only way to find the nonce is by trial-and-error, and thus, the more hash power invested by a miner, the more likely that miner will find the nonce. This process is a competitive task since all miners in the network are competing against each other to find the desirable hash value of the next block. Note that the block difficulty can be dynamically adjusted to control how often blocks are generated.
Due to the delay incurred by propagating blocks between nodes in the network (see Network Layer), other nodes might generate the next block before hearing of another competitive block that has recently announced. This leads to conflicts, known as forks, which occur when nodes have multiple, differing views of the ledger. The task of the consensus layer in blockchain systems is to resolve such conflicts. Different consensus algorithms use different rules to select which blockchain (fork) should be accepted as the global chain. For example, the PoW algorithm used by Bitcoin and Ethereum resolves the conflicts by adopting the longest chain. Other proposals such as GHOST [@sompolinsky:2015] select the fork with the heaviest work.
### Incentives Layer {#ss:BC:incentives}
The incentives layer utilizes the blockchain’s cryptocurrency to establish an incentive structure, distributing rewards among the participating miners who maintain the blockchain’s ledger. The incentive model is essential to maintain any permissionless blockchain system. Incentives should compensate miners fairly for their work and motivate them to behave honestly [@aldweesh:2018] [@alharby18survey]. The incentives also protect the blockchain system from various attacks (e.g., DDoS attacks in Ethereum [@EthAttacks]) and against malicious behaviors of the nodes (e.g., selfish mining strategies [@eyal:2018]).
In most blockchain systems rewards are associated with generating blocks and completing transactions, called block reward and transaction fees, respectively. Depending on the chain, there are subtle differences in what is rewarded, e.g., Ethereum issues a reward for stale (or uncle) blocks, even if they do not make it into the blockchain when conflicts are resolved. When a miner receives a reward (e.g., through appending a new block to the ledger), its balance will increase accordingly. The block reward, in all known blockchains, is set to a fixed amount, while the transaction fee is calculated as a variable amount of cryptocurrencies, depending on effort as well as the prize a transaction submitter is willing to pay.
Modeling and Simulation {#ss:BC:modeling}
-----------------------
A model is an abstract representation of a real system, either existing or in design. The model usually comprises mathematical expressions as well as structural and logical relationships to describe the dynamics of the system [@anderson:2015]. Simulation is a quantitative method, which ‘executes’ the model to mimic the behavior of the system [@anderson:2015]. Simulation can be used to predict and describe how different configurations and scenarios impact the behavior of the system. Thus, simulation can be used to answer ‘what-if’ questions and to experiment with new designs and policies without a need to interrupt a functioning system [@jerry:1984].
Simulation can be classified into two categories, namely, discrete-event simulation and continuous-event simulation [@haverkort:1998]. Human-made systems such as digital computer and information systems are most suitable represented as discrete-event simulation, as the systems change state at discrete moments in time [@fishman]. BlockSim utilizes the discrete-event simulation approach to design and implement the simulator.
There are two approaches to develop simulation tools, namely, general-purpose programming languages (e.g., C++, Java or Python) and special-purpose simulation languages (e.g., Arena and GPSS) [@leemis2006discrete]. The former is more flexible and familiar, while the latter provides several built-in features (e.g., statistics, event scheduler and animation) that reduce the time required to build models. As stated in [@leemis2006discrete], there is a debate and conflict about which method is preferable. Also worth noting are simulation frameworks that enable developing simulation models using general-purpose languages, for instance, OMNeT++ and SimPy for developing models in C++ and Python.
To develop and implement BlockSim, we opt for Python as a general-purpose language. We do not use its simulation framework *SimPy* because it follows a process-oriented paradigm, which differs from the approach we take in BlockSim. However, it will be useful to consider integrating features provided by SimPy with our simulator in a future version.
BlockSim Base Model {#s4:BlockSim:model}
===================
In this section, we introduce the [*Base Model*]{} underlying BlockSim, which is designed to model any kind of blockchain system, with application specific extensions as needed. We first define the design principles and goals for BlockSim: generality, extensibility and simplicity. Then, we discuss the design layer by layer: Network Layer, Consensus Layer and Incentives Layer. Within each layer we identify the key functional units (entities) and the actions or activities it executes.
Design Principles {#ss4:design:principles}
-----------------
We design a Base Model to fulfill the main design goals for BlockSim, which are:
- **Generality:** we want to be able to use BlockSim for a large set of blockchain systems, configurations and design questions.
- **Extensibility:** BlockSim should be easily manipulated by a designer or analyst to study different aspects of blockchain systems.
- **Simplicity:** the above two objectives should be met while making BlockSim easy to use, both for simulation studies and for extending it.
The art of designing a tool such as BlockSim is to find a useful trade-off between generality and extensibility on the one hand, and simplicity to achieve these two objectives on the other hand. The Base Model is critical in achieving this goal, aiming to find the optimal trade-off among the above three objectives for the domain of blockchain systems.
The Base Model identifies the key building blocks (e.g., blocks, transactions, nodes and incentives) common across all blockchains BlockSim is meant for, see Figure \[fig:entities\]. The Base Model dictates how general the model class is that is supported by BlockSim, and particularly how easy it is to build new models. The Base Model will be translated in software modules and therefore also determines if BlockSim can be extended easily, for instance, to provide more detailed models of certain processes that take place in blockchains.
Network Layer {#ss4:design:network}
-------------
This layer defines two entities *Node* and the underlying *Broadcast protocol*, as depicted in Figure \[fig:entities\]. The *Node* entity is responsible for updating the system state variables (e.g., the blockchain ledger and the transactions pool). The *Broadcast protocol* specifies how information entities (e.g., *Blocks* and *Transactions*) are propagated in the network.
Both *Blockchain ledger* and *Transactions pool* entities are part of the *Node* entity (see Figure \[fig:entities\]). That is, every node maintains and continuously updates these entities. We model nodes as objects that have different attributes such as unique ID, balance, local ledger and transactions pool. The transactions pool and the local ledger are modeled as array lists that can be extended when new transactions and blocks are received. These attributes are common across the different implementation of blockchains. It could, however, be possible to extend this by including more additional attributes, as we will show in Section \[ss4:btc\].
The propagation of information entities depends on the *Broadcast protocol* entity, which can be modeled in detail by accounting for the network configurations, the geographical distribution of the nodes and the connectivity among the nodes, or it can be modeled in an abstraction level by only considering a time delay for propagating information among the nodes. The reason for abstracting the broadcast protocol is to make our simulator as simple as possible by hiding unnecessary details. This will alleviate the user of the simulator from configuring many parameters related to the network configurations such as the broadcast protocol, the geographical distribution of the nodes and the number of connections per node. Having the propagation delay as the only configurable parameter will improve both the efficiency and the usability aspects of the simulator.
Consensus Layer {#ss4:design:consensus}
---------------
This layer aims at establishing the rules that nodes can follow to reach an agreement about the blockchain’s state. This layer includes four entities, namely, *Transaction*, *Block*, *Transactions pool* and *Blockchain ledger*, as depicted in Figure \[fig:entities\].
The *Blockchain ledger* entity depends on the *Block* entity, and the *Block* entity depends on the *Transaction* entity. That is, the blockchain ledger is composed of blocks and blocks are composed of transactions. The *Transactions pool* depends on the *Transaction* entity, as every transaction created is fed into the transactions pool. The *Node* entity maintains these four entities.
Within the consensus layer, there are several activities or actions to be executed by the entities. The creation of blocks and transactions is an example of such activities. The flow of these activities is depicted in Figure \[fig:BlockSim\]. These activities run continuously, transactions and blocks, for instance, always keep arriving in the network.
### Transaction {#ss4:con:transaction}
are one of the building blocks (entities) common across all blockchain systems. It plays a significant role in updating the blockchain’s state. The arrival of a new transaction in the network results in updating the transactions pool by inserting that transaction.
We model transactions in two different ways, namely, full and light. The full technique helps to track each transaction in the system (e.g., when a transaction has been created and included in a valid block). This technique models transactions as in any blockchain system, and it is useful if one is interested in, for instance, studying the latency of individual transactions in blockchain systems. However, this type of modeling consumes an enormous amount of computing resources and time during the simulation since each transaction has to be tracked. On the other hand, the light technique does not track each transaction. It is useful when studying the throughput of blockchain systems without caring about the confirmation time of transactions within the system.
In both techniques, we model transactions as objects that have several attributes or fields such as transaction ID, size, fee, timestamp, contents as well as the submitter and the recipient of the transaction. These attributes are almost common across all blockchains, and that some systems have more additional attributes (e.g., Ethereum has also gas-related attributes such as Gas Limit).
**Full modeling Technique**: In this technique as we discussed in Section \[ss4:design:network\], we model an individual transactions pool for each node by assigning an array list for each node as a way to abstract the pool. Each transaction created by a node is propagated to all other nodes in the network. Upon receiving the transaction, the recipient node appends it to their pool. Thus, we model transactions in three different activities labelled from 1 to 3, as depicted in Figure \[fig:BlockSim\].
- **Creating transactions:** This involves generating transactions by the participating nodes. The number of transactions to be created per unit of time can be controlled and configured.
- **Propagating transactions:** This requires the creator of the transaction to propagate it to other participating nodes. This is to notify other nodes about the newly created transactions.
- **Appending transactions:** This requires the recipient of the transaction to append it to their transactions pool.
**Light modeling Technique**: In this technique, we only model a single transactions pool to be shared among all nodes in the network. The intention behind this technique is to provide an alternative and simplified way to model transactions by omitting the propagation process as well as the needs for nodes to update their pools continuously (see Section \[ss4:con:block\]). Thus, the light technique is more efficient and faster during the simulation. However, this technique cannot be used to draw conclusions about the latency of transactions as transactions are not tracked. Nevertheless, it is useful to get indicators about the throughput in blockchain systems.
In this technique, we create a set of transactions (N) and then append it to the shared pool before the mining process, so miners can access the pool to select several transactions to include in their forthcoming block. Hence, N should be more than enough for a block, usually enough for two blocks. Once a miner has successfully generated a block, the pool is reset and then filled up with a fresh set of transactions to be included in the next block.
Both techniques could be implemented and then the user would be given a choice to select which method to adopt based on their own needs. For instance, if one is only interested in throughput, there is no need for choosing the full technique since it makes the simulator runs for a very long time.
### Block {#ss4:con:block}
are another essential building block (entity) of any blockchain system. Blocks consist of transactions. The arrival of a new block results in an update in the transactions pool and blockchain ledger. The pool is updated by removing all transactions included in the block, while the ledger is updated by appending the newly created block.
We model blocks as objects that have several attributes, namely, depth, block ID, previous block ID, timestamp, size, miner ID and transactions. The block ID is a unique identifier for the block. The block depth indicates the index of the block in the node’s blockchain. The miner ID refers to the node that created the block. Each block can accept a list of transactions as its content. These attributes are common across blockchains.
We model blocks in the consensus layer as *Block Generation* and *Block Reception*, see Figure \[fig:BlockSim\]. Block generation specifies when blocks are generated as well as which node is eligible for appending the next blocks. It covers all the common actions required by a miner to create and attach a block to the blockchain ledger. The actions embrace executing the block’s transactions, constructing and appending the block to the local blockchain and propagating the block to other nodes in the network. Block reception specifies how the network’s nodes update their blockchain ledgers upon receiving new blocks. It covers the common activities taken by a node when receiving a newly generated block. Upon receiving a valid block, the recipient node will perform three actions, which are updating local blockchain if necessary, appending the block to the local blockchain and updating the transactions pool.
The consensus algorithm is responsible for selecting a miner to build the next block. The methodology used to choose a miner varies among blockchains, depending on the adopted consensus protocol. In PoW, for instance, miners are selected based on solving a mathematical task. Once a miner is chosen to construct and append a new block to the ledger, the miner would undertake the following actions. Hence, these actions are common across all blockchain systems, and that some specific systems may include other activities (e.g., including uncle blocks in a future block as in Ethereum)
- **Executing and adding transactions to the block:** This requires the miner to select several pending transactions to be executed and included in the next block. Often, miners first sort those pending transactions based on their associated fees. Then, miners select the best transaction according to their ranking criteria, execute it if and only if there is a space in the block. The transaction will then be recorded in the block. After that, miners will select the next transaction and continue until the block is full or there is no pending transaction.
- **Constructing and appending the block to the local blockchain:** After preparing the block content (e.g., transactions), the miner would construct the block after which the block will be appended to the miner’s local blockchain.
- **Propagating the block to other nodes:** This is to propagate the block to other nodes in the network. This is to notify the network’s nodes about the newly generated block.
Once a node has received a new block, it will check its validity. The block is considered valid if it was constructed correctly and all embedded transactions were correctly executed. Beside the block validity, the block must point to the last block in the ledger (the block’s depth should be higher than that of the last block). We only model the block depth, and thus, we abstract the validity of the block. If the depth of the received block is not higher than that of the last block, the block will be discarded. Otherwise, the node will perform the following actions.
- **Updating local blockchain:** This requires the recipient node to update its local blockchain, where necessary, before appending the newly received block. This is because sometimes the received block is built on different preceding blocks (a different chain branch) compared to the ones the recipient node has or because it is built on missing blocks. Therefore, the node has to update all the preceding blocks (and fetch all missing blocks if any) according to the ones the received block is following.
- **Appending the block to local blockchain:** This is to append the received block to the local copy of the blockchain.
- **Updating transactions pool:** This requires the recipient node to update its transactions pool, where necessary, upon appending the newly received block. This is to remove all the transactions that have already been executed in the received block from the node’s pool.
### Transactions Pool and Blockchain Ledger {#ss4:con:pool:ledger}
are also important building blocks (entities) since they represent the state of blockchain systems. The transactions pool is updated upon the arrival of a new transaction or block, while the blockchain ledger is only updated once a block has arrived, as discussed in Sections \[ss4:con:transaction\] and \[ss4:con:block\]. Nodes are responsible for updating both the pool and the ledger, as every node in the blockchain network maintains a local copy of them (see Section \[ss4:design:network\]).
**The rule of updating the ledger in the case of forks:** Nodes at some point in time may have different views of the blockchain ledger due to the network’s propagation delay. A significant role of the consensus layer is to define the rules that can be used by the nodes to resolve the forks. For instance, Bitcoin and Ethereum use the longest-chain rule to resolve the forks. That is, nodes update their ledgers every time they receive a block that follows a chain that is longer than their local chains. By doing so, nodes will have the same view of the blockchain ledger. Other systems, however, use different rules (e.g., GHOST [@sompolinsky:2015]).
Incentives Layer {#ss4:design:incentives}
----------------
The incentives layer is responsible for designing the underlying incentive model by defining the rewarded elements (e.g., blocks and transactions) as well as distributing the rewards among the participating miners. This layer has the *reward* entity, which depends on the *Block* entity (see Figure \[fig:entities\]). That is, the rewards are only given to the miners upon appending new blocks to the ledger. The calculation and the distribution of such rewards are considered as actions.
We model the basic incentive model used by most blockchain systems such as Bitcoin. Our model provides a reward for generating a valid block (block reward) and a reward for all transactions included in a block (transaction fee). The block reward is modeled as a fixed amount of cryptocurrency that can be configured and changed by the end-user. The transaction fee is calculated as the multiplication of its size and its prize, where the prize is the amount of money the submitter of the transaction is willing to pay per unit of size. The size and the prize for transactions can also be configured as fixed or variable (random) values. However, it is possible to extend the current model to include different rewards (e.g., rewards for uncle blocks) or change the way how the fee for transactions is calculated. We model the distribution of rewards by increasing the balance of each miner after having a valid block attached to the ledger.
BlockSim Implementation {#s4:BlockSim:imp}
=======================
We present the implementation of the BlockSim simulator using Python 3.6.4[^1]. The main modules are given in Figure \[fig:modules\]. The Simulator Module implements the core engine of the simulator, in particular the event scheduler, which we explain in Section \[ss4:imp:event\]. The main topic of discussion in that section is the granularity at which events are handled, since it heavily impacts the performance of the simulator. This simulation engine module is complemented by the Configuration Module, to be described in Section \[ss4:imp:conf\], which provides the user with ways to configure the simulation model and experiments. Section \[ss4:imp:model\] explains the implementation of the Base Model, subdivided according to the main layers: Network Module, Consensus Module and Incentives Module.
BlockSim Simulation Engine and Event Scheduler {#ss4:imp:event}
----------------------------------------------
As depicted in Figure \[fig:modules\], the main Simulation Module contains four classes, which are Event, Scheduler, Statistics and Main. We start with explaining our design choices for the event scheduling.
We provide event scheduling at two abstraction levels, the first one considers blocks as the event ‘unit’, the second considers transactions as the event ‘unit’. We explain the block-level events. The class *Event* defines the structure of events in our simulator. In the case of a block-level event it has four attributes: *type*, *nodeID*, *time* and *block*. The attribute *type* indicates how to handle the event, in particular whether the event at hand is to *create a new block* or to *receive an existing block*. The *nodeID* and *time* attributes specify the node that handles the event and the time at which the event takes place. The *block* attribute contains the necessary information for the block to be handled.
*Scheduler class* is responsible for scheduling future events and record them in the *Queue*. *Queue* is an array list that maintains all future events, and it is continuously updated during the simulation by either inserting new events or removing existing ones. At the block-level, for instance, once a block is created through a *block creation* event, the *Scheduler* class schedules *block reception* events for other nodes to receive the block. Also, it schedules a new *block creation* event by selecting a miner to propose and generate a new block on top of the last one.
The function of the *Main* and *Statistics* classes is as one would expect. *Main* runs the simulator. It prepares the setup and then triggers the *Scheduler* class to schedule some initial events. The setup includes the creation of transactions as well as the creation of the first (genesis) block, an empty block that will be attached to the local blockchain for all the nodes in the network. Then, it keeps going through all the events and executes them one by one until the *Queue* is empty or the pre-specified simulation time is reached. *Statistics* maintains the results and calculates the statistics of the final output of the simulation, including block statistics (number of blocks included in the ledger and percentage of discarded blocks), throughput and mining profits.
Base Model Modules {#ss4:imp:model}
------------------
We discuss the implementation of simulation classes that represent the Base Model of Section \[s4:BlockSim:model\] using the same three layers as before.
**Network Module:** We implement the network module in two different classes, namely, Node and Network. *Node class* defines the structure of nodes in our simulator. We implement each node as an object in which each node is given a unique ID and a balance. For each node, we assign two array lists to model the local blockchain and the transactions pool. It is worth noting that each node maintains a transactions’ pool only if the full transaction technique is applied. Otherwise, a common pool will be shared by all the nodes. *Network class* implements the network latency for propagating both blocks and transactions between the nodes. Currently, we implement the latency as a time delay that can be configured by the user of the simulator in the configuration module. Hence, it could be possible to extend this class to implement a particular broadcast protocol.
**Consensus Module:** We implement the consensus module in different classes, namely, Transaction, Block and Consensus. *Transaction class* defines the structure of transactions in our simulator. We implement each transaction as an object that has seven attributes, namely, ID, timestamp, submitter ID, recipient ID, value, size and fee. The end-user can configure the size and fee of transactions in the configuration module. This class also implements both full and light techniques for modeling transactions, as we discussed in Section \[ss4:con:transaction\]. *Block class* defines the structure of blocks in our simulator. We implement each block as an object that has seven attributes, namely, depth, ID, previous ID, timestamp, size, miner ID and transactions. This class also implements the processes required by the nodes to generate and receive blocks, as discussed in Section \[ss4:con:block\]. *Consensus class* implements the consensus algorithm as well as the fork resolution rule. It also implements the process of selecting leaders, aka miners, to generate and append new blocks to the ledger. This class is structured to be easy to implement any consensus protocol of interest. For instance, to implement PoW algorithm with the longest-chain rule to resolve potential forks as the case in Bitcoin and Ethereum.
**Incentives Module:** This module is responsible for setting the rewarded elements as well as calculating the rewards. Also, it distributes the rewards among the participating nodes by increasing the balance of each node after calculating the rewards. It is, however, possible to extend this module by adding more rewarded elements or changing the way the awards are calculated if required. To make it easier for the end-user, the rewards (e.g., block rewards) can be configured and changed in the configuration module.
Configuration Module {#ss4:imp:conf}
--------------------
**Type** **Parameter** **Description**
-------------- --------------- ----------------------------------------------
Blocks B~interval~ Average time to generate a block in seconds
B~size~ Block size in Megabyte (MB)
B~delay~ Propagation delay of blocks in seconds
B~reward~ Block generation reward
Transactions hasTrans Enable/Disabled transactions
T~technique~ Technique for modeling transactions
T~n~ Rate at which transactions can be created
T~delay~ Propagation delay of transactions in seconds
T~fee~ Transaction fee
T~size~ Transaction size in MB
Nodes N~n~ Total number of nodes in the network
Simulation Sim~time~ Length of the simulation time
Runs Number of simulation runs
: Input parameters for the simulator.[]{data-label="t:inputs"}
This module serves as the main user interface, in which users can select from the available models as well as configuring various parameters related to the participating nodes, blocks, transactions, consensus, incentives and the simulation setups. Table \[t:inputs\] summarizes the input parameters to be configured before running the simulator. We can, for instance, configure the number of nodes, the block interval time, the volume of transactions to be created per second and other parameters. Besides, our simulator allows disabling transactions if they are not of interest. This can be done by only setting the parameter *hasTrans* to be “False", without modifying the code of the simulator. Furthermore, it allows selecting a suitable technique (either full or light) for modeling transactions. If we extend the simulator by, for example, including new consensus protocols, this would be reflected in this module to allow the user of the simulator to choose the desired protocol.
BlockSim Case Studies {#S4:BlockSim:cases}
=====================
BlockSim is designed to be used for any type of blockchain, and to demonstrate this we apply the Base Model of BlockSim to simulate Bitcoin as well as Ethereum. We also discuss how to extend the BlockSim implementation of the Base Model to support any consensus algorithm of interest.
Bitcoin in BlockSim {#ss4:btc}
-------------------
To simulate Bitcoin we introduce the following modifications and extensions to the core implementation of BlockSim discussed in Section \[s4:BlockSim:imp\].
**Network Layer:** For Bitcoin we abstract the underlying broadcast protocol by modeling the propagation of transactions and blocks as a time delay, as indicated in Section \[ss4:design:network\]. To parameterize the model one can use DSN Bitcoin Monitoring to obtain the propagation delay of information. The Node module is extended with an attribute for a node’s hash power, which we add to the configuration module for the user to set as an input parameter. To distinguish between regular nodes and miners, we can assign zero as the hash power for regular nodes to indicate that the node cannot build blocks (only create and propagate transactions).
**Consensus Layer:** Bitcoin uses PoW with the longest-chain rule to resolve the forks. As discussed in Section \[ss:BC:consensus\], in PoW miners compete against each other to be allowed to create the next block. They repeatedly draw a random number, combine it with info from the new block and generate a hash. If the hash fulfills some property, the block can be added to the blockchain and forwarded to other nodes. That is, miners execute what amount to a Bernouilli trial, and therefore the time until the trial succeed is exponentially distributed. In the configuration module, one can set the block difficulty through the *B~interval~* parameter, which is the time interval (in seconds) between two consecutive blocks. If multiple chains have the same depth, Bitcoin uses the longest chain to reach a global view of the blockchain ledger by resolving the forks.
**Incentives Layer:** The incentives in Bitcoin for generating blocks and executing transactions is the same as that of the Base Model. In our main BlockSim implementation, all rewards will be distributed to miners at the end of each simulation run. If needed, the Incentives module can be modified to distribute rewards in run-time. The miner of a block that is finalized and is part of the longest chain receives the block reward and the fees for all transactions included in that block. The rewards can be set in the configuration module.
Ethereum in BlockSim {#ss4:eth}
--------------------
Ethereum is very similar to Bitcoin but introduces a few additional elements associated with the handling of uncle blocks as well as attributes required for incentives associated with smart contracts.
**Network and Consensus Layers:** Ethereum allows attaching uncle blocks to a valid block and rewards miners for this. Therefore, we extend the Bitcoin Node module with an unclechain attribute. The unclechain for a node is modeled as an array list storing all chains with uncle blocks that occur during the simulation run. Ethereum allows miners to include a maximum of 2 uncle blocks within the last seven block generations (e.g., an uncle block with a depth 10 can be referenced in a block with a depth less than or equal to 17). We include this logic in the configuration module and allow configuring the maximum number of uncle blocks per block, the number of generations in which an uncle block can be included as well as disabling uncle inclusion mechanism if it is not of interest.
Similarly, we extend the Node module when receiving a block. If the block has a smaller depth or index, the block is appended to the recipient’s unclechain as an uncle block to be referenced in a future block. Also, when receiving and appending a valid block to the local blockchain ledger, the miner updates its local unclechain, where necessary, by removing all the uncle blocks that have already been included in the received block.
**Incentives Layer:** The incentive model of Ethereum, similar to that of Bitcoin, includes block reward and transactions fee. Yet, Ethereum uses the Gas mechanism to calculate the fee for transactions with smart contracts. To determine the fee for transactions and blocks, we therefore require some additional attributes related to the gas model. For transactions, we add Gas Limit, Used Gas and Gas Price attributes. For blocks, we include the attributes of Gas Limit and Used Gas. We refer to the literature, e.g., [@wood2019; @alharby:2017] for details, but in short, Used Gas multiplied by the Gas Price corresponds to the fee the miner receives, where Used Gas depends on the computational requirements of the smart contract [@aldweesh:2018], but never exceeds Gas Limit.
Ethereum also introduces rewards for uncle blocks. The uncle reward is distributed between the miner who generated the uncle and the miner who included it in his block, as follows [@wood2019]. The miner who generated the uncle gets a variable reward depending on when the uncle has been referenced in a main block. The sooner the uncle is referenced in a block, the higher the uncle reward (R~uncle~): $$% \begin{aligned}
R\textsubscript{uncle}= \big( D\textsubscript{uncle} + (G\textsubscript{uncle}+1) - D\textsubscript{block}) * \frac{ R\textsubscript{block}} {G\textsubscript{uncle}+1}
% \end{aligned}$$ where D~uncle~ is the depth of the uncle, G~uncle~ is the number of generations in which the uncle can be included, D~block~ is the depth of the block and R~block~ is the block reward. The miner who included the uncle in his block will get a fixed reward, which is calculated as $\frac{1}{32}$ \* R~block~. All this is implemented in the incentives module, but the amount of rewards can be set in the configuration module, if required.
Different Consensus Protocols in BlockSim
-----------------------------------------
Thus far we have mainly considered PoW as consensus protocol, but there are many other, including Proof of Stake (PoS), Proof of Authority or message-based consensus algorithms such as Practical Byzantine Fault Tolerance and its many variants [@angelis:2018].
A significant difference between these protocols and PoW is that in PoW miners are not directly selected by the consensus protocol, but instead, miners continuously invest their computing power to create the subsequent blocks. In PoS, for instance, miners would be selected by the protocol based on the amount of stake or cryptocurrencies they hold. The more cryptocurrencies a miner deposited in the system, the more chance they would be selected to generate the next block. Other protocols select miners in a round-robin manner such as Tendermint [@kwon2014tendermint] or based on different metrics [@angelis:2018].
To support approaches such as PoS, we modify the consensus class by changing how miners are being selected to generate the next blocks. Other consensus elements (e.g., transactions, blocks and fork resolution) and modules (simulation, network and incentives) remain unchanged. In general, as long as the output metrics can be truthfully simulated with events scheduled at the granularity of blocks, BlockSim can be extended in a natural matter. The time consumed by the consensus algorithm would then be represented by a delay. However, if one wants to analyse the impact of specific message sequences on the performance of PBFT style consensus protocols, BlockSim is a less obvious candidate. For efficient (i.e., fast) simulation, one would study such consensus protocols through simulation tools that operate at message-level and not mix different levels of abstractions and time granularity.
BlockSim Validation {#ss4:BlockSim:validation}
===================
A nice feature of the blockchain design is that it offers invariants (such as the block creation interval) and plenty of publicly available data to validate the results of any simulator. First we compare BlockSim with existing blockchain systems (Section \[ss4:real:validation\]), then we compare with various peer-reviewed studies (Section \[ss4:studies:validation\]).
Comparison with Measurements {#ss4:real:validation}
----------------------------
We compare the results from BlockSim with the most popular public blockchains, Bitcoin and Ethereum. These provide certain ‘invariants’ that we know to be true, such as the frequency of generating blocks and the proportionality between the miner’s hashing share and the probability to win the Proof of Work competition. Bitcoin and Ethereum also provide ample public data to validate our simulator.
Parameters Bitcoin Ethereum
------------- --------- --------------
B~interval~ 596s 12.42s
B~delay~ 0.42s 2.3s
B~size~ 0.83MB 7,997,148Gas
T~size~ 546Byte Distribution
: Data gathered from Bitcoin and Ethereum, serves as input to the simulation runs used as validation.[]{data-label="t:realData"}
[**Validation of block and transaction metrics. **]{} We use the following metrics for validation: number of blocks created, number of uncle or stale blocks (blocks that will not be part of the final chain), and the number of transactions completed per time unit. The results obtained from our simulator and that from the actual systems are reported in Table \[t:validation\]. We report both the average and the 95% confidence interval values, for a run of the simulation that corresponds to a full month of real time. From Table \[t:validation\], we see that our simulator’s confidence interval contains the result from the measurements. However, our simulator shows a slightly higher throughput for Ethereum compared to the real data observed. We believe that this is either due to the small sample of transactions retrieved or the fitted frequency distribution.
Bitcoin Measured Simulated
-------------------- ------------------- -------------------
B~included~ $146\pm4$ $143\pm5$
Stale (uncle) rate $0.025\pm0.051\%$ $0.049\pm0.069\%$
Throughput $2.69\pm0.09$ $2.66 \pm 0.09$
Ethereum Measured Simulated
B~included~ $6083\pm27$ $6079\pm25$
Stale (uncle) rate $12.56\pm0.43\%$ $12.55\pm 0.14\%$
Throughput $5.99\pm0.18$ $6.96\pm0.03$
: Validation of the simulator results by comparison with measurements from Bitcoin and Ethereum. B~included~ is the number of blocks included in the main blockchain per day, the stale (or uncle) rate per day are blocks not in the main chain, and throughput is the number of transactions processed per second. []{data-label="t:validation"}
To obtain the above results, Table \[t:realData\] shows the data gathered from both Bitcoin and Ethereum used as input to the validation runs. That is, we use the values from Table \[t:realData\] for the relevant input parameters given in Table \[t:inputs\]. We gather the Bitcoin’s data from blockchain.info [^2], while the Ethereum’s data comes from etherscan.io [^3]. We collect one month of data for each system as of October 2018. From these sources, we were able to directly collect all the necessary data, apart from the block propagation delay and the transactions’ size in Ethereum. However, we obtain the block delay using DSN Bitcoin Monitoring[^4] and ETHstats[^5]. To obtain the size of transactions in Ethereum, we implement a python script that makes use of etherscan.io APIs to retrieve transactions information. We retrieve the data for the latest 5,000 transactions and then fit a frequency distribution for transactions’ size to be used as input in our simulator. For the sake of this experiment, we fit a straightforward frequency distribution with the limited collected data.
[**Validation of PoW. **]{} An invariant we can use for validation is the share of blocks each miner generates since it is known that share is equal to the miner’s share of the overall hashing power. For instance, if a miner controls 40% of the network’s hash power, it should generate 40% of the total blocks. To validate PoW, we collect the estimated hash power as well as the fraction of blocks contributed by Bitcoin miners and miner pools from blockchain.info and input this into our simulator. That is, the simulation is with miners that have the same share of the hashing power as various existing Bitcoin miners.
Figure \[fig:pow\] shows the results. We simulate four days of the Bitcoin network, a total of 1000 times and obtain the average fraction of blocks generated by each miner. The *x*-axis of Figure \[fig:pow\] shows the name of the miners and the *y*-axis shows the fraction of blocks contributed by the miners for both the real Bitcoin network (the green bars) and the simulation results (the gray bars). From Figure \[fig:pow\] we see that the simulation results are very close to that of the real Bitcoin network.
Comparison with Peer-reviewed Studies {#ss4:studies:validation}
-------------------------------------
-------------------------- ------------- ---------- ---------- ------------------------
B~interval~ B~delay~ Measured Simulated
Bitcoin [@gervais:2016] 600s 14.7s 1.51% $1.69\% \pm 0.08\% $
Bitcoin [@decker:2013] 600s 12.6s 1.68% $ 1.73\% \pm 0.09\% $
Litecoin [@gervais:2016] 150s 4.18s 1.82% $ 1.88\% \pm 0.11\% $
Dogecoin [@gervais:2016] 60s 2.08s 2.15% $ 2.38\% \pm 0.08\% $
-------------------------- ------------- ---------- ---------- ------------------------
: A comparison between BlockSim and previous studies in terms of the stale rate observed.[]{data-label="t:staleTable"}
We also compare the simulator results for the stale rate with that of previous peer-reviewed studies. Decker et al. [@decker:2013] run an experiment on the Bitcoin blockchain by listening to 10,000 blocks. They found the average block propagation delay is 12.6 seconds and the stale rate is 1.69%. Gervais et al. [@gervais:2016] run some simulation experiments using the configurations of different blockchain systems such as Bitcoin, Litecoin and Dogecoin. They found that their simulation results matched that of the actual systems. To validate our simulator against these studies, we use the same configurations of the block interval (B~interval~) and block propagation delay (B~delay~) as reported in these studies. We simulate each configuration for a total of 10,000 blocks and report the average results obtained from 10 independent runs, see Table \[t:staleTable\]. From Table \[t:staleTable\], we see that the stale rates obtained from our simulator are close to the ones reported in previous studies, with a difference of less than 10%.
BlockSim Simulation Results {#ss4:BlockSim:results}
===========================
To show the applicability of our simulator, we conduct a simulation experiment to investigate the impact of different consensus and network parameters on the security, performance and mining ecosystem of blockchain systems. We use very similar metrics as in the validation, but for a wider range of parameter values. The main discussion in this section is about how the stale block rate impacts mining decentralization and how Ethereum’s approach to reward uncle blocks improves mining decentralization.
More precisely, we study the impact of different combinations of block interval and block propagation delay on the stale rate, throughput and mining decentralization. Stale rate is a security indicator of a blockchain system, and the lower the rate, the better for the security of the system [@gervais:2016]. Throughput represents the number of transactions that can be processed per second, thus directly indicating how well the system performs. Mining decentralization indicates that the fraction of blocks a miner includes in the main ledger is proportional to the hash power of that miner. In other words, mining decentralization means each miner gets a fair reward compared to its hash power.
Table \[t:results\] shows the results for 25 different combinations of different block interval B~interval~ and block delay B~delay~. For ease of presentation, we consider only five miners (M1, M2, ..., M5) with hash powers ranging from 5% to 40%. The hash power for a miner is a configurable parameter (see Section \[ss4:btc\]). For all configurations, we set the block size to be 1MB and the average transaction size to be 546 bytes (as in the Bitcoin network). We simulate each configuration for a total of 10,000 blocks and report the average results from 10 independent runs. The confidence intervals are not reported here, but are all within 10% of the average values.
**Stale rate:** From the stale rate results reported in Table \[t:results\], we observe the following. First, reducing the block interval, i.e., the time between successive blocks being created, leads to higher stale rates, especially when the block interval is already small. For instance, reducing the block interval from 12 to 1 second in the case of 0.5 second block delay will result in an increase of the stale rate by about sevenfold. When the block interval is small, other nodes could manage to find the next block before hearing of other competitive blocks due to the network latency, leading to conflicts. Also, increasing the block propagation delay leads to higher stale rates. For instance, the stale rate increases about tenfold when increasing the delay from 0.5 to 16 seconds in the case of 12 seconds block interval. The block delay includes the block’s transmission time as well as the verification of the block and its embedded transactions [@decker:2013]. Thus, the bigger the block size, the more time required to transmit and verify the block. Hence, increasing the block size will result in higher stale rates. Furthermore, to ensure the lowest stale rate the block delay should be as small as possible and the block interval as large as possible. For instance, in the case of 600 seconds block interval, the stale rates are minimal since the block delay is only a tiny fraction of the block interval.
**Throughput:** From the throughput results reported in Table \[t:results\], we observe the following. First, reducing the block interval leads to higher throughput. This is because more blocks will be generated, and thus, more transactions will be processed. We also observe that the block delay could reduce the throughput significantly, especially when the block interval is small. The number of transactions that can be processed per second is reduced from 147 to 92 when increasing the block delay from 0.5 to 16 seconds in the case of 12 seconds block interval. Furthermore, the block delay does not have a significant impact on the throughput if the delay is too small compared to the block interval. For instance, in the case of 600 seconds block interval, the throughput achieved is almost the same even when the block delay is increased from 0.5 to 16 seconds.
**Mining decentralization:** From the mining decentralization results reported in Table \[t:results\], we observe the following. First and most importantly, we observe a correlation between stale rates and mining decentralization. The smaller the stale rates the better the mining decentralization and vice versa. In the discussion about stale rates, we observe that reducing the block interval or increasing the block delay can lead to a higher stale rate. That is, reducing the block interval leads to poor mining decentralization. In the case of 1 second block interval, for instance, miners with a large hash power (e.g., M1) have a higher fraction of blocks included in the main ledger, and thus gain higher profit, compared to their hash power invested. On the contrary, small miners have a small fraction of blocks included in the ledger, and thus gain less profit, compared to their hash power invested. Similarly, increasing the block delay negatively impacts the decentralization of the mining process. For a better mining decentralization, the stale rate should be reduced by having the block interval relatively larger than the block delay.
**Bitcoin throughput:** The current implementation of Bitcoin compromises of 596 seconds block interval and 0.42 second block delay, as reported in Table \[t:realData\]. That means the Bitcoin network experiences a low stale rate as well as a good mining decentralization. However, it suffers from poor throughput as the number of transactions processed per second is only about 3. We argue that we could securely reduce the block interval of Bitcoin to 60 seconds to improve the throughput by about a factor 10, without any significant impact on the stale rate or mining decentralization.
**Ethereum mining decentralization through uncle inclusion:** The current implementation of Ethereum compromises of 12.42 seconds block interval and 2.3 seconds block delay, as reported in Table \[t:realData\]. This results in a stale rate of about 12.56% and imperfect mining decentralization, but a better throughput than the Bitcoin blockchain. To eliminate the negative impact on the stale rate and mining decentralization, Ethereum uses an uncle inclusion mechanism, where stale blocks are included in the main ledger as uncle blocks and the miners of such blocks are rewarded. However, this does not guarantee that miners will receive fair rewards compared to their hash power invested (e.g., a miner with a hash power of 20% should receive 20% of the total rewards distributed in the network). This is especially true as miners get a lower reward for uncle blocks compared to main blocks as well as they are not rewarded for the transactions included in the uncle blocks.
---- ------------------------- ---------------------- --------
without uncle inclusion with uncle inclusion
M1 40% 41.32% 40.2%
M2 30% 30.28% 30.18%
M3 15% 14.47% 14.91%
M4 10% 9.34% 9.85%
M5 5% 4.6% 4.86%
---- ------------------------- ---------------------- --------
: The fraction of rewards gained by each miner (M1,M2,...,M5), with and without uncle inclusion mechanism.[]{data-label="t:ETHprofit"}
We use the same parameters as currently in Ethereum to further explore whether the fraction of rewards a miner would receive with uncle inclusion mechanism is proportional to its hash power. We execute 10 independent simulation runs of 10,000 blocks and report the average results in Table \[t:ETHprofit\]. From Table \[t:ETHprofit\], we see that the fraction of rewards gained by the miners with uncle inclusion mechanism is closer to their hash power than in the case where the uncle mechanism is not applied. Thus, Ethereum indeed achieves a better mining decentralization using its uncle inclusion mechanism.
Discussion: Evaluation of BlockSim against Design Objectives {#ss4:eval}
============================================================
We evaluate our simulator against the design criteria mentioned in Section \[ss4:design:principles\], which are generality, extensibility and simplicity.
**Generality:** Generality refers to the ability to use BlockSim for a variety of analysis questions and for a variety of blockchains. The key technology to achieve generality is the BlockSim Base Model, which has been designed in such a way that many blockchain systems and analysis questions can be answered. The Base Model covers all common building blocks of blockchains such as nodes, transactions, blocks, blockchain ledger, fork resolution and incentive models. We have demonstrated the application of blockchain to analyze Bitcoin and Ethereum, and arguably BlockSim is well-suited for the full class of permissionless blockchain systems. Furthermore, BlockSim achieves generality by supporting different properties and metrics such as performance (both throughput and latency), functionality metrics such as stale rates and system properties such as mining decentralization and mining incentives. To further support this criterion, however, we aim to model and implement different consensus protocols (e.g., Proof-of-Stake) as well as different generic broadcast protocols for the Network layer in a later version of BlockSim.
**Extensibility:** Extensibility refers to the ability of the BlockSim tool to be extended in a natural manner for various systems and analysis problems. This boils down to the design of the software, which is through modules that can easily be manipulated and extended to investigate different properties or problems of interest. One can achieve this by only modifying the relevant modules, instead of building everything from scratch. For example, we show how to extend the classes that implements the Base Model of BlockSim to support Bitcoin and Ethereum.
As another example, we will briefly explain how to extend BlockSim to support different malicious behaviors of the nodes (e.g., selfish mining strategies). The current implementation of BlockSim assumes that all nodes are honest. To support such behaviors, we can extend the Node module by introducing a new attribute (e.g., selfish) for each behavior. Hence, each behavior needs to be adequately defined (e.g., by writing a function or a separate class that specifies the procedures involved in this behavior).
To establish selfish mining behavior for a node, for instance, we configure that node to work on its fork without propagating the blocks it generates to other nodes in the network. Once the behaviors are defined, the user of the simulator has only to access the configuration module and choose which type of behaviors to be studied when defining the nodes, without modifying the underlying code of the simulator. Similar to this is the study of the uncertainty miners face during the selection of transactions [@alharby2018impact] and the analysis of the Ethereum Verifier’s Dilemma [@alharby2020datadriven].
**Simplicity:** BlockSim achieves this criterion as it has been implemented in different modules as well as it provides a user interface (a configuration module) that allows the end-user to set up the input parameters for the simulator. This makes BlockSim easy to use and understand. Besides, the current version of BlockSim hides and abstracts many details. For example, it abstracts all the details of the network layer by only introducing a configurable time delay for information propagation to model this layer. Also, it hides details about the validation process of blocks and transactions. By doing so, BlockSim becomes simple and easy to use and understand. Although hiding and abstracting details can result in an incomplete model, it is possible to extend BlockSim to incorporate these details if required.
Related Work {#s4:related}
============
In the literature, there are some attempts to utilize simulation models to evaluate various aspects of blockchain systems. In [@yasa:2017], the authors use architectural modeling and simulation to measure the latency in blockchain systems under different configurations. In [@alharby2018impact], the authors propose a simulation model to investigate the impact of profit uncertainty in the Ethereum blockchain. They found that miners in Ethereum are not able to make informed decisions about which transactions to include in their blocks to maximize their revenue. In [@neudecker:2015], the authors propose a simulation model to analyze and evaluate attacks on the Bitcoin network. In [@gobel:2016], the authors use discrete-event simulation to study the behavior of Bitcoin miners (including selfish-mining strategies) when there is a delay in propagating information among miners. Besides these proposals, there are some blockchain simulators proposed in the literature. In [@gervais:2016], the authors propose a Bitcoin simulator to analyze the security and performance of different configurations in both the consensus and network layers.
Several others Bitcoin-like network simulators are proposed in the literature such as [@aoki:2019], [@miller2015shadow] and [@stoykov2017vibes]. However, these proposals utilize simulation-based models to study specific aspects of blockchain systems. They neither cross different layers nor cover all common functional building blocks (e.g., blocks and transactions) for blockchain systems. For instance, neither of these proposals model transactions in the blockchain system nor capture the incentives layer in the same detail as BlockSim.
With BlockSim we provide a general-purpose, widely usable, simulation tool for blockchains, to assist in answering a variety of design and deployment questions. Our discrete-event simulator generalizes on the ones proposed in the related literature by integrating different layers of the blockchain system to gain a more comprehensive insight into different aspects such as performance, security and incentives. In BlockSim, for instance, we take a step further by considering the functional blocks common across the different implementation of blockchain systems. We design and structure BlockSim to cross different layers of blockchains. Furthermore, we model transactions in two different ways, each of which for specific purposes as well as modeling both Bitcoin and Ethereum blockchains.
Conclusion {#s4:conclusion}
==========
This paper proposes BlockSim, a discrete-event simulation framework for blockchain systems, capturing network, consensus and incentives layers of blockchain systems. The simulation tool is implemented in Python and is available for general use. We introduce the design and evaluate it against the design objectives of generality, extensibility and simplicity.
BlockSim’s results have been validated by comparing it with design properties and measurement studies available from real-life blockchains such as Bitcoin and Ethereum. We also demonstrated the use of BlockSim in a study of stale rate, throughput and mining decentralization across a variety of blockchain configurations.
Future work should further demonstrate the extensibility of BlockSim by implementing additional variants of blockchain systems, such as those based on Proof of Stake as well as blockchains augmented with channels. In addition, one can build on the current version of BlockSim and extend it with additional reusable classes that represent other important system aspects and mechanisms, in particular mining pools and channels.
[^1]: https://github.com/maher243/BlockSim.
[^2]: https://www.blockchain.com/explorer
[^3]: https://etherscan.io/
[^4]: https://dsn.tm.kit.edu/bitcoin/
[^5]: https://ethstats.net/
|
---
abstract: 'We show that the clogging susceptibility and flow of particles moving through a random obstacle array can be controlled with a transverse or longitudinal ac drive. The flow rate can vary over several orders of magnitude, and we find both an optimal frequency and an optimal amplitude of driving that maximizes the flow. For dense arrays, at low ac frequencies a heterogeneous creeping clogged phase appears in which rearrangements between different clogged configurations occur. At intermediate frequencies a high mobility fluidized state forms, and at high frequencies the system reenters a heterogeneous frozen clogged state. These results provide a technique for optimizing flow through heterogeneous media that could also serve as the basis for a particle separation method.'
author:
- 'C. Reichhardt and C.J.O. Reichhardt'
title: ' Controlled Fluidization, Mobility and Clogging in Obstacle Arrays Using Periodic Perturbations '
---
Particle transport through heterogeneous media is relevant to flows in porous media [@1; @2], transport of colloidal particles on ordered or disordered substrates [@3; @4; @5; @6; @7], clogging phenomena [@8; @9; @10; @11; @12; @13], filtration [@14; @15; @16], and active matter motion in disordered environments [@17; @18; @19; @20]. It also has similarities to systems that exhibit depinning phenomena when driven over random or ordered substrates [@21]. Recent work has focused on clogging effects for particle motion through obstacle arrays, where the onset of clogging is characterized by the formation of a heterogeneously dense state [@11; @12; @13]. Such clogging is relevant for the performance of filters or for limiting the amount of flow through disordered media, so understanding how to avoid clog formation or how to optimize the particle mobility in obstacle arrays is highly desirable. Clogging also occurs for particle flow through hoppers or constrictions, where there can be a transition from a flowing to a clogged state as the aperture size decreases or the flow rate increases [@22; @23; @24; @25; @26]. The clogging susceptibility in such systems can be reduced with periodic perturbations or vibrations [@27; @28; @29]. Applied perturbations generally produce enhanced flows in disordered systems [@30; @31; @N1; @32; @33]; however, there are examples where the addition of perturbations or noise can decrease the flow or induce jamming, such as the freezing by heating phenomenon [@34; @35] or the appearance of a reentrant high viscosity state in vibrated granular matter [@36]. A natural question is whether clogging and mobility for particle flows through obstacles can be controlled or optimized with applied perturbations in the same way as hopper flow. The situation is more complex for two-dimensional (2D) disordered obstacle arrays than for hopper geometries since shaking can be applied in either the longitudinal or transverse direction, and one type of shaking may be more effective than the other.
In this work we numerically examine particle flow though a disordered obstacle array where the particles experience both a dc drive and ac shaking. In the absence of the ac shaking, there is a well defined clogging transition at a critical obstacle density $\phi_c^{dc}$ above which the flux of particles drops to zero. We find that application of a transverse or longitudinal ac drive above $\phi_c^{dc}$ unclogs the system and permits flow to occur, while the mobility drops back to zero at a higher second critical obstacle density $\phi_c^{ac}$. We identify an optimal ac frequency for mobility and find that at low frequencies, the system forms a nearly immobile heterogeneous creeping clogged state in which particle rearrangements produce transitions between different clogged configurations. At intermediate frequencies, a more uniform flowing fluidized state appears, and at high frequencies a heterogeneous frozen clogged state emerges in which there are no particle rearrangements. The mobility for fixed frequency and changing ac amplitude is also nonmonotonic. For obstacle densities below $\phi_{c}^{dc}$, the ac drive still strongly affects the flow rate, and we find an optimal frequency that maximizes the flow as well as a local minimum in the mobility produced by a resonance effect of the ac motion with the average spacing between obstacles. In most cases, transverse ac drives produce higher mobility than longitudinal ac drives; however, at low obstacle densities the transverse ac drive reduces the flow. We show that these effects are robust for a wide range of particle densities, and we map a dynamic phase digram describing the fluid regime, the creeping clogged phase, and the frozen clogged state.
[*Simulation and System—*]{} We simulate a 2D system of non-overlapping repulsive particles in the form of disks interacting with a random array of obstacles where the particles are subjected to a dc drift force and an ac shaking force. The sample is of size $L \times L$ with $L = 100$ and we impose periodic boundary conditions in the $x$ and $y$ directions. Interactions between pairs of disks $i$ and $j$ are given by the repulsive harmonic force ${\bf F}^{ij}_{dd} = k(r_{ij} -2R_{d})\Theta(r_{ij} -2R_{d}){\hat {\bf r}}_{ij}$, where the disk radius $R_d=0.5$, $r_{ij} = |{\bf r}_{i} - {\bf r}_{j}|$, $ {\hat {\bf r}}_{ij} = ({\bf r}_{i} - {\bf r}_{j})/r_{ij}$, and $\Theta$ is the Heaviside step function. The spring stiffness $k = 200$ is large enough that disks overlap by less than one percent, placing us in the hard disk limit as confirmed by previous works [@11; @12; @37]. The obstacles are modeled as immobile disks with the same radius and disk-disk interactions as the mobile particles. There are $N_m$ mobile particles with an area coverage of $\phi_{m} = N_{m}\pi R^{2}_{d}/L^2$, while the area coverage of the $N_{\rm obs}$ obstacles is $\phi_{\rm obs} = N_{\rm obs}\pi R^2_{d}/L^2$ and the total area coverage is $\phi_{\rm tot} = \phi_{m} + \phi_{\rm obs}$. For monodisperse disks the system forms a triangular solid at $\phi_{\rm tot} = 0.9$ [@37]. The obstacles are placed in a dense lattice and randomly diluted until the desired $\phi_{obs}$ is reached, so that the minimum spacing between obstacle centers is $d_{\rm min}=2.0$. The particle dynamics are governed by the following overdamped equation of motion: $\eta d {\bf r}_i/dt = {\bf F}_{\rm inter}^i + {\bf F}_{\rm obs}^i + {\bf F}_{dc} + {\bf F}_{ac}$. Here ${\bf F}_{\rm inter}^i=\sum_{j=0}^{N_m}{\bf F}_{dd}^{ij}$ are the particle-particle interactions, ${\bf F}_{\rm obs}^i=\sum_{k=0}^{N_{\rm obs}}{\bf F}_{dd}^{ik}$ are the particle-obstacle interactions, and ${\bf F}_{dc} = F_{dc}{\hat {\bf x}}$ is the dc drift force applied in the positive $x$-direction, where $F_{dc}=0.05$. Each simulation time step is of size $dt=0.002$. We apply a sinusoidal ac drive that is either transverse (perpendicular) to the dc drive, ${\bf F}_{ac}=F_{ac}^{\perp}{\hat {\bf y}}$, or longitudinal (parallel) to the dc drive, ${\bf F}_{ac}=F_{ac}^{||}{\hat {\bf x}}$ . We measure the time average of the velocity per particle in the dc drift direction, $\langle V_{x}\rangle = N_m^{-1}\sum^{N_m}_{i =1}{\bf v}_{i}\cdot {\hat {\bf x}}$, where ${\bf v}_i$ is the velocity of particle $i$. We define the mobility as $M = \langle V_{x}\rangle/ \langle V^{0}_{x}\rangle $, where $\langle V^{0}_{x}\rangle$ is the obstacle-free drift velocity, so that in the free flow limit, $M = 1.0$. We wait at least $10^7$ simulation time steps before taking measurements to ensure that the system has reached a steady state.
. []{data-label="fig:1"}](Fig1.ps){width="\columnwidth"}
[*Results –*]{} In Fig. \[fig:1\](a) we illustrate the positions of the particles and obstacles in a sample with $F_{dc} = 0.05$, $\phi_{\rm tot}=0.275$, and $\phi_{\rm obs}=0.1256$ under a transverse drive of magnitude $F^{\perp}_{ac} = 0.5$ in what we define as the low frequency limit of $\omega = 10^{-7}$, where the mobility is very small, $M=0.01$. The particles assemble into high density clogged regions separated by large void areas. There are slow rearrangements of the particles but little net motion in the direction of the dc drift, so the system is effectively transitioning between different clogged configurations. At $\omega=10^{-4}$ in Fig. \[fig:1\](b), the mobility of the same sample reaches its maximum value of $M = 0.27$. Here the clustering is reduced compared to what occurs at lower frequencies, and the system is in a partially fluidized state. For the high frequency of $\omega=10^{-1}$ in Fig. \[fig:1\](c), a completely frozen clogged state with $M = 0$ appears. In Fig. \[fig:1\](d), when $\omega=10^{-1}$ but the obstacle density is reduced to $\phi_{\rm obs}=0.047$, the system is in a flowing state.
 at $\phi_{\rm tot} = 0.275$, $\phi_{\rm obs} = 0.1256$, and $F_{ac}= 0.5$ for transverse (blue circles) and longitudinal (red squares) ac driving showing a low frequency clogged state, an intermediate frequency flowing state, and a high frequency clogged state. The letters a, b, c mark the frequencies at which the images in Fig. \[fig:1\](a–c) were obtained. (c) $M$ vs $F^{\perp}_{ac}$ for the system in (b) under transverse driving with $\omega = 10^{-4}$ (blue), $10^{-3}$ (green), $10^{-2}$ (gold), and $10^{-1}$ (red). (d) M vs $F_{ac}^{||}$ at the same frequencies as in (c) under longitudinal driving. []{data-label="fig:2"}](Fig2.ps){width="\columnwidth"}
In Fig. \[fig:2\](a) we plot $M$ versus obstacle density $\phi_{\rm obs}$ for a system with $\phi_{\rm tot} = 0.275$ for zero ac drive, a transverse ac drive of $F^{\perp}_{ac} = 0.5$ at $\omega = 10^{-4}$, and a transverse dc drive with $F^{||}_{ac}=0.5$ and $\omega=10^{-4}$. A clogged state with $M=0$ appears for $\phi_{\rm tot}>0.115$ under no ac drive, for $\phi_{\rm tot}>0.2$ under transverse ac driving, and for $\phi_{\rm tot}>0.155$ under longitudinal driving, so there is a wide range of frequencies over which the transverse ac drive is the most effective at reducing clogging. For $\phi_{\rm obs} < 0.07$, the transverse ac drive produces a lower mobility $M$ than either the longitudinal or zero ac driving.
In Fig. \[fig:2\](b) we plot $M$ versus ac frequency $\omega$ for the system from Fig. \[fig:1\](a–c) with $\phi_{\rm tot} = 0.275$ and $\phi_{\rm obs} = 0.1256$ for transverse and longitudinal ac driving of magnitude $F_{ac}=0.5$. We find a low mobility state for $\omega < 10^{-6}$ and a zero mobility state for $\omega \geq 10^{-2}$. The optimal mobility occurs at a frequency of $\omega \approx 2.5 \times 10^{-4}$. Both directions of ac driving produce the same dynamic states, but the maximum value of $M$ for longitudinal driving is less than half that found for transverse driving, and the window of unclogged states is narrower for longitudinal driving. Additionally, the low frequency states with $\omega < 10^{-5}$ are fully clogged with $M = 0$ for longitudinal driving, but have a small finite mobility for transverse driving. These results indicate that there are two different types of clogged states separated by an intermediate fluidized state in which the mobility reaches its optimum value.
In Fig. \[fig:2\](c) we plot $M$ versus $F^{\perp}_{ac}$ for a system with $\phi_{\rm tot} = 0.275$ and $\phi_{\rm obs} = 0.1256$ at the optimal frequency of $\omega=10^{-4}$ and at $\omega=10^{-3}$, $10^{-2}$, and $10^{-1}$. For each driving frequency, there is an optimal value of $F^{\perp}_{ac}$ that maximizes $M$. Figure \[fig:2\](d) shows $M$ versus $F^{||}_{ac}$ for the same system at the same driving frequencies. At $\omega = 10^{-4}$, $M$ initially increases with $F^{||}_{ac}$ but it then decreases until the system reaches a clogged state with $M = 0$ for $F^{||}_{ac} > 1.5$. Previous studies of particles moving over randomly placed obstacles under a purely dc drive have shown that negative differential conductivity or a zero mobility state can appear at high dc drives [@38; @39; @40; @41]. In our system we find a similar effect under large longitudinal ac drives, so that in general the system reaches a clogged state for high $F^{||}_{ac}$. For $\omega = 10^{-3}$ in Fig. \[fig:2\](d), $M$ increases monotonically over the range of $F^{||}_{ac}$ shown; however, $M$ does decrease for much larger values of $F^{||}_{ac}$. In general, $M$ is higher for transverse ac driving since the transverse shaking permits the particles to more easily move around obstacles, whereas for longitudinal ac driving, the particles are pushed toward the obstacles and $M$ is reduced.
![(a) $M$ vs $\phi_{\rm tot}$ for $\phi_{\rm obs} = 0.1256$ and $F^{\perp}_{ac} = 0.5$ at $\omega = 5.0\times 10^{-6}$ (black squares), $10^{-4}$ (red circles), $10^{-2}$ (green diamonds) and $10^{-1}$ (blue triangles). Over the entire range of $\phi_{\rm tot}$, the $\omega=10^{-4}$ curve has the highest values of $M$. (b) $M$ vs $\phi_{\rm tot}$ in the same system for transverse (blue circles) and longitudinal (red squares) ac driving at $\omega=10^{-4}$, where transverse ac driving produces the highest values of $M$. []{data-label="fig:3"}](Fig3.ps){width="\columnwidth"}
In Fig. \[fig:3\](a) we plot $M$ versus $\phi_{\rm tot}$ for samples with $\phi_{\rm obs} = 0.1256$ and $F^{\perp}_{ac} = 0.5$ at $\omega = 5.0\times 10^{-6}$, $10^{-4}$, $10^{-2}$, and $10^{-1}$. $M$ is always small at low $\phi_{\rm tot}$, increases to a local maximum at $\phi_{\rm tot} = 0.5$, and decreases to zero as $\phi_{\rm tot}$ approaches $\phi_{\rm tot}=0.85$, corresponding to the density at which the system starts to form a crystallized solid state [@37; @42]. We find the highest mobility for $\omega = 10^{-4}$, particularly for $0.66 < \phi_{\rm tot} < 0.85$ where $M$ is close to zero for $\omega = 10^{-2}$ and $10^{-1}$. In Fig. \[fig:3\](b) we show $M$ versus $\phi_{\rm tot}$ at $\omega = 10^{-4}$ for transverse and longitudinal ac driving, where we again find that the transverse ac driving gives higher values of $M$ for all $\phi_{\rm tot}$ and where the local maximum in $M$ falls at $\phi_{\rm tot} = 0.5$ for both ac driving directions.
![ (a) $M$ vs $\omega$ for samples with $\phi_{\rm tot} = 0.275$ and $F^{\perp}_{ac} = 0.5$ at $\phi_{\rm obs} = 0.00157$, 0.031416, 0.047124, 0.062831, 0.07754, 0.09424, 0.1099, 0.12566, 0.14137, and $0.157$, from top to bottom. (b) Dynamic phase diagram as a function of $\phi_{\rm obs}$ vs $\omega$ for transverse driving with $F^{\perp}_{ac}=0.5$. I: flowing fluidized state; II: creeping clogged state; III: frozen clogged state. []{data-label="fig:4"}](Fig4.ps){width="\columnwidth"}
In Fig. \[fig:4\](a) we plot $M$ versus $\omega$ in samples with $\phi_{\rm tot}=0.275$ and $F^{\perp}_{ac}=0.5$ at $\phi_{\rm obs} = 0.00157$ to $0.157$. For $\phi_{\rm obs} > 0.1099$, the system reaches a fully clogged state with $M = 0$. We define the onset of the low frequency clogged state as the point at which $M < 0.02$. A local maximum in $M$ appears near $\omega = 2.5 \times 10^{-4}$ and shifts to slightly lower frequencies as $\phi_{\rm obs}$ decreases. A local minimum near $\omega = 10^{-3}$ develops when $\phi_{\rm obs} < 0.1099$, and this minimum also shifts to lower frequencies with decreasing $\phi_{\rm obs}$. Both of the local extrema are correlated with characteristic length scales of the system. The local maximum at $\phi_{\rm obs} = 0.1256$ falls at a value of $\omega$ for which the distance $d_{\tau}=\omega^{-1}dt(F_{ac}^{\perp}/\sqrt{2}+F_{dc})$ a particle moves during a single ac cycle matches the average spacing $1/\sqrt{\phi_{\rm obs}}$ between obstacles. As this average spacing decreases for increasing $\phi_{\rm obs}$, the frequency at which the maximum value of $M$ occurs decreases as well. The frequency at which the local minimum appears for $\phi_{\rm obs}<0.1099$ corresponds to the point at which $d_{\tau}$ matches the minimum transverse surface-to-surface obstacle spacing of $d_{\rm min}-2R_d$. At this matching frequency, the particles preferentially collide with the obstacles rather than moving between them or around them, reducing the mobility. The two resonant frequencies are separated by a factor of 10 since $F^{\perp}_{ac}/F_{dc} = 10$.
In Fig. \[fig:4\](b) we plot a dynamic phase diagram as a function of $\phi_{\rm obs}$ versus $\omega$ for samples with $F^{\perp}_{ac}=0.5$. Here phase I is the flowing fluidized state, phase II is the low frequency creeping clogged state, and phase III is the frozen clogged state. For $\phi_{\rm obs}> 0.165$, the spacing between obstacles becomes so small that the system is in a frozen state for all values of $\omega$. The fluidized state is of maximum extent between $\omega = 10^{-5}$ and $\omega=10^{-4}$. The dynamic phase diagram for longitudinal ac driving (not shown) is similar; however, the extent of phase I is reduced.
Our results resemble what has been found in recent experiments on the viscosity of vibrated granular matter, where the system is in a jammed state for low vibration frequencies, enters a low viscosity fluid state at intermediate frequencies, and shows a reentrant jammed state at high frequencies [@36]. Other studies have also revealed optimal frequencies for dynamic resonances in granular matter, where the speed of sound is the lowest at intermediate frequencies when the grains are the least jammed [@43].
Our results show that the clogging and flow of particulate matter moving through heterogeneous media can be controlled with ac driving, which could be applied to colloidal particles moving through disordered or porous media. Since the mobility is a function of the driving frequency, ac driving could be used to separate different particle species when one species is in a low mobility or clogged state for a given frequency while the other is in a high mobility state. These results can be generalized to the depinning dynamics in many other systems such as active matter, vortices in superconductors, or frictional systems, where there is a competition between the collective interactions of the particles and quenched disorder in the substrate.
[*Summary—*]{} We have examined the clogging and flow of particles moving through random obstacle arrays under a dc drift and an additional transverse or longitudinal ac drive. At zero ac driving, there is a well defined obstacle density above which the system reaches a clogged state. When ac driving is added, this clogging transition shifts to much higher obstacle densities. For large obstacle densities, we find a low frequency creeping clogged state where the particles undergo rearrangements from one clogged configuration to another with a drift mobility that is nearly zero. At intermediate frequencies, the particles form a high mobility fluidized state, while at high frequencies, a zero mobility frozen clogged state appears, so that there is an optimal mobility at intermediate frequencies. The mobility is also nonmonotonic as a function of ac driving amplitude for fixed ac driving frequency. In most cases the transverse ac driving is more effective at increasing the mobility than longitudinal ac driving. When the ac amplitude and frequency are both fixed, we find that there is an optimal disk density that maximizes the mobility, while for high disk densities the system enters a low mobility jammed state. At low obstacle densities the system is always in a flowing state; however, for transverse ac driving we find a resonant frequency with reduced flow when the magnitude of the transverse oscillations matches the minimum transverse spacing of the obstacles. We map a dynamic phase diagram showing the locations of the flowing state, creeping clogged state, and frozen clogged state. Our results suggest that ac driving could be used to avoid clogging and to optimize particle flows in disordered media, and this technique could also be used as a method for separating different species of particles. Our results can be generalized for controlling flows in a wide class of collectively interacting particle systems in heterogeneous environments, including colloids, bubbles, granular matter, vortices in superconductors, and skyrmions in chiral magnets.
We gratefully acknowledge the support of the U.S. Department of Energy through the LANL/LDRD program for this work. This work was carried out under the auspices of the NNSA of the U.S. DoE at LANL under Contract No. DE-AC52-06NA25396 and through the LANL/LDRD program.
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---
abstract: 'We study the rest-frame optical properties of 74 luminous ($L_{\rm bol}=10^{46.2-48.2}\,{\rm erg\,s^{-1}}$), $1.5<z<3.5$ broad-line quasars with near-IR ($JHK$) slit spectroscopy. Systemic redshifts based on the peak of the [\[O[III]{}\]$\lambda$5007]{} line reveal that redshift estimates from the rest-frame UV broad emission lines (mostly ) are intrinsically uncertain by $\sim 200\,{\rm km\,s^{-1}}$ (measurement errors accounted for). The overall full-width-at-half-maximum of the narrow line is $\sim 1000\,{\rm km\,s^{-1}}$ on average. A significant fraction of the total flux ($\sim 40\%$) is in a blueshifted wing component with a median velocity offset of $\sim 700\,\kms$, indicative of ionized outflows within a few kpc from the nucleus; we do not find evidence of significant flux beyond $\sim 10\,{\rm kpc}$ in our slit spectroscopy. The line is noticeably more asymmetric and weaker than that in typical less luminous low-$z$ quasars. However, when matched in quasar continuum luminosity, low-$z$ quasars have similar profiles and strengths as these high-$z$ systems. Therefore the exceptionally large width and blueshifted wing, and the relatively weak strength of in high-$z$ luminous quasars are mostly a luminosity effect rather than redshift evolution. The [H[$\beta$]{}]{}- region of these high-$z$ quasars displays a similar spectral diversity and Eigenvector 1 correlations with anti-correlated and optical strengths, as seen in low-$z$ quasars; but the average broad [H[$\beta$]{}]{} width is larger by 25% than typical low-$z$ quasars, indicating more massive black holes in these high-$z$ systems. These results highlight the importance of understanding in the general context of quasar parameter space in order to understand quasar feedback in the form of outflows. The calibrated one-dimensional near-IR spectra are made publicly available, along with a composite spectrum.'
author:
- 'Yue Shen$^{1}$'
title: 'Rest-frame Optical Properties of Luminous $1.5<z<3.5$ Quasars: the [H[$\beta$]{}]{}- region'
---
Introduction {#sec:intro}
============
Recent development of high-throughput near-IR spectrographs on moderate- to large-aperture telescopes has enabled the study of the rest-frame optical properties of high-redshift ($z>1.5$) quasars [e.g., @McIntosh_etal_1999; @Yuan_Wills_2003; @Shemmer_etal_2004; @Netzer_etal_2004; @Sulentic_etal_2004; @Sulentic_etal_2006]. Over the past decade or so, the sample of high-redshift quasars with near-IR spectroscopy has grown considerably in size, and started to explore the statistical properties of the narrow-line regions (NLRs) and rest-frame optical broad-line regions (BLRs) of high-redshift quasars and their possible evolution from their low-redshift counterparts.
Near-IR spectroscopy of high-$z$ quasars provides a broad range of important applications, from estimating their black hole (BH) masses using the single-epoch virial BH mass estimators [for a recent review, see @Shen_2013] based on the most reliable Balmer lines [e.g., @Dietrich_etal_2002; @Dietrich_Hamann_2004; @Netzer_etal_2004; @Sulentic_etal_2004; @Sulentic_etal_2006; @Dietrich_etal_2009; @Greene_etal_2010; @Assef_etal_2011; @Ho_etal_2012; @Shen_Liu_2012; @Runnoe_etal_2013b; @Zuo_etal_2015; @Brotherton_etal_2015; @Plotkin_etal_2015; @Shemmer_Lieber_2015; @Saito_etal_2015], to studying the sizes and kinematics of the NLR (usually utilizing the strong [\[O[III]{}\]$\lambda\lambda$4959,5007]{} line) at $z>1.5$ [e.g., @Netzer_etal_2004; @Nesvadba_etal_2008; @Kim_etal_2013; @Brusa_etal_2015; @Perna_etal_2015; @Carniani_etal_2015]. A wide wavelength coverage combining optical and near-IR spectroscopy of quasars is also valuable for constraining the spectral energy distribution (SED) of quasars and for testing accretion disk models [e.g., @Capellupo_etal_2015].
The sizes and kinematics of the NLR of low-$z$ Seyfert galaxies and quasars have been studied extensively in the past [e.g., @Mulchaey_etal_1996; @Bennert_etal_2002; @Schmitt_etal_2003; @Bennert_etal_2006a; @Bennert_etal_2006b]. These studies suggest that the NLR size increases with quasar luminosity, although there is evidence for an upper limit of $\sim 10\,{\rm kpc}$ on the NLR size for the most luminous quasars [e.g., @Netzer_etal_2004; @Hainline_etal_2014]. In recent years, interests on NLR gas distributions and kinematics have been revived in the context of quasar-driven outflows and feedback from BH accretion. The emission in low-$z$ Seyferts and quasars often shows significant blueshifted velocity components indicative of outflows [e.g., @Heckman_etal_1981; @Whittle_1985; @Antonucci_2002], and spatial extensions beyond $\sim {\rm kpc}$ scales, as inferred from spatially resolved slit or integral-field-unit (IFU) spectroscopy [e.g., @Stockton_MacKenty_1987; @Crenshaw_Kraemer_2000; @Nelson_etal_2000; @Fu_Stockton_2009; @Fischer_etal_2010; @Villar-Martin_etal_2011; @Greene_etal_2011; @Shen_etal_2011b; @Liu_etal_2013a; @Liu_etal_2013b; @Husemann_etal_2013]. Such kinematic studies of the NLR have been recently extended to $z>1.5$ for small samples with spatially-resolved near-IR spectroscopy [e.g., @Nesvadba_etal_2008; @Harrison_etal_2012; @Brusa_etal_2015; @Perna_etal_2015; @Carniani_etal_2015], which suggest that quasar-driven outflows in ionized gas are common in luminous high-$z$ quasars.
On the other hand, the rest-frame UV-to-optical regime of quasars displays a well-organized spectral diversity, which is ultimately connected to the fundamental properties of BH accretion. The most prominent feature of the quasar spectral diversity is a collection of quantities that all correlate with the strength of the optical emission known as Eigenvector 1 (EV1), discovered by @Boroson_Green_1992. In the optical, one of the most important EV1 correlations is the anti-correlation between the strengths of and , even when quasar luminosity is fixed. EV1 has been the focus of quasar phenomenology for the past two decades [e.g., @Boroson_Green_1992; @Wang_etal_1996; @Boller_etal_1996; @Brotherton_1996; @Laor_1997; @Laor_2000; @Wills_etal_1999; @Sulentic_etal_2000a; @Marziani_etal_2001; @Boroson_2002; @Shang_etal_2003; @Netzer_etal_2007; @Sulentic_etal_2007; @Dong_etal_2011; @Shen_Ho_2014; @Sun_Shen_2015], as it provides important clues to understanding quasar accretion and feedback processes. It has been suggested that the main physical driver of EV1 is the Eddington ratio of the BH accretion [e.g., @Boroson_Green_1992; @Sulentic_etal_2000a; @Boroson_2002; @Dong_etal_2011; @Shen_Ho_2014; @Sun_Shen_2015], although other effects (such as orientation) may still play a minor role in affecting the observed strength of the lines [e.g., @Risaliti_etal_2011]. In addition to the EV1 correlation, the line also shows a Baldwin effect [@Baldwin_1977], i.e., the restframe equivalent width (REW) of decreases as continuum luminosity increases [e.g., @Brotherton_1996; @Zhang_etal_2011; @Stern_Laor_2012a; @Stern_Laor_2012b], often accompanied by increasing flux in the blueshifted wing [e.g., @Zhang_etal_2013]. @Shen_Ho_2014 presented a comprehensive analysis of the properties in low-$z$ quasars, and showed that the strength of the core component decreases with quasar luminosity and optical strength faster than the wing component, leading to overall broader and more blueshifted profiles as luminosity and strength increases. However, the blueshifted component appears to be a ubiquitous feature among quasars at different luminosities [see figs. 2, E1 and E2 in @Shen_Ho_2014].
Building on these results on low-$z$ quasars, a natural step forward is to extend such studies to $z>1.5$ with near-IR spectroscopy to cover the rest-frame optical regime, and to investigate if these correlations involving and other optical lines exist at earlier times. However, compared with optical spectroscopy, near-IR spectroscopy of faint high-$z$ targets is much more expensive, and hence most of the earlier near-IR spectroscopic studies of high-$z$ quasars are still limited either by small sample statistics (of order ten objects or less) or by low spectral quality (i.e., low S/N, low spectral resolution, and/or limited spectral coverage). To enable a robust statistical study, high-quality near-IR spectroscopy for a large sample of high-$z$ quasars is therefore desirable.
We have been conducting a near-IR spectroscopic survey of $1.5<z<3.5$ quasars to study their rest-frame optical properties. @Shen_Liu_2012 presented a study on the correlations among virial mass estimators based on the UV broad lines and optical broad Balmer lines using 60 quasars at $z\sim 1.5-2.2$ from this survey. Here we present 14 additional quasars at $z\sim 3.5$ with new near-IR spectroscopy, and use a total of 74 quasars to study the rest-frame optical properties of these high-$z$ quasars, focusing on properties and EV1 correlations that involve the [H[$\beta$]{}]{}- region. In §\[sec:data\] we describe the sample and spectral measurements, with a discussion on the systemic redshift estimation. We present our results in §\[sec:results\] and discussions in §\[sec:disc\], and conclude in §\[sec:con\]. Throughout the paper we adopt a flat $\Lambda$CDM cosmology with $\Omega_0=0.3$ and $H_0=70\,{\rm km\,s^{-1}Mpc^{-1}}$. We use the REW to refer to the strength of a particular emission line.
Data and Spectral Measurements {#sec:data}
==============================
Sample
------
![Distribution of our near-IR quasar sample (open symbols) in the redshift-luminosity plane. For comparison, we show the distribution of the low-$z$ SDSS sample from @Shen_etal_2011 with [H[$\beta$]{}]{}- coverage. []{data-label="fig:dist"}](logL5100_z.eps){width="49.00000%"}
NOTE. — Summary of the 14 new SDSS quasars at $z\sim 3.3$ for which we have obtained near-infrared spectroscopy. Columns (4)-(6): plate, fiber and MJD of the optical SDSS spectrum for each object; (7): systemic redshift determined from the near-IR spectrum (see §\[sec:spec\_mea\]); (8): SDSS $i$-band PSF magnitudes; (9)-(11): 2MASS magnitudes (Vega) and UKIDSS [@Lawrence_etal_2007] magnitudes (Vega) in the parentheses when available; (12): instrument for the near-IR spectroscopy; (13): UT dates of the near-IR observations. Note that here the 2MASS magnitudes were taken from @Schneider_etal_2010, where aperture photometry was performed upon 2MASS images to detect faint objects, hence these near infrared data go beyond the 2MASS All-Sky and “$6\times$” point source catalogs [see @Schneider_etal_2010 for details]; zero values indicate non-detections.
Our sample of quasars with near-IR spectroscopy consists of the 60 quasars at $z\sim 1.5-2.2$ from @Shen_Liu_2012, and 14 additional quasars at $z\sim 3.3$ for which we have obtained near-IR spectroscopy with the Folded-port InfraRed Echellett [FIRE; @Simcoe_etal_2010] on the 6.5 m Magellan-Baade telescope during two runs in May and Dec, 2013. Table \[table:basic\] summarizes the basic information of the 14 new $z\sim 3.3$ quasars. The data reduction and flux calibration of the new FIRE spectroscopy followed the same procedure as described in @Shen_Liu_2012. All 74 quasars have simultaneous $JHK$ coverage in the near-IR. These quasars were selected from the SDSS DR7 quasar catalog [@Shen_etal_2011] with good optical spectra covering the [C[IV]{}]{} line and in redshift windows of $z\sim 1.5, 2.1, 3.3$ such that the [H[$\beta$]{}]{}- region can be covered in the $JHK$ bands in the near-IR. Requiring them to have good quality (S/N per pixel$\gtrsim 10$) SDSS spectra preferentially selects high-luminosity quasars at these redshifts, but the resulting sample still covers a range of spectral diversities in the emission line properties. Most of these quasars are radio-quiet [@Shen_etal_2011].
Fig. \[fig:dist\] shows the distribution of our sample in the redshift-luminosity plane, compared to the low-$z$ quasar sample drawn from SDSS DR7 [@Shen_etal_2011] with optical spectroscopy covering the [H[$\beta$]{}]{}- region. We have applied an average correction for host contamination in the rest-frame 5100Å luminosities for the low-$z$ comparison sample, using the empirical formula in @Shen_etal_2011 [eqn. 1]. There is no need to apply this correction for the luminous quasars in our near-IR sample.
The reduced and calibrated 1d near-IR spectra for all 74 quasars used in this study are available in ASCII format in the online version of the paper.
Spectral Measurements {#sec:spec_mea}
---------------------
We use functional fits to measure the continuum and emission line properties of our near-IR quasar sample following earlier work [e.g., @Shen_etal_2008; @Shen_etal_2011] in the [H[$\beta$]{}]{}- region covered by near-IR spectroscopy. In short, we fit a local power-law continuum plus an empirical optical template [@Boroson_Green_1992] to the wavelength regions just outside the [H[$\beta$]{}]{}- complex to form a pseudo-continuum. We then subtract this pseudo-continuum model from the original spectrum, and fit a number of Gaussian functions (in logarithmic wavelength space) to model the broad and narrow emission lines. We used 3 Gaussians to describe the broad [H[$\beta$]{}]{} and 1 Gaussian to describe the narrow [H[$\beta$]{}]{}. The [\[O[III]{}\]$\lambda\lambda$4959,5007]{} doublet was each modeled by two Gaussians, one for the “core” component and one for the blueshifted “wing” component. During the fit, the velocity shift and dispersion of the narrow [H[$\beta$]{}]{} component are tied to those of the “core” component of .
We determine the systemic redshift $z_{\rm sys}$ using the model peak of the full [\[O[III]{}\]$\lambda$5007]{} profile; in cases where is not covered or if its S/N is poor, we use the model peak from the [H[$\beta$]{}]{} line. Fig. \[fig:zdiff\] compares these systemic redshifts with those reported by @Hewett_Wild_2010 ($z_{\rm HW}$), which were based on cross-correlations of rest-frame UV lines (mostly for most of our objects) with quasar templates and empirical corrections for the velocity offsets between UV lines and the systemic redshifts. There is a small mean offset of $\sim 100\,{\rm km\,s^{-1}}$ between $z_{\rm HW}$ and $z_{\rm sys}$ and a dispersion of $\sim 280\,{\rm km\,s^{-1}}$ between the two redshifts. The reported measurement uncertainties in the Hewett & Wild redshifts are typically $\sim 180\,{\rm km\,s^{-1}}$, while the typical measurement uncertainty in our systemic redshift estimates is $\sim 60\,{\rm km\,s^{-1}}$. Subtracting the typical measurement uncertainties, there is a residual difference of $\sim 200\,{\rm km\,s^{-1}}$ in the two sets of redshifts, which reflects the systematic uncertainty in estimating the systemic redshifts based on rest-frame UV lines (mostly for the bulk of our near-IR sample). This systematic uncertainty in -based redshifts is consistent with that inferred from comparing and -based redshifts in low-$z$ SDSS quasars using the spectral measurements in @Shen_etal_2011. The refined systemic redshifts for the near-IR sample are important for deriving the composite spectrum and studying the average line profile of these objects.
We note that for the bulk of the population the peak of the full is consistent to within $\sim 50\,{\rm km\,s^{-1}}$ with the systemic redshifts based on stellar absorption lines [e.g., @Hewett_Wild_2010; @Bae_Woo_2014] or low-ionization lines such as [\[S[II]{}\]]{} [e.g., @Zhang_etal_2011] or [\[O[II]{}\]]{} [e.g., @Shen_Ho_2014], although rare individual objects could have a large blueshifted velocity offset in the peak [e.g., @Zamanov_etal_2002; @Boroson_2005; @Komossa_etal_2008]. However, if a single Gaussian were fit to the overall profile, the potential blueshifted “wing” component could bias the redshift estimation based on the Gaussian peak.
On the other hand, @Hewett_Wild_2010 suggested that, based on the analysis of low-$z$ quasar spectra, the centroid measured above the 50 per cent peak-height level[^1], is on average blueshifted from systemic (defined by Ca II K line at 3934.8Å) by $\sim 45\,{\rm km\,s^{-1}}$. Taken this mean offset of into account, the comparison shown in Fig. \[fig:zdiff\] suggests that the -based redshifts in @Hewett_Wild_2010 only mildly overestimate the systemic redshifts by $\sim 50\,{\rm km\,s^{-1}}$ on average for our quasars.
![Histogram of the differences (with measurement errors) between the HW redshifts based on rest-frame UV lines (mostly ) and the systemic redshifts based on (or [H[$\beta$]{}]{}) for our near-IR sample. The red line is a best-fit Gaussian to the distribution, with the mean and dispersion shown. The HW redshifts are on average larger than the ([H[$\beta$]{}]{})-based systemic redshifts by $\sim 100\,{\rm km\,s^{-1}}$.[]{data-label="fig:zdiff"}](zdiff.eps){width="49.00000%"}
![Median composite spectra for our near-IR sample (red) and for SDSS quasars from @VandenBerk_etal_2001 [black]. Other than the apparently broader emission line profiles and the bluer continuum (see text), the composite spectrum for the high-$z$ quasars is similar to that of general SDSS quasars. The full composite spectrum for the near-IR sample is tabulated in Table \[table:composite\]. []{data-label="fig:fullspec"}](full_composite.eps){width="48.00000%"}
{width="47.00000%"} {width="47.00000%"}
We measure the continuum luminosity at rest-frame 5100Å and the emission line properties (such as the rest-frame equivalent width REW and FWHM) using the model fits. To estimate measurement errors, we use a Monte Carlo approach [e.g., @Shen_etal_2008; @Shen_etal_2011]: for each object we perturb the original spectrum by adding artificial noise using the reported spectral error array to generate a mock spectrum and perform the same fit on it; we repeat the process for 50 realizations of mock spectra and record the measurements for each realization; the nominal 1$\sigma$ measurement errors are then estimated as the semi-amplitude of the range enclosing the 16th and 84th percentiles of the distribution from the mock spectra. We provide a fits catalog of the spectral measurements in the [H[$\beta$]{}]{}- region for our near-IR sample and documented the catalog in Table \[table:fits\]. Additional properties of these quasars can be found in the @Shen_etal_2011 catalog.
![Median composite spectra of the $z>1.5$ near-IR quasar sample (black lines) and comparison with the composite spectra of the most luminous low-$z$ SDSS quasars in different luminosity bins. *Top:* Using the estimated systemic redshifts $z_{\rm sys}$ for the near-IR sample. *Bottom:* Using the redshifts reported by @Hewett_Wild_2010 for the near-IR sample. The Hewett & Wild redshifts appear to be overestimated on average for the near-IR sample, and have an additional scatter relative to the systemic redshifts from (and [H[$\beta$]{}]{}). []{data-label="fig:coadd"}](comp_oiii_median_zsys.eps "fig:"){width="49.00000%"} ![Median composite spectra of the $z>1.5$ near-IR quasar sample (black lines) and comparison with the composite spectra of the most luminous low-$z$ SDSS quasars in different luminosity bins. *Top:* Using the estimated systemic redshifts $z_{\rm sys}$ for the near-IR sample. *Bottom:* Using the redshifts reported by @Hewett_Wild_2010 for the near-IR sample. The Hewett & Wild redshifts appear to be overestimated on average for the near-IR sample, and have an additional scatter relative to the systemic redshifts from (and [H[$\beta$]{}]{}). []{data-label="fig:coadd"}](comp_oiii_median_zHW.eps "fig:"){width="49.00000%"}
Results {#sec:results}
=======
NOTE. — Median composite spectrum for our near-IR quasar sample. Wavelengths are in units of Å. Flux and flux error units are arbitrary. The last column indicates how many objects contributed to the median composite at each wavelength pixel.
NOTE. — Format of the fits table containing the [H[$\beta$]{}]{}- region spectral measurements of our sample. The full table is available in FITS format in the online version of the paper.
Composite Spectrum of the Near-IR Sample {#sec:composite}
----------------------------------------
We create a composite spectrum of our near-IR sample combining optical SDSS and near-IR spectroscopy. We follow @VandenBerk_etal_2001 to create a median composite spectrum, which better preserves the relative strengths of emission lines. The full composite spectrum is shown in Fig. \[fig:fullspec\] and compared to the SDSS quasar composite in @VandenBerk_etal_2001. Our composite is slightly bluer than the @VandenBerk_etal_2001 composite, which may be caused by our selection of objects with relatively blue continua typical of classical quasars, but a more likely explanation is that our luminous quasars are much less affected by host contamination than the low-$z$ and low-luminosity quasars used in the Vanden Berk et al. composite at rest-frame optical wavelengths [see discussions in @Shen_etal_2011]. One advantage of our composite is that all objects contributed to the wavelength coverage, whereas the Vanden Berk et al. composite used high-$z$/high-luminosity quasars to cover the rest-frame UV and low-$z$/low-luminosity quasars to cover the rest-frame optical. This explains why the two composite spectra have similar UV broad-line properties but different optical line properties. The use of the Vanden Berk et al. composite for high-$z$ quasars should caution on the potential impact of host contamination in the composite.
Although we only have a small number of objects contributing to the rest-frame optical regime, we clearly detect several weak narrow lines such as and [\[S[II]{}\]]{}, which will be used to probe the physical properties of the NLRs of these high-$z$ quasars in future work. These detected weak narrow lines appear to be substantially broader than those in the composite spectrum generated for low-$z$ quasars. In addition, the broad [H[$\beta$]{}]{} and [H[$\alpha$]{}]{} lines also appear broader than the low-$z$ composite. These results are consistent with the fact that our near-IR sample represents the most luminous quasars and thus likely more massive hosts than low-$z$ quasars, hence both the broad lines and the narrow lines have larger widths than their low-$z$ counterparts with less massive BHs and hosts.
The full composite spectrum for our near-IR sample is tabulated in Table \[table:composite\].
Strength and Kinematics of
----------------------------
We were able to measure [\[O[III]{}\]$\lambda$5007]{} for all but one of our objects, although for $\sim 35\%$ of them [\[O[III]{}\]$\lambda$5007]{} is not detected at $>3\sigma$ significance. Only for one object (J$0412-0612$) [\[O[III]{}\]$\lambda$5007]{} is not covered in our near-IR spectroscopy. Fig. \[fig:oiii\] (left) shows the [\[O[III]{}\]$\lambda$5007]{} REW as a function of continuum luminosity, where we also plot the low-$z$ SDSS quasars for comparison. The scatter in the [\[O[III]{}\]$\lambda$5007]{} REW is large for our high-$z$ quasar sample, but there is no obvious indication that the distribution is bimodal. The average [\[O[III]{}\]$\lambda$5007]{} strength is consistent with earlier near-IR spectroscopic studies on smaller samples of quasars with similar luminosities and redshifts as studied here [e.g., @Sulentic_etal_2004; @Netzer_etal_2004].
The average trend of decreasing [\[O[III]{}\]$\lambda$5007]{} REW with luminosity as shown in Fig. \[fig:oiii\] (left) is known as the Baldwin effect [e.g., @Baldwin_1977; @Brotherton_1996; @Zhang_etal_2013; @Stern_Laor_2012a; @Stern_Laor_2012b]. @Shen_Ho_2014 showed that the [\[O[III]{}\]$\lambda$5007]{} Baldwin effect is primarily driven by the flux reduction in the “core” component of [\[O[III]{}\]$\lambda$5007]{} when quasar luminosity increases. A byproduct of this effect is the increase in the overall [\[O[III]{}\]$\lambda$5007]{} width, as the broad “wing” component becomes more prominent at higher luminosities [e.g., @Shen_Ho_2014].
Recent near-IR spectroscopy of $z>1.5$ luminous quasars (unobscured or obscured) often report exceptionally large widths, with FWHM$\gtrsim 1000\,{\rm km\,s^{-1}}$ [e.g., @Netzer_etal_2004; @Kim_etal_2013; @Brusa_etal_2015]. Fig. \[fig:oiii\] (right) shows the FWHM of the full [\[O[III]{}\]$\lambda$5007]{} profile as a function of continuum luminosity and compares with the low-$z$ SDSS quasar sample. The median [\[O[III]{}\]$\lambda$5007]{} FWHM is $\sim 1000\,{\rm km\,s^{-1}}$ for our near-IR sample, confirming the large FWHM values found in earlier work [e.g., @Netzer_etal_2004; @Kim_etal_2013; @Brusa_etal_2015]. However, despite the large scatter in our sample, they tend to follow the luminosity trend extrapolated from less luminous objects at lower redshifts. This suggests that the NLR kinematics of $z>1.5$ quasars is not significantly different from those at lower redshifts with similar quasar luminosities.
To strengthen the above point, we use the median composite spectrum generated for our near-IR sample (§\[sec:composite\]), and compare it to the median composite spectra of low-$z$ SDSS quasars in different luminosity bins in Fig. \[fig:coadd\]. When luminosity is matched, our high-$z$ sample shows a similar average profile as the most luminous low-$z$ quasars, suggesting there is limited redshift evolution in terms of properties when luminosity is matched. For comparison, in the bottom panel of Fig. \[fig:coadd\], we show the resulting median composite spectrum using the Hewett & Wild redshifts for our high-$z$ quasars. The composite shows an additional broadening due to the systematic uncertainty in the Hewett & Wild redshifts based on broad UV lines. In addition, the peak of in the Hewett & Wild composite is blueshifted from that in the composite based on systemic redshifts by $\sim 2$Å (rest-frame), consistent with the result in §\[sec:spec\_mea\] and Fig. \[fig:zdiff\] that the Hewett & Wild redshifts (mostly -based) on average overestimate the -based systemic redshifts by $\sim 100\,{\rm km\,s^{-1}}$ for our near-IR sample.
High-luminosity quasars display a strong blueshifted component in their emission [e.g., @Shen_Ho_2014]. We have used a simple decomposition method to decompose the emission into a core component and a blueshifted wing component. The median fraction of the blueshifted wing component to the total flux is $\sim 40\%$. The wing component for our near-IR sample shows a broad range of blueshift velocities of up to $\sim 1200\,\kms$, and has a median blueshift of $\sim 700\,{\rm km\,s^{-1}}$, much larger than the $\sim 200\,{\rm km\,s^{-1}}$ average blueshift of the wing component in much less luminous low-$z$ quasars ($L_{5100}\sim 10^{44}\,{\rm erg\,s^{-1}}$) [e.g., @Zhang_etal_2011; @Shen_Ho_2014]. However, @Shen_Ho_2014 showed that the blueshift velocity of the wing component increases with quasar luminosity, and the observed $\sim 700\,\kms$ velocity at the high luminosity regime sampled by our quasars ($L_{5100}\sim 10^{46}\,{\rm erg\,s^{-1}}$) is consistent with simple extrapolation from the luminosity trend found by @Shen_Ho_2014 [their Fig. E2].
Aperture Effects and NLR Sizes
------------------------------
![The rectified 2D spectrum in the [\[O[III]{}\]$\lambda$5007]{} region for J1220$+$0004 from FIRE. Vertical is the slit direction, which covers $\sim 6\arcsec$ along the slit. Horizontal is the wavelength direction, covering rest-frame 4970–5050Å. The local continuum has been subtracted using a crude polynomial fit to pixels outside the [\[O[III]{}\]$\lambda$5007]{} region. Most of the flux is concentrated within the central $\sim 1\arcsec$. For this particular object there is evidence (the fluffy feature above the central emission) that some extended flux exists beyond the central $\sim 1\arcsec$, which, however, does not contribute to the total flux significantly. []{data-label="fig:2d_oiii"}](atv){width="45.00000%" height="10.00000%"}
Our near-IR spectra were obtained using two different instruments: TripleSpec [@Wilson_etal_2004] on the ARC 3.5m telescope (38 quasars), and FIRE on the 6.5m Magellan-Baade telescope (36 quasars including the 14 quasars at $z\sim 3.3$). The TripleSpec observations used a slit width of either 1.1 or 1.5, and the FIRE observations used a slit width of 0.6. In both cases the slit was positioned at the parallactic angle in the middle of the observation. Since the major axis of the NLR is at random with respect to the slit position, our near-IR spectra on average enclose a physical region with comparable sizes to the slit width. Given the angular scale of $\sim 8.5\,{\rm kpc}/$ (at $z\sim 1.6$) and $\sim 7.5\,{\rm kpc}/$ (at $z\sim 3.3$), our near-IR spectra enclose all flux within $\sim 5-10\,{\rm kpc}$. Therefore the blueshifted wing component revealed in our spectra, if originated from quasar-driven outflows, is constrained to be within a few kpc from the nucleus. Future adaptive optics (AO) assisted near-IR IFU observations will improve the constraints on the spatial extent of this blueshifted component in luminous $z>1.5$ quasars.
We also expect that aperture losses due to the finite slit widths used are not important for our near-IR quasars. @Netzer_etal_2004 showed that there is little flux beyond $\sim 1$ radius ($\sim 7\,{\rm kpc}$) in their slit near-IR spectra of high-$z$ quasars, which have similar luminosities and redshifts as our objects. With their careful analysis on the long-slit optical spectroscopy of $0.4<z<0.7$ type 2 quasars, @Hainline_etal_2014 showed that there is an upper limit of $\sim 7\,{\rm kpc}$ on the size of the NLR in luminous quasars. Since the surface brightness of emission typically decreases with distance [e.g., @Liu_etal_2013a], it is reasonable to expect that our aperture already encloses most of the flux, and the observed Baldwin effect is intrinsic and not subject to increasing aperture losses as quasar luminosity increases.
To further ensure that we are not missing significant flux beyond our slit aperture, we visually inspected all 2D spectra for our objects. We found that in essentially all but 2–3 cases the flux is concentrated within the central $\sim 1\arcsec$ in the slit direction, and is consistent with being unresolved under the seeing conditions (e.g., similar spatial profiles for and continuum emission). In these 2–3 exceptions we see evidence that some extended emission exist beyond the central $\sim 1\arcsec$, but even in such cases the contribution of these extended emission to the total flux is negligible. An example of extended emission in our high-$z$ quasars is shown in Fig. \[fig:2d\_oiii\]. We therefore confirm the earlier result in @Netzer_etal_2004 that most of the flux in luminous $z>1.5$ quasars is within $\sim 10\,{\rm kpc}$ from the nucleus. However, to perform a detailed analysis on the spatial extent of the emission using our slit spectroscopy requires more careful spectral reductions/calibrations and a proper treatment of seeing effects, and is beyond the scope of the current work.
Eigenvector 1 Relations
-----------------------
As mentioned in §\[sec:intro\], another important spectral quantity that regulates the strength and width of [\[O[III]{}\]$\lambda$5007]{} is the optical strength (e.g., the EV1 correlations). These spectral correlations are governed by simple underlying physical processes of the BH accretion. Early near-IR spectroscopic studies on small samples already hinted that similar EV1 correlations may exist in high-redshift quasars [e.g., @Sulentic_etal_2004; @Sulentic_etal_2006; @Runnoe_etal_2013b].
Fig. \[fig:rfe\_oiii\_ew\] shows the anti-correlation between strength and optical strength, defined as $R_{\rm FeII}\equiv {\rm REW_{FeII}}/{\rm REW_{H\beta,broad}}$. Our near-IR quasars, although focused on the most luminous quasars, still shows a broad range of strength, and they fall consistently on the relation defined by the low-$z$ quasars (black lines), albeit with slightly lower REW given the Baldwin effect discussed earlier. Therefore we confirm earlier results [e.g., @Sulentic_etal_2004; @Sulentic_etal_2006] that similar EV1 correlations already exist at $z>1.5$ with our much larger sample.
Fig. \[fig:ev1\] shows the locations of our near-IR quasars in the 2D EV1 plane defined by the strength $R_{\rm FeII}$ and the broad [H[$\beta$]{}]{} FWHM [e.g., @Boroson_Green_1992; @Sulentic_etal_2000a; @Shen_Ho_2014]. Notably our high-luminosity near-IR quasars show a systematic offset in the broad [H[$\beta$]{}]{} FWHM, compared to the low-$z$ SDSS quasars (contours). The median broad [H[$\beta$]{}]{} FWHM is $\sim 5000\,{\rm km\,s^{-1}}$ for our high-$z$ sample and $\sim 4000\,{\rm km\,s^{-1}}$ for low-$z$ SDSS quasars. This is consistent with our findings with the composite spectrum, and again indicates that these most luminosity quasars have larger BH masses than their low-$z$ and low-luminosity counterparts. We note that the vertical dispersion of objects in Fig. \[fig:ev1\] is primarily an orientation effect, based on the results of low-$z$ quasars [e.g., @Shen_Ho_2014; @Sun_Shen_2015 also see Marziani [et al.]{} 2001]. Such an orientation bias in the broad [H[$\beta$]{}]{} FWHM is also argued based on radio-loud quasars, where the orientation of the radio jet (hence the accretion disk and BLR orientation) can be inferred from the radio core dominance [e.g., @Wills_Browne_1986; @Runnoe_etal_2013; @Brotherton_etal_2015b].
![The anti-correlation between the EW and the optical strength (the EV1 relation). Symbol notations are the same as in Fig. \[fig:oiii\]. The gray points are for the low-$z$ SDSS quasars in the @Shen_etal_2011 catalog, and the black lines indicate the median, 16th and 84th percentiles of the distribution. Our $z>1.5$ quasars spread a similar range in optical strength, and follow the same EV1 trend defined by low-$z$ quasars. []{data-label="fig:rfe_oiii_ew"}](Rfe_OIII_ew.eps){width="49.00000%"}
![Distribution of our near-IR sample in the EV1 plane defined by $\Rfe$ and the broad [H[$\beta$]{}]{} FWHM. The contours are for the low-$z$ and low-luminosity SDSS quasars based on the measurements in @Shen_etal_2011. The high-$z$ near-IR sample shows a similar wedged distribution, but the broad [H[$\beta$]{}]{} FWHMs are offset to systematically larger values, which is consistent with the scenario that these most luminous quasars also have more massive BHs than their low-luminosity counterparts. Symbol notations are the same as in Fig. \[fig:dist\].[]{data-label="fig:ev1"}](nir_EV1.eps){width="49.00000%"}
Discussion {#sec:disc}
==========
The strong blueshifted components and the exceptionally broad FWHMs observed in our broad-line quasars are consistent with recent studies of different types of AGN (e.g., ultraluminous infrared galaxies ULIRGs/AGN and obscured quasars, radio-loud and radio-quiet quasars) at similar redshifts and luminosities [e.g., @Harrison_etal_2012; @Kim_etal_2013; @Brusa_etal_2015]. In particular, @Harrison_etal_2012 observed eight $z=1.4-3.4$ ULIRGs hosting AGN activity with near-IR integral field spectroscopy, and found evidence of broad emission on several kpc scales with velocity offsets of up to $\sim 850\,{\rm km\,s^{-1}}$. The FWHM, velocity offset and spatial extent of the emitting gas in their ULIRG/AGN sample are consistent with those inferred from our slit spectroscopy for luminous broad-line quasars (see §\[sec:results\]). Our results, along with these recent studies, suggest that kpc-scale outflows in ionized gas are common among the most luminous high-redshift actively accreting SMBHs.
@Brusa_etal_2015 presented near-IR spectroscopy for 8 obscured quasars at $z\sim 1.5$ and measured properties for 6 of them. Their obscured quasars have bolometric luminosities $\sim 10^{45-46.5}\, {\rm erg\,s^{-1}}$, about a factor of $\sim 10$ lower than those for our quasars, but are still among the luminous quasar population. Consistent with our results here, they reported large widths and velocity offsets for their small sample. They also performed a comprehensive comparison of the FWHM among different populations of active SMBHs, and found that their $z\sim 1.5$ obscured quasars have substantially larger FWHMs than those in $z<0.6$ type 2 quasars with similar luminosities. However, one potential caveat is that the properties in quasars (strength and profile) are also strong functions of continuum luminosity and EV1 [e.g., @Shen_Ho_2014], so systems with matched luminosities may still have different physical properties such as quasar continuum luminosity or Eddington ratio.[^2] When matching the quasar continuum luminosity for our high-$z$ quasars and for low-$z$ SDSS quasars, we do not observe difference in their properties, suggesting negligible redshift evolution in the properties, at least in the most luminous unobscured broad-line quasars.
We note that while significant flux beyond $\sim 10$ kpc is rare for our objects, there are exceptions at low redshift, such as extended emission at tens of kpc in both radio-loud and radio-quiet quasars (obscured and unobscured) [e.g., @Fu_Stockton_2009; @Greene_etal_2011; @Shen_etal_2011b; @Fu_etal_2012; @Husemann_etal_2013], some of which may be due to mergers. Since our slit spectroscopy may miss such extended emission along other directions, a systematic search for extended emission in the general population of $z>1.5$ quasars with near-IR IFU observations is highly desirable.
It is interesting to note that a recent study of cool gas in quasar hosts at $z\sim 1$ traced by absorption imprinted on background quasar spectra [@Johnson_etal_2015] also revealed a luminosity dependence of the -absorption gas covering fraction and velocity offset. Although the spatial extent of the -absorption gas in these low-$z$ quasars is much larger than that of the emission probed by our near-IR spectroscopy, and the luminosities of these $z\sim 1$ quasars are much lower than those of our high-$z$ quasars, it is possible that there is a connection between gas outflows on $\sim {\rm kpc}$ scales and on larger scales, as suggested by the similar luminosity trends seen in the two studies with different gas tracers.
Conclusions {#sec:con}
===========
We have performed a detailed study on the rest-frame optical properties (focusing on the [H[$\beta$]{}]{}- region) of $1.5<z<3.5$ luminous ($L_{\rm bol}=10^{46.2-48.2}\, {\rm erg\,s^{-1}}$) broad-line quasars, using a large sample of 74 objects with our own near-IR spectroscopy. The findings from this study are the following:
1. The redshifts of these high-$z$ quasars based on the UV broad lines (mostly ) are uncertain by $\sim 200\,{\rm km\,s^{-1}}$ compared to the more reliable systemic redshifts from the peak of the narrow lines. In addition, the improved redshifts for SDSS-DR7 quasars by @Hewett_Wild_2010 using broad UV lines are systematically biased high by $\sim 100\,{\rm km\,s^{-1}}$ from the -based redshifts for our quasars.
2. The strength is lower than that for typical SDSS quasars at $z<1$, with a median REW of $\sim 13\,$Å. Our high-$z$ objects tend to follow the same Baldwin effect of decreasing REW with quasar continuum luminosity as defined by low-$z$ quasars.
3. The profile of these luminous quasars is highly asymmetric, with $\sim 40\%$ of the total flux in a blueshifted wing component on average. The wing component is on average blueshifted by $\sim 700\,{\rm km\,s^{-1}}$ from the systemic velocity. The overall width is exceptionally large, with a median ${\rm FWHM}\sim 1000\,{\rm km\,s^{-1}}$. These results confirm earlier observations with smaller near-IR spectroscopic samples at these redshifts [e.g., @Netzer_etal_2004].
4. However, we found that the strength and profile of of these high-$z$ luminous quasars are similar to those of their low-$z$ counterparts with comparable quasar continuum luminosity, and they follow the extrapolated trends with luminosity defined by the less luminous low-$z$ quasars. Therefore we conclude that the extreme properties of in these high-$z$ quasars are mainly driven by quasar luminosity rather than redshift evolution.
5. Even within the limited dynamic range in quasar luminosity of our high-$z$ sample, we observe a similar spectral diversity in terms of the optical strength and the well known EV1 correlations for low-$z$ quasars. This suggests that the same physical processes that drive the diversity of quasars are already in place in these earlier active SMBHs. On the other hand, the average broad [H[$\beta$]{}]{} FWHM is larger than that of the low-$z$ and lower-luminosity quasars, reflecting the larger BH masses in these high-$z$ quasars.
6. Our slit spectroscopy suggests that most of the flux in our objects is within the central $\sim 10$ kpc, and the blueshifted wing component must also originate from below such spatial scales. We only found a handful of objects showing evidence of extended (but insignificant) emission beyond the central $\sim 10\,{\rm kpc}$ covered in our slit spectra, which will be good targets for spatially-resolved follow-up observations (such as adaptive optics assisted near-IR IFU observations).
The average values and spread in the REW and FWHM in luminous high-$z$ quasars presented here serve as a useful reference for planning near-IR spectroscopy to cover the region in high-$z$ quasars (e.g., to obtain a reliable redshift estimate based on ). The relatively weaker (due to the Baldwin effect) and broader width of of these high-luminosity quasars compared to typical low-$z$ and low-luminosity quasars means that it is more difficult to detect and measure accurately for these luminous objects.
We have concluded that these luminous $1.5<z<3.5$ quasars are not different from their low-$z$ counterparts at similar quasar continuum luminosities, in terms of the properties. The diversities in strength and kinematics are already clearly seen in recent studies using $z<1$ SDSS quasars [e.g., @Stern_Laor_2012a; @Stern_Laor_2012b; @Zhang_etal_2011; @Zhang_etal_2013; @Shen_Ho_2014]. In particular, we point out that blueshifted components are not unique to the most luminous quasars – they are ubiquitous among quasars, with their properties (e.g., the fraction to total flux, velocity offset and width) correlated with quasar parameters [luminosity and Eddington ratio, e.g., @Shen_Ho_2014 figs. E1 and E2]. Many recent studies use the kinematics of in different types of active galaxies to argue for AGN-driven outflows and feedback. Therefore it is important to understand the properties of emission in the general context of quasar parameter space, in order to understand the physical mechanisms driving these outflows. For example, the correlations of profile with both quasar continuum luminosity and optical strength [see fig. 2 and fig. E2 in @Shen_Ho_2014] suggest that simple accretion parameters (luminosity and Eddington ratio) may play the primary role in regulating the behaviors of outflows.
I thank the referee for comments that led to improvement of the manuscript, and Xin Liu and Luis Ho for useful discussions. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
[*Facilities*]{}: Sloan, Magellan:Baade (FIRE), ARC (TripleSpec)
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[^1]: This is slightly different from our approach of measuring the peak velocity from model fits.
[^2]: Different selections, e.g., -based low-$z$ type 2 quasar selection versus X-ray selected high-$z$ obscured quasars, may also introduce additional complications in the comparison of their sample properties.
|
---
abstract: 'We study Nash equilibria for a sequence of symmetric $N$-player stochastic games of finite-fuel capacity expansion with singular controls and their mean-field game (MFG) counterpart. We construct a solution of the MFG via a simple iterative scheme that produces an optimal control in terms of a Skorokhod reflection at a (state-dependent) surface that splits the state space in [*action*]{} and [*inaction*]{} region. We then show that a solution of the MFG of capacity expansion induces approximate Nash equilibria for the $N$-player games with approximation error $\varepsilon$ going to zero as $N$ tends to infinity. Our analysis relies entirely on probabilistic methods and extends the well-known connection between singular stochastic control and optimal stopping to a mean-field framework.'
address:
- 'L. Campi: Department of Statistics, London School of Economics, Houghton Street, WC2A 2AE London, UK'
- 'T. De Angelis: School of Mathematics, University of Leeds, Woodhouse Lane, LS2 9JT Leeds, UK.'
- 'M. Ghio and G. Livieri: Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy'
author:
- Luciano Campi
- Tiziano De Angelis
- Maddalena Ghio
- Giulia Livieri
bibliography:
- 'goodwillBib.bib'
title: 'Mean-Field Games of Finite-Fuel Capacity Expansion with Singular Controls'
---
Introduction {#sec:intro}
============
Mean field games (MFGs) were first introduced in [@huang2006large] and [@lasry2007mean] as an elegant and tractable way to study approximate Nash equilibria in nonzero-sum symmetric stochastic differential games for a large population of players with a mean-field interaction, i.e., each player interacts with the rest of the population through its empirical distribution. Exploiting the underlying symmetry while passing to the limit with the number of players tending to infinity, the sequence of games converges in some sense to an optimisation problem for a ‘representative player’. Such representative player responds optimally to the distribution of the population which, at equilibrium, coincides with the distribution of the optimally controlled state variable. Once the limit problem is solved, its solution can typically be implemented in the $N$-player game and provides an approximate Nash equilibrium with vanishing error as $N \to \infty$.
From a practical point of view this approximation result can be very useful since tackling directly the $N$-player game is often unfeasible when $N$ is very large due the curse of dimensionality. In the past few years many authors from different mathematical backgrounds have studied this class of games. Two approaches have been adopted to tackle the study of MFGs: an analytic approach as in, e.g., the initial paper [@lasry2007mean], and a probabilistic one. Here we follow the latter, which has been developed in a series of papers by Carmona, Delarue, and their co-authors (see, e.g., [@carmona2013probabilistic; @carmona2013mean; @carmona2015probabilistic; @carmona2016mean]). We refer the reader to the lecture notes by [@cardaliaguet2012notes] and the two-volume monograph by [@carmona2018probabilistic] for comprehensive presentations of MFG theory and its applications from analytic and probabilistic perspective, respectively.
The literature on MFGs is rapidly growing. Most of the papers deal with games where players use ‘regular controls’ in order to optimise their payoffs. Here by regular controls we mean those having a bounded impact on the velocity of the underlying dynamics. Only few papers have studied the case of MFGs with singular controls, which is a larger class of controls allowing for unbounded changes in the velocity of the underlying process and possible discontinuities in the state trajectories. More specifically, [@fu2017mean] established an abstract existence result for solutions of a general MFG with singular control, using the notion of relaxed solutions. The same approach was also applied in [@fu2019extended] to extend the previous results to MFG with interaction through the controls as well. In both papers, the issue of finding approximate Nash equilibria in the $N$-player games is left aside. To the best of our knowledge, only the works of [@cao2017approximation] and of [@guo2019stochastic] tackle simultaneously MFGs and $N$-player games with singular controls. Their analysis is based on verification theorems and quasi-variational inequalities specifically designed for their settings and not amenable to simple extensions. For completeness, we also mention the two papers [@hu2014singular], [@zhang2012relaxed], which use a maximum principle approach to solve singular control problems with mean-field dynamics for the state variables. A class of controls closely related to singular controls is that of impulses, which has also received attention recently within MFG theory. We mention the two papers [@basei2019nonzero] and [@zhou2017mean], where MFGs with impulse controls are considered and solved using an approach based on quasi-variational inequalities and exploiting the stationarity properties of their settings. Finally, the article [@bertucci2020fokker] provides a variational characterization of the density of jumping particles coming from an impulse control problem.
Model description {#sec:model}
-----------------
In this article, we study Nash equilibria for a class of symmetric $N$-player stochastic differential games, for large $N$, and we characterize the solutions of the associated MFG. Specifically, we consider a class of finite-fuel capacity expansion games with *singular controls*.
In order to set out our main results, we provide here a short description of the $N$-player game of capacity expansion (see Section \[subsec:Nplayer\] for a full account). The game is set over a finite-time horizon $T$ given and fixed. We consider a complete probability space $(\Omega,\cF,\P)$ equipped with a right-continuous filtration $\mathbb F:=(\cF_t)_{t\in[0,T]}$ which is augmented with all the $\P$-null sets. There are $N$ players in the game and the $i$-th player $i=1,\ldots, N$ chooses a strategy $\xi^{N,i} = (\xi^{N,i} _t)_{t \in [0,T]}$ from the set of all right-continuous non-decreasing adapted processes, affecting their own private state variables $(X^{N,i},Y^{N,i})$. Given a drift $a\,:\,\mathbb{R} \times [0,1] \rightarrow \mathbb{R}$ and a volatility $\sigma\,:\,\mathbb{R}\rightarrow\mathbb{R}^{+}$, the private states have dynamics
$$\begin{split}
& X_{t}^{N,i}= X_0 ^{i} + \int_{0}^{t}\,a(X_s^{N,i},m_s^{N})\,\ud s + \int_0^t\,\sigma(X_s^{N,i})\,\ud W_s^{i},\\
& Y_{t}^{N,i} = Y_{0-}^{i}+ \xi^{N,i}_t,\quad\quad t \in [0,T],
\end{split}
\label{eq:dynamics_Nplayer_0}$$
where $(W^{1},\ldots,W^{N})$ is a $N$-dimensional Brownian motion. The initial conditions $(X_0^{i}, Y_{0-}^{i})$ are i.i.d. random variables with common distribution $\nu \in \mathcal{P}(\Sigma)$, where $\mathcal{P}(\Sigma)$ is the space of all probability measures on $\Sigma := \mathbb{R} \times [0,1]$. The players interact through the mean-field term $m_t^{N}$ appearing in the drift and given by $$m_t^{N} = \frac{1}{N}\sum_{i = 1}^{N} Y_t^{N,i} = \int_{\Sigma}\,y\,\mu_t^{N}(\ud x,\ud y), \quad t \in [0,T],\label{emp-mean}$$ where $\mu_t^{N} = \frac{1}{N}\sum_{i = 1}^{N} \delta_{(X_t^{N,i}, Y_{t}^{N,i})}$ denotes the empirical probability measure of the players’ states with $\delta_{z}$ the Dirac delta mass at $z \in \Sigma$. In , each $\xi^{N,i}$ represents the investment in additional capacity made by the $i$-th player. Each player aims at maximizing an expected payoff of the form
$$J^{N,i} := \E \left[\int_{0}^{T} e^{-rt} f(X_t^{N,i}, Y_t^{N,i})\,\ud t - \int_{[0,T]} e^{-r t}\,c_0\,\ud\xi_{t}^{N,i}\right],
\label{eq:cost_Nplayer_0}$$
for a fixed discount rate $r > 0$, some cost $c_0 > 0$ and some running payoff $f$ (the same for all players). The optimisation is subject to the so-called *finite-fuel constraint*: $Y_{0-} + \xi_t ^{N,i} \in [0,1]$ for all $t \in [0,T]$ and all $i=1,\ldots, N$. We are interested in computing (approximate) Nash equilibria of this $N$-player game via the MFG approach. This requires to pass to the limit as $N \to \infty$ and to identify the limiting MFG. The latter must be solved (as explicitly as possible) and the associated optimal control is then implemented in the $N$-player game for sufficiently large $N$, as a proxy for the equilibrium strategy.
Singular control problems with finite (and infinite) fuel find numerous applications in the economic literature and originated from the engineering literature in the late 60’s (see [@bather1967sequential] for a seminal paper and, for example, [@benevs1980some; @karatzas1985; @karoui1988] for early contributions to the finite fuel case). Game versions of these problems are a natural extension of the single agent set-up and allow to model numerous applied situations. Here in particular we make assumptions on the structure of the interaction across players that are suitable to model the so-called [*goodwill*]{} problem (see, e.g., [@marinelli2007stochastic; @jack2008singular] in a stochastic environment and [@buratto2002new] in a deterministic one). Specifically, players can be interpreted as firms that produce the same good (e.g., mobile phones) and must decide how to advertise it over a finite time horizon. The $i$-th firm’s product has a market price that depends on the particular type/brand (e.g., Apple, Huawei, etc.) and we model that by the process $X^{N,i}$. Each firm can invest in marketing strategies in order to raise the profile of their product and its popularity. The $i$-th firm’s cumulative amount of investment that goes towards advertising is modelled by the process $Y^{N, i}$, where the finite-fuel feature naturally incorporates the idea that firms set a maximum budget for advertising over the period $[0,T]$. All firms measure their performance in terms of future discounted revenues: they use a running profit function $(x,y)\mapsto f(x,y)$ and deduct the (proportional) cost of advertising $c_0 \ud \xi$. A typical example is $f(x,y)=x \cdot y^\alpha$, $\alpha\in(0,1)$, where profits are linear in the product’s price and increasing and concave as function of the total investment made towards advertising.
From the point of view of the $i$-th firm, investing $\Delta \xi^{N,i}>0$ has a cost $c_0\Delta \xi^{N,i}$ and produces two effects. First of all it increases the popularity of the $i$-th firm’s product, hence increasing the running profit to the level $f(x,y+\Delta\xi^{N,i})$ (we are tacitly assuming $y\mapsto f(x,y)$ increasing). Secondly, it has a broader impact on the visibility of the type of product (e.g., mobile phones) and will stimulate the public’s demand for that good. This has a knock-on effect on the trend of the prices of all the firms that produce the same good. We model this fact via the interaction term $m^N_t$ in the price dynamics and we will assume that the drift function increases with the average spending in advertising across all companies, i.e., $m\mapsto a(x,m)$ is non-decreasing.
Our contribution and methodology
--------------------------------
We focus on the construction of approximate Nash equilibria for the $N$-player game through solutions of the corresponding MFG. First, we formulate the MFG of capacity expansion, i.e., the limit model corresponding to the above $N$-player games as $N\rightarrow\infty$ (Section \[sec:setup\]). Then, under mild assumptions on the problem’s data we construct a solution in feedback form of the MFG of capacity expansion (Section \[sec:existence\]). Our constructive approach, based on an intuitive iterative scheme, allows us to determine the optimal control in the MFG in terms of an optimal boundary $(t,x)\mapsto c(t,x)$ that splits the state space $[0,T]\times\mathbb R\times [0,1]$ into an [*action*]{} region and an [*inaction*]{} region; see Theorem \[teo:existenceSolMFG\] in Section \[sec:setup\]. The optimal strategy prescribes to keep the controlled dynamics underlying the MFG inside the closure of the inaction region by Skorokhod reflection. Finally, whenever the optimal boundary in the MFG is Lipschitz continuous in its second variable we can show that it induces a sequence of approximate $\varepsilon_N$-Nash equilibria for the $N$-player games with vanishing approximation error at rate $O(1/\sqrt{N})$ as $N$ tends to infinity; see Theorem \[teo:approximation\] in Section \[sec:setup\]. While Lipschitz regularity of optimal boundaries is in general a delicate issue, we provide sufficient conditions on our problem data that guarantee such regularity. Since our conditions are far from being necessary, in Section \[subsec:gbm\] we also illustrate an example with a clear economic interpretation which violates those conditions and yet yields a Lipschitz boundary.
The proof of Theorem \[teo:existenceSolMFG\] on the existence of a feedback solution for the limit MFG hinges on a well-known connection between singular stochastic control and optimal stopping (e.g., [@baldursson1996irreversible; @karatzaShreve1984; @karatzasShreve1985]), which we apply in the analysis of our iterative scheme. This approach allows us to overcome the usual difficulties arising from fixed-point arguments often employed in the literature on MFGs. Moreover, as a byproduct we also obtain that a connection between singular control problems of capacity expansion and problems of optimal stopping holds in the setting of our MFG. The finite-fuel condition is not a structural condition and it is clear that our choice of $y\in[0,1]$ is not restrictive: indeed, we could equally consider $y\in[0,\bar y]$ for any $\bar y>0$ (see Remark \[rem:fuel\]). This is suggestive that our results may be extended to the infinite-fuel setting by considering sequences of problems with increasing fuel and limiting arguments. However, since this extension is non-trivial, we leave it for later work and here focus on the finite-fuel case.
Organization of the paper
-------------------------
In Section \[sec:setup\], we formulate the MFG of capacity expansion, we state the standing assumptions on the coefficients in the underlying dynamics and on the profit function, and we state the existence result for the MFG (Theorem \[teo:existenceSolMFG\]). In Section \[sec:existence\] we prove Theorem \[teo:existenceSolMFG\] by constructing a solutions of the MFG in feedback form via an iterative scheme. In particular, we characterise the optimal control in terms of the solution to a Skorokhod reflection problem at an optimal boundary (surface) that splits the state space $[0,T]\times\Sigma$ into [*action*]{} and [*inaction*]{} regions. Finally, in Section \[sec:approximation\] we formally introduce the symmetric $N$-player game of capacity expansion and we show that the solution of the MFG found in Section \[sec:existence\] induces approximate Nash equilibria for the $N$-player games, with vanishing error of order $O(1/\sqrt{N})$ as $N\to\infty$.
Frequently used notations {#sec:notation}
-------------------------
We conclude the Introduction with a summary of notations that will be used throughout the paper.
Let $\Sigma := \mathbb{R} \times [0,1]$ and let $\mathcal{P}(\Sigma)$ denote the set of all probability measures on $\Sigma$ equipped with the Borel $\sigma$-field $\mathcal B(\Sigma)$. Let $\mathcal P_2 (\Sigma)$ be the subset of $\mathcal P(\Sigma)$ of probability measures with finite second moment. The set $\Sigma$ and the $N$-fold product space $\Sigma^N$ are the state spaces for the controlled processes $(X,Y)$ and $(X^{N},Y^{N})$ that are underlying the MFG and the $N$-player game, respectively. Since our problems are set on a finite-time horizon, we also consider ‘time’ as a state variable and will use the state space $[0,T]\times\Sigma$. Given a set $A\subset [0,T]\times\Sigma$ we denote its closure by $\overline A$.
Given a filtered probability space $\Pi:=(\Omega, \mathcal{F}, \mathbb{F} =(\mathcal{F}_{t})_{t \geq 0}, \P)$ satisfying the usual conditions and a $\cF_0$-measurable random variable $Z\in[0,1]$, we denote
$$\begin{aligned}
\Xi^\Pi(Z):= \big \{ \xi:&\: \text{$(\xi_t)_{t\ge 0}$ is $\mathbb F$-adapted, non-decreasing, right-continuous,}\\
&\text{ with $\xi_{0-}=0$ and $Z+\xi_t\in [0,1]$ for all $t\in[0,T]$, $\P$-a.s.} \big \}.\end{aligned}$$
The set $\Xi^\Pi(Z)$ will be the set of admissible strategies for the players in our games. The random variable $Z$ will be replaced by the initial value of the process $Y$ (for the MFG) or $Y^{N,i}$ (for the $i$-th player in the $N$-player game). Often we will drop the dependence of $\Xi$ on the probability space $\Pi$ and the random variable $Z$, as no confusion shall arise.
Finally, the parameters $c_0>0$ and $r\ge 0$ are fixed throughout and describe the cost of exerting control and the discount factor, respectively.
The mean-field game: setting and main results {#sec:setup}
=============================================
In this section we set-up the mean-field game associated with the $N$-player game described above and we state the main result concerning the existence and structure of the optimal control for this game; see Theorem \[teo:existenceSolMFG\]. Later, in Section \[sec:approximation\] we will link the MFG to the $N$-player game. Below we use the notation introduced in Section \[sec:notation\].
The mean-field game {#subsec:mfg}
-------------------
Let $\Pi=(\Omega, \mathcal{F}, \mathbb{F} =(\mathcal{F}_{t})_{t \geq 0}, \P)$ be a filtered probability space satisfying the usual conditions and supporting a one-dimensional $\mathbb F$-Brownian motion $W$. Notice that the initial $\sigma$-field $\mathcal F_0$ is not necessarily trivial.
Let $(X_0, Y_{0-})$ be a two-dimensional $\mathcal F_0$-measurable random variable with joint law $\nu \in \mathcal{P}(\Sigma)$ and let $\xi\in\Xi^\Pi(Y_{0-})$ be an admissible strategy. Then, given a bounded Borel measurable function $m : [0,T] \rightarrow [0,1]$, for all $t \in [0,T]$ we define the 2-dimensional, degenerate, controlled dynamics $$\label{eq:dynamics_mfg_01}
\begin{split}
X_t &= X_0 + \int_{0}^{t} a(X_s, m(s))\,\ud s + \int_{0}^{t} \sigma(X_s) \ud W_s,\\
Y^\xi_t &= Y_{0-} + \xi_t.
\end{split}$$ The goal of the “representative player” consists of maximizing over the set of all admissible strategies $\xi \in \Xi^\Pi(Y_{0-})$ the following objective functional $$\label{eq:J0}
J(\xi)=\E\left[\int_0^{T}e^{-rt}f(X_t,Y^\xi_t)\ud t-\int_{[0,T]} e^{-rt}c_0\ud\xi_t\right],$$ where $f$ is some running payoff function and we recall that $c_0 > 0$ is some cost and $r\ge 0$. Assumptions on all the coefficients appearing in the state variables’ dynamics and in the objective functional will be given below. The integral with respect to the positive random measure $\ud \xi$ includes possible atoms at the initial and terminal time (corresponding to possible jumps of $\xi$).
\[rem:fuel\] The choice $Y \in [0,1]$ in the definition of the set $\Xi$ of admissible strategies is with no loss of generality and we could equally consider $Y \in [0,\bar y]$ for $\bar y >0$. The assumption of finite fuel is consistent with real-world applications, where a firm would set aside a certain budget to be spent over a given period $[0,T]$.
Since we are interested in the MFG that arises from the $N$-player game -, in the limit as $N\to \infty$, it is natural to seek for an admissible optimal strategy $\xi$ (given $m$) such that the following consistency condition holds $$\begin{aligned}
\label{eq:cons0}
m(t)=\E[\,Y^\xi_t],\quad t\in[0,T].\end{aligned}$$ The precise definition of MFG solution will be given in Definition \[def:solMFG\] below. In order to develop our methodology, it is convenient to state a version of the MFG starting from any time $t\in[0,T]$ and any realization $(x,y) \in \Sigma$ of the states $(X_t, Y_{t-})$. Therefore, let us consider the dynamics conditional on the initial data $(t,x,y) \in [0,T]\times\Sigma$, i.e., $$\label{eq:dynamics_mfg_03}
\begin{split}
&X_{t+s}^{t,x} = x + \int_{0}^{s} a(X_{t+u}^{t,x}, m(t+u))\,\ud u + \int_{0}^{s} \sigma(X_{t+u}^{t,x}) \ud W_{t+u},\\
&Y_{t+s}^{t,x,y;\,\xi} = y + (\xi_{t+s}-\xi_{t-}),\quad s \in [0,T-t],
\end{split}$$ where $\ud W_{t+u}=\ud (W_{t+u}-W_t)$. Since the increments of the control $\xi\in\Xi^\Pi(Y_{0-})$, after time $t$, may in general depend on $(t,x,y)$, we account for that dependence by denoting $Y^{t,x,y;\,\xi}$ (and $\xi^{t,x,y}$ if necessary). Instead, given a bounded measurable function $m$, the dynamics of $X$ only depends on the initial condition $X_t=x$, which motivates the use of the notation $X^{t,x}$. For the original case of the process started at time zero (i.e., $t=0$), we use the simpler notation $(X^x_s,Y^{x,y;\,\xi}_s)_{s\in[0,T]}$.
The notation introduced above is somewhat cumbersome and we will often use $\P_{t,x,y}(\,\cdot\,)=\P(\,\cdot\,|X_t=x,Y_{t-}=y)$ for simplicity. So for any bounded measurable function $g$ and any stopping time $\tau\in[0,T-t]$ we have $$\E\left[g(t+\tau,X^{t,x}_{t+\tau},Y^{t,x,y;\,\xi}_{t+\tau})\right]=\E_{t,x,y}\left[g(t+\tau,X_{t+\tau},Y^{\xi}_{t+\tau})\right],$$ and, moreover, we use $\P_{x,y}=\P_{0,x,y}$ for the special case $t=0$.
It is clear that given $\xi\in\Xi^\Pi(Y_{0-})$ the process $\hat \xi_s:=\xi_{t+s}-\xi_{t-}$ is right continuous, non-decreasing and adapted with $\hat \xi_{0-}=0$. Moreover, $y+\hat\xi\in[0,1]$, $\P_{t,x,y}$ a.s. (i.e., conditionally on $(X_t,Y^\xi_{t-})=(x,y)$) because $\xi\in\Xi^\Pi(Y_{0-})$. Then, it is useful to introduce the set $$\begin{aligned}
\Xi^\Pi_{t,x}(y):= \big \{ \xi:&\: \text{$(\xi_s)_{s\ge 0}$ is $(\mathcal F_{t+s})_{s\ge 0}$-adapted, non-decreasing, right-continuous,}\\
&\quad\text{ with $\xi_{0-}=0$ and $y+\xi_s\in [0,1]$ for all $s\in[0,T-t]$, $\P_{t,x,y}$-a.s.} \big \}.\end{aligned}$$ Clearly $\Xi^\Pi_{0,x}(y)=\Xi^\Pi(y)$. Here $\Pi$ is fixed, so we can drop the superscript in the definition of the set of admissible controls. We will sometimes drop also the subscript $x$ and just write $\Xi_t(y)=\Xi_{t,x}(y)$. Furthermore, when no confusion shall arise we write $\xi\in\Xi_t(y)$ although we refer to $\hat\xi\in\Xi_t(y)$ with $\hat \xi_s=\xi_{t+s}-\xi_{t-}$.
Assuming that the mapping $(x,y)\mapsto \E_{x,y}[Y^{\xi}_t]$ is measurable for any admissible $\xi$, we can express the consistency condition as $$m(t) =\int_\Sigma\E_{x,y}[Y^{\xi}_t]\nu(\ud x,\ud y) = \int_\Sigma \int_{\Sigma} y' \mu^{x,y;\,\xi}_{t}(\ud x', \ud y') \nu(\ud x,\ud y),$$ where $\mu^{x,y;\,\xi}_t := \mathcal{L}(X^{x}_t, Y^{x,y;\,\xi}_t) \in \mathcal{P}(\Sigma)$ is the law of the pair $(X^{x}_t, Y^{x,y;\,\xi}_t)$ and the integral with respect to $\nu(\ud x,\ud y)$ accounts for the fact that $(X_0,Y_{0-})\overset{d}\sim\nu$.
Turning our attention to the optimisation problem, we have that the maximal expected payoff associated with a condition $(t,x,y) \in [0,T]\times\Sigma$ is given by $$\label{eq:P0}
\begin{split}
v(t,x,y) &:= \sup_{\xi \in \Xi_{t,x}(y)}\,J(t,x,y ; \xi)\quad\text{with}\\
J(t,x, y; \xi) &:= \E_{t,x,y}\left[\int_{0}^{T-t} e^{- r s} f(X_{t+s}, Y_{t+s}^{\xi})\,\ud s - \int_{[0,T-t]} e^{-r s} c_0 \ud\xi_s\right].
\end{split}$$ The initial objective function in and the optimisation problem in are easily linked by averaging the latter over the initial condition $(X_0 , Y_{0-})\overset{d}{\sim}\nu \in \mathcal{P}(\Sigma)$. That is $$\label{Vnu}
V^{\nu} := \sup_{\xi \in \Xi}\,J(\xi)\quad\text{with}\quad J(\xi) := \int_{\Sigma} J(0,x,y; \xi)\nu(\ud x, \ud y),$$ where we write $\Xi=\Xi(Y_{0-})$ for simplicity.
Now we define solutions of the MFG of capacity expansion.
A solution of the MFG of capacity expansion with initial condition $\nu\in\mathcal P_2(\Sigma)$ is a pair $(m^* , \xi^*)$ with $m^* :[0,T]\rightarrow[0,1]$ a measurable function and $\xi^* \in \Xi$ such that:
1. [*(Optimality property)*]{}. $\xi^*$ is optimal, i.e., $$J(\xi^* )=V^{\nu}=\sup_{\xi \in\Xi}\E\left[\int_0^{T}e^{-rt}f(X^*_t,Y^\xi_t)\ud t-\int_{[0,T]} e^{-rt}c_0\ud \xi_t\right],
\nonumber$$ where $(X^*,Y^\xi)$ is a solution of Eq. associated to $(m^*,\xi)$.
2. [*(Mean-field property)*]{}. Letting $(X^* , Y^* )$ be the solution of Eq. associated to $(m^*, \xi^*)$, the consistency condition holds, i.e., $$m^* (t)=\int_{\Sigma}\E_{x,y}[Y^{*}_t]\nu(\ud x,\ud y),$$ for each $t\in[0,T]$.
We will say that a solution $\xi^*$ of the MFG is in *feedback form* if we have $\xi^*_t=\eta(t,X,Y_{0-})$, $t\in[0,T]$, for some non-anticipative mapping $$\eta:[0,T]\times C([0,T]; \mathbb R)\times[0,1] \to [0,1]$$ (i.e., such that $\eta(t,X,Y_{0-})=\eta(t,(X_{s\wedge t})_{s\in[0,T]},Y_{0-})$). \[def:solMFG\]
We observe that the definition of MFG solution above mimics the structure of a Nash equilibrium (NE) in classical game theory. Indeed, for a NE we first need to compute the best response of each player while keeping the strategies of her competitors fixed, and then we obtain the equilibrium as a fixed point of the best response map. Likewise, the optimality condition (i) corresponds to computing the best response against a given behaviour of the population described by $m^*$; condition (ii) is a fixed point condition, stating that $m^*$ has to be consistent with the best response of the ‘representative player’.
Assumptions and main result {#sub:assumption}
---------------------------
Before stating our main result regarding the existence and structure of the solution to the MFG, we list below the assumptions needed in our approach.
\[ass:SDE\] For the functions $a:\Sigma\rightarrow\R$ and $\sigma:\R\rightarrow\R_+$ the following hold:
- $a$ and $\sigma$ are Lipschitz continuous with constant $L>0$, i.e., for all $x,x'\in\R$ and $m,m'\in[0,1]$, we have $$\left|a(x,m)-a(x',m')\right|+\left|\sigma(x)-\sigma(x')\right|\leq L(\left| x-x' \right|+\left| m- m' \right|).$$
- The mapping $m\mapsto a(x,m)$ is non-decreasing on $[0,1]$ for all $x\in\mathbb R$.
Part (i) of the assumption guarantees that given any Borel measurable function $m:[0,T]\to [0,1]$ the first equation in admits a unique strong solution (see, e.g., [@karatzasShreve], Theorem 5.2.9). Moreover, by a well-known application of Kolmogorov-Chentsov’s continuity theorem, there exists a modification $\tilde X$ of $X$ which is continuous as a random field, i.e., $(t,x,s)\mapsto \tilde X^{t,x}_{t+s}$ is continuous $\P$-a.s. (see, e.g., [@karatzasShreve], pp. 397-398, or [@baldi], Theorem 9.9). From now on we tacitly assume that we always work with such modification and we denote it again by $X$.
Part (i) of the assumption could be relaxed but at the cost of additional technicalities in the proofs. In principle we only need sufficient regularity on the coefficients to guarantee existence of a unique strong solution for $X$ which is also continuous with respect to its initial datum $(t,x)$. Part (ii) instead is instrumental in our construction of the optimal control in the MFG and will be used later for a comparison result (Lemma \[lem:comparisonTime\]). Notice that (ii) is well-suited for the application to the [*goodwill*]{} problem described in Section \[sec:model\] in the Introduction. Typical examples that we have in mind for the drift are $a(x,m)=(m-x)$ (mean-reverting), $a(x,m)=m x$ (geometric Brownian motion) and $a(x, m)=m$ (arithmetic Brownian motion).
Next we give assumptions on the running profit appearing in the optimisation problem.
\[ass:f\] The running profit $f:\Sigma\rightarrow[0,\infty)$ is continuous and the partial derivatives $\partial_y f$ and $\partial_{xy} f$ exist and are continuous on $\mathbb R\times (0,1)$. Furthermore, we have
- Monotonicity: $x\mapsto f(x,y)$, $y\mapsto f(x,y)$ and $x\mapsto\partial_y f(x,y)$ are increasing, with $$\label{ass:partial-y-f}
\lim_{x\rightarrow -\infty}\partial_yf(x,y)< rc_0 < \lim_{x\rightarrow +\infty}\partial_yf(x,y);$$
- Concavity: $y\mapsto f(x,y)$ is strictly concave for all $x\in\mathbb{R}$.
- The mixed derivative is strictly positive, i.e., $\partial_{xy}f> 0$ on $\mathbb R\times(0,1)$.
The set of assumptions above is in line with the literature on irreversible investment and is fulfilled for example by profit functions of Cobb-Douglas type (i.e., $f(x,y)=x^\alpha y^\beta$ with $\alpha\in[0,1]$, $\beta\in(0,1)$ and $x>0$).\
We conclude with some standard integrability conditions that guarantee that the problem is well-posed and will allow us to use dominated convergence theorem in some of the technical steps in the proofs.
\[ass:int\] There exists $p>1$ such that, given any Borel measurable $m:[0,T]\to [0,1]$ and letting $X$ be the associated solution of the SDE , we have $$\begin{aligned}
\E_{t,x,y}\left[\int_0^{T-t}e^{-rs}\Big(\left|f(X_{t+s},y)\right|^{p}+\left|\partial_yf(X_{t+s},y)\right|^{p}\Big)\ud s\right]<\infty,\end{aligned}$$ for all $(t,x,y)\in[0,T]\times\R\times[0,1]$. Finally, $\nu\in\mathcal P_{2}(\Sigma)$.
\[rem:statespace\] For specific applications it may be convenient to restrict the state space of the process $X$ to the positive half-line $[0,\infty)$ or to a generic (possibly unbounded) interval $(\underline x,\overline x)$. In those cases the assumptions above and the further ones we will make in the next sections can be adapted in a straightforward manner. In particular the limits in are amended by letting $x$ tend to the endpoints of the relevant domain. If the end-points of the domain are inaccessible to the process $X$ all our arguments of proof continue to hold up to trivial changes in the notation. For a more general boundary behaviour of the process some tweaks may be needed on a case by cases basis.
We are now ready to state the main results concerning the MFG described above. The proof requires a number of technical steps and hinges on a iterative method whose details are provided in Section \[sec:existence\].
\[teo:existenceSolMFG\] Suppose Assumptions \[ass:SDE\], \[ass:f\] and \[ass:int\] hold. Then, there exists a upper-semi continuous function $c: [0,T] \times \R \to [0,1]$, with $t\mapsto c(t,x)$ and $x\mapsto c(t,x)$ both non-decreasing, such that the pair $(m^*, \xi^*)$ with $$\xi_t ^* := \sup_{0\le s\le t} (c(s,X^* _s)-Y_{0-})^+, \quad m^* (t) := \int_\Sigma \E_{x,y} \left[Y_t ^* \right] \nu(\ud x,\ud y), \quad t \in [0,T],$$ is a solution of the MFG as in Definition \[def:solMFG\].
Differently from the vast majority of papers that analyse MFGs here we are able not only to prove existence of a solution but also to characterise the optimal control in terms of a upper semi-continuous, monotone surface in the state space $[0,T]\times\Sigma$. Moreover, the iterative scheme that we devise for the proof of the theorem suggests a procedure to actually construct the optimal boundary numerically.
The second key result in this paper shows that the optimal control $\xi^*$ solution of the MFG can be used (under mild additional assumptions) to construct an ${\varepsilon}$-Nash equilibrium in the $N$-player game. The statement and proof of this fact are given in Section \[sec:approximation\] below, whereas in the next section we prove Theorem \[teo:existenceSolMFG\].
Construction of the solutions to the MFG {#sec:existence}
========================================
In this section, we provide the complete proof of Theorem \[teo:existenceSolMFG\] together with an intuitive description of the iterative scheme that underpins it. Some of the auxiliary results used along the way can be found in the Appendix as indicated.
Description of the iterative scheme
-----------------------------------
The idea is to start an iterative scheme based on singular control problems that are analogue to the one in the MFG but without consistency condition in the mean-field interaction.
We initialise the scheme by setting $m^{[-1]}(t)\equiv 1$, for $t \in [0,T]$. At the $n$-th step, $n \ge 0$, assume a non-decreasing, right-continuous function $m^{[n-1]}:[0,T]\to [0,1]$ is given and fixed and consider the dynamics $$\begin{aligned}
X_{t+s}^{[n];t,x}&= x+\int_0^{s} a(X_{t+u}^{[n];t,x},m^{[n-1]}(t+u))\ud u+\int_0^{s}\sigma(X_{t+u}^{[n];t,x}) \ud W_{t+u},\label{eq:Xn}\\
Y_{t+s}^{[n];t,x,y}&=y+(\xi_{t+s}-\xi_{t-}),
\label{Yn}\end{aligned}$$ for $(x,y)\in\Sigma$, $s\in[0,T-t]$, $t\in[0,T]$ and where $\xi\in \Xi(Y_{0-})$. As already noticed we have $\xi_{t+\,\cdot}-\xi_{t-}\in\Xi_t(y)$ and we define the singular control problem $\textbf{SC}^{[n]}_{t,x,y}$ as: $$\begin{aligned}
v_n(t,x,y)&:= & \sup_{\xi\in\Xi_{t}(y)}J_n(t,x,y;\xi)\qquad\text{with}\label{SCn-1}\\
J_n(t,x,y;\xi)&:= &\E_{t,x,y}\left[\int_0^{T-t}e^{-rs}f(X^{[n]}_{t+s},y + \xi_{s})\ud s-\int_{[0,T-t]}e^{-rs}c_0\ud\xi_s\right].\label{SCn-2}\end{aligned}$$ Now, in order to define the $(n+1)$-th step of the algorithm, let us assume that we can find an optimal control $\xi^{[n]*}$ for problem $\textbf{SC}^{[n]}_{0,x,y}$ for each $(x,y)\in\Sigma$. Set $Y^{[n]*}:=y+\xi^{[n]*}$ and assume that $(x,y)\mapsto\E_{x,y}\big[ Y^{[n]*}_t\big]$ is measurable for all $t\in[0,T]$. Then, we define $$m^{[n]}(t):=\int_\Sigma\E_{x,y}\left[Y^{[n]*}_t\right]\nu(\ud x,\ud y).$$ The map $t\mapsto m^{[n]}(t)$ is non-decreasing and right-continuous (by dominated convergence) with values in $[0,1]$, so we can use it to define $(X^{[n+1]}, Y^{[n+1]})$ and $v_{n+1}$ by iterating the above construction.
It is well-known in singular control theory that since $y\mapsto f(x,y)$ is concave and the dynamics of $X^{[n]}$ is independent of the control $\xi$, then the $y$-derivative of $v_n(t,x,y)$ corresponds to the value function of an optimal stopping problem. While we will re-derive this fact in Proposition \[th:theoremConnection\] for completeness, here we state the optimal stopping problem that should be associated to $\textbf{SC}^{[n]}_{t,x,y}$ above. For $(t,x,y)\in[0,T]\times\Sigma$ we define the stopping problem $\textbf{OS}^{[n]}_{t,x,y}$ as $$\begin{aligned}
u_n(t,x,y)&:= & \inf_{\tau\in\T_t}U_n(t,x,y;\tau)\qquad\text{with}\label{eq:un}\\
U_n(t,x,y;\tau)&:= &\E_{t,x}\left[\int_0^{\tau}e^{-rs}\partial_yf(X_{t+s}^{[n]},y)\ud s+c_0e^{-r\tau}\right],\quad\text{for $\tau\in\T_t$}\label{eq:Un}\end{aligned}$$ and where $\T_t$ is the set of stopping times for the filtration generated by the Brownian motion in , with values in $[0,T-t]$. Since $W_{t+u}-W_{t}= W_u$ in law, it is convenient for the analysis of the stopping problems (and there is no loss of generality) to use always the same Brownian motion in the dynamics of the process $X^{[n];t,x}$, irrespectively of $t\in[0,T]$. With this convention we have the useful fact that $\T_{t_2}\subset \T_{t_1}$ for $t_1<t_2$. This stopping problem is standard (see, e.g., [@peskirShyriyaev], Chapter I, Section 2, Theorem 2.2): thanks to Assumption \[ass:int\] and continuity of the gain process $$u\mapsto \int_0^{u}e^{-rs}\partial_yf(X_{t+s}^{[n]},y)\ud s+c_0e^{-ru}$$ we know that the smallest optimal stopping time is $$\begin{aligned}
\label{tau*n}
\tau^{[n]}_*(t,x,y)=\inf\{s\in[0,T-t]: u_n(t+s,X^{[n];t,x}_{t+s},y)=c_0\}.\end{aligned}$$ Letting $$\begin{aligned}
Z^{[n]}_s:=e^{-rs}u_n(t+s,X^{[n]}_{t+s},y)+\int_0^se^{-ru}\partial_y f(X^{[n]}_{t+u},y)\ud u\end{aligned}$$ we have that, under $\P_{t,x,y}$, $$\begin{aligned}
\label{eq:mart}
\text{$(Z^{[n]}_s)_{s\in[0,T-t]}$ is a submartingale and $\big(Z^{[n]}_{s\wedge\tau^{[n]}_*}\big)_{s\in[0,T-t]}$ is a martingale}.\end{aligned}$$ Accordingly, we define the continuation region, $\mathcal{C}^{[n]}$, and the stopping region, $\mathcal{S}^{[n]}$, of the optimal stopping problem as $$\begin{aligned}
\mathcal{C}^{[n]}&:= &\lbrace (t,x,y) \in [0,T] \times \Sigma\,:\,u_n(t,x,y)<c_0 \rbrace,\nonumber\\
\mathcal{S}^{[n]}&:= &\lbrace (t,x,y) \in [0,T] \times \Sigma\,:\,u_n(t,x,y)=c_0 \rbrace.
\nonumber\end{aligned}$$ Finally, we introduce an auxiliary set, which will be used in our analysis $$\mathcal{H}:=\lbrace(x,y)\in\R\times[0,1]:\,\partial_yf(x,y)-rc_0<0\rbrace. \label{def-H}$$ Notice that condition in Assumption \[ass:f\] implies that $\mathcal H$ is not empty. This will be needed to prove that the continuation and stopping regions are not empty either.
The rest of our algorithm of proof for Theorem \[teo:existenceSolMFG\] goes as follows:
- Using a probabilistic approach we study in detail continuity and monotonicity of the value function $u_n$, for a generic $n\ge 0$.
- Thanks to the results in step 1 we construct a solution to $\textbf{OS}^{[n]}_{t,x,y}$ by determining the geometry of the stopping region $\cS^{[n]}$. In particular we need to prove regularity properties of the optimal stopping boundary $\partial\cC^{[n]}$ that guarantee that we can construct a process $Y^{[n]*}$ so that the couple $(X^{[n]},Y^{[n]*})$ is bound to evolve in the closure $\overline \cC^{[n]}$ of the continuation set, by Skorokhod reflection.
- We confirm that $Y^{[n]*}$ is optimal in the singular control problem $\textbf{SC}^{[n]}_{t,x,y}$ and that $v_n$ can be constructed by integrating $u_n$ with respect to $y$ (as already shown in the existing literature).
- We prove that the sequence $(u_n)_{n\ge 0}$ is decreasing and use this fact to prove that the iterative scheme converges to the MFG, in the sense that $(X^{[n]},Y^{[n]*},m^{[n]})$ converges to $(X^*,Y^*,m^*)$ from Definition \[def:solMFG\] and that $(Y^*,m^*)$ are expressed as in Theorem \[teo:existenceSolMFG\].
Solution of the $n$-th stopping problem {#subsec:OCOS}
---------------------------------------
Here we construct the solution to problem $\textbf{OS}^{[n]}_{t,x,y}$ for a generic $n\ge 0$. In particular, $t \mapsto m^{[n-1]}(t)$ is a given right-continuous, non-decreasing function bounded between zero and one. First we state a simple but useful comparison result.
\[lem:comparisonTime\] Let Assumption \[ass:SDE\] hold and recall that $m^{[n-1]}:\,[0,T]\rightarrow[0,1]$ is non-decreasing. Then, for any $t\leq t'$ we have $$\label{eq:comp}
\P\left(X^{[n];t,x}_{t+s}\leq X^{[n];t',x}_{t'+s},\:\: \forall s\in[0,T-t']\right)=1,$$ under the dynamics in .
It suffices to compare the drift coefficients in the SDEs for $X^{[n];t,x}$ and $X^{[n];t',x}$ and then apply the comparison result in [@karatzasShreve Proposition 5.2.18] (that proof does not use time-continuity of the drift and it is the same as the proof of Proposition 5.2.13 therein). Set $A(x,s):= a(x,m(t+s))$ and $A'(x,s):= a(x,m(t'+s))$. Since both $t\mapsto m(t)$ and $m\mapsto a(x,m)$ are non-decreasing (Assumption \[ass:SDE\]-(ii)), we have $A(x,s)\leq A'(x,s)$ for all $(x,s)\in\R\times[0,T-t']$. Therefore, applying [@karatzasShreve Proposition 5.2.18] we obtain .
Next we prove continuity and monotonicity of the value function.
\[prop:OSvalueFunc\] Let Assumptions \[ass:SDE\]–\[ass:int\] hold. Then the value function of the optimal stopping problem [*$\textbf{OS}^{[n]}_{t,x,y}$*]{} has the following properties:
- $0 \le u_n(t,x,y)\leq c_0$;
- the map $x\mapsto u_n(t,x,y)$ is non-decreasing for each fixed $(t,y)\in[0,T]\times[0,1]$ and $y\mapsto u_n(t,x,y)$ is non-increasing for each $(t,x)\in[0,T]\times\R$;
- the map $t\mapsto u_n(t,x,y)$ is non-decreasing for each fixed $(x,y)\in\Sigma$;
- the value function is continuous, i.e., $u_n\in C([0,T]\times\Sigma; \R )$.
[*(i)*]{}. The upper bound is due to $u_n(t,x,y)\le U_n(t,x,y;0)=c_0$. For the lower bound it is enough to recall that $\partial_y f\ge 0$ by Assumption \[ass:f\]-(i).
[*(ii)*]{}. Fix $(t,y)\in[0,T]\times[0,1]$. Let $x_2>x_1$ and set $\tau_2:= \tau^{[n]}_*(t,x_2,y)$ as in , which is optimal in $u_n(t,x_2,y)$. Then $$u_n(t,x_2,y)-u_n(t,x_1,y)\geq \E\left[\int_0^{\tau_2}e^{-rs}\left(\partial_yf(X^{[n];t,x_2}_{t+s},y)-\partial_yf(X^{[n];t,x_1}_{t+s},y)\right)\ud s\right]\geq 0
\nonumber$$ because $X^{[n];t,x_2}_{t+s}\geq X^{[n];t,x_1}_{t+s}$ by uniqueness of the solution to and $x\mapsto\partial_y f(x,y)$ is increasing by Assumption \[ass:f\]-(i). By a similar argument we also obtain monotonicity in $y$, since $y\mapsto\partial_y f(x,y)$ is decreasing by Assumption \[ass:f\]-(ii).
[*(iii)*]{}. For this part of the proof we use Lemma \[lem:comparisonTime\]. Fix $(x,y)\in\Sigma$ and take $t_2>t_1$ in $[0,T]$. Then let $\tau_2= \tau^{[n]}_*(t_2,x,y)$ be optimal in $u_n(t_2,x,y)$ and notice that the stopping time is also admissible for $u_n(t_1,x,y)$ because $\mathcal{T}_{t_2}\subset\mathcal{T}_{t_1}$. Then $$u_n(t_2,x,y)-u_n(t_1,x,y)\geq \E\left[\int_0^{\tau_2}e^{-rs}\left(\partial_yf(X^{[n];t_2,x}_{t_2+s},y)-\partial_yf(X^{[n];t_1,x}_{t_1+s},y)\right)\ud s\right]\geq 0,
\nonumber$$ where the final inequality uses that $X^{[n];t_2,x}_{t_2+s}\geq X^{[n];t_1,x}_{t_1+s}$ for $s\in[0,T-t_2]$, $\P$-a.s. by Lemma \[lem:comparisonTime\] and $x\mapsto \partial_yf(x,y)$ is non-decreasing by Assumption \[ass:f\]-(i).
[*(iv)*]{}. Joint continuity of the value function can be deduced by separate continuity in each variable and monotonicity (see, e.g., [@kruse]). Thanks to (ii) and (iii), it suffices to show that $u_n$ is continuous separately in each variable.
Fix $(t,x,y)\in[0,T]\times\Sigma$. Let $x_k\to x$ as $k\to \infty$ and let $\tau_*=\tau^{[n]}_*(t,x,y)$ be optimal for $u_n(t,x,y)$. First we show right-continuity of $u_n(t,\,\cdot\,,y)$ and assume that $x_k\downarrow x$. For each $k$, using monotonicity proven in (ii) we have $$\begin{aligned}
0\leq &u_n(t,x_k,y)-u_n(t,x,y)\label{RHS}\\
\leq &\E\left[\int_0^{\tau_*}e^{-rs}\left(\partial_yf(X^{[n];t,x_k}_{t+s},y)-\partial_yf(X^{[n];t,x}_{t+s},y)\right)\ud s\right]\nonumber\\
\leq &\E\left[\int_0^{T-t}e^{-rs}\left|\partial_yf(X^{[n];t,x_k}_{t+s},y)-\partial_yf(X^{[n];t,x}_{t+s},y)\right|\ud s\right].\nonumber\end{aligned}$$ Taking limits as $k\to\infty$, Assumption \[ass:int\] allow us to use dominated convergence so that we only need $$\lim_{k\to\infty}\left|\partial_yf(X^{[n];t,x_k}_{t+s},y)-\partial_yf(X^{[n];t,x}_{t+s},y)\right|= 0,\quad\P-a.s.$$ The latter holds by continuity of $\partial_y f$ and continuity of the flow $x\mapsto X^{[n];t,x}$ (which is guaranteed by Assumption \[ass:SDE\]).
We can prove left-continuity by analogous arguments. Letting $x_k\uparrow x$ and, for each $k$, selecting the stopping time $\tau_k=\tau^{[n]}_*(t,x_k,y)$ which is optimal for $u_n(t,x_k,y)$ we get $$\begin{aligned}
0\leq &u_n(t,x,y)-u_n(t,x_k,y)\nonumber\\
\leq &\E\left[\int_0^{\tau_k}e^{-rs}\left(\partial_yf(X^{[n];t,x}_{t+s},y)-\partial_yf(X^{[n];t,x_k}_{t+s},y)\right)\ud s\right].\end{aligned}$$ Then we can conclude as in . Completely analogous arguments allow to prove continuity of the value function with respect to $y$ and we omit them here for brevity.
Continuity in time only requires a small adjustment to the argument above. Let $t_k\to t$ as $k\to\infty$, with $(t,x,y)\in[0,T]\times\Sigma$ fixed. First let us consider $t_k\downarrow t$ and set $\tau_*=\tau^{[n]}_*(t,x,y)$, which is optimal for $u_n(t,x,y)$. Then $\tau_*\wedge(T-t_k)$ is admissible for $u_n(t_k,x,y)$ and, by the monotonicity proven in (iii), we have $$\begin{aligned}
0 \leq& u(t_k,x,y)-u(t,x,y)\nonumber\\
\leq &\E\left[\int_0^{\tau_*\wedge(T-t_k)}e^{-rs}\left(\partial_yf(X^{[n];t_k,x}_{t_k+s},y)-\partial_yf(X^{[n];t,x}_{t+s},y)\right)\ud s\right]\nonumber\\
& \quad +\E\left[\int_{\tau_*\wedge(T-t_k)}^{\tau_*}e^{-rs}\partial_yf(X^{[n];t,x}_{t+s},y)\ud s\right]\nonumber\\
\leq& \E\left[\int_0^{T-t_k}e^{-rs}\left|\partial_yf(X^{[n];t_k,x}_{t_k+s},y)-\partial_yf(X^{[n];t,x}_{t+s},y)\right|\ud s\right] \nonumber\\
& \quad +\E\left[\int_{T-t_k}^{T-t}e^{-rs}\left|\partial_yf(X^{[n];t,x}_{t+s},y)\right|\ud s\right].\end{aligned}$$ Now we can let $k\to\infty$ and use dominated convergence (thanks to Assumption \[ass:int\]), continuity of the stochastic flow $t\mapsto X^{t,x}_{t+\cdot}$ and continuity of $\partial_yf$ (Assumption \[ass:f\]) to obtain right-continuity of $u_n(\,\cdot\,,x,y)$. An analogous argument allows to prove left-continuity as well.
Thanks to the properties of the value function we can easily determine the shape of the continuation region $\cC^{[n]}$, whose boundary $\partial\cC^{[n]}$ turns out to be a surface with ‘nice’ monotonicity properties, that we will subsequently use to obtain a solution of the singular control problem $\textbf{SC}^{[n]}$. Part of the proof is based on the following equivalent representation of the value function: $$u_n(t,x,y)=c_0+ \inf_{\tau\in\T_t}\E_{t,x}\left[\int_0^{\tau}e^{-rs}\left(\partial_yf(X^{[n]}_{t+s},y)-rc_0\right)\ud s\right].
\label{eq:valueFuncRepresetation2}$$
\[prop:OSboundary\] Under Assumptions \[ass:SDE\]–\[ass:int\], the continuation and stopping regions, $\cC^{[n]}$ and $\mathcal{S}^{[n]}$, are non-empty. The boundary of $\cC^{[n]}$ can be expressed as a function $c_n:[0,T]\times \R\rightarrow[0,1]$, such that $$\begin{aligned}
&\cC^{[n]}=\{(t,x,y)\!\in\![0,T]\!\times\!\Sigma: y>c_n(t,x) \},
&\cS^{[n]}=\{(t,x,y)\!\in\![0,T]\!\times\!\Sigma: y\le c_n(t,x) \}.\end{aligned}$$ The map $(t,x)\mapsto c_n(t,x)$ is upper semi-continuous with $t\mapsto c_n(t,x)$ and $x\mapsto c_n(t,x)$ non-decreasing (hence $c_n(\,\cdot\,,x)$ and $c_n(t,\,\cdot\,)$ are right-continuous).
Thanks to (ii) in Proposition \[prop:OSvalueFunc\], for any $(t,x)\in[0,T]\times \R$ we can define $$\begin{aligned}
\label{def:cn}
c_n(t,x):=\inf\{y\in[0,1]:u_n(t,x,y)<c_0\}=\inf\{y\in[0,1]:(t,x,y)\in\cC^{[n]}\}\end{aligned}$$ with the convention that $\inf\varnothing=1$. Since $x\mapsto u_n(t,x,y)$ and $t\mapsto u_n(t,x,y)$ are non-decreasing we have, for any ${\varepsilon}>0$ $$(t,x,y)\in\cS^{[n]}\implies (t,x+{\varepsilon},y)\in\cS^{[n]}$$ and $$(t,x,y)\in\cS^{[n]}\implies (t+{\varepsilon},x,y)\in\cS^{[n]}.$$ Then, $c_n$ is non-decreasing in both $t$ and $x$.
To show upper semi-continuity we fix $(t,x)$ and take a sequence $(t_k,x_k)_{k\ge 1}$ that converges to $(t,x)$. Then $(t_k,x_k,c_n(t_k,x_k))\in\cS^{[n]}$ for all $k$’s and, since the stopping region is closed, in the limit we get $$\limsup_{k\to\infty}\,(t_k,x_k,c_n(t_k,x_k))=(t,x,\limsup_{k\to\infty}c_n(t_k,x_k))\in\cS^{[n]}.$$ Then, by definition of $c_n$ it must be $$\limsup_{k\to\infty}c_n(t_k,x_k)\le c_n(t,x).$$
It only remains to show that $\cC^{[n]}$ and $\cS^{[n]}$ are both non-empty. A standard argument implies that $[0,T)\times\mathcal{H}\subset \cC^{[n]}$ with $\mathcal H$ the open set in . Indeed, starting from $(t,x,y)\in[0,T)\times \mathcal H$ and taking the suboptimal strategy $$\tau_{\mathcal{H}}:=\inf\{s\in[0,T-t]:(X^{[n];t,x}_{t+s},y)\notin\mathcal H\}$$ we easily obtain $u_n(t,x,y)\le U_n(t,x,y;\tau_{\mathcal H})<c_0$ by continuity of paths of $X^{[n]}$ and since $\P_{t,x,y}(\tau_{\mathcal H}>0)=1$. So $\cC^{[n]}\neq\varnothing$ because $\mathcal H\neq\varnothing$ thanks to in Assumption \[ass:f\]. We conclude with an argument by contradiction. Assume that $\mathcal{S}^{[n]}=\varnothing$. Then, given any $(t,x,y)\in[0,T)\times\Sigma$ we have $$u_n(t,x,y)=c_0+\E\left[\int_0^{T-t}e^{-rs}\left(\partial_yf(X^{[n];t,x}_{t+s},y)-rc_0\right)\ud s\right],
\nonumber$$ thanks to . Taking limits as $x\rightarrow\infty$ and using monotone convergence to pass it under the expectation and the integral (Assumption \[ass:f\]-(i)) we get $$\lim_{x\rightarrow\infty}u_n(t,x,y)-c_0=\E\left[\int_0^{T-t}e^{-rs}\left(\lim_{x\rightarrow\infty}\partial_yf(X^{[n];t,x}_{t+s},y)-rc_0\right)\ud s\right]>0
\nonumber$$ thanks to . This contradicts $u_n(t,x,y)\leq c_0$, hence $\cS\neq\varnothing$.
Solution of the $n$-th singular control problem {#sec:nSC}
-----------------------------------------------
Here we follow a well-trodden path to show that the boundary $c_n$ obtained in the section above is actually all we need to construct the optimal control in the singular control problem $\textbf{SC}^{[n]}$. First we provide the candidate optimal control in the next lemma.
\[lem:SK\] Fix $(t,x,y)\in[0,T]\times\Sigma$ and let $\xi^{[n]*}$ be defined $\P_{t,x,y}$-almost surely as $$\xi^{[n]*}_{t+s}:=\sup_{0\le u\le s}\left(c_n(t+u,X^{[n]}_{t+u})-y\right)^+\quad\text{with}\quad \xi^{[n]*}_{t-}=0.$$ Then, $\xi^{[n]*}\in\Xi_{t,x}(y)$ and realises $\P_{t,x,y}$-almost surely the Skorokhod reflection of the process $(X^{[n]},Y^{[n]*})$ inside the continuation region $\cC^{[n]}$, where $Y^{[n]*}=y+\xi^{[n]*}$. That is, $\P_{t,x,y}$-almost surely we have
- $(X^{[n]}_{t+s},Y^{[n]*}_{t+s})\in\overline\cC^{[n]}$ for all $s\in[0,T-t]$ (recall that $\overline\cC^{[n]}$ is the closure of $\cC^{[n]}$);
- Minimality condition: $$\begin{aligned}
\label{eq:minSK}
\int_{[t,T]}\mathbf{1}_{\{Y^{[n]*}_{s-}>c_n(s,X^{[n]}_s)\}}\ud\xi^{[n]*}_{s}=\sum_{t<s\le T}\int_{Y^{[n]*}_{s-}}^{Y^{[n]*}_{s}}\mathbf{1}_{\{Y^{[n]*}_{s-}+z>c_n(s,X^{[n]}_s)\}}\ud z=0.\end{aligned}$$
Clearly $\xi^{[n]*}$ is non-decreasing, adapted and bounded by $1-y$. So if we prove that it is also right-continuous we have shown that it belongs to $\Xi_{t,x}(y)$. The proof of right-continuity uses ideas as in [@de2017optimal]. For any ${\varepsilon}>0$ we have $$\xi^{[n]*}_{t+s}\le \xi^{[n]*}_{t+s+{\varepsilon}}=\xi^{[n]*}_{t+s}\vee\sup_{0< u\le {\varepsilon}}\left(c_n(t+s+u,X^{[n]}_{t+s+u})-y\right)^+.$$ By upper semi-continuity of the boundary and continuity of the trajectories of $X^{[n]}$ we have $$\begin{aligned}
&\lim_{{\varepsilon}\to 0}\sup_{0< u\le {\varepsilon}}\left(c_n(t+s+u,X^{[n]}_{t+s+u})-y\right)^+\\
&=\limsup_{u\to 0}\left(c_n(t+s+u,X^{[n]}_{t+s+u})-y\right)^+\le\left(c_n(t+s,X^{[n]}_{t+s})-y\right)^+ \leq \xi^{[n]*}_{t+s}\end{aligned}$$ Then, combining the above expressions we get $\xi^{[n]*}_{t+s}=\lim_{{\varepsilon}\to 0} \xi^{[n]*}_{t+s+{\varepsilon}}$ as needed.
Next we show the Skorokhod reflection property. By construction we have $$Y^{[n]*}_{t+s}=y+\xi^{[n]*}_{t+s}\ge c_n(t+s,X^{[n]}_{t+s})$$ so that $(X^{[n]}_{t+s},Y^{[n]*}_{t+s})\in\overline\cC^{[n]}$ for all $s\in[0,T-t]$ as claimed in $(i)$. For the minimality condition (ii) fix $\omega\in\Omega$ and let $s\in[t,T]$ be such that $Y^{[n]*}_{s-}(\omega)>c_n\big(s,X^{[n]}_s(\omega)\big)$. Then by definition of $Y^{[n]*}$ and by upper semi-continuity of $c_n$ we have $$\begin{aligned}
\label{eq:SKpr}
\sup_{t\le u <s}\left(c_n\big(u,X^{[n]}_u(\omega)\big)-y\right)^+> c_n\big(s,X^{[n]}_s(\omega)\big)-y,\end{aligned}$$ which implies $Y^{[n]*}_{s-}(\omega)=Y^{[n]*}_{s}(\omega)$. The latter and imply that there exists $\delta>0$ such that $$\begin{aligned}
\label{eq:sup}
\left(c_n\big(s,X^{[n]}_s(\omega)\big)-y\right)^+\le \sup_{t\le u \le s}\left(c_n\big(u,X^{[n]}_u(\omega)\big)-y\right)^+-\delta.\end{aligned}$$ By upper semi-continuity of $s\mapsto c_n\big(s,X^{[n]}_s(\omega)\big)$ there must exist $s'>s$ such that $$\left(c_n\big(u,X^{[n]}_u(\omega)\big)-y\right)^+ \le \left(c_n\big(s,X^{[n]}_s(\omega)\big)-y\right)^++\frac{\delta}{2}$$ for all $u\in[s,s')$. The latter and imply $Y^{[n]*}_{s-}(\omega)=Y^{[n]*}_{u}(\omega)$ for all $u\in[s,s')$. Hence $\ud \xi^{[n]*}(\omega)=0$ on $[s,s')$ as needed to show that the first term in is zero. For the second term, it is enough to notice that by the explicit form of $\xi^{[n]*}$ we easily derive $\{\Delta\xi^{[n]*}_s>0\}=\{Y^{[n]*}_{s-}< c(s,X^{[n]}_s)\}$ for any $s\in[t,T]$. Therefore $$\begin{aligned}
Y^{[n]*}_{s-}+\Delta\xi^{[n]*}_s=&Y^{[n]*}_{s-}+\xi^{[n]*}_{s-}\vee\left(c_n(s,X^{[n]}_s)-y\right)^+-\xi^{[n]*}_{s-}\\
=&Y^{[n]*}_{s-}+\left(c_n(s,X^{[n]}_s)-Y^{[n]*}_{s-}\right)^+=Y^{[n]*}_{s-}\vee c_n(s,X^{[n]}_s),\end{aligned}$$ as needed (i.e., any jump of the control $\xi^{[n]*}$ will bring the controlled process to the boundary of the continuation set).
Using the lemma we can now establish optimality of $\xi^{[n]*}$ and obtain $v_n$ as the integral of $u_n$. The proof of the next proposition follows very closely the proof of Theorem 5.1 in [@de2017optimal], except that here we have a finite-fuel problem (see also [@baldursson1996irreversible; @karoui1991new] for earlier similar proofs). So we move it to the appendix for completeness.
\[th:theoremConnection\] Let Assumptions \[ass:SDE\]–\[ass:int\] hold. For any $(t,x,y)\in[0,T]\times\Sigma$ we have $$\begin{aligned}
v_n(t,x,y)=\Phi_n(t,x)-\int_y^1u_n(t,x,z)\ud z,\end{aligned}$$ with $$\Phi_n(t,x):=\E_{t,x}\left[\int_0^{T-t}e^{-rs}f(X^{[n]}_{t+s},1)\ud s\right].$$ Moreover, $\xi^{[n]*}$ as in Lemma \[lem:SK\] is optimal, i.e., $v_n(t,x,y)=J_n(t,x,y;\xi^{[n]*})$.
Limit of the iterative scheme
-----------------------------
Now that we have characterised the solution of the $n$-th singular control problem, we turn to the study of convergence of the iterative scheme. First we show monotonicity of the scheme in terms of the sequence of value functions $(u_n)_{n\ge 0}$ of the stopping problems.
\[teo:monotonicity\] Under Assumptions \[ass:SDE\]–\[ass:int\] we have $u_n\ge u_{n+1}$ on $[0,T]\times\Sigma$ and $c_n\ge c_{n+1}$ on $[0,T]\times\mathbb R$. Moreover, for any $(t,x,y)\in[0,T]\times\Sigma$ we also have $$\begin{aligned}
\label{eq:comp-Xn}
\text{$X^{[n]}_{t+s}\ge X^{[n+1]}_{t+s}$ and $Y^{[n]*}_{t+s}\ge Y^{[n+1]*}_{t+s}$ for $s\in[0,T-t]$, $\P_{t,x,y}$-a.s.}\end{aligned}$$ Finally, $m^{[n]}\ge m^{[n+1]}$ on $[0,T]$.
We argue by induction and assume that for some $n\ge 0$ we have $m^{[n-1]}\ge m^{[n]}$ on $[0,T]$. Then, by monotonicity of the drift coefficient (Assumption \[ass:SDE\]-(ii), we have $a(x,m^{[n]}(t))\le a(x,m^{[n-1]}(t))$ for all $(t,x)\in[0,T]\times \mathbb R$. It follows from comparison results for SDEs [see, e.g., @karatzasShreve Proposition 5.2.18] and that $X^{[n]}_{t+s}\ge X^{[n+1]}_{t+s}$ for all $s\in[0,T-t]$, $\P_{t,x}$-a.s., for all $(t,x)\in[0,T]\times \mathbb R$. By monotonicity of the profit function (Assumption \[ass:f\]-(i)) we have $\partial_yf(X^{[n+1]}_{t+s},y)\le \partial_y f(X^{[n]}_{t+s},y)$ and therefore and imply $u_{n+1}\le u_n$ on $[0,T]\times\Sigma$. The latter and the definition of the optimal boundary in give us $c_{n+1}\le c_n$ on $[0,T]\times\mathbb R$. Now, using the definition of the optimal control in Lemma \[lem:SK\] we have $\P_{t,x,y}$-a.s. $$\begin{aligned}
\xi^{[n+1]*}_{t+s}=&\sup_{0\le u\le s}\left(c_{n+1}(t+u,X^{[n+1]}_{t+u})-y\right)^+\le\sup_{0\le u\le s}\left(c_{n}(t+u,X^{[n+1]}_{t+u})-y\right)^+\\
\le &\sup_{0\le u\le s}\left(c_{n}(t+u,X^{[n]}_{t+u})-y\right)^+=\xi^{[n]*}_{t+s},\end{aligned}$$ where the first inequality is due to $c_n\ge c_{n+1}$ and the second one to $X^{[n]}\ge X^{[n+1]}$, since $x\mapsto c_n(t,x)$ is non-decreasing (Proposition \[prop:OSboundary\]). Monotonicity of the optimal controls implies monotonicity of the optimally controlled processes $Y^{[n]*}_{t+s}\ge Y^{[n+1]*}_{t+s}$ for all $s\in[0,T-t]$ and from the latter we obtain $$m^{[n+1]}(t)=\int_\Sigma \E_{x,y}\left[Y^{[n+1]*}_t\right]\nu(\ud x,\ud y)\ge\int_\Sigma \E_{x,y}\left[Y^{[n]*}_t\right]\nu(\ud x,\ud y)=m^{[n]}(t).$$ So the argument is complete once we show that we can find $n\ge 0$ such that $m^{[n-1]}\ge m^{[n]}$ on $[0,T]$. The latter is true in particular for $n=0$ since $m^{[-1]}\equiv 1$ and $m^{[0]}\le 1$ on $[0,T]$.
It is clear that by construction $0\le c_{n}(t,x)\le 1$ and $0\le m^{[n]}(t)\le 1$ for all $(t,x)\in[0,T]\times \mathbb R$ and all $n\ge 0$. Moreover, $a(x,0)\le a(x,m^{[n]}(t))\le a(x,1)$ for all $(t,x)\in[0,T]\times \mathbb R$ and all $n\ge 0$, so that by the comparison principle $\bar X^0_{t+s}\le X^{[n]}_{t+s}\le X^{[0]}_{t+s}$, for all $s\in[0,T-t]$, $\P_{t,x,y}$-a.s. for all $n\ge 0$ and with $\bar X^0$ the solution of associated to $a(x,0)$.
By monotonicity of the sequences $(u_n)_{n\ge 0}$, $(c_n)_{n\ge 0}$ and $(m^{[n]})_{n\ge 0}$ we can define the functions $$\begin{aligned}
\label{eq:lims}
u(t,x,y):=&\,\lim_{n\to\infty}u_n(t,x,y),\quad c(t,x):=\lim_{n\to\infty}c_n(t,x)\\
&\:\:\text{and}\:\: \widetilde m(t):=\lim_{n\to\infty}m^{[n]}(t),\notag\end{aligned}$$ for all $(t,x,y)\in[0,T]\times\Sigma$. Pointwise limit preserves the monotonicity of $\widetilde{m}$, $c$ and $u$ with respect to $(t,x,y)$. Moreover, since $u_n$ is continuous and $c_n$, $m^{[n]}$ are upper semi-continuous for all $n\ge 0$ we have that $$\begin{aligned}
\label{eq:usc-lim}
\textit{the functions $u$, $\widetilde{m}$ and $c$ are upper semi-continuous}\end{aligned}$$ on their respective domains as decreasing limit of upper semi-continuous functions. Since $\widetilde{m}$ is also non-decreasing, then it must be right-continuous.
Notice that for each $n\ge 0$ the null set in depends on $n$ and $(t,x,y)$ so we denote it by $N^n_{t,x,y}$. Then we can define a universal null set $N_{t,x,y}:=\cup_{n\ge 0}N^n_{t,x,y}$ and for any $(t,x,y)\in[0,T]\times\Sigma$ and all $\omega\in\Omega\setminus N_{t,x,y}$ we define the processes $\widetilde X$ and $\widetilde \xi$ as $$\begin{aligned}
\label{eq:lim-Xi}
\widetilde X_{t+s}(\omega):=\lim_{n\to\infty}X^{[n]}_{t+s}(\omega)\quad\text{and}\quad \widetilde\xi_{t+s}(\omega):=\lim_{n\to\infty}\xi^{[n]*}_{t+s}(\omega),\end{aligned}$$ for all $s\in[0,T-t]$. We can then set $\widetilde X\equiv0$ and $\widetilde\xi\equiv0$ on $N_{t,x,y}$ and recall that the filtration is completed with $\P_{t,x,y}$-null sets, so that the limit processes are adapted. Of course we also have $$\widetilde Y_t:=y+\widetilde{\xi}_t=\lim_{n\to\infty} Y^{[n]*}_t$$ and thanks to monotone convergence we can immediately establish $$\begin{aligned}
\label{eq:m-lim}
\widetilde{m}(t)=\lim_{n\to\infty}\int_\Sigma\E_{x,y}\big[Y^{[n]*}_t\big]\nu(\ud x,\ud y)=\int_\Sigma\E_{x,y}\big[\,\widetilde Y_t\,\big]\nu(\ud x,\ud y).\end{aligned}$$ Notice that here we are using that $(x,y)\mapsto\E_{x,y}[\xi^{[n]*}_t]$ is measurable, thanks to the explicit expression of $\xi^{[n]*}$ and measurability of $c_n$. Therefore $(x,y)\mapsto\E_{x,y}[\,\widetilde \xi_t\,]$ is measurable too as pointwise limit of measurable functions.
We now derive the dynamics of $\widetilde X$ and show that $\widetilde \xi\in\Xi$.
\[lem:convXY\] Suppose Assumptions \[ass:SDE\]–\[ass:int\] hold. For any $(t,x,y)\in[0,T]\times\Sigma$ the process $\widetilde X$ is the unique strong solution of $$\widetilde X_{t+s}=x+\int_0^s a\big(\widetilde X_{t+u},\widetilde{m}(t+u)\big)\ud u+\int_0^s\sigma\big(\widetilde X_{t+u}\big) \ud W_{t+u},\quad s\in[0,T-t], \label{limit-SDE}$$ and the process $\widetilde \xi$ belongs to $\Xi_{t,x}(y)$.
Fix $(t,x,y)\in[0,T]\times\Sigma$. The first observation is that $\widetilde X$ and $\widetilde \xi$ are $(\cF_{t+s})_{s\ge 0}$-adapted processes as pointwise limit of adapted processes on $\Omega\setminus N_{t,x,y}$ and by $\P_{t,x,y}$-completeness of the filtration. Since $\widetilde\xi$ is decreasing limit of right-continuous non-decreasing processes (hence upper semi-continuous), then it is also non-decreasing and upper-semi continuous. The latter two properties imply right-continuity of the limit process $\widetilde\xi$ as well. Since $\xi^{[n]*}_{t-}=0$ and $\xi^{[n]*}_{T}\le 1-y$ for all $n\ge 0$ we also have $\widetilde\xi_{t-}=0$ and $\widetilde\xi_T\le 1-y$. Hence $\widetilde \xi\in\Xi_{t,x}(y)$.
Let us now prove . Denote by $X'$ the unique strong solution of and let us show that $\widetilde X=X'$. By standard estimates and using Lipschitz continuity of the drift $a(\,\cdot\,)$ (Assumption \[ass:SDE\]-(i))we have $$\begin{aligned}
&\E_{t,x}\left[\sup_{0\le s\le T-t}\left|X^{[n]}_{t+s}-X'_{t+s}\right|^2\right]\\
&\le 2\,\E_{t,x}\left[L\cdot T\int_0^{T-t}\left(\big|X^{[n]}_{t+s}-X'_{t+s}\big|^2+\big|m^{[n]}(t+s)-\widetilde{m}(t+s)\big|^2\right)\ud s\right]\\
&\quad\quad+2\,\E_{t,x}\left[\sup_{0\le s\le T-t}\left|\int_0^{s}\big(\sigma(X^{[n]}_{t+s})-\sigma(X'_{t+s})\big)\ud W_{t+s}\right|^2\right].\end{aligned}$$ Since $\sigma$ enjoys linear growth and $X^{[n]}$ and $X'$ are solutions of SDEs with Lipschitz coefficients, then $$s\mapsto \int_0^{s}\big(\sigma(X^{[n]}_{t+s})-\sigma(X'_{t+s})\big)\ud W_{t+s}$$ is a martingale on $[0,T-t]$ and we can use Doob’s inequality to get $$\begin{aligned}
&\E_{t,x}\left[\sup_{0\le s\le T-t}\left|\int_0^{s}\big(\sigma(X^{[n]}_{t+s})-\sigma(X'_{t+s})\big)\ud W_{t+s}\right|^2\right]\\
&\le 4\,\E_{t,x}\left[\int_0^{T-t}\big(\sigma(X^{[n]}_{t+s})-\sigma(X'_{t+s})\big)^2\ud s\right]\le 4 L^2 \E_{t,x}\left[\int_0^{T-t}\big|X^{[n]}_{t+s}-X'_{t+s}\big|^2\ud s\right],\end{aligned}$$ Combining the estimates above and using Gronwall’s inequality we obtain $$\E_{t,x}\left[\sup_{0\le s\le T-t}\left|X^{[n]}_{t+s}-X'_{t+s}\right|^2\right]\le c\int_0^{T-t}\big|m^{[n]}(t+s)-\widetilde{m}(t+s)|^2\ud s,$$ for some constant $c>0$. Letting $n\to\infty$ and using bounded convergence and the definition of $\widetilde{m}$ we conclude.
Next we connect $u(\,\cdot\,)$ and $c(\,\cdot\,)$ with an optimal stopping problem for $\widetilde X$. Recall that $u$ and $c$ are upper semi-continuous by and enjoy the same monotonicity properties of $u_n$ and $c_n$.
\[lem:limitOS\] Suppose Assumptions \[ass:SDE\]–\[ass:int\] hold. Then, for all $(t,x,y)\in[0,T]\times\Sigma$ we have $$\label{eq:uOS}
\begin{split}
u(t,x,y)&=\inf_{\tau\in\mathcal T_t}U(t,x,y;\tau)\qquad \text{with}\\
U(t,x,y;\tau)&:= \E_{t,x}\left[\int_0^\tau e^{-rs}\partial_y f(\widetilde X_{t+s},y)\ud s+c_0e^{-r\tau}\right]
\end{split}$$ and $$c(t,x)=\inf\{y\in[0,1]:u(t,x,y)<c_0\}\quad\text{with $\inf\varnothing = 1$}.$$ In particular $c$ is the boundary of the set $$\begin{aligned}
\label{cC}
\cC:=\{(t,x,y)\in[0,T]\times\Sigma: u(t,x,y)<c_0\}\end{aligned}$$ and, moreover, both $\cC$ and $\cS:=([0,T]\times\Sigma)\setminus \cC$ are not empty.
Since $X^{[n]}\ge \widetilde X$ for all $n\ge 0$ and $x\mapsto\partial_y f(x,y)$ is non-decreasing, for any $\tau\in\T_t$ we have $U_n(t,x,y;\tau)\ge U(t,x,y;\tau)$ and therefore $$u(t,x,y)=\lim_{n\to\infty}\inf_{\tau\in\T_t}U_{n}(t,x,y;\tau)\ge\inf_{\tau\in\T_t}U(t,x,y;\tau).$$ Now, given ${\varepsilon}>0$ we can find a stopping time $\tau_{\varepsilon}\in\T_t$ such that $$\inf_{\tau\in\T_t}U(t,x,y;\tau)+{\varepsilon}\ge U(t,x,y;\tau_{\varepsilon}).$$ Moreover, by dominated convergence (Assumption \[ass:int\]) and continuity of $\partial_y f$ we have $$U(t,x,y;\tau_{\varepsilon})=\E_{t,x}\left[\int_0^{\tau_{\varepsilon}}e^{-r s}\lim_{n\to\infty}\partial_y f(X^{[n]}_{t+s},y)\ud s + c_0 e^{-r \tau_{\varepsilon}}\right]= \lim_{n\to\infty} U_n(t,x,y;\tau_{\varepsilon}).$$ So combining the above we get $$\inf_{\tau\in\T_t}U(t,x,y;\tau)+{\varepsilon}\ge\lim_{n\to\infty} U_n(t,x,y;\tau_{\varepsilon})\ge \lim_{n\to\infty}u_n(t,x,y)=u(t,x,y)$$ and since ${\varepsilon}>0$ was arbitrary we conclude $$u(t,x,y)\le \inf_{\tau\in\T_t}U(t,x,y;\tau)$$ as needed for the first claim.
Let us next prove that $c$ coincides with the optimal stopping boundary for the limit problem. Since $u\le u_n$ for all $n\ge 0$ we have $$c_n(t,x)=\inf\{y\in[0,1]: u_n(t,x,y)<c_0\}\ge \inf\{y\in[0,1]: u(t,x,y)<c_0\}$$ so that $$c(t,x)\ge \inf\{y\in[0,1]: u(t,x,y)<c_0\}.$$ For the reverse inequality, let us fix $(t,x)\in[0,T]\times \R$, take $\eta\in[0,1]$ such that $$\begin{aligned}
\label{eta}
\eta > \inf\{y\in[0,1]: u(t,x,y)<c_0\}.\end{aligned}$$ Then there must be $\delta>0$ such that $u(t,x,\eta)\le c_0-\delta$. By pointwise convergence, there exists $n_\delta\ge 0$ such that $u_n(t,x,\eta)\le u(t,x,\eta)+\delta/2$ for all $n\ge n_\delta$ and therefore, $u_n(t,x,\eta)\le c_0-\delta/2$ for all $n\ge n_\delta$. Hence, $\eta>c_n(t,x)$ for all $n\ge n_\delta$ and $\eta>c(t,x)$ too. The result holds for any $\eta\in[0,1]$ such that is true and therefore $$c(t,x)\le \inf\{y\in[0,1]: u(t,x,y)<c_0\}.$$ Since $y\mapsto u(t,x,y)$ is decreasing it is clear that $c$ is the boundary of the set $\cC$ defined in .
The exact same arguments as in the proof of Proposition \[prop:OSboundary\] apply to the stopping problem with value $u$ and allow us to show that $\cC\neq\varnothing$ and $\cS\neq\varnothing$ thanks to in Assumption \[ass:f\].
Thanks to the probabilistic representation of $u$ we can use the same arguments as in the proof of Proposition \[prop:OSvalueFunc\] to show that $u$ indeed fulfils the same properties as $u_n$.
\[cor:reg-u\] Under Assumptions \[ass:SDE\]-\[ass:int\] the function $u$ satisfies (i)–(iv) in Proposition \[prop:OSvalueFunc\].
In what follows we let $$\begin{aligned}
\label{eq:tau*}
\tau_*(t,x,y)=\inf\{s\in[0,T-t]: u(t+s,\widetilde X^{t,x}_{t+s},y)=c_0\},\end{aligned}$$ which is optimal for the limit problem with value $u(t,x,y)$. Continuity of the value function allows a simple proof of convergence of optimal stopping times. The result is of independent interest and might be used for numerical approximation of the optimal stopping rule $\tau_*$. We state the result here but put its proof in the appendix as it will not be needed in the rest of the paper.
\[lem:tau-n\] For all $(t,x,y)\in[0,T]\times\Sigma$ we have $\tau^{[n]}_*\uparrow \tau_*$, $\P_{t,x,y}$-a.s., as $n\to\infty$.
Since the dynamics of $(\widetilde X_t)_{t\in[0,T]}$ is fully specified and we have obtained a solution of the optimal stopping problem with value $u$ (Lemma \[lem:limitOS\]), we can state a result similar to Proposition \[th:theoremConnection\].
\[prop:ex-uniq\] Let Assumptions \[ass:SDE\]–\[ass:int\] hold and let $\widetilde X$ be specified as in Lemma \[lem:convXY\]. Define $$\label{eq:hat-v}
\begin{split}
\hat v(t,x,y)&:= \sup_{\xi\in\Xi_{t,x}(y)}\hat J(t,x,y;\xi)\qquad\text{with}\\
\hat J(t,x,y;\xi)&:= \E_{t,x}\left[\int_0^{T-t}e^{-rs}f(\widetilde X_{t+s},y + \xi_{s})\ud s-\int_{[0,T-t]}e^{-rs}c_0\ud\xi_s\right].
\end{split}$$ Then, for any $(t,x,y)\in[0,T]\times\Sigma$ we have $$\begin{aligned}
\hat v(t,x,y)=\Phi(t,x)-\!\int_y^1\!u(t,x,z)\ud z\quad\text{with}\quad\Phi(t,x):=\E_{t,x}\left[\int_0^{T-t}\!\!e^{-rs}f(\widetilde X_{t+s},1)\ud s\right].\end{aligned}$$ Moreover, $$\begin{aligned}
\label{eq:xi*}
\xi^{*}_{t+s}:=\sup_{0\le u\le s}\left(c(t+u,\widetilde X_{t+u})-y\right)^+\quad\text{with}\quad \xi^{*}_{t-}=0,\end{aligned}$$ is the unique optimal control in , up to indistinguishability.
Before passing to the proof we would like to emphasise that at this stage we are not claiming that $(\widetilde{m},\xi^*)$ is a solution of the MFG because $\widetilde{m}$ is specified in and a priori the consistency condition may not hold. Hence, a priori $\hat v$ is not the function $v$ defined in . Of course uniqueness of the optimal control also holds in Proposition \[th:theoremConnection\] albeit not stated there.
We only need to prove uniqueness of the optimal control as all remaining claims are obtained by repeating verbatim the proof of Proposition \[th:theoremConnection\]. As usual, uniqueness follows by strict concavity of the map $y\mapsto f(x,y)$, convexity of the set $\Xi_{t,x}(y)$ of admissible controls and an argument by contradiction.
For notational simplicity and with no loss of generality we take $t=0$. Assume that $\eta\in \Xi_{0,x}(y)$ is another optimal control. Since, $\eta$ and $\xi^*$ are both right-continuous they are indistinguishable as soon as they are modifications, i.e. $\P_{x,y}(\xi^*_{s}=\eta_{s})=1$ for all $s\in[0,T]$. Arguing by contradiction assume there exists $s_0\in[0,T)$ such that $3 p:=\P_{x,y}(\xi^*_{s_0}\neq \eta_{s_0})>0$. Then, there also exists ${\varepsilon}>0$ such that $\P_{x,y}(|\xi^*_{s_0}- \eta_{s_0}|\ge {\varepsilon})\ge 2p$ and, by right-continuity of the paths, there exists $s_1>s_0$ such that $$\P_{x,y}\left(\inf_{s_0\le u\le s_1}|\xi^*_{u}- \eta_{u}|\ge {\varepsilon}\right)\ge p>0.$$ Let us denote $$\begin{aligned}
\label{A0}
A_{0}:=\left\{\omega:\inf_{s_0\le u\le s_1}|\xi^*_{u}(\omega)- \eta_{u}(\omega)|\ge {\varepsilon}\right\}.\end{aligned}$$
For any $\lambda\in(0,1)$, since $\eta$ and $\xi^*$ are both optimal, we have $$\begin{aligned}
\hat v(0,x,y)=&\lambda \hat J(0,x,y;\eta)+(1-\lambda)\hat J(0,x,y;\xi^*)\\
=&\E_{x}\left[\int_0^{T}e^{-rs}\left[\lambda f(\widetilde X_{s},y + \eta_{s})+
(1-\lambda)f(\widetilde X_{s},y + \xi^*_{s})\right]\ud s\right]\\
&-\E_{x}\left[\int_{[0,T]}e^{-rs}c_0(\lambda\ud \eta_s+(1-\lambda)\ud\xi^*_s)\right].\end{aligned}$$ Now, letting $\zeta^\lambda:=\lambda\eta+(1-\lambda)\xi^*$ it is immediate to check that $\zeta^\lambda\in\Xi_{0,x}(y)$ and, by strict concavity of $y\mapsto f(x,y)$ (and joint continuity of $f$), we have $$\begin{aligned}
&\mathds{1}_{A_0}\left[\lambda f(\widetilde X_{s},y + \eta_{s})+(1-\lambda)f(\widetilde X_{s},y + \xi^*_{s})\right]<\mathds{1}_{A_0} f( \widetilde X_{s},y + \zeta^\lambda_{s}),\quad\text{for $s\in[s_0,s_1]$}\end{aligned}$$ with $\mathds 1_{A_0}$ the indicator of the event in . Since $\P_{0,x}(A_0)>0$ and $s_0<s_1$, the strict inequality holds for the expected values as well. Hence we reach the contradiction $$\hat v(0,x,y)<\hat J(0,x,y;\zeta^\lambda),$$ which concludes the proof.
Solution of the MFG
-------------------
In this section we first show that $\widetilde \xi$ obtained in the previous section (see ) is optimal for the control problem in Proposition \[prop:ex-uniq\] and then conclude that $(\widetilde m,\widetilde \xi)$ solves the MFG.
Let Assumptions \[ass:SDE\]–\[ass:int\] hold, take $\widetilde \xi$ as in , $\widetilde m$ as in and $\widetilde X$ as in Lemma \[lem:convXY\]. Then $$\hat v(t,x,y)=\hat J(t,x,y;\widetilde \xi),\quad \text{for any $(t,x,y)\in[0,T]\times\Sigma$}$$ and $\widetilde \xi$ is indistinguishable from $\xi^*$ as in .
We only need to prove optimality of $\widetilde \xi$ as the rest follows by uniqueness of the optimal control (Proposition \[prop:ex-uniq\]).
Recall the value function $v_n$ of $\textbf{SC}^{[n]}$ (see –) and its expression from Proposition \[th:theoremConnection\]. Using dominated convergence we obtain $$\begin{aligned}
&\lim_{n\to\infty}\left(\Phi_n(t,x)-\int_y^1 u_n(t,x,z)\ud z\right)\\
&=\E_{t,x}\left[\int_0^{T-t}e^{-rs}\lim_{n\to\infty}f(X^{[n]}_{t+s},1)\ud s\right]-\int_y^1 \lim_{n\to\infty}u_n(t,x,z)\ud z=\hat v(t,x,y),\end{aligned}$$ where the final equality is due to , and Proposition \[prop:ex-uniq\]. Therefore we have $$\lim_{n\to\infty}v_n(t,x,y)=\hat v(t,x,y).$$ Since $v_n(t,x,y)=J_n(t,x,y;\xi^{[n]*})$, if we can show that $$\lim_{n\to\infty}J_n(t,x,y;\xi^{[n]*})=\hat J(t,x,y;\widetilde \xi),$$ the proof is complete. The latter is not difficult, indeed by integration by parts and dominated convergence we have $$\begin{aligned}
&\lim_{n\to\infty}J_n(t,x,y;\xi^{[n]*})\\
&=\lim_{n\to\infty}\E_{t,x}\left[\int_0^{T-t}\!\!e^{-r s}f(X^{[n]}_{t+s},y+\xi^{[n]*}_{t+s})\ud s-c_0e^{-r(T-t)}\xi^{[n]*}_T-rc_0\int_0^{T-t}\!\!e^{-rs}\xi^{[n]*}_{t+s}\ud s\right]\\
&=\E_{t,x}\bigg[\int_0^{T-t}\!\!e^{-r s}\lim_{n\to\infty}f(X^{[n]}_{t+s},y+\xi^{[n]*}_{t+s})\ud s\\
&\qquad\qquad-c_0e^{-r(T-t)}\lim_{n\to\infty}\xi^{[n]*}_T-rc_0\int_0^{T-t}\!\!e^{-rs}\lim_{n\to\infty}\xi^{[n]*}_{t+s}\ud s\bigg]\\
&=\E_{t,x}\left[\int_0^{T-t}\!\!e^{-r s}f(\widetilde X_{t+s},y+\widetilde \xi_{t+s})\ud s-c_0e^{-r(T-t)}\widetilde \xi_T-rc_0\int_0^{T-t}\!\!e^{-rs}\widetilde \xi_{t+s}\ud s\right]\\
&=\hat J(t,x,y;\widetilde{\xi}\,),\end{aligned}$$ where the penultimate equality comes from and the final one is obtained by undoing the integration by parts.
By construction $\widetilde Y$ and $\widetilde m$ fulfil the consistency condition hence we have a simple corollary.
The pair $(\widetilde m,\widetilde \xi)$ is a solution of the MFG as in Definition \[def:solMFG\]. Since $\widetilde\xi$ is indistinguishable from $\xi^*$ in then Theorem \[teo:existenceSolMFG\] holds with $$X^*=\widetilde X,\quad Y^*=Y_{0-}+\widetilde\xi=Y_{0-}+\xi^* \quad\text{and}\quad m^*=\widetilde{m}.$$
As a byproduct of this result and of Proposition \[prop:ex-uniq\] we also have that the classical connection between singular stochastic control and optimal stopping still holds in our specific mean-field game.
Approximate Nash equilibria for the $N$-player game {#sec:approximation}
===================================================
The $N$-player game: setting and assumptions {#subsec:Nplayer}
--------------------------------------------
Here we start with a formal description of the $N$-player game sketched in the introduction.
Let $\Pi:=(\Omega, \mathcal{F}, \mathbb{F} = (\mathcal{F}_t)_{t \geq 0}, \P)$ be a filtered probability space satisfying the usual conditions, supporting an infinite sequence of independent one-dimensional $\mathbb{F}$-Brownian motions $(W^{i})_{i=1}^\infty$, as well as i.i.d. $\mathcal{F}_0$-measurable initial states $(X_0^{i}, Y_0^{i})_{i = 1}^\infty$ with common distribution $\nu \in \mathcal{P}(\Sigma)$, independent of the Brownian motions. For each $N \ge 1$, define $\mathbb{F}^N = (\mathcal F_0\vee\mathcal{F}^N_t)_{t \geq 0}$, where $(\mathcal F^N_t)_{t\ge 0}$ is the augmented filtration generated by the Brownian motions $(W^{i})_{i=1}^N$. Then the filtered probability spaces $\Pi^N:=(\Omega, \mathcal{F}, \mathbb{F}^N, \P)$ satisfy the usual conditions. These are the spaces on which we define strong solutions of the $N$-player systems.
Each player $i = 1, \ldots, N$ observes/controls her own private state process $(X^{N,i}, Y^{N,i})$, whose dynamics is $$\begin{split}
& X_{t}^{N,i}= X_0^{i} + \int_{0}^{t}\,a(X_s^{N,i},m_s^{N})\,ds + \int_0^t\,\sigma(X_s^{N,i})\,\ud W_s^{i},\\
& Y_{t}^{N,i} = Y_{0-}^{i} + \xi^{N,i}_t,\quad\quad t \in [0,T],
\end{split}
\label{eq:dynamics_Nplayer_01}$$ where $\xi^{N,i}\in\Xi^{\Pi^N}(Y^{i}_{0-})$ is the strategy chosen by the $i$-th player, while $m^{N}$ is the mean-field interaction term given by $$\label{emp-mean2}
m_t^{N} = \frac{1}{N}\sum_{i = 1}^{N} Y_t^{N,i} = \int_{\Sigma}\,y\,\mu_t^{N}(\ud x,\ud y),\quad \mu_t^{N} = \frac{1}{N}\sum_{i = 1}^{N} \delta_{(X_t^{N,i}, Y_{t}^{N,i})}.$$ The process $\mu_t ^N$ above denotes the empirical distribution of the players’ states, with $\delta_{z}$ the Dirac delta mass at $z \in \Sigma$.
In the rest of this section we use the notation $$\Xi^N(Y_{0-})=\{(\xi^i)_{i=1}^N:\:\xi^i\in\Xi^{\Pi^N}(Y^{i}_{0-})\:\:\text{for all $i=1,\ldots N$ }\}.$$ We will also consider the dynamics of $(X^N,Y^N)$ conditionally on specific initial conditions $({\bf x},{\bf y}):=(x^{i}, y^{i})_{i = 1}^{N}$ drawn independently from the common initial distribution $\nu$. Therefore, the dynamics of the state variables under $\P_{{\bf x},{\bf y}}$ reads $$\begin{split}
& X_{t}^{N,i}= x^{i} + \int_{0}^{t}\,a(X_s^{N,i},m_s^{N})\,\ud s + \int_0^t\,\sigma(X_s^{N,i})\,\ud W_s^{i},\\
& Y_{t}^{N,i} = y^{i} + \xi^{N,i}_t,\quad\quad t \in [0,T].
\end{split}
\label{eq:dynamics_Nplayer_02}$$ Accordingly, since the initial conditions $({\bf x},{\bf y})=(x^{i}, y^{i})_{i=1}^N$ are drawn from the $N$-fold product of the measure $\nu$, denoted $\nu^{N}$, the expected payoff of the $i$-th player is given by $$J^{N,i}(\xi^{N}) := \int_{\Sigma^{N}} J^{N,i}({\bf x}, {\bf y}; \xi^{N}) \nu^{N}(\ud {\bf x}, \ud {\bf y}),
\nonumber$$ where $$J^{N,i}({\bf x}, {\bf y}; \xi^{N}):=\E_{{\bf x},{\bf y}}\left[\int_0^T e^{-rt}f\left(X^{N,i}_t,Y^{N,i}_t\right)\ud t-\int_{[0,T]}e^{-r t}c_0\ud \xi^{N,i}_t\right].$$
Given $\varepsilon \ge 0$, an admissible strategy vector $\xi^{\varepsilon}\in\Xi^N(Y_{0-})$ is called *$\varepsilon$-Nash equilibrium* for the $N$-player game of capacity expansion if for every $i = 1,\ldots,N$ and for every admissible individual strategy $\xi^i\in\Xi^{\Pi^N}(Y^{i}_{0-})$, we have $$J^{N,i}(\xi^{\varepsilon})\geq J^{N,i}([\xi^{{\varepsilon}, -i},\xi^i])-\varepsilon,\quad
\nonumber$$ where $[\xi^{{\varepsilon},-i},\xi^i]$ denotes the $N$-player strategy vector that is obtained from $\xi^{{\varepsilon}}$ by replacing the $i$-[th]{} entry with $\xi^i$. \[def:nashEq\]
In order to construct ${\varepsilon}$-Nash equilibria using the optimal control obtained in the MFG it is convenient to make an additional set of assumptions on the profit function.
\[ass:f2\] The running payoff $f$ is locally Lipschitz, i.e. $$|f(x,y)-f(x',y')|\leq \Lambda(x,x')(|x-x'|+|y-y'|),\quad (x,y),(x',y')\in\Sigma\nonumber.$$ Moreover, there exists $q>1$ such that the function $\Lambda: \mathbb{R} \times \mathbb{R}\rightarrow\R_+$ satisfies the integrability condition $$\label{eq:int-Lambda}
C(\Lambda,q):=\sup_{N\in\N}\,\sup_{\xi^{N,1}}\E\left[\int_0^T\Lambda^q(X_t^{N,1},X^*_t)\ud t\right]<\infty$$ where $X^{N,1}$ is the solution of , $ X^*=\widetilde X$ is the solution of obtained in the MFG (see also ) and the supremum is taken over all admissible controls $\xi^{N,1}\in\Xi^{\Pi^N}(Y^1_{0-})$ and all $N\in\N$.
The integrability condition is redundant if $f$ is Lipschitz continuous. Since $Y^{N,1}\in[0,1]$ there is no loss of generality in taking $\Lambda$ independent of $y$ and the supremum over $\xi^{N,1}$ is not restrictive either. If $\Lambda$ has polynomial growth of order $p\ge 1$, then holds thanks to the Lipschitz continuity of the coefficients $a(x,m)$ and $\sigma(x)$ as soon as $\E[(X_0)^{p\cdot q}]<\infty$. Later in Section \[subsec:gbm\] we will consider and example where $\Lambda$ has exponential growth and holds.
The next is an assumption on the optimal boundary found in Theorem \[teo:existenceSolMFG\].
\[ass:Lip-c\] The optimal boundary $(t,x)\mapsto c(t,x)$ of Theorem \[teo:existenceSolMFG\] is uniformly Lipschitz continuous in $x$ with constant $\theta_c>0$, i.e. $$\sup_{0\le t\le T}|c(t,x)-c(t,x')|\le \theta_c|x-x'|,\quad x,x'\in\R.$$
Lipschitz regularity of optimal stopping/control boundaries is a delicate issue in general. However, the question can be addressed by analytical methods (see, e.g., [@soner1991free]) or by probabilistic methods as in [@deangelisStabile]. In Section \[sec:Lip\] we show how ideas from the latter paper can be used in our context to prove that Assumption \[ass:Lip-c\] indeed holds in a large class of examples.
Approximate Nash equilibria
---------------------------
Here we prove that the MFG solution constructed in Theorem \[teo:existenceSolMFG\] induces approximate Nash equilibria in the $N$-player game of capacity expansion, when $N$ is large enough.
\[teo:approximation\] Suppose Assumptions \[ass:SDE\]–\[ass:Lip-c\] hold. Recall the feedback solution $(m^*,\xi^*)$ of the MFG of capacity expansion constructed in Theorem \[teo:existenceSolMFG\]. Recall also that $$\xi^*_t= \eta^*(t,X^*,Y_{0-}),\quad t\in[0,T],
\nonumber$$ with $\eta^*:[0,T]\times C([0,T]; \mathbb R)\times[0,1]\to [0,1]$ the non-anticipative mapping defined by $$\eta^*(t,\varphi,y):= \sup_{0\le s\le t}\Big(c(s,\varphi(s))-y\Big)^+
\nonumber$$ and with $X^*$ the dynamics in associated to $m^*$. Setting $\hat \xi^{N,i}_t:= \eta^*(t,X^{N,i},Y^i_{0-})$, the vector $\hat \xi^{N}$ is a $\varepsilon_N $-Nash equilibrium for the $N$-player game of capacity expansion according to Definition \[def:nashEq\] with $\varepsilon_N \rightarrow 0$ as $N\rightarrow\infty$. Further, if $q=2$ in of Assumption \[ass:f2\], then the rate of convergence is of order $N^{-1/2}$.
For each Brownian motion $W^i$ we introduce the following auxiliary dynamics, which are the analogues of the solution $(X^*,Y^*)$ of with $(\xi^*,m^*)$ as in Theorem \[teo:existenceSolMFG\]: $$\label{X-limN}
\begin{split}
& X^i_t=X^i_0+\int_0^t a(X^i_s,m^*(s))ds+\int_0^t\sigma(X^i _s) \ud W^i_s,\\
& Y^i_t=Y^i_{0-}+\zeta^i_t:= Y_{0-}+\eta^*(t,X^i,Y^i_{0-}),\quad t\in[0,T],\quad i\in\lbrace 1,\ldots,N\rbrace.
\end{split}$$ Notice that the initial conditions above are the same as in the dynamics of $(X^{N,i},Y^{N,i})$. Moreover, $(X^i_t,Y_t^i)_{i=1}^\infty$ is a sequence of i.i.d. random variables with values in $\mathbb R\times [0,1]$, so that in particular the law of large numbers (LLN) holds. The rest of the proof is structured in three steps:
- We prove that $J^{N,1}(\hat \xi^N)\rightarrow J(\xi^*)$ as $N\rightarrow\infty$.
- Recalling the notation $[\hat \xi^{N,-1},\xi]=(\xi,\hat \xi^{N,2},\ldots,\hat \xi^{N,N})$ introduced in Definition \[def:nashEq\] we prove $$\limsup_{N\to\infty} \sup_{\xi\in\Xi^{\Pi^N}(Y^1_{0-})} J^{N,1}([\hat \xi^{N,-1},\xi])\leq J(\xi^*)=V^\nu.
\nonumber$$
- Combining (i) and (ii), for any $\varepsilon>0$ there exists $N_{\varepsilon}\in\N$ such that $$J^{N,1}(\hat \xi^N)\geq \sup_{\xi \in \Xi^{\Pi^N}(Y^{1}_{0-})}J^{N,1}([\hat\xi^{N,-1}, \xi])-\varepsilon$$ for all $N\geq N_{\varepsilon}$.
In the three steps above we singled out the first player with no loss of generality since the $N$-player game is symmetric.
\(i) Let us start by observing that $(X^*,Y^*,\xi^*)$ from Theorem \[teo:existenceSolMFG\] and $(X^1,Y^1,\zeta^1)$ defined above have the same law, so that $$J(\xi^*)=\E\left[\int_0^T e^{-r s}f(X^1_t,Y^1_t)\ud t-c_0\int_{[0,T]}e^{-r t}\ud \zeta^1_t\right].$$ By triangular inequality we get $$\begin{aligned}
\label{eq:J-J}
|J^{N,1}(\hat \xi^N)- J(\xi^*)| & \le \E\left[\int_0^Te^{-rt}\Big|f(\hat X_t^{N,1},\hat Y_t^{N,1})-f(X_t^1,Y_t^1)\Big| \ud t \right] \\
& \quad + c_0\E\left[\left|\int_{[0,T]}e^{-rt}\ud \big(\hat \xi^{N,1}_t-\zeta^1_t\big)\right|\right],\notag\end{aligned}$$ where we use $(\hat X^{N,i},\hat Y^{N,i})$ for the state process of the $i$-th player when all players use the control vector $\hat \xi^N$. Similarly we denote by $\hat m^N$ the empirical average of the processes $\hat Y^{N,i}$.
We estimate the first term on the right-hand side using Assumption \[ass:f2\] and obtain $$\begin{aligned}
&\E\left[\int_0^T e^{-rt}\Big|f(\hat X_t^{N,1},\hat Y_t^{N,1})-f(X_t^1,Y_t^1)\Big|\ud t \right]\nonumber\\
&\leq \E\left[\int_0^Te^{-rt}\Lambda(\hat X_t^{N,1},X_t^1)\big(|\hat X_t^{N,1}-X_t^1|+|\hat Y_t^{N,1}-Y_t^1|\big)\ud t \right]\nonumber\\
&\leq C_{1}\E\left[\int_0^T\!\!\Lambda^q(\hat X_t^{N,1},X_t^1)\ud t\right]^{\frac{1}{q}}\E\left[\int_0^T\big(|\hat X_t^{N,1}-X_t^1|^{p}+|\hat Y_t^{N,1}-Y_t^1|^{p}\big)\ud t\right]^{\frac{1}{p}}\nonumber\\
&\leq C_{1} C(\Lambda,q)\E\left[\int_0^T \big(|\hat X_t^{N,1}-X_t^1|^{p}+|\hat Y_t^{N,1}-Y_t^1|^{p}\big)\ud t\right]^{\frac{1}{p}},\end{aligned}$$ for some positive constant $C_{1}=C_1(T,q)$, using Hölder’s inequality with $p=q/(q-1)$ and $q>1$ as in Assumption \[ass:f2\]. For the remaining term in we use integration by parts and $\hat \xi^{N,1}_{0-}=\zeta^1_{0-}=0$ to obtain $$\begin{aligned}
\E\left[\left|\int_{[0,T]}e^{-rt}\ud \big(\hat \xi^{N,1}_t-\zeta^1_t\big)\right|\right]=\E\left[\left|\int_0^T e^{-rt}\big(\hat \xi^{N,1}_t-\zeta_t^1\big)\ud t \right|\right]\le \E\left[\int_0^Te^{-rt}\Big|\hat \xi^{N,1}_t-\zeta_t^1\Big|\ud t\right].
\nonumber\end{aligned}$$ Recall that $\hat \xi^{N,1}_t=\eta^*(t,\hat X^{N,1},Y^{1}_{0-})$ and $\zeta^1_t=\eta^*(t,X^{1},Y^{1}_{0-})$. Then using Assumption \[ass:Lip-c\] we obtain $$\begin{aligned}
\label{lip0}
\Big|\hat \xi^{N,1}_t-\zeta_t^1\Big|\le \sup_{0\le s\le t}\big|c(s,\hat X^{N,1}_s)-c(s,X^1_s)\big|\le \theta_c \sup_{0\le s\le t}\big|\hat X^{N,1}_s-X^1_s\big|\end{aligned}$$ and the same bound also holds for $|\hat Y^{N,1}_t-Y^1_t|$. Then, combining the above estimates we arrive at $$\begin{aligned}
\label{eq:JJN}
|J^{N,1}(\hat \xi^N)- J(\xi^*)|\le&\, C_{1} C(\Lambda,q)T(1+\theta_c)\E\Big[\sup_{0\le t\le T}|\hat X_t^{N,1}-X_t^1|^{p}\Big]^{\frac{1}{p}}\\
&+c_0\theta_c T\,\E\left[\sup_{0\le s\le t}\big|\hat X^{N,1}_s-X^1_s\big|\right].\notag\end{aligned}$$ Since $p>1$ it remains to show that $$\begin{aligned}
\label{limXXN}
\lim_{N\to\infty}\E\Big[\sup_{0\le t\le T}|\hat X_t^{N,1}-X_t^1|^{p}\Big]=0.\end{aligned}$$ Repeating the same estimates as those in the proof of Lemma \[lem:convXY\] but with $(\hat X^{N,1},X^1)$ instead of $(X^{[n]},X')$ and with $(\hat m^N,m^*)$ instead of $(m^{[n]},\widetilde m)$ we obtain $$\begin{aligned}
\label{XXN-0}
\E\Big[\sup_{0\le t\le T}|\hat X_t^{N,1}-X_t^1|^{p}\Big]\le C\,\E\left[\int_0^T\left|\hat m^N_t-m^*(t)\right|^p\ud t\right],\end{aligned}$$ for some constant $C>0$ depending on $p$, $T$ and the Lipschitz constant of the coefficients $a(\,\cdot\,)$ and $\sigma(\,\cdot\,)$.
Now, observe that $$\label{eq:epsN}
\varepsilon_{p,N}:= \int_0^T\E\left[\,\left|\frac{1}{N}\sum_{i=1}^N Y^i_t-m^*(t)\right|^{p}\,\right]\ud t\rightarrow 0, \quad N \to \infty,$$ by the LLN and the bounded convergence theorem, since $(Y^i_t)_{i=1}^N$ are i.i.d. with mean $m^*(t)$ (recall that $\eta^*$ is the feedback map of the optimal control in the MFG). Hence, we have $$\begin{aligned}
\label{eq:limN}
\E\left[\int_0^T|\hat m^N_t-m^*(t)|^{p}\ud t\right] \le& 2^{p-1}\int_0^T\E\left[\,\left|\frac{1}{N}\sum_{i=1}^N(\hat Y^{N,i}_t-Y^i_t)\right|^{p}\,\right]\ud t+2^{p-1}\varepsilon_{p,N}\nonumber\\
\leq &2^{p-1}\int_0^T\frac{1}{N}\sum_{i=1}^N\E\left[|\hat Y^{N,i}_t-Y^i_t|^{p}\right]\ud t+2^{p-1}\varepsilon_{p,N}\\
=&2^{p-1}\int_0^T\E\left[\,\left|\hat Y^{N,1}_t-Y^1_t\right|^{p}\,\right]\ud t +2^{p-1}\varepsilon_{p,N}\nonumber\\
\le& 2^{p-1}\theta^p_c \int_0^T\E\left[\,\sup_{0\le s\le t}\left|\hat X^{N,1}_s-X^1_s\right|^{p}\,\right]\ud t +2^{p-1}\varepsilon_{p,N}\nonumber\end{aligned}$$ where the first inequality uses $|a+b|^p\le 2^{p-1}(|a|^p+|b|^p)$, the second inequality follows by Jensen’s inequality ($p>1$), the equality by the fact that the processes $(\hat Y^{N,i}-Y^i)_{i=1}^N $ are exchangeable and the final inequality uses applied to $|\hat Y^{N,1}_t-Y^1_t|$.
Plugging the latter estimate back into and applying Gronwall’s lemma once again we obtain $$\E\Big[\sup_{0\le t\le T}|\hat X_t^{N,1}-X_t^1|^{p}\Big]\le C' \varepsilon_{p,N},$$ for a suitable constant $C'>0$ depending on $T$ and the other constants above. Thanks to we obtain .
(ii). This part of the proof is similar to the above but now the first player deviates by choosing a generic admissible control $\xi$ while all remaining players pick $\hat \xi^{N,i}$, $i=2,\ldots, N$; we denote this strategy vector $\beta^N=[\hat \xi^{N,-1},\xi]$. In particular we notice that the empirical average associated to this strategy reads $$\frac{1}{N}\left(Y^{1}_{0-}+\xi_t+\sum_{i=2}^N(Y^i_{0-}+\hat \xi^{N,i}_t)\right)=\bar m^N_t+\frac{1}{N}(\xi_t-\hat \xi^{N,1}_t),$$ where $\bar m^N_t:=N^{-1}\sum_{i=1}^N(Y^i_{0-}+\hat \xi^{N,i}_t)$. One should be careful here that $\bar m^N$ is different to $\hat m^N$ used in the proof of (i) above, because the deviation of player 1 from the strategy vector $\hat \xi^N$ causes a knock-on effect on the dynamics of $\hat \xi^{N,i}$ for all $i$’s through the non-anticipative mapping $\eta^*(t,X^{N,i;\beta},Y^i_{0-})$. To keep track of this subtle aspect we use the notations $\hat \xi^{N,i;\beta}_t=\eta^*(t,X^{N,i;\beta},Y^i_{0-})$ and $\bar Y^{N,i;\beta}_t=Y^i_{0-}+\hat \xi^{N,i;\beta}_t$, for $i=1,\ldots, N$, in the calculations below. Accordingly, the state process of the first player reads $$\begin{aligned}
&X^{N,1;\beta}_t=X^{1}_0+\int_0^t a\big(X^{N,1;\beta}_s,\bar m^N_s+N^{-1}(\xi_s-\hat \xi^{N,1;\beta}_s)\big)\ud s+\int_0^t\sigma(X^{N,1;\beta}_s)\ud W^1_s\\
&Y^{N,1;\beta}_t=Y^{1}_{0-}+\xi_t,\qquad t\in[0,T].\end{aligned}$$
Using the above expression for $X^{N,1;\beta}$ and the same arguments as in we obtain $$\begin{aligned}
&\E\Big[\sup_{0\le t\le T}|X_t^{N,1;\beta}-X_t^1|^{p}\Big]\\
&\le C\,\E\left[\int_0^T\left|\bar m^N_t+N^{-1}(\xi_t-\hat \xi^{N,1;\beta}_t)-m^*(t)\right|^p\ud t\right]\\
&\le 2^{p-1}C\,\E\left[\int_0^T\left|\bar m^N_t-m^*(t)\right|^p\ud t\right]+2^p 2^{p-1} C T N^{-p},\end{aligned}$$ where the final inequality uses $|a+b|^p\le 2^{p-1}(|a|^p+|b|^p)$ and $|\xi_t-\hat \xi^{N,1;\beta}_t|\le 2$ (by the finite-fuel condition), and $C>0$ is a suitable constant. Repeating the same steps as in we have $$\begin{aligned}
\E\left[\int_0^T|\bar m^N_t-m^*(t)|^{p}\ud t\right] \le& 2^{p-1}\int_0^T\E\left[\,\left|\frac{1}{N}\sum_{i=1}^N(\bar Y^{N,i;\beta}_t-Y^i_t)\right|^{p}\,\right]\ud t+2^{p-1}\varepsilon_{p,N}\nonumber\\
\le&2^{p-1}\int_0^T\E\left[\,\left|\bar Y^{N,1;\beta}_t-Y^1_t\right|^{p}\,\right]\ud t +2^{p-1}\varepsilon_{p,N}\nonumber\\
\le& 2^{p-1}\theta^p_c \int_0^T\E\left[\,\sup_{0\le s\le t}\left| X^{N,1;\beta}_s-X^1_s\right|^{p}\,\right]\ud t +2^{p-1}\varepsilon_{p,N}\nonumber\end{aligned}$$ where we have used that $(\bar Y^{N,i;\beta}-Y^i)_{i=1}^N$ are exchangeable by construction. Combining the two estimates above and using Gronwall’s inequality we obtain a bound which is uniform with respect to $\xi\in\Xi^{\Pi^N}(Y^1_{0-})$. In particular we have $$\begin{aligned}
\label{limXb}
\lim_{N\to\infty}\sup_{\xi\in\Xi^{\Pi^N}(Y^1_{0-})}\E\Big[\sup_{0\le t\le T}|X_t^{N,1;\beta}-X_t^1|^{p}\Big]\le C'\lim_{N\to \infty}{\varepsilon}_{p,N} =0,\end{aligned}$$ where $C'>0$ is the constant appearing from Gronwall’s inequality. Since any $\xi\in\Xi^{\Pi^N}(Y^1_{0-})$ is admissible but suboptimal in the MFG with state process $X^1$ as in we get $$\begin{aligned}
&\sup_{\xi\in\Xi^{\Pi^N}(Y^1_{0-})}J^{N,1}([\hat \xi^{N,-1},\xi])-V^\nu\\
&\le \sup_{\xi\in\Xi^{\Pi^N}(Y^1_{0-})}\left(J^{N,1}([\hat \xi^{N,-1},\xi])-J(\xi)\right)\\
&\le \sup_{\xi\in\Xi^{\Pi^N}(Y^1_{0-})}\E\left[\int_0^T e^{-r s}\Big(f\big(X_t^{N,1;\beta}, Y^{1}_{0-}+\xi_t\big)-f\big(X_t^{1}, Y^{1}_{0-}+\xi_t\big)\Big)\ud t\right]\\
&\le \sup_{\xi\in\Xi^{\Pi^N}(Y^1_{0-})}\E\left[\int_0^Te^{-rt}\Lambda\big(X^{N,1;\beta}_t,X^1_t\big)\big|X^{N,1;\beta}_t-X^{1}_t\big|\ud t\right]\end{aligned}$$ where in the final inequality we used Assumption \[ass:f2\]. Now, arguing as in (i) and using and we obtain $$\begin{aligned}
\limsup_{N\to\infty}\sup_{\xi\in\Xi^{\Pi^N}(Y^1_{0-})}J^{N,1}\big([\hat \xi^{N,-1},\xi]\big)\le V^\nu=J(\xi^*),\end{aligned}$$ where the final equality holds by optimality of $\xi^*$ in the MFG.
(iii). This step follows from the previous two. With no loss of generality we consider only the first player as the game is symmetric. Given ${\varepsilon}>0$, thanks to (ii) there exists $N_{{\varepsilon}}>0$ sufficiently large that for any $\xi\in\Xi^{\Pi^N}(Y^{1}_{0-})$ $$J^{N,1}([\hat \xi^{N,-1},\xi])\le V^\nu +\frac{{\varepsilon}}{2}\quad\text{for all $N>N_{{\varepsilon}}$}.$$ From (i), with no loss of generality we can also assume $N_{{\varepsilon}}>0$ sufficiently such large that $$J^{N,1}(\hat \xi^N)\ge V^\nu-\frac{{\varepsilon}}{2}\quad\text{for all $N>N_{{\varepsilon}}$}.$$ Combining the two inequalities above we obtain that for all $\xi\in\Xi^{\Pi^N}(Y^1_{0-})$ holds $$J^{N,1}(\hat \xi^N)\ge J^{N,1}([\hat \xi^{N,-1},\xi])-{\varepsilon}\quad\text{for all $N>N_{{\varepsilon}}$}.$$
The final claim on the speed of convergence can be verified by taking $q=p=2$ in the above estimates. The leading term in the convergence of is $\sqrt{{\varepsilon}_{2,N}}$ (see ). Since reads $$\varepsilon_{2,N}= \int_0^T\text{Var}\left(\frac{1}{N}\sum_{i=1}^N Y^i_t\right)\ud t=\frac{1}{N}\int_0^T\text{Var}(Y^1_t)\,\ud t$$ upon noticing that $\E\big[N^{-1}\sum_{i=1}^N Y^i_t\big]=\E[Y^1_t]=m^*(t)$ since $(Y^i)_{i=1}^N$ are i.i.d., the claim follows.
Conditions for a Lipschitz continuous optimal boundary {#sec:Lip}
------------------------------------------------------
Here we complement results from [@deangelisStabile] to provide sufficient conditions under which Assumption \[ass:Lip-c\] holds. Notice that our problem is parabolic and degenerate as there is no diffusive dynamics in the $y$-direction. Therefore classical PDE results cannot be applied. Moreover, we extend [@deangelisStabile] by considering non-constant diffusion coefficients in the the dynamics of $X^*$. Thanks to Lemma \[lem:limitOS\], the question reduces to finding sufficient conditions on the data of the optimal stopping problem that guarantee a Lipschitz stopping boundary. In the dynamics of $\widetilde X$ was obtained from Lemma \[lem:convXY\] and corresponds to the dynamics of $X^*$ in the MFG. In the rest of this section we always use such $X^*$.
We make some additional assumptions on the coefficients of the SDE.
\[ass:dX\] We have $x\mapsto a(x,m)$ and $x\mapsto \sigma(x)$ continuously differentiable with $\partial_x\sigma(x)\ge 0$ and $\partial_x a(x,m)\le \bar a$ for some $\bar a>0$.
Thanks to this assumption we have that the stochastic flow $x\mapsto X^{*;\,t,x}(\omega)$ is continuously differentiable. The dynamics of $Z^{t,x}:=\partial_xX^{*;\,t,x}$ is given by [see @protter2005stochastic Chapter V.7] $$\begin{aligned}
Z_{t+s}^{t,x} = 1 + \int_{0}^{s} \!\partial_{x} a(X_{t+u}^{*;\,t,x}, m^*(t\!+\!u))Z_{t+u}^{t,x}\ud u+\!\int_{0}^{s}\! \partial_x \sigma(X_{t+u}^{*;\,t,x})Z_{t+u}^{t,x}\,\ud W_{t+u},
\label{eq:dynamic_derivative}\end{aligned}$$ for all $(t,x)\in[0,T]\times\R$ and $s\in[0,T-t]$. The solution of is explicit in terms of $X^*$ and it reads $$\begin{aligned}
Z_{t+s}^{t,x} = \exp\Big[\int_{0}^{s}\!\Big(\partial_{x} a(X_{t+u}^{*;\,t,x}, m^*(t+u))\! -\! \frac{1}{2}\partial_{x}\sigma(X_{t+u}^{*;\,t,x})^2\Big)\,\ud u + \int_{0}^{s}\!\partial_{x}\sigma(X_{t+u}^{t,x})\,\ud W_{t+u} \Big],
\label{eq:representation_of_Z}\end{aligned}$$ for $(t,x)\in[0,T]\times\R$ and $s\in[0,T-t]$. Thanks to this explicit formula we can deduce that $(t,x)\mapsto Z^{t,x}$ is a continuous flow, by continuity of the flow $(t,x)\mapsto X^{*;\,t,x}$.
Later on we will perform a change of measure using $Z$ and for that we also require:
\[ass:Z\] For all $(t,x)\in[0,T]\times\R$ we have $$\begin{aligned}
\label{ass:Z2}
\E_{t,x}\left[\int_0^{T-t}\Big(\partial_x \sigma(X^*_{t+u})Z_{t+u}\Big)^2\ud u\right]<+\infty.\end{aligned}$$
Then $$\begin{aligned}
\label{eq:Q}
\frac{\ud \Q}{\ud \P}\Big|_{\cF_T}:=Z_T \exp\left(-\int_{0}^{T}\partial_{x} a(X^*_{t}, m^*(t))\ud t\right)\end{aligned}$$ defines the Radon-Nikodym derivative of the absolutely continuous change of measure from $\P$ to $\Q$.
Next we assume some extra conditions on the profit function.
\[ass:f3\] We have $f\in C^2(\R\times(0,1))$ and either $\sigma(x)=\sigma$ is constant or we have $x\mapsto \partial_{xy}f(x,y)$ non-increasing. Moreover, the integrability condition below holds: $$\begin{aligned}
\sup_{(t,x,y)\in K}\E_{t,x,y}\left[\int_0^{T-t}\!\!\!e^{-rs}\Big(\left|\partial_{yy}f(X^*_{t+s},y)\right|+(1+Z_{t+s})\left|\partial_{xy}f(X^*_{t+s},y)\right|\Big)\ud s\right]<\infty,\end{aligned}$$ for any compact $K\subset [0,T]\times\Sigma$.
Notice that $f(x,y)=x^\alpha y^\beta$ with $\alpha\in(0,1]$ and $\beta\in(0,1)$ satisfies Assumption \[ass:f3\] combined with Assumption \[ass:SDE\]. The next proposition provides sufficient conditions for Lipschitz continuity of the optimal boundary.
\[prop:Lip\] Let Assumptions \[ass:SDE\]–\[ass:int\] and Assumptions \[ass:dX\]–\[ass:f3\] hold. If either of the two conditions below holds:
- there exist $\alpha,\gamma>0$ such that $$|\partial_{yy}f|\ge \alpha>0\quad\text{and}\quad|\partial_{xy}f|\le \gamma (1+|\partial_{yy}f|)\quad\text{on $\Sigma$};$$
- there exists $\gamma>0$ such that $|\partial_{xy}f|\le \gamma|\partial_{yy}f|$ on $\Sigma$;
then Assumption \[ass:Lip-c\] holds.
The proof of the proposition uses the next lemma concerning the optimal stopping time defined in , whose slightly technical proof we move to the appendix.
\[lem:ST\] The mapping $(t,x,y)\mapsto \tau_*(t,x,y)$ is $\P$-almost surely continuous on $[0,T]\times\Sigma$ with $\tau_*(t,x,y)=0$, $\P$-a.s. for $(t,x,y)\in\partial\cC$.
The idea of the proof combines ideas from [@deangelisStabile] and [@deangelisPeskir]. First, for $\delta>0$ we define $$c_\delta(t,x):=\inf\{y\in[0,1]: u(t,x,y)<c_0-\delta\}$$ with $\inf\varnothing=1$. Then it is clear that $c_\delta(\,\cdot\,)>c_{\delta'}(\,\cdot\,)>c(\,\cdot\,)$ for all $0<\delta'<\delta$ by monotonicity of $y\mapsto u(t,x,y)$. Since $u$ is continuous then $$\lim_{\delta\downarrow 0}c_\delta(t,x)=c(t,x)\quad(t,x)\in[0,T]\times\R$$ and if we can prove that $x\mapsto c_\delta(t,x)$ is Lipschitz with a constant independent of $\delta$ we can conclude. By continuity of $u$ we know that $$u\big(t,x,c_\delta(t,x)\big)=c_0-\delta$$ so that by the implicit function theorem, whose use is justified in step 2 below, we have $$\begin{aligned}
\label{eq:IFth}
\partial_x c_\delta(t,x)=-\frac{\partial_{x}u\big(t,x,c_\delta(t,x)\big)}{\partial_y u\big(t,x,c_\delta(t,x)\big)},\quad(t,x)\in[0,T]\times\R.\end{aligned}$$ Thanks to Corollary \[cor:reg-u\] we have $\partial_x c_\delta(t,x)\ge 0$. In step 2 below we will find an upper bound so that $|\partial_x c_\delta|\le \theta_c$ on $[0,T]\times\R$, for a suitable constant $\theta_c>0$. This concludes the proof.
[*Step 1:*]{} (Gradient estimates). We fix an arbitrary $(t,x,y)\in[0,T]\times\Sigma$ and let $\tau_*=\tau_*(t,x,y)$. Then for any ${\varepsilon}>0$ we have $$\begin{aligned}
&u(t,x,y+{\varepsilon})-u(t,x,y)\\
&\le \E\left[\int_0^{\tau_*}e^{-rs}\Big(\partial_y f(X^{*;\,t,x}_{t+s},y+{\varepsilon})-\partial_y f(X^{*;\,t,x}_{t+s},y)\Big)\ud s\right]\\
&=\int_0^{\varepsilon}\E\left[\int_0^{\tau_*}e^{-rs}\partial_{yy} f(X^{*;\,t,x}_{t+s},y+z)\ud s\right]\ud z,\end{aligned}$$ where we used Fubini’s theorem in the final equality. Dividing by ${\varepsilon}$, letting ${\varepsilon}\to 0$ and using the integrability condition from Assumption \[ass:f3\] we conclude $$\limsup_{{\varepsilon}\to 0}\frac{u(t,x,y+{\varepsilon})-u(t,x,y)}{{\varepsilon}}\le \E\left[\int_0^{\tau_*}e^{-rs}\partial_{yy} f(X^{*;\,t,x}_{t+s},y)\ud s\right].$$ Taking $\tau^{\varepsilon}_*:=\tau_*(t,x,y+{\varepsilon})$ in the first expression above we have $$\begin{aligned}
&u(t,x,y+{\varepsilon})-u(t,x,y)\\
&\ge \E\left[\int_0^{\tau^{\varepsilon}_*}e^{-rs}\Big(\partial_y f(X^{*;\,t,x}_{t+s},y+{\varepsilon})-\partial_y f(X^{*;\,t,x}_{t+s},y)\Big)\ud s\right]\\
&=\int_0^{\varepsilon}\E\left[\int_0^{\tau^{\varepsilon}_*}e^{-rs}\partial_{yy} f(X^{*;\,t,x}_{t+s},y+z)\ud s\right]\ud z.\end{aligned}$$ Dividing again by ${\varepsilon}>0$ and letting ${\varepsilon}\to 0$, we can now invoke Lemma \[lem:ST\] to justify that $\tau^{\varepsilon}_*\to\tau_*$ and obtain $$\liminf_{{\varepsilon}\to 0}\frac{u(t,x,y+{\varepsilon})-u(t,x,y)}{{\varepsilon}}\ge \E\left[\int_0^{\tau_*}e^{-rs}\partial_{yy} f(X^{*;\,t,x}_{t+s},y)\ud s\right].$$
So, in conclusion we have shown that $\partial_y u$ exists in $[0,T]\times\Sigma$ and it reads $$\partial_y u(t,x,y)=\E\left[\int_0^{\tau_*}e^{-rs}\partial_{yy} f(X^{*;\,t,x}_{t+s},y)\ud s\right].$$ Further, in light of the fact that $(t,x,y)\mapsto\tau_*(t,x,y)$ and $(t,x,y)\mapsto (X^{*;\,t,x}_{t+s},y)$ are $\P$-a.s. continuous, we deduce that $\partial_y u$ is also continuous on $[0,T]\times\Sigma$, by dominated convergence and Assumption \[ass:f3\]. Finally, since $\partial_{yy}f<0$ (Assumption \[ass:f\]-(ii)), we have that $$\begin{aligned}
\label{eq:uy<0}
\partial_{y} u\big(t,x,c_\delta(t,x)\big)<0,\quad \text{for all $(t,x)\in[0,T]\times\Sigma$,}\end{aligned}$$ because $\big(t,x,c_\delta(t,x)\big)\in\cC$ and $\tau_*>0$ at those points.
Next we obtain a similar result for $\partial_x u$. With the same notation as above we have $$\begin{aligned}
&u(t,x+{\varepsilon},y)-u(t,x,y)\\
&\le
\E\left[\int_0^{\tau_*}e^{-r s}\Big(\partial_y f(X^{*;\,t,x+{\varepsilon}}_{t+s},y)-\partial_y f(X^{*;\,t,x}_{t+s},y)\Big)\ud s\right]\\
&=\int_x^{x+{\varepsilon}}\E\left[\int_0^{\tau_*}e^{-r s}\partial_{xy} f(X^{*;\,t,\eta}_{t+s},y)Z^{t,\eta}_{t+s}\ud s\right]\ud \eta.\end{aligned}$$ Dividing by ${\varepsilon}$ and letting ${\varepsilon}\to 0$, we use dominated convergence (Assumption \[ass:f3\]) and continuity of the flows $x\mapsto\big(X^{*;\,t,x},Z^{t,x}\big)$ to conclude $$\limsup_{{\varepsilon}\to 0}\frac{u(t,x+{\varepsilon},y)-u(t,x,y)}{{\varepsilon}}\le \E\left[\int_0^{\tau_*}e^{-r s}\partial_{xy} f(X^{*;\,t,x}_{t+s},y)Z^{t,x}_{t+s}\ud s\right].$$ By a symmetric argument and continuity of the optimal stopping time we also obtain the reverse inequality and therefore conclude $$\partial_x u(t,x,y)= \E\left[\int_0^{\tau_*}e^{-r s}\partial_{xy} f(X^{*;\,t,x}_{t+s},y)Z^{t,x}_{t+s}\ud s\right].$$ Also in this case continuity of $(t,x,y)\mapsto \tau_*(t,x,y)$, due to Lemma \[lem:ST\], and $(t,x)\mapsto\big(X^{*;\,t,x},Z^{t,x}\big)$, combined with dominated convergence, imply that $\partial_x u$ is continuous on $[0,T]\times \Sigma$.
[*Step 2:*]{} (Bound on $\partial_x c_\delta$). Since $\partial_yu$ and $\partial_xu$ are continuous and holds, the equation in is fully justified as an application of the implicit function theorem. In this step we use the probabilistic representations of $\partial_x u$ and $\partial_y u$ to obtain an upper bound on $\partial_x c_\delta$. First we recall the change of measure induced by $Z$ (see ) and we use it to write $$\partial_x u(t,x,y)= \E^{\Q}_{t,x}\left[\int_0^{\tau_*}e^{-r s+\int_0^s a(X^*_{t+u},m^*(t+u))\ud u}\partial_{xy} f(X^{*}_{t+s},y)\ud s\right].$$ We want to find an upper bound for $\partial_x u$ in terms of the process under the original measure $\P$. Under the measure $\Q$ we have, by Girsanov theorem, that $X^*$ evolves according to $$\ud X^*_{t+s}=\big[a(X^*_{t+s},m^*(t+s))+\sigma(X^*_{t+s})\partial_x\sigma(X^*_{t+s})\big]\ud s+\sigma(X^*_{t+s})\ud W^\Q_{t+s},$$ where $W^\Q_{t+s}=W_{t+s}-\int_0^s\partial_x\sigma(X^*_{t+u})\ud u $ defines a Brownian motion under $\Q$. Analogously, under the original measure $\P$ we can define a process $\bar X$ with the same dynamics, i.e., $$\ud \bar X_{t+s}=\big[a(\bar X_{t+s},m^*(t+s))+\sigma(\bar X_{t+s})\partial_x\sigma(\bar X_{t+s})\big]\ud s+\sigma(\bar X_{t+s})\ud W_{t+s},$$ and denote $$\bar\tau_*:=\inf\{s\in[0,T-t]: c(t+s,\bar X_{t+s})\ge y\}.$$ Then we have that the processes and stopping times are equal in law, i.e. $$\text{Law}^\Q\big(X^*,\tau_*\big)=\text{Law}^\P\big(\bar X,\bar \tau_*\big)$$ and we can express $\partial_x u$ in terms of the original measure as $$\begin{aligned}
\label{eq:lip-p}
\partial_x u(t,x,y)= \E_{t,x}\left[\int_0^{\bar \tau_*}e^{-r s+\int_0^s a(\bar X_{t+u},m^*(t+u))\ud u}\partial_{xy} f(\bar X_{t+s},y)\ud s\right].\end{aligned}$$
Let us first consider $x\mapsto \sigma(x)$ not constant. By comparison principles we have $\bar X\ge X^*$ since $\partial_x \sigma\ge 0$ (Assumption \[ass:dX\]), therefore $\partial_{xy} f(\bar X,y)\le \partial_{xy} f(X^*,y)$ by Assumption \[ass:f3\]. Since $x\mapsto c(t,x)$ is non-decreasing as pointwise limit of non-decreasing functions (recall Proposition \[prop:OSboundary\]), then $c(t+s,\bar X_{t+s})\ge c(t+s,X^*_{t+s})$, hence implying $\bar \tau_*\le \tau_*$, $\P$-a.s. Recalling that $\partial_{xy}f>0$ from Assumption \[ass:f\] and combining these few facts we have $$\partial_x u(t,x,y)\le e^{\bar a(T-t)}\E_{t,x}\left[\int_0^{\tau_*}e^{-r s}\partial_{xy} f(X^*_{t+s},y)\ud s\right],$$ where we also used $\partial_x a\le \bar a$ (Assumption \[ass:dX\]). If instead $\sigma(x)=\sigma$ is constant then $X^*=\bar X$ by uniqueness of the SDE and therefore the above estimate follows directly from . Plugging this bound in and recalling that $\partial_{yy}f<0$ we obtain $$0\le \partial_xc_\delta(t,x)\le e^{\bar a(T-t)}\frac{\E_{t,x}\left[\int_0^{\tau_*}e^{-r s}\partial_{xy} f\big(X^*_{t+s},c_\delta(t,x)\big)\ud s\right]}{\E_{t,x}\left[\int_0^{\tau_*}e^{-r s}\big|\partial_{yy} f\big(X^*_{t+s},c_\delta(t,x)\big)\big|\ud s\right]}.$$ Now, if condition (i) holds we obtain $$0\le \partial_xc_\delta(t,x)\le e^{\bar a(T-t)}\left(\frac{\gamma}{\alpha}+\gamma\right), \quad\text{for all $(t,x)\in[0,T]\times\R$ and any $\delta>0$,}$$ whereas if condition (ii) holds we obtain $$0\le \partial_xc_\delta(t,x)\le e^{\bar a(T-t)}\gamma, \quad\text{for all $(t,x)\in[0,T]\times\R$ and any $\delta>0$}.$$ So in the first case Assumption \[ass:Lip-c\] holds with $$\theta_c=e^{\bar a T}\left(\frac{\gamma}{\alpha}+\gamma\right),$$ and in the second case with $\theta_c=\gamma \exp (\bar a T)$.
Next we provide a couple of examples meeting the requirements of Proposition \[prop:Lip\].
Let $a(x,m):= \alpha (m-x)$ for some $\alpha>0$ and $\sigma(x)\equiv \sigma$ for some $\sigma>0$. Given a Borel function $m:[0,T]\to[0,1]$ the dynamics of $X$ from reads $$X_t=X_0+\int_0^t\alpha (m(s)-X_s)\ud s+\sigma W_t,\quad t\in[0,T].$$ Let $f(x,y):= e^xy^\beta$ for some $\beta\in(0,1)$ and for all $(x,y)\in\Sigma$. Finally assume that $\E[\exp (qX_0)]<\infty$ for some $q\ge 1$.
We check the assumptions of Proposition \[prop:Lip\]. Assumptions \[ass:SDE\] and Assumption \[ass:dX\] on the dynamics’ coefficients are trivially satisfied. The profit function $f$ has the monotonicity required by Assumption \[ass:f\]-(i) and it is strictly concave (Assumption \[ass:f\]-(ii)). Also, $\partial_{xy}f(x,y)= \beta e^xy^{\beta-1}>0$ (Assumption \[ass:f\]-(iii)) and is satisfied since $$\begin{aligned}
\lim_{x\to -\infty} \frac{\beta e^x}{y^{1-\beta}}=0< rc_0<\lim_{x\to \infty} \frac{\beta e^x}{y^{1-\beta}}=+\infty
\nonumber\end{aligned}$$ for any $y\in(0,1)$ fixed. The integrability Assumption \[ass:int\] is satisfied by the Ornstein-Uhlenbeck dynamics with initial condition as above. Assumption \[ass:Z\] is another integrability assumption that reduces to finiteness of the second moment of the exponential martingale $Z$ (which is satisfied by boundedness of $\partial_x a$ and $\partial_x\sigma$). Assumption \[ass:f3\] holds because $\sigma$ is constant. Finally Assumption (ii) in Proposition \[prop:Lip\] is satisfied with any $\gamma\ge \frac{1}{1-\beta}$ since $$|\partial_{xy}f(x,y)|=\frac{\beta e^x}{y^{1-\beta}}\quad\text{and}\quad
|\partial_{yy}f(x,y)|=\frac{\beta(1-\beta)e^x}{y^{2-\beta}}.
\nonumber$$ We also notice that Assumption \[ass:f2\] holds so that Theorem \[teo:approximation\] can be applied.
Let $a(x,m):= \alpha m x $ for some $\alpha>0$ and $\sigma(x):=\sigma x$ for some $\sigma>0$. Given a Borel function $m:[0,T]\to[0,1]$, the dynamics of $X$ from reads $$X_t=X_0+\int_0^t\alpha X_s m(s)\ud s+\int_0^t\sigma X_s \ud W_s,\quad t\in[0,T].$$ Non-negativity of the trajectories reduces the state space $\Sigma$ to $[0,\infty)\times[0,1]$ (see Remark \[rem:statespace\]). Let $f(x,y):= (1+x)(1+y)^\beta$ for some $\beta\in(0,1)$ and for all $(x,y)\in\Sigma$. Finally assume that $\nu\in\mathcal{P}_2(\Sigma)$ and that $rc_0>\beta$.
Let us check the assumptions of Proposition \[prop:Lip\]. Assumptions \[ass:SDE\] and Assumption \[ass:dX\] on the dynamics’ coefficients are trivially satisfied. The profit function $f$ has the monotonicity required by Assumption \[ass:f\]-(i) and is strictly concave (Assumption \[ass:f\]-(ii)). Also, $\partial_{xy}f(x,y)= \beta(1+y)^{\beta-1}>0$ (Assumption \[ass:f\]-(iii)) and Eq. is satisfied since $$\begin{aligned}
\lim_{x\to 0} \frac{\beta(1+x)}{(1+y)^{1-\beta}}=\frac{\beta}{(1+y)^{1-\beta}}<\beta< rc_0<\lim_{x\to \infty} \frac{\beta(1+x)}{(1+y)^{1-\beta}}=+\infty
\nonumber\end{aligned}$$ for any $y\in(0,1)$ fixed. The integrability Assumption \[ass:int\] is satisfied with $p=2$ (or higher provided the initial condition has finite $p$-th moment) thanks to sub-linearity of the logarithm and standard estimates on the GBM dynamics. Assumption \[ass:Z\] is another integrability assumption that reduces to finiteness of the second moment of the exponential martingale $Z$ (which is satisfied by boundedness of $\partial_x a$ and $\partial_x\sigma$). Assumption \[ass:f3\] holds because $x\mapsto\partial_{xy}f(x,y)$ is decreasing. Finally Assumption (ii) in Proposition \[prop:Lip\] is satisfied with any $\gamma\ge \frac{2}{1-\beta}$ since $$|\partial_{xy}f(x,y)|=\frac{\beta}{(1+y)^{1-\beta}}\quad\text{and}\quad
|\partial_{yy}f(x,y)|=\frac{\beta(1-\beta)(1+x)}{(1+y)^{2-\beta}}.
\nonumber$$ We also notice that Assumption \[ass:f2\] holds so that Theorem \[teo:approximation\] can be applied.
We would like to emphasise that the conditions of Proposition \[prop:Lip\] are far from being necessary. While it would be overly complicated to state a general theorem in this sense, we provide below an example with a clear economic interpretation for which Proposition \[prop:Lip\] is not directly applicable.
A model with Geometric Brownian motion {#subsec:gbm}
--------------------------------------
Let us assume that $a(x,m)=(\mu + m)x$ and $\sigma(x)=\sigma\,x$ for some $\mu\in\R$ and $\sigma\in\R_+$. Let us also assume that $f(x,y)= x g(y)$ with $g\in C^2([0,1])$, $g> 0$, strictly concave and strictly increasing. This specification corresponds to the classical model of the goodwill problem in which firms produce a good whose price evolves as a geometric Brownian motion and revenues are linear in the price of the good and increasing and concave in the amount of investment that goes towards advertising.
On the one hand, Assumptions \[ass:SDE\]–\[ass:int\] are easily verified and Theorem \[teo:existenceSolMFG\] holds (i.e., our construction of the solution to the MFG holds). On the other hand, neither (i) or (ii) in Proposition \[prop:Lip\] hold, so we cannot apply directly the result on Lipschitz continuity of the boundary which is needed for the approximation result in Theorem \[teo:approximation\]. However, we shall now see how an alternative argument of proof can be applied to prove that Assumption \[ass:Lip-c\] remains valid.
First of all we change our coordinates by letting $\psi:=\ln x$, so that the value function of the optimal stopping problem can be written as $$\widetilde u(t,\psi,y):=u(t,e^\psi,y)=\inf_{\tau\in\T_t}\E_{t,\psi}\left[\int_0^\tau e^{-rs}g'(y)e^{\Psi_{t+s}}\ud s+e^{-r\tau}c_0\right],$$ where $\Psi_{t+s}:=\ln X^*_{t+s}$ is just a Brownian motion with drift, i.e., $$\Psi^{t,\psi}_{t+s}=\psi+\int_0^s\big(\mu-\sigma^2/2+m^*(t+s)\big)\ud s+\sigma\ud (W_{t+s}-W_t).$$ The optimal boundary can also be expressed in terms of $(t,\psi)$ by putting $\widetilde c(t,\psi)=c(t,e^\psi)$. Then the mean-field optimal control reads $$\xi^*_{t}=\sup_{0\le s\le t}\Big(\widetilde c(s,\Psi_s)-y\Big)^+,\quad t\in[0,T]$$ whereas the optimal stopping time for the value $\widetilde u (t,\psi,y)$ reads $$\tau_*=\inf\{s\in[0,T-t]:\, \widetilde c\,(t+s,\Psi_{t+s})\ge y\}.$$ Now we show that the optimal boundary $\widetilde c(\,\cdot\,)$ is indeed Lipschitz with respect to $\psi$ and therefore the proof of Theorem \[teo:approximation\] can be repeated with $\Psi$ instead of $X^*$ so that the theorem holds as stated. Since $\partial_\psi \Psi^{t,\psi}_{t+s}\equiv 1$ for $s\in[0,T-t]$ and Assumption \[ass:f3\] holds, we can use the same arguments as in step 1 of the proof of Proposition \[prop:Lip\] to obtain $$\partial_y \widetilde u(t,\psi,y)=g''(y)\E_{t,\psi}\left[\int_0^{\tau_*}e^{-rs +\Psi_{t+s}}\ud s\right]$$ and $$\partial_\psi \widetilde u(t,\psi,y)=g'(y)\E_{t,\psi}\left[\int_0^{\tau_*}e^{-rs +\Psi_{t+s}}\ud s\right].$$ Then, by the same arguments as in step 2 of the proof of Proposition \[prop:Lip\] we obtain $$\partial_\psi \widetilde c_\delta(t,\psi)=-\frac{\partial_x\widetilde w\big(t,\psi,\widetilde c_\delta(t,\psi)\big)}{\partial_y\widetilde w\big(t,\psi,\widetilde c_\delta(t,\psi)\big)}=\frac{g'\big(\widetilde c_\delta(t,\psi)\big)}{|g''\big(\widetilde c_\delta(t,\psi)\big)|}\le \kappa,$$ for some $\kappa>0$, where the final inequality holds because $g\in C^2([0,1])$ and strictly concave. Therefore for the optimal boundary we have $$\sup_{0\le t\le T}|\widetilde c(t,\psi_1)-\widetilde c(t,\psi_2)|\le \kappa|\psi_1-\psi_2|,\quad\psi_1,\psi_2\in\R,$$ as needed. In conclusion, the result of Theorem \[teo:approximation\] remains valid, even though the optimal boundary in the original parametrisation of the problem is not uniformly Lipschitz.
Appendix
========
In this appendix we collect a number of technical results used in the paper.
[**Proof of Proposition \[th:theoremConnection\]**]{}. Fix $(t,x,y) \in [0,T] \times \Sigma$. Take any admissible control $\zeta\in\Xi_{t,x}(y)$ and define, for $q \geq 0$, its right-continuous inverse (see, e.g., [@revuz2013continuous Ch. 0, Sec. 4]) $\tau^{\zeta}(q):=\inf\{s \in [t,T): \,\,\zeta_{s-t} > q\} \wedge T$. The process $\tau^{\zeta} :=(\tau^{\zeta}(q))_{q \geq 0}$ has increasing right-continuous sample paths, hence it admits left limits $\tau_{-}^{\zeta}(q):=\inf\left\{s \in [t,T): \,\zeta_{s-t} \geq q\right\} \wedge T$, for $q \geq 0$. It can be shown that both $\tau^{\zeta}(q)$ and $\tau^{\zeta}_{-}(q)$ are $(\mathcal{F}_{t+s})$-stopping times for any $q \geq 0$.
Let now $q = z - y$ for $z \geq y$ and consider the function $w$ defined as $$\begin{aligned}
w(t, x,y)&:=\Phi_n(t, x) -\int_{y}^{1} u_n(t, x, z)\,\ud z.\end{aligned}$$ Since $\tau^\zeta(z-y)$ is admissible for $u_n(t,x,z)$ we have $$\begin{aligned}
&w(t, x,y) - \Phi_n(t, x)
&\geq - \int_{y}^{1} \E_{t,x}\left[ c_0 e^{-r \tau^{\zeta}(z-y)}+\int_{t}^{\tau^{\zeta}(z-y)} e^{-r s}\partial_y f(X^{[n]}_s,z)\ud s\right]\ud z.\end{aligned}$$ In order to compute the integral with respect to $\ud z$ we observe that for $t\le s<T$ we have $$\{\zeta_{s-t} < z- y \}\subseteq\{ s < \tau^{\zeta}(z-y)\} \subseteq \{ \zeta_{s-t}\leq z - y\}$$ by right-continuity and monotonicity of the process $s\mapsto \zeta_{s-t}$. The left-most and right-most events above are the same up to $\ud z$-null sets. Then, applying Fubini’s theorem more than once we obtain $$\begin{aligned}
&w(t, x,y) - \Phi_n(t, x) \nonumber\\
&\geq \E_{t,x}\left[ - \int_{y}^{1} e^{-r \tau^{\zeta}(z-y)} c_0 \,\ud z - \int_{t}^{T} e^{-r s}\int_{y}^{1}\partial_y f(X^{[n]}_s, z){\mathbf 1}_{\{s<\tau^{\zeta}(z-y)\}}\,\ud z\,\ud s \right] \nonumber\\
& = \E_{t, x}\left[- \int_{y}^{1} e^{-r \tau^{\zeta}(z-y)} c_0 \,\ud z - \int_{t}^{T} e^{-r s}\int_{y}^{1}\partial_y f(X^{[n]}_s, z) {\mathbf 1}_{\left\{\zeta_{s-t} < z - y\right\}}\ud z\,\ud s \right] \nonumber\\
& = \E_{t, x}\left[- \int_{y}^{1} e^{-r \tau^{\zeta}(z-y)} c_0 \,\ud z - \int_{t}^{T} e^{-r s}[f(X^{[n]}_s,1) - f(X^{[n]}_s, y + \zeta_{s-t})]\,\ud s \right] \nonumber\\
& = J_n(t, x, y; \zeta) - \Phi_n(t, x),
\nonumber\end{aligned}$$ where the final equality uses the well-known change of variable formula [see, e.g., @revuz2013continuous Ch. 0, Proposition 4.9] $$\int_{y}^{1} e^{-r \tau^{\zeta}(z-y)}\ud z=\int_{[t,T]}e^{-rs}\ud\zeta_{s-t} .$$ By the arbitrariness of $\zeta\in\Xi_{t,x}(y)$ we conclude $w_n(t, x,y)\geq v_n(t,x,y)$.
For the reverse inequality we take $\zeta_{s}=\xi^{[n]*}_{t+s}$ as defined in Lemma \[lem:SK\]. Recall that $$\tau^{[n]}_*(t,x,z) = \inf\big\{s \in [0,T-T]:\, z \leq c_n(t+s, X_{t+s}^{[n];\,t,x}) \big\}.$$ and since $s\mapsto c_n(s, X_s^{[n];\,t, x})-z$ is upper semi-continuous, it attains a maximum over any compact interval in $[t,T)$. In particular, for $s\in[t,T)$ $$\begin{aligned}
\text{$\tau^{[n]}_*(t,x,z)\leq s \iff $there exists $\theta\in[t,t+s]$ such that $c_n(\theta, X^{[n];\,t,x}_{\theta})\ge z$}.\end{aligned}$$ For any $y<z$, the latter is also equivalent to $$\begin{aligned}
\xi^{[n]*}_{t+s}= \sup_{0\leq u \leq s}\left(c_n(t+u, X^{[n];\,t,x}_{t+u})- y\right)^{+} \geq z-y\end{aligned}$$ and, therefore, it is also equivalent to $\tau_{-}^{\xi^{[n]*}}(z-y) \leq s$. Since $s\in[t,T]$ was arbitrary the chain of equivalences implies that $\tau_{-}^{\xi^{[n]*}}(z-y) = \tau^{[n]}_*(t,x,z)$, $\P$-a.s. for any $z>y$. However, we have already observed that for a.e. $z > y$ it must be $\tau^{\xi^{[n]*}}_{-}(z-y) = \tau^{\xi^{[n]*}}(z-y)$, $\P$-a.s., hence $\tau^{\xi^{[n]*}}(z-y) = \tau^{[n]}_*(t,x,z)$ as well. The latter, in particular implies $$\begin{aligned}
&w(t, x,y) - \Phi_n(t, x)
=- \int_{y}^{1} \E_{t,x}\Big[ c_0 e^{-r \tau^{\xi^{[n]*}}(z-y)}+\int_{t}^{\tau^{\xi^{[n]*}}(z-y)} e^{-r s}\partial_y f(X^{[n]}_s,z)\ud s\Big]\ud z,\end{aligned}$$ by optimality of $\tau^{[n]}_*(t,x,z)$ in $u_n(t,x,z)$. Repeating the same steps as above we then find $$w(t,x,y)=J_n(t,x,y;\xi^{[n]*}),$$ which combined with $v_n\le w$ concludes the proof.$\square$
[**Proof of Lemma \[lem:tau-n\]**]{}. We have $\cC^{[n]}\subset \cC^{[n+1]}\subset \cC$ because the sequence $(c_n)_{n\ge 0}$ is decreasing. Then, the sequence $(\tau^{[n]}_*)_{n\ge 0}$ is increasing and $\lim_{n\to\infty}\tau^{[n]}_*\le \tau_*$, $\P_{t,x,y}$-a.s. for any $(t,x,y)\in[0,T]\times\Sigma$. In order to prove the reverse inequality, first we observe that $t\mapsto X^{[k]}_t(\omega)$ is continuous for all $\omega\in\Omega\setminus N_k$ with $\P(N_k)=0$, for all $k\ge 0$. Moreover, $t\mapsto X_t(\omega)$ is continuous for all $\omega\in\Omega\setminus N$ with $\P(N)=0$. Let us set $N_0:=(\cup_{k}N_k)\cup N$ and $\Omega_0:=\Omega\setminus N_0$ so that $\P(\Omega_0)=1$. Fix $(t,x,y)\in[0,T]\times\Sigma$ and $\omega\in\Omega_0$. Let $\delta>0$ be such that $\tau_*(\omega)>\delta$ (if no such $\delta$ exists, then $\tau_*(\omega)=0$ and $\tau^{[n]}_*(\omega)\ge \tau_*(\omega)$ for all $n\ge 0$). Then, since $s\mapsto u(t+s,\widetilde X_{t+s}(\omega),y)$ is continuous, there must exist ${\varepsilon}>0$ such that $$\begin{aligned}
\label{eq:st0}
\sup_{0\le s\le \delta}\left(u(t+s,\widetilde X_{t+s}(\omega),y)-c_0\right)\le -{\varepsilon}.\end{aligned}$$ At the same time we also notice that $s\mapsto u_n(t+s,X^{[n]}_{t+s}(\omega),y)$ is continuous and moreover $$u_n(t+s,X^{[n]}_{t+s}(\omega),y)\ge u_{n+1}(t+s,X^{[n]}_{t+s}(\omega),y)\ge u_{n+1}(t+s,X^{[n+1]}_{t+s}(\omega),y)$$ by monotonicity of the sequences $(u_n)_{n\ge 0}$ and $(X^{[n]})_{n\ge 0}$ and of the map $x\mapsto u_n(t,x,y)$. So we have that $u_n(t+\cdot,X^{[n]}_\cdot(\omega),y)$ is a decreasing sequence of continuous functions of time and since the limit is also continuous, the convergence is uniform on $[0,\delta]$. Then, there exists $n_0\ge 0$ sufficiently large that $$\sup_{0\le s\le \delta}\left|u(t+s,\widetilde X_{t+s}(\omega),y)-u_n(t+s,X^{[n]}_{t+s}(\omega),y)\right|\le -\frac{{\varepsilon}}{2}, \quad\text{for $n\ge n_0$}.$$ Using this fact and we have $$\begin{aligned}
\sup_{0\le s\le \delta}\left(u_n(t+s,X^{[n]}_{t+s}(\omega),y)-c_0\right)\le -\frac{{\varepsilon}}{2}\end{aligned}$$ and $\tau^{[n]}_*(\omega)>\delta$, for all $n\ge n_0$. Since $\delta>0$ was arbitrary, we obtain $$\lim_{n\to\infty}\tau^{[n]}_*(\omega)\ge \tau_*(\omega).$$ Recalling that $\omega\in\Omega_0$ was also arbitrary we obtain the desired result.$\square$
[**Proof of Lemma \[lem:ST\]**]{}. The proof is divided in two steps: first we show that $(t,x,y)\mapsto \tau_*(t,x,y)$ is lower semi-continuous and then that it is upper semi-continuous. Both parts of the proof rely on continuity of the flow $(t,x,s)\mapsto X^{*;\,t,x}_{t+s}(\omega)$. The latter holds for all $\omega\in\Omega\setminus N$ where $N$ is a universal set with $\P(N)=0$. For simplicity, in the rest of the proof we just write $X$ instead of $X^*$.
[*Step 1.*]{} (Lower semi-continuity). This part of the proof is similar to that of Lemma \[lem:tau-n\]. Fix $(t,x,y)\in[0,T]\times\Sigma$ and take a sequence $(t_n,x_n,y_n)_{n\ge 1}$ that converges to $(t,x,y)$ as $n\to \infty$. Denote $\tau_*=\tau_*(t,x,y)$ and $\tau_n:=\tau_*(t_n,x_n,y_n)$ and fix an arbitrary $\omega\in\Omega\setminus N$. If $(t,x,y)\in \cS$ then $\tau_*(\omega)=0$ and $\liminf_n \tau_n(\omega)\ge \tau_*(\omega)$ trivially. Let $\delta>0$ be such that $\tau_*(\omega)>\delta$. Then by continuity of the value function $u$ and of the trajectory $s\mapsto X^{t,x}_{t+s}(\omega)$ there must exist ${\varepsilon}>0$ such that $$\sup_{0\le s\le \delta}\big(u(t+s,X^{t,x}_{t+s}(\omega),y)-c_0\big)\le -{\varepsilon}.$$ Thanks to continuity of the stochastic flow there is no loss of generality in assuming that $(t_n+s,X^{t_n,x_n}_{t_n+s}(\omega),y_n)$ lies in a compact $K$ for all $n\ge 1$ and $s\le \delta$. Then there must exits $n_{\varepsilon}>0$ such that $$\sup_{0\le s\le \delta}\big|u(t+s,X^{t,x}_{t+s}(\omega),y)-u(t_n+s,X^{t_n,x_n}_{t_n+s}(\omega),y)\big|\le {\varepsilon}/2$$ for all $n\ge n_{\varepsilon}$ (by uniform continuity). Combining the above we get $$\sup_{n\ge n_{\varepsilon}}\sup_{0\le s\le \delta}\big(u(t_n+s,X^{t_n,x_n}_{t_n+s}(\omega),y_n)-c_0\big)\le -{\varepsilon}/2,$$ which implies $\tau_n(\omega)>\delta$ for all $n\ge n_{\varepsilon}$. Hence $\liminf_n\tau_n(\omega)>\delta$ and since $\delta$ and $\omega$ were arbitrary we conclude $\liminf_n\tau_n(\omega)>\tau_*(\omega)$, for all $\omega\in\Omega\setminus N$.
[*Step 2.*]{} (Upper semi-continuity). For this part of the proof we need to introduce the [*hitting time*]{} $\sigma_*^\circ$ to the interior of the stopping set $\cS^\circ:=\text{int}(\cS)$ (which is not empty thanks to the argument of proof of Lemma \[lem:limitOS\]), i.e., $$\sigma^\circ_*(t,x,y):=\inf\{s\in(0,T-t]\,:\, (t+s,X^{t,x}_{t+s},y)\in\cS^\circ\}.$$ Assume for a moment that $$\begin{aligned}
\label{usc}
\P_{t,x,y}(\tau_*=\sigma^\circ_*)=1\quad\text{for all $(t,x,y)\in[0,T]\times\Sigma$}.\end{aligned}$$ Then we can invoke Lemma 4 in [@deangelisPeskir] (see Eq. (3.7) therein) to conclude that $(t,x,y)\mapsto \sigma^\circ_*(t,x,y)$ is upper semi-continuous. Hence, given $(t,x,y)\in[0,T]\times\Sigma$ and any sequence $(t_n,x_n,y_n)_{n\ge 1}$ converging to $(t,x,y)$ as $n\to\infty$, setting $\tau_n=\tau_*(t_n,x_n,y_n)$ and $\sigma_n^\circ=\sigma^\circ_*(t_n,x_n,y_n)$, we have $\tau_n(\omega)=\sigma^\circ_n(\omega)$ for all $\omega\in\Omega_n$ with $\P(\Omega_n)=1$ for each $n\ge 1$; therefore letting $\bar\Omega:=\cap_{n\ge 1}\Omega_n$ we have $\P(\bar \Omega)=1$ and $$\limsup_{n\to\infty}\tau_n(\omega)=\limsup_{n\to\infty}\sigma^\circ_n(\omega)\le \sigma_*^\circ(\omega)=\tau_*(\omega),$$ with $\tau_*=\tau_*(t,x,y)$ and $\sigma^\circ_*=\sigma^\circ_*(t,x,y)$, for all $\omega\in\bar\Omega\cap\{\tau_*=\sigma^\circ_*\}$ where $\P(\bar\Omega\cap\{\tau_*=\sigma^\circ_*\})=1$.
Let us now prove . We introduce the generalised left-continuous inverse of $x\mapsto c(t,x)$, i.e. $$b(t,y)=\sup\{x\in\R: c(t,x)<y\}.$$ Then it is easy to check that $t\mapsto b(t,y)$ is non-increasing. This implies that $\P_{t,x,y}(\tau_*=\sigma^\circ_*)=1$ for all $(t,x,y)\in\cS^\circ$ by continuity of the paths of $X$. Moreover, for $(t,x,y)\in\cC$ we have $\sigma^\circ_*=\tau_*+\sigma^\circ_*\circ\theta_{\tau_*}$, where $\{\theta_t,\,t\ge 0\}$ is the shift operator, i.e., $(t,X_t(\omega))\circ\theta_s=(t+s,X_{t+s}(\omega))$. Then, $\tau_*=\sigma^\circ_*$ if and only if $\sigma^\circ_*\circ\theta_{\tau_*}=0$. Since $\sigma^\circ_*\circ\theta_{\tau_*}$ is the hitting time to $\cS^\circ$ after the process $(t,X,y)$ has reached the boundary $\partial\cC$ of the continuation set, the previous condition is implied by $\P_{t,x,y}(\sigma^\circ_*=0)=1$ for $(t,x,y)\in\partial\cC$. So we now focus on proving the latter.
We claim that $$\cS^\circ=\{(t,x,y):\,x>b(t,y)\}$$ and will give a proof of this fact in Lemma \[lem:app\] below. Then by the law of iterated logarithm and non-increasing $t\mapsto b(t,y)$ we immediately obtain $\P_{t,x,y}(\sigma^\circ_*=0)=1$ for $(t,x,y)\in\partial\cC$ because $(t,x,y)\in\partial\cC$ if and only if $x\ge b(t,y)$.$\square$
\[lem:app\] We have $$\begin{aligned}
\label{claim1}
\cS^\circ=\{(t,x,y):\,x>b(t,y)\}.\end{aligned}$$
While the claim may seem obvious, since $y\mapsto b(t,y)$ is non-decreasing, one should notice that for it to hold we must rule out the case $b(t,y)<b(t,y+)$ for all $(t,y)\in[0,T]\times[0,1)$. Indeed, if the latter occurs for some $(t_0,y_0)$ we have $\{t_0\}\times (b(t_0,y_0),b(t_0,y_0+))\times\{y_0\}\in\partial\cC$ and fails.
Here we use an argument by contradiction inspired to [@de2015note]. Assume that there exists $(t_0,y_0)\in[0,T]\times[0,1]$ such that $x^0_1:=b(t_0,y_0)<b(t_0,y_0+)=:x^0_2$. Then we proceed in two steps.
[*Step 1*]{}. (A PDE for the value function). Since $(t,x)\mapsto a(x,m^*(t))$ is not continuous in general we cannot immediately apply standard PDE arguments that guarantee that $$\partial_t u+\frac{\sigma^2(\,\cdot\,)}{2}\partial_{xx} u + a(\,\cdot,m^*(\,\cdot\,))\partial_x u-r u =-\partial_y f,\quad\text{for $(t,x,y)\in\cC$}$$ [see @peskirShyriyaev Chapter III]. However, given $\delta>0$ and letting $\cO_\delta:=(t_0-\delta,t_0)\times(x^0_1,x^0_2)$ we have $\cO_\delta\times(y_0,y_0+\delta)\subset\cC$. Moreover, with no loss of generality we can assume $\delta>0$ sufficiently small and such that $m^*_-(t):=\lim_{{\varepsilon}\downarrow 0}m^*(t-{\varepsilon})$ is continuous on $(t_0-\delta,t_0]$ (recall that $m^*$ is non-decreasing and right-continuous). Since $m^*$ is non-decreasing it has at most countably many jumps on any compact and therefore replacing $m^*$ with $m^*_-$ in the dynamics of $X$ (recall that $m^*=\widetilde m$) we obtain a new process $X'$ which is indistinguishable from the original one. Then, starting from $(t,x,y)\in\cO_\delta\times(y_0,y_0+\delta)$ and letting $\tau_\cO$ be the first exit time of $(t+s,X'_{t+s})$ from $\cO_\delta$ we have that $$s\mapsto e^{-r (s\wedge\tau_\cO)}u(t+s\wedge\tau_\cO, X'_{s\wedge\tau_\cO},y)+\int_0^{s\wedge\tau_\cO}e^{-r u}\partial_y f(X'_{t+u},y)\ud u$$ is a continuous martingale. By standard arguments [see @peskirShyriyaev Chapter III] this translates to the fact that [*for each $y\in(y_0,y_0+\delta)$*]{} the value function $u(\,\cdot\,,y)$ is the unique solution in $C^{1,2}(\cO_\delta)\cap C(\overline\cO_\delta)$ of the boundary value problem $$\begin{aligned}
\label{eq:PDE}
\partial_t w+\frac{\sigma^2(\,\cdot\,)}{2}\partial_{xx} w + a(\,\cdot,m^*_-(\,\cdot\,))\partial_x w-r w =-\partial_y f(\,\cdot\,,y),\quad\text{on $\cO_\delta$}\end{aligned}$$ with $w(\,\cdot\,)=u(\,\cdot\,,y)$ at $\partial_P\cO_\delta$, where $\partial_P\cO_\delta$ is the parabolic boundary of $\cO_\delta$ (notice that now the claim is correct because $(t,x)\mapsto a(x,m^*_-(t))$ is continuous in $\cO_\delta$).
[*Step 2*]{}. (Contradiction). Thanks to the result in step 1 we can now find a contradiction. Pick an arbitrary $\varphi\in C^{\infty}_c((x^0_1,x^0_2))$, $\varphi \geq 0$ and multiply by $\varphi$ (with $w(\,\cdot\,)=u(\,\cdot\,,y)$). Since $t\mapsto u(t,x,y)$ is non-decreasing we have $\partial_t u(\,\cdot\,,y)\ge 0$ on $\cO_\delta$ and therefore, for each $y\in(y_0,y_0+\delta)$ we have $$\begin{aligned}
\varphi(\,\cdot\,)\left[\frac{\sigma^2(\,\cdot\,)}{2}\partial_{xx}u(\,\cdot\,,y) + a(\,\cdot\,, m^*_-(\,\cdot\,))\partial_{x}u(\,\cdot\,, y)-r u(\,\cdot\,, y)\right]\leq -\varphi(\,\cdot\,)\partial_{y} f(\,\cdot\,, y),\quad\text{on $\cO_\delta$.}\end{aligned}$$ In the inequality above we fix $t_0$ and integrate over $(x^0_1,x^0_2)$. By using integration-by-parts formula we obtain[^1] $$\begin{aligned}
&\int_{x^0_1}^{x^0_2} \left( \tfrac{1}{2}\partial_{xx}\big[\sigma^2(x)\varphi (x)\big] - \partial_x\big[a(x, m(t_0))\varphi(x)\big]-r\varphi(x) \right)u(t_0, x, y) \ud x \\
& \leq - \int_{x^0_1}^{x^0_2} \partial_y f(x, y)\varphi(x)\ud x. \end{aligned}$$ Now, letting $y\downarrow y_0$, using dominated convergence and $u(t_0,x,y_0)=c_0$ for $x\in(x^0_1,x^0_2)$, and undoing the integration by parts we obtain $$\label{final}
\int_{x^0_1}^{x^0_2}(\partial_yf(x,y_0)-rc_0)\varphi(x)\ud x\leq 0.$$ Hence $\partial_yf(x,y_0)-rc_0\leq 0$ for all $x\in(x^0_1,x^0_1)$ by arbitrariness of $\varphi\geq 0$ and continuity of $x\mapsto\partial_yf(x,y_0)-rc_0$. However, since $\cS\subseteq [0,T]\times(\Sigma\setminus\cH)$ (recall ), then it must be $\partial_yf(x,y_0)-rc_0= 0$ for all $x\in(x^0_1,x^0_1)$, which contradicts $\partial_{xy}f>0$ (Assumption \[ass:f\]-(iii)).
[^1]: Notice that the argument seems to require $\sigma\in C^2(\R)$. However, given that the final estimate does not depend on $\sigma$ we can apply a smoothing of $\sigma$, if necessary, and then pass to the limit at the end.
|
---
abstract: 'Data from the newly-commissioned *Transiting Exoplanet Survey Satellite* ([[*TESS*]{}]{}) has revealed a “hot Earth” around [LHS3844]{}, an M dwarf located 15 pc away. The planet has a radius of [$1.32\pm0.02$]{} $R_\oplus$ and orbits the star every 11 hours. Although the existence of an atmosphere around such a strongly irradiated planet is questionable, the star is bright enough ($I=11.9$, $K=9.1$) for this possibility to be investigated with transit and occultation spectroscopy. The star’s brightness and the planet’s short period will also facilitate the measurement of the planet’s mass through Doppler spectroscopy.'
author:
- 'Roland K. Vanderspek, Chelsea X. Huang, Andrew Vanderburg, George R. Ricker, David W. Latham, Sara Seager, Joshua N. Winn, Jon M. Jenkins, Jennifer Burt, Jason Dittmann, Elisabeth Newton, Samuel N. Quinn, Avi Shporer, David Charbonneau, Jonathan Irwin, Kristo Ment, Jennifer G. Winters, Karen A. Collins, Phil Evans, Tianjun Gan, Rhodes Hart, Eric L.N. Jensen, John Kielkopf, Shude Mao, William Waalkes, François Bouchy, Maxime Marmier, Louise D. Nielsen, Gaël Ottoni, Francesco Pepe, Damien Ségransan , Stéphane Udry, Todd Henry, Leonardo A. Paredes, Hodari-Sadiki James, Rodrigo H. Hinojosa, Michele L. Silverstein, Enric Palle, Zachory Berta-Thompson, Misty D. Davies, Michael Fausnaugh, Ana W. Glidden, Joshua Pepper, Edward H. Morgan, Mark Rose, Joseph D. Twicken, Jesus Noel S. Villaseñor, and the TESS Team'
bibliography:
- 'refs.bib'
title: '[[*TESS*]{}]{} Discovery of an Ultra-Short-Period Planet Around the Nearby M Dwarf LHS3844'
---
Introduction {#sec:intro}
============
The [*Transiting Exoplanet Survey Satellite*]{} ([[*TESS*]{}]{}) is a NASA Explorer mission that was launched on April 18, 2018. The mission’s primary objective is to discover hundreds of transiting planets smaller than Neptune, around stars bright enough for spectroscopic investigations of planetary masses and atmospheres [@ricker:2015]. Using four 10cm refractive CCD cameras, [[*TESS*]{}]{} obtains optical images of a rectangular field spanning 2300 square degrees. The field is changed every 27.4 days (two spacecraft orbits), allowing the survey to cover most of the sky in 2 years. [[*TESS*]{}]{} is a wider-field, brighter-star successor to the successful space-based transit surveys CoRoT [@Barge:2008; @haywood2014] and [*Kepler*]{} [@boruckikoi].
Another way in which [[*TESS*]{}]{} differs from the previous space missions is that M dwarfs constitute a larger fraction of the stars being searched, mainly because of a redder observing bandpass (600–1000 nm). Compared to solar-type stars, M dwarfs are advantageous for transit surveys because the signals are larger for a given planet size, and because the transits of planets in the “habitable zone” are geometrically more likely and repeat more frequently [see, e.g., @gould:2003; @Charbonneau:2007; @Latham:2012]. We also know that close-orbiting planets are very common around M dwarfs, based on results from the [*Kepler*]{} survey [@dc15; @muirhead:2015]. By focusing on nearby M dwarfs, the pioneering ground-based transit surveys MEarth and TRAPPIST have discovered four of the most remarkable planetary systems known today: GJ1214 [@charbonneau09], GJ1132 [@bertathompson2015], LHS1140 [@dittmann2017; @ment2018], and TRAPPIST-1 [@gillon16].
Simulations have shown that [[*TESS*]{}]{} should be capable of detecting hundreds of planets around nearby M dwarfs [@Sullivan:2015; @Bouma:2017; @ballard2018; @muirhead2018; @barclay:2018; @huang:2018]. Here, we report the first such detection, based on data from the first month of the survey. The planet is $1.32\pm 0.02$ times larger than the Earth, and orbits the M dwarf [LHS3844]{} every 11 hours. The star, located 15 parsecs away, has a mass and radius that are about 15% and 19% of the Sun’s values. The proximity and brightness of the star make this system a good candidate for follow-up Doppler and atmospheric spectroscopy.
This [*Letter*]{} is organized as follows. Section \[sec:obs\] presents the data from [[*TESS*]{}]{} along with follow-up observations with ground-based telescopes. Section \[sec:analysis\] describes our method for determining the system parameters. This section also explains why the transit-like signal is very likely to represent a true planet and not an eclipsing binary or other types of “false positives.” Section \[sec:dis\] compares LHS3844b with the other known transiting planets, and discusses some possibilities for follow-up observations.
Observations and Data analysis {#sec:obs}
==============================
TESS {#sec:tess}
----
[[*TESS*]{}]{} observed [LHS3844]{} between 2018 Jul 25 and Aug 22, in the first of 26 sectors of the two-year survey. The star appeared in CCD 2 of Camera 3. The CCDs produce images every 2 seconds, which are summed onboard the spacecraft into images with an effective exposure time of 30 minutes. In addition, 2-minute images are prepared for subarrays surrounding pre-selected target stars, which are chosen primarily for the ease of detecting transiting planets. LHS3844 was prioritized for 2-minute observations on account of its brightness in the [[*TESS*]{}]{} bandpass ($T=11.877$), small stellar radius, and relative isolation from nearby stars [@Stassun:2017; @muirhead2018].
The 2-minute data consist of 11 by 11 pixel subarrays. They were reduced with the Science Processing Operations Center (SPOC) pipeline, originally developed for the [*Kepler*]{} mission at the NASA Ames Research Center [@Jenkins:2015; @Jenkins:2016]. For [LHS3844]{}, the signal-to-noise ratio of the transit signals was 32.4. The 30-minute data were analyzed independently with the MIT Quick Look Pipeline (QLP; Huang et al., in prep). A transit search with the Box Least Square algorithm [BLS, @Kovacs2002] led to a detection with a signal-to-noise ratio of 31.6.
{width="\linewidth"}
{width="0.95\linewidth"}
For subsequent analysis, we used the 2-minute Pre-search Data Conditioning light curve from the SPOC pipeline [@stumpe], which was extracted from the photometric aperture depicted in the lower right panel of Figure \[fig:image\]. The resulting light curve is shown in the top panel of Figure \[fig:lc\]. To filter out low-frequency variations, we fitted a basis spline to the light curve (excluding the transits and 3$\sigma$ outliers) and divided the light curve by the best-fit spline. The result is shown in the second panel of Figure \[fig:lc\]. The interruption in the middle of the time series occurred when the spacecraft was at perigee, when it reorients and downlinks the data. There was also a 2-day interval when the data were compromised by abnormally unstable spacecraft pointing. In addition, we omitted the data collected in the vicinity of “momentum dumps,” when the thrusters are used to reduce the speed of the spacecraft reaction wheels. These lasted 10–15 minutes and took place every 2.5 days.
Ground-based Photometry {#sec:groundphot}
-----------------------
[LHS3844]{} was observed by the ground-based MEarth-South telescope array as part of normal survey operations [@Irwin2015; @Dittmann2017a]. A total of 1935 photometric observations were made between 2016 Jan 10 and 2018 Aug 25. No transits had been detected prior to the [[*TESS*]{}]{} detection, but when the data were revisited, a BLS search identified a signal with a period and amplitude consistent with the [[*TESS*]{}]{} signal (Figure \[fig:lc\]). The MEarth data also reveal the stellar rotation period to be 128 days, based on the method described by @Newton:2016 [@Newton:2018].[^1]
{width="\linewidth"}
Additional ground-based transit observations were performed as part of the [[*TESS*]{}]{} Follow-up Observing Program (TFOP). A full transit was observed on UT 2018 Sep 06 in the $I_C$ band, using the El Sauce Observatory Planewave CDK14 telescope located in El Sauce, Chile. Five more transits were observed in the Sloan $i'$ band using telescopes at the Cerro Tololo International Observatory (CTIO) node of the Las Cumbres Observatory.$\footnote{https://lco.global}$ The transit of UT 2018 Sep 08 was observed with a 0.4m telescope, and the transits of UT 2018 Sep 08, 09, 10 and 16 were observed with a 1.0m telescope. The data are shown in the lower panels of Figure \[fig:lc\]. Together, they confirm the fading events are occurring and localize the source to within $2\arcsec$ of [LHS3844]{}.
High-Resolution Spectroscopy {#sec:spec}
----------------------------
[lrrl]{} $2458287.9183$ & $-10.626$ & $0.056$ & CHIRON\
$2458287.9393$ & $-10.667$ & $0.108$ & CHIRON\
$2458369.6266$ & $-10.719$ & $0.031$ & CHIRON\
$2458369.6476$ & $-10.732$ & $0.046$ & CHIRON\
$2458369.6686$ & $-10.724$ & $0.048$ & CHIRON\
$2458371.653268$ & $-10.600$ & $0.090$ & CORALIE\
$2458372.780028$ & $-10.540$ & $0.068$ & CORALIE\
\[-2ex\]
We obtained optical spectra on UT 2018 Jun 18 and Sep 08 using the CTIO HIgh ResolutiON (CHIRON) spectrograph on the 1.5m telescope of the CTIO Small and Moderate Aperture Research Telescope System [@Tokovinin+2013]. We used the image slicer mode, giving a resolution of about 80[,]{}000. The first observation was a pair of 30min exposures centered at an orbital phase of $0.355$. The second observation comprised three 30min exposures centered at phase $0.880$. The data were analyzed as described by @Winters(2018a), using a spectrum of Barnard’s Star as a template. The spectra show no evidence of additional lines from a stellar companion, no sign of rotational broadening, no detectable H$\alpha$ emission, and no radial velocity variation.
Additional spectroscopy was performed with the CORALIE spectrograph [@queloz:2000; @pepe:2017] on the Swiss Euler 1.2m telescope at La Silla Observatory in Chile. Spectra were obtained on UT 2018 Sep 10 and 11, at phases 0.211 and 0.645, near the expected radial-velocity extrema. Radial-velocity calibration was performed with a Fabry-Pérot device. With exposure times of 45 and 60min, the signal-to-noise ratio per pixel was about 3 in the vicinity of 600 nm. Cross-correlations were performed with a weighted M2 binary mask from which telluric and interstellar lines were removed [@pepe:2002]. Only a single peak was detected. The difference in radial velocities was $60 \pm 110$ [$\rm m\,s^{-1}$]{}, i.e., not statistically significant.
To place an upper limit on the radial velocity variation using both datasets (see Table \[tab:rvs\]), we fitted for the amplitude of a sinusoidal function with a period and phase specified by the [[*TESS*]{}]{} transit signal. The free parameters were the amplitude $K$, and two additive constants representing the zeropoints of the CHIRON and CORALIE velocity scales. The result was $K = -28^{+64}_{-60}\ {\ensuremath{\rm m\,s^{-1}}}$, which can be interpreted as a 3$\sigma$ upper limit of 0.96 [$M_{\rm Jup}$]{} on the mass of the transiting object.
Analysis {#sec:analysis}
========
Stellar parameters {#sec:stellar_parameters}
------------------
Using an empirical relationship between mass and $K_s$-band absolute magnitude [@Benedict:2016], and the parallax from Data Release 2 of the [*Gaia*]{} mission [@GaiaDR2:2018; @Lindegren:2018], the mass of [LHS3844]{} is $0.151 \pm 0.014$ [$M_\sun$]{}. The uncertainty is dominated by the scatter in the mass-$K_s$ relation. Based on this mass determination, and the empirical mass-radius relationship of @Boyajian:2012, the stellar radius is $0.188\pm0.006$ [$R_\sun$]{}. These results are consistent with the empirical relationship between radius and absolute $K_s$ magnitude presented by @Mann:2015, which gives $0.189 \pm 0.004$ [$R_\sun$]{}. The bolometric luminosity is $(2.72\pm0.04)\times 10^{-3}$ [$L_\sun$]{}, based on the observed $V$ and $J$ magnitudes and the bolometric correction from Table 3 of @Mann:2015. Based on these determinations of $R_\star$ and $L_\star$, the Stefan-Boltzmann law gives an effective temperature of $3036 \pm 77$ K. The spectral type is M4.5 or M5, based on a comparison with MEarth survey stars of known spectral types on a color-magnitude diagram ([*Gaia*]{} $G$ versus $H-K_s$).
Light curve modeling {#sec:lcmodeling}
--------------------
We jointly analyzed the light curves from [[*TESS*]{}]{}, MEarth, and TFOP, using the formalism of @MandelAgol:2002 as implemented by @Kreidberg(2015). We assumed the orbit to be circular, and placed Gaussian prior constraints on the two parameters of a quadratic limb-darkening law [$u_1 = 0.145\pm0.05$ and $u_2 = 0.54\pm0.05$; @Claret:2018]. Because of the differing bandpasses of [[*TESS*]{}]{}, MEarth, and the TFOP instruments, each dataset was allowed to have different values for the limb-darkening parameters. We imposed a prior constraint on the mean stellar density based on the results of Section \[sec:stellar\_parameters\]. The model was evaluated with 0.4 min sampling and averaged as appropriate before comparing with the data. We used the [emcee]{} Monte Carlo Markov Chain code of @ForemanMackey:2012 to determine the posterior distributions for all the model parameters. The results are given in Table \[tab:stellar\]. Figure \[fig:lc\] shows the best-fitting model.
We also performed a fit to the [[*TESS*]{}]{} data only, without any prior constraint on the mean stellar density, in order to allow for a consistency check between the two density determinations. Based on the stellar parameters derived in Section \[sec:stellar\_parameters\], the mean density is $31.36\pm0.23$ [$\rm g\,cm^{-3}$]{}, while the light-curve solution gives $30.0_{-2.8}^{+7.4}$ [$\rm g\,cm^{-3}$]{}. The agreement between these two results is a sign that the transit signal is from a planet, and is not an astrophysical false positive. A related point is that the ratio $\tau/T$ between the ingress/egress duration and the total duration is $0.11^{+0.01}_{-0.02}$, and less than 0.14 with 99% confidence. This information is used in Section \[sec:lcmodeling\] to help rule out the possibility that the fading events are from an unresolved eclipsing binary. In general, $\tau/T
\geq R_p/R_\star$, even when the photometric signal includes the constant light from an unresolved star [@morris2018].
Photocenter motion {#sec:centroids}
------------------
Many transit-like signals turn out to be from eclipsing binaries that are nearly along the same line of sight as the intended target star, such that the light from the binary is blended together with the constant light of the intended target star. These cases can often be recognized by measuring any motion of the center of light (“centroid”) associated with the fading events [@Wu:2010]. To do so, we modeled the time series of the $X$ and $Y$ coordinates of the center of light as though they were light curves, after removing long-timescale trends by fitting out a cubic spline. Based on the fitted depths of the “centroid transits” we were able to put $3\,\sigma$ upper limits on centroid shifts of $\Delta X < 2\times10^{-4}$ and $\Delta Y < 6\times10^{-4}$ pixels, corresponding to 4.4 and 13.2mas. Thus, there is no evidence for photocenter motion.
Possible False Positives
------------------------
As mentioned previously, not all transit-like signals are from transiting planets. Below, we consider the usual alternatives to a transiting planet, and explain how the available data render them very unlikely.
1. [*The signal is an instrumental artifact.*]{} This is ruled out by the detection of the transit signals with ground-based telescopes (Section \[fig:lc\]).
2. [*LHS3844 is an eclipsing binary star.*]{} This is ruled out by the upper limit on radial-velocity variations, corresponding to a secondary mass of 0.96 [$M_{\rm Jup}$]{} (Section \[sec:spec\]). In addition, the absence of detectable phase variations in the [[*TESS*]{}]{} light curve requires that any companion be sub-stellar. An 80[$M_{\rm Jup}$]{} companion would have produced ellipsoidal variations of order 0.1% [see, e.g., @Shporer:2017], which can be excluded.
3. [*Light from a distant eclipsing binary, or a distant star with a transiting planet, is blended with that of LHS3844.*]{} We can rule out this possibility thanks to the star’s high proper motion (800 mas yr$^{-1}$). Because the star moves quickly relative to background stars, images from previous wide-field surveys allow us to check for faint stars along the current line of sight. No sources are detected within 6 mag of LHS3844, the brightness level that would be required to produce 0.4% flux dips. In addition, the ground-based observations require the fading source to be within $2\arcsec$ of LHS3844 (Section \[sec:groundphot\]), and the [[*TESS*]{}]{} images reveal no detectable motion of the stellar image during transits (Section \[sec:centroids\]).
4. [*LHS3844 is physically associated with an eclipsing binary star.*]{} The light-curve analysis (Section \[sec:lcmodeling\]) requires the eclipsing object to be smaller than 14% of the size of the eclipsed star. Since the spectrum is that of an M4-5 dwarf, any secondary star would need to be of that size or smaller, implying that the eclipsing object is smaller than $0.14 \times 0.15 {\ensuremath{R_\sun}}$ or 2.3 ${\ensuremath{R_\earth}}$. This rules out a stellar binary.
5. [*LHS3844 is a binary star and the transiting planet is around the secondary star.*]{} The companion would have to be faint and close to LHS3844 in order to escape detection by *Gaia* [@ziegler; @rizzuto]. Another indication that any secondary star needs to be faint is that the [*Gaia*]{} parallax and apparent magnitude are consistent with the properties of a single M dwarf. However, if the transiting planet is around such a faint companion, then the true transit depth must be less than about 2% in order for the transiting object to be smaller than about 14% the size of the eclipsed star. Thus, in order to produce the 0.4% transit we observe, a secondary star would have to contribute at least 20% of the total flux in the TESS aperture while still escaping detection by *Gaia* and seeing-limited imaging.
Thus, almost all of these scenarios are ruled out, except for the possibility that the planet is actually orbiting a low-luminosity secondary star. This scenario seems contrived, and is [*a priori*]{} unlikely because of the low companion fraction for mid M dwarfs [@Winters(2018a)], the small parameter space for companions which could produce the transits we observe, the lower probability for an M dwarf to host a larger planet compared to a 1.3 $R_\oplus$ planet [@berta2013; @mulders], and the tendency of selection effects to favor finding transits around primary stars [@Bouma+2018]. Probably the only way to rule out this possibility, or more exotic scenarios, is through precise Doppler monitoring.
{width="\linewidth"}
Discussion {#sec:dis}
==========
[LHS3844b]{} is one of the closest known planets, both in terms of its distance from the Earth, and its distance from its host star (see Figure \[zachstyle\]). It joins the small club of transiting planets around the Sun’s nearest M dwarf neighbors, which also includes GJ1214b [@charbonneau09], GJ1132b [@bertathompson2015], TRAPPIST-1b-h [@gillon16], and LHS1140 b-c [@dittmann2017; @ment2018]. [LHS3844b]{} is also the most easily studied example of an ultra-short-period (USP) planet, defined by the simple criterion $P<1$ day [@SanchisOjeda+2014; @Winn+2018]. It has the largest transit depth of any known sub-Jovian USP planet, and is brighter and closer to Earth than the other well-known systems CoRoT-7 [@leger2009], Kepler-10 [@Batalha+2011], Kepler-42 [@Muirhead+2012], and Kepler-78 [@SanchisOjeda+2013].
As such, [LHS3844b]{} provides interesting opportunities for atmospheric characterization through transit and occultation (secondary eclipse) spectroscopy. With an equilibrium temperature of about 805K, and an orbital distance amounting to only 7.1 stellar radii, it is unclear what type of atmosphere the planet might have, if any. If the planet formed at or near this location, its primordial atmosphere could have been completely stripped away during the host star’s youth, when it was much more luminous and chromospherically active. The observed radius function of the short-period [*Kepler*]{} planets has a dip at around 1.8 $R_\oplus$ that has been interpreted as a consequence of atmospheric loss. Planets smaller than 1.8 $R_\oplus$ seem to have lost their primordial hydrogen-helium atmospheres due to photoevaporation [@Fulton+2017; @LopezFortney2013; @OwenWu2013]. With a radius of 1.32 $R_\oplus$, we might expect [LHS3844b]{} to have suffered this process, too. In this case, transit spectroscopy would show no variation in the planetary radius with wavelength, although occultation spectroscopy could still be used to measure the emission spectrum of the planet’s surface.
Indeed, of all the known planets smaller than 2 $R_\oplus$, [LHS3844b]{} has perhaps the most readily detectable occultations. This is based on a ranking of the 907 planets in the NASA Exoplanet Archive by a crude signal-to-noise metric, $${\rm S/N} \propto \sqrt{F_\star}~\frac{R_p^2 T_p}{R_\star^2 T_\star},$$ which assumes that the star and planet are both radiating as blackbodies in the Rayleigh-Jeans limit. Here, $F_\star$ is the star’s $K$-band flux, $T_p$ is the planet’s equilibrium temperature, and $T_\star$ is the star’s effective temperature[^2]. According to this metric, [LHS3844b]{} ranks second, closely trailing HD 219134 b, which orbits a much brighter, but larger, star. Even then, [LHS3844b]{} will likely be easier to observe than HD 219134 b thanks to its significantly deeper secondary eclipse, which should avoid observational systematic noise floors, and the planet’s ultra-short period, which simplifies scheduling observations.
The ultra-short period will also facilitate the measurement of the planet’s mass through Doppler spectroscopy. Short periods lead to stronger signals: assuming the planet’s mass is 2.8 $M_\oplus$, as it would be for a terrestrial composition, the expected semiamplitude of the Doppler signal is 8 [$\rm m\,s^{-1}$]{}, which is unusually high for a rocky planet. The orbital period is short enough for the signal to be measured in its entirety in just a few nights. The orbital period is also 280 times shorter than the stellar rotation period, allowing for a clear separation of timescales between the orbital motion and any spurious Doppler signals related to stellar activity.
The discovery of a terrestrial planet around a nearby M dwarf during the first [[*TESS*]{}]{} observing sector suggests that the prospects for future discoveries are bright. It is worth remembering that 90% of the sky has not yet been surveyed by either [[*TESS*]{}]{} or [[*Kepler*]{}]{}.
This work makes use of observations from the LCOGT network. Work by JNW was partially supported by the Heising-Simons Foundation. We acknowledge the use of [[*TESS*]{}]{} Alert data, which is currently in a beta test phase, from the [[*TESS*]{}]{} Science Office. Funding for the [[*TESS*]{}]{} mission is provided by NASA’s Science Mission directorate. The MEarth team acknowledges funding from the David and Lucile Packard Fellowship for Science and Engineering (awarded to D.C.). This material is based on work supported by the National Science Foundation under grants AST-0807690, AST-1109468, AST-1004488 (Alan T. Waterman Award) and AST-1616624. Acquisition of the CHIRON data was made possible through the support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. J.A.D. acknowledges support by the Heising-Simons Foundation as a 51 Pegasi b postdoctoral fellow. E.R.N. is supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship under award AST-1602597. We thank the Geneva University and the Swiss National Science Foundation for their continuous support for the Euler telescope. This research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. CXH acknowledges support from MIT’s Kavli Institute as a Torres postdoctoral fellow. AV’s work was performed under contract with the California Institute of Technology / Jet Propulsion Laboratory funded by NASA through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute. [*Facilities:*]{} , , , ,
[lcr]{} R.A. (h:m:s) & 22:41:59.089 & Gaia DR2\
Dec. (d:m:s) & -69:10:19.59 & Gaia DR2\
Epoch & 2015.5 & Gaia DR2\
Parallax (mas) & $67.155 \pm 0.051$ & Gaia DR2\
$\mu_{ra}$ (mas yr$^{-1}$) & $334.357 \pm 0.083$ & Gaia DR2\
$\mu_{dec}$ (mas yr$^{-1}$) & $-726.974 \pm 0.086$ & Gaia DR2\
Gaia DR2 ID & 6385548541499112448 &\
TIC ID & 410153553 &\
LHS ID & 3844 &\
TOI ID & 136 &\
$TESS$ (mag)& 11.877 & TIC V7\
$Gaia$ (mag)& 13.393 & Gaia DR2\
Gaia RP (mag)& 12.052 & Gaia DR2\
Gaia BP (mag)& 15.451 & Gaia DR2\
$V_J$ (mag)& 15.26$\pm$0.03 & RECONS\
$R_{KC}$ (mag)& 13.74$\pm$0.02 & RECONS\
$I_{KC}$ (mag)& 11.88$\pm$0.02 & RECONS\
$J$ (mag)& 10.046$\pm$0.023 & 2MASS\
$H$ (mag)& 9.477 $\pm$0.023& 2MASS\
$K_s$ (mag)& 9.145$\pm$0.023 & 2MASS\
${\ensuremath{M_\star}}$ (${\ensuremath{M_\sun}}$)& $0.151\pm $0.014& Parallax +@Benedict:2016\
${\ensuremath{R_\star}}$ (${\ensuremath{R_\sun}}$)& $0.189\pm$0.006 & Parallax +@Mann:2015\
${\ensuremath{\log{g_{\star}}}}$ (cgs)& $5.06\pm0.01$ & empirical relation + LC\
${\ensuremath{L_\star}}$ (${\ensuremath{L_\sun}}$)& $0.00272\pm0.0004$ & @Mann:2015\
${\ensuremath{T_{\rm eff\star}}}$ (K)& $3036\pm77$ &\
$M_V$ (mag)& $14.39\pm$0.02 & Parallax\
$M_K$ (mag)& $8.272\pm$0.015 & Parallax\
Distance (pc)& [$14.9\pm0.01$]{}& Parallax\
[$\rho_\star$]{}([$\rm g\,cm^{-3}$]{})& [$31.69\pm0.37$]{}& empirical relation + LC\
$P$ (days) & [$0.46292792\pm0.0000016$]{} &\
$T_c$ (${\rm BJD} - 2457000$) & [$1325.72568\pm0.00025$]{} &\
$T_{14}$ (min) & [$31.08\pm0.25$]{} &\
$T_{12} = T_{34}$ (min) & [$3.73_{-0.7}^{+0.4}$]{} &\
${\ensuremath{a/{\ensuremath{R_\star}}}}$ & [$7.1059\pm0.028$]{} &\
${\ensuremath{R_{p}}}/{\ensuremath{R_\star}}$ & [$0.0640\pm0.0007$]{} &\
$b \equiv a \cos i/{\ensuremath{R_\star}}$ & [$0.221\pm0.038$]{} &\
$i$ (deg) & [$88.22\pm0.30$]{} &\
$c_1$, MEarth (linear term) & $0.13\pm0.03 $ &\
$c_2$, MEarth (quadratic term) & $0.53\pm0.03$ &\
$c_1,TESS$ & $0.52\pm0.04$ &\
$c_2,TESS$ & $0.46\pm0.01$ &\
$c_1,i$ & $0.19\pm0.04$ &\
$c_2,i$ & $0.56\pm0.04$ &\
${\ensuremath{R_{p}}}$ (${\ensuremath{R_\earth}}$) & [$1.32\pm0.02$]{} &\
$a$ (AU) & [$0.00623\pm0.00015$]{} &\
$T_{\rm eq}$ (K) & [$805\pm20$]{} &\
$\langle F_j \rangle$ ($10^{9}$ [$\rm erg\,s^{-1}\,cm^{-2}$]{}) & [$0.0954\pm0.00070$]{} &\
\[-1.5ex\]
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[^1]: Although @Newton:2018 did not detect rotational modulation, subsequent data have allowed for a “Grade B” detection, in the rating system described in that work.
[^2]: See also work by @kempton2018
|
---
abstract: 'We report on the observation of confinement-induced resonances in strongly interacting quantum-gas systems with tunable interactions for one- and two-dimensional geometry. Atom-atom scattering is substantially modified when the s-wave scattering length approaches the length scale associated with the tight transversal confinement, leading to characteristic loss and heating signatures. Upon introducing an anisotropy for the transversal confinement we observe a splitting of the confinement-induced resonance. With increasing anisotropy additional resonances appear. In the limit of a two-dimensional system we find that one resonance persists.'
author:
- Elmar Haller
- 'Manfred J. Mark'
- Russell Hart
- 'Johann G. Danzl'
- Lukas Reichsöllner
- Vladimir Melezhik
- Peter Schmelcher
- 'Hanns-Christoph Nägerl'
title: 'Confinement-Induced Resonances in Low-Dimensional Quantum Systems'
---
Low-dimensional systems have recently become experimentally accessible in the context of ultracold quantum gases. For a two-dimensional (2D) geometry, the Berezinskii-Kosterlitz-Thouless (BKT) transition has been observed [@Hadzibabic2006], and in one dimension the strongly-correlated Tonks-Girardeau (TG) [@Girardeau1960; @Kinoshita2004; @Paredes2004; @Syassen2008; @Haller2009] and super-Tonks-Girardeau (sTG) gases [@Haller2009] have been realized. In these experiments steep optical potentials freeze out particle motion along one or two directions and restrict the dynamics to a plane or to a line. Such quasi-2D or quasi-1D systems can be realized with ultracold gases when the kinetic and the interaction energy of the particles are insufficient to transfer the particles to transversally excited energy levels. Whereas the confinement removes motional degrees of freedom, it also provides an additional structure of discrete energy levels that can be used to modify scattering along the unconfined direction and by this to effectively control the interaction properties of the low-dimensional system [@Olshanii1998; @Petrov2000; @Bergeman2003]. In this Letter, we investigate the few-body scattering processes that give rise to the capability to tune interactions and hence to drastically alter the properties of low-dimensional many-body quantum systems [@Haller2009].
In three-dimensional (3D) geometry magnetically-induced Feshbach resonances (FBRs) [@Chin2009] allow tuning of the inter-particle interaction strength. A FBR occurs when the scattering state of two atoms is allowed to couple to a bound molecular state. Typically, scattering state and bound state are brought into degeneracy by means of the magnetically tunable Zeeman interactions. For particles in 1D and 2D geometry a novel type of scattering resonance occurs. Coupling between the incident channel of two incoming particles and a transversally excited molecular bound state generates a so-called confinement-induced resonance (CIR) [@Olshanii1998; @Petrov2000; @Bergeman2003; @LowDSystems; @Kim2005; @Naidon2007]. A CIR occurs when the 3D scattering length $a_{\text{3D}}$ approaches the length scale that characterizes the transversal confinement, i.e. the harmonic oscillator length $a_{\perp} = \sqrt{\hbar /(m \omega_{\perp})}$ for a particle with mass $m$ and transversal trapping frequency $\omega_{\perp}$. This causes the 1D coupling parameter $g_\mathrm{1D} = \frac{2\hbar^2 a_{\text{3D}}}{m a_{\perp}^2} \frac{1}{1-C a_{\text{3D}}/a_{\perp}}$ to diverge at $a_{\perp} = C a_{\text{3D}}$, where $C=1.0326$ is a constant [@Olshanii1998; @Bergeman2003]. The CIR allows tuning of interactions from strongly repulsive to strongly attractive and thus represents a crucial ingredient for the control of interactions in a low-dimensional system. Modification of scattering properties due to confinement has been measured near a FBR for fermions [@Guenter2005], and, recently, a CIR has been observed for a strongly-interacting 1D quantum gas of bosonic Cs atoms and was used to drive the crossover from a TG gas with strongly repulsive interactions to an sTG gas with strongly attractive interactions [@Haller2009]. Here, for an ultracold quantum gas of Cs atoms with tunable interactions, we study the properties of CIRs by measuring particle loss and heating rate and, in particular, confirm the resonance condition $a_{\perp} = C a_{\text{3D}}$ for symmetric 1D confinement. For the case of transversally anisotropic confinement we find that the CIR splits and, to our surprise, persists for positive $a_{\text{3D}}$ even when the anisotropy reaches the limit of a 2D system.
Figure \[fig1\](a) reviews the basic mechanism that causes a CIR for zero collisional energy in 1D [@Bergeman2003]. It is assumed that in 3D the scattering potential supports a single universal bound state for strong repulsive interactions (dotted line) [@Chin2009]. The point where the incoming channel of two colliding atoms and the universal dimer state are degenerate marks the position of a 3D FBR (triangle). In 1D, strong transversal confinement shifts the zero-energy of the incoming channel (middle dashed line) and introduces a transversally excited state (upper dashed line). As a result of the strong confinement, which modifies the long-range part of the molecular potential, the universal dimer state with binding energy $E_\mathrm{B}$ (lower solid line) exists also for attractive interactions [@Moritz2003] whereas the original 3D FBR has disappeared. Instead, there is a CIR (star) when the incoming scattering channel becomes degenerate with the transversally excited molecular bound state (upper solid line). It is assumed that the binding energy of this state is also $E_\mathrm{B}$, shifted by $2\hbar\omega_\perp$[@Olshanii1998]. In more detail, as depicted in Fig. \[fig1\](b), we assume that the energy levels of non-interacting atoms, as a result of cylindrically symmetric transversal confinement, can be approximated by those of a 2D harmonic oscillator with $E_{n_1,n_2} = \hbar \omega_{\perp}(n_1 + n_2 + 1)$ and quantum numbers $n_1$ and $n_2$ belonging to the two Cartesian directions. Scattering atoms [@Separation] in the transversal ground state $(0,0)$ can couple to the excited states $(n_1, n_2)$ if the parity of the total wave function is preserved [@Kim2005]. The energetically lowest allowed excited states are threefold degenerate with an energy $E=3 \hbar\omega_{\perp}$ and with quantum numbers $(1,1)$, $(2,0)$ and $(0,2)$. For the transversally symmetric confinement, they contribute towards a single CIR [@Bergeman2003]. However, the contribution of the state (1,1) is negligible due to the zero contact probability of the atoms and the short-range character of the interatomic interaction. Unequal transversal trapping frequencies $\omega_1$ and $\omega_2 = \omega_1 + \Delta \omega$ lift this degeneracy and shift the energy levels according to $ E_{n_1,n_2} = \hbar \omega_1(n_1 + n_2 + 1) + \hbar \Delta \omega (n_2+ 1/2)$. One thus expects a splitting of the CIR.
![(color online) (a) Illustration of the mechanism responsible for a CIR, see Ref.[@Bergeman2003] and text for details. The energy levels near a scattering resonance are plotted as a function of $1/a_{\text{3D}}$. The CIR occurs for $C a_{\text{3D}}=a_\perp$ when scattering atoms are allowed to couple to transversally excited bound states. (b) indicates the shift and splitting for anisotropic confinement characterized by $\Delta \omega$. (c) Experimental configuration. Two laser beams create an optical lattice that confines the atoms to an array of approximately $3000$ independent, horizontally-oriented elongated 1D tubes. (d) Tuning of $a_{\text{3D}}$ is achieved by means of a FBR with a pole at $B=47.78(1)$ G [@Lange2009].[]{data-label="fig1"}](figure1.pdf){width="8.5cm"}
We start from a tunable Bose-Einstein condensate (BEC) of $1.0$ to $1.4\times10^5$ Cs atoms in the energetically lowest hyperfine sublevel [@Kraemer2004] confined in a crossed-beam optical dipole trap and levitated against gravity by a magnetic field gradient of $|\nabla B| \approx 31.1$ G/cm. Tunability of $a_{\text{3D}}$ is given by a FBR as shown in Fig. \[fig1\](d) with its pole at $B_0=47.78(1)$ G and a width of $164$ mG [@Kraemer2004; @Lange2009]. The resonance resides on top of a slowly varying background that allows tuning of $a_{\text{3D}}$ from $0$ to values of about $1000 \ a_0$, where $a_0$ is Bohr’s radius. Using the FBR, we can further tune to values up to $a_{\text{3D}} \approx 6000 \ a_0$ given our magnetic field control with an uncertainty of $\Delta B \approx 10$ mG. We convert $B$ into $a_{\text{3D}}$ using the FBR parameters from Ref.[@Lange2009]. The BEC is produced at $a_{\text{3D}} \approx 290 \ a_0$. We load the atoms within $300$ ms into an optical lattice, which is formed by two retro-reflected laser beams at a wavelength of $\lambda=1064.49(1)$ nm and with a beam waist of approximately $350 \ \mu$m, one propagating vertically and one propagating horizontally as illustrated in Fig. \[fig1\](c). These lattice beams confine the atoms to an array of approximately $3000$ horizontally oriented, elongated 1D tubes with a maximum occupation of $60$ atoms at a linear peak density of approximately $n_\text{1D} \approx 2 / \mu$m. Weak longitudinal confinement results from the Gaussian-shaped intensity distribution of the beams. We raise the lattice to a depth of typically $V = 30 \ E_R$, where $E_R=h^{2}/(2 m \lambda^{2})$ is the photon recoil energy. At this depth, the resulting transversal and longitudinal trap frequencies are $\omega_\perp = 2 \pi \times 14.5$ kHz and $ \omega_\parallel = 2 \pi \times 16$ Hz and we then have $a_{\perp} \approx 1370 \ a_0$. After loading we slowly ramp down $|\nabla B|$ in $50$ ms and adiabatically increase $a_{\text{3D}}$ to $915$ a$_0$ in $100$ ms to create a TG gas with well-defined starting conditions near the CIR [@Haller2009]. To detect the CIR as a function of $B$, manifested by a loss resonance, we quickly set $B$ in less than $200 \ \mu$s to the desired value, wait for a hold time of typically $\tau=200$ ms, and then measure the number $N$ of remaining atoms by absorption imaging. For this, we re-levitate the atoms, ramp down the lattice beams adiabatically with respect to the lattice band structure, and allow for $50$ ms of levitated expansion and $2$ ms time-of-flight. Note that $\tau$ is chosen to be much longer than the lifetime of the sTG phase [@Haller2009].
![(color online) Particle loss and heating rates in the vicinity of a CIR. (a) The number $N$ of remaining atoms after $\tau=200$ ms shows a distinct drop (“edge”) when $B$ is scanned across the CIR. A clear shift of the position of the edge to lower values for $B$ can be observed when the transversal confinement is stiffened, $\omega_{\perp} = 2 \pi \times (0.84,0.95,1.05)\times 14.2(2)$ kHz (circles, squares, triangles). (b) Position of the edge (circles) as determined from the intersection point of a second-order polynomial fit to the minimum for $N$ and the initial horizontal baseline as shown in (a), converted into values for $a_{\text{3D}}$. For comparison, the position of the minimum (triangles) is also shown. The solid line is given by $C a_{\text{3D}} = a_\perp$ with the predicted value $C=1.0326$. (c) Heating rates near the CIR (circles). For comparison, $N$ is also shown (triangles). For this measurement, $\omega_{\perp}=2\pi \times 12.0(2)$ kHz. All error bars reflect $1\sigma$ statistical uncertainty.[]{data-label="fig2"}](figure2.pdf){width="8.5cm"}
We observe the CIR in the form of an atomic loss signature as shown in Fig. \[fig2\]. We attribute the loss near the resonance to inelastic three-body [@Weber2003] or higher-order collisions [@Ferlaino2009], which lead to molecule formation and convert binding energy into kinetic energy, causing trap loss and heating, similar to the processes observed near a FBR [@Chin2009]. In our case, inelastic two-body processes can be ruled out for energetic reasons and single-particle loss occurs on the timescale of tens of seconds. In Fig. \[fig2\](a) the CIR can be identified as a distinct “edge” for the atom number $N$. Initially, in the TG regime of strong repulsive interactions, here for $B<47.35$ G, losses are greatly suppressed, but increase rapidly on the attractive side of the CIR. $N$ drops to a minimum when $B$ is increased and then recovers somewhat. A clear shift of the loss signature to lower values for $B$ and hence lower values for $a_\text{3D}$ can be discerned when the confinement is stiffened. When we identify the position of the edge with the position of the CIR, we find good agreement with the analytical result $C a_\text{3D} = a_{\perp}$ as shown in Fig. \[fig2\](b). As we have no theoretical description of the detailed shape of the loss resonance, we also plot, for comparison, the position of the minimum, which is shifted accordingly.
In Fig. \[fig2\](c) we juxtapose the loss and the heating rate that we measure in the vicinity of the CIR. For this, we measure the increase of the release energy within the first $100$ ms. After holding the atoms for time $\tau$ at a given value of $B$, we decrease $a_{\text{3D}}$ back to $250$ $a_0$ in $20$ ms, switch off the lattice potential and determine the release energy in the direction of the tubes from the momentum distribution in free space expansion. We observe an increase for the heating rate when the CIR is crossed. From a low value of $10$ nK/s in the TG regime it rises to a maximum of approximately $150$ nK/s and then drops to settle at some intermediate value. The position of the maximum agrees well with the maximum for atom loss. We check that the system’s increase in energy is sufficiently small so that its 1D character is not lost. The release energy, even at maximal heating, remains below $k_B \times 30$ nK, which is far below the energy spacing of the harmonic oscillator levels, $\hbar \omega_{\perp}\approx k_B \times 600$ nK.
![(color online) Splitting of a CIR for a 1D system with transversally anisotropic confinement. (a) As the horizontal confinement is stiffened, $\omega_2/\omega_1 = 1.00, 1.10, 1.18 $ (circles, diamonds, triangles) for $\omega_1=2 \pi \times 13.2(2)$ kHz, the CIR splits into CIR$_1$ and CIR$_2$. (b) Position of CIR$_1$ ($a_{\mathrm{3D},1}$, circles) and CIR$_2$ ($a_{\mathrm{3D},2}$, squares) as a function of the frequency ratio $\omega_2/\omega_1$. (c) Binding energy difference $\Delta E_\mathrm{B}$ as determined from the implicit equation (see text) in comparison to the expectation from the simple harmonic oscillator model (solid line).[]{data-label="fig3"}](figure3.pdf){width="8.5cm"}
We now examine 1D systems with transversally anisotropic confinement. Starting from a lattice depth of $V=25\ E_R$ along both transversal directions, yielding $\omega_\perp = \omega_1 = \omega_2 = 2\pi \times 13.2(2)$ kHz, we increase the horizontal confinement to frequencies up to $\omega_2 = 2 \pi \times 16.5(2)$ kHz, corresponding to a lattice depth of $39\ E_R$, while keeping the depth of the vertical confinement constant. Fig. \[fig3\](a) shows a distinct splitting of the original CIR into two loss resonances, CIR$_1$ and CIR$_2$. The splitting increases as the anisotropy is raised. In Fig. \[fig3\](b) we plot the 3D scattering length values $a_{\mathrm{3D},1}$ and $a_{\mathrm{3D},2}$ that we associate with the positions of CIR$_1$ and CIR$_2$ as a function of the frequency ratio $\omega_2/\omega_1$. For this, as it becomes difficult to assign an edge to both of them, we simply determine the associated atom number minima and subtract a constant offset of $88(7) \ a_0$ as determined from the measurement shown in Fig. \[fig2\](b). One of the resonances, CIR$_2$, exhibits a pronounced shift to smaller values for $a_\mathrm{3D}$ as the horizontal confinement is stiffened. The second resonance, CIR$_1$, shows a slight shift towards higher values for $a_\mathrm{3D}$. We now use the lifting of the degeneracy for the energy levels as indicated in Fig. \[fig1\](b) to model the observed splitting of the CIR. We assume that the implicit equation $\zeta(1/2,-E_\mathrm{B}/(2\hbar\omega_{\perp}) + 1/2) = - a_{\perp}/a_{3D}$ for the binding energy $E_\mathrm{B}$ [@Bergeman2003] remains approximately valid for sufficiently small $\Delta \omega$, taking $\omega_{\perp}= \omega_1$. Here, $\zeta$ is the Hurwitz zeta function. We translate the scattering length values $a_{\mathrm{3D},1}$ and $a_{\mathrm{3D},2}$ into binding energies and calculate the energy difference $\Delta E_\mathrm{B} = E_\mathrm{B}(a_\mathrm{3D,1}) - E_\mathrm{B}(a_\mathrm{3D,2})$, shown in Fig. \[fig3\](c). While this model does not explain the upward deviation seen for CIR$_1$, the difference $\Delta E_\mathrm{B}$ is in reasonable agreement with the expected energy shift caused by the shifts of the excited harmonic oscillator states $(E_{0,2}-E_{2,0}) = 2\hbar \Delta \omega$ (solid line in Fig. \[fig3\](c)). We thus attribute CIR$_2$ to the stiffened confinement along the horizontal direction and hence to state $(0,2)$, while CIR$_1$ corresponds to the unchanged confinement along the vertical direction and hence to state $(2,0)$.
. The position of the CIR as determined from the edge (circles) and, alternatively, from the minimum in atom number (triangles) shifts to lower values for $a_\text{3D}$ as the confinement is stiffened and $a_{\perp,\text{2D}}$ is reduced. The solid line is a linear fit according to $C_\text{2D} a_{\text{3D}}=a_{\perp,\text{2D}}$ with $C_\text{2D}=1.19(3)$.[]{data-label="fig4"}](figure4.pdf){width="8.5cm"}
We observe the appearance of additional structure in the measured loss curves when we increase the transversal anisotropy by weakening the confinement along one axis, here along the vertical direction. Fig. \[fig4\](a) shows the atom number after $\tau=300$ ms for trapping frequency ratios $\omega_1/\omega_2$ from $0.67$ to $0.45$. Multiple loss resonances appear close to the position of CIR$_1$. The number of resonances increases and the positions shift continuously as the confinement is weakened. We speculate that those resonances are a result of a coupling to additional excited states, resulting in a multi-channel scattering situation. Also the weakening of the confinement could induce sufficient anharmonicity to allow for violation of the parity rule [@Peano2005].
Surprisingly, we find that one of the CIRs persists in the limit of a 2D system. Previous theoretical studies on 2D systems have predicted the appearance of a CIR for negative $a_\text{3D}$, but not for positive $a_\text{3D}$ [@Petrov2001; @Naidon2007]. In the experiment, we reduce the horizontal confinement while keeping the vertical confinement constant to probe the transition from the array of tubes to a stack of pancake-shaped, horizontally-oriented 2D systems. Trapping in the horizontal direction is still assured, now by the Gaussian profile of the vertically propagating laser beam, for which $\omega_2 = 2\pi\times 11$ Hz. Fig. \[fig4\](b) shows that the CIR associated with the tight confinement shifts to lower values for $B$ and hence for $a_\text{3D}$ as the horizontal confinement is weakened. In the limit of 2D confinement, one of the CIRs, and in fact all the additional structure observed above, have disappeared, but one resonance persists. To check that the observed resonance is indeed the result of the 2D confinement, we vary the confinement along the tight vertical direction. Fig. \[fig4\](c) plots the positions of edge and minimum of the loss signature as a function of $a_{\perp,\text{2D}}$, the confinement length associated with this direction. When we again associate the edge with the pole of the resonance, we obtain $ C_\text{2D} a_{\text{3D}}=a_{\perp,\text{2D}}$ with $C_\text{2D}=1.19(3)$, where $C_\text{2D}$ is a scaling factor similar to $C$ for the 1D case. Further scattering experiments are needed to elucidate the energy dependence of this 2D scattering resonance.
In summary, we have investigated the properties of CIRs, which appear in low-dimensional quantum systems as a result of tight confinement and which replace “conventional” 3D Feshbach resonances to tune the effective atomic interaction strength. We observed a splitting of the CIR for anisotropic transversal confinement, the appearance of multiple resonances for strongly anisotropic confinement, and the survival of one resonance for positive $ a_\text{3D} $ in the limit of 2D confinement. We expect that CIRs will not only be used in 1D geometry to tune the effective interaction strength as recently demonstrated [@Haller2009], but also in 2D geometry and mixed dimensions [@Lamporesi2010] for the study of strongly-interacting quantum systems.
We thank W. Zwerger for discussions and R. Grimm for generous support. We acknowledge funding by the Austrian Ministry of Science and Research and the Austrian Science Fund and by the European Union within the framework of the EuroQUASAR collective research project QuDeGPM. R.H. is supported by a Marie Curie Fellowship within FP7. P.S. acknowledges financial support by the Deutsche Forschungsgemeinschaft. Financial support by the Heisenberg-Landau Program is appreciated by P.S. and V.S.M.
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|
---
abstract: |
We present an unbiassed near-IR selected AGN sample, covering 12.56 square degrees down to $K_s \sim 15.5$, selected from the Two Micron All Sky Survey (2MASS). Our only selection effect is a moderate color cut ($J-K_s>1.2$) designed to reduce contamination from galactic stars. We observed both point-like and extended sources. Using the brute-force capabilities of the 2dF multi-fiber spectrograph on the Anglo-Australian Telescope, we obtained spectra of 65% of the target list: an unbiassed sub-sample of 1526 sources.
80% of the 2MASS sources in our fields are galaxies, with a median redshift of 0.15. The remainder are K- and M-dwarf stars.
We find tentative evidence that Seyfert-2 nuclei are more common in our IR-selected survey than in blue-selected galaxy surveys. We estimate that $5.1 \pm 0.7$% of the galaxies have Seyfert-2 nuclei with H$\alpha$ equivalent widths $> 0.4$nm, measured over a spectroscopic aperture of radius $\sim 2.5$kpc. Blue selected galaxy samples only find Seyfert-2 nuclei meeting these criteria in $\sim 1.5$% of galaxies.
$1.2 \pm 0.3$% of our sources are broad-line (Type-1) AGNs, giving a surface density of $1.0 \pm 0.3$ per square degree, down to $K_s <15.0$. This is the same surface density of Type-1 AGNs as optical samples down to $B<18.5$. Our Type-1 AGNs, however, mostly lie at low redshifts, and host galaxy light contamination would make $\sim 50$% of them hard to find in optical QSO samples.
We conclude that the Type-1 AGN population found in the near-IR is not dramatically different from that found in optical samples. There is no evidence for a large population of AGNs that could not be found at optical wavelengths, though we can only place very weak constraints on any population of dusty high-redshift QSOs. In contrast, the incidence of Type-2 (narrow-line) AGNs in a near-IR selected galaxy sample seems to be higher than in a blue selected galaxy sample.
author:
- 'Paul J. Francis'
- 'Brant O. Nelson and Roc M. Cutri'
title: An Unbiassed Census of Active Galactic Nuclei in the Two Micron All Sky Survey
---
Introduction
============
To date, nearly all complete Active Galactic Nuclei (AGN) samples are flux limited at blue optical wavelengths. Such surveys are highly efficient, and can be very complete [eg. @mey01], picking up all AGNs [ *down to their blue flux limit*]{}. Unfortunately, any survey with a blue flux limit will be relatively insensitive to objects whose emission peaks at any other wavelength.
How seriously does this blue flux limit bias AGN samples? The situation is somewhat different for QSO searches (searches for AGNs which are considerably brighter than their host galaxy) and Seyfert galaxy searches (searches for less luminous AGNs).
Luminous QSOs
-------------
There has long been speculation that there might exist a substantial population of luminous QSOs with red colors in the optical/near-IR. These red colors could be caused by small quantities of dust, or the QSOs could be intrinsically red. Given the steepness of the luminosity function for luminous QSOs, most will lie close to the magnitude limit of a survey, so even small amounts of extinction will eliminate them from a blue-selected sample (Fig \[dust\]).
QSOs live in the nuclei of galaxies, which are dusty places. It should therefore be no surprise that our sight-line to the centers of many QSOs is obscured by dust. What is surprising is that the dust seems to either completely obscure our view of the central engine of the QSO (Type-2 AGN), or not to obscure it at all (Type-1 AGN). There seem to be very few QSOs that are partially obscured by dust, so that we still see a nuclear QSO spectrum, albeit a reddened one. Our sight-line seems either to intersects a giant molecular cloud or no dust at all. This contrasts with sight-lines from the Earth out of our galaxy, most of which intersect small quantities of optically thin dust [@sch98]. Is this a selection effect, or does AGN activity expel or destroy optically thin dust, as suggested by @dop98?
A few red QSOs have now been found. Many radio-selected quasars are quite red, though this redness may be caused by synchrotron emission or weak blue bump emission, rather than dust [@web95; @bak95; @whi01; @fra00; @fra01]. At least a few radio-selected quasars, however, show unmistakable evidence of severe dust reddening [@mal97; @cou98; @gre02]. A handful of red AGNs have also been identified in other surveys [eg. @mcd89; @bro98].
To accurately determine the population of red QSOs, and to better characterize their apparently diverse nature, a QSO sample with a magnitude limit at some wavelength unaffected by dust would be ideal. Radio surveys only pick up the small fraction of QSOs that are radio-loud, which are probably not representative. Hard X-ray surveys [eg. @mus00; @ale01] are unaffected by dust, but many hard X-ray sources are so faint at optical wavelength that follow-up spectroscopy is very difficult, even with large telescopes. It does, however, seem clear that dusty AGNs are a major contributer to the X-ray background. Far-IR selection [eg. @low88; @mat02] is biased [*towards*]{} dusty sources, but discriminating between QSOs and starburst galaxies has proven extremely hard.
Complete near-IR selected surveys are still somewhat biased against dusty QSOs (Fig \[dust\]). They have the major advantage that most QSOs found in a near-IR limited survey will be bright enough for relatively easy follow-up spectroscopy. Surveys with $i$-band magnitude limits, such as the Sloan Digital Sky Survey (SDSS) QSO survey [@ric02], are an improvement on $B$-band limited surveys, but Fig \[dust\] makes it clear that going still further to the red should yield big gains.
Can we construct a complete $K$-band limited QSO sample? @war00 showed that by combining optical and near-IR photometry, it should be possible to construct such a sample. Unfortunately, suitable photometry does not yet exist over larger areas of the sky, though the technique has been successfully applied in one small region [@cro01].
Seyfert Nuclei
--------------
The situation is somewhat different for less luminous AGNs. These cannot be found by color selection, as the host galaxy light dominates their broad-band colors. They are normally found by getting spectra of the nuclear regions of large samples of galaxies [eg. @huc92; @ho97]. To date, these galaxy samples have been magnitude limited in the blue. This may well introduce a bias: the blue light from galaxies is dominated by young stars, and is hence an indication of recent star formation.
The near-IR light from galaxies is coming from an older stellar population, and hence correlates with the total stellar mass rather than the recent star formation rate. Near-IR selected galaxy samples are dominated by elliptical galaxies, unlike blue selected samples which are dominated by spirals.
We might thus expect the population of AGNs in an IR-selected galaxy sample to differ from that in a blue-selected sample for many reasons. The black hole masses, which are known to correlate with the bulge stellar mass, should be larger. If accretion onto the black hole correlates with star formation, we might be looking at lower accretion rates. Dust properties may be quite different, altering the ratios of obscured (Type-2) and unobscured (Type-1) AGNs.
Searching for AGNs in 2MASS\[search\]
-------------------------------------
By far the largest near-IR survey to date is the Two Micron All Sky Survey [2MASS, @skr97]. There have already been several studies of the different AGN populations within 2MASS. @cut02 have shown that 2MASS sources with extremely red near-IR colors ($J-K_s>2$) are mostly an unusual type of Type-1 AGN [@smi02; @wil02]. $K_s$ is a filter similar to $K$ but cutting off at a shorter red wavelength to minimize thermal emission [@skr97]. @bar01 have studied the 2MASS colors of QSOs identified at other wavelengths, and @gre02 identified some very unusual and dusty QSOs by cross-correlating the 2MASS database with a radio sample. While these various papers clearly show that 2MASS imaged large numbers of AGNs, none of them made any pretense at giving an unbiassed picture of the AGN population within 2MASS.
In this paper, we assemble a relatively unbiassed sample of 2MASS AGNs. We use brute force: we apply only a very weak color selection, and then use multi-object spectroscopy to pick out the few AGNs from the large contamination of other objects.
Our only selection criteria were that an object had to be detected in all three 2MASS bands, and that it had $J-K_s>1.2$. This color cut was designed to eliminate halo giant and disk dwarf stars (the peaks at $J-K_s \sim 0.45$, $0.75$ in Fig \[jkhist\]), but still be sensitive to most galaxies and QSOs. Nearly all QSOs with redshifts below $\sim 0.5$, selected at other wavelengths, have $J-K_s>1.2$ [@fra00; @bar01; @cut02]. At higher redshifts, the near-IR flux excess [@san89] is shifted out of the $K_s$-band, causing the average $J-K_s$ color of known AGN to become bluer, but even at these higher redshifts, at least 10% of AGN have $J-K_s>1.2$.
The color cut eliminates some galaxies from our sample. The median $J-K_s$ color of galaxies in the 2MASS Extended Source Catalog (XSC) is $\sim 1.1$. The mean color of galaxies shifts rapidly to the red with increasing redshift due to k-corrections, though. At the magnitude limit of the 2MASS Point Source Catalogue (our input catalog), most galaxies will be unresolved and will lie at redshifts well above 0.1 where the median galaxy color is redder than $J-K_s = 1.2$. We estimate our incompleteness to galaxies by counting the number of 2MASS XSC sources with $J-K_s < 1.2$ in our survey areas, and by cross-correlating the Sloan Digital Sky Survey Early Release galaxy catalog with the 2MASS PSC: fewer than 23% of galaxies would fail to meet our color cut. The missing galaxies will be predominantly at redshifts less than 0.1 and resolved by 2MASS.
The bias of our survey towards low redshift AGN does limit our ability to find dusty QSOs. The luminosity of most QSOs found in the local universe is only a little greater than that of their host galaxies. Even small amounts of dust extinction will thus reduce the AGN light below the host galaxy light, causing the source to be classified as a Type-2 AGN rather than a dusty Type-I AGN. Spectacularly reddened Type-1 AGN should thus only be found in high redshift, high luminosity samples, such as that of @gre02.
Our target selection, observations and data reduction are described in § \[obsred\], and the spectral classification of our sources in § \[class\]. Our results are presented in § \[results\] and discussed in § \[discuss\]. Finally, conclusions are drawn in § \[conclude\]
Observations and Reduction\[obsred\]
====================================
Target Selection
----------------
Targets were selected from the 2MASS Point Source Catalog. Note that this catalog includes extended sources. All cataloged sources with $J-K_s>1.2$ and detections in all three bands were potential targets, regardless of optical magnitude or morphology. No attempt was made to exclude previously observed sources.
We observed spectra of sources in four fields. Each field was circular, and one degree in radius. The fields were centered at 09:44$+$00:00, 12:44$+$00:00, 13:00$-$25:00 and 14:15$-$26:00 (J2000). The first two fields were chosen to overlap with the imaging data from the early data release of the Sloan Digital Sky Survey [SDSS, @sto02]. All fields lie at galactic latitudes greater than 30 degrees.
Observations were carried out with the Two Degree Field (2dF) spectrograph on the Anglo-Australian Telescope [AAT, @lew02]. This spectrograph has 400 fibers, spread over a circular field of radius one degree, located at the prime focus of the AAT. Each fiber has a projected diameter of 2 on the sky. A small number of fibers were set aside to measure the sky spectrum. The remaining fibers were allocated to targets using the [*configure*]{} program [@lew02]. The program was set to allocate fibers to the brightest $K_s$-band sources first, and then progressively to the fainter ones. We were able to allocate fibers to all 69 sources with $K_s <14.0$, 677 of the 873 sources with $14.0 < K_s < 15.0$, but only 780 of the 1407 sources with $15.0 < K_s < 15.5$. The incompleteness in the $14.0 < K_s < 15.0$ range is mostly due to fiber positioning constraints, while the incompleteness at fainter magnitudes is due to the limited number of fibers. The incompleteness that this introduces should be random in every parameter except K-band magnitude. The magnitude and color distribution of the sources for which we obtained spectra are shown in Fig \[complete\].
Observations and Reduction
--------------------------
Spectra were taken of sources in our four fields on the nights of 2002 March 5 – 7. Conditions were partially cloudy at times, and the seeing was typically around 1.8. Each field was observed with two different fiber configurations: one for the brightest $\sim 30$ sources and the other for the remaining $\sim 350$. This technique was chosen to minimize scattered light problems. Exposure times were 600-900 sec for the bright object configurations, and 10,036 – 10,800 sec for the faint object configurations. Bright sources in the 1415$-$2600 field were not observed, due to cloud. The 300R and 316R gratings were used in the two spectrographs, giving a spectral resolution of 10Å and a wavelength coverage of 4500 - 8500Å.
The data were reduced using the [*2dfdr*]{} software [@lew02], using standard settings. All galaxy spectra were averaged (in the observed frame) to provide a template atmospheric absorption spectrum. The individual spectra were divided through by this template, which did a reasonably good job of correcting for these absorption bands. The spectra are not of spectrophotometric quality.
Spectra were obtained for a total of 1526 sources. 59 of these spectra were of such poor quality that regardless of the nature of the source, no spectral classification was possible. Fig \[complete\] shows that the unusable quality spectra are predominantly those of the brighter sources. This was mainly caused by poor weather: the bright objects in one of the four fields were never observed while in two other fields they were observed through significant cloud cover. Observations of the fainter sources were not as badly affected.
Classification\[class\]
=======================
An initial classification was attempted for all spectra using the software developed for the 2dF Galaxy Redshift Survey [@col01]. This software is optimized for measuring galaxy redshifts from 2dF spectra of comparable quality to our own, and uses template fitting, line-fitting and cross-correlation techniques to classify spectra and to measure redshifts. All classifications were checked manually and assigned a quality flag. The program produced high quality classifications and (where relevant) redshifts for around 80% of our spectra. It showed excellent performance for galaxies in our sample, but was less reliable for broad-line AGN and stars. The remaining spectra were checked by eye, and in many cases secure identifications could be made interactively.
Emission line diagnostics\[eline\]
----------------------------------
[lcc]{} H$\beta$ & 474.0 – 484.0, 488.0 – 494.0 & 484.5 – 488.0\
$[$$]$ & 488.0 – 494.0, 503.0 – 510.0 & 499.0 – 502.5\
H$\alpha$ & 640.0 – 652.0, 663.0 – 670.0 & 655.0 – 657.2\
& 640.0 – 652.0, 663.0 – 670.0 & 657.2 – 659.4\
$[$$]$ & 663.0 – 670.0, 674.5 – 684.5 & 670.0 – 674.5\
Galaxies showing H$\alpha$ and/or H$\beta$ emission lines with velocity widths (full width at half maximum height: FWHM) greater than 1000${\rm km\ s}^{-1}$ were classified as Type-1 AGNs. Emission-line ratios were measured automatically for the remaining galaxies, by interpolating a straight-line continuum under them and summing the flux above this continuum. Wavelength regions used to define the continuum and over which line fluxes were summed are shown in Table \[linewave\]. The effect of the underlying stellar absorption lines was corrected for by measuring the mean absorption-line equivalent width of the line-less galaxies in the sample, and adding this to the measured emission-line equivalent widths. This assumes that the underlying stellar continua of emission-line and non-emission-line galaxies are the same, which will only be true to first approximation. These corrections are small: 0.05 nm for H$\alpha$, 0.18 nm for H$\beta$, 0.15nm for \[, -0.07nm for and 0.005 nm for . To check that these corrections did not significantly affect our results, we repeated our classification without making them. This did not alter the classification of any of our galaxies, principally because the most affected sources were Seyfert-2 galaxies with very weak H$\beta$ emission, and these lie a long way from the selection boundary. All line measurements were checked by eye and awarded a quality flag: 6% of spectra were too poor at the relevant wavelength to obtain a good measurement of H$\alpha$. This was usually caused by H$\alpha$ falling on a strong sky line or atmospheric absorption band.
We estimate our equivalent width limit by looking at the dispersion in H$\alpha$ equivalent width measurements in galaxies with no detectable line emission (Fig \[hahist\]). We estimate that we are sensitive to all galaxies with rest-frame H$\alpha$ equivalent widths of $>0.4$nm. This excludes the 6% of galaxy spectra which were too poor at the relevant wavelength.
All galaxies with adequate quality data in all emission lines were classified using the diagnostic diagrams of @kew01. The results are shown in Fig \[classplot\]. The galaxies split cleanly between AGN and starbursts. The AGN have the line ratios of Seyfert-II galaxies and not of LINERS. The 23 sources lying above both classification lines were classified as Type II AGN, and the 34 lying below as Starburst galaxies. One source lay above one line and below the other: we classified it as an unknown emission-line galaxy.
Unfortunately, while many galaxies had good quality data for the lines near H$\alpha$ ( and $[$$]$), the shorter wavelength lines (H$\beta$ and $[$$]$) were often too weak for us to be able to calculate their position along the y-axis of Fig \[classplot\]. We note, however, a reasonably strong correlation between the x- and y-axes in the classification plots. If this correlation holds for the galaxies with weaker short wavelength lines, we can use it to tentatively classify at least some of these sources. All otherwise unclassified emission-line galaxies with $\log_{10}($$ / H \alpha ) > -0.2$ and $\log_{10}([$$]/ H \alpha ) > -0.35$ were classified as probable AGN, while sources with $\log_{10}($$ / H \alpha ) < -0.3$ and $\log_{10}([$$] / H \alpha ) < -0.4$ were classified as probable starbursts. This yielded another 12 probable Seyfert II galaxies, and 65 probable starburst galaxies. All other galaxies with H$\alpha$ equivalent widths greater than 0.4nm were classified as unknown emission-line galaxies.
Sloan Digital Sky Survey Data\[sdss\]
-------------------------------------
We extracted data from the SDSS early data release for the 739 of our targets lying within its region of coverage. For all but two of these sources, a SDSS cataloged source was found within 1.6of the 2MASS position. The median positional offset between the 2MASS and SDSS coordinates was 0.269.
Two 2MASS sources did not have SDSS cataloged sources within 5of the 2MASS position: 2MASS 0943007$-$000955 (an M-dwarf star) and 2MASS 0946501$+$002050 (a galaxy at redshift 0.1414). As we obtained good spectra of both sources, using the 2.0 diameter 2dF fibers centered at the 2MASS coordinates, the error must not lie in the 2MASS catalog.
All the 477 sources we classified as galaxies (on the basis of their spectra) were classified as extended sources by SDSS, while 147/156 stars were classified as point sources. Of the 68 sources with SDSS data that we were unable to classify based on their spectra, 54 (80%) were classified by SDSS as extended sources. A visual inspection of their spectra confirms that most are probably galaxies without strong emission or absorption lines at wavelengths with good data.
Results\[results\]
==================
We obtained 1526 spectra, or which 1467 were of usable quality. We were able to obtain secure classifications for 1298 of these spectra (88% completeness). As noted in Section \[sdss\], the remaining unclassified sources are predominantly galaxies without strong absorption or emission lines at wavelengths for which we have good data. If any of these unclassified sources with usable quality spectra were AGNs with H$\alpha$ equivalent widths of $> 0.4$nm, we would have detected their emission lines. The unclassified sources are concentrated in the fields observed through cloud: our successful classification rate is much higher in the fields observed in clear weather.
[cccccc]{} 1.2 – 1.4 & 707 & 33% & 67% & 0.3% & 1.7%\
1.4 – 1.6 & 357 & 19% & 81% & 1.4% & 2.5%\
1.6 – 1.8 & 174 & 15% & 85% & 1.7% & 1.7%\
$>$ 1.8 & 66 & 9% & 91% & 6.0% & 0.0%\
$>$ 2.0 & 664 & 1% & 99% & 58% & 15%\
330 (25%) of the classified objects are stars. Around 20% are late K-dwarfs, and the remainder are M-dwarfs. The stars with SDSS data have a median $R=19.22$. The fractions of objects with various classifications as a function of $J-K_s$ color are shown in Table \[tclass\].
[lcccccl]{} 2MASS 09403186-0028433 & 09:40:31.86 -00:28:43.3 & 18.2 & 15.38 & 1.39 & 0.153 &\
2MASS 09441580+0011015 & 09:44:15.80 +00:11:01.5 & 17.2 & 14.61 & 1.51 & 0.128 & sdss J094415.78+001101.2\
2MASS 09452492+0041448 & 09:45:24.92 +00:41:44.8 & 17.8 & 15.04 & 1.65 & 0.200 &\
2MASS 09460212+0035186 & 09:46:02.12 +00:35:18.6 & 18.1 & 14.85 & 1.73 & 0.649 & sdss J094602.11+003518.7\
2MASS 12420264+0012191 & 12:42:02.64 +00:12:29.1 & 16.9 & 14.63 & 1.46 & 1.216 & LBQS 1239+0028\
2MASS 12442311+0027160 & 12:44:23.11 +00:27:16.0 & 17.8 & 15.02 & 1.49 & 0.165 & sdss J124423.07+002715.9\
2MASS 12452459-0009379 & 12:45:24.59 -00:09:37.9 & 17.6 & 15.42 & 1.22 & 2.077 & LBQS 1242+0006\
2MASS 12461313-0042330 & 12:46:13.13 -00:42:33.0 & 16.7 & 14.46 & 1.49 & 0.649 & LBQS 1243-0026\
2MASS 13025113-2428552 & 13:02:51.13 -24:28:55.2 & 17.5 & 14.50 & 2.17 & 0.246 &\
2MASS 13031854-2435071 & 13:03:18.54 -24:35:01.7 & 17.4 & 14.69 & 1.84 & 2.255 & HB1300-243\
2MASS 14161423-2607468 & 14:16:14.23 -26:07:46.8 & 17.5 & 14.25 & 1.70 & 0.220 &\
2MASS 14165581-2524134 & 14:16:55.81 -25:24:13.4 & 15.3 & 12.73 & 1.82 & 0.236 & CTS0025\
2MASS 14180763-2548430 & 14:18:07.63 -25:48:43.0 & 16.8 & 14.73 & 2.02 & 0.494 &\
2MASS 14183782-2540138 & 14:18:37.82 -25:40:13.8 & 16.8 & 14.55 & 1.48 & 0.155 &\
The remaining 968 sources (75%) are galaxies of various types. 14 have broad emission lines and are hence Type-1 AGN (Table \[type1\]). 23 are definite Type-2 AGNs (Seyfert 2 galaxies) while another 12 are probable Type-2 AGNs (as described in Section \[eline\]). The Type-2 AGNs are listed in Table \[type2\]. There are 106 galaxies whose line ratios make them definite or probable starburst galaxies, and a further 71 galaxies with H$\alpha$ rest-frame equivalent widths greater than 0.4nm , but which we couldn’t classify. The galaxies with SDSS data have a median $R=17.65$. Redshifts and spectral classifications for all galaxies with adequate data are shown in Table \[allgal\].
[lcccccl]{}
2MASS 09444446+0035446 & 9:44:44.46 +00:35:44.6 & 17.9 & 15.15 & 1.52 & 0.1659 &\
2MASS 09412757-0020337 & 9:41:27.57 -00:20:33.7 & 16.2 & 15.31 & 1.21 & 0.1487 &\
2MASS 09443759+0034107 & 9:44:37.59 +00:34:10.7 & 17.5 & 14.75 & 1.52 & 0.1447 &\
2MASS 09410612-0028238 & 9:41:06.12 -00:28:23.8 & 16.4 & 14.81 & 1.47 & 0.1481 &\
2MASS 09472633-0005562 & 9:47:26.33 -00:05:56.2 & 17.4 & 14.96 & 1.47 & 0.1260 &\
2MASS 09415313+0009185 & 9:41:53.13 +00:09:18.5 & 16.5 & 14.13 & 1.75 & 0.1221 &\
2MASS 09422430-0000051 & 9:42:24.30 -00:00:05.1 & 16.1 & 14.16 & 1.80 & 0.1465 &\
2MASS 09425917+0031414 & 9:42:59.17 +00:31:41.4 & 16.3 & 14.36 & 1.39 & 0.0633 &\
2MASS 09430377+0008076 & 9:43:03.77 +00:08:07.6 & 17.0 & 14.93 & 1.33 & 0.1240 &\
2MASS 09452964-0021547 & 9:45:29.64 -00:21:54.7 & 13.2 & 14.00 & 1.24 & 0.0515 &\
2MASS 09451196-0007119 & 9:45:11.96 -00:07:11.9 & 12.4 & 13.64 & 1.37 & 0.0306 &\
2MASS 09441489+0018082 & 9:44:14.89 +00:18:08.2 & 17.1 & 14.75 & 1.47 & 0.1223 &\
2MASS 09443030+0045287 & 9:44:30.30 00:45:28.7 & 17.0 & 14.88 & 1.32 & 0.1237 &\
2MASS 12432177+0015370 & 12:43:21.77 00:15:37.0 & 17.1 & 14.86 & 1.38 & 0.1433 &\
2MASS 13031198-2447024 & 13:03:11.98 -24:47:02.4 & 17.8 & 15.22 & 1.31 & 0.1252 &\
2MASS 12593796-2523148 & 12:59:37.96 -25:23:14.8 & 15.6 & 13.87 & 1.50 & 0.0728 &\
2MASS 12595227-2516420 & 12:59:52.27 -25:16:42.0 & 12.8 & 13.01 & 1.36 & 0.0486 &\
2MASS 14143610-2546458 & 14:14:36.10 -25:46:45.8 & 16.6 & 15.27 & 1.55 & 0.1653 &\
2MASS 14132413-2615549 & 14:13:24.13 -26:15:54.9 & 16.0 & 14.54 & 1.72 & 0.1717 &\
2MASS 14140715-2528597 & 14:14:07.15 -25:28:59.7 & 16.2 & 14.66 & 1.33 & 0.1695 &\
2MASS 14143799-2520079 & 14:14:37.99 -25:20:07.9 & 17.2 & 14.85 & 1.35 & 0.1391 &\
2MASS 14154583-2518245 & 14:15:45.83 -25:18:24.5 & 16.5 & 14.50 & 1.53 & 0.0750 &\
2MASS 14155435-2557332 & 14:15:54.35 -25:57:33.2 & 16.8 & 14.73 & 1.33 & 0.1681 &\
2MASS 14155854-2544131 & 14:15:58.54 -25:44:13.1 & 16.2 & 14.44 & 1.48 & 0.1697 &\
2MASS 09430751-0002492 & 9:43:07.51 -00:02:49.2 & 17.6 & 15.37 & 1.33 & 0.1247 &\
2MASS 09452796+0051041 & 9:45:27.96 +00:51:04.1 & 16.1 & 14.52 & 1.32 & 0.1431 &\
2MASS 12445756-0016176 & 12:44:57.56 -00:16:17.6 & 16.8 & 14.50 & 1.30 & 0.1186 &\
2MASS 12451295-0040566 & 12:45:12.95 -00:40:56.6 & 15.8 & 14.49 & 1.28 & 0.1043 &\
2MASS 12452522-0046579 & 12:45:25.22 -00:46:57.9 & 15.7 & 14.85 & 1.45 & 0.0806 &\
2MASS 12595666-2439065 & 12:59:56.66 -24:39:06.5 & 17.4 & 15.24 & 1.23 & 0.1053 &\
2MASS 13015692-2533499 & 13:01:56.92 -25:33:49.9 & 17.7 & 15.12 & 1.55 & 0.1859 &\
2MASS 12583172-2508040 & 12:58:31.72 -25:08:04.0 & 16.9 & 15.27 & 1.28 & 0.1045 &\
2MASS 13005827-2423430 & 13:00:58.27 -24:23:43.0 & 15.3 & 14.27 & 1.24 & 0.0993 &\
2MASS 12591093-2446418 & 12:59:10.93 -24:46:41.8 & 15.3 & 13.63 & 1.73 & 0.1125 &\
2MASS 12594636-2535568 & 12:59:46.36 -25:35:56.8 & 12.9 & 13.66 & 1.49 & 0.0639 & ARP 1257-251\
2MASS 14175559-2538535 & 14:17:55.59 -25:38:53.5 & 17.4 & 14.84 & 1.47 & 0.1209 &\
2MASS 14143476-2522301 & 14:14:34.76 -25:22:30.1 & 17.3 & 14.79 & 1.64 & 0.1159 &\
SDSS classifies 91% of the galaxies without measurable H$\alpha$ emission as elliptical galaxies (ie. a De Vaucouleurs profile fits significantly better than an exponential profile). For emission-line galaxies (excluding AGNs) the fraction is 50%. The redshift histograms are shown in Fig \[gal\_zhist\]. The galaxies without emission lines are quite strongly clustered: the peak seen at redshift 0.14 is due to one such cluster. The emission-line galaxies lie at a lower mean redshift than those without emission lines. This is probably because the emission-line galaxies are late type while those without emission lines are massive luminous early type galaxies, and hence seen to larger distances.
The Type 2 AGNs have a redshift distribution indistinguishable from that of other emission-line galaxies in the survey (Fig \[AGN\_zhist\]). Most Type 1 AGNs also lie at low redshifts, but there is a tail to very high redshifts.
Discussion\[discuss\]
=====================
The QSO Sample\[kband\]
-----------------------
We find 12 Type-1 AGNs with $1.2 < J-K_s < 2.0$. Allowing for our incomplete spectroscopy of the faintest sources, this implies a surface density of $1.7 \pm 0.5$ Type-1 AGNs per square degree, down to our selection limits (3-band detection by 2MASS in J, H and K). Many of our sources, however, are fainter than the nominal completeness limit of the 2MASS survey. We estimated this completeness limit for our fields by comparing our $K_s$-band galaxy counts against the compilation of @hua01. Down to $K_s=15$, our target list appear to be highly complete: the statistical error due to our small sample of AGNs is much greater than any error caused by sample incompleteness. At this limit, and correcting for the unobserved targets, we are finding $1.0 \pm 0.3$ Type-1 AGNs per square degree.
Are we missing many AGNs with $J-K_s<1.2$? @bar01 showed that essentially all low redshift ($z<0.5$) AGNs discovered by other techniques have $J-K_s>1.2$, and should hence have been found in our survey. At higher redshifts, however, most AGNs have bluer $J-K_s$ colors. This is probably a $k$-correction effect: most AGN show a sharp rise in flux between rest-frame 1 and 2 microns, perhaps due to hot dust emission [eg. @san89], and at redshifts above 0.5, this is redshifted out of the K-band. Only $\sim 10$% of high-z ($z>0.5$) AGNs detected by other techniques have $J-K_s>1.2$, except in the redshift range $2.2 < z < 2.5$, in which the H$\alpha$ emission line lies within the K-band. This is consistent with our redshift distribution (Fig \[AGN\_zhist\]).
Another way to test our completeness is to see whether we recovered previously know AGNs in our fields. The NASA Extragalactic Database (NED) lists 99 AGNs in our field, of which only nine meet our magnitude limits. Only one of the nine has $J-K_s<1.2$. We recovered six of the remaining nine objects: we did not put fibers on the other two sources.
We can therefore place a lower limit on the surface density of Type-1 AGN of $1.0 \pm 0.3$ per square degree, down to $K=15$. This matches the surface density of optically selected AGNs down to $B \sim 18.5$ [@mey01]. Given that a typical quasar has $B-K \sim 3.5$ [@fra00], this suggests that we may be seeing the same population sampled by optical surveys. This comparison should be treated with caution, however, as most optically selected QSOs down to $B=18.5$ lie at redshifts to which we are largely insensitive, and most of our AGN have such low luminosities that they would be discarded from most optical samples due to host galaxy contamination (§ \[find\]).
Could these QSOs be found by conventional techniques?\[find\]
-------------------------------------------------------------
As Table \[type1\] shows, only 57% of the Type-1 AGNs, and none of the Type-2 AGNs had been previously identified as AGNs. All 5 Type-1 AGNs with redshifts above 0.5 had been previously identified.
In Fig \[colourplot\], we compare the optical colors of our AGNs with the colors of field stars and galaxies. Only sources overlapping with the SDSS early data release are shown. Our AGNs are clearly separated from the stellar locus. Half the Type-1 AGNs are also well separated from the galactic locus, but the other half are not, and the Type-2 AGNs also lie well within the galactic locus. This explains why so many of our sources were not previously identified: they are spatially resolved and have galaxy-like optical colors. Adding near-IR photometry doesn’t help: their spectra energy distributions are indistinguishable from those of inactive galaxies all the way from the $U$ to the $K_s$ band.
Why do the colors of so many of our sources resemble galaxies? The spectra of all AGNs with galaxy-like colors show strong stellar absorption lines, so their colors are probably dominated by the host galaxy, and not by nuclear emission. Furthermore, Fig \[kmag\] shows that the $K_s$-band magnitudes of these AGNs are comparable to those of inactive galaxies at the same redshifts. We thus conclude that the host galaxy, rather than the nucleus, is dominating the observed continuum flux at all wavelengths.
While these AGNs would not be identified as such on the basis of their broad-band colors, their strong broad H$\alpha$ emission lines would make them detectable in objective prism surveys, and large galaxy redshift surveys should contain thousands of them.
Dusty QSOs?
-----------
Is there a large population of dusty red QSOs? We are unable to determine whether our low redshift AGNs are dust-reddened, as the host galaxies dominate their broad-band colors, and as our observations are not spectrophotometric, we cannot determine the reddening by looking at line ratios. It is possible that their spectra are dominated by host galaxy light precisely because they are dusty. Alternatively, their nuclei could be intrinsically less luminous, or their host galaxies intrinsically more luminous than those of optically selected AGNs with the same $K_s$-band luminosities.
This leaves the small number of high redshift QSOs. SDSS colors are available for four of these sources. All four are quite blue: their mean $g-K_s$ color is 2.86, which is very comparable to that seen in optically selected QSO samples [@fra00]. None are more than 0.7 mag redder than the mean in $g-K$, corresponding to $A_V=0.8$ (for dust with an optical depth inversely proportional to wavelength).
We thus see no evidence for a population of dusty red QSOs. Our sample is, however, too small to place strong constraints. Let us define a red QSO as one with $g-K_s>3.5$, corresponding to $A_V > 0.8$. The fact that none of the four high-z QSOs with SDSS data meet this definition allows us to say with 95% confidence that no more than 50% of QSOs down to our magnitude limit are red. As shown in Fig \[dust\], however, imposing even a K-band magnitude limit will suppress the numbers of red AGNs by a factor of $\sim 5$. Our limit is thus a weak one: no more than 80% of QSOs can be red. This is quite consistent with the limits derived from radio surveys [@fra01; @gre02].
The Fraction of Galaxies with Active Nuclei
-------------------------------------------
What fraction of the galaxies in our sample contained active nuclei, and how does this compare to the fraction found in blue galaxy samples?
We are only sensitive to AGNs with H$\alpha$ rest-frame equivalent widths of greater than 0.4nm, within our aperture (our fibers are 2in diameter, which for the median redshift of the sample (0.15) corresponds to a physical diameter of 5kpc). @ho97 showed that we will miss most AGNs at this equivalent width limit, with our large aperture. In particular, we are mainly sensitive to Seyfert-2 galaxies and not to LINERs (low-ionization nuclear emission-line regions).
We obtained 1467 usable spectra, of which 330 were stars. Another 169 showed no significant emission or absorption features and are probably inactive galaxies. The remaining 968 are galaxies for which we were able to measure secure redshifts. 6% of these galaxies had something wrong with their spectra at the wavelength of H$\alpha$. 187 of the remaining galaxies had H$\alpha$ equivalent widths of $>0.4$nm. Of these, we were able to spectrally classify 116. We identified 14 Type-1 AGNs and 35 Type-2 definite or probable AGNs.
There are 71 galaxies with narrow H$\alpha$ lines above our selection threshold but with the signal-to-noise ratio too poor to allow classification. If we assume that these galaxies have the same relative proportions of Type-2 AGNs and Starburst galaxies as the 116 we could classify, then there are 58 Type-2 AGNs in the sample.
The sample population in which we could have seen Type-2 AGN activity is $968+169=1137$ galaxies. We thus estimate that $58/1137 = 5.1 \pm 0.9$% of galaxies in our sample were Type-2 AGNs, down to our H$\alpha$ equivalent width limit (Poisson errors). If none of the unclassifiable emission-line galaxies were AGNs, which seems unlikely, the fraction would be $35/1137 = 3.1$%. This gives a lower limit on the fraction. For Type-1 AGNs, the fraction is $1.2 \pm 0.3$%. Note also that our 35 definite or probable AGNs included several classified on the basis of their red emission lines only, as discussed in Section \[eline\].
How does this compare with the AGN fraction in blue-selected galaxy samples? @huc92 only found Seyfert activity (of any type) in 1.3% of a sample of 2399 nearby galaxies. Unfortunately, they do not list their equivalent width threshold, so this number cannot be compared to our figure. @ho97, however, find that nearly 50% of nearby blue-selected galaxies are AGNs, but they used nuclear spectra and were sensitive to much weaker lines than we are.
We defined a sub-sample of the @ho97 sources that would have been classified as Seyfert galaxies by our criteria. Firstly, we had to correct for the different spectroscopic aperture. They typically measured the spectrum over a region of radius $< 200$ pc, compared to our physical radii of $\sim 2500$pc. To see how much difference this typically makes, we obtained archival CCD images of six Seyfert-2 galaxies from their sample, using the NASA Extragalactic Database. We measured the broad-band optical flux for each galaxy in a 200pc aperture and a 2500pc aperture. The fluxes in the larger apertures were greater by a factor of between 4 and 25.
Thus for one of their galaxies to have made it into our sample (if it were at the median redshift of our galaxies), it would need a nuclear H$\alpha$ equivalent width exceeding 1.6nm ($4 \times 0.4$nm). @ho97 also use slightly different diagnostics to identify AGN but this makes little difference to the final numbers.
We find that $1.5 \pm 0.6$ of their galaxies contained Type-2 AGNs (meeting our selection criteria) and $1 \pm 0.4$% contained Type-1 AGNs. There is thus no significant difference in the fraction of Type-1 AGNs, but we are finding a significantly higher fraction of Type-2 AGNs.
Why do we find a higher fraction of galaxies with AGNs? Finding an AGN requires the presence of a black hole, a suitable accretion rate of mass on to it, gas to be ionized by the nucleus and a dust geometry that both allows this ionization to take place and allows us to see the resultant narrow line emission. Many of these factors could be different in an IR-selected sample. As black hole masses are correlated with the stellar masses of the bulge [eg @mag98], this may indicate that our sample of galaxies contain larger nuclear black holes than those found in blue-selected samples. On the other hand, blue galaxies typically have more gas and a higher star formation rate than red ones. Note also that our galaxies lie at higher redshifts than the @ho97 sample, and are on average more massive and luminous. We may simply be looking at a tendency for AGNs to be found in the most massive galaxies.
Could the difference simply be because the near-IR selected galaxies have less continuum flux at the wavelength of H$\alpha$? Our galaxies have a median $r-K_s = 2.77$, while blue-selected galaxies at similar magnitude limits have a median $r-K_s \sim 1.5$. Thus the younger stellar populations in the blue selected galaxies are increasing the continuum flux per unit stellar mass by a factor of $\sim 3$. Even allowing for this, the fraction of Seyfert-2 galaxies in @ho97 would only rise to $\sim 3$% .
This result should be considered tentative. Secure line diagnostics are only available for around half of the Seyfert-2 population in our sample, and the comparison with the very different @ho97 sample involves large corrections for the different AGN detection thresholds.
Conclusions\[conclude\]
=======================
We have selected a small sample of AGNs in the near-IR, using the brute-force power of the 2dF spectrograph to minimize selection biases. Perhaps the most surprising thing about this sample is how similar it looks to conventional blue-selected AGN samples. While many of our Type-1 AGN would not have been found by optical techniques, in all cases this seems to be due to host galaxy contamination. Large galaxy surveys, such as the 2dF Galaxy Redshift Survey [@col01], are probably the best way to find such AGNs. Our sample of high redshift QSOs was too small to usefully constrain the population of dusty red QSOs.
We tentatively conclude that the fraction of galaxies in our sample with AGN emission is greater than that found in the blue selected galaxy sample of @ho97. There are many possible reasons for this difference and discriminating between them will be difficult.
Finally, we can extrapolate from our data to estimate the number of active galactic nuclei in the 2MASS point source catalog. There should be $\sim$ 50,000 Type-1 AGNs and $\sim$ 200,000 Type-2 AGNs that meet our selection criteria.
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.
Funding for the creation and distribution of the SDSS Archive has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society.
The SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, Princeton University, the United States Naval Observatory, and the University of Washington.
This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
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[ccccccccccc]{} 9:40:06.62 -0:05:14.2 & 0.0629 & 5 & sb? & 14.0 & 1.22 & 0.3 & & & 0.3 & 0.1\
9:40:08.85 0:15:07.8 & 0.0497 & 5 & gem & 14.9 & 1.66 & 1.6 & & & 1.1 & 0.6\
9:40:09.09 -0:14:41.1 & 0.1724 & 5 & gal & 14.9 & 1.37 & & & & &\
9:40:10.60 0:13:12.6 & 0.0625 & 5 & sb? & 13.8 & 1.40 & 0.5 & & & 0.3 & 0.1\
9:40:11.62 -0:11:50.5 & 0.2045 & 5 & gal & 15.4 & 1.36 & & & & &\
9:40:14.73 -0:19:46.7 & 0.1256 & 5 & gal & 15.0 & 1.28 & & & & &\
9:40:19.15 0:04:25.4 & 0.0904 & 5 & sb & 14.9 & 1.72 & 5.0 & 0.7 & 0.4 & 1.7 & 2.0\
9:40:20.75 0:23:32.0 & 0.0162 & 4 & gal & 15.0 & 1.27 & & & & &\
9:40:28.31 0:05:23.3 & 0.2468 & 5 & gal & 15.2 & 1.45 & & & & &\
9:40:31.86 -0:28:43.3 & 0.1542 & 5 & qso & 15.4 & 1.39 & & & & &\
|
---
abstract: 'This contribution reviews the present status on the available constraints to the nuclear equation of state (EoS) around saturation density from nuclear structure calculations on ground and collective excited state properties of atomic nuclei. It concentrates on predictions based on self-consistent mean-field calculations, which can be considered as an approximate realization of an exact energy density functional (EDF). EDFs are derived from effective interactions commonly fitted to nuclear masses, charge radii and, in many cases, also to pseudo-data such as nuclear matter properties. Although in a model dependent way, EDFs constitute nowadays a unique tool to reliably and consistently access bulk ground state and collective excited state properties of atomic nuclei along the nuclear chart as well as the EoS. For comparison, some emphasis is also given to the results obtained with the so called [*ab initio*]{} approaches that aim at describing the nuclear EoS based on interactions fitted to few-body data only. Bridging the existent gap between these two frameworks will be essential since it may allow to improve our understanding on the diverse phenomenology observed in nuclei. Examples on observations from astrophysical objects and processes sensitive to the nuclear EoS are also briefly discussed. As the main conclusion, the isospin dependence of the nuclear EoS around saturation density and, to a lesser extent, the nuclear matter incompressibility remain to be accurately determined. Experimental and theoretical efforts in finding and measuring observables specially sensitive to the EoS properties are of paramount importance, not only for low-energy nuclear physics but also for nuclear astrophysics applications.'
author:
- |
X. Roca-Maza$^{1}$ and N. Paar$^2$\
$^1$Dipartimento di Fisica, Università degli Studi di Milano\
and INFN, Sezione di Milano, 20133 Milano, Italy.\
$^2$Department of Physics, Faculty of Science,\
University of Zagreb, Bijeni$\check{\rm c}$ka c. 32, 10000 Zagreb, Croatia.\
bibliography:
- 'bibliography.bib'
title: Nuclear Equation of State from ground and collective excited state properties of nuclei
---
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the Croatian Science Foundation under the project Structure and Dynamics of Exotic Femtosystems (IP-2014-09-9159) and the QuantiXLie Centre of Excellence, a project co-financed by the Croatian Government and European Union through the European Regional Development Fund - the Competitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004). Funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 654002 is also acknowledged.
|
---
author:
- 'F. T. Watson'
- 'L. Fletcher'
- 'S. Marshall'
bibliography:
- 'SunspotCataloguePaperRefs.bib'
date: 'Received ; accepted '
title: Evolution of sunspot properties during solar cycle 23
---
[The long term study of the Sun is necessary if we are to determine the evolution of sunspot properties and thereby inform modeling of the solar dynamo, particularly on scales of a solar cycle.]{} [We aim to determine a number of sunspot properties over cycle 23 using the uniform database provided by the SOHO Michelson Doppler Imager data. We focus in particular on their distribution on the solar disk, maximum magnetic field and umbral/penumbral areas. We investigate whether the secular decrease in sunspot maximum magnetic field reported in Kitt Peak data is present also in MDI data.]{} [We have used the Sunspot Tracking And Recognition Algorithm (STARA) to detect all sunspots present in the SOHO Michelson Doppler Imager continuum data giving us 30084 separate detections. We record information on the sunspot locations, area and magnetic field properties as well as corresponding information for the umbral areas detected within the sunspots, and track them through their evolution.]{} [We find that the total visible umbral area is 20-40% of the total visible sunspot area regardless of the stage of the solar cycle. We also find that the number of sunspots observed follows the Solar Influences Data Centre International Sunspot Number with some interesting deviations. Finally, we use the magnetic information in our catalogue to study the long term variation of magnetic field strength within sunspot umbrae and find that it increases and decreases along with the sunspot number. However, if we were to assume a secular decrease as was reported in the Kitt Peak data and take into account sunspots throughout the whole solar cycle we would find the maximum umbral magnetic fields to be decreasing by 23.6 $\pm$ 3.9 Gauss per year, which is far less than has previously been observed by other studies (although measurements are only available for solar cycle 23). If we only look at the declining phase of cycle 23 we find the decrease in sunspot magnetic fields to be 70 Gauss per year.]{}
Introduction
============
Sunspots are dark areas on the solar surface and are associated with strong magnetic fields. The magnetic field inhibits the convective flow of plasma in the region and as this is the primary mechanism for heat transport at the surface, the sunspot is cooler and darker. Study of sunspots started around the early 1600s although there are records of observations in China going back for 2000 years [@Yau1988; @Eddy1989]. Since the discovery of the magnetic field in sunspots [@Hale1908] they have been a primary indicator of solar activity and detailed records have been kept. By studying the evolution of sunspot characteristics (area, field strength, etc), on timescales of days we can gain insight into their formation and dispersal, while studies on longer timescales (months and years) can reveal the longer-term behaviour of the Sun’s large-scale magnetic field, naturally of great importance for constraining models of the solar dynamo. For example, the North-South asymmetry of sunspot numbers and areas is well-established and has been studied for many decades and may indicate a phase lag between the magnetic activity in the northern and southern hemispheres, possibly hinting at non-linear behaviour, such as random fluctuations of the dynamo terms and strong high order terms .
The sunspot cycle variation of many solar parameters is of course well established, however it was reported by @Penn2006 that Zeeman splitting observations of the strongest fields in sunspot umbrae show a secular decrease between 1998 and 2005, apparently without a clear cyclic variation. This goes hand-in-hand with an increase in the umbral brightness. Such a secular change, if verified, would have striking implications for the coming sunspot cycles - @2010arXiv1009.0784P suggest that if the trend continues there would be virtually no sunspots at the time of cycle 25. It is one of the main goals of the present study to automatically examine the MDI data for such behaviour. In creating the dataset necessary to do this we also obtain and report on the cycle-dependent behaviour of sunspot areas and locations. In particular, the total projected area of sunspots present on the visible disk is of interest in solar spectral irradiance studies [@1982JGR....87.4319W; @1985SoPh...97...21P; @1997SoPh..173..427F] where it enters as a parameter in spectral irradiance calculations.
We are fortunate now to have long and consistent series of solar observations from which such parameters can be extracted, and the computational capacity to do it automatically. Image processing and feature recognition/tracking in solar data is now a very active field [@2010SoPh..262..235A], and sunspot detection is a well-defined image processing problem that has been studied by several authors [@2005SoPh..228..377Z; @2008SoPh..248..277C; @2008SoPh..250..411C; @Watson2009]. It is the purpose of this article to detail some physical properties of sunspots detected in the continuum images from the SOHO/MDI instrument [@Scherrer1995] and how they vary throughout solar cycle 23. We have used an image processing algorithm based on mathematical morphology [@Watson2009].
The article proceeds with section 2 detailing the generation of the sunspot catalogue and the results of looking at evolution in sunspot area and locations over solar cycle 23. Then, section 3 details the evolution of magnetic fields in sunspots, particularly in the umbra where the fields are strongest. Finally, in section 4 we finish with our discussion and conclusions.
Creating a catalogue of sunspots
================================
In order to analyse the sunspots over solar cycle 23, the STARA (Sunspot Tracking and Recognition Algorithm) code developed by [@Watson2009] was used, and readers are referred there for information on the method and its testing. This is an automated system for detecting and tracking sunspots through large datasets and also records physical parameters of the sunspots detected. It involves using techniques from the field of morphological image processing to detect the outer boundaries of sunspot penumbrae. This is achieved by means of the top-hat transform which allows us to remove any limb-darkening profile from the data and to perform the detections in one step. In addition to the method given in @Watson2009 the code had to be developed further to separate the umbra and penumbra of spots as we would be looking at the magnetic fields present in the umbra. When visually inspecting the data there is a clear intensity difference between the umbra and penumbra in sunspots. This difference is due to the magnetic structure of susnpots. The umbra has a higher density of magnetic flux which inhibits convection more than in the penumbral region. This causes the umbra to be cooler and therefore appear darker. However, as sunspots move towards the limb both the umbra and penumbra are limb-darkened. For this reason, we cannot use a single threshold value to define the outer edge of the umbra. The algorithm we use removes all limb darkening effects at the same time as sunspot detection, greatly increasing speed as these two steps are carried out together. This problem has been approached by other authors using different techniques, for example the inflection point method of [@1997SoPh..171..303S], the cumulative histogram method of [@1997SoPh..175..197P], the fuzzy logic approach of [@2009SoPh..260...21F], and the morphological approach of [@2005SoPh..228..377Z]. Our method begins with the sunspots (which includes umbrae and penumbrae) detected by STARA, and then produces a histogram of sunspot pixel intensities for each spot. This clusters in two peaks, the local minimum between which corresponds to the intensity value at the edge of the umbra. A similar histogram-based approach was implemented by [@2009SoPh..260...21F] who then used concepts from fuzzy logic to assign membership to umbra or penumbra; they showed that particularly the pixel membership of the penumbra can vary significantly (tens of percent) depending on a parameter known as the membership function, but this is apparently less of a problem for low-resolution data, in which brightness variations within the penumbra are smeared out. We have not adopted such a method, but have instead identified the local minimum for each sunspot’s histogram, and created a mask for umbral pixels. We normally find that the umbra region of sunspots has an MDI pixel value of less than 7000 - 8000. However, our algorithm does have the benefit of being applied consistently across the entire data series, and being able to deal with the varying intensity across the solar disk due to limb darkening which eases the problems of sunspot detection and area estimation that occur if a straightforward intensity threshold is used.
The data used in this study are taken from the MDI instrument [@Scherrer1995] on the SOHO spacecraft. We use the level 1.8 continuum data as well as the level 1.8 magnetograms to analyse magnetic fields present in the spots. Our dataset uses 15 years of data and we analyse daily measurements taken at 0000UT when co-temporal continuum images and magnetograms are recorded. The STARA code takes around 24 hours to process the approximately 5000 days of data available to generate the sunspot catalogue used in this article and holds 30084 separate sunspot detections. The same sunspot will be detected in many different images and tracked from image to image allowing them to be associated with one another. The physical parameters obtained from this analysis are the sunspot total area and ‘centre of mass’ location, number and area of umbrae; mean, maximum and minimum magnetic fields in the umbrae and penumbra; total and excess flux in the umbrae and penumbra and the information relating to the observation itself such as time and instrument used.
Number of sunspots
------------------
The trend of sunspot number throughout a solar cycle is well documented and generally rises rapidly at the start of a solar cycle before a slower decrease towards the end of the cycle. The Solar Influences Data Center (SIDC, <http://www.sidc.be/sunspot-data/>) keeps records on the sunspot index and so we compare the results of our detections with the findings of the SIDC as an initial test. It must be noted that both indicators are not measuring the same thing as the international sunspot number recorded by the SIDC weights the sunspots seen in groups so that it becomes a stronger proxy for solar activity whereas STARA only gives us the raw number of observed sunspots. However, it is beneficial to see if the same trends are present. The data used here are the smoothed monthly sunspot number [@SIDC] and so our daily measurements have been treated in the same way to give a fair comparison.
In Fig. \[fig:emergences\] we can see that both curves share several features. The STARA output has been scaled up to the same level as the International Sunspot Number around 2001 - 2003 when sunspot count rates were higher and the general trends are more important here than absolute values due to the differences in counting methods (this scaling is permissible due to the somewhat arbitrary factors present in the SIDC sunspot numbers - see Equation \[eq:1\].) We see that both datasets exhibit the same patterns of increasing and decreasing at the same time and the agreement is very good in the declining phase of the cycle. This also continues into cycle 24 shown at the right hand side of the plot with both curves rising at the same time and we will continue to track the agreement of these further into the next cycle.
The SIDC data [@Clette2007], shown as a dashed line on the plot has a smooth rise up to the first maximum sometime in the year 2000 and falls before reaching a second maximum in 2002. This ‘double maximum’ feature, separated by the ‘Gnevyshev gap’ [@1967SoPh....1..107G] is also seen in the STARA output although the first maximum is weaker when compared to the second, in contrast with the SIDC data in which the first maximum is larger than the second. However, both sets of data scale well with one another after this second maximum with very little deviation and this continues from 2002 up to the current day.
The differences in the first peak, and indeed in the rise before that are most likely due to the method of counting sunspots as mentioned previously. In fact, the SIDC sunspot number is calculated using the formula
$$\label{eq:1}
T = k ( 10g + s )$$
where $T$ is the total sunspot number for that measurement, $g$ is the number of sunspot groups observed and $s$ is the number of individual sunspots observed. It is based on the assumption that sunspot groups have an average of 10 sunspots in them and so even in poor observing conditions, this would be a good substitute. The coefficient $k$ is a number that represents the seeing conditions from the observing site and is usually less than 1.
What Fig. \[fig:emergences\] suggests is that the SIDC observers are either detecting more sunspots than STARA in the first half of the cycle, or that they are detecting groups that have fewer than 10 sunspots in them, on average. This second explanation is more likely. Inspecting the STARA data we find it is rare to see a sunspot group with as many as ten spots in this stage of the cycle, which would account for the SIDC number being an overestimate for the actual sunspot number at this time. This in itself has interesting implications for the solar cycle, suggesting that very complex magnetic groups - and the heightened activity that accompanies them - are more likely to appear in the second part of the overall solar maximum.
Sunspot locations
-----------------
The locations of sunspots were also recorded by the STARA code and this allows us to produce a butterfly diagram of sunspot locations. The ‘butterfly’ shape is produced by the pattern of sunspot emergences seen in each cycle. At the start of a cycle sunspots tend to appear at high latitudes, between 20 and 40 degrees above and below the solar equator. But as the cycle progresses, the spot emergences are observed closer to the equator. The cycle then ends before the sunspots are seen to emerge at the equator and as a result of this it is very rare to see a sunspot forming within a few degrees of the solar equator. @Zharkov2007 have observed a ‘standard’ butterfly pattern in sunspot emergences in cycle 23 and our results are shown in Fig. \[fig:butterfly\]
The butterfly shape can be clearly seen as can some other features. There are gaps in 1998 as the SOHO spacecraft was lost for some time and no data were recorded. Also, the vertical line in early 1999 corresponds to the failure of the final gyroscope onboard and a rescue using gyroless control software. This caused the spacecraft to roll and so all data recorded at this time does not have a consistent sun orientation. These artifacts have been left in the figure (although corrected for in our subsequent analysis) to illustrate some of the potential problems with using long term data sets.
To enable the continuation of the mission the spacecraft is rotated approximately every three months to allow the high gain antenna to point at the Earth as it can no longer be moved. This means that the data are rotated and this introduces further small errors in position detection as the roll angle is not known exactly but the algorithm assumes that the data is either ’north up’ or ’south up’.
We can see from Fig. \[fig:butterfly\] that the end of solar cycle 23 exhibited asymmetric behaviour with very few spots appearing on the north hemisphere compared to the south. @Hathaway2010 shows that a north-south asymmetry in sunspot area during a cycle is very common but he also states that any systematic trend in the asymmetry during a solar cycle is found to change in the next cycle and so is not particularly useful for predictions of activity or for solar dynamo modelling. This asymmetry was studied in more detail by @Carbonell1993 using a variety of statistical methods and they found that a random component was dominant in determining the trend of hemispheric asymmetry in sunspots.
Sunspot areas
-------------
As was the case with the number of sunspots detected, the area of the largest visible sunspot also follows the activity of the solar cycle with a clear rising phase and a slower declining phase. When calculating the area of a sunspot or umbra the number of pixels within the spot or umbral boundary is corrected to take into account the geometrical foreshortening effects that change the observed area relative to its position on the solar disk. We show this in Fig. \[fig:areas\]. The variation is larger as sunspot sizes have a larger range than the number of spots that are present. Again, this has been smoothed to give a fair comparison to the international sunspot number calculated by the SIDC. An interesting feature of this plot is that at the start of cycle 24 there is no significant increase in the areas of observed spots so we can say that there are more spots beginning to appear but the spot magnetic fields are still weak.
[ l ]{}\
![We show the total sunspot (solid line) and umbra (crosses) area here as a percentage of the area of the projected solar disk. The data are smoothed over a three month period.[]{data-label="fig:deproj_areas"}](total_deproj_area_w_error.eps){width="45.00000%"}
In addition to looking at the largest sunspot areas observed, we are also able to examine the total area of the solar surface covered by sunspots at any one time. This is shown in Fig. \[fig:ratio\_areas\]. Both the total sunspot and umbral areas are shown and, yet again, they both follow the overall trend of the solar cycle with increases and decreases at the same times. More interesting than this however, is the ratio of umbral area to sunspot area, shown in the bottom panel. We observe that the umbral area is 20-40% of the total observed sunspot area and the ratio stays within this range throughout the cycle. Even though a large variety of sunspot shapes and configurations are seen, the fractional area of associated umbra does not show high amplitude fluctuations unlike the maximum sunspot area observed - the dominant characteristic is a relatively smooth variation. Note that this does not hold for individual sunspots due to the variety of configurations seen, only to the large scale distribution of sunspots over time. There are also interesting features present, most of all the dip in the year 1999. At this time, the sunspot area is increasing more quickly than the area of the associated umbrae. This soon changes and the umbral areas start to occupy more of the sunspot again, rising by a few percent by 2004 before starting to drop off again. During the first peak in solar activity in 2000 we see that the umbra is occupying a lower fraction of the sunspot and from Fig. \[fig:emergences\] this is when the International Sunspot Number was higher than the STARA sunspot count. This could indicate that there are sunspot groups with lower than ten sunspots present in them. This suggests that there is more space in these groups for the sunspot penumbrae to grow. In comparison to this, in the second peak of activity in 2002 we see that the fraction of sunspot area occupied by umbrae has grown and that the STARA count rate is above the International Sunspot Number. This suggests that we are seeing sunspot groups with more than ten spots in them. These would be very complex groups and so it may be the case that the sunspots have multiple umbrae present within them which would likely increase the fractional umbral area.
In Fig. \[fig:areas\] and Fig. \[fig:ratio\_areas\] we show the error in the areas measured as a shaded band surrounding the line representing the data points. Estimating the errors involved is done by examining the output of the STARA algorithm. When detecting sunspots and sunspot umbrae, the centroid of the region is determined with good accuracy. However, when defining the perimeter of the region, we believe that there is an error of 1 pixel both towards and away from the centre of the region. This means that large sunspots will have a smaller fractional error than small spots, even though the absolute value of the error will be greater for large spots.
We also show the percentage of the projected solar disk covered by sunspots from the viewpoint of the SOHO spacecraft in Fig. \[fig:deproj\_areas\]. The trend is very similar to that of the absolute total area of sunspots looked at previously. We see the fraction of the solar disk covered by sunspots rise to about 0.35% at the peak of activity in cycle 23 which is equivalent to 3500 MSH (millionths of a solar hemisphere). This is comparable to some of the largest sunspots ever detected. There are significant short-term fluctuations in this series, in addition to the overall solar cycle variation.
The evolution of sunspot magnetic fields
========================================
As the detection algorithm is directly linked with the MDI magnetograms recorded at the same time, we are also able to track the evolution of the magnetic field present in sunspots throughout the cycle. We assume that the magnetic field within sunspot umbrae is in the local vertical direction. As the MDI data only gives the line of sight magnetic field we apply a cosine correction to account for this. The amplification of magnetic field strength due to the cosine correction becomes very large as sunspots approach the limb, so making an incorrect assumption about the field being vertical can lead to vastly wrong B values at the limb. To minimise these effects we only include sunspots with a value of $\mu > 0.95$ where $\mu$ is the cosine of the angle between the local solar vertical and the observers line of sight. In addition to this, the observed line of sight field is corrected with the assumption that the true field direction is perpendicular to the local photosphere. As we are looking at the strongest fields in sunspot umbrae this is a reasonable approximation.
Fig. \[fig:rawspots\] shows the maximum sunspot umbral fields measured daily from 1996 - 2010. The first thing to notice is the spread of magnetic fields measured. We also see that the majority of measurements fall between 1500 and 3500 Gauss. It is very difficult to see any kind of trend in the data due to the spread of values but we can observe a lack of strong sunspots from 2008 - 2010 when the most recent solar minimum occured.
A similar study has been undertaken by @Penn2006 using the McMath-Pierce telescope on Kitt Peak which includes umbra measurements going further back, to 1991. The method is different as they use the Zeeman splitting of the Fe I line (1564.8nm) to infer a magnetic field strength at the location of the measurement. Measurements are made in the darkest part of the umbra, where this is identified in the image using a brightness meter. The Zeeman splitting identified at that location is used to determine the true magnetic field as the splitting of the spectral line observed is not dependent on the angle between the magnetic field and the observers line of sight. Very small spots were excluded from their dataset, as the small size of the umbra increases the risk of scattering of penumbral radiation into the umbral area, and consequent distortion of the line profile. Pore fields correspond to the range 1600-2600 G, with a mean of 2100 G. When this dataset of maximum measured umbral field is binned and averaged by year, and plotted as a function of time, a decrease is visible which can be fitted with a linear trend equivalent to around -52 Gauss per year. We repeat the analysis carried out by [@Penn2006] on our dataset, both including and excluding all spots with a vertical magnetic field component below 1500 G to minimise the possible effects of pores being included in the analysis, for a direct comparison with the [@Penn2006] result. The results are shown in Fig. \[fig:spottrend\].
![Maximum sunspot umbra field from 1996-2010. Measurements are taken daily.[]{data-label="fig:rawspots"}](raw_spot_data.eps){width="45.00000%"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![The data shown in Fig. \[fig:rawspots\] have been binned by year and the mean of each bin is plotted here. Top panel: all data from Fig. \[fig:rawspots\] are included. Bottom panel: only measurements with a field above 1500 Gauss are included. The error bars correspond to the standard error on the mean. The solid line shows the evolution of the international sunspot number over the same period for reference. Assuming a linear trend gives a gradient of -23.6 $\pm $ 3.9 Gauss per year and -22.3 $\pm$ 3.9 Gauss per year respectively.[]{data-label="fig:spottrend"}](mean_stdev_all_spots.eps "fig:"){width="45.00000%"}
![The data shown in Fig. \[fig:rawspots\] have been binned by year and the mean of each bin is plotted here. Top panel: all data from Fig. \[fig:rawspots\] are included. Bottom panel: only measurements with a field above 1500 Gauss are included. The error bars correspond to the standard error on the mean. The solid line shows the evolution of the international sunspot number over the same period for reference. Assuming a linear trend gives a gradient of -23.6 $\pm $ 3.9 Gauss per year and -22.3 $\pm$ 3.9 Gauss per year respectively.[]{data-label="fig:spottrend"}](mean_stdev_penn_spots.eps "fig:"){width="45.00000%"}
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The top panel includes all sunspots detected whereas the bottom panel excludes any spots with a maximum field strength of less than 1500 Gauss. The error bars are calculated as the standard error on the mean of all measurements in the bin.
The data are in line with a picture in which the umbral fields are simply following a cyclical variation pattern, as the increases and decreases follow the international sunspot number. This cannot be confirmed with the current data and we will need to wait until the next cycle is well under way to see if the trends continues to be present. If we do a straight line fit as in @Penn2006, then the gradient of the best fitting line gives a decrease in umbral fields of 23.6 $\pm$ 3.9 Gauss per year which, although still decreasing, is a far slower decline than seen by Penn and Livingston. Repeating the analysis excluding sunspots with fields below 1500 Gauss gives a long term decrease in field strength of 22.4 $\pm$ 3.9 Gauss per year. This is even further from the result they observed, although as the sunspots with fields below 1500 Gauss make up such a small fraction of the population we observe, we would not expect a significant change in the result. Other studies have also cast doubt on the long term decrease of umbral magnetic fields. The @Penn2006 article suggests that a decrease of 600 Gauss over a solar cycle would cause a change in mean umbral radius as a relationship between these two quantities has been shown by @Kopp1992 and @Schad2010 but follow up observations by @Penn2007 could not see this in their data. It has also been suggested by @Mathew2007 that a small sunspot sample may introduce a bias into results if the size distribution of sunspots used is not calculated in advance.
However, the long term decline in sunspot magnetic fields does agree with the lack of an increase in sunspot area as shown in Fig. \[fig:areas\]. If the magnetic field is now weaker than at the same time in the last cycle we would expect sunspots to be smaller and this is currently what is observed.
Interestingly, if the data from only the declining phase of the cycle (from 2000 to 2010) are used, then the maximum umbral field strengths are seen to decrease by around 70 Gauss per year which is far greater than the @Penn2006 study.
This then leads to the question of how valid this comparison is. In fact, instruments such as MDI and the new Helioseismic and Magnetic Imager on SDO do not measure the true value of magnetic field strength in a pixel. The value they return is an average magnetic field strength with a resolution determined by pixel size. However, if the filling factor of spatially unresolved magnetic elements within the pixel is close to unity, then the pixel value is a good approximation for the true line of sight magnetic field strength. This is thought to be the case deep in the umbrae of strong sunspots and so for these measurements we can say that our observations are good approximations for the true line of sight magnetic fields. In addition to this, we have only used sunspots with $\mu > 0.95$ which corresponds to 18.2 degrees from solar disk centre in an effort to minimise any corrections to the magnetic field measurements but still assume that the field in the core of sunspot umbrae is perpendicular to the local photosphere.
Also, MDI has problems with saturation in magnetic field measurements with a peak value of between 3000 and 3500 Gauss depending on when the observation was made (the saturation value has lowered as the instrument degrades). This has a greater effect on measurements made at solar maximum and so has the effect of reducing the long term field strength decrease. However, this does not fully account for the discrepancy between our value of the rate of long term field decrease and that of other studies.
We have not only compared the trends seen but also the data points used in calculating these trends. The latest Livingston and Penn data is kept up to date by Leif Svalgaard and can be viewed at his own website (see [www.leif.org/research](www.leif.org/research)). With the exception of a single data point in 1994, the Livingston and Penn yearly averages are similar to ours. Sadly, there are no yearly averages in the Livingston and Penn data between 1994 and 2001 to better compare the two studies.
Discussion and Conclusions
==========================
Using a catalogue of sunspot detections created by the STARA code provides a reliable way to analyse the long term variation of certain physical parameters relating to sunspots. We found that the number of sunspots detected compared very well with the international sunspot number, even through the period of 2008-2010 when sunspot detections have been more sparse and difficult due to the decreased magnetic field strengths that are causing them. When looking at the locations of sunspots a traditional butterfly pattern is seen which also shows the end of cycle 22 as well as the period of almost no sunspots from late 2008 to early 2010 before cycle 24 started. Fig. \[fig:butterfly\] also shows some of the problems of a long term observing run, such as spikes in early 1999 caused by failure of the gyroscopes onboard SOHO. In addition to this, the high gain antenna on SOHO malfunctioned in mid 2003.
The area of sunspots was then examined with the maximum spot area being first observed. The rough pattern of an initial steep rise and gradual fall associated with a solar cycle was seen but with many other features present. However, when the total observable sunspot area was plotted, a much smoother evolution was seen. The same smooth evolution was also present in the total observable umbral area. We also found that throughout the whole of solar cycle 23, if smoothed over a three month period, the area of umbra visible was between 20 and 40% of the visible sunspot area once corrections for geometric foreshortening had been applied.
We then continued to show the evolution of magnetic fields in sunspot umbrae and Fig. \[fig:rawspots\] shows the large spread of sunspot magnetic fields observed. Once the spot magnetic field data had been binned by year, a long term cyclical trend could be observed but it is yet unknown whether this is a cyclical variation around a long term linear decrease as suggested by other studies. Our data supports stronger fields near solar maximum and weaker fields at solar minimum. When compared with other similar studies, the rate of magnetic field decrease is very different and is likely due to the wide range of sunspot fields. The next solar cycle should bring a more definitive answer to the question of whether a secular trend in sunspot fields exists over multiple solar cycles. We will continue to track this for as long as SOHO still flies and also plan to incorporate data from the new Helioseismic and Magnetic Imager on the Solar Dynamics Observatory spacecraft which serves as the successor to SOHO.
F.T.W. acknowledges the support of an STFC Ph. D. studentship. This work was supported by the European Commission through the SOLAIRE Network (MRTN-CT-2006-035484) and by STFC rolling grant STFC/F002941/1. SOHO is a project of international cooperation between ESA and NASA. We acknowledge useful discussions with M. Hendry and would like to thank our anonymous referee for their thought provoking comments. Thanks also to Leif Svalgaard for allowing us to use his plots for comparing with Livingston and Penn data.
|
---
abstract: 'The viscous Gilbert damping parameter governing magnetization dynamics is of primary importance for various spintronics applications. Although, the damping constant is believed to be anisotropic by theories. It is commonly treated as a scalar due to lack of experimental evidence. Here, we present an elaborate angle dependent broadband ferromagnetic resonance study of high quality epitaxial La$_{0.7}$Sr$_{0.3}$MnO$_{3}$ films. Extrinsic effects are suppressed and we show convincing evidence of anisotropic damping with twofold symmetry at room temperature. The observed anisotropic relaxation is attributed to the magnetization orientation dependence of the band structure. In addition, we demonstrated that such anisotropy can be tailored by manipulating the stain. This work provides new insights to understand the mechanism of magnetization relaxation.'
author:
- Qing Qin
- 'Shikun He[\*]{}'
- Haijun Wu
- Ping Yang
- Liang Liu
- Wendong Song
- Stephen John Pennycook
- 'Jingsheng Chen[\*]{}'
title: 'Anisotropic Gilbert damping in perovskite La$_{0.7}$Sr$_{0.3}$MnO$_{3}$ thin film '
---
introduction
============
The magnetization relaxation process determines the speed of magnetization relaxation and the energy required for current-induced magnetization reversal [@Slonczewski_STT_1996; @Katine_STT_GMR; @Ikeda_2010_Namat; @Ralph2008; @Brataas_ST_2012; @FukamiShapeMTJ]. Understanding the mechanism and controlling of magnetization relaxation [@Brataas2008a; @spinPumping_PRL; @Schoen2016; @heinrich_2011_YIG_4_cavity; @Okada2017; @damping_breathMode_PRL], including intrinsic Gilbert damping and extrinsic effects, pave the way for ultra-low power and high performance spintronic devices based on spin transfer and spin orbit torques [@STT_block_2014; @Urazhdin2014; @SOT_YIG2017]. It has been demonstrated that Gilbert damping constant ($\alpha$) can be tuned effectively by engineering the density of states and spin orbit coupling (SOC) [@Schoen2016; @Kubota2009; @PhysRevLett.110.077203; @2017LowDamping]. In addition, magnetization relaxations subjected to finite size and interfacial effects have also been extensively investigated [@spinPumping_PRL; @Moriyama2008; @ModeSizeDependent]. However, it is still an open question that if magnetic damping is anisotropic. In principle, $\alpha$ is magnetization orientation dependent and should be a 3$\times$3 tensor in the phenomenological Gilbert equation [@Gilbert1955; @Steiauf2005], yet it is often treated as a scalar (isotropic). In the case of polycrystalline thin films prepared by sputtering, such treatment is reasonable due to the smearing of long range structural order. Whereas for single crystal thin films, it is still difficult to draw a conclusion due to the lack of convincing experimental evidence. From the view of theories, the Gilbert damping is determined by two scattering processes, the interband resistivity-like scattering and the intraband conductivity-like scattering [@damping_breathMode_PRL]. Both terms vary with temperature through their dependence on electron relaxation time. The interband scattering which dominates damping in most ferromagnets becomes isotropic at room temperature [@Gilmore2010]. Therefore, anisotropic linewidth in 3d magnetic metals was only observed at low temperature[@Rudd1985]. From the aspect of experimental technique, Seib et al. have predicted that the precession trajectory of magnetization in a ferromagnetic resonance (FMR) measurement (standard technique for measuring damping) may partially average out the anisotropy [@Seib2009]. Hence, detecting the anisotropy in Gilbert damping is extremely difficult. Furthermore, the existence of several angle dependent extrinsic contributions to damping in most materials further hinders the determination of a possible weak anisotropic damping [@PhysRevB.69.184417; @Arias-PRB-1999; @rippleTMS; @Okada2017]. We note that in a ferromagnet with nearly half-metallic band structure, the isotropic interband term is suppressed [@Butler_lowDamping] and the damping can be dominated by the anisotropic intraband contribution[@Gilmore2010]. Recent reports have claimed the observation of anisotropic damping in half-metallic Heusler alloy[@Kasatani2014; @Yilgin2007]. However, unavoidable chemical disorder [@Wen2014; @PhysRevB.96.224425]of Heusler alloy introduces extrinsic effects such as spin wave scattering hence complicates the verification procedure of such anisotropy.
La$_{0.7}$Sr$_{0.3}$MnO$_{3}$ (LSMO) is an oxide perovskite material exhibited half-metallic band structure and ultra-low damping at room temperature [@LSMO_halfMetal_nature; @Qin2017]. In this work, we studied the magnetization relaxation of LSMO films deposited on $\textrm{NdGaO}{}_{3}$ (NGO) (110) substrates using angle-resolved broadband ferromagnetic resonance. The purpose of choosing NGO (110) substrates is to utilize its non-equal $a$ and $b$ axis value. Such asymmetry will potentially lead to non-spherical Fermi surface. Two types of high quality samples with different static magnetic anisotropies were investigated. The normal LSMO film (hereafter denoted as S-LSMO) exhibited weak uniaxial magnetic anisotropy whereas the other with modulated strain relaxation mode (hereafter denoted as W-LSMO) have both uniaxial and cubic anisotropy fields. The angle dependence of the in-plane intrinsic Gilbert damping showed two-fold symmetry in both type of samples. Strikingly, the orientation of minimum damping differs 90 degree. This work provided strong evidence of anisotropic nature of magnetization relaxation and demonstrated the tuning of anisotropy in damping through stress relaxation engineering.
Results
=======
Epitaxial growth of LSMO {#epitaxial-growth-of-lsmo .unnumbered}
------------------------
Pulsed laser deposition (PLD) was used to deposit LSMO thin films with a thickness of 25$\,$nm on (110) NGO substrates. The energy and repetition frequency of KrF laser (248$\,$nm) were 225$\,$mJ and 2$\,$Hz, respectively. During deposition, the substrate temperature was fixed at 950$^{\circ}C$. The oxygen pressure was 225$\,$mTorr for S-LSMO and 200$\,$mTorr for W-LSMOAfter deposition, S-LSMO was cooled down to room temperature at 10K/min under the oxygen pressure of 1 Torr, whereas W-LSMO at 5$\,$K/min under the oxygen pressure of 100 Torr in order to promote the modification of strain hence micro-structurestructure.
Crystalline quality analysis {#crystalline-quality-analysis .unnumbered}
----------------------------
The crystallographic structures of the films were characterized by synchrotron high resolution X-ray diffraction. Reciprocal space maps (RSMs) taken at room temperature around {013}$_{\textrm{pc}}$ (here the subscript pc stands for pseudocubic) reflections confirm the epitaxial growth of LSMO layers on the NGO substrate as shown in \[fig:XRD\_Main\] (a). The vertical alignment of LSMO and NGO reciprocal lattice point clearly shows that the LSMO film is completely strained on the NGO substrate. Lattice mismatch along [\[]{}100[\]]{}$_{\textrm{pc}}$ and [\[]{}010[\]]{}$_{\textrm{pc}}$ are 1.03% and 0.8%, respectively. Considering the position of the LSMO reciprocal lattice point in the {013}$_{\textrm{pc}}$ mappings, equal $L$ values of (103)$_{\textrm{pc}}$ and (-103)$_{\textrm{pc}}$ indicates the perpendicular relation between vector $a$ and $c$ in the lattice, whereas different L values for (013)$_{\textrm{pc}}$ and (0-13)$_{\textrm{pc}}$ shows that the angle between $b$ and $c$ is not equal to 90°. Thus, the LSMO is monoclinic phase which is consistent with previous reports [@Vailionis2011]. The good crystalline quality was further verified by aberration-corrected scanning transmission electron microscopy (AC-STEM). \[fig:XRD\_Main\] (b, c) are the simultaneously acquired high angle annular dark field (HAADF) and annular bright field (ABF) images of S-LSMO along [\[]{}100[\]]{}$\textrm{pc}$ direction, while \[fig:XRD\_Main\] (d, e) are for [\[]{}010[\]]{}$_{\textrm{pc}}$ direction. The measurement directions can be differentiated from the diffraction of NGO substrate: 1/2[\[]{}010[\]]{} superlattices for [\[]{}100[\]]{}$_{\textrm{pc}}$ direction (inset of \[fig:XRD\_Main\](c)) and 1/2[\[]{}101[\]]{} superlattices for [\[]{}100[\]]{}$_{\textrm{pc}}$ direction (inset of \[fig:XRD\_Main\](e)). High quality single crystalline films are essential for the present purposes because high density of defects will result in spin wave scattering [@PhysRevB.69.184417].
Magnetic anisotropy fields {#magnetic-anisotropy-fields .unnumbered}
--------------------------
The magnetic dynamic properties were investigated by a home-built angle-resolved broadband FMR with magnetic field up to 1.5$\,$T. All measurements were performed at room temperature. Shown in \[fig:Main\_anisotropy\](a) is the color-coded plot of the transmission coefficient S21 of the S-LSMO sample measured at 10$\,$GHz. $\varphi_{H}$ is the in-plane azimuth angle of the external magnetic field counted from [\[]{}010[\]]{}$_{\textrm{pc}}$ direction (\[fig:Main\_anisotropy\](b)). This relative orientation was controlled by a sample mounting manipulator with a precision of less than 0.1$^{\circ}$. The olive shape of the color region indicates the existence of anisotropy field, whereas the very narrow field region of resonances is an evidence of low damping. Three line cuts at $\varphi_{H}$=0, 45 and 90 degrees are plotted in \[fig:Main\_anisotropy\](c), showing the variation of both FMR resonance field ($\mathit{H}_{\textrm{res}}$) and line shape with $\varphi_{H}$. All curves are well fitted hence both the $H{}_{\textrm{res}}$ and resonance linewidth $\Delta$H are determined. The $\varphi_{H}$ dependence of H$_{\textrm{res}}$ at two selected frequencies (20 and 40 GHz) are shown in \[fig:Main\_anisotropy\](d) for S-LSMO. The angle dependencies of the resonance field $H{}_{\textrm{res}}(\varphi_{H})$ is calculated starting from the total energy [@Farle_review_1998]: $$\begin{array}{l}
E=-MH\left[\cos\theta_{H}\cos\theta_{M}{\rm +}\sin\theta_{H}\sin\theta_{M}\cos(\varphi_{M}-\varphi_{H})\right]+2\pi M^{2}\cos^{2}\theta_{M}-\frac{1}{2}MH_{2\bot}\cos^{2}\theta_{M}\\
-\frac{1}{4}MH_{4\bot}\cos^{4}\theta_{M}-\frac{1}{2}MH_{2\parallel}\sin^{2}\theta_{M}\cos^{2}(\varphi_{M}-\phi_{2IP})-\frac{1}{4}MH_{4\parallel}\frac{3+\cos4(\varphi_{M}-\phi_{4IP})}{4}\sin^{4}\theta_{M}
\end{array}\label{eq:Total_Energy}$$
where $\theta_{M}$ and $\varphi_{M}$ are the polar angle and the azimuth angle of the magnetization ($M$), $H_{2\bot}$, $H_{4\bot}$, $H_{2\parallel}$, $H_{4\parallel}$ are the uniaxial and cubic out-fo-plane and in-plane anisotropy fields. The easy axes of in-plane anisotropies are along $\phi_{2\textrm{IP}}$ and $\phi_{4\textrm{IP}}$, respectively. According to Smit-Beljers equation the resonance condition for $\theta_{M}=\mbox{\ensuremath{\pi}/2}$ is [@Smit1955]: $$2\pi f{\rm =}\frac{\gamma}{M\sin\theta}\sqrt{E_{\theta\theta}E_{\varphi\varphi}}\label{eq:FMR_peak}$$ Here, $E_{\theta\theta}=H_{{\rm res}}\cos(\varphi_{M}-\varphi_{H})+4\pi M_{{\rm eff}}-H_{2\parallel}\cos^{2}(\varphi_{M}-\phi_{2\textrm{IP}})+H_{4\parallel}(3+\cos4(\varphi_{M}-\phi_{4\textrm{IP}})/4)$ and $E_{\varphi\varphi}=H_{{\rm res}}\cos(\varphi_{M}-\varphi_{H})+H_{2\parallel}\cos2(\varphi_{M}-\phi_{2\textrm{IP}})+H_{4\parallel}\cos4(\varphi_{M}-\phi_{4\textrm{IP}})$ are second partial derivatives of the total energy with respect to the polar and azimuth angles. $\gamma$=1.76$\times$10$^{7}$s$^{-1}$G$^{-1}$ denotes the gyromagnetic ratio, $4\pi M_{{\rm eff}}=4\pi M-H_{2\bot}$ is the effective magnetization. The resonance field of S-LSMO shows pronounced minimum at $\varphi_{H}=n\cdot\pi$, indicating the existence of uniaxial magnetic anisotropy with easy axis along $\phi_{2\textrm{IP}}=0$ or [\[]{}010[\]]{}$_{\textrm{pc}}$ direction. Cubic anisotropy is negligible hence $H_{4\parallel}$=0. Such uniaxial anisotropy observed in S-LSMO is consistent with previous reports [@Boschker2009a], which is attributed to anisotropic strain produced by the NGO(110) substrate [@Suzuki1998; @BOSCHKER20112632; @Tsui2000]. Compared to the resonance fields in our measurement, the magnetic anisotropy fields are orders of magnitude smaller. Therefore, the calculated difference between $\varphi_{H}$ and $\varphi_{M}$ are always smaller than $1^{\circ}$ and $\varphi=\varphi_{H}=\varphi_{M}$ is assumed in the following discussion.
Magnetization orientation dependence of Gilbert damping {#magnetization-orientation-dependence-of-gilbert-damping .unnumbered}
-------------------------------------------------------
In order to study the symmetry of magnetization relaxation of the sample. The FMR linewidth $\Delta H$ for a matrix of parameter list (72 field orientations and 36 frequency values) are extracted. The results are shown by 3-D plots in \[fig:damping\](a) . Here, $z$ axis is $\Delta H$ and $x$, $y$ axes are $f\cdot\cos\varphi$ and $f\cdot\sin\varphi$, respectively. The figure clearly shows that the linewidth depends on magnetization orientation. At a given frequency, the linewidth is maximum (minimum) at $\varphi=0$ ($\varphi=\pi/2$) for S-LSMO. \[fig:damping\](c) shows the $\Delta H$ versus frequency for three field orientations. The FMR linewidth due to intrinsic magnetic damping scales linearly with frequency ${\rm \Delta}H_{GL}=4\pi\alpha f/\gamma{\rm cos}\left(\varphi_{M}-\varphi_{H}\right)$ according to Laudau-Lifshitz-Gilbert phenomenological theory [@NTU_FMR; @heinrich_inhomo_1991]. However, a weak non-linearity in the low frequency range can be identified. In general, extrinsic linewidth contributions such as inhomogeneity and magnon scattering will broaden the FMR spectrum hence result in additional linewidth contributions scales non-linearly with frequency [@Schoen2016; @Okada2017]. The interfacial magnon scattering is suppressed due to relative large film thickness (25$\ $nm) and the bulk magnon scattering contribution to the linewidth is negligible in our samples with very good atomic order. However, the static magnetic properties of the thin film may vary slightly in the millimeter scale. Since the FMR signal is an averaged response detected by the coplanar waveguide (5$\,$mm long), a superposition of location resonance modes broadens the FMR spectrum. Such well-known contribution to linewidth, defined as ${\rm \Delta}H_{{\rm inhom}}$, are generally treated as a constant [@Schoen2016; @heinrich_inhomo_1991; @Shaw2011]. However, it is frequency dependent for in-plane configuration and need to be treated carefully for samples with ultra-low damping. Here, we fit the data with ${\rm \Delta}H={\rm \Delta}H_{{\rm GL}}+{\rm \Delta}H_{{\rm inhom}}$, taking into account the frequency and orientation dependence of ${\rm \Delta}H_{{\rm inhom}}$. As can be seen from \[fig:damping\](c), the data are well reproduced for every field orientations. Hence, the magnetization orientation dependence of intrinsic damping constant is determined and plotted in \[fig:damping\](e). Remarkably, the damping constant shows two-fold symmetry. The lowest damping of S-LSMO with in-plane magnetization, observed at $\varphi=0$ and $\varphi=\pi$, is $\left(8.4\pm0.3\right)\times10^{-4}$ and comparable to the value measured under a perpendicular field (**\[tab:SummaryValues\]**). The maximum damping at $\varphi=\pi/2$ and $\varphi=3\pi/2$ is about 25% higher.
Since the magnetization damping and resonance field of the S-LSMO sample exhibited identical symmetry (\[fig:Main\_anisotropy\] (d) and \[fig:Main\_anisotropy\](e)), it seems that the observed anisotropic damping is directly related to crystalline anisotropy. Therefore, we prepared the W-LSMO sample with slightly different structure and hence modified static magnetic anisotropy properties. The W-LSMO sample exhibited 1D long range atomic wave-like modulation [@Vailionis2011] (twining domain motif) along [\[]{}100[\]]{}$_{\textrm{pc}}$ axis near the interface between substrate and film. Due to different strain relaxation mechanism as compared to S-LSMO, the $\varphi_{H}$ dependence of $H_{\textrm{res}}$ for the W-LSMO have additional features and can only be reproduced by including both $H_{2\parallel}(13.9\pm0.9$ Oe) and $H_{4\parallel}(11.8\pm1.2$ Oe) terms. The easy axis of the uniaxial anisotropy ($\phi{}_{2\textrm{IP}}$=0 ) is the same as S-LSMO whereas the additional cubic anisotropy is minimum at $\phi{}_{4\textrm{IP}}$=45°. The magnetization orientation dependence of the FMR linewidth for W-LSMO is significantly different (\[fig:damping\](b)) as compared to S-LSMO. Such change in trend can be clearly identified from the frequency dependence of linewidth for selected magnetization orientations shown in \[fig:damping\](d). Magnetization damping values are extracted using the same procedure as S-LSMO because the spin wave contribution is excluded. The damping constant again showed two-fold in-plane symmetry. However, in contrast to S-LSMO, the maximum damping value of W-LSMO is observed at $\varphi=0$ and $\varphi=\pi$.
Discussion
==========
Anisotropy in linewidth at low temperatures have been reported decades ago, however, data in most early publications were taken at a fixed frequency in a cavity-based FMR [@Rudd1985; @Vittoria1967]. Due to lack of frequency dependence information, it is not clear if the anisotropy in linewidth is due to intrinsic damping or extrinsic effects [@Dubowik2011; @PhysRevB.84.054461; @PhysRevB.58.5611]. In this study, besides wide range of frequencies, we also adopted samples with effective anisotropy orders of magnitude smaller than the external field. Therefore, the field dragging effect and mosaicity broadening, both of which are anisotropic in natur e[@Mosacity2007], are negligibly small and the Gilbert damping constant is determined reliably. Furthermore, the mechanism in this simple system is different from previous reports related to interfacial exchange coupling and spin pumping[@LeGraet2010; @BakerAnisotropy2016]. Since both S-LSMO and W-LSMO exhibited in-plane uniaxial magnetic anisotropy, the opposite trends observed in these two samples exclude the existence of a direct link between anisotropic damping and effective field. Both magnetic anisotropy and damping are related to the band structure but in quite different ways. According to perturbation theory, the magnetic anisotropy energy is determined by the matrix elements of the spin-orbit interaction between occupied states. Hence, the contributions from all the filled bands must be considered to calculate the absolute value of magnetic anisotropy. On the other hand, the magnetic damping is related to the density of states at the Fermi level.
$4\pi M_{\textrm{eff}}$ (T) $H_{2\parallel}$(Oe) $H_{4\parallel}$(Oe) $\alpha_{\perp}$ $\alpha$($\varphi=0$) $\alpha(\varphi=\pi/2)$
-------- ----------------------------- ---------------------- ---------------------- ---------------------------- ---------------------------- ----------------------------
S-LSMO 0.3280$\pm$0.0011 37$\pm4$ 0 $(8.6\pm0.5)\times10^{-4}$ $(8.4\pm0.3)\times10^{-4}$ $(11\pm0.6)\times10^{-4}$
W-LSMO 0.3620$\pm$0.0025 13.9$\pm0.9$ 11.8$\pm1.2$ $(4.7\pm0.7)\times10^{-4}$ $(6.5\pm0.3)\times10^{-4}$ $(5.3\pm0.3)\times10^{-4}$
: Summary of the parameters for S-LSMO and W-LSMO samples. \[tab:SummaryValues\]
The damping term in the Landau-Lifshitz-Gilbert equation of motion is $\frac{\alpha}{{\rm |}M{\rm |}}\left(M\times\frac{dM}{dt}\right)$, therefore, anisotropy in damping can have two origins, one related to the equilibrium orientation of magnetization $M$ (orientation anisotropy) and the other depends on the instantaneous change in magnetization $dM/dt$ (rotational anisotropy). In FMR experiments the magnetization vector rotates around its equilibrium position, therefore, the rotational anisotropy may be smeared out [@Seib2009]. The orientation anisotropy is described by both interband and intraband scattering process. According to Gilmore et al.[@Gilmore2010], the latter is isotropic at sufficiently high scattering rates at room temperature. We suspect that the anisotropic damping in LSMO is due to its half-metallic band structure. As a result of high spin polarization, interband scattering is suppressed and the room temperature damping is dominated by intraband scattering. The intraband contribution to damping exhibit anisotropy for all scattering rates [@Gilmore2010] which agree well with our experiments. The suppression of interband scattering is evidenced by the ultra-low damping in the order of $10^{-4}$. Notably, the absolute value of the observed anisotropy, 2.6$\times$10$^{-4}$ for S-LSMO and 1.2$\times$10$^{-4}$ for W-LSMO, is so small that could not be identified reliably for a material with typical damping values between 5$\times$10$^{-3}$ to 2$\times$10$^{-2}$.
In a microscopic picture, the Gilbert damping is proportional to the square of SOC constant ($\xi$) and density of states at the Fermi level, $\mbox{\ensuremath{\alpha\sim\xi^{2}D(E_{F})}}$. The shape of the Fermi surface depends on the orientation of the magnetization due to SOC. Hence, the anisotropy can be attributed to the angle dependence of $D(E_{F})$ which is in turn induced by the substrate. The trend reversal in the damping anisotropy of the two LSMO samples can be explained by the modification of the Fermi surface and thus $D(E_{F})$ by strain relaxation. During the preparation of this paper, we noticed a similar work in ultra-thin Fe layers deposited on GaAs substrate[@NatPhy2018]. There, the anisotropy is attributed to interfacial SOC. This work suggests that anisotropic damping can exist in bulk samples.
The research is supported by the Singapore National Research Foundation under CRP Award No. NRF-CRP10-2012-02. P. Yang is supported from SSLS via NUS Core Support C-380-003-003-001. S.J.P is grateful to the National University of Singapore for funding.
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![**Structure characterization of S-LSMO sample.** (a) and (b) XRD profiles around S-LSMO (00L) reflections (L=1,2,3,4) with the incident beam aligned along the [\[]{}100[\]]{}$_{\textrm{pc}}$ and [\[]{}010[\]]{}$_{\textrm{pc}}$ , respectively. (b) and (c) STEM-HAADF/ABF lattice images of S-LSMO along [\[]{}100[\]]{}$_{\textrm{pc}}$ direction. (d) and (e) STEM-HAADF/ABF images of S-LSMO along [\[]{}010[\]]{}$_{\textrm{pc}}$ direction. the insets are the intensity profile and FFT image; The red dashed line indicates the interface.\[fig:XRD\_Main\]](Main_Fig1){width="14cm"}
![**Magnetic anisotropy characterization.** (a) The 2D polar color plot of the FMR spectra of S-LSMO. The frequency is 10GHz. (b) Schematics of the FMR setup and the definition field orientation. (c) FMR spectra for $\varphi_{H}$=0, 45 and 90 degrees for S-LSMO. (d) Field orientation ($\varphi_{H}$) dependence of the resonance fields ($H_{\textrm{res}}$) of the S-LSMO sample at f=20 and 40GHz. The solid lines in (c) and (d) are calculated values. \[fig:Main\_anisotropy\]](Main_Fig2C){width="14cm"}
![**Anisotropic linewidth and damping:** (a)-(b) 3-D plot of frequency and in-plane field orientation dependence of FMR linewidth. (c)-(d) frequency dependence of FMR linewidth for seleted field orientations. Solid symbols are experimental data and the lines are calculated value. (e)-(f) Damping constant as a function of $\varphi$. (a),(c), (e) are for S-LSMO and (b),(d), (f) are for W-LSMO.\[fig:damping\]](Main_Fig3){width="14cm"}
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abstract: 'In the present work we systematically study $\mathcal{\alpha}$ decay half-lives of $Z>51$ nuclei using the modified Gamow-like model which includes the effects of the centrifugal potential and electrostatic shielding. For the case of even-even nuclei, this model contains two adjustable parameters: the parameter $a$ related to the screened electrostatic barrier and the radius constant $r_0$, while for the case of odd-odd and odd-A nuclei, it is added a new parameter i.e. hindrance factor $h$ which is used to describe the effect of an odd-proton and/or an odd-neutron. Our calculations can well reproduce the experimental data. In addition, we use this modified Gamow-like model to predict the $\mathcal{\alpha}$-decay half-lives of seven even-even nuclei with $Z=120$ and some un-synthesized nuclei on their $\mathcal{\alpha}$ decay chains.'
address:
- 'School of Nuclear Science and Technology, University of South China, 421001 Hengyang, People’s Republic of China'
- 'School of Math and Physics, University of South China, 421001 Hengyang, People’s Republic of China'
- 'Cooperative Innovation Center for Nuclear Fuel Cycle Technology $\&$ Equipment, University of South China, 421001 Hengyang, People’s Republic of China'
- 'Key Laboratory of Low Dimensional Quantum Structures and Quantum Control, Hunan Normal University, 410081 Changsha, People’s Republic of China'
- 'School of Science, Qingdao Technological University, 266000 Qingdao, People’s Republic of China'
author:
- 'Jun-Hao Cheng'
- 'Jiu-Long Chen'
- 'Jun-Gang Deng'
- 'Xi-Jun Wu'
- 'Xiao-Hua Li'
- 'Peng-Cheng Chu'
title: 'Systematic study of $\mathcal{\alpha}$ decay half-lives based on Gamow–like model with a screened electrostatic barrier'
---
[$\mathcal{\alpha}$ decay]{},Gamow-like model,electrostatic shielding ,un-synthesized nuclei
Introduction
============
$\mathcal{\alpha}$ decay, the spontaneous emission of a $^{4}$He by the nucleus and the formation of a new nuclides, was first defined by Rutherford in 1899. Since then, great efforts have been made in the realm of both theory and experiment, e.g., from the discovery of the atomic nucleus by $\mathcal{\alpha}$ scattering to the Geiger-Nuttall law describing a relationship between $\mathcal{\alpha}$ decay half-life and decay energy [@PhysRevC.85.044608; @Dong2005; @VIOLA1966741; @PhysRevLett.103.072501], from the barrier tunneling theory based on the quantum mechanics to the investigation of superheavy nuclei(SHN) [@Gamow1928; @0034-4885-78-3-036301; @RevModPhys.84.567; @PhysRevLett.110.242502; @SOBICZEWSKI2007292; @RevModPhys.70.77; @PhysRevLett.104.142502]. $\mathcal{\alpha}$ decay, as an important tool to investigate SHN, provides abundant information about the nuclear structure and stability of SHN[@0034-4885-78-3-036301; @Hofmann2016; @PhysRevC.85.044608; @Yang2015]. Nowadays, there are many theoretical models used to study $\mathcal{\alpha}$ decay including the cluster model [@PhysRevLett.65.2975; @PhysRevC.74.014304; @XU2005303], the unified model for $\mathcal{\alpha}$ decay and $\mathcal{\alpha}$ capture [@PhysRevC.73.031301; @PhysRevC.92.014602], the liquid drop model [@0305-4616-5-10-005; @PhysRevC.48.2409; @0954-3899-26-8-305; @PhysRevC.74.017304; @GUO2015110], the two-potential approach [@1674-1137-41-1-014102; @PhysRevC.94.024338; @PhysRevC.93.034316; @PhysRevC.95.014319; @PhysRevC.95.044303; @PhysRevC.96.024318; @PhysRevC.97.044322], the empirical formulas [@PhysRevC.85.044608; @0954-3899-39-1-015105; @PhysRevC.80.024310; @0954-3899-42-5-055112] and others [@PhysRevLett.59.262; @SANTHOSH201528; @PhysRevC.87.024308; @0954-3899-31-2-005; @PhysRevC.81.064318; @QI2014203].
Recently, K. Pomorski *et al.* proposed a Gamow-like model which is a simple phenomenological model based on the Gamow theory for the evaluations of half-life for $\mathcal{\alpha}$ decay [@PhysRevC.87.024308; @1402-4896-2013-T154-014029], while the nuclear potential is chosen as the square potential well, the centrifugal potential is ignored and the Coulomb potential is taken as the potential of a uniformly charged sphere with radius $R$ defined as Eq.(\[eq5\]). They also extended this model to study the proton radioactivity[@Zdeb2016], for the proton radioactivity shares the same mechanism as the $\mathcal{\alpha}$ decay. In 2016, Niu Wan *et al.* systematically calculated the screened $\mathcal{\alpha}$ decay half-lives of the $\mathcal{\alpha}$ emitters with proton number $Z=52-105$ by considering the electrons in different external environments such as neutral atoms, a metal, and so on. They found that the decay energy and the interaction potential between $\mathcal{\alpha}$ particle and daughter nucleus are both changed due to the electrostatic shielding effect. And the electrostatic shielding effect is found to be closely related to the decay energy and its proton number[@Wan2016]. In 2017, R. Budaca and A. I. Budaca proposed a simple analytical model based on the WKB approximation for the barrier penetration probability which includes the centrifugal and overlapping effects besides the electrostatic repulsion[@Budaca2017]. In their model, there is only one parameter $a$ which is used to describe the electrostatic shielding effect of Coulomb potential by using the Hulthen potential[@Budaca2017]. They systematically calculated the half-lives of proton emission for $Z\geq 51$ nuclei. The results can well reproduce the experimental data. Combining these points, in this work we modify the Gamow-like model proposed by K. Pomoski *et al.*, considering the shielding effect of the Coulomb potential and the influence of the centrifugal potential, to systematically study the $\mathcal{\alpha}$ decay half-lives. All the database are taken from the latest atomic nucleus parameters from NUBASE 2016 [@1674-1137-41-3-030001]. We also extend our model to predict the $\mathcal{\alpha}$ decay half-lives of seven even-even superheavy nuclei with $Z=120$ and some un-synthesized nuclei on their $\mathcal{\alpha}$ decay chains.
This article is organized as follows. In Sec. II the theoretical framework for $\mathcal{\alpha}$ decay half-life is described in detail including Gamow-like model and other models such as Coulomb potential and Proximity potential model (CPPM) with Bass73 formalism, the Viola–Seaborg–Sobiczewski (VSS) empirical formula, the Universal curve (UNIV), Royer formula, the Universal decay law (UDL) and the Ni-Ren-Dong-Xu empirical formula (NRDX). In Sec. III, the detailed calculations, discussion and predictions are provided. A brief summary is given in Sec. IV.
THEORETICAL FRAMEWORK
=====================
Gamow-like model
----------------
$\mathcal{\alpha}$ decay half-life $T_\frac{1}{2}$, an important indicator of nuclear stability, can be calculated by the $\mathcal{\alpha}$ decay constant $\mathcal{\lambda}$ as $$\label{1}
T_\frac{1}{2}=\frac{ln2}{\lambda}10^h,$$ where $h$ is the so-called hindrance factor of $\mathcal{\alpha}$ decay due to the effect of an odd-proton and/or an odd-neutron. For the even-even nuclei, $h$ = 0, while for nuclei with an odd number of nucleons i.e. even-$N$, odd-$Z$ nuclei or odd-$N$, even-$Z$ $h=h_p=h_n$, odd-$N$, odd-$Z$ nuclei $2h=h_{np}$. The $\mathcal{\alpha}$ decay constant $\mathcal{\lambda}$ is given by [@PhysRevC.83.014601] $$\label{2}
\lambda=\nu S_\alpha P,$$ where $S_{\alpha}$ represents the preformation probability of $\mathcal{\alpha}$ particles in $\mathcal{\alpha}$ decay. According to Ref. [@PhysRevC.87.024308], it can be known that the value of the preformation probability $S_{\alpha}$ can be changed by adjusting the radius constant $r_{0}$ appropriately. The results show that the best fitting result can be obtained with $S_{\alpha}$ =1, meanwhile $r_0\approx 1.2$fm, which also confirms the conclusion of Refs. [@PhysRevC.83.014601; @0954-3899-17-S-045]. Then we choose $S_{\alpha}$=1 in this work.
$P$ given in Eq. (\[2\]) represents the penetration probability of the $\mathcal{\alpha}$ particle crossing the barrier, calculated by the classical WKB approximation. Its concrete representation in the Gamow-like model is expressed as $$\label{3}
P=\exp\! [- \frac{2}{\hbar} \int_{R}^{b} \sqrt{2\mu (V(r)-E_\text{k})}\, dr],$$ here $E_\text{k}={Q_\alpha}{\frac{A-4}{A}}$ is the kinetic energy of $\mathcal{\alpha}$ particle emitted during $\mathcal{\alpha}$ decay. $Q_\alpha$ and $A$ are $\mathcal{\alpha}$ decay energy and the mass number of the parent nucleus, respectively. $b$ is the classical turning point. It satisfies the condition $V(b)=E_\text{k}$. $\mu=\frac{M_\text{d}M_{\alpha}}{(M_\text{d}+M_{\alpha})}$ is the reduced mass of the $\mathcal{\alpha}$ particle and the daughter nucleus in the center-of-mass coordinate with $M_{\text{d}}$ and $M_\alpha$ being masses of the daughter nucleus and $\mathcal{\alpha}$ particle. $V(r)$ is the total $\alpha$–daughter nucleus interaction potential.
In general, the $\alpha$–daughter nucleus electrostatic potential is by default of the Coulomb type as $$\label{eq6}
V_C(r)=Z_{\alpha}Z_de^2/r,$$ where $Z_{\alpha}$ and $Z_d$ are the proton numbers of $\alpha$ particle and daughter nucleus. Whereas, in the process of $\alpha$ decay, for the superposition of the involved charges, movement of the emitted $\alpha$ particle which generates a magnetic field and the inhomogeneous charge distribution of the nucleus, the emitted $\alpha$-daughter nucleus electrostatic potential behaves as a Coulomb potential at short distance and drop exponentially at large distance i.e. the screened electrostatic effect[@Budaca2017]. This behavior of electrostatic potential can be described as the Hulthen type potential which is widely used in nuclear, atomic, molecular and solid state physics[@doi:10.1063/1.4995175; @PhysRevC.91.034614] and defined as $$\label{eq7}
V_{h}(r)=\frac{a Z_\text{d} Z_\alpha e^2}{e^{ar}-1},$$ where $a$ is the screening parameter. In this framework, the total $\alpha$–daughter nucleus interaction potential $V(r)$ is given by $$V(r)=\left\{
\begin{array}{rcl}
-V_0,& & {0 \leq r \leq R,}\\
V_h(r)+V_l(r), & & {r \ge R,}
\end{array} \right.$$ where $V_0$ is the depth of the square well. $V_h(r)$ and $V_l(r)$ are the Hulthen type of screened electrostatic Coulomb potential and centrifugal potential, respectively. The spherical square well radius $R$ is equal to the sum of the radii of both daughter nucleus and $\mathcal{\alpha}$ particle, it is expressed as $$\label{eq5}
R=r_0({A_\text{d}}^\frac{1}{3}+{A_\alpha}^\frac{1}{3}),$$ where $A_{\text{d}}$ and $A_{\alpha}$ are the mass number of the daughter nucleus and $\mathcal{\alpha}$ particle, respectively. $r_{0}$, the radius constant, is the adjustable parameter in our model.
Because $l(l + 1) \rightarrow (l + 1/2)^2$ is a necessary correction for one-dimensional problem [@Gur31], the centrifugal potential $V_{l}(r)$ is chose as the Langer modified form in this work. It can be expressed as $$\label{eq10}
V_{l}(r)=\frac{\hbar^2(l+\frac{1}{2})^2}{2{\mu}r^2},$$ where $l$ is the orbital angular momentum taken away by the $\mathcal{\alpha}$ particle. $l = 0$ for the favored $\mathcal{\alpha}$ decays, while $l\ne 0$ for the unfavored decays. Based on the conservation laws of party and angular momentum [@PhysRevC.82.059901], the minimum angular momentum $l_{\text{min}}$ taken away by the $\mathcal{\alpha}$ particle can be determined by $$\
l_{\text{min}}=\left\{\begin{array}{llll}
{\Delta}_j,&\text{for even${\Delta}_j$ and ${\pi}_p$= ${\pi}_d$},\\
{\Delta}_j+1,&\text{for even${\Delta}_j$ and ${\pi}_p$$\ne$${\pi}_d$},\\
{\Delta}_j,&\text{for odd${\Delta}_j$ and ${\pi}_p$$\ne$${\pi}_d$},\\
{\Delta}_j+1,&\text{for odd${\Delta}_j$ and ${\pi}_p$= ${\pi}_d$},
\end{array}\right.
\label{6}$$ where ${\Delta}_j= |j_p-j_d|$. $j_p$, ${\pi}_p$, $j_d$, ${\pi}_d$ represent spin and parity values of the parent and daughter nuclei, respectively.
The $\nu$ represents the collision frequency of $\mathcal{\alpha}$ particle in the potential barrier. It can be calculated with the oscillation frequency $\omega$ and expressed as [@PhysRevC.81.064309] $$\label{9}
\nu=\omega/2\pi=\frac{(2n_\text{r}+l+\frac{3}{2})\hbar}{2\pi \mu {R_\text{n}}^2}=\frac{(G+\frac{3}{2})\hbar}{1.2\pi \mu{R_\text{0}}^2},$$ where $R_\text{n}= \sqrt{3/5}R_\text{0}$ is the nucleus root-mean-square (rms) radius and $R_0 = 1.28A^{1/3}-0.76+0.8A^{-1/3}$ is the radius of the parent nucleus. $G=2n_\text{r}+l$ is the main quantum number with $n_\text{r}$ and $l$ being the radial quantum number and the angular quantity quantum number, respectively. In the work of Ref. [@PhysRevC.69.024614], for $\mathcal{\alpha}$ decay, $G$ can be obtained by $$G=2n_\text{r}+l=\left\{
\begin{array}{rcl}
18, & & {N \leq 82,}\\
20, & & {82 < N \leq 126,}\\
22, & & {N > 126.}
\end{array} \right.$$
Other models
------------
### Coulomb potential and Proximity potential model with proximity potential Bass73 formalism (CPPM-Bass73)
In CPPM, the $\mathcal{\alpha}$ decay half-life $T_\frac{1}{2}$ is related to the decay constant $\mathcal{\lambda}$ as $$T_\frac{1}{2}=\frac{ln2}{\lambda},$$ where the decay constant $\mathcal{\lambda}$ can be obtained by $$\lambda=\nu P.$$ The assault frequency $\nu$ can be calculated with the oscillation frequency $\omega$ and expressed as $$\nu=\omega/2\pi=2E_v/h,$$ where $h$ is the Planck constant. The zero-point vibration energy $E_v$ can be calculated with $Q_\alpha$ and expressed as [@Poenaru1986] $$\
E_v=\left\{\begin{array}{llll}
0.1045Q_\alpha,\text{for even-even nuclei},\\
0.0962Q_\alpha,\text{for even-$N$, odd-$Z$ nuclei},\\
0.0907Q_\alpha,\text{for odd-$N$, even-$Z$ nuclei},\\
0.0767Q_\alpha,\text{for odd-odd nuclei}.
\end{array}\right.$$ $P$ denote the semiclassical WKB barrier penetration probability, which is expressed as $$P=\exp\! [- \frac{2}{\hbar} \int_{R_{in}}^{R_{out}} \sqrt{2\mu (V(r)-E_\text{k})}\, dr],$$ where $R_{in}$ and $R_{out}$ are the classical turning points which satisfy the conditions $V(R_{in})=V(R_{out})=Q_\alpha$. The total interaction potential $V (r)$, between the emitted proton and daughter nucleus, including nuclear, Coulomb and centrifugal potential barriers. It can be expressed as $$V(r)=V_N(r)+V_C(r)+V_l(r)$$ $V_l(r)$ are same as Eq.(\[eq10\]), $V_C(r)$ can be expressed as
$$\
V_C(r)=\left\{\begin{array}{ll}
\frac{Z_{\alpha}Z_de^2}{2R}[3-(\frac{r}{R})^2],&r<R,\\
\frac{Z_{\alpha}Z_de^2}{r},&r>R.
\end{array}\right.
\label{subeq:1}$$
We select proximity potential Bass73 to calculate the nuclear potential $V_N(r)$ [@BASS1973139; @BASS197445], which is given by $$V_N(r)=-4\pi\gamma\frac{dR_1R_2}{R}exp(-\frac{\xi}{d})=\frac{-da_s{A_\text{d}^\frac{1}{3}}{{A_\alpha}^\frac{1}{3}}}{R}exp(-\frac{r-R}{d}),$$ where the $d=1.35$ fm is the range parameter, and the surface term in the liquid drop model mass formula $a_s=17.0$ MeV. The $\gamma$ is the specific surface energy of the liquid drop model. $R=r_0({A_\text{d}}^\frac{1}{3}+{A_\alpha}^\frac{1}{3})$ represents the the sum of the half-maximum density radii with $r_0 = 1.07$ fm.
### The Viola–Seaborg–Sobiczewski (VSS) semi-empirical relationship
The Viola–Seaborg–Sobiczewski semi-empirical relationship, one of the commonly used formulas for calculating the half-life of $\alpha$ decay, is proposed by Viola and Seaborg and the value given by Sobiczewski instead of the original value given by Viola and Seaborg [@VIOLA1966741]. It can be expressed as $$log_{10}(T_\frac{1}{2})=(aZ+b)Q^{-1/2}+cZ+d+h_{log},$$ where $Z$ is the atomic number of the parent nucleus and $h_{log}$ is hindrance factor. The values of parameters are $a = 1.66175, b=-8.5166, c=-0.20228, d=-33.9069$ and $$\
h_{log}=\left\{\begin{array}{llll}
0 ,&\text{for even-even nuclei},\\
0.772,&\text{for even-$N$, odd-$Z$ nuclei},\\
1.066,&\text{for odd-$N$, even-$Z$ nuclei},\\
1.114,&\text{for odd-odd nuclei}.
\end{array}\right.$$
### The Universal curve (NUIV)
Poenaru et al. proposed the Universal (UNIV) curve for calculating the decay half-lives by extending a fission theory to larger asymmetry, which can be expressed as [@PhysRevC.83.014601; @PhysRevC.85.034615] $$log_{10}T_\frac{1}{2}=-log_{10}P-log_{10}S_\alpha+[log_{10}(ln2)-log_{10}\nu].$$ The penetrability of an external Coulomb barrier $P$ may be obtained analytically as[@SANTHOSH201833] $$-log_{10}P=0.22873\sqrt{\mu Z_d{Z_\alpha}R_b} \times [arccos {\sqrt{r}}-\sqrt{(r(1-r))}],$$ where $r=R_a/R_b$ fm with $R_a=1.2249({A_\text{d}}^\frac{1}{3}+{A_\alpha}^\frac{1}{3})$ fm and $R_b=1.43998Z_d{Z_\alpha}/Q_\alpha$ fm being the two classic turning points. The logarithmic form of the pre-formation factor is given by $$log_{10}S_\alpha=-0.598(A_\alpha-1).$$ $C=[-log_{10}\nu+log_{10}(ln2)]=-22.16917$ is the additive constant [@PhysRevC.83.014601; @PhysRevC.85.034615].
### Royer formula
Royer proposed the analytical formula for determining $\alpha$ decay half-lives by fitting $\alpha$ emitters experimental data [@0954-3899-26-8-305]. It can be written as $$log_{10}T_\frac{1}{2}=a+bA^{1/6}Z^{1/2}+\frac{cZ}{{Q_\alpha}^{1/2}}.$$ The parameters a, b and c are given by $$\
\left\{\begin{array}{llll}
a=-25.31,b=-1.1629,c=1.5864,\text{for even-even nuclei},\\
a=-25.68,b=-1.1423,c=1.5920,\text{for even-$N$, odd-$Z$ nuclei},\\
a=-26.65,b=-1.0859,c=1.5848,\text{for odd-$N$, even-$Z$ nuclei},\\
a=-29.48,b=-1.1130,c=1.6971,\text{for odd-odd nuclei}.
\end{array}\right.$$
### The Universal decay law (UDL)
Qi et al. given a new universal decay law (UDL) for describing $\alpha$-decay and cluster decay modes starting from $\alpha$-like $R$-matrix theory and the microscopic mechanism of the charged-particle emission[@PhysRevLett.103.072501; @PhysRevC.80.044326]. It can be expressed as
$$log_{10}T_\frac{1}{2}=a\chi'+b\rho'+c,$$
where $\chi'=a{Z_\alpha}Z_d\sqrt{\frac{\mu}{Q_\alpha}}$ and $\rho'=\sqrt{A{Z_\alpha}Z_d({A_\text{d}}^\frac{1}{3}+{A_\alpha}^\frac{1}{3})}$. Here the parameters $a=0.4314, b=-0.4087$ and $c=-25.7725$ are determined by fitting to experiments of $\alpha$ and cluster decays.[@PhysRevLett.103.072501; @PhysRevC.80.044326]
### The Ni-Ren-Dong-Xu empirical formula (NRDX)
Ni et al. proposed a new general formula with three parameters for determining half-lives and decay energies of $\alpha$ decay and cluster radioactivity [@PhysRevC.78.044310].This new formula is directly deduced from the WKB barrier penetration probability with some approximations. Their calculations by using this formula show excellent agreement between the experimental data and the calculated values. It can be given by, $$log_{10}T_\frac{1}{2}=a\sqrt{\mu}{Z_\alpha}Z_d{Q}^{-1/2}+b\sqrt{\mu}(Z_dZ_\alpha)^{1/2}+c,$$ The parameters a, b and c are given by $$\
\left\{\begin{array}{lll}
a=0.39961,\\
b=-1.31008,\\
c_{e-e}=-17.00698.
\end{array}\right.$$ This formula successfully combines the phenomenological laws of $\alpha$ decay and cluster radioactivity.
RESULTS AND DISCUSSION
======================
In this work, we use the least squares principle to fit the adjustable parameters, while the database are taken from the latest evaluated nuclear properties table NUBASE2016 [@1674-1137-41-3-030001]. At first, for the parameter $h$ being used to describe the effect of an odd-proton and/or an odd-neutron, we choose the experimental data of $\mathcal{\alpha}$ decay half-lives of 169 even-even nuclei as the database to obtain the parameters $a$ and $r_0$, while $h=0$. Then choosing the experimental data of $\mathcal{\alpha}$ decay half-lives of 132 odd-$N$, even-$Z$ nuclei, 94 even-$N$, odd-$Z$ nuclei and 66 doubly-odd nuclei as the database to determine the parameter $h$, while fixed the parameters $a$ and $r_0$, using the relationship $h_n=h_p=\frac{1}{2}h_{np}=h$. The values of 3 adjustable parameters are given as $$\ r_0=1.14\text{fm}, a=7.8\times10^{-4},\\
\ h=0.3455.$$
The value of $a$ is small but it observably impacts on the classical turning point $b$, whereas the $\mathcal{\alpha}$ decay half-life is sensitive to $b$. For intuitively display the effects, in Fig. \[fig6\] we show the different kinetic energy $E_\text{k}$ values correspond to difference in $b$ values for the pure Coulomb and Hulthen potential, i.e., no centrifugal potential contribution, where $b_c$ and $b_h$ represent the $b$ value calculated using Coulomb and using the Hulthen potential, respectively. From this figure, we can find that the smaller decay energy and larger proton number of the daughter nucleus are, the greater difference in the $b$ value between the pure Coulomb and the Hulthen potential be.
![The difference between $b_c$ and $b_h$ obtained by $V(r)=E_\text{k}$ only considering the Coulomb potential. For obtained $b_c$, Coulomb potential is taken as the potential of a uniformly charged sphere expressed as Eq. (\[eq6\]), while for $b_h$ Coulomb potential is taken as Hulthen potential with $a=7.8\times10^{-4}$ expressed as Eq. (\[eq7\]).[]{data-label="fig6"}](rcrh1){width="9.0cm"}
Using our modified Gamow–like model, we systematically calculate the $\mathcal{\alpha}$ decay half-lives of even-even, odd-odd, odd-$A$ nuclei. The detailed results are shown in the Fig. \[fig7\] – \[fig10\]. In Fig. \[fig7\], we show the 169 $\mathcal{\alpha}$ decay experimental data of even-even nuclei and the theoretical values of $\mathcal{\alpha}$ decay half-live calculated by different methods. The X-axis represents the mass number in the corresponding $\mathcal{\alpha}$ decay, the Y-axis represents the logarithmic of the $\mathcal{\alpha}$ decay half-life. The three coordinate points represent logarithmic form of the experimental $\mathcal{\alpha}$ decay half-lives, logarithmic forms of the calculated $\mathcal{\alpha}$ decay half-lives in this work denoted as ${\text{lg}T_{1/2}^{\text{cal1}}}$ and by the theoretical model and parameters in Ref. [@PhysRevC.87.024308] denoted as ${\text{lg}T_{1/2}^{\text{cal2}}}$, respectively. The cases of even-$Z$, odd-$N$ nuclei, odd-$Z$, even-$N$ nuclei and odd-$N$, odd-$Z$ nuclei are shown in Fig. \[fig8\], Fig. \[fig9\] and Fig. \[fig10\], respectively. The meanings of each coordinate in Fig. \[fig8\] – \[fig10\] is same as Fig. \[fig7\].
![The calculation of the $\mathcal{\alpha}$ decay half-life of the even-even nuclei. ${\text{lg}T_{1/2}^{\text{cal1}}}$ is the logarithmic form of the $\mathcal{\alpha}$ decay half-life calculated in this work, and ${\text{lg}T_{1/2}^{\text{cal2}}}$ is the logarithmic form of the $\mathcal{\alpha}$ decay half-life calculated by the theoretical model and parameters in Ref.[@PhysRevC.87.024308]. The experimental $\mathcal{\alpha}$ decay half-lives and decay energies are taken from the latest evaluated nuclear properties table NUBASE2016 [@1674-1137-41-3-030001] and evaluated mass number table AME2016 [@1674-1137-41-3-030003]. []{data-label="fig7"}](EE1){width="14.0cm"}
![The same as Fig. \[fig7\], but for the case of even-$Z$, odd-$N$ nuclei.[]{data-label="fig8"}](EO1){width="14.0cm"}
![The same as Fig. \[fig7\], but for the case of odd-$Z$, even-$N$ nuclei.[]{data-label="fig9"}](OE1){width="14.0cm"}
![The same as Fig. \[fig7\], but for the case of odd-$Z$, odd-$N$ nuclei.[]{data-label="fig10"}](OO1){width="14.0cm"}
As can be seen from the Fig. \[fig7\] – \[fig10\], the ${\text{lg}{T^{\text{cal1}}_{1/2}}}$ can better reproduce with experimental data than ${\text{lg}{T^{\text{cal2}}_{1/2}}}$. In order to intuitively compare ${{T^{\text{cal1}}_{1/2}}}$ with ${{T^{\text{cal2}}_{1/2}}}$, we calculate the standard deviation $\sigma=\sqrt{\sum ({\text{lg}{T^{\text{expt}}_{1/2}}}-{\text{lg}{T^{\text{cal}}_{1/2}}})^2/n}$ between $\mathcal{\alpha}$ decay half-lives of calculations and experimental data. The results $\sigma_1$, $\sigma_2$ represent standard deviations between ${\text{lg}{T^{\text{cal1}}_{1/2}}}$, ${\text{lg}{T^{\text{cal2}}_{1/2}}}$ and ${\text{lg}{T^{\text{expt}}_{1/2}}}$, which are given in the Table \[table 6\]. From this table, we can clearly see that for the cases of even-even, odd-proton, odd-neutron and doubly-odd nuclei, our calculations ${\text{lg}{T^{\text{cal1}}_{1/2}}}$ improve $\frac{0.487-0.348}{0.487}\approx28.5\%$, $\frac{0.967-0.681}{0.967}\approx29.6\%$, $\frac{0.789-0.598}{0.789}\approx24.2\%$ and $\frac{1.235-0.748}{1.235}\approx39.4\%$ compared to ${\text{lg}{T^{\text{cal2}}_{1/2}}}$, respectively. It is shown that ${T^{\text{cal1}}_{1/2}}$ can better reproduce with experimental data than ${T^{\text{cal2}}_{1/2}}$ by considering the shielding effect of the Coulomb potential and the centrifugal potential in this work. And for even-even nuclei, ${T^{\text{cal2}}_{1/2}}$ are calculated by Gamow-like model proposed by K. Pomorski [@PhysRevC.87.024308] which contain only one parameter $r_0$, and ${T^{\text{cal1}}_{1/2}}$ are calculated by our improved Gamow-like model with two parameters $r_0$ and $a$. So the addition of the parameter $a$ makes the ${T^{\text{cal1}}_{1/2}}$ more consistent with the experimental data than ${T^{\text{cal2}}_{1/2}}$. In many of the Fig. \[fig7\] – \[fig10\] we can see dips and peaks in half-lives, it is because $Z/N=50$, $Z/N=82$ and $N=126$ is the magic core, and the nucleons in the core play an essential role on the $\alpha$ preformation probability[@PhysRevC.94.024338].
[ccccc]{}
&n&$h$&$\sigma_1$&$\sigma_2$\
\
e-e&169&[–]{}&[0.348]{}&[0.487]{}\
\
e-o&132&0.3455&[0.681]{}&[0.967]{}\
\
o-e&94&0.3455&[0.598]{}&[0.789]{}\
\
o-o&66&0.691&[0.748]{}&[1.235]{}\
The synthesis and research of SHN have became a hot topic in nuclear physics [@PhysRevC.97.064609; @1674-1137-41-7-074106; @PhysRevC.98.014618]. Now we extend our model to predict the $\alpha$ decay half-lives of nuclei $Z=120$ i.e. $^{296}120$, $^{298}120$,$^{300}120$,$^{302}120$,$^{304}120$,$^{306}120$ as well as $^{308}120$ and some un-synthesized nuclei on their $\mathcal{\alpha}$ decay chains. From the conclusion of decay properties for SHN in Ref. [@SANTHOSH201833], we can obtain the $\mathcal{\alpha}$ decay chains of these nuclei, which are $^{296}120\to^{292}\text{Og}\to^{288}\text{Lv}\to^{284}\text{Fl}\to^{280}\text{Cn}\to^{276}\text{Ds}\to^{272}\text{Hs}\to^{268}\text{Sg}$, $^{298}120\to^{294}\text{Og}\to^{290}\text{Lv}\to^{286}\text{Fl}\to^{282}\text{Cn}\to^{278}\text{Ds}\to^{274}\text{Hs}$, $^{300}120\to^{296}\text{Og}\to^{292}\text{Lv}\to^{288}\text{Fl}\to^{284}\text{Cn}$, $^{302}120\to^{298}{\text{Og}}\to^{294}\text{Lv}\to^{290}\text{Fl}$, $^{304}120\to^{300}\text{Og}\to^{296}\text{Lv}\to^{294}\text{Fl}$, $^{306}120\to^{302}\text{Og}\to^{298}\text{Lv}$, and $^{308}120\to^{304}\text{Og}\to^{300}\text{Lv}$. In our previous studies of the superheavy nucleus[@1674-1137-42-4-044102; @1674-1137-41-12-124109], the $\mathcal{\alpha}$ decay energy is one key input for calculating the $\mathcal{\alpha}$ decay half-life. Meanwhile Sobiczewski [@PhysRevC.94.051302] discovered that the calculation taking $\mathcal{\alpha}$ decay energy from WS3+ [@PhysRevC.84.051303] can best reproduce experimental $\mathcal{\alpha}$ decay half-life. In the present work, we use $\mathcal{\alpha}$ decay energy from WS3+ to calculate the half-life of even-even nuclide with proton number $Z=120$ and nuclei on their $\mathcal{\alpha}$ decay chains except the five known nuclei i.e. $^{294}$Og, $^{290}$Lv, $^{286}$Fl, $^{292}$Lv and $^{288}$Fl are taken from NUBASE2016 [@1674-1137-41-3-030001].
For comparatively, we also systematically calculate the $\mathcal{\alpha}$ decay half-lives of even-even nuclei of proton numbers $Z=120$ and nuclei on their $\mathcal{\alpha}$ decay chain using Coulomb potential and Proximity potential model with proximity potential Bass73 formalism (CPPM-Bass73) [@BASS197445], the Viola-Seaborg-Sobiczewski (VSS) empirical formula [@VIOLA1966741], the Universal (UNIV) curve [@PhysRevC.83.014601; @PhysRevC.85.034615], Royer formula [@0954-3899-26-8-305], the Universal decay law (UDL) [@PhysRevLett.103.072501; @PhysRevC.80.044326] and the Ni-Ren-Dong-Xu (NRDX) empirical formula [@PhysRevC.78.044310], respectively. The logarithmic forms of calculated $\mathcal{\alpha}$ decay half-lives are listed in Table \[table 7\]. In this tables, the first two columns represent the parent nucleus of the $\mathcal{\alpha}$ decay and the $\mathcal{\alpha}$ decay energy, the next seven columns represent the theoretical $\mathcal{\alpha}$ decay half-lives calculated by CPPM-Bass73, VSS, UNIV, Royer, UDL, NRDX and our improved Gamow-like model denoted as ${\text{lg}{T^{\text{CPPM-Bass73}}_{1/2}}}$(s), ${\text{lg}{T^{\text{VSS}}_{1/2}}}$(s), ${\text{lg}{T^{\text{UNIV}}_{1/2}}}$(s), ${\text{lg}{T^{\text{Royer}}_{1/2}}}$(s), ${\text{lg}{T^{\text{UDL}}_{1/2}}}$(s), ${\text{lg}{T^{\text{NRDX}}_{1/2}}}$(s) and ${\text{lg}{T^{\text{This work}}_{1/2}}}$(s), respectively. The last column represents logarithmic form of the experimental $\mathcal{\alpha}$ decay half-lives taken from NUBASE2016 [@1674-1137-41-3-030001]. It can be seen from Table \[table 7\] that for the $\mathcal{\alpha}$ decay of the same parent nuclear, the logarithmic form of theoretical $\mathcal{\alpha}$ decay half-life of all models are not much different, and the $\mathcal{\alpha}$ decay theoretical half-lives of the CPPM-Bass73 model is smaller than other models. For the parent nuclei with known experimental half-life, the maximum difference between the logarithmic forms of experimental half-life value and the logarithmic forms of theoretical half-life obtained from the model of this work is less than 0.65. To make a more intuitive comparison of these theoretical predictions, the theoretical half-life of $\mathcal{\alpha}$ decay calculated using this seven theoretical models are plotted in Fig. \[fig1\]. In this figure, decay chains begin with an nucleus with a proton number $Z=120$, the nucleus at the end of each decay chain is spontaneous fission, and the decay of the remaining nucleus is $\mathcal{\alpha}$ decay. The X-axis represents the mass number in the corresponding $\mathcal{\alpha}$ decay chain, the Y-axis represents the logarithmic of the $\mathcal{\alpha}$ decay half-life.
[cccccccccc]{} Nucleus &$Q_{\alpha}$ (MeV)&${\text{lg}{T^{\text{CPPM-Bass73}}_{1/2}}}$(s)&${\text{lg}{T^{\text{VSS}}_{1/2}}}$(s)&${\text{lg}{T^{\text{UNIV}}_{1/2}}}$(s)&${\text{lg}{T^{\text{Royer}}_{1/2}}}$(s)&${\text{lg}{T^{\text{UDL}}_{1/2}}}$(s)&${\text{lg}{T^{\text{NRDX}}_{1/2}}}$(s)&${\text{lg}{T^{\text{This work}}_{1/2}}}$(s)&$\text{lg}{T^{\text{expt}}_{1/2}}$ (s)\
\
$^{ 296 }$ 120 & 13.187 & -6.189 & -5.613 & -5.884 & -5.774 & -5.842 & -5.398 & -5.662 & –\
$^{ 292 }$ Og & 12.015 & -4.264 & -3.662 & -4.044 & -3.842 & -3.809 & -3.516 & -3.832 & –\
$^{ 288 }$ Lv & 11.105 & -2.698 & -2.082 & -2.527 & -2.275 & -2.165 & -1.994 & -2.326 & –\
$^{ 284 }$ Fl & 10.666 & -2.202 & -1.568 & -2.018 & -1.767 & -1.647 & -1.515 & -1.831 & –\
$^{ 280 }$ Cn & 10.911 & -3.471 & -2.797 & -3.183 & -2.999 & -2.975 & -2.744 & -3.006 & –\
$^{ 276 }$ Ds & 10.976 & -4.259 & -3.555 & -3.891 & -3.76 & -3.802 & -3.508 & -3.723 & –\
$^{ 272 }$ Hs & 9.54 & -1.077 & -0.406 & -0.823 & -0.603 & -0.477 & -0.425 & -0.678 & –\
\
$^{ 298 }$ 120 & 12.9 & -5.643 & -5.032 & -5.371 & -5.231 & -5.258 & -4.826 & -5.148 & –\
$^{ 294 }$ Og & 11.835 & -3.889 & -3.254 & -3.688 & -3.471 & -3.408 & -3.115 & -3.474 & -2.939\
$^{ 290 }$ Lv & 11.005 & -2.482 & -1.832 & -2.319 & -2.062 & -1.932 & -1.748 & -2.12 & -2.097\
$^{ 286 }$ Fl & 10.365 & -1.431 & -0.771 & -1.28 & -1.008 & -0.833 & -0.731 & -1.097 & -0.456\
$^{ 282 }$ Cn & 10.106 & -1.375 & -0.695 & -1.186 & -0.934 & -0.776 & -0.677 & -1.014 & –\
$^{ 278 }$ Ds & 10.31 & -2.601 & -1.882 & -2.315 & -2.122 & -2.057 & -1.862 & -2.15 & –\
\
$^{ 300 }$ 120 & 13.287 & -6.461 & -5.811 & -6.13 & -6.045 & -6.116 & -5.591 & -5.907& –\
$^{ 296 }$ Og & 11.561 & -3.279 & -2.612 & -3.109 & -2.867 & -2.759 & -2.484 & -2.895 & –\
$^{ 292 }$ Lv & 10.775 & -1.922 & -1.243 & -1.784 & -1.51 & -1.338 & -1.168 & -1.587 & -1.602\
$^{ 288 }$ Fl & 10.065 & -0.624 & 0.06 & -0.506 & -0.214 & 0.017 & 0.087 & -0.325 & -0.125\
\
$^{ 302 }$ 120 & 12.878 & -5.671 & -4.986 & -5.391 & -5.259 & -5.273 & -4.781 & -5.166 & –\
$^{ 298 }$ Og & 12.118 & -4.607 & -3.893 & -4.358 & -4.182 & -4.148 & -3.741 & -4.144 & –\
$^{ 294 }$ Lv & 10.451 & -1.083 & -0.379 & -0.981 & -0.683 & -0.453 & -0.319 & -0.785 & –\
\
$^{ 304 }$ 120 & 12.745 & -5.43 & -4.71 & -5.162 & -5.019 & -5.011 & -4.509 & -4.937 & –\
$^{ 300 }$ Og & 11.905 & -4.162 & -3.414 & -3.935 & -3.741 & -3.672 & -3.27 & -3.719 & –\
$^{ 296 }$ Lv & 10.777 & -2.002 & -1.248 & -1.853 & -1.588 & -1.406 & -1.172 & -1.654 & –\
\
$^{ 306 }$ 120 & 13.823 & -7.59 & -6.836 & -7.169 & -7.175 & -7.296 & -6.595 & -6.949 & –\
$^{ 302 }$ Og & 11.995 & -4.404 & -3.618 & -4.16 & -3.98 & -3.92 & -3.47 & -3.944 & –\
\
$^{ 308 }$ 120 & 13.036 & -6.102 & -5.309 & -5.784 & -5.689 & -5.709 & -5.096 & -5.559 & –\
$^{ 304 }$ Og & 13.104 & -6.789 & -5.96 & -6.389 & -6.354 & -6.434 & -5.769 & -6.178 & –\
![The different cases of $\mathcal{\alpha}$ decay half-lives calculated by different theories. The abscissa A represents the mass of the nucleus, and the ordinate is the theoretical value of the $\mathcal{\alpha}$ decay half-life, the different color lines represent the calculations using different theoretical models.[]{data-label="fig1"}](296.eps){height="5.0cm" width="6.0cm"}
The case of $ ^{296}120$
![The different cases of $\mathcal{\alpha}$ decay half-lives calculated by different theories. The abscissa A represents the mass of the nucleus, and the ordinate is the theoretical value of the $\mathcal{\alpha}$ decay half-life, the different color lines represent the calculations using different theoretical models.[]{data-label="fig1"}](298.eps){height="5.0cm" width="6.0cm"}
The case of $ ^{298}120$
![The different cases of $\mathcal{\alpha}$ decay half-lives calculated by different theories. The abscissa A represents the mass of the nucleus, and the ordinate is the theoretical value of the $\mathcal{\alpha}$ decay half-life, the different color lines represent the calculations using different theoretical models.[]{data-label="fig1"}](300.eps){height="5.0cm" width="6.0cm"}
The case of $ ^{300}120$
![The different cases of $\mathcal{\alpha}$ decay half-lives calculated by different theories. The abscissa A represents the mass of the nucleus, and the ordinate is the theoretical value of the $\mathcal{\alpha}$ decay half-life, the different color lines represent the calculations using different theoretical models.[]{data-label="fig1"}](302304.eps){height="5.0cm" width="6.0cm"}
The case of $ ^{302}120$ and $ ^{304}120$
![The different cases of $\mathcal{\alpha}$ decay half-lives calculated by different theories. The abscissa A represents the mass of the nucleus, and the ordinate is the theoretical value of the $\mathcal{\alpha}$ decay half-life, the different color lines represent the calculations using different theoretical models.[]{data-label="fig1"}](306308.eps){height="5.0cm" width="6.0cm"}
The case of $ ^{306}120$ and $ ^{308}120$
From Fig. \[fig1\], we can clearly see that theoretical calculations of $\mathcal{\alpha}$ decay half-lives by different models of the same nucleus are different due to the model dependent. But all theoretically calculated $\mathcal{\alpha}$ decay half-life curves have the same trend. In order to intuitively compare different theories, we calculate the standard deviation $\Delta=\sqrt{\sum ({\text{lg}{T^{\text{expt}}_{1/2}}}-{\text{lg}{T^{\text{cal}}_{1/2}}})^2/n}$ between $\mathcal{\alpha}$ decay half-lives of calculations and experimental data of different theories in Table \[table 8\].
[cccccccc]{}
&$\Delta_{CPPM-bass73}$&$\Delta_{VSS}$&$\Delta_{UNIV}$&$\Delta_{Royer}$&$\Delta_{UDL}$&$\Delta_{NRDX}$&$\Delta_{This -work}$\
\
298&0.817& 0.299& 0.656& 0.443& 0.360& 0.276&0.482\
300&0.419& 0.286& 0.299& 0.091& 0.212& 0.342&0.142\
We can clearly see that NRDX model reproduces experimental half lives well in superheavy region in case $^{298}120$, Royer formula reproduces experimental half lives well in superheavy region in case $^{300}120$. In particular, our calculations are sandwiched in other decay chains, and closed to the known experimental data for the $\mathcal{\alpha}$ half-lives, which shows that the model and calculated parameters of present work are believable.
Summary
=======
In summary, we modify the Gamow-like model by considering the effects of screened electrostatic for Coulomb potential and the centrifugal potential and use this model systematically to study $\mathcal{\alpha}$ decay half-lives for $Z>51$ nuclei. In addition, we extend this model to the superheavy nuclei, and predict the half-lives of seven even-even nuclei with a proton number $Z=120$ and some un-synthesized nuclei on their $\mathcal{\alpha}$ decay chains. This work is useful for the future research of superheavy nuclei. \[section 4\]
Acknowledgements {#acknowledgements .unnumbered}
================
This work is supported in part by the National Natural Science Foundation of China (Grant No. 11205083), the construct program of the key discipline in Hunan province, the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 15A159 and 18A237), the Natural Science Foundation of Hunan Province, China (Grant No. 2015JJ3103 and No. 2015JJ2123), the Innovation Group of Nuclear and Particle Physics in USC, the double first class construct program of USC.
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---
abstract: 'By considering generalized logarithm and exponential functions used in nonextensive statistics, the four usual algebraic operators : addition, subtraction, product and division, are generalized. The properties of the generalized operators are investigated. Some standard properties are preserved, e.g., associativity, commutativity and existence of neutral elements. On the contrary, the distributivity law and the opposite element is no more universal within the generalized algebra.'
author:
- |
L. Nivanen, A. Le Méhauté and Q.A. Wang\
Institut Supérieur des Matériaux du Mans,\
44, Avenue F.A. Bartholdi, 72000 Le Mans, France\
Email: [email protected]
title: Generalized algebra within a nonextensive statistics
---
[PACS : 02.10.-v, 02.70.Rr, 02.90.+p]{}
[Keywords : Algebra, nonextensivity, generalized statistics]{}
Introduction
============
Although scientists apply Boltzmann-Gibbs statistics (BGS), or its exponential factor of probability distribution, to systems having long range correlation or finite size[@Conform], this classical statistical theory, from the usual point of view, remains an additive theory in the thermodynamic limits, i.e., the extensive thermodynamic quantities are proportional to its volume or to the number of its elements.
As far as we know, there is no direct experimental evidence of nonadditive energy. However, theoretical and numerical work has disclosed many models characterized by nonadditive energy or entropy with, e.g., black hole[@Hayw98; @Frol96] and some magnetic and fluid models including long range interaction[@Jund95; @Cann96; @Ruff95; @Grig96; @Ruff01]. Interesting numerical results also showed that, in the case of above long range (nonintegrable) interactions, in the thermodynamic limits, the systems show complex behaviors at the edge of chaos with non-gaussian distribution, anomalous diffusion and dynamic correlation in the phase space[@Lato02]. For this kind of non-equilibrium systems at stationary state[@Letters], a possible way out has been suggested on the basis of a nonadditive statistical mechanics (NSM)[@Tsal88; @Cura91; @Penni; @Tsal99; @Wang01; @Wang02; @Wang02b; @Wang02c].
NSM introduces[@Tsal88; @Tsal94] generalized distribution functions called $q$-exponential given by $exp_q(x)=[1+(1-q)x]^{1/(1-q)}$ or $exp_q^*(x)=[1+(q-1)x]^{1/(q-1)}$ according to different formalisms related to incomplete information[@Wang01; @Wang02] and complete information hypothesis[@Tsal88; @Cura91; @Penni; @Tsal99]. The above functions are the inverse functions of the generalized logarithm $\ln_q(x)=\frac{x^{1-q}-1}{1-q}$ or $\ln_q^*(x)=\frac{x^{q-1}-1}{q-1}$ which can be used as a generalization of Hartley logarithmic information measure[@Wang01]. In NSM, we require that the $[.]$ in $q$-exponential be positive to assure physically significant probability distribution. This requirement may lead to high or low energy cutoff depending on the value of $q$. In what follows, we will write the generalized functions as follows : $\ln_a(x)=\frac{x^a-1}{a}$ and $e_a^x=exp_a(x)=[1+ax]^{1/a}$. On the one hand, these forms are simpler to treat mathematically. On the other hand, this choice ensures the generality of the present results which should be independent of the physical circumstances with different formalisms of NSM. $e_a(x)$ and $\ln_a(x)$ tend to the conventional counterparts whenever $a\rightarrow 0$.
Although defined originally from the physical viewpoint, these functions present important mathematical interests. Some of their important properties have been studied recently in [@Yama02; @Agui03]. They have also been used to generalize hyperbolic function and algebra[@Borges]. In this paper, we discuss several properties of the new algebra generated by the $q$-logarithm and $q$-exponential. This algebra can be viewed as a generalization of the conventional additive algebra related to the normal logarithm and exponential, but characterized by a nonadditive additional factor. A similar generalization of the conventional algebra has been proposed by Kaniadakis on the basis of the $\kappa$-deformed logarithm and exponential[@Kaniadakis].
The paper is organized as follows. In the following section we briefly discuss the essential aspects of NSM. The generalized algebra is discussed in the third section.
Some nonadditive relations of NSM
=================================
We know that, for a nonadditive thermodynamic system, the existence of thermodynamic stationarity may be used as a constraint upon the form of the nonextensivity of physical quantities[@Abe01]. For the entropy $S$ and the internal energy $U$, this can be discussed rigorously by using the conventional BGS method for the statistical interpretation of the zeroth law. It has been shown[@Abe01] that the simplest nonextensivities prescribed by the thermodynamic stationarity condition were generated by the following relationships : $S(A)=\frac{h(A)-1}{\lambda_S}$, $S(B)=\frac{h(B)-1}{\lambda_S}$, $S(A+B)=\frac{h(A+B)-1}{\lambda_S}$ and $h(A+B)=h(A)h(B)$, where $A$ and $B$ are two subsystems of an isolated system $A+B$, $\lambda_S$ is a constant, $h(A)$ or $h(B)$ is the factor depending on $A$ or $B$ in the derivative $\frac{\partial S(A+B)}{\partial S(B)}$ or $\frac{\partial S(A+B)}{\partial S(A)}$. It was also found that these relationships are also valid if $S$ is replaced by $U$ and $\lambda_S$ by $\lambda_U$[@Wang02d]. So the nonextensivities should be the following : $S(A+B)=S(A)+S(B)+\lambda_SS(A)S(B)$ and $U(A+B)=U(A)+U(B)+\lambda_UU(A)U(B)$. We have shown[@Wang02] that only these two relationships allow to interpret the zeroth law of thermodynamics within NSM and define a physical temperature without any approximation.
The above nonextensivity of entropy, according to some authors[@Santos], uniquely leads to the entropy $$\label{1}
S=-k\frac{\sum_ip_i-\sum_ip_i^q}{1-q},$$ which can be shown[@Wang01; @Wang02] to be the following expectation value $S=\sum_ip_i^qI_i$ of a generalized information measure : $$\label{1a}
I_i=k\ln_{1-q}(p_i)$$ where $q=1+\lambda_S$ is a real parameter[@Tsal88], $k$ is Boltzmann constant and $p_i$ the probability that the system is found at the state $i$, if it is supposed that $p_{ij}(A+B)=p_i(A)p_j(B)$. This product law of probability has always been considered as a result of the “statistical independence” of $A$ and $B$ and considered by many as a “reason” for accepting additive energy $U(A+B)=U(A)+U(B)$ within NSM. As shown with the counter-example, this is not necessarily true and entails several fundamental and practical problems which have been discussed in detail[@Wang02]. We observed[@Wang02e] that the product law of probability could be considered as a consequence of the entropy in Eq.(\[1\]) and the nonextensivity imposed by the condition of the existence of equilibrium. In this way, the composite energy is freed from the constraint of the independence of the subsystems and NSM is then entitled to treat nonadditive systems having the nonadditive entropy postulated in Eq.(\[1\]) (or the information measure postulated in Eq.(\[1a\]) and the nonadditive energy prescribed by the thermal equilibrium.
The maximization of the above entropy leads to the $q$-exponential probability distribution. The currently improved coherence of this statistics and the investigations of the relationships between this statistics and the chaotic or fractal phenomena[@Wang03] tell us that $q$-logarithm and $q$-exponential have solid physical background and may play a major role in the nonadditive physics. So it is worthwhile to investigate them from pure mathematical point of view and especially according to the related algebra associated with nonadditive properties. In what follows, we present the generalized nonadditive algebra related to these functions.
Generalized operations
======================
From now on, the mathematical notion of morphism will be used to exhibit generalized expressions for the four fundamental algebraic operators : addition, subtraction, product and division. In the fourth part, the reader will see that the algebraic properties of these generalized operators are not always identical to those of the classical operators. New features will be pointed out, for instance, the distributive law is no more verified.
The standard exponential and logarithm functions present some remarkable properties. For instance, the exponential function is a morphism from $(\mathbf{R}, +)$ to $(\mathbf{R}_+^*,\times)$. Through reciprocity, the logarithm function is a morphism from $(\mathbf{R}_+^*,\times)$ to $(\mathbf{R}, +)$. As a result : $e^{x+y}=e^x e^y$ and $\ln
xy = \ln x+\ln y$. Although these properties cannot be generalized directly by a simple substitution of $e^x$ by $e_a^x$ (or $\ln x$ by $\ln_a x$, the following slightly more complicated relationships are verified :
$$\label{2}
e_a^x e_a^y=e_a^{x+y+axy}\neq e_a^{x+y},$$
$$\label{3}
e_a^{x+y}=[(e_a^x)^a+(e_a^y)^a-1]^{1/a}\neq e_a^x e_a^y,$$
$$\label{4}
\ln_a x +\ln_a y=\ln_a (x^a+y^a-1)^{1/a}\neq \ln_a xy,$$
$$\label{5}
\ln_a xy = \ln_a x + \ln_a y + a\ln_a x \ln_a y\neq \ln_a x + \ln_a y.$$
We see that with standard addition and product operators, the concept of morphism cannot be applied to generalized functions. But if we $define$ generalized addition (denoted by $+_a$) and product ($\times_a$) operators that depend on the parameter $a$ as follows :
$$\label{a}
x+_ay = x+y+axy$$
and $$\label{b}
x\times_ay = (x^a+y^a-1)^{1/a},$$ then Eqs.(\[2\]) to (\[5\]) can be recast into :
$$\label{2a}
e_a^{x+_ay}=e_a^x e_a^y,$$
$$\label{3a}
e_a^{x+y}=e_a^x \times_a e_a^y,$$
$$\label{4a}
\ln_a x\times_ay=\ln_a x +\ln_a y,$$
$$\label{5a}
\ln_a xy = \ln_a x +_a \ln_a y .$$
The two standard morphisms have split into four ones. Let us denote the definition set of the function $e_a^x$ by $D_a$. From Eq.(\[2a\]), it comes that the generalized exponential function is a morphism from $({D_a}, {+_a})$ to $(\mathbf{R}_+^*,\times)$. But from Eq.(\[3a\]), we note that this function is also a morphism from $({D_a}, +)$ to $({\mathbf{R}}_+^*,{{\times}_a})$. From Eq.(\[4a\]), it comes that the generalized logarithm function is a morphism from $({\mathbf{R}}_+^*,{{\times }_a})$ to $({D_a}, +)$. But from Eq.(\[5a\]), we note that this function is also a morphism from $(\mathbf{R}_+^*,\times)$ to $({D_a}, {+_a})$. It is worth noticing that a standard operator and a generalized one are present in each of the Eqs.(\[2a\]) to (\[5a\]). This is because that the generalized operators are defined with the help of the standard ones in Eqs.(\[a\]) and (\[b\]).
In the same way, the generalized subtraction $-_a$ and division $/_a$ operators can be defined. If we write :
$$\label{aa}
x-_ay = \frac{x-y}{1+ay}$$
and $$\label{bb}
x/_ay = (x^a-y^a+1)^{1/a}$$ then the following relationships exist :
$$\label{2b}
e_a^{x-_ay} = e_a^x /e_a^y$$
$$\label{3b}
e_a^{x-y}=e_a^x /_a e_a^y$$
$$\label{4b}
\ln_a(x /_a y) = \ln_a x - \ln_a y$$
$$\label{5b}
\ln_a(x/y)=\ln_a (x) {-_a} \ln_a y.$$
which recover the standard ones $e^{x-y}=e^x/e^y$ and $\ln(x/y)=\ln x -\ln y$ when $a\rightarrow 0$.
Properties of generalized operators
===================================
Additivity
----------
The generalized addition operator has following properties :
1. Associativity : $(x+_ay)+_az=x+_a(y+_az)$.
2. Commutativity : $x+_ay=y+_ax$.
3. 0 is the neutral element, i.e., $x+_a0=0+_ax=x$.
4. If $x\neq x_0=-1/a$, it has an opposite element $-_ax$, i.e., $-_ax = -x/(1 + ax)$. The fact that $x_0$ has no opposite element is a new feature. It should be noticed that the above definition of opposite element is compatible with the definition of substraction : $x {-_a} y=x +_a (-_ay)$.
5. The sign rules : ${-_a}({-_a}x)={+_a}({+_a}x)=x $ and ${-_a}({+_a}x)={+_a}({-_a}x)={-_a}x$.
6. Generating role of $+_1$ : if we note $Z=x+_ay$, then $aZ=(ax)+_1(ay)$.
Multiplication
--------------
The product operator $\times_a$ has more complicated features. For a given $a$, we see from Eq.(\[4\]) that, if $x,y > 0$, the product $x\times_ay$ is defined unambiguously and the inequality ${x^a} + {y^a} > 1$ is verified. We notice following properties :
1. Associativity : $(x\times_ay)\times_az=x\times_a(y\times_az)$.
2. Commutativity : $x\times_ay=y\times_ax$.
3. 1 is the neutral element, i.e., $x\times_a 1=1\times_a x=x$.
4. There is no absorbing element (like zero in the usual case). We cannot find a real number $y$ such that, for arbitrary real number $x$, we have $x \times_a y = y$.
5. x has an inverse element noted $1{/_a} x = (2 - x^a)^{1/a}$. In particular, we see that if $a > 0$, then $1{/_a} 0 = 2^{1/a}$ and $1/_a 2^{1/a} = 0$. So 0 can have a finite inverse in the generalized algebra.
6. Generating role of $\times_1$ : if we note $Z=x\times_ay$, then $Z^a=(x^a)\times_1(y^a)$.
We notice that the new operators of this generalized algebra have more complex properties than the usual ones. Unlike the standard case, the existence of an opposite element is not automatic. On the other hand, 0 can now be inverted. Now with the new operators, the infinity concept may be taken into account with finite numbers. There is a strong analogy between this property and the role of hyperbolic space in metric topology[@Bear]. It is natural to find these two distinct mathematical tools in the study of physical phenomena taking place on complex media characterized by a set of singularities in a compact space[@Leme1; @Leme2].
Another important feature of the new algebra with the operators defined previously is the disparition of the law of distributivity, i.e., $x y +x z = x (y + z)$ within the usual algebra. With the above generalized operators, no combination is possible, except for a particular case like x = 1. It is straightforward to show that, [*in general*]{}, we have :
$$\label{6}
x {{\times }_a} y + x {{\times }_a} z \neq x {{\times }_a} (y + z)$$
$$\label{7}
x y {+_a} x z \neq x (y {+_a} z)$$
$$\label{8}
x {{\times }_a} y {+_a} x {{\times }_a} z \neq x {{\times }_a} (y {+_a} z)$$
This non distributivity has important consequences for the manipulation of analytical expressions. The most important one is the impossibility to develop or factorize expressions.
On the other hand, distributivity can be recovered by defining different generalized addition and product operators as has been done in [@Kaniadakis] :
$$\label{9}
\ln[e_a^{x{\times^a}y}] = \ln e_a^x\ln e_a^y$$
and $$\label{10}
e^{[\ln_a[x{+^a}y]} = e^{\ln_ax} + e^{\ln_a y}.$$ The distributivity can be established with either $\times^a$ and $+_a$ or $\times_a$ and $+^a$, i.e. : $$\label{11}
x {\times^a} y +_a x{\times^a}z = x {{\times }^a} (y +_ a z)$$ or $$\label{12}
x {\times_a} y +^a x{\times_a}z = x {\times_ a} (y +^a z).$$ It should be noted that the distributivity does not exist neither between the $\times^a$ and $+ ^a$ defined in Eqs.(\[9\]) to (\[10\]), i.e. : $$\label{13}
x {\times^a} y +^a x{\times^a}z \neq x {\times^a} (y +^a z),$$ nor between these two operators and the ordinary operators.
The operators $\times^a$ and $+^a$ are respectively given by $$\label{14}
x {\times^a} y = \frac{e^{\ln(1+ax)\ln(1+ay)/a}-1}{a}$$ and $$\label{15}
x {+^a} y = [a\ln(e^{x^a/a}+e^{y^a/a})]^{1/a},$$ which are different from $\times_a$ and $+_a$ and have different properties. For example, we have here $x {+^a}0\neq x$, $x {\times^a} 1\neq x$ and $x {\times^a} 0 = 0$.
Examples of simple application
==============================
It is obvious that, if we use this generalized algebra with the operators defined in Eqs.(\[a\]) to (\[5a\]), the nonadditive statistical mechanics and thermodynamics will be able to be expressed in additive form just as in BGS. In one of our papers[@Wang02c], NSM is given in additive form by deforming the nonadditive energy and entropy. The same formalism can be given with generalized algebra without deforming the nonadditive physical quantities. That is, for a total system composed of two subsystems $A$ and $B$, we can write for entropy $S(A+B)=S(A)+_{a_S}S(B)$ and for energy $U(A+B)=U(A)+_{a_U}U(B)$ where $a_S=-\lambda_S=(1-q)/k$ and $a_U=-\lambda_U=(1-q)/kT$ for NSM.
Let us consider the temporal evolution of the price of a product at successive 1st of January. It costs respectively $x_0 = 100\$$ in 2001, ${x_1} = 110\$$ in 2002 and ${x_2}=
121\$$ in 2003. The annual evolution ratio is ${y_0} = ({x_1} - {x_0} )/{x_0} = 0,1$ and ${y_1} = ({x_2} - {x_1} )/{x_1} = 0.1$, or $10\%$ each time. The global evolution ratio over two years is $y'_0 = ({x_2} - {x_0} )/{x_0} = 0.21$, that is $21\%$. Of course $y'_0\neq y_0+y_1$ because the ratios are not based upon the same denominator. However, simple calculation leads to $y'_0=y_0+y_1+y_0 y_1$. This expression is rewritten with the help of the generalized addition operator defined in Eq.(\[a\]) : $y'_0 = y_0 +_1 y_1$. It means that combinations of ratios can be considered as additive by using the generalized addition. In this case, $a=1$ is a universal value. This can be shown for whatever ratios as follows. From Eq.(\[a\]), we have $a=(\frac{x_2-x_0}{x_0}-\frac{x_1-x_0}{x_0}-\frac{x_2-x_1}{x_1})/\frac{(x_1-x_0)(x_2-x_1)}{x_0x_1}
=\frac{x_1-x_0}{x_1-x_0}=1$. This calculation can be useful for other temporal process.
Conclusion
==========
In this paper some algebraic aspects of the nonadditive statistics have been studied. The four classical operators of the usual algebra have been generalized. The new operators allow to preserve the morphism properties of the exponential and logarithm functions. Some properties (associativity, commutativity, existence of neutral element and the sign rules) of the standard operators can be extended to the generalized ones. But others cannot, e.g. the opposite element does not exist for arbitrary element; and the generalized additions and multiplications are not always distributive. Another interesting point should be noted : in the generalized formalism, “0" is no more the absorbing element of multiplication but can be inverted just like other element.
We hope that this generalized algebra can be helpful for the understanding and the development of nonadditive physics which seems inevitable in view of the complex phenomena that cannot be described within the additive statistical theory.
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---
abstract: 'This paper describes the application of a Progressive Hedging (PH) algorithm to the least-cost var planning under uncertainty. The method PH is a scenario-based decomposition technique for solving stochastic programs, i.e., it decomposes a large scale stochastic problem into s deterministic subproblems and couples the decision from the s subproblems to form a solution for the original stochastic problem. The effectiveness and computational performance of the proposed methodology will be illustrated with var planning studies for the IEEE 24-bus system (5 operating scenarios), the 200-bus Bolivian system (1,152 operating scenarios) and the 1,600-bus Colombian system (180 scenarios).'
author:
- Igor Carvalho
- Tiago Andrade
- Joaquim Dias Garcia
- Maria de Lujan Latorre
bibliography:
- 'myref.bib'
date: 'April 17, 2020'
subtitle:
title: Application of Progressive Hedging to Var Expansion Planning Under Uncertainty
---
Introduction {#Intro}
============
The fast worldwide insertion of variable renewable energy (VRE) sources such as wind and solar has brought substantial economic and environmental benefits. However, it has also created some technical challenges, for instance, the need for additional generation reserve and/or storage to manage the VRE intermittency. In many systems, VRE sources also disrupted the usual power flow patterns (around which the transmission networks were designed), creating congestions and voltage issues. The solution of these network problems may require substantial investments in new lines, FACTS devices and var sources. Due to the complexity of real networks (size, nonconvexity, discrete variables, uncertainty on load and generator production etc.), transmission planners usually apply a two-step hierarchical approach.
The first step determines the least-cost reinforcements of circuits and transformers that eliminate overloads for many operational scenarios (robust optimization). This planning step uses a linearized active optimal power flow (OPF) representation. Because the linearized OPF is convex, it is also possible to use Benders decomposition [@latorre2019stochasticrobust] to handle the multiple operational scenarios (see Fig. \[fig: first\_step\]).
![First step of the planning process – circuit/transformer reinforcements (Benders decomposition). Source: [@latorre2019stochasticrobust][]{data-label="fig: first_step"}](fig_first_step.png){width="5in"}
Given the active power reinforcements from the first step, the second step - which is the focus of this paper - determines the least cost var reinforcements ensuring that voltage levels remain within their limits for the same set of operational scenarios used previously [@vaahedi2001dynamic]. This var planning step is arguably more challenging than the first step for two reasons: (i) it requires the solution of AC OPFs, which are nonlinear nonconvex optimization problems; (ii) because of nonconvexities, many decomposition techniques cannot be directly applied.
This paper describes the application of a Progressive Hedging (PH) algorithm [@wets1989aggregation] to the least-cost var planning under uncertainty. Similarly to Benders decomposition [@benders1962partitioning], PH is based on the separate solutions of AC OPF [@granville1988integrated] subproblems for each scenario. The difference is that, instead of an investment module, there is an update in the OPF objective functions. The PH has already been applied successfully to other related problems such as unit commitment [@ryan2013toward; @gade2016obtaining], transmission and generation planning [@munoz2015scalable], and a dynamic deterministic OPF [@ruppert2018dynamic], but to the best of our knowledge, it has not been applied yet to the stochastic var expansion problem.
Although global optimality cannot be guaranteed in the case of nonconvex subproblems, PH has attracted our interest for the following reasons: (i) it is straightforward to implement; (ii) each subproblem is a (slightly modified) OPF, which can be solved by effective techniques such as nonlinear interior point methods [@granville1994optimal]; (iii) the OPF subproblems can be solved in parallel; (iv) it has recently become possible to calculate lower bounds to the optimal solution through convex relaxations; and (v) as will be illustrated in this paper, the resulting var plans are near-optimal. Fig. \[fig: PH\] shows the proposed PH scheme.
![PH procedure for var planning under uncertainty.[]{data-label="fig: PH"}](fig_PH_uncertainty.png){width="5in"}
The proposed algorithm is be detailed described in Section \[sec: PH\]. Then we present a case study in Section \[sec: study\], where the methodology is initially illustrated for the IEEE 24-bus test case, and then applied to realistic studies of the Bolivian and Colombian power systems. Finally, conclusions are drawn in Section \[sec: conclusions\].
Progressive Hedging Algorithm {#sec: PH}
=============================
The PH scheme is implemented in the following steps:
1. Initialize the PH linear and quadratic terms
2. Solve the modified OPF problems for scenarios $s \in 1,\dots, S$. For completeness, we highlight the non-linear OPF model: \[step: Ph\_b\]
**Objective function**: $$\begin{aligned}
\min & \sum_k [I_k^R Q_{k,(s)}^{C,INV} + I_k^C Q_{k,(s)}^{C,INV} \\
& + w_{k,(s)}^{R} (Q_{k,(s)}^{R,INV} - Q_{k}^{R,AVE}) + w_{k,(s)}^{C} (Q_{k,(s)}^{C,INV} - Q_{k)}^{C,AVE}) \\
& + \rho_k^{R} (Q_{k,(s)}^{R,INV} - Q_{k}^{R,AVE})^2 /2 + \rho_k^{C} (Q_{k,(s)}^{C,INV} - Q_{k}^{C,AVE})^2 /2]
\end{aligned}$$
where:
$I_k^R$ and $I_k^C$ are, respectively, the unit investment costs (\$/Mvar) of reactor and capacitor banks in bus k;
$Q_{k,(s)}^{R,INV}$ and $Q_{k,(s)}^{C,INV}$ are the investment decisions: reactor and capacitor sizes (Mvar) for each scenario s.;
$w_{k,(s)}^R$ and $w_{k,(s)}^C$ are the weights of the linear PH terms;
$\rho_k^R$ and $\rho_k^R$ are the weights of the quadratic PH terms.
As seen in the equation above, the PH terms are related to the linear and quadratic differences between the var investment decisions for each scenario $s$, $Q_{k,(s)}^{R,INV}$ and $Q_{k,(s)}^{C,INV}$, and “target” investments $Q_{k)}^{R,AVE}$ and $Q_{k}^{C,AVE}$ (known values).
**Capacity limits on var injections**: $$\begin{aligned}
0 \leq Q_{k,(s)}^{R,INJ} \leq Q_{k,(s)}^{R,INV} \\
0 \leq Q_{k,(s)}^{C,INJ} \leq Q_{k,(s)}^{C,INV}
\end{aligned}$$
where $Q_{k,(s)}^{R,INJ}$ and $Q_{k,(s)}^{R,INJ}$ are the operational decisions. They are the decision variables to choose how much power will be injected in each bus $k$ for each scenario $s$. They are limited by how much was invested.
**AC power flow equations and constraints**: [ $$\begin{aligned}
P_{k,(s)}^{GEN} - P_{k,(s)}^{DEM} - \sum_{j \in \Omega_k} P_{k,j,(s)}^{FLW} (v_k, v_j, \theta_k, \theta_j, t_{k,j}, \phi_{k,j}) = 0 \\
Q_{k,(s)}^{GEN} - Q_{k,(s)}^{DEM} - \sum_{j \in \Omega_k} Q_{k,j,(s)}^{FLW} (v_k, v_j, \theta_k, \theta_j, t_{k,j}, \phi_{k,j}) + Q_{k,(s)}^{C,INJ} - Q_{k,(s)}^{R,INJ} = 0
\end{aligned}$$ ]{} $$\begin{aligned}
\sqrt{{P_{k,j,(s)}^{FLW}}^2 + {Q_{k,j,(s)}^{FLW}}^2} \leq \bar S_{k,j}
\end{aligned}$$
**Variable bounds**: $$\begin{aligned}
\underline Q_{k,(s)}^{GEN} & \leq Q_{k,(s)}^{GEN} \leq \bar Q_{k,(s)}^{GEN} \\
\underline v_k & \leq v_k \leq \bar v_k \\
\underline t_k & \leq t_k \leq \bar t_k \\
\underline \phi_k & \leq \phi_k \leq \bar \phi_k
\end{aligned}$$
where:
$P_{k,(s)}^{GEN}$ is the active power generation at bus $k$ (known value, taken from the probabilistic simulation of system operation for scenario $s$, see Fig. \[fig: PH\]);
$Q_{k,(s)}^{GEN}$, reactive power generation at k (decision variable, with limits $\underline Q_{k,(s)}^{GEN}$ and $\bar Q_{k,(s)}^{GEN}$);
$P_{k,(s)}^{DEM}$ and $P_{k,(s)}^{DEM}$ active and reactive loads at bus k, also known values, taken from the same simulation;
$P_{k,j,(s)}^{FLW}$ and $Q_{k,j,(s)}^{FLW}$ active and reactive power flows in circuit k-j, with limit $\bar S_{k,j}$;
$\theta_k$ and $\theta_j$ voltage angles at buses $k$ and $j$;
$t_{kj}$, tap position value for transformer $k-j$;
$v_k$, voltage at bus $k$, with limits $\bar v_k$ and $\underline v_k$;
$\phi_{kj}$ phase shifting angle in circuit $k-j$, with limits $\bar \phi_{kj}$ and $\underline \phi_{kj}$;
3. Let $ \{(\tilde q_k^{rs},\tilde q_k^{cs}), k \in 1,\dots,K \}$ for $s \in 1,\dots, S$ be the optimal solutions of the modified OPF problems of step \[step: Ph\_b\]; carry out the following updates:
$$\begin{aligned}
Q_{k}^{R,AVE} \leftarrow \sum_s Pr(s) Q_{k,(s)}^{R,INV} \\
Q_{k}^{C,AVE} \leftarrow \sum_s Pr(s) Q_{k,(s)}^{C,INV}
\end{aligned}$$
$$\begin{aligned}
w_{k,(s)}^R \leftarrow w_{k,(s)}^R + \delta_{k)}^{R} \rho_k^r (Q_{k,(s)}^{R,INV} - Q_{k}^{R,AVE}) \\
w_{k,(s)}^C \leftarrow w_{k,(s)}^C + \delta_{k)}^{C} \rho_k^r (Q_{k,(s)}^{C,INV} - Q_{k}^{C,AVE})
\end{aligned}$$
where $\delta_{k)}^{R}$ and $\delta_{k)}^{C}$ are user-defined step sizes, and $Pr(s)$ is vector of the occurrenc probability of each scenario s. It is important to highlight that this step size is a contribution to this paper since the standard PH use an implicit step fixed at 1. In the present work, all scenarios were considered as equiprobable. If the probabilities of the scenarios were different, the method would work in the same way, where the difference consists of the decision variables tending to the value resulting from the weighted average with different weights.
4. In case of convergence, stop; otherwise go to \[step: Ph\_b\].
Penalty Values Calculations
---------------------------
The PH performance is based on the chosen values for $\rho_k^R$ and $\rho_k^C$. With larger values, convergence is faster; on the other hand, there may be an oscillatory behavior. Conversely, smaller values lead to a steadier, but slower, convergence. In this work, $\rho_k^R$ and $\rho_k^C$ were made proportional to the respective investment costs $I_k^R$ and $I_k^C$.
$$\begin{aligned}
\rho_k^R = K \cdot I_k^R \\
\rho_k^C = K \cdot I_k^C
\end{aligned}$$
where $K$ is a constant chosen to implement variations on the penalty value for each simulation.
Convergence
-----------
Two methods are used to measure convergence in this paper. The first one is the usage of a normalized average per-scenario deviation (NApSD) [@watson2011progressive] from the reference value detects the proximity to the standard convergence of the method:
$$\begin{aligned}
NApSD_{(i)}^{R} = \frac{1}{s}\sum_{k,s | Q_{k}^{R,AVE} > 0} \frac{| Q_{k,(s),(i)}^{R,INV} - Q_{k,(i-1)}^{R,AVE} |}{Q_{k,(i-1)}^{R,AVE}} \\
NApSD_{(i)}^{C} =\frac{1}{s}\sum_{k,s | Q_{k,(i-1)}^{C,AVE} > 0} \frac{| Q_{k,(s),(i)}^{C,INV} - Q_{k,(i-1)}^{C,AVE} |}{Q_{k,(i-1)}^{C,AVE}}\end{aligned}$$
The NApSD is calculated for each decision variable and varies at each iteration. Note that the index $i$ indicates the iteration and $i-1$ the previously iteration. As shown above, the NApSD is the average of the relative differences between the investment for scenario $s$ and the average of the investments from the previous iteration. The second method to detect convergence is a relative duality gap computed with valid upper ($UB$) and lower bounds ($LB$). The duality gap is defined as $\frac{UB - LB}{UB}$.
The upper bound is constructed by the union of the investments of all scenarios since an over investment does not harm feasibility. Moreover, Gade et al. [@gade2016obtaining] showed how to obtain a lower bound using the PH that is equivalent to the bound provided by the Lagrangian relaxation of the problem that we use in this paper. Since the problem is nonconvex, a zero duality gap could be assured only in special cases [@lavaei2011zero]. The average of these problems is equivalent to a Lagrangian relaxation of the full space problem, thus, generating a lower bound (LB). There are alternative methods to obtain a LB, such as [@boland2018combining]. The method described in [@gade2016obtaining] was implemented since it is simpler. The LB is obtained by solving the PH problem replacing the objective function by the following expression and taking the average for all scenarios.
$$\begin{aligned}
\min & \sum_k [I_k^R Q_{k,(s)}^{C,INV} + I_k^C Q_{k,(s)}^{C,INV} \\
& + w_{k,(s)}^{R} (Q_{k,(s)}^{R,INV} - Q_{k}^{R,AVE}) + w_{k,(s)}^{C} (Q_{k,(s)}^{C,INV} - Q_{k}^{C,AVE})]
\end{aligned}$$
Two additional stop criteria were also considered. The first one consists on verifying the OPF’s convergence for all scenarios under analysis. If any OPF has not converged, the iterative process must stop. The model checks information from the solver if each one of the OPFs has converged or not at each iteration of the method. The other one is for exceeding the maximum number of iterations as established before the method’s execution (timeout). The analysis of simulations with the system is required before defining a good value for this parameter.
Case study {#sec: study}
==========
The present research considers the application of the methodology described for the IEEE 24-bus test case (10 simulations), and also for the electrical power systems from two southamerican countries: Bolivia (5 simulations) and Colombia (1 simulation).
PSR’s software Optflow [@optflowmanual] was used to run the optimal power flow simulations and served as a basis for the PH implementation. The OptFlow model is based on the primal-dual interior point algorithm [@granville1994optimal], which have recognized efficiency to solve non convex problems with a large number of variables and constraints. The computer used to run the 10 simulations for the present work consists of an Intel® Core ™ i7-7700K CPU 4.20 GHz with 64 GB of installed RAM.
Artelys Knitro [@nocedal2006knitro] is the local nonlinear solver chosen to obtain the solutions from the OPF previously presented. It solves nonlinear problems by considering integer or continuous variables and is prepared for multiple objective functions and nonlinear constraints.
Since the continuous problem is already complex to solve due to its nonconvexity, we avoided including integer variables into the model. The required investment is decided by choosing how much Mvar is needed at each bus of the system. After the continuous problem is solved, a post-processing is done to choose which shunt equipment should be installed with predetermined capacities and costs.
The list of candidate buses was previously obtained from a process of shunt allocation. Initially, the model is used to identify candidate buses for reactive support, minimizing the reactive power injections in the system, where we discover which buses are in need of reactive support. Results from simulations with IEEE 24-Bus case illustrated the importance of this step.
The cost considered for each var equipment came from a report from Ente Operador Regional (EOR) [@eor]. The values considered are illustrated in Table \[tab: var cost\].
------------------- ----------------
Equipament Cost \[kUS\$\]
5 Mvar Capacitor 313
10 Mvar Capacitor 362
15 Mvar Capacitor 418
15 Mvar Reactor 1,810
60 Mvar Reactor 2,171
Connection Bay 3,217
------------------- ----------------
: Var sources cost[]{data-label="tab: var cost"}
IEEE 24-Bus System
------------------
The list of parameters considered for simulations with the present system is:
- Maximum shunt allocation for each bus: 500 Mvar;
- 5 different generation/load set-points considered at each iteration;
- Maximum number of iterations: 100;
- Convergence gap: 1%
With the IEEE 24-Bus database, the first two simulations were done considering ($\delta_k^{rs}$,$\delta_k^{cs}$) = 0 and K = 1:
- **Simulation 1:** Reactive investment is allowed at all buses from the system (Full set of shunt candidates);
- **Simulation 2:** Reactive investment is allowed only at the buses that presented reactive investment in iteration 0 (Reduced set of shunt candidates)
The comparison of the reactive requirement between these two simulations is presented in Fig. \[fig: 24-bus Mvar\].
![Mvar requirement reduction for simulations 1 and 2[]{data-label="fig: 24-bus Mvar"}](fig_simulation_1_2_v2.png){width="5in"}
It is observed in the Fig. \[fig: 24-bus Mvar\] that, with the reduced set of candidates, the total amount of shunt requirement reaches slightly smaller values than those from **Simulation 1**. As the results presented were better for the simulations with a reduced set of candidates, for the next simulations presented in this paper, this reduced set will always be considered.
The comparison of the total system’s reactive requirement along the iterative process for the five simulations above is illustrated in Fig. \[fig: 24-bus Mvar 2-6\]. It is observed that for **Simulation 5** and **Simulation 6** (step-sizes 0.1 and 1, respectively), the problem becomes maximizing instead of minimizing in the early first iterations. It occurs because the linear term in the objective function becomes negative in such a way that all objective function becomes negative. Then all reactive investment possible (4,000 MVAr) is allocated. For better comparison between the simulations from this set, **Simulation 5** and **Simulation 6** were both excluded from Fig. \[fig: 24-bus Mvar 2-6\].
![Mvar requirement reduction for simulations 2-6[]{data-label="fig: 24-bus Mvar 2-6"}](fig_simulation_2_6.png){width="5in"}
Fig. \[fig: 24-bus Mvar 2-4\] illustrates the simulations which reduced the system’s requirement from iteration zero. It is observed that **Simulation 4** stops before the timeout, because it converges from the duality gap criterium, as illustrated in Fig. \[fig: 24-bus LB UB sim4\].
![Mvar requirement reduction for simulations 2-4[]{data-label="fig: 24-bus Mvar 2-4"}](fig_simulation_2_4.png){width="5in"}
![LB-UB Convergence scheme – IEE 24-Bus.[]{data-label="fig: 24-bus LB UB sim4"}](fig_ieee24_gap_sim4.png){width="5in"}
Note that using duality criterion convergence is reached (1% duality gap) while the NApSD criterion is 40% as illustrated in Fig. \[fig: 24-bus LB UB sim4\]. **Simulation 4** converges from the duality gap, and **Simulation 2** does not converge from NApSD. As step-size is null at **Simulation 2**, it is not possible to compute lower bounds, and therefore, the duality gap is not applicable for this simulation. About **Simulation 3**, the convergence is slower, as illustrated in Fig. \[fig: 24-bus LB UB sim3\], and it is concluded that this step-size is, on the other hand, too small.
![LB-UB Convergence scheme – IEE 24-Bus.[]{data-label="fig: 24-bus LB UB sim3"}](fig_ieee24_gap_sim3.png){width="5in"}
From the previous analyses, it is concluded that the step-size equals to 0.01 is the best option to proceed with further simulations and analyses. However, to reinforce the idea, the same analyses with the step size parameter will be repeated with the Bolivian system in the next section.
Fig. \[fig: 24-bus of\] highlights the reductions on total amount invested compared to the solution obtained without PH (iteration 0). Some oscillatory behavior is observed from variations with weights at each iteration. It is essential to highlight that all solutions are feasible ones, and at the end of the process, the one with the smallest total system’s var requirement is chosen.
![Reduction in objective function, **Simulation 4**– IEEE 24-Bus[]{data-label="fig: 24-bus of"}](fig_ieee24_reduction.png){width="5in"}
The comparison of the best total system’s reactive requirement found so far along the iterative process for the five simulations above is illustrated in Fig. \[fig: 24-bus Mvar 4 7-10\]. It is observed that K = 0.5 or 1 results in similar total system’s var requirement. As the constant K rise from 1 to 5, the local optimal solution found is slightly better. The final solution obtained from simulations 11 and 12 (K = 5 and K = 10, respectively) were also similar.
![Mvar requirement reduction for simulations 4, 7-10[]{data-label="fig: 24-bus Mvar 4 7-10"}](fig_simulation_4_7-10.png){width="5in"}
Table \[tab: CPU Time - IEEE 24\] summarizes the total CPU time for the whole iterative process from the 10 simulations analyzed for this system.
------------ ------------- ------- ----- ---------------------- ----------------------- ------------------
Simulation Candidates t K Number of iterations Total var requirement CPU Time \[min\]
1 Full set 0 1 100 1,009 4.1
2 Reduced set 0 1 100 1,006 4.1
3 Reduced set 0.001 1 100 1,007 8.3
4 Reduced set 0.01 1 34 1,006 2.8
5 Reduced set 0.1 1 100 1,129 8.1
6 Reduced set 1 1 100 1,129 8.1
7 Reduced set 0.01 0.5 52 1,006 4.3
8 Reduced set 0.01 2 24 1,004 2.0
9 Reduced set 0.01 5 26 1,002 2.1
10 Reduced set 0.01 10 36 1,002 3.0
------------ ------------- ------- ----- ---------------------- ----------------------- ------------------
: Total CPU Time - IEEE 24 Bus[]{data-label="tab: CPU Time - IEEE 24"}
From Table \[tab: CPU Time - IEEE 24\] it is observed that considering a full set or a reduced set of candidates makes no visible change in computational effort. It is also observed that with the introduction of the step size, the CPU time after 100 iterations doubles. It happens because another problem is required to be solved to compute lower bounds from the duality gap. Then, with a step size different than zero, two problems are solved at each iteration. With the null step size, only one problem is solved, which explains the rise in the CPU time observed in Table \[tab: CPU Time - IEEE 24\]. The simulations considering a value different than zero for the step-size which presented a CPU time much lower than the others, it is because it has converged from the duality gap criterium in the iteration indicated. It is also observable that the value of the penalty parameter seems to have no interference in the CPU Time required to solve the problem with the method’s application.
Bolivian Electrical Power System
--------------------------------
The list of parameters considered for simulations with the present system is:
- Maximum shunt allocation for each bus: 500 Mvar;
- 1,152 different generation/load set-points considered at each iteration;
- Maximum number of iterations: 30;
- Convergence gap: 1%
With the Bolivian system database, two sets of simulations were done in order to evaluate the impact of the step-size and the penalty parameter, as previously done with the IEEE 24-bus system. Then, as stated before, all simulations with the Bolivian system considered a reduced set of shunt candidates. First, in order to evaluate the impact of the step-size, three simulations were done with K=1. The set to compare now is:
- **Simulation 11:** ($\delta_k^{rs}$,$\delta_k^{cs}$) = 0.005;
- **Simulation 12:** ($\delta_k^{rs}$,$\delta_k^{cs}$) = 0.01;
- **Simulation 13:** ($\delta_k^{rs}$,$\delta_k^{cs}$) = 0.1;
The comparison of the reactive requirement between these three simulations is presented in Fig. \[fig: bol\_stepanalysis\].
![Mvar requirement reduction in % for simulations 11-13 – Bolivia[]{data-label="fig: bol_stepanalysis"}](fig_bolivia_stepanalysis.png){width="5in"}
It is observed in Fig. \[fig: bol\_stepanalysis\] that with a larger step-size (**Simulation 13**), the weight vectors become negative and higher than the cost in absolute value. It turns the objective function to maximize instead of minimizing the investment in shunt equipment, as observed in the analyses with IEEE 24 bus system. For better comparison, **Simulation 13** was excluded from the Fig. \[fig: bol\_stepanalysis\], and the “simulations with minimization” are illustrated in Fig. \[fig: bol\_stepanalysis2\]. It is observed that the smaller step-sizes used for **Simulation 11** and **Simulation 12** resulted in quite similar amount reductions, where the step-size of 0.01 from **Simulation 12** presented slightly better results.
![Mvar requirement reduction in % for simulations 11-12 – Bolivia[]{data-label="fig: bol_stepanalysis2"}](fig_bolivia_stepanalysis2.png){width="5in"}
In the first iterations of the method, great reductions on the total system’s var requirement are obtained. Fig. \[fig: bol\_reduction\] illustrates the reductions on total amount invested along the iterative process compared to the solution without progressive hedging (iteration 0) for **Simulation 12**.
![Reductions in objective function – Bolivia[]{data-label="fig: bol_reduction"}](fig_bolivia_reduction.png){width="5in"}
Fig. \[fig: bol\_gap\] illustrates the convergence of **Simulation 12**. Note that duality gap is 85% while the NApSD criterion is 93%. The approximation between LB and UB are very slow, which wait for convergence would take days of simulation, if it reaches convergence. An approach to accelerate the process aiming the convergence is the parallelization of the simulations or better, the clusterization of the 1,152 operative scenarios considered, reducing the number of scenarios to be solved at each iteration.
![LB-UB Convergence scheme – Bolivia[]{data-label="fig: bol_gap"}](fig_bolivia_gap.png){width="5in"}
The comparison of the reactive requirement between these three simulations is presented in Fig. \[fig: bol\_penaltyanalisys\]. It is observed that as the parameter K grows, the more significant is the reduction in the total amount of var required for the first iterations. However, the three simulations seem to stabilize in the same “reduction amount” by the end of the iterative process.
![Mvar requirement reduction in % for simulations 12, 14 and 15 – Bolivia[]{data-label="fig: bol_penaltyanalisys"}](fig_bolivia_penaltyanalysis.png){width="5in"}
The Table \[tab: CPU Time - Bolivia\] summarizes the total CPU time for the whole iterative process from the 5 simulations analyzed for this system. It is observed that all simulations took basically the same computational effort and stopped by the timeout criterium, except **Simulation 13**, which stopped before 30 iterations because one of the OPFs in the 23rd iteration did not converge. As the number of iterations rise, the CPU time rises as expected, and compared to the simulations with the IEEE 24-Bus, it is observable that the rise in number of scenarios and size of the system makes the total CPU time to rise. 30 iterations for 5 scenarios with the IEEE 24-Bus system took something between 2 and 3 minutes and here takes more than 10 hours. It is also observable that varying the step-size and k seems to have no interference in the CPU Time required to solve the problem with the method’s application.
------------ ------- ----- ---------------------- ----------------------- ------------------
Simulation t k Number of iterations Total var requirement CPU Time \[min\]
11 0.005 1 30 261 634
12 0.01 1 30 258 638
13 0.1 1 22 486 459
14 0.01 0.5 30 261 637
15 0.01 2 30 256 635
------------ ------- ----- ---------------------- ----------------------- ------------------
: Total CPU Time - Bolivian system[]{data-label="tab: CPU Time - Bolivia"}
Colombian Electrical Power System
---------------------------------
The list of parameters considered for simulations with the present system is:
- Maximum shunt allocation for each bus: 1500 Mvar;
- 180 different generation/load set-points considered at each iteration;
- Maximum number of iterations: 4;
- Convergence gap: 1%
The database for the present simulation came from a real and complete system expansion study. First, a generation expansion plan was obtained for the horizon (2018-2040) [@latorre2019stochasticrobust; @pereira1991multi; @campodonico2003expansion; @fern2019stochastic]. Then, it was obtained the transmission expansion plan for the active power transport (reinforcements of transmission lines and transformers). Finally, the reactive power expansion could be then realized after generation and transmission expansions were concluded. For the present simulation, it was considered the analysis for var expansion system for year 2021 (first year of var planning analysis from the study).
The simulation with the Colombian system considered a reduced set of candidates, ($\delta_k^{rs}$,$\delta_k^{cs}$) = 0.01, K = 1 and stopped by timeout criterium (4 iterations in 114 minutes of model’s execution). From PH’s application, it is observed that in the first iteration of the method there is a reduction of approximately 27%, which remains practically constant for the next iterations. Table \[tab: col solution\] presents the var requirement at each bus as the convex hulls from the solutions of iterations 0 and 1.
--------------------------- ------------- -------------
Equipment Iteration 0 Iteration 1
Reactor at SE Rio Grande 10 Mvar 0 Mvar
Reactor at SE Santa Rosar 11.8 Mvar 11.8 Mvar
Capacitor at SE Mompox 10 Mvar 10 Mvar
Capacitor at SE Tumaco 9.3 Mvar 9.3 Mvar
--------------------------- ------------- -------------
: Solutions for the simulation with the Colombian Eletrical Power System[]{data-label="tab: col solution"}
From iteration 0 to iteration 1, it was no longer required the investment in 10 Mvar shunt reactor at SE Rio Grande. This requirement was only in two scenarios (approx. 1% of the scenarios). Since the PH method induces the variables to approach the reference value set at each iteration, the solution for these two scenarios was the reallocation of the reactor requirement to SE Santa Rosa.
Therefore, PH method’s application resulted in big savings (5 MUS\$), reducing the total cost of the var expansion plan by 29%.
Conclusions {#sec: conclusions}
===========
Currently, due to the high complexity and non-convexity of the optimal nonlinear power flow problem, the analysis of the var expansion planning is done individually for each scenario. In this way, the final investment decision is subject to a superposition of the individual solutions resulting from these scenarios, leading to high expansion costs for the system.
Although it does not guarantee global optimality, the methodology proposed in this work finds a solution that couples investment decisions, referring to each deterministic problem, by imposing a reference value for final decision making. As can be seen from the results, the coupling of the decisions and the proposed methodology resulted in positive results for the var expansion planning problem by reducing the total amount of investments required in the network, meeting all operational requirements.
From simulations with IEEE 24-Bus database, it was concluded, for this particular case, that considering a reduced set of shunt candidates, led to faster convergence and no change in the optimal solution. It was observed that with the introduction of the step-size, the computational effort required doubles, as other optimization problem is required to calculate the lowerbound at each iteration.
It was also observed from simulations with IEEE 24-Bus and Bolivian system that the chosen parameters can change the way until an optimal solution is achieved (when achieved). With ($\delta_k^{rs}, \delta_k^{rs}$) = 1, as it is usually seen in the literature, the term $w_k^{rs}$ ($q_k^{rs}$ - $q_k^r$) + $w_k^{cs}$ ($q_k^{cs}$ - $q_k^c$) becomes negative in such way that all objective function is negative, and the problem starts to maximize shunt investment instead of minimizing it.
Finally, the simulation from Colombian system presented real gains from the method, where the decision of investing in a 15 Mvar reactor was dismissed, saving 5 MUS\$.
|
---
author:
- |
Jonah Brown-Cohen\
[University of California, Berkeley]{}\
[California, Berkeley]{}\
`[email protected]`
- |
Prasad Raghavendra [^1]\
[University of California, Berkeley]{}\
[California, Berkeley]{}\
`[email protected]`
bibliography:
- 'bib/papers.bib'
title: Combinatorial Optimization Algorithms via Polymorphisms
---
=1 [ ]{}
\[sec:linearity\]
\[sec:corr-decay\]
Approximate Polymorphisms for CSPs {#sec:maxcsps}
==================================
[^1]: Supported by NSF Career Award and Alfred. P. Sloan Fellowship
|
---
author:
- 'G. W. Fuchs, H. M. Cuppen, S. Ioppolo, C. Romanzin, S. E. Bisschop, S. Andersson, E. F. van Dishoeck, and H. Linnartz'
date: 'Received ; accepted '
subtitle: 'a combined experimental/theoretical approach'
title: Hydrogenation reactions in interstellar CO ice analogues
---
[Hydrogenation reactions of CO in inter- and circumstellar ices are regarded as an important starting point in the formation of more complex species. Previous laboratory measurements by two groups on the hydrogenation of CO ices resulted in controversial results on the formation rate of methanol (2002, ApJ, 577, 265 and 2002, ApJL, 571, L173). ]{} [Our aim is to resolve this controversy by an independent investigation of the reaction scheme for a range of H-atom fluxes and different ice temperatures and thicknesses. In order to fully understand the laboratory data, the results are interpreted theoretically by means of continuous-time, random-walk Monte Carlo simulations. ]{} [Reaction rates are determined by using a state-of-the-art ultra high vacuum experimental setup to bombard an interstellar CO ice analog with room temperature H atoms. The reaction of CO + H into H$_2$CO and subsequently CH$_3$OH is monitored by a Fourier transform infrared spectrometer in a reflection absorption mode. In addition, after each completed measurement a temperature programmed desorption experiment is performed to identify the produced species according to their mass spectra and to determine their abundance. Different H-atom fluxes, morphologies, and ice thicknesses are tested. The experimental results are interpreted using Monte Carlo simulations. This technique takes into account the layered structure of CO ice. ]{} [ The formation of both formaldehyde and methanol via CO hydrogenation is confirmed at low temperature ($T = 12-20$ K). We confirm, as proposed by Hidaka et al. (2004, ApJ, 614, 1124), that the discrepancy between the two Japanese studies is mainly due to a difference in the applied hydrogen atom flux. The production rate of formaldehyde is found to decrease and the penetration column to increase with temperature. [Temperature-dependent ]{}reaction barriers and diffusion rates are inferred using a Monte Carlo physical chemical model. The model is extended to interstellar conditions to compare with observational H$_2$CO/CH$_3$OH data.]{}
Introduction
============
In recent years an increasing number of experimental and theoretical studies have been focussing on the characterisation of solid state astrochemical processes. These studies were triggered by the recognition that many of the simple and more complex molecules in the interstellar medium are most likely formed on the surfaces of dust grains. Astronomical observations along with detailed laboratory studies and recent progress in UHV surface techniques have made possible an experimental verification of the initial surface reaction schemes introduced by Tielens, Hagen and Charnley [@Tielens:1982; @Tielens:1997]. Very recently the formation of water was shown in hydrogenation schemes starting from solid molecular oxygen [@Miyauchi:2008; @Ioppolo:2008] and that of ethanol from acetaldehyde [@Bisschop:2007I]. The first solid state astrochemical laboratory studies focused on the formation of formaldehyde and methanol by H-atom bombardment of CO ice. Methanol is abundantly observed in interstellar ices and is considered to be a resource for the formation of more complex molecules via surface reactions and after evaporation in the gas phase [@Charnley:1992]. The hydrogenation scheme for the solid state formation of methanol was proposed as $$\rm
CO \xrightarrow{H} HCO \xrightarrow{H} H_2CO \xrightarrow{H} H_3CO \xrightarrow{H} CH_3OH$$
The past laboratory studies of H-atom bombardment of CO ice have been performed independently by two groups [@Hiraoka:2002; @Watanabe:2002]. [@Hiraoka:2002] observed only formaldehyde formation, whereas [@Watanabe:2002] also found efficient methanol production. In a series of papers these conflicting results have been discussed [@Hiraoka:2002; @Watanabe:2003; @Watanabe:2004] and the existing discrepancy has been proposed as a consequence of different experimental conditions, most noticeable the adopted H-atom flux [@Hidaka:2004]. Understanding the solid state formation route to methanol became even more pressing with the recent experimental finding that the gas-phase formation route via ion-neutral reactions is less efficient than thought before and cannot explain the observed interstellar abundances [@Geppert:2005; @Garrod:2006].
Recently, also deuteration experiments were performed on CO ice which confirmed the formation of both fully deuterated formaldehyde and methanol, but with substantially lower reaction rates [@Nagaoka:2005; @Watanabe:2006]. It was suggested that in the presence of both hydrogen and deuterium first the normal methanol forms which then gradually converts to the deuterated species by exchange reactions.
The present paper strongly supports the flux argument given by [@Hidaka:2004]. It furthermore presents a systematic study of the physical dependencies involved in the CO-ice hydrogenation with the aim to put previous work in a context that allows an extension of solid state astrochemical processes to more complex species. Special emphasis is put on the flux and temperature dependence of the formation rate. An analysis of the spectral changes of CO ice during hydrogenation is included to give insight in the structure of the reactive layer. Furthermore, Monte Carlo simulations are presented that allow to interpret the experimental results in more detail and to vary parameters that are hard to study independently by experiment. We conclude with a simulation of H$_2$CO/CH$_3$OH formation under interstellar conditions, in particular for low H-atom fluxes. The outcome is compared with astronomical observations.
Experimental procedure
======================
The experiments are performed under UHV conditions. The room temperature base pressure of the vacuum system is better than 3 $\times$ 10$^{-10}$ mbar. Figure \[setup\] shows a schematic representation of the setup. (See [@Ioppolo:2008] for additional information) Amorphous CO ices ranging from a few to several monolayers are grown on a gold coated copper substrate that is located in the centre of the main chamber and mounted on the tip of a cold finger of a 10 K He cryostat. The temperature of the ice is controlled between 12.0 K and 300 K with 0.5 K relative precision between experiments. The absolute accuracy is better than 2 K. During deposition the layer thickness is monitored by simultaneous recording of reflection absorption infrared (RAIR) spectra. In order to exclude the effect of potential pollutions, ices are grown using CO, $^{13}$CO or C$^{18}$O isotopologues.
The ice layers are exposed to a hydrogen atom beam. The atoms are produced by a well characterised commercial thermal cracking source [@Tschersich:1998; @Tschersich:2000] that provides H-atom fluxes on the sample surface between 10$^{12}$ and 10$^{14}$ atoms cm$^{-2}$s$^{-1}$. For comparison, the Hiraoka group used fluxes below 10$^{13}$ atoms cm$^{-2}$s$^{-1}$ and the Watanabe group worked in the 10$^{14}$-10$^{15}$ atoms cm$^{-2}$s$^{-1}$ regime. The hot ($\sim$ 2000 K) hydrogen atoms are cooled down to room temperature via surface collisions in a nose-like shape quartz pipe between the atomic source and the ice sample. In this way hot hydrogen atoms cannot affect the ice directly. H-atom recombination in this connecting pipe results in a lower final flux. [Details about the flux determination are given in Appendix A. The absolute fluxes are estimated to be within a factor of two, the relative fluxes within 50 %.]{}
[The relatively high temperature of the incident atoms of 300 K]{} does not affect the process; previous experiments with colder H atoms did not show any substantial temperature dependence because the atoms are immediately thermalized on the surface [@Watanabe:2002]. It is argued that the surface is covered with a thin layer of hydrogen molecules under these conditions. These molecules are either formed on the surface or originate from the partially dissociated beam. Since the incoming atoms have to penetrate this cold H$_2$ layer, they are thermally adjusted to the surface temperature once they come in contact with the CO molecules.
Information about the reaction products is obtained using two complementary techniques. During the H-atom bombardment reactants and products are monitored by recording RAIR spectra. The RAIR spectra are recorded using a Fourier transform infrared spectrometer with 1 and 4 cm$^{-1}$ resolution and covering the spectral region in which CO (2143 (s) cm$^{-1}$), formaldehyde (1732 (s), 1479 and 2812 (m), and 1246, 1175, 2991, 2880, and 2812 (mw) cm$^{-1}$) and methanol (1035 (s) and 1125 (w) cm$^{-1}$) exhibit strong (s), medium (m) or weak (w) absorptions. The intensity of spectral features is directly related to the density in the ice. The products are monitored mass spectrometrically using temperature programmed desorption (TPD) once a hydrogenation experiment is completed.
Experimental results
====================
A sample experiment
-------------------
To illustrate the experimental method we start by discussing a sample experiment in which a CO ice of 8$\times$10$^{15}$ molecules cm$^{-2}$ is bombarded with H atoms with a flux of 5$\times$10$^{13}$ cm$^{-2}$ s$^{-1}$ for three hours at a surface temperature of 12.0 K. This corresponds to a fluence of 5.4$\times$10$^{17}$ cm$^{-2}$. Figure \[IR\_hiflux\] shows the RAIR difference spectrum [($\Delta Abs$)]{} after these three hours of exposure (after $-$ before). Indicated are the CO, the H$_2$CO and the CH$_3$OH spectral signatures with respect to the spectrum recorded before the H-atom bombardment started. The CO appears as a negative band indicating its use-up and the other bands are positive, indicating the formation of H$_2$CO and CH$_3$OH. Neither the intermediate species, HCO and H$_3$CO, nor more complex species are observed.
{width="90.00000%"}
The column density $N_{X}$ (molecules cm$^{-2}$) of species $X$ in the ice is calculated using $$N_{X}=\frac{\int A(\nu) \textrm{d} \nu}{S_{X}} \label{N_X}$$ where $A(\nu)$ is the wavelength dependent absorbance. Since literature values of transmission band strengths cannot be used in reflection measurements, an apparent absorption band strength, $S_{X}$ of species $X$ is calculated from a calibration experiment in which an ice layer of species $X$ desorbs at constant temperature until the sub-monolayer regime. This is illustrated in Fig. \[Calibration\] that shows the decrease in integrated absorbance of CO and CH$_3$OH during such an experiment. The arrows in the graph indicate the deviation onset from constant desorption which marks the transition point from multi- to sub-monolayer regime. The thus obtained apparent absorption band strengths of CO and CH$_3$OH (1035 cm$^{-1}$) are setup specific. The corresponding uncertainty in the band strengths remains within 50 %. The ratio between $S_{\rm CO}$ and $S_{\rm CH_3OH}$ in our reflection experiment is similar to the transmittance ratio, 0.85. The value for $S_{\rm H_2CO}$ is obtained by assuming mass balance $$N_{\rm CO}(t) + N_{\rm CH_3OH}(t) = - \frac{\int A(\nu) \textrm{d} \nu}{S_{\rm H_2CO}}$$ for a set of different experiments. In addition, the results discussed in the present paper are all in a regime where the proportionality relation [@Teolis:2007] still holds ($<$ 3$\times 10^{16}$ molecules cm$^{-2}$).
The CO band shape can change when more molecules other than CO are formed. Figure \[COchange\] shows the 2143 cm$^{-1}$ IR peak before and after the H-atom exposure. A clear decrease of the peak height can be observed due to the use-up of CO during the experiment, as is expected. However, an additional peak appears at 2135 cm$^{-1}$ (see inset Fig. \[CH3OHmixture\]), which is due to a CH$_3$OH-CO ice interaction. Transmission IR spectra of a CH$_3$OH:CO mixture show a band at 2136 cm$^{-1}$ [@Bisschop:thesis; @Palumbo:1993]. When the methanol bands grow also the band at 2135 cm$^{-1}$ increases. Figure \[CH3OHmixture\] shows how the peak position of CO shifts with the methanol content in the reflection spectra. The RAIR spectra on which this graph is based, are taken of ice layers that are formed by co-deposition of CO and CH$_3$OH of known ratio. The CO stretching mode in H$_2$O:CO and NH$_3$:CO mixtures shows similar behaviour [@Standford:1988; @Bouwman:2007]. Like H$_2$O and NH$_3$, CH$_3$OH is able to form hydrogen bonds and these hydrogen bonds most likely cause the redshift of the CO band. By comparing the position of the peak in Fig. \[COchange\] at 2135 cm$^{-1}$ to Fig. \[CH3OHmixture\], we conclude on the methanol fraction in the top layers assuming that the formed CH$_3$OH:CO mixture has the same spectral behaviour as the deposited mixtures. The observed data after three hours correspond to a CH$_3$OH:CO mixture of at least 90 %. This means that the top layer of the ice is completely converted to H$_2$CO and CH$_3$OH and that no or very little additional mixing with CO occurs. For the H$_2$CO and CH$_3$OH band no spectral changes are observed during the experiments.
[In order to quantify the use-up of CO and the formation of new products, we have to assume that the apparent absorption band strength is constant during an experiment, *i.e.* independent of the ice composition. [@Bouwman:2007] found that indeed the band strength of the 2143 cm$^{-1}$ CO feature is not affected within the experimental error by water content in H2O:CO-ice mixtures up to 4:1. The band strength is expected to behave similarly for a CO:CH$_3$OH-mixture. Furthermore, if the band strength would be strongly affected by the ice composition, the total ice thickness determined using a constant band strength would vary in time, whereas the real thickness is constant. Since this does not occur, we estimate that the change in band strength due to changing ice composition is negligible and well within our error bars.]{}
Figure \[dNdt\] (a) shows the time evolution of the integrated CO, H$_2$CO and CH$_3$OH signals in symbols. It shows how the amount of CO decreases as the abundance of H$_2$CO grows for four different temperatures. After bombardment with 1$\times$10$^{17}$ H atoms cm$^{-2}$ the formation of methanol kicks off at the expense of the growth of the H$_2$CO abundance. Similar abundance evolutions as a function of fluence have been reported by [@Watanabe:2006]. This indicates that the fluence is determined with relatively high accuracy since in both experiments different atomic sources (Tschersich vs. microwave induced plasma) and different calibration methods are used.
![RAIR difference spectrum of a CO ice at 12.0 K exposed to 5.4$\times$10$^{17}$ cm$^{-2}$ H atoms at a flux of 5$\times$10$^{13}$ cm$^{-2}$ s$^{-1}$. [The spectrum after CO deposition is used as the reference spectrum.]{} Note that the CO peak reaches an absorbance difference of -0.006.[]{data-label="IR_hiflux"}](IR_hifluxII.eps){width="45.00000%"}
![The decrease in integrated absorbance of CO and CH$_3$OH (1035 cm$^{-1}$) following desorption at a constant temperature of 29 and 135 K, respectively. The arrows indicate the transition points from the multi- to sub-monolayer regime.[]{data-label="Calibration"}](Calibration.eps){width="45.00000%"}
![Spectral change of the CO 2143 cm$^{-1}$ RAIR band before and after H-atom bombardment. The inset shows the corresponding difference spectrum.[]{data-label="COchange"}](COchange2.eps){width="45.00000%"}
![[CO RAIR band position as a function of CH$_3$OH content in a CO:CH$_3$OH mixed ice obtained by codeposition experiments.]{}[]{data-label="CH3OHmixture"}](CH3OHmixture.eps){width="45.00000%"}
![Time evolution of the surface abundance (in molecules cm$^{-2}$) of CO, H$_2$CO and CH$_3$OH during H-atom bombardment of CO ice with a H-atom flux of 5$\times$10$^{13}$ cm$^{-2}$ s$^{-1}$ at [surface temperatures of ]{}12.0 K (a), 13.5 K (b), 15.0 K (c), and 16.5 K (d). Experimental data (symbols) and Monte Carlo simulation results (solid lines) are shown as well. []{data-label="dNdt"}](dNdt3.eps){width="45.00000%"}
Flux dependence
---------------
As mentioned in the introduction, the apparent discrepancy between the results by [@Hiraoka:2002] and [@Watanabe:2002] was attributed to a difference in the H-atom flux used in the respective experiments. The setup in our laboratory is able to cover the entire flux range from 10$^{12}$ to 10$^{14}$ cm$^{-2}$s$^{-1}$. For high flux, both formaldehyde and methanol are formed as can be seen in Figs. \[IR\_hiflux\] and \[dNdt\] and in the corresponding work of [@Watanabe:2002].
A difference spectrum of a similar experiment but with a much lower flux of 10$^{12}$ cm$^{-2}$s$^{-1}$ is plotted in Fig. \[IR\_lowflux\]. The exposure time here is four hours to obtain better statistics, but the total fluence of $1\times 10^{16}$ cm$^{-2}$ is still significantly less than the sample experiment shown in Fig. \[IR\_hiflux\]. Note that the vertical scales in Figs. \[IR\_hiflux\] and \[IR\_lowflux\] are the same. For longer exposures surface contamination will become a problem, but methanol features will eventually become detectable. As Fig. \[IR\_lowflux\] clearly shows, much less CO is transformed to H$_2$CO and the sensitivity of the RAIR spectrometer is not high enough to confirm the formation of CH$_3$OH at these circumstances. TPD, however, is more sensitive as a diagnostics tool, although harder to use for a quantitative or time resolved analysis. Figure \[TPD\_lowflux\] plots several TPD spectra. It shows a small methanol desorption peak around 150 K. We have experimentally checked that the carrier of this peak is indeed formed in the ice during the hydrogen exposure and that the observed CH$_3$OH is not a contaminant in the UHV chamber. This is a strong indication that the formation mechanism of formaldehyde and methanol does not fundamentally change with varying flux. [The H$_2$O desorption at 20-30 K originates from frozen background water on the surrounding parts of the cryohead. ]{}
Arrows in Fig. \[dNdt\]a indicate the corresponding fluences for the low and high flux experiments, respectively shown in Figs. \[IR\_lowflux\] and \[IR\_hiflux\]. From this it is immediately apparent that only a limited amount of methanol can be formed under low flux circumstances. Note that [@Hiraoka:2002] probably used an even lower fluence since their exposure time was four times shorter than in our experiment. In addition, they used a slightly lower temperature of 10 K.
![RAIR spectrum of a CO ice at 12.0 K exposed to 1$\times$10$^{16}$ H atoms cm$^{-2}$. []{data-label="IR_lowflux"}](IR_lowfluxII.eps){width="45.00000%"}
![The TPD spectra corresponding to Fig. \[IR\_lowflux\].[]{data-label="TPD_lowflux"}](TPD_lowflux.eps){width="45.00000%"}
Thickness dependence
--------------------
The effect of the initial layer thickness on the formation yield of H$_2$CO and CH$_3$OH is investigated by repeating the sample experiment for different CO layer thicknesses. Figure \[Yield\] shows the absolute reaction yield after a fluence of $5.4\times 10^{17}$ H atoms cm$^{-2}$ as a function of the layer thickness. A steady state value for H$_2$CO is reached for this fluence in all cases. The figure clearly shows that for CO layers thicker than 4$\times$10$^{15}$ molecules cm$^{-2}$ the absolute yield is layer thickness independent and the results are reproducible within the measurement error. The combined H$_2$CO and CH$_3$OH yield of $2\times10^{15}$ molecules cm$^{-2}$ is lower than the $4\times10^{15}$ molecules cm$^{-2}$ penetration column. From these experiments we conclude that the penetration column of the H atoms into the CO ice is at most $4\times10^{15}$ molecules cm$^{-2}$ at 12.0 K. This corresponds to 4 monolayers (ML) of solid (bulk) CO molecules. At least half of the CO molecules in the active layer is converted to H$_2$CO and CH$_3$OH. The determination of the penetration column by this experiment is only an upper limit due to the low thickness resolution in Fig. \[Yield\]. It is however in agreement with the previous estimate of nearly 100 % conversion.
![The absolute reaction yield of H$_2$CO and CH$_3$OH after a fluence of 5.4$\times$10$^{17}$ H atoms cm$^{-2}$ as a function of the layer thickness for experiments at 12.0 K.[]{data-label="Yield"}](Yield.eps){width="45.00000%"}
Temperature dependence
----------------------
Several experiments for different surface temperatures have been performed. The initial layer thickness and flux values are comparable to the values used in the sample experiment. Figures \[dNdt\] (b)-(d) show the results for hydrogenation experiments at 13.5, 15.0, and 16.5 K, respectively. These clearly indicate the very different evolution of CO, H$_2$CO, and CH$_3$OH abundance with temperature. Table \[yieldtab\] gives the initial formation rate of formaldehyde (slope at $t=0$) and the final H$_2$CO and CH$_3$OH yields. It is also indicated whether or not a steady state is reached. The table shows that at early times the formation rate of H$_2$CO is much lower for higher temperatures as compared to 12.0 K. We will come back to this later. The final yield of CH$_3$OH is however larger at 13.5 and 15.0 K. For $T>15$ K the production rate of H$_2$CO is simply so low that a steady state is not reached. [Minimal amounts of formed methanol were also detected in experiments at 18.0 and 20.0 K, but since some CO desorption and redeposition occurs at these temperatures, they are not presented here for a quantitive discussion.]{}
The appearance of the extra CO band at 2135 cm$^{-1}$ indicates that for the temperatures from 12.0 to 15.0 K a nearly pure methanol layer is formed. We expect a similar behaviour for formaldehyde. This means that the active CO layer involved in the reactions can be determined directly from the steady state yield of H$_2$CO and CH$_3$OH. This active layer increases with temperature indicating that the penetration column of H atoms into CO ice increases with temperature as one would expect. The CO molecules in the ice are more mobile at higher temperatures making it easier for H atoms to penetrate the CO ice, [since the ice becomes less rigid]{}. Note that the absolute temperature calibration in the set-up of Watanabe and ours appears to differ by 1-2 K (comparing Fig. 3 in [@Watanabe:2006] and Fig. \[dNdt\] here), but the observed trends are identical.
------ --------------------------- ------------------------------ ------------------------------ -------------- ------------------------------
$T$ Rate(H$_2$CO)$_{t=0}$ Yield (H$_2$CO) Yield (CH$_3$OH) Steady state Calc. pen. column
(K) (10$^{-3}$ molec./H atom) (10$^{15}$ molec. cm$^{-2}$) (10$^{15}$ molec. cm$^{-2}$) (10$^{15}$ molec. cm$^{-2}$)
12.0 9.0 1.2 0.8 yes 2.0
13.5 7.3 1.0 1.4 yes 2.4
15.0 3.2 0.9 1.6 yes 2.5
16.5 1.1 0.8 0.6 no
18.0 1.0 0.5 0.2 no
20.0 0.9 0.4 0.1 no
------ --------------------------- ------------------------------ ------------------------------ -------------- ------------------------------
Monte Carlo simulations
=======================
The method
----------
To infer the underlying mechanisms leading to the formation of methanol a detailed physical-chemical model is required. The present section discusses an approach based on the continuous time, random-walk Monte Carlo simulation. This method is different from previous studies based on rate equations and enables the study of surface processes in more detail. In addition, it gives a better understanding about what occurs physically on the surface. In contrast to an analysis using rate laws, the Monte Carlo method determines the H surface abundance by taking into account the layered structure of the ice, the H-atom flux, diffusion, reaction and desorption. This allows an extension of the results to conditions with much lower fluxes like in the interstellar medium (ISM). For a detailed description of method and program, see [@Cuppen:2007].
During a simulation a sequence of processes - hopping, desorption, deposition and reaction - is performed where this sequence is chosen by means of a random number generator in combination with the rates for the different processes. First, an initial ice layer is created by deposition of CO on a surface. The resulting surface roughness of this layer depends on temperature and flux. [For the experimental conditions that are simulated here, the CO ice is compact with a maximum height difference across the surface of only 2-3 monolayers.]{} Hydrogen atoms and hydrogen molecules are subsequently deposited according to their relative abundance in the H-atom flux [with an angle perpendicular to the surface to mimic the experimental conditions.]{} They move, react and desorb according to rates with a similar form as used in gas-grain models $$R_x = A \exp\left(-\frac{E_x}{T}\right),$$ where $E_x$ is the activation energy for process $X$ and $A$ is the pre-exponential factor for which a constant number of $\nu$ $\sim$ $kT/h$ = $2\times
10^{11}$ s$^{-1}$ is used. The activation energies are not well determined *ab initio* or by experiment. The desorption energies are determined by the binding energy as explained below and depend on an energy parameter $E$. The barriers for reaction are used as a parameter to fit the data. The barrier for hopping (diffusion) from site $i$ to $j$ is obtained by $$E_{\rm hop}(i,j) = \xi E + \frac{\Delta E_{\rm bind}(i,j)}{2}.
\label{Ehop}$$ [This expression ensures microscopic reversibility between the different types of sites.]{} The parameter $\xi$ is another input parameter, which is varied between simulations. Little quantitative information is available about diffusion rates on these kind of surfaces which makes the value of $\xi$ uncertain.
Diffusion into the ice is also considered. Minimum energy path calculations suggest that CO and H can swap position enabling an H atom to penetrate into the CO ice (see Appendix B). The barrier for this process strongly depends on the layer in which the H atom is situated. In the simulations the barrier for this event is (350 + 2($z_1 + z_2$)) K for an H atom to swap between layer $z_1$ and $z_2$. [This compares to a hopping barrier of $E_{\rm hop}^{\rm H, flat\rightarrow flat} = 256$ K and a desorption energy of $E_{\rm bind}^{\rm H, flat} = 320$ K (see next section).]{} [@Hiraoka:1998] found that hydrogen atoms can relatively easily diffuse through the CO ice. Moreover, the current experiments show that hydrogen atoms can penetrate into a maximum of four monolayers for 12.0 K. Hydrogen atoms are also allowed to swap with formaldehyde and methanol, but here the initial barrier is chosen to be higher (450 and 500 K) since these species are heavier and are more strongly bound in the ice matrix.
The CO ice layer
----------------
Although the experimental CO layers are probably amorphous [@Kouchi:1990], crystalline layers are used in the Monte Carlo simulations discussed here. In this way a lattice-gas Monte Carlo method can be used which enables much longer simulation times than in off-lattice methods. We expect the crystalline assumption to be reasonable since the local structure of the CO layers is probably close to crystalline. The energy that is released during deposition may help the molecules to rearrange slightly during deposition leading to micro-crystalline domains. The $\alpha$-CO structure [@Vegard:1930] is used with layers in the (110) orientation. The dominant faces on a CO crystal will have this crystallographic orientation. The CO surface consists of alternating carbon and oxygen terminated bi-layers. In the bulk configuration each CO molecule has 14 nearest neighbours: [five in layers below, five in layers above and four in the same layer. ]{} The additive energy contribution of these neighbours is $2E$ for the layers below and $E$ for the neighbours in the same layer or with lower $z$, the depth in layers with respect to the top layer. The different treatment for sites below the particle is to add a contribution for longer range interactions from the ice layer. For atomic hydrogen, $E$ is chosen to be 32 K and for CO to be 63 K. This leads to a binding energy of $E_{\rm bind}^{\rm H, flat} = 320$ K for H on top of flat CO ice layer and $E_{\rm bind}^{\rm H, layer} = 448$ K and $E_{\rm bind}^{\rm CO, layer} = 882$ K for, respectively, H and CO embedded in a CO layer. These values agree very well with binding energies obtained by calculations with accurate H-CO and CO-CO potentials of 320, 440, and 850 K, respectively (see Appendix B).
Comparison to the experiment
----------------------------
The solid lines in Fig. \[dNdt\] represent the results from the Monte Carlo calculations. The exact mechanisms included in these simulations are discussed in more detail in the following sections. The resulting time evolution series are in very good agreement for 12.0 K. The agreement for 13.5, 15.0 and 16.5 K is much less. This is probably due to missing mechanisms that promote the penetration into the ice. In the current simulations only swapping of species is included. Because of thermal motion of the CO molecules, “real” penetration in which the H atoms penetrate in the CO matrix may be possible as well. The shape of the curves is reproduced and only the H$_2$CO abundance levels off at too low yields.
The main parameters varied to fit the experimental data are the reaction barriers and the diffusion rates. The best fitting barriers are summarized in Table \[Energies\]. Since the intermediate species HCO and H$_3$CO are not experimentally detected, the barriers for hydrogenation of these species are significantly lower than for the other two reactions, presumably even zero. The HCO and H$_3$CO abundances stay below detectable levels in the simulations. The reaction barriers for H + CO and H + H$_2$CO are temperature dependent and increase with temperature. Our values are in good absolute agreement with the barriers found by [@Awad:2005], who also found a similar temperature behaviour. Their values were obtained using a rate equation analysis for $T=10$, 15 and 20 K using the data from [@Watanabe:2006]. The temperature dependence suggests that there is a clear tunnelling component for the reaction at low temperature. The two barriers for forming H$_2$CO and CH$_3$OH show different temperature dependencies. The formation of methanol becomes relatively more important for higher temperature. Note that the Monte Carlo method automatically treats a reaction in competition with desorption and hopping. This is in contrast with gas-grain codes where it has to be included explicitly. In order to describe the chemical processes properly one should introduce this competition in the gas-grain model.
------ -------------- -------------------- -------------- --------------------
$T$
barrier rate barrier rate
(K) (K) (s$^{-1}$) (K) (s$^{-1}$)
12.0 390 $\pm$ 40 $2 \times 10^{-3}$ 415 $\pm$ 40 $2 \times 10^{-4}$
13.5 435 $\pm$ 50 $2 \times 10^{-3}$ 435 $\pm$ 50 $1 \times 10^{-3}$
15.0 480 $\pm$ 60 $5 \times 10^{-3}$ 470 $\pm$ 60 $1 \times 10^{-3}$
16.5 520 $\pm$ 70 $4 \times 10^{-3}$ 500 $\pm$ 70 $1 \times 10^{-3}$
------ -------------- -------------------- -------------- --------------------
: Reaction rates and barriers for CO + H and H$_2$CO + H for different temperatures. \[Energies\]
[The errors in the energy barriers reflect the errors due to the uncertainties in the sticking probability, H-atom flux, diffusion and exact structure of the CO ice.]{}
[Molecular hydrogen is formed on the surface with efficiencies ranging from 3 % ($T =16.5$ K) to 70 % ($T =12.0$ K). However, because of the large excess energy of the formation reaction the majority of the formed H$_2$ molecules leaves the surface and the H$_2$ surface abundance is predominantly determined by impinging H$_2$ molecules.]{}
Effect of diffusion
-------------------
Since the diffusion rates are uncertain, this section discusses its effect in more detail. Minimum energy path calculations of the diffusion of a single hydrogen atom on a CO (110) surface (see Appendix B) results in energy barriers ranging from 70 to 170 K ($\xi$ = 2-5.3) depending on the direction of diffusion. The Monte Carlo program only considers one type of diffusion between “flat” sites. This corresponds better to the isotropic nature of an amorphous surface. Amorphous surfaces are usually more corrugated than crystalline surfaces which increases the hopping barrier. The second term in Eq. \[Ehop\] ensures microscopic reversibility. Figure \[Xi\] shows the influence of the diffusion parameter $\xi$ on the H$_2$CO and CH$_3$OH production. The simulations are carried out in the presence of H$_2$ for 12.0 K (top) and 15.0 K (bottom). The difference in diffusion appears to have a larger effect for 15.0 K than for 12.0 K. Faster diffusion (smaller $\xi$) clearly results in less CH$_3$OH and H$_2$CO production, since the H atoms are more likely to find each other and to react away to H$_2$. Slower diffusion allows the H atoms more time per CO encounter to cross the reaction barrier and to form HCO. In the simulations presented in Figs. \[dNdt\] and \[ISM\], we use $\xi = 8$ to reduce the simulation time. This parameter choice results in a ratio $E_{\rm hop}({\rm flat},{\rm flat})/E_{\rm
bind} ({\rm flat})$ of 0.78, which is in agreement with the experimentally found ratio for H atoms on olivine and amorphous carbon [@Katz:1999]. The amorphocity of the surface may be responsible for such a high ratio.
![Monte Carlo simulations of the time evolution of the surface abundance of CO, H$_2$CO and CH$_3$OH during H-atom bombardment of CO ice at 12.0 K (top) and 15.0 K (bottom). Reaction barriers for H + CO and H + H$_2$CO can be found in Table \[Energies\]. The diffusion is varied via the parameter $\xi$ (Eq. 5). []{data-label="Xi"}](Xi.eps){width="45.00000%"}
Effect of H$_2$ molecules on the hydrogenation
----------------------------------------------
All simulations include deposition of both H atoms and H$_2$ molecules, which results from the undissociated H$_2$ molecules in the H-beam. If the H$_2$ molecules are excluded from the simulations, the formation of H$_2$CO and CH$_3$OH is affected in only a limited set of cases: fast diffusion and high temperature. The presence of H$_2$ appears to have mainly two effects. It limits the penetration into the ice and it slows down the H atoms, since they move through a “sea” of H$_2$. The first has a negative effect on the production rate, the latter depends on the reaction barrier.
The experimental results at temperatures higher than 12.0 K show non-first-order behaviour at early times (exponential decay of CO). The H$_2$CO production rate increases until 30 and 50 minutes of exposure for $T$ = 13.5 and 15.0 K, respectively. After this time, the H$_2$CO and CH$_3$OH follow the expected first order behaviour. None of the simulations in Fig. \[Xi\] shows this trend. The only mechanism which is able to describe this phenomenon is an increasing effective H-atom flux with time. This increasing effective flux can be due to an increased sticking of atomic hydrogen on the surface. Since the incoming H atoms are relatively warm, they need to dissipate this extra energy to the surface in order to stick. Because CO is relatively heavy as compared to the H atoms, this energy dissipation will be inefficient and most of the H atoms will scatter back into the gas phase. Once the surface abundance of the much lighter H$_2$ molecules increases, the sticking of the H atoms to the surface will increase as well. We assume a 1 % sticking for H atoms and H$_2$ molecules on a bare CO surface and a 65 % sticking of H atoms on a surface which is fully H$_2$ covered . The sticking probability is further assumed to grow linearly with H$_2$ coverage. The H$_2$ surface abundance reaches a contant level of 0.39 ML after a few minutes for $T$ = 12.0 K. This results in a sticking of H atoms to the surface of 26 %. For higher temperatures it takes noticeably longer to equilibrate, explaining the non-linear behaviour at early times and indicating a lower final sticking probability. The solid lines in Fig. \[dNdt\] include this mechanism.
As mentioned earlier, [@Watanabe:2002] concluded that the temperature of the beam has little effect on the hydrogenation process, which seems to contradict our H$_2$ argument. However, their experiments were carried out at 10 K, where the surfaces will be covered with hydrogen atoms early on in the experiment because of the enhanced sticking at low temperatures. They further reported an unknown flux difference between the cold and warm beam, which makes quantifying the sticking probability using these experiments not possible. In conclusion, the temperature of the beam can affect the effective flux of H atoms landing on the surfaces, but it does not introduce additional energetic effects which influence the crossing of the barrier.
CO hydrogenation under interstellar conditions
==============================================
Based on the fitting results in the previous section, the Monte Carlo routine can now be used to simulate CO hydrogenation reactions under interstellar conditions. An important ingredient is the H-atom density in the cloud. Just as in our laboratory beam, the gas in dense clouds consists of a mix of H and H$_2$. Under steady-state conditions, the balance of the rates of H$_2$ formation on grains and H$_2$ destruction by cosmic rays gives an H-atom density around 1 cm$^{-3}$ [@Hollenbach:1971]. This H-atom number density is independent of the total density because both the formation and destruction rates scale with density. Before steady-state is reached, however, the H-atom density may be higher because the time scale for H to H$_2$ conversion is long ($\sim 10^7$ yr) starting from a purely atomic low density cloud [@Goldsmith:2007]. Our model assumes a constant H-atom density of 10 cm$^{-3}$. Our other model parameters are a gas temperature of 20 K and dust temperatures of 12.0 and 16.5 K. A CO surface is then simulated for $2\times 10^5$ yr, which corresponds to a fluence of $10.8\times 10^{17}$ atoms cm$^{-2}$. Note that half of this fluence was actually realized in our experiments. Because the H-atom velocities are low, the sticking of H atoms to the CO ice is kept constant at 100 %.
[ The starting configuration for the simulations is a layer of pure CO ice. This is believed to be representative for the top layers of the grain mantles in the center of a high-density collapsing cloud. Here, the ice layer is observed to consist of predominantly CO ice as the result of “catastrophic” CO freeze-out [@Pontoppidan:2006; @Pontoppidan:2008]. More heterogeneous ice layers are formed at lower densities, where CO and H$_2$O are mixed, or towards the center of proto-stellar envelops or proto-planetary disks where the dust has been heated and CO has desorbed from the top layers. ]{}
Figure \[ISM\] (top) shows the resulting time evolution of CO, H$_2$CO, and CH$_3$OH ice (thick lines) for 12.0 K. The thin lines in Fig. \[ISM\] represent the direct scaling of the simulations of the experiment to interstellar time scales. The H$_2$CO/CH$_3$OH ratio for the low flux simulation is very different from the scaled experimental simulation. [The reason for this is that in the laboratory environment twice as many hydrogen atoms react with each other to form H$_2$ than are involved in the four CO hydrogenation reactions since the surface density is relatively high. For interstellar conditions the CO hydrogenation reactions are dominant and only $<5$ % of the reacting H atoms are converted to H$_2$. ]{} A second effect that changes the time evolution in the ISM is the difference in sticking. Under laboratory conditions the sticking probability is much lower since the incoming H atoms with room temperature cannot release their energy very efficiently to the CO ice. The presence of H$_2$ on the surface may have a positive effect on the sticking probability. In the ISM the incoming atoms are much colder and energy dissipation will not be a limiting factor for the sticking of H atoms to CO ice. This can be modelled using the Monte Carlo simulations but only after deriving the energy barriers by fitting the laboratory data.
The bottom panel in Fig. \[ISM\] shows similar trends for 16.5 K. Again the onset of H$_2$CO and CH$_3$OH formation is at much lower fluences as compared to the experiment. At the end of the simulation nearly all H$_2$CO has been converted to CH$_3$OH. This is in contrast with the 12.0 K simulations where a constant non-zero amount of H$_2$CO remains after $2\times 10^5$ yr. The crossover point from H$_2$CO-rich to CH$_3$OH-rich ice occurs at slightly later times at 16.5 K compared to 12.0 K. This can clearly be seen in Fig. \[H2CO/CH3OH\] which plots the H$_2$CO/CH$_3$OH ratio for both temperatures. At early times this ratio is similar for 12.0 and 16.5 K. At $t > 10^3$ yr, the ratio starts to level off for 12.0 K, while it still decreases rapidly for 16.5 K. The noise in the curve for 16.5 K below $t = 5\times 10^3$ yr is due to the low abundances of H$_2$CO and CH$_3$OH.
In space, the H$_2$CO/CH$_3$OH ice ratio has been determined directly for only three high-mass young stellar objects (YSOs): W33A, NGC 7538 IRS9 and AFGL 70009S, with inferred ratios ranging from 0.09 to 0.51 [@Keane:2001; @Gibb:2004]. The laboratory curves for the H$_2$CO and CH$_3$OH production show that H$_2$CO is more or equally abundant during most of our experiments. Thus, values as low as 0.09-0.51 cannot easily be reproduced in the experiments. However, the Monte Carlo simulations for interstellar conditions have a crossover from H$_2$CO-rich to CH$_3$OH-rich ice at significantly earlier times than the experimental curves and a H$_2$CO/CH$_3$OH ratio of 0.51 is obtained after $5\times 10^3$ yr at $T_{\rm dust}$=12 K. Grains at higher temperatures will have this crossover at even earlier times and for grains with $T_{\rm dust}$=16.5 K a H$_2$CO/CH$_3$OH ratio of even 0.09 is obtained after $2\times 10^4$ yr. Thus, the observed ratios are in agreement with the models discussed above for chemical time scales $>2 \times 10^4$ yr, which is consistent with the estimated ages of these high-mass protostars of a few $10^4-10^5$ yr [@Hoare:2007].
CH$_3$OH ice has also been detected toward low-mass YSOs with abundances ranging from $<$1 % to more than 25 % with respect to H$_2$O ice [@Pontoppidan:2003; @Boogert:2008]. An interesting example is the Class 0 protostar , for which a particularly high CH$_3$OH abundance of 28 % with respect to H$_2$O ice has been deduced for the outer envelope [@Pontoppidan:2004]. The upper limit on the H$_2$CO-ice abundance implies a H$_2$CO/CH$_3$OH ratio $ < $0.18, implying an age $> 1 \times 10^4$ yr at 16.5 K. This is consistent with the estimated time scale for heavy freeze out in low-mass YSOs of $10^{5\pm 0.5}$ yr, including both the pre-stellar and proto-stellar phases [@Jorgensen:2005I].
Other observational constraints come from sub-millimetre observations of the gas in a sample of massive hot cores, where a constant ratio of H$_2$CO/CH$_3$OH of 0.22$\pm0.05$ was found [@Bisschop:2007III]. If both the observed H$_2$CO and CH$_3$OH have just evaporated freshly off the grains and if they have not been affected by subsequent gas-phase chemistry, the observed ratio should reflect the ice abundances. This ratio is roughly consistent with the asymptotic value that is reached in the 12 K model. This remarkably constant abundance ratio would imply very similar physical conditions (dust temperatures, H-atom abundances, ...) during ice formation.
In contrast, the fact that the CH$_3$OH ice abundance with respect to that of H$_2$O is known to vary by more than an order of magnitude suggests the opposite: that local conditions and time scales do play a role. Note, however, that for CH$_3$OH abundances as large as 25 % (columns as large as $10^{18}$ cm$^{-2}$), the CH$_3$OH layer is approximately 25 ML thick ( $0.25 \times n({\rm H_2O}) / (n_{\rm dust} \times \textrm{$<$binding sites per grain$>$}) = 0.25 \times 10^{-4} / (10^{-12} \times 10^6) = 25$ ML), much more than can be produced from just the upper 4 ML of the CO ice. Thus, conversion of CO to CH$_3$OH ice must in these cases occur simultaneously with the freeze-out and building up of the CO layer. Pure CO ice can also easily desorb as soon as the protostar heats up. This complicates the use of CH$_3$OH/CO ice as an evolutionary probe. A proper model of interstellar CH$_3$OH ice formation should therefore include the changing CO-ice abundances and dust temperatures in the pre- and protostellar phases, taking into account the time scales for CH$_3$OH-ice formation compared with those of CO adsorption and desorption. This paper provides the necessary molecular data to work on such a model.
![Monte Carlo simulations of CO-ice hydrogenation at 12.0 (top) and 16.5 K (bottom). A constant atomic hydrogen gas phase density of 10 cm$^{-3}$ and a gas temperature of 20 K is assumed. Thick lines represent interstellar conditions, thin lines are the scaled experimental simulations. The results are shown as the change in column density compared with t=0 yr.[]{data-label="ISM"}](ISM2.eps){width="45.00000%"}
![The H$_2$CO/CH$_3$OH ratio as a function of time obtained from the Monte Carlo simulations of CO hydrogenation at 12.0 and 16.5 K under ISM conditions (see Fig. \[ISM\]). The gray box indicates Spitzer ice observations, the black box gas phase observations.[]{data-label="H2CO/CH3OH"}](H2CO_CH3OH.eps){width="45.00000%"}
Conclusion
==========
The present paper shows that the formation of methanol via successive hydrogenation of CO and H$_2$CO is efficient under various laboratory conditions covering $T_{\rm surf} = 12- 20 $ K, ice thicknesses between 1$\times 10^{15}$ and 1$\times 10^{16}$ molecules cm$^{-2}$ equivalent to 1 and 10 ML bulk CO, and H-atom fluxes between $1\times 10^{12}$ and 5$\times 10^{13}$ cm$^{-2}$s$^{-1}$. [Our results show that the discrepancy between [@Hiraoka:2002] and [@Watanabe:2002] was indeed mainly due to the use of different H-atom fluxes and we agree with the latter that CH$_3$OH is formed at low temperature.]{} *On the basis of this, the surface hydrogenation of CO can now be safely used to explain the majority of the formed methanol in the interstellar medium*, where it serves as a key molecule in the synthesis of more complex molecules.
Energy barriers for the H + CO and H$_2$CO + H reactions are obtained by fitting Monte Carlo simulation results to the experimental data. Using these barriers the methanol production is simulated for interstellar conditions. The obtained H$_2$CO and CH$_3$OH abundances do not scale directly with fluence due to a different relative importance of H$_2$ production and CO hydrogenation in space compared with the laboratory, [as can be clearly seen by comparing the thick and thin lines in Fig. \[ISM\]]{}. But the laboratory experiments are required to derive the necessary rates that serve as input for the Monte Carlo program. The obtained H$_2$CO/CH$_3$OH ratios for the interstellar simulations are in closer agreement with observational limits than direct translation of the experimental observations.
Monte Carlo simulations of the hydrogenation process show that the presence of H$_2$ has three effects: it promotes the sticking of the warm H atoms, it limits the penetration into the ice and it slows down the diffusion of H atoms. The first effect will be negligble under interstellar conditions since the incoming H atoms will be cold already and the sticking probability will therefore be high regardless of the substrate. The latter two effects will be important and are similar to the conditions in the laboratory with also a high H$_2$ abundance.
The experiments show that the hydrogenation process is thickness independent for layers thicker than 4 $\times 10^{15}$ cm$^{-2}$ and that the active layer, which contains only a limited amount of CO after steady state is reached, becomes slightly thicker with temperature. For temperatures higher than 15.0 K, a clear drop in the production rate of methanol is observed. This is probably due to two effects: the desorption of H atoms becomes important and the sticking of H atoms is reduced due to the low H$_2$ surface abundance. Both effects cause the H surface abundance to drop substantially at those temperatures and therefore reduce the probability of hydrogenation reactions to occur in the laboratory. Simulations of CO hydrogenation in space show a strong temperature dependence of the H$_2$CO/CH$_3$OH ratio over several orders of magnitude. The CH$_3$OH abundance changes with time, temperature and fluence.
Part of this work was supported by the Netherlands Research School for Astronomy, NOVA, and Netherlands Organisation for Scientific Research (NWO) through a VENI grant. We thank Stephan Schlemmer and Helen Fraser for their contribution during the first construction phase and Gijsbert Verdoes, Martijn Witlox and Ewie de Kuyper from the Fijn Mechanische Dienst for their support. Ayman Al-Halabi, Lou Allamandola, Eric Herbst, Xander Tielens, Klaus Pontoppidan, and Zainab Awad have contributed to this work through long and inspiring discussions.
Absolute and relative H-atom flux determination
===============================================
Absolute flux determination
---------------------------
The (accuracy of the) absolute value of the H-atom flux at the ice surface is obtained by estimating lower and upper limits in two independent ways. We exemplify here the H-atom flux determination for the case of our standard values with an H$_2$ pressure in the chamber of $p_{\rm H_2}= 1\times 10^{-5}$ mbar and a filament temperature of $T = 2300$ K.
The lower limit on the absolute flux is directly available from the experimental results presented in [@Ioppolo:2008]. That paper discusses the H$_2$O$_2$ and H$_2$O production from H-atom bombardment of O$_2$-ice in time using the same setup and settings. During the first hour, H$_2$O$_2$ and H$_2$O are produced with an almost constant production rate of $6.0 \times 10^{12}$ molecules cm$^{-2}$ s$^{-1}$. Since both molecules contain two hydrogen atoms, this means that the H-atom flux should be at least twice this value. Assuming a conservative sticking probability of hydrogen atoms at 300 K to O$_2$-ice at 12-28 K of at most 50 %, we determine a lower limit on the flux results in $2.4 \times 10^{13}$ cm$^{-2}$ s$^{-1}$.
The determination of the upper limit on the H-atom flux is more elaborate and involves several steps. Fig. \[setup\] shows that the hydrogen atoms travel from the source through the atomic-line chamber to a quartz pipe where the atoms are collisionally cooled and then through the main chamber onto the substrate. The final H-atom flux is then determined by $$\phi_{\rm H} = \frac{N_{\rm H, source} k_1 k_2 p r}{A},$$ where $N_{\rm H, source}$ is the number of hydrogen atoms leaving the source per second, $k_1$ is the coupling efficiency between the source and quartz pipe, $k_2$ is the coupling efficiency between the quartz pipe and the ice surface, $p$ accounts for the pressure drop between the two chambers, $r$ for the loss in H-atoms because of recombinations in the quartz pipe and $A$ is the surface area that is exposed by the H-atom beam.
Our specific hydrogen source, used in the experiments described here, has been tested prior to delivery at the Forschungszentrum in Jülich where the flux, the solid angle and dissociation rate have been measured for a wide range of H$_2$ pressures and filament temperatures. The set-up used for these calibration experiments is described in [@Tschersich:1998]. These measurements confirmed that there is little variation between individual instruments, since nearly identical rates have been obtained in the publications by [@Tschersich:1998] and [@Tschersich:2000] and later by [@Tschersich:2008] for different H-atom sources of the same type. From the flux and dissociation rate measured in Jülich, $N_{\rm H, source}$ can be obtained as well as $k_1$ using the solid angle information. In our example case $4.1 \times 10^{16}$ H-atoms s$^{-1}$ leave the H-atom source and 44 % of these atoms enter the quartz pipe which is located at a distance of 1.5 cm.
The pipe has been designed such that the atoms cannot reach the substrate directly and that the number of hydrogen recombinations is kept to a minimum. This is achieved by using a short pipe with a high diameter/length ratio and choosing quartz which is known to have a low recombination efficiency. Following [@Walraven:1982] a theoretical estimate of the number of recombinations in the pipe can be determined, considering the specific shape and material. This reduces the H-atom flux by another 27 %. The pipe ends in close proximity of the cryogenic surface. The use of a pipe instead of a pinhole or a slit results in a focused H-atom beam for which the flux can be determined with relatively low uncertainty. From geometric considerations a minimum solid angle can be estimated. This will suffice, since our aim is to obtain an upper limit for the flux. The H-atom beam covers $A=4.9$ cm$^2$ of the substrate that is located 3 cm behind the quartz pipe. This spot falls completely on the surface and $k_2$ can readily be assumed to be unity.
Finally, the pressure drop between the source and the main chamber can be determined in two ways: by a calculation using the conductance of the pipe and the pumping speed and by measuring the pressures in both chambers using undissociated beams. Both results are in reasonable agreement leading to $p = 3.2 \times 10^{-2}$.
Our upper limit for the flux is now $$\phi_{\rm H} = \frac{4.1 \times 10^{16}\cdot 0.44 \cdot 1 \cdot 3.2 \times 10^{-2} \cdot 0.73}{4.9} = 8.6 \times 10^{13} {\rm cm^{-2} s^{-1}}.$$ Deviations from this upper limit are expected to be due to a smaller $k_1$ value, because of misalignments between the source and the entrance of the quartz pipe, an underestimation of the solid angle of the exiting beam from the quartz pipe (lower $k_1$ and higher $A$), more recombinations in the pipe or backscattering of atoms from the quartz pipe to the chamber of the H-atom source.
The value for the flux adopted in the present paper is the resulting intermediate H-atom flux of $5 \times 10^{13}$ cm$^{-2}$ s$^{-1}$, which is within a factor of 2 of the upper and lower limits. It should be noted that this is a conservative error, since the actual lower and higher flux limits are likely to be higher and lower, respectively.
Relative flux determination
---------------------------
The accuracy in the relative flux is particularly important for the conclusion presented in this paper, more than the absolute value. For this we use the CO-hydrogenation data obtained from the experiments. Figure \[flux\] shows the CO, H$_2$CO and CH$_3$OH evolution as a function of fluence for three different fluxes. The fluences are calculated using the flux determination as described above. The three curves clearly overlap, which means that the accuracy of the relative fluxes is well within our error bars. We conclude that the accuracy in the relative flux is substantially higher than the accuracy of the absolute flux, well below 50 %. One of the main conclusions of the paper, that the discrepancy between the two Japanese groups is due to a difference in flux, as envisaged by [@Hidaka:2004], is therefore solid.
Finally, reproducing the same experiments on different days over the course of several months showed that reproducibility over periods from day-to-day to months is excellent, within a few percent.
![Time evolution of the surface abundance (in molecules cm$^{-2}$) of CO, H$_2$CO and CH$_3$OH during H-atom bombardment of CO ice at 12.0 K with three different H fluxes of 5$\times$10$^{13}$, 3$\times$10$^{13}$, and 7$\times$10$^{12}$ cm$^{-2}$. []{data-label="flux"}](COhydroFlux.eps){width="45.00000%"}
Binding energy calculations
===========================
To calculate binding energies and barriers to diffusion, recently-developed CO–CO and H–CO potentials are used. Takahashi and van Hemert (in prep.) have fitted high level electronic structure (coupled cluster) calculations on the CO–CO dimer to an analytic potential consisting of partial charges on the atoms and the centres of mass of the CO molecules, atom-based Lennard-Jones type interactions, and Morse potentials for the intramolecular C–O interaction. In the work by Andersson et al. (in prep.) a potential for the interaction between a hydrogen atom and CO has been calculated through fitting damped dispersion and exponential repulsion potentials to coupled cluster calculations.
Using the CO–CO potential a CO (110) surface has been created consisting of 528 CO molecules in 11 monolayers in a cell with dimensions 33.8 [Å]{} $\times$ 31.8 [Å]{} in the surface plane. By applying periodic boundary conditions an infinite surface is created. Binding energies have been calculated by performing energy minimisations for H atoms at different sites on top of and inside the CO surface and comparing to the energy with the hydrogen far away from the surface. In the same manner the binding energy for a CO molecule in the top layer has been calculated. In all instances the top 3 monolayers of the ice have been allowed to relax.
To calculate energy barriers to diffusion on and into the surface, initially the Nudged Elastic Band (NEB) method [@Jonsson:1998] has been used to map out the minimum energy path (MEP) connecting two potential minima. To fine-tune the barrier height, the Lanczos method is used to optimize the saddle point of the potential energy [@Olsen:2004].
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---
address:
- '4 Chome 11-16-502, Shimomeguro, Meguro, Tokyo, 153, Japan'
- |
Department of Physics, SUNY, Stony Brook, New York 11794, USA.\
e-mail: [email protected]
author:
- Hidenaga YAMAGISHI
- Ismail ZAHED
title: TWO TOPICS IN CHIRAL EFFECTIVE LAGRANGIANS
---
=cmr8
1.5pt \#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
Introduction
============
Effective Lagrangians have been an important tool in our understanding of hadronic processes. If we consider processes without baryons or other heavy particles for simplicity, then the loop expansion of the effective Lagrangian is equivalent to an expansion in the momentum $p$. Georgi and Manohar [@GEORGI1] have suggested that the quantitative expansion parameter in this case is $p^2/16\pi^2 f_{\pi}^2 \sim (p/1.2 {\rm GeV})^2$, where the factor of $16\pi^2$ arises from a one-loop Feynman diagram.
For $p=m_{\pi}$, this gives 1.4 % accuracy. However, it is seldom in chiral physics to achieve an accuracy of this order. The standard tree level result for $\pi\pi\rightarrow \pi\pi$ gives scattering lengths which are off by 40 % and 25 %, depending on the isospin and angular momentum [@EXPERIMENT]. In $\gamma\gamma\rightarrow \pi^0\pi^0$ where the Born term is absent, the one-loop result is off by 25-30 % [@BIJNENS].
In this note, we show that the true expansion parameter must be closer to $p^2/4\pi f_{\pi}^2\sim (p/0.33 {\rm GeV})^2$ through a consideration of partial wave unitarity for $\pi\pi$ scattering. For $\pi N$ scattering, we first show that a pion-nucleon Lagrangian can be organized in terms of the Tani-Foldy-Wouthuysen transformation, and then observe that partial wave unitarity in the S31 channel yields a bound that is close to the $\pi\pi$ one.
Partial Wave Unitarity
======================
In the chiral limit, the invariant amplitude $T^I (s, t)$ with isospin $I$ can be decomposed into partial waves as T\^I (s, t) = 32\_l (2l+1) P\_l ([cos]{} ) \_l\^I e\^[i\_l\^I]{} [sin]{} \_l\^I \[1\] where $s$ and $t$ are Mandelstam variables, $P_l$ Legendre polynomials, $\theta$ the scattering angle in the center of mass frame, $\eta_l^I$ the partial-wave inelasticities, and $\delta_l^I$ the partial-wave phase shifts. Projecting out the S-wave gives the bound |T\_0\^I| 32\[2\] On the other hand the tree result for massless pions is T\^0 (s, t) = T\_0\^0 (s) = \[3\] where $k$ is the pion momentum in the center of mass frame. It follows that the unitarity bound for massless pions is k\^2 4 f\_\^2 \[4\] and accordingly, the chiral loop expansion parameter must be closer to $p^2/4\pi f_{\pi}^2$.
For massive pions, Eqs. (\[2\]) and (\[3\]) are modified to |T\_0\^I (s) | 32(1+)\^[12]{} \[5\] T\^0 (s, t) = T\_0\^0 (s) = \[6\] Since Eq. (\[5\]) is monotonically decreasing and Eq. (\[6\]) monotonically increasing, we obtain the bound k\^2 5.2 m\_\^2 \~(0.32[GeV]{})\^2 \[7\] which is numerically close to Eq. (\[4\]).
It follows that the loop expansion of conventional chiral perturbation theory [@GASSER] must break down before 300 MeV as far as $\pi\pi$ scattering is concerned, in agreement with a similar observation in [@HOLSTEIN].
Tani-Foldy-Wouthuysen Transformation
====================================
For purely pionic processes such as previously discussed, the loop expansion of the effective Lagrangian is equivalent to an expansion in the momentum $p$. This is no longer true when nucleons are put in Gasser et al. [@GSS]. To deal with this situation, heavy baryon chiral perturbation theory was proposed [@MANOHAR], and used as a $1/m_N$ expansion [@ALL]. In all of this work, the projected fields [@GEORGI] \^\_v (x) = &&e\^[im vx]{} 12 (1/[v]{})(x) \[I1\] with $v^2=1$, were employed.
In the nonrelativistic reduction of the Dirac equation, it is perhaps more convenient to work with the TFW transformation [@TFW], rather than Eq. (\[I1\]). In this note, we apply the TFW transformation to the pion-nucleon effective Lagrangian and obtain terms up to ${\cal O} (p^4)$.
We take the model relativistic pion-nucleon Lagrangian as [^1] =&&+ 4 [Tr]{} (\_ U\^ U\^ ) + 14 f\_\^2m\_\^2 [Tr]{} (U+U\^)\
&&+\^ (i\_0 - [H]{} )\
[H]{} = &&+( +i) -i\_0\
&&-ig\_A\^i\_i -ig\_A \_5\_0 + m\_N\
&&+ ((U+U\^ -2) +\_5 (U\^-U)) \[I2\] where $U=\xi^2$ is the chiral field and \_ =12 \[\^,\_\]\_ =12 \^ (\_ U) \^ \[I3\] $\Gamma_{\mu}$ and $\Delta_{\mu}$ count as ${\cal O} (p)$ and $m_{\pi}^2/\Lambda$ as ${\cal O} (p^2)$. The Lagrangian is standard except for the term proportional to $m_{\pi}^2/\Lambda$, which is the pion-nucleon sigma term at tree level. In QCD the quark mass term generates both the pion mass and the sigma term, so the two should go together [@STEELE].
The idea of TFW is to perform a series of unitary transformations so that the upper components of $\psi$ are decoupled from the lower components to a given order in $1/m_N$. This makes the evaluation of the fermion determinant (i\_0 -[H]{}) = (id\^4x \^ (i\_0- [H]{} )) \[I4\] straightforward. Since we are interested in terms of ${\cal O} (p^4)$ in ${\cal H}$, we must work to ${\cal O}(1/m_N^3)$.
Operators which do not connect the upper and lower components will be called even; operators which connect upper components only with lower components will be called odd. Algebraically, an even operator ${\cal E}$ obeys ${\cal
E}\beta=\beta{\cal E}$, and an odd operator ${\cal O}$ obeys ${\cal
O}\beta=-\beta{\cal O}$. We may write = &&m\_N + [E]{} + [O]{}\
[E]{} = &&-i\_0 -ig\_A\^i\_i + (U+U\^ -2)\
[O]{} = &&(+i)-ig\_A\_5\_0 + \_5 (U\^-U) \[I5\] We now apply the unitary transformation $\psi=e^{-iS}\psi'$, where $S$ is taken as ${\cal O}(1/m_N)$. Expanding the exponential to the desired order &&\^ (i\_0 -[H]{})= [’]{}\^(i\_0 -[H]{}’ ) ’\
&&[H]{}’=[H]{}+ i \[S, [H]{}\] -12 \[S, \[S, [H]{}\]\]\
&&-i6 \[S, \[S, \[S, [H]{}\]\]\] +1[24]{} \[S, \[S, \[S, \[S, [m\_N]{}\]\]\]\]\
&&--i2 \[S, \] + 16 \[S,\[S, \]\] \[I6\] where the dot denotes the time derivative. To cancel the odd term to ${\cal O}
(m_N^0)$, we choose $S=-i\beta {\cal O}/2m_N$, which is consistent with our initial assumption.
Substitution into Eq. (\[I6\]) gives ’ = &&m\_N + [E]{}’ + [O]{}’\
[E]{}’= &&[E]{} +1[2m\_N]{} \^2 -1[8m\_N\^2]{} \[[O]{}, \[[O]{}, [E]{} \]\]\
&&-1[8m\_N\^3]{} \^4 -i[8m\_N\^2]{} \[[O]{}, \]\
[O]{}’=&&1[2m\_N]{} - 1[3m\_N\^2]{} [O]{}\^3 -1[48m\_N\^3]{} \[, \[[O]{}, \[[O]{}, [E]{}\]\]\]\
&&+i[2m\_N]{} -i[48m\_N\^3]{} \[, \[[O]{}, \]\] \[I7\] The odd term is now ${\cal O}(1/m_N)$. Applying a second unitary transformation $\psi'={\rm exp} (-\beta{\cal O}'/2m_N)\psi''$ ” = &&m\_N + [E]{}” + [O]{}”\
[E]{}”= &&[E]{}’ -1[4m\_N\^3]{}(\[[O]{}, [E]{}\] + i)\^2\
[O]{}”=&& 1[2m\_N]{} +i[2m\_N]{} ’ \[I8\] The odd term is now ${\cal O} (1/m_N^2)$. Applying a third unitary transformation $\psi''={\rm exp} (-\beta{\cal O}''/2m_N)\psi'''$ ”’ = &&m\_N + [E]{}” + [O]{}”’\
[O]{}”’= &&1[2m\_N]{} + i[2m\_N]{} ” \[I9\] The odd term is now ${\cal O}(1/m_N^3)$. Applying a fourth unitary transformation $\psi'''={\rm exp} (-\beta{\cal O}'''/2m_N)\psi''''$ ”” = m\_N + [E]{}” + [O]{} (1[m\_N\^4]{}) \[I10\] so we have an even Hamiltonian to the desired order.
We may note that the TFW transformations preserve charge conjugation symmetry, so the Hamiltonian Eq. (\[I10\]) can be used for the nonrelativistic $N\overline{N}$ system, in contrast with previous work.
There is one worry, namely that the functional Jacobian of the transformations. However, one can plausibly argue they are 1. In the case of the chiral anomaly [@FUJIKAWA], the natural basis for expanding $\psi$ was the eigenfunctions of the massless Dirac operator, which anticommuted with the generator of axial transformations $\gamma_5$. In our case the natural basis for expanding $\psi$ should be $\beta$ diagonal, since it anticommutes with the generators $\beta {\cal O}$, $\beta {\cal O}'$, $\beta {\cal O}''$, and $\beta {\cal O}'''$. However, $\beta$ has no zero modes, so the anomaly should vanish.
Counterterms necessary to absorb loop divergences may be derived in the standard manner, by using the BPHZ scheme for instance with on-shell subtractions. In this way, the nucleon mass appearing in the expansion at tree level is the renormalized mass. Incidentally, one observes that the relativistic one-loop calculations yield terms of order $p^2{\rm ln} m_N/m_N^n$ [@GSS; @ALL]. How these terms are generated after nonrelativistic reduction deserves further investigation. Furthermore, the ${\cal O} (m_N^0)$ term from Eq. (\[I10\]) gives \_0 = -i\_0 -ig\_A\^i\_i + (U+U\^ -2) \[I11\] as the improved version of the static model. Does it account for the $\Delta$ ?
In terms of the present construction, the $\pi N$ scattering amplitude can be constructed and partial wave unitarity tested [@STEELE]. Explicit calculations using the tree results show that the S31 wave gives a bound on the pion momentum to be $k\leq 0.28\,\,{\rm GeV}$. This is close to the bound established above using $\pi\pi$ scattering.
Conclusions
===========
In $\pi\pi$ and $\pi N$ scattering, if we are to reach 300 MeV and beyond, there are three possible courses of action. Use generalized chiral perturbation theory [@GENERALIZED]. However, this tends to decrease predictive power. Try unitarization [@UNITARIZATION]. However, this breaks crossing symmetry. The one we favor is the master formula approach developed recently [@MASTER]. In this approach, the $\pi\pi$ scattering amplitude is reduced to a sum of Green’s functions and form factors, some of which are measurable. By making educated guesses about the unknown pieces, one may test chiral symmetry even at $\rho$ energies.
In a way, in $\pi\pi$ scattering say, one should not be worried if lowest order predictions of chiral symmetry are off by 20 % rather than 1.4 % for $p\sim 140$ MeV. However, one should be aware that the predictions can be significantly off already for $p\sim 300$ MeV.
Acknowledgements {#acknowledgements .unnumbered}
================
The results in this work are dedicated to Mannque Rho for his sixtieth birthday. Mannque has inspired us throughout our careers, and we take this opportunity to thank him for his friendship and support. This work was supported in part by the US DOE grant DE-FG-88ER40388.
References {#references .unnumbered}
==========
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T.N. Truong, [*Phys. Rev. Lett.*]{} [22]{} (1988) 2526; D. Morgand and M.R. Pennington, [*Phys. Lett.*]{} [B272]{} (1991) 134.
H. Yamagishi and I. Zahed, [*Phys. Rev.*]{} [D53]{} (1996) 2288; [*Ann. Phys.*]{} [247]{} (1996) 292.
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[^1]: The general case can be found in Steele et al. [@STEELE], for which the present arguments also apply.
|
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abstract: 'We derive upper and lower limits for the basic physical parameters (mass-radius ratio, anisotropy, redshift and total energy) for arbitrary anisotropic general relativistic matter distributions in the presence of a cosmological constant. The values of these quantities are strongly dependent on the value of the anisotropy parameter (the difference between the tangential and radial pressure) at the surface of the star. In the presence of the cosmological constant, a minimum mass configuration with given anisotropy does exist. Anisotropic compact stellar type objects can be much more compact than the isotropic ones, and their radii may be close to their corresponding Schwarzschild radii. Upper bounds for the anisotropy parameter are also obtained from the analysis of the curvature invariants. General restrictions for the redshift and the total energy (including the gravitational contribution) for anisotropic stars are obtained in terms of the anisotropy parameter. Values of the surface redshift parameter greater than two could be the main observational signature for anisotropic stellar type objects.'
author:
- 'C. G. Böhmer'
- 'T. Harko'
title: Bounds on the basic physical parameters for anisotropic compact general relativistic objects
---
Introduction
============
It is generally believed that for smooth equations of state no stable stellar configurations with central densities above that corresponding to the limiting mass of neutron stars is stable against acoustical vibrational modes [@1]. The maximum allowed gravitational mass of a neutron star has been derived by using the properties of neutron matter at density ranges where they can be accurately predicted and imposing a minimum number of constraints at densities exceeding a higher fiducial density, $\rho _{0}$, e.g. subluminal sound velocity and thermodynamic stability . Following this approach it has been rigorously proved that the mass of a stable neutron star becomes maximum for the stiffest possible equation of state that is consistent with the fundamental physical constraints [@2]. As a result a maximum neutron star mass of $3.2M_{\odot }$ has been found. On the other hand, by using the static spherically symmetric gravitational field equations, Buchdahl [@3] has obtained an absolute constraint of the maximally allowable mass $M$-radius $R$ ratio for isotropic fluid spheres of the form $2M/R<8/9$ (in the present paper we use natural units so that $%
c=G=1$).
The study of the maximum mass and mass-radius ratio for compact stars has been done mainly for isotropic stellar objects, in which the tangential pressure equals the radial one. But, as suggested by Ruderman [@4], theoretical investigations of more realistic stellar models show that the stellar matter may be anisotropic at least in certain very high density ranges ($\rho >10^{15}$ g/cm$^{3}$), where the nuclear interactions must be treated relativistically. According to these views in such massive stellar objects the radial pressure $p_{r}$ may not be equal to the tangential one $%
p_{\perp }$, $p_{r}\neq p_{\perp }$. No celestial body is composed of purely perfect fluid. Anisotropy in fluid pressure could be introduced by the existence of a solid core or by the presence of type 3A superfluid [@5], different kinds of phase transitions [@6], pion condensation [@7] or by other physical phenomena. A slowly rotating system can be formally described as a static anisotropic fluid [@8]. The mixture of two gases (e.g., monatomic and molecular hydrogen, or ionized hydrogen and electrons) can also be interpreted as an anisotropic fluid [@9]. For a review of the appearance of local anisotropy in self-gravitating systems and of its main physical consequences see [@HeSa97].
For arbitrarily large anisotropy, in principle there is neither limiting mass nor limiting redshift [@10]. Semi-realistic equations of state lead to a mass of $3-4M_{\odot }$ for neutron stars with an anisotropic equation of state [@10]. Bowers and Liang [@11] have analytically obtained the maximum equilibrium mass and surface redshift in the case of incompressible neutron matter. They also numerically investigated models with a special form of anisotropy, founding that specific models lead to increases in the redshift proportional to the deviations from isotropy.
Bondi [@12] considered the relation between redshift and the ratio of the trace of the pressure tensor to local density. When anisotropic pressures are allowed considerably larger redshift values can be obtained. Several classes of solutions of the gravitational field equations for anisotropic matter distributions have been obtained in [@HaMa02] and [@MaHa03].
The value of the bound $M/R$ is an important problem in relativistic astrophysics since “the existence of such a bound is intriguing because it occurs well before the appearance of an apparent horizon at $M=R/2$” [@13]. In [@13] the upper limit of $M/R$ for compact general relativistic configurations has been re-investigated by assuming that inside the star the radial stress $p_{r}$ is different from the tangential one $p_{\perp }$. If the density is monotonically decreasing and $p_{r}\geq p_{\perp
}$ then the upper bound $8/9$ is still valid to the entire bulk if $m$ is replaced by the quasi-local mass. This bound cannot be recovered if $p_{\perp }\geq $ $%
p_{r}$ and / or the density is not a monotonic function.
The maximum value of the redshift for anisotropic stars was derived in [@Iv02]. For realistic anisotropic star models the surface redshift cannot exceed the values $3.842$ or $5.211$ when the tangential pressure satisfies the strong or the dominant energy condition, respectively. Both values are higher than 2, the bound in the perfect fluid case. Several bounds on the important physical parameters for the anisotropic stars have been derived in [@BaHaGl03]. If the radial pressure is larger than the tangential pressure, then the radial pressure is also larger than the corresponding pressure for a fiducial isotropic model with the same mass function and total mass, while the opposite holds if the tangential pressure is larger than the radial one. By imposing an energy condition, the value for the maximum possible redshift at the surface of the star can be obtained.
Several bounds on the mass-radius ratio and anisotropy parameter have also been found, for models in which the anisotropy increases as $r^2$, in [@MaDoHa02].
One of the most important results of modern cosmology is the observational evidence for the existence of the cosmological constant. The first pressing piece of data involved the study of Type Ia Supernovae. Observations of Type Ia Supernovae with redshift up to about $z\sim 1$ provided evidence that we may live in a low mass-density Universe, with the contribution of the non-relativistic matter (baryonic plus dark) to the total energy density of the Universe of the order of $\Omega _{m}\sim 0.3$ [@Ri98; @Pe98]. The value of $\Omega _{m}$ is significantly less than unity [@OsSt95], and consequently either the Universe is open or there is some additional energy density $\rho $ sufficient to reach the value $\Omega_{\mathrm{total}}=1$, predicted by inflationary theory.
The existence of the cosmological constant modifies the allowed ranges for various physical parameters, like, for example, the maximum mass of compact stellar objects, thus leading to modifications of the “classical” Buchdahl limit [@MaDoHa00].
On the other hand, we cannot exclude *a priori* the possibility that the cosmological constant, as a manifestation of vacuum energy, may play an important role not only at galactic or cosmological scales, but also at the level of elementary particles. With the use of the generalized Buchdahl identity [@MaDoHa00], it can be rigorously proven that the existence of a non-negative $\Lambda $ imposes a lower bound on the mass $M$ and density $%
\rho $ of general relativistic objects of radius $R$, which is given by [@BoHa05] $$2M\geq \frac{8\pi \Lambda }{6}R^{3},\qquad \rho =\frac{3M}{4\pi
R^{3}}\geq \frac{\Lambda }{2}=:\rho _{\min }. \label{minm}$$
Therefore, the existence of the cosmological constant implies the existence of an absolute minimum mass and density in the universe. No object present in relativity can have a density that is smaller than $\rho_{\min }$. For $\Lambda >0$ this result also implies a minimum density for stable fluctuations in energy density.
There are some other astrophysical systems which may be modeled, at least at a qualitative level, by using an effective “cosmological constant”. For example, there is the possibility that scalar fields present in the early universe could condense to form the so named boson stars [@Ry97; @MiSc00; @ScMi03]. The simplest kind of boson star is made up of a self-interacting complex scalar field $\Phi $ describing a state of zero temperature [@ScMi03; @SeSu90; @Ku91]. If we suppose that in the star’s interior regions and for some field configurations $\Phi$ is a slowly varying function of $r$, so that it is nearly a constant, then in the gravitational field equations the scalar field will play the role of a cosmological constant, which could also describe a mixture of ordinary matter and bosonic particles.
It is the purpose of the present paper to obtain the minimum and the maximum allowable mass-radius ratio in the case of anisotropic compact general relativistic objects, in the presence of a cosmological constant, as a function of the anisotropy parameter. We found that for anisotropic compact general relativistic bodies an upper limit (different from half) does not exist in general. Consequently there are no limits on the red shift of the radiation coming from this type of objects. On the other the presence of the anisotropy induces a minimum mass-radius ratio even in the absence of the cosmological constant. Upper bounds on the anisotropy are also derived by using the properties of the linear and quadratic scalars formed out of the curvature tensor (the Ricci invariants).
The present paper is organized as follows. The Buchdahl limit for anisotropic general relativistic objects in the presence of a cosmological constant is derived in Section II. In Section III we consider the problem of the minimum mass of anisotropic compact objects. Upper limits for the surface anisotropy of relativistic stars are obtained in Section IV. We discuss our results and conclude our paper in Section V.
The Buchdahl limit for anisotropic stars
========================================
For a static general relativistic spherically symmetric matter configuration the interior line element is given by $$ds^{2}=e^{\nu }dt^{2}-e^{\lambda }dr^{2}-r^{2}\left (d\theta
^{2}+\sin ^{2}\theta d\varphi ^{2}\right ).$$
We assume that the star consists of an anisotropic fluid distribution in the presence of a cosmological constant. For such a system the components of the energy-momentum tensor are $$T_{0}^{0}=\rho +\Lambda ,\quad T_{1}^{1}=-p_{r}+\Lambda ,\quad
T_{2}^{2}=T_{3}^{3}=-p_{\perp }+\Lambda ,$$ where $\rho $ is the energy density and $\Lambda $ is the cosmological constant.
We suppose that inside the star $p_{r}\neq p_{\perp }$, $\forall
r\neq 0$. We define the anisotropy parameter as $\Delta =p_{\perp
}-p_{r}$. $\Delta $ is a measure of the deviations from isotropy. If $\Delta >0,\forall r\neq 0$ the body is tangential pressure dominated while $\Delta <0$ indicates that $%
p_{r}>p_{\perp }$. In realistic physical models for compact stars, $\Delta $ should be finite, positive and should satisfy the dominant energy condition (DEC) $\Delta \leq \rho $ and the strong energy condition (SEC) $2\Delta +p_{r}\leq \rho $. These conditions may be written together as $\Delta \leq n\rho $, where $n=1$ for DEC and $n=1/2$ for SEC, if the realistic condition for the positivity of $p_{r}$ in the interior is accepted [@Iv02].
The properties of the anisotropic compact object can be completely described by the gravitational structure equations, which are given by: $$\label{5}
\frac{dm}{dr}=4\pi \rho r^{2},$$ $$\label{6}
\frac{dp_{r}}{dr}=-\frac{\left (\rho +p_{r}\right )\left [m+4\pi
r^{3}\left
(p_{r}-\frac{2\Lambda }{3}\right )\right ]}{r^{2}\left (1-\frac{2m}{r}-\frac{%
8\pi }{3}\Lambda r^2\right )}+\frac{2\Delta }{r},$$ $$\label{7}
\frac{d\nu }{dr}=-\frac{2}{\rho +p_{r}}\frac{dp_{r}}{dr}+\frac{4\Delta }{%
r\left (\rho +p_{r}\right )},$$ where $m(r)$ is the mass inside radius $r$ .
In the Newtonian limit and in the absence of the cosmological constant Eq. (\[6\]) reduces to the expression [@HeSa97] $$\frac{dp_{r}}{dr}=-\frac{m\rho }{r^{2}}+\frac{2\Delta }{r}.$$
Hence the anisotropy term is Newtonian in origin [@11]. A solution of Eqs. (\[5\])-(\[7\]) is possible only when boundary conditions have been imposed. As in the isotropic case we require that the interior of any matter distribution be free of singularities, which imposes the condition $%
m(r)\rightarrow 0$ as $r\rightarrow 0$. Assuming that $p_{r}$ is finite at $%
r=0$, we have $\nu ^{\prime }\rightarrow 0$ as $r\rightarrow 0$. Therefore the gradient $dp_{r}/dr$ will be finite at $r=0$ only if $\Delta $ vanishes at least as rapidly as $r$ when $r\rightarrow
0$. This requires that the anisotropy parameter satisfies the boundary condition $$\lim _{r\rightarrow 0}\frac{\Delta \left (r\right )}{r}=0.$$
At the center of the star the other boundary conditions for Eqs. (\[5\])-(\[7\]) are $p_{r}(0)=p_{\perp }(0)=p_{c}$ and $\rho (0)=\rho _{c}$, where $%
\rho _{c}$ and $p_{c}$ are the central density and pressure, respectively. The radius $R$ of the star is determined by the boundary condition $%
p_r\left (R\right )=0$. We do not necessarily require that the tangential pressure $p_{\perp }$ vanishes for $r=R$. Therefore at the surface of the star the anisotropy parameter satisfies the boundary condition $\Delta (R)=p_{\perp }(R)-p_{r}(R)=p_{\perp
}(R)\geq 0$. To close the field equations the equations of state of the radial pressure $p_r=p_r\left (\rho \right )$ and of the tangential pressure $p_{\perp }=p_{\perp }\left (\rho \right )$ must also be given.
With the use of Eqs. (\[5\])-(\[7\]) it is easy to show that the function $\zeta =e^{\nu /2}>0,\forall r\in \lbrack 0,R\rbrack
$, obeys the equation $$\label{9} \frac{y}{r}\frac{d}{dr}\left [\frac{y}{r}\frac{d\zeta
}{dr}\right ]=\frac{\zeta }{r}\left
[\frac{d}{dr}\frac{m(r)}{r^{3}}+\frac{8\pi \Delta }{r}\right ],$$ where we denoted $$\alpha (r)=1+\frac{4\pi }{3}\Lambda \frac{r^3}{m(r)}, \qquad y(r)
= \sqrt{1-\frac{2\alpha(r)m(r)}{r}}.$$
For $\Delta =0$ and $\Lambda =0$ Eq. (\[9\]) reduces to the isotropic equation considered in [@14]. Since the density $\rho $ does not increase with increasing $r$, the mean density of the matter $\langle \rho \rangle =3m(r)/4\pi r^{3}$ inside radius $r$ does not increase either.
Therefore we assume that inside a compact general relativistic object the condition $$\frac{d}{dr}\frac{m(r)}{r^{3}}<0,$$ holds independently of the equation of state of dense matter. By defining a new function $$\label{10}
\eta(r) = 8\pi \int _{0}^{r} \frac{r'}{y(r')} \left \{ \int
_{0}^{r'} \frac{\Delta(r'')}{y(r'')} \frac{\zeta(r'')}{r''} dr''
\right \}dr',$$ denoting $$\Psi =\zeta -\eta ,$$ and introducing a new independent variable $$\xi =\int _{0}^{r} \frac{r'}{y(r')} dr',$$ from Eq. (\[9\]) we obtain the basic result that all stellar type general relativistic matter distributions with negative density gradient obey the condition $$\label{12}
\frac{d^{2}\Psi }{d\xi ^{2}}<0,\quad \forall r\in \left [0,R\right
].$$
Using the mean value theorem we conclude [@14] $$\frac{d\Psi }{d\xi }\leq \frac{\Psi \left (\xi \right )-\Psi
(0)}{\xi },$$ or, taking into account that $\Psi (0)>0$, we find $$\Psi ^{-1}\frac{d\Psi }{d\xi }\leq \frac{1}{\xi }.$$
In the initial variables we have $$\begin{gathered}
\label{15}
\frac{y(r)}{r} \left(\frac{1}{2}\frac{d\nu }{dr}e^{\nu (r)/2}-8\pi
\frac{r}{y(r)} \int _{0}^{r}\frac{\Delta(r') e^{\nu(r')/2}}{y(r')
r'} dr' \right)\\ \leq \frac{e^{\nu (r)/2}-8\pi \int _{0}^{r}
\frac{r'}{y(r')} \left( \int _{0}^{r'} \frac{\Delta \left
(r''\right )e^{\nu(r'')/2}}{y(r'') r''} dr'' \right) dr'}{\int
_{0}^{r} \frac{r'}{y(r')} dr'}.\end{gathered}$$
Since for stable stellar type compact objects $m/r^{3}$ does not increase outwards, the condition $$\frac{m(r^{\prime })}{r^{\prime }}\geq \frac{m(r)}{r}\left (\frac{r^{\prime }%
}{r}\right )^{2},\quad \forall r^{\prime }\leq r,$$ holds for all points inside the star [@14]. Moreover, we assume that in the presence of a cosmological constant, the condition $$\frac{\alpha \left (r^{\prime }\right )m\left (r^{\prime }\right )}{%
r^{\prime }}\geq \frac{\alpha \left (r\right )m\left (r\right )}{r}\left (%
\frac{r^{\prime }}{r}\right )^{2},$$ or, equivalently, $$\left [1+\frac{4\pi }{3}\Lambda \frac{r^{\prime 3}}{m\left
(r^{\prime }\right )}\right ]\frac{m\left (r^{\prime }\right
)}{r^{\prime }}\geq \left
[1+\frac{4\pi }{3}\Lambda \frac{r^{3}}{m\left (r\right )}\right ]\frac{%
m\left (r\right )}{r}\left (\frac{r^{\prime }}{r}\right )^{2},
\label{cond1}$$ holds inside the compact object. In fact Eq. (\[cond1\]) is satisfied for all values of the cosmological constant $\Lambda $ and is valid for all decreasing density compact matter distributions.
In the following we assume that the anisotropy function satisfies the general condition $$\label{16}
\frac{\Delta (r^{\prime \prime })e^{\frac{\nu \left (r^{\prime
\prime
}\right )}{2}}}{r^{\prime \prime }}\geq \frac{\Delta (r^{\prime })e^{\frac{%
\nu \left (r^{\prime }\right )}{2}}}{r^{\prime }}\geq \frac{\Delta (r)e^{%
\frac{\nu (r)}{2}}}{r},r^{\prime \prime }\leq r^{\prime }\leq r.$$
This condition is quite natural, taking into account that, since the matter satisfies the dominant and strong energy conditions, the anisotropy parameter is a monotonically decreasing function inside the star.
Therefore we can evaluate the denominator in the RHS of Eq. (\[15\]) as follows: $$\begin{aligned}
\label{17}
\int _{0}^{r} \frac{r'}{y(r')}dr^{\prime }\geq
\int _{0}^{r}r^{\prime }\left [1-\frac{2\alpha \left (r\right )m(r)}{r^{3}}%
r^{\prime 2}\right ]^{-1/2}dr^{\prime }=\frac{r^{3}}{2\alpha \left
(r\right )m(r)}\left(1-y(r)\right).\end{aligned}$$
For the second term in the bracket of the LHS of Eq. (\[15\]) we find: $$\begin{gathered}
\label{19}
\int _{0}^{r} \frac{\Delta \left (r'\right )e^{\nu(r')/2}}{y(r')
r'}dr' \geq
\frac{\Delta(r)e^{\nu (r)/2}}{r}\int _{0}^{r}\left [1-\frac{2\alpha \left (r\right )m(r)%
}{r^{3}}r^{\prime 2}\right ]^{-1/2}dr^{\prime } \\ = \Delta
(r)e^{\nu (r)/2}\left [\frac{2\alpha \left (r\right
)m(r)}{r}\right
]^{-1/2} \arcsin\left (\sqrt{\frac{2\alpha \left (r\right )m(r)}{r}}%
\right ).\end{gathered}$$
The second term in the nominator of the RHS of Eq. (\[15\]) gives: $$\begin{gathered}
\label{20} \int _{0}^{r}\frac{r'}{y(r')} \left
\{\int_{0}^{r'}\frac{\Delta(r'')
e^{\nu(r'')/2}}{y(r'')r''}dr''\right \}dr' \\ \geq \int
_{0}^{r}r^{\prime 2}\frac{\Delta \left (r^{\prime
}\right )e^{\nu \left (y(r')r^{\prime }\right )/2}}{r^{\prime }}\left [\frac{%
2\alpha \left (r^{\prime }\right )m(r^{\prime })}{r^{\prime
}}\right ]^{-1/2} \arcsin \left (\sqrt{\frac{2\alpha \left
(r^{\prime }\right )m(r^{\prime })}{r^{\prime }}}\right
)dr^{\prime } \\ \geq \frac{\Delta (r)e^{\nu (r)/2}}{r}\int
_{0}^{r}r^{\prime 2} \left [1-\frac{2\alpha \left (r\right
)m(r)}{r^{3}}r^{\prime 2}\biggl/ \frac{2\alpha \left (r\right
)m(r)}{r^{3}}r^{\prime 2}\right ]^{-1/2} \arcsin\left
[\sqrt{\frac{2\alpha \left (r\right )m(r)}{r^{3}}}r^{\prime}\right
] dr^{\prime } \\ =
\Delta (r)e^{\nu (r)/2}r^{2}\left [\frac{2\alpha \left (r\right )m(r)}{r}%
\right ]^{-3/2}\left \{\sqrt{\frac{2\alpha \left (r\right
)m(r)}{r}}- y(r) \arcsin\left [\sqrt{\frac{2\alpha \left (r\right
)m(r)}{r}}\right ]\right \}.\end{gathered}$$
In order to obtain (\[20\]) we have also used the property of monotonic increase of the function $\arcsin x/x$ for $x\in \left
[0,1\right ]$.
Using Eqs. (\[17\])-(\[20\]), Eq. (\[15\]) becomes: $$\begin{gathered}
\label{21}
\left \{1-\left [1-\frac{2\alpha \left (r\right )m(r)}{r}\right
]^{1/2}\right \}\frac{m(r)+4\pi r^{3}\left (p_{r}-\frac{2\Lambda }{3}\right )%
}{r^{3}\sqrt{1-\frac{2\alpha \left (r\right )m(r)}{r}}} \\ \leq
\frac{2\alpha \left (r\right )m(r)}{r^{3}}+8\pi \Delta (r)\left
\{\frac{
\arcsin\left [\sqrt{\frac{2\alpha \left (r\right )m(r)}{r}}\right ]}{%
\sqrt{\frac{2\alpha \left (r\right )m(r)}{r}}}-1\right \}.\end{gathered}$$
Eq. (\[21\]) is valid for all $r$ inside the star. It does not depend on the sign of $\Delta $.
Consider first the isotropic case $\Delta =0$ and $\Lambda =0$. By evaluating (\[21\]) for $r=R$ we obtain $$\frac{1}{\sqrt{1-\frac{2M}{R}}}\leq 2\left [1-\left (1-\frac{2M}{R}\right )^{%
\frac{1}{2}}\right ]^{-1},$$ leading to the well-known result $2M/R\leq 8/9$ [@3], [@14].
By taking $\Delta =0$ but considering $\Lambda \neq 0$ we obtain the following upper limit for the mass-radius ratio of a compact object [@MaDoHa00]: $$\frac{2M}{R}\leq \left (1-\frac{8\pi }{3}\Lambda R^{2}\right )\left [1-\frac{%
1}{9}\frac{\left (1-2\Lambda /\langle\rho\rangle \right )^{2}%
}{1-\frac{8\pi }{3}\Lambda R^{2}}\right ].$$
Next consider the case $\Delta \neq 0$ and $\Lambda \neq 0$. We denote $$f\left (M,R,\Lambda ,\Delta \right )=2\frac{\Delta \left (R\right
)}{
\langle \rho \rangle }\left \{\frac{ \arcsin\left [\sqrt{\frac{%
2\alpha \left (R\right )M}{R}}\right ]}{\sqrt{\frac{2\alpha \left (R\right )M%
}{R}}}-1\right \}.$$
Then Eq. (\[21\]) leads to the following restriction on the mass-radius ratio for compact anisotropic stars in the presence of a cosmological constant: $$\label{24}
\frac{2M}{R}\leq \left (1-\frac{8\pi }{3}\Lambda R^{2}\right )\left [1-\frac{%
1}{9}\frac{\left (1-2\Lambda /\langle \rho \rangle \right )^{2}%
}{\left (1-\frac{8\pi }{3}\Lambda R^{2}\right )\left (1+f\right
)^{2}}\right].$$
For a static general relativistic object the condition $1-2M/R-8\pi \Lambda
R^2/3\geq 0$ must hold for all $R$, $M$ and $\Lambda $. Therefore from Eq. (\[24\]) we obtain that $\Delta (R)$ must obey the general condition $(1-2\Lambda /\langle \rho \rangle)^{2}/(1+f)^2>0$, which holds for all $\Delta $. Hence generally we cannot obtain any limiting value for $\Delta (R)$ from Eq. (\[24\]). But for a monotonically decreasing anisotropy several upper bounds for the anisotropy parameter can be derived, as will be shown in Section IV.
The minimum mass of the anisotropic general relativistic objects
================================================================
On the vacuum boundary of the anisotropic star, corresponding to $r=R$, Eq. (\[21\]) takes the equivalent form $$\label{an1}
\sqrt{1-\frac{2M}{R}-\frac{8\pi }{3}\Lambda R^{2}}\geq \frac{1}{3}\left (1-%
\frac{2\Lambda }{\langle \rho \rangle }\right )\frac{1}{1+f\left
(M,R,\Delta ,\Lambda \right )}.$$
For small values of the argument the function $\arcsin x/x-1$ which appears in the definition of $f$ can be approximated as $\arcsin x/x-1\approx x^{2}/6$. Therefore, Eq. (\[an1\]) can be written as $$\label{an2}
\sqrt{1-\frac{2M}{R}-\frac{8\pi }{3}\Lambda R^{2}}\geq \frac{M-\frac{8\pi }{3%
}\Lambda R^{3}}{3M+\frac{4\pi }{3}\Delta (R)R^{2}\left (2M+\frac{8\pi }{3}%
\Lambda R^3\right )}.$$
By introducing a new variable $u$ defined as $$u=\frac{M}{R}+\frac{4\pi }{3}\Lambda R^{2},$$
Eq. (\[an2\]) takes the form $$\sqrt{1-2u}\geq \frac{u-a}{bu-a}, \label{an3}$$ where we denoted $a=4\pi \Lambda R^{2}$ and $b=3+8\pi \Delta \left
(R\right )R^{2}/3$, respectively. Then, by squaring we can reformulate the condition given by Eq. (\[an3\]) as $$u\left [2b^{2}u^{2}-\left (b^{2}+4ab-1\right )u+2a\left
(a+b-1\right )\right ]\leq 0,$$ or, equivalently, $$\label{cond2}
u\left (u-u_{1}\right )\left (u-u_{2}\right )\leq 0,$$ where $$u_{1}=\frac{b^{2}+4ab-1-\left (1-b\right
)\sqrt{(1+b)^{2}-8ab}}{4b^{2}},$$ and $$u_{2}=\frac{b^{2}+4ab-1+\left (1-b\right
)\sqrt{(1+b)^{2}-8ab}}{4b^{2}},$$ respectively.
In the following we keep only the first order terms in both $\Lambda $ and $%
\Delta $. Since $u\geq 0$, Eq. (\[cond2\]) is satisfied if $u\leq u_{1}$ and $u\geq u_{2}$, or $u\geq u_{1}$ and $u\leq
u_{2}$. However, the condition $u\geq u_{1}$ contradicts the upper bound given by Eq. (\[24\]).
Therefore, Eq. (\[cond2\]) is satisfied if and only if for all values of the physical parameters the condition $u\geq u_{2}$ holds. This is equivalent to the existence of a minimum bound for the mass-radius ratio of compact anisotropic objects, which is given by $$u\geq \frac{2a}{1+b},$$and explicitly written out using $a,b$ and $u$ as defined above yields $$\frac{2M}{R}\geq \frac{8\pi \Lambda }{6}R^{2}\left( \frac{1-\frac{4\pi }{3}%
\Delta R^{2}}{1+\frac{2\pi }{3}\Delta R^{2}}\right) . \label{min}$$
The presence of the anisotropy weakens the lower bound on the mass, however, there still exists an absolute minimal mass in nature. In the case $\Delta \equiv 0$, we recover the lower bound for the minimum mass and density for isotropic general relativistic objects, obtained in [@BoHa05]. In this case the existence of a minimum mass is determined by the presence of the cosmological constant only. For $\Lambda \equiv 0$, the presence of an anisotropic pressure distribution reduces to the requirement of the positivity of $M$, $M\geq 0$.
For isotropic systems with $\Delta \equiv 0$ and for a value of the cosmological constant of the order of $\Lambda \approx 3\times 10^{-56}$ cm$%
^{-2}$, the numerical value of minimum density following from Eq. (\[min\]) is $\rho _{\min }\approx 8\times 10^{-30}$ g/cm$^{3}$. By assuming the existence in nature of an absolute minimum length of the order of the Planck length $l_{Pl}$ it follows that the corresponding absolute minimum mass is of the order of $1.4\times
10^{-127}$ g. However, by combining the minimum mass condition with energy stability conditions objects with masses as high as $10^{55}$ g can also be obtained (for a detailed discussion of the properties of the minimum mass particles see [@BoHa06]).
Bounds on the surface anisotropy of compact objects
===================================================
Curvature is described by the tensor field $R^{l}{}_{ijk}$. It is well known that if one uses singular behavior of the components of this tensor or its derivatives as a criterion for singularities, one gets into trouble since the singular behavior of components could be due to singular behavior of the coordinates or tetrad basis rather than that of the curvature itself. To avoid this problem, one should examine the linear and quadratic scalars formed out of curvature.
In order to find a general restriction for $\Delta (R)$ we shall consider the behavior of the Ricci invariants $$r_{0}=R_{i}^{i}=R,$$ $$r_{1}=R_{ij}R^{ij},$$ and $$r_{2}=R_{ijkl}R^{ijkl},$$ respectively.
If the static line element is regular, satisfying the conditions $e^{\nu (0)}=\mathrm{constant}\neq 0$ and $e^{\lambda (0)}=1$ , then the Ricci invariants are also non-singular functions throughout the star. In particular for a regular space-time the invariants are non-vanishing at the origin $r=0$ . For the invariant $r_{2}$ we find $$\begin{aligned}
\label{a}
&r_{2}=\left (16\pi \Delta +8\pi \rho +8\pi p_{r}-\frac{4m}{r^{3}}-\frac{%
16\pi \Lambda }{3}\right )^{2}+2\left (8\pi p_{r}-\frac{16\pi \Lambda }{3}+%
\frac{2m}{r^{3}}\right )^{2}+ \notag \\
&2\left (8\pi \rho +\frac{16\pi \Lambda
}{3}-\frac{2m}{r^{3}}\right )^{2}+4\left
(\frac{2m}{r^{3}}+\frac{8\pi \Lambda }{3}\right )^{2}.\end{aligned}$$
For a monotonically decreasing and regular anisotropy parameter $\Delta $, the function $r_{2}$ is also regular and monotonically decreasing throughout the star. Therefore it satisfies the condition $r_{2}(R)<r_{2}(0),$ leading to the following general constraint on $\Delta (R)$: $$\begin{aligned}
\Delta (R)&\leq \frac{\langle \rho \rangle +\Lambda
}{3}-\frac{\rho _s}{3}+
\notag \\
\frac{\rho _c}{6}&\sqrt{18\frac{p_c^2}{\rho _c^2}+15+12\frac{p_c}{\rho _c}%
\left (1-\frac{2\Lambda }{\rho _c}\right )+\left (3+2\frac{\langle
\rho
\rangle }{\rho _c}-6\frac{\rho _s}{\rho _c}\right )\frac{\Lambda }{\rho _c}%
-2\left (9\frac{\rho _s^2}{\rho _c^2}-6\frac{\langle \rho \rangle }{\rho _c}%
\frac{\rho _s}{\rho _c}+4\frac{\langle \rho \rangle ^2}{\rho
_c^2}\right )}, \label{b}\end{aligned}$$ where $\rho _{s}$ is the value of the density at the surface of the star, $%
\rho _{s}=\rho (R)$.
Another condition on $\Delta (R)$ can be obtained from the study of the scalar $$\label{e}
r_{1}=64\pi ^{2}\left [\left (\rho +\Lambda \right )^{2}+3\left
(p_{r}-\Lambda \right )^{2}+2\Delta \left (\Delta +2p_{r}-2\Lambda
\right )\right ].$$
Under the same assumptions of regularity and monotonicity for the functions $%
r_{1}$ and $\Delta (r)$ and considering a non-vanishing surface density $%
\rho _{s}\neq 0$ we find for anisotropy parameter at the surface of the star the upper limit $$\label{f}
\Delta (R)\leq \rho _c\sqrt{\frac{1}{2}\left
(1+3\frac{p_{c}^{2}}{\rho
_{c}^{2}}-\frac{\rho _s^2}{\rho _c^2}\right )-\left (3\frac{p_c}{\rho _c}+%
\frac{\rho _s}{\rho _c}-1\right )\frac{\Lambda }{\rho _c}+\frac{\Lambda ^2}{%
\rho _c^2}}+\Lambda .$$
The invariant $$r_{0}=-8\pi \left (\rho -3p_r-2\Delta +4\Lambda \right ),$$ leads to the following bound for the anisotropy: $$\label{bound1}
\Delta (R)\leq \frac{\rho _{c}}{2}\left [3\frac{p_{c}}{\rho
_{c}}+\frac{\rho _{s}}{\rho _{c}}-1\right ],$$ which is absolute in the sense that it does not depend on the value of the cosmological constant.
In the case of isotropic ($\Delta =0$) and stable regular fluid spheres the condition of monotonic decrease of the scalar $r_{2}$ is always satisfied. By assuming that $\rho _{s}=0$ (a condition which, for example, is readily satisfied by polytropic equations of state) from Eq. (\[b\]) we obtain the following upper bound for the mean density of the isotropic star: $$\label{c}
\langle \rho \rangle \leq \rho _c\sqrt{1+\frac{1}{4}\left (1+3\frac{p_c}{%
\rho _c}\right )^2+\left (1-3\frac{p_c}{\rho _c}\right
)\frac{\Lambda }{\rho _c}+\frac{3}{2}\left (\frac{\Lambda }{\rho
_c}\right )^2}.$$
If the central pressure of the star satisfies an equation of state of the form $p_c=\rho _c$, and in the absence of the cosmological constant ($%
\Lambda =0$), we obtain the following upper bound for the mean density of the star: $$\langle \rho \rangle \leq \sqrt{5}\rho _c.$$
For a radiation-like equation of state at the center, $p_c=\rho
_c/3$, the mean density of the star must satisfy the constraint $$\langle \rho \rangle \leq \sqrt{2}\rho _c.$$
These constraints are physically justified since we have assumed a monotonically decreasing density inside the compact general relativistic object. The conditions on the anisotropy and mean density obtained here have been derived only from the study of the behavior of the curvature invariants, without explicitly solving the gravitational field equations.
Discussions and final remarks
=============================
The existence of a limiting value of the mass-radius ratio leads to upper bounds for other physical quantities of observational interest. One of these quantities is the surface red shift $z$, defined according to $$z=\left [1-\frac{2\alpha \left (R\right )M}{R}\right ]^{-1/2}-1.$$
In the isotropic case $\Delta =0$ and in the absence of the cosmological constant, $\Lambda =0$, Eq. (\[21\]) leads to the well-known constraint $%
z\leq 2$ [@3; @14]. For an anisotropic star in the presence of a cosmological constant the surface red shift must obey the general restriction $$\label{46}
z\leq \frac{2+3f\left (M,R,\Lambda ,\Delta \right )+2\Lambda
/\langle \rho \rangle }{1-2\Lambda /\langle \rho \rangle }.$$
By keeping only the first order terms in $\Delta $ and $\Lambda $ Eq. ([46]{}) can be written as $$z\leq 2+3f\left (M,R,\Lambda ,\Delta \right )+6\frac{\Lambda
}{\langle \rho \rangle }.$$
Therefore much higher surface red shifts than $2$ could be observational criteria indicating the presence of anisotropic ultra-compact matter distributions.
By taking into account that the function $\arcsin x/x$ reaches its maximum value at $x=1$, it follows that the maximum value $f_{\max
}$ of the function $f$ can be approximated as $f_{\max }\approx
\Delta(R)/\langle \rho \rangle $. Therefore we obtain the following absolute upper bound for the redshift $$z\leq 2+\frac{3}{\langle \rho \rangle }\left (\Delta (R)+2\Lambda
\right ).$$
With the use of Eq. (\[bound1\]) we obtain the following general restriction on the redshift of anisotropic stars: $$z\leq 2+\frac{3}{2}\frac{\rho _{c}}{\langle \rho \rangle }\left
(3\frac{p_{c}}{\rho _{c}}+\frac{\rho _{s}}{\rho _{c}}-1\right )+6\frac{%
\Lambda }{\langle \rho \rangle }.$$
For high density compact general relativistic objects the mean density can be approximated by the central density. Thus we have $\rho _{c}\approx \langle \rho \rangle $. In the limit of high densities the equation of state of dense matter satisfies the Zeldovich equation of state $%
p=\rho $. We choose to assume that matter actually behaves in this manner at densities above about ten times nuclear, that is at densities greater than $10^{17}$ g/cm$^3$, or temperatures $T>(\rho /\sigma)^{1/4}$, where $\sigma $ is the radiation constant [@14]. Therefore, by neglecting the surface density ($\rho _{s}/\rho _{c}\approx 0$) and the effect of the cosmological constant, it follows that the maximum redshift of anisotropic stars must satisfy the condition $$z\leq 5,$$ value which is consistent with the bound $z\leq 5.211$ obtained by Ivanov [@Iv02].
However, if the equation of state of the compact matter at the center of the star satisfies a radiation-type equation of state, $p=\rho /3$, the redshift of the anisotropic stars satisfies the same upper bound as the isotropic general relativistic objects, $z\leq 2$. Therefore the surface redshift is strongly dependent on the physical conditions at the center of the star. By assuming that the density of the star is slowly varying, so that $\rho_c\approx \langle \rho \rangle \approx \rho _s$ and furthermore the equation of state of the dense matter at the center of the star satisfies the stiff Zeldovich equation of state $p_c=\rho _c$, then the surface redshift of anisotropic stars is constrained by $z\leq 7.5$. Hence very large values of the redshift may be the main observational signature of anisotropic stars.
As another application of the obtained upper mass-radius ratios we shall derive an explicit limit for the total energy of the compact general relativistic star. The total energy (including the gravitational field contribution) inside an equipotential surface $S$ can be defined to be [@15] $$E=E_{M}+E_{F}=\frac{1}{8\pi }\xi _{s}\int_{S}\left[ K\right] dS,
\label{47}$$where $\xi ^{i}$ is a Killing field of time translation, $\xi
_{s}$ its value at $S$ and $\left[ K\right] $ is the jump across the shell of the trace of the extrinsic curvature of $S$, considered as embedded in the 2-space $t=\mathrm{constant}$. $E_{M}=\int_{S}T_{i}^{k}\xi ^{i}\sqrt{-g}%
dS_{k}$ and $E_{F}$ are the energy of the matter and of the gravitational field, respectively. This definition is manifestly coordinate invariant. In the case of a static spherically symmetric matter distribution from Eq. ([47]{}) we obtain the following exact expression [@15]: $$E=-re^{\nu /2}\left[ e^{-\lambda /2}\right] .$$
Hence the total energy of a compact general relativistic object is $$E=R\left (1-\frac{2M}{R}-\frac{8\pi }{3}\Lambda R^2\right
)^{1/2}\left [1-\left (1-\frac{2M}{R}-\frac{8\pi }{3}\Lambda
R^2\right )^{1/2}\right ].$$
With the use of Eq. (\[21\]) we immediately find the following upper limit for the total energy of the star: $$E\leq 2R\frac{1+\Lambda /\langle \rho \rangle +3f\left
(M,R,\Lambda ,\Delta \right )/2}{1-2\Lambda /\langle \rho \rangle
} \left (1-\frac{2M}{R}-\frac{8\pi }{3}\Lambda R^{2}\right ).$$
For an isotropic matter distribution $\Delta =0$ and $$E\leq 2R\frac{1+\Lambda /\langle \rho \rangle }{1-2\Lambda
/\langle \rho \rangle }\left (1-\frac{2M}{R}-\frac{8\pi }{3}%
\Lambda R^{2}\right ).$$
In the case of a vanishing cosmological constant we obtain the upper bound $$E\leq 2R\left (1-\frac{2M}{R}\right ).$$
All the previous results on the mass-radius ratio for anisotropic stellar objects have been obtained by assuming the basic conditions (\[12\]) and (\[16\]). But for an arbitrary large anisotropy parameter $\Delta
$ we can not exclude in principle the situation in which these conditions do not hold. If, for example $\eta (r)>\Psi (r)$ ,$\forall r\neq 0$ or $$\frac{8\pi \Delta (r)}{r}+\frac{d}{dr}\frac{m}{r^{3}}>0,$$ then for a star with monotonically decreasing density instead of the condition (\[12\]) we must have $$\frac{d^{2}\Psi }{d\xi ^{2}}>0,\quad\forall r.$$
This situation corresponds to a tangential pressure dominated stellar structure with $\Delta (r)>0$. In this case we obtain a restriction on the minimum mass-radius ratio of the compact object of the form $$\left (1-\frac{8\pi }{3}\Lambda R^{2}\right )\left
[1-\frac{1}{9}\frac{\left
(1-2\Lambda /\langle \rho \rangle \right )^{2}}{\left (1-\frac{%
8\pi }{3}\Lambda R^{2}\right )\left (1+f\right )^{2}}\right
]<\frac{2M}{R}<1.$$
For this hypothetically ultra-compact anisotropic star $4/9$ is a lower bound for the mass-radius ratio.
In the present paper we have considered the mass-radius ratio bound for anisotropic compact general relativistic objects. Also in that case it is possible to obtain explicit inequalities involving $2M/R$ as an explicit function of the anisotropy parameter $\Delta $. Contrary to the isotropic case we have not found a universal limit (different from half) for this type of (possible) astrophysical objects. The surface red shift and the total energy (including the gravitational one) are strongly modified due to the presence of anisotropies in the pressure distribution inside the compact object. The mass-radius ratio depends very sensitively on the value of the anisotropy parameter at the surface of the star and different physical models can lead to very different mass-radius relations. A general feature of the behavior of physical parameters of anisotropic compact stars is that the increase in mass, red shift or total energy is proportional to the deviations from isotropy. Therefore there are no theoretical restrictions for these stellar type structures to extend up to the apparent horizon and achieve masses of the order of $M\leq R/2$ .
The authors would like to thank to the two anonymous referees for comments and suggestions that significantly improved the manuscript. C.G.B. wishes to thank the Department of Physics of the Hong Kong University where parts of this work has been performed. The work of C.G.B. is supported by research grant BO 2530/1-1 of the German Research Foundation (DFG). The work of T. H. was supported by a Seed Funding Programme for Basic Research of the Hong Kong Government.
[99]{}
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abstract: 'We have studied the energy dependence of the $pp$ elastic scattering data and the pion-photoproduction data at 90$^\circ$ c.m. angle in light of the new generalized counting rule derived for exclusive processes. We show that by including the helicity-nonconserving amplitudes and their interference with the Landshoff amplitude, we are able to reproduce the energy dependence of all the $pp$ elastic cross-section and spin-correlation (A$_{NN}$) data available above the resonance region. The pion-photoproduction data can also be described by this approach, however, data with much finer energy spacing is needed to confirm the oscillations about the scaling behavior. This study strongly suggests an important role for helicity-nonconserving amplitudes related to quark orbital angular momentum and for the interference of these amplitudes with the Landshoff amplitude at GeV energies.'
author:
- |
D. Dutta, H. Gao\
[*Triangle Universities Nuclear Laboratory and\
Department of Physics, Duke University, Durham, NC 27708, USA*]{}
title: '**The Generalized Counting Rule and Oscillatory Scaling**'
---
\#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{}
The transition between perturbative and non-perturbative regimes of Quantum Chromo Dynamics (QCD) is of long-standing interest in nuclear and particle physics. Exclusive processes play a central role in studies trying to map out this transition. The differential cross sections for many exclusive reactions [@white] at high energies and large momentum transfers appear to obey dimensional scaling laws [@brodsky] (also called quark counting rules). In recent years, the onset of this scaling behavior has been observed at a hadron transverse momentum of $\sim$ 1.2 (GeV/c) in deuteron photo-disintegration [@schulte; @rossi] and in pion photoproduction from nucleon [@zhu]. On the other hand, these models also predict hadron helicity conservation in exclusive processes [@hhc], and experimental data in similar energy and momentum regions tend not to agree with these helicity conservation selection rules [@krishni]. Although contributions from non-zero parton orbital angular momenta are power suppressed, as shown by Lepage and Brodsky [@lepage], they could break hadron helicity conservation rule [@gousset]. Interestingly recent re-analysis of quark orbital angular momenta seems to contradict the notion of power suppression [@rj_hh]. Furthermore, Ref [@isgur] argues that non-perturbative processes could still be important in some kinematic regions even at high energies. Thus the transition between the perturbative and non-perturbative regimes remains obscure and makes it essential to understand the exact mechanism governing the early onset of scaling behavior.
Towards this goal, it is important to look closely at claims of agreement between the differential cross section data and the quark counting rule prediction. Deviations from the quark counting rules have been found in exclusive reactions such as elastic proton-proton ($pp$) scattering [@ppdata; @hendry]. In fact, the re-scaled 90$^\circ$ center-of-mass $pp$ elastic scattering data, $s^{10}{\frac{d\sigma}{dt}}$ show substantial oscillations about the power law behavior. Oscillations are not restricted to the $pp$ elastic scattering channel; they are seen in elastic $\pi p$ fixed angle scattering [@pidata] and hints of oscillation about the $s^{-7}$ scaling have also been reported in the recent data [@zhu] from Jefferson Lab (JLab) on photo-pion production above the resonance region. In addition to violations of the scaling laws, spin correlations in polarized $pp$ elastic scattering also show significant deviations from perturbative QCD (pQCD) expectations [@crabb; @correlations]. Several sets of arguments have been put forward to account for these deviations from scaling laws and the unexpected spin correlations. Brodsky and de Teramond [@brodsky_de] explain the $pp$ scattering data in terms of the opening up of the charm channel and excitation of $c\bar{c}uuduud$ resonant states. Alternatively the deviations are said to be an outcome of the interference between the pQCD (short distance) and the long distance Landshoff amplitude (arising from multiple independent scattering between quark pairs in different hadrons) [@ralston]. Gluonic radiative corrections to the Landshoff amplitude give rise to an energy dependent phase [@sen] and thus the energy dependent oscillation. Carlson, Chachkhunashvili, and Myhrer [@carlson] have also applied a similar interference concept to explain the $pp$ polarization data. The QCD re-scattering calculation of the deuteron photo-disintegration process by Frankfurt, Miller, Sargsian and Strikman [@sargsian] predicts that the additional energy dependence of the differential cross-section, beyond the $\frac{d\sigma}{dt} \propto s^{-11}$ scaling, arises primarily from the $n-p$ scattering in the final state. In this scenario the oscillations may arise due to QCD final state interaction. If these predictions are correct, such oscillatory behavior may be a general feature of high energy exclusive photo-reactions.
Recently, a number of new developments have generated renewed interest in this topic. Zhao and Close [@close] have argued that a breakdown in the locality of quark-hadron duality (dubbed as “restricted locality” of quark-hadron duality) results in oscillations around the scaling curves predicted by the counting rule. They explain that the smooth behavior of the scaling laws arise due to destructive interference between various intermediate resonance states in exclusive processes at high energies. However, at lower energies this cancellation due to destructive interference breaks down locally and gives rise to oscillations about the smooth behavior. On the other hand, Ji [*et al.*]{} [@ji-scaling] have derived a generalized counting rule based on a pQCD inspired model, by systematically enumerating the Fock components of a hadronic light-cone wave function. Their generalized counting rule for hard exclusive processes include parton orbital angular momentum and hadron helicity flip, thus they provide the scaling behavior of the helicity flipping amplitudes. The interference between the different helicity flip and non-flip amplitudes offers a new mechanism to explain the oscillations in the scaling cross-sections and spin correlations. The counting rule for hard exclusive processes has also been shown to arise from the correspondence between the anti-de Sitter space and the conformal field theory [@cft] which connects superstring theory to conformal gauge theory. Brodsky [*et al.*]{} [@brodsky_new] have used this anti-de Sitter/Conformal Field Theory correspondence or string/gauge duality to compute the hadronic light front wave functions. This yields an equivalent generalized counting rule without the use of perturbative theory. Moreover, pQCD calculations of the nucleon formfactors including quark orbital angular momentum [@Ji_ff; @rj_ff] and those computed from light-front hadron dynamics [@brodsky_new] both seem to explain the $\frac{1}{Q^2}$ fall-off of the proton form-factor ratio, $G_{E}^{p}(Q^2)/G_{M}^{p}(Q^2)$, measured recently at JLab in polarization transfer experiments [@poltar].
In this letter we examine the role of the helicity flipping amplitudes in the oscillatory scaling behavior of $pp$ scattering and charged photo-pion production from nucleons and the oscillations in the spin correlations observed in polarized $pp$ scattering. We have used the generalized counting rule of Ji [*et al.*]{} [@ji-scaling] to obtain the scaling behavior of the helicity flipping amplitudes.
It is well known that $pp$ scattering can be described by five independent helicity amplitudes [@hel_amp_ref]. According to the dimensional as well as the generalized counting rules the three helicity-conserving amplitudes, $M(+,+ ; +,+), M(+,- ; +,-)$ and $M(-,+ ; +,-)$, have an energy dependence of $\sim 1/s^4$. On the other hand the simple constituent quark interchange models [@hel_amp_ref] assume the two helicity flipping (nonconserving) amplitudes, $M(+,+ ; +,-)$($NC1$) and $M(-,- ; +,+)$ ($NC2$) to be zero. Later analysis by Lepage and Brodsky [@lepage] have shown these amplitudes to be non-zero but power suppressed. The new generalized counting rule predicts their energy dependence to be $\sim 1/s^{4.5}$ and $\sim 1/s^{5}$ respectively [@ji-scaling]. Thus the generalized counting, rule which includes the helicity flipping amplitudes and the interference between them, gives rise to additional energy dependence beyond the $s^{-10}$ scaling predicted by dimensional scaling.
In addition to these short distance amplitudes, Landshoff [@landshoff] has shown that there can be contributions from three successive on-shell quark-quark scattering. Although each scattering process is itself a short distance process, different independent scatterings can be far apart, limited only by the hadron size. The Landshoff amplitude also carries an energy dependent phase arising from gluonic radiative corrections which are calculable in pQCD [@sen] and has a known energy dependence, similar to the renormalization-group evolution: $\phi(s) = \frac{\pi}{0.06}lnln(s/\Lambda_{QCD}^2)$. This effect is believed to be analogous to the coulomb-nuclear interference that is observed in low-energy charged-particle scattering. It has been shown that this energy dependence of the phase occurs at medium energies [@botts] and becomes independent of energy at asymptotically high energies [@botts], [@mueller]. In Ref. [@ralston]; Ralston and Pire have used the helicity-conserving amplitudes, the Landshoff amplitude with an energy dependent phase and the interference between them to reproduce the oscillations in the $pp$ scattering data at 90$^\circ$ c.m. angle (a similar method was used by Carlson [*et. al*]{} [@carlson] to describe oscillation in the cross-section as well as the spin-correlation). They write the two amplitudes as $M = M_{S} + e^{i\phi(s) +i\delta}M_{L}$, where $M_{S}\sim 1/s^4$ represents the three helicity-conserving short distance amplitudes, $M_{L}\sim 1/s^{3.5}$ is the Landshoff amplitude and $\phi(s)$ is the energy dependent phase, $\delta$ is an arbitrary energy independent phase. By fitting to the existing $pp$ scattering data at 90$^\circ$ c.m. angle, they find that the ratio of $M_{L}$ to $M_{S}$ is 1:0.04 for an energy dependent phase given by $\phi(s) = \frac{\pi}{0.06}lnln(s/\Lambda_{QCD}^2)$, where $\Lambda_{QCD}$ = 100 MeV. It has been argued that the asymptotic leading limit used to calculate this energy dependence phase of the Landshoff amplitude is not entirely valid [@kundu] and thus the Landshoff term is better parametrized as, $$\begin{aligned}
\label{kundu_eq}
M_{L} & = & b_j s^{-3.5}\frac{e^{ic_j[lnln(s/\Lambda_{QCD})]+i\delta_j}}{[\log(s)]^{d_j}},\end{aligned}$$ where $b_j$, $c_j$, $d_j$ and the energy independent phase $\delta_j$ are now parameters which are not exactly calculable. Fig. \[ralston\_fit\]a shows the fit of Ref. [@ralston] compared to the world data, and Fig. \[ralston\_fit\]b is a fit using the more general parametrization of the Landshoff described above. Both these fits deviate drastically from the data at $s<$ 10 GeV$^2$ and are not sensitive to the different parameterizations of the Landshoff amplitude. Since the Landshoff amplitude is expected to be significant only at high energies, it is not unreasonable that the above formalism does not describe the data at low energies.
[![(a) The fit to $pp$ scattering data at $\theta_{cm}= 90^\circ$ of Ralston and Pire [@ralston], this fit had two parameters; the overall normalization $A_{1}$ and the arbitrary phase $\delta$. (b) The same data fitted with the new more general parametrization of the Landshoff amplitude, this fit includes the 3 additional parameters $b_1, c_1$ and $d_1$ mentioned in Eq. \[kundu\_eq\]. The data are from Ref. [@ppdata][]{data-label="ralston_fit"}](gsrfig1.eps "fig:"){width="9.0cm" height="9.0cm"}]{}
As the interference between the Landshoff and the short distance amplitudes fail to describe the data at low energies, it is possible that the helicity flip amplitudes and their interference may play an important role at these energies. The helicity flip amplitudes arising from the parton orbital angular momentum are non-negligible when the parton transverse momentum can not be neglected compared with the typical momentum scale in the exclusive processes at relatively low energies. Thus one would expect the helicity flip amplitudes to be a significant contribution to the cross-section at low energies. Moreover, the generalized counting rule of Ji [*et al.*]{} [@ji-scaling] predicts a much faster fall-off with energy for the helicity flip amplitudes as expected. We have refitted the world data by including the two helicity-nonconserving amplitudes according to the generalized counting rule of Ji [*et al.*]{} [@ji-scaling]. The two different forms for the energy dependence of the phase in the Landshoff amplitude, described above, were employed in the fits to examine their sensitivity to them. The three helicity-conserving amplitudes combined as one amplitude and the two helicity flipping amplitudes, along with the Landshoff contributions, can be written as; $$\begin{aligned}
\label{new_eq}
M_{HC} &=& s^{-4}(a_{1} + b_{1}s^{0.5}e^{i\phi_{1}(s)}) \nonumber\\
M_{NC1} &=& s^{-4}(a_{2}s^{-0.5} + b_{2}s^{0.5}e^{i\phi_{2}(s)}) \nonumber \\
M_{NC2} &=& s^{-4}(a_{3}s^{-1} + b_{3}s^{0.5}e^{i\phi_{3}(s)}),\end{aligned}$$ where $\phi_{j}(s)$ is the energy dependent phase. Two different forms for the phase $\phi_{j}(s)$ were used in our fits; $\phi_{j}(s)=\frac{\pi}{0.06}lnln(s/\Lambda_{QCD}^2) + \delta_j$ and $\phi_{j}(s)=c_j\frac{lnln(s/\Lambda_{QCD}^2)+\delta_j}{(\log(s))^{d_j}}$. We have neglected the helicity flipping Landshoff contributions. The scaled cross-section is then given by, $$\label{new_eq2}
R = s^{10}\frac{d\sigma}{dt} \propto |M_{HC}|^2 + 4|M_{NC1}|^2 + M_{NC2}|^2,$$ The factor of four associated with the $NC1$ helicity flipping amplitude arises because of the two possible configurations of this single spin flip amplitude [@hel_amp_ref].
Fig \[newfit\] shows the results of our fit and also shows the explicit contributions from the $s^{-11}$ and $s^{-12}$ term for this approach. The value of $\Lambda_{QCD}$ was fixed at 100 MeV for all fits. This new fit is in much better agreement with the data. The helicity flip amplitudes (mostly the term $\sim s^{-4.5}$) are significant at low energies and seem to help in describing the data at low energies. It is interesting to note that among the helicity flip amplitudes the one with the lower angular momentum dominates. These are very promising results and should be examined for other reactions.
[![(a) The fit to $pp$ scattering data at $\theta_{cm}= 90^\circ$ when helicity flip amplitudes are included as described in Eq. \[new\_eq\]. The parameters for the energy dependent phase was kept same as the earlier fit of Ralston and Pire [@ralston]. The solid line is the fit result, the dotted line is contribution from the helicity flip term $\sim s^{-11}$, the dot-dashed line is contribution from the helicity flip term $\sim s^{-12}$. The $\sim s^{-12}$ contribution has been multiplied by 100 for display purposes.(b)The same data fitted to the form described in Eq. \[new\_eq\] but with the new more general parametrization of the Landshoff amplitude which includes the 3 additional parameters per term, $b_j, c_j$ and $d_j$ ($j$=1,2,3) as mentioned in Eq. \[kundu\_eq\].[]{data-label="newfit"}](gsrfig2.eps "fig:"){width="9.0cm" height="9.0cm"}]{}
As mentioned earlier the $A_{NN}$ spin-correlation in polarized $pp$ elastic scattering also shows large deviations [@correlations] from the expectations of pQCD (assuming hadron helicity is conserved). In terms of the helicity amplitudes $A_{NN}$ is given by [@hel_amp_ref]; $$\begin{aligned}
\label{ann_eqn}
R A_{NN} &=& 2{\mbox{Re}}[M^*(++;++)M(--;++)] \nonumber \\
&+& 2{\mbox{Re}}[M^*(+-;+-)M(-+;+-)] \nonumber \\
&+& 4|M(++;+-)|^2, \end{aligned}$$ where $R$ has been defined in Eq. \[new\_eq2\]. At $\theta_{cm}= 90^\circ$ the ratio of the three helicity non-flip amplitudes is $2:1:1$ [@hel_amp_ref]. Taking this into account we have fit the $A_{NN}$ data by including the helicity flipping amplitudes. Fig. \[ann\_fig\]a shows the results for the case where the helicity flip amplitude is neglected and only the interference between short distance amplitude and the Landshoff amplitude is used (in this case the expression for $A_{NN}$ simplifies to $R A_{NN} = 2{\mbox{Re}}[M^*(+-;+-)M(-+;+-)]$). These results are similar to those obtained by Carlson [*et. al*]{} [@carlson] and they described the $A_{NN}$ data at high energies but fail to describe the low energy data using this idea of interference between short distance and Landshoff terms. Fig. \[ann\_fig\]b shows the results of our fit when the helicity flipping amplitudes are included. It is clear that this method is a better fit to a larger fraction of the data which includes some low energy data. This suggests that even in case of the spin correlation $A_{NN}$ in polarized $pp$ elastic scattering the helicity flip amplitudes play an important role at low energies ($s < $ 10 GeV$^2$).
[![(a) The fit to $A_{NN}$ from polarized $pp$ scattering data at $\theta_{cm}= 90^\circ$ with the helicity non-flip and Landshoff amplitudes only. (b) Fit to the same data when the helicity flip amplitudes are included. The data are from Ref. [@crabb; @correlations]. The solid line is the fit and the dashed line is the expectation assuming hadron helicity conservation.[]{data-label="ann_fig"}](gsrfig3.eps "fig:"){width="9.0cm" height="7.0cm"}]{}
Recently some precision data on pion-photoproduction from nucleons above the resonance region has become available from JLab [@zhu]. These data show hints of oscillation about the $s^{-7}$ scaling predicted by the quark counting rule. In pion-photoproduction from nucleons the helicity non-flip amplitudes has an energy dependence of $s^{-2.5}$, and there is just one helicity flip amplitude which according to the generalized counting rule has an energy dependence of $s^{-3}$ [@ji-scaling]. There are no leading order Landshoff terms in pion-photoproduction since the initial state has a single hadron. However, the Landshoff process can contribute at sub-leading order [@farrar2] (i.e. $\sim s^{-3}$ instead of $\sim s^{-2}$). In principle, the fluctuation of a photon into a $q\bar q$ in the initial state can contribute an independent scattering amplitude at sub-leading order. But, experimentally it has been shown that vector-meson dominance diffractive mechanism is suppressed in vector meson photoproduction at large values of $t$ [@hallb]. On the other hand such independent scattering amplitude can contribute in the final state if more than one hadron exist in the final state, as is the case in nucleon photo-pion production reactions. Thus an unambiguous confirmation of such an oscillatory behavior in exclusive photoreactions with hadrons in the final state at large $t$ may provide a signature of QCD final state interaction.
We have fit the pion-photoproduction data at $\theta_{cm}= 90^\circ$ including the helicity flip amplitude and the Landshoff amplitude at sub leading order with an energy dependent phase. The Landshoff amplitude was parametrized according to the ansatz given in Ref. [@kundu]. The amplitudes for $\gamma p \rightarrow \pi^{+} n$ and $\gamma n \rightarrow \pi^{-} p$ and the respective Landshoff contribution to each amplitude can be written as; $$\begin{aligned}
\label{new_eq3}
M_{HC} &=& s^{-2.5}(a_{1} + b_{1}s^{-0.5}\frac{e^{ic_1\phi(s)+i\delta_1}}{(\log(s))^{d_1}}) \nonumber\\
M_{NC1} &=& s^{-2.5}(a_{2}s^{-0.5} + b_{2}s^{-0.5}\frac{e^{ic_2\phi(s)+i\delta_2}}{(\log(s))^{d_2}}),\end{aligned}$$ and the scaled cross-section is given by;\
$s^7\frac{d\sigma}{dt} \propto |M_{HC}|^2 + |M_{NC1}|^2$, where $\phi(s) = lnln(s/\Lambda^2)$. As seen in Fig \[pipfit\] the existing data can be fit quite well with this form. However, the data are rather coarsely distributed in energy and so these results are not a conclusive evidence for oscillations in pion-photoproduction. This underscores the point that a fine scan of energies above the resonance region is urgently needed. This is exactly the issue that will be addressed in the JLab experiment E02010 [@e02010] in the near future.
[![(a) The fit to $\gamma p \rightarrow \pi^+ n$ scattering data at $\theta_{cm}= 90^\circ$ when helicity flip and sub-leading order Landshoff amplitudes are included (b) Fit to $\gamma n \rightarrow \pi^- p$ scattering data at $\theta_{cm}= 90^\circ$. The data are from Ref. [@pidata; @zhu].[]{data-label="pipfit"}](gsrfig4.eps "fig:"){width="9.0cm" height="8.0cm"}]{}
We have shown that the generalized counting rule of Ji [*et al.*]{} [@ji-scaling] along with the Landshoff terms and associated interferences does a better job of describing the oscillations about the quark counting rule, in the $pp$ elastic scattering data at $\theta_{cm}= 90^\circ$. This is specially true in the low energy region ($s<$ 10 GeV$^2$). The contributions from helicity flipping amplitudes which are related to quark orbital angular momentum, seem to play an important role at these low energies, which is reasonable given that the quark orbital angular momentum is non-negligible compared to the momentum scale of the scattering process. Similarly the spin-correlation $A_{NN}$ in polarized $pp$ elastic scattering data can be better described by including the helicity flipping amplitude along with the Landshoff amplitude and their interference. The photo-pion production data from nucleons at large angles can also be described similarly; however, because of the coarse energy spacing of the data, the results are not as illustrative. This points to the urgent need for more data on pion-photoproduction above the resonance region with finer energy spacing. We expect that our experiment at JLab which is approved for running will help address this need in the near future.
We acknowledge fruitful discussions with X. Ji and S. J. Brodsky. This work is supported by the U.S. Department of Energy under contract number DE-FG02-03ER41231.
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abstract: 'We have measured the radiant of the Leonids meteor storm in November 2001 by using new observational and analysis techniques. The radiant was measured as the intersections of lines which were detected and extrapolated from images obtained at a single observing site (Akeno Observatory, Japan). The images were obtained by two sets of telephoto lenses equipped with cooled CCD cameras. The measured radiant, (R.A., Dec.)=(154$^\circ$.35, 21$^\circ$.55) (J2000), is found to be in reasonable agreement with the theoretical prediction by McNaught and Asher (2001), which verifies their dust trail theory.'
author:
- 'Ken’ichi <span style="font-variant:small-caps;">Torii</span> and Mitsumiro <span style="font-variant:small-caps;">Kohama</span>'
- 'Toshifumi <span style="font-variant:small-caps;">Yanagisawa</span>'
- 'Kouji [Ohnishi]{}'
title: The radiant of the Leonids meteor storm in 2001
---
Introduction
============
Recent progress of numerical celestial mechanics has made it possible to accurately predict the occurrence time, position, and rate of meteoric activities (e.g., [@mcnaught1999]; [@lyytinen]; [@mcnaught]). These theories predict that meteor storms occur when the Earth passes through the dense regions of meteoroids, or dust trails. The dust tubes are produced along the trails of the comet near perihelion passages and are kept in narrow tubes by combined effects of gravitational perturbations from planets and solar radiation pressure. The development of dust trail theories is important not only from celestial mechanics or interplanetary physics points of view but also from planning the strategies for protecting artificial satellites or manned missions against collisions of meteoroids (e.g., [@pawlowski]; [@brown]).
Although these theories have successfully predicted the peak time of meteoric activities, no extensive verifications have been made from other aspects. Since the radiant of meteors is directly related to the dust trails, the observational measurements of the radiant can be critical test for the dust trail theories. For 2001 November activity of Leonids meteor streams, McNaught and Asher (2001) predicted the presence of two strong outbursts (ZHR$\sim$ 2000 and 8000) over East Asian longitudes each of which is due to meteoroids released from the comet 55P/Tempel-Tuttle around its 1699 (9 revolutions ago) and 1866 (4 revolutions ago) return, respectively.
We have thus made observations for verifying the dust trail theories ([@ohnishi]; [@yanagisawa]). Our method differs from conventional ones in several ways. We use cooled CCD cameras with relatively narrow field of view optics. These instruments enable accurate ($\leq$10 arcseconds) measurement of each meteor with reference to background fixed stars. We have also employed a new image processing technique ([@yanagisawa][^1]) which effectively picks up faint line shapes of meteors superposed on the background fixed stars. Our aim here is not to measure the orbit of each meteor as conventionally studied but to measure the apparent radiant point projected on the celestial sphere. We therefore did not make the observations from multiple observing sites. Two sky positions which subtend nearly right angle to the predicted radiant were observed. Radiant is obtained as the intersections of orthogonal lines detected from the two cameras.
Observations
============
Observations were made on 2001 November 18 UT on the premises of Akeno Observatory (35$^\circ$47$'$N, 138$^\circ$30$'$E, 900m altitude) of Institute for Cosmic Ray Research, University of Tokyo. We used unfiltered telephoto lenses with the focal lengths of 180mm (Nikkor) and cooled CCD cameras as summarized in table 1. Camera 1 is Apogee’s AP7p with the backside illuminated CCD SITe SI-032AB (512$\times$512 pixels of $24\mu$m$\times24\mu$m). Camera 2 is Apogee’s AP6E with the Kodak’s CCD KAF-1001E (1024$\times$1024 pixels of $24\mu$m$\times24\mu$m). The exposures were continuously made with integration times of 20-s. Readout times are 10-s and 3-s for the cameras 1 and 2, respectively, which result in $\sim 33$% and $\sim 13$% of dead time in the observation. The pixel scales were $28''$/pixel and $27''$/pixel for the cameras 1 and 2, respectively. Limiting magnitudes for background fixed stars were $\sim 12-13$ mag for a single frame. These cameras were placed on an equatorial mount and tracked at the sidereal rate. The pointing positions (centers of the field of view), as summarized in table 1, subtended $\sim90^\circ$ to the expected radiant. Camera 1 pointed at a position about $-20^\circ$ (minus sign means westward) away from the radiant along right ascension while the camera 2 pointed at a position about $+40^\circ$ (plus sign means northward) along the declination. The position angles of the cameras were slightly rotated from the north so that the line shapes of the meteors do not become parallel to the column or row of the CCD pixels. This setting reduces the false events due to instrumental effects (e.g., column defects of the chip). Table 2 shows the start times of exposures for the first and last frames for the two cameras as well as the total number of frames. The weather condition was very good and no significant clouds bothered our abbreviations.
We detect linear shapes of meteors from the obtained images and extrapolate the lines toward the radiant. The radiant is thus determined as intersections of many lines detected by the two cameras. Systematic errors for the determination of lines come from several factors. First, the fixed stars revolve $15\,T\, cos(\delta)$ arcseconds during the integration time of $T\, {\rm [s]}$, which leads to uniform error within $\pm 2'.5$ for the current observation ($T=20\, {\rm [s]}$). In the current configuration, an error in the angle determination of a line corresponding to 1 pixel is magnified to $1'.7$ and $3'.4$ at the radiant. These two factors are combined to make total systematic error of $\sim 4'$. This value is marginally smaller than the expected separation ($\sim 0^\circ.1$) of the two radiants corresponding to the dust tubes of 4-revolutions and 9-revolutions ago ([@mcnaught]). Figure \[fig1\] shows some bright meteors as observed by the current system. This image was created by stacking 65 frames from camera 2.
Camera Optics Field of view Center of the Field (J2000)
-------- ------------- -------------------------------- -----------------------------
1 180mm f/2.8 3.9$^\circ$$\times$$3.9^\circ$ (08 48 25, +19 21 38)
2 180mm f/2.8 7.8$^\circ$$\times$$7.8^\circ$ (09 23 00, +65 06 02)
: Observing Instruments.[]{data-label="tab:first"}
Camera Start time \[UT\] End time \[UT\] Exposure \[s\] Total number of frames
-------- ---------------------------- ---------------------------- ---------------- ------------------------
1 2001 November 18, 15:11:43 2001 November 18, 20:52:31 20 671
2 2001 November 18, 16:34:38 2001 November 18, 20:21:04 20 601
: Observation Log.[]{data-label="tab:second"}
(80mm,80mm)[fig1.eps]{}
Analyses
========
The data were stored in the local hard disk and later processed offline. After dark frame subtraction and flat fielding, astrometric measurement was made for each frame with reference to USNO-A 2.0 catalog ([@monet]). The softwares PIXY[^2] and imwcs [^3] were used and the field center and the rotation angle were determined for each frame. The accuracy of the astrometry (field center) is typically better than the pixel scale and that for the position angle is typically better than $\simeq 0.01^\circ$.
We have used the new method ([@yanagisawa]) for detecting meteors from the observed frames. This technique was originally developed for detecting trails of space debris or artificial satellites from CCD images while it can be generally used for detecting line shapes on two dimensional images in the presence of point-like backgrounds. The details of the algorithm is described in [@yanagisawa] and the method is briefly explained here. Each image is rotated around its center by a trial angle $\theta_{rot}$. Then the central square region is extracted and the median values of each row is calculated and stored. In the absence of a line in the image, the median values are randomly distributed as background levels. If a line is present and the rotation correctly puts the line along the row, the median value becomes higher than those of adjacent rows, due to the systematic shift in the distribution of the pixel values toward higher side. Since the presence of point-like stellar images does not systematically shift the distribution, this technique effectively picks up line shapes from the background stars and noise fluctuations. For detecting a line of unknown angle, we make trials with different rotation angles $\theta_{rot}$ with small steps. In the current analysis, we subtracted the $(i-1)$-th image (image of the previous exposure) from the $i$-th image so that the effects of fixed stars and small imperfection of flat fielding are further reduced.
### Results
We have examined the appropriate thresholds to discern real events (lines) from background fluctuations. To do this, we have created histograms of rotation angles at which the lines were detected. For bright real events, the angles are concentrated toward the radiant, while the background events are uniformly distributed. We set the thresholds so that the signal to noise ratios are more than 2 in the histogram. Consequently, the limiting magnitude of the current observation is estimated to $\sim
7$ mag for meteors by the cross calibration with the wide-field TV observation at the same site. As the results of the line detection analyses, we detected 9 and 80 lines from the camera 1 and 2. The small number of detected line for the camera 1 is partly due to the presence of a stellar cluster (M 44 = NGC 2632) and a bright star ($\delta$ Cnc, $\sim$3.9 mag) within the field of view which made background higher than that for the camera 2.
Based on the astrometry of fixed stars, we have converted the positions of each line to the celestial coordinates (right ascension and declination) and extrapolated (extended) back to their origin on the spherical coordinates. The apparent position of radiant moves due to the combined effects of diurnal aberration and zenithal attraction. These effects smear out the apparent radiant in a short time of interval and makes it difficult to resolve the radiant structures. We therefore calculated the shift as a function of time and corrected the positions of each line to cancel the effect. The reference time for this correction was set to November 18 18:13 UT which was the predicted peak time for the 4-rev trail encounter ([@mcnaught]). This procedure makes it possible to combine data of long duration to improve the statistics. The result is shown in figure \[fig2\] and \[fig3\]. The lines are distributed around the expected position while we can clearly see the concentration at around ($\alpha$, $\delta$)=(154$^\circ$.35, 21$^\circ$.55).
To clearly see the radiant, we have examined the concentration of the lines in the following way. For each grid point near the radiant, we calculated the distance between the position and the lines. If the number of lines within the threshold distance $r_{th}$ is more than the threshold numbers, that positions is considered as the correct radiant. This procedure has thus three free parameters, $r_{th}$, $n_1$, and $n_2$ which are not given apriori. We use $r_{th}\leq 4'$, $n_1 \geq 3$, and $n_2\geq15$. This value of $r_{th}$ is chosen so that the value is comparable to the systematic error as estimated above. The values of $n_1$ and $n_2$ are determined by the number of detected lines. Particularly, the value of $n_1$ had to be set to as small as 3 due to the small number of detected lines from the camera 1. The value of $n_2/n_1$ may be reasonable, taking into account the effective area and exposures of the two cameras. Figure \[fig4\] shows the radiant structure as finally determined.
The position derived in the current work is in reasonable agreement with the theoretical prediction of [@mcnaught]. The measured position in declination looks displaced by $\sim -0^\circ.1$ from that predicted (Figure \[fig4\]). However, we may not conclude that they are inconsistent, taking into account the small number of east-west lines used to constrain the declination. The better constrained right ascention is consistent with the prediction both in the central position and the extension. Although we expected to resolve the two radiants from the two dust trails, they could not be resolved partly due to the small number of the east-west lines and partly due to the relatively large systematic error of the current study. We find weak evidence of shift of right ascension from RA$\sim 154^\circ.4$ to RA$\sim 154^\circ.3$ by the time resolved analysis. Although it is as expected from the peak times of the two dust trails, limited statistics does not allow us to conclude that they unambiguously come from the two distinct points or from relatively extended ($\sim 0.^\circ1$) region.
(80mm,80mm)[fig2\_gray.eps]{}
(71mm,71mm)[fig3.eps]{}
(80mm,80mm)[fig4.eps]{}
The measured position is also found to be in reasonable agreement with the complementary result by using longer (530 mm) focal length optics at the same observing site ([@yanagisawa2]). Since all the hardwares and softwares are independent between the two works, this agreement ensures the adequacy of the whole procedures of the two analyses.
Conclusions
===========
We have shown that the new method presented here can be a powerful diagnostic tool for studying the radiant structure of meteor storms. The measured position is found to be in reasonable agreement with the prediction of [@mcnaught] based on their dust trail theory.
In 2002, Leonids meteor storm is expected to be observed in North America and in Europe. The observation and analysis method presented herein will be useful to further resolve the profile structure of the dust tubes. Cameras with large area, fast read-out CCDs such as ROTSE ([@akerlof]), LOTIS ([@park]),and RAPTOR ([@borozdin], [@vestrand]) may be most useful for detecting a large number of meteors in a short time. If enough number of meteors could be detected in a short time, the three dimensional (time resolved) structure of dust tubes may be obtained with the current method.
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors are grateful to Prof. Masahiro Teshima and all the staffs of Akeno Observatory of University of Tokyo for their kind support during our observations.
Akerlof, C., Balsano, R., Barthelmy, S., Bloch, J., Butterworth, P., Casperson, D., Cline, T., Fletcher, S., et al. 2000, ApJ, 532, L25
Brown, P., Cooke, B. 2001, MNRAS, 326, L19
Borozdin, K., Brumby, S., Galassi, M., McGowan, K., Starr, D., Vestrand, W.T., White, R., Wozniak, P., Wren, J. 2002, astro-ph/0210008
Lyytinen, E.J., Flandern, T.V. 2000, EMP, 82, 149
McNaught, R.H., Asher, D.J. 2001, JIMO, 27, 85
McNaught, R.H., Asher, D.J. 2001, JIMO, 29, 156
Monet, D., Bird A., Canzian, B., Dahn, C., Guetter, H., Harris, H., Henden, A., Levine, S., et al., 1998, USNO-A 2.0, A catalog of astrometric standards (U.S. Naval Observatory Flagstaff Station)
Ohnishi, K., Yanagisawa, T., Torii, K., Kohama, M., Kawai, N., Sato, R., Hatsukade, I., Chaya M., et al. 2002, The Institute of Space and Astronautical Science Report SP No. 20, November 2002
Park, H-S., Elden, A., Barthelmy, S.D., Bionta, R.M., Ott, L.L., Parker, E.L., Williamsm, G.G. 1998, SPIE, 3355, 658
Pawlowski, J.F., Hebert, T.T. 2000, EMP, 82, 249
Vestrand, W.T., Borozdin, K., Brumby, S., Casperson, D., Fenimore, E., Galassi, M., McGowan, K., Perkins, S., et al. 2002, astro-ph/0209300
Yanagisawa, T., Ohnishi, K., Torii, K., Kohama, M., Nakajima, A., Asher, D. 2002a, The Institute of Space and Astronautical Science Reprot SP No. 20, November 2002
Yanagisawa, T., Ohnishi, K., Torii, K., Kohama, M., Nakajima, A., Asher, D. 2002b, PASJ, This volume
[^1]: The technique is patent pending.
[^2]: http://www.aerith.net/misao/pixy1/
[^3]: http://tdc-www.harvard.edu/software/wcstools/imwcs/
|
Introduction {#sec:intro}
============
Conventional semimetals like the pentavalent elements As, Sb and Bi can be well understood using the approximation of independent electrons. Having an even number of electrons per primitive cell, these solids indeed come very close to being insulators. They are not, because the occupied band nearest to the Fermi energy overlaps slighty with the unoccupied band lowest in energy. Therefore a few holes and a few electrons in these two bands can contribute to transport leading to a very small number of charge carriers $n$, as seen in the measured Hall coefficient $R_{\rm H} = 1/n{\rm ec}$ of these materials. However, the independent electron picture is often not appropiate to describe transition-metal and rare-earth compounds. Prominent examples are the transition-metal oxides like $\rm V_2O_3$, $\rm MnO$ or $\rm
NiO$. These oxides are insulators although they contain partially filled $d$ bands. Already in 1949, Mott identifies the Coulomb interaction of the conduction electrons as reason for the insulating behavior [@Mott49]. Meanwhile, the term Mott insulator has become a synonym for solids with localized charge carriers due to their mutual Coulomb repulsion. In the early transition-metal oxides the movement of the charge carriers is completely blocked due to their mutual Colomb repulsion. It seems likely that there also exist compounds in which the movement of the charge carriers is only nearly blocked and which exhibit therefore not insulating but semimetallic behavior. The aforementioned conventional semimetals As, Sb and Bi are solids which come close to being band insulators. There might exist transition-metal or rare-earth compounds which are semimetallic because they come close to being a Mott insulator. In Sec. \[sec:exp\] we will discuss a possible example, the semimetal $\rm Yb_4As_3$.
In this paper we want to investigate a simple model which is able to mimic a semimetal coming close to being a Mott insulator. Therefore we are heading for a model which can describe the transition from a Mott insulator to a metal by varying some control parameter. The appealing idea is that a metal near a quantum transition to an insulator behaves like a semimetal. This is so, because the conductivity vanishes for an insulator. Therefore resistivity and Hall coefficient should become very large if one approach the insulating state from the metallic side. It is important to note, that a non vanishing Hall coefficient can be only obtained in a non particle-hole-symmetric model. This excludes one possible candiate, the well-known Hubbard model with one electron per site. Here the control parameter would be the ratio between the kinetic and the Coulomb energy of the electrons which can be changed by applying pressure. A suited model system for our purpose is the self-doped Hubbard model introduced in Ref. . Here additional orbitals are coupled to a half-filled Hubbard model via hybridization. The additional electron states serve as a generic reservoir of charge. The hybridization drives the desired insulator-to-metal transition.
The self-doped Hubbard model is not based on purely academic considerations. It is inspired by the experimental observation of charge order in some transition-metal and rare-earth compounds like $\rm Yb_4As_3$ and $\rm NaV_2O_5$. At high temperatures these compounds are mixed-valent systems. In $\rm Yb_4As_3$ for example the Yb-ions fluctuate between a twofold and threefold positive charged state leading to a formal valency of $\rm Yb^{2.25+}$. At low temperatures the charges avoid their mutual Coulomb repulsion and stay as far apart as possible. In $\rm Yb_4As_3$ this leads to a static charge ordered state in which the nearest-neighbor sites of an $\rm
Yb^{3+}$-ion are only occupied by $\rm Yb^{2+}$-ions and not by other $\rm Yb^{3+}$-ions [@Fulde95; @Schmidt96a; @Schmidt96b; @Schmidt96c; @Kohgi97; @Rams96]. Therefore, the $\rm Yb^{3+}$-ion are confined on a subsystem of all possible sites, here chains in $\langle 111 \rangle$-direction. A charge lattice is superposed on the crystal lattice. In $\rm
Yb_4As_3$ the charge order leads to a structural phase transition at $T_c \approx 295$K. The foremost equivalent Yb-sites split up in $\rm
Yb^{2+}$-sites with a crystal environment of lower symmetry and $\rm
Yb^{3+}$-site with one of higher symmetry. The superposed charge lattice manifest itself in the distorted crystal structure below 295K. It is important to note, that charge ordering drive a compound towards a Mott-insulating state because the confinement of the charges to a sublattice reduces their kinetic energy. Transition-metal or rare-earth compounds whith a charge order transition are therefore promising candidates of systems coming close to being a Mott insulator. Indeed, $\rm Yb_4As_3$ is a semimetal [@Ochiai90]. A very simplified model have to contain sites preferential occupied by the charges representing the sublattice superposed by charge order. The charges are able to leave this sites to gain some kinetic energy. Therefore, the sites lower in energy have to be coupled to an empty reservoir of charges representing all other accessible but predominantly unoccupied sites of the crystal. The self-doped Hubbard model is a possible realization of this idea. To take advantage of sophisticated many-body techniques, namely the dynamical mean-field theory [@Pruschke95; @Georges96], the empty reservoir of charges in the self-doped Hubbard model is simply represented by unoccupied orbitals hybridizing purely local with the ones of the occupied sites.
The temperature behavior of the resistivity and the Hall coefficient is anomalous in $\rm Yb_4As_3$. Both quantities show a non-monotonous temperature dependence and exhibit a maximum at a characteristic temperature. The characteristic temperatures differ for the two properties. Therefore, not only the maximum value of the resistivity and the Hall coefficient of the self-doped Hubbard model will be of interest but also the temperature dependence of the two properties. In this paper we calculate the density of states, resistivity and Hall coefficient of the self-doped Hubbard model in the vicinity of the metal-to-insulator transition using dynamical mean-field theory. In the next section we introduce the model and discuss the mapping of this multi-band model to an impurity model within the dynamical mean-field theory. The impurity model is studied numerically by an extension of the non-crossing approximation to a two-orbital impurity. We present the results in Sec. \[sec:results\], discuss the applicability of the self-doped Hubbard model to $\rm Yb_4As_3$ in Sec. \[sec:exp\] and finally conclude in Sec. \[sec:conc\].
Model and Method {#sec:model}
================
The self-doped Hubbard model is given by $$\begin{aligned}
\label{eq.smsk}
H & = & -(\Delta+\mu) \sum_{\vec{k},\sigma} {f_{\vec{k}\sigma}^{\,\dagger}}{f_{\vec{k}\sigma}^{\,\phantom{\dagger}}}+ \sum_{\vec{k},\sigma} {\epsilon_{\vec{k}}^{\,\phantom{\dagger}}}\, {f_{\vec{k}\sigma}^{\,\dagger}}{f_{\vec{k}\sigma}^{\,\phantom{\dagger}}}-\mu \sum_{\vec{k},\sigma} {c_{\vec{k}\sigma}^{\,\dagger}}{c_{\vec{k}\sigma}^{\,\phantom{\dagger}}}\nonumber\\
& & + V \sum_{\vec{k},\sigma} \left(\,\,{f_{\vec{k}\sigma}^{\,\dagger}}{c_{\vec{k}\sigma}^{\,\phantom{\dagger}}}+ {\rm h.c.} \right)
+ U \sum_i {n_{i\uparrow}^{\,f}}{n_{i\downarrow}^{\,f}}\; \; ,\end{aligned}$$
The model consists of a lattice of $f$ orbitals described by the Hubbard Hamiltonian with on-site Coulomb repulsion $U$. Locally the $f$ orbitals hybridize with additional orbitals called $c$. The hybridization strength is $V$. For an electron it is energetic favourable to occupy an $f$ orbital. The energy difference between an occupied $c$ orbital and an occupied $f$ orbital is given by the charge-transfer energy $\Delta$. The filling is one electron per site, i.e., the system is quarter-filled. For $V=0$ the self-doped model reduces to a half-filled one-band Hubbard model. The Coulomb repulsion of the electrons is large, $U>U_{\rm c}$, and the system is a Mott insulator. Here we have assumed a finite critical value of the Mott-Hubbard transition. For $V \neq 0$ the mean occupation of $f$ orbitals is smaller than one. We may say that for $V=0$ the electrons order in the sense that they only occupy the $f$ subsystem. The charge ordered state is an insulator. A finite hybridization hinders the order to be perfect.
In Ref. we have shown that for $V=\infty$ the self-doped Hubbard model again reduces to a half-filled one. However, the effective Coulomb repulsion is reduced to $U/4$ and the effective hopping to $t/2$. In a regime of $U_{\rm c} < U < 2 U_{\rm c}$ we therefore expect an insulator-to-metal transition to take place with increasing value of $V$. Unfortunately simple approximation schemes like slave-boson mean-field and alloy-analog approximation fail to reproduce this quantum transition. In this paper we apply the dynamical mean-field theory to the model.
Though we are aiming at the resistivity and the Hall coefficient of the model we first calculate the one-particle Green’s function $G(\vec{k},\omega)$. Given the noninteracting Green’s function, $G_0(\vec{k},\omega)$, of the self-doped Hubbard model $$G_0^{-1}(\vec{k},\omega) = \left(
\begin{array}{cc}
\omega+\mu+\Delta-\epsilon_{\vec{k}} & -V \\
-V & \omega+\mu
\end{array}
\right)$$ the self-energies are defined by Dyson’s equation $$G(\vec{k},\omega) = \left( G_0^{-1}(\vec{k},\omega) -
\Sigma(\vec{k},\omega) \right)^{-1}.$$ The dynamical mean-field theory assumes a momentum-independent self-energy $\Sigma(\vec{k},\omega) \rightarrow \Sigma(\omega)$. In general the self-energy can be considered as a functional [@Baym62] of the full Green’s function $G(\vec{k},\omega)$. In particular the local approximated self-energy is only a functional of the local Green’s function $$\label{eq:local}
G(\omega) = \frac{1}{N} \sum_{\vec{k}} G(\epsilon_{\vec{k}},\omega).$$ The functional dependence is generated purely by the interaction term in the Hamiltonian. It is thus the same for an impurity model with the same interactions. Therefore, lattice and impurity model have the same self-energy provided we identify the Green’s function ${\cal
G}(\omega)$ of the impurity model with $G(\omega)$. In the case of the self-doped Hubbard model the corresponding impurity model reads
[cl]{}
------------------------------------------------------------------------
\(i) & $G_{ff}(\omega) = \sum_{nm}
|P_{\,1}^{\,\sigma}(n,m)|^2\, {\langle\langle\,X_{nm}\, ; \,X_{mn} \,
\rangle\rangle \! {\atop\omega}}$\
(ii) & ${\langle\langle\,X_{nm}\, ; \,X_{mn} \,
\rangle\rangle \! {\atop\omega}} = \frac{1}{Z_{\rm Imp}}
{\int_{-\infty}^{\infty} {\rm d}\epsilon}\,e^{-\beta\epsilon}
\{p_n(\epsilon)\,P_m^{\phantom{*}}(\epsilon+\omega)
-P_n^{*}(\epsilon-\omega)\,p_m(\epsilon)\}$\
(iii) & $Z_{\rm\,Imp} = \sum_m
{\int_{-\infty}^{\infty} {\rm d}\epsilon}\,e^{-\beta\epsilon}\,p_m(\epsilon)$\
(iv) & $p_m(\epsilon) = -\frac{1}{\pi}\,{\rm Im}\,P_m(\epsilon+{\rm
i}0^{+})\;,$ $\;\;\;P_m(z)={\int_{-\infty}^{\infty} {\rm d}\epsilon}\,\frac{p_m(\epsilon)}{z-\epsilon}$\
(v) & $P_n(z) = \frac{1}{z-E_n-\Sigma_n^{(2)}(z)}$\
(vi) & $\Sigma_n^{(2)}(z) = -\frac{1}{\pi} \sum_{m\sigma} {\int_{-\infty}^{\infty} {\rm d}\epsilon}\,{\rm Im}
{\cal J}(\epsilon+{\rm i}0^{+}) \left\{ |P_{\,1}^{\,\sigma}(n,m)|^2
f(\epsilon) P_m(z+\epsilon)+|P_{\,1}^{\,\sigma}(m,n)|^2 f(-\epsilon)
P_m(z-\epsilon) \right\}$
$$\begin{aligned}
\label{eq:imp}
H_{\rm eff} & = & H_{\rm cell} + H_{\rm med}\\[0.2cm]
H_{\rm cell} & = &
-(\Delta+\mu) \sum_{\sigma} {n_{\sigma}^{\,f}}+ U \, {n_{\uparrow}^{\,f}}{n_{\downarrow}^{\,f}}- \mu \sum_{\sigma} {n_{\sigma}^{\,c}}\nonumber \\
& & + V \sum_{\sigma} \left(\,\,{f_{\sigma}^{\,\dagger}}{c_{\sigma}^{\,\phantom{\dagger}}}+ {\rm h.c.} \right) \\
H_{\rm med} & = &
\sum_{\vec{k}\sigma} E_{\vec{k}}^{\,\phantom{\dagger}} \,
d_{\vec{k}\sigma}^{\,\dagger} d_{\vec{k}\sigma}^{\,\phantom{\dagger}}
+ \sum_{\vec{k}\sigma}
\left(\,\,\Gamma_{\vec{k}}^{\,\phantom{\dagger}} \,
d_{\vec{k}\sigma}^{\,\dagger} {f_{\sigma}^{\,\phantom{\dagger}}}+ {\rm h.c.} \right)\;.
\nonumber\end{aligned}$$
Note, $H_{\rm eff}$ just embeds a single unit-cell of the original lattice model in an effective medium. Following the discussions in Ref. and it is easy to show that the matrix equation $$\label{eq:selfcon}
G(\omega)={\cal G}(\omega)$$ can be fulfilled by choosing a single function $${\cal J}(\omega) = \sum_{\vec{k}} \frac{|\Gamma_{\vec{k}}|^2}
{\omega-E_{\vec{k}}}$$ which describes the coupling of the $f$ orbital to the bath. The integration in Eq. (\[eq:local\]) can be performed analytically if we choose a semielliptic density of states $$\rho_0(\epsilon) = \frac{1}{N} \sum_{\vec{k}} {\delta(\epsilon-{\epsilon_{\vec{k}}^{\,\phantom{\dagger}}})}
=\frac{2}{\pi\,W^2} \,\sqrt{W^2-\epsilon^2}\; .$$ The self-consistency (\[eq:selfcon\]) then reduces to $${\cal J}(\omega) = \frac{W^2}{4^{\phantom{2}}} \, G_{ff}(\omega) \; .$$ In all our calculation we choose $W=1$ as unit of energy.
What remains is the calculation of the one-particle Green’s function of the impurity problem (\[eq:imp\]). Here we make use of an extension of the non-crossing approximation [@Bickers87a] (NCA) to the case of more than two ionic propagators. Note, that the “impurity” in (\[eq:imp\]) has 16 eigenstates $$\label{eq:dia}
H_{\rm cell} = \sum_{m=1}^{16} E_m \, X_{mm}$$ and not only two as in the case of the $U=\infty$ impurity Anderson model. In Eq. (\[eq:dia\]) we have introduced the Hubbard operators $X_{nm} = {\left| n \,\left\rangle \right\langle m \right|}$. The generalized NCA has been applied successfully to the finite-$U$ impurity Anderson model [@Pruschke89], the Emery model in the dynamical mean-field theory [@Lombardo96], and the Anderson-Hubbard [@Schork97] as well as the Kondo-Hubbard model [@Schork98] in this approximation. We just summarize the basic equations in Tab. \[tab:nca\] and refer the reader for further details to the literature, e.g. see Refs. . The coupled integral equations (v) and (vi) in Tab. \[tab:nca\] are solved numerically by introducing defect propagators [@Bickers87b] and making use of the fast Fourier transformation [@Lombardo96]. The $f$ Green’s function of the impurity model gives a new function ${\cal J}(\omega)$ via Eq. (\[eq:selfcon\]) and the calculations are iterated until self-consistency is achieved.
We are now turning to the calculation of the resistivity and the Hall coefficient. In the self-doped Hubbard model the electrons can hop only from one $f$ orbital on site $i$ to an $f$ orbital on a neighboring site $i+\delta_\alpha$ ($\alpha = x,y,z$ denotes the direction in space). The current operator of the model is therefore given by $$\label{eq:current}
j_{i+\frac{\delta_{\alpha}}{2}}^{\,\alpha} = \frac{{\rm
i}\,}{{\rm \hbar}}\,a\,t\, \sum_{\sigma}
(f_{i+\delta_{\alpha}\,\sigma}^{\,\dagger} {f_{i\sigma}^{\,\phantom{\dagger}}}- {f_{i\sigma}^{\,\dagger}}f_{i+\delta_{\alpha}\,\sigma}^{\,\phantom{\dagger}})\;,$$ where $a$ is the lattice spacing. In momentum space the current operator reads $$\label{eq:currentq}
j_{\vec{q}}^{\,\alpha} = \frac{1}{{\rm \hbar}} \sum_{\vec{p}\sigma}
\left( {\frac{{\rm \partial} \epsilon_{\vec{p}}}{{\rm \partial} p^{\alpha}}}\right)
f_{\vec{p}-\frac{\vec{q}}{2}\,\sigma}^{\,\dagger}
f_{\vec{p}+\frac{\vec{q}}{2}\,\sigma}^{\,\phantom{\dagger}}\;.$$
Given the current operator we can straightforward repeat the calculation of the conductivity in the case of the Hubbard model. It is not surprising that we end with the same expressions as in the case of the Hubbard model [@Pruschke95] just inserting the $f$ spectral density $$\begin{aligned}
\label{eq:leitxx}
\sigma^{xx} & = & \frac{{\rm e^2} \pi}{6 {\rm \hbar} a}
\int_{-\infty}^{\infty} {\rm d}\omega \left(-{\frac{{\rm \partial} f}{{\rm \partial} \omega}}\right)
\int_{-1}^{1} {\rm d}\epsilon\, \rho_{0}(\epsilon)\,
\rho_f^2(\epsilon,\omega) \\[0.2cm]
\label{eq:leitxy}
\sigma_{\rm H}^{xyz} & = & \frac{{\rm |e|^3} \pi^2 a}{27 {\rm \hbar}^2
{\rm c}}
\int_{-\infty}^{\infty} {\rm d}\omega \left(-{\frac{{\rm \partial} f}{{\rm \partial} \omega}}\right)
\int_{-1}^{1} {\rm d}\epsilon\, \rho_{0}(\epsilon)\,
\epsilon\, \rho_f^3(\epsilon,\omega)\;.\end{aligned}$$ Here $f(\omega) = \left[ \exp(\beta \omega) + 1 \right]^{-1}$ is the Fermi function. The derivation involves two steps. First, we consider the noninteracting case like in Ref. . Second, we mark that vertex corrections in the linear response diagrams vanish. Here the fact enters that the one-particle self-energy is momentum independent. In addition the special $\vec{k}$-dependence of the free propagators via $\epsilon_{\vec{k}}$ and of the vertices via ${\frac{{\rm \partial} \epsilon_{\vec{k}}}{{\rm \partial} k^{\alpha}}}$ (see Eq. (\[eq:currentq\])) is needed. [@Pruschke95] Because the vertex corrections vanish the free propagators in the expression for the noninteracting case can simply be replaced by full ones.
Given the conductivity $\sigma^{xx}$ and the Hall conductivity $\sigma_{\rm H}^{xyz}$ we know the resistivity $\rho_{xx} =
1/\sigma^{xx}$ and the Hall coefficient $R_{\rm H}=\sigma_{\rm
H}^{xyz}/(\sigma^{xx})^2$. The units of the two transport properties are given by $[\rho_{xx}] \approx [\hbar a/{\rm e^2}]
\approx \rm 0.1 m\Omega cm$ and $[R_{\rm H}] \approx [a^3/{\rm e c}]
\approx \frac{1}{\rm c} 10^{-5} \rm cm^3/C$, respectively. Note, the conductivities are given by the one-particle spectral function only $$\rho_f(\epsilon,\omega) = -\frac{1}{\pi}\;{\rm Im}\,
G_{ff}(\epsilon,w+i0^+)\;.$$ In particular, if $\rho_f(\epsilon,\mu)$ vanishes for all energies $\epsilon$, i.e., $\rho_f(\mu)=0$, than also the conductivity vanishes for low temperatures. In this case the system is an insulator.
Results {#sec:results}
=======
When discussing the results we will frequently refer to experimental findings for the semimetal $\rm Yb_4As_3$. In the next Section we will discuss the applicability of the self-doped Hubbard model to this rare-earth compound.
Heading for the metal-insulator diagram of the self-doped Hubbard model we first concentrate on the value of the $f$ spectral function at the chemical potential $\rho_f(\mu)$ for different values of the hybridization. In Fig. \[fig:spectra1\] we display the evolution of the spectral function of the model with increasing $V$ and fixed values for $U$ and $\Delta$. The $f$ and $c$ spectral functions are obtained by the treatment outlined in Sec. \[sec:model\]. Within our approach the critical Coulomb repulsion of the Mott-Hubbard transition in the one-band Hubbard model is $U_{\rm c} = 1.77(5)$. The chosen value of $U=3$ fullfills $U_{\rm c} < U < 2 U_{\rm c}$. As expected for this value of $U$ the model undergoes an insulator-to-metal transition. For $V < 3$ we obtain $\rho_f(\mu)=0$ and the system is an insulator. For $V \geq 3$ we find a finite value $\rho_f(\mu) \neq 0$ and the model is a metal. Increasing the resolution in $V$ we obtain the value $V_{\rm c} \approx 2.7$ for the critical hybridization of the transition. Next, we consider the case of a very large Coulomb repulsion $U=5$, i.e., $U>2U_{\rm c}$. In Fig. \[fig:spectra2\] we demonstrate that the gap between the filled band and the lowest empty band never closes. The density of states at the chemical potential vanishes for all values of the hybridization. This shows clearly that the insulator-to-metal transition is restricted to a parameter region $U_{\rm c} < U < 2 U_{\rm c}$. In Fig. \[fig:phas\] we display the resulting metal-insulator diagram of the self-doped Hubbard model in the $(U,V)$-plane for two different values of the charge transfer energy.
Qualitatively a phase diagram of this form was already predicted in Ref. . Indeed, the bands in the spectral function of the lattice model evolve like the transition energies of the single two-orbital impurity ($H_{\rm cell}$ in Eq. (\[eq:imp\])). Therefore no sophisticated calculations are required to determine roughly the position of the centers of bands. We denote the possible transitions of the single two-orbital impurity by $(f^1c^0) \rightarrow (f^nc^m)$. The correspondence of the respective transition energies and the band energies allows us to classify the occupied band below the chemical potential as $(f^0c^0)$-band and the unoccupied bands above $\mu$ as $(f^1c^1)_{\rm s}$-, $(f^1c^1)_{\rm
t}$- and $(f^2c^0)$-bands (here s and t refers to singlet and triplet). Lattice effects show up in the ratio between $f$ and $c$ weight constituting a single band. For example, the occupation number of $c$ orbitals is larger in the lattice model than in the two-orbital impurity marking a larger $c$ weight in the occupied $(f^0c^0)$-band. This is a simple consequence of quantum fluctuations between different impurity configurations in the ground state of the lattice model.
Nevertheless, the most prominent feature of the lattice model is the appearance of a sharp Kondo-like resonance close to the chemical potential. The temperature dependence of the resonance is shown in . The peak arises below a characteristic temperature $T_0$. Regarding the hybridization dependence of the resonance we state that it becomes narrower and shows up at a lower temperature $T_0(V)$ when we approach the insulating state from the metallic side, i.e., when we regard the limit $V \rightarrow
V_{\rm c}^+$. Note, that for temperatures just above $T_0$ $\rho_f(\mu)$ is a monotonously decreasing function in this limit. Therefore the observed behavior of $T_0$ is very similar to the well-known dependence of the Kondo temperature on the density of states $T_{\rm K} \sim \exp{\left(-b/\rho(\mu)\right)}$. The usual Kondo effect can be interpreted in terms of the resonant-level model.[@Hewson] The conduction electrons with energies close to $\mu$ are virtually bound to localized electrons due to resonance scattering. In a similar way one may assign the resonance seen in our calculations to a band-Kondo effect where the conduction electrons are virtually bound not to localized but to moving electrons. This “band-Kondo effect” is a typical feature of strongly correlated metals treated within the dynamical mean-field theory. It occurs also in the one-band Hubbard model. It is still a matter of dispute if the effect is physical or a shortcoming of the mapping of the lattice model to an impurity model as done within the dynamical mean-field theory. But at least in the case of the Hubbard model there are indications that the effect can be seen also by other methods. [@Bulut94; @Preuss95]
The strong temperature dependence of the spectral density close to the chemical potential causes a strong temperature dependence of the resistivity. In Fig. \[fig:res\] we show this dependence as obtained by the treatment outlined in Sec. \[sec:model\]. First we focus on the magnitude of the resistivity. The resistivity increases when we decrease the hybridization, i.e., when we approach the insulating regime. In fact we expect a diverging resistivity in the limit $V
\rightarrow V_{\rm c}^+$ because the conductivity vanishes in an insulator. This is the key idea. In the vicinity of a quantum transition to an insulator we can obtain large values for the resistivity and the Hall coefficient implying semimetallic behavior. In Fig. \[fig:res\] the order of magnitude of the resistivity is $\rm 1 m\Omega cm$, i.e., the same as observed in $\rm Yb_4As_3$. Note, that large values of $\rho$ and $R_{\rm H}$ do not imply that the occupation number of the $c$ orbitals $n^c=1-n^f$ have to be small. Large values are possible for every $n_c$ in striking contrast to a simple Drude picture when assuming the charge carriers in the system are given by the missing electrons in the $f$ subsystem. We stress this point because in $\rm Yb_4As_3$ there is experimental evidence [@Kohgi97] that the number of missing holes in the $\langle 111
\rangle$-chains is not identical with the low carrier concentration obtained from the large Hall coefficient.
As function of temperature the resistivity shows a maximum at a characteristic temperature $T_0$. This temperature is identical with the one discussed above, i.e., the temperature where the sharp resonance arises in the spectral function. As seen in Fig. \[fig:res\] and as discussed before $T_0$ is larger for parameter values deeper in the metallic regime of the self-doped Hubbard system. $\rm Yb_4As_3$ can be tuned to a more insulating or metallic state (indicated by the value of the resistivity) by applying pressure [@Okunuki95] or substituting P or Sb for As. [@Ochiai97; @Aoki97] Indeed, the experimentally observed characteristic temperature shows the expected behavior. In samples with a lower (more metallic) resistivity $T_0$ becomes larger as compared to samples with a higher resistivity. Note, that the resonance appears also in the $c$ spectral function (see Fig. \[fig:spectra1\] or Fig. \[fig:reso\]). Concerning $\rm Yb_4As_3$ we may interpret the c orbitals as As $p$ band with zero bandwidth. However, in $\rm Yb_4As_3$ the $p$ band is broad and especially the holes in this band should contribute to transport [@Antonov98]. We may conjecture that also the $p$ holes reveal the described low temperature scale $T_0$ (see also the discussion in the next Section). This is known for the case of a periodic Anderson model where also a non-monotonous temperature dependence of the resistivity is obtained. [@Czycholl93]
The enhanced $f$ spectral function close to the chemical potential indicates heavy masses of the charge carriers at low temperatures. Unfortunately, our approach does not allow us to perform the limit $T \rightarrow 0$. It is well-known [@Cox88] that the non-crossing approximation underestimates the absolute value of the imaginary part of the self-energy of the single-impurity Anderson model. This failure will be enhanced in the self-consistent adjustment of the hosting bath of conduction electrons leading to an unphysical change of sign of the self-energy at low temperatures. [@Pruschke93] Therefore we cannot proof the relation $\rho(T) = A
T^2$ and we cannot calculate the coefficient $A$. However, note that in Fig. \[fig:res\] the onset of a $T^2$-behavior can be seen at least for the case of $V=0.6$.
We now turn to the temperature dependence of the Hall coefficient as displayed in . For discussion we want to compare qualitatively the calculated Hall coefficient with the one measured in $\rm Yb_4As_3$. First of all the Hall coefficient of the self-doped Hubbard model is negative whereas it is positive in $\rm
Yb_4As_3$. From our point of view this has to be expected. In the model electrons order in the sense that they prefer to occupy the $f$ subsystem. In $\rm Yb_4As_3$ however holes in the $4f$-shells of the Yb-ions are predominantly confined to the $\langle 111
\rangle$-chains. So there is a different sign for the charge carriers which should show up in the Hall coefficient (see also the modified model in the next Section). Second, the magnitude of the Hall coefficient seems to be wrong. In Fig. \[fig:hall\] it is still five orders of magnitude too small. This rather small value of the Hall coefficient is caused by the still very symmetric spectral function close to the chemical potential (see Fig. \[fig:temp\]). In this respect the chosen values of the hybridization (about $25\%$ of the unperturbed $f$ bandwidth $2W$) are too large. Nevertheless, as we have stressed before the value of the Hall coefficient can be tuned by choosing parameter values of the model closer to the metal-to-insulator transition. Close to the transition arbitrary large values for the Hall coefficient may be obtained.
Keeping this in mind we find indeed an anomalous temperature dependence of the Hall coefficient as in experiment with a characteristic temperature $T_{\rm Max}$. In the case of $V=0.6$ in Fig. \[fig:hall\] this characteristic temperature is smaller than the one found in the resistivity (see Fig. \[fig:res\]). The relation $T_0/T_{\rm Max} \approx 2.13$ is close to the experimental one $T_0/T_{\rm Max} \approx 1.75$. As we will point out in the next Section, the self-doped Hubbard model is not a model simplifying the band structure of $\rm Yb_4As_3$ close to the chemical potential. Therefore, we only want to discuss possible implications of the Coulomb interaction of the charge carriers on the transport properties. If we would e.g. identify the temperature $T_0 = 0.0425$ with the experimental one of 140K we obtain for the $f$ bandwidth $2W
\approx 0.6 \rm eV$. The LSDA+$U$ approach which will be reviewed in the next Section gives a value of $2W=0.007 \rm eV$. But even qualitatively there is still an essential difference. Surprisingly $T_{\rm Max}$ decreases with increasing value of the hybridization. This behavior of $T_{\rm Max}$ is just opposite to the one of $T_0$. In experiment both $T_0$ and $T_{\rm Max}$ behave the same. Especially the ratio $\rho(T)/R_{\rm H}(T)$, i.e., the inverse Hall mobility of the charge carriers, is independent of the applied pressure. It is not clear if our finding for the behavior of $T_{\rm Max}$ in the model depends on the parameter regime and possibly change closer to the metal-to-insulator transition or not.
The semimetal {#sec:exp}
==============
In the preceding Section we have presented the semimetal $\rm
Yb_4As_3$ as possible candidate for a compound which can be described by the self-doped Hubbard model. In this Section we will describe the properties of this compound in more detail. We want to argue that it is indeed justified to compare the transport properties of the self-doped Hubbard model and of $\rm Yb_4As_3$ as we have already done.
$\rm Yb_4As_3$ is an example of a low-carrier heavy-fermion system [@Ochiai90]. Below 100K a linear specific heat is found with a large coefficient $\gamma \approx 200\,{\rm mJ/(molK^2)}$. The Sommerfeld-Wilson ratio is of order unity. The large Hall coefficient at low temperatures indicates an extremely small concentration of $\delta = 0.3 \times 10^{-3}$ positive charge carriers/Yb-atom. The large Hall coefficient and the large residual resistivity of $1\,{\rm
m\Omega cm}$ at low temperatures classify $\rm Yb_4As_3$ as a semimetal. In $\rm Yb_4As_3$, all $\rm Yb$-atoms are aligned on four families of chains. Assuming trivalent As the ratio of Yb-ions is $\rm
Yb^{3+}:Yb^{2+} = 1:3$. A $\rm Yb^{3+}$-ion has one hole in the $f$ shell ($4f^{13}$) and shows a magnetic moment due to Hund’s rule coupling. In a sequence of papers [@Fulde95; @Schmidt96a; @Schmidt96b; @Schmidt96c] Schmidt, Thalmeier and Fulde suggests that at low temperatures the $\rm Yb^{3+}$-ions are confined to one of the four chain systems, e.g., parallel $\langle 111
\rangle$-chains. The low energy excitations of these quasi-one-dimensional spin chains explain the specific heat including the large $\gamma$-coefficient. However, due to the strong Coulomb repulsion of the holes and the large distance of the $\rm
Yb^{3+}$-sites within one chain a perfect ordered state would imply an insulator. Because $\rm Yb_4As_3$ is a semimetal a small fraction of holes have to be redistributed either on the As-atoms or on the other three chain systems, i.e., the $\langle 111 \rangle$-chains have to be “self-doped”. It is tempting to identify this small fraction of holes with the low concentration of charge carriers seen in the Hall coefficient. The charge ordering of the $\rm Yb^{3+}$-ions is meanwhile experimentally confirmed. [@Kohgi97; @Rams96] However, the number of missing holes in the $\langle 111 \rangle$-chains cannot be exactly measured. Note, that the polarized neutron diffraction experiment suggest that this number is not small but of the order of several percent.
During the completion of our work, the electronic structure of $\rm
Yb_4As_3$ has been investigated using energy band calculations within the so-called LSDA+$U$ approach [@Antonov98]. The calculations take into account the possibility of hole occupation on the Yb-sites of the $\langle 111 \rangle$-chains and on the As-sites but not on the remaining three chains system. In result a very narrow $\rm Yb^{3+}$ $4f$ hole band is obtained close to the top of a broad As $p$ band. The Fermi energy is pinned to the bottom of the $4f$ hole band. From this band structure indeed a very small occupation number of holes on the As-sites results comparable to the small carrier number estimated from the Hall coefficient. This finding is independent from the chosen value of $U$ over a wide range of several eV. One should mention that the possible hole distributions in the used approach are too restrictive to deal with valence fluctuating systems like $\rm
Yb_4Bi_3$ and $\rm Yb_4Sb_3$ which do not show a charge ordering transition. Only the $p$ holes should contribute to transport properties because there is no direct $ff$-hopping. This explain correctly the sign and magnitude of the Hall coefficient.
Some questions remain intriguing. In agreement with Fermi-liquid behavior the resistivity is found to be of the form $\rho(T) = \rho_0
+ AT^2$ with a large coefficient $A \approx 0.84 \mu\Omega{\rm
cm/K^2}$. It is hard to understand which scattering mechanism of the $p$ holes can lead to such high effective masses in transport. Not understood at all is the temperature dependence of the resistivity and the Hall coefficient of $\rm Yb_4As_3$. Both quantities show a non-monotonous temperature dependence and exhibit a maximum at a characteristic temperature. The characteristic temperatures differ for the two properties.
Two comments seem to be appropiate. First of all, the band-structure results question strongly the validity of the self-doped Hubbard model for the compound under consideration $\rm Yb_4As_3$. In this compound the transport seems to be purely due to the few charge carriers in the nearly empty charge reservoir, e.g., the holes on the As-sites. In the self-doped Hubbard model however the electrons in the $c$-states do not contribute to the transport properties of the model at all. Moreover, the perfect charge ordered state, e.g., all holes confined to the $\langle 111 \rangle$-chains, is a Mott insulator of a very extreme type because the kinetic energy of the holes on the Yb-sites is nearly zero. Therefore, a suitable model Hamiltonian is still of the form of Eq. (\[eq.smsk\]) but with two important changes. The filling should be three electrons per site and the Coulomb repulsion should be suffered by the $c$-electrons. In this modified version the $f$-subsystem (and not $c$) represents the As sites with nearly no holes and the $c$-subsystem represents the $\rm
Yb^{3+}$-ions occupied by nearly one hole per site. In a future work we will investigate this model. Nevertheless, we believe that the modified model behaves quite similar to the self-doped Hubbard model. Following the arguments outlined in Ref. an insulator-to-metal transition as function of the hybridization have to be expected for both, the self-doped Hubbard model and the modified model. In Fig. \[fig:reso\] we show the $f$ and $c$ spectral functions of the self-doped Hubbard model in the metallic regime. A sharp peak close to the chemical potential is seen in both, the $f$ and the $c$ spectral function. The interaction induced many-body effects of the correlated subsystem are carried over to the system of the uncorrelated orbitals. Therefore, the similarity of the self-doped Hubbard model with the proper model of $\rm Yb_4As_3$ holds despite the fact that the conductivity of the modified model is given by the spectral function of the uncorrelated orbitals. In conclusion, it is justified to compare here some findings for the self-doped Hubbard model with transport measurements of $\rm Yb_4As_3$.
We can argue in a slightly different way. In the charge ordered phase of $\rm Yb_4As_3$ the Coulomb repulsion constrains the holes in the $4f$-shell of the Yb-ions to be localised. The long range part of the repulsion restricts the holes to chains in $\langle
111\rangle$-direction. The on-site part hinders the holes to move along a single chain. This implies that the $4f$-spectral density is split into bands separated by gaps which we may call charge order and Mott-Hubbard gaps, respectively. The charge order gap will be of the order of the transition temperature $T_c \approx 295$K and is much smaller than the Mott-Hubbard gap of several eV. Therefore, the relevant gap for electron excitations is the charge order gap and not the Mott-Hubbard gap. However, the charge order gap in the $4f$-spectral density is a true many-particle effect and cannot be explained by the change of the unit cell between the charge disorderd and the charge ordered phase. Note, the size of the unit cell is unchanged in the charge order transition. The cell is only distorted, changing from cubic to trigonal. The distortion of the unit cell has only little influence on the electronic structure of $\rm Yb_4As_3$, as shown in Ref. . We believe that the charge order gap resembles much more a Mott-Hubbard gap than it resembles a gap in an usual band insulator. The physics of the charge ordering is not included in the self-doped Hubbard model. But what is considered in the used model is the effect of hybridization between a correlated (Hubbard like) band and an uncorrelated band.
The second comment concerns the temperature dependence of the resistivity and the Hall coefficient of $\rm Yb_4As_3$. On the first sight it resembles the one found in other heavy-fermion compounds like $\rm UPt_3$. In $\rm UPt_3$ the anomalous Hall conductivity arises from skew-scattering of the charge carriers [@Kontani94]. As a consequence, the anomalous Hall coefficient should be proportional to the square of the resistivity for temperatures smaller than $T_{\rm
Max}$, the temperature for which a maximum in the Hall coefficient is observed. However, in $\rm Yb_4As_3$ the relation $R_{\rm H} \sim
\rho^2$ is not fullfilled in the temperature regime from 80K down to 4K. Moreover, the zero-temperature value of the Hall coefficient in $\rm Yb_4As_3$ is at least three orders of magnitude larger than in other heavy-fermion compounds with a positive Hall coefficient. We therefore exclude skew-scattering as possible mechanism to explain the experimetal data.
Conclusion {#sec:conc}
==========
In conclusion, we extended the Hubbard model by coupling one additional orbital per site via hybridization to the Hubbard orbitals. The Coulomb repulsion is supposed to be large, $U>U_{\rm c}$, where $U_{\rm c}$ denotes the critical Coulomb repulsion for the Mott-Hubbard transition in the one-band Hubbard model. We calculated the spectral function, resistivity and Hall coefficient of the “self-doped” Hubbard model using dynamical mean-field theory. To this end the lattice model is mapped onto an impurity model in which a unit cell of the lattice is embedded self-consistently in a bath of free electrons. The impurity model is studied numerically by an extension of the non-crossing approximation to a two-orbital impurity.
The self-doped Hubbard model is an insulator only in a restricted parameter regime. The hybridization with the added orbitals drives an insulator-to-metal transition, provided the Coulomb repulsion fullfills the constraint $U_{\rm c} < U < 2 U_{\rm c}$. This is a correlation driven metal-insulator transition in a non particle-hole symmetric case. In the vicinity of the transition to an insulator the resistivity and the Hall coefficient become very large. Therefore the model serves as a model for semimetallic behavior in systems where the Coulomb interactions of the charge carriers nearly block their movement. The number of missing electrons in the Hubbard subsystem, i.e., the occupation number of the additional orbitals, can not be interpreted as number of charge carriers seen in the transport properties. A simple Drude analysis fails for this type of mechanism leading to semimetallic behavior.
In the semimetallic regime the resistivity and the Hall coefficient show an unusual temperature dependence. Both, $\rho(T)$ and $|R_{\rm
H}(T)|$, exhibit a maximum at a characteristic temperature. The characteristic temperatures for the resistivity $T_0$ differs from the one of the Hall coefficient $T_{\rm Max}$. Depending on the parameter regime either $T_0 < T_{\rm Max}$ or $T_0 > T_{\rm Max}$ is possible. The characteristic temperatures change considerably when varying the hybridization and tuning the system into a more insulating or metallic state. At $T_0$ a resonance arises in the spectral function close to the chemical potential. This resonance can be assigned to a Kondo-like effect for band electrons. The enhanced spectral weight at the chemical potential indicates heavy masses for the charge carriers.
We compared our findings with measurements of the resistivity and the Hall coefficient for the semimetal $\rm Yb_4As_3$ including pressure and substitution experiments. Although the self-doped Hubbard model is not a simplified model extracted from band structure calculations we found surprising similarities. We take this as indication that the anomalous transport properties of $\rm Yb_4As_3$ can be indeed assigned to the Coulomb interaction of the charge carriers which is not treated adequate in standard band structure calculations. This should be true even when the transport properties are dominated by As $p$ holes. At least in the model the interaction induced many-body effects of the Hubbard subsystem are carried over to the system of the additional orbitals. Nevertheless, to obtain final conclusive results the band structure of $\rm Yb_4As_3$ have to be used as input for a simple model which still can be treated within the dynamical mean-field theory. This is left for future work.
We would like to thank P. Fulde, B. Schmidt and P. Thalmeier for useful discussions and J. Schmalian for numerical advice.
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---
abstract: 'Direct photon emission in heavy-ion collisions is calculated within a relativistic micro+macro hybrid model and compared to the microscopic transport model UrQMD. In the hybrid approach, the high-density part of the collision is calculated by an ideal 3+1-dimensional hydrodynamic calculation, while the early (pre-equilibrium-) and late (rescattering-) phase are calculated with the transport model. We study both models with $Au+Au$-collisions at $\sqrt{s_{\rm NN}} = 200$ GeV and compare the results to experimental data published by the PHENIX collaboration.'
address: 'Frankfurt Institute for Advanced Studies, Ruth-Moufang-Straße 1, 60438 Frankfurt am Main, Germany'
author:
- B Bäuchle and M Bleicher
title: 'Calculations of direct photon emission in Heavy Ion Collisions at $\sqrt{s_{\rm NN}} = 200$ GeV'
---
Introduction
============
Creating and studying high-density and -temperature nuclear matter is the major goal of heavy-ion experiments. A state of quasi-free partonic degrees of freedom, the Quark-Gluon-Plasma (QGP) [@Harris:1996zx; @Bass:1998vz] may be formed, if the energy density reached in the reaction is high enough. Strong jet quenching, large elliptic flow and other observations made at the Relativistic Heavy Ion Collider (BNL-RHIC) suggest the successful creation of a strongly coupled QGP (sQGP) at these energies [@Adams:2005dq; @Back:2004je; @Arsene:2004fa; @Adcox:2004mh], and possible evidence for the creation of this new state of matter has also been put forward by collaborations at the Super Proton Synchrotron (CERN-SPS), as for instance the step in the mean transverse mass excitation function of protons, kaons and pions and the enhanced $K^+/\pi^+$-ratio [@:2007fe].
Electromagnetic probes provide a unique insight into the early stages of heavy-ion collisions, since they have the advantage of negligible rescattering cross-sections. Therefore, they leave the production region without rescattering and carry the information from this point to the detector. Besides single- and dileptons, direct photon emission can therefore be used to study the early hot and dense, possibly partonic, stages of the reaction.
Unfortunately, most photons measured in heavy-ion collisions come from hadronic decays. The experimental challenge of obtaining spectra of only direct photons has been gone through by several experiments; WA98 (CERN-SPS) [@Aggarwal:2000ps] and PHENIX (BNL-RHIC) [@Adler:2005ig] have published explicit data points for direct photons.
On the theory side, the elementary photon production cross-sections are known since long, see e.g. Kapusta [*et al.*]{} [@Kapusta:1991qp] and Xiong [*et al.*]{} [@Xiong:1992ui]. The major problem is the difficulty to describe the time evolution of the produced matter, for which first principle calculations from Quantum Chromodynamics (QCD) cannot be done. Well-developed dynamical models are therefore needed to describe the space-time evolution of nuclear interactions.
Among the approaches used are relativistic transport theory [@Geiger:1997pf; @Bass:1998ca; @Bleicher:1999xi; @Ehehalt:1995is; @Molnar:2004yh; @Xu:2004mz; @Lin:2004en; @Burau:2004ev; @Bass:2007hy] and relativisitc fluid- or hydrodynamics [@Cleymans:1985wp; @McLerran:1986nc; @PHRVA.D34.794; @Kataja:1988iq; @Srivastava:1991ju; @Srivastava:1991nc; @Srivastava:1991dm; @Srivastava:1992gh; @Cleymans:1992zc; @Rischke:1995mt; @Hirano:2001eu; @Huovinen:2001wx; @Huovinen:2002im; @Kolb:2003dz; @Nonaka:2006yn; @Frodermann:2007ab]. For both models, approximations have to be made, and in both models, the restrictions imposed by these approximations can be loosened. For transport theory, the necessary approximations include the restriction of scattering processes to two incoming particles, which limits the applicability to low particle densities. For hydrodynamics, on the other hand, matter has to be in local thermal equilibrium (for ideal, non-viscous hydrodynamic calculations) or at least close to it (for viscous calculations) [@Dusling:2007gi; @Baier:2006um; @Song:2008si].
From this, it is clear where the advantages for both models are: While in transport, non-equilibrium matter, which is present in the beginning of the heavy-ion reaction, and dilute matter, which is present in the late phase, can be described, hydrodynamics may be better suited to describe the intermediate stage, which is supposed to be dense, hot and thermalized. In addition, the transition between two phases of matter, such as Quark Gluon Plasma (QGP) and Hadron Gas (HG) can be easily described in hydrodynamics, while this is not (yet) possible for transport models, since the microscopic details of this transition are not known.
The Model {#sec:model}
=========
UrQMD v2.3 (Ultra-relativistic Quantum Molecular Dynamics) is a microscopic transport model [@Bass:1998ca; @Bleicher:1999xi; @Petersen:2008kb]. It includes all hadrons and resonances up to masses $m \approx 2.2~{\rm GeV}$ and at high energies can excite and fragment strings. The cross-sections are either parametrized, calculated via detailed balance or taken from the additive quark model (AQM), if no experimental values are available. At high parton momentum transfers, PYTHIA [@Sjostrand:2006za] is employed for pQCD scatterings.
UrQMD differentiates between two regimes for the excitation and fragmentation of strings. Below a momentum transfer of $Q < 1.5$ GeV a maximum of two longitudinal strings are excited according to the LUND picture, at momentum transfers above $Q > 1.5$ GeV hard interactions are modelled via PYTHIA. For detailed information on the inclusion of PYTHIA, the reader is referred to Section II of [@Petersen:2008kb]. In the UrQMD framework, propagation and spectral functions are calculated as in vacuum.
In the following, we compare results from this microscopic model to results obtained with a hybrid model description [@Petersen:2008dd]. Here, the high-density part of the reaction is modelled using ideal 3+1-dimensional fluid-dynamics. The unequilibrated initial state and the low-density final state are described by UrQMD. In the hydrodynamic intermediate stage we use a Hadron Gas Equation of State (HG-EoS) which includes the same degrees of freedom as are present in the transport phase. This allows to explore the effects due to the change of the kinetic description.
To connect the initial transport phase with the high-density fluid phase, the baryon-number-, energy- and momentum-densities are smoothed and put into the hydrodynamic calculation after $t = 0.6$ fm. Temperature, chemical potential, pressure and other macroscopic quantities are determined from the densities by the Equation of State used in the current calculation. During this transition, the system is forced into an equilibrated state. Particles with high rapidity $y > 2$, however, are excluded from the hydrodynamic grid and propagated in the cascade without interaction to the hydrodynamic medium.
In non-central collisions, the spectators are propagated in the cascade. After the local rest frame energy density has dropped below a threshold value of $\epsilon_{\rm crit} \approx 5 \epsilon_0$, particles are created on a hyper-surface from the densities by means of the Cooper-Frye formula and propagation is continued in UrQMD.
The transition from hydrodynamic to cascade description used in the calculations presented here is gradual. I.e. each transverse slice (constant $z$) is transferred to the cascade at the same time, when the condition is met throughout that slice. This represents a pseudo-eigentime condition.
Photon emission is calculated perturbatively in both models, hydrodynamics and transport, because the evolution of the underlying event is not altered by the emission of photons due to their very small emission probability. The channels considered for photon emission may differ between the hybrid approach and the binary scattering model. Emission from a Quark-Gluon-Plasma can only happen in the hydrodynamic phase, and only if the equation of state used has partonic or chirally restored degrees of freedom. Photons from baryonic interactions are neglected in the present calculation.
For emission from the transport part of the model, we use the well-established cross-sections from Kapusta [*et al.*]{} [@Kapusta:1991qp], and for emission from the hydrodynamic phase, we use the parametrizations by Turbide, Rapp and Gale and Arnold [*et al.*]{} [@Turbide:2003si] (the latter for QGP-emission). For detailed information on the emission process, the reader is referred to Bäuchle and Bleicher [@arXiv:0905.4678].
Results
=======
Photon emission has been calculated for minimum bias (0-92 %) and central (0-10 %) $Au+Au$-collions at $\sqrt{s_{\rm NN}} = 200$ GeV with both the pure cascade and the cascade-hydrodynamic hybrid models. We note that the high-$p_\bot$ data obtained by PHENIX [@Adler:2005ig] can be described very well with pQCD-calculations by Gordon and Vogelsang [@Gordon:1993qc], therefore we investigate the contributions of our model to low-$p_\bot$ photons only.
In Figure \[fig:phenix\_low\], we show the direct photons emitted by the cascade- and hybrid models and compare them to the aforementioned pQCD-calculations and data from the PHENIX-collaboration.
While our hadronic model fails to predict the amount of direct photons emitted at intermediate transverse momentum, the disagreement is smaller at low $p_\bot$. The spectra obtained with an intermediate hydrodynamic part are significantly higher than those with pure cascade calculations. In any case, the spectra are negligible compared to the pQCD contributions predicted by Gordon and Vogelsang, therefore, an enhancement with respect to those data cannot be explained by hadronic sources.
In Figure \[fig:stages\], we explore the contribution of the different stages to the hybrid model spectra. The contribution of the intermediate hydro phase turns out to be dominant at low transverse momenta $p_\bot <
3$ GeV. The contributions of the initial and final stages are lower than the complete spectrum from the cascade calculations, so that we can assume the intermediate stage in the cascade calculation to be on the order of the final stage contribution in the hybrid model calculations.
The difference between the hydrodynamic and transport descriptions of the intermediate stage has not been observed for collisions with lower energies $E_{\rm lab} = 158$ AGeV (see [@arXiv:0905.4678]) and $E_{\rm lab} =
45$ AGeV (see [@arXiv:1003.5454]).
Summary
=======
In this article, we have applied UrQMD and the UrQMD+Hydro hybrid model to calculate photon spectra from central and minimum bias Au+Au-collions at $\sqrt{s_{\rm NN}} = 200$ GeV. The comparison of the (hadronic) calculations to data from the PHENIX collaboration suggest that a significant contribution to the measured spectra comes from non-hadronic sources. The low-$p_\bot$-excess over pQCD-predictions seen by the PHENIX-collaboration [@:2008fqa] cannot be explained by hadronic sources; partonic sources such as a Quark-Gluon-Plasma are therefore very likely to be responsible for this excess.
The comparison of transport and hybrid calculations show that the conclusion drawn at lower energy, which suggested that there is no difference between the spectra obtained with or without intermediate hydrodynamic stage, is not valid at these high energies. The excess of the hybrid model calculations over the transport calculations are visible in both central and minimum bias collision samples, and the magnitude of this excess is similar in both models. This suggests that the excess depends only the collision energy, not on the system size.
Outlook
=======
The results shown here suggest the need for further studies. Calculations of direct photon spectra have to be done with the current model for other centrality selections, and with different Equations of State such as a Bag Model EoS and an EoS with a chirally restored phase. In order to investigate the difference between the hybrid model and the cascade model, an energy dependent investigation of the excess is advisable.
Different systems measured at the RHIC-facility, such as Cu+Cu-collisions and collisions at smaller center-of-mass energy, will also be calculated.
The parameters of the model – the conditions for switching to and from the hydrodynamic description and the scenario for the latter transition – will be investigated in the future.
This work has been supported by the Frankfurt Center for Scientific Computing (CSC), the GSI and the BMBF. The authors thank Hannah Petersen for providing the hybrid- and Dirk Rischke for the hydrodynamic code. B.Bäuchle gratefully acknowledges support from the Deutsche Telekom Stiftung, the Helmholtz Research School on Quark Matter Studies and the Helmholtz Graduate School for Hadron and Ion Research. This work was supported by the Hessian LOEWE initiative through the Helmholtz International Center for FAIR.
The authors thank Elvira Santini, Pasi Huovinen and Rene Bellwied for valuable discussions and Klaus Reygers for experimental clarifications.
References {#references .unnumbered}
==========
[99]{}
|
---
abstract: 'A thermal squeezed state representation of inflaton is constructed for a flat Friedmann-Robertson-Walker background metric and the phenomenon of particle creation is examined during the oscillatory phase of inflaton, in the semiclassical theory of gravity. An approximate solution to the semiclassical Einstein equation is obtained in thermal squeezed state formalism by perturbatively and is found obey the same power-law expansion as that of classical Einstein equation. In addition to that the solution shows oscillatory in nature except on a particular condition. It is also noted that, the coherently oscillating nonclassical inflaton, in thermal squeezed vacuum state, thermal squeezed state and thermal coherent state, suffer particle production and the created particles exhibit oscillatory behavior. The present study can account for the post inflation particle creation due to thermal and quantum effects of inflaton in a flat FRW universe.'
address: 'School of Physics, University of Hyderabad, Hyderabad-500 046.India.'
author:
- P K Suresh
title: Particle creation in the oscillatory phase of inflaton
---
Introduction
============
According to the simplest version of the inflationary scenario, the universe in the past expanded exponentially with time, while its energy density was dominated by the effective potential energy density of a scalar field, called the inflaton. Sooner or later, inflation terminated and the inflaton field started quasiperiodic motion with slowly decreasing amplitude. The universe was empty of particles after inflation and particles of various kinds created due to the quasiperiodic evolution of the inflaton field. The universe became hot again due the oscillations and decay of the created particles of various kinds. Form on,it can be described by the hot big bang theory.
The standard cosmology provides reliable and tested account of the history of the universe from about 0.01sec after the big bang until today, some 15 billions years later. Despite its success, the hot big bang model left many features of the universe unexplained. The most important of these are horizon problem, singularity problem, flatness problem, homogeneity problem, structure formation problem, monopole problem and so on. All these problems are very difficult and defy solution within the standard cosmology. Most of these problems have been either completely resolved or considerably relaxed in the context inflationary scenario [@1]. At present there are different versions [@2]-[@4] of the inflationary scenario. The main feature of all these versions is known as the inflationary paradigm. Inflationary cosmology is also widely accepted because of its success in explaining cosmological observations [@5].
Most of the inflationary scenarios are based on the classical gravity of the Friedmann equation and the scalar field equation in the Friedmann-Robertson-Walker (FRW) universe, assuming its validity even at the very early stage of the universe. However, quantum effects of matter fields and quantum fluctuations are expected to play a significant role in this regime, though quantum gravity effects are still negligible. Therefore, the proper description of a cosmological model can be studied in terms of the semiclassical gravity of the Friedmann equation with quantized matter fields as the source of gravity. The semiclassical quantum gravity seems to be a viable method throughout the whole non-equilibrium quantum process from the pre-inflation period of hot plasma in thermal equilibrium to the inflation period and finally to the matter-dominated period.
Recently, the study of quantum properties of inflaton has been received much attention in semiclassical theory of gravity and inflationary scenarios. [@6; @7]. In the new inflation scenario [@8] quantum effects of the inflaton were partially taken into account by using one-loop effective potential and an initial thermal condition. In the stochastic inflation [@9] scenario the inflaton was studied quantum mechanically by dealing with the phase-space quantum distribution function and the probability distribution [@10]. The aforementioned studies show that results obtained in classical gravity are quite different from those in semiclassical gravity. Such studies reveal that quantum effects and quantum phenomena play an important role in inflation scenario and the related issues. Recently, it has been found that nonclassical state formalisms are quite useful to deal with quantum effects in cosmology [@11]-[@18], particularly squeezed sates and coherent state formalism of quantum optics [@19].
The above mentioned squeezed sates and coherent state formalisms are zero temperature states. There exist a thermal counterparts of coherent and squeezed sates and are useful to deal with finite temperature effects and quantum effects. From the cosmological point of view it would be more natural to consider the temperature effects on the background of FRW metric. Therefore this motivates the study of thermal squeezed states and thermal coherent states in cosmology.
The goal of the present paper is to study quantum and finite temperature effects of minimally coupled massive inflaton in the FRW universe. Hence to examine the thermal and quantum particle creation, in the oscillatory phase, of the inflaton in thermal coherent and thermal squeezed state formalisms, in the semiclassical theory of gravity. For the present study we follow the unit system $ c= G= \hbar $=1.
Thermal squeezed states and thermal coherent states
===================================================
The thermo field dynamic [@20] formalism can be use to get the thermal counterparts of coherent and squeezed states. The main feature of thermo field dynamics is the thermal Bogoliubov transformation that maps the theory from zero to finite temperature. One can construct a thermal vacuum $ \mid 0(\beta) \rangle $ annihilated by thermal annihilation operators and can express the average value of any observable $A$ as the expectation value in the thermal [@20] vacuum $$Z(\beta)^{-1}\tr[\rho {A}]= \langle 0(\beta)\mid {A} \mid 0(\beta) \rangle ,$$ where $\rho$ is the distribution function, $\beta= {1\over k T}$ and $k$ Boltzmann’s constant, and T the temperature. In order to fulfill the requirement (2.1), the vacuum should belong to the direct space between the original Fock space by an identical copy of it denoted by a tilde. Therefore $$\begin{aligned}
\mid 0(\beta) \rangle &=& e^{-iM} \mid 0, \tilde{0} \rangle,
M= -i\theta(\beta) ( a^{\dagger} \tilde{a}^{\dagger}-a\tilde{a} ),\end{aligned}$$ where $a$, $a^{\dag}$ are the annihilation and creation operators in original Fock space and $\tilde{a}$, $\tilde{a}^{\dag}$ are the same for the tilde space, and are obeying boson commutation relations $[a,a^{\dag}]=[\tilde{a},\tilde{a}^{\dag}]=1$, the other combinations are zero.
The density matrix approach usually gives us a convenient method for incorporating finite temperature effects. Hence, various definitions of thermal coherent states (tcs) can be summarized by giving its density matrix and it can be written for the single mode case as [@21] $$\begin{aligned}
\rho_{tcs} = D^{\dagger}(\alpha) e^{-\beta \omega
a^{\dagger} a} D(\alpha),\end{aligned}$$ where $\alpha$ is a complex number specifying the coherent state, $\omega $ is the energy of the mode, and $$\begin{aligned}
D(\alpha)&=& \exp{ \left(\alpha a^{\dagger}- \alpha^* a\right)}. \end{aligned}$$
The characteristic function for single mode thermal coherent state, ${\cal{Q}}_{tcs}$, is defined [@21] by $$\begin{aligned}
{\cal{Q}}_{tcs}(\eta ,\eta^*) &=& \exp [ - f(\beta) {\mid \eta \mid}^2
+ \eta^* \alpha - \eta \alpha^*],\end{aligned}$$ where $\eta$ and $\eta^*$ are as independent variables and, $$\begin{aligned}
f(\beta)& =& { 1 \over e^{\beta \omega} - 1}.\end{aligned}$$
Similarly the density matrix for a single mode thermal squeezed states (tss) is given [@21]by $$\begin{aligned}
\rho_{tss} = D^{\dagger}(\alpha) S^{\dagger} (\xi) e^{- \beta {a^{\dagger}} a
} S(\xi) D(\alpha),\end{aligned}$$ where $$\begin{aligned}
S(\xi) = \exp [( \xi {a^{\dagger} }^2 - \xi^* a^2 )/2],
\xi = r e^{i \vartheta}.\end{aligned}$$ Here $ r $ is the squeezing parameter and $ \vartheta $ is the squeezing angle.
The characteristic function of a single mode thermal squeezed state is given by $$\begin{aligned}
\fl \eqalign{
{\cal{Q}}_{tss}(\eta ,\eta^*)=& \exp [ -{\mid \eta \mid}^2
\left ( \sinh^{2} r \coth {\beta \omega \over 2} + f(\beta) \right ) \\
& - {\cosh r \sinh r \over 2} \coth { \beta \omega \over 2}
\left (
e^{-i\varphi} \eta^2 +e^{i \varphi} {\eta^*}^2 \right ) - \eta \alpha^*
+ \eta^* \alpha ].}\end{aligned}$$ The density matrix for a single mode thermal squeezed vacuum (tsv)is given by $$\begin{aligned}
\rho_{tsv}& =& S^{\dagger} (\xi) e^{- \beta {a^{\dagger}} a
} S(\xi),\end{aligned}$$ and the characteristic function is $$\begin{aligned}
\eqalign{
{\cal{Q}}_{tsv}(\eta ,\eta^*)=& \exp [ -{\mid \eta \mid}^2
\left ( \sinh^{2} r \coth {\beta \omega \over 2} + f(\beta) \right ) \\
& - {\cosh r \sinh r \over 2} \coth { \beta \omega \over 2}
\left (
e^{-i\varphi} \eta^2 +e^{i \varphi} {\eta^*}^2 \right )
].}\end{aligned}$$ Though the space is direct product between the original space and identical copy of it, the observational quantities are the expectation values of $ a, a^{\dagger}, a^2, {a^{\dagger}}^2 $ [@21] etc. Those quantities can be computed in thermal coherent state, thermal squeezed state and thermal squeezed vacuum state formalisms by applying their corresponding characteristic function in the following relations. $$\begin{aligned}
\eqalign{
\langle a \rangle =& {\partial {\cal{Q}} \over \partial
{{\eta ^*}}}{ \mid_{ \eta
= {\eta^*} = 0}}, \\
\langle a^{\dagger } \rangle =& -{\partial {\cal{Q}} \over \partial
{\eta}}
{\mid_{ \eta
= \eta^* = 0} }.}\end{aligned}$$ Similarly the higher order expectation values of $ a $ and $a^{\dagger}$ can be also evaluated using the same procedure of eq (2.12).
Inflaton in a flat FRW metric
==============================
Consider a flat Friedmann-Robertson-Walker spacetime with the line element $$ds^{2}=-dt^{2}+R^{2}(t) (dx^{2}+dy^{2}+dz^{2}),$$ The metric is treated as an unquantized external field.
The minimally coupled inflaton with the gravity, for the metric (3.1), can be described by the Lagrangian $$L={1\over2} R^{3} \left({\dot{\varphi}^{2}}-m^{2}\varphi^{2}\right).$$ Where overdot represents a derivative with respect to time. The equation governing the inflaton, for the metric (3.1), can be written as $$\ddot\varphi+3\frac{\dot{R}}{R}\dot\varphi+m^{2}\varphi = 0.$$ One can define the momentum conjugate to $\varphi$ as, $
\pi = \frac{\partial L}{\partial\dot{\varphi}}.
$ Thus, the Hamiltonian of the inflaton is $$H=\frac{\pi^{2}}{2R^{3}}+{1\over2}R^{3}m^{2}\varphi^{2}\,\,.$$ Therefore, $0-0 $ component the energy-momentum tensor for the inflaton takes the following form $$T_{00}=\frac{R^{3}}{2}(\dot\varphi^{2}+ m^{2}\varphi^{2})\,\,.$$ Consider the minimally coupled inflaton as the source of gravity. Therefore the classical Einstein equation becomes $$\left(\frac{\dot{R}}{R}\right)^2=\frac{8\pi}{ 3} \frac{T_{00}}{R^3}
\,\,,$$ where $T_{00}$ is the energy density of the inflaton, given by (3.5). In the cosmological context, the classical Einstein equation (3.6) means that the Hubble constant, $H=\frac{\dot{R}}{R}$, is determined by the energy density of the dynamically evolving inflaton as described by (3.3).
Thermal and quantum particle creation
======================================
Since there is no consistent quantum theory of gravity available, it would be meaningful to consider the semiclassical gravity theory to study quantum effect of matter field in a classical background metric. The semiclassical approach is also useful to deal with problems in cosmology, where quantum gravity effects are negligible. Oscillatory phase of inflaton is such a situation, where one can neglect the quantum gravity effects. Therefore the present study can be restricted in the frame work of semiclassical theory of gravity. In semiclassical theory the Einstein equation can be written as $$G_{\mu\nu}=8\pi \langle {\hat{T}}_{\mu\nu}\rangle\,\,,$$ where the quantum field, represented by a scalar field $\phi$, is governed by the time-dependent Schr$\ddot{o}$dinger equation $$i\frac{\partial \phi}{\partial t} =\hat{H}_{\phi} \phi
\,\,.$$. Consider quantum inflaton as the source, then the Friedmann equation, for the metric (3.1), in the semiclassical theory, can be written as $$\left(\frac{\dot{R}}{R}\right)^2=\frac{8\pi}{3} \frac{1}{R^3}
\langle \hat{H}_\varphi \rangle\,\,,$$ where $\langle \hat{H}_{\varphi} \rangle$ represent the expectation value of the Hamiltonian of the inflaton in a quantum state under consideration.
The inflaton can be described by the time dependent harmonic oscillator, with the Hamiltonian given in (3.4). To study, the semiclassical Friedmann equation, the expectation value the Hamiltonian (3.4) to be computed, in a quantum state under consideration. Therefore (3.4) becomes $$\langle\hat{H}_{\varphi} \rangle=\frac{1}{2R^{3}}\langle\hat{\pi}^2\rangle+
\frac{m^{2}R^{3}}{2}\langle\hat{\varphi}^2\rangle\,\, .$$ The eigenstates of the Hamiltonian are the Fock states $$a^{\dag}(t)a(t)|n,\varphi,t \rangle = n|n,\varphi,t \rangle\,\,,$$ where $$\begin{aligned}
\eqalign{
a(t)=&\varphi^*(t) \hat{\pi} - R^{3}\dot{\varphi}^*(t)
\hat{\varphi},\\
a^{\dag}(t)=&\varphi(t) \hat{\pi}-R^{3} \dot{\varphi}(t)
\hat{\varphi}\,\,. }\end{aligned}$$ As an alternative to the $n$ representation, consider the inflaton in thermal squeezed state formalism. Therefor the expectation value of the Hamiltonian (4.4) in thermal squeezed state can be computed as follow.
From (2.9), (2.12) and (4.6), we get $$\eqalign{
\langle \hat{\pi}^{2}\rangle =&
- R^{6}\dot{\varphi}^{2}\left( \alpha^{2}-e ^{i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2} \right) \\
&-R^{6}\dot{\varphi}^{\ast 2} \left( \alpha^{\ast 2}-e ^{-i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2}\right)\\
&+ R^{6} \dot{\varphi} ^{\ast}\dot{\varphi} \left( 2| \alpha|^{2}+ 2 \sinh^{2} r \coth\frac{\beta \omega}{2} + 2 f(\beta) + 1 \right),}$$ and $$\begin{aligned}
\eqalign{
\langle \hat{\varphi}^{2}\rangle=&
- \varphi^{2}\left( \alpha^{2}-e ^{i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2} \right) \\
&-\varphi^{\ast 2} \left( \alpha^{\ast 2}-e ^{-i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2}\right)\\
&+ \varphi ^{\ast}\varphi \left( 2| \alpha|^{2}+ 2 \sinh^{2} r \coth\frac{\beta \omega}{2} + 2 f(\beta) + 1 \right).}\end{aligned}$$ Substituting (4.7) and (4.8) in (4.4), and the apply the result in (4.3), then the semiclassical Friedmann equation becomes $$\begin{aligned}
\fl \eqalign{
\left(\frac{\dot{R}}{R}\right)^2 =&
\frac{4 \pi}{3} \left[ (\dot{\varphi }^{\ast} \dot{\varphi} + m^2 {\varphi }^{\ast} \varphi)
\left( 2| \alpha|^{2}+ 2 \sinh^{2} r \coth\frac{\beta \omega}{2} + 2 f(\beta) + 1 \right) \right. \\
&\left.- (\dot{ \varphi }^{2} +m^{2} \varphi^{2})
\left( \alpha^{2}-e ^{i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2} \right) \right.\\
&\left.-(\dot{\varphi}^{\ast 2}+ m^2 \varphi^{\ast 2})
\left( \alpha^{\ast 2}-e ^{-i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2}\right) \right],}\end{aligned}$$ where, $\varphi$ and $\varphi^*$ satisfy eq (3.3) and the Wronskian condition $$R^3(t)\left(\dot{\varphi^*}(t)\varphi(t)-\varphi^{*}(t) \dot{\varphi}(t) \right)=i\,\,.$$ The above boundary condition, fixes the normalization constants of the two independent solutions.
To solve the self-consistent semiclassical Einstein equation (4.9), transform the solution in the following form $$\varphi(t)=\frac{1}{R^{3\over2}}\psi(t),$$ thereby obtaining $$\ddot{\psi}(t)+\left(m^{2}-{3\over4}\left(\frac{\dot{R}(t)}{R(t)}\right)^{2}
-{3\over2}
\frac{\ddot{R}(t)}{R(t)}\right)\psi(t)=0\,\,.$$ Next, focus on the oscillatory phase of the inflaton after inflation. In the parameter region satisfying the inequality $$m^2 > \frac{3\dot{R}^2}{4R^2}+\frac{3\ddot{R}}{2R},$$ the inflaton has an oscillatory solution of the form $$\psi(t)=\frac{1}{\sqrt{2\sigma(t)}}\exp(-i\int \sigma(t)dt)\,\,.$$ With $$\sigma(t)=\sqrt{m^{2}-{3\over4}\left(\frac{\dot{R}}{R}\right)^{2}-{3\over2}
\frac{\ddot{R}}{R}+{3\over4}\left(\frac{\dot{\sigma}(t)}{\sigma(t)}\right)^{2}-
{1\over2}\frac{\ddot{\sigma}(t)}{\sigma(t)}}\,\,.$$ By applying the transform solution (4.11) in (4.9), and also using the fact $ \alpha = e^{i\delta} \alpha $, we obtain $$\begin{aligned}
\fl \eqalign{
R(t)=&\left[\frac{2 \pi}{3 \sigma} \frac{1} {(\frac{\dot{R}}{R})^2} \left[\frac{1}{4}\left(\left( 3 \frac{\dot{R}}{R} + \frac{\dot{\sigma}}{\sigma}\right)^{2}
+\sigma^{2} +m^{2}\right)\right. \right.\\
&\left. \left. \times
\left( 2 |\alpha |^{2} + 2 \sinh^2 r \coth \frac{\beta \omega}{2}
+ 2 f(\beta) + 1 \right)\right. \right.\\
&\left. \left.
-\frac{1}{4}\left(\left( 3 \frac{\dot{R}}{R} + \frac{\dot{\sigma}}{\sigma}\right)^{2}
-\sigma^{2} +m^{2}\right) \right.\right.\\
&\left.\left. \times \left( 2 \alpha^{2} \cos(2 \delta - 2 \sigma t )
- \cos(\vartheta - 2 \sigma t)
\sinh(2r) \coth \frac{\beta \omega}{2}\right) \right. \right.\\
& \left. \left.+ \sigma \left( 3 \frac{\dot{R}}{R} + \frac{\dot{\sigma}}{\sigma}\right)
\left( 2 \alpha^{2} \sin(2 \delta - 2 \sigma t )
+ \sin(\vartheta - 2 \sigma t)
\sinh(2r) \coth \frac{\beta \omega}{2}\right)\right]
\right ]^{1/3}.}\end{aligned}$$ The next order approximation solution of the eq (4.16) can be obtained by using the following approximation ansatzs $$\sigma_{0}(t)=m,$$ and $$R_{0}(t)=R_{0}t^{2\over3}\,\,.$$ Thus we get $$\begin{aligned}
\fl \eqalign{
R_{1}(t)=&\left[3 \pi m t^{2} \left[
\left( 1+ \frac{1}{2 m^2 t^2}\right)
\left( 2 |\alpha |^{2} + 2 \sinh^2 r \coth \frac{\beta \omega}{2}
+ 2 f(\beta) + 1 \right)\right. \right.\\
&\left. \left.
-\frac{1}{2 m^2 t^2} \left( 2 \alpha^{2} \cos(2 \delta - 2 m t )
- \cos(\vartheta - 2 m t)
\sinh(2r) \coth \frac{\beta \omega}{2}\right) \right. \right.\\
& \left. \left.+ \frac{2}{m t^2}
\left( 2 \alpha^{2} \sin(2 \delta - 2 m t )
+ \sin(\vartheta - 2 m t)
\sinh(2r) \coth \frac{\beta \omega}{2}\right)\right]
\right ]^{1/3}.}\end{aligned}$$ When $2 \delta = \vartheta= 2 m t$, then (4.19) becomes $$\begin{aligned}
\fl \eqalign{
R_{1}(t)=&\left[3 \pi m t^{2} \left[
\left( 1+ \frac{1}{2 m^2 t^2}\right)
\left( 2 |\alpha |^{2} + 2 \sinh^2 r \coth \frac{\beta \omega}{2}
+ 2 f(\beta) + 1 \right)\right. \right.\\
&\left. \left.
-\frac{1}{2 m^2 t^2} \left( 2 \alpha^{2}
-
\sinh(2r) \coth \frac{\beta \omega}{2}\right) \right]
\right ]^{1/3}.}\end{aligned}$$ Next, consider the particle production of the inflaton, in thermal squeezed states formalisms, in semiclassical theory of gravity. First, consider the Fock space which has a one parameter dependence on the cosmological time $t$. The number of particles at a later time $t$ produced from the vacuum at the initial time $t_{0}$ is given by $$N_0(t,t_0)=\langle 0,\varphi,t_0\mid\hat{N}(t)\mid 0,\varphi, t_0\rangle ,$$ here, $\hat{N}(t)=a^{\dag}a$ and its expectation value and can be calculated by using (4.6). Therefor, $$\langle \hat{N}(t) \rangle = R^6 \dot{\varphi} \dot{\varphi^*} \langle \hat{\varphi^2} \rangle
+ \varphi \varphi^* \langle \hat{\pi}^2 \rangle - R^3 \varphi
\dot{\varphi}^* \langle \hat{\pi} \hat{\varphi} \rangle-R^3 \dot{\varphi} \varphi^* \langle
\hat{\varphi} \hat{\pi}
\rangle .$$ Again using (4.6) we get $$\eqalign{
\langle\hat{\varphi}^2\rangle&=\varphi^* \varphi , \\
\langle\hat{\pi}^2\rangle&=R^6\dot{\varphi}^*\dot{\varphi}, \\
\langle\hat{\pi}\hat{\varphi}\rangle&=R^3 \dot{\varphi} \varphi^*, \\
\langle\hat{\varphi}\hat{\pi}\rangle&=R^3\varphi\dot{\varphi}^* . }$$ Therefore, substituting (4.23), in (4.22), we get $$N_0(t,t_0)=R^{6}|\varphi(t)\dot{\varphi}(t_0)-\dot{\varphi}(t)\varphi(t_0)|^{2}\,\,.$$ Using the approximation ansatzs (4.17),(4.18) and (4.24), the number of particles created at a later time $t$ from the vacuum state at the initial time $t_0$ in the limit $mt_0$, $mt>1$ can be computed and is \[7\] given by $$\begin{aligned}
\nonumber N_0(t,t_0)&=&
\frac{1}{4 \sigma(t) \sigma(t_0)}\left(\frac{R(t)}{R(t_0)}\right)^3
\left[\frac{1}{4}\left(3\frac{\dot{R}(t)}{R(t)}-3\frac{\dot{R}(t_0)}{R(t_0)} \right.\right.\\
\nonumber && \left.\left.
-\frac{\dot{\sigma}(t)}{\sigma(t)}+\frac{\dot{\sigma}(t_0)}{\sigma(t_0)}\right)^2
+ (\sigma(t)-\sigma(t_0))^2\right] \\
&\simeq &\frac{ {( t-t_{0})}^2 } { 4 m^2 t_{0}^4} \, .\end{aligned}$$ To compute the particle creation in thermal squeezed state, the expectation values of the $\langle\hat{\pi}^2\rangle , \langle\hat{\varphi}^2\rangle, \langle\hat{\pi} \hat{\varphi} \rangle $ and $\langle \hat{\varphi} \hat{\pi}\rangle$ in the thermal squeezed state are required. And are respectively obtained by using eqs (2.9),(2.12) and (4.6) as follow $$\begin{aligned}
\nonumber \fl \langle\hat{\pi}^2\rangle_{tss} = - R^{6} \dot{\varphi}^{2}(t_{0})\left( \alpha^{2}-e ^{i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2} \right) \\
\nonumber -R^{6}\dot{\varphi}^{\ast 2}(t_{0}) \left( \alpha^{\ast 2}-e ^{-i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2}\right) \\
\nonumber + R^{6} \dot{\varphi} ^{\ast}(t_{0})\dot{\varphi} (t_{0})\left( 2| \alpha|^{2}+ 2 \sinh^{2} r \coth\frac{\beta \omega}{2} + 2 f(\beta) + 1 \right)
, \\
\nonumber \fl \langle\hat{\varphi}^2\rangle_{tss}= - \varphi^{2}(t_{0})\left( \alpha^{2}-e ^{i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2} \right) \\
\nonumber -\varphi^{\ast 2} (t_{0})\left( \alpha^{\ast 2}-e ^{-i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2}\right)\\
\nonumber + \varphi ^{\ast}(t_{0})\varphi (t_{0})\left( 2| \alpha|^{2}+ 2 \sinh^{2} r \coth\frac{\beta \omega}{2} + 2 f(\beta) + 1 \right), \\
\eqalign{
\fl \langle\hat{\pi}\hat{\varphi}\rangle_{tss}= -R^3
\varphi(t_0)\dot{\varphi}(t_0)
\left( \alpha^{2}-e ^{i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2} \right)\\
-R^3\varphi^*(t_0)
\dot{\varphi}^*(t_0) \left( \alpha^{\ast 2}-e ^{-i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2}\right) \\
+R^3 (
\varphi(t_0)\dot{\varphi}^*(t_0)
+ \varphi^*(t_0)\dot{\varphi}(t_0) )
\left( 2| \alpha|^{2}+ 2 \sinh^{2} r \coth\frac{\beta \omega}{2} + 2 f(\beta) + 1 \right)
,\\
\fl \langle\hat{\varphi}\hat{\pi}\rangle_{tss} = -R^3
\dot{\varphi}(t_0)\varphi(t_0)
\left( \alpha^{2}-e ^{i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2} \right)\\
-R^3
\dot{\varphi}^*(t_0)\varphi^*(t_0) \left( \alpha^{\ast 2}-e ^{-i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2}\right) \\
+R^3 (\dot{\varphi}^*(t_0)
\varphi(t_0)
+ \dot{\varphi}(t_0) \varphi^*(t_0) )
\left( 2| \alpha|^{2}+ 2 \sinh^{2} r \coth\frac{\beta \omega}{2} + 2 f(\beta) + 1 \right). }\end{aligned}$$ Substituting (4.26) in (4.22), we get $$\begin{aligned}
\eqalign{
N_{tss}(t,t_{0})=& \frac{1}{16}\frac{1}{\sigma(t)}\frac{1}{\sigma (t_0)}\left(\frac{R(t)}{R(t_0)}\right)^3 \\
& \times \left [ \left [\left( 3 \frac{\dot{R}(t)}{R(t)}
- 3 \frac{\dot{R}(t_0)}{R(t_0)}
+ \frac{\dot{\sigma}(t)}{\sigma(t)}
- \frac{\dot{\sigma}(t_0)}{\sigma(t_0)}\right)^2 + \sigma (t)^{2}-\sigma(t_0)^{2}\right] \right. \\
& \left.
\times \left( 2| \alpha|^{2}+ 2 \sinh^{2} r \coth\frac{\beta \omega}{2} + 2 f(\beta) + 1 \right) \right.\\
& \left. -\left( 3 \frac{\dot{R}(t)}{R(t)}
- 3 \frac{\dot{R}(t_0)}{R(t_0)} + \frac{\dot{\sigma}(t)}{\sigma(t)}
- \frac{\dot{\sigma}(t_0)}{\sigma(t_0)}\right)^2 \right.\\
&\left. \times \left[
\left( \alpha^{2}-e ^{i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2} \right) e^{-2 i \sigma(t_0) t_0}\right. \right.\\
&\left.\left. + \left( \alpha^{\ast2}-e ^{-i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2} \right) e^{2 i \sigma(t_0) t_0}\right]
\right ].}\end{aligned}$$ Which is the number of particles produced in thermal squeezed state, at a later time $t$ from the initial time $t_0$.
By using (4.17) and (4.18) the above equation (4.27) can be rewritten as follows $$\begin{aligned}
\eqalign{
N_{tss} \simeq &\frac{1}{4}\frac{1}{m^{2}}\frac{(t-t_0)^2}{t_0 ^4} \left[
\left( 2| \alpha|^{2}+ 2 \sinh^{2} r \coth\frac{\beta \omega}{2} + 2 f(\beta) + 1 \right) \right.\\
&\left.-
\left( \alpha^{2}-e ^{i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2} \right) e^{-2 i m t_0} \right.\\
&\left.- \left( \alpha^{\ast2}-e ^{-i\vartheta}\cosh r \sinh r \coth \frac{\beta \omega}{2} \right) e^{2 i m t_0}\right].}\end{aligned}$$ By taking $ \alpha = e^{i\delta} \alpha $, eq (4.28) becomes $$\begin{aligned}
\eqalign{
N_{tss} \simeq& N_0(t,t_0)
\left[
2|\alpha|^{2}+ 2 \sinh^{2} r \coth\frac{\beta \omega}{2} + 2 f(\beta) +1 \right.\\
&\left. -2 \alpha^{2} \cos (2\delta-2 m t_0)+ \cos (\vartheta -2 m t_0) \sinh (2r) \coth \frac{\beta \omega}{2} \right],}\end{aligned}$$ where $N_0(t,t_0)$ is given by (4.25).
When $\alpha = 0$ eq (4.29) leads to $$\begin{aligned}
\eqalign{
N_{tcs} \simeq& N_0(t,t_0)
\left[
2|\alpha|^{2} + 2 f(\beta) +1
-2 \alpha^{2} \cos (2\delta-2 m t_0) \right]}.
\end{aligned}$$ Which is particle creation in thermal coherent state. The same result can be also obtained by using eqs (2.5), (2.12), (4.6), (4.17), (4.18) and (4.22).
When $r=0$, eq (4.29) becomes $$\begin{aligned}
\eqalign{
N_{tsv} \simeq& N_0(t,t_0)
\left[
2 \sinh^{2} r \coth\frac{\beta \omega}{2} + 2 f(\beta) +1 \right.\\
&\left.+ \cos (\vartheta -2 m t_0) \sinh (2r) \coth \frac{\beta \omega}{2} \right].}\end{aligned}$$ The eq (4.31) can be also obtained by using eqs (2.11), (2.12), (4.6), (4.17), (4.18) and (4.22), and is the particle production due thermal squeezed vacuum state.
When $2\delta=2 m t_0$ and $\vartheta =2 m t_0$, eqs (4.29), (4.30) and (4.31) respectively become $$\begin{aligned}
\eqalign{ N_{tss} \simeq &N_0(t,t_0)
\left[
2| \alpha|^{2}+ 2 \sinh^{2} r \coth\frac{\beta \omega}{2} + 2 f(\beta) +1 \right.\\
&\left. -2
\alpha^{2} + \sinh (2r) \coth \frac{\beta \omega}{2}
\right],}\end{aligned}$$ $$\begin{aligned}
N_{tcs} \simeq &N_0(t,t_0)
\left[
2| \alpha|^{2}+ 2 f(\beta) +1
-2 \alpha^{2}
\right],\end{aligned}$$ and $$\begin{aligned}
N_{tsv} \simeq N_0(t,t_0)
\left[
2 \sinh^{2} r + 2 f(\beta) +1+
\sinh (2r) \coth\frac{\beta \omega}{2}
\right].\end{aligned}$$ When $r= \alpha =0$, then eq (4.29) take the following form $$\begin{aligned}
N_{th} \simeq &N_0(t,t_0)
\left[
2 f(\beta) +1
\right].\end{aligned}$$ Which is the particle creation due to purely thermal effects.
Conclusions
============
In this paper, we studied particle production of the coherently oscillating inflaton, after the inflation, in thermal coherent states and thermal squeezed states formalisms, in the frame work of semiclassical theory of gravity. The number of particles at a later time $t$, produced from the thermal coherent state, at the initial time $t_0$, in the limit $mt_0$, $mt > 1$ calculated. It shows, the particle production depends on the coherent state parameter and finite temperature effects. The particle creation in thermal squeezed vacuum state in the limit $mt_0 >mt >1$ is also computed, it is found that the particle production depending on the associated squeezing parameter and temperature. Similarly the number of particles produced in thermal squeezed state also computed. It is observed that, when $r=0$, the result agree with the number of particles produced in the thermal coherent state and when $\alpha = 0$, the result equal to the number of particles created in thermal squeezed vacuum state.
The approximate leading solution obtained for the Einstein equation, in the thermal squeezed sates shows oscillatory behavior except when the condition, $ 2\delta = \vartheta = 2 mt$, satisfies. Though both classical and quantum inflaton in the oscillatory phase of the inflaton lead the same power law expansion, the correction to the expansion does not show any oscillatory behavior in semiclassical gravity in contrast to the oscillatory behavior seen in classical gravity only when $ 2\delta = \vartheta = 2 mt$. It is also noted that, the coherently oscillating inflaton, in thermal squeezed vacuum, thermal squeezed and thermal coherent states representation, suffer particle creation and created particle exhibit oscillations. The oscillation of the created particles is necessary to preheat the universe to hot again after the inflation. The present study can account for the post inflation particle creation due to thermal and quantum effects of inflaton in a flat FRW universe. Since the created particle oscillate, we hope that this kind of study can light on preheating issues of post inflationary scenario.
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|
---
abstract: 'A graph is closed when its vertices have a labeling by $[n]$ with a certain property first discovered in the study of binomial edge ideals. In this article, we prove that a connected graph has a closed labeling if and only if it is chordal, claw-free, and has a property we call *narrow*, which holds when every vertex is distance at most one from all longest shortest paths of the graph.'
address:
- 'Department of Mathematics and Statistics, Amherst College, Amherst, MA 01002-5000, USA'
- 'Department of Mathematics and Statistics, Amherst College, Amherst, MA 01002-5000, USA'
author:
- 'David A. Cox'
- Andrew Erskine
title: On Closed Graphs I
---
Introduction {#intro}
============
In this paper, $G$ will be a simple graph with vertex set $V(G)$ and edge set $E(G)$.
\[closeddef\] A *labeling* of $G$ is a bijection $V(G) \simeq [n] =
\{1,\dots,n\}$, and given a labeling, we typically assume $V(G) =
[n]$. A labeling is *closed* if whenever we have distinct edges $\{j,i\}, \{i,k\} \in E(G)$ with either $j > i < k$ or $j < i > k$, then $\{j,k\} \in E(G)$. Finally, a graph is *closed* if it has a closed labeling.
A labeling of $G$ gives a direction to each edge $\{i,j\} \in E(G)$ where the arrow points from $i$ to $j$ when $i < j$, i.e., the arrow points to the bigger label. The following picture illustrates what it means for a labeling to be closed: $$\label{closedpicture}
\begin{array}{ccc}
\begin{tikzpicture}
\node[vertex] (n1) at (2,1) {$i\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n2) at (1,3) {$\rule[-2.5pt]{0pt}{10pt}j$};
\node[vertex] (n3) at (3,3) {$k\rule[-2.5pt]{0pt}{10pt}$};
\foreach \from/\to in {n1/n2,n1/n3}
\draw[->] (\from)--(\to);;
\foreach \from/\to in {n2/n3}
\draw[dotted] (\from)--(\to);;
\end{tikzpicture}&\hspace{30pt}&
\begin{tikzpicture}
\node[vertex] (n1) at (2,1) {$i\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n2) at (1,3) {$j\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n3) at (3,3) {$k\rule[-2.5pt]{0pt}{10pt}$};
\foreach \from/\to in {n2/n1,n3/n1}
\draw[->] (\from)--(\to);;
\foreach \from/\to in {n2/n3}
\draw[dotted] (\from)--(\to);;
\end{tikzpicture}
\end{array}$$ Whenever the arrows point away from $i$ (as on the left) or towards $i$ (as on the right), closed means that $j$ and $k$ are connected by an edge.
Closed graphs were first encountered in the study of binomial edge ideals. The *binomial edge ideal* of a labeled graph $G$ is the ideal $J_G$ in the polynomial ring $\mathsf{k}[x_1,\dots,x_n,y_1,\dots,y_n]$ ($\mathsf{k}$ a field) generated by the binomials $$f_{ij} = x_iy_j - x_jy_i$$ for all $i,j$ such that $\{i,j\} \in E(G)$ and $i < j$. A key result, discovered independently in [@H] and [@O], is that the above binomials form a Gröbner basis of $J_G$ for lex order with $x_1 >
\cdots > x_n > y_1 > \cdots > y_n$ if and only if the labeling is closed. The name “closed” was introduced in [@H].
Binomial edge ideals are explored in [@E] and [@S], and a generalization is studied in [@R]. The paper [@CR] characterizes closed graphs using the clique complex of $G$, and closed graphs also appear in [@E3; @E4; @E2].
The goal of this paper is to characterize when a graph $G$ has a closed labeling in terms of properties that can be seen directly from the graph. Our starting point is the following result proved in [@H].
\[Hprop\] Every closed graph is chordal and claw-free.
“Claw-free” means that $G$ has no induced subgraph of the form $$\label{ex1}
\begin{array}{c}
\begin{tikzpicture}
\node[vertex] (k) at (3,6) {$\bullet$};
\node[vertex] (j) at (2.1,3.9){$\bullet$};
\node[vertex] (l) at (3.9,3.9) {$\bullet$};
\node[vertex] (i) at (3,5){$\bullet$};
\foreach \from/\to in {i/l,i/k,i/j}
\draw (\from) -- (\to);
\end{tikzpicture}
\end{array}$$
Besides being chordal and claw-free, closed graphs also have a property called *narrow*. The *distance* $d(v,w)$ between vertices $v,w$ of a connected graph $G$ is the length of the shortest path connecting them, and the *diameter* of $G$ is $\mathrm{diam}(G) = \max\{d(v,w) \mid v,w \in E(G)\}$. Given vertices $v,w$ of $G$ satisfying $d(v,w) = \mathrm{diam}(G)$, a shortest path connecting $v$ and $w$ is called a *longest shortest path* of $G$.
\[narrowdef\] A connected graph $G$ is *narrow* if for every $v \in V(G)$ and every longest shortest path $P$ of $G$, either $v \in V(P)$ or $\{v,w\} \in E(G)$ for some $w \in V(P)$.
Thus a connected graph is narrow if every vertex is distance at most one from every longest shortest path. Here is a graph that is chordal and claw-free but not narrow: $$\label{ex2}
\begin{array}{c}
\begin{tikzpicture}
\node[vertex] (n1) at (3,1) {$A\rule[-2pt]{0pt}{10pt}$};
\node[vertex] (n2) at (2,3) {$B\rule[-2pt]{0pt}{10pt}$};
\node[vertex] (n3) at (4,3) {$C\rule[-2pt]{0pt}{10pt}$};
\node[vertex] (n4) at (3,5){$E\rule[-2pt]{0pt}{10pt}$};
\node[vertex] (n5) at (5,5){$F\rule[-2pt]{0pt}{10pt}$};
\node[vertex] (n6) at (1,5){$D\rule[-2pt]{0pt}{10pt}$};
\foreach \from/\to in {n1/n2,n1/n3,n2/n3,n2/n4, n3/n4, n3/n5,
n4/n5,n2/n6,n4/n6}
\draw (\from)--(\to);;
\end{tikzpicture}
\end{array}$$ Narrowness fails because $D$ is distance two from the longest shortest path $ACF$.
We can now state the main result of this paper.
\[mainthm\] A connected graph is closed if and only if it is chordal, claw-free, and narrow.
This theorem is cited in [@E3; @E4; @E2]. Since a graph is closed if and only if its connected components are closed [@CR], we get the following corollary of Theorem \[mainthm\].
\[cormainthm\] A graph is closed if and only if it is chordal, claw-free, and its connected components are narrow.
The independence of the three conditions (chordal, claw-free, narrow) is easy to see. The graph is chordal and narrow but not claw-free, and the graph is chordal and claw-free but not narrow. Finally, the $4$-cycle $$\begin{tikzpicture}
\node[vertex] (A) at (2,1) {$\bullet$};
\node[vertex] (B) at (4,1) {$\bullet$};
\node[vertex] (C) at (4,3) {$\bullet$};
\node[vertex] (D) at (2,3) {$\bullet$};
\foreach \from/\to in {A/B,B/C,C/D,D/A}
\draw (\from)--(\to);;
\end{tikzpicture}$$ is claw-free and narrow but not chordal.
The paper is organized as follows. In Section \[properties\] we recall some known properties of closed graphs and prove some new ones, and in Section \[algorithm\] we introduce an algorithm for labeling connected graphs. Section \[characterize\] uses the algorithm to prove Theorem \[mainthm\].
In a subsequent paper [@closed2] we will explore further properties of closed graphs.
Properties of Closed Labelings {#properties}
==============================
Directed Paths
--------------
A path in a graph $G$ is $P = v_0v_1\cdots v_{\ell-1}v_\ell$ where $\{v_j,v_{j+1}\} \in E(G)$ for $j = 0,\dots,\ell-1$. A single vertex is regarded as a path of length zero. When $G$ is labeled, we assume as usual that $V(G) =
[n]$. Then a path $P = i_0i_1\cdots i_{\ell-1}i_\ell$ is *directed* if either $i_j < i_{j+1}$ for all $j$ or $i_j >
i_{j+1}$ for all $j$. Here is a result from [@H].
\[directed\] A labeling on a graph $G$ is closed if and only if for all vertices $i,j \in V(G) = [n]$, all shortest paths from $i$ to $j$ are directed.
Neighborhoods and Intervals
---------------------------
Given a vertex $v \in V(G)$, the *neighborhood* of $v$ in $G$ is $$N_G(v) = \{ w \in V(G) \mid \{v,w\} \in E(G)\}.$$ When $G$ is labeled and $i \in V(G) = [n]$, we have a disjoint union $$N_G(i) = N_G^>(i) \cup N_G^<(i),$$ where $$N_G^>(i) = \{j \in N_G(i) \mid j > i\},\
N_G^<(i) = \{j \in N_G(i) \mid j < i\}.$$ This is the notation used in [@CR], where it is shown that a labeling is closed if and only if $N_G^>(i)$ and $N_G^<(i)$ are complete for all $i \in V(G) = [n]$.
Vertices $i,j \in [n]$ with $i \le j$ give the *interval* $[i,j]
= \{k \in [n] \mid i \le k \le j\}$. Here is a characterization of when a labeling of a connected graph is closed.
\[nbdinterval\] A labeling on a connected graph $G$ is closed if and only if for all $i \in V(G) = [n]$, $N_G^>(i)$ is complete and equal to $[i+1,i+r]$, $r = |N_G^>(i)|$.
Assume that the labeling is closed. Then Definition \[closeddef\] easily implies that $N_G^>(i)$ is complete. It remains to show that $N_G^>(i)$ is an interval of the desired form.
Pick $j \in N_G^>(i)$ and $k\in[n]$ with $i<k<j$. A shortest path $P=i_0i_1 i_2\cdots i_m$ from $i = i_0$ to $k = i_m$ is directed by Proposition \[directed\]. Since $i<k$, we have $i=i_0<i_1<i_2<\cdots<i_m=k$. Thus $i_1\in {N_G^>}(i)$ and hence $\{i_1,j \}\in E(G)$ since ${N_G^>}(i)$ is complete. Since $i_1<j$, we have $j\in {N_G^>}(i_1)$.
We now prove by induction that $j \in {N_G^>}(i_u)$ for all $u =
1,\dots,m$. The base case is proved in the previous paragraph. Now assume $j\in {N_G^>}(i_u)$. Then $\{j,i_{u+1}\}\in E(G)$ since $\{i_u,i_{u+1}\}\in E(G)$ and the labeling is closed. This completes the induction. Since $k = i_m$, it follows that $j\in {N_G^>}(k)$. Then we have $\{i,j\},\{k,j\}\in E(G)$ with $i<j>k$. Thus $\{i,k\}\in
E(G)$ since the labeling is closed, so $k\in {N_G^>}(i)$ since $i<k$. Hence $N_G^>(i)$ is an interval of the desired form.
Conversely, suppose that $N_G^>(i)$ is complete and $N_G^>(i) =
[i+1,\dots,i+r]$, $r = |N_G^>(i)|$, for all $i \in V(G)$. Take $\{j,i\}, \{i,k\} \in E(G)$ with $j > i < k$ or $j < i > k$. The former implies $\{j,k\} \in E(G)$ since ${N_G^>}(i)$ is complete. For the latter, assume $j<k$. Then $j < k < i$ with $i \in {N_G^>}(j)$. Since ${N_G^>}(j)$ is an interval containing $j+1$ and $i$, ${N_G^>}(j)$ also contains $k$. Hence $\{j,k\} \in E(G)$.
Layers
------
The following subsets of $V(G)$ will play a key role in what follows.
\[layerdef\] Let $G$ be a connected graph labeled so that $V(G)
= [n]$. Then the *$N^{\mathit{th}}$ layer of $G$* is the set $$L_N = \{i \in [n] \mid d(i,1) = N\}.$$
Thus $L_N$ consists of all vertices that are distance $N$ from the vertex $1$. Note that $L_0 = \{1\}$ and $L_1 = {N_G}(1) = {N_G^>}(1)$. Furthermore, since $G$ is connected, we have a disjoint union $$V(G) = L_0 \cup L_1 \cup \cdots \cup L_h,$$ where $h = \max\{d(i,1) \mid i \in [n]\}$. We omit the easy proof of the following lemma.
\[layerlem\] Let $G$ be a connected graph labeled so that $V(G) = [n]$. Then:
1. If $i \in L_N$ and $\{i,j\} \in E(G)$, then $j \in L_{N-1}$, $L_N$, or $L_{N+1}$.
2. If $P$ is a path in $G$ connecting $i \in L_N$ to $j \in L_M$ with $N \le M$, then for every integer $N \le m \le M$, there exists $k \in V(P)$ with $k \in L_m$.
\[layerprop\] Let $G$ be a connected graph with a closed labeling satisfying $V(G) =
[n]$. Then:
1. Each layer $L_N$ is complete.
2. If $d = \max\{L_N\}$, then $L_{N+1} = {N_G^>}(d)$.
We first show that $$\label{nextlayer}
r \in L_M,\ s \in L_{M+1},\ \{r,s\} \in E(G) \Longrightarrow r < s.$$ To see why, take a shortest path from 1 to $r \in L_M$. This path has length $M$, so appending the edge $\{r,s\}$ gives a path of length $M+1$ to $s$. Since $s \in L_{M+1}$, this is a shortest path and hence is directed by Proposition \[directed\]. Thus $r<s$.
For (1), we use induction on $N \ge 0$. The base case is trivial since $L_0=\{1\}$. Now assume $L_N$ is complete and take $i,j\in L_{N+1}$ with $i\neq j$. A shortest path $P_1$ from $1$ to $i\in L_{N+1}$ has a vertex $k \in
L_N$ adjacent to $i$, and a shortest path $P_2$ from $1$ to $j\in
L_{N+1}$ has a vertex $l\in L_{N}$ adjacent to $j$. Then $k<i$ and $l<j$ by .
If $k=l$, then $i>k<j$, which implies $\{i,j\}\in E(G)$ since the labeling is closed. If $k\neq l$, then $\{l,k\}\in E(G)$ since $L_N$ is complete. Assume $l>k$. Then $l>k<i$ and closed imply $\{l,i\}\in
E(G)$. Since $l \in L_N$ and $i \in L_{N+1}$, we have $l<i$ by . Then $i>l<j$ and closed imply $\{i,j\}\in E(G)$. Hence $L_{N+1}$ is complete.
We now turn to (2). To prove $L_{N+1} \subseteq {N_G^>}(d)$, $d =
\max\{L_N\}$, take $i \in L_{N+1}$. A shortest path from $1$ to $i$ will have a vertex $k\in L_N$ such that $\{k,i\}\in E(G)$. Then $k<i$ by , hence $i\in {N_G^>}(k)$. Also, $k\leq d$ since $d=\max(L_N)$. If $k=d$, then $i\in {N_G^>}(d)$. If $k<d$, then $\{k,d\}\in E(G)$ since $L_N$ is complete. Then $i>k<d$ and closed imply $\{d,i\}\in E(G)$, and then $d<i$ by . Thus $i\in {N_G^>}(d)$.
To prove the opposite inclusion, take $i\in {N_G^>}(d)$. Since $\{d,i\}\in E(G)$ and $d \in L_N$, we have $i \in L_M$ for $M = N-1,N,
N+1$ by Lemma \[layerlem\]. If $i\in L_{N-1}$, then would imply $i < d$, contradicting $i \in
{N_G^>}(d)$. If $i\in L_N$, then $i \le \max\{L_N\} = d$, again contradicting $i \in {N_G^>}(d)$. Hence $i\in L_{N+1}$.
Longest Shortest Paths
----------------------
When the labeling of a connected graph is closed, the diameter of the graph determines the number of layers as follows.
\[diameters\] Let $G$ be a connected graph with a closed labeling. Then:
1. ${\mathrm{diam}}(G)$ is the largest integer $h$ such that $L_h\neq
\emptyset$.
2. If $P$ is a longest shortest path of $G$, then one endpoint of $P$ is in $L_0$ or $L_1$ and the other is in $L_h$, where $h={\mathrm{diam}}(G)$.
For (1), let $h$ be the largest integer with $L_h\neq \emptyset$. Since points in $L_h$ have distance $h$ from $1$, we have $h \le
{\mathrm{diam}}(G)$.
For the opposite inequality, it suffices to show that $d(i,j) \le h$ for all $i,j \in V(G)$ with $i \ne j$. We can assume $G$ has more than one vertex, so that $h\geq 1$. Suppose $i\in L_N$ and $j\in L_M$ with $N\leq M$. If $N=0$, then $i=1$ and $d(i,j) = d(1,j) = M \le h$ since $j \in L_M$. Also, if $M=N$, then $i,j\in L_N$, so that $d(i,j) =
1 \le h$ since $L_N$ is complete by Proposition \[layerprop\]. Finally, if $0<N<M$, let $d_u=\max(L_u)$ for each integer $u$. By Proposition \[layerprop\], we know that $j \in {N_G^>}(d_{M-1})$. Hence, if $i\neq d_N$, then $P=id_Nd_{N+1}\cdots d_{M-2}d_{M-1}j$ is a path of length $M-N+1$. If $i=d_N$, then $P=id_{N+1}\cdots d_{M-1}j$ is a path of length $M-N$. Thus we have a path from $i$ to $j$ of length at most $M-N+1$, so that $d(i,j) \le M-N+1 \le M \le h$.
For (2), let $i$ and $j$ be the endpoints of the longest shortest path $P$ with $i\in L_N$, $j\in L_M$ and $N\leq M$. If $0 < N < M$, then the previous paragraph implies $${\mathrm{diam}}(G) = d(i,j) \le M-N+1 \le M \le h = {\mathrm{diam}}(G),$$ which forces $N = 1$ (so $i \in L_1$) and $M = h$ (so $j \in L_h$). The remaining cases $N = 0$ and $N = M$ are straightforward and are left to the reader.
Recall from Definition \[narrowdef\] that a connected graph $G$ is narrow when every vertex is distance at most one from every longest shortest path. Narrowness is a key property of connected closed graphs.
\[narrowthm\] Every connected closed graph is narrow.
Let $G$ be a connected graph with a closed labeling. Pick a vertex $i\in V(G)$ and a longest shortest path $P$. Since $G$ is connected, $i\in L_N$ for some integer $N$. By Proposition \[diameters\], the endpoints of $P$ lie in $L_0$ or $L_1$ and $L_h$, $h = {\mathrm{diam}}(G)$. Then Lemma \[layerlem\] implies that $P$ has a vertex $i_M$ in $L_M$ for every $1 \le M \le h$.
If $N \ge 1$, then either $i = i_N \in V(P)$ or $i \ne i_N$, in which case $\{i,i_N\} \in E(G)$ since $L_N$ is complete by Proposition \[layerprop\]. On the other hand, if $N = 0$, then $i\in L_0$, hence $i=1$. Then $\{i,i_1\} = \{1,i_1\} \in E(G)$ since $i_1 \in L_1 = {N_G}(1)$. In either case, $i$ is distance at most one from $P$.
A Labeling Algorithm {#algorithm}
====================
We introduce Algorithm \[alg:Labeling\], which labels the vertices of a connected graph. This algorithm will play a key role in the proof of Theorem \[mainthm\].
\[alg:input\] \[alg:output\] $i:=1$ $j:=0$ $v_0:=$ endpoint of a longest shortest path with minimal degree\[alg:possible1\] label $v_0$ as $i$ \[alg:label1\] $l(i):=j$\[alg:function1\] $i:=i+1$ $J:=\{v_0\}$ $j:=j+1$\[alg:labelj1\]
The algorithm works as follows. Among the endpoints of all longest shortest paths, we select one of minimal degree and label it as $1$. We then go through the vertices in ${N_G}(1)$ and label them $2,3,\ldots$, first labeling vertices with the fewest number of edges connected to unlabeled vertices. This process is repeated for the unlabeled vertices connected to vertex $2$, and vertex $3$, and so on until every vertex is labeled. Furthermore, every vertex will be labeled because we first label everything in ${N_G^>}(1)$, then label everything in ${N_G^>}(2)$ not already labeled, and so on. Since the input graph is connected, this process must eventually reach all of the vertices. Hence we get a labeling of $G$.
The following lemma explains the function $l$ that appears in Algorithm \[alg:Labeling\].
\[claim:meaningOfFunction\] Let $G$ be a connected graph with the labeling from Algorithm \[alg:Labeling\]. Then:
1. $l(1)=0$, and for every $i\in [n]$ with $i>1$, $l(i)=\min({N_G}(i))$.
2. If $l(t)<l(s)$, then $t<s$.
Algorithm \[alg:Labeling\] defines $l(1)=0$. Now assume $i>1$ and let $v$ be the vertex assigned the label $i$. By lines \[alg:label2\] and \[alg:function2\] of the algorithm, we need to show that when the label $i$ is assigned to $v$, the variable $j$ equals $\min({N_G}(i))$. This follows because for any smaller value $j' < j$, line \[alg:secondLoop\] implies that everything in the neighborhood of $j'$ is labeled before $j'$ is incremented. However, lines \[alg:Sset\]–\[alg:beginSecondLoop\] show that $v$ is adjacent to $j$ and unlabeled at the start of the loop on line \[alg:secondLoop\]. Hence $v$ cannot link to any smaller value of $j$, and since $v$ has label $i$, $j = \min({N_G}(i))$ follows.
\(2) Suppose that $s,t\in [n]$ satisfy $l(t)<l(s)$. Since $l(t)$ (resp. $l(s)$) is the value of $j$ when the label $t$ (resp. $s$) was assigned in Algorithm \[alg:Labeling\], $l(t)<l(s)$ implies that the label $s$ was assigned later than $t$ in the algorithm. Since the labels are assigned in numerical order, we must have $t < s$.
The labeling produced by Algorithm \[alg:Labeling\] allows us to define the layers $L_N$. These interact with the function $l$ as follows:
\[claim:orderingClaim\] Let $G$ be a connected graph with the labeling from Algorithm \[alg:Labeling\]. Then:
1. If $t\in L_N$, then $l(t)\in L_{N-1}$ if $N>0$.
2. If $t\in L_N$ and $s\in L_M$ with $N<M$, then $t<s$.
We prove (1) and (2) simultaneously by induction on $N \ge 1$ (the case $N = 0$ of (2) is trivially true). The first time Algorithm \[alg:Labeling\] gets to Line \[alg:Sset\], we have $S =
{N_G}(1)\setminus J = {N_G}(1) = L_1$. Every vertex in $S = L_1$, is labeled during the loop starting on Line \[alg:firstLoop\], so $l(t)
= 1$ for all $t \in L_1$. Hence (1) holds when $N =1$. Also, if $s\in L_M$ with $1<M$, then the vertex $s$ is not labeled at this stage. Since labels are assigned in numerical order, we must have $t<s$ for all $t \in L_1$. Hence (2) holds when $N = 1$.
Now assume that (1) and (2) hold for $M$ and every $N\leq N_0$. Given $t\in L_{N_0+1}$, a shortest path from 1 to $t$ gives $v\in L_{N_0}$ with $v \in {N_G}(t)$. Since $l(t)=\min({N_G}(t))$ by Lemma \[claim:meaningOfFunction\](1), we have $l(t)\leq v$. We have $l(t) \in L_u$ for some $u$. If $u > N_0$, then the inductive hypothesis for (2) would imply $l(t) > v$, which contradicts $l(t)\leq v$. Hence $l(t) \in L_u$ for some $u
\le N_0$. But $t \in L_{N_0+1}$ and $\{t,l(t)\} \in E(G)$ imply $l(t)
\in L_u$ for $u \ge N_0$ by Lemma \[layerlem\](1). Hence $l(t) \in
L_{N_0}$, proving (1) for $N_0+1$.
Turning to (2), pick $t\in L_{N_0+1}$ and $s\in L_M$ with $N_0+1<M$. We just showed that $l(t)\in L_{N_0}$, and Lemma \[layerlem\](1) implies that $l(s)\in L_{u}$, $u \ge M-1$, since $s \in L_M$. Then $N_0 < M-1 \le u$, so our inductive hypothesis, applied to $l(t)\in
L_{N_0}$ and $l(s) \in L_u$, implies $l(t)<l(s)$. Then $t<s$ by Lemma \[claim:meaningOfFunction\](2), proving (2) for $N_0+1$.
Proof of the Main Theorem {#characterize}
=========================
We now turn to the main result of the paper. Theorem \[mainthm\] from the Introduction states that a connected graph is closed if and only if it is chordal, claw-free and narrow. One direction is now proved, since closed graphs are chordal and claw-free by Proposition \[Hprop\], and connected closed graphs are narrow by Theorem \[narrowthm\].
The proof of converse is harder. The key idea that the labeling constructed by Algorithm \[alg:Labeling\] is closed when the input graph is chordal, claw-free and narrow. Thus the proof of Theorem \[mainthm\] will be complete once we prove the following result.
\[converse\] Let $G$ be a connected, chordal, claw-free, narrow graph. Then the labeling produced by Algorithm \[alg:Labeling\] is closed.
By Proposition \[nbdinterval\], it suffices to show that the labeling produced by Algorithm \[alg:Labeling\] has the property that for all $m \in V(G) = [n]$, $$\label{toprove}
{N_G^>}(m) \text{ is complete and }
{N_G^>}(m) = [i+m,i+r_m] \text{ for } r_m = |{N_G^>}(m)|.$$ We will prove this by induction on $m$. In below, we show that holds for $m = 1$, and in below, we show that if holds for all $1\leq u<m$, then it also holds for $m$. Thus, we will be done after proving and .
The Base Case
-------------
After Algorithm \[alg:Labeling\] runs on a chordal, claw-free and narrow graph $G$, the base case of the induction in the proof of Theorem \[converse\] is the following assertion: $$\label{basecase}
{N_G^>}(1)=[2,1+r],\ r = |{N_G^>}(1)|,\ \text{and } {N_G^>}(1) \text{ is
complete} .$$ We will first show that ${N_G^>}(1)=[2,1+r]$, $r = |{N_G^>}(1)|$. The first time through the the loop beginning on Line \[alg:firstLoop\] in Algorithm \[alg:Labeling\], $j=1$ and $i=2$ and $S={N_G}(1)$. For each vertex in $S$, the loop beginning on Line \[alg:secondLoop\] labels that vertex $i$, removes it from $S$, and increments $i$. This continues until $S=\emptyset$, at which point every vertex in $S$ has been labeled $2,3,\ldots,1+r$, where $r$ is the initial size of $S$. Hence ${N_G^>}(1)={N_G}(1) = [2,1+r]$.
To prove that ${N_G^>}(1)$ is complete, there are several cases to consider. Pick distinct vertices $s,t\in {N_G^>}(1)$ and assume that $\{s,t\}\notin E(G)$. Note that $s,t\in L_1$ are distance $2$ apart and therefore $h={\mathrm{diam}}(G)\geq2$. Our choice of vertex $1$ guarantees that there is a longest shortest path $P$ with 1 as an endpoint. Let $z\in V(G)$ be the other, so that $P=v_0v_1\cdots v_h$, $1 = v_0$ and $v_h = z$. Since $v_1 \in V(P)$ is the only vertex of $P$ in $L_1$, $s$ and $t$ cannot both lie on $P$.
Therefore, either $s\in V(P)$, $t\in V(P)$, or $s,t\notin V(P)$. We will show that each possibility leads to a contradiction, proving that $\{s,t\}\in E(G)$.
plus 1 pt minus 1 pt
[**Case 1.**]{} Both $s,t\notin V(P)$. If $s$ has distance $h-1$ from $z$, then appending the edge $\{1,s\}$ to a shortest path from $s$ to $z$ gives a longest shortest path $P'$ from $1$ to $z$ that contains $s$. Replacing $P$ with $P'$, we get $s \in V(P)$, which is Case 2 to be considered below. Similarly, if $t$ has distance $h-1$ from $z$, then replacing $P$ allows us to assume $t \in
V(P)$, which is also covered by Case 2 below.
Thus we may assume that neither $s$ nor $t$ has distance $h-1$ from $z$. Since $d(s,z) < h-1$ would imply $d(1,z) < h$, we conclude that $s$ has distance $h$ from $z$, and the same holds for $t$. It follows that $\{s,v_2\},\{t,v_2\} \notin E(G)$, since otherwise there is a path shorter than length $h$ from $s$ or $t$ to $z$.
Since the subgraph induced on vertices $1,s,t,v_1$ cannot be a claw, either $\{v_1,s\}\in E(G)$ or $\{v_1,t\}\in E(G)$ or both. We consider each possibility separately.
plus 1 pt minus 1 pt
[**Case 1A.**]{} Both $\{v_1,s\}$, $\{v_1,t\}\in E(G)$, as shown in Figure [1]{}(a) on the next page. Then the subgraph induced on $v_2,v_1,s,t$ is a claw, contradicting our assumption of claw-free.
-10pt $$\begin{array}{ccc}
& \begin{tikzpicture}
\node[vertex] (n1) at (2,1) {$1\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (s) at (3.5,3) {$s\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (t) at (.5,3) {$t\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (v1) at (2,3) {$v_1\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (v2) at (2,5) {$v_2\rule[-2.5pt]{0pt}{10pt}$};
\foreach \from/\to in {n1/v1,n1/t,n1/s,v1/v2,s/v1,t/v1}
\draw (\from)--(\to);
\node [right, font = \large] at (4.2,3) {$L_1$};
\node [right, font=\large] at (4.2,5) {$L_2$};
\node [right, font = \large] at (4.2,1) {$L_0$};
\node [right] at (2.5,.3) {(a)};
\end{tikzpicture} &\hspace{25pt}\begin{tikzpicture}
\node[vertex] (n1) at (2,1) {$1\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (s) at (3.5,3.2) {$s\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (t) at (.5,3.2) {$t\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (v1) at (2,3.2) {$v_1\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (v2) at (2,5.4) {$v_2\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (t2) at (.5,5.4) {$t_2\rule[-2.5pt]{0pt}{10pt}$};
\foreach \from/\to in {n1/s,n1/t,n1/v1, v1/v2, s/v1,t/t2}
\draw (\from)--(\to);
\node [right, font = \large] at (4.3,3.2) {$L_1$};
\node [right, font=\large] at (4.3,5.4) {$L_2$};
\node [right, font = \large] at (4.3,1) {$L_0$};
\node [right] at (2.5,.3) {(b)};
\end{tikzpicture}
\end{array}$$ -8pt
plus 1 pt minus 1 pt
[**Case 1B.**]{} Exactly one of $\{s,v_1\}, \{t,v_1\}$ is in $E(G)$. Without loss of generality, we may assume $\{s,v_1\}\in E(G)$ and $\{t,v_1\}\notin E(G)$, as shown in Figure [1]{}(b). Recall that $\{t,v_2\}\notin E(G)$ and $t$ is distance $h$ to $z$.
Since $t$ and $1$ are both endpoints of longest shortest paths, Line \[alg:possible1\] of Algorithm \[alg:Labeling\] implies that $\deg(1) \le \deg(t)$. Since $v_1$ is adjacent to $1$ but not $t$, there must be at least one $t_2$ adjacent to $t$ but not $1$, i.e., $t_2\in {N_G}(t)$ with $t_2\notin {N_G}(1)$.
For this $t_2$, it follows that $t_2\in L_2$. We also have $t_2\neq
v_2$ since $\{t,v_2\}\notin E(G)$. Furthermore, $\{t_2,s\}\notin
E(G)$, since otherwise we would have the $4$-cycle $t_2s1tt_2$ with no chords as $\{t_2,1\}$, $\{t,s\}\notin E(G)$. Similarly, $\{t_2,v_1\}\notin E(G)$ or else we would have the $4$-cycle $t_2v_1\hskip-.9pt 1tt_2$ with no chords since $\{t_2,1\}$, $\{t,v_1\}\notin E(G)$. Note also that $\{t_2,v_2\}\notin E(G)$, since otherwise we would have the $5$-cycle $t_2v_2v_1\hskip-.9pt
1tt_2$ with no chords as $\{1,v_2\}$, $\{1,t_2\}$, $\{t,v_2\}$, $\{t,v_1\}$, $\{t_2,v_1\}\notin E(G)$, contradicting chordal. Hence $t_2$ gives Figure [1]{}(b) as an induced subgraph.
Since $G$ is narrow, either $t_2\in V(P)$ or $t_2$ is adjacent to a vertex of $P$. However, $t_2\in V(P)$ would imply $t_2=v_2$ since both lie in $L_2$, contradicting $t_2\neq v_2$. Thus $\{t_2,v_u\}\in
E(G)$ for some $u>1$. Since $t_2 \in L_2$ and $v_u \in L_u$, we have $u \le 3$ by Lemma \[layerlem\](1). We just proved $\{t_2,v_2\}\notin E(G)$, so we must have $\{t_2,v_3\} \in E(G)$. This gives the $6$-cycle $t_2v_3v_2v_1\hskip-.9pt 1tt_2$. Since Figure [1]{}(b) is an induced subgraph, the only possible chords are $\{1,v_3\}$, $\{t,v_3\}$, $\{v_1,v_3\}$, but by Lemma \[layerlem\](1) none of these are in $E(G)$ since $v_3 \in
L_3$ and $1,t,v_1 \in L_0\cup L_1$. Hence the $6$-cycle has no chords, contradicting chordal.
[**Case 2.**]{} $s \in V(P)$ or $t \in V(P)$. We may assume $s=v_1$. Arguing as in Case 1B, there is $t_2\in {N_G}(t)$ with $t_2\notin {N_G}(1)$ and $t_2\in L_2$. We also have $\{t,v_2\}\notin
E(G)$, since otherwise the $4$-cycle $1sv_2t1$ has no chords as $\{t,s\}$, $\{1,v_2\}\notin E(G)$.
Since $G$ is narrow, $t_2$ must either be in $P$ or be adjacent to a vertex in $P$. However, $t_2 \in V(P)$ would imply $t_2 = v_2$ since $t_2,v_2 \in L_2$, and the latter would give $\{t,v_2\} = \{t,t_2\}
\in E(G)$, which we just showed to be impossible. Hence $t_2\notin
V(P)$, so that $\{t_2,v_u\}\in E(G)$ for some $u$. Note that $u < 4$ by Lemma \[layerlem\](1). We claim that $u = 3$.
To see why, first note that $\{t_2,s=v_1\}\notin E(G)$, since otherwise we would have the $4$-cycle $1tt_2v_1\hskip-.9pt 1$ with no chords as $\{t,s\}$, $\{t_2,1\}\notin E(G)$. We also know that $\{t_2,v_2\}\notin E(G)$, as otherwise we would have the $5$-cycle $t_2v_2s1tt_2$ with no chords since $\{t_2,s\}$, $\{t_2,1\}$, $\{s,t\}$, $\{t,v_2\}$, $\{v_2,1\}\notin E(G)$. See Figure [2]{}(a).
Thus we must have $\{t_2,v_3\}\in E(G)$. However, this gives a $6$-cycle $t_2v_3v_2s1tt_2$ with the same impossible chords as before along with $\{t,v_3\}$, $\{1,v_3\}$, $\{s,v_3\}$, $\{v_2,t_2\}\notin
E(G)$, as in Figure [2]{}(b). This contradicts chordal, and follows.
$$\begin{array}{ccc}
\begin{tikzpicture}
\node[vertex] (n1) at (2,1) {$1\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n2) at (1,3) {$t\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n3) at (3,3) {$s\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n4) at (1,4.5){$t_2\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n5) at (3,4.5){$v_2\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n6) at (3,6){$v_3\rule[-2.5pt]{0pt}{10pt}$};
\foreach \from/\to in {n1/n2,n1/n3,n2/n4, n3/n5, n6/n5}
\draw (\from)--(\to);;
\foreach \from/\to in {n4/n5}
\draw[dotted] (\from)--(\to);
\node [right, font = \large] at (3.7,3) {$L_1$};
\node [right, font=\large] at (3.7,4.5) {$L_2$};
\node [right, font = \large] at (3.7,1) {$L_0$};
\node [right, font = \large] at (3.7,6) {$L_3$};
\node [right] at (2.5,.3) {(a)};
\end{tikzpicture}&\hspace{25pt}&
\begin{tikzpicture}
\node[vertex] (n1) at (2,1) {$1\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n2) at (1,3) {$t\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n3) at (3,3) {$s\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n4) at (1,4.5){$t_2\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n5) at (3,4.5){$v_2\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n6) at (3,6){$v_3\rule[-2.5pt]{0pt}{10pt}$};
\foreach \from/\to in {n1/n2,n1/n3,n2/n4, n3/n5, n6/n5}
\draw (\from)--(\to);;
\foreach \from/\to in {n4/n6}
\draw[dotted] (\from)--(\to);
\node [right, font = \large] at (3.7,3) {$L_1$};
\node [right, font=\large] at (3.7,4.5) {$L_2$};
\node [right, font = \large] at (3.7,1) {$L_0$};
\node [right, font = \large] at (3.7,6) {$L_3$};
\node [right] at (2.5,.3) {(b)};
\end{tikzpicture}
\end{array}$$ -8pt
The Inductive Step
------------------
After Algorithm \[alg:Labeling\] runs on a chordal, claw-free and narrow graph $G$, we now prove that the resulting labeling satsifies the inductive step in the proof of Theorem \[converse\]: $$\label{indstep}
\begin{aligned}
&\text{If ${N_G^>}(u)=[u+1,u+r_u]$, $r_u = |{N_G^>}(u)|$, ${N_G^>}(u)$ is
complete, $1\leq u<m$,} \\ &\text{then ${N_G^>}(m) = [m+1,m+r_m]$,
$r_m = |{N_G^>}(m)|$, and ${N_G^>}(m)$ is complete.}
\end{aligned}$$
For the first assertion of , we know that ${N_G^>}(m-1)=[m,m-1+r_{m-1}]$ is complete, which implies that $m+1,\ldots, m-1+r_{m-1}\in {N_G^>}(m)$. By analyzing the loop beginning on Line \[alg:secondLoop\] at this stage of Algorithm \[alg:Labeling\], one finds that every vertex in $S$ will be labeled with consecutive integers, starting at $i=m+r_{m-1}$ and continuing until the final vertex in ${N_G}(m)$ is labeled $i=m+r_{m-1}+r-1$, where $r$ is the original size of $S$. It follows that ${N_G}(m)$ is an interval of the desired form.
To show that ${N_G^>}(m)$ is complete, pick $s \ne t$ in ${N_G^>}(m)$. Let $P= v_0v_1\cdots v_{q-1}v_q$ be a shortest path from $1 = v_0$ to $v_q = m$, with $v_u \in L_u$ for all $u$. Lemmas \[layerlem\](1) and \[claim:orderingClaim\](2) imply that $s,t \in L_q \cup L_{q+1}$. Hence, $s$ and $t$ are either both distance $q+1$ from 1, both distance $q$ from 1, or one of $s$ and $t$ is distance $q$ from 1 and the other is distance $q+1$ from 1. We consider each case separately.
plus3pt minus2pt
[**Case 1.**]{} $s,t\in L_{q+1}$. Then $\{s,v_{q-1}\}$, $\{t,v_{q-1}\}\notin E(G)$ by Lemma \[layerlem\](1). Since the subgraph induced on $s,t,m,v_{q-1}$ cannot be a claw, we must have $\{s,t\} \in E(G)$.
plus3pt minus2pt
[**Case 2.**]{} $s,t\in L_q$. We can assume $s<t$ and choose a shortest path $P_1=w_0w_1\cdots w_q$ from $1=w_0$ to $w_q = t$ with $w_u\in L_u$. Then $w_{q-1}<m$ by Lemma \[claim:orderingClaim\](2), giving $w_{q-1}<m<s<t$. Since $t\in {N_G^>}(w_{q-1})$ and ${N_G^>}(w_{q-1})$ is an interval by hypothesis, we have $s\in
{N_G^>}(w_{q-1})$. But then $\{s,t\}\in E(G)$ since we are also assuming that ${N_G^>}(w_{q-1})$ is complete.
plus3pt minus2pt
**Case 3.**
We can assume $s\in L_q$ and $t\in L_{q+1}$, so $s<t$ by Lemma \[claim:orderingClaim\](2). We also have $l(m)\leq l(s)$ by Lemma \[claim:meaningOfFunction\](2) since $m<s$. We will consider separately the two possibilities that $l(m)<l(s)$ and $l(m)=l(s)$.
plus3pt minus2pt
[**Case 3A.**]{} Suppose that $l(m) < l(s)$. Then $\{l(m),s\}\notin E(G)$ since $l(s)=\min({N_G}(s))$ by Lemma \[claim:meaningOfFunction\](1). We also have $\{l(m),t\}\notin
E(G)$, for otherwise we would have $l(t)\leq l(m)$ since $l(t)=\min({N_G}(t))$. Then $l(t) \le l(m) < l(s)$, which implies $t<s$ by Lemma \[claim:meaningOfFunction\](2), contradicting $s<t$. Since the subgraph induced on $l(m),m,s,t$ cannot be a claw, we must have $\{s,t\} \in E(G)$.
plus3pt minus2pt
[**Case 3B.**]{} Suppose that $l(m)=l(s)$. We will assume $\{s,t\} \notin E(G)$ and derive a contradiction. The equality $l(m)=l(s)$ means that $m$ and $s$ were both labeled when $j=l(m)=l(s)$ in the loop starting on Line \[alg:firstLoop\] of Algorithm \[alg:Labeling\]. Consider the moment in the algorithm when the label $m$ is assigned. Since $m < s$ and $j=l(m)=l(s)$, this happens during an iteration of the loop on Line \[alg:secondLoop\] for which $m,s\in S$. Line \[alg:beginSecondLoop\] guarantees that the vertices assigned the labels $m$ and $s$ satisfy $|{N_G}(m)\setminus J| \leq |{N_G}(s)\setminus J|$. Since $s$ is not yet labeled at this point and $s<t$, $t$ is also not yet labeled and therefore $t\notin J$. It follows that $t\in {N_G}(m)\setminus J$ and $t\notin {N_G}(s)\setminus J$. But, in order for $|{N_G}(m)\setminus
J| \leq |{N_G}(s)\setminus J|$ to hold, there must be $s_2\in
{N_G}(s)$ with $s_2>m$ and $s_2\notin J$ and $s_2\notin {N_G}(m)$.
Let us study $s_2$. If $\{s_2,l(m)\}\in E(G)$, then $s_2\in
{N_G^>}(l(m))$. But we also have $m \in {N_G^>}(l(m))$. Since $l(m)<m$, ${N_G^>}(l(m))$ is complete by the hypothesis of , so we would have $\{m,s_2\}\in E(G)$. This contradicts our choice of $s_2$. Hence $\{s_2,l(m)\}\notin E(G)$. We also have $\{s_2,t\}\notin E(G)$, since otherwise the $4$-cycle $s_2tmss_2$ would have no chords as $\{s_2,m\},\{s,t\}\notin E(G)$. Also, since $m\in L_q$, Lemma \[claim:orderingClaim\](1) implies that $j=l(m)=l(s)\in L_{q-1}$.
We claim that $s_2 \in L_{q+1}$. Lemma \[layerlem\](1), $s \in
L_q$, and $s_2 > m \in L_q$ imply that $s_2 \in L_q$ or $L_{q+1}$. If $s_2 \in L_q$, then $l(s_2) \in L_{q-1}$ by Lemma \[claim:orderingClaim\](2). From here, $m \in L_q$ implies $m
> l(s_2)$ by Lemma \[claim:orderingClaim\](2). Hence we have $l(s_2) < m < s_2$. The hypothesis of implies that ${N_G^>}(l(s_2))$ is complete and is an interval. Since $s_2 \in
{N_G^>}(l(s_2))$, it follows that $m \in {N_G^>}(l(s_2))$, which contradicts our choice of $s_2$. Hence $s_2 \in L_{q+1}$ and we have Figure [3]{}(a).
$$\begin{array}{ccc}
\begin{tikzpicture}
\node[vertex] (n1) at (2,1) {$j\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n2) at (1,3) {$m\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n3) at (3,3) {$s\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n4) at (1,4.5){$t\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n5) at (3,4.5){$s_2\rule[-2.5pt]{0pt}{10pt}$};
\foreach \from/\to in {n1/n2,n1/n3,n2/n3,n2/n4, n3/n5}
\draw (\from)--(\to);;
\node [vertex, font = \large] at (4,3) {$L_q$};
\node [vertex, font=\large] at (4,4.5) {$L_{q+1}$};
\node [vertex, font = \large] at (4,1) {$L_{q-1}$};
\node [right] at (2.5,.3) {(a)};
\end{tikzpicture}&\hspace{25pt}&
\begin{tikzpicture}
\node[vertex] (n1) at (2,1) {$j\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n2) at (1,3) {$m\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n3) at (3,3) {$s\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n4) at (1,4.5){$t\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n5) at (3,4.5){$s_2\rule[-2.5pt]{0pt}{10pt}$};
\node[vertex] (n6) at (1,6){$w_{q+2}$};
\foreach \from/\to in {n1/n2,n1/n3,n2/n3,n2/n4, n3/n5, n4/n6, n6/n5}
\draw (\from)--(\to);;
\node [right, font = \large] at (4,3) {$L_q$};
\node [right, font=\large] at (4,4.5) {$L_{q+1}$};
\node [right, font = \large] at (4,1) {$L_{q-1}$};
\node [right, font = \large] at (4,6) {$L_{q+2}$};
\node [right] at (2.5,.3) {(b)};
\end{tikzpicture}
\end{array}$$ -10pt
Let $z$ be a vertex of distance $h={\mathrm{diam}}(G)$ from $1$ and pick a longest shortest path $P_2=w_0w_1\cdots w_h$ from $1=w_0$ to $w_h =z$, so $w_u\in L_u$. Since $G$ is narrow, $t$ and $s_2$ must each either be in $P_2$ or be adjacent to a vertex in $P_2$. We will consider each of these cases.
*First*, suppose that $t\in V(P_2)$. Then $t\in L_{q+1}$ implies that $t=w_{q+1}$. Since $l(m) \in L_{q-1}$, there is a path of length $q-1$ connecting $1$ to $l(m)$. Using $t = w_{q+1}$, it follows that $P_3=1\cdots l(m) m t w_{q+2}\cdots z$ is a path of length $h={\mathrm{diam}}(G)$. Since $G$ is narrow, $s_2$ must be adjacent to some vertex $P_3$. Then $\{s_2,t\}$, $\{s_2,m\},\notin E(G)$ and Lemma \[layerlem\](1) imply that $\{s_2,w_{q+2}\} \in E(G)$. This gives the $5$-cycle $mss_2w_{q+2}tm$ with no chords since $\{s_2,t\}$, $\{m,s_2\}$, $\{s,t\}\notin E(G)$ and $\{w_{q+2},m\}$, $\{w_{q+2},s\}\notin E(G)$ since $w_{q+2}\in L_{q+2}$ but $s,m\in
L_q$. See Figure [3]{}(b). Hence we have a contradiction since $G$ is chordal.
*Second*, suppose that $s_2\in V(P_2)$. Then $s_2=w_{q+1}$. Arguing as in the *First*, we arrive at Figure [3]{}(b) with the same $5$-cycle with no chords, again a contradiction.
*Third*, suppose that $s_2,t\notin V(P_2)$. First note that $P_2$ was an arbitrary longest shortest path starting at $1$. Thus the above *First* and *Second* give a contradiction whenever $s_2$ or $t$ are on *any* longest shortest path starting at $1$. Hence we may assume that $s_2$ and $t$ are not on any shortest path of length $h$ starting at $1$.
Since $G$ is narrow, $s_2 \in L_{q+1}$ is adjacent to a vertex of $P_2$, which must be $w_q$, $w_{q+1}$, or $w_{q+2}$ by Lemma \[layerlem\](1). However, if $\{s_2,w_{q+2}\} \in E(G)$, then we would get a path of length $h$ from $1$ to $z$ by taking any shortest path from 1 to $s_2$, followed by $\{s_2,w_{q+2}\}$, and then continuing along $P_2$ from $w_{q+2}$ to $z$. This longest shortest path starts at $1$ and contains $s_2$, contradicting the previous paragraph. Hence $\{s_2,w_{q+2}\} \notin E(G)$ and $s_2$ must be adjacent to $w_q$ or $w_{q+1}$, and the same is true for $t$ by a similar argument.
In fact, we must have $\{s_2,w_q\} \in E(G)$, since otherwise $\{s_2,w_{q+1}\} \in E(G)$ and the subgraph induced on $w_q,w_{q+1},w_{q+2},s_2$ would be a claw. A similar argument shows that $\{t,w_q\} \in E(G)$. Since $w_{q-1}\in L_{q-1}$ and $s_2,t\in
L_{q+1}$, this implies that the subgraph induced on $t,s_2,w_q,w_{q-1}$ is a claw, again contradicting claw-free. This final contradiction completes the proof of , and Theorem \[converse\] is proved.
\[remark:Ching\] In and , the chordal hypothesis is applied only to cycles of length $4$, $5$, or $6$. Hence, in Theorem \[mainthm\] and Corollary \[cormainthm\], we can replace chordal with the weaker hypothesis that all cycles of length $4$, $5$, or $6$ have a chord.
Acknowledgements {#acknowledgements .unnumbered}
================
Theorem \[mainthm\] is based on the senior honors thesis of the second author, written under the direction of the first author. We are grateful to Amherst College for the Post-Baccalaureate Summer Research Fellowship that supported the writing of this paper. Thanks also to Michael Ching for Remark \[remark:Ching\].
[10]{}
D. Cox and A. Erskine, *On closed graphs II*, in preparation.
M. Crupi and G. Rinaldo, *Binomial edge ideals with quadratic Gröbner bases*, Electron. J. Combin. [**18**]{} (2011), Paper 211, 13pp. V. Ene, J. Herzog and T. Hibi, *Cohen-Macaulay binomial edge ideals*, Nagoya Math. J. [**204**]{} (2011), 57–68. V. Ene, J, Herzog and T. Hibi, *Koszul binomial edge ideals*, arXiv:1310.6426 \[math.AC\]. V. Ene, J. Herzog and T. Hibi, *Linear flags and Koszul filtrations*, arXiv:1312.2190 \[math.AC\]. V. Ene and A. Zarojanu, *On the regularity of binomial edge ideals*, arXiv:1307.2141 \[math.AC\]. J. Herzog, T. Hibi, F. Hreinsdóttir, T. Kahle and J. Rauh, *Binomial edge ideals and conditional independence statements*, Adv. in Appl. Math. [**45**]{} (2010), 317–333. M. Ohtani, *Graphs and ideals generated by some 2-minors*, Commun. Algebra [**39**]{} (2011), 905–917. J. Rauh, *Generalized binomial edge ideals*, Adv. in Appl. Math. [**50**]{} (2013), 409–414. S. Saeedi Madani and D. Kiani, *Binomial edge ideals of graphs*, Electron. J. Combin. [**19**]{} (2012), Paper 44, 6pp.
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---
abstract: 'Mixtures of ideal polymers with hard spheres whose diameters are smaller than the radius of gyration of the polymer, exhibit extensive immiscibility. The interfacial tension between demixed phases of these mixtures is estimated, as is the barrier to nucleation. The barrier is found to scale linearly with the radius of the polymer, causing it to become large for large polymers. Thus for large polymers nucleation is suppressed and phase separation proceeds via spinodal decomposition, as it does in polymer blends.'
author:
- |
[**Richard P. Sear**]{}\
\
Department of Physics, University of Surrey,\
Guildford, Surrey GU2 7XH, United Kingdom\
email: [email protected]
title: Interfacial tension and nucleation in mixtures of colloids and long ideal polymer coils
---
[2]{}
Introduction
============
In earlier work [@sear01] immiscibility in mixtures of colloidal hard spheres and long ideal polymer molecules was studied. Long means that the root-mean-square end-to-end separation of the polymer molecules, $R_E$, is larger than the diameter $\sigma$ of the hard spheres. A mixture of spheres and long polymers was found to demix at comparable number densities of polymer molecules and spheres, both densities scale with $R_E$ as $1/R_E^2\sigma$, for $R_E>\sigma$. This scaling comes directly from the leading order term in the second virial coefficient for the sphere-polymer interaction. The interaction and hence the virial coefficient must be extensive in the number of monomers for large $R_E$ and hence must scale as $R_E^2$ for our ideal polymers. The requirement that it has the dimensions of a volume then imposes the scaling $R_E^2\sigma$ as $\sigma$ is the only other relevant length in the problem: the monomer size is assumed to be much less than $\sigma$ and so is irrelevant. Once the mixture has demixed we have two coexisting phases: one with a high density of colloidal particles and a low density of polymer molecules, and one with a high density of polymer molecules and a low density of colloidal particles. There is an interface between these two coexisting phases. Here we determine the scaling of the interfacial tension $\gamma$ of this interface, and use it to show that when a mixed sample of polymer and colloid is prepared and quenched into the two-phase region, the dynamics of the separation into two phases starts off with spinodal decomposition not nucleation. The fact that the phase separation starts off with spinodal decomposition makes mixtures of the long polymers like polymer blends but unlike simple mixtures, e.g., mixtures of oil and water. Thus, we can apply much of what we have learned of spinodal decomposition in systems like polymer blends, to mixtures of colloidal spheres and much larger polymer molecules.
The colloid-polymer interactions in and bulk thermodynamics of, mixtures of colloids and large ideal polymers have both been studied, see Refs. [@eisenriegler96; @hanke99; @meijer94; @sear01; @chatterjee98; @odijk00; @tuinier00], but this is the first study of the interfacial tension and nucleation of these systems. The opposite limit to that of interest here, i.e., where the polymer molecules are smaller than the colloidal spheres, has been considered extensively, see Refs. [@gast83; @lekkerkerker92; @meijer94; @dijkstra99] for work on the bulk phase behaviour and Refs. [@vrij97; @brader00; @evans01] for work on interfaces. The following two sections deal with the interfacial tension, and with nucleation. Throughout, the objective will be to determine the scaling of the behaviour with the ratio of the size of the polymer to that of the sphere. Also note that here the polymers are always ideal, mixtures of polymers with strong excluded volume interactions and spheres, are very different [@maassen01; @bolhuis01; @fuchs02; @tobe].
Interfacial tension between the demixed phases
==============================================
The interfacial tension between the demixed phases, one colloid-rich, the other polymer rich, can be estimated using just dimensional analysis. The tension $\gamma$ is an energy per unit area. It is obtained by multiplying the free energy per unit volume, which is $kT/(R_E^2\sigma)$ [@sear01], by the width of the interface. This width will be of the order of the polymer size $R_E$. Thus, $\gamma\sim kT/R_E\sigma$. Note that the energy scale has to be the thermal energy $kT$ as there are no other relevant energy scales in the problem. The mixture is athermal, there are no attractive interactions or soft repulsions to provide another energy scale. The free-energy density is then of order $kT$ times the number density, which is of order $1/R_E^2\sigma$ for both the polymer molecules and the colloidal spheres when they demix. This is just a simple scaling argument so we confirm it by determining the scaling of $\gamma$ within a standard square-gradient or Cahn-Hilliard theory for the interface [@cahn58; @evans79; @chaikin; @binder87; @debenedetti].
We apply this theory to the system in the semigrand ensemble of Ref. [@sear01] where the characteristic thermodynamic potential is the semigrand potential $\omega$, which is a function of the number density of colloidal particles, $\rho_C$, and the activity $z$ of the polymer molecules. As we are specifying the activity not the density of the polymer our system is equivalent to a single component system whose thermodynamic state depends on the density and on the activity of the polymer $z$; $\ln z$ acts as an inverse temperature in the sense that the larger it is the stronger is the effect of the attractions. Thus we can apply the standard square-gradient expression for the interfacial tension of a single component system, which is [@cahn58; @evans79; @debenedetti; @chaikin; @brader00] $$\gamma=\int{\rm d}x\left[\Psi+
\kappa\left(\frac{{\rm d}\rho_C}{{\rm d}x}\right)^2\right],
\label{sq}$$ where $$\Psi=\omega(\rho_C(x))-\omega^{(b)}-\mu_C(\rho_C(x)-\rho_C^{(b)}),$$ is the excess grand potential at a point. $\omega^{(\alpha)}$ and $\rho_C^{(\alpha)}$ are the semigrand potential and density in either one of the coexisting phases. The superscript $\alpha=C,P$ for the colloid-rich and polymer-rich phases respectively. $\mu_C$ is the chemical potential of the colloid. The interface is normal to the $x$-axis. The coefficient $\kappa$ of the gradient term is assumed to be density independent. The equilibrium profile is obtained by minimising Eq. (\[sq\]). Then standard manipulations enable a simpler expression for the equilibrium interfacial tension to be derived [@cahn58] $$\gamma=2\int_{\rho_C^{(P)}}^{\rho_C^{(C)}}
{\rm d}\rho_C\left[\kappa\Psi\right]^{1/2}.
\label{sq_eq}$$
If we require that the functional Eq. (\[sq\]) be consistent with linear response theory [@evans79] we obtain an expression for the coefficient $\kappa$ of the gradient term $$\kappa=\frac{kT}{12}\int{\rm d}{\bf r}r^2c_2(r;\rho_C,z),
\label{kappa}$$ where $c_2(r;\rho_C,z)$ is the direct correlation function of the fluid of colloidal hard spheres in the presence of polymer. For our systems the most basic assumption is to use the low density approximation to the direct correlation function. This replaces $c$ with the Mayer f function for the effective sphere-sphere interaction in the presence of the polymer [@evans79]. For two spheres with centres separated by less than $\sigma$, the interaction energy is infinite and the Mayer f function equals $-1$. For separations larger than $\sigma$ the only interaction is that due to the polymer. This interaction is known [@hanke99], and is long-range and weak thus we linearise the Mayer f function. Adding this altogether we obtain $$c(r;\rho_C,z)\sim\left\{\begin{array}{cc}
-1 & r<\sigma \\
zR_E^2\sigma\left(\sigma/r\right) & \sigma<r\lesssim R_E \\
0 & r\gg R_E \\
\end{array},\right.$$ the ideal polymer induces an attraction which decays as $1/r$ for separations less than the radius of the polymer and roughly exponentially beyond this. Putting our approximate $c$ into Eq. (\[kappa\]) we obtain an estimate of this coefficient $$\begin{aligned}
\kappa &\sim & kTzR_E^2\sigma^2\int_0^{R_E}{\rm d}rr^3\\
&\sim& kTz\sigma^2R_E^6.
\label{kscale}\end{aligned}$$
We now return to Eq. (\[sq\_eq\]) for the interfacial tension and determine its scaling with $R_E$. We note that the density difference $\rho_C^{(C)}-\rho_C^{(P)}\sim 1/R_E^2\sigma$, $\kappa$ scales as given by Eq. (\[kscale\]), $\Psi\sim kT/R_E^2\sigma$ and the polymer activity is of order the polymer number density $z\sim 1/R_E^2\sigma$. Putting this all together we see that we recover the scaling $\gamma\sim kT/R_E\sigma$ obtained earlier by dimensional analysis. Also, from Eq. (\[sq\]) we see that the characteristic length-scale for the interface must be $(\kappa/\Psi)^{1/2}\rho_C^{(\alpha)}\sim R_E$, as we assumed earlier. Earlier work by Vrij [@vrij97], and by Brader and Evans [@brader00] on the interfacial tension between demixed colloid-rich and polymer-rich phases when the colloid and polymer were of comparable sizes, $R_E\sim\sigma$, found that, as expected. $\gamma\sim kT/\sigma^2\sim kT/R_E^2$. This is consistent with experimental findings [@hoog99].
The interfacial tension $\gamma$ will be of order $kT/R_E\sigma$ only if we are not too close to the critical point of the polymer-colloid demixing. In general we have $\gamma=(kT/R_E\sigma)s(z/z_c-1)$, where $s$ is a scaling function and $z_c$ is the polymer activity at the critical point. We have been assuming that we are not very close to the critical point, i.e., that $z/z_c-1=O(1)$, and for these values of its argument $s=O(1)$ and we return to $\gamma$ being of order $kT/R_E\sigma$. However, as the critical point is approached, $z/z_c-1\ll 1$, we have that the scaling function $s(x)=s_0x^{\mu}$ for $x\ll1$, where $s_0$ is a dimensionless constant and $\mu$ is the (positive) critical exponent of the interfacial tension [@widom85]. The interfacial tension tends to 0 as the critical point of demixing is approached, and near the critical point it varies as a power law. See the review of Widom [@widom85] for an excellent introduction to interfaces near critical points. Sufficiently close to the critical point the scaling of the interfacial tension will be dominated by fluctuations and then the exponent $\mu$ will take the value for the Ising model in three dimensions, $\mu=1.26$ [@widom85; @chaikin]. However, for very long polymers $R_E\gg\sigma$ the effective interaction is long-ranged and long-range interactions suppress fluctuations and make the system mean-field–like. The mean-field value of the exponent $\mu$ is $3/2$ [@widom85]. Which value of the exponent, Ising or mean-field, is observed is determined by whether or not the Ginzburg criterion is obeyed or not; see Ref. [@chaikin] or any introduction to critical phenomena for a definition of the Ginzburg criterion. Note that Eq. (\[sq\_eq\]), belonging as it does to a mean-field theory, will yield an interfacial tension which tends to 0 with an exponent $\mu=3/2$, its mean-field value.
Nucleation and other fluctuations
=================================
Now consider a single phase mixture of spheres and polymer quenched into the two-phase coexistence region. For definiteness assume that the single phase is the polymer-rich one. Then in order for the second, colloid-rich, phase to form and coexist with the polymer-rich one, this second phase must form. The dynamics of the formation of a new phase fall into two broad categories: nucleation then growth, and spinodal decomposition. See Refs. [@debenedetti; @binder87; @chaikin] for an introduction to the dynamics of first-order phase transitions. For example a mixture of simple liquids such as water and an alcohol phase separate via nucleation of the new water-rich or alcohol-rich phase followed by growth of the nuclei, whereas polymer blends phase separate via spinodal decomposition. Here we show that for large ideal polymers and spheres, nucleation becomes very difficult, so mixtures of large ideal polymers and much smaller spheres will start to phase separate via spinodal decomposition.
The rate of nucleation $N_n$ can be estimated using classical nucleation theory, see the book of Debenedetti [@debenedetti] for a comprehensive discussion, see also Refs. [@chaikin; @binder87]. $N_n$ is the number of nuclei crossing the barrier per unit time per unit volume. The classical nucleation theory expression for the rate $N_n$ is $$N_n=\Gamma\exp(-\Delta F^*/kT),
\label{cnt}$$ where $\Gamma$ is an attempt frequency per unit volume, generally slowly varying, and $\Delta F^*$ is the free energy barrier that must be crossed in order for a new phase to nucleate. The variation in the rate is generally dominated by that in $\Delta F^*$ so we focus on this. The free energy barrier comes from the free energy needed to form a microscopic droplet of the new phase, here a colloid-rich phase. This droplet is the nucleus of the new phase. Within classical nucleation theory the free energy of formation of a microscopic droplet is the sum of two terms, a bulk term and a surface term, $$\Delta F=\frac{4}{3}\pi R^3\Psi_n+4\pi R^2\gamma,
\label{df1}$$ where $R$ is the radius of the droplet and $$\Psi_n=\omega(\rho_C^{(n)})-\omega(\rho_C)-\mu_C(\rho_C^{(n)}-\rho_C)
\label{psin1}$$ is the difference between the grand potential inside the nucleus and the grand potential of the phase in which the nucleus forms, $\rho_C$ is the density of colloid in the phase in which the nucleus forms, and $\rho_C^{(n)}$ is the density of colloid inside the nucleus. So long as we do not approach the spinodal too closely, we can express $\Psi_n$ as a Taylor expansion in chemical potential, around the chemical potential of the colloid at coexistence $\mu_{co}$. Truncating the Taylor series after the linear term, we get $$\Psi_n\simeq \Psi'\left(\mu_C-\mu_{co}\right),
\label{psin2}$$ as $\Psi_n=0$ at coexistence, with $$\Psi'=\left(\frac{\partial \Psi_n}
{\partial\mu_C}\right)_{\mu_C=\mu_{co}},
\label{psid}$$ the derivative of $\Psi_n$ at coexistence.
The barrier is given by the free energy of the droplet whose free energy is highest, which occurs when the two terms in Eq. (\[df1\]) are comparable, $R^3\Psi_n\sim R^2\gamma$. Thus, $R\sim \gamma/(\Psi'(\mu_C-\mu_{co}))$, and $$\Delta F^* \sim\frac{\gamma^3}
{\left(\Psi'\left(\mu_C-\mu_{co}\right)\right)^2}.
\label{df_s1}$$ From Eq. (\[psid\]), $\Psi'$ scales as $1/R_E^2\sigma$. Using this scaling together with that of $\gamma$, we find that at the top of the barrier the size of the nucleus is of order $R_E$, for $\mu_C-\mu_{co}$ not too much less than $kT$. Using these same scalings in Eq. (\[df\_s1\]), we obtain the principle result of this section, the scaling of the nucleation barrier $\Delta F^*$, $$\frac{\Delta F^*}{kT} \sim\left(\frac{\mu_{co}}
{\mu_C-\mu_{co}}\right)^2
\frac{R_E}{\sigma},
\label{df_scale}$$ the barrier scales as $R_E/\sigma$ and so increases as the size of the polymer molecules relative to that of the colloidal spheres increases. The factor in parentheses is a dimensionless measure of the supersaturation: how far we are into the two-phase region.
For sufficiently large $R_E$ of the polymers the nucleation barrier will become so large that nucleation cannot occur. The mixture will be metastable up to very close to the spinodal [@herrmann82; @binder83; @binder84; @binder87]. Thus, the mixture will only start to demix when quenched beyond the spinodal, where the phase separation will start with spinodal decomposition. This is precisely analogous to polymer blends and systems of particles in which the particle-particle attractions are long ranged. In these systems the nucleation barrier scales as $N^{1/2}$ and as $r^3$, where $N$ is the length of the polymer and $r$ is the range of the attraction [@binder84]. The phase transition dynamics of systems of polymers and of particles with long-range attractions, were studied extensively in the early 1980s by Binder and Klein and their coworkers [@herrmann82; @binder83; @binder84]. Many of the conclusions of that work also apply to the demixing of mixtures of hard spheres and much larger ideal polymers.
Finally, we note that our finding that nucleation is suppressed is equivalent to saying that our mixture satisfies the Ginzburg criterion for the irrelevance of fluctuations [@chaikin]. Essentially, nucleation [*is*]{} a fluctuation so when fluctuations are weak nucleation is suppressed and vice versa, again this was found for polymer blends/particles with long-range attractions [@degennes; @binder84]. For our mixtures, the Ginzburg criterion is essentially that the root-mean-square (rms) fluctuations in the number of colloidal spheres or of polymer molecules, in a volume $R_E^3$, are much less than the mean number in that volume. The volume $R_E^3$ is the volume over which a pair of spheres or polymer interact (the sphere-sphere interaction is mediated by the polymer molecules, and the polymer-polymer interaction is mediated by the spheres). The rms fluctuations scale as the square root of the number of spheres in a volume $R_E^3$, which is $\sim (R_E/\sigma)^{1/2}$, whereas the mean number scales as $R_E/\sigma$. Thus for large values of the ratio of the sizes, $R_E/\sigma$, the rms fluctuations in the number of spheres (or polymer molecules) inside the interaction volume is a small fraction of the mean number. Fluctuations about the mean are small and so mean-field theory applies and nucleation, which is a fluctuation, has a very high free-energy cost.
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|
---
abstract: 'Empirically determining the averaged variations of the orbital parameters of the stars orbiting the Supermassive Black Hole (SBH) hosted by the Galactic center (GC) in Sgr A$^{\ast}$ is, in principle, a valuable tool to test the General Theory of Relativity (GTR), in regimes far stronger than those tested so far, and certain key predictions of it like the no-hair theorems. We analytically work out the long-term variations of all the six osculating Keplerian orbital elements of a test particle orbiting a nonspherical, rotating body with quadrupole moment $Q_2$ and angular momentum $\bds S$ for a generic spatial orientation of its spin axis $\kap$. This choice is motivated by the fact that, basically, we do not know the position in the sky of the spin axis of the SBH in Sgr A$^{\ast}$ with sufficient accuracy. We apply our results to S2, which is the closest star discovered so far having an orbital period $P_{\rm b}=15.98$ yr, and to a hypothetical closer star $X$ with $P_{\rm b}=0.5$ yr. Our calculations are quite general, not being related to any specific parameterization of $\kap$, and can be applied also to astrophysical binary systems, stellar planetary systems, and planetary satellite geodesy in which different reference frames, generally not aligned with the primary’s rotational axis, are routinely used.'
author:
- 'L. Iorio'
title: 'Perturbed stellar motions around the rotating black hole in Sgr A$^{\ast}$ for a generic orientation of its spin axis'
---
\#1[Eq. (\[\#1\])]{} \#1\#2[Eq. (\[\#1\])-Eq. (\[\#2\])]{} \#1[Eq. (\[\#1\])]{} \#1\#2[Eq. (\[\#1\])-Eq. (\[\#2\])]{} \#1\#2 \#1\#2 \#1[“\#1”]{} \#1\#2[\^[\#2]{}\_[.\#1]{}]{} \#1\#2\#3 \#1\#2[[\#1\#2]{}]{} \#1[\[\#1\]]{} \#1\#2\#3
\#1
Introduction
============
There is nowadays wide consensus [@Genzel; @nigri; @Ghez] that the Galactic Center (GC) hosts a Supermassive Black Hole (SBH) [@Wol; @Falke] whose position coincides with that of the radio-source Sagittarius A$^{\ast}$ (Sgr A$^{\ast}$) [@Bal; @Reid07] at $d=8.28\pm 0.44$ kpc from us [@Gille]; for a popular overview of such an object, see, e.g., Ref. [@Melia07]. The Galactic SBH is surrounded by a number of recently detected main-sequence stars of spectral class B [@Pau; @Gille]. They have been revealed and tracked in the near infrared since 1992 at the 8.2 m Very Large Telescope (VLT) on Cerro Paranal, Chile and the 3.58 m New Technology Telescope (NTT) on La Silla, Chile [@Ecka], and since 1995 at the Keck 10 m telescope on Mauna Kea, Hawaii [@Ghez98]. They are dubbed SN, or S0-N in the Keck nomenclature, where N is a progressive order number. Their relatively fast orbital motions, characterized by orbital periods $P_{\rm b}\gtrsim 16$ yr, high eccentricities $e\gtrsim 0.2$, random orientations $i$ of their orbits in the sky and average distances from the SBH $\overline{r} \gtrsim 2\times 10^4 r_g$, where $r_g$ denotes the Schwarzschild radius, allowed to dynamically infer a mass of about $M\approx 4\times 10^6 M_{\odot}$ [@Ghez; @Gille; @Gille2] for it.
The direct access to such S/S0 stars, and of other closer objects which may hopefully be discovered in the future, has induced several researchers to investigate various predictions that the General Theory of Relativity (GTR) directly makes for their orbital motions along with other competing effects from standard Newtonian gravity which may mask the relativistic ones [@Jaro; @Fragile; @Rubi; @Weinb; @Kran; @Nuci; @Will08; @Preto; @Khan; @Merr010; @Iorio011; @Sade011]. Concerning several effects related to propagating electromagnetic waves in connection with the stellar orbital motions like, e.g., relativistic redshifts, see Ref. [@Zuck06; @Ange010a; @Ange010b; @Ange011]. In fact, although the currently known stars, in a strict sense, cannot probe the strong field regime of GTR because of their relatively large distance from the SBH, on the other hand they yield a unique opportunity to put on the test GTR in the strongest field regime ever probed so far. Indeed, even in the double binary pulsar PSR J0737-3039A/B [@pulsar; @Lyne] $r_g/\overline{r}$ is one order of magnitude smaller than for S2, which is the closest SBH star discovered so far [@Gille].
In this paper we analytically work out the averaged variations of all the six standard osculating Keplerian parameters of a test particle caused by the rotation of the central object endowed with angular momentum $\bds S=S\kap$ and quadrupole moment $Q_2$. Note that the stars orbiting the SBH can safely be considered test particles: their masses are about $m\lesssim 10^{-5} M$, and relativistic corrections to their internal structures are assumed to be too small to yield noticeable effects on their orbital motions. No assumptions about any specific spatial orientation for $\kap$ are made. Thus, our calculations are not restricted to a particular reference frame, and are valid also for different scenarios like, e.g., stellar planetary systems and planetary satellite geodesy in which natural and/or artificial test bodies are employed for testing GTR. Moreover, in order to keep our calculations as general as possible, we will not adopt any particular representation for $\kap$ in terms of specific angular variables in the sense that we will refer it to the global reference frame adopted; for a different approach, see Ref. [@Will08] in which $\kap$ is referred to the orbital plane of each star. Concerning the SBH in Sgr A$^{\ast}$, the orientation of its spin axis is substantially unknown, despite the attempts by different groups [@spin1; @spin2; @mmvlbi2] to constrain it using different parameterizations which yielded quite loose bounds. A strategy to partially overcome such an obstacle have been recently put forward in Ref. [@Hio]; it is based on the possible observation of the apparent shape of the shadow cast by the BH on the plane of the sky, and would allow the measurement of $S$ and the angle $i^{'}$ between $\kap$ and the line-of-sight.
The GTR prediction for the standard 1PN periastron precession, which is analogous to Mercury’s well known perihelion precession of $42.98$ arscec cty$^{-1}$ and is independent of $\kap$, amounts to about \_[S2]{}\^[(1PN)]{}=4510 [arcsec yr]{}\^[-1]{}for S2; the quoted uncertainty comes from the errors in the parameters of both the SBH and S2 entering the GTR formula: they are displayed in Table \[tavola\]. The result of , computed in a frame with the SBH at its origin, corresponds to a precession $\dot\xi_{\rm S2} = 27\pm 6$ microarcseconds per year ($\mu$as yr$^{-1}$ in the following). At present, it is still undetectable from the currently available direct astrometric measurements in terms of right ascension $\alpha$ and declination $\delta$ which barely cover just one full orbital period of S2. Indeed, according to Table 1 of Ref. [@Gille], the present-day error in the periastron is $\sigma_{\omega}=0.81\ {\rm deg}=2916$ arcsec over about 16 yr, from which an uncertainty of about $\sigma_{\dot\omega} \simeq 182$ arcsec yr$^{-1}$ in the periastron precession may naively be inferred: it corresponds to a limiting accuracy of $\sigma_{\dot\xi}=110$ $\mu$as yr$^{-1}$ in monitoring angular rates as seen from the Earth. As we will see, the sizes of the other precessions of S2 due to $S$ and $Q_2$ may be smaller by about $2$ and $4-5$ orders of magnitude, respectively for a moderate rotation of the SBH. Concerning future perspectives, according to Ref. [@Eise09] future astrometric measurements of S2 may bring its 1PN periastron rate into the measurability domain; indeed, the periastron advance would indirectly be inferred from the corresponding apparent position shift in the recorded orbit. Moreover, the ASTrometric and phase-Referenced Astronomy (ASTRA) project [@Eisner], to be applied to the Keck interferometer, should be able to monitor stellar orbits with an accuracy of [@Pott] $\sigma_{\Delta\xi}\approx 30$ $\mu$as as seen from the Earth. The GRAVITY instrument [@Gille010], devoted to enhance the capabilities of the VLT interferometer (VLTI), aims to reach an accuracy of $\sigma_{\Delta\xi}\approx 10$ $\mu$as [@Gille010] in measuring astrometric shifts $\Delta\xi$ as seen from the Earth, which, among other things, would allow exploration of the innermost stable circular orbits around the SBH [@Vinc011].
About testing GTR in the SBH scenario, we make the following general considerations. In order to meaningfully compare theoretical predictions for a given effect to its empirically determined counterpart, we need to know some of the key ambient parameters entering the predictions independently from the effects themselves we are looking for. In the specific case, the mass $M$, the spin $\bds S$ and the quadrupole $Q_2$ of the SBH should be known, if possible, independently of the precessions we are going to consider. Concerning the SBH mass $M$, the values which we presently have for it can be thought as inferred from the third Kepler law used in conjunction with the directly measured orbital period $P_{\rm b}$, and the semimajor axis $a$ empirically determined by modeling the recorded stellar orbit in the plane of the sky with an ellipse (see Fig. 2 of Ref. [@Gille]). Such a determination of $M$ would be, in principle, by GTR itself since it implies a correction to the third Kepler law, but it is far too small with respect to the present-day accuracy in determining $P_{\rm b}$. Indeed, it is $\sigma_{P_{\rm b}}\simeq 10^{-1}$ yr [@Ghez; @Gille2], while the 1PN GTR correction to the Keplerian orbital period is [@Dam; @Soffel] $\Delta P_{\rm b}^{(\rm 1PN)}\propto (3\pi/c^2)\sqrt{GM a}\simeq 10^{-3}$ yr for S2. The same holds also if $M$ is straightforwardly inferred, in a perhaps less transparent manner, as a solve-for quantity from multiparameter global fits of all the stars’ data: modeling or not GTR at 1PN level has not yet statistically significant influence in its estimated values, as shown by Table 2 of Ref. [@Gille]. We stress that, when such an approach is followed to test GTR, it is intended that different dynamical models, with and without GTR, are fitted to the same data sets to see if statistically significant differences occur in the solve-for estimated parameters. The quadrupole moment $Q_2$ of the SBH in Sgr A$^{\ast}$ may be measured, e.g., by means of imaging observations with Very Long Baseline Interferometry (VLBI) in the strong field regime; see Ref. [@bambi; @bambib; @Joha] for recent reviews and other proposals. In regard to the spin $\bds S$ of Sgr A$^{\ast}$, one tries to gain independent information about $S$ from the interpretation of some measured Quasi-Periodic Oscillations (QPOs) in the X-ray spectrum of the electromagnetic radiation emitted by the gas orbiting in the accretion disk close to its inner edge [@Kato; @Genznat]. More recent observations conducted with the Millimeter Very Long Baseline Interferometry (mm-VLBI), probing the immediate vicinity of the horizon, have been able to get information on $S$ [@spin2; @mmvlbi2]. In interpreting such measurements, the validity of the Kerr metric [@Kerr] as predicted by GTR is assumed, thus inferring $S$ from, say, the radius of the inner edge extracted from the X-ray diagnostics. It is worth pointing out that the mere fact of obtaining a good fit of the Kerr metric [@Kerr] to a certain empirically determined quantity like, e.g., the X-ray spectrum, getting a reasonable value for $S$ as a least-square adjustable parameter, cannot be considered in itself as a test of the rotation-related predictions of GTR, also because other competing mechanisms to explain, say, the QPOs, whose physics is still rather disputed, exist. Independent empirical determinations of different effects connected with $\bds S$ are required, and the stellar orbital precessions would be just what we need. The greater the number of precessions empirically determined, the greater the number of GTR tests which can be performed. In principle, more than five precessions are required since $M,S,Q_2$ and two components of $\kap$ must be determined; see also the discussion in Ref. [@Will08]. Thus, the need for more than one star is apparent. Such a number of necessary orbital rates may be reduced if some of the aforementioned parameters are somehow reliably obtained from other sources. Of course, also the accuracy with which the precessions can be determined plays a role, in the sense that the previous reasoning holds in the ideal case in which all the three dynamical effects considered are detectable. Basically, it is the same logic behind the usual tests in the binary systems hosting at least one active radio-pulsar [@Damo]. Indeed, in that case the interpretation of just two empirically determined post-Keplerian effects in terms of their 1PN-GTR predictions is not sufficient since it only allows to obtain the masses $m_1$ and $m_2$ of the system, which are a priori unknown. In the binary pulsar systems the effects which can, actually, be inferred from the data are not limited just to the post-Keplerian periastron precession. Genuine tests of GTR are made when more than two post-Keplerian parameters are empirically determined, and the additional ones are interpreted with GTR by using in their analytical predictions just the previously obtained values for $m_1$ and $m_2$ [@Damo]. The plan of the paper is as follows. In Sec. \[due\] we review basic facts of standard perturbation theory which will be applied in Sec. \[orbisa\] to $Q_2$ (Sec. \[tre\]) and $\bds S$ (Sec. \[quattro\]). In Sec. \[onda\] it is briefly remarked that also gravitational waves with ultralow frequency traveling from the outside could be constrained by the orbital precessions of the stars in Sgr A$^{\star}$. In Sec. \[compa\] we compare our results to those obtained by Will [@Will08]. Numerical evaluations of the effects worked out in Sec. \[orbisa\] are presented in Sec. \[cinque\]. Sec. \[sei\] is devoted to summarizing our findings and to the conclusions.
Overview of the method adopted
==============================
Here we deal with a generic perturbing acceleration $\bds A$ induced by a given dynamical effect which can be considered as small with respect to the main Newtonian monopole $A_{\rm Newton}=-GM/r^2$, where $G$ is the Newtonian constant of gravitation and $r$ is the mutual particle-body distance.
First, $\bds A$ has to be projected onto the radial, transverse and normal orthogonal unit vectors $\bds{\hat{R}},\bds{\hat{T}},\bds{\hat{N}}$ of the comoving frame of the test particle orbiting the central object. Their components, in Cartesian coordinates of a reference frame centered in the primary, are [@Monte] =(
[c]{} u -iu\
u + iu\
iu\
) =(
[c]{} -u-iu\
-u+iu\
iu\
) =(
[c]{} i\
-i\
i\
). In -, . In this specific case, we will choose the unit vector $\bds{\hat{\rho}}$ of the line-of-sight, pointing from the object to the observer, to be directed along the positive $z$ axis, so that the $\{x,y\}$ plane coincides with the usual plane of the sky which is tangential to the celestial sphere at the position of the BH. With such a choice, corresponding to the frame actually used in data reduction [@Eise; @Ghez], $i$ is the inclination of the orbital plane to the plane of the sky ($i=90$ deg corresponds to edge-on orbits, while $i=0$ deg implies face-on orbits), and $\Om$ is an angle in it counted from the reference $x$ direction; it is such a node which is actually determined from the observations [@Ghez; @Gille; @Gille2], and, in general, it is not referred to the SBH’s equator. Subsequently, the projected components of $\bds A$ have to be evaluated onto the Keplerian ellipse r=, pa(1-e\^2),where $p$ is the semilatus rectum and $a,e$ are the semimajor axis and the eccentricity, respectively. The Cartesian coordinates of the Keplerian motion in space are [@Monte] $${\begin{array}{lll}
x &=& r\left(\cos\Om\cos u\ -\cos i\sin\Om\sin u\right),\\ \\
%
y &=& r\left(\sin\Om\cos u + \cos i\cos\Om\sin u\right),\\ \\
%
z &=& r\sin i\sin u.
%
\end{array}}\lb{xyz}$$ Then, $A_R,A_T,A_N$ are to be into the right-hand-sides of the Gauss equations for the variations of the osculating Keplerian orbital elements [@Roy; @Soffel]. , computed for the perturbing accelerations of the dynamical effect considered, have to be inserted into the analytic expression of the time variation $d\Ps/dt$ of the osculating Keplerian orbital element $\Ps$ of interest. Then, it must be averaged over one orbital revolution by means of
Calculation of the long-term orbital effects
============================================
The long-term precessions caused by the quadrupole mass moment of the central body for an arbitrary orientation of its spin axis
--------------------------------------------------------------------------------------------------------------------------------
The acceleration experienced by a test particle orbiting a nonspherical central mass rotating about a generic direction $\kap$ is A\^[(Q\_2)]{}=-{+2()},where $Q_2$ is the quadrupole moment of the body, with $[Q_2]={\rm L}^5 {\rm T}^{-2}$. A dimensionless quadrupole parameter $J_2$ can be introduced by posing $Q_2\rightarrow GM \mathcal{R}_e^2 J_2$, where $\mathcal{R}_e$ is the equatorial radius of the rotating body. According to the or uniqueness theorems of GTR [@hair1; @hair2], an electrically neutral BH is completely characterized by its mass $M$ and angular momentum $S$ only. As a consequence, all the multipole moments of its external spacetime are functions of $M$ and $S$ [@multi1; @multi2]. In particular, the quadrupole moment of the BH is Q\_2=-.The spatial orientation of the BH’s spin axis can be considered as unknown. Thus, looking for a more direct connection with actually measurable quantities, we will retain a generic orientation for $\kap$ in the ongoing calculation, i.e., we will not align it to any of axes of the reference frame used. After having computed the $R-T-N$ components of by means of - as $${\begin{array}{lll}
A_R^{(Q_2)}=\bds{A}^{(Q_2)}\cdot\bds{\hat{R}}, \\ \\
%
A_T^{(Q_2)}=\bds{A}^{(Q_2)}\cdot\bds{\hat{T}}, \\ \\
%
A_N^{(Q_2)}=\bds{A}^{(Q_2)}\cdot\bds{\hat{N}},
%
\end{array}}\lb{GaussQ}$$ to be evaluated onto the unperturbed Keplerian ellipse, it is possible to obtain for the semimajor axis and the eccentricity, as in the standard calculations in which $\kap$ is usually aligned with the $z$ axis.
Instead, the inclination $i$ undergoes a long-term variation given by = -(,i;),with
(,i;)(+).
If $\kx=\ky=0$, as in the usual calculation [@Roy], $i$ stays constant.
Concerning the node $\Om$, its long-term variation is = (,i;),with
(,i;)22ii(-) +.
Notice that $\kx=\ky=0$ in yields the standard secular precession [@Roy] with (i)=-2.
The long-term change of the argument of pericenter $\omega$ is a little more cumbersome. It is = (,i;),with
[lll]{} (,i;)&& 8-11\^2-11\^2-2\^2+(\^2-\^2)2-\
\
&-& 2(i -53 ii)(-)-22+\
\
&+&52 i.
In the case $\kx=\ky=0$ reduces to (i)=2(3+52 i)=4(4-5\^2 i),which yields the standard expression for the secular precession of the pericenter [@Roy].
The longitude of the pericentre experiences a long-term variation given by = (,i;),with
[lll]{} (,i;)&&
8-11\^2-11\^2-2\^2 +(\^2+\^2-2\^2)(4-52 i)-\
\
&-& 4(\^2-\^2)(3+5)\^2()2- 2()+\
\
&+& 2()- 8\^2()(3+5)2.
For $\kx=\ky=0$ reduces to (i)= 2=4(4-5\^2 i-2),which yields the usual expression for the secular rate of $\varpi$ [@Roy].
Finally, the long-term change of the mean anomaly is =-(,i;), with
[lll]{} (,i;)&& -8+9\^2+9\^2+6\^2+3(\^2+\^2-2\^2)2 i +6(\^2-\^2)\^2 i2+\
\
&+& 12.
Also in this case, for $\kx=\ky=0$ the standard secular precession [@Roy] is recovered since reduces to (i)=-2(1+32 i)=-4(2-3\^2 i).
Incidentally, we remark that the field of applicability of - is not limited just to the BH arena, being generally valid also for astrophysical binary systems, stellar planetary systems, and planetary satellite geodesy. In particular, they could be useful when satellite-based tests of GTR are performed or designed (See Sec. V).
The Lense-Thirring long-term precessions for a generic orientation of the spin axis of the central body
-------------------------------------------------------------------------------------------------------
According to GTR, the gravitomagnetic Lense-Thirring acceleration felt by a test particle moving with velocity $\bds v$ around a rotating body with angular momentum $\bds S=S\kap$ at large distance from it is A\^[(LT)]{}=-2()B\_g.In the gravitomagnetic field $\bds B_g$, far from the central object where the Kerr metric [@Kerr] reduces to the Lense-Thirring one, is B\_g=-.Concerning $S$, the existence of the horizon in the Kerr metric [@Kerr] implies a maximum value for the angular momentum of a spinning BH [@Bar1; @Mel1], so that $ S=\chi_g S_{\rm max},$ withS\_[max]{} = .If $\chi_g>1$, the Kerr metric [@Kerr] would have a naked singularity without a horizon. Thus, closed timelike curves could be considered, implying a causality violation [@Chan]. Although not yet proven, the cosmic censorship conjecture [@Pen69] asserts that naked singularities cannot be formed via the gravitational collapse of a body. If the limit of is actually reached or not by astrophysical BHs depends on their accretion history [@Barde]. In the case of Sgr A$^{\ast}$, it may be $\chi_g\approx 0.44-0.52$ [@Genznat; @Kato] or even less [@spin2; @mmvlbi2]. Contrary to BHs, no theoretical constraints on the value of $\chi_g$ exist for stars. For main-sequence stars, $\chi_g$ depends sensitively on the stellar mass, and can be much larger than unity [@Kraft1; @Kraft2; @Dicke; @stella]. The case of compact stars was recently treated in Ref. [@cinesi], showing that for neutron stars with $M\gtrsim 1 M_{\odot}$ it should be $\chi_g\lesssim 0.7$, independently of the Equation Of State (EOS) governing the stellar matter. Hypothetical quark stars may have $\chi_g>1$, strongly depending on the EOS and the stellar mass [@cinesi].
In the standard derivations of the Lense-Thirring effect [@LT] existing in literature the reference $\{x,y\}$ plane was usually chosen coincident with the equatorial plane of the rotating mass. In principle, the Lense-Thirring orbital precessions for a generic orientation of $\bds S$ could be worked out with the Gauss equations in the same way as done for $Q_2$. Anyway, they were recently worked out [@Iorio010], in a different framework, with the less cumbersome Lagrange planetary equations [@Roy]. For the , we display here the final result
$$\begin{array}{lll}
\textcolor{black}{\left\langle \dert a t\right\rangle} & = & 0, \\ \\
%
\textcolor{black}{\left\langle\dert e t\right\rangle} & = & 0, \\ \\
%
\textcolor{black}{\left\langle\dert i t\right\rangle} & = & \rp{2GS\left(\kx \cos\Om+\ky\sin\Om\right)}{c^2 a^3(1-e^2)^{3/2}}, \\ \\
%
\textcolor{black}{\left\langle\dert\Om t\right\rangle} & = & \rp{2GS\left[ \kz + \cot i\left(\ky\cos\Om -\kx \sin\Om\right)\right]}{c^2 a^3(1-e^2)^{3/2}}, \\ \\
%
\textcolor{black}{\left\langle\dert\omega t\right\rangle} & = & -\rp{GS\left[6\kz\ci +\left(3\cos 2 i -1\right)\csc i\left(\ky\cO-\kx\sO\right)\right]}{c^2 a^3(1-e^2)^{3/2}}, \\ \\
%
\textcolor{black}{\left\langle\dert\varpi t\right\rangle} & = & -\rp{ GS \left\{4\left[\kz\cos i+\sin i\left(\kx\sin\Om-\ky\cos\Om \right) \right]
-2\left[\kz\sin i+\cos i\left(\ky\cos\Om-\kx\sin\Om \right) \right]\tan(i/2)
\right\}}{c^2 a^3(1-e^2)^{3/2}}, \\ \\
%
\textcolor{black}{\left\langle\dert{\mathcal{M}} t\right\rangle} & = & 0.
\end{array}\lb{piccololt}$$
Notice that yields just the usual Lense-Thirring secular rates [@LT; @Soffel] for $\kx=\ky=0$. Contrary to such a scenario, the inclination $i$ experiences a long-term gravitomagnetic change for an arbitrary orientation of $\bds S$: it is independent of the inclination $i$ itself.
A comparison with a different approach
--------------------------------------
Will [@Will08] refers $\kap$ to the orbital plane of a generic star by choosing $\bds{e}_p,\bds{e}_q,\bds{h}$ as orthonormal vectors: $\bds{e}_p$ is directed along the line of the nodes, $\bds{e}_q$ lies in the orbital plane perpendicularly to $\bds{e}_p$, and $\bds h$ is directed along the orbital angular momentum. Thus, one has $$\begin{array}{lll}
\kx &=& \kp\cO+\left(\kh\si-\kq\ci\right)\sO, \\ \\
%
\ky &=& \kp\sO-\left(\kh\si-\kq\ci\right)\cO, \\ \\
%
\kz &=& \kh\ci+\kq\si, \\ \\
%
\end{array}\lb{kpkqkh}$$
Inserting into the equations of Sec. \[tre\] and Sec. \[quattro\] allows us to obtain Eq. (2a), Eq. (2b), and Eq. (2c) of Ref. [@Will08] after some algebra.
Stellar orbital perturbations caused by ultralow frequency gravitational waves
------------------------------------------------------------------------------
The stars orbiting the SBH in Sgr A$^{\ast}$ could also be used, in principle, as probes for detecting or constraining plane gravitational waves of ultralow frequency ($\nu\approx 10^{-8}-10^{-10}$ Hz or less) impinging on the system from the outside. Indeed, the passage of such waves through the orbits of the closest stars would cause long-term variations of all their Keplerian orbital elements, apart from the semimajor axis $a$. They have recently been worked out in Ref. [@Iorio011b] for general orbital configurations, i.e., without making a-priori assumptions on their inclinations and eccentricities of the perturbed test particle, and arbitrary directions of incidence for the wave. Conversely, gravitational waves can be generated within the stellar system of Sgr A$^{\ast}$, as discussed in Ref. [@Freitag].
Numerical evaluations
=====================
In Table \[tavola\] we quote the relevant physical and orbital parameters for the SBH-S2 system. The orbital period of S2 is $P_{\rm b}=15.98$ yr, so that the astrometric measurements currently available cover a full revolution of it.
------------------------ ------------------------- ------------------------ ---------- ---------------------- ---------- ----------- -------------
$\mu$ (m$^3$ s$^{-2}$) $S$ (kg m$^2$ s$^{-1}$) $Q_2$ (m$^5$ s$^{-2}$) $\chi_g$ $a$ (m) $e$ $i$ (deg) $\Om$ (deg)
$5.70\times 10^{26}$ $8.46\times 10^{54}$ $-6.22\times 10^{45}$ $0.52$ $1.54\times 10^{14}$ $0.8831$ $134.87$ $226.53$
$6.6\times 10^{25}$ $4.66\times 10^{54}$ $6.58\times 10^{45}$ $ 0.26$ $8\times 10^{12}$ $0.0034$ $0.78$ $0.72$
------------------------ ------------------------- ------------------------ ---------- ---------------------- ---------- ----------- -------------
The quadrupole-induced precessions of - are all linear combinations of the products of the components of $\kap$ plus, sometimes, a term independent of $\kap$: they can be cast into the form
= D\_0(Q\_2,a,e,i,)+\_[s,l]{} D\_[sl]{}(Q\_2,a,e,i,) \_s\_l, s,l=x,y,z, =i,,,.
The numerical values of the coefficients $D_0$ and $D_{sl}=D_{ls}$ for S2, in $\mu$as yr$^{-1}$, are quoted in Table \[tavolaQ2\].
--------------- --------- ----------- ----------- ----------- ---------- ---------- ----------
$D_0$ $D_{x^2}$ $D_{y^2}$ $D_{z^2}$ $D_{xy}$ $D_{xz}$ $D_{yz}$
$i$ $0$ $406$ $-406$ $0$ $43$ $558$ $588$
$\Omega$ $0$ $427$ $384$ $-810$ $-809$ $-5$ $5$
$\omega$ $-1149$ $1568$ $1584$ $294$ $293$ $-1254$ $1189$
$\mathcal{M}$ $-539$ $595$ $616$ $406$ $405$ $-587$ $556$
--------------- --------- ----------- ----------- ----------- ---------- ---------- ----------
: \[tavolaQ2\]Coefficients of the quadrupole precessions of S2, in $\mu$as yr$^{-1}$, according to Table . GTR was assumed for $Q_2$, with $\chi_g=0.52$.
The largest effects occur for $\omega$ and $\mathcal{M}$ because of $D_0$, which is of the order of $\approx 1$ milliarcsec yr$^{-1}$ (mas yr$^{-1}$). The other terms are damped by the square of the components of $\kap$. Moreover, partial mutual may occur depending on the orientation of the SBH spin axis.
The Lense-Thirring precessions of are all linear combinations of the components of $\kap$: they can be cast into the form = \_j C\_j(S,a,e,i,) \_j, j=x,y,z, =i,,.The numerical values of the coefficients $C_j$ for S2, in arcsec yr$^{-1}$, are listed in Table \[tavolaLT\].
---------- --------- --------- --------
$C_x$ $C_y$ $C_z$
$i$ $-0.14$ $-0.15$ $0$
$\Omega$ $-0.15$ $0.14$ $0.21$
$\omega$ $0.11$ $-0.10$ $0.45$
---------- --------- --------- --------
: \[tavolaLT\]Coefficients of the Lense-Thirring precessions of S2, in arcsec yr$^{-1}$, according to Table . In particular, $\chi_g=0.52$ was used for the spin of the SBH.
They are of the order of about $10^{-1}$ arcsec yr$^{-1}$, i.e., orders of magnitude larger than the quadrupole precessions of Table \[tavolaQ2\]: also in this case, partial mutual may occur depending on $\kap$, thus impacting the detectability of the gravitomagnetic rates.
The figures of Table \[tavolaQ2\] and Table \[tavolaLT\] can be compared with the present-day accuracies in empirically determining the orbital precessions of S2 listed in Table \[tavola\_errori\].
-------------------- ----------------------- ----------------------- ------------------------------
$\sigma_{\dot{i}}$ $\sigma_{\dot\Omega}$ $\sigma_{\dot\omega}$ $\sigma_{\dot{\mathcal{M}}}$
$176$ $163$ $182$ $1203$
-------------------- ----------------------- ----------------------- ------------------------------
: \[tavola\_errori\]Naive evaluations of the uncertainties in the secular variations of the S2 osculating Keplerian orbital elements, in arcsec yr$^{-1}$, obtained by dividing the errors in the elements from Table 1 of Ref. by a time interval $\Delta T\approx P_{\rm b}$. Concerning the mean anomaly, its uncertainty was evaluated from that of the time of periastron passage $t_{\rm p}$, released in Ref. , according to the expression for the mean anomaly at the epoch of periastron passage $\mathcal{\mathcal{M}}_0=-n t_{\rm p}$; also the errors coming from $a$ and $\mu$ through $n$ were taken into account.
They are of the order of $10^2-10^3$ arcsec yr$^{-1}$. The Lense-Thirring precessions of S2 (Table \[tavolaLT\]) are about three orders of magnitude smaller than the current accuracy, while the quadrupole effects of Table \[tavolaQ2\] are negligibly small.
By considering a fictitious star $X$ with, say, the same orbital parameters of S2 apart from the semimajor axis $a$, assumed to be one order of magnitude smaller so that its orbital period would just be $P_{\rm b}=0.5$ yr, it turns out that its 1PN GTR periastron precession would be as large as 4 deg yr$^{-1}$, while its Lense-Thirring and quadrupole precessions would be of the order of about $\approx 10^2$ arcsec yr$^{-1}$ and $\approx 1$ arcsec yr$^{-1}$, respectively. If, as expected, angular shifts of $\Delta\xi\approx 10$ $\mu$as, as seen from the Earth, will really become measurable in future thanks to GRAVITY and ASTRA, this would imply an accuracy of the order of $\Delta\Ps\approx \left(d/a\right)\Delta\xi=16$ arcsec for S2, and 160 arsec for a star one order of magnitude closer to the SBH. If such targets will be discovered, their Lense-Thirring shifts should become detectable after some years, while the $Q_2-$induced perturbations would still remain hard to measure, for $e\approx 0.9$.
Summary and conclusions
=======================
We analytically worked out the long-term, i.e., averaged over one full revolution, variations of all the six osculating Keplerian orbital elements of a test particle orbiting a nonspherical, spinning body endowed with angular momentum $\bds S$ and quadrupole moment $Q_2$ for a generic spatial orientation of its spin axis $\kap$. We did not restrict ourselves to any specific orbital configuration . Here we applied our results to the stars orbiting the SBH in Sgr A$^{\ast}$: those identified so far are moving along highly elliptical trajectories with periods $P_{\rm b}\geq 16$ yr. The current level of accuracy in empirically determining the precessions of the angular orbital elements of S2, having $P_{\rm b}=16$ yr, can be evaluated to be of the order of $\approx 10^2-10^3$ arcsec yr$^{-1}$. The predicted 1PN GTR periastron precession of S2, which is independent of the orientation of the spin axis of the SBH, is $40\pm 10$ arcsec yr$^{-1}$. The predicted GTR spin and quadrupole-induced precessions of S2 are of the order of $\approx 10^{-1}$ arcsec yr$^{-1}$ and $\approx 10^2-10^3\mu$as yr$^{-1}$, respectively: they depend on $\kap$, and partial among their components may occur, thus reducing their magnitude. Concerning hypothetical stars with orbital periods of less than 1 yr, not yet discovered, the 1PN GTR periastron precessions would be as large as some deg yr$^{-1}$, while the $S$ and $Q_2$ effects would be of the order of $\approx 10^2$ arcsec yr$^{-1}$ and $\approx 1$ arcsec yr$^{-1}$, respectively. Planned improvements of the infrared telescopes used so far aim to reach an accuracy level of $\approx 10$ $\mu$as at best in measuring angular shifts as seen from the Earth corresponding to stellar orbital shifts of about $1.6\times 10^1-10^2$ arcsec for S2 and stars closer than it by one order of magnitude, respectively.
Acknowledgements {#acknowledgements .unnumbered}
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I thank S. Gillessen for useful correspondence.
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|
---
abstract: 'Domain wall, wormhole, particlelike, and cosmic string general relativistic solutions supported by two interacting phantom or ordinary scalar fields with 4th-, 6th-, and 8th-order potentials are studied. Numerical calculations indicate that regular finite energy solutions exist only for specific values of two free parameters of the potentials. By solving nonlinear eigenvalue problems for some fixed sets of values of the free parameters and of boundary conditions, it is shown that the presence or absence of the solutions depends on a particular symmetry of the problem, on the type of the scalar fields (ordinary or phantom), and on the form of the potential.'
author:
- Vladimir Dzhunushaliev
- Vladimir Folomeev
- Arislan Makhmudov
- Ainur Urazalina
title: 'Static general relativistic solutions supported by phantom and ordinary scalar fields with higher-order potentials'
---
Introduction
============
In recent years interest in obtaining solutions with various scalar fields has grown considerably, primarily because of the discovery of the accelerated expansion of the present Universe. It is now widely believed that such acceleration is caused by the presence of a special form of matter – dark energy, whose key feature is that it violates various energy conditions.
In the most extreme case, the violation of the so-called null energy condition can occur. In hydrodynamical language, this corresponds to the fact that the effective pressure of matter filling the Universe, $p$, is negative, and its modulus is greater than the energy density $\varepsilon$, i.e. $p<-\varepsilon$. Such a substance is referred to as exotic matter. As a model of exotic matter, one can consider phantom (or ghost) scalar fields, i.e., fields with the opposite sign in front of the kinetic term of the scalar field Lagrangian density. Such fields are widely used both in describing the current accelerated expansion of the Universe [@AmenTsu2010] and in modelling various localized objects (see below). The possible existence of phantom scalar fields in nature is indirectly supported by the observed accelerated expansion of the present Universe (see, e.g., Refs. [@Sullivan:2011kv; @Ade:2015xua], from which one may conclude that to explain the recent observational data one should take exotic matter into consideration).
In the present paper we consider regular solutions to Einstein’s gravitational equations supported by two ordinary or phantom scalar fields. In Ref. [@Dzhunushaliev:2016], we have obtained plane symmetric (domain walls), spherically symmetric (wormholes and phantom balls), and cylindrically symmetric (cosmic strings) solutions supported by two interacting phantom scalar fields with a 4th-order potential. Here we extend those results and study the possibility of obtaining such solutions with 6th- and 8th-order potential terms. Also, we compare the obtained results with those found earlier for the 4th-order potential of Ref. [@Dzhunushaliev:2016].
The aforementioned configurations are well known in the bulk of literature. Cosmic strings are extended objects that could be formed in the early Universe under phase transitions associated with spontaneous symmetry breaking [@Vilen; @Bran2]. In their modelling various types of scalar fields are employed [@Baze; @Sant], including two interacting scalar fields [@BezerradeMello:2003ei; @Dzhunushaliev:2007cs].
Another category of extended objects are plane symmetric domain walls, which are topological defects that arise in both particle physics and cosmology [@Bran2; @Cvet]. In particular, domain wall solutions may exist in theories where a scalar field potential has isolated minima, and a domain wall is a surface that separates those minima [@thin_dom]. In such a case a scalar field changes over space and tends asymptotically to two different minima. The region where the scalar field changes rapidly corresponds to the domain wall. In the thin-wall approximation, the change in the scalar field energy density is localized on the surface of the domain wall, and it is replaced by a delta function [@thick_dom]. In the case where all fields are constant on each side of the wall, i.e., when they are at the potential minimum, the domain walls are called vacuum domain walls.
Finally, one can consider a situation where scalar fields are localized on relatively small scales comparable to the sizes of stars. In this case, they may create spherically symmetric configurations possessing both trivial and nontrivial spacetime topologies. As an example of systems with a trivial topology, one can consider boson stars consisting of various ordinary scalar fields [@Schunck:2003kk; @Liebling:2012fv]. In turn, the use of phantom scalar fields permits obtaining solutions of the Einstein-matter equations describing configurations both with a trivial [@Dzhunushaliev:2008bq] and a nontrivial wormholelike topology (for a recent review on the subject, see, e.g., Ref. [@Bronnikov:2018vbs]), including configurations supported by complex ghost scalar fields [@Dzhunushaliev:2017syc].
In this paper, we consider all four types of configurations (domain walls, particlelike systems, wormholes, and cosmic strings) constructed from two interacting phantom or ordinary scalar fields with higher-order potentials. Systems with two ordinary scalar fields are well known from quantum field theory [@rajaraman]. In the presence of a gravitational field, such systems were also repeatedly considered in the cosmological and astrophysical contexts [@2_fields_syst]. In our previous papers we have obtained a number of solutions with two scalar fields (both ordinary and phantom ones) which can be used both in describing astrophysical objects and when considering cosmological problems: regular spherically and cylindrically symmetric solutions [@2_fields_our; @Dzhunushaliev:2007cs; @Dzhunushaliev:2015sla]; cosmological solutions [@Dzhunushaliev:2006xh; @Folomeev:2007uw]; thick brane solutions supported by ordinary and phantom scalar fields [@2_fields_brane]. In the present paper we proceed with research in this direction by considering scalar fields with different potentials.
General equations {#GE}
=================
We consider compact gravitating configurations consisting of two real scalar fields $\phi$ and $\chi$. The modeling is carried out within the framework of Einstein’s general relativity. The corresponding Lagrangian of the system is (hereafter, we work in units where $c=\hbar=1$) $$\label{lagrangian}
L=-\frac{R}{16\pi G}+\epsilon\left[
\frac{1}{2}\partial_\mu \phi \partial^\mu
\phi + \frac{1}{2}\partial_\mu \chi \partial^\mu \chi - V(\phi,\chi)
\right],$$ where $R$ is the scalar curvature, $G$ – the Newtonian gravitational constant, $\mu, \nu=0,1,2,3$, and $\epsilon=+1$ or $-1$ corresponds to ordinary or phantom fields, respectively. Using this Lagrangian, the gravitational and scalar field equations can be written in the form: $$\begin{aligned}
\label{EinstEQ}
R^k _i-\frac{1}{2}\delta^k_i R &=& \kappa T_i^k ,
\\
\frac{1}{\sqrt{-g}}\frac{\partial}{\partial x^i} \left[
\sqrt{-g} g^{i k} \frac{\partial (\phi,\chi)}{\partial x^k}
\right] &=& - \frac{\partial V}{\partial (\phi,\chi)} ~,
\label{scalar_eqs}\end{aligned}$$ where $\kappa = 8\pi G$. In the present paper we assume that the interacting scalar fields have a potential in one of the forms: $$\begin{aligned}
\label{pot1}
V(\phi,\chi) &=& \frac{\lambda_1}{4}(\phi^2 - m_1^2)^2 + \frac{\lambda_2}{4}(\chi^2 - m_2^2)^2 + \phi^2
\chi^2 - V_0,
\\
\label{pot2}
V(\phi,\chi) &=& \frac{\lambda_1}{2}\phi^2(\phi^2 - m_1^2)^2 + \frac{\lambda_2}{2}\chi^4(\chi^2 - m_2^2)^2+\frac{1}{2}\phi^2
\chi^2 - V_0,
\\
\label{pot3}
V(\phi,\chi) &=& \frac{\lambda_1}{4}\phi^4(\phi^2 - m_1^2)^2 + \frac{\lambda_2}{2}\chi^4(\chi^2 - m_2^2)^2 + \frac{1}{2}\phi^2
\chi^2 - V_0.\end{aligned}$$ Here, $m_1$ and $m_2$ are some free parameters, $\lambda_1 $ and $\lambda_2 $ – self-interaction constants, and $V_0$ – a constant whose value can be chosen from the statement of the problem.
The corresponding energy-momentum tensor entering the right-hand side of Eq. is $$\label{emt}
T_{\mu}^\nu = \epsilon \left\{
\partial_\mu \phi \partial^\nu \phi+
\partial_\mu \chi \partial^\nu \chi-
\delta_{\mu}^\nu \left[
\frac{1}{2}\partial_\rho \phi \partial^\rho
\phi + \frac{1}{2}\partial_\rho \chi \partial^\rho \chi - V(\phi,\chi)
\right]
\right\}.$$
Domain walls {#DW}
=============
![Phantom domain wall: profiles of the phantom ($\epsilon=-1$) scalar fields $\phi(x), \chi(x)$, the metric function $a^\prime(x)/a(x)$, and the energy density $T^0_0(x)$ are shown. The curve 1 corresponds to the 4th-order potential , the curve 2 – to the 6th-order potential , the curve 3 – to the 8th-order potential . The labeling of the curves is also valid for all other figures presented below.[]{data-label="fig_dom_wall_phantom"}](fig_dom_wall_ph){width="1\linewidth"}
In considering plane symmetric domain walls solutions, we choose the metric in the form: $$\label{metric_wall}
ds^2=a^2(x) (dt^2-dy^2-dz^2)-dx^2,$$ where $x,y,z$ are Cartesian coordinates. Then Eqs. - yield $$\begin{aligned}
\label{ein_wall_1}
3\left(\frac{a^\prime}{a}\right)^2 &=& -\epsilon\left[-\frac{1}{2}\left(
\phi^{\prime 2} + \chi^{\prime 2}\right) + V\right] ~,
\\
\frac{a^{\prime \prime}}{a} - \left(\frac{a^\prime}{a}\right)^2 &=& -\frac{\epsilon}{2}\left(\phi^{\prime 2} + \chi^{\prime 2}\right) ~,
\label{ein_wall_2} \\
\label{field_wall_1}
\phi^{\prime \prime} + 3 \frac{a^\prime}{a}\phi^\prime &=& \phi\left[2\chi^2 + \lambda_1(\phi^2 - m_1^2)\right] ~,
\\
\chi^{\prime \prime} + 3 \frac{a^\prime}{a}\chi^\prime &=&
\chi\left[2\phi^2 + \lambda_2(\chi^2-m_2^2)\right] ~,
\label{field_wall_2}\end{aligned}$$ where Eqs. and are the $\left(_1^1\right)$ and $\left[\left(_0^0\right)-\left(_1^1\right)\right]$ components of the Einstein equations, respectively, and the prime denotes differentiation with respect to $x$. The results of numerical calculations for the phantom ($\epsilon=-1$) and ordinary $(\epsilon=+1)$ scalar fields are shown in Figs. \[fig\_dom\_wall\_phantom\] and \[fig\_dom\_wall\_ord\], respectively. The corresponding eigenvalues of the parameters $m_{1,2}$ for the potentials - and $\epsilon = \pm 1$ are given in Table \[eignvlsDW\].
![Ordinary domain wall: profiles of the ordinary ($\epsilon=+1$) scalar fields $\phi(x), \chi(x)$, the metric function $a^\prime(x)/a(x)$, and the energy density $T^0_0(x)$ are shown. []{data-label="fig_dom_wall_ord"}](fig_dom_wall_ord){width="1\linewidth"}
\# Potentials $\epsilon$ $m_1$ $m_2$
---- ------------ ------------ --------------- ----------------
1 4th-order -1 1.77426601 1.80400455
2 6th-order -1 1.30901092 1.73766048
3 8th-order -1 1.4251234264 1.7965336329
4 4th-order +1 2.05880064139 1.720175382122
5 6th-order +1 1.42405708294 1.61615084819
6 8th-order +1 no solution no solution
: Eigenvalues of the parameters $m_1, m_2$ for the phantom/ordinary domain wall solutions with the 4th-, 6th-, and 8th-order potentials -. The boundary conditions at the center $x=0$ are $\phi_0=1, \chi_0=0.7, a_0=1, \phi^\prime_0 =\chi^\prime_0= a^\prime_0=0$. The values of the free parameters $\lambda_1=0.15, \lambda_2=1.1$.[]{data-label="eignvlsDW"}
Phantom balls
=============
![ Phantom ball: profiles of the scalar fields $\phi(r), \chi(r)$, the metric functions $A(r), B(r)$, and the energy density $T^0_0(r)$ are shown. []{data-label="fig_phant_ball"}](fig_phant_ball){width="1\linewidth"}
Let us now consider particlelike solutions supported by phantom fields. For this case, we choose the spherically symmetric line element in Schwarzschild coordinates $$\label{metric_sph}
ds^2 = B(r)dt^2 - A(r)dr^2 - r^2(d\theta^2 + \sin^2\theta d\varphi^2),$$ where $r,\theta,\varphi$ are spherical coordinates. The Einstein and scalar field equations and together with Eq. give for the phantom case $$\begin{aligned}
\label{einst1_sph}
\frac{1}{r}\frac{A^\prime}{A^2} + \frac{1}{r^2}\left(1 -
\frac{1}{A}\right) &=&
- \frac{1}{2A}\left(\phi^{\prime 2} + \chi^{\prime 2}\right) -
V(\phi,\chi),
\\
\frac{1}{r}\frac{B^\prime}{A B} -
\frac{1}{r^2}\left(1 - \frac{1}{A}\right) &=&
- \frac{1}{2A}\left(\phi^{\prime 2} +\
\chi^{\prime 2}\right)+V(\phi,\chi),
\label{einst2_sph} \\
\frac{B^{\prime \prime}}{B} -
\frac{1}{2}\left(\frac{B^\prime}{B}\right)^2 - \frac{1}{2}\frac{A^\prime}{A}\frac{B^\prime}{B} - \frac{1}{r}\left(\frac{A^\prime}{A} -
\frac{B^\prime}{B}\right) &=& 2A\left[\frac{1}{2A}\left(\phi^{\prime 2} + \chi^{\prime 2}\right) + V(\phi,\chi)\right],
\label{einst3_sph}\end{aligned}$$ $$\begin{aligned}
\label{sfe1_sph}
\phi^{\prime \prime} + \left(\frac{2}{r} + \frac{B^\prime}{2B} - \frac{A^\prime}{2A}\right)\phi^\prime &=&
A\phi\left[2\chi^2+\lambda_1(\phi^2-m_1^2)\right]~,
\\
\chi^{\prime \prime} + \left(\frac{2}{r} + \frac{B^\prime}{2B} - \frac{A^\prime}{2A}\right)\chi^\prime &=&
A\chi\left[2\phi^2 + \lambda_2(\chi^2-m_2^2)\right]~,
\label{sfe2_sph}\end{aligned}$$ where the prime denotes differentiation with respect to $r$. These equations describe spherically symmetric objects that can be called phantom balls [@Dzhunushaliev:2016]. The results of numerical calculations are shown in Fig. \[fig\_phant\_ball\]. Notice that the solutions for the 6th- and 8th-order potentials are practically coincide. Table \[eignvls\_bs\] shows the eigenvalues of the parameters $m_{1,2}$ for the potentials -.
\# Potentials $m_1$ $m_2$
---- ------------ ------------ ------------
1 4th-order 1.54248223 1.89958804
2 6th-order 1.04506272 4.1962616
3 8th-order 1.050035 4.2023521
: Eigenvalues of the parameters $m_1, m_2$ for the phantom ball solutions with the 4th-, 6th-, and 8th-order potentials -. The boundary conditions at the center $r=0$ are $\phi_0=1, \chi_0=0.7, A_0=1, B_0=1, \phi^\prime_0 =\chi^\prime_0= B^\prime_0=0$. The values of the free parameters $\lambda_1=0.15, \lambda_2=1.1$. []{data-label="eignvls_bs"}
In the case of ordinary ($\epsilon=+1$) scalar fields and for the values of the parameters $\phi_0=1, \chi_0=0.7, \lambda_1=0.15, \lambda_2=1.1$, we did not find solutions with the potentials -.
Wormhole solutions
==================
Here, it is convenient to choose the metric in polar Gaussian coordinates $$\label{metric}
ds^2 = B(r) dt^2-dr^2-A(r)(d\theta^2+\sin^2\theta d\varphi^2),$$ where $r,\theta,\varphi$ are spherical coordinates. Then one derives the following set of Einstein’s and scalar field equations describing a traversable wormhole supported by the phantom fields $\phi, \chi$: $$\begin{aligned}
\label{Einstein_a}
\frac{A^{\prime \prime}}{A} -
\frac{1}{2}\left(\frac{A^{\prime}}{A}\right)^2 -
\frac{1}{2}\frac{A^{\prime}}{A}\frac{B^{\prime}}{B} &=&
\phi^{\prime 2} + \chi^{\prime 2}~,
\\
\label{Einstein_b}
\frac{A^{\prime \prime}}{A} + \frac{1}{2}\frac{A^{\prime}}{A}\frac{B^{\prime}}{B} - \frac{1}{2}\left(\frac{A^{\prime}}{A}\right)^2 -
\frac{1}{2}\left(\frac{B^{\prime}}{B}\right)^2 +
\frac{B^{\prime \prime}}{B} &=& 2\left[\frac{1}{2}(\phi^{\prime 2} + \chi^{\prime 2}) + V\right] ,
\\
\label{Einstein_c}
\frac{1}{4}\left(\frac{A^{\prime}}{A}\right)^2 - \frac{1}{A} + \frac{1}{2}\frac{A^{\prime}}{A}\frac{B^{\prime}}{B} &=&
- \frac{1}{2}(\phi^{\prime 2} + \chi^{\prime 2}) + V ,
\\
\label{field_a}
\phi^{\prime \prime} + \left(\frac{A^\prime}{A} + \frac{1}{2}\frac{B^\prime}{B}\right)\phi^\prime &=&
\phi \left[2\chi^2 + \lambda_1(\phi^2 - m_1^2)\right] ,
\\
\label{field_b}
\chi^{\prime \prime} + \left(\frac{A^\prime}{A} + \frac{1}{2}\frac{B^\prime}{B}\right)\chi^\prime &=&
\chi \left[2\phi^2 + \lambda_2(\chi^2 - m_2^2)\right],\end{aligned}$$ where the prime denotes differentiation with respect to $r$. The results of numerical calculations are shown in Fig. \[fig\_wormhole\] for the eigenvalues of the parameters $m_{1,2}$ given in Table \[eignvls\_WH\].
![Traversable wormhole: profiles of the phantom scalar fields $\phi(r), \chi(r)$, the metric functions $A(r), B(r)$, and the energy density $T^0_0(r)$ are shown. []{data-label="fig_wormhole"}](fig_wormhole){width="1\linewidth"}
\# Potentials $m_1$ $m_2$
---- ------------ ------------ --------------
1 4th-order 1.82729811 1.7869422825
2 6th-order 1.32067169 1.7205753
3 8th-order 1.45731329 1.7806672
: Eigenvalues of the parameters $m_1, m_2$ for the wormhole solutions with the 4th-, 6th-, and 8th-order potentials -. The boundary conditions at the throat $r=0$ are $\phi_0=1, \chi_0=0.7, A_0=-1/V\left(\phi_0,\chi_0\right), B_0=1, \phi^\prime_0 =\chi^\prime_0= A^\prime_0= B^\prime_0=0$. The values of the free parameters $\lambda_1=0.15, \lambda_2=1.1$. []{data-label="eignvls_WH"}
Cosmic strings {#DW}
==============
![Phantom cosmic string: profiles of the scalar fields $\phi(\rho), \chi(\rho)$, the metric functions $\gamma(\rho), \psi(\rho)$, and the energy density $T^0_0(\rho)$ are shown. []{data-label="fig_string"}](fig_string){width="1\linewidth"}
In describing such cylindrically symmetric objects, we use the metric $$\label{metric_string}
ds^2 = e^{2 \nu(\rho)} dt^2 - e^{2 (\gamma(\rho) - \psi(\rho))} d\rho^2 -
e^{2 \psi(\rho)} dz^2 - \rho^2 e^{-2 \psi(\rho)} d\varphi^2 ,$$ where $\rho,z,\varphi$ are cylindrical coordinates. In this case, one has the following Einstein and phantom scalar field equations: $$\begin{aligned}
\frac{\gamma^\prime}{\rho} - {\psi^\prime}^2 &=&
- \kappa \left(
\frac{1}{2} {\phi^\prime}^2 + \frac{1}{2} {\chi^\prime}^2 +
e^{2 (\gamma - \psi)} V(\phi, \chi)
\right) ,
\label{ef_1} \\
\frac{\nu^\prime + \psi^\prime}{\rho} - {\psi^\prime}^2 &=&
- \kappa \left(
\frac{1}{2} {\phi^\prime}^2 + \frac{1}{2} {\chi^\prime}^2 -
e^{2 (\gamma - \psi)} V(\phi, \chi)
\right) ,
\label{ef_2} \\
\psi^{\prime \prime } - \nu^{\prime \prime} - \psi^\prime \gamma^\prime +
\nu^\prime \gamma^\prime - {\nu^\prime}^2 +
\frac{\psi^\prime + \gamma^\prime - \nu^\prime}{\rho} &=&
\kappa \left(
- \frac{1}{2} {\phi^\prime}^2 - \frac{1}{2} {\chi^\prime}^2 -
e^{2 (\gamma - \psi)} V(\phi, \chi)
\right) ,
\label{ef_3} \\
- \psi^{\prime \prime } - \nu^{\prime \prime} +
\psi^\prime \gamma^\prime +
\nu^\prime \gamma^\prime -
2 {\psi^\prime}^2 - 2\psi^\prime \nu^\prime -
{\nu^\prime}^2 &=& \kappa \left(
- \frac{1}{2} {\phi^\prime}^2 - \frac{1}{2} {\chi^\prime}^2 -
e^{2 (\gamma - \psi)} V(\phi, \chi)
\right) ,
\label{ef_4}\end{aligned}$$ $$\begin{aligned}
\phi^{\prime \prime} + \phi^\prime \left(
\frac{1}{\rho} - \gamma^\prime + \psi^\prime + \nu^\prime
\right) &=& e^{2 (\gamma - \psi)} \phi \left[
2 \chi^2 + \lambda_1 \left( \phi^2 - m_1^2 \right)
\right] ,
\label{ef_5} \\
\chi^{\prime \prime} + \chi^\prime \left(
\frac{1}{\rho} - \gamma^\prime + \psi^\prime + \nu^\prime
\right) &=& e^{2 (\gamma - \psi)} \chi \left[
2 \phi^2 + \lambda_2 \left( \chi^2 - m_2^2 \right)
\right] .
\label{ef_6}\end{aligned}$$
To simplify them, let us make an additional assumption that two of the metric functions are equal, i.e., $\nu = \psi$. After some algebraic manipulations and performing the rescaling $\rho/\sqrt \kappa \rightarrow \rho$, $\phi \sqrt \kappa \rightarrow \phi$, $\chi \sqrt \kappa \rightarrow \chi$, and $m_{1,2} \sqrt \kappa \rightarrow m_{1,2} $, we get the following equations for the metric functions $\gamma(\rho), \psi(\rho)$ and the phantom $(\epsilon=-1)$ scalar fields $\phi(\rho), \chi(\rho)$: $$\begin{aligned}
\frac{\gamma^\prime}{\rho} - {\psi^\prime}^2 &=&
- \left(
\frac{1}{2} {\phi^\prime}^2 + \frac{1}{2} {\chi^\prime}^2 +
e^{2 (\gamma - \psi)} V(\phi, \chi)
\right) ,
\label{cosmic_string_1} \\
2 \frac{\psi^\prime}{\rho} - {\psi^\prime}^2 &=&
- \left(
\frac{1}{2} {\phi^\prime}^2 + \frac{1}{2} {\chi^\prime}^2 -
e^{2 (\gamma - \psi)} V(\phi, \chi)
\right) ,
\label{cosmic_string_1a} \\
\psi^{\prime \prime } + \frac{\psi^\prime}{\rho} &=&
e^{2 (\gamma - \psi)} \left(
1 - 2 \rho \psi^\prime
\right) V(\phi, \chi) ,
\label{cosmic_string_2} \\
\phi^{\prime \prime} + \phi^\prime \left(
\frac{1}{\rho} - \gamma^\prime + 2 \psi^\prime
\right) &=& e^{2 (\gamma - \psi)} \phi \left[
2 \chi^2 + \lambda_1 \left( \phi^2 - m_1^2 \right)
\right] ,
\label{cosmic_stringl_3} \\
\chi^{\prime \prime} + \chi^\prime \left(
\frac{1}{\rho} - \gamma^\prime + 2 \psi^\prime
\right) &=& e^{2 (\gamma - \psi)} \chi \left[
2 \phi^2 + \lambda_2 \left( \chi^2 - m_2^2 \right)
\right] ,
\label{cosmic_string_4}\end{aligned}$$ where the prime denotes differentiation with respect to the rescaled radial coordinate $\rho$.
The results of numerical calculations are shown in Fig. \[fig\_string\] for the eigenvalues of the parameters $m_{1,2}$ given in Table \[eignvlsCS\]. It is seen that the profiles for $\chi$ and for the energy densities of the 6th- and 8th-order potential cases are practically coincide.
\# Potentials $m_1$ $m_2$
---- ------------ -------------- ---------------
1 4th-order no solution no solution
2 6th-order 1.154579476 2.30250731
3 8th-order 1.1926167892 2.32316842475
: Eigenvalues of the parameters $m_1, m_2$ for the phantom cosmic string solutions with the 6th- and 8th-order potentials and . The boundary conditions at $\rho=0$ are $\phi_0=1, \chi_0=0.7, \psi_0=\gamma_0= \phi^\prime_0 =\chi^\prime_0= \psi^\prime_0=0$. The values of the free parameters $\lambda_1=0.15, \lambda_2=1.1$. []{data-label="eignvlsCS"}
In the case of ordinary ($\epsilon=+1$) scalar fields and for the values of the parameters $\phi_0=1, \chi_0=0.7, \lambda_1=0.15, \lambda_2=1.1$, we did not find solutions with the potentials -.
Summarizing the results, we have obtained plane, cylindrically, and spherically (particlelike and wormhole) symmetric static general relativistic solutions supported by two interacting phantom/ordinary scalar fields with 4th-, 6th-, and 8th-order potentials of the form -. All the solutions have been obtained numerically for the fixed central values of the scalar fields $\phi_0=1, \chi_0=0.7$ and for the free parameters $\lambda_1=0.15, \lambda_2=1.1$. In doing so, we have solved nonlinear eigenvalue problems for the parameters $m_1, m_2$. It was shown that solutions may exist (or not exist) depending on a particular symmetry of the problem, on the type of the scalar fields (ordinary or phantom), and on the form of the potential.
Acknowledgments {#acknowledgments .unnumbered}
===============
V.D. and V.F. gratefully acknowledge support provided by Grant No. BR05236494 in Fundamental Research in Natural Sciences by the Ministry of Education and Science of the Republic of Kazakhstan. We are also grateful to the Research Group Linkage Programme of the Alexander von Humboldt Foundation for the support of this research.
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|
---
abstract: 'The emergent phenomenon of collective neutrino oscillations arises from neutrino-neutrino interactions in environments with very large number of neutrinos. Since such environments are likely sites of the heavy-element synthesis, understanding all aspects of collective neutrino oscillations seems to be necessary for a complete accounting of nucleosynthesis. I briefly summarize some of the salient features along with recent work on the properties and astrophysical applications of the collective neutrino oscillations.'
author:
- 'A.B. Balantekin'
bibliography:
- 'sample.bib'
nocite: '[@*]'
title: Collective Neutrino Oscillations and Nucleosynthesis
---
1.5cm
INTRODUCTION
============
All astrophysical environments where synthesis of heavier elements is supposed to occur, may they be core-collapse supernovae, merging neutron stars, or gamma-ray burst emitting objects, contain numerous neutrinos. For example a proto-neutron star, formed following the emission of a core-collapse supernova, cools down by emitting neutrinos with the total energy of $\sim 10^{59}$ MeV. If the average energy of each neutrino is $\sim 10$ MeV, this corresponds to $10^{58}$ neutrinos streaming out of the core in a short time. In such environments a proper description of the neutrino transport must include neutrino-neutrino interactions, a Standard Model process usually omitted in all other applications of weak interactions. When the neutrino transport is dominated by the coherent forward scattering of neutrinos, the effect is proportional to $G_F$ (as opposed to $G_F^2$ for collisions when it can be safely ignored). In the case of coherent forward scattering the Hamiltonian which describes neutrino transport is given by $$\label{total}
\hat{H}
= \left(
\sum_p\frac{\delta m^2}{2p}\mathbf{B}\cdot\mathbf{J}_p - \sqrt{2} G_F
N_e J_p^0 \right)
+ \frac{\sqrt{2}G_{F}}{V}\sum_{\mathbf{p} \neq\mathbf{q}}\left(1-
\cos\vartheta_{\mathbf{p}\mathbf{q}}\right)\mathbf{J}_{\mathbf{p}}\cdot\mathbf{J}_{\mathbf{q}}$$ where an auxiliary vector quantity $$\mathbf{B} = (\sin2\theta,0,-\cos2\theta)$$ is parameterized in terms of the neutrino mixing angle $\theta$. In writing these equations, for simplicity we ignored antineutrinos and assumed only two flavors. It is straightforward to relax this assumption [@Balantekin:2006tg]. Eq. (\[total\]) is written using the neutrino flavor isospin algebras: $$\begin{aligned}
J^+_{{\bf p}} &=& a_x^\dagger({\bf p}) a_e({\bf p}), \> \> \>
J^{\>-}_{{\bf p}}=a_e^\dagger({\bf p}) a_x({\bf p}), \nonumber \\
J^0_{{\bf p}} &=& \frac{1}{2}\left(a_x^\dagger({\bf p})a_x({\bf p})-a_e^\dagger({\bf p})a_e({\bf p})
\right). \label{su2}\end{aligned}$$ where neutrino creation and annihilation operators, $a_e, a_e^\dagger, a_x, a_x^\dagger$ are introduced. These operators span SU(2) algebras labeled by each neutrino momenta. In Eq. (\[total\]) $\cos\vartheta_{\mathbf{p}\mathbf{q}}$ is the angle between neutrino momenta $\mathbf{p}$ and $\mathbf{q}$ and $N_e$ is the electron density in the medium. Unlike the one-body Hamiltonian of the matter-enhanced neutrino oscillations where neutrinos interact with a mean-field (generated by the background particles other than neutrinos), the Hamiltonian describing the many-neutrino gas in a core-collapse supernova contains both one- and two-body terms, making it technically much more challenging. Inclusion of the neutrino-neutrino interaction terms leads to very interesting collective effects (for review articles see Refs. [@Duan:2010bg] and [@Duan:2009cd]). Such collective oscillations of neutrinos represent emergent nonlinear flavor evolution phenomena instigated by neutrino-neutrino interactions in astrophysical environments with sufficiently high neutrino densities.
Sometimes the term containing the angle between neutrino momenta is averaged over in an approximation known as the [*single angle approximation*]{}: $$\label{satotal}
\hat{H}_{\mbox{\tiny SA}}
= \left(
\sum_p\frac{\delta m^2}{2p}\mathbf{B}\cdot\mathbf{J}_p - \sqrt{2} G_F
N_e J_p^0 \right)
+ \frac{\sqrt{2}G_{F}}{V} \left(\langle 1-
\cos\vartheta_{\mathbf{p}\mathbf{q}} \rangle \right) \sum_{\mathbf{p} \neq \mathbf{q}} \mathbf{J}_{\mathbf{p}}\cdot\mathbf{J}_{\mathbf{q}} .$$ A further approximation, which can be applied to either the multi-angle Hamiltonian of Eq. (\[total\]) or the single-angle Hamiltonian of Eq. (\[satotal\]), is the mean field approximation. In this approximation the two-body term is replaced by a one-body term: $$\mathbf{J}_{\mathbf{p}}\cdot\mathbf{J}_{\mathbf{q}} \rightarrow \langle \mathbf{J}_{\mathbf{p}} \rangle \cdot\mathbf{J}_{\mathbf{q}}
+ \mathbf{J}_{\mathbf{p}}\cdot \langle \mathbf{J}_{\mathbf{q}} \rangle$$ where the averaging is done over an appropriately chosen state. Exactly how this state is chosen determines the physics that is emphasized. The evolution of the system under the many-body Hamiltonian of Eq. (\[total\]) can be formulated as a coherent-state path integral, and a possible mean-field approximation represents the saddle-point solution of the path integral for this many-body system [@Balantekin:2006tg]. This is the most commonly used mean field approximation. One can also consider the contribution of neutrino-antineutrino pairing to the mean field [@Serreau:2014cfa]. Such a mean field would be proportional to the neutrino masses and could play a role in anisotropic environments [@Cirigliano:2014aoa].
EXACT SOLUTIONS
===============
Bethe Ansatz
------------
Exact solutions for the eigenstates of the Hamiltonian of Eq. (\[satotal\]) are easier to calculate in the mass basis $$\label{totalinmass}
\hat{H}
=
\sum_p \omega_p {\cal J}_p^0 + \mu \sum_{\mathbf{p} \neq\mathbf{q}} \mathbf{\cal J}_{\mathbf{p}}\cdot\mathbf{\cal J}_{\mathbf{q}}$$ where we introduced the neutrino flavor isospin operators in the mass basis: $$\begin{aligned}
{\cal J}^+_{{\bf p}} &=& a_2^\dagger({\bf p}) a_1({\bf p}), \> \> \>
{\cal J}^{\>-}_{{\bf p}}=a_1^\dagger({\bf p}) a_2({\bf p}), \nonumber \\
{\cal J}^0_{{\bf p}} &=& \frac{1}{2}\left(a_2^\dagger({\bf p})a_2({\bf p})-a_1^\dagger({\bf p})a_1({\bf p})
\right),
\label{su2mass}\end{aligned}$$ with $$\omega_p = \frac{m_2^2 - m_1^2}{2p} = \frac{\delta m^2}{2p}$$ and $$\mu = \frac{\sqrt{2}G_{F}}{V} \left\langle \left(1-
\cos\vartheta_{\mathbf{p}\mathbf{q}}\right) \right\rangle .$$ In writing Eq. (\[totalinmass\]) we ignored the electron background since we would like focus on regions where neutrino-neutrino interactions dominate. Note that the second term in this equation has the same form either in the mass or the flavor basis. Eigenvalues and eigenvectors of the Hamiltonian in Eq. (\[totalinmass\]) can be calculated from the solutions of the Bethe ansatz equations [@Pehlivan:2011hp]: $$\label{bethe}
-\frac{1}{\mu} - \sum_p \frac{j_p}{\omega_p - x_i} = \sum_{j \neq i}^N \frac{1}{x_i-x_j}$$ In this equation $j_p$ is the SU(2) quantum number associated with the algebra describing neutrinos with momentum $p$. There are $2j_p+1$ such neutrinos. Once the solutions of the coupled equations in Eq. (\[bethe\]) are identified, the eigenvectors are given by $$\label{eigenstate}
\left( \sum_p \frac{a^{\dagger}_2(p) a_1(p)}{\omega_p - x_1} \right) \left( \sum_p \frac{a^{\dagger}_2(p) a_1(p)}{\omega_p - x_2} \right) \cdots \left( \sum_p \frac{a^{\dagger}_2(p) a_1(p)}{\omega_p - x_N} \right) \left( \prod_p a_1^{\dagger}(p) \right) | 0 \rangle$$ with $|0\rangle$ being the fermion vacuum, and the eigenvalues are given by $$E = 2 \sum_{p\neq q} j_p j_q - \sum_p \omega_p j_p- N\mu \sum_p j_p - \frac{\mu}{2} N(N-1) + \sum_{i=1}^N x_i .$$
0.5cm
Spectral splits
---------------
One interesting effect resulting from the collective neutrino oscillations is spectral swappings or splits, on the final neutrino energy spectra: at a particular energy these spectra are almost completely divided into parts of different flavors [@Raffelt:2007cb; @Duan:2008za]. In the single-angle limit of the full many-body Hamiltonian it was shown that the spectral split of a neutrino ensemble, which initially consists of single flavor neutrinos, is analogous to the crossover from the BCS to the Bose-Einstein condensate limits [@Pehlivan:2016lxx]. When one uses the mean field approximation for the two-flavor case, total neutrino number is no longer conserved and needs to be enforced using a Lagrange multiplier. This Lagrange multiplier can be interpreted as the critical energy where the spectral swap/split takes place [@Pehlivan:2011hp]. These swaps may be independent of the mean field approximation: In Ref. [@Pehlivan:2016voj] the adiabatic evolution of a full many-body state of 250 electron neutrinos with inverted hierarchy distributed in a thermal energy spectrum was followed as the value of $\mu$ decreased from a very high value down to zero. It was found that the resulting split energy is the same as that was obtained in the mean-field approximation.
To illustrate the spectral split behavior in the full-many body case, we consider a simple toy model with two neutrinos ($j_1 = j_2 = 1/2$) with momenta yielding values $\omega_1$ and $\omega_2$. For this simple case the Bethe ansatz equations, Eq. (\[bethe\]) can be solved providing two different solutions for $N=1$: $$\label{N1sol}
x_{\pm} = \frac {\omega_1 + \omega_2}{2} + \frac{1}{2} \mu \pm \frac{1}{2} \sqrt{(\omega_1-\omega_2)^2 + \mu^2} ,$$ and a pair of solutions for $N=2$: $$\label{N2sol}
x_{1,2} = \frac {\omega_1 + \omega_2}{2} + \frac{1}{2} \mu \pm \frac{1}{2} \sqrt{(\omega_1-\omega_2)^2 - \mu^2} .$$ Note that the latter solutions become complex as $\mu$ gets very large. Let us assume that $\omega_1 > \omega_2>0$ and consider $N=1$ case.
\[f:1\]
![Solutions of the Bethe ansatz equations given in Eq. (\[N1sol\]). Upper two lines are $x_+$ for $\omega_2/\omega_1 = 0.7$ (upper thick solid line) and $\omega_2/\omega_1 = 0.3$ (upper thin solid line) wheras lower two lines are $x_-$ for $\omega_2/\omega_1 = 0.7$ (lower thick dashed line) and $\omega_2/\omega_1 = 0.3$ (lower thin dashed line).](roots){height=".3\textheight"}
In Figure 1 we show solutions given in Eq. (\[N1sol\]): upper two lines denote $x_+$ for $\omega_2/\omega_1 = 0.7$ (upper thick solid line) and $\omega_2/\omega_1 = 0.3$ (upper thin solid line) wheras lower two lines denote $x_-$ for $\omega_2/\omega_1 = 0.7$ (lower thick dashed line) and $\omega_2/\omega_1 = 0.3$ (lower thin dashed line). This figure illustrates what seems to be a generic property of the solutions of the Bethe ansatz equations: there is one solution that increases with increasing $\mu$ eventually becoming infinite whereas the other solution remains finite. The solution which grows with $\mu$ is the solution which starts as $\omega_1$ at $\mu=0$ (other solution starts as $\omega_2$ at $\mu=0$). Note that $\omega_1$ is the larger of the two $\omega$ values and corresponds to the smaller of the two values of neutrino momenta. For example, in the case of the adiabatic expansion of the neutrino gas formed in a core-collapse supernova $\mu \rightarrow \infty$ represents the neutrinosphere region and $\mu \sim 0$ represents the outer shells. Hence the behavior of the solutions described above gives us a hint on how neutrinos may exchange energy as they travel away from the neutrinosphere. Here to be specific let us consider the adiabatic eigenstates. The normalized eigenstates are given by $$| \xi_+ \rangle = \frac{1}{\cal N} \left( \frac{2 a_2^\dagger({\bf p_1}) a_1({\bf p_1})}{\eta - \mu- \sqrt{\eta^2 + \mu^2}} - \frac{2 a_2^\dagger({\bf p_2}) a_1({\bf p_2})}{\eta + \mu +\sqrt{\eta^2 + \mu^2}} \right) a_1^{\dagger} (p_1) a_1^{\dagger}(p_2) | 0 \rangle$$ with $\eta = \omega_1 - \omega_2$ and $${\cal N}^2 = 4 \left( \frac{1}{(\eta - \mu- \sqrt{\eta^2 + \mu^2})^2} + \frac{1}{\eta + \mu +\sqrt{\eta^2 + \mu^2}} \right).$$ One can write a similar expression for $|\xi_-\rangle$. One can easily show that as $\mu \rightarrow 0$ $$\begin{aligned}
\lim_{\mu \rightarrow 0} |\xi_+ \rangle &=& {\cal J}^+ (p_1) |0 \rangle, \nonumber \\
\lim_{\mu \rightarrow 0} |\xi_- \rangle &=& {\cal J}^+(p_2) |0 \rangle \end{aligned}$$ which are the eigenstates of the first term in Hamiltonian of Eq. (\[totalinmass\]) as expected. Similarly $$\begin{aligned}
\lim_{\mu \rightarrow \infty} |\xi_+ \rangle &=& \frac{1}{\sqrt{2}} \left( {\cal J}^+ (p_1) + {\cal J}^+ (p_2)\right) |0 \rangle, \nonumber \\
\lim_{\mu \rightarrow \infty} |\xi_- \rangle &=& \frac{1}{\sqrt{2}} \left( {\cal J}^+ (p_1) - {\cal J}^+ (p_2) \right) |0 \rangle. \end{aligned}$$ As an example let us consider the state $$| \psi (\mu) \rangle = \frac{1}{\sqrt{2}} \left( | \xi_+ \rangle + | \xi_- \rangle \right) .$$ From the expressions above we see that near the neutrinosphere such a state would be $$| \psi (\mu \rightarrow \infty) \rangle = a_2^{\dagger} (p_1) a_1^{\dagger} (p_2) | 0 \rangle,$$ i.e. the neutrino in the mass eigenstate $2$ has momentum $p_1$ and the neutrino in the mass eigenstate $1$ has momentum $p_2$. Far away from the proto-neutron star this state would take the form $$| \psi (\mu \rightarrow 0) \rangle = \frac{1}{\sqrt{2}} \left( a_2^{\dagger} (p_1) a_1^{\dagger} (p_2) \> | 0 \rangle - a_2^{\dagger} (p_2) a_1^{\dagger} (p_1) \> | 0 \rangle \right) ,$$ i.e. either mass eigenstate has equal probability to carry momenta $p_1$ and $p_2$. One should emphasize that, since the above arguments assume $\omega_1 > \omega_2$, the evolution will be opposite in the inverted hierarchy case (when the signs of $\omega$s change) than in the normal hierarchy case.
0.5cm
Conserved quantities
--------------------
It can be shown that there are additional conserved quantities commuting with the Hamiltonian in Eq. (\[totalinmass\]). They can be written in terms of quantity [@Pehlivan:2011hp] $$\label{hpinv}
\hat{h}_p = {\cal J}_p^0 + \mu \sum_{q, \mathbf{q} \neq\mathbf{p}}
\frac{\mathbf{\cal J}_{\mathbf{p}}\cdot\mathbf{\cal J}_{\mathbf{q}}}{\omega_p - \omega_q} .$$ In fact the Hamiltonian can be written in terms of these quantities: $$\hat{H} = \sum_p \omega_p \hat{h}_p.$$ The eigenvalues of $\hat{h}_p$ when acted on the state in Eq. (\[eigenstate\]) are given by $$\epsilon_p = \mu \sum_{q, q\neq p} \frac{j_pj_q}{\omega_p - \omega_q} - j_p -\mu j_p \sum_{i=1}^N \frac{1}{\omega_p - x_i} .$$
0.5cm
Three flavors and CP Violation
------------------------------
In the previous sections we ignored the third flavor state as well as antineutrinos. A third flavor can be incorporated by introducing SU(3) as the neutrino flavor isospin algebras for each momenta and antineutrinos require a second set of the SU(3) algebras. One can investigate the symmetries of the problem in the full three flavor mixing scheme and in the exact many-body formulation, and in addition include the effects of CP violation and neutrino magnetic moments [@Pehlivan:2014zua]. One finds that, similar to what was discussed above, several dynamical symmetries exist for three flavors in the single-angle approximation if the net electron background in the environment and the effects of the neutrino magnetic moment are negligible. These dynamical symmetries are present even in the presence of CP-violating phases. One can explicitly write down the constants of motion through which these dynamical symmetries manifest themselves in terms of the generators of the SU(3) flavor transformations. In this case the effects due to the CP-violating Dirac phase factor out of the many-body evolution operator and evolve independently of nonlinear flavor transformations if neutrino electromagnetic interactions with external magnetic fields are ignored [@Pehlivan:2014zua].
COLLECTIVE OSCILLATIONS FOR THE $\nu$p PROCESS
==============================================
An example of the application of multi-angle three-flavor calculations was recently given in Ref. [@Sasaki:2017jry] where the effects of collective neutrino oscillations on $\nu$p process nucleosynthesis in proton-rich neutrino driven winds were studied by coupling neutrino transport calculations with nucleosynthesis network calculations. Collective neutrino oscillations transform the spectra of all neutrino species, but of particular importance is the modification of the electron neutrino and electron antineutrino energy distributions: the capture of $\nu_e$ and $\bar{\nu}_e$ on free nucleons determine the neutron-to-proton ratio, hence the yields of nucleosynthesis.
In simple spectral split scenarios motivated by the single-angle collective oscillations abundances of p-nuclei are enhanced when the outflows are proton rich [@MartinezPinedo:2011br]. However the single-angle approximation, since it ignores angular correlations, changes the onset of the collective oscillations. For example, nucleosynthesis yields are drastically different between single-angle and multi-angle calculations of the supernova r-process [@Duan:2010af]. A complete spectral swap as obtained in calculations with the single-angle approximation do not seem to emerge from the multi-angle calculations: in both hierarchies the onset of collective oscillations are delayed as compared with that in the single angle approximation.
In Ref. [@Sasaki:2017jry] calculations were carried out for two proton-rich neutrino-driven winds at $t=0.6$ s and $t=1.1$ s after the core bounce in a one-dimensional explosion simulation model. For the early wind trajectory collective neutrino oscillations in the inverted mass hierarchy increase the energetic $\nu_e$ flux in the region where the $\nu$p nucleosynthesis takes place. However, in the later wind trajectory oscillations increase energetic $\bar{\nu}_e$s leading to an enhancement of the $\nu$p process, about up to 20 times larger than that was obtained in Ref. [@MartinezPinedo:2011br]. The fact that the enhancement is dominated by the later wind suggests that model dependence of these results may not be too strong.
Of course uncertainties of the initial neutrino fluxes as well as hydrodynamic quantities such as the wind velocity could alter these conclusions. If the luminosities and the energies of the neutrino fluxes for different flavors are similar, then the effects of the neutrino oscillations would be minimized. However, it is clear that collective neutrino oscillations [*can*]{} significantly influence the $\nu$p process yields and further systematic studies of neutrino physics and hydrodynamics input would be very useful.
ACKNOWLEDGMENTS
===============
This work was supported in part by the US National Science Foundation Grant No. PHY-1514695 and and in part by the University of Wisconsin Research Committee with funds granted by the Wisconsin Alumni Research Foundation. I would like to thank National Astronomical Observatory of Japan for its hospitality and acknowledge my collaborators T. Kajino, Y. Pehlivan, T. Hayakawa, H. Sasaki, T. Takiwaki, and T. Yoshida who contributed to much of the work reported here.
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---
abstract: 'It is shown that embedding a four-dimensional flipped $SU(5)$ model in a five-dimensional $SO(10)$ model, preserves the best features of both flipped $SU(5)$ and $SO(10)$. The missing partner mechanism, which naturally achieves both doublet-triplet splitting and suppression of $d=5$ proton decay operators, is realized as in flipped $SU(5)$, while the gauge couplings are unified as in $SO(10)$. The masses of down quarks and charged leptons, which are independent in flipped $SU(5)$, are related by the $SO(10)$. Distinctive patterns of quark and lepton masses can result. The gaugino mass $M_1$ is independent of $M_3$ and $M_2$, which are predicted to be equal.'
author:
- 'S.M. Barr'
- Ilja Dorsner
title: 'Unifying flipped $SU(5)$ in five dimensions'
---
Introduction
============
A beautiful feature of flipped $SU(5)$ [@Barr:1981qv; @Derendinger:1983aj; @Antoniadis:1987dx] is that it provides a natural setting for the missing partner mechanism. This mechanism, when implemented in flipped $SU(5)$, not only solves the doublet-triplet splitting problem but also allows one to avoid entirely the Higgsino-mediated proton decay that is such a difficulty for supersymmetric grand unified theories (SUSY GUTs). On the other hand, flipped $SU(5)$ gives up one of the most attractive features of grand unification, namely unification of gauge couplings, because it is based on the group $SU(5) \times U(1)$. Another drawback of flipped $SU(5)$ models is that the masses of down quarks and charged leptons come from different operators, so that one does not obtain the relation $m_b(M_{GUT}) = m_{\tau}(M_{GUT})$. The unification of gauge couplings and relations between down quark masses and charged lepton masses could be recovered by embedding the group $SU(5) \times U(1)$ in the simple group $SO(10)$. However, in that case, the missing partner mechanism no longer works, since the partner that was missing from the $SU(5)$ multiplets is present in the larger $SO(10)$ multiplets.
One thus has somewhat of a quandary. The point of this paper is that a way out of this quandary is provided by unification in five dimensions. We show that if the group $SO(10)$ in five dimensions is broken by orbifold compactification to the group $SU(5) \times U(1)$ in four dimensions it is possible to have at the same time the good features of both flipped $SU(5)$ and of $SO(10)$. The essential reason is that if $SO(10)$ is broken by the orbifold compactification then the fields of the effective four-dimensional theory need not be in complete $SO(10)$ multiplets. This means that at the four-dimensional level the famous missing partners can still be missing and the doublet-triplet splitting can be achieved without the dangerous Higgsino-mediated proton decay. On the other hand, because there is $SO(10)$ at the five-dimensional level, there is approximate unification of gauge couplings, and there is also the possibility of getting $SO(10)$-like Yukawa couplings for the quarks and leptons.
By now there are many models that use orbifold compactification of extra dimensions to break grand unified symmetries. The first such models [@Kawamura:1999nj; @Kawamura:2000ev; @Kawamura:2000ir; @Altarelli:2001qj; @Kobakhidze:2001yk; @Hall:2001pg; @Hebecker:2001wq] showed that with one extra dimension it is possible to construct $SU(5)$ models which have natural doublet-triplet splitting and no problem with the $d=5$ proton decay operators that plague four-dimensional SUSY GUTs. The breaking of grand unified symmetries by orbifold compactification of a single extra dimension does not reduce the rank of the group [@Hebecker:2001jb]. Thus to break $SO(10)$ all the way to the Standard Model by orbifold compactification requires at least two extra dimensions. Interesting six-dimensional $SO(10)$ models have been constructed in several papers [@Asaka:2001eh; @Hall:2001xr; @Haba:2002ek]. However, it is also possible that the breaking from the grand unified group to the Standard Model is achieved by a combination of orbifold compactification and the conventional four-dimensional Higgs mechanism. That allows the construction of realistic $SO(10)$ models with only a single extra dimension, as was shown by Derm' išek and Mafi [@Dermisek:2001hp]. In their model the theory in the five-dimensional bulk has $\mathcal{N}=1$ supersymmetry and gauge group $SO(10)$. Orbifolding breaks $SO(10)$ to the Pati-Salam [@Pati:1974yy] symmetry $SU(4)_c \times SU(2)_L \times SU(2)_R$. The orbifold has two inequivalent fixed points $O$ and $O'$. On $O$ there is a full $SO(10)$ symmetry, but on $O'$ only the Pati-Salam group. On the brane at $O$ the conventional Higgs mechanism breaks $SO(10)$ down to $SU(5)$. Thus the unbroken symmetry in the low-energy theory in four dimensions is the intersection of $SU(5)$ and the Pati-Salam group, i.e. the Standard Model group.
The model we shall present is similar in some ways to that of Derm' išek and Mafi but differs from it in several important respects. Whereas they use orbifold compactification to break to the Pati-Salam group and Higgs fields on the brane $O$ to break to $SU(5)$, we shall use orbifold compactification to break to $SU(5) \times U(1)$ and Higgs fields in the bulk to break the rest of the way to the Standard Model. They use orbifold breaking to split the doublets from the triplets, whereas we use the four-dimensional flipped-$SU(5)$ missing partner mechanism.
Missing partners in four dimensions
===================================
Before we consider higher dimensional theories we shall briefly review the missing partner mechanism in four-dimensional theories, showing why it works in flipped $SU(5)$ but not in $SO(10)$.
Flipped SU(5)
-------------
First recall what happens in ordinary (i.e. Georgi-Glashow) $SU(5)$ [@Georgi:sy]. In ordinary $SU(5)$ the two Higgs doublets of the MSSM, which we shall denote ${\bf 2}$ and $\overline{{\bf 2}}$, have color-triplet partners, which we shall denote ${\bf 3}$ and $\overline{{\bf 3}}$. (We use this shorthand notation for Standard Model representations: ${\bf 2} \equiv (1, 2, -\frac{1}{2})$, $\overline{{\bf 2}}
\equiv (1,2, \frac{1}{2})$, ${\bf 3} \equiv (3,1, \frac{1}{3})$, $\overline{{\bf 3}} \equiv (\overline{3}, 1, -\frac{1}{3})$.) These are contained in fundamental and anti-fundamental multiplets of $SU(5)$: ${\bf 5} = {\bf 2} + {\bf 3}$ and $\overline{{\bf 5}} = \overline{{\bf 2}} + \overline{{\bf 3}}$. A combination of an $SU(5)$-singlet mass term and a Yukawa coupling to a Higgs in the adjoint representation, can (with suitable fine-tuning) give GUT-scale mass to the triplet partners while leaving the MSSM Higgs doublets light. This can be represented schematically as
(6,1.4) (2.8,1.165)[(1,0)[0.42]{}]{}
where the solid horizontal line represents a large Dirac mass $M_3$ connecting the colored Higgsinos ${\bf 3}$ and $\overline{{\bf 3}}$. It is well-known that the exchange of these colored Higgsinos gives a dangerous $d=5$ proton-decay operator, as shown in Fig. \[dimension5\]. From the figure one sees that this proton decay amplitude is proportional to the mass connecting ${\bf 3}$ to $\overline{{\bf 3}}$. Suppressing this proton decay therefore requires severing this connection. This can be done by introducing an extra pair of Higgs multiplets ${\bf 5}'
+ \overline{{\bf 5}}'$, so that the triplets in the unprimed multiplets get mass not with each other but with the triplets in the primed multiplets as shown in the following diagram
(6,1.4) (2.20,1.165)[(1,0)[0.41]{}]{} (3.35,1.165)[(1,0)[0.41]{}]{} (2.76,0.89)(0.09,0)[5]{}[(1,0)[0.04]{}]{} (2.76,1.165)(0.09,0)[5]{}[(1,0)[0.04]{}]{}
If the MSSM Higgs doublets are the unprimed ones, then one sees that their colored partners are not connected to each other by a mass term, so that the $d=5$ proton-decay amplitude vanishes. Unfortunately, however, there is an extra pair of doublets that remains light, namely the primed ones. The effect of these on the renormalization group equations would destroy gauge coupling unification. To give the needed superheavy mass to these doublets one could introduce a term $M \overline{{\bf 5}}' {\bf 5}'$; however, this would give mass terms connecting not only ${\bf 2}'$ to ${\bf 2}$ but also ${\bf 3}'$ to $\overline{{\bf 3}}'$ (indicated by dotted lines in the previous diagram) and thus indirectly (after the primed triplets were integrated out) reconnecting ${\bf 3}$ to $\overline{{\bf 3}}$ and bringing back the dangerous $d=5$ proton decay amplitude.
Now let us turn to flipped $SU(5)$ and see how it avoids these problems very elegantly [@Antoniadis:1987dx]. In flipped $SU(5)$ models one has Higgs fields in the following representations of $SU(5) \times U(1)$: $h = {\bf 5}^{-2}$, $\overline{h} = \overline{{\bf 5}}^2$, $H = {\bf 10}^1$, and $\overline{H} = \overline{{\bf 10}}^{-1}$. Under the Standard Model group these decompose as follows, $h = \overline{{\bf 2}} + {\bf 3}$, $\overline{h} = {\bf 2} + \overline{{\bf 3}}$, $H = \overline{{\bf 3}} + (3,2, \frac{1}{6}) + (1,1, 0)$, and $\overline{H} = {\bf 3} + (\overline{3}, 2, -\frac{1}{6})
+ (1,1, 0)$. The Higgs superpotential contains the terms $h \; H \; H + \overline{h} \; \overline{H} \; \overline{H}$. When the Standard Model singlets $(1,1, 0)$ in the $H$ and $\overline{H}$ acquire vacuum expectation values (VEVs) they break $SU(5) \times U(1)$ down to the Standard Model group and they also give mass to the triplet Higgs. Schematically,
(6,1.4) (1.95,1.165)[(1,0)[0.55]{}]{} (3.47,1.165)[(1,0)[0.55]{}]{} (3,0.7)(0,0.2)[4]{}[(0,1)[0.1]{}]{}
where, for simplicity, $(3,2, \frac{1}{6}) + (1,1, 0) \equiv {\rm other}$. The triplets in $h$ and $\overline{h}$ get mass with those in $H$ and $\overline{H}$. However the doublets in $h$ and $\overline{h}$ remain massless because there are no doublets in $H$ and $\overline{H}$ for them to mate with — thus the name “missing partner mechanism".
At first glance one might worry that the same problem arises here as in the ordinary $SU(5)$ case discussed previously. The multiplets ${\bf 5}'$ and $\overline{{\bf 5}}'$ there played the same role as the multiplets $H$ and $\overline{H}$ here. And we saw that one could not give mass to the doublets in ${\bf 5}'$ and $\overline{{\bf 5}}'$ without reintroducing the dangerous proton decay amplitude. This leads to the question whether there is not an analogous difficulty in giving mass to some of the components of $H$ and $\overline{H}$, and specifically to the $(3,2, \frac{1}{6}) + (1,1,0) + ( \overline{3}, 2, -\frac{1}{6}) +
(1,1,0)$, since here also an explicit mass term $M \overline{H} H$ would reintroduce the problem of proton decay. The beautiful answer is that these “other" components of $H$ and $\overline{H}$ do not have to get mass. Indeed, they [*must not*]{} get mass, because they are the goldstone modes that get eaten when $SU(5) \times U(1)$ breaks to $SU(3) \times SU(2) \times U(1)$. In other words, the fact that $SU(5) \times U(1)$ breaks to the Standard Model group guarantees that there is no mass connecting $H$ and $\overline{H}$ and therefore guarantees the absence of the $d=5$ proton decay amplitude.
SO(10)
------
Now let us see why embedding flipped $SU(5)$ in $SO(10)$ in four dimensions destroys the beautiful missing partner solution to the doublet-triplet splitting and proton decay problems that we have just reviewed.
In $SO(10)$ the simplest possibility is that the terms $h \; H \; H + \overline{h} \; \overline{H} \; \overline{H}$ come from the terms ${\bf 10} \;{\bf 16} \;{\bf 16} + {\bf 10} \; \overline{{\bf 16}}
\; \overline{{\bf 16}}$, where ${\bf 10} = \overline{h} + h$, ${\bf 16} = H + \overline{h}' + {\bf 1}^5$, and $\overline{{\bf 16}} = \overline{H} + h' + {\bf 1}^{-5}$. Here $h' = {\bf 5}^3$ and $\overline{h}' = \overline{{\bf 5}}^{-3}$. The problem is that the doublet partners that were missing from $H$ and $\overline{H}$ are now present in $\overline{h}'$ and $h'$.
The terms ${\bf 10} \;{\bf 16} \;{\bf 16} + {\bf 10} \;\overline{{\bf 16}}
\;\overline{{\bf 16}}$ contain not only $h \; H \langle H \rangle + \overline{h} \; \overline{H} \langle
\overline{H} \rangle$ but also $\overline{h} \; \overline{h}' \langle H
\rangle + h \; h' \langle \overline{H} \rangle$. These latter terms mate the doublet Higgs in $h$ and $\overline{h}$ with those in $\overline{h}'$ and $h'$, destroying the solution of the doublet-triplet splitting problem.
A possible remedy to this difficulty suggests itself. One can have $h$ and $\overline{h}$ come from different ${\bf 10}$s of $SO(10)$. Let us examine what happens in this case, since it will be directly relevant to what we shall do in five dimensions later. Suppose there are two vector Higgs representations, denoted ${\bf 10}_1$ and ${\bf 10}_2$, with couplings ${\bf 10}_1 \; {\bf 16} \; {\bf 16} + {\bf 10}_2 \; \overline{{\bf 16}}
\; \overline{{\bf 16}}$. We write ${\bf 10}_1 = h_1 + \overline{h}_1$ and ${\bf 10}_2 = h_2 + \overline{h}_2$. Suppose that the two light doublets of the MSSM lie in $h_1$ and $\overline{h}_2$; then the triplet partners of these light doublets will obtain mass from the terms $h_1 H \langle H \rangle + \overline{h}_2 \overline{H} \langle
\overline{H} \rangle$. The terms that give superlarge mass to doublets, and which correspond to those we found troubling before, are $\overline{h}_1 \overline{h}' \langle H \rangle +
h_2 h' \langle \overline{H} \rangle$. These do [*not*]{} give superlarge mass to the MSSM doublets, but to the doublets in $\overline{h}_1$ and $h_2$. Thus, we would appear to have satisfactory doublet-triplet splitting with no dangerous $d=5$ proton decay, just as in flipped $SU(5)$.
However, this is not so, for the question arises how the triplets in $\overline{h}_1$ and $h_2$ are to acquire superheavy mass. It would seem that the only way is through a mass term connecting them. But that would have to come from a term $M \overline{h}_1 h_2$, which in turn comes from $M {\bf 10}_1 {\bf 10}_2$, and this would also give $M h_1 \overline{h}_2$ and thus superlarge mass to the MSSM doublets.
We see, then, that the missing partner mechanism does not work in four-dimensional $SO(10)$ theories. However, we shall see that it can work in five-dimensional $SO(10)$ theories. The crucial difference will be that orbifold breaking of $SO(10)$ can split the $SO(10)$ Higgs representations. In particular, in the example we just looked at the troublesome triplets in $\overline{h}_1$ and $h_2$ can be given Kaluza-Klein masses by the orbifold compactification while leaving the MSSM doublets in $h_1$ and $\overline{h}_2$ light.
An $SO(10)$ model in five dimensions
====================================
We now present an $SO(10)$ supersymmetric model in five dimensions compactified on an $S^1/(Z_2 \times Z'_2)$ orbifold that yields a realistic supersymmetric flipped $SU(5)$ model in four dimensions. The breaking of $SU(5) \times U(1)$ down to the Standard Model gauge group, the doublet-triplet splitting, and the solution to the problem of $d=5$ proton-decay operators will all be as in conventional four-dimensional flipped $SU(5)$ models. Moreover, there will be distinctive flipped $SU(5)$ relationships among gaugino masses. However, the gauge couplings will be unified (with some threshold corrections, that can be argued to be small [@Hall:2001pg]) because of the underlying five-dimensional $SO(10)$ symmetry. And the Yukawa couplings of the quarks and leptons can have relationships that are similar to what is found in ordinary $SU(5)$ and $SO(10)$ models rather than in flipped $SU(5)$.
As already elaborated in Refs. [@Kawamura:1999nj; @Kawamura:2000ev; @Kawamura:2000ir; @Altarelli:2001qj; @Hall:2001pg; @Kobakhidze:2001yk; @Hebecker:2001wq], the fifth dimension, being the circle with coordinate $y$ and circumference $2 \pi R$, is compactified through the reflection $y \rightarrow -y$ under $Z_2$ and $y' \rightarrow -y'$ under $Z'_2$ where $y'=y+\pi R/2$. This identification procedure leaves two fixed points $O$ and $O'$ of $Z_2$ and $Z'_2$ respectively and reduces the physical region to the interval $y \in [-\pi R/2,0]$. Point $O$ at $y=0$ is the “visible brane" while point $O'$ at $y'=0$ is the “hidden brane". The compactification scale $1/R \equiv M_C$ is assumed to be close to the scale at which the gauge couplings unify, i.e. the GUT scale $M_{GUT} \sim 10^{16}$ GeV.
The generic bulk field $\phi(x^\mu,y)$, where $\mu=0,1,2,3$, has definite parity assignment under $Z_2 \times Z'_2$ symmetry. Taking $P=\pm 1$ to be parity eigenvalue of the field $\phi(x^\mu,y)$ under $Z_2$ transformation and $P'=\pm 1$ to be parity eigenvalue under the $Z'_2$ transformation, a field with $(P,P')=(\pm,\pm)$ can be denoted $\phi^{PP'}(x^\mu,y)=\phi^{\pm \pm}(x^\mu,y)$. The Fourier series expansion of the fields $\phi^{\pm \pm}(x^\mu,y)$ yields
\[Fourier\] $$\begin{aligned}
\phi^{++}(x^\mu,y)&=&\frac{1}{\sqrt{2^{\delta_{n0}}\pi R}}
\sum^{\infty}_{n=0}
\phi^{++(2n)}(x^\mu) \cos \frac{2ny}{R},\\
\phi^{+-}(x^\mu,y)&=&\frac{1}{\sqrt{\pi R}} \sum^{\infty}_{n=0}
\phi^{+-(2n+1)}(x^\mu) \cos \frac{(2n+1)y}{R},\\
\phi^{-+}(x^\mu,y)&=&\frac{1}{\sqrt{\pi R}} \sum^{\infty}_{n=0}
\phi^{-+(2n+1)}(x^\mu) \sin \frac{(2n+1)y}{R},\\
\phi^{--}(x^\mu,y)&=&\frac{1}{\sqrt{\pi R}} \sum^{\infty}_{n=0}
\phi^{--(2n+2)}(x^\mu) \sin \frac{(2n+2)y}{R}.\end{aligned}$$
In the effective theory in four dimensions all the fields in Eqs. (\[Fourier\]) have masses of order $M_C$ except the Kaluza-Klein zero mode $\phi^{++(0)}$ of $\phi^{++}(x^\mu,y)$, which remains massless. Moreover, fields $\phi^{-\pm}(x^\mu,y)$ vanish on the visible brane and fields $\phi^{\pm-}(x^\mu,y)$ vanish on the hidden brane.
In our model, we assume that gauge fields and gauge-non-singlet Higgs fields exist in the five-dimensional bulk, while the quark and lepton fields and certain gauge-singlet Higgs fields exist only on the visible brane at $O$. The gauge fields in the bulk are of course in a vector supermultiplet of 5d supersymmetry that is an adjoint representation of $SO(10)$. We will denote it by ${\bf 45}_g$, where the subscript ‘$g$’ stands for ‘gauge’. The gauge-non-singlet Higgs fields in the bulk are in hypermultiplets of 5d supersymmetry and consist of two tens of $SO(10)$, denoted ${\bf 10}_{1H}$ and ${\bf 10}_{2H}$, and a spinor-antispinor pair of $SO(10)$ denoted ${\bf 16}_H$ and $\overline{{\bf 16}}_H$. The subscript ‘$H$’ indicates a Higgs field.
The vector supermultiplet ${\bf 45}_g$ decomposes into a vector multiplet $V$ and a chiral multiplet $\Sigma$ of $\mathcal{N}= 1$ supersymmetry in four dimensions. Each hypermultiplet splits into two left-handed chiral multiplets $\Phi$ and $\Phi^c$, having opposite gauge quantum numbers. Under the $SU(5) \times U(1)$ subgroup the $SO(10)$ representations decompose as follows: ${\bf 45} \rightarrow {\bf 24}^0+{\bf 10}^{-4}+\overline{{\bf 10}}^4+
{\bf 1}^0$; ${\bf 10} \rightarrow {\bf 5}^{-2}+\overline{{\bf 5}}^2$; ${\bf 16} \rightarrow {\bf 10}^1 + \overline{{\bf 5}}^{-3} + {\bf 1}^5$; and $\overline{{\bf 16}} \rightarrow \overline{{\bf 10}}^{-1} +
{\bf 5}^3 + {\bf 1}^{-5}$. With these facts in mind we shall now discuss the transformation of the various fields under the $Z_2 \times Z'_2$ parity transformations.
The first $Z_2$ symmetry (the one we denote as unprimed) is used to break supersymmetry to $\mathcal{N}=1$ in four-dimensions. ($\mathcal{N}=1$ in five dimensions is equivalent to $\mathcal{N}=2$ in four dimensions; so we are breaking half the supersymmetries.) To do this we assume that under $Z_2$ the $V$ is even, $\Sigma$ is odd, $\Phi$ are even, and $\Phi^c$ are odd. The $Z_2'$ is used to break $SO(10)$ down to $SU(5) \times U(1)$. The ${\bf 24}^0$ and ${\bf 1}^0$ of $V$ are taken to be even under $Z_2'$, while the ${\bf 10}^{-4}$ and $\overline{{\bf 10}}^4 $ are taken to be odd. In ${\bf 10}_{1H}$ the ${\bf 5}^{-2}$ are taken to be even and the $\overline{{\bf 5}}^2$ odd, whereas in ${\bf 10}_{2H}$ the parities are taken to be the reverse, ${\bf 5}^{-2}$ odd and $\overline{{\bf 5}}^2$ even. All told we have
\[parity\] $$\begin{aligned}
{\bf 45}_g & = & V^{++}_{{\bf 24}^0} + V^{++}_{{\bf 1}^0} +
V^{+-}_{{\bf 10}^{-4}} + V^{+-}_{\overline{{\bf 10}}^4} + \Sigma^{-+}_{{\bf 24}^0} + \Sigma^{-+}_{{\bf 1}^0} +
\Sigma^{--}_{{\bf 10}^{-4}} + \Sigma^{--}_{\overline{{\bf 10}}^4} \\
{\bf 10}_{1H} & = & \Phi^{++}_{{\bf 5}_1^{-2}} +
\Phi^{+-}_{\overline{{\bf 5}}_1^2} + \Phi^{c --}_{\overline{{\bf 5}}_1^2} + \Phi^{c-+}_{{\bf 5}_1^{-2}} \\
{\bf 10}_{2H} & = & \Phi^{+-}_{{\bf 5}_2^{-2}} +
\Phi^{++}_{\overline{{\bf 5}}_2^2} + \Phi^{c -+}_{\overline{{\bf 5}}_2^2} + \Phi^{c--}_{{\bf 5}_2^{-2}} \\
{\bf 16}_H & = & \Phi^{++}_{{\bf 10}^1} + \Phi^{+-}_{\overline{{\bf 5}}^{-3}}
+ \Phi^{+-}_{{\bf 1}^5} + \Phi^{c--}_{\overline{{\bf 10}}^{-1}} + \Phi^{c-+}_{{\bf 5}^3}
+ \Phi^{c-+}_{{\bf 1}^{-5}} \\
\overline{{\bf 16}}_H & = &
\Phi^{++}_{\overline{{\bf 10}}^{-1}} + \Phi^{+-}_{{\bf 5}^3}
+ \Phi^{+-}_{{\bf 1}^{-5}} + \Phi^{c--}_{{\bf 10}^1} + \Phi^{c-+}_{\overline{{\bf 5}}^{-3}}
+ \Phi^{c-+}_{{\bf 1}^5}\end{aligned}$$
Massless zero modes of the Kaluza-Klein towers exist only for fields with $Z_2 \times Z_2'$ parity $++$. This includes $\Phi^{++}_{{\bf 5}^{-2}_1}$, $\Phi^{++}_{\overline{{\bf 5}}^2_2}$, $\Phi^{++}_{{\bf 10}^1}$, and $\Phi^{++}_{\overline{{\bf 10}}^{-1}}$. We will call the zero modes of these components $h_1$, $\overline{h}_2$, $H$, and $\overline{H}$, respectively, using the same notation we used in the last section. The $h_1$ and $\overline{h}_2$ contain the two Higgs doublets of the MSSM and their colored partners.
To understand these parity assignments, we observe the invariance of the action for the bulk fields [@Arkani-Hamed:2001tb] given by $$\begin{aligned}
\label{action}
\nonumber
S_5&=&\int{{\rm d}^5x}\Bigg\{\frac{1}{4 k g^2}
{\rm Tr}\bigg[\int{{\rm d}^2\theta W^\alpha W_\alpha}+h.c.\bigg]
\qquad\qquad\qquad\\
\nonumber
&+&\frac{1}{k g^2} \int{{\rm d}^4\theta}{\rm Tr}\bigg[\Big((\sqrt{2}\partial_5+\overline{\Sigma})
{\rm e}^{-V}(-\sqrt{2}\partial_5+\Sigma){\rm e}^V+
\partial_5 {\rm e}^{-V}\partial_5 {\rm e}^V\Big)\bigg]\\
\nonumber
&+&\sum_{i=1}^4 \int{{\rm d}^4\theta}\bigg[\Phi_i^c {\rm e}^V \overline{\Phi}_i^c +
\overline{\Phi}_i {\rm e}^{-V}\Phi_i\bigg]\\
&+&\sum_{i=1}^4 \bigg[\int{{\rm d}^2\theta}\Phi_i^c(\partial_5-\frac{1}{\sqrt{2}}
\Sigma)\Phi_i+{\rm h.c.}\bigg]\Bigg\}\end{aligned}$$ under $y \rightarrow -y$ reflection with the superfields transforming as
\[fields\] $$\begin{aligned}
V^a(x^\mu,-y) T^a & = & V^a(x^\mu,y) P T^a P\\
\Sigma^a(x^\mu,-y) T^a & = &- \Sigma^a(x^\mu,y) P T^a P\\
\Phi_i(x^\mu,-y) & = & \pm P \Phi_i(x^\mu,-y)\\
\Phi^c_i(x^\mu,-y) & = & \mp P^T \Phi^c_i(x^\mu,-y)\end{aligned}$$
where $V=V^a T^a$, and $\Sigma=\Sigma^a T^a$. The $T^a$s are the generators of $SO(10)$ in the appropriate representation with normalization ${\rm Tr}[T^a T^b]=k \delta^{ab}$, and $P=P^{-1}$ is the parity operator. The replacement $y \rightarrow y'$ and $P \rightarrow P'$ in Eqs. (\[fields\]) specifies the transformation of the superfields under $y' \rightarrow -y'$ reflection. Finally, defining $P$ and $P'$ through their action on the ${\bf 10}$ of $SO(10)$, we associate $P=\sigma_0 \otimes I$ with the $Z_2$ and $P'=\sigma_2 \otimes I$ with $Z'_2$, where $I$ and $\sigma_0$ are $5 \times 5$ and $ 2 \times 2$ identity matrices and $\sigma_2$ is the usual Pauli matrix.
Having done with the parity assignment for the bulk fields we can turn our attention towards the brane physics. On the brane at $O$ we put the three families of quarks and leptons. Since the gauge symmetry on this brane is $SO(10)$, these are contained in three chiral superfields that are spinors of $SO(10)$, which we denote ${\bf 16}_i$, where $i = 1,2,3$ is the family index. Later for various reasons we shall introduce some gauge-singlet superfields on the brane at $O$, but let us first discuss the interactions of the fields introduced up to this point.
The $Z_2$ parity of fields in the ${\bf 16}_i$ must be positive. The $Z'_2$ parity, determined by the content of Eqs. (\[parity\]), is ${\bf 16} \rightarrow
{\bf 10}^{1 \pm} + \overline{{\bf 5}}^{-3 \mp} +{\bf 1}^{5 \mp}$, where ${\bf 10}_i=(Q,D,N)_i$, $\overline{{\bf 5}}_i=(U,L)_i$, and ${\bf 1}_i=(E)_i$. The action for the coupling of the matter fields, residing on the visible brane, with the Higgs fields, coming from the bulk, is $$\begin{aligned}
\label{mattera}
\nonumber
S^{matter}_5 & = & \int{{\rm d}^5x} \, \frac{1}{2} \left[\delta(y)+\delta(y-\pi R)\right] \sqrt{2 \pi R}
\, \int{{\rm d}^2\theta} \, A_{ij} {\bf 16}_i {\bf 16}_j {\bf 10}_{1H} \\
& + & \int{{\rm d}^5x} \, \frac{1}{2} \left[\delta(y)-\delta(y-\pi R)\right] \sqrt{2 \pi R}
\, \int{{\rm d}^2\theta} \, B_{ij} {\bf 16}_i {\bf 16}_j {\bf 10}_{2H} + {\rm h.c.},\end{aligned}$$ where $A_{ij}$ and $B_{ij}$ are Yukawa couplings. Integrating over the fifth dimension $y$ using the Eqs. (\[Fourier\]), and keeping only the terms that involve the Yukawa couplings of the MSSM Higgs doublets and their triplet partners we obtain the following Lagrangian in four dimensions $$\begin{aligned}
\label{matterb}
\nonumber
\mathcal{L}^{matter}_4 & = & \sum^{\infty}_{n=0}\int{{\rm d}^2\theta} \sqrt{\frac{2}{2^{\delta_{n0}}}}
\bigg\{ A_{ij}
\Big[ Q_i D_j \overline{d}^{(2n)}_{1H} + L_i E_j \overline{d}^{(2n)}_{1H} +
\frac{1}{2}Q_i Q_j t^{(2n)}_{1H} + U_i E_j t^{(2n)}_{1H} \Big] \\
&+&B_{ij} \Big[ Q_i U_j d^{(2n)}_{2H} + L_i N_j d^{(2n)}_{2H} +
Q_i L_j \overline{t}^{(2n)}_{2H} + U_i E_j \overline{t}^{(2n)}_{2H} \Big] \bigg\} +{\rm h.c.}\end{aligned}$$ where $\overline{d}_{1H}^{(2n)}$ and $t_{1H}^{(2n)}$ are the doublet and triplet contained in $\Phi^{++}_{{\bf 5}^{-2}_1}$ (whose zero mode is $h_1$) and $d_{2H}^{(2n)}$ and $\overline{t}_{2H}^{(2n)}$ are the doublet and triplet contained in $\Phi^{++}_{\overline{{\bf 5}}^2_2}$ (whose zero mode is $\overline{h}_2$). All the remaining terms coming from Eq. (\[mattera\]) are found by the replacement $A_{ij} \leftrightarrow B_{ij}$, $(1H) \leftrightarrow (2H)$, $(2n) \rightarrow (2n+1)$, and $\delta_{n0}
\rightarrow 1$ in Eq. (\[matterb\]).
This represents a minimal set of Yukawa terms, and would lead to the following relations among the quark and lepton mass matrices: $M_L = M_D \propto A$ and $M_{\nu}^{Dirac} = M_U \propto B$, with $A$ and $B$ being completely independent symmetric matrices. This is different from the relations that arise with a minimal set of Yukawa terms in four-dimensional models based on $SO(10)$ or flipped $SU(5)$. In four-dimensional flipped $SU(5)$, the minimal Yukawa terms give $M_{\nu}^{Dirac} = M_U^T$, where these matrices are not predicted to be symmetric, and no relation for $M_L$ and $M_D$. In four-dimensional $SO(10)$, the minimal Yukawa terms give $M_L = M_D \propto M_{\nu}^{Dirac} = M_U$, with these matrices predicted to be symmetric.
The Higgs fields, though defined in the bulk, will also couple to each other on the branes. We assume that the Higgs coupling on the visible brane is of the form $$\begin{aligned}
\label{higgs}
\nonumber
S^{higgs}_5 & = & \int{{\rm d}^5x} \, \frac{1}{2} \left[\delta(y)+\delta(y-\pi R)\right] \sqrt{2 \pi R}
\, \int{{\rm d}^2\theta} \, {\bf 10}_{1H} {\bf 16}_H {\bf 16}_H\\
& + & \int{{\rm d}^5x} \, \frac{1}{2} \left[\delta(y)-\delta(y-\pi R)\right] \sqrt{2 \pi R}
\, \int{{\rm d}^2\theta} \, {\bf 10}_{2H} {\bf 16}_H {\bf 16}_H
+ {\rm h.c.}.\end{aligned}$$ There could also be terms of the form ${\bf 10}_{iH} {\bf 10}_{jH}$, which would directly produce a GUT-scale $\mu$ term and destroy the gauge hierarchy. These must be forbidden by a symmetry. This is not a novel requirement introduced by the fact that there are extra dimensions. Terms that would directly produce a GUT-scale $\mu$ term must also be forbidden in four-dimensional unified theories. For example, in four-dimensional $SU(5)$ theories as well as four-dimensional flipped $SU(5)$ theories, there are Higgs multiplets in $\overline{{\bf 5}}$ and ${\bf 5}$, and these must be prevented from obtaining a superheavy mass term together. Similarly, in four-dimensional $SO(10)$ theories the light Higgs doublets are typically in a ${\bf 10}$ of Higgs, which must be prevented from acquiring a superheavy self-mass term [@Babu:1993we]. The same problem arises also in GUTs in higher dimensions. Generally, some symmetry must be imposed to protect the gauge hierarchy from such dangerous terms. We shall assume here that there is a $U(1)'$ of the Peccei-Quinn type under which the quark and lepton spinors ${\bf 16}_i$ have charge $+1$, the Higgs fields ${\bf 10}_{1H}$ and ${\bf 10}_{2H}$ have charge $-2$, and the Higgs fields ${\bf 16}_H$ and $\overline{{\bf 16}}_H$ have charge $+1$. This approach of using a vector-like symmetry to prevent a large direct $\mu$ term is used in Ref. [@Dermisek:2001hp]. A drawback of using that method here, as will be seen later, is that to generate large Majorana mass terms for the neutrinos without too large a $\mu$ term being generated by higher-dimension operators, it will be necessary to assume a hierarchy of $10^{-4}$ between the $U(1)'$ breaking scale and $M_{GUT}$.
Another way of suppressing direct GUT-scale $\mu$ terms is by means of a continuous $U(1)_R$ symmetry as in Ref. [@Hall:2001pg]. In that paper it was found that $\mu$ and $\mu B$ parameters of the order of the weak scale could be generated, without any fine-tuning, through the Giudice-Masiero mechanism [@Giudice:1988yz]. We do not pursue other approaches such as that here.
The most general effective action of our theory should also include brane-localized kinetic terms for the modes of the bulk fields that have non-vanishing wavefunction on the branes. Since the symmetry that survives on the hidden brane differs from the symmetry that governs the interactions on the visible brane and in the bulk, one might worry that the hidden-brane kinetic terms with the arbitrary coefficients for the gauge fields would spoil the gauge coupling unification, and that the hidden-brane kinetic terms for the Higgs fields could affect the mass matrix prediction that stems from Eq. (\[mattera\]).
As it turns out, the gauge kinetic terms on the hidden brane do not spoil the gauge coupling unification if the volume of the extra dimension is large enough [@Hall:2001pg]. In that case the arbitrary coefficients of the gauge kinetic terms on the hidden and the visible brane get diluted and their contribution to the gauge couplings of the Standard Model can be neglected. The dominant contribution comes from the universal coefficient that belongs to the gauge kinetic term in the bulk obeying the full symmetry of the theory.
The hidden brane kinetic terms for the Higgs fields do not affect the mass relations $M_L = M_D \propto A$ and $M_{\nu}^{Dirac} = M_U \propto B$. These hidden-brane terms violate $SO(10)$ but respect $SU(5) \times U(1)$, and so will have the effect of changing the relative normalization of the $\overline{{\bf 5}}$ and ${\bf 5}$ of Higgs that are inside the same ${\bf 10}$ of $SO(10)$. However, the ${\bf 5}$ of Higgs and the $\overline{{\bf 5}}$ of Higgs that contribute to quark and lepton masses in this model come from different ${\bf 10}$’s of Higgs anyway. The former comes from ${\bf 10}_{1H}$, while the latter comes from ${\bf 10}_{2H}$. While the matrices $A$ and $B$ will be differently affected by the hidden-brane kinetic terms, the predictions that $M_L = M_D \propto A$ and $M_{\nu}^{Dirac} = M_U \propto B$ are not affected by that. The essential point is that these predictions depend only on the $SU(5)$ that is respected by the hidden-brane kinetic terms and not on the full $SO(10)$.
As noted earlier, the only massless modes of the Higgs fields are $h_1 \subset \Phi^{++}_{{\bf 5}^{-2}_1} \subset
{\bf 10}_{1H}$, $\overline{h}_2 \subset \Phi^{++}_{\overline{{\bf 5}}^2_2}
\subset {\bf 10}_{2H}$, $H \subset \Phi^{++}_{{\bf 10}^1} \subset
{\bf 16}_H$, and $\overline{H} \subset \Phi^{++}_{\overline{{\bf 10}}^{-1}}
\subset \overline{{\bf 16}}_H$. Therefore, after integrating over the fifth dimension, the terms in Eq. (\[higgs\]) yield in the superpotential of the low-energy effective theory the terms $h_1 \; H \; H + \overline{h}_2 \; \overline{H} \; \overline{H}$. These are just the same terms that are present in conventional four-dimensional flipped $SU(5)$ models to do the doublet-triplet splitting.
We assume that the $H$ and $\overline{H}$ acquire superlarge vacuum expectation values that break $SU(5) \times U(1)$ down to the Standard Model group. The tree-level scalar potential generated by the terms $h_1 H H + \overline{h}_2 \overline{H} \overline{H}$ is flat in this direction. However, as noted in [@Antoniadis:1987dx], this flatness can be lifted by radiative effects after supersymmetry is broken. It is also possible that additional terms in the Higgs superpotential on the visible brane can lead to a tree-level superpotential that produces the required symmetry breaking, as we shall see later.
Besides breaking the gauge symmetry from $SU(5) \times U(1)$ down to $SU(3) \times SU(2) \times U(1)$, the vacuum expectation values of the fields $H \subset {\bf 16}_H$ and $\overline{H} \subset
\overline{{\bf 16}}_H$ allow masses for the right-handed neutrinos. Such masses come from effective operators of the form ${\bf 16}_i {\bf 16}_j \overline{{\bf 16}}_H \overline{{\bf 16}}_H$. However, this product of fields has charge $+4$ under the symmetry $U(1)'$. Consequently, this symmetry must be spontaneously broken. It must be broken in such a way as to permit sufficiently large right-handed neutrino masses without at the same time allowing too large a $\mu$ parameter (which is the coefficient of the term ${\bf 10}_{1H}
{\bf 10}_{2H}$). This can be done in the following way (which we do not claim to be unique). Suppose that there are fields $S$ and $\overline{S}$ living on the brane at $O$ that are singlets under $SO(10)$ and that have $U(1)'$ charges $+1$ and $-1$ respectively. In the superpotential on the brane at $O$ there can be terms of the form $(\overline{S} S - M^2) X$, where $M = \epsilon M_{GUT}$, with $\epsilon \ll 1$. These terms force $\langle S \rangle =
\langle \overline{S} \rangle = M$. Let us suppose that on the brane at $O$ there are, in addition to the quark and lepton families in ${\bf 16}_i$, some leptons ${\bf 1}_i$ ($i=1,2,3$) that are $SO(10)$ singlets but have charge $-1$ under $U(1)'$. Then the following terms in the superpotential at $O$ are possible: $C_{ij} {\bf 16}_i {\bf 1}_j \overline{{\bf 16}}_H
\overline{S}/M_{*} + F_{ij} {\bf 1}_i {\bf 1}_j S^2/M_{*}$, where the dimensionless coefficients $C_{ij}$ and $F_{ij}$ are assumed to be of order one. The mass $M_{*}$ is an ultraviolet cutoff that specifies the scale at which new physics (eg. other dimensions beyond five, strings) become important. We take $M_{*}$ to be close to $M_{GUT}$ but, of course, somewhat larger. These terms give a mass matrix for the neutrinos that has the form $$(\nu_i \quad N^c_i \quad {\bf 1}_i) \left( \begin{array}{ccc}
0 & (M_{\nu}^{Dirac})_{ij} & 0 \\ (M_{\nu}^{Dirac})_{ji}
& 0 & C_{ij} \epsilon \overline{M} \\
0 & C_{ji} \epsilon \overline{M} & F_{ij} \epsilon^2 \overline{M}
\end{array} \right)
\left( \begin{array}{c} \nu_j \\ N^c_j \\ {\bf 1}_j \end{array}
\right),$$ where $\overline{M} \equiv M_{GUT}^2/M_{*}$. (Note that we have taken $\langle {\bf 16}_H \rangle = M_{GUT}$.) It is clear that the six superheavy neutrinos have masses of order $\epsilon \overline{M}$, whereas the three light (left-handed) neutrinos have masses of order $(M_{\nu}^{Dirac})^2/\overline{M}$. Taking the largest neutrino mass $m_3$ to be about $6 \times 10^{-2}$ eV, as suggested by the atmospheric neutrino data, and its Dirac mass to be $m_c
\cong 174$ GeV, as suggested by the relation $M_{\nu}^{Dirac} = M_U$ (which would hold in a minimal $SO(10)$ model), one has that $\overline{M} \sim 10^{15}$ GeV. This accords well with the assumption that $M_{*}$ is slightly larger than the GUT scale $M_{GUT} \sim 10^{16}$ GeV.
The reason that we have assumed that the parameter $\epsilon \equiv
\langle S \rangle/M_{GUT}$ is much smaller than one is that it suppresses certain dangerous operators. For example, $U(1)'$ allows the effective operator ${\bf 16}_i {\bf 16}_j {\bf 16}_k {\bf 16}_{\ell}
\overline{S}^4/M_{*}^5$. This gives a $d=5$ proton decay operator with coefficient of order $\epsilon^4 (1/M_{*})$. Sufficient suppression of proton decay requires that $\epsilon \sim 10^{-3}$ to $10^{-4}$. Similarly, $U(1)'$ allows the operator ${\bf 10}_{1H} {\bf 10}_{2H} S^4/M_{*}^3$. This gives a $\mu$ parameter for the MSSM doublet Higgs fields that is of order $\epsilon^4 M_{*}$. Requiring that this be no larger than the weak scale requires that $\epsilon$ be less than about $3 \times 10^{-4}$. This corresponds to right-handed neutrino masses of order $3 \times 10^{11}$ GeV. Such intermediate mass scales for $M_R$ are good from the point of view of leptogenesis [@leptogenesis].
The same singlet Higgs field $S$ can play a role in generating the vacuum expectation value for the spinor Higgs fields ${\bf 16}_H$ and $\overline{{\bf 16}}_H$. Such VEVs, as we have already noted, can arise due to radiative effects after SUSY breaking. But they can also arise at tree level from a term in the superpotential on the brane at $O$ of the form $(\lambda \overline{{\bf 16}}_H {\bf 16}_H - S^2) Y$, where $Y$ is a singlet superfield with $U(1)'$ charge of $-2$, and $\lambda \sim \epsilon^2$. Note that the $F$-terms of the fields ${\bf 16}_H$ and $\overline{{\bf 16}}_H$ force $\langle Y \rangle
= 0$, meaning that there is no mass term linking $\overline{{\bf 16}}_H$ to ${\bf 16}_H$ and thus $\overline{H}$ to $H$. The $U(1)'$ charge assignments allow the higher dimensional term $\overline{S}^2 \overline{{\bf 16}}_H {\bf 16}_H/M_{*}$. This will shift the VEV of $Y$, but the $F$-terms of the fields ${\bf 16}_H$ and $\overline{{\bf 16}}_H$ still enforce the condition that there is no mass term linking $\overline{{\bf 16}}_H$ to ${\bf 16}_H$.
Let us now examine the doublet-triplet splitting and proton decay problems. The terms $h_1 H \langle H \rangle + \overline{h}_2
\overline{H} \langle \overline{H} \rangle$ will couple the triplets in $h_1$ and $\overline{h}_2$ to those in $H$ and $\overline{H}$. The doublets in $h_1$ and $\overline{h}_2$ remain light and are the two doublets of the MSSM. There is no problem with $d=5$ proton decay, because the triplet partners of the MSSM Higgs doublets are not connected to each other. The triplets in $h_1$ and $H$ have no mass terms with the triplets in $\overline{h}_2$ and $\overline{H}$. Moreover, there are no unwanted light states contained in the Higgs multiplets ${\bf 10}_{1H}$, ${\bf 10}_{2H}$, ${\bf 16}_H$, $\overline{{\bf 16}}_H$. In the zero modes ($h_1$, $\overline{h}_2$, $H$, and $\overline{H}$), the doublets remain light, the triplets become superheavy by coupling to the VEVs of $H$ and $\overline{H}$, and the other gauge-non-singlet fields get eaten by the Higgs mechanism when $SU(5) \times U(1)$ breaks to the Standard Model group. All the non-zero modes, of course, have superheavy Kaluza-Klein masses. This is the crucial difference with four-dimensional theories in which flipped $SU(5)$ is embedded in $SO(10)$. In four dimensions, as we saw in the last section, the $SO(10)$ Higgs multiplets ${\bf 10}_{1H}$ and ${\bf 10}_{2H}$ when decomposed under $SU(5) \times U(1)$ contain not only $h_1$ and $\overline{h}_2$ but also $\overline{h}_1$ and $h_2$; and these multiplets have triplets that cannot be given mass without destroying the gauge hierarchy. Here, however, these extra pieces are all made heavy by the orbifold compactification, since they do not have parity $++$. Thus it is the fact that the unification of $SU(5) \times U(1)$ into $SO(10)$ occurs only in higher dimensions that allows the missing partner mechanism to be implemented.
We have seen that with what may be called the minimal Yukawa couplings ${\bf 16}_i {\bf 16}_j (A_{ij} {\bf 10}_{1H}
+ B_{ij} {\bf 10}_{2H})$ this model gives distinctive relations among mass matrices that are different from those that result in four dimensional models with minimal Yukawa couplings in either $SO(10)$ or flipped $SU(5)$. In particular, $M_L = M_D$, and $M_{\nu}^{Dirac} =
M_U$, with all these matrices being symmetric. This does give the desired relation $m_b = m_{\tau}$ at the unification scale, a result of the fact that flipped $SU(5)$ is embedded in $SO(10)$. However, this minimal set of Yukawa terms is clearly not enough to give a realistic model of quark and lepton masses.
Recently it has been found that realistic and simple models of quark and lepton masses can be constructed using so-called “lopsided" mass matrices [@Babu:1995hr; @Sato:1997hv; @Albright:1998vf; @Irges:1998ax]. The essential feature of such models is that the mass matrices of the down quarks and charged leptons are highly asymmetric and that $M_L \sim M_D^T$. Such a relationship between $M_L$ and $M_D^T$ is typical of models with an ordinary $SU(5)$, not flipped $SU(5)$. However, as we shall now see, because the flipped $SU(5)$ is here embedded in $SO(10)$ at the five-dimensional level, it is possible to obtain such a lopsided structure.
Suppose that one introduces on the visible brane not only spinors of quarks and leptons, but $SO(10)$ vectors as well. And suppose that there is in the bulk a spinor Higgs field ${\bf 16}'_H$ that has a weak-doublet component that contributes to the breaking of the electroweak symmetry. Then a diagram like that shown in Fig. \[masses\](a) may be possible. When decomposed under the $SU(5) \times U(1)$ subgroup, this diagram contains the two diagrams shown in Figs. \[masses\](b) and \[masses\](c). It is easy to see that these give contributions to $M_L$ and $M_D$ that are asymmetric and that are transposes of each other, just as needed to build “lopsided" models. The reason for this is that the diagram in Fig. \[masses\](a) directly depends only on the GUT-scale breaking done by the ${\bf 16}_H$ and not on that coming from orbifold compactification. The point is that the ${\bf 16}_H$ VEV by itself would only break $SO(10)$ down to the Georgi-Glashow $SU(5)$. (It is the orbifold compactification that breaks $SO(10)$ to the flipped $SU(5) \times U(1)$ group.) That is why this diagram leads to the kind of mass contributions that one expects from ordinary Georgi-Glashow $SU(5)$. This reasoning also shows that in order to introduce into the mass matrices contributions that break Georgi-Glashow $SU(5)$ it is necessary that the mass-splittings produced by the orbifold compactification be involved. For example, by mixing quarks and leptons that are on the visible brane with quarks and leptons in the bulk, it should be possible to break the (bad) minimal $SU(5)$ relations $m_s = m_{\mu}$ and $m_d = m_e$.
Gaugino mediated supersymmetry breaking
=======================================
In this section we address the issue of how to break $\mathcal{N}=1$ supersymmetry of our model below the compactification scale $M_C$. As it turns out, the solution allows the construction of viable SUSY breaking model that can easily satisfy present experimental constraints.
It is well known that the models with visible and hidden branes separated by extra dimension(s) naturally accommodate breaking of supersymmetry via gaugino mediation [@Kaplan:1999ac; @Chacko:1999mi]. The basic idea behind gaugino mediation in the models based on the orbifold compactification is as follows. The source of the SUSY breaking is localized at the hidden brane. It couples directly to the gauginos at that brane providing them with nonzero masses. If the gauge symmetry at the hidden brane is reduced with respect to the bulk gauge symmetry this coupling induces non-universal gaugino masses. For example, if the bulk symmetry is $SO(10)$ and hidden brane symmetry is flipped $SU(5)$ one obtains $M_3=M_2\neq M_1$. Here, $M_1$, $M_2$, and $M_3$ represent gaugino masses of the MSSM.
Following in the footsteps of [@Dermisek:2001hp], we take the source of the SUSY breaking to be a flipped $SU(5)$ singlet chiral superfield $Z$ localized on the hidden brane with the VEV $$\langle Z \rangle=\theta^2 F_Z.$$ The gaugino masses originate from the non-renormalizable operators at the hidden brane of the form $$\mathcal{L}^{Z}_5=\frac{1}{2}[\delta(y-\pi R/2)+\delta(y+\pi R/2)]
\int{{\rm d}^2\theta}\Big(\lambda_5^\prime\frac{Z}{M^2_*}W^{i\alpha}
W^i_\alpha+\lambda_1^\prime\frac{Z}{M^2_*}W^\alpha W_\alpha+{\rm h.c.}\Big),$$ where the first and the second term under the integral represent the $SU(5)$ and $U(1)$ part of the gauge group respectively. Corresponding gaugino masses generated in this way are $$\label{gauginos}
M_{SU(5)}=\frac{\lambda_5^\prime F_Z M_c}{M^2_*}, \qquad M_{U(1)}=\frac{\lambda_1^\prime F_Z M_c}{M^2_*},$$ which translates into the following MSSM gaugino masses (we normalize the generators of $SO(10)$ demanding that $k=1/2$) $$\label{gauginos1}
\frac{M_1}{g_1^2}=\frac{1}{25} \frac{M_{SU(5)}}{g_{SU(5)}^2}+\frac{24}{25} \frac{M_{U(1)}}{g_{U(1)}^2},
\qquad M_2=M_{SU(5)},\qquad M_3=M_{SU(5)}.$$ Here $g_{SU(5)}$, and $g_{U(1)}$ are gauge coupling constants of the $SU(5)$ and $U(1)$ gauge groups respectively, while $g_1$ represents the $U(1)_Y$ gauge coupling constant of the Standard Model (normalized as in GUTs). The relations of Eq. (\[gauginos1\]), which is valid at the compactification scale $M_C$, show that the gaugino mass $M_1$ can in general be completely different from the mass $M_2=M_3$ due to their different origins. Namely, the mass $M_1$ is dominated by the $U(1)$ sector of the theory as oppose to the masses $M_2$ and $M_3$ that have their origin in the $SU(5)$ part of the theory. We will later see that this feature of non-universality of gaugino masses allows the construction of the theory of SUSY breaking that leads to the realistic mass spectrum.
At this point we note that the natural scale for $\sqrt{F_Z}$ is the cutoff scale $M_*$. (For the reasons that have to do with gauge coupling unification we take $(M_C \sim 10^{16}$ GeV$) < (M_{GUT} = 1.2
\times 10^{16}$ GeV$) < (M_* \sim 10 M_C)$ [@Dermisek:2001hp].) This implies that masses in Eq. (\[gauginos\]) are close to the compactification scale $M_C$ if the dimensionless coefficients $\lambda_1^\prime$ and $\lambda_5^\prime$ are taken to be of order one. To obtain SUSY breaking masses that are in the TeV range we need to decrease the value of $F_Z$ in a way that does not involve any fine-tuning. To do that we propose to use the shining mechanism [@Arkani-Hamed:1998; @Arkani-Hamed:1999pv] which can reduce the natural scale of $F_Z$ by an exponential factor.
The shining mechanism requires the existance of a source $J$ that is localized on the visible brane and a massive hypermultiplet in the bulk. The hypermultiplet of mass $m$ is taken to be a gauge singlet and has couplings with both the source and the superfield $Z$. These couplings can be arranged in a manner that leaves the superfield $Z$ with the nonzero F-term $F_Z \sim J {\rm exp}(-\pi m R/2)$ after the massive hypermultiplet is integrated out [@Arkani-Hamed:1999pv]. If the mass $m$ is taken to be of order $M_*$ the $\sqrt{F_Z}$ will be of order $10^{12}$ GeV which gives desired TeV scale masses for gauginos in Eq. (\[gauginos\]).
The matter fields in our model reside on the visible brane. Thus, due to the spatial separation between the branes the soft SUSY breaking scalar masses and trilinear couplings are negligible at the compactification scale. This is good because the number of the soft SUSY breaking parameters one has to consider is reduced with respect to the usual set.
There are two additional positive features of the gaugino mediated supersymmetry breaking models with the non-universal gaugino masses. Firstly, the renormalization group running of scalar masses and trilinear couplings between $M_C$ and electroweak scale is significantly affected by gaugino masses but these contributions, being flavor blind, do not cause any disastrous flavor violating effects. Secondly, non-universality opens up the possibility for the deviation from the experimentally disfavored prediction of the models with universal gaugino mass of stau being the lightest supersymmetric particle (LSP). (The last statement holds for $M_C<M_{GUT}$ which is exactly the case we have.)
The class of models with non-universal gaugino mediated supersymmetry breaking has been studied in more details by Baer et al. [@Baer:2002by]. Their numerical study of the allowed region of SUSY parameter space shows that viable models with acceptable mass spectrum and neutral LSP particle can be obtained. The study includes the case of completely independent $M_3$, $M_2$, and $M_1$, as well as the case where $M_1$ is a definite linear combination (determined by group theory) of $M_2$ and $M_3$. (The former case can be seen as a consequence of orbifold reduction of $SU(5)$ down to the Standard Model group on the hidden brane as in Ref. [@Hall:2001pg] and the latter one follows from the reduction of $SO(10)$ down to the Pati-Salam group as in Ref. [@Dermisek:2001hp].) We have an intermediate scenario where $M_1$ is independent of $M_2$ and $M_3$ which are made equal due to the $SU(5)$ part of the flipped $SU(5)$. (This possibility was considered in Ref. [@Hall:2001xr] in the context of a six dimensional $SO(10)$ model.)
It is not difficult to adapt the analysis of Baer et al. to our model to conclude that for large enough $M_1$ (i.e. $|M_1| >|M_2|,M_2=M_3)$ at the compactification scale $M_C$ a viable region of parameter space opens up regardless of $\tan \beta $ value yielding realistic mass spectrum with the LSP being wino-like or a mixture of higgsino and bino. An example of this behavior is shown in Fig. \[region\].
At the end we observe that if we had the case of $SO(10)$ being reduced on the hidden brane to the Georgi-Glashow $SU(5)$ with an extra $U(1)$ symmetry we would not only be prevented from using the simple form of the missing partner mechanism but would also obtain universal gaugino masses $M_1=M_2=M_3$.
Conclusions
===========
We have seen that by embedding a four-dimensional flipped $SU(5)$ model into a five-dimensional $SO(10)$ model the advantages of flipped $SU(5)$ can be maintained while avoiding its well-known drawbacks. The two main drawbacks are the loss of unification of gauge couplings and the loss of the possibility of relating down quark masses to charged lepton masses, and therefore of obtaining desirable predictions such as $m_b = m_{\tau}$ and realistic quark and lepton mass schemes such as those based on “lopsided" mass matrices. By embedding $SU(5) \times U(1)$ in $SO(10)$, the unification of gauge couplings is restored. There are corrections to this unification, coming for example from gauge kinetic terms on the hidden brane; however, these have been argued to be small [@Hall:2001pg]. The embedding in $SO(10)$ also yields relationships between the charged lepton and down quark mass matrices. We have also found that interesting patterns of quark and lepton masses are possible that are different from those encountered in four-dimensional grand unified theories, for example $M_L = M_D \neq M_{\nu}^{Dirac} = M_U$.
Embedding flipped $SU(5)$ in $SO(10)$ in four dimensions is well known to destroy the missing partner mechanism for doublet-triplet splitting, which is one of the most elegant features of flipped $SU(5)$. However, when the unification in $SO(10)$ takes place in higher dimensions and the breaking to $SU(5) \times U(1)$ is achieved through orbifold compactification, then the missing partner mechanism can still operate, as we have shown. One of the advantages of the missing partner mechanism in flipped $SU(5)$ is that it kills the dangerous $d=5$ proton decay operators that plague supersymmetric grand unified theories.
Thus in extra dimensions it is possible to have the best of both worlds, the best of $SO(10)$ combined with the best of flipped $SU(5)$. One of the distinctive predictions of the flipped $SU(5)$ scheme that we have presented is that the gaugino masses will have the pattern $M_3 = M_2 \neq M_1$. The fact that $M_1$ is independent of $M_2$ and $M_3$ allows a much larger viable region of parameter space for the MSSM.
Acknowledgments
===============
I. D. thanks R. Derm' išek for discussion.
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![\[dimension5\] The kind of graph that gives rise to $d=5$ proton decay operators.](dimension5.eps){width="3in"}
![\[masses\] (a) A diagram that can give operators producing “lopsided" contributions to $M_D$ and $M_L$. (b) A term in its $SU(5) \times U(1)$ decomposition that contributes to $M_D$. (c) A term in its $SU(5) \times U(1)$ decomposition that contributes to $M_L$.](massabc.eps){width="3in"}
![\[region\] This diagram represents the results of numerical analysis of Baer et al. [@Baer:2002by] for the case of gaugino mediated SUSY breaking scenario in the flipped $SU(5)$ setting ($M_2=M_3 \neq M_1$) for $\tan \beta = 30$ and $\mu > 0 $. The allowed region in $M_1$ vs. $M_2=M_3$ plane is shown in dotted light gray. The excluded regions are white (due to presence of tachyonic particles in mass spectrum), light gray (due to lack of radiative breakdown of EW symmetry), gray (due to LEP constraint), dark gray (due to LEP2 constraint), and crossed gray (due to the fact that charged particle is LSP). Vertical black line is where $M_H=114$ GeV. For a full discussion of numerical methods and assumptions used in the analysis see Ref. [@Baer:2002by].](Untitled-2.eps){width="5in"}
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---
abstract: 'Oblique incidence of a $p$-polarized laser beam on a fully ionized plasma with a low density plasma corona is investigated numerically by Particle-In-Cell and Vlasov simulations in two dimensions. A single narrow self-focused current jet of energetic electrons is observed to be projected into the corona nearly normal to the target. Magnetic fields enhance the penetration depth of the electrons into the corona. A scaling law for the angle of the ejected electrons with incident laser intensity is given.'
address:
- 'Theoretische Quantenelektronik, TU Darmstadt, Hochschulstrasse 4A, 64289 Darmstadt, Germany'
- 'Institute of Laser Engineering, Osaka University, 2-6 Yamada-oka, Suita Osaka, 565, Japan'
- '$^{(1)}$Department of Electromagnetic Energy Engineering and Institute of Laser Engineering, Osaka University, 2-6 Yamada-oka, Suita Osaka, 565, Japan'
author:
- 'H. Ruhl[^1]'
- 'Y. Sentoku[^2], K. Mima, K.A. Tanaka$^{(1)}$, and R. Kodama'
title: 'Collimated Electron Jets by Intense Laser Beam-Plasma Surface Interaction under Oblique Incidence'
---
The availability of tabletop high intensity laser systems has lead to the investigation of novel regimes of short pulse laser-plasma interaction. Recently the emission of collimated electron jets under specular angles with respect to the density normal direction have been observed for an obliquely incident laser beam on a steep density plasma [@bastiani].
When a target is irradiated by an intense laser pulse above the field ionization threshold it quickly ionizes [@bauer]. For sufficiently long laser pulse irradiation the plasma present on the surface gradually expands into the vacuum with the ion acoustic speed. Hence, a plasma corona is formed. For short laser pulses there is not enough time for hydrodynamic expansion. Short pulse simulations show however, that an ion shelf is formed on a typical time scale $t_{\text{s}}=\omega_{\text{0}}^{-1}
(m_{\text{i}}/Zm_{\text{e}})^{1/2}$ due to the generation of a strong electric field at the plasma-vacuum interface [@gibbon0]. This ion shelf represents a low density plasma corona.
There are different mechanisms which can lead to collimated electron jets when an intense laser pulse interacts with a vastly over-dense, steep density plasma that has a low density plasma corona. One effect that plays a role in the interaction is the Brunel effect [@brunel] which works for oblique incidence. Here the electrons are accelerated into the vacuum as well as into the target by the electric field present along the density gradient. The coronal plasma is expected to collimate and enhance the range of the ejected electrons in two (2D) or three (3D) spatial dimensions by quasi-steady magnetic field generation. The collimating effect of quasi-steady fields has recently been addressed in [@gorbunov; @pukhov] in a different context.
To investigate the phenomenon just outlined in more detail we perform Particle-In-Cell (PIC) and Vlasov (VL) simulations both in two spatial dimensions (2D). In both approaches we do not simulate the evolution of the corona self-consistently but treat it parametrically with ions fixed. In our PIC simulations we rotate the target and in our VL simulations we boost to the frame of normal incidence to model oblique incidence of the laser beam. The boost frame method is well established in 1D [@bourdier; @ruhl1; @gibbon1]. However, in 2D using the boost frame is very helpful in establishing the physics of the underlying laser-plasma interaction.
We investigate the interaction of a $p$-polarized laser beam incident under angles of $30^{\circ}$ (PIC) and $45^{\circ}$ (VL) on a preformed fully ionized target with an under-dense plasma corona in front of it. In both simulations the laser beam has a duration of about $100 \text{fs}$. For the PIC case the laser beam intensity is $2.0 \cdot 10^{18} \text{W/cm}^{2}$ and for the VL case $1.0 \cdot
10^{17} \text{W/cm}^{2}$. The laser wavelength in the simulations is $1 \mu \text{m}$ with beam diameters of $8 \; \mu \text{m}$ (PIC) and $5 \; \mu \text{m}$ (VL) at full-width-half-maximum. The coordinates of the simulation box are $x$ and $y$, respectively. The size of the simulation box is $23 \; \mu \text{m} \times 23 \; \mu
\text{m}$ for the PIC simulations and $6 \; \mu \text{m} \times 12 \;
\mu \text{m}$ for the VL simulations. In our PIC simulations we assume mirror-reflecting boundary conditions for the electrons in $x$- and $y$-directions. In the VL simulations we use periodic boundary conditions in $y$-direction. Electrons leaving the simulation box at $x=6\mu \text{m}$ are replaced by thermal electrons and at $x=0 \mu \text{m}$ they are allowed to escape. We note that mirror-reflecting boundary conditions in our PIC simulations force us to increase the simulation box for long simulation times. The distribution functions for the electrons and ions needed for the VL simulations have two momentum directions $p_{\text{x}}$ and $p_{\text{y}}$ in addition. The quasi-particle number per cell used in the PIC simulations is $50$ for each species. The fully ionized plasma density is $4 n_{\text{c}}$ (PIC) and $8 n_{\text{c}}$ (VL). In both simulations we assume a uniform low density plasma corona with a density of $0.1 n_{\text{c}}$ in front of the target.
Plot (a) of Figure \[fig:field1\] gives the quasi-steady magnetic field $B_{\text{z}}$ in front of the target inclined at $30^{\circ}$ , as obtained by PIC simulations. The peak magnitude of the normalized magnetic field is $0.62$, which corresponds to approximately $30 \;
\text{MG}$. It changes polarity very rapidly along the density gradient revealing the presence of a very localized self-focused current jet. The low density plasma corona guarantees quasi-neutrality and helps to generate the magnetic field in front of the target. Plot (b) of the same figure shows the electron energy density. We find a collimated electron jet which coincides with the quasi-steady magnetic field from plot (a). For the parameters of plot (b) the ejection angle is approximately $17^{\circ}$ form the target normal. There are also fast electrons injected into the over-dense plasma. We again observe that they are almost normal to the target surface. Figure \[fig:field1\] (c) shows the instantaneous plot of the electron energy density with over-plotted positive $B_{\text{z}}$ field indicating the phase of the laser field. It is clearly seen that the outgoing electrons are generated on the target surface once per laser cycle by the Brunel absorption mechanism [@brunel; @ruhl1] and are bunched on the scale of the laser wavelength consequently. The range of the electrons is enhanced. A similar result we obtain from our VL simulations which make use of boost frame coordinates.
To illustrate how the boost frame approach for oblique incidence in 2D works we briefly derive the correct boundary conditions for the laser pulse in the boosted frame. We start by defining an arbitrary pulse envelope function $z(x,y,t)$ in the lab-frame. Next we perform a Lorentz rotation of electromagnetic fields about $(x_{\text{0}},y_{\text{0}})$. In the final step we boost the latter to the frame of normal incidence for which the longitudinal field $E_{\text{x}}$ disappears. We obtain
$$\begin{aligned}
\label{boost_ep}
E^{B}_x &=& 0 \; , \qquad
E^{B}_y = \frac{1}{\bar{\gamma}} \;
z \left( x_r,y_r,t \right) \; , \qquad
B^{B}_z = \frac{1}{c\bar{\gamma}} \;
z \left( x_r,y_r,t \right) \; , \end{aligned}$$
where
$$\begin{aligned}
\label{rotation_x}
x_r&=&\frac{1}{\bar \gamma} (x-x_0) + (y-y_0) \bar \beta \; , \qquad
y_r=\frac{1}{\bar \gamma} (y-y_0) - (x-x_0) \bar \beta \; ,\end{aligned}$$
with
$$\begin{aligned}
\label{boost_trafo_ct}
t &=& \frac{\bar{\gamma} \bar{\beta}}{c} y^B \; , \qquad
x = -ct^B \; , \qquad
y = \bar{\gamma} y^B \; . \end{aligned}$$
The function $z$ is the same function as in the lab-frame. For the relativistic factors we have $\bar \beta=\sin \theta$ and $\bar \gamma = 1/\cos \theta$, where $\theta$ is the angle of incidence. Plot (a) of Figure \[fig:field2\] illustrates the incident time resolved electromagnetic field $E_{\text{y}}$ for a Gaussian pulse envelope. Plot (b) of the same figure gives the incident time resolved electromagnetic field $E_{\text{y}}$ of the simulations.
Plot (a) of Figure \[fig:field3\] gives the quasi-steady magnetic field in the plasma corona in front of the over-dense plasma target. Plot (b) of the same figure gives the quasi-steady magnetic field with the quasi-steady $B^2_{\text{z}}$ over-plotted (red solid lines). Plot (c) of Figure \[fig:field3\] gives the quasi-steady magnetic field with the quasi-steady longitudinal current density $j_{\text{xe}}$ over-plotted (red dashed lines).
Since the current density $j_{\text{xe}}$ is invariant under Lorentz boosts along $y$ it may serve as a quantity from which to determine the direction of the electron jets. We now introduce the coordinates $\chi =x^{\text{B}}$ and $\xi =y^{\text{B}}+\bar \beta
c t^{\text{B}}$ which move along with the background plasma current present in the boosted frame. Since the time-averaged current density $\left< j^{\text{B}}_{\text{xe}} \right>$ in the co-moving coordinates varies slowly with time we obtain $\left< j^{B}_{xe} \right> \left(
x^{B}, y^{B},t^{B} \right) = \left< j^{B}_{xe} \right> \left( \chi,
\xi \right)$. This yields
$$\begin{aligned}
\label{trafo}
\left< j^{L}_{xe} \right> \left( \chi, \xi \right) = \left<
j^{B}_{xe} \right> \left( \chi, \bar{\gamma} \xi \right) \; .\end{aligned}$$
The direction of the collimated electron jets in the lab frame can now be calculated from the direction of the current density in the boosted frame. Plot (c) of Figure \[fig:field3\] gives $\left<
j^{B}_{xe} \right>$. The direction of the emitted electrons is indicated by the white solid line plotted in the figure. We obtain a mean emission angle of $20^{\circ}$ in the boosted frame and $14^{\circ}$ in the lab frame. We note that the lab frame is dilated in transverse direction when viewed from the boosted frame and hence the emission angle in the boost frame is larger by a factor of $\bar \gamma =1/\sqrt{1-\bar \beta^2}$ as indicated by Equation (\[trafo\]).
In boost frame coordinates we may easily analyze the physical mechanism that leads to the large areal quasi-steady magnetic field and the direction of the ejected electrons. We recall that in the boosted frame we have a constant background fluid velocity $u_{\text{B}}=c \sin \theta$ which approaches speed of light for large angles of incidence. In this frame the polarization of the magnetic field vector of the incident laser beam is normal to the $xy$-plane and to the flow direction of the background current. If the laser intensity is small enough as in [@bastiani] and the angle of incidence sufficiently large the boost velocity exceeds the laser quiver velocity. The driving force under these conditions is exerted predominantly by the oscillating magnetic field of the laser beam (see the red solid contour lines of $B^2_{\text{z}}$ plotted over $B_{\text{z}}$ in plot (b) of Figure \[fig:field3\] for the location of the force). The resulting force is ${\bf F}=-e \; {\bf u}_{\text{B}} \times {\bf B}$ and is capable of ejecting electrons out of the surface at a rate of once per laser cycle. This is the Brunel mechanism [@brunel]. The quasi-steady magnetic field in the plasma corona is generated by the electron current emitted from the target. The polarization of the magnetic field is such that it collimates the electrons propagating through the plasma corona.
To derive an approximate criterion for the angle range under which the fast electrons are emitted from the target surface we assume that the laser target interaction in the boosted frame is quasi-one-dimensional. Since the full-width-half-maximum of the laser beams in our simulations is at least $5 \; \mu \text{m}$ and the intensities are sufficiently low to prevent target imprinting we believe that this assumption is justified. We next rewrite the Vlasov equation in the boosted frame [@ruhl3] and solve it for an initial Maxwellian. We approximate the plasma-vacuum interface by a step-like density profile with $n(x)=n_{\text{0}}$ for $x>0$ and treat the ions as immobile. We obtain for the distribution function
$$\begin{aligned}
\label{df}
f(t)&=& \frac{n_0}{\sqrt{2\pi}^3 m^3 v^3_{th}} \;
\exp \left( -\frac{p_x^2(0)+p_z^2(0)}{2m^2 v_{th}^2} \right) \;
\exp \left( -\frac{(p_y(0)+\bar{\beta} \bar{\gamma} mc)^2}
{2\bar{\gamma}^2 m^2 v_{th}^2} \right) \; , \end{aligned}$$
and for the equations of motion
$$\begin{aligned}
\label{x}
x(\tau)&=&x-\int^t_{\tau} d\eta \; v_x(\eta) \; , \\
\label{px}
p_x(\tau)&=&p_x+e\int^t_{\tau} d\eta \; \left[ E_x(x(\eta),\eta)+v_y(\eta)
\; \partial_x A_y(x(\eta),\eta) \right] \; , \\
\label{py}
p_y(\tau)&=&p_y+e\left[ A_y(x(\tau),\tau)-A_y(x,t) \right] \; , \\
\label{trajectories_pz}
\label{pz}
p_z(\tau)&=&p_z \; , \end{aligned}$$
with
$$\begin{aligned}
\label{vx}
v_{x/y}(\tau)&=&\frac{cp_x(\tau)}
{\sqrt{m^2c^2+p^2_x(\tau)+p^2_y(\tau)+p^2_z(\tau)}} \; . \end{aligned}$$
Equations (\[py\]) and (\[pz\]) indicate lateral canonical momentum conservation in boost frame coordinates. We now assume that $A_{\text{y}}$ has a harmonic time dependence. Making use of Equations (\[df\]) and (\[py\]) and assuming $v_{\text{x}} \ll c$ or $v_{\text{x}} \approx c$ we obtain $\langle p_{\text{y}} \rangle \approx -\bar{\beta} \bar{\gamma}
mc$. The quantity $\langle p_{\text{y}} \rangle$ denotes the ensemble and time averaged transverse momentum. Treating $\langle p_{\text{x}} \rangle$ as a free parameter and transforming back to the lab frame yields
$$\begin{aligned}
\label{angle}
\langle p^L_y \rangle&=&\bar{\gamma}^2 \bar{\beta}mc
\left( \sqrt{1+\frac{\langle p^2_x \rangle}
{\bar{\gamma}^2m^2c^2}} -1 \right) \; , \qquad
\langle p^L_x \rangle = \langle p_x \rangle \; .\end{aligned}$$
The ejection angle is now given by $\tan \theta^{'}=
\langle p^L_{\text{y}} \rangle/\langle p^L_{\text{x}} \rangle$. For $\langle p_x \rangle \rightarrow \infty$ we obtain $\tan \theta^{'}
= \bar{\beta}\bar{\gamma} = \tan \theta$. This means that only ultrarelativistic electrons are ejected at very close to specular direction. For smaller longitudinal momenta $\langle p_x \rangle$ we expect that the electrons are emitted at angles that are smaller than the angle for specular emission as observed in our simulations. Assuming that the mean fast electron momentum in $x$-direction is given by $\langle p_x \rangle / \bar{\gamma} mc \approx
\sqrt{ \alpha \; I\lambda^2}$ we thus obtain
$$\begin{aligned}
\label{angle1}
\tan \theta^{'}&=&\frac{\sqrt{1+\alpha I\lambda^2}-1}{\sqrt{\alpha
I\lambda^2}} \; \tan \theta \; .\end{aligned}$$
Equation (\[angle1\]) looses validity as soon as target deformations start to become significant. The validity also depends on the accuracy of the mean longitudinal momentum given as a function of intensity. For $I\lambda^2=1.0 \cdot 10^{17} \mbox{Wcm}^{-2} \mu \text{m}^2$ we obtain an ejection angle of $\theta^{'}=14^{\circ}$ and for $I\lambda^2=2.0 \cdot
10^{18} \mbox{Wcm}^{-2} \mu \text{m}^2$ we obtain $\theta^{'}=17^{\circ}$ from the simulations. This yields $\alpha^{-1} \approx
8.0 \cdot 10^{17} \text{Wcm}^{-2} \mu \text{m}^2$.
In conclusion, we have demonstrated with the help of two different simulation techniques that collimated electrons with enhanced range can be emitted from an over-dense target if a low density plasma corona is present. In addition, we have shown that fast electrons are injected into the over-dense plasma. Both, the ejection and injection directions are almost along the density normal direction for $p$-polarized light. By a transformation to the moving frame in which the laser pulse appears to be normally incident we were able to give a criterion for the angle range of the emitted electrons with ejection momentum. We find that for a planar interaction interface only speed of light electrons can be emitted at specular direction for $p$-polarized light. Less energetic electrons appear under almost normal emission angles due to a lack of lateral momentum transfer. This analytical result is in qualitative agreement with our numerical observations. We note that in addition to the mechanism outlined in this paper other mechanisms of fast electron generation like wake-field acceleration in the corona may exist leading to different emission angles.
S. Bastiani [*et al.*]{}, Phys. Rev. [**E 56**]{}, 7179 (1997). D. Bauer [*et al.*]{}, Phys. Rev. [**E 58**]{}, 2436 (1998). Paul Gibbon, Phys. Rev. Lett. [**73**]{}, 664 (1994). F. Brunel, Phys. Rev. Lett [**59**]{}, 52 (1987). L. Gorbunov [*et al.*]{}, Phys. Plasmas [**4**]{}, 4358 (1997). A. Pukhov and J. Meyer-ter-Vehn [**79**]{}, 2686 (1997). A. Bourdier, Phys. Fluids [**26**]{}, 1804 (1983). H. Ruhl and P. Mulser, Phys. Lett. A [**205**]{}, 388 (1995). Paul Gibbon, Phys. Rev. Lett. [**73**]{}, 664 (1994). H. Ruhl and A. Cairns, Phys. Plasmas [**4**]{}, 2246 (1997).
[^1]: [email protected]
[^2]: [email protected]
|
---
abstract: 'In the ALICE experiment, at forward rapidity ($2.5 < y < 4$), the production of heavy quarkonium states is measured via their $\mu^+\mu^-$ decay channels. We present the first measurement of inclusive J/$\psi$ production, down to $p_{\rm T} = 0$, from Pb-Pb data collected at the LHC at $\sqrt{s_{\rm NN}}=2.76$ TeV. Preliminary results on the nuclear modification factor ($R_{\rm AA}$) and the central to peripheral nuclear modification factor ($R_{\rm CP}$) show J/$\psi$ suppression with no significant centrality dependence and an integrated $R_{\rm AA}^{0-80\%}=0.49\pm0.03({\rm stat.})\pm0.11(\rm syst.)$.'
address: 'Subatech (Université de Nantes, Ecole des Mines and CNRS/IN2P3), Nantes, France'
author:
- 'P. Pillot, for the ALICE Collaboration'
---
Heavy quarkonium states have long been proposed as a sensitive probe of the strongly-interacting deconfined medium expected to be formed in the early stages of high-energy heavy-ion collisions [@Satz86]. In particular, a measurement of the dissociation probability of different quarkonium states is expected to provide an estimate of the initial temperature of the medium. Suppression of J/$\psi$ production beyond that expected from cold nuclear matter (CNM) effects (nuclear absorption, has indeed been observed at SPS and RHIC energies [@NA60; @PHENIX07; @PHENIX11], but several questions are left open. At the LHC, larger suppression might be expected, due to the higher initial temperature. However, according to regeneration scenarios [@PBM00], with $\sim$ 10 times more $c\overline c$ pairs produced in central Pb-Pb collisions compared to RHIC, an enhancement of the J/$\psi$ production could also be observed. In the ALICE experiment, at forward rapidity ($2.5 < y < 4$), the inclusive production of heavy quarkonium states is measured down to $p_{\rm T} = 0$ via their $\mu^+\mu^-$ decay channels in the muon spectrometer, as describe in [@ALICEpp].
In fall 2010, the LHC delivered the first Pb-Pb collisions at a center of mass energy $\sqrt{s_{\rm NN}}=2.76$ TeV. ALICE collected data with a minimum bias (MB) trigger, defined as the logical AND between signals from the pixel detector (SPD) covering the range $|\eta|<2$ and the two scintillator arrays of the VZERO detector covering the ranges $2.8<\eta<5.1$ and $-3.7<\eta<-1.7$, in coincidence with two beam pick-up counters, one on each side of the interaction region. The centrality of the collision has been determined from the amplitude of the VZERO signal [@Alberica]. The total data sample available for physics analysis amounts to $17\cdot10^6$ MB events. Additional cuts have then been applied to improve the purity of the muon sample, in particular by requiring both particles in the pair to be detected in the muon trigger stations. Furthermore, we perform the cuts $-4 < \eta < -2.5$ and $17.6 < R_{\rm abs} < 89$ cm, where $R_{\rm abs}$ is the radial coordinate of the track at the end of the hadronic absorber located in front of the spectrometer, to select muons in the geometrical acceptance of the detector.
![\[fig:mass\]Invariant mass distribution for opposite-sign muon pairs in the centrality class 0-10% before (left) and after (right) mixed-event combinatorial-background subtraction, with the result of the corresponding fits.](figure1Left.eps "fig:"){width="7.8cm"} ![\[fig:mass\]Invariant mass distribution for opposite-sign muon pairs in the centrality class 0-10% before (left) and after (right) mixed-event combinatorial-background subtraction, with the result of the corresponding fits.](figure1Right.eps "fig:"){width="7.8cm"}
After these selections, the data sample has been divided into four centrality classes: , 10–20%, 20–40% and 40–80% of the inelastic Pb-Pb cross section. In each of these sub-samples, the number of J/$\psi$ particles has been extracted by using two different methods. We have first fitted the opposite-sign dimuon invariant mass distribution with a Crystal Ball (CB) function (a gaussian with polynomial tails) to reproduce the J/$\psi$ shape and a sum of two exponentials to describe the underlying continuum (Fig. ). Alternatively, we have subtracted the combinatorial background using the event-mixing technique and fitted the resulting mass distribution with a CB function and an exponential or a straight line to describe the remaining background (Fig. \[fig:mass\]-right). The width of the J/$\psi$ mass peak depends on the resolution of the spectrometer which could, $a~priori$, be affected by the detector occupancy which increases with centrality. This effect has been evaluated by embedding simulated J/$\psi$ decays to muons into real events and no modification has been observed. Such a conclusion has been confirmed by a direct measurement of the tracking chamber resolution versus centrality using reconstructed tracks. Therefore, the same CB line shape can be used for all centrality classes. Several parameters have been tested for the CB tails, either fixing them to the values obtained in p-p collisions (where the signal over background ratio is more favorable) or in simulations. For each of these choices, the mean and width of the gaussian part have been obtained by fitting the integrated mass distribution in the centrality range and then varying the width by $\pm$ 1 standard deviation to account for uncertainties (varying the mean has turned out to have negligible effect in comparison). The number of J/$\psi$ particles in each centrality class $i$ ($N_{\rm J/\psi}^i$), as well as the ratios $N_{\rm J/\psi}^i / N_{\rm J/\psi}^{40-80\%}$, have been determined as the average of the results obtained with the two fitting approaches and the various CB parameterizations, while the corresponding systematic uncertainties have been defined as two times the RMS of these results, which also approximately corresponds to the maximum difference with respect to the mean value. The largest uncertainties have been obtained for the most central class and amount to 19% and 12% for $N_{\rm J/\psi}^{0-10\%}$ and $N_{\rm J/\psi}^{0-10\%} / N_{\rm J/\psi}^{40-80\%}$ respectively.
In order to extract the inclusive J/$\psi$ yield, $N_{\rm J/\psi}^i$ has been normalized to the number of MB events in the corresponding centrality class and further corrected for the branching ratio of the dimuon decay channel and the acceptance times efficiency ($A\times\epsilon$) of the detector. This latter quantity has been determined from MC simulations. The generated J/$\psi$ $p_{\rm T}$ and $y$ distributions have been interpolated from existing measurements [@Bossu], including shadowing effects from EKS98 calculations [@EKS]. The efficiencies of the muon trigger chambers have been measured directly from data then applied to the simulations. For the tracking apparatus, the time-dependent status of each electronic channel and their run-by-run evolution have been taken into account as well as the residual misalignment of the detection elements. We thus obtained a run-averaged value of $A\times\epsilon=19.4\%$, with a 7% relative systematic uncertainty. The dependence of the tracking efficiency with the detector occupancy has also been evaluated using the embedding technique. A small decrease of this efficiency when increasing the centrality ($-2\%$ in the most central class) has been observed. This variation has been confirmed by a measurement of the tracking efficiency performed directly from data, and is included in the systematic uncertainties.
To measure the nuclear modification factors ($R_{\rm AA}^i$), the J/$\psi$ yield in the centrality class $i$ has been normalized to the inclusive J/$\psi$ cross-section measured in p-p collisions in the same rapidity domain at the same energy ($\sigma_{\rm J/\psi}^{\rm inclusive} = 3.46\pm0.13({\rm stat.})\pm0.32({\rm syst.})\pm0.28({\rm syst.lumi.})\mu {\rm b}$) [@Roberta] and scaled by the corresponding nuclear overlap function ($T_{\rm AA}^i$) calculated using the Glauber model, while the ratios $N_{\rm J/\psi}^i / N_{\rm J/\psi}^{40-80\%}$ have been normalized to the ratios $T_{\rm AA}^i / T_{\rm AA}^{40-80\%}$ to extract the central to peripheral nuclear modification factors ($R_{\rm CP}^i$). The systematic uncertainties on the Glauber model calculations are 4% and 6% for $T_{\rm AA}^{0-10\%}$ and $T_{\rm AA}^{0-10\%} / T_{\rm AA}^{40-80\%}$ respectively.
![\[fig:RAARCP\]Left: J/$\psi$ $R_{\rm AA}$ as a function of $\langle N_{\rm part} \rangle$ compared with PHENIX results in Au-Au collisions at $\sqrt{s_{\rm NN}}=200$ GeV. Right: J/$\psi$ $R_{\rm CP}$ as a function of centrality compared with ATLAS results. Error bars represent the statistical uncertainties, open boxes represent the centrality-dependent systematic uncertainties while the centrality independent uncertainties are shown by filled boxes.](figure2Left.eps "fig:"){width="7.9cm"} ![\[fig:RAARCP\]Left: J/$\psi$ $R_{\rm AA}$ as a function of $\langle N_{\rm part} \rangle$ compared with PHENIX results in Au-Au collisions at $\sqrt{s_{\rm NN}}=200$ GeV. Right: J/$\psi$ $R_{\rm CP}$ as a function of centrality compared with ATLAS results. Error bars represent the statistical uncertainties, open boxes represent the centrality-dependent systematic uncertainties while the centrality independent uncertainties are shown by filled boxes.](figure2Right.eps "fig:"){width="7.9cm"}
The inclusive J/$\psi$ $R_{\rm AA}$ are shown in Fig. \[fig:RAARCP\]-left as a function of the average number of nucleons participating to the collision ($\langle N_{\rm part} \rangle$) calculated using the Glauber model. To account for the bias due to our large centrality bins, $\langle N_{\rm part} \rangle$ has been weighted by the number of binary nucleon-nucleon collisions ($N_{\rm coll}$), which is expected to be the scaling variable of the J/$\psi$ production cross-section in A-A, in absence of nuclear matter effects. This correction is small, except for the most peripheral bin where $\langle N_{\rm part} \rangle$ = 46 while the weighted value is 70. These results show no significant dependence on centrality, and the integrated $R_{\rm AA}^{0-80\%}=0.49\pm0.03({\rm stat.})\pm0.11({\rm syst.})$. The comparison with the PHENIX measurements at $\sqrt{s_{\rm NN}}=200$ GeV [@PHENIX07; @PHENIX11] shows that the inclusive J/$\psi$ $R_{\rm AA}$ at 2.76 TeV in the rapidity domain $2.5 < y < 4$ are clearly above those measured at 200 GeV in $1.2 < |y| < 2.2$, while they are closer to the midrapidity values at 200 GeV (except in the most central collisions). The contribution from the B feed down to the J/$\psi$ production in our rapidity and $p_{\rm T}$ domain has been measured to be $\approx10\%$ in p-p collisions at $\sqrt{s_{\rm NN}}=7$ TeV [@LHCb]. Therefore, the difference between the prompt J/$\psi$ $R_{\rm AA}$ and our inclusive measurement is expected to be $\approx11\%$ if the $b$ production scales with $N_{\rm coll}$ or smaller if it is suppressed by nuclear effects (shadowing, ...). Finally, the comparison of our J/$\psi$ $R_{\rm CP}$ results to the ATLAS measurements in the same centrality classes [@ATLAS] indicates that the J/$\psi$ mesons measured at forward rapidity down to $p_{\rm T}=0$ are less suppressed than the high-$p_{\rm T}$ J/$\psi$ mesons at midrapidity (80% of the J/$\psi$ particles measured by ATLAS have a $p_{\rm T}$ larger than 6.5 GeV/$c$).
In summary, these results show a significant suppression of the inclusive J/$\psi$ production in Pb-Pb collisions at $\sqrt{s_{\rm NN}}=2.76$ TeV. The comparison with PHENIX results suggests that re-generation mechanisms could play a role. In order to provide tight constraints to suppression/regeneration models, a better knowledge of CNM effects is required, which can be obtained with a measurement of J/$\psi$ production in p-A collisions at the LHC.
References {#references .unnumbered}
==========
[99]{} T. Matsui and H. Satz, , 416 (1986). R. Arnaldi (NA60 Collaboration), [*Nucl. Phys. A*]{} [**830**]{}, 345c (2009), arXiv:0907.5004. A. Adare et al. (PHENIX Collaboration), , 232301 (2007), nucl-ex/0611020. A. Adare et al. (PHENIX Collaboration), arXiv:1103.6269 (2011). P. Braun-Munzinger and J. Stachel, , 196 (2000). K. Aamodt et al. (ALICE Collaboration), arXiv:1105.0380 (2011). A. Toia (ALICE Collaboration), these proceedings. F. Bossu et al., arXiv:1103.2394 (2011). K.J. Eskola, V.J. Kolhinen, C.A. Salgado, [*Eur. Phys. J. C*]{} [**9**]{}, 61 (1999). R. Arnaldi (ALICE Collaboration), these proceedings. R. Aaij et al. (LHCb Collaboration), [*Eur. Phys. J. C*]{} [**71**]{}, 1645 (2011), arXiv:1103.0423. G. Aad et al. (ATLAS Collaboration), [*Phys. Lett. B*]{} [**697**]{}, 294 (2011), arXiv:1012.5419.
|
---
abstract: 'Warm dark matter is consistent with the observations of the large-scale structure, and it can also explain the cored density profiles on smaller scales. However, it has been argued that warm dark matter could delay the star formation. This does not happen if warm dark matter is made up of keV sterile neutrinos, which can decay into X-ray photons and active neutrinos. The X-ray photons have a catalytic effect on the formation of molecular hydrogen, the essential cooling ingredient in the primordial gas. In all the cases we have examined, the overall effect of sterile dark matter is to facilitate the cooling of the gas and to reduce the minimal mass of the halo prone to collapse. We find that the X-rays from the decay of keV sterile neutrinos facilitate the collapse of the gas clouds and the subsequent star formation at high redshift.'
address:
- |
Korea Astronomy and Space Science Institute, Daejeon 305-348, Korea, and\
Institute of Physics, Jagiellonian University, 30-059 Kraków, Poland
- |
Max-Planck Institute for Radioastronomy, Bonn, D-53121, Germany,\
Department of Physics and Astronomy, University of Bonn, D-53121, Germany, and Department of Physics and Astronomy, University of Alabama, AL 35487, Tuscaloosa, USA
- 'Department of Physics and Astronomy, University of California, CA 90095-1547, Los Angeles, USA'
author:
- Jaroslaw Stasielak
- 'Peter L. Biermann'
- Alexander Kusenko
title: '**[Sterile neutrinos and structure formation]{}[^1]**'
---
Introduction
============
Both cold and warm dark matter models agree with the observed structure on the large scales. However, there are several inconsistencies between the predictions of the cold dark matter (CDM) model and the observations [@cdm_problems]. The low cutoff in dark matter contents of dwarf spheroids, the smoothness of our dark matter halo, and the old globular clusters (observed in Fornax) resisting the infall into the center by dynamical friction [@cdm_problems], all can be explained by warm dark matter (WDM) because it suppresses the structure on scales that are smaller than the free-streaming length.
While the suppression of the small-scale structure is desirable, it has been argued that “generic” WDM (for example, gravitino) can slow down structure formation and delay reionization of the universe, which can lead, in turn, to an inconsistency with the reionization redshift obtained by the WMAP [@Yoshida]. This problem can be alleviated in the case of the WDM in the form of sterile neutrinos with mass of several keV and a small mixing angle with the ordinary neutrino [@bier; @stas; @stas07] because such sterile neutrinos can decay and produce photons that catalyze the formation molecular hydrogen and speed up the star formation.
In the absence of metals, gas cooling is mainly due to the collisional excitation of H$_{2}$, its subsequent spontaneous de-excitation, and photon emission. In the primordial gas clouds, hydrogen molecules can be formed only in reactions involving $e^{-}$ or H$^+$ as a catalyst. Thus, an X-ray radiation can increase the production of the H$_{2}$ by enhancing the ionization fraction, which subsequently leads to speed up of the gas cooling and star formation. Although sterile neutrinos are stable on cosmological time scales, they nevertheless decay. The decay channel important for us is that of decay into one active neutrino and one photon, i.e., $\nu_s \rightarrow \nu_a \gamma$, where the photon energy is half of the sterile neutrino mass, $E_{0} \approx m_{s}c^2/2$. These decays produce an X-ray background radiation that increases the production of molecular hydrogen and can induce a rapid and prompt star formation at high redshift.
Sterile dark matter has a firm motivation from particle physics [@dw; @Kusenko:2006rh]. The discovery of the neutrino masses implies the existence of right-handed gauge-singlet fields, all or some of which can be lighter than the electroweak scale. These sterile neutrinos can be produced in the early universe by different mechanisms, for example, from neutrino oscillations [@dw] or from the Higgs decays [@Kusenko:2006rh], or from the couplings to a low-scale inflaton [@Shaposhnikov:2006xi]. The same particles, produced in a supernova, could account for the supernova asymmetries and the pulsar kicks [@kus], and can play a role in the formation of super-massive black holes in the early universe [@puzzle].
We will examine the thermal evolution of the gas clouds, taking into account both effects of the sterile neutrino decays, namely, the ionization and heating of the gas. We follow the evolution of the baryonic top-hat overdensity, the gas temperature and the H$_{2}$ and $e^{-}$ fraction. In order to perform the calculation we have incorporated to our previous code [@stas], the effects of sterile neutrino decays within collapsing halos and absorption of the X-ray background from sterile neutrinos by He atoms in the intergalactic medium. Our goal is to juxtapose the evolution of the gas temperature in the primordial clouds in the CDM model and the WDM model with keV sterile neutrinos and estimate of the minimal mass able to collapse at a given redshift.
Description of the code {#ana}
=======================
The top-hat overdensity evolution in a single-zone approximation [@tegmark; @stas] is described by the following equation [@Padmanabhan] $$\delta=\frac{9}{2}\frac{\left(\alpha-\sin
\alpha\right)^{2}}{\left(1-\cos \alpha\right)^{3}}-1 \textrm{,} \label{th1}$$ where the parameter $\alpha$ is related to the redshift $z$ and the redshift of virialization $z_{vir}$ through $$\frac{1+z_{vir}}{1+z}=\left(\frac{\alpha- \sin \alpha}{2
\pi}\right)^{2/3}. \label{th2}$$ According to these equations, the virial value of overdensity ($\delta \approx 18 \pi^2$) is reached at the redshift $z_{3\pi/2}=1.06555\left(1+z_{vir}\right)-1$.
Further evolution of $\delta$ depends on the type of the matter in the overdense region. If it is the dark matter, then after virialization its density remains constant forever. The situation is different in the case of baryons. If cooling is efficient enough, then the density gradually increases. Otherwise, the density remains constant, and there is no star formation in the halo. Following [@tegmark], we assume that the density of the halo stays constant after redshift of $z_{3\pi/2}$, i.e., it is equal to the value $\rho = 18\pi^{2}\Omega_{0}\rho_{0}\left(1+z_{vir}\right)^{3}$, where $\rho_{0}$ is the present critical density of the universe. This assumption is sufficient for our purposes.
The evolution of the gas temperature is governed by the equation [@stas] $$\begin{aligned}
\frac{dT}{dz}&=& \left(\gamma-1\right)\frac{T}{n_{p}}\frac{dn_{p}}{dz}+\gamma\frac
{T}{\mu}\frac{d\mu}{dz}+\frac{T}{\left(\gamma-1\right)}\frac{d\gamma}{dz} \nonumber \\
&+& \frac{\left(\gamma-1\right)\Lambda}{n_{p}k
H_{0}\left(1+z\right)\sqrt{\Omega_{\Lambda}+\Omega_{0}\left(1+z\right)^{3}}}
\textrm{,} \label{tz}\end{aligned}$$ where $T$, $\gamma$, $n_{p}$, $\mu$, $k$ and $\Lambda$ are the temperature of the gas, the adiabatic index, the number density of non-dark matter particles, the molecular weight, the Boltzmann constant and the cooling/heating function, respectively.
If we simply integrate equation (\[tz\]) to the redshift $z_{vir}$ then, for primordial clouds with masses greater than a critical value, the gas temperature will be much lower than the virial temperature $T_{vir}$ [@Barkana] $$T_{vir}\approx 12.3\times10^{3}\textrm{ K
}\left(\frac{\mu_{vir}}{0.6}\right)\left(\frac{M}{10^{8}h^{-1}M_{\odot}}
\right)^{2/3} \left( \frac{1+z_{vir}}{10} \right)
\textrm{,} \label{virial}$$ where $\mu_{vir}$ and $M$ are the mean molecular weight during virialization and the halo mass (combined mass of dark and baryonic constituents). Therefore, we must take into account shocks and increase the gas temperature to the virial value. We assume that the evolution of the gas temperature is linear between the redshifts $z_{3\pi/2}$ and $z_{vir}$.
To calculate the cooling/heating function $\Lambda$ in equation (\[tz\]) we follow the number density evolution of different baryonic components and take into account all relevant chemical and thermal processes, e.g, heating and cooling due to the sterile neutrino decays. The detailed list of all included processes and their coefficients can be found in [@stas].
Let us define the number fraction of component $i$ as $ x_{i}=n_{i} / n$, where $n_{i}$ and $n=n_{H}+n_{H^{+}}+2n_{H_{2}}$ are the number density of the $i$th component and hydrogen species, respectively. The time evolution of number fraction can be described by the kinetic equation: $${dx_i \over dt} = n \sum_{l} \sum_{m} k_{lmi} x_l x_m +
\sum_{j} k_{ji} x_j \textrm{,}
\label{chemia}$$ where the first component on the right-hand side describes the chemical reactions and the other one accounts for photoionization/photodissociation processes. Coefficients $k_{lmi}$ and $k_{ji}$ are reaction rates and photoionization/photodissociation rates multiplied by the numbers equal to 0, $\pm 1$ or $\pm 2$ depending on the reaction. In our calculations, we have considered the following five species: H, H$^{+}$, ${\rm H}_{2}$, $e^{-}$, and $\rm He$ with the mass fraction $Y=0.244$ [@Izotov]. We used simplified molecular hydrogen chemistry similar to the approach presented in [@tegmark].
In order to take into account the effects of sterile neutrino decays inside the collapsing halo, we have solved the radiative transfer equation for the spherically symmetric clouds with uniform density (see Section \[rad\] and also [@stasielak05; @mihalas]). In addition, we have included absorption of the X-rays from the sterile neutrino decays by both $H$ and $He$.
The photons from the decay of the sterile neutrinos are mainly absorbed by neutral helium and hydrogen atoms leading to their ionization. The ionization rate due to these photons is enhanced almost 100 times due to additional ionization by the secondary electrons, which deposit almost 1/3 of their energy into ionization. The energy of the absorbed photons partially goes into ionization and partially into heating and excitations. We have adopted the approximation [@steen; @stas], in which the ionization rate (in units s$^{-1}$) and heating (in units erg s$^{-1}$ cm$^{-3}$) due to the photons from the decay of the sterile neutrinos are respectively equal to $$\begin{aligned}
k\left(z\right) &=& \left[ \int_{\nu^{H}_{th}}^{\infty}4\pi
\sigma_{H}\left(\nu \right)
\frac{I_{\nu}\left(z\right)}{h\nu} \left(\frac{h \nu -h \nu_{th}^{H}}{h
\nu_{th}^{H}} \right) d\nu \right. \nonumber \\
&+& \left. \frac{Y}{4X} \int_{\nu^{He}_{th}}^{\infty}4\pi
\sigma_{He}\left(\nu \right)
\frac{I_{\nu}\left(z\right)}{h\nu} \left(\frac{h \nu -h \nu_{th}^{He}}{h
\nu_{th}^{H}} \right) d\nu +\frac{\Lambda_{int}\left(z\right)}{h \nu_{th}^H n_{H}} \right] \nonumber \\
&\times& C_{i}\left(1-x_{e}^{a_{i}}\right)^b_{i} +
\int_{\nu_{th}^{H}}^{\infty}4\pi\sigma_{H}\left(\nu\right)\frac{I_{\nu}
\left(z\right)}{h\nu} d\nu \label{ion} \textrm{,}\end{aligned}$$ $$\begin{aligned}
\Gamma_{s}\left(z\right) &=& \left[ \int_{\nu^{H}_{th}}^{\infty}4\pi
\sigma_{H}\left(\nu \right)
\frac{I_{\nu}\left(z\right)}{h\nu} \left(h \nu -h \nu_{th}^{H} \right) d\nu
\right. \nonumber \\
&+& \left. \frac{Y}{4X} \int_{\nu^{He}_{th}}^{\infty}4\pi
\sigma_{He}\left(\nu \right)
\frac{I_{\nu}\left(z\right)}{h\nu} \left(h \nu -h \nu_{th}^{He} \right) d\nu
+\frac{\Lambda_{int}\left(z\right)}{n_{H}}\right] \nonumber \\
&\times& C_{h}\left[ 1-\left(1-x_{e}^{a_{h}}\right)^b_{h} \right] n_{H}
\label{heat} \textrm{,}\end{aligned}$$ where $h$, $\nu$, $\sigma_{H}\left(\nu \right)$, $h \nu_{th}^{H}=13.6$ eV, $\sigma_{He} \left( \nu \right)$, $h \nu_{th}^{He}=24.6$ eV and $X$ are the Planck constant, photon frequency, the cross section and energy threshold for $H$ and $He$ ionization, and hydrogen mass ratio, respectively. The coefficients $C_{i}=0.3908$, $a_{i}=0.4092$, $b_{i}=1.7592$, $C_{h}=0.9971$, $a_{h}=0.2663$ and $b_{h}=1.3163$ are taken from [@steen]. The function $\Lambda_{int}\left(z\right)$ (in units erg s$^{-1}$ cm$^{-3}$) is the energy absorption rate of the photons from the sterile neutrino decays inside the collapsing cloud and is given by equation (\[lambdainsidefin\]). Finally, $I_{\nu}\left(z\right)$ (in units of erg cm$^{-2}$ s$^{-1}$ sr$^{-1}$ Hz$^{-1}$) is the specific intensity of the X-ray background from the sterile neutrino decays, which takes into account absorption by H and He in the intergalactic medium and which can be calculated in a similar way to the specific intensity given in [@stas].
Since the cross section for absorption of the X-ray photons by He is much larger than the same cross section for H, a large amount of energy can be accumulated in free electrons due to He ionization. Therefore, changes in the He ionization can strongly affect the absorbed energy, and we must take into account their effects to the heating/cooling function and the ionization by the secondary electrons, as in equations (\[ion\]) and (\[heat\]).
To estimate the minimal mass of the primordial halo able to collapse at a given redshift we use the following criterion of successful collapse [@tegmark]: $$T \left( \eta z_{vir} \right) \leq \eta T \left(z_{vir} \right) \label{collaps} \textrm{,}$$ where we take $\eta=0.75$. It means that the cloud is considered to collapse if its temperature drops substantially within a Hubble time, which roughly corresponds to the redshift dropping by a factor $2^{2/3}$.
Radiative transfer inside the collapsing cloud {#rad}
==============================================
In order to derive the energy absorption rate of the photons from the sterile neutrino decays inside the collapsing gas clouds $\Lambda_{int}\left(z\right)$, let us assume that these halos have spherically symmetric shape and their densities are uniform. In that case, it will be convenient to consider the radiative transport equation in a system of coordinates $\left(r, p\right)$ defined by the transformation formula: $\left(r, \mu \right) \rightarrow \left(r, p = r\sqrt{1-\mu^{2} }\right)$ for $-1\leq \mu \leq 1$ [@hu71; @stasielak05], where $r$ is a radial coordinate, $\mu = \cos \theta$, and $\theta$ is the angle between the outward normal and the photon direction.
For a given radius $r$, the “impact” parameter $p$ can vary between 0 and $r$. Because the parameter $p$ cannot distinguish between $\mu >0$ and $\mu < 0$, the radiation intensity $I_{\nu}$ has to be separated into outward $I^{+}_{\nu}$ and inward $I^{-}_{\nu}$ directed intensity, respectively.
Now, we can cast the time-independent, non-relativistic equation for radiation transport in spherical geometry into the form $$\begin{aligned}
\frac{\partial j_{\nu} \left(\tau_{\nu},p\right)}{\partial \tau_{\nu}}
&=& h_{\nu}\left(\tau_{\nu},p\right) \label{jjj-i}
\label{eqq1} \\
\frac{\partial h_{\nu} \left(\tau_{\nu},p\right)}{\partial \tau_{\nu}}
&=& j_{\nu}\left(\tau_{\nu},p\right)-S_{\nu} \
\textrm{,} \label{eqq2}\end{aligned}$$ where $$\begin{aligned}
j_{\nu} \left(r,p\right) &=& \frac{1}{2} \left( I_{\nu}^{+}
\left(r,p\right) + I_{\nu}^{-} \left(r,p\right) \right) \label{fe1} \textrm{,}
\\
h_{\nu} \left(r,p\right) &=& \frac{1}{2} \left( I_{\nu}^{+}
\left(r,p\right) - I_{\nu}^{-} \left(r,p\right) \right) \textrm{,} \label{fe2} \end{aligned}$$ and $d \tau_{\nu} = - \chi_{\nu} d\left( r \mu \right)$, is the optical depth at the radius $r$. The term $S_{\nu} = \eta_{\nu}/\chi_{\nu}$ is a source function, where $\eta_{\nu}$ (in units of erg s$^{-1}$ cm$^{-3}$ sr$^{-1}$ Hz$^{-1}$) and $\chi_{\nu}$ (in units of cm$^{-1}$) are the total emissivity and opacity at frequency $\nu$, respectively.
We consider sterile neutrino decays inside the spherically symmetric cloud, thus, the photon flux from this process will be peaked around the frequency corresponding to the energy of $E_{0}=m_{s}c^2/2$. Since the density of the cloud is much higher than its surroundings, there will be only few photons at this energy impinging upon the outer boundary, which is set by the radius $R$ of the cloud. The external radiation field also consists of the X-ray photons emitted due to the sterile neutrino decays at large distances from the cloud, however, as they have been emitted at earlier times, they will be redshifted to lower energies. Thus, they will not give the contribution to the photon flux at the energy of $E_{0}$.
Assuming, that there is no external radiation field at the energy of $E_{0}$, we can write the boundary conditions as follows [@stasielak05] $$\begin{aligned}
h_{\nu} \left(p,p\right) &=& 0 \hskip 1.5cm 0 \leq p \leq R \label{con1} \textrm{,} \\
j_{\nu} \left(R, p \right) &=& h_{\nu}\left(R,p\right) \hskip 1cm 0<p<R \textrm{.}\label{w2}\end{aligned}$$
Since we are interested only in the photon flux at the energy of $E_{0}$, from now on, we drop the $\nu$ dependence for clarity. It means that we have to multiply all of the quantities by the Dirac delta $\delta \left(E_{0}/h\right)$ or understand them as they have been already integrated over frequency. We will use the latter interpretation. According to the emissivity and opacity definition, we have $$\begin{aligned}
\eta &=& \frac{d n_{\gamma}}{d t} \frac{E_{0}}{4\pi } = \frac{\Omega_{dm} c^2 \varrho }{8\pi \tau_{s}} \textrm{,} \label{kap22} \\
\chi &=& n_{H}\sigma_{H} \left(E_{0}\right)+ n_{He}\sigma_{He}\left(E_{0}\right) \textrm{,} \label{kap1}\end{aligned}$$ where $\varrho$ is the total dark and baryonic matter mean density of the collapsing cloud, $n_{\gamma}$ is the number density of emitted photons due to the sterile neutrino decays. We assume that all of the dark matter in the collapsing halo consists of the sterile neutrinos and that the number density of sterile neutrinos do not change with time. The latter assumption can be justified by the fact that the inverse width of the sterile neutrino radiative decay, $\tau_{s}$, is much longer than the age of the universe. In addition, we neglect the stimulated emission.
Equations (\[eqq1\]) and (\[eqq2\]) can be rewritten as $$\begin{aligned}
\frac{\partial^2 j \left(\tau,p\right)}{\partial \tau^2} &=& j \left(\tau,p\right)-S \textrm{,} \\
\frac{\partial^2 h \left(\tau,p\right)}{\partial \tau^2} &=& h \left(\tau,p\right) \textrm{,}\end{aligned}$$ which with the boundary conditions (\[con1\]) and (\[w2\]) have the following solution $$\begin{aligned}
h\left(\tau,p\right) &=& -S e^{T\left(p\right)} \sinh \tau \textrm{,} \label{hhhh} \\
j\left(\tau,p\right) &=& -S e^{T\left(p\right)} \cosh \tau +S \textrm{,}\end{aligned}$$ where $T\left(p\right)=-\chi R \mu$ is the optical depth at the cloud boundary derived for the given impact parameter $p$. The luminosity of the collapsing cloud is given by $$L = 16 \pi^{2} R^{2} \int_{0}^{1} h \left( R,\mu \right) \mu d\mu \label{luminosity} \textrm{,}$$ whereas if we neglect absorption it would be equal to $$L_{s} = \frac{16}{3} \pi^{2} R^{3} \eta \label{luminositys} \textrm{.}$$ The fraction of the energy absorbed by the cloud is given by $1-L/L_{s}$, thus the energy absorption rate (in units of erg cm$^{-3}$ s$^{-1}$) is equal to $$\Lambda_{int}=\left( 1-\frac{L}{L_{s}}\right) L_{s} \frac{3}{4\pi R^3} \label{lambdainside} \textrm{.}$$
Using equations (\[kap22\]), (\[kap1\]), (\[hhhh\]), (\[luminosity\]) - (\[lambdainside\]), the definition of the source function $S$, and doing some algebra we get $$\Lambda_{int}\left(z\right)= \frac{\Omega_{dm} m_{H} c^2}{2\Omega_{b}\tau_{s} X} f \left( \alpha \right) n\left(z\right) \label{lambdainsidefin} \textrm{,}$$ where $$\begin{aligned}
f\left( \alpha \right) &=& 1- \frac{3}{4}\frac{1}{\alpha^3}\left[\alpha^2-\frac{1}{2}+\left(\alpha+\frac{1}{2}\right) e^{-2\alpha}\right] \textrm{,} \label{lll} \\
\alpha \left( z \right) &=& \left( \frac{3 \pi \Omega_{b} X M}{4 m_{H} n\left(z\right)} \right)^{1/3}
\left[ n_{H}\left(z\right) \sigma_{H}\left(E_{0}\right)+ n_{He}\left(z\right) \sigma_{He}\left(E_{0}\right) \right]
\textrm{.}\end{aligned}$$ The term $m_{H}$ and $M$ denotes hydrogen mass and the total mass of the cloud, respectively.
Results {#res}
=======
![*Top left*, *top right* and *bottom left*: Evolution of ionization fraction, $H_{2}$ fraction and temperature with redshift for different models. In each case, the mass of the primordial cloud is equal to $M=10^6 M_{\odot}$ and virialization redshift to $z_{vir}=12$. *Bottom right*: Dependence of the minimal mass of primordial halo able to collapse on its virialization redshift. Models we have used in calculation are following: $m_{s}=25$ keV and $\sin^2 \theta=3 \times 10^{-12}$ (WDM1), $m_{s}=15$ keV and $\sin^2 \theta=3 \times 10^{-12}$ (WDM2), $m_{s}=3.3$ keV and $\sin^2 \theta=3 \times 10^{-9}$ (WDM3), and CDM.[]{data-label="fig:f1"}](f1.eps){width="100.00000%"}
We have performed a detailed analysis of the cooling and collapse of the primordial gas clouds in the model with warm dark matter, taking into account both the increase in the fraction of molecular hydrogen and the heating due to the sterile neutrino decays. To illustrate these effects, we have performed the analysis for some benchmark cases which arise in realistic scenarios [@stas]. The effect on the largest gas clouds is negligible, whereas smaller clouds will be affected: for the largest clouds, the additional molecular hydrogen makes no difference, but for the smaller ones, the increase in the X-ray background makes the collapse possible in cases where it could not occur in the absence of sterile neutrino decays.
Our results presented in Fig. \[fig:f1\] show that the overall effect of sterile neutrino decays is to enhance the $H_{2}$ fraction and to speed up the cooling of the gas in the primordial halos. The minimal mass of the cloud able to collapse is reduced in all WDM models we have examined. We note that a more detailed treatment of the sterile neutrino free-streaming may affect our results: we did not take into account the filamentary star formation [@Gao:2007yk]; some other effects may also be important [@ripamonti].
In summary, the X-ray photons from sterile neutrino decays could play an important role in the formation of the first stars because they increase the fraction of molecular hydrogen.
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[^1]: Presented at the XLVII Cracow School of Theoretical Physics, Zakopane, Poland, June 2007.
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---
abstract: 'We have studied the insulator-superconductor transition (IST) by tuning the thickness in quench-condensed $Bi$ films. The resistive transitions of the superconducting films are smooth and can be considered to represent “homogeneous" films. The observation of an IST very close to the quantum resistance for pairs, $R_{\Box}^N \sim h/4e^2$ on several substrates supports this idea. The relevant length scales here are the localization length, and the coherence length. However, at the transition, the localization length is much higher than the superconducting coherence length, contrary to expectation for a “homogeneous" transition. This suggests the invalidity of a purely fermionic model for the transition. Furthermore, the current-voltage characteristics of the superconducting films are hysteretic, and show the films to be granular. The relevant energy scales here are the Josephson coupling energy and the charging energy. However, Josephson coupling energies ($E_J$) and the charging energies ($E_c$) at the IST, they are found to obey the relation $E_J < E_c$. This is again contrary to expectation, for the IST in a granular or inhomogeneous, system. Hence, a purely bosonic picture of the transition is also inconsistent with our observations. We conclude that the IST observed in our experiments may be either an intermediate case between the fermioinc and bosonic mechanisms, or in a regime of charge and vortex dynamics for which a quantitative analysis has not yet been done.'
address: 'Department of Physics, Indian Institute of Science, Bangalore 560 012, India'
author:
- 'G. Sambandamurthy, K. Das Gupta, and N. Chandrasekhar'
title: 'Effect of granularity on the insulator-superconductor transition in ultrathin Bi films'
---
PACS Numbers : 73.50.-h, 74.40.+k, 74.80.Bj
Introduction
============
The interplay between disorder and superconductivity in two dimensions has been an active field of study during the last decade. Weak localization in two dimensions [@c1tvr] is a phenomenon where electronic states are localized by any arbitrary amount of disorder in the absence of interaction, resulting in non-metallic behavior. Superconductivity is an effect in the opposite extreme, in which phase coherence is established due to electron-electron interaction, across the entire length of the sample. The interplay between these two opposing phenomena has led to various interesting results [@c1gold].
When quench condensed, the properties of many elements change drastically. Some elements (e.g. $Ga$) show enhanced superconducting transition temperatures, whereas others (e.g. $Bi$) are found to be superconducting only in amorphous form [@c1buchil]. $Ge$ and $Sb$ which are not superconducting in amorphous or crystalline forms, show signs of superconductivity when mixed with materials such as $Au$, $Cu$ and $Ag$. Signatures of superconducting transitions have been observed in noble metal thin films when deposited at room temperatures on $Ge$ substrates also. Strongin and co-workers first studied quench condensed films grown on thin $Ge$ underlayers [@c1stro1]. A variety of materials and substrates were investigated, the important finding being that films grown on $Ge$, $SiO$ showed measurable conductance even when they were a few monolayers thick, but films on glass, $LiF$ showed measurable conductance only at higher thicknesses. Therefore, films quench condensed on underlayers such as $Ge$ are classified as “homogeneous" and films grown directly on substrates, “granular".
There are some important differences between these “homogeneous" and “granular" films, which have been reported [@c1gold]. Granular superconducting films show “local superconductivity" i.e., a drop in resistance at the bulk transition temperature (bulk $T_c$) value, but develop an upturn at lower temperatures. Thicker films eventually go completely superconducting. This behavior has been explained by the fact that individual grains go superconducting at the bulk $T_c$ and phase coherence is achieved locally but not across the entire sample. Competition between Josephson coupling energy $E_J$ and charging energy $E_c$ is known to be the driving force for zero temperature phase transitions between the superconducting and insulating states in artificial arrays, films and bulk materials. Due to Cooper pairing, there are no free electrons in these systems, and conduction is due to the tunneling of Cooper pairs from one superconducting grain to another. In such films, the inter-grain capacitance is usually larger than the capacitance of the grain to ground. This charging energy opposes this tunneling, so that the pairs may become localized. This mechanism is referred to as the bosonic mechanism of the suppression of superconductivity.
On the other hand, homogeneous films show smooth transitions to the zero resistance state. However, in homogeneous films, $T_c$ is greatly suppressed from the bulk value as the films are made thinner. Conduction in these ultrathin films is completely different from bulk materials and transport mechanisms such as hopping conduction, classical percolation dominate the properties. In these materials, screening is reduced due to the disorder,resulting in a decrease of the attractive interaction required for Cooper pairing. This reduces the transition temperature. This mechanism is completely different from the previous one, since the key idea here is the complete disappearance of Cooper pairs.
The insulator-superconductor (I-S) transition has been extensively investigated over the last decade, in a variety of systems such as thin films [@c1havi; @c1beas], single Josephson junction [@c1pent], arrays [@c1van] and one dimensional wires [@c1tink]. The transition can be tuned by changing a parameter such as disorder [@c1havi], carrier concentration [@c1hebpa] or applied magnetic field [@c1sitb]. At very low temperatures this transition can be considered a continuous quantum phase transition [@c1shah]. A putative film with a $T_c$ = 0, separates the films showing insulating behavior from those showing superconducting behavior.
Experiment and Observations
===========================
In this paper, we report results on the insulator-superconductor transition (IST), tuned by changing the thickness of quench-condensed $Bi$ films. The resistive transitions of the superconducting films are smooth and can be considered to represent “homogeneous" films. The IST occurs very close to the quantum resistance for pairs, $R_{\Box}^N \sim h/4e^2$ on several substrates. The IST in homogeneous films can be imagined as the point where the effect on the transport properties, by the localization of electrons and superconducting coherence become comparable [@c1loccoh; @c1taop]. However, at the transition, the localization length is found to be much higher than the superconducting coherence length, contrary to expectation for a “homogeneous" transition. This suggests the invalidity of a purely fermionic model for the transition. Furthermore, the current-voltage characteristics of the superconducting films are hysteretic, and show the films to be granular. The relevant energy scales here are the Josephson coupling energy and the charging energy. However, Josephson coupling energies ($E_J$) and the charging energies ($E_c$) at the IST, are found to obey the relation $E_J < E_c$. This is again contrary to expectation, for the IST in a granular or inhomogeneous, system. Hence, a purely bosonic picture of the transition also appears inconsistent with our observations.
The experiments were done in a UHV cryostat, custom designed for [*in-situ*]{} experiments and is described in [@gsmssc]. Pumping is provided by a turbomolecular pump, backed by an oil-free diaphragm pump. A completely hydrocarbon free vacuum $\le 10^{-8}$ Torr can be attained. The substrate is amorphous quartz of size2.5cm X 2.5cm and is mounted on a copper cold finger whose temperature can be maintained down to 1.8K by pumping on the liquid helium bath. The material ($Bi$) is evaporated from a Knudsen-cell with a pyrolytic Boron Nitride crucible, of the type used in Molecular Beam Epitaxy (MBE). $Bi$ is evaporated from the cell at $650^{o}$ C, into a 4-probe resistivity measurement pattern by using a metal mask in front of the substrate. Successive liquid helium and liquid nitrogen cooled jackets surrounding the substrate reduce the heat load on the substrate and provide cryo-pumping. This produces an ultimate pressure $\sim 10^{-10}$ Torr in the system. The metal flux reaching the substrate is controlled using a carefully aligned mechanical shutter in the nitrogen shield. The thickness of the film is increased by small amounts by opening the shutter for a time interval corresponding to the desired increase in thickness. A quartz crystal thickness monitor measures the nominal thickness of the film. Electrical contacts to the film are provided through pre-deposited platinum contact pads ($\sim 50 \AA$ thick). $Ge$ underlayers are deposited on one side of the substrate (a-quartz) before loading the substrate into the cryostat. Separate electrical connections to films on $Ge$ and on bare a-quartz allow us to study both the films simultaneously. I-V’s and electrical resistance measurements are done using a standard d.c.current source (Keithley model 220/224) and nanovoltmeter (Keithley model 182) and elctrometer (Keithley model 6514).
Figure 1. shows the I-S transition in Bi films quench-condensed on 10 $\AA$ of $Ge$ underlayer. Even though there are variations in the value of $R_{\Box}^N$ and transition temperature $T_c$ between films of same thickness quench condensed on different substrates, the value of $R_c$ is found to be close to $h/4e^2$ for $Bi$ films quench condensed on a-quartz, a-quartz with 10 $\AA$ $Ge$ underlayer and films quench condensed on a solid inert layer of $Xe$ in our experiments. The evolution of $R_\Box$ vs T for all the films look similar to the ones shown for $Ge$ underlayer.
The superconducting transition temperatures of thin films decrease as the thickness $d$ is reduced ($R_{\Box}^N$ is increased). Strong disorder (high $R_{\Box}^N$ or low thickness $d$ with $k_Fl\ll 1$) localizes electron wave functions, increases inelastic scattering rate, suppresses $N(E)$ (the density of states near Fermi energy $E_F$) and ultimately causes a metal-insulator transition. The destructive effect of increasing normal state sheet resistance on superconductivity in 2D has been treated theoretically as a competition between disorder and interaction effects. It has been shown within the frame work of the BCS theory that weak localization of electrons leads to an effective increase in Coulomb repulsion, and corresponding decrease in transition temperature $T_c$. Finkel’shtein [@c3fink] has shown that the lowering of $T_c$ from the bulk value $T_{co}$ follows the relation,$$\label{c3fink}\frac{T_c}{T_{co}}=exp\left(-\frac{1}{\gamma}\right)\left[ \frac{\left(1+\frac{(r/2)^{1/2}}{\gamma-r/4}\right)}{\left(1-\frac{(r/2)^{1/2}}{\gamma-r/4}\right)}\right]^{1/\sqrt{2r}}$$where r is the reduced film sheet resistance ($r = R_{\Box}^Ne^2/2\pi^2\hbar$, measured in units of $\approx 81 k\Omega$) and $\gamma = 1/ln (kT_{co}\tau/2\pi\hbar$) characterises the ratio of the bulk critical temperature $T_{co}$ and the elastic scattering frequency $\tau^{-1}$. $Bi$ in bulk, crystalline form does not superconduct and the “bulk" critical temperature is normally taken to be the thick film value of 6.10K. The important observation from this equation is that the reduction of $T_c$ from its bulk value does not depend on any intrinsic material property and only depends on $\tau$.
From this equation we can calculate the reduction in $T_c$ (from the bulk value $T_{co}$) for two types of films, films on $Ge$ underlayers and for films on bare a-quartz. The purpose of showing the data for two different substrates will be discussed later. This is plotted against the reduced r (of equation 3.8) in Fig. 2. The solid lines show the function for different values of $\tau$. From the figure it is clear that except for a small region for the films on bare quartz, the results are not consistent with the Finkel’shtein theory. Finkel’shtein’s theory was based on a homogenous two-dimensional disordered system with uniform thickness. If it is believed that the presence of a $Ge$ underlayer facilitates a homogenous film growth, at least these films should have shown reasonable fit to the theory. But we find that such is not the case. This can be understood by the fact that the “homogeneity" of the films might be at length scales smaller than the typical thermal length scale, important for electron - electron interaction ($\sqrt{D\hbar/\pi kT}$), which is of the order of a few hundreds of $\AA$ at these low temperatures [@c1adk1]. For the films on bare quartz, in the small region where Finkel’shtein’s theory seems to fit, a value of $-\frac{1}{\gamma}$ = 9 gives a mean free path of $\sim 10\AA$, considering a free electron model.
The I-V characteristics were obtained at 2.25K (which is well below the $T_c$) for all the films. They are plotted in Fig. 3. When current is increased fromthe zero value, the voltage jumps to the normal state value at the critical current ($I_c$). Upon reducing the current from the normal state, the voltage returns to zero not at $I_c$, but at a much lower value $I_{min}$. These observations are consistent with the I-V characteristics of an underdamped resistively and capacitively shunted Josephson junction (RCSJ). [@BP; @ST; @MC; @PFC] In our case, due to the large area of the film, the realization is that of a Josephson junction array, with values of $E_J$ and $E_C$ that are characterized by some distribution, the characteristics of which are determined by the morphology of the film. Our analysis of these I-V’s in terms of an RCSJ model has been published. [@gsmssc] This suggests that the film is granular. Further evidence for its granular character is discussed below.
Discussion
==========
We now discuss these results in the framework of existing models/theories for the IST. Before moving on to this discussion, we present results on the structure and morphology of the films, as inferred from reflection high energy electron diffraction (RHEED). Structure and morphology are important parameters, that can influence the properties of the films, and therefore, conceivably the IST as well. RHEED studies on Bi films, grown on various substrates and underlayers, show that the Bi is almost amorphous. A RHEED picture is shown in Fig. 4. Based on the Scherrer formula for the peak broadening, we estimate that films thicker than 10 $\AA$ are composed of clusters that vary in size from 25 $\AA$ to 100 $\AA$ [@gsmssc]. Since the information obtained is in reciprocal space, it is difficult to comment on the real space surface morphology. Our RHEED observations are consistent with previous STM work. [@vall] We therefore assign an average size to the clusters of $50 \AA$. This yields a spacing between clusters of approximately $150 \AA$ (considering hemispherical clusters, and using conservation of deposited material) for a film close to the IST, which has a normal state resistance of $6.25 k\Omega$. It turns out that these parameters, the size of a grain or island, and the average distance between islands are important parameters, a fact that is obvious for the bosonic mechanism.
Our observations suggest that the film can be considered as a random array of Josephson junctions, which are shunted by a resistance. Consequently the resistively and capacitively shunted junction (RCSJ) model [@BP] can be used to describe the hysteretic behaviour of the I-V curves, with the capacitance being the intrinsic capacitance of the junction. From the ratio of $I_{min}/I_c$, the value of the admittance ratio ($\beta$) can be calculated [@ST; @MC; @PFC]. Here $\beta = \omega_{c}C/G$, where $\omega_{c}$ is ($2e/\hbar)I_c R_s$. C is the intergranular capacitance and G the normal state conductance of the array. We wish to point out that these are lumped parameters, which characterize the whole array. From the values of $\beta$, the intergrain capacitance is calculated. The charging energy $E_c (= e^2/2C)$ and the Josephson coupling energy $E_J = \hbar I_c/2e$ are calculated for all the film thicknesses studied. These values are calculated using single value of C and G, which correspond to capacitance and conductance of the array. C and G will have a range of values, the distribution of these values and the moments of the distribution will of course depend on film thickness. We measure the critical current $I_c$ for different films at T = 2.0K (lower than $T_c$ for all the thicknesses studied) and calculate the relevant energy parameters such as the charging energy ($E_c$), Josephson coupling energy ($E_J$) etc. Fig. 5. shows the ratio of the Josephson coupling energy to the charging energy vs the sheet resistance for the films quench condensed on $Ge$ underlayer at the temperature where the I-V’s were acquired, T=2.25 K. We find that even though the IST occurs near $R_c$, the relevant energy scales become equal at a much higher thickness. This suggests that the a purely bosonic mechanism may be an incorrect picture for understanding the destruction of superconductivity in these films. We next investigate the validity of the fermionic mechanism.
To check whether the fermionic mechanism is a good representation of the physical mechanism, we estimate the electron localization length from the high temperature resistance data. In strongly disordered films, the temperature dependence of the resistivity is of the form $R = R_0 exp[(T_o/T)^{\alpha}]$, where $\alpha$ varies from 0.75 for collective variable range hopping, 0.5 for hopping dominated by Coulomb interactions (Efros-Shklovskii correlated hopping) and 0.33 for Mott variable range hopping. In the weak-localization regime, the conductivity shows a logarithmic dependence on temperature,$$\sigma = \sigma_0 + e^2p lnT/(\pi h)$$ where p is a coefficient determined by the scattering mechanism for electrons. In estimating the localization length, we neglect interactions. In previous work, which involved studies of quench condensed films on different substrates whose dielectric constants varied from 1.5 (for solid Xe underlayers) to 15 (for Ge underlayers), we have demonstrated that the IST is robust and unaffected by the dielectric constant of the substrate [@prb1]. This is our justification for neglect of interactions. We use the theory of Wölfle and Vollhardt [@wv] which describes the transition from weak to strong localization, neglecting interaction. Their result is $$\hbar/(e^2R_{\Box}) = 1/(2\pi^2) [ln(1+y^2)](1+y)exp(-y)$$where $y = L/\xi_{loc}$. $\xi_{loc}$ is a localization length, related to the elastic mean free path l by $\xi_{loc} = l exp(\pi k_F l/2)$, where $k_F$ is theFermi wave vector. Here the sample size is regarded as a cutoff length due to inelastic scattering $L = D T^{-p}$, where D is a diffusion constant for electrons. Knowing the resistivity at a suitably high temperature (where there is no observable temperature dependence) and its variation with temperature (at lower temperatures), the various parameters can be determined.
We determine the superconducting coherence length from upper critical field data, which has been presented in a separate publication [@prbrap]. We have determined $B_{c2}(T)$ of our films, from the resistive transition in a perpendicular magnetic field. The convention that we have followed is to define $B_{c2}$ as the field at which the sheet resistance is half its normal state value, $R_{\Box}^N$. We then use the Ginzburg-Landau definition of $B_{c2}(T) = \Phi_0/(2\pi\xi^2)$, to determine the coherence length. Fig. 6 shows the variation of the superconducting coherence length and the localization length with sheet resistance of the films. We note here that the coherence lengths of the thinnest films which show a decreasing resistance with temperature, cannot be measured experimentally, since the lowest temperatures accessible to us in our apparatus is 1.6 K. We therefore extrapolate the coherence lengths to higher sheet resistances. From the behavior of $\xi$ at lower sheet resistances, this approximation is clearly justified. As is evident from Fig. 6, the IST occurs at a point where the localization length is much larger than the coherence length, $\xi_{loc}$ is $800 \AA$, whereas $\xi$ is only $25 \AA$. The ratio $\xi_{loc}/\xi$ is 32 in our study of quench condensed Bi films. This is to be contrasted with the results of Kagawa [*et al.*]{} [@c1loccoh], who found a ratio of two for Pb films. Whether this difference is due to the different materials studied, or the differing deposition geometry, is unclear to us at the present time.
Other models for the destruction of superconductivity in such granular systems have been considered for proximity arrays of varying geometry (ratio of the separation of the superconducting regions to their size). [@lark1; @lark2] Although these papers consider a mechanism that appears to be intermediate between the bosonic and fermionic mechanisms discussed above, there are several constraints on the sample geometry and physical properties. The ratio of the spacing between islands $b$, to the island size $d$ should obey the relation $ln(b/d) \ge 3$. In our films, this condition is clearly not met, since we have $b/d \sim 3$, and $ln(b/d) \sim 1$. Further, the authors consider a case where there is a fairly large tunneling conductance between the substrate and each superconducting island. This large tunneling conductance can help couple the islands, so that the conventional Coulomb blockade effect is suppressed. This is the physical reason for a mechanism for the IST that is not purely bosonic. The tunneling conductance between the islands and the substrate are expected to be quite different for Bi films on Ge and a-quartz, since the dielectric constants of the two materials differ by a factor of three. In Fig. 1, we show fits to Finkel’shtein’s theory for these two substrates. We reiterate the point that the fits are reasonable only over a small region, for the quartz substrate. Since we expect the tunneling conductances to differ, we expect to be in a regime where the model of Feigel’man [*et al.*]{} may be relevant.The large tunneling conductances between the substrate and thesuperconducting islands results in disorder enhanced multiple Andreev reflection. In our studies, we have used several substrates, ranging from solid xenon to Ge, but we still observe a robust IST close to a thickness of $25 \AA$ and a $R_{\Box}$ of $h/4e^2$ [@prb1]. Over this range of dielectric constants, it is natural to expect the island-substrate (or underlayer) tunneling conductance to vary substantially. However the robust nature of the IST suggests that Andreev reflection and the associated physics is not relevant in our case.
Considerations of phase fluctations in arrays of regular as well as random Josephson junctions have shown that both vortex-unbinding transition, as well as the charge unbinding transition can occur [@fazio]. Both of these transitions can occur, depending on the ratio between the charging energy and the Josephson coupling energy. Fazio and Schon have set a limit for $E_J/E_c$ of $2/\pi^2$ for the boundary between the insulator and the superconductor. In our case, the transition occurs at a small $E_J/E_c \simeq 10^{-3}$. At such small values, Cooper pairs are expected to remain frozen (no long range phase order), but the single electron dynamics is the same as in the normal state, resulting in an insulating state [@fazio]. In our experiments, we are in a regime where the normal state conductance is close to 1, the films are granular, and $E_J/E_c$ is very small. This is a regime, for which a quantitative analysis has not yet been done. Since an experimental realization of this limit has now been observed, we hope that this work would stimulate such an analysis. In this regime, where the normal state conductance in high, single electron processes other than Andreev reflection, may be important.
Conclusions
===========
In conclusion, we find an IST very close to the quantum resistance for pairs, $R_{\Box}^N \sim h/4e^2$ for Bi films on several substrates, however, at the transition, the localization length is much higher than the superconducting coherence length. This is contrary to expectation for the IST in a “homogeneous" film. Therefore, we explore models other than a purely fermionic model for the transition. The current-voltage characteristics of the superconducting films are hysteretic, and can be fitted to an RCSJ model, suggesting that the films are granular. In this case, the relevant energy scales here are the Josephson coupling energy ($E_J$) and the charging energy ($E_c$). However, at the IST, we find that $E_J < E_c$. This is in conflict with the conventional model for the IST in a granular or inhomogeneous system. A simple bosonic picture of the transition is also inconsistent with the reported observations. In the experiments reported here, the dimensionless conductance is nearly unity, and the ratio of Josephson energy to charging energy very small. This is a regime which has not been investiagated theoretically, due to the difficult nature of the analysis. Our observations suggest that such a regime merits further investigation, since this is the most likely experimental realization not only in our study, but in several earlier studies as well. [@c1gold; @c1havi; @c1hebpa]
[**Acknowledgemnets**]{} This work was supported by DST, Government of India. KDG thanks CSIR, New Delhi for the financial support through a Senior Research Fellowship.
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\(1) Fig. 1. Insulator - Superconductor transition for a set of $Bi$ films quench condensed at 15K on quartz substrates pre-deposited with 15 $\AA Ge$ underlayer.\
(2) Fig. 2. $T_c/T_{co}$ for films on $Ge$ and a-quartz, plotted against the reduced resistance. The solid lines show the perdictions of Finkel’shtein’s theory for different values of $\tau$. See text for details.\
(3) Fig. 3. Set of I-V curves for superconducting $Bi$ films which shows hysteretic behavior. The number beside each I-V is the film thickness in $\AA$.\
(4) Fig. 4. The RHEED picture from a typical superconducting Bi film during growth. At least one diffuse ring is clearly visible.
\(5) Fig. 5. The variation of the ratio of the Josephson coupling energy to the charging energy ($E_J/E_c$) with the normal state sheet resistance $R_N$ for the films in Fig.1. The arrow indicates $R_N = h/4e^2$.\
(6) Fig. 6. Variation of the localization lengths and coherence lengths with the normal state sheet resistance $R_N$ for the films in Fig.1. The arrow indicates $R_N = h/4e^2$.\
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abstract: 'We develop and experimentally validate a novel artificial neural network (ANN) design framework for silicon photonics devices that is both practical and intuitive. As case studies, we train ANNs to model both strip waveguides and chirped Bragg gratings using a small number of simple input and output parameters relevant to designers. Once trained, the ANNs decrease the computational cost relative to traditional design methodologies by more than 4 orders of magnitude. To illustrate the power of our new design paradigm, we develop and demonstrate both forward and inverse design tools enabled by the ANN. We use these tools to design and fabricate several integrated Bragg grating devices. The ANN’s predictions match very well with the experimental measurements and do not require any post-fabrication training adjustments.'
author:
- 'Alec M. Hammond'
- 'Ryan M. Camacho'
bibliography:
- 'paper\_ANN\_RMC.bib'
title: Designing Silicon Photonic Devices using Artificial Neural Networks
---
Introduction
============
Silicon photonics has become a viable technology for integrating a large number of optical components in a chip-scale format. Driven primarily by telecommunications applications, a growing number of CMOS fabrication facilities dedicated to silicon photonics are now in operation and available for researchers and engineers to submit photonic integrated circuit (PIC) designs to be fabricated [@chrostowski_silicon_2015].
Designing the silicon photonics components and circuits, however, remains a major bottleneck. Current design flows are complicated by computational tractability and the need for designers with extensive experience [@bogaerts_silicon_2018]. Unlike their electronic counterparts, photonic integrated circuits require computationally expensive simulation routines to accurately predict their optical response functions. The typical time to design integrated photonic devices now often exceeds the time to manufacture and test them.
To address this challenge, we propose and experimentally validate a new design paradigm for silicon photonics that leverages artificial neural networks (ANN) in an intuitive way and is at least four orders of magnitude faster than traditional simulation methods. Our design paradigm only requires a small number of input *and* output neurons corresponding to descriptive parameters relevant to the designer.
This new approach provides benefits such as rapid prototyping, inverse design, and efficient optimization, but requires a more sophisticated ANN than previous approaches. We demonstrate practical confidence in our method’s accuracy by fabricating and measuring devices that experimentally validate the ANN’s predictions. As illustrative examples, we design, fabricate, and test chirped integrated Bragg gratings. Their large parameter spaces and nonlinear responses are typical for devices that are computationally prohibitive using other techniques. The experimental results show remarkable agreement with the ANN’s predictions, and to the authors’ knowledge represent the first experimental validation of photonic devices designed using ANN’s.
This work builds on recent theoretical results showing that it is possible to model nanophotonic structures using ANNs. For example, Ferreira et al. [@da_silva_ferreira_towards_2018] and Tahersima et al. [@tahersima_deep_2018] demonstrated that ANNs could assist with the numerical optimization of waveguide couplers and integrated photonic splitters respectively. In both cases the input parameter space was the entire 2D array of grid points, showing the power of ANNs in blind “black box” approach, though limiting the designer’s ability to intuitively adjust input parameters. A related approach has also been applied to periodic photonic structures, which are often difficult to efficiently model [@inampudi_neural_2018; @ferreira_computing_2018].
Two recent theoretical papers by Zhang et al [@zhang_spectrum_2018], and Peurifoy et al. [@peurifoy_nanophotonic_2018] go beyond simply optimizing over a large parameter space, and used ANNs to calculate complicated spectra using a smaller number of intuitive, smoothly varying input parameters. Even though in both cases each wavelength point in the calculated spectra required its own ANN output neuron, the usefulness of using ANNs to model systems with intuitive input parameters was demonstrated.
To illustrate the power of our new design paradigm, we demonstrate both forward and inverse design tools that use a chirped Bragg grating ANN as a computational backend. The forward design tool is interactive, and was used to design our fabricated circuits. The inverse design tool quickly constructs a temporal pulse compressing chirped Bragg grating within specified design constraints — a task typically too computationally expensive for traditional methods.
Results
=======
Overview {#overview .unnumbered}
--------
To motivate our approach, we first describe a neural network that models the effective index of a silicon photonic strip waveguide with various widths and thicknesses. While waveguide simulation is already straightforward from a designer’s perspective, the model illustrates the advantages of our approach and is a key building block for more advanced ANN models described below which are less straightforward using existing techniques. These advantages include the ANN’s computational speedup of over 4 orders of magnitude, and the simplification and speedup of other complicated simulation routines that rely on effective index calculations. A more advanced example that is computationally intractable via traditional methods is then given, in which we demonstrate an ANN that models the complex relationship between a chirped silicon photonic Bragg grating’s design parameters and its corresponding spectral response. Many designers leverage silicon photonic chirped Bragg gratings to equalize optical amplifier gain [@rochette_gain_1999], compensate for semiconductor laser dispersion [@tan_chip-scale_2008; @strain_design_2010], and enable nonlinear temporal pulse compression [@tan_monolithic_2010; @b._j._eggleton_g._lenz_n._m._litc_optical_2000].
Figure \[fig:flowChart\] illustrates the new design methodology. First, we iterated between generating an appropriate dataset and training the ANN until the model adequately characterized the device. Next, we used the ANN to simulate circuits and solve inverse design problems. Finally, we fabricated devices to validate the results.
![The process overview describing the new design methodology. First, datasets are generated using traditional numerical methods (described in Methods). From this dataset, a neural network is trained to characterize the device under consideration. Figures \[fig:NN\_neff\_err\] & \[fig:panelFig\_BG\] illustrate this process for a strip waveguide and a chirped grating respectively. Often, the designer iterates between these two steps until an appropriate model is developed. Once the model is ready, several design applications, like circuit simulations and inverse design solutions, are available. The designs are then fabricated to validate the model’s results. From here, the model can be shared and extended. []{data-label="fig:flowChart"}](figs/flowchart_wide_mod.png)
Waveguide Neural Network {#waveguide-neural-network .unnumbered}
------------------------
We first report on a simple waveguide neural network capable of estimating the effective index of any arbitrary silicon photonic waveguide geometry for a variety of modes. Specifically, we modeled the relationship between the waveguide’s width, thickness, and operating wavelength and the effective index for the first two TE and TM modes. We note that including wavelength as an input parameter is a unique and enabling strategy not previously adopted (see Discussion section below). Figure \[fig:NN\_neff\_err\] (f) compares the ANN’s predicted effective index to a its corresponding simulation. The first TE and TM mode for any silicon photonic waveguide with a width between 350 nm and 1000 nm and a thickness between 150 nm and 350 nm are demonstrated. The network estimates a smooth response for both modes simultaneously, even for data points outside of its training set. The ANN’s smooth output also produces smooth analytic derivatives, which are essential for calculating group index profiles and for gradient-based optimization routines.
We implemented various tests to validate the network’s accuracy. First, we split the initial dataset into a training set and a validation set. While the network evaluated both sets after each epoch (i.e. training iteration) only the training set’s results were used to update the network’s weights. We monitored the validation set’s results to assess overfitting. To better understand the network’s performance after each iteration, we recorded each epoch’s mean-square-error (MSE) and coefficient of determination ($R^2$). Figure \[fig:NN\_neff\_err\] (a) and (b) illustrate the MSE and $R^2$ respectively after each epoch. To prevent overfitting, we stopped training at 100 epochs, where the MSE and $R^2$ appear to converge. At this point, the network demonstrated a MSE of $1.323 \times 10^{-4}$ for the training set and $7.490 \times 10^{-5}$ for the validation set. The final $R^2$ values for the training data and validation data were $0.9996$ and $0.9997$ respectively. The MSE and $R^2$ evolution for both the training set and validation set converge well, indicating little to no overfitting. Figure \[fig:NN\_neff\_err\] (c) illustrates the relative error for both the training and validation sets after the final epoch. Both the training set errors and validation set errors are similarly distributed and tightly bounded between $-1\%$ and $1\%$, once again indicating little to no overfitting.
With confidence in the waveguide neural network’s prediction accuracy, we benchmarked its speed and found that a single neural network evaluation was $10^4$ times faster on average than the corresponding finite difference eigenmode simulation. Figure \[fig:NN\_neff\_err\] (d) compares the computation speed for the ANN to the eigenmode solver, Meep Photonic Bands (MPB). This significant speedup enables many simulation techniques, like the layered dielectric media transfer matrix method (LDMTMM) [@helan_comparison_2006] or the eigenmode expansion method (EMM) [@gallagher_eigenmode_2003], where photonic components are discretized into individual waveguides. Using the ANN, a transfer matrix for each waveguide can be quickly generated and cascaded to formulate a fairly accurate response for the device. In addition, modeling fabrication variations is now much quicker since existing Monte Carlo sampling routines can leverage the ANN’s speed.
![Waveguide artificial neural network training results demonstrated by the training convergence with reference to the mean square error (a), the coefficient of determination (b), and the residual errors after training (c). Panel (d) compares the computational cost for the ANN and the eigenmode solver thas is used to simulate the mode profiles (e). Panel (f) exhibits the effective index profiles as a function of a waveguide geometry at 1550 nm for the first TE and TM modes.[]{data-label="fig:NN_neff_err"}](figs/panelFig_wg.png)
Bragg Grating Neural Network {#bragg-grating-neural-network .unnumbered}
----------------------------
Modeling the relationship between a Bragg grating’s physical design parameters and its corresponding responses is difficult since no one-to-one mapping exists. Consequently, many designers resort to black-box optimization routines that strategically search the parameter space for viable design options. As a result, inverse design problems — where a simulation needs to run each iteration — become intractable for even modest size gratings. If a full 3D FDTD simulation is performed, for example, each optimization iteration can take between 8-12 hours on typical desktop computing systems. In addition, the optimization routines tend to inefficiently simulate redundant test scenarios for different design problems. We train and demonstrate a Bragg ANN, however, that can predict a grating’s response on the order of milliseconds on the same system, enabling much faster solutions to more complex design problems. We fabricate various test devices and validate our neural network’s predictions.
Using the waveguide neural network, we generated a dataset to train our Bragg grating neural network to predict the reflection spectrum and group delay response of a silicon photonic, sidewall-corrugated, linearly chirped Bragg grating, as illustrated in Figure \[fig:panelFig\_BG\] (d). We note that generating the training dataset was approximately 2 orders of magnitude faster using the waveguide ANN reported above rather than traditional methods. To smooth apodization dependent ringing, we pre-processed the training data. More information regarding this step is provided in the Supplementary Material. We parameterized the gratings by length of the first grating period ($a_0$), length of the last grating period ($a_1$), number of grating periods $NG$, and grating corrugation width difference( $\Delta w = w_1 - w_0$). We designed the network to receive these four parameters along with a single wavelength point as inputs. The network has two outputs: reflected optical power and group delay.
Similar to the waveguide network, we divided the dataset into a training set and validation set. We tracked both the MSE and the $R^2$ metrics after each epoch. The Bragg training set was much larger than the waveguide training set, owing to the larger parameter space. Consequently, the MSE converged within the first few epochs and we stopped training after just five epochs to prevent overfitting. The final MSE for the training and validaton sets was $1.845 \times 10^{-4}$ and $1.677\times10^{-4}$ respectively. The final $R^2$ was 0.9975 and 0.9977 respectively. Once again, the MSE and $R^2$ evolution for both the training set and validation set converge well, indicating little to no overfitting. Figure \[fig:panelFig\_BG\] (a-b) illustrates the network’s MSE and $R^2$ evolution. Figure \[fig:panelFig\_BG\] (c) illustrates the absolute error for both the training sets and validation sets. We calculated the absolute error because several training samples were at or near zero and skewed the relative error.
We note that calculating Bragg grating response with the ANN is much more computationally efficient than previously demonstrated methods. This is because the Bragg ANN linearly increases in computation complexity with added grating parameters, while LDMTMM and all other methods known to these authors increase at least quadratically.
![Bragg grating artificial neural network training results demonstrated by the training convergence with reference to the mean square error (a), the coefficient of determination (b), and the absolute error after training (c). (d) illustrates the different adjustable grating parameters and (e) illustrates the interrogation circuit used to extract the reflection, transmission, and group delay profiles simultaneously from the chirped Bragg grating. A grating coupler (GC) feeds light into various Y-branches (YB) and directional couplers (DC) such that the transmission and reflection spectra can both be extracted from the chirped Bragg grating (BG). Half of the reflected signal is sent through a Mach-Zehner Interferometer (MZI). The output of which is used to extract the group delay. .[]{data-label="fig:panelFig_BG"}](figs/panelFig_BG.png)
To validate the Bragg ANN, we fabricated and measured several silicon photonic Bragg gratings with different chirping patterns and compared their transmission, reflection, and group delay spectra to the neural network’s predictions. The gratings were arranged in one of two configurations: (1) a simple circuit that only measured the Bragg grating’s transmission spectra and (2) a more complicated interrogation circuit capable of measuring the reflection, transmission, and group delay profiles from the same device simultaneously. Figure \[fig:panelFig\_BG\] illustrates the interrogation circuit used to measure all three responses simultaneously. In both configurations, grating couplers were used to route light on and off the chip. While the simpler circuit required less de-embedding, the full interrogator circuit allowed for a more comprehensive device characterization.
The transmission-only gratings were designed with various grating period bandwidths from 5 nm to 20 nm, each with 600 periods and a corrugation width of 50 nm. The initial design parameters produced ANN predictions that match the measured data extremely well. Small discrepancies in the grating responses are largely attributed to the grating’s apodization profile and detector noise. Figure \[fig:lukasData\] illustrates the comparison between the ANN’s predictions and the measured data.
We designed the remaining gratings using a much smaller chirp bandwidth of 3 nm with 750 grating periods and a 30 nm corrugation width. We mirrored the orientation of half the gratings in order to measure both positive and negative sloped group delay profiles. Once measured, we normalized the data by de-embedding the responses from the various Y-branches, directional couplers, and grating couplers that complicate the measurement data. The process is explained in the Supplementary Material. Even with the rather complex transfer function, the transmission, reflection, and group delay profiles match the ANN’s corresponding predictions well except for occasional resonant features caused by fabrication defects. These defects are expected since the narrow bandwidth devices have a grating pitch with a fine discretization that approaches the e-beam raster grid resolution. Small changes in grating pitch that don’t align with the raster grid occasionally produce weak Fabry-Perot resonance conditions visible in the data. These raster-induced defects also account for a small lateral shift (\~1 nm) in the responses. Even with these fabrication challenges, it is notable that the ANN successfuly predicts the transmission, reflection, and group delay profiles simultaneously. In fact, the ability to do so in noisy fabrication environments is one of the key advantages of the ANN and may allow for efficient parameter extraction where other methods fail.
![Fabrication data compared to corresponding ANN predictions. (a1-a4) Measured transmission responses for gratings with a period chirp of 5 nm (a1), 10 nm (a2), 15 nm (a3), and 20 nm (a4). (b1-b2) Transmission and reflection responses for two different Bragg gratings. Both gratings share the same design parameters, and have an identical but opposite linear chirp. The result of the mirrored chirping is seen in both the normalized MZI interference patterns (c1 , c2) and the extracted group delay responses (d1 , c2).[]{data-label="fig:lukasData"}](figs/fabrication_results.png)
Forward design {#forward-design .unnumbered}
--------------
The neural network’s speed and flexibility enable forward design exploration. For example, Figure \[fig:forward\_design\] illustrates a graphical user interface (GUI) built with slider bars to adjust the Bragg grating’s design parameters (i.e. corrugation widths, grating length, chirp pattern, etc). The plots dynamically update, calling the neural network every time the user modifies the input, and display the corresponding reflection and group delay profiles. Because wavelength is included as an input to the ANN rather than an output, arbitrary wavelength sampling within the domain is allowed. Computing these responses in real time is not possible using traditional techniques. This capability is valuable and allows even novice designers the ability to rapidly gain device intuition without necessarily understanding the underlying numerical techniques.
![Graphical user interface used to explore the design space of a chirped Bragg grating. The slider bars on the left control physical parameters like grating length (NG), grating corrugation (dw), and the grating chirp (a1 and a2). Any time the user adjusts these parameters, the program calls the ANN and reproduces the expected reflection and group delay profiles for that particular grating. Due to the ANN’s speed, the program is extremely responsive.[]{data-label="fig:forward_design"}](figs/combinedGUI.png)
Inverse design {#inverse-design .unnumbered}
--------------
This new approach also enables an entirely new set of inverse design problems. For example, we used the neural network in conjunction with a truncated Newtonian optimization algorithm to design a temporal pulse compressor. Designers often rely on dispersive Bragg gratings to generate short, optical pulses for high-capacity communications [@tan_monolithic_2010]. In this particular case study, we assumed an arbitrary source generates a 20 ps wide chirped pulse with 4 nm of bandwidth. Figure \[fig:inverse\_problem\] illustrates the optimization routine’s evolution, the resulting grating response, and the pulse shape before and after the Bragg grating. Such optimization algorithms run much quicker than previously known methods, owing to the accelerated cost function. The agnostic nature of the neural network interface works well with typical optimization routines, especially since any arbitrary wavelength sampling is allowed. Depending on the cost function formulation, gradient-based methods could directly evaluate the Jacobian and Hessian tensors from the ANN without any extra sampling or discretization.
![ANN-assisted design of a monolithic temporal pulse compressor using a silicon photonic chirped Bragg grating. A truncated Newton algorithm was tasked with constructing a grating that compressed an arbitrary chirped pulse by a factor of 2. After 340 grating simulations, the optimizer sufficiently minimized a cost function (right) that compared the new pulse’s width to the old pulse. The resulting grating is demonstrated below and the input, output, and desired pulses for iterations 1, 140, and 340 are demonstrated on the left.[]{data-label="fig:inverse_problem"}](figs/inverseDesign_v2.png)
Discussion
==========
Our method demonstrates a new, viable platform for silicon photonic circuit design. With a single global parameter fit, we successfully modeled silicon photonic waveguides and silicon photonic chirped Bragg gratings with arbitrary bandwidths, chirping patterns, lengths, and corrugation widths and allowed arbitrary wavelength sampling. Future work could explore new network architectures (e.g. different activation functions, layer connections, etc.) and training algorithms. Several other devices, like ring resonators [@bogaerts_silicon_2012], can also be modeled. One could subsequently cascade several ANNs that model the scattering parameters of different devices, opening the door to large-scale optimization problems.
An important feature of this work is the choice to model the wavelength as a continuous input parameter rather than fix each output neuron at a specific wavelength point as done in all previous work known to the authors. The waveguide ANN, for example, outputs effective index values and the Bragg ANN outputs reflection and group delay values across the entire input spectrum. This approach, while more difficult to train, is more convenient for the designer. For example, an optimization routine tasked with designing a Bragg filter can focus more on parameters like the bandwidth and shape, rather than an arbitrarily sampled wavelength profile. Furthermore, this method doesn’t require the training spectra to have the same sampling. Training sets for structures like ring resonators, whose features may require finer wavelength resolution than other devices, can now be strategically simulated to highlight these features. Assuming the network is trained correctly across a suitable domain, the ANN will seamlessly interpolate between both design parameters and wavelength points without any additional routines.
Unlike traditional simulation methods, training an arbitrary device ANN requires large datasets that are too time-intensive for most typical computers. With the growing availability of vast cloud-based computational resources, however, several million training simulations can now be run in hours or days. [@vecchiola_high-performance_2009]. Once trained, a neural network can reliably interpolate between training data, is compact and easily shared with the community, and can even continue to learn on new datasets via transfer learning [@pan_survey_2010]. Thus, the computational complexity inherent in designing integrated photonic devices can be moved to the front end of the design process, allowing individual designers to work with abstracted components whose optical response can be rapidly calculated.
As with all deep learning applications, the network’s utility is limited by biases introduced in the training set, the network architecture, or even the training process itself [@srivastava_dropout:_nodate]. Fortunately, we can anticipate these biases by extracting the model’s prediction uncertainty without modifying our network architecture. Dropout inference techniques leverage models that rely on dropout layers to mitigate over-fitting (a form of network bias) [@gal_dropout_2015]. Even pre-trained networks can use dropout inference to extract prediction uncertainties without any modifications to the network. This particular network design methodology opens the door to many more applications, like training on fabricated device data. Foundry’s that develop process design kits (PDKs), for example, can use this technique to model their fabrication processes while preserving their trade secrets.
Methods
=======
Training data generation and preprocessing {#training-data-generation-and-preprocessing .unnumbered}
------------------------------------------
We generated our waveguide neural network’s training set on a high performance computing cluster (HPC) using MEEP Photonic Bands (MPB) [@johnson_block-iterative_2001], a finite difference eigenmode solver. The solver simulated 31 different waveguide widths from 350 nm to 1500 nm and 31 waveguide thicknesses ranging from 150 nm to 400 nm resulting in 961 different geometries. The solver simulated 200 distinct wavelength points in the range of 1400 nm to 1700 nm. The total number of training samples fed into the neural network was 192,200. 70% of the data set was used as training samples and the remaining 30 % was used as validation samples. Each sample had three inputs (width, thickness, and wavelength) and four corresponding outputs (effective indices for the first two TE and TM modes). No postprocessing was performed on the waveguide training data.
On the same HPC, we generated our Bragg training set by simulating 104,131 different gratings with the layered dielectric media transfer matrix method (LDMTMM) [@helan_comparison_2006]. The LDMTMM method models each individual section of the Bragg grating as an ideal waveguide and cascades each sections’s corresponding transfer matrix to estimate the grating’s response for each wavelength point of interest. We calculated each individual waveguide’s effective index using the waveguide neural network. Our simulations swept through 10 different corrugation widths from 10 nm to 100 nm, 11 different grating lengths from 100 periods to 2000 periods, and 32 different chirping patterns.
Once the grating spectra were generated, we fit the results to a generalized skewed Gaussian (see Supplementary Material) to reduce ringing and to generalize the grating’s response to arbitrary apodization profiles. We found that without fitting, the resulting oscillations significantly complicate the training process and restrict the network’s domain to a single apodization. We fit both the reflection spectrum and group delay responses to generalized Gaussians and resampled the results with 250 wavelength points from 1.45 $\mu$m to 1.65 $\mu$m. Since the nonlinear fitting routine occasionally failed, not all of the simulated gratings were appropriate for testing. After filtering through the results, we generated a database of 26,032,750 training samples.
Neural network design and training {#neural-network-design-and-training .unnumbered}
----------------------------------
Both neural networks were trained on the same HPC cluster using Keras [@chollet_keras_2015] and Tensorflow [@martin_abadi_tensorflow:_2015]. Several hundred different architectures were tested. To gauge the effectiveness of each architecture, the mean-squared-error and coefficient of determination ($R^2$) metrics were used. The waveguide neural network that worked best had 4 hidden layers with 128 neurons, 64, neurons, 32 neurons, and 16 neurons. Each neuron used a hyperbolic tangent activation function. The Bragg grating neural network was designed with 10 hidden layers and 128 neurons each. RELU activation functions were used. Both networks were trained with 16 sample batch sizes. While the waveguide neural network was trained with 100 epochs, the Bragg grating neural network only needed about 5 epochs to reach sufficient results, primarily due to the large training set. The Bragg training set was normalized to improve the network’s expressive capabilities.
Simulation benchmarks {#simulation-benchmarks .unnumbered}
---------------------
We performed all benchmarks using a quad-core Intel(R) i5-2400 CPU clocked at 3.10 GHz with 12 GB of RAM. To evaluate the waveguide ANN’s speed, we simulated various waveguide parameters in serial using both the ANN and MPB. To evaluate the BG ANN’s speed, we simulated various grating’s in serial using both the ANN and the LDMTMM. We linearly fit each method’s results and compared the slopes to examine the speedup.
Device fabrication {#device-fabrication .unnumbered}
------------------
The silicon photonic Bragg gratings were fabricated by Applied Nanotools Inc (Edmonton, Canada) using a direct-write 100 keV electron beam lithographic process. Silicon-on-Insulator wafers with 220 nm device thickness and 2 $\mu$m thick insulator layer were used. The devices were patterned with a raster step of 5 $\mu$m and etched with a ICP-RIE etch process. A 2.2 $\mu$m oxide cladding was deposited using a plasma-enhanced chemical vapour deposition (PECVD) process.
Device measurement {#device-measurement .unnumbered}
------------------
Each device was measured using an automated process at the University of British Colombia (UBC). An Agilent 81600B tunable laser was used as the input source and Agilent 81635A optical power sensors as the output detectors. The wavelength was swept from 1500 to 1600 nm in 10 pm steps. A polarization maintaining (PM) fibre was used to maintain the polarization state of the light, to couple the TE polarization in and out of the grating couplers. Several dembedded test structures were used to normalize out the coupler profiles.
Data Availability {#data-availability .unnumbered}
=================
The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Lukas Chrostowski for useful discussions relating to the Bragg structures and for facilitating the SiEPIC fabricating process, as well as David Buck for supplying additional fabrication data.
Author Contributions {#author-contributions .unnumbered}
====================
AMH & RMC conceived the idea, AMH designed and trained the ANN, AMH & RMC designed the devices, AMH & RMC evaluated and interpreted the data, AMH & RMC wrote the manuscript.
Competing Interests {#competing-interests .unnumbered}
===================
The authors declare no competing interests.
Materials & Correspondence {#materials-correspondence .unnumbered}
==========================
Please send all correspondence and data requests to [email protected]
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bibliography:
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$*$ Department of Physics, University of Perugia, and Istituto Nazionale di Fisica Nucleare, Sezione di Perugia, Via A. Pascoli, I–06100 Perugia, Italy\
$\dagger$ Sapporo Gakuin University, Bunkyo-dai 11, Ebetsu 069, Hokkaido, Japan
A linked cluster expansion for the calculation of ground state observables of complex nuclei with realistic interactions has been used to calculate the ground state energy, density and momentum distribution of $^{16}O$ and $^{40}Ca$. Using the same cluster expansion and the wave function and correlation parameters obtained from the energy calculation, we have evaluated the semi inclusive reaction $A(e,e'p)X$ taking final state interaction (FSI) into account by a Glauber type approach; the comparison between the distorted and undistorted momentum distributions provides an estimate of the transparency of the nuclear medium to the propagation of the hit proton. The effect of color transparency is also included by considering the Finite Formation Time (FFT) that the hit hadron needs to reach its asymptotic physical state.
Introduction
============
The exclusive, $A(e,e'p)B$, and semi-inclusive, $A(e,e'p)X$, electro-disintegration of nuclei represent a powerful tool to investigate various aspects of nuclear structure (e.g. single-particle motion and mean field effects, nuclear correlations), as well as QCD motivated effects (e.g. color transparency). The accuracy of recent and forthcoming experimental data require realistic theoretical calculations to be performed, based as much as possible on a description of the nucleus stemming from *first principle* calculations, which means that the nuclear wave functions which appear in the calculation of various matrix elements, used either to predict or interpret the experimental data, should in principle result from many body calculations and realistic interactions. The problem has been solved in the case of few body system, for which realistic wave functions are currently being used in the interpretation of electro-disintegration processes, but this problem still needs a solution in case of complex nuclei. As a matter of fact, calculations of ground state observables for complex nuclei represent still a hard task, and even in those cases when approximate many body calculations for the ground state energy can be performed, the structure of the wave function is so complicated that its use for calculations of matrix elements of operators, different from the potential and kinetic energies ones, is very involved. For such a reason, a simpler, but still realistic, method which would allow one to calculate various kinds of matrix elements with nuclear wave functions which correctly incorporate the most relevant features of a realistic wave function, in particular its correlation structure resulting from the main features of modern two-nucleon interactions, would be extremely useful. Cluster expansion techniques, when the expectation value of different operators can be calculated to a certain order, may represent a valid and practicable alternative to the full complex “exact” solution of the many body problem. It is the aim of the present paper to illustrate a cluster expansion approach to the calculation of ground state energy properties (energy, density and momentum distribution) and various types of electro-disintegration processes.
Cluster expansion and the nuclear wave function
===============================================
In our linked-cluster expansion approach, the expectation value of a certain operator $\hat{\mathcal{O}}$ $$\label{omedio1}
\langle\hat{\mathcal{O}}\rangle\,=\,\frac{\langle\Psi_A|\,\hat{\mathcal{O}}
\,|\Psi_A\rangle}{\langle\Psi_A | \Psi_A\rangle}$$ is evaluated with correlated wave functions of the following “classical” form $$\label{psi1}
\Psi_A\,=\,\hat{F}({\bf r}_1,...,{\bf r}_A)\,\Phi_A({\bf r}_1,...,{\bf r}_A)\,,$$ where $\Phi_A$ is a mean field (Slater determinant) wave function, and $\hat{F}$ a symmetrized (by the symmetrization operator $\hat{S}$) correlation operator which generates $\textit{correlations}$ into the mean field wave function; it has the following general form $$\label{corre1}
\hat{F}\,=\,{{\hat{S}}}\prod^A_{i<j}\hat{f}(r_{ij}) \,\,$$ with $$\hat{f}(r_{ij})=\sum_p\,{\hat
f}^{(p)}(r_{ij})\;\;\hspace{2cm}{\hat f}^{(p)}(r_{ij})=
f^{(p)}(r_{ij})\,\hat{O}^{(p)}_{ij}\,
\label{effecorr}$$ where the operators $\hat{O}^{(p)}$ are the same which appear in the two-nucleon interaction, having the form ( e.g. in case of a $V8$-type interaction) $$\label{operator}
{\hat{O}}^{p=1,8}_{ij}=\left[1,\, {\bf \sigma}_i
\cdot{{\bf \sigma}}_j,\,S_{ij},\,({\bf L} \cdot {\bf S})_{ij}\right]\otimes
\left[1,\,{\bf \tau}_i\cdot\ {\bf \tau}_j \right]\,$$ The central parts $f^{(p)}(r_{ij})$’s of the correlation function ${\hat f}^{(p)}$, reflect the radial behaviour of the various components and their actual form is determined either by the minimization of the ground state energy, or by other criteria.
The cluster expansion of Eq.\[omedio1\] is carried out in terms of the quantity $\hat{\eta}_{ij}={\hat{f}}^2_{ij}-1$, whose integral plays the role of a small expansion parameter; by expanding the numerator and the denominator the terms $\hat{\mathcal{O}}_n$ of the same order $n$ in $\eta_{ij}$, are collected obtaining $\langle\hat{\mathcal{O}}\rangle=
\mathcal{O}_0+\mathcal{O}_1+\mathcal{O}_2+...$, with $$\begin{aligned}
\label{eta1}
\mathcal{O}_0&=&\langle\hat{\mathcal{O}}\rangle\, \nonumber\hspace{2.73cm}
\mathcal{O}_1\,=\,\langle\sum_{ij}
\hat{\eta}_{ij}\,\hat{\mathcal{O}}\rangle\,
-\,\mathcal{O}_0\,\langle\sum_{ij}\,
\hat{\eta}_{ij}\rangle\,\nonumber\\
\nonumber\\
\mathcal{O}_2&=&\langle\sum_{ij<kl}\hat{\eta}_{ij}\,\hat{\eta}_{kl}\,
\hat{\mathcal{O}}\rangle\,
-\,\langle\sum_{ij}\hat{\eta}_{ij}\,\hat{\mathcal{O}}\rangle\,
\langle\sum_{ij}\hat{\eta}_{ij}\rangle\,+\nonumber\\
& &\hspace{3cm}+\,\mathcal{O}_0\,\left(\langle\sum_{ij<kl}\hat{\eta}_{ij}\,
\hat{\eta}_{kl}\,\rangle\,-\langle\sum_{ij}\hat{\eta}_{ij}\rangle^2
\right)\,;\end{aligned}$$ where $\langle [...] \rangle \equiv \langle \Phi_A \left|[...] \right|\Phi_A
\rangle$. From now on, our approach will consist in obtaining the parameters characterizing the correlation functions and the mean-field single-particle wave function which correspond to an acceptable value of the ground state energy, we will then use the obtained wave function $\Psi_A$ to calculate the transition matrix elements entering in the theoretical description of electro-disintegration processes using the same cluster expansion employed to calculate the energy. We have calculated the ground state energy of $^{16}O$ and $^{40}Ca$ using the Argonne $V8'$ [@pud01] potential and adopting, as in Ref. [@fab01], the so called $f_6$ approximation consisting in considering only the first six components of Eq. \[operator\]. The expectation value of the many body non relativistic Hamiltonian of the nucleus was obtained by calculating the average values of the kinetic and potential energies, i.e. $$\label{kin1}
\langle\hat{T}\rangle\,=\,-\frac{{\hbar}^2}{2m}\,\int\,d{\bf k}\,k^2\,n({k})\,,$$ where $n(k)$ is the nucleon momentum distribution ($k \equiv |{\bf k}|$), $$\label{momdis1}
n(k)\,=\,\frac{1}{(2\pi)^3}\,\int\,d{\bf r}_1\,d{\bf r}_1'
\,e^{-i\,{\bf k}\cdot({\bf r}_1-{\bf r}_1')}\,\rho^{(1)}({\bf r}_1,{\bf r}_1'),$$ and $$\label{pot1}
\langle\hat{V}\rangle\,=\,\frac{1}{2}\sum_{i<j}\langle \hat {v}_{ij}\rangle
=\,\frac{A(A-1)}{2}\sum_p\;\int\;d{\bf r}_1 d
{\bf r}_2\;v^{(p)}(r_{12})\rho^{(2)}_{(p)}({\bf r}_1,{\bf r}_2)\,.$$ The calculations have been performed by cluster expanding the expectation value of the non diagonal one-body, $\hat{\rho}^{(1)}$, and diagonal two-body, $\hat{\rho}^{(2)}({\bf r}_1,{\bf r}_2)$ density matrix operators. The six correlation functions $f^{(p)}(r_{ij})$ have been borrowed from Ref. [@fab01], whereas Harmonic Oscillator (HO) and Saxon-Woods (SW) spwf’s have been used to describe the mean field. As in Ref. [@fab01], we found that the charge densities corresponding to the minimum of the energy, appreciably disagree with the corresponding experimental quantities, therefore, in view of the mild dependence of the energy around the minimum upon the mean-field parameters, following Ref. [@fab01] we have changed the latter to obtain agreement between theoretical and experimental charge densities. The results for the charge densities and momentum distributions, which are shown in Figs. \[density-opt\] and \[momdisHO16-opt\], deserve the following comments:
1. [the agreement between our cluster expansion and FHC/SOC result of Ref. [@fab01] is very good]{};
2. [both approaches predict momentum distributions which do not appreciably differ from the ones obtained in Ref. [@pan01], where the Variational Monte Carlo method and the $AV18$ interaction have been used;]{}
3. [the high momentum part of $n(k)$ is almost entirely exhausted by *non-central*, *long-range* correlations, with the *central*, *short-range*, Jastrow correlations under-predicting the high momentum part of $n(k)$ by about one order of magnitude;]{}
4. [the dominant non-central correlations are the isospin, $f_4=f^{(4)}(r_{ij}) {\bf \tau}_i\cdot{\bf \tau}_j$, and isospin-tensor, $f_6=f^{(6)}(r_{ij}) {\bf \tau}_i\cdot{\bf \tau}_j S_{ij}$, correlations]{};
=5.9cm
The final state interaction in $A(e,e'p)X$ reactions off complex nuclei: Glauber approach
=========================================================================================
Using the results obtained in the previous Section, we have calculated the semi-inclusive $A(e,e'p)X$ process in which an electron with 4-momentum $k_1\equiv\{{\bf k}_1,i\epsilon_1\}$, is scattered off a nucleus with 4-momentum $P_A\equiv\{{\bf 0},iM_A\}$ to a state $k_2\equiv\{{\bf k}_2,i\epsilon_2\}$ and is detected in coincidence with a proton $p$ with 4-momentum $p\equiv\{{\bf p},iE_p\}$; the final $(A-1)$ nuclear system with 4-momentum $P_X\equiv\{{\bf P}_X,iE_X\}$ is undetected. The cross section for the exclusive process $A(e,e'p)B$ can be written as follows $$\frac{d\sigma}{dQ^2 d\nu d{\bf p}}=K\sigma_{ep}P_D(E_m,{\bf p}_m)
\label{sezione}$$ where $K$ is a kinematical factor, $\sigma_{ep}$ the off-shell electron-nucleon cross section, and $Q^2=|{\bf q}|^2-\nu^2$ the four momentum transfer. The quantity $P_D(E_m,{\bf p}_m)$ is the distorted nucleon spectral function which depends upon the observable *missing momentum* ${\bf p}_m={\bf q}-{\bf p}$ (${\bf p}_m={\bf k}$ when the FSI is absent) and *missing energy* $E_m=\nu-T_p -T_{A-1}$. In the semi-inclusive $A(e,e'p)X$ process, the cross section (\[sezione\]) is integrated over the missing energy $E_m$, at fixed value of ${\bf p}_m$ and becomes directly proportional to the *distorted* momentum distribution $$n_D({\bf p}_m)={(2 \pi)^{-3}} \int e^{i {\bf p}_m({\bf r}_1 -{\bf r}_1')}
\rho_D ({\bf r}_1,{\bf r}_1') d{\bf r}_1 d{\bf r}_1'
\label{nd}$$ where $$\begin{aligned}
\rho_D ({\bf r}_1,{\bf r}_1')= \frac {\langle\Psi_A\,S^{\dagger}
\,\hat{O}({\bf r}_1,{\bf r}_1')\,S'\,{\Psi_A}'\rangle}{\langle\Psi_A\Psi_A\rangle}
\label{rodi}\end{aligned}$$ is the distorted one-body mixed density matrix, $S$ the S-matrix describing FSI, and the primed quantities have to be evaluated at ${\bf r}_i'$ with $i=1, ...,A$. The integral of $n_D({\bf p}_m)$ gives the nuclear transparency $T$ $$T = \frac{\int n_D({\bf p}_m) d{\bf p}_m}{\int n(k) d{\bf k}} =
% (2 \pi)^{-3}\int \rho_D ({\bf r}_1,{\bf r}'_1)
% d{\bf r}_1 d{\bf r}'_1 \int e^{i {\bf p}_m({\bf r}_1 -{\bf r}'_1)}
% d{\bf p}_m\\ \nonumber
\int \rho_D ({\bf r})d{\bf r} = 1+ \Delta T
\label{intnd}$$ where $\rho_D({\bf r})=\rho_D ({\bf r}_1={\bf r}'_1\equiv {\bf r})$ and $\Delta T$ originates from the FSI. In Ref. [@cio01] Eq. \[nd\] has been evaluated using a Glauber representation for the scattering matrix $S$, *viz* $$S \rightarrow S_G({\bf r}_1\dots{\bf r}_A)=\prod_{j=2}^AG({\bf r}_1,{\bf r}_j)
\equiv \prod_{j=2}^A\bigl[1-\theta(z_j-z_1)\Gamma({\bf b}_1-{\bf b}_j)\bigr]
\label{SG}$$ where ${\bf b}_j$ and $z_j$ are the transverse and the longitudinal components of the nucleon coordinate ${\bf r}_j\equiv({\bf b}_j,z_j)$, ${\mit\Gamma}({\bf b})$ the Glauber profile function for elastic proton nucleon scattering, and the function $\theta(z_j-z_1)$ takes care of the fact that the struck proton “1” propagates along a straight-path trajectory so that it interacts with nucleon “$j$” only if $z_j>z_1$. The same cluster expansion described in Section II has been used taking Glauber rescattering exactly into account at the given order $n$, and using the approximation $|\Psi_{A-3}|^2 =
\prod_3^A \rho(i)$. Using the mean-field and correlation parameters obtained from the energy calculation, we have obtained the *distorted* nucleon momentum distributions $n_D({\bf p}_m)=n_D(p_m, \theta)$, where $\theta$ is the angle between ${\bf q}$ and ${\bf p}_m$; the results for $^{16}O$ and $^{40}Ca$ are presented in Fig. \[momdisto01\].
=5.9cm =5.9cm
Finite formation time effects
=============================
Recently [@bra01] the effects of color transparency in quasi-elastic lepton scattering off nuclei, have been introduced by explicitly considering the finite formation time (FFT) that the hit hadron needs to evolve to its asymptotic physical state. It has been shown that at the values of the Bjorken scaling variable $x=Q^2/2m\nu \simeq 1$, FFT effects can be treated in a simple way, i.e. by replacing the Glauber operator (Eq. \[SG\]) with
=5.9cm =5.6cm
$$\label{smatrix}
S_{FFT}({\bf r}_1,...,{\bf r}_A)=\,
\,\prod^A_{j=2}\Big(1-J(z_1-z_j)\Gamma({\bf b}_1-{\bf b}_j)\Big)\,,$$
where $$J(z)\,=\,\theta(z)\,e^{-\,z\,\frac{x\,m\,M^2}{Q^2}}\,$$ $m$ being the nucleon mass and $M^2=m^{*\,2}-m^2$ is a parameter describing the average excitation energy of the ejectile. It can be seen that at sufficiently high values of $Q^2$, $J \rightarrow 1$ and the FSI vanishes. The effects of FFT on the distorted momentum distribution is shown in Fig. \[transpa02\].
Summary and conclusions
=======================
We have obtained fully correlated wave functions by calculating the average value of the nuclear Hamiltonian by means of a linked cluster expansion and using realistic two-nucleon interactions. The wave functions have been used to obtain the ground state density and momentum distribution. By introducing FSI effect by a Glauber-type approach, the distorted momentum distributions appearing in the semi-inclusive $A(e,e'p)X$ processes have been calculated. By such a procedure, a consistent treatment of initial-state correlations and final-state interactions has been achieved. Color transparency effects have also been investigated by the Finite Formation Time approach. Comparison with available experimental data are in progress and will be reported elsewhere.\
This work is part of the research activity of M. A. performed, under the supervision of C. d. A., for the fulfillment of the PhD title. H. M. thanks the Department of Physics, University of Perugia, and INFN, Sezione di Perugia, for hospitality and support. Partial support by the Italian Ministero dell’Istruzione, Universitá e Ricerca (MIUR), through the funds COFIN01, is acknowledged. We are indebted to A. Fabrocini for providing the correlation functions obtained in Ref. [@fab01].
[3]{} B.S. Pudliner, V.R. Pandharipande, J. Carlson, S.C. Pieper and R.B. Wiringa, *Phys. Rev.* **C56**, 1720 (1997); A. Fabrocini, G. Co’, *Phys. Rev.* **C61**, 044302 (2000) and Private Communication; S.C. Pieper, R.B. Wiringa and V.R. Pandharipande, *Phys. Rev.* **C46**, 1741 (2000); H. de Vries, C.W. de Jager and C. de Vries, *Atom. Data Nucl. Data Tabl.* **36**, 495 (1987) C.Ciofi degli Atti, D. Treleani *Phys. Rev.* **C60**, 024602 (1999); M.A. Braun, C.Ciofi degli Atti, D. Treleani, *Phys. Rev.* **C62**, 034606 (2000); H. Morita, M.A. Braun, C.Ciofi degli Atti, D. Treleani *Nucl. Phys.* **A699**, 328c (2002); M. Alvioli, C.Ciofi degli Atti, H. Morita *(in preparation)*
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---
author:
- 'G. Testor$^{**}$, J.L. Lemaire,[^1] , M. Heydari-Malayeri, L. E. Kristensen, S. Diana, D. Field'
date: 'Received ...; accepted ...'
title: 'VLT/NACO near-infrared imaging and spectroscopy of N88A in the SMC [^2]'
---
[We present near-infrared imaging and spectroscopic high spatial resolution observations of the SMC region N88 containing the bright, excited, extincted and compact H II region N88A of size $\sim$1 pc.]{} [To investigate its stellar content and reddening, N88 was observed using spectroscopy and imagery in the JHKs- and L’-band at a spatial resolution of $\sim$ 0.1-0.3 $\arcsec$, using the VLT UT4 equipped with the NAOS adaptive optics system. In order to attempt to establish if the origin of the infra-red (IR) excess is due to bright nebulosity, circumstellar material and/or local dust, we used Ks vs J-K colour-magnitude (CM) and JHK colour-colour (CC) diagrams, as well as L’ imagery.]{} [Our IR-data reveal in the N88 area an IR-excess fraction of $\geq$30 per cent of the detected stars, as well as an unprecedently detailed morphology of N88A. It consists of an embedded cluster of $\sim$ 3.5$\arcsec$ ($\sim$ 1 pc) in diameter, of at least thirteen resolved stars superposed with an unusual bright continuum centered on a very bright star. The four brightest stars in this cluster lie red-ward of H-K $\geq$ 0.45 mag, and could be classified as young stellar object (YSO) candidates. Four other probable YSO candidates are also detected in N88 along a south-north bow-shaped thin H$_2$ filament at $\sim$ 7$\arcsec$ east of the young central bright star. This star, that we assume to be the main exciting source, could also be complex. At 0.2$\arcsec$ east of this star, a heavily embedded core is detected in the L’-band. This core with L’ $\sim$ 14 mag and L’-K $\geq$ 4.5 mag could be a massive class I protostar candidate. The 2.12 $\mu$m H$_2$ image of N88A resembles a shell of diameter $\sim$ 3$\arcsec$ ($\sim$ 0.9 pc) centered on the bright star. This shell is formed of three bright components, of which the brightest one superposes the ionization front. The line ratios of H$_{2}$ 2-1 S(1) and 1-0 S(0) relative to 1-0 S(1), as well as the presence of high v lines, are indicative of photodissociation regions, rather than shocks.]{}
Introduction
============
The Small Magellanic Cloud (SMC) is rich in H II regions and young OB associations. Because of its known and relatively small distance ($\sim$ 65 kpc) (Kovacs 2000), its face-on position relatively free from foreground extinction (McNamara $\&$ Feltz 1980), and low internal extinction (Westerlund 1997), it is well suited to study both individual stars and very compact objects, as well as global structures. It is an ideal laboratory for studying the formation and evolution of massive stars in a low metallicity environment. Understanding the characteristics of massive stars and their interaction with their environment is a key problem in astrophysics. We have made some progress concerning the early stages of massive star formation in the galaxy, but the current knowledge about the early stages of massive star evolution in other galaxies is mediocre at best. The main reason is that the earliest stages of massive star evolution are deeply enshrouded, inaccessible in the optical wavelengths. Another reason is that these stars are often members of very crowded regions. Today, high-spatial near-infra-red (NIR) resolution observations using NACO attached to the VLT, are able to overcome these obstacles in the SMC. Our search for the youngest massive stars in the Magellanic Clouds (MCs) (Heydari-Malayeri $\&$ Testor 1982) led to the discovery of a distinct and very rare class of H II regions that we called high-excitation compact H II “blobs” (hereafter HEBs) listed in Testor (2001). In contrast to the ordinary H II regions of the MCs, which are extended structures spanning several arcminutes on the sky ($>$ 50 pc) and are powered by a large number of hot stars, HEBs are very dense small regions, usually 2$\arcsec$ to 8$\arcsec$ in diameter (0.8 to 3 pc), ionized by one or a few massive stars and affected by local dust. Two other HII regions, MA 1796 and MG 2 (less than 1pc across) heavily extincted and ionized by a small young cluster, have been found by Stanghellini et al. (2003) in the SMC.
In the present paper we focus on the peculiar HEB LHA 115-N88A, hereafter labelled N88A of diameter $\sim$1 pc (Testor $\&$ Pakull 1985) in the extended H II region LHA 115-N88 or N88 (Henize 1956) of diameter $\sim$ 10 pc. N88 lies in the Shapley Wing of the SMC and contains the young cluster HW 81 ($\sim$ 0.6$\arcmin$) (Hodge $\&$ Wright 1977). It is known that the SMC is made of four H I layers with different velocities (McGee & Newton 1982). This situation complicates the study, particularly in the region of the Shapley wing. However, the available H I observations provide helpful data for the study of this region. In particular, the N88 region lying at about 35$\arcmin$ ($\sim$ 700 pc) west of N83/N84 is not apparently associated with the H I cloud of these H II regions (Heydari-Malayeri et al. 1988). N88A should be associated with the H I component of velocity +134 kms$^{-1}$ (McGee & Newton 1982).
N88A is the brightest and the most excited HEB in the MCs. It is also the most reddened H II region in these galaxies of low-metallicity (Heydari-Malayeri et al. 2007). Numerous optical detailed studies have been made on this object (Testor $\&$ Pakull 1985, Wilcot 1994, Heydari-Malayeri et al. 1999 (hereafter HM99), Kurt et al. 1999, Testor et al. 2003). Nevertheless, many uncertainties remain to understand the true nature of N88A, such as its exciting source, that still remains unidentified, as well as the nature of the reddening.
Israel $\&$ Koorneef (1988, 1991) have detected the presence of molecular hydrogen in N88 through H$_{2}$ emission which is either shock-excited on a small scale of 0.46$\arcsec$ by stars embedded in the molecular cloud, or radiatively excited on a large scale (3$\arcsec$-60$\arcsec$). However, their low spectral (R=50) and spatial (7.5-10$\arcsec$ aperture) resolutions did not allow discrimination of these different processes. They described N88A as a strong NIR source dominated by nebular emission containing a strong hot dust component and noticed that N88A has an unusual blue J-H colour. In Testor et al. (2005), at low spatial resolution, a pure H$_{2}$ emission is detected in N88A as well as along a south-north diffuse long filament at $\sim$ 6-8$\arcsec$ to the east. In N66 (Henize 1956) a giant HII region in the SMC, Schmeja et al. (2009) have reported that most of the H$_{2}$ emission peaks coincide with the bright component of the ionized gas and with compact embedded young clusters where candidate YSOs have been identified. Using SEST, Israel et al. (2003) detected a CO molecular cloud of 1.5$\arcmin$ x 1.5$\arcmin$ in the region, reporting spectra and maps of the $^{12}$CO lines J=1-0 and J=2-1. Stanimirovic et al. (2000) found that the highest values of the dust-to-gas mass ratio and dust temperature in the SMC are found in N88A.
IR studies of a similar size young star formation region like the Trapezium region in Orion ($\sim$0.75 x 0.75 pc) (Lada et al. 2000) and the more extended 30 Doradus in the LMC (Maercker $\&$ Burton 2005) showed that during star formation, YSOs are associated with the circumstellar material inducing IR-excess emission, and also that the use of JHK CC diagrams are useful tools to detect this emission. However, for young massive stars generally found in embedded clusters, their surrounding material destruction time scale is short, making their observation difficult (Bik et al. 2005, 2006). The most suitable wavelength to determine the nature of the IR-excess is the L-band, increasing the IR-excess and reducing the contribution of extended emission from reflection nebulae and H II regions (Lada et al. 2000). IR-excess can be useful to determine the origin of the reddening in embedded young clusters. Martin-Hernandez et al. (2008) found in N88A, from a mid-IR high spatial resolution Spitzer-IRS spectrum, a rising dust continuum and PAH bands, typical characteristics in H II regions. Using radio observations, Indebetouw et al. (2004) found that N88A is ionized by an O5 type star.
In the present paper we present the results of JHK- and L’-band high spatial resolution observations of N88A and its surroundings, using adaptive optics at the VLT. Section 2 discusses the instrumentation employed during these observations, and the data acquisition and reduction procedures used. Section 3 describes the results and analysis of our observations, and Section 4 gives our conclusions.
{width="14.0cm"}
{width="18.0cm"}
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[llccccc]{} Id. & Filter & Expo. &Mode & Date & Seeing & FWHM\
& &t(s) x n & & &($\arcsec$)&($\arcsec$)\
N88A &J 1.27 $\mu$m &20 x 30 & - & - & - & 0.35\
&H 1.66 $\mu$m &20 x 30 & - & - & & 0.27\
&Ks 2.18 $\mu$m &30 x 30 & S54 &9/10/2004 & 0.6-0.9 & 0.21\
&L’ 3.8 $\mu$m &0.2 x 26 & S27 &10/10/2004 & 0.9-1.2 & 0.10\
&NB-2.12 &200x 20 & S54 & - & & 0.19\
&NB-2.24 &150x 20 & S54 & - & & 0.25\
&Ks &60 x 30 & S27 &11/10/2004 & & 0.10\
{width="18.0cm"}
=0.04cm
[lcccccc]{} Id. & Date & Slit & Exposures & Mode &$\lambda$/$\delta$$\lambda$& Seeing\
& & (mas) & t(s) x n & & &($\arcsec$)\
S1 &9/10/2004 &172 & 100 x 10 &S54-4-SHK & 400 & 0.6-0.9\
S2 &10/10/2004 & - & - & - & - & -\
S3 &11/10/2004 & - & - & - & - & -\
& & & & & &\
hip8485 &9/10/2004 & - & 2.5 x 4 & - & - & -\
hip29968 &9/10/2004 & - & 2 x 4 & - & - & -\
hip103087 &11/10/2004 & - & 1.8 x 4 & - & - & -\
Observations and data reduction
===============================
NIR observations of N88A were obtained at the ESO Very Large Telescope (VLT) during October 2004. Images and spectra were taken using NACO on UT4, composed of the Nasmyth Adaptive Optics System (NAOS) and the High Resolution IR Camera and Spectrometer (CONICA). The detector was a 1026 x 1024 SBRC InSb Alladin 3 array. The cameras S54 and S27 were used in the range 1.0-2.5 $\mu$m and the L27 camera in the range 2.5-5.0 $\mu$m.
The field-of-view (FOV) of the S54 camera was 54$\arcsec$ x 54$\arcsec$ with a pixel size of 0.05274$\arcsec$, corresponding to 0.015 pc at the distance modulus of 19.05 for the SMC (Kovacs 2000). The FOV of the S27 and L27 camera was 27.15$\arcsec$ x 27.15$\arcsec$ with a pixel size of 0.02637$\arcsec$, corresponding to 0.0075 pc. For spectroscopy we used the S54 camera.
As adaptive optics (AO) reference source for wavefront sensing we used the object itself (N88A) (Testor et al. 2005). The conditions were photometric and the seeing ranging from 0.65$\arcsec$ to 1$\arcsec$ in the visible. After subtraction of the average dark frame, each image was divided by the normalized flat field image. The data were reduced mainly with the ESO software packages MIDAS and ECLIPSE.
Imaging
-------
Images through J, H, Ks broad-band and 2.12$\mu$m, 2.24$\mu$m narrow band filters were obtained with the S54 camera. A composite JHKs colour image of the observed field is shown in Fig. \[fig:S54\]. Images with higher spatial resolution, in the Ks and L’ large band, were also obtained with the S27 camera. The log of NIR imaging observations is given in Table 1. The AutoJitter mode was used: that is, at each exposure, the telescope moves according to a random pattern in a 6$\arcsec$ x 6$\arcsec$ box. Table 1 lists the stellar full-width-at-half-maximum (FWHM) in final images of different observed bands of the star at J2000 coordinates ($\alpha$, $\delta$) = (1$^{h}$,24$^{m}$,8.88$^{s}$, -73$^{o}$,8’,56.2") of the 2MASS survey (Cutri et al. 2003). The AO image is affected by anisoplanatism and leads to degradation of the point spread function (PSF) becoming more elongated as the angular offset from the guide star increases. This has been taken into consideration for the photometric measurements, as explained in Sect. 2.3.
Spectroscopy
------------
Spectroscopy was performed in the S54-4-SHK mode (broad-band filter), giving a linear dispersion of 1.94 nm/pixel and a spatial scale of 53 mas/pixel. Three long-slit spectra S1, S2, and S3 were chosen from the IR images given by NACO. S1 (PA = 115$\degr$) and S2 (PA = 89.3$\degr$) cross the central bright star. S3 (PA = 18.9$\degr$) crosses the stars $\#$37 and $\#$47 (see Fig. \[fig:S27inset\]). The slit width was 172 mas and the spectral resolution $\sim$ 400. For each exposure the detector integration time (DIT) was 100s. Ten exposures were obtained in the Autonod on slit mode, which allows spectroscopy of moderately extended objects. The log of spectroscopic observations is given in Table 2. In order to remove telluric absorption features, stars with a similar airmass were observed as telluric standards. Spectroscopy was reduced with the MIDAS software package LONG. =0.1cm
[llllllrrr]{} Id. & $\alpha$(2000) &$\delta$(2000)&mag-J&mag-H&Mag-K&J-K&J-H&H-K\
1 & 1 24 11.42 & 73 9 26.15& 18.24& 17.65& 17.50& 0.74& 0.59& 0.15\
2 & 1 24 11.86 & 73 9 25.85& 17.63& 17.71& 17.65& -0.02& -0.08& 0.06\
3 & 1 24 8.67 & 73 9 23.08& 19.08& 18.88& 18.84& 0.24& 0.20& 0.04\
4 & 1 24 13.84 & 73 9 22.77& 18.46& 18.48& 18.45& 0.01& -0.02& 0.03\
5 & 1 24 7.69 & 73 9 20.63& 17.89& 17.61& 17.34& 0.55& 0.28& 0.27\
6 & 1 24 11.02 & 73 9 20.11& 15.76& 15.85& 15.82& -0.06& -0.09& 0.03\
7 & 1 24 6.85 & 73 9 19.56& 19.16& 18.71& 18.51& 0.66& 0.45& 0.21\
8 & 1 24 13.75 & 73 9 18.85& 15.17& 15.37& 15.38& -0.21& -0.20& -0.01\
9 & 1 24 8.72 & 73 9 18.79& 18.22& 18.18& 18.02& 0.20& 0.04& 0.16\
10 & 1 24 7.03 & 73 9 18.05& 18.80& 18.79& 18.71& 0.09& 0.01& 0.08\
11 & 1 24 12.53 & 73 9 15.61& 16.27& 16.45& 16.47& -0.20& -0.18& -0.02\
12 & 1 24 7.30 & 73 9 15.25& 16.72& 16.87& 16.87& -0.16& -0.15& -0.00\
13 & 1 24 10.75 & 73 9 14.84& 18.67& 18.53& 18.06& 0.61& 0.14& 0.47\
14 & 1 24 5.18 & 73 9 14.70& 16.27& 16.44& 16.31& -0.04& -0.17& 0.13\
15 & 1 24 7.45 & 73 9 14.32& 17.31& 17.48& 17.47& -0.15& -0.17& 0.01\
16 & 1 24 5.16 & 73 9 14.07& 16.19& 16.40& 16.26& -0.07& -0.21& 0.14\
17 & 1 24 12.57 & 73 9 13.33& 17.44& 17.59& 17.40& 0.04& -0.15& 0.19\
18 & 1 24 8.97 & 73 9 11.82& 19.78& 19.65& 19.52& 0.26& 0.12& 0.14\
19 & 1 24 9.30 & 73 9 10.17& 18.44& 18.21& 17.87& 0.57& 0.23& 0.34\
20 & 1 24 9.13 & 73 9 10.09& 18.30& 18.23& 18.19& 0.11& 0.08& 0.03\
21 & 1 24 9.58 & 73 9 9.84& 19.11& 18.51& 17.66& 1.45& 0.60& 0.84\
22 & 1 24 5.76 & 73 9 9.34& 18.89& 18.59& 17.93& 0.96& 0.30& 0.66\
23 & 1 24 8.58 & 73 9 9.01& 19.65& 19.34& 18.95& 0.70& 0.31& 0.39\
24 & 1 24 9.93 & 73 9 8.68& 18.66& 18.36& 17.80& 0.86& 0.31& 0.55\
25 & 1 24 11.52 & 73 9 8.03& 18.89& 18.73& 18.51& 0.38& 0.15& 0.22\
26 & 1 24 10.13 & 73 9 7.70& 19.48& 19.46& 19.30& 0.17& 0.02& 0.15\
27 & 1 24 9.45 & 73 9 7.50& 18.37& 18.18& 17.97& 0.40& 0.19& 0.20\
28 & 1 24 4.96 & 73 9 7.42& 16.32& 16.48& 16.42& -0.10& -0.16& 0.06\
29 & 1 24 9.25 & 73 9 7.17& 18.21& 18.11& 17.88& 0.33& 0.10& 0.24\
30 & 1 24 6.14 & 73 9 6.35& 17.85& 17.51& 17.33& 0.52& 0.34& 0.18\
31 & 1 24 9.09 & 73 9 5.99& 15.74& 15.73& 15.51& 0.23& 0.01& 0.16\
32 & 1 24 7.57 & 73 9 5.80& 19.04& 19.09& 18.91& 0.13& -0.05& 0.18\
33 & 1 24 9.98 & 73 9 5.77& 19.46& 18.76& 17.83& 1.63& 0.70& 0.93\
36 & 1 24 5.82 & 73 9 5.09& 17.36& 17.43& 17.26& 0.10& -0.07& 0.17\
& & & & & & & &\
37$^{n}$ & 1 24 7.84 & 73 9 4.24& 15.01& 15.10& 14.28& 0.73& -0.09& 0.81\
41$^{n}$ & 1 24 7.96 & 73 9 3.74& 14.68& 14.56& 13.82& 0.86& 0.12& 0.74\
42$^{n}$ & 1 24 7.85 & 73 9 3.60& 15.05& 15.05& 14.44& 0.61& 0.00& 0.61\
47$^{n}$ & 1 24 7.96 & 73 9 3.00& 15.55& 15.36& 14.88& 0.67& 0.19& 0.48\
& & & & & & & &\
46 & 1 24 10.08 & 73 9 3.05& 19.65& 18.93& 18.18& 1.47& 0.72& 0.75\
48 & 1 24 11.02 & 73 9 2.86& 18.93& 18.90& 18.82& 0.12& 0.03& 0.09\
50$^{n}$ & 1 24 7.53 & 73 9 2.09& 17.61& 17.62& 17.16& 0.45& -0.01& 0.46\
52 & 1 24 7.01 & 73 9 1.79& 18.80& 18.98& 18.57& 0.23& -0.18& 0.41\
53 & 1 24 10.35 & 73 9 0.80& 19.91& 19.51& 18.95& 0.96& 0.40& 0.56\
54 & 1 24 12.17 & 73 8 59.81& 17.75& 17.59& 17.29& 0.46& 0.16& 0.30\
55 & 1 24 8.53 & 73 8 57.45& 20.05& 20.01& 19.65& 0.40& 0.04& 0.36\
56 & 1 24 9.27 & 73 8 56.87& 17.57& 17.42& 17.29& 0.28& 0.15& 0.13\
57 & 1 24 9.07 & 73 8 56.24& 15.36& 14.88& 14.59& 0.77& 0.48& 0.29\
58 & 1 24 10.81 & 73 8 55.78& 18.45& 18.43& 18.26& 0.19& 0.02& 0.17\
59 & 1 24 9.86 & 73 8 54.07& 16.51& 16.67& 16.64& -0.13& -0.16& 0.03\
60 & 1 24 3.68 & 73 8 53.33& 18.51& 18.03& 17.59& 0.92& 0.48& 0.44\
61 & 1 24 8.36 & 73 8 53.17& 19.27& 18.65& 18.44& 0.83& 0.62& 0.21\
62 & 1 24 3.59 & 73 8 50.89& 18.06& 17.65& 17.43& 0.63& 0.41& 0.22\
$^{n}$ Stars in N88A analyzed using the NSTAR routine.
[llllllllll]{} Id. && $\alpha$(2000)&$\delta$(2000)&K$^{dao}$& err &K$^{psf}$ &L$^{ap}$&K-L\
57 && 1 24 8.88 & -73 8 56.00& 14.64 & 0.003& &14.36$\pm$0.05&\
34 && 1 24 7.94 & -73 9 5.69 & 18.45 & 0.070& & &\
35 && 1 24 7.64 & -73 9 5.42 & 18.73 & 0.050& & &\
37 && 1 24 7.83 & -73 9 4.56& 16.43 & 0.032& 16.66$\pm$0.15& &\
38 && 1 24 8.30 & -73 9 3.85& 18.46 & 0.060& & &\
40 && 1 24 8.21 & -73 9 3.80& 18.40 & 0.065& & &\
41 && 1 24 7.93 & -73 9 3.99& 14.99 & 0.010& 15.05$\pm$0.10&14.1$\pm$0.10 & 0.95\
42 && 1 24 7.85 & -73 9 3.82& 16.11 & 0.047& 16.60$\pm$0.30& &\
43 && 1 24 8.21 & -73 9 3.30& 17.62 & 0.066& & &\
44 && 1 24 8.23 & -73 9 3.05& 17.76 & 0.060& & &\
45 && 1 24 8.16 & -73 9 3.14& 17.12 & 0.044& & &\
47 && 1 24 7.93 & -73 9 3.19& 15.62 & 0.036& 15.99$\pm$0.20& &\
49 && 1 24 8.16 & -73 9 2.70& 17.41 & 0.061& & &\
51 && 1 24 7.49 & -73 9 2.29& 18.27 & 0.040& & &\
52 && 1 24 7.88 & -73 9 1.90& 18.65 & 0.061& & &\
L1-C && 1 24 7.95 & -73 9 3.80&$\geq$18.5& 0.024& &13.98$\pm$0.20& $\geq$4.52\
$^{dao}$, $^{psf}$ and $^{ap}$ magnitudes derived using DAOPHOT, the PSF of star $\#$57, and an aperture respectively.
Photometry
-----------
In Fig. \[fig:S54inset\] we present the N88 region observed with the S54 camera (Field 1) through the J-band filter. The instrumental magnitudes of the elongated stars (see Sect. 2.1) outside the central region, were derived with DAOPHOT (Stetson 1987), using concentric aperture photometry to integrate all the flux of each star. Although PSF photometry is better adapted for crowded fields, we could not use it. Indeed the stars in our field were too faint and/or crowded to obtain the number of PSF stars necessary to use the photometric analysis elaborated by Pugliese et al (2002) taking into account the AO anisoplanatism effect.
The detected stars are identified by a number referring to Column 1 of Table 3 that gives the astrometry and photometry. The central object N88A is not affected by anisoplanatism, so the JHK instrumental magnitudes of the stars were derived using the DAOPHOT’s multiple-simultaneous-profile-fitting photometry routine (NSTAR), well adapted for photometry in crowded fields. The detected stars are shown in the inset of Fig. \[fig:S54inset\] and are also listed in Table 3. Almost all the stars of Field 1 (Fig. \[fig:S54inset\]) have photometric uncertainties in the J-, H- and K-band, less than 0.03 mag for stars with K $<$ 16 mag, less than 0.06 for stars with 16 $<$ K $<$ 18 mag and greater than 0.1 for stars 18 $<$ K $<$ 20 mag.
Fig. \[fig:S27inset\] (Field 2) shows only a part of N88 observed with the S27 camera through the Ks-band filter. On this figure the directions of the spectra S1, S2 and S3 are plotted. This camera, with a pixel two times smaller than S54, has a better spatial resolution, so the analysis of the crowded field of N88A with the routine NSTAR allows a more accurate star detection. In Fig. \[fig:S27zoom\]a the stars of N88A are identified by a number referring to Column 1 of Table 4. In this table, the average photometric errors of the stars reported by DAOPHOT are $\sim$0.04 mag in J and 0.06 in H and Ks for stars of magnitude $\leq$ 16 except for the bright isolated star $\#$57 outside N88A ($\sim$0.03 and $\sim$0.05 mag). This star will be used as PSF.
The photometric calibration was obtained using the isolated 2MASS star at J2000 ($\alpha$, $\delta$) = (1$^{h}$,24$^{m}$,8.88$^{s}$, -73$^{o}$,8’,56.2") corresponding to our star $\#$57. The conversion of pixel coordinates to $\alpha$ and $\delta$ was derived using the same star and the relative positions of our stars are accurate to better than 0.1$\arcsec$.
In the core of N88A the determination of the sky aperture parameters used in NSTAR is very sensitive, even with the S27 camera. Indeed, the wings of the stars superpose the wings of the strong continuum. The distribution of this continuum resembles a gaussian profile (Fig. \[fig:PSF\]). The error on the magnitude of these stars, due to a steeply sloping continuum background, is greater than the error given by DAOPHOT. Because of this situation, the K magnitude of the central star labelled $\#$41 at low and at higher spatial resolution (Field 1 and Field 2) is respectively 13.82 and 14.99 mag. Therefore, the K magnitudes of the stars $\#$37, $\#$41, and $\#$47 (Field 2) were remeasured by subtracting a one-dimensional (1-D) profile corresponding to the PSF crossing the center of the isolated reference star $\#$ 57. The magnitude of the PSF was multiplied by a factor in such a way that only the continuum remains visible. In this case its magnitude corresponds to the magnitude of the star. An example is given in Fig. \[fig:PSF\]. The K magnitudes of the stars obtained with this method are listed in col. 7 of Table 4. In this table the magnitude of $\#$41 is in agreement with the magnitude obtained with NSTAR (col 5), while for $\#$37, $\#$42 and $\#$47 the Ks magnitude is greater. Several faint stars under the detection level, or slightly extended, are not detected or rejected by DAOPHOT (Fig. \[fig:S27zoom\]b).
In our underexposed L’-band frame of N88, not shown here, contrary to the JHK-band, no star is found except star $\#$57. However, in N88A the bright star $\#$41 is visible as well as the stars $\#$37, $\#$42, $\#$47 and several unresolved features. A peculiar bright core labelled L1-C (see Sect. 3. below) located 0.2$\arcsec$ east of $\#$41 is also found. This core coincides approximatively with the HST absorption lane (HM99) and has a very faint counterpart in the Ks-band (Fig. \[fig:KS\_L\]). In order to derive the L’ magnitude of these objects, we referred to the L’ photometry of Israel $\&$ Koorneef (1991), obtained through a 7$\arcsec$ aperture (Table 5). The integration of N88A in our sky subtracted Ks image, using a 4$\arcsec$ aperture, gives a magnitude of 11.08, in agreement with Israel $\&$ Koorneef (1991) and other authors (Table 5). In an aperture of 4$\arcsec$ we have integrated the L’ flux of N88A that was then calibrated with L’ = 8.92 mag given by Israel $\&$ Koorneef (1991). The L’ magnitudes of the stars $\#$57, $\#$41 and the core L1-C (continuum subtracted) were obtained using an aperture of 0.35$\arcsec$ and are listed in Table 4.
=0.005cm
Results and discussion
======================
The HEB N88A
------------
The HST H$\alpha$ (F656N) and continuum Stromgren y (F547M) images of N88A described in HM99, show two inhomogeneous wings separated by a north-south absorption lane (Fig. \[fig:S27zoom\]c). The western wing is much brighter and contains two faint stars, $\#$1 and $\#$2, corresponding to our stars $\#$41 and $\#$42, as well as a ‘dark spot’ to the south at the location of our star $\#$37. Fig. \[fig:supKsHST\] shows the intensity distribution plots in the Stromgren y- and Ks-band in the direction of the slit S1 (Fig. \[fig:S27inset\]) crossing N88B and the star $\#$41. In Fig. \[fig:supKsHST\] the stars $\#$41 and $\#$42 of magnitude 14.99 and 16.11 respectively (Table 4) coincide with the two faint stars $\#1$ and $\#$2 (Fig. \[fig:S27zoom\]c) of y = 18.2 mag and 18.3 mag (HM99).
Through the Ks filter (Figs. \[fig:S27zoom\]a and b), N88A appears as a circular nebular region of $\sim$ 3.4$\arcsec$ diameter centered on the relatively bright star $\#$41. This star coincides with the 2MASS point source 012407.92-730904.1 of K = 11.18 mag (Cutri et al. 2003). In a diameter of $\sim$ 3.6$\arcsec$ our N88A image exhibits a small embedded cluster labelled N88A-cl of at least thirteen stars (Fig. \[fig:S27zoom\]b). In Figure \[fig:S27zoom\]b the usual Digital Development Process (DDP) introduced by Okano was applied to enhance the faint stars by compressing the range of brightnesses between the bright and dim portions of the image. The K photometry of these stars is listed in Table 4. These stars mainly concentred to the north and east, superpose numerous nebular structures. Interestingly these stars are aligned in the direction of the interface between the HII regions N88A and N88B (HM99).Through the L’-band, N88A seems to be essentially formed by four distinct components labelled L1, L2, L3 and L4 (Fig. \[fig:S27zoom\]d). L1, that contains the core labelled L1-C (Fig. \[fig:supKsHST\]), is the brightest and most compact component. In the Ks-band L1-C (Figs. \[fig:S27zoom\]a and \[fig:KS\_L\]) shows a very faint counterpart (L-Ks $\geq$ 4.2). L2 and L3 are more diffuse and coincide with the stars $\#$47 and $\#$37. L4 is bright, extended and formed by two east-west elongated subcomponents spanning between stars $\#$47 and $\#$41. In the L’-band the star $\#$41 is relatively bright (L’ $\sim$ 14.1 mag) and well resolved (Fig. \[fig:S27zoom\]d). All these components superpose a diffuse nebular continuum. On the y (F547M) continuum image (Fig. \[fig:S27zoom\]c) the ‘dark spot’ corresponding to our component L3 is very bright. The y continuum structures located between stars $\#$41 and $\#$37, as well as north to the ‘dark spot’ (Fig. \[fig:S27zoom\]c), are not seen in the L’-band (Fig. \[fig:S27zoom\]d).
The N88B region
---------------
At the optical wavelengths, HM99 found that the central star of N88B, corresponding to our star $\#$31, has an integrated magnitude of y = 16.57 and consists of at least three components. Our high spatial resolution Ks-band image also shows that star $\#$31 of magnitude K = 15.74 and J-K = 0.23 is complex and formed of at least three components visible in the inset of Fig. \[fig:S27inset\]. Two of them oriented approximatively south-north are relatively bright, whereas the one to the north-east corresponds to a faint diffuse feature. To the east of N88B lies a red bow-shaped filament with a curvature radius of $\sim$ 3$\arcsec$ (Fig. \[fig:S54\]) centered on N88B (see Sect. 3.6). This filament coincides with the narrow filament north-east of component B detected in the H$\alpha$-band (HM99).
JHK CM and CC diagrams
----------------------
Fig. \[fig:HR\] shows a Ks vs. J-K CM diagram of the N88 region. The magnitudes of the stars (Field 1) belonging to N88A are systematically underestimated (see Sect. 2.3). This difference is visible on the diagram where the stars of N88A (Field 2) analysed using the PSF of star $\#$57 (Table 4) are overplotted. However, taking into account the PSF given by DAOPHOT for JHK of about 0.2-0.3$\arcsec$ we assume that the J-K colour values as well as J-H and H-K are correct within the uncertainties. The color excess E(B-V) towards N88 derived for hot stars from the Magellanic Clouds Photometric Survey (Zaritsky et al. 2002) in a radius of 1 $\arcmin$, is small $\sim$ 0.15. On Fig. \[fig:HR\] the reddening track for O stars is plotted, assuming a total visual extinction Av = 5.8E(J-K) (Tapia et al. 2003) and Ak = 0.112Av (Rieke $\&$ Lebosky 1985). It was derived using as reference the star $\#$8 of type O9.5 in Wilcots 1994b. For this star we adopted a (J-K)$_{0}$ of -0.15 mag (Lejeune $\&$ Shaerer 2001).
Several isochrones with different ages corresponding to Z = 0.004 are overplotted (Fig. \[fig:HR\]). The diagram appears to reveal two populations. The first one is a young population of dwarf and massive O stars which appears to be fitted with the 3 Myr isochrone. The second one could be a clump of red giant stars of K magnitude in the range of 17-19.5 expanding in the age 300 Myr-10 Gyr. The stars lying beyond the 10 Gyr isochrones are likely to represent embedded stars situated deeper in the molecular cloud, young stars with circumstellar material or evolved stars surrounded by dust.
Fig. \[fig:HKJH\] shows the H-K vs. J-H CC diagram. In this figure the solid line represents the reddening vector up to Av = 5 mag. All stars that lie on the right side of the reddening vector should have IR-excess. Due to uncertainties on H-K colour we take into account only the stars beyond $\sim$ 0.1 mag to the right of the reddening vector. Hence, the number of stars with IR-excess extracted from the CC diagram corresponds to at least 30$\%$ of the measured stars. On the CC diagram, obtained after integration in an aperture of 4$\arcsec$ diameter (Table 5), N88A is overplotted and is found to the extreme right (Fig. \[fig:HKJH\]). The plot shows a red J-H colour of 0.33 mag which contrasts with the blue J-H colour given by Israel $\&$ Koorneef (1991).
=0.08cm
[lcccccccl]{} Reference & J & H & K & L’ & J-K & Ks-L’ & FWHM &Av\
& & & & & & & ($\arcsec$) &\
Denis$^{d}$ & 12.19 & &11.11 & &1.08 & & 4 & 7.3\
2MASS$^{m}$ & 12.31 &11.98 &11.18 & &1.13 & & 4 & 7.3\
Israel$^{i}$ & & &11.05 &8.92 & &2 & 10 &\
this paper & 12.15 &11.92 &11.08 & & & & 4 & 7.1\
& & & & & & & &\
$^{d}$ Cioni et al. (2000), $^{m}$ Cutri et al. (2003), $^{i}$ Israel $\&$ Koorneef (1991). From the Infrared Array Camera (IRAC) archive, Charmandaris et al. (2008) using an aperture of $\sim$7$\arcsec$, give for the wavelength 3.6$\mu$m (L-band), 4.5$\mu$m, 5.8$\mu$m and 8$\mu$m a magnitude for N88A of 9.52, 8.29, 7.1 and 5.58 mag respectively.
Search for YSO candidates in N88A and N88
-----------------------------------------
Due to their IR-excess emission, YSOs are positioned in the redder parts of the CM and CC diagrams. We first examined low spatial resolution ($\sim$2.5$\arcsec$) mid-IR Spitzer data of N88A (Charmandaris et al. 2008), and then the near-IR stellar content of both the regions N88A and N88 obtained with the high spatial resolution allowed by NACO ($\sim$0.10 - 0.3$\arcsec$). On the CM plot \[3.6\]-\[8\] versus \[8\], presented by Charmandaris et al. (2008), N88A lies at the border of the box representing the domain of Class II YSOs. N88A appears also inside the H II region domain. Similarly, on the CC \[5.8\]-\[8\] versus \[3.6\]-\[8\] diagram N88A is located near the H II region domain, but outside Class I and Class II YSO areas. These observations are explained by the fact that N88A is above all a very bright H II region with strong nebular emission lines and affected by heavy extinction from local dust. In fact the Spitzer data represent flux integrations over the whole H II region ($\sim$1 pc$^{2}$). Therefore, detecting a YSO inside the H II region seems hazardous unless the YSO is the dominant source inside N88A, which obviously is not the case. Note that although on the \[5.8\]-\[8\] versus \[3.6\]-\[8\] colour diagram, based on model calculations (Whitney et al. 2004), N88A appears among Class 0 and Class I data points. This diagram is not applicable to the case of N88A for the reasons explained earlier.
In order to probe the presence of YSOs inside N88A we used our high resolution JHK data. The stars $\#$37, $\#$41, $\#$42 and $\#$47 in N88A-cl are located at the upper part of the CM diagram (Fig. \[fig:HR\]). In the JHK CC diagram (Fig. \[fig:HKJH\]) these stars exhibit an H-K color ranging from 0.48 to 0.81 mag, and can be YSO candidates according to the JHK CC diagram of Maercker & Burton (2005). However, their positions on the J-K versus K diagram (Fig. \[fig:HR\]) suggest heavily reddened main-sequence (MS) massive stars of masses between 15$M_{\sun}$ and 30 $M_{\sun}$. Their positions between the 300 Myr and 1 Gyr isochrones are also compatible with supergiants. If confirmed as supergiants, these stars would not be physically associated with N88A, which is very young. The assumption of reddened MS massive stars or massive YSOs seem more plausible. It is very difficult to distinguish between these possibilities. Consequently, caution must be applied, using only JHK band observations to infer circumstellar material fraction in strong nebulous environments. The high spatial, but low spectral resolution S1, S2 and S3 spectra crossing the stars $\#$37, $\#$41, $\#$42 and $\#$47 (see Fig. 10) do not allow to analyze accurately the Br$\gamma$〓 line emission profile, which is characteristic for YSO sources, and new spectroscopic observations are needed to clarify the nature of these stars. In N88A, L1 presents a relatively bright peak (L1-C) in the L’-band (Fig. 6). With K-L’ $\ge$4.5, L1-C can be interpreted as a deeply embedded protostar (Lada et al. 2000). L1-C with a FWHM slightly larger than the PSF should still be in its contraction phase, surrounded by a dust shell. With a magnitude of 14 and a strong IR-excess (Table 4) L1-C could be a massive protostar of Class I.
On what concerns the region outside N88A, from our high spatial resolution data the JHK CC diagram (Fig. \[fig:HKJH\]), taking into account the uncertainties excluding some stars close to the reddening vector, we show hereabove that in the extended N88 region, at least 30% of the faint detected stars have an IR-excess. These reddened stars seem to belong to a cluster of faint stars labeled N88cl (Fig. 1) coinciding with the young cluster HW81 (Hodge & Wright 1977) formed of bright stars, situated towards N88 and not affected by dust (HM99). The JHK CC diagram (Fig. \[fig:HKJH\]) shows that the bright stars $\#$6, $\#$8, $\#$11, $\#$12, $\#$15, $\#$28 and $\#$59 in HW81 have no IR-excess and their masses spread in the range of 15M$\sun$ to 30M$\sun$. In N88-cl most of the reddened stars have a mass $\le$ 12M$\sun$ (Fig. \[fig:HR\]) and are probably intermediate-mass YSO candidates. We assume that N88-cl belongs to N88 (Fig. 1). The stars $\#$21, $\#$33 and $\#$46 of N88-cl aligned on the east filament of N88B (Fig. 1) with an H-K excess $>$ 0.7 mag, could be good YSO candidates (Fig. \[fig:HKJH\]). Their alignment suggests that their formation may be triggered by the expansion of the shell around N88B.
![ [**a)**]{} Relative intensity distributions along slit S1 aligned on stars $\#$31, $\#$41 and $\#$42 (Fig. \[fig:S54inset\]). [**b)**]{} Relative intensity distributions along slit S2 crossing the star $\#$41. [**c)**]{} Relative intensity distributions along slit S3 aligned on stars $\#$37 and $\#$47. The red line represents the intensity distribution of the He I 2.058 $\mu$m emission line. The dashed red line represents the He I 2.113 $\mu$m and the green one the H$_{2}$ 2.121 $\mu$m, both are multiplied by a factor 5. The continuum/Br$\gamma$ ratio multiplied by a factor 100 is plotted (dashed line). The position of each 1-D spectrum is indicated by a small horizontal segment (solid line). The plot range corresponds to 10.6 $\arcsec$ (1pix = 0.05273$\arcsec$). []{data-label="fig:S1S2S3"}](testor_Fig12.pdf){width="13cm" height="15cm"}
The ionizing sources
--------------------
### N88A
At 3cm radio emission, Indebetouw et al. (2004) found for N88A a Lyman continuum flux of log N$_{Lyc}$ = 49.5. Using the spectral classification of Smith et al. (2002) we estimate, from this flux, the type of the ionizing source of N88A to range from O4 to O5. The type of the ionizing source derived by HM99 using the H$\beta$ flux corresponds to an O6 star. The extracted 1-D spectra at different positions along the slits S1, S2 and S3 crossing N88A (Fig. \[fig:S27inset\]) are listed in Table 6. In each position the rows are averaged and the corresponding 1-D spectra are shown in Fig. \[fig:spectresstar\] for the stars and Fig. \[fig:spectresH2\] for nebular emission. The positions of these spectra are represented by horizontal line segments on the plots corresponding to the distribution of the Br$\gamma$, He I 2.058, 2.113 and H$_{2}$ 2.121 $\mu$m emission lines as well as the continuum emission (Figures \[fig:S1S2S3\]a, b and c). The length of the segment is proportional to the number of lines integrated along the slit. All the emission lines are continuum subtracted. This figure also indicates the H$_{2}$ components C1, C2 and C3 (see Sect. 3.5) of which the distribution intends to clarify the presence of the structures seen in Figure 13 (see Sects. 3.6 and 3.7). From the spectra presented in Figs. \[fig:spectresstar\] and Fig. \[fig:spectresH2\] we derive a ratio He I 2.113$\mu$/Br$\gamma$ lines of mean value 0.06 indicating a hot O star of T$_{eff}$ $\geq$ 40000 K (Hanson et al. 2002). Table 6 (col. 8) shows that this ratio is fairly constant across N88A.
In the spectra of stars $\#$37, $\#$41, $\#$42 and $\#$47 (Fig. \[fig:spectresstar\]) the He II 2.185$\mu$m absorption is not detected, if present. The detection is difficult because of our low signal/noise and our low spectral resolution. The NIII 2.115$\mu$m is not detected either. When the He II 2.185$\mu$m absorption line is not present (Bik et al. 2005), the spectral type of a star should be later than O8 V, which is the case for our four resolved stars. As seen above, the spectral type of the ionizing source of the whole nebula derived from radio and H$\beta$ flux ranges between O4 and O6. We will adopt a type O5 for our computation. Its comparison with the type O8 V derived from our spectroscopy for the bright central star $\#$2-41 clearly shows that other massive stars must contribute to the ionization of N88A. The flux excess between the ionizing star $\#$41 of type O8 V and the O5 type derived from the flux could be produced by at least five O8 V stars. The massive stars $\#$37, $\#$42, and $\#$47 (Fig. \[fig:S27zoom\]) located in the upper part of the CM diagram could be good candidates for the ionization of N88A. The 3cm radio peak centered at ($\alpha$, $\delta$) = (1$^h$24$^m$7$^s$.9, -73$^o$9$\arcmin$4$\arcsec$) and our images show that the radio peak coincides perfectly with the central bright star $\#$41 (1$^h$24$^m$7$^s$.95, -73$^o$9$\arcmin$3$\arcsec$8) (Table 4). This strong radio emission superposing the NIR emission Br$\gamma$ line 2.16$\mu$m line (Figs. \[fig:S1S2S3\]a and b) is characteristic for an H II region.
### The N88A-cl cluster
On the JHK image of Lada et al. (2000) the Trapezium region of size of $\sim$0.75 x 0.75 pc located at a distance of 450 pc, contains four bright central massive stars and a plethora of low-mass stars with IR-excess. In our Ks-band N88A has approximatively a similar diameter (Fig. \[fig:S27zoom\]b) and contains also the cluster N88A-cl. This cluster contains other resolved stars not identified by DAOPHOT as well as unresolved stars in crowded groups (Fig. \[fig:S27zoom\]b). Among these stars the four brightest ones analyzed using the JHK bands also exhibit IR-excess. N88A with its cluster appears morphologically comparable with the Trapezium region, and other compact star formation regions of similar size, like SH2 269 in our galaxy, and N159-5 in the LMC (Testor et al. 2007). N88A-cl can also be compared with the pre-main-sequence (PMS) clusters, candidate YSOs, of size 0.24 pc to 2.4 pc found in SMC-N66 (Gouliermis et al. 2008). Like N88A-cl these PMS clusters are found coinciding with \[OIII\], H$\alpha$ and H$_{2}$ emission peaks (see Sect. 3.7). Their clustering properties are similar to the star forming region Orion despite its higher metallicity (Hennekemper et al. 2008).
{width="15cm"}
### The bright central star $\#$41
In the Trapezium, the four ionizing bright stars lie within a diameter of $\sim$ 0.05 pc. At the distance of the SMC our spatial resolution is $\sim$ 0.03 pc ($\sim$ 6900 AU). With this relative lower spatial resolution it is not excluded that the ionizing star $\#$41 could also be a tight young cluster. This assumption is strengthened by the photometry. The magnitude of this star obtained using the PSF is 15.05 (Table 4). Its dereddened magnitude derived with A$_{K}$ = 0.58 mag corresponds to a mass of $\sim$ 40 $M_{\sun}$ (Fig. \[fig:HR\]). Using the parameters for O stars of Vacca (1996), we classify $\#$41 as a O6.5 V type star instead of O8 V derived by spectrocopy, and it could also be multiple.
=0.06cm
[l l l l l l l c ]{} Id & HeI & HeI & H$_{2}$ 1-0 S(1) &Br$\gamma$ &H$_{2}$ 1-0 S(0) &H$_{2}$ 2-1 S(1) & I(He 2.113$\mu$m)/I(Br$\gamma$)\
&2.058$\mu$m&2.113$\mu$m &2.121$\mu$m &2.166$\mu$m &2.223$\mu$m &2.247$\mu$m &\
S1-41 & 3.18 & 0.375 & 0.121 & 6.07 & 0.099 & 0.031 &0.062\
S1-42 & 3.36 & 0.337 & 0.075 & 5.94 & 0.092 & 0.044 &0.057\
S2-41 & 2.38 & 0.257 & 0.094 & 4.76 & 0.068 & 0.038 &0.054\
S3-37 & 0.76 & 0.065 & 0.028 & 1.21 & 0.018 & 0.009 &0.054\
S3-47 & 0.60 & 0.055 & 0.066 & 1.01 & 0.033 & 0.006 &0.054\
S1-1 & 2.43 & 0.24 & 0.199 & 4.30 & 0.126 & 0.069 &0.054\
S1-2 & 0.17 & 0.017 & 0.094 & 0.31 & 0.054 & 0.035 &0.057\
S2-1 & 1.49 & 0.16 & 0.150 & 3.24 & 0.093 & 0.048 &0.049\
S3-1 & 0.67 & 0.065 & 0.049 & 1.11 & 0.033 & 0.009 &0.059\
S3-2 & 0.58 & 0.057 & 0.026 & 0.95 & 0.018 & 0.009 &0.060\
S3-3 & 0.02 & 0.002 & 0.017 & 0.04 & 0.010 & 0.008 &0.056\
S3-4 & 0.02 & & 0.019 & 0.02 & 0.009 & 0.009 &\
S3-p1 & 0.74 & 0.092 & 0.020 & 1.42 & 0.014 & 0.009 &0.065\
S3-p2 & 0.89 & 0.090 & 0.036 & 1.55 & 0.034 & 0.008 &0.058\
The continuum dust emission
---------------------------
Through the JHK-band, N88A presents a relatively bright nebular continuum emission centered on star $\#$41 (Figs. \[fig:S27zoom\]a and \[fig:PSF\]). In the L’-band the continuum is less homogeneous, due to the brightness of L1, L2, L3 and L4. Although the signal to noise ratio is not high, L4 appears very faint in the y-band image (Fig. \[fig:S27zoom\]c). The nature of this continuum is not clear. The relatively strong L-band excess of K-L’ = 2 mag derived from the integration of the whole region N88A (Table 4) supposes that the continuum could come from the emission of circumstellar material (Lada et al. 2000) around resolved and unresolved young stars. These stars could be located mainly at the positions L1, L2, L3 and L4 (Fig. \[fig:S27zoom\]d). L4 shows two peaks P1 and P2 visible on the continuum plot of S3 (Fig. \[fig:S1S2S3\]c). However, the continuum emission could also be formed by interstellar dust associated with the gas of the CO cloud (Testor et al. 1985, HM99, Stanamirovic et al. 2000, Israel et al. 2003). In Fig. \[fig:S27zoom\]c the strong optical emission at the position of the ‘dark spot’ could be explained by strong dust scattering reflecting the light of at least star $\#$37. The nature of the continuum emission of N88A should be a combination of the two possibilities: emission of circumstellar material and/or dust associated with the gas. Along the slits S1, S2 and S3 the intensity distribution of the continuum near the Br$\gamma$ line shows a strong continuum/Br$\gamma$ ratio of 0.30-0.4 over a range of 6$\arcsec$ around star $\#$41 (Figs. \[fig:S1S2S3\]ab). In these figures the broadness of the continuum and Br$\gamma$ distribution are similar (FWHM $\sim$ 1.8$\arcsec$).
H$_2$ emission
--------------
In Testor et al. (2005) the profile of the H$_2$ emission along the slit corresponding to our spectrum S1 appears in the form of two blended profiles, due to the low spatial resolution spectroscopy. Thanks to the high spatial resolution of our new data, the complex morphology of the H$_2$ emission in N88A is revealed both by imagery and spectroscopy. Fig. \[fig:dif\] shows a bidimensional image of the H$_2$ emission (v=1-0 S(1) line). This image is achieved by subtracting the image in the 2.24 $\mu$m filter, which allows the passage only of the continuum radiation from the image in the 2.13 $\mu$m filter. In this H$_2$ image, N88A resembles a circular shell (a) of diameter $\sim$ 3$\arcsec$ with three maxima labelled C1, C2 and C3 (Fig. \[fig:dif\]). Within (a) there is a cavity suggesting radiation pressure, especially from the four central stars. The structure C2 is very bright and extended along the direction northeast-southwest and coincides with the ionization front detected by HM99. C2 has a sharp extension in the direction of star $\#$47. The H$_2$-band image also shows that the bow-shaped filament located east of N88A seems to belong to a second shell (b) of diameter 7$\arcsec$ centered on N88B (Fig. \[fig:dif\]). The shells (a) and (b) seem to be in interaction approximatively at the position C1.
Unlike the spectra obtained with ISAAC (Testor et al. 2005) the high-spatial resolution long-slit spectra S1, S2 and S3 allow to resolve the inner structures and stars of N88A. In the direction of the slit S1, the two H$_2$ emission structures C1 and C2 are well resolved (Fig. \[fig:S1S2S3\]a) and separated by $\sim$ 2$\arcsec$. C1 distant of $\sim$ 0.5 $\arcsec$ from $\#$41 coincides with the absorption lane observed in optical images by Kurt et al. (1999) and HM99. In the direction of the slit S2, only the structure C1 is seen east of star $\#$41 (Fig. \[fig:S1S2S3\]b). In the direction of the slit S3, the well seen structures C2 and C3 (Fig. \[fig:S1S2S3\]c) coincide with the stars $\#$47 and $\#$37 respectively. According to Rubio et al. (2000) massive star formation could be taking place in dense H$_{2}$ knots associated with molecular clumps. According to Gouliermis et al. (2008) PMS clusters could be candidate YSOs. These results strengthen our assumption that N88A-cl could contain YSOs.
From Israel $\&$ Koorneef (1988), the molecular hydrogen emission may be caused either by shock excitation due to embedded stars, or by fluorescence of molecular material in the ultraviolet radiation field of the OB stars exciting the H II region in the molecular cloud. They conclude that in the MCs, shock excitation of H$_{2}$ is only expected very close to (i.e. 0.15pc) stars embedded in a molecular cloud. At a larger distance, radiative excitation of H$_{2}$ by the UV radiation field of the OB stars is the only mechanism. Their spectrophotometry with a large aperture (10$\arcsec$) made difficult a precise determination of the brightness of the lines 2-1(S1), 1-0(S1) and 1-0(S0) usually considered between shocks and radiative excitation. These lines deblended when necessary have been derived from our low-spectral-resolution spectra S1, S2 and S3 crossing the zones C1, C2 and C3 as well as the stars $\#$41, $\#$42, $\#$37 and $\#$47 and their intensities are shown in Table 6. None of these lines suffers from atmospheric absorption, considering a V$_{lsr}$ of 147 km s$^{-1}$ (Israel et al. 2003), as it can be derived from the solar spectrum atlas (Livingston & Wallace 1991) with the help of a useful home-made software$^{1}$. The lines may suffer from differential reddening. Mathis (1990) estimates that the effect for Galactic Sources follows a power-law in the J-, H-, and K-bands: $I_{1}/I_{2} = (\lambda_{1}/\lambda_{2})^{-1.7}$. The effect on the v=1-0 S(0) and v=2-1 S(1) lines would be that they are overestimated by 10%. However, this is based entirely on Galactic Sources. To the best of our knowledge, a differential reddening law has not been determined for Extra-Galactic Sources. Moreover, since the effect is already within our observational uncertainty, we choose to ignore it. In Figure \[fig:tableline\] the strengths of the 2.247$\mu$m 2-1(S1) and 2.223$\mu$m 1-0(S1) lines are shown relative to the 2.121$\mu$m 1-0(S1) line for all the objects in Table 6. For radiative excitation (PDRs), the usual criteria are that these ratios should range from 0.5 to 0.6 and from 0.4 to 0.7 respectively (Black $\&$ van Dishoeck 1987), while for shock-excitation with T=2000K they should be 0.08 and 0.21 respectively (Shull $\&$ Hollenbach 1978). We show in Figure \[fig:tableline\] the results from Draine & Bertoldi (1996) reported by Hanson et al. (2002) for high density (n$_H$=10$^6$ and $\chi$=10$^4$ and 10$^5$) and low density (n$_H$=10$^4$ and $\chi$=10$^2$) PDRs. We also show in this figure results for more recent and elaborated PDR models (Le Petit et al. 2006) \[for n$_H$ ranging from 10$^4$ to 10$^7$ and $\chi$ from 10$^3$ to 10$^7$\] as well as for shock models (Flower $\&$ Pineau des Forêts 2003) \[for the same range of n$_H$, v$_S$ from 10 to 60 km s$^{-1}$, the magnetic scaling factor b from 0 to 10 and an ortho/para ratio of 3\]. None of these models, either PDRs or shocks really fit with our observations, with the exception of the objects S3-p1 and p2 as well as S3-1, well inside the nebulosity, which could fit with shocks (relatively high velocity v$_S$, 30 to 50 km s$^{-1}$) and low b $\sim$ 0.1. Nevertheless, apart from the three H$_{2}$ lines mentioned above, four additional ones are observed: 2-1 S(2), 2-1 S(3), 3-2 S(1) and especially 3-2 S(2) at respectively 2.154, 2.073, 2.386 and 2.286$\mu$m. These lines are more sensitive to absorption by atmospheric lines, depending in fact on the accuracy of the v$_{lsr}$. The first one is the less affected by positive or negative velocity shift, whilst the two following are slightly absorbed up to 180 km s$^{-1}$ but may suffer a 50 $\%$ absorption at 140 km s$^{-1}$. The last one is free of absorption between 115 and 150 km s$^{-1}$ and does not appear to be blended with any other lines. Possible turbulence in the emitting region may broaden the lines, then lowering the effect due to atmospheric absorption. In any case the concomitant appearance of lines emanating from high v or J as the 3-2 S(2) H$_2$ line shows a clear trend in favor of the major presence of PDR excitation for most of the observed objects, without nevertheless totally excluding the additional presence of shock excitation. Clearly higher, both spatial and spectral resolutions are required to progress in the knowledge of these objects.
{width="14cm" height="10cm"}
Conclusions
===========
We present high spatial resolution imaging of FWHM $\sim$ 0.12$\arcsec$ - 0.25$\arcsec$ in the JHKL’-band of the HEB N88A and its immediate environment, and the main results are as follows:
N88A is associated with a cluster that contains at least thirteen stars centered approximatively on the bright central star $\#$41, that could be multiple.
N88A coincides perfectly with the 3cm radio peak and should be ionized not only by the star $\#$41 classified of type O8 V, but also by other low to high-mass stars.
From analysis of the JHK CC diagram we found four possible MYSO star candidates in the N88A cluster, as well as three probable YSOs in the red bow east of N88A. In N88 at least 30$\%$ of the detected stars have an IR-excess.
From the K-L excess we found that the core L1-C in N88A should be a heavily embedded, high mass protostar of Class I.
The continuum emission at the position of $\#$41 is very bright and represents about 30$\%$ of the Br$\gamma$ emission peak.
The H$_2$ infrared emission in N88A resembles a shell formed mainly by three peaks of which one coincides with the ionization front.
We show that the excitation mechanism may be caused predominantly by PDRs, without excluding combination with shocks.
The morphology of N88A could be comparable with galactic regions such as the nearby Trapezium region in the Orion nebula.
Future JHK band imaging data, using higher spatial resolution and longer wavelengths, as L’ and M’ provided by the NACO S13 camera are still needed to disentangle the IR-excess origin in N88A. Higher spectral resolution spectra are also required to obtain a better analysis of the different spectral lines like the Br$\gamma$ emission. These new observations should allow to investigate more thoroughly the HEB N88A. This object, which is the brightest, the most excited and reddened of the MCs, presents a unique opportunity to progress in the knowledge of newborn massive stars in regions of low metallicity.
We would like to thank the Directors and Staff of the ESO-VLT for making possible these observations and particularly the NACO team for their excellent support. JLL, LK and SD would like to acknowledge the support of the French PCMI program “Physico Chimie du Milieu Interstellaire”, funded by the CNRS. This research has made use of the Simbad database, VizieR and Aladin operated at CDS, Strasbourg, France, and the NASA’s Astrophysics Data System Abstract Service.
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[^1]: visiting astronomer at VLT Paranal
[^2]: Based on observations obtained at the European Southern Observatory, El Paranal, Chile
|
---
abstract: 'Gamma-ray burst (GRB) 050904 is the most distant X-ray source known, at $z=6.295$, comparable to the farthest AGN and galaxies. Its X-ray flux decays, but not as a power-law; it is dominated by large variability from a few minutes to at least half a day. The spectra soften from a power-law with photon index $\Gamma=1.2$ to 1.9, and are well-fit by an absorbed power-law with possible evidence of large intrinsic absorption. There is no evidence for discrete features, in spite of the high signal-to-noise ratio. In the days after the burst, GRB050904 was by far the brightest known X-ray source at $z>4$. In the first minutes after the burst, the flux was $>10^{-9}$ergcm$^{-2}$s$^{-1}$ in the 0.2–10keV band, corresponding to an apparent luminosity $>10^5$ times larger than the brightest AGN at these distances. More photons were acquired in a few minutes with *Swift*-XRT than XMM-*Newton* and *Chandra* obtained in $\sim300$ks of pointed observations of $z>5$ AGN. This observation is a clear demonstration of concept for efficient X-ray studies of the high-$z$ IGM with large area, high-resolution X-ray detectors, and shows that early-phase GRBs are the only backlighting bright enough for X-ray absorption studies of the IGM at high redshift.'
author:
- 'D. Watson, J. N. Reeves, J. Hjorth, J. P. U. Fynbo, P. Jakobsson, K. Pedersen, J. Sollerman, J. M. Castro Cerón, S. McBreen, and S. Foley'
title: |
Outshining the quasars at reionisation:\
The X-ray spectrum and lightcurve of the redshift 6.29 $\gamma$-Ray Burst GRB050904
---
Introduction\[introduction\]
============================
The promise of $\gamma$-ray bursts (GRBs) as cosmic lighthouses to rival quasars is being fulfilled in the areas of GRB-DLAs , as tracers of star-formation , and as early warning of SNe . Central to this promise is the belief that GRBs from early in the universe can be detected [$z\sim10$, e.g. @2003ApJ...591L..91M]. But while the highest redshifts of AGN and galaxies increased, for 5 years the highest GRB redshift was $z=4.50$ . Now, a GRB at $z>6$ has finally been detected: GRB050904 at $z=6.295\pm0.002$ (@2005astro.ph.12052K [[email protected]], see also @2005astro.ph..9660H [[email protected]], @2005astro.ph..9766T [[email protected]], and @2005astro.ph..9697P [[email protected]]). To date, X-ray observations of $z>5$ AGN with *Chandra* and XMM-*Newton* have obtained bare detections [@2002ApJ...571L..71S; @2002ApJ...569L...5B; @2002ApJ...570L...5M; @2003AJ....125.2876V; @2003ApJ...588..119B], and from the most luminous, spectra with a few hundred counts using long exposures [@2004ApJ...611L..13F; @2005astro.ph..1521G; @2005ApJ...630..729S], allowing contraints to be placed on AGN evolution up to the edge of reionisation. In this *Letter* we examine the X-ray spectra and lightcurve of GRB050904 from *Swift*-XRT. Uncertainties quoted are at the 90% confidence level unless otherwise stated. A cosmology where $H_0=70$kms$^{-1}$Mpc$^{-1}$, $\Omega_\Lambda = 0.7$ and $\Omega_{\rm m}=0.3$ is assumed throughout.
Observations and data reduction\[observations\]
===============================================
GRB050904 triggered *Swift*-BAT at 01:51:44UT. The BAT and XRT data were obtained from the archive and reduced in a standard way using the most recent calibration files. The BAT spectrum is well-fit with a single power-law with photon index $\Gamma = 1.26\pm0.04$ and 15–150keV fluence = $5.1\pm0.2\times10^{-6}$ergcm$^{-2}$, consistent with early results [@2005GCN..3910....1C; @2005GCN..3918....1P] that also suggested a duration $T_{90} = 225\pm10$s. An upper limit to the peak energy of the burst, $E_{\rm peak} > 130$keV was found by fitting a cut-off power-law model to the spectrum and deriving the 3$\sigma$ limit on the cut-off energy. The *Swift*-XRT rapidly localised a bright source [@2005GCN..3920....1M] and began observations in windowed timing (WT) mode at $\sim170$s after the trigger and photon counting (PC) mode at $\sim580$s.
Results\[results\]
==================
The XRT lightcurve (Fig. \[fig:lightcurve\])
fades by $>1000$ over the first day. But the lightcurve does not decay as a power-law as in many afterglows [@2005astro.ph..8332N; @2005astro.ph..7708D; @2005astro.ph..7710G]. Instead, the afterglow flares at $446\pm3$s, doubling the flux. This flaring is similar to that observed in other GRBs at early times [@2005Sci...309.1833B], but the lightcurve does not settle into a power-law decay, continuing to be dominated by large variability (up to a factor of ten). The WT lightcurve is poorly fit by a power-law plus a single Gaussian emission peak ($\chi^2$/dof = 195.7/78). Allowing a second peak improves the fit (but is still poor, $\chi^2$/dof = 125.7/75), giving central times of $468\pm3$ and $431^{+5}_{-7}$s. Dividing the data into hard (2–10keV) and soft bands (0.5–2.0keV) it is clear that the later peak is harder, and the earlier peak softer (Fig. \[fig:hardness\]).
A two-peak fit to the soft band is acceptable ($\chi^2$/dof = 41.6/34) and gives different peak times than the fit to the full band. There is considerable scatter around this model when fit to the hard band data, giving an unacceptable $\chi^2$/dof (93.5/34), which suggests greater variability in the hard band on timescales of $\sim10$s.
The spectra (Fig. \[fig:spectra\])
------ ------------------------ --------------- ------------------------- -- -- --
Mode Time since trigger (s) $\Gamma$ $N_{\rm H}$ at $z=6.29$
($10^{22}$cm$^{-2}$)
WT 174–374 $1.23\pm0.05$ $3.3\pm1.5$
WT 374–594 $1.62\pm0.06$ $3.6\pm1.4$
PC 594–1569 $1.68\pm0.08$ $<1.6$
PC 9080–63480 $1.88\pm0.04$ $2.9\pm0.8$
------ ------------------------ --------------- ------------------------- -- -- --
: Spectral evolution of GRB050904[]{data-label="tab:spectra"}
can be fit by a hard power-law with Galactic absorption . The spectrum softens appreciably during the observation, reaching $\Gamma\sim1.9$ in the 10–50ks after the GRB (Table \[tab:spectra\]). There is no evidence for discrete emission or absorption features. (6.97keV) and (8.10keV) at $z=6.29$ have respective restframe equivalent widths $<43$ and $<44$eV in the WT spectra and $<27$ and $<137$eV in the PC spectra. There is some evidence of absorption above the Galactic level: the best fit gives $N_{\rm H}=8.3\pm0.8\times10^{20}$cm$^{-2}$. This excess ($N_{\rm
H}=3.4\times10^{20}$cm$^{-2}$) is statistically required (significant at a level $>5\sigma$ using the $f$-test). Typical variations in the hydrogen column density at scales $\lesssim1\deg$ at high Galactic latitudes are too small to explain this excess . Without discrete features, the redshift of the absorption is essentially unconstrained. Because of the high redshift of the GRB, to observe even a modest column at $z=0$ requires a high column at $z=6.29$; in this case the best-fit excess column density at $z=6.29$ is $2.8\pm0.8\times10^{22}$cm$^{-2}$. Such a high column could not be considered entirely unexpected—a column density nearly as high as this has been detected before in a GRB [e.g. @2005astro.ph.10368W]. Nonetheless it is intriguing at such an early time in the star-formation history of the universe, especially since the absorption is dominated primarily by oxygen and other $\alpha$-chain elements. However, it should be noted that the combination of the uncertainties in the Galactic column density and the current calibration uncertainty of the XRT response at low energies must render one cautious about the detection of excess absorption in this case.
Discussion\[discussion\]
========================
The BAT-detected emission overlaps the start of XRT observations and has a power-law photon index close to that observed in WT mode ($\Gamma=1.3$). It is likely that we are observing part of the prompt emission with the XRT at these times given the similarity with the BAT spectrum, the rapid decay, the flaring, and a spectrum that softens considerably over the first few hundred seconds in the restframe. This may not be surprising considering the restframe energy band extends to nearly $73$keV. The fact that we are observing higher restframe energies in this GRB does not seem to contribute much to the remarkable variability of the lightcurve, since the soft band (0.5–2.0keV) has similar overall variability (Fig. \[fig:hardness\]). The amplitude of these variations seems to indicate continued energy injection from the central engine at least for the first few hundred seconds. Interestingly, the large variability continues as late as 45ks (Fig. \[fig:lightcurve\]), and the spectrum remains hard ($\Gamma<2.0$), suggesting that significant energy output from the central engine is likely to be continuing at these times, corresponding to $\sim6000$s in the restframe. While continued energy injection at observed times of up to a few hours has been indicated since the launch of *Swift* [@2005Sci...309.1833B; @2005astro.ph..8332N], energy injection from the remnant at times of more than half a day was proposed to explain the late-appearing X-ray line emission in GRB030227 [@2003ApJ...595L..29W; @2000ApJ...545L..73R]. The maximum heights of the later variations in GRB050904 also seem to decay exponentially, indicating that if accretion onto the remnant is responsible for these variations, that the accretion rate is decaying in the same way.
A power spectral density analysis of the lightcurve shows no significant periodicity independent of the period of the data gaps in the range $10^{-3}-10^{-4}$Hz. The large flaring amplitude and lack of a periodic signal is reminiscent of typical prompt phase emission from GRBs. However, the total duration of the flaring ($\gtrsim45$ks) and the individual rise times (a few thousand seconds) are much longer . The overall decay envelope observed here is not typical of prompt emission either, although there are a few cases where such an overall decay is seen (BATSE triggers 678, 2891, 2993, 2994, 7766) and it has been speculated that these continuous decays of the prompt emission result from spin-down of a black hole by magnetic field torques .
Is GRB050904 Different?
-----------------------
Assuming an upper limit to the redshift of GRB formation of $z=20$ [@2004NewA....9..353B], the likely maximum age of the GRB progenitor is $\lesssim650$Myr, consistent with a massive star progenitor [@1998Natur.395..670G; @2003Natur.423..847H; @2003ApJ...591L..17S; @2005astro.ph..8175W]. At this early time in the universe, the question arises whether GRB050904 could have a different progenitor than GRBs at lower redshift; for instance, a star formed in pristine gas may be one of the massive populationIII stars.
Assuming the relation between total energy ($E_\gamma$) and $E_{\rm peak}$ [@2004ApJ...616..331G], the high restframe $E_{\rm peak}$ ($>940$keV) implies a very high $E_\gamma$ [$>2\times10^{51}$erg, consistent with a possible jet-break in the near-infrared, @2005astro.ph..9766T]. This high $E_\gamma$ and the large isotropic equivalent energy suggests that GRB050904 was intrinsically highly energetic. The persistence of the flaring in the X-ray lightcurve, is also different from typical GRB X-ray afterglows after a few hours [@2005astro.ph..7710G; @2005astro.ph..7708D]. Both the high intrinsic energy output and the large amplitude, long duration flaring are notable differences between GRB050904 and typical GRBs, and might hint at an unusual progenitor. On the other hand, the X-ray flux of the afterglow at 10 hours, $\sim10^{-11}$ergcm$^{-2}$s$^{-1}$, implies a k-corrected equivalent isotropic luminosity of $5\times10^{46}$ergs$^{-1}$, well within the typical range [@2003ApJ...590..379B]. Although if the beaming correction is relatively small, as suggested by the high value of $E_{\rm
peak}$, the energy inferred for the X-ray afterglow would also be large.
High-$z$ Warm IGM Studies with GRBs
-----------------------------------
Access to the edge of the reionisation epoch using GRBs has begun with the observation of GRB050904 at $z=6.295$. Optical studies of the intervening matter at early times have used quasars [e.g. @2001AJ....122.2850B; @2001ApJ...560L...5D; @2005ApJ...628..575W], but may be affected by the quasar’s significant influence on its surroundings. GRBs are therefore potent tools in this study at optical wavelengths. With X-rays, the warm intergalactic medium (IGM) can be probed. Such work has also just begun to bear fruit with very bright, nearby sources (e.g. @2005ApJ...629..700N [[email protected]], see also, @2002ApJ...572L.127F [[email protected]] and @2003ApJ...582...82M [[email protected]]). This is because millions of X-ray photons are required to make these absorption line measurements reliably [@2000ApJ...539..532F; @1998ApJ...509...56H]. The blazar Mkn421 ($z=0.03$) has a bright, intrinsically featureless continuum which provides an easily-modelled spectrum against which to detect intervening absorption features. Long exposures ($\sim250$ks) with the gratings on *Chandra* provided $\sim7.5\times10^6$ photons from this source, mostly when the blazar was in extremely bright flaring states. This allowed @2005ApJ...629..700N to detect absorption from ionised C, N, O, and Ne from IGM filaments at $z=0.011$ and $z=0.027$. The spectra of GRBs, in prompt or afterglow emission, are usually dominated by a featureless power-law [although see @2003ApJ...595L..29W; @2003ApJ...597.1010B; @2000Sci...290..955P; @2002Natur.416..512R; @2003ApJ...591L..91M], as observed in this case, which makes them ideal for studies of intervening matter in a way analogous to blazars [@2000ApJ...544L...7F; @2005astro.ph..4594K].
The rapid response of *Swift* to GRB050904 yielded high signal-to-noise ratio X-ray spectra in spite of the relatively modest aperture of the XRT. This contrasts favourably with observations with *Chandra* and XMM-*Newton* of AGN at redshifts $z>5$ that have so far yielded many fewer counts, even in aggregate, in a total exposure time of 300ks [e.g. @2001AJ....121..591B; @2002ApJ...571L..71S; @2002ApJ...569L...5B; @2002ApJ...570L...5M; @2003AJ....125.2876V; @2003ApJ...588..119B; @2004AJ....128.1483S; @2004ApJ...611L..13F], *excluding* the deep field observations and in spite of the far larger collecting areas of both instruments.
The $>10^{-9}$ergcm$^{-2}$s$^{-1}$ X-ray continuum detected in the first minutes after GRB050904, demonstrates the power of GRBs to probe the universe in X-rays to the highest redshifts. Follow-up observations of GRBs with XMM-*Newton*, and *Chandra*, have shown that in practice the typical fluxes for observations made more than $\sim6$ hours after the burst [@2005astro.ph..7710G] are too low to detect the ionised IGM [c.f. @2000ApJ...544L...7F]. For instance, GRB020813, with one of the highest average fluxes, provided about 5000 counts in the *Chandra* gratings over 100ks [@2003ApJ...597.1010B]; out of more than thirty observations over the past five years, the average observed fluxes are $\lesssim2\times10^{-12}$ergcm$^{-2}$s$^{-1}$. It is now clear that observations of GRB afterglows with instruments not possessing a very rapid response cannot provide grating spectra with anywhere near 100000 counts, as had been speculated [@2000ApJ...544L...7F]. It is also now clear that a good detection of the IGM requires a flux high enough to provide in excess of $10^6$ counts at moderate spectral resolution. It would be feasible to obtain enough photons in 50–100ks with the Narrow Field Instruments on the proposed *XEUS* mission, if it began observing up to about 6 hours after the burst; with the brighter bursts this might also be possible with *Constellation-X*. But this is clearly not the most efficient way to study the IGM with GRBs. It was suggested as an alternative, that a high resolution instrument with small effective area could make rapid observations of GRB afterglows in their early phases [@2000ApJ...544L...7F]. To exploit the huge fluence provided by the high state and flares in the first few minutes after the GRB, a small area detector could routinely provide 10000 counts, but this is insufficient for IGM studies [@2005ApJ...629..700N]. A very rapid response, similar to *Swift*’s, to a GRB like GRB050904 with a large area detector with good spectral resolution and fast readout times (e.g.*Constellation-X* or *XEUS*) would reliably yield several to tens of millions of photons in an exposure of only a few minutes. Such short observations would allow studies of the high-redshift universe along many different sightlines; observations that would each require months of *effective* exposure time observing a high-$z$ AGN. Such rapid observations are demanding, but this is a technique now demonstrated in practice by *Swift*, and the short observations would be highly efficient as well as providing superb spectra. A sample of such observations could allow us to fix the fraction of baryonic dark matter, determine the metallicity and density evolution of the IGM, and put strong constraints on structure formation at high redshifts.
The Dark Cosmology Centre is funded by the DNRF. We acknowledge benefits from collaboration within the EU FP5 Research Training Network, ‘Gamma-Ray Bursts: An Enigma and a Tool’. We are indebted to S. Larsson for the power spectral density analysis.
{#section .unnumbered}
After submission of this *Letter*, a paper by @2005astro.ph..9737C, appeared on the arXiv preprints servers (astro-ph/0509737) based on the analysis of the XRT and BAT data from GRB050904. Their findings are similar to those reported here.
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|
---
abstract: 'Evidence is presented that 2MASS J03202839$-$0446358, a late-type dwarf with discrepant optical (M8:) and near-infrared (L1) spectral types, is an as-yet unresolved stellar/brown dwarf binary with late-type M dwarf and T dwarf components. This conclusion is based on low-resolution, near-infrared spectroscopy that reveals a subtle but distinctive absorption feature at 1.6 $\micron$. The feature, which is also present in the combined light spectrum of the M8.5 + T6 binary SCR 1845$-$6357, arises from the combination of FeH absorption from an M8.5 primary and pseudo-continuum flux from a T5$\pm$1 secondary, as ascertained from binary spectral templates constructed from empirical data. The binary templates provide a far superior match to the overall near-infrared spectral energy distribution of 2MASS J0320$-$0446 than any single comparison spectra. Laser guide star adaptive optics (LGS AO) imaging observations, including the first application of LGS AO aperture mask interferometry, fail to resolve a faint companion, restricting the projected separation of the system to less than 8.3 AU at the time of observation. 2MASS J0320$-$0446 is the second very low mass binary to be identified from unresolved, low-resolution, near-infrared spectroscopy, a technique that complements traditional high resolution imaging and spectroscopic methods.'
author:
- 'Adam J. Burgasser, Michael C. Liu, Michael J. Ireland, Kelle L. Cruz, and Trent J. Dupuy'
title: 'Subtle Signatures of Multiplicity in Late-type Dwarf Spectra: The Unresolved M8.5 + T5 Binary 2MASS J03202839$-$0446358'
---
Introduction
============
The optical and near-infrared spectral energy distributions of very low mass stars and brown dwarfs—late-type M, L and T dwarfs—are distinctly non-blackbody. Overlapping molecular bands and strong line emission produce a rich array of spectral diagnostics for classification and characterization of physical properties. Considerable effort is now being devoted toward decrypting the spectral fingerprints of late-type dwarfs to determine masses, ages, metallicities and other fundamental parameters (e.g., @luh99 [@gor03; @metgrav; @all07; @liu07]). In some cases, spectral peculiarities arise when an observed source is in fact an unresolved multiple system, with components of different masses, effective temperatures, and other spectral properties. While several classes of stellar multiples are recognized on the basis of their unusual spectral or photometric properties (U Geminorum stars, M dwarf + white dwarf systems, etc.), identifying such cases amongst late-type dwarfs is complicated by the influence of other physical effects. Delineation of spectral peculiarities that arise purely from multiplicity as opposed to other physical effects is essential if we hope to unambiguously characterize the physical properties of the lowest luminosity stars and brown dwarfs.
Very low mass multiple systems are important in their own right, as they enable mass and occasionally radius measurements (e.g., @lan01 [@zap04; @sta06; @liu08]), provide constraints for star/brown dwarf formation scenarios (e.g., @clo03 [@pall07; @luh07ppv]) and facilitate detailed studies of atmospheric properties (e.g., @mehst2 [@liu06; @mar06]). Of the roughly 90 very low mass multiple systems currently known,[^1] the majority have been identified through high angular resolution imaging, using the [*Hubble Space Telescope*]{} ([*HST*]{}; e.g., @mar99a [@bou03; @mehst2; @rei06]), ground-based adaptive optics systems (e.g., @clo03 [@cha04; @sie03; @sie05; @kra05; @liu06; @loo08]) and more recently aperture masking interferometry (e.g., @ire08 [@kra08]). However, as the vast majority of very low mass binaries have small separations ($>$90% have $\rho$ $<$ 20 AU; @me06ppv), expanding the population of known binaries to greater distances requires either finer angular sampling or the identification of systems that are unresolved. The frequency of nearby, tightly-bound binaries is also essential for a complete assessment of the overall very low mass dwarf binary fraction, since imaging studies provide only a lower limit to this fundamental statistic. Such systems are also more likely to eclipse, enabling radius measurements and fundamental tests of evolutionary models (e.g., @sta06). While searches for radial velocity variability via high resolution spectroscopy can be useful in this regime (e.g., @bas99 [@ken03; @bas06; @bla07; @joe07]), in many cases very low luminosity and/or distant late-type dwarfs are simply too faint to be followed up in this manner.
Recently, @me0805 demonstrated that in certain cases the presence of an unresolved companion can be inferred directly from the morphology of a source’s low-resolution near-infrared spectrum. In particular, it was shown that the spectrum of the peculiar L dwarf SDSS J080531.84+481233.0 (hereafter SDSS J0805+4812; @haw02 [@kna04]), which has highly discrepant optical and near-infrared spectral classifications, could be accurately reproduced as a combination of “normal” L4.5 + T5 components. Indeed, the binary hypothesis provides a far simpler and more consistent explanation for the unusual optical, near-infrared and mid-infrared properties of SDSS J0805+4812 than other alternatives (e.g., @kna04 [@fol07; @leg07]). The identification of unresolved multiples like SDSS J0805+4812 by low-resolution near-infrared spectroscopy is a potential boon for low-mass multiplicity studies, as this method is not subject to the same physical or projected separation limitations inherent to high-resolution imaging and spectroscopic techniques.
This article reports the discovery of a second unresolved very low mass binary system, [[2MASS J03202839$-$0446358]{}]{} (hereafter [[2MASS J0320$-$0446]{}]{}), identified by the morphology of its low-resolution, near-infrared spectrum. The spectroscopic observations leading to this conclusion are described in $\S$ 2, as are laser guide star adaptive optics (LGS AO) imaging observations aimed at searching for a faint companion. Analysis of the spectral data using the binary template matching technique described in @me0805 is presented in $\S$ 3. $\S$ 4 discusses the viability of [[2MASS J0320$-$0446]{}]{} being a binary, with specific comparison to the known M dwarf + T dwarf system SCR 1845$-$6357. We also constrain the projected separation of the [[2MASS J0320$-$0446]{}]{} system based on our imaging observations, and discuss overall limitations on the variety of unresolved M dwarf + T dwarf binaries that can be identified from composite near-infrared spectroscopy. Conclusions are summarized in $\S$ 5.
Observations
============
Previous Observations of [[2MASS J0320$-$0446]{}]{}
---------------------------------------------------
[[2MASS J0320$-$0446]{}]{} was originally discovered by @cru03 and @wil03 in the Two Micron All Sky Survey (2MASS; @skr06), and classified M8: (uncertain) and L0.5 on the basis of optical and near-infrared spectroscopy, respectively. The M8: optical classification is uncertain because of the low signal-to-noise of the spectral data, and is not due to any specific spectral peculiarity. @cru03 estimate a distance of 26$\pm$4 pc for this source based on its classification and empirical $M_J$/spectral type relations. @dea05, using $I$-band plate data from the SuperCosmos Sky Survey (SSS; @ham01a [@ham01b; @ham01c]), report a relatively high proper motion of 0$\farcs$68$\pm$0$\farcs$04 yr$^{-1}$ at position angle 191$\degr$ for this source. Figure \[fig\_chart\] shows the field around [[2MASS J0320$-$0446]{}]{} imaged by $R$ and $I$ photographic plates. A faint source is seen in the 1955 Palomar Sky Survey I [@abe59] $R$-band image roughly at the offset position indicated by the @dea05 proper motion. By including this source position along with additional astrometry drawn from the SSS and 2MASS catalogs, an improved proper motion measurement of 0$\farcs$562$\pm$0$\farcs$005 yr$^{-1}$ at position angle 205.9$\pm$0.5$\degr$ was determined. This proper motion and the estimated distance indicates a rather large tangential space velocity for [[2MASS J0320$-$0446]{}]{}, [[$V_{tan}$]{}]{} = 69$\pm$11 [[km s$^{-1}$]{}]{}, suggesting that it could be an older disk star. None of the previous studies of [[2MASS J0320$-$0446]{}]{} report the presence of a faint companion.
Near-Infrared Spectroscopy
--------------------------
Low resolution near-infrared spectral data for [[2MASS J0320$-$0446]{}]{} were obtained on 2007 September 16 (UT) using the SpeX spectrograph [@ray03] mounted on the 3m NASA Infrared Telescope Facility (IRTF). The conditions on this night were poor with patchy clouds, cirrus and average seeing (0$\farcs$8 at $J$-band), and [[2MASS J0320$-$0446]{}]{} was observed as a bright backup target ($J$ = 12.13$\pm$0.03). The 0$\farcs$5 slit was used to obtain 0.7–2.5 $\micron$ spectroscopy with resolution [[$\lambda/{\Delta}{\lambda}$]{}]{} $\approx 120$ and dispersion across the chip of 20–30 [Å]{} pixel$^{-1}$. To mitigate the effects of differential refraction, the slit was aligned to the parallactic angle. Six exposures of 90 s each were obtained in an ABBA dither pattern along the slit. The A0 V star HD 18571 was observed immediately after [[2MASS J0320$-$0446]{}]{} and at a similar airmass (1.21) for flux calibration. Internal flat field and argon arc lamps were observed after both target and flux standard observations for pixel response and wavelength calibration. Data were reduced with the IDL SpeXtool package, version 3.4 [@cus04; @vac03], using standard settings. A detailed description of the reduction procedures is given in @me0805.
The near-infrared spectrum of [[2MASS J0320$-$0446]{}]{} is shown in Figure \[fig\_nirspec\], compared to equivalent data for the optical spectral standards VB 10 (M8; @bie61 [@kir91]) and 2MASS J14392836+1929149 (L1, hereafter 2MASS J1439+1929; @kir99). Despite the poor observing conditions, the data for [[2MASS J0320$-$0446]{}]{} have exceptionally good signal-to-noise, $\gtrsim$150 in the $JHK$ flux peaks and $\sim$50 in the bottom of the 1.4 and 1.8 $\micron$ [[H$_2$O]{}]{} bands. Color biases due to telluric cloud absorption do not appear to be present, as indicated by comparison of 2MASS photometry and synthetic $J-H$, $H-K_s$ and $J-K_s$ colors computed from the spectral data, which agree to within the photometric uncertainties.
The morphology of the near-infrared spectrum of [[2MASS J0320$-$0446]{}]{} is typical of a late-type M or early-type L dwarf, with bands of TiO and VO absorption at red optical wavelengths ($\lambda < 1$ ); prominent [[H$_2$O]{}]{} absorption at 1.4 and 1.8 $\micron$; FeH absorption at 0.99, 1.2 and 1.55 $\micron$; and line absorption in the 1.0-1.3 $\micron$ region; weak lines at 2.2 $\micron$; and strong CO bandheads at 2.3–2.4 $\micron$. For the most part, the spectrum of [[2MASS J0320$-$0446]{}]{} is more consistent with that of 2MASS J1439+1929; note in particular the similarities in the overall shape of the 1.0–1.35 $\micron$ $J$-band flux peak and the deep 1.4 $\micron$ [[H$_2$O]{}]{} band. However, TiO and VO bands are more similar to (but weaker than) those seen in the spectrum of VB 10, while the weak 2.2 $\micron$ lines are rarely seen in L dwarf spectra (e.g., @mcl03). The near-infrared spectrum of [[2MASS J0320$-$0446]{}]{} is also somewhat bluer than that of 2MASS J1439+1929, in line with their respective colors ($J-K_s$ = 1.13$\pm$0.04 versus 1.21$\pm$0.03).
The similarities to 2MASS J1439+1929 suggests an L1 near-infrared spectral type for [[2MASS J0320$-$0446]{}]{}, which is confirmed by examination of the spectral indices and index/spectral type relations defined by @rei01a and @geb02. The average subtype for the four indices K1 (measuring the shape of the $K$-band flux peak; @tok99), [[H$_2$O]{}]{}-A, [[H$_2$O]{}]{}-B and [[H$_2$O]{}]{}-1.5 (all measuring the strength of the 1.4 $\micron$ [[H$_2$O]{}]{} band) yields a near-infrared classification of L1 ($\pm$0.6 subtypes), consistent with the L0.5 near-infrared classification reported by @wil03.
This classification is fully three subtypes later than the M8: optical spectral type reported by @cru03. However, such discrepancies are not altogether uncommon amongst late-type dwarfs. @geb02 and @kna04 have found disagreements of up to 1.5 subtypes between optical (based on the @kir99 scheme) and near-infrared classifications (based on their own scheme) for several L dwarfs. @me1126 have discussed a subclass of unusually blue L dwarfs whose optical classifications are consistently 2-3 subtypes earlier than their near-infrared classifications. Such discrepancies have been variously attributed to surface gravity, metallicity, condensate cloud or multiplicity effects (e.g., @kna04 [@chi06; @cru07; @fol07; @me1126]). The large [[$V_{tan}$]{}]{} of [[2MASS J0320$-$0446]{}]{}, indicating that this source may be somewhat older, suggests that high surface gravity and/or slightly subsolar metallicity could explain the discrepant optical and near-infrared spectral classifications.
However, [[2MASS J0320$-$0446]{}]{} exhibits one unusual feature not seen in the comparison spectra in Figure \[fig\_nirspec\], a slight dip at 1.6 $\micron$, that suggests multiplicity is relevant in this case. The 1.6 $\micron$ feature is nearly coincident with the 1.57-1.64 $\micron$ FeH absorption band commonly observed in L dwarf near-infrared spectra [@wal01; @cus03]. Yet its morphology is clearly different, with a cup-shaped depression as opposed to the flat plateau seen in the comparison spectra of Figure \[fig\_nirspec\]. More importantly, this feature has the same morphology and is centered at the same wavelength as the peculiar feature noted in the spectrum of SDSS J0805+4812 [@me0805]. In that case, the 1.6 $\micron$ feature and other spectral peculiarities were attributed to the presence of a mid-type T dwarf companion. Given the similar discrepancy in optical and near-infrared classifications for SDSS J0805+4812 (L4 and L9.5, respectively), it is reasonable to consider whether [[2MASS J0320$-$0446]{}]{} also harbors a faint T dwarf companion.
High Angular Resolution Imaging
-------------------------------
In an attempt to search for faint companions, [[2MASS J0320$-$0446]{}]{} was imaged on 2008 January 15 (UT) with the sodium LGS AO system [@wiz06; @van06] and facility near-infrared camera NIRC2 on the 10m Keck Telescope. Conditions were photometric with average/below-average seeing. The narrow field-of-view camera of NIRC2 was utilized, providing an image scale of $9.963\pm0.011$ mas/pixel [@pra06] over a $10.2\arcsec \times 10.2\arcsec$ field of view. All observations were conducted using the MKO[^2] $K_s$-band filter. The LGS provided the wavefront reference source for AO correction, while tip-tilt aberrations and quasi-static changes were measured contemporaneously by monitoring the $R=16.7$ mag field star USNO-B1.0 0852$-$0031783 [@mon03], located 14 away from [[2MASS J0320$-$0446]{}]{}. The LGS, with an equivalent brightness of a $V\approx10.4$ mag star, was pointed at the center of the NIRC2 field-of-view for all observations.
[[2MASS J0320$-$0446]{}]{} was imaged using two different methods in order to probe the widest possible range of projected separations: (1) direct imaging and (2) aperture mask interferometry. In the first case, a series of 3 dithered 60-second images was obtained, offsetting the telescope by a few arcseconds between exposures, for a total integration time of 180 seconds. Raw frames were reduced using standard procedures. Normalized flat field frames were constructed from the differences of images of the telescope dome interior with and without continuum lamp illumination. A master sky frame was created from the median average of the bias-subtracted, flat-fielded images and subtracted from the individual exposures. Individual frames were registered and stacked to form a final mosaic imaged. The observations achieved a point spread function (PSF) full-width at half-maximum of 007 and a Strehl ratio of 0.21. With the exception of the primary target, no sources were detected in a $6\arcsec \times 6\arcsec$ region centered on [[2MASS J0320$-$0446]{}]{}.
Aperture mask observations were also obtained with the LGS AO+NIRC2 instrumental setup. In this method, a 9-hole aperture mask is placed in a filter wheel near a re-imaged pupil plane within the NIRC2 camera. The mask has non-redundant spacing, so each Fourier component of the recorded image corresponds to a unique pair of patches on the Keck primary mirror. The primary interferometric observables of squared visibility and closure-phase are therefore calibrated much better than images using the full aperture. This technique has a long history of achieving the full diffraction limit of a telescope (e.g. @mic20 [@bal86; @nak89; @tut00]) and has been recently applied to natural guide star AO observations at Keck [@ire08; @kra08]. This is the first application of aperture mask interferometry to LGS AO observations that we are aware of.
[[2MASS J0320$-$0446]{}]{} was observed in this setup using a two-point dither pattern, with five 50-second integrations at each dither position. The nearby field star 2MASS J03381363$-$0332508, which has a similar $K_s$-band brightness and tip-tilt star asterism as [[2MASS J0320$-$0446]{}]{}, was contemporaneously observed to calibrate both instrumental closure phase and visibility. Images of the interferograms formed by the mask were recorded by the NIRC2 detector, and squared visibilities and closure-phases were extracted from the image Fourier transforms. Raw visibility amplitudes were $\sim$0.05 on the longest baselines. The closure phases for this calibrator star were subtracted from those of [[2MASS J0320$-$0446]{}]{}, while the calibrator’s squared visibilities were divided into those of [[2MASS J0320$-$0446]{}]{}. The one-sigma scatter in the calibrated closure phase was 5$\degr$. Using standard analysis techniques (e.g., @kra08), we found no evidence of a binary solution in the data.
Upper limits on the presence of a faint companion to [[2MASS J0320$-$0446]{}]{} were computed separately for the direct imaging and aperture mask observations. For the direct imaging data, upper limits were determined by first smoothing the final mosaic with an analytical representation of the PSF’s radial profile, modeled as the sum of multiple gaussians. We then measured the standard deviation of flux counts in concentric annuli out to 3$\arcsec$ in radius centered on the science target, normalized by the peak flux of the science target. We considered 10$\times$ these values as the flux ratio limits for any companions, as visually verified by inserting fake sources into the image using translated and scaled versions of the science target. For the aperture mask data, detection limits at 99% confidence were calculated in three annuli spanning 0$\farcs$020–0$\farcs$16 in separation (the lower limit corresponding to the diffraction limit of the aperture mask) using the Monte-Carlo approach described in @kra08.
Figure \[fig\_ao\] displays the resulting flux ratio limits for a faint companion as a function of separation for both datasets. At separations $\lesssim$0$\farcs$25, the aperture mask data exclude any companions with $\Delta{K_s} \lesssim$ 3 mag for separations down to 0$\farcs$04. Note that better seeing, as opposed to longer integrations, would have provided greater improvement in sensitivity in this range. The direct imaging observations exclude any companions with $\Delta{K_s} \lesssim$ 7 mag at separations ${\gtrsim}0{\farcs}7$, with the floor set primarily by sky shot noise and detector read noise. These limits are discussed further in $\S$ 4.1.
Binary Template Matching
========================
Spectral Sample
---------------
As an alternative method to identify and characterize a possible companion to [[2MASS J0320$-$0446]{}]{}, a variant of the binary spectral template matching technique described in @me0805 was applied to the near-infrared spectral data.[^3] In this method, the spectrum of a late-type source is compared to a large set of binary spectral templates constructed from empirical data for M, L and T dwarfs. The component spectra of each binary template were scaled according to empirical absolute magnitude/spectral type relations. To minimize systematic effects, source and template spectra are required to have the same resolution and wavelength coverage, which is facilitated in this case by using a sample of nearly 200 SpeX prism spectra of M5–T8 dwarfs drawn from the literature[^4] and our own unpublished observations. Spectral types for the sources in this sample were assigned according to published classifications,[^5] based either on the optical classification schemes of @kir91 and @kir99 for M5–L8 dwarfs or the near-infrared classification scheme of @meclass2 for L9–T8 dwarfs (M and L dwarfs with only near-infrared classifications reported were not included here). The initial spectral sample was purged of low signal-to-noise data as well as spectra of those sources known to be binary or noted as peculiar in the literature (e.g., low surface gravity brown dwarfs, subdwarfs, etc.). This left a sample of 132 spectra of 125 sources, listed in Table \[tab\_templates\].
Single Template Fits
--------------------
To ascertain whether an unresolved binary truly provides a better fit to the spectrum of [[2MASS J0320$-$0446]{}]{}, comparisons were first made to individual sources in the SpeX sample. All spectra were initially normalized to their peak flux in the 1.2–1.3 $\micron$ band. The statistic $\sigma^2$ was then computed between the [[2MASS J0320$-$0446]{}]{} ($f_{\lambda}(0320)$) and template spectra ($f_{\lambda}(T)$), where $$\sigma^2 \equiv \sum_{\{ \lambda\} }\frac{[f_{\lambda}(0320)-f_{\lambda}(T)]^2}{f_{\lambda}(0320)}$$ (see @me0805). The summation is performed over the wavelength ranges $\{\lambda\}$ = 0.95–1.35 $\micron$, 1.45–1.8 $\micron$ and 2.0–2.35 $\micron$ in order to avoid regions of strong telluric absorption. The denominator provides a rough estimate of shot noise in the spectral data, which is dominant in the highest signal-to-noise spectra, and therefore makes $\sigma^2$ a rough approximation of the $\chi^2$ statistic.[^6] To eliminate normalization biases, each template spectrum was additionally scaled by a multiplicative factor in the range 0.5–1.5 to minimize $\sigma^2$.
Figure \[fig\_single\] displays the four best single template matches, all having $\sigma^2 < 0.6$. The three best-fitting sources—LEHPM 1-6333 (M8), 2MASS J1124+3808 (M8.5) and LEHPM 1-6443 (M8.5)—have optical spectral types consistent with the optical type of [[2MASS J0320$-$0446]{}]{}. The fourth-best fit, the L1 2MASS J1493+1929, was shown to provide an adequate match to the spectrum of [[2MASS J0320$-$0446]{}]{} in Figure \[fig\_nirspec\]. The LEHPM[^7] sources have large proper motions ($\mu > 0\farcs$4 yr$^{-1}$), notably similar to [[2MASS J0320$-$0446]{}]{}. All four sources shown in Figure \[fig\_single\] provide reasonably good matches to the broad near-infrared spectral energy distribution of [[2MASS J0320$-$0446]{}]{}, but with two key discrepancies: an absence of the 1.6 $\micron$ feature (inset boxes in Figure \[fig\_single\]) and a shortfall in the peak spectral flux at 1.27 $\micron$. In the first case, FeH absorption bands are clearly seen in the comparison spectra but do not produce the distinct dip seen in the spectrum of [[2MASS J0320$-$0446]{}]{}. In the second case, the spectrum of [[2MASS J0320$-$0446]{}]{} is consistently brighter in the 1.2–1.35 $\micron$ range as compared to the (appropriately scaled) late-type M dwarf templates. As demonstrated below, both of these discrepancies can be resolved by the addition of a T dwarf component.
Binary Template Fits
--------------------
Binary spectral templates from the SpeX prism sample were constructed by first flux-calibrating each spectrum according to established absolute magnitude/spectral type relations. For M5-L5 dwarfs, the 2MASS $M_J$/spectral type relation of @cru03 was used. For L5-T8 dwarfs, both of the MKO $M_K$/spectral type relations defined in @liu06 were considered. The Liu et al. relations are based on a sample of L and T dwarfs with measured pallaxes and MKO photometry, but one relation (“bright”) was constructed after rejecting known (resolved) binaries while the other relation (“faint”) was constructed after rejecting all known and [*candidate*]{} binaries as described in that study. As illustrated in Figure 3 of @meltbinary, these two relations envelope the $M_K$ values of currently measured sources (including components of resolved binaries), but diverge by as much as $\sim$1 mag for spectral types L8–T5. Nevertheless, the @liu06 relations represent our current best constraints on the absolute magnitude/spectral type relation across the L dwarf/T dwarf transition. In all cases, synthetic magnitudes to scale the data were calculated directly from the spectra. Binary templates were then constructed by adding together the calibrated spectra of source pairs whose types differ by at least 0.5 subclasses, producing a total of 8248 unique combinations. The binary templates were then normalized to their peak flux in the 1.2–1.3 $\micron$ band and compared to the spectrum of [[2MASS J0320$-$0446]{}]{} in the same manner as the single source templates; i.e., with additional scaling to minimize $\sigma^2$.
Figure \[fig\_double\] displays the best fitting binary templates constructed from the primaries shown in Figure \[fig\_single\] and using the “faint” $M_K$/spectral type relation of @liu06. For all four cases, the addition of a mid-type T dwarf secondary spectrum considerably improves the spectral template match. In particular, the 1.6 $\micron$ spectral dip is very well reproduced, while the flux peaks at 1.27 $\micron$ in the binary templates are more consistent with the spectrum of [[2MASS J0320$-$0446]{}]{}. Even detailed alkali line and FeH features in the 0.9–1.3 $\micron$ region are better matched with the binary templates.
Figure \[fig\_double2\] displays the best fitting binary templates using the “bright” $M_K$/spectral type relation of @liu06. There is a small degree of improvement in these fits over those using the “faint” $M_K$ relation, although the differences are very subtle due to the very small contribution of light by the T dwarf secondaries ($\Delta{J} \approx$ 3.5 mag, depending on the components). This result is fortuitous, as it indicates that the better fits provided by the binary templates are only weakly dependent on the absolute magnitude relation assumed over a spectral type range in which such relations are currently most uncertain.
Besides the best-fit comparisons shown in Figures \[fig\_double\] and \[fig\_double2\], there were many excellent matches ($\sigma^2 < 0.1$) found among binaries templates which had LEHPM 1-6333 or 2MASS J1124+3808 as primaries: 30 for the “faint” $M_K$/spectral type relation and 58 for the “bright” relation. The average primary and secondary spectral types for the combinations in this well-matched sample are M8.5$\pm$0.3 and T5.0$\pm$0.9, respectively, with no significant differences between analyses using the “faint” or “bright” $M_K$/spectral type relations. The mean relative magnitudes of the primary and secondary components were $\Delta{J} = 3.5{\pm}0.2$ mag, $\Delta{H} = 4.3{\pm}0.3$ mag, $\Delta{K} = 4.9{\pm}0.3$ mag for the “faint” relation and $\Delta{J} = 3.1{\pm}0.4$ mag, $\Delta{H} = 3.8{\pm}0.5$ mag, $\Delta{K} = 4.3{\pm}0.6$ mag for the “bright” relation, as calculated directly from the flux-calibrated spectral templates. There is a large difference in the relative magnitudes between these two relations. If resolved photometry is eventually obtained for this system, such measurements could provide a means of distinguishing which of the absolute magnitude relations proposed in @liu06 accurately characterize mid-type T dwarfs.
The origin of the 1.6 $\micron$ feature in the spectrum of [[2MASS J0320$-$0446]{}]{} is clearly revealed in Figures \[fig\_double\] and \[fig\_double2\]: it is a combination of FeH absorption in the M dwarf primary and [[CH$_4$]{}]{} absorption in the T dwarf secondary. Specifically, the relatively sharp $H$-band flux peak in the spectrum of the T dwarf secondary blueward of the 1.6 $\micron$ [[CH$_4$]{}]{} band contributes light to the 1.55–1.6 $\micron$ spectrum of the composite system. This is on the blue end of the 1.55-1.65 $\micron$ FeH absorption band, producing a distinct “dip” feature. Similarly, the apparently brighter 1.2–1.35 $\micron$ flux in the spectrum of [[2MASS J0320$-$0446]{}]{} can be attributed to the T dwarf companion, which exhibits a narrow $J$-band peak between strong 1.1 $\micron$ and 1.4 $\micron$ [[H$_2$O]{}]{} and [[CH$_4$]{}]{} bands. Both spectral features are therefore unique to binaries containing late-type M and L dwarf primaries (in which FeH is prominent) and T dwarf secondaries.
Discussion
==========
Is [[2MASS J0320$-$0446]{}]{} an M dwarf plus T dwarf Binary?
-------------------------------------------------------------
It may be concluded from the analysis above that the near-infrared spectrum of [[2MASS J0320$-$0446]{}]{}, and in particular the subtle feature observed at 1.6 $\micron$, can be accurately reproduced by assuming that this source is an unresolved M8.5 + T5 binary. But does this mean that [[2MASS J0320$-$0446]{}]{} actually is a binary? Our LGS AO imaging observations failed to detect any faint secondaries near [[2MASS J0320$-$0446]{}]{} to the limits displayed in Figure \[fig\_ao\]. Based on the “bright” MKO $M_K$/spectral type relation of @liu06 and the $K_s$/$K$ filter transformations of @ste04, the measured upper limits rule out a T5 companion wider than a projected separation of $\sim$0$\farcs$33, or roughly 8.3 AU at the estimated distance of [[2MASS J0320$-$0446]{}]{} (see below). This is a relatively weak constraint given that less than 25% of known very low mass binaries have projected separations at least this wide [@me06ppv]. Furthermore, [[2MASS J0320$-$0446]{}]{} could have been observed in an unfortunate geometry, as was originally the case for the L dwarf binary Kelu 1 [@mar99a; @liu05; @gel06]. On the other hand, if the physical separation of the [[2MASS J0320$-$0446]{}]{} system is significantly smaller than indicated by the imaging observations, high resolution spectroscopic monitoring could potentially reveal radial velocity signatures, although this depends critically on the component masses of this system. Indeed, the determination of a spectroscopic orbit in combination with the component spectral types deduced here would provide both mass and age constraints for this system, making it a potentially powerful benchmark test for evolutionary models.
An alternative test of the binary hypothesis for [[2MASS J0320$-$0446]{}]{} is to identify similar spectral traits in a comparable binary system. Fortunately, one such system is known: the M8.5 + T6 binary SCR 1845$-$6357 [@ham04; @bil06; @mon06]. This nearby (3.85$\pm$0.02 pc; @hen06), well-resolved binary (angular separation of 1$\farcs$1) has individually classified components based on resolved spectroscopy [@kas07]. More importantly, the relative near-infrared magnitudes of this system ($\Delta{J} = 3.68{\pm}0.03$ mag, $\Delta{H} = 4.20{\pm}0.04$ mag, $\Delta{K} = 5.12{\pm}0.03$ mag; @kas07) are somewhat larger than but consistent with the estimated relative magnitudes of the putative [[2MASS J0320$-$0446]{}]{} system. Figure \[fig\_scr1845\] displays the component spectra of this system, scaled to their relative $H$-band magnitudes,[^8] as well as the sum of the component spectra. The composite spectrum shows a relative increase in spectral flux as compared to the primary in both the 1.2–1.35 $\micron$ and 1.55–1.6 $\micron$ regions. Indeed, the latter gives rise to the same “dip” feature observed in the $H$-band spectrum of [[2MASS J0320$-$0446]{}]{}, particularly when the SCR 1845$-$6357AB data are reduced in resolution to match that of the SpeX prism data (inset box in Figure \[fig\_scr1845\]). The presence of this feature in the composite spectrum of a known M dwarf plus T dwarf binary lends some confidence to the conclusion that [[2MASS J0320$-$0446]{}]{} is itself an M dwarf plus T dwarf binary.
Assuming then that [[2MASS J0320$-$0446]{}]{} is a system with M8.5 and T5 dwarf components, it is possible to characterize the physical properties of these components in some detail based on the analysis in $\S$ 3.3. Synthetic component $JHK$ magnitudes on the MKO system assuming the “bright” $M_K$/spectral type relation of @liu06 were computed from the best-fitting binary templates ($\sigma^2 < 0.1$) and are listed in Table \[tab\_component\]. The M dwarf primary is only slightly fainter than the composite source, while the T dwarf companion is exceptionally faint, $J$ = 16.4$\pm$0.4 mag. The low luminosity of the secondary, [[$\log_{10}{L_{bol}/L_{\sun}}$]{}]{} = -5.0$\pm$0.3 dex based on its inferred spectral type [@gol04; @meltbinary], suggests that [[2MASS J0320$-$0446]{}]{} could have a relatively low system mass ratio ($q \equiv$ M$_2$/M$_1$). However, the mass ratio depends critically on the age of the system, for which the analysis presented above provides no robust constraints. Using the evolutionary models of @bur97 and component luminosities as listed in Table \[tab\_component\], primary and secondary mass estimates for ages of 1, 5 and 10 Gyr were derived. If [[2MASS J0320$-$0446]{}]{} is an older system, as suggested by its large [[$V_{tan}$]{}]{}, its inferred mass ratio $q$ $>$ 0.8 is consistent with the typical mass ratios of very low mass binaries in the field (e.g., @pall07). Based on the primary’s photometry and spectral type, and the $M_J$/spectral type relation of @cru03, a distance of 25$\pm$3 pc is estimated for the [[2MASS J0320$-$0446]{}]{} system.
On the Identification of M dwarf plus T dwarf Binaries from Composite Near-Infrared Spectra
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The subtlety of the peculiar features present in the composite spectra of [[2MASS J0320$-$0446]{}]{} and SCR 1845$-$6357 is due entirely to the considerable difference in flux between their M and T dwarf components. Yet in both cases the 1.6 $\micron$ feature, indicating the presence of a T dwarf companion, can be discerned. But for how early of an primary can a binary with a T dwarf companion be identified in this manner, and what variety of T dwarf companions can be discerned in such systems? To examine these questions, Figure \[fig\_mtsim\] displays binary spectral templates for four primary types—M7, M8, M9 and L0—combined with T0–T8 dwarf secondaries. For all cases, the 1.6 $\micron$ feature is most pronounced when the secondary is a mid-type T dwarf, spectral types T3–T5. This is due to a tradeoff in the sharpness of the $H$-band flux peak in this component (i.e., the strength of 1.6 $\micron$ [[CH$_4$]{}]{} absorption, which deepens with later spectral types) and its brightness relative to the primary. Not surprisingly, the 1.6 $\micron$ feature is more pronounced in binaries with later-type primaries, making it a useful multiplicity diagnostic for L dwarf + T dwarf systems (such as SDSS J0805+4812) but far more subtle in systems with M dwarf primaries. Indeed, the spectra in Figure \[fig\_mtsim\] suggest that this feature is basically undetectable in binaries with M7 and earlier-type primaries. [[2MASS J0320$-$0446]{}]{} and SCR 1845$-$6357 probably contain the earliest-type primaries for which a T dwarf secondary could be identified solely from their composite near-infrared spectra.
It is also important to consider the other prominent spectral peculiarity caused by the presence of a T dwarf companion, the slight increase in flux at 1.3 $\micron$. This feature increases the contrast in the 1.4 $\micron$ [[H$_2$O]{}]{} band, and therefore serves to bias [[H$_2$O]{}]{} spectral indices toward later subtypes. This effect explains why the near-infrared classification of [[2MASS J0320$-$0446]{}]{} is so much later than its optical classification (the T dwarf secondary contributes negligible flux in the optical). Figure \[fig\_mtsim\] shows that the 1.3 $\micron$ flux increase can be discerned for systems with early- and mid-type T dwarf companions. While it is again more pronounced for systems with later-type primaries, it is still present (but subtle) in the spectra of systems with M7 primaries. A source with unusually strong absorption at 1.35 $\micron$, or equivalently with a near-infrared spectral type that is significantly later than its optical spectral type, may harbor a T dwarf companion. However, other physical effects, notably reduced condensate opacity (e.g., @me1126), can also give rise to this spectral peculiarity. Hence, both the contrast of the 1.4 $\micron$ [[H$_2$O]{}]{} band and the presence of the 1.6 $\micron$ dip should be considered together as indicators of an unresolved T dwarf companion.
Detecting the near-infrared spectral signature of a T dwarf companion need not be limited to low-resolution observations. While the dip feature at 1.6 $\micron$ is less pronounced in the higher-resolution composite spectrum of SCR 1845$-$6357AB from @kas07, individual [[CH$_4$]{}]{} lines may still be distinguishable amongst the many FeH and [[H$_2$O]{}]{} lines present in the same spectral region. It may also be possible to identify [[CH$_4$]{}]{} lins amongst the forest of [[H$_2$O]{}]{} lines in the 1.30-1.35 $\micron$ region (e.g. @bar06). Such detections require significantly higher resolutions, of order [[$\lambda/{\Delta}{\lambda}$]{}]{} $\approx$ 20,000 or more, due to the substantial overlap of the many molecular features present at these wavelengths (e.g., @mcl07). Furthermore, an improved line list for the [[CH$_4$]{}]{} molecule may be needed [@sha07]. Yet such observations have the potential to provide an additional check on the existence and characteristics of mid-type T dwarf companions in binaries with late-M/L dwarf primaries.
Relevant to the identification of late-type M dwarf plus T dwarf binaries from composite near-infrared spectra is the number of such systems that are expected to exist. As a rough estimate, we examined the results of the Monte Carlo mass function and multiplicity simulations presented in @meltbinary. Using the baseline assumptions of these simulations–a mass function that scales as $\frac{dN}{d{\rm M}} \propto {\rm M}^{-0.5}$, a component mass range of 0.01 $\leq$ M $\leq$ 0.1 M$_{\sun}$, a flat age distribution over 10 Gyr, the @bar03 evolutionary models, and a binary mass ratio distribution that scales as $f(q) \propto q^{1.8}$ (see @pall07)—we found that 12-14% of binaries with M8–L0 primaries are predicted to contain a T3–T5 secondary; i.e., detectable with composite near-infrared spectroscopy. These are primarily older systems whose components that just straddle the hydrogen burning minimum mass limit ($\sim$0.07 M$_{\sun}$; @cha00a). The overall binary fraction of very low mass stars and brown dwarfs has been variously estimated to lie in the 10–35% range (e.g. @bou03 [@clo03; @bas06; @mehst2; @meltbinary; @pall07; @kra08]), and is thus currently uncertain by over a factor of three. However, within this range the Monte Carlo simulations predict that 1-5% of [*all*]{} M8–L0 dwarfs harbor a T3–T5 dwarf companion. While this percentage is small, in a given magnitude-limited survey there may be a similar number of T dwarf companions in these relatively bright systems as compared to faint, isolated T dwarfs. Such companions, based on the analysis above, can be reasonably well-characterized without the need of resolved imaging.
There are many other variables that must be considered if the binary spectral template technique described here and in @me0805 is to be used to determine accurate binary statistics for very low mass stars and brown dwarfs. Component peculiarities, such as unusual surface gravities or cloud variations; intrinsic scatter in absolute magnitude/spectral type relations; magnetic- or weather-induced photometric variability; the detailed properties of the still poorly-constrained L dwarf/T dwarf transition; and the possible presence of tertiary components all contribute in constraining the variety of systems that can be identified from composite near-infrared spectroscopy. Furthermore, because brown dwarfs cool over their lifetimes, the detectability of binaries based on component spectral types does not map uniquely to the detectability of binaries based on their mass ratios and ages, resulting in complex selection biases. These issues will be addressed in a future publication.
Conclusions
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We have found that subtle peculiarities observed in the near-infrared spectrum of [[2MASS J0320$-$0446]{}]{}, in particular a characteristic bowl-shaped dip at 1.6 $\micron$, indicate the presence of a mid-type T dwarf companion. This companion is unresolved in LGS AO imaging observations (including the first application of aperture mask interferometry with LGS AO), indicating a maximum projected separation of 8.3 AU at the time of observations. The binary scenario not only provides a simple and straightforward explanation for the 1.6 $\micron$ feature—also present in the composite spectrum of the known M8.5 + T6 binary SCR 1845$-$6357—but also resolves the discrepancy between the optical and near-infrared classifications of [[2MASS J0320$-$0446]{}]{}. Furthermore, empirical binary templates composed of “normal” M dwarf plus T dwarf pairs provide a far superior match to the overall near-infrared spectral energy distribution of [[2MASS J0320$-$0446]{}]{} than any single comparison source. The hypothesis that [[2MASS J0320$-$0446]{}]{} is an unresolved binary is therefore compelling, and could potentially be verified through radial velocity monitoring observations. In addition, we estimate that roughly 1-5% of all late-type M dwarfs may harbor a mid-type T dwarf companion that could similarly be identified and characterized using low resolution near-infrared spectroscopy and binary spectral template analysis.
The authors acknowledge telescope operator Paul Sears and instrument specialist John Rayner at IRTF, and Al Conrad, Randy Campbell, Jason McIlroy, and Gary Punawai at Keck, for their assistance during the observations. We also thank Markus Kasper for providing the spectral data for SCR 1845$-$6357 and Sandy Leggett, Dagny Looper and Kevin Luhman for providing a portion of the SpeX prism spectra used in the binary spectral template analysis. Our anonymous referee provided a helpful and very prompt critique of the original manuscript. MCL and TJD acknowledge support for this work from NSF grant AST-0507833 and an Alfred P. Sloan Research Fellowship. This publication makes use of data from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center, and funded by the National Aeronautics and Space Administration and the National Science Foundation. 2MASS data were obtained from the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This research has benefitted from the M, L, and T dwarf compendium housed at DwarfArchives.org and maintained by Chris Gelino, Davy Kirkpatrick, and Adam Burgasser; the VLM Binaries Archive maintained by Nick Siegler at <http://www.vlmbinaries.org>; and the SpeX Prism Spectral Libraries, maintained by Adam Burgasser at <http://www.browndwarfs.org/spexprism>. The authors wish to recognize and acknowledge the very significant cultural role and reverence that the summit of Mauna Kea has always had within the indigenous Hawaiian community. We are most fortunate to have the opportunity to conduct observations from this mountain.
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[llllcl]{} SDSS J0000+2554 & J00001354+2554180 & & T4.5 & 15.06$\pm$0.04 & [**1**]{};2\
2MASS J0034+0523 & J00345157+0523050 & & T6.5 & 15.54$\pm$0.05 & [**3**]{};1\
2MASS J0036+1821 & J00361617+1821104 & L3.5 & L4$\pm$1 & 12.47$\pm$0.03 & [**4**]{};2,5,6\
HD 3651B & J0039191+211516 & & T7.5 & 16.16$\pm$0.03 & [**7**]{};8,9,10\
2MASS J0050$-$3322 & J00501994$-$3322402 & & T7 & 15.93$\pm$0.07 & [**11**]{};1,12\
2MASS J0103+1935 & J01033203+1935361 & L6 & & 16.29$\pm$0.08 & [**13**]{};6\
2MASS J0117$-$3403 & J01174748$-$3403258 & L2: & & 15.18$\pm$0.04 & [**56**]{};14\
SDSS J0119+2403 & J01191207+2403317 & & T2 & 17.02$\pm$0.18 & [**15**]{}\
IPMS 0136+0933 & J01365662+0933473 & & T2.5 & 13.46$\pm$0.03 & [**4**]{};16\
2MASS J0144$-$0716 & J01443536$-$0716142 & L5 & & 14.19$\pm$0.03 & [**4**]{};17\
SDSS J0151+1244 & J01514155+1244300 & & T1 & 16.57$\pm$0.13 & [**3**]{};1,18\
2MASS J0205+1251 & J02050344+1251422 & L5 & & 15.68$\pm$0.06 & [**19**]{};6\
SDSS J0207+0000 & J02074284+0000564 & & T4.5 & 16.80$\pm$0.16 & [**1**]{};18\
2MASS J0208+2542 & J02081833+2542533 & L1 & & 13.99$\pm$0.03 & [**4**]{};6\
SIPS J0227$-$1624 & J02271036$-$1624479 & L1 & & 13.57$\pm$0.02 & [**4**]{};20\
2MASS J0228+2537 & J02281101+2537380 & L0: & L0 & 13.84$\pm$0.03 & [**4**]{};14,21\
GJ 1048B & J02355993$-$2331205 & L1 & L1 & & [**4**]{};22\
2MASS J0241$-$1241 & J02415367$-$1241069 & L2: & & 15.61$\pm$0.07 & [**56**]{};14\
2MASS J0243$-$2453 & J02431371$-$2453298 & & T6 & 15.38$\pm$0.05 & [**3**]{};1,23\
SDSS J0247$-$1631 & J02474978$-$1631132 & & T2$\pm$1.5 & 17.19$\pm$0.18 & [**15**]{}\
SO 0253+1625 & J02530084+1652532 & M7 & & 8.39$\pm$0.03 & [**4**]{};24,25\
DENIS J0255$-$4700 & J02550357$-$4700509 & L8 & L9 & 13.25$\pm$0.03 & [**1**]{};26,27\
2MASS J0310+1648 & J03105986+1648155 & L8 & L9 & 16.03$\pm$0.08 & [**28**]{};1,6\
SDSS J0325+0425 & J03255322+0425406 & & T5.5 & 16.25$\pm$0.14 & [**15**]{}\
2MASS J0328+2302 & J03284265+2302051 & L8 & L9.5 & 16.69$\pm$0.14 & [**4**]{};2,6\
LP 944$-$20 & J03393521$-$3525440 & M9 & & 10.73$\pm$0.02 & [**4**]{}\
2MASS J0345+2540 & J03454316+2540233 & L0 & L1$\pm$1 & 14.00$\pm$0.03 & [**29**]{};2,30,31\
SDSS J0351+4810 & J03510423+4810477 & & T1$\pm$1.5 & 16.47$\pm$0.13 & [**15**]{}\
2MASS J0407+1514 & J04070885+1514565 & & T5 & 16.06$\pm$0.09 & [**3**]{};1\
2MASS J0415$-$0935 & J04151954$-$0935066 & T8 & T8 & 15.70$\pm$0.06 & [**3**]{};1,23,32\
2MASS J0439$-$2353 & J04390101$-$2353083 & L6.5 & & 14.41$\pm$0.03 & [**28**]{};14\
2MASS J0510$-$4208 & J05103520$-$4208140 & & T5 & 16.22$\pm$0.09 & [**33**]{}\
2MASS J0516$-$0445 & J05160945$-$0445499 & & T5.5 & 15.98$\pm$0.08 & [**4**]{};1,34\
2MASS J0559$-$1404 & J05591914$-$1404488 & T5 & T4.5 & 13.80$\pm$0.02 & [**1**]{};32,35\
2MASS J0602+4043 & J06020638+4043588 & & T4.5 & 15.54$\pm$0.07 & [**33**]{}\
LEHPM 2$-$461 & J06590991$-$4746532 & M6.5 & M7 & 13.64$\pm$0.03 & [**4**]{};36,37\
2MASS J0727+1710 & J07271824+1710012 & T8 & T7 & 15.60$\pm$0.06 & [**11**]{};23,32\
2MASS J0729$-$3954 & J07290002$-$3954043 & & T8 & 15.92$\pm$0.08 & [**33**]{}\
2MASS J0755+2212 & J07554795+2212169 & T6 & T5 & 15.73$\pm$0.06 & [**1**]{};23,32\
SDSS J0758+3247 & J07584037+3247245 & & T2 & 14.95$\pm$0.04 & [**4**]{};1,2\
SSSPM 0829$-$1309 & J08283419$-$1309198 & L2 & & 12.80$\pm$0.03 & [**38**]{};39,40\
SDSS J0830+4828 & J08300825+4828482 & L8 & L9$\pm$1 & 15.44$\pm$0.05 & [**4**]{};18,27\
SDSS J0837$-$0000 & J08371718$-$0000179 & T0$\pm$2 & T1 & 17.10$\pm$0.21 & [**33**]{};1,32,41\
2MASS J0847$-$1532 & J08472872$-$1532372 & L2 & & 13.51$\pm$0.03 & [**42**]{};14\
SDSS J0858+3256 & J08583467+3256275 & & T1 & 16.45$\pm$0.12 & [**15**]{}\
SDSS J0909+6525 & J09090085+6525275 & & T1.5 & 16.03$\pm$0.09 & [**15**]{}\
2MASS J0939$-$2448 & J09393548$-$2448279 & & T8 & 15.98$\pm$0.11 & [**1**]{};12\
2MASS J0949$-$1545 & J09490860$-$1545485 & & T2 & 16.15$\pm$0.12 & [**1**]{};12\
2MASS J1007$-$4555 & J10073369$-$4555147 & & T5 & 15.65$\pm$0.07 & [**33**]{}\
2MASS J1010$-$0406 & J10101480$-$0406499 & L6 & & 15.51$\pm$0.06 & [**19**]{}\
HD 89744B & J10221489+4114266 & L0 & L (early) & 14.90$\pm$0.04 & [**4**]{};43\
SDSS J1039+3256 & J10393137+3256263 & & T1 & 16.41$\pm$0.15 & [**15**]{}\
2MASS J1047+2124 & J10475385+2124234 & T7 & T6.5 & 15.82$\pm$0.06 & [**4**]{};1,32,44\
SDSS J1048+0111 & J10484281+0111580 & L1 & L4 & 12.92$\pm$0.02 & [**4**]{};45,46\
SDSS J1052+4422 & J10521350+4422559 & & T0.5$\pm$1 & 15.96$\pm$0.10 & [**4**]{};15\
Wolf 359 & J10562886+0700527 & M6 & & 7.09$\pm$0.02 & [**4**]{}\
2MASS J1104+1959 & J11040127+1959217 & L4 & & 14.38$\pm$0.03 & [**3**]{};14\
2MASS J1106+2754 & J11061197+2754225 & & T2.5 & 14.82$\pm$0.04 & [**33**]{}\
SDSS J1110+0116 & J11101001+0116130 & & T5.5 & 16.34$\pm$0.12 & [**11**]{};1,18\
2MASS J1114$-$2618 & J11145133$-$2618235 & & T7.5 & 15.86$\pm$0.08 & [**11**]{};1,12\
2MASS J1122$-$3512 & J11220826$-$3512363 & & T2 & 15.02$\pm$0.04 & [**1**]{};12\
2MASS J1124+3808 & J11240487+3808054 & M8.5 & & 12.71$\pm$0.02 & [**3**]{};14\
SDSS J1206+2813 & J12060248+2813293 & & T3 & 16.54$\pm$0.11 & [**15**]{}\
SDSS J1207+0244 & J12074717+0244249 & L8 & T0 & 15.58$\pm$0.07 & [**33**]{};1,45\
2MASS J1209$-$1004 & J12095613$-$1004008 & & T3 & 15.91$\pm$0.07 & [**3**]{};1,27\
SDSS J1214+6316 & J12144089+6316434 & & T3.5$\pm$1 & 16.59$\pm$0.12 & [**15**]{}\
2MASS J1217$-$0311 & J12171110$-$0311131 & T7 & T7.5 & 15.86$\pm$0.06 & [**11**]{};1,32,44\
2MASS J1221+0257 & J12212770+0257198 & L0 & & 13.17$\pm$0.02 & [**4**]{};47\
2MASS J1231+0847 & J12314753+0847331 & & T5.5 & 15.57$\pm$0.07 & [**3**]{};1\
2MASS J1237+6526 & J12373919+6526148 & T7 & T6.5 & 16.05$\pm$0.09 & [**48**]{};1,32,44\
SDSS J1254$-$0122 & J12545393$-$0122474 & T2 & T2 & 14.89$\pm$0.04 & [**3**]{};1,32,44\
2MASS J1324+6358 & J13243559+6358284 & & T2 & 15.60$\pm$0.07 & [**33**]{}\
SDSS J1346$-$0031 & J13464634$-$0031501 & T7 & T6.5 & 16.00$\pm$0.10 & [**11**]{};1,32,49\
SDSS J1358+3747 & J13585269+3747137 & & T4.5$\pm$1 & 16.46$\pm$0.09 & [**15**]{}\
2MASS J1404$-$3159 & J14044941$-$3159329 & & T2.5 & 15.60$\pm$0.06 & [**33**]{}\
LHS 2924 & J14284323+3310391 & M9 & & 11.99$\pm$0.02 & [**29**]{}\
SDSS J1435+1129 & J14355323+1129485 & & T2$\pm$1 & 17.14$\pm$0.23 & [**15**]{}\
2MASS J1439+1929 & J14392836+1929149 & L1 & & 12.76$\pm$0.02 & [**3**]{};31\
SDSS J1439+3042 & J14394595+3042212 & & T2.5 & 17.22$\pm$0.23 & [**15**]{}\
Gliese 570D & J14571496$-$2121477 & T7 & T7.5 & 15.32$\pm$0.05 & [**3**]{};1,32,50\
2MASS J1503+2525 & J15031961+2525196 & T6 & T5 & 13.94$\pm$0.02 & [**3**]{};1,32,51\
2MASS J1506+1321 & J15065441+1321060 & L3 & & 13.37$\pm$0.02 & [**28**]{};52\
2MASS J1507$-$1627 & J15074769$-$1627386 & L5 & L5.5 & 12.83$\pm$0.03 & [**28**]{};2,5,6\
SDSS J1511+0607 & J15111466+0607431 & & T0$\pm$2 & 16.02$\pm$0.08 & [**15**]{}\
2MASS J1526+2043 & J15261405+2043414 & L7 & & 15.59$\pm$0.06 & [**3**]{};6\
2MASS J1546$-$3325 & J15462718$-$3325111 & & T5.5 & 15.63$\pm$0.05 & [**4**]{};1,23\
2MASS J1615+1340 & J16150413+1340079 & & T6 & 16.35$\pm$0.09 & [**33**]{}\
SDSS J1624+0029 & J16241436+0029158 & & T6 & 15.49$\pm$0.05 & [**11**]{};1,53\
2MASS J1632+1904 & J16322911+1904407 & L8 & L8 & 15.87$\pm$0.07 & [**28**]{};1,31\
2MASS J1645$-$1319 & J16452211$-$1319516 & L1.5 & & 12.45$\pm$0.03 & [**4**]{};54\
VB 8 & J16553529$-$0823401 & M7 & & 9.78$\pm$0.03 & [**4**]{}\
SDSS J1750+4222 & J17502385+4222373 & & T2 & 16.47$\pm$0.10 & [**1**]{};2\
SDSS J1750+1759 & J17503293+1759042 & & T3.5 & 16.34$\pm$0.10 & [**3**]{};1,18\
2MASS J1754+1649 & J17545447+1649196 & & T5 & 15.81$\pm$0.07 & [**4**]{}\
SDSS J1758+4633 & J17580545+4633099 & & T6.5 & 16.15$\pm$0.09 & [**11**]{};1,2\
2MASS J1807+5015 & J18071593+5015316 & L1.5 & L1 & 12.93$\pm$0.02 & [**4**]{};14,21\
2MASS J1828$-$4849 & J18283572$-$4849046 & & T5.5 & 15.18$\pm$0.06 & [**3**]{};1\
2MASS J1901+4718 & J19010601+4718136 & & T5 & 15.86$\pm$0.07 & [**3**]{};1\
VB 10 & J19165762+0509021 & M8 & & 9.91$\pm$0.03 & [**3**]{}\
2MASS J2002$-$0521 & J20025073$-$0521524 & L6 & & 15.32$\pm$0.05 & [**4**]{};55\
SDSS J2028+0052 & J20282035+0052265 & L3 & & 14.30$\pm$0.04 & [**3**]{};45\
LHS 3566 & J20392378$-$2926335 & M6 & & 11.36$\pm$0.03 & [**3**]{}\
2MASS J2049$-$1944 & J20491972$-$1944324 & M7.5 & & 12.85$\pm$0.02 & [**3**]{}\
SDSS J2052$-$1609 & J20523515$-$1609308 & & T1$\pm$1 & 16.33$\pm$0.12 & [**4,15**]{}\
2MASS J2057$-$0252 & J20575409$-$0252302 & L1.5 & L1.5 & 13.12$\pm$0.02 & [**3**]{};14,46\
2MASS J2107$-$0307 & J21073169$-$0307337 & L0 & & 14.20$\pm$0.03 & [**3**]{};14\
SDSS J2124+0100 & J21241387+0059599 & & T5 & 16.03$\pm$0.07 & [**15**]{};1,2\
2MASS J2132+1341 & J21321145+1341584 & L6 & & 15.80$\pm$0.06 & [**59**]{};55\
2MASS J2139+0220 & J21392676+0220226 & & T1.5 & 15.26$\pm$0.05 & [**1**]{};56\
HN Peg B & J21442847+1446077 & & T2.5 & 15.86$\pm$0.03 & [**10**]{}\
2MASS J2151$-$2441 & J21512543$-$2441000 & L3 & & 15.75$\pm$0.08 & [**56**]{};55,57\
2MASS J2151$-$4853 & J21513839$-$4853542 & & T4 & 15.73$\pm$0.07 & [**4**]{};1,58\
2MASS J2154+5942 & J21543318+5942187 & & T6 & 15.66$\pm$0.07 & [**33**]{}\
2MASS J2212+1641 & J22120345+1641093 & M5 & & 11.43$\pm$0.03 & [**3**]{}\
2MASS J2228$-$4310 & J22282889$-$4310262 & & T6 & 15.66$\pm$0.07 & [**3**]{};1,34\
2MASS J2234+2359 & J22341394+2359559 & M9.5 & & 13.15$\pm$0.02 & [**3**]{}\
SDSS J2249+0044 & J22495345+0044046 & L3 & L5$\pm$1.5 & 16.59$\pm$0.13 & [**4**]{};2,18,45\
2MASS J2254+3123 & J22541892+3123498 & & T4 & 15.26$\pm$0.05 & [**3**]{};1,23\
2MASS J2331$-$4718 & J23312378$-$4718274 & & T5 & 15.66$\pm$0.07 & [**3**]{};1\
2MASS J2339+1352 & J23391025+1352284 & & T5 & 16.24$\pm$0.11 & [**1**]{};23\
LEHPM 1$-$6333 & J23515012$-$2537386 & M8 & M8 & 12.47$\pm$0.03 & [**4**]{};36,40,55\
LEHPM 1$-$6443 & J23540928$-$3316266 & M8.5 & M8 & 13.05$\pm$0.02 & [**4**]{};36,40\
2MASS J2356$-$1553 & J23565477$-$1553111 & & T5.5 & 15.82$\pm$0.06 & [**1**]{};23\
[lccc]{} Spectral Type & M8.5$\pm$0.3 & T5$\pm$0.9 &\
${J}$ (mag) & 13.25$\pm$0.03 & 16.4$\pm$0.4 & 3.1$\pm$0.4\
${H}$ (mag) & 12.61$\pm$0.03 & 16.4$\pm$0.5 & 3.8$\pm$0.5\
${K}$ (mag) & 12.13$\pm$0.03 & 16.5$\pm$0.6 & 4.3$\pm$0.6\
[[$\log_{10}{L_{bol}/L_{\sun}}$]{}]{} & -3.48$\pm$0.10 & -5.0$\pm$0.3 & 1.5$\pm$0.3\
$\mu$ ($\arcsec$ yr$^{-1}$) & 0.562$\pm$0.005 & &\
$\phi$ ($\degr$) & 205.9$\pm$0.5 & &\
$d$ (pc) & 25$\pm$3 & &\
[[$V_{tan}$]{}]{} ([[km s$^{-1}$]{}]{}) & 67$\pm$8 & &\
$\rho$ (AU) & $<$8.3 ($<$0$\farcs$33) & &\
Mass (M$_{\sun}$) at 1 Gyr & 0.081 & 0.035 & 0.44\
Mass (M$_{\sun}$) at 5 Gyr & 0.086 & 0.068 & 0.79\
Mass (M$_{\sun}$) at 10 Gyr & 0.086 & 0.074 & 0.86\
[^1]: A current list is maintained by N. Siegler at <http://www.vlmbinaries.org>.
[^2]: Mauna Kea Observatory (MKO) photometric system; @sim02 [@tok02].
[^3]: See also @me0423 [@mehst2; @me1126; @rei2252; @meltbinary; @loo07; @sie07]; and @loo08.
[^4]: See @mewide3 [@meclass2; @me1520; @cru04; @sie05; @metgrav; @me2200; @chi06; @mce06; @rei2252; @mehd3651; @meltbinary; @lie07; @loo07]; and [@luh07]. These data are available at <http://www.browndwarfs.org/spexprism>.
[^5]: A current list of L and T dwarfs with their published optical and near-infrared spectral types is maintained by C. Gelino, J. D.Kirkpatrick and A. Burgasser at <http://www.dwarfarchives.org>.
[^6]: In the near-infrared, foreground emission generally dominates noise contributions. However, given the broad range of observing conditions in which the [[2MASS J0320$-$0446]{}]{} and template data were taken, we chose not to include this term in our $\sigma^2$ statistic.
[^7]: Liverpool-Edinburgh High Proper Motion (LEHPM) Catalog of @por04.
[^8]: The $J$-band portion of the spectrum of SCR 1845$-$6357A shown here is slightly reduced relative to the $H$- and $K_s$-band spectra as shown to Figure 2 in @kas07. The relative flux calibration between spectral orders applied in that study did not account for missing data over 1.33–1.50 $\micron$, slightly inflating the flux levels in the $J$-band. A recalibration of this spectrum was made by scaling each order by a constant factor to match the SpeX prism spectrum of 2MASS J1124+3808, which has a similar $J-K_s$ color (1.14$\pm$0.03 versus 1.06$\pm$0.03 for SCR 1845$-$6357 from @kas07) and optical spectral type (M8.5). Such recalibration is not necessary for the SCR 1845$-$6357B spectrum due to the strong 1.35 $\micron$ [[CH$_4$]{}]{} and 1.4 $\micron$ [[H$_2$O]{}]{} bands in this source. The recalibration of the SCR 1845$-$6357A $J$-band spectrum does not affect the analysis presented here, which depends solely on the relative $H$-band scaling of the component spectra.
|
---
bibliography:
- '../../papers/references.bib'
---
Astro2020 Science White Paper
The Role of Machine Learning in the Next Decade of Cosmology
**Thematic Areas:** $\square$ Planetary Systems $\square$ Star and Planet Formation $\square$ Formation and Evolution of Compact Objects ${\rlap{$\square$}{\raisebox{2pt}{\large\hspace{1pt}{\ding{51}}}}\hspace{-2.5pt}}$ Cosmology and Fundamental Physics $\square$ Stars and Stellar Evolution $\square$ Resolved Stellar Populations and their Environments $\square$ Galaxy Evolution $\square$ Multi-Messenger Astronomy and Astrophysics
**Principal Author:**
Name: Michelle Ntampaka Institution: Harvard Data Science Initiative\
Center for Astrophysics $|$ Harvard & Smithsonian Email: [email protected] **Co-authors:** Camille Avestruz, Steven Boada, João Caldeira, Jessi Cisewski-Kehe, Rosanne DiStefano, Cora Dvorkin, August E. Evrard, Arya Farahi, Doug Finkbeiner, Shy Genel, Alyssa Goodman, Andy Goulding, Shirley Ho, Arthur Kosowsky, Paul La Plante, François Lanusse, Michelle Lochner, Rachel Mandelbaum, Daisuke Nagai, Jeffrey A. Newman, Brian Nord, J. E. G. Peek, Austin Peel, Barnabás Póczos, Markus Michael Rau, Aneta Siemiginowska, Dougal J. Sutherland, Hy Trac, Benjamin Wandelt
**Abstract:** In recent years, machine learning (ML) methods have remarkably improved how cosmologists can interpret data. The next decade will bring new opportunities for data-driven cosmological discovery, but will also present new challenges for adopting ML methodologies and understanding the results. ML could transform our field, but this transformation will require the astronomy community to both foster and promote interdisciplinary research endeavors.
[The Role of Machine Learning in the Next Decade of Cosmology]{}\
Machine learning permeates our daily lives performing tasks from identifying the people in a photograph to suggesting the next big purchase but will it change the way we do research? The last decade has seen a remarkable rise in interdisciplinary machine learning (ML)-based astronomy research, offering enticing improvements in the ways we can interpret data. The next decade will see a continued rise in data-driven discovery as methods improve and data volumes grow, but realizing the full potential of ML even in an era of unprecedented data volumes presents challenges.
What Are Data Science and Machine Learning? {#sec:datasci}
===========================================
Data science is the study, application, and often the *art* of creating and using sophisticated algorithms and cutting-edge data analysis techniques to extract information from data. It includes the fields of statistics, machine learning, applied mathematics, and computer science. Data science is an inherently interdisciplinary endeavor that reaches across departmental lines to produce new and innovative ways to interpret simulated data and astronomical observations. When used well, data science provides methods for extracting more information from data sets than ever before, reducing bias and scatter, identifying interesting outliers, and inexpensively generating simulated data. When used *very* well, it guides our physical interpretation of observations and can lead to great discoveries.
While it’s important to define what data science is, it’s also important to define what it is *not*. Data science is not the study of how to store or disseminate data. While data storage and dissemination are important issues facing the astronomy community in the era of large surveys such as the Large Synoptic Survey Telescope (LSST), addressing these issues is not data science. Nor is data science equivalent to data analysis. When we refer to expanded uses of data science in cosmology, we do not include solidly established and easy-to-implement foundational tools such as chi-square analysis and linear regression. Data science also is not “big data,” though the two often work side-by-side.
Data science encompasses a broad range of sophisticated data-analysis methodologies, and machine learning (ML) is just one tool in the toolbox. Machine learning research explores the development and application of algorithms that find patterns in data. In the context of astronomy, ML algorithms can be used to address a broad range of tasks including: describing complicated relationships, identifying data clusters and data outliers, reducing scatter by using complex or subtle signals, generating simulated data, classifying observations, addressing sparse data, and exploring data sets to understand the physical underpinning. Machine learning is gaining traction within the astronomy community, and compelling successful applications of ML indicate that it has the potential to be transformative in the upcoming decade.
Machine Learning Successes in Cosmology {#sec:successes}
=======================================
Recent successes illustrate the potential for sophisticated machine learning data-analysis tools to make significant strides in cosmology. These successes include the following important results:
1. Galaxy clusters are sensitive to the underlying cosmological model, and low-scatter cluster mass proxies are one essential ingredient in using these objects to constrain parameters. ML has been shown to significantly reduce scatter in cluster mass estimates compared to more traditional methods [@2015ApJ...803...50N; @2018arXiv181008430A; @2019arXiv190205950H].
2. Weak lensing maps can shed light on the fundamental nature of gravity and cosmic acceleration. ML has been used with such maps to discriminate between standard and modified gravity models that generate statistically similar observations [@2018arXiv181011030P]. Non-Gaussianities in weak lensing maps can encode cosmological information, but these are hard to measure or parameterize. ML has been shown to tighten parameter constraints by a factor of five or more by harnessing these non-Gaussianities [@2018PhRvD..97j3515G; @2018arXiv180605995R].
3. Next-generation cosmic microwave background (CMB) experiments will have increased sensitivity, enabling improved constraints on fundamental physics parameters. Achieving optimal constraints requires high signal-to-noise extraction of the projected gravitational potential from the CMB maps. ML techniques have been shown to provide competitive methods for this extraction, and are expected to excel in capturing hard-to-model non-Gaussian foreground and noise contributions [@2018arXiv181001483C].
4. $N$-body simulations are an effective approach to predicting structure formation of the universe, but are computationally expensive. ML has been used to predict structure formation of the universe, generating a full 3D $N$-body-like simulation with positions and velocities in $30ms$ [@2018arXiv181106533H]. This method outperforms traditional fast analytical approximation and accurately extrapolates far beyond its training data.
5. Estimating cosmological parameters from the large-scale structure is traditionally done by calculating summary statistics of the observed large-scale structure traced by galaxies and then compared to the analytical theory. ML can be used to estimate cosmological parameters directly from the large-scale structure field and find more stringent constraints on the cosmological parameters [@2016arXiv160905796R].
6. Observations of the Epoch of Reionization can provide information about the earliest luminous sources. ML can classify the types of sources driving reionization [@2018arXiv180703317H] and measure the duration of reionization to within 10%, given a semi-analytic model and a strong prior on the midpoint of reionization [@2018arXiv181008211L].
7. Topological data analysis (TDA) is an ML and statistical method for summarizing the shape of data. TDA has been useful for discriminating dark energy models on simulated data [@van2011alpha], isolating structures of the cosmic web [@sousbie2011persistent; @libeskind2018tracing], and defining new types of structures in the cosmic web such as filament loops [@xu2018finding]. TDA may also help constrain the sum of neutrino masses [@xu2018finding].
8. Supernova classification is a critical step in obtaining cosmological constraints from type Ia supernovae in photometric surveys such as LSST. ML has proven to be a powerful tool [e.g., @Lochner2016:1603.00882v3] and has been successfully applied to the current largest public supernova dataset [@Narayan2018:1801.07323v1]. The public has become heavily involved in developing new classification techniques [@team2018:1810.00001v1; @Malz2018:1809.11145v1].
9. Strong lensing probes cosmic structure along lines of sight. ML was the most effective method at correctly identifying strong lensing arcs in a recent data challenge, outperforming humans at this classification task [@2018MNRAS.473.3895L]. ML makes the analysis of strong lensing systems 10 million times faster than the state-of-the-art method [@2017Natur.548..555H; @2017ApJ...850L...7P].
ML will not displace standard statistical reasoning for well-modeled phenomena. However, there are many cases where our current parametric models are inadequate to fully describe a physical system. These ML successes in cosmology imply that there is great potential for data-driven discovery, particularly as data sets grow and become more complex.
Challenges for the Next Decade {#sec:challenges}
==============================
ML represents the next step in automation, driven by both rapidly increasing data volumes and the desire to prioritize human attention on tasks that require our insight and ingenuity. There is no doubt that ML techniques will become more powerful and widespread in the 2020s, transforming our ability to address previously intractable problems.
ML has demonstrated its potential to accelerate discovery in astrophysics, but challenges to more widespread adoption remain. New tools come with new failure modes, and ML poses the temptation to choose expediency over understanding. A common complaint about ML methods is that they are black boxes that cannot lead to physical understanding, but this need not be the case. Though it is difficult to understand the inner workings of a complex ML model, in many cases it is not impossible. There are ways to peer inside the box and to gain physical understanding from complex models. For example, Google’s DeepDream project [@deepdream] originated as a way to visualize inputs that maximize activation in various layers of a neural network, and recent applications to cosmology indicate that the method can be used to gain physical understanding of astronomical systems [e.g., @2018arXiv181007703N]. Other recent developments to improve ML interpretability include saliency maps [@2013arXiv1312.6034S] that reveal which parts of an input most influence the output, and the deep k-nearest neighbors approach [@2018arXiv180304765P] that shows which training examples have the most influence on a specific outcome. ML interpretability is an active area of research, and we expect further improvements in the quality and diversity of interpretation techniques in the next decade.
At the intersection of ML and cosmology lies a unique opportunity for the benefit of both fields. ML is likely to accelerate discovery in cosmology through multiple applications and modalities (classification, regression, reinforcement learning). Cosmology, in return, provides new tasks and challenges for ML researchers and well-understood data sets for testing ML methods. The challenges provided by cosmology open opportunities for breakthroughs in the fundamental understanding of ML.
More advanced analyses of cosmological data place stringent requirements on the interpretability of results. This represents a key hurdle for applying ML to cosmology: the assessment of uncertainty and the removal of bias. Integrating traditional statistical methods with modern ML models may provide a solution, but this will require cross-disciplinary collaboration among statisticians, ML researchers, and cosmologists. During the 2020s, it is plausible that we could train, characterize, and use ML with the same rigor that we bring to more conventional statistical analysis. This is a significant shift in how we approach our research, and supporting this shift will require the community’s investment in education, interdisciplinary research endeavors, and the development and transfer of methodologies from the computer science community.
Opportunities in the 2020 and Beyond {#sec:opportunities}
====================================
The assertion that “astronomy is entering the era of big data” has become cliché. And yet, we cannot help but note that upcoming data sets, both big and small, will provide rich opportunities to use machine learning for teasing out complex correlations. Here, we provide a few examples of those opportunities.\
**Big Data Opportunities With LSST:** The LSST survey [@2009arXiv0912.0201L] will provide the optical astronomical community with an unprecedented data rate. It will cover nearly the entire visible southern sky roughly every three days for a decade, providing $\sim$1,000 exposures total at each position (split across 6 passbands). With 500 petabytes of images, and a database including tens of trillions of observations of tens of billions of objects [@2008arXiv0805.2366I], LSST’s discovery potential will be enormous but standard analysis methods will not enable the community to unlock the full potential of LSST. Its high source density, as well as the transient nature of some phenomena (e.g., asteroids and supernovae), will present a new set of challenges related to source identification and classification in this colossal dataset. For example, LSST expects to identify, and subsequently classify, 10 million rapid-response transient alerts on any given night. The continued development and future implementation of carefully designed ML algorithms at both the image processing [e.g., @2018PASJ...70S..37G; @2018arXiv180710406D; @2018MNRAS.479..415A] and catalog [e.g., @Narayan2018:1801.07323v1; @Malz2018:1809.11145v1] levels have the potential for producing significant advances in our ability to efficiently extract scientifically useful information (e.g., classification, distance, morphology, and mass) from the LSST data. However, these ML methodologies will require further exploration to fully understand their feasibility and general applicability to LSST.
**Big Data Opportunities in Radio Astronomy:** The Hydrogen Epoch of Reionization Array (HERA), a radio interferometer seeking to provide the first detection of the 21 cm power spectrum from the Epoch of Reionization, is projected to produce 50 TB of data per night of observation when completed in late 2019. ML can identify radio frequency interference present in data from HERA faster and more reliably than traditional algorithms [@2019arXiv190208244K]. ML is also well-poised to replace other key aspects of the data analysis and reduction pipeline, such as the calibration of antennae and automatic identification of malfunctioning equipment. Further into the decade, the Square Kilometre Array (SKA) will feature data volumes even larger than HERA, with ML representing a viable path for analyzing and reducing these data in real time.
**Pipeline Optimization Opportunities:** ML has the potential to have a dramatic impact on the efficiency of cosmological experiments. For example, real-bogus [@2013MNRAS.435.1047B] is an ML system for determining whether a transient detected in photometric variation is a true variable object or simply an artifact. Similarly, the Dark Energy Camera Plane Survey [@2018ApJS..234...39S] uses a simple deep neural network to find images that have nebulosity, and, thus, require a separate processing algorithm. Furthermore, ML has the potential to simplify and accelerate the building of statistical inference pipelines in the context of full forward models through Likelihood-Free Inference [@2018MNRAS.477.2874A] and the automated extraction of informative features from data sets [@2018PhRvD..97h3004C]. The cosmological explorations of the 2020s will require excellent quality control while simultaneously handling unprecedented volumes of data, and ML is a strong option for pipeline optimization.
**Low Signal Opportunities With *eROSITA*:** While the most obvious targets of opportunity are observations with unprecedented data volumes, fully harnessing our smaller data sets provides rich opportunities for applying ML methodology as well. For example, the upcoming *eROSITA* mission is estimated to find more than 90,000 galaxy clusters with masses $> 10^{13.7}h^{-1}\,M_\odot$ [@2012MNRAS.422...44P]. While the mission will detect clusters out to $z\sim 2$, a significant fraction of these cluster observations will be in the low-photon regime of 100 photons or fewer [@2018MNRAS.481..613P]. Fully utilizing this sample will require developing techniques that provide low scatter mass proxies in the low signal regime, and ML is one viable option for this.
**Archival Data Opportunities:** The potential for ML to help make great scientific strides is not limited to these upcoming data sets. Research based on *Hubble* archival data, for example, outnumbers those on new observations [@hubble], showing that there is a vast, untapped potential even in the community’s archival data.\
The astronomy community already makes a significant investment in state-of-the-art instrumentation, software development to support data reduction [e.g., @astropy:2013; @2015ascl.soft10007C; @JHazelton2017; @2017ascl.soft03004M; @astropy:2018], and data management [e.g., @2015AAS...22542204S; @2017AAS...22912801D]. There remains a strong need for this community to invest in the interdisciplinary development and application of cutting-edge ML techniques to interpret our rich and complex data, and help propel us into the next decade.
We encourage the astronomy community to invest in education and interdisciplinary research efforts that will transfer knowledge and methods from the ML research community to our field. We also encourage efforts to build communities of practice for ML-based studies, especially those that profitably join simulated and observational survey data. In the 2020s and beyond, these communities could cluster around discipline-focused hubs, or science gateways[^1], offering researchers access to open-source software, relevant data products, and common analysis workflows.
Summary {#sec:summary}
=======
Recent applications of machine learning techniques to cosmological questions have made remarkable improvements in the way we interpret our data, and these compelling successes imply that ML has the potential to be transformative to our field. This transformation will require the astronomy community to cultivate and support research endeavors that cross traditional discipline boundaries, but the payoff has the potential to be steep. Machine learning will give cosmologists access to the data analysis methods that we need to fully utilize our rich data sets and make great scientific leaps forward over the next decade.
References {#references .unnumbered}
==========
[^1]: https://sciencegateways.org/
|
---
abstract: 'We consider the Ginzburg-Landau energy for a type-I superconductor in the shape of an infinite three-dimensional slab, with two-dimensional periodicity, with an applied magnetic field which is uniform and perpendicular to the slab. We determine the optimal scaling law of the minimal energy in terms of the parameters of the problem, when the applied magnetic field is sufficiently small and the sample sufficiently thick. This optimal scaling law is proven via ansatz-free lower bounds and an explicit branching construction which refines further and further as one approaches the surface of the sample. Two different regimes appear, with different scaling exponents. In the first regime, the branching leads to an almost uniform magnetic field pattern on the boundary; in the second one the inhomogeneity survives up to the boundary.'
author:
- 'Sergio Conti, Felix Otto, and Sylvia Serfaty'
bibliography:
- 'COS.bib'
title: 'Branched microstructures in the Ginzburg-Landau model of type-I superconductors'
---
Introduction
============
Superconductivity, discovered in 1911 by Kamerlingh Onnes, is a phenomenon happening at low temperature in certain materials which loose their resistivity and expel an applied magnetic field. The latter is called the Meissner effect. More precisely, when the applied magnetic field is small, the sample is everywhere superconducting and completely expels the magnetic field, while when the magnetic field becomes larger, it partially penetrates the sample via regions of normal phase where the material is not superconducting. If the magnetic field is further increased, then superconductivity is completely destroyed and the sample behaves like a normal conductor.
The standard model for describing superconductivity is the Ginzburg-Landau functional, which was introduced in the 1950’s by Landau and Ginzburg on a phenomenological basis. It was later justified based on microscopic quantum mechanical principles via the Bardeen-Cooper-Schrieffer (BCS) theory, which explains superconductivity through the appearance of “Cooper pairs" of superconducting electrons. The Ginzburg-Landau model is a formal limit of the BCS model, and this derivation was accomplished rigorously in the recent work [@FrankSeiringer2012].
The Ginzburg-Landau model describes the state of the sample via a complex-valued order parameter $u$. The squared modulus of $u$ represents the local density of the Cooper pairs of “superconducting electrons". In other words, $\rho:=|u|^2$ indicates whether one is in the normal phase $\rho\simeq 0$, or in the superconducting phase $\rho\simeq 1$. The transition between $0$ and $1$ happens within relatively thin interfacial layers (or walls). The order parameter $u$ is coupled with the magnetic vector potential $A$, which yields the magnetic field $B:=\nab \times A $ induced in the sample. The Meissner effect can be roughly understood as the fact that the magnetic field $B$ can only exist in the normal phase $\rho=0$, or in other words $$\label{apprxmeissner} \rho B\approx 0.$$ Another important property of superconductors is flux quantization. If we consider a closed circuit well inside the superconducting region (on a scale set by the penetration length $\lambda$), the contour integral of $A$ has to be an integer multiple of $2\pi$, in units of $\hbar/2e$. This arises because $\nabla u$ is very close to $iAu$ and $|u|$ is very close to 1. Correspondingly the flux of the magnetic field $B$ through any surface with such boundary is quantized, in the sense that it has to be an integer multiple of $2\pi$ (again, and for the rest of this paper, in units of $\hbar/2e$).
The Ginzburg-Landau functional in a three-dimensional region $Q_{L,T}:= (0,L)^2 \times (0,T)$ can be written, after appropriate non-dimensionalization, as $$\label{eqdefmodelintro}
{E_\mathrm{GL}}[u,A]:=\int_{Q_{L,T}} \left[ |\nabla_A u |^2
+ \frac{\kappa^2}{2} (1-|u|^2)^2\right] dx
+ \int_{Q_L\times \R} |\nabla\times A-{B_{\mathrm{ext}}}|^2 dx.$$ Here $\nab_A := \nab -i A $ denotes the covariant gradient, ${B_{\mathrm{ext}}}:={b_{\mathrm{ext}}}e_3$ is the applied magnetic field, which is assumed to be uniform and vertical. The constant $\kappa>0$, usually called the Ginzburg-Landau parameter, is the ratio of the “penetration length" (of the magnetic field in the sample) $\lambda$ and the “coherence length" $\xi$. For a general presentation of superconductivity and the Ginzburg-Landau model, we refer to the standard physics textbooks, such as [@Tinkham1996; @DeGennes; @SST]. For further mathematical reference on the Ginzburg-Landau functional, one can see for example [@SandierSerfaty2007].
Superconductors are usually classified in type-I and type-II superconductors, according to whether $\kappa<1/\sqrt{2}$ or $\kappa>1/\sqrt{2}$. In type-II superconductors, there is an intermediate regime, for low applied magnetic fields, where the penetration of the magnetic field happens along very thin vortex filaments, carrying an integer flux, and around which the sample is normal — this is called the mixed phase. The size of the vortices is only limited by the flux quantization condition, and indeed in most situations each of them carries exactly one quantum of flux. This is the regime studied in details in [@SandierSerfaty2007] in dimension 2 and in [@BJOS] in dimension 3. By contrast, in type-I superconductors the ratio of characteristic lengthscales $\kappa$ does not allow these vortex-filaments to form and larger regions of normal phase appear, separated interfaces (called walls) from the superconducting phase. Each normal region in this case carries a magnetic flux much larger than the flux quantum. We will be interested only in the latter situation, and we will assume that $\kappa$ is small enough, and also that the applied field ${b_{\mathrm{ext}}}$ is much smaller than the critical field, which in the present units is $\kappa\sqrt2$.
The pattern of the normal phase arises from the competition of different effects. On the one hand, the interfacial energy favours a coarse structure in the interior of the sample. On the other hand, the magnetic energy outside the sample favours a fine-scale mixture close to the interface. Therefore, the optimal pattern is expected to branch, as predicted by Landau back in 1938 [@Landau38; @Landau43]. This permits to combine a coarse pattern in the interior with an induced magnetic field $\nab \times A$ almost aligned with ${B_{\mathrm{ext}}}$ at the surface, see Figure \[figbranching\] for a sketch. Experimentally, this is manifested by complex patterns observed at the surface of the sample [@Prozorov2007; @ProzorovGiannetta2005; @ProzorovHobergCanfield2008; @ProzorovHoberg2009]. This phenomenon of domain branching occurs also in other areas of materials science, as for example ferromagnetic materials, where the magnetization pattern is constrained to oscillate between two opposite vectors [@Lifshitz44; @Hubert67], and in shape-memory alloys, where the strain can oscillate between finitely many values, corresponding to the different martensitic variants. The average behavior of these branched patterns can be characterized via scaling laws: one determines how the minimal energy per cross-sectional area scales with the various parameters of the system, and shows that the optimal scaling of the energy can be achieved with branching-type patterns. This is usually rigorously established by showing ansatz-free lower bounds and complementing them with the construction of explicit branching patterns whose energy is estimated to have the same order in the parameters as the lower bound. This was achieved in martensites in [@KohnMuller92; @KohnMuller94; @Conti00; @CapellaOtto2009; @CapellaOtto2012; @Zwicknagl2014; @ChanConti2015] and in magnetic materials in [@CK98; @CKO99; @ViehmannOtto2010; @Viehmanndiss].
In the case of type-I superconductors, a similar program was carried out in [@ChoksiKohnOtto2004; @ChoksiContiKohnOtto2008] for a simplified model: it is a “sharp-interface" version of the Ginzburg-Landau functional, where the order parameter $u$ is only represented via its modulus $\rho$, which in turn is only allowed to take values in $\{0,1\}$; at the same time the kinetic energy is replaced by a constant times the $BV$ norm of $\rho$, i.e., the perimeter of the set where $\rho=1$, see Section \[secsharpinterface\] below for details. This resulted in a full characterization of the phase diagram at the level of energy scaling, and in particular led to the discovery of a new phase for very small applied fields, see Figure \[figbranching\] for a sketch.
![Sketch of the flux patterns. Left: the regime with uniform flux on the surface, with energy proportional to ${b_{\mathrm{ext}}}^{2/3} \kappa^{2/3} T^{1/3}L^2$, which is the optimal scaling if ${b_{\mathrm{ext}}}\ge \kappa^{5/7}/T^{2/7}$. Right: the regime with flux concentration on the surface, with energy proportional to ${b_{\mathrm{ext}}}\kappa^{3/7} T^{3/7} L^2$, which is the optimal scaling if ${b_{\mathrm{ext}}}\le \kappa^{5/7}/T^{2/7}$.[]{data-label="figbranching"}](fig2-crop.pdf "fig:"){height="5cm"} ![Sketch of the flux patterns. Left: the regime with uniform flux on the surface, with energy proportional to ${b_{\mathrm{ext}}}^{2/3} \kappa^{2/3} T^{1/3}L^2$, which is the optimal scaling if ${b_{\mathrm{ext}}}\ge \kappa^{5/7}/T^{2/7}$. Right: the regime with flux concentration on the surface, with energy proportional to ${b_{\mathrm{ext}}}\kappa^{3/7} T^{3/7} L^2$, which is the optimal scaling if ${b_{\mathrm{ext}}}\le \kappa^{5/7}/T^{2/7}$.[]{data-label="figbranching"}](fig3-crop.pdf "fig:"){height="5cm"}
We study here the full Ginzburg-Landau model, as given in (\[eqdefmodelintro\]), and determine the scaling of the minimum energy per cross-sectional area in dependence of the problem parameters $\kappa$, ${b_{\mathrm{ext}}}$ and $T$. We prove that the energy scaling is characterized by the same two regimes which had been found for the sharp-interface functional. In fact, some of the ideas of proof of [@ChoksiKohnOtto2004; @ChoksiContiKohnOtto2008] carry over to the full model, once an appropriate splitting of the energy, involving a “Bogomoln’yi operator" has been performed (cf. Section \[secprelim\]). The treatment of the lower bound contains several additional difficulties, mainly due to the fact that the Meissner condition (\[apprxmeissner\]) is only true “on average”, in the sense that an appropriate weak norm is small. The constructions in the upper bound, at the same time, need to take into account the quantization condition, locally in each tube, and to construct an order parameter with diffuse interfaces. The relationship with the simplified model is discussed in more detail in Section \[secsharpinterface\].
We work in an infinite periodic slab geometry, which is enough to understand the main surface branching features that we are interested in, hence the choice of working in the domain $Q_{L,T}:=Q_L\times(0,T)$ with $Q_L:=(0,L)^2$, with $L$ very large, and with horizontally periodic boundary conditions. We recall that the Ginzburg-Landau functional is invariant under gauge-transformations: two configurations $(u,A) $ and $(\hat u,\hat A)$ are called gauge-equivalent if there exists $\Phi \in H^2_{\mathrm{loc}}$ such that $$\left\{\begin{array}
{l}
u( x)= \hat u(x) e^{i \Phi(x)}\\
A(x) = \hat A(x)+ \nabla \Phi(x) \, .\end{array} \right.$$ The physical quantities are gauge-invariant (i.e., invariant under a gauge-transformation). This includes the magnetic field $\nabla\times A$, the energy, the density $\rho= |u|^2$ and the superconducting current defined as $$\label{defj}
j_A:=\mathrm{Re} (-i\bar{u} \nabla_A u)=
\frac12 \left(- i\bar{u} \nab_A u + iu \overline{\nab_A {u}}\,\right)\,.$$ We will work in the space $H^1_{\mathrm{per}}(Q_{L,T}\times \R;{\mathbb{C}}\times\R^3) $ defined to be the set of $ (u,A)\in H^1_{\mathrm{loc}}$ such that ${E_\mathrm{GL}}[u,A]$ is finite and for every $\vec{k}\in \mathbb{Z}^2\times\{0\}$, $(u( \cdot + L \vec{k}), A (\cdot + L \vec{k}))$ is gauge-equivalent to $(u,A)$ (this periodic setting was rigorously formalized in [@Ode67], see also [@Du99]). All gauge-invariant quantities, such as $\rho$, $j_A$ and $B$, are then $Q_L$-periodic. We stress that periodicity is only assumed in the first two variables. We will also call such pairs $(u,A)\in H^1_{\mathrm{per}}$ admissible.
Our main result, characterizing the energy in the regime of small applied fields ${b_{\mathrm{ext}}}$ and large and thick enough samples, is
For any $\kappa,{b_{\mathrm{ext}}},L,T>0$ such that $${b_{\mathrm{ext}}}L^2\in 2\pi\Z, \quad 8{b_{\mathrm{ext}}}\le \kappa \le \frac12\,,\quad
\kappa T\ge 1$$ and $L$ is sufficiently large (in the sense of ), one has $$\label{scalingresult}
\min_{H^1_{\mathrm{per}}} {E_\mathrm{GL}}[u,A]-(\kappa\sqrt2 {b_{\mathrm{ext}}}-{b_{\mathrm{ext}}}^2)L^2T\sim \min \left\{{b_{\mathrm{ext}}}\kappa^{3/7} T^{3/7} L^2,
{b_{\mathrm{ext}}}^{2/3} \kappa^{2/3} T^{1/3}L^2\right\} \,.$$
The result will follow from Theorem \[theolowerbGL\] and Theorem \[theoupperbGL\] below using Lemma \[lemmaseparatebulk\] to separate the bulk contribution. The notation $a\sim b$ means that a universal constant $c>0$ exists, such that $c^{-1}a\le b\le ca$.
The scaling result is in the end the same as in [@ChoksiKohnOtto2004; @ChoksiContiKohnOtto2008], after some rescaling of the lengths and magnetic field intensity. As in those works, the minimum in the right-hand side reflects the fact that two types of construction are needed, one corresponding to the regime where ${b_{\mathrm{ext}}}\kappa^{3/7} T^{3/7} L^2\ll {b_{\mathrm{ext}}}^{2/3} \kappa^{2/3} T^{1/3}L^2$ and one corresponding to the opposite case. In both cases, the construction that gives the optimal scaling law is that of a self-similar branching tree of normal region (where the magnetic field penetrates) which is symmetric with respect to the $x_3=T/2$ plane, and refines further and further as $x_3 $ approaches $0$ and $T$. The optimal “opening ratio" of the self-similar tree depends on the parameters of the problem and has to be chosen differently in the two regimes above.
A finer analysis in the asymptotic regime ${b_{\mathrm{ext}}}\to0$, including $\Gamma$-convergence to a reduced model with energy concentrated on lines, will be discussed in [@ContiGoldmanOttoSerfaty].
This paper is organized as follows. In Section \[secprelim\] we show how the bulk contribution energy can be algebraically separated, via the Bogomoln’yi operator, and we define in the functional $E$ on which we shall focus for most of the paper. In Section \[seclowerb\] we prove the lower bound, first for the sharp-interface version of the problem, and then for $E$. In Section \[secupperb\] we prove the corresponding upper bounds; again we first work on the sharp-interface problem and then extend the upper bound to the full Ginzburg-Landau functional.
Preliminaries {#secprelim}
=============
Preliminary on notation
-----------------------
We use a prime to indicate the first two components of a vector in $\R^3$, and identify $\R^2$ with $\R^2\times\{0\}\subset\R^3$. Precisely, for $a\in \R^3$ we write $a'=(a_1,a_2,0)\in\R^2\subset\R^3$; given two vectors $a,b\in\R^3$ we write briefly $a'\times b'=(a\times b)_3 =
(a'\times b')_3$.
We shall denote sections of $Q_{L,T}$ by $Q(z)=Q_L\times\{z\}$, for integrals over $Q(z)$ we write $dx'$ instead of $d{\mathcal{H}}^2(x')$. In the entire paper we let $\kappa$, ${b_{\mathrm{ext}}}$, $L$, $T$ be positive parameters which obey $$\label{eqparameters}
{b_{\mathrm{ext}}}\le \frac18 \kappa\,, \hskip1cm \kappa \le \frac12\,,\hskip1cm\text{ and}\hskip1cm
\kappa T\ge 1\,.$$ In many parts we shall additionally require the quantization condition ${b_{\mathrm{ext}}}L^2\in 2\pi\Z$.
By $a{\lesssim}b$ or $b{\gtrsim}a$ we mean that a universal constant $c>0$ exists (which may change from line to line but does not depend on the parameters of the problem) such that $a\le c b$. By $a\sim b$ we mean $a{\lesssim}b$ and $b{\lesssim}a$ (with two different implicit constants).
We denote by $H^{1/2}(Q_L)$ the space of traces of $Q_L$-periodic functions $u\in H^1_{{\mathrm{loc}}}(Q_L\times(-\infty,0))$ with $\nabla u\in L^2$, and use the homogeneous norm $\|u\|_{H^{1/2}(Q_L)}:=\inf
\{\|\nabla u\|_{L^{2}(Q_L\times(-\infty,0))}\}$, where the infimum is taken over all possible extensions. We denote by $H^{-1/2}(Q_L)$ its dual space.
Separating the bulk energy
--------------------------
Our first step is to subtract the bulk contribution to ${E_\mathrm{GL}}$, which will lead us to the definition of the energy $E$, which only contains the contributions of the microstructure. The precise formula for $E$ is given in (\[eqdefE\]) below. This is done as in [@ss] via an algebraic relation which involves the operator ${\mathcal{D}}_A$ defined as follows: $$\label{DA} {\mathcal{D}}_A^ku := (\nabla_A u)_{k+2} - i (\nabla_A u)_{k+1} \,,$$ where components are understood cyclically (i.e., $a_k=a_{k+3}$). In particular, $${\mathcal{D}}_A^3u = (\partial_2 u - i A_2 u) - i (\partial_1 u
- i A_1 u)
= (\nabla_A u)_2 - i (\nabla_A u)_1 \,.$$ The operator ${\mathcal{D}}_A$ corresponds to a “creation operator" for a magnetic Laplacian in quantum mechanics. It was introduced in the context of Ginzburg-Landau by Bogomoln’yi to prove the self-duality of the Ginzburg-Landau functional at $\kappa=1/\sqrt{2}$, cf. e.g [@jaffetaubes]. His proof relies on identities similar to the next one.
\[lemmaformula\] With the notation above, one has $$|\nabla'_A u|^2 = |{\mathcal{D}}_A^3u|^2 + \rho B_3 + \nabla'\times
j_A'$$ and, for any $k=1,2,3$, $$|(\nabla_A u)^{k+1}|^2 + |(\nabla_A u)^{k+2}|^2= |{\mathcal{D}}_A^ku|^2
+ \rho B_k + (\nabla\times j_A)_k \,.$$
We only prove the first relation, the other follows by relabeling coordinates. Notice that $$\overline{\nabla_A u}=\nabla_{-A}\bar u\,.$$ We compute $$\begin{aligned}
|{\mathcal{D}}_A^3u|^2 &=&
\left| (\nabla_A u)_2 - i (\nabla_A u)_1 \right|^2\\
&=&
\left( (\nabla_A u)_2 - i (\nabla_A u)_1 \right)
\left( ( \overline{\nabla_{A} u})_2 + i (\overline{\nabla_{A} u})_1
\right) \\
&=&|(\nabla_Au)_1|^2 + |(\nabla_Au)_2|^2 -
i(\nabla_Au)_1(\overline{\nabla_A u})_2 +
i(\nabla_Au)_2(\overline{\nabla_A u})_1\\
&=& |\nabla_A'u|^2 -i (\nabla'_Au)\times (\overline{\nabla_A' u})\end{aligned}$$ and $$\begin{aligned}
\nabla'\times (-i\bar u\nabla'_Au)&=&-i(\nabla' \bar u)\times \nabla_A' u
- i \bar u\nabla'\times(\nabla_A'u)
\\
&=& -i(\nabla' \bar u+iA'\bar u)\times(\nabla_A' u)
-A'\bar u\times(\nabla_A' u)
-\bar u \nabla'\times (A'u)\\
&=&-i (\overline{\nabla_A' u}) \times(\nabla_A' u)
-A'\bar u \times(\nabla'u) -|u|^2\nabla'\times A'
- (\nabla'u)\times (A'\bar u)\\
&=&-i (\overline{\nabla_A' u}) \times(\nabla_A' u)
-|u|^2\nabla'\times A'\,.\end{aligned}$$ Since the vector product is antisymmetric, the last expression is real. Recalling the definition of $j_A$ from we obtain $$\nabla'\times j'_A
=\mathrm {Re} \nabla'\times[ -i\bar u \nabla_A'u]
=-i (\overline{\nabla_A' u}) \times(\nabla_A' u)
-|u|^2\nabla'\times A'\,.$$ Adding terms concludes the proof.
We now show that the flux of $B_3$ over every section is constant, due to the divergence-free condition.
\[lemmaHmeno12\] Let $B \in L^2_{{\mathrm{loc}}}(\R^3 ;\R^3) $ be $Q_L$-periodic and obey $\Div B = 0 $. Then the quantity $$ \varphi(z):=\int_{Q(z)} B_3\,dx'$$ does not depend on $z$. In particular, if $B - {b_{\mathrm{ext}}}e_3 \in L^2(Q_L\times \R;\R^3)$ (implied by the finiteness of ${E_\mathrm{GL}}$) then for all $z\in \R$, $$\int_{Q(z)} B_3\,dx' = {b_{\mathrm{ext}}}L^2.$$
By the periodicity condition we can test the relation $\Div B=0$ (which is true since $B = \nabla \times A$) with a function $\theta\in C^1_c(\R)$, depending on $x_3$ alone. This yields that for any $\theta \in C^1_c(\R)$ $$0= \int_{Q_L\times \R} B_3 \theta'(x_3) dx= \int_\R \varphi(z) \theta'(z) \, dz\,.$$ It follows that $\varphi$ is constant. The second assertion follows from the fact that $\int_{Q(z)} B_3^2 \, dx' \ge \varphi(z)^2/L^2$.
At this point we are ready to separate the bulk term, and define the microstructure functional we shall study below.
\[lemmaseparatebulk\] For every admisible pair $(u,A)$ and any parameter set obeying (\[eqparameters\]) one has $${E_\mathrm{GL}}[u,A]= (\kappa\sqrt2 {b_{\mathrm{ext}}}-{b_{\mathrm{ext}}}^2)L^2T + E[u,A]\,,$$ where $$\begin{aligned}
1
E[u,A]&:=
\int_{Q_{L,T}} \left[(1-\kappa \sqrt2) |\nabla_A' u |^2 + \kappa\sqrt2
|{\mathcal{D}}_A^3u|^2 +
|\partial_3 u -
i A_3 u|^2 \right]dx \nonumber \\
& + \int_{Q_{L,T}}
\left(B_3 - \frac{\kappa}{\sqrt2} (1-\rho)\right)^2dx \nonumber \\
&+ \int_{Q_L\times \R} |B'|^2 dx +
\int_{Q_L\times(\R\setminus(0,T))} |B_3 -{b_{\mathrm{ext}}}|^2dx \,,
\label{eqdefE}\end{aligned}$$ and as above $B=\nabla \times A$, $\rho=|u|^2$.
Lemma \[lemmaformula\] implies $$|\nabla_A' u |^2 = (1-\kappa \sqrt2) |\nabla_A' u |^2 + \kappa\sqrt2
|{\mathcal{D}}_A^3u|^2
+\kappa\sqrt2\rho B_3 + \kappa\sqrt2\nabla'\times j'_A\,.$$ The last term integrates to zero by the periodicity of $j_A$. By Lemma \[lemmaHmeno12\], the average of the normal component of the magnetic field $B_3$ is ${b_{\mathrm{ext}}}$. Therefore for each fixed $z$, we have $$\begin{aligned}
1
\int_{Q(z)} |\nabla_A' u |^2 dx' = & \int_{Q(z)}\left[ (1-\kappa \sqrt2) |\nabla_A' u |^2 + \kappa\sqrt2
|{\mathcal{D}}_A^3u|^2 +\kappa\sqrt2(\rho-1) B_3 + \kappa\sqrt2 {b_{\mathrm{ext}}}\right] dx'\,.\end{aligned}$$ We substitute and obtain, using $\int_{Q_{L,T}} (B_3-{b_{\mathrm{ext}}})^2dx=\int_{Q_{L,T}} (B_3^2-{b_{\mathrm{ext}}}^2)dx$, $$\begin{aligned}
1
{E_\mathrm{GL}}[u,A]&=\int_{Q_{L,T}} \left[(1-\kappa \sqrt2) |\nabla_A' u |^2 + \kappa\sqrt2
|{\mathcal{D}}_A^3u|^2 + \left(B_3 - \frac{\kappa}{\sqrt2} (1-\rho)\right)^2 \right] dx \\
& +\int_{Q_{L,T}} \left[|\partial_3 u -
i A_3 u|^2 -{b_{\mathrm{ext}}}^2 +\kappa\sqrt2 {b_{\mathrm{ext}}}\right]dx\\
&+ \int_{Q_L\times \R} |(\nabla\times A)'|^2 dx+
\int_{Q_L\times(\R\setminus(0,T))} |(\nabla\times A)_3 -{b_{\mathrm{ext}}}|^2dx\,.\end{aligned}$$ Thus, the bulk energy is $\kappa\sqrt2 {b_{\mathrm{ext}}}-{b_{\mathrm{ext}}}^2 $, and the result follows.
Construction of test functions
------------------------------
One main ingredient of the proof of the lower bound is the following concentration lemma. This can be seen as a combination of truncation and mollification, and is closely related to Lemma 3.1 from [@ChoksiContiKohnOtto2008] and Lemma 2.1 from [@ContiNiethammerOtto2006]. We formulate this lemma in generic dimension, with $Q_L^n=(0,L)^n$, for $Q_L^n$-periodic functions. In the following only the $n=2$ case is used. For $f\in L^1_{{\mathrm{loc}}}(\R^n)$ and $r>0$, we define $f_r\in C^0(\R^n)$ by averaging over $r$-balls, $$\label{eqdefaverage}
f_{r}(x):= \frac{1}{|B_{r}|}
\int_{B_{r}(x)} f(y) dy\,.$$ Notice that this operation preserves periodicity.
\[lemmatestfunction\] Let $\chi\in L^1_{{\mathrm{loc}}}(\R^n)$, $Q_L^n$-periodic, $\chi\ge 0$, $0< \ell\le \err$. Then there is $\psi\in L^1_{{\mathrm{loc}}}(\R^n)$, $Q_L^n$-periodic, such that
1. $\displaystyle \psi(x)\ge \chi_\ell (x) $;
2. $\displaystyle \sup\psi\le \sup \chi$;
3. $\displaystyle \int_{Q_L^n}\psi\, dy\le \frac{2^n\err^n}{\ell^n} \int_{Q_L^n}\chi\, dy$;
4. $\displaystyle \err\sup|\nabla\psi|{\lesssim}\sup\chi$;
5. $\displaystyle \err\int_{Q_L^n}|\nabla \psi|\, dy{\lesssim}\frac{\err^n}{\ell^n} \int_{Q_L^n} \chi\, dy$.
For future reference we remark that these estimates immediately imply $$\label{eqlemmatestfunctl2}
\|\psi\|_{L^2(Q^n_L)}^2{\lesssim}\frac{r^n}{\ell^n} \|\chi\|_{L^\infty(Q^n_L)}\|\chi\|_{L^1(Q^n_L)}
\text{ and }
\|\nabla\psi\|_{L^2(Q^n_L)}^2{\lesssim}\frac{r^{n-2}}{\ell^n} \|\chi\|_{L^\infty(Q^n_L)}\|\chi\|_{L^1(Q^n_L)}\,.$$
By homogeneity, it suffices to consider the case $\sup\chi=1$. Define $$\psi(x):=\left(\min\left\{\frac{2^n\err^n}{\ell^n} \chi_{2\err}, 1\right\}\right)_\err(x) =
\frac{1}{|B_\err|} \int_{B_\err(x)} \min\left\{\frac{2^n\err^n}{\ell^n}\chi_{2\err}(y), 1\right\} dy\,,$$ where we use the notation of (\[eqdefaverage\]). Clearly $\psi\le 1$, and (ii) follows. (iii) is immediate.
To prove (i), observe first that $\chi_\ell\le \sup\chi= 1$. Fix some $x$. Since $B_\ell(x)\subset B_{2\err}(z)$ for all $z\in
B_\err(x)$, we have $$\begin{aligned}
\chi_\ell(x) &=& \frac{1}{|B_\ell|} \int_{B_\ell(x)} \chi \, dy\\
&\le &\frac{1}{|B_\ell|} \int_{B_{2\err}(z)} \chi \, dy
= \frac{2^n\err^n}{\ell^n} \chi_{2\err}(z)\,.\end{aligned}$$Therefore $$\chi_\ell(x)\le \min\left\{\frac{2^n\err^n}{\ell^n} \chi_{2\err}(z), 1\right\} \hskip1cm
\forall z\in B_\err(x)\,.$$ Averaging over $B_\err(x)$ we obtain $$\chi_\ell(x)\le \frac{1}{|B_\err|} \int_{B_\err(x)}
\min\left\{\frac{2^n\err^n}{\ell^n} \chi_{2\err}(z), 1\right\} dz=\psi(x)\,,$$ which proves (i).
Further, for any pair $x$, $z$ one has, writing $C_{\err}(x,z):=(B_r(x)\setminus B_r(x+z)) \cup ( B_r(x+z)\setminus B_r(x))$, $$|\psi(x)-\psi(x+z)|\le \frac{|C_{\err}(x,z)|}{|B_{\err}|}
\le \frac{c_n}{\err}|z|\,,$$ which implies (iv). To prove (v), write analogously $$|\psi(x)-\psi(x+z)|\le \frac{1}{|B_r|} \int_{C_{\err}(x,z)}
f\, dy$$ where $f=\min\{2^nr^n\ell^{-n}\chi_{2r},1\}$. Integrating in $x$ and estimating as above $|C_{\err}(x,z)|{\lesssim}r^{n-1}|z|$ we get $$\int_{Q_L^n} |\psi(x)-\psi(x+z)|dx\le \int_{Q_L^n} f \frac{|C_{\err}(x,z)|}{|B_r|}dx
{\lesssim}\frac{|z|}{r} \int_{Q_L^n} f \, dx$$ and the proof is concluded.
The sharp-interface functional {#secsharpinterface}
------------------------------
In closing this preliminary section, we introduce the sharp-interface version of the functional $E$ and the corresponding function spaces. In the sharp-interface functional, a function denoted $\chi$ represents the characteristic function of the normal phase, and is constrained to take values in $\{0,1\}$. Thus $\chi$ is formally the equivalent of $1-\rho$ and the approximate Meissner effect is imposed via $$B(1-\chi)=0.$$
\[defadmisssharp\] We say that a pair $B\in L^2_{{\mathrm{loc}}}(\R^3;\R^3)$, $\chi\in BV_{{\mathrm{loc}}}(\R^3;\{0,1\})$ is admissible for the sharp-interface functional if both of them are $Q_L$-periodic and $$\label{eqsidecondF}
\Div B=0 \text{ distributionally, and }B(1-\chi)=0 \text{ a.e.}$$ The condition $\Div B=0$ is understood as $\int_{\R^3} B\cdot \nabla\theta\, dx=0$ for all test functions $\theta\in C^1_c(\R^3)$.
Given an admissible pair $(\chi,B)$ we set $$\label{eqdefF}
F[\chi,B] := \int_{Q_{L,T}} \kappa |D\chi| +\int_{Q_{L,T}} \left[|B'|^2 + \chi\left(B_3
- \frac{\kappa}{\sqrt2}\right)^2 \right]dx+
\int_{Q_L\times[\R\setminus(0,T)]} |B-{b_{\mathrm{ext}}}e_3|^2 dx\,.$$
This is the sharp-interface functional studied in [@ChoksiKohnOtto2004; @ChoksiContiKohnOtto2008]. In comparing with those papers, it is important to notice that several quantities are scaled differently. In particular, lengths are rescaled by a factor of $L$, and magnetic fields by $\kappa/\sqrt2$. Precisely, denoting by $\chi^*$, $B^*$, $E^*$, ${\varepsilon}^*$, $L^*$ and $b_a^*$ the objects used in [@ChoksiKohnOtto2004; @ChoksiContiKohnOtto2008], one has $$\chi^*(x)=1-\chi(xL)\,,\hskip1cm B^*(x)=\frac{\sqrt2}{\kappa} B(xL)$$ and correspondingly $$E^*=\frac{2F}{L^3\kappa^2}\,,\hskip5mm
{\varepsilon}^*=\frac2{\kappa L} \,,\hskip5mm
L^*=\frac{T}{L}\,,\hskip5mm
b_a^*=\frac{\sqrt2}{\kappa}{b_{\mathrm{ext}}}\,.$$
Lower bound {#seclowerb}
===========
\[secoldfunctional\]
Lower bound with sharp interfaces
---------------------------------
In order to prepare some intermediate results and to explain the strategy of the proof in a simpler context, we first prove the lower bound for the sharp-interface functional, recovering the result from [@ChoksiContiKohnOtto2008]. In proving the lower bound the fields $\chi$ and $B$ will be fixed admissible functions, in the sense of Definition \[defadmisssharp\], and we shall simply denote by $F$ the total energy, and by $F(z)$ the part of the energy localized in the surface $Q(z)$, i.e., $$F(z):=\int_{Q(z)} \kappa |D'\chi| +\int_{Q(z)} \left[ |B'|^2 + \chi\left(B_3
- \frac{\kappa}{\sqrt2}\right)^2 \right]dx'\,.$$ Clearly $\int_0^T F(z) dz \le F[\chi,B]$. In several lemmas we shall additionally focus on the case that the energy is bounded by $$\label{eqFzallgood}
F[\chi,B]\le \frac1{8} \kappa {b_{\mathrm{ext}}}L^2T\,,$$ and that a $z\in (0,L)$ is given, so that $$\label{eqFzzgood}
F(z)\le \frac1{8} \kappa {b_{\mathrm{ext}}}L^2\,.$$
The key strategy is to select a good section $Q(z)$ which has small energy. Since the energy is small, and the flux of the magnetic field is the same on every section, the magnetic field necessarily concentrates on a small subset, whose perimeter is controlled by the energy (interior estimate). At the same time, close to the surface the energy favours a uniformly distributed magnetic field (exterior term). But “moving around” the magnetic field as $z$ changes is only possible, due to the divergence-free condition, if the tangential components $B'$ are nonzero, which are also penalized by the energy (transport term). Making these three effects quantitative, and balancing them, leads to the lower bound.
### Equidistribution of the phases
We first show that the average value of $\chi$ across “good” sections is the one that relaxation theory would predict, up to a factor.
\[lemmachiloc\]If the admissible pair $(\chi,B)$ obeys (\[eqFzallgood\]) and (\[eqFzzgood\]) for some $z\in (0,T)$, then $$\label{eqintchi1-2}
\int_{Q_{L,T}} \chi \, dx\, \sim \, \frac{{b_{\mathrm{ext}}}L^2T}{\kappa}$$ and $$\label{eqintchi-2}
\int_{Q(z)} \chi \, dx'\, \sim \, \frac{{b_{\mathrm{ext}}}L^2}{\kappa}\,.
$$
We start with the second assertion. Recalling that $\chi^2=\chi$, $\int_{Q(z)} (B_3-{b_{\mathrm{ext}}})dx'=0$ (Lemma \[lemmaHmeno12\]) and $B(1-\chi)=0$, we compute $$\begin{aligned}
\left|\int_{Q(z)}\left( \frac{\kappa}{\sqrt2} \chi-{b_{\mathrm{ext}}}\right)dx'\right|&=&
\left|\int_{Q(z)} \chi \left(\frac{\kappa}{\sqrt2}-B_3\right)dx'\right|\\
&\le&
\left(\int_{Q(z)} \chi dx' \right)^{1/2}
\left(\int_{Q(z)}\chi\left(\frac{\kappa}{\sqrt2}-B_3\right)^2dx'\right)^{1/2}\\
&=&
\left(\int_{Q(z)}\frac{\kappa}{\sqrt2} \chi dx' \right)^{1/2}
\left(\frac{\sqrt2}{\kappa}\int_{Q(z)}\chi\left(\frac{\kappa}{\sqrt2}-B_3\right)^2dx'\right)^{1/2}\\
&\le&
\frac14 \left|\int_{Q(z)} \frac{\kappa}{\sqrt2}\chi dx'\right| + \frac{\sqrt2}{\kappa}F(z)\,.
\end{aligned}$$ Therefore, recalling (\[eqFzzgood\]), $$\frac34 \int_{Q(z)}\frac{\kappa}{\sqrt2} \chi \, dx' \le {b_{\mathrm{ext}}}L^2 + \frac{\sqrt2}{\kappa}F(z)
\le \frac54 {b_{\mathrm{ext}}}L^2$$ and $$\frac54\int_{Q(z)} \frac{\kappa}{\sqrt2} \chi \, dx' \ge {b_{\mathrm{ext}}}L^2 - \frac{\sqrt2}{\kappa}F(z)
\ge \frac34 {b_{\mathrm{ext}}}L^2\,.$$ This concludes the proof of (\[eqintchi-2\]). The proof of (\[eqintchi1-2\]) is analogous, integrating over $Q_{L,T}$ instead of $Q(z)$.
### Interior term
We show that on sections with small energy the magnetic field necessarily concentrates, as captured by the test function $\psi$.
\[lemmaoldinterior\] For any $r\ge \ell>0$ and $z\in(0,T)$, if (\[eqFzzgood\]) holds and $\psi$ is the function constructed via Lemma \[lemmatestfunction\] from $\chi(\cdot, z)$ then $$\frac{\kappa}{\sqrt2}\int_{Q(z)} \chi\, dx' -
\int_{Q(z)} B_3\psi\, dx'{\lesssim}\ell F(z) + \left(
\frac{r^2}{\ell^2} \frac{{b_{\mathrm{ext}}}L^2}{\kappa}\right)^{1/2} F^{1/2}(z)$$ for some universal $c>0$.
We write $$B_3=B_3\chi = \frac{\kappa}{\sqrt2} \chi +
\left(B_3-\frac{\kappa}{\sqrt2}\right)\chi\,.$$ Testing with $\psi$ we get $$\begin{aligned}
\int_{Q(z)} B_3\psi\, dx'&=& \frac{\kappa}{\sqrt2} \int_{Q(z)}\chi\psi\, dx' +
\int_{Q(z)}\left(B_3-\frac{\kappa}{\sqrt2}\right)\chi\psi\, dx'\,.\end{aligned}$$ Using Lemma \[lemmatestfunction\](i) we obtain $$\begin{aligned}
\int_{Q(z)} \chi\psi \,dx'\ge \int_{Q(z)} \chi\chi_\ell \,dx'
&=&\int_{Q(z)}\chi^2\,dx' + \int_{Q(z)}\chi(\chi_\ell-\chi)dx'\\
&\ge& \int_{Q(z)} \chi \,dx'- \int_{Q(z)}
|\chi-\chi_\ell| \,dx'\end{aligned}$$ and therefore $$\frac\kappa{\sqrt2} \int_{Q(z)} \chi \,dx'
\le \int_{Q(z)} B_3\psi\, dx'+ \frac\kappa{\sqrt2}\int_{Q(z)}
|\chi-\chi_\ell| \,dx'-
\int_{Q(z)}\left(B_3-\frac{\kappa}{\sqrt2}\right)\chi\psi\, dx'\,.$$ The second term can be estimated by $$\int_{Q(z)} |\chi-\chi_\ell| dx'\le \ell
\int_{Q(z)} |D' \chi|\le \frac{\ell}{\kappa} F(z)\,,$$ and the last one by $$\begin{aligned}
\left| \int_{Q(z)}\left(B_3-\frac{\kappa}{\sqrt2}\right)\chi\psi\,dx' \right|& \le &
\|\psi\|_{L^2(Q_L)}
\left\|\left(B_3-\frac{\kappa}{\sqrt2}\right)\chi\right\|_{L^2(Q(z))}\\
& \le&
\|\psi\|_{L^2(Q_L)} F^{1/2}(z)\,.\end{aligned}$$ Collecting terms we obtain $$\frac{\kappa}{\sqrt2} \int_{Q(z)}\chi\, dx'
\le \int_{Q(z)} B_3\psi dx'+ \ell F(z)
+\|\psi\|_{L^2(Q_L)} F^{1/2}(z)\,.$$ Using (\[eqlemmatestfunctl2\]) and (\[eqintchi-2\]), we also have $$\label{psil2}
\|\psi\|_{L^2(Q_L)}\le \|\psi\|_{L^1(Q_L)}^{1/2}{\lesssim}\left(
\frac{r^2}{\ell^2} \frac{{b_{\mathrm{ext}}}L^2}{\kappa}\right)^{1/2}\,.$$ This concludes the proof.
### Transport term {#secoldtransport}
We now relate the value of $B_3$ over different sections by exploiting the $\int |B'|^2dx$ term in the energy. Since we have $\Div B=0$ we may write $$\partial_{3} B_3 + {\Div}' B'=0$$ i.e. $B'$ can be seen as the flux transporting $B_3$. Since $B_3$ takes, approximately, only the two values 0 and $\kappa/\sqrt2$, up to a factor we can understand $B'$ as the velocity with which $B_3$ is transported. Thus we call $\int |B'|^2dx$ the transport term by analogy with the Benamou-Brenier formula for the Wasserstein distance in optimal transport.
\[LemmaMongeKanto\] Let $B \in L^2_{{\mathrm{loc}}}(\R^2\times(0,T) ;\R^3) $, $Q_L$-periodic, such that $\Div
B=0$. Then for any $z_1, z_2\in [0,T]$ one has $$\label{eqbprime}
\left|\int_{Q_L} \left(B_3(\cdot, z_1)\, -\, B_3(\cdot,z_2)\right)
\, \psi\, dx'\right|
\le \|\nabla\psi\|_{L^\infty}\,
\int_{Q_L \times (z_1,z_2)} |B'| \, dx$$ for any $\psi\in W^{1,\infty}(\R^2)$, $Q_L$-periodic. The values of $B_3$ on the sections are understood as traces, which exist since $\Div B=0$.
This is the same as [@ChoksiContiKohnOtto2008 Lemma 2.2], for completeness we give here the short argument. We can assume without loss of generality that $z_1<z_2$. We compute, using $\Div B=0$ and the $Q_L$-periodicity of $B$ and $\psi$, $$\begin{aligned}
1
\int_{Q_L} [B_3(z_2)-B_3(z_1)]\psi \,dx'
=&\int_{Q_L\times(z_1,z_2)}
\frac{\partial B_3}{\partial x_3} \psi\,dx
=-\int_{Q_L\times(z_1,z_2)}
\nabla'\cdot B' \psi\,dx\\
=&\int_{Q_L\times(z_1,z_2)}
B'\cdot\nabla\psi\,dx
\le \|\nabla\psi\|_{L^\infty(Q_L)}\int_{Q_{L}\times(z_1,z_2)} |B'|\,dx\,.
\end{aligned}$$ This, together with the same estimate with $-\psi$ in place of $\psi$, concludes the proof.
Since $B'$ is nonzero only in a small part of the volume, the embedding of $L^1$ into $L^2$ gives an additional factor proportional to ${b_{\mathrm{ext}}}$.
\[lemmatransportsharp\] Let $(\chi,B)$ be an admissible pair which fulfills (\[eqFzallgood\]), $z\in (0,T)$ such that (\[eqFzzgood\]) holds. Then the function $\psi$ from Lemma \[lemmaoldinterior\] fulfills $$\left| \int_{Q(z)} B_3\psi \, dx'-
\int_{Q(0)} B_3\psi\, dx'\right|{\lesssim}\frac1r
\left( \frac{{b_{\mathrm{ext}}}L^2T}{\kappa}\right)^{1/2}
F^{1/2}\,.$$
By Lemma \[LemmaMongeKanto\] we have $$\begin{aligned}
\left| \int_{Q(z)} B_3\psi\,dx' -
\int_{Q(0)} B_3\psi\,dx'\right| &\le&
\|\nabla\psi\|_{L^\infty} \int_{Q_{L,T}} |B'|\,dx\,.
\end{aligned}$$ We estimate $$\begin{aligned}
\int_{Q_{L,T}} |B'|\,dx=
\int_{Q_{L,T}} \chi |B'|\,dx
&\le& \left( \int_{Q_{L,T}} \chi\,dx\right)^{1/2}
\left( \int_{Q_{L,T}} |B'|^2\,dx\right)^{1/2}\\
&{\lesssim}& \left( \frac{{b_{\mathrm{ext}}}L^2T}{\kappa}\right)^{1/2}
F^{1/2}\,.
\end{aligned}$$ To conclude the proof it suffices to recall that $\|\nabla\psi\|_{L^\infty}{\lesssim}1/r$.
### Exterior term
Finally, we show that the energy outside the sample penalizes configurations with a magnetic field that oscillates strongly at the boundary, as measured by the $H^{-1/2}$ norm.
\[lemmaextsharp\] For all admissible $(\chi,B)$, and any $\psi\in H^{1/2}_{{\mathrm{loc}}}(\R^2)$, $Q_L$-periodic, it holds that $$\left| \int_{Q(0)} (B_3-{b_{\mathrm{ext}}})\psi \,dx' \right|{\lesssim}\|\psi\|_{H^{1/2}(Q_L)}F^{1/2} \,.$$ If (\[eqFzzgood\]) holds for some $z\in(0,T)$ and $r\ge \ell>0$ and $\psi$ are as in Lemma \[lemmaoldinterior\], then $$\left| \int_{Q(0)} (B_3-{b_{\mathrm{ext}}})\psi \,dx'\right|{\lesssim}\left(\frac{r {b_{\mathrm{ext}}}L^2}{\ell^2\kappa}\right)^{1/2}F^{1/2} \,.$$
Let $\hat\psi\in H^1_{{\mathrm{loc}}}(\R^3)$ be $Q_L$ periodic, such that $\hat\psi(\cdot, 0)=\psi$ in the sense of traces and $\|\psi\|_{H^{1/2}(Q_L)}=\|\nabla \hat\psi\|_{L^2(Q_L\times(-\infty,0))}$ (we recall that we are using the homogeneous $H^{1/2}$ norm). Since $\Div B=0$ on $(-\infty,0)\times Q_L$, using periodicity we obtain $$\int_{Q_L\times(-\infty,0)} (B-{B_{\mathrm{ext}}})\cdot \nabla\hat\psi\, dx =
\int_{Q(0)} (B-{B_{\mathrm{ext}}}) \hat\psi \cdot e_3 \,dx'\,.$$ Therefore $$\left| \int_{Q(0)} (B_3-{b_{\mathrm{ext}}})\psi\, dx'\right|\le
\|B-{B_{\mathrm{ext}}}\|_{L^2(Q_L\times(-\infty,0))}
\|\nabla \hat\psi\|_{L^2(Q_L\times(-\infty,0))}\le
F^{1/2} \|\psi\|_{H^{1/2}(Q_L)}\,.$$ This proves the first assertion. It remains to estimate $\|\psi\|_{H^{1/2}(Q_L)}$. By interpolation and (\[eqlemmatestfunctl2\]) we have $$\begin{aligned}
1
\|\psi\|_{H^{1/2}(Q_L)}&{\lesssim}\|\psi\|_{L^2(Q_L)}^{1/2}
\|\nabla\psi\|_{L^2(Q_L)}^{1/2}
{\lesssim}\left(\frac{r^2}{\ell^4}
\|\chi\|_{L^1(Q_L)}^2\right)^{1/4}\,.
\end{aligned}$$ Recalling we conclude $$\begin{aligned}
1
\|\psi\|_{H^{1/2}(Q_L)}&{\lesssim}\left(\frac{r {b_{\mathrm{ext}}}L^2}{\ell^2\kappa}\right)^{1/2}\,.
\end{aligned}$$ This concludes the proof.
### Derivation of the lower bound
From what precedes, we deduce the lower bound result for the sharp-interface functional.
\[theochilower1\] For any admissible pair $(\chi,B)$ (in the sense of Definition \[defadmisssharp\]) and for all ${b_{\mathrm{ext}}},\kappa,L,T>0$ such that $$8{b_{\mathrm{ext}}}\le \kappa \le \frac12\hskip1cm\text{ and }\hskip1cm
\kappa T\ge 1\,,$$ one has $$F[\chi,B]{\gtrsim}\min \left\{{b_{\mathrm{ext}}}\kappa^{3/7} T^{3/7} L^2,
{b_{\mathrm{ext}}}^{2/3} \kappa^{2/3} T^{1/3}L^2\right\} \,.$$
Let $F=F[\chi,B]$. If $F\ge {b_{\mathrm{ext}}}\kappa TL^2/8$ then, since $\kappa T\ge 1$, we obtain $F\ge {b_{\mathrm{ext}}}(\kappa T)^{3/7}L^2/8$ and the proof is concluded. We can therefore assume that (\[eqFzallgood\]) holds. By a mean-value argument, we may choose $z\in(0,T)$ such that $F(z)\le F/T$, so that (\[eqFzzgood\]) holds as well. We start by constructing $\psi$ as in Lemma \[lemmaoldinterior\], namely, with Lemma \[lemmatestfunction\] applied to $\chi(\cdot, z)$ for some $0< \ell\le r$ chosen below. One key observation is that, since $\psi$ depends only on $x'$, $$\begin{aligned}
\int_{Q(z)} B_3\psi \,dx'= &\int_{Q_L}{b_{\mathrm{ext}}}\psi \,dx'+ \left[\int_{Q(z)}B_3\psi\,dx'
-\int_{Q(0)}B_3\psi\,dx'\right]
+ \int_{Q(0)} (B_3-{b_{\mathrm{ext}}})\psi\,dx'\,.\end{aligned}$$ With Lemma \[lemmatransportsharp\] and Lemma \[lemmaextsharp\] this can be tranformed into $$\begin{aligned}
\nonumber \int_{Q(z)} B_3\psi \,dx'\le &\int_{Q_L}{b_{\mathrm{ext}}}\psi \,dx'
+ c\frac1r \left( \frac{{b_{\mathrm{ext}}}L^2T}{\kappa}\right)^{1/2} F^{1/2}
+ c \left(\frac{r {b_{\mathrm{ext}}}L^2}{\ell^2\kappa}\right)^{1/2} F^{1/2}
\,,\end{aligned}$$ Lemma \[lemmatestfunction\](iii) gives $$\int_{Q_L}{b_{\mathrm{ext}}}\psi \,dx'\le {b_{\mathrm{ext}}}\|\psi\|_{L^1(Q_L)}\le 4\frac{r^2}{\ell^2} {b_{\mathrm{ext}}}\int_{Q(z)} \chi \, dx'\,.$$ Recalling Lemma \[lemmaoldinterior\], $$\frac{\kappa}{\sqrt2}\int_{Q(z)} \chi\, dx'
\le \int_{Q(z)} B_3\psi\, dx'+
c \ell F(z) + c \left(
\frac{r^2}{\ell^2} \frac{{b_{\mathrm{ext}}}L^2}{\kappa}\right)^{1/2} F^{1/2}(z)\,.$$ Combining the last three estimates gives $$\begin{aligned}
\frac{\kappa}{\sqrt2}\int_{Q(z)} \chi\, dx'\le &
4\frac{r^2}{\ell^2} {b_{\mathrm{ext}}}\int_{Q(z)} \chi \, dx'
+ c\frac1r \left( \frac{{b_{\mathrm{ext}}}L^2T}{\kappa}\right)^{1/2} F^{1/2}
+c \left(\frac{r {b_{\mathrm{ext}}}L^2}{\ell^2\kappa}\right)^{1/2} F^{1/2}
\\
& + c \ell F(z) +c \left(
\frac{r^2}{\ell^2} \frac{{b_{\mathrm{ext}}}L^2}{\kappa}\right)^{1/2} F^{1/2}(z)\,.\end{aligned}$$ Assume now that $$\label{eqdefadmissrl}
0<\ell\le r\le \left(\frac{\kappa}{8{b_{\mathrm{ext}}}}\right)^{1/2} \ell\,,$$ so that the coefficient of $\int_{Q(z)} \chi \, dx'$ in the right-hand side is smaller than the one on the left-hand side. This is possible, since we assumed ${b_{\mathrm{ext}}}\le \kappa/8$. Then, recalling (\[eqintchi-2\]) and $F(z)\le F/T$, $$\begin{aligned}
\nonumber
{b_{\mathrm{ext}}}L^2{\lesssim}&
\frac1r \left( \frac{{b_{\mathrm{ext}}}L^2T}{\kappa}\right)^{1/2} F^{1/2}
+\left(\frac{r {b_{\mathrm{ext}}}L^2}{\ell^2\kappa}\right)^{1/2} F^{1/2}
+ \ell \frac FT
+ \left( \frac{r^2}{\ell^2} \frac{{b_{\mathrm{ext}}}L^2}{\kappa}\right)^{1/2} \frac{F^{1/2}}{T^{1/2}}\,.\end{aligned}$$ At least one of the terms in the right-hand side has to be at least one-quarter of the one on the left, and therefore for all pairs $(r,\ell)$ which obey (\[eqdefadmissrl\]), we have $$F{\gtrsim}\min\left\{
{b_{\mathrm{ext}}}L^2 \frac{r^2 \kappa}{T},
{b_{\mathrm{ext}}}L^2 \frac{\ell^2\kappa}{r},
{b_{\mathrm{ext}}}L^2\frac{T}{\ell},
{b_{\mathrm{ext}}}L^2\frac{\ell^2\kappa T}{r^2}
\right\}.$$ Equivalently, $$\label{eqfinalboundF}
F{\gtrsim}\kappa {b_{\mathrm{ext}}}L^2T \min\left\{
\frac{r^2 }{T^2},
\frac{\ell^2}{rT},
\frac{1}{\kappa\ell},
\frac{\ell^2}{r^2}
\right\}.$$ We finally have to choose $r$ and $\ell$, and check that in each case some terms give the optimal bound, and the others are irrelevant. Balancing the first three terms we obtain $$\ell=T^{4/7}\kappa^{-3/7}\,, \hskip5mm r=T^{5/7}\kappa^{-2/7}\,.$$ This choice is admissible only if (\[eqdefadmissrl\]) is satisfied, which since $\kappa T\ge1$ is equivalent to $(\kappa T)^{1/7}\le (\kappa/8{b_{\mathrm{ext}}})^{1/2}$. In this case, (\[eqfinalboundF\]) becomes $$F{\gtrsim}\kappa^{3/7}{b_{\mathrm{ext}}}L^2T^{3/7}
\min\left\{
1,1,1, (\kappa T)^{2/7}\right\}$$ and since $\kappa T\ge 1$ the assertion holds.
If instead $(\kappa T)^{1/7}\ge (\kappa/8{b_{\mathrm{ext}}})^{1/2}$, we choose $r=\ell (\kappa/8{b_{\mathrm{ext}}})^{1/2}$. Inserting this into (\[eqfinalboundF\]) and then balancing the first and third term yields, after some rearrangement, $$\begin{aligned}
F&{\gtrsim}&
\kappa^{2/3}{b_{\mathrm{ext}}}^{2/3}L^2T^{1/3} \min\left\{
1,
(\kappa T)^{1/3} \left(\frac{{b_{\mathrm{ext}}}}{\kappa}\right)^{7/6},
1,
(\kappa T)^{2/3} \left(\frac{{b_{\mathrm{ext}}}}{\kappa}\right)^{4/3}
\right\}\,.\end{aligned}$$ We observe that since $\kappa T\ge1$ and $(\kappa T)^{1/7}\ge(\kappa/8{b_{\mathrm{ext}}})^{1/2}$ all terms of the form $$(\kappa T)^{\alpha} \left(\frac{{b_{\mathrm{ext}}}}{\kappa}\right)^{\beta}$$ with $\alpha\ge\frac27\beta$ are bounded from below. This concludes the proof.
Lower bound for the Ginzburg-Landau functional {#lowerboundfull}
----------------------------------------------
The proof is structured in a similar way as the one for the sharp-interface functional, but contains several additional difficulties. In particular, when working with diffuse interfaces we can enforce the Meissner condition only in a weak sense, see Section \[secmeissneravg\]. This generates difficulties both in the interior estimate and in the transport term. Additionally, the energy does not directly control the size of the boundary of the normal phase, and a suitable estimate needs to be formulated and proven (Section \[secfullsurface\]). The only term which can be treated in the same way is the one corresponding to the external field.
We consider a pair $(u,A)\in H^1_{\mathrm{per}}$ with ${E_\mathrm{GL}}[u,A]$ finite, hence $E[u,A]$ finite. By Lemma \[lemmarhominoredi1\] below, we can assume that $\rho\le 1$ pointwise. For any $z\in(0,T)$, we denote by $E(z)$ the energy contained in the section $Q(z):=Q_L\times\{z\}$, $$\begin{aligned}
1
E(z):=&
\int_{Q(z)}\left[ (1-\kappa \sqrt2) |\nabla_A' u |^2 + \kappa\sqrt2
|{\mathcal{D}}_A^3u|^2 + \left(B_3 - \frac{\kappa}{\sqrt2} (1-\rho)\right)^2 \right] dx'\\
& + \int_{Q(z)}\left[ |B'|^2+|\partial_3 u -
i A_3 u|^2 \right] dx'\,.\end{aligned}$$ We write for brevity $E=E[u,A]$. We recall that $\rho=|u|^2$ and define $$\label{eqdefchi}
\chi := (1-\rho)^2\,.$$
### Normalization of the density
We first show that we can assume without loss of generality that $|u|\le 1$, or, equivalently, $\rho\le 1$.
\[lemmarhominoredi1\] Let $(u,A)$ be an admissible pair. Then, $$|\nabla(\rho^{1/2})|\le|\nabla_Au|\,,\hskip5mm |{\mathcal{D}}_A'u|\le\sqrt2|\nabla_Au|\,.$$ Let ${\widetilde}u:\R^2\times(0,T)\to{\mathbb{C}}$ be defined by $${\widetilde}u(x) :=
\begin{cases}
u(x) & \text{ if }|u|(x)\le 1\\
\displaystyle\frac{u}{|u|}(x) & \text{ if }|u|(x)> 1\,.
\end{cases}$$ Then $({\widetilde}u, A)$ is in $H^1_{\mathrm{per}}$ and $E[{\widetilde}u,A]\le E[u,A]$.
By density it suffices to prove all assertions under the additional assumption that $u\in C^1$, at points where $\rho\ne0$. Writing locally $u=\rho^{1/2} e^{i\theta}$ one obtains $$\nabla_Au=e^{i\theta}\left[ \nabla\rho^{1/2} + i \rho^{1/2} (\nabla\theta -A)\right]\,,$$ hence locally $$\label{naba}
|\nabla_Au|^2=|\nabla\rho^{1/2}|^2+\rho|\nabla\theta-A|^2\,.$$ The second assertion follows immediately from the definition of ${\mathcal{D}}_A$ in (\[DA\]). It also follows directly from that $|\nabla_A {\widetilde}u|\le |\nabla_A u|$, and hence ${E_\mathrm{GL}}[{\widetilde}u,A]\le {E_\mathrm{GL}}[u,A]$. Since $E$ and ${E_\mathrm{GL}}$ differ by a constant, we conclude that $E[{\widetilde}u,A]\le E[u,A]$.
### Meissner effect “on average” {#secmeissneravg}
In the reduced model we had the compatibility condition $B(1-\chi)=0$. In the true model, we expect as in that $B\rho$ is, in some sense, small. The next lemmas make this quantitative, in appropriate weak norms.
\[lemmarhoB3\] Let $(u,A)$ be admissible, and assume (\[eqparameters\]) and $\rho\le 1$. For any $z$, we have $$\label{eqsectionrhohunemnorhodue}
\left|\int_{Q(z)} \rho B_3 \,dx'\right| \le 8E(z)$$ and $$\label{eqsectionrhohunemnorho}
\left|\int_{Q(z)} \rho B_3 (1-\rho)\,dx'\right| \le 16\, E(z)\,.$$
Further, for any $\varphi\in L^\infty\cap W^{1,2}_{{\mathrm{loc}}}(\R^2\times(0,T))$, $Q_L$-periodic and such that $\varphi=0$ on $Q(0)$ and $Q(T)$ (in the sense of traces), and any $k=1,2,3$, we have $$\label{eqhcontrollatutti}
\left| \int_{Q_{L,T}} \rho B_k\varphi \,dx\right| {\lesssim}E \|\varphi\|_{L^\infty(Q_{L,T})} +
E^{1/2} \|\nabla\varphi\|_{L^2(Q_{L,T})}\,.$$
We start with (\[eqsectionrhohunemnorhodue\]). By Lemma \[lemmaformula\], $$\int_{Q(z)} \rho B_3 \,dx'=
\int_{Q(z)} \left[|\nabla_A'u|^2-\nabla'\times
j'_A-|{\mathcal{D}}_A^3u|^2\right]dx'\,.$$ The integral of $\nabla'\times j'_A$ is zero, since $j'_A$ is $Q_L$-periodic and only in-plane derivatives appear. The other terms are bounded by the energy, i.e., $$\frac12 \int_{Q(z)} |{\mathcal{D}}_A^3u|^2\,dx'\le
\int_{Q(z)} |\nabla_A'u|^2\,dx'\le \frac{E(z)}{1-\kappa\sqrt2}\le 4E(z)\,,$$ where we used $\kappa\le 1/2$. This concludes the proof of (\[eqsectionrhohunemnorhodue\]).
The argument for (\[eqsectionrhohunemnorho\]) is similar. We write $$\int_{Q(z)} \rho B_3 (1-\rho)\,dx'=
\int_{Q(z)} \left[|\nabla_A'u|^2-\nabla'\times
j'_A-|{\mathcal{D}}_A^3u|^2\right](1-\rho)\,dx'\,.$$ The terms with $\nabla_Au$ and ${\mathcal{D}}_Au$ can be estimated as above. The term with $\nabla'\times j_A'$ however needs more care. Since $j_A$ and $\rho$ are $Q_L$-periodic, $|j_A|\le \rho^{1/2}|\nabla_A u|$, $\rho \le 1$ and $|\nabla\rho^{1/2}|\le|\nabla_Au|$, an integration by parts leads to $$\left|\int_{Q(z)} (1-\rho) \nabla'\times j_A'\,dx'\right|=
\left|2\int_{Q(z)} \rho^{1/2} \nabla'\rho^{1/2} \times j_A'\,dx'\right|
\le \frac{2}{1-\kappa \sqrt2}E(z)\,.$$ This proves (\[eqsectionrhohunemnorho\]).
Finally, we consider (\[eqhcontrollatutti\]). Here we include the other components, and we required that the localization function $\varphi$ vanishes on the top and bottom boundaries (where we have no periodicity). We compute $$\begin{aligned}
\left| \int_{Q_{L,T}} \rho B_k\varphi \,dx\right| &=&
\left| \int_{Q_{L,T}} \left[
|\nabla_A^{(k+1)} u|^2+|\nabla_A^{(k+2)} u|^2 - |{\mathcal{D}}_A^{(k)}u|^2 - (\nabla\times j_A)_k
\right]\varphi\,dx\right| \\
&\le&
16 E \|\varphi\|_{L^\infty} + \int_{Q_{L,T}} |j_A|\,|\nabla\varphi|\,dx\\
&\le& 16E \|\varphi\|_{L^\infty}+
\frac{1}{(1-\kappa\sqrt2)^{1/2}}
E^{1/2} \|\nabla\varphi\|_{L^2(Q_{L,T})}\,.\end{aligned}$$
### Surface energy {#secfullsurface}
We now show how the surface energy can be recovered from the functional. This arises from the combination of a $|\nabla \rho^{1/2}|^2$ and a $(1-\rho)^2$ term, but the latter is not directly present in the energy, and needs first to be reconstructed.
Let $(u,A)$ be admissible, and assume (\[eqparameters\]) and $\rho\le 1$. For every $z\in(0,T)$ we have $$\label{eqrho6nc}
\kappa^2\int_{Q(z)} \rho(1-\rho)^2 \,dx'{\lesssim}E(z)\,.$$ The function $\chi$ defined in (\[eqdefchi\]) satisfies $$\label{eqnablaprimechi}
\kappa\int_{Q(z)} |\nabla'\chi|\,dx' {\lesssim}E(z)$$ and, for any $\ell>0$, $$\label{eqchichiell}
\int_{Q(z)} |\chi_\ell-\chi|\,dx'
{\lesssim}\frac{\ell}{\kappa} E(z)\,.$$ Here $\chi_\ell$ is the average of $\chi$ over $\ell$-balls, defined in (\[eqdefaverage\]).
We have $$\begin{aligned}
\left(B_3 - \frac{\kappa}{\sqrt2} (1-\rho)\right)^2
&\ge& \rho \left(B_3 - \frac{\kappa}{\sqrt2} (1-\rho)\right)^2\\
&\ge& \frac{\kappa^2}{2} \rho(1-\rho)^2 - \kappa \sqrt2\rho B_3(1-\rho)\,.\end{aligned}$$ Therefore, recalling (\[eqsectionrhohunemnorho\]), we have $$\begin{aligned}
\frac{\kappa^2}{2} \int_{Q(z)} \rho(1-\rho)^2 \,dx'\le E(z)+16\sqrt2 \kappa E(z)\,,\end{aligned}$$ and (\[eqrho6nc\]) is proven.
At the same time, since $|\nabla\rho^{1/2}|\le |\nabla_Au|$, we have $$\int_{Q(z)}|\nabla'\rho^{1/2}|^2 \,dx'\le 4E(z)\,.$$ Therefore $$\begin{aligned}
E(z)&{\gtrsim}&\int_{Q(z)}\left[|\nabla'\rho^{1/2}|^2 + \kappa^2 \rho(1-\rho)^2\right]dx' \\
&\ge&
2 \int_{Q(z)} \kappa \rho^{1/2}(1-\rho) |\nabla'\rho^{1/2}| \,dx'
= \frac{\kappa}{2} \int_{Q(z)} |\nabla' \chi|\,dx'\,,\end{aligned}$$ since $\nabla'\chi = 4 \rho^{1/2}(\rho-1)\nabla'\rho^{1/2}$. This proves (\[eqnablaprimechi\]).
Let now $\chi_\ell$ be defined as in (\[eqdefaverage\]). By Jensen’s inequality and the mean-value theorem we obtain $$\int_{Q(z)} |\chi_\ell-\chi|\,dx' \le
\sup_{|h'|\le \ell}
\int_{Q(z)} |\chi(x'+h')-\chi(x')|\,dx' \le
\ell \int_{Q(z)} |\nabla'\chi|\,dx'\,.$$ Notice that this still holds if $\ell>L$ (indeed, in this case the coefficient could be improved to $L$). This concludes the proof of (\[eqchichiell\]).
### Equidistribution of the phases
We show that in every section $Q(z)$ with a good energy bound the volume fraction of the normal phase is approximately “right”, in the sense that it can be obtained assuming that $B_3$ equals $\kappa/\sqrt2$ in the normal phase, and zero outside.
\[lemmaequidistribution\] Let $(u,A)$ be admissible, and assume (\[eqparameters\]) and $\rho\le 1$.
1. If $$\label{eqEzgoodequip}
E(z)\le \frac{1}{8} \kappa {b_{\mathrm{ext}}}L^2\,,$$ then $$\int_{Q(z)} \chi \,dx'= \int_{Q(z)} (1-\rho)^2 \,dx'\sim \frac{{b_{\mathrm{ext}}}}{\kappa}L^2\,.$$
2. If $$\label{eqEzgoodequiptot}
E\le \frac{1}{8} \kappa {b_{\mathrm{ext}}}L^2T\,,$$ then $$\int_{Q_{L,T}} \chi \,dx= \int_{Q_{L,T}} (1-\rho)^2 \,dx\sim \frac{{b_{\mathrm{ext}}}}{\kappa}L^2T\,.$$
We start with assertion (i). Using Lemma \[lemmaHmeno12\] we write $$\begin{aligned}
1
{b_{\mathrm{ext}}}L^2 - \frac{\kappa}{\sqrt2} \int_{Q(z)}\chi\,dx' &=
\int_{Q(z)} \left(B_3 - \frac{\kappa}{\sqrt2} (1-\rho)^2 \right)\,dx'\\
&=
\int_{Q(z)} (1-\rho)\left(B_3 - \frac{\kappa}{\sqrt2} (1-\rho)\right)\,dx'
+\int_{Q(z)} \rho B_3\,dx'
\end{aligned}$$ and estimate, using Cauchy-Schwarz, (\[eqsectionrhohunemnorhodue\]), and $\kappa\le 1/2$, $$\begin{aligned}
1
\left|{b_{\mathrm{ext}}}L^2 - \frac{\kappa}{\sqrt2} \int_{Q(z)} \chi \, dx'\right|&\le
\left( \int_{Q(z)} (1-\rho)^2\,dx'\right)^{1/2} E^{1/2}(z) +8E(z)
\\
&\le \frac14\kappa \int_{Q(z)} \chi\,dx'+ \frac{E(z)}{\kappa}+4 \frac{E(z)}{\kappa}\,.\end{aligned}$$ The conclusion follows.
The second part is proven analogously, just extending all integrals to $Q_{L,T}$.
### Interior term
The next lemma is the key ingredient of our proof. It shows that, if the localization is performed appropriately, in low-energy sections the field $B_3$ necessarily concentrates. The concentration is made quantitative by a test function constructed via Lemma \[lemmatestfunction\] starting from $\chi=(1-\rho)^2$.
\[lemmainterior\] For any admissible $(u,A)$ with $\rho\le 1$, any parameters which obey (\[eqparameters\]), any $r\ge \ell>0$, and any $z\in(0,T)$ such that (\[eqEzgoodequip\]) holds one has $$\begin{aligned}
\frac\kappa{\sqrt2} \int_{Q(z)} \chi \,dx' - \int_{Q(z)} B_3 \psi \,dx'{\lesssim}\frac{E(z)}{\kappa}+\ell E(z)+
\left(\frac{r^2{b_{\mathrm{ext}}}L^2 }{\ell^2\kappa}\right)^{1/2} E^{1/2} (z)\,.\end{aligned}$$ The function $\psi$ is the one is obtained via Lemma \[lemmatestfunction\] from the restriction to $Q(z)$ of $\chi=(1-\rho)^2$.
We first write $$\begin{aligned}
\nonumber
\kappa \int_{Q(z)} \chi \,dx'
&= \kappa \int_{Q(z)} (\chi-\chi^2) dx' + \kappa\int_{Q(z)} \chi(\chi-\chi_\ell) dx'
+ \kappa\int_{Q(z)} \chi \chi_\ell dx'\,.\end{aligned}$$ We now estimate the three terms on the right-hand side. For the first one, we compute $$\begin{aligned}
\kappa \int_{Q(z)}( \chi-\chi^2) \,dx'&\le&
\kappa \int_{Q(z)} (1-\rho)^2(2\rho-\rho^2)\,dx'\\
&\le& 2 \kappa \int_{Q(z)} \rho(1-\rho)^2\,dx'{\lesssim}\frac{E(z)}{\kappa}\end{aligned}$$ by (\[eqrho6nc\]). For the second, $$\kappa\int_{Q(z)} \chi(\chi-\chi_\ell) dx' \le
\kappa\int_{Q(z)} |\chi-\chi_\ell| dx'
{\lesssim}\ell E(z)$$ by (\[eqchichiell\]). For the third, we write, recalling Lemma \[lemmatestfunction\](i) and the definition of $\chi$ and $\chi_\ell$, $$\begin{aligned}
1
\kappa\int_{Q(z)} \chi \chi_\ell dx'
&\le \kappa\int_{Q(z)} (1-\rho) \psi dx'\,.\end{aligned}$$ Therefore $$\begin{aligned}
\label{eqchiurho2}
\kappa \int_{Q(z)} \chi \,dx'&\le \kappa\int_{Q(z)} (1-\rho) \psi\, dx'+c\frac{E(z)}{\kappa}+c\ell E(z)\,.\end{aligned}$$ At this point we write $$\int_{Q(z)} B_3 \psi \,dx'=
\int_{Q(z)} \frac{\kappa}{\sqrt2} (1-\rho) \psi\,dx'+
\int_{Q(z)}\left(B_3- \frac{\kappa}{\sqrt2} (1-\rho)\right) \psi\,dx'\,.$$ We use $$\left| \int_{Q(z)}\left(B_3- \frac{\kappa}{\sqrt2} (1-\rho)\right) \psi\,dx'
\right| \le\|\psi\|_{L^2(Q_L)} E^{1/2} (z)$$ and (\[eqchiurho2\]) to obtain $$\begin{aligned}
\frac\kappa{\sqrt2} \int_{Q(z)} \chi \,dx' \le \int_{Q(z)} B_3 \psi \,dx'+c\frac{E(z)}{\kappa}+c\ell E(z)+
\|\psi\|_{L^2(Q_L)} E^{1/2} (z)\,.\end{aligned}$$ Recalling (\[eqlemmatestfunctl2\]), $$\|\psi\|_{L^2(Q_L)}^2
{\lesssim}\frac{r^2}{\ell^2}{\|\chi\|_{L^1}(Q_L)}{\lesssim}\frac{r^2{b_{\mathrm{ext}}}L^2 }{\ell^2\kappa}\,,$$ where we used once again Lemma \[lemmaequidistribution\](i), concludes the proof.
### Transport term {#transport-term}
It remains to relate the behavior of $B_3$ in the interior with the behavior at the boundary. As in the sharp-interface case, this is done in two steps, but both steps are different than the corresponding ones in Section \[secoldtransport\]. The first estimate is easy, but does not give the optimal bound, since it is oblivious to the fact that $B'$ needs to be concentrated on a small volume (this is relevant, since we are estimating an $L^1$ term with an energy that contains the corresponding $L^2$ norm). The estimate is then improved in the following lemma.
\[lemmatransportshort\] Let $(u,A)$ be admissible, $\psi\in W^{1,2}_{{\mathrm{loc}}}(\R^2)$, $Q_L$-periodic. For any pair $z_1, z_2\in (0,T)$ we have $$\left|\int_{Q(z_1)}B_3\psi \,dx'- \int_{Q(z_2)}B_3\psi\,dx' \right|
\le |z_2-z_1|^{1/2} E^{1/2} \|\nabla'\psi\|_{L^2(Q_L)}\,.$$
We can assume without loss of generality that $z_1<z_2$. We compute, using $\Div B=0$ and the $Q_L$-periodicity of $B$ and $\psi$, $$\begin{aligned}
1
\int_{Q_L} [B_3(\cdot,z_2)-B_3(\cdot,z_1)]\psi \,dx'
=&\int_{Q_L\times(z_1,z_2)}
\frac{\partial B_3}{\partial x_3} \psi\,dx
=-\int_{Q_L\times(z_1,z_2)}
\nabla'\cdot B' \psi\,dx\\
=&\int_{Q_L\times(z_1,z_2)}
B'\cdot\nabla\psi\,dx\\
&\le\left(\int_{Q_{L}\times(z_1,z_2)} |B'|^2\,dx\right)^{1/2}
\left(|z_2-z_1| \int_{Q_{L}} |\nabla\psi|^2 \,dx'\right)^{1/2}\\
&\le |z_2-z_1|^{1/2} E^{1/2} \|\nabla\psi\|_{L^2(Q_L)}\,.
\end{aligned}$$ This concludes the proof.
\[lemmatransportlong\] Let $(u,A)$ be admissible, and assume (\[eqparameters\]) holds and $\rho\le 1$. If (\[eqEzgoodequiptot\]) holds, then for any $z\in (0,T)$ and any $\psi\in W^{1,\infty}(\R^2)$, $Q_L$-periodic, we have $$\left|\int_{Q(z)}B_3\psi\,dx' - \int_{Q(0)}B_3\psi \,dx'\right|
{\lesssim}\left(\frac{{b_{\mathrm{ext}}}L^2T}{\kappa}\right)^{1/2}
\|\nabla'\psi\|_{L^\infty}E^{1/2}+ \|\nabla'\psi\|_{L^2(Q_L)} E^{1/2} \,.
$$
If $z\le 2$, this follows immediately from Lemma \[lemmatransportshort\]. Assume $z>2$, and fix $\delta\in(0,z/2)$. Let $\eta:\R\to\R$ be defined by $$\eta(x_3):=
\begin{cases}
\displaystyle
\frac{x_3}{\delta} & \text{ if } 0<x_3<\delta\,,\\
\displaystyle
1 & \text{ if } \delta\le x_3\le z-\delta\,,\\
\displaystyle
\frac{z-x_3}{\delta} & \text{ if } z-\delta<x_3<z\,,\\
0 & \text{ otherwise.}
\end{cases}$$ We compute $$\begin{aligned}
1
\int_{Q_{L,T}} B_3(x) &\psi(x') \frac{d\eta}{d x_3}(x_3) \,dx=
-\int_{Q_{L,T}} \frac{\partial B_3}{\partial x_3}(x) \psi(x')
\eta(x_3) \,dx\\
=& \int_{Q_{L,T}} \nabla'\cdot B' \psi \eta \,dx
= -\int_{Q_{L,T}} B' \cdot (\nabla' \psi) \eta\,dx\\
=& -\int_{Q_{L,T}} \rho B' \cdot (\nabla' \psi) \eta\,dx
- \int_{Q_{L,T}} (1-\rho) B' \cdot (\nabla' \psi) \eta\,dx\,.
\end{aligned}$$ The second term can be estimated by $$\begin{aligned}
1
&\hskip-1cm \left|\int_{Q_{L,T}} (1-\rho) B' \cdot (\nabla' \psi) \eta\,dx\right|\\
& \le \left( \int_{Q_{L,T}} (1-\rho)^2\,dx\right)^{1/2}
\left( \int_{Q_{L,T}} |B'|^2\,dx\right)^{1/2}
\|\nabla'\psi\|_{L^\infty}\,.
$$ Since (\[eqEzgoodequiptot\]) holds, Lemma \[lemmaequidistribution\] gives $$\int_{Q_{L,T}} (1-\rho)^2\,dx {\lesssim}\frac{{b_{\mathrm{ext}}}L^2T}{\kappa}$$ and therefore $$\left| \int_{Q_{L,T}} (1-\rho) B' \cdot (\nabla' \psi) \eta\,dx\right|
{\lesssim}\left(\frac{{b_{\mathrm{ext}}}L^2T}{\kappa}\right)^{1/2} E^{1/2}
\|\nabla'\psi\|_{L^\infty}\,.
$$ For the first term we use Lemma \[lemmaformula\] to obtain $$\begin{aligned}
\left| \int_{Q_{L,T}} \rho B' \cdot (\nabla' \psi) \eta \,dx\right|
& {\lesssim}& \int_{Q_{L,T}} \left(|\nabla_A u|^2+|{\mathcal{D}}_Au|^2\right)
|\nabla' \psi| \eta\,dx \\
&&\hskip5mm
+\left| \int_{Q_{L,T}} ( \nabla\times j_A) \cdot (\nabla' \psi) \eta\,dx\right| \,.\end{aligned}$$ The part containing $j_A$ can be transformed according to $$\int_{Q_{L,T}} \eta\, (\nabla\times j_A)\cdot\nabla'\psi \,dx=
-\int_{Q_{L,T}} \nabla\eta\cdot j_A\times \nabla'\psi\,dx\,.$$ This is easily proven by integration by parts and using the symmetry of the triple product. Since $\|\nabla'\psi \, \frac{d \eta}{dx_3} \|_{L^2(Q_{L,T})}=\|\nabla'\psi\|_{L^2(Q_L)}
\|\frac{d \eta}{dx_3} \|_{L^2((0,T))}$ and $\|j_A\|_{L^2(Q_{L,T})}{\lesssim}E^{1/2}$, we obtain $$\left| \int_{Q_{L,T}} \rho B' \cdot (\nabla' \psi) \eta \,dx\right|
{\lesssim}E \|\nabla'\psi\|_{L^\infty} + E^{1/2}\|\nabla'\psi\|_{L^2(Q_L)}
\|\frac{d \eta}{dx_3}\|_{L^2((0,T))} \,.$$ Adding terms, and replacing $\eta$ by $-\eta$, we conclude that $$\label{eqfaverage}
\left|\int_0^T f \frac{d \eta}{dx_3} \, dx_3 \right|{\lesssim}\left(\frac{{b_{\mathrm{ext}}}L^2T}{\kappa}\right)^{1/2} E^{1/2}
\|\nabla'\psi\|_{L^\infty}+ E^{1/2}\|\nabla'\psi\|_{L^2(Q_L)}
\|\frac{d \eta}{dx_3}\|_{L^2((0,T))}$$ where we dropped the term $ E \|\nabla'\psi\|_{L^\infty} $ using $\kappa\le 1$ and the assumption (\[eqEzgoodequiptot\]) and we defined $$f(x_3):=\int_{Q(x_3)} B_3 \psi\,dx'\,.$$ This estimate controls the variation of $f$ on a scale $\delta$, and can be combined with Lemma \[lemmatransportshort\], which gives a bound on the Hölder $1/2$ norm of $f$, to obtain a pointwise estimate. Precisely, since $$\int_0^T f \frac{d \eta}{dx_3} \,dx_3 = \frac1\delta\int_0^\delta f \,dx_3 - \frac1\delta
\int_{z-\delta}^{z} f\,dx_3\,,$$ we get $$|f(z)-f(0)|\le \left|\int_0^T f \frac{ d \eta}{dx_3} \,dx_3\right| +
\sup_{x_3\in(0,\delta)} |f(x_3)-f(0)|+
\sup_{x_3\in(z-\delta,z)} |f(x_3)-f(z)|\,.$$ Therefore $$\begin{aligned}
1
\left|f(z)-f(0)\right|
{\lesssim}&
\left(\frac{{b_{\mathrm{ext}}}L^2T}{\kappa}\right)^{1/2} E^{1/2}
\|\nabla'\psi\|_{L^\infty}+ E^{1/2}\frac{\|\nabla'\psi\|_{L^2(Q_L)}}{\delta^{1/2}}\\
& +\delta^{1/2}E^{1/2}\|\nabla'\psi\|_{L^2(Q_L)}\,.
\end{aligned}$$ We finally choose $\delta=1$ and conclude the proof.
### Exterior term
\[lemmaexterior\] For all admissible $(u,A)$ we have $$\left| \int_{Q(0)} (B_3-{b_{\mathrm{ext}}})\psi \,dx'\right|{\lesssim}E^{1/2} \|\psi\|_{H^{1/2}(Q_L)} \,.$$
This is the same as in the sharp-interface case, cf. Lemma \[lemmaextsharp\].
### Proof of the lower bound
\[theolowerbGL\] Let $(u,A)$ be an admissible pair. If $${b_{\mathrm{ext}}}\le\frac18 \kappa\,,\hskip1cm
\kappa\le\frac12\,,\hskip1cm\text{ and}\hskip1cm
\kappa T\ge 1\,,$$ one has $$E[u,A]{\gtrsim}\min \left\{{b_{\mathrm{ext}}}\kappa^{3/7} T^{3/7} L^2,
{b_{\mathrm{ext}}}^{2/3} \kappa^{2/3} T^{1/3}L^2\right\} \,.$$
By Lemma \[lemmarhominoredi1\] we can assume without loss of generality that $\rho\le 1$. We can assume $B-{B_{\mathrm{ext}}}\in L^2(Q_L\times\R;\R^3)$, otherwise the energy is infinite, so that we can use Lemma \[lemmaHmeno12\].
If (\[eqEzgoodequiptot\]) does not hold then $E{\gtrsim}{b_{\mathrm{ext}}}L^2 (\kappa T)^{3/7}$, since $\kappa T\ge 1$, and the proof is concluded. Therefore we can assume that (\[eqEzgoodequiptot\]) holds. We choose $z\in(0,T)$ such that $E(z)\le E[u,A]/T$, so that in particular (\[eqEzgoodequip\]) holds. Let $\psi$ be the function constructed via Lemma \[lemmatestfunction\] from the restriction to $Q(z)$ of $\chi=(1-\rho)^2$, as in Lemma \[lemmainterior\], for some parameters $r,\ell$ still to be chosen. We start from the identity $$ \int_{Q(z)} B_3\psi \,dx'= \int_{Q_L}{b_{\mathrm{ext}}}\psi \,dx' + \left[\int_{Q(z)}B_3\psi\,dx'
-\int_{Q(0)}B_3\psi\,dx'\right]+ \int_{Q(0)} (B_3-{b_{\mathrm{ext}}})\psi\,dx'\,.$$ From Lemma \[lemmatransportlong\] $$ \left|\int_{Q(z)}B_3\psi\,dx' - \int_{Q(0)}B_3\psi \,dx'\right|
{\lesssim}\frac1r \left(\frac{{b_{\mathrm{ext}}}L^2T}{\kappa}\right)^{1/2}
E^{1/2}+ \left(\frac{{b_{\mathrm{ext}}}L^2}{\kappa \ell^2}\right)^{1/2} E^{1/2} \,,$$ where we used $\|\nabla'\psi\|_{L^\infty}{\lesssim}1/r$ and $\|\nabla'\psi\|_{L^2(Q_L)}\le
\left(\frac{{b_{\mathrm{ext}}}L^2}{\kappa \ell^2}\right)^{1/2}$ (both obtained from Lemma \[lemmatestfunction\](iv) and (v) and Lemma \[lemmaequidistribution\](i)). Analogously, Lemma \[lemmaexterior\] shows that $$ \int_{Q(0)} (B_3-{b_{\mathrm{ext}}})\psi \,dx'{\lesssim}\left(\frac{r{b_{\mathrm{ext}}}L^2}{\kappa\ell^2
}\right)^{1/2} E^{1/2}\,,$$ where we inserted the bound on the $H^{1/2}$ norm of $\psi$ : $$\begin{aligned}
\|\psi\|_{H^{1/2}(Q_L)}&\le& \|\psi\|_{L^2(Q_L)}^{1/2}
\,\, \|\nabla \psi\|_{L^2(Q_L)}^{1/2}
{\lesssim}\left(\frac{r{b_{\mathrm{ext}}}L^2}{\kappa\ell^2 }\right)^{1/2}\,.
\end{aligned}$$ Further, by Lemma \[lemmatestfunction\](iii), $$\int_{Q_L}{b_{\mathrm{ext}}}\psi \,dx'\le {b_{\mathrm{ext}}}\|\psi\|_{L^1(Q_L)}\le 4{b_{\mathrm{ext}}}\frac{r^2}{\ell^2}
\int_{Q(z)} \chi \,dx'
\,,$$ therefore $$\begin{aligned}
\int_{Q(z)} B_3\psi \,dx'\le& 4{b_{\mathrm{ext}}}\frac{r^2}{\ell^2}
\int_{Q(z)} \chi \,dx'+
c\frac1r \left(\frac{{b_{\mathrm{ext}}}L^2T}{\kappa}\right)^{1/2}
E^{1/2}\\
&+ c\left(\frac{{b_{\mathrm{ext}}}L^2}{\kappa \ell^2}\right)^{1/2} E^{1/2}+c\left(\frac{r{b_{\mathrm{ext}}}L^2}{\kappa\ell^2
}\right)^{1/2} E^{1/2}\,.\end{aligned}$$ At this point we use Lemma \[lemmainterior\], which states that $$\begin{aligned}
\frac\kappa{\sqrt2} \int_{Q(z)} \chi \,dx'
\le \int_{Q(z)} B_3 \psi \,dx'+ c\frac{E(z)}{\kappa}+c\ell E(z)
+ c\left(\frac{r^2{b_{\mathrm{ext}}}L^2 }{\ell^2\kappa}\right)^{1/2} E^{1/2} (z)\,.\end{aligned}$$ Combining the previous estimate gives $$\begin{aligned}
\frac\kappa{\sqrt2} \int_{Q(z)} \chi \,dx'
\le& 4{b_{\mathrm{ext}}}\frac{r^2}{\ell^2} \int_{Q(z)} \chi \,dx'
+c \frac1r\left(\frac{{b_{\mathrm{ext}}}L^2T}{\kappa}\right)^{1/2}
E^{1/2}+ c \left(\frac{{b_{\mathrm{ext}}}L^2}{\kappa \ell^2}\right)^{1/2} E^{1/2} \\
& +c \left(\frac{r{b_{\mathrm{ext}}}L^2}{
\kappa\ell^2}\right)^{1/2} E^{1/2}
+c\frac{E(z)}{\kappa}+c\ell E(z)+c
\left(\frac{r^2{b_{\mathrm{ext}}}L^2 }{\kappa\ell^2}\right)^{1/2} E^{1/2} (z)\,.
\end{aligned}$$ Assume now that $$\label{eqdefadmissrlc}
0<\ell\le r\le \left(\frac{\kappa}{8{b_{\mathrm{ext}}}}\right)^{1/2} \ell\,,$$ so that the first term on the right is no larger than the one on the left divided by $\sqrt2$. Recalling Lemma \[lemmaequidistribution\](i) and $E(z)\le E/T$ we conclude that for all pairs $(r,\ell)$ which obey (\[eqdefadmissrlc\]), we have $$\begin{aligned}
{b_{\mathrm{ext}}}L^2 {\lesssim}&\frac1r\left(\frac{{b_{\mathrm{ext}}}L^2T}{\kappa}\right)^{1/2}
E^{1/2}+ \left(\frac{{b_{\mathrm{ext}}}L^2}{\kappa \ell^2}\right)^{1/2} E^{1/2} \\
& + \left(\frac{r{b_{\mathrm{ext}}}L^2}{
\kappa\ell^2}\right)^{1/2} E^{1/2}
+\frac{E}{\kappa T}+\ell \frac ET+
\left(\frac{r^2{b_{\mathrm{ext}}}L^2 }{\kappa\ell^2}\right)^{1/2} \frac{E^{1/2}}{T^{1/2}}\,.
\end{aligned}$$ We remark that only the second and the fourth term are new with respect to the sharp-interface case. At least one of the six terms has to be at least one-sixth of the total, therefore $$E{\gtrsim}\min \left\{
{b_{\mathrm{ext}}}L^2\frac{r^2 \kappa}{T},
{b_{\mathrm{ext}}}L^2\kappa \ell^2 ,
{b_{\mathrm{ext}}}L^2 \frac{\kappa\ell^2}{r},
{b_{\mathrm{ext}}}L^2\kappa T,
{b_{\mathrm{ext}}}L^2\frac{T}{\ell},
{b_{\mathrm{ext}}}L^2\frac{\kappa\ell^2 T}{r^2}
\right\}\,.$$ Equivalently, $$E{\gtrsim}\kappa {b_{\mathrm{ext}}}L^2T \min\left\{
\frac{r^2 }{T^2},
\frac{\ell^2}{T},
\frac{\ell^2}{rT},
1,
\frac{1}{\kappa\ell},
\frac{\ell^2}{r^2}
\right\}.$$ The fourth term can be dropped, since the sixth one is always less than 1. Therefore we can focus on $$\label{eqfinalboundFc}
E{\gtrsim}\kappa {b_{\mathrm{ext}}}L^2T
\min\left\{
\frac{r^2 }{T^2},
\frac{\ell^2}{T},
\frac{\ell^2}{rT},
\cdot,
\frac{1}{\kappa\ell},
\frac{\ell^2}{r^2}
\right\}\,,$$ where a dot marks the term we already know to be irrelevant. In comparing with (\[eqfinalboundF\]), we see that the only new term is the second one, $\ell^2/T$. Averaging the first and the sixth we see that $\ell/T$ would be irrelevant; hence the second term is irrelevant for all choices of $\ell{\gtrsim}1$. We shall see later that this is the case.
We finally have to choose $r$ and $\ell$, and check that in each case some terms give the optimal bound, and the others are irrelevant. Since we already know the scalings, we do not need to check all possible combinations. Balancing the first, third and fifth term suggests the choice $$\ell=T^{4/7}\kappa^{-3/7}\,, \hskip5mm r=T^{5/7}\kappa^{-2/7}\,.$$ This choice is admissible only if (\[eqdefadmissrlc\]) is satisfied. The first condition is always true, since $\kappa T\ge1$; the second one is equivalent to $(\kappa T)^{1/7}\le (\kappa/8{b_{\mathrm{ext}}})^{1/2}$. Since $\kappa \le 1 \le \kappa T$, one can compute $\ell=(\kappa T)^{4/7}/\kappa\ge 1$, hence the second term can indeed be dropped. In this case, (\[eqfinalboundFc\]) becomes $$E{\gtrsim}\kappa^{3/7}{b_{\mathrm{ext}}}L^2T^{3/7} \min\left\{
1,
\cdot,
1,
\cdot,
1,
(\kappa T)^{2/7}
\right\}\,.$$ Since $\kappa T\ge1$, in the regime $(\kappa T)^{1/7}\le (\kappa/8{b_{\mathrm{ext}}})^{1/2}$ we have shown $E{\gtrsim}\kappa^{3/7}{b_{\mathrm{ext}}}L^2T^{3/7}$.
If instead $(\kappa T)^{1/7}\ge (\kappa/8{b_{\mathrm{ext}}})^{1/2}$, we need to choose $r= \ell (\kappa/8{b_{\mathrm{ext}}})^{1/2}$. Then (\[eqdefadmissrlc\]) is always satisfied. Balancing the first and fifth term in (\[eqfinalboundFc\]) with this constraint results, after some rearrangement, into $$\ell = \frac{{b_{\mathrm{ext}}}^{1/3}T^{2/3}}{\kappa^{2/3}}$$ and $$E{\gtrsim}\kappa^{2/3}{b_{\mathrm{ext}}}^{2/3}L^2T^{1/3} \min\left\{
1,
\frac{T\kappa}{\kappa} \frac{{b_{\mathrm{ext}}}}{\kappa},
(\kappa T)^{1/3} \left(\frac{{b_{\mathrm{ext}}}}{\kappa}\right)^{7/6},
\cdot,
1,
(\kappa T)^{2/3} \left(\frac{{b_{\mathrm{ext}}}}{\kappa}\right)^{4/3}
\right\}\,.$$ Again, a dot marks terms we already know to be irrelevant. We observe that since $\kappa T\ge1$ and $(\kappa T)^{1/7}\ge(\kappa/8{b_{\mathrm{ext}}})^{1/2}$ one has $$(\kappa T)^{\alpha} \left(\frac{{b_{\mathrm{ext}}}}{\kappa}\right)^{\beta}
{\gtrsim}(\kappa T)^{\alpha-\frac27\beta} {\gtrsim}1$$ whenever $\alpha\ge\frac27\beta$. This permits to show that the second, third and sixth terms do not contribute, and therefore concludes the proof.
Upper bound {#secupperb}
===========
Before presenting our construction for the energy ${E_\mathrm{GL}}$ we construct fields with optimal scaling for the sharp-interface functional $F$. We prove a refined version of the results of [@ChoksiKohnOtto2004], giving a construction which satisfies several additional properties, which will be needed in the following generalization to ${E_\mathrm{GL}}$. In particular, we need to make sure that there is an integer number of flux quanta in each flux tube and that the thickening of the tubes on the scale of the correlation length still has small volume. The construction for ${E_\mathrm{GL}}$ will then be derived from this one.
Construction with sharp interfaces {#secconstrsharp}
----------------------------------
The key point in the construction is to use in two stages a subdivision of the domain into subsets with integer flux. This is done via the domain subdivision algorithm of Lemma \[lemmasubdivideflux\]. We first subdivide the domain down to scale $L/N$, and obtain rectangles $(r_j)_{j=1\,\dots, N^2}$, which will be used to fix the microstructure in the central section of the sample, $Q_L\times \{T/2\}$. In each of these rectangles, the magnetic field will be concentrated in a smaller concentric rectangle $\hat r_j$, keeping the same flux. Then we subdivide a second time, again using Lemma \[lemmasubdivideflux\], down to smaller rectangles, which will be the ones used close to the surfaces of the sample, $Q_L\times\{0\}$ and $Q_L\times\{T\}$. At the same time, the total flux inside each rectangle is concentrated in a smaller rectangle, so that the intensity of the magnetic field is the one preferred by the energy, $\kappa/\sqrt2$ (see Figure \[figpropchilower2\]). After this setup we will make the actual branching construction, which corresponds to the subdivision generated in the second application of Lemma \[lemmasubdivideflux\].
Before stating the main result of this section we recall the definition of $F$ in (\[eqdefF\]) and introduce the notation $(\omega)_\rho$ for a $\rho$-neighbourhood of a set $\omega$. Precisely, for $\omega \subset\R^3$ and $\delta>0$, $$\label{eqdefomegarho}
(\omega)_\delta:=\{x\in\R^3: \dist(x,\omega)<\delta\}=\bigcup_{x\in \omega} B_\delta(x)$$ where the distance is interpreted $Q_L$-periodically, $\dist(x,A):=\inf\{|x-z-kL|: z\in A, k\in \Z^2\times\{0\}\}$.
\[theochiupper1\] For any ${b_{\mathrm{ext}}},\kappa, L,T>0$ such that $$2{b_{\mathrm{ext}}}\le \kappa\le \frac12\,,\hskip1cm
\kappa T\ge 1 \,,\hskip1cm{b_{\mathrm{ext}}}L^2\in 2\pi \Z\,,$$ and $$\label{eqLadmissible}
L\ge \min\left\{
\frac{8T^{2/3}}{(\kappa {b_{\mathrm{ext}}})^{1/6}},
\frac{8T^{4/7}\kappa^{1/14}}{ {b_{\mathrm{ext}}}^{1/2}}
\right\}$$ there is a pair $(\chi,B)$, admissible in the sense of Definition \[defadmisssharp\], and such that $$F[\chi,B]{\lesssim}\min \left\{{b_{\mathrm{ext}}}\kappa^{3/7} T^{3/7} L^2,
{b_{\mathrm{ext}}}^{2/3} \kappa^{2/3} T^{1/3}L^2\right\}\,.$$ The set $\omega:=\{x\in Q_{L,T}: \chi(x)=1\}$ is formed by the union of finitely many sheared parallelepipeds, with two faces normal to $e_3$, and $$\label{eqestchiuppernbd}
|(\omega)_{1/\kappa}\setminus\omega| {\lesssim}\frac{1}{\kappa}\int_{Q_{L,T}} |D\chi|\,.$$ For any $z\in (0,T)$ the flux of $B$ across each connected section of $\{x_3=z\} \cap \omega$ is an integer multiple of $2\pi$.
![Sketch of the notation used in the proof of Theorem \[theochiupper1\]. \[figpropchilower2\]](fig1-crop.pdf){width="8cm"}
Before giving the proof of Theorem \[theochiupper1\], we formulate and prove the partial results that will be needed. We start with the domain subdivision. Here we refine a flux pattern making sure that each component keeps the quantization condition. In order for the field to maintain the optimal intensity, the areas are changed. Some parts of the construction would be simpler if one would work with squares, but then the macroscopic distribution of the flux would be modified. We work with rectangles, of aspect ratio uniformly close to 1, so that on any scale the perturbation to the distribution of flux is kept to a minimum.
\[lemmasubdivideflux\] Let $R_{0,1}=(0,a)\times (0,b)$ and $B_\ast>0$ be such that $\frac 13 a\le b\le 3a$ and $ab B_\ast\in 2\pi \Z$. Then for any $k\in \N$ there are $4^k$ pairwise disjoint rectangles $\{R_{k,i}\}_{i=1,\dots, 4^k}$, $R_{k,i}=x_{k,i}+(0,a_{k,i})\times(0, b_{k,i})$, such that $\frac 13 a_{k,i}\le b_{k,i}\le 3a_{k,i}$, each $R_{k,i}$ is the union of four $R_{k+1,i}$ (up to null sets), and $ a_{k,i}b_{k,i}B_\ast\in 2\pi\Z$ for all $k,i$. Further, $|a_{k,i}b_{k,i}-4^{-k}ab|\le 4\pi/B_\ast$.
The lemma follows immediately from Lemma \[lemmasubdivideflux2\], if one ignores half of the stages. Precisely, if $(R^*_{k,i})_{i=1,\dots, 2^k}$ are the rectangles produced by Lemma \[lemmasubdivideflux2\], we set $R_{k,i}=R^*_{2k,i}$, for $i=1,\dots, 4^k=2^{2k}$.
For the proof it is more convenient to focus on the following version, in which only one subdivision of each rectangle into two is performed at each step. To simplify the notation, we say that a rectangle $R=x'+(0,a)\times (0,b)\subset\R^2$ is $B_\ast$-good, for some $B_\ast>0$, if $$\frac 13 a\le b\le 3a\hskip3mm \text{ and } \hskip3mm abB_\ast\in 2\pi \Z\,.$$
\[lemmasubdivideflux2\] Let $B_*>0$, and let $R_{0,1}$ be a $B_\ast$-good rectangle. Then for any $k\in \N$ there are $2^k$ pairwise disjoint $B_\ast$-good rectangles $\{R_{k,i}\}_{i=1,\dots, 2^k}$ such that each $R_{k,i}$ is the union of two $R_{k+1,i}$ (up to null sets) and $||R_{k,i}|-2^{-k}|R_{0,1}||\le 4\pi/B_\ast$.
The construction is iterative, starting with $R_{0,1}$. Consider one rectangle at step $k$, say, $R_{k,i}=x_{k,i}+(0,a_{k,i})\times(0, b_{k,i})$. Assume for definiteness that $a_{k,i}\le b_{k,i}$. If $R_{k,i}$ is empty, i.e., has side lengths zero, we replace $R_{k,i}$ by two empty rectangles, setting $R_{k+1,2i}:=R_{k+1,2i+1}:=\emptyset$. If $B_\ast a_{k,i}b_{k,i}=2\pi$, we replace it by a copy of itself and an empty rectangle, setting $R_{k+1,2i}:=R_{k,i}$ and $R_{k+1,2i+1}:=\emptyset$. Otherwise, we set $$y:=\frac{2\pi}{B_\ast a_{k,i}} \left\lfloor \frac{B_\ast a_{k,i}b_{k,i}}{4\pi}\right\rfloor$$ and replace $R_{k,i}$ by the two rectangles $x_{k,i}+(0,a_{k,i})\times (0,y)$ and $x_{k,i}+(0,a_{k,i})\times (y,b_{k,i})$. Here $\lfloor t\rfloor=\max\{z\in\Z: z\le t\}$.
Clearly $y\le b_{k,i}/2$. At the same time, since for any $z\in\N$ with $z\ge 2$ one has $\lfloor z/2\rfloor \ge z/3$, one has $y\ge b_{k,i}/3$. Therefore the aspect ratio of the two new rectangles is also not larger than 3 and they are $B_\ast$-good.
It remains to estimate the area. Consider one rectangle $R_{k,i}$ at stage $k$. It has been generated from $R_{0,1}$ by $k$ subdivision steps. Let $s_h$ be the area of the rectangle at stage $h$ along this subdivision path, so that $s_0:=|R_{0,1}|$. By the definition of $y$, we obtain $|2s_h-s_{h-1}|\le 4\pi/B_\ast$. Summing the series we obtain $$|2^ks_k-|R_{0,1}|| \le \sum_{h=1}^k |2^h s_h-2^{h-1} s_{h-1}|
\le \frac{4\pi}{B_\ast} \sum_{h=1}^k 2^{h-1}
\le 2^k\frac{4\pi}{B_\ast} \,,$$ which gives $|s_k-2^{-k}|R_{0,1}||\le 4\pi/B_\ast$.
We next estimate the energy of a flux configuration on the boundary. We assume that the magnetic field inside each rectangle $R_j$ is concentrated in a subrectangle $r_j$. The difference of the fields $a_j{\mathds{1}}_{r_j}-A_j{\mathds{1}}_{R_j}$ then has average zero over the larger rectangle $R_j$. Here and below we denote by ${\mathds{1}}_E$ the characteristic function of a set $E$, ${\mathds{1}}_E(x)=1$ if $x\in E$, $0$ otherwise.
\[lemmaestimatehminus12\] For $M\in\N$ let $$\{R_1,\dots, R_{M}\}\hskip5mm
\text{ and }\hskip5mm
\{r_1,\dots, r_{M}\}$$ be rectangles such that each of them has aspect ratio not larger than 3, $r_j\subset R_j\subset Q_L$, with the $R_j$ pairwise disjoint. Let $a_j, A_j\in\R$ be such that $a_j| r_j|=A_j|R_j|$. Then $$\left\| \sum_{j=1}^M a_j {\mathds{1}}_{r_j}-A_j{\mathds{1}}_{R_j}\right\|_{H^{-1/2}(Q_L)}^2 {\lesssim}\sum_{j=1}^M a_j^2 \, |r_j|^{3/2}\,.$$
Fix one index $j$, and let $g:=a_j {\mathds{1}}_{r_j} -A_j{\mathds{1}}_{R_j}$. Since $g$ has average 0 over $R_{j}$, for any $\varphi\in L^1(R_{j})$ we have $$\left| \int_{R_{j}} g \varphi\, dx' \right|=
\left| \int_{R_{j}} g (\varphi-\varphi_0) dx'\right| =
\left|\int_{r_j} a_j (\varphi-\varphi_0) dx' \right|
\le |a_j| \, \|\varphi-\varphi_0\|_{L^1(r_j)}\,,$$ where $\varphi_0$ is the average of $\varphi$ over $R_{j}$. Since the trace of a $H^1$ function (in three dimensions) belongs to $L^4$, if $\varphi\in H^1_{{\mathrm{loc}}}(R_j\times(0,\infty))$ we obtain $$\|\varphi-\varphi_0\|_{L^1(r_j)}\le
| r_j|^{3/4}\|\varphi-\varphi_0\|_{L^4( r_j)}{\lesssim}| r_j|^{3/4}\|\nabla \varphi\|_{L^{2}(R_{j}\times(0,\infty))} \,.$$ Since the aspect ratio of the rectangles is controlled, the constant is universal.
Now fix $\Phi\in H^{1}_{{\mathrm{loc}}}(Q_L\times(0,\infty))$, $Q_L$-periodic. Let $\varphi_j$ be the average of $\Phi$ over $R_j$. Then the same computation gives $$\begin{aligned}
\left| \int_{Q_L}\sum_j \left( a_j {\mathds{1}}_{r_j}-A_j{\mathds{1}}_{R_j}\right)\Phi\, dx' \right|
&\le \sum_j \left|\int_{r_j} a_j (\Phi-\varphi_j) dx' \right|
{\lesssim}\sum_j |a_j| \, |r_j|^{3/4} \|\nabla \Phi\|_{L^{2}(R_{j}\times(0,\infty))}\\
& \le \left(\sum_j a_j^2 \, |r_j|^{3/2}\right)^{1/2} \|\nabla \Phi\|_{L^{2}(Q_L\times(0,\infty))}\,,
\end{aligned}$$ where in the last step we used Cauchy-Schwarz.
Controlling the $H^{-1/2}$ norm of the normal component of $B$ on the boundary is sufficient to estimate the energetic cost of the magnetic field outside the sample. We recall this general fact in the following Lemma.
\[lemmah12\] Let $g\in L^2_{{\mathrm{loc}}}(\R^2)$, $Q_L$-periodic, with average ${b_{\mathrm{ext}}}\in\R$. Then there is $B\in L^2_{{\mathrm{loc}}}(\R^3;\R^3)$, also $Q_L$-periodic, such that $\Div B=0$, $B_3(x',x_3)=g(x')$ for $x_3\ge 0$, and $$\int_{Q_L\times(-\infty,0)} |B-{b_{\mathrm{ext}}}e_3|^2dx{\lesssim}\|g-{b_{\mathrm{ext}}}\|_{H^{-1/2}(Q_L)}^2\,.$$
It suffices to consider the case ${b_{\mathrm{ext}}}=0$. Let $\hat g(k')$ be the Fourier coefficients of $g$, so that $g(x')=\sum_{k'\in 2\pi\Z^2/L} e^{ik'\cdot x'} \hat g(k')$ and $\sum_{k'\ne0} |\hat g|^2(k')/|k'|\sim \|g\|_{H^{-1/2}}^2$. Since $g$ has average $0$, $\hat g(0)=0$. We define, for $x_3\le0$, $$B(x',x_3):=\sum_{k'\in 2\pi\Z^2/L} e^{ik'\cdot x'} \hat B(k',x_3)$$ where $$\hat B_3(k',x_3):=\hat g(k') e^{|k'|x_3}\,, \hskip5mm
\hat B'(k',x_3):=i\frac{k'}{|k'|} \hat g(k') e^{|k'|x_3}\,.$$ It is then straightforward to check that the stated properties are satisfied.
Before starting the construction in the interior region, we introduce a separate notation for the interior contribution to the energy. We define, for $\Omega\subset\R^3$ open, $\chi\in BV(\Omega)$ and $B\in L^2(\Omega;\R^3)$, $$\label{eqdefFint}
F^\Int[\chi,B,\Omega] := \int_{\Omega} \kappa |D\chi| +\int_{\Omega} \left[|B'|^2 + \chi\left(B_3
- \frac{\kappa}{\sqrt2}\right)^2 \right]dx,$$ so that $F[\chi,B]=F^\Int[\chi,B,Q_{L,T}]+ \int_{Q_L\times[\R\setminus(0,T)]} |B-{b_{\mathrm{ext}}}e_3|^2 dx$. By Lemma \[lemmah12\] it suffices to control $F^\Int[\chi,B,Q_{L,T}]+\|B-{b_{\mathrm{ext}}}\|_{H^{-1/2}(Q_L\times\{0\})}^2+\|B-{b_{\mathrm{ext}}}\|_{H^{-1/2}(Q_L\times\{T\})}^2$.
The next construction step is a procedure to generate an admissible $\chi$ and $B$ with given boundary data in a slab $\R^2\times (0,t)$. The explicit construction is done for the case that the boundary data are characteristic functions of rectangles. An extension to the case where circles are used, which gives a smaller surface energy (by a factor which does not affect the scaling), is discussed in [@ContiGoldmanOttoSerfaty].
\[lemmatransformrectintorect\] Let $r:=p+(0,a)\times (0,b)$, $\hat r:=\hat p+(0,\hat a)\times (0,\hat b)\subset\R^2$ be two rectangles with $|r|=|\hat r|$, $a\sim b$ and $\hat a\sim \hat b$. For any $t{\gtrsim}a$ there are $(\chi,B):\R^2\times[0,t]\to\{0,1\}\times \R^3$ such that $$B_3(\cdot, \cdot, 0)=\frac\kappa{\sqrt2}{\mathds{1}}_r\,,\hskip1cm
B_3(\cdot, \cdot, t)=\frac\kappa{\sqrt2}{\mathds{1}}_{\hat r}\,\,,\hskip1cm \Div B=0\,,$$ the first two in the sense of traces, and such that, defining $\omega:=\{x: B(x)\ne 0\}$, $$B_3=\frac\kappa{\sqrt2}\chi=\frac\kappa{\sqrt2}{\mathds{1}}_\omega \,,\hskip3mm
F^\Int[\chi,B,\R^2\times(0,t)]{\lesssim}\kappa a t+ \kappa a |p-\hat p|+\kappa^2
\frac{|p-\hat p|^2+ab}{t} ab\,.$$ If $R=(x_1,y_1)\times(x_2,y_2)$ is such that $r\cup\hat r\subset R$, then $\omega\subset R\times[0,t]$. For any $\delta{\lesssim}a$, $$|(\omega)_\delta\setminus \omega|{\lesssim}\delta a t$$ with $(\omega)_\delta$ defined as in (\[eqdefomegarho\]).
We define $\varphi:\R\to\R$ by $$\varphi(x_3):= \exp\left( \frac{x_3}{t} \ln\frac{\hat a}{a}\right)
= \left(\frac{\hat a}{a}\right)^{x_3/t}\,.$$ Then $\varphi(0)=1$, $\varphi(t)=\hat a/a=b/\hat b$, $|\varphi'|(x_3){\lesssim}1/t$ for $x_3\in[0,t]$. Further, we define $$v(x):=
\frac{x_3}{t}
\begin{pmatrix}
\hat p_1-p_1\\ \hat p_2-p_2\\0
\end{pmatrix}
+
\begin{pmatrix}
x_1\varphi(x_3) \\ x_2/\varphi(x_3) \\ x_3
\end{pmatrix}\,,$$ which is a diffeomorphism of $\R^2\times[0,t]$ into itself, with $\det \nabla v=1$ pointwise, and finally define $\chi$ and $B$ by $$\label{eqdefBv}
\chi(v(x)):={\mathds{1}}_r(x')\,, \hskip2mm
B_3(v(x)):=\frac\kappa{\sqrt2}{\mathds{1}}_r(x')\,, \hskip2mm
B'(v(x)):=\frac\kappa{\sqrt2}{\mathds{1}}_r(x')\partial_3 v'(x)\,.$$ Let $\theta\in C^1_c(\R^2\times (0,t))$. By a change of variables $$\int_{\R^2\times(0,t)} (\partial_3\theta\, B_3 + \nabla'\theta \cdot B' )dx
=\int_{\R^2\times(0,t)} (\partial_3\theta \circ v \, B_3\circ v + \nabla'\theta \circ v \cdot B'\circ v )dx\,.$$ Inserting the definition from (\[eqdefBv\]) this becomes $$\frac\kappa{\sqrt2} \int_{\R^2\times (0,t)} [\partial_3\theta \circ v \,{\mathds{1}}_r+ \nabla'\theta \circ v\cdot ({\mathds{1}}_r\, \partial_3 v') ]dx
=\frac\kappa{\sqrt2}\int_{\R^2\times (0,t)} {\mathds{1}}_r(x')\frac{d}{dx_3} (\theta\circ v)(x)dx=0\,.$$ Therefore the divergence condition is satisfied.
Finally, $\|B'\|_{L^\infty}\le \kappa \|{\mathds{1}}_r\partial_3 v'\|_{L^\infty}\le \kappa |p-\hat p|/t + c\kappa a/t$. Since $a\sim b$ and the volume of its support is $tab$, we obtain $$\int_{\R^2\times (0,t)} |B'|^2 dx {\lesssim}\kappa^2\frac{|p-\hat p|^2+ab}{t} ab\,.$$ From the definition of $\omega$ one easily obtains the other properties.
At this point we present the branching construction, which gives the refinement of the magnetic flux close to the boundary. We start from a prescribed interior structure, which can be obtained either by uniform subdivision of the domain and quantization of the field, or by nonuniform subdivision of the domain with a uniform field. We stress that the rectangles in which the construction is localized do not need to cover $Q_L$. Indeed, for very small fields the optimal scaling is only obtained if these rectangles cover a very small fraction of $Q_L$ (the volume fraction will be $\gamma^2$ in the proof of Theorem \[theochiupper1\] below), see Figure \[figinternal\].
![Sketch of the horizontal geometry in Lemma \[lemmaconstrinternal\]. Left panel: cross-section at $x_3=T$, with the initial rectangles $r_j$ (yellow) and $\hat r_j$ (red) shown. Right panel: cross-section at $x_3=0$, with the finer structure of $B_3(\cdot,0)$ shown.[]{data-label="figinternal"}](fig4-crop.pdf){height="6cm"}
\[lemmaconstrinternal\] Let $\kappa,L,T, d_0,\rho_0,N>0$, with $N^2\in\N$ and $\kappa\le 1$, and assume that two sets of $N^2$ rectangles are given, $$\{r_1,\dots, r_{N^2}\}\hskip5mm
\text{ and }\hskip5mm
\{\hat r_1,\dots, \hat r_{N^2}\}\,,$$ such that each of them has aspect ratio no larger than 3, $\hat r_j\subset r_j\subset Q_L$, the $r_j$ are pairwise disjoint, $|\hat r_j|\sim \rho_0^2$, $|r_j|\sim d_0^2$, with $$\frac\kappa{\sqrt2}| \hat r_j|\in 2\pi \N \hskip5mm\text{ for all $j$. }$$ Assume $$\label{eqkapparho0}
\kappa\rho_0{\gtrsim}1 \,.$$ Then there is a pair $(\chi,B)$, admissible in the sense of Definition \[defadmisssharp\], and such that $$B_3(\cdot,T)=
\frac\kappa{\sqrt2} \sum_{j=1}^{N^2} {\mathds{1}}_{\hat r_j}\hskip1cm\text{ on $Q_L$}\,,$$ $$\label{eqfintpropcostr}
F^\Int[\chi,B,Q_{L,T}]
{\lesssim}\kappa \rho_0 TN^2+\kappa^2 \frac{\rho_0^2 d_0^2N^2}{T}$$ and, with $\hat b_j:=\kappa/\sqrt2 (|\hat r_j|/|r_j|)$, $$\label{eqhm12QLd}
\left\|B_3(\cdot, 0)-\sum_{j=1}^{N^2} \hat b_j {\mathds{1}}_{r_j}
\right\|_{H^{-1/2}(Q_L)}^2
{\lesssim}{\kappa\rho_0^2N^2} + \frac{\kappa^{2}\rho_0^3N^2d_0}{T} \,.$$ The set $\omega:=\{x\in Q_{L,T}: \chi(x)=1\}$ is formed by the union of finitely many sheared parallelepipeds, with two faces normal to $e_3$, and $$\label{eqestchiuppernbd2}
|(\omega)_{1/\kappa}\setminus\omega| {\lesssim}\frac{1}{\kappa}\int_{Q_{L,T}} |D\chi|\,.$$ For any $z\in (0,T)$ the flux of $B$ across each connected component of $\{x_3=z\} \cap \omega$ is an integer multiple of $2\pi$.
We first observe that a simple construction which obeys all kinematic constraints is obtained using a pattern which does not depend on $x_3$. From the boundary data at $x_3=T$ we see that this necessarily is $$\hat B=(0,0,\frac{\kappa}{\sqrt2}\hat\chi)\,,\hskip1cm \hat\chi(x',x_3)=\sum_{j=1}^{N^2} {\mathds{1}}_{\hat r_j}(x')\,.$$ A simple computation shows that $$F^\Int[\hat\chi,\hat B, Q_{L,T}] {\lesssim}\kappa\rho_0 TN^2$$ and, using Lemma \[lemmaestimatehminus12\], $$\left\|\hat B_3(\cdot,0)-\sum_{j=1}^{N^2} \hat b_j {\mathds{1}}_{r_j}
\right\|_{H^{-1/2}(Q_L)}^2
{\lesssim}\sum_j \kappa^2|\hat r_j|^{3/2} {\lesssim}\kappa^2N^2\rho_0^3\,.$$ At the same time $|(\omega)_{1/\kappa}\setminus\omega|{\lesssim}N^2T (\rho_0/\kappa+1/\kappa^2)$. If $T{\lesssim}d_0$, the proof is concluded. We observe that if $\kappa\rho_0{\lesssim}1$ the energy estimates would also hold, but not the one on the measure of $(\omega)_{1/\kappa}$. In the following we assume $T\gg d_0$.
We choose $I\in\N$ such that $$\label{eqconstrIcond}
2^I\sim \min\left\{\kappa\rho_0, \frac{T}{d_0}\right\}\,.$$ Possibly reducing $I$ by a few units, which does not affect the statement, we can assume that $$\label{eq4iadm}
4^I \le \frac{\hat b_j|r_j|}{8\pi} = \frac{\kappa|\hat r_j|}{8\pi \sqrt2}\hskip5mm \text{ for all $j$}$$ (to see this, one observes that $\kappa|\hat r_j|\sim\kappa\rho_0^2>(\kappa\rho_0)^2$). The construction is performed independently in each set $r_j\times (0,T)$, we take both fields to vanish outside the union of these sets. Let $R_{j,i,h}$ be the rectangles given in Lemma \[lemmasubdivideflux\] for $1\le i\le I$, $1\le h\le 4^i$, starting from $r_j$, using the field $B_\ast=\hat b_j$. We denote by $i$ the refinement level, by $h$ the numbering of the rectangles at each level. By construction we have $||R_{j,i,h}|-4^{-i}|r_j||\le 4\pi/\hat b_j$ for all $j,i,h$; with (\[eq4iadm\]) we obtain $|R_{j,i,h}|\ge 4^{-i}|r_j| - 4\pi/\hat b_j
= 4^{-i}|r_j| (1- 4\pi 4^i/(|r_j|\hat b_j))
\ge \frac12 4^{-i} |r_j|$, and analogously $|R_{j,i,h}|\le \frac32 4^{-i}|r_j|$. Therefore $|R_{j,i,h}|\sim 4^{-i}|r_j|$.
We localize the magnetic field in an appropriate subset of each of the $R_{j,i,h}$. This is done by the family of inner rectangles $(\hat R_{j,i,h})_{h=1,\dots, 4^i}$ which we now define: we let $\hat R_{j,i,h}$ have the same center and aspect ratio as $R_{j,i,h}$, and area given by $$|\hat R_{j,i,h}| \frac{\kappa}{\sqrt2} = |R_{j,i,h}| \hat b_j\,.$$ We recall that $\hat b_j$ is defined so that $\hat b_j|r_j|=\kappa |\hat r_j|/\sqrt2\in 2\pi\N$, in particular $\hat b_j\le \kappa/\sqrt2$. Therefore, $\hat R_{j,i,h}\subset R_{j,i,h}$, and the locally optimal field $\kappa/\sqrt2$ carries over $\hat R_{j,i,h}$ the same flux that $\hat b_j$ carries over $R_{j,i,h}$. At the level $i=0$ this definition gives $\hat R_{0,j}=\hat r_j$.
To estimate the size of the rectangles we define $$d_i:=2^{-i}d_0\hskip5mm \text{ and }\hskip5mm
\rho_i:=\rho_0 2^{-i} \,.$$ Then $|R_{j,i,h}|\sim 4^{-i}|r_j|\sim d_i^2$ and, correspondingly, $|\hat R_{j,i,h}|
=|R_{j,i,h}|\,|\hat r_j|/|r_j|
\sim \rho_i^2$ for all $i=1,\dots,I$.
![Sketch of the vertical structure in Lemma \[lemmaconstrinternal\]. The thickness of the tubes at stage $i$ is $\rho_i$, their horizontal separation $d_i$, the distance between two vertical steps $t_i$. []{data-label="figvertbr"}](fig5-crop.pdf){height="5.5cm"}
We now define the vertical structure, see Figure \[figvertbr\]. The refinement steps, labeled by $i$ in the decomposition of the rectangles, will describe the structure at different levels, which are labeled $y_i$. Precisely, for some $\theta\in (0,1)$ chosen below, we define $$y_i:= T \theta^i, \hskip5mm
t_i:=y_i-y_{i+1}= T \theta^i (1-\theta)\,.$$ By (\[eqconstrIcond\]), if $\theta\ge 1/4$ we obtain $t_i{\gtrsim}d_i$ for all $i\le I$.
The explicit construction in $Q_L\times (y_{i+1}, y_i)$, for $i=0, \dots, I-1$, is done using Lemma \[lemmatransformrectintorect\]. At each step we interpolate between $\hat R_{j,i,h}$ and the four corresponding $\hat R_{j,i+1,h_l}$, $l=1,2,3,4$. Since the side length of $R_{j,i,h}$ is controlled by $d_i$, the side length of the inner rectangle where the field is located is controlled by $\rho_i$, and $d_i{\lesssim}t_i$, the energy in $Q_L\times (y_{i+1}, y_i)$ is bounded by $$F^\Int[\chi,B,Q_L\times (y_{i+1}, y_i)]{\lesssim}\sum_{j=1}^{N^2} \sum_{h=1}^{4^i} \left[ \kappa \rho_it_i +\kappa^2 \rho_i^2\frac{d_i^2}{t_i}
\right] {\lesssim}N^2\kappa \rho_0t_0(2\theta)^i + N^2\kappa^2 \frac{\rho_0^2 d_0^2}{t_0 (4\theta)^i} \,.$$ The series converges for all $\theta\in (1/4, 1/2)$. Choosing $\theta=1/3$ we conclude that the energy in $Q_L\times (y_I,T)$ is bounded by $$F^\Int[\chi,B,Q_L\times (y_{I}, T)]{\lesssim}\sum_{i=1}^I \sum_{j=1}^{N^2} \sum_{h=1}^{4^i}\left[ \kappa \rho_it_i +\kappa^2 \rho_i^2\frac{d_i^2}{t_i}
\right] {\lesssim}\kappa \rho_0 TN^2+\kappa^2\rho_0^2 \frac{d_0^2 N^2}{T}\,.$$ We still have to consider the region $Q_L\times(0, y_I)$. Here we set $B'=0$ and $(\chi,B)(x',x_3)=(\chi,B)(x', y_I)$. The only energy contribution comes from the surface term, and is $N^24^I\kappa\rho_I y_I$. Since $y_I{\lesssim}t_I$, this is controlled by the last summand in the series, and therefore does not change the scaling of the total energy.
We finally estimate the boundary term. Using Lemma \[lemmaestimatehminus12\] we obtain $$\|\sum_{j=1}^{N^2}\sum_{h=1}^{4^I}\left( \frac{\kappa}{\sqrt2} {\mathds{1}}_{\hat R_{j,I,h}} - \hat b_j {\mathds{1}}_{R_{j,I,h}} \right)
\|_{H^{-1/2}(Q_L)}^2 {\lesssim}\kappa^2 \sum_{j=1}^{N^2}\sum_{h=1}^{4^I} |\hat R_{j,I,h}|^{3/2}
{\lesssim}\frac{N^2\kappa^{2}\rho_0^3}{2^I} \,.$$ We observe that $\sum_h {\mathds{1}}_{R_{j,I,h}}={\mathds{1}}_{r_j}$, by the construction of the rectangles. Inserting, from the definition (\[eqconstrIcond\]), $2^{-I}\sim (\kappa\rho_0)^{-1}+d_0/T$, concludes the proof of (\[eqhm12QLd\]).
To prove (\[eqestchiuppernbd2\]) from Lemma \[lemmatransformrectintorect\] it suffices to show that $1/\kappa{\lesssim}\rho_I$, i.e., $\kappa\rho_I{\gtrsim}1$. Since $\kappa\rho_I=2^{-I}\kappa\rho_0 $, this follows immediately from (\[eqconstrIcond\]).
We fix $k\in\N$, set $N:=2^{k}$, and let $\{R_{k,j}\}_{1\le j\le N^2}$ be the rectangles given in Lemma \[lemmasubdivideflux\] applied to $R_{0,1}:=Q_L=(0,L)^2$ with $B_\ast:={b_{\mathrm{ext}}}$. We assume $$\label{eqcondNcostr}
N^2\le \frac{{b_{\mathrm{ext}}}L^2}{8\pi}$$ so that $|R_{k,j}|\sim L^2/N^2$. We first localize the flux to the central part of each of the rectangles. This is the step in which the two regimes differ. Fix a factor $\gamma\in (0,1]$, chosen below, and let $r_j$ be a rectangle with the same center and aspect ratio as $R_{k,j}$, scaled by a factor $\gamma$. We shall concentrate the entire flux over $R_{k,j}$ into the smaller rectangle $r_j$, so that the magnetic field over $r_j$ is $\hat b:={b_{\mathrm{ext}}}/\gamma^2$; from ${b_{\mathrm{ext}}}|R_{k,j}|\in 2\pi \Z$ we obtain $\hat b |r_j|\in2\pi\Z$. Since ${b_{\mathrm{ext}}}=\sum_{j=1}^{N^2} {b_{\mathrm{ext}}}{\mathds{1}}_{R_{k,j}}$, Lemma \[lemmaestimatehminus12\] yields $$\begin{aligned}
1
\|\sum_{j=1}^{N^2} \hat b {\mathds{1}}_{r_j} - {b_{\mathrm{ext}}}\|_{H^{-1/2}(Q_L)}^2 &
{\lesssim}\sum_j \hat b^2|r_j|^{3/2} {\mathds{1}}_{\gamma<1}
{\lesssim}{b_{\mathrm{ext}}}^2 \frac{L^3}{N \gamma} {\mathds{1}}_{\gamma<1}\,.\label{eqhm12QLc}
\end{aligned}$$ The factor $ {\mathds{1}}_{\gamma<1}$ represents the fact that this term is only present if $\gamma<1$. In the case $\gamma=1$, indeed, we have $\sum {\mathds{1}}_{r_j}=1$ on $Q_L$, and therefore this term vanishes.
We then use Lemma \[lemmaconstrinternal\] with the given set of $r_j$, $d_0=\gamma\frac{L}{N}$, $\kappa$, $L$, $N$ as above, and thickness $T/2$, on the set $Q_L\times(0,T/2)$ (the other half is symmetric and not discussed explicitly). The inner rectangles are chosen so that $\hat r_j$ has the same center and aspect ratio as $r_j$ and area given by $\kappa|\hat r_j|/\sqrt2=\hat b |r_j|$, correspondingly the length scale is $$\rho_0:=d_0 \left(\frac{\hat b \sqrt2}{\kappa}\right)^{1/2}
=\frac LN \left(\frac{{b_{\mathrm{ext}}}\sqrt2}{\kappa}\right)^{1/2}\,.$$ Since $\hat r_j$ needs to be a subset of $r_j$, this is possible only if $\hat b\le \kappa/\sqrt2$, therefore it is only possible if $\gamma$ is chosen such that $$\label{eqconstrgammacond}
\left(\frac{{b_{\mathrm{ext}}}\sqrt2}{\kappa}\right)^{1/2}\le\gamma \le 1\,.$$ Since ${b_{\mathrm{ext}}}\le \kappa/2$, this set is non empty. At the same time, the condition (\[eqkapparho0\]) is satisfied provided that $$\label{eqkapparho0subst}
({b_{\mathrm{ext}}}\kappa)^{1/2} \frac{L}{N} \ge1\,.$$ To estimate the boundary term we combine (\[eqhm12QLc\]) with (\[eqhm12QLd\]) $$\begin{aligned}
\| B_3(\cdot,0)-{b_{\mathrm{ext}}}\|_{H^{-1/2}(Q_L)}^2
& \le 2 \|B_3(\cdot,0)-\sum_{j=1}^{N^2} \hat b {\mathds{1}}_{r_j}\|_{H^{-1/2}(Q_L)}^2
+2\|\sum_{j=1}^{N^2} \hat b {\mathds{1}}_{r_j}-{b_{\mathrm{ext}}}\|_{H^{-1/2}(Q_L)}^2
\end{aligned}$$ and obtain $$ \|B_3(\cdot,0) - {b_{\mathrm{ext}}}\|_{H^{-1/2}(Q_L)}^2 {\lesssim}{b_{\mathrm{ext}}}^2 \frac{L^3}{N \gamma}{\mathds{1}}_{\gamma<1}+ \kappa\rho_0^2N^2 + \frac{\kappa^{2}\rho_0^3N^2d_0}{T} \,.$$ We then extend $B$ to $Q_L\times(-\infty,0)$ by Lemma \[lemmah12\], and define $\chi$ and $B$ on $Q_L\times (T/2,\infty)$ by symmetry, $(\chi,B)(x',T/2+z)=(\chi,B)(x',T/2-z)$. Recalling (\[eqfintpropcostr\]) we see that the total energy is bounded by $$F[\chi,B]{\lesssim}\kappa \rho_0 TN^2+\kappa^2 \frac{\rho_0^2 d_0^2N^2}{T}
+{b_{\mathrm{ext}}}^2 \frac{L^3}{N \gamma}{\mathds{1}}_{\gamma<1}+ \kappa\rho_0^2N^2 + \frac{\kappa^{2}\rho_0^3N^2d_0}{T}\,.$$ Since $\rho_0\le d_0$, the last term is smaller than the second one and therefore can be neglected. Dividing by the area $L^2$ and inserting the definitions $\rho_0=L {b_{\mathrm{ext}}}^{1/2}/(N \kappa^{1/2})$ and $d_0= L\gamma/N$ gives $$\label{equbfchibsdf}
\frac{F[\chi,B]}{L^2}{\lesssim}\kappa^{1/2} {b_{\mathrm{ext}}}^{1/2} \frac{TN}{L}+ \kappa {b_{\mathrm{ext}}}\frac{L^2\gamma^2}{N^2T}+
{b_{\mathrm{ext}}}^2 \frac{L}{N \gamma}{\mathds{1}}_{\gamma<1}+
{b_{\mathrm{ext}}}\,.$$ The construction is possible for all $N\ge 1$, $\gamma\in(0,1]$ which obey (\[eqcondNcostr\]), (\[eqconstrgammacond\]), and (\[eqkapparho0subst\]). To conclude the proof it suffices to choose these parameters appropriately.
[*Choice of the parameters: intermediate regime.*]{} We assume here $$\label{eqregimeint}
{b_{\mathrm{ext}}}\ge \frac{\kappa^{5/7}}{\sqrt2T^{2/7}}\,.$$ In this regime, we balance the first two terms in (\[equbfchibsdf\]) by choosing the length scale $L/N$ as $$\gamma=1 \hskip1cm \text{ and }\hskip1cm N=\inf\{2^k: k\in\N, 2^k\ge N_*\} \hskip5mm \text{ where }\hskip5mm
\frac{N_*}{L}=\frac{(\kappa {b_{\mathrm{ext}}})^{1/6}}{\alpha T^{2/3}}\,,$$ where $\alpha$ is a number of order 1 chosen below. We assume $N_*\ge 1$ so that the rounding of $N_*$ to the next power of $2$ does not modify it by more than a factor of 2, $N_*\le N\le 2N_*$. One obtains $$\begin{aligned}
\frac{F[\chi,B]}{L^2}{\lesssim}&(\kappa {b_{\mathrm{ext}}})^{2/3} T^{1/3}
+ {b_{\mathrm{ext}}}{\lesssim}(\kappa {b_{\mathrm{ext}}})^{2/3} T^{1/3} \end{aligned}$$ where we used that one term disappears because $\gamma=1$; the ${b_{\mathrm{ext}}}$ term can be dropped since ${b_{\mathrm{ext}}}/ ((\kappa {b_{\mathrm{ext}}})^{2/3} T^{1/3} )=({b_{\mathrm{ext}}}/\kappa)^{1/3} (\kappa T)^{-1/3}\le 1$.
It remains to check that the choices made are admissible. Condition (\[eqkapparho0subst\]) translates into $\alpha (\kappa {b_{\mathrm{ext}}}T^2)^{1/3}\ge 2$. Using (\[eqregimeint\]), $\kappa T\ge 1$ and assuming $\alpha\ge4$ one can easily see that it is satisfied. Since $N\le 2N_*$, condition (\[eqcondNcostr\]) translates into $$\left(\frac{{b_{\mathrm{ext}}}^2 T^4}{\kappa}\right)^{1/3} \ge \frac{32\pi}{\alpha^2} \,.$$ Using first (\[eqregimeint\]) and then $\kappa T\ge 1$ and $\kappa\le 1/2$ one can easily check that the parenthesis is at least 4, therefore it suffices to choose $\alpha=8$. Condition (\[eqconstrgammacond\]) is immediate. Finally, we check that $N_*\ge 1$. This is equivalent to $$L\ge 8\frac{T^{2/3}}{(\kappa {b_{\mathrm{ext}}})^{1/6}}\,.$$
[*Choice of the parameters: extreme regime.*]{} In this case we assume $$\label{eqregimeext}
{b_{\mathrm{ext}}}\le \frac{\kappa^{5/7}}{\sqrt2T^{2/7}}\,.$$ In this regime, we can actually make the three first terms in (\[equbfchibsdf\]) balance by choosing the length scale $L/N$and $\gamma$ according to $$\gamma=2^{1/4}\frac{T^{1/7}{b_{\mathrm{ext}}}^{1/2}}{\kappa^{5/14}}
\hskip5mm \text{ and }\hskip5mm N=\inf\{2^k: k\in\N, 2^k\ge N_*\} \hskip5mm \text{ where }\hskip5mm
\frac{N_*}{L}=\frac{{b_{\mathrm{ext}}}^{1/2}}{\alpha\kappa^{1/14}T^{4/7}}\,,$$ for some $\alpha>0$ chosen below. Again, we require $N_*\ge 1$ so that $N_*\le N\le 2N_*$. This gives $d_0\sim T^{5/7}/\kappa^{2/7}$ and $$\frac{F[\chi,B]}{L^2}{\lesssim}{b_{\mathrm{ext}}}\kappa^{3/7} T^{3/7}
+ {b_{\mathrm{ext}}}{\lesssim}{b_{\mathrm{ext}}}\kappa^{3/7} T^{3/7}$$ since $\kappa T\ge 1$.
We turn to checking that the choices made are admissible. The assumption (\[eqregimeext\]) is equivalent to $\gamma\le 1$. The other inequality in (\[eqconstrgammacond\]) is fulfilled by $\kappa T\ge 1$ (this is the reason for inserting the factors $2^{1/4}$ and $\sqrt2$ in the definition of $\gamma$ and (\[eqregimeext\])). Condition (\[eqkapparho0subst\]) becomes $\alpha (\kappa T)^{4/7}\ge 2$, which is true for any $\alpha\ge 2$. Condition (\[eqcondNcostr\]) becomes $$\frac{\alpha^2}{32\pi} \frac{(\kappa T)^{8/7}}{\kappa}\ge 1\,,$$ which again is satisfied if $\alpha=8$. The fact that $N_*\ge 1$ translates into $$L\ge8 \frac{T^{4/7}\kappa^{1/14}}{ {b_{\mathrm{ext}}}^{1/2}}\,.$$ This concludes the proof.
Construction for the Ginzburg-Landau functional
-----------------------------------------------
We finally give the upper bound construction for the Ginzburg-Landau functional. We start from the constructions given in Section \[secconstrsharp\] for the sharp-interface functional $F$.
The first step is to construct the vector potential $A$ from the magnetic field $B$. We use the following lemma, which is a variant of Hodge’s decomposition in the current geometry.
\[lemmaconstructA\] Let $B\in L^2_{{\mathrm{loc}}}(\R^3;\R^3)$, $Q_L$ periodic, such that $\Div B=0$ distributionally and $\int_{Q_L\times \R} |B|^2 dx<\infty$. Then there is $A\in W^{1,2}_{{\mathrm{loc}}}(\R^3;\R^3)$, also $Q_L$-periodic, such that $$\nabla \times A = B \,.$$
From Lemma \[lemmaHmeno12\] we obtain $$\int_{Q(z)}B_3\,dx'=0\text{ for all }z\in \R$$ (in the sense of traces). We define $$\hat B(x) := B(x) - \frac{1}{L^2} \int_{Q(x_3)} B \,dx'$$ so that all components of $\hat B$ have average zero on each cross section $Q(z)$. Notice that $\hat B_3=B_3$, and $\nabla'\hat
B=\nabla'B$. This implies $\hat B\in
L^2$, and $\Div \hat B=0$.
The rest of the proof gives a construction of $\hat A$ such that $\hat B=\nabla\times \hat A$. If one could solve directly $\Delta \Psi= \hat{B}$ with $\Div \Psi=0$, then $\nabla\times\Psi$ would work. Since we are working in an unbounded domain with mixed boundary conditions, for completeness we give an explicit construction of $\hat A$ based on Fourier series.
We Fourier transform to obtain coefficients $\hat b(k)$ such that $$\hat B(x)=\int_\R dk_3\sum_{k'\in 2\pi\Z^2/L} e^{ik\cdot x} \hat b(k)\,.$$ The transformation we just performed ensures that $\hat b(k)=0$ whenever $k'=0$ (this is the reason to consider $\hat B$ instead of $B$). Further, $k\cdot \hat b(k)=0$ for all $k$.
We define $$\hat a(k):=
\begin{cases}\displaystyle
\frac{-ik\times \hat b(k)}{k^2} & \text{ if } k'\ne 0\\
\displaystyle
0 & \text{ if }k'=0\,,
\end{cases}$$ so that for all $k$ $$ ik\times \hat a(k) =
\hat b(k)\,.$$ The family $\hat a(k)$ also corresponds to a converging Fourier series, since $|\hat a(k)|\le L|\hat b(k)|/(2\pi)$ (this is the step where it is important that we requested $k'\ne 0$, and not merely $k\ne 0$).
In real space, we set $$\hat A(x):=\int_\R dk_3\sum_{k'\in 2\pi\Z^2/L} e^{ik\cdot x} \hat
a(k)\,.$$ Clearly $\hat A\in W^{1,2}_{{\mathrm{loc}}}(\R^3;\R^3)$, it is $Q_L$-periodic and obeys $$\nabla\times \hat A = \hat B\,,$$ both sides being in $L^2(Q_L\times\R;\R^3)$. We finally set $$A(x)=\hat A(x) - \frac{1}{L^2}\int_{Q_L\times(0,x_3)} (e_3\times B)dx\,.$$ The correction depends only on $x_3$. Therefore, recalling $\int_{Q(x_3)} B_3 dx'=0$, $$\begin{aligned}
\nabla\times A &=& \nabla\times \hat A - \frac{1}{L^2}\int_{Q(x_3)} e_3\times
(e_3\times B)dx'\nonumber\\
&=& \hat B + \frac{1}{L^2}\int_{Q(x_3)} B\, dx'= B\,.\end{aligned}$$ This concludes the proof.
\[theoupperbGL\] For any ${b_{\mathrm{ext}}}, \kappa,L,T>0$ such that $$4{b_{\mathrm{ext}}}\le \kappa\le\frac12\hskip1cm\text{ and }\hskip1cm
\kappa T\ge 1\,,$$ $L$ sufficiently large (in the sense of (\[eqLadmissible\]) and $ {b_{\mathrm{ext}}}L^2\in 2\pi \Z$ there is a pair $(u,A)\in H^1_{\mathrm{per}}$, such that $$E[u,A]{\lesssim}\min \left\{{b_{\mathrm{ext}}}\kappa^{3/7} T^{3/7} L^2,
{b_{\mathrm{ext}}}^{2/3} \kappa^{2/3} T^{1/3}L^2\right\} \,.$$
Let $\chi$ and $B$ be the functions constructed in Theorem \[theochiupper1\]. Starting from $B$, we obtain $A$ from Lemma \[lemmaconstructA\]. Consider now the superconducting domain $\omega:=\{x\in Q_{L,T}: \chi(x)=0\}$. Here $A$ is a curl-free vector field, hence it is locally the gradient of some potential $\theta$. The domain is multiply connected, but the flux of $B$ across each tube is an integer multiple of $2\pi$, hence we can globally write $A$ as the gradient of a multi-valued function $\theta$, such that $\theta$ mod $2\pi$ is single-valued.
We set $\rho=0$ in the normal phase, and let it grow to 1 in the superconducting phase, on a length scale $1/\kappa$, $$\rho(x) := \min\{1, \kappa^2\,\dist^2(x,\omega)\}.$$ The distance function is understood, as usual, $Q_L$-periodic in the first two components. Finally, we set $$u(x):=\rho^{1/2}(x) e^{i\theta(x)}\,.$$
Since $\nabla \theta=A$ whenever $\rho\ne 0$, we have $$|\nabla_Au|^2 = |\nabla\rho^{1/2}|^2.$$ The $B'$ part of the energy is identical, and so is the outer field. It remains to treat the coupling term. In $\omega$ we have $\rho=0$ and $\chi=1$, hence $$\left(B_3-\frac{\kappa}{\sqrt2} (1-\rho)\right)^2 =
\chi\left(B_3-\frac{\kappa}{\sqrt2} \right)^2 \,.$$ Outside $\omega$ we have $B=0$, hence $$\left(B_3-\frac{\kappa}{\sqrt2} (1-\rho)\right)^2 =
\frac{\kappa^2}{2}(1-\rho)^2$$ The first term is exactly the one appearing in $F$. Therefore the $E[u,A]\le F[\chi,B]+E_S$, where $$E_S:= \int_{Q_{L,T}} \left[|\nabla\rho^{1/2}|^2 + \kappa^2 (\chi-(1-\rho))^2\right]dx\,.$$ Let $(\omega)_{1/\kappa}$ be a $1/\kappa$-neighbourhood of $\omega$. Then in $\omega$ we have $\chi=1=1-\rho$, outside $(\omega)_{1/\kappa}$ we have $\chi=0=1-\rho$, and recalling $|\nabla \rho^{1/2}|\le \kappa$ we obtain $$E_S \le2\kappa^2 |(\omega)_{1/\kappa}\setminus\omega|\,.$$ Recalling (\[eqestchiuppernbd\]) we conclude $E_S{\lesssim}\kappa \int_{Q_{L,T}} |D\chi|\le F[\chi,B]$. This concludes the proof.
|
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abstract: 'We investigate the transverse momentum dependent parton distributions (TMDs) in the quasi-parton-distribution framework. The long-standing hurdle of the so-called pinch pole singularity from the space-like gauge links in the TMD definitions can be resolved by the finite length of the gauge link along the hadron moving direction. In addition, with the soft factor subtraction, the quasi-TMD is free of linear divergence. We further demonstrate that the energy evolution equation of the quasi-TMD [[a.k.a.]{}]{} the Collins-Soper evolution, only depends on the hadron momentum. This leads to a clear matching between the quasi-TMD and the standard TMDs.'
author:
- Xiangdong Ji
- 'Lu-Chang Jin'
- Feng Yuan
- 'Jian-Hui Zhang'
- Yong Zhao
title: 'Transverse Momentum Dependent Quasi-Parton-Distributions'
---
Introduction
============
Transverse momentum dependent parton distributions (TMDs) are one of the major focuses in nucleon tomography studies at existing and future facilities [@Boer:2011fh]. Theoretically, they have attracted great interest starting in early 80’s, and considerable developments have been achieved in recent years [@Collins; @Collins:1981uk; @Ji:2004wu; @bbdm]. Pioneering work to compute the TMD matrix elements from lattice QCD has also been performed in Ref. [@Musch:2010ka], where the longitudinal momentum fraction $x$ for the quarks has been integrated out. Such results have generated interest in computing TMDs from lattice QCD in hadron physics community.
In the last few years, there has been great progress on computing parton physics from lattice QCD, thanks to the large momentum effective theory (LaMET) [@Ji:2013dva]. LaMET is based on the observation that parton physics defined in terms of lightcone correlations can be obtained from time-independent Euclidean correlations, now known as quasi-distributions, boosted to the infinite momentum frame. For a finite but large momentum feasible on the lattice, the two quantities are not identical, but they can be connected to each other by a perturbative matching relation, up to power corrections that are suppressed by the hadron momentum. LaMET has been applied to computing various PDFs [@Lin:2014zya; @Alexandrou:2015rja; @Chen:2016utp; @Alexandrou:2016jqi; @Chen:2017mzz; @Lin:2017ani; @Alexandrou:2017dzj] as well as meson DAs [@Zhang:2017bzy; @Chen:2017gck] (see also [@Ma:2014jla; @Ma:2017pxb] for slightly different proposals). In addition, theoretical developments have been achieved on the renormalization of the quasi-parton-distribution-functions (Q-PDFs) and on their matching to the usual PDFs [@Xiong:2013bka; @Xiong:2017jtn; @Wang:2017qyg; @Wang:2017eel; @Stewart:2017tvs; @Ji:2015qla; @Xiong:2015nua; @Ji:2015jwa; @Ishikawa:2016znu; @Chen:2016fxx; @Constantinou:2017sej; @Alexandrou:2017huk; @Chen:2017mzz; @Ji:2017oey; @Ji:2017rah; @Ishikawa:2017faj; @Green:2017xeu; @Li:2016amo; @Monahan:2016bvm; @Briceno:2017cpo; @Monahan:2017hpu; @Zhang:2018ggy; @Izubuchi:2018srq]. Unfortunately, there has been no lattice effort to compute the TMDs from the quasi-TMDs (Q-TMDs). The major hurdle is that the formulation of the TMDs is different from the integrated PDFs and, in particular, the gauge links associated with the Q-TMDs lead to the so-called pinch pole singularities. This is a generic feature of the TMDs defined with a space-like gauge link [@bbdm; @Collins:1981uk]. We have to either subtract or regulate these singularities before we can make meaningful computations of the Q-TMDs on the lattice [@Collins]. In Ref. [@Ji:2014hxa], a soft factor subtraction involving transverse gauge links has been proposed to formulate the Q-TMDs. However, this formalism may have practical difficulties for lattice computations at present.
In this paper, we will reinvestigate the TMDs in LaMET or Q-TMDs framework. We will show that, with finite length gauge links in the Q-TMDs, there will be no pinch pole singularity. This will pave the way to perform the TMD calculations on the lattice. Moreover, with an explicit one-loop calculation, we demonstrate that the energy evolution of the TMDs depends on the hadron momentum. This will clarify an important issue to match the Q-TMDs to the standard TMDs extracted from the experiments.
Our focus will be on the basics of the formalism and setting up the foundation for future numerical simulations on the lattice. Let us start with the un-subtracted Q-TMD quark distribution defined with finite length gauge links, $$\begin{aligned}
q(x_z,{\vec k}_\perp;L)|^{(unsub.)}&=&\frac{1}{2}\int\frac{dz\, d^2{\vec b}_\perp}{(2\pi)^3}e^{-ik_z z-i\vec{k}_\perp\cdot \vec{b}_\perp}\langle PS|\overline{\psi}(-\frac{{\vec b}_\perp}{2},-\frac{z}{2})
{\cal L}_{n_z(-\frac{{\vec b}_\perp}{2},-\frac{z}{2};-\frac{{\vec b}_\perp}{2},\pm L)}^\dagger\gamma^z \nonumber\\
&&\times {\cal L}_{T(-\frac{{\vec b}_\perp}{2},\pm L;\frac{{\vec b}_\perp}{2},\pm L)}^\dagger
{\cal L}_{n_z(\frac{{\vec b}_\perp}{2},\frac{z}{2};\frac{{\vec b}_\perp}{2},\pm L)}\psi(\frac{{\vec b}_\perp}{2},\frac{z}{2})|PS\rangle \ , \label{tmdq}\end{aligned}$$ where $(\vec{b}_\perp,z)$ represents the 3-dimensional coordinate space variable separated by the quark and antiquark fields, $x_z=k_z/P_z$ and the proton is moving along $+\hat z$ direction, ${\vec k}_\perp$ represents the transverse momentum of the quark. In the above definition, ${\cal L}_{n_z({\vec y}_\perp, z_1;{\vec y}_\perp, z_2)}={\cal P}\,exp\left[-ig\int_{z_2}^{z_1}
d\lambda\, n_z\cdot A(\lambda n_z+{\vec y}_\perp)\right]$ represents the gauge link along the $\hat z$ direction with the large length $L\gg |z|$, where the 4-vector $n_z$ is defined as $n_z^\mu=(0,0,0,1)$. We have also included a transverse gauge link to make the gauge links connected as shown in Fig. \[gaugelink\](a).
![Illustration of the gauge links in the un-subtracted Quasi-TMD (a) and the soft factor (b).[]{data-label="gaugelink"}](gaugelink){width="11cm"}
In the TMD formalism, it has been demonstrated that the soft factor subtraction plays an important role to properly address the relevant factorization properties [@Collins]. In this paper, we introduce the following soft factor subtraction, $$q^{(sub.)}(x_z,{\vec b}_\perp)=\frac{q^{(unsub.)}(x_z,{\vec b}_\perp;L)}{\sqrt{S^{n_z, n_z}({\vec b}_\perp;L)}}\ , \label{tmd}$$ where $q^{(unsub.)}(x_z,{\vec b}_\perp; L)$ is the un-subtracted Q-TMD in Eq. (\[tmdq\]) in the Fourier transform ${\vec b}_\perp$-space with respect to the transverse momentum ${\vec k}_\perp$, and $S^{n_z, n_z}({\vec b}_\perp; L)$ is defined as $$\begin{aligned}
S^{n_z, n_z}({\vec b}_\perp;L)&=&{\langle 0|{\cal L}_{T(\frac{{\vec b}_\perp}{2},-L;-\frac{{\vec b}_\perp}{2},-L)}^\dagger{\cal L}_{n_z(\frac{{\vec b}_\perp}{2},0;\frac{{\vec b}_\perp}{2},-L)}^\dagger
{\cal L}_{n_z(\frac{{\vec b}_\perp}{2},L;\frac{{\vec b}_\perp}{2},0)}^\dagger}\nonumber\\
&&\times {{\cal L}_{T(\frac{{\vec b}_\perp}{2},L;-\frac{{\vec b}_\perp}{2},L)} {\cal
L}_{ n_z(-\frac{{\vec b}_\perp}{2},L;-\frac{{\vec b}_\perp}{2},0)} {\cal L}_{n_z(-\frac{{\vec b}_\perp}{2},0;-\frac{{\vec b}_\perp}{2},-L)} |0\rangle }\, , \label{softg}\end{aligned}$$ with ${\cal L}_{n_z}$ being the longitudinal gauge link along the $\hat z$ direction and ${\cal L}_T$ the transverse gauge link at $z=L$ and $z=-L$, as shown in Fig. \[gaugelink\] (b). In other words, the above soft factor is just a Wilson loop.
The rest of this paper is organized as follows. In Sec. II, we will show the absence of pinch pole singularity in the Q-TMD with finite length gauge links with an explicit calculation at one-loop order. In Sec. III, we will discuss the matching between the Q-TMD and the standard TMD. We then summarize our paper in Sec. IV.
Absence of the Pinch Singularity in Q-TMDs
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To show that we do not encounter the pinch pole singularity, we will carry out a one-loop calculation. We take the example of quark Q-TMD on a quark target. In Feynman gauge, the one-loop diagrams are shown in Figs. \[pdfr\] and \[pdfv\]. The final result can also serve as a matching between the Q-TMD and the standard TMD. Because of the finite length of the gauge links, the eikonal propagator in these diagrams will be modified accordingly, $$(-ig)\frac{i n^\mu }{n\cdot k\pm i\epsilon}\Longrightarrow
(-ig)\frac{i n^\mu}{n\cdot k}\left(1-e^{\pm in\cdot k L}\right) \ ,\label{eikonal}$$ where $n^\mu$ represents the gauge link direction. In the present case $n^\mu=n_z^\mu$. In perturbative calculations, we will make use of the large length limit $|LP_z|\gg 1$. By doing that, many of previous results can be applied to our calculations. For example, in the large $L$ limit, we have the following identity: $\lim_{L\to \infty}\frac{1}{n\cdot k}e^{\pm i Ln\cdot k}=\pm i\pi \delta(n\cdot k) $.
At one-loop order, the pinch pole singularity could potentially come from the diagram $(c)$ of Fig. \[pdfr\] in the limit of infinite gauge link with $L\to \infty$, $$\begin{aligned}
&q^{(1)}(x_z,{\vec k}_\perp)|_{\ref{pdfr}(c)}^{L\to \infty}\nonumber\\
&=\frac 1 2\int\frac{dk^0 dk_z}{(2\pi)^4}\bar u(p)\gamma^z (ig t^a)(ig t^a) \frac{-i}{n\cdot(P-k)-i\epsilon}\frac{i}{n\cdot(P-k)+i\epsilon}\frac{-i}{(P-k)^2} u(p)\delta\big(k_z-x_z P_z\big)\nonumber\\
&=\frac{\alpha_s}{4\pi^2}C_F
\frac{P_z}{\sqrt{(1-x_z)^2P_z^2+{\vec k}_\perp^2}}\frac{1}{(1-x_z)P_z+i\epsilon}
\frac{1}{(1-x_z)P_z-i\epsilon}\ .\end{aligned}$$ For the case of integrated parton distributions, we integrate over ${\vec k}_\perp$ to obtain the one-loop result. However, in the current case, we have to keep the transverse momentum ${\vec k}_\perp$. In addition, we note that the above contribution is power suppressed by ${\vec k}_\perp/P_z$ for $x_z\neq1$. That means to leading power in $P_z$ this diagram only contributes to $\delta(1-x_z)$, and can be written as $$\begin{aligned}
q^{(1)}(x_z,{\vec k}_\perp)|_{\ref{pdfr}(c)}^{L\to \infty}&=&\frac{\alpha_s}{4\pi^2}C_F
\delta(1-x_z)\int\frac{dk_z}{\sqrt{k_z^2+{\vec k}_\perp^2}}\frac{1}{k_z+i\epsilon}\frac{1}{k_z-i\epsilon} \ .\end{aligned}$$ The above integral is not well-defined, because the two poles are pinched. We are forced to take the pole at $k_z=0$, which is, however, divergent. This is a common issue for parton distributions defined with gauge links along the space-like direction [@bbdm; @Collins:1981uk].
![Real diagram contributions to the Q-TMD quark distributions at one-loop order. The complex conjugate of Diagram (b) is implied. Diagram (c) would contain the pinch pole singularity with infinite length gauge links in the TMD definition. However, with a finite length $L$, this singularity is absent in Q-TMD.[]{data-label="pdfr"}](pdfr){width="10cm"}
![Virtual diagram contributions to the Q-TMD quark distributions at one-loop order. The complex conjugate is also implied. Diagram (c) requires the renormalization of the gauge link self-interaction. This follows recent examples in the collinear Q-PDF cases.[]{data-label="pdfv"}](pdfv){width="10cm"}
With finite length gauge links, the above result will be modified to $$\begin{aligned}
\label{fig2cfiniteL}
q^{(1)}(x_z,{\vec k}_\perp)|_{\ref{pdfr}(c)}&=&\frac{\alpha_s}{4\pi^2}C_F
\frac{P_z}{\sqrt{(1-x_z)^2P_z^2+{\vec k}_\perp^2}}\frac{1}{(1-x_z)P_z}\frac{1}{(1-x_z)P_z}\nonumber\\
&&\times \left(1-e^{ i(1-x_z)P_zL}\right)\left(1-e^{ -i(1-x_z)P_zL}\right)\ ,\end{aligned}$$ where we have used Eq. (\[eikonal\]). We find again that this result is power suppressed for $x_z\neq 1$. Therefore, we can simplify the above equation as $$\begin{aligned}
q^{(1)}(x_z,{\vec k}_\perp)|_{\ref{pdfr}(c)}&=&\frac{\alpha_s}{4\pi^2}C_F\delta(1-x_z)\int\frac{dk_z}{k_z^2}\frac{1}{\sqrt{k_z^2+{\vec k}_\perp^2}}
\left(1-e^{ ik_zL}\right)\left(1-e^{ -ik_zL}\right)\ .\label{e13}\end{aligned}$$ Now, the integral is well regulated around $k_z=0$. Moreover, it does not contribute to the infrared behavior of the Q-TMD at low transverse momentum, as can be seen by an explicit integration over small ${\vec k}_\perp$ in Eq. (\[e13\]), which does not yield any divergence. In particular, taking the Fourier transform with respect to ${\vec k}_\perp$, we obtain the following expression in the ${\vec b}_\perp$-space, $$\begin{aligned}
q^{(1)}(x_z,{\vec b}_\perp)|_{\ref{pdfr}(c)}&=&\frac{\alpha_s}{2\pi}C_F\delta(1-x_z) 2{\cal K}(\xi_b) \ ,\end{aligned}$$ where $\xi_b=L/|{\vec b}_\perp|$ and the function ${\cal K}$ is defined as $${\cal K}(\xi_b)=2\xi_b \tan^{-1}\xi_b-\ln(1+\xi_b^2) \ .$$ At large $\xi_b$ the above equation goes like $\pi\xi_b-2\ln\xi_b$, while at small $\xi_b$ it behaves as $\xi_b^2$.
Furthermore, with the soft factor subtraction, we will be able to eliminate the $1/|{\vec k}_\perp|$ term at small ${\vec k}_\perp$ in Eq. (\[e13\]). The subtraction term relevant to Fig. \[pdfr\](c) comes from the gluon exchange between the two longitudinal Wilson lines with length $2L$ in Fig. \[gaugelink\](b). It can be computed in the same way as that of Fig. \[pdfr\](c) and leads to the same result as Eq. (\[fig2cfiniteL\]) except that $L$ needs to be replaced by $2L$. We then have the following result after subtraction, $$\begin{aligned}
q^{(1)}(x_z,{\vec k}_\perp)|_{\ref{pdfr}(c)}^{(sub.)}&=&q^{(1)}(x_z,{\vec k}_\perp)|_{\ref{pdfr}(c)}^{(sub.)}-\frac{1}{2}q^{(1)}(x_z,{\vec k}_\perp)|_{\ref{pdfr}(c)}^{(sub.)}(L\to 2L)\nonumber\\
&=&\frac{\alpha_s}{4\pi^2}
C_F\delta(1-x_z)\int\frac{dk_z}{k_z^2}\frac{1}{\sqrt{k_z^2+{\vec k}_\perp^2}}
\frac{1}{2}\left[\left(1-e^{ ik_zL}\right)^2+\left(1-e^{ -ik_zL}\right)^2\right]\ .\label{e13p}\end{aligned}$$ In the Fourier transform ${\vec b}_\perp$-space, the above result becomes, $$\begin{aligned}
q^{(1)}(x_z,{\vec b}_\perp)|_{\ref{pdfr}(c)}^{(sub.)}&=&\frac{\alpha_s}{2\pi}C_F\delta(1-x_z) \left[2{\cal K}(\xi_b)-{\cal K}(2\xi_b)\right] \ ,\end{aligned}$$ where the second term comes from the soft factor. Clearly, the linear term of $\xi_b$ is cancelled out in the subtracted contribution, and the result goes like $\ln(\xi_b^2)$ at large $\xi_b$, whereas at small $\xi_b$ it again behaves like $\xi_b^2$.
Similarly, the contribution from Fig. \[pdfv\](c) is given by (for a finite length gauge link) $$\begin{aligned}
q^{(1)}(x_z,{\vec k}_\perp)|_{\ref{pdfv}(c)}&=\frac{-ig^2 C_F}{2}\int\frac{d^4k'}{(2\pi)^4}\frac{1}{(p-k')^2}\frac{1}{[n\cdot(p-k')]^2}\nonumber\\
&\times\left(1-e^{ in\cdot(p-k')L}\right)\left(1-e^{ -in\cdot(p-k')L}\right)\delta(k'_z-x_z P_z)\delta^{(2)}({\vec k}_\perp).\end{aligned}$$ After soft factor subtraction, it gives the following expression, $$\begin{aligned}
q^{(1)}(x_z,{\vec k}_\perp)|_{\ref{pdfv}(c)}^{(sub.)}&=&\frac{\alpha_s}{8\pi^2}
C_F\delta(1-x_z)\delta^{(2)}({\vec k}_\perp)\int\frac{dk_zd^2{\vec k}_\perp'}{k_z^2}\left(\frac{1}{\sqrt{{\vec k}_\perp^{\prime 2}}}-\frac{1}{\sqrt{k_z^2+{\vec k}_\perp^{\prime 2}}}\right)\nonumber\\
&&\times \left[\left(1-e^{ ik_zL}\right)\left(1-e^{ -ik_zL}\right)-\frac{1}{2}\left(1-e^{ i2k_zL}\right)\left(1-e^{ -i2k_zL}\right)\right] \ ,\label{e13p2}\end{aligned}$$ where we have rewritten the integral over ${\vec k}_\perp'$ in such a way that the linear divergence is manifestly absent. In addition, all $L$-dependent contributions that are not suppressed in the large $L$ limit cancel out in the full subtracted Q-TMD. The cancellation occurs either among the unsubtracted Q-TMD diagrams or with similar contributions from the soft factor. This can be easily seen from computations in coordinate space.
As there is no linear divergence associated with the gauge links after soft factor subtraction, we can work in dimensional regularization, which leads to the following contribution, $$\begin{aligned}
\label{3csub}
q^{(1)}(x_z,{\vec b}_\perp)|_{\ref{pdfv}(c)}^{(sub.)}&=&\frac{\alpha_s}{4\pi}
C_F\delta(1-x_z)\big[\ln\frac{L^2\mu^2}{4c_0^2}+2\big] \ ,\end{aligned}$$ in the Fourier transform ${\vec b}_\perp$-space, where the UV divergence has been subtracted with $\overline{\rm MS}$ scheme. In the dimensional regulation, the linear divergence is not manifest explicitly in the unsubtracted Q-TMD, and the result is the same as above with a factor of 2. However, if a cutoff scheme is chosen, there will be an explicit linear divergence in the unsubtracted Q-TMD, $$\begin{aligned}
q^{(1)}(x_z,{\vec b}_\perp)|_{\ref{pdfv}(c), tot.}^{(unsub.)}&=&\frac{\alpha_s}{2\pi}
C_F\delta(1-x_z)\left[4-\frac{2\pi L}{a}+2\ln\frac{L^2}{a^2}\right] \ ,\end{aligned}$$ whereas the linear divergence is cancelled out for the subtracted Q-TMD $$\begin{aligned}
q^{(1)}(x_z,{\vec b}_\perp)|_{\ref{pdfv}(c), tot.}^{(sub.)}&=&\frac{\alpha_s}{2\pi}
C_F\delta(1-x_z)\left[2+\ln\frac{L^2}{4a^2}\right] \ .\label{e16}\end{aligned}$$ The transverse gauge link contribution can be calculated in complete analogy and we have $$\begin{aligned}
q^{(1)}(x_z,{\vec b}_\perp)|_{\ref{pdfv}(c)}^{(unsub.)T}&=&\frac{\alpha_s}{2\pi}
C_F\delta(1-x_z)\left[2-\frac{\pi {\vec b}_\perp}{a}+ \ln\frac{{\vec b}_\perp^2}{a^2}\right] \ .\end{aligned}$$ For the subtracted Q-TMD, the transverse gauge link contribution is cancelled out completely, $$\begin{aligned}
\label{3csubT}
q^{(1)}(x_z,{\vec b}_\perp)|_{\ref{pdfv}(c)}^{(sub.)T}&=&0 \ .\end{aligned}$$
Eqs. (\[3csub\]) to (\[e16\]) are independent of ${\vec b}_\perp$, and therefore will remain the same at large or small $\xi_b$. From the results above, one can easily see that the subtracted result of Fig. \[pdfr\](c), \[pdfv\](c) has a residual logarithmic UV divergence, $$\begin{aligned}
q^{(1)}(x_z,{\vec b}_\perp)|_{\ref{pdfr}(c),\ref{pdfv}(c)}^{(sub.)}&=\frac{\alpha_s}{2\pi}
C_F\delta(1-x_z)\big[\ln\frac{L^2\mu^2}{4c_0^2}+2+2{\cal K}(\xi_b)-{\cal K}(2\xi_b)\big] \ ,\label{e13p3new}\end{aligned}$$ in the Fourier transform ${\vec b}_\perp$-space with respect to the transverse momentum ${\vec k}_\perp$, where $c_0=2e^{-\gamma_E}$. In the above equation, we have applied the dimensional regulation for the UV divergence and renormalize in the $\overline{\rm MS}$ scheme with scale $\mu$. If a lattice regulator is adopted, we will obtain the same expression with $\mu/c_0=1/a$, where $a$ is the lattice spacing parameter. Because of the above contribution, we will have an additional anomalous dimension contribution from Eq. (\[e13p3new\]) for the evolution equation of the Q-TMD.
The rest of the real diagrams in Fig. \[pdfr\] can be calculated by safely taking the large $L$ limit. For example, the contribution of Fig. \[pdfr\](b) is given by $$\begin{aligned}
\frac {-i g^2 C_F}{2}\int\frac{dk^0 dk_z}{(2\pi)^4}\bar u(p)\gamma^z \frac{1}{n\cdot(P-k)}\frac{1}{\slashed k}\gamma^z\frac{1}{(P-k)^2} u(p)
\left(1-e^{ in\cdot(P-k)L}\right)\delta\big(k_z-x_z P_z\big)\ ,\end{aligned}$$ and leads to the following result $$\begin{aligned}
q^{(1)}(x_z,{\vec k}_\perp)|_{\rm Fig.~2(b)}=&\frac{\alpha_s}{4\pi^2}C_F\left[\frac{1}{{\vec k}_\perp^2}\frac{x_z}{1-x_z}\left(\frac{P_z(1-x_z)}{\sqrt{{\vec k}_\perp^2+P_z^2(1-x_z)^2}} + \frac{P_zx_z}{\sqrt{{\vec k}_\perp^2+P_z^2x_z^2}}\right)\right.\nonumber\\
&\left.+{1\over 1-x_z}{1\over P_z^2}\left({P_z\over \sqrt{{\vec k}_\perp^2 + P_z^2x_z^2}}-{P_z\over \sqrt{{\vec k}_\perp^2 + P_z^2(1-x_z)^2}}\right)\right]\left(1-e^{ i(1-x_z)P_zL}\right) \ . \label{e6}\end{aligned}$$ However, this additional factor $e^{[i(1-x_z)P_zL]}$ does not contribute in the large $L$ limit. It is interesting to note that if we take $P_z\to \infty$ first, the above equation will lead to a divergence of $1/(1-x_z)$, which is same as the light-cone singularity in the usual TMD definition. Again, the contributions from the regions of $x_z<0$ and $x_z>1$ are power suppressed in the limit $|{\vec k}_\perp|\ll P_z$. The final result from this diagram can be written as, $$\frac{\alpha_s}{2\pi^2}C_F\frac{1}{{\vec k}_\perp^2}\left[\frac{2x_z}{(1-x_z)_+}\theta(x_z)\theta(1-x_z)+\delta(1-x_z)\ln\frac{\zeta^2}{{\vec k}_\perp^2}\right] \ ,\label{e7}$$ where $\zeta^2=x_z^2(2n_z\cdot P)^2/(-n_z^2)=4x_z^2P_z^2$ and we have applied a principal-value prescription to evaluate the second term in Eq. (\[e7\]).
Because there is no gauge link contribution from Fig. \[pdfr\] (a), its result will be the same as previously calculated in Ref. [@Ji:2014hxa] $$\begin{aligned}
q^{(1)}(x_z,{\vec k}_\perp)|_{\rm Fig.~2(a)}&=\frac{\alpha_s}{4\pi^2}C_F\frac{1-\epsilon}{{\vec k}_\perp^2}\frac{(1-x_z)\big(\sqrt{{\vec k}_\perp^2+P_z^2(1-x_z)^2}+P_z(1-x_z)\big)}{\sqrt{{\vec k}_\perp^2+P_z^2(1-x_z)^2}}.\end{aligned}$$ In the limit $|{\vec k}_\perp|\ll P_z$, the above result reduces to $$\label{e2ared}
\frac{\alpha_s}{2\pi^2}C_F\frac{1-\epsilon}{{\vec k}_\perp^2}(1-x_z)\ .$$
Similar calculations can be performed for the virtual diagrams of Fig. \[pdfv\](a,b), and the result reads [@Ji:2014hxa] $$\begin{aligned}
\label{e3ab}
q^{(1)}(x_z,{\vec b}_\perp)|_{3(a), 3(b)}&=\frac{\alpha_s}{2\pi}C_F\delta(1-x_z)\Big[-\frac{1}{\epsilon^2}-\frac{3}{2\epsilon}+\frac{1}{\epsilon}\ln\frac{\zeta^2}{\mu^2}+\ln\frac{\zeta^2}{\mu^2}-\frac{1}{2}\Big(\ln\frac{\zeta^2}{\mu^2}\Big)^2+\frac{\pi^2}{12}-2\Big].\end{aligned}$$
Finally, the total contribution of the subtracted Q-TMD quark distribution at one-loop order can be obtained from Eqs. (\[e13p3new\]), (\[e3ab\]) and the Fourier transform of (\[e7\]), (\[e2ared\]), $$\begin{aligned}
{q}_{QTMD}^{(sub.)(1)}(x_z,{\vec b}_\perp;\zeta^2)&=&\frac{\alpha_s}{2\pi}C_F\left\{\left(-\frac{1}{\epsilon}
+\ln\frac{c_0^2}{{\vec b}_\perp^2\mu^2}\right){\cal P}_{q/q}(x_z)+(1-x_z)\right.\nonumber\\
&&\left.+\delta(1-x_z)\left[\frac{3}{2}\ln\frac{{\vec b}_\perp^2\mu^2}{c_0^2}+\ln\frac{\zeta^2L^2}{4c_0^2}
-\frac{1}{2}\left(\ln\frac{\zeta^2{\vec b}_\perp^2}{c_0^2}\right)^2\right.\right.\nonumber\\
&&\left.\left.+2{\cal K}(\xi_b)-{\cal K}(2\xi_b)\right]\right\} \label{oneloop}\end{aligned}$$ in ${\vec b}_\perp$-space, where $\mu$ is the renormalization scale in the $\overline{\rm MS}$ scheme, and ${\cal P}_{q/q}(x_z)=\left(\frac{1+x_z^2}{1-x_z}\right)_+$ is the usual splitting kernel for the quark. We would like to emphasize a number of important points here. First, the Q-TMDs only have contributions in the region $0<x_z<1$. This is because, as mentioned above, we are taking the physical limit for TMD, i.e., $P_z\gg |{\vec k}_\perp|$. In this limit, the contributions in the region $x_z>1$ and $x_z<0$ are power suppressed. Second, similar to the previous formalisms for the TMDs, the Q-TMDs contain the double logarithms as indicated in the above equation. From the explicit calculations, we find that these double logarithms depend on the hadron momentum $P_z$ in the Q-PDF framework. Therefore, the associated energy evolution, i.e., the Collins-Soper evolution, will depend on $P_z$ not $L$. Finally, as expected, the Q-TMD at one-loop order contains infrared divergence, which corresponds to the collinear splitting of the quark.
Comparing to the result in Ref. [@Ji:2014hxa], we find an additional term from the soft factor subtraction in the Q-TMD. This term will lead to a different matching between the Q-TMD and the standard TMD.
Matching to the Standard TMDs
=============================
With the above one-loop result for the Q-TMD quark distribution, we can match to the usual TMDs at this order following the procedure of Ref. [@Ji:2013dva]. However, there is scheme dependence in the usual TMDs to regulate the relevant light-cone singularities [@Collins]. Therefore, a direct matching to the various TMDs will introduce the scheme dependence as well. On the other hand, as demonstrated in Refs. [@Catani:2000vq; @Catani:2013tia; @Prokudin:2015ysa], all TMD schemes lead to the same result after resumming the large logarithms. Therefore, it is more appropriate to carry out the matching between the Q-TMDs and the standard TMDs after the resummation has been performed.
This resummation is carried out by solving the associated evolution equations [@Collins]. For the Q-TMD quark distribution, the relevant Collins-Soper evolution can be derived [@Ji:2014hxa], and the complete resummation result can be expressed in terms of the integrated parton distributions [@Collins], $$\begin{aligned}
\label{tmdaspdf}
{q}_{QTMD}(x_z,{\vec b}_\perp;\zeta^2)&=& e^{-{ {S}^q(\zeta,{\vec b}_\perp)}}e^{-{ {S}_w^q(\zeta,\mu_L)}}\int\frac{dx'}{x'}f_q(x',\mu_b)\nonumber\\
&\times&\left\{
\delta(1-\xi)\left[1+\frac{\alpha_s}{2\pi}C_F(2{\cal K}(\xi_b)-{\cal K}(2\xi_b))\right]+\frac{\alpha_s}{2\pi}C_F(1-\xi)\right\}\ ,\end{aligned}$$ where $f_q$ represents the integrated quark distribution, the Sudakov factors resum the logarithmically enhanced contributions with the following form $$\begin{aligned}
S^q(\zeta,{\vec b}_\perp)&=&\int_{\mu_b^2}^{\zeta^2}\frac{d{\bar\mu}^2}{{\bar\mu}^2}\left[A\ln\frac{\zeta^2}{{\bar\mu}^2}+B\right]\ ,\\
S^q_w(\zeta,\mu_L)&=&\int_{\mu_L^2}^{\zeta^2}\frac{d{\bar\mu}^2}{{\bar\mu}^2}\gamma_w \ .\end{aligned}$$ In the above equation, we have chosen the factorization scale $\mu=\zeta$, $\xi=x_z/x'$, $\mu_b=c_0/|{\vec b}_\perp|$, $\mu_L=2c_0/L$. $A$ and $B$ are perturbatively calculable coefficients with $A=\sum_{i=1}A^{(i)}(\alpha_s/\pi)^i$ and $B=\sum_{i=1}B^{(i)}(\alpha_s/\pi)^i$, and the one-loop order coefficients can be read off from Eq. (\[oneloop\]) as $A^{(1)}=C_F/2$ and $B^{(1)}=-3C_F/4$. The additional Sudakov factor $S_w^q$ comes from the soft factor subtraction, and the anomalous dimension at one-loop is given by $\gamma_w=-C_F\frac{\alpha_s}{2\pi}$, as can be read off from the coefficient of the $\ln\frac{\zeta^2L^2}{4c_0^2}$ term in Eq. (\[oneloop\]). Note that we have set the scale for the Wilson line renormalization as $\mu=\zeta$ as well. In practice, it may depend on how the Wilson lines are renormalized for the lattice computations. The hard coefficient in the second row of Eq. (\[tmdaspdf\]) contains all remaining one-loop contributions in Eq. (\[oneloop\]).
In order to carry out the matching to the usual TMDs, we compute the TMD quark distribution in the standard scheme [@Collins; @Catani:2000vq; @Catani:2013tia; @Prokudin:2015ysa] as well, $$\begin{aligned}
{q}_{TMD}(x_z,{\vec b}_\perp;\zeta^2)&=& e^{-{ {S}^q(\zeta,{\vec b}_\perp)}}\int\frac{dx'}{x'}f_q(x',\mu_b)\left\{
\delta(1-\xi)\left[1+{\cal O}(\alpha_s^2)\right]+\frac{\alpha_s}{2\pi}C_F(1-\xi)\right\}\ ,\nonumber\\\end{aligned}$$ where $\zeta^2$ represents the hard momentum scale for the TMDs extracted from the experiments, for example, the invariant mass of lepton pair in the Drell-Yan lepton pair production process. We can also define the above standard TMD as that in the Collins 2011 scheme [@Collins]. We would like to emphasize that the Sudakov factor is the same as above. Notice that in the standard TMD scheme (or Collins 2011 scheme), the hard coefficient vanishes at one-loop order. Comparing the above two equations, we can read out the matching between the Q-TMD quark distribution and the standard TMD quark distribution as $$\begin{aligned}
{q}_{QTMD}(x_z,{\vec b}_\perp;\zeta^2)&=&e^{-{ {S}_w^q(\zeta,\mu_L)}}{q}_{TMD}(x_z,{\vec b}_\perp;\zeta^2)\left[1+\frac{\alpha_s}{2\pi}C_F(2{\cal K}(\xi_b)-{\cal K}(2\xi_b))\right] \ . \label{final}\end{aligned}$$ The above equation indicates that the Q-TMD computed on the lattice can be interpreted as the TMD for phenomenological applications.
Discussions and Summary
=======================
Our final result as shown in Eq. (\[final\]) has a number of interesting features. First, because the gauge links in the unsubtracted and subtracted TMD contain Wilson line renormalization, we have additional scale evolution expressed in term of $e^{-{ {S}_w^q(\zeta,\mu_L)}}$. If different renormalization is chosen, we will have a different factor. For example, for the cutoff scheme in the lattice calculation, we will have different factor. In practical calculations, we may not need to perform resummation for this term at all.
In the matching coefficient, we have a functional dependence on $\xi_b$. Its contribution depends on the relative size between $L$ and ${\vec b}_\perp$. In the non-perturbative region with ${\vec b}_\perp\gg L$, this is a power correction and can be safely ignored. On the other hand, in perturbative region of ${\vec b}_\perp\ll L$, it could lead to a large logarithm. This, however, will be dominated over by the Sudakov logs of $e^{-S^q}$. We do no need worry too much on its contribution. Of course, in the non-perturbative region of $L\gg {\vec b}_\perp\sim \Lambda_{\rm QCD}$, this term may become important and needs to be carefully handled. If we can vary the gauge link length $L$ in such way, we may be able to avoid this region. Therefore, this additional term does not cause any problem.
To illustrate the above point, we have investigated the behavior of the term $2{\cal K}(\xi_b)-{\cal K}(2\xi_b)$ by plotting itas a function of ${\vec b}_\perp$ for different choices of $L$, which implies that an optimal choice of $L$ would be around $2\sim 3/P_z$ for a reasonable range of $P_z$.
To summarize, we have laid out the basic procedure to compute the TMDs from lattice QCD using LaMET or Q-TMDs. We have shown that the finite length of gauge links plays a crucial role to regulate the so-called pinch pole singularities associated with space-like gauge links in the Q-TMDs. Additional soft factor subtraction improves the theoretical convergence, especially that it cancels out the linear divergence completely. This paves the way to correctly interpret the numerical results in lattice calculations of the TMDs. We have also shown that the energy evolution equation for the Q-TMDs comes from the large momentum of the hadron $P_z$. At one-loop order, a double logarithm depending on $P_z$ is found in the Q-TMD calculations. The relevant evolution equation and resummation can be performed following the TMD formalism. In particular, our results show that the energy evolution does not depend on the gauge link length $L$.
Our results may provide a justification of the technique set up in previous attempts to calculate the TMDs on the lattice [@Musch:2010ka]. However, we would like to emphasize that the Q-TMD depends on longitudinal momentum fraction $x$. Integral over $x$ may induce difficulties to interpret the results from lattice calculations.
Further developments shall follow along the direction outlined in this paper. In particular, we would like to apply our method to a realistic calculation of the Q-TMDs on the lattice. This will be considered in future work. Extensions to the Wigner distributions and other nucleon tomography observables are desirable to follow up as well.
We thank Markus Ebert and Iain Stewart for interesting conversations related to the subject of this paper. This material is based upon work partially supported by the LDRD program of Lawrence Berkeley National Laboratory, the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, under contract number DE-AC02-05CH11231, and within the framework of the TMD Topical Collaboration, and a grant from National Science Foundation of China (X.J.). JHZ is supported by the SFB/TRR-55 grant “Hadron Physics from Lattice QCD” and by a grant from National Science Foundation of China (No. 11405104). YZ is also supported by the the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, from DE-SC0011090.
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---
abstract: 'Although recent work in numerical relativity has made tremendous strides in quantifying the gravitational wave luminosity of black hole mergers, very little is known about the electromagnetic luminosity that might occur in immediate conjunction with these events. We show that whenever the heat deposited in the gas near a pair of merging black holes is proportional to its total mass, and the surface density of the gas in the immediate vicinity is greater than the (quite small) amount necessary to make it optically thick, the characteristic scale of the luminosity emitted in direct association with the merger is the Eddington luminosity [*independent*]{} of the gas mass. The duration of the photon signal is proportional to the gas mass, and is generally rather longer than the merger event. At somewhat larger distances, dissipation associated with realigning the gas orbits to the new spin orientation of the black hole can supplement dissipation of the energy gained from orbital adjustment to the mass lost in gravitational radiation; these two heat sources can combine to augment the electromagnetic radiation over longer timescales.'
author:
- 'Julian H. Krolik'
bibliography:
- '../references.bib'
title: Estimating the Prompt Electromagnetic Luminosity of a Black Hole Merger
---
Introduction
============
The merger of two supermassive black holes has been a topic of lively astrophysical speculation for many years [@BBR80]. Recent developments in galaxy formation theory have made the prospect more plausible and suggest an environment for such events: the centers of galaxies that underwent major mergers a few hundred million years in the past [@HaehKauff02; @Volont03]. Mergers may be particularly likely when the galaxy contains a relatively rich supply of interstellar gas, which may help binary black holes overcome the “last parsec problem" and approach each other close enough for gravitational wave emission to compress the orbit to merger within a Hubble time [@GR00; @AN02; @Kazantz04; @Escala05; @Cuadra09; @Dotti09] (but for a contrary view see [@Lodato09]). The presence of sizable quantities of interstellar gas in the parsec-scale environment then raises the question of how much gas might find itself even closer to the pair at their moment of merger.
This is a subject of great uncertainty. It has been argued, for example, that there should be little gas closer to the merging pair than $\sim 100$–$1000r_g$ ($r_g \equiv GM/c^2$, where $M$ is the total mass of the system) because eventually the timescale for shrinkage of the binary orbit by gravitational wave radiation becomes shorter than the timescale for mass inflow due to internally generated fluid stresses [@MP05]. On the other hand, the mass of such a circumbinary disk might be as large as $\sim 100 M_{\odot}$ or more [@MP05; @AN02; @Rossi09; @Corrales09]; if even $1 M_{\odot}$ were close enough to the merging black holes to be given heat equal to $1\%$ of its rest mass, the total energy—$\sim 10^{50}$ erg—might well be large enough to produce observable radiation. It is therefore a worthwhile exercise to estimate what sort of light might be generated if even a small fraction of the surrounding gas were able to make its way in close to the merging black holes.
Because the amount of mass near the merging black holes is so difficult to estimate at present, the plan of this paper is to explore prompt electromagnetic radiation in a way that is scaled to whatever gas mass is there. Thus, we will first estimate the heat per unit mass that might be deposited in this gas, then, in order to find the luminosity, estimate the timescale on which the energy is radiated. Next, the more model-dependent subject of the spectrum will be broached. Lastly, having seen how the light emitted depends on gas mass, we will discuss the issues related to whether an “interesting" amount of mass may be present. In order to avoid additional complications, we will ignore any luminosity due to accretion through the circumbinary disk.
Efficiency of Energy Deposition
===============================
Let us begin, then, with the supposition that immediately before a merger of two supermassive black holes there is at least some gas orbiting over a range of distances not too far from the system’s center of mass. To discuss the effect of the merger on this relatively nearby gas, it is useful to distinguish two regions: the inner gas ($r/r_g \lesssim 10$) and gas farther away ($10 < r/r_g < 10^3$). These regions are distinguished both by the magnitude of the heating they are likely to experience and by the time at which it occurs.
Because the inner gas is in the “near-field" regime, its gravitational environment during the merger is better described as a nonlinear time-dependent distortion of spacetime, rather than a passage of gravitational waves. The amplitude and extent of the distortions are, in some sense, proportional to the binary mass ratio, reaching a maximum when the two black holes have similar mass. Strong shearing can deflect fluid orbits, provoking shocks, and can also stretch magnetic field lines, driving MHD waves whose energy can ultimately be dissipated. Because the characteristic timescale of the spacetime variability is $\sim O(10)r_g/c$ [@BCP07], while the dynamical timescale of particle orbits in this region is only a little longer ($\sim (r_g/c)(r/r_g)^{3/2}$), the fluctuations can be very efficient in transferring energy to the gas (cf. the test-particle calculations of @vanMeter09). The same near-coincidence of merger timescale with the gas’s dynamical timescale means that the dissipation timescale (via shocks, etc. which develop on a dynamical timescale) should also be comparable to the merger duration.[^1] Unfortunately, without detailed general relativistic MHD calculations, it is very difficult to make a quantitative prediction of just how much energy might thus be given to the gas.
Nonetheless, to the extent that gravitational dynamics dominate, the Equivalence Principle suggests that the energy left in the gas should be proportional to its mass. It is therefore convenient to scale the gas heating in familiar rest-mass efficiency terms. For a fiducial value, one might imagine that this efficiency $\epsilon$ could be as large as $\sim 0.1$ for matter subjected to a truly nonlinear dynamical spacetime. Gravitational shear acting on magnetic fields may supplement this energy deposition [@MK04; @Duez05; @Pal09].
At greater distances, the perturbations to spacetime are much smaller and the contrast between the wave frequency and the dynamical frequency is much greater. In addition, there can be a considerable delay between the time of the merger and the time at which the principal heating occurs. Three mechanisms can cause heating at these larger distances, one resulting from the sudden loss of mass from the merged black hole due to its emission of gravitational waves [@BodePh07], another due to its sudden loss of angular momentum, and a third the result of the merged black hole’s recoil as a result of asymmetric gravitational wave radiation [@LFH08]. We stress, however, that the radial scales whose radiation is under consideration here are [*not*]{} those responsible for the longer-term afterglow that has been the focus of prior work [@MP05; @LFH08; @SB08; @SK08; @Rossi09; @Corrales09]; the afterglow is due to heating of mass in the circumbinary disk proper, at the relatively large radii (at least $\sim 100 r_g$: [@MP05]) where conventional accretion dynamics were able to bring it during the time when gravitational wave evolution of the binary was faster than the typical inflow rate. Although the relevant heating mechanisms in the radial range considered here are very similar to those acting at larger radii, the focus of this work is on matter interior to the disk proper, where any gas present arrived as the result of angular momentum loss faster than that acting on the bulk of the circumbinary disk. Note, too, that because we restrict our attention to $r/r_g < 10^3$, all the gas remains bound to the merged black hole for even the largest of recoil velocities.
As a result of the mass lost by the black hole in gravitational wave radiation, the binding energy of orbiting matter is immediately reduced by $(\Delta M/M)(r/r_g)^{-1}$, where the fractional black hole mass loss $\Delta M/M \sim 1$–$10\%$ [@Berti07; @Rezz08; @LCZ09] with the exact number depending on the mass ratio and spins of the merging pair. Relative to the dynamical timescale, this change in energy occurs almost instantaneously because the merger duration is so much shorter than an orbital period when $r \gg r_g$. However, any heating due to this change in energy is delayed by $\sim (r_g/c)(r/r_g)^{3/2}(\Delta M/M)^{-1}$ because the eccentricity induced in the orbits is only $\sim \Delta M/M$ [@SK08; @ONeill09]. In addition, it is possible that only a fraction $\Delta M/M$ of the orbital energy gained is dissipated, as most of the energy may be used simply to expand the gas orbits [@ONeill09; @Corrales09; @Rossi09].
There can also be dissipation due to the sudden change in angular momentum. Torques driven by the binary’s quadrupolar mass distribution acting on matter surrounding the pre-merger binary will cause any obliquely-orbiting gas to precess around the direction of the binary’s total angular momentum (at large separation, the total angular momentum is almost exactly the orbital angular momentum); dissipation between intersecting fluid orbits should then lead to alignment of the gas’s orbital angular momentum with the binary’s angular momentum [@LubowOg00]. If there is any misalignment between either of the two spin directions and the orbital angular momentum, the angular momentum of the merged black hole could be in a different direction from the original total angular momentum [@Schn04; @Bogdan07; @Rezz08]. After the merger, Lense-Thirring torques will act in a fashion closely analogous to the Newtonian torques acting during the binary phase, and dissipation should then reorient orbiting gas into the new equatorial plane [@BP75]. Because the kinetic energy of motion out of the new equatorial plane is a fraction $\sim \sin^2(\Delta\theta)$ of the orbital energy for misalignment angle $\Delta\theta$, the amount of energy dissipated can be an order unity fraction of the orbital energy. The delay from the time of merger to when this mechanism acts will be of order the precession time, $\sim (r/r_g)^3 (r_g/c)$, which is rather longer than the delay before dissipating the orbital energy gained from mass-loss wherever $r/r_g > (\Delta M/M)^{-2/3}$. However, the degree of misalignment is highly uncertain and may depend strongly on details of the environment; for example, accretion during the inspiral may align both black hole spins with the orbital angular momentum, eliminating a change in angular momentum direction as a result of the merger [@Bogdan07]. Close alignment of both spins with the orbital angular momentum will, however, have the compensating effect of producing an especially large $\Delta M/M$ [@LCZ09]. Because both the mass-loss and the Lense-Thirring effects scale with the local binding energy, we will combine them, writing their efficiency as $\epsilon \simeq \eta (r/r_g)^{-1}$, where we expect $\eta \lesssim 0.01$.
Lastly, response to the black hole recoil adds an energy per unit mass to the disk matter $\sim v_{\rm recoil}^2$. Even if only a fraction $\Delta M/M$ of the mass-loss energy leads to heating, the recoil energy becomes larger than the heating due to mass-loss only at radii $$r > \left(\frac{\Delta M/M}{v_{\rm recoil}/c}\right)^2
\simeq 10^3 \left(\frac{\Delta M/M}{0.03}\right)^2
\left(\frac{v_{\rm recoil}}{300\hbox{~km~s$^{-1}$}}\right)^{-2} ,$$ [@SK08; @Rossi09], so we neglect it in the estimates presented here.
Cooling Time and Luminosity
===========================
Whatever heat is deposited in the gas, the rate at which this energy is carried away by radiation is determined by how rapidly electrons can generate photons and then by how rapidly those photons can make their way outward through the opacity presented by the material itself. We will consider the first issue later (§ \[sec:spectrum\]) when we discuss the complications of estimating the radiating gas’s temperature. For the time being we will assume that photon diffusion is the slower of the two processes. Unless $\epsilon$ is extremely small, the gas is likely to be so thoroughly ionized that electron scattering dominates its opacity, so the optical depth is simply proportional to the surface density.
To be optically thin, the surface density $\Sigma$ must then be very small: $\Sigma < 3$ gm cm$^{-2} \simeq 3 \times 10^{-9} M_7^2 (M_{\odot}/r_g^2)$, where $M_7$ is the total black hole mass in units of $10^7 M_{\odot}$. In this case, if the gas is able to convert heat into photons at least as fast as the heat is delivered, the luminosity per logarithmic radius is the ratio of energy deposited to the time in which it is dissipated, roughly the dynamical time: $$\frac{dL}{d\ln r} \simeq 2.5 \times 10^{44} \epsilon(r) \Sigma (r/r_g)^{1/2} M_7
\hbox{~erg~s$^{-1}$}$$ for $\Sigma$ in gm cm$^{-2}$. The duration of such a flare should be only $\sim 50 (r/r_g)^{3/2}M_7$ s.
A more interesting regime is presented by the case of an optically thick region. Under optically thick conditions, if the vertical scaleheight of the disk is $h$, the cooling time $t_{\rm cool} \sim \tau h/c \sim \tau (h/r)(r/r_g)(r_g/c)$, where the optical depth from the midplane outward is $\tau = \kappa \Sigma/2$ for opacity per unit mass $\kappa$. Because the duration of the merger event is only $\sim O(10) r_g/c$, unless the disk is able to stay very geometrically thin (which the following argument will demonstrate is unlikely), the heat given the gas during the merger will be radiated over a time much longer than the merger event proper.
The disk thickness is controlled by two elements: the initial heat content of the disk and the heat deposited as a result of the merger. We will neglect the former partly because it is so uncertain, partly because it seems plausible that it will be outweighed by the latter, and partly because this represents in some respects a conservative assumption.
Suppose, then, to begin that the optical depth is large enough to make the cooling time longer than the orbital time (i.e., $\tau (h/r) > (r/r_g)^{1/2}$). There will then be time for the gas to achieve a dynamical equilibrium (if one is possible). Radiation pressure will likely dominate gas pressure because the photon escape time is much longer than the photon radiation time (the reasoning behind this assertion is discussed in § \[sec:spectrum\]). Put another way, nearly all the thermal energy density initially given the matter is rapidly converted into photons; when the cooling time is longer than the dynamical time, they are still present for many dynamical times. Consequently, their pressure becomes much greater than the gas pressure. The force exerted by the (slowly) diffusing photons is proportional to their flux times the opacity; because the flux is the energy per unit area per cooling time, we can determine $h$ by matching the vertical radiation force to the vertical component of gravity. Here, in order to obtain a rough estimate of the disk thickness, we suppose that $h \ll r$ and that Newtonian gravity applies: $$\frac{\kappa\epsilon \Sigma c^2}{2 \tau h} \simeq h \Omega^2,$$ which leads to $$h/r \simeq \epsilon^{1/2} (r/r_g)^{1/2}.$$ In other words, the geometric profile of optically thick gas immediately post-merger depends only on the heating efficiency $\epsilon$ (if it were optically thin, $h/r \simeq (\tau \epsilon r/r_g)^{1/2}$). Moreover, if this equilibrium is achieved, the criterion that the cooling time exceed the dynamical time is easily achieved in optically thick disks if $\epsilon$ is not too small, for all that is required is $\tau > \epsilon^{-1/2}$.
Close to the black hole, where $\epsilon$ may be as much as $\sim O(0.1)$, the disk may be almost spherical. It is imaginable that $\epsilon$ is so large that no hydrostatic equilibrium is possible (i.e., $h/r > 1$); in that case, the radiation flux would drive the gas away from the black hole merger remnant. For the purposes of this order-of-magnitude treatment, we ignore that possibility; we also ignore further order-unity corrections that might result from time-dependent photon diffusion effects. Farther from the black hole, where $\epsilon \sim \eta (r/r_g)^{-1}$, the disk should be thinner: $h/r \sim \eta^{1/2}$. In these more distant regions, cancellation of the radial scalings leaves the thickness to be determined by the details of local heat dissipation (i.e., the effectiveness of dissipating the energy gain due to mass-loss and whatever disk re-orientation takes place).
With the disk thickness estimated, the cooling time immediately follows: $$t_{\rm cool} = \tau h/c \simeq 50 \tau \epsilon^{1/2} (r/r_g)^{3/2} M_7\hbox{~s}.$$ Equivalently, it is $\simeq (\tau/2\pi)\epsilon^{1/2}$ orbital periods.
Although the radiating timescale is proportional to the surface density, the luminosity is [*independent*]{} of it so long as the heating time is shorter than the cooling time. This is simply because the energy to be radiated is proportional to the surface density, while the cooling time is likewise: $$\label{eq:lum}
\frac{dL}{d\ln r} \simeq 1.5 \times 10^{45} \epsilon^{1/2} (r/r_g)^{1/2}
M_7 \hbox{~erg~s$^{-1}$} = \epsilon^{1/2} (r/r_g)^{1/2} L_E .$$ The luminosity scale is the Eddington luminosity because the time to cross the heated (and inflated) radiating region is proportional to how well the radiation flux can resist gravity. How close the luminosity approaches to Eddington is determined by the efficiency.
Farther from the black hole, the efficiency $\epsilon = \eta (r/r_g)^{-1}$, so if $t_{\rm cool} > t_{\rm heat}$ and $\tau > 1$, $$\frac{dL}{d\ln r} \simeq \eta^{1/2} L_E.$$ That is, the luminosity from regions where $r/r_g \gg 1$ should scale with the Eddington luminosity, but may be a fairly small fraction of it.
However, in these more distant regions (i.e., $10 \lesssim r/r_g < 10^3$), the heating time can be so extended that it might be longer than the cooling time, particularly if the optical depth is not very large: $$\frac{t_{\rm cool}}{t_{\rm heat}} \sim \tau \eta^{1/2}\begin{cases}
(r/r_g)^{-1/2} & \text{mass-loss}\\
(r/r_g)^{-2} & \text{Lense-Thirring}\end{cases}.$$ When the dynamical response of the disk is so slow that it exceeds the cooling time, the luminosity becomes $$\frac{dL}{d\ln r} = \eta \tau L_E\begin{cases}
(r/r_g)^{-1/2} & \text{mass-loss} \\
(r/r_g)^{-2} & \text{Lense-Thirring}\end{cases}.$$
To find the total observed luminosity, we must assemble the luminosity from different regions, making proper allowance for their different time-dependences. Initially, the radiative output will be dominated by the inner radii, so that $L \simeq (\epsilon r_{\rm in}/r_g)^{1/2}L_E$, where $r_{\rm in}$ is the scale of the region subject to the truly dynamical spacetime. After a time $50 \tau \epsilon^{1/2} (r_{\rm in}/r_g)^{3/2} M_7$ s, this light decays, to be replaced over longer timescales by the signals due to mass-loss and the Lense-Thirring reorientation: $\sim 50 (r/r_g)^{3/2} M_7 (\Delta M/M)^{-1}$ s for the former, $\sim 50 (r/r_g)^3 M_7$ s for the latter. As estimated above, the luminosity from these regions at somewhat larger radius is likely rather less than the luminosity issuing from the innermost region.
Temperature and Spectrum {#sec:spectrum}
========================
Prediction of the output spectrum is much more model-dependent. If there is enough absorptive opacity to thermalize the radiation, its characteristic temperature would be $$T \sim 1 \times 10^6 \epsilon^{1/8} (r/r_g)^{-3/8} M_7^{-1/4} \hbox{~K},$$ similarly universal, decreasing only slowly with increasing black hole mass.
Whether thermalization can be achieved, however, may be sensitive to conditions. The effectiveness of free-free opacity can be enhanced by the additional path-length to escape imposed upon the photons by scattering. At the temperature just estimated, the effective optical depth (i.e., the geometric mean of the free-free and Thomson optical depths) for photons near the thermal peak is $$\tau_{eff} \simeq 3 \times 10^{-4} \tau^{3/2} \epsilon^{-15/32}
(r/r_g)^{-3/32} M_7^{-1/16}.$$ Thus, if free-free is the only absorption mechanism, $\tau \gtrsim 100 (\epsilon/0.1)^{5/16}$ is required to achieve thermalization. On the other hand, where the temperature is $\sim 10^5$ K or less, atomic transitions substantially enhance the absorptive opacity, making thermalization much more thorough. Thus, the detailed character of the emitted spectrum could vary considerably from case to case.
With an estimate of the temperature, we can now estimate the timescale on which electrons radiate most of the heat given the gas. Considering only free-free emission, it is $$t_{\rm rad} \simeq 60 \epsilon^{1/2} (r/r_g)^{3/2} M_7 \tau^{-1} T_5^{1/2}\hbox{~s}.$$ In other words, $t_{\rm rad}/t_{\rm cool} \simeq 1.2 \tau^{-2} T_5^{1/2}$, so that thermal balance at a temperature $\lesssim 10^5$ K is a self-consistent condition for an optically thick region.
However, unless $\epsilon$ is exceedingly small, the gas’s temperature immediately upon being heated will be far higher than $10^5$ K. If the initial shocks driven by the dynamical spacetime have speeds comparable to the orbital speed, the post-shock electron energies will more likely be well in excess of 1 MeV, and the characteristic radiation rate will be much slower. In this initial high-temperature state, the free-free radiation time is $$t_{\rm rad} \simeq 1 \times 10^4 \epsilon^{1/2} (r/r_g) M_7 \tau^{-1}
\left( \ln \Theta\right)^{-1}\hbox{~s},$$ where $\Theta$ is the electron energy in rest-mass units (assumed to be $> 1$ in this expression), and we have estimated the disk aspect ratio $h/r \simeq \epsilon^{1/2}$ (by assumption, at this stage radiation pressure is not yet important). Comparing this estimate of $t_{\rm rad}$ to $t_{\rm cool}$, we find that $$\label{eq:freefreerad}
\frac{t_{\rm rad}}{t_{\rm cool}} = 200 \tau^{-2} \left( \ln \Theta\right)^{-1},$$ entirely independent of $h/r$ (as well as $r/r_g$ and $M$). Thus, if free-free radiation is the only photon-creation mechanism, optical depths $\gtrsim O(10)$ would be required in order to create photons carrying most of the heat in a time shorter than it takes for those photons to escape.
There are, however, other processes that can also likely contribute. Suppose, for example, that the magnetic field energy density is a fraction $q$ of the plasma pressure. Synchrotron radiation would then cool the gas on a timescale $$t_{\rm rad} \sim 50 q^{-1}\tau^{-1}\Theta^{-2} \epsilon^{1/2}(r/r_g)M_7\hbox{~s}$$ wherever $\Theta > 1$. Relative to the cooling time, this photon production timescale is $$\frac{t_{\rm rad}}{t_{\rm cool}} \sim q^{-1}\tau^{-2}\Theta^{-2}.$$ Inverse Compton radiation would be equally effective if the energy density in photons of energy lower than $\sim \Theta m_e c^2$ is that same fraction $q$ of the plasma pressure. As shown by equation \[eq:freefreerad\], relativistic electron free-free radiation is able to radiate at least $\sim 10^{-2}$ of the heat during a photon diffusion time if $\tau > 1$; we therefore expect the $q$ in photons to be at least this large. Thus, even a small initial cooling by free-free radiation, particularly when supplemented by a modest magnetic field, should provide enough seeds for inverse Compton cooling to allow the gas to radiate the great majority of its heat content in a time shorter than the photon diffusion time. All that is required is $\tau > 10 (q/0.01)^{-1/2}\Theta^{-1}$.
Even if the majority of the dissipated energy is given to the ions, so that their temperature is larger than the electrons’, rapid radiation is still likely to occur. The ratio between the timescale for thermal coupling between ions and electrons by Coulomb collisions and the photon diffusion time is $$\frac{t_{\rm ep}}{t_{\rm cool}} \simeq 60 \tau^{-2} \Theta^{3/2}.$$ Thus, provided $\tau > 8 \Theta^{3/4}$, the plasma should achieve a one-temperature state more rapidly than the photons can leave.
What is $\Sigma(r)$?
====================
Lastly, we turn to the hardest question to answer at this stage: how much gas there should be as a function of radius, here parameterized as $\tau(r)$. Particularly in the inner region, it would take very little gas to create a large optical depth: even integrated out to $r/r_g=100$, a disk with constant $\tau = 100$ would require only $10^{-4}M_7^2 M_{\odot}$.
Even though there is no reason to think the disk is anywhere near a conventional state of inflow equilibrium, one could use the optical depth of such a disk as a standard of comparison. The thermodynamics of equilibrium disks creates a characteristic scale for the surface density: the maximum at which thermal equilibrium can be achieved. One of the predictions of the [@SS73] model is that in a steady-state disk in which the vertically-integrated $r$-$\phi$ stress is $\alpha$ times the vertically-integrated total pressure, the accretion rate at any particular radius increases as the surface density increases, but only up to a point. Larger surface density (and accretion rate) lead to a larger ratio of radiation to gas pressure. If radiation pressure exceeds gas pressure, increasing accretion rate can only be accommodated by a [*decreasing*]{} surface density. In other words, there is a [*maximum*]{} possible surface density. Although recent work on explicit simulation of disk thermodynamics under the influence of MHD turbulence driven by the magneto-rotational instability has shown that this phenomenological model’s prediction about the thermal stability of disks is wrong [@Hirose09a], they also show that vertically-integrated disk properties averaged over times long compared to a thermal time [*do*]{} match those predicted by the $\alpha$ model [@Hirose09b]: when radiation pressure dominates, the surface density and accretion rate are inversely related. The Thomson optical depth corresponding to this maximum surface density is $$\tau \simeq 2.5 \times 10^4 (\alpha/0.1)^{-7/8} M_7^{1/8}(r/10r_g)^{3/16},$$ where we have scaled the stress/pressure ratio to 0.1. Close to the black hole, it occurs at a comparatively low accretion rate in Eddington units: $$\dot m \simeq 1.4 \times 10^{-3} (\alpha/0.1)^{-1/8} (r/10r_g)^{21/16} M_{7}^{-1/8}.$$ Such a state might be consistent with a gas supply rate at large radius capable of feeding an AGN (i.e., $\dot m \sim 0.1$), but reduced two orders of magnitude by the effects of binary torques and the inability of internal stresses in the disk to drive its inner edge inward as fast as gravitational wave emission compresses the black hole binary. It is significant in this respect that even such a strong suppression of accretion still yields an inner disk optical depth that is quite large. A smaller accretion rate would produce a smaller optical depth, but only $\propto \dot m^{3/5}$, when gas pressure dominates and the disk remains radiative.
These estimates can also serve as a springboard to gain a sense of what might occur in states of inflow non-equilibrium. For example, if gas accretes at larger radii but is held back by binary orbital torques at radii several times the binary separation, its surface density at the point where it is held back should be larger than what would be expected on the basis of equilibrium inflow at the accretion rate farther out. In such a case, although the preceding estimates might be reasonable at larger radii, the optical depth could be substantially enhanced closer to the merging binary.
Another uncertainty is presented by the question of whether the optical depth in a given region remains the same over the entire radiating period. In the inner disk, for example, there could be significant radial motions, both inward or outward, engendered directly by the dynamical spacetime during the merger event. Because the cooling time in the inner disk is rather longer than an orbital period, there might be time for the magneto-rotational instability to stir MHD turbulence that could drive accretion and restore some of the merger heat carried off by radiation. If the optical depth is relatively small, so that $\tau \epsilon^{3/2} \alpha (r_{\rm in}/r_g) < 1$ and the accretion time in the inner disk is longer than the cooling time, the luminosity would gradually taper off as the accretion luminosity extends the bright period, but disk cooling causes the accretion rate to diminish. Alternatively, if the optical depth is larger, the inflow time would actually be shorter than the cooling time. In this case, the luminosity would be greater than earlier estimated, but the lifetime of bright emission from the inner disk would be reduced to the inflow time. In either case, the total energy released due to accretion could increase the total emission by a factor of order unity or more because the radiative efficiency of accretion near a black hole is generically $\sim O(0.1)$.
Summary
=======
To summarize, when a pair of supermassive black holes merge, provided only that the gas very close to the merging pair has at least a small electron scattering optical depth, we expect the prompt EM signal to likely have a luminosity comparable to the Eddington luminosity of the merged system, $\sim 10^{45}\epsilon^{1/2} M_7$ erg s$^{-1}$. The duration of this bright phase is proportional to the mass of gas at $\sim 10r_g$ from the merged black hole, a quantity that is at present extremely difficult to estimate. If there is only the very small amount of gas necessary to be optically thick, the duration would be only $\sim 10M_7$ min; on the other hand, quantities several orders of magnitude larger are well within the range of plausibility. Gas at somewhat greater distance ($10 < r/r_g < 10^3$) should continue to radiate at a level possibly as high as $\sim 0.1 L_E$, but for a longer time.
Because gravitational wave observatories like the [*Laser Interferometric Space Antenna*]{} (LISA) are expected to give approximate source localization days or even weeks in advance of merger [@LH09], one could hope for synergistic observing campaigns that might catch the entire EM signal. Alternatively, it may be feasible to use EM surveys with large solid-angle coverage to search for source flaring having the characteristics predicted here in order to identify candidate black hole merger systems before any gravitational wave detectors are ready.
The spectrum of this light is, at present, much more difficult than its luminosity to predict with any degree of confidence. It may peak in the ultraviolet, but if it does, it is likely to be rather bluer than the familiar UV-peaking spectra of AGN (if the spectrum does peak in the UV, extinction in the host galaxy may obscure some number events). While the inner region still shines, its luminosity will likely be greater than that from greater radii, making the spectrum (to the degree it is thermalized) closer to that of a single-temperature system. If it is only incompletely thermalized, a still harder spectrum might be expected. At later times, when the outer disk dominates, the temperature corresponding to a given radius is $\propto \eta^{1/8} (r/r_g)^{-1/2} M_7^{1/4}$. Compared to the temperature profile of an accretion disk in inflow equilibrium ($T \propto (r/r_g)^{-3/4}$), this is a slower decline outward, suggesting a spectrum that might be softer than during the initial post-merger phase, but still harder than that of typical AGN. In addition, over time its high frequency cut-off will move to lower frequencies. Because our understanding of what determines the spectra of accretion disks around black holes is far less solid than our understanding of their gross energetics (consider, for example, the still-mysterious fact that ordinary AGN radiate a significant fraction of their bolometric luminosities in hard X-rays), these spectral predictions are far shakier than the predicted bolometric luminosity scale.
After the merger heat has been radiated, the disk should revert to more normal accretion behavior and, beginning at the smallest radii, should display standard AGN properties [@MP05]. Unfortunately, the pace of fading as a function of radius depends on the run of surface density with radius, which at the moment is highly uncertain. It is therefore far beyond the scope of even this speculative paper to guess how rapidly that may occur.
This work was partially supported by NSF grant AST-0507455 and NASA Grant NNG06GI68G. I am grateful to Omer Blaes, John Hawley, and Jeremy Schnittman for comments on the manuscript, and to both Manuela Campanelli and Jeremy Schnittman for stimulating my interest in this subject.
[^1]: [@KocLoeb08] suggested that the dissipation should be related to shear in the way characteristic of steady-state accretion disks. However, in the present case, the shear varies much more rapidly than the saturation time for the MHD turbulence whose dissipation is relevant to steady-state disks. The classical relationship is therefore unlikely to apply.
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abstract: 'We present a first, consistent combination of measurements from top-quark and $B$ physics to constrain top-quark properties within the Standard Model Effective Field Theory (SMEFT). We demonstrate the feasibility and benefits of this approach and detail the ingredients required for a proper combination of observables from different energy scales. Specifically, we employ measurements of the $t\bar t\gamma$ cross section together with measurements of the $\bar B\rightarrow X_s\gamma$ branching fraction to test the Standard Model and look for new physics contributions to the couplings of the top quark to the gauge bosons within SMEFT. We perform fits of three Wilson coefficients of dimension-six operators considering only the individual observables as well as their combination to demonstrate how the complementarity between top-quark and $B$ physics observables allows to resolve ambiguities and significantly improves the constraints on the Wilson coefficients. No significant deviation from the Standard Model is found with present data.'
author:
- 'Stefan Bi[ß]{}mann'
- Johannes Erdmann
- Cornelius Grunwald
- Gudrun Hiller
- Kevin Kröninger
date: 'Received: date / Accepted: date'
subtitle: 'DO-TH 19/17'
title: 'Constraining top-quark couplings combining top-quark and $B$ decay observables'
---
Introduction
============
The experiments at the Large Hadron Collider (LHC) conduct various searches for physics beyond the Standard Model (BSM). The searches for direct production of new particles have not yet resulted in any discovery of BSM physics. A complementary approach are indirect searches, where precise measurements of total rates and kinematic distributions are compared to their Standard Model (SM) predictions. If the new particles are heavier than the experimental energy scale, the Standard Model Effective Field Theory (SMEFT) can be applied to parametrize potential deviations from the SM in a model-independent way [@Weinberg:1978kz; @Buchmuller:1985jz; @Grzadkowski:2010es]. For energies below the scale of BSM physics, $\Lambda$, effects of new particles and interactions can be described in a series of higher-dimensional operators constructed from SM fields.
The top quark plays a special role in SMEFT analyses and a large number of precision measurements regarding top-quark physics have been performed at the LHC. As the top quark is the only fermion with an $\mathcal{O}(1)$ Yukawa coupling, it is of special interest in BSM scenarios explaining the origin of electroweak symmetry breaking (EWSB). For these reasons, numerous SMEFT analyses in the top-quark sector have been performed during the recent past [@Degrande:2018fog; @Chala:2018agk; @Durieux:2014xla; @AguilarSaavedra:2010zi; @DHondt:2018cww; @Durieux:2018ggn; @Buckley:2015nca; @Buckley:2015lku; @deBeurs:2018pvs; @Brown:2019pzx; @AguilarSaavedra:2018nen; @Hartland:2019bjb; @Maltoni:2019aot; @Durieux:2019rbz]. In particular, first global studies have been presented in Refs. [@Buckley:2015nca; @Buckley:2015lku; @AguilarSaavedra:2018nen; @Brown:2019pzx; @Hartland:2019bjb; @Durieux:2019rbz].
Additional constraints on BSM contributions to top-quark physics come from $B$ physics (see e.g. Refs. [@Fox:2007in; @Grzadkowski:2008mf; @Drobnak:2011aa]). Especially flavor-changing neutral currents are excellent probes of BSM physics due to suppression by the Fermi constant, small CKM matrix elements and loop factors. The Weak Effective Field Theory (WET) Lagrangian describing $b\rightarrow s$ transitions is not invariant under the full SM gauge group due to EWSB at the scale $v$. Since the scale $\Lambda$ has to be above $v$, BSM physics needs to be integrated out before EWSB. To constrain SMEFT coefficients using low-energy observables, the effective Lagrangian must be matched onto the WET Lagrangian by integrating out all particles heavier than the $b$ quark [@Aebischer:2015fzz; @Fox:2007in; @Drobnak:2011aa; @Grzadkowski:2008mf].
Matching and renormalization group equation (RGE) evolution enable to combine measurements at different energy scales in one analysis that allows to investigate the impact of measurements from top-quark and $B$ physics on the top-quark sector of SMEFT.
In this paper, we consider $t\bar t\gamma$ production cross sections and the $\bar B\rightarrow X_s\gamma$ branching fraction to perform a first consistent fit of SMEFT Wilson coefficients using a combination of top-quark and $B$ physics observables that have a common set of relevant dimension-six operators. We present the steps necessary for such a combined analysis of BSM contributions to top-quark interactions and highlight possible pitfalls in this procedure. We determine the dependence of the observables on the Wilson coefficients and compare our computations to results obtained with existing tools. We estimate the gain in the sensitivity for BSM contributions when considering top-quark and $B$ physics observables in a combined fit.
The outline of this paper is as follows. In Sec. \[Sec:SMEFT\] we introduce the SMEFT and WET Lagrangians and introduce conventions used throughout this paper. In Sec. \[Sec:Match\] we discuss the steps necessary to calculate low energy observables in dependence of SMEFT Wilson coefficients. The measurements used to constrain the SMEFT Wilson coefficients are presented in Sec. \[Sec:Measurements\]. In Sec. \[Sec:Modelling\] we describe the corresponding computations of the SM and BSM contributions. In Sec. \[Sec:Fit\] we determine constraints on the SMEFT Wilson coefficients. We investigate the individual impact of top-quark and $B$ observables and demonstrate how the combination of these observables improves the constraints. In Sec. \[Sec:Conclusion\] we conclude. Auxiliary information is given in several appendices.
Effective field theories at different scales {#Sec:SMEFT}
============================================
In this section we describe the effective field theory approach to $t\bar t \gamma$ production and transitions, for which a set of common dimension-six operators exists. In Sec. \[Sec:SMEFT\_Top\] we give the SMEFT operators considered in our analysis. In Sec. \[Sec:WET\] we introduce the effective theory for $b\rightarrow s\gamma$ transitions.
Effective Lagrangian for $t\bar{t}\gamma$ production {#Sec:SMEFT_Top}
----------------------------------------------------
The effects of heavy BSM particles with mass scale $\Lambda$ can be described at lower energies $E \ll \Lambda$ in a basis of effective operators with mass dimension $d>4$ [@Weinberg:1978kz; @Buchmuller:1985jz]. Such higher-dimensional operators are constructed from SM fields and are required to be Lorentz invariant and in accord with SM gauge symmetries. The SMEFT Lagrangian $\mathcal{L}_\textmd{SMEFT}$ is an expansion in powers of $\Lambda^{-1}$. Higher-dimensional operators $O_i^{(d)}$ of dimension $d$ are added to the SM Lagrangian together with the corresponding Wilson coefficients $C_i^{(d)}$ and a factor $\Lambda^{d-4}$. The effective Lagrangian reads $$\begin{aligned}
\mathcal{L}_\textmd{SMEFT}=\mathcal{L}_\textmd{SM}+\sum_i\frac{C^{(6)}_i}{\Lambda^2}O_i^{(6)}+\mathcal{O}\left(\Lambda^{-4}\right)\,.
\label{Glg:L_eff}\end{aligned}$$ Operators of dimension $d=5$ and $d=7$ are not considered in this work since they violate lepton and baryon number conservation [@Degrande:2012wf; @Kobach:2016ami]. In the following, we only consider operators with mass dimension $d=6$, which are the leading BSM contributions to LHC physics.
A complete basis containing 59 independent operators for one generation (2499 for three generations [@Alonso:2013hga]) of fermions is presented in Ref. [@Grzadkowski:2010es] in the *Warsaw basis*, which is used in the following. Fortunately, for any class of observables only a small subset of operators has to be considered.
{width="30.00000%"} {width="30.00000%"} {width="30.00000%"}\
{width="30.00000%"} {width="30.00000%"} {width="30.00000%"}
We study the dimension-six operators affecting $t\bar{t}\gamma$ production at the LHC. Examples for lowest order Feynman diagrams with both gluons and quarks as initial states are shown in Fig. \[Fig:Feynman\_ttbary\]. We consider only operators involving third-generation quarks and bosonic fields, including the Higgs field. The corresponding operators can be written as $$\begin{aligned}
\begin{aligned}
O_{uB}&=\left(\bar{q}_L\sigma^{\mu\nu}u_R\right)\tilde{\varphi}B_{\mu\nu} \,, \\
O_{uG}&=\left(\bar{q}_L\sigma^{\mu\nu}T^{A}u_R\right)\tilde{\varphi}G_{\mu\nu}^{A} \,, \\
O_{uW}&=\left(\bar{q}_L\sigma^{\mu\nu}\tau^{I}u_R\right)\tilde{\varphi}W_{\mu\nu}^{I} \,,
\label{Eq:TopOperator}
\end{aligned}\end{aligned}$$ with $q_L$ the $SU(2)$ doublet, $u_R$ the up-type $SU(2)$ singlet, the gauge field strength tensors $B_{\mu\nu}$, $W^I_{\mu\nu}$ and $G^A_{\mu\nu}$ of $U(1)_Y$, $SU(2)_L$ and $SU(3)_C$ and the generators $T^A$ and $\tau^I$ of $SU(3)_C$ and $SU(2)_L$, respectively. Generally, the effective operators in Eq. (\[Eq:TopOperator\]) are non-hermitian which leads to complex-valued Wilson coefficients. In this analysis, we assume all Wilson coefficients to be real valued. Four-quark operators can in principle also affect $t\bar{t}\gamma$ production. As $t\bar t$ production at the LHC is dominated by the $gg$ channel ($\sim 75\,\%$ and $\sim90\,\%$ at $8\,\si{\tera\electronvolt}$ and $13\,\si{\tera\electronvolt}$, respectively [@Buckley:2015lku]), we neglect contributions from four-quark operators.
Effective Lagrangian for $\bar B \rightarrow X_s \gamma $ decays {#Sec:WET}
----------------------------------------------------------------
Rare $b\rightarrow s\gamma$ processes can be described by the Weak Effective Field Theory (WET) Lagrangian [@Chetyrkin:1996vx] $$\mathcal{L}_\text{WET}=\frac{4G_F}{\sqrt{2}}V_{ts}^ *V_{tb}\sum_{i=1}^8\bar C_iQ_i\,,
\label{Eq:WETLagrangian}$$ where $V_{ij}$ are elements of the CKM matrix, $G_F$ is the Fermi coupling constant, $Q_i$ are effective operators and $\bar C_i$ are the corresponding Wilson coefficients including both SM and BSM contributions. The effective operators relevant for the processes considered here are the four-fermion operators $$\begin{aligned}
Q_1&=(\bar s_L\gamma_\mu T^ac_L)(\bar c_L\gamma^\mu T^a b_L)\,,\\
Q_2&=(\bar s_L\gamma_\mu c_L)(\bar c_L\gamma^\mu b_L)\,,\\
Q_3&=(\bar s_L\gamma_\mu b_L)\sum_q(\bar q\gamma^\mu q)\,,\\
Q_4&=(\bar s_L\gamma_\mu T^ab_L)\sum_q(\bar q\gamma^\mu T^a q)\,,\\
Q_5&=(\bar s_L\gamma_\mu\gamma_\nu\gamma_\sigma b_L)\sum_q(\bar q\gamma^\mu\gamma^\nu\gamma^\sigma q)\,,\\
Q_6&=(\bar s_L\gamma_\mu\gamma_\nu\gamma_\sigma T^ab_L)\sum_q(\bar q\gamma^\mu\gamma^\nu\gamma^\sigma T^aq)\,,
\end{aligned}$$ as well as the dipole operators $$\begin{aligned}
Q_7&=\frac{e}{16\pi^2}m_b(\bar s_L\sigma^{\mu\nu}b_R)F_{\mu\nu}\,,\\
Q_8&=\frac{g_s}{16\pi^2}m_b(\bar s_L\sigma^{\mu\nu}T^ab_R)G^a_{\mu\nu}\,,
\end{aligned}$$ with chiral left (right) projectors $L$ ($R$) and the field strength tensor of the photon $F_{\mu\nu}$. We neglect contributions proportional to the small CKM matrix element $V_{ub}$ and to the strange-quark mass.
Matching at one-loop level {#Sec:Match}
==========================
{width="80.00000%"}
To describe BSM physics at energies below the electroweak scale $\mu_W$, the SMEFT Lagrangian in Eq. (\[Glg:L\_eff\]) has to be matched onto the WET Lagrangian as illustrated in Fig. \[fig:energyscale\]. Top-quark measurements allow to constrain the values of Wilson coefficients at the scale $\mu_t\sim m_t$. At the scale $\mu_b\sim m_b$, $B$ measurements can be used to constrain the values of the WET coefficients. To express $B$ observables in terms of SMEFT Wilson coefficients at the scale $\mu_t$, the following steps have to be performed, extending the procedure described in Ref. [@Aebischer:2015fzz]: First, RGE evolution of the SMEFT Wilson coefficients from the scale $\mu_t$ to $\mu_W$ has to be performed. As a next step, $\mathcal{L}_\textmd{SMEFT}$ has to be matched onto $\mathcal{L}_\textmd{WET}$. Finally, the RGE evolution of the WET Wilson coefficients from $\mu_W$ to $\mu_b$ has to be carried out. These three steps allow the computation of observables, such as BR($\bar B\rightarrow X_s\gamma$), at the scale $\mu_b$ in dependence of the SMEFT Wilson coefficients $C_i(\mu_t)$ at the scale $\mu_t$. In the following, we describe each of the three steps for the $b \rightarrow s \gamma $ process considered in this work.
RGE evolution in SMEFT
----------------------
The computation of the RGEs in SMEFT is based on Refs. [@Alonso:2013hga; @Jenkins:2013zja; @Jenkins:2013wua]. To describe the RGE evolution of the operators in Eq. (\[Eq:TopOperator\]) at $\mathcal{O}(\alpha_s)$, the following SMEFT operators have to be included due to mixing: $$\begin{aligned}
O_{u\varphi}&=\left(\varphi^\dagger\varphi\right)\left(\bar q_L u_R \tilde \varphi\right)\,,\\
O_{\varphi G}&=\left(\varphi^\dagger\varphi\right)G^A_{\mu\nu}G^{A\mu\nu}\,,
\\
O_{\varphi \tilde G}&=\left(\varphi^\dagger\varphi\right)\tilde G^A_{\mu\nu}G^{A\mu\nu}\,,\quad
\end{aligned}$$ with $\tilde G^A_{\mu\nu} = \frac{1}{2}\epsilon_{\mu\nu\alpha\beta}G^{A\alpha\beta}$ ($\epsilon_{0123}=+1$). To compute the anomalous dimension matrix at $\mathcal{O}(\alpha_s)$, the effective operators have to be rescaled [@Jenkins:2013sda]: $$\begin{aligned}
O^\prime_{uB}&= yg^\prime\left(\bar{q}_L\sigma^{\mu\nu}u_R\right)\tilde{\varphi}B_{\mu\nu} \,, \\
O^\prime_{u\varphi}&=y\left(\varphi^\dagger\varphi\right)\left(\bar q_L u_R \tilde \varphi\right)\,,\\
O^\prime_{uG}&=yg_s\left(\bar{q}_L\sigma^{\mu\nu}T^{A}u_R\right)\tilde{\varphi}G_{\mu\nu}^{A} \,, \\
O^\prime_{\varphi G}&=g_s^2\left(\varphi^\dagger\varphi\right)G^A_{\mu\nu}G^{A\mu\nu}\,,\\
O^\prime_{uW}&=yg\left(\bar{q}_L\sigma^{\mu\nu}\tau^{I}u_R\right)\tilde{\varphi}W_{\mu\nu}^{I} \,,\\
O^{\prime}_{\varphi \tilde G}&=g_s^2\left(\varphi^\dagger\varphi\right)\tilde G^{A}_{\mu\nu}G^{A\mu\nu}\,,
\end{aligned}$$ where $g^\prime$, $g$ and $g_s$ are the coupling constants corresponding to $U(1)_Y$, $SU(2)_L$ and $SU(3)_C$, respectively, and $y$ denotes a Yukawa coupling. The Wilson coefficients change with inverse powers of the couplings. In terms of the rescaled coefficients, the RGEs in SMEFT read $$\frac{d}{d\ln\mu}\begin{pmatrix}
C_{uG}^\prime\\C_{uW}^\prime\\C_{uB}^\prime\\C^\prime_{u\varphi}\\C^\prime_{\varphi G}\\C^{\prime}_{\varphi \tilde G}
\end{pmatrix}
=\frac{\alpha_s}{4\pi} \frac{4}{3}\begin{pmatrix}
1 & 0 & 0 & 0 & -3 & -3i\\
2 & 2 & 0 & 0 & 0 & 0 \\
\frac{10}{3} & 0 & 2 & 0 & 0 & 0\\
-24 & 0 & 0 & -6 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}\begin{pmatrix}
C_{uG}^\prime\\C_{uW}^\prime\\C_{uB}^\prime\\C^\prime_{u\varphi}\\C^\prime_{\varphi G}\\C^{\prime}_{\varphi \tilde G}
\end{pmatrix}
\,.
\label{Eq:SMEFTRGE}$$
This matrix is not closed at $\mathcal{O}(\alpha_s)$: The operators $O^\prime_{\varphi G}$ and $O^\prime_{\varphi \tilde G}$ give contributions to the running of $$\begin{aligned}
O^\prime_{dG}=yg_s\left(\bar{q}_L\sigma^{\mu\nu}T^{A}d_R\right){\varphi}G_{\mu\nu}^{A}\end{aligned}$$ and $O^\prime_{u G}$ contributes to the running of the four-quark operators $$\begin{aligned}
O^{\prime(1)}_{quqd}&=(q_L^iu_R)\epsilon_{ij}(q_L^jd_R)\,,\\ O^{\prime(8)}_{quqd}&=(q_L^iT^Au_R)\epsilon_{ij}(q_L^jT^Ad_R)\,,\end{aligned}$$ where $i,j$ are isospin indices and $\epsilon_{12}=+1$. These contributions are suppressed by small down-type Yukawa couplings and neglected in Eq. (\[Eq:SMEFTRGE\]). Further more, we see from Eq. (\[Eq:SMEFTRGE\]) that $C^\prime_{\varphi G}$ and $C^{\prime}_{\varphi \tilde G}$ do not change their values due to running at $\mathcal{O}(\alpha_s)$. Since $O_{\varphi G}$ and $O_{\varphi \tilde G}$ have no sizable effect on $t\bar t \gamma$ production [@Buckley:2015lku] and $b\rightarrow s\gamma $ transitions, we neglect $O^\prime_{\varphi G}$ and $O^\prime_{\varphi \tilde G}$ under the assumption that only operators including the top quark are generated at the scale $\Lambda$. The operator $ O^\prime_{u\varphi}$ does not directly affect the observables we study but is needed to absorb the UV divergence in the top-quark mass corrections from $O^\prime_{uG}$ in SMEFT NLO computations [@Zhang:2014rja]. We compute the BSM contributions at LO QCD and neglect $ O^\prime_{u\varphi}$.
Matching SMEFT onto WET
-----------------------
![Examples of one-loop diagrams for $b\rightarrow s\gamma$ and $b\rightarrow sg$ transitions. The black dots denote the insertion of a SMEFT operator[]{data-label="Fig:MatchingSMEFTWET"}](draft-figure0.pdf "fig:"){width="23.00000%"} ![Examples of one-loop diagrams for $b\rightarrow s\gamma$ and $b\rightarrow sg$ transitions. The black dots denote the insertion of a SMEFT operator[]{data-label="Fig:MatchingSMEFTWET"}](draft-figure1.pdf "fig:"){width="23.00000%"} ![Examples of one-loop diagrams for $b\rightarrow s\gamma$ and $b\rightarrow sg$ transitions. The black dots denote the insertion of a SMEFT operator[]{data-label="Fig:MatchingSMEFTWET"}](draft-figure2.pdf "fig:"){width="23.00000%"} ![Examples of one-loop diagrams for $b\rightarrow s\gamma$ and $b\rightarrow sg$ transitions. The black dots denote the insertion of a SMEFT operator[]{data-label="Fig:MatchingSMEFTWET"}](draft-figure3.pdf "fig:"){width="23.00000%"}
In Fig. \[Fig:MatchingSMEFTWET\] we give examples for one-loop diagrams including contributions from operators in Eq. (\[Eq:TopOperator\]) to $\mathcal{L}_\text{WET}$. The matching conditions have been calculated in Ref. [@Aebischer:2015fzz] and read $$\begin{aligned}
\begin{aligned}
\Delta \bar C_{7}^{(0)}&=\frac{\sqrt{2}m_t}{m_W}\bigg[\tilde C_{uW}E_{7}^{uW}(x_t)+\tilde C_{uW}^*F_{7}^{uW}(x_t)
\\&+\frac{\cos\theta_w}{\sin\theta_w}\left(\tilde C_{uB}E_{7}^{uB}(x_t)+\tilde C_{uB}^*F_{7}^{uB}(x_t)\right)\bigg]\,, \label{Eq:MatchC7}
\end{aligned}\end{aligned}$$ $$\begin{aligned}
\begin{aligned}
\Delta \bar C_{8}^{(0)}&=\frac{\sqrt{2}m_t}{m_W}\bigg[\tilde C_{uW}E_{8}^{uW}(x_t)+\tilde C_{uW}^*F_{8}^{uW}(x_t)\\
&-\frac{g}{g_s}\left(\tilde C_{uG}E_8^{uG}(x_t)+\tilde C_{uG}^*F_8^{uG}(x_t)\right)\bigg]\,,
\label{Eq:MatchC8}
\end{aligned}\end{aligned}$$ where $x_t=m_t^2/m_W^2$ and $\Delta \bar C_i^{(0)}$ denotes BSM contributions at order $\alpha_s^0$ to the coefficients in $\mathcal{L}_{\textmd{WET}}$. The $\tilde C_i$ denote rescaled Wilson coefficients $$\tilde C_i =C_i \frac{v^2}{\Lambda^2}\,,$$ where $v=\SI{246}{\GeV}$ is the Higgs vacuum expectation value. Explicit expressions for the $x_t$-dependent functions $E_{7}^{uW}$, $F_{7}^{uW}$, $E_{8}^{uW}$ and $F_{8}^{uW}$ can be found in Ref. [@Aebischer:2015fzz] and are given in \[App:Match\].
RGE evolution in WET
--------------------
At the scale $\mu_W$, both the SM and BSM contributions are matched onto $\mathcal{L}_\textmd{WET}$. The RGEs are then used to evolve the coefficients $\bar C_i$ from $\mu_W$ to $\mu_b$. By doing so, large logarithms are resummed to all orders in perturbation theory. Instead of the original coefficients $\bar C_i$ it is convenient to use the effective coefficients [@Buras:1993xp; @Greub:1996jd] $$C_i^\textmd{eff}=\begin{cases}
\bar{C}_i & \textmd{for }i=1,...,6\\
\bar C_7+\sum_{j=1}^6y_j\bar C_j & \textmd{for }i=7\\
\bar C_8+\sum_{j=1}^6z_j\bar C_j & \textmd{for }i=8\\
\end{cases}\,.$$ One finds $y=(0,0,-1/3,-4/9,-20/3,-80/9)$ and $z=(0,0,1,-1/6,20,-10/3)$ [@Chetyrkin:1996vx] in the $\overline{MS}$ scheme with fully anticommuting $\gamma_5$. The RGEs for the effective coefficients read $$\frac{d}{d\ln\mu}C_i^\textmd{eff}(\mu)= \gamma^\textmd{eff}_{ji}(\mu)C_j^\textmd{eff}(\mu)\,,
\label{Eq:WETRGE}$$ with the anomalous dimension matrix $\gamma^\textmd{eff}$. The perturbative expansion of this matrix is given as $$\gamma^\textmd{eff}(\mu)=\frac{\alpha_s(\mu)}{4\pi}\gamma^{(0)\textmd{eff}}+\frac{\alpha_s^2(\mu)}{(4\pi)^2}\gamma^{(1)\textmd{eff}}+\frac{\alpha_s^3(\mu)}{(4\pi)^3}\gamma^{(2)\textmd{eff}}+...\ .
\label{Eq:ADMWET}$$ The matrices $\gamma^{(0)\textmd{eff}}$ and $\gamma^{(1)\textmd{eff}}$ are given in Ref. [@Chetyrkin:1996vx]. The matrix $\gamma^{(2)\textmd{eff}}$ is specified in Ref. [@Czakon:2006ss]. Analogously, the coefficients expanded in powers of $\alpha_s$ read $$\begin{aligned}
\begin{aligned}
C_i^\textmd{eff}(\mu)&=C_i^{(0)\textmd{eff}}(\mu)+\frac{\alpha_s(\mu)}{4\pi}C_i^{(1)\textmd{eff}}(\mu)\\&+\frac{\alpha_s^2(\mu)}{(4\pi)^2}C_i^{(2)\textmd{eff}}(\mu)+...\ .
\end{aligned}\end{aligned}$$ The SM values of the effective coefficients at the scale $\mu_W$ are known at NNLO QCD [@Czakon:2015exa; @Bobeth:1999mk; @Misiak:2004ew].
Obviously, performing the matching of $\tilde C_i$ to $\Delta \bar C^{(0)}_i$ without running in SMEFT and WET only by setting $\mu_W=\mu_b$ in Eq. (\[Eq:MatchC7\]) and Eq. (\[Eq:MatchC8\]) leads to a completely different dependence of the SMEFT coefficients. The impact of the $\tilde C_i$ on $\Delta \bar C^{(0)}_i$ can become larger by factors up to $\approx 40$ and contributions due to mixing are not included.
Measurements {#Sec:Measurements}
============
In this section, the measurements of the $t\bar t \gamma$ production cross section and of the $\bar B \rightarrow X_s \gamma $ branching fraction that we use for constraining the Wilson coefficients are described.
Measurements of the $t\bar{t}\gamma$ cross section {#tta_measurements}
--------------------------------------------------
Cross sections of $t\bar{t}\gamma$ production have been measured at different center-of-mass energies by the ATLAS [@Aad:2015uwa; @Aaboud:2017era; @ATLAS_13] and CMS [@Sirunyan:2017iyh] experiments. For our fits, we consider the cross sections determined in the analysis performed by the ATLAS collaboration using 2015 and 2016 LHC data corresponding to an integrated luminosity of [@ATLAS_13]. In this , the $t\bar{t}\gamma$ production cross section is reported as a fiducial cross section for final states containing one or two leptons (in the following referred to as single-lepton or dilepton channel, respectively), where the leptons can be either electrons or muons (or their corresponding antiparticles). The fiducial regions for both channels are defined in Sec. 7.1 of Ref. [@ATLAS_13]. The measured values of the single-lepton and dilepton fiducial cross sections are reported as $$\begin{aligned}
\sigma^\mathrm{fid}_\mathrm{ATLAS}(t\bar{t}\gamma, 1\ell) &= 521 \pm 9 \, \text{(stat.)} \pm 41 \, \text{(syst.)}\, \si{\femto\barn}\,,\nonumber\\
\sigma^\mathrm{fid}_\mathrm{ATLAS}(t\bar{t}\gamma, 2\ell) &= 69 \pm 3 \, \text{(stat.)} \pm 4 \, \text{(syst.)}\, \si{\femto\barn}\, .\nonumber\end{aligned}$$ Within uncertainties, the measurements agree well with the SM predictions at NLO QCD [@ATLAS_13; @Melnikov:2011ta]: $$\begin{aligned}
\sigma^\mathrm{fid}_\mathrm{SM, NLO}(t\bar{t}\gamma, 1\ell) &= 495 \pm \SI{99}{\femto\barn}\,,\nonumber\\
\sigma^\mathrm{fid}_\mathrm{SM, NLO}(t\bar{t}\gamma, 2\ell) &= 63 \pm \SI{9}{\femto\barn}\, .\nonumber\end{aligned}$$
Measurements of BR($ \bar B \rightarrow X_s \gamma $) {#Sec:Meas_B}
-----------------------------------------------------
For the branching fraction of $\bar B \rightarrow X_s \gamma $ multiple measurements, performed by the BaBar [@Aubert:2007my; @Lees:2012ym; @Lees:2012wg], Belle [@Limosani:2009qg; @Saito:2014das; @Belle:2016ufb] and CLEO [@Chen:2001fja] experiments, are available. A combination of these measurements has been performed by the Heavy Flavor Averaging Group (HFLAV) [@HFLAV16], taking into account the different minimum photon energy requirements applied in the respective analyses. The differences are corrected for by performing an extrapolation according to the method described in Ref. [@Buchmuller:2005zv]. For our fits we use the most recent result of the combination of $\text{BR}(\bar B \rightarrow X_s \gamma)$ measurements [@HFLAV_19], $$\begin{aligned}
\text{BR}(\bar B \rightarrow X_s \gamma) = (332 \pm 15)\times 10^{-6}\,,\end{aligned}$$ with a minimum photon energy requirement of $E_\gamma >\SI{1.6}{\GeV}$. This value agrees well with the NNLO SM prediction [@Misiak:2015xwa] $$\begin{aligned}
\text{BR}_\text{SM}(\bar B \rightarrow X_s \gamma) = (336 \pm 23)\times 10^{-6}\,.\end{aligned}$$
Modeling observables {#Sec:Modelling}
====================
In the following we describe the computation of the SM and BSM contributions to the observables. In Sec. \[Sec:CStty\] we discuss how to model the fiducial $t\bar{t}\gamma$ cross section and in Sec. \[Sec:BRBXsy\] we describe the computation of $\text{BR} (\bar B \rightarrow X_s \gamma )$.
Computation of $\sigma(t\bar t\gamma)$ {#Sec:CStty}
--------------------------------------
The $t\bar{t}\gamma$ production cross section can be computed at LO QCD for any given configuration of Wilson coefficients using Monte Carlo (MC) simulations. Since the MC simulations take too long to be directly interfaced to the fit of Wilson coefficients, we determine a parametrization of $\sigma(t\bar t\gamma)$ in terms of the Wilson coefficients. By squaring the matrix element of processes including dimension-six operators, the cross section in the presence of Wilson coefficients $\tilde C_i$ can be expressed as $$\sigma = \sigma^\mathrm{SM} + \sum_i \tilde C_i\sigma_i^\text{interf.} + \sum_{i \leq j} \tilde C_i \tilde C_j \sigma_{ij}^\text{BSM}\,,
\label{Eq:interpol}$$ where $\sigma_i^\text{interf.}$ are terms coming from the interference between SM and EFT diagrams and $\sigma_{ij}^\text{BSM}$ are purely BSM contributions. Using cross sections computed with MC simulations for different configurations of Wilson coefficients as sampling points, an interpolation to Eq. (\[Eq:interpol\]) can be performed, yielding numerical values for the $\sigma_i$ terms and thus a parametrization of the cross section as a function of the Wilson coefficients that can be used in the fit.
To parametrize the influence of the dimension-six operators $O_{uB}$, $O_{uG}$ and $O_{uW}$ on the $t\bar t\gamma$ production cross section, we perform simulations using [<span style="font-variant:small-caps;">MadGraph5</span>\_aMC@NLO ]{}[@MG5] with the `dim6top_LO` UFO model [@AguilarSaavedra:2018nen]. We generate MC samples similar to the signal sample described in Ref. [@ATLAS_13] to make sure that the simulations are suitable for a fit to the fiducial measurements. The samples are generated using $2 \rightarrow 7$ processes for both, the single-lepton and the dilepton channel. For the BSM contributions only one insertion of a dimension-six operator is allowed at a time and the BSM energy scale is set to $\Lambda =\SI{1}{\TeV}$. The dimension-six operators we consider in this paper are $O_{uB}$, $O_{uG}$ and $O_{uW}$, as given in Eq. (\[Eq:TopOperator\]). In the `dim6top_LO` UFO model different degrees of freedom are chosen than in this analysis, so that it is not possible to directly specify the value of the coefficient $\tilde C_{uB}$ but only the value of the linear combination $$\tilde C_{uZ} = \cos\theta_W \tilde C_{uW} - \sin\theta_W \tilde C_{uB}\,, \label{Eq:ctz}$$ where $\theta_W$ is the Weinberg angle (in the notation of Ref. [@AguilarSaavedra:2018nen] $C_{tZ}$ is used instead of $C_{uZ}$). Thus, we generate sampling points in the space of the Wilson coefficients $\tilde C_{uG}$, $\tilde C_{uW}$ and $\tilde C_{uZ}$ and use the equivalent representation in terms of $\tilde C_{uB}$, $\tilde C_{uG}$ and $\tilde C_{uW}$ for determining constraints on the coefficients hereinafter. We choose 201 different sampling points, where up to two Wilson coefficients at a time can take non-zero values. For each of the sampling points, events are generated. Comparing the SM value obtained with the cross section of the LO signal sample described in Ref. [@ATLAS_13], we find good agreement with a relative deviation of less than .
We determine the parametrization of the $t \bar t \gamma$ cross sections as a function of the Wilson coefficients $\tilde{C}_{uG}\, ,\ \tilde{C}_{uW}$ and $\tilde{C}_{uZ}$ by performing an interpolation according to Eq. (\[Eq:interpol\]). For the interpolation we apply a least squares fit with the Levenberg–Marquardt algorithm provided by the `LsqFit.jl` package [@LsqFit].
The sampling points and the result of the interpolation are shown in Fig. \[Fig:Interpolation\_total\] as slices of the phase space where only one Wilson coefficient is varied at a time, while the others are set to zero. We find that the simulated cross sections are well described by the interpolation, as the relative differences between the simulated values and the interpolation, calculated at all sampling points, have a standard deviation of only .
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To obtain fiducial acceptances, we apply parton showering to the events using `PYTHIA8` [@PYTHIA8] and perform a particle-level event selection with `MadAnalysis` [@Conte:2012fm; @Conte:2014zja; @Dumont:2014tja]. For the clustering of particle jets, the anti-$k_t$ algorithm [@Cacciari:2008gp] with a radius parameter $R=0.4$ is applied using `FastJet` [@Cacciari:2011ma]. At each sampling point we determine the fiducial acceptances for the single-lepton and dilepton channels using an event selection that is similar to the definition of the fiducial regions described in Ref. [@ATLAS_13]. Comparisons of the fiducial acceptances for the SM sampling point with the values given in Ref. [@ATLAS_13] show that we obtain the same fiducial acceptance for the dilepton channel and only a small deviation of for the single-lepton channel.
It should be noted that performing a parton-level simulation and applying the fiducial cuts at this level, which might be considered as a first approximation, is not sufficient as the resulting LO fiducial cross sections deviate from the LO SM predictions in Ref. [@ATLAS_13] by about for the single-lepton and for the dilepton channel.
The dependence of the fiducial acceptance $A$ on the Wilson coefficients $\tilde C_i$ can be parametrized as $$A = \frac{A^\mathrm{SM}\sigma^\mathrm{SM} + \sum_i \tilde{C}_i A_i^\mathrm{interf.} \sigma_i^\mathrm{interf.} + \sum_{i \leq j} \tilde C_i \tilde C_j A_{ij}^\text{BSM} \sigma_{ij}^\text{BSM}}{\sigma^\mathrm{SM} + \sum_i \tilde C_i\sigma_i^\text{interf.} + \sum_{i \leq j} \tilde C_i \tilde C_j \sigma_{ij}^\text{BSM}}\,,
\label{Eq:acceptance}$$ where the denominator is the parametrization of the cross section $\sigma$ as given in Eq. (\[Eq:interpol\]). The acceptances $A_i$ account for changes in kinematics due to BSM contributions. With the parameters $\sigma_i$ already determined in the previous interpolation of the cross section, we perform a least squares fit of the fiducial acceptances to Eq. (\[Eq:acceptance\]) in each channel using the acceptances from the event selection as sampling points. The result of the interpolation and the sampling points for the fiducial acceptance of the single-lepton channel are shown in Fig. \[Fig:acceptance\]. It is observable that the Wilson coefficients $\tilde{C}_{uW}$ and $\tilde{C}_{uZ}$ have a stronger influence on the acceptance than $\tilde{C}_{uG}$. Compared to the SM value, the former coefficients can both change the acceptance by up to a factor of 2.5, while the latter changes it only by up to a factor of 1.3. For the fiducial acceptances of the dilepton channel a comparable behavior can be observed. The corresponding plots are shown in \[Sec:app\_acc\]. In both channels, fluctuations in the simulated acceptances are present. The standard deviation of the relative difference between simulation and interpolation is in the single-lepton channel and in the dilepton channel, indicating that both interpolations are sufficient.
We obtain the dependence of the fiducial cross sections on the Wilson coefficients by multiplying the interpolation of the total cross section with the interpolations of the fiducial acceptances. As our simulations are performed at LO QCD and NLO calculations of the SM fiducial cross sections are available, we apply a SM $k$-factor by setting the SM contributions to the according values of the NLO predictions presented in Sec. \[tta\_measurements\].
{width="49.00000%"} {width="49.00000%"}
In Fig. \[Fig:tta\] the resulting parametrizations of the fiducial $t \bar t \gamma$ cross sections as functions of the Wilson coefficients $\tilde{C}_{uB}\, ,\ \tilde{C}_{uG}$ and $\tilde{C}_{uZ}$ are shown for the single-lepton and dilepton channels. The dependence on $\tilde{C}_{uB}$ is determined using Eq. (\[Eq:ctz\]). Shown are slices of the phase space where only one Wilson coefficient is varied at a time, while the others are set to zero. In both channels, we observe a comparable behavior of the fiducial cross sections and similar sensitivities to the Wilson coefficients.
Computation of BR($\bar B \rightarrow X_s \gamma $) {#Sec:BRBXsy}
---------------------------------------------------
The most recent estimate of the $\bar B \rightarrow X_s \gamma $ branching fraction at NNLO QCD has been presented in Ref. [@Misiak:2015xwa], following the algorithm described in Ref. [@Czakon:2015exa]. We adapt this procedure in our computation of BR($\bar B \rightarrow X_s \gamma $) and extend it to LO BSM contributions. Applying the notation of Ref. [@Misiak:2006ab], the branching fraction can be expressed as $$\begin{aligned}
\begin{aligned}
\textmd{BR}(\bar B \rightarrow X_s \gamma)=&\textmd{BR}(\bar B \rightarrow X_c e\bar\nu)_\textmd{exp} \\
&\times \left|\frac{V_{ts}^*V_{tb}}{V_{cb}}\right|^2\frac{6\alpha_{e}}{\pi C}(P(E_0)+N(E_0))\,,
\end{aligned}\end{aligned}$$ where $\alpha_{e}$ is the fine structure constant, $E_0=1.6\,\si{\giga\electronvolt}$ is the photon energy cut and $P(E_0)$ and $N(E_0)$ denote perturbative and non-perturbative corrections, respectively. The factor $C$ is given as $$C=\left|\frac{V_{ub}}{V_{cb}}\right|^2\frac{\Gamma(\bar B \rightarrow X_c e\bar\nu)}{\Gamma(\bar B \rightarrow X_u e\bar\nu)}\,,$$ with an experimental value $C_\textmd{exp}=0.568\pm0.007\pm0.01$ [@Alberti:2014yda]. The quantity $P(E_0)$ is given as $$P(E_0)=\sum_{i,j=1}^8C^\textmd{eff}_i(\mu_b)C^\textmd{eff}_j(\mu_b)K_{ij}(E_0,\mu_b)\,,
\label{Eq:PE0}$$ where the matrix $K(E_0,\mu_b)$ expanded in $\alpha_s$ reads: $$\begin{aligned}
\begin{aligned}
K_{ij}(E_0,\mu_b)=&\delta_{i7}\delta_{j7}+\frac{\alpha_s(\mu_b)}{4\pi}K^{(1)}_{ij}\\
&+\frac{\alpha_s^2(\mu_b)}{(4\pi)^2}K^{(2)}_{ij}+\mathcal{O}(\alpha_s^3(\mu_b))\,.
\label{Eq:Ki_expand}
\end{aligned}\end{aligned}$$ The coefficients $K^{(1)}_{ij}$ can be derived from the NLO results given in Ref. [@Buras:2002tp]. For the computation of $P(E_0)$ at approximate NNLO we include the effects of charm and bottom masses in $K^{(2)}_{77}$ [@Asatrian:2006rq], $K^{(2)}_{78}$ [@Ewerth:2008nv] and $K^{(2)}_{1(2)7}$ [@Boughezal:2007ny] as well as the complete computation of $K^{(2)}_{78}$ [@Asatrian:2010rq] and the NNLO computation of $K^{(2)}_{1(2)7}$ [@Czakon:2015exa]. Contributions of three-body and four-body final states to $K^{(2)}_{88}$ [@Ferroglia:2010xe; @Misiak:2010tk] and $K^{(2)}_{1(2)8}$[@Misiak:2010tk] are included in the Brodsky–Lepage–Mackenzie (BLM) approximation [@Brodsky:1982gc].
For the computation of non-perturbative corrections we include results from [@Benzke:2010js; @Ewerth:2009yr; @Alberti:2013kxa]. The scales are chosen to be $\mu_W=m_W$ and $\mu_b=\SI{2}{\giga\electronvolt}$. For the SM central value we find BR$_\textmd{SM}(\bar B\rightarrow X_s\gamma)=336\times 10^{-6}$, matching the results in Ref. [@Misiak:2015xwa].
In Fig. \[Fig:BRSmeftCoeff\] we give the dependence of BR($\bar B \rightarrow X_s \gamma $) on the SMEFT coefficients at the scale $\mu=m_t$. Only one coefficient is varied while the other two are set to zero. We also indicate the averaged measurements described in Sec. \[Sec:Meas\_B\]. The branching fraction BR($\bar B \rightarrow X_s \gamma $) shows the strongest dependence on $\tilde C_{uB}$, whereas the dependence on $\tilde C_{uG}$ and $\tilde C_{uW}$ is significantly weaker in comparison. This is expected since $\tilde C_{uB}$ gives the largest contribution to $\Delta \bar C^{(0)}_7$.
![Dependence of BR($\bar B \rightarrow X_s \gamma $) on the SMEFT coefficients . Only one coefficient is varied at a time while the other two are set to zero. The grey band denotes the experimental average[]{data-label="Fig:BRSmeftCoeff"}](BR.pdf){width="49.00000%"}
As a cross check for our computation, we apply `flavio` [@Straub:2018kue] together with `wilson` [@Aebischer:2018bkb] and Eq. (\[Eq:MatchC7\]) and Eq. (\[Eq:MatchC8\]) to compute the branching fraction. Since `wilson` provides only tree-level matching between SMEFT and WET, the matching conditions in Eq. (\[Eq:MatchC7\]) and Eq. (\[Eq:MatchC8\]) are not included. We therefore apply `wilson` only for the RGE evolution in WET. For the SM prediction we find good agreement with the result obtained using `flavio`, BR$_\text{flavio}(\bar B \rightarrow X_s \gamma)= (326\pm23)\times 10^{-6}$. The deviation of the central value is only and thus smaller than the theory uncertainties. For the dependence on the Wilson coefficients we find very similar behavior and obtain only deviations smaller than the theory uncertainties in the range $-1\leq \tilde C_i \leq 1$.
Constraining Wilson coefficients {#Sec:Fit}
================================
With the parametrizations of the $t\bar t \gamma$ cross sections and of the branching fraction determined in Sec. \[Sec:Modelling\], we perform fits to the measurements described in to constrain the Wilson coefficients $\tilde{C}_{uB}$, $\tilde{C}_{uG}$ and $\tilde{C}_{uW}$. We use a new implementation of the EFT*fitter* tool [@Castro:2016jjv] based on the *Bayesian Analysis Toolkit - BAT.jl* [@BAT; @BAT.jl]. This allows to perform fits of Wilson coefficients in a Bayesian reasoning, yielding (marginalized) posterior probability distributions of the parameters.
We include both the experimental uncertainties and the SM theory uncertainties given in Sec. \[Sec:Measurements\] in the fit. Focusing on the combination of observables from different energy scales, we make the simplifying assumption that the uncertainties of the measurements included are uncorrelated. This assumption seems reasonable for the correlations between top-quark and $B$ physics measurements and also for the correlation between the statistical uncertainties of the two channels contributing to $\sigma(t\bar t \gamma)$. The systematic and theoretical uncertainties of both channels can in principle be correlated in a non-negligible manner. As no information about the correlations is available, we investigate their influence afterwards by performing several fits varying the corresponding correlation coefficients.
To illustrate the benefit of combining observables from top-quark and $B$ physics, we first constrain the Wilson coefficients using only one set of measurements at a time (Secs. \[sec:b\], \[sec:t\]) before performing the combined fit (Sec. \[sec:tplusb\]).
$B$ physics only \[sec:b\]
--------------------------
Considering only BR$(\bar B \rightarrow X_s \gamma)$, we perform a fit to the HFLAV average described in Sec. \[Sec:Meas\_B\] using the description of the branching fraction given in Sec. \[Sec:BRBXsy\]. Treating $\tilde{C}_{uB}$, $\tilde{C}_{uG}$ and $\tilde{C}_{uW}$ as free parameters of the fit and providing no prior knowledge about their distributions, we assign uniform prior probability distributions in the range of \[-1, 1\] to them. Larger values of the rescaled Wilson coefficients $\tilde C$ would not be reasonable and would lead to a breakdown of the EFT expansion.
When performing the fit, we observe that only $\tilde{C}_{uB}$ can be constrained using this setup. No constraints on the other two coefficients can be obtained, as the resulting marginalized posterior probabilities of $\tilde{C}_{uG}$ and $\tilde{C}_{uW}$ are uniformly distributed. As can be seen from Fig. \[Fig:BRSmeftCoeff\], $\tilde{C}_{uB}$ is the Wilson coefficient with the largest influence on the $\bar B \rightarrow X_s \gamma $ branching fraction, thus receiving stronger constraints than $\tilde{C}_{uG}$ and $\tilde{C}_{uW}$ in a fit with three free parameters and a single observable.
![Marginalized posterior probability distribution of $\tilde{C}_{uB}$ from the fit of all three Wilson coefficients to BR($\bar B \rightarrow X_s \gamma $) only. The smallest interval containing of the posterior probability and the SM value (dashed line) are indicated[]{data-label="Fig:CuB_onlyB"}](CuB_marginalized.pdf){width="49.00000%"}
The marginalized posterior distribution of $\tilde{C}_{uB}$ is shown in Fig. \[Fig:CuB\_onlyB\]. Two regions for $\tilde{C}_{uB}$ are favored by the fit. Comparing with Fig. \[Fig:BRSmeftCoeff\], the two regions with the highest probability at about $\tilde{C}_{uB} \approx -0.5$ and $\tilde{C}_{uB} \approx 0.0$ are reasonable since the quadratic shape of BR($\bar B \rightarrow X_s \gamma $) as a function of $\tilde{C}_{uB}$ leads to an agreement with the measurement in these two regions. Apparently, without further information, neither of them can be rejected. Indeed, as is well-known, this ambiguity can be resolved by studies of semileptonic $ b \to s \ell^+ \ell^-$ decays [@Ali:1999mm], notably, angular distributions thereof, whose measurements support the close-to-the-SM branch [@Aaij:2013iag]. Since the purpose of this work is to demonstrate complementarity and feasibility of a joint bottom and top SMEFT-analysis rather than performing a most global fit, we leave the study of further observables beyond BR($\bar B \rightarrow X_s \gamma $) and $\sigma(t\bar t \gamma)$ for future work.
Top physics only \[sec:t\]
--------------------------
We perform a fit of the Wilson coefficients using $\sigma(t\bar t \gamma)$ only. We apply the parametrizations of the single-lepton and dilepton channel fiducial cross sections obtained in Sec. \[Sec:CStty\] and fit to the corresponding measurements described in Sec. \[tta\_measurements\]. Again, all three Wilson coefficients are free parameters of the fit, having uniform prior probability distributions within the range \[-1, 1\]. The resulting marginalized posterior distribution of $\tilde{C}_{uB}$ and the smallest area containing of the posterior probability of the 2D marginalized distribution of $\tilde{C}_{uG}$ vs. $\tilde{C}_{uW}$ are shown in Fig. \[Fig:results\_onlyT\]. With a fit to $\sigma(t\bar t \gamma)$ all three Wilson coefficients can be constrained to a similar extent. The posterior probability distributions of the coefficients have similar shapes and the intervals are of comparable size. These results are compatible with what is observed in the parabolas shown in Fig. \[Fig:tta\]. When performing the fit considering only the single-lepton or only the dilepton channel measurements as a cross check, very similar results are obtained. This is also expected from Fig. \[Fig:tta\] as it indicates that both channels have similar sensitivity to the Wilson coefficients.
{width="49.00000%"} {width="49.00000%"}
Combined analysis \[sec:tplusb\]
--------------------------------
For the combined fit, we apply the same uniform priors as in the individual fits and constrain $\tilde{C}_{uB}$, $\tilde{C}_{uG}$ and $\tilde{C}_{uW}$ using both BR($\bar B \rightarrow X_s \gamma $) and $\sigma(t\bar t \gamma)$. The resulting smallest areas containing of the posterior probability are shown in Fig. \[Fig:comparison2d\] for the 2D marginalized distributions. The plots also include the corresponding regions from the previously described fits including only one set of observables at a time.
{width="49.00000%"} {width="49.00000%"}\
{width="49.00000%"}
In Fig. \[Fig:comparison2d\] it is noticeable that the ambiguity in $\tilde{C}_{uB}$, which is observed in the fit including only the BR($\bar B \rightarrow X_s \gamma $) measurement, is resolved in the combined fit. It is recognizable that even though the branching fraction measurement alone constrains only $\tilde{C}_{uB}$, in the combination with the $t\bar t \gamma$ cross sections the constraints on all three Wilson coefficients improve as the sizes of the areas containing of the probability decrease in all plots. The area of the fit using only BR($\bar B \rightarrow X_s \gamma $) in the upper left plot of Fig. \[Fig:comparison2d\] has a size of of the total parameter space $\tilde{C}_{uB} \in [-1, 1]$ and $\tilde{C}_{uG} \in [-1, 1]$ specified by the priors. For the fit considering only $\sigma(t\bar t \gamma)$ the corresponding area is of a similar size, taking up about of the allowed space. Due to the orthogonality of the observables, combining top and bottom measurements gives, on the other hand, a posterior region reduced by more than an order of magnitude, yielding an area that corresponds to only about of the allowed parameter space. The same numbers apply also for the upper right plot of $\tilde C_{uB}$ vs. $\tilde C_{uW}$. Even in the bottom plot of Fig. \[Fig:comparison2d\], which does not directly depend on $\tilde{C}_{uB}$ and is thus not directly constrained by the branching fraction measurement, the area is reduced. In combination with the BR($\bar B \rightarrow X_s \gamma $) measurement, the area decreases by a factor of 1.9 compared to the fit considering only the $\sigma(t\bar t \gamma)$ measurements. This is a consequence of the reduction of allowed regions in the three-dimensional parameter space.
A different representation of the same fit results is given in the left plot of Fig. \[Fig:comparison\], where the smallest intervals containing probability of the 1D marginalized posterior distributions are shown for the combined fit as well as for the fits using only one of the measurements.
In the right plot of Fig. \[Fig:comparison\] the smallest intervals containing probability of the 1D marginalized posterior distribution are shown for individual fits in which only one of the Wilson coefficients is allowed to vary at a time, while the other two are fixed to zero. Overall, a similar behaviour of the results can be observed compared to the fits with three free parameters. As there are fewer degrees of freedom in the fits, stronger constraints on the Wilson coefficients can be obtained. It is noticeable that in the individual fits not only the ambiguity in $\tilde{C}_{uB}$ can be resolved by the $t\bar t \gamma$ measurement but that also an ambiguity in the top-measurements interval of $\tilde{C}_{uW}$ can be resolved by BR($\bar B \rightarrow X_s \gamma $).
{width="49.00000%"} {width="49.00000%"}
As mentioned above, we study the influence of correlations between the systematic and theoretical uncertainties of the single-lepton and dilepton channels of $\sigma(t\bar t \gamma)$. For this purpose, we perform the combined fit assuming different correlations between the two channels for these uncertainties. We vary the correlation coefficient of the systematic uncertainties between values of $-0.9$ and $ 0.9$ as negative correlations are conceivable. The correlation coefficient of the theory uncertainties is varied up to a value of $0.9$ since we do not expect negative correlations for these uncertainties. When comparing the sizes of the areas containing of the marginalized posterior probability to the results assuming uncorrelated uncertainties, we observe only minor changes for the two distributions of $\tilde{C}_{uB}$ vs. $\tilde{C}_{uG}$ and $\tilde{C}_{uB}$ vs. $\tilde{C}_{uW}$. We find relative changes in the size of the areas of about at maximum and no changes in the general shape or positions compared to the combination shown in the two upper plots of Fig. \[Fig:comparison2d\]. As the distribution of $\tilde{C}_{uG}$ vs. $\tilde{C}_{uW}$ is dominantly constrained by the $\sigma(t\bar t \gamma)$ measurements, we observe larger changes due to variations of the correlation coefficients. The size of the area can change by up to for this distribution. Again, the general shape and the positions are not affected but only the width of the ring in the bottom plot of Fig. \[Fig:comparison2d\] varies. Therefore, we conclude that even in the presence of correlations between the systematic or theoretical uncertainties of the single-lepton and dilepton channels our previously presented findings are valid.
It should be noted that our focus is to demonstrate how observables from $B$ and top-quark physics can be combined in a single fit of the SMEFT Wilson coefficients. Using only two observables, we do not obtain the most stringent constraints on the coefficients considered. Including further observables would certainly improve the constraints. For example, the Wilson coefficients $\tilde C_{uG}$ and $\tilde C_{uW}$ are strongly constrained by the $t\bar t$ production cross section and $W$-boson helicity-fraction measurements, respectively [@Buckley:2015lku; @Hartland:2019bjb], whereas measurements of semileptonic $b\rightarrow s \ell^+\ell^-$ decays, especially $B\rightarrow K^{*}\mu^+\mu^-$ angular distributions [@Aaij:2013iag], exclude values $\tilde{C}_{uB}\approx -0.5$ which are allowed by BR($\bar B\rightarrow X_s \gamma$).
Conclusions {#Sec:Conclusion}
===========
Effective theories provide a systematic toolbox to exploit multi-observable systems and probe the SM in a model-independent way. The SMEFT-framework allows to combine data from the precision flavor and the high energy frontiers. We exploited synergies between top-quark and $B$-physics measurements from the LHC and precision flavor factories.
Specifically, we performed an exploratory study combining data on the $\bar B \rightarrow X_s \gamma $ branching ratio and on fiducial $t\bar t \gamma$ production cross sections within SMEFT, after detailing the ingredients required to connect measurements from different energy scales. We pointed out that for the processes considered in this work it is necessary to perform a dedicated matching that goes beyond the tree-level matching that is currently available in tools. Using MC simulations and a particle-level event selection, we performed interpolations of the total $t\bar t \gamma$ production cross section and the fiducial acceptances to parametrize the dependence of the fiducial cross sections on the Wilson coefficients.
We demonstrated that due to the different sensitivities of the observables to the SMEFT operators, a combination of the fiducial $t\bar t \gamma$ cross section with the $\bar B \rightarrow X_s \gamma $ branching fraction improves the constraints on the Wilson coefficients (Sec. \[Sec:Fit\]). The complementarity of the different observables used in the fit allows to resolve ambiguities and to reduce posterior regions in the marginalized parameter space by up to an order of magnitude.
Further, more global analyses of combined top-quark and flavor physics measurements should be pursued in the future with more precise data expected from LHCb [@Cerri:2018ypt] and Belle II [@Kou:2018nap] and the high-$p_T$-experiments [@Atlas:2019qfx], to decipher physics at higher energies and pursue the quest for BSM physics.
C.G. is supported by the doctoral scholarship program of the *German Academic Scholarship Foundation*.
Parameters and experimental input {#App:Parameters}
=================================
The parameters used for numerical computations are given in Ref. [@Tanabashi:2018oca] $$\begin{aligned}
&m_{t}=(173.1\pm0.4)\,\si{\giga\electronvolt}\,,\\
&m_t(m_t)=\left(160^{+5}_{-4}\right)\,\si{\giga\electronvolt}\,,\\
&m_b(m_b)=\left(4.18^{+0.04}_{-0.03}\right)\,\si{\giga\electronvolt}\,,\\
&m_c(m_c)=\left(1.275^{+0.025}_{-0.035}\right)\,\si{\giga\electronvolt}\,,\\
&m_s(2\,\si{\giga\electronvolt})=\left(0.095^{+0.009}_{-0.008}\right)\,\si{\giga\electronvolt}\,,\\
&m_{Z}=91.188\,\si{\giga\electronvolt}\,,\\
&m_{W}=80.4\,\si{\giga\electronvolt}\,,\\
&\alpha_s(m_Z)=0.1181\,,\\
&\alpha_{e}=7.29735257\times10^{-3}\,,\\
&\sin^2\theta_w(m_Z)=0.2313\,,\\
&G_F=1.166379\times10^{-5}\,\si{\giga\electronvolt}^{-2}\,.\end{aligned}$$ The relevant CKM Matrix elements are given in Refs. [@Bona:2006ah; @UTfit] $$\begin{aligned}
V_{tb}&=0.999097\pm0.000024\,,\\
V_{ts}&=(-0.04156\pm0.00056)\exp[(1.040\pm0.035)\si{\degree}]\,,\\
V_{cb}&=0.04255\pm0.00069\,.\end{aligned}$$ The experimental input for the computation of BR($\bar B \rightarrow X_s \gamma $) reads [@Alberti:2014yda; @Aubert:2004aw] $$\begin{aligned}
&C=0.568\pm0.007\pm0.01 \ \,,\\
&\textmd{BR}(\bar B \rightarrow X_c e\bar\nu)_\textmd{exp}=0.1061\pm0.0017 \ \,.\end{aligned}$$
Matching condition {#App:Match}
==================
The functions $E_{7}^{uW}$, $F_{7}^{uW}$, $E_{8}^{uW}$ and $F_{8}^{uW}$ are given by $$\begin{aligned}
E_{7}^{uW}(x_t)&=
\frac{-9 x_{t}^3+63 x_{t}^2-61 x_{t}+19}{48 \left(x_{t}-1\right)^3}\\
&+\frac{\left(3 x_{t}^4-12 x_{t}^3-9 x_{t}^2+20 x_{t}-8\right) \ln \left(x_{t}\right)}{24 \left(x_{t}-1\right)^4}\\
&+\frac{1}{8}\ln\left(\frac{m_W^2}{\mu_W^2}\right)\,,\\
F_{7}^{uW}(x_t)&=\frac{x_{t} \left(2-3 x_{t}\right) \ln \left(x_{t}\right)}{4 \left(x_{t}-1\right)^4}
-\frac{3 x_{t}^3-17 x_{t}^2+4 x_{t}+4}{24 \left(x_{t}-1\right)^3}\,,\\
E_{7}^{uB}(x_t)&=-\frac{1}{8} \ln \left(\frac{m_W^2}{\mu_W^2}\right)-\frac{\left(x_{t}+1\right)^2}{16 \left(x_{t}-1\right)^2}\\
&-\frac{x_{t}^2 \left(x_{t}-3\right) \ln \left(x_{t}\right)}{8 \left(x_{t}-1\right)^3}\,,\\
F_{7}^{uB}(x_t)&=-\frac{1}{8}\,,\\
E_8^{uW}(x_t)&=\frac{3 x_{t}^2-13 x_{t}+4}{8 \left(x_{t}-1\right)^3}
+\frac{\left(5 x_{t}-2\right) \ln \left(x_{t}\right)}{4 \left(x_{t}-1\right)^4}\,,\\
F_8^{uW}(x_t)&=\frac{x_{t}^2-5 x_{t}-2}{8 \left(x_{t}-1\right)^3}+\frac{3 x_{t} \ln \left(x_{t}\right)}{4 \left(x_{t}-1\right)^4}\,,\\
E_8^{uG}(x_t)&=E_7^{uB}(x_t)\,,\\
F_8^{uG}(x_t)&=F_7^{uB}(x_t)\,.\end{aligned}$$
Fiducial acceptance of the dilepton channel {#Sec:app_acc}
===========================================
![Sampling points and interpolation result for the fiducial acceptance of the dilepton channel $A(2\ell)$, represented as slices of the phase space where only one of the Wilson coefficient is varied at a time, while the others are set to zero.[]{data-label="Fig:acceptance_2l"}](2l_acc_CuG.pdf "fig:"){width="35.00000%"} ![Sampling points and interpolation result for the fiducial acceptance of the dilepton channel $A(2\ell)$, represented as slices of the phase space where only one of the Wilson coefficient is varied at a time, while the others are set to zero.[]{data-label="Fig:acceptance_2l"}](2l_acc_CuW.pdf "fig:"){width="35.00000%"} ![Sampling points and interpolation result for the fiducial acceptance of the dilepton channel $A(2\ell)$, represented as slices of the phase space where only one of the Wilson coefficient is varied at a time, while the others are set to zero.[]{data-label="Fig:acceptance_2l"}](2l_acc_CuZ.pdf "fig:"){width="35.00000%"}
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title: |
Concurrent Transmission Scheduling for Perceptual Data Sharing\
in mmWave Vehicular Networks
---
Sharing perceptual data (e.g., camera and LiDAR data) with other vehicles enhances the traffic safety of autonomous vehicles because it helps vehicles locate other vehicles and pedestrians in their blind spots. Such safety applications require high throughput and short delay, which cannot be achieved by conventional microwave vehicular communication systems. Therefore, millimeter-wave (mmWave) communications are considered to be a key technology for sharing perceptual data because of their wide bandwidth. One of the challenges of data sharing in mmWave communications is broadcasting because narrow-beam directional antennas are used to obtain high gain. Because many vehicles should share their perceptual data to others within a short time frame in order to enlarge the areas that can be perceived based on shared perceptual data, an efficient scheduling for concurrent transmission that improves spatial reuse is required for perceptual data sharing. This paper proposes a data sharing algorithm that employs a graph-based concurrent transmission scheduling. The proposed algorithm realizes concurrent transmission to improve spatial reuse by designing a rule that is utilized to determine if the two pairs of transmitters and receivers interfere with each other by considering the radio propagation characteristics of narrow-beam antennas. A prioritization method that considers the geographical information in perceptual data is also designed to enlarge perceivable areas in situations where data sharing time is limited and not all data can be shared. Simulation results demonstrate that the proposed algorithm doubles the area of the cooperatively perceivable region compared with a conventional algorithm that does not consider mmWave communications because the proposed algorithm achieves high-throughput transmission by improving spatial reuse. The prioritization also enlarges the perceivable region by a maximum of 20%.
mmWave communications, VANET, data sharing, directional antenna, concurrent transmission scheduling
Introduction
============
Millimeter-wave (mmWave) vehicular adhoc networks (VANETs) are expected to be an enabler of numerous safety applications for autonomous vehicles that require high-throughput transmission capability [@MmWaveVanetSurvey; @eband; @PathLossPrediction; @perfecto2017millimeter; @wu2017cooperative]. As vehicles become increasingly automated, the number of sensors equipped on vehicles increases and an increasingly massive amount of data are generated while driving. Sharing these sensor data, such as camera and LiDAR data, would help extend a vehicle’s perceptual range to cover its blind spots or locate hidden objects. However, a sufficient data rate for sharing sensor data cannot be provided by currently standardized vehicular communication systems (e.g., IEEE 802.11p/dedicated short range communications (DSRC) and cellular vehicle-to-everything (C-V2X), standardized in the third generation partnership project (3GPP) Release 14 [@molina2017lte]) because of their limited bandwidth. Therefore, mmWave communications, which provide high-throughput communication by leveraging huge bandwidth and efficient spatial reuse, have been attracting much attention for vehicular communications.
One of the most important traffic safety applications facilitated by mmWave communications is cooperative perception, which enables autonomous vehicles to perceive their blind spots by sharing perceptual data, such as camera, LiDAR, and radar data, with other vehicles. For example, see-through systems provide following vehicles with front views of the leader of platooning vehicles and bird’s-eye-view systems generate top views of surrounding areas by aggregating perceptual data of multiple vehicles [@kim2015multivehicle; @li2011multi]. Such techniques are particularly important at intersections with poor visibility to avoid car crash. By sharing information regarding their surroundings, the region that autonomous vehicles can perceive is enlarged based on the shared information. Computer vision systems enable vehicles to recognize other vehicles, pedestrians, and traffic signs, even if they cannot be seen directly because buildings or other obstacles block the line of sight. To cover the entire area surrounding an intersection, vehicles near the intersection should send their massive data to the other vehicles within a short period, in particular 100ms for safety applications [@VehNetworking]. For example, assume 20 vehicles attempt to share compressed camera images within 100ms. The image sizes range from 1–9Mbit because they are generated at rates of 10–90Mbit/s [@MmWaveVANET]. Therefore, 20–180Mbit of data must be transmitted within 100ms by 20 vehicles, meaning each datum must be transmitted at a rate of 0.2–1.8Gbit/s. Such a high-throughput system is difficult to be realized by DSRC or C-V2X owing to their limited bandwidth.
Although mmWave communications enable high-throughput transmission, it is difficult to broadcast data to all vehicles compared with microwave communications because few vehicles can receive transmitted signals because of narrow-beam directional antennas and severe attenuation by the blockage effect. Therefore, an efficient mechanism to share perceptual data in mmWave multihop networks should be developed. As mentioned above, vehicles are required to share perceptual data and obtain data of as wide region as possible within 100ms. To meet these requirements, concurrent transmission and routing with cached data are promising approaches. Concurrent transmission, where many transmitters send data to different receivers at the same time, promotes efficient spatial reuse, which is realized by leveraging antenna directionality and high attenuation. On the other hand, routing using cached data reduces redundant transmissions for data sharing in multihop networks because each datum is requested to be sent to many different vehicles. In multihop networks, if relay vehicles store the forwarded data, the source vehicles do not need to transmit the same data many times. Leveraging the geographical information in perceptual data also helps to enlarge perceivable regions.
There have been a few studies on concurrent transmissions in mmWave VANET. For example, [@perfecto2017millimeter] proposed a beam-width-controlling scheme to reduce beam-alignment delay by considering signal-to-interference plus noise power ratio (SINR). Most concurrent transmission protocols for mmWave communications are found not in VANETs, but in wireless sensor networks (WSNs) [@qiao2012stdma; @niu2015blockage; @wang2014throughput]. However, such protocols do not adopt data caching because their objectives are not data sharing. In data sharing, the same data are sent from the source vehicles to different vehicles and thus, the same data might be transmitted redundantly without data caching. Additionally, their algorithms do not consider the geographical information in transmitted data. There have been many studies on data dissemination methods for DSRC-based VANETs, some of which utilize the geographical information in disseminated data. [@wischhof2005information; @bronsted2006specification] proposed the data aggregation of the geographical information to suppress redundant data broadcasts. [@yamada2017data] proposed controlling the frequency of broadcasting. However, these studies did not discuss concurrent transmission or multihop routing with directional antennas.
Concurrent dissemination with data caching was proposed in [@coopDataSched], where the authors presented a system to realize a road-side-unit (RSU)-controlled concurrent dissemination by two communication mode: vehicle-to-infrastructure (V2I) and vehicle-to-vehicle (V2V) communications. [@coopDataSched] proposed a graph-based algorithm, where potential transmissions (from which, to which, and which data should be transmitted) and their conflicts (e.g. half duplex and interference constraints) are represented as a graph, i.e., two transmissions are connected when they cannot be operated at the same time. Each vertex has weight that represents the priority of receiver vehicles. Then, the optimal concurrent transmission schedule for multihop dissemination can be obtained by solving the maximum weighted independent set (MWIS) problem on the graph. Although the MWIS problem is one of the NP-hard problems, a greedy algorithm with a performance guarantee to maximize the total vertex weights can be utilized. However, [@coopDataSched] does not assume mmWave communications and thus, it cannot be used directly for mmWave communications. There is also room to utilize the geographical information in the transmitted data for cooperative perceptions.
In this paper, we propose a mmWave data sharing algorithm where vehicles share perceptual data with each other and enlarge the perceivable regions. The proposed algorithm is based on [@coopDataSched] to realize concurrent transmission with data caching. Because the algorithm in [@coopDataSched] optimizes concurrent transmissions considering not only pair selection of the transmitter and receiver but also which data to transmit among the currently cached data, it effectively reduces redundant transmission in data sharing, where the same data are transmitted to different vehicles. However, the original algorithm is based on microwave communications, meaning it must be modified for mmWave communications. [@coopDataSched] designed a conflict rule, which is utilized to decide which pair of transmission vertices of the graph should be connected, considering radio interference among omnidirectional antennas. We design a new rule for mmWave communications by estimating interference among narrow-beam directional antennas, which are utilized for mmWave communications to obtain high gain. Because the conflict rule should be defined between two transmissions, we develop an interference approximation scheme that can calculate the interference between two transmissions without summing all possible interferences. By using the newly designed rule, near-optimal concurrent transmission in mmWave networks can be realized.
We also design a prioritization method in order to enlarge the perceivable region for situations where data sharing time is limited. Although [@coopDataSched] gave high priority to receiver vehicles that soon run out of the service area of the RSU, such a prioritization does not fit for cooperative perceptions at an intersection. We give high priority to data corresponding to regions far from an intersection to enlarge the perceivable area based on shared data because regions near the center of the intersection are covered by many vehicles, meaning it is desirable to transmit data far from the intersection. Such a prioritization scheme can be realized by customizing the weight function of the MWIS problem.
The main contributions of this paper are summarized as follows: (1) We propose a data sharing algorithm for cooperative perception, which improves spatial reuse by considering interference among narrow-beam directional antennas and increases the perceivable region by prioritizing the data to be forwarded based on geographical information, even if not all data can be collected. In order to realize concurrent transmission, we employ the algorithm presented in [@coopDataSched]. (2) We prove that if the data sharing time is sufficiently long and the vehicular network is represented as a connected graph, the proposed data sharing algorithm guarantees that all data are shared with all vehicles.
Related Works {#sec:related}
=============
Data sharing for cooperative perceptions should achieve a large perceivable area within a short period, in particular 100ms for safety applications. Key techniques to meet this requirement are concurrent transmission with directional antennas for improving system throughput, efficient routing with cached data for reducing redundant transmission, and leveraging geographical information in transmitted data.
Dissemination algorithms with directional antennas for VANETs have been studied by many researchers. [@li2012collaborative] presented theoretical analysis of content dissemination time in vehicular networks with directional antennas and demonstrated that directional antennas accelerate content propagation. [@li2007distance] proposed a broadcast protocol for directional antennas in VANETs. In this protocol, the furthest receiver forwards data packets along road segments and a directional repeater forwards the data in multiple directions at intersections. In contrast to the protocol in [@li2007distance], which considers the positions of transmitters, our algorithm considers the positions where data are obtained to achieve a large perceivable region.
Dissemination algorithms for local information were proposed in [@wischhof2005information; @bronsted2006specification; @yamada2017data]. In [@wischhof2005information], a scalable dissemination protocol, called segment-oriented data abstraction and dissemination (SODAD), and its application, self-organizing traffic-information system (SOTIS), were proposed. SOTIS is a mechanism for gathering traffic information sensed by vehicles. It aggregates the received traffic information from road segments and sends only up-to-date information to vehicles. In [@bronsted2006specification], Zone Flooding and Zone Diffusion were proposed to suppress redundant data broadcasting. In Zone Flooding, only vehicles in a flooding zone forward received packets. Zone Diffusion is a data aggregation method considering geographical information, where vehicles merge road environment data as it is received and broadcast only merged data. [@yamada2017data] proposed controlling the frequency of information broadcasting and selecting the data to send to reduce communication traffic. Although these studies considered the geographical information in each datum, they did not focus on concurrent transmission.
The authors of [@perfecto2017millimeter] proposed vehicle pairing and beam-width controlling for mmWave VANETs. In the protocol in [@perfecto2017millimeter], pairs of transmitters and receivers are selected based on matching theory and beam widths are determined via particle swarm optimization. This protocol successfully improves throughput and reduces delay by considering SINR. Other concurrent transmission methods for mmWave communications have been proposed for WSN, rather than VANETs [@qiao2012stdma; @niu2015blockage; @wang2014throughput]. The authors of [@qiao2012stdma] formulated the concurrent transmission scheduling problem as an optimization problem to maximize the number of flows to satisfy the quality-of-service requirements of each flow. In [@niu2015blockage], relay selection and spatial reuse were jointly optimized to improve network throughput and a blockage robust algorithm was proposed. The authors of [@wang2014throughput] minimized transmission time by solving an optimization problem. Although these algorithms for concurrent transmissions presented in [@perfecto2017millimeter; @qiao2012stdma; @niu2015blockage; @wang2014throughput] achieved efficient spatial reuse, redundant data were transmitted because their primary objective was not data sharing and thus, they did not consider situations where the same data are sent to different receivers. Additionally, they did not consider geographical information.
The authors of [@coopDataSched] proposed an RSU-controlled scheduling that maximizes system throughput in hybrid V2I/V2V communications. This algorithm realizes concurrent dissemination based on the graph theory. It also adopts a data caching mechanism. The algorithm proposed in [@coopDataSched] generates graphs for dissemination scheduling, where the set of vertices represents potential transmissions consisting of a transmitter, receiver, and data, and the set of edges represents pairs of transmissions that cannot be performed at the same time. The authors of [@coopDataSched] proved that optimal scheduling can be obtained by solving the MWIS problem for a generated graph. However, because the algorithm in [@coopDataSched] assumes omnidirectional antennas, interference calculations must be extended for mmWave communications, where narrow-beam directional antennas are utilized. Additionally, there is still room to improve the efficiency of data transmissions for cooperative perception by leveraging the geographical information in perceptual data. Thus, a data sharing algorithm in mmWave vehicular networks that increases perceivable regions should be developed for traffic safety, especially when data sharing time is limited.
![System model (Top view).[]{data-label="fig:systemmodel"}](images/systemmodel.pdf){width="45.00000%"}
System Model {#sec:system}
============
Figure \[fig:systemmodel\] shows our system model. At an intersection, there are vehicles equipped with mmWave communication devices for data transmission via V2V channels and microwave communication devices for control signal transmission via V2I channels. Vehicles participating in cooperative perception are selected among vehicles within tens of meters from the center of the intersection considering stopping distance. The number of participants is also limited to ${N_\mathrm{v}}$ vehicles because it is difficult to complete data sharing owing to the time limit when the number of participants is large. The vehicles perceive the surrounding environment utilizing their sensors, such as LiDARs or cameras. We assume that the vehicle sensors cover a surrounding rectangular region (on road segments) or cross-shaped region (at the intersection), bounded by the buildings along the roads and their sensor range ${r_\mathrm{s}}$. The data generated by vehicle $v_i$ is denoted as $d_i$. We assume the sizes of $d_i$ are approximately the same among vehicles for simplicity. The vehicles share the data with each other to obtain information regarding the intersection and then perform cooperative perception.
Data are transmitted through mmWave V2V channels to reduce the pressure on V2I channels. However, control signals, which must be broadcasted to all vehicles, are transmitted through microwave V2I channels. We assume there is an RSU (or an eNodeB) that covers all vehicles near the intersection on the microwave channel and performs scheduling based on vehicle positions and mmWave V2V link topology. While a large amount of sensor data are transmitted over the mmWave V2V channels, control signals and position information, which are relatively small, can be broadcasted by the RSU utilizing DSRC or C-V2X.
Figure \[fig:time\] shows the time frame for data sharing. The vehicles perform sensing at an interval of ${T}$ and generate $d_i$. The data update interval consists of the scheduling period and data sharing period. In the scheduling period, data sharing scheduling is determined by the RSU. Vehicle position information obtained from global positioning system (GPS) is sent to the RSU, which then estimates the mmWave connectivity between vehicles and determines the preferred data to be shared.
![Time frame for data sharing. All vehicles generate their perceptual data at the beginning of each data update interval. They transmit their data during the data sharing period.[]{data-label="fig:time"}](images/timeslot.pdf){width="48.00000%"}
In the data sharing period, vehicles share their data through mmWave V2V channels. The data sharing period consists of ${\tau_\mathrm{max}}$ time slots, each of which is sufficiently long to transmit one datum. Let $\tau\in\{0,1,\dots,{\tau_\mathrm{max}}\}$ denote the index of a time slot. ${\tau_\mathrm{max}}$ is limited by the transmitted data volume and data rate.
By sharing data $d_i$, the perceivable region is enlarged. Let ${\bm{d}}_{i,\tau}$ and ${R}_{i,\tau}$ denote the dataset possessed by vehicle $v_i$ and the perceivable region of the dataset ${\bm{d}}_{i,\tau}$, respectively. ${R}_{i,\tau}$ is defined as ${R}_{i,\tau} \coloneqq \bigcup_{d \in {\bm{d}}_{i,\tau}}{R}(d)$, where ${R}(d)$ denotes the perceivable region of $d$ (i.e., the region covered by the sensor of a single vehicle). At the beginning of the data sharing period, the datasets are initialized as ${\bm{d}}_{i,0} \leftarrow \{d_i\}$. When vehicle $v_i$ transmits $d_k \in {\bm{d}}_{i,\tau}$ to vehicle $v_j$, $v_j$ updates its dataset ${\bm{d}}_{j,\tau}$ as follows: $$\begin{aligned}
{\bm{d}}_{j,\tau+1} \leftarrow {\bm{d}}_{j,\tau} \cup \{d_k\}. \label{eq:updatedataset}\end{aligned}$$ Subsequently, the area of region ${R}_{j,\tau}$ is enlarged. We evaluate system performance based on the normalized perceivable area, which is defined as follows: $$\begin{aligned}
{\hat{S}_{\tau}} &\coloneqq {S}({R}_{i,\tau}) / {S}({R_\mathrm{all}}), \label{eq:nrm}\end{aligned}$$ where ${S}({R})$ and ${R_\mathrm{all}}$ denote the area of region ${R}$ and the area covered by all data, defined as ${R_\mathrm{all}}\coloneqq \bigcup_{i=1}^{{N_\mathrm{v}}}{R}(d_i)$, respectively.
At the end of the data sharing period, the vehicles regenerate $d_i$ by sensing. Then, the RSU collects vehicle position information and determines scheduling for sharing new perceptual data in the following scheduling period.
Data Sharing Algorithm {#seq:proposed}
======================
In the scheduling period, the RSU selects transmitters, receivers, and data to be transmitted during each time slot. First, the RSU constructs a vehicular network graph that represents the network topology of the mmWave vehicular network by estimating the connectivity between vehicles based on their positions and a propagation loss model. Second, a graph that is utilized to determine concurrent transmission behavior is constructed from the vehicular network graph for each time slot. Because the vertices of the graph represent transmissions, each of which consists of a transmitter, receiver, and data to be transmitted, and the edges of the graph represent conflicts between two transmissions, independent sets in the graph represent sets of transmissions that do not conflict with each other. Therefore, by solving the MWIS problem for the graph, which we call a scheduling graph, the optimal concurrent transmission can be found for each time slot.
Although our algorithm is based on that proposed in [@coopDataSched], our system model is quite different from that in [@coopDataSched]. First, we consider a short period (i.e., 100ms), while long-span dissemination was discussed in [@coopDataSched]. Because [@coopDataSched] designed a prioritization based on vehicle mobility over a long period, we modify the prioritization design to enlarge perceivable regions within a short period. Second, our objective is to share data generated by vehicles with each other, while [@coopDataSched] assumed that each vehicle requests data that is stored in the RSU. We prove that data sharing can be completed by our algorithm if the vehicular network graph is connected and there are enough time slots. Third, we utilize mmWave communications for data transmission and thus, we redesign how to construct the scheduling graph. Especially, conflict rules between two potential transmissions are modified to reflect the mmWave propagation characteristics. Finally, data are transmitted through V2V channels in our system to reduce the pressure on V2I channels, while [@coopDataSched] utilized both V2I and V2V channels for data transmission. The following subsections describe the details of constructing a vehicular network graph and scheduling graph.
Vehicular Network Graphs
------------------------
The RSU estimates the connectivity between each pair of vehicles and defines the vehicular network graph ${\bm{G}_{\mathrm{v}}}$ as follows: $$\begin{aligned}
{\bm{G}_{\mathrm{v}}}&\coloneqq {\left(}{\mathcal{V}}, {\mathcal{L}}{\right)}, \label{eq:vehgraph} \\
{\mathcal{L}}&\coloneqq \{\{v_i, v_j\} \mid v_i, v_j \in {\mathcal{V}}, {\mathit{LOSS}}(v_i, v_j) \leq {\theta}\}, \label{eq:link}\end{aligned}$$ where ${\mathcal{V}}$, ${\mathcal{L}}$, ${\mathit{LOSS}}(v_i,v_j)$, and ${\theta}$ denote the set of vehicles, set of vehicle connections, mmWave propagation loss between $v_i$ and $v_j$, and a threshold that indicates that mmWave communications are possible, respectively. The mmWave propagation loss can be obtained from the path loss models proposed in [@PathLossPrediction]. The authors of [@PathLossPrediction] measured the propagation loss of 60-GHz mmWave channels when there were one, two, or three vehicles between the transmitter and receiver. For scenarios with more than three blockers, [@MmWaveVanetSurvey] provided an extension to the path loss model. Another approach for predicting mmWave propagation loss was proposed in [@RNNbasedRSSPred], where the authors predicted received signal power based on perceptual data. The threshold ${\theta}$ is calculated based on the Shannon capacity as follows: $$\begin{aligned}
&{B}\log_2 {\left(}1 + \frac{{P_\mathrm{t}}{G_\mathrm{t}}{G_\mathrm{r}}/ {\mathit{LOSS}}(v_i, v_j)}{{B}{N}}{\right)}\geq {\mathit{Rate}}, \\
&{\theta}\coloneqq \frac{{P_\mathrm{t}}{G_\mathrm{t}}{G_\mathrm{r}}}{{B}{N}{\left(}2^{{\mathit{Rate}}/{B}} - 1 {\right)}},\end{aligned}$$ where ${B}$, ${P_\mathrm{t}}$, ${G_\mathrm{t}}$, ${G_\mathrm{r}}$, ${N}$, and ${\mathit{Rate}}$ denote the bandwidth, transmission power, transmitter and receiver antenna gain, thermal noise power spectral density, and rate requirements, respectively. When calculating the vehicle connectivity, the antenna directions of the transmitter and receiver point at each other. We also assume that ${\bm{G}_{\mathrm{v}}}$ does not change within the data update interval because the interval is very short (less than 100ms), meaning the mobility of the vehicles is negligible.
${\mathcal{T}}_\tau \leftarrow {\mathcal{T}}_\tau \cup \{{t}_{ijk}\}$ ${\bm{G}_{\mathrm{s},\tau}}$
Initialize ${\bm{d}}_i \leftarrow \{d_i\}$ for all $i$ Obtain ${\bm{G}_{\mathrm{s},0}}$ from Algorithm \[alg:sched\_graph\] with ${\bm{G}_{\mathrm{v}}}, {\bm{d}}_{i,0}$ $\tau \leftarrow 0$ ${\bm{t}}_\tau \leftarrow \text{ MWIS of } {\bm{G}_{\mathrm{s},\tau}}$ Perform ${\bm{t}}_\tau$ and update ${\bm{d}}_{i,\tau+1}$ Obtain ${\bm{G}_{\mathrm{s},\tau+1}}$ from Algorithm \[alg:sched\_graph\] with ${\bm{G}_{\mathrm{v}}}, {\bm{d}}_{i,\tau+1}$ $\tau \leftarrow \tau+1$
Scheduling Graph and Data Sharing Scheduling
--------------------------------------------
For every time slot $\tau$, the RSU selects transmitter and receiver vehicles from ${\mathcal{V}}$, as well as data $d_k \in {\bm{d}}_{i,\tau}$ to send for each transmitter vehicle $v_i$. This selection is calculated by solving the MWIS problem for the scheduling graphs, which are constructed as follows:
Algorithm \[alg:sched\_graph\] is utilized to construct scheduling graphs for each time slot ${\bm{G}_{\mathrm{s},\tau}}\coloneqq{\left(}{\mathcal{T}}_\tau,{\mathcal{C}}_\tau,{W}{\right)}$ from ${\bm{G}_{\mathrm{v}}}$, where ${\mathcal{T}}_\tau$, ${\mathcal{C}}_\tau$, and ${W}$ denote the set of vertices, set of edges, and vertex weighting function such that ${W}\colon{\mathcal{T}}_\tau\to\mathbb{R}^{+}$, respectively. $\mathbb{R}^{+}$ is the set of positive real numbers. A transmission ${t}_{ijk} \in {\mathcal{T}}_\tau$ represents a set containing transmitter $v_i$, receiver $v_j$, and data $d_k$, meaning $v_i$ transmits $d_k$ to $v_j$. Each element in ${\mathcal{C}}_\tau$ represents a conflict between two transmissions, meaning they cannot be performed concurrently. Further details are explained in Section \[sec:conflict\]. The weight of vertex ${W}({t}_{ijk})$ represents the priority of each transmission, and its definition is described in Section \[sec:priority\]. Figure \[fig:sched\_graph\] presents an example of a scheduling graph constructed from the vehicular network graph shown in Fig. \[fig:veh\_graph\]. From lines 2–10 in Algorithm \[alg:sched\_graph\], a set of transmissions ${\mathcal{T}}_\tau$ is obtained by listing all directly connected pairs of vehicles (i.e., neighbors in ${\bm{G}_{\mathrm{v}}}$) and data not possessed by receivers. Next, the conflict between each pair of transmissions is calculated in lines 11–15.
Algorithm \[alg:scheduling\] is a scheduling algorithm utilizing ${\bm{G}_{\mathrm{s},\tau}}$. In each time slot, a set of transmissions ${\bm{t}}_\tau \subset {\mathcal{T}}_\tau$ is selected. After the transmissions are performed, the datasets ${\bm{d}}_{i,\tau+1}$ are updated utilizing (\[eq:updatedataset\]), and ${\bm{G}_{\mathrm{s}}}$ is recalculated based on the updated ${\bm{d}}_{i,\tau+1}$. We describe our method for selecting transmissions in the following subsection. After the scheduling for all time slots in the data update interval is completed, each vehicle follows the determined schedule during the data sharing period.
Priority of Transmissions {#sec:priority}
-------------------------
To perform scheduling, the controller calculates the MWIS of ${\bm{G}_{\mathrm{s},\tau}}$ to increase the number of transmissions performed in each time slot. Independent sets of ${\bm{G}_{\mathrm{s},\tau}}$ represent sets of non-conflicting transmissions and thus, maximum transmissions that can be performed concurrently can be obtained by solving the maximum independent set (MIS) problem. The MIS problem is a special case of MWIS, where the weight function ${W}$ is a constant function (i.e., ${W}({t}_{ijk}) = 1, \forall {t}_{ijk} \in {\mathcal{T}}_\tau$). We refer to the data sharing algorithm with MIS as max transmission scheduling. Although max transmission scheduling maximizes the number of transmissions, it does not consider the perceivable region represented by the perceptual data. When ${\tau_\mathrm{max}}$ is small due to the limit of the data sharing period, the algorithm stops before all data are shared with all vehicles. In such cases, vehicles perceive their environments based on limited information that covers only a limited area of the intersection.
To increase the perceivable area in such cases, prioritization for transmitted data can be implemented. Data from near the intersection tend to overlap with each other, because the vehicle density near intersections is higher than that far from intersections. Therefore, it is inefficient to forward data representing areas near intersections. We propose assigning a high priority to data that represent areas far from the intersection and lower priority to data that represent areas near the intersection. We refer to the algorithm with a priority function as max distance scheduling. Distance priority can be represented as a weight function ${W}$ of transmissions ${t}_{ijk}$ as follows: $$\begin{aligned}
{W}({t}_{ijk}) &\coloneqq {\mathit{dist}}(p_k, {p_\mathrm{c}}),\end{aligned}$$ where $p_k$, ${p_\mathrm{c}}$, and ${\mathit{dist}}(a,b)$ denote the position of $v_k$, center of the intersection, and distance between positions $a$ and $b$, respectively.
Although the MIS and MWIS problem are NP-hard problems, it has been proven that a simple greedy algorithm can approximately solve these problems with a guaranteed performance ratio of greater than or equal to $1/\Delta$, where $\Delta$ denotes the maximum degree of any vertex in the graph [@noteMWIS]. Thus, we adopt a greedy approach in our proposed scheduling algorithm.
Conflict Rule of Interference {#sec:conflict}
-----------------------------
When constructing ${\bm{G}_{\mathrm{s},\tau}}$, conflicts between transmissions ${t}_{ijk} \in {\mathcal{T}}_\tau$ must be determined to obtain the set of edges ${\mathcal{C}}_\tau$. The rules used to determine the conflicts are referred to as conflict rules. The basic conflict rules are defined as follows:
- A transmitter cannot transmit different data at the same time or transmit data to different receivers because of the use of a narrow-beam directional antenna: ${t}_{ijk}$ conflicts with ${t}_{i'j'k'}$ if $i=i'$.
- A receiver cannot receive data from multiple transmitters: ${t}_{ijk}$ conflicts with ${t}_{i'j'k'}$ if $j=j'$.
- A vehicle cannot transmit and receive data simultaneously because of half-duplex communication: ${t}_{ijk}$ conflicts with ${t}_{i'j'k'}$ if $i=j' \lor j=i'$.
Because the original algorithm proposed in [@coopDataSched] assumed a DSRC channel and omnidirectional antennas, the following conflict rule was added to the basic rules:
- A receiver near a transmitter cannot receive data from other transmitters: ${t}_{ijk}$ conflicts with ${t}_{i'j'k'}$ if $v_j \in {\mathcal{N}_{{\bm{G}_{\mathrm{v}}}}}(v_i') \lor v_j' \in {\mathcal{N}_{{\bm{G}_{\mathrm{v}}}}}(v_i)$, where ${\mathcal{N}_{{\bm{G}_{\mathrm{v}}}}}(v_i)$ denotes the neighbors of $v_i$.
However, this conflict rule does not match our problem because we assume mmWave V2V communications.
Considering the narrow beam width and high attenuation of mmWave communications, it seems that the radio interference between two transmissions is negligible. In this case, the conflict set ${\mathcal{C}}_\tau$ is defined only by the basic rules: (a), (b), and (c). However, interference sometimes occurs when an interferer is near a receiver or the transmission direction of the desired and interfering signal are nearly parallel.
In order to overcome interference, we design a conflict rule that reflects mmWave radio characteristics. However, it is difficult to estimate SINR during scheduling because interference cannot be calculated before all the transmitters and their antenna directions are determined. We propose an approximation method that can be adopted for our conflict rules, which are defined for only two transmissions and utilized when constructing the scheduling graphs.
Considering the narrow beam width and high attenuation of mmWave radio signals, we assume that the largest interference signal is the main factor of SINR. Therefore, SINR can be approximated as follows: $$\begin{aligned}
{\mathit{SINR}}_{i,j} &\coloneqq \frac{{P_{\mathrm{r}}}^{(i,j)}}{{B}{N}+ \displaystyle{\sum_{k\neq i,j}{I}_k}} \\
&\approx \frac{{P_{\mathrm{r}}}^{(i,j)}}{{B}{N}+ \displaystyle{\max_{k\neq i,j} {I}_k}},\end{aligned}$$ where ${\mathit{SINR}}_{i,j}$, ${P_{\mathrm{r}}}^{(i,j)}$, and ${I}_k$ denote the SINR at vehicle $v_i$, whose desired signal comes from $v_j$, received signal strength of desired signals from $v_j$ at vehicle $v_i$, and interference power from vehicle $v_k$, respectively. Although knowledge regarding all interference signals seems to be required when calculating $\max_{k\neq i,j} {I}_k$, a conflict rule can be designed between pairs of transmissions by assuming that the currently considered interferer is the largest one. The conflict rule reflecting interference is designed as follows:
- A receiver cannot receive data when interfering signals are large: ${t}_{ijk}$ conflicts with ${t}_{i'j'k'}$ if ${\mathit{sinr}}({t}_{ijk},{t}_{i'j'k'}) \leq \Theta \lor {\mathit{sinr}}({t}_{i'j'k'},{t}_{ijk}) \leq \Theta$, where ${\mathit{sinr}}({t}_{ijk},{t}_{i'j'k'}) \coloneqq {P_{\mathrm{r}}}^{(j,i)}/({B}{N}+{I}_{i'})$ and $\Theta \coloneqq 2^{{\mathit{Rate}}/{B}} - 1$.
Consider the interfering signals from $v_{{j_\mathrm{t}}}$ and $v_{{k_\mathrm{t}}}$ to $v_{{i_\mathrm{r}}}$, where ${I}_{{j_\mathrm{t}}} < {I}_{{k_\mathrm{t}}}$. $v_{{j_\mathrm{t}}}$ and $v_{{k_\mathrm{t}}}$ attempt to transmit signals to $v_{{j_\mathrm{r}}}$ and $v_{{k_\mathrm{r}}}$, respectively, and $v_{{i_\mathrm{r}}}$ receives signals from $v_{{i_\mathrm{t}}}$. If ${\mathit{sinr}}({t}_{{i_\mathrm{t}}{i_\mathrm{r}}a},{t}_{{k_\mathrm{t}}{k_\mathrm{r}}b}) \leq \Theta$, then the concurrent transmission of ${t}_{{i_\mathrm{t}}{i_\mathrm{r}}a}$ and ${t}_{{k_\mathrm{t}}{k_\mathrm{r}}b}$ cannot be scheduled by the proposed algorithm. Therefore, when calculating the interference from $v_{{j_\mathrm{t}}}$ to $v_{{i_\mathrm{r}}}$, we do not need to consider interference from $v_{{k_\mathrm{t}}}$, and the interference from $v_{{j_\mathrm{t}}}$ is assumed to be the largest. This assumption can be extended inductively for more than three transmitters.
Required Time Slot for Complete Data Sharing
--------------------------------------------
In this section, we discuss the situation where sufficient time slots are available to complete data sharing. First, we prove that the proposed scheduling algorithm terminates in finite time and that all data are shared with all vehicles if the data sharing time is not limited and the vehicular network graph ${\bm{G}_{\mathrm{v}}}$ is connected. We then discuss the bounds for the required number of time slots.
If ${\bm{G}_{\mathrm{v}}}$ is connected, the proposed data sharing algorithm terminates in finite time and all the data initially possessed by vehicles are shared with all vehicles when the algorithm terminates, which can be expressed as follows: $$\begin{aligned}
\forall i, {\bm{d}}_{i,{\tau_\mathrm{end}}} = \{d_1, \dots, d_{{N_\mathrm{v}}} \}, \label{eq:data_end}
\end{aligned}$$ where ${\tau_\mathrm{end}}$ denotes the step count at the end of Algorithm \[alg:scheduling\].
First, we prove that the proposed algorithm terminates in finite time and then, we prove that all vehicles possess all data at the end of the algorithm.
Let $n_\tau$ denote the total size of the dataset ${\bm{d}}_{i,\tau}$, defined as $n_\tau \coloneqq \sum_{i=1}^{{N_\mathrm{v}}} |{\bm{d}}_{i,\tau}|$, where $|\cdot|$ represents the cardinality of a set. When ${t}_{ijk}$ is performed, meaning vehicle $v_j$ receives data $d_k$, $n_\tau$ is updated as $n_{\tau+1} \leftarrow n_\tau + 1$ because ${\mathcal{T}}_\tau$ is constructed from all elements in ${t}_{ijk}$ that satisfy $d_k \notin {\bm{d}}_{j,\tau}$ (lines 5–7 in Algorithm \[alg:sched\_graph\]). Let $m_\tau \coloneqq |{\bm{t}}_\tau|$ denote the number of transmissions selected by the RSU. Then, $n_\tau$ is updated as $n_{\tau+1} \leftarrow n_\tau + m_\tau$ in each time slot. Meanwhile, the maximum value of $n_\tau$ is bounded by ${N_\mathrm{v}}^2$. Therefore, the algorithm terminates in finite time if at least one transmission is selected in each time slot.
Next, we prove that if there exists a vehicle that does not possess all data, at least one transmission can be performed. Assume $v_i$ does not possess $d_j$, which means $d_j \notin {\bm{d}}_i$. Then, there exists a connected pair $\{v_\alpha, v_\beta\} \in {\mathcal{L}}$ on the paths between $v_i$ and $v_j$ that satisfies $d_j \in {\bm{d}}_\alpha \land d_j \notin {\bm{d}}_\beta$ because at least $v_j$ possesses $d_j$. Note that the paths between $v_i$ and $v_j$ exist because ${\bm{G}_{\mathrm{v}}}$ is connected. Now, we have ${t}_{\alpha \beta j} \in {\mathcal{T}}_\tau$ because $v_\alpha$ possesses $d_j$ and $v_\beta$ does not possess $d_j$, meaning ${\mathcal{T}}_\tau \neq \emptyset$. An independent set of a graph is not $\emptyset$ if a vertex set of the graph is not $\emptyset$. Therefore, ${\bm{G}_{\mathrm{s},\tau}}$ has an independent set whose size is greater than zero.
If not all vehicles possess all data, a transmission can be performed and thus, the algorithm does not terminate. When the algorithm terminates, all vehicles possess all data. Because it is guaranteed that the algorithm always terminates in finite time, all data can be shared with all vehicles in finite time.
Next, we reveal the bounds of ${\tau_\mathrm{end}}$.
If the vehicular network graph ${\bm{G}_{\mathrm{v}}}$ is connected, ${\tau_\mathrm{end}}$ is bounded as, $$\begin{aligned}
\frac{{N_\mathrm{v}}^2 - {N_\mathrm{v}}}{\lfloor{N_\mathrm{v}}/2\rfloor} \leq {\tau_\mathrm{end}}\leq {N_\mathrm{v}}^2 - {N_\mathrm{v}}, \label{eq:bound}
\end{aligned}$$ where $\lfloor\cdot\rfloor$ represents the floor function.
At the beginning of the algorithm, we have $n_0 = {N_\mathrm{v}}$. From (\[eq:data\_end\]), we have $n_{{\tau_\mathrm{end}}} = {N_\mathrm{v}}^2$. Meanwhile, $n_{{\tau_\mathrm{end}}}$ is also written as $n_{{\tau_\mathrm{end}}} = n_0 + \sum_{\tau=0}^{{\tau_\mathrm{end}}-1} m_\tau$. Because vehicles cannot transmit and receive data simultaneously, $m_\tau$ satisfies $m_\tau \leq \lfloor{N_\mathrm{v}}/2\rfloor$. Additionally, $m_\tau$ also satisfies $m_\tau \geq 1$ because at least one transmission is performed in each time slot. Therefore, we have $\frac{{N_\mathrm{v}}^2 - {N_\mathrm{v}}}{\lfloor{N_\mathrm{v}}/2\rfloor} \leq {\tau_\mathrm{end}}\leq {N_\mathrm{v}}^2 - {N_\mathrm{v}}$.
On one hand, ${\tau_\mathrm{end}}$ is equal to the upper bound if only one vehicle transmits data in every time slot. On the other hand, ${\tau_\mathrm{end}}$ achieves the lower bound if half of the vehicles send data in every time slot, which is the optimal case under the constraint that each vehicle cannot send and receive data simultaneously. In Section \[sec:results\], simulation results demonstrate that ${\tau_\mathrm{end}}$ is near the lower bound in many cases.
Parameters Values
------------------------------------- ------------------------
Number of lanes 4
Lane width 3.5m
Sidewalk width 4m
Sensor range ${r_\mathrm{s}}$ 50m
Bandwidth ${B}$ 2.16GHz
Thermal noise ${N}$ -174dBm/Hz
Data rate ${\mathit{Rate}}$ 1Gbit/s
Transmission power ${P_\mathrm{t}}$ 10dBm
Antenna beam width 15$^\circ$, 30$^\circ$
: Simulation parameters[]{data-label="tbl:simparams"}
Simulation Results {#sec:results}
==================
We evaluated our algorithm through simulations. In our simulations, the distribution of inter-vehicle distance followed an exponential distribution. This assumption was confirmed in [@RoutingSparseVANET], where the authors demonstrated that the distribution of inter-vehicle distance follows an exponential distribution based on empirical data collected in a real environment. We assumed the average distance between vehicles ${l_\mathrm{avg}}$ in each lane was approximately the same as the stopping distance for traffic safety. We evaluated the situations where ${l_\mathrm{avg}}=20\,\mathrm{m}$ and 40m, which are slightly larger than the stopping distances when the velocity is 40km/h and 60km/h, respectively [@world2008speed]. We evaluated an intersection with two roads and four buildings assuming an urban area. Each road had four lanes and sidewalks on both sides. Each vehicle was modeled as a rectangle with a size of 1.7m $\times$ 4.4m. The ${N_\mathrm{v}}$ vehicles closest to the center of the intersection shared their data with each other. The number of participants ${N_\mathrm{v}}$ was fixed to 20 and 40 when ${l_\mathrm{avg}}$ was 20m and ${N_\mathrm{v}}$ was fixed to 10 and 20 when ${l_\mathrm{avg}}$ was 40m. If ${\left(}{l_\mathrm{avg}}, {N_\mathrm{v}}{\right)}={{\left(}20\,\mathrm{m},20{\right)}}$ and ${{\left(}40\,\mathrm{m},10{\right)}}$, the perceivable region with shared data covers $(25\,\mathrm{m}+{r_\mathrm{s}}/2)$ from the center of the intersection, because vehicles within approximately 25m of the center of the intersection participate in data sharing. When the number of vehicles is doubled, perceivable region covers $(50\,\mathrm{m}+{r_\mathrm{s}}/2)$ from the center of the intersection. The achievable coverage is sufficient for some applications, e.g., accident or congestion detection system with which a driver or self-driving system gets alerts and stops or slows down the vehicle when an accident or congestion is detected at the intersection. We drew lines from the transmitter to the receiver vehicles and counted the number of blocking vehicles on the lines. The number of blockers was used to calculate the pass loss based on the model proposed in [@PathLossPrediction] and construct ${\bm{G}_{\mathrm{v}}}$ from (\[eq:vehgraph\]) and (\[eq:link\]). We also assumed that vehicles could not communicate with each other if the buildings blocked their line-of-sight path. The antenna gain was calculated from the model in [@maltsev2010channel]. The other parameters are listed in Table \[tbl:simparams\].
In our simulations, the RSU first determined the scheduling. Next, vehicles transmitted data based on the scheduling. When interference occurred, a receiver failed to receive data. If a transmitter was scheduled to transmit data that it did not possess, it did not transmit any data during that time slot.
![Normalized perceivable area as a function of the number of time slots when ${\left(}{l_\mathrm{avg}},{N_\mathrm{v}}{\right)}={{\left(}40\,\mathrm{m},20{\right)}}$ and beam width is 15$^\circ$. The normalized perceivable area is enlarged by utilizing the conflict rule (d’).[]{data-label="fig:maxtrans"}](images/dat/greedy_vnum20_bw15.pdf){width="41.00000%"}
![Empirical CDF of normalized perceivable area when ${\left(}{l_\mathrm{avg}},{N_\mathrm{v}}{\right)}={{\left(}40\,\mathrm{m},20{\right)}}$. Nearly all vehicles achieve 90% of the perceivable area at $\tau=40$ when the conflict rule (d’) is utilized.[]{data-label="fig:cdf"}](images/dat/cdf_vnum20_bw15.pdf){width="41.00000%"}
Figure \[fig:maxtrans\] presents the normalized perceivable area ${\hat{S}_{\tau}}$ defined in Section \[sec:system\] as a function of the number of time slots $\tau$ when ${\left(}{l_\mathrm{avg}},{N_\mathrm{v}}{\right)}={{\left(}40\,\mathrm{m},20{\right)}}$. When the proposed data sharing algorithm was used, the perceivable areas ${\hat{S}_{\tau}}$ were enlarged by data sharing at first. Then, ${\hat{S}_{\tau}}$ saturated when $\tau\geq 40$ because the entire region ${R_\mathrm{all}}$ was covered by the shared perceptual data. In other words, transmitted data after $\tau\geq 40$ did not contribute to enlarging the perceivable area because of overlap. Finally, the algorithm terminated at $\tau=61$ when all scheduled transmissions were completed. The proposed algorithm achieved approximately twice the perceivable area compared with the conventional algorithm at $\tau=40$. This is because the conventional algorithm assumed microwave communications and thus, few vehicles could transmit data concurrently because of the conflict rule (d), which was designed for microwave communications. In contrast, the proposed method achieved efficient concurrent transmission because its conflict rules reflect mmWave radio characteristics. Additionally, adopting the conflict rule (d’) enlarged the perceivable area because when this rule is not adopted, certain interferences cannot be avoided and data sharing cannot be completed owing to transmission failure.
Figure \[fig:cdf\] shows the empirical cumulative distribution function (CDF) of the normalized perceivable area ${\hat{S}_{\tau}}$. When utilizing the conflict rule (d’), nearly all vehicles achieved 90% of the normalized perceivable area at $\tau=40$, whereas only 86% of the vehicles achieved 90% of the normalized perceivable area where interference was not considered when determining the scheduling.
![Normalized perceivable area when ${\left(}{l_\mathrm{avg}},{N_\mathrm{v}}{\right)}={{\left(}20\,\mathrm{m},40{\right)}},$ ${{\left(}20\,\mathrm{m},20{\right)}}, {{\left(}40\,\mathrm{m},20{\right)}}, {{\left(}40\,\mathrm{m},10{\right)}}$ and the beam width is 15$^\circ$. The differences between the performances with and without the mmWave interference conflict rule are smaller when the number of participants is small or inter-vehicle distance is large. Averages of normalizing factors ${S}({R_\mathrm{all}})$ in (\[eq:nrm\]) for ${\left(}{l_\mathrm{avg}},{N_\mathrm{v}}{\right)}={{\left(}20\,\mathrm{m},40{\right)}},{{\left(}20\,\mathrm{m},20{\right)}},{{\left(}40\,\mathrm{m},20{\right)}},$ and ${{\left(}40\,\mathrm{m},10{\right)}}$ are 6,161m$^2$, 3,949m$^2$, 5,793m$^2$, and 3,616m$^2$, respectively. []{data-label="fig:vnum"}](images/dat/greedy_bw15.pdf){width="41.00000%"}
![Normalized perceivable area when ${\left(}{l_\mathrm{avg}},{N_\mathrm{v}}{\right)}={{\left(}40\,\mathrm{m},20{\right)}}$ and the beam width is 15$^\circ$ and 30$^\circ$. The differences between the beam widths of 15$^\circ$ and 30$^\circ$ are larger when the mmWave interference conflict rule is not adopted compared with when the rule (d’) is adopted.[]{data-label="fig:bw"}](images/dat/greedy_vnum20.pdf){width="41.00000%"}
The normalized perceivable areas with different inter-vehicle distances and number of vehicles are shown in Fig. \[fig:vnum\]. The beam width was 15$^\circ$. When the vehicle average inter-vehicle distance was the same, the smaller the number of vehicles, the faster the data sharing terminated. This is confirmed by (\[eq:bound\]). It is also shown that the superiority of adopting conflict rule (d’) does not depend on ${l_\mathrm{avg}}$ and ${N_\mathrm{v}}$. The performance gain of adopting conflict rule (d’) is larger when the number of participants is large or the inter-vehicle distance is small because interference is more likely to occur in these cases.
Figure \[fig:bw\] presents the normalized perceivable areas with beam widths of 15$^\circ$ and 30$^\circ$, when ${\left(}{l_\mathrm{avg}}, {N_\mathrm{v}}{\right)}={{\left(}40\,\mathrm{m},20{\right)}}$. The differences between the beam widths of 15$^\circ$ and 30$^\circ$ were larger when the conflict rule for mmWave interference was not adopted compared with when the rule (d’) is adopted. This is because interferences occur more frequently with a wider beam width, but the proposed conflict rule (d’) can successfully avoid interference.
Figure \[fig:comp\_weight\] presents differences between the two priority designs when ${\left(}{l_\mathrm{avg}}, {N_\mathrm{v}}{\right)}={{\left(}40\,\mathrm{m},20{\right)}}$. The max distance scheduling design achieved a larger perceivable area than the max transmission scheduling design when $\tau\leq 36$. ${\hat{S}_{\tau}}$ of the max distance design was 20% larger than that of the max transmission design when $\tau=8$. Based on the max distance scheduling, data far from the center of the intersection were transmitted at first and thus, the overlapping regions tended to be smaller than those in the algorithm without such priority control. Therefore, ${\hat{S}_{\tau}}$ became larger than that in the max transmission scheduling design. When the number of time slot was limited, such that ${\tau_\mathrm{max}}\leq 36$, the max distance scheduling design provided a larger perceivable area during every data update interval.
![Normalized perceivable area with the different priority designs, when ${\left(}{l_\mathrm{avg}},{N_\mathrm{v}}{\right)}={{\left(}40\,\mathrm{m},20{\right)}}$ and beam width is 15$^\circ$. The conflict rule (d’) is adopted. The perceivable area can be enlarged by prioritizing data far from the intersection when the number of time slot is limited.[]{data-label="fig:comp_weight"}](images/dat/weight_vnum20_bw15.pdf){width="41.00000%"}
Figure \[fig:step\_cdf\] shows empirical CDF of the number of time slots required to share all data, denoted ${\tau_\mathrm{end}}$, when ${\tau_\mathrm{max}}$ is much larger than ${\tau_\mathrm{end}}$. Inter-vehicle distance was assumed to be 40m. From (\[eq:bound\]), the lower bounds of ${\tau_\mathrm{end}}$ were calculated as 18, 30, and 38, for ${N_\mathrm{v}}=10,15,20$. The lower bounds are depicted as black vertical lines in Fig. \[fig:step\_cdf\]. In most cases, ${\tau_\mathrm{end}}$ were closer to the lower bounds than the upper bounds of 90, 210, and 380. In very few cases, as shown in Fig. \[fig:step\_cdf\], the data sharing algorithm terminated after fewer iterations than the lower bounds. This is because the vehicular network graphs ${\bm{G}_{\mathrm{v}}}$ were disconnected in such cases and thus, the data sharing algorithm terminated before all data were shared. The differences between the protocols with and without prioritization can be observed when ${N_\mathrm{v}}=20$. The max transmission scheduling design achieved efficient data sharing because it maximized the number of transmitted data at each time slot, meaning the algorithm terminated faster than the max distance scheduling design when data sharing time was not limited.
![Empirical CDF of the number of time slots required to share all data. The conflict rule (d’) is adopted. The beam width is 15$^\circ$ and inter-vehicle distance is 40m. Black vertical lines represent lower bounds. The proposed algorithm achieves near-optimal scheduling.[]{data-label="fig:step_cdf"}](images/dat/step_cdf_exgraph.pdf){width="41.00000%"}
Conclusion {#sec:conclusion}
==========
We proposed a data sharing scheduling method with concurrent transmission for mmWave VANETs for cooperative perception. We modified the algorithm in [@coopDataSched] by designing a conflict rule that represents mmWave communication characteristics and a weight function that prioritizes data to be forwarded to enlarge the perceivable area. Simulation results demonstrated that the proposed conflict rule for scheduling graphs achieved a larger perceivable area compared with the original rules, which did not consider directional antennas. Priority control methods also enlarged the perceivable region by sharing data that covered areas far from an intersection at first. The priority control worked efficiently in situations where the number of time slots was limited. We also proved that the proposed algorithms terminate in finite time and all data can be shared with all vehicles if a vehicular network is represented as a connected graph and there are sufficient time slots.
For future work, we will develop a data-aggregation and vehicle-selection method for reducing redundant data transmissions. The algorithm proposed in this paper transmits data without considering overlapping regions covered by multiple data. To suppress the transmission of data representing overlapping regions can reduce data traffic without reducing the perceivable area. The data-aggregation method that aggregates some overlapping data to a single datum also reduces the amount of data to be transmitted. When the vehicles densely located, to select vehicles generating data considering their sensor coverage can reduce data transmissions including the same regions.
Another interest is to develop a scalable distributed scheduling method. In the proposed algorithm, an RSU, which act as a central controller, determines the schedule based on information from all vehicles participating in cooperative perception. When the number of vehicles is large, it is difficult for a central controller to obtain accurate information from all vehicles and to perfectly control whole schedules because mmWave channels vary rapidly. Therefore, distributed scheduling including hybrid schemes of centralized and distributed scheduling should be developed to reduce the amount of vehicle information transmitted to the RSU and to allow vehicles to decide scheduling autonomously using their own information.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work was supported in part by JSPS KAKENHI Grant Number JP17H03266, KDDI Foundation, and Tateisi Science and Technology Foundation.
|
---
abstract: 'In this paper, based on a proposed notion of generalized conjugate harmonic pairs in the framework of complex Clifford analysis, necessary and sufficient conditions for the solvability of inhomogeneous perturbed generalized Moisil-Teodorescu systems in higher dimensional Euclidean spaces are proved. As an application, we derive corresponding solvability conditions for the inhomogeneous Maxwell’s equations.'
author:
- 'Juan Bory-Reyes$^{(1)}$ and Marco Antonio Pérez-de la Rosa$^{(2)}$'
date: |
$^{(1)}$ ESIME-Zacatenco. Instituto Politécnico Nacional. CD-MX. 07738. México.\
E-mail: [email protected]\
$^{(2)}$ Department of Actuarial Sciences, Physics and Mathematics, Universidad de las Américas Puebla. San Andrés Cholula, Puebla. 72810. México.\
Email: [email protected]
title: 'Solutions of inhomogeneous perturbed generalized Moisil-Teodorescu system and Maxwell’s equations in Euclidean Space'
---
**Keywords.** Clifford analysis; Moisil-Teodorescu system; Maxwell’s equations; Conjugate harmonic pairs.\
**AMS Subject Classification (2010):** 30G35, 47F05, 47G10, 35Q60.
Introduction
============
Clifford analysis is the study of properties of solutions of the first-order, vector-valued Dirac operator $\partial_x$ acting on functions defined on Euclidean spaces $\mathbb R^{m+1}$ ($m\geq 2$) with values in the corresponding real or complex Clifford algebra, that will be denoted below by $\mathbb{R}_{0,m+1}$ and $\mathbb{C}_{0,m+1}$ respectively. Thereby, this function theory may be considered as an elegant way of extending the theory of holomorphic functions in the complex plane to higher dimension and it provides at the same time a refinement of the theory of harmonic functions.
Clifford analysis is centered around the notion of monogenic function, i.e. a null solution of $\partial_x$. It is, however, often important (and interesting) to consider special types of solutions obtained by considering functions taking values in suitable subspaces of the real or complex Clifford algebras (so be the case).
As is established in [@BDS2], there exists a isomorphism between the Cartan algebra of differential forms and the algebra of multivector functions in Clifford analysis. In particular, the action of the operator $d-d^*$, where $d$ and $d^*$ are the differential and codifferential operators (the standard de Rham differential and its adjoint) respectively, on the space of smooth $k$-forms is identified with the action (on the right) of the Dirac operator, which plays the role of the Cauchy-Riemann operator on the space of smooth $k$-vector fields. Meanwhile the action of the operator $d+d^*$ is identified with the action (on the left) of the Dirac operator. A smooth differential form belonging to the kernel of $d+d^*$ was called in [@C; @Ma] self-conjugate differential form.
The Moisil-Teodorescu elliptic system of equations of first order in $\mathbb R^{3}$ is a vector valued analogue of Cauchy-Riemann system [@MT].
In the context of real Clifford analysis a generalized Moisil-Teodorescu systems of type $(r,p,q)$ was introduced in [@RBDS], where some general properties of solutions to this system have been investigated. Afterwards, there was a growing interest in the study and better understanding of properties of solutions of generalized Moisil-Teodorescu systems, see for instance [@BD1; @D1; @D2; @D3; @DLS; @FDSc; @La; @SiHe; @So1; @So2]. Some special cases of these systems are well known and well understood.
Following the identification mentioned before, a subsystem of generalized Moisil-Teodorescu systems leads to a subsystem of self-conjugate differential forms and vice versa.
To deal with the inhomogeneous generalized Moisil-Teodorescu systems in [@BoPe] the authors embedded the systems in an appropriate real Clifford analysis setting. Necessary and sufficient conditions for the solvability of inhomogeneous systems are provided and its general solution described and consequently some results in the literature are re-obtained, such as those given in [@RBMS; @RB; @BAPS; @PSV1; @PSV2].
The present paper is devoted to give explicit general solution of the inhomogeneous generalized perturbed Moisil-Teodorescu system in the framework of complex Clifford analysis upon the usage of a notion of generalized harmonic conjugates pairs.
A wealth of information about the subject of the nontrivial connection between Maxwell’s electrodynamics, Clifford algebras and, in fact, expressing the monogenicity of a certain Clifford algebra-valued function can be found in the literature, see for instance [@Chi; @F; @I; @J; @K; @Ma; @McMi; @Mi1; @Mi2; @Mi3; @MoWiA; @Se; @Sp]. The common feature of the method is to represent the Maxwell’s equations in a Dirac like form.
To illustrate the application of the main result, we establish solvability conditions for the inhomogeneous Maxwell’s equations. Our study is based on the complex Clifford algebra-based form of the Dirac equation and the Maxwell’s equations proposed in [@McMi].
Rudiments of Clifford analysis
==============================
The section provides a brief exposition of the basic notions and terminology of Clifford analysis aimed at readers who are unfamiliar with this function theory. Standard references are the monographs [@BDS1; @DSS; @GM; @GS; @GHS].
Let $\mathbb R^{0,m+1}$ be the real vector space $\mathbb R^{m+1}$ equipped with a quadratic form of signature $(0, m+ 1)$ and let $e_0, e_1,e_2,\dots, e_m$ be an orthogonal basis of $\mathbb R^{0,m+1}$.
The real Clifford algebra ${{\mathbb R}}_{0,m+1}$ with generators $e_0, e_1,e_2,\dots, e_m$, subject to the basic multiplication rules $$e_i^2=-1,\quad e_ie_{j}=-e_{j}e_i,\quad i,j=1,2,\dots m,\quad i<j,$$ is a real linear associative but non-commutative algebra with identity 1, having dimension $2^{m+1}$ and containing ${{\mathbb R}}$ and ${{\mathbb R}}^{m+1}$ as subspaces.
The complex Clifford algebra $\mathbb{C}_{0,m+1}$ constructed over $\mathbb R^{0,m+1}$, as a linear associative algebra over $\mathbb{C}$, has dimension $2^{m+1}$ meaning that one takes the same standard basis as for ${{\mathbb R}}_{0,m+1}$, with the same multiplication rules, however allowing for complex constants. Indeed, an element of $\mathbb{C}_{0,m+1}$ may be written as $a=\sum_{A} a_A e_A$, where $a_A$ are complex constants and $A$ runs over all the possible ordered sets $$A=\{i_{1},\dots,i_{s}\},\quad 0\le i_{1}<i_{2}<\dots<i_{s}\le m,\;{\mbox{or}}\; A=\emptyset,$$ and $$(e_{A}: |A|=s,\; s=0,1,\dots,m+1),\;e_A=e_{i_1}e_{i_2}\cdots e_{i_s},\;e_0=e_\emptyset=1.$$
One of the basic properties relied upon in building up the $\mathbb{C}_{0, m+1}$-valued continuously differentiable function theory in domains of $\mathbb R^{m+1}$ is the fact that the Dirac operator $\partial_x$ in $\mathbb R^{m+1}$ factorizes the Laplacian $\Delta_x$ through the relation $\partial_x^2=-\Delta_x$, where $\partial_x=\sum_{i=0}^{m}e_i\partial_{x_i}$, $x=(x_0,x_1,\dots ,x_m)\in\mathbb R^{m+1}$.
For technical reasons to become clear below (see [@McMi]), we embed everything into a larger Clifford algebra, say $\mathbb{R}^{0,m+1}\subseteq\mathbb{C}_{0,m+1}\subseteq\mathbb{C}_{0,m+2}$. Fix $\alpha\in\mathbb{C}$ and set $$\partial_{x,\alpha}=\partial_x+\alpha e_{m+1},$$ then $-\partial_{x,\alpha}^2=\Delta_x+\alpha^2$, the Helmholtz operator.
If $F$ is a $\mathbb{C}_{0,m+2}$-valued function defined in an open subset $\Omega\subset\mathbb R^{m+1}$, set $$\partial_{x,\alpha}F:=\sum_{i=0}^{m}e_i\frac{\partial F}{\partial x_i}+\alpha e_{m+1}F,$$ and $$F \partial_{x,\alpha}:=\sum_{i=0}^{m}\frac{\partial F}{\partial x_i}e_i+\alpha F e_{m+1}.$$
Let $F:\Omega\to\mathbb{C}_{0, m+2}$, whose components are of class $C^1$ in $\Omega$. Then $F$ is called left (right, two-sided) $\alpha$-monogenic in $\Omega$ if $\partial_{x,\alpha}F$ ($F \partial_{x,\alpha}$, or both $\partial_{x,\alpha}F$ and $F \partial_{x,\alpha}$) $=0$ in $\Omega$. Observe that each component of a $\alpha$-monogenic function is annihilated by the Helmholtz operator $\Delta_x+\alpha^2$.
An important example of a function which is both right and left $\alpha$-monogenic is the fundamental solution of the perturbed Dirac operator (see [@Mi1]), given for $x\in{{\mathbb R}}^{m+1}\setminus\{0\}$ by $$E_{\alpha}(x)=
\begin{cases}\displaystyle \frac{1}{\sigma_{m+1}}\frac{\bar{x}}{|x|^{m+1}}-\alpha e_{m+1}\Lambda_{\alpha}(x)+O\left(|x|^{-m+2}\right)&\text{as}\;|x|\to0,\\\\
\displaystyle O\left(\text{exp}\{-\text{Im}\alpha|x|\}\right)&\text{as}\;|x|\to+\infty,
\end{cases}$$ where $\sigma_{m+1}$ is the area of the sphere in ${{\mathbb R}}^{m+1}$, $\bar{x}$ is the conjugate of $x$ defined below and for $x\neq0$ $$\Lambda(x):=-\frac{1}{(4\pi)^{\frac{m+1}{2}}}\int_0^{+\infty}\text{exp}\left(\alpha^2 t-\frac{|x|^2}{4t}\right)\frac{dt}{t^{\frac{m+1}{2}}},$$ is the fundamental solution of the Helmholtz operator $\Delta_x+\alpha^2$. The function $E_{\alpha}(x)$ plays the same role in Clifford analysis as the Cauchy kernel does in complex analysis, for this reason it is also called the Cauchy kernel in ${{\mathbb R}}^{m+1}$.
Writing $E_{\alpha}(x)=E_1(x)+e_{m+1}E_2(x)$ with $E_1(x):=\displaystyle\frac{1}{\sigma_{m+1}}\frac{\bar{x}}{|x|^{m+1}}$ and $E_2(x):=-\alpha \Lambda_{\alpha}(x)$ by letting $|x|\to 0$, we have that $E_1$ is $\mathbb{C}_{0,m+2}^{(1)}$–valued while $E_2$ is $\mathbb{C}_{0,m+2}^{(0)}$–valued.
If $S$ is a subspace of $\mathbb{C}_{0,m+2}$, then ${\cal E}(\Omega,S)$; ${\cal M}(\Omega,S)$ and ${\cal H}(\Omega,S)$ denote, respectively, the spaces of smooth $S$–valued functions, left monogenic and harmonic $S$–valued functions in $\Omega$. Clearly, we have that ${\cal M}(\Omega,S)\subset{\cal H}(\Omega,S)\subset{\cal E}(\Omega,S)$.
Also, recall that the space $\mathbb{C}_{0,m+2}^{(s)}$ of $s$-vectors in $\mathbb{C}_{0,m+2}$ ($0\le s\le m+2$) is defined by $$\label{1.4}
\mathbb{C}_{0,m+2}^{(s)}={\mbox{span}}_\mathbb{C}(e_A:|A|=s).$$ Notice, in particular, that for $s=0$, $\mathbb{C}^{(0)}_{0,m+2}\cong\mathbb{C}$.
For $0\le s\le m+2$ fixed, the space $\mathbb{C}_{0,m+2}^{(s)}$ of $s$-vectors lead to the decomposition $$\label{2.1}
\mathbb{C}_{0,m+2} = \sum^{m+2}_{s=0} \bigoplus \mathbb{C}^{(s)}_{0,m+2},$$ and the associated projection operators $[\,\,]_s:\mathbb{C}_{0,m+2}\mapsto\mathbb{C}^{(s)}_{0,m+2}$.
An element $x=(x_0,x_1,\dots,x_{m+1})\in{{\mathbb R}}^{m+2}$ is usually identified with $x=\sum_{i=0}^{m+1}e_ix_i\in{{\mathbb R}}^{0,m+2}$.
For $x,y\in\mathbb{C}^{(1)}_{0,m+2}$, the product $xy$ splits in two parts, namely $$\label{2.2}
xy=x\bullet y+x\wedge y,$$ where $x\bullet y=[xy]_0$ is the scalar part of $xy$ and $x\wedge y=[xy]_2$ is the 2-vector or bivector part of $xy$ which are given by $$x\bullet y=-\sum_{i=0}^{m+1}x_iy_i,$$ and $$x\wedge y=\sum_{i<j}e_ie_j(x_iy_j-x_jy_i).$$
More generally, for $x\in\mathbb{C}^{(1)}_{0,m+2}$ and $\upsilon\in\mathbb{C}^{(s)}_{0,m+2}$, ($0<s<m+2$), we have that the product $x\upsilon$ decomposes into $$x\upsilon=x\bullet\upsilon+x\wedge\upsilon,$$ where $$\label{bullet}
x\bullet\upsilon=[x\upsilon]_{s-1}=\frac{1}{2}(x\upsilon-(-1)^s\upsilon x),$$ and $$\label{wedge}
x\wedge\upsilon=[x\upsilon]_{s+1}=\frac{1}{2}(x\upsilon+(-1)^s\upsilon x).$$
Finally, a conjugation is defined as the unique linear morphism of $\mathbb{C}^{(0)}_{0,m+2}$ with $\bar{e}_0=e_0$, $\bar{e}_j=-e_j$, $j=1,...,m+1$, while for $x,y\in\mathbb{C}^{(0)}_{0,m+2}$, $$\overline{(xy)}=\bar{y}\,\bar{x}.$$ Notice that for any basic element $e_{A}$ with $|A|=s$, $\bar{e}_{A}=(-1)^{\frac{s(s+1)}{2}}e_{A}.$
Generalized perturbed Moisil-Teodorescu systems
===============================================
Let $r, p, q\in\mathbb N$ with $0\le r\le m+2$, $0\le p\le q$ and $r+2q\le m+2$. Exploring further the multivector structure of $\mathbb{C}_{0, m+2}^{(r,p,q)}$ one may also write $$\mathbb{C}_{0, m+2}^{(r,p,q)}=\sum_{j=p}^q\bigoplus\mathbb{C}_{0, m+2}^{(r+2j)}.$$
If a $\mathbb{C}_{0, m+2}^{(r+1,p,q)}\bigoplus e_{m+1}\mathbb{C}_{0, m+2}^{(r,p,q)}$–valued smooth function $F$ defined in an open subset $\Omega\subset\mathbb R^{m+1}$ is decomposed following $$\begin{aligned}
F&=\sum_{j=p}^q\bigoplus F^{(r+2j+1)}+e_{m+1}\sum_{j=p}^q\bigoplus F^{(r+2j)}\\
&=\sum_{j=p}^q\bigoplus F^{(r+2j+1)}+\sum_{j=p}^q\bigoplus (-1)^{r+2j}F^{(r+2j)}e_{m+1},\end{aligned}$$ then in $\Omega$: $$\partial_{x,\alpha}F=0\quad\text{if and only if},$$ $$\label{MT}
\begin{cases}
\displaystyle \partial_{x}^{-}F^{r+2p+1}-\alpha F^{r+2p}=0,\\
\displaystyle \partial_{x}^{+}F^{r+2j+1}+\partial_{x}^{-}F^{r+2(j+1)+1}-\alpha F^{r+2(j+1)}=0,\qquad j=p,\dots,{q-1};\\
\displaystyle \partial_{x}^{+}F^{r+2q+1}=0,\\
\displaystyle \partial_{x}^{-}(-1)^{r+2p}F^{r+2p}=0,\\
\displaystyle \partial_{x}^{+}(-1)^{r+2j}F^{r+2j}+\partial_{x}^{-}(-1)^{r+2(j+1)}F^{r+2(j+1)}+\alpha (-1)^{r+2j+1}F^{r+2j+1}=0,\\
\displaystyle \hfill j=p,\dots,{q-1};\\
\displaystyle \partial_{x}^{+}(-1)^{2+2q}F^{r+2q}+\alpha (-1)^{r+2q+1}F^{r+2q+1}=0,
\end{cases}$$ where the differential operators $\partial_x^+$ and $\partial_x^-$ act on smooth $\mathbb{C}_{0,m+2}^{(s)}$–valued functions $F^s$ in $\Omega$ as $$\label{d+}
\partial_x^+F^s=\frac{1}{2}(\partial_xF^s-(-1)^sF^s\partial_x),$$ and $$\label{d-}
\partial_x^-F^s=\frac{1}{2}(\partial_xF^s+(-1)^sF^s\partial_x).$$ It is perhaps worth remarking that $\partial^+_{x} F^s$ is $\mathbb{C}^{(s+1)}_{0,m+2}$–valued while $\partial^-_{x} F^s$ is $\mathbb{C}^{(s-1)}_{0,m+2}$–valued.
The system (\[MT\]) generalizes that of [@BoPe] and is called generalized perturbed Moisil-Teodorescu system of type $(r,p,q)$.
In [@BoPe] is pointed out that for $\alpha=0$ the system (\[MT\]) includes some basic systems of first order linear partial differential equations as particular cases. For example, if $p=0$, $q=1$ and $0\le r\le m+1$ fixed, the system (\[MT\]) reduces to the Moisil-Teodorescu system in ${{\mathbb R}}^{m+1}$ introduced in [@BD1]. If $p=q=0$ and $0<r<m+1$ fixed, the system (\[MT\]) reduces to the generalized Riesz system $\partial_xF^r=0$; its solutions are called harmonic multi-vector fields. If $r=0$, $p=0$ and $m+1=3$ then $q=1$, the original Moisil-Teodorescu system introduced in [@Shap] is re-obtained. If $r=0$, $p=0$ and $m+1=4$ then $q=2$ and one obtains the Fueter system in ${{\mathbb R}}^4$ for so-called left regular functions of quaternion variable; it lies at the basis of quaternionic analysis (see [@Fu; @Sub]).
The inhomogeneous Dirac equation
================================
From now on we assume $\Omega$ to be a Lipschitz domain in ${{\mathbb R}}^{m+1}$, i.e., a domain whose boundary $\Gamma$ is given locally by the graph of a real valued Lipschitz function, after an appropriate rotation of coordinates.
The fundamental tool for solving the inhomogeneous perturbed Dirac equation (commonly called the $\overline\partial$-problem) $$\label{inh sys Cliff}
\partial_{x,\alpha} F=G,$$ is the Borel-Pompeiu integral formula (see below) which is named after the French and Romanian mathematicians Émile Borel (1871-1956) and Dimitrie Pompeiu (1873-1954), respectively.
For bounded $F\in C^0(\Omega;\mathbb{C}_{0,m+2})$, we consider the Teodorescu and the Cauchy-type operators associated to the Cauchy kernel $E_{\alpha}$, i.e., $$T_{\Omega}[F](x):=\int_{\Omega}E_{\alpha}(x-y)F(y)dy,\quad x\in{{\mathbb R}}^{m+2},$$ and by $$C_{\Gamma}[F](x):=-\int_{\Gamma}E_{\alpha}(x-y)n(y)F(y)d\Gamma_y,\quad x\notin\Gamma,$$ where $n(y)=\sum_{i=0}^{m+1}e_i n_i(y)$ is the outward pointing unit normal to $\Gamma$ at $y\in\Gamma$.
Let $F\in C^1(\Omega;\mathbb{C}_{0,m+2})\cap C^0(\Omega\cup \Gamma;\mathbb{C}_{0,m+2})$. Then we have
- $$\label{BPCliff}
C_{\Gamma}[F](x)+T_{\Omega}[\partial_{x,\alpha} F](x)=\begin{cases}
F(x),&x\in\Omega,\\
0,&x\in{{\mathbb R}}^{m+1}\setminus(\Omega\cup \Gamma).
\end{cases}$$
- $$\label{RightInv}
\partial_{x,\alpha} T_{\Omega}[F](x)=\begin{cases}
F(x),&x\in\Omega,\\
0,&x\in{{\mathbb R}}^{m+1}\setminus(\Omega\cup \Gamma).
\end{cases}$$
- $$\partial_{x,\alpha} C_{\Gamma}[F](x)=0,\quad x\in\Omega\cup\left({{\mathbb R}}^{m+1}\setminus(\Omega\cup \Gamma)\right).$$
For a proof of the Borel-Pompeiu formula (\[BPCliff\]) see, e.g. [@Zhen].
The Borel-Pompeiu formula (\[BPCliff\]) solves the inhomogeneous perturbed Dirac equation (\[inh sys Cliff\]) in the standard way and the general solution is given by $$\label{inh sys sol Cliff}
F=T_{\Omega}[G]+H,$$ where $H\in\mathcal M(\Omega; \mathbb{C}_{0,m+2})$.
Generalized conjugate harmonics pairs
=====================================
The notion of conjugate harmonic functions in the complex plane is well-known. This concept has been generalized to higher dimensional setting in the framework of Clifford analysis, see for instance [@BoPe; @FDSo; @GM; @Nol; @Shap].
In this section, we introduce a new generalization of notion of conjugate harmonic functions in a Clifford setting, based on a certain splitting of the Clifford algebra.
Let $$F_1=\sum_{j=p}^q F^{r+2j+1}+e_{m+1}\sum_{j=p}^q F^{r+2j},$$ in ${\cal H}\left(\Omega;\mathbb{C}_{0, m+2}^{(r+1,p,q)}\bigoplus e_{m+1}\mathbb{C}_{0, m+2}^{(r,p,q)}\right)$. An $$F_2=\left(F^{r+2p-1}+F^{r+2q+3}\right)+e_{m+1}\left(F^{r+2p-2}+F^{r+2q+2}\right),$$ in ${\cal H}\left(\Omega;\left(\mathbb{C}^{(r+2p-1)}_{0,m+2} \bigoplus \mathbb{C}^{(r+2q+3)}_{0,m+2}\right)\bigoplus e_{m+1}\left(\mathbb{C}^{(r+2p-2)}_{0,m+2} \bigoplus \mathbb{C}^{(r+2q+2)}_{0,m+2}\right)\right)$ is called hyper-conjugate harmonic to $F_1$ if $$F_1+F_2\in {\cal M}(\Omega;\mathbb{C}_{0,m+2}).$$ The pair $(F_1,F_2)$ is then called a pair of hyper-conjugate harmonic functions.
Main result
===========
We are in condition to state and proof our main result
\[maintheo\] Let $\displaystyle G\in C\left(\Omega;\sum_{j=p}^{q+1} \bigoplus \mathbb{C}^{(r+2j)}_{0,m+2}+\sum_{j=p}^{q+1} \bigoplus \mathbb{C}^{(r+2j-1)}_{0,m+2}e_{m+1}\right)$. The inhomogeneous generalized perturbed Moisil-Teodorescu system
$$\label{general system}
\begin{cases}
\displaystyle \partial_{x}^{-}F^{r+2p+1}-\alpha F^{r+2p}=G^{r+2p},\\
\displaystyle \partial_{x}^{+}F^{r+2j+1}+\partial_{x}^{-}F^{r+2(j+1)+1}-\alpha F^{r+2(j+1)}=G^{r+2j+2},\qquad j=p,\dots,{q-1};\\
\displaystyle \partial_{x}^{+}F^{r+2q+1}=G^{r+2q+2},\\
\displaystyle \partial_{x}^{-}(-1)^{r+2p}F^{r+2p}=G^{r+2p-1},\\
\displaystyle \partial_{x}^{+}(-1)^{r+2j}F^{r+2j}+\partial_{x}^{-}(-1)^{r+2(j+1)}F^{r+2(j+1)}+\\
\displaystyle\quad+\alpha (-1)^{r+2j+1}F^{r+2j+1}=G^{r+2j+1}, \hfill j=p,\dots,{q-1};\\
\displaystyle \partial_{x}^{+}(-1)^{2+2q}F^{r+2q}+\alpha (-1)^{r+2q+1}F^{r+2q+1}=G^{r+2q+1},
\end{cases}$$
where $G^s\in C^1\left(\Omega;\mathbb{C}^{(s)}_{0,m+2}\right)\cap C^0\left(\Omega\cup \Gamma;\mathbb{C}^{(s)}_{0,m+2}\right)$, has a solution if and only if for the $\left(\mathbb{C}^{(r+2p-1)}_{0,m+2} \bigoplus\mathbb{C}^{(r+2q+3)}_{0,m+2}\right)\bigoplus e_{m+1}\left(\mathbb{C}^{(r+2p-2)}_{0,m+2} \bigoplus \mathbb{C}^{(r+2q+2)}_{0,m+2}\right)$–valued function $$\begin{aligned}
P:&=\left(\int E_1\bullet G^{r+2p}dy+(-1)^{r+2p}\int E_2\,G^{r+2p-1}dy+\right.\\
&\quad\left.+\int E_1\wedge G^{r+2q+2}dy\right)+\\
&\quad+e_{m+1}\left((-1)^{r+2p}\int E_1\bullet G^{r+2p-1}dy+\int E_1\wedge G^{r+2q+1}dy+\right.\\
&\quad\left.+\int E_2\,G^{r+2q+2}dy\right),
\end{aligned}$$
- either is identically zero;
- or has a hyper-conjugate harmonic function.
If it is true, then the general solution of (\[general system\]) is given by:
- either
$$\begin{aligned}
F=&\sum_{j=p}^{q}\left[\left(\int E_1\wedge G^{r+2j}dy+\int E_1\bullet G^{r+2j+2}dy+\right.\right.\notag\\
&+\left.(-1)^{r+2j+2}\int E_2\,G^{r+2j+1}dy\right)+\notag\\
&+e_{m+1}\left(\int E_1\wedge G^{r+2j-1}dy+(-1)^{r+2j+2}\int E_1\bullet G^{r+2j+1}dy+\right.\notag\\
&\left.\left.+\int E_2\,G^{r+2j}dy\right)\right] +\hat{H}_1,\label{sol A*}
\end{aligned}$$
- or
$$\begin{aligned}
F=&\sum_{j=p}^{q}\left[\left(\int E_1\wedge G^{r+2j}dy+\int E_1\bullet G^{r+2j+2}dy+\right.\right.\notag\\
&+\left.(-1)^{r+2j+2}\int E_2\,G^{r+2j+1}dy\right)+\notag\\
&+e_{m+1}\left(\int E_1\wedge G^{r+2j-1}dy+(-1)^{r+2j+2}\int E_1\bullet G^{r+2j+1}dy+\right.\notag\\
&\left.\left.+\int E_2\,G^{r+2j}dy\right)\right] +\hat{H}_1+\tilde{H}_1,\label{sol B*}
\end{aligned}$$
where $\tilde{H}_1$ is a harmonic hyper-conjugate of $-P$, and $\hat{H}_1$ is an arbitrary monogenic function.
[**Proof:**]{} First, notice that system (\[general system\]) is a restriction of (\[inh sys Cliff\]), for $$F=\sum_{j=p}^q F^{r+2j+1}+e_{m+1}\sum_{j=p}^q F^{r+2j},$$ and with $$G=\sum_{j=p}^{q+1} G^{r+2j}+\sum_{j=p}^{q+1} G^{r+2j-1}e_{m+1}.$$ Let $F$ be such a solution. Looking at the components of (\[inh sys sol Cliff\]) one has: $$\label{system sol ICS}
\begin{cases}
\displaystyle 0=\int E_1\bullet G^{r+2p}dy+(-1)^{r+2p}\int E_2\,G^{r+2p-1}dy+H^{r+2p-1},\\
\displaystyle F^{r+2j+1}=\int E_1\wedge G^{r+2j}dy+\int E_1\bullet G^{r+2j+2}dy+\\
\displaystyle\qquad\qquad+(-1)^{r+2j+2}\int E_2\,G^{r+2j+1}dy+H^{r+2j+1},\qquad j=p,...,q,\\
\displaystyle 0=\int E_1\wedge G^{r+2q+2}dy+H^{r+2q+3},\\
\displaystyle 0=(-1)^{r+2p}\int E_1\bullet G^{r+2p-1}dy+H^{r+2p-2},\\
\displaystyle F^{r+2j}=\int E_1\wedge G^{r+2j-1}dy+(-1)^{r+2j+2}\int E_1\bullet G^{r+2j+1}dy+\\
\displaystyle\qquad\qquad+\int E_2\,G^{r+2j}dy+H^{r+2j},\hfill j=p,...,q,\\
\displaystyle 0=\int E_1\wedge G^{r+2q+1}dy+\int E_2\,G^{r+2q+2}+H^{r+2q+2},
\end{cases}$$ where $(H_1,H_2)$ is a hyper-conjugate harmonic pair with $$H_1:=\sum_{j=p}^q H^{r+2j+1}+e_{m+1}\sum_{j=p}^q H^{r+2j},$$ and $$H_2:=\left(H^{r+2p-1}+H^{r+2q+3}\right)+e_{m+1}\left(H^{r+2p-2}+H^{r+2q+2}\right).$$ For the case of $P$ being the zero function, one obtains that $\hat{H}_1$ becomes also the zero function, and both the necessity of (A) and the formula (\[sol A\*\]) is proved.
For the case of $P$ being identically zero one has $$\begin{aligned}
H_2:&=-\left(\int E_1\bullet G^{r+2p}dy+(-1)^{r+2p}\int E_2\,G^{r+2p-1}dy+\right.\\
&\quad\left.+\int E_1\wedge G^{r+2q+2}dy\right)-\\
&\quad-e_{m+1}\left((-1)^{r+2p}\int E_1\bullet G^{r+2p-1}dy+\int E_1\wedge G^{r+2q+1}dy+\right.\\
&\quad\left.+\int E_2\,G^{r+2q+2}dy\right),\end{aligned}$$ meaning the existence of the hyper-conjugate harmonic function to $$\begin{aligned}
&-\left(\int E_1\bullet G^{r+2p}dy+(-1)^{r+2p}\int E_2\,G^{r+2p-1}dy+\right.\\
&+\left.\int E_1\wedge G^{r+2q+2}dy\right)-\\
&-e_{m+1}\left((-1)^{r+2p}\int E_1\bullet G^{r+2p-1}dy+\int E_1\wedge G^{r+2q+1}dy+\right.\\
&\left.+\int E_2\,G^{r+2q+2}dy\right),\end{aligned}$$ which (whenever it exists) will be denote by $\tilde{H}_1$. Thus, the necessity of (B) is also proved together with formula (\[sol B\*\]).
Finally, for the “sufficiency part” one has to reverse the reasoning. If (A) is true the one can take $H_2$ as the zero function implying $H_1$ to be an arbitrary monogenic function; thus we arrive at (\[sol A\*\]). If (B) is true then one can take in (\[system sol ICS\]) the function $H_2$ to be $H_2=-P$; hence, one can write $H_1$ as $H_1=\hat{H}_1+\tilde{H}_1$ for any arbitrary monogenic $\hat{H}_1$; thus, one obtains (\[sol B\*\]), and the theorem is proved. $\hfill \square$
Our approach provides a rather new generalization and certain consolidation of the hyper-complex method (by means of techniques based on harmonic conjugates) described in [@RBMS; @RB; @BAPS; @BoPe; @CLSSS; @DP; @DK; @PSV1; @PSV2] concerning the solvability of the $\overline\partial$-problem in spaces of functions satisfying different systems in the framework of (complex) Clifford and Quaternionic analysis.
The inhomogenous Maxwell’s equations
====================================
To keep the exposition self-contained let us repeat key observations from [@McMi pag. 1613]. Let $E$ be a $r$–vector and $H$ a $(r+1)$–vector with $0\leq r\leq m+1$, defined in some open subset of $\mathbb{R}^{m+1}$. To these, we associate an $\mathbb{C}_{0,m+2}$–valued function $M$ by setting $$\label{EHfield}
M:=H-ie_{m+1}E=H+i(-1)^{r+1}Ee_{m+1}.$$
Observe that $$\partial_{x,\alpha}M=i(-1)^{r+1}\left(\partial_{x}^{+}E+\partial_{x}^{-}E-i\alpha H\right)e_{m+1}+\left(\partial_{x}^{+}H+\partial_{x}^{-}H+i\alpha E\right).$$
The function $M$ defined in (\[EHfield\]) is (left or right) $\alpha$-monogenic if and only if $E$ and $H$ satisfy the Maxwell’s equations $$\label{Max}
\begin{cases}
\displaystyle \partial_{x}^{+}E-i\alpha H=0,\\
\displaystyle \partial_{x}^{-}E=0,\\
\displaystyle \partial_{x}^{-}H+i\alpha E=0,\\
\displaystyle \partial_{x}^{+}H=0.
\end{cases}$$ Moreover, for $x\in\mathbb{R}^{m+1}\setminus\{0\}$ the function $M$ satisfies the radiation condition $$\left(1-ie_{m+1}\frac{x}{|x|}\right)M(x)=o\left(|x|^{-m/2}\right)\quad\text{as}\quad |x|\to\infty,$$ if and only if $E$ and $H$ satisfy the Silver-Müller-type radiation conditions $$\begin{aligned}
\frac{x}{|x|}\wedge E-H&=o\left(|x|^{-m/2}\right)\quad\text{as}\quad |x|\to\infty,\\
\frac{x}{|x|}\bullet H-E&=o\left(|x|^{-m/2}\right)\quad\text{as}\quad |x|\to\infty.\end{aligned}$$
If $\alpha$ is non-zero then, because $(\partial_{x}^{+})^2=(\partial_{x}^{-})^2=0$, the equations $\partial_{x}^{-}E=0$ and $\partial_{x}^{+}H=0$ become superfluous. Nonetheless, as for $\alpha=0$ Maxwell’s equations decouple (i.e. $E$ and $H$ become unrelated), it is precisely this case for which these two equations are relevant. Also, when $m+1=3$, $r=1$ (and $\alpha\neq0$), the formulae (\[Max\]) reduce to the more familiar system of equations $$\label{Max2}
\begin{cases}
\displaystyle \nabla\times E-i\alpha H=0,\\
\displaystyle \nabla\times H+i\alpha E=0.
\end{cases}$$
\[maintheoMax\] Let $\displaystyle G\in C\left(\Omega;\sum_{j=0}^{1} \bigoplus \mathbb{C}^{(r+2j)}_{0,m+2}+\sum_{j=0}^{1} \bigoplus \mathbb{C}^{(r+2j-1)}_{0,m+2}e_{m+1}\right)$. The inhomogeneous Maxwell’s equations
$$\label{InhMax}
\begin{cases}
\displaystyle \partial_{x}^{+}E-i\alpha H=G^{r+1},\\
\displaystyle \partial_{x}^{-}E=G^{r-1},\\
\displaystyle \partial_{x}^{-}H+i\alpha E=G^{r},\\
\displaystyle \partial_{x}^{+}H=G^{r+2},
\end{cases}$$
where $G^s\in C^1\left(\Omega;\mathbb{C}^{(s)}_{0,m+2}\right)\cap C^0\left(\Omega\cup \Gamma;\mathbb{C}^{(s)}_{0,m+2}\right)$ has a solution if and only if for the $\left(\mathbb{C}^{(r-1)}_{0,m+2} \bigoplus \mathbb{C}^{(r+3)}_{0,m+2}\right)\bigoplus e_{m+1}\left(\mathbb{C}^{(r-2)}_{0,m+2} \bigoplus \mathbb{C}^{(r+2)}_{0,m+2}\right)$–valued function $$\begin{aligned}
P:&=\left(\int E_1\bullet G^{r}dy+(-1)^{r}\int E_2\,G^{r-1}dy+\int E_1\wedge G^{r+2}dy\right)+\\
&+e_{m+1}\left((-1)^{r}\int E_1\bullet G^{r-1}dy+\int E_1\wedge G^{r+1}dy+\int E_2\,G^{r+2}dy\right),
\end{aligned}$$
- either is identically zero;
- or has a hyper-conjugate harmonic function.
If it is true, then the general solution of (\[InhMax\]) is given by:
- either $$\begin{aligned}
M&=H-ie_{m+1}E=H+i(-1)^{r+1}Ee_{m+1}\notag\\
&=\left(\int E_1\wedge G^{r}dy+\int E_1\bullet G^{r+2}dy+(-1)^{r+2}\int E_2\,G^{r+1}dy\right)+\notag\\
&+e_{m+1}\left(\int E_1\wedge G^{r-1}dy+(-1)^{r+2}\int E_1\bullet G^{r+1}dy+\int E_2\,G^{r}dy\right)+\notag\\
&+\hat{H}_1,\label{sol A*2}
\end{aligned}$$
- or $$\begin{aligned}
M&=H-ie_{m+1}E=H+i(-1)^{r+1}Ee_{m+1}\notag\\
&=\left(\int E_1\wedge G^{r}dy+\int E_1\bullet G^{r+2}dy+(-1)^{r+2}\int E_2\,G^{r+1}dy\right)+\notag\\
&+e_{m+1}\left(\int E_1\wedge G^{r-1}dy+(-1)^{r+2}\int E_1\bullet G^{r+1}dy+\int E_2\,G^{r}dy\right)+\notag\\
&+\hat{H}_1+\tilde{H}_1,\label{sol B*2}
\end{aligned}$$
where $\tilde{H}_1$ is a harmonic hyper-conjugate of $-P$, and $\hat{H}_1$ is an arbitrary $\alpha$-monogenic function.
[**Proof:**]{} It is a direct consequence of the Theorem \[maintheo\] by choosing $p=q=0$ and $F=M$. $\hfill \square$
Observe that, under the assumptions of Theorem 1, it is possible to obtain both the electric field and the magnetic field separately, say if $P$ is identically zero, then $$\begin{aligned}
H&=\int E_1\wedge G^{r}dy+\int E_1\bullet G^{r+2}dy+(-1)^{r+2}\int E_2\,G^{r+1}dy+\left[\hat{H}_1\right]_{r+1},\\
E&=i\left(\int E_1\wedge G^{r-1}dy+(-1)^{r+2}\int E_1\bullet G^{r+1}dy+\int E_2\,G^{r}dy+\left[\hat{H}_1\right]_{r}\right),\end{aligned}$$ while if $P$ has a hyper-conjugate harmonic function, then $$\begin{aligned}
H&=\int E_1\wedge G^{r}dy+\int E_1\bullet G^{r+2}dy+(-1)^{r+2}\int E_2\,G^{r+1}dy+\\
&\quad+\left[\hat{H}_1\right]_{r+1}+\left[\tilde{H}_1\right]_{r+1},\\
E&=i\left(\int E_1\wedge G^{r-1}dy+(-1)^{r+2}\int E_1\bullet G^{r+1}dy+\int E_2\,G^{r}dy+\right.\\
&\quad+\left.\left[\hat{H}_1\right]_{r}+\left[\tilde{H}_1\right]_{r}\right).\end{aligned}$$
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors wish to thank Instituto Politécnico Nacional and Universidad de las Américas Puebla, for partial financial support.
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---
abstract: 'Dark matter (DM) is currently searched for with a variety of detection strategies. Accelerator searches are particularly promising, but even if Weakly Interacting Massive Particles (WIMPs) are found at the Large Hadron Collider (LHC), it will be difficult to prove that they constitute the bulk of the DM in the Universe $ \Omega_{\rm \tiny{DM}} $. We show that a significantly better reconstruction of the DM properties can be obtained with a combined analysis of LHC and direct detection (DD) data, by making a simple Ansatz on the WIMP local density $ \rho_{\neut} $, i.e., by assuming that the local densiy scales with the cosmological relic abundance, $ (\rho_{\neut}/\rho_{\rm \tiny{DM}})=(\relic/\Omega_{\rm \tiny{DM}} $). We demonstrate this method in an explicit example in the context of a 24-parameter supersymmetric model, with a neutralino LSP in the stau co-annihilation region. Our results show that future ton-scale DD experiments will allow to break degeneracies in the SUSY parameter space and achieve a significantly better reconstruction of the neutralino composition and its relic density than with LHC data alone.'
author:
- 'G. Bertone$^{1,2}$ D.G. Cerdeño$^{3}$ M. Fornasa$^{4}$ R. Ruiz de Austri$^{5}$ and R. Trotta$^{6}$'
title: Identification of Dark Matter particles with LHC and direct detection data
---
Introduction {#sec:introduction}
============
Identifying the nature of the dark matter (DM) remains one of the central unsolved problems in modern particle physics and cosmology. A generic Weakly Interacting Massive Particle (WIMP) is among the best-motivated possibilities since it can be thermally produced in the early Universe in the right amount to account for the observed DM density. Indeed, many theories for Physics beyond the Standard Model contain viable WIMP candidates, as is the case of Supersymmetry (SUSY) when the lightest SUSY particle (LSP) is the lightest neutralino (a linear superposition of the supersymmetric partners of the gauge and Higgs bosons) [@Jungman:1995df; @Munoz:2003gx; @Bertone:2004pz; @book].
DM can be searched for in various ways. One possibility is attempting a [*direct*]{} detection, through its scattering off nuclei inside an underground detector. Many experiments have been running or are under construction which are mostly sensitive to the spin-independent part of the WIMP-nucleus cross section, $ \crosssection $. Among these, the DAMA/LIBRA collaboration reported a possible DM signal [@Bernabei:2008yi; @Bottino:2008mf]. However, its interpretation in terms of the elastic scattering of a WIMP with a mass around $ 10-100 \mbox{ GeV} $ and $ \crosssection \sim 10^{-3}-10^{-5}\, \mbox{pb} $ has been challenged by other experiments, such as the CoGeNT [@Aalseth:2008rx; @Aalseth:2010vx], CDMS [@Ahmed:2008eu; @Ahmed:2009zw], XENON [@Angle:2007uj] and ZEPLIN [@Lebedenko:2009xe]. The CoGeNT collaboration has itself recently reported an irreducible excess of low-energy events which could also be understood as due to the scattering of a very light WIMP [@Aalseth:2010vx] (see also Ref.[@Chang:2010yk]), but this intepretation has in turn be put under pressure by the XENON-100 results, obtained with a fiducial target mass of 40 kg and 11 days of exposure [@Aprile:2010um]. Finally, the recent results from the CDMS-II collaboration show two events compatible with a WIMP signal, although these results are still statistically inconclusive [@Ahmed:2009zw].
The future increase of the sensitivity may clarify the situation, but it is becoming clear that several independent pieces of evidence will be necessary to claim discovery of DM. In fact, even if in principle the WIMP mass and scattering cross section can be determined with some accuracy after its direct detection in one direct detection experiment, provided that the measured event rate is large and the WIMP mass is small [@Green:2007rb; @Green:2008rd], a second direct detection with a different target would actually allow a much more precise determination of the WIMP mass [@Drees:2008bv], and if the new target is sensitive to the spin-dependent contribution of the WIMP-nucleus cross section it could even be used to discriminate among WIMP candidates [@Bertone:2007xj].
Another possibility consists in looking for the products of DM annihilation (e.g., high energy neutrinos, gamma-rays or antimatter) and thus [*indirectly*]{} reveal the presence of the DM [@Bertone:2004pz; @book]. We leave the discussion of this search strategy to a forthcoming work, where we will present the constraints that can be set on the DM parameter space from the observation (or non-observation, see also [@Scott:2009jn]) of DM annihilation radiation [@Bertone:2010].
Finally, collider experiments, most notably the Large Hadron Collider (LHC), will explore the nature of Physics at the TeV scale, where many of the extensions of the SM that propose DM candidates would manifest themselves. The detection of new Physics in particle colliders can provide crucial information about DM. For example, the mass and spin of the LSP could be determined through the study of kinematic variables [@Cho:2007qv; @Cho:2008tj]. However, to prove that the newly discovered particles account for all (or most) of the DM in the Universe, is a challenging task. In fact, although particle accelerators can provide some information about the neutralino relic density [@Baltz:2006fm], it was found that in many cases the LHC would be unable to determine the precise composition of the neutralino, leading to an unreliable prediction of its relic abundance or to the occurrence of multiple solutions spanning several orders of magnitude, thus not allowing to establish whether or not it is the DM (see also Ref. [@Nath:2010zj] and references therein).
One possibility is to build a new collider, such as the proposed International Linear Collider (ILC), that would allow a much more precise evaluation of the supersymmetric masses and couplings, and a better determination of the inferred relic density, as argued by the authors of Ref.[@Baltz:2006fm]. However, this machine will not be available in the near future, and it is therefore crucial to devise strategies that can be implemented as soon as new particles are discovered at the LHC. Fortunately, direct detection experiments are expected to greatly improve their sensitivity in the next few years and start probing interesting regions of the supersymmetric parameter space. In case of discovery, it will certainly be reassuring if the mass reconstructed from direct detection experiments matched the value obtained from accelerator measurements, since it would prove the existence of a particle which is stable over cosmological timescales. The error on the mass reconstructed from direct detection experiments depends on the DM particle parameters, and on the experimental setup, and the interested reader can find a detailed analysis in Refs.[@Green:2007rb; @Green:2008rd]. But one can do much more than checking the compatibility of the two mass determinations. We show here that a combined analysis of the two data sets will allow a much better reconstruction of the DM properties, and a convincing identification of DM particles.
Although the strategy discussed here is model-independent, we work out an explicit example in the context of a 24-parameters supersymmetric model, with a neutralino LSP in the stau co-annihilation region.
Theoretical framework and LHC data
==================================
We work within the framework of the minimal supersymmetric extension of the Standard Model (MSSM), for which we adopt a low energy parametrization in terms of 24 parameters, corresponding to its CP-conserving version. The input parameters are the coefficients of the trilinear terms for the three generations, the mass terms for gauginos (for which no universality assumption is made), right-handed and left-handed squarks and leptons, the mass of the pseudoscalar Higgs, the Higgsino mass parameter $ \mu $, and finally the ratio between the vacuum expectation values of the two Higgs bosons $ \tan\beta $.
If searches for new Physics at the LHC are consistent with a SUSY scenario, the study of different kinematical variables will allow us to determine some properties of the SUSY spectrum. In particular, the masses of several particles or mass-splittings between them could be extracted, with a precision that obviously depends on the properties of the specific point of the parameter space. These measurements can then be used as constraints on the 24-dimensional SUSY model, in order to determine the regions of the MSSM parameter space which are consistent with such a measurement. This can be done by applying Bayes’ theorem $$p(\mathbf{x}|\mathbf{d})=
\frac{p(\mathbf{d}|\mathbf{x})p(\mathbf{x})}{p(\mathbf{d})},
\label{eqn:Bayesian_theorem}$$ which updates the so-called prior probability density $ p(\mathbf{x}) $, encapsulating the knowledge of the 24-dimensional space before taking into account the experimental constraints, $ \mathbf{d} $, into the posterior probability function (pdf) $ p(\mathbf{x}|\mathbf{d}) $. The latter describes the probability density assigned to a generic 24-dimensional point $ \mathbf{x} $ once the data have been taken into account via the likelihood function $ p(\mathbf{d}|\mathbf{x})$. Furthermore, on the RHS of Eq., $ p(\mathbf{d}) $ is the Bayesian evidence which, in our case, can be dropped since it simply plays the role of a normalization constant for the posterior in this context (see [@Trotta:2008qt] for further details).
The marginal pdf of a particular subset (as e.g. only one) of the 24 parameters defining $ \mathbf{x} $ can be obtained by integrating over the remaining directions: $$p(x^i|\mathbf{d})=\int_{[1,24]\backslash \{i\}}p(\mathbf{x}|\mathbf{d})
dx^1... dx^{i-1}dx^{i+1}...dx^{24}.
\label{eqn:1dim_pdf}$$
The posterior encodes both the information contained in the priors and in the experimental constraints, but, ideally, it should be largely independent of the choice of priors, so that the posterior inference is dominated by the data contained in the likelihood. If some residual dependence on the prior $ p(\mathcal{\mathbf{x}}) $ remains this should be considered as a sign that the experimental data employed are not constraining enough to override completely different plausible prior choices and therefore the resulting posterior should be interpreted with some care, as it might depend on the prior assumptions. The probability distribution for any observable that is a function of the 24 SUSY parameters $ f(\mathbf{x})$ can also be obtained since $ p(f|\mathbf{d})=\delta(f-f(\mathbf{x})) p(\mathbf{x}|\mathbf{d}) $.
For the practical implementation of the Bayesian analysis sketched above we employed the `SuperBayeS` code [@SuperBayeS], extending the publicly available version 1.35 to handle the 24 dimensions of our SUSY parameter space. To scan in an efficient way the SUSY parameter space we have upgraded the MultiNest [@Feroz:2008xx; @MN] algorithm included in SuperBayeS to the latest MultiNest release (v 2.7). MultiNest is a multi-modal implementation of the nested sampling algorithm, which is used to produce a list of samples in parameter space whose density is proportional to the posterior pdf of Eq. . For further information on nested sampling we refer the reader to the appendix of Ref.[@Trotta:2008bp] and references therein.
For the present work we have chosen a specific benchmark point in the MSSM parameter space, corresponding to the low-energy extrapolation of model LCC3 defined in Ref.[@Baltz:2006fm]. This benchmark is representative of SUSY models in the co-annihilation region, where the lightest neutralino is almost degenerate in mass with the lightest stau. In this region, co-annihilation effects reduce the neutralino relic abundance down to values compatible with the results from the WMAP satellite [@Komatsu:2008hk], and therefore, the mass difference between the neutralino and the lightest stau is a fundamental parameter for the reconstruction of the relic density. It has been shown [@Baltz:2006fm] that for this benchmark point LHC would be able to provide a measurement of the masses of a good part of the SUSY spectrum, including the two lightest neutralinos (see Ref. [@Khotilovich:2005gb] for an extension of this analysis to the case of the ILC). However the masses of some particles (most notably the two heaviest neutralinos and both charginos) would not be measured. The set of measurements that we use as constraints in our analysis corresponds to that in Table 6 of Ref.[@Baltz:2006fm] [^1], which assumes an integrated luminosity of 300 fb$^{-1}$. Furthermore, as pointed out in Ref.[@Arnowitt:2008bz], the neutralino-stau mass difference can be measured with an accuracy of 20% with 10 fb$^{-1}$ luminosity in models where the squark masses are much larger than those of the lightest chargino and second-lightest neutralino, as is our case. We therefore also include a measurement of the neutralino-stau mass difference in our likelihood. For convenience, we summarize in Table \[tab:constraints\] the set of LHC measurements on which we build our likelihood. Each of the constraints listed in Table \[tab:constraints\] is implemented in the likelihood as an independent Gaussian distributed measurement around the true value $\mu$ for that observable, with standard deviation $\sigma$, as given in Table \[tab:constraints\].
Mass Benchmark value, $\mu$ LHC error, $\sigma$
--------------------------------- ------------------------ ---------------------
$ m(\s\chi^0_1) $ 139.3 14.0
$ m(\s\chi^0_2) $ 269.4 41.0
$ m(\s e_1) $ 257.3 50.0
$ m(\s \mu_1) $ 257.2 50.0
$ m(h) $ 118.50 0.25
$ m(A) $ 432.4 1.5
$ m(\s \tau_1)- m(\s\chi^0_1) $ 16.4 2.0
$ m(\s u_R) $ 859.4 78.0
$ m(\s d_R) $ 882.5 78.0
$ m(\s s_R) $ 882.5 78.0
$ m(\s c_R) $ 859.4 78.0
$ m(\s u_L) $ 876.6 121.0
$ m(\s d_L) $ 884.6 121.0
$ m(\s s_L) $ 884.6 121.0
$ m(\s c_L) $ 876.6 121.0
$ m(\s b_1) $ 745.1 35.0
$ m(\s b_2) $ 800.7 74.0
$ m(\s t_1) $ 624.9 315.0
$ m(\s g) $ 894.6 171.0
$ m(\s e_2) $ 328.9 50.0
$ m(\s \mu_2) $ 328.8 50.0
: Sparticle spectrum (in GeV) for our benchmark SUSY point and relative estimated measurements errors at the LHC (standard deviation $\sigma$).[]{data-label="tab:constraints"}
Compared with previous Bayesian studies in which only precision tests of the Standard Model are considered as experimental constraints [@deAustri:2006pe; @Trotta:2008bp; @AbdusSalam:2009qd; @Cabrera:2009dm], we are assuming here a scenario in which LHC reports a quite stringent collection of measurements. For this reason our posterior constraints are quite tight and we expect the prior dependence of our results to be very mild. This is confirmed by the inspection of the profile likelihood, which agrees well with the posterior pdf (see [@Trotta:2008bp] for a detailed discussion). This indicates that volume effects from the prior are unlikely to be playing a major role given the strong constraints we assume for our benchmark point.
Future direct detection data
============================
In the simulation of a direct detection experiment we assume a future signal giving a WIMP detection, namely a certain number of events $ N $ and a corresponding set of recoil energies $ \{ E_i \}_{i=1,...,N} $. The total number $ N $ of simulated events is the sum of both background events (mainly interactions of detector nuclei with neutrons from surrounding rock, from residual contaminants or from spallation of cosmic muons) and recoils due to DM. For concreteness, we will exemplify the method in the case of an experiment akin to the 1-ton scale SuperCDMS experiment [@CDMS]. We simulated the differential number of background events as in Ref.[@Bernal:2008zk]. Since the capability of a simulated direct detection experiment to reconstruct the DM properties (see Refs. [@Green:2007rb; @Green:2008rd; @Bernal:2008zk] for more details) is worse in the case of a constant background distribution than for an exponential one, we only consider the case of energy-independent background recoil spectrum in order to be conservative. Therefore, we adopt a constant background differential spectrum $ (dN_{\rm back}/dE) = {\rm const}$ which is normalized so that, when binning the spectrum in 9 bins of 10 keV width (from $ E_{\rm th}=10 \mbox{ keV} $ to $ E_{\rm max}=100 \mbox{ keV}$) the number of background events in the first bin is the same as the number of DM signal events there.
The expected number of events $ \lambda $ for our benchmark model and for an exposure $ \epsilon=300 \mbox{ ton day} $ is obtained by integrating the sum of the differential rate of WIMP and background events $$\lambda=\epsilon \int_{E_{\rm th}}^{E_{\rm max}}
\frac{dR_{\chi}}{dE}+\frac{dR_{\rm back}}{dE} \, dE.
\label{eqn:number_of_events}$$
The dependency of the WIMP event rate on the physical quantities in the problem becomes apparent in the following parametrization [@Lewin:1995rx] $$\frac{dR_{\chi}}{dE} = c_1R_0 e^{-E/(E_0c_2)} F^2(E)\, ,$$ where $$\label{eq:R0}
R_0=\frac{\crosssection \rho_\chi A^2 c^2 (m_\chi+m_p)^2}
{\sqrt{\pi} m^3 m_p^2 v_0}\,,$$ and $$E_0=\frac{2m_\chi^2v_0^2 Am_p}{(m_\chi+Am_p)^2c^2}\,.$$ Here, $ \rho_\chi $ is the local WIMP density, $ A $ is the mass number of the target nuclei ($ A=73 $ in the case of Germanium), $ m_p $ is the proton mass, $ v_0 $ is the characteristic WIMP velocity and $ F^2(E) $ denotes the nuclear form factor. A discussion on the values of the parameters $ c_1 $ and $ c_2 $ and the functional form of $ F(E) $ can be found in Refs. [@Green:2007rb; @Green:2008rd; @Lewin:1995rx]. The specific values of the quantities for our case study are summarized in Table \[tab:DD\_parameters\].
Target A $ \epsilon $ $ E_{\rm th} $ $ E_{\rm max} $ $ \rho_\chi $ $ \lambda $
-------- ---- -------------------------- ---------------- ----------------- ---------------------- -------------
Ge 73 $ 300\, \mbox{ton day} $ 10 keV 100 keV 0.385 GeV cm$^{-3} $ 638
: Relevant quantities for a SuperCDMS-like direct detection experiment. The quantity $\lambda$ gives the expected number of WIMP recoils for our SUSY benchmark model.[]{data-label="tab:DD_parameters"}
In order to combine the result of a direct detection experiment with LHC data, we run an additional scan of the SUSY parameter space including in the likelihood function an additional Poisson-distributed term that compares the number of events and their spectral shape predicted in each point in parameter space with the recoil spectrum corresponding to the benchmark value of Table \[tab:DD\_parameters\]. The overall background rate and its spectral shape are assumed to be known.
As shown by Eqs.-, the number of detected events is proportional to the product of the WIMP-proton cross section and the local DM density $ \lambda \propto \crosssection \rho_\chi $. Therefore, unless one specifies the value of $ \rho_\chi $, any information on the number of events leaves the scattering cross section practically unconstrained.
We propose two different strategies to specify $ \rho_\chi $:
1. [*Consistency check:*]{} we impose that $$\rho_\chi=\rho_{\rm \tiny{DM}} \, ,$$ and we adopt for this quantity the value obtained in a recent paper by Catena and Ullio [@Catena:2009mf], through a careful analysis of dynamical observables in the Galaxy, namely $ \rho_\chi=0.385 \mbox{ GeV} \mbox{ cm}^{-3} $ (see also [@Strigari:2009zb; @Salucci:2010qr; @Weber:2009pt; @Pato:2010yq]). Although this assumption completely removes the degeneracy between $ \crosssection $ and $ \rho_\chi $, it forces the identification of neutralino with the DM particle, irrespectively of the value of its thermal relic density. This is therefore equivalent to assuming that, a non-standard cosmological history of the Universe can correct any excess or deficit in the thermal relic density and make it agree with the WMAP result, for example, either by invoking late injection of entropy, non-thermal production through late-decaying particles (such as a modulus or a gravitino [@Moroi:1999zb]), scenarios with a low-reheating temperature [@Giudice:2000ex] (see also Ref.[@Fornengo:2002db]) or a faster expansion rate [@Salati:2002md; @Profumo:2003hq]. For these reasons this Ansatz must be considered as a [*consistency check*]{} rather than a proof of the identification of DM particles.
2. [*Scaling Ansatz:*]{} we assume that the local density of the neutralino scales with the cosmological abundance. More precisely, we propose the following [*Ansatz*]{} $$\rho_{\neut}/\rho_{\rm \tiny{DM}}=\relic/\Omega_{\rm \tiny{DM}}.$$ This Ansatz is strictly valid in the reasonable case where the distribution of neutralinos in large structures, and in particular in the Galaxy, traces the cosmological distribution of the DM. This Ansatz is obviously true if neutralinos contribute all the DM in the Universe, but is also valid in the case where the neutralino is a subdominant component of DM, provided that DM behaves, as expected, as a cold collisionless particle. As shown below, this simple assumption is powerful tool to remove degeneracies in the parameter space.
The reconstruction of the neutralino relic density is shown in Fig.\[fig:relic\_density\]. The left panel corresponds to the case where [*only*]{} LHC constraints are considered. Consistently with previous analyses [@Baltz:2006fm], multiple peaks can be observed, as a consequence of degeneracies in the SUSY parameters space that the LHC constraints are unable to break. In particular, the two observed peaks correspond to neutralinos with different composition: mostly Wino and mostly Bino, from left to right. This is a consequence of the fact that the LHC is assumed to be able to measure only the two lightest neutralino states, but not the two more massive ones or the charginos. The true value of the relic density for our benchmark point ($\relic h^2 = 0.176$), represented by a diamond in Fig. \[fig:relic\_density\], is indeed inside the peak corresponding to mostly Bino dark matter. Although this value is about 60% larger than the relic abundance measured by the WMAP satellite [@Komatsu:2008hk], we expect our results to remain qualitatively correct for other points in the co-annihilation region leading to the correct cosmological relic abundance. As commented above and already pointed out in previous works [@Baltz:2006fm] the better reconstruction of sparticle masses at the ILC could allow a more precise determination of the neutralino relic abundance, potentially removing some of these degeneracies. However, this information would only be available after a much longer period of time.
The constraints from LHC only data are also shown in the left panel of Fig.\[fig:omega\_vs\_sigma\], in the plane $ \crosssection $ vs $ \relic h^2 $, where the true value of those quantities is given by ($\relic h^2 = 0.176$, $\crosssection = 7.1\times10^{-8}$ pb). The leftmost region corresponds to a neutralino which has a leading Wino component, thereby displaying a smaller relic abundance, whereas the region towards larger relic abundance corresponds to Bino-like neutralinos, for which the scattering cross section is also slightly smaller.
In the central and right panels of the two figures, we show the impact of adding information from direct detection experiments. These plots have been obtained by statistical posterior re-sampling of the LHC only scan, adding the relevant Ansätze and the likelihood function of a direct detection experiment as specified above. The central panels correspond to the assumption 1, or [*consistency check*]{}. This amounts to fixing the local neutralino density, and therefore we expect that only regions along a direction of constant $\crosssection$ to survive after direct detection data are implemented. This can be understood as follows: for a given number of measured events, and a fixed local density, there is only a range of values of $\crosssection$ that are compatible with the measurement. Notice that, as explained above, this Ansatz does not further constrain the neutralino thermal relic density. In this case the pdf for the neutralino relic density still displays the two maxima, corresponding to the two peaks in Fig.\[fig:relic\_density\] and the two “islands” in Fig.\[fig:omega\_vs\_sigma\]. This is due to the fact that the neutralino can have a similar scattering cross section for both compositions and therefore (if the fact that it might be a subdominant DM component is not properly taken into account) could account for the same detected rate.
The most interesting case is the one that corresponds to our assumption 2, namely the [*scaling Ansatz*]{}, which represents the most important result of this paper. When the appropriate scaling of the local density is applied, the Ansatz cuts the parameter space along a direction $$\crosssection \propto \relic^{-1},$$ due to the fact that for a fixed number of events $ \crosssection \propto \rho_\chi^{-1} $ and that under the scaling Ansatz $ \rho_\chi \propto \relic $. The dramatic consequences of this simple Ansatz are shown in the right-most panels of both figures. Models corresponding to a low relic density are essentially ruled out, because under the scaling Ansatz they correspond to a low local density. Given a number of observed events in direct detection searches, a low local density would require a larger scattering cross section, which is incompatible with LHC constraints. As a consequence, the parameter space region corresponding to a neutralino that is mostly Wino can now be ruled out with high confidence, thereby leading to a much better reconstruction of the DM composition than it would be possible under the consistency check Ansatz.
We note that if the reconstructed relic density matches the observational determination of $\Omega_{\rm \tiny{DM}}$, this procedure also validates the standard cosmological history, and constrains deviations from the standard expansion rate at the epoch of DM freeze-out. Conversely, a mismatch between the reconstructed relic density and $\Omega_{\rm \tiny{DM}}$ would point towards a multi-component DM sector, or a non-standard expansion rate (see e.g. Ref. [@Gelmini:2008sh] and references therein).
Discussion and conclusions
==========================
We have investigated the effect of combining information from accelerator searches with data from direct DM detection, assuming realistic measurements at the LHC and in a Germanium detector with an exposure of 300 ton days.
An interesting question is whether the systematic and statistical errors on this quantity and on other relevant physical quantities entering in the calculation of the event rate for direct detection experiments can spoil the reconstruction procedure presented here. For instance, we have assumed a Maxwellian distribution for the velocity dispersion of DM particles, but a more refined analysis should keep into account the uncertainties on this quantity. Fortunately, recent estimates based on numerical simulations, suggest that the small measured deviations from a Maxwell-Boltzmann distribution lead to errors of 10% or less on the recoil rate, and they are therefore subdominant with respect to other uncertainties, such as the error on the nuclear form factor [@Ellis:2008hf], and especially the error on the observed DM local density. The most important effect of such uncertainties, once marginalized over, would be to widen the pdf’s of Fig. \[fig:omega\_vs\_sigma\] in the vertical direction by less than 10%, if one considers only the statistical error on $\rho_{\rm \tiny{DM}}$ derived in Ref. [@Catena:2009mf], and by up to a factor of two if one also considers the systematic error due to halo triaxiality [@Pato:2010yq]. Since the vertical thickness of the contours in the 2D posterior of Fig.\[fig:omega\_vs\_sigma\] is approximately equal to a factor of 2, we expect that including these uncertainties would not modify qualitatively the marginal posterior distribution for $\relic h^2$, and our results would still apply. A more detailed discussion of these effects is beyond the scope of this paper, and we leave it to a separate upcoming work. However we explicitly studied the effect of varying the value for the mass of the top, including it as a nuisance parameter in the likelihood. The variation in the reconstructed pdf for the neutralino relic abundance and neutralino-nucleon scattering cross section is negligible.
We stress once more the importance of combining different types of experiments. The specific case discussed here shows that when reasonable assumptions are made to link the local density to the relic abundance, [*a combined analysis of data from accelerators and direct detection experiments allows a significantly better reconstruction of the DM properties*]{}.
This is true in the co-annihilation region discussed here, but it will provide important information for any SUSY scenario, and more in general for any new physics scenario. Even in cases where the LHC data are sufficient to pinpoint the underlying DM scenario, direct detection experiments can corroborate the results, and they can also be used to identify deviations from the standard expansion rate of the Universe at freeze-out that would appear as an inconsistency between the $\relic $ inferred from LHC data and cosmological measurements.
[*Acknowledgements.*]{} D.G.C. is supported by the Ramón y Cajal program of the Spanish MICINN, by the Spanish grants FPA2009-08958, HEPHACOS S2009/ESP-1473 and by the EU network PITN-GA-2009-237920. R.T. would like to thank the EU FP6 Marie Curie Research and Training Network “UniverseNet” (MRTN-CT-2006-035863) for partial support, the Instituto de Fisica Teorica (Madrid) and the Institut d’Astrophysique de Paris for hospitality. The work of R. RdA has been supported in part by MEC (Spain) under grant FPA2007-60323, by Generalitat Valenciana under grant PROMETEO/2008/069 and by the Spanish Consolider Ingenio-2010 program PAU (CSD2007-00060). We also thank the support of the spanish MICINN’s Consolider-Ingenio 2010 Programme under grant MultiDark CSD2009-00064.
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|
---
author:
- 'F. Ambrosino'
- 'A. Antonelli'
- 'M. Antonelli'
- 'F. Archilli'
- 'G. Bencivenni'
- 'C. Bini'
- 'C. Bloise'
- 'S. Bocchetta'
- 'F. Bossi'
- 'P. Branchini'
- 'G. Capon'
- 'T. Capussela'
- 'F. Ceradini'
- 'P. Ciambrone'
- 'A. De Angelis'
- 'E. De Lucia'
- 'M. De Maria A. De Santis'
- 'P. De Simone'
- 'G. De Zorzi'
- 'A. Denig'
- 'A. Di Domenico'
- 'C. Di Donato'
- 'B. Di Micco'
- 'M. Dreucci'
- 'G. Felici'
- 'S. Fiore'
- 'P. Franzini'
- 'C. Gatti'
- 'P. Gauzzi'
- 'S. Giovannella'
- 'E. Graziani'
- 'M. Jacewicz'
- 'V. Kulikov'
- 'J. Lee-Franzini'
- 'M. Martini'
- 'P. Massarotti'
- 'S. Meola'
- 'S. Miscetti'
- 'M. Moulson'
- 'S. Müller'
- 'F. Murtas'
- 'M. Napolitano'
- 'F. Nguyen'
- 'M. Palutan'
- 'A. Passeri'
- 'V. Patera'
- 'P. Santangelo'
- 'B. Sciascia'
- 'A. Sibidanov'
- 'T. Spadaro'
- 'C. Taccini'
- 'L. Tortora'
- 'P. Valente'
- 'G. Venanzoni'
- 'R. Versaci'
date: 'Received: date / Revised version: date'
subtitle: The KLOE Collaboration
title: Precision Measurement of Meson Lifetime with the KLOE detector
---
Introduction {#sec:intro}
============
We have collected very large samples, 109 events;, of slow $K$-mesons of well known momentum, with the KLOE detector at . Kaons originate from the decay of -mesons produced in collisions. We have used the above samples to measure many properties of kaons such as masses, branching ratios and lifetimes, refs. through . The ultimate motivation was the determination of the quark mixing parameter $V_{us}$, see ref. . KLOE had not however attempted to measure the lifetime. We present a precise measurement of the lifetime based on a sample of about 20 million decays corresponding to an integrated luminosity of 0.4 fb$^{-1}$.
The reaction chain , (unobserved), , with $p_\phi$ = 13 MeV in the horizontal plane, is geometrically and kinematically overdetermined. We can therefore, event by event, determine the -meson vector momentum [**pK**]{}, the kaon production point P and its decay point D. From $p$K, P and D we obtain the decay proper time of the . A fit to the proper time distribution gives the -meson lifetime. The vast available statistics allows us to select some 20 million decays with favorable configuration to provide the most accurate and least biased measurement of time. Averaging over the sample gives a statistical accuracy of 2 $\mu$m in the measurement of the kaon mean decay length. For consistency we use our value of the kaon mass, $M_K$=(497.583 0.021) MeV, ref. .
The KLOE detector has been described in all the references mentioned above, see also refs. , , , . In particular ref. summarizes the use of the KLOE detector in collecting kaon data and reconstructing all decay channels.
Data reduction {#sec:ana}
==============
Data were collected in 2004 with the KLOE detector at , the Frascati . is an collider operating at a center of mass energy $\sqrt s$1020 MeV, the -meson mass. Beams collide at an angle of $\pi$-0.025 rad. For each run of about 2 hours, we measure the CM energy $\sqrt s$, ${\bf p}_\phi$ and the average position of the beams interaction point P using Bhabha scattering events. Data are combined into 34 run periods each corresponding to an integrated luminosity of about 15 pb. For each run set, we generate a sample of Monte Carlo (MC) events of 3 equivalent statistics. We use a coordinate system with the $z$-axis along the bisector of the external angle of the beams, the so called beam axis, the $y$-axis pointing upwards and the $x$-axis toward the collider center.\
decays are reconstructed from two opposite sign tracks which must intersect at a point D with $r_{\rm D}\!<\,$10 cm and $|z_{\rm D}|\!<\,$20 cm, where $x\!=\!y\!=\!z\!=0$ is the average collision point. The invariant mass of the two tracks, assumed to be pions, must satisfy $|M_{\pi\pi}-M_{K^0}|\!<\!5$ MeV. D is taken as the decay point. The kaon momentum ${\bf p}_K$ can be obtained from the sum of the pion momenta and also from the kaon direction with respect to the known, fixed momentum ${\bf p}_\phi$. We call the latter value ${\bf p}^\prime_K$. The magnitude of the two values of the kaon momentum must agree to within 10 MeV. If the two tracks intersect in more than one point satisfying the above requirements, the one closest to the origin is retained as the decay point. We refer to the finding of D as vertexing.
The above procedure selects a sample almost 100% pure. For each event we need the kaon production point P. In fact only the $z$-coordinate of P is required since the interaction region is 2-3 cm long while the other dimensions are negligible and the $x,\ y$ coordinates well known. P lies on the beam axis and is taken as the point of closest approach to the path as determined by the tracks. The resolution in $z_{\,\rm P}$ is about 2 mm. Events with $|z_{\,\rm P}|\!>\,$2 cm are rejected. From the length of PD and $p^\prime_K$ we compute the proper time in units of a reference value of , the lifetime value used in our MC, =89.53 ps. Its distribution is shown in fig. \[fig:reso\] top, histogram a.
The distribution has an rms spread of 0.86 and is not symmetric. Time resolution can be improved discarding events with poor vertexing resolution. From MC we observe that bad vertex reconstruction is correlated with large values of ${\char1}p$ = $p_K-p^{\,\prime}_K$, the difference in magnitude of ${\bf p}_K$ and ${\bf p}^\prime_K$. Fig.\[fig:reso\] bottom shows the ${\char1}p$ distribution for data and MC. We therefore retain events with $\cos\alpha_{\pi\pi}<-0.87$, $0.5<|\alpha^\perp_{\pi^+K}|<2.2$ rad, $|M_{\pi\pi}-M_K|\!<\,$ 2 MeV and events with $-0.5<\cos\theta(\pi^\pm)<+0.5$. $\alpha_{\pi\pi}$ is the opening angle of the pion pair. The definition of $\alpha^\perp_{\pi^+K}$ is slightly more complicated. Information about the angle between the positive pion and the kaon at the decay point D is required. We must also distinguish between the two ’V’ configurations illustrated in fig. \[tplg\].
Calling [**r**]{} and [**s**]{} the projections of kaon and positive pion on the $\{x, y\}$ plane, $\alpha^\perp_{\pi^+K}$ is defined as $$\alpha^\perp_{\pi^+K}={\rm{sign}}\,\left(({\bf r}\times{\bf s})_z\right)\arccos\left(\frac{{\bf r}\cdot{\bf s}}{rs}\right).$$ The angle $\alpha^\perp_{\pi^+K}$ is defined in $\{-\pi,\ \pi\}$. Positive sign corresponds to the configuration of fig. \[tplg\], left. All angles are in the laboratory system.
After applying the cuts above, only 1/3 of the events survive while the rms time spread is reduced to 0.63 . Another significant improvement is obtained performing a geometrical fit of each event to obtain the production point P and the decay point D. We chose a new point P on the beam axis and a new decay point D on a line through P, parallel to the kaon path, so as to minimize the $\chi^2$ function $${|{\bf r}_{\rm D^{\,\prime}}-{\bf r}_{\rm D}|^2\over\sigma^2_{r_{\rm D}}} + {(z_{\rm P^{\,\prime}}-z_{\rm P})^2\over\sigma^2_z}.$$ The proper time distribution, after all cuts and the fit, is shown in fig. \[fig:reso\] top, curve b. The rms spread in $t$ is 0.32. We check the correctness of the direction using a sample of -mesons reaching the calorimeter, where they are detected by nuclear interactions. The interaction point in the calorimeter together with the known $\phi$ momentum gives the direction with good resolution. Comparison with the direction as obtained from pions shows a negligible difference. The final efficiency for detection is shown in fig. \[fig:eff\] as a function of proper time.
The average efficiency depends on the direction, is almost flat and in average is 9%. Errors in the reconstruction of the pion tracks can bias the position of P and D. In fact, the value of lifetime differs by 6% for events with $\alpha^\perp_{\pi^+K}>0$ and $<0$, where the sign distinguishes the topologies of the di-pion ‘V’. see fig. \[tplg\].
We do correct for this effect. From MC we obtain the correction, ${\char1}\ell_K$, to be applied to the decay length, as a function of ${\char1}p$. The correction is applied event by event to the data. The procedure is repeated for each run period. After applying this correction the 6% difference mentioned above is reduced to $10^{-3}$, although the average result is only 2$\sigma$ (0.1%) different from the result before applying it.
Proper time distribution fit {#sec:fit}
============================
MC and data, see fig. \[fig:reso\] top, studies show that the time resolution is well described by the sum of two Gaussians. We write the resolution function, normalized to unity, as $$\eqalign{&r(t,\tau,\sigma_1,\sigma_2,\alpha)=\cr
&\kern3mm{\alpha\over\sigma_1\sqrt{2\pi}}\,\exp \left(-{t^2\over2\,\sigma_1^2}\right)+ {1-\alpha\over\sigma_2\sqrt{2\pi}} \,\exp\left(-{t^2\over2\,\sigma_2^2}\right)\cr}$$ and the decay function, for a lifetime $\tau$, as: $$d(t)={1\over\tau}\x\exp\left(-{t\over\tau}\right)\x\theta(t).$$ The expected decay curve, normalized to unity, is given by the convolution $$g(t)=\int_{-\infty}^{\infty}d(\eta)\,r(t-\eta)\,\dif\eta.$$ Allowance must be still be made for small mistakes in the reconstruction of the decay and production position, D and P. A shift $\delta$ in the proper time is therefore introduced. Thus the function which we use for fitting the observed distribution is $$f(t,\tau,\sigma_1,\sigma_2,\alpha,\delta)=g(t-\delta).$$ The four parameters, $\sigma_1$, $\sigma_2$, $\alpha$, $\delta$ in $f(t)$ depend on colatitude and azimuth, $\theta$ and $\phi$, of the kaon and it is not realistic to attempt to obtain them from MC. We divide the data in a 2018 grid in $\cos\theta,\phi$ and fit each data set for the lifetime $\tau$ with the above parameters free. In order to improve the result stability, we retain only events with $|\cos\theta|<0.5$ and 0$\,<\!\phi\!<\,$360, discarding in this way only 8% of the events. We therefore perform 180 independent fits only to events in a 1018 grid. The fit range, 1 to 6.5 , is divided in 15 proper time bins. The kaon lifetime is obtained as the weighted average of the 180 $\tau_i$ values $$\tau(\ks)=\langle\tau\rangle=\sum_{i}{\tau_i\over\sigma^2(\tau_i)}\left/ \sum_{i}{1\over\sigma^2(\tau_i)}\right..$$ The corresponding $\chi^2$ value is $\chi^2 = \sum_i(\tau_i-\langle\tau\rangle)^2/\sigma^2(\tau_i)$. We find $\chi^2/\rm{dof}$ = 202/179 for a confidence level, CL, of 11.4%. The normalized residuals of the 180 fit values $\tau_i$ have an rms spread of 1.1. Tab. \[tab:correl\] gives the average correlations between fit parameters and fig. \[fig:fit\] top shows a fit example.
$\sigma_1$ $\sigma_2$ $\alpha$ $\delta$
------------ ------------ ------------ ---------- ----------
$\tau_S$ 0.18 0.09 0.11 0.62
$\sigma_1$ 0.50 0.75 0.28
$\sigma_2$ 0.69 0.11
$\alpha$ 0.16
: Correlation of fit parameters (averaged values).[]{data-label="tab:correl"}
The resolution () versus $\{\theta,\phi\}$ isshown in fig. \[fig:fit\] bottom. The resolution varies from 0.22 to 0.27 over the accepted $\{cos\theta,\phi\}$ range with an average of 0.24 . The $\delta_i$ values show a dependence on $\phi$ with period $2\pi$ corresponding to a shift of the position of P of $-10\,\mu$m in $y$ and 50$\,\mu$m in $x$. In addition, a very small, $10^{-4}$, eccentricity of the drift chamber is evident. All these effects are consistent with mechanical and surveying inaccuracies. To ensure that the lifetime evaluation is correct to the 10 level, we correct the value of $\langle\tau\rangle$ obtained above by the factor $\tt/\tau_{\rm fit}^{\rm MC}$=1.000360.00019, where $\tau_{\rm fit}^{\rm MC}$ is the result of fitting the MC data with the procedure described above.
Systematics and result {#sec:syst}
======================
Changes in analysis cuts and FV corresponding to a60% change in efficiency result in a lifetime shift of 0.024ps. Varying the fit range gives a shift of 0.012ps. As mentioned in Sec.\[sec:ana\], we use the KLOE value of the kaon mass in the kinematic determination of the momentum and the calculation of $\beta_K$. The measurement of $\beta_K$ and the decay position are independent. The uncertainty on the calibration of $p^\prime_K$ gives an uncertainty of 0.033ps. The uncertainty due to mass is 0.004ps. All fits are then performed assuming uniform efficiency versus proper time, resulting in an uncertainty of 0.005ps. Table \[tab:syst\] summarizes all systematic errors.
-------- -----------------------------------
source absolute value ([ps]{})\
cuts & FV & 0.024\
fit range & 0.012\
$p^\prime_K$ calibration & 0.033\
kaon mass & 0.004\
efficiency & 0.005\
total & 0.043\
-------- -----------------------------------
: Systematic error contributions.[]{data-label="tab:syst"}
The result is stable across the entire data taking period. As said before, without applying the vertex correction the result still remains within $2\,\sigma$ of the final result, but stability with the run period is lost. Our result for lifetime is: $$\tau(\ks) = 89.562 \pm 0.029_{\rm{stat}} \pm 0.043_{\rm{syst}}\ \, {\rm{ps}.}
\label{eq:life}$$ Subdividing the data in 9 $\phi$ intervals and summing over $\cos\theta$ the $\phi$ dependence of the lifetime becomes quite obvious. The average of the 9 $\tau(\ks)$ values are of course exactly as eq. \[eq:life\] but $\chi^2$/dof=24/8 for a CL of 0.2%. Enlarging the statistical error by a factor $\sqrt{24/8}$ restores $\chi^2$=8 (CL=43%) and corresponds to $\tau(\ks)$=89.5620.050, an error very close to $\sqrt{0.029^2+0.043^2}$=0.052 confirming our estimate of the systematic error in eq. \[eq:life\].
The result of eq. \[eq:life\] is in agreement with recent measurements, ref. , as shown in fig.\[fig:market\].
wave;6.;
Including the present measurement, the new world average for the lifetime is $\tau_S$=89.5670.039 ps, with $\chi^2/\rm{dof}$ = 0.5/3, or CL92%.
In KLOE we can measure the lifetime for kaons traveling in different directions. We choose three orthogonal directions, the first being $\{\ell_1,\ b_1\}$ = {264, 48} in galactic coordinates. This is the direction of the dipole anisotropy of the cosmic microwave background (CMB), ref. . The other two directions are taken as $\{\ell_2,\ b_2\}$ = {174, 0} and $\{\ell_3,\ b_3\}$ = {264, -42}. After transforming the kaon momentum to the above systems, we retain only events with [**p**]{}K inside a cone of 30 opening angle, parallel (+) and antiparallel () to the chosen directions and evaluate the kaon lifetime. The 6 results are consistent with eq.\[eq:life\]. Defining the asymmetry ${\cal A}=(\tau^+_S - \tau^-_S)/(\tau^+_S +\tau^-_S)$, we obtain the results of tab.\[tab:cmbresult\].
$\{\ell,\ b\}$ ${\cal A}\times10^3$
---------------- ----------------------
{264, 48} 0.2 1.0
{174, 0} 0.21.0
{264,-42} 0.00.9
: Observed asymmetry. Errors are dominated by statistics.[]{data-label="tab:cmbresult"}
Systematic errors are strongly reduced when evaluating the asymmetry. Results in tab.\[tab:cmbresult\] show all the asymmetries values are well consistent with zero.
A further check has been performed using all KLOE data sample (about 2 fb$^{-1}$). The result for the asymmetry in the direction of CMB anisotropy, consistent with that given in tab.\[tab:cmbresult\], is $(\minus 0.13 \plm 0.40_{\rm{stat}})\times 10^{-3}$. No estimate of systematic error has been performed.\
. We thank the team for their efforts in maintaining low background running conditions and their collaboration during all data-taking. We want to thank our technical staff: G.F. Fortugno and F. Sborzacchi for their dedication in ensuring efficient operation of the KLOE computing facilities; M. Anelli for his continuous attention to the gas system and detector safety; A. Balla, M. Gatta, G. Corradi and G. Papalino for electronics maintenance; M. Santoni, G. Paoluzzi and R. Rosellini for general detector support; C. Piscitelli for his help during major maintenance periods. This work was supported in part by EURODAPHNE, contract FMRX-CT98-0169; by the German Federal Ministry of Education and Research (BMBF) contract 06-KA-957; by the German Research Foundation (DFG),‘Emmy Noether Programme’, contracts DE839/1-4.
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|
---
abstract: 'We present an XML schema for marking up gauge configurations called QCDml. We discuss the general principles and include a tutorial for how to use the schema.'
address:
- ' School of Physics, University of Edinburgh, Edinburgh EH9 3JZ, UK'
- ' John von Neumann Institute NIC / DESY Zeuthen, D-15738 Zeuthen, Germany'
author:
- 'C.M. Maynard, D. Pleiter,'
title: ' QCDml: First milestone for building an International Lattice Data Grid '
---
REPORT FROM THE METADATA WORKING GROUP
======================================
To achieve ILDG’s aim of sharing gauge field configurations world-wide a standardised description of configurations is mandatory. XML (EXtensible Markup Language) is the language of choice for metadata since it is designed to describe data. These metadata documents will be both human readable, since XML is verbose, and easy to parse by computers. Finally, standards on the structure and contents of XML documents can be enforced by using XML schemata.[^1]
The ILDG metadata working group [@ildg-mdwg] addressed in recent years the task of defining an XML schema. During the 2003 lattice conference [@ildg-lat03] the group presented an initial proposal. Since then the strategy for marking-up the physics parameters has been revised. However, whilst the contents remained unchanged, the usability has been significantly improved. The working group presented at this conference the first working version of the schema, QCDml.
Many lattice practitioners, who are typically not familiar using XML yet, might ask whether the proposed strategy is too complicated. However, using XML is much easier than many might expect. A large number of software tools exists for creating and parsing XML documents. When looking at the proposed schema, it should be realised that it’s complexity originates from the large variety of different simulations being carried out within the lattice community. Metadata documents will only contain information on one particular simulation. All metadata documents will have to conform to the schema. It is the schema which contains the complexity which allows the many different actions being used for simulating QCD with dynamical fermions.
During the design process three general requirements have been taken into account. Firstly, the schema has to be *extensible* as parameters of future simulations cannot be anticipated. This has to be done in such a way that any metadata document which conforms to the current schema will also conform to any future extended schema. The long-term validity of all metadata documents published by users of ILDG is a definite design goal of the schema. Secondly, the mark-up of simulation parameters has to be *unique* to avoid, e.g., the same action being described in two different ways. This would otherwise spoil the possibility to search for certain configurations. Finally, the schema has been kept *general* enough to allow the description of data other than gauge configurations (propagators, correlators, etc.) in the future.
Overview on the xml schemata
----------------------------
Gauge configurations are generated by a Markov chain. All configurations from one chain share many properties. Therefore the metadata can be split into two documents. The *ensemble XML* document contains all parameters which remain unchanged for the whole Markov chain. Other parameters are specific to one or a set of consecutive Markov steps and will be stored in a *configuration XML* document. A Universal Resource Indicator (URI) is used to link these two documents as well as the Logical File Name (LFN) to link the configuration XML document and the gauge configuration itself (see Fig. \[fig:lfn\]). For both types of XML documents corresponding schemata have been developed which can be downloaded from the working group’s web-site [@ildg-mdwg].
An example for parameters which will be the same for all configurations of an ensemble are the physics parameters. The corresponding parts of the ensemble XML document consists of information about the lattice size and a mark-up of the action. The description of the action is most critical for preserving uniqueness and extensibility of the schema. The metadata working group adopted the following strategy which is visualised in Fig. \[fig:hierarchy\]:
- Each action can be split into a gauge and a fermion action.
- The ensemble XML schema contains an element `<generalGluonAction>` and an optional element `<generalQuarkAction>` which will substituted by the actually used action.
- Actions which contain a structure which is the same as for a simpler action are ordered by an inheritance tree. For example, the clover fermion action is equivalent to the standard Wilson fermion action plus an improvement term.
- Actions which have the same structure in common are grouped. For instance, the Iwasaki and the Symanzik improved gauge actions only differ by the choice of the couplings.
This inheritance tree of possible actions is obviously extensible. Any action will be included into the schema only once to ensure uniqueness.
The description of each action is organised in three parts (See Fig. \[fig:action\]). Firstly, an array of `<couplings>` allows to store the names and values of all couplings and, in case of the fermion action, the number of flavours. Secondly, a description of the fields is required to store information, e.g., about the used normalisation or boundary conditions. Finally, any further information can be stored in a *glossary*.
The element `<glossary>` contains a URL to a document provided by the contributors. This document does not have to conform to any schema, it may even be not an XML but rather a human readable document, e.g. a TeX file. This gives the contributors the freedom to store all kind of information with regard to the used action, for instance information on the particular choice of couplings. Nevertheless, some guidelines will be needed to ensure that these documents contain all relevant information in a comprehensive form.
The variety of algorithms being used in lattice simulations is even larger than the number of different actions. The parameters of the algorithms are therefore essentially unconstrained. It should be noted that as a consequence such parameters are in practise not searchable. The only constrained element `<exact>` provides information on whether the algorithm being used is exact or not.
It will be mandatory to provide a reference to a publication on the used algorithm and an URL to a glossary document. Furthermore, all submitters are strongly encouraged to provide a full list of all algorithmic parameters used in their simulations. The names of the parameters should be chosen in such a way that they can be uniquely related with the algorithmic parameters described in the publication and the glossary file. Unlike the physics parameters the algorithmic parameters might change when generating a Markov chain. For instance, the step size of the HMC algorithm might be adjusted during a run. While the ensemble XML document will contain most of the information on the used algorithm, the submitter can store those parameters which might change within an ensemble into the configuration XML document.
As an matter of good scientific research practise, the generation of each configuration should be fully and comprehensively documented. Therefore submitters will have to provide information which machine and what code has been used to generate a particular configuration. Each machine can be identified by machine (or partition) name, the hosting institution and the machine type. Additional information can be stored as an optional comment. Concerning the simulation program submitters have to ensure that it can be identified by a name, a version string (e.g. a CVS tag), and the date of compilation. Again an optional comment allows to add further information, e.g. on compile time variables. All these parameters are not constrained and therefore not searchable. Only the information on the precision used to generate configurations will be searchable, as users might care about the used machine precision, in particular when quark masses become light.
The metadata will also include information about who submitted a configuration to ILDG within which project. This information can be stored in the management section which is foreseen in both the ensemble and the configuration XML document. Within this section also information will be stored which allows the user of a configuration to check the integrity of the downloaded data. To do so he can verify the checksum for the binary files, which will however not be preserved when transforming the gauge configuration into a different format. The user can still perform another test by recalculating the plaquette value and comparing this with the value stored in the configuration XML document. It should however be noticed that this test is less strong as both values will only agree within rounding errors and because the plaquette value is preserved by various transformations of a gauge field configuration.
All operations affecting an ensemble or just a particular configuration should be documented. Possible actions include the insertion and modification of an ensemble and the insertion, replacement or even the revocation of a configuration. The last two actions might be necessary if for example the computer or the code which was used to generate a configuration turned out to be broken. It should be noted that the submitters of configurations might not have to generate this information themselves, as the user interfaces to be developed for performing such actions could take care of patching the ensemble and configuration XML documents accordingly.
QCDML TUTORIAL {#sec:tutorial}
==============
The purpose of this section is to demonstrate how to mark up configurations according to the XML schema QCDml. We start with some Frequently Asked Questions (FAQ) about XML schema.
XML Schema FAQ
--------------
- What is XML Schema?
- XML schema is a collection of rules for XML documents
- An XML schema is itself an XML instance Document (ID)
- Why do we need an XML schema?
- So that computers can read and understand XML IDs
- e.g. `<length>16</length>`
- The meaning of length is context dependent, the schema makes this information explicit
- Do users need to learn XML schema?
- No. XML schema makes it easier to write XML IDs
Getting started
---------------
QCDml1.1 is available for use and can be downloaded along with documentation and example XML IDs from the ILDG website [@ILDG] by following the links in the metadata section. In QCDml1.1 the metadata is split into two parts. Metadata which is common to all configurations in an ensemble lives in the namespace of the ensemble, and only one XML ID for the whole ensemble is required.
An XML namespace is defined by W3C [@w3C:XML] consortium as [*a collection of names identified with a URI reference*]{}. Metadata which is specific to each configuration lives in a separate namespace and an XML ID is required for each configuration. Below is an XML chunk, it is the start of an example QCDml ID.
<?xml version="1.0" encoding="UTF-8"?>
<markovChain xmlns="http://www.lqcd.org/
#ildg/QCDml/ensemble1.1"
xmlns:xsi="http://www.w3.org/2001/
#XMLSchema-instance"
xsi:schemaLocation="http://www.lqcd.org/
#ildg/QCDml/ensemble1.1
#www.ph.ed.ac.uk/ukqcd/community/
#the_grid/QCDml1.1/
#QCDml1.1Ensemble.xsd">
<markovChainURI>
www.lqcd.org/ildg/ukqcd/ukqcd1
</markovChainURI>
+<management/>
+<physics/>
+<algorithm/>
</markovChain>
The “+” symbol is used to show that there is substructure below the element, and the `#` symbol is used to indicate line continuation. The element `<markovChain/>` is the root of the XML ID. The rest of the first line is the URI which identifies the namespace of the ensemble metadata. This has no prefix to identify elements which belong to this namespace as it is the default namespace. The second line is the namespace of XML schema itself. The third and fourth lines give the location of the file which contains the schema. The attribute `xsi:schemaLocation` is used to link the URI which identifies the namespaces with a URL which is the file which contains the schema. This could be a URL which is the URI of the namespace but it doesn’t have to be.
The element `<markovChainURI/>` which follows `<markovChain/>` is the URI which identifies this ensemble. Each configuration XML ID which belongs to this ensemble is linked to it using this URI.
If an XML ID conforms to the rules of a particular schema it is said to be [*valid*]{}. A software application which verifies that an XML ID is valid is unsurprisingly called a validator. Schema aware applications can then read and use valid XML IDs. One can write XML IDs in an editor such as `vi` or `emacs`, however, other tools are available. XMLspy is commercial software which can be used for schema and XML ID manipulation, it can, for instance, generate an XML ID from the schema. There are many other XML manipulation tools, links can be found at [@W3C:Schema].
Physics and Actions
-------------------
The element `<physics/>` contains two elements, `<size/>` and `<action/>`. The former is rather self explanatory and contains the size of the system.
Most searches of metadata will be on the action, consequently a lot of thought has gone into marking up the actions. Some of the object oriented features of XML schema have been employed in the schema to categorise actions, such as inheritance and the substitution group. This enables the XML IDs to be relatively simple. The general structure is shown in figure \[fig:action\]. The action has been split into two parts, gluon and quark. These [*general*]{} elements encapsulate the general properties of the actions, such as the fields and the glossary document. The glossary contains information such as the mathematical definitions of the actions and a reference to a paper where the action is discussed. However, this type of information is not suitable to being marked up in XML, it is essentially unconstrained and as such is not really searchable by a computer.
Specific quark and gluons inherit their properties from the general actions. These actions, such as `<wilsonQuarkAction/>` have specific couplings, in this case `<kappa/>`. The `<cloverQuarkAction/>` is an extension of this action, as it is a Wilson action, but has an extra coupling, `<cSW/>`. This is shown in figure \[fig:npClover\]. An inheritance tree for various actions can be built up in this way.
The metadata working group (MDWG) has not set up inheritance trees for all possible actions, but the schema is extensible so that further actions can be added without existing XML IDs having to be modified. Actions that have been added to QCDml are shown at the ILDG metadata web pages, and an example of which is shown in figure \[fig:actionsInheritance\].
For the gauge actions the metadata working group adopted a particular convention for `<sixLinkGluonActions/>` $$S_g^{\rm 6 link} = \beta\times\left( c_0 {\mathcal P} + c_1 {\mathcal R}
+ c_2 {\mathcal C} + c_3{\mathcal X}\right)$$ Where $\mathcal{P}$ is the Plaquette Wilson loop, $\mathcal{R}$ the six-link rectangle, $\mathcal{C}$ the six-link chair and $\mathcal X$ the three dimensional Wilson loop. The values of some of the couplings can be restricted to certain ranges or specific values. For example, in the Iwasaki RG action, the couplings are constrained, $c_2=c_3=0$, $c_0=(1-8c_1)$ and $c_1=-0.331$.
The quark action coupling has an integer valued element `<numberOfFlavours/>`. This labels how many flavours have these couplings, [*i.e.*]{} how many degenerate flavours. The element `<couplings/>` is array valued, that is this part of the action can be repeated but with different couplings. This is useful for marking up non-degenerate quark flavours.
An XML chunk for the $n_f=2$ non-perturbative clover action is shown below.
<npCloverQuarkAction>
<glossary>
www.lqcd.org/ildg/
#npCloverQuarkAction.xml
</glossary>
+<quarkField/>
<couplings>
<numberOfFlavours>
2
</numberOfFlavours>
<kappa>0.1350</kappa>
<cSW>2.0171</cSW>
</couplings>
</npCloverQuarkAction>
This is quite a short XML chunk, as the hierarchy npCloverQuarkAction $\rightarrow$ CloverQuarkAction $\rightarrow$ WilsonQuarkAction $\rightarrow$ GeneralQuarkAction is contained in the schema.
A rather technical point is that in the XPath 1.0 [@W3C:XPath] specification, there is no support for substitution groups which means that a search for WilsonQuarkAction elements would not return any cloverQuarkAction elements, although this can be achieved with a boolean “or” such as ` [/action/quark/npCloverQuarkAction | /action/quark/WilsonQuarkAction]`. However, the specification for XPath 2.0 is nearing completion [@W3C:XPath2], and this issue is beginning to be addressed.
An XML chunk for the $n_f=2+1$ AsqTad Kogut-Susskind quark action is shown below.
<asqTadQuarkAction>
<glossary>
www.lqcd.org/lqcd/
#asqTadQuarkAction.xml
</glossary>
+<quarkField/>
<couplings>
<numberOfFlavours>
2
</numberOfFlavours>
<mass>0.02</mass>
<cNaik>-0.05713116</cNaik>
<c1Link>0.625</c1Link>
<c3Link>-0.08569673</c3Link>
<c5LinkChair>
0.02937572
</c5LinkChair>
<c7LinkTwist>
-0.006713076
</c7LinkTwist>
<cLepage>-0.1175029</cLepage>
</couplings>
<couplings>
<numberOfFlavours>
1
</numberOfFlavours>
<mass>0.05</mass>
<cNaik>-0.05713116</cNaik>
<c1Link>0.625</c1Link>
<c3Link>-0.08569673</c3Link>
<c5LinkChair>
0.02937572
</c5LinkChair>
<c7LinkTwist>
-0.006713076
</c7LinkTwist>
<cLepage>-0.1175029</cLepage>
</couplings>
</asqTadQuarkAction>
The structure is the same, and all the couplings are clearly shown. The non-degenerate quark masses result in a second `<couplings/>` element, but with different number of flavours and different mass. It is easy to distinguish between $n_f=2+1$ and $n_f=3$.
Management
----------
This metadata gives the status of the data that is registered with the ILDG. In that sense it is created when the data is made public. In principal this would be generated or “stamped” by some ILDG middleware. As this application does not yet exist, it will have to be generated “by hand”. Below is an example of the management chunk of XML.
<management>
<revisions>1</revisions>
<collaboration>UKQCD</collaboration>
<projectName>Clover NF=2</projectName>
<archiveHistory>
<elem>
<revision>1</revision>
<revisionAction>
add
</revisionAction>
<numberConfigs>
829
</numberConfigs>
<participant>
<name>Chris Maynard</name>
<institution>
University of Edinburgh
</institution>
</participant>
<date>
2004-04-04T16:20:10Z
</date>
<comment>
This is the time of addition
</comment>
</elem>
</archiveHistory>
</management>
The `<archiveHistory/>` element can have several revisions. `<revision/>` is array valued. An ensemble could have configurations added to it, replaced or even removed, if a mistake has been found. So the allowed values of `<revisionAction>` are an enumeration of `{add,remove,replace}`. To discover how many configurations are in an ensemble, it is relatively easy to construct an XPath query to find the number of revisions and then the number of configurations for each revision.
Algorithm
---------
Algorithmic metadata is split between the ensemble and configuration documents, as it is possible, for instance, to have different stopping requirements for the inverter across the ensemble. The algorithmic metadata is in the form of unconstrained `<name/> <value/>` pairs. For example
<algorithm>
<name>GHMC</name>
<glossary>
www.ph.ed.ac.uk/ukqcd/
#community/GHMC.xml
</glossary>
<reference>
Phys.Rev.D65:054502,2002
</reference>
<exact>true</exact>
<parameters>
<name>stepSize</name>
<value>0.00625</value>
</parameters>
</algorithm>
It would be very difficult to create a hierarchical structure for algorithms, and especially difficult to make such hierarchy extensible. Again there is a glossary document which contains the free text, or mathematical definition of the algorithm, and a reference to a paper which describes the algorithm. There is also the boolean valued element `<exact/>` which denotes whether or not the algorithm is exact.
Configuration XML
-----------------
The configuration XML follows along similar lines. However, it is much shorter and so in principle could be directly output from the code that produced the configuration. Below is an example configuration XML ID. Again we start with a set of namespace declarations, which whilst the default namespace for configuration is separate from that of the ensemble, it still follows the same pattern.
The management section is very similar to that of the ensemble, however, there is an important addition: there is a “zeroth” revision which is [*generate*]{}. There is important metadata of when the gauge configuration was generated, and not just when it is submitted to the ILDG catalogue. As noted above ILDG middleware will eventually create the management part of the metadata when it is added to the ILDG catalogue, but this has yet to be written. The second important difference between the ensemble and configuration metadata is the `<crcCheckSum/>` which can be used to verify the data has been copied correctly.
<?xml version="1.0" encoding="UTF-8"?>
<gaugeConfiguration
xmlns="http://www.lqcd.org/ildg/QCDml/
#config1.1"
xmlns:xsi="http://www.w3.org/2001/
#XMLSchema-instance"
xsi:schemaLocation="http://
#www.lqcd.org/ildg/QCDml/config1.1
www.ph.ed.ac.uk/ukqcd/community/
#the_grid/QCDml1.1/QCDml1.1Config.xsd">
<management>
<revisions>1</revisions>
<crcCheckSum>
2632843688
</crcCheckSum>
<archiveHistory>
<elem>
<revision>0</revision>
<revisionAction>
generate
</revisionAction>
<participant>
<name>Chris Maynard</name>
<institution>
Edinburgh
</institution>
</participant>
<date>
1998-04-24T10:25:52Z
</date>
</elem>
<elem>
<revision>1</revision>
<revisionAction>
add
</revisionAction>
<participant>
<name>Chris Maynard</name>
<institution>
University of Edinburgh
</institution>
</participant>
<date>
2002-04-24T10:25:52Z
</date>
</elem>
</archiveHistory>
</management>
<implementation>
<machine>
<name>T3E-900</name>
<institution>
epcc Edinburgh
</institution>
<machineType>
Alpha processor
</machineType>
</machine>
<code>
<name>
UKQCD FORTRAN
</name>
<version>16.8.3.1</version>
<date>
1997-04-04T16:20:10Z
</date>
</code>
</implementation>
<algorithm>
<parameters>
<name>targetResidue</name>
<value>1e-07</value>
</parameters>
</algorithm>
<precision>single</precision>
<markovStep>
<markovChainURI>www.lqcd.org/
#ildg/ukqcd/ukqcd1</markovChainURI>
<series>1</series>
<update>010170</update>
<avePlaquette>
0.53380336E+00
</avePlaquette>
<dataLFN>
D52C202K3500U010170
</dataLFN>
</markovStep>
</gaugeConfiguration>
The next element is `<implementation/>` which holds information such as code versions, and machine version. Both of these entries are really only important for bug tracking, but if ever a bug is found then they are vital for tracking down the effected configurations. This metadata section is best written by the code that generated the configuration, as it is quite easy for this metadata to become lost.
The `<algorithm/>` element is the same as that of the ensemble, e.g. a name value pair for each algorithmic parameter that is specific to that configuration. The `<precision/>` element is also algorithmic in nature. it is the precision in which the configuration was computed, not in which the data is stored. It is an enumeration of `{single,double,mixed}`, it is possible to have some parts of gauge configuration generation code in single precision and some in double.
The final segment `markovStep` is the most immediately useful. `<markovChainURI/>` is the URI of the Markov Chain to which this configuration belongs. This links the ensemble and the configuration XML IDs together. `<series/>` and `<update/>` locate the configuration in the Markov Chain. The average Plaquette is useful for checking that downloads, copies or data reads have all worked correctly, not least as this metadata is data format independent. Finally `<dataLFN/>` is the logical filename of the data on the grid. This links the metadata to the data. In QCDgrid (UKQCD’s data grid) the data submission tool reads this element from the metadata and then uses this as the logical file name.
This tutorial hopefully gives a flavour of how to mark up gauge configurations in QCDml1.1. The ILDG website contains more detailed documentation on the schema along with example XML IDs. The website will be updated regularly as changes and extensions occur, but this should still serve as a guide.
FUTURE PROGRESS
===============
The MDWG along with the middleware working group is actively considering the issue of data and file formats, but this is discussed elsewhere. Completing the hierarchy tree for all commonly used actions is another task to be finished. Gauge configurations are not the only data that could be shared by ILDG members, for instance quark propagators and hadron correlator. The MDWG is considering how to extend QCDml to such data.
[1]{} `http://www.lqcd.org/ildg/tiki-index.php? page=MetaData` A.C. Irving, R.D. Kenway, C.M. Maynard and T. Yoshié, Nucl. Phys. B(Proc. Suppl.) 129 (2004) 159 http://www.lqcd.org/ildg
http://www.w3.org/XML
http://www.w3.org/XML/Schema
http://www.w3.org/TR/xpath
http://www.w3.org/TR/xpath20/
[^1]: See section \[sec:tutorial\] for references and further details.
|
---
abstract: |
It appears inevitable that reionization processes would have produced large-scale temperature fluctuations in the intergalactic medium. Using toy temperature models and detailed heating histories from cosmological simulations of [He[ ii]{}]{} reionization, we study the consequences of inhomogeneous heating for the Ly$\alpha$ forest. The impact of temperature fluctuations in physically well-motivated models can be surprisingly subtle. In fact, we show that temperature fluctuations at the level predicted by our reionization simulations do not give rise to detectable signatures in the types of statistics that have been employed previously. However, because of the aliasing of small-scale density power to larger scale modes in the line-of-sight Ly$\alpha$ forest power spectrum, earlier analyses were not sensitive to $3$D modes with $\gtrsim 30~$comoving Mpc wavelengths – scales where temperature fluctuations are likely to be relatively largest. The ongoing Baryon Oscillation Spectroscopic Survey (BOSS) aims to measure the $3$D power spectrum of the Ly$\alpha$ forest, $P_F$, from a large sample of quasars in order to avoid this aliasing. We find that physically motivated temperature models can alter $P_F$ at an order unity level at $k \lesssim
0.1~$comoving Mpc$^{-1}$, a magnitude that should be easily detectable with BOSS. Fluctuations in the intensity of the ultraviolet background can also alter $P_F$ significantly. These signatures will make it possible for BOSS to study the thermal impact of [He[ ii]{}]{} reionization at $2 < z < 3$ and to constrain models for the sources of the ionizing background. Future spectroscopic surveys could extend this measurement to even higher redshifts, potentially detecting the thermal imprint of hydrogen reionization.
author:
- |
\
\
$^{1}$ Department of Astronomy, University of California, Berkeley, CA 94720, USA\
$^{2}$ Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA\
$^3$ Department of Physics and Astronomy, University of Pennsylvania; Philadelphia, PA 19104, USA\
$^{4}$ Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA\
bibliography:
- 'References.bib'
title: 'The Signatures of Large-scale Temperature and Intensity Fluctuations in the Lyman-$\alpha$ Forest'
---
\[firstpage\]
cosmology: theory – cosmology: large-scale structure – quasars: absorption lines – intergalactic medium
Introduction
============
The temperature of the intergalactic medium (IGM) is largely determined by how and when the cosmic hydrogen and helium were reionized. Measurements of the mean temperature of the IGM at redshifts $2 <z< 6$ appear to be consistent with the paradigm that hydrogen was reionized at $z\sim 10$ and helium was doubly ionized at $z\sim 3$ [@2000ApJ...534...41R; @2000MNRAS.318..817S; @zaldarriaga01c; @mcdonald01b; @hui03; @lidz09; @bolton10; @becker10]. However, not all of these studies agree on the trends nor on their interpretation.
In addition to raising the mean temperature of the IGM, reionization processes would have heated the intergalactic gas inhomogeneously. Once imprinted, an intergalactic temperature fluctuation would fade away over roughly a Hubble time [@hui03]. Inhomogeneities in the temperature of the IGM alter the Ly$\alpha$ forest absorption because the ionization state of hydrogen depends on the temperature, because the gas distribution on $\lesssim 100$ kpc scales is smoothed by thermal pressure, and because of thermal broadening of the absorption features. However, the vast majority of studies of the Ly$\alpha$ forest have assumed that there is an approximately power-law relationship between temperature and density. Our study investigates the detectability of realistic models for temperature inhomogeneities.
The level of intergalactic temperature fluctuations has been constrained by previous analyses of the forest. @2002MNRAS.332..367T and @lidz09 placed upper limits on the allowed level of these fluctuations in the context of toy inhomogeneous heating models. In addition, the standard model for the Ly$\alpha$ forest has been successful at explaining the statistical properties of this absorption [@miralda96; @hernquist96; @katz96; @dave99; @mcdonald05b]. This picture posits that the fluctuations in the $2 \lesssim z \lesssim 5$ forest were driven primarily by density inhomogeneities, that the gas temperature can be approximated as a power law in density, and that the photoionization rate was nearly spatially invariant. The success of this model implies that deviations from it cannot be large. Several theoretical studies of the Ly$\alpha$ forest have examined the impact of temperature inhomogeneities on various statistics. @lai06 investigated the effect of temperature fluctuations on the Ly$\alpha$ forest line-of-sight power spectrum. Surprisingly, they found that models with ${\cal O}(1)$ fluctuations in the temperature ($\Delta T/T \sim 1$) that correlated over tens of comoving Mpc altered this statistic at *only* the few percent-level for $k < 5~$comoving Mpc$^{-1}$ compared to models with power-law temperature-density relations.
Other statistics have been found to be more sensitive to such inhomogeneities. @meiksin00, @theuns00, and @zaldarriaga02 advocated the use of wavelets filters as a means to spatially identify temperature inhomogeneities, and they showed that wavelets are sensitive to $10$s of comoving Mpc fluctuations with $\Delta T/T \approx 1$ when applied to $\approx 10$ high-resolution Ly$\alpha$ forest spectra. @lee10 found that threshold clustering functions (a popular statistic in the material sciences) could discriminate between different allowed thermal states. @fang04 argued that a three-point statistic which correlates the large-scale flux with the small-scale power could place a strong constraint on deviations from a power-law temperature-density relation. Lastly, @white10 showed that large temperature fluctuations could have a significant effect on two- and three-point functions estimated from correlating multiple Ly$\alpha$ forest sightlines.
Here, we quantify the impact of $10$s of comoving Mpc temperature fluctuations on a diverse set of Ly$\alpha$ forest statistics. We use both toy models as well as the [He[ ii]{}]{} reionization simulations presented in @mcquinn09 to understand these effects. Section \[sec:background\] discusses the different nonstandard contributions to fluctuations in the forest, concentrating on inhomogeneities in the temperature. Section \[sec:methodology\] describes the methods of analysis used in our study. Section \[sec:statistics\] (and Appendix A) quantifies the impact of models for temperature inhomogeneities on different Ly$\alpha$ forest statistics. Section \[sec:Gamma\] discusses how fluctuations in the ionizing background could also alter Ly$\alpha$ forest statistics.
We adopt a flat $\Lambda$CDM cosmological model consistent with the most recent cosmological constraints [@komatsu10], and we henceforth will use “Mpc” as shorthand for “comoving Mpc.” All of our Ly$\alpha$ forest calculations normalize to the mean flux, $\langle F \rangle$, values of @faucher07, which measured $\langle F \rangle = 0.69$ at $z=3$ and $\langle F \rangle = 0.39$ at $z=4$.
Background {#sec:background}
==========
In the Sobolev approximation, the optical depth for a photon to be absorbed as it redshifts across the Ly$\alpha$ resonance of hydrogen is $$\tau_{\rm Ly\alpha} \approx 1.1 \, \Delta_b^2 \; T_4^{-0.7}\; \Gamma_{-12}^{-1} \; \left(\frac{1 +z}{4}\right)^{{9}/{2}} \; \frac{H(z) /(1+z)}{dv/dx},
\label{eqn:tauHI_GP}$$ where $\Delta_b$ is the gas density in units of the cosmic mean, $T_4$ is the temperature in units of $10^4$ K, $H(z)$ is the Hubble parameter, and $dv/dx$ is the line-of-sight velocity gradient (which is equal to $H(z)/(1+z)$ in the absence of peculiar velocities). Equation (\[eqn:tauHI\_GP\]) assumes that the hydrogen is in photoionization equilibrium with a photoionization rate of $\Gamma_{-12}$, expressed in units of $10^{-12}\,$s$^{-1}$. Measurements find $\Gamma_{-12} \approx 1$ (e.g., @faucher08). The $T_4^{-0.7}$ factor arises because of the temperature dependence of the [H[ i]{}]{} fraction in photoionization equilibrium, and many of the effects studied here derive from this dependence.
The amount of Ly$\alpha$ absorption is determined by inhomogeneities in the gas density, the peculiar velocity field, the photoionization rate, and the gas temperature. The first two sources are the standard contributions that were included in most previous studies and that are primarily responsible for the statistical properties of the Ly$\alpha$ forest. Almost all Ly$\alpha$ forest studies also have assumed that the temperature follows a power-law relation in density parametrized as $T(\Delta_b) = T_0 \, \Delta^{\gamma - 1}$ (or they have effectively assumed a power-law relation by using simulations with a spatially uniform ionizing background). This study focusses on the impact of temperature fluctuations around $T(\Delta_b)$, and it also considers fluctuations in the photoionization rate. In addition to modulating the amplitude of $\tau_{\rm Ly\alpha}$, inhomogeneous heating alters $10$s of km s$^{-1}$ absorption features via its impact on the gas pressure and on the thermal widths of absorption lines. Large-scale temperature fluctuations should be present in the IGM as a relic of reionization processes. The reionization of hydrogen and helium were spatially inhomogeneous, with some regions ionized earlier and some later depending on their proximity to the ionizing sources. Models for these processes predict order unity fluctuations in the ionization fraction of the species being reionized on $\sim 10$ Mpc scales and accompanying order unity fluctuations in the temperature (e.g., @trac08 [@mcquinn09]). Temperature fluctuations should have been imprinted on the IGM during reionization because: (1) different regions in the IGM would have been ionized by different energy photons, and (2) a gas element’s instantaneous temperature depends on its ionization history. Once a typical element was reionized and heated, it would have subsequently cooled adiabatically owing to the expansion of the Universe (with Compton cooling off of the CMB and line cooling playing a minor role). This cooling proceeded until the gas element reached the temperature at which heating from ionizations of the residual bound electrons balanced the cooling from expansion [@miralda94; @hui97; @theunsschaye02]. At this point, the temperature difference with initially cooler elements would have been erased. At $z <6$, it takes on the order of a Hubble time to erase temperature fluctuations.[^1]
The Ly$\alpha$ forest indicates that hydrogen was likely reionized at $z >6$. For any physical radiation spectrum that reionized the hydrogen, the intergalactic helium would have been at least singly ionized simultaneously with the hydrogen. At $z < 4$, the relic temperature fluctuations from $z > 6$ reionization processes would have largely faded away [@hui97; @trac08]. However, a hard source of ionizing photons is required to doubly ionize helium (normal stars cannot do it). The current paradigm is that the intergalactic helium was doubly ionized by the radiation from quasars at $z \sim 3$ [@davidsen96; @2000MNRAS.318..817S; @agafonova07; @mcquinnGP; @becker10; @shull10; @furlanettodixon; @dixonfurlanetto]. Models predict that at $z\approx 3$ [He[ ii]{}]{} reionization would have imprinted large inhomogeneities in the temperature, much larger than those remaining from hydrogen reionization [@furlanetto07a; @mcquinn09].
Simulations of the IGM Temperature
----------------------------------
Using radiative transfer simulations, @mcquinn09 made predictions for the level of intergalactic temperature fluctuations expected during [He[ ii]{}]{} reionization. These simulations are employed extensively in our new study. In these simulations, the first regions in which the [He[ ii]{}]{} was reionized were closest to quasars and, therefore, were ionized by softer photons and heated by less than $10^4~$K. These regions subsequently cooled with the expansion of the Universe. Whereas, the last regions to be ionized were more significantly heated by ionizations from a hardened radiation background. Thus, the temporal extent of this process resulted in large fluctuations in the intergalactic temperature. See @abel99 for discussion of relevant radiative transfer effects.
Figure \[fig:big\_box\] shows slices through three snapshots of a $430~$Mpc [He[ ii]{}]{} reionization simulation from @mcquinn09. The left panels are the [He[ iii]{}]{} fraction, the middle panels are the temperature, and the right panels are the change in the Ly$\alpha$ forest transmission relative to the case with the same mean relation between $T$ and $\Delta_b$, but without temperature fluctuations. By the end of [He[ ii]{}]{} reionization in this simulation, the temperature at the cosmic mean density fluctuated between $10$ and $30~$kiloK, resulting in $\approx 10\%$ fluctuations in the Ly$\alpha$ forest transmission. We briefly summarize the characteristics of the temperature fluctuations in these simulations, but see @mcquinn09 for additional details. The solid curves in Figure \[fig:Tdeltadist\] show the mean temperature as a function of $\Delta_b$ at different times in a higher resolution version of the simulation in Figure \[fig:big\_box\]. All of these curves can be approximated by a power-law temperature density-relation with $\langle \gamma - 1 \rangle \approx 0.35$. However, there is significant scatter around this relation, with the scatter increasing as [He[ ii]{}]{} reionization proceeds. Moreover, the scatter is larger at lower $\Delta_b$, a property that affects the observability of these fluctuations (Section \[sec:statistics\]). The contours in Figure \[fig:Tdeltadist\] enclose $33, ~67, ~90,~ 99$, and $99.9\%$ of the grid cells.
The top panel in Figure \[fig:fluctuations\] plots the dimensionless $3$D power spectrum of temperature fluctuations, calculated from different times during the [He[ ii]{}]{} reionization simulation. We use the notation $\Delta_X^2 \equiv k^3P_X(k)/2\pi^2$, where $P_X$ is the power spectrum of the overdensity in $X$. The power law-like scaling of $\Delta_T^2$ at $k \gtrsim 1~$Mpc$^{-1}$ owes to the temperature fluctuations being highly correlated with the small-scale density fluctuations because of adiabatic cooling and heating. The shape of $\Delta_T^2$ at $k \lesssim 1~$Mpc$^{-1}$ owes to the large-scale inhomogeneous heating during [He[ ii]{}]{} reionization (top panel, Figure \[fig:fluctuations\]). Fluctuations at the $\sim 10\%$ level are present even for the largest modes in our $430~$Mpc box during the bulk of [He[ ii]{}]{} reionization.
The bottom panel in Figure \[fig:fluctuations\] compares $\Delta_T^2$ from the snapshot with $x_{\rm HeII} = 0.5$ (dotted curve) with other relevant sources of fluctuations. A common toy model for IGM temperature fluctuations takes half of the IGM to be at $10~$kiloK and the other half to be at $20~$kiloK. The curve labeled “Toy T Model” is such a model, in which $30~$Mpc, $20~$kiloK bubbles are placed randomly until they fill half of the volume. The ringing behavior owes to the artificial top-hat bubble morphology. The fluctuations in this model are a few times larger than in the simulation. Thus, a similar level of temperature fluctuations as in the simulations should be more difficult to detect than in this model. This is an important point because @2002MNRAS.332..367T and @lidz09 found they could rule out this toy model using a wavelet analysis (Section \[ss:wavelet\]). The curve labeled “Density” is $k^3 \,P_{\Delta}(k)/2\pi^2$ at $z=3$, where the non-linear density power-spectrum $P_{\Delta}(k)$ is calculated using the @peacock96 fitting function. The temperature fluctuations in the simulation can be as large as the density fluctuations at the smallest $k$ captured, which suggests that such temperature fluctuations would have a significant effect on the large-scale transmission fluctuations. [He[ ii]{}]{} reionization also produces additional free electrons, increasing the electron abundance (and $\tau_{\rm Ly\alpha}$) by $8\%$ in [He[ iii]{}]{} regions. The power spectrum of the electron fraction from the $\bar{x}_{\rm HeII} = 0.5$ snapshot is represented by the dashed green curve in the bottom panel in Figure \[fig:fluctuations\]. The temperature fluctuations in the simulation are almost an order of magnitude larger than the fluctuations in $x_e$.
An inhomogeneous ultraviolet background (which modulates $\Gamma_{-12}$) is the final potential nonstandard source of transmission fluctuations in the forest and is also the most studied [@zuo92; @zuo92b; @meiksin04; @croft04; @mcdonald05; @furlanettoJfluc; @2009MNRAS.400.1461M; @white10]. The curve labeled “Intensity" represents an empirically motivated model for intensity fluctuations at $z=3$ (see Section \[sec:Gamma\]). Thus, the intensity power can also be comparable to that in the density at the smallest $k$.
Methodology {#sec:methodology}
===========
We use two of the [He[ ii]{}]{} reionization simulations presented in @mcquinn09. The radiative transfer calculation in these simulations was run on a $256^3$ grid in post-processing using the density field from either a $190$ or $430~$Mpc cosmological $N$-body simulation. Both simulations employ the same model for the sources. In addition to the [He[ ii]{}]{} reionization simulations, we use three supplementary simulations, all initialized with different random seeds. We use a $4000^3$ $N$-body simulation described in @white10. This simulation has dimensions of $750~$Mpc/h and the dynamics were softened at $100~$kpc/h to approximate pressure smoothing for $\sim 10^4~$K gas. The other two supplementary simulations are $25~$Mpc/h Gadget-2 runs [@springel05] described in @lidz09 with either $512^3$ or $1024^3$ dark matter and SPH particles. The supplementary simulations are used to overcome the resolution and sample variance limitations of the [He[ ii]{}]{} reionization simulations, as well as to include hydrodynamical effects absent in the reionization simulations. All of these simulations were initialized with cosmological parameters that are consistent with the measurement of @komatsu10.
We add fluctuations in the temperature to the supplementary simulations using both toy models and a method that employs the temperature information from the [He[ ii]{}]{} reionization simulations. The latter method relies on the property that locally the $T$–$\Delta_b$ relation is well-approximated by a power law even though globally a power law is a poor approximation. A local power law holds because neighboring cells have roughly the same thermal history. Specifically, this method obtains $T_0$ and $\gamma$ for a cell in the [He[ ii]{}]{}reionization simulation by fitting this power-law model to $T$ and $\Delta_b$ in the surrounding $5^3$ cells, and it does this along one long diagonal skewer that cycles through the [He[ ii]{}]{} reionization simulation box. It then uses the resulting $T_0$ and $\gamma$ to transfer the temperature field to density and velocity skewers extracted from the supplementary simulations (which we make periodic with length equal to the box size).
Figure \[fig:Tdelta\_local\] examines the validity of this method. The three sets of $5^3$ points with different markers represent the values of $T$ and $\Delta_b$ for the cells in three randomly selected regions of size $3.6$ Mpc in the fiducial [He[ ii]{}]{} reionization simulation. The lines are the best-fit power law to each set of points. We find that $\Delta T/T \approx 0.1-0.2$ is typical of the error around the best-fit $T-\Delta_b$ relation, which is much smaller than the global dispersion in the temperature. Thus, the local power-law approximation appears to be justified. However, this method assumes that the correlations between the large-scale temperature and density modes are unimportant. This is motivated by the fact that the correlation between temperature and large-scale density is weak in the reionization simulations owing to the rare nature of the sources. We will elaborate on the applicability of this approximation later.
We will often use our method to add the [He[ ii]{}]{} reionization simulation temperature field at one redshift to a snapshot in the supplementary simulations from another redshift. This makes possible visual comparison of how the temperature fluctuations affect the statistic in question throughout the reionization process without the confusion of cosmological evolution in the density field. In addition, there is uncertainty in exactly when [He[ ii]{}]{} reionization by quasars would have occurred: @lidz09 argued from the inferred temperature history that the end was at $z\approx 3.5$. Whereas, @mcquinnGP and @shull10 argued that it was not complete until at least $z=2.7$, citing the presence of [He[ ii]{}]{} Ly$\alpha$ Gunn-Peterson troughs down to this redshift.
Statistics {#sec:statistics}
==========
Small-scale Line-of-Sight Power Spectrum {#ss:sslos}
----------------------------------------
The line-of-sight power spectrum of transmission fluctuations is the principle statistic that has been used to derive cosmological constraints from the Ly$\alpha$ forest. Here we investigate how temperature fluctuations affect this statistic at $k \sim 0.1$ s km$^{-1}$, and Section \[ss:LSC\] examines larger scales. Studies of the forest marginalize over $T_0$, $\gamma$, and (in some cases) the reionization redshift to derive cosmological constraints, but the impact of the thermal history has the potential to be more complicated.
Thermal broadening is the most important temperature-dependent effect on the small-scale power spectrum. Pressure smoothing of the gas can also smooth out the small-scale transmission fluctuations. However, pressure effects are not included self-consistently in any of the simulations analyzed here. There are physical arguments for why pressure smoothing is less important to include than thermal broadening:
1. @gnedin98 showed analytically that in an expanding universe the scales at which pressure damps the growth of linear modes are a couple times smaller than the Jeans wavelength and smaller than the scale at which thermal broadening erases fluctuations. Both @gnedin98 and @peeples09a confirmed with simulations that thermal broadening dominates the exponential damping of the small-scale power in the forest.
2. The timescale for a typical Ly$\alpha$ forest absorber to relax to equilibrium after a heating event is comparable to the Hubble time [@gnedin98]. During [He[ ii]{}]{} reionization, the additional pressure smoothing from the associated heating would not have had a significant effect on the statistics of the forest if this heating occurred within a redshift interval of $\Delta z \approx 2$ (see Fig. 29 in @lidz09).
Heating will also induce velocity gradients that broaden the majority of absorption systems [@2000MNRAS.315..600T]. The effect of this broadening mechanism on the Ly$\alpha$ forest power-spectrum has not been quantified in as much detail. This process is unlikely to lead to an exponential cutoff in the small-scale power like thermal broaden because systems with zero width in redshift-space after including this process can still exist.
Thus, this study ignores the dynamical response of the gas, a common approximation in such analyzes. However, this may bias the interpretation in this section and that in Section \[ss:wavelet\] towards favoring higher temperatures. A treatment that includes this effect would be challenging.
The shape of small-scale Ly$\alpha$ forest power spectrum is sensitive to the average temperature as well as the temperature distribution. If $f_1$ of the volume is at temperature $T_1$ and the rest is at temperature $T_2$, thermal broadening results in the small-scale power spectrum achieving the limiting form at high $k$ of $$P_F^{\rm los}(k) \rightarrow f_1\exp \left(-a \, T_1 \, k^2 \right) + \left(1-f_1 \right) \exp \left(-a \, T_2 \, k^2 \right),$$ where $a = 2 k_b/m_p$. In the limit that $T_2 > T_1$ and $a \, T_2 \, k^2 \gg 1$, the power spectrum is solely determined by the regions with $T_1$.
An additional consideration is that the small-scale power at $z\sim3$ is dominated by slightly overdense regions because these are the regions that contribute the narrowest absorption lines. At $z=3$, simple tests show that power on the exponential tail of $P_F^{\rm los}$ is primarily from regions with $\Delta_b \approx 2-3$, and this characteristic density decreases with increasing redshift. The dispersion in the temperature resulting from an inhomogeneous reionization process should be smaller in overdense regions than in underdense ones. This trend is evident in the simulations (Fig. \[fig:Tdeltadist\]), and it acts to reduce the impact of temperature fluctuations on the small-scale shape of $P_F^{\rm los}$.
Figures \[fig:pk\] and \[fig:pk2\] plot the small-scale line-of-sight power spectrum of $\delta_F$ for several temperature models at $z=3$ and $z=4$, respectively. All curves were calculated from the $25/h$ Mpc Gadget-2 simulations (Section \[sec:methodology\]), and the points with error bars are the measurements of @mcdonald00 and @croft02. In the top panels, the green dashed (blue dotted) curves are for an IGM with $\gamma = 1.3$ and $T_0 = 10~$kiloK ($20~$kiloK). If $\gamma = 1.3$ and half of the IGM had $T_0 = 10~$kiloK and the other half $T_0 = 20~$kiloK, then the small-scale power spectrum would be given by the magenta dotted curve, which has a slightly flatter shape than the single $T_0$–$\gamma$ models. The the shape of this curve appears to be inconsistent with the @mcdonald00 data. However, a toy model for the temperature field that is better motivated by the [He[ ii]{}]{} reionization simulations is half of the IGM at $T_0 = 15~$kiloK with $\gamma = 1.5$ and half at $T_0 = 25~$kiloK with $\gamma =1.2$. The difference in the small-scale power spectrum between these two states is negligible. (Compare the teal dot-dashed and grey quadruple-dotted curves in the top panels.) Therefore, the fluctuations in this toy model would not be detectable in $P_F^{\rm los}$. The bottom panels in Figures \[fig:pk\] and \[fig:pk2\] are the power spectrum of models that use the temperature field of different snapshots from the $190~$Mpc [He[ ii]{}]{} reionization simulation. The IGM is gradually heated throughout [He[ ii]{}]{} reionization, which results in the small-scale power decreasing with time, or equivalently, with [He[ iii]{}]{} fraction. The amount of evolution is sensitive to the simulation’s initial conditions, especially with decreasing $\bar{x}_{\rm HeIII}$. This simulation was initialized with $T_0 = 10~$kiloK and $\gamma = 1.3$ at the starting redshift of $z=6$. If the simulation were instead initialized with $T_0 = 15~$kiloK, there would have been less evolution.
At the end of [He[ ii]{}]{} reionization, the simulation temperature reaches values that are slightly hotter than suggested by the $z=3$ measurements of @mcdonald00 and @croft02. The $x_{\rm HeIII}=0.94$ curve in Figure \[fig:pk\] corresponds to $z=3$ in the simulation. The measurement of @mcdonald00 at $z=4$ in Figure \[fig:pk2\] appears to be more consistent with *slightly* cooler temperatures than their $z= 3$ measurement. For reference, the $x_{\rm HeIII} = 0.4$ output is at $z=4$ in the reionization simulation.
There are several systematics that may bias these interpretations. The observations are biased in the direction of having extra power by instrumental noise and especially by metal lines, and the simulations are also biased in the direction of extra power as previously described because pressure smoothing is not properly captured and the simulation gas temperatures (prior to our post-processing) are on the low side. @lidz09 estimated that pressure smoothing could alter the power at $k\sim 0.1~$s km$^{-1}$ by a maximum of $20\%$ (see their Fig. 29), but its impact should be significantly smaller for the [He[ ii]{}]{} reionization scenarios considered here where the heating has been more recent than in this case in @lidz09. Uncertainties in the mean flux are also important for the comparison at $z=4$ [@lidz09].
In conclusion, measurements of the small-scale Ly$\alpha$ forest power spectrum reveal that $15 < T_0 < 25~$kiloK at $z=3$, with tentative evidence for an decrease in temperature from $z=3$ to $4$ in the @mcdonald00 measurement. However, finer determinations would be challenging because the systematics are at the level of the model differences. We also find that it would be difficult to detect physically motivated models for temperature fluctuations with this statistic.
Appendix A considers the impact of the temperature models considered here on wavelet statistics. Wavelets are also a measure of the small-scale power in the forest, and are better suited than the small-scale behavior of $P_F^{\rm los}$ for detecting temperature inhomogeneities. Despite this advantage, Appendix A reaches similar conclusions to this section.
Flux Probability Distribution {#ss:FPDF}
-----------------------------
Another statistic that is often applied to the Ly$\alpha$ forest is the probability distribution function of the normalized flux (FPDF). A distribution of temperatures at a given $\Delta_b$ will broaden the FPDF relative to the case of a single $T-\Delta_b$ relation. In fact, measurements of the FPDF have claimed to find too many hot voids at $2 < z < 3$ compared to cosmological hydrodynamic simulations with standard, homogeneous thermal histories. This led @bolton08 and @2009MNRAS.tmpL.290V to argue that an inverted $T-\Delta_b$ relation ($\gamma - 1 < 0$) was needed to reconcile this discrepancy. [He[ ii]{}]{} reionization by the observed population of quasars cannot generate an inverted relation [@mcquinn09]. However, perhaps an inverted relation is not required to explain the data, but may rather be accounted for by additional dispersion in the temperature as conjectured in @mcquinn09. Dispersion would have made some of the voids hotter and, as a result, more transparent to Ly$\alpha$ photons.
The top panel in Figure \[fig:PDF\] plots the FPDF at $z=3$. The points with errorbars are the measured FPDF from @kim07. The curves are this statistic calculated from $1000$ skewers through the $25/h~$Mpc simulation. The red solid curve represents a toy model in which $\gamma = 1.3$ and where half the IGM has $T_0 = 10~$kiloK and the other half $T_0= 20~$kiloK. The green dashed curve was calculated using the temperature at the end of the [He[ ii]{}]{} reionization simulation (where $x_{\rm HeIII} = 0.94$), and corresponds to the simulation snapshot in which the dispersion in temperature is approximately maximized. The blue dotted curve represents a model with $\gamma = 1.3$ and with $T_0$ equal to the average in the $x_{\rm HeIII} = 0.94$ snapshot. However, the dependence of this curve on $T_0$ is weak. The bottom panel in Figure \[fig:PDF\] plots the relative difference between the $\gamma =1.3$ model and the other temperature models featured in the top panel.
We next estimate the magnitude by which temperature fluctuations alter the FPDF. To proceed, we write the FPDF, ${\cal P}$, as a Taylor expansion in $\delta_{Tp7}$: $$\begin{aligned}
{\cal P}(F) &\approx&\tilde{{\cal P}}(F) + A \, \langle \delta_{Tp7} \rangle + B \, \langle \delta_{Tp7}^2 \rangle + ...,\\
A &\equiv& \frac{d\tilde{\cal P}}{dF} \, \tilde{F} \, \tau - {\cal P}\left(1 - \tau \right), \nonumber \\
B &\equiv& \frac{d^2 \tilde{\cal P}}{dF^2} \, \tau^2 \, \tilde{F}^2 - \frac{d\tilde{\cal P}}{dF} \; \tilde{F} \, \left(4\,\tau - 3\,\tau^2 \right) + {\cal P} \left(2 - \tau \, \left[4 -\tau \right] \right), \nonumber \end{aligned}$$ where $\tilde{\cal P}$ is the FPDF without temperature fluctuations, $\delta_{Tp7}$ is the fluctuation in $T^{-0.7}$ around the average $T$-$\Delta_b$ relation, $\tilde{F}$ is flux for the average $T$-$\Delta_b$ relation, $\langle...\rangle$ represents an average at $\tilde{F}$ (roughly corresponding to fixed $\Delta_b$), and $\tau \equiv -\log(\tilde{F})$. Both $A$ and $B$ are of the order of ${\cal P}(F)$, and we find $A \approx 0.3$ and $B \approx -0.1$ at $F = 0.7$. The leading order contribution from temperature fluctuations *about the mean $T-\Delta_b$ relation* is $B \, \langle \delta_{Tp7}^2 \rangle$. Plugging in $\langle \delta_{Tp7}^2 \rangle \sim 0.2^2$, which is characteristic of the simulation temperature model at $F \approx 0.7$, we estimate that temperature fluctuations should alter ${\cal P}(F)$ at the several percent level.
The size of the residuals in the bottom panel of Figure \[fig:PDF\] qualitatively agree with the above estimate that temperature fluctuations should produce a percent-level change in ${\cal P}(F)$. The predictions are not in quantitative agreement (and sometimes has the incorrect sign) because we have not included the normalization to a single mean flux in our analytic expressions, which fixes the first moment of the FPDF. We conclude that a different effect is required to create the $\approx 20\%$ *suppression* of the FPDF that @bolton08 finds at $F \approx 0.7$. Additional dispersion in the temperature (even with a much larger amplitude than in our [He[ ii]{}]{} reionization simulations) is unable to explain the discrepancy between the observed and simulated FPDFs.
Changing $\gamma$ can have a larger effect on the FPDF than adding dispersion to the temperature because, in this case, the leading order term is $A \, \langle \delta_{Tp7} \rangle$. (Compare the $\gamma =1.3$ with the $\gamma =1.0$ curve in Figure \[fig:PDF\].) Performing a more detailed comparison than considered here, @bolton08 found that an inverted relation with $\gamma \approx 0.5$ provided the best fit to the @kim07 FPDF measurement. However, an inverted relation may not be the only explanation and would require an unknown heating mechanism. The FPDF is especially sensitive to the accuracy of continuum fitting [@2011arXiv1103.2780L] and the efficacy of metal-line removal.
Large-Scale Correlations {#ss:LSC}
------------------------
The remainder of this paper discusses large-scale correlations in the Ly$\alpha$ forest. At scales where $\delta_F \equiv F/\bar{F} - 1$ and $\delta_T$ are small, the $3$D power spectrum of $\delta_F $ can be expressed as bias factors times the power spectra of the different sources of fluctuations. In particular, $$\begin{aligned}
P_{F}({{\boldsymbol{k}}}) &\approx & b^2 \bigg(G^2 \, P_{\Delta}(k) + 2\,G \epsilon^{-1} \, \left[P_{\Delta \, Tp7}(k) - P_{\Delta \, J}(k) \right] \nonumber \\
& &+ \epsilon^{-2} \, \left[P_{Tp7}(k) + P_{J}(k)\right] \bigg),
\label{eqn:Pklg2}\end{aligned}$$ where ${{\boldsymbol{k}}}$ is the Fourier wavevector, $k = | {{\boldsymbol{k}}}|$, $Tp7$ is shorthand for the temperature field to the $-0.7$ power, $J$ represents intensity, $P_{\Delta \, X}(k)$ is the cross power spectrum between density and the overdensity in $X$, $\mu = \hat{{{\boldsymbol{k}}}}\cdot \hat{n}$ where $\hat{n}$ is the line-of-sight direction, and $\epsilon \approx 2 - 0.7\,(\gamma-1)$. The $G \equiv ( 1 + g \mu^2)$ factors arise from peculiar velocities [@kaiser87], and $g$ reflects that these distortions have a different bias than density fluctuations. Our simulations require $g \approx 1$ (Appendix B).
The $\epsilon$ and $\epsilon^2$ suppression of $T$- and $J$-dependent terms is approximate and results because the flux depends on $\Delta_b$, $T$, and $J$ via the combination $\Delta_F^\epsilon \, T^{-0.7} \, J^{-1}$. This suppression holds in linear theory, but can be violated by higher order correlations. Equation (\[eqn:Pklg2\]) suggests that at scales where either $P_{Tp7}$ or $P_J$ are comparable to $P_{\Delta}$, these fluctuations can have a large effect on $P_F$.
### Line-of-sight Correlations {#ss:los}
Previous Ly$\alpha$ forest correlation analyses have primarily focused on the line-of-sight power spectrum and neglected correlations between sightlines. The line-of-sight power spectrum $P_{F}^{\rm los}$ can be expressed in terms of the $3$D power spectrum $P_F$ as $$P_{F}^{\rm los}(k_{\parallel}) = \int_{k_{\parallel}}^\infty \frac{dk}{2\pi} k \,P_{F} (k, k_{\parallel}/k),
\label{eqn:Plos}$$ where $k_{\parallel}$ is a line-of-sight wavevector. Because $\mu \equiv k_{\parallel}/k$, the projection to $1$D suppresses the impact of large-scale peculiar velocities (and the $P_{\Delta X}$ terms; c.f. eqn. \[eqn:Pklg2\]). In addition, $3$D wavevectors with even $k \gg k_{\parallel}$ still contribute to $P_{F}^{\rm los}$ at $k_{\parallel}$. Figure \[fig:aliasing\] illustrates this aliasing effect, plotting at several $k_{\parallel}$ the fractional contribution per $\log k$ to $P_F^{\rm los}$ from different $3$D modes with wavevector $k$. The red thick dashed curve represents this for $k_{\parallel} = 10^{-1}~$Mpc$^{-1}$ and $g=0$ (the red thin dashed this for $g=1$), approximately the value of the *3D* wavevector where $P_T$ is maximized relative to $P_\Delta$ in some of our models (Fig. \[fig:fluctuations\]). Thus, $P_{F}^{\rm los}(k_{\parallel})$ receives a significant contribution from approximately two decades in $k$. This aliasing dilutes the impact of large-scale temperature fluctuations.[^2]
Figure \[fig:los\] quantifies the impact of temperature fluctuations on the line-of-sight power spectrum. The curves are $P_{F}^{\rm los}$, calculated using $22,500$ skewers of side-length $750$ Mpc/h from the $z=2.8$ snapshot of Run 1 in @white10 and using different temperature models. Each curve represents a different model for the temperature. However, the values of $P_F^{\rm los}$ in all of the considered models differ by less than $10\%$. These small differences qualitatively agree with the results of @lai06.
The points with errorbars in Figure \[fig:los\] are the SDSS measurement from @mcdonald05b. The differences between the plotted models are comparable or smaller than the @mcdonald05b errorbars. Thus, the impact of these temperature models almost certainly could not be detected in $P_{F}^{\rm los}$ once one marginalizes over the cosmology. Temperature fluctuations could bias cosmological parameter determinations from $P_{F}^{\rm los}$, but at no more than the few percent-level in the considered models.
The calculations in Figure \[fig:los\] implicitly assume that the large-scale temperature fluctuations do not correlate strongly with the large-scale density modes due to our method for adding temperature fluctuations. This is not always a good approximation for the calculation that uses the [He[ ii]{}]{} reionization simulation temperature field. Accounting for these correlations would increase the difference with the D5 curve by approximately a factor of $2$, as can be inferred from the discussion in the Section \[ss:HI\_3D\].
The next section discusses the $3$D power spectrum of the flux, a statistic which avoids the aliasing issues in $P_{F}^{\rm los}$. We show that temperature models that have almost no effect on $P_{F}^{\rm los}$ can alter the $3$D power spectrum of the flux at an order unity level.
### $3$D Correlations {#ss:HI_3D}
The Sloan Digital Sky Survey III’s Baryon Oscillation Spectroscopic Survey (BOSS) aims to observe the Ly$\alpha$ forest in the spectra of $1.6\times 10^5$ quasars over an area of $8000$ deg$^2$. A future survey, BigBOSS, will push a magnitude fainter, increasing this sample by a factor of a few.[^3] BOSS’s high density of quasars will allow the first measurement of the $3$D Ly$\alpha$ forest power spectrum. A major goal of this effort is to detect the baryon acoustic oscillation (BAO) features. The $3$D power spectrum will also be useful for studying temperature and other nonstandard sources of fluctuations in the forest because it avoids the aliasing issues that diminish the impact of large-scale correlations in $P_F^{\rm los}$.
Figure \[fig:3dspectrum\] quantifies the effect of the temperature fluctuations in the [He[ ii]{}]{} reionization simulations on $P_F$. The thick set of curves are calculated from the $190$ Mpc simulation, and the thin (shown in three of the panels) are from the $430~$Mpc one. The different normalization of the curves in the two simulations results primarily because of the different redshifts of the snapshots. The redshift of the $430~$Mpc simulation curves is $\Delta z \approx 0.3$ higher than the quoted redshift because [He[ ii]{}]{} reionization occurs slightly earlier in this simulation (owing to fewer recombinations in this simulation). When calculating these curves, we normalize to the observed mean flux at the respective redshift. The solid curves are $\Delta_F^2$ computed using the temperature field from the [He[ ii]{}]{} reionization simulation at the quoted $x_{\rm HeIII}$ and, for comparison, the dotted curves show $\Delta_F^2$ for a power-law $T-\Delta_b$ relation with $\gamma = 1.3$. Changing $\gamma$ or $T_0$ has little effect on the latter curve.
Figure \[fig:3dspectrum\] illustrates that temperature fluctuations can significantly alter $\Delta_F^2$ on large scales, changing its amplitude by as much as a factor of $2$. In addition, the relative impact of temperature fluctuations at fixed $x_{\rm HeIII}$ largely agree between the small- and large-box simulations. This agreement is suggestive that the impact of temperature fluctuations on these scales is not affected by how well the small-scale features in the forest are resolved. The bottom right panel shows for comparison $\Delta_F^2$ predicted for our toy model with $30$ Mpc bubbles at twice the temperature filling half of space in the larger box (cyan thin dashed curve). Interestingly, even though the temperature fluctuations are much larger in this model than in the simulations (see Fig. \[fig:fluctuations\]), their effect on $\Delta_F^2$ is comparable to that in the simulation temperature models.
The BOSS quasar survey is forecast to measure $\Delta_F^2$ at $z\approx 3$ to $\sim 10\%$ accuracy in bins with $\Delta k \sim k$ in the range $10^{-2} < k < 1~$Mpc$^{-1}$, precision beyond what would be required to detect this extra power in some of our models. The bottom panels in Figure \[fig:3dspectrum\] includes an estimate discussed in @mcquinnwhite for the measurement error of BOSS *multiplied by 5*, assuming $8000~$deg$^{-2}$ over a region of depth $500~$Mpc. These curves are calculated assuming $\bar n_{\rm eff} = 3\times 10^{-4}~$Mpc$^{-2}$ for the $z=3$ panel and $\bar n_{\rm eff} = 2\times 10^{-4}~$Mpc$^{-2}$ for the $z=3.3$ (corresponding to $\approx 2-3~$per deg$^{2}$), where $n_{\rm eff}$ is a noise weighted number density of quasars on the sky and defined in @mcquinnwhite. The sensitivity scales as $n_{\rm eff}^{-1}$. These number densities take the quoted magnitude limit for BOSS of $m=22$, assuming that all quasars below this magnitude are targeted and that $S/N =1$ in $1~$Å pixels at $m=22$. With these assumptions, BOSS will be several times more sensitive to $P_F$ at $z=2-2.5$ compared to at $z=3$. In addition, the vertical line segments in this figure mark the centers of the three BAO features that the BOSS and other upcoming quasar surveys are targeting as a standard ruler. Temperature fluctuations have the potential to bias the measurement of this feature unless a fairly general functional shape is assumed for the continuum on which these features sit.
BigBOSS aims to obtain a denser sample of quasars, and we forecast that it will be able to constrain $\Delta_F^2$ to $z \approx 4$ if its survey strategy is similar to that of BOSS, and possibly to higher redshifts if this strategy is optimized to go deeper [@mcquinnwhite]. The dot dashed curves in the top panels in Figure \[fig:fluctuations\] show what BigBoSS can achieve, assuming that it obtains spectra for quasars that are a magnitude fainter than BOSS such that $\bar n_{\rm eff} = 5\times 10^{-4}~$Mpc$^{-2}$ at $z=2.6$ and $1\times10^{-4}~$Mpc$^{-2}$ at $z=4.2$.
Peculiar velocities are not included in the calculation of the curves in Figure \[fig:3dspectrum\] and the gas is assumed to be in the Hubble flow. Peculiar velocities reduce the importance of temperature fluctuations. However, peculiar velocities do not contribute to the power in modes transverse to the line of sight, and, thus, we expect that the differences seen in Figure \[fig:3dspectrum\] should be representative of the effect on transverse modes and the monopole component of $\Delta_F^2$. Observations will be most sensitive to the $\Delta_F^2$ monopole (Section \[sec:angular\]).
Figure \[fig:modelcomp\] demonstrates that equation (\[eqn:Pklg2\]) provides a reasonable description of the impact of temperature fluctuations in the reionization simulations. This figure compares the predictions for $\Delta_{F}^2$ from equation (\[eqn:Pklg2\]) (red dashed curve) with that of the full numerical result (black solid curve). The red dashed curve is constructed by adding each contribution that appears in equation (\[eqn:Pklg2\]) separately, and the value of $b^2$ in this equation was derived by taking the ratio of $\tilde{P}_F$ to $P_\Delta$. The blue dotted curve is $\Delta_{F}^2$ for a power-law $T-\Delta_b$ relation with $\gamma = 1.3$, the green long dashed curve and thick cyan dot-dashed are respectively $(b/\epsilon)^2 \, \Delta_{Tp7}^2$ and $2 b^2/\epsilon \, \Delta_{Tp7 \Delta}^2$. (The thin cyan curves are $-2 b^2/\epsilon \, \Delta_{Tp7 \Delta}^2$.) The field $\delta_{Tp7}$ is taken to be the overdensity in temperature around the mean $T-\Delta_b$ relation, a slightly different definition than used previously and that is motivated in Appendix C. The sum of the blue dotted, green long dashed, and $\pm$ the cyan dot-dashed curve yields the red dashed curve, which is remarkably similar to the black solid curve. (See Appendix C for a complementary analytic description of the impact of temperature fluctuations.)
The history of $P_{Tp7 \Delta}$ in the [He[ ii]{}]{} reionization simulation is complicated. This function is less than zero at the beginning of [He[ ii]{}]{} reionization and at the largest-scales because quasars are heating their immediate surroundings (top left panel, Fig. \[fig:modelcomp\]). Later, it becomes positive as the large-scale void regions are ionized. Constraining this nontrivial evolution from observations of $\Delta_F^2$ would inform [He[ ii]{}]{} reionization models. The contribution from $P_{Tp7 \Delta}$ to $\Delta_F^2$ is of comparable import to the contribution from $P_{Tp7}$ in the four cases considered in Figure \[fig:modelcomp\], despite $P_{Tp7 \Delta}$ being smaller than $P_{Tp7}$. This occurs because $P_{Tp7 \Delta}$ is enhanced relative to $P_{Tp7}$ by a factor of $2 \, \epsilon$ in equation (\[eqn:Pklg2\]). This enhancement explains why the temperature fluctuations in the simulations produce a comparable change in $\Delta_F^2$ to that in our toy temperature model, despite the toy model having a much larger $P_T$. If the [He[ ii]{}]{} ionizers were more abundant than assumed here (as would be the case if the faint end of the quasar luminosity function were steeper; @furlanetto07b), $P_{Tp7 \Delta}$ could be significantly larger, increasing the impact of temperature fluctuations on $P_F$. In conclusion, temperature fluctuations could have a significant effect on the $3$D power spectrum in the forest and would be detectable in future $3$D Ly$\alpha$ forest analyses such as with BOSS. Our simple analytic description for their impact is extremely successful.
Intensity Fluctuations {#app:Gfluc}
======================
\[sec:Gamma\]
Several studies have investigated the impact of intensity ($J$) fluctuations on the Ly$\alpha$ forest [@zuo92; @zuo92b; @meiksin04; @croft04; @mcdonald05; @furlanettoJfluc; @2009MNRAS.400.1461M]. Here we attempt to understand how intensity fluctuations could affect $3$D correlations in the forest. @meiksin04 and @croft04 concluded that $J$ fluctuations have a small effect at $10-100$ km s$^{-1}$ separations in the Ly$\alpha$ forest correlation function, but found that they could change this statistic at the $10$s of percent-level on larger scales. Interestingly, @white10 found that intensity fluctuations could have an order-unity impact on $3$D correlations. A detection of intensity fluctuations would constrain the rarity of the sources of ionizing photons, and it would constrain the contribution of quasars versus that of galaxies to the metagalactic ionizing background.
Intensity fluctuations (equivalent to $\Gamma_{-12}$ fluctuations) alter the $3$D Ly$\alpha$ forest power spectrum on linear scales as $$P_F({{\boldsymbol{k}}}) \approx \tilde{P}_F({{\boldsymbol{k}}}) + b^2 \left[\epsilon^{-2} \, P_{J}(k) - 2 \, \epsilon^{-1} (1+g\mu^2)\, P_{\Delta J}(k) \right],
\label{eqn:PFJ}$$ (c.f. eqn. \[eqn:Pklg2\]) where $\tilde{P}_F({{\boldsymbol{k}}})$ is the flux power-spectrum without $J$ fluctuations. If we assume a Euclidean space in which all [H[ i]{}]{} ionizing photons experience a single attenuation length $\lambda$ and treat the quasars as continuously shining lightbulbs, then $$P_J = \left( \frac{\arctan(\lambda k)}{\lambda k} \right)^2 \left(\frac{\langle L^2 \rangle}{\bar{n} \, \langle L \rangle^2 } + b_q^2 \, P_{\Delta}(k) \right), \label{eqn:PJ}$$ (which is similar to the expression for $P_J$ in @2009MNRAS.400.1461M) and $$P_{\Delta J} = \left( \frac{\arctan(\lambda k)}{\lambda k} \right)\, b_q\, P_{\Delta}(k).
\label{eqn:PDJ}$$ Here, $P_J$, is the power spectrum of intensity fluctuations, $P_{\Delta J}$ is the cross correlation between intensity and density, $b_q$ and $\bar{n}$ are respectively the luminosity-weighted bias and $3$D number density of the sources, and $(\lambda k)^{-1}\arctan(\lambda k)$ is (up to a constant factor) the Fourier transform of $r^{-2} \exp[-r/\lambda]$.[^4]
Finite quasar lifetimes alter $P_J$ when the lifetime $t_{\rm q}$ is comparable to or shorter than $\lambda/c$, or $300$ Myr for $\lambda = 100$ proper Mpc. The variance of $J$ does not depend on $t_{\rm q}$ when normalizing to a single luminosity function, but finite lifetimes can reduce large-scale $J$ correlations because $J$ becomes uncorrelated in regions separated by distances greater than $>c \, t_{\rm q}$. These effects can be included in equation (\[eqn:PFJ\]) by substituting a more complicated window function for $(\lambda k)^{-1}\arctan(\lambda k)$. However, the primary effect is for $P_J$ to become white at wavelengths greater than $c \, t_{\rm q}$ rather than at $\sim\lambda$. Thus, detecting intensity fluctuations on $100~$Mpc scales would place interesting constraints on quasar lifetimes. In addition, light travel effects alter the isotropy of the $J$ fluctuations because the emission will have travelled further in the line-of-sight direction. This would result in additional $\mu$-dependent terms in equation (\[eqn:PJ\]).
The top panel in Figure \[fig:Jfluc\] plots the Poisson and clustering components of $\Delta_J \equiv [k^3 \, P_J/2\pi^2]$ (the first and second terms in eqn. \[eqn:PJ\]) for infinite lifetimes. The thick dashed curves are the Poisson component for $\lambda = 500, 300, 150,$ and $70~$Mpc (in order of decreasing amplitude). For reference, these $\lambda$ correspond roughly to its measured value at $z=2, 3,4$ and $5$ [@faucher08; @prochaska09]. Our calculations assume that the quasar luminosity function has the form $\Phi(L) \sim ([L/L_*]^{\alpha} + [L/L_*]^{\beta})^{-1}$ with a cutoff $3$ decades above and below $L_*$, where $\alpha = 1.5$ and $\beta = 3.5$. These choices result in $\langle L^2 \rangle/\langle L \rangle^2 \approx 30$. Changing both the upper and lower cutoff by a factor of $10$ and in opposite directions results in a factor of $3$ change in $\langle L^2 \rangle/\langle L \rangle^2$. Furthermore, the curves in the top panel in Figure \[fig:Jfluc\] use $\bar{n} = 10^{-4}~$Mpc$^{-3}$, representative of quasars at $z = 2-3$. The @hopkins07a quasar luminosity function yields $\bar{n}= \{1.2\times 10^{-4},~ 9\times 10^{-5},~5\times 10^{-5}, ~2\times 10^{-5} \}$ Mpc$^{-3}$ at $z = \{2.5, ~3, ~4, ~5\}$.
For three of the four cases featured in the top panel in Figure \[fig:Jfluc\], the power in the Poisson component is *always* larger than the power in the density field (the black solid curve). However, the normalization of the Poissonian component of $\Delta_J$ is highly uncertain. The Poisson curves in the top panel of Figure \[fig:Jfluc\] are roughly a factor of $3$ below the corresponding estimate in @furlanettoJfluc in part because @furlanettoJfluc used $\beta = 2.9$ for which $\langle L^2 \rangle/\langle L \rangle^2 \approx 100$ with our luminosity cutoffs. Most investigations of $J$ fluctuations on the forest have used values for $\beta$ that are more similar to the value used here [@meiksin04; @croft04; @mcdonald05]. However, @furlanettoJfluc argued that $\beta = 2.9$ provides a better fit to the @hopkins07a luminosity function.
The thin dashed curves in the top panel in Figure \[fig:Jfluc\] represent the contribution to $\Delta_J$ from source clustering at $z=3$ for $b_q=3$ and for the same $\lambda$ as the corresponding thick curve. These curves fall below the Poisson component with the same $\lambda$. However, the clustering component of intensity fluctuations will likely be the dominant source of $J$ fluctuations if galaxies are the source of ionizing photons (as several studies have argued might be the case at $z\gtrsim 4$; e.g. @faucher08). Because the clustered component also affects the flux field via $P_{\Delta J}$, this will enhance its effect beyond its contribution to $P_J$ and also allow this contribution to be separated via its distinct angular dependence (Section \[sec:angular\]). In addition, the power in the clustering component of $J$ is always larger than that in the density — the standard source of fluctuations in the forest — at $k \lesssim \lambda^{-1}$ since $b_q > 1$. These scales become observable in the Ly$\alpha$ forest at $z \gtrsim 3$.
Effect of Intensity Fluctuations on $P_F$
-----------------------------------------
The previous section showed that $P_J$ can be comparable to $P_\Delta$ in the Ly$\alpha$ forest. If our simple analytic expression is correct (eqn. \[eqn:PFJ\]), this implies that $J$ fluctuations significantly increase $P_F$ beyond $\tilde{P}_F$. However, previous numerical studies of the impact of intensity fluctuations on $P_F^{\rm los}$ have considered models where the power in $J$ was comparable to or larger than the power in $\Delta_b$, and these studies found that these fluctuations had a small effect on $P_F^{\rm los}$ for $k > 0.1~$Mpc$^{-1}$ (e.g., @croft04 [@mcdonald05]). Here we try try to resolve this apparent discrepancy.
Figure \[fig:PFJ\] features numerical calculations of the impact of intensity fluctuations on $\Delta_F^2$. The thin black dashed curve is the contribution to the total power from intensity fluctuations using equation (\[eqn:PFJ\]) and the fiducial model for $J$ fluctuations with $\lambda = 300~$Mpc. To generate the $J$ field, we have convolved a random distribution of quasars which have the said luminosity function with the function $r^{-2}\, \exp[-r/\lambda]$. This ignores the clustering contribution, which Figure \[fig:Jfluc\] suggests is smaller but non-negligible in this model. The solid red curve is $\Delta_F^2$ without intensity fluctuations, and the dashed green is the fully numerical calculation of the effect of intensity fluctuations in this model. The blue dotted curve is the prediction of equation (\[eqn:PFJ\]) – the addition of the dashed black and solid red curves. The analytic model does poorly at capturing the impact of intensity fluctuations at $k > 0.1~$Mpc$^{-1}$, but does better at smaller wavevectors. The teal dot-dashed curve is the same as the green dashed but for a model with $\lambda = 100~$Mpc such that the intensity fluctuations are larger and thus have a larger impact on $\Delta_F^2$.
The poor performance of the analytic model at $k > 0.1~$Mpc$^{-1}$ owes to the diverging character of $J$ in quasar proximity regions. Within the proximity region of a quasar (within the distance $r_p(L) \equiv [4 \pi \langle L \rangle/L \,\bar{n} \, \lambda]^{-1/2}$), the effect on $\Delta_F^2$ will be suppressed beyond the prediction in equation (\[eqn:PFJ\]) by the exponential transformation from $\tau_{\rm Ly\alpha}$ to transmission. Because $\tau_{\rm Ly\alpha} \sim (1 + [r_p/r]^2)^{-1}$, the small-scale transmission power from proximity regions is damped exponentially with a kernel similar to that of a Gaussian with s.d. $\approx r_p^{-1}/\sqrt{2}$. The bottom panel in Figure \[fig:Jfluc\] is the same as the top panel, but where for the Poisson term in $P_J$ given by $k^{3/2} \arctan( \lambda k)/\lambda k$ has been forced to transition to a constant function of $k$ with the transition occurring at $k = r_p(L)^{-1}/\sqrt{2}$. This operation is meant to approximate the convolution of $\arctan( \lambda k)/\lambda k$ with the Gaussian-like shape of the proximity region. The power in $J$ is significantly affected by this operation at the largest $k$ that are shown. The $k$ where the proximity-region damping occurs are roughly $k \gtrsim r_p(\langle L^2 \rangle^{1/2})^{-1}$, which correspond to $k \gtrsim 0.1~$Mpc$^{-1}$ in our fiducial model. Smaller $k$ correspond to where the analytic model provides a good description in Figure \[fig:PFJ\].
Density fluctuations become larger than those in $J$ at $k > 0.3~$Mpc$^{-1}$ in three of the Poissonian models in the bottom panel of Figure \[fig:Jfluc\] (and the intensity power is also suppressed by an additional factor of $\epsilon^2$ in $\Delta_F^2$). Wavevectors with $k > 0.3~$Mpc$^{-1}$ correspond roughly to those where $P_F^{\rm los}$ has been measured. As with temperature fluctuations, the impact of intensity fluctuations are further diluted by aliasing in $P_F^{\rm los}$ such that $3$D fluctuations are better suited for detecting them. Thus, it is unlikely that the intensity fluctuations in three of these models would have produced a detectable imprint on $P_F^{\rm los}$. In fact, we have calculated the impact of our fiducial $J$ model on $P_F^{\rm los}$ in the $25/h~$Mpc box and, because of this small-scale damping, found no appreciable effect owing to these damping and aliasing effects.
At $z\gtrsim4$, it is conceivable that intensity fluctuations contribute a substantial portion of the power even in $P_F^{\rm los}$ because both $\lambda$ and $\bar{n}$ are smaller. Interestingly, @mcdonald05b constrained the $z=4$ mean flux from power spectrum measurements to be $10-20\%$ higher than the more direct measurement in @faucher07. Such a disparity could potentially occur if an unaccounted source of fluctuations were contributing to $P_F^{\rm los}$. Some of the intensity models considered in @mcdonald05b \[their Fig. 5\] and more generally in models where $\langle L^2 \rangle /[\langle L \rangle^2 \, \lambda^2 \, \bar{n}] \gtrsim 10^2~$Mpc are able to produce a tens of percent change in the power at all scales.
There are several possibilities for how the impact of intensity fluctuations on $P_F$ can be distinguished from the fluctuations in the temperature. Temperature fluctuations should peak in amplitude at $z=2.5-3$ if [He[ ii]{}]{} reionization were ending around these redshifts. In contrast, intensity fluctuations are likely to have increased monotonically with redshift. In addition, the amplitude of $P_J$ is set by the factor $\langle L^2 \rangle /[\langle L \rangle^2 \,\lambda^2 \, \bar{n} _{\rm 3D}]$, which can potentially be constrained by ancillary observations.
Separating the Components of $P_F$ {#sec:angular}
==================================
The signal we are interested in has the form $P_F = P_{0} + 2 g \, \mu^2 P_{1} + g^2 \mu^4 P_{2}$ in linear theory, where the $P_i$ are functions of only $k \equiv | \mathbf{k} |$ and are given by $P_{0} = b^2 \,P_{\Delta} + b^2 \,\epsilon^{-2} \, [P_{J} + P_{Tp7}]$, $P_1 = 2 \, b^2\, P_{\Delta} + 2 \,b^2 \,\epsilon^{-1} \, [P_{\Delta J} + P_{\Delta Tp7}]$, and $P_2 = b^2 \,P_{\Delta}$. Thus, temperature and intensity fluctuations affect $P_{0}$ and, to the extent they correlate with density, $P_{1}$. (Light-travel effects can generate additional $\mu^{2n}$ terms that we do not consider.) Here we investigate the sensitivity of a survey to the terms in this angular decomposition. Detecting these angular components may prove easier in the Ly$\alpha$ forest than it has for galaxy surveys because the velocity bias $g$ is likely to be larger in the Ly$\alpha$ forest than in galaxy surveys (for which $g = 1/b_g$, where $b_g$ is the galaxy bias).
The Fisher matrix for the parameters $P_{i}$ is $${{\mathbf{F}}}_{\rm i j} = \sum_{\rm k-shell} \frac{1}{{\rm var}[P_F({{\boldsymbol{k}}})]} \, \frac{dP_F({{\boldsymbol{k}}})}{dP_i} \frac{dP_F({{\boldsymbol{k}}})}{dP_j},$$ where the sum is over ${\cal N}_{k}$ independent elements in a shell in $k$-space centered at distance $|{{\boldsymbol{k}}}|$ from the origin. In addition, ${\rm var}[P_F({{\boldsymbol{k}}})]$ is the variance on an estimate of $P_F$ in a pixel centered on ${{\boldsymbol{k}}}$.
Let us first approximate ${\rm var} [P_F({{\boldsymbol{k}}})]$ as isotropic, which is the relevant limit for BOSS and, more generally, for all surveys with less than $100$ Ly$\alpha$ forest spectra per degree [@white10; @mcquinnwhite]. In this limit, the error on a measurement of $P_i$ is $$\boldsymbol{\sigma}(k) \equiv {[{{\mathbf{F}}}^{-1}]^{1/2}}_{ii} = \frac{{\rm var} [P_F(k)]^{1/2}}{\sqrt{{\cal N}_{k}}} \; (1.5, \; 4.0 \, g^{-1}, \;10.7 \, g^{-2}).$$ Also informative is the cosmic variance limit in which ${\rm var} [P_F({{\boldsymbol{k}}})] = 2\, b^4 P_\Delta(k)^2 \; (1+ g\mu^2)^4$, and for $g=1$ results in $$\boldsymbol{\sigma}(k) = \sqrt{\frac{2}{{\cal N}_{k}}}\, b^2 \, P_\Delta (k) \; (2.1, \;20.4, 28.2).$$ In both limits, a survey detects $P_0$ at the highest significance. $P_1$ reveals how the temperature and photoionization fluctuations correlate with those of density. Because BOSS can constrain $P_F$ in bins of $\Delta k \sim k$ at $z\sim 2.5$ at the $1\%$ level, it should also be able to place $10\%$-level constraints on $P_1$.
Conclusions
===========
Significant temperature fluctuations with $\Delta T/T \approx 1$ were likely imprinted in the high-redshift IGM by reionization processes, and they would have lasted for roughly a Hubble time thereafter. Temperature fluctuations would have been imprinted at $z\sim 3$ if [He[ ii]{}]{} reionization were ending near this redshift and, thereby, affected the majority of Ly$\alpha$ forest spectra. In addition, if hydrogen reionization ended at $z\approx 6$, remnant temperature fluctuations could still be observable in the forest from redshifts as low as $z =4$ (e.g., @cen09). This paper investigated the detectability of these fluctuations. Previous studies have shown that the $z=2-4$ IGM cannot have been half filled with $\sim 10~$Mpc patches of temperature $20~$kiloK and the rest at $10$ kiloK [@2002MNRAS.332..367T; @lidz09]. However, we showed that half with $T_0 = 25~$kiloK and half with $T_0 = 15~$kiloK is consistent with recent Ly$\alpha$ forest analyzes. Such a temperature distribution is also close to what is found near the end of [He[ ii]{}]{} reionization in the radiative transfer simulations of @mcquinn09. If fact, we showed that the temperature history in the fiducial simulation of [He[ ii]{}]{} reionization in @mcquinn09 is grossly consistent with previous Ly$\alpha$ forest measurements, although the observations tentatively favor less evolution in the mean temperature. The temperature fluctuations in this simulation do not alter the small-scale Ly$\alpha$ forest power spectrum (Section \[ss:sslos\]), the PDF of the normalized flux (Section \[ss:FPDF\]), the large-scale line-of-sight power spectrum (Section \[ss:LSC\]), the wavelet PDF (Appendix A), and the three-point statistic proposed in @zaldarriaga01b (Appendix A) in a manner that is distinguishable from a power-law $T-\Delta_b$ relation. Thus, the temperature fluctuations produced during a $z\sim3$ [He[ ii]{}]{} reionization by quasars would probably have evaded previous searches and also likely would evade future searches using these standard statistics.
Interestingly, we find that realistic models for large-scale temperature fluctuations could have a significant effect on the $3$D Ly$\alpha$ forest power spectrum. This statistic can be measured by cross-correlating multiple quasar sightlines. In the line-of-sight power spectrum, the aliasing of small-scale $3$D density power to larger scale line-of-sight modes dramatically suppresses the prominence of temperature fluctuations for physically motivated models. However, we found that physically motivated temperature models could impart an order unity increase in the $3$D power at $k \sim 0.1~$Mpc$^{-1}$. We showed that the impact of temperature fluctuations on the $3$D power spectrum could be understood with a simple analytic model.
Intensity fluctuations could also alter the large-scale correlations in the Ly$\alpha$ forest. These too would have been hidden in the line-of-sight power spectrum by aliasing effects. At sufficiently large scales, intensity fluctuations will be the dominant source of fluctuations in the forest, and the fluctuations will become more prominent with increasing redshift. The impact of intensity fluctuations on $P_F$ is likely to be distinguishable spectrally from the impact of temperature inhomogeneities.
BOSS is forecasted to measure the $3$D flux power spectrum $P_F$ with percent-level accuracy at $z\approx 2.5$ in a $k$-space shell with $\Delta k \sim k$. Such a measurement would constrain the effects discussed here, and we anticipate that these extra sources of fluctuations can be distinguished from other uncertainties in $P_F$ (such as in the cosmology) when their impact is larger than $\sim 10\%$. In addition, BOSS is capable of placing interesting constraints on this statistic to redshifts as high as $z\approx 3$ and BigBOSS to $z\approx 4$. Deeper surveys on an $8-10~$m class telescope can push this measurement to even higher redshifts [@mcquinnwhite]. Furthermore, $P_F$ can be separated into different angular components with BOSS and future quasar surveys, which will facilitate the separation of the density, intensity, and temperature contributions. The cross correlation of the Ly$\alpha$ forest with another tracer of large-scale structure (such as a high-z galaxy survey) could also enable this separation [@2011MNRAS.410.1130G; @mcquinnwhite].
The impact of temperature and intensity fluctuations will complicate attempts to constrain cosmic distances using the BAO features in the $3$D Ly$\alpha$ forest. Because there is not one unique template for the spectrum of $T$ and $J$ fluctuations, marginalizing over their potential impact would likely require a fairly general parameterization for the continuum on which the BAO sits. Such a marginalization procedure would reduce the sensitivity to cosmological parameters. Temperature and intensity fluctuations may also impart (via their correlation with density) a scale dependence to the BAO amplitude.
@cen09 found that temperature fluctuations from models of [H[ i]{}]{} reionization had a larger effect on the line-of-sight Ly$\alpha$ forest power spectrum than we have found in our temperature models. They examined $P_{F}^{\rm los}$ at $z=4$ and $z=5$, using simulations of hydrogen reionization. They found that temperature fluctuations resulted in an increase in $P_{F}^{\rm los}$ by $5-10\%$ at $k \sim 10^{-2}~$s km$^{-1}$ and $20-30\%$ at $k \sim 10^{-3}~$s km$^{-1}$ – the latter being roughly an order of magnitude larger than what our temperature models produce at $z=3$. The larger effect found in @cen09 likely owes to two reasons. First, if the hydrogen were reionized by numerous dwarf galaxies as is assumed in the @cen09 simulations, there would have been a stronger anti-correlation after this process completed between $T$ and the large-scale density at relevant scales than in our [He[ ii]{}]{} reionization simulations. The effect of temperature fluctuations on $P_{F}^{\rm los}$ could be significantly enhanced by such an anti-correlation, as can be noted from equation (\[eqn:Pklg2\]). Second, the density power decreases with increasing redshift, which results in the temperature fluctuations (which tend to have $\Delta T/T \sim 1$ because of the photoionization physics) becoming relatively larger. The impact of temperature fluctuations in the @cen09 models should be even more dramatic on the $3$D Ly$\alpha$ forest power spectrum.\
We would like to thank Steven Furlanetto, Shirley Ho, Avi Loeb, and Nic Ross for useful discussions, and Martin White for useful comments on the manuscript. We thank Claude-Andr[é]{} Faucher-Gigu[è]{}re for providing the hydrodynamic simulations used in some of our calculations. MM is supported by the NASA Einstein Fellowship.
Other Line-of-Sight Statistics
==============================
Wavelets {#ss:wavelet}
--------
Several studies have suggested using wavelet functions to search for temperature fluctuations in the Ly$\alpha$ forest [@meiksin00; @2002MNRAS.332..367T; @zaldarriaga02]. This method uses the wavelet property that they are localized in configuration in addition to Fourier space. The idea is to convolve the Ly$\alpha$ forest transmission field with a wavelet that is sensitive to the amount of small-scale power to search for spatial variations in the power and, thus, temperature fluctuations.
We compare the wavelet predictions of different temperature models with the recent measurement by @lidz09. This study applied this technique to $40$ high resolution, high S/N Ly$\alpha$ forest spectra spanning $2 \lesssim z \lesssim 4.5$. The wavelet filter that @lidz09 used is $$\Psi(x) = C\, \exp \left({\it i} k_0 x \right) \; \exp \left[-\frac{x^2}{2 \,s_n^2} \right].$$ @lidz09 chose $C$ such that $\Sigma_{i=1}^N \Delta x \, \Psi(x)^2/N = 1$ for $\Delta x = 4.4~$km s$^{-1}$. @lidz09 found $s_n = 69.7$ km s$^{-1}$ and $k_0 \, s_n = 6$ to be a good compromise between maximizing sensitivity to temperature while minimizing the impact of instrumental noise. Conveniently, the Fourier transform of $\Psi(x)$, which we denote as $\tilde{\Psi}(k)$, is a Gaussian centered around $ k = k_0$ with standard deviation $s_n^{-1}$. The full width half maximum of $\tilde{\Psi}(k)$ spans $0.06 < k < 0.1~$s km$^{-1}$, and $\tilde{\Psi}(k)$ is plotted in the top panels in Figures \[fig:pk\] and \[fig:pk2\] (thin curves, arbitrarily normalized). The wavelet filter that was used in other wavelet studies of the forest is qualitatively similar to the @lidz09 filter, and we expect our conclusions are robust to the exact filter choice (such as the curvature statistic of @becker10). @lidz09 primarily analyzed the PDF of $$A_L (x) = \frac{1}{2\, L} \int_{-L}^{L} dx \, \left | \Psi(x) \circ \delta_F(x) \right |^2,$$ where $``\circ"$ denotes a convolution, and the integral averages the convolved signal over $L = 500~$km s$^{-1}$ in order to reduce the noise. Thus, their $A_L$ was the average of the wavelet power in a $\approx 10~$Mpc region. The mean of this PDF is sensitive to the average temperature (because it is a measure of the average power within the wavelet bandpass), whereas the width is a measure of the spatial variance in the temperature. However, cosmic variance in the forest is the primary determinant of the width of this PDF, and temperature fluctuations would manifest as an excess in the width over what is expected from simple models for the IGM thermal state.
Figures \[fig:wavelet\] and \[fig:waveletZ4\] plot the predictions for the wavelet PDF at $z=3$ and $z=4.2$ using different temperature models. These curves were calculated from the same simulation skewers as the $P_F^{\rm los}$ curves in Figures \[fig:pk\] and \[fig:pk2\]. The points with error bars are the measured values from @lidz09. The top panel in both figures explore toy power-law $T-\Delta_b$ models. (See @lidz09 for a more extensive comparison of such models.)
The wavelet PDF of the $\gamma = 1.3$, $T_0 = 10~$kiloK model is quite discrepant with that of the $\gamma = 1.3$, $20~$kiloK model (Fig. \[fig:wavelet\]). The data favors the latter of these two models, and would also be inconsistent with a $50\%$ mix of both temperatures (which would be the average of these two curves). Section \[sec:background\] suggested that a more realistic toy fluctuating-temperature model is half the volume with $T_0 = 15~$kiloK and $\gamma = 1.5$ and the other half with $T_0 = 25~$kiloK and $\gamma = 1.2$. Unfortunately, the top panel illustrates that $T_0 = 15~$kiloK and $\gamma = 1.5$ produces a very similar wavelet PDF to the $T_0 = 25~$kiloK and $\gamma = 1.2$ case. Thus, as we found for $P_F^{\rm los}$ in Section \[ss:sslos\], a $50\%$ mix of these two models would be difficult to distinguish from a single $T_0$ and $\gamma$ model.
The red solid curves in Figures \[fig:wavelet\] and \[fig:waveletZ4\] represent the simulation result near the end of [He[ ii]{}]{} reionization. These curves have a very similar width to the other curves in their respective panel despite the fact that these curves include dispersion in the temperature. The bottom panels show the wavelet PDF at different times during the simulation. Interestingly, none of these PDFs are noticeably broader than the power-law $T-\Delta_b$ case. Over the course of [He[ ii]{}]{} reionization in the simulation, the mean of the PDF shifts to smaller values owing to the heating of the simulated IGM. The teal and red curves represent the $z=3$ and $z=4$ outputs in the simulation. The mean of the PDF in the simulation appears to evolve slightly more between these redshifts than the data, consistent with what we found in Section \[ss:sslos\].
In conclusion, measurements of the wavelet PDF constrain the mean temperature and place limits on temperature fluctuations at the level of $\Delta T/T \approx 1$ for slightly overdense gas. However, temperature fluctuations at the smaller level found in the simulations of @mcquinn09 do not significantly effect the width of the wavelet PDF and would be extremely difficult to detect with this statistic.
Small-Scale Power – Large-Scale Flux Correlation {#sec:3point}
------------------------------------------------
@zaldarriaga01b and @fang04 discussed a particular three-point statistic that they argued provides an excellent test of the paradigm that gravitational instability shapes the transmission fluctuations in the Ly$\alpha$ forest. They advocated the statistic $$C(k) \equiv \frac{P_{h \,\delta_F} (k)} {[P_{h}(k) \, P_{\delta_F} (k)]^{1/2}},$$ where $P_{\delta_F}(k)$ is the power-spectrum of ${\delta}_F$, $P_{h}(k)$ is this of ${h}$, and $P_{h \delta_F}(k)$ is the cross power between ${h}$ and ${\delta}_F$. Here, $h(x) \equiv \delta_H(x)^2$ and $\tilde{\delta}_H \equiv \tilde{\delta}_F(k) W_{k_1, k_2}(k)$, where $W_{k_1, k_2}(k)$ is a band-pass filter that transmits at $k_1 < k < k_2$ and tildes distinguish the Fourier dual. Thus, $h$ is the square of the band pass-filtered flux field, making it a measure of the bandpass power. This statistic is normalized such that a perfect correlation between $h$ with $\delta_F$ yields $C =1$. @fang04 showed that this statistic could place strong constraints on the level of temperature fluctuations.
As in @zaldarriaga01b and @fang04, we take $W_{k_1, k_2}(k)$ to be unity for $k_1 < k < k_2$ and zero otherwise. The statistic $C(k)$ is a measure of the correlation between ${\delta}_F$ and the small-scale power in the high pass-filtered Ly$\alpha$ transmission field. If gravity dominates the fluctuations in the forest, there will be more structure in a large-scale overdense (opaque) region, driving $C(k)$ negative. Predictions for $C(k)$ are shown in the top panel of Figure \[fig:zald3piont\] for different power-law $T-\Delta_b$ models. The curves are calculated from the $1000$, $25/h~$Mpc random skewers drawn from the $z=3$ snapshot of the hydrodynamic simulation and with $k_1 = 0.1$ and $k_2 = 0.2~$s km$^{-1}$. These curves all have $-0.6 < C < -0.5$ at $k \lesssim 0.01~$km s$^{-1}$, and the curves with the largest $\gamma$ have slightly smaller $|C|$. This trend results because, the larger the deviation from an isothermal relation, the more decorrelated a large-scale density mode is from the small-scale density power [@zaldarriaga01b].
Temperature fluctuations from reionization could also decorrelate the small-scale density power from the large-scale flux. However, temperature fluctuations could also enhance the negative correlation because a large-scale hot region has increased transmission and also less small-scale power. We find that the former effect is most important in our reionization simulations: These temperature fluctuations decrease $|C|$, but not significantly. The red solid curve in the top panel of Figure \[fig:zald3piont\] uses the temperature fluctuations from the $x_{\rm HeIII} = 0.94$ snapshot, and we find that $|C|$ is slightly smaller for this case compared to the power-law $T-\Delta_b$ models. The bottom panel in Figure \[fig:zald3piont\] plots $C$ using the temperature field from different snapshots of the [He[ ii]{}]{} reionization simulation.
@fang04 showed that adding a lognormal dispersion at the Jeans scale around the mean $T-\Delta_b$ relation with standard deviation $0.2$ results in $|C|$ becoming significantly smaller. They used this result to constrain the standard deviation in this relation to be $< 0.2$ from a single quasar sightline. The temperature fluctuations in our [He[ ii]{}]{} reionization simulations have a standard deviation of $\sigma_T \approx 0.1$ (Fig. \[fig:Tdeltadist\]), but the simulations’ temperature fluctuations are correlated over $10$s of Mpc: In a large-scale hot or cold region, the dispersion in $T-\Delta_b$ is much smaller (Fig. \[fig:Tdelta\_local\]). Likely because of these large-scale correlations, we find a significantly smaller suppression than in @fang04. @fang04 found the most dramatic effect for a higher pass filter than used in this section, in particular with $k_1 = 0.2$ and $k_2 = 0.3~$s km$^{-1}$. We find that if we use this filter choice, $|C|$ is reduced by a larger factor than for our fiducial filter (to a value as low as $-0.3$) but never to zero. However, even the choice $k_1 = 0.1$ and $k_2 = 0.2~$s km$^{-1}$ is pushing the limit on what can be applied to even the highest quality Ly$\alpha$ forest data [@lidz09].
Peculiar Velocities {#app:lognormal}
===================
This Appendix quantifies the impact of peculiar velocities on large-scale flux correlations. Peculiar velocities have two effects: (1) The large-scale peculiar velocity field results in a redshift-space compression so that more systems appear in regions with converging flows. (2) The nonlinear peculiar velocity compresses or dilates Jeans-scale dense regions in absorption space (and dilates Jeans-scale voids). On large scales, the latter effect enters by altering the bias of the forest. The former effect produces the redshift-space anisotropy of this signal. It affects $\tau_{\rm Ly\alpha}$ via the factor $(H \, a + dv/dx)^{-1}$, where $v$ and $x$ are the peculiar velocity and comoving distance along the line of sight. In Fourier space this factor is simpler and approximately equal to $1+ \Omega_m(z)^{0.6} \mu^2 \delta$, where $\mu = \hat{k} \cdot \hat{n}$ [@kaiser87]. Thus, on linear scales the redshift-space power spectrum of $\delta_F$ is likely to have the form $$P_{F} \approx b^2 \left[ G^2 \, P_\Delta + 2\, G \, \epsilon^{-1} \, P_{\Delta X} + \epsilon^{-2} \, P_{X} \right],
\label{eqn:Pklgnormal_largescales}$$ where $G = (1 + g \mu^2)$, $g$ is the large-scale bias of velocity fluctuations, and $X$ is a placeholder for $T$ or $-J$ fluctuations. The average of $G^2$ over solid angle is $28/15$. In linear theory, $g \approx \Omega_m(z)^{0.6} \, \epsilon^{-1}$. However, $g$ will depart from the linear theory prediction in part because the absorption saturates in regions. In the limit that all the absorption is from saturated lines with bias $b$, $g = 1/b$. Interestingly, @mcdonald03 measured $g=1.5$ at $z=2$, larger than the linear theory prediction of $g\approx 0.5$. @slosar09 found a value closer to unity from large box, low resolution simulations. We have also measured $g$ from our $25/h~$Mpc, $2\times 512^3$ hydrodynamic simulation and find values that are consistent with $g \approx 0.5-1$ (Fig. \[fig:pkani\]), although a larger box size is needed for a more precise determination.
A Second Derivation of Effect of Intensity Fluctuations on $P_F$ {#sec:oldcalc}
================================================================
This section develops a more sophisticated understanding for how $\Delta_F^2$ is altered by large-scale temperature fluctuations. Let us assume $\xi_{Tp7}(r, \Delta_b)$ does not depend on $\Delta_b$, where $\xi_{Tp7}(r, \Delta_b)$ is defined as the correlation function of $T_\Delta^{-0.7}/\langle T_\Delta^{-0.7}\rangle-1$ and $T_\Delta$ is the mean temperature at a given $\Delta_b$. This differs somewhat from our previous definition of the $\delta_{Tp7}$ field as the fluctuation field, but the distinction makes little difference in practice.
We define $\tau_0$ as the Gunn-Peterson optical depth at $\Delta_b= 1$, and the value of $\tau_0$ is set by the mean flux normalization. We also define $X \equiv \Delta^{2-.7\gamma}$ and the normalized flux as $F \equiv \exp[-\tau_0 X (1 + \delta_{T})]$. Finally, we neglect peculiar velocities, and we assume that the temperature fluctuations are uncorrelated with those in density and that they are Gaussian. While we found in Section \[ss:HI\_3D\] that the correlations with density are important and it is likely that the temperature fluctuations are not Gaussian, the results of this model are illustrative and generalize beyond these assumptions.
With these assumptions and definitions, the correlation between the normalized flux in a region with density $\Delta_1$ and a region separated by a distance $r$ with $\Delta_2$ is $$\begin{aligned}
\langle F_1 F_2 \rangle_T &=& \int d \delta_{T_1} d \delta_{T_2} ~e^{-\tau_0 \left(X_1 (1 + \delta_{T_1})+ X_2 (1 + \delta_{T_2}) \right)} \nonumber \\
& &\times \, \frac{e^{-\frac{1}{2}(\delta_{T_1} \delta_{T_2}) {{\mathbf{C}}}^{-1} (\delta_{T_1} \delta_{T_2})^T}}{\sqrt{(2 \pi)^2 \det {{\mathbf{C}}}}} , \\
&=& e^{-\tau_0 (X_1+ X_2)} \times \exp[\frac{\tau_0^2 \sigma_{Tp7}^2}{2} \, (X_1^2 + X_2^2)] \nonumber \\
& & \times \exp\left[\tau_0^2 \,X_1 X_2 \, \xi_{Tp7}(r)) \right], \label{eqn:prevcorr}\\
&\approx & \tilde{F}_1 \tilde{F}_2 \left(1 + \tau_0^2 X_1 X_2 \,\xi_{Tp7}(r)\right), \label{eqn:corr}\end{aligned}$$ where $\tilde{F} \equiv \exp[-\tau_0 X]$, ${{\mathbf{C}}}\equiv (1,~\xi_{Tp7}/\sigma_{Tp7}^2;~ \xi_{Tp7}/\sigma_{Tp7}^2,~ 1)$, and $\langle ... \rangle_Y$ represents an ensemble average with respect to $Y$. To go from equation (\[eqn:prevcorr\]) to equation (\[eqn:corr\]), we used that $\xi_{Tp7}(r) \ll 1$ to expand the exponential. Furthermore, we set the second exponential term in equation (\[eqn:prevcorr\]) to unity. Since regions that dominate the flux correlation function have $\tau_0 X \sim 1$, the error from dropping this term is of order $\sigma_{Tp7}^2 \ll 1$. While this is comparable to the temperature term that we kept, it is less interesting because it does not depend on $\xi_{Tp7}(r)$. In addition, most of its effect is re-absorbed in the renormalization to a single mean flux.
To calculate the flux correlation function, we average $\langle F_1 F_2 \rangle_T$ over $\Delta_1$ and $\Delta_2$, which yields $$\xi_F \equiv \langle F_1 F_2 \rangle_{T \Delta} \approx \xi_{\tilde{F}(r)} + K^2 \, \xi_T(r) + \tau_0^2 \xi_{\tilde{F}}(r) \, \xi_T(r),
\label{eqn:tempmod}$$ where $\xi_{\tilde{F}}(r)$ is the unperturbed flux correlation function and $$\begin{aligned}
K &=& \tau_0 \, \langle F_1 X_1 \rangle_{\Delta_1} , \\
&=&\tau_0 \int d\Delta \, \Delta^{2-.7\gamma} \, \exp[-\tau_0 \Delta^{2-.7\gamma}] \, p
(\Delta).\end{aligned}$$ The function $p(\Delta_b)$ is the volume-weighted density probability distribution. We have dropped terms in equation (\[eqn:tempmod\]) that involve additional $\xi(r)$ factors as well as the connected moments that are not incorporated in $K^2 \, \xi_T(r)$.
We are interested in the impact of temperature fluctuations on $\gtrsim 10$ Mpc correlations. At these scales, it is justified to drop the last term in equation (\[eqn:tempmod\]). With these simplifications, the power spectrum of fluctuations in the normalized flux is given by $$P_{F}(k) \approx P_{\tilde{F}}(k) + \frac{K^2}{\bar{F}^2} P_{T}(k).
\label{eqn:3Dpower}$$ Thus, the correction proportional to $\xi_T(r)$ has bias $K/\bar{F}$ in this model. We calculate $K^2/\bar{F}^2 = 0.18$ at $z=4$ and $0.04$ at $z=3$, assuming $T_0 = 2 \times 10^4~$K, $\gamma = 1.3$, and $\Gamma_{\rm HI} = 10^{-12}~$s$^{-1}$, using the @miralda00 fitting function for $p(\Delta)$. It turns out that $K^2/\bar{F}^2$ is almost identical numerically to the corresponding factor that appears in equation (\[eqn:Pklg2\]), $b^2/\epsilon^2$.
[^1]: To quantify this, we have solved for the evolution of $T_0$ as in @hui97 after hydrogen and helium reionization, assuming a flat specific intensity at the [H[ i]{}]{}, [He[ i]{}]{}, and [He[ ii]{}]{} ionization edges (but the results depend weakly on this assumption). If half of the gas were at $T_0= 15,000~$K and half at $T_0= 25,000~$K at $z=3$, by $z=2$ the amplitude of fluctuations in $T_0$ would have been $70\%$ of what they had been, half at $17,000~$K and half at $12,000~$K. If instead half the IGM were at $T_0= 15,000~$K and half at $T_0= 25,000~$K at $z=4$, the amplitude of temperature fluctuations at $z=2$ would have been reduced to $40\%$ of its initial value.
[^2]: The calculations in Figures \[fig:aliasing\] assume the simple form $P_F(k, k_{\parallel}) = b^2 \, [1+ g \, (k_{\parallel}/k)^2]^2 \, P_{\Delta}^{\rm lin}(k) \, \exp[-k_b T k_{\parallel}^2/m_p]$, where $P_{\Delta}^{\rm lin}$ is the linear theory overdensity power. We take $g=1$ and $T = 20,000~$K. We find that this functional form provides a decent approximation to the spectrum of $P_F$ in our simulations.
[^3]: <http://cosmology.lbl.gov/BOSS/>, <http://bigboss.lbl.gov/index.html>
[^4]: A commonly used formula for the correlation function of intensity fluctuations, $\xi_{J}$, is $$\xi_J(r) = \frac{1}{3 \, N} \; \frac{\langle L^2 \rangle}{\langle L \rangle^2} \; \frac{\lambda}{r} ~ I_J(\frac{r}{\lambda}),$$ with $$I_J(u) = 2 \int_0^\infty dv \, \frac{v }{\sinh v} \, \exp \left[ - u \, \frac{1 + e^{-v}}{1- e^{-v}} \right],$$ and $N = 4 \pi \lambda^3 \bar{n}/3$. Note that this formula for $\xi_J$ (derived by fairly complicated means in @zuo92b) is just the convolution of $r^{-2} \,\exp[-r/\lambda]$ with itself (times a factor that depends on the luminosity function). Put another way, it is the Fourier transform of the Poissonian term in equation (\[eqn:PJ\]).
|
---
abstract: 'Oil and gas drilling is based, increasingly, on operational technology, whose cybersecurity is complicated by several challenges. We propose a graphical model for cybersecurity risk assessment based on Adversarial Risk Analysis to face those challenges. We also provide an example of the model in the context of an offshore drilling rig. The proposed model provides a more formal and comprehensive analysis of risks, still using the standard business language based on decisions, risks, and value.'
author:
- Aitor Couce Vieira Siv Hilde Houmb
- David Rios Insua
bibliography:
- 'ARAOGgramsec.bib'
title: A Graphical Adversarial Risk Analysis Model for Oil and Gas Drilling Cybersecurity
---
Introduction
============
Operational technology (OT) refers to hardware and software that detects or causes a change through the direct monitoring and/or control of physical devices, processes and events in the enterprise [@gartner]. It includes technologies such as SCADA systems. Implementing OT and information technology (IT) typically lead to considerable improvements in industrial and business activities, through facilitating the mechanization, automation, and relocation of activities in remote control centers. These changes usually improve the safety of personnel, and both the cost-efficiency and overall effectiveness of operations.
The oil and gas industry (O&G) is increasingly adopting OT solutions, in particular offshore drilling, through drilling control systems (drilling CS) and automation, which have been key innovations over the last few years. The potential of OT is particularly relevant for these activities: centralizing decision-making and supervisory activities at safer places with more and better information; substituting manual mechanical activities by automation; improving data through better and near real-time sensors; and optimizing drilling processes. In turn, they will reduce rig crew and dangerous operations, and improve efficiency in operations, reducing operating costs (typically of about \$300,000 per day).
Since many of the involved OT employed in O&G are currently computerized, they have become a major potential target for cyber attacks [@Shauk2013c], given their economical relevance, with large stakes at play. Indeed, we may face the actual loss of large oil reserves because of delayed maneuvers, the death of platform personnel, or potential large spills with major environmental impact with potentially catastrophic consequences. Moreover, it is expected that security attacks will soon target several production installations simultaneously with the purpose of sabotaging production, possibly taking advantage of extreme weather events, and attacks oriented towards manipulating or obtaining data or information. Cybersecurity poses several challenges, which are enhanced in the context of operational technology. Such challenges are sketched in the following section.
Cybersecurity Challenges in Operational Technology
--------------------------------------------------
Technical vulnerabilities in operational technology encompass most of those related with IT vulnerabilities [@byres2004myths], complex software [@DoDDefSci2013], and integration with external networks [@giani2009viking]. There are also and specific OT vulnerabilities [@zhu2011taxonomy; @brenner2013eyes]. However, OT has also strengths in comparison with typical IT systems employing simpler network dynamics.
Sound organizational cybersecurity is even more important with OT given the risks that these systems bring in. Uncertainties are considerable in both economical and technical sense [@anderson2010security]. Therefore better data about intrusion attempts are required for improving cybersecurity [@pfleeger2008cybersecurity], although gathering them is difficult since organizations are reluctant about disclosing such information [@ten2008vulnerability].
More formal approaches to controls and measures are needed to deal with advanced threat agents such as assessing their attack patterns and behavior [@hutchins2011intelligence] or implementing intelligent sensor and control algorithms [@cardenas2008research]. An additional problem is that metrics used by technical cybersecurity to evaluate risks usually tell little to those evaluating or making-decisions at the organizational cybersecurity level. Understanding the consequences of a cyber attack to an OT system is difficult. They could lead to production losses or the inability to control a plant, multimillion financial losses, and even impact stock prices [@byres2004myths]. One of the key problems for understanding such consequences is that OT systems are also cyber-physical systems (CPS) encompassing both computational and complex physical elements [@thomas2013bad].
Risk management is also difficult in this context [@mulligan2011doctrine]. Even risk standards differ on how to interpret risk: some of them assess the probabilities of risk, others focus on the vulnerability component [@hutchins2011intelligence]. Standards also tend to present oversimplifications that might alter the optimal decision or a proper understanding of the problem, such as the well-known shortcomings of the widely employed risk matrices [@cox2008matrix].
Cyber attacks are the continuation of physical attacks by digital means. They are less risky, cheaper, easier to replicate and coordinate, unconstrained by distance [@cardenas2009challenges], and they could be oriented towards causing high impact consequences [@DoDDefSci2013]. It is also difficult to measure data related with attacks such as their rate and severity, or the cost of recovery [@anderson2010security]. Examples include Stuxnet [@brenner2013eyes], Shamoon [@brenner2013eyes], and others [@cardenas2008research]. Non targeted attacks could be a problem also.
Several kinds of highly skilled menaces of different nature (e.g., military, hacktivists, criminal organizations, insiders or even malware agents) can be found in the cyber environment [@DoDDefSci2013], all of them motivated and aware of the possibilities offered by OT [@byres2004myths]. Indeed, the concept Advanced Persistent Threat (APT) has arisen to name some of the threats [@Ltd2011]. The diversity of menaces could be classified according their attitude, skill and time constraints [@dantu2007classification], or by their ability to exploit, discover or even create vulnerabilities on the system [@DoDDefSci2013]. Consequently, a sound way to face them is profiling [@atzeni2011here] and treating [@li2009botnet] them as adversarial actors.
Related Work Addressing the Complexities of Cybersecurity Challenges
--------------------------------------------------------------------
Several approaches have been proposed to model attackers and attacks, including stochastic modelling [@muehrcke2010behavior; @sallhammar2007stochastic], attack graph models [@kotenko2006attack] and attack trees [@mauw2006foundations], models of directed and intelligent attacks [@ten2008vulnerability]; models based on the kill chain attack phases [@hutchins2011intelligence], models of APT attack phases [@Ltd2011], or even frameworks incorporating some aspects of intentionality or a more comprehensive approach to risk such as CORAS [@lund2011model] or ADVISE [@Advise2013].
Game theory has provided insights concerning the behavior of several types of attackers such as cyber criminal APTs and how to deal with them. The concept of incentives can unify a large variety of agent intents, whereas the concept of utility can integrate incentives and costs in such a way that the agent objectives can be modeled in practice [@liu2005incentive]. Important insights from game theory are that the defender with lowest protection level tends to be a target for rational attackers [@Johnson2011], that defenders tend to under-invest in cybersecurity [@amin2011interdependence], and that the attackers target selection is costly and hard, and thus it needs to be carefully carried on [@florencio2013all]. In addition to such general findings, some game-theoretic models exist for cybersecurity or are applicable to it, modelling static and dynamic games in all information contexts [@roy2010survey]. However, game-theoretic models have their limitations [@hamilton2002challenges; @roy2010survey] such as limited data, the difficulty to identify the end goal of the attacker, the existence of a dynamic and continuous context, and that they are not scalable to the complexity of real cybersecurity problems in consideration. Moreover, from the conceptual point of view, they require common knowledge assumptions that are not tenable in this type of applications.
Additionally, several Bayesian models have been proposed for cybersecurity risk management such as a model for network security risk analysis [@xie2010using]; a model representing nodes as events and arcs as successful attacks [@dantu2007classification]; a dynamic Bayesian model for continuously measuring network security [@frigault2008measuring]; a model for Security Risk Management incorporating attacker capabilities and behavior [@dantu2009network]: or models for intrusion detection systems (IDS) [@balchanos2012probabilistic]. However, these models require forecasting attack behavior which is hard to come by.
Adversarial Risk Analysis (ARA) [@rios2009adversarial] combine ideas from Risk Analysis, Decision Analysis, Game-Theory, and Bayesian Networks to help characterizing the motivations and decisions of the attackers. ARA is emerging as a main methodological development in this area [@merrick2011comparative], providing a powerful framework to model risk analysis situations with adversaries ready to increase our threats. Applications in physical security may be seen in [@sevillano2012adversarial].
Our Proposal
------------
The challenges that face OT, cybersecurity and the O&G sector create a need of a practical, yet rigorous approach, to deal with them. Work related with such challenges provides interesting insights and tools for specific issues. However, more formal but understandable tools are needed to deal with such problems from a general point of view, without oversimplifying the complexity underlying the problem. We propose a model for cybersecurity risk decisions based on ARA, taking into account the attacker behavior. Additionally, an application of the model in drilling cybersecurity is presented, tailored to decision problems that may arise in offshore rigs employing drilling CS.
Model
=====
Introduction to Adversarial Risk Analysis
-----------------------------------------
ARA aims at providing one-sided prescriptive support to one of the intervening agents, the Defender (she), based on a subjective expected utility model, treating the decisions of the Attacker (he) as uncertainties. In order to predict the Attackers actions, the Defender models her decision problem and tries to assess her probabilities and utilities but also those of the Attacker, assuming that the adversary is an expected utility maximizer. Since she typically has uncertainty about those, she models it through random probabilities and uncertainties. She propagates such uncertainty to obtain the Attacker’s optimal random attack, which she then uses to find her optimal defense.
ARA enriches risk analysis in several ways. While traditional approaches provide information about risk to decision-making, ARA integrates decision-making within risk analysis. ARA assess intentionality thoroughly, enabling the anticipation and even the manipulation of the Attacker decisions. ARA incorporates stronger statistical and mathematical tools to risk analysis that permit a more formal approach of other elements involved in the risk analysis. It improves utility treatment and evaluation. Finally, an ARA graphical model improves the understandability of complex cases, through visualizing the causal relations between nodes.
The main structuring and graphical tool for decision problems are Multi-Agent Influence Diagrams (MAID), a generalization of Bayesian networks. ARA is a decision methodology derived from Influence Diagrams, and it could be structured with the following basic elements:
- *Decisions or Actions*. Set of alternatives which can be implemented by the decision makers. They represent what one can do. They are characterized as decision nodes (rectangles).
- *Uncertain States*. Set of uncontrollable scenarios. They represent what could happen. They are characterized as uncertainty nodes (ovals).
- *Utility and Value*. Set of preferences over the consequences. They represent how the previous elements would affect the agents. They are characterized as value nodes (rhombi).
- *Agents*. Set of people involved in the decision problem: decision makers, experts and affected people. In this context, there are several agents with opposed interests. They are represented through different colors.
We describe now the basic MAID that may serve as a template for cybersecurity problems in O&G drilling CS, developed using GeNIe [@genie].
Graphical Model
---------------
Our model captures the Defender cybersecurity main decisions prior to an attack perpetrated by an APT, which is strongly business-oriented. Such cyber criminal organization behavior suits utility-maximizing analysis, as it pursues monetary gains. A sabotage could also be performed by this type of agents, and they could be hired to make the dirty job for a foreign power or rival company. We make several assumptions in the Model, to make it more synthetic:
- We assume one Defender. The Attackers nodes do not represent a specific attacker, but a generalization of potential criminal organizations that represent business-oriented APTs, guided mostly by monetary incentives.
- We assume an atomic attack (the attacker makes one action), with several consequences, as well as several residual consequences once the risk treatment strategy is selected.
- The Defender and Attacker costs are deterministic nodes.
- We avoid detection-related activities or uncertainties to simplify the Model. Thus, the attack is always detected and the Defender is always able to respond to it.
- The scope of the Model is an assessment activity prior to any attack, as a risk assessment exercise to support incident handling planning.
- The agents are expected utility maximizers.
- The Model is discrete.
By adapting the proposed template in Figure 1, we may generalize most of the above assumptions to the cases at hand.
**Figure 1**. MAID of the ARA Model for O&G drilling cybersecurity.
### Defender Decision and Utility Nodes
The Defender nodes, in white, are:
- *Protect* (*DP*) decision node. The Defender selects among security measures portfolios to increase protection against an Attack, e.g., access control, encryption, secure design, firewalls, or personal training and awareness.
- *Forensic System* (*DF*) decision node. The Defender selects among different security measures portfolios that may harm the Attacker, e.g., forensic activities that enable prosecution of the Attacker.
- *Residual Risk Treatment* (*DT*) decision node. This node models Defender actions after the assessment of other decisions made by the Defender and the Attacker. They are based on the main risk treatment strategies excluding risk mitigation, as they are carried on through the Protect and the Respond and Recovery nodes: avoiding, sharing, or accepting risk. This node must be preceded by the Protect defender decision node, and it must precede the Attack uncertainty node (the residual risk assessment is made in advance).
- *Respond and Recovery* (*DR*) decision node. The Defender selects between different response and recovery actions after the materialization of the attack, trying to mitigate the attack consequences. This will depend on the attack uncertainty node.
- *Defender Cost* (*DC*) deterministic node. The costs of the decisions made by the Defender are deterministic, as well as the monetary consequences of the attack (the uncertainty about such consequences is solved in the Monetary Consequences node). In a more sophisticated model, most of the costs could be modeled as uncertain nodes. This node depends on all decision nodes of the Defender and the Monetary Consequences uncertainty node.
- *Value Nodes* (*DCV* and *DHV*). The Defender evaluates the consequences and costs, taking into account her risk attitude. They depend on the particular nodes evaluated at each Value node.
- *Utility Nodes* (*DU*). This node merges the Value nodes of the Defender. It depends on the Defenders Value nodes.
The Decision nodes are adapted to the typical risk management steps, incorporating ways of evaluating managing sound organizational cybersecurity strategy, which takes into account the business implications of security controls, and prepare the evaluation of risk consequences. Related work (Section 1.2) on security costs and investments could incorporate further complexities underlying the above nodes.
### Attacker Decision and Utility Nodes
The Attacker nodes, in black, are:
- *Perpetrate* (AP) decision node. The [\[]{}generic[\]]{} Attacker decides whether he attacks or not. It could be useful to have a set of options for a same type of attack (e.g., preparing a quick and cheap attack, or a more elaborated one with higher probabilities of success). It should be preceded by the Protect and Residual Risk Treatment decision nodes, and might be preceded by the Contextual Threat node (in case the Attacker observes it).
- *Attacker Cost* (*AC*) deterministic node. Cost of the Attacker decisions. Preceded by the Perpetrate decision node.
- *Value Nodes* (*AMV* and *ACV*). The Attacker evaluates the different consequences and costs, taking into account his risk attitude. They depend on the deterministic or uncertainty nodes evaluated at each Value node.
- *Utility Nodes* (*AU*). It merges the Value nodes of the Attacker to a final set of values. It must depend on the Attackers Value nodes.
These nodes help in characterizing the Attacker, avoiding the oversimplification of other approaches. Additionally, the Defender has uncertainty about the Attacker probabilities and utilities. This is propagated over their nodes, affecting the Attacker expected utility and optimal alternatives, which are random. Such distribution over optimal alternatives is our forecast for the Attacker’s actions.
### Uncertainty Nodes
The uncertainty nodes in grey are:
- *Contextual Threats* (*UC*) uncertainty node. Those threats (materialized or not) present during the Attack. The Attacker may carry out a selected opportunistic Attack (e.g. hurricanes or a critical moment during drilling).
- *Attack* (*UA*) uncertainty node. It represents the likelihood of the attack event, given its conditioning nodes. It depends on the Perpetrate decision node, and on the Protect decision node.
- *Consequences* (*UM* and *UV*) uncertainty node. It represents the likelihood of different consequence levels that a successful attack may lead to. They depend on the Attack and Contextual Threat uncertainty nodes, and on the Respond and Recovery decision node.
- *Residual Consequences* (*URH*) uncertainty node. It represents the likelihood of different consequence levels after applying residual risk treatment actions. They depend on the Consequence node modelling the same type of impact (e.g., human, environmental, or reputation).
- *Counter-Attack* (*UCA*) uncertainty node. Possibility, enabled by a forensic system, to counter-attack and cause harm to the Attacker. Most of the impacts may be monetized. It depends on the Forensic System decision node.
Dealing with the uncertainties and complexities and obtaining a probability distribution for these nodes could be hard. Some of the methodologies and findings proposed in the sections 1.1 and 1.2 are tailored to deal with some of these complexities. Using them, the Model proposed in this paper could lead to limit the uncertainties in cybersecurity elements such as vulnerabilities, controls, consequences, attacks, attacker behavior, and risks. This will enable achieving simplification, through the proposed Model, without limiting the understanding of the complexities involved, and a sounder organizational cybersecurity.
Example
=======
We present a numerical example of the previous Model tailored to a generic decision problem prototypical of a cybersecurity case that may arise in O&G offshore rig using drilling CS. The model specifies a case in which the driller makes decisions to prevent and respond to a cyber attack perpetrated by a criminal organization with APT capabilities, in the context of offshore drilling and drilling CS. The data employed in this example are just plausible figures helpful to provide an overview of the problems that drilling cybersecurity faces. Carrying on the assessment that the Model enables may be helpful for feeding a threat knowledge base, incident management procedures or incident detection systems.
The context is that of an offshore drilling rig, a floating platform with equipment to drill a well through the seafloor, trying to achieve a hydrocarbon reservoir. Drilling operations are dangerous and several incidents may happen in the few months (usually between 2 or 4) that the entire operation may last. As OT, drilling CS may face most of the challenges presented in Section 1.1 (including being connected to Enterprise networks, an entry path for attackers) in the context of high-risk incidents that occur in offshore drilling.
Agent Decisions
---------------
### Defender Decisions
The Defender has to make three decisions in advance of the potential attack. In the Protect decision node (DP), the Defender must decide whether she invests in additional protection: if the Defender implements additional protective measures, the system will be less vulnerable to attacks. In the Forensic System decision node (DF), the Defender must decide whether she implements a forensic system or not. Implementing it enables the option of identifying the Attacker and pursuing legal or counter-hacking actions against him. The Residual Risk Treatment decision node (DT) represents additional risk treatment strategies that the Defender is able to implement: avoiding (aborting the entire drilling operation to elude the attack), sharing (buying insurance to cover the monetary losses of the attack), and accepting the risk (inheriting all the consequences of the attack, conditional on to the mitigation decisions of DP, FD, and DR).
Additionally, the Respond and Recovery decision node (DR) represents the Defenders decision between continuing and stopping the drilling operations as a reaction to the attack. Continuing the drilling may lead to worsen the consequences of the attack, whereas stopping the drilling will incur in higher costs due to holding operations. This is a major issue for drilling CS. In general, critical equipment should not be stopped, since core operations or even the safety of the equipment or the crew may be compromised.
### Attacker Decisions
For simplicity, in the Perpetrate decision node (AP) the Attacker decides whether he perpetrates the attack or not, although further attack options could be added. In this example, the attack aims at manipulating the devices directly under control of physical systems with the purpose of compromising drilling operations or harming equipment, the well, the reservoir, or even people.
Threat Outcomes and Uncertainty
-------------------------------
### Outcomes and Uncertainty during the Incident
The Contextual Threats uncertainty node (UC) represents the existence of riskier conditions in the drilling operations (e.g., bad weather or one of the usual incidents during drilling), which can clearly worsen the consequences of the attack. In this scenario, the Attacker is able to know, to some extent, these contextual threats (e.g., a weather forecast, a previous hacking in the drilling CS that permits the attacker to read what is going on in the rig).
The Attack uncertainty node (UA) represents the chances of the Attacker of causing the incident. If the Attacker decides not to execute his action, no attack event will happen. However, in case of perpetration, the chances of a successful attack will be lower if the Defender invests in protective measures (DC node). An additional uncertainty arises in case of materialization of the attack: the possibility to identify and counter-attack the node, represented by the Counter-Attack uncertainty node (UCA).
If the attack happens, the Defender will have to deal with different consequence scenarios. The Monetary (UM) and Human Consequences (UH) nodes represent the chances of different consequences or impact levels that the Defender may face. The monetary consequences refer to all impacts that can be measured as monetary losses, whereas human consequences represent casualties that may occur during an incident or normal operations. However, the Defender has the option to react to the attack by deciding whether she continues or stops the drilling (DR node). If the Defender decides to stop, there will be lower chances of casualties and lower chances of worst monetary consequences (e.g., loss of assets or compensations for injuries or deaths), but she will have to assume the costs of keeping the rig held (one day in our example) to deal with the cyber threat.
### Outcomes and Uncertainty in Risk Management Process
The previous uncertainties appear after the Attacker’s decision to attack or not. The Defender faces additional relevant uncertainties. She must make a decision between avoiding, sharing, or accepting the risk (DT node). Such decision will determine the final or residual consequences. The final monetary consequences are modeled through the Defender Cost deterministic node (DC node), whose outcome represents the cost of different Defender decisions (nodes DP, DF, DT, and DR). In case of accepting or sharing the risk, the outcome of the DC node will also inherit the monetary consequences of the attack (UM node). Similarly, the outcome of the Residual Human Consequences uncertainty node (URH) is conditioned by the risk treatment decisions (DC node) and, in case of accepting or sharing the risk, it will inherit the human consequences of the attack (UH node). If the Defender decides to avoid the risk, she will assume the cost of avoiding the entire drilling operations and will cause that the crew face a regular death risk rather than the higher death risk of offshore operations. If the Defender shares the risk, she will assume the same casualties as in UH and a fixed insurance payment, but she will avoid paying high monetary consequences. Finally, in case the Defender accepts the risk, she will inherit the consequences from the UM and UH nodes.
The Attacker Cost deterministic node (AC) provides the costs (non-uncertain by assumption) of the decision made by the Attacker. Since he only has two decisions (perpetrate or not), the node has only two outcomes: cost or not. This node could be eliminated, but we keep it to preserve the business semantics within the graphical model.
Agent Preferences
-----------------
The Defender aims at maximizing her expected utility, with the utility function being additive, through the Defender Utility node (DU). The Defender key objective is minimizing casualties, but he also considers minimizing his costs (in this example we assume she is risk-neutral). Each objective has its own weight in the utility function.
The objective of the Attacker is to maximize his expected utility, represented by an additive utility function, through the Attacker Utility node (AU). The Attacker key objective is maximizing the monetary consequences for the Defender. We assume that he is risk-averse towards this monetary impact (he prefers ensuring a lower impact than risking the operations trying to get a higher impact). He also considers minimizing his costs (i.e., being identified and perpetrating the attack). Each of these objectives has its own weight in the utility function, and its own value function. The Attacker does not care about eventual victims.
Uncertainty about the Opponent Decisions
----------------------------------------
The Attacker is able to know to some extent the protective decisions of the defender (DP node), gathering information while he tries to gain access to the drilling CS. While knowing if the Defender avoided the risk (avoiding all the drilling operations) is easy, knowing if the Defender chose between sharing or accepting the risk is difficult. The most important factor, the decision between continue or stop drilling in case of an attack, could be assessed by observing the industry or company practices. The Defender may be able to assess also how frequent similar attacks are, or how attractive the drilling rig is for this kind of attacker. In ARA, and from the Defender perspective, the AP node would be an uncertainty node whose values should be provided by assessing the probabilities of the different attack actions, through analyzing the decision problem from the Attacker perspective and obtaining his random optimal alternative.
Example Values
--------------
An annex provides the probability tables of the different uncertainty nodes employed to simulate the example in Genie (Tables 1 to 7). It also provides the different parameters employed in the utility and value functions (Tables 8 to 10). Additionally, the risk-averse values for AMV are obtained with $AMV=\sqrt[3]{\frac{DC}{10^{7}}}$; the risk-neutral values for DCV are obtained with $DCV=1-\frac{DC}{10^{7}}$; and, the values for DHV are 0 in case of victims and 1 in case of no victims.
Evaluation of Decisions
-----------------------
Based on the solution of the example, we may say that the Attacker should not perpetrate his action in case he believes the Defender will avoid or share the risk. However, the Attacker may be interested in perpetrating his action in case he believes that the Defender is accepting the risk. Additionally, the less preventive measures the Defender implements (DP and DT nodes), the more motivated the Attacker would be (if he thinks the Defender is sharing the risk). The Attackers expected utility is listed in Table 11 in the Annex. The Defender will choose in this example not to implement additional protection (DP node) without a forensic system (DF node). If the Defender believes that she is going to be attacked, then she would prefer sharing the risk (DT node) and stop drilling after the incident (DR node). In case she believes that there will be no attack, she should accept the risk and continue drilling. The Defenders expected utilities are listed in Table T12 in Annex.
Thus, the Defender optimal decisions create a situation in which the Attacker is more interested in perpetrating the attack. Therefore, to affect the Attackers behavior, the Defender should provide the image that her organization is concerned with safety, and especially that it is going to share risks. On the other hand, if the Attacker perceives that the Defender pays no attention to safety or that she is going to accept the risk, he will try to carry on his attack. The ARA solution for the Defender is the following:
1. Assess the problem from the point of view of the Attacker. The DT and DR nodes are uncertainty nodes since that Defender decisions are uncertain for the Attacker. The Defender must model such nodes in the way that she thinks the Attacker models such uncertainties. In general, perpetrating an attack is more attractive in case the Attacker strongly believes that the Defender is going to accept the risk or is going to continue drilling.
2. Once forecasted the Attackers decision, the Defender should choose between sharing and accepting the risk. Accepting the risk in case of no attack is better than sharing the risk, but accepting the risk in case of attack is worse.
Thus, the key factor for optimizing the decision of the Defender are her estimations on the uncertainty nodes that represent the DT and DR nodes for the attacker. Such nodes will determine the Attacker best decision, and this decision the Defender best decision.
Conclusions and Further Work
============================
We have presented the real problem and extreme consequences that OT cybersecurity in general, and drilling cybersecurity in particular, are facing. We also explained some of the questions that complicate cybersecurity, especially in OT systems. The proposed graphical model provides a more comprehensive, formal and rigorous risk analysis for cybersecurity. It is also a suitable tool, able of being fed by, or compatible with, other more specific models such as those explained in Section 1.
Multi-Agent Influence Diagrams provide a formal and understandable way of dealing with complex interactive issues. In particular, they have a high value as business tools, since its nodes translate the problem directly into business language: decisions, risks, and value. Typical tools employed in widely used risk standards, such as risk matrices, oversimplify the problem and limit understanding. The proposed ARA-based model provides a business-friendly interpretation of a risk management process without oversimplifying its underlying complexity.
The ARA approach permits us to include some of the findings of game theory applied to cybersecurity, and it also permits to achieve new findings. The model provides an easier way to understand the problem but it is still formal since the causes and consequences in the model are clearly presented, while avoiding common knowledge assumptions in game theory.
Our model presents a richer approach for assessing risk than risk matrices, but it still has the security and risk management language. In addition, it is more interactive and modular, nodes can be split into more specific ones. The proposed model can still seem quite formal to business users. However, data can be characterized using ordinal values (e.g., if we only know that one thing is more likely/valuable than other), using methods taken from traditional risk management, employing expert opinion, or using worst case figures considered realistic. The analysis would be poorer but much more operational.
Using the nodes of the proposed model as building blocks, the model could gain in comprehensiveness through adding more attackers or attacks, more specific decision nodes, more uncertainty nodes, or additional consequence nodes, such as environmental impact or reputation. Other operations with significant business interpretation can be done, such as sensitivity analysis (how much the decision-makers should trust a figure) or strength of the influence analysis (which are the key elements).
Its applicability is not exempt of difficulties and uncertainties, but in the same way than other approaches. Further work is needed to verify and validate the model and its procedures (in a similar way to the validation of other ARA-based models[@RiosInsua2013]), and to identify the applicability and usability issues that may arise. The model could gain usability through mapping only the relevant information to decision-makers (roughly, decisions and consequences) rather than the entire model.
[^1]
Appendix: Tables with Example Data {#appendix-tables-with-example-data .unnumbered}
==================================
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[^1]: **Acknowledgments**\
- Work supported by the EU’s FP7 Seconomics project 285223\
- David Rios Insua grateful to the support of the MINECO, Riesgos project and the Riesgos-CM program
|
---
abstract: 'We determine the set of catenary degrees, the set of distances, and the unions of sets of lengths of the monoid of nonzero ideals and of the monoid of invertible ideals of orders in quadratic number fields.'
address:
- 'Institut für Mathematik und wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, NAWI Graz, Heinrichstra[ß]{}e 36, 8010 Graz, Austria'
- 'Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003, United States'
author:
- 'Johannes Brantner, Alfred Geroldinger,'
- Andreas Reinhart
title: On monoids of ideals of orders in quadratic number fields
---
[^1]
Introduction {#1}
============
Factorization theory for Mori domains and their semigroups of ideals splits into two cases. The first and best understood case is that of Krull domains (i.e., of completely integrally closed Mori domains). The arithmetic of a Krull domain depends only on the class group and on the distribution of prime divisors in the classes, and it can be studied – at least to a large extent – with methods from additive combinatorics. The link to additive combinatorics is most powerful when the Krull domain has a finite class group and when each class contains at least one prime divisor (this holds true, among others, for rings of integers in number fields). Then sets of lengths, sets of distances and of catenary degrees of the domain can be studied in terms of zero-sum problems over the class group. Moreover, we obtain a variety of explicit results for arithmetical invariants in terms of classical combinatorial invariants (such as the Davenport constant of the class group) or even in terms of the group invariants of the class group. We refer to [@Ge09a] for a description of the link to additive combinatorics and to the recent survey [@Sc16a] discussing explicit results for arithmetical invariants.
Let us consider Mori domains that are not completely integrally closed but have a nonzero conductor towards their complete integral closure. The best investigated classes of such domains are weakly Krull Mori domains with finite $v$-class group and C-domains. For them there is a variety of abstract arithmetical finiteness results but in general there are no precise results. For example, it is well-known that sets of distances and of catenary degrees are finite but there are no reasonable bounds for their size. The simplest not completely integrally closed Mori domains are orders in number fields. They are one-dimensional noetherian with nonzero conductor, finite Picard group, and all factor rings modulo nonzero ideals are finite. Thus they are weakly Krull domains and C-domains. Although there is recent progress for seminormal orders, for general orders in number fields there is no characterization of half-factoriality (for progress in the local case see [@Ka05b]) and there is no information on the structure of their sets of distances or catenary degrees (neither for orders nor for their monoids of ideals).
In the present paper we focus on monoids of ideals of orders in quadratic number fields and establish precise results for their set of distances $\Delta (\cdot)$ and their set of catenary degrees ${\rm Ca}(\cdot)$. Orders in quadratic number fields are intimately related with quadratic irrationals, continued fractions, and binary quadratic forms and all these areas provide a wealth of number theoretic tools for the investigation of orders. We refer to [@HK13a] for a modern presentation of these connections and to [@Co-Ma-Ok17a; @Pe-Za16a] for recent progress on the arithmetic and ideal theoretic structure of quadratic orders.
Let $\mathcal{O}$ be an order in a quadratic number field, $\mathcal{I}^*(\mathcal{O})$ be the monoid of invertible ideals, and $\mathcal{I}(\mathcal{O})$ be the monoid of nonzero ideals (note that $\mathcal{I}(\mathcal{O})$ is not cancellative if $\mathcal{O}$ is not maximal). Since $\mathcal{I}^*(\mathcal{O})$ is a divisor-closed submonoid of $\mathcal{I}(\mathcal{O})$, the set of catenary degrees and the set of distances of $\mathcal{I}^*(\mathcal{O})$ are contained in the respective sets of $\mathcal{I}(\mathcal{O})$. We formulate a main result of this paper and then we compare it with related results in the literature.
\[theorem 1.1\] Let $\mathcal{O}$ be an order in a quadratic number field $K$ with discriminant $d_K$ and conductor $\mathfrak{f}=f\mathcal{O}_K$ for some $f\in\mathbb{N}_{\geq 2}$.
1. The following statements are equivalent[:]{}
1. $\mathcal{I}(\mathcal{O})$ is half-factorial.
2. $\mathsf{c}\big(\mathcal{I}(\mathcal{O})\big)=2$.
3. $\mathsf{c}\big(\mathcal{I}^*(\mathcal{O})\big)=2$.
4. $\mathcal{I}^*(\mathcal{O})$ is half-factorial.
5. $f$ is squarefree and all prime divisors of $f$ are inert.
2. Suppose that $\mathcal{I}^*(\mathcal{O})$ is not half-factorial.
1. If $f$ is squarefree, then ${\rm Ca}\big(\mathcal{I}(\mathcal{O})\big)=[1,3]$, ${\rm Ca}\big(\mathcal{I}^*(\mathcal{O})\big)=[2,3]$,
$\Delta\big(\mathcal{I}(\mathcal{O})\big)=\Delta\big(\mathcal{I}^*(\mathcal{O})\big)=\{1\}$.
2. Suppose that $f$ is not squarefree.
1. If ${\rm v}_2\left(f\right)\not\in\{2,3\}$ or $d_K\not\equiv 1\mod 8$, then ${\rm Ca}\big(\mathcal{I}(\mathcal{O})\big)=[1,4]$,
${\rm Ca}\big(\mathcal{I}^*(\mathcal{O})\big)=[2,4]$, and $\Delta\big(\mathcal{I}(\mathcal{O})\big)=\Delta\big(\mathcal{I}^*(\mathcal{O})\big)=[1,2]$.
2. If ${\rm v}_2\left(f\right)\in\{2,3\}$ and $d_K\equiv 1\mod 8$, then ${\rm Ca}\big(\mathcal{I}(\mathcal{O})\big)=[1,5]$,
${\rm Ca}\big(\mathcal{I}^*(\mathcal{O})\big)=[2,5]$, and $\Delta\big(\mathcal{I}(\mathcal{O})\big)=\Delta\big(\mathcal{I}^*(\mathcal{O})\big)=[1,3]$.
We say that a cancellative monoid $H$ is [*weakly Krull*]{} if $\bigcap_{P\in\mathfrak{X}(H)} H_P=H$ and $\{P\in\mathfrak{X}(H)\mid a\in P\}$ is finite for each $a\in H$ (where $\mathfrak{X}(H)$ denotes the set of height-one prime ideals of $H$). Moreover, a cancellative monoid $H$ is called [*weakly factorial*]{} if every nonunit of $H$ is a finite product of primary elements of $H$. Let all notation be as in Theorem \[theorem 1.1\], and recall that $\mathcal{I}^*(\mathcal{O})$ is a weakly factorial C-monoid, and that for every atomic monoid $H$ with $\Delta(H)\ne\emptyset$ we have $\min\Delta(H)=\gcd\Delta(H)$.
There is a characterization (due to Halter-Koch) when the order $\mathcal{O}$ is half-factorial ([@Ge-HK06a Theorem 3.7.15]). This characterization and Theorem \[theorem 1.1\] or [@Ph12b Corollary 4.6] show that the half-factoriality of $\mathcal{O}$ implies the half-factoriality of $\mathcal{I}^*(\mathcal{O})$. Consider the case of seminormal orders whence suppose that $\mathcal{O}$ is seminormal. Then $f$ is squarefree (this follows from an explicit characterization of seminormal orders given by Dobbs and Fontana in [@Do-Fo87 Corollary 4.5]). Moreover, $\mathcal{I}^*(\mathcal{O})$ is seminormal and if $\mathcal{I}^*(\mathcal{O})$ is not half-factorial, then its catenary degree equals three by [@Ge-Ka-Re15a Theorems 5.5 and 5.8]. Clearly, this coincides with 2.(a) of the above theorem. Among others, Theorem \[theorem 1.1\] shows that the sets of distances and of catenary degrees are intervals and that the minimum of the set of distances equals $1$. We discuss some analogous results and some results which are in sharp contrast to this. If $H$ is a Krull monoid with finite class group, then $H$ is a weakly Krull C-monoid and if there are prime divisors in all classes, then the sets ${\rm Ca}(H)$ and $\Delta (H)$ are intervals ([@Ge-Zh19a Theorem 4.1]). On the other hand, for every finite set $S\subset\mathbb{N}$ with $\min S=\gcd S$ (resp. every finite set $S\subset\mathbb{N}_{\ge 2}$) there is a finitely generated Krull monoid $H$ such that $\Delta (H)=S$ (resp. ${\rm Ca}(H)= S$) ([@Ge-Sc17a] resp. [@Fa-Ge17a Proposition 3.2]). Just as the monoids of ideals under discussion, every numerical monoid is a weakly factorial C-monoid. However, in contrast to them, the set of distances need not be an interval ([@Co-Ka17a]), its minimum need not be $1$ ([@B-C-K-R06 Proposition 2.9]), and a recent result of O’Neill and Pelayo ([@ON-Pe18a]) shows that for every finite set $S\subset\mathbb{N}_{\ge 2}$ there is a numerical monoid $H$ such that ${\rm Ca}(H)=S$.
We proceed as follows. In Section \[2\] we summarize the required background on the arithmetic of monoids. In Section \[3\] we do the same for orders in quadratic number fields and we provide an explicit description of (invertible) irreducible ideals in orders of quadratic number fields (Theorem \[theorem 3.6\]). In Section \[4\] we give the proof of Theorem \[theorem 1.1\]. Based on this result we establish a characterization of those orders $\mathcal{O}$ with $\min\Delta(\mathcal{O})>1$ (Theorem \[theorem 4.14\]) which allows us to give the first explicit examples of orders $\mathcal{O}$ with $\min\Delta(\mathcal{O})>1$. Our third main result (given in Theorem \[theorem 5.2\]) states that unions of sets of lengths of $\mathcal{I}(\mathcal{O})$ and of $\mathcal{I}^*(\mathcal{O})$ are intervals.
Preliminaries on the arithmetic of monoids {#2}
==========================================
Let $\mathbb{N}$ be the set of positive integers, $\mathbb P\subset\mathbb{N}$ the set of prime numbers, and for every $m\in\mathbb{N}$, we denote by $$\varphi (m)=\big| (\mathbb{Z}/m\mathbb{Z})^{\times}\big|\quad\text{\it Euler's $\varphi$-function}\,.$$ For $a,b\in\mathbb{Q}\cup\{-\infty,\infty\}$, $[a,b]=\{x\in\mathbb{Z}\mid a\le x\le b\}$ denotes the discrete interval between $a$ and $b$. Let $L,L'\subset
\mathbb{Z}$. We denote by $L+L'=\{a+b\mid a\in L,\,b\in L'\}$ their [*sumset*]{}. A positive integer $d\in\mathbb{N}$ is called a [*distance*]{} of $L$ if there exists a $k\in L$ such that $L\cap [k,k+d]=\{k,k+d\}$, and we denote by $\Delta (L)$ the [*set of distances*]{} of $L$. If $\emptyset\not=L\subset\mathbb{N}$, we denote by $\rho (L)=\sup L/\min L\in\mathbb{Q}_{\ge 1}\cup\{\infty\}$ the [*elasticity*]{} of $L$. We set $\rho (\{0\})=1$ and $\max\emptyset=\min\emptyset=\sup\emptyset=0$. All rings and semigroups are commutative and have an identity element.
**Monoids.** {#Monoids}
------------
Let $H$ be a multiplicatively written commutative semigroup. We denote by $H^{\times}$ the group of invertible elements of $H$. We say that $H$ is reduced if $H^{\times}=\{1\}$ and we denote by $H_{\text{\rm red}}=\{aH^{\times}\mid a\in H\}$ the associated reduced semigroup of $H$. An element $u \in H$ is said to be cancellative if $au=bu$ implies that $a=b$ for all $a, b \in H$. The semigroup $H$ is said to be
- [*cancellative*]{} if every element of $H$ is cancellative.
- [*unit-cancellative*]{} if $a,u\in H$ and $a=au$ implies that $u\in H^{\times}$.
By definition, every cancellative semigroup is unit-cancellative. All semigroups of ideals, that are studied in this paper, are unit-cancellative but not necessarily cancellative.
*Throughout this paper, a monoid means a*
*commutative unit-cancellative semigroup with identity element.*
Let $H$ be a monoid. A submonoid $S\subset H$ is said to be [*divisor-closed*]{} if $a\in S$ and $b\in H$ with $b\mid a$ implies that $b\in S$. An element $u\in H$ is said to be
- [*prime*]{} if $u\notin H^{\times}$ and, for all $a,b\in H$, $u\mid ab$ and $u\nmid a$ implies $u\mid b$.
- [*primary*]{} if $u\notin H^{\times}$ and, for all $a,b\in H$, $u\mid ab$ and $u\nmid a$ implies $u\mid b^n$ for some $n\in\mathbb{N}$.
- [*irreducible*]{} (or an [*atom*]{}) if $u\notin H^{\times}$ and, for all $a,b\in H$, $u=ab$ implies that $a\in H^{\times}$ or $b\in H^{\times}$.
The monoid $H$ is said to be [*atomic*]{} if every $a\in H\setminus H^{\times}$ is a product of finitely many atoms. If $H$ satisfies the ACC (ascending chain condition) on principal ideals, then $H$ is atomic ([@F-G-K-T17 Lemma 3.1]).
**Sets of lengths.** {#Sets of lengths}
--------------------
For a set $P$, we denote by $\mathcal{F}(P)$ the free abelian monoid with basis $P$. Every $a\in\mathcal{F}(P)$ is written in the form $$a=\prod_{p\in P} p^{\mathsf v_p (a)}\text{ with }\mathsf v_p (a)\in\mathbb{N}_0\quad\text{and}\quad\mathsf v_p (a)= 0\text{ for almost all $p\in P$}\,.$$ We call $|a|=\sum_{p\in P}\mathsf v_p (a)$ the length of $a$ and ${\rm supp}(a)=\{p\in P\mid\mathsf v_p(a)> 0\}\subset P$ the support of $a$. Let $H$ be an atomic monoid. The free abelian monoid $\mathsf{Z}(H)=\mathcal{F} (\mathcal{A}(H_{\text{\rm red}}))$ denotes the [*factorization monoid*]{} of $H$ and $$\pi\colon\mathsf{Z}(H)\to H_{\text{\rm red}}\quad\text{satisfying}\quad\pi(u)=u\text{ for all } u\in\mathcal{A}(H_{\text{\rm red}})$$ denotes the [*factorization homomorphism*]{} of $H$. For every $a\in H$, $$\begin{aligned}
\mathsf{Z}_H(a)=\mathsf{Z}(a)&=\pi^{-1}(aH^{\times})\quad\text{is the {\it set of factorizations} of $a$ and }\\
\mathsf{L}_H(a)=\mathsf{L}(a)&=\{|z|\mid z\in\mathsf{Z}(a)\}\quad\text{is the {\it set of lengths} of $a$}\,.
\end{aligned}$$ For a divisor-closed submonoid $S\subset H$ and an element $a\in S$, we have $\mathsf{Z}(S)\subset\mathsf{Z}(H)$ whence $\mathsf{Z}_S (a)=\mathsf{Z}_H (a)$, and $\mathsf{L}_S(a)=\mathsf{L}_H(a)$. We denote by
- $\mathcal{L}(H)=\{\mathsf{L}(a)\mid a\in H\}$ the [*system of sets of lengths*]{} of $H$ and by
- $\Delta(H)=\bigcup_{L\in\mathcal{L}(H)}\Delta(L)\subset\mathbb{N}$ the [*set of distances*]{} of $H$.
The monoid $H$ is said to be [*half-factorial*]{} if $\Delta(H)=\emptyset$ and if $H$ is not half-factorial, then $\min\Delta(H)=\gcd\Delta(H)$.
**Distances and chains of factorizations.** {#Distances}
-------------------------------------------
Let two factorizations $z,z'\in\mathsf{Z}(H)$ be given, say $$z=u_1\cdot\ldots\cdot u_{\ell}v_1\cdot\ldots\cdot v_m\quad\text{and}\quad z'=u_1\cdot\ldots\cdot u_{\ell} w_1\cdot\ldots\cdot w_n\,,$$ where $\ell,m,n\in\mathbb{N}_0$ and all $u_i,v_j,w_k\in\mathcal{A}(H_{\text{\rm red}})$ such that $v_j\ne w_k$ for all $j\in [1,m]$ and all $k\in [1,n]$. Then $\mathsf{d}(z,z')=\max\{m,n\}$ is the [*distance*]{} between $z$ and $z'$. If $\pi(z)=\pi(z')$ and $z\ne z'$, then $$\label{equation 1}
1+\bigl||z |-|z'|\bigr|\le\mathsf{d}(z,z')\text{ resp. } 2+\bigl||z |-|z'|\bigr|\le\mathsf{d}(z,z')\text{ if $H$ is cancellative}$$ (see [@F-G-K-T17 Proposition 3.2] and [@Ge-HK06a Lemma 1.6.2]). Let $a\in H$ and $N\in\mathbb{N}_0$. A finite sequence $z_0,\ldots,z_k\in\mathsf{Z}(a)$ is called an $N$-chain of factorizations (concatenating $z_0$ and $z_k$) if $\mathsf{d}(z_{i-1},z_i)\le N$ for all $i\in [1,k]$. For $z ,z'\in\mathsf{Z}(H)$ with $\pi(z)=\pi(z')$, we set $\mathsf{c}(z,z')=\min\{N\in\mathbb{N}_0\mid z$ and $z'$ can be concatenated by an $N$-chain of factorizations from $\mathsf{Z}\big(\pi(z)\big)\}$. Then, for every $a\in H$, $$\mathsf{c}(a)=\sup\{\mathsf{c}(z,z')\mid z,z'\in\mathsf{Z}(a)\}\in\mathbb{N}_0\cup\{\infty\}\quad\text{is the {\it catenary degree} of $a$}.$$ Clearly, $a$ has unique factorization (i.e., $|\mathsf{Z}(a)|=1$) if and only if $\mathsf{c}(a)=0$. We denote by $${\rm Ca}(H)=\{\mathsf{c}(a)\mid a\in H,\mathsf{c}(a)>0\}\subset\mathbb{N}\quad\text{the {\it set of catenary degrees} of $H$},$$ and then $$\mathsf{c}(H)=\sup{\rm Ca}(H)\in\mathbb{N}_0\cup\{\infty\}\quad\text{is the {\it catenary degree} of $H$}.$$ We use the convention that $\sup\emptyset=0$ whence $H$ is factorial if and only if $\mathsf{c}(H)=0$. Note that $\mathsf{c}(a)=0$ for all atoms $a\in H$. The restriction to positive catenary degrees in the definition of ${\rm Ca}(H)$ simplifies the statement of some results whence it is usual to restrict to elements with positive catenary degrees. If $H$ is cancellative, then Equation implies that min ${\rm Ca}(H)\ge 2$ and $$2+\sup\Delta (H)\le\mathsf{c}(H)\quad\text{if $H$ is not factorial}\,.$$ If $H=\coprod_{i\in I}H_i$, then a straightforward argument shows that $$\label{equation 2}
{\rm Ca}(H)=\bigcup_{i\in I}{\rm Ca}(H_i)\quad\text{whence}\quad\mathsf{c}(H)=\sup\{\mathsf{c}(H_i)\mid i\in I\}\,.$$
**Semigroups of ideals.** {#Semigroups of ideals}
-------------------------
Let $R$ be a domain. We denote by $\mathsf{q}(R)$ its quotient field, by $\mathfrak{X}(R)$ the set of minimal nonzero prime ideals of $R$, and by $\overline R$ its integral closure. Then $R \setminus \{0\}$ is a cancellative monoid,
- $\mathcal{I}(R)$ is the semigroup of nonzero ideals of $R$ (with usual ideal multiplication),
- $\mathcal{I}^*(R)$ is the subsemigroup of invertible ideals of $R$, and
- ${\rm Pic}(R)$ is the Picard group of $R$.
For every $I\in\mathcal{I}(R)$, we denote by $\sqrt{I}$ its radical and by $\mathcal{N}(I)=(R\negthinspace :\negthinspace I)=|R/I|\in\mathbb{N}\cup\{\infty\}$ its norm.
Let $S$ be a Dedekind domain and $R\subset S$ a subring. Then $R$ is called an [*order*]{} in $S$ if one of the following two equivalent conditions hold:
- $\mathsf{q}(R)=\mathsf{q}(S)$ and $S$ is a finitely generated $R$-module.
- $R$ is one-dimensional noetherian and $\overline R=S$ is a finitely generated $R$-module.
Let $R$ be an order in a Dedekind domain $S=\overline R$. We analyze the structure of $\mathcal{I}^*(R)$ and of $\mathcal{I}(R)$.
Since $R$ is noetherian, Krull’s Intersection Theorem holds for $R$ whence $\mathcal{I}(R)$ is unit-cancellative ([@Ge-Re18d Lemma 4.1]). Thus $\mathcal{I}(R)$ is a reduced atomic monoid with identity $R$ and $\mathcal{I}^*(R)$ is a reduced cancellative atomic divisor-closed submonoid. For the sake of clarity, we will say that an ideal of $R$ is an ideal atom if it is an atom of the monoid $\mathcal{I}(R)$. If $I,J\in\mathcal{I}^*(R)$, then $I\mid J$ if and only if $J\subset I$. The prime elements of $\mathcal{I}^*(R)$ are precisely the invertible prime ideals of $R$. Every ideal is a product of primary ideals belonging to distinct prime ideals (in particular, $\mathcal{I}^*(R)$ is a weakly factorial monoid). Thus every ideal atom (i.e., every $I\in\mathcal{A}(\mathcal{I}(R)$) is primary, and if $\sqrt{I}=\mathfrak{p}\in\mathfrak{X}(R)$, then $I$ is $\mathfrak p$-primary. Since $\overline R$ is a finitely generated $R$-module, the conductor $\mathfrak{f}=(R\negthinspace :\negthinspace\overline R)$ is nonzero, and we set $$\mathcal{P}=\{\mathfrak p\in\mathfrak{X}(R)\mid\mathfrak{p}\not\supset\mathfrak{f}\} \quad \text{and} \quad \mathcal{P}^*=\mathfrak{X}(R)\setminus\mathcal{P} \,.$$ Let $\mathfrak{p}\in\mathfrak{X}(R)$. We denote by $$\mathcal{I}^*_{\mathfrak{p}}(R)=\{I\in\mathcal{I}^*(R)\mid\sqrt{I}\supset\mathfrak{p}\} \quad \textnormal{ and } \quad \mathcal{I}_{\mathfrak{p}}(R)=\{I\in\mathcal{I}(R)\mid\sqrt{I}\supset\mathfrak{p}\}$$ the set of invertible $\mathfrak{p}$-primary ideals of $R$ and the set of $\mathfrak{p}$-primary ideals of $R$. Clearly, these are monoids and, moreover, $$\mathcal{I}_{\mathfrak{p}}(R) \subset \mathcal{I}(R), \quad \mathcal{I}^*_{\mathfrak{p}}(R) \subset \mathcal{I}_{\mathfrak{p}}(R), \quad \text{and} \quad \mathcal{I}^*_{\mathfrak{p}}(R) \subset \mathcal I^* (R)$$ are divisor-closed submonoids. Thus $\mathcal{I}^*_{\mathfrak{p}}(R)$ is a reduced cancellative atomic monoid, $\mathcal{I}_{\mathfrak{p}}(R)$ is a reduced atomic monoid, and if $\mathfrak{p}\in\mathcal{P}$, then $\mathcal{I}^*_{\mathfrak{p}}(R)=\mathcal{I}_{\mathfrak{p}}(R)$ is free abelian. Since $R$ is noetherian and one-dimensional, $$\label{equation 3}
\alpha:\mathcal{I}(R)\rightarrow\coprod_{\mathfrak{p}\in\mathfrak{X}(R)}\mathcal{I}_{\mathfrak{p}}(R), \quad \text{ defined by} \quad \alpha(I)=(I_{\mathfrak{p}}\cap R)_{\mathfrak{p}\in\mathfrak{X}(R)}$$ is a monoid isomorphism which induces a monoid isomorphism $$\label{equation 4}
\alpha_{\mid\mathcal{I}^*(R)}:\mathcal{I}^*(R)\rightarrow\coprod_{\mathfrak{p}\in\mathfrak{X}(R)}\mathcal{I}^*_{\mathfrak{p}}(R) \,.$$
Orders in quadratic number fields {#3}
=================================
The goal of this section is to prove Theorem \[theorem 3.6\] which provides an explicit description of (invertible) ideal atoms of an order in a quadratic number field. These results are essentially due to Butts and Pall (see [@Bu-Pa72] where they are given in a different style), and they were summarized without proof by Geroldinger and Lettl in [@Ge-Le90]. Unfortunately, that presentation is misleading in one case (namely, in case $p=2$ and $d_K\equiv 5\mod 8$). Thus we restate the results and provide a full proof.
First we put together some facts on orders in quadratic number fields and fix our notation which remains valid throughout the rest of this paper. For proofs, details, and any undefined notions we refer to [@HK13a]. Let $d\in\mathbb{Z}\setminus\{0,1\}$ be squarefree, $K=\mathbb{Q}(\sqrt{d})$ be a quadratic number field, $$\omega=\begin{cases}
\sqrt{d},&\text{if $d\equiv 2,3\mod 4$;}\\
\frac{1+\sqrt{d}}{2},&\text{if $d\equiv 1\mod 4$.}
\end{cases}
\quad\text{and}\quad
d_K=\begin{cases}
4d,&\text{if $d\equiv 2,3\mod 4$;}\\
d,&\text{if $d\equiv 1\mod 4$.}
\end{cases}$$ Then $\mathcal{O}_K=\mathbb{Z}[\omega]$ is the ring of integers and $d_K$ is the discriminant of $K$. For every $f\in\mathbb{N}$, we define $$\varepsilon\in\{0,1\}\text{ with }\varepsilon\equiv f d_K\mod 2\,,\quad\eta=\frac{\varepsilon - f^2d_K}{4}\,,\quad\text{and}\quad\tau=\frac{\varepsilon+f\sqrt{d_K}}{2}\,.$$ Then $$\mathcal{O}_f=\mathbb{Z}\oplus f\omega\mathbb{Z}=\mathbb{Z}\oplus\tau\mathbb{Z}$$ is an order in $\mathcal{O}_K$ with conductor $\mathfrak{f}=f\mathcal{O}_K$, and every order in $\mathcal{O}_K$ has this form. With the notation of Subsection \[Semigroups of ideals\] we have $$\mathcal{P}^*=\{\mathfrak p\in\mathfrak X (\mathcal{O}_f)\mid\mathfrak p\supset\mathfrak f \}= \{p\mathbb{Z}+f\omega\mathbb{Z}\mid p\in\mathbb{P},p\mid f\}\,.$$ If $\alpha=a+b\sqrt{d}\in K$, then $\overline{\alpha}=a-b\sqrt{d}$ is its conjugate, $\mathcal{N}_{K/\mathbb{Q}}(\alpha)=\alpha\overline{\alpha}=a^2-b^2d$ is its norm, and ${\rm tr}(\alpha)=\alpha+\overline{\alpha}=2a$ is its trace. For an $I \in \mathcal I ( \mathcal{O}_f)$, $\overline{I} = \{\overline{\alpha} \mid \alpha \in I\}$ denotes the conjugate ideal. A simple calculation shows that $$\mathcal{N}_{K/\mathbb{Q}}(r+\tau)=r^2+\varepsilon r+\eta\quad\text{for each}\ r\in\mathbb{Z}\,.$$ If $\mathcal{O}$ is an order and $I\in\mathcal{I}^*(\mathcal{O})$, then $(\mathcal{O}_K\negthinspace :\negthinspace I\mathcal{O}_K)=(\mathcal{O}\negthinspace :\negthinspace I)$ and if $a\in\mathcal{O}\setminus\{0\}$, then $$(\mathcal{O}\negthinspace :\negthinspace a\mathcal{O})=(\mathcal{O}_K\negthinspace :\negthinspace a O_K)=|\mathcal{N}_{K/\mathbb{Q}}(a)|$$ (see [@Ge-HK-Ka95 Pages 99 and 100] and note that the factor rings $\mathcal{O}_K/I\mathcal{O}_K$ and $\mathcal{O}/I$ need not be isomorphic). For $p\in\mathbb{P}$ and for $a\in\mathbb{Z}$ we denote by $\left(\frac{a}{p}\right)\in\{-1,0,1\}$ the [*Kronecker symbol*]{} of $a$ modulo $p$. A prime number $p\in\mathbb{Z}$ is called
- [*inert*]{} if $p\mathcal{O}_K\in {\rm spec}(\mathcal{O}_K)$.
- [*split*]{} if $p\mathcal{O}_K$ is a product of two distinct prime ideals of $\mathcal{O}_K$.
- [*ramified*]{} if $p\mathcal{O}_K$ is the square of a prime ideal of $\mathcal{O}_K$.
An odd prime $$p\text{ is }\begin{cases}
\text{inert}\ &\text{ if }\left(\frac{d_K}{p}\right)=-1;\\
\text{split}\ &\text{ if }\left(\frac{d_K}{p}\right)=1;\\
\text{ramified}\ &\text{ if }\left(\frac{d_K}{p}\right)=0\,.
\end{cases}
\,\text{and}\ 2\text{ is }
\begin{cases}
\text{inert}\ &\text{ if}\ d_K\equiv 5\mod 8;\\
\text{split}\ &\text{ if}\ d_K\equiv 1\mod 8;\\
\text{ramified}\ &\text{ if}\ d_K\equiv 0\mod 2\,.
\end{cases}$$
\[proposition 3.1\] Let $p$ be a prime divisor of $f$, $\mathcal{O}=\mathcal{O}_f$, and $\mathfrak{p}=p\mathbb{Z}+f\omega\mathbb{Z}$.
1. The primary ideals with radical $\mathfrak{p}$ are exactly the ideals of the form $$\mathfrak{q}=p^\ell(p^m\mathbb{Z}+(r+\tau)\mathbb{Z})$$ with $\ell,m\in\mathbb{N}_0$, $\ell+m\geq1$, $0\leq r<p^m$ and $\mathcal N_{K/\mathbb{Q}}(r+\tau)\equiv 0\mod p^m$. Moreover, $\mathcal{N}(\mathfrak{q})=p^{2\ell+m}$.
2. A primary ideal $\mathfrak{q}=p^\ell(p^m\mathbb{Z}+(r+\tau)\mathbb{Z})$ is invertible if and only if $$\mathcal N_{K/\mathbb{Q}}(r+\tau)\not\equiv 0\mod p^{m+1}.$$
1\. Let $\mathfrak{q}$ be a $\mathfrak{p}$-primary ideal in $\mathcal{O}$. By [@HK13a Theorem 5.4.2] there exist nonnegative integers $\ell,m,r$ such that $\mathfrak{q}=\ell(m\mathbb{Z}+(r+\tau)\mathbb{Z})$, $r<m$ and $\mathcal N_{K/\mathbb{Q}}(r+\tau)\equiv 0\mod m$. Since $\mathfrak{q}$ is nonzero and proper, we have $\ell m>1$. We prove, that $\ell m$ is a power of $p$. First observe that $\mathfrak{q}\subset\sqrt{\mathfrak{q}}=\mathfrak{p}$ implies that $p\mid\ell m$. Assume to the contrary that there exists another rational prime $p'\not=p$ dividing $\ell m$, say $\ell m=p's$. But then $p's\in\mathfrak{q}$, $s\not\in\mathfrak{q}$ and $p'\not\in\mathfrak{p}=\sqrt{\mathfrak{q}}$. A contradiction to $\mathfrak{q}$ being primary. Conversely, assume that $\mathfrak{q}=p^\ell(p^m\mathbb{Z}+(r+\tau)\mathbb{Z})$ for integers $\ell,m\in\mathbb{N}_0,\ell+m\geq 1,0\leq r<p^m$ and $\mathcal N_{K/\mathbb{Q}}(r+\tau)\equiv 0\mod p^m.$ By [@HK13a Theorem 5.4.2], $\mathfrak{q}$ is an ideal of $\mathcal{O}$. Since $p\in\sqrt{\mathfrak{q}}$ and $\mathfrak{p}$ is the only prime ideal in $\mathcal{O}$ containing $p$ we obtain that $\sqrt{\mathfrak{q}}=\bigcap_{\substack{\mathfrak{a}\in {\rm spec}(\mathcal{O}),\mathfrak{a}\supset\mathfrak{q}}}\mathfrak{a}=\mathfrak{p}$. The nonzero prime ideal $\mathfrak{p}$ is maximal, since $\mathcal{O}$ is one-dimensional. Therefore, $\mathfrak{q}$ is $\mathfrak{p}$-primary. It follows from [@HK13a Theorem 5.4.2] that $\mathcal{N}(\mathfrak{q})=p^{2\ell+m}$.
2\. By [@HK13a Theorem 5.4.2], $\mathfrak{q}=p^\ell(p^m\mathbb{Z}+(r+\tau)\mathbb{Z})$ is invertible if and only if $\gcd(p^m,2r+\varepsilon,\frac{\mathcal N_{K/\mathbb{Q}}(r+\tau)}{p^m})=1$. Since $p\mid f$ and $\mathcal{N}_{K/\mathbb{Q}}(r+\tau)=\frac{1}{4}((2r+\varepsilon)^2-f^2d_K)$, this is the case if and only if $p\nmid\frac{\mathcal N_{K/\mathbb{Q}}(r+\tau)}{p^m}$, that is $\mathcal N_{K/\mathbb{Q}}(r+\tau)\not\equiv 0\mod p^{m+1}$.
If $x\in\mathbb{Z}$ and $y\in\mathbb{N}$, then let ${\rm rem}(x,y)$ be the unique $z\in [0,y-1]$ such that $y\mid x-z$. Let $p$ be a prime divisor of $f$. Note that ${\rm v}_p(0)=\infty$, and if $\emptyset\not=A\subseteq\mathbb{N}_0$, then $\min(A\cup\{\infty\})=\min A$. We set $$\begin{aligned}
P_{f,p}=p\mathbb{Z}+f\omega\mathbb{Z},\quad\mathcal{I}^*_p(\mathcal{O}_f) &=\mathcal{I}^*_{P_{f,p}}(\mathcal{O}_f), \mathcal{I}_p(\mathcal{O}_f)=\mathcal{I}_{P_{f,p}}(\mathcal{O}_f),\quad\text{and}\\
\mathcal{M}_{f,p} &=\{(x,y,z)\in\mathbb{N}_0^3\mid z<p^y,{\rm v}_p(z^2+\varepsilon z+\eta)\geq y\}
\end{aligned}$$ Let $\ast:\mathcal{M}_{f,p}\times\mathcal{M}_{f,p}\rightarrow\mathcal{M}_{f,p}$ be defined by $(u,v,w)\ast (x,y,z)=(a,b,c)$, where $$\begin{aligned}
&a=u+x+g,\textnormal{ }b=v+y+e-2g,\\
&c={\rm rem}\left(h-t\frac{h^2+\varepsilon h+\eta}{p^g},p^b\right),\textnormal{ }g=\min\{v,y,{\rm v}_p(w+z+\varepsilon)\},\\
&e=\min\{g,{\rm v}_p(w-z),{\rm v}_p(w^2+\varepsilon w+\eta)-v,{\rm v}_p(z^2+\varepsilon z+\eta)-y\},\\
&t\in\mathbb{Z}\textnormal{ is such that }t\frac{w+z+\varepsilon}{p^g}\equiv 1\mod\textnormal{ }p^{\min\{v,y\}-g},\textnormal{ and }h=\begin{cases} z &\textnormal{ if }y\geq v\\ w &\textnormal{ if }v>y\end{cases}.\end{aligned}$$ Let $\xi_{f,p}:\mathcal{M}_{f,p}\rightarrow\mathcal{I}_p(\mathcal{O}_f)$ be defined by $\xi_{f,p}(x,y,z)=p^x(p^y\mathbb{Z}+(z+\tau)\mathbb{Z})$.
\[proposition 3.2\] Let $p$ be a prime divisor of $f$ and $I,J\in\mathcal{I}_p(\mathcal{O}_f)$.
1. $(\mathcal{M}_{f,p},\ast)$ is a reduced monoid and $\xi_{f,p}$ is a monoid isomorphism.
2. If $w,z\in\mathbb{Z}$ are such that ${\rm v}_p(w^2+\varepsilon w+\eta)>0$ and ${\rm v}_p(z^2+\varepsilon z+\eta)>0$, then ${\rm v}_p(w+z+\varepsilon)>0$ and ${\rm v}_p(w-z)>0$.
3. $\mathcal{N}(I)\mathcal{N}(J)\mid\mathcal{N}(IJ)$ and $\mathcal{N}(IJ)=\mathcal{N}(I)\mathcal{N}(J)$ if and only if $I$ is invertible or $J$ is invertible. If $I$ and $J$ are proper, then $IJ\subset p\mathcal{O}_f$.
4. If $I\in\mathcal{A}(\mathcal{I}_p(\mathcal{O}_f))$, then there is some $I^{\prime}\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $\mathcal{N}(IJ)\mid\mathcal{N}(I^{\prime}J)$. If $I\in\mathcal{A}(\mathcal{I}_p(\mathcal{O}_f))$ is not invertible, then $\mathcal{N}(I)\mid\mathcal{N}(I^{\prime})$ and $\mathcal{N}(I)<\mathcal{N}(I^{\prime})$ for some $I^{\prime}\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$.
5. If $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$, then $\overline{I}\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ and $I\overline{I}=\mathcal{N}(I)\mathcal{O}_f$.
1\. Let $(u,v,w),(x,y,z)\in\mathcal{M}_{f,p}$. Set $g=\min\{v,y,{\rm v}_p(w+z+\varepsilon)\}$ and $e=\min\{g,{\rm v}_p(w-z),{\rm v}_p(w^2+\varepsilon w+\eta)-v,{\rm v}_p(z^2+\varepsilon z+\eta)-y\}$. Note that ${\rm gcd}(p^{\min\{v,y\}},w+z+\varepsilon)=p^g$, and hence there are some $s,t\in\mathbb{Z}$ such that $sp^{\min\{v,y\}}+t(w+z+\varepsilon)=p^g$. This implies that $t\frac{w+z+\varepsilon}{p^g}\equiv 1\mod\textnormal{ }p^{\min\{v,y\}-g}$. Set $a=u+x+g$, $b=v+y+e-2g$ and let $h=z$ if $y\geq v$ and $h=w$ if $v>y$. Finally, set $c={\rm rem}(h-t\frac{h^2+\varepsilon h+\eta}{p^g},p^b)$. First we show that $c$ does not depend on the choice of $t$. Let $t^{\prime}\in\mathbb{Z}$ be such that $t^{\prime}\frac{w+z+\varepsilon}{p^g}\equiv 1\mod\textnormal{ }p^{\min\{v,y\}-g}$. Then $p^{\min\{v,y\}-g}\mid t-t^{\prime}$. Note that $\min\{v,y\}+{\rm v}_p(h^2+\varepsilon h+\eta)\geq v+y+e$, and hence $p^b\mid (t-t^{\prime})\frac{h^2+\varepsilon h+\eta}{p^g}$. Consequently, $c={\rm rem}(h-t^{\prime}\frac{h^2+\varepsilon h+\eta}{p^g},p^b)$.
Next we show that $(a,b,c)\in\mathcal{M}_{f,p}$. It is clear that $(a,b,c)\in\mathbb{N}_0^3$ and $c<p^b$. It remains to show that ${\rm v}_p(c^2+\varepsilon c+\eta)\geq b$. Without restriction we can assume that $v\leq y$. Then $h=z$. Set $k=z-t\frac{z^2+\varepsilon z+\eta}{p^g}$. There is some $r\in\mathbb{Z}$ such that $c=k+rp^b$. Since $c^2+\varepsilon c+\eta=k^2+\varepsilon k+\eta+mp^b$ for some $m\in\mathbb{Z}$, it is sufficient to show that ${\rm v}_p(k^2+\varepsilon k+\eta)\geq b$.
Observe that $k^2+\varepsilon k+\eta=\frac{z^2+\varepsilon z+\eta}{p^{2g}}(p^{2g}-tp^g(2z+\varepsilon)+t^2(z^2+\varepsilon z+\eta))=\frac{z^2+\varepsilon z+\eta}{p^{2g}}(sp^{v+g}+tp^g(w-z)+t^2(z^2+\varepsilon z+\eta))$. Note that $g+{\rm v}_p(w-z)=\min\{v+{\rm v}_p(w-z),{\rm v}_p(w+z+\varepsilon)+{\rm v}_p(w-z)\}=\min\{v+{\rm v}_p(w-z),{\rm v}_p(w^2+\varepsilon w+\eta-(z^2+\varepsilon z+\eta))\}\geq\min\{v+{\rm v}_p(w-z),{\rm v}_p(z^2+\varepsilon z+\eta),{\rm v}_p(w^2+\varepsilon w+\eta)\}\geq v$. Moreover, we have ${\rm v}_p(z^2+\varepsilon z+\eta)\geq y+e$. Therefore, ${\rm v}_p(k^2+\varepsilon k+\eta)\geq {\rm v}_p(z^2+\varepsilon z+\eta)-2g+\min\{v+g,g+{\rm v}_p(w-z),{\rm v}_p(z^2+\varepsilon z+\eta)\}\geq y+e-2g+v=b$.
Now we prove that $p^u(p^v\mathbb{Z}+(w+\tau)\mathbb{Z})p^x(p^y\mathbb{Z}+(z+\tau)\mathbb{Z})=p^a(p^b\mathbb{Z}+(c+\tau)\mathbb{Z})$. (Note that this can be shown by using [@HK13a Theorem 5.4.6].) Set $I=p^u(p^v\mathbb{Z}+(w+\tau)\mathbb{Z})p^x(p^y\mathbb{Z}+(z+\tau)\mathbb{Z})$. Without restriction let $v\leq y$. Note that $(w+\tau)(z+\tau)=wz-\eta+(w+z+\varepsilon)\tau$. Set $\alpha=p^v(z+\tau)$ and $\beta=wz-\eta+(w+z+\varepsilon)\tau$. We infer that $I=p^{u+x}(p^{v+y}\mathbb{Z}+p^y(w+\tau)\mathbb{Z}+\alpha\mathbb{Z}+\beta\mathbb{Z})$.
Moreover, $p^y(w+\tau)\mathbb{Z}+\alpha\mathbb{Z}=p^y(w-z)\mathbb{Z}+\alpha\mathbb{Z}$. Observe that $s\alpha+t\beta=p^gz-t(z^2+\varepsilon z+\eta)+p^g\tau$. Set $k=z-t\frac{z^2+\varepsilon z+\eta}{p^g}$. Then $s\alpha+t\beta=p^g(k+\tau)$. We have $\alpha-p^v(k+\tau)=tp^{v-g}(z^2+\varepsilon z+\eta)$ and $(w+z+\varepsilon)(k+\tau)-\beta=sp^{v-g}(z^2+\varepsilon z+\eta)$. Set $r=p^{v-g}(z^2+\varepsilon z+\eta)$. Consequently, $\alpha\mathbb{Z}+\beta\mathbb{Z}=sr\mathbb{Z}+tr\mathbb{Z}+p^g(k+\tau)\mathbb{Z}=r\mathbb{Z}+p^g(k+\tau)\mathbb{Z}$, since ${\rm gcd}(s,t)=1$. Putting these facts together gives us $I=p^{u+x}(p^{v+y}\mathbb{Z}+p^y(w-z)\mathbb{Z}+r\mathbb{Z}+p^g(k+\tau)\mathbb{Z})$.
We have ${\rm gcd}(p^{v+y},p^y(w-z),r)=p^{\ell}$ with $\ell=\min\{v+y,y+{\rm v}_p(w-z),v-g+{\rm v}_p(z^2+\varepsilon z+\eta)\}$ and $p^{v+y}\mathbb{Z}+p^y(w-z)\mathbb{Z}+r\mathbb{Z}=p^{\ell}\mathbb{Z}$. Note that $\ell=v+y-g+\min\{g,{\rm v}_p(w-z)-v+g,{\rm v}_p(z^2+\varepsilon z+\eta)-y\}$ and ${\rm v}_p(w-z)-v+g=\min\{{\rm v}_p(w-z),{\rm v}_p(w-z)+{\rm v}_p(w+z+\varepsilon)-v\}=\min\{{\rm v}_p(w-z),{\rm v}_p(w^2+\varepsilon w+\eta-(z^2+\varepsilon z+\eta))-v\}$, and hence $\ell=v+y-g+\min\{g,{\rm v}_p(w-z),{\rm v}_p(w^2+\varepsilon w+\eta-(z^2+\varepsilon z+\eta))-v,{\rm v}_p(z^2+\varepsilon z+\eta)-y\}$.
CASE 1: ${\rm v}_p(w^2+\varepsilon w+\eta)\geq {\rm v}_p(z^2+\varepsilon z+\eta)$. Then ${\rm v}_p(w^2+\varepsilon w+\eta)-v\geq {\rm v}_p(z^2+\varepsilon z+\eta)-y$ and ${\rm v}_p(w^2+\varepsilon w+\eta-(z^2+\varepsilon z+\eta))-v\geq {\rm v}_p(z^2+\varepsilon z+\eta)-y$.
CASE 2: ${\rm v}_p(z^2+\varepsilon z+\eta)>{\rm v}_p(w^2+\varepsilon w+\eta)$. Then ${\rm v}_p(w^2+\varepsilon w+\eta-(z^2+\varepsilon z+\eta))-v={\rm v}_p(w^2+\varepsilon w+\eta)-v$.
In any case we have $\min\{{\rm v}_p(w^2+\varepsilon w+\eta-(z^2+\varepsilon z+\eta))-v,{\rm v}_p(z^2+\varepsilon z+\eta)-y\}=\min\{{\rm v}_p(w^2+\varepsilon w+\eta)-v,{\rm v}_p(z^2+\varepsilon z+\eta)-y\}$. Obviously, $\ell=v+y+e-g$ and $I=p^{u+x+g}(p^{v+y+e-2g}\mathbb{Z}+(z-t\frac{z^2+\varepsilon z+\eta}{p^g}+\tau)\mathbb{Z})$. Consequently, $I=p^a(p^b\mathbb{Z}+(c+\tau)\mathbb{Z})$.
So far we know that $\ast$ is an inner binary operation on $\mathcal{M}_{f,p}$. It follows from Proposition \[proposition 3.1\].1 that $\xi_{f,p}$ is surjective. It follows from [@HK13a Theorem 5.4.2] that $\xi_{f,p}$ is injective. It is clear that $(\mathcal{I}_p(O_f),\cdot)$ is a reduced monoid. We have shown that $\xi_{f,p}$ maps products of elements of $\mathcal{M}_{f,p}$ to products of elements of $\mathcal{I}_p(O_f)$. It is clear that $(0,0,0)$ is an identity element of $\mathcal{M}_{f,p}$ and $\xi_{f,p}(0,0,0)=\mathcal{O}_f$. Therefore, $(\mathcal{M}_{f,p},\ast)$ is a reduced monoid and $\xi_{f,p}$ is a monoid isomorphism.
2\. Let $w,z\in\mathbb{Z}$ be such that ${\rm v}_p(w^2+\varepsilon w+\eta)>0$ and ${\rm v}_p(z^2+\varepsilon z+\eta)>0$. Then $p\mid z^2+\varepsilon z+\eta=\frac{1}{4}((2z+\varepsilon)^2-f^2d_K)$, and hence $p\mid 2z+\varepsilon$. Moreover $p\mid w^2+\varepsilon w+\eta-(z^2+\varepsilon z+\eta)=(w+z+\varepsilon)(w-z)$, and thus $p\mid w+z+\varepsilon$ or $p\mid w-z$. Since $p\mid 2z+\varepsilon$, we infer that $p\mid w+z+\varepsilon$ if and only if $p\mid w-z$. Consequently, $\min\{{\rm v}_p(w+z+\varepsilon),{\rm v}_p(w-z)\}>0$.
3\. By 1., there are $(u,v,w),(x,y,z),(a,b,c)\in\mathcal{M}_{f,p}$ such that $I=p^u(p^v\mathbb{Z}+(w+\tau)\mathbb{Z})$, $J=p^x(p^y\mathbb{Z}+(z+\tau)\mathbb{Z})$ and $IJ=p^a(p^b\mathbb{Z}+(c+\tau)\mathbb{Z})$ with $a=u+x+g$, $b=v+y+e-2g$, $g=\min\{v,y,{\rm v}_p(w+z+\varepsilon)\}$ and $e=\min\{g,{\rm v}_p(w-z),{\rm v}_p(w^2+\varepsilon w+\eta)-v,{\rm v}_p(z^2+\varepsilon z+\eta)-y\}$. It follows by Proposition \[proposition 3.1\].1 that $\mathcal{N}(I)=p^{2u+v}$, $\mathcal{N}(J)=p^{2x+y}$ and $\mathcal{N}(IJ)=p^{2a+b}=p^{2(u+x)+v+y+e}$. It is obvious that $\mathcal{N}(I)\mathcal{N}(J)\mid\mathcal{N}(IJ)$. Moreover, $\mathcal{N}(IJ)=\mathcal{N}(I)\mathcal{N}(J)$ if and only if $e=0$. We infer by 2. that $e=0$ if and only if $v=0$ or $y=0$ or ${\rm v}_p(w^2+\varepsilon w+\eta)=v$ or ${\rm v}_p(z^2+\varepsilon z+\eta)=y$, which is the case if and only if $I$ is invertible or $J$ is invertible by Proposition \[proposition 3.1\].2. If $I$ and $J$ are proper, then $u+v>0$ and $x+y>0$, and hence $a>0$ by 2. This implies that $IJ\subset p(p^b\mathbb{Z}+(c+\tau)\mathbb{Z})\subset p\mathcal{O}_f$.
4\. Let $I\in\mathcal{A}(\mathcal{I}_p(\mathcal{O}_f))$. Without restriction let $I$ be not invertible. We have $I=p^b\mathbb{Z}+(r+\tau)\mathbb{Z}$ for some $(0,b,r)\in\mathcal{M}_{f,p}$ and $b<{\rm v}_p(r^2+\varepsilon r+\eta)$. Set $c={\rm v}_p(r^2+\varepsilon r+\eta)$ and $I^{\prime}=p^c\mathbb{Z}+(r+\tau)\mathbb{Z}$. Then $I^{\prime}\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$, $\mathcal{N}(I)\mid\mathcal{N}(I^{\prime})$, and $\mathcal{N}(I)<\mathcal{N}(I^{\prime})$ by Proposition \[proposition 3.1\]. There is some $(x,y,z)\in\mathcal{M}_{f,p}$ such that $J=p^x(p^y\mathbb{Z}+(z+\tau)\mathbb{Z})$. Then $\mathcal{N}(I^{\prime}J)=p^{c+2x+y}$ and $\mathcal{N}(IJ)=p^{b+2x+y+e}$ with $e=\min\{b,y,{\rm v}_p(r+z+\varepsilon),{\rm v}_p(r-z),c-b,{\rm v}_p(z^2+\varepsilon z+\eta)-y\}\leq c-b$. Therefore, $\mathcal{N}(IJ)\mid\mathcal{N}(I^{\prime}J)$.
5\. Let $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$. If $I=p\mathcal{O}_f$, then $\overline{I}=p\mathcal{O}_f$ and $\mathcal{N}(I)=p^2$ by Proposition \[proposition 3.1\].1. Therefore, $I\overline{I}=\mathcal{N}(I)\mathcal{O}_f$. Now let $I\not=p\mathcal{O}_f$. There is some $(0,m,r)\in\mathcal{M}_{f,p}$ such that $I=p^m\mathbb{Z}+(r+\tau)\mathbb{Z}$. Set $s=p^m-r-\varepsilon$. It follows that $\overline{I}=p^m\mathbb{Z}+(r+\overline{\tau})\mathbb{Z}=p^m\mathbb{Z}+(r+\varepsilon-\tau)\mathbb{Z}=p^m\mathbb{Z}+(s+\tau)\mathbb{Z}$. Observe that $s^2+\varepsilon s+\eta=r^2+\varepsilon r+\eta+p^m(p^m-(2r+\varepsilon))$. Since $p\mid r^2+\varepsilon r+\eta=\frac{1}{4}((2r+\varepsilon)^2-f^2d_K)$, we have ${\rm v}_p(2r+\varepsilon)>0$, and hence ${\rm v}_p(p^m(p^m-(2r+\varepsilon)))>m$. Since ${\rm v}_p(r^2+\varepsilon r+\eta)=m$, we infer that ${\rm v}_p(s^2+\varepsilon s+\eta)=m$, and thus $(0,m,s)\in\mathcal{M}_{f,p}$. Therefore, $\overline{I}\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$. Note that $\min\{m,{\rm v}_p(r+s+\varepsilon)\}=m$, and thus $I\overline{I}=p^m\mathcal{O}_f=\mathcal{N}(I)\mathcal{O}_f$ by 1. and Proposition \[proposition 3.1\].1.
\[proposition 3.3\] Let $p$ be a prime divisor of $f$ and $f^{\prime}=p^{{\rm v}_p(f)}$. Set $\mathcal{O}=\mathcal{O}_f$, $\mathcal{O}^{\prime}=\mathcal{O}_{f^{\prime}}$, $P=P_{f,p}$ and $P^{\prime}=P_{f^{\prime},p}$. For $g\in\mathbb{N}$ let $\varphi_{g,p}:\mathcal{I}_p(\mathcal{O}_g)\rightarrow\mathcal{I}(({\mathcal{O}_g})_{P_{g,p}})$ be defined by $\varphi_{g,p}(I)=I_{P_{g,p}}$ and $\zeta_{g,p}:\mathcal{I}(({\mathcal{O}_g})_{P_{g,p}})\rightarrow\mathcal{I}_p(\mathcal{O}_g)$ be defined by $\zeta_{g,p}(J)=J\cap\mathcal{O}_g$.
1. $\mathcal{O}_P=\mathcal{O}^{\prime}_{P^{\prime}}$.
2. $\varphi_{f,p}$ and $\zeta_{f,p}$ are mutually inverse monoid isomorphisms.
3. There is a monoid isomorphism $\delta:\mathcal{I}_p(\mathcal{O})\rightarrow\mathcal{I}_p(\mathcal{O}^{\prime})$ such that $\delta(p\mathcal{O})=p\mathcal{O}^{\prime}$ and $\delta_{\mid\mathcal{I}^*_p(\mathcal{O})}:\mathcal{I}^*_p(\mathcal{O})\rightarrow\mathcal{I}^*_p(\mathcal{O}^{\prime})$ is a monoid isomorphism.
1\. It is clear that $\mathcal{O}\subset\mathcal{O}^{\prime}$ and $P^{\prime}\cap\mathcal{O}=P$. Therefore, $\mathcal{O}_P\subset\mathcal{O}^{\prime}_{P^{\prime}}$. Observe that $\mathcal{O}\setminus P=(\mathbb{Z}\setminus p\mathbb{Z})+f\omega\mathbb{Z}$ and $\mathcal{O}^{\prime}\setminus P^{\prime}=(\mathbb{Z}\setminus p\mathbb{Z})+f^{\prime}\omega\mathbb{Z}$. It remains to show that $\{f^{\prime}\omega\}\cup\{x^{-1}\mid x\in (\mathbb{Z}\setminus p\mathbb{Z})+f^{\prime}\omega\mathbb{Z}\}\subset\mathcal{O}_P$. Since $\frac{f}{f^{\prime}}f^{\prime}\omega=f\omega\in\mathcal{O}$ and $\frac{f}{f^{\prime}}\in\mathbb{Z}\setminus p\mathbb{Z}\subset\mathcal{O}\setminus P$, we have $f^{\prime}\omega\in\mathcal{O}_P$. Therefore, $\mathcal{O}^{\prime}\subset\mathcal{O}_P$. Now let $a\in\mathbb{Z}\setminus p\mathbb{Z}$ and $b\in\mathbb{Z}$. Observe that $a+bf^{\prime}\overline{\omega}\in\mathcal{O}^{\prime}\subset\mathcal{O}_P$. Since $\omega+\overline{\omega},\omega\overline{\omega}\in\mathbb{Z}$, we have $(a+bf^{\prime}\omega)(a+bf^{\prime}\overline{\omega})=a^2+abf^{\prime}(\omega+\overline{\omega})+b^2(f^{\prime})^2\omega\overline{\omega}\in\mathbb{Z}\setminus p\mathbb{Z}\subset\mathcal{O}\setminus P$. Therefore, $\frac{1}{a+bf^{\prime}\omega}=\frac{a+bf^{\prime}\overline{\omega}}{(a+bf^{\prime}\omega)(a+bf^{\prime}\overline{\omega})}\in\mathcal{O}_P$.
2\. It is clear that $\varphi_{f,p}$ is a well-defined monoid homomorphism. Note that $\zeta_{f,p}$ is a well-defined map (since every nonzero proper ideal $J$ of $\mathcal{O}_P$ is $P_P$-primary, and hence $J\cap\mathcal{O}$ is $P$-primary). Moreover, $\zeta_{f,p}(\mathcal{O}_P)=\mathcal{O}$. Now let $J_1,J_2\in\mathcal{I}(\mathcal{O}_P)$. Observe that $J_1J_2\cap\mathcal{O}$ and $(J_1\cap\mathcal{O})(J_2\cap\mathcal{O})$ coincide locally (note that both are either $P$-primary or not proper). Therefore, $J_1J_2\cap\mathcal{O}=(J_1\cap\mathcal{O})(J_2\cap\mathcal{O})$, and hence $\zeta_{f,p}$ is a monoid homomorphism. If $J\in\mathcal{I}(\mathcal{O}_P)$, then $(J\cap\mathcal{O})_P=J$. Therefore, $\varphi_{f,p}\circ\zeta_{f,p}={\rm id}_{\mathcal{I}(\mathcal{O}_P)}$. If $I$ is a $P$-primary ideal of $\mathcal{O}$, then $I_P\cap\mathcal{O}=I$. This implies that $\zeta_{f,p}\circ\varphi_{f,p}={\rm id}_{\mathcal{I}_p(\mathcal{O})}$.
3\. Set $\delta=\zeta_{f^{\prime},p}\circ\varphi_{f,p}$. Then $\delta:\mathcal{I}_p(\mathcal{O})\rightarrow\mathcal{I}_p(\mathcal{O}^{\prime})$ is a monoid isomorphism by 1. and 2. Furthermore, we have by 1. that $\delta(p\mathcal{O})=\zeta_{f^{\prime},p}(\varphi_{f,p}(p\mathcal{O}))=\zeta_{f^{\prime},p}(p\mathcal{O}_P)=\zeta_{f^{\prime},p}(p\mathcal{O^{\prime}}_{P^{\prime}})=p\mathcal{O}^{\prime}_{P^{\prime}}\cap\mathcal{O}^{\prime}=p\mathcal{O}^{\prime}$.
Since $\mathcal{O}$ is noetherian, we have $\mathcal{I}^*_p(\mathcal{O})$ is the set of cancellative elements of $\mathcal{I}_p(\mathcal{O})$. It follows by analogy that $\mathcal{I}^*_p(\mathcal{O}^{\prime})$ is the set of cancellative elements of $\mathcal{I}_p(\mathcal{O}^{\prime})$. Therefore, $\delta(\mathcal{I}^*_p(\mathcal{O}))=\mathcal{I}^*_p(\mathcal{O}^{\prime})$, and hence $\delta_{\mid\mathcal{I}^*_p(\mathcal{O})}$ is a monoid isomorphism.
\[lemma 3.4\] Let $p$ be a prime number, let $k\in\mathbb{N}_0$, let $c,n\in\mathbb{N}$ be such that ${\rm gcd}(c,p)=1$ and for each $\ell\in\mathbb{N}$ let $g_{\ell}=|\{y\in [0,p^{\ell}-1]\mid y^2\equiv c\mod p^{\ell}\}|$.
1. If $p\not=2$, then $p^kc$ is a square modulo $p^n$ if and only if $k\geq n$ or $(k<n$, $k$ is even and $(\frac{c}{p})=1)$.
2. $2^kc$ is a square modulo $2^n$ if and only if one of the following conditions holds.
1. $k\geq n$.
2. $k$ is even and $n=k+1$.
3. $k$ is even, $n=k+2$ and $c\equiv 1\mod 4$.
4. $k$ is even, $n\geq k+3$ and $c\equiv 1\mod 8$.
3. If $\ell\in\mathbb{N}$, then $g_{\ell}=\begin{cases} 4 &{\it if}\textnormal{ }p=2,\ell\geq 3,c\equiv 1\mod 8\\ 2 &{\it if}\textnormal{ }(p\not=2,(\frac{c}{p})=1)\textnormal{ }{\it or}\textnormal{ }(p=2,\ell=2,c\equiv 1\mod 4)\\ 1 &{\it if}\textnormal{ }p=2,\ell=1\\ 0 &{\it else}\end{cases}$.
Note that $p^kc$ is a square modulo $p^n$ iff $k\geq n$ or $(k<n$, $k$ is even and $c$ is a square modulo $p^{n-k}$).
1\. Let $p\not=2$. It remains to show that if $\ell\in\mathbb{N}$, then $c$ is a square modulo $p^{\ell}$ if and only if $(\frac{c}{p})=1$. If $\ell\in\mathbb{N}$ and $c$ is a square modulo $p^{\ell}$, then $c$ is a square modulo $p$, and hence $(\frac{c}{p})=1$. Now let $(\frac{c}{p})=1$. It suffices to show by induction that $c$ is a square modulo $p^{\ell}$ for all $\ell\in\mathbb{N}$. The statement is clearly true for $\ell=1$. Now let $\ell\in\mathbb{N}$ and let $x\in\mathbb{Z}$ be such that $x^2\equiv c\mod p^{\ell}$. Without restriction let ${\rm v}_p(x^2-c)=\ell$. Note that $p\nmid x$, and hence $2bx\equiv -1\mod p$ for some $b\in\mathbb{Z}$. Set $y=x+b(x^2-c)$. Then $y^2\equiv c\mod p^{\ell+1}$.
2\. It remains to show that if $\ell\in\mathbb{N}$, then $c$ is a square modulo $2^{\ell}$ if and only if $\ell=1$ or $(\ell=2$ and $c\equiv 1\mod 4)$ or $(\ell\geq 3$ and $c\equiv 1\mod 8)$. Let $\ell\in\mathbb{N}$ and let $c$ be a square modulo $2^{\ell}$. If $\ell=2$, then $c$ is a square modulo $4$ and $c\equiv 1\mod 4$. Moreover, if $\ell\geq 3$, then $c$ is a square modulo $8$ and $c\equiv 1\mod 8$.
Clearly, if $\ell=1$ or ($\ell=2$ and $c\equiv 1\mod 4$), then $c$ is a square modulo $2^{\ell}$. Now let $c\equiv 1\mod 8$. It is sufficient to show by induction that $c$ is a square modulo $2^{\ell}$ for each $\ell\in\mathbb{N}_{\geq 3}$. The statement is obviously true for $\ell=3$. Now let $\ell\in\mathbb{N}_{\geq 3}$ and let $x\in\mathbb{Z}$ be such that $x^2\equiv c\mod 2^{\ell}$. Without restriction let ${\rm v}_2(x^2-c)=\ell$. Set $y=x+2^{\ell-1}$. Then $y^2\equiv c\mod 2^{\ell+1}$.
3\. Let $\ell\in\mathbb{N}$. By 1. and 2., it is sufficient to consider the case $g_{\ell}>0$. Let $g_{\ell}>0$. Observe that $g_{\ell}=|\{y\in [0,p^{\ell}-1]\mid y^2\equiv 1\mod p^{\ell}\}|=|\{y\in (\mathbb{Z}/p^{\ell}\mathbb{Z})^{\times}\mid {\rm ord}(y)\leq 2\}|$. If $p=2$ and $\ell=1$, then $(\mathbb{Z}/p^{\ell}\mathbb{Z})^{\times}$ is trivial, and hence $g_{\ell}=1$. If ($p=2$, $\ell=2$ and $c\equiv 1\mod 4$) or ($p\not=2$ and $(\frac{c}{p})=1$), then $(\mathbb{Z}/p^{\ell}\mathbb{Z})^{\times}$ is a cyclic group of even order, and thus $g_{\ell}=2$. Finally, if $p=2$, $\ell\geq 3$ and $c\equiv 1\mod 8$, then $(\mathbb{Z}/2^{\ell}\mathbb{Z})^{\times}\cong\mathbb{Z}/2\mathbb{Z}\times\mathcal{C}_{2^{\ell-2}}$ is the product of two cyclic groups of even order. Consequently, $g_{\ell}=4$.
\[lemma 3.5\] Let $p$ be a prime number, $a,m\in\mathbb{N}$, $c=\frac{a}{p^{{\rm v}_p(a)}}$, $M=\{x\in [0,p^m-1]\mid {\rm v}_p(x^2-a)=m\}$, $N=|M|$ and for each $\ell\in\mathbb{N}$ let $g_{\ell}=|\{y\in [0,p^{\ell}-1]\mid y^2\equiv c\mod p^{\ell}\}|$.
1. If $m<{\rm v}_p(a)$, then $N=\begin{cases}\varphi(p^{m/2}) &{\it if}\textnormal{ }m\textnormal{ }{\it is}\textnormal{ }{\it even}\\ 0 &{\it if}\textnormal{ }m\textnormal{ }{\it is}\textnormal{ }{\it odd}\end{cases}$.
2. Let $m={\rm v}_p(a)$.
1. If $a$ is a square modulo $p^{m+1}$, then $N=\begin{cases}p^{m/2-1}(p-2) &{\it if}\textnormal{ }p\not=2\\ 2^{{m/2}-1} &{\it if}\textnormal{ }p=2\end{cases}$.
2. If $a$ is not a square modulo $p^{m+1}$, then $N=p^{\lfloor m/2\rfloor}$.
3. If $m>{\rm v}_p(a)$ and $a$ is not a square modulo $p^m$, then $N=0$.
4. If $k\in\mathbb{N}$ is such that $m=k+{\rm v}_p(a)$ and $a$ is a square modulo $p^m$, then $N=p^{{\rm v}_p(a)/2-1}(pg_k-g_{k+1})$.
1\. Let $m<{\rm v}_p(a)$. Observe that $M=\{x\in [0,p^m-1]\mid 2{\rm v}_p(x)=m\}$. Clearly, if $m$ is odd, then $N=0$. Now let $m$ be even. We have $M=\{p^{m/2}y\mid y\in [0,p^{m/2}-1],{\rm gcd}(y,p)=1\}$, and thus $N=|\{y\in [0,p^{m/2}-1]\mid {\rm gcd}(y,p)=1\}|=\varphi(p^{m/2})$.
2\. Note that $M=\{x\in [0,p^m-1]\mid 2{\rm v}_p(x)\geq m,x^2\not\equiv a\mod p^{m+1}\}$ and $|\{x\in [0,p^m-1]\mid 2{\rm v}_p(x)\geq m\}|=p^{\lfloor m/2\rfloor}$. Set $M^{\prime}=\{x\in [0,p^m-1]\mid x^2\equiv a\mod p^{m+1}\}$. Then $M^{\prime}=\{x\in [0,p^m-1]\mid 2{\rm v}_p(x)\geq m,x^2\equiv a\mod p^{m+1}\}$ and $N=p^{\lfloor m/2\rfloor}-|M^{\prime}|$. If $a$ is not a square modulo $p^{m+1}$, then $M^{\prime}=\emptyset$, and hence $N=p^{\lfloor m/2\rfloor}$. Now let $a$ be a square modulo $p^{m+1}$. Then $M^{\prime}\not=\emptyset$, and thus $m$ is even. Observe that $M^{\prime}=\{x\in [0,p^m-1]\mid 2{\rm v}_p(x)=m,x^2\equiv a\mod p^{m+1}\}=\{p^{m/2}y\mid y\in [0,p^{m/2}-1],y^2\equiv c\mod p\}$. Therefore, $|M^{\prime}|=|\{y\in [0,p^{m/2}-1]\mid y^2\equiv c\mod p\}|=p^{m/2-1}|\{y\in [0,p-1]\mid y^2\equiv c\mod p\}|$.
If $p\not=2$, then $N=p^{\lfloor m/2\rfloor}-|M^{\prime}|=p^{m/2}-2p^{m/2-1}=p^{m/2-1}(p-2)$ by Lemma \[lemma 3.4\].3. Moreover, if $p=2$, then $N=2^{\lfloor m/2\rfloor}-|M^{\prime}|=2^{m/2}-2^{m/2-1}=2^{m/2-1}$ by Lemma \[lemma 3.4\].3.
3\. This is obvious.
4\. Let $k\in\mathbb{N}$ be such that $m=k+{\rm v}_p(a)$ and let $a$ be a square modulo $p^m$. It follows by Lemma \[lemma 3.4\] that ${\rm v}_p(a)$ is even. Set $r={\rm v}_p(a)/2$ and for $\theta\in\{0,1\}$ set $M_{\theta}=\{x\in [0,p^m-1]\mid 2{\rm v}_p(x)={\rm v}_p(a),x^2\equiv a\mod p^{m+\theta}\}$. Then $M=\{x\in [0,p^m-1]\mid {\rm v}_p(x)=r,{\rm v}_p(x^2-a)=m\}=M_0\setminus M_1$. Since $\{x\in [0,p^m-1]\mid {\rm v}_p(x)=r\}=\{p^ry\mid y\in [0,p^{k+r}-1],{\rm gcd}(y,p)=1\}$, we infer that $M_{\theta}=\{p^ry\mid y\in [0,p^{k+r}-1],y^2\equiv c\mod p^{k+\theta}\}$. Therefore, $|M_{\theta}|=|\{y\in [0,p^{k+r}-1]\mid y^2\equiv c\mod p^{k+\theta}\}|=p^{r-\theta}|\{y\in [0,p^{k+\theta}-1]\mid y^2\equiv c\mod p^{k+\theta}\}|=p^{r-\theta}g_{k+\theta}$. This implies that $N=|M_0|-|M_1|=p^rg_k-p^{r-1}g_{k+1}=p^{r-1}(pg_k-g_{k+1})$.
\[theorem 3.6\] Let $\mathcal{O}$ be an order in a quadratic number field $K$ with conductor $\mathfrak{f}=f\mathcal{O}_K$ for some $f\in\mathbb{N}_{\geq 2}$, $p$ be a prime divisor of $f$, and $\mathfrak{p}=P_{f,p}$.
1. The primary ideals with radical $\mathfrak{p}$ are exactly the ideals of the form $$\mathfrak{q}=p^\ell(p^m\mathbb{Z}+(r+\tau)\mathbb{Z})$$ with $\ell,m\in\mathbb{N}_0$, $\ell+m\geq1$, $0\leq r<p^m$, and $\mathcal N_{K/\mathbb{Q}}(r+\tau)\equiv 0\mod p^m$. Moreover, $\mathcal{N}(\mathfrak{q})=p^{2\ell+m}$.
2. A primary ideal $\mathfrak{q}=p^\ell(p^m\mathbb{Z}+(r+\tau)\mathbb{Z})$ is invertible if and only if $$\mathcal N_{K/\mathbb{Q}}(r+\tau)\not\equiv 0\mod p^{m+1}.$$
3. A primary ideal $\mathfrak{q}$ with radical $\mathfrak{p}$ is an ideal atom if and only if $\mathfrak{q}=p\mathcal{O}$ or $\mathfrak{q}=p^m\mathbb{Z}+(r+\tau)\mathbb{Z}$ with $m\in\mathbb{N}$ and $p^m\mid \mathcal N_{K/\mathbb{Q}}(r+\tau)$.
4. Table \[table1\] gives the number of invertible ideal atoms of the form $p^m\mathbb{Z}+(r+\tau)\mathbb{Z}$ with norm $p^m$; this number is $0$ if $m$ is not listed in the table.
----------------------------- ----------------------------------- ------------------------------------------------- ------------------------------ -------------------------------
$m$ $2h$ $2{\rm v}_p\left(f\right)$ $2{\rm v}_p\left(f\right)+1$ $>2{\rm v}_p\left(f\right)+1$
$1\leq h<{\rm v}_p\left(f\right)$
$p$ [is inert]{.nodecor} $p^{{\rm v}_p\left(f\right)}$
$p$ [is ramified]{.nodecor} $p^{{\rm v}_p\left(f\right)}$
$p$ [splits]{.nodecor} $p^{{\rm v}_p\left(f\right)-1}\left(p-2\right)$
----------------------------- ----------------------------------- ------------------------------------------------- ------------------------------ -------------------------------
: Number of nontrivial invertible $\mathfrak{p}$-primary ideal atoms[]{data-label="table1"}
5. The number of ideal atoms with radical $\mathfrak{p}$ is finite if and only if the number of invertible ideal atoms with radical $\mathfrak{p}$ is finite if and only if $p$ does not split.
1\. and 2. are an immediate consequence of Proposition \[proposition 3.1\].
3\. In 1. we have seen, that all $\mathfrak{p}$-primary ideals of $\mathcal{O}$ are of the form $\mathfrak{q}=p^\ell(p^m\mathbb{Z}+(r+\tau)\mathbb{Z})$. If both $\ell$ and $m$ are greater than $0$, then $\mathfrak{q}$ is not an ideal atom. Indeed, $\mathfrak{q}=(p\mathcal{O})^\ell(p^m\mathbb{Z}+(r+\tau)\mathbb{Z})$ is a nontrivial factorization. It remains to be proven, that $p\mathcal{O}$ and $p^m\mathbb{Z}+(r+\tau)\mathbb{Z}$ are ideal atoms.
Assume that there exist proper ideals $\mathfrak{a}_1,\mathfrak{a}_2$ of $\mathcal{O}$ such that $p\mathcal{O}=\mathfrak{a}_1\mathfrak{a}_2$. Since $p\mathcal{O}$ is $\mathfrak{p}$-primary, we have $\mathfrak{a}_1$ and $\mathfrak{a}_2$ are $\mathfrak{p}$-primary. Using this information, we deduce, that $p\mathcal{O}\subset\mathfrak{p}^2$, implying $$p\in p\mathcal{O}\subset\mathfrak{p}^2=(p^2,pf\omega,f^2\omega^2)=p(p,f\omega,\frac{f}{p}\omega f\omega)=p(p,f\omega)=p\mathfrak{p}.$$ Therefore, $1\in\mathfrak{p}$, a contradiction.
Assume that there exist proper ideals $\mathfrak{a}_1,\mathfrak{a}_2$ of $\mathcal{O}$ such that $p^m\mathbb{Z}+(r+\tau)\mathbb{Z}=\mathfrak{a}_1\mathfrak{a}_2$. Note that $\mathfrak{a}_1$ and $\mathfrak{a}_2$ are $\mathfrak{p}$-primary. By Proposition \[proposition 3.2\].3, it follows that $p^m\mathbb{Z}+(r+\tau)\mathbb{Z}\subset p\mathcal{O}$, a contradiction to $r+\tau\not\in p\mathcal{O}$.
4\. By 1. and 3., the nontrivial $\mathfrak{p}$-primary ideal atoms of norm $p^m$ are all $\mathfrak{q}=p^m\mathbb{Z}+(r+\tau)\mathbb{Z}$ with $m\in\mathbb{N}$, $0\leq r<p^m$ and $\mathcal N_{K/\mathbb{Q}}(r+\tau)\equiv 0\mod p^m$. By 2., an ideal of this form is invertible if and only if $\mathcal N_{K/\mathbb{Q}}(r+\tau)\not\equiv 0\mod p^{m+1}$.
Thus if we want to count the number of invertible $\mathfrak{p}$-primary ideal atoms of the form $\mathfrak{q}=p^m\mathbb{Z}+(r+\tau)\mathbb{Z}$ we have to count the number of solutions $r\in[0,p^m-1]$ of the equation $$\label{equation 5}
{\rm v}_p(\mathcal N_{K/\mathbb{Q}}(r+\tau))=m.$$ Set $N=|\{r\in[0,p^m-1]\mid {\rm v}_p(\mathcal N_{K/\mathbb{Q}}(r+\tau))=m\}|$ and $a=\begin{cases} (\frac{f}{2})^2d_K &\textnormal{if }p=2\\ f^2d_K &\textnormal{if }p\not=2\end{cases}$. Next we show that $N=|\{r\in[0,p^m-1]\mid {\rm v}_p(r^2-a)=m\}|$. Note that $\mathcal N_{K/\mathbb{Q}}(r+\tau)=\frac{(2r+\varepsilon)^2-f^2d_K}{4}$ for each $r\in [0,p^m-1]$. If $p=2$, then $\varepsilon=0$, and hence $\mathcal N_{K/\mathbb{Q}}(r+\tau)=r^2-a$. Now let $p\not=2$. Then ${\rm v}_p(\mathcal N_{K/\mathbb{Q}}(r+\tau))={\rm v}_p((2r+\varepsilon)^2-a)$ for each $r\in [0,p^m-1]$. Let $f:\{r\in[0,p^m-1]\mid {\rm v}_p(r^2-a)=m\}\rightarrow\{r\in[0,p^m-1]\mid {\rm v}_p((2r+\varepsilon)^2-a)=m\}$ and $g:\{r\in[0,p^m-1]\mid {\rm v}_p((2r+\varepsilon)^2-a)=m\}\rightarrow\{r\in[0,p^m-1]\mid {\rm v}_p(r^2-a)=m\}$ be defined by $f(r)=\begin{cases}\frac{r-\varepsilon}{2} &\textnormal{if }r-\varepsilon\textnormal{ is even}\\\frac{r+p^m-\varepsilon}{2} &\textnormal{if }r-\varepsilon\textnormal{ is odd}\end{cases}$ and $g(r)={\rm rem}(2r+\varepsilon,p^m)$ for each $r\in [0,p^m-1]$. Observe that $f$ and $g$ are well-defined injective maps. Therefore, $N=|\{r\in[0,p^m-1]\mid {\rm v}_p(r^2-a)=m\}|$ in any case. Set $c=\frac{a}{p^{{\rm v}_p(a)}}$ and for $\ell\in\mathbb{N}$ set $g_{\ell}=|\{y\in [0,p^{\ell}-1]\mid y^2\equiv c\mod p^{\ell}\}|$. If $m<{\rm v}_p(a)$, then the statement follows immediately by Lemma \[lemma 3.5\].1. Therefore, let $m\geq {\rm v}_p(a)$. In what follows we use Lemmas \[lemma 3.4\] and \[lemma 3.5\] without further citation.
CASE 1: $p=2$ and $2$ is inert. We have ${\rm v}_2(a)=2{\rm v}_2(f)-2$, $c\equiv d_K\equiv 5\mod 8$, $g_1=1$, $g_2=2$ and $g_3=0$. If $m={\rm v}_2(a)$, then $a$ is a square modulo $2^{m+1}$, and hence $N=2^{m/2-1}=\varphi(2^{m/2})$. If $m={\rm v}_2(a)+1$, then $a$ is a square modulo $2^m$, and thus $N=2^{{\rm v}_2(a)/2-1}(2g_1-g_2)=0$. If $m={\rm v}_2(a)+2$, then $a$ is a square modulo $2^m$, whence $N=2^{{\rm v}_2(a)/2-1}(2g_2-g_3)=2^{{\rm v}_2(a)/2+1}=2^{{\rm v}_2(f)}$. Finally, let $m\geq {\rm v}_2(a)+3$. Then $a$ is not a square modulo $2^m$, and hence $N=0$.
CASE 2: $p=2$ and $2$ is ramified. Note that ${\rm v}_2(a)\in\{2{\rm v}_2(f),2{\rm v}_2(f)+1\}$. First let ${\rm v}_2(a)=2{\rm v}_2(f)$. Then $a=f^2d$ with $c\equiv d\equiv 3\mod 4$, $g_1=1$ and $g_{\ell}=0$ for each $\ell\in\mathbb{N}_{\geq 2}$. If $m={\rm v}_2(a)$, then $a$ is a square modulo $2^{m+1}$, and thus $N=2^{m/2-1}=2^{{\rm v}_2(f)-1}=\varphi(2^{{\rm v}_2(f)})$. If $m={\rm v}_2(a)+1$, then $a$ is a square modulo $2^m$, and hence $N=2^{{\rm v}_2(a)/2-1}(2g_1-g_2)=2^{{\rm v}_2(f)}$. Finally, let $m\geq {\rm v}_2(a)+2$. Then $a$ is not a square modulo $2^m$, and thus $N=0$.
Now let ${\rm v}_2(a)=2{\rm v}_2(f)+1$. If $m={\rm v}_2(a)$, then $a$ is not a square modulo $2^{m+1}$, and hence $N=2^{\lfloor m/2\rfloor}=2^{{\rm v}_2(f)}$. If $m>{\rm v}_2(a)$, then $a$ is not a square modulo $2^m$, and thus $N=0$.
CASE 3: $p=2$ and $2$ splits. Observe that ${\rm v}_2(a)=2{\rm v}_2(f)-2$, $c\equiv d_K\equiv 1\mod 8$, $g_1=1$, $g_2=2$ and $g_{\ell}=4$ for each $\ell\in\mathbb{N}_{\geq 3}$. If $m={\rm v}_2(a)$, then $a$ is a square modulo $2^{m+1}$, and hence $N=2^{m/2-1}=\varphi(2^{m/2})$. Now let $m>{\rm v}_2(a)$ and set $k=m-{\rm v}_2(a)$. Note that $a$ is a square modulo $2^m$, and hence $N=2^{{\rm v}_2(a)/2-1}(2g_k-g_{k+1})$. If $m<{\rm v}_2(a)+3$, then $N=0$. Finally, let $m\geq {\rm v}_2(a)+3$. Then $N=2^{{\rm v}_2(a)/2+1}=2^{{\rm v}_2(f)}=2\varphi(2^{{\rm v}_2(f)})$.
CASE 4: $p\not=2$ and $p$ is inert. We have ${\rm v}_p(a)=2{\rm v}_p(f)$, $(\frac{c}{p})=(\frac{d_K}{p})=-1$ and $g_{\ell}=0$ for each $\ell\in\mathbb{N}$. If $m={\rm v}_p(a)$, then $a$ is not a square modulo $p^{m+1}$, and hence $N=p^{\lfloor m/2\rfloor}=p^{{\rm v}_p(f)}$. If $m>{\rm v}_p(a)$, then $a$ is not a square modulo $p^m$, and thus $N=0$.
CASE 5: $p\not=2$ and $p$ is ramified. It follows that ${\rm v}_p(a)=2{\rm v}_p(f)+1$. If $m={\rm v}_p(a)$, then $a$ is not a square modulo $p^{m+1}$, and thus $N=p^{\lfloor m/2\rfloor}=p^{{\rm v}_p(f)}$. If $m>{\rm v}_p(a)$, then $a$ is not a square modulo $p^m$, and thus $N=0$.
CASE 6: $p\not=2$ and $p$ splits. Note that ${\rm v}_p(a)=2{\rm v}_p(f)$, $(\frac{c}{p})=(\frac{d_K}{p})=1$ and $g_{\ell}=2$ for each $\ell\in\mathbb{N}$. If $m={\rm v}_p(a)$, then $a$ is a square modulo $p^{m+1}$, and hence $N=p^{m/2-1}(p-2)=p^{{\rm v}_p(f)-1}(p-2)$. If $m>{\rm v}_p(a)$, then $a$ is a square modulo $p^m$, and thus $N=p^{{\rm v}_p(a)/2-1}(pg_k-g_{k+1})=2p^{{\rm v}_p(f)-1}(p-1)=2\varphi(p^{{\rm v}_p(f)})$.
5\. It is an immediate consequence of 4. that the number of invertible ideal atoms with radical $\mathfrak{p}$ is finite if and only if $p$ does not split. It remains to show that $\mathcal{A}(\mathcal{I}_p(\mathcal{O}))$ is finite if and only if $\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ is finite. It follows from [@An-Mo92 Theorem 4.3] that $\mathcal{I}(\mathcal{O}_{\mathfrak{p}})$ is a finitely generated monoid if and only if $\mathcal{I}^*(\mathcal{O}_{\mathfrak{p}})$ is a finitely generated monoid. Therefore, Proposition \[proposition 3.3\].2 implies that $\mathcal{I}_p(\mathcal{O})$ is a finitely generated monoid if and only if $\mathcal{I}^*_p(\mathcal{O})$ is a finitely generated monoid. Observe that $\mathcal{I}_p(\mathcal{O})$ and $\mathcal{I}^*_p(\mathcal{O})$ are atomic monoids. Therefore, $\mathcal{A}(\mathcal{I}_p(\mathcal{O}))$ is finite if and only if $\mathcal{I}_p(\mathcal{O})$ is a finitely generated monoid if and only if $\mathcal{I}^*_p(\mathcal{O})$ is a finitely generated monoid if and only if $\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ is finite.
Sets of distances and sets of catenary degrees {#4}
==============================================
The goal in this section is to prove Theorem \[theorem 1.1\]. The proof is based on the precise description of ideals given in Theorem \[theorem 3.6\]. We proceed in a series of lemmas and propositions and use all notation on orders as introduced at the beginning of Section \[3\]. In particular, $\mathcal{O} =\mathcal O_f$ is an order in a quadratic number with conductor $f\mathcal O_K$ for some $f\in\mathbb{N}_{\ge 2}$.
\[proposition 4.1\] Let $H$ be a reduced atomic monoid and suppose there is a cancellative atom $u\in\mathcal{A}(H)$ such that for each $a\in H\setminus H^{\times}$ there are $n\in\mathbb{N}_0$ and $v\in\mathcal{A}(H)$ such that $a=u^nv$.
1. For all $n,m\in\mathbb{N}_0$ and $v,w\in\mathcal{A}(H)$ such that $u^nv=u^mw$, it follows that $n=m$ and $v=w$.
2. For all $n\in\mathbb{N}_0$ and $v\in\mathcal{A}(H)$, it follows that $\max\mathsf{L}(u^nv)=n+1$.
3. $\mathsf{c}(H)=\sup\{\mathsf{c}(w\cdot y,u^n\cdot v)\mid n\in\mathbb{N}$ and $v,w,y\in\mathcal{A}(H)$ such that $wy=u^nv\}$.
4. If $H$ is half-factorial, then $\mathsf{c}(H)\leq 2$.
5. $\sup\Delta(H)=\sup\{\ell-2\mid\ell\in\mathbb{N}_{\geq 3}$ such that $\mathsf{L}(vw)\cap [2,\ell]=\{2,\ell\}$ for some $v,w\in\mathcal{A}(H)\}$.
1\. Let $n,m\in\mathbb{N}_0$ and $v,w\in\mathcal{A}(H)$ be such that $u^nv=u^mw$. Without restriction let $n\leq m$. Since $u$ is cancellative, we infer that $v=u^{m-n}w$. Since $v\in\mathcal{A}(H)$, we have $n=m$, and thus $v=w$.
2\. It is clear that $n+1\in\mathsf{L}(u^nv)$ for all $n\in\mathbb{N}_0$ and $v\in\mathcal{A}(H)$. Therefore, it is sufficient to show by induction that for all $n\in\mathbb{N}_0$ and $v\in\mathcal{A}(H)$, $\max\mathsf{L}(u^nv)\leq n+1$. Let $n\in\mathbb{N}_0$ and $v\in\mathcal{A}(H)$. If $n=0$, then the assertion is obviously true. Now let $n>0$ and $z\in\mathsf{Z}(u^nv)$. Then there are some $z^{\prime},z^{\prime\prime}\in\mathsf{Z}(H)\setminus\{1\}$ such that $z=z^{\prime}\cdot z^{\prime\prime}$. There are some $m^{\prime},m^{\prime\prime}\in\mathbb{N}_0$ and $w^{\prime},w^{\prime\prime}\in\mathcal{A}(H)$ such that $\pi(z^{\prime})=u^{m^{\prime}}w^{\prime}$ and $\pi(z^{\prime\prime})=u^{m^{\prime\prime}}w^{\prime\prime}$. There are some $\ell\in\mathbb{N}$ and $y\in\mathcal{A}(H)$ such that $w^{\prime}w^{\prime\prime}=u^{\ell}y$. We infer that $u^nv=u^{m^{\prime}+m^{\prime\prime}+\ell}y$, and thus $n=m^{\prime}+m^{\prime\prime}+\ell$ by 1. Since $m^{\prime},m^{\prime\prime}<n$, it follows by the induction hypothesis that $|z^{\prime}|\leq m^{\prime}+1$ and $|z^{\prime\prime}|\leq m^{\prime\prime}+1$. Consequently, $|z|\leq m^{\prime}+m^{\prime\prime}+2\leq m^{\prime}+m^{\prime\prime}+\ell+1=n+1$.
3\. Set $k=\sup\{\mathsf{c}(w\cdot y,u^n\cdot v)\mid n\in\mathbb{N}_0$ and $v,w,y\in\mathcal{A}(H)$ such that $wy=u^nv\}$. Since $\mathsf{c}(H)=\sup\{\mathsf{c}(z,z^{\prime})\mid a\in H, z,z^{\prime}\in\mathsf{Z}(a)\}$, it is obvious that $k\leq\mathsf{c}(H)$. It remains to show by induction that for all $n\in\mathbb{N}_0$ and $v\in\mathcal{A}(H)$, it follows that $\mathsf{c}(u^nv)\leq k$. Let $n\in\mathbb{N}_0$ and $v\in\mathcal{A}(H)$. Since $\mathsf{c}(v)=0$, we can assume without restriction that $n>0$. Since $\mathsf{c}(u^nv)=\sup\{\mathsf{c}(z,u^n\cdot v)\mid z\in\mathsf{Z}(u^nv)\}$, it remains to show that $\mathsf{c}(z,u^n\cdot v)\leq k$ for all $z\in\mathsf{Z}(u^nv)$. Let $z\in\mathsf{Z}(u^nv)$.
CASE 1: For all $w,y\in\mathcal{A}(H)\setminus\{u\}$, we have $w\cdot y\nmid z$. There are some $m\in\mathbb{N}$ and $w\in\mathcal{A}(H)$ such that $z=u^m\cdot w$. We infer by 1. that $z=u^n\cdot v$, and thus $\mathsf{c}(z,u^n\cdot v)=0\leq k$.
CASE 2: There are some $w,y\in\mathcal{A}(H)\setminus\{u\}$ such that $w\cdot y\mid z$. Set $z^{\prime}=\frac{z}{w\cdot y}$. There exist $m\in\mathbb{N}$ and $a\in\mathcal{A}(H)$ such that $wy=u^ma$. We infer that $m\leq n$ and $u^nv=\pi(z)=\pi(w\cdot y)\pi(z^{\prime})=u^ma\pi(z^{\prime})$, and thus $a\pi(z^{\prime})=u^{n-m}v$. Observe that $\mathsf{c}(z,u^m\cdot a\cdot z^{\prime})\leq\mathsf{c}(w\cdot y,u^m\cdot a)\leq k$. Since $n-m<n$, it follows by the induction hypothesis that $\mathsf{c}(u^m\cdot a\cdot z^{\prime},u^n\cdot v)\leq\mathsf{c}(a\cdot z^{\prime},u^{n-m}\cdot v)\leq k$, and hence $\mathsf{c}(z,u^n\cdot v)\leq k$.
4\. Let $H$ be half-factorial, $n\in\mathbb{N}$ and $v,w,y\in\mathcal{A}(H)$ be such that $wy=u^nv$. We infer that $n=1$, and thus $\mathsf{c}(w\cdot y,u^n\cdot v)\leq\mathsf{d}(w\cdot y,u\cdot v)\leq 2$. Therefore, $\mathsf{c}(H)\leq 2$ by 3.
5\. Set $N=\sup\{\ell-2\mid\ell\in\mathbb{N}_{\geq 3}$ such that $\mathsf{L}(vw)\cap [2,\ell]=\{2,\ell\}$ for some $v,w\in\mathcal{A}(H)\}$. It is obvious that $N\leq\sup\Delta(H)$. It remains to show that $k\leq N$ for each $k\in\Delta(H)$. Let $k\in\Delta(H)$. Then there are some $a\in H$ and $r,s\in\mathsf{L}(a)$ such that $r<s$, $\mathsf{L}(a)\cap [r,s]=\{r,s\}$, and $k=s-r$. Let $z\in\mathsf{Z}(a)$ with $|z|=r$ be such that ${\rm v}_u(z)=\max\{{\rm v}_u(z^{\prime})\mid z^{\prime}\in\mathsf{Z}(a) \ \text{with} \ |z^{\prime}|=r\}$. Since $r<\max\mathsf{L}(a)$, it follows by 2., that there are some $v,w\in\mathcal{A}(H)\setminus\{u\}$ such that $v\cdot w\mid z$. There are some $n\in\mathbb{N}$ and $y\in\mathcal{A}(H)$ such that $vw=u^ny$. Since ${\rm v}_u(z)$ is maximal amongst all factorizations of $a$ of length $r$, we have $n\geq 2$. Consequently, there is some $\ell\in\mathsf{L}(vw)$ such that $2<\ell\leq n+1$ and $\mathsf{L}(vw)\cap [2,\ell]=\{2,\ell\}$. Note that $r+\ell-2\in\mathsf{L}(a)$, and thus $s\leq r+\ell-2$. This implies that $k\leq\ell-2\leq N$.
Theorem \[theorem 3.6\] implies that, for all prime divisors $p$ of $f$, $\mathcal{I}^*_p(\mathcal{O}_f)$ and $\mathcal{I}_p(\mathcal{O}_f)$ are reduced atomic monoids satisfying the assumption in Proposition \[proposition 4.1\].
\[lemma 4.2\] Let $p$ be a prime divisor of $f$.
1. $\mathsf{Z}(pP_{f,p})=\{A\cdot P_{f,p}\mid A=P_{f,p}$ or $A\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $\mathcal{N}(A)=p^2\}$ and $1\in {\rm Ca}(\mathcal{I}_p(\mathcal{O}_f))$.
2. If $I,J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ are such that $\mathcal{N}(I)=p^2$ and $\mathcal{N}(J)>p^2$, then $IJ=pL$ for some $L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$.
3. $2\in {\rm Ca}(\mathcal{I}^*_p(\mathcal{O}_f))$.
1\. Note that $\{I\in\mathcal{I}_p(\mathcal{O}_f)\mid\mathcal{N}(I)=p\}=\{P_{f,p}\}$. First we show that $\mathsf{Z}(pP_{f,p})=\{A\cdot P_{f,p}\mid A=P_{f,p}$ or $A\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $\mathcal{N}(A)=p^2\}$.
Let $z\in\mathsf{Z}(pP_{f,p})$. It follows from Proposition \[proposition 4.1\].2 that $|z|\leq 2$, and hence $|z|=2$. Consequently, $z=A\cdot B$ for some $A,B\in\mathcal{A}(\mathcal{I}_p(\mathcal{O}_f))$. By Proposition \[proposition 3.2\].1 there are some $(u,v,w),(x,y,t)\in\mathcal{M}_{f,p}$ such that $A=p^u(p^v\mathbb{Z}+(w+\tau)\mathbb{Z})$ and $B=p^x(p^y\mathbb{Z}+(t+\tau)\mathbb{Z})$. Set $g=\min\{v,y,{\rm v}_p(w+t+\varepsilon)\}$ and $e=\min\{g,{\rm v}_p(w-t),{\rm v}_p(w^2+\varepsilon w+\eta)-v,{\rm v}_p(t^2+\varepsilon t+\eta)-y\}$. We infer by Proposition \[proposition 3.2\].1 that $u+x+g=1$ and $v+y+e-2g=1$. Note that $g\in\{0,1\}$. If $g=0$, then $u+x=v+y=1$, and thus ($A=p\mathcal{O}_f$ and $B=P_{f,p}$) or ($A=P_{f,p}$ and $B=p\mathcal{O}_f$). Now let $g=1$. Then $u=x=0$, $v,y\geq 1$, $v+y+e=3$, and $e\in\{0,1\}$. If $e=1$, then $v=y=1$, and thus $A=B=P_{f,p}$. Now let $e=0$. Then ($v=1$ and $y=2$) or ($v=2$ and $y=1$). Without restriction let $v=2$ and $y=1$. Then $B=P_{f,p}$, $\mathcal{N}(A)=p^v=p^2$, and $\mathcal{N}(A)\mathcal{N}(B)=p^3=\mathcal{N}(pP_{f,p})=\mathcal{N}(AB)$. Since $B$ is not invertible, it follows by Proposition \[proposition 3.2\].3 that $A$ is invertible.
To prove the converse inclusion note that $P_{f,p}=p\mathbb{Z}+(r+\tau)\mathbb{Z}$ for some $(0,1,r)\in\mathcal{M}_{f,p}$. By Proposition \[proposition 3.2\].1 we have $P_{f,p}^2=p^a(p^b\mathbb{Z}+(c+\tau)\mathbb{Z}$ with $(a,b,c)\in\mathcal{M}_{f,p}$, $a=\min\{1,{\rm v}_p(2r+\varepsilon)\}$ and $b=2+e-2a$ with $e=\min\{a,{\rm v}_p(r^2+\varepsilon r+\eta)-1\}$. By Proposition \[proposition 3.2\].3 we have $a>0$, and thus $a=b=e=1$. Consequently, $P_{f,p}^2=pP_{f,p}$. Now let $A\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ be such that $\mathcal{N}(A)=p^2$. It follows by Proposition \[proposition 3.2\].3 that $\mathcal{N}(AP_{f,p})=\mathcal{N}(A)\mathcal{N}(P_{f,p})=p^3$ and $AP_{f,p}=pI$ for some $I\in\mathcal{I}_p(\mathcal{O}_f)$. We infer that $\mathcal{N}(I)=p$, and hence $I=P_{f,p}$.
Observe that $\mathsf{d}(z^{\prime},z^{\prime\prime})\leq 1$ for all $z^{\prime},z^{\prime\prime}\in\mathsf{Z}(pP_{f,p})$ and $(p\mathcal{O}_f)\cdot P_{f,p}$ and $P_{f,p}^2$ are distinct factorizations of $pP_{f,p}$. Therefore, $1=\mathsf{c}(pP_{f,p})\in {\rm Ca}(\mathcal{I}_p(\mathcal{O}_f))$.
2\. Let $I,J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ be such that $\mathcal{N}(I)=p^2$ and $\mathcal{N}(J)>p^2$. Without restriction we can assume that $I\not=p\mathcal{O}_f$. There are some $(0,2,r),(0,k,s)\in\mathcal{M}_{f,p}$ such that $I=p^2\mathbb{Z}+(r+\tau)\mathbb{Z}$ and $J=p^k\mathbb{Z}+(s+\tau)\mathbb{Z}$. Since $I$ and $J$ are invertible, we have ${\rm v}_p(r^2+\varepsilon r+\eta)=2$ and ${\rm v}_p(s^2+\varepsilon s+\eta)=k>2$. Therefore, ${\rm v}_p(r+s+\varepsilon)+{\rm v}_p(r-s)={\rm v}_p(r^2+\varepsilon r+\eta-(s^2+\varepsilon s+\eta))=2$, and thus ${\rm v}_p(r+s+\varepsilon)=1$, by Proposition \[proposition 3.2\].2. Therefore, $\min\{2,k,{\rm v}_p(r+s+\varepsilon)\}=1$, and hence $IJ=pL$ for some $L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ by Proposition \[proposition 3.2\].1.
3\. We distinguish two cases.
CASE 1: $p\not=2$ or ${\rm v}_p(f)\geq 2$ or $d\not\equiv 1\mod 8$. It follows from Theorem \[theorem 3.6\] that there is some $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $\mathcal{N}(I)=p^2$ and $I\not=p\mathcal{O}_f$. We have $I\overline{I}=(p\mathcal{O}_f)^2$, and hence $\mathsf{L}(I\overline{I})=\{2\}$. Since $I\cdot\overline{I}$ and $(p\mathcal{O}_f)\cdot (p\mathcal{O}_f)$ are distinct factorizations of $I\overline{I}$, we have $2=\mathsf{c}(I\overline{I})\in {\rm Ca}(\mathcal{I}^*_p(\mathcal{O}_f))$.
CASE 2: $p=2$, ${\rm v}_p(f)=1$ and $d\equiv 1\mod 8$. By Proposition \[proposition 3.3\].3 we can assume without restriction that $f=2$. By Theorem \[theorem 3.6\] there is some $I\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ such that $\mathcal{N}(I)=8$. There is some $(0,3,r)\in\mathcal{M}_{f,2}$ such that $I=8\mathbb{Z}+(r+\tau)\mathbb{Z}$. We have ${\rm v}_2(r^2-d)=3$, and hence ${\rm v}_2(r)=0$. Therefore, $\min\{3,{\rm v}_2(2r)\}=1$, and thus $I^2=2J$ for some $J\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$. Consequently, $\mathsf{L}(I^2)=\{2\}$. Since $I\cdot I$ and $(2\mathcal{O}_f)\cdot J$ are distinct factorizations of $I^2$, it follows that $2=\mathsf{c}(I^2)\in {\rm Ca}(\mathcal{I}^*_p(\mathcal{O}_f))$.
\[proposition 4.3\] Let $p$ be an odd prime divisor of $f$ such that ${\rm v}_p(f)\geq 2$.
1. There is a $C\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $\mathsf{L}(C^2)=\{2,3\}$ whence $1\in\Delta(\mathcal{I}^*_p(\mathcal{O}_f))$ and $3\in {\rm Ca}(\mathcal{I}^*_p(\mathcal{O}_f))$. Moreover, if $(p\not=3$ or $d\not\equiv 2\mod 3$ or ${\rm v}_p(f)>2)$, then there are $I,J,L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $I^2=p^2J$ and $J^2=p^2L$.
2. If $|{\rm Pic}(\mathcal{O}_f)|\leq 2$ and $(p\not=3$ or $d\not\equiv 2\mod 3$ or ${\rm v}_p(f)>2)$, then there is a nonzero primary $a\in\mathcal{O}_f$ such that $2,3\in\mathsf{L}(a)$ whence $1\in\Delta(\mathcal{O}_f)$.
1\. By Proposition \[proposition 3.3\].3 there is a monoid isomorphism $\delta:\mathcal{I}^*_p(\mathcal{O}_f)\rightarrow\mathcal{I}^*_p(\mathcal{O}_{\frac{f}{2^{{\rm v}_2(f)}}})$ such that $\delta(p\mathcal{O}_f)=p\mathcal{O}_{\frac{f}{2^{{\rm v}_2(f)}}}$. Therefore, we can assume without restriction that $f$ is odd.
CLAIM: $\mathsf{L}(I^2)=\{2,3\}$ for some $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$, $1\in\Delta(\mathcal{I}^*_p(\mathcal{O}_f))$, $3\in {\rm Ca}(\mathcal{I}^*_p(\mathcal{O}_f))$ and if ${\rm v}_p(p^4+f^2d)=4$, then $I^2=p^2J$ and $J^2=p^2L$ for some $I,J,L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$.
For $r\in\mathbb{N}_0$ set $k={\rm v}_p(\mathcal{N}_{K/\mathbb{Q}}(r+\tau))$ and $I=p^k\mathbb{Z}+(r+\tau)\mathbb{Z}$. Let $k>0$ and $r<p^k$. Then $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$. Moreover, $I^2=p^a(p^b\mathbb{Z}+(c+\tau)\mathbb{Z})$ with $a=\min\{k,{\rm v}_p(2r+\varepsilon)\}$, $b=2(k-a)$ and $c={\rm rem}(r-t\frac{\mathcal{N}_{K/\mathbb{Q}}(r+\tau)}{p^a},p^b)$ for each $t\in\mathbb{Z}$ with $t\frac{2r+\varepsilon}{p^a}\equiv 1\mod p^{k-a}$. Set $J=p^b\mathbb{Z}+(c+\tau)\mathbb{Z}$. Then $I^2=p^aJ$ and if $b>0$, then $J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$. In particular, if $a=2$ and $b>0$, then $I,J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ and $\mathsf{L}(I^2)=\{2,3\}$, and hence $1\in\Delta(I^2)\subseteq\Delta(\mathcal{I}^*_p(\mathcal{O}_f))$ and $3=\mathsf{c}(I^2)\in {\rm Ca}(\mathcal{I}^*_p(\mathcal{O}_f))$. Observe that $J^2=p^{a^{\prime}}(p^{b^{\prime}}\mathbb{Z}+(c^{\prime}+\tau)\mathbb{Z})$ with $a^{\prime}=\min\{b,{\rm v}_p(2c+\varepsilon)\}$, $b^{\prime}=2(b-a^{\prime})$ and $c^{\prime}\in\mathbb{N}_0$ such that $c^{\prime}<p^{b^{\prime}}$. Set $L=p^{b^{\prime}}\mathbb{Z}+(c^{\prime}+\tau)\mathbb{Z}$. Then $J^2=p^{a^{\prime}}L$ and if $b^{\prime}>0$, then $L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$.
CASE 1: $d\not\equiv 1\mod 4$. Set $r=p^2$. We have $\mathcal{N}_{K/\mathbb{Q}}(r+\tau)=p^4-f^2d$, $k\geq 4$, $a=2$, $b=2(k-2)>0$, $r<p^k$, and $t=\frac{p^{k-2}+1}{2}$ satisfies the congruence. Therefore, $c={\rm rem}(p^2-\frac{(p^{k-2}+1)(p^4-f^2d)}{2p^2},p^{2(k-2)})=\frac{p^4+f^2d+p^{k-2}f^2d-p^{k+2}+2\ell p^{2(k-1)}}{2p^2}$ for some $\ell\in\mathbb{Z}$. For the rest of this case let ${\rm v}_p(p^4+f^2d)=4$. It follows that ${\rm v}_p(c)=2$, and hence $a^{\prime}=\min\{2(k-2),{\rm v}_p(2c)\}=2$ and $b^{\prime}=4(k-3)>0$.
CASE 2: $d\equiv 1\mod 4$. Set $r=\frac{p^2-1}{2}$. Observe that $\mathcal{N}_{K/\mathbb{Q}}(r+\tau)=\frac{p^4-f^2d}{4}$, $k\geq 4$, $a=2$, $b=2(k-2)>0$, $r<p^k$, and $t=1$ satisfies the congruence. Consequently, $2c+\varepsilon=2{\rm rem}(\frac{p^2-1}{2}-\frac{p^4-f^2d}{4p^2},p^{2(k-2)})+1=\frac{p^4+f^2d+4\ell p^{2(k-1)}}{2p^2}$ for some $\ell\in\mathbb{Z}$. For the rest of this case let ${\rm v}_p(p^4+f^2d)=4$. We infer that $a^{\prime}=\min\{2(k-2),{\rm v}_p(2c+\varepsilon)\}=2$. Moreover, $b^{\prime}=4(k-3)>0$. This proves the claim.
Note that if $g\in\mathbb{N}$ with ${\rm v}_p(g)={\rm v}_p(f)$, then there is a monoid isomorphism $\alpha:\mathcal{I}^*_p(\mathcal{O}_f)\rightarrow\mathcal{I}^*_p(\mathcal{O}_g)$ such that $\alpha(p\mathcal{O}_f)=p\mathcal{O}_g$ by Proposition \[proposition 3.3\].3. By the claim it remains to show that if $(p\not=3$ or $d\not\equiv 2\mod 3$ or ${\rm v}_p(f)>2)$, then there is some odd $g\in\mathbb{N}$ such that ${\rm v}_p(g)={\rm v}_p(f)$ and ${\rm v}_p(p^4+g^2d)=4$.
Let $(p\not=3$ or $d\not\equiv 2\mod 3$ or ${\rm v}_p(f)>2)$. Furthermore, let ${\rm v}_p(p^4+f^2d)>4$. This implies that ${\rm v}_p(f)=2$ and $p\nmid d$. Without restriction we can assume that ${\rm v}_p(p^4+(p^2)^2d)>4$. We have ${\rm v}_p(1+d)>0$, and hence $p\not=3$. Set $g=(p-2)p^2$. Then ${\rm v}_p(g)={\rm v}_p(f)$. Assume that ${\rm v}_p(p^4+g^2d)>4$. Then $p^5\mid p^4+(p-2)^2p^4d-p^4(1+d)$, and thus $p\mid (p-2)^2-1=p^2-4p+3$. It follows that $p=3$, a contradiction.
2\. Let $|{\rm Pic}(\mathcal{O}_f)|\leq 2$ and let $p\not=3$ or $d\not\equiv 2\mod 3$ or ${\rm v}_p(f)>2$. By 1. there are some $I,J,L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $I^2=p^2J$ and $J^2=p^2L$. We infer that $I^2$ is principal, and hence $J$ and $L$ are principal. Consequently, there are some $u,v\in\mathcal{A}(\mathcal{O}_f)$ such that $J=u\mathcal{O}_f$, $L=v\mathcal{O}_f$ and $u^2=p^2v$. Note that $u^2$ is primary. Since $p\in\mathcal{A}(\mathcal{O}_f)$, we have $2,3\in\mathsf{L}(u^2)$. Therefore, $1\in\Delta(\mathcal{O}_f)$.
\[proposition 4.4\] Let $p$ be a prime divisor of $f$ such that ${\rm v}_p(f)\geq 2$. Then there are $I,J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $\mathsf{L}(IJ)=\{2,4\}$ whence $2\in\Delta(\mathcal{I}^*_p(\mathcal{O}_f))$ and $4\in {\rm Ca}(\mathcal{I}^*_p(\mathcal{O}_f))$.
CASE 1: $p\not=2$ or ${\rm v}_p(f)>2$ or $d\not\equiv 1\mod 8$. By Theorem \[theorem 3.6\] there is some $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $\mathcal{N}(I)=p^4$. Set $J=\overline{I}$. We infer that $IJ=(p\mathcal{O}_f)^4$, and hence $\{2,4\}\subset\mathsf{L}(IJ)\subset\{2,3,4\}$. Assume that $3\in\mathsf{L}(IJ)$. Then there are some $A,B,C\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $IJ=ABC$ and $\mathcal{N}(A)\leq\mathcal{N}(B)\leq\mathcal{N}(C)$. Again by Theorem \[theorem 3.6\] we have $\mathcal{N}(L)\in\{p^2\}\cup\{p^n\mid n\in\mathbb{N}_{\geq 4}\}$ for all $L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$. This implies that $\mathcal{N}(A)=\mathcal{N}(B)=p^2$ and $\mathcal{N}(C)=p^4$. It follows by Lemma \[lemma 4.2\].2 that $ABC=p^2L$ for some $L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$. Consequently, $L=p^2\mathcal{O}_f$, a contradiction. We infer that $\mathsf{L}(IJ)=\{2,4\}$ whence $2\in\Delta(\mathcal{I}^*_2(\mathcal{O}_f))$ and $4\in {\rm Ca}(\mathcal{I}^*_2(\mathcal{O}_f))$.
CASE 2: $p=2$, ${\rm v}_p(f)=2$ and $d\equiv 1\mod 8$. Since $\mathcal{I}^*_2(\mathcal{O}_4)\cong\mathcal{I}^*_2(\mathcal{O}_f)$ by Proposition \[proposition 3.3\].3, we can assume without restriction that $f=4$. We set $$w=\begin{cases}6 &\textnormal{if }d\equiv 1\mod\textnormal{ } 16\\2 &\textnormal{if }d\equiv 9\mod\textnormal{ } 16\end{cases}\quad\textnormal{and}\quad
z=\begin{cases}18 &\textnormal{if }d\equiv 1\mod\textnormal{ } 32\\22 &\textnormal{if }d\equiv 9\mod\textnormal{ } 32\\2 &\textnormal{if }d\equiv 17\mod\textnormal{ } 32\\6 &\textnormal{if }d\equiv 25\mod\textnormal{ } 32\end{cases}.$$ In any case, we have ${\rm v}_2(\mathcal{N}_{K/\mathbb{Q}}(w+\tau))=5$ and ${\rm v}_2(\mathcal{N}_{K/\mathbb{Q}}(z+\tau))=6$. Set $I=32\mathbb{Z}+(w+\tau)\mathbb{Z}$ and $J=64\mathbb{Z}+(z+\tau)\mathbb{Z}$. Then $I,J\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_4))$ and Proposition \[proposition 3.2\].1 implies that $IJ=2^a(2^b\mathbb{Z}+(c+\tau)\mathbb{Z})$ with $a=\min\{5,6,{\rm v}_2(w+z)\}$, $b=5+6-2a$ and $c\in\mathbb{N}_0$ such that $c<2^b$. Observe that ${\rm v}_2(w+z)=3$, and thus $a=3$ and $b=5$. Set $L=32\mathbb{Z}+(c+\tau)\mathbb{Z}$. Then $L\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_4))$ and $IJ=(2\mathcal{O}_4)^3L$. We infer that $\{2,4\}\subset\mathsf{L}(IJ)\subset\{2,3,4\}$, by Proposition \[proposition 4.1\].2.
Assume that $3\in\mathsf{L}(IJ)$. Then there are some $A,B,C\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_4))$ such that $IJ=ABC$ and $\mathcal{N}(A)\leq\mathcal{N}(B)\leq\mathcal{N}(C)$. It follows by Theorem \[theorem 3.6\] that $\mathcal{N}(U)\in\{4\}\cup\{2^n\mid n\geq 5\}$ for all $U\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_4))$. Since $\mathcal{N}(A)\mathcal{N}(B)\mathcal{N}(C)=\mathcal{N}(I)\mathcal{N}(J)=2048$, we infer that $\mathcal{N}(A)=\mathcal{N}(B)=4$ and $\mathcal{N}(C)=128$. It follows by Lemma \[lemma 4.2\].2 that $ABC=4D$ for some $D\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_4))$. This implies that $D=2L$, a contradiction. Consequently, $\mathsf{L}(IJ)=\{2,4\}$, and thus $2\in\Delta(\mathcal{I}^*_2(\mathcal{O}_4))$ and $4=\mathsf{c}(IJ)\in {\rm Ca}(\mathcal{I}^*_2(\mathcal{O}_4))$.
\[proposition 4.5\] Suppose that one of the following conditions hold[:]{}
1. ${\rm v}_2(f)\geq 5$ or $({\rm v}_2(f)=4$ and $d\not\equiv 1\mod 4)$.
2. ${\rm v}_2(f)=3$ and $d\equiv 2\mod 4$.
3. ${\rm v}_2(f)=2$ and $d\equiv 1\mod 4$.
Then there are $I,J\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ with $\mathsf{L}(IJ)=\{2,3\}$ whence $1\in\Delta(\mathcal{I}^*_2(\mathcal{O}_f))$ and $3\in {\rm Ca}(\mathcal{I}^*_2(\mathcal{O}_f))$. If $|{\rm Pic}(\mathcal{O}_f)|\leq 2$, then there is a nonzero primary $a\in\mathcal{O}_f$ with $2,3\in\mathsf{L}(a)$ whence $1\in\Delta(\mathcal{O}_f)$.
CASE 1: ${\rm v}_2(f)\geq 5$ or (${\rm v}_2(f)=4$ and $d\not\equiv 1\mod 4$). We show that there are some $A,B,I,J,L\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ such that $A^2=32I$, $B^2=16J$ and $IJ=4L$. Set $k={\rm v}_2(\mathcal{N}_{K/\mathbb{Q}}(16+\tau))$ and $A=2^k\mathbb{Z}+(16+\tau)\mathbb{Z}$. Then $k\geq 8$, $A\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ and $A^2=32(2^{2k-10}\mathbb{Z}+(c+\tau)\mathbb{Z})$ with $(5,2k-10,c)\in\mathcal{M}_{f,2}$ and ${\rm v}_2(c)\geq 3$. Set $I=2^{2k-10}\mathbb{Z}+(c+\tau)\mathbb{Z}$. Then $I\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$. Set $B=64\mathbb{Z}+(8+\tau)\mathbb{Z}$. Then $B\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ and $B^2=16(16\mathbb{Z}+(4+\tau)\mathbb{Z})$. Set $J=16\mathbb{Z}+(4+\tau)\mathbb{Z}$. Then $B^2=16J$, $J\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ and $IJ=4L$ with $L\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$.
CASE 2: ${\rm v}_2(f)=3$ and $d\equiv 2\mod 4$. We show that $AB=2I$, $AC=2I^{\prime}$, $BC=8I^{\prime\prime}$, $B^2=16J$, $IJ=4L$, $I^{\prime}J=4L^{\prime}$, $I^{\prime\prime}J=4L^{\prime\prime}$ for some $A,B,C,I,I^{\prime},$ $I^{\prime\prime},J,L,L^{\prime},L^{\prime\prime}\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$. By Proposition \[proposition 3.3\].3, we can assume without restriction that $f=8$. Set $A=4\mathbb{Z}+(2+\tau)\mathbb{Z}$, $B=64\mathbb{Z}+(8+\tau)\mathbb{Z}$ and $C=128\mathbb{Z}+\tau\mathbb{Z}$. Then $A,B,C\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$, $AB=2(64\mathbb{Z}+(40+\tau)\mathbb{Z})$, $AC=2(128\mathbb{Z}+(64+\tau)\mathbb{Z})$, $B^2=16(16\mathbb{Z}+(12+\tau)\mathbb{Z})$ and $BC=8(128\mathbb{Z}+(c+\tau)\mathbb{Z})$ with $(3,7,c)\in\mathcal{M}_{f,2}$ and ${\rm v}_2(c)=4$. Furthermore, $(64\mathbb{Z}+(40+\tau)\mathbb{Z})(16\mathbb{Z}+(12+\tau)\mathbb{Z})=4(64\mathbb{Z}+(56+\tau)\mathbb{Z})$, $(128\mathbb{Z}+(64+\tau)\mathbb{Z})(16\mathbb{Z}+(12+\tau)\mathbb{Z})=4(128\mathbb{Z}+(r+\tau)\mathbb{Z})$ with $(2,7,r)\in\mathcal{M}_{f,2}$ and $(128\mathbb{Z}+(c+\tau)\mathbb{Z})(16\mathbb{Z}+(12+\tau)\mathbb{Z})=4(128\mathbb{Z}+(s+\tau)\mathbb{Z})$ with $(2,7,s)\in\mathcal{M}_{f,2}$. Set $J=16\mathbb{Z}+(12+\tau)\mathbb{Z}$. In particular, if $I\in\{64\mathbb{Z}+(40+\tau)\mathbb{Z},128\mathbb{Z}+(64+\tau)\mathbb{Z},128\mathbb{Z}+(c+\tau)\mathbb{Z}\}$, then $I,J\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ and $IJ=4L$ for some $L\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$.
CASE 3: ${\rm v}_2(f)=2$ and $d\equiv 1\mod 4$. We show that $A^2=4I$ and $I^2=4L$ for some $A,I,L\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$. By Proposition \[proposition 3.3\].3, we can assume without restriction that $f=4$. First let $d\equiv 1\mod 8$. If $d\equiv 1\mod 16$, then set $A=32\mathbb{Z}+(6+\tau)\mathbb{Z}$ and if $d\equiv 9\mod 16$, then set $A=32\mathbb{Z}+(2+\tau)\mathbb{Z}$. In any case, we have $A\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ and $A^2=4(64\mathbb{Z}+(c+\tau)\mathbb{Z})$ with $(2,6,c)\in\mathcal{M}_{f,2}$ and ${\rm v}_2(c)=1$. Set $I=64\mathbb{Z}+(c+\tau)\mathbb{Z}$. Then $I\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$, $A^2=4I$ and $I^2=4(256\mathbb{Z}+(r+\tau)\mathbb{Z})$ with $(2,8,r)\in\mathcal{M}_{f,2}$.
Now let $d\equiv 5\mod 8$. Set $A=16\mathbb{Z}+(2+\tau)\mathbb{Z}$. Then $A\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ and $A^2=4(16\mathbb{Z}+(c+\tau)\mathbb{Z})$ with $(2,4,c)\in\mathcal{M}_{f,2}$ and ${\rm v}_2(c)=1$. Set $I=16\mathbb{Z}+(c+\tau)\mathbb{Z}$. Then $A^2=4I$ and $I^2=4(16\mathbb{Z}+(z+\tau)\mathbb{Z})$ with $(2,4,z)\in\mathcal{M}_{f,2}$.
Using the case analysis above we can find $I,J,L\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ such that $IJ=4L$. In particular, $\mathsf{L}(IJ)=\{2,3\}$, $1\in\Delta(\mathcal{I}^*_p(\mathcal{O}_f))$ and $3=\mathsf{c}(IJ)\in {\rm Ca}(\mathcal{I}^*_p(\mathcal{O}_f))$. Now let $|{\rm Pic}(\mathcal{O}_f)|\leq 2$. Observe that if $A,B,C\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$, then $A^2$ is principal and $\{AB,AC,BC\}$ contains a principal ideal of $\mathcal{O}_f$. In any case we can choose $I,J,L$ to be principal. There are some $u,v,w\in\mathcal{A}(\mathcal{O}_f)$ such that $I=u\mathcal{O}_f$, $J=v\mathcal{O}_f$, $L=w\mathcal{O}_f$ and $uv=4w$. Note that $uv$ is primary. Since $2\in\mathcal{A}(\mathcal{O}_f)$, we have $2,3\in\mathsf{L}(uv)$, and thus $1\in\Delta(\mathcal{O}_f)$.
\[proposition 4.6\] Let $p$ be a prime divisor of $f$. Then the following statements are equivalent[:]{}
1. $\mathcal{I}^*_p(\mathcal{O}_f)$ is half-factorial.
2. $\mathcal{I}_p(\mathcal{O}_f)$ is half-factorial.
3. $\mathsf{c}(\mathcal{I}^*_p(\mathcal{O}_f))=2$.
4. $\mathsf{c}(\mathcal{I}_p(\mathcal{O}_f))=2$.
5. ${\rm v}_p(f)=1$ and $p$ is inert.
\(a) $\Rightarrow$ (e) If ${\rm v}_p(f)>1$ or $p$ is not inert, then there is some $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $\mathcal{N}(I)>p^2$ by Theorem \[theorem 3.6\].4. Set $k={\rm v}_p(\mathcal{N}(I))$. Then $k\geq 3$ and $I\overline{I}=(p\mathcal{O}_f)^k$ by Proposition \[proposition 3.2\].5. Since $\overline{I}\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$, we have $2,k\in\mathsf{L}(I\overline{I})$.
\(e) $\Rightarrow$ (b) Observe that $\mathcal{N}(A)\in\{p,p^2\}$ for each $A\in\mathcal{A}(\mathcal{I}_p(\mathcal{O}_f))$, and thus $\mathcal{A}(\mathcal{I}_p(\mathcal{O}_f))=\{P_{f,p}\}\cup\{A\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))\mid\mathcal{N}(A)=p^2\}$. Let $I\in\mathcal{I}_p(\mathcal{O}_f)\setminus\{\mathcal{O}_f\}$. There are some $k\in\mathbb{N}_0$ and $J\in\mathcal{A}(\mathcal{I}_p(\mathcal{O}_f))$ such that $I=p^kJ$. Let $z\in\mathsf{Z}(I)$. Then $z=(\prod_{i=1}^n I_i)\cdot P_{f,p}^{\ell}$ with $\ell,n\in\mathbb{N}_0$ and $I_i\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ for each $i\in [1,n]$. Note that $|z|=n+\ell$. It is sufficient to show that $n+\ell=k+1$.
CASE 1: $I$ is invertible. Then $J$ is invertible and $\ell=0$. It follows that $p^{2n}=\mathcal{N}(\prod_{i=1}^n I_i)=\mathcal{N}(I)=\mathcal{N}(p^kJ)=p^{2k+2}$ by Proposition \[proposition 3.2\].3, and thus $n+\ell=n=k+1$.
CASE 2: $I$ is not invertible. Then $J=P_{f,p}$ and $\ell>0$. It follows from Lemma \[lemma 4.2\] that $P_{f,p}^{\ell}=p^{\ell-1}P_{f,p}$. Consequently, $$p^{2(n+\ell)-1}=\mathcal{N}(\prod_{i=1}^n I_i)\mathcal{N}(p^{\ell-1}P_{f,p})=\mathcal{N}(I)=\mathcal{N}(p^kP_{f,p})=p^{2k+1}$$ by Proposition \[proposition 3.2\].3, and hence $n+\ell=k+1$.
\(b) $\Rightarrow$ (d) Since $\mathcal{I}^*_p(\mathcal{O}_f)$ is a cancellative divisor-closed submonoid of $\mathcal{I}_p(\mathcal{O}_f)$ and not factorial, we infer by Proposition \[proposition 4.1\].4 that $$2\leq\mathsf{c}(\mathcal{I}^*_p(\mathcal{O}_f))\leq\mathsf{c}(\mathcal{I}_p(\mathcal{O}_f))\leq 2.$$
\(d) $\Rightarrow$ (c) Note that $\mathcal{I}^*_p(\mathcal{O}_f)$ is a divisor-closed submonoid of $\mathcal{I}_p(\mathcal{O}_f)$, and thus $\mathsf{c}(\mathcal{I}^*_p(\mathcal{O}_f))\leq\mathsf{c}(\mathcal{I}_p(\mathcal{O}_f))=2$. Since $\mathcal{I}^*_p(\mathcal{O}_f)$ is not factorial, we infer that $\mathsf{c}(\mathcal{I}^*_p(\mathcal{O}_f))=2$.
\(c) $\Rightarrow$ (a) Since $\mathcal{I}^*_p(\mathcal{O}_f)$ is cancellative and not factorial, it follows that $2+\sup\Delta(\mathcal{I}^*_p(\mathcal{O}_f))\leq\mathsf{c}(\mathcal{I}^*_p(\mathcal{O}_f))=2$, and thus $\sup\Delta(\mathcal{I}^*_p(\mathcal{O}_f))=0$. Consequently, $\Delta(\mathcal{I}^*_p(\mathcal{O}_f))=\emptyset$, and hence $\mathcal{I}^*_p(\mathcal{O}_f)$ is half-factorial.
\[lemma 4.7\] Let $p$ be a prime divisor of $f$, $|{\rm Pic}(\mathcal{O}_f)|\leq 2$, $I,J,L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$.
1. If $J$ is principal and $IJ=p^2L$, then $1\in\Delta(\mathcal{O}_f)$.
2. If $I$ and $J$ are not principal and $IJ=pL$, then $1\in\Delta(\mathcal{O}_f)$.
Note that if $|{\rm Pic}(\mathcal{O}_f)|>1$, then it follows from [@Ge-HK06a Corollary 2.11.16] that there is some invertible prime ideal $P$ of $\mathcal{O}_f$ that is not principal. Observe that $p\in\mathcal{A}(\mathcal{O}_f)$. Also note that if $I$ is not principal, then $PI$ is principal, and hence $PI$ is generated by an atom of $\mathcal{O}_f$, since $PI$ has no nontrivial factorizations in $\mathcal{I}^*(\mathcal{O}_f)$.
1\. Let $J$ be principal and $IJ=p^2L$. There is some $v\in\mathcal{A}(\mathcal{O}_f)$ such that $J=v\mathcal{O}_f$.
CASE 1: $I$ is principal. Then $L$ is principal, and hence there are some $u,w\in\mathcal{A}(\mathcal{O}_f)$ such that $I=u\mathcal{O}_f$, $L=w\mathcal{O}_f$ and $uv=p^2w$. We infer that $2,3\in\mathsf{L}(uv)$, and thus $1\in\Delta(\mathcal{O}_f)$.
CASE 2: $I$ is not principal. Then $L$ is not principal and $|{\rm Pic}(\mathcal{O}_f)|>1$, and thus there are some $u,w\in\mathcal{A}(\mathcal{O}_f)$ such that $PI=u\mathcal{O}_f$, $PL=w\mathcal{O}_f$ and $uv=p^2w$. It follows that $2,3\in\mathsf{L}(uv)$, and thus $1\in\Delta(\mathcal{O}_f)$.
2\. Let $I$ and $J$ not be principal and $IJ=pL$. Then $L$ is principal and $|{\rm Pic}(\mathcal{O}_f)|>1$, and hence there are some $u,v,w,y\in\mathcal{A}(\mathcal{O}_f)$ such that $PI=u\mathcal{O}_f$, $PJ=v\mathcal{O}_f$, $P^2=w\mathcal{O}_f$, $L=y\mathcal{O}_f$ and $uv=pwy$. Therefore, $2,3\in\mathsf{L}(uv)$, and hence $1\in\Delta(\mathcal{O}_f)$.
\[proposition 4.8\] Let $p$ be a prime divisor of $f$.
1. If ${\rm v}_p(f)\geq 2$ or $p$ is not inert, then there are $I,J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $\mathsf{L}(IJ)=\{2,3\}$ whence $1\in\Delta(\mathcal{I}^*_p(\mathcal{O}_f))$ and $3\in {\rm Ca}(\mathcal{I}^*_p(\mathcal{O}_f))$.
2. Suppose that $\mathcal{O}_f$ is not half-factorial and that one of the following conditions holds[:]{}
1. $|{\rm Pic}(\mathcal{O}_f)|\geq 3$ or ${\rm v}_p(f)\geq 2$ or $p$ does split.
2. $p$ is inert and there is some $C\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ that is not principal.
3. $p$ is ramified and there is some principal $C\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $\mathcal{N}(C)=p^3$.
4. $f$ is a squarefree product of inert primes.
Then $1\in\Delta(\mathcal{O}_f)$.
We prove 1. and 2. simultaneously. Set $G={\rm Pic}(\mathcal{O}_f)$. Let $\mathcal{B}(G)$ be the monoid of zero-sum sequences of $G$. It follows by [@Ge-HK06a Theorem 6.7.1.2] that if $|G|\geq 3$, then $1\in\Delta(\mathcal{B}(G))$. We infer by [@Ge-HK06a Proposition 3.4.7 and Theorems 3.4.10.3 and 3.7.1.1] that there exists an atomic monoid $\mathcal{B}(\mathcal{O}_f)$ such that $\Delta(\mathcal{B}(\mathcal{O}_f))=\Delta(\mathcal{O}_f)$ and $\mathcal{B}(G)$ is a divisor-closed submonoid of $\mathcal{B}(\mathcal{O}_f)$. In particular, if $|G|\geq 3$, then $1\in\Delta(\mathcal{O}_f)$. Thus, for the second assertion we only need to consider the case $|G|\leq 2$. By Propositions \[proposition 4.3\] and \[proposition 4.5\] we can restrict to the following cases.
CASE 1: $p=2$ and $(({\rm v}_2(f)\in\{3,4\}$ and $d\equiv 1\mod 4)$ or $({\rm v}_2(f)\in\{2,3\}$ and $d\equiv 3\mod 4))$. If $({\rm v}_2(f)=4$ and $d\equiv 1\mod 4)$ or $({\rm v}_2(f)=3$ and $d\equiv 3\mod 4)$, then set $I=16\mathbb{Z}+(4+\tau)\mathbb{Z}$. If ${\rm v}_2(f)=3$ and $d\equiv 1\mod 4$, then set $I=16\mathbb{Z}+\tau\mathbb{Z}$. Finally, if ${\rm v}_2(f)=2$ and $d\equiv 3\mod 4$, then there is some $I\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ such that $\mathcal{N}(I)=32$ by Theorem \[theorem 3.6\]. In any case, it follows that $I\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$.
It is a consequence of Proposition \[proposition 3.2\].1 and Theorem \[theorem 3.6\] that there are some $A,J\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ and $\ell\in\mathbb{N}$ such that $A^2=\ell J$ with values according to the following table. Let $k\in\{1,3,5,7\}$ be such that $d\equiv k\mod 8$. Note that $I=2^a\mathbb{Z}+(r+\tau)\mathbb{Z}$ and $J=2^b\mathbb{Z}+(s+\tau)\mathbb{Z}$ with $(0,a,r),(0,b,s)\in\mathcal{M}_{f,2}$.
${\rm v}_2(f)$ $k$ $\mathcal{N}(A)$ $\ell$ $\mathcal{N}(J)$ ${\rm v}_2(r)$ ${\rm v}_2(s)$
---------------- ------------ ------------------ -------- ------------------ ---------------- ----------------
$4$ $1$ $512$ $16$ $1024$ $2$ $3$
$4$ $5$ $256$ $16$ $256$ $2$ $3$
$3$ $1$ $128$ $8$ $256$ $\infty$ $2$
$3$ $5$ $64$ $8$ $64$ $\infty$ $2$
$3$ $3$ or $7$ $128$ $16$ $64$ $2$ $\geq 4$
$2$ $3$ or $7$ $32$ $8$ $16$ $2$ $\geq 3$
Since ${\rm v}_2(r+s)=2$ in any case, we infer that $IJ=4L$ for some $L\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$. Now let $|G|\leq 2$. We have $J$ is principal, and hence $1\in\Delta(\mathcal{O}_f)$ by Lemma \[lemma 4.7\].1.
CASE 2: $p=2$, ${\rm v}_2(f)=2$ and $d\equiv 2\mod 4$. Set $A=32\mathbb{Z}+\tau\mathbb{Z}$ and $B=32\mathbb{Z}+(8+\tau)\mathbb{Z}$. Then $A,B\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ and $AB=8I$ for some $I\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ with $I=16\mathbb{Z}+(r+\tau)\mathbb{Z}$, $(0,4,r)\in\mathcal{M}_{f,2}$, and ${\rm v}_2(r)=2$. Therefore, we have $AI=4J$ and $BI=4L$ for some $J,L\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$. Now let $|G|\leq 2$. Since $\{A,B,I\}$ contains a principal ideal of $\mathcal{O}_f$, we infer by Lemma \[lemma 4.7\].1 that $1\in\Delta(\mathcal{O}_f)$.
CASE 3: $p=3$, ${\rm v}_3(f)=2$ and $d\equiv 2\mod 3$. First let $d\not\equiv 1\mod 4$. Set $I=81\mathbb{Z}+\tau\mathbb{Z}$ and $J=81\mathbb{Z}+(9+\tau)\mathbb{Z}$. Then $I,J\in\mathcal{A}(\mathcal{I}^*_3(\mathcal{O}_f))$ and $IJ=9L$ for some $L\in\mathcal{A}(\mathcal{I}^*_3(\mathcal{O}_f))$ with $L=81\mathbb{Z}+(r+\tau)\mathbb{Z}$, $(0,4,r)\in\mathcal{M}_{f,3}$, and ${\rm v}_3(r)=2$. It follows that $IL=9A$ for some $A\in\mathcal{A}(\mathcal{I}^*_3(\mathcal{O}_f))$.
Now let $d\equiv 1\mod 4$. By Proposition \[proposition 3.3\].3 we can assume without restriction that $f$ is odd. Set $I=81\mathbb{Z}+(4+\tau)\mathbb{Z}$ and $J=81\mathbb{Z}+(13+\tau)\mathbb{Z}$. Then $I,J\in\mathcal{A}(\mathcal{I}^*_3(\mathcal{O}_f))$ and $IJ=9L$ for some $L\in\mathcal{A}(\mathcal{I}^*_3(\mathcal{O}_f))$. There is some $(0,4,r)\in\mathcal{M}_{f,3}$ such that $L=81\mathbb{Z}+(r+\tau)\mathbb{Z}$. Since ${\rm v}_3(2r+1)\geq 2$, we have $IL=9A$ for some $A\in\mathcal{A}(\mathcal{I}^*_3(\mathcal{O}_f))$ or $JL=9A$ for some $A\in\mathcal{A}(\mathcal{I}^*_3(\mathcal{O}_f))$.
In any case if $|G|\leq 2$, then $\{I,J,L\}$ contains a principal ideal of $\mathcal{O}_f$, and hence $1\in\Delta(\mathcal{O}_f)$ by Lemma \[lemma 4.7\].1.
CASE 4: ${\rm v}_p(f)=1$ and $p$ splits. By Theorem \[theorem 3.6\] there is some $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $\mathcal{N}(I)=p^3$. There is some $(0,3,r)\in\mathcal{M}_{f,p}$ such that $I=p^3\mathbb{Z}+(r+\tau)\mathbb{Z}$. Observe that ${\rm v}_p(2r+\varepsilon)=1$. We infer that $I^2=pJ$ for some $J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ and $I\overline{I}=p^2L$ with $\overline{I}\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ and $L=p\mathcal{O}_f\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$. Now let $|G|\leq 2$. We infer by Lemma \[lemma 4.7\] that $1\in\Delta(\mathcal{O}_f)$.
CASE 5: ${\rm v}_p(f)=1$ and $p$ is ramified. By Theorem \[theorem 3.6\] there is some $C\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $\mathcal{N}(C)=p^3$. Note that $C\overline{C}=p^3\mathcal{O}_f$ and $\overline{C}\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$. Now let $C$ be principal. It follows by Lemma \[lemma 4.7\].1 that $1\in\Delta(\mathcal{O}_f)$.
Cases 1-5 show that there are some $I,J,L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $IJ=p^2L$. In particular, $\mathsf{L}(IJ)=\{2,3\}$, $1\in\Delta(\mathcal{I}^*_p(\mathcal{O}_f))$ and $3=\mathsf{c}(IJ)\in {\rm Ca}(\mathcal{I}^*_p(\mathcal{O}_f))$. This proves 1. For the rest of this proof let $\mathcal{O}_f$ be not half-factorial and $|G|\leq 2$.
CASE 6: ${\rm v}_p(f)=1$, $p$ is inert and there is some $C\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ that is not principal. We have $C^2=pL$ for some $L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$, and thus $1\in\Delta(\mathcal{O}_f)$ by Lemma \[lemma 4.7\].2.
CASE 7: $f$ is a squarefree product of inert primes. Then $\mathcal{I}^*_p(\mathcal{O}_f)$ is half-factorial by Proposition \[proposition 4.6\]. If $G$ is trivial, then $\mathcal{O}_f$ is half-factorial, a contradiction. Note that $\mathcal{O}_f$ is seminormal by [@Do-Fo87 Corollary 4.5]. It follows from [@Ge-Ka-Re15a Theorem 6.2.2.(a)] that $1\in\Delta(\mathcal{O}_f)$.
\[lemma 4.9\] Let $p$ be a prime divisor of $f$, $k\in\mathbb{N}_{\geq 2}$, and $N=\sup\{{\rm v}_p(\mathcal{N}(A))\mid A\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))\}$. If $\ell\in\mathbb{N}$ and $A\in\mathcal{I}_p(\mathcal{O}_f))$ is both a product of $k$ atoms and a product of $\ell$ atoms, then $\ell\leq\frac{kN}{2}$.
Let $\ell\in\mathbb{N}$ and suppose that a product of $k$ atoms can be written as a product of $\ell$ atoms and set $P=P_{f,p}$. There are some $a,b\in\mathbb{N}_0$, $I_i\in\mathcal{A}(\mathcal{I}_p(\mathcal{O}_f))\setminus\{P\}$ for each $[1,b]$ and $J_j\in\mathcal{A}(\mathcal{I}_p(\mathcal{O}_f))$ for each $j\in [1,k]$ such that $\ell=a+b$ and $\prod_{j=1}^k J_j=P^a\prod_{i=1}^b I_i$. Note that $p^2\mid\mathcal{N}(I_i)$ for each $i\in [1,b]$.
CASE 1: $a=0$. Then $b=\ell$. It follows by induction from Proposition \[proposition 3.2\].4 that there are $J^{\prime}_j\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ for each $j\in [1,k]$ such that $\mathcal{N}(\prod_{j=1}^k J_j)\mid\mathcal{N}(\prod_{j=1}^k J^{\prime}_j)$. Set $M={\rm lcm}\{\mathcal{N}(J^{\prime}_j)\mid j\in [1,k]\}$. Then $p^{2\ell}\mid\prod_{i=1}^{\ell}\mathcal{N}(I_i)\mid\mathcal{N}(\prod_{i=1}^{\ell} I_i)=\mathcal{N}(\prod_{j=1}^k J_j)\mid\mathcal{N}(\prod_{j=1}^k J^{\prime}_j)=\prod_{j=1}^k\mathcal{N}(J^{\prime}_j)\mid M^k$. This implies that $2\ell\leq k{\rm v}_p(M)\leq kN$, and thus $\ell\leq\frac{kN}{2}$.
CASE 2: $a>0$. By Lemma \[lemma 4.2\] we have $P^a=p^{a-1}P$, and thus $\mathcal{N}(P^a)=p^{2a-1}$. Note that $\prod_{j=1}^k J_j$ is not invertible, and hence one member of the product, say $J_1$, is not invertible. Observe that ${\rm v}_p(\mathcal{N}(J_1))\leq N-1$ by Proposition \[proposition 3.2\].4. We infer by induction from Proposition \[proposition 3.2\].4 that there are $J^{\prime}_j\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ for each $j\in [2,k]$ such that $\mathcal{N}(\prod_{j=1}^k J_j)\mid\mathcal{N}(J_1\prod_{j=2}^k J^{\prime}_j)$. Set $M={\rm lcm}\{\mathcal{N}(J^{\prime}_j)\mid j\in [2,k]\}$. Then $p^{2\ell-1}\mid\mathcal{N}(P^a)\prod_{i=1}^b\mathcal{N}(I_i)\mid\mathcal{N}(P^a\prod_{i=1}^b I_i)=\mathcal{N}(\prod_{j=1}^k J_j)\mid\mathcal{N}(J_1\prod_{j=2}^k J^{\prime}_j)=\mathcal{N}(J_1)\prod_{j=2}^k\mathcal{N}(J^{\prime}_j)\mid\mathcal{N}(J_1)M^{k-1}$. This implies that $2\ell-1\leq {\rm v}_p(\mathcal{N}(J_1))+(k-1){\rm v}_p(M)\leq kN-1$, and hence $\ell\leq\frac{kN}{2}$.
\[lemma 4.10\] Let $p$ be a prime divisor of $f$. For every $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$, we set ${\rm v}_I={\rm v}_p(\mathcal{N}(I))$, and let $\mathcal{B}=\{{\rm v}_A\mid A\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))\}$.
1. For all $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$, we have $\mathsf{c}(I\cdot\overline{I},(p\mathcal{O}_f)^{{\rm v}_I})\leq 2+\sup\Delta(\mathcal{B})$.
2. Let $p=2$, $d\equiv 1\mod 8$, and ${\rm v}_p(f)\geq 4$. Then $\mathsf{c}(I\cdot\overline{I},(p\mathcal{O}_f)^{{\rm v}_I})\leq 4$ for all $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$.
1\. It is sufficient to show by induction that for all $n\in\mathbb{N}_{\geq 2}$ and $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ with ${\rm v}_I=n$, it follows that $\mathsf{c}(I\cdot\overline{I},(p\mathcal{O}_f)^n)\leq 2+\sup\Delta(\mathcal{B})$. Let $n\in\mathbb{N}_{\geq 2}$ and $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ be such that ${\rm v}_I=n$. If $n=2$, then $\mathsf{c}(I\cdot\overline{I},(p\mathcal{O}_f)^2)\leq\mathsf{d}(I\cdot\overline{I},(p\mathcal{O}_f)^2)\leq 2\leq 2+\sup\Delta(\mathcal{B})$. Now let $n>2$. Note that $2={\rm v}_{p\mathcal{O}_f}\in\mathcal{B}$, and hence there is some $k\in\mathcal{B}$ such that $2\leq k<n$ and $\mathcal{B}\cap [k,n]=\{k,n\}$. Observe that $n-k\in\Delta(\mathcal{B})$. Furthermore, there is some $J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $k={\rm v}_J$. Note that $J\overline{J}=(p\mathcal{O}_f)^k$, and thus $I\overline{I}=(p\mathcal{O}_f)^{n-k}J\overline{J}$. By the induction hypothesis, we infer that $c((p\mathcal{O}_f)^{n-k}\cdot J\cdot\overline{J},(p\mathcal{O}_f)^n)\leq c(J\cdot\overline{J},(p\mathcal{O}_f)^k)\leq 2+\sup\Delta(\mathcal{B})$. Since $\mathsf{d}(I\cdot\overline{I},(p\mathcal{O}_f)^{n-k}\cdot J\cdot\overline{J})\leq 2+(n-k)\leq 2+\sup\Delta(\mathcal{B})$, it follows that $\mathsf{c}(I\cdot\overline{I},(p\mathcal{O}_f)^n)\leq 2+\sup\Delta(\mathcal{B})$.
2\. By Proposition \[proposition 3.3\].3 we can assume without restriction that $f=2^{{\rm v}_2(f)}$. We show by induction that for all $n\in\mathbb{N}_{\geq 2}$ and $I\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ with ${\rm v}_I=n$, we have $\mathsf{c}(I\cdot\overline{I},(2\mathcal{O}_f)^n)\leq 4$. Let $n\in\mathbb{N}_{\geq 2}$ and $I\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ be such that ${\rm v}_I=n$. If $n=2$, then $\mathsf{c}(I\cdot\overline{I},(2\mathcal{O}_f)^2)\leq\mathsf{d}(I\cdot\overline{I},(2\mathcal{O}_f)^2)\leq 2\leq 2+\sup\Delta(\mathcal{B})$. Next let $n>2$. Observe that $2={\rm v}_{2\mathcal{O}_f}\in\mathcal{B}$, and hence there is some $k\in\mathcal{B}$ such that $2\leq k<n$ and $\mathcal{B}\cap [k,n]=\{k,n\}$. There is some $J\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ such that $k={\rm v}_J$. Note that $J\overline{J}=(2\mathcal{O}_f)^k$, and hence $I\overline{I}=(2\mathcal{O}_f)^{n-k}J\overline{J}$. By the induction hypothesis, we have $c((2\mathcal{O}_f)^{n-k}\cdot J\cdot\overline{J},(2\mathcal{O}_f)^n)\leq c(J\cdot\overline{J},(2\mathcal{O}_f)^k)\leq 4$.
CASE 1: $n\not=2{\rm v}_2(f)+1$. It follows from Theorem \[theorem 3.6\] that $n-k\leq 2$. Since $\mathsf{d}(I\cdot\overline{I},(2\mathcal{O}_f)^{n-k}\cdot J\cdot\overline{J})\leq 4$, we infer that $\mathsf{c}(I\cdot\overline{I},(2\mathcal{O}_f)^n)\leq 4$.
CASE 2: $n=2{\rm v}_2(f)+1$. By Theorem \[theorem 3.6\] we have $n-k=3$. Set $A=16\mathbb{Z}+(4+\tau)\mathbb{Z}$, $B=2^{n-3}\mathbb{Z}+(2^{n-5}+\tau)\mathbb{Z}$, and $C=2^{n-3}\mathbb{Z}+(2^{n-4}+\tau)\mathbb{Z}$. Then $A,B,C\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ and $ABC=2^{n-5}A(16\mathbb{Z}+(12+\tau)\mathbb{Z})=(2\mathcal{O}_f)^{n-1}$. Observe that $\mathsf{d}(I\cdot\overline{I},(2\mathcal{O}_f)\cdot A\cdot B\cdot C)\leq 4$ and $\mathsf{d}((2\mathcal{O}_f)\cdot A\cdot B\cdot C,(2\mathcal{O}_f)^{n-k}\cdot J\cdot\overline{J}))\leq 4$. Therefore, $\mathsf{c}(I\cdot\overline{I},(2\mathcal{O}_f)^n)\leq 4$.
\[proposition 4.11\] Let $p$ be a prime divisor of $f$ and set $\mathcal{B}=\{{\rm v}_p(\mathcal{N}(\mathcal{A}))\mid A\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))\}$.
1. $\sup\Delta(\mathcal{I}_p(\mathcal{O}_f))\leq\sup\Delta(\mathcal{B})$ and $\mathsf{c}(\mathcal{I}_p(\mathcal{O}_f))\leq 2+\sup\Delta(\mathcal{B})$.
2. Let $p=2$, $d\equiv 1\mod 8$, and ${\rm v}_p(f)\geq 4$. Then $\sup\Delta(\mathcal{I}_2(\mathcal{O}_f))\leq 2$ and $\mathsf{c}(\mathcal{I}_2(\mathcal{O}_f))\leq 4$.
1\. First we consider the case that ${\rm v}_p(f)=1$ and $p$ is inert. It follows from Theorem \[theorem 3.6\] that $\sup\Delta(\mathcal{B})=0$. Proposition \[proposition 4.6\] implies that $\sup\Delta(\mathcal{I}_p(\mathcal{O}_f))=0$ and $\mathsf{c}(\mathcal{I}_p(\mathcal{O}_f))=2$. Now let ${\rm v}_p(f)\geq 2$ or $p$ not inert. Observe that $\sup\Delta(\mathcal{B})\geq 1$ by Theorem \[theorem 3.6\]. Let $I,J\in\mathcal{A}(\mathcal{I}_p(\mathcal{O}_f))$. There are some $n\in\mathbb{N}$ and $L\in\mathcal{A}(\mathcal{I}_p(\mathcal{O}_f))$ such that $IJ=p^nL$.
By Proposition \[proposition 4.1\], it remains to show that $\mathsf{c}(I\cdot J,(p\mathcal{O}_f)^n\cdot L)\leq 2+\sup\Delta(\mathcal{B})$ and if $\ell\in\mathbb{N}_{\geq 3}$ is such that $\mathsf{L}(IJ)\cap [2,\ell]=\{2,\ell\}$, then $\ell-2\leq\sup\Delta(\mathcal{B})$. Set $N=\sup\mathcal{B}$. Since a product of two atoms of $\mathcal{I}_p(\mathcal{O}_f)$ can be written as a product of $n+1$ atoms, Lemma \[lemma 4.9\] implies that $n+1 \le N$. If $n=1$, then $\mathsf{d}(I\cdot J,(p\mathcal{O}_f)\cdot L)\leq 2\leq 2+\sup\Delta(\mathcal{B})$ and there is no $\ell\in\mathbb{N}_{\geq 3}$ with $\mathsf{L}(IJ)\cap [2,\ell]=\{2,\ell\}$. Now let $n\geq 2$ and $\ell\in\mathbb{N}_{\geq 3}$ be such that $\mathsf{L}(IJ)\cap [2,\ell]=\{2,\ell\}$.
CASE 1: $n\in\mathcal{B}$. Then $A\overline{A}=(p\mathcal{O}_f)^n$ for some $A\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$. Therefore, $\mathsf{c}(A\cdot\overline{A}\cdot L,(p\mathcal{O}_f)^n\cdot L)\leq\mathsf{c}(A\cdot\overline{A},(p\mathcal{O}_f)^n)\leq 2+\sup\Delta(\mathcal{B})$ by Lemma \[lemma 4.10\].1. Moreover, $\mathsf{d}(I\cdot J,A\cdot\overline{A}\cdot L)\leq 3\leq 2+\sup\Delta(\mathcal{B})$, and thus $\mathsf{c}(I\cdot J,(p\mathcal{O}_f)^n\cdot L)\leq 2+\sup\Delta(\mathcal{B})$ and $\ell-2=1\leq\sup\Delta(\mathcal{B})$.
CASE 2: $n\not\in\mathcal{B}$. Note that $n\geq 3$. It follows by Theorem \[theorem 3.6\] that ${\rm v}_p(f)\geq 2$ and $\sup\Delta(\mathcal{B})\geq 2$.
CASE 2.1: $p\not=2$ or $d\not\equiv 1\mod 8$ or $n\not=2{\rm v}_p(f)$. Since $n\leq N$, it follows from Theorem \[theorem 3.6\] that $n-1=\mathcal{N}(A)$ for some $A\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$, and hence $A\overline{A}=(p\mathcal{O}_f)^{n-1}$. We infer that $\mathsf{c}((p\mathcal{O}_f)\cdot A\cdot\overline{A}\cdot L,(p\mathcal{O}_f)^n\cdot L)\leq\mathsf{c}(A\cdot\overline{A},(p\mathcal{O}_f)^{n-1})\leq 2+\sup\Delta(\mathcal{B})$ by Lemma \[lemma 4.10\].1. Moreover, we have $\mathsf{d}(I\cdot J,A\cdot\overline{A}\cdot (p\mathcal{O}_f)\cdot L)\leq 4\leq 2+\sup\Delta(\mathcal{B})$, and thus $\mathsf{c}(I\cdot J,(p\mathcal{O}_f)^n\cdot L)\leq 2+\sup\Delta(\mathcal{B})$ and $\ell-2\leq 2\leq\sup\Delta(\mathcal{B})$.
CASE 2.2: $p=2$, $d\equiv 1\mod 8$ and $n=2{\rm v}_p(f)$. We infer by Theorem \[theorem 3.6\] that $\sup\Delta(\mathcal{B})=3$. By Theorem \[theorem 3.6\] there is some $A\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ such that $n-2=\mathcal{N}(A)$, and thus $A\overline{A}=(2\mathcal{O}_f)^{n-2}$. This implies that $\mathsf{c}((2\mathcal{O}_f)^2\cdot A\cdot\overline{A}\cdot L,(2\mathcal{O}_f)^n\cdot L)\leq\mathsf{c}(A\cdot\overline{A},(2\mathcal{O}_f)^{n-2})\leq 2+\sup\Delta(\mathcal{B})$ by Lemma \[lemma 4.10\].1. Observe that $\mathsf{d}(I\cdot J,A\cdot\overline{A}\cdot (2\mathcal{O}_f)^2\cdot L)\leq 5=2+\sup\Delta(\mathcal{B})$, and hence $\mathsf{c}(I\cdot J,(2\mathcal{O}_f)^n\cdot L)\leq 2+\sup\Delta(\mathcal{B})$ and $\ell-2\leq 3=\sup\Delta(\mathcal{B})$.
2\. By Proposition \[proposition 3.3\].3 we can assume without restriction that $f=2^{{\rm v}_2(f)}$. Let $I,J\in\mathcal{A}(\mathcal{I}_2(\mathcal{O}_f))$. There are some $n\in\mathbb{N}$ and $L\in\mathcal{A}(\mathcal{I}_2(\mathcal{O}_f))$ such that $IJ=2^nL$. It follows from Lemma \[lemma 4.9\] that $n+1\leq\sup\mathcal{B}$. By Proposition \[proposition 4.1\], it is sufficient to show that $\mathsf{c}(I\cdot J,(2\mathcal{O}_f)^n\cdot L)\leq 4$ and if $\ell\in\mathbb{N}_{\geq 3}$ is such that $\mathsf{L}(IJ)\cap [2,\ell]=\{2,\ell\}$, then $\ell-2\leq 2$. The assertion is trivially true for $n=1$. Let $n\geq 2$ and let $\ell\in\mathbb{N}_{\geq 3}$ be such that $\mathsf{L}(IJ)\cap [2,\ell]=\{2,\ell\}$.
CASE 1: $n\in\mathcal{B}$. There is some $A\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ such that $A\overline{A}=(2\mathcal{O}_f)^n$. It follows by Lemma \[lemma 4.10\].2 that $\mathsf{c}(A\cdot\overline{A}\cdot L,(2\mathcal{O}_f)^n\cdot L)\leq\mathsf{c}(A\cdot\overline{A},(2\mathcal{O}_f)^n)\leq 4$. Furthermore, $\mathsf{d}(I\cdot J,A\cdot\overline{A}\cdot L)\leq 3$, and thus $\mathsf{c}(I\cdot J,(2\mathcal{O}_f)^n\cdot L)\leq 4$ and $\ell-2\leq 1$.
CASE 2: $n\not\in\mathcal{B}$ and $n\not=2{\rm v}_2(f)$. It follows by Theorem \[theorem 3.6\] that there is some $A\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ such that $A\overline{A}=(2\mathcal{O}_f)^{n-1}$. We infer by Lemma \[lemma 4.10\].2 that $\mathsf{c}((2\mathcal{O}_f)\cdot A\cdot\overline{A}\cdot L,(2\mathcal{O}_f)^n\cdot L)\leq \mathsf{c}(\cdot A\cdot\overline{A},(2\mathcal{O}_f)^{n-1})\leq 4$. Furthermore, $\mathsf{d}(I\cdot J,(2\mathcal{O}_f)\cdot A\cdot\overline{A}\cdot L)\leq 4$, and thus $\mathsf{c}(I\cdot J,(2\mathcal{O}_f)^n\cdot L)\leq 4$ and $\ell-2\leq 2$.
CASE 3: $n=2{\rm v}_2(f)$. By Theorem \[theorem 3.6\] there is some $D\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ such that $D\overline{D}=(2\mathcal{O}_f)^{n-2}$. Set $A=16\mathbb{Z}+(4+\tau)\mathbb{Z}$, $B=2^{n-2}\mathbb{Z}+(2^{n-4}+\tau)\mathbb{Z}$ and $C=2^{n-2}\mathbb{Z}+(2^{n-3}+\tau)\mathbb{Z}$. Then $A,B,C\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ and $ABC=2^{n-4}A(16\mathbb{Z}+(12+\tau)\mathbb{Z})=(2\mathcal{O}_f)^n$. This implies that $\mathsf{c}((2\mathcal{O}_f)^2\cdot D\cdot\overline{D}\cdot L,(2\mathcal{O}_f)^n\cdot L)\leq\mathsf{c}(D\cdot\overline{D},(2\mathcal{O}_f)^{n-2})\leq 4$ by Lemma \[lemma 4.10\].2. Moreover, $\mathsf{d}(A\cdot B\cdot C\cdot L,(2\mathcal{O}f)^2\cdot D\cdot\overline{D}\cdot L)\leq 4$ and $\mathsf{d}(I\cdot J,A\cdot B\cdot C\cdot L)\leq 4$. Consequently, $\mathsf{c}(I\cdot J,(2\mathcal{O}_f)^n\cdot L)\leq 4$ and $\ell-2\leq 2$.
\[proposition 4.12\] Let ${\rm v}_2(f)\in\{2,3\}$ and $d\equiv 1\mod 8$. Then $3\in\Delta(\mathcal{I}^*_2(\mathcal{O}_f))$ and $5\in {\rm Ca}(\mathcal{I}^*_2(\mathcal{O}_f))$.
We distinguish two cases.
CASE 1: ${\rm v}_2(f)=2$. By Theorem \[theorem 3.6\] there is some $I\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ such that $\mathcal{N}(I)=32$. Set $J=\overline{I}$. Then $IJ=32\mathcal{O}_f$, and hence $\{2,5\}\subset\mathsf{L}(IJ)\subset [2,5]$. Again by Theorem \[theorem 3.6\] we have $\mathcal{N}(L)\in\{4\}\cup\{2^n\mid n\in\mathbb{N}_{\geq 5}\}$ for all $L\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$. Note that if $A,B,C,D\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$, then $\mathcal{N}(ABCD)\in\{256\}\cup\mathbb{N}_{\geq 2048}$. Since $\mathcal{N}(IJ)=1024$, we have $4\not\in\mathsf{L}(IJ)$. Assume that $3\in\mathsf{L}(IJ)$. Then there are some $A,B,C\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ such that $IJ=ABC$ and $\mathcal{N}(A)\leq\mathcal{N}(B)\leq\mathcal{N}(C)$. Therefore, $\mathcal{N}(A)=\mathcal{N}(B)=4$ and $\mathcal{N}(C)=64$. We infer by Lemma \[lemma 4.2\].2 that $ABC=4L$ for some $L\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$, and hence $L=8\mathcal{O}_f$, a contradiction. We have $\mathsf{L}(IJ)=\{2,5\}$, and thus $3\in\Delta(\mathcal{I}^*_2(\mathcal{O}_f))$ and $5=\mathsf{c}(IJ)\in {\rm Ca}(\mathcal{I}^*_2(\mathcal{O}_f))$.
CASE 2: ${\rm v}_2(f)=3$. By Proposition \[proposition 3.3\].3 we can assume without restriction that $f=8$. By Theorem \[theorem 3.6\] there are some $I,J\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ such that $\mathcal{N}(I)=128$ and $\mathcal{N}(J)=16$. We have $I\overline{I}=128\mathcal{O}_f$ and $J\overline{J}=16\mathcal{O}_f$, and hence $I\overline{I}=8J\overline{J}$. This implies that $\{2,5\}\subset\mathsf{L}(I\overline{I})$. It follows from Theorem \[theorem 3.6\] that $\mathcal{N}(L)\in\{4,16\}\cup\{2^n\mid n\in\mathbb{N}_{\geq 7}\}$ for all $L\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$.
First assume that $3\in\mathsf{L}(I\overline{I})$. Then there exist $A,B,C\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ such that $I\overline{I}=ABC$, and $\mathcal{N}(A)\leq\mathcal{N}(B)\leq\mathcal{N}(C)$. Therefore, $(\mathcal{N}(A),\mathcal{N}(B),\mathcal{N}(C))\in\{(4,16,256),(4,4,1024)\}$. If $(\mathcal{N}(A),\mathcal{N}(B),\mathcal{N}(C))=(4,16,256)$, then it follows by Lemma \[lemma 4.2\].2 that $AB=2D$ for some $D\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ with $\mathcal{N}(D)=16$. We infer that $DC=64\mathcal{O}_f$, and hence $C=4\overline{D}$, a contradiction. Now let $(\mathcal{N}(A),\mathcal{N}(B),\mathcal{N}(C))=(4,4,1024)$. Then $ABC=4D$ for some $D\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ by Lemma \[lemma 4.2\].2, and thus $D=32\mathcal{O}_f$, a contradiction. Consequently, $3\not\in\mathsf{L}(I\overline{I})$.
Next assume that $4\in\mathsf{L}(I\overline{I})$. Then there exist $A,B,C,D\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ such that $I\overline{I}=ABCD$, and $\mathcal{N}(A)\leq\mathcal{N}(B)\leq\mathcal{N}(C)\leq\mathcal{N}(D)$.
Then $(\mathcal{N}(A),\mathcal{N}(B),\mathcal{N}(C),\mathcal{N}(D))\in\{(4,4,4,256),(4,16,16,16)\}$.
If $(\mathcal{N}(A),\mathcal{N}(B),\mathcal{N}(C),\mathcal{N}(D))=(4,4,4,256)$, then $ABCD=8E$ for $E\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ by Lemma \[lemma 4.2\].2, and hence $E=16\mathcal{O}_f$, a contradiction. Now let $(\mathcal{N}(A),\mathcal{N}(B),\mathcal{N}(C),\mathcal{N}(D))=(4,16,16,16)$. By Lemma \[lemma 4.2\].2 there is some $E\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ with $\mathcal{N}(E)=16$ such that $AB=2E$. Therefore, $ECD=64\mathcal{O}_f$, and hence $CD=4\overline{E}$. There are some $(0,4,r),(0,4,s)\in\mathcal{M}_{f,2}$ such that $C=16\mathbb{Z}+(r+\tau)\mathbb{Z}$ and $D=16\mathbb{Z}+(s+\tau)\mathbb{Z}$. We have ${\rm v}_2(r^2-16d)={\rm v}_2(s^2-16d)=4$. Since $d\equiv 1\mod 8$, this implies that ${\rm v}_2(r),{\rm v}_2(s)\geq 3$. Therefore, $\min\{4,{\rm v}_2(r+s+\varepsilon)\}\in\{3,4\}$, and hence $CD=8F$ for some $F\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$. We infer that $\overline{E}=2F$, a contradiction. Consequently, $4\not\in\mathsf{L}(I\overline{I})$.
Therefore, $2$ and $5$ are adjacent lengths of $I\overline{I}$, and hence $3\in\Delta(\mathcal{I}^*_2(\mathcal{O}_f))$. Note that $\mathsf{c}(\mathcal{I}^*_2(\mathcal{O}_f))\leq 5$ by Proposition \[proposition 4.11\].1 and Theorem \[theorem 3.6\]. Moreover, since $\mathcal{I}^*_2(\mathcal{O}_f)$ is a cancellative monoid, we have $5\leq 2+\sup\Delta(\mathsf{L}(I\overline{I}))\leq\mathsf{c}(I\overline{I})\leq 5$, and thus $5=\mathsf{c}(I\overline{I})\in {\rm Ca}(\mathcal{I}^*_2(\mathcal{O}_f))$.
\[lemma 4.13\] Let $H\in\{\mathcal{I}(\mathcal{O}_f),\mathcal{I}^*(\mathcal{O}_f)\}$. For every prime divisor $p$ of $f$, we set $H_p=\mathcal{I}_p(\mathcal{O}_f)$ if $H=\mathcal{I}(\mathcal{O}_f)$ and $H_p=\mathcal{I}^*_p(\mathcal{O}_f)$ if $H=\mathcal{I}^*(\mathcal{O}_f)$.
1. $H$ is half-factorial if and only if $H_p$ is half-factorial for every $p\in\mathbb{P}$ with $p\mid f$.
2. If $H$ is not half-factorial, then $\sup\Delta(H)=\sup\{\sup\Delta(H_p)\mid p\in\mathbb{P}$ with $p\mid f\}$.
3. $\mathsf{c}(H)=\sup\{\mathsf{c}(H_p)\mid p\in\mathbb{P}$ with $p\mid f\}$.
By Equations \[equation 3\] and \[equation 4\], we have $$\mathcal{I}^*(\mathcal{O}_f)\cong\coprod_{P\in\mathfrak{X}(\mathcal{O}_f)}\mathcal{I}^*_P(\mathcal{O}_f) \quad \text{ and } \quad \mathcal{I}(\mathcal{O}_f)\cong\coprod_{P\in\mathfrak{X}(\mathcal{O}_f)}\mathcal{I}_P(\mathcal{O}_f) \,.$$ Thus the assertions are easy consequences (see [@Ge-HK06a Propositions 1.4.5.3 and 1.6.8.1]).
1\. This is an immediate consequence of Proposition \[proposition 4.6\] and Lemma \[lemma 4.13\].
2\. First, suppose that $f$ is squarefree. By 1., we have $f$ is not a product of inert primes. It follows from Lemma \[lemma 4.13\], Proposition \[proposition 4.11\].1 and Theorem \[theorem 3.6\] that $\mathsf{c}(\mathcal{I}^*(\mathcal{O}))\leq\mathsf{c}(\mathcal{I}(\mathcal{O}))\leq 3$ and $\sup\Delta(\mathcal{I}^*(\mathcal{O}))\leq\sup\Delta(\mathcal{I}(\mathcal{O}))\leq 1$. By Lemma \[lemma 4.2\] and Proposition \[proposition 4.8\].1, it follows that $1\in\Delta(\mathcal{I}^*(\mathcal{O}))$, $1\in {\rm Ca}(\mathcal{I}(\mathcal{O}))$ and $[2,3]\subset {\rm Ca}(\mathcal{I}^*(\mathcal{O}))$, and thus ${\rm Ca}(\mathcal{I}(\mathcal{O}))=[1,3]$, ${\rm Ca}(\mathcal{I}^*(\mathcal{O}))=[2,3]$, and $\Delta(\mathcal{I}(\mathcal{O}))=\Delta(\mathcal{I}^*(\mathcal{O}))=\{1\}$.
Now we suppose that $f$ is not squarefree and we distinguish two cases.
CASE 1: ${\rm v}_2\left(f\right)\not\in\{2,3\}$ or $d_K\not\equiv 1\mod 8$. By Lemma \[lemma 4.13\], Proposition \[proposition 4.11\] and Theorem \[theorem 3.6\] it follows that $\mathsf{c}(\mathcal{I}^*(\mathcal{O}))\leq\mathsf{c}(\mathcal{I}(\mathcal{O}))\leq 4$ and $\sup\Delta(\mathcal{I}^*(\mathcal{O}))\leq\sup\Delta(\mathcal{I}(\mathcal{O}))\leq 2$. We infer by Lemma \[lemma 4.2\] and Propositions \[proposition 4.4\] and \[proposition 4.8\] that $[1,2]\subset\Delta(\mathcal{I}^*(\mathcal{O}))$, $1\in {\rm Ca}(\mathcal{I}(\mathcal{O}))$, and $[2,4]\subset {\rm Ca}(\mathcal{I}^*(\mathcal{O}))$, and hence ${\rm Ca}(\mathcal{I}(\mathcal{O}))=[1,4]$, ${\rm Ca}(\mathcal{I}^*(\mathcal{O}))=[2,4]$, and $\Delta(\mathcal{I}(\mathcal{O}))=\Delta(\mathcal{I}^*(\mathcal{O}))=[1,2]$.
CASE 2: ${\rm v}_2\left(f\right)\in\{2,3\}$ and $d_K\equiv 1\mod 8$. We infer by Lemma \[lemma 4.13\], Proposition \[proposition 4.11\].1 and Theorem \[theorem 3.6\] that $\mathsf{c}(\mathcal{I}^*(\mathcal{O}))\leq\mathsf{c}(\mathcal{I}(\mathcal{O}))\leq 5$ and $\sup\Delta(\mathcal{I}^*(\mathcal{O}))\leq\sup\Delta(\mathcal{I}(\mathcal{O}))\leq 3$. Lemma \[lemma 4.2\] and Propositions \[proposition 4.4\], \[proposition 4.8\] and \[proposition 4.12\] imply that $[1,3]\subset\Delta(\mathcal{I}^*(\mathcal{O}))$, $1\in {\rm Ca}(\mathcal{I}(\mathcal{O}))$ and $[2,5]\subset {\rm Ca}(\mathcal{I}^*(\mathcal{O}))$. Consequently, ${\rm Ca}(\mathcal{I}(\mathcal{O}))=[1,5]$, ${\rm Ca}(\mathcal{I}^*(\mathcal{O}))=[2,5]$, and $\Delta(\mathcal{I}(\mathcal{O}))=\Delta(\mathcal{I}^*(\mathcal{O}))=[1,3]$.
Based on the results of this section we derive a result on the set of distances of orders. Let $\mathcal{O}$ be a non-half-factorial order in a number field. Then the set of distances $\Delta(\mathcal{O})$ is finite. If $\mathcal{O}$ is a principal order, then it is easy to show that $\min\Delta(\mathcal{O})=1$ (indeed much stronger results are known, namely that sets of lengths of almost all elements – in a sense of density – are intervals, see [@Ge-HK06a Theorem 9.4.11]). The same is true if $|{\rm Pic}(\mathcal{O})|\ge 3$ or if $\mathcal{O}$ is seminormal ([@Ge-Zh16c Theorem 1.1]). However, it was unknown so far whether there exists an order $\mathcal{O}$ with $\min\Delta(\mathcal{O})>1$. In the next result of this section we characterize all non-half-factorial orders in quadratic number fields with $\min\Delta(\mathcal{O})>1$ which allows us to give the first explicit examples of orders $\mathcal{O}$ with $\min\Delta(\mathcal{O})>1$. A characterization of half-factorial orders in quadratic number fields is given in [@Ge-HK06a Theorem 3.7.15].
Let $\mathcal{O}$ be an order in a quadratic number field $K$ with conductor $f\in\mathbb{N}_{\ge 2}$. Then the class numbers $|{\rm Pic}(\mathcal{O}_K)|$ and $|{\rm Pic}(\mathcal{O})|$ are linked by the formula ([@HK13a Corollary 5.9.8]) $$\label{equation 6}
|{\rm Pic}(\mathcal{O})|=|{\rm Pic}(\mathcal{O}_K)|\frac{f}{(\mathcal{O}_K^{\times}:\mathcal{O}^{\times})}\prod_{p\in\mathbb{P},p\mid f}\left(1-\Big(\frac{d_K}{p}\Big) p^{-1}\right),$$
and $|{\rm Pic}(\mathcal{O})|$ is a multiple of $|{\rm Pic}(\mathcal{O}_K)|$.
Since the number of imaginary quadratic number fields with class number at most two is finite (an explicit list of these fields can be found, for example, in [@Ri01a]), shows that the number of orders in imaginary quadratic number fields with $|{\rm Pic}(\mathcal{O})|=2$ is finite. The complete list of non-maximal orders in imaginary quadratic number fields with $|{\rm Pic}(\mathcal{O})|=2$ is given in [@Kl12a page 16]. We refer to [@HK13a] for more information on class groups and class numbers and end with explicit examples of non-half-factorial orders $\mathcal{O}$ satisfying $\min\Delta(\mathcal{O})>1$.
\[theorem 4.14\] Let $\mathcal{O}$ be a non-half-factorial order in a quadratic number field $K$ with conductor $f\mathcal{O}_K$ for some $f\in\mathbb{N}_{\geq 2}$. Then the following statements are equivalent[:]{}
- $\min\Delta(\mathcal{O})>1$.
- $|{\rm Pic}(\mathcal{O})|=2$, $f$ is a nonempty squarefree product of ramified primes times a $($possibly empty$)$ squarefree product of inert primes, and for every prime divisor $p$ of $f$ and every $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$, $I$ is principal if and only if $\mathcal{N}(I)=p^2$.
If these equivalent conditions are satisfied, then $K$ is a real quadratic number field and $\min\Delta(\mathcal{O})=2$.
CLAIM: If $|{\rm Pic}(\mathcal{O})|=2$, $p$ is a ramified prime with ${\rm v}_p(f)=1$, and every $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ with $\mathcal{N}(I)=p^3$ is not principal, then every $L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ with $\mathcal{N}(L)=p^2$ is principal.
Let $|{\rm Pic}(\mathcal{O})|=2$, let $p$ be a ramified prime with ${\rm v}_p(f)=1$, and suppose that every $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ with $\mathcal{N}(I)=p^3$ is not principal. By Theorem \[theorem 3.6\] we have $\{\mathcal{N}(J)\mid J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))\}=\{p^2,p^3\}$. There is some $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ such that $\mathcal{N}(I)=p^3$. If $J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ with $\mathcal{N}(J)=p^3$, then $IJ=p^2L$ for some $L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ with $\mathcal{N}(L)=p^2$ (since there are no atoms with norm bigger than $p^3$). It follows by Theorem \[theorem 3.6\] that $|\{J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))\mid\mathcal{N}(J)=p^3\}|=|\{L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))\mid\mathcal{N}(L)=p^2\}|=p$ (note that $\mathcal{N}(p\mathcal{O})=p^2$). Let $g:\{J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))\mid\mathcal{N}(J)=p^3\}\rightarrow\{L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))\mid\mathcal{N}(L)=p^2\}$ be defined by $g(J)=L$ where $L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ is such that $\mathcal{N}(L)=p^2$ and $IJ=p^2L$. Then $g$ is a well-defined bijection. Now let $L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ with $\mathcal{N}(L)=p^2$. There is some $J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ such that $\mathcal{N}(J)=p^3$ and $IJ=p^2L$. Since $|{\rm Pic}(\mathcal{O})|=2$ and $I$ and $J$ are not principal, we have $IJ$ is principal, and hence $L$ is principal. This proves the claim.
\(a) $\Rightarrow$ (b) Observe that if $p$ is an inert prime such that ${\rm v}_p(f)=1$, then $\{\mathcal{N}(J)\mid J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))\}=\{p^2\}$ by Theorem \[theorem 3.6\]. Also note that if $p$ is a ramified prime such that ${\rm v}_p(f)=1$, then $\{\mathcal{N}(J)\mid J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))\}=\{p^2,p^3\}$ by Theorem \[theorem 3.6\]. The assertion now follows by the claim and Proposition \[proposition 4.8\].2.
\(b) $\Rightarrow$ (a) Assume to the contrary that $\min\Delta(\mathcal{O})=1$. Let $\mathcal{H}$ be the monoid of nonzero principal ideals of $\mathcal{O}$. There is some minimal $k\in\mathbb{N}$ such that $\prod_{i=1}^k U_i=\prod_{j=1}^{k+1} U^{\prime}_j$ with $U_i\in\mathcal{A}(\mathcal{H})$ for each $i\in [1,k]$ and $U^{\prime}_j\in\mathcal{A}(\mathcal{H})$ for each $j\in [1,k+1]$.
Set $\mathcal{Q}_1=\{P\in\mathfrak{X}(\mathcal{O})\mid P$ is principal$\}$, $\mathcal{Q}_2=\{P\in\mathfrak{X}(\mathcal{O})\mid P$ is invertible and not principal$\}$, $\mathcal{L}=\{p\in\mathbb{P}\mid p\mid f,p$ is ramified$\}$ and $\mathcal{K}=\{\{p,q\}\mid p,q\in\mathcal{L},p\not=q\}$. For every prime divisor $p$ of $f$ set $\mathcal{A}_p=\{V\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))\mid\mathcal{N}(V)=p^2\}$, $a_p=|\{i\in [1,k]\mid U_i\in\mathcal{A}_p\}|$ and $a^{\prime}_p=|\{j\in [1,k+1]\mid U^{\prime}_j\in\mathcal{A}_p\}|$. For $p\in\mathcal{L}$ set $\mathcal{D}_p=\{V\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))\mid\mathcal{N}(V)=p^3\}$, $\mathcal{B}_p=\{PV\mid P\in\mathcal{Q}_2$ and $V\in\mathcal{D}_p\}$, $b_p=|\{i\in [1,k]\mid U_i\in\mathcal{B}_p\}|$ and $b^{\prime}_p=|\{j\in [1,k+1]\mid U^{\prime}_j\in\mathcal{B}_p\}|$. Set $\mathcal{C}=\{PQ\mid P,Q\in\mathcal{Q}_2\}$, $c=|\{i\in [1,k]\mid U_i\in\mathcal{C}\}|$ and $c^{\prime}=|\{j\in [1,k+1]\mid U^{\prime}_j\in\mathcal{C}\}|$. If $z\in\mathcal{K}$ is such that $z=\{p,q\}$ with $p,q\in\mathcal{L}$ and $p\not=q$, then set $\mathcal{E}_z=\{VW\mid V\in\mathcal{D}_p,W\in\mathcal{D}_q\}$, $e_z=|\{i\in [1,k]\mid U_i\in\mathcal{E}_z\}|$ and $e^{\prime}_z=|\{j\in [1,k+1]\mid U^{\prime}_j\in\mathcal{E}_z\}|$.
Since $|{\rm Pic}(\mathcal{O})|=2$, we have $\mathcal{A}(\mathcal{H})\subset (\mathcal{A}(\mathcal{I}^*(\mathcal{O}))\cap\mathcal{H})\cup\{VW\mid V,W\in\mathcal{A}(\mathcal{I}^*(\mathcal{O})),V$ and $W$ are not principal$\}$. As shown in the proof of the claim, $VW\not\in\mathcal{A}(\mathcal{H})$ for all $p\in\mathcal{L}$ and $V,W\in\mathcal{D}_p$. We infer that $\mathcal{A}(\mathcal{H})=\mathcal{Q}_1\cup\bigcup_{p\in\mathbb{P},p\mid f}\mathcal{A}_p\cup\bigcup_{p\in\mathcal{L}}\mathcal{B}_p\cup\mathcal{C}\cup\bigcup_{z\in\mathcal{K}}\mathcal{E}_z$.
Since $k$ is minimal, we have $U_i,U^{\prime}_j\not\in\mathcal{Q}_1$ for all $i\in [1,k]$ and $j\in [1,k+1]$. Again since $k$ is minimal and $\mathcal{I}^*_p(\mathcal{O})$ is half-factorial for all inert prime divisors $p$ of $f$ by Proposition \[proposition 4.6\], we have $a_p=a^{\prime}_p=0$ for all inert prime divisors $p$ of $f$. Therefore,
$$k=\sum_{p\in\mathcal{L}} (a_p+b_p)+c+\sum_{z\in\mathcal{K}} e_z\textnormal{ and }k+1=\sum_{p\in\mathcal{L}} (a^{\prime}_p+b^{\prime}_p)+c^{\prime}+\sum_{z\in\mathcal{K}} e^{\prime}_z.$$
If $i\in [1,k]$, then $\sum_{P\in\mathcal{Q}_2} {\rm v}_P(U_i)=\begin{cases} 2 & \textnormal{ if } U_i\in\mathcal{C}\\ 1 & \textnormal{ if } U_i\in\bigcup_{p\in\mathcal{L}}\mathcal{B}_p\\ 0 &\textnormal{ else}\end{cases}$. This implies that $\sum_{P\in\mathcal{Q}_2} {\rm v}_P(\prod_{i=1}^k U_i)=\sum_{i=1}^k\sum_{P\in\mathcal{Q}_2}{\rm v}_P(U_i)=\sum_{p\in\mathcal{L}} b_p+2c$. It follows by analogy that $\sum_{P\in\mathcal{Q}_2} {\rm v}_P(\prod_{j=1}^{k+1} U^{\prime}_j)=\sum_{p\in\mathcal{L}} b^{\prime}_p+2c^{\prime}$. Therefore, $\sum_{p\in\mathcal{L}} b_p+2c=\sum_{p\in\mathcal{L}} b^{\prime}_p+2c^{\prime}$. Let $r\in\mathcal{L}$.
If $i\in [1,k]$, then ${\rm v}_r(\mathcal{N}((U_i)_{P_{f,r}}\cap\mathcal{O}))=\begin{cases} 3 & \textnormal{ if } U_i\in\mathcal{B}_r\cup\bigcup_{q\in\mathcal{L}\setminus\{r\}}\mathcal{E}_{\{r,q\}}\\ 2 & \textnormal{ if } U_i\in\mathcal{A}_r\\ 0 &\textnormal{ else}\end{cases}$. Consequently,
$${\rm v}_r(\mathcal{N}((\prod_{i=1}^k U_i)_{P_{f,r}}\cap\mathcal{O}))=\sum_{i=1}^k {\rm v}_r(\mathcal{N}((U_i)_{P_{f,r}}\cap\mathcal{O}))=2a_r+3b_r+3\sum_{q\in\mathcal{L}\setminus\{r\}} e_{\{r,q\}}.$$
By analogy we have ${\rm v}_r(\mathcal{N}((\prod_{j=1}^{k+1} U^{\prime}_j)_{P_{f,r}}\cap\mathcal{O}))=2a^{\prime}_r+3b^{\prime}_r+3\sum_{q\in\mathcal{L}\setminus\{r\}} e^{\prime}_{\{r,q\}}$. This implies that $2a_r+3b_r+3\sum_{q\in\mathcal{L}\setminus\{r\}} e_{\{r,q\}}=2a^{\prime}_r+3b^{\prime}_r+3\sum_{q\in\mathcal{L}\setminus\{r\}} e^{\prime}_{\{r,q\}}$. We infer that $$\begin{aligned}
&\sum_{p\in\mathcal{L}} (a^{\prime}_p-a_p+b^{\prime}_p-b_p)+c^{\prime}-c+\sum_{z\in\mathcal{K}} (e^{\prime}_z-e_z)=1,\textnormal{ }\sum_{p\in\mathcal{L}} (b^{\prime}_p-b_p)=2(c-c^{\prime})\\
&\textnormal{and }2\sum_{p\in\mathcal{L}} (a^{\prime}_p-a_p)+3\sum_{p\in\mathcal{L}} (b^{\prime}_p-b_p)+3\sum_{p\in\mathcal{L}}\sum_{q\in\mathcal{L}\setminus\{p\}} (e^{\prime}_{\{p,q\}}-e_{\{p,q\}})=0.\end{aligned}$$
Note that $\sum_{p\in\mathcal{L}}\sum_{q\in\mathcal{L}\setminus\{p\}} (e^{\prime}_{\{p,q\}}-e_{\{p,q\}})=2\sum_{z\in\mathcal{K}} (e^{\prime}_z-e_z)$, and hence $\sum_{p\in\mathcal{L}} (a^{\prime}_p-a_p)=3(c^{\prime}-c)-3\sum_{z\in\mathcal{K}} (e^{\prime}_z-e_z)$. Consequently, $$\begin{aligned}
1&=\sum_{p\in\mathcal{L}} (a^{\prime}_p-a_p+b^{\prime}_p-b_p)+c^{\prime}-c+\sum_{z\in\mathcal{K}} (e^{\prime}_z-e_z)\\
&=3(c^{\prime}-c)-3\sum_{z\in\mathcal{K}} (e^{\prime}_z-e_z)+2(c-c^{\prime})+c^{\prime}-c+\sum_{z\in\mathcal{K}} (e^{\prime}_z-e_z)\\
&=2(c^{\prime}-c-\sum_{z\in\mathcal{K}} (e^{\prime}_z-e_z)),\end{aligned}$$ a contradiction.
Now let the equivalent conditions be satisfied. Assume to the contrary that $K$ is an imaginary quadratic number field. Since $\mathcal{O}$ is a non-maximal order with $|{\rm Pic}(\mathcal{O})|=2$, it follows from [@Kl12a page 16] that $(f,d_K)\in\{(2,-8),(2,-15),(3,-4),(3,-8),(3,-11),(4,-3),(4,-4),(4,-7),(5,-3),(5,-4),(7,-3)\}$.
Since $f$ is squarefree and divisible by a ramified prime, we infer that $f=2$ and $d_K=-8$. Therefore, $\mathcal{O}=\mathbb{Z}+2\sqrt{-2}\mathbb{Z}$. Set $I=8\mathbb{Z}+2\sqrt{-2}\mathbb{Z}$. Observe that $I\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}))$ and $\mathcal{N}(I)=8$. Moreover, $I=2\sqrt{-2}\mathcal{O}$ is principal, a contradiction. Consequently, $K$ is a real quadratic number field.
It remains to show that $\min\Delta(\mathcal{O})=2$. There is some ramified prime $p$ which divides $f$ and there is some $J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ with $\mathcal{N}(J)=p^3$. As shown in the proof of the claim, $J^2=p^2L$ for some $L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$. By [@Ge-HK06a Corollary 2.11.16], there is some invertible prime ideal $P$ of $\mathcal{O}$ that is not principal. Observe that $J$ is not principal. We have $PJ$, $P^2$ and $L$ are principal, and hence there are some $u,v,w\in\mathcal{A}(\mathcal{O})$ such that $PJ=u\mathcal{O}$, $P^2=v\mathcal{O}$, $L=w\mathcal{O}$, and $u^2=p^2vw$. Therefore, $\{2,4\}\subseteq\mathsf{L}(u^2)$, and since $\min\Delta(\mathcal{O})>1$, we infer that $\min\Delta(\mathcal{O})=2$.
\[proposition 4.15\] Let $\mathcal{O}$ be an order in the quadratic number field $K$ with conductor $f\mathcal{O}_K$ for some $f\in\mathbb{N}_{\geq 2}$ such that $\min\Delta(\mathcal{O})>1$, let $g$ be the product of all inert prime divisors of $f$ and let $\mathcal{O}^{\prime}$ be the order in $K$ with conductor $g\mathcal{O}_K$. Then $\mathcal{O}^{\prime}$ is half-factorial and, in particular, $g\in\{1\}\cup\mathbb{P}\cup\{2p\mid p\in\mathbb{P}\setminus\{2\}\}$.
Set $\mathcal{Q}_1=\{P\in\mathfrak{X}(\mathcal{O}^{\prime})\mid P$ is principal$\}$ and $\mathcal{Q}_2=\{P\in\mathfrak{X}(\mathcal{O}^{\prime})\mid P$ is invertible and not principal$\}$. Observe that $\mathcal{N}(I)=|\mathcal{O}/I|=|\mathcal{O}^{\prime}/I\mathcal{O}^{\prime}|=\mathcal{N}(I\mathcal{O}^{\prime})$ for all $I\in\mathcal{I}^*(\mathcal{O})$. Note that for all inert prime divisors $p$ of $f$ and all $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ and $J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}^{\prime}))$, we have $\mathcal{N}(I)=\mathcal{N}(J)=p^2$. Moreover, for all ramified prime divisors $p$ of $f$, we have $\{\mathcal{N}(I)\mid I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))\}=\{p^2,p^3\}$. In this proof we will use Theorem \[theorem 4.14\] without further citation.
CLAIM 1: For all prime divisors $p$ of $g$ and all $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}^{\prime}))$, it follows that $I$ is principal. Let $p$ be a prime divisor of $g$ and let $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}^{\prime}))$. Set $P=P_{f,p}$ and $P^{\prime}=P_{g,p}$. It follows by Proposition \[proposition 3.3\] that $\mathcal{O}_P=\mathcal{O}^{\prime}_{P^{\prime}}$ and that $\delta:\mathcal{I}^*_p(\mathcal{O})\rightarrow\mathcal{I}^*_p(\mathcal{O}^{\prime})$ defined by $\delta(J)=J_P\cap\mathcal{O}^{\prime}$ for all $J\in\mathcal{I}^*_p(\mathcal{O})$ is a monoid isomorphism. In particular, we have $\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}^{\prime}))=\{J_P\cap\mathcal{O}^{\prime}\mid J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))\}$. Therefore, there is some $J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ such that $J_P\cap\mathcal{O}^{\prime}=I$. Note that $\mathcal{N}(I)=p^2=\mathcal{N}(J)=\mathcal{N}(J\mathcal{O}^{\prime})$. Since $J\mathcal{O}^{\prime}\subseteq J\mathcal{O}^{\prime}_{P^{\prime}}\cap\mathcal{O}^{\prime}=J\mathcal{O}_P\cap\mathcal{O}^{\prime}=I$, we infer that $I=J\mathcal{O}^{\prime}$. Since $J$ is a principal ideal of $\mathcal{O}$, it follows that $I$ is principal. This proves Claim 1.
CLAIM 2: If $P\in\mathcal{Q}_2$, $p$ is a ramified prime divisor of $f$ such that $P\cap\mathbb{Z}=p\mathbb{Z}$ and $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ with $\mathcal{N}(I)=p^3$, then $P^2$ is principal and $I\mathcal{O}^{\prime}=P^3$. Let $P\in\mathcal{Q}_2$, $p$ a ramified prime divisor of $f$ such that $P\cap\mathbb{Z}=p\mathbb{Z}$ and $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ with $\mathcal{N}(I)=p^3$. Since $p$ is ramified, there is some $A\in\mathfrak{X}(\mathcal{O}_K)$ such that $p\mathcal{O}_K=A^2$. Observe that $\mathcal{N}(A^2)=p^2$, and thus $\mathcal{N}(A)=p$. We have $A\cap\mathcal{O}^{\prime}=P$, $P\mathcal{O}_K=A$ and $\mathcal{N}(P)=\mathcal{N}(A)=p$. Note that since $P$ is invertible, it follows that every $P$-primary ideal of $\mathcal{O}^{\prime}$ is a power of $P$. Therefore, $p\mathcal{O}^{\prime}=P^k$ for some $k\in\mathbb{N}$, and hence $p^k=\mathcal{N}(P^k)=\mathcal{N}(p\mathcal{O}^{\prime})=p^2$. Consequently, $k=2$ and $P^2$ is principal. Clearly, $I\mathcal{O}^{\prime}$ is a $P$-primary ideal of $\mathcal{O}^{\prime}$, and thus $I\mathcal{O}^{\prime}=P^m$ for some $m\in\mathbb{N}$. We infer that $p^m=\mathcal{N}(P^m)=\mathcal{N}(I\mathcal{O}^{\prime})=\mathcal{N}(I)=p^3$, and thus $m=3$ and $I\mathcal{O}^{\prime}=P^3$. This proves Claim 2.
CLAIM 3: $PQ$ is principal for all $P,Q\in\mathcal{Q}_2$. Let $P,Q\in\mathcal{Q}_2$.
CASE 1: $P\cap\mathcal{O}$ and $Q\cap\mathcal{O}$ are invertible. Note that $P=(P\cap\mathcal{O})\mathcal{O}^{\prime}$, $Q=(Q\cap\mathcal{O})\mathcal{O}^{\prime}$ and $P\cap\mathcal{O}$ and $Q\cap\mathcal{O}$ are not principal. Since $|{\rm Pic}(\mathcal{O})|=2$, we have $(P\cap\mathcal{O})(Q\cap\mathcal{O})$ is a principal ideal of $\mathcal{O}$, and thus $PQ=(P\cap\mathcal{O})(Q\cap\mathcal{O})\mathcal{O}^{\prime}$ is principal.
CASE 2: ($P\cap\mathcal{O}$ is invertible and $Q\cap\mathcal{O}$ is not invertible) or ($P\cap\mathcal{O}$ is not invertible and $Q\cap\mathcal{O}$ is invertible). Without restriction let $P\cap\mathcal{O}$ be invertible and let $Q\cap\mathcal{O}$ be not invertible. Observe that $P=(P\cap\mathcal{O})\mathcal{O}^{\prime}$. Moreover, there is some ramified prime $q$ that divides $f$ such that $Q\cap\mathbb{Z}=q\mathbb{Z}$ and there is some $J\in\mathcal{A}(\mathcal{I}^*_q(\mathcal{O}))$ with $\mathcal{N}(J)=q^3$. Observe that $P\cap\mathcal{O}$ and $J$ are not principal. Since $|{\rm Pic}(\mathcal{O})|=2$, it follows that $(P\cap\mathcal{O})J$ is a principal ideal of $\mathcal{O}$. Note that $PQ^3=(P\cap\mathcal{O})J\mathcal{O}^{\prime}$ by Claim 2, and thus $PQ^3$ is principal. Since $Q^2$ is principal by Claim 2, we infer that $PQ$ is principal.
CASE 3: $P\cap\mathcal{O}$ and $Q\cap\mathcal{O}$ are not invertible. There are ramified primes $p$ and $q$ that divide $f$ such that $P\cap\mathbb{Z}=p\mathbb{Z}$ and $Q\cap\mathbb{Z}=q\mathbb{Z}$. There are some $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ and $J\in\mathcal{A}(\mathcal{I}^*_q(\mathcal{O}))$ with $\mathcal{N}(I)=p^3$ and $\mathcal{N}(J)=q^3$. Since $|{\rm Pic}(\mathcal{O})|=2$ and $I$ and $J$ are not principal, we have $IJ$ is a principal ideal of $\mathcal{O}$. It follows that $P^3Q^3=IJ\mathcal{O}^{\prime}$ by Claim 2, and hence $P^3Q^3$ is principal. Since $P^2$ and $Q^2$ are principal by Claim 2, we have $PQ$ is principal. This proves Claim 3.
Finally, we show that $\mathcal{O}^{\prime}$ is half-factorial. Set $\mathcal{C}=\{PQ\mid P,Q\in\mathcal{Q}_2\}$ and let $\mathcal{H}$ denote the monoid of nonzero principal ideals of $\mathcal{O}^{\prime}$. It is an immediate consequence of Claim 1 and Claim 3 that $\mathcal{A}(\mathcal{H})=\mathcal{Q}_1\cup\mathcal{C}\cup\bigcup_{p\in\mathbb{P},p\mid g}\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}^{\prime}))$.
Let $k,\ell\in\mathbb{N}$ and $I_i,I_j^{\prime}\in\mathcal{A}(\mathcal{H})$ for each $i\in [1,k]$ and $j\in [1,\ell]$ be such that $\prod_{i=1}^k I_i=\prod_{j=1}^{\ell} I_j^{\prime}$. It remains to show that $k=\ell$. Set $b=|\{i\in [1,k]\mid I_i\in\mathcal{Q}_1\}|$, $b^{\prime}=|\{j\in [1,\ell]\mid I_j^{\prime}\in\mathcal{Q}_1\}|$, $c=|\{i\in [1,k]\mid I_i\in\mathcal{C}\}|$, $c^{\prime}=|\{j\in [1,\ell]\mid I_j^{\prime}\in\mathcal{C}\}|$ and for each prime divisor $p$ of $g$ set $a_p=|\{i\in [1,k]\mid I_i\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}^{\prime}))\}|$ and $a^{\prime}_p=|\{j\in [1,\ell]\mid I_j^{\prime}\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}^{\prime}))\}|$. If $p$ is a prime divisor of $g$, then $\mathcal{I}^*_p(\mathcal{O}^{\prime})$ is half-factorial by Proposition \[proposition 4.6\], and hence $a_p=a^{\prime}_p$ by Claim 1. We have $b=\sum_{i=1}^k\sum_{P\in\mathcal{Q}_1}{\rm v}_P(I_i)=\sum_{P\in\mathcal{Q}_1}{\rm v}_P(\prod_{i=1}^k I_i)=\sum_{P\in\mathcal{Q}_1}{\rm v}_P(\prod_{j=1}^{\ell} I_j^{\prime})=\sum_{j=1}^{\ell}\sum_{P\in\mathcal{Q}_1}{\rm v}_P(I_j^{\prime})=b^{\prime}$.
Moreover, $2c=\sum_{P\in\mathcal{Q}_2}{\rm v}_P(\prod_{i=1}^k I_i)=\sum_{P\in\mathcal{Q}_2}{\rm v}_P(\prod_{j=1}^{\ell} I_j^{\prime})=2c^{\prime}$. Therefore, $k=b+c+\sum_{p\in\mathbb{P},p\mid g} a_p=b^{\prime}+c^{\prime}+\sum_{p\in\mathbb{P},p\mid g} a^{\prime}_p=\ell$.
The remaining assertion follows from [@Ge-HK06a Theorem 3.7.15].
\[remark 4.16\] Let $\mathcal{O}$ be an order in the quadratic number field $K$ with conductor $f\mathcal{O}_K$ for some $f\in\mathbb{N}$ such that $|{\rm Pic}(\mathcal{O})|=2$ and let $p$ be an odd ramified prime such that ${\rm v}_p(f)=1$ and $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ such that $\mathcal{N}(I)=p^3$ and $I$ not principal. Then every $J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ with $\mathcal{N}(J)=p^3$ is not principal.
Set $\mathcal{L}=\{J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))\mid\mathcal{N}(J)=p^3\}$ and $\mathcal{K}=\{L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))\mid\mathcal{N}(L)=p^2\}$. It follows by the claim in the proof of Theorem \[theorem 4.14\] that for all $J\in\mathcal{L}$ and $L\in\mathcal{K}$, there is a unique $A\in\mathcal{L}$ such that $AJ=p^2L$. By Theorem \[theorem 3.6\] we have $|\mathcal{L}|=|\mathcal{K}|=p$, and hence $|\{(A,J)\in\mathcal{L}^2\mid AJ=p^2L\}|=p$ for all $L\in\mathcal{K}$. Since $p$ is odd, we infer that for each $L\in\mathcal{K}$ there is some $A\in\mathcal{L}$ such that $A^2=p^2L$. Consequently, every $L\in\mathcal{K}$ is principal. Now let $J\in\mathcal{L}$. There is some $B\in\mathcal{K}$ such that $IJ=p^2B$, and thus $IJ$ is principal. Therefore, $J$ is not principal.
Next we show that the assumption that $p$ is odd in Remark \[remark 4.16\] is crucial.
\[example 4.17\] Let $\mathcal{O}=\mathbb{Z}+2\sqrt{-2}\mathbb{Z}$ be the order in the quadratic number field $K=\mathbb{Q}(\sqrt{-2})$ with conductor $2\mathcal{O}_K$. Let $I=8\mathbb{Z}+2\sqrt{-2}\mathbb{Z}$ and $J=8\mathbb{Z}+(4+2\sqrt{-2})\mathbb{Z}$. Then $2$ is ramified, $|{\rm Pic}(\mathcal{O})|=2$, $I,J\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}))$, $\mathcal{N}(I)=\mathcal{N}(J)=8$, $I$ is principal and $J$ is not principal.
It is clear that $J\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}))$ and $\mathcal{N}(J)=8$. By the proof of Theorem \[theorem 4.14\], it remains to show that $J$ is not principal. Assume that $J$ is principal. Then there are some $a,b\in\mathbb{Z}$ such that $J=(8a+4b+2\sqrt{-2}b)\mathcal{O}$, and hence $8=\mathcal{N}(J)=|\mathcal{N}_{K/\mathbb{Q}}(8a+4b+2\sqrt{-2}b)|=|(8a+4b)^2+8b^2|$. Therefore, $2(2a+b)^2+b^2=1$. It is clear that $|b|\leq 1$. If $b=0$, then $8a^2=1$, a contradiction. Therefore, $|b|=1$ and $2a+b=0$, a contradiction.
\[lemma 4.18\] Let $d\in\mathbb{N}_{\geq 2}$ be squarefree, let $K=\mathbb{Q}(\sqrt{d})$, let $\mathcal{O}$ be the order in $K$ with conductor $f\mathcal{O}_K$ for some $f\in\mathbb{N}_{\geq 2}$, and let $p$ be a ramified prime with ${\rm v}_p(f)=1$. If $(p\equiv 1\mod 4$ and $(\frac{d/p}{p})=-1)$ or $((\frac{p}{q})=-1$ for some prime $q$ with $q\equiv 1\mod 4$ and $q\mid df)$, then each $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ with $\mathcal{N}(I)=p^3$ is not principal.
Note that if $p$ is odd, then $\{I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))\mid\mathcal{N}(I)=p^3\}=\{p^3\mathbb{Z}+(p^2k+\frac{\varepsilon p^2+f\sqrt{d_K}}{2})\mathbb{Z}\mid k\in [0,p-1]\}$. Moreover, if $p=2$ and $d$ is odd, then $\{I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))\mid\mathcal{N}(I)=p^3\}=\{8\mathbb{Z}+(2k+f\sqrt{d})\mathbb{Z}\mid k\in\{1,3\}\}$. Furthermore, if $p=2$ and $d$ is even, then $\{I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))\mid\mathcal{N}(I)=p^3\}=\{8\mathbb{Z}+(2k+f\sqrt{d})\mathbb{Z}\mid k\in\{0,2\}\}$.
CASE 1: $p\equiv 1\mod 4$ and $(\frac{d/p}{p})=-1$. Let $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ be such that $\mathcal{N}(I)=p^3$. Since $p$ is odd, we have $I=p^3\mathbb{Z}+(p^2k+\frac{\varepsilon p^2+f\sqrt{d_K}}{2})\mathbb{Z}$ for some $k\in [0,p-1]$. Assume that $I$ is principal. Then there are some $a,b\in\mathbb{Z}$ such that $I=(p^3a+p^2bk+\frac{\varepsilon p^2+f\sqrt{d_K}}{2}b)\mathcal{O}$. We infer that $p^3=\mathcal{N}(I)=|\mathcal{N}_{K/\mathbb{Q}}(p^3a+p^2bk+\frac{\varepsilon p^2+f\sqrt{d_K}}{2}b)|=\frac{1}{4}|p^4(2pa+2bk+\varepsilon b)^2-f^2b^2d_K|$, and hence $\frac{f^2}{p^2}b^2\frac{d_K}{p}\equiv 4\beta\mod p$ for some $\beta\in\{-1,1\}$. Since $p\equiv 1\mod 4$, we have $(\frac{-1}{p})=1$, and thus $(\frac{d/p}{p})=(\frac{d_K/p}{p})=(\frac{f^2b^2d_K/p^3}{p})=(\frac{4\beta}{p})=1$, a contradiction.
CASE 2: There is some prime $q$ such that $q\equiv 1\mod 4$, $q\mid df$ and $(\frac{p}{q})=-1$. Let $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ be such that $\mathcal{N}(I)=p^3$. First let $p$ be odd. Then $I=p^3\mathbb{Z}+(p^2k+\frac{\varepsilon p^2+f\sqrt{d_K}}{2})\mathbb{Z}$ for some $k\in [0,p-1]$. Assume that $I$ is principal. Then there are some $a,b\in\mathbb{Z}$ such that $I=(p^3a+p^2bk+\frac{\varepsilon p^2+f\sqrt{d_K}}{2}b)\mathcal{O}$. This implies that $p^3=\mathcal{N}(I)=|\mathcal{N}_{K/\mathbb{Q}}(p^3a+p^2bk+\frac{\varepsilon p^2+f\sqrt{d_K}}{2}b)|=\frac{1}{4}|p^4(2pa+2bk+\varepsilon b)^2-f^2b^2d_K|$, and thus $\ell^2\equiv 4\beta p^3\mod q$ for some $\ell\in\mathbb{Z}$ and $\beta\in\{-1,1\}$. Since $q\equiv 1\mod 4$, we have $(\frac{-1}{q})=1$, and hence $(\frac{p}{q})^3=(\frac{4\beta p^3}{q})=1$. Therefore, $(\frac{p}{q})=1$, a contradiction.
Now let $p=2$. Then $I=8\mathbb{Z}+(2k+f\sqrt{d})\mathbb{Z}$ for some $k\in [0,3]$. Assume that $I$ is principal. Then there are some $a,b\in\mathbb{Z}$ such that $I=(8a+2bk+bf\sqrt{d})\mathcal{O}$. Consequently, $8=\mathcal{N}(I)=|(8a+2bk)^2-b^2f^2d|$, and thus $\ell^2\equiv 8\beta\mod q$ for some $\ell\in\mathbb{Z}$ and $\beta\in\{-1,1\}$. This implies that $(\frac{2}{q})^3=(\frac{8\beta}{q})=1$. Therefore, $(\frac{2}{q})=1$, a contradiction.
\[proposition 4.19\] Let $d\in\mathbb{N}_{\geq 2}$ be squarefree, let $K=\mathbb{Q}(\sqrt{d})$, and let $\mathcal{O}$ be the order in $K$ with conductor $f\mathcal{O}_K$ such that $f$ is a nonempty squarefree product of ramified primes times a squarefree product of inert primes and $|{\rm Pic}(\mathcal{O})|=|{\rm Pic}(\mathcal{O}_K)|=2$. If for every ramified prime divisor $p$ of $f$, we have $(p\equiv 1\mod 4$ and $(\frac{d/p}{p})=-1)$ or $((\frac{p}{q})=-1$ for some prime $q$ with $q\equiv 1\mod 4$ and $q\mid df)$, then $\min\Delta(\mathcal{O})=2$.
It follows by Lemma \[lemma 4.18\] that for every ramified prime divisor $p$ of $f$ and every $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ with $\mathcal{N}(I)=p^3$, we have $I$ is not principal. It follows by the claim in the proof of Theorem \[theorem 4.14\] that $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$ is principal if and only if $\mathcal{N}(I)=p^2$. Now let $p$ be an inert prime divisor of $f$ and let $J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}))$. Since $|{\rm Pic}(\mathcal{O})|=|{\rm Pic}(\mathcal{O}_K)|$, it follows that the group epimorphism $\theta:{\rm Pic}(\mathcal{O})\rightarrow {\rm Pic}(\mathcal{O}_K)$ defined by $\theta([L])=[L\mathcal{O}_K]$ for all $L\in\mathcal{I}^*(\mathcal{O})$ is a group isomorphism. Set $P=p\mathcal{O}_K$. Then $J\mathcal{O}_K$ is a $P$-primary ideal of $\mathcal{O}_K$, and hence $J\mathcal{O}_K$ is a principal ideal of $\mathcal{O}_K$. Since $\theta$ is an isomorphism, we infer that $J$ is a principal ideal of $\mathcal{O}$. Now it follows by Theorem \[theorem 4.14\] that $\min\Delta(\mathcal{O})=2$.
Next we provide two counterexamples that show that the additional assumption on the ramified prime divisors of $f$ in Proposition \[proposition 4.19\] is important.
\[example 4.20\] There is some real quadratic number field $K$ and some order $\mathcal{O}$ in $K$ with conductor $p\mathcal{O}_K$ for some ramified prime $p$ such that $p\equiv 1\mod 4$, $|{\rm Pic}(\mathcal{O})|=|{\rm Pic}(\mathcal{O}_K)|=2$, and $\min\Delta(\mathcal{O})=1$.
Let $\mathcal{O}=\mathbb{Z}+5\sqrt{30}\mathbb{Z}$ be the order in the real quadratic number field $K=\mathbb{Q}(\sqrt{30})$ with conductor $5\mathcal{O}_K$. Observe that $5$ is ramified, $5\equiv 1\mod 4$, $|{\rm Pic}(\mathcal{O}_K)|=2$ and $\alpha=11+2\sqrt{30}$ is a fundamental unit of $\mathcal{O}_K$. Since $\alpha\not\in\mathcal{O}$ and $(\mathcal{O}_K^{\times}:\mathcal{O}^{\times})\mid 5$, we infer that $(\mathcal{O}_K^{\times}:\mathcal{O}^{\times})=5$, and hence $|{\rm Pic}(\mathcal{O})|=|{\rm Pic}(\mathcal{O}_K)|\frac{5}{(\mathcal{O}_K^{\times}:\mathcal{O}^{\times})}=2$. Let $I=125\mathbb{Z}+5\sqrt{30}\mathbb{Z}$. Then $I\in\mathcal{A}(\mathcal{I}^*_5(\mathcal{O}))$ with $\mathcal{N}(I)=125$. Since $I=(12625+2305\sqrt{30})\mathcal{O}$ is principal, we infer by Theorem \[theorem 4.14\] that $\min\Delta(\mathcal{O})=1$.
\[example 4.21\] There is some real quadratic number field $K=\mathbb{Q}(\sqrt{d})$ with $d\in\mathbb{N}_{\geq 2}$ squarefree and some order $\mathcal{O}$ in $K$ with conductor $p\mathcal{O}_K$ for some odd ramified prime $p$ such that $(\frac{d/p}{p})=-1$, $|{\rm Pic}(\mathcal{O})|=|{\rm Pic}(\mathcal{O}_K)|=2$, and $\min\Delta(\mathcal{O})=1$.
Let $\mathcal{O}=\mathbb{Z}+7\sqrt{42}\mathbb{Z}$ be the order in the real quadratic number field $K=\mathbb{Q}(\sqrt{42})$ with conductor $7\mathcal{O}_K$. Note that $7$ is an odd ramified prime, $(\frac{42/7}{7})=-1$, $|{\rm Pic}(\mathcal{O}_K)|=2$ and $\alpha=13+2\sqrt{42}$ is a fundamental unit of $\mathcal{O}_K$. We have $\alpha\not\in\mathcal{O}$ and $(\mathcal{O}_K^{\times}:\mathcal{O}^{\times})\mid 7$. Therefore, $(\mathcal{O}_K^{\times}:\mathcal{O}^{\times})=7$, and thus $|{\rm Pic}(\mathcal{O})|=|{\rm Pic}(\mathcal{O}_K)|\frac{7}{(\mathcal{O}_K^{\times}:\mathcal{O}^{\times})}=2$. Set $I=343\mathbb{Z}+7\sqrt{42}\mathbb{Z}$. Then $I\in\mathcal{A}(\mathcal{I}^*_7(\mathcal{O}))$, $\mathcal{N}(I)=343$, and $I=(825601+127393\sqrt{42})\mathcal{O}$ is principal. Consequently, $\min\Delta(\mathcal{O})=1$ by Theorem \[theorem 4.14\].
Finally, we provide the examples of orders $\mathcal{O}$ in quadratic number fields with $\min\Delta(\mathcal{O})=2$.
\[example 4.22\] Let $K$ be a quadratic number field and $\mathcal{O}$ the order in $K$ with conductor $f\mathcal{O}_K$ such that $(f,d_K)\in\{(2,60),(3,60),(5,60),(6,60),(10,60),(15,60),(30,60),(10,85),(35,40),(195,65),(30,365)\}$.
1. If $(f,d_K)\in\{(2,60),(3,60),(5,60)\}$, then $f$ is a ramified prime.
2. If $(f,d_K)\in\{(6,60),(10,60),(15,60)\}$, then $f$ is the product of two distinct ramified primes.
3. If $(f,d_K)=(30,60)$, then $f$ is the product of three distinct ramified primes.
4. If $(f,d_K)\in\{(10,85),(35,40)\}$, then $f$ is the product of an inert prime and a ramified prime.
5. If $(f,d_K)=(195,65)$, then $f$ is the product of an inert prime and two distinct ramified primes.
6. If $(f,d_K)=(30,365)$, then $f$ is the product of two distinct inert primes and a ramified prime.
7. $\min\Delta(\mathcal{O})=2$.
It is straightforward to prove the first six assertions. We prove the last assertion in the case that $d_K=60$ and $f\in\mathbb{N}_{\geq 2}$ is a divisor of $30$. The remaining cases can be proved in analogy by using Proposition \[proposition 4.19\]. It is clear that $2$, $3$, and $5$ are ramified primes. Note that $|{\rm Pic}(\mathcal{O}_K)|=2$ (e.g., [@HK13a page 22]) and $\alpha=4+\sqrt{15}$ is a fundamental unit of $\mathcal{O}_K$.
We have $\alpha^2=31+8\sqrt{15}$, $\alpha^3=244+63\sqrt{15}$, and $\alpha^5=15124+3905\sqrt{15}$. Moreover, $\alpha^6=119071+30744\sqrt{15}$, $\alpha^{10}=457470751+118118440\sqrt{15}$, and $\alpha^{15}=13837575261124+3572846569215\sqrt{15}$. Set $k=(\mathcal{O}_K^{\times}:\mathcal{O}^{\times})$. Then $k$ is a divisor of $f$ by . Observe that $\alpha\not\in\mathbb{Z}+2\sqrt{15}\mathbb{Z}$, $\alpha\not\in\mathbb{Z}+3\sqrt{15}\mathbb{Z}$, $\alpha\not\in\mathbb{Z}+5\sqrt{15}\mathbb{Z}$, $\alpha^2,\alpha^3\not\in\mathbb{Z}+6\sqrt{15}\mathbb{Z}$, $\alpha^2,\alpha^5\not\in\mathbb{Z}+10\sqrt{15}\mathbb{Z}$, $\alpha^3,\alpha^5\not\in\mathbb{Z}+15\sqrt{15}\mathbb{Z}$, and $\alpha^6,\alpha^{10},\alpha^{15}\not\in\mathbb{Z}+30\sqrt{15}\mathbb{Z}$. This implies that $k=f$, and hence $|{\rm Pic}(\mathcal{O})|=\frac{f}{k}|{\rm Pic}(\mathcal{O}_K)|=|{\rm Pic}(\mathcal{O}_K)|=2$ by . We have $5\equiv 1\mod 4$ and $(\frac{15/5}{5})=(\frac{3}{5})=(\frac{2}{5})=-1$. We infer by Proposition \[proposition 4.19\] that $\min\Delta(\mathcal{O})=2$.
Unions of sets of lengths {#5}
=========================
The goal of this section is to show that all unions of sets of lengths of the monoid of (invertible) ideals in orders of quadratic number fields are intervals (Theorem \[theorem 5.2\]). To gather the background on unions of sets of lengths, let $H$ be an atomic monoid with $H\ne H^{\times}$ and $k\in\mathbb{N}_0$. Then $$\begin{aligned}
\mathcal{U}_k (H) & =\bigcup_{k\in L\in\mathcal{L}(H)} L \qquad\text{denotes the {\it union of sets of lengths} containing $k$ and } \\
\rho_k (H) & = \sup \mathcal U_k (H) \qquad \text{is the {\it $k$th elasticity} of $H$} \,.
\end{aligned}$$ Then, for the [*elasticity*]{} $\rho (H)$ of $H$, we have ([@F-G-K-T17 Proposition 2.7]), $$\rho(H)=\sup\{\rho (L) \mid L\in\mathcal{L}(H)\} = \lim_{k\to\infty}\frac{\rho_k (H)}{k}\,.$$ Clearly, $\mathcal{U}_0(H)=\{0\}$, $\mathcal{U}_1(H)=\{1\}$ and $\mathcal{U}_k(H)$ is the set of all $\ell\in\mathbb{N}_0$ with the following property:
- There are atoms $u_1,\ldots,u_k,v_1,\ldots,v_{\ell}$ in $H$ such that $u_1\cdot\ldots\cdot u_k= v_1\cdot\ldots\cdot v_{\ell}$.
Let $d\in\mathbb{N}$ and $M\in\mathbb{N}_0$. A subset $L\subset\mathbb{Z}$ is called an AAP (with difference $d$ and bound $M$) if $$L=y+\big( L'\cup L^*\cup L''\big)\subset y+d\mathbb{Z}\,,$$ where $y\in\mathbb{Z}$, $L^*$ is a non-empty arithmetical progression with difference $d$ and $\min L^*=0$, $L'\subset [-M,-1]$, and $L''\subset\sup L^*+[1,M]$ (with the convention that $L''=\emptyset$ if $L^*$ is infinite). We say that $H$ satisfies the [*Structure Theorem for Unions*]{} if there are $d\in\mathbb{N}$ and $M\in\mathbb{N}_0$ such that $\mathcal U_k (H)$ is an AAP with difference $d$ and bound $M$ for all sufficiently large $k\in\mathbb{N}$. If $\Delta (H)$ is finite and the Structure Theorem for Unions holds for some parameter $d\in\mathbb{N}$, then $d = \min \Delta (H)$ ([@F-G-K-T17 Lemma 2.12]).
The Structure Theorem for Unions holds for a wealth of monoids and domains (see [@Ba-Sm18; @Fa-Tr18a; @Tr18a] for recent contributions and see [@F-G-K-T17 Theorem 4.2] for an example where it does not hold). Since it holds for C-monoids ([@Ga-Ge09b]), it holds for the monoid of invertible ideals of orders in number fields. In some special cases (including Krull monoids having prime divisors in all classes) all unions of sets of lengths are intervals, in other words the Structure Theorem for Unions holds with $d=1$ and $M=0$ ([@Ge09a Theorem 3.1.3], [@Ge-Ka-Re15a Theorem 5.8], [@Sm13a]). In Theorem \[theorem 5.2\] we show that the same is true for the monoids of (invertible) ideals of orders in quadratic number fields.
\[proposition 5.1\] Let $p$ be a prime divisor of $f$ and let $N=\sup\{{\rm v}_p(\mathcal{N}(A))\mid A\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))\}$.
1. If $p$ splits, then $\mathcal{U}_{\ell}(\mathcal{I}_p(\mathcal{O}_f))=\mathcal{U}_{\ell}(\mathcal{I}^*_p(\mathcal{O}_f))=\mathbb{N}_{\geq 2}$ for all $\ell\in\mathbb{N}_{\geq 2}$.
2. If $p$ does not split, then $\mathcal{U}_{\ell}(\mathcal{I}_p(\mathcal{O}_f))\cap\mathbb{N}_{\geq\ell}=\mathcal{U}_{\ell}(\mathcal{I}^*_p(\mathcal{O}_f))\cap\mathbb{N}_{\geq\ell}=[\ell,\lfloor\frac{\ell N}{2}\rfloor]$ for all $\ell\in\mathbb{N}_{\geq 2}$.
We prove 1. and 2. simultaneously. By Proposition \[proposition 3.3\].3 we can assume without restriction that $f=p^{{\rm v}_p(f)}$. First we show that both assertions are true for $\ell=2$. It follows from Theorem \[theorem 3.6\] that $[2, N]=[2,2{\rm v}_p(f)]\cup\{{\rm v}_p(\mathcal{N}(A))\mid A\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))\}$. It is obvious that $\mathcal{U}_2(\mathcal{I}^*_p(\mathcal{O}_f))\subset\mathcal{U}_2(\mathcal{I}_p(\mathcal{O}_f))$. It follows from Lemma \[lemma 4.9\] that $\mathcal{U}_2(\mathcal{I}_p(\mathcal{O}_f))\subset [2, N]$.
Let $k\in [2, N]$. It remains to show that $k\in\mathcal{U}_2(\mathcal{I}^*_p(\mathcal{O}_f))$. If $k>2{\rm v}_p(f)$, then there is some $I\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $\mathcal{N}(I)=p^k$. It follows by Proposition \[proposition 3.2\].5 that $I\overline{I}=(p\mathcal{O}_f)^k$, and hence $k\in\mathcal{U}_2(\mathcal{I}^*_p(\mathcal{O}_f))$. Now let $k\leq 2{\rm v}_p(f)$. By Proposition \[proposition 4.8\].1 we can assume without restriction that ${\rm v}_p(f)\geq 2$ and $k\geq 4$.
CASE 1: $d\not\equiv 1\mod 4$ or $(d\equiv 1\mod 4$, $p=2$ and $k\leq 2({\rm v}_2(f)-1))$. We set $a={\rm v}_p(\mathcal{N}_{K/\mathbb{Q}}(p^{k-2}+\tau))$ and $b={\rm v}_p(\mathcal{N}_{K/\mathbb{Q}}(p^{k-2}(p-1)+\tau))$. Observe that if $d\not\equiv 1\mod 4$, then $a,b\geq\min\{2k-4,2{\rm v}_p(f)\}\geq k$. Moreover, if $d\equiv 1\mod 4$, $p=2$ and $k\leq 2({\rm v}_2(f)-1)$, then $a,b\geq\min\{2k-4,2({\rm v}_2(f)-1)\}\geq k$. Set $I=p^a\mathbb{Z}+(p^{k-2}+\tau)\mathbb{Z}$ and $J=p^b\mathbb{Z}+(p^{k-2}(p-1)+\tau)\mathbb{Z}$. Then $I,J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$, $\min\{a,b,{\rm v}_p(p^{k-2}+p^{k-2}(p-1)+\varepsilon)\}=k-1$, and $a+b-2(k-1)>0$. Therefore, there is some $L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $IJ=p^{k-1}L$, and hence $k\in\mathsf{L}(IJ)\subset\mathcal{U}_2(\mathcal{I}^*_p(\mathcal{O}_f))$.
CASE 2: $d\equiv 1\mod 4$ and $p\not=2$. We set $a={\rm v}_p(\mathcal{N}_{K/\mathbb{Q}}(\frac{p^{k-2}-1}{2}+\tau))$ and $b={\rm v}_p(\mathcal{N}_{K/\mathbb{Q}}(\frac{p^{k-2}(p^2+p-1)-1}{2}+\tau))$. Note that $a,b\geq\min\{2k-4,2{\rm v}_p(f)\}\geq k$. Set $I=p^a\mathbb{Z}+(\frac{p^{k-2}-1}{2}+\tau)\mathbb{Z}$ and $J=p^b\mathbb{Z}+(\frac{p^{k-2}(p^2+p-1)-1}{2}+\tau)\mathbb{Z}$. Then $I,J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$, $\min\{a,b,{\rm v}_p(\frac{p^{k-2}-1}{2}+\frac{p^{k-2}(p^2+p-1)-1}{2}+\varepsilon)\}=k-1$, and $a+b-2(k-1)>0$. Consequently, there is some $L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $IJ=p^{k-1}L$, and thus $k\in\mathsf{L}(IJ)\subset\mathcal{U}_2(\mathcal{I}^*_p(\mathcal{O}_f))$.
CASE 3: $d\equiv 1\mod 8$, $p=2$ and $k\in\{2{\rm v}_2(f)-1,2{\rm v}_2(f)\}$. Set $h={\rm v}_2(f)$. If $h=2$, then $k=4$, and hence $k\in\mathcal{U}_2(\mathcal{I}^*_2(\mathcal{O}_f))$ by Proposition \[proposition 4.4\]. Now let $h\geq 3$. Note that $2$ splits. By Theorem \[theorem 3.6\] there are some $I,J,L\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ such that $\mathcal{N}(I)=2^{2h+1}$, $\mathcal{N}(J)=2^{2h+2}$ and $\mathcal{N}(L)=16$. By Proposition \[proposition 3.2\].5 we have $L\overline{L}=16\mathcal{O}_f$, $I\overline{I}=2^{2h+1}\mathcal{O}_f=2^{2h-3}L\overline{L}$ and $J\overline{J}=2^{2h+2}\mathcal{O}_f=2^{2h-2}L\overline{L}$. We infer that $k\in\{2h-1,2h\}\subset\mathcal{U}_2(\mathcal{I}^*_2(\mathcal{O}_f))$.
CASE 4: $d\equiv 5\mod 8$, $p=2$ and $k\in\{2{\rm v}_2(f)-1,2{\rm v}_2(f)\}$. Set $h={\rm v}_2(f)$. If $h=2$, then $k=4$, and thus $k\in\mathcal{U}_2(\mathcal{I}^*_2(\mathcal{O}_f))$ by Proposition \[proposition 4.4\]. Now let $h\geq 3$. Set $A=2^{2h}\mathbb{Z}+(2^{h-1}+\tau)\mathbb{Z}$, $B=2^{2h}\mathbb{Z}+(2^{2h-2}-2^{h-1}+\tau)\mathbb{Z}$, and $C=2^{2h}\mathbb{Z}+(2^{2h-1}-2^{h-1}+\tau)\mathbb{Z}$. Then $A,B,C\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$, $AB=2^{2h-2}I$ and $AC=2^{2h-1}J$ for some $I,J\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$. Therefore, $k\in\{2h-1,2h\}\subset\mathcal{U}_2(\mathcal{I}^*_2(\mathcal{O}_f))$.
So far we have proved that both assertions are true for $\ell=2$. If $p$ splits, then we have $N=\infty$ by Theorem \[theorem 3.6\], and hence $\mathcal{U}_2(\mathcal{I}_p(\mathcal{O}_f))=\mathcal{U}_2(\mathcal{I}^*_p(\mathcal{O}_f))=\mathbb{N}_{\geq 2}$. The first assertion now follows easily by induction on $\ell$. Now let $p$ not split. Then $N<\infty$. Next we show that 2. is true for $\ell=3$.
Since $[3,N+1]=\{1\}+\mathcal{U}_2(\mathcal{I}^*_p(\mathcal{O}_f))\subset\mathcal{U}_3(\mathcal{I}^*_p(\mathcal{O}_f))\cap\mathbb{N}_{\geq 3}\subset\mathcal{U}_3(\mathcal{I}_p(\mathcal{O}_f))\cap\mathbb{N}_{\geq 3}\subset [3,\lfloor\frac{3N}{2}\rfloor]$ by Lemma \[lemma 4.9\] and $N\in\{2{\rm v}_p(f),2{\rm v}_p(f)+1\}$, it remains to show that $N+m\in\mathcal{U}_3(\mathcal{I}^*_p(\mathcal{O}_f))$ for all $m\in [2,{\rm v}_p(f)]$. Let $m\in [2,{\rm v}_p(f)]$. It is sufficient to show that there are some $I,J,L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ such that $IJ=p^mL$ and $\mathcal{N}(L)=p^N$, since then $IJ\overline{L}=p^{N+m}\mathcal{O}_f$ by Proposition \[proposition 3.2\].5, and thus $N+m\in\mathcal{U}_3(\mathcal{I}^*_p(\mathcal{O}_f))$.
CASE 1: $p$ is inert. Observe that $N=2{\rm v}_p(f)$ by Theorem \[theorem 3.6\]. Let $m\in [2,{\rm v}_p(f)]$. First let $p\not=2$. If $d\not\equiv 1\mod 4$, then set $I=p^{2m}\mathbb{Z}+(p^m+\tau)\mathbb{Z}$ and $J=p^{2{\rm v}_p(f)}\mathbb{Z}+(p^{2{\rm v}_p(f)-m}+\tau)\mathbb{Z}$. If $d\equiv 1\mod 4$, then set $I=p^{2m}\mathbb{Z}+(\frac{p^m-1}{2}+\tau)\mathbb{Z}$ and $J=p^{2{\rm v}_p(f)}\mathbb{Z}+(\frac{p^{2{\rm v}_p(f)-m}-1}{2}+\tau)\mathbb{Z}$. In any case we have $I,J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ and $IJ=p^mL$ for some $L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ with $\mathcal{N}(L)=p^N$.
Next let $p=2$. Since $2$ is inert, it follows that $d\equiv 5\mod 8$. If $m<{\rm v}_2(f)-1$, then set $I=2^{2m}\mathbb{Z}+(2^m+\tau)\mathbb{Z}$. If $m={\rm v}_2(f)-1$, then set $I=2^{2m}\mathbb{Z}+\tau\mathbb{Z}$. Finally, if $m={\rm v}_2(f)$, then set $I=2^{2m}\mathbb{Z}+(2^{m-1}+\tau)\mathbb{Z}$. Set $J=2^{2{\rm v}_2(f)}\mathbb{Z}+(2^{{\rm v}_2(f)-1}+\tau)\mathbb{Z}$. Observe that $I,J\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ and $IJ=2^mL$ for some $L\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ with $\mathcal{N}(L)=2^N$.
CASE 2: $p$ is ramified. It follows that $N=2{\rm v}_p(f)+1$ by Theorem \[theorem 3.6\]. Let $m\in [2,{\rm v}_p(f)]$. First let $p\not=2$. Since $p$ is ramified, we have $p\mid d$. If $d\not\equiv 1\mod 4$, then set $I=p^{2m}\mathbb{Z}+(p^m+\tau)\mathbb{Z}$ and $J=p^{2{\rm v}_p(f)+1}\mathbb{Z}+(p^{{\rm v}_p(f)+1}+\tau)\mathbb{Z}$. If $d\equiv 1\mod 4$, then set $I=p^{2m}\mathbb{Z}+(\frac{p^m-1}{2}+\tau)\mathbb{Z}$ and $J=p^{2{\rm v}_p(f)+1}\mathbb{Z}+(\frac{p^{{\rm v}_p(f)+1}-1}{2}+\tau)\mathbb{Z}$. We infer that $I,J\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ and $IJ=p^mL$ for some $L\in\mathcal{A}(\mathcal{I}^*_p(\mathcal{O}_f))$ with $\mathcal{N}(L)=p^N$ in any case.
Now let $p=2$. Since $2$ is ramified, we have $d\not\equiv 1\mod 4$. If $d$ is even or $m<{\rm v}_2(f)$, then set $I=2^{2m}\mathbb{Z}+(2^m+\tau)\mathbb{Z}$. If $d$ is odd and $m={\rm v}_2(f)$, then set $I=2^{2m}\mathbb{Z}+\tau\mathbb{Z}$. If $d$ is even, then set $J=2^{2{\rm v}_2(f)+1}\mathbb{Z}+\tau\mathbb{Z}$. If $d$ is odd, then set $J=2^{2{\rm v}_2(f)+1}\mathbb{Z}+(2^{{\rm v}_2(f)}+\tau)\mathbb{Z}$. In any case we have $I,J\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ and $IJ=2^mL$ for some $L\in\mathcal{A}(\mathcal{I}^*_2(\mathcal{O}_f))$ with $\mathcal{N}(L)=2^N$.
Finally, we prove the second assertion by induction on $\ell$. Let $\ell\in\mathbb{N}_{\geq 2}$ and let $H\in\{\mathcal{I}_p(\mathcal{O}_f),\mathcal{I}^*_p(\mathcal{O}_f)\}$. Without restriction we can assume that $\ell\geq 4$. We infer by the induction hypothesis that $(\mathcal{U}_{\ell-2}(H)\cap\mathbb{N}_{\geq\ell-2})+\mathcal{U}_2(H)=[\ell-2,\lfloor\frac{(\ell-2)N}{2}\rfloor]+[2,N]=[\ell,\lfloor\frac{\ell N}{2}\rfloor]$. Observe that $(\mathcal{U}_{\ell-2}(H)\cap\mathbb{N}_{\geq\ell-2})+\mathcal{U}_2(H)\subset\mathcal{U}_{\ell}(H)\cap\mathbb{N}_{\geq\ell}$. It follows by Lemma \[lemma 4.9\] that $\mathcal{U}_{\ell}(H)\cap\mathbb{N}_{\geq\ell}\subset [\ell,\lfloor\frac{\ell N}{2}\rfloor]$, and thus $\mathcal{U}_{\ell}(H)\cap\mathbb{N}_{\geq\ell}=[\ell,\lfloor\frac{\ell N}{2}\rfloor]$.
\[theorem 5.2\] Let $\mathcal{O}$ be an order in a quadratic number field $K$ with conductor $f\mathcal{O}_K$ for some $f\in\mathbb{N}_{\geq 2}$.
1. If $f$ is divisible by a split prime, then $\mathcal{U}_k(\mathcal{I}(\mathcal{O}))=\mathcal{U}_k(\mathcal{I}^*(\mathcal{O}))=\mathbb{N}_{\geq 2}$ for all $k\in\mathbb{N}_{\geq 2}$.
2. Suppose that $f$ is not divisible by a split prime and set $M=\max\{{\rm v}_p(f)\mid p\in\mathbb{P}\}$. Then $\mathcal{U}_k(\mathcal{I}(\mathcal{O}))=\mathcal{U}_k(\mathcal{I}^*(\mathcal{O}))$ is a finite interval for all $k\in\mathbb{N}_{\geq 2}$, and for their maxima we have[:]{}
1. If ${\rm v}_q(f)=M$ for a ramified prime $q$, then $\rho_k(\mathcal{I}(\mathcal{O}))=\rho_k(\mathcal{I}^*(\mathcal{O}))=kM+\lfloor\frac{k}{2}\rfloor$ for all $k\in\mathbb{N}_{\geq 2}$ and $\rho(\mathcal{I}(\mathcal{O}))=\rho(\mathcal{I}^*(\mathcal{O}))=M+\frac{1}{2}$.
2. If ${\rm v}_q(f)<M$ for all ramified primes $q$, then $\rho_k(\mathcal{I}(\mathcal{O}))=\rho_k(\mathcal{I}^*(\mathcal{O}))=kM$ for all $k\in\mathbb{N}_{\geq 2}$ and $\rho(\mathcal{I}(\mathcal{O}))=\rho(\mathcal{I}^*(\mathcal{O}))=M$.
1\. Let $f$ be divisible by a split prime $p$ and let $k\in\mathbb{N}_{\geq 2}$. Since $\mathcal{I}^*_p(\mathcal{O})$ is a divisor-closed submonoid of $\mathcal{I}^*(\mathcal{O})$ and $\mathcal{I}_p(\mathcal{O})$ is a divisor-closed submonoid of $\mathcal{I}(\mathcal{O})$, it follows from Proposition \[proposition 5.1\].1 that $\mathcal{U}_k(\mathcal{I}(\mathcal{O}))=\mathcal{U}_k(\mathcal{I}^*(\mathcal{O}))=\mathbb{N}_{\geq 2}$.
2\. Let $k\in\mathbb{N}_{\geq 2}$ and $\ell\in\mathcal{U}_k(\mathcal{I}(\mathcal{O}))$. There are $I_i\in\mathcal{A}(\mathcal{I}(\mathcal{O}))$ for each $i\in [1,k]$ and $J_j\in\mathcal{A}(\mathcal{I}(\mathcal{O}))$ for each $j\in [1,\ell]$ such that $\prod_{i=1}^k I_i=\prod_{j=1}^{\ell} J_j$. Note that $\sqrt{I_i},\sqrt{J_j}\in\mathfrak{X}(\mathcal{O})$ for all $i\in [1,k]$ and $j\in [1,\ell]$. For $P\in\mathfrak{X}(\mathcal{O})$ set $k_P=|\{i\in [1,k]\mid\sqrt{I_i}=P\}|$ and $\ell_P=|\{j\in [1,\ell]\mid\sqrt{J_j}=P\}|$. If $p$ is a prime divisor of $f$, then set $k_p=k_{P_{f,p}}$ and $\ell_p=\ell_{P_{f,p}}$. Observe that $k=\sum_{P\in\mathfrak{X}(\mathcal{O})} k_P$ and $\ell=\sum_{P\in\mathfrak{X}(\mathcal{O})}\ell_P$. Recall that the $P$-primary components of $\prod_{i=1}^k I_i$ are uniquely determined, and thus $\ell_P\in\mathcal{U}_{k_P}(\mathcal{I}_P(\mathcal{O}))$ for all $P\in\mathfrak{X}(\mathcal{O})$. If $P\in\mathfrak{X}(\mathcal{O})$ does not contain the conductor, then $\mathcal{I}_P(\mathcal{O})$ is factorial, and hence $\ell_P=k_P$. Also note that if $P\in\mathfrak{X}(\mathcal{O})$ and $k_P\leq 1$, then $\ell_P=k_P$. If $p$ is an inert prime that divides $f$, then it follows from Proposition \[proposition 5.1\].2 and Theorem \[theorem 3.6\] that $\rho_r(\mathcal{I}_p(\mathcal{O}))=\rho_r(\mathcal{I}^*_p(\mathcal{O}))=r{\rm v}_p(f)$ for all $r\in\mathbb{N}_{\geq 2}$. We infer again by Proposition \[proposition 5.1\].2 and Theorem \[theorem 3.6\] that $\rho_r(\mathcal{I}_p(\mathcal{O}))=\rho_r(\mathcal{I}^*_p(\mathcal{O}))=r{\rm v}_p(f)+\lfloor\frac{r}{2}\rfloor$ for all ramified primes $p$ that divide $f$ and all $r\in\mathbb{N}_{\geq 2}$.
CASE 1: ${\rm v}_q(f)=M$ for some ramified prime $q$. If $P\in\mathfrak{X}(\mathcal{O})$, then $\ell_P\leq k_PM+\lfloor\frac{k_P}{2}\rfloor$.
Consequently, $\ell=\sum_{P\in\mathfrak{X}(\mathcal{O})}\ell_P\leq (\sum_{P\in\mathfrak{X}(\mathcal{O})} k_P)M+\sum_{P\in\mathfrak{X}(\mathcal{O})}\lfloor\frac{k_P}{2}\rfloor\leq kM+\lfloor\frac{k}{2}\rfloor$. In particular, $\rho_k(\mathcal{I}(\mathcal{O}))\leq kM+\lfloor\frac{k}{2}\rfloor=\max\{\rho_k(\mathcal{I}^*_p(\mathcal{O}))\mid p\in\mathbb{P},p\mid f\}\leq\rho_k(\mathcal{I}^*(\mathcal{O}))\leq\rho_k(\mathcal{I}(\mathcal{O}))$. This implies that $\rho_k(\mathcal{I}(\mathcal{O}))=\rho_k(\mathcal{I}^*(\mathcal{O}))=\max\{\rho_k(\mathcal{I}^*_p(\mathcal{O}))\mid p\in\mathbb{P},p\mid f\}=kM+\lfloor\frac{k}{2}\rfloor$.
CASE 2: ${\rm v}_q(f)<M$ for all ramified primes $q$. Note that $\ell_p\leq k_p {\rm v}_p(f)+\lfloor\frac{k_p}{2}\rfloor\leq k_pM$ for all ramified primes $p$ that divide $f$. Therefore, $\ell_P\leq k_PM$ for all $P\in\mathfrak{X}(\mathcal{O})$. This implies that $\ell=\sum_{P\in\mathfrak{X}(\mathcal{O})}\ell_P\leq (\sum_{P\in\mathfrak{X}(\mathcal{O})} k_P)M=kM$. We infer that $\rho_k(\mathcal{I}(\mathcal{O}))\leq kM=\max\{\rho_k(\mathcal{I}^*_p(\mathcal{O}))\mid p\in\mathbb{P},p\mid f\}\leq\rho_k(\mathcal{I}^*(\mathcal{O}))\leq\rho_k(\mathcal{I}(\mathcal{O}))$, and thus $\rho_k(\mathcal{I}(\mathcal{O}))=\rho_k(\mathcal{I}^*(\mathcal{O}))=\max\{\rho_k(\mathcal{I}^*_p(\mathcal{O}))\mid p\in\mathbb{P},p\mid f\}=kM$.
By Proposition \[proposition 5.1\].2, we obtain that $\mathcal{U}_k(\mathcal{I}(\mathcal{O}))\cap\mathbb{N}_{\geq k}=\mathcal{U}_k(\mathcal{I}^*(\mathcal{O}))\cap\mathbb{N}_{\geq k}$ is a finite interval. Since the last assertion holds for every $k\in\mathbb{N}_{\geq 2}$, we infer that $\mathcal{U}_k(\mathcal{I}(\mathcal{O}))=\mathcal{U}_k(\mathcal{I}^*(\mathcal{O}))$ is a finite interval for all $k\in\mathbb{N}_{\geq 2}$. If ${\rm v}_q(f)=M$ for some ramified prime $q$, then $$\rho(\mathcal{I}(\mathcal{O}))=\rho(\mathcal{I}^*(\mathcal{O}))=\lim_{k\rightarrow\infty}\frac{\rho_k(\mathcal{I}(\mathcal{O}))}{k}=\lim_{k\rightarrow\infty} M+\frac{1}{k}\left\lfloor\frac{k}{2}\right\rfloor=M+\frac{1}{2}.$$ Finally, let ${\rm v}_q(f)<M$ for all ramified primes $q$. Then $$\rho(\mathcal{I}(\mathcal{O}))=\rho(\mathcal{I}^*(\mathcal{O}))=\lim_{k\rightarrow\infty}\frac{\rho_k(\mathcal{I}(\mathcal{O}))}{k}=\lim_{k\rightarrow\infty}\frac{kM}{k}=M. \qedhere$$
In a final remark we gather what is known on further arithmetical invariants of monoids of ideals of orders in quadratic number fields.
\[remark 5.3\] Let $\mathcal{O}$ be an order in a quadratic number field $K$ with conductor $f\mathcal{O}_K$ for some $f\in\mathbb{N}_{\geq 2}$.
1\. The monotone catenary degree of $\mathcal I^*(\mathcal{O})$ is finite by [@Ge-Re18d Corollary 5.14]. Precise values for the monotone catenary degree are available so far only in the seminormal case ([@Ge-Ka-Re15a Theorem 5.8]).
2\. The tame degree of $\mathcal I^*(\mathcal{O})$ is finite if and only if the elasticity is finite if and only if $f$ is not divisible by a split prime. This follows from Equations \[equation 3\] and \[equation 4\], Theorem \[theorem 5.2\], and from [@Ge-HK06a Theorem 3.1.5]. Precise values for the tame degree are not known so far.
3\. For an atomic monoid $H$, the set $\{\rho(L)\mid L\in\mathcal{L}(H)\}\subset\mathbb{Q}_{\ge 1}$ of all elasticities was first studied by Chapman et al. and then it found further attention by several authors (e.g., [@Ba-Ne-Pe17a; @Ch-Ho-Mo06], [@Ge-Sc-Zh17b Theorem 5.5], [@Ge-Zh19a; @Zh19a]). We say that $H$ is [*fully elastic*]{} if for every rational number $q$ with $1 < q < \rho (H)$ there is an $L\in\mathcal{L}(H)$ with $\rho(L)=q$. Since $\mathcal I^*(\mathcal{O})$ is cancellative and has a prime element, it is fully elastic by [@B-C-C-K-W06 Lemma 2.1]. Since $\mathcal I^*(\mathcal{O})\subset\mathcal I(\mathcal{O})$ is divisor-closed and $\rho(\mathcal{I}(\mathcal{O}))=\rho(\mathcal{I}^*(\mathcal{O}))$ by Theorem \[theorem 5.2\], it follows that $\mathcal I(\mathcal{O})$ is fully elastic.
4\. For an atomic monoid $H$, let $$\daleth^*(H)=\{\min(L\setminus\{2\})\mid 2\in L\in\mathcal{L}(H)\ \text{with}\ |L|>1\}\subset\mathbb{N}_{\ge 3}\,.$$ By definition, we have $\daleth^*(H)\subset 2+\Delta (H)$ and in [@Fa-Ge17a; @Ge-Zh19a] the invariant $\daleth^*(H)$ was used as a tool to study $\Delta(H)$. Proposition \[proposition 4.1\].4 shows that, both for $H=\mathcal I(\mathcal{O})$ and for $H=\mathcal I^*(\mathcal{O})$, we have $\max\daleth^*(H)=2+\max\Delta(H)$.
[**Acknowledgements.**]{} We would like to thank the referee for carefully reading the manuscript and for many suggestions and comments that improved the quality of this paper and simplified the proof of Theorem \[theorem 3.6\].
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[^1]: This work was supported by the Austrian Science Fund FWF, Project Numbers J4023-N35 and P28864-N35
|
---
abstract: 'We conduct a joint X-ray and weak-lensing study of four relaxed galaxy clusters (Hydra A, A478, A1689 and A1835) observed by both [*Suzaku*]{} and Subaru out to virial radii, with an aim to understand recently-discovered unexpected feature of the intracluster medium (ICM) in cluster outskirts. We show that the average hydrostatic-to-lensing total mass ratio for the four clusters decreases from $\sim 70\%$ to $\sim 40\%$ as the overdensity contrast decreases from 500 to the virial value. The average gas mass fraction from lensing total mass estimates increases with cluster radius and agrees with the cosmic mean baryon fraction within the virial radius, whereas the X-ray-based gas fraction considerably exceeds the cosmic values due to underestimation of the hydrostatic mass. We also develop a new advanced method for determining normalized cluster radial profiles for multiple X-ray observables by simultaneously taking into account both their radial dependence and multivariate scaling relations with weak-lensing masses. Although the four clusters span a range of halo mass, concentration, X-ray luminosity and redshift, we find that the gas entropy, pressure, temperature and density profiles are all remarkably self-similar when scaled with the weak-lensing $M_{200}$ mass and $r_{200}$ radius. The entropy monotonically increases out to $\sim 0.5r_{200}\sim r_{1000}$ following the accretion shock heating model $K(r)\propto r^{1.1}$, and flattens at ${\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}0.5r_{200}$. The universality of the scaled entropy profiles indicates that the thermalization mechanism over the entire cluster region ($>0.1r_{200}$) is controlled by gravitation in a common way for all clusters, although the heating efficiency in the outskirts needs to be modified from the standard $r^{1.1}$ law. The bivariate scaling functions of the gas density and temperature reveal that the flattening of the outskirts entropy profile is caused by the steepening of the temperature, rather than the flattening of the gas density.'
author:
- 'N. <span style="font-variant:small-caps;">Okabe</span>, K. <span style="font-variant:small-caps;">Umetsu</span>, T. <span style="font-variant:small-caps;">Tamura</span>, Y. <span style="font-variant:small-caps;">Fujita</span>, M. <span style="font-variant:small-caps;">Takizawa</span>, Y. -Y. <span style="font-variant:small-caps;">Zhang</span>, K. <span style="font-variant:small-caps;">Matsushita</span>, T. <span style="font-variant:small-caps;">Hamana</span>, Y. <span style="font-variant:small-caps;">Fukazawa</span>, T. <span style="font-variant:small-caps;">Futamase</span>, M. <span style="font-variant:small-caps;">Kawaharada</span>, S. <span style="font-variant:small-caps;">Miyazaki</span>, Y. <span style="font-variant:small-caps;">Mochizuki</span>, K. <span style="font-variant:small-caps;">Nakazawa</span>, T. <span style="font-variant:small-caps;">Ohashi</span>, N. <span style="font-variant:small-caps;">Ota</span>, T. <span style="font-variant:small-caps;">Sasaki</span>, K. <span style="font-variant:small-caps;">Sato</span>, and S. I. <span style="font-variant:small-caps;">Tam</span>'
bibliography:
- 'my.bib'
- 'hydraa.bib'
title: 'Universal Profiles of the Intracluster Medium from Suzaku X-Ray and Subaru Weak Lensing Obesrvations [^1]'
---
\[firstpage\]
Introduction
============
Recent studies with the [*Suzaku*]{} X-ray satellite [@Mitsuda07] have reported detections of very faint X-ray emission in the outskirts of galaxy clusters thanks to its low and stable particle background. These Suzaku observations revealed unexpected observational features of the intracluster medium (ICM) in the cluster outskirts [e.g. @Fujita08; @Bautz09; @Kawaharada10; @Hoshino10; @Simionescu11; @Sato12; @Walker12a; @Walker12b; @Walker13; @Ichikawa13; @Reiprich13]: The observed gas temperature sharply declines beyond about half the cluster virial radius. In the cluster outskirts, the gas temperature is at most 20-50% of those at intermediate radii (from one-fourth to half the virial radius). The gas entropy profile $K(r)=k_B T(r)/n_e^{2/3}(r)$ flattens beyond about half the virial radius, in contrast to predictions of accretion shock-heating models [e.g. @Tozzi01; @Ponman03]. If all kinetic energy of the infalling gas is instantly thermalized by the accretion shock, the entropy profile is predicted to increase with cluster radius as $K(r)\propto r^{1.1}$ [@Tozzi01], as supported by [*Chandra*]{} and [*XMM-Newton*]{} observations of cluster central regions [e.g. @Cavagnolo09; @Pratt10]. In fact, the entropy in the interior region is enhanced compared to the $r^{1.1}$, as also shown in [@Walker13]. The outskirts entropy from [*Suzaku*]{} is systematically lower than these model predictions as well as those extrapolated from observations in the central regions. These pieces of evidence indicate that the majority of electrons in the cluster outskirts are not yet thermalized, so that the thermal pressure in the outskirts is not sufficient to fully balance with the total gravity of the cluster. Furthermore, [@Kawaharada10] and [@Ichikawa13] showed that the anisotropic distributions of gas temperature and entropy in the outskirts are correlated with large-scale structure of galaxies outside the central clusters. [@Sato12] found that the gas density distribution is correlated with large-scale structure of galaxies. These spatial correlations between thermodynamic properties of the ICM and large-scale environments indicate that the physical processes in the cluster outskirts are influenced by surrounding cosmological environments.
Several interpretations have been proposed to explain the presence of low entropy gas in the ICM. For example, hot ions could provide main thermal-pressure support [@Hoshino10] if the electron in the outskirts are not yet fully thermalized. This could be possible because the timescale of thermal equilibrium of the electrons by Coulomb collisions is much longer than that of the ions. However, this scenario ignores the fact that the timescale for electrons to achieve thermal equilibrium governed by wave-particle interactions of plasma kinetic instabilities is much shorter than that by Coulomb collisions. Alternatively, the kinetic energy of bulk and/or turbulent motions could contribute to some fraction of the total pressure to fully balance with the gravity [@Kawaharada10]. [@Lapi10] and [@Cavaliere11] proposed that the outer slope of the potential becomes shallower due to the acceleration of cosmic expansion and hence the efficiency of accretion shock heating is weakened. [@Fujita13b] proposed that the accretion of cosmic-rays at cluster formation shocks consumes kinetic energy of infalling gas and decreases the entropy of the downstream gas. A high degree of gas clumpiness in cluster outskirts could lead to an overestimate of the observed gas density, causing the apparent flattening of the derived entropy profile [@Nagai11]. @Simionescu11 found that the gas mass fraction in the Perseus cluster based on hydrostatic mass estimation exceeds the cosmic baryon fraction within the virial radius, and attributed this apparent excess to gas clumpiness. However, their X-ray-only analysis suffers from the inherent assumption of hydrostatic equilibrium. In particular, in light of the observed low gas temperature and entropy in cluster outskirts, the ICM there is expected to be out of hydrostatic equilibrium. Therefore, accurate and direct cluster mass determinations without the hydrostatic equilibrium assumption are essential for understanding the physical state of the ICM in cluster outskirts.
Weak gravitational lensing techniques are complementary to X-ray observations because weak-lensing mass estimates do not require any assumptions on cluster dynamical states. A coherent distortion pattern of background galaxy shapes caused by the gravitational potential of clusters enables us to recover the cluster mass distribution [e.g., @Bartelmann01]. A comparison of X-ray and weak-lensing cluster mass estimates provides a stringent test of the level of hydrostatic equilibrium [e.g., @Zhang10; @Kawaharada10; @Mahdavi13], which is important for empirically understanding the systematic bias in cluster mass estimates and for constructing well-calibrated cluster mass-observable relations for cluster cosmology [e.g. @Vikhlinin09b]. In particular, multi-wavelength cluster datasets of high quality covering the entire cluster region are crucial for a diagnostic of the ICM states out to the virial radius.
In the present paper we compile a sample of four massive clusters (Hydra A, A478, A1689, A1835) which have been deeply observed out to the virial radius with the [*Suzaku*]{} X-ray satellite and the Subaru Telescope, and perform a joint X-ray and weak-lensing analysis to study the relations between the total mass and ICM in this cluster sample. The four clusters have different properties of the X-ray luminosity ($L_X$), average temperature ($\langle k_B
T\rangle$) and redshift ($z$), as summarized in Table \[tab:target\]. The X-ray luminosites differ by one order of magnitude and the redshifts are at $0.05{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}r{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}0.25$. The typical integration time with [*Suzaku*]{} to detect faint X-ray emission from cluster outskirts is much longer than that for cluster central regions and multi-pointing Subaru observations are required to cover the entire region of nearby [*Suzaku*]{} targets. Therefore our current sample is limited only to four objects. Weak-lensing and X-ray analyses are briefly described in Section \[sec:data\]. In Section \[sec:compare\], we compare our weak-lensing mass measurements with X-ray properties of the ICM as function of the cluster overdensity radius. The results are summarized in Section \[sec:sum\]. In the paper, we use $\Omega_{m,0}=0.27$, $\Omega_{\Lambda}=0.73$ and $H_0=70h_{70}~{\rm km\,s^{-1}Mpc^{-1}}$.
Data Analysis {#sec:data}
=============
Subaru lensing Analysis
-----------------------
We carried out weak-lensing analyses of individual clusters in our sample using wide-field multi-band observations taken with the Suprime-Cam [@Miyazaki02] at the prime focus of the 8.2-m Subaru Telescope. We securely selected background source galaxies in order to avoid contamination by unlensed cluster galaxies. We measure the mass $M_\Delta$ at overdensities of $\Delta=2500,1000,500,200$ and the virial overdensity $\Delta_{\rm vir}\sim100-110$ [@Nakamura97]. Here, $M_\Delta$ represents the mass enclosed within a sphere of radius $r_\Delta$ inside which the mean interior density is $\Delta$ times the critical mass density, $\rho_{\rm cr}(z)$, at the redshift, $z$.
For A1689 ($z=0.1832$), we employ the nonparametrically-deprojected spherical mass model from a joint strong-lensing, weak-lensing shear and magnification analysis described in [@Kawaharada10] and @Umetsu08. We fit the tangential (reduced) shear profile for the other three clusters with a parametrized mass model. Here, the tangential distortion signal, the mean ellipticity of background galaxies tangential to the cluster center, is obtained as a function of projected distance from the brightest cluster galaxy (BCG).
Our weak-lensing analysis with new Subaru observations of Hydra A ($z=0.0538$) and A478 ($z=0.0881$) is described in details in [@Okabe14b]. As for A1835 ($z=0.25280$), we have reanalyzed the Subaru data using new background selection of [@Okabe13], and measured the mass profile in combination with strong lensing data [@Richard10]. We employ the universal mass profile of @NFW96 [@NFW97] which is empirically motivated by numerical simulations of collsionless cold dark matter. The NFW density profile is given by the following form: $$\rho_{\rm NFW}(r)=\frac{\rho_s}{(r/r_s)(1+r/r_s)^2},
\label{eq:rho_nfw}$$ where $\rho_s$ is the central density parameter and $r_s$ is the scale radius. The halo concentration is defined by $c_\Delta=r_\Delta/r_s$. The resulting $M_{200}$ is listed in Table \[tab:target\].
Since the uncertainty of mass estimates by joint strong- and weak-lensing measurements is much smaller than those by weak-lensing-only measurements, we shall apply unweighted averaging in our cluster ensemble analysis.
--------- ---------- -------------------------- ------------------------- --------------------------------- ----------------- --------------------------------
Name $z$ $L_X$ $\langle k_B T \rangle$ $M_{200}$ [*Suzaku*]{} [*XMM-Newton*]{}/[*Chandra*]{}
\[$10^{45}$ergs$^{-1}$\] \[keV\] \[$h_{70}^{-1}10^{14}M_\odot$\]
Hydra A $0.0538$ $0.27$ $3.0$ $3.72_{-1.44}^{+2.11}$ [@Sato12] [@David01]
A 478 $0.0881$ $0.72$ $7.0$ $13.05_{-3.30}^{+4.12}$ [@Mochizuki14] [@Sanderson05]
A 1689 $0.1832$ $1.25$ $9.3$ $16.73_{-3.44}^{+4.88}$ [@Kawaharada10] [@Zhang07]
A 1835 $0.2528$ $1.97$ $8.0$ $10.35_{-2.40}^{+2.80}$ [@Ichikawa13] [@Zhang07]
--------- ---------- -------------------------- ------------------------- --------------------------------- ----------------- --------------------------------
Suzaku X-ray Analysis
---------------------
The [*Suzaku*]{} studies for Hydra A, A478, A1689 and A1835 are described in details in [@Sato12], [@Mochizuki14], [@Kawaharada10] and [@Ichikawa13], respectively. We here briefly summarize the analyses. Low and stable particle background of [*Suzaku*]{} is powerful to detect diffuse faint X-ray emission beyond about half of the virial radius. On the other hand, high angular resolutions of [*Chandra*]{} and [*XMM-Newton*]{} have advantages to measure the ICM properties within about half of the virial radius. A joint X-ray study, combined with these datasets in different sensitivities and resolutions, well constrains the temperature and density profiles from the cores out to virial radii [@Sato12; @Ichikawa13; @Mochizuki14]. We also found that the X-ray observables (the density, temperature, pressure and entropy) derived by [*Suzaku*]{}, [*XMM-Newton*]{} and [*Chandra*]{} agree with each other at overlapping radii [@Zhang07; @David01; @Sanderson05]. The X-ray surface brightness profile of [*Suzaku*]{} is consistent with the [*ROAST*]{} flux. The thermal pressure out to virial radius measured by [*Suzaku*]{} is also in good agreement with the [*Planck*]{} flux of the Sunyaev-Zel’dovich (SZ) effect on the cosmic microwave background (CMB) [@Ichikawa13; @Mochizuki14]. We assume the spherical distribution to derive hydrostatic equilibrium masses and gas masses from best-fit functions of the temperature and gas density profiles, as described in our earlier papers [@Kawaharada10; @Sato12; @Ichikawa13; @Mochizuki14]. In the cluster outskirts, we use azimuthal averages of [*Suzaku*]{} X-ray observables for A478, A1689 and A1835. As for Hydra A, we use unweighted averages of observables in two directions of an over-dense filamentary structure of galaxies and a low density void environment outside the cluster. In the central regions, X-ray observables measured by the [*XMM-Newton*]{} or [*Chandra*]{} are added to the [*Suzaku*]{} data. The references are listed in Table \[tab:target\]. We take into account only statistical errors.
Comparison of Lensing and X-ray measurements {#sec:compare}
============================================
Comparison of Hydrostatic Equilibrium Mass and Weak-lensing Mass {#subsec:Mratio}
----------------------------------------------------------------
Weak-lensing mass measurements do not require the hydrostatic equilibrium assumption, and are complementary to X-ray measurements. It is of critical importance to compare these two independent mass estimates for understanding the physical state of the ICM as well as for examining the degree of hydrostatic equilibrium [@Kawaharada10; @Sato12; @Ichikawa13; @Mochizuki14]. Our unique dataset of wide-field [*Suzaku*]{} and Subaru observations enables us to directly compare X-ray and lensing masses from the cluster core to the virial radius.
Figure \[fig:Mratio\_ave\] shows the hydrostatic-to-lensing total mass ratio as a function of overdensity $\Delta$ with $\Delta=2500,1000,500,200$ and $\Delta_{\rm vir}$. To avoid aperture-induced errors in enclosed-mass measurements, we calculate the X-ray mass inside the same radius determined by lensing analysis. Overall, the mass-ratio profiles decrease outward in a similar manner. We show in the figure that the unweighted average of the mass ratios (large black circles) monotonically decreases as the overdensity $\Delta$ decreases. We fit the average profile with the functional form of $\ln(\langle M_{\rm
H.E.}/M_{\rm WL}\rangle )=A+B\ln(\Delta)$. The best-fit slope is $B=0.22\pm0.07$. The overdensity dependence is thus detected at the $3\sigma$ level. The mass discrepancy is negligible at a high overdensity of $\Delta=2500$. However, we find that the X-ray hydrostatic mass can only account for $\sim70$% and $\sim40$% of the lensing mass at $\Delta=500$ and $\Delta_{\rm vir}$, respectivly.
Similar results were found inside $r_{500}$ by previous observational studies and numerical simulations. [@Mahdavi13] compared their weak-lensing masses with X-ray masses for 50 clusters, finding that the X-ray to weak-lensing total mass ratio for their full sample is $0.92\pm0.05$ at $\Delta=2500$, $0.89\pm0.05$ at $\Delta=1000$, and $0.88\pm0.05$ at $\Delta=500$, respectively, Hydrodynamical numerical simulations [@Nagai07; @Piffaretti08; @Lau09; @Lau13; @Nelson14] found that the hydrostatic-to-true total mass ratio within $r_{500}$ is more or less comparable with our results. On the other hand, the hydrostatic-to-true mass ratio within $r_{200}$ is found to be $\sim80-90\%$, which is much larger than our results. Hence, there is a substantial discrepancy between our results and numerical simulations in the cluster outskirts at low overdensities $\Delta{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}200$. Such a large bias can be caused if the hydrostatic cumulative mass unphysically decreases with radius at cluster outskirts [e.g. @Kawaharada10; @Sato12]. We will discuss the possible deviations of X-ray observables in Section \[subsec:joint\].
Measuring and quantifying any bias in hydrostatic mass estimates is one of the key issues for cluster cosmology. Lensing observations suffer from noisy projection effects, but can provide unbiased cluster mass estimates in a statistical sense if one can avoid an orientating bias [e.g., @Meneghetti2014]. In particular, stacked cluster lensing measurements [@Okabe10b; @Okabe13; @Oguri12; @Umetsu11; @Umetsu14], which are insensitive to systematics due to projection effects, allow us to determine the representative mass profile for a cluster sample.
Our cluster sample is small but solely defined by the current availability of both [*Suzaku*]{} and Subaru observations. We note that, although A1689 is a well-known strong-lensing cluster, the overall trends in the observed mass ratios are common to all clusters. Our results suggest that the degree of breakdown of the hydrostatic equilibrium assumption can vary substantially with cluster radius, indicating that an accurate characterization of the radial-dependent mass bias is crucial for cluster cosmology. This has direct consequences for the origin and degree of the apparent [*tension*]{} between the number counts of SZ clusters detected by [*Planck*]{} compared to those predicted by the [*Planck*]{} CMB cosmology [@Planck13ClusterCosmology]. In their cluster cosmology analysis, @Planck13ClusterCosmology adopted [*XMM-Newton*]{}-based hydrostatic mass estimates and calibrated their cluster masses $M_{500}$ assuming a constant bias of 20%. Their assumed bias is smaller than but compatible with our results.[^2] Note that, since the size of our sample is small, it is essential to conduct further systematic studies with larger samples of X-ray and weak-lensing data covering the entire cluster region.
![X-ray hydrostatic to weak-lensing total mass ratios as a function of the density contrast $\Delta$. Red diamonds, blue triangle, green squares, magenta circles and large black circles denote the mass ratios for A478, Hydra A, A1689, A1835 and the unweighted average of the 4 clusters, respectively. From left to right, the data points with error bars represent the mass ratios at $\Delta=2500,1000,500,200$ and $\Delta_{\rm vir}$, respectively. The mass ratios for the clusters are horizontally offset for visual clarity. The black-solid curve and gray-solid area are the best-fit profile and $1\sigma$ uncertainty. []{data-label="fig:Mratio_ave"}](fig1.eps){width="\hsize"}
Gas Fraction
------------
We have computed the gas mass fractions for the four clusters, $f_{\rm gas}(<r)=M_{\rm gas}(<r)/M_{\rm WL}(<r)$, as a function of the density contrast $\Delta$ (Figure \[fig:fgas\_ave\]). Here, the gas mass $M_{\rm gas}(<r)$ is measured inside the aperture radius $r_\Delta$ determined by weak-lensing analysis. Compared to the total mass ratio, intrinsic scatter in gas fraction for individual clusters is large.
Using the lensing total mass estimates, we find that the gas fractions within the virial radius are lower than or comparable to the cosmic mean baryon fraction [@WMAP09; @Planck13Cosmology], finding no evidence for the excess gas fraction relative to the cosmic value.
An apparent baryon excess within $r_{200}$ was reported in the Perseus cluster on the basis of [*Suzaku*]{} hydrostatic mass estimates [@Simionescu11]. However, as discussed in Section \[subsec:Mratio\], since the X-ray hydrostatic masses are underestimated especially at large cluster radii, the X-ray-based gas fractions can be largely overestimated and exceed the cosmic mean baryon fraction [e.g., @Sato12; @Ichikawa13; @Mochizuki14]. Weak-lensing mass determinations are needed to avoid such systematics.
Figure \[fig:fgas\_ave\] shows that the unweighted average of the gas fractions increases as the overdensity decreases. We fit the average gas fraction profile with the functional form of $\langle f_{\rm gas}\rangle=A+B\ln(\Delta)$. The best-fit normalization and slope are $A=0.250\pm0.065$ and $B=-0.018\pm0.009$, respectively. The average gas fraction within the virial radius agrees within errors with the cosmic mean baryon fractions from the [*WMAP*]{} [@WMAP09] and [*Planck*]{} [@Planck13Cosmology] experiments.
![Gas mass fraction, $M_{\rm gas}/M_{\rm WL}$, from weak-lensing total mass estimates, shown as a function of the density contrast $\Delta$. Red diamonds, blue triangle, green squares, magenta circles and large black circles denote the mass ratios for A478, Hydra A, A1689, A1835 and the unweighted average of the 4 clusters, respectively. The data points for the 4 clusters at each density are horizontally offset for visual clarity. The horizontal-solid lines with dotted error bars are the cosmic mean baryon fractions from [*WMAP*]{} [@WMAP09] and [*Planck*]{} [@Planck13Cosmology] with their respective $1\sigma$ uncertainties. The black-solid curve and gray-solid area show the best-fit profile and the $1\sigma$ uncertainty interval.[]{data-label="fig:fgas_ave"}](fig2.eps){width="\hsize"}
Outskirts Entropy {#subsec:Kout}
-----------------
The gas entropy profiles for relaxed clusters are observed to be fairly universal over a wide radial range [@Walker12b; @Sato12], increasing following a power-law ($\propto r^{1.1}$) out to intermediate radii and then flattening off from $\sim 0.5r_{\rm vir}$ to $r_{\rm vir}$. Accordingly, the observed entropy in the outskirts is significantly lower than that extrapolated with the power-law form of $K(r)\propto r^{1.1}$. The scaled entropy profiles from [*Suzaku*]{} observations are well fitted by a universal function in which the entropy flattening and turnover are characterized by the virial radius or $r_{200}$ [@Walker12b; @Sato12], suggesting that there is a physical correlation between the outskirts entropy and the virial mass.
Here we investigate a correlation between the outskirts entropy and the virial mass by using our lensing-derived virial mass estimates. We adopt the average entropy $K(r)=k_B T/n_e^{2/3}$ in the radial range $r_{500}-r_{\rm vir}$ as the outskirts entropy, $K_{\rm out}$. Since the observed entropy profiles in the outskirts are fairly flat, the results here are insensitive to the choice of the radial range. We fit the functional form $\ln(K_{\rm out})=A+B\ln(M_{\rm vir}E(z))$ for our sample of four clusters, where $E(z)=(\Omega_{m,0}(1+z)^3+\Omega_\Lambda)^{1/2}$ is the dimensionless Hubble expansion rate, $A$ is the normalization and $B$ is the slope. The best-fit normalization and slope are $A=5.40\pm0.57$ and $B=0.69\pm0.25$, respectively. We find a tight correlation between $K_{\rm out}$ and $E(z)M_{\rm vir}$ as shown in left panel of Figure \[fig:K\_ave\]. Similarly, we also compare the relationship between the outskirts entropy and $M_{200}$, finding again a tight correlation ($A=5.57\pm0.50$ and $B=0.69\pm0.24$). Our results suggest that the gravity of the cluster has an important effect on the thermalization process even in the outskirts. However, since the outskirts entropy is lower than predicted by the accretion shock heating model [@Tozzi01], the shock heating is not sufficient to explain the observations.
A possible mechanism for weakening of accretion shock heating in cluster outskirts has been proposed by [@Lapi10] and [@Cavaliere11]. According to their scenarios, the outer slope of the gravitational potential becomes progressively shallower in the accelerating universe, and the background gas density is decreased at late times accordingly. Then the entropy production is reduced by the slowdown in the growth of outskirts. Their model predicts that the outskirts entropy is anti-correlated with the halo concentration at a fixed mass because the concentration is related to the formation epoch. The two clusters A478 and A1689 have similar virial masses, albeit different concentration parameters: $c_{\rm vir}\sim4$ for the former [@Okabe14b] and $c_{\rm vir}\sim13$ for the latter [@Umetsu08]. We do not find any significant difference in the outskirts entropy between the two clusters. However, since the concentration parameter is sensitive to halo triaxiality and orientation [e.g. @Oguri04b] and the cluster redshifts are different, the uncertainty from such other factors is large. Therefore, a further systematic study with a large statistical sample is required to investigate this possible correlation.
[@Nagai11] proposed that clumpy gas structures in cluster outskirts lead to an overestimate of the gas density, so that the outskirts entropy can be underestimated from X-ray observations. If high gas clumpiness is a dominant source of the flat entropy, a correlation between the outskirts entropy and the gas fraction is expected. However, in their simulations, gas clumpiness becomes dominant only beyond $r_{200}$. From [*Suzaku*]{} observations [@Simionescu11] found an extremely-high gas fraction within the virial radius using their hydrostatic mass estimate for the Perseus cluster, and suggested that this is due to high gas clumpiness in the cluster outskirts. Here we argue that the cumulative gas fraction used by [@Simionescu11] is not a good quantity to discuss the degree of gas clumpiness which is locally defined. Nevertheless, following [@Simionescu11], we compare the outskirts entropy with the cumulative gas fractions within the virial radii, in the middle panel of Figure \[fig:K\_ave\]. Based on [@Simionescu11], it is expected that the gas fraction is anti-correlated with the outskirts entropy. We find no clear correlation between $K_{\rm out}$ and $f_{\rm
gas,vir}\equiv f_{\rm gas}(<r_{\rm vir})$ for our three high-mass clusters. Since there is a possibility that the scatter in $K_{\rm out}-f_{\rm gas,vir}$ plane is caused by the mass dependence on the entropy (left panel), we show in the right panel a scaled version of the entropy normalized using the scaling relation. This shows that the scaled entropy does not correlate with the gas fraction, indicating that one cannot attribute the high gas fraction and low gas entropy to the gas clumping.
{width="\hsize"}
Universal Entropy Profile {#subsec:K}
-------------------------
@Walker12b and @Sato12 showed that the entropy profiles for relaxed clusters have a universal form in the radial range from $\sim 0.1r_{200}$ to $\sim r_{200}$. In their analyses, they first normalize entropy profiles to unity at a certain pivot point (e.g., $0.3r_{200}$) and scale the cluster aperture radii by $r_{200}$ which is determined according to the mass–temperature relation. Subsequently, fitting is performed to obtain the best-fit universal function. However, such scaling operations make it difficult to correctly propagate errors in the normalizations. Since we use the weak-lensing mass $M_{\Delta}$ and the aperture radius $r_{\Delta}$ for the normalizations, their errors should be explicitly taken into account.
To properly account for the errors in the weak-lensing and X-ray observables, we avoid two-steps procedures of @Walker12b and @Sato12, and perform a simultaneous ensemble fit of the observed entropy profiles as a function of cluster radius and weak-lensing mass. The log-likelihood function is defined by $$\begin{aligned}
-2\ln {\mathcal L}&=&\sum_{i,j}\ln(\delta_{\ln K,ij}^2+\delta_{\ln f,ij}^2+\sigma_{\ln K}^2) \nonumber \\
& &+ \frac{(\ln(K_i(r_j))-\ln(f_K(M_i,r_j)))^2}{\delta_{\ln K,ij}^2+\delta_{\ln f,ij}^2+\sigma_{\ln K}^2},\end{aligned}$$ where $i$ and $j$ denote the $i$-th cluster and $j$-th radial bin, respectively. $\delta_{\ln K}$ is the fractional error of the entropy, $\delta_{\ln f}$ is the fractional error in the function $f_K$, through its dependence on the total mass $M_{\Delta}$ and the aperture radius $r_{\Delta}$ from weak-lensing, and $\sigma_{\ln K}$ denotes intrinsic scatter of the entropy in the mass scaling relation. Here we have introduced the function $f_K$, which takes into account the flattening of the outskirts entropy profile, given by $$\begin{aligned}
f_K(M_{\Delta},\tilde{r})&=&K_0E(z)^{-4/3}\left(\frac{M_{\Delta}E(z)}{10^{14}\h70Msol}\right)^{a} \nonumber \\
&
&\times(\tilde{r}/\tilde{r}_0)^\alpha\left(1+(\tilde{r}/\tilde{r}_0)^\beta\right)^{-\alpha/\beta}
\label{eq:K}, \end{aligned}$$ where $K_0$ is the normalization factor, $\tilde{r}=r/r_{\Delta}$ is the aperture radius in units of $r_{\Delta}$, $a$ is the mass slope, and $r_0$ denotes a characteristic scale radius at which the logarithmic entropy gradient changes. The asymptotic behavior is $K(r)\propto r^{\alpha}$ and $K(r)\propto
{\rm constant}$ for $r\ll r_0$ and $r\gg r_0$, respectively. All errors in $M_{\Delta}$, $r_{\Delta}$ and $K$ are considered.
We perform fitting of two X-ray datasets. The first dataset is composed only of the [*Suzaku*]{} data for the four clusters. The second one includes the [*XMM-Newton*]{} and [*Chandra*]{} as well as [*Suzaku*]{} data. We restrict our X-ray data to $r>0.1r_{200}$ to excise the core regions. The fitting is performed both with and without intrinsic scatter.
Since the [*Suzaku*]{} data alone cannot constrain the $\beta$ parameter, we fix $\beta=4$ without intrinsic scatter. The resulting Bayesian estimates of the model parameters are listed in Table \[tab:best-fit\_K\]. The results do not change significantly when we change $\beta$ by $\pm2$. The inner slope of the entropy profile, $\alpha=1.18^{+0.93}_{-0.44}$, agrees with the power-law slope ($K\propto r^{1.1}$) of the accretion shock heating model [@Tozzi01].
Next, we fit the full [*Suzaku*]{}, [*XMM-Newton*]{} and [*Chandra*]{} data with and without intrinsic scatter. The constraint on the inner slope, $\alpha=1.11_{-0.13}^{+0.17}$, is significantly improved. The best-fit entropy function, which was obtained without intrinsic scatter taken into account, gives an excellent description of the scaled entropy profiles of the four clusters (Figure \[fig:K\_prof\]). The mass dependence of the entropy is consistent with the self-similar model ($a=2/3$). The inner radial slope $\alpha$ is in a remarkably good agreement with the shock heating model [@Tozzi01] and with results from previous X-ray studies [e.g. @Ponman03]. The flattening of the entropy profile occurs at $r\sim 0.5r_{200}\sim
r_{1000}$, where the averaged hydrostatic to weak-lensing mass ratio is less than unity (Section \[subsec:Mratio\]).
We note that the four clusters studied here span a range of halo mass, concentration, X-ray luminosity and redshift, but exhibit a remarkable self-similarity in the scaled entropy profiles over the entire cluster ($>0.1r_{200}$). In other words, not only the $r^{1.1}$-law’s entropy profile but also the outskirts flattening entropy depends on cluster $M_{200}$ masses and $r_{200}$ radius. The results indicate that the thermalization process in the ICM outside X-ray cores could be governed by the gravity of the cluster. The efficiency of gravitational heating from outside the core to $0.5r_{200}$ follows the accretion shock heating model [@Tozzi01], whereas the efficiency at $r{\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}0.5r_{200}$ is lower than the model prediction. The universality, independent of cluster properties, indicates that the thermalization mechanism at work in the ICM could be a common physical process to all clusters, perhaps controlled by the growth of large sale structure surrounding the cluster, although the heating efficiency in the outskirts may need to be modified from the standard shock-heating mechanism [@Tozzi01].
We also fit our data with an alternative functional form of $K\propto \tilde{r}^A\exp(B(1-\tilde{r}))$ proposed by [@Lapi10] and [@Cavaliere11], by including our normalization parameter and using $\tilde{r}=r/r_{\rm vir}$. We find this model also gives a good fit to the data. To statistically distinguish these two models, we need a larger sample of cluster observations with a wide radial coverage beyond the virial radius.
------------------------------------------------------------------------------------------------------------------------------------------------------ ---------- ---------------- ----- ---------- --------- ------------------ ------------------
Dataset $\Delta$ $K_0$ $a$ $\alpha$ $\beta$ $r_0$ $\sigma_{\ln K}$
\[keV cm$^2$\] \[$r_{\Delta}$\]
*[Suzaku]{} & $200$ & $283.52_{-80.71}^{+102.98}$ & $0.71_{-0.11}^{+0.14}$ & $1.18_{-0.44}^{+0.93}$ & $4$ (fixed) & $0.41_{-0.16}^{+0.25}$ & $-$\
Full & $200$ & $359.61_{-75.47}^{+92.00}$ & $0.63_{-0.08}^{+0.09}$ & $1.11_{-0.13}^{+0.17}$ & $5.97_{-2.87}^{+2.49}$ & $0.48_{-0.08}^{+0.10}$ & $-$\
Full & $200$ & $380.49_{-71.61}^{+81.79}$ & $0.62_{-0.07}^{+0.08}$ & $1.10_{-0.13}^{+0.15}$ & $5.97$ (fixed) & $0.49_{-0.07}^{+0.10}$ & $<0.06$\
*
------------------------------------------------------------------------------------------------------------------------------------------------------ ---------- ---------------- ----- ---------- --------- ------------------ ------------------
![Normalized entropy profile as a function of the radius scaled by $r_{200}$. Red diamonds, blue triangle, green squares and magenta circles denote the normalized entropy for A478, Hydra A, A1689, and A1835, respectively. The normalization is $K_*(M_{200},z)=K_0E(z)^{-4/3}\left(M_{200}E(z)/10^{14}\h70Msol \right)^{a}$, where $K_0=359.61\,[{\rm keV cm^2}]$ and $a=0.63$. The black solid curve represents the best-fit universal entropy profile. The entropy profile at $r{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}0.5r_{200}$ follows $r^{1.1}$ as predicted by the standard model [@Tozzi01] and becomes flat at $r{\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}0.5r_{200}$. The black-dashed curve shows the best-fit profile for the [@Lapi10] model. []{data-label="fig:K_prof"}](fig4.eps){width="\hsize"}
Universal Pressure Profile {#subsec:P}
--------------------------
It was shown by [@PlanckSZPprof] that the stacked pressure profile $P_e=n_ek_BT$ constructed from [*Planck*]{} SZ observations of 62 clusters is well described by a generalized NFW pressure profile [@Nagai07b] out to $3\times r_{500}$. It is of great importance to compare our independent [*Suzaku*]{} X-ray pressure measurements to the [*Planck*]{} SZ observations [@Walker12a].
Here, we derive the average electron pressure profile scaled with the weak-lensing mass $M_\Delta$ following the same procedure described in Section \[subsec:K\]. We consider the universal pressure function [@Nagai07b; @PlanckSZPprof] of the following form, by simultaneously taking into account the scaling relation between the electron pressure and total mass $M_\Delta$: $$\begin{aligned}
f_P(M_{\Delta},\tilde{r})&=&P_0E(z)^{2}\left(\frac{M_{\Delta}E(z)}{10^{14}\h70Msol}\right)^{b} \nonumber \\
& &\times (\tilde{r}/\tilde{r}_0)^{-\gamma}\left(1+(\tilde{r}/\tilde{r}_0)^\beta\right)^{(\gamma-\delta)/\beta}, \label{eq:P}\end{aligned}$$ where $P_0$ is the normalization factor, $\tilde{r}=r/r_{\Delta}$ is the aperture radius in units of $r_{\Delta}$, and the mass slope $b$ accounts for the dependence of the pressure normalization on the halo mass $M_\Delta$ determined from the lensing analysis. The asymptotic entropy slopes are $P\propto r^{-\gamma}$ and $r^{-\delta}$ for $r\ll r_0$ and $r\gg r_0$, respectively. The inverse of the scale radius, ${\tilde r}_0^{-1}=r_{\Delta}/r_0$, is equivalent to the concentration parameter, $c_\Delta$, in the definition of [@PlanckSZPprof].
First, we fit our pressure profiles at the reference overdensity $\Delta=500$ to make a fair comparison with [@PlanckSZPprof]. We fix $r_0$ and $\beta$ by the [*Planck*]{} results ($c_{500}={\tilde
r}_{0}^{-1}=1.81$ and $\beta=1.33$), because it is difficult to constrain them from our data. The best-fit parameters are listed in Table \[tab:best-fit\_P\]. The normalized pressure profiles derived from the full [*Suzaku*]{}, [*XMM-Newton*]{} and [*Chandra*]{} dataset are displayed in Figure \[fig:P\_prof\]. The logarithmic gradient of the pressure profile progressively steepens from $-1$ to $-4$.
For comparison, we calculate the normalized [*Planck*]{} pressure profiles using the equations (7), (10) and (11) of [@PlanckSZPprof]. We use the best-fit parameters of the stacked pressure profile and determine the normalization by two approaches, namely using our weak-lensing or X-ray hydrostatic mass estimates, instead of their original mass estimates using the $Y_X=M_{\rm gas}T_X$ mass proxy. Figure \[fig:P\_prof\] shows that the [*Planck*]{} average pressure profiles agree with the [*Suzaku*]{} results. The outer slope $\delta$ is in excellent agreement with the average slope of the [*Planck*]{} SZ pressure profile ($\delta=4.1$). Although our sample size is much smaller than that of the [*Planck*]{} sample, our results show good consistency between the observationally independent X-ray and SZ profiles of individual clusters [@Walker12a; @Mochizuki14].
Next, we fix $r_0=0.48r_{200}$ and $\beta=5.97$ using the best-fit entropy profile (Section \[subsec:K\]). Now we derive scaled profiles at a density contrast of $\Delta=200$. The best-fit pressure profile, converted to $r_{500}$ units, is shown in Figure \[fig:P\_prof\]. The resulting pressure profile is compatible with our and the average [*Planck*]{} pressure profiles. We also measure intrinsic scatter with $b=2/3$ fixed (Table \[tab:best-fit\_P\]). Some parameters, including $b$, $\gamma$, and $\delta$, are slightly changed depending on the choice of $\beta$ and $r_0$. A more definitive determination of these parameters requires a further systematic study.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------- ------------------------ ----- --------- ---------- ---------- ------------------ ------------------
Dataset $\Delta$ $\log_{10}(P_0)$ $b$ $\beta$ $\gamma$ $\delta$ $r_0$ $\sigma_{\ln P}$
\[log(keV cm$^{-3}$)\] \[$r_{\Delta}$\]
*[Suzaku]{} & $500$ & $-2.41_{-0.17}^{+0.17}$ & $0.43_{-0.10}^{+0.10}$ & $1.33$ (fixed) & $1.08_{-0.25}^{+0.25}$ & $3.41_{-0.38}^{+0.42}$ & $1/1.81$ (fixed) & $-$\
Full & $500$ & $-2.00_{-0.14}^{+0.15}$ & $0.08_{-0.05}^{+0.07}$ & $1.33$ (fixed) & $0.81_{-0.21}^{+0.22}$ & $4.07_{-0.35}^{+0.36}$ & $1/1.81$ (fixed) & $-$\
Full & $500$ & $-2.35_{-0.21}^{+0.2}$ & $0.15_{-0.08}^{+0.10}$ & $1.33$ (fixed) & $1.24_{-0.38}^{+0.38}$ & $4.07_{-0.35}^{+0.36}$ & $1/1.81$ (fixed) & $0.24_{-0.06}^{+0.10}$\
Full & $200$ & $-3.14_{-0.16}^{+0.16}$ & $0.32_{-0.14}^{+0.14}$ & $5.97$ (fixed) & $1.84_{-0.15}^{+0.17}$ & $3.45_{-0.36}^{+0.43}$ & $0.48$ (fixed) & $-$\
Full & $200$ & $-2.85_{-0.36}^{+0.38}$ & $2/3$ (fixed) & $5.54_{-2.84}^{+2.81}$ & $1.26_{-0.79}^{+0.44}$ & $2.67_{-0.21}^{+0.30}$ & $0.24_{-0.07}^{+0.11}$ & $0.28_{-0.09}^{+0.10}$\
*
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------- ------------------------ ----- --------- ---------- ---------- ------------------ ------------------
![Normalized pressure profile as a function of cluster radius scaled by $r_{500}$. Red diamonds, blue triangle, green squares and magenta circles represent the normalized pressure for A478, Hydra A, A1689, and A1835, respectively. The normalization is $P_*(M_{500},z)=P_0E(z)^{1/2}\left(M_{500}E(z)/10^{14}\h70Msol
\right)^{b}$. The black-solid and red-dotted lines are the best-fit profiles using $r_0$ and $\beta$ from the [*Planck*]{} results [@PlanckSZPprof] and our best-fit entropy profile, respectively. The blue-dashed and green-dotted-dashed lines are the average SZ pressure profiles [@PlanckSZPprof], obtained using the weak-lensing and hydrostatic masses, respectively. []{data-label="fig:P_prof"}](fig5.eps){width="\hsize"}
Joint Fit with the Number Density and Temperature Profiles {#subsec:joint}
----------------------------------------------------------
In Sections \[subsec:K\] and \[subsec:P\], we have shown that the scaled entropy and pressure profiles of the four clusters are well fitted with respective universal functions, by taking into account the uncertainties in the weak-lensing mass and X-ray measurements.
Here we simultaneously fit the gas density and temperature profiles to our X-ray observations for determining the respective universal functions. From equations (\[eq:K\]) and (\[eq:P\]), we obtain the following expressions for the gas number density and temperature: $$\begin{aligned}
f_n &=&n_0E(z)^{2}\left(\frac{M_{\Delta}E(z)}{10^{14}\h70Msol}\right)^{\frac{3}{5}(b-a)}\nonumber \\
&& \times(\tilde{r}/\tilde{r}_0)^{-\frac{3}{5}(\alpha+\gamma)}\left(1+(\tilde{r}/\tilde{r}_0)^\beta\right)^{-\frac{3}{5}(\delta-\gamma-\alpha)/\beta},\label{eq:n}\\
f_T&=&T_0\left(\frac{M_{\Delta}E(z)}{10^{14}\h70Msol}\right)^{\frac{3}{5}a+\frac{2}{5}b} \nonumber \\
&& \times(\tilde{r}/\tilde{r}_0)^{\frac{3}{5}\alpha-\frac{2}{5}\gamma}\left(1+(\tilde{r}/\tilde{r}_0)^\beta\right)^{-(\frac{2}{5}\delta-\frac{2}{5}\gamma+\frac{3}{5}\alpha)/\beta}, \label{eq:T}\end{aligned}$$ where $M_{\Delta}$ is the total cluster mass from lensing measurements, $n_0$ and $T_0$ are the normalization factors for the gas density and temperature profiles, respectively. Here we have assumed that the gas density and temperature profiles have the same scale radius $r_0$ and $\beta$.
The joint likelihood for the number density and the temperature profiles is given by $$\begin{aligned}
-2\ln {\mathcal L}&=&\sum_{i,j}\ln(\det(\mbox{\boldmath $C$}_{ij})) + \mbox{\boldmath $v$}_{ij}^T\mbox{\boldmath $C$}_{ij}^{-1}\mbox{\boldmath $v$}_{ij}, \\
\mbox{\boldmath $v$}&=&\left(
\begin{array}{ccc}
\ln(n(\tilde{r}))-\ln(f_n(M_{\Delta},\tilde{r})) \\
\ln(T(\tilde{r}))-\ln(f_T(M_{\Delta},\tilde{r})) \\
\end{array}
\right), \nonumber \\
\mbox{\boldmath $C$}&=&\mbox{\boldmath $C$}_{\rm stat}+\mbox{\boldmath $C$}_{\rm int} \nonumber \\
\mbox{\boldmath $C$}_{\rm stat}&=&\left(
\begin{array}{ccc}
\delta_{\ln n}^2+\delta_{\ln fn}^2 & \delta_{\ln fn,\ln fT} \\
\delta_{\ln fn,\ln fT} & \delta_{\ln T}^2+\delta_{\ln fT}^2 \\
\end{array}
\right). \nonumber \\
\mbox{\boldmath $C$}_{\rm int}&=&\left(
\begin{array}{ccc}
\sigma_{\ln n}^2 & \rho\sigma_{\ln n}\sigma_{\ln T} \\
\rho\sigma_{\ln n}\sigma_{\ln T} & \sigma_{\ln T}^2 \\
\end{array}
\right). \nonumber\end{aligned}$$ Here, $i$ and $j$ denote the $i$-th cluster and $j$-th radial bins, respectively. $\mbox{\boldmath $C$}_{\rm stat}$ is the error covariance matrix for the data vector , $\delta_{\ln n}$ and $\delta_{\ln T}$ are the fractional errors of the gas density and temperature, $\sigma_{\ln f_n}$ and $\sigma_{\ln f_T}$ are the fractional errors in the functions $f_n$ and $f_T$ (see equations (\[eq:n\]) and (\[eq:T\])) through their weak-lensing mass dependence. We assume that the error correlation between the number density and the temperature is negligible. Note that the off-diagonal elements in the covariance matrix cannot be ignored because both the gas density and temperature depend on the weak-lensing mass. The intrinsic covariance $\mbox{\boldmath $C$}_{\rm int}$ consists of intrinsic scatter of the number density $\sigma_{\ln n}$ and the temperature $\sigma_{\ln T}$. By definition the correlation coefficient $\rho$ is in the range $-1\leq \rho\leq 1$.
Since the number density and temperature profiles are equivalent to the entropy and pressure profiles which have been investigated in Sections \[subsec:K\] and \[subsec:P\], we shall focus on the results of the intrinsic covariance based on the self-similar solution ($a=b=2/3$). For this, we use the full [*Suzaku*]{}, [*XMM-Newton*]{} and [*Chandra*]{} dataset. The best-fit profiles of gas density and temperature are shown in the top panel of Figure \[fig:all\_prof\]. The pressure and entropy profiles corresponding to the best-fit parameters (Table \[tab:best-fit\]) are shown in the bottom panel of Figure \[fig:all\_prof\]. The normalization factors for the gas density, temperature, pressure, and entropy are $n_*=n_0E(z)^2\left(M_{200}E(z)\right)^{3(b-a)/5}$,$T_*=T_0\left(M_{200}E(z)\right)^{3b/5+2a/5}$, $P_*=n_0T_0E(z)^2\left(M_{200}E(z)\right)^{b}$ and $K_*=T_0n_0^{-2/3}E(z)^{-4/3}\left(M_{200}E(z)\right)^{a}$, respectively.
The best-fit density profile describes well the observations as shown in the top-left panel of Figure \[fig:all\_prof\]. The residual deviation $\Delta_{\rm dev}= n/f_n-1$ from the best-fit profile is shown in the lower subpanel. Compared to A478 and A1689, the deviations for Hydra A and A1835, corresponding to the clusters with high gas fractions (Figure \[fig:fgas\_ave\]), are high but constant with radius $r/r_{200}$, showing that the observed scaled density profiles follow the universal trend within intrinsic scatter. The logarithmic gradient of the gas density slightly changes from $d\ln n_e/d \ln r=-3(\alpha+\gamma)/5\sim-1.9$ at $r\ll r_0$ to $-3\delta/5\sim-1.6$ at $r\gg r_0$. The gas density slope outside the X-ray cores ($r>0.1r_{200}$) is thus shallower than the outer asymptotic slope of the NFW density profile ($-3$). Similar results were reported in previous studies of [*Chandra*]{} and [*XMM-Newton*]{} observations [e.g. @Vikhlinin06; @Zhang08]. A possible interpretation of the apparent shallow outer slope is that the degree of gas clumpiness is increased in the cluster outskirts [@Nagai11].
The X-ray temperature profiles are nearly constant at $0.1r_{200}{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}r{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}0.5r_{200}$, which is consistent with previous studies [e.g. @Vikhlinin06; @Zhang08]. The logarithmic slope of the temperature drastically changes at $\sim0.5r_{200}$ from $3\alpha/5-2\gamma/5\sim0$ to $-2\delta/5\sim-1.1$. The deviation $\Delta_{\rm dev}=T/f_T-1$ is constant with radius and close to zero, showing high similarity of the temperature profiles over a wide radial range.
For the gas pressure, the logarithmic gradient steepens from $d\ln{P}/dln{r}=-\gamma\sim-1.8$ to $-\delta\sim-2.7$ over the full radial range (see Section \[subsec:P\] for the case of $\Delta=500$). The deviations $P/f_P-1$ of Hydra A and A1835 are higher than those of A1689 and A478, which reflects the large deviations in the gas density.
The entropy profile increases with radius as $\propto r^{1.16}$ at $r{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}0.5r_{200}$, and then flattens at radii greater than $\sim 0.5r_{200}$. The inner slope is consistent with the gravitational shock heating model [$\propto r^{1.1}$; @Tozzi01]. Our results show that the entropy flattening in the outskirts is caused not by the shallow outer density slope but by the steep temperature drop. We find the density slope changes with radius only by $\sim +0.3$, corresponding to a slope change of $\sim -0.2$ for the entropy, $K\propto n^{-2/3}$. On the other hand, the steepening of the temperature slope by $\sim-1.1$ significantly affects the entropy slope because $K\propto T$.
Including the intrinsic covariance matrix in our analysis, we can constrain the intrinsic scatter between the X-ray observables and weak-lensing masses. We find the intrinsic scatter of the gas density $\sigma_{\ln n}$ is greater than that of the temperature $\sigma_{\ln T}$ for our sample of the clusters. This is qualitatively consistent with the trends in the deviation profiles shown in Figure \[fig:all\_prof\].
From Bayesian inference, we obtain a $1\sigma$ lower limit on the correlation coefficient, $\rho>0.47$. On the other hand, we find a maximum-likelihood estimate of $\rho=0.96$, which is high and close to the upper bound ($\rho=1$) of the parameter range. We thus computed the probability $\mathcal{P}(\geq |\rho |)$ that the correlation coefficient of two random variables for a sample size of 4 is greater than $0.96$, finding $\mathcal{P}(\geq |\rho |)=0.04$, which is very small and rules out the possibility that the high correlation coefficient is randomly generated.
Since the intrinsic scatter of the quantity $X=Tn^p$ is expressed as $\sigma_{\ln X}^2=\sigma_{\ln T}^2+p^2\sigma_{\ln n}^2 + 2 p\rho\sigma_{\ln n}\sigma_{\ln T}$, the observed positive coefficient indicates that the third terms for the entropy ($p=-2/3$) and pressure ($p=1$) are negative and positive, respectively. Thus, the intrinsic scatter of the entropy is smaller than that of the pressure, which is consistent with the results from individual analyses of the pressure and entropy (Sections \[subsec:K\] and \[subsec:P\]). The positive correlation between the gas density and temperature can also be seen in their deviation profiles in Figure \[fig:all\_prof\]. Due to this positive correlation, the deviation amplitude for the pressure is larger than that for the entropy. Similarly, a joint X-ray and weak-lensing analysis of @Okabe10c derived bivariate $M$-$T$ and $M$-$M_{\rm gas}$ scaling relations for 12 clusters at $\Delta=500$, finding that the intrinsic scatter between the gas mass and the temperature is positively correlated.
Theoretical predictions for the intrinsic correlation between the gas mass and pressure are rather controversial. Using an adaptive-mesh refinement code, [@Kravtsov06] found from their simulations that the gas-temperature deviations from the $M$-$T$ relation are anti-correlated with the gas-mass deviations from the $M$-$M_{\rm gas}$ relation. On the other hand, [@Stanek10] showed that the temperature and gas-mass deviations are positively correlated with each other. Therefore, a larger sample is required to constrain the intrinsic correlations between cluster properties, and it will allow us to investigate the functional form of the radial profile with the lowest scatter and optimal mass proxies based on the principal component analysis [@Okabe10c].
Now we examine the validity of the hydrostatic-equilibrium assumption: $$\begin{aligned}
\frac{1}{\rho_g}\frac{dP_g}{dr}=-\frac{GM}{r^2},\end{aligned}$$ where $\rho_g$ and $P_g$ are the gas mass density and thermal pressure, respectively. Since $\rho_g\propto n_e$ and $P_g\propto P_e$, the best-fit parameters (Table \[tab:best-fit\]) in equations (\[eq:P\]) and (\[eq:n\]) allow us to determine the radius beyond which the enclosed hydrostatic mass unphysically decreases (i.e., $dM/dr<0$). We find that the breakdown of the assumption occurs at $r\sim0.84r_{200}\sim1.3r_{500}$, which is consistent with previous studies [e.g. @Mochizuki14; @Kawaharada10; @Ichikawa13]. The breakdown of the hydrostatic assumption underestimates hydrostatic mass estimates at $r{\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}1.3r_{500}$. Indeed, Figure \[fig:Mratio\_ave\] shows that the hydrostatic masses are on average much lower than the weak-lensing masses at $r>r_{500}$.
{width="\hsize"}
The methodology we have applied here, which simultaneously fits X-ray observable profiles taking into account multivariate scaling relations, stems from our earlier work on the multivariate scaling relations [@Okabe10c]. In the traditional method, one obtains scaling relations between the total mass and X-ray observables at a given reference overdensity (e.g., $\Delta=200$ or 500). Furthermore, we have also established the average scaled radial profiles and intrinsic scatters for X-ray observables. In contrast to the traditional method, this new method enables us to simultaneously constrain the shape of the radial profile, normalization, and its intrinsic scatter, where the normalization of the profile corresponds to the mass-observable scaling relation. Importantly, this method does not require an iterative procedure to reconstruct mass-observable scaling relations [@Vikhlinin09a] because the scaling parameters at specific overdensities are determined by lensing and hence independent of X-ray observables.
We note that this method is complementary to stacking approaches [@PlanckSZPprof; @Okabe13; @Umetsu14], which allow us to derive ensemble-averaged cluster profiles in a model-independent way. This new method to measure the characteristic shapes and normalizations of profiles will provide us a powerful means to establish the mass proxy for cluster cosmology studies, especially for the forthcoming large-sky X-ray surveys (e.g. [*eROSITA*]{}).
------------------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
$\log_{10}(n_0)$ $T_0$ $a$ $b$ $\alpha$ $\beta$
$\log_{10}(1$cm$^{-3}$) \[keV\]
$-3.60_{-0.14}^{+0.15}$ $1.27_{-0.19}^{+0.24}$ $2/3$ (fixed) $2/3$ (fixed) $1.16_{-0.12}^{+0.17}$ $5.52_{-2.64}^{+2.87}$
$\gamma$ $\delta$ $r_0$ $\sigma_{\ln n}$ $\sigma_{\ln T}$ $\rho$
\[$r_{200}$\]
$1.82_{-0.30}^{+0.28}$ $2.72_{-0.35}^{+0.34}$ $0.45_{-0.07}^{+0.08}$ $0.22_{-0.04}^{+0.05}$ $0.07_{-0.04}^{+0.06}$ $>0.49$
------------------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------
Summary {#sec:sum}
=======
We have performed a joint X-ray and weak-lensing analysis of a sample of four relaxed clusters (Hydra A, A478, A1689, and A1835), which had been deeply observed to date by both [*Suzaku*]{} and Subaru out to virial radii. We have shown that the X-ray hydrostatic mass estimates are systematically underestimated, where the average hydrostatic-to-lensing mass ratio decreases from $\sim 70\%$ at $r_{500}$ to $\sim 30\%$ at $r_{\rm vir}$. This radial dependence is detected at the $3\sigma$ significance level. The average gas mass fraction from weak-lensing mass estimates increases with radius, and agrees with the cosmic baryon fraction within the virial radius. The outskirts entropy is shown to be tightly correlated with the total cluster mass from lensing, but not with the gas mass fraction within the virial radii.
We have developed a new advanced method for determining normalized cluster radial profiles for multiple X-ray observables by simultaneously taking into account both their radial dependence and multivariate scaling relations with weak-lensing masses. This method stems from the techniques for determining the multivariate scaling relations [@Okabe10c] and is complementary to stacking approaches. In the paper, we first used this method to individually reconstruct each universal X-ray observable function (Sections \[subsec:K\] and \[subsec:P\]). We then applied it to simultaneously determine two X-ray observable functions by taking into account their intrinsic covariance (Section \[subsec:joint\]). A combination of complementary data sets of the weak-lensing masses and radii and X-ray observables is essential to this method.
We find the gas entropy, pressure, and density profiles are all remarkably self-similar when scaled with the weak-lensing $M_{200}$ mass and $r_{200}$ radius. The entropy monotonically increases out to $\sim 0.5r_{200}\sim r_{1000}$ following the accretion shock heating model $K(r)\propto r^{1.1}$ [@Tozzi01], and flattens at ${\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}0.5r_{200}$. The logarithmic gradient of the gas density becomes slightly shallower at $r\sim 0.5r_{200}$. A possible interpretation for this is that the degree of gas clumpiness is increased in the outskirts. The temperature profile is constant at $0.1r_{200}<r<{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}0.5r_{200}$, and sharply drops off outside $\sim 0.5 r_{200}$. The bivariate scaling functions of the gas density and temperature reveal that the flatness of the outskirts entropy profile is caused by the steepening of the temperature, rather than the flattening of the gas density. Thus, gas clumpiness alone cannot be responsible for all of the flatness of the outskirts entropy. The pressure profile exhibits a steep outer slope, in good agreement with the averaged [*Planck*]{} Sunyaev-Zel’dovich pressure profile. The assumption of hydrostatic equilibrium breaks down beyond $\sim0.84r_{200}\sim1.3r_{500}$.
Our cluster sample may not be representative of a homogeneous class of actual clusters because the clusters were selected solely by the current availability of both [*Suzaku*]{} and Subaru observations. The sample clusters span a range of halo mass, concentration, X-ray luminosity and redshift. Nevertheless, we find the universality of the scaled entropy profiles. This indicates that the thermalization mechanism in the ICM over the entire region ($>0.1r_{200}$) is controlled by gravitation in a common way for all clusters. The entropy flattening in cluster outskirts appears to be a common phenomena, not limited to a special class of clusters. This demonstrates that the heating efficiency in the outskirts needs to be modified from the standard $r^{1.1}$ law. Since the current sample is small and limited by the availability of data, a further systematic multi-wavelength study of cluster outskirts is vitally important to understand the physical state of the ICM.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. N. Okabe (26800097), M. Takizawa (26400218), and K. Sato (25800112) are supported by a Grant-in-Aid from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. K. Umetsu acknowledges partial support from the National Science Council of Taiwan (grant NSC100-2112-M-001-008-MY3). Y. -Y. Zhang acknowledges support by the German BMWi through the Verbundforschung under grant 50OR1304.
\[lastpage\]
[^1]: Based on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan.
[^2]: Again, the X-ray mass has been measured within the weak-lensing determined aperture radius in our study.
|
---
abstract: 'Proximal algorithms are well-suited for nonsmooth and constrained large-scale optimization problems and therefore suitable for applications in many areas of science. There are essentially four proximal algorithms based on fixed-point iterations currently known: forward-backward splitting, forward-backward-forward or Tseng splitting, Douglas-Rachford, and the Davis-Yin three operator splitting. In addition, the alternating direction method of multipliers (ADMM) is also closely related. In this paper we show that all of these algorithms can be derived as different discretizations of a single differential equation, namely the simple gradient flow. This is achieved through splitting methods for differential equations. Moreover, employing similar discretization schemes to a particular second-order differential equation, which we refer to as the accelerated gradient flow, results in accelerated variants of each respective proximal algorithm; we simultaneously consider two types of acceleration, although other choices are also possible. For instance, we propose accelerated variants of Davis-Yin and Tseng splitting, as well as accelerated extensions of ADMM. Interestingly, we show that ADMM and its accelerated variants correspond to rebalanced splittings, which is a recent technique designed to preserve steady states of the underlying differential equation. We show that all derived algorithms are valid first-order integrators under suitable assumptions. Our results strengthen the connections between optimization and continuous dynamical systems, offer a unified perspective on accelerated algorithms, and provide new accelerated algorithms.'
bibliography:
- 'accelerated\_splitting\_2019.bib'
---
[**Gradient Flows and\
Accelerated Proximal Splitting Methods** ]{}
[ Guilherme França,$^{\!a}$[^1] Daniel P. Robinson,$^{\!b}$ and Ren' e Vidal$^{\,a}$ ]{}
[ *${}^{a}$Mathematical Institute for Data Science, Johns Hopkins University\
${}^{b}$Department of Industrial and Systems Engineering, Lehigh University* ]{}
0.02in 0.1in
Introduction
============
Acceleration strategies in the context of optimization have proved to be powerful. One example, termed *heavy ball acceleration*, was introduced by Polyak [@Polyak:1964], while perhaps the most influential form of acceleration was introduction by Nesterov [@Nesterov:1983] and is often called *Nesterov acceleration*. Although neither are intuitive in their precise design, it has recently been shown that both can be obtained as explicit Euler discretizations of a certain second-order ordinary differential equation (ODE), and correspond to accelerated variants of gradient descent. This perspective has helped demystify the “magic" of acceleration techniques in optimization. There is an increasing interest in understanding the connections between optimization and continuous dynamical systems, especially for accelerated gradient based methods [@Candes:2016; @Wibisono:2016; @Krichene:2015; @Attouch:2016; @Wilson:2016; @Maddison:2018; @Maddison:2019; @Jordan:2019; @Franca:2019]. More recently, extensions of these connections to nonsmooth settings using proximal methods have started to be considered [@Attouch:2018; @Attouch:2018b; @May:2016; @Attouch:2018c; @Attouch:2018d; @Franca:2018; @Franca:2018b]. Proximal algorithms play an important role in optimization since they can enjoy improved stability and be applied under weaker assumptions, and in many cases the associated proximal operators have simple closed form expressions. The majority of known proximal algorithms fall into the following types:
- forward-backward splitting [@Lions:1979; @Passty:1979; @Han:1988];
- forward-backward-forward or Tseng splitting [@Tseng:2000];
- Douglas-Rachford [@Douglas:1956; @Lions:1979];
- Davis-Yin three operator splitting [@Davis:2017]; and
- alternating direction method of multipliers (ADMM) [@Glowinsky:1975; @Gabay:1976].
Many more sophisticated methods are extensions or variations of these themes. The first three were the only known methods for quite a long time. Only recently have Davis and Yin [@Davis:2017] solved the problem of obtaining a three operator splitting that cannot be reduced to any of the existing two operator splitting schemes. Such proximal methods are based on fixed-point iterations of nonexpansive monotone operators. A different technique based on projections onto separating sets has recently been proposed but will not be considered in this paper; see [@Johnstone:2018] and the references therein. ADMM dates back to the 70’s and has gained popularity due to its effectiveness in solving large-scale problems with sparse and low rank regularizations [@Boyd:2011]. We will focus on proximal algorithms that have previously been introduced from an *operator splitting* approach.[^2] The literature on operator splitting is huge, so here we simply mention that these methods have origins in functional analysis and differential equations [@Browder:1963; @Browder:1963b; @Browder:1963c], and were later explored in convex analysis and optimization; see [@Rockafellar:1976; @Combettes:2004; @Combettes:2011; @Combettes:2005; @Guler:1991; @Eckstein:1992]. (See also [@BauschkeBook; @Ryu:2016] for an introduction and historical account.)
Summary of main results {#main_results}
-----------------------
We consider algorithms for solving $$\label{optimization}
\min_{x\in\mathbb{R}^n} \{ F(x) \equiv f(x) + g(x) + w(x) \}$$ where $f$, $g$ and $w$ are functions from $\mathbb{R}^{n}$ into $\mathbb{R}$ obeying the following condition.
\[assump1\] The functions $f$ and $g$ are proper, closed, and convex. The function $w$ is differentiable.
This assumption will be used throughout the paper. The convexity requirements ensure that the proximal operators associated to $f$ and $g$ are well-defined, i.e., have a unique solution. For simplicity, in the discussion below we also assume that $f$ and $g$ are differentiable, however this condition can be relaxed.
Perhaps surprisingly, we show that *all* of the above mentioned (bulleted) proximal algorithms can be obtained as different discretizations (more precisely first-order integrators) of the simple *gradient flow* introduced by Cauchy [@Cauchy:1847] and given by $$\label{gradflow}
\dot{x} = -\nabla F(x),$$ where $x=x(t) \in \mathbb{R}^n$, $t$ is the time variable, and $\dot{x} \equiv dx / dt$. Our approach makes connections between optimization algorithms with *ODE splitting methods* [@McLachlan:2002; @MacNamara:2016], which is a powerful approach for designing algorithms. Interestingly, within this approach, ADMM emerges as a *rebalanced splitting* scheme, which is a recently introduced technique designed to preserve steady states of the underlying ODE [@Speth:2013]. We show that the dual variable associated with ADMM is precisely the *balance coefficient* used in this approach. We show that the other algorithms also preserve steady states of the gradient flow, but for different reasons, which in turn sheds light on the connections between ODE splitting ideas and operator splitting techniques from convex analysis.
The emphasis of this paper is on accelerated algorithms. We show that by employing similar discretization strategies to the *accelerated gradient flow* given by $$\label{secode}
\ddot{x} + \eta(t)\dot{x} = -\nabla F(x),$$ where $\ddot{x} \equiv d^2 x/dt^2$ and $\eta(t)$ is a damping coefficient chosen to be either $$\label{nagdamp}
\eta(t) = r/t \qquad \mbox{(decaying damping with $r\ge 3$)}$$ or $$\label{hbdamp}
\eta(t) = r \qquad \mbox{(constant damping $r > 0$),}$$ one obtains *accelerated variants* of the respective algorithms. The vanishing damping is related to the ODE from which Nesterov’s accelerated gradient method may be obtained [@Candes:2016], while the constant damping is related to the ODE associated with Polyak’s heavy ball method [@Polyak:1964]. We will refer to algorithms arising as discretizations of derived with as accelerated variants with “decaying damping,” and those derived with as accelerated variants with “constant damping.” In our analysis we treat both types in a unified manner. We also note that choices other than and are possible and would be automatically covered by our framework. To the best of our knowledge, the accelerated frameworks that we introduce are new, although some recover known methods as special cases. We show that they are all *first-order integrators* that *preserve steady states* of the accelerated gradient flow .
We also show how this continuous-to-discrete analysis can be extended to monotone operators using two approaches: *differential inclusions* (nonsmooth dynamical systems) and the combination of ODEs and Yosida regularized operators.
Overall, this paper brings an alternative perspective on the design of proximal optimization algorithms by establishing tight relationships with continuous dynamical systems and numerical analyses of ODEs. Since the dynamical systems and play a central role in the paper, we summarize their convergence rates in Table \[convergence\]. One expects that a suitable discretization would, at least up to a small error, preserve the same rates. However, a discrete analysis is still necessary to formalize such results.
convex strongly convex
--------------------------- ------------------------------ ----------------------------------------
gradient flow $\Order\left(t^{-1}\right)$ $\Order\left(e^{-m t }\right)$
accelerated gradient flow $\Order\left(t^{-2}\right)$ $\Order\left( t^{-2 r/3} \right)$
accelerated gradient flow $\Order\left( t^{-1}\right)$ $\Order\big( e^{- \sqrt{m} t } \big)$
: \[convergence\] Convergence rates of gradient flow vs. accelerated gradient flow with decaying damping and constant damping . Let $F^\star \equiv \inf_{x\in\mathbb{R}^n} F(x)$ and $x^\star$ be the unique minimizer of an $m$-strongly convex function $F$. We show rates for $F(x(t)) - F^\star$ when $F$ is convex, and for $\| x(t) - x^\star\|^2$ when $F$ is strongly convex. Note the tradeoff between decaying and constant damping for convex vs. strongly convex functions. These rates follow as special cases of the analysis in [@Franca:2018b].
Basic building blocks
---------------------
Standard discretizations of the derivatives appearing in the first- and second-order systems and , respectively, are $$\begin{aligned}
\dot{x}(t_k) &= \pm(x_{k\pm1}-x_k)/h + \Order(h), \label{eulerfirst} \\
\ddot{x}(t_k) &= (x_{k+1}-2x_k + x_{k-1})/h^2 + \Order(h),
\label{eulerfirst2}\end{aligned}$$ where $t_k = kh$ for $k\in\{0,1,2,\dotsc\}$, $x_k$ is an approximation to the true trajectory $x(t_k)$, and $h >0$ is the step size. To discretize it will be convenient to define $$\label{xhat}
\hat{x}_k \equiv x_k + \gamma_k(x_k - x_{k-1})
\quad \text{with} \quad
\gamma_k = \begin{cases} \tfrac{k}{k+r} & \mbox{for decaying damping~\eqref{nagdamp},} \\ 1 - r h & \mbox{for constant damping~\eqref{hbdamp}.} \end{cases}$$ Using this notation and the “minus" choice in , one can verify that $$\label{eulersec}
\ddot{x}(t_k) + \eta(t_k) \dot{x}(t_k) = (x_{k+1}-\hat{x}_k)/h^2 +
\Order(h).$$ As usual, we will often keep the leading order term and neglect $\Order(h)$ terms. We note that although we consider , other choices of damping $\eta(t)$ in are possible, which would lead to the same discretization but with a different $\gamma_k$ in .
Since this paper focuses on proximal methods, the *resolvent* operator plays a central role, and from a dynamical systems perspective so do *implicit* discretizations. For example, consider the implicit Euler discretization of given by $$\label{gd_implicit}
(x_{k+1} - x_{k} ) / h = -\nabla F(x_{k+1}).$$ This nonlinear equation in $x_{k+1}$ can be solved using the *resolvent*,[^3] which for an operator $A$ and spectral parameter $\lambda \in \mathbb{C}$ is defined as $$\label{resolvent}
J_{\lambda A} \equiv \left( I + \lambda A \right)^{-1}.$$ We are interested in cases where $A$ is a maximal monotone operator (see Section \[nonsmooth\]) and will always use $\lambda > 0$ as a real number related to the discretization step size. For instance, when $A = \nabla F$, $\lambda > 0$, and the *proximal operator* $\prox_{\lambda F}$ of $F$ is well-defined, it follows that the resolvent is equal to[^4] $$\label{prox}
J_{\lambda \nabla F}(v) \equiv
\prox_{\lambda F}(v) \equiv \argmin_x \left(
F(x) + \tfrac{1}{2\lambda}\| x - v\|^2\right).$$ It follows from , , and that $$\label{proximal_point}
x_{k+1}
= J_{h \nabla F}(x_k) = \prox_{h F}(x_k),$$ which is the *proximal point algorithm* [@Martinet1; @Martinet2] (see [@Rockafellar:1976b; @Rockafellar:1976c; @Guler:1991] for a thorough analysis and generalizations). When $F$ is convex, the proximal point algorithm is known to converge when a minimizer exists. In practice, it is often more stable than gradient descent and allows for a more aggressive choice of step size, as is common for implicit discretizations. Although the proximal point algorithm has the cost of computing the operator in , it can be employed even when $F$ is nondifferentiable. The result below follows from for the case of gradient flow, and from , , and for the case of accelerated gradient flow. This lemma is a key building block for the design of the algorithmic frameworks in this paper.
\[implilemma\] An implicit Euler discretization of the gradient flow and the accelerated gradient flow with stepsize $h > 0$ yields, respectively, the updates $$\label{impli12}
x_{k+1} =
\begin{cases}
J_{h \nabla F}(x_k) & \text{for the gradient flow \eqref{gradflow},} \\
J_{h^2 \nabla F}(\hat{x}_k) & \text{for the accelerated gradient flow \eqref{secode},}
\end{cases}$$ where $\hat{x}_k$ is defined in based on which type of damping is used.
The update is a proximal point computation associated with either $x_k$ or $\hat{x}_k$, depending on whether acceleration is used. In the accelerated case, since these updates result from the discretization of , we obtain an intuitive interpretation: in the case of the parameter $r$ controls the amount of friction (dissipation) in the system, and for the first choice the friction is decaying over time, while for the second choice the friction is constant. Other choices are also possible.
Outline
-------
In Section \[splitting\_methods\], we introduce the balanced and rebalanced splitting approaches recently proposed in [@Speth:2013]. However, we propose two modifications to this scheme that will enable us to make connections with existing optimization algorithms as well as propose new ones. In Section \[proximal\_splitting\], we derive accelerated extensions of ADMM, accelerated extensions of the Davis-Yin method (forward-backward and Douglas-Rachford follow as special cases), and accelerated extensions of Tseng’s method from an ODE splitting approach. We also show that all discretizations considered in this paper are proper first-order integrators that preserve steady states of the underlying ODE. In Section \[nonsmooth\], we argue that our analysis extends to the nonsmooth setting, and to maximal monotone operators more generally. Finally, numerical results in Section \[applications\] illustrate the speedup achieved by our accelerated variants.
Splitting Methods for ODEs {#splitting_methods}
==========================
Assume that solving or simulating the ODE $$\label{first_order}
\dot{x} = \varphi(x)$$ is an intractable problem, i.e., the structure of $\varphi$ makes the problem not computationally amenable to an iterative numerical procedure. We denote the flow map of by $\Phi_t$. The idea is then to split the vector field $\varphi:\mathbb{R}^n \to \mathbb{R}^n$ into parts, each integrable or amenable to a feasible numerical approximation. For simplicity, consider $$\label{splitphi}
\varphi = \varphi^{(1)} + \varphi^{(2)}$$ and suppose that both $$\label{spliteq}
\dot{x} = \varphi^{(1)}(x),
\qquad \dot{x} = \varphi^{(2)}(x) ,$$ are feasible, either analytically or numerically, with respective flow maps $\Phi_t^{(1)}$ and $\Phi_t^{(2)}$. For step size $h$, it can be shown that the simplest composition [@Hairer] $$\label{composition}
\hat{\Phi}_h = \Phi_h^{(2)}\circ\Phi_h^{(1)}$$ provides a first-order approximation in the following sense.
\[order\_def\] Consider a map $\hat{\Phi}_h : \mathbb{R}^n \to \mathbb{R}^n$, with step size $h > 0$, which approximates the true flow $\Phi_t$ of the ODE with vector field $\varphi : \mathbb{R}^n \to \mathbb{R}^n$. Then $\hat{\Phi}_h$ is said to be an integrator of order $p$ if, for any $x \in \mathbb{R}^n$, it holds that $$\| \Phi_h(x) - \hat{\Phi}_{h}(x) \| = \Order(h^{p+1}) .$$ This implies a global error $\| \Phi_{t_k}(x) - (\hat{\Phi}_h)^{k}(x)\|=\Order(h)$ for a finite interval $t_k = hk$.
There are different ways to compose individual flows, each resulting in a different method. For instance, one can use a preprocessor map $\chi_h:\mathbb{R}^n \to \mathbb{R}^n$ such that $$\label{preprocessor}
\tilde{\Phi}_h = \chi_h^{-1}\circ \hat{\Phi}_h \circ \chi_h$$ is more accurate than $\hat{\Phi}_h$ with little extra cost. There are many interesting ideas in splitting methods for ODEs, some quite sophisticated. We mention these options to highlight that exploration beyond that considered in this paper is possible [@McLachlan:2002; @Blanes:2008; @Hairer]. Naturally, more accurate methods are more expensive since they involve extra computation of individual flows. A good balance between accuracy and computational cost are methods of order $p=2$. Here we will focus on the simple first-order scheme , which suffices to make connections with many optimization methods.
Balanced splitting {#bal_splitting}
------------------
In general, splittings such as do not preserve steady states of the system . Recently, an approach designed to preserve steady states was proposed under the name of *balanced splitting* [@Speth:2013]. The idea is to introduce a balance coefficient $c = c(t)$ into by writing $\dot{x} = \varphi(x) + c - c$ and then perform a splitting as above, which results in the pair of ODEs $$\label{balancedsplit}
\dot{x} = \varphi^{(1)}(x) + c,
\qquad \dot{x} = \varphi^{(2)}(x) - c.$$ We now show how this may be used to preserve steady states of the system .
First, assume $x_\infty$ is a steady state of the system so that $x_\infty = \lim_{t\to\infty}x(t)$ satisfies $\varphi^{(1)}(x_\infty) + \varphi^{(2)}(x_\infty) = 0$. If $c_\infty = \lim_{t\to\infty} c(t)$ is found to satisfy $$\label{cinfty}
c_\infty = \tfrac{1}{2}\big(\varphi^{(2)}(x_\infty) -
\varphi^{(1)}(x_\infty)\big)$$ then $x_\infty$ is also a stationary state for both ODEs in since
\[steadystate\] $$\begin{aligned}
\varphi^{(1)}(x_\infty) + c_\infty
&= \tfrac{1}{2}\big(\varphi^{(1)} + \varphi^{(2)}\big)(x_\infty) = 0, \\ \varphi^{(2)}(x_\infty) - c_\infty
&= \tfrac{1}{2}\big(\varphi^{(2)} + \varphi^{(1)}\big)(x_\infty) = 0.\end{aligned}$$
To establish a result for the other direction, now assume that $x_\infty$ is a steady state of both ODEs in . It follows that $$\label{cinfty.2}
c_\infty
= \varphi^{(2)}(x_\infty)
= - \varphi^{(1)}(x_\infty)
= \tfrac{1}{2}\big(\varphi^{(2)}(x_\infty) -
\varphi^{(1)}(x_\infty)\big).$$ From and the fact that $x_\infty$ is stationary for both systems in gives
\[steadystate.all\] \[steadystate.2\] $$\begin{aligned}
0 &= \varphi^{(1)}(x_\infty) + c_\infty
= \tfrac{1}{2}\big(\varphi^{(1)} + \varphi^{(2)}\big)(x_\infty), \\ 0 &= \varphi^{(2)}(x_\infty) - c_\infty
= \tfrac{1}{2}\big(\varphi^{(2)} + \varphi^{(1)}\big)(x_\infty),\end{aligned}$$
so that both equations in imply that $x_\infty$ is stationary for . Motivated by , this can be implemented by computing the updates $c_k = \tfrac{1}{2}\big(\varphi^{(2)}(x_k) - \varphi^{(1)}(x_k) \big)$ during the numerical method, together with discretizations of the ODEs in . Note that this approach requires the explicit computation of $\varphi^{(i)}$. In optimization, one might have $\varphi^{(i)} = -\nabla f^{(i)}$, in which case it is not well-defined when $f^{(i)}$ is nonsmooth. We address this concern in the next section.
Rebalanced splitting {#rebal_splitting_sec}
--------------------
The *rebalanced splitting* approach was proposed by [@Speth:2013], and claimed to be more stable than the balanced splitting of Section \[bal\_splitting\]. Importantly, for our purposes, it allows for the computation of a balance coefficient using only the previous iterates so that, in particular, no evaluation of $\varphi^{(i)}$ is needed. Let $t_k = kh$ for $k\in\{0,1,2,\dotsc\}$ and step size $h>0$. We then integrate $\dot{x} = \varphi^{(1)}(x) + c_k$ with initial condition $x(t_k) = x_k$ over the interval $[t_k, t_k+h]$ to obtain $x_{k+1/2}$, and then integrate $\dot{x} = \varphi^{(2)}(x) - c_k$ over the interval $[t_k, t_k+h]$ with initial condition $x(t_k) = x_{k+1/2}$ to obtain $x_{k+1}$ (note that $c_k$ is kept fixed during this procedure). The resulting integrals are given by
\[rebal\] $$\begin{aligned}
x_{k+1/2} &= x_k +
\int_{t_k}^{t_k + h}\big(\varphi^{(1)}(x(t))+c_k\big)dt, \label{rebal1}\\
x_{k+1} &= x_{k+1/2} +
\int_{t_k}^{t_k + h}\big(\varphi^{(2)}(x(t))-c_k\big)dt. \label{rebal2}\end{aligned}$$
In light of , two reasonable ways of computing $c_{k+1}$ are given by the *average* of either $\tfrac{1}{2}(\varphi^{(1)} - \varphi^{(2)})$ or $\varphi^{(2)}$ over the time step, which with gives, respectively,
\[crebal\] $$\begin{aligned}
c_{k+1}
&= \dfrac{1}{h}\int_{t_k}^{t_k + h}\!\! \dfrac{ \varphi^{(2)}(x(t)) - \varphi^{(1)}(x(t)) }{2} dt
= c_k + \dfrac{1}{h}\left( \dfrac{x_{k+1} + x_k}{2} - x_{k+1/2} \right) , \label{crebal.1} \\
c_{k+1}
&= \dfrac{1}{h}\int_{t_k}^{t_k+h} \!\! \varphi^{(2)}(x(t)) dt = c_k + \dfrac{1}{h}\left(x_{k+1}-x_{k+1/2} \right). \label{crebal.2}\end{aligned}$$
In contrast to the balanced case in Section \[bal\_splitting\], both of these options need not compute $\varphi^{(i)}$ to obtain $c_{k+1}$ as shown in . Thus, the above approaches are better suited for nonsmooth optimization since they do not require explicit gradient computations. Both options in are slight variations of the approach proposed in [@Speth:2013]. To motivate the potential usefulness of compared to , let us first remark that the updates to the balance coefficient play an important role in the stability of the numerical method [@Speth:2013]. For example, if $\varphi^{(2)}$ is much more “stiff” than $\varphi^{(1)}$, the method may be unstable for large step sizes. In an optimization context where $\varphi^{(2)}$ may be related to a regularizer (e.g., $\varphi^{(2)}(x) = \partial \| x \|_1$), it may be desirable to preserve steady states only through the first system in , which leads to the choice . In Section \[extension\_admm\], we show that this type of rebalanced splitting is related to the ADMM algorithm since the dual variable is precisely the balance coefficient in .
Proximal Algorithms from ODE Splittings {#proximal_splitting}
=======================================
We now use the previous ideas to construct implicit discretizations of both the gradient flow and the accelerated gradient flow . However, our emphasis is on the latter since the analysis is more involved and can be easily adapted to the former. In addition to Assumption \[assump1\], throughout this section we assume the following conditions.
\[assump2\] The functions $f$, $g$ and $w$ in the optimization problem are continuous differentiable with Lipschitz continuous gradients.
Lipschitz continuity of the gradients ensures uniqueness of solutions of the ODEs [@Butcher]. Finally, in continuous-time both the gradient flow and the accelerated gradient flow , with damping as in or , asymptotically solve the optimization problem since these systems are stable and their trajectories tend to lower level sets of $F$ [@Franca:2018b; @Franca:2018]. In the following we construct suitable discretizations of these ODEs.
Accelerated extensions of ADMM {#extension_admm}
------------------------------
Let us introduce a balance coefficient $c=c(t)$ and write as a first-order system: $$\label{secbal}
\dot{x} = v , \qquad \dot{v} = \underbrace{-\eta(t) v - \nabla f(x) -\nabla w(x)}_{\varphi^{(1)}} \underbrace{-\nabla g(x)}_{\varphi^{(2)}} +c - c .$$ Splitting the second ODE above as indicated, we obtain the two independent systems $$\label{admm_twosplit_pre1}
\begin{cases}
\dot{x} &= v \\ \dot{v} &= -\eta(t) v - \nabla f(x) - \nabla w(x) + c ,
\end{cases} \qquad \begin{cases}
\dot{x} &= v \\
\dot{v} &= -\nabla g(x) - c .
\end{cases}$$ Note that we are only splitting the second equation in . It will be convenient to treat each of these respective systems in their equivalent second-order forms: $$\label{admm_twosplit}
\ddot{x} + \eta(t) \dot{x} = -\nabla f(x) - \nabla w(x) + c , \qquad \ddot{x} = -\nabla g(x) - c.$$ We now discretize these systems using the results from Lemma \[implilemma\], although we introduce some intermediary steps that will be justified by our analysis later. To this end, let us choose a step size parametrized as $h \equiv \sqrt{\lambda}$, and then use (after dropping the $O(h)$ error term) and a semi-implicit discretization on the first equation of to obtain the equation $$\label{admm_disc1}
x_{k+1/2} - \hat{x}_k = -\lambda \nabla f(x_{k+1/2}) -
\lambda \nabla w(\hat{x}_k) + \lambda c_k .$$ This can now be solved with the resolvent (i.e., in an analogous way as the second relation in ) to obtain the equation $$\label{admm_up1}
x_{k+1/2} = J_{\lambda \nabla f}\big( \hat{x}_k - \lambda \nabla w(\hat{x}_k) + \lambda c_k \big).$$ For the second ODE in we use (after dropping the $O(h)$ term) and an implicit discretization to obtain the equation $$\label{xtilde}
\tilde{x}_{k+1} -2 x_{k+1/2} + \hat{x}_k = -\lambda \nabla g(x_{k+1}) -
\lambda c_k$$ where $$\label{xtilde1}
\tilde{x}_{k+1} = x_{k+1} + (x_{k+1/2} - \hat{x}_k).$$ Note that the endpoint $\tilde{x}_{k+1}$ is related to the other endpoint $x_{k+1}$ via the momentum term $(x_{k+1/2} - \hat{x}_k)$ based on the first splitting, and together this results in[^5] $$\label{admm_disc2}
x_{k+1} - x_{k+1/2} = -\lambda \nabla g(x_{k+1}) - \lambda c_k.$$ This implicit equation can again be solved with the resolvent yielding $$\label{admm_up2}
x_{k+1} = J_{\lambda \nabla g}\big( x_{k+1/2} - \lambda c_k \big).$$
For the balance coefficient $c$, we use the update based on $\varphi^{(2)} = -\nabla g$ (see ). An implicit discretization is equivalent to approximating the integral by its upper limit, which in this cases results in $$\label{baladmm1}
c_{k+1} = \dfrac{1}{h}\int_{t_k}^{t_k+h} \varphi^{(2)} (x(t)) dt =
- \nabla g(x_{k+1}) + \Order(h).$$ Using , and neglecting $\Order(h)$ terms, we thus obtain $$\label{admm_up3}
c_{k+1} = c_k + \lambda^{-1}\left( x_{k+1}-x_{k+1/2} \right).$$ Collecting the updates , , , and we obtain Algorithm \[agenadmmrebal\].
Choose $\lambda > 0$, and initialize $c_0$ and $\hat{x}_0$. Choose $r \geq 3$ if decaying damping , or $r > 0$ if constant damping . $x_{k+1/2} \leftarrow J_{\lambda \nabla f}\left(
\hat{x}_k - \lambda \nabla w(\hat{x}_k) + \lambda c_k\right)$ $x_{k+1} \leftarrow J_{\lambda \nabla g}( x_{k+1/2} - \lambda c_k)$ $c_{k+1} \leftarrow c_k + \lambda^{-1}
\left( x_{k+1}-x_{k+1/2}\right)$ Using $h=\sqrt{\lambda}$ and $r$, compute $\gamma_{k+1}$ and $\hat{x}_{k+1}$ from .
We would like to stress some important aspects of Algorithm \[agenadmmrebal\].
- The standard ADMM [@Gabay:1976; @Glowinsky:1975] is recovered from Algorithm \[agenadmmrebal\] when $w=0$ in problem and no acceleration is used, i.e., when $\gamma_k=0$ so that $\hat{x}_k = x_k$ for all $k$. Algorithm \[agenadmmrebal\] extends ADMM to handle the case $w \neq 0$ in problem by incorporating $\nabla w$ in the update to $x_{k+1/2}$ in .
- The dual vector update in ADMM is here represented by the update to the balance coefficient $c_k$, which as described earlier aims to preserve critical points of the underlying ODE. This brings new meaning to the dual vector.
- Acceleration through the update to $\hat{x}_k$ based on vanishing and constant damping in have been considered. However, one is free to consider other damping functions $\eta(t)$ in the dynamical system as well. By a suitable discretization, this would lead to a new update to $\gamma_k$ in . For example, choosing $\eta(t) = r_1/t + r_2$ for constants $\{r_1,r_2\} \subset(0,\infty)$ yields $$\label{damping_combination}
\gamma_k = k/(k+r_1) + r_2.$$ This observation is valid for every accelerated algorithm derived in this paper.
- When decaying damping is chosen in and $w=0$ in problem , Algorithm \[agenadmmrebal\] is similar to the Fast ADMM proposed in [@Goldstein:2014]. They differ in that the latter also “accelerates” the dual variable $c$ (i.e., the Lagrange multiplier update). Connections between Fast ADMM and continuous dynamical systems was recently considered in [@Franca:2018; @Franca:2018b] and corresponds to system with $w=0$. However, in this case the discretization is not a rebalanced splitting.
- The choice of discretization leading to , which involves relating $\tilde{x}_{k+1}$ to $x_{k+1}$ (recall ), is motivated by obtaining updates similar to ADMM. This choice is formally justified by Theorem \[admm\_first\_order\], which shows that the discretization has a local error of $\Order(h^2)$ compared to the continuous trajectory.
\[admm\_grad\_flow\] Although we focus on accelerated algorithms, similar (and easier) analyses apply to the gradient flow which lead to non-accelerated variants of the respective algorithm. For example, as in , one can introduce a balance coefficient $c$ and split the system into $$\dot{x} = - \nabla f(x) - \nabla w(x) + c, \qquad \dot{x} = - \nabla g(x) - c.$$ Then, as for , a semi-implicit discretization of the first equation yields $$\label{admmu1}
x_{k+1/2} = J_{\lambda \nabla f}(x_k - \lambda \nabla w(x_k) + \lambda c_k),$$ where now $h \equiv \lambda $ in . An implicit discretization of the second equation yields $x_{k+1} - x_{k+1/2} = - \lambda \nabla g(x_{k+1}) - \lambda c_k$, so that $x_{k+1}$ can be computed as $$\label{admmu2}
x_{k+1} = J_{\lambda \nabla g}(x_{k+1/2} - \lambda c_k).$$ The balance coefficient update is obtained in an analogous manner as before. Note that updates and , together with , are precisely the ADMM algorithm in the particular case $w=0$.
The next result shows that the above discretization is justified since it yields a first-order integrator for the underlying ODE.
\[admm\_first\_order\] The following hold true:
1. Algo. \[agenadmmrebal\] is a first-order integrator to the accelerated gradient flow ;
2. Algo. \[agenadmmrebal\] with $\gamma_k = 0$ for all $k \geq 0$ is a first-order integrator to the gradient flow .
For $f$ satisfying Assumption \[assump1\] and Assumption \[assump2\] it holds that $y = J_{ \lambda \nabla f}(x)$ if and only if $y = x - \lambda \nabla f(y)$ (see ). Thus, Assumption \[assump2\] gives $$\label{proxappro}
y = J_{\lambda \nabla f}(x) = x - \lambda \nabla f(x - \lambda \nabla f(y))
= x - \lambda \nabla f(x) + \Order(\lambda^2),$$ where the $\|\nabla f(y)\|$ that normally appears in the $\Order$ term is suppressed because it is bounded independently of $\lambda$ for all $\lambda$ on a compact set as a consequence of $y = J_{\lambda \nabla f}(x)$ and Assumption \[assump2\]. (A similar convention is used in and below.) Thus, this equality and a Taylor expansion on $\nabla f$ in the first update of Algo. \[agenadmmrebal\] give $$x_{k+1/2} = \hat{x}_k - \lambda \nabla w(\hat{x}_k) + \lambda c_k -
\lambda \nabla f(\hat{x}_k) + \Order(\lambda^2). \label{limm}$$ Similarly, the second update of Algo. \[agenadmmrebal\] leads to $$\label{limmm}
\begin{split}
x_{k+1} &= x_{k+1/2} - \lambda c_k -\lambda \nabla g(x_{k+1/2}) + \Order(\lambda^2) \\
&= \hat{x}_k - \lambda \nabla F(\hat{x}_k) + \Order(\lambda^2)
\end{split}$$ where to derive the second equality we used and a Taylor expansion of $\nabla g$. Recall and note that $\gamma_k = 1- \eta(t) h$ for constant damping $(\eta(t)=r)$, while $$\gamma_k = \dfrac{k}{k+r} = 1 - \dfrac{r}{k+r} = 1 - \dfrac{r h }{t_k} \left(1+ \dfrac{rh}{t_k}\right)^{-1}
= 1 - \eta(t_k) h + \Order(h^2)$$ for decaying damping ($\eta(t) = r/t$). Thus, in either case, we conclude that $$\label{xhat_ord}
\begin{split}
\hat{x}_k
&= x_k + h \big( 1 - \eta(t_k)h \big) v_k + \Order(h^3) = x_k + \Order(h)
\end{split}$$ where we have defined the velocity variable $$\label{velocity}
v_k \equiv \left( x_k - x_{k-1} \right)/h,$$ which is finite even in the limit $h \to 0$. Using , both equalities in , , and recalling that $\lambda \equiv h^2$, we conclude that $$\label{admm_first_final}
\begin{split}
v_{k+1} &= v_k - h \eta(t_k) v_k - h \nabla F(x_k) + \Order(h^2), \\
x_{k+1} &= x_k + h v_{k+1} = x_k + h v_k + \Order(h^2).
\end{split}$$
Now, consider the ODE , i.e., $\dot{x} = v$ and $\dot{v} = -\eta(t) v - \nabla F(x)$. Combining this with Taylor expansions we have $$\label{xtplush}
\begin{split}
v(t+h) &= v(t) + h \dot{v}(t) + \Order(h^2) = v(t) - h \eta(t) v(t) - h \nabla F(x(t)) + \Order(h^2), \\
x(t+h) &= x(t) + h \dot{x}(t) + \Order(h^2) = x(t) + h v(t) + \Order(h^2) .
\end{split}$$ Therefore, by comparison with we conclude that $$\label{admm_first_order_final}
v(t_{k+1}) = v_{k+1} + \Order(h^2), \qquad x(t_{k+1}) = x_{k+1} + \Order(h^2) ,$$ i.e., in one step of the algorithm the discrete trajectory agrees with the continuous trajectory up to $\Order(h^2)$. This means that Definition \[order\_def\] is satisfied with $p=1$.
The above argument can be adapted to Algo. \[agenadmmrebal\] with $\gamma_k=0$ in relation to the gradient flow . The derivation is simpler and is thus omitted.
Accelerated extensions of Davis-Yin {#accel_dy}
-----------------------------------
We now split the accelerated gradient flow as in , but without introducing a balance coefficient, to obtain $$\varphi^{(1)} = -\eta(t) v - \nabla f(x), \qquad \varphi^{(2)} = - \nabla g(x) - \nabla w(x).$$ Hence, instead of , we obtain the following two individual ODEs: $$\label{ady2flow}
\ddot{x} + \eta(t)\dot{x} = -\nabla f(x), \qquad \ddot{x} = -\nabla g(x) - \nabla w(x).$$ An implicit discretization of the first system is $$\label{ady.disc1}
x_{k+1/4} - \hat{x}_k = -\lambda \nabla f(x_{k+1/4})$$ where $h \equiv \sqrt{\lambda}$, which as a result of Lemma \[implilemma\] leads to $$\label{ady.u1}
x_{k+1/4} \equiv \Phi_{h}^{(1)}(\hat{x}_k) = J_{\lambda \nabla f}(\hat{x}_k) .$$ Next, to “inject momentum” in the direction of $\nabla f$, we define the translation operator $$\mathcal{T}_h(z) \equiv z - \lambda \nabla f(x_{k+1/4})$$ for any vector $z$. Thus, the next point in the discretization is defined to be $$\label{ady.u2}
x_{k+1/2} \equiv \mathcal{T}_{h}(x_{k+1/4}) = x_{k+1/4} - \lambda \nabla f(x_{k+1/4}) = 2x_{k+1/4} - \hat{x}_k$$ where we used to obtain the last equality. Next, we can use to obtain a semi-implicit discretization of the second system in given by $$\label{adyfirst}
x_{k+3/4}-2x_{k+1/4}+\hat{x}_k = - \lambda \nabla g(x_{k+3/4}) - \lambda \nabla w(x_{k+1/4}).$$ This allows us to solve the implicit equation in the form $$\label{ady.u3}
x_{k+3/4} \equiv \Phi_h^{(2)}(\hat{x}_k) = J_{\lambda \nabla g}\left(x_{k+1/2} - \lambda \nabla w(x_{k+1/4})
\right).$$ Finally, we apply the inverse $\mathcal{T}^{-1}_h(z) \equiv z + \lambda \nabla f(x_{k+1/4} )$ and use to obtain $$\label{ady.u4}
x_{k+1} \equiv \mathcal{T}^{-1}_{h}(x_{k+3/4})
= x_{k+3/4} + \lambda \nabla f(x_{k+1/4})
= x_{k+3/4}-(x_{k+1/4}-\hat{x}_k).$$ The collection of , , , and results in Algo. \[ady\], and the entire discretization procedure is illustrated in Fig. \[discpic2\].
Choose $\lambda > 0$, and initialize $\hat{x}_0$. Choose $r \ge 3$ if decaying damping , or $r > 0$ if constant damping . $x_{k+1/4} \leftarrow J_{\lambda \nabla f}(\hat{x}_k)$ $x_{k+1/2} \leftarrow 2x_{k+1/4} - \hat{x}_k$ $x_{k+3/4} \leftarrow J_{\lambda \nabla g}\big(x_{k+1/2} - \lambda \nabla w(x_{k+1/4})\big)$ $x_{k+1} \leftarrow \hat{x}_k + x_{k+3/4} - x_{k+1/4}$ Using $h = \sqrt{\lambda}$ and $r$, compute $\gamma_{k+1}$ and $\hat{x}_{k+1}$ from .
![ \[discpic2\] An illustration of the discretization underlying Algo. \[ady\], consistent with the mapping , that gives the accelerated Davis-Yin method. We indicate the points at which the gradients act that define the implicit/explicit discretization. ](davis_yin_discretization.pdf)
The following comments concerning Algo. \[ady\] are appropriate.
- Algo. \[ady\] reduces to the Davis-Yin method [@Davis:2017] when $\gamma_k = 0$ in , so that $\hat{x}_k = x_k$ for all $k$, i.e., when no acceleration is used. In this case, it has been shown for convex functions that the method has a convergence result of $\Order( 1/k)$ in an average or ergodic sense, and when all functions are strongly convex and satisfy some regularity conditions that linear convergence holds [@Davis:2017]. In the non-accelerated case, the algorithm corresponds to the application of a similar discretization of the gradient flow with splitting $\dot{x} = -\nabla f(x)$ and $\dot{x} = -\nabla g(x) - \nabla w(x)$ (also see Remark \[admm\_grad\_flow\]).
- Algo. \[ady\] is equivalent to the composition $$\label{dycompos}
\hat{\Phi}_h =
\mathcal{T}^{-1}_h\circ\Phi_h^{(2)}\circ\mathcal{T}_h\circ\Phi_h^{(1)} .$$ Thus, comparing with we see that $\mathcal{T}_h$ is actually a preprocessor map.
- In Theorem \[dy\_first\_order\] we show that the above procedure yields a first-order integrator. Furthermore, in Theorem \[dy\_critical\] we show that this discretization preserves critical points of the underlying ODE.
\[dy\_first\_order\] The following hold true:
1. Algo. \[ady\] is a first-order integrator to the accelerated gradient flow ;
2. Algo. \[ady\] with $\gamma_k=0$ for all $k \geq 0$ is a first-order integrator to the gradient flow .
The arguments are very similar to the proof of Theorem \[admm\_first\_order\]. From and Taylor expansions, the first three updates of Algo. \[ady\] yield
$$\begin{aligned}
x_{k+1/4} &= \hat{x}_k - \lambda \nabla f(\hat{x}_k) + \Order(\lambda^2), \\
x_{k+1/2} &= \hat{x}_k - 2 \lambda \nabla f(\hat{x}_k) + \Order(\lambda^2), \\
x_{k+3/4} &= \hat{x}_k - 2\lambda \nabla f(\hat{x}_k) -
\lambda \nabla g(\hat{x}_k) - \lambda \nabla w(\hat{x}_k) + \Order(\lambda^2).\end{aligned}$$
Hence, the fourth update of Algo. \[ady\] becomes $$x_{k+1} = \hat{x}_k - \lambda \nabla F(\hat{x}_k) + \Order(\lambda^2),$$ which is exactly the same as equation . Therefore, the remaining steps of the proof follow exactly as in the proof of Theorem \[admm\_first\_order\], and establishes that Algo. \[ady\] is an integrator with $p=1$ according to Definition \[order\_def\].
The proof related to the gradient flow (i.e., with $\gamma_k=0$ in Algo. \[ady\]) is similar, but easier, and therefore omitted.
In order to show that Algo. \[ady\] preserves critical points of the underlying ODE, we require the following technical result.
\[caleylemma\] It holds that $(\nabla f + \nabla g + \nabla w)(\bar{x}) = 0$ if and only if $\mathcal{P}(x) = x$ with $$\label{Top}
\mathcal{P} \equiv \tfrac{1}{2}I +
\tfrac{1}{2}C_{\lambda \nabla g}\circ \left( C_{\lambda \nabla f} -\lambda \nabla w \circ J_{\lambda \nabla f} \right) - \tfrac{1}{2} \lambda \nabla w \circ J_{\lambda \nabla f},$$ $\bar{x} = J_{\lambda \nabla f}(x)$, and $C_{\lambda \nabla f} \equiv 2 J_{\lambda \nabla f} - I$ is the Cayley operator.
The first equation in the statement of the theorem is equivalent to $(I+\lambda \nabla g)(\bar{x}) = (I-\lambda \nabla f -
\lambda \nabla w)(\bar{x})$ since $\lambda > 0$. Using the resolvent this yields $$\label{lem1id}
\bar{x} = J_{\lambda \nabla g}\circ\left( I-\lambda \nabla f -
\lambda \nabla w \right) (\bar{x}).$$ Now, making use of the identity $$C_{\lambda \nabla f}\circ(I+\lambda \nabla f) = \left(
2(I+\lambda \nabla f)^{-1}-I\right)\circ
(I + \lambda \nabla f) = I -\lambda \nabla f$$ and substituting $J_{\lambda \nabla g} = \tfrac{1}{2}(C_{\lambda \nabla g} +I)$ into we obtain $$\label{cinter}
\bar{x} = \tfrac{1}{2}\left( C_{\lambda \nabla g} + I \right)
\circ \left\{ C_{\lambda \nabla f}\circ(I + \lambda \nabla f)
-\lambda \nabla w\right\}(\bar{x}).$$ Since $x \equiv (I + \lambda \nabla f)(\bar{x})$, or equivalently $\bar{x} = J_{\lambda \nabla f}(x)$, it follows from that $$\label{iff1}
J_{\lambda \nabla f} (x) = \tfrac{1}{2} C_{\lambda \nabla g}\circ\left(
C_{\lambda \nabla f} - \lambda \nabla w \circ J_{\lambda \nabla f}\right)(x)
+ \tfrac{1}{2} C_{\lambda \nabla f}(x) -
\tfrac{1}{2}\lambda \nabla w\circ J_{\lambda \nabla f}(x),$$ which by the definition of the Cayley operator is equivalent to $$\tfrac{1}{2} x = \tfrac{1}{2} C_{\lambda \nabla g}\circ\left(
C_{\lambda \nabla f} - \lambda \nabla w \circ J_{\lambda \nabla f}\right)(x)
-
\tfrac{1}{2}\lambda \nabla w\circ J_{\lambda \nabla f}(x).$$ Adding $x/2$ to each side of the previous equality yields $x = \mathcal{P}(x)$, as claimed.
\[dy\_critical\] If the iterate sequence $\{x_k\}$ generated by Algo. \[ady\] satisfies $\{x_k\}\to x_\infty$ for some $x_\infty$, then the vector $\bar{x}_\infty \equiv J_{\lambda \nabla f}(x_\infty)$ satisfies $$\{x_{k+1/4}\} \to \bar{x}_\infty,
\qquad (\nabla f + \nabla g + \nabla w)(\bar{x}_\infty) = 0,$$ i.e., $\bar{x}_\infty$ is a solution of and a steady state of the accelerated gradient flow . When $\gamma_k=0$ for all $k \geq 0$, $\bar{x}_\infty$ is a steady state of the gradient flow .
It follows from the updates in Algo. \[ady\], the definition of $C_{\lambda \nabla f}$ in the statement of Lemma \[caleylemma\], and the definition of $\mathcal{P}$ in that $x_{k+1} = \hat{\Phi}_h(\hat{x}_k)$ where $$\label{dycaley}
\begin{split}
\hat{\Phi}_h &=
I + J_{\lambda \nabla g}\circ\left(2 J_{\lambda \nabla f} - I - \lambda \nabla w
\circ J_{\lambda \nabla f}\right) - J_{\lambda \nabla f} \\
&= I + J_{\lambda \nabla g}
\circ\left( C_{\lambda \nabla f} - \lambda \nabla w \circ
J_{\lambda \nabla f} \right) - \tfrac{1}{2}(C_{\lambda \nabla f} + I) \\
&= \tfrac{1}{2}I - \tfrac{1}{2} C_{\lambda \nabla f} +
\tfrac{1}{2}\left(C_{\lambda \nabla g} + I\right)\circ
\left( C_{\lambda \nabla f} -
\lambda \nabla w\circ J_{\lambda \nabla f} \right) \\
&= \tfrac{1}{2}I +\tfrac{1}{2}C_{\lambda \nabla g}\circ\left( C_{\lambda \nabla f}
- \lambda \nabla w \circ J_{\lambda \nabla f} \right) - \tfrac{1}{2} \lambda \nabla w \circ
J_{\lambda \nabla f} \\
&= \mathcal{P},
\end{split}$$ i.e., that $x_{k+1} = \mathcal{P}(\hat{x}_k)$. (Note that $\mathcal{P}$ is the operator $\mathcal{T}$ studied in [@Davis:2017] and proved to be an “averaged operator", which means that $\mathcal{P}$ is a nonexpansive operator and thus continuous [@BauschkeBook Remark 4.34].) Combining this with shows that $$\label{P-update-DY}
\hat{x}_{k+1}
= x_{k+1} + \gamma_{k+1}(x_{k+1} - x_k)
= \mathcal{P}(\hat{x}_k) + \gamma_{k+1}(x_{k+1} - x_k).$$ By assumption there exists some $x_\infty$ such that $\{x_k\} \to x_\infty$, which combined with shows that $\{\hat{x}_k\} \to x_\infty$. Combining these facts with and continuity of $\mathcal{P}$ establishes that $\mathcal{P}(x_\infty) = x_\infty$. It now follows from Lemma \[caleylemma\] that $\bar{x}_\infty = J_{\lambda \nabla f}(x_\infty)$ satisfies $(\nabla f + \nabla g + \nabla w)(\bar{x}_\infty) = 0$. Moreover, from the update for $x_{k+1/4}$ in Algo. \[ady\] and continuity of $J_{\lambda \nabla f}$ we can conclude that $$\lim_{k\to\infty} x_{k+1/4}
= \lim_{k\to\infty} J_{\lambda \nabla f}(\hat{x}_k)
= J_{\lambda \nabla f}(x_\infty)
= \bar{x}_\infty.$$ This completes the proof for this case once we recall that $(\nabla f + \nabla g + \nabla w)(\bar{x}_\infty) = 0$. The same argument applies when $\gamma_k = 0$ (i.e., when $\hat{x}_k = x_k$ for all $k \geq 0$), in which case Algo. \[ady\] is a discretization of .
### Accelerated extensions of Douglas-Rachford {#dougrach}
When Algo. \[ady\] with $\gamma_k = 0$ for all $k$ is applied to problem with $w=0$, one obtains the well-known Douglas-Rachford algorithm [@Douglas:1956; @Lions:1979], which has been extensively studied in the literature (for recent results see [@Moursi:2017] and the references therein). Therefore, Douglas-Rachford is a discretization of the gradient flow with $w=0$. Also, from Algo. \[ady\] one obtains accelerated variants of Douglas-Rachford that are discretizations of the accelerated gradient flow . For instance, when decaying damping in is used, the resulting method was studied in [@Patrinos:2014]. We are not aware of previous work on accelerated variants with constant damping, or other choices such as .
### Accelerated extensions of forward-backward {#fbsec}
When Algo. \[ady\] with $\gamma_k=0$ for all $k$ is applied to problem with $f=0$, one obtains the forward-backward splitting method [@Lions:1979; @Passty:1979; @Han:1988], i.e., $x_{k+1} = J_{\lambda \nabla g}(x_{k}-\lambda \nabla w(x_k))$. This corresponds to a first-order integrator to the gradient flow . Similarly, for nonzero $\gamma_k$, Algo. \[ady\] gives accelerated variants of forward-backward, i.e., $x_{k+1} = J_{\lambda \nabla g}\big( \hat{x}_k -
\lambda \nabla w(\hat{x}_k) \big)$. From an ODE perspective, this is not a splitting method but rather a semi-implicit discretization; the first equation in is absent (also see Fig. \[tsengfig\] (left) for an illustration). In any case, Theorem \[dy\_first\_order\] ensures that the iterates correspond to first-order integrators, and Theorem \[dy\_critical\] shows that critical points are preserved, where the operator in reduces to $J_{\lambda \nabla g}\circ(I-\lambda \nabla w)$.
Accelerated extensions of Tseng’s splitting {#subsec:tseng}
-------------------------------------------
The final proximal-based method to be considered is the forward-backward-forward splitting proposed by Tseng [@Tseng:2000], which consists of a modification (a perturbation) of the forward-backward splitting discussed above. In order to propose accelerated extensions of Tseng’s scheme, we consider the accelerated gradient flow with $f=0$ written as $$\dot{x} = v , \qquad \dot{v} = \underbrace{-\eta(t) v -\nabla g(x) - \nabla w(x)}_{\varphi^{(1)}}
+ \underbrace{\nabla w(x) - \nabla w(x)}_{\varphi^{(2)}}.$$ Splitting this system leads to the two independent ODEs $$\label{tseng_split}
\ddot{x} + \eta(t) \dot{x} = -\nabla g(x)-\nabla w(x), \qquad \ddot{x} = \nabla w(x)-\nabla w(x).$$ Using $h \equiv \sqrt{\lambda}$, , and a forward-backward discretization of the first equation gives $$\label{tseng.u1}
x_{k+1/2} = J_{\lambda \nabla g}\left( \hat{x}_k - \lambda
\nabla w (\hat{x}_k) \right).$$ This is the same step as in the forward-backward method. For the second equation in we use in the form $\tilde{x}_{k+1} - 2 x_{k+1/2} +
\hat{x}_k = \lambda \nabla w(\hat{x}_k) - \lambda \nabla w(x_{k+1/2})$, where $\tilde{x}_{k+1}$ is given by . This gives $$\label{tseng.u2}
x_{k+1} = x_{k+1/2} - \lambda \big( \nabla w(x_{k+1/2}) - \nabla w(\hat{x}_k) \big).$$ By combining and we arrive at Algo. \[atseng\] (also see Fig. \[tsengfig\]).
Choose $\lambda > 0$, and initialize $\hat{x}_0$. Choose $r\ge 3$ if decaying damping , or $r > 0$ if constant damping . $x_{k+1/2} \leftarrow J_{\lambda \nabla g}\left(\hat{x}_k-
\lambda \nabla w(\hat{x}_k)\right)$ $x_{k+1} \leftarrow x_{k+1/2} -\lambda \left( \nabla w(x_{k+1/2})-
\nabla w(\hat{x}_k) \right)$ Using $h = \sqrt{\lambda}$ and $r$, compute $\gamma_{k+1}$ and $\hat{x}_{k+1}$ from .
![\[tsengfig\] *Left:* Illustration of the accelerated forward-backward method, which is a semi-implicit Euler discretization. *Right:* Illustration of accelerated Tseng splitting, which adds a perturbation to the forward-backward method (see ). ](fb_discretization.pdf "fig:")![\[tsengfig\] *Left:* Illustration of the accelerated forward-backward method, which is a semi-implicit Euler discretization. *Right:* Illustration of accelerated Tseng splitting, which adds a perturbation to the forward-backward method (see ). ](tseng_discretization.pdf "fig:")
The original method proposed in [@Tseng:2000] is recovered from Algo. \[atseng\] by setting $\gamma_k=0$ for all $k$ (i.e., without acceleration), in which case the algorithm is a discretization of the gradient flow . We believe that the accelerated variants in Algo. \[atseng\] have not previously been considered in the literature.
The next result shows that Algo. \[atseng\] is a first-order integrator.
\[tsengcritical\] The following hold true:
1. Algo. \[atseng\] is a first-order integrator to the accelerated gradient flow ;
2. Algo. \[atseng\] with $\gamma_k=0$ for all $k \geq 0$ is a first-order integrator to the gradient flow .
The results may be proved using the similar arguments as those used to establish Theorem \[admm\_first\_order\] and Theorem \[dy\_first\_order\].
Let $f = 0$. If $\lambda$ is sufficiently small and the iterate sequence $\{x_k\}$ generated by Algo. \[atseng\] satisfies $\{x_k\} \to x_\infty$ for some $x_\infty$, then $$(\nabla g + \nabla w)(x_\infty) = 0,$$ i.e., $x_\infty$ is a solution of problem and a steady state of the accelerated gradient flow . When $\gamma_k = 0$ for all $k \geq 0$, $x_\infty$ is a steady state of the gradient flow .
From the updates in Algo. \[atseng\], it follows that $x_{k+1} = \hat{\Phi}_h(\hat{x}_k)$ where $$\label{tsengop}
\hat{\Phi}_h
= (I - \lambda \nabla w)\circ
J_{\lambda \nabla g}\circ(I - \lambda \nabla w) + \lambda \nabla w.$$ Combining this with shows that $$\label{xhat-eq-tseng}
\hat{x}_{k+1}
= x_{k+1} + \gamma_{k+1}(x_{k+1} - x_{k})
= \hat{\Phi}_h(\hat{x}_k) + \gamma_{k+1}(x_{k+1} - x_{k}).$$ Combining $\{x_k\} \to x_\infty$ with the first equality in and yields $\{\hat{x}_k\} \to x_\infty$. Combining these facts with and continuity of $\hat{\Phi}_h$ shows that $$x_\infty
= \lim_{k\to\infty} \hat{\Phi}_h(\hat{x}_k)
= \hat{\Phi}_h(x_\infty).$$ Subtracting $\lambda \nabla w(x_\infty)$ from both sides of the previous inequality gives $$(I-\lambda \nabla w)(x_\infty)
= (I - \lambda \nabla w)\circ
J_{\lambda \nabla g}\circ(I - \lambda \nabla w)(x_\infty).$$ Next, applying the operator $(I - \lambda \nabla w)^{-1}$, which exists for $\lambda$ sufficiently small, yields $$x_\infty
= J_{\lambda \nabla g}\circ(I - \lambda \nabla w)(x_\infty),$$ which is itself equivalent to $$(I+\lambda\nabla g)(x_\infty)
= (I - \lambda \nabla w)(x_\infty).$$ Since $\lambda > 0$, the previous inequality shows that $(\nabla g + \nabla w)(x_\infty) = 0$, as claimed.
The same argument applies when $\gamma_k = 0$ (i.e., when $\hat{x}_k = x_k$ for all $k \geq 0$), in which case Algo. \[atseng\] is a discretization of .
Monotone Operators {#nonsmooth}
==================
As indicated by Assumption \[assump2\], all previously considered operators associated with the resolvent were single-valued. Since proximal algorithms can be generalized to the more abstract level of monotone operators, in this section we discuss how our previous analysis applies in this context.
Let us recall some concepts about monotone operators (see [@BauschkeBook] for more details). Let $H$ be a Hilbert space with inner product $\langle \cdot, \cdot \rangle:
H \times H \to \mathbb{C}$. A multi-valued map $A: H \rightrightarrows H$ with $\dom A \equiv \{ x \in H \, \vert \, Ax \ne \emptyset \}$ is *monotone* if and only if $$\langle A y - Ax, y - x\rangle \ge 0 \qquad \mbox{ for all
$x,y\in\dom A$}.$$ A monotone operator is said to be *maximal* if no enlargement of its graph is possible. Every monotone operator admits a maximal extension. Hence, from now on, every operator $A$ is assumed to be maximal monotone. The resolvent of $A$ with parameter $\lambda$ is defined by and can be shown to be a single-valued map, i.e., $J_{\lambda A}: H \to H$. Moreover, $x^\star \in \zer(A) \equiv \{x\in H \, \vert \, 0\in Ax \}$ if and only if $J_{\lambda A}(x^\star) = x^\star$. An important concept is the *Yosida regularization* of $A$ with parameter $\mu > 0$: $$\label{yosida}
A_\mu \equiv \mu^{-1} (I - J_{\mu A}).$$ This operator is *single-valued*, since $J_{\mu A}$ is single-valued, and Lipschitz continuous. Moreover, $0 \in A x^{\star}$ if and only if $0=A_{\mu} x^\star$, thus $A$ and $A_\mu$ have the same zeros. It can be shown that in the limit $\mu \downarrow 0$ one has $A_{\mu} x \to A_0 x$, where $A_0 x \in A x$ is the element of minimal norm. Often, one is faced with the resolvent of the Yosida regularization $J_{\lambda A_{\mu}} = (I + \lambda A_{\mu})^{-1}$, which can be expressed in terms of the operator $A$ by using $$\label{resolvent3}
J_{\lambda A_\mu} = (\mu + \lambda)^{-1}\left( \mu I + \lambda J_{(\mu + \lambda)A} \right).$$ Importantly, in the limit $\mu \downarrow 0$ we see that recovers the resolvent $J_{\lambda A}$.
Differential inclusions
-----------------------
Consider the following two differential inclusions (we refer to [@CellinaBook] for background on nonsmooth dynamical systems): $$\begin{aligned}
\label{firstmon}
\dot{x} & \in -A x - B x - Cx , \\ \label{secmon}
\ddot{x} + \eta(t)\dot{x} & \in -A x - B x - Cx ,\end{aligned}$$ with $\eta(t)$ given by or , and under the following assumption.
\[max\_mon\_assump\] The operators $A,B : H \rightrightarrows H$ are maximal monotone. The operator $C : H \to H$ is maximal monotone and single-valued.
Under Assumption \[max\_mon\_assump\], the differential inclusions and have a unique solution [@CellinaBook]. The previous discretizations of the gradient flow and the accelerated gradient flow extend naturally to and , respectively. This is a consequence of the resolvent being a single-valued map, which we illustrate through an example. Consider the procedure of Section \[accel\_dy\] that led to Algo. \[ady\]. Using a similar procedure, we use a splitting of to obtain the differential inclusions $$\ddot{x} + \eta(t) \dot{x} \in -A x , \qquad \ddot{x} + C x\in -B x.$$ An implicit discretization of the first inclusion yields $(I + \lambda A)(x_{k+1/4}) \ni \hat{x}_k$, while a semi-implicit discretization of the second inclusion together with the definition yield $(I + \lambda B)(x_{k+3/4}) \ni x_{k+1/2} - \lambda C x_{k+1/4}$. Since under Assumption \[max\_mon\_assump\] the resolvents of $A$ and $B$ are single-valued, we can invert these relations to obtain
\[ady.mon\] $$\begin{aligned}
x_{k+1/4} &= J_{\lambda A}(\hat{x}_k), \\
x_{k+1/2} &= 2 x_{k+1/4} - \hat{x}_k, \\
x_{k+3/4} &= J_{\lambda B}(x_{k+1/2} - \lambda C x_{k+1/4}), \\
x_{k+1} &= x_{k+3/4} - (x_{k+1/4} - \hat{x}_k), \\
\hat{x}_{k+1} &= x_{k+1} + \gamma_k (x_{k+1} - x_{k}),\end{aligned}$$
where the second to last update follows from , and the last update from . The algorithm given by updates is the “operator analog” of Algo. \[ady\] that aims to find a vector $x^*$ satisfying $0 \in (A + B + C)(x^\star)$.
Setting $C = 0$ into one obtains the operator analog of the accelerated Douglas-Rachford (see Section \[dougrach\]), while setting $A = 0$ one obtains the operator analog of the accelerated forward-backward method (see Section \[fbsec\]). Operator analogs of Algo. \[agenadmmrebal\] and Algo. \[atseng\] follow in a similar manner. One can also remove acceleration by setting $\gamma_k = 0$ for all $k \geq 0$, in which case these algorithms are discretizations of the first-order differential inclusion (also see Remark \[admm\_grad\_flow\]).
We mention a subtlety regarding the order of accuracy of a discretization such as to a differential inclusion such as or . In the smooth case we concluded that such a discretization is a first-order integrator; see Definition \[order\_def\] and Theorems \[admm\_first\_order\], \[dy\_first\_order\], and \[tsengcritical\]. An important ingredient was the Taylor approximation of the resolvent . However, for a maximal monotone operator $A$ only the following weaker approximation is available [@BauschkeBook Remark 23.47]: $$\label{taylor_res}
J_{\lambda A} = I - \lambda A_0 + \smallO(\lambda)$$ where the action of $A_0 = \lim_{\mu \downarrow 0}A_\mu$ on a vector $x\in \dom(A)$ gives the minimal norm element of the set $Ax$. Thus, by the same argument used in the proofs of Theorems \[admm\_first\_order\] and \[dy\_first\_order\], but now using and assuming that one can expand $A_0(x + \Order(\lambda)) = A_0(x) + \Order(\lambda)$ and similarly for $B_0$ and $C_0$, one concludes that the discrete and continuous trajectories agree up to $\smallO(h)$. This is in contrast with the $\Order(h^2)$ approximation that we proved for the smooth setting considered in Section \[proximal\_splitting\].
Regularized ODEs
----------------
There is an alternative to considering the differential inclusions and , which is to consider the respective ODEs $$\begin{aligned}
\label{firstreg}
\dot{x} &= -A_\mu x - B_\mu x - Cx , \\ \label{secreg}
\ddot{x} + \eta(t)\dot{x} &= -A_\mu x - B_\mu x - Cx,\end{aligned}$$ with the multivalued operators replaced by their respective Yosida regularization , which are single-valued and Lipschitz continuous. Thus, both ODEs admit a unique global solution. Note, however, that $$\zer(A + B + C) \ne \zer( A_\mu+ B_\mu + C)$$ so that the ODEs and do not have steady states that are compatible with zeros of the operator sum $A+B+C$. Nevertheless, after discretizing these regularized ODEs, one can take the limit $\mu \downarrow 0$ to recover iterates aimed at finding zeros of $A+B+C$. We stress that this procedure will give exactly the same updates as if one discretizes the original differential inclusions because of the identity ; let us illustrate with an example. Consider the discretization procedure of Section \[accel\_dy\] but applied to the ODE . Similarly to –, together with the accelerated variable $\hat{x}_k$ in , we immediately obtain with the help of the updates
\[ady.reg\] $$\begin{aligned}
x_{k+1/4} &= (\mu+\lambda)^{-1}( \mu I + \lambda J_{(\mu+\lambda) A})(\hat{x}_k), \\
x_{k+1/2} &= 2x_{k+1/4} - \hat{x}_k, \\
x_{k+3/4} &= (\mu+\lambda)^{-1}(\mu I + \lambda J_{(\mu + \lambda) B})(x_{k+1/2} - \lambda C x_{k+1/4}), \\
x_{k+1} &= x_{k+3/4} - (x_{k+1/4} - \hat{x}_k), \\
\hat{x}_{k+1} &= x_{k} + \gamma_k(x_{k+1}-x_k).\end{aligned}$$
Taking the limit $\mu \downarrow 0$ above results in the updates , which we can recall as a discretization of the differential inclusion .
It is possible to generalize Lemma \[caleylemma\] to general maximal monotone operators, i.e., $0 \in (A + B +C)(\bar{x})$ if and only if $ \mathcal{P}(x) = x$ where $$\mathcal{P} \equiv \tfrac{1}{2}I + \tfrac{1}{2}
C_{\lambda B}\circ \left( C_{\lambda A} - \lambda C \circ
J_{\lambda A} \right) - \tfrac{1}{2}\lambda C \circ J_{\lambda A}$$ and $\bar{x} = J_{\lambda A}x$. The proof is similar to the one of Lemma \[caleylemma\], although one needs to be careful in replacing equalities by appropriate inclusions and using the fact that the resolvent of a maximal monotone operator is single-valued. Therefore, by the same arguments as in Theorem \[dy\_critical\], the updates preserve zeros of $A + B + C$.
To place the above concepts within an optimization context, consider the case where $A = \partial f$ is the subdifferential of a nonsmooth convex function $f$. Then, the Yosida regularization becomes $$\nabla f_{\mu}(x) = \mu^{-1}(x - \prox_{\mu f}(x)),$$ which is the gradient of the Moreau envelope $f_\mu(x) \equiv \min_{y}\big( f(y) + \tfrac{1}{2\mu}\| y-x\|^2\big)$. The Moreau envelope is always differentiable and has the same minimizers as $f$.
Numerical Experiments {#applications}
=====================
Based on the continuous rates of Table \[convergence\], we expect that the accelerated algorithms that we have introduced will converge faster than their non-accelerated counterparts. In this section we investigate this numerically.
LASSO
-----
Consider the well-known LASSO regression problem $$\label{lassoprob}
\min_{x\in\mathbb{R}^n} \left\{ F(x) = \tfrac{1}{2}\| A x - b\|^2 + \alpha \| x\|_1 \right\}$$ where $A \in \mathbb{R}^{m\times n}$ is a given matrix, $b \in \mathbb{R}^m$ is a given signal, and $\alpha > 0$ is a weighting parameter. We generate data by sampling $A \sim \mathcal{N}(0,1)$, where $\mathcal{N}(\mu, \sigma)$ denotes a normal distribution with mean $\mu$ and standard deviation $\sigma$, and then normalizing its columns to have unit two norm. We sample $x_{\bullet} \in \mathbb{R}^{n}\sim \mathcal{N}(0,1)$ with sparsity level $95\%$ (i.e., only $5\%$ of its entries are nonzero), and then add noise to obtain the observed signal $b = A x_{\bullet} + e$, where the entries of $e$ are chosen i.i.d. from a normal distribution with mean zero and standard deviation $10^{-3}$. We choose $m=500$ and $n=2500$. The resulting signal-to-noise ratio is on the order of $250$, and $x_\bullet$ has $125$ nonzero entries. The parameter $\alpha$ is set as $\alpha= 0.1 \alpha_{\textnormal{max}}$ where $\alpha_{\textnormal{max}} = \| A^T b \|_{\infty}$ is the maximum value for $\alpha$ such that admits a nontrivial solution. We evaluate methods by computing the relative error $|F_k - F^\star| / F^\star$ where $F_k \equiv F(x_k)$ and $F^\star$ is the value of the objective obtained with the default implementation of CVXPY. We compare four frameworks: ADMM and Douglas-Rachford (DR) (these correspond to Algo. \[agenadmmrebal\] and Algo. \[ady\], respectively, with $f(x) = \tfrac{1}{2} \|Ax-b\|_2^2$, $g(x) = \alpha\|x\|_1$, and $w(x) = 0$), and forward-backward (FB) splitting and Tseng splitting (these correspond to Algo. \[ady\] and Algo. \[atseng\], respectively, with $f(x) = 0$, $g(x) = \alpha \|x\|_1$, $w(x) = \tfrac{1}{2} \|Ax-b\|_2^2$). For each of these four frameworks, we consider three variants: no acceleration (i.e., setting $\gamma_k = 0$ for all $k$), acceleration based on decaying damping (i.e., setting $\gamma_k$ by with damping coefficient defined in ), and acceleration based on constant damping (i.e., setting $\gamma_k$ by with damping coefficient defined in ). This results in a total of twelve algorithms that we denote by ADMM, ADMM-decaying, ADMM-constant, DR, DR-decaying, DR-constant, FB, FB-decaying, FB-constant, Tseng, Tseng-decaying, Tseng-constant. In all cases, we choose a step size of $\lambda = 0.1$ for the proximal operators. For decaying damping we choose $r=3$, and for constant damping $r=0.5$.
![\[lasso\] Performance of our twelve tested algorithm variants on problem . We perform 10 Monte-Carlo runs and show the mean and standard deviation of the relative error between $F_k = F(x_k)$ and $F^*$, where $F^\star$ is the solution obtained by CVXPY. ](lasso1.pdf "fig:")![\[lasso\] Performance of our twelve tested algorithm variants on problem . We perform 10 Monte-Carlo runs and show the mean and standard deviation of the relative error between $F_k = F(x_k)$ and $F^*$, where $F^\star$ is the solution obtained by CVXPY. ](lasso2.pdf "fig:")
In Fig. \[lasso\] we report the mean and standard deviation (errorbars) across 10 randomly generated instances of problem for various methods. The figure shows that the accelerated variants of each method improve over the non-accelerated variant. In particular, the constant damping accelerated variant is the fastest in this example.
Nonnegative matrix completion
-----------------------------
We now consider a matrix completion problem where the entries of the matrix are constrained to lie in a specified range. Suppose that for a low-rank matrix $M \in \mathbb{R}^{n\times m}$, we only have access to certain entries whose ordered pairs are collected in a set $\Omega$; let the operator $\PP:\mathbb{R}^{n\times m} \to \mathbb{R}^{n\times m}$ denote the projection onto these observable entries. The observable data matrix is given by $\Mobs = \PP(M)$ where $[\PP(M)]_{ij} = M_{ij}$ if $(i,j) \in \Omega$ and $[\PP(M)]_{ij} = 0$ otherwise. The goal is then to estimate the missing entries of $M$. One popular approach is to solve the convex optimization problem $\min \{ \| X \|_{*} \, \vert \, \PP(X) = \PP(M)\}$, where $\| X \|_{*}$ is the nuclear norm of $X$ [@Cai:2010]. We consider a modification of this approach by imposing constraints of the form $a \le X_{ij} \le b$ for given constants $a$ and $b$. Specifically, we solve $$\label{matcomp}
\min_{X\in\mathbb{R}^{n\times m}} \bigg\{ F(X) = \underbrace{\alpha \| X\|_*}_{f(X)} +
\underbrace{\mathbb{I}_{[a,b]}(X)}_{g(X)} +
\underbrace{\tfrac{1}{2} \| \PP(X) - \PP(M) \|_{F}^{2}}_{w(X)} \bigg\}$$ where $\| \cdot \|_F$ denotes the Frobenius norm, $\mathbb{I}_{[a,b]}(X) = 0$ if $ a \le X_{ij} \le b$ for all $(i,j)$ and $\mathbb{I}_{[a,b]}(X) = \infty$ otherwise, and $\alpha > 0$ is a weighting parameter such that larger values of $\alpha$ promote lower rank solutions [@Cai:2010] in problem .
We generate the low-rank matrix as $M = L_1 L_2^T$ where $\{L_1,L_2\} \subset \mathbb{R}^{100 \times 5}$ with entries chosen i.i.d. from $\mathcal{N}(3, 1)$. This ensures $M$ has rank $5$ (with probability one) and that each entry is positive with high probability (each test instance was verified to have positive entries). We sample $s n^2$ entries of $M$ uniformly at random, with a sampling ratio $s=0.4$, i.e., $40\%$ of the matrix $M$ is observed in $M_\textnormal{obs}$. We choose $$\begin{split}
a &= \min\{ [M_{\textnormal{obs}}]_{ij} \, \vert \, (i,j)\in\Omega\} - \sigma/2 , \qquad \\ b &= \max\{ [M_{\textnormal{obs}}]_{ij} \, \vert \, (i,j)\in\Omega \} + \sigma/2
\end{split}$$ where $\sigma$ is the standard deviation of all entries of $M_\textnormal{obs}$. We compare two frameworks: Davis-Yin (DY) (see Algo. \[ady\]) and ADMM (see Algo. \[agenadmmrebal\]). For each of these two frameworks, we consider the same three variants discussed in the previous section: no acceleration, acceleration based on decaying damping, and acceleration based on constant damping. These six algorithms are denoted by DY, DY-decaying, DY-constant, ADMM, ADMM-decaying, and ADMM-constant. Problem can be solved using these algorithms with the proximal operator $ J_{\tau \partial \| \cdot \|_*}(X) = U D_{\tau }(\Sigma) V^T$, where $X = U \Sigma V^T$ is the singular value decomposition of $X$ and $[D_{\tau}(\Sigma)]_{ii} = \max\{ \Sigma_{ii} - \tau, 0\}$; see [@Cai:2010] for details. The proximal operator of $g$ is just the projection $\big[J_{\lambda \partial \mathbb{I}_{ [a,b] } }(X)\big]_{ij}
= \max\{a, \min(X_{ij}, b)\}$. Finally, $\nabla w(X) = \PP(X - M)$. In terms of algorithm parameters, we choose a step size of $\lambda=1$ (for all variants), $r=3$ for decaying damping, and $r=0.1$ for constant damping. To evaluate algorithm performance, we use the relative error measure $$\| M_k - M \|_F \big/ \| M\|_F$$ where $M_k$ is the solution estimate obtained during the $k$th iteration. The stopping criteria for the optimization algorithms is $
\| M_{k+1} - M_k \|_F \big / \| M_k\|_F \le 10^{-10},
$ which was satisfied for every problem instance even though it is a relatively tight tolerance.
In Fig. \[mc1\] we report the mean and standard deviation (errorbars) across 10 randomly generated instances of problem with $\alpha = 3.5$ for the above algorithm variants. All methods terminate successfully and recover a matrix with the correct rank of five and a final relative error of $\approx 5\times 10^{-3}$. The total number of iterations performed by each method are also shown in Fig. \[mc1\].
![Performance of algorithms on problem with $\alpha = 3.5$. We perform $10$ Monte Carlo runs and indicate the mean and standard deviation for the relative error between the ground truth matrix $M$ and the $k$th iterate $M_k$ (left), and the number of iterations needed by the method to reach the termination tolerance (right).[]{data-label="mc1"}](mc_iteration1)
Motivated by the relatively large final relative error achieved for the single value of $\alpha$ in the previous paragraph, next we consider an annealing schedule on $\alpha$ that improves the relative error of the computed solutions. We follow the procedure of [@Goldfarb:2011] as follows. Given a sequence $\alpha_1 > \alpha_2 > \dotsm > \alpha_{L} = \bar{\alpha} > 0$ for some $\bar{\alpha}$, we run each algorithm with $\alpha_j$ and then use its solution as a starting point for the solution to the next run with $\alpha_{j+1}$; all other parameters are kept fixed. Such an approach has been used in compressed sensing [@Hale:2010] and matrix completion [@Goldfarb:2011]. Starting with $\alpha_0 = \delta \| \Mobs \|_F$ for some $\delta\in(0,1)$, we use the schedule $\alpha_{j+1} = \max\{\delta \alpha_j, \bar{\alpha}\}$ until reaching $\bar{\alpha}$. In our tests we choose $\delta = 0.25$ and $\bar{\alpha}=10^{-8}$. We use the same algorithm parameters as those used in creating Fig. \[mc1\], except that for the constant damping variants we now use $r=0.5$ since it performs better. In Fig. \[mc2\] we report the mean and standard deviation (errorbars) across $10$ randomly generated instances of problem . All methods successfully reach the termination tolerance, as for the previous test, but now achieve a much better reconstruction accuracy (compare Fig. \[mc1\] and Fig. \[mc2\]). The total number of iterations for each method are also shown in Fig. \[mc2\]. In this example the decaying damping variants do not improve over the non-accelerated method, but the constant damping variants still provide a speedup. We believe these findings can be explained by the fact that the accelerated gradient flow with constant damping attains exponential convergence on strongly convex problems, as opposed to the decaying damping (see Table \[convergence\]).
![Performance of algorithms on problem when annealing is used for $\alpha$. We perform $10$ Monte Carlo runs and indicate the mean and standard deviation for the relative error between the ground truth matrix $M$ and the $k$th iterate $M_k$ (left), and the number of iterations needed to reach the termination tolerance (right).[]{data-label="mc2"}](mc_iteration2)
Final Remarks {#conclusion}
=============
We showed that four types of proximal algorithms, namely forward-backward, Tseng splitting, Douglas-Rachford, and Davis-Yin, correspond to different discretizations of the gradient flow . We also showed that several accelerated variants of each of these methods arise from a similar discretization to the accelerated gradient flow . Such algorithms are steady-state-preserving first-order integrators to the associated ODE. Moreover, we showed that ADMM and its accelerated variants correspond to a rebalanced splitting, which is a technique recently introduced in the literature [@Speth:2013] to obtain discretizations that preserve steady states.
The new accelerated frameworks (see Algos. \[agenadmmrebal\], \[ady\], and \[atseng\]), which are new in general, reduce to known methods as special cases. Our frameworks provide different types of acceleration depending on the choice of damping strategy such as or , although other choices are also possible. Our derivations provide a new perspective on the important class of “operator splitting methods” by establishing tight connections with splitting methods for ODEs. Such an approach endows the gradient flow and the accelerated gradient flow with a unifying character for optimization since they capture the leading order behavior of several known algorithms. However, a complete understanding of a particular algorithm requires a more refined analysis and is an interesting problem.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We would like to thank Patrick Johnstone for discussions. This work was supported by grants ARO MURI W911NF-17-1-0304 and NSF 1447822.
[^1]: [email protected]
[^2]: One should not confuse operator splitting in convex analysis with splitting methods for ODEs.
[^3]: The resolvent was introduced by Fredholm in the late 19th century to study integral equations related to partial differential equations. This name was coined by Hilbert who used it extensively to develop the theory of linear operators. Usually, the resolvent is defined as $R(\lambda) \equiv (A - \lambda I)^{-1}$ when studying the spectral decomposition of $A$. However, in convex analysis the resolvent is defined as , but both are related via $J_{\lambda A} = \lambda^{-1} R(-\lambda^{-1})$.
[^4]: This holds for nondifferentiable functions as well where $A = \partial F$ is the subdifferential of $F$. In this case the differential equation is replaced by a differential inclusion (see Section \[nonsmooth\]). Thus, although we often denote the proximal operator by $J_{\lambda \nabla F}$ due to the connection with ODEs, the reader should keep in mind that this applies to nonsmooth functions as well and the resulting algorithm does not require differentiability. Recall also that $\partial F(x) = \{ \nabla F(x)\}$ when $F$ is differentiable.
[^5]: Note that $\tilde{x}_{k+1}$ in is a little further away from $x_{k+1}$, which makes the algorithm “look ahead” and implicitly introduces dependency on the curvature of $g$ in the resulting update.
|
---
abstract: 'To improve our understanding of connected systems, different tools derived from statistics, signal processing, information theory and statistical physics have been developed in the last decade. Here, we will focus on the graph comparison problem. Although different estimates exist to quantify how different two networks are, an appropriate metric has not been proposed. Within this framework we compare the performances of different networks distances (a topological descriptor and a kernel-based approach) with the simple Euclidean metric. We define the performance of metrics as the efficiency of distinguish two network’s groups and the computing time. We evaluate these frameworks on synthetic and real-world networks (functional connectomes from Alzheimer patients and healthy subjects), and we show that the Euclidean distance is the one that efficiently captures networks differences in comparison to other proposals. We conclude that the operational use of complicated methods can be justified only by showing that they out-perform well-understood traditional statistics, such as Euclidean metrics.'
author:
- 'Johann H. Martínez'
- Mario Chavez
title: 'In defence of the simple: Euclidean distance for comparing complex networks'
---
Despite the success of complex networks modeling and analysis, some methodological challenges are still to be tackled to describe and compare different interconnected systems. Identifying and quantifying dissimilarities among networks is a challenging problem of practical importance in many fields of science. Given two graphs $\{G,\ G^{'}\}$, we aim at finding a real-valued function $f$ that maps $G\times G^{'}\to\mathbb{R} \ \forall\ \{G,\ G^{'}\}$. Functions $f(G,G^{'})$ that quantify the (dis)similarity between two networks have been been studied in several areas such as chemistry, protein structures, social networks up to neuroscience, among others [@Borgwardt2005; @Deshpande2005; @Ralaivola2005]. Without an $f$ uniqueness, different approaches have been proposed including isomorphisms, distances based on divergences, spectral parameters, kernels, or different combinations of the previous [@Donnat2018; @Wegner2018; @Hammond2013; @Schieber2017; @Bai2015].
In this work, we consider three classes of the function $f$: the first class, who is the large bunch in the literature, quantifies local changes via structural differences. These metrics may range from the simplest Euclidean distance [@Higham2002; @Golub1996; @Real1996] to more elaborated algorithms that assign costs of different operations to map nodes/edges of $G$ to their $G^{'}$ counterparts [@Sanfeliu1983]. Another distance class considers topological descriptors that map each graph into a feature vector (e.g. degree distribution, nodes centrality, etc.). These vectors are compared with any multivariate statistical distance to compute the graph dissimilarity [@Basseville1999; @Runber1998; @Schieber2017]. We notice that considering one type of feature may imply to lose topological information from others parameters, and the price of complet caracterisation may be paid with more runtime. The last class considered here includes kernel-based approaches that compare global substructures (i.e. walks, paths, etc). These methods capture global information of networks (e.g. the graph Laplacian) considered in a metric space, where a defined inner product directly estimates its dissimilarity. Kernel methods, however, often integrates over local neighborhoods, which renders these approaches less sensitive to small or local perturbations [@Donnat2018].
In this work we show than the use of a simple Euclidean metric may provides good performances to asses graph differences, when compared to other more complicated functions. We propose a framework for measuring the performance of functions $f$’s applied on undirected-binary graphs of equal sizes. We define the $f$’s performance in terms of “discriminability” and “runtime”. The former is the capability of $f$ for discriminating two sets of network associated to two different groups. The latter is simply the computing time.
In what follows, we compare the performance of the standard Euclidean distance ($D_f$), the dissimilarity measure ($D_d$) defined in Ref [@Schieber2017], and the graph diffusion kernel distance ($D_k$) [@Hammond2013], from each of the classes mentioned above. For these algorithms, we evaluate the discriminability and runtime in different synthetic and real-world brain networks. We show that the Euclidean distance substantially outperforms other methods to capture differences between networks of the same size.
*Euclidean distance.–* Assuming that $\{A_1, A_2\}$ are the adjacency matrix representations of graphs $\{G_1, G_2\}$, we have the Euclidean distance defined by: $$\label{dE}
D_f = \|A_1-A_2\|_F$$ where $\|\cdot\|_F$ denotes the Frobenius norm.
*Network structural dissimilarity.–* This dissimilarity measure captures several topological descriptors [@Schieber2017]: network distance distributions $\mu_{\{A_1,A_2\}}$, node-distance distribution functions $NND_{\{A_1,A_2\}}$ (local connectivity of each node), $\alpha$-centrality distributions $P_{\alpha \{A_1,A_2\}}$, the equivalent for their graph complements $P_{\alpha \{A_1^c,A_2^c\}}$ and several tuning parameters $\{\alpha, w_1,w_2,w_3\}$. The network distance is obtained via the Jensen-Shannon divergence $\Gamma$ between different feature vectors.
$$\begin{aligned}
D_d = &w_1 \sqrt{\frac{\Gamma(\mu_{A_1}, \mu_{A_2})}{log 2}}+ w_2|\sqrt{NND(A_1)}-\sqrt{NND(A_2)}|\nonumber \\
& +\frac{w_3}{2}\Big(\sqrt{\frac{\Gamma(P_{\alpha A_1},P_{\alpha A_2})}{log 2}}+\sqrt{\frac{\Gamma(P_{\alpha A_1^c},P_{\alpha A_2^c})}{log 2}}\Big)\end{aligned}$$
*Kernel-based distance.–* A recently proposed distance is based on diffusion kernels [@Hammond2013]. This method estimates the differences between diffusion patterns of two networks undergoing a continuous node-thermal diffusion. A set of distances at different scales $t$ can be obtained by means of the Laplacian exponential kernels $e^{-t\mathcal{L}_{\{A_1,A_2\}}}$. The kernel-based distance is obtained by: $$D_k = \|\exp(-t\mathcal{L}_1)-\exp(-t\mathcal{L}_2) \|_F$$ where $\mathcal{L}_k$ denotes the graph Laplacian of network $k$.
To assess the performances of these functions to capture network’s differences, we consider a network $A$ and a set of perturbed networks $\{A_p\}$ generated with an incremental rewiring probability $p$ of original network $A$. We evaluate $f$’s by computing the differences between perturbed versions $\{A_p\}$ and its original configuration $A$. For low values of $p$ networks are very similar. Network differences are expected to increase with $p$.
#### Benchmark tests.– {#benchmark-tests. .unnumbered}
We build binary Barabasi-Albert (BA) and Strogatz-Watts (SW) [@Newman2010] models with $L$ links and $N=100$. For SW model, the number of initial neighbors is $K=4$ for a $L=N*K$ edges and $\langle k\rangle=2K$. For each model we recreate a continuous perturbation process by reshuffling their links with and incremental rewiring probability step $p=0.001$. This allows us to create a set of $||\{A_p\}||=1000$ connected networks, each of them with $L*p$ rewired links.
Let $\delta_{p,f}$ be the network-distance vector that contains all differences between perturbed networks $\{A_p\}$ and A measured for a given metric $f$. We compute the averaged profiles $\langle\delta_{p,f}\rangle$ as well as the 5$^{th}$-95$^{th}$ percentiles (Fig. \[fig:01\]). All the averaged profiles display monotonically increasing curves that reach out certain saturation around $p=10^{-1}$. Results suggest that all the measures (including the Euclidean distance) are sensitive to small structural changes (10% of reshuffled links), and reflect well the structural perturbations. Beyond this threshold ($p> 10^{-1}$), however, all functions cannot distinguish between a graph $A$ and its perturbed version $\{A_p\}$.
#### Assessment of performances.– {#assessment-of-performances. .unnumbered}
To assess the metrics’ performances we quantify the “discriminability” and the “runtime”. Discriminability assesses whether a given function $f$ is sensitive at certain perturbation $p$, and whether it is suitable for distinguish two different group of networks at a given $p$. Discriminability is defined as the percentage of times a function $f$ distinguishes the differences of each group of networks at certain perturbation level. The more times $f$ distinguishes two different datasets, the better the $f$ discriminability is. In addition, runtime simply measures the $f$ execution time. The faster a given function $f$ estimates the differences, the better the corresponding metric is. For the sake of applicability we tested the performance of different $f$’s in real networks.
#### Real networks.– {#real-networks. .unnumbered}
In this work, we use a recently published brain connectivity dataset, which includes functional connectivity matrices estimated from magnetoencephalographic (MEG) signals recorded from 23 Alzheimer patients ($P$) and a set of controls subjects ($C$) during a condition of resting-state with eyes-closed [@Guillon2017]. Alzheimer disease is caracterised by anatomical brain deteriorations, which are reflected in an abnormal brain connectivity. MEG activity was reconstructed on the cortical surface by using a source imaging technique [@Guillon2017]. Connectivity matrices were obtained from $N=148$ regions of interest by means of the spectral coherence between activities in the band of 11-13 Hz. We specifically focused on this frequency band, which is particularly activated during resting activity with closed eyes, and it reflects the main functional connectivity changes accompanying the disease [@Stam2002]. All the recording parameters and pre-processing details of connectivity matrices are explained in Ref. [@Guillon2017].
Following the procedure of Ref. [@DeVicoFallani2017], we thresholded each connectivity matrix by recovering its minimum spanning tree and then filling the network up with the strongest links until to reach a mean degree of three. This method is useful for optimizing the balance between the network efficiency and its rewiring cost [@DeVicoFallani2017]. The resulting connectivity networks are binary adjacency matrices with $N=148$ nodes with $L=222$ links.
A direct comparison of connectivity matrices between the graphs of two groups $A \in \{P \lor C\}$ does not not allow to distinguish them. This result agrees with a previous studies that found group differences related to very local changes in connectivity [@Stam2002; @Guillon2017]. Authors in Ref. [@Guillon2017] for instance, found that only 3% and 4% of the nodes accounts for the connectivity differences between groups, when different frequency bands are combined in the analysis.
We propose an approach that allows to detect global network differences between those groups. For this, each connectivity graph $A$ is firstly perturbed by randomly choosing $l$ links $\forall\ l=1,2,\dots,L$ and reshuffling them such that the graph remains connected. We get thus a set of $||\{A_l\}||=222$ perturbed networks. We then compute the network differences between all pairs $(A,A_l)$. We finally repeat this procedure for 20 independent realizations. The distances profile $\delta_{l,f}^s$ results from the average of the network differences across realizations for a given subject $S$. The set of $||\{\delta_{l,f}^s\}||=23$ distances profiles per group (one for each subject) is used to compare the differences captured by $f$ when $l$ links are rewired. A function $f$ distinguishes two populations $\{\delta_{l,f}^s\}^P \wedge \{\delta_{l,f}^s\}^C$ at certain level $l$, if the group differences are statistically different at that perturbation level. Discriminability is defined as the hits percentage along all $L$ perturbations, i.e. the number of times the null hypothesis $H_o$ of no difference between the two groups is rejected. To assess significant differences, we used a non parametric permutation test allowing 500 permutations for each $l$ and we reject $H_o$ at $p \leq 0.05$ (corrected by a Bonferroni method).
The mean distance profiles $\langle\delta_{l,f}^s\rangle$ for each $f$ are plotted in Fig. \[fig:02\]. As in synthetic models, profiles show a monotonically increasing behaviour. At low rewiring percentages ($\leq 11\%$) there is no significant differences at group level. For small perturbation levels, functions $f$ can not distinguishes connectivity between groups. Something similar is observed when links perturbation are above $\approx 70\%$. On the other hand, $D_f$ appears as the one with the highest discriminability closely followed by $D_k$, while $D_d$ appears with lowest one. Results clearly suggest that Euclidean distance distinguishes better the two groups of networks considered here.
Finally, we assessed the execution time for computing a distances profile for each subject (we use MATLAB R2017a algorithms ran in an OS 10.12.6, with a 4GHz Intel dual core i7 processor, with 32GB of memory). Fig. \[fig:03\] shows the relatives orders of magnitude in seconds that each metric takes to compute the networks differences. Average times obtained are: $t_f = 6.83\times10^{-5}$, $t_d = 2.68\times10^{-2}$, $t_k = 1.90\times10^{-1}$ for the Euclidean distance, the dissimilarity metric and the kernel-based method, respectively. The results clearly show Euclidean distance as the fastest method in comparison with the others two. Clearly, $D_f$ is 3 (4) orders of magnitude faster than $D_d$ ($D_k$).
Runtime finally determines which measure has the best performance when computing graphs distances. While the discriminability of $D_k$ is close to that of $D_f$, its runtime is four orders of magnitude slower than $D_f$ due to the fact that $D_k$ needs to search into several scales to find the highest difference. $D_d$ runtime is three orders of magnitude slower than $D_f$, because $D_d$ looks for many topological properties under several tuning parameters. In summary, the Euclidean distance emerges as the metrics with the best performances when computing graph differences. $D_f$ highlights for both: highest discriminability for distinguish groups of networks, as well as fastest computation, which is something really important when one manage large datasets.
#### Concluding remarks.– {#concluding-remarks. .unnumbered}
Finding an accurate graph distance is a difficult task, and many metrics have been described without a framework to properly benchmark such proposals. Here we make a call of the simple Euclidean distance as the one with the best tradeoff between good and fast performances in contrast to more elaborated algorithms. In this work, we propose a method to detect global network differences with high efficiency and fast computation time. We also propose a simple framework to assess any metric’s performance in terms discriminability and runtime. Results indicate that, for comparing binary networks of the same size, the Euclidean distance’s discriminating capabilities outperform those of graph dissimilarity and diffusion kernel distance. More elaborated network models (e.g. multi-layer, weighted, signed or time-varying networks) might, however, need more elaborated tools to account for interdependencies of interacting units, and make their comparisons more robust.
#### Acknowledgements.– {#acknowledgements. .unnumbered}
We are indebted to X. Navarro and M. Dovergine for their valuable comments.
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|
---
abstract: 'We consider functions $f(T,R)$ of pairs of noncommuting contractions on Hilbert space and study the problem for which functions $f$ we have Lipschitz type estimates in Schatten–von Neumann norms. We prove that if $f$ belongs to the Besov class ${\big(B_{\be,1}^1\big)_+(\T^2)}$ of nalytic functions in the bidisk, then we have a Lipschitz type estimate for functions $f(T,R)$ of pairs of not necessarily commuting contractions $(T,R)$ in the Schatten–von Neumann norms $\bS_p$ for $p\in[1,2]$. On the other hand, we show that for functions in ${\big(B_{\be,1}^1\big)_+(\T^2)}$ there are no Lipschitz such type estimates for $p>2$ as well as in the operator norm.'
author:
- 'A.B. Aleksandrov and V.V. Peller'
title: Functions of perturbed pairs of noncommuting contractions
---
[^1] [^2]
[****]{}
**Introduction**
================
\[In\]
The purpose of this paper is to study the behavior of functions $f(T,R)$ of (not necessarily commuting) contractions $T$ and $R$ under perturbation. We are going to obtain Lipschitz type estimates in the Sachatten–von Neumann norms $\bS_p$, $1\le p\le2$, for functions $f$ in the Besov class ${\big(B_{\be,1}^1\big)_+(\T^2)}$ of analytic functions. Note that functions $f(T,R)$ of noncommuting contractions can be defined in terms of double operator integrals with respect to semi-spectral measures, see §\[dvoitro\] below.
This paper can be considered as a continuation of the results of [@Pe1]–[@Pe7], [@AP1]–[@AP4], [@AP6], [@APPS], [@NP], [@ANP], [@PS] and [@KPSS] for functions of perturbed self-adjoint operators, contractions, normal operators, dissipative operators, functions of collections of commuting operators and functions of collections of noncommuting operators.
Recall that a Lipschitz function $f$ on $\R$ does not have to be [*operator Lipschitz*]{}, i.e., the condition $|f(x)-f(y)|\le\const|x-y|$, $x,\,y\in\R$, does not imply that $$\|f(A)-f(B)\|\le\const\|A-B\|$$ for arbitrary self-adjoint operators (bounded or unbounded, does not matter) $A$ and $B$. This was first established in [@F].
It turned out that functions in the (homogeneous) Besov space $B_{\be,1}^1(\R)$ are operator Lipschitz; this was established in [@Pe1] and [@Pe3] (see [@Pee] for detailed information about Besov classes). We refer the reader to the recent survey [@AP4] for detailed information on operator Lipschitz functions. In particular, [@AP4] presents various sufficient conditions and necessary conditions for a function on $\R$ to be operator Lipschitz. It is well known that if $f$ is an operator Lipschitz function on $\R$, and $A$ and $B$ are self-adjoint operators such that the difference $A-B$ belongs to the Schatten–von Neumann class $\bS_p$, $1\le p<\be$, then $f(A)-f(B)\in\bS_p$ and $\|f(A)-f(B)\|_{\bS_p}\le\const\|A-B\|_{\bS_p}$. Moreover, the constant on the right does not depend on $p$. In particular, this is true for functions $f$ in the Besov class $B_{\be,1}^1(\R)$, i.e., \[LiShNOL\] f(A)-f(B)\_[\_p]{}f\_[B\_[,1]{}\^1]{}A-B\_[\_p]{},1p.
However, it was discovered in [@AP1] (see also [@FN]) that the situation becomes quite different if we replace the class of Lipschitz functions with the class $\L_\a(\R)$ of Hölder functions of order $\a$, $0<\a<1$. Namely, the inequality $|f(x)-f(y)|\le\const|x-y|^\a$, $x,\,y\in\R$, implies that $$\|f(A)-f(B)\|\le\const\|A-B\|^\a$$ for arbitrary self-adjoint operators $A$ and $B$. Moreover, it was shown in [@AP2] that if $A-B\in\bS_p$, $p>1$, and $f\in\L_a(\R)$, then $f(A)-f(B)\in\bS_{p/\a}$ and $$\|f(A)-f(B)\|_{\bS_{p/a}}\le\const\|A-B\|_{\bS_p}^\a$$ for arbitrary self-adjoint operators $A$ and $B$.
Analogs of the above results for functions of normal operators, functions of contractions, functions of dissipative operators and functions of commuting collections of self-adjoint operators were obtained in [@Pe2], [@AP3], [@APPS], [@NP].
Note that it was shown in [@PS] that for $p\in(1,\be)$, inequality holds for arbitrary [*Lipschitz*]{} (not necessarily operator Lipschitz) functions $f$ with constant on the right that depends on $p$. An analog of this result for functions of commuting self-adjoint operators was obtained in [@KPSS].
In [@ANP] similar problems were considered for functions of two noncommuting self-adjoint operators (such functions can be defined in terms of double operator integrals, see [@ANP]). It was shown in [@ANP] that for functions $f$ on $\R^2$ in the (homogeneous) Besov class $B_{\be,1}^1(\R^2)$ and for $p\in[1,2]$, the following Lipschitz type estimate holds: $$\|f(A_1,B_1)-f(A_2,B_2)\|_{\bS_p}\le\const
\max\big\{\|A_1-A_2\|_{\bS_p},\|B_1-B_2\|_{\bS_p}\big\}$$ for arbitrary pairs $(A_1,B_1)$ and $(A_2,B_2)$ of (not necessarily commuting) self-adjoint operators.
However, it was shown in [@ANP] that for $p>2$ there is no such Lipschitz type estimate in the $\bS_p$ norm as well as in the operator norm. Moreover, it follows from the construction given in [@ANP] that for $p\in(2,\be]$ and for positive numbers $\e,\,\s,\,M$, there exists a function $f$ in $L^\be(\R^2)$ with Fourier transform supported in $[-\s,\s]\times[-\s,\s]$ such that $$\max\big\{\|A_1-A_2\|_{\bS_p},\|B_1-B_2\|_{\bS_p}\big\}<\e$$ while $$\|f(A_1,B_1)-f(A_2,B_2)\|_{\bS_p}>M.$$ Here we use the notation $\|\cdot\|_{\bS_\be}$ for operator norm.
This implies that unlike in the case of commuting operators, there cannot be any Hölder type estimates in the norm of $\bS_p$, $p>2$, for Hölder functions $f$ of order $\a$. Moreover, for $p>2$, there cannot be any estimate for $\|f(A_1,B_1)-f(A_2,B_2)\|_{\bS_p}$ for functions in the Besov class $B_{\be,q}^s(\R)$ for any $q>0$ and $s>0$.
On the other hand, it was observed by the anonymous referee of [@ANP] that unlike in the case of commuting self-adjoint operators, there is no Lipschitz type estimates for $\|f(A_1,B_1)-f(A_2,B_2)\|_{\bS_2}$ for [*Lipschitz functions*]{} $f$ on $\R^2$, see [@ANP].
Finally, let us mention that in the case of functions of triples of noncommuting operators there are no such Lipschitz type estimates for functions in the Besov class $B_{\be,1}^1(\R^3)$ in the norm of $\bS_p$ for any $p\in[1,\be]$. This was established in [@Pe7].
In §\[dvoitro\] we give an introduction to double and triple operator integrals and we define functions $f(T,R)$ of noncommuting contractions. We define the Haagerup and Haagerup-like tensor products of three copies of the disk-algebra ${{\rm C}_{\rm A}}$ and we define triple operator integrals whose integrands belong to such tensor products.
Lipschitz type estimates in Schatten–von Neumann norm will be obtained in §\[LiptysvN\]. We show that for $p\in[1,2]$ and for a function $f$ on $\T^2$ in the analytic Besov space ${\big(B_{\be,1}^1\big)_+(\T^2)}$, the following Lipschitz type inequality holds: $$\big\|f(T_1,R_1)-f(T_0,R_0)\big\|_{\bS_p}\le\const
\max\big\{\|T_1-T_0\|_{\bS_p},\|R_1-R_0\|_{\bS_p}\big\}$$ for arbitrary pairs $(T_0,T_1)$ and $(R_0,R_1)$ of contractions. Recall that similar inequality was established in [@ANP] for functions of self-adjoint operators. However, to obtain this inequality for functions of contractions, we need new algebraic formulae. Moreover, to obtain this inequality for functions of contractions, we offer an approach that does not use triple operator integrals. To be more precise, we reduce the inequality to the case of analytic polynomials $f$ and we integrate over finite sets, in which case triple operator integrals become finite sums. We establish explicit representations of the operator differences $f(T_1,R_1)-f(T_0,R_0)$ for analytic polynomials $f$ in terms of finite sums of elementary tensors which allows us to estimate the $\bS_p$ norms.
However, we still use triple operator integrals to obtain in §\[pretrioi\] explicit formulae for the operator differences for arbitrary functions $f$ in ${\big(B_{\be,1}^1\big)_+(\T^2)}$.
In §\[differ\] we study differentiability properties in Schatten–von Neumann norms of the function $$t\mapsto f\big(T(t),R(t)\big)$$ for $f$ in ${\big(B_{\be,1}^1\big)_+(\T^2)}$ and contractive valued functions $t\mapsto T(t)$ and $t\mapsto R(t)$. We obtain explicit formulae for the derivative in terms of triple operator integrals. Again, to prove the existence of the derivative, we do not need triple operator integrals.
As in the case of functions of pairs self-adjoint operators (see [@ANP]), there are no Lipschitz type estimates in the norm of $\bS_p$, $p>2$, for functions of pairs of not necessarily commuting contractions $f(T,R)$, $f\in{\big(B_{\be,1}^1\big)_+(\T^2)}$. This will be established in §\[counter\]. Note that the construction differs from the construction in the case of self-adjoint operators given in [@ANP].
In §\[zada\] we state some open problems and in §\[Bes\] we give an introduction to Besov classes on polydisks.
We use the notation $\m$ for normalized Lebesgue measure on the unit circle $\T$ and the notation $\m_2$ for normalized Lebesgue measure on $\T^2$.
For simplicity we assume that we deal with separable Hilbert spaces.
**Besov classes of periodic functions** {#Bes}
=======================================
In this section we give a brief introduction to Besov spaces on the torus.
To define Besov spaces on the torus $\T^d$, we consider an infinitely differentiable function $w$ on $\R$ such that $$w\ge0,\quad\supp w\subset\left[\frac12,2\right],\quad\mbox{and} \quad w(s)=1-w\left(\frac s2\right)\quad\mbox{for}\quad s\in[1,2].$$ Let $W_n$, $n\ge0$, be the trigonometric polynomials defined by $$W_n(\z)\df\sum_{j\in\Z^d}w\left(\frac{|j|}{2^n}\right)\z^j,\quad n\ge1,
\quad W_0(\z)\df\sum_{\{j:|j|\le1\}}\z^j,$$ where $$\z=(\z_1,\cdots,\z_d)\in\T^d,\quad j=(j_1,\cdots,j_d),\quad\mbox{and}\quad
|j|=\big(|j_1|^2+\cdots+|j_d|^2\big)^{1/2}.$$ For a distribution $f$ on $\T^d$ we put \[f=sumfn\] \[fnWn\] f\_n=f\*W\_n,n0. It is easy to see that \[fSigmafn\] f=\_[n0]{}f\_n; the series converges in the sense of distributions. We say that $f$ belongs the [*Besov class*]{} $B_{p,q}^s(\T^d)$, $s>0$, $1\le p,\,q\le\be$, if \[Bperf\] {2\^[ns]{}f\_n\_[L\^p]{}}\_[n0]{}\^q. The analytic subspace $\big(B_{p,q}^s\big)_+(\T^d)$ of $B_{p,q}^s(\T^d)$ consists of functions $f$ in $B_{p,q}^s(\T^d)$ for which the Fourier coefficients $\widehat f(j_1,\cdots,j_d)$ satisfy the equalities: \[anafunapo\] f(j\_1,,j\_d)=0\_[1kd]{}j\_k<0. We refer the reader to [@Pee] for more detailed information about Besov spaces.
**Double and triple operator integrals\
with respect to semi-spectral measures**
========================================
\[dvoitro\]
In this section we give a brief introduction to double and triple operator integrals with respect to semi-spectral measures. Double operator integrals with respect to spectral measures are expressions of the form \[dvooipolu\] (x,y)dE\_1(x)QdE\_2(y), where $E_1$ and $E_2$ are spectral measures, $Q$ is a linear operator and $\Phi$ is a bounded measurable function. They appeared first in [@DK]. Later Birman and Solomyak developed in [@BS1]–[@BS3] a beautiful theory of double operator integrals.
Double operator integrals with respect to semi-spectral measures were defined in [@Pe2], see also [@AP4] (recall that the definition of a [*semi-spectral measure*]{} differs from the definition of a spectral measure by replacing the condition that it takes values in the set of orthogonal projections with the condition that it takes values in the set of nonnegative contractions, see [@AP4] for more detail).
For the double operator integral to make sense for an arbitrary bounded linear operator $T$, we have to impose an additional assumption on $\Phi$. The natural class of such functions $\Phi$ is called the class of [*Schur multipliers*]{}, see [@Pe1]. There are various characterizations of the class of Schur multipliers. In particular, $\Phi$ is a Schur multiplier if and only if it belongs to the Haagerup tensor product $L^\be(E_1)\otimes_{\rm h}\!L^\be(E_2)$ of $L^\be(E_1)$ and $L^\be(E_2)$, i.e., it admits a representation of the form \[tenzpre\] (x,y)=\_j\_j(x)\_j(y), where the $\f_j$ and $\psi_j$ satisfy the condition \[Haatepro\] \_j|\_j|\^2L\^(E\_1)\_j|\_j|\^2L\^(E\_2). In this case \[fladlya doi\] (x,y)dE\_1(x)QdE\_2(y)= \_j(\_jdE\_1)Q(\_jdE\_2); the series converges in the weak operator topology. The right-hand side of this equality does not depend on the choice of a representation of $\Phi$ in .
One can also [*consider double operator integrals of the form [**]{} in the case when $E_1$ and $E_2$ are semi-spectral measures*]{}. In this case, as in the case of spectral measures, formula still holds under the same assumption .
It is easy to see that if $\Phi$ belongs to the projective tensor product $L^\be(E_1)\widehat\otimes L^\be(E_2)$ of $L^\be(E_1)$ and $L^\be(E_2)$, i.e., $\Phi$ admits a representation of the form with $\f_j$ and $\psi_j$ satisfying $$\sum_j\|\f_j\|_{L^\be(E_1)}\|\psi_j\|_{L^\be(E_2)}<\be,$$ then $\Phi$ is a Schur multiplier and holds.
[**3.2. The semi-spectral measures of contractions.**]{} Recall that if $T$ is a contraction (i.e., $\|T\|\le1$) on a Hilbert space $\h$, then by the Sz.-Nagy dilation theorem (see [@SNF]), $T$ has a unitary dilation, i.e., there exist a Hilbert space $\K$ that contains $\h$ and a unitary operator $U$ on $\K$ such that $$T^n=P_\h U^n\big|\h,\quad n\ge0,$$ where $P_\h$ is the orthogonal projection onto $\h$.
Among all unitary dilations of $T$ one can always select a [*minimal*]{} unitary dilation (in a natural sense) and all minimal unitary dilations are isomorphic, see [@SNF].
The existence of a unitary dilation allows us to construct the natural functional calculus $f\mapsto f(T)$ for functions $f$ in the disk-algebra ${{\rm C}_{\rm A}}$ defined by $$f(T)=P_\h f(U)\big|\h=P_\h\left(\int_\T f(\z)\,dE_U(\z)\right)\Big|\h,\quad f\in{{\rm C}_{\rm A}}.$$ where $E_U$ is the spectral measure of $U$.
Consider the operator set function $\E_T$ defined on the Borel subsets of the unit circle $\T$ by $$\E_T(\D)=P_\h E_U(\D)\big|\h,\quad\D\subset\T.$$ Then $\E_T$ is a [*semi-spectral measure*]{}. It can be shown that it does not depend on the choice of a unitary dilation. The semi-spectral measure $\E_T$ is called the [*semi-spectral measure*]{} of $T$.
[**3.3. Functions of noncommuting contractions.**]{} Let $f$ be a function on the torus $\T^2$ that belongs to the Haagerup tensor product ${{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}$, i.e., $f$ admits a representation of the form $$f(\z,\t)=\sum_j\f_j(\z)\psi_j(\t),\quad\z,~\t\in\T,$$ where $\f_j$, $\psi_j$ are functions in ${{\rm C}_{\rm A}}$ such that $$\sup_{\z\in\T}\sum_j|\f_j(\z)|^2<\be
\quad\mbox{and}\quad
\sup_{\t\in\T}\sum_j|\psi_j(\t)|^2<\be.$$ For a pair $(T,R)$ of (not necessarily commuting contractions), the operator $f(T,R)$ is defined as the double operator integral $$\iint_{\T\times\T}f(\z,\t)\,d\E_T(\z)\,d\E_R(\t)=
\iint_{\T\times\T}f(\z,\t)\,d\E_T(\z)I\,d\E_R(\t).$$
Note that if $f\in{\big(B_{\be,1}^1\big)_+(\T^2)}$, then $f\in{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}$, and so we can take functions $f(T,R)$ of contractions for an arbitrary function $f$ in ${\big(B_{\be,1}^1\big)_+(\T^2)}$. Indeed, if $f$ is an analytic polynomial in two variables of degree at most $N$ in each variable, then we can represent $f$ in the form $$f(\z,\t)=\sum_{j=0}^N\z^j\left(\sum_{k=0}^N\widehat f(j,k)\t^k\right).$$ Thus $f$ belongs to the projective tensor product ${{\rm C}_{\rm A}}\hat\otimes{{\rm C}_{\rm A}}$ and \[tenzprepol\] f\_[[[C]{}\_[A]{}]{}[[C]{}\_[A]{}]{}]{}\_[j=0]{}\^N\_|\_[k=0]{}\^Nf(j,k)\^k| (1+N)f\_[L\^]{} It follows easily from that every function $f$ of Besov class ${\big(B_{\be,1}^1\big)_+(\T^2)}$ belongs to ${{\rm C}_{\rm A}}\hat\otimes{{\rm C}_{\rm A}}$, and so the operator $f(T,R)$ is well defined. Clearly, \[opfunnekom\] f(T,R)=\_[n0]{}\_[j=0]{}\^[2\^[n+1]{}]{}T\^j (\_[k=0]{}\^[2\^[n+1]{}]{}f\_n(j,k)R\^k), where $f_n$ is the polynomial defined by . It follows immediately from and that the series converges absolutely in the operator norm. Note that formula can be used as a definition of the functions $f(T,R)$ of noncommuting contractions in the case when $f\in{\big(B_{\be,1}^1\big)_+(\T^2)}$.
[**3.4. Triple operator integrals. Haagerup tensor products.**]{} There are several approaches to multiple operator integrals. Triple operator integrals are expressions of the form $$W_\Phi\df\iiint\Phi(x,y,z)\,dE_1(x)X\,dE_2(y)Y\,dE_3(z),$$ where $\Phi$ is a bounded measurable function, $E_1$, $E_2$ and $E_3$ are spectral measures, and $X$ and $Y$ are bounded linear operators on Hilbert space.
In [@Pe4] triple (and more general, multiple) operator integrals were defined for functions $\Phi$ in the integral projective product . For such functions $\Phi$, the following Schatten–von Neumann properties hold: $$\left\|\iiint\Phi\,dE_1X\,dE_2Y\,dE_3\right\|_{\bS_r}
\le\|\Phi\|_{L^\be\otimes_{\rm i}L^\be\otimes_{\rm i}L^\be}
\|X\|_{\bS_p}\|Y\|_{\bS_q},\quad \frac1r=\frac1p+\frac1q,$$ whenever $1/p+1/q\le1$. Later in [@JTT] triple (and multiple) operator integrals were defined for functions $\Phi$ in the Haagerup tensor product . However, it turns out that under the assumption $\Phi\in L^\be\!\otimes_{\rm h}\!L^\be\!\otimes_{\rm h}\!L^\be$, the conditions $X\in\bS_p$ and $Y\in\bS_q$ imply that $\iiint\Phi\,dE_1X\,dE_2Y\,dE_3\in\bS_r$, $1/r=1/p+1/q$, only under the conditions that $p\ge2$ and $q\ge2$, see [@AP5] (see also [@ANP]). Moreover, the following inequality holds: $$\left\|\iiint\Phi\,dE_1X\,dE_2Y\,dE_3\right\|_{\bS_r}
\le\|\Phi\|_{L^\be\otimes_{\rm h}L^\be\otimes_{\rm h}L^\be}
\|X\|_{\bS_p}\|Y\|_{\bS_q},\quad \frac1r=\frac1p+\frac1q,$$ whenever $p\ge2$ and $q\ge2$, see [@AP5].
Note also that to obtain Lipschitz type estimates for functions of noncommuting self-adjoint operators in [@ANP], we had to use triple operator integrals with integrands $\Phi$ that do not belong to the Haagerup tensor product $L^\be\!\otimes_{\rm h}\!L^\be\!\otimes_{\rm h}\!L^\be$. That is why we had to introduce in [@ANP] Haagerup-like tensor products of the first kind and of the second kind.
In this paper we are going to use triple operator integrals with integrands being continuous functions on $\T^3$ that belong to Haagerup and Haagerup-like tensor products of three copies of the disk-algebra ${{\rm C}_{\rm A}}$. We briefly define such tensor products and discuss inequalities we are going to use in the next section.
[**Definition 1.**]{} We say that a continuous function $\Phi$ on $\T^3$ belongs to the [*Haagerup tensor product*]{} ${{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}$ if $\Phi$ admits a representation \[Haatepre\] (,,)=\_[j,k0]{}\_j()\_[jk]{}()\_k(), ,,, where $\a_j$, $\b_{jk}$ and $\g_k$ are functions in ${{\rm C}_{\rm A}}$ such that \[noHaatepr\] \_(\_[j0]{}|\_j()|\^2)\^[1/2]{} \_{\_[jk]{}()}\_[j,k0]{}\_ \_(\_[k0]{}|\_k()|\^2)\^[1/2]{}<. Here $\|\cdot\|_{\mathcal{B}}$ stands for the operator norm of a matrix (finite or infinite) on the space $\ell^2$ or on a finite-dimensional Euclidean space. By definition, the norm of $\Phi$ in ${{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}$ is the infimum of the left-hand side of over all representations of $\Phi$ in the form of .
Suppose that $\Phi\in{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}$ and both and hold. Let $T_1$, $T_2$ and $T_3$ be contractions with semi-spectral measures $\E_{T_1}$, $\E_{T_2}$ and $\E_{T_3}$. Then for bounded linear operators $X$ and $Y$, we can define the triple operator integral \[tropin\] W\_=d\_[T\_1]{}Xd\_[T\_2]{}Yd\_[T\_3]{} as $$\begin{aligned}
W_\Phi&\df\sum_{j,k}
\Big(\int\a_j(\z)\,d\E_{T_1}(\z)\Big)X
\Big(\int\b_{jk}(\t)\,d\E_{T_2}(\t)\Big)Y
\Big(\int\g_k(\vk)\,d\E_{T_3}(\vk)\Big)\\[.2cm]
&=\sum_{j,k}\a_j(T_1)X\b_{jk}(T_2)Y\g_k(T_3).\end{aligned}$$ It is easy to verify that the series converges in the weak operator topology if we consider partial sums over rectangles. It can be shown in the same way as in the case of triple operator integrals with respect to spectral measures that the sum on the right does not depend on the choice of a representation of $\Phi$ in the form of , see Theorem 3.1 of [@ANP].
We are going to use Lemma 3.2 of [@AP5]. Suppose that $\{Z_j\}_{j\ge0}$ is a sequence of bounded linear operators on Hilbert space such that \[nrvadlyaAj\] \_[j0]{}Z\_j\^\*Z\_j\^[1/2]{}M\_[j0]{}Z\_jZ\_j\^\*\^[1/2]{}M. Let $Q$ be a bounded linear operator. Consider the row ${\rm R}_{\{Z_j\}}(Q)$ and the column ${\rm C}_{\{Z_j\}}(Q)$ defined by $${\rm R}_{\{Z_j\}}(Q)\df\big(Z_0Q\:Z_1Q\:Z_2Q\:\cdots\big)$$ and $${\rm C}_{\{Z_j\}}(Q)\df\left(\begin{matrix}QZ_0\\QZ_1\\QZ_2\\\vdots\end{matrix}\right).$$ Then by Lemma 3.2 of [@AP5], for $p\in[2,\be]$, the following inequalities hold: \[RCAjSp\] \_[{Z\_j}]{}(Q)\_[\_p]{}MQ\_[\_p]{}\_[{Z\_j}]{}(Q)\_[\_p]{}MQ\_[\_p]{} whenever $Q\in\bS_p$.
It is easy to verify that under the above assumptions \[WPhi\] W\_=[R]{}\_[{\_j(T\_1)}]{}(X)B[C]{}\_[{\_j(T\_3)}]{}(Y), where $B$ is the operator matrix $\{\b_{jk}(T_2)\}_{j,k\ge0}$.
\[norBjk\] Under the above hypotheses, $$\|B\|\le\sup_{\t\in\T}\big\|\{\b_{jk}(\t)\}_{j,k\ge0}\big\|_{{\mathcal{B}}}.$$
Let $U$ be a unitary dilation of the contraction $T_2$ on a Hilbert space $\K$, $\K\supset\h$. Clearly, we can consider the space $\ell^2(\h)$ as a subspace of $\ell^2(\K)$. It is easy to see that $$\{\b_{jk}(T_2)\}_{j,k\ge0}=P_{\ell^2(\h)}\{\b_{jk}(U)\}_{j,k\ge0}\big|\ell^2(\h),$$ where $P_{\ell^2(\h)}$ is the orthogonal projection onto $\ell^2(\h)$. The result follows from the inequality $\|\{\b_{jk}(U)\}_{j,k\ge0}\|\le\sup_{\t\in\T}\big\|\{\b_{jk}(\t)\}_{j,k\ge0}\big\|_{{\mathcal{B}}}$, which is a consequence of the spectral theorem. $\bl$
It follows from Lemma 3.2 of [@Pe8] that under the above assumptions, inequalities hold for $Z_j=\a_j(T_1)$, $j\ge0$, with $M=\sup_{\z\in\T}\left(\sum_{j\ge0}|\a_j(\z)|^2\right)^{1/2}$ and for $Z_j=\g_j(T_3)$, $j\ge0$, with $M=\sup_{\z\in\T}\left(\sum_{j\ge0}|\g_j(\z)|^2\right)^{1/2}$. This together with Lemma and inequalities implies that under the above assumptions, $$\begin{gathered}
\label{nervodlyapr}
\big\|{\rm R}_{\{\a_j(T_1)\}}(X)\,B\,{\rm C}_{\{\g_j(T_3)\}}(Y)\big\|_{\bS_r}
\\[.2cm]
\le
\sup_{\z\in\T}\left(\sum_{j\ge0}|\a_j(\z)|^2\right)^{1/2}
\sup_{\t\in\T}\big\|\{\b_{jk}(\t)\}_{j,k\ge0}\big\|_{{\mathcal{B}}}\,
\sup_{\vk\in\T}\left(\sum_{k\ge0}|\g_k(\vk)|^2\right)^{1/2}\end{gathered}$$ whenever $p\ge2$, $q\ge2$ and $1/r=1/p+1/q$.
The following theorem is an analog of the corresponding result for triple operator integrals with respect to spectral measures, see [@AP5]. It follows immediately from .
\[troipolume\] Let $T_1$, $T_2$ and $T_3$ be contractions, and let $X\in\bS_p$ and $Y\in\bS_q$, $2\le p\le\be$, $2\le q\le\be$. Suppose that $\Phi\in{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}$. Then $W_\Phi\in\bS_r$, $1/r=1/p+1/q$, and
$$\left\|\iiint\Phi\,d\E_{T_1}X\,d\E_{T_2}Y\,d\E_{T_3}\right\|_{\bS_r}
\le\|\Phi\|_{{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}}
\|X\|_{\bS_p}\|Y\|_{\bS_q}.$$ Recall that by $\bS_\be$ we mean the class of bounded linear operators.
[**3.5. Haagerup-like tensor products.**]{} We define here Haagerup-like tensor products of disk-algebras by analogy with Haagerup-like tensor products of $L^\be$ spaces, see [@ANP].
[**Definition 2.**]{} A continuous function $\Phi$ on $\T^3$ is said to belong to the [*Haagerup-like tensor product ${{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}\!\otimes^{\rm h}\!{{\rm C}_{\rm A}}$ of the first kind*]{} if it admits a representation \[tensprpe\] (,,)=\_[j,k0]{}\_j()\_[k]{}()\_[jk]{}(), ,,, where $\a_j$, $\b_k$ and $\g_{jk}$ are functions in ${{\rm C}_{\rm A}}$ such that $$\sup_{\z\in\T}\left(\sum_{j\ge0}|\a_j(\z)|^2\right)^{1/2}
\sup_{\t\in\T}\left(\sum_{k\ge0}|\b_k(\t)|^2\right)^{1/2}\,
\sup_{\vk\in\T}\big\|\{\g_{jk}(\vk)\}_{j,k\ge0}\big\|_{{\mathcal{B}}}<\be.$$
Clearly, $\Phi\in{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}\!\otimes^{\rm h}\!{{\rm C}_{\rm A}}$ if and only if the function $$(z_1,z_2,z_3)\mapsto\Phi(z_3,z_1,z_2)$$ belongs to the Haagerup tensor product ${{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}$.
Similarly, we can define the Haagerup-like tensor product ${{\rm C}_{\rm A}}\!\otimes^{\rm h}\!{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}$ of the second kind.
[**Definition 3.**]{} A continuous function $\Phi$ on $\T^3$ is said to belong to the [*Haagerup-like tensor product ${{\rm C}_{\rm A}}\!\otimes^{\rm h}\!{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}$ of the second kind*]{} if it admits a representation \[tensprvt\] (,,)=\_[j,k0]{}\_[jk]{}()\_[j]{}()\_k(), ,,, where $\a_{jk}$, $\b_j$ and $\g_k$ are functions in ${{\rm C}_{\rm A}}$ such that $$\sup_{\z\in\T}\big\|\{\a_{jk}(\z)\}_{j,k\ge0}\big\|_{{\mathcal{B}}}\,
\sup_{\t\in\T}\left(\sum_{j\ge0}|\b_j(\t)|^2\right)^{1/2}\,
\sup_{\vk\in\T}\left(\sum_{k\ge0}|\g_k(\vk)|^2\right)^{1/2}<\be.$$
Let us first consider the situation when $\Phi$ is defined by or by with summation over a finite set. In this case triple operator integrals of the form can be defined for arbitrary bounded linear operators $X$ and $Y$ and for arbitrary contractions $T_1$, $T_2$ and $T_3$.
Suppose that \[Fitipa1\] (,,)=\_[jF\_1]{}\_[kF\_2]{}\_j()\_[k]{}()\_[jk]{}(), ,,, \_j,\_k,\_[jk]{}[[C]{}\_[A]{}]{}, where $F_1$ and $F_2$ are finite sets. We put \[konsumpeti\] d\_[T\_1]{}Xd\_[T\_2]{}Yd\_[T\_3]{}\_[jF\_1]{}\_[kF\_2]{}\_j(T\_1)X\_[k]{}(T\_2)Y\_[jk]{}(T\_3). Suppose now that \[Fitipa2\] (,,)=\_[jF\_1]{}\_[kF\_2]{}\_[jk]{}()\_[j]{}()\_k(), ,,, \_[jk]{},\_j,\_k[[C]{}\_[A]{}]{}, where $F_1$ and $F_2$ are finite sets. Then we put \[konsumvtti\] d\_[T\_1]{}Xd\_[T\_2]{}Yd\_[T\_3]{}\_[jF\_1]{}\_[kF\_2]{}\_[jk]{}(T\_1)X\_[j]{}(T\_2)Y\_k(T\_3).
The following estimate is a very special case of Theorem \[pervto\] below. However, we have stated it separately because its proof is elementary and does not require the definition of triple operator integrals with integrands in Haagerup-like tensor products.
\[otsHaapevto\] Let $X$ and $Y$ be bounded linear operators and let $T_1$, $T_2$ and $T_3$ are contractions. Suppose that $F_1$ and $F_2$ are finite sets. The following statements hold:
[*(i)*]{} Let $\Phi$ be given by [**]{}. Suppose that $q\ge2$ and $1/r\df1/p+1/q\in[1/2,1]$. If $X\in\bS_p$ and $Y\in\bS_q$, then the sum on the right of [**]{} belongs to $\bS_r$ and $$\begin{gathered}
\left\|\sum_{j\in F_1}\sum_{k\in F_2}\a_j(T_1)X\b_{k}(T_2)Y\g_{jk}(T_3)\in\bS_r\right\|_{\bS_r}\le\\[.2cm]
\;\;\;\sup_{\z\in\T}\left(\sum_{j\in F_1}|\a_j(\z)|^2\right)^{1/2}
\!\!\sup_{\t\in\T}\left(\sum_{k\in F_2}|\b_k(\t)|^2\right)^{\1/2}
\!\!\sup_{\vk\in\T}\big\|\big\{\g_{jk}(\vk)\big\}_{j\in F_1,k\in F_2}\big\|_{\mathcal{B}}\|X\|_{\bS_p}\|Y\|_{\bS_q}.\end{gathered}$$
[*(ii)*]{} Let $\Phi$ be given by [**]{}. Suppose that $q\ge2$ and $1/r\df1/p+1/q\in[1/2,1]$. If $X\in\bS_p$ and $Y\in\bS_q$, then the sum on the right of [**]{} belongs to $\bS_r$ and $$\begin{gathered}
\left\|\sum_{j\in F_1}\sum_{k\in F_2}\a_{jk}(T_1)X\b_{j}(T_2)Y\g_k(T_3)\in\bS_r\right\|_{\bS_r}\le\\[.2cm]
\sup_{\z\in\T}\big\|\big\{\a_{jk}(\z)\big\}_{j\in F_1,k\in F_2}\big\|_{\mathcal{B}}\sup_{\t\in\T}\left(\sum_{j\in F_1}|\b_j(\t)|^2\right)^{1/2}
\!\!\sup_{\vk\in\T}\left(\sum_{k\in F_2}|\g_k(\vk)|^2\right)^{\1/2}
\!\!\|X\|_{\bS_p}\|Y\|_{\bS_q}.\end{gathered}$$
Let us prove (i). The proof of (ii) is the same. We are going to use a duality argument. Suppose that $Q\in\bS_{r'}$ and $\|Q\|_{\bS_{r'}}\le1$, $1/r+1/r'=1$. We have $$\begin{gathered}
\sup_Q\left|
\trace\left(Q\sum_{j\in F_1}\sum_{k\in F_2}\a_{jk}(T_1)X\b_{j}(T_2)Y\g_k(T_3)
\right)\right|\\[.2cm]
=
\sup_Q\left|
\trace\left(\sum_{j\in F_1}\sum_{k\in F_2}\g_k(T_3)Q\a_{jk}(T_1)X\b_{j}(T_2)
\right)Y\right|\\[.2cm]
\le\|Y\|_{\bS_q}\sup_Q
\left\|\sum_{j\in F_1}\sum_{k\in F_2}
\g_k(T_3)Q\a_{jk}(T_1)X\b_{j}(T_2)\right\|_{\bS_{q'}}.\end{gathered}$$ Th result follows now from and . $\bl$
[**3.6. Triple operator integrals with integrands in Haagerup-like tensor products.**]{} We define triple operator integrals with integrands in ${{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}\!\otimes^{\rm h}\!{{\rm C}_{\rm A}}$ by analogy with triple operator integrals with respect to spectral measures, see [@ANP] and [@AP5]. Let $\Phi\in{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}\!\otimes^{\rm h}\!{{\rm C}_{\rm A}}$ and let $p\in[1,2]$. Suppose that $T_1$, $T_2$ and $T_3$ are contractions. For an operator $X$ of class $\bS_p$ and for a bounded linear operator $Y$, we define the triple operator integral \[WPhi1\] \_[@]{}(,,)d\_[T\_1]{}()Xd\_[T\_2]{}()Yd\_[T\_3]{}() as the following continuous linear functional on $\bS_{p'}$, $1/p+1/p'=1$ (on the class of compact operators in the case $p=1$): $$Q\mapsto
\trace\left(\left(
\iiint
\Phi(\z,\t,\vk)\,dE_{T_2}(\t)Y\,dE_{T_3}(\vk)Q\,dE_{T_1}(\z)
\right)X\right).$$ Note that the triple operator integral $\iiint\Phi(\z,\t,\vk)\,dE_{T_2}(\t)Y\,dE_{T_3}(\vk)Q\,dE_{T_1}(\z)$ is well defined as the integrand belongs to the Haagerup tensor product ${{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}$.
Again, we can define triple operator integrals with integrands in ${{\rm C}_{\rm A}}\!\otimes^{\rm h}\!{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}$ by analogy with the case of spectral measures, see [@ANP] and [@AP5]. Let $\Phi\in{{\rm C}_{\rm A}}\!\otimes^{\rm h}\!{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}$ and let $T_1$, $T_2$ and $T_3$ be contractions. Suppose that $X$ is a bounded linear operator and $Y\in\bS_p$, $1\le p\le2$. The triple operator integral \[WPhi2\] \_[@]{}(,,)d\_[T\_1]{}()Xd\_[T\_2]{}()Yd\_[T\_3]{}() is defined as the continuous linear functional $$Q\mapsto
\trace\left(\left(
\iiint\Phi(\z,\t,\vk)\,dE_3(\vk)Q\,dE_1(\z)X\,dE_2(\t)
\right)Y\right)$$ on $\bS_{p'}$ (on the class of compact operators if $p=1$).
As in the case of spectral measures (see [@AP5]), the following theorem can be proved:
\[pervto\] Suppose that $T_1$, $T_2$ and $T_3$ are contractions, and let $X\in\bS_p$ and $Y\in\bS_q$. The following statements hold:
[*(1)*]{} Let $\Phi\in{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}\!\otimes^{\rm h}\!{{\rm C}_{\rm A}}$. Suppose $q\ge2$ and $1/r\df1/p+1/q\in[1/2,1]$. If $X\in\bS_p$ and $Y\in\bS_q$, then the operator ${\stackrel{1}{W}}_\Phi$ in [**]{} belongs to $\bS_r$ and \[rpq\] \_\_[\_r]{}\_[[[C]{}\_[A]{}]{}\_[h]{}[[C]{}\_[A]{}]{}\^[h]{}[[C]{}\_[A]{}]{}]{} X\_[\_p]{}Y\_[\_q]{};
[*(2)*]{} Let $\Phi\in{{\rm C}_{\rm A}}\!\otimes^{\rm h}\!{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}$. Suppose that $p\ge2$ and $1/r\df1/p+1/q\in[1/2,1]$. If $X\in\bS_p$ and $Y\in\bS_q$, then the operator ${\stackrel{2}{W}}_\Phi$ in [**]{} belongs to $\bS_r$ and $$\Big\|{\stackrel{2}{W}}_\Phi\Big\|_{\bS_r}\le\|\Phi\|_{{{\rm C}_{\rm A}}\!\otimes^{\rm h}{{\rm C}_{\rm A}}\!\otimes_{\rm h}{{\rm C}_{\rm A}}}
\|X\|_{\bS_p}\|Y\|_{\bS_q}.$$
**Lipschitz type estimates in Schatten–von Neumann norms**
==========================================================
\[LiptysvN\]
In this section we obtain Lipschitz type estimates in the Schatten–von Neumann classes $\bS_p$ for $p\in[1,2]$ for functions of contractions. To obtain such estimates, we are going to use an elementary approach and obtain elementary formulae that involve only finite sums.
Later we will need explicit expressions for operator differences, which will be obtained in the next section in terms of triple operator integrals. Such formulae will be used in §\[differ\] to obtain formulae for operator derivatives.
Suppose that $f$ is a function that belongs to the Besov space $\big(B_{\be,1}^1\big)_+(\T^2)$ of analytic functions (see §\[Bes\]). As we have observed in Subsection 3.3, we can define functions $f(T,R)$ for (not necessarily commuting) contractions $T$ and $R$ on Hilbert space by formula . For a differentiable function $f$ on $\T$, we use the notation ${\frak D}f$ for the divided difference: $$({\frak D}f)(\z,\t)\df
\left\{\begin{array}{ll}
\displaystyle{\frac{f(\z)-f(\t)}{\z-\t}},&\z\ne\t\\[.4cm]f'(\z),&\z=\t,
\end{array}\right.
\qquad\z,\;\t\in\T.$$ For a differentiable function $f$ on $\T^2$, we define the divided differences ${\frak D}^{[1]}f$ and ${\frak D}^{[2]}f$ by $$\big({\frak D}^{[1]}f\big)(\z_1,\z_2,\t)\df
\left\{\begin{array}{ll}
\displaystyle{\frac{f(\z_1,\t)-f(\z_2,\t)}{\z_1-\z_2}},&\z_1\ne\z_2,\\[.4cm]
\displaystyle{\frac{\partial f}{\partial\z}\Big|_{\z=\z_1}},&\z_1=\z_2,
\end{array}\right.
\qquad\z_1,\;\z_2,\:\t\in\T,$$ and $$\big({\frak D}^{[2]}f\big)(\z,\t_1,\t_2)\df
\left\{\begin{array}{ll}
\displaystyle{\frac{f(\z,\t_1)-f(\z,\t_2)}{\t_1-\t_2}},&\t_1\ne\t_2,\\[.4cm]
\displaystyle{\frac{\partial f}{\partial\t}\Big|_{\t=\t_1}},&\t_1=\t_2,
\end{array}\right.
\qquad\z,\;\t_1,\:\t_2\in\T.$$
We need several elementary identities.
Let $\Pi_m$ be the set of $m$th roots of 1: $$\Pi_m\df\{\xi\in\T:\xi^m=1\}$$ and let $$\Upsilon_m(\z)\df\frac{\z^{m}-1}{m(\z-1)}=\frac1{m}\sum_{k=0}^{m-1}\z^k,\quad\z\in\T.$$
The following elementary formulae are well known. We give proofs for completeness.
\[136\] Let $f$ and $g$ be analytic polynomials in one variable of degree less than $m$. Then $$\int_\T f\ov g\,d\m=\frac1m\sum_{\xi\in\Pi_m}f(\xi)\ov{g(\xi)}.$$ In particular, $$\int_\T |f|^2\,d\m=\frac1m\sum_{\xi\in\Pi_m}|f(\xi)|^2.$$
It suffices to consider the case where $f(z)=z^j$ and $g(z)=z^k$ with $0\le j, k<m$. Then $-m<j-k<m$ and $$\sum_{\xi\in\Pi_m}\xi^j\,\ov\xi^k=\left\{\begin{array}{ll}0,&j\ne k\\[.2cm]
m,&j=k.
\end{array}\right.\quad\bl$$
\[summodkv\] \_[\_m]{}|\_m(|)|\^2=1, .
In the same way we can obtain similar formulae for polynomials in several variables. We need only the case of two variables.
\[137\] Let $f$ and $g$ be polynomials in two variables of degree less than $m$ in each variable. Then $$\int_{\T^2} f\ov g\,d\m_2=\frac1{m^2}\sum_{\xi,\eta\in\Pi_m}f(\xi,\eta)\ov{g(\xi,\eta)}.$$ In particular, $$\int_{\T^2} |f|^2\,d\m_2=\frac1{m^2}\sum_{\xi,\eta\in\Pi_m}|f(\xi,\eta)|^2.$$
It suffices to consider the case when $f(\z,\t)=\z^{j_1}\t^{j_2}$ and $g(\z,\t)=\z^{k_1}\t^{k_2}$ with $0\le j_1, j_2, k_1, k_2<m$. Then $-m<j_1-k_1, j_2-k_2<m$ and \[summirj1j2k1k2\] \_[,\_m]{}\^[j\_1]{}\^[j\_2]{}\^[k\_1]{}\^[k\_2]{}={
[ll]{}0,&(j\_1,j\_2)(k\_1,k\_2)\
m\^2,&(j\_1,j\_2)=(k\_1,k\_2).
.
Suppose now that $(T_0,R_0)$ and $(T_1,R_1)$ are pairs of not necessarily commuting contractions.
\[flydlyaraz\] Let $f$ be an analytic polynomial in two variable of degree at most $m$ in each variable. Then $$\begin{aligned}
\label{perra}
f(T_1,R_1)-f(T_0,R_1)
=\sum_{\xi,\eta\in\Pi_m}\Upsilon_m(\ov\xi T_1)(T_1-T_0)\,\Upsilon_m(\ov\eta T_0)\,({\frak D}^{[1]}f)(\xi,\eta,R_1)\end{aligned}$$ and $$\begin{aligned}
\label{vtorra}
f(T_0,R_1)-f(T_0,R_0)
=\sum_{\xi,\eta\in\Pi_m}({\frak D}^{[2]}f)(T_0,\xi,\eta)\,\Upsilon_m(\ov\xi R_1)(R_1-R_0)\,\Upsilon_m(\ov\eta R_0).\end{aligned}$$
We are going to establish . The proof of is similar.
We need the following lemma.
\[odnape\] Let $\f$ be an analytic polynomial in one variable of degree at most $m$. Then $$\f(T_1)-\f(T_0)=
\sum_{\xi,\eta\in\Pi_m}\Upsilon_m(\ov\xi T_1)(T_1-T_0)\,\Upsilon_m(\ov\eta T_0)({\frak D}\f)(\xi,\eta).$$
[**Proof of the lemma.**]{} Let $0\le j, j_0, k, k_0<m$. Then $$\sum_{\xi,\eta\in\Pi_m}(\ov\xi T_1)^{j_0}\,(\ov\eta T_0)^{k_0}\xi^j\eta^k=
\left\{\begin{array}{ll}\displaystyle{m^2T_1^jT_0^k},&(j_0,k_0)=(j,k),\\[.4cm]
0,&(j_0,k_0)\ne(j,k).
\end{array}\right.$$ Thus, $$\sum_{\xi,\eta\in\Pi_m}\Upsilon_m(\ov\xi T_1)\,\Upsilon_m(\ov\eta T_0)\xi^j\eta^k=T_1^jT_0^k$$ if $0\le j, k<n$. Hence, $$\sum_{\xi,\eta\in\Pi_m}\Upsilon_m(\ov\xi T_1)T_1\,\Upsilon_m(\ov\eta T_0)\xi^j\eta^k=T_1^{j+1}T_0^k$$ and $$\sum_{\xi,\eta\in\Pi_m}\Upsilon_m(\ov\xi T_1)\,T_0\Upsilon_m(\ov\eta T_0)\xi^j\eta^k=T_1^{j}T_0^{k+1}.$$ It follows that $$\sum_{\xi,\eta\in\Pi_m}\Upsilon_m(\ov\xi T_1)(T_1-T_0)\,\Upsilon_m(\ov\eta T_0)\xi^j\eta^k=T_1^j(T_1-T_0)T_0^k$$ whenever $0\le j, k<m$.
Let $\f=\sum\limits_{s=0}^ma_sz^s$. It is easy to see that $$({\frak D}\f)(z,w)=\sum_{j,k\ge0, j+k<m}a_{j+k+1}z^jw^k.$$ Hence, $$\begin{gathered}
\sum_{\xi,\eta\in\Pi_m}\Upsilon_m(\ov\xi T_1)(T_1-T_0)\,\Upsilon_m(\ov\eta T_0)
({\frak D}\f)(\xi,\eta)\\
=\sum_{j,k\ge0, j+k<m}a_{j+k+1}\sum_{\xi,\eta\in\Pi_m}\Upsilon_m(\ov\xi T_1)(T_1-T_0)\,\Upsilon_m(\ov\eta T_0)\xi^j\eta^k\\
=\sum_{j,k\ge0, j+k<m}a_{j+k+1}T_1^j(T_1-T_0)T_0^k=\f(T_1)-\f(T_0).\quad\bl\end{gathered}$$
[**Proof of Theorem \[flydlyaraz\].**]{} Clearly, it suffices to prove in the case when $f(z_1,z_2)=\f(z_1)z_2^j$, where $\f$ is a polynomial of one variable of degree at most $n$ and $0\le j\le m$. Clearly, in this case $$f(T_1,R_1)-f(T_0,R_1)=\big(\f(T_1)-\f(T_0)\big)R_1^j.$$ On the other hand, $$({\frak D}f^{[1]})(\xi,\eta,R_1)=({\frak D}\f)(\xi,\eta)R_1^j.$$ Identity follows now from Lemma \[odnape\]. $\bl$
For $K\in L^2(\T^2)$, we denote by $\I_K$ the integral operator on $L^2(\T)$ with kernel function $K$, i.e., $$(\I_K\f)(\z)=\int_\T K(\z,\t)\f(\t)\,d\m(\t),\quad \f\in L^2(\T).$$ The following lemma allows us to evaluate the operator norm $\|\I_K\|_{{\mathcal{B}}(L^2)}$ of this operator for polynomials $K$ of degree less than $m$ in each variable in terms of the operator norms of the matrix $\{K(\z,\eta)\}_{\z,\eta\in\Pi_m}$.
\[intopera\] Let $K$ be an analytic polynomial in two variables of degree less than $m$ in each variable. Then $$\|\{K(\xi,\eta)\}_{\xi,\eta\in\Pi_m}\|_{{\mathcal{B}}}=m\|\I_K\|_{{\mathcal{B}}(L^2)}.$$
It is easy to see that \_K\_[(L\^2)]{}=\_[\_[L\^2]{}1,\_[L\^2]{}1]{}|\_K(,)d()d()|\
=\_[\_[L\^2]{}1,\_[L\^2]{}1]{}|\_K(,)d()d()|, where $\varphi_m(z)=\sum\limits_{k=0}^{m-1}\widehat\varphi(k)z^k$ and $\psi_m(z)=\sum\limits_{k=0}^{m-1}\widehat\psi(k)z^k$. Hence, $$\|\I_K\|_{{\mathcal{B}}(L^2)}=
\sup\left|\iint_{\T\times\T}K(\z,w)\,\ov{\varphi(z)\psi(w)}\,d\m(\z)\,d\m(\t)\right|,
$$ where the supremum is taken over all polynomials $\varphi$ and $\psi$ in one variable of degree less than $m$ and such that $\|\varphi\|_{L^2}\le1$, $\|\psi\|_{L^2}\le1$. Next, by Lemma \[137\], for arbitrary polynomials $\varphi$ and $\psi$ with $\deg\varphi<m$ and $\deg\psi<m$, we have $$\iint_{\T\times\T}K(\z,\t)\,\ov{\varphi(z)\psi(w)}\,d\m(\z)\,d\m(\t)=\frac1{m^2}\sum_{\xi,\eta\in\Pi_m}K(\xi,\eta)\,\ov{\varphi(\xi)\psi(\eta)}.$$ It remains to observe that by Lemma \[136\], $\|\varphi\|_{L^2}\le1$ if and only if $\sum\limits_{\xi\in\Pi_m}|\varphi(\xi)|^2\le m$ and the same is true for $\psi$. $\bl$
\[matrazra\] Let $g$ be a polynomial in one variable of degree at most $m$. Then $$\|\{({\frak D}g)(\xi,\eta)\}_{\xi,\eta\in\Pi_m}\|_{{\mathcal{B}}}\le m\|g\|_{L^\be}.$$
The result follows from Lemma \[intopera\] and the inequality $$\|\I_{{\frak D}g}\|_{{\mathcal{B}}(L^2)}\le\|g\|_{L^\be},$$ which is a consequence of the fact that $\|\I_{{\frak D}g}\|_{{\mathcal{B}}(L^2)}$ is equal to the norm of the Hankel operator $H_{\bar g}$ on the Hardy class $H^2$, see [@Pe5], Ch. 1, Th. 1.10. $\bl$
\[sleu\] Let $f$ be a trigonometric polynomial of degree at most $m$ in each variable and let $p\in[1,2]$. Suppose that $T_1,\,R_1,T_0,\,R_0$ are contractions such that $T_1-T_0\in\bS_p$ and $R_1-R_0\in\bS_p$. Then $$\|f(T_1,R_1)-f(T_0,R_0)\|_{\bS_p}\le2m\|f\|_{L^\be}
\max\big\{\|T_1-T_0\|_{\bS_p},\|R_1-R_0\|_{\bS_p}\big\}.$$
Let us estimate $\|f(T_1,R_1)-f(T_0,R_1)\|_{\bS_p}$. The norm $\|f(T_0,R_1)-f(T_0,R_0)\|_{\bS_p}$ can be estimated in the same way. The result is a consequence of formula , Theorem \[otsHaapevto\], Theorem \[matrazra\] and Corollare \[summodkv\]. $\bl$
Corollary \[sleu\] allows us to establish a Lipschitz type inequality for functions in $\big(B_{\be,1}^1\big)_+(\T^2)$.
\[p2unit\] Let $1\le p\le2$ and let $f\in \big(B_{\be,1}^1\big)_+(\T^2)$. Suppose that $T_1,\,R_1,T_0,\,R_0$ are contractions such that $T_1-T_0\in\bS_p$ and $R_1-R_0\in\bS_p$. Then $$\|f(T_1,R_1)-f(T_0,R_0)\|_{\bS_p}\le\const\|f\|_{B_{\be,1}^1}
\max\big\{\|T_1-T_0\|_{\bS_p},\|R_1-R_0\|_{\bS_p}\big\}.$$
Indeed, the result follows immediately from Corollary \[sleu\] and inequality . $\bl$
**A representation of operator differences\
in terms of triple operator integrals**
===========================================
\[pretrioi\]
In this section we obtain an explicit formula for the operator differences $f(T_1,R_1)-f(T_0,R_0)$, $f\in{\big(B_{\be,1}^1\big)_+(\T^2)}$, in terms of triple operator integrals.
\[razdraquasziHaa\] Let $f\in{\big(B_{\be,1}^1\big)_+(\T^2)}$. Then $${\frak D}^{[1]}f\in{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}\!\otimes^{\rm h}\!{{\rm C}_{\rm A}}\quad\mbox{and}\quad
{\frak D}^{[2]}f\in{{\rm C}_{\rm A}}\!\otimes^{\rm h}\!{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}.$$
\[predg1idg2\] Let $f$ be an analytic polynomial in two variables of degree at most $m$ in each variable. Then \[dg1f\] ([D]{}\^[\[1\]]{}f)(\_1,\_2,)= \_[,\_m]{}\_m(\_1)\_m(\_2) ([D]{}\^[\[1\]]{}f)(,,) and \[dg2f\] ([D]{}\^[\[2\]]{}f)(,\_1,\_2)=\_[,\_m]{} ([D]{}\^[\[2\]]{}f)(,,) \_m(\_1)\_m(\_2).
Both formulae and can be verified straightforwardly. However, we deduce them from Theorem \[flydlyaraz\].
Formula follows immediately from formula if we consider the special case when $T_0$, $T_1$ and $R_1$ are the operators on the one-dimensional space of multiplication by $\z_2$, $\z_1$ and $\t$. Similarly, formula follows immediately from formula . $\bl$
\[otsquasiHaanorm\] Under the hypotheses of Lemma [*\[predg1idg2\]*]{}, $$\big\|{\frak D}^{[1]}f\big\|_{{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}\!\otimes^{\rm h}{{\rm C}_{\rm A}}}
\le m\|f\|_{L^\be}\quad\mbox{and}\quad
\big\|{\frak D}^{[2]}f\big\|_{{{\rm C}_{\rm A}}\!\otimes^{\rm h}{{\rm C}_{\rm A}}\!\otimes_{\rm h}\!{{\rm C}_{\rm A}}}
\le m\|f\|_{L^\be}.$$
The result is a consequence of Lemma \[predg1idg2\], Theorem \[matrazra\], Corollary \[summodkv\] and Definitions 2 and 3 in §\[dvoitro\]. $\bl$
[**Proof of Theorem \[razdraquasziHaa\].**]{} The result follows immediately from Corollary \[otsquasiHaanorm\] and inequality . $\bl$
\[predraquasiHaa\] Let $p\in[1,2]$. Suppose that $T_0,\,R_0,\,T_1,\,R_1$ are contractions such that $T_1-T_0\in\bS_p$ and $R_1-R_0\in\bS_p$. Then for $f\in{\big(B_{\be,1}^1\big)_+(\T^2)}$, the following formula holds: $$\begin{aligned}
\label{osnfuni}
f(T_1,R_1)&-f(T_0,R_0)\nonumber\\[.2cm]
&=
\iint\!\!{\DOTSI\upintop\ilimits@}\big({\frak D}^{[1]}f\big)(\z_1,\z_2,\t)
\,dE_{T_1}(\z_1)(T_1-T_0)\,dE_{T_2}(\z_2)\,dE_{R_1}(\t),\nonumber\\[.2cm]
&+{\DOTSI\upintop\ilimits@}\!\!\!\iint\big({\frak D}^{[2]}f\big)(\z,\t_1,\t_2)
\,dE_{T_2}(\z)\,dE_{R_1}(\t_1)(R_1-R_0)\,dE_{R_2}(\t_2).\end{aligned}$$
Suppose first that $f$ is an analytic polynomial in two variables of degree at most $m$ in each variable. In this case equality is a consequence of Theorem \[flydlyaraz\], Lemma \[predg1idg2\] and the definition of triple operator integrals given in Subsection 3.5.
In the general case we represent $f$ by the series and apply to each $f_n$. The result follows from . $\bl$
**Differentiability properties**
================================
\[differ\]
In this section we study differentiability properties of the map \[fT1T0R1R0\] tf(T(t),R(t)) in the norm of $\bS_p$, $1\le p\le2$, for functions $t\mapsto T(t)$ and $t\mapsto R(t)$ that take contractive values and are differentiable in $\bS_p$.
We say that an operator-valued function $\Psi$ defined on an interval $J$ is [*differentiable*]{} in $\bS_p$ if $\Phi(s)-\Phi(t)\in\bS_p$ for any $s,\,t\in J$, and the limit $$\lim_{h\to\0}\frac1h\big(\Psi(t+h)-\Psi(t)\big)\df\Phi'(t)$$ exists in the norm of $\bS_p$ for each $t$ in $J$.
Let $p\in[1,2]$ and let $f\in{\big(B_{\be,1}^1\big)_+(\T^2)}$. Suppose that $t\mapsto T(t)$ and $t\mapsto R(t)$ are operator-valued functions on an interval $J$ that take contractive values and are differentiable in $\bS_p$. Then the function [**]{} is differentiable on $J$ in $\bS_p$ and $$\begin{aligned}
\frac d{dt}&f\big(T(t),R(t)\big)\Big|_{t=s}
\\[.2cm]
&=
\iint\!\!{\DOTSI\upintop\ilimits@}\big({\frak D}^{[1]}f\big)(\z_1,\z_2,\t)
\,dE_{T(s)}(\z_1)T'(s)\,dE_{T(s)}(\z_2)\,dE_{R(s)}(\t)\nonumber\\[.2cm]
&+{\DOTSI\upintop\ilimits@}\!\!\!\iint\big({\frak D}^{[2]}f\big)(\z,\t_1,\t_2)
\,dE_{T(s)}(\z)\,dE_{R(s)}(\t_1)R'(s)\,dE_{R(s)}(\t_2),\end{aligned}$$ $s\in J$.
As before, it suffices to prove the result in the case when $f$ is an analytic polynomial of degree at most $m$ in each variable. Suppose that $f$ is such a polynomial. Put $F(t)\df f\big(T(t),R(t)\big)$. We have $$\begin{aligned}
F(s+h)&-F(s)\\[.2cm]
&=\sum_{\xi,\eta\in\Pi_m}\Upsilon_m\big(\ov\xi T(s+h)\big)\big(T(s+h)-T(s)\big)\Upsilon_m\big(\ov\eta T(s)\big)\big({\frak D}^{[1]}f\big)\big(\xi,\eta,R(s+h)\big)\\[.2cm]
&+
\sum_{\xi,\eta\in\Pi_m}\big({\frak D}^{[2]}f\big)\big(T(s),\xi,\eta\big)
\Upsilon_m\big(\ov\xi R(s+h)\big)
\big(R(s+h)-R(s)\big)\Upsilon_m\big(\ov\eta R(s)\big).\end{aligned}$$ Clearly, $$\lim_{h\to0}\frac1h\big(T(s+h)-T(s)\big)=T'(s)\quad\mbox{and}\quad
\lim_{h\to0}\frac1h\big(R(s+h)-R(s)\big)=R'(s)$$ in the norm of $\bS_p$. On the other hand, it is easy to see that $$\lim_{h\to0}\Upsilon_m\big(\ov\xi T(s+h)\big)=\Upsilon_m\big(\ov\xi T(s)\big),\quad
\lim_{h\to0}\big({\frak D}^{[1]}f\big)\big(\xi,\eta,R(s+h)\big)
=\big({\frak D}^{[1]}f\big)\big(\xi,\eta,R(s)\big)$$ and $$\lim_{h\to0}\Upsilon_m\big(\ov\xi R(s+h)\big)=\Upsilon_m\big(\ov\xi R(s)\big)$$ in the operator norm. Hence, $$\begin{aligned}
F'(s)
&=\sum_{\xi,\eta\in\Pi_m}\Upsilon_m\big(\ov\xi T(s)\big)T'(s)\Upsilon_m\big(\ov\eta T(s)\big)\big({\frak D}^{[1]}f\big)\big(\xi,\eta,R(s)\big)\\[.2cm]
&+
\sum_{\xi,\eta\in\Pi_m}\big({\frak D}^{[2]}f\big)\big(T(s),\xi,\eta\big)
\Upsilon_m\big(\ov\xi R(s)\big)
R'(s)\Upsilon_m\big(\ov\eta R(s)\big).\end{aligned}$$ It follows now from Lemma \[predg1idg2\] and from the definition of triple operator integrals given in §\[dvoitro\] that the right-hand side is equal to $$\begin{gathered}
\iint\!\!{\DOTSI\upintop\ilimits@}\big({\frak D}^{[1]}f\big)(\z_1,\z_2,\t)
\,dE_{T(s)}(\z_1)T'(s)\,dE_{T(s)}(\z_2)\,dE_{R(s)}(\t)\\[.2cm]
+{\DOTSI\upintop\ilimits@}\!\!\!\iint\big({\frak D}^{[2]}f\big)(\z,\t_1,\t_2)
\,dE_{T(s)}(\z)\,dE_{R(s)}(\t_1)R'(s)\,dE_{R(s)}(\t_2)\end{gathered}$$ which completes the proof. $\bl$
**The case $\bs{p>2}$**
=======================
\[counter\]
In this section we show that unlike in the case $p\in[1,2]$, there are no Lipschitz type estimates in the norm of $\bS_p$ in the case when $p>2$ for functions $f(T,R)$, $f\in{\big(B_{\be,1}^1\big)_+(\T^2)}$, of not noncommuting contractions. In particular, there are no such Lipschitz type estimates for functions $f\in{\big(B_{\be,1}^1\big)_+(\T^2)}$ in the operator norm. Moreover, we show that for $p>2$, such Lipschitz type estimates do not hold even for functions $f$ in ${\big(B_{\be,1}^1\big)_+(\T^2)}$ and for pairs of noncommuting [*unitary operators*]{}.
Recall that similar results were obtained in [@ANP] for functions of noncommuting self-adjoint operators. However, in this paper we use a different construction to obtain results for functions of unitary operators.
\[inter\] For each matrix $\{a_{\xi\, \eta}\}_{\xi, \eta\in\Pi_m}$, there exists an analytic polynomial $f$ in two variables of degree at most $2m-2$ in each variable such that $f(\xi,\eta)=a_{\xi\, \eta}$ for all $\xi, \eta\in\Pi_m$ and $\|f\|_{L^\be(\T^2)}\le \sup\limits_{\xi, \eta\in\Pi_m}|a_{\xi\, \eta}|$.
Put $$f(z,w)\df\sum_{\xi, \eta\in\Pi_m}a_{\xi\, \eta}\Upsilon_m^2(z\ov\xi)\Upsilon_m^2(w\ov\eta).$$ Clearly, $f(\xi,\eta)=a_{\xi\, \eta}$ for all $\xi, \eta\in\Pi_m$ and $$\begin{aligned}
|f(z,w)|&\le\sup_{\xi, \eta\in\Pi_m}|a_{\xi\, \eta}|\sum_{\xi, \eta\in\Pi_m}|\Upsilon_m(z\ov\xi)|^2|\Upsilon_m(w\ov\eta)|^2\\[.2cm]
&=
\sup_{\xi, \eta\in\Pi_m}|a_{\xi\, \eta}|\sum_{\xi\in\Pi_m}|\Upsilon_m(z\ov\xi)|^2\sum_{\eta\in\Pi_m}|\Upsilon_m(w\ov\eta)|^2
=\sup_{\xi, \eta\in\Pi_m}|a_{\xi\, \eta}|\end{aligned}$$ by Corollary \[summodkv\]. $\bl$
\[fU1U2V\] For each $m\in\Bbb N$, there exists an analytic polynomial $f$ in two variables of degree at most $4m-2$ in each variable, and unitary operators $U_1$, $U_2$ and $V$ such that $$\|f(U_1,V)-f(U_2,V)\|_{\bS_p}>\pi^{-1}m^{\frac32-\frac1p} \|f\|_{L^\be(\T^2)}\|U_1-U_2\|_{\bS_p}$$ for every $p>0$.
One can select orthonormal bases $\{g_\xi\}_{\xi\in\Pi_m}$ and $\{h_\eta\}_{\eta\in\Pi_m}$ in an $m$-dimensional Hilbert space $\h$ such that $|(g_\xi,h_\eta)|=m^{-\frac12}$ for all $\xi,\eta\in\Pi_m$. Indeed, let $\h$ be the subspace of $L^2(\T)$ of analytic polynomials of degree less than $m$. We can put $g_\xi\df\sqrt{m}\Upsilon_m(z\ov\xi)$ and $h_\eta=z^k$, where $\eta=e^{2\pi{\rm i}k/m}$, $0\le k\le m-1$.
Consider the rank one projections $\{P_\xi\}_{\xi\in\Pi_m}$ and $\{Q_\eta\}_{\eta\in\Pi_m}$ defined by $P_\xi v = (v,g_\xi)g_\xi$, $\xi\in\Pi_m$, and $Q_\eta v = (v,h_\eta)h_\eta$, $\eta\in\Pi_m$. We define the unitary operators $U_1$, $U_2$, and $V$ by $$U_1=\sum_{\xi\in\Pi_m}\xi P_\xi,\quad U_2=e^{\frac{\pi{\rm i}}m}U_1
\quad\mbox{and}\quad
V=\sum_{\eta\in\Pi_m}\eta Q_\eta.$$
By Lemma \[inter\], there exists an analytic polynomial $f$ in two variables of degree at most $4m-2$ in each variable such that $f(\xi,\eta)=\sqrt m(g_\xi,h_\eta)$ for all $\xi, \eta\in\Pi_m$, $f(\xi,\eta)=0$ for all $\xi\in\Pi_{2m}\setminus \Pi_m$, $\eta\in\Pi_m$ and $\|f\|_{L^\be(\T^2)}=1$. Clearly, $f(U_2,V)=\0$ and $f(U_1,V)=\sum\limits_{\xi,\eta\in\Pi_m}f(\xi,\eta)P_\xi Q_\eta$. We have $$( f(U_1,V)h_\eta, g_\xi)=f(\xi,\eta)(h_\eta, g_\xi)=\frac1{\sqrt m}.$$ Hence, $\rank f(U_1,V)=1$ and $$\|f(U_1,V)-f(U_2,V)\|_{\bS_p}=\|f(U_1,V)\|_{\bS_p}=\|f(U_1,V)\|_{\bS_2}=\sqrt m.$$ It remains to observe that $\|U_1-U_2\|_{\bS_p}=
\big|1-e^{\frac{\pi{\rm i}}m}\big|m^{\frac1p}<\pi m^{\frac1p-1}$. $\bl$
[**Remark.**]{} If we replace the polynomial $f$ constructed in the proof of Lemma \[fU1U2V\] with the polynomial $g$ defined by $$g(z_1,z_2)=z_1^{4m-2}z_2^{4m-2}f(z_1,z_2),$$ it will obviously satisfy the same inequality: \[gU1U2V\] g(U\_1,V)-g(U\_2,V)\_[\_p]{}> \^[-1]{}m\^[32-1p]{} g\_[L\^(\^2)]{}U\_1-U\_2\_[\_p]{}.
It is easy to deduce from that for such polynomials $g$ $$c_1m\|g\|_{L^\be(\T^2)}\le\|g\|_{B^\be_{\be,1}}\le c_2m\|g\|_{L^\be(\T^2)}$$ for some constants $c_1$ and $c_2$.
This together with implies the following result:
Let $M>0$ and $2<p\le\be$. Then there exist unitary operators $U_1$, $U_2$, $V$ and an analytic polynomial $f$ in two variables such that $$\|f(U_1,V)-f(U_2,V)\|_{\bS_p}>M\|f\|_{B^1_{\infty,1}(\T^2)}\|U_1-U_2\|_{\bS_p}.$$
**Open problems**
=================
\[zada\]
In this section we state open problems for functions of noncommuting contractions.
[**Functions of triples of contractions.**]{} Recall that it was shown in [@Pe7] that for $f\in B_{\be,1}^1(\R)$, there are no Lipschitz type estimates in the norm of $\bS_p$ for any $p>0$ for functions $f(A,B,C)$ of triples of noncommuting self-adjoint operators. We conjecture that the same must be true in the case of functions of triples of not necessarily commuting contractions. Note that the construction given in [@Pe7] does not generalize to the case of functions of contractions.
[**Lipschitz functions of noncommuting contractions.**]{} Recall that an unknown referee of [@ANP] observed that for [*Lipschitz*]{} functions $f$ on the real line there are no Lipschitz type estimates for functions $f(A,B)$ of noncommuting self-adjoint operators in the Hilbert–Schmidt norm. The construction is given in [@ANP]. We conjecture that the same result must hold in the case of functions of noncommuting contractions.
[**Lipschitz type estimates for $\bs{p>2}$ and Hölder type estimates.**]{} It follows from results of [@ANP] that in the case of functions of noncommuting self-adjoint operators for any $s>0$, $q>0$ and $p>2$, there exist pairs of self-adjoint operators $(A_0,A_1)$ and $(B_0,B_1)$ and a function $f$ in the homogeneous Besov space $B_{\be,q}^s(\R)$ such that $\|f(A_1,B_1)-f(A_0,B_0)\|_{\bS_p}$ can be arbitrarily large while $\max\{\|A_1-A_0\|_{\bS_p},\|B_1-B_0\|_{\bS_p}\}$ can be arbitrarily small. In particular, the condition $f\in B_{\be,q}^s(\R)$ does not imply any Lipschitz or Hölder type estimates in the norm of $\bS_p$, $p>2$, for any positive $s$ and $q$.
It is easy to see that in the case of contractions the situation is different: for any $q>0$ and $p\ge1$, there exists $s>0$ such that the condition $f\in B_{\be,q}^s$ guarantees a Lipschitz type estimate for functions of not necessarily commuting contractions in $\bS_p$.
It would be interesting to find optimal conditions on $f$ that would guarantee Lipschitz or Hölder type estimates in $\bS_p$ for a given $p$.
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----------------------------------- ---------------------------------
A.B. Aleksandrov V.V. Peller
St.Petersburg Branch Department of Mathematics
Steklov Institute of Mathematics Michigan State University
Fontanka 27, 191023 St.Petersburg East Lansing, Michigan 48824
Russia USA
and
Peoples’ Friendship University
of Russia (RUDN University)
6 Miklukho-Maklaya St., Moscow,
117198, Russian Federation
----------------------------------- ---------------------------------
[^1]: The research of the first author is supposed by RFBR grant 17-01-00607. The publication was prepared with the support of the RUDN University Program 5-100
[^2]: Corresponding author: V.V. Peller; email: [email protected]
|
---
abstract: |
Andreev’s Problem states the following: Given an integer $d$ and a subset of $S \subseteq {\mathbb{F}}_q \times {\mathbb{F}}_q$, is there a polynomial $y = p(x)$ of degree at most $d$ such that for every $a \in {\mathbb{F}}_q$, $(a,p(a)) \in S$? We show an $\text{AC}^0[\oplus]$ lower bound for this problem.
This problem appears to be similar to the list recovery problem for degree $d$-Reed-Solomon codes over ${\mathbb{F}}_q$ which states the following: Given subsets $A_1,\ldots,A_q$ of ${\mathbb{F}}_q$, output all (if any) the Reed-Solomon codewords contained in $A_1\times \cdots \times A_q$. For our purpose, we study this problem when $A_1, \ldots, A_q$ are random subsets of a given size, which may be of independent interest.
author:
- 'Aditya Potukuchi [^1]'
bibliography:
- 'references.bib'
title: 'On the $\text{AC}^0[\oplus]$ complexity of Andreev’s Problem'
---
Introduction
============
For a prime power $q$, let us denote by ${\mathbb{F}}_q$, the finite field of order $q$. Let us denote the elements of ${\mathbb{F}}_q = \{a_1,\ldots, a_q\}$. One can think of $a_1,\ldots,a_q$ as some ordering of the elements of ${\mathbb{F}}_q$. Let $\mathcal{P}_d = \mathcal{P}_d^{q}$ be the set of all univariate polynomials of degree at most $d$ over ${\mathbb{F}}_q$. Let us define the problem which will be the main focus of this paper:
- **Input:** A subset $S \subset {\mathbb{F}}_q^2$, and integer $d$.
- **Output:** Is there a $p \in \mathcal{P}_q^d$ such that $\{(a_i,p(a_i)) {\;\ifnum\currentgrouptype=16 \middle\fi|\;}i\in [q]\} \subseteq S$?
The problem of proving $\text{NP}$-hardness of the above function seems to have been first asked in [@J86]. It was called ‘Andreev’s Problem’ and still remains open. One may observe that above problem is closely related to the *List Recovery of Reed-Solomon codes*. In order to continue the discussion, we first define Reed-Solomon codes:
\[def:RS\] The degree $d$ Reed-Solomon over ${\mathbb{F}}_q$, abbreviated as $\operatorname{RS}[q,d]$ is the following set: $$\operatorname{RS}[q,d] = \{(p(a_1),\ldots,p(a_q)) {\;\ifnum\currentgrouptype=16 \middle\fi|\;}p \in \mathcal{P}_{d}^q\}$$
Reed-Solomon codes are one of the most widely (if not the most widely) studied families of error-correcting codes. It can be checked that $\operatorname{RS}[q,d]$ is a $d+1$-dimensional subspace of ${\mathbb{F}}_q^q$ such that every non-zero vector has at least $q - d$ non-zero coordinates. In coding theoretic language, we say that $\operatorname{RS}[q,d]$ is a *linear code* of *block length* $q$, *dimension* $d+1$ and *distance* $q - d$. The *rate* of the code is given by $\frac{d}{q}$. The set of relevant facts about Reed-Solomon codes for this paper may be found in Appendix \[sec:RS\]
The *List Recovery* problem for a code $\mathcal{C} \subset {\mathbb{F}}_q^n$ is defined as follows:
- - **Input:** Sets $A_1,\ldots, A_n \subseteq {\mathbb{F}}_q$.
- **Output:** $\mathcal{C} \cap (A_1 \times\cdots \times A_n)$
Given the way we have defined these problems, one can see that Andreev’s Problem is essentially proving $\text{NP}$-hardness for the List Recovery of Reed-Solomon codes where one just has to output a Boolean answer to the question $$\mathcal{C} \cap (A_1 \times\cdots \times A_n) \neq \emptyset?$$
Indeed, let us consider a List Recovery instance where the code $\mathcal{C}$ is $\operatorname{RS}[q,d]$, and the input sets are given by $A_1,\ldots,A_q$. Let us identify $(A_1,\ldots, A_q)$ with the set $$S = \bigcup _{i \in [q]}\{(a_i,z) {\;\ifnum\currentgrouptype=16 \middle\fi|\;}z \in A_i\} \subseteq {\mathbb{F}}_q^2$$ and let us identify every codeword $w = (w_1,\ldots,w_q) \in \mathcal{C}$, with a set $w_{\text{set}} = \{(a_i,w_i) {\;\ifnum\currentgrouptype=16 \middle\fi|\;}i\in [q]\}$. Clearly, we have that $w \in A_1\times \cdots \times A_q$ if and only if $w_{\text{set}} \subseteq S$. Often, we will drop the subscript on $w_{\text{set}}$ and refer to $w$ both as a codeword, and as the set of points it passes through. Further identifying ${\mathbb{F}}_q^2$ with $[q^2]$, and and parameterizing the problem by $r = \frac{d}{q}$, we view Andreev’s Problem as a Boolean function $\text{AP}_r:\{0,1\}^{q \times q} \rightarrow \{0,1\}$.
The main challenge here is to prove (or at least conditionally disprove) $\text{NP}$-hardness for Andreev’s Problem, which has been open for over $30$ years. Another natural problem one could study is the circuit complexity for $\text{AP}_r$. This is the main motivation behind this paper, and we will study the $\text{AC}^0[\oplus]$ complexity of $\text{AP}_r$. We shall eventually see that even this problem needs relatively recent results about the power of $\text{AC}^0[\oplus]$ in our proof. Informally, $\text{AC}^0$ is the class of Boolean functions computable by circuits of constant depth, and polynomial size, using $\land$, $\lor$, and $\lnot$ gates. $\text{AC}^0[\oplus]$ is the class of Boolean functions computable by circuits of constant depth, and polynomial size, using $\land$, $\lor$, $\lnot$, and $\oplus$ ($\text{MOD}_2$) gates. The interested and unfamiliar reader is referred to [@AB09] (Chapter $14$) for a more formal definition and further motivation behind this class. We show that $\text{AP}_r$ cannot be computed by $\text{AC}^0$ circuits for a constant $r$. This type of result is essentially motivated by a similar trend in the study of the complexity of *Minimum Circuit Size Problem*. Informally, the Minimum Size Circuit Problem (or simply $\text{MCSP}$) takes as input a truth table of a function on $m$ bits, and an integer $s$. The output is $1$ if there is a Boolean circuit that computes the function with the given truth table and has size at most $s$. It is a *major* open problem to show the $\text{NP}$-hardness of $\text{MCSP}$. A lot of effort has also gone into understanding the circuit complexity of $\text{MCSP}$. Allender et al. [@ABKMR06] proved a superpolynomial $\text{AC}^0$ lower bound, and Hirahara and Santanam [@HS17] proved an almost-quadratic formula lower bound for $\text{MCSP}$. A recent result by Golonev et al. [@GIIKKT19] extends [@ABKMR06] and proves an $\text{AC}^0[\oplus]$ lower bound for $\text{MCSP}$. Thus one can seek to answer the same question about $\text{AP}_r$.
We now state our main theorem:
\[thm:AC0\] For any prime power $q$, and $r \in (0,1)$, we have that any depth $h$ circuit with $\land$, $\lor$, $\lnot$, and $\oplus$ gates that computes $\text{AP}_r$ on $q^2$ bits must have size at least $\exp\left(\tilde{\Omega}\left(hq^{\frac{c^2}{h-1}} \right) \right)$.
We make a couple of comments about the theorem. The first, most glaring aspect is that $r \in (0,1)$ is more or less a limitation of our proof technique. Of course, as $r$ gets *very* small, i.e., $r = O\left(\frac{1}{q}\right)$, one can find depth $2$ circuits of size $q^{O(rq)} = q^{O(1)}$. But, we do not know that the case where, for example, $r = \Theta\left( \frac{1}{\log q}\right)$ is any easier for $\text{AC}^0$. Secondly, we are not aware of a different proof of an $\text{AC}^0$ (as opposed to an $\text{AC}^0[\oplus]$) lower bound.
Some notation and proof ideas
-----------------------------
For a $p \in (0,1)$, let $X_1, X_2, \ldots$ denote independent $\operatorname{Ber}(p)$ random variables. For a family of Boolean functions $f:\{0,1\}^n \rightarrow \{0,1\}$, we use $f^{(n)}(p)$ to denote the random variable $f(X_1,\ldots, X_n)$.
For a monotone family of functions $f$, we say that $f$ has a *sharp threshold* at $p$ if for every $\epsilon > 0$, there is an $n_0$ such that for every $n > n_0$, we have that ${\mathbb{P}}(f^{(n)}(p(1 - \epsilon)) = 0) \geq 0.99$, and ${\mathbb{P}}(f^{(n)}(p(1 + \epsilon)) = 1) \geq 0.99$.
Henceforth, we shall assume that $q$ is a very large prime power. So, all the high probability events and asymptotics are as $q$ grows. Where there is no ambiguity, we also just use $f(p)$ to mean $f^{(n)}(p)$ and $n$ growing.
One limitation of $\text{AC}^0[\oplus]$ that is exploited when proving lower bounds (including in [@GIIKKT19]) for monotone functions is that $\text{AC}^0[\oplus]$ cannot compute functions with ‘very’ sharp thresholds. For a quantitative discussion, let us call the smallest $\epsilon$ in the definition above the *threshold interval*. It is known that $\text{AC}^0$ (and therefore, $\text{AC}^0[\oplus]$) can compute (some) functions with threshold interval of $O\left(\frac{1}{\log n}\right)$, for example, consider the following function on Boolean inputs $z_1,\ldots,z_n$: Let $Z_1 \sqcup \cdots \sqcup Z_{\ell}$ be an equipartition of $[n]$, such that each $|Z_i| \approx \log n - \log \log n $. Consider the function given by
$$f(z_1,\ldots, z_n) = \bigvee_{i \in [\ell]} \left( \bigwedge_{j \in Z_i} z_j \right).$$
This is commonly known as the *tribes* function and is known to have a threshold interval of $O\left( \frac{1}{\log n}\right)$. This is clearly computable by an $\text{AC}^0$ circuit. A construction from [@LSSTV19] gives an $\text{AC}^0$ circuit (in $n$ inputs) of size $n^{O(h)}$ and depth $h$ that has a threshold interval $\tilde{O}\left( \frac{1}{(\log n)^{h - 1}}\right)$. A remarkable result from [@LSSTV19] and [@CHLT19] (Theorem 13 from [@CHLT19] and Lemma $3.2$ in [@A19]) says that this is in some sense, tight. Formally,
\[thm:LSSTV\] Let $n$ be any integer and $f :\{0,1\}^n \rightarrow \{0,1\}$ be a function with threshold interval $\delta$ at $\frac{1}{2}$. Any depth $h$ circuit with $\land$, $\lor$, $\lnot$, and $\oplus$ gates that computes $f$ must have size at least $\exp\left(\Omega\left(h \left(1/\delta\right)^{\frac{1}{h - 1}} \right)\right)$.
In [@LSSTV19], this was studied as the *Coin Problem*, which we will also define in Section \[sec:AC0\]. Given the above theorem, a natural strategy suggests itself. If we could execute the following two steps, then we would be done:
- Establish Theorem \[thm:LSSTV\] for functions with thresholds at points other than $\frac{1}{2}$.
- Show that $\text{AP}_r$ has a sharp threshold at $q^{-r}$ with a suitably small threshold interval, i.e., $\frac{1}{\operatorname{poly} q}$.
The first fact essentially reduces to approximating $p$-biased coins by unbiased coins in constant depth. Though we are unable to find a reference for this, this is relatively straightforward, and is postponed to Appendix \[sec:asec4\]. Understanding the second part, naturally leads us to study $\text{AP}_r(p)$ for some $p = p(q)$. Let $A_1,\ldots, A_q \subset {\mathbb{F}}_q$ be independently chosen random subsets where each element is included in $A_i$ with probability $p$. Let $\mathcal{C}$ be the $\operatorname{RS}[q,rq]$ code. We have $|\mathcal{C}| = q^{rq + 1}$. Let us denote
$$X := |(A_1\times \cdots \times A_q) \cap \mathcal{C}|.$$
For $w \in \mathcal{C}$, let $X_w$ denote the indicator random variable for the event $\{w \in A_1 \times \cdots \times A_q\}$. Clearly, $X = \sum_{w \in \mathcal{C}}X_w$, and for every $w \in \mathcal{C}$, we have ${\mathbb{P}}(X_w = 1) = p^{q}$. We first note that for $\epsilon = \omega\left(\frac{\log q}{q}\right)$, and $p = q^{-r}(1 - \epsilon)$, we have, using linearity of expectation,
$$\begin{aligned}
{{\rm I\kern-.3em E}}[X] & = \sum_{w \in \mathcal{C}}{{\rm I\kern-.3em E}}[X_w] \\
& = |\mathcal{C}| \cdot (q^{-r}(1 - \epsilon))^q \\
& = q^{rq + 1}\left(q^{-r}(1 - \epsilon\right))^q \\
& = q\cdot (1 - \epsilon)^q \\
& \leq q \cdot e^{-\epsilon q} \\
& = o(1).\end{aligned}$$
When $p = q^{-r}(1 + \epsilon)$, using a similar calculation as above, we have
$${{\rm I\kern-.3em E}}[X] = q \cdot (1 + \epsilon)^q \geq q.$$
To summarize, for $\epsilon = \omega \left( \frac{\log q}{q}\right)$, and $p = q^{-r}(1 - \epsilon)$, ${{\rm I\kern-.3em E}}[X] \rightarrow 0$, and for $p = q^{-r}(1 + \epsilon)$, ${{\rm I\kern-.3em E}}[X] \rightarrow \infty$.
\[lem:smallp\] For $\epsilon = \omega\left(\frac{\log q}{q}\right)$, we have $${\mathbb{P}}(\text{AP}_r(q^{-r}(1 - \epsilon)) = 1) \leq \exp\left(- \Omega(\epsilon q)\right).$$
This is just Markov’s inequality. We have ${\mathbb{P}}(\text{AP}_r(p(1- \epsilon)) = 0) = {\mathbb{P}}(X \geq 1) \leq {{\rm I\kern-.3em E}}[X] \leq q \cdot e^{-\epsilon q} = \exp\left(-\Omega(\epsilon q)\right)$.
This counts for half the proof of the sharp threshold for $\text{AP}_r$. The other half forms the main technical contribution of this work. We show the following:
\[thm:sharpthreshold\] Let $q$ be a prime power, $r = r(q)$ and $\epsilon = \epsilon(q)$ be real numbers such that $q^{-r} \geq \frac{\log q}{q}$ and $\epsilon = \omega\left(\max\left\{q^{-r}, \sqrt{q^{r - 1} \log \left(q^{1 - r}\right)}\right\}\right)$.
Let $A_1,\ldots,A_q$ be independently chosen random subsets of ${\mathbb{F}}_q$ with each point picked independently with probability $q^{-r}(1 + \epsilon)$. Then $${\mathbb{P}}((A_1 \times \cdots \times A_q) \cap \operatorname{RS}[q,rq] = \emptyset) = o(1).$$
There is a technical condition on $\epsilon$ that can be ignored for now, and will be addressed before the proof. The only relevant thing to observe is that when $r$ is bounded away from $0$ and $1$, then $\epsilon = \frac{1}{\operatorname{poly} (q)}$ suffices. The condition to focus on here is that $q^{-r} \geq \frac{\log q}{q}$. Indeed, one can see that this condition is necessary to ensure that w.h.p, all the $A_i$’s are nonempty. So, for example, if the dimension of $\mathcal{C}$ is $q - 1$, then setting $p = q^{-1}(1 + \epsilon)$ is enough for ${{\rm I\kern-.3em E}}[X] = \omega(1)$ but this does not translate to there almost surely being a codeword in $A_1 \times \cdots \times A_q$.
Lemma \[lem:smallp\] and Theorem \[thm:sharpthreshold\] together give us that $\text{AP}_r$ has a sharp threshold at $\max \left\{ q^{-r}, \frac{\log q}{q} \right\}$ whenever $1 - \frac{1}{q} \geq r \gg \frac{1}{\log p}$. For the sake of completeness one could ask if $\text{AP}_r$ has a threshold for all feasible values of $r$, and we show that the answer is yes. More formally,
\[thm:fullrange\] For every $r = r(q)$, there is a critical $p = p(r,q)$ such that for every $\epsilon >0$,
1. ${\mathbb{P}}\left(\text{AP}_{r}(p(1 - \epsilon)) = 1\right) = o(1)$.
2. ${\mathbb{P}}\left(\text{AP}_{r}(p(1 + \epsilon)) = 1\right) = 1 - o(1)$.
The case that is not handled by Theorem \[thm:sharpthreshold\] is when $r = O\left(\frac{1}{\log q} \right)$ (since in this case, Theorem \[thm:sharpthreshold\] requires $\epsilon = \Omega(1)$). This corresponds to the case where $q^{-r}$ is a number bounded away from $0$ and $1$.
What doesn’t work, and why
--------------------------
One obvious attempt to prove Theorem \[thm:sharpthreshold\] is to consider the second moment of $X (= |\mathcal{C} \cap (A_1\times\cdots\times A_q)|)$ and hope that ${{\rm I\kern-.3em E}}[X^2] = (1 + o(1)){{\rm I\kern-.3em E}}^2[X]$. Unfortunately, ${{\rm I\kern-.3em E}}[X^2]$ is too large. Through a very careful calculation using the weight distribution of Reed Solomon codes which we do not attempt to reproduce here, we have ${{\rm I\kern-.3em E}}[X^2] = \Omega\left( e^{\frac{1}{p}}{{\rm I\kern-.3em E}}^2[X]\right)$. So in the regime where, for example, $p = q^{-\Omega(1)}$, this approach is unviable.
To understand this (without the aforementioned involved calculation) in an informal manner, let us fix $p = q^{-r}$ for some fixed constant $r$. Let us identify the tuple of sets $(A_1,\ldots, A_q)$ with the single set $S = \cup_{i \in [q]}\{(a_i,z) {\;\ifnum\currentgrouptype=16 \middle\fi|\;}z \in A_i\}$. So, we are choosing a random subset $S \subset {\mathbb{F}}_q^2$ of size $\approx q^{2-r}$. On the other hand, the objects we are looking for, i.e., codewords, have size $q$. This is much larger than the standard deviation of $|S|$, which is of the order of $q^{1 - (r/2)}$. Thus, conditioning on the existence of some codeword $w \subset {\mathbb{F}}_q^2$, the distribution of $S$ changes significantly. One way to see this is the following: Using standard Chernoff bounds, one can check that the size of $S$ is almost surely $q^{2 - r} \pm O\left(q^{1 - (r/2)} \log q \right)$. However, conditioned on $w \in A_1 \times \cdots \times A_q$, the size of $S$ is almost surely $q + q^{-r}(q^2 - q) \pm O\left(q^{1 - (r/2)}\log q\right)$ (the additional $q$ comes from the points that make up $w$). This is much larger than before when $r$ is relatively large. On the other hand, the main point behind (successful) applications of the second moment method is that the distribution does not significantly change after such a conditioning.
One possible way to circumvent the above problem is to pick a uniformly random set $S \subset {\mathbb{F}}_q^2$ of size $q^{2 - r}$, instead of every point independently with probability $q^{-r}$. This is a closely related distribution, and it is often the case that Theorems in this model are also true in the above ‘i.i.d.’ model. This fact can be also be made formal (see, for example [@JLR00] Corollary $1.16$). Here, when one conditions on the existence of some codeword $w$, at least $|S|$ does not change. Thus the second moment method is not ruled out right at the start. However, it seems to be much more technically involved and it is unclear if it is possible to obtain the relatively small threshold interval that is required for Theorem \[thm:AC0\] in this way.
What works and how
------------------
Here, we sketch the proofs of the Theorem \[thm:sharpthreshold\] and Theorem \[thm:fullrange\], which can be considered the two main technical contributions of this work.
### Proof sketch of Theorem \[thm:sharpthreshold\]
The key idea in the proof of this theorem is to count the number of polynomials in the ‘Fourier basis’. Let us consider $f :{\mathbb{F}}_q^q \rightarrow \{0,1\}$ to be the indicator of $\mathcal{C}$. For $i \in [q]$, let $g_i : {\mathbb{F}}_q \rightarrow \{0,1\}$ denote the indicator of $A_i$.
For an extremely brief and informal discussion, what we what we want is essentially $\langle f, \prod_{i \in [q]}g_i \rangle$, which, by Plancharel’s identity (see Fact \[fact:Plancherel\]) is $\sum_{\alpha}\widehat{f} \cdot \widehat{\prod_ig_i}(\alpha)$. Since $\mathcal{C}$ is a vector space, we have that $\widehat{f}$ is supported on $\mathcal{C}^{\perp}$. Moreover, $\widehat{g_i}(\alpha_i)$ is much larger when $\alpha_i = 0$ than when $\alpha_i \neq 0$ if $A_i$ is random. This combined with the fact that most points in $\mathcal{C}^{\perp}$ have large weight, and a bit more Fourier analysis means that the inner product, $\langle f,\prod_i g_i\rangle$ is dominated by $\widehat{f}(0) \prod_{i \in [q]}\widehat{g_i}(0)$ which is the expected number of codewords in $A_1 \times \cdots \times A_q$.
Now we give a slightly less informal overview. What we are trying to estimate is exactly
$$\begin{aligned}
X = |\mathcal{C} \cap (A_1\times \cdots \times A_q)| = \sum_{(x_1,\ldots,x_q) \in {\mathbb{F}}^q}f(x)\left( \prod_{i \in [q]}g_i(x_i)\right).\end{aligned}$$
Using Fourier analysis over ${\mathbb{F}}_q$, one can show that
$$\begin{aligned}
\frac{q^q}{|\mathcal{C}|} \cdot X & = \sum_{(\alpha_1,\ldots,\alpha_q) \in \mathcal{C}^{\perp}}\prod_{i \in [q]}\widehat{g_i}(\alpha_i) \\
& \geq \prod_{i \in [q]}\widehat{g_i}(0) - \left|\sum_{(\alpha_1,\ldots, \alpha_q) \in \mathcal{C}^{\perp}}\left(\prod_{i \in [q]}\widehat{g_i}(\alpha_i)\right)\right|.\end{aligned}$$
Using the fact that $\mathcal{C}$ is an $\operatorname{RS}[q,rq]$ code, one has (see Fact \[fact:dual\]) that $\mathcal{C}^{\perp}$ is an $\operatorname{RS}[q, q - rq - 1]$ code. What will eventually help in the proof is that the weight distribution of Reed Solomon codes (and so in particular, $\mathcal{C}^{\perp}$) is well understood (see Theorem \[thm:MDSwtdist\]).
Now clearly, it suffices to understand the term $\sum_{(\alpha_1,\ldots, \alpha_q) \in \mathcal{C}^{\perp}}\left(\prod_{i \in [q]}\widehat{g_i}(\alpha_i)\right) =: R$. One way to control $|R|$ is to control $|R|^2 = R \overline{R}$. Here, one can use the fact that the $A_i$’s are randomly and independently chosen to establish cancellation in many terms of ${{\rm I\kern-.3em E}}[|R|^2]$. More formally, one can prove that
$${{\rm I\kern-.3em E}}[|R|^2] = \sum_{(\alpha_1,\ldots,\alpha_q) \in \mathcal{C}^{\perp} \setminus \{\overline{0}\}}\prod_{i \in [q]}{{\rm I\kern-.3em E}}[|\widehat{g_i}(\alpha_i)|^2].$$
It is a more or less standard fact that if $A_i$ is a uniformly random set of size $pq =: t$, then $${{\rm I\kern-.3em E}}[|\widehat{g_i}(0)|^2] \sim \left( \frac{t}{q}\right)^2$$
and
$${{\rm I\kern-.3em E}}[|\widehat{g_i}(\alpha_i)|^2] \sim \frac{t}{q^2}$$
for $\alpha_i \neq 0$. This difference, will be the reason why $|R|$ is typically much smaller than $\prod_{i \in [q]}\widehat{g_i}(0)$. To continue, let us believe the heuristic that most polynomials over ${\mathbb{F}}_q$ of degree $\Theta(q)$ have very few ($o(q)$) zeroes, we can use the rough estimate: $$\begin{aligned}
{{\rm I\kern-.3em E}}[|R|^2] & \approx |\mathcal{C}^{\perp}|\left( \frac{t}{q^2}\right)^{q - o(q)} \\
& \approx q^{q - rq}\left(\frac{p}{q}\right)^{q - o(q)} \\
& \approx q^{-rq}p^{q - o(q)}.
\end{aligned}$$
And so, Markov’s Inequality gives that $|R|$ is unlikely to be much greater than $q^{\frac{rq}{2}}p^{\frac{q}{2} + o(q)}$. On the other hand, with high probability, $$\begin{aligned}
\prod_{i \in [q]} \widehat{g_i}(0) & \approx \left(\frac{t}{q}\right)^q \\
& \approx p^{q}.
\end{aligned}$$
Thus if $p \geq q^{-r + o(1)}$, we have that $(q^q / |\mathcal{C}|)\cdot X \geq \prod_{i \in [q]}\widehat{g_i}(0) - |R| > 0$, and so in particular, $X > 0$. The proof of Theorem \[thm:sharpthreshold\] is essentially a much tighter, and more formal version of the above argument, and is postponed to Appendix \[sec:sharpthreshold\].
### Proof sketch of Theorem \[thm:fullrange\]
The starting point of Thoerem \[thm:fullrange\] is noticing that the only case not covered by Theorem \[thm:sharpthreshold\] is $p \in (0,1)$ is some fixed constant, or equivalently $r = O\left( \frac{1}{\log q}\right)$. Here we have a somewhat crude weight distribution result for Reed Solomon codes (Proposition \[prop:RSwtdist\]) to compute the second moment. We first show that ${{\rm I\kern-.3em E}}[X^2] = O\left( e^{\frac{1}{p}} {{\rm I\kern-.3em E}}^2[X]\right)$. Using, for example the Paley-Zygmund Inequality (\[eqn:PaleyZygmund\]), this means that ${\mathbb{P}}(X > 0) \geq \Omega(e^{-\frac{1}{p}})$. Thus we have that $\{X > 0\}$ with at least some (possibly small) constant probability. But what we need is that ${\mathbb{P}}(X > 0) \geq 0.99$. For this, we now use the fact that $\text{AP}_r$ is monotone, and transitive-symmetric, which informally means that any two variables of $\text{AP}_r$ look the same (see Definition \[defn:transitive\] for a formal definition). Standard applications of hypercontractivity for the Boolean hypercube (see Theorem \[thm:FK\]) gives that for $p' = p + O\left( \frac{1}{\log q}\right)$, we have that ${\mathbb{P}}(\text{AP}_r(p') = 1) \geq 0.99$.
The details of this proof are postponed to Appendix \[sec:fullrange\].
One thing to note is that our definition of sharp threshold only makes sense when the critical probability $p_r$ is bounded away from $1$ (since otherwise trivially there is some function $\epsilon = \epsilon(q) = o(1)$ such that $p\cdot (1 + \epsilon) = 1$). So, we will restrict ourselves to the regime where $r = \Omega\left( \frac{1}{\log q}\right)$. Also, it is to be understood that all the statements above (and below) only make sense when $rq$ is an integer, and thus we shall restrict ourselves to this case.
Finally, we address the question of random list recovery with errors as another application of Theorem \[thm:sharpthreshold\].
Random list recovery with errors
--------------------------------
Given a random subset of points in $S \subseteq {\mathbb{F}}_q^2$, what is the largest fraction of any degree $d = \Theta(q)$ polynomial that is contained in this set? Using the Union Bound, it is easy to see that no polynomial of degree $d$ has more than $d \log_{\frac{1}{p}}q + o(q)$ points contained in $S$ (formal details are given in Section \[sec:expagreement\]). We show that perhaps unsurprisingly, this is the truth. Formally,
\[corr:errors\] Let $S$ be a randomly chosen subset of ${\mathbb{F}}_q^2$ where each point is picked independently with probability $p$. Then with probability $1 - o(1)$, $$\max_{w \in \operatorname{RS}[q,d]}|w \cap S| = d \log_{\frac{1}{p}} q - O \left( \frac{q}{\log \left( \frac{1}{p}\right)}\right).$$
We restrict our attention to the case when $d = \Theta(q)$, where the above statement is nontrivial. This is the content of Section \[sec:expagreement\]. However, we believe that the statement should hold for all rates, and error (in general) better than $O\left( \frac{q}{\log q}\right)$.
We make two final comments before proceeding to the proofs: (1) In Theorem \[thm:sharpthreshold\], each $A_i$ is chosen by including each point independently. However, the same proof works if $A_i$ is a uniformly random set with a prescribed size. (2) Although we only state the lower bound for $\text{AC}^0[\oplus]$, one can check that all the tools (and, therefore, the lower bound) still work when we replace the $\oplus$ gates with any $\oplus_p$ ($\text{MOD}_p$) for any small prime $p$.
$\text{AC}^0[\oplus]$ lower bound for $\text{AP}_r$ {#sec:AC0}
===================================================
We prove the lower bound by showing that $\text{AP}_r$ solves a biased version of the *Coin Problem*, and use the lower bounds known for such kinds of functions, obtained by [@LSSTV19], [@CHLT19].
We say that a circuit $C = C^{n}$ on $n$ inputs solves the $(p,\epsilon)$-coin problem if
- For $X_1,\ldots, X_n \sim \operatorname{Ber}(p(1 - \epsilon))$, $${\mathbb{P}}(\text{C}(X_1,\ldots,X_n) = 0) \geq 0.99$$
- For $X_1,\ldots, X_n \sim \operatorname{Ber}(p(1 + \epsilon))$, $${\mathbb{P}}(\text{C}(X_1,\ldots,X_n) = 1) \geq 0.99$$
We shall abbreviate the $(p,\epsilon)$-coin problem on $n$ variables as $\text{CP}^n(p,\epsilon)$. We observe that a function $f:\{0,1\}^n \rightarrow \{0,1\}$ *solves* $\text{CP}^n\left(p,\epsilon \right)$ if it has a sharp threshold at $p$ with threshold interval at most $\epsilon$. The one obstacle we have to overcome in using Theorem \[thm:LSSTV\] is that $\text{AP}_r$ has a sharp threshold at $p^{-c} \ll \frac{1}{2}$. However, we will show how to simulate biased Bernoulli r.v’s from almost unbiased ones. Let $\text{C}(s,d)$ to denote the class of functions on $n$ variables which have circuits of size $O(s) = O(s(n))$ and depth $d = d(n)$ using $\land$, $\lor$, $\lnot$, and $\oplus$ gates. Here, we make the following simple observation about the power of $\text{AC}^0[\oplus]$ circuits to solve biased and unbiased $\epsilon$-coin problem. First, we observe that it is possible to simulate a biased coin using an unbiased one.
\[lem:bias\] Let $s$ be such that $\frac{1}{2^s} \leq p \in (0,1)$, and $\epsilon \leq \frac{1}{s^K}$ for a large constant $K$. Then, there is a CNF $F_p$ on $t \leq s^2$-variables such that for inputs $X_1 \ldots, X_t \in \operatorname{Ber}\left(\frac{1}{2} + \epsilon \right)$, $${\mathbb{P}}\left(F_p(X_1,\ldots, X_t) = 1\right) = p(1 + \Omega(\epsilon L))$$ and for inputs $X_1 \ldots, X_t \in \operatorname{Ber}\left(\frac{1}{2} - \epsilon \right)$, $${\mathbb{P}}\left(F_p(X_1,\ldots, X_t) = 1\right) = p\left(1 + \frac{1}{2^{\Omega(\sqrt{t})}} - \Omega(\epsilon L)\right)$$ where $L = \lfloor\log_2(1/p)\rfloor$.
The idea is essentially that the $\texttt{AND}$ of $k$ unbiased coin is a $2^{-k}$-biased coin. However, some extra work has to be done if we want other biases (say, $(0.15) \cdot 2^k$). The proof of this lemma is postponed to the Appendix \[sec:asec4\]. This lemma now gives us the following:
\[lem:biasedcoin\] Let $z \in (0,1)$ be a fixed constant. If $\text{CP}^n\left(\frac{1}{n^{z}},o(\epsilon \log n)\right) \in \text{C}^n(s,h)$, then there is a $t \leq \log^2n$ such that $\text{CP}^{nt}\left(\frac{1}{2}, \epsilon \right) \in \text{C}^{nt}(z s\log n, h+2)$.
Let $C$ be a circuit for $\text{CP}^n\left(\frac{1}{n^{z}},\delta\right)$-coin problem. Replace each input variable with the CNF $F_{\left(\frac{1}{n^{z}}\right)}$ from Lemma \[lem:bias\] on $t = O(\log^4 n) $ independent variables. Call this circuit $\text{C}'$, on $tn$ variables. If the bias of each of these input variables is $\frac{1}{2} + \epsilon$, then the guarantee of Lemma \[lem:bias\] is that output of the and gate is $1$ with probability at least $\frac{1}{n^{z}}(1 + \Omega(\epsilon \log n))$. A similar computation gives that if the bias of the inputs are $\left( \frac{1}{2} - \epsilon \right)$, then the bias of the output is at most $\frac{1}{n^{z}}(1 - \Omega(\epsilon \log n))$. Therefore, $\text{C}'$ solves $\text{CP}^{nt}\left(\frac{1}{2}, \epsilon\right)$, and has size at most $s \log n$, and depth $h+2$.
Theorem \[thm:sharpthreshold\] and Lemma \[lem:smallp\], together, now give us the following corollary:
Let $q$ be a large enough prime power. Then $AP_r$ on $q^2$ inputs solves the $\left(q^{-r}, \epsilon\right)$ coin problem, for $\epsilon = \omega\left(\max\left\{q^{-r}, q^{\frac{r - 1}{3}}\right\}\right)$
As a result, Theorem \[thm:LSSTV\], and Lemma \[lem:bias\], and Lemma \[lem:biasedcoin\] together, give us the following bounded depth circuit lower bound for $\text{AP}_r$:
For any $r \in (0,1)$, and $h \in {\mathbb{N}}$, we have that $$\text{AP}_{r} \not \in \text{C}\left(\exp\left\{\tilde{\Omega}\left(hq^{\frac{r^2}{h-1}} \right) \right\},h\right).$$
Random list recovery with errors {#sec:expagreement}
================================
In this section, we shall again consider Reed-Solomon codes $\operatorname{RS}[q,rq]$ where $r$ is some constant between $0$ and $1$. Let us slightly abuse notation, as before, and think of a codeword $w \in \operatorname{RS}[q,rq]$ corresponding to a polynomial $p(X)$ as the set of all the zeroes of the polynomial $Y = p(X)$. That is, for a codeword $w = (w_1,\ldots, w_q)$ associated with polynomial $p$, we think of $w$ as a subset $\{(a_i,p(a_i)) {\;\ifnum\currentgrouptype=16 \middle\fi|\;}i \in [q]\}$ (recall that ${\mathbb{F}}= \{a_1,\ldots, a_q\}$). For a set of points $S \subset {\mathbb{F}}_q^2$ and a codeword $w$ we say the *agreement between $w$ and $S$* to denote the quantity $|w \cap S|$. For a code $\mathcal{C}$, we say that the *agreement between $\mathcal{C}$ and S* to denote $\max_{w \in \mathcal{C}}|m \cap S|$.
We are interested in the following question: For a set $S \subset {\mathbb{F}}_q^2$. What is the smallest $\ell$ such that there exists a $w \in \operatorname{RS}[q,rq]$ such that $|w \cap S| \geq q - \ell$? In other words, what is the largest agreement between $\operatorname{RS}[q,rq]$ and $S$? This is (very close to) the list recovery problem for codes *with errors*. Naturally, we seek to answer this question when $S$ is chosen randomly in an i.i.d. fashion with probability $p$. Theorem \[thm:sharpthreshold\] gives asymptotically tight bounds in a relatively straightforward way for constant error rate.
One can observe that the only properties about Reed-Solomon codes that was used in Theorem \[thm:sharpthreshold\] was the weight distribution in the dual space of codewords. However, (see Appendix \[sec:RS\]) these are also true for punctured Reed-Solomon codes codes. So, an analogus theorem also holds for punctured Reed Solomon codes. Formally,
Let $q,n,d$ be integers such that $q$ is a prime power and $n = \omega(\log q)$, and $q^{-\frac{d}{n}} \geq \frac{\log n}{q}$ and let $\epsilon = \omega \left(\max\left\{q^{- \frac{d}{n}} \sqrt{q^{1 - \frac{d}{n}}\log \left(q^{1 - \frac{d}{n}}\right)}\right\}\right)$. Let $\mathcal{C}$ be an $\operatorname{RS}[q,d]|_n$ code.
Let $A_1,\ldots,A_n$ be independently chosen random subsets of ${\mathbb{F}}_q$ with each point picked independently with probability $q^{-\frac{d}{n}}(1 + \epsilon)$. Then $${\mathbb{P}}((A_1 \times \cdots \times A_n) \cap \mathcal{C} = \emptyset) = o(1).$$
We do not repeat the proof but is it the exact same as that of Theorem \[thm:sharpthreshold\]. Let $\mathcal{E}_a$ denote the event that the agreement between $S$ and $\operatorname{RS}[q,rq]$ is $a$. Union bound gives us that
$$\label{eqn:expagreement}
{\mathbb{P}}(\mathcal{E}_{q - \ell}) \leq \binom{q}{\ell}q^{rq+1}p^{q - \ell }.$$
So if $\ell$ is such that the RHS of \[eqn:expagreement\] is $o(1)$. Then the agreement is almost surely less than $q - \ell$. For the other direction, we have the following corollary:
\[corr:punctured\] Let $\epsilon \geq \max\left\{10q^{- \frac{d}{q - \ell}}, \sqrt{q^{1 - \frac{d}{q - \ell}} \cdot \log q }\right\}$. Let $S$ be a randomly chosen subset of ${\mathbb{F}}_q^2$ with each point picked independently with probability at least $q^{-\frac{d}{q - \ell}}(1 + \epsilon)$, then with probability at least $1 - o(1)$, the agreement between $S$ and $\operatorname{RS}[q,d]$ is at least $q - \ell $.
For $i \in [q - \ell]$, let us denote $$S_i:= \{j {\;\ifnum\currentgrouptype=16 \middle\fi|\;}(i,j) \in S\}.$$ Let us use $S' := S_1\times\cdots \times S_{q - \ell }$. Let us denote $\mathcal{C} = \operatorname{RS}[q,d]|_{q - \ell }$. Formally, for a codeword $w \in \operatorname{RS}[q,d]$, denote $p_w$ to be the polynomial corresponding to $m$. We have $$\mathcal{C} = \{(i, p_w(i)) {\;\ifnum\currentgrouptype=16 \middle\fi|\;}i \in [q - \ell])\}$$ We observe that the conditions in Theorem \[thm:sharpthreshold\] hold, so $${\mathbb{P}}(C' \cap S' = \emptyset) = o(1)$$ as desired.
Given a random subset $S \subseteq {\mathbb{F}}_q^2$ where each point is picked with probability $p$, then with probability at least $1 - o(1)$, the largest agreement $\operatorname{RS}[q,d]$ with $S$ is $ d \log_{\frac{1}{p}} q - O \left( \frac{q}{\log \left( \frac{1}{p}\right)}\right)$.
Let $a$ be an integer that denotes the maximum agreement between $S$ and $\operatorname{RS}(q,d)$. Suppose that $a \leq d \log_{\frac{1}{p}} q$, then setting $\ell = q - a$, and noting that the conditions for Corollary \[corr:punctured\] are satisfied, we get that with probability at least $1 - o(1)$, there is a polynomial that agrees with the set $S$ in the first $q - \ell$ coordinates. On the other hand, if $a \geq d\log_{ \frac{1}{p}}q + 4\frac{q}{\log \left( \frac{1}{p}\right)}$, again, setting $\ell = q - a$, Union Bound gives us:
$$\begin{aligned}
{\mathbb{P}}(\mathcal{E}_{q - \ell}) & \leq \sum_{w \in \mathcal{C}} \sum_{\substack{P \subset {\mathbb{F}}_q \\ |P| = q - \ell}} {\mathbb{P}}(w|_P \subseteq S)\\
& = \binom{q}{\ell}q^{d+1}p^{q - r} \\
& \leq \left( e\frac{q}{\ell}\right)^{\ell} q^{d+1}p^{q - \ell} \\
& \leq e^qq^{d+1}p^{q - \ell} \\
& \ll \frac{1}{q} .\end{aligned}$$
And so we have that with probability at least $1 - o(1)$, the agreement of $\operatorname{RS}(q,d)$ with $S$ is $d \log_{\frac{1}{p}} q - O\left( \frac{q}{\log \left( \frac{1}{p}\right)}\right)$.
Conclusion
==========
We started off by attempting to prove a bounded depth circuit lower bound for Andreev’s Problem. This led us into (the decision version of the) random List Recovery of Reed-Solomon codes. Here we show a sharp threshold for a wide range of parameters, with nontrivial threshold intervals in some cases. However, one of the unsatisfactory aspects about Theorem \[thm:fullrange\] is that it is proved in a relatively ‘hands-off’ way possibly resulting in a suboptimal guarantee on $\epsilon$. The obvious open problem that is the following:
#### Open Problem:
Is Theorem \[thm:fullrange\] with a better bound on $\epsilon$?
If it is true with a much smaller $\epsilon$, it would extend in a straightforward way to the $\text{AC}^0[\oplus]$ lower bound as well. Another point we would like to make is that the only thing stopping us from proving Theorem \[thm:fullrange\] for general $\text{MDS}$ codes is the lack of Proposition \[prop:symm\]
#### Acknowledgements
I am extremely grateful to Amey Bhangale, Suryateja Gavva, and Mary Wootters for the helpful discussions. I am especially grateful to Nutan Limaye and Avishay Tal for explaining [@LSSTV19] and [@CHLT19] respectively to me. I am grateful to Partha Mukhopadhyay for suggesting the $\text{AC}^0[\oplus]$ lower bound problem for list recovery. I am also extremely grateful to Swastik Kopparty for the discussions that led to Corollary \[corr:errors\], and to Bhargav Narayanan for the discussions that led to Theorem \[thm:fullrange\].
More preliminaries {#sec:prelims}
==================
Properties of Reed-Solomon codes {#sec:RS}
--------------------------------
The first fact we will use is that the dual vector space of a Reed-Solomon code is also a Reed-Solomon code.
\[fact:dual\] Let $\mathcal{C} := \operatorname{RS}[q,d]$. Then $\mathcal{C}^{\perp} = \operatorname{RS}[q,q - d - 1]$.
For $t \neq 0$, let $W_t$ be the number of codewords of weight $t$ in $\operatorname{RS}[q,d]$. This is a relatively well understood quantity.
\[thm:MDSwtdist\] We have: $$|W_{q-i}| = \binom{q}{i}\sum_{j = 0}^{d - i }(-1)^j\binom{q-i}{j}(q^{d - i - j +1} - 1).$$
However, we just need the following slightly weaker bound that is easier to prove:
\[prop:RSwtdist\] We have $W_{q-i} \leq \frac{q^{d+1}}{i!} $.
We have that $\mathcal{C}$ is a $d+1$-dimensional subspace of ${\mathbb{F}}^q$. Add $i$ extra constraints by choosing some set of $i$ coordinates and restricting them to $0$. As long as $i < d$, these new constraints strictly reduce the dimension of $\mathcal{C}$. There are exactly $\binom{q}{i}$ ways to choose the coordinates, and the resulting space has dimension $d+1 - i$. Therefore, the number of codewords of weight at most $q - i$ is at most $q^{d+1 -i} \cdot \binom{q}{i}\cdot \leq \frac{q^{d+1}}{i!}$.
The above bound is asymptotically only a factor of $e$ away for small values of $i$.
### Punctured Reed-Solomon codes
All of the statements above when instead of Reed-Solomon codes, one considers *punctured* Reed-Solomon codes. For a $w = (w_1,\ldots,w_n) \in {\mathbb{F}}_q^n$, and a set $S \subset [n']$, let us define $$w|_S = (w_i)_{i \in S}.$$ For a subset $\mathcal{C} \subset {\mathbb{F}}_q^{n'}$, let us define $$\mathcal{C}|_S : = \{w|_S{\;\ifnum\currentgrouptype=16 \middle\fi|\;}w \in \mathcal{C}\}$$
We call $\operatorname{RS}[q,d]|_S$ the $S$-punctured $\operatorname{RS}[q,d]$ code. Let $\mathcal{C}$ denote the $\operatorname{RS}[q,d]|_n$ code. Since the properties we will care about are independent of the specific set $S$, let is just parametrize this by $|S| =: n$. The following properties hold
1. $\mathcal{C}^{\perp} = \operatorname{RS}[q, q - d - 1]_n$.
2. Let $W_{i}$ be the number of codewords in $\mathcal{C}$ code of weight $i$. Then we have $$W_{n-i} \leq \frac{q^{k - i}n^i}{i!} .$$
Both facts can be easily checked.
Basic probability inequalities {#sec:probability}
------------------------------
We will use the standard (multiplicative) Chernoff bound for sums of i.i.d. Bernoulli random variables. Let $X_1,\ldots,X_n$ be independent $\operatorname{Ber}(p)$ random variables. Let $X := \sum_{i \in [n]}X_i$and denote $\mu = {{\rm I\kern-.3em E}}[X] = np$. Then for any $\epsilon \in (0,1)$, we have:
$$\label{eqn:Chernoff}
{\mathbb{P}}\left(|X - \mu|\geq \epsilon\mu \right) \leq e^{\frac{\epsilon^2\mu}{2}}.$$
We also have (a special case of) the Paley-Zygmund inequality, which states that for a nonnegative random variable $X$, $$\label{eqn:PaleyZygmund}
{\mathbb{P}}(X > 0)\geq \frac{{{\rm I\kern-.3em E}}^2[X]}{{{\rm I\kern-.3em E}}[X^2]}.$$
Fourier analysis over ${\mathbb{F}}_q$ {#sec:Fourier}
--------------------------------------
For functions $u,v: {\mathbb{F}}_q^n \rightarrow {\mathbb{C}}$, we have a normalized inner product $\langle u,v \rangle := \frac{1}{q^n}\sum_{s \in {\mathbb{F}}_q^n}u(s)\overline{v(s)}$. Consider any symmetric, non-degenerate bi-linear map $\chi:{\mathbb{F}}_q^n \times {\mathbb{F}}_q^n \rightarrow {\mathbb{R}}/{\mathbb{Z}}$ (such a map exists). For an $\alpha \in {\mathbb{F}}_q^n$, the *character* function associated with $\alpha$, denoted by $\chi_{\alpha}:{\mathbb{F}}_q^n \rightarrow {\mathbb{C}}$ is given by $\chi_{\alpha}(x) = e^{-2\pi i \chi(\alpha, x)}$.
We have that for all distinct $\alpha, \beta \in {\mathbb{F}}_q$, we have that $\langle\chi_{\alpha},\chi_{\beta}\rangle = 0$, and every function $f :{\mathbb{F}}_q \rightarrow {\mathbb{C}}$ can be written in a unique way as $f(x) = \sum_{\alpha \in {\mathbb{F}}_q}\widehat{f}(\alpha)\chi_{\alpha}(x)$. Here the $\widehat{f}(\alpha)$’s are called the *Fourier coefficients*, given by $$\widehat{f}(\alpha) = \langle f,\chi_{\alpha} \rangle.$$
We will state some facts that we will use in the proof of Theorem \[thm:sharpthreshold\]. The interested reader is referred to the excellent book of Tao and Vu [@TV06] (chapter 4) for further details.
For ${\mathbb{F}}_q^n \ni \alpha \neq 0$, we have: $$\langle 1,\chi_{\alpha} \rangle = 0.$$
\[fact:Plancherel\] For functions $f,g:{\mathbb{F}}^n \rightarrow {\mathbb{C}}$, we have $$\langle f,g\rangle = \sum_{\alpha}\hat{f}(\alpha) \hat{g}(\alpha).$$
\[fact:conv\] Suppose $g: {\mathbb{F}}_q^n \rightarrow {\mathbb{C}}$ can be written as a product $g(x) = \prod_{i \in [t]}g_i(x)$, then we have the Fourier coefficients of $g$ given by:
$$\begin{aligned}
\widehat{g}(\alpha) & = \left(\widehat{g_1}\ast \cdots \ast \widehat{g_t}\right)(\alpha)\\
& = \sum_{\beta_1,\ldots ,\beta_{t-1}} \widehat{g_1}(\beta_1)\cdots \widehat{g_{t-1}}(\beta_{t-1}) \widehat{g_t}(\alpha - \sum_{i \in [q-1]}\beta_i).\end{aligned}$$
\[prop:linspace\] If $g : {\mathbb{F}}_q^n \rightarrow {\mathbb{C}}$ is the indicator of a linear space $\mathcal{C}$, we have:
$$\widehat{g}(\alpha)=
\begin{cases}
\frac{|\mathcal{C}|}{|{\mathbb{F}}|^n},& \text{if } \alpha \in \mathcal{C}^{\perp}\\
0, & \text{otherwise}.
\end{cases}$$
Hypercontractivity and sharp thresholds {#sec:FK}
---------------------------------------
Here we state some tools from the analysis of Boolean function that we will use:
\[defn:transitive\] We say that a function $f :\{0,1\}^n\rightarrow \{0,1\}$ is *transitive-symmetric* if for every $i,j \in [n]$, there is a permutation $\sigma \in \mathfrak{S}_n$ such that:
- $\sigma(i) = j$
- $f(x_{\sigma(1)},\ldots,x_{\sigma_{n}}) = f(x)$ for all $x \in \{0,1\}^n$.
Let $f : \{0,1\} \rightarrow \{0,1\}$ be a monotone function. We will state an important theorem by Friedgut and Kalai, as stated in the excellent reference [@O14], regarding sharp thresholds for balanced symmetric monotone Boolean functions. This will be another important tool that we will use.
\[thm:FK\] Let $f: \{0,1\}^n\rightarrow \{0,1\}$ be a nonconstant, monotone, transitive-symmetric function and let $F:[0,1]\rightarrow[0,1]$ be the strictly increasing function defined by $F(p)={\mathbb{P}}(f(p)=1)$. Let $p_{\text{crit}}$ be the critical probability such that $F(p_{\text{crit}})=1/2$ and assume without loss of generality that $p_{\text{crit}}\leq1/2$. Fix $0<\epsilon<1/4$ and let $$\eta=B\log(1/\epsilon) \cdot \frac{\log(1/p_{\text{crit}})}{\log n},$$
where $B>0$ is a universal constant. Then assuming $\eta\leq1/2$, $$F(p_{\text{crit}} \cdot(1 - \eta))\leq\epsilon,~~~F(p_{\text{crit}}\cdot(1+\eta)) \geq1-\epsilon.$$
We will use an immediate corollary of the above theorem.
\[corr:FK\] Let $f:\{0,1\}^n\rightarrow \{0,1\}$ be a nonconstant, monotone, transitive-symmetric function. Let $F:[0,1]\rightarrow[0,1]$ be the strictly increasing function defined by $F(p)=\Pr(f(p)=1)$. Let $p$ be such that $F(p) \geq \epsilon$, and let $\eta = B\log(1/\epsilon) \cdot \frac{\log(1/p_c)}{\log n}$. Then $F(p(1 + 2\eta)) \geq 1 - \epsilon$.
In particular, in the above corollary, if for some $\epsilon \in (0,1)$ we have that $F^{-1}(\epsilon) \in (0,1)$, then the function $f$ has a sharp threshold.
One easy observation that will allow us to use Theorem \[thm:FK\] is the following:
\[prop:symm\] The Boolean function $\text{AP}_{r}:\{0,1\}^{q \times n} \rightarrow \{0,1\}$ is transitive-symmetric.
For a pair of coordinates indexed by $(i_1,j_1)$ and $(i_2,j_2)$, it is easy to see that the map $(x,y) \mapsto (x + i_2 - i_1,y+j_2-j_1)$ gives us what we need since the set of polynomials is invariant under these operations.
Proof of Theorem \[thm:sharpthreshold\] {#sec:sharpthreshold}
=======================================
First, we restate the theorem that we will prove in order to make a few more remarks:
Let $q$ be a prime power, and $r = r(q)$ and $\epsilon = \epsilon(q)$ be such that $q^{-r} \geq \frac{\log q}{q}$, and let $\epsilon \geq \omega\left(\max\left\{q^{-r}, \sqrt{q^{r - 1} \log \left(q^{1 - r}\right)}\right\}\right)$.
Let $A_1,\ldots,A_q$ be independently chosen random subsets of ${\mathbb{F}}$ with each point picked independently with probability least $q^{-r}(1 + \epsilon)$. Then $${\mathbb{P}}((A_1 \times \cdots \times A_q) \cap \operatorname{RS}[q,rq] = \emptyset) = o(1).$$
#### Remarks
Before we proceed to the proof, we first make some simple observations that hopefully make the technical conditions on $q,c,\epsilon$ above seem more natural.
1. We need $q$ to be a prime power for the existence of ${\mathbb{F}}_q$.
2. If $r$ is too large, i.e., if $q^{-r} \leq \frac{\log q}{q}(1 - \delta)$, for some $\delta>0$, then we will almost surely not contain *any* codeword. Indeed, we will almost surely have some $i \in [n]$ such that $A_i = \emptyset$.
3. The reason for $\epsilon = \sqrt{q^{r - 1} \log \left(q^{1 - r}\right)}$ is more or less the same reason as above in that this helps us prove that w.h.p., $|A|$ is not much smaller than expected, as in Claim \[claim:mainterm\]. This is probably not the best dependence possible, and we make no attempt to optimize. But as $q^{-r}$ gets closer to $\frac{\log q}{q}$, then this condition gets closer to the truth.
We now proceed to the proof.
Let us abbreviate ${\mathbb{F}}= {\mathbb{F}}_q$, denote the subspace $\mathcal{C} \leq {\mathbb{F}}^q$ to be the $\operatorname{RS}[q,rq]$ code. Let $f:{\mathbb{F}}^q \rightarrow {\mathbb{C}}$ be the indicator of $\mathcal{C}$, i.e., $f(x) = 1$ iff $x \in \mathcal{C}$ and $0$ otherwise. Let $A_i \subset {\mathbb{F}}$ for $i \in [n]$ and let $g_i : {\mathbb{F}}\rightarrow \{0,1\}$ be the indicator for $A_i$. Let us slightly abuse notation and also think of $g_i:{\mathbb{F}}^q \rightarrow {\mathbb{C}}$ which depends only on the $i$’th variable.
We will estimate the quantity $|\mathcal{C} \cap \left(A_1 \times \cdots \times A_q \right)| = q^q\langle f, g \rangle$. Setting $|\widehat{f}(0)| =: \rho$, standard steps yield:
$$\begin{aligned}
\rho^{-1}\langle f, g \rangle & = \rho^{-1}\sum_{\alpha}\widehat{f}(\alpha) \widehat{g}(\alpha) \nonumber \\
& = \sum_{\alpha \in \mathcal{C}^{\perp}}\widehat{g}(\alpha) \nonumber \\
& = \sum_{\alpha \in \mathcal{C}^{\perp}}\sum_{\beta_1,\ldots ,\beta_{q-1}} \widehat{g_1}(\beta_1)\cdots \widehat{g_{n-1}}(\beta_{q-1}) \widehat{g_q}(\alpha - \sum_{i \in [q-1]}\beta_i) \nonumber \\
& = \sum_{\beta_1,\ldots ,\beta_{q-1}} \widehat{g_1}(\beta_1)\cdots \widehat{g_{q-1}}(\beta_{q-1})\sum_{\alpha \in \mathcal{C}^{\perp}} \widehat{g_q}(\alpha - \sum_{i \in [q-1]}\beta_i) \nonumber \\
& = \sum_{(\alpha_1,\ldots, \alpha_q) \in \mathcal{C}^{\perp}}\prod_{i \in [q]}\widehat{g_i}(\alpha_i) \nonumber \\
& \geq \prod_{i \in [q]}\widehat{g_i}(0) - \left|\sum_{(\alpha_1,\ldots, \alpha_q) \in \mathcal{C}^{\perp} \setminus \{\overline{0}\}} \left(\prod_{i \in [q]}\widehat{g_i}(\alpha_i) \right) \right|. \label{eqn:main}\end{aligned}$$
where the first equality is due to Plancherel’s identity, the third inequality is using Fact \[fact:conv\], the and last equality is because of the fact that $\widehat{g_i}(\beta_i)$ is nonzero only if $\operatorname{supp}(\beta_i) \subseteq \{i\}$. Let us denote
$$R := \sum_{(\alpha_1,\ldots, \alpha_q) \in \mathcal{C}^{\perp} \setminus \{\overline{0}\}} \left(\prod_{i \in [q]}\widehat{g_i}(\alpha_i) \right).$$
For $\alpha \in {\mathbb{F}}^q$, let us define $M(\alpha) = \prod_{i \in [n]}\widehat{g_i}(\alpha_i)$, to be the ‘monomial’ corresponding to $\alpha$. So, we have $R = \sum_{\mathcal{C}^{\perp} \setminus \{\overline{0}\}}M(\alpha)$, and $|R|^2 = \sum_{\alpha,\beta \in \mathcal{{\mathbb{C}}}^{\perp} \setminus \{\overline{0}\}}M(\alpha) \overline{M(\beta)}$. By linearity of expectation, we have
$$\begin{aligned}
{{\rm I\kern-.3em E}}[|R|^2] & = \sum_{\alpha, \beta \in \mathcal{C}^{\perp} \setminus \{\overline{0}\}} {{\rm I\kern-.3em E}}[M(\alpha)\overline{M(\beta)}] \\
&= \sum_{\alpha \in \mathcal{C}^{\perp} \setminus \{\overline{0}\}}{{\rm I\kern-.3em E}}[M(\alpha)\overline{M(\alpha)}] + \sum_{\alpha \neq \beta}{{\rm I\kern-.3em E}}[M(\alpha)\overline{M(\beta)}].\end{aligned}$$
For $\alpha \neq \beta$, let $t$ be a coordinate such that $\alpha_t \neq \beta_t$. We have:
$$\begin{aligned}
q^2 \cdot {{\rm I\kern-.3em E}}[\widehat{g_i}(\alpha_i) \overline{\widehat{g_i}(\beta_i)}] & = {{\rm I\kern-.3em E}}\left[\left(\sum_{x \in {\mathbb{F}}} g_i(x)\chi_{\alpha_i}(x)\right)\left(\overline{\sum_{y \in {\mathbb{F}}} g_i(x)\chi_{\beta_i}(y)}\right) \right] \\
& = {{\rm I\kern-.3em E}}\left[\sum_{x,y}g_i(x)g_i(y) \chi_{\alpha_i}(x) \overline{\chi_{\beta_i}(y)} \right] \\
& = \sum_{x,y}\left({{\rm I\kern-.3em E}}\left[g_i(x)g_i(y) \chi_{\alpha_i}(x) \overline{\chi_{\beta_i}(y)} \right]\right) \\
& = \sum_x \left({{\rm I\kern-.3em E}}\left[g_i(x) \ \chi_{\alpha_i}(x) \overline{\chi_{\beta_i}(x)} \right]\right) + \sum_{x \neq y} \left({{\rm I\kern-.3em E}}\left[g_i(x)g_i(y) \chi_{\alpha_i}(x) \overline{\chi_{\beta_i}(y)} \right]\right) \\
& = p \cdot \sum_{x}\left( \chi_{\alpha_i - \beta_i}(x) \right) + p^2 \cdot \sum_{x \neq y} \left( \chi_{\alpha_i}(x) \overline{\chi_{\beta_i}(y)} \right) \\
& = (p - p^2)\cdot \sum_{x}\left( \chi_{\alpha_i - \beta_i}(x) \right) + p^2 \cdot \sum_{x , y} \left( \chi_{\alpha_i}(x) \overline{\chi_{\beta_i}(y)} \right) \\
& = (p - p^2)\cdot \sum_{x}\left( \chi_{\alpha_i - \beta_i}(x) \right) + p^2 \left(\sum_x \chi_{\alpha_i}(x) \right)\left(\overline{\sum_y\chi_{\beta_i}(y)}\right)\\
& = 0.\end{aligned}$$
The last equality is because at least one of $\alpha_i$ or $\beta_i$ is nonzero, and $\alpha_i - \beta_i$ is nonzero. Therefore, for $\alpha \neq \beta$, we have:
$$\begin{aligned}
{{\rm I\kern-.3em E}}[M(\alpha)\overline{M(\beta)}] & = {{\rm I\kern-.3em E}}\left[\left(\prod_{i}\widehat{g_i}(\alpha_i)\right)\left(\overline{\prod_i \widehat{g_i}(\beta_i)} \right) \right] \\
& = \prod_i \left( {{\rm I\kern-.3em E}}[\widehat{g_i}(\alpha_i) \overline{\widehat{g_i}(\beta_i)}] \right) \\
& = 0.\end{aligned}$$
Where the second equality is because $A_1,\ldots A_q$ are chosen independently. For $\alpha = \beta$, it is easy to see that ${{\rm I\kern-.3em E}}[M(\alpha)\overline{M(\alpha)}] = \prod_{i \in [q]}|\widehat{g_i}(\alpha_i)|^2$. So, we have the identity:
$$\begin{aligned}
{{\rm I\kern-.3em E}}[|R|^2] & = \sum_{\alpha \in \mathcal{C}^{\perp} \setminus \{0\}}{{\rm I\kern-.3em E}}\left[\prod_{i \in [q]}|\widehat{g_i}(\alpha_i)|^2\right] \nonumber \\
& = \sum_{\alpha \in \mathcal{C}^{\perp} \setminus \{0\}}\prod_{i \in [q]}{{\rm I\kern-.3em E}}\left[|\widehat{g_i}(\alpha_i)|^2\right]. \label{eqn:ER^2}\end{aligned}$$
The following two identities are easy to check:
$$\begin{aligned}
{{\rm I\kern-.3em E}}\left[|\widehat{g_i}(0)|^2\right] & = p^2 + \frac{p(1-p)}{q} \label{eqn:Eg0}\\
{{\rm I\kern-.3em E}}\left[|\widehat{g_i}(\alpha_i)|^2\right] & = \frac{p(1-p)}{q} \text{ for } \alpha_i \neq 0. \label{eqn:Egalpha}\end{aligned}$$
Equipped with \[eqn:ER\^2\], \[eqn:Eg0\], \[eqn:Egalpha\], and Proposition \[prop:RSwtdist\], we have:
$$\begin{aligned}
{{\rm I\kern-.3em E}}\left[|R|^2\right] & = \sum_{i = 0}^{q-1}W_{q-i}p^{2i}\left(1 + \frac{1-p}{pq}\right)^{2i}p^{q - i}\left(\frac{1-p}{q}\right)^{q-i} \\
& \leq \cdot \left(\frac{1 - p}{q} \right)^{q} \cdot \sum_{i = 0}^{q-1}\frac{q^{q - rq}}{i!}{p^{q+i}} \left( \frac{q}{1 - p}\right)^i\left( 1 + \frac{1-p}{pq}\right)^{2i}\\
& = \cdot \left(\frac{1 - p}{q} \right)^{q} \cdot q^{q-rq }p^{q} \sum_{i = 0}^{q-1}\frac{1}{i!}\left(\frac{pq}{1 - p} \right)^i \left( 1 + \frac{1-p}{pq}\right)^{2i}\\
& \leq e \cdot(1 - p)^q \cdot q^{q- rq }\left(\frac{p}{q}\right)^{q}e^{\left( \frac{2pq}{1 - p}\right)}.\end{aligned}$$
Markov’s inequality gives us:
$$\label{eqn:Rterm}
{\mathbb{P}}\left(|R| \geq \left(qe \cdot q^{q-rq }\left(\frac{p}{q}\right)^{q}e^{\left( \frac{2pq}{1 - p}\right)}(1 - p)^q\right)^{\frac{1}{2}}\right) \leq \frac{1}{q}.$$
On the other hand, we have:
\[claim:mainterm\] For $q,c$ as given above, let $\epsilon = \omega\left(\sqrt{q^{r -1} \log \left(q^{1- r}\right)} \right)$, and $p = q^{-r}(1 + \epsilon)$. Then we have:
$${\mathbb{P}}\left(\prod_{i \in [q]}\widehat{g_i}(0) \leq q^{-rq}(1 + 0.9\epsilon)^{0.9q}\right) = o(1).$$
The proof is postponed to the Appendix.
So, using \[eqn:main\], Claim \[claim:mainterm\], and \[eqn:Rterm\], and setting $p = q^{-r}(1 + \epsilon)$ we have that with probability at least $1 - o(1)$,
$$\begin{aligned}
\rho^{-1}\langle f,g \rangle & \geq q^{-rq}(1 + 0.9\epsilon)^{0.9q} - q\cdot eq^{-rq}(1 + \epsilon)^{\frac{q}{2}}e^{\frac{2pq}{2(1- p)}}(1 - p)^{\frac{q}{2}} \\
& \geq q^{-rq}(1 + \epsilon)^{\frac{q}{2}}\left((1 + (\epsilon/2))^{0.4q} - eq \cdot e^{\frac{2pq}{2(1 - p)}}(1 - p)^{\frac{q}{2}}\right).\end{aligned}$$
Where the inequality follows from the fact that for $x \in [0,1]$, $$\frac{(1 + 0.9x)^{0.9}}{(1+x)^{0.5}} \geq (1 + 0.5x)^{0.4} .$$
It remains to check that if $\epsilon = \omega\left(q^{-r} \right)$, then $\rho^{-1}\langle f,g \rangle > 0$, which completes the proof.
Proof of Theorem \[thm:fullrange\] {#sec:fullrange}
==================================
Here, we address the case when $r = O\left(\frac{1}{\log q}\right)$. In this case, we observe that Theorem \[thm:sharpthreshold\] does not give us a sharp threshold for the random list recovery since in this case, $p \in (0,1)$. However, this case can be handled by the second moment method and Theorem \[thm:FK\].
Let us use $X$ to denote the number of codewords of $\text{RS}[q,rq]$ contained in a randomly chosen set $S$. Linearity of expectation gives us
$$\label{eqn:EX}
{{\rm I\kern-.3em E}}[X] = q^{rq+1}p^q$$
Again, we have that if $p = q^{-r}(1 - \epsilon)$, we have that ${{\rm I\kern-.3em E}}[X] = (1 - \epsilon)^q$. And so, by Union Bound, we have that with high probability $X = 0$. On the other hand, if $p = q^{-r}(1 + \epsilon)$, we will show that with high probability, $X > 0$. We start off by computing the second moment.
\[lem:easysecondmoment\] We have $${{\rm I\kern-.3em E}}[X^2] \leq {{\rm I\kern-.3em E}}[X] + e^{\frac{1}{p}}{{\rm I\kern-.3em E}}^2[X]$$
Let us denote $\mathcal{C} = \operatorname{RS}[q,rq]$. For every $w \in \mathcal{C}$, let us denote $X_w$ for the indicator random variable for the event $\{w \subset S\}$. We have: $$\begin{aligned}
{{\rm I\kern-.3em E}}[X^2] & = \sum_{w \in \mathcal{C}}{{\rm I\kern-.3em E}}[X_w] + \sum_{\substack{ w_1,w_2 \in \mathcal{C} \\ w_1 \neq w_2}}{{\rm I\kern-.3em E}}[X_{w_1}X_{w_2}] \\
& = q^{rq+1}p^q + \sum_{i = 0}^{rq+1}\sum_{\substack{w_1,w_2 \in \mathcal{C}\\ \Delta(w_1,w_2) = q - i}}p^{2q - i} \\
& = q^{rq+1}p^q + q^{rq+1}\sum_{i = 0}^{n - rq}|W_{q-i}|p^{2q - i} \\
& \leq q^{rq+1}p^q + q^{rq+1}\sum_{i = 0}^{rq+1}\frac{q^{rq+1}}{i!}p^{2q - i} \\
& \leq q^{k}p^q + e^{\frac{1}{p}}\left( q^{rq+1} p^{q}\right)^2 \\
& = {{\rm I\kern-.3em E}}[X] + e^{\frac{1}{p}}{{\rm I\kern-.3em E}}^2[X].\end{aligned}$$
When ${{\rm I\kern-.3em E}}[X] = \omega(1)$, one can bound second moment by $2e^{\frac{1}{p}}{{\rm I\kern-.3em E}}^2[X]$. The Paley-Zygmund inequality \[eqn:PaleyZygmund\] immediately gives:
$${\mathbb{P}}(X \geq 0) \geq \frac{1}{2e^{\frac{1}{p}}}.$$
Now, Corollary \[corr:FK\] gives us that if $p \geq q^{-r}\left(1 + \omega_{p}\left(\frac{1}{\log q} \right)\right)$, then ${\mathbb{P}}(X > 0) = 1 - o(1)$.
This, combined with Lemma \[lem:smallp\] and Theorem \[thm:sharpthreshold\] finishes the proof.
Technical lemmas {#sec:appendix}
================
Proofs from Section \[sec:sharpthreshold\] {#sec:asec3}
------------------------------------------
For $i \in [n]$, let define the indicator random variables $$Y_i = \mathbbm{1}\left[\{\widehat{g_i}(0) < (1 + 0.9\epsilon)q^{-r}\}\right]$$ and $$Z_i = \mathbbm{1}\left[\{\widehat{g_i}(0) = 0\}\right].$$
Let $Y = \sum_iY_i$, and $Z = \sum_iZ_i$. First off, Chernoff bound gives us that there is a constant $C < \frac{1}{200}$ such that ${\mathbb{P}}(Y_i = 1) \leq e^{- C\epsilon^2 pq}$, so we have ${{\rm I\kern-.3em E}}[Y] \leq ne^{- C\epsilon^2pq}$. Since all the $Y_i$’s are independent, Chernoff bound again gives us that $$\label{eqn:A}
{\mathbb{P}}\left(Y \geq 2q e^{-C\epsilon^2 pq}\right) = o(1).$$ Moreover, since we have $q^{-r} \geq \frac{\log q}{q}$, we have $$\label{eqn:B}
{\mathbb{P}}(Z > 0) = o(1)$$
Let us abbreviate $t = 2q e^{-C\epsilon^2 pq}$. Taking \[eqn:A\] and \[eqn:B\] into consideration, we have that with probability at least $1 - o(1)$,
$$\begin{aligned}
\prod_{i \in [q]}\widehat{g_i}(0) & \geq p^{q}(1 + 0.9\epsilon)^{q - t}e^{-t\log q} \\
& \geq (1 + 0.9\epsilon)^{q - t - \frac{2t \log q}{\epsilon}} \end{aligned}$$
It remains to check that if $\epsilon \gg \sqrt{\frac{\log \left(pq\right)}{pq}}$, then $t + \frac{2t \log q}{\epsilon} \leq 0.1q$. With this in mind, we compute
$$\begin{aligned}
t + \frac{2t \log q}{\epsilon} & \leq \frac{3t \log q}{\epsilon} \\
& \leq 6q \cdot \frac{1}{(pq)^{\omega(1)}}\cdot\sqrt{pq} \log q \\
& = o(q)\end{aligned}$$
which finishes the proof.
Proofs from Section \[sec:AC0\] {#sec:asec4}
-------------------------------
Consider the sequence of integers $\{k_i\}_{i \in {\mathbb{N}}}$ such that for every $i$, $k_i$ is the largest such that
$$\prod_{j = 1}^i\left(1 - \frac{1}{2^j}\right)^{k_j} \geq p.$$
We make a basic observation:
\[obs:bias\] We have that $k_1 = \lfloor \log_2 (1/p) \rfloor \leq s$ and for all $j \geq 2$, we have that $k_j \leq 3$.
Let $\ell$ be the largest such that $k_{\ell} > 0$ and $\sum_{i \in [\ell]}i\cdot k_i < s^2$. Let $t = \sum_{i \in [\ell]}i\cdot k_i $. Consider the $CNF$ given by $$C_p = \bigwedge_{j \in [\ell]}\left( \bigwedge_{i \in k_j}C_i^j\right)$$
where the clause $C_i^j$ is an $\lor$ of $j$ independent variables. We first estimate $p_{\frac{1}{2}} := {\mathbb{P}}(C_p = 1)$ when $X_1,\ldots, X_n \sim \operatorname{Ber}\left(\frac{1}{2}\right)$.
Using the fact that $k_1 \leq s$ and Observation \[obs:bias\], we have $\ell = \Omega(\sqrt{t})$. Therefore $$p \leq {\mathbb{P}}(C_p = 1) \leq p\left( 1 - \frac{1}{2^{{\ell}+1}}\right)^{-4} \leq p\left(1 + \frac{4}{2^{\Omega(\sqrt{t})}} \right).$$
And so for $X_1,\ldots, X_n \sim \operatorname{Ber}\left(\frac{1}{2} + \epsilon \right)$ we bound
$$\begin{aligned}
{\mathbb{P}}(C_p = 1) & = \prod_{j = 1}^\ell\left(1 - \left(\frac{1}{2} - \epsilon\right)^j\right)^{k_j} ~~~~~~~~= \prod_{j = 1}^\ell\left(1 - \frac{1}{2^j}\left(1 - 2\epsilon\right)^j\right)^{k_j} \\
& \geq \prod_{j = 1}^\ell\left(1 - \frac{1}{2^j}\left(1 - \epsilon j\right)\right)^{k_j} ~~~~~~~\geq \prod_{j = 1}^\ell\left(1 - \frac{1}{2^j}\right)^{k_j}\left(1 + (\epsilon j/2^{j-1})\right)^{k_j} \\
& \geq p_{\frac{1}{2}}(C_p = 1)\left(1 + \epsilon \sum_{j = 1}^{\ell}\frac{jk_j}{2^{j-1}} \right) \geq p(1 + \epsilon k_1 ).\end{aligned}$$
Similarly, for $X_1,\ldots, X_n \sim \operatorname{Ber}\left(\frac{1}{2} - \epsilon \right)$:
$$\begin{aligned}
{\mathbb{P}}(C_p = 1) & = \prod_{j = 1}^\ell\left(1 - \left(\frac{1}{2} + \epsilon\right)^j\right)^{k_j} ~~= \prod_{j = 1}^\ell\left(1 - \frac{1}{2^j}\left(1 + 2\epsilon\right)^j\right)^{k_j} \\
& \leq \prod_{j = 1}^\ell\left(1 - \frac{1}{2^j}\left(1 + 4\epsilon j\right)\right)^{k_j} \leq \prod_{j = 1}^\ell\left(1 - \frac{1}{2^j}\right)^{k_j}\left(1 - \frac{\epsilon j}{2^{j}}\right)^{k_j} \\
& \leq p_{\frac{1}{2}}(C_p = 1)(1 - \Omega(k_1 \epsilon ) ) ~~\leq p\left(1 + \frac{1}{2^{\Omega(\sqrt{t})}} - \Omega(\epsilon k_1)\right).\end{aligned}$$
[^1]: Department of Computer Science, Rutgers University. [[email protected]]{}. Research supported in part by NSF grant CCF-1514164
|
**ON THE BOCHNER CURVATURE TENSOR**
**IN AN ALMOST HERMITIAN MANIFOLD [^1]**
OGNIAN T. KASSABOV
Let $M$ be a $2n$-dimensional almost Hermitian manifolds, $n \ge 3$, with metric tensor $g$ and almost complex structure $J$ and let $\nabla$ be the covariant differentiation on $M$. The curvature tensor $R$ is defined by $$R(X,Y,Z,U) = g(\nabla_X\nabla_Y Z - \nabla_Y\nabla_X Z - \nabla_{[X,Y]} Z,U)$$ for $X,\, Y,\, Z,\, U \in \mathfrak X(M)$. The manifold is said to be an RK-manifold [@V] if $$R(X,Y,Z,U) = R(JX,JY,JZ,JU)$$ for all $X,Y,Z,U \in \mathfrak X(M)$. In this paper we trate for simplicity only the case of an RK-manifold although one cane made analogous considerations for an arbitrary almost Hermitian manifold.
Let $ E_i ,\, i=1,\hdots,2n$ be a local orthonormal frame field. The Ricci tensor $S$ and the scalar curvature $\tau(R)$ are defined by $$S(X,Y) = \sum_{i=1}^{2n} R(X,E_i,E_i,Y) \ , \quad \tau(R) = \sum_{i=1}^{2n} S(E_i,E_i).$$ Analogously we set $$S'(X,Y) = \sum_{i=1}^{2n} R(X,E_i,JE_i,JY) \ , \quad \tau'(R) = \sum_{i=1}^{2n} S'(E_i,E_i).$$ We note that $S$ and $S'$ are symmetric and $S(X,Y)=S(JX,JY)$, $S'(X,Y)=S'(JX,JY)$.
The Bochner curvature tensor $B$ [@TV] for $M$ is defined by $$B=R-\frac 1{8(n+2)}(\varphi + \psi)(S+3S')-\frac 1{8(n-2)}(3\varphi - \psi)(S-S')$$ $$+\frac{\tau(R)+3\tau'(R)}{16(n+1)(n+2)}(\pi_1+\pi_2)
+\frac{\tau(R)-\tau'(R)}{16(n-1)(n-2)}(3\pi_1-\pi_2),$$ where $\varphi,\ \psi,\, \pi_1$ and $\pi_2$ are defined by $$\varphi(Q)(X,Y,Z,U) = g(X,U)Q(Y,Z) - g(X,Z)Q(Y,U)$$ $$+ g(Y,Z)Q(X,U) - g(Y,U)Q(X,Z)\, ,$$ $$\psi(Q)(X,Y,Z,U) = g(X,JU)Q(Y,JZ) - g(X,JZ)Q(Y,JU)$$ $$+ g(Y,JZ)Q(X,JU) - g(Y,JU)Q(X,JZ)\, ,$$ $$-2g(X,JY)Q(Z,JU) -2g(Z,JU)Q(X,JY)\, ,$$ $$\pi_1(X,Y,Z,U) = g(X,U)g(Y,Z) - g(X,Z)g(Y,U)\, ,$$ $$\pi_2(X,Y,Z,U) = g(X,JU)g(Y,JZ) - g(X,JZ)g(Y,JU) -2g(X,JY)g(Z,JU)\, .$$
By a plane we mean a 2-dimensional linear subspace of the tangent space $T_p(M)$ of $M$ in $p$. A plane $\alpha$ is said to be holomorphic (resp. antiholomorphic) if $J\alpha = \alpha$ (resp. $J\alpha$ is perpendicular to $\alpha$).
A tensor field $T$ of type (0,4) is said to be an LC-tensor if it has the properties:
1\) $ T(X,Y,Z,U) = - T(Y,X,Z,U)$,
2\) $ T(X,Y,Z,U) = - T(X,Y,U,Z)$,
3\) $ T(X,Y,Z,U) + T(Y,Z,X,U) + T(Z,X,Y,U) = 0$.
We need the following lemma.
L e m m a [@G].
*Let $M$ be a $2n$-dimensional almost Hermitian manifold, $n \ge 2$. Let $T$ be an $LC$-tensor, satisfying the conditions:*
1\) $ T(X,Y,Z,U) = T(JX,JY,JZ,JU)$,
2\) $ T(x,y,y,x) = 0$, where $\{ x,y \}$ is a basis of any holomorphic or antiholomorphic plane.
Then $T=0$.
In section 2 we shall prove the following theorem.
T h e o r e m.
*Let $M$ be a $2n$-dimensional RK-manifold, $n\ge 3$, which satisfies $$\lambda R(x,y,y,x) + \mu (S(x,x)+S(y,y)) + \nu (S'(x,x)+S'(y,y)) = c(p) \leqno (1.1)$$ for each point $p \in M$ and for all unit vectors $x,y \in T_p(M)$ with $g(x,y)=g(x,Jy)=0$, where $\lambda,\, \mu,\, \nu$ are constants, $(\lambda,\mu,\nu) \ne (0,0,0)$ and $c(p)$ does not depend on $x,y$. Then*
1\) if $\lambda = 0$, then $$\mu S+\nu S'= \frac{\mu \tau(R) + \nu \tau'(R)}{2n} g\ ;$$
2\) if $\lambda \ne 0$, then $M$ has vanishing Bochner curvature tensor and the tensor\
$((n+1)\lambda + 2(n^2-4)\mu)S + (2(n^2-4)\nu - 3\lambda)S'$ is proportional to the metric tensor
: $$((n+1)\lambda + 2(n^2-4)\mu)S + (2(n^2-4)\nu - 3\lambda)S'$$ $$=\frac{1}{2n}\{ ((n+1)\lambda + 2(n^2-4)\mu)\tau(R) + (2(n^2-4)\nu - 3\lambda)\tau'(R) \} g \ .$$
An almost Hermitian manifold $M$ is said to be of pointwise constant antiholomorphic sectional curvature if for each point $p \in M$ the curvature of an arbitrary antiholomorphic plane $\alpha$ in $T_p(M)$ does not depend on $\alpha$.
C o r o l l a r y. [*Let $M$ be a $2n$-dimensional RK-manifold, $n\ge 3$. Then $M$ has pointwise constant antiholomorphic sectional curvature if and only if $M$ has vanishing Bochner curvature tensor and*]{} $$(n+1)S-3S'= \frac{1}{2n}((n+1)\tau(R) - 3\tau'(R))g \ .$$
This is an analogue of a well known theorem of S c h o u t e n and S t r u i k [@S], see also [@T].
Let $ e_i ,\, Je_i,\, i=1,\hdots,2n$ be an orthonormal basis of $T_p(M), \, p \in M$. In (1.1) we put $X=e_1, \, Y=e_i$ or $Y=Je_i$, $i=2,...,n$. Adding on $i$ we find $$\lambda R(e_1,Je_1,Je_1,e_1) - (\lambda +2(n-2)\mu)S(e_1,e_1) - 2(n-2)\nu S'(e_1,e_1)$$ $$=\mu\tau(R)(p) + \nu\tau'(R)(p) - 2(n-1)c(p)$$ and since we can take for $e_1$ an arbitrary unit vector in $T_p(M)$ we have $$\begin{array}{c}
\lambda H(x) - (\lambda +2(n-2)\mu)S(x,x) - 2(n-2)\nu S'(x,x) \\
=\mu\tau(R)(p) + \nu\tau'(R)(p) - 2(n-1)c(p)
\end{array} \leqno(2.1)$$ for each unit vector $x \in T_p(M)$, where $H(x)$ is the curvature of the holomorphic plane spanned by $x,\, Jx$, i.e. $H(x)=R(x,Jx,Jx,x)$.
If $\lambda = 0$ (2.1) takes the form $$\mu S(x,x) + \nu S'(x,x)
=\frac{2(n-1)c(p)-\mu\tau(R)(p) - \nu\tau'(R)(p)}{2(n-2)} \ .$$
We put $ x=e_i,\, x=Je_i$ and adding on $i$ we obtain $$c=\frac{\mu\tau(R) + \nu\tau'(R)}{n}$$ and case 1) is proved.
If $\lambda \ne 0$, (1.1) and (2.1) take the form $$R(x,y,y,x) + \mu_1 (S(x,x)+S(y,y)) + \nu_1 (S'(x,x)+S'(y,y)) = c_1(p) \, ,\leqno (2.2)$$ $$\begin{array}{c}
H(x) - (1 +2(n-2)\mu_1)S(x,x) - 2(n-2)\nu_1 S'(x,x) \\
=\mu_1\tau(R)(p) + \nu_1\tau'(R)(p) - 2(n-1)c_1(p) \, ,
\end{array} \leqno(2.3)$$ where $\mu_1=\mu/\lambda$, $\nu_1=\nu/\lambda$, $c_1=c/\lambda$.
From (2.2) $R(x,y,y,x) = R(x,Jy,Jy,x)$ and consequently $$R\left( \frac{x+y}{\sqrt 2},\frac{x-y}{\sqrt 2},\frac{x-y}{\sqrt 2},\frac{x+y}{\sqrt 2} \right)=
R\left( \frac{x+y}{\sqrt 2},\frac{Jx-Jy}{\sqrt 2},\frac{Jx-Jy}{\sqrt 2},\frac{x+y}{\sqrt 2} \right)$$ which gives $$H(x)+H(y)=4R(x,y,y,x)-2R(x,Jy,Jy,x)+2R(x,Jx,Jy,y)+2R(x,Jy,Jx,y) \ .$$
Hence it is easy to find $$(n+2)H(x) + \sum_{i=1}^{n} H(e_i) = S(x,x) +3S'(x,x) \leqno(2.4)$$ and $$\sum_{i=1}^{n} H(e_i) =\frac{\tau(R)(p) +3\tau'(R)(p)}{4(n+1)} \, . \leqno (2.5)$$
From (2.4) and (2.5) we obtain $$H(x) -\frac{1}{n+2}(S(x,x)+3S'(x,x))=-\frac{\tau(R)(p) +3\tau'(R)(p)}{4(n+1)(n+2)} \, .\leqno (2.6)$$
Using (2.3) and (2.6) we get $$\begin{array}{c}
{\displaystyle}\left( 2(n-2)\mu_1 - \frac{n+1}{n+2} \right) S(x,x)
+\left( 2(n-1)\nu_1-\frac{3}{n+2} \right)S'(x,x) \\
=2(n-1)c_1(p) - \mu_1\tau(R)(p) -\nu_1\tau'(R)(p) -
{\displaystyle}\frac{\tau(R)(p) + 3\tau'(R)(p)}{4(n+1)(n+2)} \ .
\end{array} \leqno (2.7)$$
Hence by a simple calculation we obtain $$c_1 = \left( \frac{\mu_1}{n} + \frac{2n+1}{8n(n^2-1)} \right) \tau(R)
+\left( \frac{\nu_1}{n} - \frac{3}{8n(n^2-1)} \right) \tau'(R) \ . \leqno (2.8)$$
The substitution of (2.8) in (2.7) gives $$\begin{array}{c}
{\displaystyle}\left( \mu_1 + \frac{n+1}{2(n^2-4)} \right) S(x,x)
+\left( \nu_1-\frac{3}{2(n^2-4)} \right)S'(x,x) \\
= {\displaystyle}\frac{1}{2n}\left\{\left( \mu_1 +\frac{n+1}{2(n^2-4)}\right) \tau(R)(p) +
\left( \nu_1 - \frac{3}{2(n^2-4)} \right) \tau'(R)(p)\right\} \ .
\end{array} \leqno (2.9)$$
From (2.2), (2.8) and (2.9) it follows $$\begin{array}{c}
{\displaystyle}R(x,y,y,x) -\frac{n+1}{2(n^2-4)}(S(x,x)+S(y,y))
+\frac{3}{2(n^2-4)} (S'(x,x) + S'(y,y)) \\
{\displaystyle}=-\frac{2n^2+3n+4}{8(n^2-1)(n^2-4)}\tau(R)(p) +
\frac{9n}{8(n^2-1)(n^2-4)} \tau'(R)(p) \ .
\end{array} \leqno (2.10)$$
According to the lemma, (2.6) and (2.10) imply that the Bochner curvature tensor $B$ for $M$ vanishes. The rest of the theorem follows from (2.2) and (2.10).
[9]{}
G. G a n č e v. Almost Hermitian manifolds similar to the complex space forms. [*C. R. Acad. bulg. Sci.*]{}, [**32**]{}, 1979, 1179-1182.
J. S c h o u t e n, D. S t r u i k. On some properties of general manifolds relating to Einstein’s theory of gravitation. [*Amer. J. Math.*]{}, [**43**]{}, 1921, 213-216.
S. T a c h i b a n a. On the Bochner curvature tensor. [*Nat. Sc. Rep. Ochanomizu Univ.*]{}, [**18**]{}, 1967, 15-19.
F. T r i c e r r i, L. V a n h e c k e. Curvature tensors on almost Hermitian manifolds. [*Trans. Amer. Math. Soc.*]{}, [**267**]{}, 1981, 365-398.
L. V a n h e c k e. Almost Hermitian manifolds with $J$-invariant Riemann curvature tensor. [*Rend. Sem. Matem. Torino*]{}, [**34**]{}, 1975-76, 487-498.
*Centre for Mathematics and Mechanics Received 26.5.1981*
Sofia 1090 P. O. Box 373
[^1]: *SERDICA Bulgaricae mathematicae publicationes. Vol. 9, 1983, p. 168-171.*
|
---
abstract:
- 'This short example shows a contrived example on how to format the authors’ information for [*IJCAI–19 Proceedings*]{} using LaTeX.'
- 'With the recently rapid development in deep learning, deep neural networks have been widely adopted in many real-life applications. However, deep neural networks are also known to have very little control over its uncertainty for test examples, which potentially causes very harmful and annoying consequences in practical scenarios. In this paper, we are particularly interested in designing a higher-order uncertainty metric for deep neural networks and investigate its performance on the out-of-distribution detection task proposed by [@hendrycks2016baseline]. Our method first assumes there exists an underlying higher-order distribution $\mathbb{P}(z)$, which generated label-wise distribution $\mathbb{P}(y)$ over classes on the K-dimension simplex, and then approximate such higher-order distribution via parameterized posterior function $p_{\theta}(z|x)$ under variational inference framework, finally we use the entropy of learned posterior distribution $p_{\theta}(z|x)$ as uncertainty measure to detect out-of-distribution examples. However, we identify the overwhelming over-concentration issue in such a framework, which greatly hinders the detection performance. Therefore, we further design a log-smoothing function to alleviate such issue to greatly increase the robustness of the proposed entropy-based uncertainty measure. Through comprehensive experiments on various datasets and architectures, our proposed variational Dirichlet framework with entropy-based uncertainty measure is consistently observed to yield significant improvements over many baseline systems.'
author:
- 'First Author$^1$[^1]'
- Second Author$^2$
- |
Third Author$^{2,3}$Fourth Author$^4$\
$^1$First Affiliation\
$^2$Second Affiliation\
$^3$Third Affiliation\
$^4$Fourth Affiliation\
{first, second}@example.com, [email protected], [email protected]
bibliography:
- 'ijcai19.bib'
title: 'IJCAI–19 Example on typesetting multiple authors'
---
Introduction
============
Recently, deep neural networks [@lecun2015deep] have surged and replaced the traditional machine learning algorithms to demonstrate its potentials in many real-life applications like speech recognition [@hannun2014deep], image classification [@deng2009imagenet; @he2016deep], and machine translation [@wu2016google; @vaswani2017attention], reading comprehension [@rajpurkar2016squad], etc. However, unlike the traditional machine learning algorithms like Gaussian Process, Logistic Regression, etc, deep neural networks are very limited in their capability to measure their uncertainty over the unseen test cases and tend to produce over-confident predictions. Such over-confidence issue [@amodei2016concrete; @zhang2016understanding] is known to be harmful or offensive in real-life applications. Even worse, such models are prone to adversarial attacks and raise concerns in AI safety [@DBLP:journals/corr/GoodfellowSS14; @moosavi2016deepfool]. Therefore, it is very essential to design a robust and accurate uncertainty metric in deep neural networks in order to better deploy them into real-world applications. Recently, An out-of-distribution detection task has been proposed in [@hendrycks2016baseline] as a benchmark to promote the uncertainty research in the deep learning community. In the baseline approach, a simple method using the highest softmax score is adopted as the indicator for the model’s confidence to distinguish in- from out-of-distribution data. Later on, many follow-up algorithms [@liang2017enhancing; @lee2017training; @shalev2018out; @devries2018learning] have been proposed to achieve better performance on this benchmark. In ODIN [@liang2017enhancing], the authors follow the idea of temperature scaling and input perturbation [@pereyra2017regularizing; @hinton2015distilling] to widen the distance between in- and out-of-distribution examples. Later on, adversarial training [@lee2017training] is introduced to explicitly introduce boundary examples as negative training data to help increase the model’s robustness. In [@devries2018learning], the authors proposed to directly output a real value between \[0, 1\] as the confidence measure. The most recent paper [@shalev2018out] leverages the semantic dense representation into the target labels to better separate the label space and uses the cosine similarity score as the confidence measure.
These methods though achieve significant results on out-of-distribution detection tasks, they conflate different levels of uncertainty as pointed in [@malinin2018predictive]. For example, when presented with two pictures, one is faked by mixing dog, cat and horse pictures, the other is a real but unseen dog, the model might output same belief as {cat:34%, dog:33%, horse:33%}. Under such scenario, the existing measures like maximum probability or label-level entropy [@liang2017enhancing; @shalev2018out; @hendrycks2016baseline] will misclassify both images as from out-of-distribution because they are unable to separate the two uncertainty sources: whether the uncertainty is due to the data noise (class overlap) or whether the data is far from the manifold of training data. More specifically, they fail to distinguish between the lower-order (aleatoric) uncertainty [@gal2016uncertainty], and higher-order (episdemic) uncertainty [@gal2016uncertainty], which leads to their inferior performances in detecting out-domain examples.
![An intuitive explanation of higher-order distribution and lower-order distribution and their uncertainty measures.[]{data-label="fig:higher-lower"}](uncertainty){width="1.0\linewidth"}
In order to resolve the issues presented by lower-order uncertainty measures, we are motivated to design an effective higher-order uncertainty measure for out-of-distribution detection. Inspired by Subjective Logic [@DBLP:books/sp/Josang16; @yager2008classic; @sensoy2018evidential], we first view the label-wise distribution $\mathbb{P}(y)$ as a K-dimensional variable $z$ generated from a higher-order distribution $\mathbb{P}(z)$ over the simplex $\mathbb{S}_k$, and then study the higher-order uncertainty by investigating the statistical properties of such underlying higher-order distribution. Under a Bayesian framework with data pair $D=(x, y)$, we propose to use variational inference to approximate such “true" latent distribution $\mathbb{P}(z)=p(z|y)$ by a parameterized Dirichlet posterior $p_{\theta}(z|x)$, which is approximated by a deep neural network. Finally, we compute the entropy of the approximated posterior for out-of-distribution detection. However, we have observed an overwhelming over-concentration problem in our experiments, which is caused by over-confidence problem of the deep neural network to greatly hinder the detection accuracy. Therefore, we further propose to smooth the Dirichlet distribution by a calibration algorithm. Combined with the input perturbation method [@liang2017enhancing; @krizhevsky2009learning], our proposed variational Dirichlet framework can greatly widen the distance between in- and out-of-distribution data to achieve significant results on various datasets and architectures.
The contributions of this paper are described as follows:
- We propose a variational Dirichlet algorithm for deep neural network classification problem and define a higher-order uncertainty measure.
- We identify the over-concentration issue in our Dirichlet framework and propose a smoothing method to alleviate such problem.
Model
=====
In this paper, we particularly consider the image classification problem with image input as $x$ and output label as $y$. By viewing the label-level distribution $\mathbb{P}(y)=[p(y=\omega_1), \cdots, p(y=\omega_k)]$ as a random variable $z=\{z \in \mathbb{R}^k:\sum_{i=1}^k z_i=1\}$ lying on a K-dimensional simplex $\mathbb{S}_k$, we assume there exists an underlying higher-order distribution $\mathcal{P}(z)$ over such variable $z$. As depicted in , each point from the simplex $\mathbb{S}_k$ is itself a categorical distribution $\mathbb{P}(y)$ over different classes. The high-order distribution $\mathbb{P}(z)$ is described by the probability over such simplex $\mathbb{S}_k$ to depict the underlying generation function. By studying the statistical properties of such higher-order distribution $\mathbb{P}(z)$, we can quantitatively analyze its higher-order uncertainty by using entropy, mutual information, etc. Here we consider a Bayesian inference framework with a given dataset $D$ containing data pairs $(x, y)$ and show the plate notation in , where $x$ denotes the observed input data (images), $y$ is the groundtruth label (known at training but unknown as testing), and $z$ is latent variable higher-order variable. We assume that the “true" posterior distribution is encapsulated in the partially observable groundtruth label $y$, thus it can be viewed as $\mathbb{P}(z)=p(z|y)$. During test time, due to the inaccessibility of $y$, we need to approximate such “true" distribution with the given input image $x$. Therefore, we propose to parameterize a posterior model $p_{\theta}(z|x)$ and optimize its parameters to approach such “true" posterior $p(z|y)$ given a pairwise input $(x, y)$ by minimizing their KL-divergence $D_{KL}(p_{\theta}(z|x)||p(z|y))$. With the parameterized posterior $p_{\theta}(z|x)$, we are able to infer the higher-order distribution over $z$ given an unseen image $x^*$ and quantitatively study its statistical properties to estimate the higher-order uncertainty.
[l]{}[0.25]{} {width="1.0\linewidth"}
In order to minimize the KL-divergence $D_{KL}(p_{\theta}(z|x)||p(z|y))$, we leverage the variational inference framework to decompose it into two components as follows (details in appendix): $$\begin{aligned}
D_{KL}(p_{\theta}(z|x)||p(z|y)) = -\loss(\theta) + \log p(y)\end{aligned}$$ where $\loss(\theta)$ is better known as the variational evidence lower bound, and $\log p(y)$ is the marginal likelihood over the label $y$. $$\begin{aligned}
\loss(\theta) = \expect{z \sim p_{\theta}(z|x)}[\log p(y|z)] - D_{KL}(p_{\theta}(z|x)||p(z))\end{aligned}$$ Since the marginal distribution $p(y)$ is constant w.r.t $\theta$, minimizing the KL-divergence $D_{KL}(p_{\theta}(z|x)||\true)$ is equivalent to maximizing the evidence lower bound $\loss(\theta)$. Here we propose to use Dirichlet family to realize the higher-order distribution $p_{\theta}(z|x) = Dir(z|\alpha)$ due to its tractable analytical properties. The probability density function of Dirichlet distribution over all possible values of the K-dimensional stochastic variable $z$ can be written as: $$\begin{aligned}
Dir(z|\alpha) = \begin{cases}
\frac{1}{B(\alpha)}\prod_{i=1}^K z_i^{\alpha_i - 1} \quad &for \quad z \in \mathbb{S}_k\\
0 \qquad & otherwise,
\end{cases}\end{aligned}$$ where $\alpha$ is the concentration parameter of the Dirichlet distribution and $B(\alpha)=\frac{\prod_i^K \Gamma(\alpha_i)}{\Gamma(\sum_i^k \alpha_i)}$ is the normalization factor. Since the LHS (expectation of log probability) has a closed-formed solution, we rewrite the empirical lower bound on given dataset $D$ as follows: $$\begin{aligned}
\loss(\theta) = \sum_{(x, y) \in D} [\psi(\alpha_y) - \psi(\alpha_0) - D_{KL}(Dir(z|\alpha)||p(z))]\end{aligned}$$ where $\alpha_0$ is the sum of concentration parameter $\alpha$ over K dimensions. However, it is in general difficult to select a perfect model prior to craft a model posterior which induces an the distribution with the desired properties. Here, we assume the prior distribution is as Dirichlet distribution $Dir(\hat{\alpha})$ with concentration parameters $\hat{\alpha}$ and specifically talk about three intuitive prior functions in .
![An intuitive explanation of different prior functions.[]{data-label="fig:priors_funcs"}](priors){width="1.0\linewidth"}
The first uniform prior aggressively pushes all dimensions towards 1, while the \*-preserving priors are less strict by allowing one dimension of freedom in the posterior concentration parameter $\alpha$. This is realized by copying the value from $k_{th}$ dimension of posterior concentration parameter $\alpha$ to the uniform concentration to unbind $\alpha_k$ from KL-divergence computation. Given the prior concentration parameter $\hat{\alpha}$, we can obtain a closed-form solution for the evidence lower bound as follows: $$\begin{gathered}
\loss(\theta) = \sum_{(x, y) \in D} [\psi(\alpha_y) - \psi(\alpha_0) - \log \frac{B(\hat{\alpha})}{B(\alpha)} - \sum_{i=1}^k ({\alpha}_i - \hat{\alpha}_i)(\psi(\alpha_i) - \psi(\alpha_0))\end{gathered}$$ $\Gamma$ denotes the gamma function, $\psi$ denotes the digamma function. We write the derivative of $\loss(\theta)$ w.r.t to parameters $\theta$ based on the chain-rule:$\frac{\partial \loss}{\partial \theta} = \frac{\partial \loss}{\partial \alpha} \odot \alpha \boldsymbol{\cdot} \frac{\partial f_{\theta}(x)}{\partial \theta}$, where $\odot$ is the Hardamard product and $\frac{\partial f_{\theta}(x)}{\partial \theta}$ is the Jacobian matrix. In practice, we parameterize $Dir(z|\alpha)$ via a neural network with $\alpha = f_{\theta}(x)$ and re-weigh the two terms in $\loss(\theta)$ with a balancing factor $\eta$. Finally, we propose to use mini-batch gradient descent to optimize the network parameters $\theta$ as follows: $$\begin{aligned}
\frac{\partial \loss}{\partial \alpha} = \sum_{(x, y) \in B(x, y)} [\frac{\partial [\psi(\alpha_y) - \psi(\alpha_0)]}{\partial \alpha} + \eta \frac{\partial D_{KL}(Dir(z|\alpha)||Dir(z|\hat{\alpha}))}{\partial \alpha}]\end{aligned}$$ where $B(x,y)$ denotes the mini-batch in dataset $D$. During inference time, we use the marginal probability of assigning given input $x$ to certain class label $i$ as the classification evidence: $$p(y=i|x)=\int_{z}p(y=i|z)p_{\theta}(z|x) dz= \frac{\alpha_i}{\sum_{j=1}^k \alpha_j}$$ Therefore, we can use the maximum $\alpha$’s index as the model prediction class during inference $\hat{y} = \argmax_i p(y=i|x) = \argmax_i \alpha_i$.
Uncertainty Measure
===================
After optimization, we obtain a parametric Dirichlet function $p_{\theta}(z|\alpha)$ and compute its entropy $E$ as the higher-order uncertainty measure. Formally, we write the such metric as follows: $$\begin{aligned}
E(\alpha) = -C(\alpha) = -\int_{z} z Dir(z|\alpha)dz = \log B(\alpha) + (\alpha_0 - K) \psi(\alpha_0) - \sum_i^k (\alpha_i - 1)\psi(\alpha_i)\end{aligned}$$ where $\alpha$ is computed via the deep neural network $f_{\theta}(x)$. Here we use negative of entropy as the confidence score $C(\alpha)$. By investigating the magnitude distribution of concentration parameter $\alpha$ for in-distribution test cases, we can see that $\alpha$ is either adopting the prior $\alpha=1.0$ or adopting a very large value $\alpha \gg 1.0$. In order words, the Dirichlet distribution is heavily concentrated at a corner of the simplex regardless of whether the inputs are from out-domain region, which makes the model very sensitive to out-of-distribution examples leading to compromised detection accuracy. In order to resolve such issue, we propose to generally decrease model’s confidence by smoothing the concentration parameters $\alpha$, the smoothing function can lead to opposite behaviors in the uncertainty estimation of in- and out-of-distribution data to enlarge their margin.
#### Concentration smoothing
In order to construct such a smoothing function, we experimented with several candidates and found that the log-smoothing function $\hat{\alpha} = \log(\alpha + 1)$ can achieve generally promising results. By plotting the histogram of concentration magnitude before and after log-scaling in , we can observe a very strong effect in decreasing model’s overconfidence, which in turn leads to clearer separation between in- and out-of-distribution examples (depicted in ). In the experimental section, we detail the comparison of different smoothing functions to discuss its impact on the detection accuracy.
![Concentration and confidence distribution before and after smoothing for CIFAR10 under VGG13 architecture with iSUN as out-of-distribution dataset.[]{data-label="fig:smoothing_dir"}](smoothed_dirichlet){width="1.0\linewidth"}
#### Input Perturbation
Inspired by fast gradient sign method [@DBLP:journals/corr/GoodfellowSS14], we propose to add perturbation in the data before feeding into neural networks: $$\begin{aligned}
\hat{x} = x - \epsilon * sign(\nabla_x [\psi(\alpha_0) - \psi(\alpha_y)])\end{aligned}$$ where the parameter $\epsilon$ denotes the magnitude of the perturbation, and $(x, y)$ denotes the input-label data pair. Here, similar to [@liang2017enhancing] our goal is also to improve the entropy score of any given input by adding belief to its own prediction. Here we make a more practical assumption that we have no access to any form of out-of-distribution data. Therefore, we stick to a rule-of-thumb value $\epsilon=0.01$ throughout our experiments.
#### Detection
For each input $x$, we first use input perturbation to obtain $\hat{x}$, then we feed it into neural network $f_{\theta}(\hat{x})$ to compute the concentration $\alpha$, finally we use log-scaling to calibrate $\alpha$ and compute $C(\hat{\alpha})$. Specifically, we compare the confidence $C(\hat{\alpha})$ to the threshold $\delta$ and say that the data $x$ follows in-distribution if the confidence score $C(\hat{\alpha})$ is above the threshold and that the data $x$ follows out-of-distribution, otherwise.
Experiments
===========
In order to evaluate our variational Dirichlet method on out-of-distribution detection, we follow the previous paper [@hendrycks2016baseline; @liang2017enhancing] to replicate their experimental setup. Throughout our experiments, a neural network is trained on some in-distribution datasets to distinguish against the out-of-distribution examples represented by images from a variety of unrelated datasets. For each sample fed into the neural network, we will calculate the Dirichlet entropy based on the output concentration $\alpha$, which will be used to predict which distribution the samples come from. Finally, several different evaluation metrics are used to measure and compare how well different detection methods can separate the two distributions.
In-distribution and Out-of-distribution dataset
-----------------------------------------------
These datasets are all available in Github [^2].
- In-distribution: CIFAR10/100 [@krizhevsky2009learning] and SVHN [@netzer2011reading], which are both comprised of RGB images of $32 \times 32$ pixels.
- Out-of-distribution: TinyImageNet [@deng2009imagenet], LSUN [@yu2015lsun] and iSUN [@xiao2010sun], these images are resized to $32 \times 32$ pixels to match the in-distribution images.
Before reporting the out-of-distribution detection results, we first measure the classification accuracy of our proposed method on the two in-distribution datasets in , from which we can observe that our proposed algorithm has minimum impact on the classification accuracy.
Training Details
----------------
In order to make fair comparisons with other out-of-distribution detectors, we follow the same setting of [@liang2017enhancing; @DBLP:conf/bmvc/ZagoruykoK16; @devries2018learning; @shalev2018out] to separately train WideResNet [@DBLP:conf/bmvc/ZagoruykoK16] (depth=16 and widening factor=8 for SVHN, depth=28 and widening factor=10 for CIFAR100), VGG13 [@simonyan2014very], and ResNet18 [@he2016deep] models on the in-distribution datasets. All models are trained using stochastic gradient descent with Nesterov momentum of 0.9, and weight decay with 5e-4. We train all models for 200 epochs with 128 batch size. We initialize the learning with 0.1 and reduced by a factor of 5 at 60th, 120th and 180th epochs. we cut off the gradient norm by 1 to prevent from potential gradient exploding error. We save the model after the classification accuracy on validation set converges and use the saved model for out-of-distribution detection.
Experimental Results
--------------------
We measure the quality of out-of-distribution detection using the established metrics for this task [@hendrycks2016baseline; @liang2017enhancing; @shalev2018out]. (1) FPR at 95% TPR (lower is better): Measures the false positive rate (FPR) when the true positive rate (TPR) is equal to 95%. (2) Detection Error (lower is better): Measures the minimum possible misclassification probability defined by $\min_{\delta} \{0.5P_{in}(f(x) \leq \delta) + 0.5P_{out}(f(x) > \delta)\}$. (3) AUROC (larger is better): Measures the Area Under the Receiver Operating Characteristic curve. The Receiver Operating Characteristic (ROC) curve plots the relationship between TPR and FPR. (4) AUPR (larger is better): Measures the Area Under the Precision-Recall (PR) curve, where AUPR-In refers to using in-distribution as positive class and AUPR-Out refers to using out-of-distribution as positive class.
We report our VGG13’s performance in and ResNet/WideResNet’s performance in under groundtruth-preserving prior, where we list the performance of Baseline [@hendrycks2016baseline], ODIN [@liang2017enhancing], Bayesian Neural Network [@gal2016uncertainty][^3], Semantic-Representation [@shalev2018out] and Learning-Confidence [@devries2018learning]. The results in both tables have shown remarkable improvements brought by our proposed variational Dirichlet framework. For CIFAR datasets, the achieved improvements are very remarkable, however, the FPR score on CIFAR100 is still unsatisfactory with nearly half of the out-of-distribution samples being wrongly detected. For the simple SVHN dataset, the current algorithms already achieve close-to-perfect results, therefore, the improvements brought by our algorithm is comparatively minor.
Ablation Study
--------------
In order to individually study the effectiveness of our proposed methods (entropy-based uncertainty measure, concentration smoothing, and input perturbation), we design a series of ablation experiments in . From which, we could observe that concentration smoothing has a similar influence as input perturbation, the best performance is achieved when combining these two methods.
![Ablation experiments for VGG13 architecture to investigate the impact of our proposed smoothing with CIFAR10 as in-distribution dataset and iSUN/LSUN as out-of-distribution dataset.[]{data-label="fig:ablation_strat"}](ablation_strategy){width="1.0\linewidth"}
Here we mainly experiment with four different priors and depict our observations in . From which, we can observe that the non-informative uniform prior is too strong assumption in terms of regularization, thus leads to inferior detection performances. In comparison, giving model one dimension of freedom is a looser assumption, which leads to generally better detection accuracy. Among these two priors, we found that preserving the groundtruth information can generally achieve slightly better performance, which is used through our experiments.
![Impact of different prior distributions. The network architecture is VGG13 with CIFAR10 as in-distribution dataset and iSUN/LSUN as out-of-distribution dataset. []{data-label="fig:priors"}](ablation_prior){width="1.0\linewidth"}
We also investigate the impact of different smoothing functions on the out-of-distribution detection accuracy. For smoothing functions, we mainly consider the following function forms: $\sqrt{x}$, $\sqrt[3]{x}$, $log(1+x)$, $x$, $x^2$, $Sigmoid(x)$ and $SoftSign(x)$. Here we use $Sigmoid(x), SoftSign(x)$ (range=$[0, 1]$) as baselines to investigate the impact of the range of smoothing function on the detection accuracy, and use $x, x^2$ as baselines to investigate the impact of compression capability of smoothing function on the detection accuracy. From , we can observe that the first three smoothing functions greatly outperforms the baseline functions. Therefore, we can conclude that the smoothing function should adopt two important characteristics: 1) the smoothing function should not be bounded, i.e. the range should be $[0, \infty]$. 2) the large values should be compressed.
![Impact of different smoothing functions. The network architecture is VGG13 with in-distribution CIFAR10 dataset and out-of-distribution iSUN/LSUN dataset.[]{data-label="fig:calibration"}](calibration.pdf){width="1.0\linewidth"}
Related Work
============
The novelty detection problem [@pimentel2014review] has already a long-standing research topic in traditional machine learning community, the previous works [@vincent2003manifold; @ghoting2008fast; @schlegl2017unsupervised] have been mainly focused on low-dimensional and specific tasks. Their methods are known to be unreliable in high-dimensional space. Recently, more research works about detecting an anomaly in deep learning like [@akcay2018ganomaly] and [@lee2017training], which propose to leverage adversarial training for detecting abnormal instances. In order to make the deep model more robust to abnormal instances, different approaches like [@bekker2016training; @xiao2015learning; @li2017learning; @lathuiliere2018deepgum] have been proposed to increase deep model’s robustness against outliers during training. Another line of research is Bayesian Networks [@gal2016dropout; @gal2015bayesian; @gal2016uncertainty; @kingma2015variational], which are powerful in providing stochasticity in deep neural networks by assuming the weights are stochastic. However, Bayesian Neural Networks’ uncertainty measure like variational ratio and mutual information rely on Monte-Carlo estimation, where the networks have to perform forward passes many times, which greatly reduces the detection speed.
Conclusion
==========
In this paper, we aim at finding an effective way for deep neural networks to express their uncertainty over their output distribution. Our variational Dirichlet framework is empirically demonstrated to yield better results, but its detection accuracy on a more challenging setup like CIFAR100 is still very compromised. We conjecture that better prior Dirichlet distribution or smoothing function could help further improve the performance. In the future work, we plan to apply our method to broader applications like natural language processing tasks or speech recognition tasks.
Derivation
==========
Here we want to approximate the parameterized function towards to true distribution over latent variable $z$, which we write as: $$\begin{aligned}
\begin{split}
&D_{KL}(p_{\theta}(z|x)||\mathbb{P}(z))\\
=&D_{KL}(p_{\theta}(z|x)||p(z|y)\\
=& \int_z p_{\theta}(z|x)\log \frac{p_{\theta}(z|x)}{p(z|y)} dz\\
=& \int_z p_{\theta}(z|x)\log \frac{p_{\theta}(z|x)p(y)}{p(z, y)} dz\\
=& \int_z p_{\theta}(z|x)\log \frac{p_{\theta}(z|x)}{p(z, y)} dz + \log p(y)\\
=& \int_z p_{\theta}(z|x)\log \frac{p_{\theta}(z|x)}{p(z)p(y|z)} dz + \log p(y)\\
=& \int_z p_{\theta}(z|x)\log \frac{p_{\theta}(z|x)}{p(z)}dz - \int_z p_{\theta}(z|x) \log p(y|z)dz + \log p(y)\\
=& KL(p_{\theta}(z|x)||p(z)) -\expect{z \sim p_{\theta}(z|x)} [\log p(y|z)] + \log p(y)\\
=& -[\expect{z \sim p_{\theta}(z|x)} [\log p(y|z)] - KL(p_{\theta}(z|x)||p(z))] + \log p(y)\\
=& -\loss(\theta) + \log p(y)
\end{split}\end{aligned}$$
Detailed Dataset
================
- CIFAR10/100 (in-distribution): The CIFAR-10 and CIFAR100 dataset [@krizhevsky2009learning] consists of RGB images of $32 \times 32$ pixels. Each image is classified into 10/100 classes, such as dog, cat, automobile, or ship. The training split for both datasets is comprised of 50,000 images, while the test split is comprised of 10,000 images.
- SVHN (in-distribution): The Street View Housing Numbers (SVHN) dataset [@netzer2011reading] consists of colored housing number pictures ranging from 0 to 9. Images are also with a resolution of $32 \times 32$. The official training split is comprised of 73,257 images, and the test split is comprised of 26,032 images.
- TinyImageNet (out-of-distribution): The TinyImageNet dataset2 is a subset of the ImageNet dataset [@deng2009imagenet]. The test set for TinyImageNet contains 10,000 images from 200 different classes for creating the out-of-distribution dataset, it contains the original images, downsampled to $32 \times 32$ pixels.
- LSUN (out-of-distribution): The Large-scale Scene UNderstanding dataset (LSUN) [@yu2015lsun] has a test set consisting of 10,000 images from 10 different scene classes, such as bedroom, church, kitchen, and tower. We downsample LSUN’s original image and create $32 \times 32$ images as an out-of-distribution dataset.
- iSUN (out-of-distribution): The iSUN dataset [@xiao2010sun] is a subset of the SUN dataset, containing 8,925 images. All images are downsampled to $32 \times 32$ pixels.
Effects on KL-divergence
========================
Here we investigate the impact of KL-divergence in terms of both classification accuracy and detection errors. By gradually increasing the weight of KL loss (increasing the balancing factor $\eta$ from 0 to 10), we plot their training loss curve in . With a too strong KL regularization, the model’s classification accuracy will decrease significantly. As long as $\eta$ is within a rational range, the classification accuracy will become stable. For detection error, we can see from that adopting either too large value or too small value can lead to compromised performance. For the very small value $\eta \rightarrow 0$, the variational Dirichlet framework degrades into a marginal log-likelihood, where the concentration parameters are becoming very erratic and untrustworthy uncertainty measure without any regularization. For larger $\eta > 1$, the too strong regularization will force both in- and out-of-distribution samples too close to prior distribution, thus erasing the difference between in- and out-of-distribution becomes and leading to worse detection performance. We find that adopting a hyper-parameter of $\eta=0.01$ can balance the stability and detection accuracy.
![The training loss curve under ResNet18 on CIFAR10 dataset for different $\eta$ is demonstrated on the left side, the accuracy and out-of-distirbution detection results on the right side.[]{data-label="fig:training_loss"}](Training_loss){width="1.0\linewidth"}
Results on CIFAR100
===================
Here we particularly investigate the out-of-distribution detection results of our model on CIFAR100 under different scenarios. Our experimental results are listed in .
[^1]: Contact Author
[^2]: <https://github.com/ShiyuLiang/odin-pytorch>
[^3]: We use the variational ratio as uncertainty measure to perform out-of-distribution detection, specifically, we forward Bayesian deep network 100 times for each input sample for Monte-Carlo estimation.
|
---
abstract: |
One proves the uniqueness of distributional solutions to nonlinear Fokker–Planck equations with monotone diffusion term and derive as a consequence (restricted) uniqueness in law for the corresponding McKean–Vlasov stochastic differential equation (SDE).\
[**Mathematics Subject Classification (2000):**]{} 60H30, 60H10, 60G46, 35C99.\
[**Keywords:**]{} Fokker–Planck equation, mild solution, distributional solution.
author:
- 'Viorel Barbu[^1]'
- 'Michael Röckner[^2] [^3]'
title: 'Uniqueness for nonlinear Fokker–Planck equations and weak uniqueness for McKean-Vlasov SDEs'
---
Introduction {#s1}
============
Consider the nonlinear equation $$\label{e1.1}
\barr{ll}
u_t-\Delta\beta(u)+{\rm div}(b(x,u)u)=0\mbox{ in }\cald'((0,\9)\times\rr^d),\vsp
u(0,x)=u_0(x),
\earr$$ where $\b:\rr\to\rr$ and $b:\rrd\times\rr\to\rrd$ satisfy the following assumptions
- [*$\beta(0)=0$, $\b\in C^1(\rr)$, and $$\g_0|r_1-r_2|^2\le(\b(r_1)-\b(r_2))(r_1-r_2),\ r_1,r_2\in\rr,\label{e1.2}$$ where $0<\g_0<\9.$* ]{}
- $b^i\in C_b(\rrd\times\rr)\cap C^1(\rrd\times\rr)$, $b^i(x,0)\equiv0$, $x\in\rrd$, $$\mbox{$\sup\{|b^i_r(x,r)|;x\in\rrd,i=1,2,$ $|r|\le M\}\le C_M,$ $\ff M>0,$}$$[*and, for $$\delta(r):=\sup\{|b_x(x,r)|;\ x\in\rrd\},$$we have*]{} $\delta\in C_b(\rr).$
Here$$b(x,u)=\{b^i(x,u)\}^d_{i=1}\mbox{ and }b^i_r=\frac{\pp b^i}{\pp r}.$$ By a distributional solution (in the sense of Schwartz) with initial condition $u_0\in L^1$ we mean a function $u:[0,\9)\to L^1(\rrd)$ such that $(u(t,\cdot)dx)_{t\in[0,T]}$ is narrowly continuous, that is, $$\begin{aligned}
&&\lim_{t\to s}\int_{\rrd}u(t,x)\psi(x)dx=\int_{\rrd}u(s,x)\psi(x)dx,\ \ff\psi\in C_b(\rrd),\ s\ge0,\qquad\label{e1.3}\\[2mm]
&&\dd\int^\9_0\int_{\rr^d}(u(t,x)\vf_t(t,x)+\b(u(t,x))\D\vf(t,x)\label{e1.4}\\
&&\qquad+b(x,u(t,x))u(t,x))\cdot\nabla_x\vf(t,x))dt\,dx=0,\nonumber\\
&&\qquad\qquad\qquad\qquad\qquad\qquad\ff\vf\in C^\9_0((0,\9)\times\rr^d)\nonumber\\[2mm]
&&u(0,x)=u_0(x),\mbox{ a.e. }x\in\rr^d.\label{e1.5}\end{aligned}$$ (In the following, we shall use the notation $b^*(x,u)=b(x,u)u.$)
In [@1] it was proved, in particular, that, if (i)–(ii) hold and, in addition, for $\Phi(u)\equiv\frac{\beta(u)}u,$ $u\in\rr,$ we have $\Phi\in C^2(\rr)$, then there is a mild solution $u\in C([0,\9);L^1(\rr^d))$ for each $u_0\in L^1(\rr^d)$. The mild solution $u$ is defined as $$u(t)=\lim_{h\to0}u_h(t)\mbox{ in }L^1(\rr^d),\ \ff t\ge0,$$where $$\label{e1.6}
\barr{l}
u_h(t)=u^i_h\mbox{ for }t\in[ih,(i+h)h],\ i=0,1,...,Nh=T,\vsp
u^{i+1}_h-h\D\b(u^{i+1}_h)+h\,{\rm div}(b(x,u^{i+1}_h)u^{i+1}_h)=u^i_h\mbox{ in }\cald'(\rr^d),\\\hfill i=0,1,...,\vsp
u^0_h=u_0.\earr$$Moreover, $S(t)u_0=u(t),\ t\ge0,$ is a strongly continuous semigroup of nonexpansive mappings in $L^1(\rrd)$.
As easily seen, any mild solution is a distributional solution but the uniqueness follows in the class of mild solutions only. Here, we shall prove the uniqueness for in the class of distributional solutions and derive from this result the uniqueness in law of solutions to McKeen–Vlasov SDE $$\label{e1.7}
dX(t)=b(X(t),u(t,X(t)))dt+\frac1{\sqrt{2}}\(\frac{\b(u(t,X(t))}{u(t,X(t))}\)^{\frac12}dW(t).$$
Denote by $L^p(\rr^d)=L^p$ the space of $p$-summable functions on $L^p$, with the norm denoted $|\cdot|_p$. By $H^k(\rrd)=H^k$, $k=1,2$, and $H^{-k}(\rrd)=H^{-k},$ we denote the standard Sobolev spaces on $\rrd$ and by $C_b(\rrd)$ the space of continuous and bounded functions on $\rrd$. By $C^k(\rrd)$ we denote the space of continuously differentiable functions on $\rrd$ of order $k$, by $C^1_b(\rrd)$ the space $\left\{u\in C^1(\rrd);\frac{\pp u}{\pp y}\in C_b(\rrd),\ j=1,...,d\right\}.$ The spaces of continuous and differentiable functions on $(0,T)\times\rrd$ are denoted in a similar way and we shall simply write $$C^1_b(\rrd)=C^1_b,\ C^k(\rrd)=C^k,\ k=1,2.$$ The scalar product in $L^2$ is denoted $\<\cdot,\cdot\>_2$ and by ${}_{H^{-1}}\!\<\cdot,\cdot\>_{H^1}$ the pairing between $H^1$ and $H^{-1}.$ Of course, on $L^2\times L^2$ this coincides with $\<\cdot,\cdot\>_2.$ The scalar product $\<\cdot,\cdot\>_{-1}$ on $H\1$ is taken as $$\label{e1.8}
\<u,v\>_{-1}=((I-\D)\1u,v)_2,\ \ff u,v\in H\1$$with the corresponding norm $$\label{e1.9}
|u|_{-1}=(\<u,u\>_{-1})^{\frac12},\ u\in H\1.$$ By $\cald'((0,\9)\times\rrd)$ and $\cald'(\rrd)$ we denote the space of Schwartz distributions on $(0,\9)\times\rrd$ and $\rrd$, respectively. If $\calx$ is a Banach space, we denote by $W^{1,2}([0,T];\calx)$ the infinite dimensional Sobolev space $\{y\in L^2(0,T;\calx);$ $\frac{dy}{dt}\in L^2(0,T;\calx)\}$, where $\frac d{dt}$ is taken in the sense of vectorial distributions. We also set, for each $z\in C^1(\rrd\times\rr)$, $$z_r(x,r)=\frac\pp{\pp r}\ z(x,r),\ \ z_x=\nabla_x z(x,r).$$ We shall denote the norms on $\rrd$ and $\rr$ by the same symbol $|\cdot|$.
The main result
===============
The next result is a uniqueness theorem for distributional solutions $u$ to . In the special case $b\equiv0$, such a uniqueness result for was established earlier in [@3] for continuous and monotonically nondecreasing functions $\b$.
\[t1\] Let $T>0$ and let conditions [(i)–(ii)]{} on $\beta$ and $b$ hold. For each $u_0\in L^\9\cap L^1$, the equation has at most one distributional solution $u\in L^\9((0,T);L^1)\cap L^\9((0,T)\times\rrd).$
Let $u_1,u_2\in L^\9(0,T;L^1)\cap L^\9((0,T)\times\rrd)$ be two distributional solutions to and let $u=u_1-u_2.$ We have $$\label{e2.1}
\barr{l}
u_t-\D(\b(u_1)-\b(u_2))+{\rm div}(b^*(x,u_1)-b^*(x,u_2))=0\\\hfill\mbox{ in }\cald'((0,\9)\times\rrd)\vsp
u(0,x)=0.\earr$$ (Here, $b^*(x,r)=b(x,r)r,\ \ff x\in\rrd,\ r\in\rr.$)
It should be mentioned that, by , it follows that $u_i,\beta(u_i)\in L^2((0,T);L^2),$ $i=1,2$, and, therefore, $u\in W^{1,2}([0,T];H^{-2}).$
Consider the operator $\Gamma:H\1\to H^1$ defined by $$\Gamma u=(1-\D)^{-1}u,\ u\in H\1(\rrd)$$and note that $\Gamma$ is an isomorphism of $H\1$ onto $H^1$ and $\Gamma\in L(H^{-2},L^2).$ Since $u_i\in L^2(0,T;L^2),$ $i=1,2,$ it follows that $y=\Gamma u\in L^2(0,T;H^2)\cap W^{1,2}([0,T];L^2)$ and so, by , we have $$\label{e2.2}
\barr{l}
\dd\frac{dy}{dt}-\Gamma\D(\b(u_1)-\b(u_2))+\Gamma\,{\rm div}(b^*(x,u_1)-b^*(x,u_2))=0,\\\hfill\mbox{ a.e. }t\in(0,T),\\
y(0)=0,\earr$$ where $\frac{dy}{dt}\in L^2(0,T;L^2).$ (We note that here $\frac{dy}{dt}$ is taken in the sense of $L^2$-valued vectorial distributions on $(0,T)$ and so $y:[0,T]\to L^2$ is absolutely continuous.) Hence, $u:[0,T]\to H^{-2}$ is absolutely continuous.
Now, we take the scalar product in $L^2$ of with $u=u_1-u_2$. Taking into account that $$\<\frac{dy}{dt}\,(t),y(t)\>_2=\frac12\ \frac d{dt}\,|y(t)|^2_{2}, \mbox{ a.e. }t\in(0,T),$$we get, by – that $$\barr{l}
\dd\frac 12\ \frac d{dt}\,|u(t)|^2_{-2}+\<\b(u_1)-\b(u_2),u_1-u_2\>_2
=\<\Gamma(\b(u_1)-\b(u_2)),u_1-u_2\>_2\vsp
\qquad-\<\Gamma\,{\rm div}((b^*(x,u_1)-b^*(x,u_2)),u_1-u_2\>_2,\ \mbox{a.e. }t\in(0,T),\earr$$where $|\cdot|_{-2}$ is the norm of $H^{-2}$. By , this yields $$\label{e2.3}
\barr{r}
\dd\frac12\ \frac d{dt}\,|u(t)|^2_{-2}+\g_0|u(t)|^2_2
\le\<\b(u_1(t))-\b(u_2(t)),u_1(t)-u_2(t)\>_{-1}\vsp
\qquad-\<{\rm div}((b^*(x,u_1(t))-b^*(x,u_2(t))),u_1-u_2\>_{-1}.\earr$$ We note that $$|\Gamma f|_2\le|f|_2,\ \ff f\in L^2,$$and, therefore, $$\label{e2.4}
|f|_{-1}\le|f|_2,\ \ff f\in L^2.$$ We also have $$\label{e2.5}
|{\rm div}\,F|_{-1}\le2|F|_2,\ \ff F\in (L^2)^d.$$This yields $$\label{e2.6}
\barr{ll}
|\<\b(u_1)-\b(u_2),u_1-u_2\>_{-1}|\!\!\!&\le|\b(u_1)-\b(u_2)|_{2}|u|_{-1}\vsp
&\le\b_M|u|_2|u|_{-1}\le\beta_M|u|^{\frac32}_2\
|u|_{-2}^{\frac12}\earr$$and $$\label{e2.7}
\barr{l}
\left|\<{\rm div}\ (b^*(x,u_1)-b^*(x,u_2)),u_1-u_2\>_{-1}\right|\vsp
\qquad\le
2|(b^*(x,u_1)-b^*(x,u_2))|_2|u_1-u_2|_{-1}\vsp
\qquad\le2(|b|_\9+b_M|u_1|_\9)|u|_2|u|_{-1}\vsp
\qquad\le2(|b|_\9+b_M|u_1|_\9)|u|^{\frac32}_2|u|^{\frac12}_{-2}.
\earr$$ where $M=\max\{|u_1|_\9,|u_2|_\9)$ and $$\barr{lcl}
\b_M&=&\dd\sup\{\b'(r);\ |r|\le M\},\vsp
b_M&=&\dd\sup\left\{\frac{|b(x,u_1)-b(x,u_2)|}{|r_1-r_2|};\ x\in\rrd,\ |r_1|,|r_2|<M\right\}\vsp&\le&
\sup\{|b_r(x,r)|;|r|\le M,\ x\in\rrd\}.\earr$$ (Here, we have used the interpolation inequality $|u|_{-1}\le|u|^{\frac12}_2|u|^{\frac12}_{-2}$.)
By –, we get $$\barr{r}
\dd\frac12\ \frac d{dt}\,|u(t)|^2_{-2}+\g_0|u(t)|^2_2
\le(\b_M+2(|b|_\9+b_M|u_1|_\9)
|u(t)|^{\frac32}_2|u(t)|^{\frac12}_{-2},\\\hfill\mbox{a.e. }t\in(0,T),\earr$$where $|b|_\9=\sup\{|b(x,r)|;x\in\rrd,r\in\rr\}.$ This yields $$\frac d{dt}\,|u(t)|^2_{-2}\le C|u(t)|^2_{-2},\mbox{ a.e. }t\in(0,T).$$Since $u:[0,T]\to H^{-2}$ is absolutely continuous and narrowly continuous, we infer that $|u(t)|_{-2}=0$, $\ff t\in[0,T]$, and so $u\equiv0$, as claimed.$\Box$
As mentioned earlier, under hypotheses (i)–(ii), if $\Phi\in C^2$, where $\Phi(u)\equiv\frac{\beta(u)}u$, $u\in\rr$, then the equation , for each $u_0\in L^1$, has a unique mild solution $u\in C([0,\9);L^1).$ This mild solution is also easily checked to be a distributional solution to . As regards this solution, we also have
\[p2\] Assume that [(i)–(ii)]{} hold, and that, for $\Phi(u)\equiv\frac{\beta(u)}u$,
- $\Phi\in C^2(\rrd).$
Then, for each $u_0\in L^1\cap L^\9$, the mild solution $u$ to satisfies also $$\label{e2.8}
u\in L^\9((0,T)\times\rrd),\ \ff T>0.$$
We rewrite as $$\label{e2.9}
\barr{l}
(u-|u_0|_\9-\alpha(t))_t-\D(\b(u)-\b(|u_0|_\9+\a(t)))\vsp \qquad
+{\rm div}(b^*(x,u)-b^*(x,|u_0|_\9+\a(t)))\vsp
\qquad=-{\rm div}(b^*(x,|u_0|_\9+\a(t)))-\a'(t)\le0\mbox{ in }(0,\9)\times\rrd,
\earr$$where $\a\in C^1([0,\9))$ is chosen in such a way that $$\label{e2.10}
\hspace*{-3mm}\barr{l}
\a'(t)+\sup\{|b_x(x,|u_0|_\9+\a(t))|;x\in\rrd\}(|u_0|_\9+\a(t))=0,\ t\in(0,T),\vsp
\a(0)=0.\earr$$ We may find $\a$ of the form $\a=\eta-|u_0|_\9,$ where $\eta$ is a solution to the equation $$\label{e2.11}
\barr{l}
\eta'-\delta(\eta)\eta =0,\ t\ge0,\vsp
\eta(0)=|u_0|_\9,\earr$$ $\delta(r)=\sup\{|b_x(x,r)|;x\in\rrd\},$ $r\in\rr.$ Clearly, has such a solution $\eta\in C^1([0,\9)),$ $\eta\ge0,$ on $[0,\9)$ because $\delta\in C_b(\rr).$
Formally, if we multiply by ${\rm sign}(u-|u_0|_\9-\a)^+$, integrate over $\rrd$ and use the monotonicity of $\b$, we get by that $$\label{e2.11a}
\frac d{dt}\,|(u(t)-|u_0|_\9-\a(t))^+|_1\le0,\ \mbox{a.e. }t\in(0,T).$$This yields $u(t)\le|u_0|_\9+\a(t)$, $\ff t\ge0$, and similarly it follows that $u(t)\ge-|u_0|_\9-\a(t).$ Hence, $u\in L^\9((0,T)\times\rrd),$ as claimed.
The above formal argument can be made rigorous if $u$ is a strong solution to (which is not the case here). Then (see the detailed argument in [@1]) $$\label{e2.11aa}
\hspace*{-2mm}\barr{l}
\dd\lim_{\delta\to0}\frac1\delta
\int_{[0<(\beta(u)-\beta(|u_0|_\9+\a(t))^+)\le\delta]}
|b^*(x,u)-b^*(x,|u_0|_\9+\a(t))|\,|\nabla u|dx\vsp
\qquad=\dd\lim_{\delta\to0}\frac1\delta
\int_{[0<(\beta(u)-\beta(|u_0|_\9+\a(t))^+)\le\delta]}
(|b(x,u)-b(x,|u_0|_\9+\a(t))|\,|u|
\vsp\qquad+|b(x,|u_0|_\9+\a(t))|\,|u-|u_0|_\9-\a(t)|)|\nabla u|dx=0,\ \ff t\in(0,T),\earr$$which is true if $\nabla u\in L^2(0,T;L^2)$ and $b(x,\cdot)\in{\rm Lip}(\rr)$ uniformly in $x$ (which is the case if $b_u\in C_b(\rrd\times\rr))$. In order to be in such a situation, we approximate by $$\label{e2.11aaa}
\barr{l}
u_t-\D(\b(u)+\vp\b(u)+{\rm div}(b_\vp(x,u)u))=0\mbox{ in }(0,T)\times\rrd,\vsp
u(0,x)=u_0(x),\earr$$where $\vp>0$ and $b_\vp\in C^1_b(\rrd\times\rr)$ is a smooth approximation of $b$. (For instance, $b_\vp=b*\rho_\vp$, where $\rho_\vp$ is a standard mollifier.) Then, as proved earlier in [@1], [@2az], [@2], equation has a unique solution $u_\vp\in L^2(0,T;H^1)\cap C([0,T];L^1)\cap W^{1,2}([0,T];H\1)$ and $u_\vp\to u$ in $C([0,T];L^1)$ as $\vp\to0$. An easy way to prove this is to apply the Trotter–Kato theorem to the family of $m$-accretive operators in $L^1$ $$\barr{lcl}
A_\vp u&=&-\D\b(u)+\vp\b(u)+{\rm div}(b_\vp(x,u)u),\vsp
D(A_\vp)&=&\{u\in L^1;-\Delta\beta(u)+\vp\beta(u)+{\rm div}(b_\vp(x,u)u)\in L^1\}.\earr$$ (See the argument in [@2].) Then, we replace by $$\label{e2.14a}
\barr{l}
(u_\vp-|u_0|_\9-\a(t))_t-\D(\b(u_\vp)-\b(|u_0|_\9+\a(t)))\vsp
+\vp(\b(u)-\b(|u_0|_\9+\a(t)))+{\rm div}(b^*_\vp(x,u_\vp)-b^*_\vp(x,|u_0|_\9+\a(t))\vsp
=-b^*_\vp(x,|u_0|_\9+\a(t))-\a'(t)-\vp\beta(|u_0|_\9+\a(t))\le0,\vsp\hfill \mbox{ a.e. in }(0,T)\times\rrd,\earr$$ where $b^*_\vp(u)=b_\vp(u)u.$
Let $\calx_\delta\in{\rm Lip}(\rr)$ be the following approximation of the signum function $$\calx_\delta(r)=\left\{\barr{rll}
1&\mbox{for }&r\ge\delta,\vsp
\dd\frac r\delta&\mbox{for }&|r|<\delta,\vsp
-1&\mbox{for }r<-\delta,\earr\right.$$where $\delta>0.$ If we multiply by $\calx_\delta((\beta(u_\vp)-\beta(|u_0|_\9+\a))^+)$ and integrate over $\rrd$, we get $$\!\!\!\barr{l}
\dd\int_{\rrd}(u_\vp-|u_0|_\9-\a)_t\calx_\delta((\beta(u_\vp)-\beta(|u_0|_\9+\a))^+)dx\\
\qquad\le\dd\frac1\delta
\int_{[0<(\beta(u_\vp)-\beta(|u_0|_\9+\a))^+\le\delta]}
(b^*(x,u_\vp)u_\vp-b^*_\vp(x,|u_0|_\9+\a))\cdot\nabla u_\vp\,dx,\\\hfill \ff t\in(0,T),\earr$$because $\beta$ is monotonically increasing and $$\barr{r}
\dd\nabla(\beta(u_\vp)-\beta(|u_0|_\9+\a)\cdot\nabla\calx_\delta
((\beta(u_\vp)-\beta(|u_0|_\9+\a))^+)\ge0\mbox{ in } (0,T)\times\rrd.\earr$$ Then, by , we get, for $\delta\to0$, $$\int_\rrd(u_\vp-|u_0|_\9-\a(t))^+_t\,dx\le0,\ \ff t\in(0,T),$$and this yields $$u_\vp(t,x)-|u_0|_\9-\a(t)\le0,\mbox{ a.e. on }(0,T)\times\rrd,$$and so, $u_\vp\le|u_0|_\9+\a$, a.e. on $(0,T)\times\rrd$. Then, we pass to the limit $\vp\to0$ to get the claimed inequality.$\Box$
By Theorem \[t1\] and Proposition \[p2\], we therefore get the following existence and uniqueness result for .
\[t4\] Under hypotheses [(i)–(iii)]{}, for each $u_0\in L^1\cap L^\9$, equation has a unique distributional solution $$\label{e2.12}
u\in L^\9((0,T);L^1)\cap L^\9((0,T)\times\rrd),\ \ff T>0.$$
The uniqueness of the linearized equation
=========================================
Consider a distributional solution of the linearized equation corresponding to , that is, $$\label{e3.1}
\barr{l}
v_t-\D(\Phi(u)v+{\rm div}(b(x,u)v)=0\mbox{ in }\cald'((0,\9)\times\rrd,\vsp
v(0,x)=v_0(x),\earr$$ where $u\in L^\9((0,T)\times\rrd),\ \ff T>0$. By (i)–(ii), we have $$b(x,u),\Phi(u)=\frac{\b(u)}{u}\in L^\9((0,\9)\times\rrd).$$Moreover, we have $$\label{e3.2}
\Phi(u)\ge\g_0>0,\mbox{ a.e. in }(0,\9)\times\rrd.$$
In the following, we denote $\Phi(u(t,x))$ by $\Psi(t,x),$ $(t,x)\in(0,\9)\times\rrd.$
\[t5\] [**(Linearized uniqueness)**]{} Under hypotheses [(i)–(ii)]{}, for each $v_0\in L^1\cap L^\9$ and $T>0$, equation has at most one distributional solution $v\in C([0,T];L^1)\cap L^\9((0,T)\times\rrd).$
We shall proceed as in the proof of Theorem \[t1\]. Namely, we set $v_1-v_2=v$ for two solutions $v_1,v_2$ of and get $$\label{e3.3}
\barr{l}
v_t-\D(\Psi v)+{\rm div}(b(x,u)v)=0,\mbox{ a.e. }t\in(0,T),\vsp
v(0)=0.\earr$$ For $y=\Gamma v$, we get $$\label{e3.4}
\barr{l}
\dd\frac d{dt}\ y-\Gamma\D(\Psi v)+\Gamma\ {\rm div}(b(x,u)v)=0\vsp
y(0)=0\earr$$ and multiplying scalarly in $L^2$ with $v$, we get as above that $$\label{e3.5}
\barr{l}
\dd\frac12\ \frac d{dt}\ |v(t)|^2_{-2}+\g_0|v(t)|^2_2\le|\Psi|_\9|v(t)|_{-2}|v(t)|_2\vsp
\qquad+|b|_\9|v(t)|_2|v(t)_{-2}
\le(|\Psi|_\9+|b|_\9)|v(t)|^{\frac32}_2|v(t)|^{\frac12}_{-2},\mbox{ a.e. }t\in(0,T).\earr$$ This yields $$\frac d{dt}\ |v(t)|^2_{-2}\le|v(t)|^2_{-2}\mbox{ a.e. }t\in(0,T),$$ and, therefore, $v\equiv0$, as claimed.
Uniqueness in law of the McKean–Vlasov stochastic differential equations (SDEs)
===============================================================================
Consider for $T\in(0,\9)$ and $u_0\in L^1\cap L^\9$ the McKean–Vlasov stochastic differential equation (SDE) $$\label{e4.1}
\barr{l}
dX(t)=b(X(t),u(t,X(t)))dt
+\dd\frac1{\sqrt{2}}\(\frac{\b(u(t,X(t)))}{u(t,X(t))}\)^{\frac12}dW(t),\vsp
\hfill 0\le t\le T,\\
u(0,\cdot)=\xi_0,\earr$$on $\rrd$. Here, $W(t),\ t\ge0,$ is an $(\calf_t)$-Brownian motion on a probability space $(\Omega,\calf,\mathbb{P})$ with normal filtration $\calf_t$, $t\ge0,$ $\xi_0:\Omega\to\rrd$ is $\calf_0$-measurable such that $$\mathbb{P}\circ\xi^{-1}_0(dx)=u_0(x)dx,$$ and $u(t,x)=\frac{d\mathcal{L}_{X(t)}}{dx}\,(x)$ is the Lebesgue density of the marginal law $\mathcal{L}_{X(t)}=\mathbb{P}\circ X(t)\1$ of the solution process $X(t)$, $t\ge0$. Here, a solution process means an $(\calf_t)$-adapted process with $\pas$ continuous sample paths in $\rrd$ solving .
\[t4.1\] Let $0<T<\9$ and let the above conditions [(i)–(ii)]{} on $b$ and $\b$ hold. Let $X(t)$, $t\ge0$, and $\wt X(t),$ $t\ge0$, be two solutions to such that, for $$u(t,\cdot):=\frac{d\mathcal{L}_{X(t)}}{dx},\ \ \wt u(t,\cdot):=\frac{d\mathcal{L}_{\wt X(t)}}{dx},$$we have $$\label{e4.2}
u,\wt u\in L^\9((0,T)\times\rrd).$$Then $X$ and $\wt X$ have the same laws, i.e., $\mathbb{P}\circ X\1=\mathbb{P}\circ\wt X^{-1}.$
By Itô’s formula, both $u$ and $\wt u$ satisfy the (nonlinear) equation in the sense of Schwartz distributions. Hence, by Theorem \[t1\], $u=\wt u$. Furthermore, again by Itô’s formula, $\mathbb{P}\circ X\1$ and $\mathbb{P}\circ\wt X\1$ satisfy the martingale problem with the initial condition $u_0dx$ for the linear Komogorov operator $$L_u:=\Phi(u)\D+b(\cdot,u)\cdot\nabla,$$where $\Phi(u)=\frac{\b(u)}u,$ $u\in\rr.$ Hence, by Theorem \[t5\], the assertion follows by Lemma 2.12 in [@4].
Here, for $s\in[0,T]$, the set $\mathcal{R}_{[s,T]}$, which appears in that lemma, is chosen to be the set of all narrowly continuous, probability measure-valued solutions of having for each $t\in[s,T]$ a density with respect to Lebesgue measure such that $\Box$
\[e4.2\]We note that, by the narrow continuity, implies that, for every $t\in[0,T],$ $u(t,\cdot),\wt u(t,\cdot)\in L^\9$. This fact was used in the above proof.
This work was supported by the DFG through CRC 1283.
[nn]{}
Barbu, V., Röckner, M., From nonlinear equations to solutions of distribution dependent SDE, arXiv:1808.107062\[math.PR\].
Barbu, V., Röckner, M., Probabilistic representation for solutions to nonlinear equation, [*SIAM J. Math. Anal.*]{}, 50 (4) (2018), 4246-4260.
Barbu, V., Röckner, M., The evolution to equilibrium of solutions to nonlinear equations, arXiv:1904.082-91\[math.PR\].
Brezis, H., Crandall, M.G., Uniqueness of solutions of the initial-value problem for $u_t-\D\b(u)=0$, [*J. Math. Pures et Appl.*]{}, 58 (1979), 153-163.
Trevisan, D., Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients, [*Electron. J. of Probab.*]{}, Volume 21 (2016), paper no. 22, 41 pp.
[^1]: Octav Mayer Institute of Mathematics of the Romanian Academy, Iaşi, Romania. Email: [email protected]
[^2]: Fakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany. Email: [email protected]
[^3]: Academy of Mathematic and System Sciences, CAS, Beijing, China
|
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abstract: |
We consider proton diffraction dissociation in the dipole Pomeron model, where the Pomeron is represented by a double pole in the $J-$plane, and show that unitarity can be satisfied without decoupling of the triple Pomeron vertex. Differential and total diffractive cross sections for the reaction $\bar{p}+p \to \bar{p}+X$ are analyzed and reproduced in this model.
PACS numbers: 12.40.Nn, 13.85.Ni.
---
-1cm
=0.3cm
0.3cm
**TRIPLE POMERON AND PROTON DIFFRACTION DISSOCIATION $~^\diamond$**
0.7cm
R. Fiore$^{a\dagger}$, A. Flachi$^{b\ddagger}$, L.L. Jenkovszky$^{c\S}$, F. Paccanoni$^{d\ast}$, A. Papa$^{a\dagger}$
.3cm
$^{a}$ *Dipartimento di Fisica, Università della Calabria,*
*Istituto Nazionale di Fisica Nucleare, Gruppo collegato di Cosenza*
*Arcavacata di Rende, I-87036 Cosenza, Italy*
.3cm
$^{b}$ *Physics Department, University of Newcastle upon Tyne,*
*Newcastle upon Tyne, NE1 7RU, United Kingdom*
.3cm
$^{c}$ *Bogoliubov Institute for Theoretical Physics,*
*Academy of Sciences of the Ukrain*
*252143 Kiev, Ukrain*
.3cm
$^{d}$ *Dipartimento di Fisica, Università di Padova,*
*Istituto Nazionale di Fisica Nucleare, Sezione di Padova*
*via F. Marzolo 8, I-35131 Padova, Italy*
0.3cm
.3cm
------------------------------------------------------------------------
.1cm
$^{\ast}$[*Work supported by the Ministero italiano dell’Università e della Ricerca Scientifica e Tecnologica and by the INTAS.*]{} $
\begin{array}{ll}
^{\dagger}\mbox{{\it e-mail address:}} &
\mbox{FIORE,[email protected]} \\
^{\ddagger}\mbox{{\it e-mail address:}} &
\mbox{[email protected]} \\
^{\S}\mbox{{\it e-mail address:}} &
\mbox{[email protected]} \\
^{\ast}\mbox{{\it e-mail address:}} &
\mbox{[email protected]}
\end{array}
$
Introduction
============
Diffractive high energy elastic scattering for hadrons appears to find a satisfactory explanation in the framework of the Regge theory with the exchange of the Pomeron trajectory. Rising cross sections can be accounted for by assuming a Pomeron intercept slightly higher than one [@DL] or, in a QCD approach, by considering the Pomeron as a gluon ladder [@LIP]. The growth of total cross sections can also be described in a way compatible with the Froissart bound in the eikonal model [@CY] or by assuming that the Pomeron is represented by a double pole in the complex $J-$plane [@BJ].
In contrast with the above picture, inclusive diffractive collisions in proton-proton or antiproton-proton scattering, where one of the initial particles changes only slightly its momentum and appears in the final state isolated in rapidity, seem to require deep modifications to the standard Regge models. The basic problem with diffraction dissociation, known for long [@BW], is that the integrated cross section $\sigma_{SD}$ appears to grow faster than the total cross section $\sigma_T$, thus violating unitarity. For example, in the case of a supercritical Pomeron with $\alpha(0)=1+\delta$, $\sigma_{SD}$ grows twice as fast, $\sim
s^{2\delta}$, as the total cross section does, $\sigma_T\sim s^{\delta}$. The only way to resolve this discrepancy seemed to require the vanishing of the triple Pomeron coupling (Pomeron decoupling [@BW]), which however contradicts the experimental data [@GL2].
A number of different unitarization recipes have been proposed in order to modify the energy dependence of the predicted cross section. Eikonal corrections [@GLM] succeed in reproducing the main features of single diffraction at high energy, while the same effect can be reached by the inclusion of cuts in the Regge theory [@KPT]. Recently a different, more phenomenological, approach has been considered [@GM; @GL1; @SCH]. Renormalization [@GM; @GL1] or damping [@SCH] of the Pomeron flux, that consists in setting a limit to the probability that the proton emits a Pomeron, allow for a rising of the total diffraction cross section compatible with experimental data. All the above approaches are based on a supercritical Pomeron input with a Pomeron intercept larger than one.
Apart from the incompatibility with the experimentally rising cross sections, a unity intercept Pomeron would present analogous problems with unitarity [@BW]. If, however, the partial wave amplitude, for the Pomeron exchange, presents a simple and a double pole in the complex $J-$plane, cross sections will grow with energy and it will be possible to satisfy unitarity at the Born level, without eikonalization [@BJ]. In all (or most) of the models explored until now, the Pomeron was assumed to be an isolated single Regge pole. From QCD we know however [@LIP] that the Pomeron is not a single pole, but rather a complicated set of singularities in the $J-$plane. A simple and feasible way to approximate this complicated structure is to take the sum of a simple and a double pole (dipole). The dipole Pomeron is known [@BJ] to have unique properties since it reproduces itself under unitarization and thus one expects that it can be used also to resolve the abovementioned problem in diffraction dissociation. Obviously, the sum of a simple and double pole - like any combination of Regge singularities - looses factorizability, although each term remains factorizable. Since Regge pole factorization appears to be in conflict with experimental results [@GM], the approach we consider is favoured.
The dipole Pomeron model has been tested successfully in elastic hadron-hadron and $\gamma-$hadron reactions [@BJ; @MB; @PD1; @PD2; @PD3] and an application to single diffractive dissociation has been considered in Refs. [@JMP; @FJP]. It turns out that, in this approach, the Pomeron contribution consists of two terms, one increasing like the logarithm of the energy and the other being energy independent, multiplied by “a priori” different $t-$dependent vertex functions. This feature, and the assumption that the Pomeron couples in a different way to Pomerons and hadrons, opens the way to a unified treatment of elastic and production amplitudes.
Since the inclusive process of hadron diffraction has been discussed extensively in the literature [@BW; @RR; @PDB; @ABK; @AG; @GL2] we will start, in Section 2, from the Mueller discontinuity formula and adapt it to the chosen model. The triple Pomeron contribution will be discussed in detail and the possibility to satisfy the unitarity constraint will be investigated. While a proof of the proposed solution cannot be given in the framework of the Regge theory because the $t-$dependence of the vertices is arbitrary to a great extent, plausibility arguments can be advanced on the basis of dynamical models for the Pomeron. Section 3 will be devoted to the inclusion of secondary Regge trajectories and the final expression for the cross section will be compared with experimental data in Section 4. The conclusion of this work will be drawn in Section 5.
The triple Pomeron in diffractive dissociation
==============================================
Consider first the process $a+b\to c+X$ with the exchange of Regge trajectories $\{i\}$. From the Mueller discontinuity formula [@MUE] we get $$\pi E_c\frac{d^3\sigma}{d\vec{p}_c}=
\frac{1}{16\pi s}\sum_X\left|\sum_i\beta^i_{a\bar{c}}(t)
\xi_i(t)F^{ib\to X}(M^2,t)\left(\frac{s}{M^2}\right)^{\alpha_i
(t)}\right|^2
\label{z1}$$ in the usual Regge pole model. $M^2$ is the squared mass of the unrevealed state $X$, $\alpha_i(t)$ represents the Regge trajectory exchanged and $$\xi_i(t)=\frac{1\pm\exp(-i\pi\alpha_i(t))}{\sin(\pi\alpha_i(t))}$$ is its signature. In the following $i=P, f,\pi$ and $\omega$, where $P$ stands for the Pomeron trajectory.
Consider now the elastic scattering and suppose that, asymptotically, the absorptive part in the $s$-channel, $A(s,t)$, goes like $$A(s,t) \propto \beta_1(t) \beta_2(t) s^{\alpha(t)} [h(t) \ln s +C]~,$$ then the partial wave amplitude presents a simple and double pole in the complex $J-$plane. The amplitude for the Pomeron exchange can then be written as $$T(s,t) \propto -\frac{(-is)^{\alpha(t)}}{\sin(\pi
\alpha(t)/2)}\beta_1(t) \beta_2(t) \left[h(t) \left(\ln s -i\frac{\pi}{2}\right)
+C\right]~,
\label{z2}$$ where constant terms have been collected in $C$. The explicit form of $h(t)$ depends on the model. As an example, in a dual model, if the residue of the simple pole has the form $\beta(\alpha(t))$, the residue of the double pole will be given by $\int\beta(\alpha)\,d\alpha+const$ [@BJ]. The form of this residue is such that the coefficient of the double pole can vanish for $t=0$, if this is required from general principles.
To substantiate this possibility, we can generalize the picture of Ref. [@LN] and suppose that the Pomeron pole couples to quarks through the exchange of two gluons. In order to describe the gluon-Pomeron-gluon vertex we use, at high energy, the rules of covariant Reggeization [@GJ; @TO] for the coupling of the Pomeron pole to massive vector mesons, since the gluons are off-shell. This require the introduction of five unknown functions of $t$ (some of them will vanish because of gauge invariance) that will appear in the amplitude multiplied by polynomials in $\alpha(t)$ and by appropriate powers of $\nu$, $\nu^{\alpha(t)-n}$, where $\nu \approx s/2$ at high energy and $n=0,1,\ldots $. The final expression is complicated, but it can be easily seen that there will appear, among the leading contributions $\nu^{\alpha(t)}$, at least one term vanishing when $\alpha(t)=1$. This property remains true when we integrate over the gluon and quark momenta and, by taking the derivative with respect to $\alpha(t)$, we introduce a double pole for the Pomeron. Moreover this result does not depend on the choice of the partonic wave function of the hadron. The conclusion is that, also in elastic hadron-hadron scattering, the presence of a term vanishing with $t$, together with other terms finite at $t=0$, is highly probable in the conventional Regge residue. It could well happen that, if we consider the triple Pomeron vertex, the unitarity condition will impose constraints on the couplings such that $h(t)$ in Eq. (\[z2\]) vanishes at $t=0$.
In the dipole Pomeron approach, Eq. (\[z1\]) becomes $$\begin{aligned}
& & \frac{d^2\sigma}{dM^2\,dt}=
\nonumber \\
& & \frac{1}{16\pi s^2}\sum_X\left| \beta^{P}_{a\bar{c}}(t)
\left(-i\frac{s}{M^2}\right)^{\alpha_{P}(t)}\left[h(t)\left(\ln
\frac{s}{M^2}-i\frac{\pi}{2}\right)+C\right]F^{P b\to X}(M^2,t) \right.
\nonumber \\
& & + \left. \sum_{i\neq P} \beta^i_{a\bar{c}}\xi_i(t) F^{ib\to X}(M^2,t)
\left(\frac{s}{M^2}\right)^{\alpha_i(t)} \right|^2~.
\label{z3}\end{aligned}$$ Let us consider now the triple Pomeron contribution to Eq. (\[z3\]), neglecting for the moment all the interference terms and replacing the sum over intermediate states by a discontinuity in $M^2$, $$\begin{aligned}
& &\frac{1}{16\pi s^2}[\beta^{P}_{a\bar{c}}(t)]^2\left(\frac{s}{M^2}
\right)^{2\alpha_{P}(t)}\times
\nonumber \\
& & \left[\left(h(t)\ln\frac{s}{M^2}+C\right)^2+\frac{\pi^2}{4}h^2(t)\right]
Im\,T^{P b}(M^2,t,\alpha_{P}(t),t_{b\bar{b}}=0)~,
\label{z4}\end{aligned}$$ where, according to Eq. (\[z2\]), $$Im\,T^{P b}=\sigma_0\,(M^2)^{\alpha_{P}(0)}(\lambda+
\bar{h}(0)\ln M^2 +\lambda' (M^2)^{\alpha_f(0)-1}) g(t)~,
\label{z5}$$ $g(t)$ being the triple Pomeron coupling. A term, decreasing with $M^2$, is present in Eq. (\[z5\]) since we consider also the secondary $f$ trajectory in $P-b$ scattering. Obviously, if $h(0)$ vanishes, the same will be true for $\bar{h}(0)$. In the following, $\alpha_{P}(t)=1+\alpha't$, $\alpha'=0.25$ GeV$^{-2}$ and the standard form for the residue will be assumed: $\beta^P_{a\bar{c}}=\exp(bt)$.
By integrating Eq. (\[z5\]) over $t$ and $M^2$ we get the Pomeron contribution to the single diffractive cross-section, $\sigma_{SD}$. We will now show that the constraint $\sigma_{SD}<\sigma_T$ for all values of $s$ requires that $h(t)\propto (-t)^{\gamma}$ with $\gamma > 1/2$. Without changing the asymptotic behaviour of the Pomeron-hadron vertex, we can assume that[^1] $$h(t) \propto \left(\frac{-t}{-t+1} \right)^{\gamma}~, \mbox{~~~~~~}
\gamma \geq 0~.
\label{h(t)}$$ The proportionality constant in the expression for $h(t)$ is unessential since it can be factorized out in (\[z4\]) by properly rescaling the constant $C$.
The proof becomes simpler if, according to experimental findings [@GM; @GL2], we consider the triple Pomeron vertex $g(t)$ as constant and neglect $Im T^{Pb}$ in Eq. (\[z4\]). Then, setting $B=2 (b+\alpha'
\ln(s/M^2))$ and $y=\ln(s/M^2)$, the $t$ integral can be easily evaluated and reads $$\int_0^{\infty}\,dt\,e^{-Bt} \left[\left(\left(\frac{t}{t+1}
\right)^{\gamma}+C\right)^2+\frac{\pi^2}{4}\left(\frac{t}{t+1}
\right)^{2 \gamma} \right] =$$ $$\Gamma(2\gamma+1) \Psi (2\gamma +1,2;B) \left(y^2+\frac{\pi^2}{4}
\right) +2 \, y \, C \, \Gamma(\gamma +1) \, \Psi(\gamma
+1,2;B)+\frac{C^2}{B}~,
\label{z6}$$ where $\Psi(a,c;x)$ is a confluent hypergeometric function [@BAT].
In order to integrate over $M^2$ we transform to the variable $B$ whose upper limit is, asymptotically, proportional to $\ln s$. The integral can be evaluated exactly by using the elementary relations for the $\Psi$ function and, in the limit $B\sim \ln s \to \infty$, the behaviour of $\sigma_{SD}$ can be inferred from the large variable estimate for $\Psi$ [@BAT] $$\sigma_{SD} \: \sim \: \Gamma(2 \gamma+1)\, \frac{2}{1-\gamma} \: B^{2-2\gamma}+
\ldots + C^2 \ln B ~,
\label{z7}$$ where dots in the l.h.s. stand for terms with a less singular behaviour when $s \to \infty$.
We note that the singularity for $\gamma=1$ in Eq. (\[z7\]) is spurious; the exact result does not present singularities for $\gamma \geq 0$. Since, in the model considered, $\sigma_T \sim \ln s$ and $\sigma_{SD} <
\sigma_T$, from the first term in Eq. (\[z7\]) we must have $2-2\gamma
\leq 1$. Hence, the parameter $\gamma$, in general, must satisfy the condition $\gamma\geq 1/2$. This inequality is necessary to avoid terms, violating unitarity, that rise faster than $\ln s$. It is important to notice that the triple Pomeron contribution does not vanish at $t=0$ because of the presence of the constant $C$.
Non-leading contributions and the differential cross section
============================================================
From now on we select the hadrons participating the process: $a$ and $c$ are antiprotons ($\bar p$) and $b$ is a proton ($p$). Later, for the evaluation of the total single diffractive cross section $\sigma_{SD}$, the process $a=c=p$ and $b=\bar{p}$ will be also taken into account.
On the basis of historical fits [@RR; @FF], the $\omega$ trajectory can be neglected and, since the $\pi$ trajectory contributes in a different kinematical region with respect to $P$ and $f$, interference terms between $\pi$ and $P, f$ are suppressed. Hence, in Eq. (\[z3\]) the sum over $i$ refers only to $f$, and the $\pi$ contribution will be chosen as in [@GL2; @FF; @MBH; @UA8] $$\left. \frac{d^2\sigma}{dM^2\,dt}\right|_{\pi} \: = \:
\frac{1}{4\pi}\frac{g^2_{\pi pp}}{4\pi}\frac{(-t)}{(t-\mu^2)^2}
\left(\frac{s}{M^2}\right)^{2\alpha_{\pi}(t)-1}G^2(t)
\sigma^{\pi p}_T(M^2)~,
\label{z8}$$ where $$G(t)=\frac{2.3-\mu^2}{2.3-t}~,$$ $g^2_{\pi pp}/(4 \pi)=14.6$ and $\alpha_{\pi}(t)=0.9\,t$.
The $f$ contribution, and its interference with the Pomeron, must now be considered. The approximation suggested in [@DL2; @DL3] is based on the assumption that the $f$ couples to hadrons in just the same way as the Pomeron. This choice avoids the proliferation of free parameters and is justified from the consideration that, while the $f$ is required by the data [@GLM; @UA8; @DL2; @DL3], its contribution is small, in percentage, and can be approximated. A Pomeron-Pomeron-Reggeon term larger than $0.15 \: \sigma_{SD}$ is excluded by high energy data [@CDF] and is completely ignored in a recent analysis [@GM].
Since the model we consider for the Pomeron is different from the conventional, supercritical one, we must take care in choosing an appropriate $f$ trajectory. Fits with a double Pomeron pole [@PD1; @PD2; @PD3] require an intercept $\alpha_f(0)$ higher than the value, usually adopted, about $0.55$ [@DL]. In a recent analysis [@CKK], however, for the degenerate $a_2/f$ trajectory, the result $\alpha_+-1=-0.31\pm 0.05$ has been obtained by refitting all the experimental cross sections considered in [@DL]. For all data, with errors added in quadrature, a smaller value for $\alpha_+$ has been obtained: $\alpha_+
-1=-0.34 \pm 0.05$. An intermediate value, $-0.32$ has been used in Ref. [@UA8]. The coincidence of $\alpha_f(0)-1$ with $-0.32$ obtained in the fit of hadronic cross sections within different models for the soft Pomeron should not be surprising; in a limited energy range a behaviour $s^{\epsilon}$, for $\epsilon$ sufficiently small, can be well approximated by a term of the form $(u+v \ln s)$.
Let $a(t)$ be the difference between the $P$ and $f$ trajectories. If we set $$a(t)=\alpha_{P}(t)-\alpha_f(t)= a(0)-\delta t~,$$ then typical values, adopted in the following, are $a(0)\simeq 0.34$ and $\delta\simeq 0.65$. The $f$ contribution $$\begin{aligned}
R(s,t) &=& k\left\{\left[h(t)\ln\frac{s}{M^2}+C\right] \cos\left(\frac{\pi a(t)}{2}
\right)- \right.
\nonumber \\
& & \left. \frac{\pi h(t)}{2}\sin\left(\frac{\pi
a(t)}{2}\right)\right\} \left(\frac{s}{M^2}
\right)^{-a(t)}+k^2\left(\frac{s}{M^2}\right)^{-2a(t)}
\label{z9}\end{aligned}$$ will appear in the final form of the differential cross section: $$\begin{aligned}
\frac{d^2\sigma}{dt\,dM^2} &=& \frac{A}{M^2}
e^{2(b+\alpha'\ln(s/M^2))t}
\left[\left(h(t) \ln\frac{s}{M^2}+ C\right)^2 \right. \nonumber \\
&+ & \left. \frac{\pi^2}{4} h^2(t)+
R(s,t) \right] (1+l (M^2)^{\alpha_f(0)-1})
\nonumber \\
&+ & \frac{1}{4\pi}\frac{g^2}{4\pi M^2}\frac{(-t)}{(t-\mu^2)^2}G^2(t)
\left(\frac{s}{M^2}\right)^{2\alpha_{\pi}(t)-2} \sigma^{\pi p}_T
(M^2)~,
\label{z10}\end{aligned}$$ where all the constant factors have been collected in $A$.
In Ref. [@DL2] a value near $7.8$ is quoted for the parameter $k$, appearing in $R(s,t)$; since, however, the expression (\[z10\]) has been rescaled, $k$ is here a new parameter. As far as the other parameters are concerned, $b$ will be fixed from $p-p$ elastic scattering (e.g. $b=2.25$ GeV$^{-2}$, consistent with the slopes used in [@GM; @MB; @PD1; @PD2; @PD3; @FJP]) and $\sigma^{\pi p}_T(M^2)$ in the dipole Pomeron model can be written as $$\sigma^{\pi p}_T(M^2)=0.565+2.902 \ln(M^2)+44.388 (M^2)^{\alpha_f(0)-1}~,
\label{z11}$$ inspired by the parametrization used in [@DL]. Since the form of $h(t)$ is determined only near $t=0$, it is well possible that the $t$-dependence of the cross-section should be corrected. Hence, a different value of $b$ could be required from the experimental data, but this possibility will not be considered in the following.
Comparison with data
====================
When comparing the model with experimental data, we find two kinds of problems. The first one is related to the experimental definition of single diffraction dissociation. The great variety of phenomenological models, adopted by different experimental groups in order to extract the published data, makes the test of any new model difficult. Moreover, integrated cross-sections do not refer to the same intervals of $M^2$ and $t$, for different experimental analyses. The second kind of problem resides in our parametrization and is strongly related to the first one. The integrated cross section cannot be given in compact form and, since the overall normalization of the data has an experimental uncertainty of 15 %, it is not an easy task to determine the parameter $\gamma$ only from the $t$-dependence of the cross sections at different energies.
While the pion contribution can be fixed as in Section 3, the parameters relative to the $f$ trajectory are different with respect to those of Refs. [@DL2; @DL3], since the Pomeron contribution differs from the one proposed there. We are left with three parameters for the Pomeron and one for the $f$, plus an overall constant multiplying these contributions, while the $\pi$ term has no free parameters.
From now on, we adopt the standard variable $\xi\equiv M^2/s$, that represents the fraction of the momentum of the proton carried by the Pomeron. Using the expression (\[z10\]) for $d^2\sigma/d\xi dt$ in our model, we performed a global fit of the data at $\sqrt s = 14$ and 20 GeV of E396 [@COOL] and at $\sqrt s = 546$ and 1800 GeV of the CDF collaboration [@CDF]. All the data were taken from the compilation of Ref. [@GM] and are at fixed $t=-0.05$ GeV$^2$. The range of $\xi$ for the data of E396 has been limited to $0.0160 \: \div \: 0.1013$; in the case of the CDF data, we have considered $\xi$ in the range $0.0064 \: \div \: 0.109$ for the data at $\sqrt s = 546$ GeV and in the range $0.0033 \: \div \: 0.0918$ for those at $\sqrt s = 1800$ GeV. We have found that our proposed model nicely fits all the data for a large range of values of the parameter $\gamma$ larger than 1/2. This weak dependence on the value of the parameter $\gamma$ was not unexpected, since the fit was performed at fixed $t$. In the particular case of $\gamma=2$ (which will be justified in the following) the fit gives for the remaining parameters the following values: $C=0.9802$, $A=1.9080$, $k=0.9839$ and $l=2.3987$, with $\chi^2/\mbox{d.o.f.}
\approx 0.9$. In Fig. 1 we compare the curve resulting from the fit with $\gamma=2$ with the experimental data. We can see that our model succeeds in reproducing the experimental data at different values of $s$. We have checked that choosing a different value for $\gamma$ produces only little changes of the other parameters, but does not affect in a sizeable way the shape of the fitting curves.
We have then fixed the parameters in the expression for $d^2\sigma/d\xi dt$ in our model according to the result of the fit at $t=-0.05$ GeV$^2$ and have checked how it reproduces other sets of data, obtained at different $t$ values. We have considered the data of Ref. [@akim77] at $t=-0.015$ GeV$^2$ and those of the UA8 collaboration [@UA8] at the relatively large value of $t=-0.95$ GeV$^2$. In both cases our curves roughly reproduce the data (see Figs. 2 and 3), thus indicating that also the $t-$dependence in our model is quite reasonable.
Finally, we have considered the total single diffractive cross section $\sigma_{SD}$, for the process $p(\bar{p})+p\to p(\bar{p})+X$ as a function of $\sqrt{s}$. We have compared our model with the experimental data of [@CDF; @SCHA; @ALB; @ARM; @BER] from the compilation given in [@GM], where some data have been corrected in order to obtain the diffraction cross section for $\xi\leq 0.05$. In order to make the comparison, we have numerically integrated our expression for $d^2\sigma/d\xi dt$, with the parameters determined by the previous fit, in the region $ 1.4/s \leq \xi \leq 0.05$ and $t \leq 0$. In Fig. 4 we observe that the result of the integration, plotted as a function of $\sqrt s$, is in good agreement with the experimental data over all the range of values of $s$, including the Tevatron energies $\sqrt s =546$ and 1800 GeV. We must stress here that the choice $\gamma=2$ is essential: values of $\gamma$ lower than 2, but larger than 1/2 in order to satisfy $\sigma_{SD} < \sigma_T$, would give a too fast growth with $s$, whereas larger values of $\gamma$ would cause an undershooting of the data at large $s$.
Conclusions
===========
In this paper we have considered the proton diffraction dissociation in the dipole Pomeron model. In this model the differential cross section $d^2\sigma/d\xi dt$ can be written in the form given in Eq. (\[z10\]). From the theoretical point of view, the result in Eq. (\[z10\]) assesses two important properties that seem to be required by the data [@GM]. First, the exact factorization, typical of the Regge pole model, is lost in the dipole Pomeron approach. Second, for $t=0$ the Pomeron and pion contributions are independent of $s$ and the scaling with $M^2$ of $d^2\sigma/dM^2 dt |_{t=0}$ becomes exact if only these terms are considered. Moreover, we remark that this model respects the unitarity condition without decoupling of the triple Pomeron vertex. The total diffractive cross section rises as $\ln ( \ln s) $, i.e. slower than the total $p-\bar{p}$ cross section that, in turn, satisfies the Froissart bound.
We notice that, in Eq. (\[z10\]), the triple Pomeron coupling and the Pomeron-proton cross section are tangled in the multiplicative constant $A$ together with an unknown scale factorized from the function $h(t)$. Hence the fit of the experimental data cannot determine the aforesaid quantities but, at any rate, it represents an important test of the model. Concerning the comparison with experimental data, we have found that this model gives a satisfactory fit to the experimental data for $d^2\sigma/d\xi dt$ with regards both to the $\xi-$ and $t-$dependence. Moreover, for a suitable choice of the parameter $\gamma$, it well reproduces also the data for the total single diffractive cross section and allows to predict a value of about 11 mb at the LHC energy $\sqrt s = 14$ TeV.
We stress that in our model the one-pion contribution, parametrized in Eq. (\[z8\]), has been fixed from the beginning, differently from Ref. [@GM], where a multiplicative constant has been considered in front of it as one of the two free parameters to be fitted. As for the $f$ contribution, in our model it is well below the limit found by CDF [@CDF]. The discrepancies observed at large $\xi$ from the data of Ref. [@akim77] and of UA8 [@UA8] could arise from an underestimation of the contribution of the $\pi$ and from neglecting that of the $\omega$. According to Ref. [@UA8], the one-pion exchange contribution is only a small part of the total non-Pomeron exchange background. Also the approximated treatment of the $f$ could be responsible for the disagreement at large $\xi$. What we need is a more rigorous method for justifying the $t$-dependence of our parametrization and an extensive study of non-leading contributions. We feel that a deeper insight in these problems is important for applications of the model to other processes.
Acknowledgment
==============
One of us (L.L.J.) is grateful to the Dipartimento di Fisica dell’Università della Calabria and to the Istituto Nazionale di Fisica Nucleare - Sezione di Padova e Gruppo Collegato di Cosenza for their warm hospitality and financial support.
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[^1]: Other choices are possible as well: the function $h(t) \propto 1-\exp(\bar{\gamma} t)$ has a similar behaviour for suitable values of $\bar{\gamma}$.
|
---
address: |
Phys. Inst., University Heidelberg, Philosophenweg 12,\
69120 Heidelberg, Germany
author:
- 'R. SCHICKER'
title: THE ALICE DETECTOR AT LHC
---
Introduction {#sec:Intro}
============
The ALICE experiment is a general purpose heavy ion experiment to study the behaviour of strongly interacting matter at LHC energies. The behaviour of such matter addresses both equilibrium and non-equilibrium physics in an energy density region of $\varepsilon \sim$ 1-1000 GeV/fm$^3$. This range includes the energy densities at which phase transitions of strongly interacting matter occur representing sudden changes of the nonperturbative vacuum structure. Of particular interest are the deconfinement transition and the restoration of chiral symmetry. Even though the two transitions are related to different aspects of the QCD vacuum, both are closely connected at the critical temperature $T_{c}$. A plethora of experimental observables is expected to indicate the onset of these phase transitions. It is the goal of the ALICE experiment to establish signatures of these transitions.
Nucleus-nucleus collisions at LHC open up a new domain. The available nucleon-nucleon center of mass energy as compared to RHIC is larger by a factor of about 30. ALICE will probe parton distributions at values of Bjorken-x as low as 10$^{-5}$. At these values, the gluon density is thought to be close to phase space saturation, and strong nuclear gluon shadowing is expected.
The ALICE detector
==================
ALICE is designed as a general purpose experiment with a central barrel and a forward muon spectrometer [@Alice_prop]. The central barrel pseudorapidity acceptance is $|\eta| \le$ 0.9 with a magnetic field of 0.5 T. The central detectors track and identify particles from $\sim$ 100 MeVc$^{-1}$ to transverse momenta. Short-lived particles such as hyperons, D and B mesons are identified by their reconstructed secondary decay vertex. The detector granularity is chosen such that these tasks can be performed in a high multiplicity environment of up to 8000 charged particles per unit of rapidity. Tracking of particles is achieved by the inner tracking system (ITS) of six layers of silicon detectors, a large Time-Projection-Chamber (TPC) and a high granularity Transition-Radiation Detector (TRD). Particle identification in the central barrel is performed by measuring energy loss in the tracking detectors, transition radiation in the TRD and time-of-flight in a high-resolution TOF array. A single arm High-Momentum Particle Identification Detector (HMPID) with limited solid angle coverage extends the momentum range of identified hadrons. Photons will be measured by a crystal PbWO$_{4}$ PHOton Spectrometer (PHOS). Additional detectors close to the beam pipe define an interaction trigger. The forward muon spectrometer covers the pseudorapidity region -4.0 $\le \eta \le$ -2.5 with a low momentum cutoff of 4 GeVc$^{-1}$.
LHC experimental conditions {#sec:LHC}
===========================
The Large Hadron Collider LHC is designed to run in proton-proton, proton-nucleus and nucleus-nucleus mode. In nucleus-nucleus mode, the Pb-Pb system is considered to be most important due to the highest energy densities reached in these collisions. Lower energy densities are reached in intermediate mass systems. Taking data in proton-proton collisions has a threefold purpose. First, proton-proton collisions at LHC explore a new energy domain and contain interesting physics in its own right [@Alice_phys]. Second, detector calibration is much simpler in the low multiplicity environment of proton collisions. Third, these data serve as reference for nucleus-nucleus collisions to establish medium modifications of observed quantities.
System $\sqrt{s}$ (TeV) $\sigma_{geom}$(b) $\mathcal{L}_{0}$ (cm$^{-2}$s$^{-1}$) $\mathcal{L}/\mathcal{L}_{0}$
-------- ------------------ -------------------- --------------------------------------- -------------------------------
Pb-Pb 5.5 7.7 1.0 x 10$^{27}$ 0.44
Ar-Ar 6.3 2.7 1.0 x 10$^{29}$ 0.64
O-O 7.0 1.4 2.0 x 10$^{29}$ 0.73
p-p 14.0 0.07 5.0 x 10$^{30}$
: Maximum nucleon-nucleon center of mass energy, geometric cross section, initial and time averaged luminosity for different collision systems \[tab:exp\]
Table \[tab:exp\] lists center of mass energies and expected luminosities for some systems which will be measured by ALICE. The time averaged luminosity $\mathcal{L}$ depends on machine filling time, experiment setup time, beam lifetime and initial beam luminosity $\mathcal{L}_{0}$. The electromagnetic cross section of $\sigma \sim$ 500 b limits the lifetime of the Pb beams to about 7 or 4 hours for one or two data taking experiments, respectively. First pp collisions will be measured during the commissioning of the LHC. Pb-Pb collisions are expected at the end of the first pp run.
The ALICE physics program
=========================
The ALICE experimental program spans a wide range of physics topics pertinent to the understanding of strongly interacting matter [@Alice_phys]. Due to limitation in scope of this contribution, I have chosen three different topics in order to illustrate the ALICE physics potential.
Multiplicity distribution
-------------------------
The average charged particle multiplicity per unit of rapidity dN$_{\rm{Ch}}$/dy is one of the first observables which will be measured at LHC start-up. Since the multiplicity is related to the entropy density and hence to the energy density, it affects the calculation of most other observables. Despite the fundamental importance, there is so far no ab initio calculation of particle multiplicity starting from the QCD Lagrangian. The particle multiplicity is driven by soft non-perturbative QCD, and the relevant processes must be modelled by the new scale R$_{A} \sim$ A$^{1/3}$ fm.
The inclusive hadron rapidity density in pp $\rightarrow$ hX is defined to be $$\frac{dN_{Ch}}{dy}= \frac{1}{\sigma_{in}^{pp}(s)}
\int_{0}^{p_{t}^{max}}
dp^{2}_{t}\frac{d\sigma^{pp \rightarrow hX}}{dy dp^{2}_{t}},
\label{eq:ppmult}$$
The pp inelastic cross section $\sigma_{in}^{pp}$ grows at energies $\sqrt{s} > $ 20 GeV. The energy dependence is, however, poorly known, and can be parametrized by either logarithmic or power law behaviour. Extrapolating measured pp data by such fits results in an extrapolated rapidity density in pp collisions of about 5 at energies relevant for ALICE [@pp_mult].
The multiplicity in central nucleus-nucleus collisions at LHC energies can be estimated by dimensional arguments. A saturation scale Q$_{s}$ is assumed which represents the transverse density of all particles produced within one unit of rapidity
$$\frac{N}{R_{A}^{2}} = Q_{s}^{2}
\label{eq:aamult1}$$
where R$_{A}$ = A$^{1/3}$ fm and proportionality factors are set to unity.
In central nucleus-nucleus collisions, at scale $Q_{s}$, we have
$$N = \frac{A^{2}}{R_{A}^{2}} \frac{1}{Q_{s}^{2}}
\label{eq:aamult2}$$
with the factors $A^{2}/R_{A}^{2}$ stemming from the nuclear overlap function and $1/Q_{s}^{2}$ from the subprocess cross section.
Eqs. \[eq:aamult1\] and \[eq:aamult2\] can be combined with R$_{A}$= A$^{1/3}$ fm
$$N = A = Q_{s}^{2} R_{A}^{2} \rightarrow Q_{s} = 0.2 A^{1/6} \rm{GeV}.
\label{eq:aamult3}$$
A more refined analysis has to include the energy dependence of Q$_{s}$, for example by a power law dependence. Models of LHC particle multiplicity need to determine the constant factors which have been set to unity in this dimensional argument.
Parton energy loss
------------------
After an energetic parton is produced in a medium by a hard collision, it will radiate energy by emitting a gluon [@Baier]. Both the parton and the gluon traverse the medium of size L. The average energy loss of the parton in the limit $E_{parton} \rightarrow \infty$ due to gluon radiation with a spectrum $\frac{\omega dI}{d\omega}$ is given by
$$\Delta E = \int^{\omega_c}\frac{\omega dI}{d\omega}
\simeq \alpha_{s}\omega_{c},
\hspace{2.cm} \rm{with} \hspace{1.cm} \omega_{c} = \frac{1}{2}\hat{q} L^{2}
\label{eq:eloss}$$
The medium dependence of the energy loss is governed by the transport coefficient
$$\hat{q} \simeq \mu^{2}/\lambda \simeq \rho \int dq^{2}_{\perp} q^{2}_{\perp}
d\sigma/dq^{2}_{\perp}
\label{eq:transcoeff}$$
with $\rho$ the density of the medium and $\sigma$ the gluon-medium interaction cross section.
The coefficient $\hat{q}$ can be expresssed by the gluon structure function, i.e. for nuclear matter
$$\hat{q} = \frac{4\pi^{2}\alpha_{s} N_{c}}{N^{2}_{c}-1}\rho [xG(x,\hat{q}L)]
\label{eq:transcoeff1}$$
with $xG(x,Q^2)$ the gluon distribution for a nucleon and $\rho$ the nuclear density.
The energy loss of heavy quarks is expected to be reduced due to suppression of gluon radiation in a cone which scales with quark mass. The yield of inclusive large $p_{\perp}$ hadrons in nucleus-nucleus collisions is modified due to the medium induced radiative energy loss. This induced energy loss leads to jet quenching. The quenching of jets can be measured by comparing spectra of particles produced in nucleus-nucleus and nucleon-nucleon collisions [@dEnterria]. Moreover, particle spectra of nucleus-nucleus collisions can be analyzed with respect to the reaction plane.
Quarkonia production
--------------------
The dissociation of quarkonia states is one of the most important observable for the existence of a deconfined state. The suppression of quarkonia is due to the shielding of potential by Debye screening [@Satz]. A quantitative characterization of dissociation temperature depends on the structure of heavy quark potential which can be extracted by an analysis of heavy quark free energies on the lattice [@Karsch]. While some authors claim an abrupt $J/\Psi$ dissociation at $T = 1.9\,T_{c}$, others conclude a rather gradual $J/\Psi$ dissociation with complete disappearance only at $T = 3.0\, T_{c}$ [@Asakawa; @Datta].
In order to be reliable probes in nucleus-nucleus collisions, the cross sections for heavy quark and quarkonia production need to be known in proton-proton collisions. Predictions for the $c\overline{c}$-pair production cross section in pp collisions at 14 TeV range from 7 to 17 mb depending on the values used for the charm mass and for the factorization and renormalization scale [@HQ]. In the case of bottom pairs in pp-collisions at 14 TeV, the different parameter sets result in $b\overline{b}$ production cross sections between 0.2 and 0.7 mb.
Nuclear absorption and secondary scatterings with comovers can break up the quarkonia states and hence reduce the expected rates. The large number of $q\overline{q}$ pairs produced at LHC energies could, however, be an abundant source of final state quarkonia by coalescence due to statistical hadronization. This mechanism could result in an enhanced J/$\Psi$ production at LHC energies [@PBM].
Acknowledgments {#acknowledgments .unnumbered}
===============
[This work was supported in part by the German BMBF under project 06HD160I. ]{}
References {#references .unnumbered}
==========
[99]{}
ALICE Technical Proposal, CERN/LHCC/95-71 (1995)
ALICE Physics Performance Report, Vol.I, J.Phys.G: Nucl.Part.Phys.30 (2004) 1517
K.J.Eskola, K.Kajantie, P.V.Ruuskanen and K.Tuominen, Nucl.Phys. B570 (2000) 379
R.Baier et al., Nucl.Phys. B483 (1997) 291
D.d’Enterria, J.Phys.G:Nucl.Part.Phys.30 (2004) 767
T.Matsui and H.Satz, Phys.Lett. B 178 (1986) 416
F.Karsch, J.Phys.G:Nucl.Part.Phys.30 (2004) 887
M.Asakawa and T.Hatsuda, Phys.Rev.Lett. 92 (2004) 012001
S.Datta, F.Karsch, P.Petreczky and I.Wetzorke, Phys.Rev. D69, (2004) 094507
Heavy flavour physics, Hard probes in HI collisions at the LHC, hep-ph/0311048
P.Braun-Munzinger and J.Stachel, Phys.Lett.B 490 (2000) 196
|
---
author:
- |
Ludovic Courtès\
Bordeaux, France\
[email protected]\
title: Functional Package Management with Guix
---
=500 =500
ABSTRACT {#abstract .unnumbered}
--------
We describe the design and implementation of GNU Guix, a purely functional package manager designed to support a complete GNU/Linux distribution. Guix supports transactional upgrades and roll-backs, unprivileged package management, per-user profiles, and garbage collection. It builds upon the low-level build and deployment layer of the Nix package manager. Guix uses Scheme as its programming interface. In particular, we devise an embedded domain-specific language (EDSL) to describe and compose packages. We demonstrate how it allows us to benefit from the host general-purpose programming language while not compromising on expressiveness. Second, we show the use of Scheme to write build programs, leading to a “two-tier” programming system.
\[\] \[\] \[\]
Introduction {#chapter8267}
============
GNU Guix[^1] is a [*[purely functional]{}*]{} package manager for the GNU system \[20\], and in particular GNU/Linux. Package management consists in all the activities that relate to building packages from source, honoring the build-time and run-time dependencies on packages, installing, removing, and upgrading packages in user environments. In addition to these standard features, Guix supports transactional upgrades and roll-backs, unprivileged package management, per-user profiles, and garbage collection. Guix comes with a distribution of user-land free software packages.
Guix seeks to empower users in several ways[58]{} by offering the uncommon features listed above, by providing the tools that allow users to formally correlate a binary package and the “recipes” and source code that led to it—furthering the spirit of the GNU General Public License—, by allowing them to customize the distribution, and by lowering the barrier to entry in distribution development.
The keys toward these goals are the implementation of a purely functional package management paradigm, and the use of both declarative and lower-level programming interfaces (APIs) embedded in Scheme. To that end, Guix reuses the package storage and deployment model implemented by the Nix functional package manager \[8\]. On top of that, it provides Scheme APIs, and in particular embedded domain-specific languages (EDSLs) to describe software packages and their build system. Guix also uses Scheme for programs and libraries that implement the actual package build processes, leading to a “two-tier” system.
This paper focuses on the programming techniques implemented by Guix. Our contribution is twofold[58]{} we demonstrate that use of Scheme and EDSLs achieves expressiveness comparable to that of Nix’s DSL while providing a richer and extensible programming environment; we further show that Scheme is a profitable alternative to shell tools when it comes to package build programs. first gives some background on functional package management and its implementation in Nix. describes the design and implementation of Guix’s programming and packaging interfaces. provides an evaluation and discussion of the current status of Guix. presents related work, and concludes.
Background
==========
This section describes the functional package management paradigm and its implementation in Nix. It then shows how Guix differs, and what the rationale is.
Functional Package Management {#section8290}
-----------------------------
Functional package management is a paradigm whereby the build and installation process of a package is considered as a pure function, without any side effects. This is in contrast with widespread approaches to package build and installation where the build process usually has access to all the software installed on the machine, regardless of what its declared inputs are, and where installation modifies files in place.
Functional package management was pioneered by the Nix package manager \[8\], which has since matured to the point of managing a complete GNU/Linux distribution \[9\]. To allow build processes to be faithfully regarded as pure functions, Nix can run them in a [`chroot`]{} environment that only contains the inputs it explicitly declared; thus, it becomes impossible for a build process to use, say, Perl, if that package was not explicitly declared as an input of the build process. In addition, Nix maps the list of inputs of a build process to a statistically unique file system name; that file name is used to identify the output of the build process. For instance, a particular build of GNU Emacs may be installed in [`/nix/store/v9zic07iar8w90zcy398r745w78a7lqs-emacs-24.2`]{}, based on a cryptographic hash of all the inputs to that build process; changing the compiler, configuration options, build scripts, or any other inputs to the build process of Emacs yields a different name. This is a form of [*[on-disk memoization]{}*]{}, with the [`/nix/store`]{} directory acting as a cache of “function results”—i.e., a cache of installed packages. Directories under [`/nix/store`]{} are immutable.
This direct mapping from build inputs to the result’s directory name is basis of the most important properties of a functional package manager. It means that build processes are regarded as [*[referentially transparent]{}*]{}. To put it differently, instead of merely providing pre-built binaries and/or build recipes, functional package managers provide binaries, build recipes, and in effect a [*[guarantee]{}*]{} that a given binary matches a given build recipe.
Nix {#section8356}
---
The idea of [*[purely functional]{}*]{} package started by making an analogy between programming language paradigms and software deployment techniques \[8\]. The authors observed that, in essence, package management tools typically used on free operating systems, such as RPM and Debian’s APT, implement an [*[imperative]{}*]{} software deployment paradigm. Package installation, removal, and upgrade are all done in-place, by mutating the operating system’s state. Likewise, changes to the operating system’s configuration are done in-place by changing configuration files.
This imperative approach has several drawbacks. First, it makes it hard to reproduce or otherwise describe the OS state. Knowing the list of installed packages and their version is not enough, because the installation procedure of packages may trigger hooks to change global system configuration files \[4, 7\], and of course users may have done additional modifications. Second, installation, removal, and upgrade are not transactional; interrupting them may leave the system in an undefined, or even unusable state, where some of the files have been altered. Third, rolling back to a previous system configuration is practically impossible, due to the absence of a mechanism to formally describe the system’s configuration.
Nix attempts to address these shortcomings through the functional software deployment paradigm[58]{} installed packages are immutable, and build processes are regarded as pure functions, as explained before. Thanks to this property, it implements [*[transparent source/binary deployment]{}*]{}[58]{} the directory name of a build result encodes all the inputs of its build process, so if a trusted server provides that directory, then it can be directly downloaded from there, avoiding the need for a local build.
Each user has their own [*[profile]{}*]{}, which contains symbolic links to the [`/nix/store`]{} directories of installed packages. Thus, users can install packages independently, and the actual storage is shared when several users install the very same package in their profile. Nix comes with a [*[garbage collector]{}*]{}, which has two main functions[58]{} with conservative scanning, it can determine what packages a build output refers to; and upon user request, it can delete any packages not referenced [*via*]{} any user profile.
To describe and compose build processes, Nix implements its own domain-specific language (DSL), which provides a convenient interface to the build and storage mechanisms described above. The Nix language is purely functional, lazy, and dynamically typed; it is similar to that of the Vesta software configuration system \[11\]. It comes with a handful of built-in data types, and around 50 primitives. The primitive to describe a build process is [`derivation`]{}.
Figure \[fig:derivation-nix\] shows code that calls the [`derivation`]{} function with one argument, which is a dictionary. It expects at least the four key/value pairs shown above; together, they define the build process and its inputs. The result is a [*[derivation]{}*]{}, which is essentially the [*[promise]{}*]{} of a build. The derivation has a low-level on-disk representation independent of the Nix language—in other words, derivations are to the Nix language what assembly is to higher-level programming languages. When this derivation is instantiated—i.e., built—, it runs the command [`static-bash -c echo hello > $out`]{} in a chroot that contains nothing but the [`static-bash`]{} file; in addition, each key/value pair of the [`derivation`]{} argument is reified in the build process as an environment variable, and the [`out`]{} environment variable is defined to point to the output [`/nix/store`]{} file name.
Before the build starts, the file [`static-bash`]{} is imported under [`/nix/store/`]{}...[`-static-bash`]{}, and the value associated with [`builder`]{} is substituted with that file name. This [`${...}`]{} form on line 3 for string interpolation makes it easy to insert Nix-language values, and in particular computed file names, in the contents of build scripts. The Nix-based GNU/Linux distribution, NixOS, has most of its build scripts written in Bash, and makes heavy use of string interpolation on the Nix-language side.
All the files referenced by derivations live under [`/nix/store`]{}, called [*[the store]{}*]{}. In a multi-user setup, users have read-only access to the store, and all other accesses to the store are mediated by a daemon running as [`root`]{}. Operations such as importing files in the store, computing a derivation, building a derivation, or running the garbage collector are all implemented as remote procedure calls (RPCs) to the daemon. This guarantees that the store is kept in a consistent state—e.g., that referenced files and directories are not garbage-collected, and that the contents of files and directories are genuine build results of the inputs hashed in their name.
The implementation of the Nix language is an interpreter written in C++. In terms of performance, it does not compete with typical general-purpose language implementations; that is often not a problem given its specific use case, but sometimes requires rewriting functions, such as list-processing tools, as language primitives in C++. The language itself is not extensible[58]{} it has no macros, a fixed set of data types, and no foreign function interface.
From Nix to Guix {#nix-to-guix}
----------------
Our main contribution with GNU Guix is the use of Scheme for both the composition and description of build processes, and the implementation of build scripts. In other words, Guix builds upon the build and deployment primitives of Nix, but replaces the Nix language by Scheme with embedded domain-specific languages (EDSLs), and promotes Scheme as a replacement for Bash in build scripts. Guix is implemented using GNU Guile 2.0[^2], a rich implementation of Scheme based on a compiler and bytecode interpreter that supports the R5RS and R6RS standards. It reuses the build primitives of Nix by making remote procedure calls (RPCs) to the Nix build daemon.
We claim that using an [*[embedded]{}*]{} DSL has numerous practical benefits over an independent DSL[58]{} tooling (use of Guile’s compiler, debugger, and REPL, Unicode support, etc.), libraries (SRFIs, internationalization support, etc.), and seamless integration in larger programs. To illustrate this last point, consider an application that traverses the list of available packages and processes it—for instance to filter packages whose name matches a pattern, or to render it as HTML. A Scheme program can readily and efficiently do it with Guix, where packages are first-class Scheme objects; conversely, writing such an implementation with an external DSL such as Nix requires either extending the language implementation with the necessary functionality, or interfacing with it [*via*]{} an external representation such as XML, which is often inefficient and lossy.
We show that use of Scheme in build scripts is natural, and can achieve conciseness comparable to that of shell scripts, but with improved expressivity and clearer semantics.
The next section describes the main programming interfaces of Guix, with a focus on its high-level package description language and “shell programming” substitutes provided to builder-side code.
Build Expressions and Package Descriptions {#api}
==========================================
Our goal when designing Guix was to provide interfaces ranging from Nix’s low-level primitives such as [`derivation`]{} to high-level package declarations. The declarative interface is a requirement to help grow and maintain a large software distribution. This section describes the three level of abstractions implemented in Guix, and illustrates how Scheme’s homoiconicity and extensibility were instrumental.
Low-Level Store Operations {#section8410}
--------------------------
As seen above, [*[derivations]{}*]{} are the central concept in Nix. A derivation bundles together a [*[builder]{}*]{} and its execution environment[58]{} command-line arguments, environment variable definitions, as well as a list of input derivations whose result should be accessible to the builder. Builders are typically executed in a [`chroot`]{} environment where only those inputs explicitly listed are visible. Guix transposes Nix’s [`derivation`]{} primitive literally to its Scheme interface.
Figure \[fig:derivation-prim\] shows the example of Figure \[fig:derivation-nix\] rewritten to use Guix’s low-level Scheme API. Notice how the former makes explicit several operations not visible in the latter. First, line 1 establishes a connection to the build daemon; line 2 explicitly asks the daemon to “intern” file [`static-bash`]{} into the store; finally, the [`derivation`]{} call instructs the daemon to compute the given derivation. The two arguments on line 9 are a set of environment variable definitions to be set in the build environment (here, it’s just the empty list), and a set of [*[inputs]{}*]{}—other derivations depended on, and whose result must be available to the build process. Two values are returned (line 11)[58]{} the file name of the on-disk representation of the derivation, and its in-memory representation as a Scheme record.
The build actions represented by this derivation can then be performed by passing it to the [`build-derivations`]{} RPC. Again, its build result is a single file reading [`hello`]{}, and its build is performed in an environment where the only visible file is a copy of [`static-bash`]{} under [`/nix/store`]{}.
Build Expressions {#build-exprs}
-----------------
The Nix language heavily relies on string interpolation to allow users to insert references to build results, while hiding the underlying [`add-to-store`]{} or [`build-derivations`]{} operations that appear explicitly in Figure \[fig:derivation-prim\]. Scheme has no support for string interpolation; adding it to the underlying Scheme implementation is certainly feasible, but it’s also unnatural.
The obvious strategy here is to instead leverage Scheme’s homoiconicity. This leads us to the definition of [`build-expression->derivation`]{}, which works similarly to [`derivation`]{}, except that it expects a [*[build expression]{}*]{} as an S-expression instead of a builder. Figure \[fig:expr-derivation\] shows the same derivation as before but rewritten to use this new interface.
This time the builder on line 2 is purely a Scheme expression. That expression will be evaluated when the derivation is built, in the specified build environment with no inputs. The environment implicitly includes a copy of Guile, which is used to evaluate the [`builder`]{} expression. By default this is a stand-alone, statically-linked Guile, but users can also specify a derivation denoting a different Guile variant.
Remember that this expression is run by a separate Guile process than the one that calls [`build-expression->derivation`]{}[58]{} it is run by a Guile process launched by the build daemon, in a [`chroot`]{}. So, while there is a single language for both the “host” and the “build” side, there are really two [*[strata]{}*]{} of code, or [*[tiers]{}*]{}[58]{} the host-side, and the build-side code[^3].
Notice how the output file name is reified [*via*]{} the [`%output`]{} variable automatically added to [`builder`]{}’s scope. Input file names are similarly reified through the [`%build-inputs`]{} variable (not shown here). Both variables are non-hygienically introduced in the build expression by [`build-expression->derivation`]{}.
Sometimes the build expression needs to use functionality from other modules. For modules that come with Guile, the expression just needs to be augmented with the needed [`(use-modules ...)`]{} clause. Conversely, external modules first need to be imported into the derivation’s build environment so the build expression can use them. To that end, the [`build-expression->derivation`]{} procedure has an optional [`#58modules`]{} keyword parameter, allowing additional modules to be imported into the expression’s environment.
When [`#58modules`]{} specifies a non-empty module list, an auxiliary derivation is created and added as an input to the initial derivation. That auxiliary derivation copies the module source and compiled files in the store. This mechanism allows build expressions to easily use helper modules, as described in .
Package Declarations {#pkg-decl}
--------------------
The interfaces described above remain fairly low-level. In particular, they explicitly manipulate the store, pass around the system type, and are very distant from the abstract notion of a software package that we want to focus on. To address this, Guix provides a high-level package definition interface. It is designed to be [*[purely declarative]{}*]{} in common cases, while allowing users to customize the underlying build process. That way, it should be intelligible and directly usable by packagers will little or no experience with Scheme. As an additional constraint, this extra layer should be efficient in space and time[58]{} package management tools need to be able to load and traverse a distribution consisting of thousands of packages.
Figure \[fig:hello\] shows the definition of the GNU Hello package, a typical GNU package written in C and using the GNU build system—i.e., a [`configure`]{} script that generates a makefile supporting standardized targets such as [`check`]{} and [`install`]{}. It is a direct mapping of the abstract notion of a software package and should be rather self-descriptive.
The [`inputs`]{} field specifies additional dependencies of the package. Here line 16 means that Hello has a dependency labeled [`gawk`]{} on GNU Awk, whose value is that of the [`gawk`]{} global variable; [`gawk`]{} is bound to a similar [`package`]{} declaration, omitted for conciseness.
The [`arguments`]{} field specifies arguments to be passed to the build system. Here [`#58configure-flags`]{}, unsurprisingly, specifies flags for the [`configure`]{} script. Its value is quoted because it will be evaluated in the build stratum—i.e., in the build process, when the derivation is built. It refers to the [`%build-inputs`]{} global variable introduced in the build stratum by [`build-expression->derivation`]{}, as seen before. That variable is bound to an association list that maps input names, like [`gawk`]{}, to their actual directory name on disk, like [`/nix/store/...-gawk-4.0.2`]{}.
The code in Figure \[fig:hello\] demonstrates Guix’s use of embedded domain-specific languages (EDSLs). The [`package`]{} form, the [`origin`]{} form (line 5), and the [`base32`]{} form (line 9) are expanded at macro-expansion time. The [`package`]{} and [`origin`]{} forms expand to a call to Guile’s [`make-struct`]{} primitive, which instantiates a record of the given type and with the given field values[^4]; these macros look up the mapping of field names to field indexes, such that that mapping incurs no run-time overhead, in a way similar to SRFI-35 records \[14\]. They also bind fields as per [`letrec*`]{}, allowing them to refer to one another, as on line 8 of Figure \[fig:hello\]. The [`base32`]{} macro simply converts a literal string containing a base-32 representation into a bytevector literal, again allowing the conversion and error-checking to be done at expansion time rather than at run-time.
The [`package`]{} and [`origin`]{} macros are generated by a [`syntax-case`]{} hygienic macro \[19\], [`define-record-type*`]{}, which is layered above SRFI-9’s syntactic record layer \[13\]. Figure \[fig:package\] shows the definition of the [`<package>`]{} record type (the [`<origin>`]{} record type, not shown here, is defined similarly.) In addition to the name of a procedural constructor, [`make-package`]{}, as with SRFI-9, the name of a [*[syntactic]{}*]{} constructor, [`package`]{}, is given (likewise, [`origin`]{} is the syntactic constructor of [`<origin>`]{}.) Fields may have a default value, introduced with the [`default`]{} keyword. An interesting use of default values is the [`location`]{} field[58]{} its default value is the result of [`current-source-location`]{}, which is itself a built-in macro that expands to the source file location of the [`package`]{} form. Thus, records defined with the [`package`]{} macro automatically have a [`location`]{} field denoting their source file location. This allows the user interface to report source file location in error messages and in package search results, thereby making it easier for users to “jump into” the distribution’s source, which is one of our goals.
The syntactic constructors generated by [`define-record-type*`]{} additionally support a form of [*[functional setters]{}*]{} (sometimes referred to as “lenses” \[15\]), [*via*]{} the [`inherit`]{} keyword. It allows programmers to create new instances that differ from an existing instance by one or more field values. A typical use case is shown in Figure \[fig:inherit\][58]{} the expression shown evaluates to a new [`<package>`]{} instance whose fields all have the same value as the [`hello`]{} variable of Figure \[fig:hello\], except for the [`version`]{} and [`source`]{} fields. Under the hood, again, this expands to a single [`make-struct`]{} call with [`struct-ref`]{} calls for fields whose value is reused.
The [`inherit`]{} feature supports a very useful idiom. It allows new package variants to be created programmatically, concisely, and in a purely functional way. It is notably used to bootstrap the software distribution, where bootstrap variants of packages such as GCC or the GNU libc are built with different inputs and configuration flags than the final versions. Users can similarly define customized variants of the packages found in the distribution. This feature also allows high-level transformations to be implemented as pure functions. For instance, the [`static-package`]{} procedure takes a [`<package>`]{} instance, and returns a variant of that package that is statically linked. It operates by just adding the relevant [`configure`]{} flags, and recursively applying itself to the package’s inputs.
Another application is the [*[on-line auto-updater]{}*]{}[58]{} when installing a GNU package defined in the distribution, the [`guix package`]{} command automatically checks whether a newer version is available upstream from [`ftp.gnu.org`]{}, and offers the option to substitute the package’s source with a fresh download of the new upstream version—all at run time.This kind of feature is hardly accessible to an external DSL implementation. Among other things, this feature requires networking primitives (for the FTP client), which are typically unavailable in an external DSL such as the Nix language. The feature could be implemented in a language other than the DSL—for instance, Nix can export its abstract syntax tree as XML to external programs. However, this approach is often inefficient, due to the format conversion, and lossy[58]{} the exported representation may be either be too distant from the source code, or too distant from the preferred abstraction level. The author’s own experience writing an off-line auto-updater for Nix revealed other specific issues; for instance, the Nix language is lazily evaluated, but to make use of its XML output, one has to force strict evaluation, which in turn may generate more data than needed. In Guix, [`<package>`]{} instances have the expected level of abstraction, and they are readily accessible as first-class Scheme objects.
Sometimes it is desirable for the value of a field to depend on the system type targeted. For instance, for bootstrapping purposes, MIT/GNU Scheme’s build system depends on pre-compiled binaries, which are architecture-dependent; its [`input`]{} field must be able to select the right binaries depending on the architecture. To allow field values to refer to the target system type, we resort to [*[thunked]{}*]{} fields, as shown on line 13 of Figure \[fig:package\]. These fields have their value automatically wrapped in a thunk (a zero-argument procedure); when accessing them with the associated accessor, the thunk is transparently invoked. Thus, the values of thunked fields are computed lazily; more to the point, they can refer to [*[dynamic state]{}*]{} in place at their invocation point. In particular, the [`package-derivation`]{} procedure (shortly introduced) sets up a [`current-system`]{} dynamically-scoped parameter, which allows field values to know what the target system is.
Finally, both [`<package>`]{} and [`<origin>`]{} records have an associated “compiler” that turns them into a derivation. [`origin-derivation`]{} takes an [`<origin>`]{} instance and returns a derivation that downloads it, according to its [`method`]{} field. Likewise, [`package-derivation`]{} takes a package and returns a derivation that builds it, according to its [`build-system`]{} and associated [`arguments`]{} (more on that ). As we have seen on Figure \[fig:hello\], the [`inputs`]{} field lists dependencies of a package, which are themselves [`<package>`]{} objects; the [`package-derivation`]{} procedure recursively applies to those inputs, such that their derivation is computed and passed as the inputs argument of the lower-level [`build-expression->derivation`]{}.
Guix essentially implements [*[deep embedding]{}*]{} of DSLs, where the semantics of the packaging DSL is interpreted by a dedicated compiler \[12\]. Of course the DSLs defined here are simple, but they illustrate how Scheme’s primitive mechanisms, in particular macros, make it easy to implement such DSLs without requiring any special support from the Scheme implementation.
Build Programs {#shell}
--------------
The value of the [`build-system`]{} field, as shown on Figure \[fig:hello\], must be a [`build-system`]{} object, which is essentially a wrapper around two procedure[58]{} one procedure to do a native build, and one to do a cross-build. When the aforementioned [`package-derivation`]{} (or [`package-cross-derivation`]{}, when cross-building) is called, it invokes the build system’s build procedure, passing it a connection to the build daemon, the system type, derivation name, and inputs. It is the build system’s responsibility to return a derivation that actually builds the software.
The [`gnu-build-system`]{} object (line 10 of Figure \[fig:hello\]) provides procedures to build and cross-build software that uses the GNU build system or similar. In a nutshell, it runs the following phases by default[58]{}
1. unpack the source tarball, and change the current directory to the resulting directory;
2. patch shebangs on installed files—e.g., replace [`#!/bin/sh`]{} by [`#!/nix/store/...-bash-4.2/bin/sh`]{}; this is required to allow scripts to work with our unusual file system layout;
3. run [`./configure --prefix=/nix/store/...`]{}, followed by [`make`]{} and [`make check`]{}
4. run [`make install`]{} and patch shebangs in installed files.
Of course, that is all implemented in Scheme, [*via*]{} [`build-expression->derivation`]{}. Supporting code is available as a build-side module that [`gnu-build-system`]{} automatically adds as an input to its build scripts. The default build programs just call the procedure of that module that runs the above phases.
The [`(guix build gnu-build-system)`]{} module contains the implementation of the above phases; it is imported on the builder side. The phases are modeled as follows[58]{} each phase is a procedure accepting several keyword arguments, and ignoring any keyword arguments it does not recognize[^5]. For instance the [`configure`]{} procedure is in charge of running the package’s [`./configure`]{} script; that procedure honors the [`#58configure-flags`]{} keyword parameter seen on Figure \[fig:hello\]. Similarly, the [`build`]{}, [`check`]{}, and [`install`]{} procedures run the [`make`]{} command, and all honor the [`#58make-flags`]{} keyword parameter.
All the procedures implementing the standard phases of the GNU build system are listed in the [`%standard-phases`]{} builder-side variable, in the form of a list of phase name/procedure pairs. The entry point of the builder-side code of [`gnu-build-system`]{} is shown on Figure \[fig:gnu-build\]. It calls all the phase procedures in order, by default those listed in the [`%standard-phases`]{} association list, passing them all the arguments it got; its return value is true when every procedure’s return value is true.
The [`arguments`]{} field, shown on Figure \[fig:hello\], allows users to pass keyword arguments to the builder-side code. In addition to the [`#58configure-flags`]{} argument shown on the figure, users may use the [`#58phases`]{} argument to specify a different set of phases. The value of the [`#58phases`]{} must be a list of phase name/procedure pairs, as discussed above. This allows users to arbitrarily extend or modify the behavior of the build system. Figure \[fig:hello-custom\] shows a variant of the definition in Figure \[fig:hello\] that adds a custom build phase. The [`alist-cons-after`]{} procedure is used to add a pair with [`change-hello`]{} as its first item and the [`lambda*`]{} as its second item right after the pair in [`%standard-phases`]{} whose first item is [`configure`]{}; in other words, it reuses the standard build phases, but with an additional [`change-hello`]{} phase right after the [`configure`]{} phase. The whole [`alist-cons-after`]{} expression is evaluated on the builder side.
This approach was inspired by that of NixOS, which uses Bash for its build scripts. Even with “advanced” Bash features such as functions, arrays, and associative arrays, the phases mechanism in NixOS remains limited and fragile, often leading to string escaping issues and obscure error reports due to the use of [`eval`]{}. Again, using Scheme instead of Bash unsurprisingly allows for better code structuring, and improves flexibility.
Other build systems are provided. For instance, the standard build procedure for Perl packages is slightly different[58]{} mainly, the configuration phase consists in running [`perl Makefile.PL`]{}, and test suites are run with [`make test`]{} instead of [`make check`]{}. To accommodate that, Guix provides [`perl-build-system`]{}. Its companion build-side module essentially calls out to that of [`gnu-build-system`]{}, only with appropriate [`configure`]{} and [`check`]{} phases. This mechanism is similarly used for other build systems such as CMake and Python’s build system.
Build programs often need to traverse file trees, modify files according to a given pattern, etc. One example is the “patch shebang” phase mentioned above[58]{} all the source files must be traversed, and those starting with [`#!`]{} are candidate to patching. This kind of task is usually associated with “shell programming”—as is the case with the build scripts found in NixOS, which are written in Bash, and resort to [`sed`]{}, [`find`]{}, etc. In Guix, a build-side Scheme module provides the necessary tools, built on top of Guile’s operating system interface. For instance, [`find-files`]{} returns a list of files whose names matches a given pattern; [`patch-shebang`]{} performs the [`#!`]{} adjustment described above; [`copy-recursively`]{} and [`delete-recursively`]{} are the equivalent, respectively, of the shell [`cp -r`]{} and [`rm -rf`]{} commands; etc.
An interesting example is the [`substitute*`]{} macro, which does [`sed`]{}-style substitution on files. Figure \[fig:substitute\*\] illustrates its use to patch a series of files returned by [`find-files`]{}. There are two clauses, each with a pattern in the form of a POSIX regular expression; each clause’s body returns a string, which is the substitution for any matching line in the given files. In the first clause’s body, [`suffix`]{} is bound to the submatch corresponding to [`(.)`]{} in the regexp; in the second clause, [`line`]{} is bound to the whole match for that regexp. This snippet is nearly as concise than equivalent shell code using [`find`]{} and [`sed`]{}, and it is much easier to work with.
Build-side modules also include support for fetching files over HTTP (using Guile’s web client module) and FTP, as needed to realize the derivation of [`origin`]{}s (line 5 of Figure \[fig:hello\]). TLS support is available when needed through the Guile bindings of the GnuTLS library.
Evaluation and Discussion {#eval}
=========================
This section discusses the current status of Guix and its associated GNU/Linux distribution, and outlines key aspects of their development.
Status {#section8851}
------
Guix is still a young project. Its main features as a package manager are already available. This includes the APIs discussed in , as well as command-line interfaces. The development of Guix’s interfaces was facilitated by the reuse of Nix’s build daemon as the storage and deployment layer.
The [`guix package`]{} command is the main user interface[58]{} it allows packages to be browsed, installed, removed, and upgraded. The command takes care of maintaining meta-data about installed packages, as well as a per-user tree of symlinks pointing to the actual package files in [`/nix/store`]{}, called the [*[user profile]{}*]{}. It has a simple interface. For instance, the following command installs Guile and removes Bigloo from the user’s profile, as a single transaction[58]{}
The transaction can be rolled back with the following command[58]{}
The following command upgrades all the installed packages whose name starts with a ‘g’[58]{}
The [`--list-installed`]{} and [`--list-available`]{} options can be used to list the installed or available packages.
As of this writing, Guix comes with a user-land distribution of GNU/Linux. That is, it allows users to install packages on top of a running GNU/Linux system. The distribution is self-contained, as explained in , and available on [`x86_64`]{} and [`i686`]{}. It provides more than 400 packages, including core GNU packages such as the GNU C Library, GCC, Binutils, and Coreutils, as well as the Xorg software stack and applications such as Emacs, TeX Live, and several Scheme implementations. This is roughly a tenth of the number of packages found in mature free software distributions such as Debian. Experience with NixOS suggests that the functional model, coupled with continuous integration, allows the distribution to grow relatively quickly, because it is always possible to precisely monitor the status of the whole distribution and the effect of a change—unlike with imperative distributions, where the upgrade of a single package can affect many applications in many unpredictable ways \[7\].
From a programming point of view, packages are exposed as first-class global variables. For instance, the [`(gnu packages guile)`]{} module exports two variables, [`guile-1.8`]{} and [`guile-2.0`]{}, each bound to a [`<package>`]{} variable corresponding to the legacy and current stable series of Guile. In turn, this module imports [`(gnu packages multiprecision)`]{}, which exports a [`gmp`]{} global variable, among other things; that [`gmp`]{} variable is listed in the [`inputs`]{} field of [`guile`]{} and [`guile-2.0`]{}. The package manager [*[and]{}*]{} the distribution are just a set of “normal” modules that any program or library can use.
Packages carry meta-data, as shown in Figure \[fig:hello\]. Synopses and descriptions are internationalized using GNU Gettext—that is, they can be translated in the user’s native language, a feature that comes for free when embedding the DSL in a mature environment like Guile. We are in the process of implementing mechanisms to synchronize part of that meta-data, such as synopses, with other databases of the GNU Project.
While the distribution is not bootable yet, it already includes a set of tools to build bootable GNU/Linux images for the QEMU emulator. This includes a package for the kernel itself, as well as procedures to build QEMU images, and Linux “initrd”—the “initial RAM disk” used by Linux when booting, and which is responsible for loading essential kernel modules and mounting the root file system, among other things. For example, we provide the [`expression->derivation-in-linux-vm`]{}[58]{} it works in a way similar to [`build-expression->derivation`]{}, except that the given expression is evaluated in a virtual machine that mounts the host’s store over CIFS. As a demonstration, we implemented a derivation that builds a “boot-to-Guile” QEMU image, where the initrd contains a statically-linked Guile that directly runs a boot program written in Scheme \[5\].
The performance-critical parts are the derivation primitives discussed in . For instance, the computation of Emacs’s derivation involves that of 292 other derivations—that is, 292 invocations of the [`derivation`]{} primitive—corresponding to 582 RPCs[^6]. The wall time of evaluating that derivation is 1.1 second on average on a 2.6 GHz [`x86_64`]{} machine. This is acceptable as a user, but 5 times slower than Nix’s clients for a similar derivation written in the Nix language. Profiling shows that Guix spends most of its time in its derivation serialization code and RPCs. We interpret this as a consequence of Guix’s unoptimized code, as well as the difference between native C++ code and our interpreted bytecode.
Purity {#section8878}
------
Providing pure build environments that do not honor the “standard” file system layout turned out not to be a problem, as already evidenced in NixOS \[8\]. This is largely thanks to the ubiquity of the GNU build system, which strives to provide users with ways to customize the layout of installed packages and to adjust to the user’s file locations.
The only directories visible in the build [`chroot`]{} environment are [`/dev`]{}, [`/proc`]{}, and the subset of [`/nix/store`]{} that is explicitly declared in the derivation being built. NixOS makes one exception[58]{} it relies on the availability of [`/bin/sh`]{} in the [`chroot`]{} \[9\]. We remove that exception, and instead automatically patch script “shebangs” in the package’s source, as noted in . This turned out to be more than just a theoretical quest for “purity”. First, some GNU/Linux distributions use Dash as the implementation of [`/bin/sh`]{}, while others use Bash; these are two variants of the Bourne shell, with different extensions, and in general different behavior. Second, [`/bin/sh`]{} is typically a dynamically-linked executable. So adding [`/bin`]{} to the [`chroot`]{} is not enough; one typically needs to also add [`/lib*`]{} and [`/lib/-linux-gnu`]{} to the chroot. At that point, there are many impurities, and a great potential for non-reproducibility—which defeats the purpose of the [`chroot`]{}.
Several packages had to be adjusted for proper function in the absence of [`/bin/sh`]{} \[6\]. In particular, libc’s [`system`]{} and [`popen`]{} functions had to be changed to refer to “our” Bash instance. Likewise, GNU Make, GNU Awk, GNU Guile, and Python needed adjustment. Occasionally, occurrences of [`/bin/sh`]{} are not be handled automatically, for instance in test suites; these have to be patched manually in the package’s recipe.
Bootstrapping {#bootstrap}
-------------
{width="100.00000%"}
Bootstrapping in our context refers to how the distribution gets built “from nothing”. Remember that the build environment of a derivation contains nothing but its declared inputs. So there’s an obvious chicken-and-egg problem[58]{} how does the first package get built? How does the first compiler get compiled?
The GNU system we are building is primarily made of C code, with libc at its core. The GNU build system itself assumes the availability of a Bourne shell, traditional Unix tools provided by GNU Coreutils, Awk, Findutils, sed, and grep. Furthermore, our build programs are written in Guile Scheme. Consequently, we rely on pre-built statically-linked binaries of GCC, Binutils, libc, and the other packages mentioned above to get started.
Figure \[boot-graph\] shows the very beginning of the dependency graph of our distribution. At this level of detail, things are slightly more complex. First, Guile itself consists of an ELF executable, along with many source and compiled Scheme files that are dynamically loaded when it runs. This gets stored in the [`guile-2.0.7.tar.xz`]{} tarball shown in this graph. This tarball is part of Guix’s “source” distribution, and gets inserted into the store with [`add-to-store`]{}.
But how do we write a derivation that unpacks this tarball and adds it to the store? To solve this problem, the [`guile-bootstrap-2.0.drv`]{} derivation—the first one that gets built—uses [`bash`]{} as its builder, which runs [`build-bootstrap-guile.sh`]{}, which in turn calls [`tar`]{} to unpack the tarball. Thus, [`bash`]{}, [`tar`]{}, [`xz`]{}, and [`mkdir`]{} are statically-linked binaries, also part of the Guix source distribution, whose sole purpose is to allow the Guile tarball to be unpacked.
Once [`guile-bootstrap-2.0.drv`]{} is built, we have a functioning Guile that can be used to run subsequent build programs. Its first task is to download tarballs containing the other pre-built binaries—this is what the [`.tar.xz.drv`]{} derivations do. Guix modules such as [`ftp-client.scm`]{} are used for this purpose. The [`module-import.drv`]{} derivations import those modules in a directory in the store, using the original layout[^7]. The [`module-import-compiled.drv`]{} derivations compile those modules, and write them in an output directory with the right layout. This corresponds to the [`#58module`]{} argument of [`build-expression->derivation`]{} mentioned in .
Finally, the various tarballs are unpacked by the derivations [`gcc-bootstrap-0.drv`]{}, [`glibc-bootstrap-0.drv`]{}, etc., at which point we have a working C GNU tool chain. The first tool that gets built with these tools (not shown here) is GNU Make, which is a prerequisite for all the following packages.
Bootstrapping is complete when we have a full tool chain that does not depend on the pre-built bootstrap tools shown in Figure \[boot-graph\]. Ways to achieve this are known, and notably documented by the [*Linux From Scratch*]{} project \[1\]. We can formally verify this no-dependency requirement by checking whether the files of the final tool chain contain references to the [`/nix/store`]{} directories of the bootstrap inputs.
Obviously, Guix contains [`package`]{} declarations to build the bootstrap binaries shown in Figure \[boot-graph\]. Because the final tool chain does not depend on those tools, they rarely need to be updated. Having a way to do that automatically proves to be useful, though. Coupled with Guix’s nascent support for cross-compilation, porting to a new architecture will boil down to cross-building all these bootstrap tools.
Related Work {#related}
============
Numerous package managers for Scheme programs and libraries have been developed, including Racket’s PLaneT, Dorodango for R6RS implementations, Chicken Scheme’s “Eggs”, Guildhall for Guile, and ScmPkg \[16\]. Unlike GNU Guix, they are typically limited to Scheme-only code, and take the core operating system software for granted. To our knowledge, they implement the [*[imperative]{}*]{} package management paradigm, and do not attempt to support features such as transactional upgrades and rollbacks. Unsurprisingly, these tools rely on package descriptions that more or less resemble those described in ; however, in the case of at least ScmPkg, Dorodango, and Guildhall, package descriptions are written in an [*[external]{}*]{} DSL, which happens to use s-expression syntax.
In \[21\], the authors illustrate how the [*units*]{} mechanism of MzScheme modules could be leveraged to improve operating system packaging systems. The examples therein focus on OS services, and multiple instantiation thereof, rather than on package builds and composition.
The Nix package manager is the primary source of inspiration for Guix \[8, 9\]. As noted in , Guix reuses the low-level build and deployment mechanisms of Nix, but differs in its programming interface and preferred implementation language for build scripts. While the Nix language relies on laziness to ensure that only packages needed are built \[9\], we instead support [*ad hoc*]{} laziness with the [`package`]{} form. Nix and Guix have the same application[58]{} packaging of a complete GNU/Linux distribution.
Before Nix, the idea of installing each package in a directory of its own and then managing symlinks pointing to those was already present in a number of systems. In particular, the Depot \[3\], Store \[2\], and then GNU Stow \[10\] have long supported this approach. GNU’s now defunct package management project called ‘stut’, ca. 2005, used that approach, with Stow as a back-end. A “Stow file system”, or [`stowfs`]{}, has been available in the GNU Hurd operating system core to offer a dynamic and more elegant approach to user profiles, compared to symlink trees. The storage model of Nix/Guix can be thought of as a formalization of Stow’s idea.
Like Guix and Nix, Vesta is a purely functional build system \[11\]. It uses an external DSL close to the Nix language. However, the primary application of Vesta is fine-grain software build operations, such as compiling a single C file. It is a developer tool, and does not address deployment to end-user machines. Unlike Guix and Nix, Vesta tries hard to support the standard Unix file system layout, relying on a virtual file system to “map” files to their right location in the build environment.
Hop defines a [*[multi-tier]{}*]{} extension of Scheme to program client/server web applications \[17\]. It allows client code to be introduced (“quoted”) in server code, and server code to be invoked from client code. There’s a parallel between the former and Guix’s use of Scheme in two different strata, depicted in .
Scsh provides a complete interface to substitute Scheme in “shell programming” tasks \[18\]. Since it spans a wide range of applications, it goes beyond the tools discussed in some ways, notably by providing a concise [*[process notation]{}*]{} similar to that of typical Unix shells, and S-expression regular expressions (SREs). However, we chose not to use it as its port to Guile had been unmaintained for some time, and Guile has since grown a rich operating system interface on top of which it was easy to build the few additional tools we needed.
Conclusion
==========
GNU Guix is a contribution to package management of free operating systems. It builds on the functional paradigm pioneered by the Nix package manager \[8\], and benefits from its unprecedented feature set—transactional upgrades and roll-back, per-user unprivileged package management, garbage collection, and referentially-transparent build processes, among others.
We presented Guix’s two main contributions from a programming point of view. First, Guix [*[embeds]{}*]{} a declarative domain-specific language in Scheme, allowing it to benefit from its associated tool set. Embedding in a general-purpose language has allowed us to easily support internationalization of package descriptions, and to write a fast keyword search mechanism; it has also permitted novel features, such as an on-line auto-updater. Second, its build programs and libraries are also written in Scheme, leading to a unified programming environment made of two strata of code.
We hope to make Guix a good vehicle for an innovative free software distribution. The GNU system distribution we envision will give Scheme an important role just above the operating system interface.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author would like to thank the Guix contributors for their work growing the system[58]{} Andreas Enge, Nikita Karetnikov, Cyril Roelandt, and Mark H. Weaver. We are also grateful to the Nix community, and in particular to Eelco Dolstra for his inspiring PhD work that led to Nix. Lastly, thanks to the anonymous reviewer whose insight has helped improve this document.
References {#chapter8967}
==========
‘=1000=0em
[ - ]{}
G. Beekmans, M. Burgess, B. Dubbs. [Linux From Scratch](http://www.linuxfromscratch.org/lfs/). 2013. [[*http[58]{}//www.linuxfromscratch.org/lfs/*]{}](http://www.linuxfromscratch.org/lfs/).
A. Christensen, T. Egge. [Store—a system for handling third-party applications in a heterogeneous computer environment](http://dx.doi.org/10.1007/3-540-60578-9_22). Springer Berlin Heidelberg, 1995, pp. 263–276.
S. N. Clark, W. R. Nist. The Depot[58]{} A Framework for Sharing Software Installation Across Organizational and UNIX Platform Boundaries. In [*In Proceedings of the Fourth Large Installation Systems Administrator’s Conference (LISA ’90)*]{}, pp. 37–46, 1990.
R. D. Cosmo, D. D. Ruscio, P. Pelliccione, A. Pierantonio, S. Zacchiroli. [Supporting software evolution in component-based FOSS systems](http://dx.doi.org/10.1016/j.scico.2010.11.001). In [*Sci. Comput. Program.*]{}, 76(12) , Amsterdam, The Netherlands, December 2011, pp. 1144–1160.
L. Courtès. [Boot-to-Guile!](http://lists.gnu.org/archive/html/bug-guix/2013-02/msg00173.html). February 2013. [[*http[58]{}//lists.gnu.org/archive/html/bug-guix/2013-02/msg00173.html*]{}](http://lists.gnu.org/archive/html/bug-guix/2013-02/msg00173.html).
L. Courtès. [Down with /bin/sh!](https://lists.gnu.org/archive/html/bug-guix/2013-01/msg00041.html). January 2013. [[*https[58]{}//lists.gnu.org/archive/html/bug-guix/2013-01/msg00041.html*]{}](https://lists.gnu.org/archive/html/bug-guix/2013-01/msg00041.html).
O. Crameri, R. Bianchini, W. Zwaenepoel, D. Kostić. Staged Deployment in Mirage, an Integrated Software Upgrade Testing and Distribution System. In [*In Proceedings of the Symposium on Operating Systems Principles*]{}, 2007.
E. Dolstra, M. d. Jonge, E. Visser. [Nix[58]{} A Safe and Policy-Free System for Software Deployment](http://nixos.org/nix/docs.html). In [*Proceedings of the 18th Large Installation System Administration Conference (LISA ’04)*]{}, pp. 79–92, USENIX, November 2004.
E. Dolstra, A. Löh, N. Pierron. [NixOS[58]{} A Purely Functional Linux Distribution](http://nixos.org/nixos/docs.html). In [*Journal of Functional Programming*]{}, (5-6) , New York, NY, USA, November 2010, pp. 577–615.
B. Glickstein, K. Hodgson. [Stow—Managing the Installation of Software Packages](http://www.gnu.org/software/stow/). 2012. [[*http[58]{}//www.gnu.org/software/stow/*]{}](http://www.gnu.org/software/stow/).
A. Heydon, R. Levin, Y. Yu. [Caching Function Calls Using Precise Dependencies](http://doi.acm.org/10.1145/349299.349341). In [*Proceedings of the ACM SIGPLAN 2000 conference on Programming Language Design and Implementation*]{}, PLDI ’00, pp. 311–320, ACM, 2000.
P. Hudak. [Building domain-specific embedded languages](http://doi.acm.org/10.1145/242224.242477). In [*ACM Computing Surveys*]{}, 28(4es) , New York, NY, USA, December 1996, .
R. Kelsey. [Defining Record Types](http://srfi.schemers.org/srfi-9/srfi-9.html). 1999. [[*http[58]{}//srfi.schemers.org/srfi-9/srfi-9.html*]{}](http://srfi.schemers.org/srfi-9/srfi-9.html).
R. Kelsey, M. Sperber. [Conditions](http://srfi.schemers.org/srfi-35/srfi-35.html). 2002. [[*http[58]{}//srfi.schemers.org/srfi-35/srfi-35.html*]{}](http://srfi.schemers.org/srfi-35/srfi-35.html).
T. Morris. [Asymmetric Lenses in Scala](http://days2012.scala-lang.org/). 2012. [[*http[58]{}//days2012.scala-lang.org/*]{}](http://days2012.scala-lang.org/).
M. Serrano, É. Gallesio. [An Adaptive Package Management System for Scheme](http://doi.acm.org/10.1145/1297081.1297093). In [*Proceedings of the 2007 Symposium on Dynamic languages*]{}, DLS ’07, pp. 65–76, ACM, 2007.
M. Serrano, G. Berry. [Multitier Programming in Hop](http://doi.acm.org/10.1145/2330087.2330089). In [*Queue*]{}, 10(7) , New York, NY, USA, July 2012, pp. 10[58]{}10–10[58]{}22.
O. Shivers, B. D. Carlstrom, M. Gasbichler, M. Sperber. [Scsh Reference Manual](http://www.scsh.net/). 2006. [[*http[58]{}//www.scsh.net/*]{}](http://www.scsh.net/).
M. Sperber, R. K. Dybvig, M. Flatt, A. V. Straaten, R. B. Findler, J. Matthews. [Revised6 Report on the Algorithmic Language Scheme](http://journals.cambridge.org/article_S0956796809990074). In [*Journal of Functional Programming*]{}, 19, 7 2009, pp. 1–301.
R. M. Stallman. [The GNU Manifesto](http://www.gnu.org/gnu/manifesto.html). 1983. [[*http[58]{}//www.gnu.org/gnu/manifesto.html*]{}](http://www.gnu.org/gnu/manifesto.html).
D. B. Tucker, S. Krishnamurthi. Applying Module System Research to Package Management. In [*Proceedings of the Tenth International Workshop on Software Configuration Management*]{}, 2001.
[^1]: [[`http58//www.gnu.org/software/guix/`]{}](http://www.gnu.org/software/guix/)
[^2]: [[`http58//www.gnu.org/software/guile/`]{}](http://www.gnu.org/software/guile/)
[^3]: The term “stratum” is this context was coined by Manuel Serrano et al. for their work on Hop where a similar situation arises \[17\].
[^4]: The [`make-struct`]{} instantiates SRFI-9-style flat records, which are essentially vectors of a disjoint type. In Guile they are lightweight compared to CLOS-style objects, both in terms of run time and memory footprint. Furthermore, [`make-struct`]{} is subject to inlining.
[^5]: Like many Scheme implementations, Guile supports [*[named]{}*]{} or [*[keyword]{}*]{} arguments as an extension to the R5 and R6RS. In addition, procedure definitions whose formal argument list contains the [`#58allow-other-keys`]{} keyword ignore any unrecognized keyword arguments that they are passed.
[^6]: The number of [`derivation`]{} calls and [`add-to-store`]{} RPCs is reduced thanks to the use of client-side memoization.
[^7]: In Guile, module names are a list of symbols, such as [`(guix ftp-client)`]{}, which map directly to file names, such as [`guix/ftp-client.scm`]{}.
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3.0cm
**Thermodynamics and Phase Transition of a Gauss-Bonnet Black Hole**
**in a Cavity**
Peng Wang, Haitang Yang and Shuxuan Ying
*Center for Theoretical Physics, College of Physics*
*Sichuan University*
*Chengdu, 610065, China*
[email protected], [email protected], [email protected]
**Abstract**
Considering a canonical ensemble, in which the temperature and the charge on a wall of the cavity are fixed, we investigate the thermodynamics of a $D$-dimensional Gauss-Bonnet black hole in a finite spherical cavity. Moreover, it shows that the first law of thermodynamics is still satisfied. We then discuss the phase structure and transition in both five and six dimensions. Specifically, we show that there always exist two regions in the parameter space. In one region, the system possesses one single phase. However in the other region, there could coexist three phases and a van der Waals-like phase transition occurs. Finally, we find that there is a fairly close resemblance in thermodynamic properties and phase structure of a Gauss-Bonnet-Maxwell black hole, either in a cavity or in anti-de Sitter space.
Introduction
============
The thermodynamics of black holes has been a subject of intensive study for several decades since J. Bekenstein and S. Hawking discovered that the black hole entropy was proportional to the area of its event horizon [@Bekenstein:1973ur; @Hawking:1974rv]. A Schwarzschild black hole in asymptotically flat space has a negative specific heat and hence become unstable as a thermodynamic system. That being said, the Schwarzschild black hole radiates more when it becomes smaller. To make a black hole thermally stable, it is necessary to put the black hole in a closed system. One of the most popular way to approach this aim is to place a black hole in anti-de Sitter (AdS) space with the negative cosmological constant, in which the timelike boundary can reflected radiation back into the bulk. Hawking and Page in [@Hawking:1982dh] first studied the thermodynamic properties of a Schwarzschild black hole in AdS space and discovered the Hawking-Page phase transition. Since then, the thermodynamic properties and phase transition of various black holes in AdS space have been discussed in [@Witten:1998zw; @Chamblin:1999tk; @Chamblin:1999hg; @Caldarelli:1999xj; @Cai:2001dz; @Nojiri:2001aj; @Kubiznak:2012wp; @Hendi:2015hoa; @Hendi:2017fxp; @Wang:2018xdz].
On the other hand, another popular choice is considering a cavity in an asymptotic flat space, where the Dirichlet boundary condition is imposed on the wall of the cavity. It was discovered by York in [@York:1986it] that a Schwarzschild black hole in a cavity is thermally stable and a Hawking-Page-like transition can occur as the temperature decreases, which is quite similar to the behavior of a Schwarzschild-AdS black hole. Later, a Reissner-Nordstrom (RN) black hole in a cavity was studied in a grand canonical ensemble [@Braden:1990hw] and a canonical ensemble [@Lundgren:2006kt]. It showed that a Hawking-Page-like phase transition and a van der Waals-like one occur in the canonical ensemble and grand canonical ensemble, respectively, which is similar to the AdS case [@Carlip:2003ne]. In the following papers [@Lu:2010xt; @Wu:2011yu; @Lu:2012rm; @Lu:2013nt; @Zhou:2015yxa; @Xiao:2015bha], the phase structure of various black holes in the cavity is studied, where Hawking-Page-like or van der Waals-like phase transitions were found except for some special cases. Considering charged scalars, boson stars and hairy black holes in a cavity in [@Basu:2016srp; @Peng:2017gss; @Peng:2017squ; @Peng:2018abh], it showed that the phase structure of the gravity system in a cavity is strikingly similar to that of holographic superconductors in the AdS gravity. In [@Sanchis-Gual:2015lje; @Dolan:2015dha; @Ponglertsakul:2016wae; @Sanchis-Gual:2016tcm; @Ponglertsakul:2016anb; @Sanchis-Gual:2016ros; @Dias:2018zjg; @Dias:2018yey], the stabilities of solitons, stars and black holes in a cavity were also investigated. It is interesting to note that most studies in the literature have been consider in the framework of the Eisenstein-Maxwell theory. On the other hand, we recently found that the phase structure of a nonlinear electrodynamics black hole in a cavity can be different from that in AdS space [@Wang:2019kxp; @Liang:2019dni]. It naturally raises a question whether there exists other theory beyond the Eisenstein-Maxwell theory, in which thermodynamics of a black hole can be dependent on boundary conditions.
The Gauss-Bonnet gravity is the simplest case of Lovelock theories, which extends the general relativity theory through adding higher derivative terms into the Einstein-Hilbert action. In [@Boulware:1985wk; @Zwiebach:1985uq], it showed that the Gauss-Bonnet term is naturally consistent with the first-order $\alpha^{\prime}$ correction of closed string low energy effective action. The Gauss-Bonnet AdS black solution was first obtained in [@Cai:2001dz]. After that, the thermodynamic properties and phase structure of a Gauss-Bonnet black hole in AdS space are discussed in various scenarios [@Cvetic:2001bk; @Kim:2007iw; @Hendi:2015bna; @Sun:2016til; @Zeng:2016aly; @Sahay:2017hlq]. It is worth noting that the Gauss-Bonnet term is a topological invariant in four dimensions, and hence thermodynamics of a Gauss-Bonnet black hole is always analyzed in higher dimensions.
In this paper, we study the thermodynamic properties and phase structure of a Gauss-Bonnet-Maxwell black hole in a cavity in a canonical ensemble. We find that thermodynamics and phase structure of a Gauss-Bonnet-Maxwell black hole in a cavity bear striking resemblance to that in AdS space. This paper is organized as follows. In section \[Sec:MGBBH\], we first review the Gauss-Bonnet-Maxwell black hole solution and obtain the Euclidean action of the Gauss-Bonnet-Maxwell black hole in a cavity. In section \[Sec:thermodynamics\], we then discuss thermodynamics of the Gauss-Bonnet-Maxwell black hole in a cavity and show that the first law of thermodynamics is satisfied. In section \[Sec:phase transition\], the phase structure of the black hole in cavity is discussed in five and six dimensions. We summarize our results in section \[Sec:conclusion\]. Finally, we discuss the phase structure of a Gauss-Bonnet-Maxwell AdS black hole in the appendix. We take $G=\hbar=c=k_{B}=1$ for simplicity in this paper.
Gauss-Bonnet Black Hole in a Cavity {#Sec:MGBBH}
===================================
In this section, we briefly review the Gauss-Bonnet-Maxwell black hole solution and obtain the Euclidean action. It is worth noting that in this paper, we study the black hole thermodynamics in a canonical ensemble, in which the temperature and charge are fixed on the boundary.
Considering the Gauss-Bonnet gravity coupled to Maxwell theory on a $D$-dimensional spacetime manifold $\mathcal{M}$ with a time-like boundary $\partial\mathcal{M}$, we can write the action as $$\mathcal{S}=\mathcal{S}_{\mathrm{bulk}}+\mathcal{S}_{\mathrm{surf}}.\label{eq:action}$$ Here, the bulk action is given by $$\mathcal{S}_{\mathrm{bulk}}=\frac{1}{16\pi}\underset{\mathcal{M}}{\int}d^{D}x\sqrt{-g}\left[R+\alpha\left(R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)-F^{\mu\nu}F_{\mu\nu}\right],$$ where $\alpha$ is the Gauss-Bonnet coupling constant. Generally, $\alpha$ is positive since it is associated with string length’s square of string theory [@Boulware:1985wk]. Furthermore, $F_{\mu\nu}$ is the electromagnetic field strength tensor, which is defined as $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ in terms of the vector potential $A_{\mu}$. On the boundary $\partial\mathcal{M}$, the surface terms are $$\begin{aligned}
\mathcal{S}_{\mathrm{surf}} & = & -\frac{1}{8\pi}\underset{\mathcal{\partial M}}{\int}d^{D-1}x\sqrt{-\gamma}\left[K+2\alpha\left(J-2\widehat{G}^{\mu\nu}K_{\mu\nu}\right)-K_{0}-2\alpha\left(J_{0}-2\widehat{G}_{0}^{\mu\nu}\left(K_{0}\right)_{\mu\nu}\right)\right]\nonumber \\
& & -\frac{1}{16\pi}\underset{\mathcal{\partial M}}{\int}d^{D-1}x\sqrt{-\gamma}n_{\nu}F^{\mu\nu}A_{\mu},\end{aligned}$$ where $\gamma$ is the determinant of the induced metric on $\partial\mathcal{M}$, $K_{\mu\nu}$ is the external curvature of $\partial\mathcal{M}$, $K$ is the trace of the external curvature, $J$ is the trace of $$\begin{aligned}
J_{\mu\nu} & \equiv & \frac{1}{3}\left(2KK_{\mu\gamma}K_{\,\nu}^{\gamma}+K_{\gamma\lambda}K^{\gamma\lambda}K_{\mu\nu}-2K_{\mu\gamma}K^{\gamma\lambda}K_{\lambda\nu}-K^{2}K_{\mu\nu}\right),\end{aligned}$$ $\widehat{G}^{\alpha\beta}$ is the $D-1$ dimensional Einstein tensor on $\partial\mathcal{M}$ corresponding to induced metric $\gamma_{ab}$, and $K_{0}$, $J_{0}$, $\widehat{G}_{0}^{\alpha\beta}$ are the correlative quantities when boundary $\partial\mathcal{M}$ embedded in flat spacetime [@Myers:1987yn]. Note that the Gauss-Bonnet term is a topological invariant in four dimensions, so we will consider $D\geq5$ in what follows.
By varying the action (\[eq:action\]), we find the equations of motion $$\begin{aligned}
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+H_{\mu\nu} & = & 8\pi T_{\mu\nu},\nonumber \\
\nabla_{\mu}F^{\mu\nu} & = & 0,\end{aligned}$$ where $$\begin{aligned}
H_{\mu\nu} & = & -\frac{1}{2}\alpha\left(R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)g_{\mu\nu}\nonumber \\
& & +2\alpha\left(RR_{\mu\nu}-2R_{\mu\alpha}R^{\alpha\beta}g_{\beta\nu}-2R_{\mu\lambda\nu\sigma}R^{\lambda\sigma}+g_{\beta\nu}R_{\mu\gamma\sigma\lambda}R^{\beta\gamma\sigma\lambda}\right),\\
T_{\mu\nu} & = & \frac{1}{4\pi}\left(-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}g_{\mu\nu}+F_{\mu}^{\,\lambda}F_{\nu\lambda}\right).\nonumber\end{aligned}$$ We consider a static spherically symmetric black hole solution with the metric $$\begin{aligned}
ds^{2} & = & -f\left(r\right)dt^{2}+\frac{dr^{2}}{f\left(r\right)}+r^{2}d\Omega_{D-2}\text{,}\nonumber \\
A & = & A_{t}\left(r\right)dt\text{.}\label{eq:ansatz}\end{aligned}$$ The equations of motion then reduce to $$\begin{aligned}
0 & = & \left[\left(D-3\right)\left(1-f\left(r\right)\right)-rf^{\prime}\left(r\right)\right]r^{D-4}+2\widetilde{\alpha}f^{\prime}\left(r\right)\left(f\left(r\right)-1\right)r^{D-5}\nonumber \\
& & +\left(D-5\right)\widetilde{\alpha}\left(f\left(r\right)-1\right)^{2}r^{D-6}+\frac{4}{D-2}\left(-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+F^{rt}\partial_{r}A_{t}\left(r\right)\right)r^{D-2},\label{eq:ttEOM}\\
0 & = & \left[\left(D-4\right)\left(D-3\right)\left(1-f\left(r\right)\right)-2\left(D-3\right)rf^{\prime}\left(r\right)-r^{2}f^{\prime\prime}\left(r\right)\right]r^{D-5}\nonumber \\
& & +2\widetilde{\alpha}\left[f^{\prime}\left(r\right)^{2}+\left(f\left(r\right)-1\right)f^{\prime\prime}\left(r\right)\right]r^{D-5}+4\left(D-5\right)\widetilde{\alpha}\left(f\left(r\right)-1\right)f^{\prime}\left(r\right)r^{D-6}\nonumber \\
& & +\left(D-5\right)\left(D-6\right)\widetilde{\alpha}\left(f\left(r\right)-1\right)^{2}r^{D-7}-F^{\mu\nu}F_{\mu\nu}r^{D-3},\label{eq:thetathetaEOM}\\
0 & = & \left[r^{D-2}F^{rt}\right]^{\prime}\text{,}\label{eq:MaxwellEOM}\end{aligned}$$ where we denote $\widetilde{\alpha}\equiv\alpha\left(D-3\right)\left(D-4\right)$ for simplicity. Integrating eqns. (\[eq:ttEOM\]) and (\[eq:MaxwellEOM\]), we obtain the solution $$f\left(r\right)=1+\frac{r^{2}}{2\widetilde{\alpha}}\left[1-\sqrt{1+4\widetilde{\alpha}\left(\frac{16\pi M}{\left(D-2\right)\omega_{D-2}r^{D-1}}-\frac{32\pi^{2}Q^{2}}{\left(D-2\right)\left(D-3\right)r^{2D-4}\omega_{D-2}^{2}}\right)}\right],\label{eq:f(r)inf}$$ where $M$ is the ADM mass and $Q$ is the charge of the black hole, and $\omega_{D-2}$ is the volume of the unit $D-2$ sphere [@Boulware:1985wk]. The outer event horizon radius $r_{+}$ of the black hole satisfies $f\left(r_{+}\right)=0$. Therefore, the metric function $f\left(r\right)$ can be rewritten in terms of $r_{+}$: $$f\left(r\right)=1+\frac{r^{2}}{2\widetilde{\alpha}}\left(1-\sqrt{1+4\widetilde{\alpha}\left[\frac{r_{+}^{D-5}}{r^{D-1}}\widetilde{\alpha}+\frac{r_{+}^{D-3}}{r^{D-1}}+\frac{32\pi^{2}Q^{2}}{\left(D-2\right)\left(D-3\right)r^{D-1}\omega_{D-2}^{2}}\left(\frac{1}{r_{+}^{D-3}}-\frac{1}{r^{D-3}}\right)\right]}\right).\label{eq:f(r)}$$
The Euclidean action $\mathcal{S}^{E}$ can be related to the action $\mathcal{S}$ (\[eq:action\]): $\mathcal{S}^{E}=i\mathcal{S}$. Using the analytic continuation $t=i\tau$ and $A_{\tau}d\tau=A_{t}dt$, we can obtain
$$A_{\tau}=iA_{t}\text{,}$$ which gives $F^{r\tau}=iF^{rt}$. Suppose that the black hole lives in a spherical cavity, where the boundary $\partial\mathcal{M}$ is at $r=r_{B}$. Since the temperature $T$ is fixed on the boundary of the cavity, we can impose the boundary condition at $r=r_{B}$ in terms of the reciprocal temperature: $$\int\sqrt{f\left(r_{B}\right)}d\tau=T^{-1},$$ which identifies the Euclidean time $\tau$ as $\tau\sim\tau+\frac{1}{T\sqrt{f\left(r_{B}\right)}}$, and hence the period of $\tau$ is $\frac{1}{T\sqrt{f\left(r_{B}\right)}}$. Integrating the Euclidean action and using eqn. $\eqref{eq:f(r)}$, the Euclidean action is rewritten as $$\begin{aligned}
\mathcal{S}^{E} & = & \frac{1}{8\pi}\left(D-2\right)\frac{\omega_{D-2}r_{B}^{D-3}}{T}\left(1-\sqrt{f\left(r_{B}\right)}\right)-S\nonumber \\
& & +\frac{\widetilde{\alpha}}{12\pi}\left(D-2\right)\frac{\omega_{D-2}}{T}r_{B}^{D-5}\left(\sqrt{f\left(r_{B}\right)}f\left(r_{B}\right)-3\sqrt{f\left(r_{B}\right)}+2\right),\label{eq:EAction}\end{aligned}$$ where $S=\frac{1}{4}\omega_{D-2}r_{+}^{D-2}\left[1+2\widetilde{\alpha}\left(D-2\right)/\left(D-4\right)r_{+}^{2}\right]$ is the entropy of the black hole.
Thermodynamics {#Sec:thermodynamics}
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In the semi-classical approximation, the on-shell Euclidean action is related to the free energy $F$: $$F=-T\ln Z=T\mathcal{S}^{E}.$$ From eqn. (\[eq:EAction\]), we can express the the free energy $F$ in terms of the temperature $T$, the charge $Q$, the Gauss-Bonnet parameter $\alpha\left(\widetilde{\alpha}\right)$ , the cavity radius $r_{B}$ and the horizon radius $r_{+}$: $$\begin{aligned}
F & = & \frac{1}{8\pi}\left(D-2\right)\omega_{D-2}r_{B}^{D-3}\left(1-\sqrt{f\left(r_{B}\right)}\right)-\frac{1}{4}S_{D-2}r_{+}^{D-2}\left(1+\frac{D-2}{D-4}\frac{2\widetilde{\alpha}}{r_{+}^{2}}\right)T\nonumber \\
& & +\frac{\widetilde{\alpha}}{12\pi}\left(D-2\right)\omega_{D-2}r_{B}^{D-5}\left(\sqrt{f\left(r_{B}\right)}f\left(r_{B}\right)-3\sqrt{f\left(r_{B}\right)}+2\right).\label{eq:F}\end{aligned}$$ where $T$, $Q$ , $\alpha\left(\widetilde{\alpha}\right)$ and $r_{B}$ are parameters of the canonical ensemble and the horizon radius $r_{+}$ is the only variable,
$$F=F\left(r_{+};T,Q,\alpha,r_{B}\right).\label{eq:F(rplus)}$$
By extremizing the free energy $F\left(r_{+};T,Q,\alpha,r_{B}\right)$ with respect to $r_{+}$, we can determine the only variable $r_{+}$: $$\begin{aligned}
& & \frac{dF\left(r_{+};T,Q,\alpha,r_{B}\right)}{dr_{+}}=0\nonumber \\
& \Longrightarrow & f^{\prime}\left(r_{+}\right)=4\pi T\sqrt{f\left(r_{B}\right)}.\label{eq:frpus}\end{aligned}$$ The solution $r_{+}=r_{+}\left(T,Q,\alpha,r_{B}\right)$ of eqn. $\left(\ref{eq:frpus}\right)$ is in relevance to a locally stationary point of $F\left(r_{+};T,Q,\alpha,r_{B}\right)$. Since the Hawking temperature of the black hole is defined as $T_{h}=f^{\prime}\left(r_{+}\right)/4\pi$, eqn. (\[eq:frpus\]) can be written as $$T=\frac{T_{h}}{\sqrt{f\left(r_{B}\right)}}\text{,}\label{eq:TBlue}$$ where the Hawking temperature is $$\begin{aligned}
T_{h} & = & \frac{\left(D-5\right)\widetilde{\alpha}+\left(D-3\right)r_{+}^{2}-\frac{1}{D-2}\frac{32\pi^{2}Q^{2}}{r_{+}^{2D-8}\omega_{D-2}^{2}}}{4\pi\left(1+\frac{2\widetilde{\alpha}}{r_{+}^{2}}\right)r_{+}^{3}}.\label{eq:HT}\end{aligned}$$ So for the observer on the wall, the temperature $T$ on the cavity is blueshifted from Hawking temperature $T_{h}$.
At the locally stationary point $r_{+}=r_{+}\left(T,Q,\alpha,r_{B}\right)$, the free energy $F\left(r_{+};T,Q,\alpha,r_{B}\right)$ can be express only in terms of $T$, $Q$, $\alpha\left(\widetilde{\alpha}\right)$ and $r_{B}$:
$$F\left(T,Q,\alpha,r_{B}\right)\equiv F\left(r_{+}\left(T,Q,\alpha,r_{B}\right);T,Q,\alpha,r_{B}\right)\text{.}$$
For later convenience, $F\left(r_{+};T,Q,\alpha,r_{B}\right)$ and $F\left(T,Q,\alpha,r_{B}\right)$ can be abbreviated to $F\left(r_{+}\right)$ and $F$, respectively. Furthermore, the thermal energy of the black hole in the cavity is $$\begin{aligned}
E & = & -T^{2}\frac{\partial\left(F/T\right)}{\partial T}\nonumber \\
& = & \frac{1}{8\pi}\left(D-2\right)\omega_{D-2}r_{B}^{D-3}\left[\left(1-\sqrt{f\left(r_{B}\right)}\right)+\frac{2\widetilde{\alpha}}{3}r_{B}^{2}\left(\sqrt{f\left(r_{B}\right)}f\left(r_{B}\right)-3\sqrt{f\left(r_{B}\right)}+2\right)\right].\end{aligned}$$ where the thermal energy $E$ is expressed in terms of the entropy $S$, the charge $Q$ and the cavity radius $r_{B}$. Moreover, we can define an electric potential and thermodynamic surface pressure as $$\Phi\equiv\frac{A_{t}\left(r_{B}\right)-A_{t}\left(r_{+}\right)}{\sqrt{f\left(r_{B}\right)}},\,\lambda\equiv-\frac{\partial E}{\partial\left(S_{D-2}r_{B}^{D-2}\right)}.\label{eq:lamda}$$ It is easy to verify that the differential $E$ with respect to $S$, $Q$ and area $A$ is satisfied: $$\frac{\partial E}{\partial S}=T,\,\frac{\partial E}{\partial Q}=\Phi,\,\frac{\partial E}{\partial A}=\lambda.\label{eq:ESQ}$$ where $A\equiv S_{D-2}r_{B}^{D-2}$ is the surface area of the cavity. Using eqns. $\left(\ref{eq:ESQ}\right)$ and $\left(\ref{eq:lamda}\right)$, the first law of thermodynamics can be established as $$dE=TdS+\Phi dQ-\lambda dA.$$
To discuss the thermodynamic stability of the Gauss-Bonnet-Maxwell black hole in the cavity, we consider the specific heat at constant electric charge
$$C_{Q}=T\left(\frac{\partial S}{\partial T}\right)_{Q}=\frac{1}{4}\omega_{D-2}r_{+}^{D-3}\left(D-2\right)\left(1+\frac{2\widetilde{\alpha}}{r_{+}^{2}}\right)T\frac{\partial r_{+}\left(T,Q,r_{B},\alpha\right)}{\partial T}.\label{eq:Cq}$$
Since the system is thermally stable with $C_{Q}>0$, the black holes is thermally stable when $\partial r_{+}\left(T,Q,r_{B}\right)/\partial T>0$. From $\partial^{2}F/\partial^{2}T=-C_{Q}$, the thermally stable/unstable phases have concave downward/upward $F$-$T$ curves. The black hole phase is thermally stable/unstable when $r_{+}\left(T,Q,\alpha,r_{B}\right)$ is a local minimum/maximum of $F\left(r_{+}\right)$. Note that the physical space of $r_{+}$ has boundaries, such as $$r_{e}\leq r_{+}\leq r_{B}\text{,}$$ where $r_{e}$ is the horizon radius of the extremal black hole.
Phase Transition {#Sec:phase transition}
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In this section, we will discuss the phase transition of a Gauss-Bonnet-Maxwell black hole in a cavity for $D=5$ and $D=6$. For convenience, we express the variables in terms of $r_{B}$: $$x\equiv\frac{r_{+}}{r_{B}}\text{, }\,\bar{Q}\equiv\frac{Q}{r_{B}^{D-3}}\text{, }\,\bar{\alpha}\equiv\frac{\alpha}{r_{B}^{2}}\text{, }\,\bar{T}\equiv r_{B}T\text{, }\,\bar{F}\left(x\right)\equiv\frac{8\pi F\left(r_{+}\right)}{\left(D-2\right)S_{D-2}r_{B}^{D-3}}.$$ From eqns. $\eqref{eq:F}$ and $\eqref{eq:f(r)}$, we obtain the free energy as a function of $x$: $$\begin{aligned}
\bar{F}\left(x\right) & = & 1-\sqrt{f\left(x\right)}-\frac{2\pi}{D-2}x^{D-2}\left(1+2\left(D-2\right)\left(D-3\right)\frac{\bar{\alpha}}{x^{2}}\right)\bar{T}\nonumber \\
& & +\frac{2\left(D-3\right)\left(D-4\right)\bar{\alpha}}{3}\left(\sqrt{f\left(x\right)}f\left(x\right)-3\sqrt{f\left(x\right)}+2\right),\end{aligned}$$ where the metric function becomes as $$\begin{aligned}
f\left(x\right) & = & 1+\frac{1}{2\left(D-3\right)\left(D-4\right)\bar{\alpha}}\left\{ 1-\left[1+4\left(D-3\right)\left(D-4\right)\bar{\alpha}\right.\right.\nonumber \\
& & \left.\left.\times\left(\left(D-3\right)\left(D-4\right)\bar{\alpha}x^{D-5}+x^{D-3}+\frac{32\pi^{2}\bar{Q}^{2}}{\left(D-2\right)\left(D-3\right)\omega_{D-2}^{2}}\left(\frac{1}{x^{D-3}}-1\right)\right)\right]^{\frac{1}{2}}\right\} .\end{aligned}$$ Moreover, the Hawking temperature in eqn. $\eqref{eq:HT}$ is rewritten as $$\bar{T}_{h}\equiv r_{B}T_{h}=\frac{\left(D-3\right)\left(D-4\right)\left(D-5\right)\bar{\alpha}+\left(D-3\right)x^{2}-\frac{32\pi^{2}\bar{Q}^{2}}{\left(D-2\right)x^{2D-8}\omega_{D-2}^{2}}}{4\pi\left(1+\frac{2\left(D-3\right)\left(D-4\right)\bar{\alpha}}{x^{2}}\right)x^{3}}.\label{eq:THX}$$
Five dimensions
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When $D=5$, the thermodynamic expressions are rewritten as:
$$\begin{aligned}
\bar{F}\left(x\right) & = & 1-\sqrt{f\left(x\right)}-\frac{2\pi}{3}x^{3}\left(1+12\frac{\bar{\alpha}}{x^{2}}\right)\bar{T}+\frac{4\bar{\alpha}}{3}\left(\sqrt{f\left(x\right)}f\left(x\right)-3\sqrt{f\left(x\right)}+2\right),\label{eq:free energy}\\
\bar{T} & = & \frac{x^{2}-\frac{4\bar{Q}^{2}}{3\pi^{2}x^{4}}}{2\pi\left(1+\frac{4\bar{\alpha}}{x^{2}}\right)x^{3}\sqrt{f\left(x\right)}},\end{aligned}$$ where the metric function is simplified as $$f\left(x\right)=1+\frac{1}{4\bar{\alpha}}\left[1-\sqrt{1+8\bar{\alpha}\left(2\bar{\alpha}+x^{2}+\frac{4\bar{Q}^{2}}{3\pi^{2}x^{2}}-\frac{4\bar{Q}^{2}}{3\pi^{2}}\right)}\right].$$ When it comes to the phase structure, we need to consider the locally stationary points $r_{+}=r_{+}\left(T,Q,\alpha,r_{B}\right)$, which can be multivalued and lead to more than one phase. The globally stable phase and phase transitions can be determined by calculating the free energy.
![The two regions in the $\bar{\alpha}$-$\bar{Q}$ phase space of a $D=5$ Gauss-Bonnet black hole in a cavity, each of which possesses distinct behavior of the phase structure and transition. Varying the temperature, there is only one phase in Regions , while a van der Waals-like LBH/SBH phase transition occurs in Regions .\[fig:5D-Qa\]](VdIII)
![$\bar{\alpha}=0.01$ and $\bar{Q}=0.4$ in the Regions of FIG. \[fig:5D-Qa\]. There is no phase transition.\[fig:5D-TF-1\]](VdxTQ04 "fig:")![$\bar{\alpha}=0.01$ and $\bar{Q}=0.4$ in the Regions of FIG. \[fig:5D-Qa\]. There is no phase transition.\[fig:5D-TF-1\]](VdFTQ04 "fig:")
![$\bar{\alpha}=0.01$ and $\bar{Q}=0.03$ in the Regions of FIG. \[fig:5D-Qa\]. There is first-order phase transition.\[fig:5D-TF-2\]](VdxTQ003 "fig:") ![$\bar{\alpha}=0.01$ and $\bar{Q}=0.03$ in the Regions of FIG. \[fig:5D-Qa\]. There is first-order phase transition.\[fig:5D-TF-2\]](VdFTQ003 "fig:")
In FIG. \[fig:5D-Qa\], we show that there are two regions in the $\bar{\alpha}$-$\bar{Q}$ phase space of a Gauss-Bonnet-Maxwell black hole in a cavity, each of which possesses distinct behavior of the phase structure and transition. There is only one phase in Regions while a van der Waals-like LBH/SBH phase transition occurs in Regions . Moreover, FIG. \[fig:5D-Qa\] and the left panel of FIG. \[fig:AdS-Qa\] show that the phase structure of a Gauss-Bonnet-Maxwell black hole in a cavity is quite similar to that of a Gauss-Bonnet-Maxwell AdS black hole. Specifically for a black hole with $\bar{\alpha}=0.01$ and charge $\bar{Q}=0.4$ in Region I, we plot the radius of the black hole horizon radius and the free energy against the temperature in FIG. \[fig:5D-TF-1\], which shows that the system has a single phase structure and no phase transition. In FIG. \[fig:5D-TF-2\], we consider a black hole with $\bar{\alpha}=0.01$ and $\bar{Q}=0.03$ in Region II and plot the horizon radius and the free energy against the temperature. The left panel of FIG. \[fig:5D-TF-2\] shows that, in some ranges of the temperature, there exists more than one horizon radius of the black hole for a fixed value of temperature. This means that the system can posses a multi-phase structure, which consists of the small, intermediate and large black hole phases. From. (\[eq:Cq\]), the small and large black holes are thermally stable while the intermediate one is unstable. From the right panel of FIG. \[fig:5D-TF-1\], we find that, as the temperature increases, the system undergoes a first-order van der Waals-like phase transition from a small black hole and a large one.
Six dimensions
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![The two regions in the $\bar{\alpha}$-$\bar{Q}$ phase of a $D=6$ Gauss-Bonnet black hole in a cavity, each of which possesses distinct behavior of the phase structure and transition. Varying the temperature, there is only one phase in Regions , while a van der Waals-like phase transition occurs in Regions .\[fig:6D-Qa\]](VIdIII)
In six dimensions, the thermodynamic expressions are simplified as follow: $$\begin{aligned}
\bar{F}\left(x\right) & = & 1-\sqrt{f\left(x\right)}-\frac{\pi}{2}x^{4}\left(1+24\frac{\bar{\alpha}}{x^{2}}\right)\bar{T}+4\bar{\alpha}\left(\sqrt{f\left(x\right)}f\left(x\right)-3\sqrt{f\left(x\right)}+2\right),\\
\bar{T} & = & \frac{6\bar{\alpha}+3x^{2}-\frac{9\bar{Q}^{2}}{8\pi^{2}x^{6}}}{4\pi\left(1+\frac{12\bar{\alpha}}{x^{2}}\right)x^{3}\sqrt{f\left(x\right)}},\end{aligned}$$ where $$f\left(x\right)=1+\frac{1}{12\bar{\alpha}}\left[1-\sqrt{1+24\bar{\alpha}\left(6\bar{\alpha}x+x^{3}+\frac{9\bar{Q}^{2}}{24\pi^{2}x^{3}}-\frac{9\bar{Q}^{2}}{24\pi^{2}}\right)}\right].$$
![Left Panel $\bar{\alpha}=0.005$ and $\bar{Q}=0.4$ in the Regions of FIG. \[fig:5D-Qa\]. There is no phase transition.\[fig:6D-TF-1\]](VIdxTQ04 "fig:")![Left Panel $\bar{\alpha}=0.005$ and $\bar{Q}=0.4$ in the Regions of FIG. \[fig:5D-Qa\]. There is no phase transition.\[fig:6D-TF-1\]](VIdFTQ04 "fig:")
![$\bar{\alpha}=0.005$ and $\bar{Q}=0.01$ in the Regions of FIG. \[fig:5D-Qa\]. There is first-order phase transition.\[fig:6D-TF-2\]](VIdxTQ001 "fig:")![$\bar{\alpha}=0.005$ and $\bar{Q}=0.01$ in the Regions of FIG. \[fig:5D-Qa\]. There is first-order phase transition.\[fig:6D-TF-2\]](VIdFTQ001 "fig:")
It is worth noting that, unlike the $D=5$ case, the extremal temperature is dependent on $\bar{\alpha}$ in the $D=6$ case. The two regions in the $\bar{\alpha}$-$\bar{Q}$ phase space are plotted in FIG. \[fig:6D-Qa\]. As shown in FIG. \[fig:6D-TF-1\], there is only one phase for the black holes in Region I. On the other hand, FIG. \[fig:6D-TF-2\] shows that, for the black holes in Region II, there exists a band of temperatures where three phases coexist, and a first-order van der Waals-like phase transition occurs. Note that the phase transition structure of a $D=6$ Gauss-Bonnet-Maxwell black hole in a cavity is quite similar to that of a $D=6$ Gauss-Bonnet-Maxwell AdS black hole, which is shown in the right panel of FIG. \[fig:AdS-Qa\].
Conclusion {#Sec:conclusion}
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In this paper, we first calculated the Euclidean action of a Gauss-Bonnet-Maxwell black hole in a finite spherical cavity and obtained the corresponding free energy in a canonical ensemble by semi-classical approximation. Moreover, the first law of thermodynamics was found to be satisfied. In the rest of this paper, we mainly discussed the phase structure and transition of a Gauss-Bonnet-Maxwell black hole in a cavity for $D=5$ and $D=6$. In five dimensions, there are two regions in the $\bar{\alpha}$-$\bar{Q}$ phase space in FIG. \[fig:5D-Qa\]. In Region I, there is a one-to-one correspondence between temperature and free energy, so no phase transition occurs. In Region II, there exists a three phase coexistence, in which the small and large black holes are both stable with a positive specific heat while the intermediate black hole is unstable. As the temperature of the system increases, the system starts from a small black hole, undergoes a van der Waals-like phase transition and ends in a large black hole. In six dimensions, the two regions are presented in the $\bar{\alpha}$-$\bar{Q}$ phase space in FIG. \[fig:6D-Qa\]. Similarly, no phase transition and a van der Waals-like phase transition occur in Regions I and II, respectively. Finally, we found that the phase structure of a Gauss-Bonnet-Maxwell black hole in cavity is almost the same as that of a Gauss-Bonnet-Maxwell in AdS space, which is discussed in the appendix.
[**Acknowledgements**]{} We are grateful to Qingyu Gan, Guangzhou Guo and Houwen Wu for useful discussions and valuable comments. This work is supported in part by the NSFC (Grant No. 11875196, 11375121 and 11005016).
Gauss-Bonnet black hole in AdS space
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In this appendix, we briefly discuss phase structure of a Gauss-Bonnet-Maxwell black hole in AdS space. In [@Cai:2001dz], the metric function was obtained as $$f\left(r\right)=1+\frac{r^{2}}{2\widetilde{\alpha}}\left[1-\sqrt{1+4\widetilde{\alpha}\left(-\frac{1}{l^{2}}+\frac{16\pi M}{\left(D-2\right)\omega_{D-2}r^{D-1}}-\frac{32\pi^{2}Q^{2}}{\left(D-2\right)\left(D-3\right)r^{2D-4}\omega_{D-2}^{2}}\right)}\right],\label{eq:f(r)inf-3}$$ where $l$ is the radius of the AdS space. Using eqn. (\[eq:f(r)inf-3\]), we can express ADM mass $M$ of the black hole in terms of the horizon radius $r_{+}$, $$M=\left(\frac{\widetilde{\alpha}}{r_{+}^{2}}+1+\frac{r_{+}^{2}}{l^{2}}+\frac{32\pi^{2}Q^{2}}{\left(D-2\right)\left(D-3\right)r_{+}^{2D-6}\omega_{D-2}^{2}}\right)\frac{r_{+}^{D-3}\left(D-2\right)\omega_{D-2}}{16\pi G}.\label{eq:M}$$ Furthermore, the black hole temperature and entropy are $$\begin{aligned}
T & \equiv & \frac{f^{\prime}\left(r_{+}\right)}{4\pi}=\frac{\left(D-1\right)\frac{r_{+}^{2}}{l^{2}}+\left(D-5\right)\frac{\widetilde{\alpha}}{r_{+}^{2}}+\left(D-3\right)-\frac{32\pi^{2}Q^{2}}{\left(D-2\right)\omega_{D-2}^{2}r_{+}^{2D-6}}}{4\pi\left(1+\frac{2\widetilde{\alpha}}{r_{+}^{2}}\right)r_{+}},\label{eq:T}\\
S & \equiv & \int\frac{1}{T}\frac{\partial M}{\partial r_{+}}dr_{+}=\frac{1}{4}\omega_{D-2}r_{+}^{D-2}\left(1+\frac{D-2}{D-4}\frac{2\widetilde{\alpha}}{r_{+}^{2}}\right),\label{eq:S}\end{aligned}$$ respectively. The free energy of the black hole is defined as $F=M-TS$, which can be expressed in terms of horizon radius $r_{+}$, AdS radius $l$, black hole charge $Q$ and Gauss-Bonnet parameter $\widetilde{\alpha}$:
$$\begin{aligned}
F & = & \left(\frac{\widetilde{\alpha}}{r_{+}^{4}}+\frac{1}{r_{+}^{2}}+\frac{1}{l^{2}}+\frac{32\pi^{2}Q^{2}}{\left(D-2\right)\left(D-3\right)r_{+}^{2D-4}\omega_{D-2}^{2}}\right)\frac{r_{+}^{D-1}\left(D-2\right)\omega_{D-2}}{16\pi}\nonumber \\
& & -T\frac{1}{4}\omega_{D-2}r_{+}^{D-2}\left(1+\frac{D-2}{D-4}\frac{2\widetilde{\alpha}}{r_{+}^{2}}\right).\end{aligned}$$
We can also define $$\bar{r}_{+}\equiv\frac{r_{+}}{l}\text{, }\,\bar{Q}\equiv\frac{Q}{l^{D-3}}\text{, }\,\bar{\alpha}\equiv\frac{\alpha}{l^{2}}\text{, }\,\bar{T}\equiv Tl\text{, }\,\bar{S}\equiv\frac{S}{\omega_{D-2}l^{D-2}}\text{, }\,\bar{F}\equiv\frac{F}{\omega_{D-2}l^{D-3}}.$$ It should be noted that the Gauss-Bonnet parameter is constrained as $$0\leq\bar{\alpha}\leq\frac{1}{4\left(D-3\right)\left(D-4\right)},$$ since the square root of eqn.(\[eq:f(r)inf-3\]) should be greater than zero when $M=Q=0$. Solving eqn.(\[eq:T\]) for $r_{+}$ in terms of $T$, we can write the free energy $\bar{F}$ as a function of the temperature $\bar{T}$, the charge $\bar{Q}$, the Gauss-Bonnet parameter $\bar{\alpha}$, and the horizon radius $\bar{r}_{+}$. In the both $D=5$ and $D=6$, FIG. \[fig:AdS-Qa\] shows that there are two regions in the $\bar{\alpha}$-$\bar{Q}$ phase space. In Region I, there is a single phase and no phase transition. However in Region II, three phases can coexist for some range of temperature, and a first-order van der Waals-like phase transition occurs.
![The two regions in the $\bar{\alpha}$-$\bar{Q}$ represent the different phase structure of a Gauss-Bonnet black hole in AdS space. The left panel is for the black holes in five dimensions, while the right panel is in six dimensions. In both cases, the yellow regions (Region I) have only one phase, while a van der Waals-like phase transition occurs in the cyan regions (Region II).\[fig:AdS-Qa\]](AdSVdIII "fig:")![The two regions in the $\bar{\alpha}$-$\bar{Q}$ represent the different phase structure of a Gauss-Bonnet black hole in AdS space. The left panel is for the black holes in five dimensions, while the right panel is in six dimensions. In both cases, the yellow regions (Region I) have only one phase, while a van der Waals-like phase transition occurs in the cyan regions (Region II).\[fig:AdS-Qa\]](AdSVIdIII "fig:")
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abstract: 'The effectiveness of learning in massive open online courses (MOOCs) can be significantly enhanced by introducing personalized intervention schemes which rely on building predictive models of student learning behaviors such as some engagement or performance indicators. A major challenge that has to be addressed when building such models is to design handcrafted features that are effective for the prediction task at hand. In this paper, we make the first attempt to solve the feature learning problem by taking the unsupervised learning approach to learn a compact representation of the raw features with a large degree of redundancy. Specifically, in order to capture the underlying learning patterns in the content domain and the temporal nature of the clickstream data, we train a modified auto-encoder (AE) combined with the long short-term memory (LSTM) network to obtain a fixed-length embedding for each input sequence. When compared with the original features, the new features that correspond to the embedding obtained by the modified LSTM-AE are not only more parsimonious but also more discriminative for our prediction task. Using simple supervised learning models, the learned features can improve the prediction accuracy by up to 17% compared with the supervised neural networks and reduce overfitting to the dominant low-performing group of students, specifically in the task of predicting students’ performance. Our approach is generic in the sense that it is not restricted to a specific supervised learning model nor a specific prediction task for MOOC learning analytics.'
author:
- Mucong Ding
- Kai Yang
- 'Dit-Yan Yeung'
- 'Ting-Chuen Pong'
bibliography:
- 'references.bib'
title: Effective Feature Learning with Unsupervised Learning for Improving the Predictive Models in Massive Open Online Courses
---
<ccs2012> <concept> <concept\_id>10010147.10010257.10010258.10010260</concept\_id> <concept\_desc>Computing methodologies Unsupervised learning</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010293.10010294</concept\_id> <concept\_desc>Computing methodologies Neural networks</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010147.10010257.10010293.10010319</concept\_id> <concept\_desc>Computing methodologies Learning latent representations</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010405.10010489.10010495</concept\_id> <concept\_desc>Applied computing E-learning</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
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abstract: 'We prove that groups for which every countable subgroup is free ($\aleph_1$-free groups) are n-slender, cm-slender, and lcH-slender. In particular every homomorphism from a completely metrizable group to an $\aleph_1$-free group has an open kernel. We also show that $\aleph_1$-free abelian groups are lcH-slender, which is especially interesting in light of the fact that some $\aleph_1$-free abelian groups are neither n- nor cm-slender. The strongly $\aleph_1$-free abelian groups are shown to be n-, cm-, and lcH-slender. We also give a characterization of cm- and lcH-slender abelian groups.'
address: 'Ikerbasque- Basque Foundation for Science and Matematika Saila, UPV/EHU, Sarriena S/N, 48940, Leioa - Bizkaia, Spain'
author:
- 'Samuel M. Corson'
title: 'Automatic continuity of $\aleph_1$-free groups'
---
[^1]
[Introduction]{} Graham Higman defined a group to be *$\kappa$-free*, with $\kappa$ a cardinal number, if each subgroup generated by fewer than $\kappa$ elements is a free group [@H1]. By the Nielsen-Schreier Theorem each free group is $\kappa$-free for all cardinals $\kappa$. The additive group of the rationals $\mathbb{Q}$ is an example of an $\aleph_0$-free group of cardinality $\aleph_0$ which is not free. Higman produced an example of an $\aleph_1$-free group of cardinality $\aleph_1$ which is not free, and $\kappa$-free groups have been a focus of much study since then ([@H1], [@S], [@EkMe], [@MaS]). We prove that $\aleph_1$-free groups satisfy strong automatic continuity conditions.
Following [@CC] we define a group $H$ to be *cm-slender* if every abstract homomorphism from a completely metrizable topological group to $H$ has open kernel. Similarly $H$ is *lcH-slender* provided each abstract homomorphism from a locally compact Hausdorff topological group to $H$ has open kernel. If, for example, a group $H$ is cm-slender then the only completely metrizable topology that can be imposed on $H$ to make $H$ a topological group is the discrete topology.
A further notion of automatic continuity comes from fundamental groups: a group $H$ is *n-slender* if every abstract group homomorphism from the fundamental group ${\operatorname{HEG}}$ of the Hawaiian earring to $H$ factors through a finite bouquet of circles [@Ed] (see Section \[Automatic\]). Free (abelian) groups were shown to be cm- and lcH-slender in [@D] and free groups were shown to be n-slender in [@H2]. Many groups have since been shown to be n-, cm- and lcH-slender, and each of these notions of slenderness requires a group to be torsion-free and to not have $\mathbb{Q}$ as a subgroup (see [@CC] for more exposition).
We prove the following:
\[thebigone\] $\aleph_1$-free groups are n-slender, cm-slender, and lcH-slender.
As free groups are $\aleph_1$-free, this result is a strengthening of the classical facts that free groups are n-, cm- and lcH-slender. The fact that $\aleph_1$-free groups are cm-slender immediately implies a result of Khelif [@Kh] that an uncountable $\aleph_1$-free group is not the homomorphic image of a *Polish group* (a topological group which is separable and completely metrizable). The cm-slenderness of $\aleph_1$-free groups can be obtained by modifying Khelif’s proof. We give a different proof which is both well suited to proving all three types of slenderness and seemingly simpler.
Theorem \[thebigone\] cannot be strengthened by substituting $\aleph_0$-freeness (that is, local freeness) for $\aleph_1$-freeness. The group ${\operatorname{HEG}}$ is itself locally free and by considering the identity map we see that local freeness does not imply that a group is n-slender. The group $\mathbb{Q}$ is locally free, and using a Hamel basis of $\mathbb{R}$ over $\mathbb{Q}$ it is possible to construct a homomorphism from $\mathbb{R}$ to $\mathbb{Q}$ which is not continuous. Since $\mathbb{R}$ is both locally compact Hausdorff and completely metrizable, local freeness implies neither cm- nor lcH-slenderness.
Analogously define a group to be *$\kappa$-free abelian* if each subgroup generated by fewer than $\kappa$ elements is free abelian. A group which is $\aleph_1$-free abelian needn’t be n- or cm-slender: the countably infinite product $\prod_{\omega}\mathbb{Z}$ is $\aleph_1$-free abelian [@B]. This group has a completely metrizable topological group structure given by taking each $\mathbb{Z}$ to be discrete and giving the entire group the product topology, and so the identity map on $\prod_{\omega}\mathbb{Z}$ shows that an $\aleph_1$-free abelian group need not be cm-slender. Also there is a canonical homomorphism from ${\operatorname{HEG}}$ to $\prod_{\omega}\mathbb{Z}$ which does not have open kernel, so n-slenderness needn’t hold for an $\aleph_1$-free abelian group either. However we have the following:
\[lcHslender\] $\aleph_1$-free abelian groups are lcH-slender.
Theorem \[lcHslender\] cannot be strengthened by replacing $\aleph_1$ with $\aleph_0$ since $\mathbb{Q}$ is not lcH-slender. We prove Theorem \[lcHslender\] from the following classification (see definitions in Section \[abeliancase\]):
\[characterizationoflcHslenderabelian\] If $H$ is an abelian group then
1. $H$ is cm-slender if and only if $H$ is torsion-free, reduced and contains no subgroup which admits a non-discrete Polish topology
2. $H$ is lcH-slender if and only if $H$ is cotorsion-free
The n-slender abelian groups are already known to be precisely the slender groups [@Ed], and Theorem \[characterizationoflcHslenderabelian\] was already known for abelian groups of cardinality $<2^{\aleph_0}$ (see [@CC Theorem C]). Thus among abelian groups we have
cm-slender $\Longrightarrow$ n-slender $\Longrightarrow$ lcH-slender
For n- and cm-slenderness we need to demand a bit more from an $\aleph_1$-free abelian group (see Definition \[stronglydef\]):
\[strongly\] Strongly $\aleph_1$-free abelian groups are n-slender, cm-slender and lcH-slender.
We have already seen that the modifier “strongly” may not be dropped while concluding n- and cm-slenderness. In Section \[Automatic\] we prove Theorem \[thebigone\] and in Section \[abeliancase\] we prove Theorems \[characterizationoflcHslenderabelian\], \[lcHslender\] and \[strongly\].
[Automatic continuity in the non-abelian case]{}\[Automatic\]
We begin this section with a review of the Hawaiian earring group ${\operatorname{HEG}}$. After this we give background lemmas and prove Theorem \[thebigone\]. Start with a countably infinite set $\{a_n^{\pm 1}\}_{n\in \omega}$ which has formal inverses. We say a function $W: \overline{W} \rightarrow \{a_n^{\pm 1}\}_{n\in \omega}$ is a *word* if the domain $\overline{W}$ is a totally ordered set and for each $m$ the preimage $W^{-1}(\{a_{n}^{\pm 1}\}_{n=0}^m)$ is finite. We write $W \equiv U$ for words $W$ and $U$ provided there exists an order isomorphism $\iota: \overline{W} \rightarrow \overline{U}$ such that $W(i) = U(\iota(i))$. Let ${\mathcal{W}}$ denote a selection from each $\equiv$ class. For $m\in \omega$ let $p_m$ denote the map from ${\mathcal{W}}$ to the set of finite words given by the restriction $p_m(W) \equiv W\upharpoonright\{i\in \overline{W}\mid W(i)\in \{a_{n}^{\pm 1}\}_{n=0}^m\}$.
For $W, U \in {\mathcal{W}}$ we write $W \sim U$ if for every $m\in \omega$ the words $p_m(W)$ and $p_m(U)$ are equal as elements in the free group $F(a_0, \ldots, a_m)$. For $U \in {\mathcal{W}}$ we write $U^{-1}$ for the word whose domain is $\overline{U}$ under the reverse order satisfying $U^{-1}(i) = (U(i))^{-1}$. We concatenate two words $W, U\in {\mathcal{W}}$ by letting $\overline{WU}$ be the disjoint union $\overline{W} \sqcup \overline{U}$ under the order which preserves that of both $\overline{W}$ and $\overline{U}$ and places elements in $\overline{W}$ below those of $\overline{U}$. The map $WU$ is given by $WU(i) = \begin{cases}W(i)$ if $i\in \overline{W}\\ U(i)$ if $i\in \overline{U}\end{cases}$
The quotient set ${\operatorname{HEG}}= {\mathcal{W}}/\sim$ has a group structure given by $[W][U] = [WU]$ and $[U]^{-1} = [U^{-1}]$. The free group $F(a_0, \ldots, a_m)$ embeds naturally into ${\operatorname{HEG}}$ by considering finite words in $\{a_n\}_{n=0}^m$ as words as defined above, and we let ${\operatorname{HEG}}_m$ denote this copy of the free group. Each aforementioned map $p_m: {\mathcal{W}}\rightarrow {\mathcal{W}}$ induces a homomorphic retraction $p_m:{\operatorname{HEG}}\rightarrow {\operatorname{HEG}}_m$. For each $m$ we similarly have a word map $p^m(W) \equiv W\upharpoonright\{i\in \overline{W}\mid W(i)\notin \{a_{n}^{\pm 1}\}_{n=0}^m\}$ which defines a retraction to the subgroup ${\operatorname{HEG}}^m$ consisting of those elements of ${\operatorname{HEG}}$ which have a representative $W$ for which $W(\overline{W}) \cap \{a_{n}^{\pm 1}\}_{n=0}^m = \emptyset$. There is a natural decomposition ${\operatorname{HEG}}\simeq {\operatorname{HEG}}_m *
{\operatorname{HEG}}^m$ for each $m$ given by considering a word as a finite concatenation of words utilizing elements in $\{a_n^{\pm 1}\}_{n=0}^m$ and words which do not. The following definition is found in [@Ed]:
\[nslender\] A group $H$ is *n-slender* if for every homomorphism $\phi: {\operatorname{HEG}}\rightarrow H$ there exists $m\in \omega$ for which $\phi = \phi \circ p_m$. Equivalently $H$ is n-slender if for every homomorphism $\phi: {\operatorname{HEG}}\rightarrow H$ there exists $m\in \omega$ such that ${\operatorname{HEG}}^m \leq \ker(\phi)$.
We will make use of the following (see [@H1 Theorem 1]):
\[basic\] If $H$ is an $\aleph_1$-free group then each nondecreasing sequence $\{K_n\}_{n\in \omega}$ of finitely generated subgroups of $H$ such that $K_n$ is not contained in a proper free factor of $K_{n+1}$ must eventually stabilize. Moreover every finitely generated $H_0 \leq H$ is included in a finitely generated $H_0 \leq H_1$ such that $H_1$ is a free factor of each free subgroup of $H$ which contains it.
We call such a subgroup $H_1$ as is asserted in the second sentence of Lemma \[basic\] a *basic* subgroup [@H1].
\[nicelemma\] The following hold:
1. If $\phi: {\operatorname{HEG}}\rightarrow H$ is a homomorphism to an $\aleph_1$-free group then for every finitely generated $F \leq H$ there exists $n\in \omega$ for which $\phi({\operatorname{HEG}}^n) \cap F = \{1_H\}$.
2. If $\phi: G \rightarrow H$ is a homomorphism with $G$ either completely metrizable or locally compact Hausdorff and $H$ an $\aleph_1$-free group then for every finitely generated $F \leq H$ there exists an open neighborhood $U \subseteq G$ of $1_G$ such that $\phi(U) \cap F = \{1_H\}$.
\(1) Assume the hypotheses and let $F\leq H$ be a finitely generated free subgroup. By Lemma \[basic\] we can select a finitely generated basic subgroup $F \leq H_1 \leq H$. Fix a free generating set for $H_1$ and let $L: H_1 \rightarrow \omega$ be the associated length function.
Suppose for contradiction that $\phi({\operatorname{HEG}}^n)\cap F$ is nontrivial for all $n$. For each $n\in \omega$ select $W_n \in {\operatorname{HEG}}^n \setminus \ker(\phi)$. Let $h_n = \phi(W_n)$ and let $k_n = L(\phi(W_n))$. Let $\{U_n\}_{n\in \omega}$ be the sequence of words such that $U_n = W_n(U_{n+1})^{k_n +2}$. Intuitively we have $U_0 = W_0(W_1(\cdots)^{k_1+2})^{k_0+2}$. Let $z_n = \phi(U_n)$ for all $n\in \omega$. Let $H_2 = \langle H_1 \cup \{z_n\}_{n\in \omega}\rangle$. Since $H_2$ is a countable subgroup of $H$ we know $H_2$ is free and therefore $H_1$ is a free factor. Let $\rho: H_2 \rightarrow H_1$ be any retraction induced by selecting a complimentary free factor and projecting to $H_1$. Letting $y_n = \rho(z_n)$ we obtain the relations
$y_n = h_n(y_{n+1})^{k_n+2}$
If $y_n \neq 1_H$ then
$L(y_{n-1}) \geq L(y_n^{k_n+2}) - L(h_{n-1})$
$\geq L(y_n) + k_n +1 - L(h_{n-1})$
$= L(y_n) +1$
and so $y_{n-1}\neq 1_H$ and $L(y_{n-1}) \geq L(y_n) +1$ and arguing backwards we see that for $m\geq n$ if $y_m \neq 1_H$ then $y_n \neq 1_H$ and $L(y_n)\geq L(y_m) + (m-n)$. This implies that the $y_n$ are eventually trivial. But then for some $n$ we have $y_n = 1_H = y_{n+1}$, from which we have $\phi(W_n) = h_n = y_ny_{n+1}^{-k_n-2} = 1_H$, contrary to the choosing of $W_n \notin \ker(\phi)$.
\(2) Suppose first that $\phi: G \rightarrow H$ is a homomorphism from a completely metrizable group to an $\aleph_1$-free group and that $F \leq H$ is finitely generated. Let $d$ be a complete metric for $G$ compatible with the topology. Select a finitely generated basic subgroup $H_1 \geq F$ and let $L$ be the length function for a fixed free generating set on $H_1$. If a neighborhood $U$ as in the conclusion does not exist then we select $g_0 \in \phi^{-1}(F \setminus \{1_H\})$. Let $k_0 = L(\phi(g_0))$. Select a neighborhood $U_1$ of $1_G$ sufficiently small that $g\in U_1$ implies
$d(g_0g^{k_0+2}, g_0) \leq \frac{1}{2}$
$d(g, 1_G)\leq \frac{1}{2}$
Select $g_1\in U_1 \cap \phi^{-1}(F \setminus \{1_G\})$ and let $k_1 = L(\phi(g_1))$. Supposing that we have selected group elements $g_0, \ldots, g_n$ and neighborhoods $U_1, \ldots, U_n$ and natural numbers $k_0, \ldots, k_n$ in this way we select a neighborhood $U_{n+1}$ of $1_G$ for which $g\in U_{n+1}$ implies
$d(g_0(\cdots g_{n-1}(g_n(g)^{k_n+2})^{k_{n-1}+2} \cdots)^{k_0+2}, g_0(g_1(\cdots g_{n-1}(g_n)^{k_{n-1}+2} \cdots)^{k_1+2})^{k_0+2})\leq\frac{1}{2^{n+1}}$
$d(g_1(\cdots g_{n-1}(g_n(g)^{k_n +2})^{k_{n-1}+2} \cdots)^{k_1+2}, g_1(g_2(\cdots g_{n-1}(g_n)^{k_{n-1}+2} \cdots)^{k_2+2})^{k_1+2})\leq \frac{1}{2^{n+1}}$
$\vdots$
$d(g_{n-1}(g_n(g)^{k_n+2})^{k_{n-1}+2}, g_{n-1}(g_n)^{k_{n-1}+2})\leq\frac{1}{2^{n+1}}$
$d(g_n(g)^{k_n+2}, g_n)\leq\frac{1}{2^{n+1}}$
$d(g, 1_G)\leq \frac{1}{2^{n+1}}$
Select $g_{n+1}\in U_{n+1}\cap \phi^{-1}(F\setminus \{1_G\})$ and let $k_{n+1} = L(\phi(g_{n+1}))$. For each $n\in \omega$ the sequence $g_n(\cdots g_{m-1}(g_m)^{k_{m-1} +2} \cdots)^{k_n+2}$ is Cauchy in $m$ and therefore converges to some $j_n = \lim_{m \rightarrow \infty}g_n(\cdots g_{m-1}(g_m)^{k_{m-1} +2} \cdots)^{k_n+2}$ and by continuity of multiplication we have $j_n = g_{n+1}j_{n+1}^{k_{n+1} + 2}$. Let $z_n = \phi(j_n)$ for all $n\in \omega$. By again letting $H_2 = \langle H_1 \cup \{z_n\}_{n\in \omega}\rangle$ and $\rho$ being any retraction from $H_2$ to $H_1$ we obtain a contradiction as in part (1).
Suppose now that $\phi: G \rightarrow H$ is a homomorphism with locally compact Hausdorff domain and $\aleph_1$-free image and that for some finitely generated $F \leq H$ there is no $U$ as in the conclusion. Select $H_1 \leq H$ as in the other case and again let $L$ be the length function with respect to a fixed free generating set for $H_1$. Let $U_0$ be an open neighborhood of $1_G$ for which $\overline{U_0}$ is compact. Select $g_0 \in U_0 \cap \phi^{-1}(F \setminus \{1_G\})$. Let $k_0 = L(\phi(g_0))$. Supposing we have selected elements $g_0, \ldots, g_n$ and nesting neighborhoods $U_0, \ldots, U_n$ of $1_G$ and natural numbers $k_0, \ldots, k_n$ in this way, we select a neighborhood $U_{n+1}\subseteq U_n$ of $1_G$ such that $g\in U_{n+1}$ implies $g_ng^{k_n+2}\in U_n$. Let $g_{n+1}\in U_{n+1} \cap \phi^{-1}(F\setminus \{1_G\})$ and let $k_{n+1} = L(\phi(g_{n+1}))$.
For each $n\in \omega$ we let $K_n = g_0(g_1(\cdots g_n(\overline{U_{n+1}})^{k_n +2} \cdots)^{k_1 +2})^{k_0+2}$. The sequence $\{K_n\}_{n\in \omega}$ consists of nonempty nesting compacta and so the intersection is nonempty. Let $j_0 \in \bigcap_{n\in \omega} K_n$ and for each $n\geq 1$ we select $j_n\in \overline{U_n}$ such that $$j_0 = g_0(g_1(\cdots g_{n-1}j_n^{k_{n-1}+2} \cdots)^{k_1 +2})^{k_0 +2}$$ Let $z_n = \phi(j_n)$ for each $n\in \omega$. Since $H$ is locally free we notice that $z_n = \phi(g_n)z_{n+1}^{k_n +2}$ for all $n$. We argue as before for a contradiction.
(of Theorem \[thebigone\]) We prove n-slenderness first and the arguments of the other types of slenderness will follow the same format. Suppose $\phi: {\operatorname{HEG}}\rightarrow H$ is a map with $\aleph_1$-free codomain and imagine for contradiction that $\phi({\operatorname{HEG}}^n)$ is never trivial. Select $W_0\in {\operatorname{HEG}}\setminus \ker(\phi)$. We have $\langle \phi(W_0)\rangle$ contained in a finitely generated basic free subgroup $F_0 \leq H$. By Lemma \[nicelemma\] pick $m_1 \in \omega$ large enough that $\phi({\operatorname{HEG}}^{m_1}) \cap F_0$ is trivial. Select $W_1\in {\operatorname{HEG}}^{m_1} \setminus \ker(\phi)$. The finitely generated subgroup $\langle F_0 \cup \phi(W_1)\rangle$ is contained in a finitely generated basic subgroup $F_1$. Supposing we have selected group elements $W_0, \ldots, W_n$ and basic subgroups $F_0\leq \ldots \leq F_n$ and natural numbers $m_0< \ldots <m_n$ in this way we select $m_{n+1}>m_n$ for which $\phi({\operatorname{HEG}}^{m_{n+1}}) \cap F_n = \{1_H\}$. Pick $W_{n+1}\in {\operatorname{HEG}}^{m_{n+1}}\setminus \ker(\phi)$ and let $F_{n+1}$ be a finitely generated basic subgroup which includes $\langle F_n \cup \{\phi(W_{n+1})\}\rangle$.
Define words $U_0, U_1, \ldots$ by $U_n = W_n^2U_{n+1}^2$. Let $h_n = \phi(W_n)$ and $y_n = \phi(U_n)$ for all $n\in \omega$. We consider the subgroup $H_{\infty} = \langle \{h_n\}_{n\in \omega}\cup \{y_n\}_{n\in \omega}\rangle \leq H$.
Notice first that for each $n\in \omega$ the elements $h_0, \ldots, h_n$ freely generate a subgroup of $H$. This claim is obvious for $n = 0$. Supposing the claim is true for $n$ we have $\langle h_0, \ldots, h_n\rangle\leq F_n$ and since $h_{n+1} = \phi(W_{n+1})\notin F_n$ we see that $F_n$ is a proper free factor of the group $\langle F_n \cup \{h_{n+1}\}\rangle$. Since finitely generated free groups are Hopfian we know that if we fix a free generating set $X_n$ for $F_n$, the elements $X_n \cup \{h_{n+1}\}$ freely generate a subgroup of $H$.
Next, we claim the elements $h_0, \ldots, h_n, y_{n+1}$ freely generate a subgroup of $H$. We have already seen that $h_0, \ldots, h_n$ freely generate a subgroup of the group $F_n$. If $y_{n+1}$ is nontrivial then since evidently $y_{n+1} \in \phi({\operatorname{HEG}}^{m_{n+1}})$ we can argue as before that $h_0, \ldots, h_n, y_{n+1}$ freely generates a subgroup. Were $y_{n+1} = h_{n+1}^2y_{n+2}^2$ trivial, we would have $h_{n+1} = y_{n+2}$ since $H$ is locally free. Then $h_{n+1} = y_{n+2} \in F_{n+1}\cap \phi({\operatorname{HEG}}^{m_{n+2}}) = \{1_H\}$, contrary to how $W_{n+1}$ was chosen.
Letting $H_n = \langle h_0, \ldots, h_n, y_{n+1} \rangle$ it is easy to see that each $H_n$ is properly contained in $H_{n+1}$ and is not a free factor (one can use [@H1 Lemma 7], for example). This contradicts Lemma \[basic\]. This group $H_{\infty} = \bigcup_{n\in \omega}H_n$ was identified by Higman as being a subgroup of ${\operatorname{HEG}}$ (see the discussion following [@H2 Theorem 6]).
Suppose now that $\phi: G \rightarrow H$ is a homomorphism from a completely metrizable group to an $\aleph_1$-free group. Suppose $\ker(\phi)$ is not open. Select $g_0 \in G \setminus \ker(\phi)$. Pick a finitely generated basic subgroup $F_0$ for which $\phi(g_0) \in F_0$. By Lemma \[nicelemma\] select an $\epsilon_1>0$ such that for $g$ in the open ball $B(1_G, \epsilon_1)$ we have
$d(g_0^2(g)^2, g_0^2) \leq \frac{1}{3}$
and for $g\in B(1_G, \epsilon_1) \setminus \ker(\phi)$ that $\phi(g) \notin F_0$. Select $g_1$ such that $g_1, g_1^2 \in B(1_G, \frac{\epsilon_1}{3})\setminus \ker(\phi)$. Select a finitely generated basic free subgroup $F_2$ of $H$ for which $F_1 \geq \langle F_0 \cup \{\phi(g_1)\}\rangle$. Select $\epsilon_2>0$ such that $g\in B(1_G, \epsilon_2)$ implies
$d(g_0^2(g_1^2(g)^2)^2, g_0^2(g_1^2)^2) \leq \frac{1}{9}$
$d(g_1^2(g)^2 , g_1^2) \leq \frac{\epsilon_1}{9}$
and for $g\in B(1_G, \epsilon_2) \setminus \ker(\phi)$ that $\phi(g) \notin F_1$. Select $g_2$ so that $g_2, g_2^2\in B(1_G, \frac{\epsilon_2}{3})\setminus \ker(\phi)$. Let $F_2$ be a finitely generated basic subgroup of $H$ containing $\langle F_1\cup \{\phi(g_2)\}\rangle$. Supposing we have selected $g_0, \ldots, g_n$ and $\epsilon_1, \ldots, \epsilon_n$ and $F_0, \ldots, F_n$ in this way, we select $\epsilon_{n+1}>0$ such that $g\in B(1_G, \epsilon_{n+1})$ implies
$d(g_0^2(g_1^2(\cdots g_n^2(g)^2 \cdots)^2)^2, g_0^2(g_1^2(\cdots (g_n^2)^2 \cdots)^2)^2)\leq \frac{1}{3^{n+1}}$
$d(g_1^2(\cdots g_n^2(g)^2 \cdots)^2, g_1^2(\cdots g_n^2 \cdots)^2)\leq\frac{\epsilon_1}{3^{n+1}}$
$\vdots$
$d(g_n^2(g)^2 , g_n^2) \leq \frac{\epsilon_n}{3^{n+1}}$
and for $g\in B(1_G, \epsilon_{n+1})\setminus \ker(\phi)$ that $\phi(g) \notin F_n$. Select $g_{n+1}$ so that $g_{n+1}, g_{n+1}^2\in B(1_G, \frac{\epsilon_{n+1}}{3})\setminus \ker(\phi)$. Pick a finitely generated basic subgroup $F_{n+1}$ containing $\langle F_n \cup \{\phi(g_{n+1})\} \rangle$. Notice that for each $n\in\omega$ the sequence $g_n^2(g_{n+1}^2(\cdots g_{m-1}^2g_m^2 \cdots )^2)^2$ is Cauchy and converges to an element $j_n$. Moreover it is clear that for $n\geq 1$ we have $j_n\in B(1_G, \epsilon_n)$. The relations $j_n = g_n^2j_{n+1}^2$ are clear by continuity of multiplication.
We let $h_n = \phi(g_n)$ and $y_n = \phi(j_n)$ for all $n\in \omega$. Performing the same argument as before, we contradict Lemma \[basic\].
Finally, we suppose $\phi: G\rightarrow H$ has locally compact Hausdorff domain and $\aleph_1$-free codomain and for contradiciton suppose that $\ker(\phi)$ is not open. We inductively define nesting sequences $\{U_n\}_{n\in \omega}$ and $\{V_n\}_{n\in \omega}$ of open neighborhoods of $1_G$ such that $U_0 \supseteq V_0 \supseteq U_1\supseteq V_1 \supseteq \cdots$ and $\overline{V_n} \subseteq U_n$, as well as a sequence $\{g_n\}_{n\in \omega}$ of elements in $G$ and finitely generated basic subgroups $F_0 \subseteq \cdots$. Let $U_0 = G$ and select a neighborhood $V_0$ of $1_G$ such that $\overline{V_0}$ is compact. Select $g_0$ so that $g_0, g_0^2 \in V_0 \setminus \ker(\phi)$. Select a finitely generated basic subgroup $F_0$ which includes $\langle \phi(g_0)\rangle$. By Lemma \[nicelemma\] select $U_1 \subseteq V_1$ such that $g \in U_1$ implies
$g_0^2g^2 \in V_0$
and if $g\in U_1 \setminus \ker(\phi)$ we have $\phi(g) \notin F_0$. Pick an open neighborhood $V_1$ of $1_G$ such that $\overline{V_1} \subseteq U_1$ and select $g_1$ such that $g_1, g_1^2 \in V_1 \setminus \ker(\phi)$. Let $F_1$ be a basic finitely generated group including $\langle F_0 \cup \{ \phi(g_1)\}\rangle$.
Suppose we have selected neighborhoods $U_0, \ldots, U_n$ and $V_0, \ldots, V_n$ as well as elements $g_0, \ldots, g_n$ and basic free groups $F_0, \ldots, F_n$ in this manner. Select a neighborhood $U_{n+1}$ of $1_G$ such that $g\in U_{n+1}$ implies
$g_0^2(g_1^2(\cdots g_n^2(g)^2 \cdots)^2)^2 \in V_0$
$g_1^2(g_2^2(\cdots g_n^2(g)^2 \cdots)^2)^2 \in V_1$
$\vdots$
$g_n^2(g)^2 \in V_n$
and if $g\in U_{n+1}\setminus \ker(\phi)$ we have $\phi(g) \notin F_n$. Pick open neighborhood $V_{n+1}$ of $1_G$ such that $\overline{V_{n+1}} \subseteq U_{n+1}$ and select $g_{n+1}$ such that $g_{n+1}, g_{n+1}^2 \in V_{n+1} \setminus \ker(\phi)$. Let $F_{n+1}$ be a basic finitely generated subgroup including $\langle F_n \cup \{\phi(g_{n+1})\} \rangle$. Define compact sets $K_n$ for $n\in \omega$ by letting $K_n = g_0^2(g_1^2(\cdots g_n^2(\overline{V_{n+1}})^2 \cdots)^2)^2$. It is easy to see that $\overline{V_0} \supseteq K_0 \supseteq K_1 \supseteq \cdots$ and so we may select $j_0 \in \bigcap_{n\in \omega}K_n$. For $n \geq 1$ select $j_n\in \overline{V_{n+1}}$ such that $j_0 = g_0^2(\cdots g_n^2(j_n)^2 \cdots)^2$. Let $h_n = \phi(g_n)$ and $y_n = \phi(j_n)$. Since $y_0 = h_0^2(\cdots h_n^2(y_{n+1})^2 \cdots)^2$ for all $n \in\omega$ and $H$ is locally free we get relations $y_n = h_n^2y_{n+1}^2$. We derive a contradiction by arguing in the same manner as for cm-slenderness.
[Automatic continuity in the abelian case]{}\[abeliancase\]
To avoid confusion we continue using multiplicative group notation, unless otherwise stated, despite the fact that some groups under discussion will be abelian. We give definitions (see [@Fu]):
\[algebraicallycompactdef\] An abelian group $H$ is *algebraically compact* if $H$ is a direct summand of a Hausdorff compact abelian group.
The algebraically compact groups are closed under inverse limits, and finite abelian groups are obviously algebraically compact. For each prime $p$ we have an inverse system of abelian groups $\mathbb{Z}/p^{n+1}\mathbb{Z} \rightarrow \mathbb{Z}/p^n\mathbb{Z}$ and let $J_p$ denote the inverse limit (the *$p$-adic completion of $\mathbb{Z}$*.) We also have an inverse system of abelian groups $\mathbb{Z}/n_0\mathbb{Z} \rightarrow \mathbb{Z}/n_1\mathbb{Z}$ (here $n_1 \mid n_0$) and denote by $\hat{\mathbb{Z}}$ the inverse limit (the $\mathbb{Z}$-adic completion of $\mathbb{Z}$.) Both $J_p$ and $\hat{\mathbb{Z}}$ are algebraically compact and $J_p$ carries a natural group topology under which it is homeomorphic to the Cantor set.
An element $a$ of $\hat{\mathbb{Z}}$ has a representation of form $a = (a_1 + 2!\mathbb{Z}, a_2 + 3!\mathbb{Z}, \ldots)$ which is formally represented by the sum $\sum_{n=1}^{\infty}n!a_n$. Two formal sums $\sum_{n=1}^{\infty}n!a_n$ and $\sum_{n=1}^{\infty}n!b_n$ represent the same element in $\hat{\mathbb{Z}}$ provided for all $m \geq 1$ we have
$(m+1)! \mid \sum_{n=1}^m n!a_n - \sum_{n=1}^mn!b_n$
\[cotorsiondef\] An abelian group $H$ is *cotorsion* if it is the homomorphic image of an algebraically compact group.
\[cotorsionfreedef\] An abelian group $H$ is *cotorsion-free* if it does not contain a nontrivial cotorsion group. Equivalently $H$ is cotorsion-free if $H$ does not contain torsion, $\mathbb{Q}$, or a copy of the $p$-adic integers $J_p$ for any prime $p$ [@Fu Theorem 13.3.8].
\[reduced\] A torsion-free abelian group is *reduced* if it contains no copy of $\mathbb{Q}$.
\[Ulmsubgroup\] The *first Ulm subgroup* of an abelian group $H$ is the subgroup $U(H) = \bigcap_{n \geq 1} H^n = \{h\in H\mid (\forall n\geq 1)(\exists h_n) h =h^n\}$.
\[lineartopology\] A topology on an abelian group $H$ is *linear* if there exists a filter $\mathcal{F}$ of subgroups of $H$ such whose elements form a basis for the open neighborhoods of $1_H$.
\[slenderdef\] An abelian group $H$ is *slender* if for every homomorphism $\phi: \prod_{\omega}\mathbb{Z} \rightarrow H$ there exists some $m\in \omega$ such that $\phi = \phi \circ p_m$ where $p_m: \prod_{\omega}\mathbb{Z} \rightarrow \bigoplus_{n=0}^m\mathbb{Z} \times (0)_{n=m+1}^{\infty}$ is the retraction which projects the first $m+1$ coordinates. Equivalently $H$ is slender if $H$ does not contain torsion, $\mathbb{Q}$, $\prod_{\omega}\mathbb{Z}$ or a copy of the $p$-adic integers $J_p$ for any prime $p$ [@Fu Theorem 13.3.5]. Equivalently $H$ is slender if $H$ is torsion-free, reduced and contains no subgroup that admits a complete non-discrete metrizable linear topology [@Fu Theorem 13.3.1].
We prove a lemma which follows along the lines of [@EdFi Theorem 3.1]:
\[getoff\] If $\phi: G \rightarrow H$ has completely metrizable or locally compact Hausdorff domain and cotorsion-free abelian codomain then $\ker(\phi)$ is closed.
Suppose $\phi: G \rightarrow H$ is a homomorphism with completely metrizable domain and cotorsion-free codomain and let $d$ be a complete metric compatible with the topology on $G$. Suppose for contradiction that $\ker(\phi)$ is not closed. If $g\in \overline{\ker(\phi)}\setminus \ker(\phi)$ then for there every neighborhood $U$ of $1_G$ we have $\phi(g)\in \phi(U) \setminus \{1_H\}$. Then letting $h = \phi(g)$ and $H_{\infty} = \bigcap_{n\in \omega} \phi(B(1_G, \frac{1}{n}))$ we get $h\in H_{\infty} \setminus\{1_H\}$ and $H_{\infty}$ is easily seen to be a subgroup. We obtain a contradiction by finding a nontrivial homomorphic image of the algebraically compact $\hat{\mathbb{Z}}$ in $H$, and since a homomorphic image of an algebraically compact group is cotorsion we will be finished.
Since $H$ is torsion-free and reduced we have that the first Ulm subgroup $U(H)$ is trivial. We show that for each sequence of integers $\{a_n\}_{n\in \omega \setminus \{0\}}$ there exists a $j\in G$ for which (under additive notation) $(m+1)! \mid \phi(j) - \sum_{n=1}^m n!a_i h$ for all $m\geq 1$. Then because $U(H)$ is trivial we get a well-defined $\psi: \hat{\mathbb{Z}} \rightarrow H$ given by $\psi(\sum_{n = 1}^{\infty}n!a_i) = \phi(j)$. Since $h\in \psi(\hat{\mathbb{Z}})\setminus \{1_H\}$ we will have our nontrivial homomorphism.
Let a sequence $\{a_n\}_{n\in \omega}$ be given. Select $g_1\in G$ such that $\phi(g_1) = h^{a_1}$. Pick a neighborhood $U_2$ of $1_G$ such that $g'\in U_2$ implies $d(g_1(g')^{2!}, g_1)\leq \frac{1}{2}$. Select $g_2 \in U_2 \cap \phi^{-1}(h^{a_2})$ (this is possible since $\phi$ surjects $U_2$ onto $H_{\infty}$ and $h \in H_{\infty}$). Supposing we have selected $g_1, \ldots, g_n$ and $U_2, \ldots, U_n$ we select a neighborhood $U_{n+1}$ of $1_G$ such that $g'\in U_{n+1}$ implies
$d(g_1(g_2(\cdots g_n(g')^{(n+1)!}\cdots)^{3!})^{2!}, g_1(g_2(\cdots g_n \cdots)^{3!})^{2!})\leq \frac{1}{2^n}$
$\vdots$
$d(g_n(g')^{(n+1)!}, g_n)\leq \frac{1}{2^n}$
Select $g_{n+1}\in U_{n+1}\cap \phi^{-1}(h^{a_{n+1}})$. Fixing a $q \geq 1$ it is clear that the sequence $g_q(g_{q+1}(\cdots g_n\cdots)^{(q+2)!})^{(q+1)!}$ is Cauchy and therefore converges to, say, $j_q$. We have $j = j_1 = g_1(\cdots g_{n-1}(j_n)^{n!} \cdots)^{2!}$ for each $n\geq 1$ by continuity of multiplication. The relationship $(m+1)! \mid\phi(j) - \sum_{n=1}^mn!a_nh$ is now clear for all $m$.
Suppose now that $G$ is locally compact Hausdorff and for contradiction that $h\in H_{\infty} = \bigcap_{U \in \mathcal{U}}\phi(U)$ is nontrivial where $\mathcal{U}$ denotes the collection of open neighborhoods of $1_G$. We again show that each sequence $\{a_n\}_{n\geq 1}$ has an element. Let $V = V_1$ be a neighborhood of $1_G$ for which $\overline{V}$ is compact. Pick $g_1\in V \cap \phi^{-1}(h^{a_1})$. Supposing we have selected sequences $g_1, \ldots g_n$ and open neighborhoods $V_1, \ldots, V_n$ of $1_G$ in this way we select a neighborhood $V_{n+1}$ of $1_G$ such that $g_nV_{n+1}^{(n+1)!}\subseteq V_n$. Let $g_{n+1}\in V_{n+1}\cap \phi^{-1}(h^{a_{n+1}})$. Let $K_n = g_1(\cdots g_n(\overline{V_{n+1}})^{(n+1)!}\cdots)^{2!}$ we then select $j \in \bigcap_{n \geq 1}K_n$ and notice once again that $(m+1)! \mid\phi(j) - \sum_{n=1}^mn!a_nh$ for all $m\geq 1$.
(of Theorem \[characterizationoflcHslenderabelian\])
\(1) Suppose an abelian group $H$ is cm-slender. Then $H$ cannot contain torsion, for then $H$ would contain some cyclic group of prime order $\mathbb{Z}/p\mathbb{Z}$. The group $\prod_{\omega}\mathbb{Z}/p\mathbb{Z}$ is compact metrizable in a natural way and any homomorphism from $\bigoplus_{\omega}\mathbb{Z}/p\mathbb{Z} \leq \prod_{\omega}\mathbb{Z}/p\mathbb{Z}$ to $\mathbb{Z}/p\mathbb{Z}$ extends to a homomorphism on the entirety of $\prod_{\omega}\mathbb{Z}/p\mathbb{Z}$ by a vector space argument, so that it is quite easy to construct a homomorphism from $\prod_{\omega}\mathbb{Z}/p\mathbb{Z}$ to $\mathbb{Z}/p\mathbb{Z} \leq H$ which does not have an open kernel.
Also, $H$ cannot have a copy of $\mathbb{Q}$ since otherwise there exists a homomorphism from $\mathbb{R}$ to $\mathbb{Q} \leq H$ which does not have open kernel. Neither can $H$ have a copy of a group which admits a non-discrete Polish topology, since then the inclusion map would witness that $H$ is not cm-slender.
Supposing $H$ is a group which is torsion-free, reduced and contains no subgroup which admits a non-discrete Polish topology. Since $J_p$ has a non-discrete metrizable compact topology, we know that $H$ cannot contain any $J_p$ and so $H$ is cotorsion-free. Let $\phi: G \rightarrow H$ be a homomorphism with $G$ completely metrizable. Since $H$ is cotorsion-free, we have by Lemma \[getoff\] that $\ker(\phi)$ is closed. Supposing for contradiction that $\ker(\phi)$ is not open, we get a sequence $\{g_n\}_{n\in \omega}$ of elements of $G$ which converges to $1_G$ and such that $g_n \notin \ker(\phi)$. Letting $G_{\infty} \leq G$ be the smallest closed subgroup of $G$ containing the elements of $\{g_n\}_{n\in \omega}$, we have that $G_{\infty}$ is Polish. Also, $\ker(\phi\upharpoonright G_{\infty}) = G_{\infty}\cap \ker(\phi)$ is closed in $G_{\infty}$, and by how we selected $\{g_n\}_{n\in \omega}$ we know $\ker(\phi\upharpoonright G_{\infty})$ is not open in $G_{\infty}$. The group $G_{\infty}/\ker(\phi\upharpoonright G_{\infty})$ is again a Polish group [@Ke 2.3.iii] and not discrete by considering the cosets $g_n\ker(\phi\upharpoonright G_{\infty})$. The map $\phi$ descends to an injective homomorphism $\overline{\phi}: G_{\infty}/\ker(\phi)\rightarrow H$. Then $H$ contains a subgroup which admits a non-discrete Polish topology and we have a contradiction.
\(2) Suppose an abelian group $H$ is lcH-slender. Then $H$ cannot contain torsion, $\mathbb{Q}$ or any $J_p$ by the reasoning as in (1) and so $H$ is cotorsion-free.
Suppose on the other hand that $H$ is cotorsion-free. Let $\phi: G \rightarrow H$ be a homomorphism with locally compact Hausdorff domain. By Lemma \[getoff\] we know $\ker(\phi)$ is closed. Then $G/\ker(\phi)$ is a locally compact abelian Hausdorff group and $\phi$ passes to an injective homomorphism $\overline{\phi}: G/\ker(\phi) \rightarrow H$. By [@Mo Theorem 25] there exists an open subgroup $U$ of $G/\ker(\phi)$ which is topologically isomorphic to $\mathbb{R}^n \times K$ where $K$ is a compact group. We show $U$ is trivial, so that $G/\ker(\phi)$ is discrete and $\ker(\phi)$ is open. First of all, the superscript $n$ must be $0$ since $\overline{\phi}$ is injective and $H$ cannot contain $\mathbb{Q}$ as a subgroup. But it is clear that $K$ must be trivial as well since otherwise $\phi(K)$ would be nontrivial cotorsion.
The proof of Theorem \[lcHslender\] now follows easily. If a group $H$ is $\aleph_1$-free abelian, then it cannot contain torsion or $\mathbb{Q}$. Also $H$ cannot contain any $J_p$ since then it would also contain the additive group of $\mathbb{Z}[\frac{1}{q}]$ for every prime $q\neq p$, and hence contain a countable subgroup which is not free abelian. Thus an $\aleph_1$-free abelian group is cotorsion-free and we apply Theorem \[characterizationoflcHslenderabelian\].
We provide some definitions towards Theorem \[strongly\] (see [@Me]):
\[purity\] If $H$ is $\kappa$-free abelian we say a subgroup $M\leq H$ is *$\kappa$-pure* if $M$ is a direct summand of $\langle M \cup X\rangle$ for each set $X \subseteq H$ of cardinality $<\kappa$.
\[stronglydef\] A $\kappa$-free abelian group $H$ is *strongly $\kappa$-free abelian* if every subset $X \subseteq H$ of cardinality $<\kappa$ is contained in a $\kappa$-pure subgroup of $H$ generated by fewer than $\kappa$ elements.
(of Theorem \[strongly\]) Suppose $\phi: G \rightarrow H$ has completely metrizable domain and strongly $\aleph_1$-free codomain. Let $d$ be a complete metric compatible with the topology of $G$. Supposing that $\ker(\phi)$ is not open we select $g_n\in B(1_G, \frac{1}{n}) \setminus \ker(\phi)$. Select a countable $\aleph_1$-pure subgroup $M \supseteq \{\phi(g_n)\}_{n\in \omega}$. Fix a free abelian generating set for $M$ and let $L: M \rightarrow \omega$ be the length function. Let $k_n = L(\phi(g_n))$ for each $n\in \omega$. We define a subsequence $\{n_q\}_{q\in \omega}$ inductively. Let $n_0 = 0$ and supposing we have defined $n_0, \ldots, n_q$ we let $n_{q+1}$ be such that
$d(g_{n_0}(g_{n_1}(\ldots g_{n_q}(g_{n_{q+1}})^{k_{n_q}+1} \ldots)^{k_{n_1}+2})^{k_{n_0} +2}, g_{n_0}(g_{n_1}(\ldots g_{n_q} \ldots)^{k_{n_1}+2})^{k_{n_0} +2}) \leq \frac{1}{2^q}$
$d(g_{n_1}(g_{n_2}(\cdots g_{n_q}(g_{n_{q+1}})^{k_{n_q}+2} \cdots)^{k_{n_2}+2})^{k_{n_1}+2}, g_{n_1}(g_{n_2}(\cdots g_{n_q}\cdots)^{k_{n_2}+2})^{k_{n_1}+2}) \leq \frac{1}{2^q}$
$\vdots$
$d(g_{n_q}(g_{n_{q+1}})^{k_{n_q}+2}, g_{n_q})\leq \frac{1}{2^q}$
For each $m\in \omega$ the sequence $g_{n_m}(g_{n_{m+1}}(\cdots g_{n_q} \cdots)^{k_{n_{m+1}}+2})^{k_{n_m}+2}$ is Cauchy and therefore converges to an element $j_m$. Letting $\rho: \langle M \cup \{\phi(j_m)\}_{m\in \omega} \rangle \rightarrow M$ be a retraction, we derive a contradiction as before.
Since for abelian groups cm-slenderness implies n-slenderness and lcH-slenderness, we are done.
There is an alternative proof for the fact that strongly $\aleph_1$-free groups are n- and lcH-slender which uses infinitary logic. If $H$ is strongly $\aleph_1$-free then $H$ has the same $L_{\infty \omega_1}$ theory as free abelian groups [@Ek]. Free abelian groups are slender, and slenderness is $L_{\infty \omega_1}$ axiomatizable [@SKo], so $H$ is slender. Slender groups are n-slender [@Ed] and they are also lcH-slender by Theorem \[characterizationoflcHslenderabelian\], so we are done.
[abcdefghijk]{}
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P. Eklof, A. Mekler, Almost Free Modules: set theoretic methods, North-Holland (1990).
L. Fuchs, Abelian Groups, Springer (2015).
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[^1]: The author is supported by European Research Council grant PCG-336983.
|
---
abstract: |
5G is envisioned to support three broad categories of services: eMBB, URLLC, and mMTC. URLLC services refer to future applications which require reliable data communications from one end to another, while fulfilling ultra-low latency constraints. In this paper, we highlight the requirements and mechanisms that are necessary for URLLC in LTE. Design challenges faced when reducing the latency in LTE are shown. The performance of short processing time and frame structure enhancements are analyzed. Our proposed DCI Duplication method to increase LTE control channel reliability is presented and evaluated. The feasibility of achieving low latency and high reliability for the IMT-2020 submission of LTE is shown. We further anticipate the opportunities and technical design challenges when evolving 3GPP’s LTE and designing the new 5G NR standard to meet the requirements of novel URLLC services.
**Keywords: 3GPP, 5G, LTE, New Radio, sTTI, URLLC**.
author:
-
bibliography:
- '3gpp.bib'
- 'references.bib'
- 'RAN1\_92.bib'
- 'RAN1\_90b.bib'
title: |
URLLC Services in 5G\
Low Latency Enhancements for LTE
---
= 10000 = 10000 = 10000
Introduction
============
The emerging 5G wireless mobile networks will be as much the result of relentless and extensive improvements of 3GPP’s (3rd Generation Partnership Project) as it is a technology revolution [@path25G]. Besides the possibilities for self-contained subframes, an entirely new air interface or grant-free access, it also prompts development of numerous incremental improvements. The IMT-2020 use cases, as depicted in Fig. \[fig:ITUreq\], shall fulfill three principal dimensions of performance [@itu.m.2410; @20155GWhitePaperNGMN]. 5G will not only focus on ; but and seemingly have a similar footing in long-term visions of what 5G might ultimately become [@7980747].
New use cases demanding very low latency, very high reliability or a combination of high reliability and low latency, i.e. , have been identified as one of the key trends of future wireless cellular communications [@3GP15Timeline; @3gpp.RP-171489]. Such use cases include a rather diverse set of requirements on the combination of reliability and latency such as remote tactile or haptic control (low latency), wireless communications in industrial automation (high reliability, low/medium latency), and smart grids (high reliability, low/medium latency), just to mention a few.
Alongside , LTE technology enhancements are needed to serve such new use cases and to remain technologically competitive up to and beyond 2020. As a candidate technology, it is motivated to further enhance the LTE system, such that the IMT-2020 5G requirements [@itu.m.2410] can be met. Including those for URLLC in terms of reliability, packet loss of $10^{-5}$ for small data packets, as well as low latency of less than in one way user plane.
![Use cases for IMT-2020 and beyond [@itu.m.2410].[]{data-label="fig:ITUreq"}](./ITU_tf_v2.pdf){width="1.00\linewidth"}
The LTE Rel. 14 with its on *L2 latency reduction* and the technical report on *shortened and processing time for LTE* [@3gpp.36.881] provides solutions for L1/L2 latency reduction. These solutions enable latencies at the levels mentioned above, but new functionality is needed to improve the reliability under latency constraints required for services. Although the term URLLC targets both achieving a very low latency, as well as fulfilling a reliability constraint, the 3GPP standardization body decoupled latency and reliability aspects. Initial focus of improving system performance was on latency related aspects and is referred to as sTTI (short ) [@3gpp.RP-161299]. Reliability aspects were the target of a later under the term [@3gpp.RP-171489].
An overview on standardization activities in as an analysis of latency and simulation results on robustness for standard and using our proposed novel scheme for duplication are presented in this paper. Section II highlights the technical requirements for URLLC systems, followed by Section III which describes the technical solutions developed and standardized in 3GPP in the context of . In Section IV we present our results and performance analysis. Latency is analyzed analytically and reliability improvements of the novel duplication approach are presented. Section V anticipates , and finally a conclusion is given in Section VI.
Physical Layer Requirements: High vs. Ultra Reliable Low Latency
================================================================
The new generation radio system (5G) addresses the demands and business contexts of 2020 and beyond. In 2015, the alliance published their 5G whitepaper [@20155GWhitePaperNGMN], listing the mobile operators’ vision on 5G use cases, business models and requirements.
**Latency-related aspects:** The NGMN proposed that 5G systems shall be able to provide latency in general (referred to as HRLLC in 3GPP), and latency (URLLC) for use cases with extremely low latency requirements. E2E latency refers to the duration between the transmission of a small data packet from the application layer and successful reception at the application layer of the destination node. The over-the-air latency constitutes only one part of the latency, whereas the core network latency poses the residual part. Hence, 3GPP agreed on aiming for over-the-air latency, although is still the hard constraint [@3gpp.38.913].
**Reliability-related aspects:** 3GPP defines the reliability by the probability to successfully transmit a packet from one radio unit to another radio unit within the given time constraint required by the targeted service [@3gpp.38.913]. For the sake of convenience, we describe the reliability with the complementary probability, that is the packet failure rate. For , 3GPP defines the target packet failure rate of $10^{-5}$ within over-the-air latency. A more relaxed constraint of $10^{-4}$, has been defined for , which is a challenge for todays 4G systems. Note, 4G systems for typically operate at a target of $10^{-1}$. Thus, future LTE releases as well as clean-slate 5G systems face tough design challenges when addressing ultra-high reliability combined with a stringent latency objective. However, the feasibility of implementing URLLC with E2E latency and NR-like parameters with a Software-Defined Radio (SDR) platform was recently shown in [@2015AdvancedSDRWirth; @2016TacintPilz].
3GPP Standardization Efforts in LTE Rel. 15
===========================================
LTE Latency Reduction Mechanisms
--------------------------------
Two basic mechanisms were defined in LTE Rel. 15 to reduce latency, namely reduced processing time and the support of a shortened frame structure. The latter is referred to as .
**Reduced processing time**: For a data packet arriving at $n$, the processing time is shortened from $n+4$ down to $n+3$. With short processing time, the User Equipment’s (UE) response time from data transmission to DL and from grant to UL data transmission is reduced from $n+4$ to $n+3$ . This means that the is reduced from $n+8$ to $n+6$ for both DL and UL.
**Short TTI** reduces the transmission time by introducing shorter frame structure. Dividing the $1$ ms subframe into either 2 parts (slots) or 5-6 parts (subslots) as shown in Fig. \[fig:FS1\]. For slot duration, the latencies are calculated on the assumption that the TTI is 7 symbols, whereas for subslot configuration the latencies are calculated following the subslot layout. For slot and subslot configuration, the processing time is scaled with the TTI length. Hence, the absolute processing time is reduced by a factor of 2 for the slot, and a factor of 5-6 for the subslot configuration. Note, for slot and subslot configurations with sTTI, the processing time remains $n+4$ but scales down with the reduced TTI length.=-1
![Frame structure type 1 (FDD) in LTE [@3gpp.36.211].[]{data-label="fig:FS1"}](./sTTI.pdf){width="0.98\linewidth"}
Division Duplexing and sTTI
---------------------------
Introducing sTTI in LTE has conflicting design aspects with regards to the frame structure. Whereas further optimizations can be made for systems, the combination of sTTI and has limits.=-1
**FDD sTTI**: New features in Rel. 15 include slot and subslot configurations from Fig. \[fig:FS1\]. The pattern is signaled in the UL and is used to reduce the DMRS overhead associated with the reduced TTI length. The DMRS symbol can be moved from the front to the end of a TTI or into the subsequent TTI using the different patterns. This allows sharing of one DMRS symbol among TTIs.=-1
**TDD sTTI**: The original design of *Frame structure type 3* in LTE did not cater for URLLC services. The minimal downlink-to-uplink switch-point periodicity is therefore in the uplink-downlink configurations $0, 1, 2$ and $6$. The configurations 3, 4 and 5 only support one downlink-to-uplink switching point with a periodicity of [@3gpp.36.211]. This limits the minimal possible to two times the switching periodicity resulting in RTT. Future changes are unlikely due to backward compatibility issues. Therefore, the efforts to introduce URLLC in TDD-systems in the LTE standardization process was limited. Slot length sTTIs and reduced processing time of $n+3$ have been agreed in [@3gpp.R1-1719247]. Although the URLLC target latency of remains unattainable for TDD LTE, the requirement can be met.=-1
LTE Latency Calculation {#sec:LTElatency}
-----------------------
The ITU definition of user plane latency is the duration from L2/L3 ingress to L2/L3 egress [@itu.m.2410]. Its timeline is depicted in Fig. \[fig:latency\]. The definitions for the following delay analysis are shown in Table \[assumptions\]. It is assumed that the propagation time is significantly lower than one , and thus can be neglected. In case of , HARQ-feedback and data retransmission can be repeated several times. This results in a total latency of: $$\label{eq:T_total}
T_{\mathrm{Total}}= 2\cdot T_{\mathrm{L1/L2}}+T_{\mathrm{Align}}+ \sum{T_{\mathrm{Proc}}}+\sum{T_{\mathrm{Tx}}}.$$ When using a repetition scheme without feedback, there is no $T_{\mathrm{Tx}}$ for the feedback, and the processing time is significantly shorter, reducing the sums for processing and transmission delay [@3gpp.R1-1802882].
---------------------- ------------------------------------------------------------
$T_{\mathrm{L1/L2}}$ L1/L2 processing delay,
for Rx and Tx at UE and eNB side respectively.
$T_{\mathrm{Align}}$ Alignment delay, the time required after
being ready to transmit and the transmission can start.
Worst-case latency is assumed (max. misalignment).
$T_{\mathrm{Proc}}$ UE/eNB processing, time needed for preparing transmissions
and decoding at the other side.
$T_{\mathrm{Tx}}$ Transmission time.
---------------------- ------------------------------------------------------------
: E2E Latency Components related to Fig. \[fig:latency\].[]{data-label="assumptions"}
[.49]{}
-------- ---------------------- --------- ---------- --------- ---------
Rel. 14 Rel. 15 Rel. 15 Rel. 15
SF SF & n+3 slot subslot
**DL** initial transmission 4 4 2 0.7
1st retransmission 12 10 6 2.0
2nd retransmission 20 16 10 3.3
3rd retransmission 28 22 14 4.7
**UL** initial transmission 12 10 6 2.0
1st retransmission 20 16 10 3.3
2nd retransmission 28 22 14 4.7
3rd retransmission 36 28 18 6.0
-------- ---------------------- --------- ---------- --------- ---------
\[tab:resultsHARQ\]
[.5]{}
-------- ---------------------- --------- ---------- --------- ---------
Rel. 14 Rel. 15 Rel. 15 Rel. 15
SF SF & n+3 slot subslot
**DL** initial transmission 4 4 2 0.7
1st repetition 5 5 2.5 0.8
2nd repetition 6 6 3.0 1.0
3rd repetition 7 7 3.5 1.2
**UL** initial transmission 12 10 6 2.0
1st repetition 14 12 7 2.3
2nd repetition 16 14 8 2.7
3rd repetition 18 16 9 3.0
-------- ---------------------- --------- ---------- --------- ---------
\[tab:resultsHARQless\]
Calculated results for Downlink (DL) and Uplink (UL) for the LTE Rel. 14 SF (subframe) TTI as well as LTE Rel. 15 short processing time, slot and subslot configurations. The circles indicate the fulfillment of the () requirement and () requirement respectively.
LTE Reliability Enhancements
----------------------------
**LTE data channel** reliability in LTE is achieved by transmitting with a low code rate often split into separate transmissions. Additional redundancy is only transmitted when needed. This is also used for -services to improve spectral efficiency. The basic HARQ scheme is shown in Fig. \[fig:HARQscheme\]. In LTE, HARQ is configured with up to $k=3$ retransmissions [@Dahlman]. Alternatively, a set number of $k$ repetitions can be sent with an optional feedback at the end, Fig. \[fig:HARQless\]. This scheme has less latency with the disadvantage of transmitting unnecessary redundancy versions compared to with feedback.
![Illustration of latency components for DL () and UL () transmissions. The latency components are defined in Table \[assumptions\].[]{data-label="fig:latency"}](./latency_components_v02.pdf){width="\linewidth"}
[.5]{} ![Retransmission schemes with $k$ as the number of retransmissions[]{data-label="fig:HARQschemes"}](./HARQscheme.pdf "fig:")
[.5]{} ![Retransmission schemes with $k$ as the number of retransmissions[]{data-label="fig:HARQschemes"}](./HARQless.pdf "fig:")
**LTE control channel**: The support of high reliability for LTE’s control channel poses another design challenge. Control messages are sent as via the shared . This information is blind decoded and checked against a user specific [@3gpp.36.213]. Blind decoding may lead to false positive decoding of in the case of a coincidental but erroneous match. A false positive can lead to successive errors, since the content of a control message is wrongly interpreted, also see [@3gpp.R1-1719503]. The most serious error is buffer contamination of another transmission. The procedure of blind decoding of a scrambled is depicted in Fig. \[fig:LTE\_RNTI\].
After decoding, the bits are split into the payload $d_0$ to $d_{n}$, and the scrambled CRC $s_0$ to $s_{15}$. After de-scrambling with the UE specific RNTI ($r_0$ to $r_{15}$), the result is compared with the calculated CRC of the received payload data. If this does not match, either the decoding failed, or the payload is addressed to another UE with a different RNTI. Blind decoding can lead to the false association of decoded or wrongly decoded [@3gpp.R1-1802887] with a probability of
$$P_{\mathrm{FP}}=1-(1-2^{-16})^N.
\label{eq:FP}$$
Here, $N$ is the number of blind decoding attempts and a uniform distribution is assumed for 16-bit . With an assumption of $N=20$ blind decoding attempts, this results in a false positive rate of $P_{\mathrm{FP}}=3.05 \cdot 10^{-4}$. Note for HRLLC services, this is too high when targeting error rates below $10^{-4}$ for data transmissions. There are two technical solutions proposed in 3GPP to reduce the false positive rate: firstly, increasing the length [@3gpp.R1-1802180] and secondly duplication[@3gpp.R1-1802887].
The DCI is currently limited to contain up to 8 , which limits the codeword size and thus lower bounds the code rate. Currently it is under discussion in 3GPP to double the number of to 16 at low . However, this requires changes to the hashing function, see [@3gpp.R1-1801941]. Alternatively, two duplicate can be sent and simultaneously used to enable operation at low SNRs and improve the rejection of false positives as shown in Fig. \[fig:DCIdup\]:
- On the left, a UE receives two DCIs but misses one. Here the two are combined and the resulting combined DCI is valid.
- On the right, a random DCI is falsely decoded and passes the CRC check. Here the combination is different and not valid.
![DCI Duplication[]{data-label="fig:DCIdup"}](./DCI_duplication-cropped.pdf){width="\linewidth"}
Performance Evaluation And Analysis
===================================
Next, we analyze the available LTE configurations with respect to the or the less stringent HRLLC requirements. Eq. \[eq:T\_total\] in the previous section describes the total over-the-air latency. For LTE, the latency analysis has to differentiate between [DL]{} and [UL]{}, since initially a scheduling request has to be sent for acquiring [UL]{} resources. The latency in downlink (DL) direction for a transmission using HARQ (HA) with $k$ retransmissions can be obtained as $$T_{\mathrm{DL},\mathrm{HA}}=T_{c}+2\cdot k\cdot T_{\mathrm{Proc}}+(1+2\cdot k)\cdot T_{\mathrm{Tx}}.$$ Here, the delay caused by the higher layers and alignment remains constant with $T_{c}=2\cdot T_{\mathrm{L1/L2}}+T_{\mathrm{Align}}$. For the uplink (UL) direction, there is an additional due to the scheduling request (SR) and following uplink grant. This leads to $k_{\mathrm{UL}}=k+1$, and thus the UL delay including HARQ results to $$T_{\mathrm{UL},\mathrm{HA}}=T_{\mathrm{c}}+2\cdot k_{\mathrm{UL}}\cdot T_{\mathrm{Proc}}+(1+2\cdot k_{\mathrm{UL}})\cdot T_{\mathrm{Tx}}.$$ For HARQ-less (HL) repetitions, the absence of feedback reduces the latency. Thus for $k$ repetitions, the latency in the can be defined as $$T_{\mathrm{DL},\mathrm{HL}}=T_{\mathrm{c}}+(1+k)\cdot T_{\mathrm{Tx}},$$ and for UL it will be $$T_{\mathrm{UL},\mathrm{HL}}=T_{\mathrm{c}}+2\cdot T_{\mathrm{Proc}}+(1+2\cdot k_{\mathrm{UL}})\cdot T_{\mathrm{Tx}}.$$ Here, $k_{\mathrm{UL}}=k+1$ again results from the scheduling request and grant. For comparison, we calculate the delays to obtain quantitative results. For this, the following assumptions are made:
- $T_{\mathrm{L1/L2}}=1 \, \textrm{TTI}$,
- $T_{\mathrm{Align}}=1 \, \textrm{TTI}$, as a reception arriving just after a TTI starts needs to be delayed for one TTI,
- $T_{\mathrm{Proc}}=3 \, \textrm{TTI}s$, unless using the reduced processing time feature for which $T_{\mathrm{Proc}}=2 \, \textrm{TTI}$,
- $T_{\mathrm{Tx}}=1 \, \textrm{TTI}$, transmissions spanning 1 TTI.
Using the normal cyclic prefix (CP), each LTE subframe contains 14 OFDM symbols with a duration of . This results in the following TTI lengths for subframe (sf), slot and subslot:
- $T_{\textrm{sf TTI}}$ $=1 \, \textrm{TTI}$ $= \unit[1]{ms},$
- $T_{\textrm{slot TTI}}$ $= T_{\textrm{sf TTI}}/2$ $= \unit[0.5]{ms},$
- $T_{\textrm{subslot TTI}}$ $= T_{\textrm{sf TTI}}/6$ $= \unit[0.17]{ms}.$
------------------- --------------------------------------------
Channel Model Rayleigh fading (ideal channel estimation)
DCI payload 45 bits
CRC size 16 bits
DCI blind decodes 20
Channel Code TBCC AL 1-8
Decoder Viterbi
Chase Combining Bitwise: LLRs are combined Bitwise
Symbolwise: combining of QAM-Symbols
------------------- --------------------------------------------
: Simulation assumptions for DCI duplication[]{data-label="tab:DCIdupSim"}
Table \[tab:results\] lists the calculated latencies for 3GPP LTE Rel. 14 and 15 for subframe (SF), slot, and subslot configurations.\
It can be seen, that with HARQ, only the HRLLC requirement is within reach when using the LTE Rel. 15 subslot configuration. HARQ-less repetition improves performance and brings URLLC into reach for transmission with the LTE Rel. 15 subslot configuration. In the UL, the delay caused by the is too high. The less stringent HRLLC requirement makes UL possible for Rel. 15 slot and subslot configurations. In the DL, even LTE Rel. 14 subframe configuration fulfills the delay requirements. In LTE, UL is handled by using with pre-allocated resources thereby removing the additional delay caused by SR and grant time.
Next, we evaluate the reliability of the control channel when using the proposed DCI duplication mechanism. The performance of DCI duplication is numerically evaluated using LTE link-level simulations. Details are given in Table \[tab:DCIdupSim\]. For this, a 16-bit is added to a generated 45-bit DCI, which are then sent over a Rayleigh fading channel. For combining the two duplicate DCIs, we compare two schemes:
- Bitwise: are combined bitwise before decoding,
- Symbolwise: received symbols are combined before demodulation.
The code rate is varied by changing the which defines how many are used for transmission. Thus, a higher AL effectively decreases the code rate. For the link-level simulations, a fixed pairing of duplicate DCIs is assumed and chase combining is only performed if one of the two DCIs is correctly decoded. Thus, a DCI is missed either if both initial decodes fail, or if the combination of the two decodes does not result in the correct DCI. This is also referred to as *miss probability*.
The results are shown in Fig. \[fig:DCIdupPlots\]. As a reference, the single DCI false positive probability is shown by the gray dotted line. This is analytically calculated from Eq. \[eq:FP\] with 20 blind decoding attempts, which is also used in the link-level simulations. In addition to the false positives, the of a single DCI is compared to the of combined DCIs in Fig. \[fig:DCIdupPlots\](a) and (b). With both schemes it can be seen, that the assumption of only performing the chase combining upon the detection of at least one DCI, does not significantly affect the performance. The combined BLER and the miss probability, also considering false rejection, are the same at the targeted error rates over all . Finally, the QAM-symbolwise combining of DCIs in Fig. \[fig:DCIdupPlots\](b) suppresses false positives significantly better.
[0.47]{} ![Numerical results on DCI duplication comparing different combining methods.[]{data-label="fig:DCIdupPlots"}](./VTC_Bitwise-cropped.pdf "fig:"){width="\textwidth"}
[0.47]{} ![Numerical results on DCI duplication comparing different combining methods.[]{data-label="fig:DCIdupPlots"}](./VTC_Symwise-cropped.pdf "fig:"){width="\textwidth"}
From LTE to New Radio (NR)
==========================
’s basic frame structure design was fixed with Rel. 8. Although the control channel can vary from 1 to 3 OFDM symbols, based on the , its size and position within a radio frame are fixed. Furthermore, the control channel spans over the whole frequency domain, forcing each to perform blind decoding over the whole frequency band, e.g. the maximum is over one component carrier, which is 20 MHz bandwidth. Future LTE releases have to stay backwards compatible to the existing radio frame structure. This static design limits implementation of new URLLC or HRLLC services, especially when multiplexing services with different service requirements, such as eMBB and URLLC in the same frequency band. Especially when it comes to TDD, ’s frame structure is quite limited, with only 8 different TDD modes. With this fixed number of time slots in up- or downlink, for a closed-loop service, a constant delay in the range of milliseconds is added to each transmission.
The goal of is to overcome the design limits of LTE by defining a “forward-compatible” frame structure. The idea is to define the frame structure in such a way, that new services can easily be added in the future. The key ingredients to support service in are mini-slots, a self-contained frame structure, and grant-free radio access concepts [@takeda2017latency]. Similar to , supports a short subslot format, called mini-slots or non-slot based scheduling. The self-contained frame structure allows a UE to only decode a very short control channel organized in control-resource sets (CORESETs) with a UE specific search space prior to decoding the data channel. This reduces the processing time and allows fast feedback to the transmitter based on the decoding outcome. Furthermore, coding rates of down to $1/12$ are expected for channel coding [@3gpp.R1-163757]. Due to the expected sporadic nature of traffic, a new multiplexing concept based on pre-emption has been introduced in [@3gpp.R1-1716941]. This allows puncturing of transmissions in case that unexpected traffic arrives at least for . Since the transmission is degraded by this mechanism, a new report, so-called , is introduced to indicate punctured positions afterwards. The same concept is currently discussed for the [@3gpp.R1-1801566].
Although, timing is designed flexibly in [@3gpp.R1-1716941], the main issue of is still an open topic. To cope with this limitation, the authors of [@ldpc_subcodes] have proposed an early feedback technique, which enables to start processing during reception. Hence, the receiver can provide the feedback at an earlier stage. This early feedback is enabled by exploiting substructures of the channel code. As shown in [@ldpc_subcodes], subcode-based early HARQ achieves a reliability comparable to regular HARQ while decreasing the .
Conclusion
==========
5G is envisioned to support three broad categories of services: eMBB, URLLC, and mMTC. URLLC services refer to future applications which require secure data communications from one end to another, while fulfilling ultra-high reliability and low latency. These have been addressed in 3GPP Rel. 15 for LTE by two work items (WIs), as well as considered in the basic design of NR. This paper gives an overview of URLLC requirements and describes technical innovations and standardization efforts in 3GPP LTE. Latency reduction techniques with reduced processing time and improved frame structures, shortened TTI, are shown. Furthermore, a detailed analysis of the resulting latencies, which are feasible with LTE Rel. 15 are given. Especially, the reliability limits of LTE’s control channel are highlighted and solutions are presented. Our numerical results show that QAM-symbol combining of duplicate DCIs improves control channel robustness achieving URLLC targets. Our presented solution can also be adopted for improving the robustness of the control channel in „forward-compatible“ 5G systems.
|
---
author:
- Álefe de Almeida
- Luca Amendola
- Viviana Niro
bibliography:
- 'report.bib'
title: Galaxy rotation curves in modified gravity models
---
Introduction
=============
The observations of dynamics of the galaxies, in particular their rotation curves, constitute one of the main evidences for dark matter. However, also alternatives to Newtonian gravity have been employed to address the dark matter problem (e.g. ref.), indicating that the underlying gravity at galactic scales may be not Newtonian and a different gravity should be considered instead of a new material component. In these models, since galaxies can be approximated within the weak-field regime of relativity, the problem can be expressed as a modified Newtonian potential. In particular, most work has been performed adopting a potential with a Yukawa-like correction $$\Phi(\textbf{x})=-G\int\frac{\rho(\textbf{x}')}{|\textbf{x}-\textbf{x}'|}\left(1+\beta e^{-|\textbf{x}-\textbf{x}'|/\lambda}\right)d^{3}\textbf{x}'\;.\label{yukawa1}$$ where $\rho(x)$ is the matter density distribution. The parameter $\beta$ measures the strength of the ‘fifth-force’ interaction while the second parameter, $\lambda$, gives its range. A Yukawa-like correction to the Newtonian potential is predicted by several modified gravity theories, see e.g.). The extra interaction could be mediated by scalars [@2007JPhCS..91a2007C; @2008PhRvD..77b4041C; @2011MNRAS.414.1301C; @2011PhRvD..84l4023S; @2013PhRvD..87f4002S], massive vectors or even with two rank-2 tensor field[@2001PhRvD..63d3503D; @2010CQGra..27w5020C].
Let us review some of the previous research in this field, in order to highlight the differences to our work. Ref.[@2011PhRvD..84l4023S] investigated the sum of a repulsive and an attractive Yukawa interaction obtained as Newtonian limit of a higher-order gravity model. Ref.[@2011PhRvD..84l4023S] did not fit the Yukawa parameters to the data but fixed them to $\lambda_1=100\;\text{kpc},\lambda_2=10^{-2}\;\text{kpc}$, and $\beta_1=1/3, \beta_2=-4/3$, showing that these values provide a good fit to the Milky Way and to NGC 3198. They also find that modified gravity interacting with baryons only is not sufficient to reproduce the observed behaviour of galaxy rotation curves and some amount of dark matter is still needed. In contrast, in ref. (see also ref.[@2013PhRvD..87f4002S]) it was shown that repulsive Yukawa corrections could produce constant profiles for rotation curves at large radii without dark matter, if $-0.95\leq \beta \leq-0.92$ and $\lambda \simeq 25-50$ kpc.
Following ref., most other works did not include dark matter. While refs.[@2001PhRvD..63d3503D; @2011PhRvD..84l4023S] applied the prediction from modified gravity to just one or two galaxy, ref.[@2013MNRAS.436.1439M] performed a full comparison against observed rotation curves datasets of nine galaxies belonging to THe HI Nearby Galaxy Survey catalogue (THINGS). Different values for $\beta$ and $\lambda$ were used for the different galaxies and then the averaged values, $\beta=-0.899 \pm 0.003$ and $\lambda=23.81 \pm 2.27$ kpc, were finally employed to fit the galaxies of the Ursa Major and THINGS catalogue. This analysis was carried out without a dark matter component. The THINGS catalogue was also implemented in ref.[@2014PhRvD..89j4011R], obtaining similar results, $\beta=-0.916 \pm 0.041$ and $\lambda=16.95 \pm 8.04$ kpc, again without dark matter.
Positive values of $\beta$ (i.e. an attractive fifth force) were instead obtained in a few works. In ref.[@2011PhRvD..83h4038M], Low Surface Brightness (LSB) galaxies were used. Considering only positive $\beta$ in the fit, values from 1.83 to 11.67 were found, while the $\lambda$ values ranged from 0.349 kpc to 75.810 kpc. In ref.[@2011MNRAS.414.1301C], instead, the value of $\beta$ was kept fixed, either to 1/3 as predicted in $f(R)$ gravity models, or to $1$ as a comparison test, and the Yukawa potential was employed to analyse simulated datasets. The main aim of ref.[@2011MNRAS.414.1301C] was to investigate the bias induced by the assumption of a wrong gravitational theory: they conclude that the disc mass is underestimated and there is a high bias on the halo scale length and the halo virial mass. A strength $\beta=1/3$ has been found to be compatible also with the dynamics of clusters ([@Capozziello:2008ny]), again without dark matter.
Our work however is closer in spirit to ref.[@2003PhRvL..91n1301P], one of the few papers including dark matter. There, it was shown that a repulsive Yukawa interaction between baryons and dark matter gives a good fit for the following galaxies: UGC 4325, $\beta$=$-1.0\pm 0.25$ and $\lambda$=$1.7 \pm 0.6$ kpc; NGC 3109, $\beta$=$-1.1\pm 0.16$ and $\lambda$=$1.2 \pm 0.17$ kpc; LSBC F571-8, $\beta$=$-0.9\pm 0.18$ and $\lambda$=$1.1 \pm 0.1$ kpc; NGC 4605, $\beta$=$-1.1\pm 0.3$ and $\lambda$=$0.2 \pm 0.02$ kpc. In this paper, however, no baryon component is included and the dark matter profile is taken as a simple power law.
As in previous work, we also wish to constrain $\beta$ and $\lambda$ with galaxy rotation curves data. However, the present paper differentiates itself from most earlier research because *a*) we do not exclude dark matter, *b*) we assume that the fifth-force couples differently to dark matter and to baryons, without restriction on the coupling sign, *c*) we include baryonic gas, disk and bulge components, according to observations, separately for each galaxy, along with a NFW dark matter halo profile, *d*) we do not fit $\beta,\lambda$ individually to each galaxy, but rather look for a global fit, and finally, *e*) we adopt a much larger datasets than earlier work, namely 40 galaxies from the Spitzer Photometry $\&$ Accurate Rotation Curves (SPARC)[@2016AJ....152..157L]. In the next section we discuss some aspects of these points.
We anticipate our main result. While most previous analyses both without dark matter and with dark matter [@2003PhRvL..91n1301P], find a best fit for $\beta\approx -1$, i.e. repulsive fifth force, we find $\beta=0.34\pm0.04$, corresponding to an attractive fifth force. Moreover we find that the dark matter mass is on average 20% lower than without the Yukawa correction. It is clear however that this result cannot be straightforwardly interpreted as a rejection of standard gravity, since it is based on a set of parametrizations of the gas, bulge, disk and dark matter profiles that, although realistic, is not yet general enough to include all possibilities.
A species-dependent coupling
============================
If the fifth force is felt differently by baryons (subscript $b$) and dark matter ($dm$), one needs to introduce two coupling constants, say $\alpha_b$ and $\alpha_{dm}$. To fix the ideas, let us assume the fifth force is carried by a scalar field with canonical kinetic term and conformal coupling. Then the particles will obey geodesic equations of the form $$\begin{aligned}
T^{\mu}_{(b)\nu;\mu}&=&-\alpha _b T_{(b)} \phi_{;\nu}\\
T^{\mu}_{(dm)\nu;\mu}&=&-\alpha_{dm} T_{(dm)} \phi_{;\nu}\end{aligned}$$ where $T_{(x)\nu}^{\mu}$ is the energy-momentum tensor of component $x$ and $T_{(x)}$ its trace. The scalar field obeys instead a Klein-Gordon equation which can be written as $$T^{\mu}_{(\phi )\nu;\mu}=(\alpha_b T_{(b)}+\alpha_{dm} T_{(dm)}) \phi_{;\nu}$$ The total energy-momentum tensor is clearly conserved.
In the so-called linear quasi-static approximation, i.e. when we can disregard the propagation of $\phi$ waves, the total potential between two particles of species $x,y$ acquires a Yukawa term as in Eq. \[yukawa1\], with strength [@1992CQGra...9.2093D; @Amendola:2003wa; @2003PhRvL..91n1301P] $$\beta=\alpha_x \alpha_y$$ and universal range $\lambda=m^{-1}$, where $m$ is the scalar field mass. In a galaxy, the baryonic component follows rotation curves that, in equilibrium, are determined by the sum of the potentials produced by the baryons themselves and by the dark matter component. As a consequence, baryons feel a fifth force which is the sum of the baryon-baryon force and the baryon-dark matter one. The first is proportional to $\alpha_b^2$, while the second one to $\alpha_b\alpha_{dm}$. Local gravity experiment, however, show that $|\alpha_b|$ has to be very small, typically less than $10^{-2}$ [@2001LRR.....4....4W; @2016ChPhC..40j0001P]. We can therefore neglect the baryon-baryon fifth force, i.e. assume that baryons exert just the standard gravitational force on the other baryons. Notice that if we cannot invoke a screening mechanism to screen the fifth force in, say, the solar system, and therefore evade the local constraints, because then the same mechanism would presumably also screen stars, which then would follow standard rotation curves, rather than those modified by the fifth force. All this means that the rotation curves only depend on the Yukawa strength $\beta=\alpha_b\alpha_{dm}$. This, along with $\lambda$, is the parameter we wish to determine. Since it turns out that $\beta$ is of order unity, we conclude that $\alpha_{dm}$ must be very large, ${\cal O}(100)$. We will not discuss whether this large value is compatible with other constraints, e.g. from cosmology. However we notice that since we find $\lambda\approx 6$ kpc, any observation involving scales much larger than this will see a Yukawa force suppressed as $\exp(-r/\lambda)$, and therefore negligible. Finally, we notice that since we are assuming no screening, the values of $\beta,\lambda$ do not change from galaxy to galaxy. That is why we do not try to fit the values individually to each galaxy, but rather seek a global fit. For technical reasons, to be discussed later, however, we decided to select 40 galaxies and group them in four datasets of ten each, and find the best Monte Carlo fit for each group. Even in this way, the complexity is quite high, since we deal with a number of simultaneously-varying parameters for each group ranging from 23 up to 25.
In many previous works, as we have seen in the previous section, dark matter was not considered. Therefore, following the interpretation given above, what has been measured was the baryon-baryon strength $\beta=\alpha_b^2$. Since the fits to rotation curves have mostly provided a negative value, one has to modify the picture above by introducing either a non-canonical kinetic term (actually, a field with imaginary sound speed, which then suffers of a gradient instability), or a vector boson rather than a scalar one. In any case, a value $|\alpha_b|$ of order unity is in contrast with local gravity constraints. The only way to make these models consistent with local gravity constraints is to assume then that experiments on Earth are screened while stars are not. For example, in ref.[@2017PhRvD..95f4050B] is considered that the fifth force, at galactic scales, is produced by the symmetron scalar field [@2010PhRvL.104w1301H]. Four galaxies from the SPARC data set were used for the fit and no dark matter was considered. The deviations from standard gravity at galactic scales are screened in high density environments, e.g. solar system [@2018arXiv180505226O]. Furthermore, in ref.[@2018arXiv180505226O] it is shown that the symmetron field can explain the vertical motion of stars in the Milky Way, without dark matter.
The Yukawa correction {#section2}
======================
The Yukawa-like corrections to the Newtonian potential has the general form $$\Phi(\textbf{x})=-G\int\frac{\rho(\textbf{x}')}{|\textbf{x}-\textbf{x}'|}\left(1+\beta e^{-|\textbf{x}-\textbf{x}'|/\lambda}\right)d^{3}\textbf{x}'\;.\label{yukawa}$$
Clearly, we recover Newtonian gravity when $\beta=0$, or at scales much larger than $\lambda$. In the case of scales much smaller than $\lambda$, gravity could be stronger or weaker than Newtonian, depending on the sign of $\beta$, that we leave free in our analysis.
We will assume a spherical distribution for dark matter derived from $N$-body simulations of cold dark matter (CDM), the Navarro-Frenk-White profile (hereafter NFW)[@1996ApJ...462..563N] $$\rho_{\text{NFW}}(r)=\frac{\rho_{s}}{\frac{r}{r_{s}}(1+\frac{r}{r_{s}})^{2}}\;,\label{nfw}$$ where $\rho_{s}$ is the characteristic density and $r_{s}$ is the scale radius. In principle, these parameters are independent but several $N$-body simulations(e.g.[@2008MNRAS.391.1940M; @Maccio:2008pcd]) claims that there is a relation between them. This relation is usually parametrized by the concentration parameter $c\equiv r_{200}/r_{s}$ and $M_{200}\equiv(4\pi/3)200\rho_{\text{crit}}r_{200}^{3}$, where $\rho_{\text{crit}}$ is the critical density. Hence, the NFW profile can be written in terms of a single parameter, namely $M_{200}$. Thus, we can relate $(\rho_{s},r_{s})\rightarrow(c,M_{200})$ via[@mo2010galaxy] $$\begin{aligned}
\rho_{s} & =\frac{200}{3}\frac{c^{3}\rho_{\text{crit}}}{\ln(1+c)-\frac{c}{1+c}}\;,\label{rhos}\\
r_{s} & =\frac{1}{c}\left(\frac{3M_{200}}{4\pi200\rho_{\text{crit}}}\right)^{1/3}\;\label{rs}\end{aligned}$$ We will then assume, for galaxy-sized halos, the following $c-M_{200}$ relation[@2008MNRAS.391.1940M] $$c(M_{200})=10^{0.905}\left(\frac{M_{200}}{10^{12}h^{-1}\text{M}_{\odot}}\right)^{-0.101}\;.$$ Hence, the equation (\[rs\]) becomes $$r_{s}\approx28.8\left(\frac{M_{200}}{10^{12}h^{-1}\text{M}_{\odot}}\right)^{0.43}\text{kpc}\;,$$ where we used $\rho_{\text{crit}}=143.84\;\text{M}_{\odot}/\text{kpc}^{3}$ and $h=0.671$.
The gravitational potential can be computed inserting the equation (\[nfw\]) in the equation (\[yukawa\]). The potential $\Phi$ can be written as a sum of the usual Newtonian potential for a NFW profile, $\Phi_{\mathrm{NFW}}$, plus a modified gravity part $\Phi_\text{mg}$ ($\propto\beta$) that can be integrated analytically for NFW [@Pizzuti:2017diz] $$\begin{aligned}
\Phi_{\text{mg}}(r) & =\frac{2\pi G\beta\rho_{s}r_{s}^{3}}{r}\Bigg\{\exp\left(-\frac{r_{s}+r}{\lambda}\right)\left[\text{Ei}\left(\frac{r_{s}}{\lambda}\right)-\text{Ei}\left(\frac{r_{s}+r}{\lambda}\right)\right]+\nonumber \\[1ex]
& -\exp\left(\frac{r_{s}+r}{\lambda}\right)\text{Ei}\left(-\frac{r_{s}+r}{\lambda}\right)+\exp\left(\frac{r_{s}-r}{\lambda}\right)\text{Ei}\left(-\frac{r_{s}}{\lambda}\right)\Bigg\}\;,\end{aligned}$$ where $\text{Ei}(x)$ is defined as $$\text{Ei}(x)=-\int_{-x}^{\infty}\frac{e^{-t}}{t}dt\;,$$ In the following, we refer to the parametrization discussed in this section (specifically, the gas, disk, and bulge baryonic components, plus the NFW profile for the dark matter with mass-concentration relation, plus the Yukawa term coupled to dark matter only) simply as the Yukawa model, and we will compare it to the “standard model”, i.e. the same parametrization but without the Yukawa term.
The data sample {#section3}
===============
As already mentioned, the observational data for the rotation curves considered in this work are taken from the catalogue Spitzer Photometry & Accurate Rotation Curves (SPARC)[@2016AJ....152..157L], which contains 175 disk galaxies. This catalogue includes observations at near-infrared (3.6$\mu$m), which can trace the stellar distribution, and high-quality rotation curves from HI/H$\alpha$ measurements. The HI regions are measured by the 21-cm line hyperfine transition of the neutral atomic hydrogen while the HII regions are visible due the emission of the H$\alpha$ line of the ionized gas.
To evaluate the kinematics of disk galaxies, the components which are relevant to the analysis are: gas, disk, bulge and dark matter halo. The main physical quantity is the total circular velocity at the galactic plane, $V_{c}$, which is related to the total gravitational potential, $\Psi$, via $$V_{c}^{2}=r\frac{d\Psi}{dr}\;.\label{circularvel}$$ Here, as already discussed, we will assume that the potential which produces the acceleration for the baryonic components is the Newtonian one, while for dark matter is given by equation (\[yukawa\]). Thus, the total gravitational potential reads $$\Psi=\Phi_{\text{gas}}+\Phi_{\text{disk}}+\Phi_{\text{bulge}}+\Phi_{\text{NFW}}+\Phi_{\text{mg}}\;.$$ The linearity of equation (\[circularvel\]) allows us to split the total circular velocity in terms of each component. We have therefore the following formula $$V_{c}^{2}(r)=V_{\text{gas}}^{2}(r)+\Upsilon_{*\text{D}}V_{\text{disk}}^{2}(r)+\Upsilon_{*\text{B}}V_{\text{bulge}}^{2}(r)+V_{\text{NFW}}^{2}(r)+V_{\text{mg}}^{2}(r)\;,\label{circular}$$ where $\Upsilon_\text{*D}$ ($\Upsilon_\text{*B}$) is the stellar mass-to-light ratio for the disk (bulge) and it is equivalent to the mass $M_\text{D}$ of the disk ($M_\text{B}$ of the bulge) divided by the luminosity $L_\text{D}$ of the disk ($L_\text{B}$ of the bulge).
The rotation velocities $V_{\text{gas}},V_{\text{disk}}$ and $V_{\text{bulge}}$ are already available on SPARC for each galaxy on [astroweb.cwru.edu/SPARC](http://astroweb.cwru.edu/SPARC). As an example we show in the table \[tab:sparcex\] the values of each component at the observed radius for a single galaxy. The details about how each baryonic component was derived can be found in ref.[@2016AJ....152..157L] but we will discuss briefly here. Using the redshift or blueshift of the HI emission line for the outer region of the galaxy and also the H$\alpha$ measurements for the inner region it is possible to obtain a high-quality rotation curve data. The gaseous component, $V_\text{gas}$, is obtained from the superficial density distribution, $\Sigma_\text{gas}$, inferred also by HI observations. Then, solving the Poisson equation for $\Phi_\text{gas}$ and using equation (\[circularvel\]) we have $V_\text{gas}$. The computation of $V_\text{disk}$ and $V_\text{bulge}$ is analogous: the surface brightness, $I(r)$, of each stellar component (disk and bulge) is directly measured at the near-infrared (3.6$\mu$m) band. At this band, the surface density of each stellar component, namely $\Sigma_\text{disk}$ and $\Sigma_\text{bulge}$, is proportional to the surface brightness, i.e. $\Sigma(r)=\Upsilon_*I(r)$, where $\Upsilon_*$ is the mass-to-light ratio of the respective component. Hence, from $\Sigma$ one obtains $V_\text{disk}$ and $V_\text{bulge}$ in the same way as for the gaseous component. For practical reasons, in the derivation of $V_\text{disk}$ and $V_\text{bulge}$ it is assumed $\Upsilon_\text{*D}=\Upsilon_\text{*B}=1$, hence with this normalization of the mass-to-light ratios the problem can be rescaled trivially for any $\Upsilon_\text{*D},\Upsilon_\text{*B}
$, which are then additional free parameters.
Since we are considering that the baryons are not coupled to the fifth force, the procedure by ref.[@2016AJ....152..157L] can be directly adapted to our purpose. Finally, in order to obtain the velocities of each component at every radius $r$ we perform a cubic spline interpolation.
-------- ---------------------- ---------------------- ---------------------- -----------------------------
Radius $V_{\text{obs}}$ $V_{\text{gas}}$ $V_{\text{disk}}$ $\Sigma_{\text{disk}}$
(kpc) $(\text{km s}^{-1})$ $(\text{km s}^{-1})$ $(\text{km s}^{-1})$ $(L_{\odot}\text{pc}^{-2})$
0.42 $14.2\pm1.9$ 4.9 4.8 11.0
1.26 $28.6\pm1.8$ 13.1 10.8 5.8
2.11 $41.0\pm1.7$ 19.6 13.6 2.7
2.96 $49.0\pm1.9$ 22.4 13.3 1.0
3.79 $54.8\pm2.0$ 22.8 12.6 0.7
4.65 $56.4\pm3.1$ 21.4 12.3 0.4
5.48 $57.8\pm2.8$ 18.7 12.0 0.2
6.33 $56.5\pm0.6$ 16.7 10.6 0.0
-------- ---------------------- ---------------------- ---------------------- -----------------------------
: \[tab:sparcex\] Table for the galaxy UGCA442 emphasizing each baryonic component (there is no bulge in this galaxy ). $\Sigma_{\text{disk}}$ is the surface density for the disk.
Finally, the components $V_{\text{NFW}}$ and $V_{\text{mg}}$ are given by $$V_{\text{NFW}}^{2}(r)=\frac{4\pi Gr_{s}^{3}\rho_{s}}{r}\left[-\frac{r}{r+r_{s}}+\ln\left(1+\frac{r}{r_{s}}\right)\right]$$ and [^1] $$\begin{aligned}
V_{\text{mg}}^{2}(r) & =-\frac{2\pi G\beta\rho_{s}r_{s}^{3}}{r}\Bigg\{\frac{2r}{r_{s}+r}+\exp\left(\frac{r_{s}+r}{\lambda}\right)\left(\frac{r}{\lambda}-1\right)\text{Ei}\left(-\frac{r_{s}+r}{\lambda}\right)+\nonumber \\[1ex]
& +\exp\left(-\frac{r_{s}+r}{\lambda}\right)\left(1+\frac{r}{\lambda}\right)\left[\exp\left(\frac{2r_{s}}{\lambda}\right)\text{Ei}\left(-\frac{r_{s}}{\lambda}\right)+\text{Ei}\left(\frac{r_{s}}{\lambda}\right)-\text{Ei}\left(\frac{r+r_{s}}{\lambda}\right)\right]\Bigg\}\;.\end{aligned}$$
Fitting the rotation curves {#section4}
============================
We present now the procedure to find the best-fit values for the set of free parameters, namely $\{\Upsilon_{*\text{D}},\Upsilon_{*\text{B}},M_{200}\}$ for each galaxy plus $\{\beta,\lambda\}$, against the observational data. It is assumed here that the errors of the observed rotation curve data follow a Gaussian distribution, so that we can build the likelihood for each galaxy as follows $$\mathcal{L}_{j}(p_{j},\beta,\lambda)=(2\pi)^{-N/2}\Bigg\{\prod_{i=1}^{N}\sigma_{i}^{-1}\Bigg\}\exp\Bigg\{-\frac{1}{2}\sum_{i=1}^{N}\Bigg(\frac{V_{\text{obs},j}(r_{i})-V_{c}(r_{i},p_{j},\beta,\lambda)}{\sigma_{i}}\Bigg)^{2}\Bigg\}\label{likelihood}$$ where $p_{j}=\{\Upsilon_{*\text{D},j},\Upsilon_{*\text{B},j},M_{200,j}\}$, $N$ is the number of observational points for each galaxy, $\sigma_{i}$ is the data error, $V_{\text{obs},j}(r_{i})$ is the observed circular velocity of the $j$-th galaxy at the radius $r_{i}$ and $V_{c}(r_{i},p_{j},\beta,\lambda)$ is the total rotation curve, which was expressed in equation (\[circular\]). We use the values of $V_{\text{obs},j}(r_{i})$ as provided by the SPARC catalogue. To constrain $\beta$ and $\lambda$ with respect to some set of galaxies it is necessary to consider an overall likelihood $\mathcal{L}$. Since the observational data measurements of the galaxies are independent, the overall likelihood function can be computed just multiplying the distributions for each galaxy. Hence, the full likelihood function is given by $$\mathcal{L}(\textbf{p},\beta,\lambda)=\prod_{j=1}^{N_{g}}\mathcal{L}_{j}(p_{j},\beta,\lambda)\;,\label{fulllikelihood}$$ where $\textbf{p}=\{p_{1},...,p_{N_g}\}$ and $N_{g}$ is the total number of galaxies.
According to the Bayes theorem, the posterior distribution is proportional to the prior times the likelihood. We assume here uniform prior for each parameter. Namely, the stellar mass-to-light ratios, for disk and bulge, are restricted $0.3<\Upsilon_{*\text{D}}<0.8$ and $0.3<\Upsilon_{*\text{B}}<0.8$, in agreement with stellar population model analysis[@2014ApJ...788..144M; @2014PASA...31...36S]. A wide range for the other parameters is considered: $10^{9}<M_{200}/\text{M}_{\odot}<10^{14}$, $-2<\beta<2$ and $\overline{\lambda}_{0}<\lambda/\text{kpc}<100$, where $\overline{\lambda}_{0}$ is the mean value among the smallest observable radii when the $N_g$ galaxies are considered. The lower limit on $\lambda$ is imposed to avoid undesired divergences when $\lambda\rightarrow0$.
There are several methods in literature for sampling the parameter space, starting with the well-known Metropolis-Hastings[@1953JChPh..21.1087M] algorithm. In this work we chose the affine-invariant ensemble sampler proposed in ref.[@goodman2010ensemble], which was implemented by the stable and well tested open-source Python package `emcee`[@2013PASP..125..306F]. According ref.[@goodman2010ensemble] this affine-invariant sampler offers several advantages over traditional MCMC sampling methods e.g. high performance and a hand-tuning of few parameters compared to traditional samplers.
Analysis and Results {#section5}
====================
The SPARC catalogue contains 175 galaxies, hence a complete analysis would require 384 free parameters: $175\times(\Upsilon_{\text{*D}},M_{200})+ 32\times\Upsilon_{\text{*B}}$ (many galaxies do not show a bulge) plus $ \beta,\lambda$. Even for the affine-invariant ensemble sampler, this number of parameters is quite high and the sampling of the likelihood becomes computationally too expensive. Thus, we decided to analyse 4 random sets of 10 galaxies each. For two sets (B and D) we have then 25 free parameters each, while for the other two sets we have 23 parameters each. The calculation of one set demands roughly one day of computation on a machine with 4 CPUs and 16 gigabytes of RAM.
![\[autocorrelation\]The autocorrelation time analysis (blue solid line) for each set of galaxies respectively containing 10 objects each. When $\tau$ reaches the dotted line the convergence of the chains is achieved, see ref.[@2013PASP..125..306F].](chains "fig:"){width="7cm"} ![\[autocorrelation\]The autocorrelation time analysis (blue solid line) for each set of galaxies respectively containing 10 objects each. When $\tau$ reaches the dotted line the convergence of the chains is achieved, see ref.[@2013PASP..125..306F].](chains_set2 "fig:"){width="7cm"}\
![\[autocorrelation\]The autocorrelation time analysis (blue solid line) for each set of galaxies respectively containing 10 objects each. When $\tau$ reaches the dotted line the convergence of the chains is achieved, see ref.[@2013PASP..125..306F].](chains_set3 "fig:"){width="7cm"} ![\[autocorrelation\]The autocorrelation time analysis (blue solid line) for each set of galaxies respectively containing 10 objects each. When $\tau$ reaches the dotted line the convergence of the chains is achieved, see ref.[@2013PASP..125..306F].](chains_set4 "fig:"){width="7cm"}
A critical issue for every MCMC sampler is the question of the convergence of the chains[@cowles1996markov]. Here, we will follow the `emcee` developers recommendation in ref.[@2013PASP..125..306F] to use the autocorrelation time $\tau$ as a diagnostics of the MCMC performance. The autocorrelation estimations is very well detailed in ref.[@sokal1997monte] so that here we will just present some essential points in order to clarify the convergence diagnostic.
Let us define the normalized autocorrelation function, $\rho_f$, $$\rho_f(\tau)= \frac{C_f(\tau)}{C_f(0)}\;,$$ with $$C_f(\tau)=\frac{1}{M-\tau}\sum_{t=1}^{M-\tau}\left[f(X(t+\tau))-\langle f\rangle\right]\left[f(X(t))-\langle f\rangle\right]\;,$$ where $\langle f\rangle=\frac{1}{M}\sum_{t=1}^{M}f(X(t))$, $X(t)$ is the sampled random variable of the parameter space, and $M$ is the total length of the chain. $C_f$ is the autocovariance function of a stochastic process: it measures the covariance between samples separated by a time lag $\tau$. If at certain value of $\tau$, namely $\hat{\tau}$, we have $C_f(\hat{\tau})\rightarrow0$, we can say that independent results are obtained. Thus, $\hat\tau$ gives a threshold of how many samples of the posterior are minimally necessary for producing independent samples. The $\tau$ estimation, $\tau_{\text{est}}$, is given by $$\tau_{\text{est}}(N)=1+2\sum_{\tau=1}^{N}\rho_f(\tau)\;,$$ where $N$ starts at $N\ll M$. We have plotted $\tau_{\text{est}}$ for the sets that we considered in this work in figure \[autocorrelation\].
Set $a_{f}$ $k_\text{Y}$ $\chi_{\text{red,tot}}^{2}$ $k_\text{sg}$ $\chi_{\text{red,tot}}^{2}|_{\beta = 0}$ $N$ $\Delta\text{BIC}$ $2\log B_{12}$ CL
---------- --------- ------------------------ ------------------------- ----------------------------- --------------- ------------------------------------------ ------ -------------------- ---------------- ------- ---------- --
$\beta$ $\lambda(\text{kpc})$ $\sigma$
A 0.16 $0.34_{-0.10}^{+0.12}$ $10.27_{-3.82}^{+2.89}$ 25 0.88 23 1.11 206 32.18 31.84 $5.29$
B 0.14 $0.30\pm0.08$ $7.42_{-3.99}^{+2.94}$ 23 0.80 21 0.96 180 17.12 23.21 $4.44$
C 0.12 $0.28_{-0.08}^{+0.09}$ $8.18_{-6.31}^{+5.39}$ 25 0.83 23 1.04 163 20.52 12.41 $3.09$
D 0.15 $0.54_{-0.10}^{+0.11}$ $4.15_{-0.95}^{+0.81}$ 23 0.78 21 1.02 196 32.92 20.38 $4.12$
Combined - $0.34\pm 0.04$ $5.61\pm0.91$ 90 0.82 88 1.03 745 91.61 87.83 8.26
: \[tabelabetalambda\] The acceptance fraction $a_{f}$, the maximum likelihood estimation for $(\beta,\lambda)$, the total goodness of fit $\chi_{\text{red,tot}}^{2}$ and the one calculated fixing $\beta = 0$, for each set of 10 galaxies and for the combination of the data sets. $k_\text{Y}$ ($k_\text{sg}$) is the number of free parameters for the Yukawa model (for standard gravity), while $N$ is the number of data points. We also report the values for the $\Delta$BIC, for $2\log B_{12}$, and the confidence level (CL), see text for more details.
Thus, we ran chains increasing the number of iterations until finally it is possible to perceive a plateau for $\tau$ as we show in figure \[autocorrelation\], which indicates that $\hat\tau$ has been found. After estimating $\hat{\tau}$, we discard the number of iterations $N_{\text{disc}}\sim\hat{\tau}$ (burn-in) and compute the posterior on the parameters. The `emcee` developers suggest that a number of iterations $M>50\hat{\tau}$ is enough to achieve the convergence of the chains. We performed a second MCMC sampling to check this assumption, but now considering a shorter chain with the number of iterations $M=70\hat{\tau}$ with $N_{\text{disc}}=\hat{\tau}$ and we obtained the same results for the posterior. Hence, indeed when the chain length crosses the dotted line, $N=50\tau$, we can achieve the convergence.
Another quantity to monitor is the acceptance fraction $a_{f}$ which is the fraction of proposed steps that are accepted in the sampling process. In our analysis we obtain an acceptance that is between 0.1 and 0.2, as can be seen in table \[tabelabetalambda\].
The main results are displayed on table \[tabelabetalambda\]. We show the best-fit values for the parameters $(\beta,\lambda)$ and their $1\sigma$ error bars, the acceptance fraction $a_{f}$ and the overall goodness of fit $\chi_{\text{red,tot}}^{2}$, i.e. considering all galaxies of the same set together. It is possible to see an improvement of $\chi^2$ from the standard $\beta=0$ model to the Yukawa model for each set and also in the combination of all sets. The analysis with all sets combined takes into account that the total data, i.e. summing all data points, is 745. The total number of parameters is 90: $40\times (\Upsilon_\text{*D}, M_{200})+8\times\Upsilon_\text{*B}$ plus $\beta,\lambda$, for a total of 655 degrees of freedom.
In table \[tabelaresultado1\] we arranged the best-fit values and their $1\sigma$ error bars for the galaxy-specific parameters $\Upsilon_{*,\text{D}},\Upsilon_{*,\text{B}},\text{and }M_{200}$ and also the individual goodness of fit $\chi_{\text{red}}^{2}$. In table \[tabelaresultado2\] we have the same but for the $\beta=0$ case.
In figure \[scatterplot\] we plotted the $1\sigma$ and $2\sigma$ contours of the marginalized distribution for the parameters ($\beta,\lambda$) and their one-dimensional distributions for each set. Finally, the individual rotation curves, evaluated using the equation (\[circular\]) with the best-fit values of tables \[tabelaresultado1\] and \[tabelabetalambda\], are plotted in figures \[rotationcurveA\],\[rotationcurveB\],\[rotationcurveC\] and \[rotationcurveD\].
We have also computed the combined posterior for the parameters $\beta$ and $\lambda$ multiplying the marginalized ones of each set, the combined result is shown in figure \[scatterplot\]. The best-fit and 1$\sigma$ ranges for the parameters $\beta$ and $\lambda$ for the combined analysis are also displayed on table \[tabelabetalambda\].
![\[scatterplot\]The marginalized distribution of the $(\beta,\lambda)$ and the one-dimensional posterior distribution according the set of 10 galaxies each and the combined analysis.](betavslambda_final_lighter "fig:"){width="10cm"}\
![\[scatterplot\]The marginalized distribution of the $(\beta,\lambda)$ and the one-dimensional posterior distribution according the set of 10 galaxies each and the combined analysis.](pdfbeta_final "fig:"){width="6cm"} ![\[scatterplot\]The marginalized distribution of the $(\beta,\lambda)$ and the one-dimensional posterior distribution according the set of 10 galaxies each and the combined analysis.](pdflambda_final "fig:"){width="5.7cm"}
According to our results, we obtained an attractive Yukawa interaction, thus a decrease of the amount of dark matter necessary for a galaxy to reproduce the behaviour of rotation curves data, is expected. We quantify this decrease with the quantity $\mu_{200}\equiv \frac{M_{200}}{M_{200(\beta=0)}}$, plotted in figure \[deltaM200\]. In order to propagate the errors in $\mu_{200}$ we assumed as the standard deviation the largest value of the asymmetric error bars. We also quantify the ratios for the other parameters $\Upsilon_\text{*D}$ and $\Upsilon_\text{*B}$, namely $\gamma_\text{*D}\equiv\frac{\Upsilon_\text{*D}}{\Upsilon_\text{*D}(\beta=0)}$ and $\gamma_\text{*B}\equiv\frac{\Upsilon_\text{*B}}{\Upsilon_\text{*B}(\beta=0)}$. The errors on $\Upsilon_\text{*D}$ and $\Upsilon_\text{*B}$ are also asymmetric, hence for $\gamma_\text{*D}$ and $\gamma_\text{*B}$ we propagate the errors as we did for $\mu_{200}$. The average value of $\mu_{200}$ is $\langle\mu_{200}\rangle=0.80\pm0.02$, corresponding to a $20\%$ of reduction of dark matter due the fifth force. We also obtained the average values $\langle\gamma_\text{*D}\rangle=0.96\pm0.01$ and $\langle\gamma_\text{*B}\rangle=0.96\pm0.04$.
![\[deltaM200\]The ratio $\mu_{200}$ (black dots) and the respective error bars. The cyan dotted line is the average value of $\mu_{200}$.](m200ratios "fig:"){width="15cm"}\
It must be noted that the simple way of counting degrees of freedom adopted above is actually misleading. In fact, the observational data are not compared just to a theoretical model depending on free parameters; rather, they are compared to the sum of a theoretical model (the NFW profile) plus the [*observed*]{} baryonic component, possibly rescaled by $\Upsilon_{*,\text{D}},\Upsilon_{*,\text{B}}$. Also for this reason, we turn now to the Bayesian evidence to assess the relative probability of the two models, with and without the Yukawa correction. In the Bayesian evidence ratio, in fact, only the difference of degrees of freedom between model matters, and this difference arises only because of the Yukawa theoretical model. The evidence is given by $$E=\int \mathcal{L(\mathbf{p},\beta,\lambda)}\mathcal{P}(\mathbf{p},\beta,\lambda)d\beta d\lambda d\mathbf{p}\;,$$ where $\mathcal{P}$ is the prior distribution.The Bayes ratio between the model 1 ($\beta\neq0$) and model 2 ($\beta=0$) is defined as $$B_{12}=\frac{\int \mathcal{L}_1(\mathbf{p},\beta,\lambda)\mathcal{P}_1(\mathbf{p},\beta,\lambda)d\beta d\lambda d\mathbf{p}}{\int \mathcal{L}_2(\mathbf{p})\mathcal{P}_2(\mathbf{p}) d\mathbf{p}}\;.$$
Approximating the likelihood and the priors as Gaussian, we can compute analytically the evidence as $$E=\mathcal{L}_\text{max}\sqrt{\frac{\det\mathbf{P}}{\det\mathbf{Q}}}\exp\left[-\frac{1}{2}(\hat{\theta}_\alpha F_{\alpha\beta}\hat{\theta}_\beta+\bar{\theta}_\alpha P_{\alpha\beta}\bar{\theta}_\beta-\tilde{\theta}_\alpha Q_{\alpha\beta}\tilde{\theta}_\beta)\right]\;,$$ where $-2\ln\mathcal{L}_\text{max}=\chi^2_\text{min}$, $\theta_\alpha=\{\mathbf{p},\beta,\lambda\}$ for our general case; $\hat{\theta}_\alpha$ are the best fit values for the parameters and $\bar{\theta}_\alpha$ are the prior means. The matrix $\mathbf{Q}$ is $\mathbf{Q}=\mathbf{F}+\mathbf{P}$ and $\tilde{\theta}_\alpha=(\mathbf{Q}^{-1})_{\alpha\beta}[F_{\beta\sigma}\hat{\theta}_\sigma+P_{\beta\sigma}\bar{\theta}_\sigma]$, where $\mathbf{F}$ is the Fisher matrix and $\mathbf{P}$ is the inverse of the covariance matrix of the prior. When the prior is weak, which is our case, the expression above can be simplified and the Bayes ratio becomes $$B_{12}=e^{-\frac{1}{2}(\chi^2_\text{min,1}-\chi^2_\text{min,2})}\sqrt{\frac{\det\mathbf{P}_1\det\mathbf{F}_2}{\det\mathbf{P}_2\det\mathbf{F}_1}}\;.$$
If we assume diagonal matrices for $\mathbf{P}$ the determinant is just the product of the diagonal entries, i.e. the inverse of the squared errors. If the prior is flat, as in our case, we can take the variance of an uniform distribution as squared error. Since the models 1,2 share most of the parameters, in the ratio $\det\mathbf{P}_1/\det\mathbf{P}_2$ all terms except the $\beta,\lambda$ simplify. Hence we have $$B_{12}=e^{-\frac{1}{2}(\chi^2_\text{min,1}-\chi^2_\text{min,2})}\frac{1}{p_\beta p_\lambda}\sqrt{\frac{\det\mathbf{F}_2}{\det\mathbf{F}_1}}\;,$$ where $p_\beta,p_\lambda$ are the square root of the variance of the uniform distribution assumed for $\beta,\lambda$.
The Bayes factor for the combined sets is computed considering the combination of the Fisher matrices of the sets. Since the correlation between the sets is zero, the combined Fisher matrix, $\mathbf{F}_\text{comb}$, is a block diagonal matrix where the diagonal entries are the Fisher matrices of each set. Thus, it is straightforward to calculate the determinant of $\mathbf{F}_\text{comb}$ for models 1 and 2. We assumed the same structure for the combined $\mathbf{P}$, namely $\mathbf{P}_\text{comb}$, since the the minimum value $\lambda_0$ changes according to the set.
Once we have $B_{12}$, then the probability $\mathcal{P}_{12}$ that the right model is 1 rather than 2 is $$\mathcal{P}_{12}=\frac{B_{12}}{1+B_{12}}\;.$$
As an aside, we also computed the Bayesian Information Criterion (BIC), which gives a very simple approximation to the evidence. The expression for BIC is given by [@schwarz1978estimating] $$\text{BIC}= -2\ln \mathcal{L}_\text{max}+2k\ln N\;,$$ where $k$ is the number of free parameters and $N$ is the number of data points. The values of $k$ for both models, namely $k_\text{Y} $ for the Yukawa model and $k_\text{sg}$ for standard gravity, are reported in table \[tabelabetalambda\] together with the number of data points $N$ used in each set and in the combined analysis. In our case, the likelihoods are Gaussian and hence we have again $-2\ln\mathcal{L}_\text{max}=\chi^2_\text{min}$. The relative BIC ($\Delta\text{BIC}$) is defined as $$\Delta\text{BIC}\equiv\text{BIC}|_{\beta=0}-\text{BIC}|_{\beta\neq0}\;.$$ The relative BIC approximates then $2\ln B_{12}$ in the limit in which the variance of the parameters decreases when going from prior to posterior by a factor $N$. The $\Delta\text{BIC}$ and the confidence level (CL) associated to $\mathcal{P}_{12} $ values for each set and for the combined analysis are displayed in table \[tabelabetalambda\]. As expected, the BIC gives a rough approximation to the Gaussian evidence. Both prefer the $\beta\not= 0$ model to an extremely high significance, more than 8$\sigma$ for the combined set.
Set Galaxy $\chi^2_{\text{red}}$
----- ---------- ------------------------------- ------------------------------- ---------------------------------- ------ --
$\Upsilon_{*\text{D}}$ $\Upsilon_{*\text{B}}$ $M_{200}(10^{11}M_{\odot})$
A F568V1 $ 0.60 ^{+ 0.20 }_{- 0.10 }$ - $ 2.91 ^{+ 0.61 } _{- 0.82 }$ 0.35
A NGC0024 $ 0.79 ^{+ 0.01 }_{- 0.01 }$ - $ 1.63 ^{+ 0.21 } _{- 0.28 }$ 1.68
A NGC2683 $ 0.64 ^{+ 0.04 }_{- 0.04 }$ $ 0.52^{+ 0.15 }_{- 0.21 }$ $ 3.79 ^{+ 0.64 } _{- 0.81 }$ 1.37
A NGC2915 $ 0.32 ^{+ 0.01 }_{- 0.02 }$ - $ 0.76 ^{+ 0.10 } _{- 0.14 }$ 0.98
A NGC3198 $ 0.40 ^{+ 0.04 }_{- 0.05 }$ - $ 4.30 ^{+ 0.27 } _{- 0.29 }$ 1.31
A NGC3521 $ 0.49 ^{+ 0.01 }_{- 0.02 }$ - $ 12.00 ^{+ 2.22 } _{- 2.81 }$ 0.37
A NGC3769 $ 0.33 ^{+ 0.02 }_{- 0.03 }$ - $ 1.90 ^{+ 0.25 } _{- 0.32 }$ 0.75
A NGC3893 $ 0.46 ^{+ 0.04 }_{- 0.04 }$ - $ 8.64 ^{+ 2.58 } _{- 2.22 }$ 1.26
A NGC3949 $ 0.36 ^{+ 0.03 }_{- 0.05 }$ - $ 8.85 ^{+ 4.37 } _{- 5.81 }$ 0.45
A NGC3953 $ 0.62 ^{+ 0.07 }_{- 0.07 }$ - $ 3.39 ^{+ 1.64 } _{- 2.82 }$ 0.73
B NGC3992 $ 0.74 ^{+ 0.05 }_{- 0.03 }$ - $ 14.47 ^{+ 2.06 } _{- 2.11 }$ 0.88
B NGC4051 $ 0.40 ^{+ 0.05 }_{- 0.10 }$ - $ 2.32 ^{+ 1.23 } _{- 1.64 }$ 1.27
B NGC4088 $ 0.31 ^{+ 0.01 }_{- 0.01 }$ - $ 3.62 ^{+ 0.65 } _{- 0.76 }$ 1.09
B NGC4100 $ 0.67 ^{+ 0.03 }_{- 0.03 }$ - $ 4.70 ^{+ 0.76 } _{- 0.83 }$ 1.20
B NGC4138 $ 0.69 ^{+ 0.09 }_{- 0.05 }$ $ 0.53^{+ 0.10 }_{- 0.21 }$ $ 3.18 ^{+ 1.00 } _{- 1.44 }$ 2.67
B NGC4157 $ 0.35 ^{+ 0.02 }_{- 0.03 }$ $ 0.45^{+ 0.09 }_{- 0.15 }$ $ 7.44 ^{+ 1.24 } _{- 1.38 }$ 0.76
B NGC4183 $ 0.49 ^{+ 0.09 }_{- 0.14 }$ - $ 1.32 ^{+ 0.22 } _{- 0.25 }$ 0.19
B NGC4559 $ 0.31 ^{+ 0.01 }_{- 0.01 }$ - $ 1.85 ^{+ 0.22 } _{- 0.22 }$ 0.43
B NGC5005 $ 0.43 ^{+ 0.06 }_{- 0.11 }$ $ 0.50^{+ 0.07 }_{- 0.08 }$ $ 52.94 ^{+ 32.80 } _{- 49.57 }$ 0.08
B NGC6503 $ 0.45 ^{+ 0.02 }_{- 0.03 }$ - $ 1.99 ^{+ 0.20 } _{- 0.22 }$ 1.91
C UGC06983 $ 0.51 ^{+ 0.11 } _{- 0.16 }$ - $ 1.66 ^{+ 0.29 } _{- 0.36 }$ 0.69
C UGC07261 $ 0.53 ^{+ 0.12 } _{- 0.21 }$ - $ 0.41 ^{+ 0.11 } _{- 0.14 }$ 0.17
C UGC07690 $ 0.68 ^{+ 0.11 } _{- 0.06 }$ - $ 0.13 ^{+ 0.05 } _{- 0.05 }$ 0.89
C UGC07866 $ 0.38 ^{+ 0.06 } _{- 0.08 }$ - $ 0.02 ^{+ 0.07 } _{- 0.02 }$ 2.52
C UGC08490 $ 0.78 ^{+ 0.02 } _{- 0.01 }$ - $ 0.60 ^{+ 0.08 } _{- 0.10 }$ 0.78
C UGC08550 $ 0.49 ^{+ 0.08 } _{- 0.17 }$ - $ 0.18 ^{+ 0.04 } _{- 0.04 }$ 1.02
C UGC08699 $ 0.71 ^{+ 0.05 } _{- 0.05 }$ $ 0.67 ^{+ 0.03 }_{- 0.05 }$ $ 6.86 ^{+ 1.16 } _{- 1.34 }$ 0.86
C UGC09992 $ 0.43 ^{+ 0.10 } _{- 0.13 }$ - $ 0.03 ^{+ 0.03 } _{- 0.03 }$ 1.98
C UGC10310 $ 0.53 ^{+ 0.10 } _{- 0.22 }$ - $ 0.28 ^{+ 0.06 } _{- 0.08 }$ 1.25
C UGC12506 $ 0.78 ^{+ 0.02 } _{- 0.01 }$ - $ 17.77 ^{+ 2.53 } _{- 2.22 }$ 1.22
D NGC7331 $ 0.32 ^{+ 0.01 }_{- 0.01 }$ $ 0.49 ^{+ 0.08 } _{- 0.18 }$ $ 20.56 ^{+ 0.86 } _{- 0.79 }$ 0.87
D NGC7793 $ 0.41 ^{+ 0.05 }_{- 0.05 }$ - $ 1.01 ^{+ 0.21 } _{- 0.24 }$ 0.95
D NGC7814 $ 0.76 ^{+ 0.04 } _{- 0.03 }$ $ 0.60 ^{+ 0.03 } _{- 0.03 }$ $ 21.39 ^{+ 2.07 } _{- 2.09 }$ 0.82
D UGC02259 $ 0.72 ^{+ 0.08 } _{- 0.05 }$ - $ 0.75 ^{+ 0.09 } _{- 0.11 }$ 2.84
D UGC03546 $ 0.55 ^{+ 0.04 } _{- 0.04 }$ $ 0.38 ^{+ 0.04 } _{- 0.04 }$ $ 9.33 ^{+ 0.66 } _{- 0.64 }$ 1.05
D UGC06446 $ 0.50 ^{+ 0.09 } _{- 0.19 }$ - $ 0.56 ^{+ 0.08 } _{- 0.09 }$ 0.25
D UGC06930 $ 0.40 ^{+ 0.06 } _{- 0.10 }$ - $ 1.19 ^{+ 0.21 } _{- 0.20 }$ 0.62
D UGC06983 $ 0.40 ^{+ 0.05 } _{- 0.09 }$ - $ 1.65 ^{+ 0.21 } _{- 0.23 }$ 0.67
D UGC07261 $ 0.49 ^{+ 0.09 } _{- 0.18 }$ - $ 0.34 ^{+ 0.08 } _{- 0.10 }$ 0.11
D UGC07690 $ 0.66 ^{+ 0.13 } _{- 0.07 }$ - $ 0.10 ^{+ 0.03 } _{- 0.04 }$ 0.72
: \[tabelaresultado1\]The maximum likelihood estimation for the $\Upsilon_{\text{*D}}$,$\Upsilon_{\text{*B}}$ and $M_\text{200}$ parameters, and the goodness of fit $\chi_\text{red}^2$ for each galaxy, for the Yukawa model.
Set Galaxy $\chi^2_{\text{red}}|_{\beta=0}$
----- ---------- ------------------------------ ---------------------------------- ---------------------------------- ------ --
$\Upsilon_{*\text{D}}$ $\Upsilon_{*\text{B}}$ $M_{200}(10^{11}M_{\odot})$
A F568V1 $ 0.63 ^{+ 0.16 }_{- 0.10 }$ - $ 5.43 ^{+2.53 } _{- 3.57 }$ 0.63
A NGC0024 $ 0.79 ^{+ 0.01 }_{- 0.01 }$ - $ 2.75 ^{+ 0.23 } _{- 0.29 }$ 2.31
A NGC2683 $ 0.68 ^{+ 0.05 }_{- 0.04 }$ $ 0.52^{+ 0.11 }_{- 0.21 }$ $ 4.38 ^{+ 0.74 } _{- 0.96 }$ 1.20
A NGC2915 $ 0.32 ^{+ 0.02 }_{- 0.02 }$ - $ 1.31 ^{+ 0.12 } _{- 0.14 }$ 1.17
A NGC3198 $ 0.52 ^{+ 0.01 }_{- 0.01 }$ - $4.65 ^{+ 0.09 } _{- 0.11 } $ 1.44
A NGC3521 $ 0.51 ^{+ 0.01 }_{- 0.01 }$ - $ 17.65 ^{+ 3.59 } _{- 3.36 }$ 0.29
A NGC3769 $ 0.36 ^{+ 0.03 }_{- 0.06 }$ - $ 2.66 ^{+ 0.31 } _{- 0.39 }$ 0.68
A NGC3893 $ 0.49 ^{+ 0.04 }_{- 0.03 }$ - $ 12.05 ^{+ 2.34 } _{- 2.52 }$ 1.27
A NGC3949 $ 0.37 ^{+ 0.03 }_{- 0.06 }$ - $ 18.98 ^{+ 10.56 } _{- 14.10 }$ 0.29
A NGC3953 $ 0.65 ^{+ 0.07 }_{- 0.07 }$ - $ 3.69 ^{+ 2.24 } _{- 2.99 }$ 0.54
B NGC3992 $ 0.77 ^{+ 0.03 }_{- 0.02 }$ - $ 15.28 ^{+ 1.36 } _{- 1.60 }$ 0.82
B NGC4051 $ 0.43 ^{+ 0.07 }_{- 0.10 }$ - $ 2.72 ^{+ 1.40 } _{- 2.20 }$ 0.92
B NGC4088 $ 0.31 ^{+ 0.01 }_{- 0.01 }$ - $ 4.62 ^{+ 0.66 } _{- 0.71 }$ 0.60
B NGC4100 $ 0.72 ^{+ 0.03 }_{- 0.03 }$ - $ 5.18 ^{+ 0.57 } _{- 0.64 }$ 1.28
B NGC4138 $ 0.71 ^{+ 0.08 }_{- 0.04 }$ $ 0.53 ^{+ 0.17 }_{- 0.22 }$ $ 4.04 ^{+ 1.24 } _{- 1.60 }$ 1.50
B NGC4157 $ 0.38 ^{+ 0.03 }_{- 0.03 }$ $ 0.46 ^{+ 0.09 }_{- 0.16 }$ $ 8.21 ^{+ 1.09 } _{- 1.00 }$ 0.55
B NGC4183 $ 0.67 ^{+ 0.12 }_{- 0.06 }$ - $ 1.43 ^{+ 0.15 } _{- 0.19 }$ 0.18
B NGC4559 $ 0.33 ^{+ 0.02 }_{- 0.03 }$ - $ 2.33 ^{+ 0.15 } _{- 0.14 }$ 0.24
B NGC5005 $ 0.44 ^{+ 0.07 }_{- 0.09 }$ $ 0.51 ^{+ 0.08 }_{- 0.08 }$ $84.90 ^{+ 53.76 } _{- 81.03 }$ 0.09
B NGC6503 $ 0.53 ^{+ 0.01 }_{- 0.01 }$ - $ 2.36 ^{+ 0.04 } _{- 0.05 }$ 2.80
C UGC06983 $ 0.65 ^{+ 0.14 }_{- 0.07 }$ - $2.16 ^{+ 0.24 } _{- 0.31 } $ 0.71
C UGC07261 $ 0.57 ^{+ 0.17 }_{- 0.15 }$ - $ 0.64 ^{+ 0.17 } _{- 0.22 }$ 0.21
C UGC07690 $ 0.70 ^{+ 0.10 }_{- 0.06 }$ - $ 0.21 ^{+ 0.06 } _{- 0.08 }$ 0.67
C UGC07866 $ 0.43 ^{+ 0.09 }_{- 0.13 }$ - $ 0.03 ^{+ 0.04 } _{- 0.02 }$ 0.61
C UGC08490 $ 0.79 ^{+ 0.01 }_{- 0.01 }$ - $ 0.87 ^{+ 0.05 } _{- 0.06 }$ 1.52
C UGC08550 $ 0.63 ^{+ 0.16 }_{- 0.08 }$ - $ 0.27 ^{+ 0.03 } _{- 0.03 }$ 0.69
C UGC08699 $ 0.77 ^{+ 0.03 }_{- 0.02 }$ $ 0.67 ^{+ 0.02 }_{- 0.02 }$ $ 8.22 ^{+ 0.61 } _{- 0.68 }$ 0.69
C UGC09992 $ 0.47 ^{+ 0.10 }_{- 0.17 }$ - $ 0.03 ^{+ 0.03 } _{- 0.02 }$ 0.32
C UGC10310 $ 0.54 ^{+ 0.15 }_{- 0.23 }$ - $ 0.43 ^{+ 0.09 } _{- 0.12 }$ 0.58
C UGC12506 $ 0.79 ^{+ 0.01 }_{- 0.01 }$ - $ 19.76 ^{+ 1.70 } _{- 1.50 }$ 1.73
D NGC7331 $ 0.35 ^{+ 0.01 }_{- 0.01 }$ $ 0.48 ^{+ 0.09 } _{- 0.17 }$ $ 20.83 ^{+ 0.74 } _{- 0.72 }$ 0.83
D NGC7793 $ 0.54 ^{+ 0.04 }_{- 0.04 }$ - $ 1.33 ^{+ 0.27 } _{- 0.32 }$ 0.90
D NGC7814 $ 0.77 ^{+ 0.04 }_{- 0.02 }$ $ 0.66 ^{+ 0.03 } _{- 0.03 }$ $ 25.32 ^{+ 1.76 } _{- 1.67 }$ 1.42
D UGC02259 $ 0.77 ^{+ 0.05 }_{- 0.02 }$ - $ 1.39 ^{+ 0.08 } _{- 0.09 }$ 6.35
D UGC03546 $ 0.65 ^{+ 0.03 }_{- 0.03 }$ $ 0.37 ^{+ 0.03 } _{- 0.04}$ $ 9.15 ^{+ 0.61 } _{- 0.60 }$ 0.98
D UGC06446 $ 0.69 ^{+ 0.15 }_{- 0.08 }$ - $ 0.98 ^{+ 0.09 } _{- 0.11 }$ 0.44
D UGC06930 $ 0.51 ^{+ 0.14 }_{- 0.16 }$ - $ 1.53 ^{+ 0.27 } _{- 0.30 }$ 0.28
D UGC06983 $ 0.66 ^{+ 0.12 }_{- 0.09 }$ - $ 2.13 ^{+ 0.22 } _{- 0.27 }$ 0.71
D UGC07261 $ 0.57 ^{+ 0.16 }_{- 0.15 }$ - $ 0.62 ^{+ 0.15 } _{- 0.19 }$ 0.19
D UGC07690 $ 0.71 ^{+ 0.11 }_{- 0.07 }$ - $ 0.21 ^{+ 0.06 } _{- 0.07 }$ 0.64
: \[tabelaresultado2\]The maximum likelihood estimation for the $\Upsilon_{\text{*D}}$,$\Upsilon_{\text{*B}}$ and $M_\text{200}$ parameters, and the goodness of fit $\chi_\text{red}^2|_{\beta=0}$ for each galaxy in the case of $\beta =0$.
![\[rotationcurveA\]The rotation curves and their components: gas (dashed yellow line), disk (dashed green line which corresponds to $\sqrt{\Upsilon_{*\text{D}}V_{\text{disk}}^{2}}$), bulge (dashed red line which corresponds to $\sqrt{\Upsilon_{*\text{B}}V_{\text{bulge}}^{2}}$) and dark matter with Yukawa-like corrections (dashed blue line). The black solid line is the overall best-fit (see equation \[circular\]) and the values for the parameters are displayed on tables \[tabelaresultado1\] and \[tabelabetalambda\], the orange solid line is the dark matter component for $\beta=0$. The red dots with error bars are the observational data taken from SPARC catalogue and the grey ones are the residual of the fit. We have plotted the results for the set A.](result_F568V1_new_emcee_testing_10_galaxy "fig:"){width="5cm"} ![\[rotationcurveA\]The rotation curves and their components: gas (dashed yellow line), disk (dashed green line which corresponds to $\sqrt{\Upsilon_{*\text{D}}V_{\text{disk}}^{2}}$), bulge (dashed red line which corresponds to $\sqrt{\Upsilon_{*\text{B}}V_{\text{bulge}}^{2}}$) and dark matter with Yukawa-like corrections (dashed blue line). The black solid line is the overall best-fit (see equation \[circular\]) and the values for the parameters are displayed on tables \[tabelaresultado1\] and \[tabelabetalambda\], the orange solid line is the dark matter component for $\beta=0$. The red dots with error bars are the observational data taken from SPARC catalogue and the grey ones are the residual of the fit. We have plotted the results for the set A.](result_NGC0024_new_emcee_testing_10_galaxy "fig:"){width="5cm"}\
![\[rotationcurveA\]The rotation curves and their components: gas (dashed yellow line), disk (dashed green line which corresponds to $\sqrt{\Upsilon_{*\text{D}}V_{\text{disk}}^{2}}$), bulge (dashed red line which corresponds to $\sqrt{\Upsilon_{*\text{B}}V_{\text{bulge}}^{2}}$) and dark matter with Yukawa-like corrections (dashed blue line). The black solid line is the overall best-fit (see equation \[circular\]) and the values for the parameters are displayed on tables \[tabelaresultado1\] and \[tabelabetalambda\], the orange solid line is the dark matter component for $\beta=0$. The red dots with error bars are the observational data taken from SPARC catalogue and the grey ones are the residual of the fit. We have plotted the results for the set A.](result_NGC2683_new_emcee_testing_10_galaxy "fig:"){width="5cm"} ![\[rotationcurveA\]The rotation curves and their components: gas (dashed yellow line), disk (dashed green line which corresponds to $\sqrt{\Upsilon_{*\text{D}}V_{\text{disk}}^{2}}$), bulge (dashed red line which corresponds to $\sqrt{\Upsilon_{*\text{B}}V_{\text{bulge}}^{2}}$) and dark matter with Yukawa-like corrections (dashed blue line). The black solid line is the overall best-fit (see equation \[circular\]) and the values for the parameters are displayed on tables \[tabelaresultado1\] and \[tabelabetalambda\], the orange solid line is the dark matter component for $\beta=0$. The red dots with error bars are the observational data taken from SPARC catalogue and the grey ones are the residual of the fit. We have plotted the results for the set A.](result_NGC2915_new_emcee_testing_10_galaxy "fig:"){width="5cm"}\
![\[rotationcurveA\]The rotation curves and their components: gas (dashed yellow line), disk (dashed green line which corresponds to $\sqrt{\Upsilon_{*\text{D}}V_{\text{disk}}^{2}}$), bulge (dashed red line which corresponds to $\sqrt{\Upsilon_{*\text{B}}V_{\text{bulge}}^{2}}$) and dark matter with Yukawa-like corrections (dashed blue line). The black solid line is the overall best-fit (see equation \[circular\]) and the values for the parameters are displayed on tables \[tabelaresultado1\] and \[tabelabetalambda\], the orange solid line is the dark matter component for $\beta=0$. The red dots with error bars are the observational data taken from SPARC catalogue and the grey ones are the residual of the fit. We have plotted the results for the set A.](result_NGC3198_new_emcee_testing_10_galaxy "fig:"){width="5cm"} ![\[rotationcurveA\]The rotation curves and their components: gas (dashed yellow line), disk (dashed green line which corresponds to $\sqrt{\Upsilon_{*\text{D}}V_{\text{disk}}^{2}}$), bulge (dashed red line which corresponds to $\sqrt{\Upsilon_{*\text{B}}V_{\text{bulge}}^{2}}$) and dark matter with Yukawa-like corrections (dashed blue line). The black solid line is the overall best-fit (see equation \[circular\]) and the values for the parameters are displayed on tables \[tabelaresultado1\] and \[tabelabetalambda\], the orange solid line is the dark matter component for $\beta=0$. The red dots with error bars are the observational data taken from SPARC catalogue and the grey ones are the residual of the fit. We have plotted the results for the set A.](result_NGC3521_new_emcee_testing_10_galaxy "fig:"){width="5cm"}\
![\[rotationcurveA\]The rotation curves and their components: gas (dashed yellow line), disk (dashed green line which corresponds to $\sqrt{\Upsilon_{*\text{D}}V_{\text{disk}}^{2}}$), bulge (dashed red line which corresponds to $\sqrt{\Upsilon_{*\text{B}}V_{\text{bulge}}^{2}}$) and dark matter with Yukawa-like corrections (dashed blue line). The black solid line is the overall best-fit (see equation \[circular\]) and the values for the parameters are displayed on tables \[tabelaresultado1\] and \[tabelabetalambda\], the orange solid line is the dark matter component for $\beta=0$. The red dots with error bars are the observational data taken from SPARC catalogue and the grey ones are the residual of the fit. We have plotted the results for the set A.](result_NGC3769_new_emcee_testing_10_galaxy "fig:"){width="5cm"} ![\[rotationcurveA\]The rotation curves and their components: gas (dashed yellow line), disk (dashed green line which corresponds to $\sqrt{\Upsilon_{*\text{D}}V_{\text{disk}}^{2}}$), bulge (dashed red line which corresponds to $\sqrt{\Upsilon_{*\text{B}}V_{\text{bulge}}^{2}}$) and dark matter with Yukawa-like corrections (dashed blue line). The black solid line is the overall best-fit (see equation \[circular\]) and the values for the parameters are displayed on tables \[tabelaresultado1\] and \[tabelabetalambda\], the orange solid line is the dark matter component for $\beta=0$. The red dots with error bars are the observational data taken from SPARC catalogue and the grey ones are the residual of the fit. We have plotted the results for the set A.](result_NGC3893_new_emcee_testing_10_galaxy "fig:"){width="5cm"}\
![\[rotationcurveA\]The rotation curves and their components: gas (dashed yellow line), disk (dashed green line which corresponds to $\sqrt{\Upsilon_{*\text{D}}V_{\text{disk}}^{2}}$), bulge (dashed red line which corresponds to $\sqrt{\Upsilon_{*\text{B}}V_{\text{bulge}}^{2}}$) and dark matter with Yukawa-like corrections (dashed blue line). The black solid line is the overall best-fit (see equation \[circular\]) and the values for the parameters are displayed on tables \[tabelaresultado1\] and \[tabelabetalambda\], the orange solid line is the dark matter component for $\beta=0$. The red dots with error bars are the observational data taken from SPARC catalogue and the grey ones are the residual of the fit. We have plotted the results for the set A.](result_NGC3949_new_emcee_testing_10_galaxy "fig:"){width="5cm"} ![\[rotationcurveA\]The rotation curves and their components: gas (dashed yellow line), disk (dashed green line which corresponds to $\sqrt{\Upsilon_{*\text{D}}V_{\text{disk}}^{2}}$), bulge (dashed red line which corresponds to $\sqrt{\Upsilon_{*\text{B}}V_{\text{bulge}}^{2}}$) and dark matter with Yukawa-like corrections (dashed blue line). The black solid line is the overall best-fit (see equation \[circular\]) and the values for the parameters are displayed on tables \[tabelaresultado1\] and \[tabelabetalambda\], the orange solid line is the dark matter component for $\beta=0$. The red dots with error bars are the observational data taken from SPARC catalogue and the grey ones are the residual of the fit. We have plotted the results for the set A.](result_NGC3953_new_emcee_testing_10_galaxy "fig:"){width="5cm"}\
![\[rotationcurveB\]Same as figure \[rotationcurveA\], but for set B. ](result_NGC3992_new_emcee_testing_10_galaxy_set2 "fig:"){width="5.5cm"} ![\[rotationcurveB\]Same as figure \[rotationcurveA\], but for set B. ](result_NGC4051_new_emcee_testing_10_galaxy_set2 "fig:"){width="5.5cm"}\
![\[rotationcurveB\]Same as figure \[rotationcurveA\], but for set B. ](result_NGC4088_new_emcee_testing_10_galaxy_set2 "fig:"){width="5.5cm"} ![\[rotationcurveB\]Same as figure \[rotationcurveA\], but for set B. ](result_NGC4100_new_emcee_testing_10_galaxy_set2 "fig:"){width="5.5cm"}\
![\[rotationcurveB\]Same as figure \[rotationcurveA\], but for set B. ](result_NGC4138_new_emcee_testing_10_galaxy_set2 "fig:"){width="5.5cm"} ![\[rotationcurveB\]Same as figure \[rotationcurveA\], but for set B. ](result_NGC4157_new_emcee_testing_10_galaxy_set2 "fig:"){width="5.5cm"}\
![\[rotationcurveB\]Same as figure \[rotationcurveA\], but for set B. ](result_NGC4183_new_emcee_testing_10_galaxy_set2 "fig:"){width="5.5cm"} ![\[rotationcurveB\]Same as figure \[rotationcurveA\], but for set B. ](result_NGC4559_new_emcee_testing_10_galaxy_set2 "fig:"){width="5.5cm"}\
![\[rotationcurveB\]Same as figure \[rotationcurveA\], but for set B. ](result_NGC5005_new_emcee_testing_10_galaxy_set2 "fig:"){width="5.5cm"} ![\[rotationcurveB\]Same as figure \[rotationcurveA\], but for set B. ](result_NGC6503_new_emcee_testing_10_galaxy_set2 "fig:"){width="5.5cm"}\
![\[rotationcurveC\]Same as figure \[rotationcurveA\], but for set C. ](result_UGC06983_new_emcee_testing_10_galaxy_set3 "fig:"){width="5.5cm"} ![\[rotationcurveC\]Same as figure \[rotationcurveA\], but for set C. ](result_UGC07261_new_emcee_testing_10_galaxy_set3 "fig:"){width="5.5cm"}\
![\[rotationcurveC\]Same as figure \[rotationcurveA\], but for set C. ](result_UGC07690_new_emcee_testing_10_galaxy_set3 "fig:"){width="5.5cm"} ![\[rotationcurveC\]Same as figure \[rotationcurveA\], but for set C. ](result_UGC07866_new_emcee_testing_10_galaxy_set3 "fig:"){width="5.5cm"}\
![\[rotationcurveC\]Same as figure \[rotationcurveA\], but for set C. ](result_UGC08490_new_emcee_testing_10_galaxy_set3 "fig:"){width="5.5cm"} ![\[rotationcurveC\]Same as figure \[rotationcurveA\], but for set C. ](result_UGC08550_new_emcee_testing_10_galaxy_set3 "fig:"){width="5.5cm"}\
![\[rotationcurveC\]Same as figure \[rotationcurveA\], but for set C. ](result_UGC08699_new_emcee_testing_10_galaxy_set3 "fig:"){width="5.5cm"} ![\[rotationcurveC\]Same as figure \[rotationcurveA\], but for set C. ](result_UGC09992_new_emcee_testing_10_galaxy_set3 "fig:"){width="5.5cm"}\
![\[rotationcurveC\]Same as figure \[rotationcurveA\], but for set C. ](result_UGC10310_new_emcee_testing_10_galaxy_set3 "fig:"){width="5.5cm"} ![\[rotationcurveC\]Same as figure \[rotationcurveA\], but for set C. ](result_UGC12506_new_emcee_testing_10_galaxy_set3 "fig:"){width="5.5cm"}\
![\[rotationcurveD\]Same as figure \[rotationcurveA\], but for set D. ](result_NGC7331_new_emcee_testing_10_galaxy_set4 "fig:"){width="5.5cm"} ![\[rotationcurveD\]Same as figure \[rotationcurveA\], but for set D. ](result_NGC7793_new_emcee_testing_10_galaxy_set4 "fig:"){width="5.5cm"}\
![\[rotationcurveD\]Same as figure \[rotationcurveA\], but for set D. ](result_NGC7814_new_emcee_testing_10_galaxy_set4 "fig:"){width="5.5cm"} ![\[rotationcurveD\]Same as figure \[rotationcurveA\], but for set D. ](result_UGC02259_new_emcee_testing_10_galaxy_set4 "fig:"){width="5.5cm"}\
![\[rotationcurveD\]Same as figure \[rotationcurveA\], but for set D. ](result_UGC03546_new_emcee_testing_10_galaxy_set4 "fig:"){width="5.5cm"} ![\[rotationcurveD\]Same as figure \[rotationcurveA\], but for set D. ](result_UGC06446_new_emcee_testing_10_galaxy_set4 "fig:"){width="5.5cm"}\
![\[rotationcurveD\]Same as figure \[rotationcurveA\], but for set D. ](result_UGC06930_new_emcee_testing_10_galaxy_set4 "fig:"){width="5.5cm"} ![\[rotationcurveD\]Same as figure \[rotationcurveA\], but for set D. ](result_UGC06983_new_emcee_testing_10_galaxy_set4 "fig:"){width="5.5cm"}\
![\[rotationcurveD\]Same as figure \[rotationcurveA\], but for set D. ](result_UGC07261_new_emcee_testing_10_galaxy_set4 "fig:"){width="5.5cm"} ![\[rotationcurveD\]Same as figure \[rotationcurveA\], but for set D. ](result_UGC07690_new_emcee_testing_10_galaxy_set4 "fig:"){width="5.5cm"}\
Discussion and conclusions {#section6}
==========================
In this work we have used observational data from the SPARC catalogue to constrain the properties of modified gravity models in the presence of dark matter, and assuming that the fifth force couples weakly to baryons but with unrestricted strength to dark matter. Contrary to some previous work, our aim is not to replace dark matter with modified gravity but to see how much modified gravity can improve the rotation curve fit. Since baryons are assumed to be weakly coupled, we do not need to invoke a screening mechanism, and the Yukawa term is left free and constant for every galaxy. We considered four different sets of 10 galaxies each and we found the region in the parameter space for $\lambda$ and $\beta$ that are allowed by the data. To the best of our knowledge, this is the largest set ever analysed in the context of modified gravity. We found that in all the data sets the standard $\beta = 0$ model gives a much worse fit than a value different from zero, with preference for a positive value, corresponding to an attractive Yukawa force. We have also calculated for each galaxies the values of the parameters $\Upsilon_{*\text{D}}$, $\Upsilon_{*\text{B}}$ and $M_{200}$. We find that the presence of the attractive fifth force reduces the need for dark matter by 20% in mass, on average.
We have then combined all the data sets together to find the allowed region in the parameter space. The values for the parameters $\beta$ and $\lambda$ are: $\beta = 0.34\pm0.04$ and $\lambda = 5.61\pm0.91 $ kpc. The Bayesian evidence ratio strongly favors the Yukawa model, to more than 8$\sigma$ for the combined dataset, with respect to the $\beta=0$ case.
We notice that the $\beta$ value is remarkably close to $\beta=1/3$, the value predicted by one of the simplest modified gravity model, the $f(R)$ theory. However, as mentioned in Sec. 2, we should interpret $\beta$ as the product of a small baryon coupling times a large dark matter coupling, neither of which would be close to the $f(R)$ prediction. So the underlying model can be identified with a scalar-tensor theory with non-universal coupling, rather than the specific form $f(R)$.
It is clear that we cannot conclude that standard gravity is ruled out. Rather, we found that a model of the baryon components (gas, disk and bulge), plus a NFW profile for the dark matter, plus an attractive Yukawa term, fits much better the rotation curves of our sample than a similar model but without the Yukawa correction. The SPARC catalog contains normal galaxies as well as LSB and dwarfs. The latter two types are known to be poorly fitted by a NFW profile[@moore1994evidence; @mcgaugh1998testing; @cote2000various], so it will be interesting in a future work to try to fit them separately with different profiles. In our set of 40 galaxies, however, only 10 galaxies are of this kind so we believe our choice of NFW for all galaxies was justified. Whether this results holds assuming different modelling for the baryon or the dark matter component, remains to be seen.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the SFB-Transregio TR33 “The Dark Universe”. The work of AOFA was supported by CAPES, grant number 88881.135537/2016-01. AOFA wants to acknowledge discussions with Davi Rodrigues about mass-to-light ratio in galaxies.
[^1]: Unfortunately the version of this equation published on JCAP contained some typos.
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---
abstract: 'Standard GRB fireballs must carry free neutrons. This crucially changes the mechanism of fireball deceleration by an external medium. As the ion fireball decelerates, the coasting neutrons form a leading front. They gradually decay, leaving behind a relativistic trail of decay products mixed with the ambient medium. The ion fireball sweeps up the trail and drives a shock wave in it. Thus, observed afterglow emission is produced in the neutron trail. The impact of neutrons turns off at $\sim 10^{17}$ cm from the explosion center, and here a spectral transition is expected in GRB afterglows. Absence of neutron signatures would point to absence of baryons and a dominant Poynting flux in the fireballs.'
author:
- 'Andrei M. Beloborodov'
title: Fireballs with a Neutron Component
---
\#1[[*\#1*]{}]{} \#1[[*\#1*]{}]{} =
= ==1=1=0pt =2=2=0pt =2=2=0pt
\#1 1.25in .125in .25in
Introduction
============
Importance of neutrons in GRBs was realized recently (Derishev, Kocharovsky, & Kocharovsky 1999a,b; Bahcall & Mészáros 2000; Mészáros & Rees 2000; Fuller, Pruet, & Abazajian 2000; Pruet & Dalal 2002; Beloborodov 2003a,b). Their presence is inevitable in a standard fireball (sections 2 and 3) and they qualitatively change the explosion picture at radii of $10^{16}-10^{17}$ cm (section 4).
Neutronization of the Central Engine
====================================
GRB central engines are very dense and hot (e.g. Mészáros 2002). The central matter (or matter surrounding the central black hole) is filled with hot $e^\pm$ pairs in thermodynamic equilibrium with blackbody radiation at a temperature $kT=1-10$ MeV. Its baryonic component is made of neutrons and protons (nuclei break up into free nucleons in the unshadowed region of the $T-\rho$ plane shown in left panel of Fig. 1) and frequent $e^\pm$ captures on nucleons take place, $$e^-+p\rightarrow n+\nu, \qquad e^++n\rightarrow p+\bar{\nu}.$$ These reactions quickly convert protons to neutrons and back, and establish an equilibrium proton fraction $Y_e=n_p/(n_n+n_p)$ which is shown in Figure 1. Any plausible GRB engines belong to the $T-\rho$ region where $Y_e<0.5$. So, the neutrons are not only present in the central engines — in fact, they dominate. When the neutronized matter is ejected into a high-entropy fireball, it remains neutron-dominated because (1) expansion occurs too fast and $Y_e$ freezes out at its initial value below 0.5, or (2) the fireball partially absorbs the neutrino flux from the engine, which tends to keep $Y_e<0.5$ (see Beloborodov 2003a for details).
Ejected Neutrons Survive the Nucleosynthesis
============================================
As the fireball expands and cools, nucleons tend to recombine into $\alpha$-particles (like they do in the big bang). This process competes, however, with rapid expansion and can freeze out. For this reason, nucleosynthesis is suppressed in fireballs with a high photon-to-baryon ratio $\phi=n_\gamma/n_b$ (or, equivalently, high entropy per baryon $s/k=3.6\phi$). A minimum $\phi$ in GRBs is $\sim 10^5$, which is just marginal for nucleosynthesis (detailed calculations of nucleosynthesis in GRBs have been done recently by Lemoine 2002; Pruet, Guiles, & Fuller 2002; Beloborodov 2003a). Even in the extreme case of complete nucleon recombination there are still leftover neutrons because of the neutron excess ($Y_e<0.5$). Their minimum mass fraction is $X_n=1-2Y_e$.
In addition, synthesized helium is likely destroyed during the subsequent evolution of the explosion. This happens if (1) there appears a substantial relative bulk velocity between the neutron and ion components (as a result of neutron decoupling during the acceleration stage of the fireball) or (2) internal shocks occur and heat the ions to a high temperature (Beloborodov 2003a).
Thus, a substantial neutron component appears inevitably in the standard fireball scenario. Neutrons develop a Lorentz factor $\Gn=10^2-10^3$ at the very beginning of the explosion when the fireball is accelerated by radiation pressure: they are collisionally coupled to the ions in the early dense fireball, and decouple close to the end of the acceleration stage. Then the neutrons coast and gradually decay with a mean lifetime $\taub\approx 900$ s and a mean decay radius $\Rb=c\taub\Gn$, \[eq:H\] =0.810\^[16]{}() [ cm]{}.
Neutron-Fed Blast Wave
======================
Let us remind what happens in a relativistic explosion without neutrons. The ejected fireball with mass $\Mej$ and Lorentz factor $\Gej$ sweeps up an ambient medium with density $n_0=1-100$ cm$^{-3}$ and gradually dissipates its kinetic energy. The dissipation rate peaks at a characteristic “deceleration” radius $\Rdec\sim 10^{16}$ cm where half of the initial energy is dissipated. $\Rdec$ corresponds to swept-up mass $\mdec=\Mej/\Gej$. Further dynamics is described by the self-similar blast wave model of Blandford & McKee (1976). How does this picture change in the presence of neutrons?
At radii under consideration, $R>10^{15}$ cm, the ejected fireball is a shell of thickness $\Delta\ll R$. In contrast to neutrons, the ion component of the fireball is aware of the external medium and its Lorentz factor $\Gamma$ decreases. As $\Gamma$ decreases below $\Gamma_n$, the ions fall behind and separate from the neutrons. Thus the fireball splits into two relativistic shells which we name N (neutrons) and I (ions). The mass of the leading N-shell is decreasing because of the $\beta$-decay, \[eq:Mn\] M\_n(R)=M\_n\^0(-). The N-shell energy, $E_n=\Gn M_nc^2$, is huge compared to the ambient rest-mass $mc^2$ even at $R>\Rb$. For example, at $R=\Rdec$ we find $E_n/\mdec c^2=X_n\Gn\Gej\exp(-\Rdec/\Rb)$ where $X_n=M_n^0/\Mej$ is the initial neutron fraction of the fireball.
The neutron decay products $p$ and $e^-$ share immediately their huge momentum with ambient particles due to two-stream instability and the N-shell leaves behind a mixed trail with a relativistic bulk velocity $\beta<\bn$ (Beloborodov 2003b) \[eq:beta\] (R)=, (R)= =, where \[eq:zet\] (R)= =()\^[-1]{}, and $m(R)$ is ambient mass enclosed by radius $R$. There exists a characteristic radius $\Rtr$ where the trail becomes nonrelativistic ($\beta=0.5$). It is defined by condition $\zeta=\Gn^{-1}$, which requires about 10 e-folds of the decay (for a typical $\mb\sim\mdec\sim 10^{-5} M_n^0$). Thus, 10=0.810\^[17]{}()[ cm]{}. $\Rtr$ depends very weakly (logarithmically) on the ambient density and the initial neutron fraction of the fireball, $X_n$.
The decaying N-shell not only accelerates the medium as it passes through it. It also compresses the medium, loads with new particles, and heat to a high temperature. The rest-frame density and relativistic enthalpy of the trail are \[eq:mu\] n=n\_0(1+)(\^2+2+1)\^[1/2]{}, =, where $n_0$ is the medium density ahead of the N-shell. For $\Gn^{-1}<\zeta<\Gn$ one finds $\mu\gg 1$, i.e. the thermal energy of the trail far exceeds its rest-mass energy.
The ion fireball follows the neutron front and collects the trail. As a result, (1) the ion Lorentz factor $\Gamma$ decreases and (2) a shock wave propagates in the trail material. The shock has a Lorentz factor $\Gsh\simgt\Gamma$ and it cannot catch up with the neutron front (unless $n_0[R]$ falls off steeper than $R^{-3}$). Dynamics and dissipation in the shock are discussed in Beloborodov (2003b). We emphasize here important differences from a customary external shock: the neutron-trail shock propagates in a relativistically moving, dense, hot, and possibly magnetized medium behind the leading neutron front. The neutron impact ceases at $\Rtr\approx 10^{17}$ cm, which can leave an imprint on the observed afterglow. For example, the shock dissipation can have a second bump (Beloborodov 2003b), and a spectral transition is also possible. The arrival time of radiation emitted at $\Rtr$ is approximately $\Rtr/2\Gamma^2 c$ (counted from the arrival of first $\gamma$-rays). It may be as long as 30 days or as short as a few seconds, depending on the fireball Lorentz factor $\Gamma(\Rtr)$. Recent early observation of a GRB afterglow (GRB 021004) discovered an interesting re-brightening at $10^3$ s. Also, we do not exclude a possible relevance of neutrons to the 20 day bumps observed in a few GRBs, as the time coincides with $\Rtr/c$.
Neutron signatures should be absent if the fireball is dominated by a Poynting flux and has extremely low baryon loading. Then the neutron component decouples early, with a modest Lorentz factor $\Gn$, and decays at small radii. The upper bound on $\Gn$ due to decoupling is $\Gn\approx 300(\dM_\Omega/10^{26})^{1/3}$ where $\dM_\Omega$ \[g/s\] is the mass outflow rate per unit solid angle of the fireball (Beloborodov 2003a).
An additional piece of GRB physics is the interaction of the prompt $\gamma$-ray radiation with an ambient medium (see Beloborodov 2002 and refs. therein). It leads to a gap opening and crucially affects the early afterglow emitted at radii $R<10^{16}(E_\gamma/10^{53})^{1/2}$ cm, where $E_\gamma$ \[erg\] is the isotropic energy of the GRB. Here, we focused on larger radii where the neutron effects dominate.
|
---
abstract: 'In this paper we review Castagnino’s contributions to the foundations of quantum mechanics. First, we recall his work on quantum decoherence in closed systems, and the proposal of a general framework for decoherence from which the phenomenon acquires a conceptually clear meaning. Then, we introduce his contribution to the hard field of the interpretation of quantum mechanics: the modal-Hamiltonian interpretation solves many of the interpretive problems of the theory, and manifests its physical relevance in its application to many traditional models of the practice of physics. In the third part of this work we describe the ontological picture of the quantum world that emerges from the modal-Hamiltonian interpretation, stressing the philosophical step toward a deep understanding of the reference of the theory.'
author:
- Olimpia Lombardi
- Juan Sebastián Ardenghi
- Sebastian Fortin
- Martin Narvaja
title: 'Foundations of quantum mechanics: decoherence and interpretation'
---
Introduction
============
Anybody who has been close to Prof. Mario Castagnino, even for a short time, knows that he is an ever-eager spirit: the many different subjects treated in this issue are a clear manifestation of the wide panoply of interests that have moved him during his long academic life. Nevertheless, the present article has a peculiarity with respect to the rest of the papers of the issue: Castagnino should be one of the authors of this work. In fact, since ten years ago he has been actively engaged with the foundations and the philosophy of physics, leading an always increasing research group to which we belong. In this field we have obtained relevant results with a remarkable repercussion.
As Castagnino uses to say, he is a senior physicist but a baby philosopher. However, this fact was not an obstacle to his eager spirit, which has been involved in the foundations of so many different subjects that cannot be addressed in a single article. In the present paper we will confine our attention to Castagnino’s contributions to the foundations of quantum mechanics (QM), in order to review his main results in this area. First, we will recall his work on quantum decoherence in closed systems, and the proposal of a general framework for decoherence from which the phenomenon acquires a conceptually clear meaning. Then, we will introduce his contribution to the hard field of the interpretation of QM: our modal-Hamiltonian interpretation solves many of the interpretive problems of the theory, and manifests its physical relevance in its application to many traditional models used in the practice of physics. In the third part of this work we will describe the ontological picture of the quantum world that emerges from our interpretation; here we will stress our philosophical step toward a deep understanding of the reference of the theory, a move not usual in the contemporary discussions about the interpretation of QM. Finally, we will briefly recall Castagnino’s contributions in other areas of the foundations and the philosophy of physics.
Foundations of quantum decoherence
==================================
More than a decade ago Castagnino developed, with Roberto Laura, a formalism that explains the limit reached by expectation values in closed quantum systems with continuous spectrum,[@CL-1]$^{-}$[@CL-6] and begun to conceive that formalism in terms of decoherence. When, some years later, those works were reanalyzed in the context of our research group, we acknowledged the conceptual relevance and the fruitful perspectives of that work. So, the original proposal was further elaborated from a conceptual viewpoint, and presented in several meetings and papers.[@SID-1]$^{-}$[@SID-10] In particular, we were invited by Prof. Fred Kronz, from the University of Texas at Austin, to discuss that new view, and he suggested the name ‘self-induced decoherence’ (SID) in contrast with the orthodox ‘environment-induced decoherence’ (EID) approach.[@Paz-Zurek]$^{,}$[Zurek-2003]{}
In those first works, we presented SID as different from EID, that is, as the way in which decoherence manifests itself in closed systems. However, shortly after we realized that both approaches can be subsumed under a *General Theoretical Framework for Decoherence* (GTFD), which encompasses decoherence in open and closed systems.[@GTFD-1]$^{-}$[MPLA-random]{} According to this framework, decoherence is just a particular case of the general phenomenon of irreversibility in QM.[@Omnes-2001]$%
^{,}$[@Omnes-2002] Since the quantum state $\rho (t)$ follows a unitary evolution, it cannot reach a final equilibrium state for $t\rightarrow
\infty $. Therefore, if we want to explain the emergence of non-unitary irreversible evolutions, we must split the whole space $\mathcal{O}$ of all possible observables into a relevant subspace $\mathcal{O}_{R}\subset
\mathcal{O}$ and an irrelevant subspace. With this strategy we restrict the maximal information about the system: the expectation values $\langle
O_{R}\rangle _{\rho (t)}$ of the observables $O_{R}\in \mathcal{O}_{R}$ express that relevant information. Of course, the decision about which observables are to be considered as relevant depends on the particular purposes in each situation; but without this restriction, irreversible evolutions cannot be described. In fact, the different approaches to decoherence always select a set $\mathcal{O}_{R}$ of relevant observables in terms of which the time behavior of the system is described: gross observables in van Kampen,[@van; @Kampen] macroscopic observables of the apparatus in Daneri *et al.*,[@Daneri] observables of the open system in EID,[@Zeh-1]$^{-}$[@Zurek-2003] relevant observables in Omnés.[@Omnes-1994]$^{,}$[@Omnes-1999]
Once the essential role played by the selection of the relevant observables is clearly understood, decoherence can be explained in three general steps:
1. **First step:** The set $\mathcal{O}_{R}$ of relevant observables is defined.
2. **Second step:** The expectation value $\langle O_{R}\rangle
_{\rho (t)}$, for any $O_{R}\in \mathcal{O}_{R}$, is obtained. This step can be formulated in two different but equivalent ways:
- $\langle O_{R}\rangle _{\rho (t)}$ is computed as the expectation value of $O_{R}$ in the unitarily evolving state $\rho (t)$.
- A coarse-grained state $\rho _{G}(t)$ is defined by $\langle
O_{R}\rangle _{\rho (t)}=\langle O_{R}\rangle _{\rho _{G}(t)}$ for any $%
O_{R}\in \mathcal{O}_{R}$, and its non-unitary evolution (governed by a master equation) is computed.
3. **Third step:** It is proved that $\langle O_{R}\rangle
_{\rho (t)}=\langle O_{R}\rangle _{\rho _{G}(t)}$ reaches a final equilibrium value $\langle O_{R}\rangle _{\rho _{\ast }}$: $$\lim_{t\rightarrow \infty }\langle O_{R}\rangle _{\rho
(t)}=\lim_{t\rightarrow \infty }\langle O_{R}\rangle _{\rho _{G}(t)}=\langle
O_{R}\rangle _{\rho _{\ast }} \label{2-1}$$
where the final equilibrium state $\rho _{\ast }$ is obviously diagonal in its own eigenbasis, which turns out to be the final pointer basis. But the unitarily evolving quantum state $\rho (t)$ of the whole system *has only a* *weak limit*: $$W-\lim_{t\rightarrow \infty }\rho (t)=\rho _{\ast } \label{2-2}$$
This weak limit means that, although the off-diagonal terms of $\rho (t)$ never vanish through the unitary evolution, the system decoheres *from an observational point of view*, that is, from the viewpoint given by any relevant observable $O_{R}\in \mathcal{O}_{R}$.
This GTFD allows us to face the conceptual challenges that the EID approach still has to face. One of them comes from the fact that, since the environment may be external or internal, the EID approach offers no general criterion to decide where to place the cut between system and environment. Zurek considers this fact as a shortcoming of his proposal: *In particular, one issue which has been often taken for granted is looming big, as a foundation of the whole decoherence program. It is the question of what are the ‘systems’ which play such a crucial role in all the discussions of the emergent classicality.*[@Zurek-cut] In order to address this problem, the first step is to realize that the EID relevant observables of the closed system $U$ are those corresponding to the open system $S$: $$O_{R}=O_{S}\otimes \mathbb{I}_{E}\in \mathcal{O}_{R}\subset \mathcal{O}
\label{2-3}$$where $O_{S}\in \mathcal{O}_{S}$ of $S$ and $\mathbb{I}_{E}$ is the identity operator in $\mathcal{O}_{E}$ of $E$. The reduced density operator $\rho
_{S}(t)$ of $S$ is defined by tracing over the environmental degrees of freedom, $$\rho _{S}(t)=Tr_{E}\,\rho (t) \label{2-4}$$The EID approach studies the time-evolution of $\rho _{S}(t)$ governed by an effective master equation; it proves that, under certain definite conditions, $\rho _{S}(t)$ converges to a stable state $\rho _{S\ast }$: $%
\rho _{S}(t)\longrightarrow \rho _{S\ast }$. But we also know that the expectation value of any $O_{R}\in \mathcal{O}_{R}$ in the state $\rho (t)$ of $U$ can be computed as $$\langle O_{R}\rangle _{\rho (t)}=Tr\,\left( \rho (t)(O_{S}\otimes \mathbb{I}%
_{E})\right) =Tr\left( \rho _{S}(t)\,O_{S}\right) =\langle O_{S}\rangle
_{\rho _{S}(t)} \label{2-6}$$Therefore, the convergence of $\rho _{S}(t)$ to $\rho _{S\ast }$ implies the convergence of the expectation values: $$\langle O_{R}\rangle _{\rho (t)}=\langle O_{S}\rangle _{\rho
_{S}(t)}\longrightarrow \langle O_{S}\rangle _{\rho _{S\ast }}=\langle
O_{R}\rangle _{\rho _{\ast }} \label{2-7}$$where $\rho _{\ast }$ is a final diagonal state of $U$, such that $\rho
_{S\ast }=Tr_{E}\,\rho _{\ast }$.
>From this new conceptual perspective, we have studied the well-known spin-bath model: a closed system $U=P\cup P_{1}\cup \ldots \cup P_{N}=P\cup
(\cup _{i=1}^{N}P_{i})$, where (i) $P$ is a spin-1/2 particle represented in the Hilbert space $\mathcal{H}_{P}$, and (ii) each $P_{i}$ is a spin-1/2 particle represented in its Hilbert space $\mathcal{H}_{i}$. The Hilbert space of the composite system $U$ is, then, $$\mathcal{H}=\mathcal{H}_{P}\otimes \left( \bigotimes\limits_{i=1}^{N}%
\mathcal{H}_{i}\right) \label{2-8}$$If the self-Hamiltonians $H_{P}$ of $P$ and $H_{i}$ of $P_{i}$ are taken to be zero, and there is no interaction among the $P_{i}$, then the total Hamiltonian $H$ of the composite system $U$ is given by the interaction between the particle $P$ and each particle $P_{i}$.[@Zurek-1982] By contrast to the usual presentations, we have studied different decompositions of the whole closed system $U$ into a relevant part and its environment.[@MPLA]
#### Decomposition 1: A large environment that produces decoherence.
In the typical situation studied by the EID approach, the open system $S$ is the particle $P$, and the remaining particles $P_{i}$ play the role of the environment $E$: $S=P$ and $E=\cup _{i=1}^{N}P_{i}$. This decomposition results in the system decoherence when the number of particles in the bath is very large.
#### Decomposition 2: A large environment with no decoherence
We can conceive different ways of splitting the whole closed system $U$. For instance, we can decide to observe a particular particle $P_{j}$ of what was previously considered the environment, and to consider the remaining particles as the new environment, in such a way that $S=P_{j}$ and $E=P\cup
(\cup _{i=1,i\neq j}^{N}P_{i})$. This decomposition results in that the system does not decohere.
#### Decomposition 3: A small environment that produces decoherence
It may be the case that the measuring arrangement observes a subset of the particles of the environment, e.g., the $p$ first particles $P_{j}$. In this case, the system of interest is composed by $p$ particles, $S=\cup _{i=1}^{p}P_{i}$, and the environment is composed by all the remaining particles, $E=P\cup (\cup
_{i=p+1}^{N}P_{i})$. This decomposition results in the system decoherence when the number $p$ is very large.
We have also studied a generalization of the spin-bath model, where a whole closed system was split into an open many-spin system and its environment.[@JPA2] In this case we studied different partitions of the whole system and identified those for which the selected system does not decohere. As stressed in that work, this might help us to define clusters of particles that can be used to store q-bits.
The results obtained in both cases allowed us to argue that Zurek’s looming big problem is actually a pseudo-problem, which is simply dissolved by the fact that the split of a closed quantum system into an open subsystem and its environment is just a way of selecting a particular space of relevant observables of the whole closed system. But since there are many different spaces of relevant observables depending on the observational viewpoint adopted, the same closed system can be decomposed in many different ways: each decomposition represents a decision about which degrees of freedom are relevant and which can be disregarded in each case. And since there is no privileged or essential decomposition, there is no need of an unequivocal criterion for deciding where to place the cut between the open system and the environment. Summing up, decoherence is a phenomenon *relative* to the relevant observables selected in each particular case. The only essential physical fact is that, among all the observational viewpoints that may be adopted to study a quantum system, some of them determine subspaces of relevant observables for which the system decoheres.
Another conceptual difficulty of the EID approach relies on its definition of the pointer basis. This basis is clearly characterized in measurements situations, where the self-Hamiltonian of the system can be neglected and the evolution is completely dominated by the interaction Hamiltonian. In those cases, the pointer basis is given by the eigenstates of the interaction Hamiltonian.[@Zurek-1981] However, there are two further regimes, differing in the relative strength of the system’s self-Hamiltonian and the interaction Hamiltonian, where the pointer basis lacks a general definition.[@Paz-Zurek; @1999] Our present research is directed to the search of a general and precise definition of the pointer basis of decoherence.
Modal-Hamiltonian interpretation of quantum mechanics
=====================================================
Our work on decoherence from a closed-system perspective taught us that the decomposition of the total Hamiltonian has to be studied in detail in each case, in order to know whether the system of interest resulting from the partition decoheres or not under the action of its self-Hamiltonian and the interaction Hamiltonian. Once we acknowledged the central role played by the Hamiltonian in decoherence, the natural further step was to ask ourselves whether it plays the same central role in interpretation. This question led us to formulate our modal-Hamitonian interpretation (MHI) of QM,[@LC]$%
^{-}$[@LASC] which belongs to the modal family:[@libro; @modal] it is a realist, non-collapse interpretation, according to which the quantum state describes the possible properties of a system but not its actual properties. Here we will only recall its main interpretative postulates.
The first step is to identify the systems that populate the quantum world. By adopting an algebraic perspective, a quantum system is defined as:
**Systems postulate (SP):** *A quantum system* $\mathcal{S}$* is represented by a pair* $(\mathcal{O},\,H)$* such that (i)* $\mathcal{O}$* is a space of self-adjoint operators on a Hilbert space* $\mathcal{H}$*, representing the observables of the system, (ii)* $H\in \mathcal{O}$* is the time-independent Hamiltonian of the system* $\mathcal{S}$*, and (iii) if* $\rho
_{0}\in \mathcal{O}^{\prime }$* (where* $\mathcal{O}^{\prime }$* is the dual space of* $\mathcal{O}$*) is the initial state of* $\mathcal{S}$*, it evolves according to the Schrödinger equation in its von Neumann version.* Of course, any quantum system can be partitioned in many ways; however, not any partition will lead to parts which are, in turn, quantum systems.[Harshman-1]{}$^{,}$[@Harshman-2] On this basis, a composite system is defined as:
**Composite systems postulate (CSP):*** A quantum system represented by* $\mathcal{S}:\;(\mathcal{O}\,,\,H)$*, with initial state* $\rho _{0}\in \mathcal{O}^{\prime }$*, is composite when it can be partitioned into two quantum systems* $\mathcal{S}^{1}:\;(\mathcal{O}%
^{1},\,H^{1})$* and* $\mathcal{S}^{2}:\;(\mathcal{O}^{2}\,,\,H^{2})$* such that (i)* $\mathcal{O}=\mathcal{O}^{1}\otimes \mathcal{O}^{2}$*, and (ii)* $H=H^{1}\otimes I^{2}+I^{1}\otimes H^{2}$*, (where* $I^{1}$* and* $I^{2}$* are the identity operators in the corresponding tensor product spaces). In this case, the initial states of* $\mathcal{S}^{1}$* and* $\mathcal{S}^{2}$* are obtained as the partial traces* $\rho _{0}^{1}=Tr_{\mathrm{(}2\mathrm{)}}\rho _{0}$* and* $\rho _{0}^{2}=Tr_{\mathrm{(}1\mathrm{)}}\rho _{0}$*; we say that* $\mathcal{S}^{1}$* and* $\mathcal{S}^{2}$* are subsystems of the composite system,* $\mathcal{S}=\mathcal{S}^{1}\cup
\mathcal{S}^{2}$*. If the system is not composite, it is elemental.* Since the contextuality of QM, as implied by the Kochen-Specker theorem,[K-S]{} prevents us from consistently assigning actual values to all the observables of a quantum system in a given state, the second step is to identify the *preferred context*, that is, the set of the actual-valued observables of the system. Whereas the different rules of actual-value ascription proposed by previous modal interpretations rely on mathematical properties of the theory, our MHI places an element with a clear physical meaning, the Hamiltonian, at the heart of its rule:
**Actualization rule (AR):*** Given an elemental quantum system represented by* $\mathcal{S}:\;(\mathcal{O}\,,\,H)$*, the actual-valued observables of* $\mathcal{S}$* are* $H$* and all the observables commuting with* $H$* and having, at least, the same symmetries as* $H$*.*
This preferred context where actualization occurs is independent of time: the actual-valued observables always commute with the Hamiltonian and, therefore, they are constants of motion of the system. In other words, the observables that receive actual values are the same during all the life of the quantum system as such $-$precisely, as a closed system$-$: there is no need of accounting for the dynamics of the actual properties of the quantum system as in other modal interpretations.[@Vermaas]
The fact that the Hamiltonian always belongs to the preferred context agrees with the many physical cases where the energy has definite value. The MHI has been applied to several well-known physical situations (hydrogen atom, Zeeman effect, fine structure, etc.), leading to results consistent with experimental evidence.[@LC] Moreover, it has proved to be effective for solving the measurement problem, both in its ideal and its non-ideal versions,[@LC] solving the deep challenges that non-ideal measurements pose to other modal interpretations.[@Elby]$^{,}$[@Albert-Loewer] In particular, the MHI distinguishes between reliable and non-reliable non-ideal measurements.[@LC] Furthermore, in spite of the fact that MHI applies to closed systems, we have proved its compatibility with EID.[Manuscrito]{}$^{,}$[@PoS]
Once the MHI was clearly formulated, our further question was whether it satisfies the Galilean invariance of the theory. In fact, any continuous transformation admits two interpretations. Under the active interpretation, the transformation corresponds to a change from one system to another $-$transformed$-$ system; under the passive interpretation, the transformation consists in a change of the viewpoint $-$reference frame$-$ from which the system is described.[@Brading] Nevertheless, in both cases the validity of a group of symmetry transformations expresses the fact that the identity and the behavior of the system are not altered by the application of the transformations: in the active interpretation language, the original and the transformed systems are equivalent; in the passive interpretation language, the original and the transformed reference frames are equivalent. Then, any realist interpretation should agree with that physical fact: the rule of actual-value ascription should select a set of actual-valued observables that remains unaltered under the transformations. Since the Casimir operators of the central-extended Galilei group are invariant under all the transformations of the group, one can reasonably expect that those Casimir operators belong to the preferred context.
As we have seen, the preferred context selected by AR only depends on the Hamiltonian of the system. Then, the requirement of invariance of the preferred context under the Galilei transformations is directly fulfilled when the Hamiltonian is invariant, that is, in the case of time-displacement, space-displacement and space-rotation:$$\begin{aligned}
H^{\prime } &=&e^{iH\tau }H\,e^{-iH\tau }=H\ \text{\ (since }\left[ H,H%
\right] =0\text{)} \label{3-1} \\
H^{\prime } &=&e^{iP_{i}r_{i}}H\,e^{-iP_{i}r_{i}}=H\ \ \text{(since }\left[
P_{i},H\right] =0\text{) } \label{3-2} \\
H^{\prime } &=&e^{iJ_{i}\theta _{i}}H\,e^{-iJ_{i}\theta _{i}}=H\ \ \text{%
(since }\left[ J_{i},H\right] =0\text{)} \label{3-3}\end{aligned}$$However, it is not clear that the requirement completely holds, since the Hamiltonian is not invariant under Galilei-boosts. In fact, under a Galilei-boost corresponding to a velocity $u_{x}$, $H$ changes as$$H^{\prime }=e^{iK_{x}^{(G)}u_{x}}H\,e^{-iK_{x}^{(G)}u_{x}}\neq H\ \ \text{%
(since }\left[ K_{x}^{(G)},H\right] =iP_{x}\neq 0\text{)} \label{3-4}$$Nevertheless, when space is homogeneous and isotropic, a Galilei-boost only introduces a change in the subsystem that carries the kinetic energy of translation: the internal energy $W$ remains unaltered under the transformation. This should not sound surprising to the extent that $W$ $-$multiplied by the scalar mass $m-$ is a Casimir operator of the central-extended Galilei group. On this basis, we can reformulate AR in an explicit Galilei-invariant form in terms of the Casimir operators of the central-extended group:
**Actualization rule’ (AR’)***: Given a quantum system free from external fields and represented by* $\mathcal{S}:\;(\mathcal{O}\,,\,H)$*, its actual-valued observables are the observables* $C_{i}^{G}$* represented by the Casimir operators of the central-extended Galilei group in the corresponding irreducible representation, and all the observables commuting with the* $C_{i}^{G}$* and having, at least, the same symmetries as the* $C_{i}^{G}$*.*
Since the observables* *$C_{i}^{G}$* * $-$in the reference frame of the center of mass$-$ are $M$, $mW$ and $m^{2}S^{2}$, this new version AR’ is in agreement with the original AR when applied to a system free from external fields:[@ACL]$^{-}$[@LCA]
- The actual-valuedness of $M$ and $S^{2}$, postulated by AR’, follows from AR: these observables commute with $H$ and do not break its symmetries because, in non-relativistic QM, both are multiples of the identity in any irreducible representation.
- The actual-valuedness of $W$ might seem to be in conflict with AR because $W$ is not the Hamiltonian: whereas $W$ is Galilei-invariant, $H$ changes under the action of a Galilei-boost. However, this is not a real obstacle because a Galilei-boost transformation only introduces a change in the subsystem that carries the kinetic energy of translation, which can be considered a mere shift in an energy defined up to a constant.[@ACL]$%
^{,} $[@LCA]
Summing up, the application of AR’ leads to reasonable results, since the actual-valued observables turn out to be invariant and, therefore, objective magnitudes. The assumption of a strong link between invariance and objectivity is rooted in a natural idea: what is objective should not depend on the particular perspective used for the description; or, in group-theoretical terms, what is objective according to a theory is what is invariant under the symmetry group of the theory. This idea is not new: it was widely discussed in the context of special and general relativity with respect to the ontological status of space and time,[@Minkowski] and since then it reappeared in several works. [@Weyl]$^{-}$[@Earman-2] >From this perspective, AR says that the observables that acquire actual values are those representing objective magnitudes. On the other hand, from any realist viewpoint, the fact that certain observables acquire an actual value is an objective fact in the behavior of the system; therefore, the set of actual-valued observables selected by a realist interpretation must be also Galilean-invariant. But the Galilean-invariant observables are always functions of the Casimir operators of the Galilean group. As a consequence, one is led to the conclusion that any realist interpretation that intends to preserve the objectivity of actualization may not stand very far from the modal-Hamiltonian interpretation.
When AR is expressed in simple group terms, one can expect that it can be extrapolated to any quantum theory endowed with a symmetry group. In particular, the actual-valued observables of a system in quantum field theory would be those represented by the Casimir operators of the Poincaré group and of the internal symmetry group. On this basis, in a recent paper we presented an alternative version of the non-relativistic limit of the centrally extended Poincaré group and its consequences for interpretive problems.[@ACR]
As it is well known, the Galilei group can be recovered from the Poincaré group by means of Inönü-Wigner contraction.[@LL] It is therefore natural to ask whether such a situation can be generalized to the central-extended Galilei group, which is the relevant group in QM. However, since the Poincaré group does not admit nontrivial central extensions,[@cari] we have to define a generalized Inönü-Wigner contraction from a trivial extension of the Poincaré group whose generators are $H$, $P_{i}$, $J_{i}$ and $K_{P_{i}}$ (where the last ones are the Lorentz boosts). With this purpose, we extend the group trivially, i.e., in such a way that all the generators of Poincaré group commute with a trivial central charge $M$. The basis of the resulting new algebra $%
I^{M}SO(1,3)=ISO(1,3)\times \left\langle M\right\rangle $ is $\left\{ H\text{%
, }P_{i}\text{, }J_{i}\text{, }K_{P_{i}}\text{, }M\right\} $. Then, we perform the following change of the generators basis $\overline{H}=H-M$. In the new basis $\left\{ \overline{H}\text{, }P_{i}\text{, }J_{i}\text{, }%
K_{P_{i}}\text{, }M\right\} $ all the commutators of the Poincaré group remain the same, with the only exception of$$\left[ P_{i},K_{P_{j}}\right] =-i\delta _{ij}H=-i\delta _{ij}(\overline{H}+M)
\label{3-5}$$The contraction is determined by the rescaling transformations (in the basis $\left\{ \overline{H}\text{, }P_{i}\text{, }J_{i}\text{, }K_{P_{i}}\text{, }%
M\right\} $) defined by$$J_{i}^{\prime }=J_{i}\text{, \ \ }P_{i}^{\prime }=\varepsilon P_{i}\text{, \
\ }K_{P_{i}}^{\prime }=\varepsilon K_{P_{i}}\text{, \ \ }\overline{H}%
^{\prime }=\overline{H}\text{, \ \ \ }M^{\prime }=\varepsilon ^{2}M\text{ \
\ \ } \label{3-6}$$The space isotropy remains unchanged by this rescaling transformation and$$\left[ P_{i}^{\prime },K_{P_{j}}^{\prime }\right] =-i\delta
_{ij}(\varepsilon ^{2}\overline{H}^{\prime }+M^{\prime })\ \ \ \ \ so\ \ \ \
\ \ \ \ \underset{\varepsilon \rightarrow 0}{\lim }\left[ P_{i}^{\prime
},K_{P_{j}}^{\prime }\right] =-i\delta _{ij}M^{\prime } \label{3-7}$$Therefore, it turns out to be clear that the contracted algebra is isomorphic to the extension of the Galilei algebra. On the basis of this result, we have also proved that the Casimir operators of the trivially extended Poincaré group contract naturally to the Casimir operators of the extended Galilei group.[@ACR]
Summing up, when AR is expressed in its explicit Galilei-invariant form AR’, it leads to a physically reasonable result: the actual-valued observables are those represented by the Casimir operators of the mass central-extended Galilei group. The natural strategy is to extrapolate the interpretation to the relativistic realm by replacing the Galilei group with the Poincaré group. But when one takes into account that the relevant group of non-relativistic QM is not the Galilei group but its central extension, the mere replacement of the relevant group is not sufficient: one has to show also that the actual-valued observables in the relativistic and the non-relativistic cases are related through the adequate limit. As a consequence, the Poincaré group has to be trivially extended, in order to show that the limit between the corresponding Casimir operators holds, and this result counts in favor of the proposed extrapolation of our MHI to non-relativistic QM. Furthermore, this result is physically reasonable because mass and spin are properties supposed to be always possessed by any elemental particle,[@Haag] and they are two of the properties that contribute to the classification of elemental particles. At present we are working on a further extrapolation of the MHI to the standard model.
The ontological picture of the quantum world
============================================
In general, the discussions about modal interpretations are concerned with the traditional problems, as the measurement problem and the no-go theorems. But these are not the only relevant issues: one should not forget the ontological question about the structure of the world referred to by QM.
All modal interpretations rely on a common assumption: QM does not describe what is the case, but rather what may be the case. The problem of the nature of possibility is as old as philosophy itself. Since Aristotle’s time to nowadays, however, two general conceptions can be identified. On the one hand, *actualism* reduces possibility to actuality. This was the position of Diodorus Cronus, who defined *the possible as that which either is or will be*. [@K-K]This view survived up to 20$^{th}$ century; for instance, for Russell ‘possible’ means ‘sometimes’, whereas ‘necessary’ means ‘always’.[Russell]{} On the other hand, *possibilism* conceives possibility as an ontologically irreducible feature of reality. From this perspective, the stoic Crissipus defined possible as *that which is not prevented by anything from happening even if it does not happen*.[@Bunge] In present day metaphysics, the debate actualism-possibilism is still alive. For the actualists, the adjective ‘actual’ is redundant: non-actual possible items (objects, properties, facts, etc.) do not exist. According to the possibilists, on the contrary, not every possible item is an actual item: possible items—*possibilia*—constitute a basic ontological category.[@Menzel]
For our MHI, probabilities measure ontological propensities, which embody a possibilist, non-actualist possibility: a possible fact does not need to become actual to be real. This possibility is defined by the postulates of QM and is not reducible to actuality. This means that reality spreads out in two realms, the realm of possibility and the realm of actuality. In Aristotelian terms, being can be said in different ways: as possible being or as actual being, and none of them is reducible to the other. Moreover, the *ontological structure of the realm of possibility* is embodied in the definition of the elemental quantum system $\mathcal{S}:\;(\mathcal{O}%
\,,\,H)$, with its initial state $\rho _{0}$: (i) the space of observables $%
\mathcal{O}\,$ identifies all the *possible type-properties* (observables) with their corresponding *possible case-properties* (eigenvalues), and (ii) the initial state $\rho _{0}$ codifies the measures of the *propensities to actualization* of all the possible case-properties at the initial time, propensities that evolve deterministically according to the Schrödinger equation.
The fact that propensities belong to the realm of possibility does not mean that they do not have physical consequences in the realm of actuality. On the contrary, propensities produce definite effects on actual reality even if they never become actual. An interesting manifestation of such effectiveness is the case of the so-called *non-interacting experiments*,[@E-V]$^{,}$[Vaidman]{} where non-actualized possibilities can be used in practice, for instance, to test bombs without exploding them.[@Penrose] This shows that possibility is a way in which reality manifests itself, a way independent of and not less real than actuality.
One of the main areas of controversy in contemporary metaphysics is the problem of the nature of individual objects: is an individual a substratum supporting properties or a mere bundleof properties?[@Loux] The idea of a substratum acting as a bearer of properties has pervaded the history of philosophy: it is present under different forms in Aristotle’s primary substance, in Locke’s doctrine of substance in general or in Leibniz’s monads. Nevertheless, many philosophers belonging to the empiricist tradition, from Hume to Russell, Ayer and Goodman, have considered the posit of a characterless substratum as a metaphysical abuse. As a consequence, they adopted some version of the *bundle theory*, according to which an individual is nothing but a bundle of properties: properties have metaphysical priority over individuals and, therefore, they are the fundamental items of the ontology.
In the Hilbert space formalism, states have logical priority over observables since observables apply to states. This logical priority favors the picture of an ontology of substances and properties, with the traditional priority of substances over properties. Our MHI, on the contrary, is based on the algebraic formalism, where the basic elements are observables and states are functionals over the space of observables. Then, the MHI favors the bundle theory, that is, an ontology of properties, where the category of substance is absent.
According to the traditional versions of the bundle theory, an individual is the convergence of certain case-properties, under the assumption that all the type-properties are determined in the actual realm. For instance, a particular billiard ball is the convergence of a definite value of position, a definite shape, say round, a definite color, say white, etc. Then, the properties taken into account are always actual properties: bundle theories identify individuals with bundles of actual properties. In QM, on the contrary, the Kochen-Specker theorem prevents the assignment of case-properties (eigenvalues) to all the type-properties (observables) of the system in a non-contradictory manner. Therefore, the classical idea of a bundle of actual properties does not work for the quantum ontology.
If, from the perspective of the MHI, the quantum world unfolds into two irreducible realms, the realm of possibility has to be taken into account when deciding what kind of properties constitutes the quantum bundle. Since the quantum system is identified by its space of observables, which represent possible properties, an individual quantum system turns out to be *a bundle of possible properties*: it inhabits the realm of possibility, which is as real as the realm of actuality.
This interpretation of quantum individual systems has the advantage of being immune to the challenge represented by the Kochen-Specker theorem, since this theorem imposes no restriction on possibilities. Moreover, it seems reasonable to expect that this conception of individual supplies the basis for solving the problem of the indistinguishability of identical particles,[@French-Krause] introduced in the formalism as an *ad hoc* restriction on the set of states. At present, we are working on this problem: if the traditional assumption of substantial objects, which preserve their individuality when considered in collections, is the main obstacle to explain quantum statistics, the conception of the quantum system as a bundle of possible properties seems to offer a promising starting point in the search for a solution of the problem.
Summing up, from our interpretational perspective, the talk of individual entities as electrons or photons and their interactions can be retained only in a metaphorical sense. In fact, in the quantum framework even the number of particles is represented by an observable $N$, which is subject to the same theoretical constraints as any other observable of the system; this leads, specially in quantum field theory, to the possibility of states that are superpositions of different particle numbers.[@Butterfield] Therefore, the number of particles $N$ has an actual definite value only in some cases, but it is indefinite in others. This fact, puzzling from an ontology populated by substantial objects, is deprived of mystery when viewed from our ontological perspective. The quantum system is not a substantial individual, but a bundle of possible properties. The particle picture, with a definite number of particles, is only a contextual picture valid exclusively when the observable $N$ is picked out by the preferred context. In this case, we could metaphorically retain the idea of a composite system composed of individual particles that interact to each other. But in the remaining cases, this idea proves to be completely inadequate, even in a metaphorical sense.
Concluding remarks
==================
We hope that this journey through the main contributions of Castagnino in the field of the foundations of QM supplies an idea of the active work that he and his research group are developing. Nevertheless, we do not want to finish this review without recalling the rest of the areas of the philosophy of physics where he has fruitfully produced: time’s arrow,[@AoT-1]$^{-}$[@AoT-8] time-asymmetric QM,[@TAQM-1]$^{,}$[@TAQM-2] quantum chaos,[@QC-1]$^{,}$[@QC-2] and even philosophy of chemistry.[Chem]{} Not bad for a baby philosopher. However, this is not surprising when coming from an even-eager spirit as Mario Castagnino.
[99]{} M. Castagnino and R. Laura, *Phys. Rev*.* A*, **56,** 108, 1997.
R. Laura and M. Castagnino, *Phys. Rev.* *A*, **57**, 4140, 1998.
R. Laura and M. Castagnino, *Phys. Rev. E*, **57**, 3948, 1998.
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---
abstract: 'Spin textures of one or two electrons in a quantum dot with Rashba or Dresselhaus spin-orbit couplings reveal several intriguing properties. We show that even at the single-electron level spin vortices with different topological charges exist. These topological textures appear in the [*ground state*]{} of the dots. The textures are stabilized by time-reversal symmetry breaking and are robust against the eccentricity of the dot. The phenomenon persists for the interacting two-electron dot in the presence of a magnetic field.'
author:
- Wenchen Luo
- Amin Naseri
- Jesko Sirker
- Tapash Chakraborty
title: 'Unique Spin Vortices in Quantum Dots with Spin-orbit Couplings'
---
=1.5truecm
A variety of topological states have recently been observed in condensed matter physics. These novel states of matter are a direct consequence of spin-orbit coupling (SOC) [@TP01; @TP02] with topological insulators (TIs) being one of the most prominent examples [@TP03; @TP04]. The SOC also plays an important role in tailoring topological superconductors (TSs) where the elusive Majorana fermions might be present [@Mj01; @Mj02; @Mj03]. Both TIs and TSs display a topologically non-trivial structure in momentum space. SOC can, however, also lead to topological charges in real space. The Dzyaloshinskii-Moriya interaction [@DM01; @DM02]—microscopically based on the SOC—can, for example, give rise to spin skyrmions in helical magnets [@sky01; @sky02] and pseudospin skyrmions in bilayer graphene [@bilayerg]. Synthetic spin-orbit couplings can also be engineered in cold atomic gases and skyrmion-like spin textures have been observed [@STB01; @STB05].
Quantum dots (QDs) are of practical and fundamental interest and provide an excellent platform to control the spin and charge of a single electron [@QD00; @QD01; @QD02; @QD03; @QD04; @QD05; @QD06; @QD07]. Extensive studies on QDs with SOCs have been reported in recent years [@QDSOC01; @QDSOC02; @QDSOC03; @QDSOC04; @QDSOC05; @QDSOC06; @QDSOC07; @QDSOC08; @QDSOC09; @QDSOC10; @QDSOC11; @siranush; @QDSOC12; @QDSOC13; @QDSOC14; @QDSOC15; @QDSOC16]. Furthermore, a Berry connection [@Berr] in momentum space induced by the SOC has been studied [@QDSOC12; @QDSOC13].
In this letter, we investigate the spin textures associated with the electron density profiles in isotropic and elliptical QDs. We show that in the presence of SOC the in-plane spin texture of a single electron is a spin vortex. The QD is consequently turned into an artificial atom [@maksym] with topological features. Spin vortices often emerge in many-spin systems forming either a crystalline arrangement or vortex/anti-vortex pairs [@vor01; @vor02]. For instance, in quantum Hall systems the skyrmion is a single-particle excitation in low Landau levels and the in-plane spin texture is similar to the one we find in a QD with SOC. The skyrmion excitations in the former case are, however, induced by Coulomb interactions [@Ezawa]. In contrast, we show here that in a QD a single vortex can exist in the ground-state of a non-interacting quantum system.
We focus on the physics of the two-dimensional (2D) surface where the QD is constructed [@QD01]. We consider both the Rashba and the linear Dresselhaus SOCs which arise in materials with broken inversion symmetry. The strength of the Rashba SOC can be controlled by a gate electric field [@Rash01; @Rash02; @Rash03; @Rash04; @Rash05]. Moreover, the ratio of the Rashba SOC to the Dresselhaus SOC can be tuned over a wide range, for instance in InAs QDs, by applying an in-plane magnetic field [@Rash05]. We will show that this leads to a system where the topological charge can be dynamically controlled by external electromagnetic fields making spin vortices in QDs possible candidates for future spintronics and quantum information applications.
The SOCs can be theoretically considered as effective momentum-dependent magnetic fields [@SOC01; @SOC02; @SOC03; @SOC04]. In the absence of a confinement and an external magnetic field, the momentum is conserved and the SOC in the Hamiltonian becomes a momentum-dependent operator with a good quantum number (e.g., the helicity operator for Rashba SOC). On the other hand, the spin state is momentum-independent if both Rashba and Dresselhaus couplings have equal strength and there is no Zeeman coupling, leading to a persistent spin helix [@shx01; @shx02; @shx03]. This particular spin state persists in the presence of a confinement potential and can be obtained by exactly solving the Hamiltonian which is equivalent to a quantum Rabi model [@supplement]. If the spin is not a good quantum number then it is instructive to study the spin field in a given single-particle wavefunction $\Psi(\mathbf{r})$ of the dot $$\begin{aligned}
\sigma_{i}(\mathbf{r})
=
\Psi^{\dag}(\mathbf{r})
\sigma_{i}
\Psi(\mathbf{r}),
\label{eq_SpDen01}\end{aligned}$$ where $\sigma_{i}$ for $i=x,y,z$ are Pauli matrices. An in-plane vector field $\boldsymbol{\sigma}(\mathbf{r}) = \left(
\sigma_{x}(\mathbf{r}), \sigma_{y}(\mathbf{r}) \right)$ reveals how the spin in real space is locally affected by the effective magnetic field. In the following, we demonstrate that generic SOCs compel the spin field to rotate around the center of the QD and to develop into a spin vortex.
The Hamiltonian of an electron with effective mass $m^{\ast }$ and charge $-e
$ in a quantum dot with SOCs is given by$$H=\frac{\left( \mathbf{p+}e\mathbf{A}\right) ^{2}}{2m^{\ast }}+\frac{m^{\ast
}}{2}\left( \omega _{x}^{2}x^{2}+\omega _{y}^{2}y^{2}\right) +\frac{\Delta
\sigma _{z}}{2}+H_{SOC}, \label{hamiltonian}$$where the vector potential is chosen in the symmetric gauge $\mathbf{A}=%
\frac{1}{2}B\left( -y,x,0\right) $ with the magnetic field $B$. The confinement is anisotropic with the frequencies in two directions, $\omega _{x}$ and $\omega _{y}$, and $\Delta $ is the Zeeman coupling. We consider both the Rashba SOC, $H_R$, and the Dresselhaus SOC, $H_D$, with $$\begin{aligned}
H_{R} &=&g_{1}\left( \sigma _{x}P_{y}-\sigma _{y}P_{x}\right) , \\
H_{D} &=&g_{2}\left( \sigma _{y}P_{y}-\sigma _{x}P_{x}\right) ,\end{aligned}$$and $H_{SOC}=H_{R}+H_{D}$. $P_{i}=p_{i}+eA_{i}$ is the kinetic momentum, and $g_{1,2}$ determine the strength of each SOC. We note that Rashba and Dresselhaus terms have different rotational symmetry generators, $H_R$ commutes with $L_{z} + \hbar \sigma_{z}/2$ while $H_D$ commutes with $L_{z} -\hbar \sigma_{z}/2$, where $L_{z}$ is the $z$-component of the angular momentum operator. In the following, we will show that this difference is responsible for the different topological charges associated with the spin vortex of the dot.
It is also useful to introduce a renormalized set of frequencies $\Omega _{i}=%
\sqrt{\omega _{i}^{2}+\omega _{c}^{2}/4}$ with the cyclotron frequency $%
\omega _{c}=eB/m^{\ast }$. The natural length scales in $x$ and $y$ directions are $\ell _{i}=\sqrt{\hbar /(m^{\ast }\Omega _{i})}$ while the confinement lengths are defined as $R_{i}=\sqrt{%
\hbar /(m^{\ast }\omega _{i})}$. In the numerical calculations presented in the following the eigenvectors of $H_{0}=\frac{\mathbf{p}^{2}}{ 2m^{\ast}}+\frac{m^{\ast }}{2}
\left( \Omega _{x}^{2}x^{2}+\Omega_{y}^{2}y^{2}\right) +\frac{\Delta }{2}\sigma _{z}$, which is a two-dimensional harmonic oscillator, are used as a basis set.
No analytical solution is known for the generic Hamiltonian in Eq. due to its complexity [@ExSOC]. We can, however, analytically investigate the special case of an isotropic dot ($\Omega_{x,y}=\Omega$, $\ell_{x,y}=\ell$) without a magnetic field and with equal SOCs, $g_{1,2}=g$. The Hamiltonian is then equivalent to a two-component quantum Rabi model which has been extensively studied in quantum optics [@supplement]. The ground states in this case are a degenerate Kramers pair due to time reversal symmetry, $$\left\vert GS\right\rangle _{\pm }=\frac{1}{\sqrt{2}} e^{\pm i\sqrt{2}m^*
\left( y-x\right) g/\hbar }\left(
\begin{array}{c}
\pm e^{-i \pi/4} \\
1
\end{array}%
\right) \left\vert 0,0\right\rangle$$where $\left\vert 0,0\right\rangle$ is the ground state of the two-dimensional quantum oscillator $H_0$. A very weak magnetic field will lift the degeneracy of the Kramers pair, and the unique ground state is then given by $\left\vert GS\right\rangle=(\left\vert
GS\right\rangle_+ + \textrm{sgn}(\Delta)
\left\vert GS\right\rangle_-)/\sqrt{2}$ which minimizes the energy [@supplement]. The spin fields are consequently well defined. We note some features of the spin field: (i) There is a mirror symmetry about the line $x=\pm y$. (ii) $ \sigma_x (\mathbf{r})+ \sigma_y (\mathbf{r}) =0$, and $\sigma_x (\mathbf{r})=\sigma_y (\mathbf{r}) =0$ along the line $x=y$. (iii) $ \sigma_z (\mathbf{r})=-\frac{\textrm{sgn}(\Delta)}{\pi\ell^2}
e^{-2x^2/\ell_x} \cos \left( 4\sqrt{2}m^* xg/ \hbar \right)$ along the line $x=-y$, i.e., $\sigma_z (\mathbf{r})$ is a spiral. Its period is related to the effective mass and the strength of the SOCs. We find that the exact solution perfectly agrees with the exact diagonalization results shown in Fig. \[fig1\].
![(Color online) Numerical results for a single-electron QD with $R_x=R_y=35$nm, $B=0.1$T, and equal SOCs $\hbar g_1=\hbar g_2=20$ nm$\cdot$ meV. (a) Electron density (contours) and in-plane spin fields (arrows), (b) $\sigma_z (\mathbf{r})$ along $x=-y$, and (c) the normalized $\tilde{\sigma}_z (\mathbf{r})=\sigma_z
(\mathbf{r})/\sqrt{\boldsymbol{\sigma} (\mathbf{r})^2+\sigma_z
(\mathbf{r})^2}$ along $x=-y$.[]{data-label="fig1"}](fig1.eps){width="0.99\columnwidth"}
Similar results are found for the case $g_1=-g_2$. For large magnetic fields the exact solution for the case without field is no longer a good starting point and the spin texture starts to rotate [@supplement].
Next, we study the case of an isotropic dot in a weak magnetic field with generic strengths of the SOCs $g_1$ and $g_2$ based on a standard perturbative calculation. We find that the in-plane spin fields up to first order in $g_{1,2}$ are given by $$\begin{aligned}
\sigma_{x}(\mathbf{r})
&=&
\xi(r) (r/\ell)
\left(
\bar{g}_{2}
\sin{\theta}
-
\bar{g}_{1}
\cos{\theta}
\right),
\\
\sigma_{y}(\mathbf{r})
&=&
\xi(r) (r/\ell)
\left(
\bar{g}_{2}
\cos{\theta}
-
\bar{g}_{1}
\sin{\theta}
\right),\end{aligned}$$ and $\sigma_z(\mathbf{r})=\xi(r)/2$ with $\xi
(r)=2e^{-r^{2}/\ell^{2}}/\pi\ell^2 $, $\theta$ is the polar angle in coordinate space, and the new parameters are $$\begin{aligned}
\bar{g}_{1,2}
=
\frac{\hbar g_{1,2}}{\ell}
\frac{1 \pm \omega_{c} / (2\Omega) }{ \hbar (\Omega \pm \omega_{c}/2)- \Delta},\end{aligned}$$ where we have assumed $\Delta<0$. The in-plane spin field $\sigma(\mathbf{r})$ winds once around the origin and acquires a topological charge $q=\pm 1$ when $\bar{g}_1 \neq \bar{g}_2$. If $\bar{g}_{1} =
\bar{g}_{2}$, no vortex appears in agreement with the exact solution discussed earlier. If ${g}_{1} = 0$ or ${g}_{2}=0$, $\sigma(\mathbf{r})$ obtained perturbatively qualitatively agrees with the numerical solutions shown in Fig. \[fig2\], and the vortices even exist in a strong magnetic field beyond the perturbation calculations.
![(Color online) Single-electron QD with $R_x=R_y=15$nm, $B=0.1$T ($\Delta<0$), and (a) Rashba SOC $\hbar g_1=40$ nm$\cdot$ meV only, and (b) Dresselhaus SOC $\hbar g_2=20$ nm$\cdot$ meV only.[]{data-label="fig2"}](fig2.eps){width="0.99\columnwidth"}
We stress that the two vortex configurations are stable and representative for the regime $g_1 \gg g_2$ and $g_2 \gg g_1$, respectively [@supplement]. We further note that under $B\to -B$ the spin field changes direction, $\sigma(\mathbf{r})\to
-\sigma(\mathbf{r})$, leaving the topological charge invariant though.
Next, we analyze the rotational symmetry of the two types of SOCs in order to characterize the sign of the winding number. First, we consider the spin field of a dot when only the Rashba SOC is present. The spin field is then invariant under the rotation matrix $$\begin{aligned}
U_{R}(\vartheta)
=
\begin{pmatrix}
\cos{\vartheta}& \sin{\vartheta} \\
-\sin{\vartheta}& \cos{\vartheta}
\end{pmatrix} ,\end{aligned}$$ for $\vartheta \in [ 0, 2 \pi]$, which is rooted in the rotational symmetry of a Rashba dot under the operator $L_{z}+\hbar
\sigma_{z}/2$. Therefore, the in-plane spin rotates clockwise by $2\pi$ if we move around the center of the dot in a clockwise direction, and hence, its winding number is $q=+1$. On the other hand, the in-plane spin field of a dot with only Dresselhaus SOC being present, is invariant under the action of $U_{D}(\vartheta)=U_{R}(-\vartheta)$. Along the same line of reasoning, the in-plane spin field then rotates anticlockwise by $2\pi$ if we move around the center in a clockwise direction. Dresselhaus SOC thus leads to a winding number $q=-1$. In the absence of an external magnetic field $B$, Kramers degeneracy may cancel the spin textures, since there is a global $\pi$ phase difference between the pair. Hence, the vortices should be stabilized by breaking of time-reversal symmetry.
In summary, we find for the single-electron dot with $g_{1}=\pm g_{2}$ and without or in a very weak magnetic field, that the in-plane spin field does not form a vortex. There is, however, a spiral in $\sigma
_{z}\left(\mathbf{r}\right) $ along the line $x=\mp y$. For dominant Rashba or Dresselhaus SOC, on the other hand, the exact diagonalization results clearly show the formation of spin vortices. Rashba SOC induces a vortex with topological charge $q=+1$ while the Dresselhaus SOC induces a vortex with $ q=-1$. These topological charges associated with the spin textures are stabilized by time-reversal symmetry breaking and are robust against the ellipticity of the dot [@supplement]. If the dot is strained, the topological features are not changed, since the spin textures originate from the SOCs of the material. The total $\langle\sigma_{z}\rangle$ in the presence of SOC is no longer constant as a function of the applied magnetic field and becomes more and more polarized with increasing magnetic field. In Fig. \[fig3\] we compare $\left\langle \sigma _{z}\right\rangle $ for different cases.
![(Color online) The total $\left\langle \sigma _{z}\right\rangle $ in a single-electron dot ($R_x=R_y=15$nm) without SOC, with Rashba SOC only ($\hbar g_1=40$ nm$\cdot$ meV), and with both Rashba and Dresselhaus SOCs ($\hbar g_1=40$ nm$\cdot$ meV, $\hbar g_2=20$ nm$\cdot$ meV).[]{data-label="fig3"}](fig3.eps){width="0.7\columnwidth"}
The distinct behavior of $\left\langle \sigma _{z}\right\rangle $ when SOCs are present might be observable experimentally via magnetometry or optically pumped NMR measurements [@sean; @private].
If there is more than one electron confined in the dot, we need to also consider the Coulomb interaction. The Hamiltonian of the interaction is given by $ H_{C}=V\left( n_{1},n_{2},n_{3},n_{4}\right)
c_{n_{1}}^{\dag }c_{n_{2}}^{\dag }c_{n_{3}}c_{n_{4}}, $ where $c$ is the electron annihilation operator and $n_{i}=\left(
n_{ix},n_{iy},n_{s}\right) $ is an index combining the quantum numbers of the two-dimensional oscillator in $x,y$ direction with the spin index. The interaction matrix elements are given in the Suppl. Mat. [@supplement]. The full Hamiltonian with interaction is then $H_{I}=H+H_{C}$ with $H$ as given in Eq. . We diagonalize the interacting Hamiltonian exactly to obtain the electron and spin densities. Since the interacting system does contain very rich physics, we restrict the discussion in the following to the case of a dot with two electrons. To be concrete, we consider the case of an InAs dot here, where the effective mass is $m^* = 0.042 m_e$, Landé factor $g_L=-14$ and dielectric constant $\epsilon=14.6$. In this system it appears to be experimentally feasible to change the ratio of the SOCs $g_1/g_2$ over a wide range.
In a two-electron dot with Coulomb interactions, the spin textures can be much more complex than in the single-electron case. If there is no time reversal symmetry breaking, the texture is cancelled by the Kramers pair. In the presence of a magnetic field, the spin textures appear again with topological charge $+1$ or $-1$ if the dot is perfectly isotropic. For an anisotropic quantum dot the electron density will split into two centers in a strong magnetic field even without SOC. With SOCs the spin textures are modified by this density deformation. In the examples shown in Fig. \[fig4\], we find in both cases three vortices along the elongated $x$ axis.
![(Color online) The in-plane spin fields in an elliptic dot with two electrons, $R_x=15$nm, $R_y=10$nm at $B=5$T. The colors represent the electron density. (a) Rashba SOC only with $\hbar g_1=40$ nm$\cdot$ meV, and (b) Dresselhaus SOC only with $\hbar g_2=20$ nm$\cdot$ meV.[]{data-label="fig4"}](fig4.eps){width="0.99\columnwidth"}
In the Rashba SOC case shown in Fig. \[fig4\](a) there are two vortices with $q=1$ and one with $q=-1$, while there are two vortices with $ q=-1$ and one with $q=1$ in the Dresselhaus SOC case presented in Fig. \[fig4\](b). Hence, the total winding numbers are still $+1$ and $-1$ in a Rashba SOC and Dresselhaus SOC system, respectively, as in the single-electron dot. Indeed, the spin textures along the edges of the dot are quite similar to the single-particle case. Here interactions are less relevant and the spin textures are thus mainly induced by the SOCs.
In an isotropic two-electron dot with equal SOCs, $g_{1}=g_{2}$, we find that both the density profiles and spin textures undergo a dramatic change as a function of the applied magnetic field \[Fig. \[fig5\]\].
![(Color online) The in-plane spin fields in a two-electron dot with $R_x=R_y=15$nm, and $\hbar g_1 =\hbar g_2 =20$ nm $\cdot$meV. The colors represent the electron density. (a) $B=3.5$T, topological charge $q=-1$, and (b) $B=18$T leading to $q=+1$.[]{data-label="fig5"}](fig5.eps){width="0.99\columnwidth"}
In this case, the spin and density profiles are determined collectively by [*both*]{} the interactions and SOCs. For large magnetic fields we find, in particular, that the electron density splits mirror symmetrically along the line $x=y$ \[Fig. \[fig5\](b)\], causing also a complete rearrangement of the associated spin texture and a change of the total topological charge. This has to be contrasted with the case of an InAs dot without SOC where the angular momentum of the ground state changes from $L=-1$ to $L=3$ at about $B=17$T leading instead to a ring-shaped electron density. We further note that in a ZnO dot with stronger Coulomb interaction [@aram], the splitting of the electron density and the spin textures can be generated in a much lower magnetic field. Details will be published elsewhere. This splitting—which only occurs if both interactions and SOCs are present—could possibly be observed experimentally and would thus provide an indirect confirmation of a non-trivial spin texture in the dot.
In summary, we find that the combination of electron confinement and SOCs leads to vortex-like spin textures in the ground state even for a single-electron dot. The spin texture can be stabilized by an external magnetic field breaking the time-reversal symmetry. Interestingly, the winding number of the vortex is different for dots with dominant Rashba SOC or Dresselhaus SOC. This difference can be traced back to the different symmetries of the Hamiltonian. The Rashba SOC commutes with $L_z+\hbar \sigma_z/2$ leading to a topological charge of the spin field of $q=+1$ while the Dresselhaus SOC commutes with $L_z-\hbar \sigma_z/2$ and the topological charge is $q=-1$. Using the exact diagonalization scheme we have shown that these spin vortices do persist also in interacting multi-electron dots. For an elliptic two-electron dot we find, in particular, that more than one spin vortex can exist. In all investigated cases the total topological charge is, however, still $q=\pm 1$ as in the single-electron case. Physically, this is understood by noting that the spin configuration at the edge of the dot, where the electron density is low, is only weakly affected by the interactions. We thus conjecture that the total topological charge for a spin texture in multi-electron dots is always fixed to $q=\pm 1$.
The spin textures in QDs described in this letter are similar to the in-plane structure of (anti-)skyrmion excitations in quantum Hall systems. The locations of the skyrmions in a quantum Hall systems are, however, unknown making it difficult to observe a single skyrmion directly. The existence of skyrmions has so far only been confirmed indirectly by NMR and transport measurements. In contrast, the spin vortices in QD systems are localized at a known position. This might possibly open new avenues for spintronics and quantum information applications. Arrays of QDs have, for example, been realized experimentally [@KouwenhovenHekking; @PiqueroZulaica] and have been considered as a potential platform for quantum computation [@LossDiVincenzo; @ZanardiRossi; @Awschalom; @Nowack]. In such an array of QDs with SOCs the ratio of Rashba to Dresselhaus couplings might be tunable by gates over a sufficiently wide range to realize a system with localized and controllable topological charges $q=\pm 1$. At a minimum, such a setup would allow for an indirect probe of the spin texture by measuring the field dependence of the out-of-plane spin component \[Fig. \[fig3\]\], either by a magnetometer or in an NMR experiment [@private].
We acknowledge useful discussions with Sean Barrett (Yale). TC acknowledges support by the Canada Research Chairs Program of the Government of Canada. JS acknowledges support by the Natural Sciences and Engineering Research Council (NSERC, Canada) and by the Deutsche Forschungsgemeinschaft (DFG) via Research Unit FOR 2316. Computation time was provided by Calcul Québec and Compute Canada.
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\
\
In the supplemental material we present details of the analytical calculations for the single-electron parabolic quantum dot (QD) [@fock; @darwin; @QD_book] and additional numerical data demonstrating the stability of the topological spin textures described in the letter. In Sec. 1 the derivation of the exact eigenstates for a single-electron dot with equal Rashba and Dresselhaus spin-orbit couplings (SOCs) and without the magnetic field, as well as the lifting of the Kramers degeneracy for small fields are discussed. In Sec. 2 we present additional numerical data for the single-electron QD showing that (i) for large magnetic fields and $g_1=g_2$ the spin texture starts to rotate, (ii) the topological charges are robust in the cases $g_1\gg g_2$ and $g_2\gg g_1$ and similar to the spin textures with Rashba or Dresselhaus coupling only discussed in the main text, and (iii) the topological charges are also robust against straining of the QD. In Sec. 3 we give explicit formulas for the Coulomb matrix elements for multi-electron QDs.
1. Single-electron dot: Equal Rashba and Dresselhaus couplings {#Sec1}
==============================================================
The Hamiltonian of an electron in an isotropic dot $\Omega_{x,y}=\Omega$, $\ell_{x,y}=\ell$ and $g_1=g_2=g$ without magnetic field is equivalent to a two-component quantum Rabi model. In this case the Hamiltonian (2) in the main text reads,$$H=\frac{\mathbf{p}^{2}}{2m^{\ast }}+\frac{1}{2}m^{\ast }\omega ^{2}\left(
x^{2}+y^{2}\right) -\left( p_{x}-p_{y}\right) g\left(\sigma_x + \sigma_y
\right) .$$We now define the ladder operators of the quantum harmonic oscillator$$\begin{aligned}
a_{\mu } &=&\sqrt{\frac{m^{\ast }\omega}{2\hbar }}\left( \mu +\frac{i%
}{m^{\ast }\omega}p_{\mu }\right) ,\end{aligned}$$and use $a_{x} =-ib_{x}, a_{y} =ib_{y},$ to transform the Hamiltonian to $$\begin{aligned}
H =b_{x}^{\dag }b_{x}+b_{y}^{\dag }b_{y}+ G\left( b_{x}^{\dag
}+b_{x}+b_{y}^{\dag }+b_{y}\right) \frac{1}{\sqrt{2}}\left(\sigma_x + \sigma_y
\right) ,\end{aligned}$$where $G= g\sqrt{\hbar m^{\ast }\omega}$. Then, performing a unitary transformation $$\begin{aligned}
U &=&\frac{1}{\sqrt{2}}\left(
\begin{array}{cc}
1 & e^{-i \pi/4} \\
e^{i \pi/4} & -1%
\end{array}%
\right) ,\end{aligned}$$ leads to $$\begin{aligned}
UHU^{\dag } &=&b_{x}^{\dag }b_{x}+b_{y}^{\dag }b_{y}+ G\left(
b_{x}^{\dag }+b_{x}+b_{y}^{\dag }+b_{y}\right) \sigma_z ,\end{aligned}$$which is a two-component quantum Rabi model with zero splitting [@rabi]. The case of $g_1=-g_2$ can be solved in a similar manner.
In the main text, we show in Eq. (5) that the ground states are a degenerate Kramers pair. If the magnetic field is infinitesimal then the degeneracy is lifted. Since ${}_{\pm}\left\langle GS\right\vert L_z
\left\vert GS\right\rangle_{\pm}=0$, the unique ground state can be found to lowest order by minimizing the Zeeman energy only. We use the ansatz $$\left\vert GS\right\rangle =A\left\vert GS\right\rangle _{+}+B\left\vert
GS\right\rangle _{-}.$$with coefficients $A,B$. The Zeeman energy is then proportional to $$\begin{aligned}
\left\langle GS\right\vert \Delta \sigma _{z}\left\vert GS\right\rangle
&=&-\Delta \int^{\infty}_{-\infty} dxdy\left( AB^{\ast }e^{-i2\sqrt{2}g
\ell \frac{m^*}{\hbar }\left( x-y\right) }+A^{\ast }Be^{i2\sqrt{2}g\ell
\frac{m^*}{\hbar }\left(x-y\right) }\right) e^{-x^{2}-y^{2}} \notag \\
&=&-\Delta \left( AB^{\ast }+A^{\ast }B\right) \pi e^{-4\left( g\ell \frac{m^*
}{\hbar }\right) ^{2}}.\end{aligned}$$For $\Delta >0$ the minimization of the Zeeman energy requires $A=B=\frac{1}{\sqrt{2}}$ while it requires $A=-B=\frac{1}{\sqrt{2}}$ for $\Delta <0$. Hence,$$\left\vert GS\right\rangle =\frac{\left\vert GS\right\rangle _{+}+
\textrm{sgn}\left( \Delta \right) \left\vert GS\right\rangle _{-}}{\sqrt{2}}.$$
The other calculations presented in the main text are the standard first-order perturbative calculations.
2. Single-electron dot: Numerical results {#Sec2}
=========================================
In Fig. 1(a) of the main text we have shown that the in-plane spin texture in a weak magnetic field for the case $g_1=g_2$ is mirror symmetric around the line $x=y$, consistent with the perturbative calculation. For larger fields the perturbative ground state given above is, however, no longer a good starting point. In Fig. \[s1\], we show how the in-plane spin textures evolve with increasing magnetic field.
![(Color online) The in-plane spin field with $\hbar g_1= \hbar g_2=20$ nm$\cdot$ meV in a single-electron dot with $R_x=R_y=15$nm at magnetic fields (a) $B=3$T, (b) $B=5$T, (c) $B=10$T, and (d) $B=15$T.[]{data-label="s1"}](s1.eps){width="12cm"}
All these results show a mirror symmetry about the line $x=\pm y$. However, when the magnetic field becomes stronger the spins start to rotate leading to a spin texture similar to the case of Rashba SOC only. Note that the in-plane spin components are weaker than in the Rashba case though because the spin becomes more and more polarized along the $z$-direction.
In Fig. 2 of the main text we have shown the spin vortices in a single-electron dot if only the Rashba or the Dresselhaus SOC is present. In Fig. \[s2\] we show that these results are indeed representative for the regimes $g_1 \gg g_2$ and $g_2 \gg g_1$.
![(Color online) The in-plane spin field in an isotropic single-electron dot with $R_x=R_y=15$nm at $B=0.1$T. The SOCs are (a) $\hbar g_1= 20$ nm$\cdot$ meV and $\hbar g_2=5$nm$\cdot$ meV, and (b) $\hbar g_1= 5$nm$\cdot$ meV and $\hbar g_2=20$nm$\cdot$meV.[]{data-label="s2"}](s2.eps){width="12cm"}
We also note that even for larger magnetic fields the topological properties are not changed, although the spin textures are weakened. States with higher topological charge $|q|>1$ may exist in the excited states. Contrary to the spin textures in the ground state they are, however, fragile due to their Kramers partner.
Finally, we also show that the spin texture is robust against the eccentricity of the dot. We consider an elliptical InAs dot with $R_x=15$nm and $R_y=10$nm. From Fig. \[s3\] it is obvious that the distortion of the dot does not qualitatively change the structure of the spin vortex.
![(Color online) The in-plane spin field in a single-electron dot with $R_x=15$nm, $R_y=10$nm in $B=0.1$T. The SOCs are (a) $\hbar g_1= 20$ nm$\cdot$ meV and $\hbar g_2=0$, and (b) $\hbar g_1= 0$ and $\hbar
g_2=20$nm$\cdot$meV.[]{data-label="s3"}](s3.eps){width="12cm"}
3. Multi-electron dots: Coulomb interaction matrix elements {#Sec3}
===========================================================
Below we explicitly display the Coulomb interaction matrix elements [@QD_book] used in the exact diagonalization method for multi-electron dots $$\begin{aligned}
V\left( n_{1},n_{2},n_{3},n_{4}\right) &=&\frac{2}{\pi } \frac{e^{2}}{\epsilon \sqrt{\ell
_{x}\ell _{y}}}\phi \left( n_{1x},n_{4x}\right) \phi \left(
n_{1y},n_{4y}\right)\phi \left( n_{2x},n_{3x}\right) \phi \left( n_{2y},n_{3y}\right) \\
&& \left( -1\right) ^{\left\vert n_{2x}-n_{3x}\right\vert +\left\vert
n_{2y}-n_{3y}\right\vert } i^{\left\vert n_{1x}-n_{4x}\right\vert +\left\vert
n_{1y}-n_{4y}\right\vert +\left\vert n_{2x}-n_{3x}\right\vert +\left\vert
n_{2y}-n_{3y}\right\vert } \nonumber \\
&&\int_{0}^{\infty }dx dy\, \Phi \left( n_{1x},n_{4x},x\right) \Phi
\left( n_{2x},n_{3x},x\right) \frac{\Phi \left( n_{1y},n_{4y},y\right)
\Phi \left( n_{2y},n_{3y},y\right) }{\sqrt{\frac{\ell _{y}}{\ell _{x}}x^{2}+\frac{%
\ell _{x}}{\ell _{y}}y^{2}}}. \nonumber\end{aligned}$$Here $\epsilon$ is the dielectric constant and $$\begin{aligned}
\phi \left( n,m\right) &=&\sqrt{\frac{2^{\min \left( n,m\right) }\min
\left( n,m\right) !}{2^{\max \left( n,m\right) }\max \left( n,m\right) !}},
\\
\Phi \left( n,m,x\right) &=&x^{\left\vert n-m\right\vert }e^{-\frac{1}{4}%
x^{2}}L_{\min \left( n,m\right) }^{\left\vert n-m\right\vert }\left( \frac{%
x^{2}}{2}\right)\end{aligned}$$with the Laguerre polynomial $L$.
[10]{}
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|
---
abstract: 'We present the results of long-baseline optical interferometry observations using the Precision Astronomical Visual Observations (PAVO) beam combiner at the Center for High Angular Resolution Astronomy (CHARA) Array to measure the angular sizes of three bright [*Kepler*]{} stars: $\theta$ Cygni, and both components of the binary system 16 Cygni. Supporting infrared observations were made with the Michigan Infrared Combiner (MIRC) and Classic beam combiner, also at the CHARA Array. We find limb-darkened angular diameters of $0.753\pm0.009$mas for $\theta$ Cyg, $0.539\pm0.007$mas for 16 Cyg A and $0.490\pm0.006$mas for 16 Cyg B. The [*Kepler Mission*]{} has observed these stars with outstanding photometric precision, revealing the presence of solar-like oscillations. Due to the brightness of these stars the oscillations have exceptional signal-to-noise, allowing for detailed study through asteroseismology, and are well constrained by other observations. We have combined our interferometric diameters with Hipparcos parallaxes, spectrophotometric bolometric fluxes and the asteroseismic large frequency separation to measure linear radii ($\theta$Cyg: 1.48$\pm$0.02R$_\odot$, 16CygA: 1.22$\pm$0.02R$_\odot$, 16CygB: 1.12$\pm$0.02R$_\odot$), effective temperatures ($\theta$Cyg: 6749$\pm$44K, 16CygA: 5839$\pm$42K, 16CygB: 5809$\pm$39K), and masses ($\theta$Cyg: 1.37$\pm$0.04M$_\odot$, 16CygA: 1.07$\pm$0.05M$_\odot$, 16CygB: 1.05$\pm$0.04M$_\odot$) for each star with very little model dependence. The measurements presented here will provide strong constraints for future stellar modelling efforts.'
author:
- |
T. R. White$^{1,2,3}$[^1], D. Huber$^{1,4,9}$, V. Maestro$^{1}$, T. R. Bedding$^{1,3}$, M. J. Ireland$^{1,2,5}$, F. Baron$^{6}$, T. S. Boyajian$^{7,8}$, X. Che$^{6}$, J. D. Monnier$^{6}$, B. J. S. Pope$^{1}$, R. M. Roettenbacher$^{6}$, D. Stello$^{1,3}$, P. G. Tuthill$^{1}$, C. D. Farrington$^{7}$, P. J. Goldfinger$^{7}$, H. A. McAlister$^{7}$, G. H. Schaefer$^{7}$, J. Sturmann$^{7}$, L. Sturmann$^{7}$, T. A. ten Brummelaar$^{7}$ and N. H. Turner$^{7}$\
$^{1}$Sydney Institute for Astronomy (SIfA), School of Physics, University of Sydney, NSW 2006, Australia\
$^{2}$Australian Astronomical Observatory, PO Box 296, Epping, NSW 1710, Australia\
$^{3}$Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark\
$^{4}$NASA Ames Research Center, MS 244-30, Moffett Field, CA 94035, USA\
$^{5}$Department of Physics and Astronomy, Macquarie University, NSW 2109, Australia\
$^{6}$University of Michigan Astronomy Department, 941 Dennison Bldg, Ann Arbor, MI 48109-1090, USA\
$^{7}$Center for High Angular Resolution Astronomy, Georgia State University, PO Box 3969, Atlanta, GA 30302, USA\
$^{8}$Yale University, Department of Astronomy, 260 Whitney Ave, New Haven, CT 06520\
$^{9}$NASA Postdoctoral Program Fellow
date: 'Accepted 2013 May 7. Received 2013 May 3; in original form 2013 March 31'
title: 'Interferometric radii of bright [*Kepler*]{} stars with the CHARA Array: $\theta$ Cygni and 16 Cygni A and B'
---
\[firstpage\]
stars: oscillations – stars: individual: $\theta$ Cygni, 16 Cygni A, 16 Cygni B – techniques: interferometric
Introduction
============
Progress in understanding stellar structure and evolution is driven by ever-more precise measurements of fundamental properties such as stellar temperature, radius and mass [see, e.g. @Demarque86; @Monteiro96; @Deheuvels11; @Trampedach11; @Piau11]. Unfortunately, many methods to determine such properties are indirect and, being model-dependent themselves, are of little use in improving stellar models. We therefore look towards methods that either themselves, or in combination with other methods, have little model dependence.
One such method is asteroseismology, the study of stellar oscillations. Stars like the Sun exhibit many convectively-excited oscillation modes whose properties depend on the structure of the star. This allows stellar parameters such as mean stellar density and surface gravity to be accurately determined with little model dependence [see, e.g. @Brown94; @C-D04; @Aerts10].
Another method is long-baseline optical interferometry (LBOI), which can be used to measure the angular sizes of stars. Combining with a parallax measurement yields the linear radius, while combining with the bolometric flux provides a direct measurement of effective temperature [see, e.g. @Code76; @Baines09; @Boyajian09; @Boyajian12a; @Boyajian12b; @Creevey12b].
The combination of asteroseismology and interferometry therefore allows us to determine mass, radius and temperature with very little model dependence. While the potential value of this has long been recognised [@Cunha07], until recently the inherent difficulties in these methods had limited their application in cool stars to a few bright objects [@North07; @Bruntt10; @Bazot11]. Progress in instrumentation, such as the Precision Astronomical Visible Observations (PAVO) beam combiner [@Ireland08] at the Center for High Angular Resolution Astronomy (CHARA) Array [@tenBrummelaar05], has pushed the sensitivity limits of LBOI. Meanwhile asteroseismology has entered a ‘golden age’, thanks to data from the space telescopes [*CoRoT*]{} [@Michel08] and [*Kepler*]{} [@Koch10; @Gilliland10; @Chaplin11a].
@Huber12b recently presented interferometric observations using the PAVO beam combiner at CHARA of F, G and K stars spanning from the main sequence to the red clump in which solar-like oscillations have been detected by [*Kepler*]{} or [*CoRoT*]{}. In this paper we present results from the same instrument, of three bright [*Kepler*]{} targets, $\theta$ Cygni and 16 Cygni A & B. These targets present an excellent opportunity to combine the remarkably precise, high signal-to-noise asteroseismology data from the [*Kepler Mission*]{} with precise constraints from interferometry and other methods, for strong tests of stellar models.
Targets
=======
$\theta$ Cyg
------------
The F4V star $\theta$ Cygni (13 Cyg, HR 7469, HD 185395, KIC 11918630), magnitude $V=4.48$, is the brightest star being observed by [*Kepler*]{}. Stellar parameters available from the literature are given in Table \[tab0\]. Thus far [*Kepler*]{} has observed it in 2010 June–September ([*Kepler*]{} Quarter 6), 2011 January–March (Q8) and 2012 January–October (Q12 – Q14).
A close companion of $V\sim$12 mag has been identified as an M dwarf, with an estimated mass of 0.35M$_\odot$, separated from the primary by $\sim$2 arcsec, a projected separation of 46AU [@Desort09]. Although it has been detected several times since 1889 [@Mason01], the orbit is still very incomplete.
@Desort09 undertook a radial velocity study of $\theta$ Cyg, finding a 150d quasi-periodic variation. The origin of this variation is still not satisfactorily explained – the presence of one or two planets does not adequately explain all the observations and stellar variation of this period is unknown in stars of this type.
The limb-darkened angular diameter of $\theta$ Cyg has previously been estimated as $\theta_\mathrm{LD}$=0.760$\pm$0.021mas from spectral energy distribution fitting to photometric observations by @vanBelle08. In 2007 and 2008, @Boyajian12a made interferometric observations with the CHARA Classic beam combiner in $K^{\prime}$-band ($\lambda_0=2.14\mu$m). They measured a larger diameter, $\theta_\mathrm{LD}$=0.861$\pm$0.015mas. @Ligi12, using the VEGA beam combiner at CHARA found $\theta_\mathrm{LD}$=0.760$\pm$0.003mas, in agreement with @vanBelle08. @Ligi12 also reported excessive scatter in their measurements and speculated on diameter variability or the existence of a new close companion, possibly related to the quasi-periodic variability seen in radial velocity by @Desort09.
The location of $\theta$ Cyg in the HR diagram places it amongst $\gamma$ Dor pulsators. Analysis of Q6 [*Kepler*]{} data by @Guzik11 did not reveal $\gamma$ Dor pulsations, but clear evidence of solar-like oscillations was seen in the power spectrum between 1200 and 2500 $\mu$Hz. The characteristic large frequency separation between modes of the same spherical degree, $\Delta\nu$, is 84.0$\pm$0.2 $\mu$Hz. The oscillation modes are significantly damped resulting in large linewidths in the power spectrum, which is typical of F stars [@Chaplin09; @Baudin11; @Appourchaux12a; @Corsaro12].
$\theta$ Cyg 16 Cyg A 16 Cyg B
-------------------------------- -------------------------------------------- ------------------------------------- -------------------------------------
Spectral Type F4V G1.5V G3V
$V$ mag 4.48 5.96 6.2
$T_\mathrm{eff}$ (K) 6745$\pm$150[^2] 5825$\pm$50[^3] 5750$\pm$50$^{\displaystyle b}$
log $g$ 4.2$\pm$0.2$^{\displaystyle a}$ 4.33$\pm$0.07$^{\displaystyle b}$ 4.34$\pm$0.07$^{\displaystyle b}$
$\mathrm{[Fe/H]}$ $-$0.03$^{\displaystyle a}$ 0.096$\pm$0.026$^{\displaystyle b}$ 0.052$\pm$0.021$^{\displaystyle b}$
Parallax (mas) 54.54$\pm$0.15[^4] 47.44$\pm$0.27$^{\displaystyle c}$ 47.14$\pm$0.27$^{\displaystyle c}$
Distance (pc) 18.33$\pm$0.05 21.08$\pm$0.12 21.21$\pm$0.12
$F_\mathrm{bol}$ (pW.m$^{-2}$) 392.0$\pm$0.4[^5] 112.5$\pm$0.2$^{\displaystyle d}$ 91.08$\pm$0.14$^{\displaystyle d}$
Luminosity (L$_\odot$) 4.11$\pm$0.02 1.56$\pm$0.02 1.28$\pm$0.01
Mass (M$_\odot$) $1.39^{+0.02}_{-0.01}$[^6] 1.11$\pm$0.02[^7] 1.07$\pm$0.02$^{\displaystyle f}$
Radius (R$_\odot$) ... 1.243$\pm$0.008$^{\displaystyle f}$ 1.127$\pm$0.007$^{\displaystyle f}$
Age (Gyr) $1.13^{+0.17}_{-0.21}$$^{\displaystyle e}$ 6.9$\pm$0.3$^{\displaystyle f}$ 6.7$\pm$0.4$^{\displaystyle f}$
\[tab0\]
: Properties of Target Stars from Available Literature
16 Cyg A & B {#16Cyglit}
------------
Our other targets are the solar analogues 16 Cygni A (HR 7503, HD 186408, KIC 12069424) and B (HR 7504, HD 186427, KIC 12069449). Properties of the stars from the literature are listed in Table \[tab0\]. [*Kepler*]{} observations between June 2010 and October 2012 (Q6 – Q14) are currently available.
The separation of the A and B components on the sky is 39.56 arcsec, which enables them to be observed independently by both [*Kepler*]{} and PAVO. They also have a distant M dwarf companion, about 10 mag fainter, in a hierarchical triple system [@Turner01; @Patience02]. There are, however, no dynamical constraints on their masses due to the long orbital period, estimated at over 18,000 years [@Hauser99]. Additionally, 16 Cyg B is known to have a planet with a mass of $\sim$ 1.5 M$_\mathrm{J}$ in an eccentric 800 day orbit [@Cochran97].
Interferometric observations of 16 Cyg A and B with the CHARA Classic beam combiner have been presented previously. Observing in $K^{\prime}$-band, @Baines08 measured a limb-darkened angular diameter for 16 Cyg B of $\theta_\mathrm{LD}$=0.426$\pm$0.056mas, although their estimate from a spectral energy distribution fit was somewhat larger ($\theta_\mathrm{LD}$=0.494$\pm$0.019mas). More recently, @Boyajian13 measured both stars with Classic in $H$-band ($\lambda_0=1.65\mu$m), finding $\theta_\mathrm{LD}$=0.554$\pm$0.011mas and $\theta_\mathrm{LD}$=0.513$\pm$0.012mas for the A and B components, respectively.
[*Kepler*]{} observations clearly show solar-like oscillations in both stars, with large separations, $\Delta\nu$, of 103.4 $\mu$Hz and 117.0 $\mu$Hz, respectively [@Metcalfe12]. Asteroseismic modelling was performed by @Metcalfe12 using several different methods. The values they obtained for mass, radius and age are given in Table \[tab0\]. Promisingly, although both stars were modelled independently, the models find a common age and initial composition, which is to be expected in a binary system. However, inspecting the individual results of each model method reveals two families of solutions. Several models favour a radius of $1.24\,\mathrm{R_\odot}$ for 16 Cyg A and $1.12\,\mathrm{R_\odot}$ for 16 Cyg B, while others favour a larger radii around $1.26\,\mathrm{R_\odot}$ and $1.14\,\mathrm{R_\odot}$, respectively. For comparison, the estimated systematic uncertainties in radius are $0.008\,\mathrm{R_\odot}$ and $0.007\,\mathrm{R_\odot}$, respectively.
Observations {#obs}
============
UT Date Baseline[^8] Target No. Scans Calibrators[^9]
------------------ -------------- -------------- ----------- -----------------
2010 July 20 S2E2 16 Cyg A 1 c
2011 May 27 E2W2 $\theta$ Cyg 3 ij
2011 May 28 E2W2 $\theta$ Cyg 2 b
2011 July 4 S1W2 16 Cyg A 3 bh
16 Cyg B 3 bh
2011 September 9 S2W2 16 Cyg A 3 be
16 Cyg B 3 be
2012 August 4 S1W2 16 Cyg A 3 aeg
16 Cyg B 3 aeg
$\theta$ Cyg 3 aegi
2012 August 6 E2W2 $\theta$ Cyg 4 egi
2012 August 8 S1W2 16 Cyg A 3 afi
16 Cyg B 3 afi
$\theta$ Cyg 3 afgi
2012 August 9 S2E2 16 Cyg A 3 fgi
16 Cyg B 3 fgi
2012 August 10 S2W2 16 Cyg A 2 bdi
16 Cyg B 2 dfi
$\theta$ Cyg 3 fgi
2012 August 11 W1W2 $\theta$ Cyg 1 i
2012 August 12 E2W1 16 Cyg A 3 fik
16 Cyg B 3 fik
2012 August 14 S2E2 16 Cyg A 3 fgi
16 Cyg B 3 fgi
\[tab1\]
: Log of PAVO interferometric observations.
HD Sp. Type $V$ $V-K$ $E(B-V)$ $\theta_{V-K}$ ID
------------- ---------- ------ ---------- ---------- ---------------- ----
176626 A2V 6.85 $0.084$ 0.026 0.146 a
177003 B2.5IV 5.38 $-0.524$ 0.023 0.198 b
179483 A2V 7.21 $0.316$ 0.028 0.144 c
180681 A0V 7.50 $0.112$ 0.031 0.111 d
181960 A1V 6.23 $0.121$ 0.042 0.200 e
183142 B8V 7.07 $-0.462$ 0.060 0.093 f
184787 A0V 6.68 $0.034$ 0.017 0.154 g
188252 B2III 5.90 $-0.461$ 0.047 0.156 h
188665[^10] B5V 5.14 $-0.384$ 0.035 0.240 i
189296 A4V 6.16 $0.250$ 0.033 0.225 j
190025 B5V 7.55 $-0.230$ 0.157 0.084 k
\[tab2\]
: Calibrators used for interferometric observations.
Our interferometric observations were made with the PAVO beam combiner [@Ireland08] at the CHARA Array at Mt. Wilson Observatory, California [@tenBrummelaar05]. PAVO is a pupil-plane beam combiner, optimised for high sensitivity (limiting magnitude in typical seeing conditions of $R\sim8$) at visible wavelengths ($\sim$600 to 900nm). Two or three beams may be combined. Through spectral dispersion each scan typically produces visibility measurements in 20 independent wavelength channels. With available baselines up to 330m, PAVO at CHARA is one of the highest angular-resolution instruments operating worldwide. Further details on this instrument were given by @Ireland08. Early PAVO science results have been presented by @Bazot11, @Derekas11, @Huber12b [@Huber12a] and @Maestro12.
Most of our observations were made during several nights in August 2012, although some data were taken during previous observing seasons in 2010 and 2011. Our observations have been made using PAVO in two-telescope mode, with baselines ranging from 110 to 250m. A summary of our observations is given in Table \[tab1\]. To calibrate the fringe visibilities in our targets we observed nearby stars, which ideally would be unresolved point sources with no close companions. In practice we used stars as unresolved as possible, which in our case meant spectral types A and B. Table \[tab2\] lists the calibrators used in our analysis. We determined the expected angular diameters of the calibrators using the ($V-K$) relation of @Kervella04. We adopted $V$-band magnitudes from the Tycho catalogue [@Perryman97] and converted them into the Johnson system using the calibration by @Bessell00. $K$-band magnitudes were taken from the Two Micron All Sky Survey [2MASS; @Skrutskie06]. To de-redden the photometry we used the extinction model of @Drimmel03 to estimate interstellar reddening, and adopted the reddening law of @ODonnell94 [see also @Cardelli89].
In the case of our largest calibrator, HD188665, there is some indication that it is larger than expected from the ($V-K$) relation ($\theta_{V-K}=0.240$mas). Calibrating with smaller calibrator stars observed at similar times we find the average interferometric response of HD188665 is consistent with a uniform-disc diameter of $\theta_{UD}=0.274\pm0.008$mas.
To best account for temporal variations in system visibility due to changes in seeing, calibrators must be observed as closely spaced in time to the targets as possible. We observed $\theta$ Cyg and its calibrators in the sequence: [*calibrator 1 – $\theta$ Cyg – calibrator 2*]{}, with a scan of each object obtaining two minutes of visibility data. Such a sequence typically lasted 15minutes, including slewing. To minimise slew times when observing 16 Cyg A and B, we observed in the sequence: [*calibrator 1 – 16 Cyg A – 16 Cyg B – calibrator 2*]{}. This typically took 20minutes.
In addition to our observations with PAVO, supporting observations were also made in the infrared with two other beam combiners at the CHARA Array. Observations of $\theta$ Cyg were made using the Michigan Infrared Combiner [MIRC; @Monnier04], while observations of 16 Cyg A and B were made with the CHARA Classic combiner.
MIRC combines up to six telescope beams in the image-plane, allowing for simultaneous visibility measurements on 15 baselines and 20 closure phase measurements. Additionally, MIRC splits the $H$-band light ($\lambda_0=1.65\mu$m) into eight independent spectral channels. Further details on the MIRC instrument may be found in @Monnier04 [@Monnier06; @Monnier10] and @Che10 [@Che12]. Our MIRC observations consist of four scans of $\theta$ Cyg, made in six-telescope mode on 19 June 2012. The calibrator star used was $\sigma$ Cyg, with an assumed diameter of $\theta_{UD}=0.54\pm0.02$mas [@Barnes78].
Classic is a pupil-plane combiner operating in a two-telescope mode in either of $H$-band ($\lambda_0=1.65\mu$m) or $K^{\prime}$-band ($\lambda_0=2.14\mu$m). The Classic observations of 16 Cyg A and B were previously presented by @Boyajian13. Observations were made in $H$-band, with 23 and 24 brackets of 16 Cyg A and B, respectively, over the nights of 16, 19, 20 and 21 August 2011 using the S1E1, E1W1 and S1W1 baselines. Calibration stars were HD185414, HD191096 and HD191195, whose estimated angular diameters were taken from the SearchCal tool developed by the JMMC Working Group [@Bonneau06; @Bonneau11].
For each target we fitted a limb-darkened disc model to the visibility measurements [@HanburyBrown74], $$V = \left( \frac{1-\mu_\lambda}{2} + \frac{\mu_\lambda}{3} \right)^{-1} \qquad\qquad\qquad\qquad\qquad\qquad$$ $$\qquad{} \times \left[ (1-\mu_\lambda) \frac{J_1(x)}{x} + \mu_\lambda (\pi/2)^{1/2} \frac{J_{3/2}(x)}{x^{3/2}} \right],\label{eqn1}$$ where $$x = \pi B \theta_\mathrm{LD} \lambda^{-1}.\label{eqn2}$$ Here, $V$ is the visibility, $\mu_\lambda$ is the linear limb-darkening coefficient, $J_n(x)$ is the $n^\mathrm{th}$ order Bessel function, $B$ is the projected baseline, $\theta_\mathrm{LD}$ is the angular diameter after correction for limb-darkening, and $\lambda$ is the wavelength at which the observations was made. The quantity $B\lambda^{-1}$ is often referred to as the spatial frequency.
We determined linear limb-darkening coefficients in the $R$ and $H$ bands for our targets by interpolating the model grid by @Claret11 to the spectroscopic estimates of $T_\mathrm{eff}$, log $g$ and \[Fe/H\] given in Table \[tab0\] for a microturbulent velocity of 2 km s$^{-1}$. The uncertainties in the spectroscopic parameters were used to create 1000 realisations of the limb-darkening coefficients, from which the uncertainties were estimated. Adopted values of the linear limb-darkening coefficients are given in Table \[tab3\]. Typically the influence of the adopted limb-darkening on the final fitted angular diameter is relatively small. Detailed 3-D hydrodynamical models by @Bigot06, @Chiavassa10 and @Chiavassa12 for dwarfs and giants have shown that the differences from simple linear limb-darkening models are $\sim$1% or less in angular diameter for stars with near-solar metallicity. For a moderately-well resolved star with $V^2 \sim 0.5$, a 1% change in angular diameter would correspond to an uncertainty of less than 1% in $V^2$. For our measurements these effects may be non negligible and our results will be valuable for comparing simple 1-D to sophisticated 3-D models.
To fit the model and estimate the uncertainty in the derived angular diameters we followed the procedure outlined by @Derekas11. This involved performing Monte Carlo simulations, taking into account uncertainties in the data, adopted wavelength calibration (0.5% for PAVO, 0.25% for MIRC), calibrator sizes (5%) and limb-darkening coefficients (see Table \[tab3\]).
Star Combiner $\mu$ $\theta_\mathrm{UD}$ (mas) $\theta_\mathrm{LD}$ (mas) $R$ (R$_\odot$) $M$ (M$_\odot$) $T_\mathrm{eff}$ (K)
-------------- ---------------------- --------------- ---------------------------- ---------------------------- ----------------------- ----------------------- ---------------------- -- -- --
$\theta$ Cyg PAVO 0.47$\pm$0.04 0.720$\pm$0.004 0.754$\pm$0.009 1.49$\pm$0.02 1.37$\pm$0.04 6745$\pm$44
MIRC 0.21$\pm$0.03 0.726$\pm$0.014 0.739$\pm$0.015 1.46$\pm$0.03 1.31$\pm$0.06 6813$\pm$72
[**PAVO+MIRC**]{} ... ... [**0.753$\pm$0.009**]{} [**1.48$\pm$0.02**]{} [**1.37$\pm$0.04**]{} [**6749$\pm$44**]{}
16 Cyg A PAVO 0.54$\pm$0.04 0.513$\pm$0.004 0.539$\pm$0.006 1.22$\pm$0.02 1.07$\pm$0.04 5839$\pm$37
Classic 0.26$\pm$0.04 0.542$\pm$0.015 0.554$\pm$0.016 1.26$\pm$0.04 1.16$\pm$0.10 5759$\pm$85
[**PAVO+Classic**]{} ... ... [**0.539$\pm$0.007**]{} [**1.22$\pm$0.02**]{} [**1.07$\pm$0.05**]{} [**5839$\pm$42**]{}
16 Cyg B PAVO 0.56$\pm$0.04 0.467$\pm$0.004 0.490$\pm$0.006 1.12$\pm$0.02 1.05$\pm$0.04 5809$\pm$39
Classic 0.27$\pm$0.04 0.502$\pm$0.020 0.513$\pm$0.020 1.17$\pm$0.05 1.20$\pm$0.14 5680$\pm$112
[**PAVO+Classic**]{} ... ... [**0.490$\pm$0.006**]{} [**1.12$\pm$0.02**]{} [**1.05$\pm$0.04**]{} [**5809$\pm$39**]{}
\[tab3\]
Results
=======
Fundamental stellar properties {#4_1}
------------------------------
Combining our interferometric measurements with astrometric, asteroseismic and photometric measurements allows us to derive radii, masses and effective temperatures that are nearly model-independent.
The linear radius, $R$, is, $$R=\frac{1}{2}\theta_\mathrm{LD}D,\label{eqn3}$$ where $D$ is the distance to the star, which itself is obtained directly from the parallax.
From an estimate of the bolometric flux at Earth, $F_\mathrm{bol}$, we can find the effective temperature, $$T_\mathrm{eff}=\left(\frac{4F_\mathrm{bol}}{\sigma\theta_\mathrm{LD}^2}\right)^{1/4},\label{eqn4}$$ where $\sigma$ is the Stefan-Boltzmann constant.
Finally, to obtain the mass we use the scaling relation between the large frequency separation of solar-like oscillations, $\Delta\nu$, and the density of the star [@Ulrich86]: $$\frac{\Delta\nu}{\Delta\nu_\odot}=\left(\frac{M}{\mathrm{M}_\odot}\right)^{1/2}\left(\frac{R}{\mathrm{R}_\odot}\right)^{-3/2}.\label{eqn5}$$ It follows from this relation that we can derive the mass of the star from measurements of the angular diameter, parallax and large frequency separation. Caution is required when using this relationship because the assumption that leads to this relation, that other stars are homologous to the Sun, is not strictly valid [@Belkacem12], although it has been shown that the relation holds to within 5% in models [@Stello09; @White11a]. Particular care is needed for stars above 1.2M$_\odot$ and beyond the main-sequence, since models indicate a departure that is largely a function of effective temperature [@White11a]. We note that one should measure $\Delta\nu$ of the stars and the Sun in a self-consistent manner. We adopt a solar value of $\Delta\nu_\odot$=135.1$\mu$Hz.
$\theta$ Cyg
------------
![Histogram of MIRC closure phase measurements for $\theta$ Cyg.[]{data-label="fig2"}](thCyg_phase_hist)
Figure \[fig1\] presents the calibrated squared-visibility measurements as a function of spatial frequency for $\theta$ Cyg. We have performed limb-darkened fits to the PAVO and MIRC data both separately and together. For the combined fit we fitted a common angular diameter, but applied a different linear limb-darkening coefficient to the MIRC and PAVO data. Provided the star has a compact atmosphere, the limb-darkened angular diameter should be independent of wavelength. We also fitted uniform-disc models separately. Uniform-disc diameters are wavelength dependent due to the effects of limb darkening.
The fitted angular diameters are given in Table \[tab3\], along with the radius, mass and effective temperature (see Sec. \[4\_1\]). We note that our diameter ($\theta_\mathrm{LD}=0.753\pm0.009$mas) is consistent with that obtained recently by @Ligi12 [$\theta_\mathrm{LD}=0.760\pm0.003$mas], as well as the estimation by @vanBelle08 [$\theta_\mathrm{LD}=0.760\pm0.021$mas]. The values found by @Boyajian12a [$\theta_\mathrm{UD}=0.845\pm0.015$mas and $\theta_\mathrm{LD}=0.861\pm0.015$mas] are inconsistent with our data. Operating at higher spatial frequencies, PAVO is better able to resolve $\theta$ Cyg than Classic. With the lower resolution of Classic, calibration errors have greater impact and this could explain the discrepancy.
We note that the uncertainty in the PAVO diameter is dominated by our adopted uncertainties in the limb-darkening coefficient rather than measurement uncertainties. Uncertainties in the calibrator sizes and wavelength scale also make significant contributions to the overall error budget. For comparison, ignoring these uncertainties in fitting the PAVO data yields a fractional uncertainty of only 0.2% compared to an uncertainty of 1.2% derived from our Monte Carlo simulations. This illustrates the importance of taking into account these additional uncertainties.
To determine the mass we have used the revised scaling relation for $\Delta\nu$ proposed by @White11a, which corrects for a deviation from the original scaling relation that is dependent upon effective temperature. Without this correction we obtain a significantly lower mass for $\theta$ Cyg (1.27M$_\odot$) that is not consistent with the value of 1.39$^{+0.02}_{-0.01}$M$_\odot$ obtained from isochrones by @Casagrande11 in their re-analysis of the Geneva-Copenhagen Survey [@Nordstrom04; @Holmberg07; @Holmberg09].
For calculating the effective temperature we have used the bolometric flux determined from spectrophotometry by @Boyajian13 (see Table \[tab0\]). In addition to the formal errors quoted by @Boyajian13, we include an additional 1% uncertainty accounting for systematics present in the absolute flux calibration of photometric data [see discusion in @Bessell12]. Our measured temperature for $\theta$ Cyg (6749$\pm$44K) is in excellent agreement with the values determined by @Erspamer03 from spectroscopy (6745$\pm$150K) and @Ligi12 from interferometry (6767$\pm$87K).
Figure \[fig2\] shows a histogram of the MIRC closure phase measurements. All values are consistent with zero, indicating the source has a point-symmetric intensity distribution. @Ligi12 reported that the scatter in their measurements of $\theta$ Cyg with the VEGA beam combiner was higher than expected, leaving open the possibility of stellar variations or a close companion. As they noted, [*Kepler*]{} observations would have detected any large stellar pulsations, and so this explanation for their result is unsatisfactory.
Our MIRC closure phase measurements also appear to rule out a new close companion. Following the method used by @Kraus08 we estimated the detection threshold for a close companion as a function of separation. Briefly, this method involves Monte Carlo simulations of data sets with the same ($u$, $v$)-sampling and error properties of our MIRC observations and finding the best-fit contrast ratio within a large grid of positions and separations. The 99.9% upper limit to companion brightness within a series of annuli was determined as the contrast ratio for which 99.9% of simulations had no companion brighter than this limit anywhere within the annulus. We find that a potential close companion with a separation between 10–20mas ($\sim$0.2–0.4AU) must be at least 4.68mag fainter in $H$-band than $\theta$ Cyg to escape detection in our observations. For separations between 20–40mas ($\sim$0.4–0.7AU), the companion must be at least 3.44mag fainter.
16 Cyg A and B
--------------
![Squared visibility versus spatial frequency for 16 Cyg A (top) and B (bottom) for PAVO (blue circles) and Classic (black diamonds) data. The red lines show the fitted limb-darkened model to the combined data. The solid lines use the limb-darkening coefficients in $R$-band (PAVO) while the dashed line is for $H$-band (Classic). Note that the error bars for each star have been scaled so that the reduced $\chi^2$ equals unity.[]{data-label="fig3"}](16CygA_PAVO_Classic "fig:") ![Squared visibility versus spatial frequency for 16 Cyg A (top) and B (bottom) for PAVO (blue circles) and Classic (black diamonds) data. The red lines show the fitted limb-darkened model to the combined data. The solid lines use the limb-darkening coefficients in $R$-band (PAVO) while the dashed line is for $H$-band (Classic). Note that the error bars for each star have been scaled so that the reduced $\chi^2$ equals unity.[]{data-label="fig3"}](16CygB_PAVO_Classic "fig:")
Figure \[fig3\] shows the calibrated squared-visibility PAVO and Classic measurements as a function of spatial frequency for 16 Cyg A and B. As for $\theta$ Cyg, we provide the fitted uniform-disc and limb-darkened diameters, along with derived properties in Table \[tab3\]. Inclusion of the Classic data in the fit does not significantly change the measured diameters.
The diameters as measured individually with PAVO and Classic agree within the uncertainties. Our 16 Cyg B measurement is 1.1$\sigma$ larger than the diameter measured by @Baines08 [$\theta_\mathrm{LD}$=0.426$\pm$0.056mas], although their estimate from spectral energy distribution fitting ($\theta_\mathrm{LD}$=0.494$\pm$0.019mas) is in excellent agreement with our final value.
The larger uncertainties in our Classic diameters compared to the values reported by @Boyajian13 arise largely from the inclusion of additional uncertainty in the linear limb-darkening coefficient. It is also worth noting that whereas we determined the limb-darkening coefficient from the model grid of @Claret11, @Boyajian13 used the values from @Claret00, leading to slightly different values being used.
We are able to compare our measured radii and masses with those obtained by preliminary asteroseismic modelling by @Metcalfe12. Several different approaches were taken to model the pair, using measured oscillation frequencies and spectroscopic values as constraints. Methods varied with different stellar evolutionary and pulsation codes, nuclear reaction rates, opacities, and treatments of diffusion and convection used.
As mentioned in Section \[16Cyglit\], the best fit model of each method fell into one of two families. A low radius, low mass family favoured $R$=1.24$\mathrm{R_\odot}$, $M$=1.10$\mathrm{M_\odot}$ for 16 Cyg A and $R$=1.12$\mathrm{R_\odot}$, $M$=1.05$\mathrm{M_\odot}$ for 16 Cyg B. The high radius, high mass family favoured $R$=1.26$\mathrm{R_\odot}$, $M$=1.14$\mathrm{M_\odot}$ and $R$=1.14$\mathrm{R_\odot}$, $M$=1.09$\mathrm{M_\odot}$, respectively.
Comparison with our results in Table \[tab3\] shows a preference for the low radius, low mass family, although the high radius, high mass family cannot be completely discounted, particularly for 16 Cyg B. This brief comparison suggests that, in conjunction with more [*Kepler*]{} data that is becoming available, our interferometric results will help to significantly constrain stellar models.
When calculating the effective temperature we have again used the bolometric flux determined by @Boyajian13, once again adopting an additional 1% uncertainty to account for systematics in the absolute flux calibration. As for $\theta$ Cyg, our measured temperatures (5839$\pm$42K and 5809$\pm$39K for 16 Cyg A and B, respectively) agree well with the spectroscopically determined values [5825$\pm$50K and 5750$\pm$50K; @Ramirez09].
Comparison with asteroseismic scaling relations
-----------------------------------------------
In addition to the scaling relation for the large frequency separation, $\Delta\nu$, given in Equation \[eqn5\], there is also a widely used scaling relation for the frequency of maximum power, $\nu_\mathrm{max}$ [@Brown91; @Kjeldsen95]: $$\frac{\nu_\mathrm{max}}{\nu_\mathrm{max,\odot}}=\left(\frac{M}{\mathrm{M}_\odot}\right)\left(\frac{R}{\mathrm{R}_\odot}\right)^{-2}\left(\frac{T_\mathrm{eff}}{\mathrm{T_{eff,\odot}}}\right)^{-1/2}.\label{eqn6}$$ Equations \[eqn5\] and \[eqn6\] may be simultaneously solved for mass and radius:
$$\frac{M}{\mathrm{M}_\odot}=\left(\frac{\nu_\mathrm{max}}{\nu_\mathrm{max,\odot}}\right)^{3}\left(\frac{\Delta\nu}{\Delta\nu_\odot}\right)^{-4}\left(\frac{T_\mathrm{eff}}{\mathrm{T_{eff,\odot}}}\right)^{3/2}\label{eqn7}$$
and $$\frac{R}{\mathrm{R}_\odot}=\left(\frac{\nu_\mathrm{max}}{\nu_\mathrm{max,\odot}}\right)\left(\frac{\Delta\nu}{\Delta\nu_\odot}\right)^{-2}\left(\frac{T_\mathrm{eff}}{\mathrm{T_{eff,\odot}}}\right)^{1/2}.\label{eqn8}$$
Provided the effective temperature is known, the stellar mass and radius may be estimated directly from the asteroseismic parameters $\Delta\nu$ and $\nu_\mathrm{max}$. This is sometimes referred to as the ‘direct method’ [@Kallinger10d; @Chaplin11a; @SilvaAguirre11] in contrast to determining mass, radius and other parameters via stellar modelling [@Stello09b; @Basu10; @Kallinger10b; @Gai11].
The scaling relation for $\Delta\nu$, which we have used to derive the masses in Table \[tab3\], is better understood theoretically with tests of its validity in models finding the relation holds to within 5% [@Stello09; @White11a]. The $\nu_\mathrm{max}$ scaling relation relies on the argument that $\nu_\mathrm{max}$ should scale with the acoustic cutoff frequency [@Brown91], although the underlying physical reason for this relationship has not been clear. Only recently has the theoretical framework behind this result begun to be developed [@Belkacem11]. Understanding the validity of these scaling relations has become particularly important as they are now commonly used to determine radii for a large number of faint [*Kepler*]{} stars, including some stars with detected exoplanet candidates [see, e.g., @Borucki12; @Huber13]. We are able to test the validity of the asteroseismic scaling relations by comparing our interferometric radii with independently determined asteroseismic radii calculated using Equation \[eqn8\]. To ensure the asteroseismic radii are truly independent of our interferometric radii, in this calculation we use the spectroscopic effective temperatures given in Table \[tab0\].
We have determined the global asteroseismic properties, $\Delta\nu$ and $\nu_\mathrm{max}$, of 16 Cyg A and B using the automated analysis pipeline by @Huber09, which has been shown to agree well with other methods [@Hekker11; @Verner11]. These values are given in Table \[tab4\], along with the radii derived from the scaling relations, Equations \[eqn5\] and \[eqn6\]. We use solar values of $\Delta\nu_\odot$=135.1$\mu$Hz and $\nu_\mathrm{max,\odot}$=3090$\mu$Hz.
We do not consider $\theta$ Cyg here because the width of the oscillation envelope is very broad, which makes $\nu_\mathrm{max}$ ambiguous. This, along with large mode linewidths [@Chaplin09; @Baudin11; @Appourchaux12a; @Corsaro12], appears to be a feature of oscillations in F stars. Observations of the F subgiant Procyon showed a similarly broad envelope [@Arentoft08].
Figure \[fig4\] shows the remarkable agreement between the interferometric and asteroseismic radii. In addition to 16 Cyg A and B we also include five stars for which @Huber12b determined interferometric and asteroseismic radii using the same method (see their Figure 7). The agreement for 16 Cyg A and B is within 1$\sigma$ and at a $\sim$2% level, which makes this the most precise independent empirical test of asteroseismic scaling relations yet. However, further studies are still needed, particularly of stars that are significantly different from the Sun, to robustly test the validity of the scaling relations.
![Comparison of stellar radii measured using interferometry and calculated using asteroseismic scaling relations. Black triangles show stars measured by @Huber12b, while blue diamonds show 16 Cyg A and B.[]{data-label="fig4"}](fig07_update)
16 Cyg A 16 Cyg B
------------------------------ ----------------- -----------------
$\Delta\nu$ ($\mu$Hz) 103.5$\pm$0.1 117.0$\pm$0.1
$\nu_\mathrm{max}$ ($\mu$Hz) 2201$\pm$20 2552$\pm$20
$R$ (R$_\odot$) 1.218$\pm$0.012 1.098$\pm$0.010
\[tab4\]
: Asteroseismic Properties and Radii of 16 Cyg A and B
Conclusions
===========
We have used long-baseline interferometry to measure angular diameters for $\theta$ Cyg and 16 Cyg A and B. All three stars have been observed by the [*Kepler Mission*]{} and exhibit solar-like oscillations, allowing for detailed study of their internal structure.
For $\theta$ Cyg we find a limb-darkened angular diameter of $\theta_\mathrm{LD}=0.753\pm0.009$ mas, which, combined with the Hipparcos parallax, gives a linear radius of $R$=1.48$\pm$0.02$\mathrm{R_\odot}$. When determining the mass (1.37$\pm$0.04$\mathrm{M_\odot}$) from the interferometric radius and large frequency separation, $\Delta\nu$, we find that it is necessary to use the revised scaling relation for $\Delta\nu$ suggested by @White11a. This revision takes into account a deviation from the standard scaling relation in stars of higher temperature, without which the determined mass would be significantly lower (1.27$\mathrm{M_\odot}$) than expected from fitting to isochrones.
Closure phase measurements of $\theta$ Cyg reveal the star to be point symmetric, consistent with being a single star. This rules out the possibility of all but a very low luminosity close companion, which had previously been suggested.
For 16 Cyg A and B we have found limb-darkened angular diameters of $\theta_\mathrm{LD}=0.539\pm0.007$ and $\theta_\mathrm{LD}=0.490\pm0.006$, and linear radii of $R$=1.22$\pm$0.02$\mathrm{R_\odot}$ and $R$=1.12$\pm$0.02$\mathrm{R_\odot}$, respectively. Comparing these radii with those derived from the asteroseismic scaling relations shows good agreement at a $\sim$2% level.
Our measurements of near-model-independent masses, radii and effective temperatures will provide strong constraints when modelling these stars.
Acknowledgments {#acknowledgments .unnumbered}
===============
The CHARA Array is funded by the National Science Foundation through NSF grant AST-0606958, by Georgia State University through the College of Arts and Sciences, and by the W.M. Keck Foundation. We acknowledge the support of the Australian Research Council. Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foundation. We acknowledge the [*Kepler*]{} Science Team and all those who have contributed to the [*Kepler Mission*]{}. Funding for the [*Kepler Mission*]{} is provided by NASA’s Science Mission Directorate. T.R.W. is supported by an Australian Postgraduate Award, a University of Sydney Merit Award, an Australian Astronomical Observatory PhD Scholarship and a Denison Merit Award. DH is supported by an appointment to the NASA Postdoctoral Program at Ames Research Center, administered by Oak Ridge Associated Universities through a contract with NASA.
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\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: @Erspamer03, high-resolution spectroscopy
[^3]: @Ramirez09, high-resolution spectroscopy
[^4]: @vanLeeuwen07, revised Hipparcos parallax
[^5]: @Boyajian13, spectrophotometry
[^6]: @Casagrande11, fit to BaSTI isochrones
[^7]: @Metcalfe12, asteroseismology
[^8]: The baselines used have the following lengths:\
W1W2, 107.92m; E2W2, 156.27m; S2W2, 177.45m; S1W2, 210.97m;\
S2E2, 248.13m; E2W1, 251.34m.
[^9]: Refer to Table \[tab2\] for details of the calibrators used.
[^10]: For this star we instead use the calibrated diameter,\
$\theta_\mathrm{UD}=0.274\pm0.008$ (see text).
|
---
abstract: |
Let $G$ be a split real connected Lie group with finite center. In the first part of the paper we define and study formal elementary spherical functions. They are formal power series analogues of elementary spherical functions on $G$ in which the role of the quasi-simple admissible $G$-representations is replaced by Verma modules. For generic highest weight we express the formal elementary spherical functions in terms of Harish-Chandra series and integrate them to spherical functions on the regular part of $G$. We show that they produce eigenstates for spin versions of quantum hyperbolic Calogero-Moser systems.
In the second part of the paper we define and study special subclasses of global and formal elementary spherical functions, which we call global and formal $N$-point spherical functions. Formal $N$-point spherical functions arise as limits of correlation functions for boundary Wess-Zumino-Witten conformal field theory on the cylinder when the position variables tend to infinity. We construct global $N$-point spherical functions in terms of compositions of equivariant differential intertwiners associated with principal series representations, and express them in terms of Eisenstein integrals. We show that the eigenstates of the spin quantum Calogero-Moser system associated to $N$-point spherical functions are also common eigenfunctions of a commuting family of first-order differential operators, which we call asymptotic boundary Knizhnik-Zamolodchikov-Bernard operators. These operators are explicitly given in terms of folded classical dynamical $r$-matrices and associated dynamical $k$-matrices.
address:
- 'J.S.: KdV Institute for Mathematics, University of Amsterdam, Science Park 105-107, 1098 XG Amsterdam, The Netherlands.'
- 'N.R.: Department of Mathematics, University of California, Berkeley, CA 94720, USA & ITMO University, Kronverskii Ave. 49, Saint Petersburg, 197101, Russia & KdV Institute for Mathematics, University of Amsterdam, Science Park 105-107, 1098 XG Amsterdam, The Netherlands'
author:
- 'J.V. Stokman & N. Reshetikhin'
title: '$N$-point spherical functions and asymptotic boundary KZB equations'
---
Introduction
============
Preliminaries on $\sigma$-spherical functions
---------------------------------------------
Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\theta\in\textup{Aut}(\mathfrak{g})$ a Chevalley involution, $\mathfrak{k}=\mathfrak{g}^\theta$ its fix-point subalgebra, $\mathfrak{h}$ a Cartan subalgebra contained in the $-1$-eigenspace of $\theta$, $\mathfrak{b}$ a Borel subalgebra containing $\mathfrak{h}$, and $\mathfrak{n}=[\mathfrak{b},\mathfrak{b}]$. Let $R$ and $R^+$ be the associated set of roots and positive roots respectively, and $\mathfrak{g}_0$ a split real form of $\mathfrak{g}$ compatible with $\mathfrak{h}$ and $\theta$. Then $\theta_0:=\theta|_{\mathfrak{g}_0}$ is a Cartan involution, $\mathfrak{h}_0:=
\mathfrak{h}\cap\mathfrak{g}_0$ is a split real form of $\mathfrak{h}$, and $\mathfrak{k}_0:=\mathfrak{k}\cap\mathfrak{g}_0$, $\mathfrak{n}_0:=\mathfrak{n}\cap\mathfrak{g}_0$ are real forms of $\mathfrak{k}$ and $\mathfrak{n}$ respectively. Let $G$ be a connected Lie group with finite center and Lie algebra $\mathfrak{g}_0$, and $K\subset G$ the connected Lie subgroup with Lie algebra $\mathfrak{k}_0$. Let $N:=\exp(\mathfrak{n}_0)$, $A:=\exp(\mathfrak{h}_0)$ and denote by $A_+$ its positive Weyl chamber. By the Iwasawa decomposition, each $g\in G$ admits a unique decomposition $g=k(g)a(g)n(g)$ with the elements $k(g)\in K$, $a(g)\in A$ and $n(g)\in N$ depending smoothly on $g\in G$.
Let $(\sigma_\ell,V_\ell)$ and $(\sigma_r,V_r)$ be finite dimensional $\mathfrak{k}$-representations and denote by $\sigma:=\sigma_\ell\otimes\sigma_r^*$ the representation map of the resulting $\mathfrak{k}\oplus\mathfrak{k}$-module $V_\ell\otimes V_r^*\simeq\textup{Hom}(V_r,V_\ell)$. If it integrates to a $K\times K$-representation, then the corresponding representation map will still be denoted by $\sigma$. If this is the case, denote by $C_\sigma^\infty(G)$ the space of [*$\sigma$-spherical functions on $G$*]{}, which consists of the smooth functions $f: G\rightarrow V_\ell\otimes V_r^*$ satisfying $$\label{transfospherical}
f(k_\ell gk_r^{-1})=\sigma(k_\ell,k_r)f(g)\qquad (k_\ell,k_r\in K,\, g\in G).$$ Then $f\in C_\sigma^\infty(G)$ is called an [*elementary $\sigma$-spherical function*]{} if it is of the form $$f_{\mathcal{H}}^{\phi_\ell,\phi_r}(g):=\phi_\ell\circ\pi(g)\circ\phi_r$$ for some quasi-simple admissible representation $(\pi,\mathcal{H})$ of $G$, where $\phi_\ell\in\textup{Hom}_K(\mathcal{H},V_\ell)$ and $\phi_r\in\textup{Hom}_K(V_r,\mathcal{H})$ are two $K$-intertwiners. The elementary $\sigma$-spherical functions $f_{\mathcal{H}}^{\phi_\ell,\phi_r}$ are common eigenfunctions for the action of the center $Z(\mathfrak{g})$ of $U(\mathfrak{g})$ as biinvariant differential operators on $G$, with the eigenvalues given by the central character of $\mathcal{H}$ (see, e.g., [@HC; @CM; @W], as well as Section \[S3\]).
For $\lambda\in\mathfrak{h}^*$ let $\pi_\lambda: G\rightarrow\textup{GL}(\mathcal{H}_\lambda)$ be the quasi-simple, admissible representation of $G$ obtained by normalised induction from the multiplicative character $\eta_{\lambda}: AN\rightarrow \mathbb{C}^*$, $\eta_\lambda(an):=e^{\lambda(\log(a))}$ of the closed subgroup $AN\subset G$ ($a\in A$, $n\in N$). It is isomorphic to a finite direct sum of principal series representations. The associated elementary $\sigma$-spherical functions $f_{\mathcal{H}_\lambda}^{\phi_\ell,\phi_r}$ admits an integral representation $$\label{link1}
f_{\mathcal{H}_\lambda}^{\phi_\ell,\phi_r}(g)=E_\lambda^\sigma(g)T_\lambda^{\phi_\ell,\phi_r}$$ with $T_\lambda^{\phi_\ell,\phi_r}\in\textup{Hom}(V_r,V_\ell)$ an explicit rank one operator and $E_\lambda^\sigma(g)$ the Eisenstein integral $$E_\lambda^\sigma(g)=\int_Ka(g^{-1}x)^{-\lambda-\rho}\sigma(x,k(g^{-1}x))dx,$$ where $dx$ is the normalised Haar measure on $K$, $\rho=\frac{1}{2}\sum_{\alpha\in R^+}\alpha$, and $a^\mu:=e^{\mu(\log a)}$ for $\mu\in\mathfrak{h}^*$ and $a\in A$ (see Proposition \[relEisPrin\]). Eisenstein integrals play an important role in Harish-Chandra’s theory of harmonic analysis on $G$ (see, e.g., [@HCe; @HCI; @HCII; @W]).
Global description of the main results
--------------------------------------
In the first part of the paper we study formal elementary $\sigma$-spherical functions, which are formal analogues of $f_{\mathcal{H}_\lambda}^{\phi_\ell,\phi_r}|_{A_+}$ in which the role of $\mathcal{H}_\lambda$ is taken over by the Verma module $M_{\lambda}$ of highest weight $\lambda\in\mathfrak{h}^*$ with respect to the Borel subalgebra $\mathfrak{b}$ (or better yet, by $M_{\lambda-\rho}$, in order to have matching central characters). Their construction is as follows.
For $\mu\in\mathfrak{h}^*$ let $\xi_\mu$ be the multiplicative character $\xi_\mu(a):=a^\mu$ on $A$. For a weight $\mu$ of $M_\lambda$ (i.e., $\mu\leq\lambda$ with respect to the dominance order $\leq$ on $\mathfrak{h}^*$), write $M_\lambda[\mu]$ for the corresponding weight space in $M_\lambda$. Write $\overline{M}_\lambda$ for the $\mathfrak{n}^-$-completion of $M_\lambda$, with $\mathfrak{n}^-:=\theta(\mathfrak{n})$ the nilpotent subalgebra of $\mathfrak{g}$ opposite to $\mathfrak{b}$. Fix $\mathfrak{k}$-intertwiners $\phi_\ell\in\textup{Hom}_{\mathfrak{k}}(M_\lambda,V_\ell)$ and $\phi_r\in\textup{Hom}_{\mathfrak{k}}(V_r,\overline{M}_\lambda)$. For $\mu\in\mathfrak{h}^*$ a weight of $M_\lambda$ let $\phi_\ell^\mu\in\textup{Hom}(M_\lambda[\mu],V_\ell)$ and $\phi_r^\mu\in\textup{Hom}(V_r,M_\lambda[\mu])$ be the weight components of $\phi_\ell$ and $\phi_r$. The corresponding [*formal elementary $\sigma$-spherical function associated with $M_\lambda$*]{} is the formal series $$F_{M_\lambda}^{\phi_\ell,\phi_r}:=\sum_{\mu\leq\lambda}\bigl(\phi_\ell^\mu\circ\phi_r^\mu)\xi_\mu$$ on $A_+$.
We show that formal elementary $\sigma$-spherical functions are common (formal) eigenfunctions of commuting differential operators on $A_+$. These differential operators are Harish-Chandra’s $\sigma$-radial components $\widehat{\Pi}^\sigma(z)$ of $z\in Z(\mathfrak{g})$, viewed as biinvariant differential operators on $G$. The eigenvalues are given by the values $\zeta_\lambda(z)$ of the central character $\zeta_\lambda$ of $M_\lambda$ at $z\in Z(\mathfrak{g})$. As a consequence, for generic highest weight $\lambda\in\mathfrak{h}_{\textup{HC}}^*$ , the formal elementary $\sigma$-spherical function $F_{M_\lambda}^{\phi_\ell,\phi_r}$ can be related to the $\sigma$-Harish-Chandra series, which is the unique asymptotically free $\textup{End}(\textup{Hom}(V_r,V_\ell))$-valued formal eigenfunction $$\Phi^\sigma_\lambda=\sum_{\mu\leq\lambda}\Gamma_{\lambda-\mu}^\sigma(\lambda)\xi_\mu$$ of the radial component $\widehat{\Pi}^\sigma(\Omega)$ of the action of the quadratic Casimir $\Omega\in Z(\mathfrak{g})$ with eigenvalue $\zeta_\lambda(\Omega)$ and with leading coefficient $\Gamma_0^\sigma(\lambda)=\textup{id}_{V_\ell\otimes V_r^*}$ (see Subsection \[S5\]): $$\label{link2}
F_{M_\lambda}^{\phi_\ell,\phi_r}=\Phi_\lambda^\sigma(\cdot)(\phi_\ell^\lambda\circ\phi_r^\lambda)$$ (see Theorem \[mainTHMF\][**c**]{})[^1]. Formula allows one to control the analytic properties of the formal elementary $\sigma$-spherical functions. In particular it implies that $F_{M_\lambda}^{\phi_\ell,\phi_r}$ for $\lambda\in\mathfrak{h}_{\textup{HC}}^*$ defines a $\textup{Hom}(V_r,V_\ell)$-valued analytic function on $A_+$ that extends to a $\sigma$-spherical function on the dense open subset $G_{\textup{reg}}:=KA_+K$ of regular elements in $G$ if $\sigma$ integrates to a $K\times K$-representation and $\phi_\ell^\lambda\circ\phi_r^\lambda\in\textup{Hom}_M(V_r,V_\ell)$, with $M:=Z_K(A)$ the centraliser of $A$ in $K$.
The $\sigma$-Harish-Chandra series plays an important role in the asymptotic analysis of $\sigma$-spherical functions through the explicit expansion of the Eisenstein integral in Harish-Chandra series, see, e.g., [@HC; @HCI; @HCII; @W]. Another interesting recent application of $\sigma$-Harish-Chandra series is in the description of four-point spin conformal blocks in Euclidean conformal field theories within the conformal bootstrap program (see [@SSI; @IS; @ILLS] and references therein).
For $\mathfrak{g}=\mathfrak{sl}_2(\mathbb{C})$ the weight components of $\mathfrak{k}$-intertwiners $\phi_\ell$ and $\phi_r$ are Meixner-Pollaczek polynomials, and the $\sigma$-Harish-Chandra series can be expressed in terms of Gauss’ hypergeometric series ${}_2F_1$. Formula then provides a representation theoretic derivation of the formula expressing the Poisson kernel of Meixner-Pollaczek polynomials as a ${}_2F_1$ (see Subsection \[rankoneSection\]).
Gauged versions of the $\sigma$-radial components of the biinvariant differential operators on $G$ give rise to quantum Hamiltonians of spin[^2]. versions of quantum hyperbolic Calogero-Moser systems. Formula then gives the representation theoretic interpretation of the corresponding asymptotically free eigenstates. The theory for the compact symmetric space associated to $G/K$ yields spin versions of the quantum [*trigonometric*]{} Calogero-Moser system, with eigenstates described by vector-valued multivariable orthogonal polynomials. We will discuss this case in a separate paper.
In the second part of the paper we define and study $N$-point spherical functions and their formal analogues. The construction of $N$-point spherical functions is as follows. Let $(\tau_i,U_i)$ be finite dimensional $G$-representations ($1\leq i\leq N$) and write $\mathbf{U}:=U_1\otimes\cdots\otimes U_N$ for the associated tensor product $G$-representation with diagonal $G$-action. Let $V_\ell$ and $V_r$ be finite dimensional $K$-representations and denote the representation map of the $K\times K$-representation $(V_\ell\otimes\mathbf{U})\otimes V_r^*$ by $\sigma^{(N)}$, where $V_\ell\otimes\mathbf{U}$ is regarded as $K$-representation with respect to the diagonal $K$-action.
[*$N$-point spherical functions*]{} are elementary $\sigma^{(N)}$-spherical functions on $G$ of the form $$f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}:=
f_{\mathcal{H}_{\lambda_N}}^{(\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{D},\phi_r}$$ with
1. $\mathcal{H}_{\underline{\lambda}}:=(\mathcal{H}_{\lambda_0},\ldots,
\mathcal{H}_{\lambda_N})$,
2. $\mathbf{D}: \mathcal{H}_{\lambda_N}^\infty\rightarrow
\mathcal{H}_{\lambda_0}^\infty\otimes\mathbf{U}$ a $G$-equivariant differential operator given as the composition $$\mathbf{D}=(D_1\otimes\textup{id}_{U_2\otimes\cdots\otimes U_N})\cdots
(D_{N-1}\otimes\textup{id}_{U_N})D_N$$ of $G$-equivariant differential operators $D_i: \mathcal{H}_{\lambda_i}^{\infty}
\rightarrow \mathcal{H}_{\lambda_{i-1}}^{\infty}\otimes U_i$ ($i=1,\ldots,N$),
3. $K$-intertwiners $\phi_\ell\in\textup{Hom}_K(\mathcal{H}_{\lambda_0},V_\ell)$ and $\phi_r\in\textup{Hom}_K(V_r,\mathcal{H}_{\lambda_N})$.
The Eisenstein integral representation of $f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}$ is $$f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}(g)=
\int_Kdx\,a(g^{-1}x)^{-\lambda_N-\rho}\Bigl(\sigma_{\ell}(x)\otimes\tau_1(x)\otimes
\cdots\otimes\tau_N(x)\otimes\sigma_r^*(k(g^{-1}x))\Bigr)T_{\lambda}^{(\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{D},\phi_r}.$$
Similarly, [*formal $N$-point spherical functions*]{} $F_{M_{\underline{\lambda}}}^{\phi_\ell,\mathbf{\Psi},\phi_r}$ are formal elementary $\sigma^{(N)}$-spherical functions of the form $$F_{M_{\underline{\lambda}}}^{\phi_\ell,\mathbf{\Psi},\phi_r}:=
F_{M_{\lambda_N}}^{(\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{\Psi},\phi_r}$$ with
1. $M_{\underline{\lambda}}=(M_{\lambda_0},\ldots,M_{\lambda_N})$,
2. $\mathbf{\Psi}: M_{\lambda_N}\rightarrow
M_{\lambda_0}\otimes\mathbf{U}$ a $\mathfrak{g}$-intertwiner given as the composition $$\mathbf{\Psi}=(\Psi_1\otimes\textup{id}_{U_2\otimes\cdots\otimes U_N})\cdots
(\Psi_{N-1}\otimes\textup{id}_{U_N})\Psi_N$$ of $N$ $\mathfrak{g}$-intertwiners $\Psi_i: M_{\lambda_i}
\rightarrow M_{\lambda_{i-1}}\otimes U_i$,
3. $\mathfrak{k}$-intertwiners $\phi_\ell\in\textup{Hom}_{\mathfrak{k}}(M_{\lambda_0},V_\ell)$ and $\phi_r\in\textup{Hom}_{\mathfrak{k}}(V_r,\overline{M}_{\lambda_N})$.
In the formal setup, $(\sigma_\ell,V_\ell)$ and $(\sigma_r,V_r)$ can be taken to be finite dimensional $\mathfrak{k}$-modules, and $(\tau_i,U_i)$ finite dimensional $\mathfrak{g}$-modules ($1\leq i\leq N$). For generic $\lambda_N$, the formal $N$-point spherical function admits an expression in terms of the $\sigma^{(N)}$-Harish-Chandra series $\Phi_{\lambda_N}^{\sigma^{(N)}}$ and the highest weight components of the $\mathfrak{k}$-intertwiners $(\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{\Psi}$ and $\phi_r$ by .
The analogous construction of formal $N$-point spherical functions in the affine setup is expected to give correlation functions for boundary Wess-Zumino-Witten (WZW) conformal field theory on an elliptic curve with conformally invariant boundary conditions. From this perspective, the $\mathfrak{g}$-intertwiners $\Psi_i$ are the analogues of vertex operators, and the $\mathfrak{k}$-intertwiners $\phi_\ell$ and $\phi_r$ the analogues of boundary vertex operators.
This analogy with WZW conformal field theory predicts that the $N$-point spherical functions satisfy additional first order differential equations, obtained from trigonometric Knizhnik-Zamolodchikov-Bernard (KZB) type equations in the limit when the points are escaping to infinity and only the differential equations in the dynamical variables remain. We will derive such first order differential equations for the $N$-point spherical functions directly, and call them [*asymptotic boundary Knizhnik-Zamolodchikov-Bernard (KZB) equations*]{}. They are explicitly given in terms of $\theta$-folded classical dynamical trigonometric $r$-matrices and associated classical dynamical trigonometric $k$-matrices. Boundary KZB equations with spectral parameters will be discussed in a separate paper (for affine $\mathfrak{sl}_2$ Kolb [@K] has already derived the associated KZB-heat equation).
Gauging to the context of the spin versions of quantum Calogero-Moser systems, the first order differential equations will be rewritten as eigenvalue equations for commuting first order differential operators $\mathcal{D}_1,\ldots,\mathcal{D}_N$ called [*asymptotic boundary KZB operators*]{}. We show that the asymptotic boundary KZB operators also commute with the quantum Hamiltonians of the spin quantum Calogero-Moser system, leading to the interpretation of the quantum Calogero-Moser system as a quantum spin Calogero-Moser chain with $N+1$ quadratic quantum Hamiltonians.[^3] We describe the second-and first-order differential operators explicitly in the following subsection.
Spin quantum Calogero-Moser systems
-----------------------------------
The construction of the spin quantum Calogero-Moser systems can be done “universally”, with $U(\mathfrak{k})^{\otimes 2}$ taking over the role of the internal spin space $V_\ell\otimes V_r^*$. The key point is the fact that Harish-Chandra’s radial component map is universal, i.e., it produces for $z\in Z(\mathfrak{g})$ mutually commuting $U(\mathfrak{k})^{\otimes 2}$-valued commuting differential operators $\widehat{\Pi}(z)$ on $A$ such that $\sigma(\widehat{\Pi}(z))=\widehat{\Pi}^\sigma(z)$ for all $\mathfrak{k}\oplus\mathfrak{k}$-representations $\sigma$. In order to describe the resulting universal Schr[ö]{}dinger operator explicitly we need a bit more notations.
The Killing form $K_{\mathfrak{g}_0}$ on $\mathfrak{g}_0$ restricts to a scalar product on $\mathfrak{h}_0$, giving $A=\exp(\mathfrak{h}_0)$ the structure of a Riemannian manifold. Let $\Delta$ be the corresponding Laplace-Beltrami operator on $A$. Denote by $\mathfrak{g}_{0,\alpha}$ the root space in $\mathfrak{g}_0$ associated to the root $\alpha\in R$. Let $e_\alpha\in\mathfrak{g}_{0,\alpha}$ ($\alpha\in R$) such that $\theta_0(e_\alpha)=-e_{-\alpha}$ and $[e_\alpha,e_{-\alpha}]=t_\alpha$, where $t_\alpha\in\mathfrak{h}_0$ is the unique element such that $K_{\mathfrak{g}_0}(t_\alpha,h)=\alpha(h)$ for all $h\in\mathfrak{h}_0$. The dual space $\mathfrak{h}_0^*$ inherits the scalar product of $\mathfrak{h}_0$ via the Killing form. We extend it to a complex bilinear form $(\cdot,\cdot)$ on $\mathfrak{h}$. Consider the elements $y_\alpha:=e_\alpha-e_{-\alpha}\in\mathfrak{k}_0$ for $\alpha\in R$. Note that $\{y_\alpha\}_{\alpha\in R^+}$ is a linear basis of $\mathfrak{k}_0$, and $y_{-\alpha}=-y_\alpha$.
Set $\delta:=\xi_\rho\prod_{\alpha\in R^+}(1-\xi_{-2\alpha})^{\frac{1}{2}}$, considered as analytic function on $A_+$, and consider the gauged commuting differential operators $$H_z:=\delta\circ\widehat{\Pi}(z)\circ\delta^{-1}
\qquad (z\in Z(\mathfrak{g})),$$ which provide the quantum Hamiltonians of the spin quantum hyperbolic Calogero-Moser system. The universal Schr[ö]{}dinger operator is defined to be $$\mathbf{H}:=-\frac{1}{2}(H_\Omega+\|\rho\|^2).$$ We show in Proposition \[qH\] that $\mathbf{H}$ is explicitly given by $$\mathbf{H}=-\frac{1}{2}\Delta+V$$ with the $U(\mathfrak{k})^{\otimes 2}$-valued potential $V$ given by $$V=-\frac{1}{2}\sum_{\alpha\in R}\frac{1}{(\xi_\alpha-\xi_{-\alpha})^2}
\Bigl(\frac{\|\alpha\|^2}{2}+\prod_{\epsilon\in\{\pm 1\}}(y_\alpha\otimes 1+\xi_{\epsilon\alpha}
(1\otimes y_\alpha)\Bigr).$$
The results from the previous subsections now lead to the following eigenstates. Suppose that $\sigma_\ell$ and $\sigma_r$ integrate to $K$-representations. Define the $(V_\ell\otimes V_r^*)^M$-valued smooth function $\mathbf{f}_\lambda^{\phi_\ell,\phi_r}$ on $A_+$ by $$\mathbf{f}_\lambda^{\phi_\ell,\phi_r}(a):=\delta(a)f_{\mathcal{H}_\lambda}^{\phi_\ell,\phi_r}(a)
\qquad (a\in A_+),$$ where $(V_\ell\otimes V_r^*)^M\simeq\textup{Hom}_K(V_r,V_\ell)$ is the space of $M$-invariant vectors in $V_\ell\otimes V_r^*$ with respect to the diagonal $M$-action.
For arbitrary finite dimensional $\mathfrak{k}$-representations $\sigma_\ell$ and $\sigma_r$, define the formal $V_\ell\otimes V_r^*$-valued power series $\mathbf{F}_\lambda^{\phi_\ell,\phi_r}$ on $A_+$ by $$\mathbf{F}_\lambda^{\phi_\ell,\phi_r}:=\delta F_{M_{\lambda-\rho}}^{\phi_\ell,\phi_r},$$ where $\delta$ is expanded as scalar valued power series on $A_+$ with leading exponent $\rho$ (in particular, $\mathbf{F}_\lambda^{\phi_\ell,\phi_r}$ has leading exponent $\lambda$). It defines a $V_\ell\otimes V_r^*$-valued analytic function on $A_+$ if $\lambda+\rho\in\mathfrak{h}_{\textup{HC}}^*$. If $V_\ell$ and $V_r$ integrate to $K$-representations, then $\mathbf{F}_\lambda^{\phi_\ell,\phi_r}$ is $(V_\ell\otimes V_r^*)^M$-valued iff $\phi_\ell^{\lambda-\rho}\circ\phi_r^{\lambda-\rho}\in\textup{Hom}_M(V_r,V_\ell)$. The normalised global and formal elementary $\sigma$-spherical functions $\mathbf{f}_\lambda^{\phi_\ell,\phi_r}$ and $\mathbf{F}_\lambda^{\phi_\ell,\phi_r}$ are eigenfunctions of the quantum Hamiltonians of the quantum $\sigma$-spin Calogero-Moser system, $$\label{ee}
H_z(f_\lambda)=\zeta_{\lambda-\rho}(z)f_\lambda\qquad \forall\, z\in Z(\mathfrak{g}),$$ where $f_\lambda=\mathbf{f}_\lambda^{\phi_\ell,\phi_r}$ and $f_\lambda=\mathbf{F}_\lambda^{\phi_\ell,\phi_r}$. In particular, $\mathbf{H}(f_\lambda)=-\frac{(\lambda,\lambda)}{2}f_\lambda$.
The equations for $f_\lambda=\mathbf{F}_\lambda^{\phi_\ell,\phi_r}$ should initially be viewed as eigenvalue equations for formal $V_\ell\otimes V_r^*$-valued power series on $A_+$ with leading exponent $\lambda$. If $\lambda+\rho\in\mathfrak{h}_{\textup{HC}}^*$ then the formal eigenfunction $\mathbf{F}_\lambda^{\phi_\ell,\phi_r}$ is a normalised $\sigma$-Harish-Chandra series, and the eigenvalue equations are valid as $V_\ell\otimes V_r^*$-valued analytic functions on $A_+$.
The eigenstates $\mathbf{f}_\lambda^{\phi_\ell,\phi_r}$ serve as wave functions for the $\sigma$-spin quantum Calogero-Moser system, while $\mathbf{F}_{w\lambda}^{\phi_\ell,\phi_r}$ ($w\in W$), with $W$ the Weyl group of $R$, play the role of the asymptotically free wave functions with respect to the fundamental Weyl chamber $A_+$. The celebrated Harish-Chandra $c$-function expansion of $\mathbf{f}_\lambda^{\phi_\ell,\phi_r}$ in $\sigma$-Harish-Chandra series provides the solution of the associated asymptotic Bethe ansatz.
Some special cases of this representation theoretic construction of spin quantum Calogero-Moser systems have been described before; see [@OP] for the case when $\sigma_\ell$ and $\sigma_r$ are the trivial representation, see [@HS Chpt. 5] for the case when $\mathfrak{g}=\mathfrak{sp}_{r}(\mathbb{C})$ and $\sigma_\ell=\sigma_r$ is one-dimensional (other natural special cases will be discussed in Subsection \[vvCM\]). For certain non-split symmetric pairs, the theory of Etingof, Kirillov Jr. and Schiffmann [@EK; @ES] on generalised weighted trace functions and Oblomkov’s [@O] version for Grassmannians fit into this framework. In these cases one of the two boundary representations $\sigma_\ell$ or $\sigma_r$ is one-dimensional. The classical integrable systems underlying the $\sigma$-spin quantum trigonometric Calogero-Moser systems were considered in [@FP; @FP1; @FP2; @Re].
Asymptotic boundary KZB operators
---------------------------------
Set $\underline{\lambda}-\rho:=(\lambda_0-\rho,\ldots,\lambda_N-\rho)$. By the previous subsection, the normalised global and formal $N$-point spherical functions $$\mathbf{f}_{\underline{\lambda}}^{\phi_\ell,\mathbf{D},\phi_r}:=
\delta f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r},
\qquad
\mathbf{F}_{\underline{\lambda}}^{\phi_\ell,\mathbf{\Psi},\phi_r}:=
\delta F_{M_{\underline{\lambda}-\rho}}^{\phi_\ell,\mathbf{\Psi},\phi_r}$$ on $A_+$ of weight $\underline{\lambda}$ are common eigenfunctions of the quantum Hamiltonians of the $\sigma^{(N)}$-spin quantum hyperbolic Calogero-Moser system. In particular we have for both $\mathbf{f}_{\underline{\lambda}}:=\mathbf{f}_{\underline{\lambda}}^{\phi_\ell,\mathbf{D},\phi_r}$ and $\mathbf{f}_{\underline{\lambda}}:=\mathbf{F}_{\underline{\lambda}}^{\phi_\ell,\mathbf{\Psi},\phi_r}$, $$\label{eigen-1}
\mathbf{H}^{(N)}\bigl(\mathbf{f}_{\underline{\lambda}}\bigr)=-\frac{(\lambda_N,\lambda_N)}{2}
\mathbf{f}_{\underline{\lambda}}$$ on $A_+$, where $$\mathbf{H}^{(N)}=-\frac{1}{2}\Delta+V^{(N)}$$ and $V^{(N)}$ is the $U(\mathfrak{k})\otimes U(\mathfrak{k})^{\otimes (N)}\otimes U(\mathfrak{k})$-valued potential $$V^{(N)}=-\frac{1}{2}\sum_{\alpha\in R}\frac{1}{(\xi_\alpha-\xi_{-\alpha})^2}
\Bigl(\frac{\|\alpha\|^2}{2}+\prod_{\epsilon\in\{\pm 1\}}\bigl(y_\alpha\otimes 1^{\otimes N}\otimes 1
+1\otimes\Delta^{(N-1)}(y_\alpha)\otimes 1
+\xi_{\epsilon\alpha}
(1\otimes 1^{\otimes N}\otimes y_\alpha\bigr)\Bigr)$$ with $\Delta^{(N-1)}: U(\mathfrak{k})\rightarrow U(\mathfrak{k})^{\otimes N}$ the $(N-1)$-fold iterated comultiplication of $U(\mathfrak{k})$ (concretely, $\Delta^{(N-1)}(y_\alpha)=\sum_{i=1}^{N}(y_\alpha)_i$ with $(y_\alpha)_i$ the tensor $1\otimes\cdots\otimes 1\otimes y_\alpha\otimes 1\otimes\cdots\otimes 1$ with $y_\alpha$ placed in the $i$th tensor component). We now explicitly describe the additional first order differential equations satisfied by $\mathbf{f}_{\underline{\lambda}}$.
Let $\{x_s\}_{s=1}^r$ be an orthonormal basis of $\mathfrak{h}_0$ and $\partial_{x_s}$ the associated first order differential operator on $A$ (then $\Delta=\sum_{s=1}^r\partial_{x_s}^2$). Write $E$ for the $U(\mathfrak{g})$-valued first order differential operator $$E:=\sum_{s=1}^r\partial_{x_s}\otimes x_s$$ on $A$. Write $A_{\textup{reg}}:=G_{\textup{reg}}\cap A$ for the dense open set of regular elements in $A$. Define $\mathfrak{g}^{\otimes 2}$-valued functions $r^{\pm}$ on $A_{\textup{reg}}$ by $$\label{foldedexpressions}
\begin{split}
r^+&=\sum_{\alpha\in R}\frac{y_\alpha\otimes e_\alpha}{1-\xi_{-2\alpha}},\\
r^-&=\sum_{s=1}^rx_s\otimes x_s+\sum_{\alpha\in R}\frac{(e_\alpha+e_{-\alpha})\otimes e_\alpha}
{1-\xi_{-2\alpha}}
\end{split}$$ (in fact, $r^+$ takes values in $\mathfrak{k}\otimes\mathfrak{g})$. Define the $U(\mathfrak{k})\otimes U(\mathfrak{g})\otimes U(\mathfrak{k})$-valued function $\kappa$ on $A_{\textup{reg}}$ by $$\kappa:=\sum_{\alpha\in R}\frac{y_\alpha\otimes e_\alpha\otimes 1}{1-\xi_{-2\alpha}}
+1\otimes \kappa^{\textup{core}}\otimes 1+\sum_{\alpha\in R}\frac{1\otimes e_\alpha\otimes y_\alpha}{\xi_\alpha-\xi_{-\alpha}},$$ with the core $\kappa^{\textup{core}}$ the $U(\mathfrak{g})$-valued function $$\kappa^{\textup{core}}:=\frac{1}{2}\sum_{s=1}^rx_s^2+\sum_{\alpha\in R}\frac{e_\alpha^2}{1-\xi_{-2\alpha}}.$$
We will call the first-order $U(\mathfrak{k})\otimes U(\mathfrak{g})^{\otimes N}\otimes U(\mathfrak{k})$-valued differential operators $$\label{introbKZBoper}
\mathcal{D}_i:=E_i-\sum_{j=1}^{i-1}r_{ji}^+-\kappa_i-\sum_{j=i+1}^Nr_{ij}^-\qquad (i=1,\ldots,N)$$ on $A_{\textup{reg}}$ the [*asymptotic boundary Knizhnik-Zamolodchikov-Bernard (KZB) operators*]{}.
Here the indices $i,j$ on the right hand side of indicate in which tensor components of $U(\mathfrak{g})^{\otimes N}$ the $U(\mathfrak{g})$-components of $E$, $r^{\pm}$ and $\kappa$ are placed (note that the only nontrivial contributions to the left and right $U(\mathfrak{k})$-tensor components arise from $\kappa_i-\kappa_i^{\textup{core}}$).
Note that $r^\pm$ and $\kappa^{\textup{core}}$ can be also written as $$\label{rpm}
r^{\pm}=\pm r+(1\otimes \theta)r_{21},\qquad \kappa^{\textup{core}}=m((1\otimes \theta)r_{21}),$$ with $m$ being the multiplication map of $U(\mathfrak{g})$, and $$r:=-\frac{1}{2}\sum_{s=1}^rx_s\otimes x_s-\sum_{\alpha\in R}\frac{e_{-\alpha}\otimes e_\alpha}{1-\xi_{-2\alpha}}.$$ being the Felder’s [@F], [@ES §2] trigonometric dynamical $r$-matrix. More generally, for $a\in A_{\textup{reg}}$, $$\kappa(a)=r^+(a)\otimes 1+1\otimes\kappa^{\textup{core}}(a)\otimes 1+
1\otimes ((\textup{Ad}_{a^{-1}}\otimes\textup{id})r_{21}^+(a)).$$
In this paper we prove that the normalised $N$-point spherical functions $\mathbf{f}_{\underline{\lambda}}^{\phi_\ell,\mathbf{D},\phi_r}$ satisfy the first order differential equations $$\label{bKZBeqintro}
\mathcal{D}_i\bigl(\mathbf{f}_{\underline{\lambda}}^{\phi_\ell,\mathbf{D},\phi_r}\bigr)=\Bigl(\frac{(\lambda_i,\lambda_i)}{2}-
\frac{(\lambda_{i-1},\lambda_{i-1})}{2}\Bigr)\mathbf{f}_{\underline{\lambda}}^{\phi_\ell,\mathbf{D},
\phi_r},\qquad i=1,\ldots,N$$ on $A_+$ (Theorem \[mainthmbKZB\]). We will present two proofs (a third proof follows from generalised radial component considerations, see footnote 3, and will appear elsewhere). The starting point for both proofs is writing $(\lambda_i,\lambda_i)-(\lambda_{i-1},\lambda_{i-1})=\zeta_{\lambda_i}(\Omega)-\zeta_{\lambda_{i-1}}(\Omega)$ and replacing $\zeta_{\lambda_i}(\Omega)$ and $\zeta_{\lambda_{i-1}}(\Omega)$ in the right hand side of with the actions of the Casimir $\Omega$ on $\mathcal{H}_{\lambda_i}^{\infty}$ and $\mathcal{H}_{\lambda_{i-1}}^\infty$, placed to the left and right of the $i$th intertwiner $D_i$ in the expression of the normalised $N$-point spherical function $\mathbf{f}_{\underline{\lambda}}^{\phi_\ell,\mathbf{D},\phi_r}$.
For the first proof we then substitute an explicit Cartan-type factorisation of the Casimir $\Omega$ (see ), which is in essence equivalent to Harish-Chandra’s explicit formula for the differential operator $\widehat{\Pi}(\Omega)$ obtained as the radial component of the action of $\Omega$ as biinvariant differential operator on spherical functions. Pushing the left-and right factors from this factorisation through the intertwiners $D_j$ to the far left and right, is creating the $r^{\pm}$ contributions to the asymptotic boundary KZB equations . The remaining factors are then absorbed by the $K$-intertwiners $\phi_\ell$ and $\phi_r$, producing the contribution $\kappa_i-\kappa_i^{\textup{core}}$ to $\mathcal{D}_i$. In this proof the core $\kappa^{\textup{core}}_i$ of $\kappa_i$ is already part of the initial factorisation of the Casimir element and stays put at its initial spot throughout this procedure. The terms $r^+_{ji}$ ($j<i$) and $r^-_{ij}$ ($j>i$) appear in this proof as the expressions , not as combinations of dynamical $r$-matrices.
In the second proof we substitute the factorisation $$\label{OmegaAAA}
\Omega=\sum_{k=1}^rx_k^2+\frac{1}{2}\sum_{\alpha\in R}\left(\frac{1+a^{-2\alpha}}{1-a^{-2\alpha}}\right)t_\alpha+2\sum_{\alpha\in R}\frac{e_{-\alpha}e_\alpha}{1-a^{-2\alpha}}$$ of the quadratic Casimir element $\Omega$ for $a\in A_{\textup{reg}}$[^4], push the left and right root vectors through the intertwiners $D_j$ to the far left and right, reflect against the $K$-intertwiners $\phi_\ell$ and $\phi_r$, and push the reflected factors back to their original position, where they merge and create the core $\kappa^{\textup{core}}_i$ of $\kappa_i$. When we move components of $\Omega$ to the boundaries (to the left and to the right) the terms $r_{ji}$ or $r_{ij}$ are created by commuting with the intertwiners $D_j$. On the way back they are producing similar terms, but now involving the $\theta$-twisted $r$-matrix $(1\otimes \theta)r_{21}$. This proof naturally leads to the expressions for $r^{\pm}$ and $\kappa^{\textup{core}}$ in terms of Felder’s $r$-matrix. We also prove that the normalised formal $N$-point spherical functions $\mathbf{F}_{\underline{\lambda}}^{\phi_\ell,\mathbf{\Psi},\phi_r}$ are formal power series eigenfunctions of the asymptotic boundary KZB operators, $$\label{bKZBeqintro2}
\mathcal{D}_i\bigl(\mathbf{F}_{\underline{\lambda}}^{\phi_\ell,\mathbf{\Psi},\phi_r}\bigr)=\Bigl(\frac{(\lambda_i,\lambda_i)}{2}-
\frac{(\lambda_{i-1},\lambda_{i-1})}{2}\Bigr)\mathbf{F}_{\underline{\lambda}}^{\phi_\ell,\mathbf{\Psi},
\phi_r}\qquad i=1,\ldots,N.$$ Some additional care is needed here, since it involves manipulations with formal power series.
The asymptotic boundary KZB equations can be viewed as an analogue of the asymptotic KZB equations for generalised weighted trace functions [@ES Thm. 3.1]. The underlying symmetric space in [@ES] is $G\times G/\textup{diag}(G)$, while in our case it is $G/K$.
Fix $\underline{\lambda}=(\lambda_0,\ldots,\lambda_N)$ with $\lambda_N\in\mathfrak{h}^*_{\textup{HC}}+\rho$. We show in Subsection \[sectionBFO\] that the set of normalised formal $N$-point spherical functions $\mathbf{F}_{\underline{\lambda}}^{\phi_\ell,\mathbf{\Psi},\phi_r}$ of weight $\underline{\lambda}$ can be natural parametrised by vectors $v\otimes\mathbf{u}\otimes f$ with $v\in V_\ell$, $f\in V_r^*$ and $\mathbf{u}=u_1\otimes\cdots\otimes u_N\in\mathbf{U}$ consisting of weight vectors $u_i\in U_i$ of weight $\lambda_i-\lambda_{i-1}$. The corresponding normalised formal $N$-point spherical function of weight $\underline{\lambda}$ then has leading coefficient $\mathbf{J}_\ell(\lambda_N-\rho)(v\otimes \mathbf{u})\otimes f\in V_\ell\otimes\mathbf{U}\otimes V_r^*$, where $\mathbf{J}_\ell(\lambda_N-\rho)$ is a linear automorphism of $V_\ell\otimes\mathbf{U}$ that can be regarded as a left boundary version of the fusion operator for $\mathfrak{g}$-intertwiners ([@E; @EV]). As a consequence we obtain sufficiently many common eigenfunctions of the spin quantum Calogero-Moser Hamiltonians and the asymptotic boundary KZB operators to ensure their commutativity (see Theorem \[consistentoperators\]). In particular, $$\begin{split}
[\mathcal{D}_i,\mathcal{D}_j]&=0\qquad (i,j=1,\ldots,N),\\
[\mathcal{D}_i, \mathbf{H}^{(N)}]&=0\qquad (i=1,\ldots,N)
\end{split}$$ as $U(\mathfrak{k})\otimes U(\mathfrak{g})^{\otimes N}\otimes U(\mathfrak{k})$-valued differential operators on $A_{\textup{reg}}$. A different reasoning leading to the commutativity of the operators uses component-wise radial component maps (see footnote 3). As a consequence of the commutativity of the asymptotic boundary KZB operators we will conclude that the $r^{\pm}$ satisfy the mixed classical dynamical Yang-Baxter equations $$\begin{split}
\sum_{k=1}^r\bigl((x_k)_1\partial_{x_k}(r_{23}^{-})-
(x_k)_2\partial_{x_k}(r_{13}^{-})\bigr)&=
\lbrack r_{13}^{-},r_{12}^{+}\rbrack
+\lbrack r_{12}^{-},r_{23}^{-}\rbrack+\lbrack r_{13}^{-},
r_{23}^{-}\rbrack,\\
\sum_{k=1}^r\bigl((x_k)_1\partial_{x_k}(r_{23}^{+})-
(x_k)_3\partial_{x_k}(r_{12}^{-})\bigr)&=\lbrack r_{12}^{-},r_{13}^{+}\rbrack
+\lbrack r_{12}^{-},r_{23}^{+}\rbrack+\lbrack r_{13}^{-},
r_{23}^{+}\rbrack,\\
\sum_{k=1}^r\bigl((x_k)_2\partial_{x_k}(r_{13}^{+})-
(x_k)_3\partial_{x_k}(r_{12}^{+})\bigr)&=
\lbrack r_{12}^{+},r_{13}^{+}\rbrack
+\lbrack r_{12}^{+},r_{23}^{+}\rbrack+\lbrack r_{23}^{-},
r_{13}^{+}\rbrack
\end{split}$$ as $U(\mathfrak{g})^{\otimes 3}$-valued functions on $A_{\textup{reg}}$, and $\kappa$ is a solution of the associated classical dynamical reflection type equation $$\sum_{k=1}^r\bigl((x_k)_1\partial_{x_k}(\kappa_2+r^{+})-
(x_k)_2\partial_{x_k}(\kappa_1+r^{-})\bigr)
=
\lbrack \kappa_1+r^{-}, \kappa_2+r^{+}\rbrack$$ as $U(\mathfrak{k})\otimes U(\mathfrak{g})^{\otimes 2}\otimes U(\mathfrak{k})$-valued functions on $A_{\textup{reg}}$. In the latter equation, $r^{\pm}(a)$ for $a\in A_{\textup{reg}}$ is viewed as the element $1\otimes r^{\pm}(a)\otimes 1$ in $U(\mathfrak{k})\otimes U(\mathfrak{g})^{\otimes 2}\otimes U(\mathfrak{k})$.
Outlook
-------
Harish-Chandra’s theory of harmonic analysis on $G$ has been developed for arbitrary real connected semisimple Lie groups $G$ with finite center (more generally, for reductive $G$ in Harish-Chandra’s class). We expect that the theory of global and formal $N$-point spherical functions and associated asymptotic boundary KZB equations extend to this more general setup. The role of the Cartan subalgebra $\mathfrak{h}_0$ will then be taken over by a maximal abelian subalgebra $\mathfrak{a}_0$ of the $-1$-eigenspace of the Cartan involution $\theta_0$ of the Lie algebra $\mathfrak{g}_0$ of $G$, and the role of the root system $R$ is taken over by the associated reduced root system in $\mathfrak{a}_0^*$.
Besides the earlier mentioned extension of the theory to the affine setting, which will appear in a separate paper, it is natural to generalise the theory to the context of quantum groups using the Letzter-Kolb [@Le; @K2] theory of quantum (affine) symmetric pairs. An intriguing question is how the resulting $K$-matrices will relate with the Balagovic-Kolb [@BK] and Kolb-Yakimov [@KY] type universal $K$-matrices. This direction has many interesting applications in integrable models in statistical mechanics and quantum field theory with integrable boundary conditions, see, e.g., [@DM; @GZ; @JKKMW] and references therein.
Contents of the paper
---------------------
In Section \[sectionII\] we recall basic facts on irreducible split Riemannian pairs and establish the relevant notations. In Section \[S3\] we recall, following [@CM; @W], Harish-Chandra’s radial component map and the explicit expression of the radial component of the quadratic Casimir element. We furthermore establish the link to spin quantum hyperbolic Calogero-Moser systems (Subsection \[vvCM\]) and highlight various important special cases, such as the case that $\sigma_\ell$ and $\sigma_r$ are one-dimensional and the case that $\sigma_\ell=\sigma_r$. The latter case is naturally related to the theory of matrix-valued orthogonal polynomials in the compact picture, see, e.g., [@GPT; @HvP]. We recall the construction of the Harish-Chandra series in Subsection \[S5\], and discuss how they give rise to eigenstates for the spin quantum hyperbolic Calogero-Moser systems. In the first two subsections of Section \[SectionRepTh\] we recall fundamental results of Harish-Chandra [@HCe; @HCI; @HCII] on the principal series representations of $G$ and its associated matrix coefficients. In Subsection \[ssss3\] we discuss the algebraic principal series representations, and the description of the associated spaces of $\mathfrak{k}$-intertwiners. Section \[RTHC\] first discusses how the algebraic principal series representations can be identified with $\mathfrak{k}$-finite parts of weight completions of Verma modules, which leads to a detailed description of the $\mathfrak{k}$-intertwining spaces $\textup{Hom}_{\mathfrak{k}}(M_\lambda,V_\ell)$ and $\textup{Hom}_{\mathfrak{k}}(V_r,\overline{M}_\lambda)$. In the second half of the section we introduce formal elementary $\sigma$-spherical functions and prove their key properties (differential equations and relation to $\sigma$-Harish-Chandra series). In Section \[SectionbKZB\] we first derive asymptotic operator KZB equations for $\mathfrak{g}$-intertwiners and relate them to factorisations of the quadratic Casimir element $\Omega$. In Subsections \[S62\] & \[S63\] we describe the spaces of $G$-equivariant differential operators $\mathcal{H}_\lambda^\infty\rightarrow\mathcal{H}_\mu^\infty\otimes U$ for a finite dimensional $G$-representation $U$, and derive the asymptotic boundary KZB equations for the associated $N$-point spherical functions. In Subsection \[intertwinersection\] we derive the asymptotic boundary KZB equations for the formal $N$-point spherical functions. Subsection \[sectionBFO\] and subsection \[S66\] introduce the boundary fusion operator and establishes the integrability of the asymptotic boundary KZB operators. Finally, in Subsection \[inteq\] we establish the resulting classical mixed dynamical Yang-Baxter and reflection equations for the terms $r^{\pm}$ and $\kappa$ in the asymptotic boundary KZB operators.\
[**Acknowledgments.**]{} We thank Ivan Cherednik, Pavel Etingof, Giovanni Felder, Gert Heckman, Erik Koelink, Christian Korff, Tom Koornwinder, Eric Opdam, Maarten van Pruijssen, Taras Skrypnyk and Bart Vlaar for discussions and comments. We thank Sam van den Brink for carefully reading the first part of the paper and pointing out a number of typos. The work of N.R. was partially supported by NSF DMS-1601947. He also would like to thank ETH-ITS for the hospitality during the final stages of the work.\
\
[**Notations and conventions.**]{} We write $\textup{ad}_L: L\rightarrow \mathfrak{gl}(L)$ for the adjoint representation of a Lie algebra $L$, and $K_L(\cdot,\cdot)$ for its Killing form. Real Lie algebras will be denoted with a subscript zero; The complexification of a real Lie algebra $\mathfrak{g}_0$ with be denoted by $\mathfrak{g}:=\mathfrak{g}_0\otimes_{\mathbb{R}}\mathbb{C}$. The tensor product $\otimes_F$ of $F$-vector spaces is denoted by $\otimes$ in case $F=\mathbb{C}$. For complex vector spaces $U$ and $V$ we write $\textup{Hom}(U,V)$ for the vector space of complex linear maps $U\rightarrow V$. Representations of Lie groups are complex, strongly continuous Hilbert space representations. If $U$ and $V$ are the representation spaces of two representations of a Lie group $G$, then $\textup{Hom}_G(U,V)$ denotes the subspace of bounded linear $G$-intertwiners $U\rightarrow V$. If $U$ and $V$ are two $\mathfrak{g}$-modules for a complex Lie algebra $\mathfrak{g}$, then $\textup{Hom}_{\mathfrak{g}}(U,V)$ denotes the subspace of $\mathfrak{g}$-intertwiners $U\rightarrow V$. The representation map of the infinitesimal $\mathfrak{g}$-representation associated to a smooth $G$-representation $(\tau,U)$ will be denoted by $\tau$ again, if no confusion can arise.
Irreducible split Riemannian symmetric pairs {#sectionII}
============================================
This short section is to fix the basic notations for (split) real semisimple Lie algebras. For further reading consult, e.g., [@Kn].
Real semisimple Lie algebras
----------------------------
Let $\mathfrak{g}_0$ be a real semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}_0\subset\mathfrak{g}_0$. Let $$\label{rootspacecomplex}
\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in R}\mathfrak{g}_\alpha$$ be the root space decomposition of the associated complex semisimple Lie algebra $\mathfrak{g}$ with root system $R=R(\mathfrak{g},\mathfrak{h})\subset\mathfrak{h}^*$ and associated root spaces $$\mathfrak{g}_\alpha:=\{x\in\mathfrak{g} \,\, | \,\, \textup{ad}_{\mathfrak{g}}(h)x=\alpha(h)x\quad \forall\, h\in\mathfrak{h}\}.$$ We write $t_\lambda\in\mathfrak{h}$ for the unique element satisfying $$K_{\mathfrak{g}}(h,t_\lambda)=\lambda(h)\qquad \forall h\in\mathfrak{h}.$$ Then $[x,y]=K_{\mathfrak{g}}(x,y)t_\alpha$ for $x\in\mathfrak{g}_\alpha$ and $y\in\mathfrak{g}_{-\alpha}$, see [@Hu Prop. 8.3].
Returning to the real semisimple Lie algebra $\mathfrak{g}_0$, recall that an involutive automorphism $\theta_0$ of the real semisimple Lie algebra $\mathfrak{g}_0$ is a Cartan involution if the bilinear form $(x,y)\mapsto -K_{\mathfrak{g}_0}(x,\theta_0(y))$ on $\mathfrak{g}_0$ is positive definite. Cartan involutions exist and are unique up to conjugation by an inner automorphism of $\mathfrak{g}_0$. The Cartan decomposition is the decomposition $$\mathfrak{g}_0=\mathfrak{k}_0\oplus\mathfrak{p}_0$$ of $\mathfrak{g}_0$ in $\pm 1$-eigenspaces for the action of $\theta_0$. The $+1$-eigenspace $\mathfrak{k}_0\subset\mathfrak{g}_0$ is a Lie subalgebra of $\mathfrak{g}_0$, and the $-1$-eigenspace $\mathfrak{p}_0$ is an $\textup{ad}_{\mathfrak{g}_0}(\mathfrak{k}_0)$-submodule of $\mathfrak{g}_0$.
Let $G$ be a connected real Lie group with Lie algebra $\mathfrak{g}_0$ and finite center. Denote $K\subset G$ for the connected Lie subgroup with Lie algebra $\mathfrak{k}_0$, which is maximal compact in $G$. The Cartan involution $\theta_0$ integrates to a global Cartan involution $\Theta_0\in\textup{Aut}(G)$, and $K$ is the subgroup of elements $g\in G$ fixed by $\Theta_0$. Hence $(G,K)$ is a Riemannian symmetric pair. The map $K\times \mathfrak{p}_0\rightarrow G$, given by $(k,x)\mapsto k\exp(x)$, is a diffeomorphism, called the global Cartan decomposition of $G$.
Choose a maximally split Cartan subalgebra $\mathfrak{h}_0\subset \mathfrak{g}_0$ with respect to the Cartan involution $\theta_0$, i.e., a $\theta_0$-stable Cartan subalgebra $\mathfrak{h}_0$ such that $\mathfrak{a}_0:=\mathfrak{h}_0\cap\mathfrak{p}_0$ is maximally abelian in $\mathfrak{p}_0$. Then $\mathfrak{h}_0=\mathfrak{t}_0\oplus\mathfrak{a}_0$ with $\mathfrak{t}_0=\mathfrak{h}_0\cap \mathfrak{k}_0$. With respect to the scalar product $(x,y)\mapsto -K_{\mathfrak{g}_0}(x,\theta_0(y))$ on $\mathfrak{g}_0$, the linear operator $\textup{ad}_{\mathfrak{g}_0}(h)\in\textup{End}(\mathfrak{g}_0)$ is skew-symmetric for $h\in\mathfrak{t}_0$ and symmetric for $h\in\mathfrak{a}_0$.
Note that the restriction of $K_{\mathfrak{g}_0}(\cdot,\cdot)$ to $\mathfrak{a}_0$ is positive definite, turning $\mathfrak{a}_0$ in a Euclidean space. We write $(\cdot,\cdot)=K_{\mathfrak{g}_0}(\cdot,\cdot)|_{\mathfrak{a}_0\times\mathfrak{a}_0}$, and $\|\cdot\|$ for the resulting norm on $\mathfrak{a}_0$. We use the same notations for the induced scalar product and norm on $\mathfrak{a}_0^*$. Furthermore, the complex bilinear extensions of $(\cdot,\cdot)$ to bilinear forms on $\mathfrak{a}$ and $\mathfrak{a}^*$ will also be denoted by $(\cdot,\cdot)$.
Let $A\subset G$ be the connected Lie subgroup with Lie algebra $\mathfrak{a}_0$. It is a closed commutative Lie subgroup of $G$, isomorphic to $\mathfrak{a}_0$ through the restriction of the exponential map $\exp: \mathfrak{g}\rightarrow G$ to $\mathfrak{a}_0$. We write $\log: A\rightarrow \mathfrak{a}_0$ for its inverse. We furthermore write $\mathfrak{m}_0$ for the centralizer $Z_{\mathfrak{k}_0}(\mathfrak{a}_0)$ of $\mathfrak{a}_0$ in $\mathfrak{k}_0$.
Split real semisimple Lie algebras
----------------------------------
In this paper we restrict attention to split real semisimple Lie algebras.
A Cartan subalgebra $\mathfrak{h}_0\subset\mathfrak{g}_0$ of a real semisimple Lie algebra $\mathfrak{g}_0$ is said to be split if $\textup{ad}_{\mathfrak{g}_0}(h)\in\textup{End}(\mathfrak{g}_0)$ is diagonalizable for all $h\in\mathfrak{h}_0$. If $\mathfrak{g}_0$ admits a split Cartan subalgebra, thenl $\mathfrak{g}_0$ is called a split real semisimple Lie algebra.
Let $\mathfrak{g}_0$ be a split real semisimple Lie algebra with split Cartan subalgebra $\mathfrak{h}_0$. The root space decomposition refines to $$\label{rootspacereal}
\mathfrak{g}_0=\mathfrak{h}_0\oplus\bigoplus_{\alpha\in R}\mathfrak{g}_{0,\alpha}$$ with $\mathfrak{g}_{0,\alpha}:=\mathfrak{g}_0\cap\mathfrak{g}_\alpha$ a one-dimensional real vector space for all $\alpha\in R$. In particular, all roots $\alpha\in R$ are real-valued on $\mathfrak{h}_0$ (conversely, a Cartan subalgebra $\mathfrak{h}_0$ of $\mathfrak{g}_0$ with the property that all the roots $\alpha\in R$ are real-valued on $\mathfrak{h}_0$, is split). Let $\{\alpha_1,\ldots,\alpha_r\}$ be a set of simple roots of $R$ and write $R^{+}$ for the associated set of positive roots. For any choice of root vectors $e_j\in\mathfrak{g}_{0,\alpha_j}$ and $f_j\in\mathfrak{g}_{0,-\alpha_j}$ satisfying $[e_j,f_j]=t_{\alpha_j}$ ($1\leq j\leq r$), there exists a unique involution $\theta_0\in\textup{Aut}(\mathfrak{g}_0)$ such that $$\label{mapSerre}
\theta_0(e_j)=-f_j,\qquad 1\leq j\leq r$$ and $\theta_0|_{\mathfrak{h}_0}=-\textup{id}_{\mathfrak{h}_0}$, cf., e.g., [@Hu Prop. 14.3]. Furthermore, $\theta_0\in\textup{Aut}(\mathfrak{g}_0)$ is a Cartan involution and $\mathfrak{a}_0=\mathfrak{h}_0$. Consequently $\mathfrak{h}_0\subset\mathfrak{g}_0$ is a maximally split Cartan subalgebra with respect to $\theta_0$, and $\mathfrak{m}_0=\{0\}$. The complex linear extension $\theta\in\textup{Aut}(\mathfrak{g})$ of $\theta_0$ is a Chevalley involution of $\mathfrak{g}$.
Conversely, let $\mathfrak{g}_0$ be a split real semisimple Lie algebra and $\theta_0\in\textup{Aut}(\mathfrak{g}_0)$ a Cartan involution. Choose a maximally split Cartan subalgebra $\mathfrak{h}_0\subset\mathfrak{g}_0$ with respect to $\theta_0$. Then $\mathfrak{h}_0\subset\mathfrak{g}_0$ is a split Cartan subalgebra, and $\mathfrak{h}_0\subset\mathfrak{p}_0$. Furthermore, there exists a choice of root vectors $e_{j}\in\mathfrak{g}_{0,\alpha_j}$ and $f_j\in\mathfrak{g}_{0,-\alpha_j}$ satisfying $[e_{j},f_{j}]=t_{\alpha_j}$ and $\theta_0(e_j)=-f_j$ for $j=1,\ldots,r$.
Complexification $\mathfrak{g}_0\mapsto \mathfrak{g}$ defines a surjective map from the set of isomorphism classes of real semisimple Lie algebras to the set of isomorphism classes of complex semisimple Lie algebras. It restricts to a bijection from the set of isomorphism classes of split real semisimple Lie algebras onto the set of isomorphism classes of complex semisimple Lie algebras.
One-dimensional $\mathfrak{k}$-representations {#SSonedim}
----------------------------------------------
We fix from now on a triple $(\mathfrak{g}_0,\mathfrak{h}_0,\theta_0)$ with $\mathfrak{g}_0$ a split real semisimple Lie algebra, $\mathfrak{h}_0$ a split Cartan subalgebra and $\theta_0$ a Cartan involution such that $\theta_0|_{\mathfrak{h}_0}=-\textup{id}_{\mathfrak{h}_0}$. The associated pair $(G,K)$ is an irreducible split Riemannian symmetric pair. We fix from now on $e_\alpha\in\mathfrak{g}_{0,\alpha}$ ($\alpha\in R$) such that $[e_\alpha,e_{-\alpha}]=t_\alpha$ and $\theta_0(e_\alpha)=-e_{-\alpha}$ for all $\alpha\in R$ (the fact that this is possible follows from, e.g., [@Hu §25.2]). Note that $K_{\mathfrak{g}_0}(e_\alpha,e_{-\alpha})=1$ for $\alpha\in R$. Set $$y_\alpha:=e_\alpha-e_{-\alpha}\in\mathfrak{k}_0,\qquad \alpha\in R,$$ then $y_{-\alpha}=-y_{\alpha}$ ($\alpha\in R$) and $$\begin{split}
\mathfrak{k}_0&=\bigoplus_{\alpha\in R^+}\mathbb{R}y_\alpha,\\
\mathfrak{p}_0&=\mathfrak{h}_0\oplus\bigoplus_{\alpha\in R^+}\mathbb{R}(e_\alpha+e_{-\alpha}).
\end{split}$$
Let $\textup{ch}(\mathfrak{k}_0)$ be the space of one-dimensional real representations of $\mathfrak{k}_0$. If $\mathfrak{g}$ is simple but not of type $C_r$ ($r\geq 1$) then $\mathfrak{k}_0$ is semisimple (see, for instance, [@St §3.1]), hence $\textup{ch}(\mathfrak{k}_0)=\{\chi_0\}$ with $\chi_0$ the trivial representation. If $\mathfrak{g}_0\simeq\mathfrak{sp}(r;\mathbb{R})$ ($r\geq 1$) then $\mathfrak{k}_0\simeq\mathfrak{gl}_r(\mathbb{R})$, hence $\textup{ch}(\mathfrak{k}_0)=\mathbb{R}\chi$ is one-dimensional. Write in this case $R_s$ and $R_\ell$ for the set of short and long roots in $R$ with respect to the norm $\|\cdot\|$ (by convention, $R_s=\emptyset$ and $R_\ell=R$ for $r=1$), and set $R_s^+:=R_s\cap R^+$ and $R_l^+:=R_l\cap R^+$. Let $\chi_{\mathfrak{sp}}\in\mathfrak{k}_0^*$ be the linear functional satisfying $\chi_{\mathfrak{sp}}(y_\alpha)=0$ for $\alpha\in R_s^+$ and $\chi_{\mathfrak{sp}}(y_\alpha)=1$ for $\alpha\in R_\ell^+$. Then $$\textup{ch}(\mathfrak{k}_0)=\mathbb{R}\chi_{\mathfrak{sp}}$$ by [@St Lemma 4.3].
The Iwasawa decomposition
-------------------------
Consider the nilpotent Lie subalgebra $$\mathfrak{n}_0:=\bigoplus_{\alpha\in R^+}\mathfrak{g}_{0,\alpha}$$ of $\mathfrak{g}_0$.The decomposition $$\mathfrak{g}_0=\mathfrak{k}_0\oplus\mathfrak{h}_0\oplus\mathfrak{n}_0$$ as vector spaces is the Iwasawa decomposition of $\mathfrak{g}_0$. Let $N\subset G$ be the connected Lie subgroup with Lie algebra $\mathfrak{n}_0$. Then $N$ is simply connected and closed in $G$, and the exponential map $\exp: \mathfrak{n}_0\rightarrow N$ is a diffeomorphism. The multiplication map $$\label{mm}
K\times A\times N\rightarrow G, \qquad (k,a,n)\mapsto kan$$ is a diffeomorphism onto $G$ (the global Iwasawa decomposition). We write $$g=k(g)a(g)n(g)$$ for the Iwasawa decomposition of $g\in G$, with $k(g)\in K$, $a(g)\in A$ and $n(g)\in N$.
Since $\mathfrak{m}_0=\{0\}$, the centralizer $M:=Z_{K}(\mathfrak{a}_0)$ of $\mathfrak{a}_0$ in $K$ is a finite group. The minimal parabolic subgroup $P=MAN$ of $G$ is a closed Lie subgroup of $G$ with Lie algebra $\mathfrak{p}_0:=\mathfrak{h}_0\oplus\mathfrak{n}_0$. Note that the complexification $\mathfrak{b}$ of $\mathfrak{b}_0$ is the Borel subalgebra of $\mathfrak{g}$ containing $\mathfrak{h}$.
Radial components of invariant differential operators {#S3}
=====================================================
Throughout this section we fix a triple $(\mathfrak{g}_0,\mathfrak{h}_0,\theta_0)$ with $\mathfrak{g}_0$ a split real semisimple Lie algebra, $\mathfrak{h}_0$ a split Cartan subalgebra and $\theta_0$ a Cartan involution such that $\theta_0|_{\mathfrak{h}_0}=-\textup{id}_{\mathfrak{h}_0}$. We write $\mathfrak{h}_0^*\subset\mathfrak{h}^*$ for the real span of the roots, $G$ for a connected Lie group with Lie algebra $\mathfrak{g}_0$ and finite center, $K\subset G$ for the connected Lie subgroup with Lie algebra $\mathfrak{k}_0$, and $A\subset G$ for the connected Lie subgroup with Lie algebra $\mathfrak{h}_0$.
The radial component map
------------------------
The radial component map describes the factorisation of elements $x\in U(\mathfrak{g})$ along algebraic counterparts of the Cartan decomposition $G=KAK$. We first introduce some preliminary notations.
For $\lambda\in\mathfrak{h}^*$ the map $$\xi_\lambda: A\rightarrow\mathbb{C}^*,\quad a\mapsto a^\lambda:=e^{\lambda(\log(a))}$$ defines a complex-valued multiplicative character of $A$, which is real-valued for $\lambda\in\mathfrak{h}_0^*$. It satisfies $\xi_\lambda\xi_\mu=\xi_{\lambda+\mu}$ ($\lambda,\mu\in\mathfrak{h}^*$) and $\xi_0\equiv 1$.
The adjoint representation $\textup{Ad}: G\rightarrow \textup{Aut}(\mathfrak{g}_0)$ extends naturally to an action of $G$ on the universal enveloping algebra $U(\mathfrak{g})$ of $\mathfrak{g}$ by complex linear algebra automorphisms. We write $\textup{Ad}_g(x)$ for the adjoint action of $g\in G$ on $x\in U(\mathfrak{g})$. Note that for $a\in A$, $$\textup{Ad}_a(e_\alpha)=a^{\alpha}e_\alpha\qquad \forall\, \alpha\in R$$ and $\textup{Ad}_a$ fixes $\mathfrak{h}$ pointwise.
Each $g\in G$ admits a decomposition $g=kak^\prime$ with $k,k^\prime\in K$ and $a\in A$. The double cosets $KaK$ and $Ka^\prime K$ ($a,a^\prime\in A$) coincide iff $a^\prime\in Wa$ with $W:=N_K(\mathfrak{h}_0)/M$ the analytic Weyl group of $G$, acting on $A$ by conjugation. Note that $W$ is isomorphic to the Weyl group of $R$ since $G$ is split. Set $$A_{\textup{reg}}:=\{a\in A \,\, | \,\, a^\alpha\not=1\quad \forall \alpha\in R\}.$$ Then $\exp: \mathfrak{h}_{0,\textup{reg}}\overset{\sim}{\longrightarrow} A_{\textup{reg}}$, with $\mathfrak{h}_{0,\textup{reg}}:=\{h\in\mathfrak{h}_0 \,\, | \,\,
\alpha(h)\not=0\,\,\, \forall\, \alpha\in R\}$ the set of regular elements in $\mathfrak{h}_0$. The Weyl group $W$ acts freely on $A_{\textup{reg}}$.
Infinitesimal analogues of the Cartan decomposition of $G$ are realized through the vector space decompositions $$\label{infinitesimalKAK}
\mathfrak{g}_0=\mathfrak{h}_0\oplus\textup{Ad}_{a^{-1}}\mathfrak{k}_0\oplus \mathfrak{k}_0$$ of $\mathfrak{g}_0$ for $a\in A_{\textup{reg}}$. The decomposition follows from the identity $$\label{elementaryrel}
e_\alpha=\frac{a^{-\alpha}(\textup{Ad}_{a^{-1}}y_\alpha)-y_\alpha}{a^{-2\alpha}-1},$$ which shows that $\{\textup{Ad}_{a^{-1}}y_\alpha, y_\alpha\}$ is a linear basis of $\mathfrak{g}_{0,\alpha}\oplus\mathfrak{g}_{0,-\alpha}$ for $a\in A_{\textup{reg}}$. Set $$\mathcal{V}:=U(\mathfrak{h})\otimes U(\mathfrak{k})\otimes U(\mathfrak{k}).$$ By the Poincar[é]{}-Birkhoff-Witt-Theorem, for each $a\in A_{\textup{reg}}$ the linear map $$\Gamma_a: \mathcal{V}\rightarrow U(\mathfrak{g}),\qquad
\Gamma_a(h\otimes x\otimes y):=\textup{Ad}_{a^{-1}}(x)hy$$ is a linear isomorphism.
Extending the scalars of the complex vector space $\mathcal{V}$ to the ring $C^\infty(A_{\textup{reg}})$ of complex valued smooth functions on $A_{\textup{reg}}$ allows one to give the factorisation $\Gamma_a^{-1}(x)$ for $x\in
U(\mathfrak{g})$ uniformly in $a\in A_{\textup{reg}}$. It suffices to extend the scalars to the unital subring $\mathcal{R}$ of $C^\infty(A_{\textup{reg}})$ generated by $\xi_{-\alpha}$ and $(1-\xi_{-2\alpha})^{-1}$ for all $\alpha\in R^+$. For $a\in A_{\textup{reg}}$ the extension of $\Gamma_a$ is then the complex linear map $\widetilde{\Gamma}_a: \mathcal{R}\otimes\mathcal{V}\rightarrow U(\mathfrak{g})$ defined by $$\widetilde{\Gamma}_a(f\otimes Z):=f(a)\Gamma_a(Z),\qquad f\in\mathcal{R},\,\,\, Z\in\mathcal{V}.$$
\[infKAK\] For $x\in U(\mathfrak{g})$ there exists a unique $\Pi(x)\in\mathcal{R}\otimes\mathcal{V}$ such that $$\widetilde{\Gamma}_a\bigl(\Pi(x)\bigr)=x\qquad \forall\, a\in A_{\textup{reg}}.$$
For example, by , $$\Pi(e_\alpha)=(\xi_{-\alpha}-\xi_\alpha)^{-1}\otimes 1\otimes y_\alpha\otimes 1
-(\xi_{-2\alpha}-1)^{-1}\otimes 1\otimes 1\otimes y_\alpha.$$ The resulting linear map $\Pi: U(\mathfrak{g})\rightarrow\mathcal{R}\otimes\mathcal{V}$ is called the radial component map.
$\sigma$-Spherical functions
----------------------------
The radial component map plays an important role in the study of spherical functions. Fix a finite dimensional representation $\sigma: K\times K\rightarrow \textup{GL}(V_\sigma)$. Denote by $C^\infty(G;V_\sigma)$ the space of smooth $V_\sigma$-valued functions on $G$.
\[defispher\] We say that $f\in C^\infty(G;V_\sigma)$ is a $\sigma$-spherical function on $G$ if $$f(k_1gk_2^{-1})=\sigma(k_1,k_2)f(g)\qquad \forall\, g\in G,\,\,\, \forall k_1,k_2\in K.$$ We denote by $C^\infty_\sigma(G)$ the subspace of $C^\infty(G;V_\sigma)$ consisting of $\sigma$-spherical functions on $G$.
Let $V_\sigma^M$ be the subspace of $M$-invariant elements in $V_\sigma$, with $M$ acting diagonally on $V_\sigma$. Examples of $\sigma$-spherical functions on $G$ are $$E_\lambda^\sigma(\cdot)v\in C^\infty_\sigma(G),\qquad v\in V_\sigma$$ with $E_\lambda^\sigma: G\rightarrow\textup{End}(V_\sigma)$ ($\lambda\in\mathfrak{h}^*$) the Eisenstein integral $$\label{Eisenstein}
E_\lambda^\sigma(g):=\int_Kdx\,
\xi_{-\lambda-\rho}(a(g^{-1}x))
\sigma(x,k(g^{-1}x)).$$ Here $\rho:=\frac{1}{2}\sum_{\alpha\in R^+}\alpha\in\mathfrak{h}^*$ and $dx$ is the normalised Haar measure on $K$. The representation theoretic construction of $\sigma$-spherical functions (see, e.g., [@CM §8]) will be discussed in Section \[SectionRepTh\].
The function space $C^\infty(A;V_\sigma^M)$ is a $W$-module with $w=kM$ ($k\in N_K(\mathfrak{h}_0)$) acting by $$(w\cdot f)(a):=\sigma(k,k)f(k^{-1}ak)$$ for $a\in A$ and $f\in C^\infty(A;V_\sigma^M)$. We write $C^\infty(A;V_\sigma^M)^W$ for the subspace of $W$-invariant $V_\sigma^M$-valued smooth functions on $A$. By the Cartan decomposition of $G$, we have the following well known result
The restriction map $\vert_{A}: C^\infty(G;V_\sigma)\rightarrow C^\infty(A;V_\sigma)$, $f\mapsto f\vert_{A}$ restricts to an injective linear map from $C_\sigma^\infty(G)$ into $C^\infty(A;V_\sigma^M)^W$. Similarly, restriction to $A_{\textup{reg}}$ gives an injective linear map $\vert_{A_{\textup{reg}}}: C_\sigma^\infty(G)\hookrightarrow C^\infty(A_{\textup{reg}};V_\sigma^M)^W$.
The action of left $G$-invariant differential operators on $C^\infty_\sigma(G)$, pushed through the restriction map $\vert_{A_{\textup{reg}}}$, gives rise to differential operators on $A_{\textup{reg}}$ that can be described explicitly in terms of the radial component map $\Pi$. We describe them in the following subsection.
Invariant differential operators
--------------------------------
Denote by $\ell$ and $r$ the left-regular and right-regular representations of $G$ on $C^\infty(G)$ respectively, $$(\ell(g)f)(g^\prime):=f(g^{-1}g^\prime),\qquad (r(g)f)(g^\prime):=f(g^\prime g),$$ with $g,g^\prime\in G$ and $f\in C^\infty(G)$. Let $\mathbb{D}(G)$ be the ring of differential operators on $G$, and $\mathbb{D}(G)^G\subseteq\mathbb{D}(G)$ its subalgebra of left $G$-invariant differential operators. Differentiating $r$ gives an an isomorphism $$r_\ast: U(\mathfrak{g})\overset{\sim}{\longrightarrow} \mathbb{D}(G)^G$$ of algebras.
Let $U(\mathfrak{g})^M\subseteq U(\mathfrak{g})$ be the subalgebra of $\textup{Ad}(M)$-invariant elements in $U(\mathfrak{g})$. Embed $\mathbb{D}(G)$ into $\mathbb{D}(G)\otimes\textup{End}(V_\sigma)$ by $D\mapsto D\otimes\textup{id}_{V_\sigma}$ ($D\in\mathbb{D}(G)$). With respect to the resulting action $r_\ast$ of $U(\mathfrak{g})$ on $C^\infty(G; V_{\sigma})$, the subspace $C^\infty_\sigma(G)$ of $\sigma$-spherical functions is a $U(\mathfrak{g})^K$-invariant subspace of $C^\infty(G; V_\sigma)$.
Let $\mathbb{D}(A)$ be the ring of differential operators on $A$ and $\mathbb{D}(A)^A$ the subalgebra of $A$-invariant differential operators. Let $r^A$ be the right-regular action of $A$ on $C^\infty(A)$. Its differential gives rise to an algebra isomorphism $$\label{rhoA}
r^A_\ast: U(\mathfrak{h})\overset{\sim}{\longrightarrow}\mathbb{D}(A)^A.$$ We will write $\partial_h:=r^A_\ast(h)\in\mathbb{D}(A)^A$ for $h\in\mathfrak{h}_0$, which are the derivations $$\bigl(\partial_hf\bigr)(a)=\frac{d}{dt}\biggr\rvert_{t=0}f\bigl(a\exp_A(th)\bigr)$$ for $f\in C^\infty(A)$ and $a\in A$. We also consider $\mathbb{D}(A)^A$ as the subring of $\mathbb{D}(A_{\textup{reg}})$ consisting of constant coefficient differential operators and write $$\mathbb{D}_{\mathcal{R}}\subset\mathbb{D}(A_{\textup{reg}})$$ for the algebra of differential operators $$D=\sum_{m_1,\ldots,m_r}c_{m_1,\ldots,m_r}\partial_{x_1}^{m_1}\cdots\partial_{x_r}^{m_r}\in
\mathbb{D}(A_{\textup{reg}})$$ with coefficients $c_{m_1,\ldots,m_r}\in\mathcal{R}$, where $\{x_1,\ldots,x_r\}$ is an orthonormal basis of $\mathfrak{h}_0$ with respect to $(\cdot,\cdot)$. The algebra isomorphism now extends to a complex linear isomorphism $$\widetilde{r}_\ast^A: \mathcal{R}\otimes U(\mathfrak{h})\overset{\sim}{\longrightarrow}\mathbb{D}_{\mathcal{R}}, \quad f\otimes h\mapsto fr_\ast^A(h)$$ for $f\in\mathcal{R}$ and $h\in U(\mathfrak{h})$. Finally, $\mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{k})^{\otimes 2}$ will denote the algebra of differential operators $D=\sum_{m_1,\ldots,m_r}c_{m_1,\ldots,m_r}\partial_{x_1}^{m_1}\cdots
\partial_{x_r}^{m_r}$ on $A_{\textup{reg}}$ with coefficients $c_{m_1,\ldots,m_r}$ in $\mathcal{R}\otimes U(\mathfrak{k})^{\otimes 2}$. It acts naturally on $C^\infty(A_{\textup{reg}}; V_\sigma)$.
By the proof of [@CM Thm. 3.1] we have for $f\in C^{\infty}_\sigma(G)$, $h\in U(\mathfrak{h})$ and $x,y\in U(\mathfrak{k})$, $$\bigl(r_\ast\bigl(\textup{Ad}_{a^{-1}}(x)hy\bigr)f\bigr)(a)=
\sigma(x\otimes S(y))\bigl(r_\ast(h)f\bigr)(a)\qquad \forall\, a\in A_{\textup{reg}},$$ with $S$ the antipode of $U(\mathfrak{k})$, defined as the anti-algebra homomorphism of $U(\mathfrak{k})$ such that $S(x)=-x$ for all $x\in \mathfrak{k}$. Combined with Theorem \[infKAK\] this leads to the following result.
\[thmRAD\] With the above conventions, define the linear map $$\widehat{\Pi}: U(\mathfrak{g})\rightarrow \mathbb{D}_{\mathcal{R}}
\otimes U(\mathfrak{k})^{\otimes 2}$$ by $\widehat{\Pi}:=(\widetilde{r}^A_\ast\otimes\textup{id}_{U(\mathfrak{k})}\otimes S)
\Pi$, and set $$\widehat{\Pi}^\sigma:=(\textup{id}_{\mathbb{D}_{\mathcal{R}}}\otimes\sigma)\widehat{\Pi}:
U(\mathfrak{g})\rightarrow \mathbb{D}_{\mathcal{R}}\otimes\textup{End}(V_\sigma).$$
1. For $z\in U(\mathfrak{g})$, $$\bigl(r_\ast(z)f\bigr)\vert_{A_{\textup{reg}}}=\widehat{\Pi}^\sigma(z)\bigl(
f\vert_{A_{\textup{reg}}}\bigr)\qquad
\forall\, f\in C^{\infty}_\sigma(G).$$
2. The restrictions of $\widehat{\Pi}$ and $\widehat{\Pi}^\sigma$ to $Z(\mathfrak{g})$ are algebra homomorphisms.
[**a.**]{} This is a well-known result of Harish-Chandra, see, e.g., [@CM Thm. 3.1].
[**b.**]{} It is well-known that the differential operators $\widehat{\Pi}(z)$ ($z\in Z(\mathfrak{g})$) pairwise commute when acting on $C^\infty(A_{\textup{reg}};V_\sigma^M)$, see [@CM Thm. 3.3]. The theory of formal spherical functions which we develop in Section \[RTHC\], implies that they also commute as $U(\mathfrak{k})^{\otimes 2}$-valued differential operators. The key point is that all formal spherical functions are formal power series eigenfunctions of $\widehat{\Pi}(z)$ ($z\in Z(\mathfrak{g})$) by Theorem \[mainTHMF\][**a**]{}, which forces the differential operators $\widehat{\Pi}(z)$ ($z\in Z(\mathfrak{g})$) to commute as $U(\mathfrak{k})^{\otimes 2}$-valued differential operators by the results in Section \[S66\] for the special case $N=0$.
\[Minvariance\] By [@CM Prop. 2.5] we have $\widehat{\Pi}(z)\in\mathbb{D}_{\mathcal{R}}\otimes
U(\mathfrak{k}\oplus\mathfrak{k})^M$ for $z\in U(\mathfrak{g})^M$, where $U(\mathfrak{k}\oplus\mathfrak{k})^M$ is the space of $M$-invariance in $U(\mathfrak{k}\oplus\mathfrak{k})\simeq U(\mathfrak{k})^{\otimes 2}$ with respect to the diagonal adjoint action of $M$ on $U(\mathfrak{k}\oplus\mathfrak{k})$. In particular, $\widehat{\Pi}^\sigma(z)\in\mathbb{D}_{\mathcal{R}}\otimes\textup{End}_M(V_\sigma)$ for $z\in U(\mathfrak{g})^M$, with $M$ acting diagonally on $V_\sigma$.
The radial component of the Casimir element {#S34}
-------------------------------------------
In this subsection we recall the computation of the radial component of the Casimir element. As before, let $e_\alpha\in\mathfrak{g}_{0,\alpha}$ ($\alpha\in R$) such that $[e_\alpha,e_{-\alpha}]=t_{\alpha}$ and $\theta_0(e_\alpha)=-e_{-\alpha}$ ($\alpha\in R$), and $\{x_1,\ldots,x_r\}$ an orthonormal basis of $\mathfrak{h}_0$ with respect to $(\cdot,\cdot)$. The Casimir element $\Omega\in Z(\mathfrak{g})$ is given by $$\label{Omega}
\begin{split}
\Omega&=\sum_{j=1}^rx_j^2+\sum_{\alpha\in R}e_\alpha e_{-\alpha}\\
&=\sum_{j=1}^rx_j^2+2t_\rho+2\sum_{\alpha\in R^+}e_{-\alpha}e_{\alpha}.
\end{split}$$ By , the second line of , and by $$[y_\alpha,\textup{Ad}_{a^{-1}}y_\alpha]=(a^{-\alpha}-a^\alpha)t_\alpha\qquad
\forall\, a\in A,$$ we obtain the following Cartan factorisation of $\Omega$, $$\label{KAKomega}
\Omega=\sum_{j=1}^rx_j^2+
\frac{1}{2}\sum_{\alpha\in R}\left(\frac{a^\alpha+a^{-\alpha}}{a^\alpha-a^{-\alpha}}\right)t_\alpha
+\sum_{\alpha\in R}\left(
\frac{\textup{Ad}_{a^{-1}}(y_\alpha^2)-(a^{\alpha}+a^{-\alpha})\textup{Ad}_{a^{-1}}(y_\alpha)y_\alpha
+y_\alpha^2}{(a^{\alpha}-a^{-\alpha})^2}\right)$$ for arbitrary $a\in A_{\textup{reg}}$. It follows that $$\label{piomega}
\begin{split}
\Pi(\Omega)&= \sum_{j=1}^r1\otimes x_j^2\otimes 1\otimes 1
+\frac{1}{2}\sum_{\alpha\in R}\left(\frac{\xi_\alpha+\xi_{-\alpha}}{\xi_\alpha-\xi_{-\alpha}}\right)\otimes t_\alpha
\otimes 1\otimes 1\\
&+\sum_{\alpha\in R}\left\{\frac{1}{(\xi_{\alpha}-\xi_{-\alpha})^2}\otimes 1
\otimes (y_\alpha^2\otimes 1+1\otimes y_\alpha^2)-
\frac{(\xi_{\alpha}+\xi_{-\alpha})}{(\xi_{\alpha}-\xi_{-\alpha})^2}\otimes 1\otimes y_\alpha\otimes y_\alpha\right\}.
\end{split}$$ This gives the following result, cf., e.g., [@W Prop. 9.1.2.11].
\[corR1\] The differential operator $\widehat{\Pi}(\Omega)\in\mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{k})^{\otimes 2}$ is given by $$\label{pihat}
\widehat{\Pi}(\Omega)=\Delta+\frac{1}{2}\sum_{\alpha\in R}\left(\frac{\xi_\alpha+\xi_{-\alpha}}
{\xi_\alpha-\xi_{-\alpha}}\right)\partial_{t_\alpha}
+\sum_{\alpha\in R}\frac{1}{(\xi_{\alpha}-\xi_{-\alpha})^2}
\prod_{\epsilon\in\{\pm 1\}}(y_\alpha\otimes 1+\xi_{\epsilon\alpha}(1\otimes
y_\alpha))$$ with $\Delta:=\sum_{j=1}^r\partial_{x_j}^2$ the Laplace-Beltrami operator on $A$.
Note that the infinitesimal Cartan factorisations of $\Omega$ are parametrised by elements $a\in A_{\textup{reg}}$. In the context of boundary Knizhnik-Zamolodchikov equations (see Section \[SectionbKZB\]) these will provide the dynamical parameters.
There are various ways to factorise $\Omega$, of which and the infinitesimal Cartan decomposition are two natural ones. Another factorisation is $$\label{Aomega}
\Omega=\sum_{j=1}^rx_j^2+\frac{1}{2}\sum_{\alpha\in R}\left(\frac{1+a^{-2\alpha}}{1-a^{-2\alpha}}\right)t_\alpha+2\sum_{\alpha\in R}\frac{e_{-\alpha}e_\alpha}{1-a^{-2\alpha}}$$ for $a\in A_{\textup{reg}}$, which is a dynamical version of . This formula can be easily proved by moving in positive root vectors $e_\alpha$ ($\alpha\in R^+$) to the left and using $[e_\alpha,e_{-\alpha}]=t_\alpha$, which causes the “dynamical” dependence to drop out and reduces to the second formula of . The decomposition is the natural factorisation of $\Omega$ in the context of Etingof’s and Schiffmann’s [@ES] generalised weighted trace functions and associated asymptotic KZB equations, as we shall explain in Section \[SectionbKZB\].
$\chi$-invariant vectors {#chisection}
------------------------
Let $V$ be a $\mathfrak{g}_0$-module and fix $\chi\in\textup{ch}(\mathfrak{k}_0)$. We say that a vector $v\in V$ is $\chi$-invariant if $xv=\chi(x)v$ for all $x\in\mathfrak{k}_0$. We write $V^\chi$ for the subspace of $\chi$-invariant vectors in $V$, $$V^\chi=\{v\in V \,\, | \,\, e_\alpha v-e_{-\alpha}v=\chi(y_\alpha)v\quad \forall\, \alpha\in R^+\}.$$ In case of the trivial one-dimensional representation $\chi_0\equiv 0$, we write $V^{\chi_0}=V^{\mathfrak{k}_0}$, which is the space of $\mathfrak{k}_0$-fixed vectors in $V$. From the computation of the radial component of the Casimir $\Omega$ in the previous subsection, we obtain the following corollary.
\[corO\] Let $V$ be a $\mathfrak{g}_0$-module such that $\Omega|_V=c\,\textup{id}_V$ for some $c\in\mathbb{R}$. Fix $\chi\in\textup{ch}(\mathfrak{k}_0)$ and $v\in V^\chi$. Then $$\Bigl(\sum_{j=1}^rx_j^2+\frac{1}{2}\sum_{\alpha\in R}\Bigl(\frac{1+a^{-2\alpha}}{1-a^{-2\alpha}}\Bigr)t_\alpha
+\sum_{\alpha\in R}\frac{1}{(a^{-\alpha}-a^\alpha)^2}
\prod_{\epsilon\in\{\pm 1\}}(\textup{Ad}_{a^{-1}}(y_\alpha)-a^{-\epsilon\alpha}\chi(y_\alpha))
\Bigr)v=cv$$ for all $a\in A_{\textup{reg}}$.
If $V$ is $\mathfrak{h}_0$-diagonalisable then Corollary \[corO\] reduces to explicit recursion relations for the weight components of $v\in V^\chi$.
In the setup of the corollary, a vector $u\in V$ is a Whittaker vector of weight $a\in A_{\textup{reg}}$ if $e_\alpha u=a^\alpha u$ for all $\alpha\in R^+$. Recursion relations for the weight components of Whittaker vectors are used in [@DKT §3.2] to derive a path model for Whittaker vectors ([@DKT Thm. 3.7]), as well as for the associated Whittaker functions ([@DKT Thm. 3.9]). It would be interesting to see what this approach entails for $\sigma$-spherical functions with $\sigma$ a one-dimensional representation of $K\times K$, when the role of the Whittaker vectors is taken over by $\chi$-invariant vectors.
$\sigma$-Spin quantum hyperbolic Calogero-Moser systems {#vvCM}
-------------------------------------------------------
We gauge the commuting differential operators $\widehat{\Pi}(z)$ ($z\in Z(\mathfrak{g})$) to give them the interpretation as quantum Hamiltonians for spin generalisations (in the physical sense) of the quantum hyperbolic Calogero-Moser system. This extends results from [@Ga; @OP; @HOI] and [@HS Part I, Chpt. 5], which deal with the “spinless” cases.
Write $$A_+:=\{a\in A\,\,\, | \,\,\, a^\alpha>1\quad \forall\,\alpha\in R^+\}$$ for the positive chamber of $A_{\textup{reg}}$. Note that $\mathcal{R}$ is contained in the ring $C^\omega(A_{+})$ of analytic functions on $A_+$.
Let $\delta$ be the analytic function on $A_+$ given by $$\label{deltagauge}
\delta(a):=a^{\rho}\prod_{\alpha\in R^+}(1-a^{-2\alpha})^{\frac{1}{2}}.$$ Conjugation by $\delta$ defines an outer automorphism of $\mathbb{D}_{\mathcal{R}}$. For $z\in U(\mathfrak{g})$ we denote by $$\label{Hz}
H_z:=\delta\circ\widehat{\Pi}(z)\circ\delta^{-1}\in\mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{k})^{\otimes 2}$$ the corresponding gauged differential operator. We furthermore write $$\label{fatH}
\mathbf{H}:=-\frac{1}{2}\delta\circ\bigl(\widehat{\Pi}(\Omega)+\|\rho\|^2\bigr)\circ\delta^{-1}.$$
\[qH\] The assignment $z\mapsto H_z$ defines an algebra map $Z(\mathfrak{g})\rightarrow \mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{k}\oplus\mathfrak{k})^{M}$. Furthermore, $$\label{qHam}
\mathbf{H}=-\frac{1}{2}\Delta-\frac{1}{2}\sum_{\alpha\in R}\frac{1}{(\xi_\alpha-\xi_{-\alpha})^2}
\Bigl(\frac{\|\alpha\|^2}{2}+\prod_{\epsilon\in\{\pm 1\}}(y_\alpha\otimes 1+
\xi_{\epsilon\alpha}(1\otimes y_\alpha))\Bigr).$$
The first statement is immediate from Theorem \[thmRAD\]. The proof of follows from the well known fact that $$\delta\circ\Bigl(\Delta+\frac{1}{2}\sum_{\alpha\in R}
\Bigl(\frac{\xi_\alpha+\xi_{-\alpha}}{\xi_\alpha-\xi_{-\alpha}}\Bigr)\partial_{t_\alpha}\Bigr)\circ\delta^{-1}=
\Delta-\|\rho\|^2+\sum_{\alpha\in R}\frac{\|\alpha\|^2}{2}\frac{1}{(\xi_\alpha-\xi_{-\alpha})^2},$$ see, e.g., the proof of [@HS Part I, Thm. 2.1.1].
For $\sigma: U(\mathfrak{k})^{\otimes 2}\rightarrow\textup{End}(V_\sigma)$ a finite dimensional representation we write $$H_z^\sigma:=\bigl(\textup{id}_{\mathbb{D}_{\mathcal{R}}}\otimes\sigma\bigr)H_z
\in\mathbb{D}_{\mathcal{R}}\otimes \textup{End}(V_\sigma)\qquad (z\in U(\mathfrak{g})).$$ Then $H_z^\sigma$ ($z\in Z(\mathfrak{g})$) are commuting $\textup{End}(V_\sigma)$-valued differential operators on $A$ which, by Proposition \[qH\], serve as quantum Hamiltonnians for the $\sigma$-spin generalisation of the quantum hyperbolic Calogero-Moser system with Schr[ö]{}dinger operator $$\mathbf{H}^\sigma:=\bigl(\textup{id}_{\mathbb{D}_{\mathcal{R}}}\otimes\sigma\bigr)(\mathbf{H}).$$ We now list a couple of interesting special cases of the $\sigma$-spin quantum hyperbolic Calogero-Moser systems.\
[**The spinless case:**]{} Take $\chi^\ell,\chi^r\in\textup{ch}(\mathfrak{k}_0)$. Their extension to complex linear algebra morphisms $U(\mathfrak{k})\rightarrow\mathbb{C}$ are again denoted by $\chi^\ell$ and $\chi^r$. Define $\chi^{\ell,r}_\alpha\in C^\omega(A)$ ($\alpha\in R$) by $$\label{chilr}
\chi^{\ell,r}_\alpha(a):=\chi^\ell(y_\alpha)+a^\alpha\chi^r(y_\alpha),\qquad a\in A_+.$$ Note that $\chi_{-\alpha}^{\ell,r}(a)=-(\chi^\ell(y_\alpha)+a^{-\alpha}\chi^r(y_\alpha))$ for $\alpha\in
R$. The Schr[ö]{}dinger operator $\mathbf{H}^{\chi^\ell\otimes\chi^r}$ then becomes $$\mathbf{H}^{\chi^\ell\otimes \chi^r}=
-\frac{1}{2}\Delta-\frac{1}{2}\sum_{\alpha\in R}\frac{1}{(\xi_\alpha-\xi_{-\alpha})^2}
\Bigl(\frac{\|\alpha\|^2}{2}-\chi_\alpha^{\ell,r}\chi_{-\alpha}^{\ell,r}\Bigr).$$ The special case $$\mathbf{H}^{\chi_0\otimes\chi_0}=-\frac{1}{2}\Delta-\frac{1}{2}\sum_{\alpha\in R}\frac{\|\alpha\|^2}{2}
\frac{1}{(\xi_\alpha-\xi_{-\alpha})^2}$$ with $\chi_0\in\textup{ch}(\mathfrak{k}_0)$ the trivial representation is the quantum Hamiltonian of the quantum hyperbolic Calogero-Moser system associated to the Riemannian symmetric space $G/K$. If $\mathfrak{g}$ is simple and of type $C_r$ ($r\geq 1$) then $\chi^\ell=c_\ell\chi_{\mathfrak{sp}}$ and $\chi^r=c_r\chi_{\mathfrak{sp}}$ for some $c_\ell, c_r\in\mathbb{C}$, see Subsection \[SSonedim\]. Using the explicit description of $\chi_{\mathfrak{sp}}$ from Subsection \[SSonedim\], we then obtain $$\mathbf{H}^{\chi^\ell\otimes\chi^r}=
-\frac{1}{2}\Delta-\frac{1}{2}\sum_{\alpha\in R_s^+}\frac{\|\alpha\|^2}{(\xi_\alpha-\xi_{-\alpha})^2}\\
+\frac{1}{4}\sum_{\beta\in R_\ell^+}\frac{\frac{1}{2}\|\beta\|^2+(c_\ell-c_r)^2}{(\xi_{\beta/2}+
\xi_{-\beta/2})^2}-\frac{1}{4}\sum_{\beta\in R_\ell^+}\frac{\frac{1}{2}\|\beta\|^2+(c_\ell+c_r)^2}
{(\xi_{\beta/2}-\xi_{-\beta/2})^2},$$ hence we recover a two-parameter subfamily of the $\textup{BC}_r$ quantum hyperbolic Calogero-Moser system. This extends [@HS Part I, Thm. 5.1.7], which deals with the special case that $\chi^\ell=-\chi^r$ with $\chi^\ell\in\textup{ch}(\mathfrak{k}_0)$ integrating to a multiplicative character of $K$.\
[**The one-sided spin case:**]{} Let $\chi\in\textup{ch}(\mathfrak{k}_0)$ and $\sigma_\ell: U(\mathfrak{k})\rightarrow
\textup{End}(V_\ell)$ a finite dimensional representation. Then $$\mathbf{H}^{\sigma_\ell\otimes\chi}=-\frac{1}{2}\Delta-\frac{1}{2}\sum_{\alpha\in R}\frac{1}{(\xi_\alpha-\xi_{-\alpha})^2}
\Bigl(\frac{\|\alpha\|^2}{2}+\prod_{\epsilon\in\{\pm 1\}}(\sigma_\ell(y_\alpha)+
\xi_{\epsilon\alpha}\chi(y_\alpha))\Bigr).$$ In the special case that $\chi=\chi_0\in\textup{ch}(\mathfrak{k}_0)$ is the trivial representation the Schr[ö]{}dinger operator reduces to $$\mathbf{H}^{\sigma_\ell\otimes\chi_0}=-\frac{1}{2}\Delta-\frac{1}{2}\sum_{\alpha\in R}\frac{1}{(\xi_\alpha-\xi_{-\alpha})^2}
\Bigl(\frac{\|\alpha\|^2}{2}+\sigma_\ell(y_\alpha^2)\Bigr).$$ Finally, if $\mathfrak{g}$ is simple and of type $C_r$ ($r\geq 1$) and $\chi=c\chi_{\mathfrak{sp}}$ with $c\in\mathbb{C}$, then $$\begin{split}
\mathbf{H}^{\sigma_\ell\otimes\chi}=-\frac{1}{2}\Delta&-\frac{1}{2}\sum_{\alpha\in R^+_s}
\frac{\|\alpha\|^2+2\sigma_\ell(y_\alpha^2)}{(\xi_\alpha-\xi_{-\alpha})^2}\\
&+\frac{1}{4}\sum_{\beta\in R_\ell^+}\frac{\frac{1}{2}\|\beta\|^2+(\sigma_\ell(y_\beta)-c)^2}
{(\xi_{\beta/2}+\xi_{-\beta/2})^2}-\frac{1}{4}
\sum_{\beta\in R_\ell^+}\frac{\frac{1}{2}\|\beta\|^2+(\sigma_\ell(y_\beta)+c)^2}{(\xi_{\beta/2}-\xi_{-\beta/2})^2}.
\end{split}$$
Feh[é]{}r and Pusztai [@FP1; @FP2] obtained the classical analog of the one-sided spin quantum Calogero-Moser system by Hamiltonian reduction. This is extended to double-sided spin Calogero-Moser systems in [@Re].
[**The matrix case:**]{} The following special case is relevant for the theory of matrix-valued spherical functions [@GPT], [@HvP §7]. Let $\tau: \mathfrak{k}\rightarrow \mathfrak{gl}(V_\tau)$ be a finite dimensional representation. Consider $\textup{End}(V_\tau)$ as left $U(\mathfrak{k})^{\otimes 2}$-module by $$\label{Endaction}
\sigma_\tau(x\otimes y)T:=\tau(x)T\tau(S(y))$$ for $x,y\in U(\mathfrak{k})$ and $T\in\textup{End}(V_\tau)$. Note that $\textup{End}(V_\tau)\simeq
V_\tau\otimes V_\tau^\ast$ as $U(\mathfrak{k})^{\otimes 2}$-modules. The associated Schr[ö]{}dinger operator $\mathbf{H}^{\sigma_\tau}$ acts on $T\in C^\infty(A_{\textup{reg}};\textup{End}(V_\tau))$ by $$\begin{split}
\bigl(\mathbf{H}^{\sigma_\tau}T\bigr)(a)=&-\frac{1}{2}(\Delta T)(a)\\
&-\frac{1}{2}\sum_{\alpha\in R}\frac{\frac{\|\alpha\|^2}{2}T(a)+\tau(y_\alpha^2)T(a)
-(a^\alpha+a^{-\alpha})\tau(y_\alpha)T(a)\tau(y_\alpha)+T(a)\tau(y_\alpha^2)}
{(a^\alpha-a^{-\alpha})^2}
\end{split}$$ for $a\in A_{\textup{reg}}$.
$\sigma$-Harish-Chandra series {#S5}
------------------------------
In this subsection we recall the construction of the Harish-Chandra series following [@W Chpt. 9]. They were defined by Harish-Chandra to analyse the asymptotic behaviour of matrix coefficients of admissible $G$-representations and of the associated spherical functions (see, e.g., [@BS; @CM; @HS] and references therein).
Consider the ring $\mathbb{C}[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]$ of formal power series at infinity in $A_+$. We express elements $f\in\mathbb{C}[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]$ as $f=\sum_{\gamma\in Q_-}c_\gamma\xi_\gamma$ with $c_\gamma\in\mathbb{C}$ and $$Q_-:=\bigoplus_{j=1}^r\mathbb{Z}_{\leq 0}\,\alpha_j
\subseteq Q:=\mathbb{Z}R.$$ We consider $\mathcal{R}$ as subring of $\mathbb{C}[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]$ using power series expansion at infinity in $A_+$ (e.g., $(1-\xi_{-2\alpha})^{-1}=\sum_{m=0}^{\infty}\xi_{-2m\alpha}$ for $\alpha\in R^+$). Similarly, we view $\xi_{-\rho}\delta$ as element in $\mathbb{C}[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]$ through its power series expansion at infinity, where $\delta$ is given by .
For $A$ a complex associative algebra we write $A[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\lambda$ for the $\mathbb{C}[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]$-module of formal series $g=\sum_{\gamma\in Q_-}d_\gamma\xi_{\lambda+\gamma}$ with coefficients $d_\gamma\in A$. If $A=U(\mathfrak{k})^{\otimes 2}$ or $A=\textup{End}(V_\sigma)$ for some $\mathfrak{k}\oplus\mathfrak{k}$-module $V_\sigma$ then $A[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\lambda$ becomes a $\mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{k})^{\otimes 2}$-module.
Set $$\label{HCgeneric}
\mathfrak{h}_{\textup{HC}}^*:=\{\lambda\in\mathfrak{h}^* \,\, | \,\,
(2(\lambda+\rho)+\gamma,\gamma)\not=0\quad \forall\, \gamma\in Q_-\setminus\{0\}\}.$$ The Harish-Chandra series associated to the triple $(\mathfrak{g}_0,\mathfrak{h}_0,\theta_0)$ is the following formal $U(\mathfrak{k})^{\otimes 2}$-valued eigenfunction of the $U(\mathfrak{k})^{\otimes 2}$-valued differential operator $\widehat{\Pi}(\Omega)$.
\[cordefHC\] Let $\lambda\in
\mathfrak{h}_{\textup{HC}}^*$. There exists a unique $U(\mathfrak{k})^{\otimes 2}$-valued formal series $$\label{Psie}
\Phi_\lambda:=\sum_{\gamma\in Q_-}\Gamma_\gamma(\lambda)\xi_{\lambda+\gamma}\in
U(\mathfrak{k})^{\otimes 2}[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\lambda$$ with coefficients $\Gamma_\gamma(\lambda)\in U(\mathfrak{k})^{\otimes 2}$ and $\Gamma_0(\lambda)=1$, satisfying $$\label{evPsie}
\widehat{\Pi}(\Omega)\Phi_\lambda=(\lambda,\lambda+2\rho)\Phi_\lambda.$$
In fact, if $\lambda\in\mathfrak{h}_{\textup{HC}}^*$ then the eigenvalue equation for a formal series of the form gives recursion relations for its coefficients $\Gamma_\lambda(\gamma)$ ($\gamma\in Q_-$) which, together with the condition $\Gamma_0(\lambda)=1$, determine the coefficients $\Gamma_\gamma(\lambda)$ uniquely. We call the $\Gamma_\gamma(\lambda)\in U(\mathfrak{k})^{\otimes 2}$ ($\gamma\in Q_-$) the Harish-Chandra coefficients. The sum $U(\mathfrak{h})+\theta(\mathfrak{n})U(\mathfrak{g})$ in $U(\mathfrak{g})$ is an internal direct sum containing $Z(\mathfrak{g})$. Denote by $\textup{pr}: Z(\mathfrak{g})\rightarrow U(\mathfrak{\mathfrak{h}})$ the restriction to $Z(\mathfrak{g})$ of the projection $U(\mathfrak{h})\oplus\theta(\mathfrak{n})U(\mathfrak{g})\rightarrow U(\mathfrak{h})$ on the first direct summand. Then $\textup{pr}$ is an algebra homomorphism (see, e.g., [@CM §1]). The central character at $\lambda\in\mathfrak{h}^*$ is the algebra homomorphism $$\zeta_\lambda: Z(\mathfrak{g})\rightarrow\mathbb{C},\qquad
z\mapsto \lambda(\textup{pr}(z))$$ with $\lambda(\textup{pr}(z))$ the evaluation of $\textup{pr}(z)\in U(\mathfrak{h})\simeq S(\mathfrak{h})$ at $\lambda$. By the second expression of the Casimir element $\Omega$ in we have $\zeta_\lambda(\Omega)=(\lambda,\lambda+2\rho)$. Furthermore, by [@Di Prop. 7.4.7], $\zeta_{\lambda-\rho}=\zeta_{\mu-\rho}$ for $\lambda,\mu\in\mathfrak{h}^*$ if and only if $\lambda\in W\mu$.
\[Casimirs\] Let $\lambda\in\mathfrak{h}_{\textup{HC}}^*$. Then $$\widehat{\Pi}(z)\Phi_\lambda=\zeta_\lambda(z)\Phi_\lambda\qquad \forall\, z\in Z(\mathfrak{g})$$ in $U(\mathfrak{k})^{\otimes 2}[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\lambda$.
Write $x^\eta:=x_1^{\eta_1}\cdots x_r^{\eta_r}\in S(\mathfrak{h})$ and $\partial^\eta:=\partial_{x_1}^{\eta_1}\cdots\partial_{x_r}^{\eta_r}\in\mathbb{D}_{\mathcal{R}}$ for $\eta\in\mathbb{Z}_{\geq 0}^r$. The leading symbol of $D=\sum_{\eta\in\mathbb{Z}_{\geq 0}^r}\bigl(\sum_{\gamma\in Q_-}
c_{\eta,\gamma}\xi_\gamma\bigr)\partial^\eta\in\mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{k})^{\otimes 2}$ is defined to be $$\mathfrak{s}_\infty(D):=\sum_{\eta\in\mathbb{Z}_{\geq 0}^r}c_{\eta,0}x^\eta\in S(\mathfrak{h})
\otimes U(\mathfrak{k})^{\otimes 2}.$$ Fix $z\in Z(\mathfrak{g})$. Let $z_\lambda^\infty\in U(\mathfrak{k})^{\otimes 2}$ be the evaluation of the leading symbol $\mathfrak{s}_\infty(\widehat{\Pi}(z))$ at $\lambda$. Note that the $\xi_\lambda$-component of the formal power series $\widehat{\Pi}(z)\Phi_\lambda$ is $z_\lambda^{\infty}$. Furthermore, $\widehat{\Pi}(z)\Phi_\lambda\in
U(\mathfrak{k})^{\otimes 2}[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\lambda$ is an eigenfunction of $\widehat{\Pi}(\Omega)$ with eigenvalue $(\lambda,\lambda+2\rho)$ by Theorem \[thmRAD\] [**b**]{}.
For any $y\in U(\mathfrak{k})^{\otimes 2}$, the formal power series $$\Phi_\lambda y:=\sum_{\gamma\in Q_-}(\Gamma_\gamma(\lambda)y)\xi_{\lambda+\gamma}$$ is the unique eigenfunction of $\widehat{\Pi}(\Omega)$ of the form $\sum_{\gamma\in Q_-}\widetilde{\Gamma}_\gamma(\lambda)\xi_{\lambda+\gamma}$ ($\widetilde{\Gamma}_\gamma(\lambda)\in U(\mathfrak{k})^{\otimes 2}$) with eigenvalue $(\lambda,\lambda+2\rho)$ and leading coefficient $\widetilde{\Gamma}_0(\lambda)$ equal to $y$ (cf. Proposition \[cordefHC\]). It thus follows that $$\widehat{\Pi}(z)\Phi_\lambda=\Phi_\lambda z_\lambda^\infty.$$ By [@CM Prop. 2.6(ii)] we have $$\mathfrak{s}_\infty\bigl(\widehat{\Pi}(z)\bigr)=\mathfrak{s}_\infty\bigl(\widehat{\Pi}(\textup{pr}(z))\bigr),$$ hence $z_\lambda^\infty=\lambda\bigl(\textup{pr}(z)\bigr)1_{U(\mathfrak{k})^{\otimes 2}}=
\zeta_\lambda(z)1_{U(\mathfrak{k})^{\otimes 2}}$. This concludes the proof of the proposition.
\[Minvariance2\] By Remark \[Minvariance\] and by an argument similar to the proof of Proposition \[Casimirs\], it follows that $\Gamma_\gamma(\lambda)\in U(\mathfrak{k}\oplus\mathfrak{k})^M$ for $\lambda\in\mathfrak{h}_{\textup{HC}}^*$ and $\gamma\in Q_-$.
Fix a finite dimensional representation $\sigma: U(\mathfrak{k})^{\otimes 2}\rightarrow\textup{End}(V_\sigma)$. For $\lambda\in\mathfrak{h}_{\textup{HC}}^*$ set $$\label{Ftau}
\Phi_\lambda^\sigma:=\sum_{\gamma\in Q_-}\sigma(\Gamma_\gamma(\lambda))\xi_{\lambda+\gamma}\in \textup{End}(V_\sigma)[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\lambda,$$ We call $\Phi_\lambda^\sigma$ the $\sigma$-Harish-Chandra series, and $$\Gamma_\gamma^\sigma(\lambda):=\sigma(\Gamma_\gamma(\lambda))\in
\textup{End}(V_\sigma),\qquad \gamma\in Q_-$$ the associated Harish-Chandra coefficients.
Suppose that $V_\sigma$ integrates to a $K\times K$-representation. Let $\textup{End}_M(V_\sigma)$ be the space of $M$-intertwiners $V_\sigma\rightarrow V_\sigma$ with respect to the diagonal action of $M$ on $V_\sigma$. Then $\Gamma_\gamma^\sigma(\lambda)\in\textup{End}_M(V_\sigma)$ for all $\gamma\in Q_-$ by Remark \[Minvariance2\].
Note that $\Phi_\lambda^\sigma$ is the unique formal power series $\sum_{\gamma\in Q_-}\Gamma_\gamma^\sigma(\lambda)\xi_{\lambda+\gamma}$ with $\Gamma_\gamma^\sigma(\lambda)\in\textup{End}(V_\sigma)$ and $\Gamma_\gamma^\sigma(\lambda)=\textup{id}_{V_\sigma}$ satisfying $\widehat{\Pi}(\Omega)\Phi_\lambda^\sigma=(\lambda,\lambda+2\rho)\Phi_\lambda^\sigma$. The $\sigma$-Harish-Chandra series in addition satisfies the eigenvalue equations $\widehat{\Pi}(z)\Phi_\lambda^\sigma=\zeta_\lambda(z)\Phi_\lambda^\sigma$ for all $z\in Z(\mathfrak{g})$.
Endow $\textup{End}(V_\sigma)$ with the norm topology. The recursion relations arising from the eigenvalue equation $\widehat{\Pi}(\Omega)\Phi_\lambda^\sigma=\zeta_\lambda(\Omega)\Phi_\lambda^\sigma$ imply growth estimates for the Harish-Chandra coefficients $\Gamma_\gamma^\sigma(\lambda)$. It leads to the following result (cf. [@W] and references therein).
\[holomorphic\] Let $\lambda\in\mathfrak{h}_{\textup{HC}}^*$. Then $$\Phi_\lambda^\sigma(a):=\sum_{\gamma\in Q_-}\Gamma_\gamma^\sigma(\lambda)a^{\lambda+\gamma},
\qquad a\in A_+$$ defines an $\textup{End}(V_\sigma)$-valued analytic function on $A_+$.
\[remaHCspherplus\] Set $G_{\textup{reg}}:=KA_+K\subset G$, which is an open dense subset of $G$. We have $k_\ell a k_r^{-1}=k_\ell^\prime a^\prime k_r^\prime{}^{-1}$ for $k_\ell, k_\ell^\prime, k_r, k_r^\prime\in K$ and $a,a^\prime\in A_+$ iff $a^\prime=a$ and $k_\ell^{-1}k_\ell^\prime=k_r^{-1}k_r^\prime\in M$. If $\sigma: K\times K\rightarrow \textup{GL}(V_\sigma)$ is a finite dimensional $K\times K$-representation, then a smooth function $f: G_{\textup{reg}}\rightarrow V_\sigma$ is called a $\sigma$-spherical function on $G_{\textup{reg}}$ if $$f(k_1gk_2^{-1})=\sigma(k_1,k_2)f(g)\qquad \forall\, g\in G_{\textup{reg}},\,\, \forall\, k_1,k_2\in K.$$ Then for $\lambda\in\mathfrak{h}_{\textup{HC}}^*$ and $v\in V_\sigma^M$, $$H_{\lambda}^{v}(k_1ak_2^{-1}):=\sigma(k_1,k_2)\Phi_\lambda^\sigma(a)v\qquad
(a\in A_+,\,\, k_1,k_2\in K)$$ defines a $\sigma$-spherical function on $G_{\textup{reg}}$. It in general does not extend to a $\sigma$-spherical function on $G$.
The Harish-Chandra series immediately provide “asymptotically free” common eigenfunctions for the quantum Hamiltonians $H_z^\sigma$ ($z\in Z(\mathfrak{g})$) of the $\sigma$-spin quantum hyperbolic Calogero-Moser system.
\[normalizedHCseriesthm\] Fix $\lambda\in\mathfrak{h}_{\textup{HC}}^*+\rho$. The $\textup{End}(V_\sigma)$-valued analytic function $$\label{normalizedHCseries}
\mathbf{\Phi}_\lambda^\sigma(a):=\delta(a)\Phi_{\lambda-\rho}^\sigma(a),\qquad a\in A_+$$ has a series expansion of the form $$\mathbf{\Phi}_\lambda^\sigma=\sum_{\gamma\in Q_-}
\mathbf{\Gamma}_\gamma^\sigma(\lambda)\xi_{\lambda+\gamma}
\in\textup{End}(V_\sigma)[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_{\lambda}$$ with $\mathbf{\Gamma}_\gamma^\sigma(\lambda)\in\textup{End}(V_\sigma)$ and $\mathbf{\Gamma}_0^\sigma(\lambda)=\textup{id}_{V_\sigma}$. It satisfies the Schr[ö]{}dinger equation $$\mathbf{H}^\sigma\bigl(\mathbf{\Phi}_\lambda^\sigma\bigr)=-\frac{(\lambda,\lambda)}{2}
\mathbf{\Phi}_\lambda^\sigma$$ as well as the eigenvalue equations $$H_z^\sigma\mathbf{\Phi}_\lambda^\sigma=\zeta_{\lambda-\rho}(z)\mathbf{\Phi}_\lambda^\sigma\qquad \forall\, z\in
Z(\mathfrak{g})$$ as $\textup{End}(V_\sigma)$-analytic functions on $A_+$.
This is an immediate consequence of Proposition \[Casimirs\] and the definitions of the differential operators $\mathbf{H}^\sigma$ and $H_z^\sigma$ ($z\in Z(\mathfrak{g})$).
Principal series representations {#SectionRepTh}
================================
We keep the conventions of the previous section. In particular, $(\mathfrak{g}_0,\mathfrak{h}_0,\theta_0)$ is a triple with $\mathfrak{g}_0$ a split real semisimple Lie algebra, $\mathfrak{h}_0$ a split Cartan subalgebra and $\theta_0$ a Cartan involution such that $\theta_0|_{\mathfrak{h}_0}=
-\textup{id}_{\mathfrak{h}_0}$, and $(G,K)$ is the associated non-compact split symmetric pair. We fix throughout this section two finite dimensional $K$-representations $\sigma_\ell: K\rightarrow\textup{GL}(V_\ell)$ and $\sigma_r: K\rightarrow\textup{GL}(V_r)$. We write $(\cdot,\cdot)_{V_\ell}$ and $(\cdot,\cdot)_{V_r}$ for scalar products on $V_\ell$ and $V_r$ turning $\sigma_\ell$ and $\sigma_r$ into unitary representations of $K$. We view $\textup{Hom}(V_r,V_\ell)$ as finite dimensional $K\times K$-representation with representation map $\sigma: K\times K\rightarrow\textup{GL}(\textup{Hom}(V_r,V_\ell))$ given by $$\label{tauaction}
\sigma(k_\ell,k_r)T:=\sigma_\ell(k_\ell)T\sigma_r(k_r^{-1})$$ for $k_\ell,k_r\in K$ and $T\in \textup{Hom}(V_r,V_\ell)$. It is canonically isomorphic to the tensor product representation $V_\ell\otimes V_r^*$.
For further details on the first two subsections, see [@Kn0 Chpt. 8].
Admissible representations and associated spherical functions {#ssss1}
-------------------------------------------------------------
Let $K^\wedge$ be the equivalence classes of the irreducible unitary representations of $K$. A representation $\pi: G\rightarrow \textup{GL}(\mathcal{H})$ of $G$ on a Hilbert space $\mathcal{H}$ is called admissible if the restriction $\pi|_K$ of $\pi$ to $K$ is unitary and if the $\tau$-isotypical component $\mathcal{H}(\tau)$ of $\pi|_K$ is finite dimensional for all $\tau\in K^\wedge$.
Let $\pi: G\rightarrow\textup{GL}(\mathcal{H})$ be an admissible representation. The subspace $$\mathcal{H}^\infty:=\{v\in\mathcal{H} \,\, | \,\, G\rightarrow\mathcal{H}, \,\,
g\mapsto \pi(g)v \hbox{\,\, is } C^\infty\}$$ of smooth vectors in $\mathcal{H}$ is a $G$-stable dense subspace of $\mathcal{H}$. Differentiating the $G$-action on $\mathcal{H}^\infty$ turns $\mathcal{H}^\infty$ into a left $U(\mathfrak{g})$-module. We write $x\mapsto x_{\mathcal{H}^{\infty}}$ for the corresponding action of $x\in U(\mathfrak{g})$.
The algebraic direct sum $$\mathcal{H}^{K-\textup{fin}}:=\bigoplus_{\tau\in K^\wedge}\mathcal{H}(\tau)$$ is the dense subspace of $K$-finite vectors in $\mathcal{H}$. It is contained in $\mathcal{H}^\infty$ since $\pi$ is admissible. Furthermore, it inherits a $(\mathfrak{g},K)$-module structure from $\mathcal{H}^\infty$. The $(\mathfrak{g},K)$-module $\mathcal{H}^{K-\textup{fin}}$ is called the Harish-Chandra module of $\mathcal{H}$.
For $\phi_\ell\in\textup{Hom}_K(\mathcal{H},V_\ell)$ and $\phi_r\in\textup{Hom}_K(V_r,\mathcal{H})$ we now obtain $\sigma$-spherical functions $$f^{\phi_\ell,\phi_r}_{\mathcal{H}}\in C^\infty_\sigma(G)$$ by $$f^{\phi_\ell,\phi_r}_{\mathcal{H}}(g):=\phi_\ell\circ\pi(g)\circ\phi_r,\qquad g\in G.$$ The $\sigma$-spherical functions $f^{\phi_\ell,\phi_r}_{\mathcal{H}}$ are actually $\textup{Hom}(V_r,V_\ell)$-valued real analytic functions on $G$, see, e.g., [@Kn0 Thm. 8.7]. Furthermore, $f_{\mathcal{H}}^{\phi_\ell,\phi_r}|_{A}$ takes values in $\textup{Hom}_M(V_r,V_\ell)$.
Since $V_\ell$ and $V_r$ are finite dimensional, we have canonical isomorphisms $$\label{algvsanal}
\begin{split}
\textup{Hom}_K(\mathcal{H},V_\ell)&\simeq\textup{Hom}_{\mathfrak{k}}(\mathcal{H}^\infty,V_\ell)
\simeq \textup{Hom}_{\mathfrak{k}}(\mathcal{H}^{K-\textup{fin}},V_\ell),\\
\textup{Hom}_K(V_r,\mathcal{H})&\simeq\textup{Hom}_{\mathfrak{k}}(V_r,\mathcal{H}^\infty)\simeq \textup{Hom}_{\mathfrak{k}}(V_r,\mathcal{H}^{K-\textup{fin}}).
\end{split}$$
The $\sigma$-spherical function $f^{\phi_\ell,\phi_r}_{\mathcal{H}}$ can be expressed in terms of matrix coefficients of $\pi$ as follows. Let $\{v_i\}_i$ and $\{w_j\}_j$ be linear bases of $V_\ell$ and $V_r$, respectively. Expand $\phi_\ell\in\textup{Hom}_K(\mathcal{H},V_\ell)$ and $\phi_r\in\textup{Hom}_K(V_r,\mathcal{H})$ as $$\phi_\ell=\sum_i\langle \cdot,f_i\rangle_{\mathcal{H}}v_i,\qquad
\phi_r=\sum_j(\cdot,w_j)_{V_r}h_j$$ with $f_i,h_j\in\mathcal{H}^{K-\textup{fin}}$, where $\langle\cdot,\cdot\rangle_{\mathcal{H}}$ is the scalar product of $\mathcal{H}$. The fact that $\phi_\ell$ and $\phi_r$ are $K$-intertwiners implies that $\sum_if_i\otimes v_i$ and $\sum_jw_j\otimes h_j$ are $K$-fixed in $\mathcal{H}\otimes V_\ell$ and $V_r\otimes \mathcal{H}$, respectively. The $\sigma$-spherical function $f^{\phi_\ell,\phi_r}_{\mathcal{H}}\in C^\infty_\sigma(G)$ is then given by $$\label{sphermatrix}
f^{\phi_\ell,\phi_r}_{\mathcal{H}}(g)=\sum_{i,j}\langle\pi(g)h_j,f_i\rangle_{\mathcal{H}}(\cdot,w_j)_{V_r}v_i.$$ Clearly, for an admissible representation $(\pi,\mathcal{H})$, the subspace of $\sigma$-spherical functions spanned by $f^{\phi_\ell,\phi_r}_{\mathcal{H}}$ ($\phi_\ell\in\textup{Hom}_K(\mathcal{H}, V_\ell)$, $\phi_r\in\textup{Hom}_K(V_r,\mathcal{H})$), is finite dimensional.
Principal series representations and $K$-intertwiners {#subsectionPSR}
-----------------------------------------------------
\[ssss2\] Recall that $M:=Z_K(\mathfrak{h}_0)\subseteq K$ is a finite group, since $\mathfrak{g}_0$ is split. Furthermore, if $G$ has a complexification then $M$ is abelian (see [@Kn Thm. 7.53]). We fix a finite dimensional irreducible representation $\xi: M\rightarrow \textup{GL}(L_\xi)$. Write $\langle\cdot,\cdot\rangle_{\xi}$ for the scalar product on $L_\xi$ turning it into a unitary representation. Fix a linear functional $\lambda\in\mathfrak{h}^*$ and extend it to a representation $\eta_{\lambda}^{(\xi)}: P\rightarrow\textup{GL}(L_\xi)$ of the minimal parabolic subgroup $P=MAN$ of $G$ by $$\eta_{\lambda}^{(\xi)}(man):=a^{\lambda}\xi(m)\qquad (m\in M, a\in A, n\in N).$$ Consider the pre-Hilbert space $U^{(\xi)}_{\lambda}$ consisting of continuous, compactly supported functions $f: G\rightarrow L_\xi$ satisfying $$f(gp)=\eta_{\lambda+\rho}^{(\xi)}(p^{-1})f(g)\qquad (g\in G, p\in P)$$ with scalar product $$\langle f_1,f_2\rangle^{(\xi)}_{\lambda}:=\int_K\langle f_1(x), f_2(x)\rangle_\xi dx\qquad
(f_1,f_2\in U^{(\xi)}_{\lambda}).$$ Consider the action of $G$ on $U^{(\xi)}_{\lambda}$ by $(\pi^{(\xi)}_\lambda(g)f)(g^\prime):=f(g^{-1}g^\prime)$ for $g,g^\prime\in G$ and $f\in U^{(\xi)}_{\lambda}$. Its extension to an admissible representation $\pi^{(\xi)}_{\lambda}: G\rightarrow
\textup{GL}(\mathcal{H}^{(\xi)}_{\lambda})$, with $\mathcal{H}^{(\xi)}_{\lambda}$ the Hilbert space completion of $U^{(\xi)}_{\lambda}$, is called the [*principal series representation*]{} of $G$. The representation $\pi_\lambda^{(\xi)}$ is unitary if $\eta_\lambda^{(\xi)}$ is unitary, i.e., if $\lambda(\mathfrak{h}_0)\subset i\mathbb{R}$.
Analogously, let $\eta_\lambda: AN\rightarrow\mathbb{C}^*$ be the one-dimensional representation defined by $\eta_\lambda(an):=a^{\lambda}$ for $a\in A$ and $n\in N$, and consider the pre-Hilbert space $U_{\lambda}$ consisting of continuous, compactly supported functions $f: G\rightarrow \mathbb{C}$ satisfying $$f(gb)=\eta_{\lambda+\rho}(b^{-1})f(g)\qquad (g\in G, b\in AN)$$ with scalar product $$\langle f_1,f_2\rangle_{\lambda}:=\int_Kf_1(x)\overline{f_2(x)}\, dx\qquad
(f_1,f_2\in U_{\lambda}).$$ Turning $U_\lambda$ into a $G$-representation by $(\pi_{\lambda}(g)f)(g^\prime):=f(g^{-1}g^\prime)$ for $g,g^\prime\in G$ and completing, gives an admissible representation $\pi_{\lambda}: G\rightarrow
\textup{GL}(\mathcal{H}_{\lambda})$. Note that $\pi_\lambda|_K: K\rightarrow\textup{GL}(\mathcal{H}_\lambda)$ is isomorphic to the left regular representation of $K$ on $L^2(K)$. In particular, $\textup{dim}(\mathcal{H}_\lambda(\tau))=\textup{deg}(\tau)^2$ for all $\tau\in K^\wedge$, where $\textup{deg}(\tau)$ is the degree of $\tau$. Furthermore, $\mathcal{H}_\lambda\simeq\bigoplus_{\xi\in M^\wedge}\bigl(\mathcal{H}_\lambda^{(\xi)}\bigr)^{\oplus\textup{deg}(\xi)}$.
Define for $\phi_\ell\in\textup{Hom}_K(\mathcal{H}_\lambda,V_\ell)$ the adjoint map $\phi^*_\ell: V_\ell\rightarrow
\mathcal{H}_\lambda$ by $$(\phi_\ell(f),v)_{V_\ell}=\langle f,\phi_\ell^*(v)\rangle_\lambda\qquad \forall\, f\in\mathcal{H}_\lambda,\,\, \forall
v\in V_\ell.$$ Since $\mathcal{H}_\lambda$ is unitary as $K$-representation for all $\lambda\in\mathfrak{h}^*$, the assignment $\phi_\ell\mapsto \phi_\ell^*$ defines a conjugate linear isomorphism from $\textup{Hom}_K(\mathcal{H}_\lambda,V_\ell)$ onto $\textup{Hom}_K(V_\ell,\mathcal{H}_\lambda)$.
The $\sigma$-spherical functions $f_{\mathcal{H}_\lambda}^{\phi_\ell,\phi_r}$ obtained from the $G$-representation $\mathcal{H}_\lambda$ using the $K$-intertwiners $\phi_\ell\in\textup{Hom}_K(\mathcal{H}_\lambda,V_{\ell})$ and $\phi_r\in\textup{Hom}_K(V_{r},\mathcal{H}_\lambda)$ now admit the following explicit description in terms of the Eisenstein integral.
\[relEisPrin\] Fix $\lambda\in\mathfrak{h}^*$.
1. The map $\j_{\lambda,V_r}: \textup{Hom}_K(V_r,\mathcal{H}_\lambda)\rightarrow V_r^*
$, $$\j_{\lambda,V_r}(\phi_r)(v):=\phi_r(v)(1)\qquad (v\in V_r),$$ is a linear isomorphism.
2. For $\phi_\ell\in\textup{Hom}_K(\mathcal{H}_\lambda,V_\ell)$ let $\iota_{\lambda,V_\ell}(\phi_\ell)\in V_\ell$ be the unique vector such that $$\bigl(v,\iota_{\lambda,V_\ell}(\phi_\ell)\bigr)_{V_\ell}=\phi_\ell^*(v)(1)\qquad \forall\, v\in V_\ell.$$ The resulting map $\iota_{\lambda,V_\ell}: \textup{Hom}_K(\mathcal{H}_\lambda,V_\ell)\rightarrow
V_\ell$ is a linear isomorphism.
3. The assignment $\phi_\ell\otimes\phi_r\mapsto
T^{\phi_\ell,\phi_r}_\lambda:=
\iota_{\lambda,V_\ell}(\phi_\ell)\otimes\j_{\lambda,V_r}(\phi_r)$ defines a linear isomorphism $$\textup{Hom}_K(\mathcal{H}_\lambda,V_{\ell})\otimes\textup{Hom}_K(V_{r},\mathcal{H}_\lambda)\overset{\sim}{\longrightarrow} V_{\ell}\otimes V_{r}^*\simeq\textup{Hom}(V_{r},V_{\ell}).$$ Furthermore, $$\label{relEis}
f_{\mathcal{H}_\lambda}^{\phi_\ell,\phi_r}(g)=E_\lambda^\sigma(g)T^{\phi_\ell,\phi_r}_\lambda\qquad (g\in G)$$ for $\phi_\ell\in\textup{Hom}_K(\mathcal{H}_\lambda,V_{\ell})$ and $\phi_r\in\textup{Hom}_K(V_{r},\mathcal{H}_\lambda)$, with $E_\lambda^\sigma(g)$ the Eisenstein integral .
Our choice of parametrisation of the $\sigma$-spherical functions associated to $\pi_\lambda$, which deviates from the standard choice (see, e.g., [@Kn0 §8.2]), plays an important in Section \[SectionbKZB\] when discussing the applications to asymptotic boundary KZB equations. We provide here a proof directly in terms of our present conventions.
We will assume without loss of generality that $\sigma_\ell, \sigma_r\in K^\wedge$.\
[**a.**]{} Since $\textup{dim}(\textup{Hom}_K(V_r,\mathcal{H}_\lambda))=\textup{deg}(\sigma_r)$ it suffices to show that $\j_{\lambda,V_r}$ is injective. Fix an orthonormal basis $\{v_i\}_i$ of $V_{r}$. Let $\phi_r\in\textup{Hom}_K(V_r,\mathcal{H}_\lambda)$ and consider its expansion $\phi_r=
\sum_j(\cdot,v_j)_{V_r} h_i$ with $h_j\in\mathcal{H}_\lambda^{K-\textup{fin}}$. Then $$\label{jALT}
\j_{\lambda,V_r}(\phi_r)=\sum_jh_j(1)(\cdot,v_j)_{V_r}.$$ Furthermore, for each index $j$ we have $$\label{transfoK}
h_j(x)=\sum_ih_i(1)(v_j,\sigma_r(x)v_i)_{V_r}\qquad \forall\, x\in K$$ since $\phi_r$ is a $K$-intertwiner.
Suppose now that $\j_{\lambda,V_r}(\phi_r)=0$. Then $h_j(1)=0$ for all $j$ by . By we conclude that $h_j(kan)=a^{-\lambda-\rho}h_j(k)=0$ for $k\in K$, $a\in A$ and $n\in N$, so $\phi_r=0$.\
[**b.**]{} This immediately follows from part [**a**]{} and the fact that $$(v,\iota_{\lambda,V_\ell}(\phi_\ell))_{V_\ell}=\j_{\lambda,V_\ell}(\phi^*_\ell)(v)$$ for $v\in V_\ell$ and $\phi_\ell\in\textup{Hom}_K(\mathcal{H}_\lambda,V_\ell)$.\
[**c.**]{} The first statement immediately follows from [**a**]{} and [**b**]{}. Let $\{v_i\}_i$ be an orthonormal basis of $V_{\ell}$ and $\{w_j\}_j$ an orthonormal basis of $V_{r}$. For $\phi_\ell=\sum_i\langle \cdot,f_i\rangle_\lambda v_i\in\textup{Hom}_K(\mathcal{H}_\lambda,V_{\ell})$ and $\phi_r=\sum_j(\cdot,w_j)_{V_r}h_j\in \textup{Hom}_K(V_{r},\mathcal{H}_\lambda)$ with $f_i,h_j\in\mathcal{H}_\lambda^{K-\textup{fin}}$ a direct computation gives $$f_{\mathcal{H}_\lambda}^{\phi_\ell,\phi_r}(g)=E_\lambda^\sigma(g)\widetilde{T}^{\phi_\ell,\phi_r}_\lambda$$ with $\widetilde{T}^{\phi_\ell,\phi_r}_\lambda\in\textup{Hom}(V_r,V_\ell)$ given by $$\widetilde{T}_\lambda^{\phi_\ell,\phi_r}(w)=\sum_{i,j}h_j(1)\overline{f_i(1)}(w,w_j)_{V_r}v_i\qquad (w\in V_{r}).$$ By this can be rewritten as $$\widetilde{T}_\lambda^{\phi_\ell,\phi_r}(w)=\j_{\lambda,V_r}(\phi_r)(w)\sum_i\overline{f_i(1)}v_i
\qquad w\in V_r,$$ hence it suffices to show that $$\label{iotaALT}
\iota_{\lambda,V_\ell}(\phi_\ell)=\sum_i\overline{f_i(1)}v_i.$$ Define $\widetilde{\chi}_{\lambda,V_\ell}\in \mathcal{H}_\lambda^{K-\textup{fin}}$ by $$\widetilde{\chi}_{\lambda,V_\ell}(kan):=\textup{deg}(\sigma_\ell)a^{-\lambda-\rho}\chi_{V_\ell}(k)\qquad
(k\in K, a\in A, n\in N),$$ with $\chi_{V_\ell}$ the character of $V_\ell$. Fix $v\in V_{\ell}$. Since $\phi^*_\ell(v)\in\mathcal{H}_\lambda(\sigma_\ell)$, its restriction $\phi_\ell^*(v)|_K$ to $K$ lies in the $\sigma_\ell$-isotypical component of $L^2(K)$ with respect to the left-regular $K$-action. By the Schur orthogonality relations we then have $$\phi^*_\ell(v)(1)=\textup{deg}(\sigma_\ell)\int_Kdx\,\phi^*_\ell(v)(x)\overline{\chi_{V_\ell}(x)}=
\langle\phi^*_\ell(v),\widetilde{\chi}_{\lambda,V_\ell}\rangle_\lambda=(v,\phi_\ell(\widetilde{\chi}_{\lambda,V_\ell}))_{V_\ell}.$$ This show that $$\iota_{\lambda,V_\ell}(\phi_\ell)=
\phi_\ell(\widetilde{\chi}_{\lambda,V_\ell}).$$ Now substitute $\phi_\ell=\sum_i\langle \cdot,f_i\rangle_\lambda v_i$ and use that $f_i\in\mathcal{H}_\lambda(\sigma^*_\ell)$ with $\sigma^*_\ell$ the irreducible $K$-representation dual to $\sigma_\ell$, we get by another application of the Schur’s orthogonality relations, $$\iota_{\mu,V_\ell}(\phi_\ell)=\phi_\ell(\widetilde{\chi}_{\lambda,V_\ell})
=\sum_i\textup{deg}(\sigma_\ell)\Bigl(\int_K dx\, \chi_{V_\ell}(x)
\overline{f_i(x)}\Bigr) v_i=
\sum_i\overline{f_i(1)}v_i.$$
\[relEisPrinRem\] [**a.**]{} For $a\in A$ and $m\in M$ one has $$\sigma(m,m)E_\lambda^\sigma(a)=E_\lambda^\sigma(a).$$ In particular, $E_\lambda^\sigma(a)$ maps $\textup{Hom}(V_r,V_\ell)$ into $\textup{Hom}_M(V_r,V_\ell)$.\
[**b.**]{} For $\xi\in M^\wedge$ and intertwiners $\phi_\ell\in\textup{Hom}_K(\mathcal{H}^{(\xi)}_{\lambda},V_{\ell})$ and $\phi_r\in\textup{Hom}_K(V_{r},\mathcal{H}^{(\xi)}_{\lambda})$, write $$\phi_\ell=\sum_i\langle \cdot,f_i\rangle^{(\xi)}_\lambda v_i,\qquad
\phi_r=\sum_j(\cdot,w_j)_{V_r}h_j$$ with $f_i,h_j\in \mathcal{H}^{(\xi),K-\textup{fin}}_{\lambda}$. Then $$\label{relEisxi}
f^{\phi_\ell,\phi_r}_{\mathcal{H}_{\lambda}^{(\xi)}}(g)=E_\lambda^\sigma(g)T^{(\xi),\phi_\ell,\phi_r}_{\lambda}\qquad (g\in G)$$ with $T^{(\xi),\phi_\ell,\phi_r}_{\lambda}\in\textup{Hom}_M(V_{r},V_{\ell})$ the $M$-intertwiner $$T^{(\xi),\phi_\ell,\phi_r}_{\lambda}(w):=\sum_{i,j}\langle h_j(1),f_i(1)\rangle_{\lambda}^{(\xi)}(w,w_j)_{V_r}v_i
\qquad (w\in V_{r}).$$
Algebraic principal series representations {#ssss3}
------------------------------------------
We first introduce some general facts and notations regarding $\mathfrak{g}$-modules, following [@Di].
Let $V$ be a $\mathfrak{g}$-module with representation map $\tau: \mathfrak{g}\rightarrow
\mathfrak{gl}(V)$. The representation map of $V$, viewed as $U(\mathfrak{g})$-module, will also be denoted by $\tau$. The dual of $V$ is defined by $$(\tau^*(x)f)(v)=-f(\tau(X)v),\qquad x\in \mathfrak{g},\,\, f\in V^*,\,\, v\in V.$$
Fix a reductive Lie subalgebra $\mathfrak{l}\subseteq\mathfrak{g}$ (in this paper $\mathfrak{l}$ will either be the fix-point Lie subalgebra $\mathfrak{k}$ of the Chevalley involution $\theta$, or the Cartan subalgebra $\mathfrak{h}$). Let $\mathfrak{l}^\wedge$ be the isomorphism classes of the finite dimensional irreducible $\mathfrak{l}$-modules. For $\tau\in\mathfrak{l}^\wedge$ we write $\textup{deg}(\tau)$ for the degree of $\tau$ and $V(\tau)$ for the $\tau$-isotypical component of $V$. A $\mathfrak{g}$-module $V$ is called a Harish-Chandra module with respect to $\mathfrak{l}$ if $V=\sum_{\tau\in\mathfrak{l}^\wedge}V(\tau)$ (it is automatically a direct sum). The isotypical component $V(\tau)$ then decomposes in a direct sum of copies of $\tau$. The number of copies, denoted by $\textup{mtp}(\tau,V)$, is called the multiplicity of $\tau$ in $V$. The Harish-Chandra module $V$ is called admissible if $\textup{mtp}(\tau,V)<\infty$ for all $\tau\in\mathfrak{l}^\wedge$.
For a $\mathfrak{g}$-module $V$ let $V^{\mathfrak{l}-\textup{fin}}$ be the subspace of $\mathfrak{l}$-finite vectors, $$V^{\mathfrak{l}-\textup{fin}}:=\{v\in V \,\,\, | \,\,\, \dim\bigl(U(\mathfrak{l})v\bigr)<\infty \, \}.$$ Then $V^{\mathfrak{l}-\textup{fin}}\subseteq V$ is a $\mathfrak{g}$-submodule. In fact, $V^{\mathfrak{l}-\textup{fin}}$ is a Harish-Chandra module with respect to $\mathfrak{l}$ satisfying $V^{\mathfrak{l}-\textup{fin}}(\tau)=V(\tau)$ for all $\tau\in\mathfrak{l}^\wedge$ (see [@Di 1.7.9]).
For $\mathfrak{l}$-modules $U,V$ with $U$ or $V$ finite dimensional we identify $$\label{identify}
U\otimes V^*\simeq\textup{Hom}(V,U)$$ as vector spaces by $u\otimes f\mapsto f(\cdot)u$. With the $U(\mathfrak{l})\otimes U(\mathfrak{l}$)-module structure on $\textup{Hom}(V,U)$ defined by $$((x\otimes z)f)(u):=xf(S(z)u)$$ for $x,z\in U(\mathfrak{l})$, $u\in U$ and $f\in\textup{Hom}(V,U)$, it is an isomorphism of $U(\mathfrak{l})\otimes U(\mathfrak{l})$-modules.
Differentiating the multiplicative character $\eta_\lambda: AN\rightarrow\mathbb{C}^*$ of the previous subsection gives a one-dimensional $\mathfrak{b}$-module, whose representation map we also denote by $\eta_\lambda$. Then $\eta_\lambda: \mathfrak{b}\rightarrow\mathbb{C}$ is concretely given by $$\eta_\lambda(h+u):=\lambda(h),\qquad h\in\mathfrak{h},\,\, u\in\mathfrak{n}.$$ We write $\mathbb{C}_\lambda$ for the associated one-dimensional $U(\mathfrak{b})$-module.
Let $\lambda\in\mathfrak{h}^*$. Write $$Y_\lambda:=\textup{Hom}_{U(\mathfrak{b})}\bigl(U(\mathfrak{g}),\mathbb{C}_{\lambda+\rho}\bigr)$$ for the space of linear functionals $f: U(\mathfrak{g})\rightarrow \mathbb{C}$ satisfying $f(xz)=\eta_{\lambda+\rho}(x)f(z)$ for $x\in U(\mathfrak{b})$ and $z\in U(\mathfrak{g})$. We view $Y_\lambda$ as $\mathfrak{g}$-module by $$(yf)(z):=f(zy),\qquad y\in\mathfrak{g},\,\, z\in U(\mathfrak{g}).$$
By [@Di Chpt. 9], the Harish-Chandra module $Y_\lambda^{\mathfrak{k}-\textup{fin}}$ is admissible with $\textup{mtp}(\tau, Y_\lambda^{\mathfrak{k}-\textup{fin}})=\textup{deg}(\tau)$ for all $\tau\in\mathfrak{k}^\wedge$. Consider $K^\wedge$ as subset of $\mathfrak{k}^\wedge$. Note that the inclusion $K^\wedge\hookrightarrow\mathfrak{k}^\wedge$ is strict unless $K$ is simply connected and semisimple.
\[relAnalAlg\] For $\lambda\in\mathfrak{h}^*$ we have an injective morphism of $\mathfrak{g}$-modules $$\label{injpsr}
\mathcal{H}_\lambda^{K-\textup{fin}}\hookrightarrow Y_\lambda^{\mathfrak{k}-\textup{fin}},
\qquad f\mapsto \widetilde{f}$$ with $$\widetilde{f}(z):=\bigl(r_*(S(z))f\bigr)(1)$$ for $f\in \mathcal{H}_\lambda^{K-\textup{fin}}$ and $z\in U(\mathfrak{g})$. For $\tau\in K^\wedge$ the embedding restricts to an isomorphism $$\label{HY}
\mathcal{H}_\lambda(\tau)\overset{\sim}{\longrightarrow} Y_\lambda(\tau)$$ of $\mathfrak{k}$-modules.
Let $f\in\mathcal{H}_\lambda^{K-\textup{fin}}$. Then $f: G\rightarrow\mathbb{C}$ is analytic and satisfies $r_*(S(x))f=\eta_{\lambda+\rho}(x)f$ for all $x\in U(\mathfrak{b})$. Hence is a well defined injective linear map. A direct computation shows that intertwines the $\mathfrak{g}_0$-action. This proves the first part of the proposition.
For $\tau\in K^\wedge$ we have $\textup{dim}\bigl(\mathcal{H}_\lambda(\tau)\bigr)=
\textup{deg}(\tau)^2=\textup{dim}\bigl(Y_\lambda(\tau)\bigr)$, hence follows from the first part of the proposition.
The embedding is an isomorphism if $K$ is simply connected and semisimple. In general, the algebraic description of the $(\mathfrak{g}_0,K)$-modules $\mathcal{H}_\lambda^{K-\textup{fin}}$ and $\mathcal{H}^{(\xi),K-\textup{fin}}_\lambda$ within $Y_\lambda^{\mathfrak{k}-\textup{fin}}$ amounts to taking the direct sum of isotypical components $Y_\lambda(\tau)$ for $\tau$ running over suitable subsets of $K^\wedge$ (see [@Di §9.3]).
Let $\phi_\ell\in\textup{Hom}_K(\mathcal{H}_\lambda,V_\ell)$, $\phi_r\in\textup{Hom}_K(V_r,\mathcal{H}_\lambda)$. The associated $\sigma$-spherical function $f_{\mathcal{H}_\lambda}^{\phi_\ell,\phi_r}\in C_\sigma^\infty(G)$ is an elementary $\sigma$-spherical function, meaning that it a common eigenfunction of the biinvariant differential operators on $G$. Indeed, by Proposition \[relAnalAlg\] it suffices to note that $Y_\lambda$ admits a central character. This follows from [@Di Thm. 9.3.3], $$\label{ccY}
zf=\zeta_{\lambda-\rho}(z)f\qquad (z\in Z(\mathfrak{g}),\,\, f\in Y_\lambda).$$ This also follows from the observation that $Y_\lambda$ is isomorphic to $M_{-\lambda-\rho}^*$ (see Lemma \[relCC\]) and the fact that $\zeta_{\mu-\rho}=\zeta_{w\mu-\rho}$ for $w\in W$.
As a consequence, the restriction $f_{\mathcal{H}_\lambda}^{\phi_\ell,\phi_r}|_{A_{\textup{reg}}}$ of $f_{\mathcal{H}_\lambda}^{\phi_\ell,\phi_r}\in
C_\sigma^\infty(G)$ to $A_{\textup{reg}}$ are common eigenfunctions of $\widehat{\Pi}^\sigma(z)$ ($z\in Z(\mathfrak{g})$), $$\label{EisensteinintDE}
\widehat{\Pi}^\sigma(z)\bigl(f_{\mathcal{H}_\lambda}^{\phi_\ell,\phi_r}|_{A_{\textup{reg}}}\bigr)=\zeta_{\lambda-\rho}(z)
f_{\mathcal{H}_\lambda}^{\phi_\ell,\phi_r}|_{A_{\textup{reg}}}\qquad \forall z\in Z(\mathfrak{g})$$ (it is sometimes more natural to write the eigenvalue as $\zeta_{w_0(\lambda+\rho)}(z)$ with $w_0\in W$ the longest Weyl group element). By Proposition \[relEisPrin\] it follows that the restriction $E_\lambda^\sigma|_{A_{\textup{reg}}}$ of the Eisenstein integral to $A_{\textup{reg}}$ is an $\textup{End}(\textup{Hom}(V_r,V_\ell))$-valued smooth function on $A_{\textup{reg}}$ satisfying the differential equations $$\label{EisensteinDE}
\widehat{\Pi}^\sigma(z)\bigl(E_\lambda^\sigma|_{A_{\textup{reg}}}\bigr)=\zeta_{\lambda-\rho}(z)
E_\lambda^\sigma|_{A_{\textup{reg}}}\qquad \forall z\in Z(\mathfrak{g}).$$
The normalised smooth $\textup{End}(\textup{Hom}(V_r,V_\ell))$-valued function on $A_+$ defined by $$\begin{split}
\mathbf{E}_\lambda^\sigma(a^\prime):=\delta(a^\prime)E_\lambda^\sigma(a^\prime)=
\delta(a^\prime)\int_Kdx\,\xi_{-\lambda-\rho}(a(a^\prime{}^{-1}x))\sigma(x,k(a^\prime{}^{-1}x))
\end{split}$$ for $a^\prime\in A_+$ is a common $\textup{End}(\textup{Hom}(V_r,V_\ell))$-valued eigenfunction for the quantum Hamiltonians of the $\sigma$-spin quantum hyperbolic Calogero-Moser system, $$\mathbf{H}^\sigma\bigl(\mathbf{E}_\lambda^{\sigma}\bigr)=-\frac{(\lambda,\lambda)}{2}
\mathbf{E}_\lambda^{\sigma}$$ and $$H_z^\sigma\bigl(\mathbf{E}_\lambda^{\sigma}\bigr)=\zeta_{\lambda-\rho}(z)
\mathbf{E}_\lambda^{\sigma}\qquad (z\in Z(\mathfrak{g})).$$
For sufficiently generic $\lambda\in\mathfrak{h}^*$, the $\sigma$-Harish-Chandra series $\Phi_{w\lambda-\rho}^\sigma$ $w\in W$ exist and satisfy the same differential equations on $A_+$ as $E_\lambda^\sigma|_{A_+}$. Harish-Chandra’s [@HCe] proved for generic $\lambda\in\mathfrak{h}^*$, $$\label{cfunctionexpansion}
E_\lambda^\sigma(a)T=\sum_{w\in W}c^\sigma(w;\lambda)\Phi_{w\lambda-\rho}^\sigma(a)T$$ for $a\in A_+$ and $T\in\textup{Hom}_M(V_r,V_\ell)$, with leading coefficients $c^\sigma(w;\lambda)\in\textup{End}(\textup{Hom}_M(V_r,V_\ell))$ called $c$-functions (see [@HCe Thm. 5]). The $c$-function expansion plays an important role in the harmonic analysis on $G$.
For the left hand side of , Remark \[relEisPrinRem\][**b**]{} provides a representation theoretic interpretation in terms of the principal series representation of $G$. In the next section we obtain a similar representation theoretic interpretation for the $\sigma$-Harish-Chandra series $\Phi_{w\lambda-\rho}^\sigma$ in terms of Verma modules.
Formal elementary $\sigma$-spherical functions {#RTHC}
==============================================
We fix in this section two finite dimensional representations $\sigma_\ell: \mathfrak{k}\rightarrow\mathfrak{gl}(V_\ell)$ and $\sigma_r: \mathfrak{k}\rightarrow
\mathfrak{gl}(V_r)$. We write $\sigma$ for the $\mathfrak{k}\oplus\mathfrak{k}$-representation map $\sigma_\ell\otimes\sigma_r^*$ of the $\mathfrak{k}\oplus\mathfrak{k}$-module $V_\ell\otimes V_r^*$.
Verma modules
-------------
In this subsection we relate the algebraic principal series representations to Verma modules. Let $V$ be a $\mathfrak{g}$-module $V$. Write $$V[\mu]:=\{v\in V \,\, | \,\, hv=\mu(h)v\quad \forall\, h\in\mathfrak{h}\}$$ for the weight space of $V$ of weight $\mu\in\mathfrak{h}^*$. Then $$\overline{V}:=\prod_{\mu\in\mathfrak{h}^*}V[\mu]$$ inherits from $V$ the structure of a $\mathfrak{g}$-module as follows. Let $v=(v[\mu])_{\mu\in\mathfrak{h}^*}\in \overline{V}$ and $z_\alpha\in\mathfrak{g}_\alpha$ ($\alpha\in R\cup\{0\}$), where $\mathfrak{g}_0:=\mathfrak{h}$. Then $z_\alpha v=((z_\alpha v)[\mu])_{\mu\in\mathfrak{h}^*}$ with $$(z_\alpha v)[\mu]:=z_\alpha v[\mu-\alpha].$$ Clearly $V\subseteq \overline{V}$ as $\mathfrak{g}$-submodule. Note that $\overline{V}^{\mathfrak{h}-\textup{fin}}=V$ for $\mathfrak{h}$-semisimple $\mathfrak{g}$-modules $V$.
For $\mu\in\mathfrak{h}^*$ write $$\label{lambdaiso}
\textup{proj}^\mu_V: \overline{V}\twoheadrightarrow V[\mu],
\qquad v\mapsto v[\mu]$$ for the canonical projection, and $\textup{incl}^\mu_V: V[\mu]\hookrightarrow V$ for the inclusion map. We omit the sublabel $V$ from the notations $\textup{proj}^\mu_V$ and $\textup{incl}^\mu_V$ if the representation $V$ is clear from the context.
The Verma module $M_\lambda$ with highest weight $\lambda\in\mathfrak{h}^*$ is the induced $\mathfrak{g}$-module $$M_\lambda:=U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\mathbb{C}_\lambda.$$
The Verma module $M_\lambda$ and its irreducible quotient $L_\lambda$ are highest weight modules of highest weight $\lambda$. In particular, they are $\mathfrak{h}$-diagonalizable with finite dimensional weight spaces. The weight decompositions are $M_{\lambda}=\bigoplus_{\mu\leq\lambda}M_{\lambda}[\mu]$ and $L_{\lambda}=\bigoplus_{\mu\leq\lambda}L_{\lambda}[\mu]$ with $\leq$ the dominance order on $\mathfrak{h}^*$ with respect to $R^+$ and with one-dimensional highest weight spaces $M_\lambda[\lambda]$ and $L_\lambda[\lambda]$. We fix once and for all a highest weight vector $0\not=m_\lambda\in M_\lambda[\lambda]$, and write $0\not=\ell_\lambda\in L_\lambda[\lambda]$ for its projection onto $L_\lambda$. Note that $M_\lambda$ and $L_\lambda$ admit the central character $\zeta_\lambda$.
The set $\mathfrak{h}^*_{\textup{irr}}$ of highest weights $\lambda$ for which $M_\lambda$ is irreducible is given by $$\mathfrak{h}_{\textup{irr}}^*=\{\lambda\in\mathfrak{h}^* \,\,\, | \,\,\,
(\lambda+\rho,\alpha^\vee)\not\in\mathbb{Z}_{>0} \quad \forall\, \alpha\in R^+\},$$ with $\alpha^\vee:=2\alpha/\|\alpha\|^2$ the co-root of $\alpha$. Note that $\mathfrak{h}_{\textup{HC}}^*\subseteq \mathfrak{h}_{\textup{irr}}^*$.
For a $\mathfrak{g}$-module $V$ write ${}^\theta V$ for $V$ endowed with the $\theta$-twisted $\mathfrak{g}$-module structure $$x\ast v:=\theta(x)v,\qquad x\in \mathfrak{g},\,\, v\in V.$$
\[relCC\] Let $\lambda\in\mathfrak{h}^*$.
1. We have $$M_{\lambda}^*\overset{\sim}{\longrightarrow}Y_{-\lambda-\rho}$$ as $\mathfrak{g}$-modules, with the isomorphism $f\mapsto\widehat{f}$ given by $\widehat{f}(x):=f(S(x)m_{\lambda})$ for $f\in M_{\lambda}^*$ and $x\in U(\mathfrak{g})$.
2. If $\lambda\in\mathfrak{h}_{\textup{irr}}^*$, then $$\overline{M}_\lambda\simeq {}^\theta Y_{-\lambda-\rho}$$ as $\mathfrak{g}$-modules. In particular, $M_\lambda\simeq {}^\theta Y_{-\lambda-\rho}^{\mathfrak{h}-\textup{fin}}$ as $\mathfrak{g}$-modules.
[**a.**]{} This is immediate (it is a special case of [@Di Prop. 5.5.4]).\
[**b.**]{} The Shapovalov form is the nondegenerate symmetric bilinear form on $L_\lambda$ satisfying $$B_\lambda(xu,v)=-B_\lambda(u,\theta(x)v)$$ for $x\in\mathfrak{g}$ and $u,v\in L_\lambda$ and normalised by $B_\lambda(\ell_\lambda,\ell_\lambda)=1$. It induces an isomorphism of $\mathfrak{g}$-modules $$\label{Shapovaloviso}
{}^\theta \overline{L}_\lambda\overset{\sim}{\longrightarrow}
L_\lambda^*$$ mapping $(v[\mu])_{\mu\in\mathfrak{h}^*}\in \overline{L}_\lambda$ to the linear functional $u\mapsto\sum_{\mu\in\mathfrak{h}^*}B_\lambda(v[\mu],u)$ on $L_\lambda$. If $\lambda\in\mathfrak{h}_{\textup{irr}}^*$ then $M_\lambda=L_\lambda$ and the result follows part [**a**]{} of the lemma.
\[AA\] [**a.**]{} The dual $M^\vee$ of a module $M$ in category $\mathcal{O}$ is defined by $M^\vee:={}^\theta M^{*,\mathfrak{h}-\textup{fin}}$. The final conclusion of part [**b**]{} of the lemma corresponds to the well known fact that $L_\lambda^\vee\simeq L_\lambda$.\
[**b.**]{} Combining Proposition \[relAnalAlg\] and Lemma \[relCC\][**a**]{}, we have $\lambda\in\mathfrak{h}^*$ an embedding of $\mathfrak{g}$-modules $$\mathcal{H}_{\lambda}^{K-\textup{fin}}\hookrightarrow M_{-\lambda-\rho}^*,\qquad
f\mapsto \breve{f},$$ with $\breve{f}(xm_{-\lambda-\rho}):=(r_\ast(x)f)(1)$ for all $x\in U(\mathfrak{g})$. It restricts to an isomorphism $\mathcal{H}_\lambda(\tau)\overset{\sim}{\longrightarrow} M_{-\lambda-\rho}^*(\tau)$ for each $\tau\in K^\wedge$.
Spaces of $\mathfrak{k}$-intertwiners
-------------------------------------
In Subsection \[subsectionPSR\] we have constructed linear isomorphisms $\iota_{\lambda,V_\ell}: \textup{Hom}_K(\mathcal{H}_\lambda,V_\ell) \overset{\sim}{\longrightarrow} V_\ell$ and $\j_{\lambda,V_r}:
\textup{Hom}_K(V_r,\mathcal{H}_\lambda)\overset{\sim}{\longrightarrow} V_r^*$, with $\mathcal{H}_\lambda$ the principal series representation. For Verma modules we have the following analogous result. Write $m_\lambda^*$ for the linear functional on $\overline{M}_\lambda$ that vanishes on $\prod_{\mu<\lambda}M_\lambda[\mu]$ and maps $m_\lambda$ to $1$.
\[relEisPrinalg\] Fix $\lambda\in\mathfrak{h}^*$.
1. The map $\textup{ev}_{\lambda,V_\ell}: \textup{Hom}_{\mathfrak{k}}(M_\lambda,V_\ell)\rightarrow V_\ell$, $$\textup{ev}_{\lambda,V_\ell}(\phi_\ell):=
\phi_\ell(m_\lambda),$$ is a linear isomorphism.
2. The linear map $\textup{hw}_{\lambda,V_r}: \textup{Hom}_{\mathfrak{k}}(V_r,\overline{M}_\lambda)\rightarrow V_r^*$, defined by $$\textup{hw}_{\lambda,V_r}(\phi_r)(v):=m_\lambda^*(\phi_r(v)) \qquad (v\in V_r),$$ is a linear isomorphism when $\lambda\in\mathfrak{h}_{\textup{irr}}^*$.
We assume without loss of generality that $\sigma_\ell,\sigma_r\in\mathfrak{k}^\wedge$.\
[**a.**]{} By Lemma \[relCC\][**a**]{} we have $$\textup{Hom}_{\mathfrak{k}}(M_\lambda,V_\ell)\simeq
\textup{Hom}_{\mathfrak{k}}(V_{\ell}^*, Y_{-\lambda-\rho})$$ as vector spaces. The latter space is of dimension $\textup{deg}(\sigma_\ell)$. Hence it suffices to show that $\textup{ev}_{\lambda,V_\ell}$ is injective. This follows from $M_\lambda=U(\mathfrak{k})m_\lambda$, which is an immediate consequence of the Iwasawa decomposition $\mathfrak{g}_0=\mathfrak{k}_0\oplus\mathfrak{h}_0\oplus \mathfrak{n}_0$ of $\mathfrak{g}_0$.\
[**b.**]{} Since $\lambda\in\mathfrak{h}_{\textup{irr}}^*$ we have $\textup{Hom}_{\mathfrak{k}}(V_r,\overline{M}_\lambda)\simeq
\textup{Hom}_{\mathfrak{k}}(V_r,Y_{-\lambda-\rho})$ as vector spaces by Lemma \[relCC\], and the latter space is of dimension $\textup{deg}(\sigma_r)$. It thus suffices to show that $\textup{hw}_{\lambda,V_r}$ is injective. Let $\phi_r\in\textup{Hom}_{\mathfrak{k}}(V_r,\overline{M}_\lambda)$ be a nonzero intertwiner and consider the nonempty set $$\mathcal{P}:=\{\mu\in\mathfrak{h}^* \,\, | \,\, \textup{proj}_{M_\lambda}^\mu(\phi_r(v))\not=0 \,\,\, \textup{ for some }\, v\in V_r \}.$$ Take a maximal element $\nu\in\mathcal{P}$ with respect to the dominance order $\leq$ on $\mathfrak{h}^*$. Fix $v\in V_r$ with $\textup{proj}_{M_\lambda}^\nu(\phi_r(v))\not=0$. Suppose that $e_\alpha(\phi_r(v)[\nu])\not=0$ in $M_\lambda$ for some $\alpha\in R^+$. Then $\textup{proj}_{M_\lambda}^{\nu+\alpha}(\phi_r(y_\alpha v))\not=0$, but this contradicts the fact that $\nu+\alpha\not\in\mathcal{P}$. It follows that $\textup{proj}^\nu_{M_\lambda}(\phi_r(v))$ is a highest weight vector in $M_\lambda$ of highest weight $\nu$. This forces $\nu=\lambda$ since $M_\lambda$ is irreducible, hence $\textup{hw}_{\lambda,V_r}(\phi_r)(v)\not=0$. It follows that $\textup{hw}_{\lambda,V_r}$ is injective, which completes the proof.
\[intertwinerparametrization\] Let $\lambda\in\mathfrak{h}^*$.
1. We call $\textup{ev}_{\lambda,V}(\phi_{\ell})$ the expectation value of the intertwiner $\phi_\ell\in\textup{Hom}_{\mathfrak{k}}(M_\lambda,V_\ell)$. We write $\phi_{\ell,\lambda}^v\in\textup{Hom}_{\mathfrak{k}}(M_\lambda,V_\ell)$ for the $\mathfrak{k}$-intertwiner with expectation value $v\in V_\ell$.
2. We call $\textup{hw}_{\lambda,V_r}(\phi_r)$ the highest weight component of the intertwiner $\phi_r\in\textup{Hom}_{\mathfrak{k}}(V_r,\overline{M}_\lambda)$. If $\lambda\in\mathfrak{h}_{\textup{irr}}^*$ then we write $\phi_{r,\lambda}^f\in\textup{Hom}_{\mathfrak{k}}(V_r,\overline{M}_\lambda)$ for the intertwiner with highest weight component $f\in V_r^*$.
The exact relation with the intertwiners from Proposition \[relEisPrin\] is as follows. Consider for $\sigma_\ell: K\rightarrow\textup{GL}(V_\ell)$ a finite dimensional $K$-representation the chain of linear isomorphisms $$\textup{Hom}_K(V_\ell^*,\mathcal{H}_{-\lambda-\rho})
\overset{\sim}{\longrightarrow}\textup{Hom}_{\mathfrak{k}}(V_\ell^*,M_\lambda^*)
\overset{\sim}{\longrightarrow}\textup{Hom}_{\mathfrak{k}}(M_\lambda,V_\ell)
\overset{\sim}{\longrightarrow} V_\ell.$$ The first isomorphism is the pushforward of the map defined in Remark \[AA\], the second map is transposition and the third map is $\textup{ev}_{\lambda,V_\ell}$. Their composition is the linear isomorphism $\j_{-\lambda-\rho,V_\ell^*}: \textup{Hom}_K(V_\ell^*,\mathcal{H}_{-\lambda-\rho})\overset{\sim}{\longrightarrow} V_\ell$ defined in Proposition \[relEisPrin\]. Similarly, for $\lambda\in\mathfrak{h}_{\textup{irr}}^*$ and $\sigma_r: K\rightarrow\textup{GL}(V_r)$ a finite dimensional $K$-representation we have the chain of linear isomorphisms $$\textup{Hom}_K(V_r,\mathcal{H}_{-\lambda-\rho})
\overset{\sim}{\longrightarrow}\textup{Hom}_{\mathfrak{k}}(V_r,M_\lambda^*)
\overset{\sim}{\longrightarrow}\textup{Hom}_{\mathfrak{k}}(V_r,\overline{M}_\lambda)
\overset{\sim}{\longrightarrow} V_r^*.$$ In this case the first isomorphism is the pushforward of the map defined in Remark \[AA\], the second isomorphism is the pushforward of the $\mathfrak{g}$-intertwiner $M_\lambda^*\overset{\sim}{\longrightarrow} {}^\theta\overline{M}_\lambda$ realized by the Shapovalov form (see Lemma \[relCC\][**b**]{} and its proof), and the third map is $\textup{hw}_{\lambda,V_r}$. Their composition is the linear isomorphism $\j_{-\lambda-\rho,V_r}: \textup{Hom}_K(V_r,\mathcal{H}_{-\lambda-\rho})\overset{\sim}{\longrightarrow} V_r^*$ defined in Proposition \[relEisPrin\].
The following corollary is the analogue of Proposition \[relEisPrin\][**c**]{} for Verma modules.
\[relEisVerma\] Let $V_\ell$ and $V_r$ be finite dimensional $\mathfrak{k}$-modules. The linear map $$\textup{Hom}_{\mathfrak{k}}(M_\lambda,V_{\ell})\otimes\textup{Hom}_{\mathfrak{k}}(V_{r},\overline{M}_\lambda)\rightarrow V_{\ell}\otimes V_{r}^*\simeq \textup{Hom}(V_{r},V_{\ell})$$ defined by $\phi_\ell\otimes\phi_r\mapsto
S^{\phi_\ell,\phi_r}_\lambda:=
\textup{ev}_{\lambda,V_\ell}(\phi_\ell)\otimes\textup{hw}_{\lambda,V_r}(\phi_r)$, is a linear isomorphism when $\lambda\in\mathfrak{h}_{\textup{irr}}^*$.
In Subsection \[HCrepSec\] we will give a representation interpretation of the analytic $\textup{Hom}(V_r,V_\ell)$-valued function $a\mapsto \Phi_\lambda^\sigma(a)S_\lambda^{\phi_\ell,\phi_r}$ for $\phi_\ell\in\textup{Hom}_{\mathfrak{k}}(M_\lambda,V_{\ell})$, $\phi_r\in \textup{Hom}_{\mathfrak{k}}(V_{r},\overline{M}_\lambda)$ and $a\in A_+$, with $\sigma$ the representation map of the $\mathfrak{k}\oplus\mathfrak{k}$-module $V_\ell\otimes V_r^*\simeq \textup{Hom}(V_r,V_\ell)$.
The construction of the formal elementary spherical functions
-------------------------------------------------------------
We first introduce $(\mathfrak{h},\mathfrak{k})$-finite and $(\mathfrak{k},\mathfrak{h})$-finite matrix coefficients of Verma modules. Let $\lambda,\mu\in\mathfrak{h}^*$ with $\mu\leq\lambda$. Recall the projection and inclusion maps $\textup{incl}_{M_\lambda}^\mu$ and $\textup{proj}_{M_\lambda}^\mu$. They are $\mathfrak{h}$-intertwiners $$\textup{incl}_{M_\lambda}^\mu\in\textup{Hom}_{\mathfrak{h}}(M_\lambda[\mu],M_\lambda),
\qquad \textup{proj}_{M_\lambda}^\mu\in\textup{Hom}_{\mathfrak{h}}(
\overline{M}_\lambda,M_\lambda[\mu]).$$ Let $\lambda,\mu\in\mathfrak{h}^*$ with $\mu\leq\lambda$, and fix $\mathfrak{k}$-intertwiners $\phi_\ell\in\textup{Hom}_{\mathfrak{k}}(M_\lambda,V_\ell)$ and $\phi_r\in\textup{Hom}_{\mathfrak{k}}(V_r,\overline{M}_\lambda)$. We write $$\phi_\ell^\mu:=\phi_\ell\circ \textup{incl}_{M_\lambda}^\mu\in\textup{Hom}(M_\lambda[\mu],V_\ell),
\qquad
\phi^\mu_r:=\textup{proj}_{M_\lambda}^\mu\circ\phi_r\in\textup{Hom}(V_r,M_\lambda[\mu])$$ for the weight-$\mu$ components of $\phi_\ell$ and $\phi_r$, respectively. The map $\phi^\mu_\ell$ encodes $(\mathfrak{k},\mathfrak{h})$-finite matrix coefficients of $M_\lambda$ of type $(\sigma_\ell,\mu)$, and $\phi^\mu_r$ the $(\mathfrak{h},\mathfrak{k})$-finite matrix coefficients of $M_\lambda$ of type $(\mu,\sigma_r)$. Formal elementary spherical functions are now defined to be the generating series of the compositions $\phi_\ell^\mu\circ\phi_r^\mu$ of the weight compositions of the $\mathfrak{k}$-intertwiners $\phi_\ell$ and $\phi_r$:
\[Glambda\] Let $\lambda\in\mathfrak{h}^*$. For $\phi_\ell\in\textup{Hom}_{\mathfrak{k}}(M_\lambda,V_\ell)$ and $\phi_r\in\textup{Hom}_{\mathfrak{k}}(V_r,\overline{M}_\lambda)$ let $$F_{M_\lambda}^{\phi_\ell,\phi_r}\in (V_\ell\otimes V_r^*)[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\lambda$$ be the formal $V_\ell\otimes\mathbf{U}\otimes V_r^*$-valued power series $$F_{M_\lambda}^{\phi_\ell,\phi_r}:=\sum_{\mu\leq\lambda}(\phi_\ell\circ\phi_r^\mu)\xi_\mu
=\sum_{\mu\leq\lambda}(\phi_\ell^\mu\circ\phi_r^\mu)\xi_\mu.$$ We call $F_{M_\lambda}^{\phi_\ell,\phi_r}$ the formal elementary $\sigma$-spherical function associated to $M_\lambda$, $\phi_\ell$ and $\phi_r$.
Note that under the natural identification $\textup{Hom}(V_r,V_\ell)\simeq V_\ell\otimes V_r^*$ we have $$\phi_\ell^\lambda\circ\phi_r^\lambda=\textup{ev}_{\lambda,V_\ell}(\phi_\ell)\otimes
\textup{hw}_{\lambda,V_r}(\phi_r)=S_{\lambda}^{\phi_\ell,\phi_r},$$ hence $F_{M_\lambda}^{\phi_\ell,\phi_r}$ has leading coefficient $S_{\lambda}^{\phi_\ell,\phi_r}$. By Corollary \[relEisVerma\], if $\lambda\in\mathfrak{h}_{\textup{irr}}^*$ and $v\in V_\ell$, $f\in V_r^*$, the formal elementary $\sigma$-spherical function $F_{M_\lambda}^{\phi_{\ell,\lambda}^v,\phi_{r,\lambda}^f}$ is the unique formal elementary $\sigma$-spherical function associated to $M_\lambda$ with leading coefficient $v\otimes f$. We will denote it by $F_{M_\lambda}^{v,f}$.
Relation to $\sigma$-Harish-Chandra series {#HCrepSec}
------------------------------------------
Recall the Harish-Chandra coefficients $\Gamma_\lambda^\sigma(\mu)\in V_\ell\otimes V_r^*$ in the power series expansion of the $\sigma$-Harish-Chandra series $\Phi_\lambda^\sigma$, see Proposition \[cordefHC\]. We have the following main result of Section \[RTHC\].
\[mainTHMF\] Let $\lambda\in\mathfrak{h}^*$, $\phi_\ell\in\textup{Hom}_{\mathfrak{k}}(M_\lambda,V_\ell)$ and $\phi_r\in\textup{Hom}_{\mathfrak{k}}(V_r,\overline{M}_\lambda)$.
1. For $z\in Z(\mathfrak{g})$, $$\label{todo2}
\widehat{\Pi}^\sigma(z)F_{M_\lambda}^{\phi_\ell,\phi_r}=\zeta_\lambda(z)F_{M_\lambda}^{\phi_\ell,\phi_r}$$ as identity in $(V_\ell\otimes V_r^*)[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\lambda$.
2. For $\lambda\in\mathfrak{h}_{\textup{HC}}^*$ and $\mu\leq\lambda$ we have $$\label{mixedmatrixformula}
\phi_\ell\circ\phi_r^\mu=\phi^\mu_\ell\circ\phi_r^\mu=\Gamma_\mu^\sigma(\lambda)S_\lambda^{\phi_\ell,
\phi_r}.$$
3. For $\lambda\in\mathfrak{h}_{\textup{HC}}^*$, $v\in V_\ell$ and $f\in V_r^*$ we have $$\label{relationFspher}
F_{M_\lambda}^{v,f}=\Phi_\lambda^\sigma(\cdot)(v\otimes f).$$ In particular, $F_{M_\lambda}^{v,f}$ is a $V_\ell\otimes V_r^*$-valued analytic function on $A_+$.
Since $\mathfrak{h}_{\textup{HC}}^*\subseteq\mathfrak{h}_{\textup{irr}}^*$, part [**b**]{} and [**c**]{} of the theorem directly follow from part [**a**]{}, Proposition \[cordefHC\], Proposition \[holomorphic\] and the fact that the leading coefficient of $F_{M_\lambda}^{v\otimes f}$ is $v\otimes f$. It thus suffices to prove .
Consider the $Q$-grading $U(\mathfrak{g})=\bigoplus_{\gamma\in Q}U[\gamma]$ with $U[\gamma]\subset U(\mathfrak{g})$ the subspace consisting of elements $x\in U(\mathfrak{g})$ satisfying $\textup{Ad}_a(x)=a^\gamma x$ for all $a\in A$. Set $$\Lambda:=\{\mu\in\mathfrak{h}^* \,\,\, | \,\,\, \mu\leq\lambda \},
\qquad
\Lambda_m:=\{\mu\in\Lambda\,\,\, | \,\,\, (\lambda-\mu,\rho^\vee)\leq m\}$$ for $m\in\mathbb{Z}_{\geq 0}$, where $\rho^\vee=\frac{1}{2}\sum_{\alpha\in R^+}\alpha^\vee$. Then $(\lambda-\mu,\rho^\vee)$ is the height of $\lambda-\mu\in\sum_{i=1}^r\mathbb{Z}_{\geq 0}\alpha_i$ with respect to the basis $\{\alpha_1,\ldots,\alpha_r\}$ of $R$. We will prove that $$\label{todoformal2}
\widehat{\Pi}^{\sigma}(x)F_{M_\lambda}^{\phi_\ell,\phi_r}=\sum_{\mu\in\Lambda}\phi_\ell(x\phi_r^\mu)\xi_\mu
\qquad (x\in U[0])$$ in $\textup{Hom}(V_r,V_\ell)[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\lambda$. This implies , since $Z(\mathfrak{g})\subseteq U[0]$ and $M_\lambda$ admits the central character $\zeta_\lambda$.
Fix $x\in U[0]$ and write $\Pi(x)=\sum_{j\in J}f_j\otimes h_j\otimes y_j\otimes z_j$ with $f_j\in\mathcal{R}$, $h_j\in U(\mathfrak{h})$ and $y_j, z_j\in U(\mathfrak{k})$. By Theorem \[infKAK\] we get the infinitesimal Cartan decomposition $$\label{infKAKspecific}
x=\textup{Ad}_a(x)=\sum_{j\in J}f_j(a)y_jh_j\textup{Ad}_a(z_j)\qquad (a\in A_{+})$$ of $x\in U[0]$. We will now substitute this decomposition in the truncated version $\sum_{\lambda\in\Lambda_m}\phi_\ell(x\phi_r^\mu)\xi_\mu$ of the right hand side of .
Note that $\sum_{\lambda\in\Lambda_m}\phi_\ell(x\phi_r^\mu)\xi_\mu\in
\textup{Hom}(V_r,V_\ell)[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]\xi_\lambda$ is a trigonometric quasi-polynomial, hence it can be evaluated at $a\in A_{+}$. Substituting and using that $\phi_\ell$ is a $\mathfrak{k}$-intertwiner we obtain the formula $$\label{blob1}
\sum_{\lambda\in\Lambda_m}\phi_\ell(x\phi_r^\mu)a^\mu=
\sum_{j\in J}\sum_{\mu\in\Lambda_m}f_j(a)\sigma_\ell(y_j)
\phi_\ell(h_j\textup{Ad}_a(z_j)\phi_r^\mu)a^\mu$$ in $\textup{Hom}(V_r,V_\ell)$. Now expand $z_j=\sum_{\gamma\in I_j}z_j[\gamma]$ along the $Q$-grading of $U(\mathfrak{g})$ with $z_j[\gamma]\in U[\gamma]$ (but no longer in $U(\mathfrak{k})$). Here $I_j\subset Q$ denotes the finite set of weights for which $z_j[\gamma]\not=0$. Then implies $$\sum_{\mu\in\Lambda_m}\phi_\ell(x\phi_r^\mu)a^\mu
=\sum_{j\in J}\sum_{\gamma\in I_j}\sum_{\mu\in\Lambda_m}(\mu+\gamma)(h_j)
\sigma_\ell(y_j)\phi_\ell(z_j[\gamma]\phi_r^\mu)f_j(a)a^{\mu+\gamma}$$ for all $a\in A_{+}$. Let $\eta\geq\lambda$ such that $\lambda+\gamma\leq \eta$ for all $\gamma\in I:=\cup_{j\in J}I_j$. Then we conclude that $$\label{blob3}
\sum_{\mu\in\Lambda_m}\phi_\ell(x\phi_r^\mu)\xi_\mu=
\sum_{j\in J}\sum_{\gamma\in I_j}\sum_{\mu\in\Lambda_m}(\mu+\gamma)(h_j)\sigma_\ell(y_j)
\phi_\ell(z_j[\gamma]\phi_r^\mu)f_j\xi_{\mu+\gamma}$$ in $\textup{Hom}(V_r,V_\ell)[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\eta$ (here the $f_j\in\mathcal{R}$ are represented by their convergent power series $f_j=\sum_{\beta\in Q_-}
c_{j,\beta}\xi_\beta$ on $A_+$ ($c_{j,\beta}\in\mathbb{C}$)). We now claim that is valid with the truncated sum over $\Lambda_m$ replaced by the sum over $\Lambda$, $$\label{blob4}
\sum_{\mu\in\Lambda}\phi_\ell(x\phi_r^\mu)\xi_\mu=
\sum_{j\in J}\sum_{\gamma\in I_j}\sum_{\mu\in\Lambda}(\mu+\gamma)(h_j)\sigma_\ell(y_j)
\phi_\ell(z_j[\gamma]\phi_r^\mu)f_j\xi_{\mu+\gamma}$$ in $\textup{Hom}(V_r,V_\ell)[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\eta$.
Fix $\nu\in\mathfrak{h}^*$ with $\nu\leq \eta$. It suffices to show that the $\xi_\nu$-component of the left (resp. right) hand side of is the same as the $\xi_\nu$-component of the left (resp. right) hand side of when $m\in\mathbb{Z}_{\geq 0}$ satisfies $(\eta-\nu,\rho^\vee)\leq m$.
Choose $m\in\mathbb{Z}_{\geq 0}$ with $(\eta-\nu,\rho^\vee)\leq m$. The $\xi_\nu$-component of the left hand side of is zero if $\nu\not\in\Lambda$ and $\phi_\ell(x\phi_r^\nu)$ otherwise. Since $(\lambda-\nu,\rho^\vee)\leq (\eta-\nu,\rho^\vee)\leq m$, this coincides with the $\xi_\nu$-component of the left hand side of . The $\xi_\nu$-component of the right hand side of is $$\label{blob5}
\sum_{(j,\beta,\gamma,\mu)}(\mu+\gamma)(h_j)c_{j,\beta}\sigma_\ell(y_j)
\phi_\ell(z_j[\gamma]\phi_r^\mu)\in V_\ell\otimes V_r^*$$ with the sum over the finite set of four tuples $(j,\beta,\gamma,\mu)\in J\times Q_-\times I\times
\Lambda$ satisfying $\gamma\in I_j$ and $\mu+\gamma+\beta=\nu$. For such a four tuple we have $$(\lambda-\mu,\rho^\vee)=(\lambda+\gamma-\nu+\beta,\rho^\vee)\leq
(\eta-\nu,\rho^\vee)\leq m,$$ from which it follows that is also the $\xi_\nu$-component of the right hand side of . This concludes the proof of .
Since $\phi_r$ is a $\mathfrak{k}$-intertwiner, we have for fixed $\nu\in\mathfrak{h}^*$, $$\sum_{\stackrel{(\mu,\gamma)\in \Lambda\times I_j:}{\mu+\gamma=\nu}}
z_j[\gamma]\phi_r^\mu=\phi_r^\nu \sigma_r(z_j)$$ in $\textup{Hom}(V_r,M_\lambda[\nu])$ (in particular, it is zero when $\nu\not\in\Lambda$). Hence simplifies to $$\label{blob6}
\begin{split}
\sum_{\mu\in\Lambda}\phi_\ell(x\phi_r^\mu)\xi_\mu&=
\sum_{j\in J}\sum_{\nu\in\Lambda}\bigl(\sigma_\ell(y_j)\phi_\ell(\phi_r^\nu)\sigma_r^*(z_j)\bigr)f_j\nu(h_j)\xi_\nu\\
&=\widehat{\Pi}^\sigma(x)F_{M_\lambda}^{\phi_\ell,\phi_r}
\end{split}$$ in $\textup{Hom}(V_r,V_\ell)[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\lambda$, as desired.
Recall the normalisation factor $\delta$, defined by . We will also view $\delta$ as formal series in $\mathbb{C}[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\rho$ through its power series expansion at infinity within $A_+$.
Let $\lambda\in\mathfrak{h}^*$ and fix $\mathfrak{k}$-intertwiners $\phi_\ell\in\textup{Hom}_{\mathfrak{k}}(M_{\lambda-\rho},V_\ell)$ and $\phi_r\in\textup{Hom}_{\mathfrak{k}}(V_r,\overline{M}_{\lambda-\rho})$. We call $$\mathbf{F}_\lambda^{\phi_\ell,\phi_r}:=\delta F_{M_{\lambda-\rho}}^{\phi_\ell,\phi_r}\in
(V_\ell\otimes V_r^*)[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_{\lambda}$$ the normalised formal elementary $\sigma$-spherical function of weight $\lambda$. We furthermore write for $\lambda\in\mathfrak{h}^*_{\textup{irr}}+\rho$ and $v\in V_\ell$, $f\in V_r^*$, $$\mathbf{F}_\lambda^{v, f}:=
\delta F_{M_{\lambda-\rho}}^{v\otimes f},$$ which is the normalised formal elementary $\sigma$-spherical function of weight $\lambda$ with leading coefficient $v\otimes f$.
By Theorem \[mainTHMF\][**c**]{} we have for $\lambda\in\mathfrak{h}_{\textup{HC}}^*+\rho$, $v\in V_\ell$ and $f\in V_r^*$, $$\mathbf{F}_\lambda^{v,f}=\mathbf{\Phi}_\lambda^\sigma(\cdot)(v\otimes f).$$ In particular, $\mathbf{F}_\lambda^{v,f}$ is an $V_\ell\otimes V_r^*$-valued analytic function on $A_+$ when $\lambda\in\mathfrak{h}_{\textup{HC}}^*+\rho$.
Theorem \[normalizedHCseriesthm\] now immediately gives the interpretation of $\mathbf{F}_\lambda^{\phi_\ell,\phi_r}$ as formal eigenstates for the $\sigma$-spin quantum hyperbolic Calogero-Moser system for all weights $\lambda\in\mathfrak{h}^*$.
\[thmnorm\] Let $\lambda\in\mathfrak{h}^*$, $\phi_\ell\in\textup{Hom}_{\mathfrak{k}}(M_\lambda,V_\ell)$ and $\phi_r\in\textup{Hom}_{\mathfrak{k}}(V_r,\overline{M}_\lambda)$. The normalised formal elementary $\sigma$-spherical function $\mathbf{F}_\lambda^{\phi_\ell,\phi_r}$ of weight $\lambda$ satisfies the Schr[ö]{}dinger equation $$\mathbf{H}^\sigma(\mathbf{F}_\lambda^{\phi_\ell,\phi_r})=-\frac{(\lambda,\lambda)}{2}
\mathbf{F}_\lambda^{\phi_\ell,\phi_r}$$ as well as the eigenvalue equations $$\label{todo2gauged}
H_z^\sigma(\mathbf{F}_\lambda^{\phi_\ell,\phi_r})=\zeta_{\lambda-\rho}(z)\mathbf{F}_\lambda^{\phi_\ell,\phi_r},
\qquad z\in Z(\mathfrak{g})$$ in $(V_\ell\otimes V_r^*)[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_{\lambda}$.
This follows from the differential equations for the formal elementary $\sigma$-spherical functions, and the results in Subsection \[vvCM\].
[**a.**]{} Let $\lambda\in\mathfrak{h}_{\textup{HC}}^*$ and fix $V_\ell,V_r$ finite dimensional $K$-representations. Let $\sigma$ be the resulting tensor product representation of $K\times K$ on $V_\sigma:=V_\ell\otimes V_r^*\simeq \textup{Hom}(V_r,V_\ell)$. For $T\in\textup{Hom}_M(V_r,V_\ell)$ write $H_\lambda^T$ for the $V_\sigma$-valued smooth function on $G_{\textup{reg}}$ constructed from the Harish-Chandra series $\Phi_\lambda^\sigma$ by $$H_\lambda^T(k_1ak_2^{-1}):=\sigma(k_1,k_2)\Phi_\lambda^\sigma(a)T
\qquad (a\in A_+,\,\, k_1,k_2\in K),$$ see Remark \[remaHCspherplus\]. Then gives an interpretation of $H_\lambda^T$ as formal elementary $\sigma$-spherical function associated with $M_\lambda$. This should be compared with , which gives an interpretation of the Eisenstein integral as spherical function associated to the principal series representation of $G$.\
[**b.**]{} In [@K Thm. 4.4] Kolb proved an affine rank one analogue of Theorem \[mainTHMF\] for the pair $(\widehat{\mathfrak{sl}}_2,\widehat{\theta})$, where $\widehat{\theta}$ the Chevalley involution on the affine Lie algebra $\widehat{\mathfrak{sl}}_2$ associated to $\mathfrak{sl}_2$. The generalisation of Theorem \[mainTHMF\] to arbitrary split affine symmetric pairs will be discussed in a followup paper.
The rank one example {#rankoneSection}
--------------------
In this subsection we consider $\mathfrak{g}_0=\mathfrak{sl}(2;\mathbb{R})$ with linear basis $$H=\left(\begin{matrix} 1 & 0\\ 0 & -1\end{matrix}\right), \quad
E=\left(\begin{matrix} 0 & 1\\ 0 & 0\end{matrix}\right),\quad
F=\left(\begin{matrix} 0 & 0\\ 1 & 0\end{matrix}\right),$$ and we take $\mathfrak{h}_0=\mathbb{R}H$ as split Cartan subalgebra. Then $\theta_0(x)=-x^t$ with $x^t$ the transpose of $x\in\mathfrak{g}_0$. Note that $\frac{H}{2\sqrt{2}}\in\mathfrak{h}_0$ has norm one with respect to the Killing form. Let $\alpha$ be the unique positive root, satisfying $\alpha(H)=2$. Then $t_\alpha=\frac{H}{4}$, and we can take $e_\alpha=\frac{E}{2}$ and $e_{-\alpha}=\frac{F}{2}$. With this choice we have $\mathfrak{k}_0=\mathbb{R}y$ with $y:=y_\alpha=\frac{E}{2}-\frac{F}{2}$.
We identify $\mathfrak{h}^*\overset{\sim}{\longrightarrow} \mathbb{C}$ by the map $\lambda\mapsto \lambda(H)$. The positive root $\alpha\in\mathfrak{h}^*$ then corresponds to $2$. The bilinear form on $\mathfrak{h}^*$ becomes $(\lambda,\mu)=\frac{1}{8}\lambda\mu$ for $\lambda,\mu\in\mathbb{C}$. Furthermore, $\mathfrak{h}^*_{\textup{irr}}=\mathfrak{h}^*_{\textup{HC}}$ becomes $\mathbb{C}\setminus
\mathbb{Z}_{\geq 0}$. We also identify $A\overset{\sim}{\longrightarrow} \mathbb{R}_{>0}$ by $\exp_A(sH)\mapsto e^s$ ($s\in\mathbb{R}$). With these identifications, the multiplicative character $\xi_\lambda$ on $A$ ($\lambda\in\mathfrak{h}^*$) becomes $\xi_\lambda(a)=a^{\lambda}$ for $a\in\mathbb{R}_{>0}$ and $\lambda\in\mathbb{C}$.
For $\nu\in\mathbb{C}$ let $\chi_\nu\in\mathfrak{k}^\wedge$ be the one-dimensional representation mapping $y$ to $\nu$. We write $\mathbb{C}_\nu$ for $\mathbb{C}$ regarded as $\mathfrak{k}$-module by $\chi_\nu$. For $\lambda\in\mathbb{C}\setminus\mathbb{Z}_{\geq 0}$ and $\nu_\ell, \nu_r\in\mathbb{C}$, the scalar-valued Harish-Chandra series $\Phi_\lambda^{\chi_{\nu_\ell}\otimes \chi_{\nu_r}}$ is the unique analytic function on $A_+$ admitting a power series of the form $$\label{asymptotics}
\Phi_\lambda^{\chi_{\nu_\ell}\otimes\chi_{\nu_r}}(a)=a^\lambda \sum_{n\geq 0}
\Gamma_{-2n}^{\chi_{\nu_\ell}\otimes\chi_{\nu_r}}(\lambda)a^{-2n}\qquad
(a\in A_+)$$ with $\Gamma_0^{\chi_{\nu_\ell}\otimes\chi_{\nu_r}}(\lambda)=1$ that satisfies the differential equation $$\label{GHE}
\widehat{\Pi}(\Omega)\Phi_\lambda^{\chi_{\nu_\ell}\otimes\chi_{\nu_r}}=\frac{\lambda(\lambda+2)}{8}\Phi_\lambda^{\chi_{\nu_\ell}\otimes\chi_{\nu_r}}$$ (see Proposition \[cordefHC\]). By Corollary \[corR1\], $$\begin{split}
(\textup{id}_{\mathbb{D}_{\mathcal{R}}}&\otimes\chi_{\nu_\ell}\otimes\chi_{\nu_r})\widehat{\Pi}(\Omega)=
\frac{1}{8}\Bigl(a\frac{d}{da}\Bigr)^2+\frac{1}{4}\left(\frac{a^2+a^{-2}}{a^2-a^{-2}}\right)
a\frac{d}{da}+\frac{2(\nu_\ell+a^2\nu_r)(\nu_\ell+a^{-2}\nu_r)}{(a^2-a^{-2})^2}\\
&=\frac{1}{8}\left(\Bigl(a\frac{d}{da}\Bigr)^2+\left(
\frac{a+a^{-1}}{a-a^{-1}}+\frac{a-a^{-1}}{a+a^{-1}}\right)a\frac{d}{da}
+\frac{4(\nu_\ell+\nu_r)^2}{(a-a^{-1})^2}-\frac{4(\nu_\ell-\nu_r)^2}{(a+a^{-1})^2}\right).
\end{split}$$ The equation is the second-order differential equation that is solved by the associated Jacobi function (cf., e.g., [@Ko2 §4.2]), and $\Phi_\lambda^{\chi_{\nu_\ell}\otimes\chi_{\nu_r}}$ is the associated asymptotically free solution.
An explicit expression for $\Phi_{\lambda}^{\chi_{\nu_\ell}\otimes\chi_{\nu_r}}$ can be now derived as follows. Rewrite as a second order differential equation for $h\Phi_\lambda^{\chi_{\nu_\ell}\otimes\chi_{\nu_r}}$ with $$h(a):=(a+a^{-1})^{i(\nu_r-\nu_\ell)}(a-a^{-1})^{-i(\nu_\ell+\nu_r)}.$$ One then recognizes the resulting differential equation as the second-order differential equation [@Ko2 (2.10)] satisfied by the Jacobi function (which is the Gauss’ hypergeometric differential equation after an appropriate change of coordinates). Its solutions can be expressed in terms of the Gauss’ hypergeometric series $$\label{Gauss}
{}_2F_1(a,b;c \, \vert\, s):=\sum_{k=0}^{\infty}\frac{(a)_k(b)_k}{(c)_kk!}s^k,$$ where $(a)_k:=a(a+1)\cdots (a+k-1)$ is the Pochhammer symbol (the series converges for $|s|<1$). Then $\Phi_\lambda^{\chi_{\nu_\ell}\otimes\chi_{\nu_r}}$ corresponds to the solution [@Ko2 (2.15)] of the second-order differential equation [@Ko2 (2.10)]. Performing the straightforward computations gives the following result.
\[GaussExpres\] For $\lambda\in\mathbb{C}\setminus\mathbb{Z}_{\geq 0}$ we have $$\Phi_\lambda^{\chi_{\nu_\ell}\otimes\chi_{\nu_r}}(a)=
(a+a^{-1})^\lambda\left(\frac{a-a^{-1}}{a+a^{-1}}\right)^{i(\nu_\ell+\nu_r)}
{}_2F_1\Bigl(-\frac{\lambda}{2}+i\nu_\ell, -\frac{\lambda}{2}+i\nu_r; -\lambda \,\, | \,\,
\frac{4}{(a+a^{-1})^2}\Bigr)$$ for $a>1$.
If $\chi_{-\nu_\ell}, \chi_{-\nu_r}\in K^\wedge$ (i.e., if $\nu_\ell, \nu_r\in i\mathbb{Z}$), the restriction of the elementary spherical function $f_{\pi_\lambda}^{\chi_{-\nu_\ell},\chi_{-\nu_r}}$ to $A$ is an associated Jacobi function. It can also be expressed in terms of a single ${}_2F_1$ (see [@Ko2 §4.2] and references therein for details).
We compute now an alternative expression for $\Phi_\lambda^{\chi_{\nu_\ell}\otimes\chi_{\nu_r}}$ using its realisation as generating function for compositions of weight components of $\mathfrak{k}$-intertwiners (see Theorem \[mainTHMF\]).
The Verma module $M_\lambda$ with highest weight $\lambda\in\mathbb{C}$ is explicitly realised as $$M_\lambda=\bigoplus_{n=0}^\infty\mathbb{C}u_n$$ with $u_n:=\frac{1}{n!}F^nm_\lambda$ and $\mathfrak{g}$-action $Hu_n=(\lambda-2n)u_n$, $Eu_n=(\lambda-n+1)u_{n-1}$ and $Fu_n=(n+1)u_{n+1}$, where we have set $u_{-1}:=0$. We will identify $$\textup{Hom}_{\mathfrak{k}}(\mathbb{C}_\nu,\overline{M}_\lambda)\overset{\sim}{\longrightarrow}
\overline{M}_\lambda^{\chi_\nu},\qquad \psi\mapsto \psi(1),$$ with $\overline{M}_\lambda^{\chi_\nu}$ the space of $\chi_\nu$-invariant vectors in $\overline{M}_\lambda$ (cf. Subsection \[chisection\]). Note furthermore that $$\textup{Hom}_{\mathfrak{k}}(M_\lambda,\mathbb{C}_\nu)\simeq M_{\lambda}^{\ast, \chi_{-\nu}}.$$ To apply Theorem \[mainTHMF\] to $\Phi_\lambda^{\chi_{\nu_\ell}\otimes\chi_{\nu_r}}$, we thus need to describe the weight components of the nonzero vectors in the one-dimensional subspaces $M_{\lambda}^{\ast, \chi_{-\nu_\ell}}$ and $\overline{M}_\lambda^{\chi_{-\nu_r}}$ for $\lambda\in\mathbb{C}\setminus\mathbb{Z}_{\geq 0}$ (here we use that $\mathbb{C}_{\nu_r}\simeq\mathbb{C}_{-\nu_r}^*$ as $\mathfrak{k}$-modules). For this we need some facts about Meizner-Pollaczek polynomials, which we recall from [@KS §9.7].
Meixner-Pollaczek polynomials are orthogonal polynomials depending on two parameters $(\lambda,\phi)$, of which we only need the special case $\phi=\frac{\pi}{2}$. The monic Meixner-Pollaczek polynomials $\{p_n^{(\lambda)}(s) \,\, | \,\, n\in\mathbb{Z}_{\geq 0}\}$ with $\phi=\frac{\pi}{2}$ are given by $$p_n^{(\lambda)}(s)=(2\lambda)_n\Bigl(\frac{i}{2}\Bigr)^n{}_2F_1\bigl(-n,\lambda+is; 2\lambda \, \vert\, 2\bigr).$$ They satisfy the three-term recursion relation $$\label{3termMP}
p_{n+1}^{(\lambda)}(s)-sp_n^{(\lambda)}(s)+\frac{n(n+2\lambda-1)}{4}p_{n-1}^{(\lambda)}(s)=0,$$ where $p_{-1}^{(\lambda)}(s):=0$.
The following result should be compared with [@BW; @Ko], where mixed matrix coefficients of discrete series representations of $\textup{SL}(2;\mathbb{R})$ with respect to hyperbolic and elliptic one-parameter subgroups of $\textup{SL}(2,\mathbb{R})$ are expressed in terms of Meixner-Pollaczek polynomials.
\[r1\] Fix $\lambda\in\mathbb{C}\setminus\mathbb{Z}_{\geq 0}$ and $\nu\in\mathbb{C}$.
1. We have $$\overline{M}_\lambda^{\chi_{-\nu}}=\mathbb{C}v_{\lambda; \nu}$$ with the $(\lambda-2n)$-weight coefficient of $v_{\lambda;\nu}$ given by $$v_{\lambda;\nu}[\lambda-2n]=\frac{(-2)^np_n^{(-\lambda/2)}(-\nu)}{(-\lambda)_n}u_n,
\qquad n\in\mathbb{Z}_{\geq 0}.$$
2. We have $$M_\lambda^{\ast,\chi_{-\nu}}=\mathbb{C}\psi_{\lambda;\nu}$$ with $\psi_{\lambda;\nu}$ satisfying $$\psi_{\lambda;\nu}(u_n)=\frac{2^np_n^{(-\lambda/2)}(-\nu)}{n!},\qquad n\in\mathbb{Z}_{\geq 0}.$$
[**a.**]{} The requirement $yv=-\nu v$ for an element $v\in \overline{M}_\lambda$ with weight components of the form $$v[\lambda-2n]=\frac{(-2)^nc_n}{(-\lambda)_n}u_n$$ is equivalent to the condition that the coefficients $c_n\in\mathbb{C}$ ($n\geq 0$) satisfy the three-term recursion relation $$c_{n+1}+\nu c_n+\frac{n(n-\lambda-1)}{4}c_{n-1}=0,\qquad n\in\mathbb{Z}_{\geq 0},$$ where $c_{-1}:=0$. By , the solution of this three-term recursion relation satisfying $c_0=1$ is given by $c_n=p_n^{(-\lambda/2)}(-\nu)$ ($n\in\mathbb{Z}_{\geq 0}$).\
[**b.**]{} The proof is similar to the proof of part [**a**]{}.
Let $\nu_\ell, \nu_r\in\mathbb{C}$ and $\lambda\in\mathbb{C}\setminus\mathbb{Z}_{\geq 0}$. We obtain from Lemma \[r1\] and Theorem \[mainTHMF\] the following expression for the Harish-Chandra series $\Phi_\lambda^{\chi_{\nu_\ell}\otimes\chi_{\nu_r}}$.
Fix $\lambda\in\mathbb{C}\setminus\mathbb{Z}_{\geq 0}$ and $\nu_\ell, \nu_r\in\mathbb{C}$. We have for $a\in \mathbb{R}_{>1}$, $$\Phi_\lambda^{\chi_{\nu_\ell}\otimes\chi_{\nu_r}}(a)=a^\lambda\sum_{n=0}^{\infty}
\frac{4^{n}p_n^{(-\lambda/2)}(-\nu_\ell)p_n^{(-\lambda/2)}(-\nu_r)}{(-\lambda)_nn!}
(-a^{-2})^n.$$
Note that $v_{\lambda;\nu}[\lambda]=m_\lambda$ and $\textup{ev}_{\lambda,\mathbb{C}_{\chi_\nu}}(\psi_{\lambda;\nu})=\psi_{\lambda;\nu}(m_\lambda)=1$, hence $$S^{\psi_{\lambda;\nu_\ell},v_{\lambda; \nu_r}}_\lambda=1$$ (we identify $\textup{Hom}(\mathbb{C}_{\nu_r},\mathbb{C}_{\nu_\ell})\overset{\sim}{\longrightarrow}
\mathbb{C}$ by $T\mapsto T(1)$). By Theorem \[mainTHMF\][**b**]{} we then get $$\Phi_\lambda^{\chi_{\nu_\ell}\otimes\chi_{\nu_r}}(a)=\sum_{n=0}^{\infty}\psi_{\lambda;\nu_\ell}(v_{\lambda;\nu_r}[\lambda-2n])
a^{\lambda-2n}$$ for $a\in \mathbb{R}_{>1}$. The result now follows from Lemma \[r1\].
Combined with Proposition \[GaussExpres\], we reobtain the following special case of the Poisson kernel identity [@Er §2.5.2 (12)] for Meixner-Pollaczek polynomials.
We have for $a\in\mathbb{R}_{>1}$, $$\begin{split}
a^\lambda\sum_{n=0}^{\infty}&
\frac{4^{n}p_n^{(-\lambda/2)}(-\nu_\ell)p_n^{(-\lambda/2)}(-\nu_r)}{(-\lambda)_nn!}
(-a^{-2})^n=\\
&=(a+a^{-1})^\lambda\left(\frac{a-a^{-1}}{a+a^{-1}}\right)^{i(\nu_\ell+\nu_r)}
{}_2F_1\Bigl(-\frac{\lambda}{2}+i\nu_\ell, -\frac{\lambda}{2}+i\nu_r; -\lambda \,\, | \,\,
\frac{4}{(a+a^{-1})^2}\Bigr).
\end{split}$$
For a different representation theoretic interpretation of the Poisson kernel identity for Meixner-Pollaczek polynomials, see [@KJ Prop. 2.1].
$N$-point spherical functions {#SectionbKZB}
=============================
Factorisations of the Casimir element {#S61}
-------------------------------------
Let $M,M^\prime,U$ be $\mathfrak{g}$-modules. We will call $\mathfrak{g}$-intertwiners $M\rightarrow M^\prime\otimes U$ vertex operators. This terminology is stretching the standard representation theoretic notion of vertex operators as commonly used in the context of Wess-Zumino-Witten conformal field theory. In that case (see, e.g., [@JM]), it refers to intertwiners $M\rightarrow M^\prime\otimes U(z)$ for affine Lie algebra representations $M,M^\prime$ and $U(z)$ with $M$ and $M^\prime$ highest weight representations (playing the role of auxiliary spaces), and $U(z)$ an evaluation representation (playing the role of state space).
The space $\textup{Hom}(M,M^\prime\otimes U)$ of all linear maps $M\rightarrow M^\prime\otimes U$ admits the following left and right $U(\mathfrak{g})^{\otimes 2}$-action, $$\begin{split}
\bigl((x\otimes y)\Psi\bigr)(m):=(x\otimes y)\Psi(m),\\
\bigl(\Psi\ast (x\otimes y)\bigr)(m):=(1\otimes S(x))\Psi(ym)
\end{split}$$ for $x,y\in U(\mathfrak{g})$, $\Psi\in\textup{Hom}(M,M^\prime\otimes U)$ and $m\in M$, with $S$ the antipode of $U(\mathfrak{g})$. Here we suppress the representation maps if no confusion can arise. We sometimes also write $x_M$ for the action of $x\in U(\mathfrak{g})$ on the $\mathfrak{g}$-module $M$.
We say that a triple $(\tau^\ell,\tau^r,d)$ with $\tau^\ell,\tau^r\in\mathfrak{g}\otimes\mathfrak{g}$ and $d\in U(\mathfrak{g})$ is a factorisation of the Casimir element $\Omega\in Z(\mathfrak{g})$ if for all $\mathfrak{g}$-modules $M,M^\prime,U$ and for all vertex operators $\Psi\in\textup{Hom}_{\mathfrak{g}}\bigl(M,M^\prime\otimes U\bigr)$ we have $$\label{operatorKZBpre}
\frac{1}{2}\bigl((\Omega\otimes 1)\Psi-\Psi\Omega\bigr)=\tau^\ell\Psi-\Psi\ast \tau^r+(1\otimes d)\Psi$$ in $\textup{Hom}(M,M^\prime\otimes U)$.
Suppose that $(\tau^\ell,\tau^r,d)$ is a factorisation of $\Omega$. If $\Omega_{M}=\zeta_M\textup{id}$ and $\Omega_{M^\prime}=\zeta_{M^\prime}\textup{id}_{M^\prime}$ for some constants $\zeta_M,\zeta_{M^\prime}\in\mathbb{C}$ then we arrive at the [*asymptotic operator Knizhnik-Zamolodchikov-Bernard (KZB) equation*]{} for vertex operators $\Psi\in\textup{Hom}_{\mathfrak{g}}(M,M^\prime\otimes U)$, $$\label{operatorKZB}
\frac{1}{2}(\zeta_{M^\prime}-\zeta_M)\Psi=\tau^\ell\Psi-\Psi\ast\tau^r+(1\otimes d)\Psi$$ (compare with the operator KZ equation from [@FR Thm. 2.1]).
Consider now vertex operators $\Psi_i\in\textup{Hom}_{\mathfrak{g}}(M_i,M_{i-1}\otimes U_i)$ for $i=1,\ldots,N$. Denote $\mathbf{U}:=U_1\otimes U_2\otimes\cdots\otimes U_N$ and write $\mathbf{\Psi}\in\textup{Hom}_{\mathfrak{g}}(M_N,M_0\otimes\mathbf{U})$ for the composition $$\label{multiplevertex}
\mathbf{\Psi}:=(\Psi_1\otimes\textup{id}_{U_2\otimes\cdots\otimes U_N})
\cdots (\Psi_{N-1}\otimes\textup{id}_{U_N})\Psi_N$$ of the vertex operators $\Psi_i$. Assume that $\Omega_{M_i}=\zeta_{M_i}\textup{id}_{M_i}$ for some constants $\zeta_{M_i}\in\mathbb{C}$ ($0\leq i\leq N$). The asymptotic operator KZB equation now extends to the following system of equations for $\mathbf{\Psi}$.
\[bulkKZB\] Let $(\tau^\ell,\tau^r,d)$ be a factorisation of $\Omega$ with expansions $\tau^\ell=\sum_k \alpha_k^\ell\otimes \beta_k^\ell$ and $\tau^r=\sum_m\alpha_m^r\otimes\beta_m^r$ in $\mathfrak{g}\otimes\mathfrak{g}$. Under the above assumptions and conventions we have $$\label{multioperatorKZB}
\begin{split}
\frac{1}{2}(\zeta_{M_{i-1}}-\zeta_{M_i})\mathbf{\Psi}&=\Bigl(\sum_{j=1}^{i-1}\tau^\ell_{U_jU_i}-\sum_{j=i+1}^N
\tau^r_{U_iU_j}+d_{U_i}\Bigr)\mathbf{\Psi}\\
&+\sum_{k}(\alpha_k^\ell)_{M_0}(\beta_k^\ell)_{U_i}\mathbf{\Psi}
+\sum_m(\alpha_m^r)_{U_i}\mathbf{\Psi}(\beta_m^r)_{M_N}
\end{split}$$ for $i=1,\ldots,N$.
Write $\Psi_{M_i}:=\Psi_i\otimes\textup{id}_{U_{i+1}\otimes\cdots\otimes U_N}$. Then $$\label{Fstep}
\frac{1}{2}(\zeta_{M_{i-1}}-\zeta_{M_i})\mathbf{\Psi}=\frac{1}{2}\Psi_{M_1}\cdots\Psi_{M_{i-1}}\bigl(\Omega_{M_{i-1}}\Psi_{M_i}-
\Psi_{M_i}\Omega_{M_i}\bigr)\Psi_{M_{i+1}}\cdots\Psi_{M_N}.$$ Now gives $$\frac{1}{2}\bigl(\Omega_{M_{i-1}}\Psi_{M_i}-
\Psi_{M_i}\Omega_{M_i}\bigr)=\sum_k(\alpha_k^\ell)_{M_{i-1}}(\beta_k^\ell)_{U_i}\Psi_{M_i}+
\sum_m(\alpha_m^r)_{U_i}\Psi_{M_i}(\beta_m^r)_{M_i}+d_{U_i}\Psi_{M_i}.$$ Substitute this equation in and push the action of $\alpha_k^\ell$ on $M_{i-1}$ (resp. the action of $\beta_m^r$ on $M_i$) through the product $\Psi_{M_1}\cdots\Psi_{M_{i-1}}$ (resp. $\Psi_{M_{i+1}}\cdots\Psi_{M_N}$) of vertex operators using the fact that $$(x\otimes 1)\Psi-\Psi x=-(1\otimes x)\Psi$$ for $x\in\mathfrak{g}$ and $\Psi\in\textup{Hom}_{\mathfrak{g}}(M,M^\prime\otimes U)$. This immediately results in .
The asymptotic operator KZB equations for an appropriate factorisation of $\Omega$ give rise to boundary KZB type equations that are solved by asymptotical $N$-point correlation functions for boundary WZW conformal field theory on a cylinder. Here asymptotical means that the “positions” of the local observables in the correlation functions escape to infinity. We will define the asymptotical $N$-point correlation functions directly in Subsections \[S63\] & \[intertwinersection\], and call them (formal) $N$-point spherical functions. The discussion how they arise as limits of correlation functions is postponed to a followup paper.
We give now first two families of examples of factorisations of $\Omega$. The first family is related to the expression of $\Omega$. It leads to asymptotic KZB equations for generalised weighted trace functions (see [@ES]). As we shall see later, this family also gives rise to the asymptotic boundary KZB equations for the (formal) $N$-point spherical functions using a reflection argument. The second family is related to the Cartan decomposition of $\Omega$, and leads directly to the asymptotic boundary KZB equations. This second derivation of the asymptotic boundary KZB equations is expected to be crucial for the generalisation to quantum groups.
Felder’s [@F], [@ES §2] trigonometric dynamical $r$-matrix $r\in\mathcal{R}\otimes \mathfrak{g}^{\otimes 2}$ is given by $$\label{Felderr}
r:=-\frac{1}{2}\sum_{j=1}^rx_j\otimes x_j-\sum_{\alpha\in R}\frac{e_{-\alpha}\otimes e_\alpha}{1-\xi_{-2\alpha}}.$$ Set $$r^{\theta_1}:=(\theta\otimes \textup{id}_{\mathfrak{g}})r=
\frac{1}{2}\sum_{j=1}^rx_j\otimes x_j+\sum_{\alpha\in R}\frac{e_\alpha\otimes e_\alpha}
{1-\xi_{-2\alpha}}.$$ For $s=\sum_is_i\otimes t_i\in\mathfrak{g}\otimes\mathfrak{g}$ we write $s_{21}:=
\sum_it_i\otimes s_i$ and $s_{21}^{\theta_2}:=(1\otimes \theta)s_{21}$. Note that $r^{\theta_1}$ is a symmetric tensor, $$r_{21}^{\theta_2}=r^{\theta_1}.$$ We will write below $r_{21}^{\theta_2}$ for the occurrences of the $\theta$-twisted $r$-matrices in the asymptotic boundary KZB equations, since this is natural when viewing the asymptotic boundary KZB equations as formal limit of integrable boundary qKZB equations (this will be discussed in future work).
Define folded $r$-matrices by $$r^{\pm}:=\pm r+r_{21}^{\theta_2}.$$ Note that $r^+\in\mathcal{R}\otimes\mathfrak{k}\otimes\mathfrak{g}$ and $r^-\in
\mathcal{R}\otimes\mathfrak{p}\otimes \mathfrak{g}$. The folded $r$-matrices $r^{\pm}$ are explicitly given by $$\label{explicitsigmatauexpl}
\begin{split}
r^+&=\sum_{\alpha\in R}\frac{y_\alpha\otimes e_\alpha}{1-\xi_{-2\alpha}},\\
r^-&=\sum_{j=1}^rx_j\otimes x_j+\sum_{\alpha\in R}\frac{(e_\alpha+e_{-\alpha})\otimes e_\alpha}
{1-\xi_{-2\alpha}}.
\end{split}$$
\[propfactorization\] Fix $a\in A_{\textup{reg}}$. The following triples $(\tau^\ell,\tau^r,d)$ give factorisations of the Casimir $\Omega\in Z(\mathfrak{g})$.
1. $\tau^\ell=\tau^r=r(a)$, $d=-\frac{1}{2}\sum_{\alpha\in R^+}\Bigl(\frac{1+a^{-2\alpha}}{1-a^{-2\alpha}}
\Bigr)t_\alpha$.
2. $\tau^\ell=r^+(a)$, $\tau^r=-r^-(a)$, $d=b(a)$ with $$\label{kappa}
b:=\frac{1}{2}\sum_{j=1}^rx_j^2-\frac{1}{2}\sum_{\alpha\in R^+}
\Bigl(\frac{1+\xi_{-2\alpha}}{1-\xi_{-2\alpha}}\Bigr)t_\alpha+\sum_{\alpha\in R}
\frac{e_\alpha^2}{1-\xi_{-2\alpha}}\in\mathcal{R}\otimes U(\mathfrak{g}).$$
The factorisations are obtained from the explicit expressions and of $\Omega$ by moving Lie algebra elements in the resulting expression of $\frac{1}{2}\bigl((\Omega\otimes 1)\Psi-\Psi\Omega\bigr)$ through the vertex operator $\Psi$ following a particular (case-dependent) strategy. The elementary formulas we need are $$\label{firsteqn}
\begin{split}
(x\otimes 1)\Psi-\Psi x&=-(1\otimes x)\Psi,\\
(xy\otimes 1)\Psi-\Psi xy&=-(1\otimes x)\Psi y-(x\otimes y)\Psi
\end{split}$$ for $x,y\in \mathfrak{g}$ and $\Psi\in\textup{Hom}_{\mathfrak{g}}(M,M^\prime\otimes U)$. Note that the second formula gives an expression of $(xy\otimes 1)\Psi-\Psi xy$ with $x$ no longer acting on $M$ and $y$ no longer acting on $M^\prime$. For case [**b**]{} we also need formulas such that both $x$ and $y$ are not acting on $M^\prime$ (resp. on $M$), $$\label{secondeqn}
\begin{split}
(xy\otimes 1)\Psi-\Psi xy&=-(1\otimes x)\Psi y-(1\otimes y)\Psi x+(1\otimes yx)\Psi,\\
(xy\otimes 1)\Psi-\Psi xy&=-(x\otimes y)\Psi-(y\otimes x)\Psi-(1\otimes xy)\Psi
\end{split}$$ for $x,y\in\mathfrak{g}$ and $\Psi\in\textup{Hom}_{\mathfrak{g}}(M,M^\prime\otimes U)$. These equations are easily obtained by combining the two formulas of .\
[**a.**]{} Substitute into $\frac{1}{2}((\Omega\otimes 1)\Psi-\Psi\Omega)$ and apply to the terms. The resulting identity can be written as $\tau^\ell\Psi-\Psi\ast\tau^r+
(1\otimes d)\Psi$ with $(\tau^\ell,\tau^r,d)$ as stated.\
[**b.**]{} Use that $$\frac{1}{2}\bigl(
(\Omega\otimes 1)\Psi-\Psi\Omega\bigr)=
\frac{1}{2}\bigl((\textup{Ad}_a(\Omega)\otimes 1)\Psi-\Psi\textup{Ad}_a(\Omega)\bigr)$$ and substitute in the right hand side of this equation. For the quadratic terms $xy$ ($x,y\in\mathfrak{g}$) in the resulting formula we use the second formula of when $x\in\mathfrak{k}$ and $y\in\textup{Ad}_a(\mathfrak{k})$, the first formula of when both $x,y\in\textup{Ad}_a(\mathfrak{k})$ or both $x,y\in\mathfrak{h}$, and the second formula of when both $x,y\in\mathfrak{k}$. It results in the formula with $(\tau^\ell,\tau^r,d)$ given by $$\begin{split}
\tau^\ell&=\frac{1}{2}\sum_{\alpha\in R}y_\alpha\otimes\frac{((a^\alpha+a^{-\alpha})\textup{Ad}_a(y_\alpha)-2y_\alpha)}{(a^\alpha-a^{-\alpha})^2},\\
\tau^r&=-\sum_{j=1}^rx_j\otimes x_j+\frac{1}{2}\sum_{\alpha\in R}
\frac{(a^\alpha+a^{-\alpha})y_\alpha-2\textup{Ad}_a(y_\alpha)}{(a^\alpha-a^{-\alpha})^2}
\otimes\textup{Ad}_a(y_\alpha),\\
d&=\frac{1}{2}\sum_{j=1}^rx_j^2-\frac{1}{2}\sum_{\alpha\in R^+}\Bigl(\frac{1+a^{-2\alpha}}{1-a^{-2\alpha}}
\Bigr)t_\alpha+\frac{1}{2}\sum_{\alpha\in R}\frac{(\textup{Ad}_a(y_\alpha^2)-y_\alpha^2)}
{(a^\alpha-a^{-\alpha})^2}.
\end{split}$$ By the elementary identities $$\begin{split}
2\textup{Ad}_a(y_\alpha)-(a^\alpha+a^{-\alpha})y_\alpha&=(a^\alpha-a^{-\alpha})(e_\alpha+e_{-\alpha}),\\
\textup{Ad}_a(y_\alpha^2)-y_\alpha^2&=(a^{2\alpha}-1)e_\alpha^2+(a^{-2\alpha}-1)e_{-\alpha}^2
\end{split}$$ the above expressions for $\tau^\ell, \tau^r$ and $d$ simplify to the expressions and for $r^+(a),-r^-(a)$ and $b(a)$.
\[remarkfactorization\] The limit ${}^{\infty}r:=\lim_{a\rightarrow\infty}r(a)$ (meaning $a^{\alpha}\rightarrow\infty$ for all $\alpha\in R^+$) gives the classical $r$-matrix $${}^{\infty}r=-\frac{1}{2}\sum_{j=1}^rx_j\otimes x_j-\sum_{\alpha\in R^+}e_{-\alpha}\otimes e_\alpha
\in\theta(\mathfrak{b})\otimes\mathfrak{b}.$$ The corresponding limit for the folded $r$-matrices $r^{\pm}(a)$ gives $${}^{\infty}r^{\pm}:=\pm {}^{\infty}r+{}^{\infty}r_{21}^{\theta_2}.$$ As a consequence of Proposition \[propfactorization\] we then obtain the following two (nondynamical) factorisations of the Casimir $\Omega$,
1. $(\sigma,\tau,d)=({}^{\infty}r,{}^{\infty}r,-2t_\rho)$.
2. $(\sigma,\tau,d)=({}^{\infty}r^+,-{}^{\infty}r^-,{}^{\infty}b)$ with $${}^{\infty}b:=\frac{1}{2}\sum_{j=1}^rx_j^2-t_\rho+\sum_{\alpha\in R^+}e_\alpha^2\in U(\mathfrak{g}).$$
Differential vertex operators {#S62}
-----------------------------
In Subsection \[S63\] we apply the results of the previous subsection to $M_i=\mathcal{H}_{\lambda_i}^\infty$ with $\lambda_i\in\mathfrak{h}^*$ ($i=0,\ldots,N$) and to finite dimensional $G$-representations $U_j$ ($j=1,\ldots,N$). Before doing so, we first describe the appropriate class of vertex operators in this context, which consists of $G$-equivariant differential operators. We use the notion of vector-valued $G$-equivariant differential operators between spaces of global sections of complex vector bundles, see [@He Chpt. II] as well as [@KR §1].
Identify the $G$-space $\mathcal{H}_{\lambda}^{\infty}$ with the space of global smooth sections of the complex line bundle $\mathcal{L}_\lambda:=(G\times\mathbb{C})/\sim_\lambda$ over $G/AN\simeq K$, with equivalence relation $\sim_\lambda$ given by $$\label{sim}
(gb,\eta_{\lambda+\rho}(b^{-1})c)\sim_\lambda (g,c),\qquad (g\in G, b\in AN, c\in\mathbb{C}).$$ For $\lambda,\mu\in\mathfrak{h}^*$ and $U$ a finite dimensional $G$-representation let $\mathbb{D}(\mathcal{H}_{\lambda}^\infty,\mathcal{H}_{\mu}^{\infty}\otimes U)$ be the space of differential $G$-intertwiners $\mathcal{H}_{\lambda}^{\infty}\rightarrow\mathcal{H}_{\mu}^{\infty}\otimes U$ (note that $\mathcal{H}_\mu^\infty\otimes U$ is the space of smooth section of the vector bundle $(G\times U)/\sim_\lambda$, with $\sim_\lambda$ given by the same formula with $c\in U$). We call $D\in\mathbb{D}(\mathcal{H}_\lambda^\infty,\mathcal{H}_\mu^{\infty}\otimes U)$ a [*differential vertex operator*]{}.
Let $\mathbb{D}_0^\prime(\mathcal{L}_\lambda)$ be the $\mathfrak{g}$-module consisting of distributions on $\mathcal{L}_\lambda$ supported at $1\in K$. Note that $\mathbb{D}_0^\prime(\mathcal{L}_\lambda)$ is contained in the continuous linear dual of $\mathcal{H}_\lambda^\infty$. A straightforward adjustment of the proof of [@CS Lem. 2.4] yields a linear isomorphism $$\mathbb{D}(\mathcal{H}_{\lambda}^\infty,\mathcal{H}_{\mu}^{\infty}\otimes U)\simeq
\textup{Hom}_{\mathfrak{g}}(\mathbb{D}_0^\prime(\mathcal{L}_\mu),\mathbb{D}_0^\prime(\mathcal{L}_\lambda)\otimes U)$$ via dualisation. Furthermore, $$M_{-\lambda-\rho}\simeq \mathbb{D}_0^\prime(\mathcal{L}_\lambda)$$ as $\mathfrak{g}$-modules by Schwartz’ theorem, with the distribution $\omega$ associated to $xm_{-\lambda-\rho}$ ($x\in U(\mathfrak{g})$) defined by $$\omega(\phi):=\bigl(r_\ast(x)\phi\bigr)(1)\qquad (\phi\in\mathcal{H}_\lambda^\infty)$$ (see again the proof of [@CS Lem. 2.4]). We thus reach the following conclusion.
\[isoprop\] For $\lambda,\nu\in\mathfrak{h}^*$ and $U$ a finite dimensional $G$-representation we have $$\label{isodiff}
\mathbb{D}(\mathcal{H}_{\lambda}^\infty,\mathcal{H}_{\mu}^{\infty}\otimes U)\simeq
\textup{Hom}_{\mathfrak{g}}(M_{-\mu-\rho}, M_{-\lambda-\rho}\otimes U).$$
\[explicitinverse\] The inverse of the isomorphism can be described explicitly as follows. Fix a vertex operator $\Psi\in\textup{Hom}_{\mathfrak{g}}(M_{-\mu-\rho},
M_{-\lambda-\rho}\otimes U)$. Let $\{u_i\}_i$ be a linear basis of $U$ and write $\{u_i^*\}_i$ for its dual basis. Let $Y_i\in U(\mathfrak{k})$ be the unique elements such that $$\Psi(m_{-\mu-\rho})=\sum_iY_im_{-\lambda-\rho}\otimes u_i,$$ cf. the proof of Proposition \[relEisPrinalg\][**a**]{}. Under the isomorphism , the intertwiner $\Psi$ is mapped to the differential vertex operator $D_\Psi=\sum_iL_i\otimes u_i\in\mathbb{D}(\mathcal{H}_\lambda^\infty,\mathcal{H}_\mu^\infty\otimes U)$ with the scalar differential operators $L_i: \mathcal{H}_\lambda^\infty\rightarrow\mathcal{H}_\mu^\infty$ explicitly given by $$(L_i\phi)(g)=\sum_ju_i^*(gu_j)\bigl(r_\ast(Y_j)\phi\bigr)(g)
\qquad (\phi\in\mathcal{H}_\lambda^\infty,\, g\in G).$$
For $\Psi_V\in\textup{Hom}_{\mathfrak{g}}(M_{-\mu-\rho}, M_{-\lambda-\rho}\otimes V)$ and $\Psi_U\in\textup{Hom}_{\mathfrak{g}}(M_{-\nu-\rho}, M_{-\mu-\rho}\otimes U)$ set $$\label{compositionvertex}
\begin{split}
\Psi_{V,U}:=(\Psi_V\otimes\textup{id}_U)\Psi_U&\in\textup{Hom}_{\mathfrak{g}}(M_{-\nu-\rho}, M_{-\lambda-\rho}\otimes V\otimes U),\\
D_{U,V}:=(D_{\Psi_U}\otimes\textup{id}_V)D_{\Psi_V}&\in\mathbb{D}(\mathcal{H}_\lambda^\infty,\mathcal{H}_\nu^\infty
\otimes U\otimes V).
\end{split}$$ These two composition rules are compatible with the isomorphism from Proposition \[isoprop\]:
Let $P_{UV}: U\otimes V\rightarrow V\otimes U$ be the $G$-linear isomorphism flipping the two tensor components. Then $$D_{\Psi_{V,U}}=(\textup{id}_{\mathcal{H}_\nu^\infty}\otimes P_{UV})D_{U,V}$$ in $\mathbb{D}(\mathcal{H}_\lambda^\infty, \mathcal{H}_\nu^\infty\otimes V\otimes U)$.
This follows by a straightforward but lengthy computation using Remark \[explicitinverse\].
Next we consider the parametrisation of the spaces of vertex operators. Write $m_\mu^*\in M_\mu^*$ for the linear functional satisfying $m_\mu^*(m_\mu)=1$ and $m_\mu^*(v)=0$ for $v\in\bigoplus_{\nu<\mu}M_\mu[\nu]$.
Let $U$ be a finite dimensional $\mathfrak{g}$-module, $\lambda,\mu\in\mathfrak{h}^*$, and $\Psi\in\textup{Hom}_{\mathfrak{g}}(M_\lambda,M_\mu\otimes U)$. Then $$\langle\Psi\rangle:=(m_{\mu}^*\otimes\textup{id}_U)\Psi(m_\lambda)\in U[\lambda-\mu]$$ is called the expectation value of the vertex operator $\Psi$.
The expectation value of the associated differential vertex operators read as follows.
For $\Psi\in\textup{Hom}_{\mathfrak{g}}(M_{-\mu-\rho}, M_{-\lambda-\rho}\otimes U)$ we have $$\bigl(D_\Psi\mathbb{I}_\lambda\bigr)(1)=\langle \Psi\rangle$$ in $U[\lambda-\mu]$, where $\mathbb{I}_\lambda\in\mathcal{H}_\lambda^\infty$ is the function $$\mathbb{I}_\lambda(kan):=a^{-\lambda-\rho}\qquad (k\in K, a\in A, n\in N).$$
Using the notations from Remark \[explicitinverse\], we have $$\langle\Psi\rangle=\sum_i\epsilon(Y_i)u_i$$ with $\epsilon$ the counit of $U(\mathfrak{k})$. On the other hand, $$\bigl(D_\Psi\mathbb{I}_\lambda\bigr)(1)=\sum_i(L_i\mathbb{I}_\lambda)(1)u_i=
\sum_i\bigl(r_*(Y_i)\mathbb{I}_\lambda\bigr)(1)u_i=\sum_i\epsilon(Y_i)u_i,$$ hence the result.
By [@E Lem. 3.3] we have the following result.
\[evlem\] Let $U$ be a finite dimensional $\mathfrak{g}$-module, $\lambda\in\mathfrak{h}^*$ and $\mu\in\mathfrak{h}_{\textup{irr}}^*$. The expectation value map $\langle\cdot\rangle$ defines a linear isomorphism $$\langle\cdot\rangle: \textup{Hom}_{\mathfrak{g}}(M_\lambda,M_{\mu}\otimes U)\overset{\sim}{\longrightarrow}
U[\lambda-\mu].$$
The weights of a finite dimensional $\mathfrak{g}$-module $U$ lie in the integral weight lattice $$P:=\{\mu\in\mathfrak{h}^* \,\, | \,\, (\mu,\alpha^\vee)\in\mathbb{Z} \quad \forall\, \alpha\in R\}.$$ Hence for $\mu\in\mathfrak{h}_{\textup{irr}}^*$, the space $\textup{Hom}_{\mathfrak{g}}(M_\lambda, M_\mu\otimes U)$ of vertex operators is trivial unless $\lambda\in \mu+P$. At a later stage (see Section \[sectionBFO\]), we want to restrict to highest weights $\lambda_0\in\mathfrak{h}_{\textup{irr}}^*$ such that for any vertex operator $\mathbf{\Psi}\in\textup{Hom}_{\mathfrak{g}}(M_{\lambda_N},M_{\lambda_0}\otimes\mathbf{U})$, given as a product of vertex operators $\Psi_i\in\textup{Hom}_{\mathfrak{g}}(M_{\lambda_i},
M_{\lambda_{i-1}}\otimes U_i)$ ($i=1,\ldots,N$), has the property that $\lambda_{i-1}\in\mathfrak{h}_{\textup{irr}}^*$ for $i=1,\ldots,N$ (i.e., all vertex operators are determined by their expectation values). In that case we will restrict to highest weights from the dense open subset $$\mathfrak{h}_{\textup{reg}}^*:=\{\nu\in\mathfrak{h}^*\,\, | \,\, (\nu,\alpha^\vee)\not\in\mathbb{Z}\,\, \quad
\forall\, \alpha\in R\}$$ of $\mathfrak{h}$. The (differential) vertex operators are then denoted as follows.
\[notVO\] Let $\lambda\in\mathfrak{h}_{\textup{reg}}^*$.
1. If $U$ is a finite dimensional $\mathfrak{g}$-module and $u\in U[\lambda-\mu]$ is a weight vector of weight $\lambda-\mu$, then we write $\Psi_\lambda^u\in\textup{Hom}_{\mathfrak{g}}(M_\lambda,M_{\mu}\otimes U)$ for the unique vertex operator with expectation value $\langle\Psi_\lambda^u\rangle=u$.
2. If $U$ is a finite dimensional $G$-representation and $u\in U[\lambda-\mu]$ is a weight vector of weight $\lambda-\mu$, then we write $D_\lambda^u\in\mathbb{D}(\mathcal{H}_\lambda^\infty,\mathcal{H}_\mu^\infty\otimes U)$ for the unique differential vertex operator with $(D_\lambda^u\mathbb{I}_\lambda)(1)=u$.
The expectation value of products of vertex operators gives rise to the fusion operator. We recall its definition in Subsection \[sectionBFO\], where we also discuss boundary versions of fusion operators.
$N$-point spherical functions and asymptotic boundary KZB equations {#S63}
-------------------------------------------------------------------
Fix finite dimensional $G$-representations $U_1,\ldots,U_N$ with representation maps $\tau_{U_1},\ldots,\tau_{U_N}$, and differential vertex operators $D_i\in\mathbb{D}(\mathcal{H}_{\lambda_i}^\infty,
\mathcal{H}_{\lambda_{i-1}}^\infty\otimes U_i)$ for $i=1,\ldots,N$. Write $\underline{\lambda}=(\lambda_0,\lambda_1,\ldots,\lambda_N)$ and $\mathbf{U}:=U_1\otimes\cdots\otimes U_N$. Write $\mathbf{D}\in\mathbb{D}(\mathcal{H}_{\lambda_N}^\infty,\mathcal{H}_{\lambda_0}^{\infty}\otimes\mathbf{U})$ for the product of the $N$ differential vertex operators $D_i$ ($1\leq i\leq N$), $$\mathbf{D}=(D_1\otimes\textup{id}_{U_2\otimes\cdots\otimes U_N})\cdots
(D_{N-1}\otimes\textup{id}_{U_N})D_N,$$ which we call a differential vertex operator of weight $\underline{\lambda}$.
Fix two finite dimensional $K$-representations $V_\ell$ and $V_r$, with representation maps $\sigma_\ell$ and $\sigma_r$ respectively. Let $\sigma_\ell^{(N)}$ be the representation map of the tensor product $K$-representation $V_\ell\otimes\mathbf{U}$. We consider $(V_\ell\otimes\mathbf{U})\otimes V_r^*\simeq\textup{Hom}(V_r,V_\ell\otimes\mathbf{U})$ as $K\times K$-representation, with representation map $\sigma^{(N)}:=\sigma_\ell^{(N)}\otimes\sigma_r^*$. Note that if $\phi_\ell\in\textup{Hom}_K(\mathcal{H}_{\lambda_0},V_\ell)$ then $$\bigl(\phi_\ell\otimes\textup{id}_{\mathbf{U}}\bigr)\mathbf{D}\in
\textup{Hom}_K(\mathcal{H}_{\lambda_N},V_\ell\otimes\mathbf{U})$$ by .
\[Nsphdef\] Let $\phi_\ell\in\textup{Hom}_K(\mathcal{H}_{\lambda_0},V_\ell)$, $\phi_r\in\textup{Hom}_K(V_r,\mathcal{H}_{\lambda_N})$ and $\mathbf{D}$ a differential vertex operator of weight $\underline{\lambda}$. We call the the smooth $\sigma^{(N)}$-spherical function $$\label{Sphericalfunctionwithinsertions}
f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}(g):=(\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{D}
(\pi_{\lambda_N}(g)\phi_r)\qquad (g\in G)$$ an $N$-point $\sigma^{(N)}$-spherical function associated with the $N+1$-tuple of principal series representations $\mathcal{H}_{\underline{\lambda}}:=(\mathcal{H}_{\lambda_0},\ldots,\mathcal{H}_{\lambda_N})$.
To keep the notations manageable we write from now on the action of $U(\mathfrak{g})$ and $G$ on $\mathcal{H}_{\lambda}^{\infty}$ without specifying the representation map if no confusion can arise. For instance, for $x\in U(\mathfrak{g})$, $g\in G$ and $v\in\mathcal{H}_{\lambda_N}^{\infty}$ we write $gxv\in\mathcal{H}_{\lambda_N}^{\infty}$ for the smooth vector $\pi_{\lambda_N}(g)((x)_{\mathcal{H}_{\lambda_N}^{\infty}}v)$, and the $N$-point spherical function will be written as $$f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}(g):=(\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{D}
(g\phi_r)\qquad (g\in G).$$
In Subsection \[intertwinersection\] we define [*formal*]{} $N$-point spherical functions, which are asymptotical $N$-point correlation functions for boundary Wess-Zumino-Witten conformal field theory on the cylinder when the positions escape to infinity. The $N$-point spherical functions in Definition \[Nsphdef\] are their analogues in the context of principal series.
By Proposition \[relEisPrin\][**c**]{} the $N$-point spherical function $f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}$ admits the (Eisenstein) integral representation $$\label{EisN}
\begin{split}
f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}(g)=
\int_Kdx\, \xi_{-\lambda_N-\rho}(a(g^{-1}x))&\Bigl(\sigma_\ell(x)\otimes\tau_{U_1}(x)\otimes\cdots\\
&\quad\cdots\otimes
\tau_{U_N}(x)\otimes\sigma_r^*(k(g^{-1}x))\Bigr)
T_{\lambda_N}^{(\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{D}, \phi_r}
\end{split}$$ with the vector $T_{\lambda_N}^{(\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{D}, \phi_r}\in V_\ell\otimes\mathbf{U}\otimes V_r^*$ given by $$T_{\lambda_N}^{(\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{D}, \phi_r}=
\iota_{\lambda_N,V_\ell\otimes\mathbf{U}}((\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{D})\otimes\j_{\lambda_N,V_r}(\phi_r).$$ Theorem \[thmRAD\][**a**]{} gives the family of differential equations $$\label{diffN}
\widehat{\Pi}^{\sigma^{(N)}}(z)\bigl(f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}|_{A_{\textup{reg}}}\bigr)=
\zeta_{\lambda_N-\rho}(z)f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}|_{A_{\textup{reg}}},
\qquad z\in Z(\mathfrak{g})$$ for the restriction of $f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}$ to $A_{\textup{reg}}$. We will now show that $f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}|_{A_{\textup{reg}}}$ satisfies $N$ additional first order asymptotic boundary KZB type differential equations. Recall the factorisation $(r^+(a),-r^-(a),b(a))$ of $\Omega$ for $a\in A_{\textup{reg}}$, with $r^{\pm}$ the folded $r$-matrices and $b$ given by .
\[bKZBresult\] The $N$-point $\sigma^{(N)}$-spherical function $f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}$ satisfies $$\label{bKZBungauged}
\begin{split}
\Bigl(&\frac{(\lambda_{i-1},\lambda_{i-1})}{2}-\frac{(\lambda_i,\lambda_i)}{2}+
\sum_{j=1}^r(x_j)_{U_i}\partial_{x_j}\Bigr)f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}|_{A_{\textup{reg}}}=\\
&\quad\,=\Bigl(
r_{V_\ell U_i}^++\sum_{j=1}^{i-1}r_{U_jU_i}^++b_{U_i}+\sum_{j=i+1}^Nr_{U_iU_j}^-
+\widetilde{r}^+_{U_iV_r^*}\Bigr)
f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}|_{A_{\textup{reg}}}
\end{split}$$ for $i=1,\ldots,N$, with right boundary term $$\label{rtildeplus}
\widetilde{r}^+:=\sum_{\alpha\in R}
\frac{e_{\alpha}\otimes y_\alpha}{\xi_\alpha-\xi_{-\alpha}}\in\mathcal{R}\otimes\mathfrak{g}\otimes\mathfrak{k}$$ satisfying $\widetilde{r}^+(a)=(\textup{Ad}_{a^{-1}}\otimes 1)r_{21}^+(a)$ for $a\in A_{\textup{reg}}$.
We derive the asymptotic boundary KZB type equations in two different ways. The first proof uses Proposition \[propfactorization\][**b**]{} involving the folded versions of Felder’s dynamical $r$-matrix, the second proof uses Proposition \[propfactorization\][**a**]{} with a reflection argument. The second argument is of interest from the conformal field theoretic point of view, and provides some extra insights in the term $b$ appearing in the asymptotic boundary KZB equations.\
[**Proof 1**]{} (using the factorisation of $\Omega$ in terms of folded $r$-matrices).\
Let $a\in A_{\textup{reg}}$. By Corollary \[bulkKZB\] applied to the factorisation $(r^+(a), -r^-(a), b(a))$ of $\Omega$, we have $$\begin{split}
\frac{1}{2}(\zeta_{\lambda_{i-1}-\rho}(\Omega)-\zeta_{\lambda_i-\rho}(\Omega))f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}(a)&=
\Bigl(\sum_{j=1}^{i-1}r_{U_jU_i}^+(a)+b_{U_i}(a)+\sum_{j=i+1}^Nr_{U_iU_j}^-(a)\Bigr)
f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}(a)\\
&+\sum_k\bigl(\phi_\ell\otimes\textup{id}_{\mathbf{U}}\bigr)\bigl((\alpha_k^+)_{\mathcal{H}_{\lambda_0}^{\infty}}(\beta_k^+)_{U_i}
\mathbf{D}(a\phi_r)\bigr)\\
&-\sum_k\bigl(\phi_\ell\otimes\textup{id}_{\mathbf{U}}\bigr)\bigl((\alpha_k^-)_{U_i}\mathbf{D}
(\beta_k^-a\phi_r)\bigr)
\end{split}$$ where we have written $r^{\pm}(a)=\sum_k\alpha_k^{\pm}\otimes\beta_k^{\pm}$. Now using $r^+(a)\in\mathfrak{k}\otimes\mathfrak{g}$ and $$\bigl(1\otimes\textup{Ad}_{a^{-1}})r^-(a)=
\sum_{j=1}^rx_j\otimes x_j+\widetilde{r}^+(a)$$ with $\widetilde{r}^+(a)\in\mathfrak{g}\otimes\mathfrak{k}$ given by , the asymptotic boundary KZB type equation follows from the fact that $\phi_\ell$ and $\phi_r$ are $K$-intertwiners and $\zeta_{\lambda_{i-1}-\rho}(\Omega)-\zeta_{\lambda_i-\rho}(\Omega)=
(\lambda_{i-1},\lambda_{i-1})-(\lambda_i,\lambda_i)$.\
[**Proof 2**]{} (using a reflection argument).\
Let $a\in A_{\textup{reg}}$. Recall that the unfolded factorisation of $\Omega$ is $(r(a), r(a), d(a))$ with $$d(a):=-\frac{1}{2}\sum_{\alpha\in R}\Bigl(\frac{1+a^{-2\alpha}}{1-a^{-2\alpha}}\Bigr)t_\alpha.$$ Then it follows from a direct computation that $$\label{kappaalt}
b(a)=d(a)+m(r^{\theta_1}(a)).$$ Furthermore, $$\label{twistedcomm}
-r^{\theta_1}(a)D-D\ast r^{\theta_1}(a)=(1\otimes m(r^{\theta_1}(a)))D$$ for a differential vertex operator $D$. This follows from the fact that $r^{\theta_1}(a)$ is a symmetric tensor in $\mathfrak{g}\otimes\mathfrak{g}$ and $$-(x\otimes x)D-D\ast (x\otimes x)=(1\otimes x^2)D$$ for $x\in\mathfrak{g}$. The proof of the asymptotic boundary KZB equation using a reflection argument now proceeds as follows. Corollary \[bulkKZB\] gives $$\begin{split}
\Bigl(\frac{(\lambda_{i-1},\lambda_{i-1})}{2}-&\frac{(\lambda_i,\lambda_i)}{2}\Bigr)f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}(a)=
\Bigl(\sum_{j=1}^{i-1}r_{U_jU_i}(a)-\sum_{j=i+1}^Nr_{U_iU_j}(a)+d_{U_i}(a)\Bigr)
f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}(a)\\
&+\bigl(\phi_\ell\otimes\textup{id}_{\mathbf{U}}\bigr)\bigl(r_{\mathcal{H}_{\lambda_0}^{\infty}U_i}(a)\mathbf{D}(a\phi_r)\bigr)
+\sum_k\bigl(\phi_\ell\otimes\textup{id}_{\mathbf{U}}\bigr)((\alpha_k)_{U_i}
\mathbf{D}(\beta_ka\phi_r))
\end{split}$$ with $r(a)=\sum_k\alpha_k\otimes\beta_k$. We now apply the identity $$\bigl(\phi_\ell\otimes\textup{id}_{\mathbf{U}}\bigr)r_{\mathcal{H}_{\lambda_0}^{\infty}U_i}(a)=
r_{V_\ell U_i}^+(a)\bigl(\phi_\ell\otimes\textup{id}_{\mathbf{U}}\bigr)-
\bigl(\phi_\ell\otimes\textup{id}_{\mathbf{U}}\bigr)r^{\theta_1}_{\mathcal{H}_{\lambda_0}^{\infty}U_i}(a)$$ in $\textup{Hom}(\mathcal{H}_{\lambda_0}^{\infty}\otimes\mathbf{U},V_\ell\otimes\mathbf{U})$ to the left boundary term and the identity $$\begin{split}
\sum_k(\alpha_k)_{U_i}\mathbf{D}(\beta_ka\phi_r)&=-\sum_{j=1}^r(x_j)_{U_i}\mathbf{D}(ax_j\phi_r)\\
&+\sum_k(\theta(\alpha_k))_{U_i}\mathbf{D}(\beta_ka\phi_r)+
\widetilde{r}^+_{U_iV_r^*}(a)\mathbf{D}(a\phi_r)
\end{split}$$ in $\mathbf{U}\otimes V_r^*$ to the right boundary term. The latter equality follows from the easily verified identities $$\begin{split}
-\bigl(\textup{id}\otimes \theta\textup{Ad}_{a^{-1}}\bigr)(r(a))&=
-\sum_{j=1}^rx_j\otimes x_j+\bigl(\textup{id}\otimes\textup{Ad}_{a^{-1}}\bigr)(r^{\theta_1}(a)),\\
-\bigl(\textup{Ad}_{a^{-1}}\otimes\textup{id}\bigr)(r_{21}^+(a))&=\bigl(\textup{id}\otimes\textup{Ad}_{a^{-1}}\bigr)(r(a))+\bigl(\textup{id}\otimes \theta\textup{Ad}_{a^{-1}}\bigr)(r(a)).
\end{split}$$ We thus arrive at the formula $$\label{tttt}
\begin{split}
\Bigl(&\frac{(\lambda_{i-1},\lambda_{i-1})}{2}-\frac{(\lambda_i,\lambda_i)}{2}+\sum_{j=1}^r(x_j)_{U_i}\partial_{x_j}\Bigr)f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}(a)=\\
&=\Bigl(r_{V_\ell U_i}^+(a)+\sum_{j=1}^{i-1}r_{U_jU_i}(a)+d_{U_i}(a)-\sum_{j=i+1}^Nr_{U_iU_j}(a)
+\widetilde{r}_{U_iV_r^*}^+(a)\Bigr)f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}(a)\\
&-\bigl(\phi_\ell\otimes\textup{id}_{\mathbf{U}}\bigr)\bigl(r_{\mathcal{H}_{\lambda_0}^{\infty}U_i}^{\theta_1}(a)\mathbf{D}(a\phi_r)\bigr)
+\sum_k\bigl(\phi_\ell\otimes\textup{id}_{\mathbf{U}}\bigr)((\theta(\alpha_k))_{U_i}
\mathbf{D}(\beta_ka\phi_r)).
\end{split}$$ Now pushing $r^{\theta_1}_{\mathcal{H}_{\lambda_0}^{\infty}U_i}(a)$ through the differential vertex operators $D_j$ ($1\leq j<i$) and pushing the action of $\beta_k$ through $D_j$ ($i<j\leq N$) using and using the fact that $r^{\theta_1}(a)$ is a symmetric tensor in $\mathfrak{g}\otimes\mathfrak{g}$, the last line becomes $$\begin{split}
-\bigl(\phi_\ell\otimes\textup{id}_{\mathbf{U}}\bigr)&\bigl(r_{\mathcal{H}_{\lambda_0}^{\infty}U_i}^{\theta_1}(a)\mathbf{D}(a\phi_r)\bigr)
+\sum_k\bigl(\phi_\ell\otimes\textup{id}_{\mathbf{U}}\bigr)((\theta(\alpha_k))_{U_i}
\mathbf{D}(\beta_ka\phi_r))\\
&=\Bigl(\sum_{j=1}^{i-1}r^{\theta_1}_{U_jU_i}(a)+\sum_{j=i+1}^Nr^{\theta_1}_{U_iU_j}(a)
\Bigr)f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}(a)\\
&+\bigl(\phi_\ell\otimes\textup{id}_{\mathbf{U}}\bigr)
(D_{\mathcal{H}_{\lambda_1}^{\infty}}\cdots D_{\mathcal{H}_{\lambda_{i-1}}^{\infty}}\bigl(\widetilde{D}_i\otimes\textup{id}_{U_{i+1}\otimes\cdots\otimes U_N}\bigr)
D_{\mathcal{H}_{\lambda_{i+1}}^{\infty}}\cdots D_{\mathcal{H}_{\lambda_N}^{\infty}}(a\phi_r))
\end{split}$$ with $D_{\mathcal{H}_{\lambda_j}^{\infty}}:=D_j\otimes\textup{id}_{U_{j+1}\otimes\cdots\otimes U_N}$ and $$\widetilde{D}_i:=-r^{\theta_1}(a)D_i-D_i\ast r^{\theta_1}(a).$$ Applying now we arrive at $$\begin{split}
-\bigl(\phi_\ell\otimes\textup{id}_{\mathbf{U}}\bigr)&\bigl(r_{\mathcal{H}_{\lambda_0}^{\infty}U_i}^{\theta_1}(a)\mathbf{D}(a\phi_r)\bigr)
+\sum_k\bigl(\phi_\ell\otimes\textup{id}_{\mathbf{U}}\bigr)((\theta(\alpha_k))_{U_i}
\mathbf{D}(\beta_ka\phi_r))\\
&=\Bigl(\sum_{j=1}^{i-1}r^{\theta_1}_{U_jU_i}(a)+\bigl(m(r^{\theta_1}(a))\bigr)_{U_i}+
\sum_{j=i+1}^Nr^{\theta_1}_{U_iU_j}(a)
\Bigr)f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}(a).
\end{split}$$ Combined with and , we obtain .\
\
Write $\kappa^{\textup{core}}\in\mathcal{R}\otimes U(\mathfrak{g})$ for the element $$\label{kcore}
\kappa^{\textup{core}}:=\frac{1}{2}\sum_{j=1}^rx_j^2+\sum_{\alpha\in R}\frac{e_\alpha^2}{1-\xi_{-2\alpha}}$$ and define $\kappa\in\mathcal{R}\otimes
U(\mathfrak{k})\otimes U(\mathfrak{g})\otimes U(\mathfrak{k})$ by $$\label{kdef}
\kappa:=\sum_{\alpha\in R}\frac{y_\alpha\otimes e_\alpha\otimes 1}{1-\xi_{-2\alpha}}
+1\otimes \kappa^{\textup{core}}\otimes 1+\sum_{\alpha\in R}\frac{1\otimes e_\alpha\otimes y_\alpha}{\xi_\alpha-\xi_{-\alpha}}.$$ Furthermore, write $$\label{E}
E:=\sum_{j=1}^r\partial_{x_j}\otimes x_j\in\mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{g}).$$ The asymptotic boundary KZB operators are now defined as follows.
The first-order differential operators $$\label{bKZBoper}
\mathcal{D}_i:=E_i
-\sum_{j=1}^{i-1}r_{ji}^+-\kappa_i-\sum_{j=i+1}^Nr_{ij}^-$$ in $\mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{k})\otimes U(\mathfrak{g})^{\otimes N}
\otimes U(\mathfrak{k})$ $i\in\{1,\ldots,N\}$ are called the asymptotic boundary KZB operators. Here the subindices indicate in which tensor factor of $U(\mathfrak{g})^{\otimes N}$ the $U(\mathfrak{g})$-components of $E$, $\kappa$ and $r^{\pm}$ are placed.
Note that $\kappa^{\textup{core}}$ is the part of $\kappa$ that survives when the $U(\mathfrak{k})$-components act according to the trivial representation of $\mathfrak{k}$. Note furthermore that $$\mathcal{D}_i\in\mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{k})\otimes
\bigl(U(\mathfrak{k})^{\otimes (i-1)}\otimes U(\mathfrak{g})^{\otimes
(N-i+1)}\bigr)\otimes U(\mathfrak{k}).$$
Consider the family $H_z^{(N)}\in\mathbb{D}_{\mathcal{R}}\otimes
U(\mathfrak{k})^{\otimes (N+2)}$ ($z\in Z(\mathfrak{g})$) of commuting differential operator $$H_z^{(N)}:=\bigl(\Delta^{(N)}\otimes \textup{id}_{U(\mathfrak{k})}\bigr)H_z$$ with $\Delta^{(N)}: U(\mathfrak{k})\rightarrow U(\mathfrak{k})^{\otimes (N+1)}$ the $N$th iterate comultiplication map of $U(\mathfrak{k})$ and $H_z$ given by . Then $\mathbf{H}^{(N)}:=-\frac{1}{2}(H_\Omega^{(N)}+\|\rho\|^2)$ is the double spin quantum Calogero-Moser Hamiltonian $$\begin{split}
\mathbf{H}^{(N)}=&
-\frac{1}{2}\Delta+V^{(N)},\\
V^{(N)}:=&-\frac{1}{2}\sum_{\alpha\in R}\frac{1}{(\xi_\alpha-\xi_{-\alpha})^2}
\Bigl(\frac{\|\alpha\|^2}{2}+\prod_{\epsilon\in\{\pm 1\}}
(\Delta^{(N)}(y_\alpha)\otimes 1+\xi_{\epsilon\alpha}(1^{\otimes (N+1)}\otimes y_\alpha))
\Bigr)
\end{split}$$ by Proposition \[qH\].
\[mainthmbKZBEisenstein\] Let $\lambda\in\mathfrak{h}^*$, $\phi_\ell\in\textup{Hom}_K(\mathcal{H}_{\lambda_0},V_\ell)$, $\phi_r\in\textup{Hom}_K(V_r,\mathcal{H}_{\lambda_N})$ and $\mathbf{D}$ a differential vertex operator of weight $\underline{\lambda}=(\lambda_0,\ldots,\lambda_N)$. Consider the smooth $V_\ell\otimes\mathbf{U}\otimes V_r^*$-valued function $$\mathbf{f}_{\underline{\lambda}}^{\phi_\ell,\mathbf{D},\phi_r}:=
\delta f_{\mathcal{H}_{\underline{\lambda}}}^{\phi_\ell,\mathbf{D},\phi_r}|_{A_+}$$ on $A_+$, called the normalised $N$-point $\sigma^{(N)}$-spherical function of weight $\underline{\lambda}$, which admits the explicit integral representation $$\begin{split}
\mathbf{f}_{\underline{\lambda}}^{\phi_\ell,\mathbf{D},\phi_r}(a^\prime)
=\delta(a^\prime)
\int_Kdx\, \xi_{-\lambda_N-\rho}(a(a^\prime{}^{-1}x))&\bigl(\sigma_\ell(x)\otimes\tau_{U_1}(x)\otimes\cdots\\
&\quad\cdots\otimes
\tau_{U_N}(x)\otimes\sigma_r^*(k(a^\prime{}^{-1}x))\bigr)
T_{\lambda_N}^{(\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{D}, \phi_r}.
\end{split}$$ It satisfies the systems of differential equations $$\label{totalDEx}
\begin{split}
\mathcal{D}_i\bigl(\mathbf{f}_{\underline{\lambda}}^{\phi_\ell,\mathbf{D},\phi_r}\bigr)&=
\Bigl(\frac{(\lambda_i,\lambda_i)}{2}-
\frac{(\lambda_{i-1},\lambda_{i-1})}{2}\Bigr)\mathbf{f}_{\underline{\lambda}}^{\phi_\ell,\mathbf{D},\phi_r}\qquad
(i=1,\ldots,N),\\
\mathbf{H}^{(N)}\bigl(\mathbf{f}_{\underline{\lambda}}^{\phi_\ell,\mathbf{D},\phi_r}\bigr)&=
-\frac{(\lambda_N,\lambda_N)}{2}\mathbf{f}_{\underline{\lambda}}^{\phi_\ell,\mathbf{D},\phi_r}
\end{split}$$ on $A_+$. Furthermore, $H_z^{(N)}\bigl(\mathbf{f}_{\underline{\lambda}}^{\phi_\ell,\mathbf{D},\phi_r}\bigr)=
\zeta_{\lambda_N-\rho}(z)\mathbf{f}_{\underline{\lambda}}^{\phi_\ell,\mathbf{D},\phi_r}$ on $A_+$ for $z\in Z(\mathfrak{g})$.
The integral representation follows from . The second line of follows from . By Proposition \[bKZBresult\] we have $$\label{todo100x}
\widetilde{\mathcal{D}}_if_{\underline{\lambda}}^{\phi_\ell,\mathbf{D},\phi_r}=\Bigl(\frac{(\lambda_i,\lambda_i)}{2}-\frac{(\lambda_{i-1},\lambda_{i-1})}{2}\Bigr)f_{\underline{\lambda}}^{\phi_\ell,\mathbf{D},\phi_r}\qquad
(i=1,\ldots,N)$$ with $\widetilde{\mathcal{D}}_i=E_i-\sum_{j=1}^{i-1}r_{ji}^+-\widetilde{\kappa}_i-\sum_{j=i+1}^Nr_{ij}^-$ and $$\widetilde{\kappa}:=r^+\otimes 1+1\otimes b\otimes 1+1\otimes \widetilde{r}^+=
\kappa-\frac{1}{2}\sum_{\alpha\in R^+}\Bigl(\frac{1+\xi_{-2\alpha}}{1-\xi_{-2\alpha}}
\Bigr)t_\alpha.$$ To prove the first set of equations of it thus suffices to show that $$\label{tododelta}
\delta E(\delta^{-1})=-\frac{1}{2}\sum_{\alpha\in R^+}\Bigl(\frac{1+\xi_{-2\alpha}}{1-\xi_{-2\alpha}}
\Bigr)t_\alpha$$ in $\mathcal{R}\otimes\mathfrak{h}$. This follows from the following computation, $$\begin{split}
\delta E(\delta^{-1})&=-\sum_{j=1}^r\left(\rho(x_j)x_j-\sum_{\alpha\in R^+}\frac{\xi_{-2\alpha}\alpha(x_j)x_j}{1-\xi_{-2\alpha}}\right)\\
&=-t_{\rho}-\sum_{\alpha\in R^+}\frac{\xi_{-2\alpha}t_\alpha}{1-\xi_{-2\alpha}}=
-\frac{1}{2}\sum_{\alpha\in R^+}\Bigl(\frac{1+\xi_{-2\alpha}}{1-\xi_{-2\alpha}}
\Bigr)t_\alpha.
\end{split}$$
Formal $N$-point spherical functions {#intertwinersection}
------------------------------------
In this subsection we introduce the analogue of $N$-point spherical functions in the context of Verma modules. They give rise to asymptotically free solutions of the asymptotic boundary KZB operators.
Fix finite dimensional $\mathfrak{g}$-representations $\tau_i: \mathfrak{g}\rightarrow
\mathfrak{gl}(U_i)$ ($1\leq i\leq N$). Let $\underline{\lambda}=(\lambda_0,\ldots,\lambda_N)$ with $\lambda_i\in\mathfrak{h}^*$ and choose vertex operators $\Psi_i\in\textup{Hom}_{\mathfrak{g}}(M_{\lambda_i},M_{\lambda_{i-1}}\otimes U_i)$ for $i=1,\ldots,N$. Set $$\label{Psi}
\mathbf{\Psi}:=(\Psi_1\otimes\textup{id}_{U_2\otimes\cdots\otimes U_N})
\cdots (\Psi_{N-1}\otimes\textup{id}_{U_N})\Psi_N,$$ which is a $\mathfrak{g}$-intertwiner $M_{\lambda_N}\rightarrow M_{\lambda_0}\otimes\mathbf{U}$. Let $(\sigma_\ell,V_\ell), (\sigma_r,V_r)$ be two finite dimensional $\mathfrak{k}$-modules. Write $\sigma_\ell^{(N)}$ for the representation map of the $\mathfrak{k}$-module $V_\ell\otimes\mathbf{U}$, and $\sigma^{(N)}=\sigma_\ell^{(N)}\otimes\sigma_r^*$ for the representation map of the associated $\mathfrak{k}\oplus\mathfrak{k}$-module $(V_\ell\otimes\mathbf{U})\otimes V_r^*$.
\[defNpoint\] For $\phi_\ell\in\textup{Hom}_{\mathfrak{k}}(M_{\lambda_0},V_\ell)$, $\phi_r\in\textup{Hom}_{\mathfrak{k}}(V_r,\overline{M}_{\lambda_N})$ and vertex operator $\mathbf{\Psi}$ given by . Then $(\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{\Psi}\in\textup{Hom}_{\mathfrak{k}}(M_\lambda,V_\ell\otimes\mathbf{U})$ and the associated formal elementary $\sigma^{(N)}$-spherical function $$\label{relCFnormal}
\begin{split}
F_{M_{\underline{\lambda}}}^{\phi_\ell,\mathbf{\Psi},\phi_r}:=&
F_{M_{\lambda_N}}^{(\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{\Psi},\phi_r}\\
=&\sum_{\mu\leq\lambda_N}((\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{\Psi}\phi_r^\mu)\xi_\mu
\in (V_\ell\otimes\mathbf{U}\otimes V_r^*)[[\xi_{-\alpha_1},
\ldots,\xi_{-\alpha_r}]]\xi_{\lambda_N}
\end{split}$$ is called a formal $N$-point $\sigma^{(N)}$-spherical function associated with the $N+1$-tuple of Verma modules $(M_{\lambda_0},\ldots,M_{\lambda_N})$.
By Theorem \[mainTHMF\], the formal $N$-point $\sigma^{(N)}$-spherical function $F_{M_{\underline{\lambda}}}^{\phi_\ell,\mathbf{\Psi},\phi_r}$ is analytic on $A_+$ for $\lambda_N\in\mathfrak{h}_{\textup{HC}}^*$.
Recall the normalisation factor $\delta$ defined by (which we will view as formal power series in $\mathbb{C}[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\rho$).
\[shiftdef\] Let $\Psi_i\in\textup{Hom}_{\mathfrak{g}}(M_{\lambda_i-\rho},M_{\lambda_{i-1}-\rho}\otimes U_i)$ ($1\leq i\leq N$) and write $$\mathbf{\Psi}\in\textup{Hom}_{\mathfrak{g}}(M_{\lambda_N-\rho},
M_{\lambda_0-\rho}\otimes\mathbf{U})$$ for the resulting vertex operator . We call $$\mathbf{F}_{\underline{\lambda}}^{\phi_\ell,\mathbf{\Psi},\phi_r}:=\delta F_{M_{\underline{\lambda}-\rho}}^{\phi_\ell,\mathbf{\Psi},\phi_r}
\in (V_\ell\otimes\mathbf{U}\otimes V_r^*)[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_{\lambda_N}$$ a normalised $N$-point $\sigma^{(N)}$-spherical function of weight $\underline{\lambda}-\rho:=(\lambda_0-\rho,\lambda_1-\rho,\ldots,\lambda_N-\rho)$.
For weight $\underline{\lambda}$ with $\lambda_N\in\mathfrak{h}_{\textup{HC}}^*+\rho$ the normalised formal $N$-point $\sigma^{(N)}$-spherical function $\mathbf{F}_{\underline{\lambda}}^{\phi_\ell,\mathbf{\Psi},\phi_r}$ is an $V_\ell\otimes\mathbf{U}\otimes V_r^*$-valued analytic function on $A_+$. In terms of the normalised formal elementary $\sigma^{(N)}$-spherical functions, we have $$\label{relCF}
\mathbf{F}^{\phi_\ell,\mathbf{\Psi},\phi_r}_{\underline{\lambda}}=\mathbf{F}_{\lambda_N}^{(\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{\Psi},\phi_r},$$ and hence $$\label{CMeigenfunctions}
\begin{split}
\mathbf{H}^{(N)}\bigl(\mathbf{F}_{\underline{\lambda}}^{\phi_\ell,\mathbf{\Psi},\phi_r}\bigr)&=
-\frac{(\lambda_N,\lambda_N)}{2}\,\mathbf{F}_{\underline{\lambda}}^{\phi_\ell,\mathbf{\Psi},\phi_r},\\
H_z^{(N)}\bigl(\mathbf{F}_{\underline{\lambda}}^{\phi_\ell,\mathbf{\Psi},\phi_r}\bigr)&=
\zeta_{\lambda_N-\rho}(z)\mathbf{F}_{\underline{\lambda}}^{\phi_\ell,\mathbf{\Psi},\phi_r},\qquad
z\in Z(\mathfrak{g})
\end{split}$$ by Theorem \[thmnorm\]. We now show by a suitable adjustment of the algebraic arguments from Subsection \[S61\] that the normalised formal $N$-point $\sigma^{(N)}$-spherical functions are eigenfunctions of the asymptotic boundary KZB operators.
\[mainthmbKZB\] Let $\phi_\ell\in\textup{Hom}_{\mathfrak{k}}(M_{\lambda_0},V_\ell)$, $\phi_r\in\textup{Hom}_{\mathfrak{k}}(V_r,\overline{M}_{\lambda_N})$ and let $\mathbf{\Psi}$ be a product of $N$ vertex operators as given in Definition \[shiftdef\]. The normalised formal $N$-point $\sigma^{(N)}$-spherical function $\mathbf{F}_{\underline{\lambda}}^{\phi_\ell,\mathbf{\Psi},\phi_r}$ satisfies the system of differential equations $$\label{totalDE}
\mathcal{D}_i\bigl(\mathbf{F}_{\underline{\lambda}}^{\phi_\ell,\mathbf{\Psi},\phi_r}\bigr)=\Bigl(\frac{(\lambda_i,\lambda_i)}{2}-\frac{(\lambda_{i-1},\lambda_{i-1})}{2}\Bigr)
\mathbf{F}_{\underline{\lambda}}^{\phi_\ell,\mathbf{\Psi},\phi_r}\qquad
(i=1,\ldots,N)$$ in $(V_\ell\otimes\mathbf{U}\otimes V_r^*)[[\xi_{-\alpha_1},
\ldots,\xi_{-\alpha_r}]]\xi_{\lambda_N}$. For $\lambda_N\in\mathfrak{h}_{\textup{HC}}^*+\rho$ the differential equations are valid as analytic $V_\ell\otimes\mathbf{U}\otimes V_r^*$-valued analytic functions on $A_+$.
As in the proof of Theorem \[mainthmbKZBEisenstein\], the differential equations are equivalent to $$\label{todo100}
\widetilde{\mathcal{D}}_i\bigl(F_{M_{\underline{\lambda}}}^{\phi_\ell,\mathbf{\Psi},\phi_r}\bigr)=\frac{1}{2}(\zeta_{\lambda_i}(\Omega)-
\zeta_{\lambda_{i-1}}(\Omega))F_{M_{\underline{\lambda}}}^{\phi_\ell,\mathbf{\Psi},\phi_r}\qquad
(i=1,\ldots,N)$$ with $\widetilde{\mathcal{D}}_i=E_i-\sum_{j=1}^{i-1}r_{ji}^+-\widetilde{\kappa}_i-\sum_{j=i+1}^Nr_{ij}^-$ and $$\widetilde{\kappa}:=\kappa-\frac{1}{2}\sum_{\alpha\in R^+}\Bigl(\frac{1+\xi_{-2\alpha}}{1-\xi_{-2\alpha}}
\Bigr)t_\alpha=
r^+\otimes 1+1\otimes b\otimes 1+1\otimes \widetilde{r}^+$$ (here $b$ is given by ).
Write $\Lambda=\{\mu\in\mathfrak{h}^*\,\, | \,\, \mu\leq\lambda_N\}$ and $\Lambda_m:=\{\mu\in\Lambda \,\, | \,\, (\lambda_N-\mu,\rho^\vee)\leq m\}$ ($m\in\mathbb{Z}_{\geq 0}$). Consider the $V_\ell\otimes\mathbf{U}\otimes V_r^*$-valued quasi-polynomial $$F_{M_{\underline{\lambda}},m}^{\phi_\ell,\mathbf{\Psi},\phi_r}:=\sum_{\mu\in\Lambda_m}
((\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{\Psi}\phi_r^\mu)\xi_\mu$$ for $m\in\mathbb{Z}_{\geq 0}$. Fix $a\in A_+$. Then we have $$\begin{split}
(\zeta_{\lambda_{i-1}}(\Omega)-&\zeta_{\lambda_i}(\Omega))F_{M_{\underline{\lambda}},m}^{\phi_\ell,\mathbf{\Psi},
\phi_r}(a)=\\
&=\sum_{\mu\in\Lambda_m}
(\phi_\ell\otimes\textup{id}_{\mathbf{U}})(\Psi_{M_{\lambda_1}}\cdots
\Psi_{M_{\lambda_{i-1}}}\widetilde{\Psi}_i
\Psi_{M_{\lambda_{i+1}}}\cdots
\Psi_{M_{\lambda_N}}\phi_r^\mu)a^\mu
\end{split}$$ with $\Psi_{M_{\lambda_i}}:=\Psi_i\otimes\textup{id}_{U_{i+1}\otimes\cdots\otimes U_N}$ and $\widetilde{\Psi}_i:=\Omega_{M_{\lambda_{i-1}}}\Psi_{M_{\lambda_i}}-
\Psi_{M_{\lambda_i}}\Omega_{M_{\lambda_i}}$. By the asymptotic operator KZB equation applied to the factorisation $(r^+(a), -r^-(a), b(a))$ of $\Omega$ (see Proposition \[propfactorization\][**b**]{}), we get $$\begin{split}
\Bigl(\Bigl(\frac{\zeta_{\lambda_{i-1}}(\Omega)}{2}-&\frac{\zeta_{\lambda_i}(\Omega)}{2}+E_{U_i}
-\sum_{j=1}^{i-1}r_{U_jU_i}^+-r_{V_\ell U_i}^+-b_{U_i}-\sum_{j=i+1}^Nr_{U_iU_j}^-\Bigr)
F_{M_{\underline{\lambda}},m}^{\phi_\ell,\mathbf{\Psi},\phi_r}\Bigr)(a)=\\
&\qquad\quad=-\sum_{\mu\in\Lambda_m}\sum_{\alpha\in R}\frac{(e_\alpha+e_{-\alpha})_{U_i}}{(1-a^{-2\alpha})}
(\phi_\ell\otimes\textup{id}_{\mathbf{U}})(\mathbf{\Psi}(e_\alpha)_{M_{\lambda_N}}\phi_r^\mu)a^\mu.
\end{split}$$ This being valid for all $a\in A_+$, hence we get $$\label{trunkm}
\begin{split}
\Bigl(\frac{1}{2}(\zeta_{\lambda_{i-1}}(\Omega)-&\zeta_{\lambda_i}(\Omega))+E_{U_i}
-\sum_{j=1}^{i-1}r_{U_jU_i}^+-r_{V_\ell U_i}^+-b_{U_i}-\sum_{j=i+1}^Nr_{U_iU_j}^-\Bigr)
F_{M_{\underline{\lambda}},m}^{\phi_\ell,\mathbf{\Psi},\phi_r}=\\
&=-\sum_{\mu\in\Lambda_m}\sum_{\alpha\in R}(e_\alpha+e_{-\alpha})_{U_i}
(\phi_\ell\otimes\textup{id}_{\mathbf{U}})(\mathbf{\Psi}(e_\alpha)_{M_{\lambda_N}}\phi_r^\mu)\frac{\xi_\mu}{(1-\xi_{-2\alpha})}
\end{split}$$ viewed as identity in $(V_\ell\otimes\mathbf{U}\otimes V_r^*)[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_{\lambda_N}$ (so $(1-\xi_{-2\alpha})^{-1}=\sum_{n=0}^{\infty}\xi_{-\alpha}^{2n}$ if $\alpha\in R^+$ and $(1-\xi_{-2\alpha})^{-1}=-\sum_{n=1}^{\infty}\xi_\alpha^{2n}$ if $\alpha\in R^-$, and analogous expansions for the coefficients of $r^{\pm}$ and $\widetilde{\kappa}$). We claim that the identity is also valid when the summation over $\Lambda_m$ is replaced by summation over $\Lambda$: $$\label{notrunk}
\begin{split}
\Bigl(\frac{1}{2}(\zeta_{\lambda_{i-1}}(\Omega)-&\zeta_{\lambda_i}(\Omega))+E_{U_i}
-\sum_{j=1}^{i-1}r_{U_jU_i}^+-r_{V_\ell U_i}^+-b_{U_i}-\sum_{j=i+1}^Nr_{U_iU_j}^-\bigr)
F_{M_{\underline{\lambda}}}^{\phi_\ell,\mathbf{\Psi},\phi_r}=\\
&=-\sum_{\mu\in\Lambda}\sum_{\alpha\in R}(e_\alpha+e_{-\alpha})_{U_i}
(\phi_\ell\otimes\textup{id}_{\mathbf{U}})(\mathbf{\Psi}(e_\alpha)_{M_{\lambda_N}}\phi_r^\mu)
\frac{\xi_\mu}{(1-\xi_{-2\alpha})}
\end{split}$$ in $(V_\ell\otimes\mathbf{U}\otimes V_r^*)[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_{\lambda_N}$. Take $\eta\in\Lambda$ and let $m\in\mathbb{N}$ such that $(\lambda_N-\eta,\rho^\vee)\leq m$. Then the $\xi_\eta$-coefficient of the left hand side of is the same as the $\xi_\eta$-coefficient of the left hand side of since the coefficients of $r^{\pm}$ and $\widetilde{\kappa}$ are in $\mathbb{C}[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]$. Exactly the same argument applies to the $\xi_\mu$-coefficients of the right hand sides of and , from which the claim follows. Now rewrite the right hand side of as $$\begin{split}
-\sum_{\mu\in\Lambda}\sum_{\alpha\in R}&(e_\alpha+e_{-\alpha})_{U_i}
(\phi_\ell\otimes\textup{id}_{\mathbf{U}})(\mathbf{\Psi}(e_\alpha)_{M_{\lambda_N}}\phi_r^\mu)
\frac{\xi_\mu}{(1-\xi_{-2\alpha})}=\\
=&-\sum_{\alpha\in R}\sum_{\nu\in\Lambda}(e_\alpha+e_{-\alpha})_{U_i}(\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{\Psi}(
\textup{proj}_{M_{\lambda_N}}^\nu(e_\alpha)_{M_{\lambda_N}}\phi_r)\frac{\xi_{\nu}}{(\xi_\alpha-\xi_{-\alpha})}
\end{split}$$ and use that $$\sum_{\alpha\in R}\frac{(e_\alpha+e_{-\alpha})\otimes e_\alpha}{\xi_\alpha-\xi_{-\alpha}}=
\sum_{\alpha\in R}\frac{e_\alpha\otimes y_\alpha}{\xi_\alpha-\xi_{-\alpha}}$$ to obtain $$\begin{split}
-\sum_{\mu\in\Lambda}\sum_{\alpha\in R}&(e_\alpha+e_{-\alpha})_{U_i}
(\phi_\ell\otimes\textup{id}_{\mathbf{U}})(\mathbf{\Psi}(e_\alpha)_{M_{\lambda_N}}\phi_r^\mu)
\frac{\xi_\mu}{(1-\xi_{-2\alpha})}=\\
=&-\sum_{\alpha\in R}\sum_{\nu\in\Lambda}(e_\alpha)_{U_i}(\phi_\ell\otimes\textup{id}_{\mathbf{U}})\mathbf{\Psi}(
\textup{proj}_{M_{\lambda_N}}^\nu(y_\alpha)_{M_{\lambda_N}}\phi_r)
\frac{\xi_{\nu}}{(\xi_\alpha-\xi_{-\alpha})}\\
=&\widetilde{r}_{U_iV_r^*}^+F_{M_{\underline{\lambda}}}^{\phi_\ell,\mathbf{\Psi},\phi_r},
\end{split}$$ where we used that $\phi_r$ is a $\mathfrak{k}$-intertwiner, as well as the explicit formula for $\widetilde{r}^+$. Substituting this identity in , we obtain .
Boundary fusion operators {#sectionBFO}
-------------------------
In Subsection \[S62\] we introduced the expectation value of (differential) vertex operators. Recall the parametrisation of the vertex operators introduced in Definition \[notVO\]. The expectation value of products of vertex operators gives rise to the fusion operator:
[@E Prop. 3.7]. Let $\lambda\in\mathfrak{h}_{\textup{reg}}^*$. The fusion operator $\mathbf{J}_{\mathbf{U}}(\lambda)$ is the $\mathfrak{h}$-linear automorphism of $\mathbf{U}$ defined by $$u_1\otimes\cdots\otimes u_N\mapsto
(m_{\lambda_0}^*\otimes\textup{id}_{\mathbf{U}})
(\Psi_{\lambda_1}^{u_1}\otimes\textup{id}_{U_2\otimes\cdots\otimes U_N})
\cdots (\Psi_{\lambda_{N-1}}^{u_{N-1}}\otimes\textup{id}_{U_N})\Psi_{\lambda_N}^{u_N}(m_\lambda)$$ for $u_i\in U_i[\mu_i]$ ($\mu_i\in P$), where $\lambda_i:=\lambda-\mu_{i+1}\cdots -\mu_N$ for $i=0,\ldots,N-1$ and $\lambda_N=\lambda$.
We suppress the dependence on $\mathbf{U}$ and denote $\mathbf{J}_{\mathbf{U}}(\lambda)$ by $\mathbf{J}(\lambda)$ if no confusion is possible (in fact, a universal fusion operator exists, in the sense that its action on $\mathbf{U}$ reproduces $\mathbf{J}_{\mathbf{U}}(\lambda)$ for all finite dimensional $\mathfrak{g}^{\oplus N}$-modules, see, e.g., [@E Prop. 3.19] and references therein).
Lemma \[evlem\] shows that for $\mathbf{u}:=u_1\otimes\cdots\otimes u_N$ with $u_i\in U[\mu_i]$ and $\lambda\in\mathfrak{h}_{\textup{reg}}^*$ we have $$\label{fusionmeaning}
\Psi_\lambda^{\mathbf{J}(\lambda)\mathbf{u}}=
(\Psi_{\lambda_1}^{u_1}\otimes\textup{id}_{U_2\otimes\cdots\otimes U_N})
\cdots (\Psi_{\lambda_{N-1}}^{u_{N-1}}\otimes\textup{id}_{U_N})\Psi_{\lambda_N}^{u_N}$$ in $\textup{Hom}_{\mathfrak{g}}(M_\lambda,M_{\lambda_0}\otimes\mathbf{U})$.
Fix from now on a finite dimensional $\mathfrak{k}$-module $V_\ell$. Recall the parametrisation of $\mathfrak{k}$-intertwiners $\phi_{\ell,\lambda}^v\in\textup{Hom}_{\mathfrak{k}}(M_\lambda,V_\ell)$ by their expectation value $v\in V_\ell$ as introduced in Definition \[intertwinerparametrization\][**a**]{}.
\[boundaryfusion\] Let $\lambda\in\mathfrak{h}_{\textup{reg}}^*$. The linear operator $\mathbf{J}_{\ell,\mathbf{U}}(\lambda)\in\textup{End}(V_\ell\otimes\mathbf{U})$, defined by $$\mathbf{J}_{\ell,\mathbf{U}}(\lambda)(v\otimes\mathbf{u}):=
(\phi_{\ell,\lambda-\mu}^v\otimes\textup{id}_{\mathbf{U}})\Psi_\lambda^{\mathbf{J}(\lambda)\mathbf{u}}(m_\lambda)$$ for $v\in V_\ell$ and $\mathbf{u}\in \mathbf{U}[\mu]$ $\mu\in P$, is a linear automorphism.
For $v\in V_\ell$ and $\mathbf{u}\in\mathbf{U}[\mu]$ we have $$\Psi_\lambda^{\mathbf{J}(\lambda)\mathbf{u}}(m_\lambda)\in m_{\lambda-\mu}\otimes
\mathbf{J}(\lambda)\mathbf{u}+\bigoplus_{\nu>\mu}M_{\lambda-\mu}\otimes
\mathbf{U}[\nu],$$ and consequently we get $$\mathbf{J}_{\ell,\mathbf{U}}(\lambda)(v\otimes\mathbf{u})\in v\otimes \mathbf{J}(\lambda)\mathbf{u}
+\bigoplus_{\nu>\mu}V_\ell\otimes \mathbf{U}[\nu].$$ Choose an ordered tensor product basis of $V_\ell\otimes\mathbf{U}$ in which the $\mathbf{U}$-components consist of weight vectors. Order the tensor product basis in such a way that is compatible with the dominance order on the weights of the $\mathbf{U}$-components of the basis elements. With respect to such a basis, $\mathbf{J}_{\ell,\mathbf{U}}(\lambda)(\textup{id}_{V_\ell}\otimes\mathbf{J}(\lambda)^{-1})$ is represented by a triangular operator with ones on the diagonal, hence it is invertible.
We call $\mathbf{J}_{\ell,\mathbf{U}}(\lambda)$ the (left) boundary fusion operator on $V_\ell\otimes\mathbf{U}$ (we denote $\mathbf{J}_{\ell,\mathbf{U}}(\lambda)$ by $\mathbf{J}_\ell(\lambda)$ if no confusion is possible). A right version $\mathbf{J}_r(\lambda)$ of the left boundary fusion operator $\mathbf{J}_\ell(\lambda)$ can be constructed in an analogous manner. We leave the straightforward details to the reader.
Let $\lambda\in\mathfrak{h}_{\textup{reg}}^*$. By Proposition \[relEisPrinalg\][**a**]{} and Lemma \[boundaryfusion\], the map $$V_\ell\otimes \mathbf{U}\rightarrow\textup{Hom}_{\mathfrak{k}}(M_\lambda,V_\ell\otimes\mathbf{U}),
\qquad
v\otimes \mathbf{u}\mapsto \phi_{\ell,\lambda}^{\mathbf{J}_\ell(\lambda)(v\otimes\mathbf{u})}$$ is a linear isomorphism. The $\mathfrak{k}$-intertwiner $\phi_{\ell,\lambda}^{\mathbf{J}_\ell(\lambda)(v\otimes\mathbf{u})}$ admits the following alternative description.
\[vertexint\] Let $\lambda\in\mathfrak{h}_{\textup{reg}}^*$, $\mu\in P$ and $v\in V_\ell$. For $\mathbf{u}\in\mathbf{U}[\mu]$ we have $$\label{boundaryfusionformula}
\phi_{\ell,\lambda}^{\mathbf{J}_\ell(\lambda)(v\otimes\mathbf{u})}=
(\phi_{\ell,\lambda-\mu}^v\otimes\textup{id}_{\mathbf{U}})
\Psi_\lambda^{\mathbf{J}(\lambda)\mathbf{u}}.$$ For $\mathbf{u}=u_1\otimes\cdots\otimes u_N$ with $u_i\in U_i[\mu_i]$ $1\leq i\leq N$ we furhermore have $$\label{boundaryfusionformula2}
\phi_{\ell,\lambda}^{\mathbf{J}_\ell(\lambda)(v\otimes\mathbf{u})}=
(\phi_{\ell,\lambda_0}^v\otimes\textup{id}_{\mathbf{U}})(\Psi_{\lambda_1}^{u_1}\otimes\textup{id}_{U_2\otimes\cdots\otimes U_N})
\cdots (\Psi_{\lambda_{N-1}}^{u_{N-1}}\otimes\textup{id}_{U_N})\Psi_{\lambda_N}^{u_N},$$ with $\lambda_i:=\lambda-\mu_{i+1}\cdots-\mu_N$ $i=0,\ldots,N-1$, and $\lambda_N:=\lambda$.
The result follows immediately from Proposition \[relEisPrinalg\][**a**]{}, Lemma \[boundaryfusion\] and .
Let $V_\ell,V_r$ be finite dimensional $\mathfrak{k}$-modules. Let $\underline{\lambda}=(\lambda_0,\ldots,\lambda_N)$ with $\lambda_N\in\mathfrak{h}_{\textup{reg}}^*$ and with $\mu_i:=\lambda_i-\lambda_{i-1}\in P$ for $i=1,\ldots,N$. Let $v\in V_\ell$, $f\in V_r^*$ and $\mathbf{u}=u_1\otimes\cdots\otimes
u_N$ with $u_i\in U_i[\mu_i]$. We write $$F_{M_{\underline{\lambda}}}^{v,\mathbf{u},f}:=
F_{M_{\underline{\lambda}}}^{\mathbf{J}_\ell(\lambda_N)(v\otimes\mathbf{u})\otimes f}$$ for the formal $N$-point spherical function with leading coefficient $\mathbf{J}_\ell(\lambda_N)(v\otimes\mathbf{u})\otimes f$, and $$\mathbf{F}_{\underline{\lambda}}^{v,u,f}:=
\delta F_{M_{\underline{\lambda}-\rho}}^{v,\mathbf{u},f}=
\mathbf{F}_{\underline{\lambda}}^{\mathbf{J}_\ell(\lambda_N-\rho)(v\otimes\mathbf{u})\otimes f}$$ for its normalised version.
Note that $$F_{M_{\underline{\lambda}}}^{v,\mathbf{u},f}=
F_{M_{\underline{\lambda}}}^{\phi_{\ell,\lambda_0}^v,\mathbf{\Psi}_{\lambda_N}^{\mathbf{J}(\lambda_N)\mathbf{u}},\phi_{r,\lambda_N}^f}$$ by Lemma \[vertexint\]. Written out as formal power series we thus have the following three expressions for $F_{M_{\underline{\lambda}}}^{v,\mathbf{u},f}$, $$\begin{split}
& F_{M_{\underline{\lambda}}}^{v,\mathbf{u},f}=\sum_{\mu\leq\lambda_N}
\phi_{\ell,\lambda_N}^{\mathbf{J}_\ell(\lambda_N)\mathbf{u}}(\textup{proj}_{M_{\lambda_N}}^\mu\phi_{r,\lambda_N}^f)\xi_\mu\\
&\qquad\quad=\sum_{\mu\leq\lambda_N}(\phi_{\ell,\lambda_0}^v\otimes\textup{id}_{\mathbf{U}})
\Psi_{\lambda_N}^{\mathbf{J}(\lambda_N)\mathbf{u}}(\textup{proj}_{M_{\lambda_N}}^\mu\phi_{r,\lambda_N}^f)\xi_\mu\\
&\quad=\sum_{\mu\leq\lambda_N}(\phi_{\ell,\lambda_0}^v\otimes\textup{id}_{\mathbf{U}})(\Psi_{\lambda_1}^{u_1}\otimes\textup{id}_{U_2\otimes\cdots\otimes U_N})
\cdots (\Psi_{\lambda_{N-1}}^{u_{N-1}}\otimes\textup{id}_{U_N})\Psi_{\lambda_N}^{u_N}
(\textup{proj}_{M_{\lambda_N}}^\mu\phi_{r,\lambda_N}^f)\xi_\mu.
\end{split}$$ The main results of the previous subsection for $\lambda\in\mathfrak{h}_{\textup{reg}}^*$ can now be reworded as follows.
\[corMAINkzb\] Let $\lambda\in\mathfrak{h}_{\textup{reg}}^*$. Let $\underline{\lambda}=(\lambda_0,\ldots,\lambda_N)$ with $\lambda_N\in\mathfrak{h}_{\textup{reg}}^*$ and with $\mu_i:=\lambda_i-\lambda_{i-1}\in P$ for $i=1,\ldots,N$. Let $v\in V_\ell$, $f\in V_r^*$ and $\mathbf{u}=u_1\otimes\cdots\otimes
u_N$ with $u_i\in U_i[\mu_i]$ $i=1,\ldots,N$. Then we have for $i=1,\ldots,N$, $$\label{eigenvalueeqn}
\begin{split}
\mathcal{D}_i\bigl(\mathbf{F}_{\underline{\lambda}}^{v,\mathbf{u},f}\bigr)&=\Bigl(\frac{(\lambda_i,\lambda_i)}{2}-
\frac{(\lambda_{i-1},\lambda_{i-1})}{2}\Bigr)\mathbf{F}_{\underline{\lambda}}^{v,\mathbf{u},f},\\
\mathbf{H}^{(N)}\bigl(\mathbf{F}_{\underline{\lambda}}^{v,\mathbf{u},f}\bigr)&=
-\frac{(\lambda_N,\lambda_N)}{2}\mathbf{F}_{\underline{\lambda}}^{v,\mathbf{u},f}
\end{split}$$ and $H_z^{(N)}\bigl(\mathbf{F}_{\underline{\lambda}}^{v,\mathbf{u},f}\bigr)=\zeta_{\lambda_N-\rho}(z)
\mathbf{F}_{\underline{\lambda}}^{v,\mathbf{u},f}$ for $z\in Z(\mathfrak{g})$. This holds true as $V_\ell\otimes\mathbf{U}\otimes V_r^*$-valued analytic functions on $A_+$ when $\lambda_N\in\mathfrak{h}_{\textup{HC}}^*\cap\mathfrak{h}_{\textup{reg}}^*$.
Integrability of the asymptotic boundary KZB operators {#S66}
------------------------------------------------------
In this subsection we show that the asymptotic boundary KZB operators $\mathcal{D}_i$ ($1\leq i\leq N$) pairwise commute in $\mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{k})\otimes
U(\mathfrak{g})^{\otimes N}\otimes U(\mathfrak{k})$, and that they also commute with the quantum Hamiltonians $H_z^{(N)}\in \mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{k})^{\otimes (N+2)}$ for $z\in Z(\mathfrak{g})$ (and hence also with $\mathbf{H}^{(N)}$). We begin with the following lemma.
Let $V$ be a finite dimensional $U(\mathfrak{k})\otimes U(\mathfrak{g})^{\otimes N}\otimes
U(\mathfrak{k})$-module and suppose that $\lambda\in\mathfrak{h}_{\textup{reg}}^*$. Then the asymptotic boundary KZB operators $\mathcal{D}_i$ $1\leq i\leq N$ and the quantum Hamiltonians $H_z^{(N)}$ $z\in Z(\mathfrak{g})$ pairwise commute as linear operators on $V[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\lambda$.
It suffices to prove the lemma for $V=V_\ell\otimes\mathbf{U}\otimes V_r^*$ with $V_\ell, V_r$ finite dimensional $\mathfrak{k}$-modules and $\mathbf{U}=U_1\otimes\cdots\otimes U_N$ with $U_1,\ldots,U_N$ finite dimensional $\mathfrak{g}$-modules.
Define an ultrametric $d$ on $(V_\ell\otimes\mathbf{U}\otimes V_r^*)[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\lambda$ by the formula $d(f,g):=2^{-\varpi(f-g)}$ with, for $\sum_{\mu\leq\lambda}e_\mu\xi_\mu\in
(V_\ell\otimes\mathbf{U}\otimes V_r^*)[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\lambda$ nonzero, $$\varpi\bigl(\sum_{\mu\leq\lambda}e_\mu\xi_\mu\bigr):=
\textup{min}\{ (\lambda-\mu,\rho^\vee) \,\, | \,\, \mu\leq\lambda: e_\mu\not=0\},$$ and $\varpi(0)=\infty$. Consider $(V_\ell\otimes\mathbf{U}\otimes V_r^*)[[\xi_{-\alpha_1},\ldots,
\xi_{-\alpha_r}]]\xi_\lambda$ as topological space with respect to the resulting metric topology. Note that $\mathcal{D}_i$ ($1\leq i\leq N$) and $H_z^{(N)}$ ($z\in Z(\mathfrak{g})$) are continuous linear operators on $(V_\ell\otimes\mathbf{U}\otimes V_r^*)[[\xi_{-\alpha_1},\ldots,
\xi_{-\alpha_r}]]\xi_\lambda$ since their scalar components lie in the subring $\mathcal{R}
\subseteq\mathbb{C}[[\xi_{-\alpha_r},\ldots,\xi_{-\alpha_r}]]$. It thus suffices to show that $(V_\ell\otimes\mathbf{U}\otimes V_r^*)[[\xi_{-\alpha_1},\ldots,
\xi_{-\alpha_r}]]\xi_\lambda$ has a topological linear basis consisting of common eigenfunctions for the differential operators $\mathcal{D}_i$ ($1\leq i\leq N$) and $H_z^{(N)}$ ($z\in Z(\mathfrak{g})$).
Fix linear basis $\{v_i\}_{i\in I}$, $\{\mathbf{b}_j\}_{j\in J}$ and $\{f_s\}_{s\in S}$ of $V_\ell$, $\mathbf{U}$ and $V_r^*$ respectively. Take the basis elements $\mathbf{b}_j$ of the form $\mathbf{b}_j=u_{1,j}\otimes\cdots\otimes u_{N,j}$ with $u_{k,j}$ a weight vector in $U_k$ of weight $\mu_k(\mathbf{b}_j)$ ($1\leq k\leq N$). For $q\in \sum_{k=1}^r\mathbb{Z}_{\geq 0}\alpha_k$ write $$\underline{\lambda}(\mathbf{b}_j)-q:=(\lambda-q-\mu_1(\mathbf{b}_j)-\cdots-\mu_N(\mathbf{b}_j),
\ldots,\lambda-q-\mu_N(\mathbf{b}_j),\lambda-q).$$ We then have $$\mathbf{F}_{\underline{\lambda}(\mathbf{b}_j)-q}^{v_i,\mathbf{b}_j,f_s}=
\bigl(\mathbf{J}_\ell(\lambda-q)(v_i\otimes\mathbf{b}_j)\otimes f_s\bigr)\xi_{\lambda-q}+
\sum_{\mu<\lambda-q}e_{i,j,s;q}(\mu)\xi_\mu$$ for certain vectors $e_{i,j,s;q}(\mu)\in V_\ell\otimes\mathbf{U}\otimes V_r^*$. Lemma \[boundaryfusion\] then implies that $$\bigl\{\mathbf{F}_{\underline{\lambda}(\mathbf{b}_j)-q}^{v_i,\mathbf{b}_j,f_s}\,\,\, | \,\,\, (i,j,s)\in I\times J\times S,\,\, q\in \sum_{k=1}^r\mathbb{Z}_{\geq 0}\alpha_k \bigr\}$$ is a topological linear basis of $(V_\ell\otimes\mathbf{U}\otimes V_r^*)[[\xi_{-\alpha_1},\ldots,
\xi_{-\alpha_r}]]\xi_\lambda$. Finally Corollary \[corMAINkzb\] shows that the basis elements $\mathbf{F}_{\underline{\lambda}(\mathbf{b}_j)-q}^{v_i,\mathbf{b}_j,f_s}$ are simultaneous eigenfunctions of $\mathcal{D}_k$ ($1\leq k\leq N$) and $H_z^{(N)}$ ($z\in Z(\mathfrak{g})$).
We can now show the universal integrability of the asymptotic boundary KZB operators, as well as their compatibility with the quantum Hamiltonians $H_z^{(N)}$ ($z\in Z(\mathfrak{g})$).
\[consistentoperators\] In $\mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{k})\otimes U(\mathfrak{g})^{\otimes N}
\otimes U(\mathfrak{k})$ we have $$[\mathcal{D}_i,\mathcal{D}_j]=0,\qquad [\mathcal{D}_i,H_z^{(N)}]=0,\qquad [H_z^{(N)},H_{z^\prime}^{(N)}]=0$$ for $i,j=1,\ldots,N$ and $z,z^\prime\in Z(\mathfrak{g})$.
By the previous lemma, it suffices to show that if the differential operator $$L\in\mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{k})\otimes U(\mathfrak{g})^{\otimes N}\otimes U(\mathfrak{k})$$ acts as zero on $V[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\lambda$ for all finite dimensional $U(\mathfrak{k})\otimes U(\mathfrak{g})^{\otimes N}\otimes U(\mathfrak{k})$-modules $V$ and all $\lambda\in\mathfrak{h}_{\textup{reg}}^*$, then $L=0$.
We identify the algebra $\mathbb{D}(A)^A$ of constant coefficient differential operators on $A$ with the algebra $S(\mathfrak{h}^*)$ of complex polynomials on $\mathfrak{h}^*$, by associating $\partial_h$ ($h\in \mathfrak{h}_0$) with the linear polynomial $\lambda\mapsto \lambda(h)$. For $p\in S(\mathfrak{h}^*)$ we write $p(\partial)$ for the corresponding constant coefficient differential operator on $A$.
Now we can expand $$L=\sum_{i,j}f_ia_{ij}p_{ij}(\partial)$$ (finite sum), with $\{f_i\}_i$ a linear independent set in $\mathcal{R}\subset\mathbb{C}[[\xi_{-\alpha_1},
\ldots,\xi_{-\alpha_r}]]$, $\{a_{ij}\}_{i,j}$ a linear independent set in $U(\mathfrak{k})\otimes U(\mathfrak{g})^{\otimes N}\otimes U(\mathfrak{k})$ and $p_{ij}\in S(\mathfrak{h}^*)$. Then $$0=L(v\xi_\lambda)=\sum_i\bigl(\sum_jp_{ij}(\lambda)a_{ij}(v)\bigr)f_i\xi_\lambda$$ in $V[[\xi_{-\alpha_1},\ldots,\xi_{-\alpha_r}]]\xi_\lambda$ for $v\in V$ and $\lambda\in\mathfrak{h}_{\textup{reg}}^*$, where $V$ is an arbitrary finite dimensional $U(\mathfrak{k})\otimes
U(\mathfrak{g})^{\otimes N}\otimes U(\mathfrak{k})$-module. Since the $\{f_i\}_i$ are linear independent, we get $$\sum_jp_{ij}(\lambda)a_{ij}(v)=0$$ in $V$ for all $i$, for all $\lambda\in\mathfrak{h}_{\textup{reg}}^*$ and all $v\in V$, with $V$ any finite dimensional $U(\mathfrak{k})\otimes
U(\mathfrak{g})^{\otimes N}\otimes U(\mathfrak{k})$-module. Then [@Di Thm. 2.5.7] implies that $$\sum_jp_{ij}(\lambda)a_{ij}=0$$ in $U(\mathfrak{k})\otimes U(\mathfrak{g})^{\otimes N}\otimes U(\mathfrak{k})$ for all $i$ and all $\lambda\in\mathfrak{h}_{\textup{reg}}^*$. By the linear independence of $\{a_{ij}\}_{i,j}$, we get $p_{ij}(\lambda)=0$ for all $i,j$ and all $\lambda\in \mathfrak{h}_{\textup{reg}}^*$, hence $p_{ij}=0$ for all $i,j$. This completes the proof of the theorem.
Folded dynamical trigonometric $r$-and $k$-matrices {#inteq}
---------------------------------------------------
We end this section by discussing the reformulation of the commutator relations $$[\mathcal{D}_i,\mathcal{D}_j]=0$$ in $\mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{k})\otimes U(\mathfrak{g})^{\otimes N}
\otimes U(\mathfrak{k})$ for $1\leq i,j\leq N$ in terms of explicit consistency conditions for the constituents $r^{\pm}$ and $\kappa$ of the asymptotic boundary KZB operators $\mathcal{D}_i$ (see ).
Before doing so, we first discuss as a warm-up the situation for the usual asymptotic KZB equations (see [@ES] and references therein), which we will construct from an appropriate “universal” version of the operators that are no longer integrable. Recall that $\Delta^{(N-1)}: U(\mathfrak{g})\rightarrow U(\mathfrak{g})^{\otimes N}$ is the $(N-1)$th iterated comultiplication of $U(\mathfrak{g})$.
Fix $\widehat{r}\in \mathcal{R}\otimes U(\mathfrak{g})^{\otimes 2}$ satisfying the invariance property $$\label{invprop}
\lbrack\Delta(h), \widehat{r}\,\rbrack=0\qquad \forall\, h\in\mathfrak{h}.$$ For $N\geq 2$ and $1\leq i\leq N$ write, $$\widehat{\mathcal{D}}_i^{(N)}:=E_i-
\sum_{s=1}^{i-1}\widehat{r}_{si}+\sum_{s=i+1}^N\widehat{r}_{is}\in
\mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{g})^{\otimes N}$$ with $E=\sum_{k=1}^r\partial_{x_k}\otimes x_k$, see . The following two statements are equivalent.
1. For $N\geq 2$ and $1\leq i\not=j\leq N$, $$\label{gencomm}
\lbrack \widehat{\mathcal{D}}_i^{(N)},\widehat{\mathcal{D}}_j^{(N)}\rbrack=
-\sum_{k=1}^r\partial_{x_k}(\widehat{r}_{ij})\Delta^{(N-1)}(x_k)$$ in $\mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{g})^{\otimes N}$.
2. $\widehat{r}$ is a solution of the classical dynamical Yang-Baxter equation, $$\label{cdYBe}
\begin{split}
\sum_{k=1}^r\bigl((x_k)_3\partial_{x_k}(\widehat{r}_{12})-
&(x_k)_2\partial_{x_k}(\widehat{r}_{13})+(x_k)_1\partial_{x_k}(\widehat{r}_{23})\bigr)\\
&\qquad+\lbrack \widehat{r}_{12},\widehat{r}_{13}\rbrack
+\lbrack \widehat{r}_{12},\widehat{r}_{23}\rbrack+\lbrack \widehat{r}_{13},\widehat{r}_{23}
\rbrack=0
\end{split}$$ in $\mathcal{R}\otimes U(\mathfrak{g})^{\otimes 3}$.
By direct computations, $$\lbrack \widehat{\mathcal{D}}_1^{(2)},\widehat{\mathcal{D}}_2^{(2)}\rbrack=
-\sum_{k=1}^r\partial_{x_k}(\widehat{r})\Delta(x_k)$$ in $\mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{g})^{\otimes 2}$ and $$\begin{split}
\lbrack \widehat{\mathcal{D}}_1^{(3)},\widehat{\mathcal{D}}_2^{(3)}\rbrack=
&-\sum_{k=1}^r\partial_{x_k}(\widehat{r}_{12})\Delta^{(2)}(x_k)
+ \lbrack \widehat{r}_{12},\widehat{r}_{13}\rbrack
+\lbrack \widehat{r}_{12},\widehat{r}_{23}\rbrack+\lbrack \widehat{r}_{13},\widehat{r}_{23}
\rbrack\\
&+
\sum_{k=1}^r\bigl((x_k)_3\partial_{x_k}(\widehat{r}_{12})-
(x_k)_2\partial_{x_k}(\widehat{r}_{13})+(x_k)_1\partial_{x_k}(\widehat{r}_{23})\bigr)
\end{split}$$ in $\mathbb{D}_{\mathcal{R}}\otimes U(\mathfrak{g})^{\otimes 3}$. Hence [**a**]{} implies [**b**]{}.
It is a straightforward but tedious computation to show that classical dynamical Yang-Baxter equation implies for all $N\geq 2$ and all $1\leq i\not=j\leq N$.
For instance, $\widehat{r}(h):=r(h/2)$ ($h\in\mathfrak{h}$) with $r$ Felder’s $r$-matrix satisfies the classical dynamical Yang-Baxter equation as well as the invariance condition . The same holds true for $\widehat{r}=2r$.
Let $N\geq 2$. Let $U_1,\ldots,U_N$ be finite dimensional $\mathfrak{g}$-modules and write $\mathbf{U}:=U_1\otimes\cdots\otimes U_N$ as before. Let $\widehat{r}\in \mathcal{R}\otimes U(\mathfrak{g})^{\otimes 2}$ be a solution of the classical dynamical Yang-Baxter equation satisfying the invariance property . Define differential operators $\widehat{\mathcal{D}}_i^{\mathbf{U}}\in
\mathbb{D}_{\mathcal{R}}\otimes \textup{End}\bigl(\mathbf{U}[0]\bigr)$ for $i=1,\ldots,N$ by $$\widehat{\mathcal{D}}_i^{\mathbf{U}}:=\widehat{\mathcal{D}}_i^{(N)}|_{\mathbf{U}[0]}.$$ Then $\lbrack \widehat{\mathcal{D}}_i^{\mathbf{U}},\widehat{\mathcal{D}}_j^{\mathbf{U}}\rbrack=0$ in $\mathbb{D}_{\mathcal{R}}\otimes \textup{End}\bigl(\mathbf{U}[0]\bigr)$ for $i,j=1,\ldots,N$.
\[ESremark\] Let $\lambda\in\mathfrak{h}_{\textup{reg}}^*$. Let $\mathbf{u}=u_1\otimes\cdots\otimes u_N\in\mathbf{U}[0]$ with $u_i\in U_i[\mu_i]$ and $\sum_{j=1}^N\mu_j=0$, and write $\lambda_i:=\lambda-\mu_{i+1}\cdots-\mu_N$ ($i=1,\ldots,N-1$) and $\lambda_N:=\lambda$. By [@ES; @E], the weighted trace of the product $\Psi_\lambda^{\mathbf{J}(\lambda)\mathbf{u}}$ of the $N$ vertex operators $\Psi_{\lambda_i}^{u_i}\in\textup{Hom}_{\mathfrak{g}}(M_{\lambda_i}, M_{\lambda_{i-1}}\otimes U_i)$ are common eigenfunctions of the asymptotic KZB operators $\widehat{\mathcal{D}}_i^{\mathbf{U}}$ $1\leq i\leq N$ with $\widehat{r}(h)=r(h/2)$ and $r$ Felder’s $r$-matrix .
Now we prove the analogous result for asymptotic boundary KZB type operators. This time the universal versions of the asymptotic boundary KZB operators themselves will already be integrable. This is because we are considering asymptotic boundary KZB operators associated to [*split*]{} Riemannian symmetric pairs $G/K$ (note that the representation theoretic context from Remark \[ESremark\] relates the asymptotic KZB operators to the group $G$ viewed as the symmetric space $G\times G/\textup{diag}(G)$, with $\textup{diag}(G)$ the group $G$ diagonally embedding into $G\times G$).
\[bKZBopergeneral\] Let $A_\ell$ and $A_r$ be two complex unital associative algebras. Let $\widetilde{r}^{\,\pm}\in
\mathcal{R}\otimes U(\mathfrak{g})^{\otimes 2}$ and $\widetilde{\kappa}\in
\mathcal{R}\otimes A_\ell\otimes U(\mathfrak{g})\otimes A_r$ and suppose that $$\label{invpropb}
\lbrack h\otimes 1, \widetilde{r}^{\,+}\rbrack=\lbrack 1\otimes h, \widetilde{r}^{\,-}\rbrack\qquad
\forall\, h\in\mathfrak{h}.$$ Write for $N\geq 2$ and $1\leq i\leq N$, $$\widetilde{\mathcal{D}}_i^{(N)}:=E_i-
\sum_{s=1}^{i-1}\widetilde{r}_{si}^{\,+}-\widetilde{\kappa}_i
-\sum_{s=i+1}^N\widetilde{r}_{is}^{\,-}
\in\mathbb{D}_{\mathcal{R}}\otimes A_\ell\otimes U(\mathfrak{g})^{\otimes N}\otimes A_r$$ with $E$ given by and with the indices indicating in which tensor components of $U(\mathfrak{g})^{\otimes N}$ the $U(\mathfrak{g})$-components of $\widetilde{r}^{\,\pm}$ and $\widetilde{\kappa}$ are placed. The following statements are equivalent.
1. For all $N\geq 2$ and all $1\leq i,j\leq N$, $$\lbrack \widetilde{\mathcal{D}}_i^{(N)},\widetilde{\mathcal{D}}_j^{(N)}\rbrack=0$$ in $\mathbb{D}_{\mathcal{R}}\otimes A_\ell\otimes U(\mathfrak{g})^{\otimes N}\otimes A_r$.
2. $\widetilde{r}^+$ and $\widetilde{r}^-$ are solutions of the following three mixed classical dynamical Yang-Baxter equations, $$\label{mixedcdYBe}
\begin{split}
\sum_{k=1}^r\bigl((x_k)_1\partial_{x_k}(\widetilde{r}_{23}^{\,-})-
(x_k)_2\partial_{x_k}(\widetilde{r}_{13}^{\,-})\bigr)&=
\lbrack \widetilde{r}_{13}^{\,-},\widetilde{r}_{12}^{\,+}\rbrack
+\lbrack \widetilde{r}_{12}^{\,-},\widetilde{r}_{23}^{\,-}\rbrack+\lbrack \widetilde{r}_{13}^{\,-},
\widetilde{r}_{23}^{\,-}\rbrack,\\
\sum_{k=1}^r\bigl((x_k)_1\partial_{x_k}(\widetilde{r}_{23}^{\,+})-
(x_k)_3\partial_{x_k}(\widetilde{r}_{12}^{\,-})\bigr)&=\lbrack \widetilde{r}_{12}^{\,-},\widetilde{r}_{13}^{\,+}\rbrack
+\lbrack \widetilde{r}_{12}^{\,-},\widetilde{r}_{23}^{\,+}\rbrack+\lbrack \widetilde{r}_{13}^{\,-},
\widetilde{r}_{23}^{\,+}\rbrack,\\
\sum_{k=1}^r\bigl((x_k)_2\partial_{x_k}(\widetilde{r}_{13}^{\,+})-
(x_k)_3\partial_{x_k}(\widetilde{r}_{12}^{\,+})\bigr)&=
\lbrack \widetilde{r}_{12}^{\,+},\widetilde{r}_{13}^{\,+}\rbrack
+\lbrack \widetilde{r}_{12}^{\,+},\widetilde{r}_{23}^{\,+}\rbrack+\lbrack \widetilde{r}_{23}^{\,-},
\widetilde{r}_{13}^{\,+}\rbrack
\end{split}$$ in $\mathcal{R}\otimes U(\mathfrak{g})^{\otimes 3}$, and $\widetilde{\kappa}$ is a solution of the mixed classical dynamical reflection equation $$\label{mixedcdRe}
\sum_{k=1}^r\bigl((x_k)_1\partial_{x_k}(\widetilde{\kappa}_2+\widetilde{r}^{+})-
(x_k)_2\partial_{x_k}(\widetilde{\kappa}_1+\widetilde{r}^{-})\bigr)
=
\lbrack\widetilde{\kappa}_1+\widetilde{r}^{-}, \widetilde{\kappa}_2+\widetilde{r}^{+}\rbrack$$ in $\mathcal{R}\otimes A_\ell\otimes U(\mathfrak{g})^{\otimes 2}\otimes A_r$.
By direct computations, $\lbrack \widetilde{\mathcal{D}}_1^{(2)},\widetilde{\mathcal{D}}_2^{(2)}\rbrack=0$ is equivalent to the mixed dynamical reflection equation and $\lbrack \widetilde{\mathcal{D}}_i^{(3)},\widetilde{\mathcal{D}}_j^{(3)}\rbrack=0$ for $(i,j)=(1,2), (1,3), (2,3)$ is equivalent to the three mixed classical dynamical Yang-Baxter equations. Hence [**a**]{} implies [**b**]{}. Conversely, a direct but tedious computation shows that the three mixed classical dynamical Yang-Baxter equations and the mixed classical dynamical reflection equation imply $\lbrack \widetilde{\mathcal{D}}_i^{(N)},\widetilde{\mathcal{D}}_j^{(N)}\rbrack=0$ for $N\geq 2$ and $1\leq i,j\leq N$.
Applied to the asymptotic boundary KZB operators $\mathcal{D}_i$ ($1\leq i\leq N$) given by , we obtain from Theorem \[consistentoperators\] with $A_\ell=U(\mathfrak{k})=A_r$ the following main result of this subsection.
\[thmDEF\] The folded dynamical $r$-matrices $r^{\pm}\in\mathcal{R}\otimes \mathfrak{g}^{\otimes 2}$ see and the dynamical $k$-matrix $\kappa\in\mathcal{R}\otimes
\mathfrak{k}\otimes U(\mathfrak{g})\otimes\mathfrak{k}$ see satisfy the mixed classical dynamical Yang-Baxter equations in $\mathcal{R}\otimes U(\mathfrak{g})^{\otimes 3}$ and the mixed classical dynamical reflection equation in $\mathcal{R}\otimes U(\mathfrak{k})\otimes U(\mathfrak{g})^{\otimes 2}\otimes U(\mathfrak{k})$.
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[^1]: Note here the remarkable fact, well known to specialists in harmonic analysis, that for generic $z\in Z(\mathfrak{g})$ and $\lambda\in\mathfrak{h}^*$ the differential equation $\widehat{\Pi}^\sigma(z)f=\zeta_\lambda(z)f$ for formal $\textup{End}(V_\ell\otimes V_r^*)$-valued power series $f=\sum_{\mu\leq\lambda}f_{\lambda-\mu}\xi_\mu$ will uniquely define the coefficients $f_\gamma\in \textup{End}(V_\ell\otimes V_r^*)$ in terms of $f_0\in \textup{End}(V_\ell\otimes V_r^*)$. This in particular holds true for $z=\Omega$ and $\lambda\in\mathfrak{h}_{\textup{HC}}^*$ . The quadratic Casimir $\Omega$ is a natural choice since $\widehat{\Pi}^\sigma(\Omega)$ is an explicit second-order differential operator that produces the Schr[ö]{}dinger operator for a spin quantum Calogero-Moser system, solvable by asymptotic Bethe ansatz.
[^2]: Throughout this paper we use “spin” in the sense how this term is used in physics as the description of internal degrees of freedom of one-dimensional quantum particles with their relative positions defined by the $\xi_\alpha$ ($\alpha\in R$) and their spin degrees of freedom given by vectors in $V_l\otimes V_r^*$
[^3]: Define for a $\sigma^{(N)}$-spherical functions $f$, $$\widetilde{f}(g_0,\ldots,g_N):=\bigl(\tau_1(g_0^{-1})\otimes
\tau_2(g_1^{-1}g_0^{-1})\otimes\cdots\otimes\tau_N(g_{N-1}^{-1}\cdots g_1^{-1}g_0^{-1})\bigr)
f(g_0\cdots g_N).$$ It satisfies $\widetilde{f}(k_\ell g_0h_1^{-1}, h_1g_1g_2^{-1},
\ldots,h_Ng_Nk_r^{-1})=(\sigma_\ell(k_\ell)\otimes\tau_1(h_1)\otimes\cdots
\otimes\tau_N(h_N)\otimes\sigma_r^*(k_r))\widetilde{f}(g_0,\ldots,g_N)$ for $(k_\ell,h_1,\ldots,h_N,k_r)\in K\times G^{\times N}\times K$. The action of $Z(\mathfrak{g})$ by biinvariant differential operators on the $j$th argument of $\widetilde{f}$ gives, through the restriction map $\widetilde{f}\mapsto f|_A$, commuting differential operators $\widehat{\Pi}_j(z)$ ($z\in Z(\mathfrak{g})$, $0\leq j\leq N$) on $A$. Up to an appropriate gauge, the quadratic quantum Hamiltonians then are $-\frac{1}{2}\widehat{\Pi}_j(\Omega)$ ($0\leq j\leq N$), and the asymptotic boundary KZB operators $\mathcal{D}_i$ are $\frac{1}{2}(\widehat{\Pi}_{i}(\Omega)-\widehat{\Pi}_{i-1}(\Omega))$ ($1\leq i\leq N$). We will show this in a separate paper.
[^4]: This factorisation can be used to derive the asymptotic KZB equations for Etingof’s and Schiffmann’s generalised weighted trace functions (see [@ES]) in a manner similar to the one as described above for $N$-point spherical functions (weighted traces are naturally associated to the symmetric space $G\times G/\textup{diag}(G)$, with $\textup{diag}(G)$ the group $G$ diagonally embedded into $G\times G$).
|
---
abstract: 'We consider the expectation values of chiral primary operators in the presence of the interface in the 4 dimensional ${\mathcal{N}}=4$ super Yang-Mills theory. This interface is derived from D3-D5 system in type IIB string theory. These expectation values are computed classically in the gauge theory side. On the other hand, this interface is a holographic dual to type IIB string theory on AdS$_5\times$S$^5$ spacetime with a probe D5-brane. The expectation values are computed by GKPW prescription in the gravity side. We find non-trivial agreement of these two results: the gauge theory side and the gravity side.'
---
OU-HET 749
1.5cm [ Koichi Nagasaki[^1] and Satoshi Yamaguchi[^2] ]{} .5cm [*Department of Physics, Graduate School of Science,\
Osaka University, Toyonaka, Osaka 560-0043, Japan*]{}
Introduction and summary
========================
The AdS/CFT correspondence [@Maldacena:1997re] is an interesting duality between a gravity theory and a gauge theory. However it is very difficult to check this duality since unprotected quantities are calculable only in the small ’t Hooft coupling $\lambda$ regime in the gauge theory side, while they are calculable in the large $\lambda$ regime in the gravity side.
There are several ways to overcome this difficulty. One of them is to introduce another large parameter as in [@Berenstein:2002jq]. In [@Berenstein:2002jq], the R-charge $J$ (the angular momentum in the gravity side) has been taken to be large and the effective expansion parameter has become $\lambda/J^2$. By virtue of this change of the effective coupling, the conformal dimension of such operators have been successfully compared to the energy of the stringy excited states in the pp-wave geometry. This result has given a non-trivial evidence of the AdS/CFT correspondence. Other examples of similar phenomena are found in surface operators [@Drukker:2008wr] (see also [@Koh:2008kt]) and the interface [@Nagasaki:2011ue].
An interface is a wall in the space-time which connect two different (or the same) quantum field theories. A partial list of related references are [@Sethi:1997zza; @Ganor:1997jx; @Kapustin:1998pb; @Karch:2000gx; @DeWolfe:2001pq; @Bachas:2001vj; @Bak:2003jk; @D'Hoker:2006uu; @D'Hoker:2006uv; @Gomis:2006cu; @D'Hoker:2007xy; @D'Hoker:2007xz; @Gaiotto:2008sa; @Gaiotto:2008sd; @Gaiotto:2008ak]. See also [@Kirsch:2004km] and references there in. The interface considered in this paper is so called a “Nahm pole” which connects SU$(N)$ gauge theory and SU$(N-k)$ gauge theory. The boundary condition is determined by the fuzzy funnel solution [@Constable:1999ac]. In this interface a parameter $k$ is introduced, and taken to be large in this paper as done in [@Nagasaki:2011ue]. This interface is described by the intersecting D3-D5 system where $k$ D3-branes end on the D5-brane. Thus the gravity dual is given by the near horizon limit of the supergravity solution for the D3-branes with the probe D5-brane with $k$ units of magnetic flux [@Karch:2000gx].
In this paper we study the expectation values of chiral primary operators in the presence of the above interface. In the gauge theory side the expectation values are evaluated by just substituting the classical solution of the fuzzy funnel solution. On the other hand they are calculated by GKPW prescription [@Gubser:1998bc; @Witten:1998qj]. Usually these two results cannot be compared to each other because the gauge theory result is only valid in the small $\lambda$ regime, while the gravity result is only valid in the large $\lambda$ regime. However in our case we can take $k\to\infty$ limit and make $\lambda/k^2$ small even if $\lambda$ is large in the gravity side. In this limit we find perfect agreement between the gauge theory result and the gravity result. This is a quite non-trivial evidence of the AdS/CFT correspondence.
The construction of this paper is as follows. In section \[sec:gauge\], we review the 4-dimensional ${\mathcal{N}}=4$ SYM theory and the interface, and show the calculation of the expectation values of the chiral primary operators in the presence of the interface. In section \[sec:gravity\], we turn to the calculation in the gravity side using GKPW prescription. In section \[sec:comparison\], the above two results are compared and the perfect agreement is found in the leading order. The next-to-leading term is predicted from the gravity side.
Gauge theory side {#sec:gauge}
=================
We consider the 4-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theory in this section. We review the action of this theory and classical solutions. After that we calculate the expectation values of the chiral primary operators in the presence of the interface.
Fields and Action
-----------------
We consider here the $\mathcal{N}$=4 super Yang-Mills theory with the gauge group $SU(N)$. We use the same convention as [@Nagasaki:2011ue]. The action is given by $$S=\frac{2}{g^2}\int d^4x \text{tr}\left[
-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
-\frac{1}{2}D_\mu\phi_iD^\mu\phi^i
+\frac{i}{2}\bar{\psi}\Gamma^\mu D_\mu \psi
+\frac{1}{2}\bar{\psi}\Gamma^i [\phi_i,\psi]
+\frac{1}{4}[\phi_i,\phi_j][\phi^i,\phi^j]
\right],$$ where $F_{\mu\nu}$, $\mu=0,\cdots,3$, are the field strength of the gauge field $A_{\mu}$, which is expressed as $F_{\mu\nu}={\partial}_{\mu}A_{\nu}-{\partial}_{\nu}A_{\mu}-i[A_{\mu},A_{\nu}]$. While $\psi$ is a fermion field and $\phi_i$, $i=4,\cdots,9$, are scalar fields. All these fields are in the adjoint representation of SU$(N)$, in other words, $N\times N$ hermitian traceless matrices. These scalar fields play a crucial role in the one-point function we want to calculate in this paper.
This action is invariant under the following supersymmetry transformation with the spinor parameter $\epsilon$. $$\begin{aligned}
&&\delta A_\mu = i\bar\epsilon\Gamma_\mu \psi,\\
&&\delta \phi_i = i\bar\epsilon\Gamma_i \psi,\\
&&\delta\psi = \frac{1}{2}F_{\mu\nu}\Gamma^{\mu\nu}\epsilon+D_\mu\phi_i\Gamma^{\mu i}\epsilon -\frac{i}{2}[\phi_i,\phi_j]\Gamma^{ij}\epsilon.\end{aligned}$$
Interface {#FuzzyFunnelSol}
---------
We introduce here a wall-like object called interface. This object separates a whole space into two regions where gauge theories with different gauge groups live. One has gauge group SU$(N)$ and the other has SU$(N-k)$. This interface is defined by a classical solution known as a fuzzy funnel solution [@Constable:1999ac]. This solution plays a crucial role in our calculation. The interface is defined by a boundary condition between two different gauge theories and leads to a non-trivial classical vacuum solution. $$\label{ClVacuumsol}
A_\mu=0,\hspace{1cm}
\phi_i=\phi_i(x_3),\:(i=4,5,6),\hspace{1cm}
\phi_i=0,\:(i=7,8,9).$$ The solution $\phi_i=\phi_i(x_3),\:(i=4,5,6),$ is called a fuzzy funnel solution [@Constable:1999ac]. The solution of scalar fields are given by $$\phi_i=-\frac{1}{x_3}t_i\oplus 0_{(N-k)\times(N-k)}\hspace{1cm} (x_3> 0),$$ where $t_i,\: i=4,5,6$, are $k\times k$ matrices which denote the generators of SU(2) algebra of the $k$-dimensional irreducible representation. The following relation is useful for our calculation. $$\label{ConvenientRelations}
\phi_4^2+\phi_5^2+\phi_6^2=\frac{1}{4x_3^2}(k^2-1){1}_{k\times k}\oplus{0}_{(N-k)\times(N-k)}.$$
One-point function
------------------
In this section we consider the one-point functions of chiral primary operators. The chiral primary operators are defined as $$\mathcal{O}_\Delta(x):=\frac{(8\pi^2)^{\Delta/2}}{\lambda^{\Delta/2}\sqrt{\Delta}}C^{I_1I_2\cdots I_\Delta}\text{Tr} (\phi_{I_1}(x)\phi_{I_2}(x)\cdots \phi_{I_\Delta}(x)),$$ where $\Delta$ denotes the conformal dimension and $C^{I_1I_2\cdots I_\Delta}$ is a traceless symmetric tensor normalized as $C^{I_1I_2\cdots I_\Delta}C^{I_1I_2\cdots I_\Delta}=1$. The normalization of the operator is determined so that the two point function without interface becomes $$\begin{aligned}
&\langle \mathcal{O}_\Delta(x)\mathcal{O}_\Delta(y)\rangle=\frac{1}{|x-y|^{2\Delta}}.\end{aligned}$$ See [@Lee:1998bxa] for the detail.
We would like to calculate the one-point function of this operator. Let us insert this operator at a point $x_3=\xi$ and consider the expectation value $\langle \mathcal{O}_{\Delta}(\xi)\rangle$. For calculating the classical expectation value of this operator we substitute the fuzzy funnel solution introduced in the above section \[FuzzyFunnelSol\]. Since our fuzzy funnel solution preserves SO$(3)\times$SO$(3)$ symmetry, only SO$(3)\times$SO$(3)$ invariant chiral primary operators can have non-vanishing expectation values. As shown in Appendix \[app:harmonics\], $\Delta$ must be even and is denoted as $\Delta=2\ell$. Moreover there is only one such chiral primary operator for each $\Delta=2\ell,\ \ell=0, 1,2,3,\cdots$.
The traceless symmetric tensors $C^{I_1\cdots I_{\Delta}}$ are related to the spherical harmonics (see Appendix \[app:harmonics\]). $$C^{I_1I_2\cdots I_\Delta}x_{I_1}\cdots x_{I_\Delta}=Y_\ell (\psi),\hspace{1cm}
\sum_{i=4}^{6} x_i^2=\sin^2\psi,\hspace{1cm}
\sum_{j=7}^{9} x_j^2=\cos^2\psi$$ Spherical harmonics is expressed as eq. $$\begin{aligned}
Y_\ell (\psi)
= C_\ell F(-\ell,\ell+2,\frac{3}{2};\cos^2\psi)=C_\ell (1+\cos^2\psi P(\cos^2\psi)),\label{inhomogeneous}\end{aligned}$$ where $P(\cos^2\psi)$ is an inhomogeneous polynomial of $\cos^2\psi$. The normalization $C_{\ell}$ is determined so that $C^{I_1I_2\cdots I_\Delta}C^{I_1I_2\cdots I_\Delta}=1$ is satisfied, or equivalently eq. . We can express this spherical harmonics by a homogeneous polynomial of $\sin^2 \psi$ and $\cos^2 \psi$. This is because if we have a inhomogeneous term, we can replace $1$ by some power of $\sin^2\psi+\cos^2\psi$. In particular we can replace the first term $1$ in the paren in eq. by $(\sin^2\psi+\cos^2\psi)^{\ell}$ and get homogeneous expression $$\begin{aligned}
Y_{\ell}= C_\ell (\sin^{2\ell}\psi+\cos^2\psi\; Q(\sin^2\psi,\cos^{2}\psi)),\end{aligned}$$ where $Q(\sin^2\psi,\cos^{2}\psi)$ is a homogeneous polynomial of $\sin^2\psi$ and $\cos^2\psi$. Then replacing $\sin^2\psi$ by $\sum_{i=4}^{6}\phi_i^2$ and $\cos^2\psi$ by $\sum_{j=7}^{9}\phi_j^2$, we obtain the relation[^3] $$C^{I_1\cdots I_\Delta}\phi_{I_1}\cdots \phi_{I_\Delta}=C_\ell \left\{\left(\sum_{i=4}^{6}\phi_i^2\right)^\ell +\left(\sum_{j=7}^{9}\phi_j^2\right) Q\left(\sum_{i=4}^{6}\phi_i^2,\sum_{j=7}^{9}\phi_j^2\right)\right\}.$$ Substituted the solution , all terms except the first one vanish since $\phi_7=\phi_8=\phi_9=0$. Using the relations we obtain the following result. $$\begin{aligned}
\left<\mathcal{O}_{2\ell}(\xi)\right>_\text{classical}
&=& \frac{(8\pi^2)^{\Delta/2}}{\lambda^{\Delta/2}\sqrt{\Delta}}C_\ell \text{Tr}\left[\left(\frac{1}{4\xi^2}(k^2-1)\right)^\ell \bold{1}_{k\times k}\right]\nonumber\\
&=& C_\ell\frac{(2\pi^2)^\ell}{\sqrt{2\ell}\lambda^\ell}(k^2-1)^{\ell}k\frac{1}{\xi^{2\ell}}.
\label{gauge-result}\end{aligned}$$ The behavior $1/\xi^{2\ell}$ is determined by the conformal symmetry and does not change by the quantum correction. The non-trivial part is the coefficient, which will change by the quantum correction. We compare this result with the gravity side calculation.
Gravity side {#sec:gravity}
============
In this section we calculate the expectation values of the chiral primary operators in the gravity side. The AdS/CFT correspondence is a duality between the $\mathcal{N}=4$ super Yang-Mills theory we discussed in the previous section and type IIB superstring theory on AdS$_5\times$S$^5$. How this gravity side is modified when the interface is inserted? The object which corresponds to our interface is a probe D5-brane with $k$ units of magnetic flux [@Karch:2000gx]. This gravity dual is obtained by the following way. We consider a D5-brane where $k$ D3-branes end. Then SU($N$) gauge theory is realized in the side where there are $N$ D3 branes and SU($N-k$) gauge theory is realized in the other side as low energy effective theories. This D5-brane is pulled by $k$ D3-branes which end on it and become funnel shape with $k$ units of magnetic flux. If we consider the supergravity solution of D3-branes and take the near horizon limit, we obtain the gravity dual mentioned above. Here we make a remark on the value $k$. Although we take $k$ large, it is still much smaller than $N$ in order not to modify the supergravity background.
The Gubser-Klebanov-Polyakov-Witten relation
--------------------------------------------
The correlation functions in the AdS/CFT correspondence are calculated by GKPW prescription [@Gubser:1998bc; @Witten:1998qj]. Due to GKPW there is one-to-one correspondence between local operators in the gauge theory and fields in the gravity theory. Let ${\mathcal{O}}$ be a scalar operator in the gauge theory, and $s$ be the scalar field in the gravity theory which corresponds to ${\mathcal{O}}$. GKPW claims that the relation $$\left<\text{e}^{\int d^4xs_0(x)\mathcal{O}(x)}\right>_\text{CFT}
=
\text{e}^{-S_\text{cl}(s_0)}$$ is satisfied in the classical gravity limit. In this equation $s_0$ is a boundary condition of $s$ up to a certain factor, $S_\text{cl}(s_0)$ is the action evaluated by the classical solution with the boundary condition given by $s_0$. Using this relation the one-point function is calculated as follows. $$\left<\mathcal{O}(x)\right>
=\left. \frac{-\delta S_\text{cl}(s_0)}{\delta s_0(x)}\right|_{s_0=0}.
\label{GKPW1pt}$$ We employ the normalization $\langle 1 \rangle=1$.
If no interface or other defects are inserted, this one-point function vanishes due to the conformal invariance. In terms of the gravity theory, this one-point function vanishes since the background is a solution of the equation of motion and thus any variation of the action vanishes at this background. In our case this one-point function does not vanish in general because the interface is inserted as we have seen in the previous section. In the gravity side, this one-point function does not vanish because we have, in addition to the supergravity, a probe D5-brane which gives non-vanishing contribution.
Background
----------
We consider here type IIB superstring theory as the gravity theory. The near horizon limit of the supergravity solution of $N$ coincident D3-branes is AdS$_5\times$S$^5$. The coordinates of AdS${}_5$ are denoted by $y,x^\mu,\mu=0,1,2,3$. The metric on this space is given by $$ds^2_\text{AdS$_5\times$S$^5$}=\frac{1}{y^2}(dy^2+\eta_{\mu\nu}dx^\mu dx^\nu)+ds^2_\text{S$^5$}.$$ In this paper we choose the unit in which the radius of AdS${}_5$ is $1$. Thus the string coupling constant $g_s$ and the slope parameter $\alpha'$ are related as $$\lambda:=4\pi g_s N =\alpha'^{-2}. \label{unit}$$ Furthermore the RR 4-form is also excited $$C_4=-\frac{1}{y^4}dx^0dx^1dx^2dx^3+\cdots.$$
In addition to the D3-brane configuration discussed earlier, we introduce a D5-brane in order to study the corresponding theory of the interface CFT. The D5-brane action is the usual DBI+WZ action. $$S=T_5 \int d^6\zeta\sqrt{\det (G+{\mathcal{F}})}+iT_5\int \mathcal{F}\wedge C_4,$$ where $T_5=(2\pi)^{-5}\alpha'^{-2}g_s^{-1}$ is the tension of the D5-brane, $\zeta$’s are the world-volume coordinates, $G$ and ${\mathcal{F}}$ denote the induced metric and the field strength of the world-volume gauge field respectively. The AdS${}_4\times$S${}^2$ solution is obtained by [@Karch:2000gx]. We use the convention of [@Nagasaki:2011ue]. AdS$_{4}$ part is embedded in AdS${}_5$ and expressed by the equation $$\label{D5inAdS}
x_3=\kappa y$$ with a constant parameter $\kappa$. S$^2$ is embedded in S${}^5$ as a great sphere. We denote world-volume coordinates of D5 by $(y,x_0,x_1,x_2,\theta,\phi)$; $(y,x_0,x_1,x_2)$ are coordinates of AdS${}_4$ and $(\theta,\phi)$ are ones of S${}^2$. The induced metric and the gauge field are summarized by a matrix $H=G+ \mathcal{F}$. $H$ takes the following form in this solution. $$H=
\left(\begin{array}{cccc|cc}
(1+\kappa^2)y^{-2} & & & & & \\
& y^{-2} & & & & \\
& & y^{-2} & & & \\
& & & y^{-2} & & \\\hline
& & & & 1 & -\kappa\sin\theta \\
& & & & \kappa\sin\theta &\sin^2\theta \end{array}\right).$$ Actually the parameter $\kappa$ is related with $k$ as $\kappa = \frac{\pi}{\sqrt{\lambda}}k$.
One-point function from gravity theory
--------------------------------------
Now let us turn to the calculation of the one point function. The scalar fields which correspond to the chiral primary operators are identified in [@Kim:1985ez; @Lee:1998bxa]. These scalar fields come from the fluctuation of the metric and the RR 4-form as $$\begin{aligned}
&h^\text{AdS}_{\mu\nu} =-\frac{2\Delta(\Delta-1)}{\Delta+1}sg_{\mu\nu}+\frac{4}{\Delta+1}\nabla_\mu\nabla_\nu s ,\label{Fluc1}\\
&h^\text{S}_{\alpha\beta} =2\Delta sg_{\alpha\beta} , \label{Fluc2}\\
&a^\text{AdS}_{\mu\nu\rho\sigma}=4i\sqrt{g^\text{AdS}}\epsilon_{\mu\nu\rho\sigma\eta}\nabla^\eta s,\label{Fluc3}\end{aligned}$$ where $h^\text{AdS}_{\mu\nu} $, $h^\text{S}_{\alpha\beta}$ and $a^\text{AdS}_{\mu\nu\rho\sigma}$ are the fluctuation of AdS${}_5$ part of the metric, S${}^5$ part of the metric and AdS${}_5$ part of the RR 4-form, respectively. $\Delta=2\ell$ corresponds to the conformal dimension of the operator in the gauge theory.
The classical solution of $s$ with the boundary condition can be written as $$\begin{aligned}
&s(y,x,\theta,\phi,\psi,\cdots)=\int d^4x' c_\Delta \frac{y^\Delta}{K(y,x,x')^\Delta}s_0(x')Y_{\Delta/2}(\psi),\\
&K(y,x,x'):=|x-x'|^2+y^2,\\
&c_{\Delta}=\frac{\Delta+1}{2^{2-\Delta/2}N \sqrt{\Delta}}.\label{classical-solution}
\end{aligned}$$ where $Y_{\Delta/2}$ is the spherical harmonics obtained in appendix \[app:harmonics\]. The normalization factor $c_{\Delta}$ is the correct one obtained in [@Freedman:1998tz; @Lee:1998bxa]. It is determined so that the coefficient of the two point function is unity.
The first order fluctuation of the action is $$\begin{aligned}
S^{(1)}&=\frac{T_5}{2}\int d^6\zeta \sqrt{\det H}(H^{-1}_{\text{sym}})^{ab}\partial_aX^M \partial_bX^N h_{MN}
+iT_5\int\mathcal{F}\wedge a_4,
\label{S1}\end{aligned}$$ where $h_{\mu\nu}$ and $a_4$ are the fluctuation of the metric and the RR 4-form given in eqs. -. $H^{-1}_{\text{sym}}$ denotes the symmetric part of the inverse matrix of $H$.
The one-point function can be calculated by using eq. . The classical action $S_{\text{cl}}$ in eq. can be replaced by $S^{(1)}$ in eq. $$\begin{aligned}
\langle {\mathcal{O}}(x)\rangle =-\frac{\delta S^{(1)}(s_0)}{\delta s_0(x)}.\end{aligned}$$ The detailed calculation of the fluctuation $S^{(1)}$ is shown in the appendix \[app:D5braneAction\]. The final result of gravity side is given by eq. $$\begin{aligned}
-\frac{\delta S_\text{cl}}{\delta s_0(\xi)}
&= C_\ell\frac{\sqrt{\lambda}2^{\ell}\Gamma(2\ell+1/2)}{\pi^{3/2}\sqrt{2\ell}\Gamma(2\ell)} \frac{1}{\xi^{2\ell}}
\int_0^\infty d u
\frac{u^{2\ell -2}}{\Big[(1-\kappa u)^2+u^2\Big]^{2\ell+1/2}}\label{GravityResult}.\end{aligned}$$ Here $\xi$ is the distance between the interface and the point where the chiral primary operator is inserted.
In eq. , the dependence of $\xi$ is $1/\xi^{2\ell}$ and this is determined by the conformal symmetry. We will compare the coefficient with the gauge theory side in the next section.
Discussion {#sec:comparison}
==========
In the previous sections, \[sec:gauge\] and \[sec:gravity\], we calculated the one-point function in the gauge theory side and the gravity side. Our goal is to confirm the correspondence between the gauge theory and the gravity theory. Let us compare these results in this section. We consider the limit $k\gg 1$ and $\lambda/k^2 \ll 1$, and compare the leading terms.
Gauge theory
------------
Since we consider the limit $k\gg 1$ the gauge theory result becomes $$\begin{aligned}
\left<\mathcal{O}_{2\ell}\right>_\text{classical}
&= C_\ell\frac{(2\pi^2)^\ell}{\sqrt{2\ell}\lambda^\ell}(k^2-1)^{\ell}k\frac{1}{\xi^{2\ell}}\nonumber\\
&\approx C_\ell\frac{(2\pi^2)^\ell}{\sqrt{2\ell}\lambda^\ell}k^{2\ell+1}\frac{1}{\xi^{2\ell}}.\label{approxGauge}\end{aligned}$$ This result is compared with the gravity side.
Gravity theory
--------------
We consider the behavior of the gravity side result in the limit $\epsilon:=\frac{1}{\kappa^2+1}\rightarrow 0$, $\kappa = \frac{\pi}{\sqrt{\lambda}}k\gg 1$. The following expression of the Dirac delta function is convenient[^4]. $$\delta(x)=\lim_{\epsilon\to 0}\frac{1}{\sqrt{\pi}}\frac{\Gamma(n)}{\Gamma(n-1/2)}\frac{\epsilon^{2n-1}}{(x^2+\epsilon^2)^n}.$$ Using this formula the integrand of the equation can be approximated by the Dirac delta function. $$\frac{1}{\big((1-\kappa u)^2+u^2\big)^{2\ell+1/2}}\longrightarrow
\frac{1}{\epsilon^{4\ell}}
\frac{\Gamma(2\ell)\Gamma(\frac{1}{2})}{\Gamma(2\ell+\frac{1}{2})}
\delta(u-\kappa\epsilon).$$ After integration we obtain the result $$-\frac{\delta S^{(1)}}{\delta s_0(\xi)}=
C_\ell \frac{(2\pi^2)^\ell}{\lambda^\ell\sqrt{2\ell}}
k^{2\ell+1}\frac{1}{\xi^{2\ell}}.\label{approxGravity}$$ Comparing and , we can conclude that these two quantities completely agree in the leading order of $\lambda/k^2$ series.
We can go to next-to-leading order in the gravity side. Actually the integral in eq. can be rewritten as $$\begin{aligned}
I:&=\int_0^{\infty}du \frac{u^{2\ell-2}}{[(1-\kappa u)^2+u^2]^{2\ell +1/2}}\nonumber\\
&=\kappa^{2\ell+1}\left(1+\frac{1}{\kappa^2}\right)^{3/2}
\int_{-\arctan \kappa}^{\pi/2}d\theta (\cos \theta)^{4\ell-1}\left(1+\frac{1}{\kappa}\tan\theta\right)^{2\ell-2},\end{aligned}$$ by the change of variable as $\tan\theta=(1+\kappa^2)u-\kappa$. This function can expanded around $\kappa\to \infty$ as [^5] $$\begin{aligned}
I&=\kappa^{2\ell+1}\frac{\Gamma(2\ell)\Gamma(1/2)}{\Gamma(2\ell+1/2)}
\left(1+\frac{1}{\kappa^2}I_{1}+O(\frac{1}{\kappa^4})\right),\\
I_1&=\frac{3}{2}+\frac{(2\ell-2)(2\ell-3)}{4(2\ell-1)}.\end{aligned}$$ Using this $I_1$ the gravity result up to next-to-leading order is $$-\frac{\delta S^{(1)}}{\delta s_0(x)}=
C_\ell \frac{(2\pi^2)^\ell}{\lambda^\ell\sqrt{2\ell}}
k^{2\ell+1}\frac{1}{\xi^{2\ell}}
\left(
1+\frac{\lambda}{\pi^2k^2}I_1
+\cdots \right).\label{correction1}$$ These corrections are formally a positive power series of $\lambda/k^2$. The expansion eq. indicates the reason why we can compare the gravity side and the gauge theory side. In the gravity side $\lambda/k^2$ can be small even though $\lambda$ is large because $k^2$ can be larger. Thus one can suppress the sub-leading terms by sending $\lambda/k^2\to 0$ which has superficially the same effects as $\lambda\to 0$. A heuristic arguments of $\lambda/k^2$ scaling in the gauge theory side is given in the discussion section of [@Nagasaki:2011ue].
An interesting future work is to compare the prediction of the 1-loop correction in eq. from the gravity side to the 1-loop calculation in the gauge theory side.
Acknowledgments {#acknowledgments .unnumbered}
---------------
We would like to thank Hiroaki Tanida for discussions and comments. S.Y. was supported in part by KAKENHI 22740165.
Spherical harmonics {#app:harmonics}
===================
SO(3) $\times$ SO(3) invariant ansatz {#InvAnsatz}
-------------------------------------
The interface in this paper preserves SO(3)$\times $SO(3) symmetry out of SO(6) R-symmetry. Thus only SO(3)$\times$SO(3) invariant operators can have non-vanishing expectation values. We would like to introduce SO(3)$\times$ SO(3) invariant spherical harmonics on $S^5$.\
$S^5$ is described as a hypersurface in 6-dimensional Euclidean space whose coordinates are $(x_4,\dots,x_9)$. $S^5$ is defined by the equation $$x_4^2+\dots +x_9^2=1.$$ We introduce a parameter $\psi$, $0\le \psi \le \frac{\pi}{2}$ and reexpress this $S^5$ as the following way. $$x_4^2+x_5^2+x_6^2=\sin^2\psi,\:\:
x_7^2+x_8^2+x_9^2=\cos^2\psi.$$ Then the metric is written as $$ds^2=d\psi^2+\cos^2\psi d\tilde{\Omega}_2^2+\sin^2\psi d{\Omega}_2^2,$$ where $d\tilde{\Omega}_2^2$ and $d{\Omega}_2^2$ are line elements of unit $S^2$.
The SO$(3)\times$SO$(3)$ invariant spherical harmonics only depends on the coordinate $\psi$. Let $Y$ be such a function of $\psi$; $Y=Y(\psi)$. The Laplacian operating on this $Y$ is written as $$\Box Y
=\frac{1}{\sqrt{g}}\partial_i\sqrt{g}g^{ij}\partial_jY
=\frac{1}{\cos^2\psi\sin^2\psi}\frac{d}{d\psi}\cos^2\psi\sin^2\psi\frac{d}{d\psi}Y(\psi).$$ After changing the variable $z:=\cos^2\psi$, the Laplacian is rewritten as $$\Box Y=4z(1-z)\partial_z^2Y+(6-12z)\partial_zY.$$ Then the eigenvalue equation, $\Box Y=-EY$, reads $$\label{EigenEqY}
z(1-z)\partial_z^2Y+\bigg(\frac{3}{2}-3z\bigg) \partial_zY+\frac{E}{4}Y=0.$$ This is a hypergeometric differential equation.
In general a hypergeometric differential equation is given by $$z(1-z)\partial^2_z F+(c-(a+b+1)z)\partial_zF-abF=0,$$ where $a,b,c$ are real parameters. The solution which is regular at $z=0$ is the hypergeometric function given by an infinite power series $$F(a,b,c;z)=\sum^{\infty}_{n=0}\frac{(a)_n(b)_n}{(c)_n}\frac{z^n}{n!}.$$ Here the Pochhammer symbol $(a)_n=\Gamma(a+n)/\Gamma(a)$ is used.
Since we need the smooth solution on whole $S^5$, the solution of eq. must be regular not only at $z=0$ but $z=1$. Then the solution must be a hypergeometric function with $a=-\ell,b=\ell+2,c=3/2,\ (\ell=0,1,2,3,\dots)$ and the eigenvalue $E=2l(2l+4)$ is obtained. Therefore the solution of the equation is expressed in terms of hypergeometric function. $$Y_\ell(\psi)=C_\ell F(-\ell,2+\ell,3/2;\cos^2\psi),\label{Cl}$$ where the normalization factor $C_\ell$ is determined by $$\int_{S^5}\sqrt{g}|Y_\ell|^2
=\frac{\pi^3}{2^{2\ell-1}(2\ell+1)(2\ell+2)}.\label{Ynormalization}$$ The conformal dimension $\Delta$ of the corresponding chiral primary operator is $\Delta =2\ell$.
Detailed calculation {#app:D5braneAction}
====================
Fluctuations $h$ and $a$
------------------------
In this appendix we show the detailed calculations of fluctuations $h$ and $a$ defined by the scalar field $s(x)$ as , and . Actually it is enough to calculate them when $s_0$ is a delta function as $$\begin{aligned}
s_0(x)=\delta^{4}(x-x').\label{sdelta}\end{aligned}$$ In this case the classical solution becomes $$\begin{aligned}
s(y,x,\theta,\phi,\psi)=c_{\Delta}\frac{y^{\Delta}}{K(y,x,x')^{\Delta}}Y_{\Delta/2}(\psi).\end{aligned}$$
We use the convention for the covariant derivative and totally anti-symmetric tensor $$\begin{aligned}
&&\nabla_i T_{j_1\cdots j_n}:=\partial_i T_{j_1\cdots j_n} -\sum^n_{l=1}\Gamma_{ij_l}^kT_{j_1\cdots j_{l-1}k j_{l+1}\cdots j_n}, \\
&&\epsilon_{y0123}=1,\end{aligned}$$ where Christoffel symbols are $\Gamma^i_{jk}:=\frac{1}{2}g^{il}(\partial_j g_{lk}+\partial_k g_{lj}-\partial_l g_{jk})$.
The first derivatives and the second derivatives of $s$ are $$\begin{aligned}
\frac{\partial_ys}{s}&=\Delta(\frac{1}{y}-\frac{2y}{K}),\\
\frac{\partial_is}{s}&=-\Delta\frac{2(x-x')_i}{K},\end{aligned}$$ $$\begin{aligned}
&\frac{\nabla_y\nabla_ys}{s}=\frac{\Delta^2}{y^2}+4\Delta(\Delta+1)\left(
-\frac{1}{K}+\frac{y^2}{K^2}\right),\\
&\frac{\nabla_y\nabla_is}{s}=\Delta(\Delta+1)\left(
+4y\frac{(x-x')_i}{K^2}-2\frac{(x-x')_i}{yK}\right),\\
&\frac{\nabla_i\nabla_js}{s}=-\Delta\frac{\delta_{ij}}{y^2}
+4\Delta(\Delta+1)\frac{(x-x')_i(x-x')_j}{K^2}.\end{aligned}$$ Using these results and the definition of $h$ in AdS the expression of fluctuations are $$\begin{aligned}
&\frac{h^\text{AdS}_{yy}}{\Delta s}
=\frac{2}{y^2}-\frac{16}{K}+\frac{16}{K^2},\\
&\frac{h^\text{AdS}_{yi}}{\Delta s}
=16y\frac{(x-x')_i}{K}-8\frac{(x-x')_i}{yK},\\
&\frac{h^\text{AdS}_{ij}}{\Delta s}
=-2\frac{\delta_{ij}}{y^2}+\frac{16(x-x')_i(x-x')_j}{K^2},\end{aligned}$$ and in 2-sphere $$\begin{aligned}
\frac{h^\text{S}_{\theta\theta}}{\Delta s}=2,\:\:
\frac{h^\text{S}_{\phi\phi}}{\Delta s}=2\sin^2\theta.\end{aligned}$$
D5-brane action
---------------
When we give fluctuation to the metric and the RR 4-form, the D5-brane action is deformed as follows in the first order. We use the notation $v_i=x_i-x'_i$ and $p,q$ run $0,1,2$. The first order fluctuation is calculated as follows. $$\begin{aligned}
S^{(1)}&=&\frac{T_5}{2}\int d^6\zeta \sqrt{\det H}(H^{-1}_{\text{sym}})^{ab}\partial_aX^M \partial_bX^N h_{MN}
+iT_5\int\mathcal{F}\wedge a_4\nonumber\\
&=&T_5\int d^6\zeta (\mathcal{L}^{(1)}_\text{DBI}+\mathcal{L}^{(1)}_\text{WZ}).
\label{DBI+WZ(1)}\end{aligned}$$ In this equation we need the explicit form of the symmetric part of $H^{-1}$. $$\begin{aligned}
H^{-1}_{\text{sym}}=
\begin{pmatrix}
(1+\kappa^2)^{-1}y^{2}&&&&&\\
&y^{2}&&&&\\
&&y^{2}&&&\\
&&&y^{2}&&\\
&&&&(1+\kappa^2)^{-1} &\\
&&&&& [\sin^2\theta(1+\kappa^2)]^{-1}
\end{pmatrix}.\end{aligned}$$ Eq. is calculated as follows. $$\begin{aligned}
\mathcal{L}^\text{(1)}_\text{DBI}
:=&\frac{1}{2}\sqrt{\det H}(H^{-1}_{\text{sym}})^{ab}\partial_aX^M\partial_bX^Nh_{MN}\nonumber\\
=& \frac{(1+\kappa^2)\sin^2\theta}{2y^4}
\{H^{yy}\partial_y X^M \partial_y X^N h^\text{AdS}_{MN}
+H^{ij}\partial_i X^M \partial_j X^N h^\text{AdS}_{MN}\nonumber\\
&\qquad +H^{\theta\theta}\partial_\theta X^M \partial_\theta X^N h^\text{S}_{MN}
+H^{\phi\phi}\partial_\phi X^M \partial_\phi X^N h^\text{S}_{MN}
\}\nonumber\\
=& \frac{\Delta s\sin\theta}{y^4K^2}\{-8y^2v_3^2+\kappa(16y^3v_3-8yv_3K)+\kappa^2(8y^2(v_pv_p+v_3^2)-4K^2)\}.\end{aligned}$$
$$\begin{aligned}
\mathcal{L}^\text{(1)}_\text{WZ}
:=&i\mathcal{F}_{\theta\phi}\frac{1}{4!}\epsilon^{abcd}(Pa)_{abcd}\nonumber\\
=& i2\kappa \sin\theta(a_{y012}+\kappa a_{3012})\nonumber\\
=& i2\kappa\sin\theta\left\{
\Delta 4s\frac{1}{y^3}\frac{2v_3}{K}
+\kappa\Delta 4s\frac{1}{y^3}
\Big(\frac{1}{y}-\frac{2y}{K}\Big)
\right\}\nonumber\\
=& \frac{i\sin\theta\Delta s}{y^4K^2}
\left\{\kappa(16v_3yK)+\kappa^2(8\kappa^2-16y^2K)\right\}.\end{aligned}$$
$S^\text{(1)}$ is the sum of these two terms $$\begin{aligned}
S^\text{(1)}
&=& T_5\int d^6 \zeta \Big(\mathcal{L}^\text{(1)}_\text{DBI}+\mathcal{L}^\text{(1)}_\text{WZ}\Big)\nonumber\\
&=& -8T_5\int d^6 \zeta
\frac{\sin\theta\cdot \Delta s}{y^2K^2}(v_3-\kappa y)^2\nonumber\\
&=& -8T_5\int d^6 \zeta
\frac{\sin\theta\cdot \Delta s}{y^2K^2}x_3'^2.\end{aligned}$$ This formula with the classical solution $s_{0}(x)=\delta^4(x-x')$ is the functional derivative $\delta S^{(1)}/\delta s_0(x')$. This functional derivative evaluated at $x_3'=\xi$ is the quantity we want. Notice that the D5-brane sits at $\psi=\pi/2$, thus the spherical harmonics should be evaluated at this surface. This value is given by (see eq. ) $$\begin{aligned}
Y_{\ell}(\psi=\pi/2)=C_{\ell}.\end{aligned}$$ Putting all these things together, we obtain $$\begin{aligned}
-\frac{\delta S^{(1)}}{\delta s_0(\xi)}
=& 32T_5\pi\Delta c_\Delta C_\ell \int^\infty_0dy\int dx^0dx^1dx^2
\frac{y^{\Delta-2}\xi^2}{((\kappa y-\xi)^2+x^p x^p+y^2)^{\Delta+2}}\nonumber\\
=& 32T_5\pi^{5/2}\Delta c_\Delta C_\ell \frac{\Gamma(\Delta+1/2)}{\Gamma(\Delta+2)}
\xi^2\int^\infty_0dy \frac{y^{\Delta-2}}{((\kappa y-\xi)^2+y^2)^{\Delta+1/2}}.
\label{Integrated}\end{aligned}$$ In the above calculation we used the formula. $$\int d^Dx\frac{1}{(x^2+A)^\alpha}=\frac{\Gamma(-D/2+\alpha)}{\Gamma(\alpha)}\frac{\pi^{D/2}}{A^{-D/2+\alpha}}.$$ In our unit the D5-brane tension is written as $T_5=\frac{2N\sqrt{\lambda}}{(2\pi)^4}$. Finally by substituting $T_5$, $c_{\Delta}$ and $\Delta=2\ell$ to eq. , and the change of valuable as $y=\xi u$, we obtain $$\begin{aligned}
-\frac{\delta S_\text{cl}}{\delta s_0(\xi)}
&= C_\ell\frac{\sqrt{\lambda}2^{\ell}\Gamma(2\ell+1/2)}{\pi^{3/2}\sqrt{2\ell}\Gamma(2\ell)} \frac{1}{\xi^{2\ell}}
\int_0^\infty du
\frac{u^{2\ell -2}}{\Big[(1-\kappa u)^2+u^2\Big]^{2\ell+1/2}}\label{AppendixGravityResult}.\end{aligned}$$ \[2\][\#2]{}
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[^1]: nagasaki \[at\] het.phys.sci.osaka-u.ac.jp
[^2]: yamaguch \[at\] het.phys.sci.osaka-u.ac.jp
[^3]: Precisely speaking the right hand side is symmetrized product.
[^4]: The case $n=1$ is well known.
[^5]: This expansion is correct for $\ell\ge 2$
|
---
author:
- '[Alexander Litvinenko]{}'
- '[Hermann G. Matthies]{} [^1]'
bibliography:
- '\\thebib/jabbrevlong.bib'
- '\\thebib/matthies\_BU\_paper-1.bib'
- '\\thebib/phys\_D.bib'
- '\\thebib/fa.bib'
- '\\thebib/risk.bib'
- '\\thebib/fuq-new.bib'
- '\\thebib/highdim.bib'
title: '[Inverse problems and uncertainty quantification]{}[^2]'
---
\[2003/12/01\]
[ `` ]{}
[^1]: Corresponding author: D-38092 Braunschweig, Germany, e-mail: `[email protected]`
[^2]: Partly supported by the Deutsche Forschungsgemeinschaft (DFG) through SFB 880.
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abstract: 'We revisit the calculation of nonfactorizable corrections induced by charm-quark loops in exclusive FCNC $B$-decays. For the sake of clarity, we make use of a field theory with scalar particles: this allows us to focus on the conceptual issues and to avoid technical complications related to particle spins in QCD. We perform a straightforward calculation of the appropriate correlation function and show that it requires the knowledge of the full generic three-particle distribution amplitude with non-aligned arguments, $\langle 0|\bar s(y)G_{\mu\nu}(x)b(0)|B(p)\rangle$. Moreover, the dependence of this quantity on the variable $(x-y)^2$ is essential for a proper account of the $\left(\Lambda_{\rm QCD}m_b/m_c^2\right)^n$ terms in the amplitudes of FCNC $B$-decays.'
author:
- 'Anastasiia Kozachuk$^{a}$ and Dmitri Melikhov$^{a,b,c}$'
title: 'Revisiting nonfactorizable charm-loop effects in exclusive FCNC $B$-decays'
---
Introduction {#Sec_introduction}
============
The interest in the contribution of virtual charm loops in rare FCNC semileptonic and radiative leptonic decays of the $B$-mesons is two-fold: (i) although CKM-suppressed, the effect of the virtual charm-quark loops, including the narrow charmonia states which appear in the physical region of the $B$-decay, has a strong impact on the $B$-decay observables [@neubert] thus providing an unpleasant “noise” for the analysis of possible new physics effects; (ii) it is known that in the charmonia region, nonfactorizable gluon exchanges dominate the amplitudes posing a challenging QCD problem.
A number of theoretical analyses of nonfactorizable effects induced by charm-quark contributions has been published in the literature. We will mention here only those that are directly related to the discussion of this letter: In [@voloshin], an effective gluon-photon local operator describing the charm-quark loop has been calculated for the real photon as an expansion in inverse charm-quark mass $m_c$ and applied to inclusive $B\to X_s\gamma$ decays; Ref. [@buchalla] obtained a nonlocal effective gluon-photon operator for the virtual photon (i.e. without expanding in inverse powers of $m_c$) and applied it to inclusive $B\to X_s l^+l^-$ decays. In [@khod1997] nonfactorizable corrections in exclusive FCNC $B\to K^*\gamma$ decays using local OPE have been studied; in [@zwicky1; @zwicky3], these corrections have been analyzed with light-cone sum rules using local OPE for the photon-gluon operator and three-particle light-cone distribution amplitudes of $K^*$-meson. As emphasized in [@voloshin; @buchalla; @ligeti; @paz; @hidr], local OPE for the charm-quark loop leads to a power series in $\Lambda_{\rm QCD} m_b/m_c^2$. This parameter is of order unity for the physical masses of $c$- and $b$-quarks and thus corrections of this type require resummation. The authors of [@hidr] derived a different form of the nonlocal photon-gluon operator compared to [@buchalla] and evaluated its effect at small values of $q^2$ ($q$ momentum of the lepton pair) making use of light-cone 3-particle DA (3DA) of the $B$-meson with the aligned arguments, $\langle 0|\bar s(y)G_{\mu\nu}(uy) b(0)|B_s(p)\rangle$.
The goal of this letter is to emphasize that the full consistent resummation of $\left(\Lambda_{\rm QCD} m_b/m_c^2\right)^n$ terms in the nonfactorizable amplitude requires a more complicated object, $\langle 0| \bar s(y)G_{\mu\nu}(x) b(0)|B_s(p)\rangle$, i.e., a generic 3DA with non-aligned coordinates.
We perform the analysis using a field theory with scalar quarks/gluons which is technically very simple and allows one to focus on the conceptual issues; the generalization of our analysis for QCD is straightforward. We calculate nonfactorizable corrections directly, keeping control over all approximations. We adopt the counting scheme in which the parameter $\Lambda_{\rm QCD}m_b/m_c^2$ is kept of order unity, and show that the full 3DA is necessary in order to resum properly the $(\Lambda_{\rm QCD}m_b/m_c^2)^n$ corrections: the dominant contribution to the $B$-decay amplitude are generated not only by the light-cone terms $y^2=0$ and $x^2=0$, but also by terms of order $\sim (xy)^n$. Therefore, the dominant contributions to the $B$-decay amplitude come from the configurations when both $x$ and $y$ lie on the light cone, but on the different axes: if $x$ is aligned along the $(+)$-axis, then $y$ is aligned along the $(-)$-axis.
Expressing the $B$-decay amplitude via the standard 3DA with the aligned arguments, one can resum only a part of the $(\Lambda_{\rm QCD}m_b/m_c^2)^n$ corrections, whereas another source of the corrections of the same order remains unaccounted.
Nonfactorizable corrections in a field theory with scalar particles {#Sec_model}
===================================================================
In order to exemplify the details of the calculation, we consider nonfactorizable effects for the case of spinless particles. We shall use the standard QCD notations for spinor fields and assume that $m_b\gg m_c \gg m_s\sim \Lambda_{\rm QCD}$, but the parameter $\Lambda_{\rm QCD}m_b/m_c^2$ is of order unity.
We study the amplitude $$\begin{aligned}
\label{Apqoriginal}
A(p,q)=i\,\int dz e^{i q z}\langle 0|T\{c^\dagger(z)c(z),s^\dagger(0)s(0)\}|B_s(p\rangle, \end{aligned}$$ which involves weak interactions. We want to study nonfactorizable corrections due to a soft-gluon exchange between the charm-quark loop and the $B$-meson loop. To lowest order, the corresponding amplitude is given by the diagram of Fig. \[Fig:1\]: $$\begin{aligned}
\label{Apqscalar}
A(p,q)=i\,\int dz e^{iq z}\langle 0|T\{c^\dagger(z)c(z),
\,i\int dy' \,L_{\rm weak}(y'),
\,i\int dx\, L_{Gcc}(x),
\,s^\dagger(0)s(0)\}|B_s(p\rangle, \end{aligned}$$ where the effective Lagrangian that mimics weak four-quark interaction is taken in the form $$\begin{aligned}
L_{\rm weak}=\frac{G_F}{\sqrt2}\, s^\dagger b\,c^\dagger c,\end{aligned}$$ and the scalar gluon field $G(x)$ couples to the scalar $c$-quarks via the interaction $$\begin{aligned}
L_{\rm Gcc}=G\,c^\dagger c,\end{aligned}$$ i.e., $G$ involves the quark-gluon coupling.
First, we consider the charm-quark loop with the emission of a soft scalar gluon. We use the gluon field in momentum representation, related to the gluon field in coordinate representation as $$\begin{aligned}
G(x)=\frac{1}{(2\pi)^4}\int d\kappa \,\tilde G(\kappa)\,e^{i\kappa x},\quad \tilde G(\kappa)=\int dx \,G(x)\,e^{-i\kappa x}. \end{aligned}$$ Then the effective operator describing the gluon emission from the charm quark loop may be written as $$\begin{aligned}
\label{t1}
{\cal O}(q)=\int d\kappa\, \tilde G(\kappa)\,\Gamma_{cc}(\kappa,q), \end{aligned}$$ where $\Gamma_{cc}(\kappa,q)$ stands for the contribution of two triangle diagram with the charm quark running in the loop. The momenta $\kappa$ and $q$ are outgoing from the charm-quark loop, whereas the momentum $q'=q+\kappa$ is emitted from the $b\to s$ vertex. $p'$ is the momentum of the outgoing $s^\dagger s$ current and $p$ is the momentum of the $B$-meson, $p=p'+q$.
In terms of the gluon field operator in coordinate space, we can rewrite (\[t1\]) as $$\begin{aligned}
\label{t2}
{\cal O}(q)=\int d\kappa\, e^{-i\kappa x}dx\,G(x)\Gamma_{cc}(\kappa, q), \end{aligned}$$
By virtue of (\[t2\]), the amplitude Eq. (\[Apqscalar\]) takes the form $$\begin{aligned}
\label{A}
A(q,p)&=&\frac{1}{(2\pi)^8}\int \frac{dk}{m_s^2-k^2}
\int dy e^{-i(k-p')y}
\int dx e^{-i\kappa x}d\kappa\,
\Gamma_{cc}(\kappa, q)\,\langle 0|\bar s(y)G(x) b(0)|B_s(p)\rangle. \end{aligned}$$ Here, we encounter the $B$-meson three-particle amplitude with three different (non-aligned) arguments, for which we may write down the following decomposition: $$\begin{aligned}
\label{3DAnew}
\langle 0|s^\dagger(y)G(x) b(0)|B_s(p)\rangle=
\int d\lambda e^{-i \lambda y p}
\int d\omega e^{-i \omega x p}\,
\left[\Phi(\lambda,\omega)+O\left(x^2,y^2,(x-y)^2\right)\right], \end{aligned}$$ where $\lambda$ and $\omega$ are dimensionless variables. Making use of the properties of Feynman diagrams, it may be shown that they should run from 0 to 1. However, if one of the constituents is heavy, it carries the major fraction of the meson momentum and as the result the function $\Phi(\lambda,\omega)$ is strongly peaked in the region $$\begin{aligned}
\label{peaking}
\lambda, \omega=O(\Lambda_{\rm QCD}/m_b). \end{aligned}$$ So, effectively one can run the $\omega$ and $\lambda$ integrals from 0 to $\infty$; the latter integration limits emerge in the DAs within heavy-quark effective theory [@hidr; @japan]. We emphasize that for the results presented below only peaking of the DAs in the region (\[peaking\]) is essential. Notice also that the function $\Phi(\lambda,\omega)$ in (\[3DAnew\]) coincides with the same function that appears in the “standard” 3-particle distribution amplitude with the aligned arguments, $x=u y$, discussed in [@japan].
Light-cone contribution
-----------------------
First, let us calculate the contribution to $A(q,p)$ from the term given by $\Phi(\lambda,\omega)$ in the 3DA (\[3DAnew\]), i.e. corresponding to $x^2=y^2=(x-y)^2=0$. After inserting (\[3DAnew\]) into (\[A\]) we can perform the $x-$ and $y-$integrals $$\begin{aligned}
\label{xyint}
&&\int dx\to\delta(\kappa +\omega p), \qquad \nonumber\\
&&\int dy\to\delta(k+\lambda p-p'). \end{aligned}$$ The next step is easy: the $\delta$-functions above allow us to take integrals over $k$ and $\kappa$, and we find $$\begin{aligned}
\label{Aqp}
A(q,p)=\int_0^\infty d\lambda \int_0^\infty d\omega\, \Phi(\lambda,\omega)
\Gamma_{cc}\left(-\omega p, q \right)\frac{1}{m_s^2-(\lambda p-p')^2}. \end{aligned}$$ For the sum of two triangle diagrams with the charm quark running in the loop we may use the representation $$\begin{aligned}
\label{tFeyn}
\Gamma_{cc}(\kappa, q)=\frac{1}{8\pi^2}
\int\limits_{0}^{1}du \int\limits_{0}^{1-u}dv \,
\frac{1}{m_c^2-2uv \kappa q -u(1-u)\kappa^2-v(1-v)q^2}. \end{aligned}$$ Now, we must take into account that the $\omega$-integral is peaked at $\omega\sim \Lambda_{\rm QCD}/m_b$ so the gluon is soft: $\kappa=- \omega p$ and $\kappa^2\sim O(\Lambda_{\rm QCD}^2)\ll m_c^2$. The momentum transferred in the weak-vertex is $q'=q+\kappa=q-\omega p$, such that $$\begin{aligned}
q'^2=(q-\omega p)^2=q^2-\omega (1-\omega )M_B^2-q^2\omega +p'^2 \omega =q^2-\omega(1-\omega)M_B^2.\end{aligned}$$ By virtue of the $y$-integration in (\[xyint\]), the $s$-quark propagator takes the form $$\begin{aligned}
m_s^2-(\lambda p-p')^2=m_s^2-\lambda q^2+(1-\lambda)(\lambda M_B^2-{p'}^2).\end{aligned}$$ Therefore, in the bulk of the $\lambda$-integration the virtuality of the $s$-quark propagator is large, $O(M_B)$. Let us notice that the $q^2$-dependence of the $s$-quark propagator is very mild and can be neglected; the main $q^2$-dependence of the amplitude $A(q,p)$ comes from the charm-quark loop.
Deviations from the light-cone
------------------------------
We now turn to the calculation of the contributions to $A(q,p)$ generated by terms $\sim x^2,y^2,(x-y)^2$ in the 3DA (\[3DAnew\]). The terms containing powers of 4-vectors $y$ and $x$ in the integral (\[A\]) can be calculated by parts integration leading to additional factors under the integrals: $$\begin{aligned}
y_\alpha\to \frac{k_\alpha}{\Lambda_{\rm QCD}m_b},\qquad
x_\alpha\to \frac{\{q_\alpha,\kappa_\alpha\}}{m_c^2}.\end{aligned}$$ Taking into account the results (\[xyint\]), we find the following relative contributions of the terms containing different powers of the coordinate variables: $$\begin{aligned}
y^2 &\to& \frac{k^2}{\Lambda_{\rm QCD}^2 m_b^2}\sim \frac{1}{\Lambda_{\rm QCD}m_b},\nonumber\\
x^2&\to& \frac{q\kappa}{m_c^4}\sim \frac{\Lambda_{\rm QCD} m_b}{m_c^4},\nonumber\\
xy&\to& \frac{(p'-\lambda p )(q-\omega p)}{\Lambda_{\rm QCD}m_b m_c^2}\sim \frac{m_b}{\Lambda_{\rm QCD}m_c^2}.\end{aligned}$$ Clearly, all terms containing powers of $x^2$ and/or $y^2$ in the 3DA lead to the suppressed contributions to $A(q,p)$ and may be neglected within the considered accuracy. However, the terms containing powers of $xy$ lead to the contributions containing powers of ${\Lambda_{\rm QCD}m_b}/{m_c^2}$, i. e., of order unity within the adopted counting rules. The kinematics of the process is thus rather simple: the vectors $x$ and $y$ are directed along the light-cone \[e.g., $x$ along the $(+)$ axis, and $y$ along the $(-)$ axis\], but the 4-vector $x-y$ is obviously not directed along the light cone. Therefore, the full dependence of 3DA on the variable $(x-y)^2$ is needed in order to properly resum corrections of order $\left(\Lambda_{\rm QCD}m_b/m_c^2\right)^n$.
Conclusions
===========
We have revisited the calculation of nonfactorizable charm-loop effects in rare FCNC $B$-decays. To put emphasis on the conceptual aspects and to make the discussion clearer, we have considered the case of all scalar particles, avoiding in this way conceptually unimportant technical details. Our conclusions are as follows:
- The relevant object that arises in the calculation of the nonfactorizable corrections is the three-particle DA with non-aligned coordinates: $$\begin{aligned}
\label{res1}
\langle 0|s^\dagger(y)G(x)b(0)|B_s(p)\rangle=
\int d\lambda e^{-i \lambda y p}
\int d\omega e^{-i \omega x p}\left[\Phi(\omega,\lambda)+O\left(x^2,y^2,(x-y)^2\right)\right]. \end{aligned}$$ The function $\Phi(\omega,\lambda)$ here is precisely the same function that parameterizes the standard 3DA with the aligned arguments, $x=uy$, discussed in [@japan]. At small $q^2\le m_c^2$, terms of order $\sim x^2,y^2$ yield the suppressed contributions to the nonfactorizable amplitude of $B$-decay compared to the contribution of the light-cone term in the three-particle DA: for terms $O(x^2)$ the suppression parameter is $\Lambda_{\rm QCD}^2/m_c^2$, and for terms $O(y^2)$ the suppression parameter is $\Lambda_{\rm QCD}/m_b$. However, terms $\sim (xy)^n$ in the 3DA yield the contributions of order $\left(\Lambda_{\rm QCD}m_b/m_c^2\right)^n$ in the $B$-meson amplitude, i. e., to the unsuppressed contributions. These contributions have the same order as the difference between the local OPE [@voloshin] and the light-cone OPE [@hidr] and should be properly resummed. The kinematics of the process looks simple: the 4-vectors $x$ and $y$ are directed along the light-cone \[e.g., $x$ along the $(+)$-axis, and $y$ along the $(-)$-axis\], but the 4-vector $x-y$ is obviously not directed along the light cone; therefore, the full dependence of the 3DA (\[res1\]) on the variable $(x-y)^2$ is needed in order to properly resum corrections of order $\left({\Lambda_{\rm QCD}m_b}/{m_c^2}\right)^n$.
- Evidently, a consistent treatment of nonfactorizable charm-loop effects in FCNC $B$-decays in QCD also requires the consideration of generic $B$-meson three-particle distribution amplitudes with non-aligned coordinates, $$\begin{aligned}
\label{3DA_QCD}
\langle 0|\bar s(y)G_{\mu\nu}(x)b(0)|B_s(p)\rangle.\end{aligned}$$ The Wilson lines between the field operators, making this quantity gauge-invariant, are implied. Notice that in QCD the complications of the generic 3DA (\[3DA\_QCD\]) compared to the 3DA with the aligned arguments are two-fold: (i) first, similar to the case of scalar constituents considered above, in each Lorentz structure that parametrizes (\[3DA\_QCD\]) one has to take into account terms $\sim (xy)^n$ that yield the unsuppressed contributions to the $B$-decay amplitude; (ii) second, (\[3DA\_QCD\]) contains additional Lorentz structures $\sim (x-y)_\alpha$ and $\sim (x+y)_\alpha$ compared to the 3DA with the aligned arguments (see, e.g., Eq. (4.7) of [@hidr]), and, respectively, new distribution amplitudes.
- It seems plausible to infer that when considering non-factorizable gluon corrections in meson-to-vacuum transition amplitudes of the type $\langle 0|T \{j_1(z) j_2(0)\}|B\rangle$ (or similar amplitudes with light meson in the final state), one encounters two distinct kinds of processes:
I. The amplitude of the process involves only one quark loop, i.e., the external boson is emitted from the same quark loop that contains valence quarks of the initial and the final mesons. In this case, non-factorizable corrections are light-cone dominated, i.e. may be expressed via light-cone three-particle distribution amplitude of the initial or of the final meson. For instance, weak form factors of $B$-meson decays treated within the method of light-cone sum rules (see e.g. [@lcsr1; @lcsr2]) belong to this kind of processes.
II\. The amplitude of the process involves two separate quark loops (one quark-loop involving valence quarks of the initial and the final mesons and another quark loop that emits the external boson). In this case, the soft gluon from the initial or the final meson vertex is absorbed by a quark in a different loop. In this case, the description of non-factorizable soft-gluon corrections requires the full three-particle DA with non-aligned coordinates of the type of (\[res1\]). Non-factorizable corrections to FCNC decays due to $c$- or $u$-quark loops belong to this kind of processes.
A more detailed investigation of the general properties of non-factorizable corrections seems worthwhile.
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---
abstract: 'Shallow water wave phenomena find their analogue in optics through a nonlocal nonlinear Schrödinger (NLS) model in $(2+1)$-dimensions. We identify an analogue of surface tension in optics, namely a single parameter depending on the degree of nonlocality, which changes the sign of dispersion, much like surface tension does in the shallow water wave problem. Using multiscale expansions, we reduce the NLS model to a Kadomtsev-Petviashvilli (KP) equation, which is of the KPII (KPI) type, for strong (weak) nonlocality. We demonstrate the emergence of robust optical antidark solitons forming Y-, X- and H-shaped wave patterns, which are approximated by colliding KPII line solitons, similar to those observed in shallow waters.'
author:
- 'Theodoros P. Horikis'
- 'Dimitrios J. Frantzeskakis'
title: 'Light meets water in nonlocal media: Surface tension analogue in optics'
---
Many physically different contexts can be brought together through common modeling and mathematical description. A common (and rather unlike) example is water waves and nonlinear optics. Two models are inextricably linked with both subjects: the universal Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations [@MJA1]. Furthermore, these models can be reduced from one to the other [@zakharov], thus suggesting that phenomena occurring in water waves may also exist in optics. Here, using such reductions for a nonlocal NLS, we find that surface tension –which causes fluids to minimize the area they occupy– has a direct analogue in optics. Key to our findings are solitons, i.e., robust localized waves that play a key role in numerous studies in physics [@dp], applied mathematics [@BlackBook] and engineering [@hasko]. A unique property of solitons is that they feature a particle-like character, i.e., they interact elastically, preserving their shapes and velocities after colliding with each other. Such elastic collisions, as well as pertinent emerging wave patterns, can sometimes be observed even in everyday life. A predominant example is the one pertaining to flat beaches: in such shallow water wave settings, two line solitons merging at proper angles give rise to patterns of X-, H-, or Y-shaped waves, as well as other more complicated nonlinear waveforms [@PRE]. All these shallow water wave structures are actually exact analytical multi-dimensional line soliton solutions of the Kadomtsev-Petviashvilli (KP) equation (which generalizes KdV to two dimensions (2D) [@MJA1; @BlackBook]) of the KPII type – a key model in the theory of shallow water waves with weak surface tension [@MJA1]. The relative equation with strong surface tension is referred to as KPI.
Here we show that such patterns can also be observed in a quite different physical setting, i.e., the one related to optical beam propagation in media with a spatially nonlocal defocusing nonlinearity. Such media include thermal nonlinear optical media [@rot; @krol1], partially ionized plasmas [@litvak; @plasma], nematic liquid crystals [@ass0; @ass1], and dipolar bosonic quantum gases [@dipolar]. It is shown that approximate solutions of the nonlocal NLS model satisfy, at proper scales, equations that appear in the context of water waves: a Boussinesq or Benney-Luke (BL) [@BL], as well as a KP equation. For a relatively strong (weak) nonlocality, or background amplitude, the latter is found to be a KPII (KPI), similarly to the water wave problem, where a KPII (KPI) results in the case of small (large) surface tension [@MJA1]. Our results thus suggest an analogue of surface tension in optics. Direct numerical simulations show that approximate antidark line soliton solutions of the nonlocal NLS, constructed from the KPII line soliton solutions, form patterns observable in shallow water [@PRE]. Pertinent Y-, X-, or H-wave patterns may be realized in an experimental setup similar to the one used for the observation of antidark solitons [@segev2].
The evolution of optical beams in nonlinear defocusing media is governed by the following paraxial wave equation (cf. Ref. [@kivshar_book] for derivation and relevant adimensionalizations): $$\begin{aligned}
i u_{t} + \frac{1}{2} \Delta u-nu =0,
\label{NLS1}\end{aligned}$$ where subscripts denote partial derivatives, $u$ is the complex electric field envelope, $\Delta\equiv\partial_x^2+\partial_y^2$ is the transverse Laplacian, and real function $n$ denotes the nonlinear, generally nonlocal, medium response. For instance, in optics, $n$ is the nonlinear change of the refractive index depending on the intensity $I=|u|^2$ [@rot; @krol1], in plasmas is the relative electron temperature perturbation [@litvak; @plasma], in liquid crystals is the optically induced angle perturbation [@ass0; @ass1], and so on. Here, we consider that $n$ obeys the following diffusion-type equation: $$\begin{aligned}
d^2\Delta n - n + |u|^2 = 0,
\label{NLS2}\end{aligned}$$ where $d$ is a spatial scale (setting the diffusion length) that measures the degree of nonlocality. Note that for $d = 0$, Eqs. (\[NLS1\])-(\[NLS2\]) reduce to the defocusing 2D NLS equation [@kivshar_book]. Importantly, Eqs. (\[NLS1\])-(\[NLS2\]), the nonlocal NLS model of principal interest herein, has been used satisfactorily to model experiments on liquid solutions exhibiting thermal nonlinearities [@liq1; @liq3], while it has also been used in studies of plasmas [@litvak; @plasma] and nematic liquid crystals [@ass0; @ass1].
The steady-state solution of Eqs. (\[NLS1\])-(\[NLS2\]) is composed by the continuous wave (cw), $u=u_0\exp(-i|u_0|^2 t)$ ($u_0$ being an arbitrary complex constant), and the constant function $n=|u_0|^2$. Considering small perturbations of this solution behaving like $\exp[i(\boldsymbol{k}
\cdot \boldsymbol{r} -\omega t)]$, with $\boldsymbol{r}=(x,~y)$, we find that the perturbations’ wavevector $\boldsymbol{k}=(k_x,~k_y)$ and frequency $\omega$ obey the dispersion relation: $$\omega^2=|\boldsymbol{k}|^2 C^2 \left(1+d^2|\boldsymbol{k}|^2\right)^{-1}
+(1/4)|\boldsymbol{k}|^4,
\label{dr}$$ where $C^2=|u_0|^2$ is the wave velocity. Here it is important to observe the following. First, since $\omega \in \mathbb{R}$ $\forall ~\boldsymbol{k}$, the steady-state solution is modulationally stable. Second, in the long-wavelength limit ($|\boldsymbol{k}|^2 \ll1$), Eq. (\[dr\]) becomes $\omega^2 \approx |\boldsymbol{k}|^2 C^2+(1/4) \alpha |\boldsymbol{k}|^4$, where $\alpha = 1-4d^2|u_0|^2$. This approximate dispersion relation features a striking similarity to the corresponding (approximate) one for shallow water waves, namely [@MJA1]: $\omega^2 \approx |\boldsymbol{k}|^2 c_0^2+(1/3) (3\hat{T}-1) c_0^2 h^2 |\boldsymbol{k}|^4$, where $c_0^2 = gh$ is the velocity, $g$ is the acceleration of gravity, $h$ the depth of water at rest, and $\hat{T}=T/(\rho g h^2)$, with $\rho$ being the density and $T$ the surface tension. Comparing these dispersion relations, the following correspondence is identified: $3\hat{T} \rightarrow 4d^2|u_0|^2$, implying that there exists a surface tension analogue in our problem, $\propto d^2|u_0|^2$. This effective surface tension is negative, as is also implied by the fact that the term $\propto d$ in the Hamiltonian $\mathcal{H}=(1/2) \int_{\mathbb{R}^2}
\left(|\boldsymbol{\nabla} u|^2-d^2(\boldsymbol{\nabla} n)^2-n^2+2n|u|^2\right)
{\rm d}\boldsymbol{r}$ of Eqs. (\[NLS1\])-(\[NLS2\]) decreases the potential energy of the system, oppositely to the water wave case where surface tension increases the respective potential energy [@zakh].
These arguments can further be solidified by analyzing the fully nonlinear problem: similarly to water waves [@MJA1], we will derive KPI and KPII equations, depending on the strength of the effective surface tension, i.e., the parameter $\alpha$, which sets the dispersion coefficient in KP. This can already be identified from the linear theory as follows. Using $|\boldsymbol{k}|^2 =k_x^2+k_y^2$, the long-wavelength limit of Eq. (\[dr\]) reads: $\omega= \pm C k_x\left[1+(k_y/k_x)^2\right]^{1/2} \left[1+(\alpha/4C^2) k_x^2+
\mathcal{O}(k_y^2)\right]$, with $\pm$ corresponding to right- and left-going waves. Assuming $|k_y/k_x|\ll 1$ and $k_y^2 \sim \mathcal{O}(k_x^4)$, we find: $(1/C)\omega k_x = \pm \left[ k_x^2 + (\alpha/4C^2) k_x^4 +(1/2)k_y^2\right]$. Then, using $\omega \rightarrow i\partial_t$, $k_{x,y} \rightarrow -i\partial_{x,y}$, the linear PDE associated to this dispersion relation is: $\partial_x [\pm q_t+ C q_x - (\alpha/8C)q_{xxx}] +(C/2)q_{yy}=0$. This is a linear KP equation, with a dispersion coefficient depending on the effective surface tension through $\alpha$, similarly to shallow water waves, where the respective dispersion coefficient depends on $\hat{T}$ [@MJA1]. To derive the full nonlinear version of the KP equation, we resort to multiple scales. We thus consider small-amplitude slowly-varying modulations of the steady state, and seek solutions of Eqs. (\[NLS1\])-(\[NLS2\]) in the form of the asymptotic expansions: $$\begin{aligned}
u&=&u_0\sqrt{\rho}\exp(-i|u_0|^2 t + i \epsilon^{1/2}\Phi),
\label{u} \\
\rho&=& 1+\sum_{j=1}^{\infty}\epsilon^j \rho_j, \quad
n=|u_0|^2 + \sum_{j=1}^{\infty}\epsilon^j n_j,
\label{n}\end{aligned}$$ where $0<\epsilon \ll1$ is a formal small parameter, while phase $\Phi$ and amplitudes $\rho_j$ and $n_j$ are unknown real functions of the slow variables $X=\epsilon^{1/2}x$, $Y=\epsilon^{1/2}y$ and $T=\epsilon^{1/2}t$. Substituting the expansions (\[u\])-(\[n\]) into Eqs. (\[NLS1\])-(\[NLS2\]), we obtain the following results. First, the real part of Eq. (\[NLS1\]) and Eq. (\[NLS2\]) yield the leading-order equations, at $\mathcal{O}(\epsilon^{3/2})$ and $\mathcal{O}(\epsilon)$: $$\begin{aligned}
\rho_{1T} +\tilde{\Delta}\Phi=0, \quad n_1=|u_0|^2\rho_1,
\label{lead}\end{aligned}$$ and the first-order equations, at $\mathcal{O}(\epsilon^{5/2})$ and $\mathcal{O}(\epsilon^2)$: $$\begin{aligned}
\rho_{2T}+\tilde{\boldsymbol{\nabla}} \cdot (\rho_1 \tilde{\boldsymbol{\nabla}}\Phi)=0,
\quad
d^2 \tilde{\Delta}n_1-n_2 +|u_0|^2\rho_2=0,
\label{first}\end{aligned}$$ connecting the amplitudes $\rho_{1,2}$ and $n_{1,2}$ with the phase $\Phi$; here, $\tilde{\Delta}
\equiv \partial_X^2 + \partial_Y^2$ and $\tilde{\boldsymbol{\nabla}}\equiv (\partial_X,~\partial_Y)$. Second, the imaginary part of Eq. (\[NLS1\]), combined with Eqs. (\[lead\])-(\[first\]), yields: $$\begin{aligned}
&&\Phi_{TT}-C^2\tilde{\Delta}\Phi + \epsilon
\left[ \frac{1}{4}\alpha \tilde{\Delta}^2\Phi
+ \frac{1}{2}\partial_T(\tilde{\boldsymbol{\nabla}}\Phi)^2
\right. \nonumber \\
&&\left.
+ \tilde{\boldsymbol{\nabla}} \cdot (\Phi_T \tilde{\boldsymbol{\nabla}}\Phi)
\right]=\mathcal{O}(\epsilon^2),
\label{B}\end{aligned}$$ Equation (\[B\]) incorporates 4th-order dispersion and quadratic nonlinear terms, resembling the Boussinesq and Benney-Luke [@BL] equations, which describe bidirectional shallow water waves [@MJA1]. Similarly to the water wave problem, we now use a multiscale expansion method to derive the KP equation, under the additional assumptions of quasi-two-dimensionality and unidirectional propagation. In particular, we introduce the asymptotic expansion $\Phi=\Phi_0+\epsilon \Phi_1 +\cdots$, where functions $\Phi_{\ell}$ ($\ell=0,1,\ldots$) depend on the variables $\xi=X-CT$, $\eta=X+CT$, $\mathcal{Y}=\epsilon^{1/2}Y$, and $\mathcal{T}=\epsilon T$. Substituting this expansion into Eq. (\[B\]), at the leading-order in $\epsilon$, we obtain the wave equation $\Phi_{0\xi
\eta}=0$, implying that $\Phi_0$ can be expressed as a superposition of a right-going wave, $\Phi_0^{(R)}$, depending on $\xi$, and a left-going one, $\Phi_0^{(L)}$, depending on $\eta$, namely: $$\Phi_{0}=\Phi_0^{(R)}(\xi,\mathcal{Y},\mathcal{T})+\Phi_0^{(L)}(\eta,\mathcal{Y},\mathcal{T}).
\label{rl}$$ In addition, at order $\mathcal{O}(\epsilon)$, we obtain the equation: $$\begin{aligned}
&&4C^2\Phi_{1\xi\eta} = -C\left(\Phi_{0\xi\xi}^{(R)}\Phi_{0\eta}^{(L)}
-\Phi_{0\xi}^{(R)}\Phi_{0\eta\eta}^{(L)} \right) \nonumber \\
&&+\left[\partial_{\xi}
\left(-2C\Phi_{0\mathcal{T}}^{(R)} +\frac{\alpha}{4}\Phi_{0\xi\xi\xi}^{(R)}
-\frac{3C}{2}\Phi_{0\xi}^{(R)2}\right)
-C^2\Phi_{0\mathcal{Y}\mathcal{Y}}^{(R)}\right]
\nonumber \\
&&+ \left[\partial_{\eta}\left(2C\Phi_{0\mathcal{T}}^{(L)}
+\frac{\alpha}{4}\Phi_{0\tilde{\eta}\tilde{\eta}\tilde{\eta}}^{(L)}
+\frac{3C}{2}\Phi_{0\tilde{\eta}}^{(L)2}
\right)-C^2\Phi_{0\mathcal{Y}\mathcal{Y}}^{(L)}
\right].
\nonumber $$ When integrating this equation, secular terms arise from the square brackets, which are functions of $\xi$ or $\eta$ alone, not both. Removal of these secular terms leads to two uncoupled nonlinear evolution equations for $\Phi_0^{(R)}$ and $\Phi_0^{(L)}$. Then, using $\Phi_T=-n_1$, obtained from the leading-order part of Eq. (\[B\]) together with Eq. (\[first\]), it is found that the amplitude $\rho_1$ can also be decomposed to a left- and a right-going wave, i.e., $\rho_1 =
\rho_1^{(R)}+\rho_1^{(L)}$, which satisfy the following KP equations: $$\begin{aligned}
\partial_{\mathcal{X}}\left(\pm \rho_{1\mathcal{T}}^{(R,L)}
-\frac{\alpha}{8C}\rho_{1\mathcal{X}\mathcal{X}\mathcal{X}}^{(R,L)}
+\frac{3C}{4}\rho_{1}^{(R,L)2} \right)
+\frac{C}{2}\rho_{1\mathcal{Y}\mathcal{Y}}^{(R,L)}=0,
\nonumber
$$ where $\mathcal{X}=\xi$ ($\mathcal{X}=\eta$) for the right- (left-) going wave. Next, for right-going waves, we use the transformations $ \mathcal{T} \rightarrow -(\alpha/8C)\mathcal{T}$, $\mathcal{Y} \rightarrow \sqrt{3|\alpha|/4C^2}\mathcal{Y}$, and $\rho_1^{(R)}=-(\alpha/2C^2) U$, and express KP in its standard dimensionless form [@BlackBook; @MJA1]: $$\begin{aligned}
\partial_{\mathcal{X}}\left(U_{\mathcal{T}}+6UU_{\mathcal{X}}
+U_{\mathcal{X}\mathcal{X}\mathcal{X}}\right)
+3\sigma^2 U_{\mathcal{Y}\mathcal{Y}}=0,
\label{usKP}\end{aligned}$$ where $\sigma^2=-\operatorname{sgn}{\alpha}=\operatorname{sgn}(4d^2|u_0|^2-1)$. Importantly, Eq. (\[usKP\]) includes both versions of the KP equation, KPI and KPII [@BlackBook]. Indeed, for $\sigma^2=1\Rightarrow\alpha<0$, i.e., for $4d^2 |u_0|^2>1$, Eq. (\[usKP\]) is a KPII equation; on the other hand, for $\sigma^2=-1\Rightarrow\alpha>0$, i.e., $4d^2 |u_0|^2<1$, Eq. (\[usKP\]) is a KPI equation. Thus, for a fixed cw intensity $|u_0|^2$, a strong (weak) nonlocality $d^2$, as defined by the above inequalities, corresponds to KPII (KPI); the same holds for a fixed degree of nonlocality $d^2$, and a larger (smaller) cw intensity $|u_0|^2$. Thus, both our linear and nonlinear analysis establishes a “homeomorphism” between optics and shallow water waves: in this latter context, weak surface tension (typical for water waves) corresponds to $\sigma^2=1$ in Eq. (\[usKP\]) (i.e., to KPII), while strong surface tension is pertinent to $\sigma^2=-1$ (i.e., to KPI) [@MJA1; @BlackBook]. Based on the above analysis, we now utilize the exact solutions of the KP, Eq. (\[usKP\]), and construct approximate solutions of the original system of Eqs. (\[NLS1\])-(\[NLS2\]); such solutions read: $$\begin{aligned}
&u\approx u_0 \left(1-\epsilon \frac{\alpha}{2|u_0|^2}U \right)^{1/2} \exp\left(-i|u_0|^2 t\right)
\nonumber \\
&\times\exp\left(\frac{i}{2} \alpha \epsilon^{-1/2} \int_0^\mathcal{T} U {\rm
d}\mathcal{T'}\right), \quad
n \approx |u_0|^2 - \frac{1}{2}\alpha U \label{apsol2}.\end{aligned}$$ Clearly, for $\alpha<0$ ($\alpha>0$), i.e., for solutions satisfying KPII (KPI), $u$ in Eq. (\[apsol2\]) has the form of a hump (dip) on top (off) of the cw background and is, thus, a antidark (dark) soliton. Notice that in the local limit of $d=0$ (i.e., $\alpha=1$), we solely obtain the KPI model, in which line solitons are unstable: as was shown in plasma physics and hydrodynamics [@zakhpr], line solitons develop undulations and eventually decay into lumps [@infeld]. In the same venue, but now in optics, the asymptotic reduction of the defocusing 2D NLS to KPI [@kuz1; @peli1], and the instability of the line solitons of the latter, was used to better understand the transverse instability of rectilinear dark solitons: indeed, these structures also develop undulations and eventually decay into vortex pairs [@peli1; @pelirev].
Here, we focus on the stable soliton solutions of the KP equations, namely the (antidark) line solitons of the KPII equation and the (dark) lump of KPI. The one-line soliton solution, travelling at an angle to the $\mathcal{Y}$-axis, is: $$U(\mathcal{X},\mathcal{Y},\mathcal{T})= 2\kappa^2 {\rm sech}^2(\mathcal{Z}),$$ where $\mathcal{Z} \equiv \kappa \left[ \mathcal{X}+ \lambda \mathcal{Y} -
\left(4\kappa^2+3\lambda^2 \right) \mathcal{T} +\delta \right]$, with $\kappa$, $\lambda$ and $\delta$ being free parameters. On the other hand, the two-line soliton can be expressed in the following form:
$$\begin{gathered}
U= 2 \partial^2_\mathcal{X}\ln\left( 1+e^{\mathcal{Z}_1}+e^{\mathcal{Z}_2}
+e^{\mathcal{Z}_1+\mathcal{Z}_2+A_{12}}\right),
\\
\exp(A_{12})=\frac{4(\kappa_1-\kappa_2)^2-(\lambda_1-\lambda_2)^2}
{4(\kappa_1+\kappa_2)^2-(\lambda_1-\lambda_2)^2},\end{gathered}$$
where $\mathcal{Z}_i \equiv \kappa_i \left[ \mathcal{X}+ \lambda_i \mathcal{Y} -
\left(4\kappa_i^2+3\lambda_i^2 \right) \mathcal{T} +\delta_i \right]$.
As was shown and observed in the context of shallow water waves [@PRE], when two line solitons of the KPII intersect, a plethora of patterns can emerge. We focus here on the ones most frequently observed in shallow waters. To do this, fix $\epsilon=0.2$, $d^2=1/3$ and $u_0=1$ and choose two line solitons with specific parameters, so that the angle of interaction will lead to different patterns. We evolve these initial waves up to $t=600$ so that they have enough time to interact. In Fig. \[y\] we show the resonant interaction of two line solitons resulting into a Y-type wave. The parameters leading to these interactions are summarized in the figure captions.
![A Y-type interaction for $2\kappa_1=\kappa_2=1$ and $\lambda_1=3\lambda_2=\frac{1}{4}$.[]{data-label="y"}](y.jpg)
![(Color online) X-type interactions. Top: short stem, for $\kappa_1=\kappa_2=\frac{1}{2}$ and $\lambda_1=-\lambda_2=\frac{2}{3}$. Middle: intermediate stem, for $\kappa_1=\kappa_2=\frac{1}{2}$, $\lambda_1=-\frac{1}{4}-10^{-2}$, and $\lambda_2=\frac{3}{4}$. Bottom: long stem, for $\kappa_1=\kappa_2=\frac{1}{2}$ and $\lambda_1=-\lambda_1+10^{-10}=\frac{1}{2}$.[]{data-label="x"}](x.jpg "fig:") ![(Color online) X-type interactions. Top: short stem, for $\kappa_1=\kappa_2=\frac{1}{2}$ and $\lambda_1=-\lambda_2=\frac{2}{3}$. Middle: intermediate stem, for $\kappa_1=\kappa_2=\frac{1}{2}$, $\lambda_1=-\frac{1}{4}-10^{-2}$, and $\lambda_2=\frac{3}{4}$. Bottom: long stem, for $\kappa_1=\kappa_2=\frac{1}{2}$ and $\lambda_1=-\lambda_1+10^{-10}=\frac{1}{2}$.[]{data-label="x"}](stem1.jpg "fig:") ![(Color online) X-type interactions. Top: short stem, for $\kappa_1=\kappa_2=\frac{1}{2}$ and $\lambda_1=-\lambda_2=\frac{2}{3}$. Middle: intermediate stem, for $\kappa_1=\kappa_2=\frac{1}{2}$, $\lambda_1=-\frac{1}{4}-10^{-2}$, and $\lambda_2=\frac{3}{4}$. Bottom: long stem, for $\kappa_1=\kappa_2=\frac{1}{2}$ and $\lambda_1=-\lambda_1+10^{-10}=\frac{1}{2}$.[]{data-label="x"}](stem2.jpg "fig:")
X-type waves can also emerge, as shown in Fig. \[x\]. These structures are essentially discriminated by their “stems”: a short, intermediate and a long stem are respectively depicted in the top, middle and bottom panels of Fig. \[x\]; notice that the long stem’s height is higher than that of the incoming line solitons. In addition, we can produce long stem interactions where the stem height is lower than the tallest incoming line soliton, cf. Fig. \[h\]. We refer to these patterns as H-type.
![An H-type interaction with $2\kappa_1=\kappa_2=1$, $\lambda_1=\frac{1}{2}-10^{-7}$, and $\lambda_2=0$.[]{data-label="h"}](h.jpg)
While we chose the above patterns as they appear more frequently in water, other more exotic, web-like structures are also supported by the KPII equation [@sarby3; @sarby2; @sarby1], and may –in principle– also be produced in optics. Furthermore, these solutions, while approximate, also hold well beyond the small-amplitude limit: similar results (not shown here) were obtained even for $\epsilon =0.7$, with the only additional effect being emission of noticeable radiation. This is due to the robustness of KPII solitons, which is also verified by the observation of these patterns in shallow water, even after the waves break [@PRE]. From the viewpoint of experiments, observations of antidark solitons were reported in Refs. [@segev2; @Tang:16]. The Y-, X- and H-waves may be observed experimentally using a setup similar to that of Ref. [@segev2]. In particular, one may employ at first a cw laser beam, which is split into two parts via a beam-splitter. One branch goes through a cavity system to form a pulse (as happens in typical pulsed lasers); this pulse branch undergoes phase-engineering, i.e., passes through a phase mask so that the characteristic phase jump of the antidark soliton is inscribed. Then, the cw and the phase-engineered pulse are incoherently coupled inside the nonlocal medium, e.g., a nematic liquid crystal, described by Eqs. (\[NLS1\])-(\[NLS2\]). This process forms one antidark soliton, as in Ref. [@segev2]. To observe Y-, X- or H-patterns predicted above, two such antidark solitons have to be combined inside the crystal. The angle between the two incident beams, which should be appropriately chosen so that a specific pattern be formed, can be controlled by a rotating mirror in one of the branches.
Finally, let us consider the KPI case ($\alpha>0$). KPI also exhibits line soliton solutions, as above, which are however unstable; it is thus most known for its solution that decays algebraically in both spatial coordinates, i.e., the lump: $$\begin{gathered}
U(\mathcal{X},\mathcal{Y},\mathcal{T}) =
\nonumber \\
4 \frac{-(\mathcal{X} + a \mathcal{Y} + 3(a^2 - b^2)\mathcal{T})^2 + b^2 (\mathcal{Y}+6a \mathcal{T})^2
+ 1/b^2}
{[(\mathcal{X} + a \mathcal{Y} + 3(a^2 - b^2)\mathcal{T})^2 + b^2 (\mathcal{Y}+6a \mathcal{T})^2 + 1/b^2]^2}, \nonumber\end{gathered}$$ where $a,b$ are free real parameters. Lumps have not yet been observed in water. In Fig. \[lump\], we show a direct simulation for the dark lump, and we refer the reader to the recent work [@baronio.kp] for details on multi-lump solutions and their interactions.
![A typical lump solution with $a=0$, $b=1$.[]{data-label="lump"}](lump_new.jpg)
Concluding, we have established a homeomorphism between nonlocal nonlinear media and shallow water waves. In particular, we have shown that there exists a surface tension analogue in optics, depending on the nonlocality strength. This was identified from the linear theory and was rigorously analyzed in the fully nonlinear regime, by the asymptotic reduction of a nonlocal NLS system to the KP equations. We demonstrated that, by depending on the effective surface tension, i.e., the degree of nonlocality, novel structures can appear in optical nonlocal media. Thus, fascinating phenomena that appear in water waves can also be observed in optics; an example studied here is the emergence of X-, H-, or Y-shaped waves resulting from the resonant interactions of stable line antidark solitons that were found to exist in strongly nonlocal media. For weakly nonlocal ones, dark lumps were also predicted to occur. The structures predicted in this work may in principle be experimentally observed, either by choosing a material of specific nonlocality or in a specific material altering the magnitude of the cw background, in a setup similar to the one used for he observation of antidark solitons [@segev2].
We thank M. J. Ablowitz, P. J. Ioannou and E. P. Fitrakis for many useful discussions.
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|
---
author:
- |
Jan Lorenz\
Department of Mathematics\
University of Bremen\
Nienburgerstr. 42, 28205 Bremen, Germany\
email: [email protected]\
date: 'In Proceedings of IASTED Conference “Modelling, Simulation and Optimization MSO 2005 ”, August 29-31, 2005, Oranjestad, Aruba'
title: 'Continuous Opinion Dynamics: Insights through Interactive Markov Chains'
---
[**ABSTRACT**]{}
[**KEY WORDS**]{}
Introduction
============
Consider a certain number of agents discussing a certain issue. Each agent has an opinion about that issue and they try to find a consensus. The opinion should be representable as a real number. Thus, agents can compromise by averaging their opinions. We label such dynamics as [*continuous opinion dynamics*]{}. ’Continuous’ refers to the type of opinions not to the time. This is in contrast to several models of opinion dynamics which cover binary opinions where agents have to decide ’yes’ or ’no’.
Examples for continuous opinions are prices or the continuum form left to right in politics. Clusters of agents in the continuous opinion space may represent e.g. low-budget vs. luxury shoppers or political parties.
There are several more or less complex models concerning continuous opinion dynamics [@DeGroot1974; @JAP:chatterjee77; @Friedkin1990; @Flache2004]. The recently most discussed models are the models of Weisbuch-Deffuant (WD) [@Weisbuch2002] and Hegselmann-Krause (HK) [@Hegselmann2002]. Both models use very similar heuristics. We will present both models and focus on the dynamical differences in computer simulation.
In the third section we will define interactive Markov Chains for both models, which lead us to ’discrete’ bifurcation diagrams that will give us deeper insights for the agent-based models.
Two Agent-based Models
======================
Opinion dynamics in the WD and the HK model is agent-based and driven by repeated averaging under bounded confidence. The models differ in their communication regime.
Agent-based means that the number of agents $n$ is the dimension of the system. The opinion of agent $i \in {\underline{n}}:= \{1,\dots,n\}$ at the time $t\in{\mathbb N}$ is represented by $x_i(t) \in {\mathbb R}$. The vector $x(t) \in {\mathbb R}^n$ is called the *opinion profile* at time step $t$. Both models propose a *bound of confidence* ${\varepsilon}\in
{\mathbb R}_{>0}$. Agents compromise with other agents only if the difference between their opinions is less or equal than ${\varepsilon}$. We will define for both models the process of continuous opinion dynamics as a sequence of opinion profiles.
Given an initial profile $x(0) \in {\mathbb R}^n$ and a bound of confidence ${\varepsilon}\in{\mathbb R}_{>0}$ we define the *Weisbuch-Deffuant process of opinion dynamics* as the random process $(x(t))_{t\in{\mathbb N}_0}$ that chooses in each time step $t\in{\mathbb N}_0$ two random[^1] agents $i,j \in {\underline{n}}$ which perform the action $$\begin{aligned}
x_i(t+1) &=& \left\{
\begin{array}{ll}
\frac{x_i(t)+x_j(t)}{2} & \hbox{if $|x_i(t)-x_j(t)|\leq{\varepsilon}$} \\
x_i(t) & \hbox{otherwise.} \\
\end{array}
\right.\\\end{aligned}$$ The same for $x_j(t+1)$ with $i$ and $j$ interchanged.[^2]
The dynamics in the WD model are driven by pairwise compromising restricted by bounded confidence.
Given a bound of confidence ${\varepsilon}\in{\mathbb R}_{>0}$ we define for an opinion profile $x\in{\mathbb R}^n$ the *confidence matrix* $A(x)$ as $$a_{ij}(x,{\varepsilon}) :=
\left\{ \begin{array}{cl}
\frac{1}{\#I(i,x)} \quad & \textrm{if } j\in I(i,x) \\
0 & \textrm{otherwise,}
\end{array} \right. \\$$ with $I(i,x) := \{j \in {\underline{n}}\,|\, |x_{i} - x_{j}| \leq {\varepsilon}\}$. Given a starting opinion profile $x(0) \in {\mathbb R}^n$, we define the *Hegselmann-Krause process of opinion dynamics* as a sequence of opinion profiles $(x(t))_{t\in{\mathbb N}_0}$ recursively defined through $$x(t+1) = A(x(t)) x(t)$$ (“$\#$” stands for the number of elements.)
In the HK model each agent builds the arithmetic mean of all the opinions which are closer than ${\varepsilon}$ to his own.
![Example processes for the WD and HK model. With initial profile of 100 agents with random equally distributed opinions within $[0,1]$.[]{data-label="fig1"}](figure1.eps){width="\columnwidth"}
Figure \[fig1\] shows example processes for both models. In both models the processes converge to a stabilized opinion profile, where opinions have divided into a certain number of clusters. The stabilization can be proved analytically, even for a bigger class of models [@Lorenz2005]. The dynamical behavior can be described in the same way: The contractive force of compromising brings opinions together, but bounded confidence forces the agents to form clusters where higher local agent densities occur, thus confidence to other agents gets lost.
We will briefly summarize the known results, which we consider in this paper. At first, results about the WD model. The number of expected major clusters is roughly the integer part of $1/2{\varepsilon}$ [@Weisbuch2002]. But not all agents join the major clusters, some remain outliers or form minor clusters (like in figure \[fig1\] at the extremes).
In the HK model there are slightly lower numbers of expected clusters as in the WD model, for details see [@Urbig2004]. Convergence to consensus may take very long, due to very few agents remaining in the middle and attracting all other agents very slowly (like in figure \[fig1\] but it can be more drastic for larger $n$).
Both models were extended in several ways. The WD model to relative agreement, smooth bounded confidence, dynamics on small world networks [@Amblard2004], heterogeneous bounds of confidence, dynamics of the bounds of confidence and vector opinions [@Weisbuch2002]. Special starting distributions were used to model a drift to the extremes [@Deffuant2002]. The HK model has been extended to asymmetric [@Hegselmann2002] or heterogeneous bounds of confidence [@Lorenz2003], multidimensional opinions [@Lorenz2003b]. Some further proposals are in [@Hegselmann2004].
We show some statistical simulation results in figure \[fig2\] to give a chance to look into more details. For the graphic of added stabilized profiles we divided the opinion interval $[0,1]$ into 25 subintervals of equal size and count the number of agents in each subinterval. In the graphic at the bottom left dotted lines represent frequencies of 1 cluster and 2 clusters where we count only clusters with more then 5 agents, thus outliers and minor clusters are ignored.
![Added stabilized profiles (top), frequencies of the number of evolving clusters (bottom) for the WD (left hand) and the HK (right hand) model. Basic data: 100 initial profiles with 200 agents and random opinions between 0 and 1; Graphics data: stabilized profiles computed for ${\varepsilon}=
0,\stackrel{+0.01}{\dots},0.4$. (Dotted lines: without minor clusters.)[]{data-label="fig2"}](figure2_1.eps "fig:"){width="\columnwidth"}\
![Added stabilized profiles (top), frequencies of the number of evolving clusters (bottom) for the WD (left hand) and the HK (right hand) model. Basic data: 100 initial profiles with 200 agents and random opinions between 0 and 1; Graphics data: stabilized profiles computed for ${\varepsilon}=
0,\stackrel{+0.01}{\dots},0.4$. (Dotted lines: without minor clusters.)[]{data-label="fig2"}](figure2_2.eps "fig:"){width="\columnwidth"}
The number of agents $n$ is a parameter which is not much studied due to computing time problems. One idea of the following analysis studies with interactive Markov chains is to get a feeling about the dynamics we will approach with greater $n$.
Interactive Markov Chains
=========================
In this section we want to reformulate the models of WD and HK as interactive Markov chains. Thus, we switch from $n$ agents to an idealized infinite population, which we divide into $n$ classes of opinions. Instead of an opinion profile $x(t) \in {\mathbb R}^n$ we consider a *discrete opinion distribution* $p(t) \in S^n :=
\{p\in [0,1]^{1\times n}$ with $\sum_{i=1}^n p_i = 1\}$ as the state of our system at time $t\in{\mathbb N}$. $S^n$ is the $(n-1)$-dimensional unit simplex of row vectors. A discrete opinion distribution is a row vector while the opinion profile is a column vector[^3]. For convenience we define $p_i = 0$ for all $i\not\in {\underline{n}}$.
We will define *transition probabilities* $b_{ij}(p) \in
[0,1]$ for agents in class $i$ to go to class $j$. For all $i\in
{\underline{n}}$ should hold $\sum_{j=1}^n b_{ij}(p) = 1$. Thus the *transition matrix* $B(p) = [b_{ij}(p)]$ is row-stochastic. The *interactive Markov chain* for an initial opinion distribution $p(0)$ is the sequence of opinion distributions $(p(t))_{t\in{\mathbb N}_0}$ recursivly defined through $$p(t+1) = p(t)B(p(t)). \label{eq:mc}$$ This Markov chain is called interactive (according to Conlisk [@Conlisk1976]) because the transition matrix depends on the state of the system in the actual time step.
In analogy to the bound of confidence ${\varepsilon}$ we define a *discrete bound of confidence* $k \in {\mathbb N}$, which defines that the transitions of opinions of class $i$ can only be influenced by opinions of the classes ${i-k,\dots,i+k}$.
If we imagine that the opinions $1,\dots,n$ are representatives for an equidistant partition of the interval $[0,1]$ in the way $i
\leftrightarrow [\frac{i-1}{n},\frac{i}{n}]$, we draw a heuristic analogy $\frac{k}{n} \leftrightarrow {\varepsilon}$. Thus, if we go $k,n\to\infty$ with $\frac{k}{n} \to {\varepsilon}$ we can ’converge’ to an agent-based model with $\infty$ agents, opinions in $[0,1]$ and bound of confidence ${\varepsilon}$.
In this setting we may consider $n$ as a parameter how accurate a continuous opinion can be communicated, e.g. how many steps do we allow on the continuous scale from minimum to maximum opinion. (The topic of accuracy is discussed for agent-based models in [@Urbig2003] in the context of opinion versus attitude dynamics.)
Calculation of the interactive Markov chains will be interesting regarding the following questions.
1. Does the Markov chain produce the same dynamical behavior as the agent-based models?
2. What conclusions can be drawn from the results about the idealized infinite populations in the Markov Chains to the finite cases of agent-based systems?
3. What is the effect of the accuracy of the Markov chains $n$?
We will start with the definiton of the WD Markov chain.
We define the *WD transition matrix* for an opinion distribution $p\in S^n$ and a discrete bound of confidence $k \in
{\mathbb N}$ as $$b_{ij}(p,k) := \left\{\begin{array}{ll}
\frac{\pi^i_{2j-i-1}}{2} + \pi^i_{2j-i} + \frac{\pi^i_{2j-i+1}}{2}, & \hbox{if $i\neq j$, } \\
q_{i}, & \hbox{if $i=j$.} \\
\end{array}\right.$$ with $q_i = 1 - \sum_{j\neq i, j=1}^n b(p,k)_{ij}$ and $$\pi^i_l := \left\{\begin{array}{ll}
p_l, & \hbox{if $|i-l|\leq k$} \\
0, & \hbox{otherwise} \\
\end{array}\right.$$ (Attention $i$ in $\pi^i$ is another index not an exponent!) Let $p(0)\in S^n$ be an initial opinion distribution. The Markov chain (\[eq:mc\]) with WD transition matrix is called *interactive WD Markov chain* with discrete bound of confidence $k$.
Remember that we defined $p_i = 0$ for all $i\not\in {\underline{n}}$. By the founding idea of the WD model an agent with opinion $i$ moves to the new opinion $j$ if he compromises with an agent with opinion $i + 2(j-i) = 2j-i$. The probability to communicate with an agent with opinion $2j-i$ is of course $p_{2j-i}$. Thus, the heuristic of random pairwise interaction is represented. The terms $\frac{\pi^i_{2j-i-1}}{2},\frac{\pi^i_{2j-i+1}}{2}$ stand for the case when agents with opinion $i$ communicate with agents with opinion $l$, but the distance $|i-l|$ is odd. In this case the population should go with probability $\frac{1}{2}$ to one of the two possible opinion classes $\lfloor\frac{i+l}{2}\rfloor,\lceil\frac{i+l}{2}\rceil$. [^4]
(This definition is inspired by the rate equation in [@Ben-Naim2003]. But they used continuous time. Further on we introduced the discrete bound of confidence.)
The interactive Markov chain for the HK model looks as follows.
Let $p\in S^n$ be an opinion distribution and $k\in{\mathbb N}$ be a discrete bound of confidence. The [*$k$-local mean*]{} of $p$ at $i\in{\underline{n}}$ is $$M_i(p,k) := \frac{\sum\limits_{m\in{\underline{n}},|i-m|\leq k} m
p_m}{\sum\limits_{m\in{\underline{n}},|i-m|\leq k} p_m}.$$ We define (with abbreviation $M_i := M_i(p,k)$ the *HK transition matrix*) as $$b_{ij}(p,k) := \left\{\begin{array}{ll}
1 & \hbox{if $j = M_i$,} \\
\lceil M_i\rceil - M_i & \hbox{if $j = \lfloor M_i \rfloor$, $j\neq M_i$,} \\
M_i - \lfloor M_i\rfloor & \hbox{if $j = \lceil M_i \rceil$, $j\neq M_i$,} \\
0 & \hbox{otherwise.} \\
\end{array}\right.$$ Let $p(0)\in S^n$ be an initial opinion distribution. The Markov chain (\[eq:mc\]) with HK transition matrix is called *interactive HK Markov chain* with discrete bound of confidence $k$.
Each row of the HK transition matrix $B(p,k)$ contains only one or two adjacent positive entries. The population with opinion $i$ goes completely to the $k$-local mean opinion if this is an integer. Otherwise they distribute to the two adjacent opinions. The fraction which goes to the lower (upper) opinion class depends on how close the $k$-local mean lies to it. Thus, the heuristic of overall averaging in a local area is represented here.
For both interactive Markov chains we define a stabilized distribution $p^\ast \in S^n$ as a fixed point (that means that it holds $p^\ast = p^\ast B(p^\ast)$). Both models converge to stabilized distributions for every initial distribution, while the set of possible stabilized distributions is huge. This properties are stated by observation of example Markov processes. There is no formal proof at this time.
In a first analysis step we will focus on the initial distribution $p(0) \in S^n$ which is equally distributed $p_1 = \dots = p_n =
\frac{1}{n}$. This coincides with the agent-based model if we start with $x\in [0,1]^n$ where the $x_i$ are chosen at random and equally distributed.
Figure \[fig3\] shows processes for both models with $n=101,
k=9$ (coincides with ${\varepsilon}\approx 0.089$) at characteristic time steps[^5]. We see clustering dynamics as in the agent-based models. Five major clusters emerging in the WD model and three major clusters emerging in the HK model.
![WD and HK process for equally distributed initial distribution with $n=101, k=9$ at characteristic time steps. []{data-label="fig3"}](figure3.eps "fig:"){width="\columnwidth"}\
![Same processes as in figure \[fig3\] with finer $y$-scale at certain time steps to show minor clusters after stabilization for WD and minor clusters attracting major clusters for HK.[]{data-label="fig31"}](figure3_1.eps "fig:"){width="\columnwidth"}\
The finer scaled plots in figure \[fig31\] give more insights. It shows for the WD model four minor clusters in the stabilized distribution. (The distribution at $t=150$ is not totally stabilized, but very close to the distribution it converges to. E.g. the big central cluster that we see in the blow-up figure \[fig31\] will converge slowly to a one bin cluster. The first two off-central clusters will converge each to a two bin cluster as we see it already in figure \[fig3\].) We got minor clusters at both extremes and two minor clusters between major clusters, which will survive forever. The minor clusters are only visible in the blow-up figure. There are no minor clusters between the central and his adjacent major clusters.
For the HK model we see the possibility of very slow convergence in the HK model on the right side in figure \[fig3\], which can be explained by figure \[fig31\]. There we see two small clusters which hold contact between the central cluster and the both adjacent major clusters. The dynamic reaches a stabilized distribution with three major clusters at $t=327$. We call the states from $t=10$ to $t=326$ ’meta-stable’ because they look like stable.
Thus, both special features that we mentioned for the agent-based models (minor clusters for WD model and very slow convergence in HK model) also occur in the aggregated models of the interactive Markov chains. This effects may be seen as artifacts by simulators but they should be regarded as intrinsic to the dynamic’s behavior (at least for great numbers of agents).
To get a complete overview about the random equally distributed initial distribution we will calculate a ’discrete’ bifurcation diagram for both models. We will calculate the stabilized distributions for all processes for $n=1001$ and $k=50,\stackrel{+1}{\dots},300$ (thus ${\varepsilon}\approx
0.05,\dots,0.3$). Our bifurcation diagram is called ’discrete’ because the analyzed bifurcation parameter $k$ is discrete, which does not fit into the usual terms of bifurcation theory. In our setting a bifurcation is an abrupt change in the stabilized profile in one discrete step of the parameter $k$.
![Bifurcation diagram for the WD model. $n=1001,
k=50,\dots,300$.[]{data-label="bifWD"}](bifWD.eps "fig:"){width="\columnwidth"}\
Figure \[bifWD\] and \[bifHK\] show discrete bifurcation diagrams for the interactive WD and HK Markov chains. The gray scale symbolizes the masses at the specific positions: White is zeros, black is fair above zero and grey is slightly above zero. We see the major clusters as black lines. (Notice that a fully converged cluster normally consists of two adjacent non-zero entries in $p$ surrounded by zero entries. You can not see this in the figure.)
We describe the bifurcation diagram for the WD process. At first we have to notice again that the diagram is not fully converged due to computation time. Going further on in time the gray areas would converge slowly to gray lines of minor clusters. Going form $k=300$ down to 50 we see at first one central cluster and minor clusters at the extremes (that will converge to the total extremes), then the central cluster bifurcates into two major clusters at roughly $k \approx 270$, then a small central cluster nucleates ($k \approx 220$), the new central cluster is growing and the two former clusters drift away from the center. At $k
\approx 150$ two minor clusters emerge. At roughly 125 the central cluster bifurcates again into two clusters, while the minor clusters vanish for a short phase. The same bifurcation and nucleation procedure repeats on a shorter $k$-scale from 125 to 80 and so on.
The existence or absence of minor clusters between major clusters can be explained with the accuracy $n$. Only if the accuracy leaves enough classes between two major clusters a minor cluster can survive in between.
A similar bifurcation diagram for the differential equation describing the same heuristics is in [@Ben-Naim2003].
![Bifurcation diagram for the HK model. $n=1001,
k=50,\dots,300$.[]{data-label="bifHK"}](bifHK.eps "fig:"){width="\columnwidth"}\
In figure \[bifHK\] we see the bifurcation diagram for the HK model. In contrast to the WD diagram the processes in this diagram have converged completely. Going form $k=300$ down to 50 we see one central cluster, then at $k=190$ a bifurcation into two major clusters occurs while very little mass remains in a central cluster. Then (k=175) we get into an interesting phase where consensus strikes back, but not for all $k$, sometimes we end up in three clusters. The outer clusters begin to drift outwards away from their old position. Consensus in this $k$-phase happens due to very slow convergence to the center like in figure \[fig3\]. The last consensus appears at $k=153$ (where convergence lasts 21424 time steps). Then the existence of the two outer clusters and their outward drift is stable until the central cluster bifurcates again on a shorter time scale (two major clusters $k=100,99,98$, consensus strikes back $k=97,\dots,89$, then stable 5 clusters) and so on. Remarkably is that we reach a consensus at the HK Markov chain for ${\varepsilon}= k/n > 0.19$ but in the tested agent-based model (figure \[fig2\]) with 200 agents the chance for consensus is only for ${\varepsilon}\geq 0.23$ fair.
The interactive Markov chains raises the question: Is it important to have an even or odd number of opinion classes? In other words, is it important to have a central opinion class? The impact of the existence of a central opinion class is very high for the HK model as we see in figure \[bifHKeven\]. For the WD model the bifurcation diagram for $n=1000$ looks just as figure \[bifWD\].
![Bifurcation diagram for the HK model. $n=1000,
k=50,\dots,300$.[]{data-label="bifHKeven"}](bifHKeven.eps "fig:"){width="\columnwidth"}\
There are two differences between the HK bifurcation diagrams in figures \[bifHK\] and \[bifHKeven\] which are generic for other other pairs $n,n+1$. First, the bifurcation from consensus to polarization occurs earlier for even $n$ (at $k=200$) and the central clusters disappears totally at that bifurcation and nucleates again some $k$-steps later. Second the ’consensus strikes back’ $k$-phase changes into a polarization phase, but with major clusters closer to the center. As for $n=1001$ this is not stable in that $k$-phase, for some values we reach two outer majors and a central cluster.
Both phenomena can be explained through the meta-stable state that we reach after the first time steps (in figure \[fig3\] at $t=10$). For an odd number of opinion classes we have a central cluster consisting of only one opinion class, while for an even number of opinion classes the central cluster contains two opinion classes.
Conclusion
==========
We summarize. The basic dynamical feature of both models is: Stabilization to a fixed point and fixed points are opinion distributions where the mass clusters in pairs of adjacent classes which have distances greater than $k$ to all other pairs. This coincides with the agent-based models (neglecting the clustering in pairs of opinion classes). The differences in the agent based models (number of clusters at specific ${\varepsilon}$, existence of outliers) appear also and even more drastic in the Markov chain models. We show that two properties which may be seen as artifacts are intrinsic to the dynamical behavior.
The WD model leads for equally distributed initial distributions to major clusters which lie as far from each other, that it is possible for small cluster to survive between them, but only for great $n$ and not too low $k$. The existence of minor clusters depends also on $\frac{k}{n}$.
The HK model leads to major clusters only (except the central cluster, which may be small). In the center we can reach a meta-stable state with two major clusters, a central cluster and between major and central cluster small clusters, which attract the big ones very slowly to the center. An even number of opinions lowers the chances for central consensus because the central mass can split.
In future research we should ask how robust the results are with other initial distributions. A first but not systematic calculation leads to the hypothesis that the basic features (clustering in pairs, space for minor clusters for WD and meta stable states with long convergence for HK) also characterize the dynamics for other initial distributions. Further on stabilization should be proved.
The heuristics of these opinion dynamic models is not based on quantitative data. Thus, they can not predict quantitative opinion clustering. Conclusions about real opinion dynamics should be drawn in a qualitative manner that links the heuristics of the model to the dynamical outcome, in a way like: Bounded confidence in opinion dynamics about ’continuous’ topics leads to clustering. Pairwise communication produces minorities while averaging over all acceptable opinions may lead to meta-stable situations where consensus is possible but reaching it will take very long.
#### Acknowledgment
The author thanks the Friedrich-Ebert-Stiftung, Bonn, Germany for financial funding.
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[^1]: With ’random’ we mean ’random and equally distributed in the respective space’.
[^2]: This is a basic version of the model in [@Weisbuch2002] with $\mu=0.5$.
[^3]: This is for matrix theoretical reasons: $A(x)$ and $B(p)$ are both row-stochastic.
[^4]: $\lfloor\cdot\rfloor$ ($\lceil\cdot\rceil$) is rounding to the lower (upper) integer.
[^5]: The programming of the HK model is numerically vulnerable. Programming it like it is may lead for equally distributed initial distributions to asymmetric distributions. This is theoretically impossible for symmetric initial distributions. This problem is circumvented in this calculation, by making the distribution symmetric with $p=(p+\mathrm{flip}(p))/2$ after each time step. ($\mathrm{flip}(p)_i := p_{n+1-i}$ for all $i\in{\underline{n}}$)
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---
abstract: 'In literature, it is known that any solution of Painlevé VI equation governs the isomonodromic deformation of a second order linear Fuchsian ODE on $\mathbb{CP}^{1}$. In this paper, we extend this isomonodromy theory on $\mathbb{CP}^{1}$ to the moduli space of elliptic curves by studying the isomonodromic deformation of the generalized Lamé equation. Among other things, we prove that the isomonodromic equation is a new Hamiltonian system, which is equivalent to the elliptic form of Painlevé VI equation for generic parameters. For Painlevé VI equation with some special parameters, the isomonodromy theory of the generalized Lamé equation greatly simplifies the computation of the monodromy group in $\mathbb{CP}^{1}$. This is one of the advantages of the elliptic form.'
address:
- 'Center for Advanced Study in Theoretical Sciences (CASTS), National Taiwan University, Taipei 10617, Taiwan '
- 'Taida Institute for Mathematical Sciences (TIMS), National Taiwan University, Taipei 10617, Taiwan '
- 'Taida Institute for Mathematical Sciences (TIMS), Center for Advanced Study in Theoretical Sciences (CASTS), National Taiwan University, Taipei 10617, Taiwan '
author:
- Zhijie Chen
- 'Ting-Jung Kuo'
- 'Chang-Shou Lin'
title: Hamiltonian system for the elliptic form of Painlevé VI equation
---
Introduction
============
The isomonodromic deformation plays an universal role to connect many different research areas of mathematics and physics. Our purpose of this paper is to develop an isomonodromy theory for the generalized Lamé equation on the moduli space of elliptic curves.
Painlevé VI in elliptic form
----------------------------
Historically, the discovery of Painlevé equations was originated from the research on complex ODEs from the middle of 19th century up to early 20th century, led by many famous mathematicians including Painlevé and his school. The aim is to classify those nonlinear ODEs whose solutions have the so-called *Painlevé property*. We refer the reader to [@Babich-Bordag; @DIKZ; @Dubrovin-Mazzocco; @Guzzetti; @Halphen; @Hit1; @Hit2; @Inaba-Iwasaki-Saito; @GP; @Kawai; @Lisovyy-Tykhyy; @Y.Manin; @Okamoto2; @Okamoto1; @Okamoto; @Poole; @Sasaki] and references therein for some historic account and the recent developments. Painlevé VI with four free parameters $(\alpha,\beta,\gamma,\delta)$ can be written as$$\begin{aligned}
\frac{d^{2}\lambda}{dt^{2}}= & \frac{1}{2}\left( \frac{1}{\lambda}+\frac
{1}{\lambda-1}+\frac{1}{\lambda-t}\right) \left( \frac{d\lambda}{dt}\right)
^{2}-\left( \frac{1}{t}+\frac{1}{t-1}+\frac{1}{\lambda-t}\right)
\frac{d\lambda}{dt}\label{46}\\
& +\frac{\lambda \left( \lambda-1\right) \left( \lambda-t\right) }{t^{2}\left( t-1\right) ^{2}}\left[ \alpha+\beta \frac{t}{\lambda^{2}}+\gamma \frac{t-1}{\left( \lambda-1\right) ^{2}}+\delta \frac{t\left(
t-1\right) }{\left( \lambda-t\right) ^{2}}\right] .\nonumber\end{aligned}$$ In the literature, it is well-known that Painlevé VI (\[46\]) is closely related to the isomonodromic deformation of either a $2\times2$ linear ODE system of first order (under the non-resonant condition, the isomonodromic equation is known as the Schlesinger system; see [@Jimbo-Miwa]) or a second order Fuchsian ODE (under the non-resonant condition, the isomonodromic equation is a Hamiltonian system; see [@Fuchs; @Okamoto2]). This associated second order Fuchsian ODE is defined on $\mathbb{CP}^{1}$ and has five regular singular points $0,1,t,\lambda(t)$ and $\infty$. Among them, $\lambda(t)$ (as a solution of Painlevé VI) is an apparent singularity. This isomonodromy theory on $\mathbb{CP}^{1}$ was first discovered by R. Fuchs [@Fuchs], and later generalized to the $n$-dimensional Garnier system by K. Okamoto [@Okamoto2]. We will briefly review this classical isomonodromy theory in Section 4.
Throughout the paper, we use the notations $\omega_{0}=0,\omega_{1}=1,\omega_{2}=\tau$, $\omega_{3}=1+\tau$, $\Lambda_{\tau}=\mathbb{Z+\tau Z}$, and $E_{\tau}\doteqdot \mathbb{C}/\Lambda_{\tau}$ where $\tau \in \mathbb{H}=\left \{ \tau|\operatorname{Im}\tau>0\right \} $ (the upper half plane). We also define $E_{\tau}\left[ 2\right] \doteqdot \left \{ \frac{\omega_{i}}{2}|i=0,1,2,3\right \} $ to be the set of 2-torsion points in the flat torus $E_{\tau}$. From the Painlevé property of (\[46\]), any solution $\lambda \left( t\right) $ is a multi-valued meromorphic function in $\mathbb{C}\backslash \left \{ 0,1\right \} $. To avoid the multi-valueness of $\lambda \left( t\right) $, it is better to lift solutions of (\[46\]) to its universal covering. It is known that the universal covering of $\mathbb{C}\backslash \left \{ 0,1\right \} $ is $\mathbb{H}$. Then $t$ and the solution $\lambda \left( t\right) $ can be lifted to $\tau$ and $p\left(
\tau \right) $ respectively through the covering map by$$t\left( \tau \right) =\frac{e_{3}(\tau)-e_{1}(\tau)}{e_{2}(\tau)-e_{1}(\tau
)}\text{ and }\lambda(t)=\frac{\wp(p(\tau)|\tau)-e_{1}(\tau)}{e_{2}(\tau)-e_{1}(\tau)}, \label{tr}$$ where $\wp \left( z|\tau \right) $ is the Weierstrass elliptic function defined by$$\wp \left( z|\tau \right) =\frac{1}{z^{2}}+\sum_{\omega \in \Lambda_{\tau
}\backslash \left \{ 0\right \} }\left[ \frac{1}{\left( z-\omega \right)
^{2}}-\frac{1}{\omega^{2}}\right] ,$$ and $e_{i}=\wp \left( \frac{\omega_{i}}{2}|\tau \right) $, $i=1,2,3$. Then $p\left( \tau \right) $ satisfies the following elliptic form$$\frac{d^{2}p\left( \tau \right) }{d\tau^{2}}=\frac{-1}{4\pi^{2}}\sum
_{i=0}^{3}\alpha_{i}\wp^{\prime}\left( p\left( \tau \right) +\frac
{\omega_{i}}{2}|\tau \right) , \label{124}$$ where $\wp^{\prime}\left( z|\tau \right) =\frac{d}{dz}\wp \left(
z|\tau \right) $ and $$\left( \alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3}\right) =\left(
\alpha,-\beta,\gamma,\frac{1}{2}-\delta \right) . \label{126}$$ This elliptic form was already known to Painlevé [@Painleve]. For a modern proof, see [@Babich-Bordag; @Y.Manin].
The advantage of (\[124\]) is that $\wp(p(\tau)|\tau)$ is single-valued for $\tau \in \mathbb{H}$, although $p(\tau)$ has a branch point at those $\tau_{0}$ such that $p(\tau_{0})\in E_{\tau_{0}}[2]$ (see e.g. (\[asymp\]) below). We take $(\alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3})=(\frac{1}{8},\frac{1}{8},\frac{1}{8},\frac{1}{8})$ for an example to explain it. Painlevé VI with this special parameter has connections with some geometric problems; see [@Chen-Kuo-Lin; @Hit1]. In the seminal work [@Hit1], N. Hitchin discovered that, for a pair of complex numbers $(r,s)\in \mathbb{C}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$, $p(\tau)$ defined by the following formula:$$\wp \left( p(\tau)|\tau \right) =\wp \left( r+s\tau|\tau \right) +\frac
{\wp^{\prime}\left( r+s\tau|\tau \right) }{2\left( \zeta \left( r+s\tau
|\tau \right) -r\eta_{1}(\tau)-s\eta_{2}(\tau)\right) }, \label{513}$$ is a solution to (\[124\]) with $\alpha_{k}=\frac{1}{8}$ for all $k$. Here $\zeta \left( z|\tau \right) \doteqdot-\int^{z}\wp(\xi|\tau)d\xi$ is the Weierstrass zeta function and has the quasi-periods$$\zeta \left( z+1|\tau \right) =\zeta \left( z|\tau \right) +\eta_{1}(\tau)\text{ and }\zeta \left( z+\tau|\tau \right) =\zeta \left(
z|\tau \right) +\eta_{2}(\tau). \label{40-2}$$ [By (\[513\]), Hitchin could construct an Einstein metric with positive curvature if ]{}$r\in \mathbb{R}$ and $s\in i\mathbb{R}$, and an Einstein metric with negative curvature if $r\in i\mathbb{R}$ and $s\in \mathbb{R}$. It follows from (\[513\]) that $\wp \left( p(\tau)|\tau \right) $ is a single-valued meromorphic function in $\mathbb{H}$. However, each $\tau_{0}$ with $p(\tau_{0})\in E_{\tau_{0}}[2]$ is a branch point of order $2$ for $p(\tau)$.
Motivated from Hitchin’s solutions, we would like to extend the beautiful formula (\[513\]) to Painlevé VI with other parameters. But it is not a simple matter because it invloves complicated derivatives with respect to the moduli parameter $\tau$. For example, for Hichin’s solutions, it seems not easy to derive (\[124\]) with $\alpha_{k}=\frac{1}{8}$ for all $k$ directly from the formula (\[513\]). We want to provide a systematical way to study this problem. To this goal, the first step is to develop a theory in the moduli space of tori which is analogous to the Fuchs-Okamoto theory on $\mathbb{CP}^{1}$. The purpose of this paper is to derive the Hamiltonian system for the elliptic form (\[124\]) by developing such an isomonodromy theory in the moduli space of tori. The key issue is *what the linear Fuchsian equation in tori is such that its isomonodromic deformation is related to the elliptic form* (\[124\]).
Generalized Lamé equation
-------------------------
Motivated from our study of the surprising connection of the mean field equation and the elliptic form (\[124\]) of Painlevé VI in [@Chen-Kuo-Lin], our choice of the Fuchsian equation is the generalized Lamé equation defined by (\[505\]) below. More precisely, let us consider the following mean field equation$$\Delta u+e^{u}=8\pi \sum_{i=0}^{3}n_{k}\delta_{\frac{\omega_{k}}{2}}+4\pi \left( \delta_{p}+\delta_{-p}\right) \text{ in }E_{\tau},\label{501}$$ where $n_{k}>-1$, $\delta_{\pm p}$ and $\delta_{\frac{\omega_{k}}{2}}$ are the Dirac measure at $\pm p$ and $\frac{\omega_{k}}{2}$ respectively. By the Liouville theorem, any solution $u$ to equation (\[501\]) could be written into the following form:$$u(z)=\log \frac{8|f^{\prime}\left( z\right) |^{2}}{(1+|f\left( z\right)
|^{2})^{2}},\label{502}$$ where $f\left( z\right) $ is a meromorphic function in $\mathbb{C}$. Conventionally $f\left( z\right) $ is called a developing map of $u$. We could see below that there associates a 2nd order complex ODE reducing from the nonlinear PDE (\[501\]). Indeed, it follows from (\[501\]) that outside $E_{\tau}\left[ 2\right] \cup \left \{ \pm p\right \} $, $$\begin{aligned}
\left( u_{zz}-\frac{1}{2}u_{z}^{2}\right) _{\bar{z}} & =\left( u_{z\bar
{z}}\right) _{z}-u_{z}u_{z\bar{z}}\\
& =\left( -\frac{1}{4}e^{u}\right) _{z}+\frac{1}{4}e^{u}u_{z}=0.\end{aligned}$$ So $u_{zz}-\frac{1}{2}u_{z}^{2}$ is an elliptic function on the torus $E_{\tau}$ with singularties at $E_{\tau}\left[ 2\right] \cup \left \{ \pm
p\right \} $. Since the behavior of $u$ is fixed by the RHS of (\[501\]), for example, $u(z)=2\log|z-p|+O(1)$ near $p$, we could compute explicitly the dominate term of $u_{zz}-\frac{1}{2}u_{z}^{2}$ near each singular point. Let us further assume that $u(z)$ is *even*, i.e., $u(z)=u(-z)$. Then we have$$\begin{aligned}
& u_{zz}-\frac{1}{2}u_{z}^{2}\label{503}\\
= & -2\left[
\begin{array}
[c]{l}\sum_{k=0}^{3}n_{k}\left( n_{k}+1\right) \wp \left( z+\frac{\omega_{k}}{2}\right) +\frac{3}{4}\left( \wp \left( z+p\right) +\wp \left( z-p\right)
\right) \\
+A\left( \zeta \left( z+p\right) -\zeta \left( z-p\right) \right) +B
\end{array}
\right] \nonumber \\
\doteqdot & -2I\left( z\right) ,\nonumber\end{aligned}$$ where $A,B$ are two (unknown) complex numbers.
On the other hand, we could deduce from (\[502\]) that the Schwarzian derivative $\{f;z\}$ of $f$ can be expressed by$$\{f;z\} \doteqdot \left( \frac{f^{\prime \prime}}{f^{\prime}}\right) ^{\prime
}-\frac{1}{2}\left( \frac{f^{\prime \prime}}{f^{\prime}}\right) ^{2}=u_{zz}-\frac{1}{2}u_{z}^{2}=-2I\left( z\right) .\label{504}$$ From (\[504\]), we connect the developing maps of an even solution $u$ to (\[501\]) with the following generalized Lamé equation$$y^{\prime \prime}(z)=I(z)y(z)\text{ in }E_{\tau},\label{505}$$ where the potential $I(z)$ is given by (\[503\]). In the classical literature, the 2nd order ODE$$y^{\prime \prime}(z)=\left( n(n+1)\wp(z)+B\right) y(z)\text{ \ in \ }E_{\tau
}\label{503-1}$$ is called the Lamé equation, and has been extensively studied since the 19th century, particularly for the case $n\in \mathbb{Z}/2$. See [@Chai-Lin-Wang; @Halphen; @Poole; @Whittaker-Watson] and the references therein. In this paper, we will prove that (\[503-1\]) appears as a limiting equation of (\[505\]) under some circumstances (see Theorem \[thm-II-9 copy(1)\]).
From (\[502\]) and (\[504\]), any two developing maps $f_{i}$, $i=1,2$ of the same solution $u$ must satisfy$$f_{2}(z)=\alpha \cdot f_{1}(z)\doteqdot \frac{af_{1}(z)+b}{cf_{1}(z)+d}$$ for some $\alpha=\left(
\begin{matrix}
a & b\\
c & d
\end{matrix}
\right) \in PSU(2)$. From here, we could define a projective monodromy representation $\rho:\pi_{1}(E_{\tau}\backslash(E_{\tau}\left[ 2\right]
\cup \{ \pm p\}),q_{0})\rightarrow PSU(2)$, where $q_{0}\not \in E_{\tau}\left[
2\right] \cup \{ \pm p\}$ is a base point. Indeed, any developing map $f$ might be multi-valued. For any loop $\ell \in \pi_{1}(E_{\tau}\backslash(E_{\tau
}\left[ 2\right] \cup \{ \pm p\}),q_{0})$, $\ell^{\ast}f$ denotes the analytic continuation of $f$ along $\ell$. Since $\ell^{\ast}f$ is also a developing map of the same $u$, there exists $\rho(\ell)\in PSU(2)$ such that $\ell
^{\ast}f=\rho(\ell)\cdot f$. Thus, the map $\ell \mapsto \rho(\ell)$ defines a group homomorphism from $\pi_{1}(E_{\tau}\backslash(E_{\tau}\left[ 2\right]
\cup \{ \pm p\}),q_{0})$ to $PSU(2)$. When $n_{k}\in \mathbb{N}\cup \left \{
0\right \} $ for all $k$, the developing map is a single-valued meromorphic function defined in $\mathbb{C}$, and $\left \{ \pm p\right \} $ would become apparent singularties. Thus, the projective monodromy representation would be reduced to a homomorphism from $\pi_{1}\left( E_{\tau},q_{0}\right) $ to $PSU(2)$. This could greatly simplify the computation of the monodromy group. The deep connection of the mean field equation (\[501\]) and the elliptic form (\[124\]) has been discussed in detail in [@Chen-Kuo-Lin]. This is our motivation to study the isomonodromic deformation of the generalized Lamé equation (\[505\]) in this paper.
Isomonodromic deformation and Hamiltonian system
------------------------------------------------
We are now in a position to state our main results. Recall the generalized Lamé equation (\[505\]):$$y^{\prime \prime}=\left[
\begin{array}
[c]{l}\sum_{k=0}^{3}n_{k}\left( n_{k}+1\right) \wp \left( z+\frac{\omega_{k}}{2}\right) +\frac{3}{4}\left( \wp \left( z+p\right) +\wp \left( z-p\right)
\right) \\
+A\left( \zeta \left( z+p\right) -\zeta \left( z-p\right) \right) +B
\end{array}
\right] y. \label{89-0}$$ Equation (\[89-0\]) has no solutions with logarithmic singularity at $\frac{\omega_{k}}{2}$ unless $n_{k}\in \frac{1}{2}+\mathbb{Z}$. Therefore, we need to assume the non-resonant condition: $n_{k}\not \in \frac{1}{2}+\mathbb{Z}$ for all $k$. Observe that the exponent difference of (\[89-0\]) at $\pm p$ is $2$. Here the singular points $\pm p$ are always assumed to be *apparent*. Under this assumption, the coefficients $A$ and $B$ together satisfy (\[101-0\]) below. Our first main result is following.
\[theorem1-2\] Let $$\alpha_{k}=\frac{1}{2}\left( n_{k}+\frac{1}{2}\right) ^{2}\text{ with }n_{k}\not \in \frac{1}{2}+\mathbb{Z},\ k=0,1,2,3. \label{125}$$ Then $p\left( \tau \right) $ is a solution of the elliptic form (\[124\]) if and only if there exist $A\left( \tau \right) $ and $B\left( \tau \right)
$ such that the generalized Lamé equation (\[89-0\]) with apparent singularities at $\pm p\left( \tau \right) $ preserves the monodromy while $\tau$ is deforming.
Our method to prove Theorem \[theorem1-2\] consists of two steps: the first is to derive the isomonodromic equation, a Hamiltonian system, in the moduli space of tori for (\[89-0\]) under the non-resonant condition $n_{k}\not \in \frac{1}{2}+\mathbb{Z}$. The second is to prove that this isomonodromic equation (the Hamiltonian system) is equivalent to the elliptic form (\[124\]). To describe the isomonodromic equation for (\[89-0\]), we let the Hamiltonian $K\left( p,A,\tau \right) $ be defined by$$\begin{aligned}
K\left( p,A,\tau \right) & =\frac{-i}{4\pi}(B+2p\eta_{1}\left(
\tau \right) A)\label{143-0}\\
& =\frac{-i}{4\pi}\left(
\begin{array}
[c]{l}A^{2}+\left( -\zeta \left( 2p|\tau \right) +2p\eta_{1}\left( \tau \right)
\right) A-\frac{3}{4}\wp \left( 2p|\tau \right) \\
-\sum_{k=0}^{3}n_{k}\left( n_{k}+1\right) \wp \left( p+\frac{\omega_{k}}{2}|\tau \right)
\end{array}
\right) .\nonumber\end{aligned}$$ Consider the Hamiltonian system$$\left \{
\begin{array}
[c]{l}\frac{dp\left( \tau \right) }{d\tau}=\frac{\partial K\left( p,A,\tau \right)
}{\partial A}=\frac{-i}{4\pi}\left( 2A-\zeta \left( 2p|\tau \right)
+2p\eta_{1}\left( \tau \right) \right) \\
\frac{dA\left( \tau \right) }{d\tau}=-\frac{\partial K\left( p,A,\tau
\right) }{\partial p}=\frac{i}{4\pi}\left(
\begin{array}
[c]{l}\left( 2\wp \left( 2p|\tau \right) +2\eta_{1}\left( \tau \right) \right)
A-\frac{3}{2}\wp^{\prime}\left( 2p|\tau \right) \\
-\sum_{k=0}^{3}n_{k}\left( n_{k}+1\right) \wp^{\prime}\left( p+\frac
{\omega_{k}}{2}|\tau \right)
\end{array}
\right)
\end{array}
\right. .\label{142-0}$$ Then our first step leads to the following result:
\[=Theorem \[thm4-2\]\]\[theorem1-3\]Let $n_{k}\not \in \frac{1}{2}+\mathbb{Z}$, $k=0,1,2,3$. Then $(p(\tau),A(\tau))$ satisfies the Hamiltonian system (\[142-0\]) if and only if equation (\[89-0\]) with $(p(\tau),A(\tau),B(\tau))$ preserves the monodromy, where $$B=A^{2}-\zeta \left( 2p\right) A-\frac{3}{4}\wp \left( 2p\right) -\sum
_{k=0}^{3}n_{k}\left( n_{k}+1\right) \wp \left( p+\frac{\omega_{k}}{2}\right) .\label{101-0}$$
And the second step is to prove
\[=Theorem \[thm4-1\]\]\[theorem1-4\]The elliptic form (\[124\]) is equivalent to the Hamiltonian system (\[142-0\]), where $\alpha_{k}=\frac
{1}{2}\left( n_{k}+\frac{1}{2}\right) ^{2},$ $k=0,1,2,3$.
Clearly Theorem \[theorem1-2\] follows from Theorems \[theorem1-3\] and \[theorem1-4\] directly.
In [@Y.Manin], Manin rewrote the elliptic form (\[124\]) into an obvious time-dependent Hamiltonian system:$$\frac{dp\left( \tau \right) }{d\tau}=\frac{\partial H}{\partial q},\text{
}\frac{dq\left( \tau \right) }{d\tau}=-\frac{\partial H}{\partial p},
\label{HM}$$ where$$H=H\left( \tau,p,q\right) \doteqdot \frac{q^{2}}{2}+\frac{1}{4\pi^{2}}\sum_{i=0}^{3}\alpha_{i}\wp \left( p\left( \tau \right) +\frac{\omega_{i}}{2}|\tau \right) .$$ However, it is not clear whether the Hamiltonian system (\[HM\]) governs isomonodromic deformations of any Fuchsian equations in $E_{\tau}$ or not. Different from (\[HM\]), our Hamiltonian system (\[142-0\]) governs isomonodromic deformations of the generalized Lamé equation for generic parameters.
Both Theorems \[theorem1-3\] and \[theorem1-4\] are proved in Section 2. It seems that the generalized Lamé equation (\[89-0\]) looks simpler than the corresponding Fuchsian ODE on $\mathbb{CP}^{1}$, and it is the same for the Hamiltonian system (\[142-0\]), compared to the corresponding one on $\mathbb{CP}^{1}$. From the second equation of (\[142-0\]), $A\left(
\tau \right) $ can be integrated so that we have the following theorem:
\[theorem1-5\]Suppose $(p(\tau),A(\tau))$ satisfies the Hamiltonian system (\[142-0\]). Define $$F(\tau)\doteqdot A(\tau)+\frac{1}{2}(\zeta(2p(\tau)|\tau)-2\zeta(p(\tau
)|\tau)).\label{F}$$ Then$$\begin{aligned}
F\left( \tau \right) & =\theta_{1}^{\prime}(\tau)^{\frac{2}{3}}\exp \left \{
\frac{i}{2\pi}\int^{\tau}\left( 2\wp(2p(\hat{\tau})|\hat{\tau})-\wp
(p(\hat{\tau})|\hat{\tau})\right) d\hat{\tau}\right \} \times \nonumber \\
& \left( \int^{\tau}\frac{-\frac{i}{4\pi}\theta_{1}^{\prime}(\hat{\tau
})^{-\frac{2}{3}}\left( \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp^{\prime}(p(\hat{\tau
})+\frac{\omega_{k}}{2}|\hat{\tau})\right) }{\exp \left \{ \frac{i}{2\pi}\int^{\hat{\tau}}\left( 2\wp(2p(\tau^{\prime})|\tau^{\prime})-\wp
(p(\tau^{\prime})|\tau^{\prime})\right) d\tau^{\prime}\right \} }d\hat{\tau
}+c_{1}\right) \label{514}$$ for some constant $c_{1}\in \mathbb{C}$, where $\theta_{1}^{\prime}(\tau
)=\frac{d\vartheta_{1}\left( z;\tau \right) }{dz}|_{z=0}$ and $\vartheta
_{1}\left( z;\tau \right) $ is the odd theta function defined in (\[ccc\]). In particular, for $n_{k}=0,\forall k$, we have$$F(\tau)=c\theta_{1}^{\prime}(\tau)^{\frac{2}{3}}\exp \left \{ \frac{i}{2\pi
}\int^{\tau}\left( 2\wp(2p(\hat{\tau})|\hat{\tau})-\wp(p(\hat{\tau})|\hat{\tau})\right) d\hat{\tau}\right \} \label{515}$$ for some constant $c\in \mathbb{C}\backslash \left \{ 0\right \} $.
\[remk-1\]Let $\eta(\tau)$ be the Dedekind eta function: $\eta
(\tau)\doteqdot q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^{n})$, where $q=e^{2\pi i\tau}$ for $\tau \in \mathbb{H}$. Then $\theta_{1}^{\prime}(\tau)=2\pi \eta^{3}(\tau)$.
The elliptic form (\[124\]) and our results above could be applied to understand the phenomena of collapsing two singular points $\pm p(\tau)$ to $\frac{\omega_{k}}{2}$ in the generalized Lamé equation (\[89-0\]). In general, when $p(\tau)\rightarrow \frac{\omega_{k}}{2}$ as $\tau \rightarrow
\tau_{0}$, the generalized Lamé equation might not be well-defined. However, when $p(\tau)$ is a solution of the elliptic form (\[124\]), by using the behavior of $p(\tau)$ near $\tau_{0}$, the following result shows that the generalized Lamé equation will converge to the classical Lamé equation (\[503-1\]).
Observe that if $p(\tau)$ is a solution of the elliptic form (\[124\]), then $p(\tau)-\frac{\omega_{k}}{2}$ is also a solution of (\[124\]) (maybe with different parameters). Therefore, we only need to study the case $p(\tau)\rightarrow0$. More precisely, we have:
\[=Theorem \[thm-II-9\]\]\[thm-II-9 copy(1)\]Suppose that $n_{k}\not \in \frac{1}{2}+\mathbb{Z}$, $k=0,1,2,3$, and (\[101-0\]) holds. Let $(p(\tau),A(\tau))$ be a solution of the Hamiltonian system (\[142-0\]) such that $p(\tau_{0})=0$ for some $\tau_{0}\in \mathbb{H}$. Then$$p(\tau)=c_{0}(\tau-\tau_{0})^{\frac{1}{2}}(1+\tilde{h}(\tau-\tau_{0})+O(\tau-\tau_{0})^{2})\text{ as }\tau \rightarrow \tau_{0},\label{asymp}$$ where $c_{0}^{2}=\pm i\frac{n_{0}+\frac{1}{2}}{\pi}$ and $\tilde{h}\in \mathbb{C}$ is some constant. Moreover, the generalized Lamé equation (\[89-0\]) as $\tau \rightarrow \tau_{0}$ converges to$$y^{\prime \prime}=\left[ \sum_{j=1}^{3}n_{j}\left( n_{j}+1\right) \wp \left(
z+\frac{\omega_{j}}{2}\right) +m(m+1)\wp(z)+B_{0}\right] y\text{ in }E_{\tau_{0}}$$ where$$m=\left \{
\begin{array}
[c]{l}n_{0}+1\text{ \ if \ }c_{0}^{2}=i\frac{n_{0}+\frac{1}{2}}{\pi},\\
n_{0}-1\text{ \ if \ }c_{0}^{2}=-i\frac{n_{0}+\frac{1}{2}}{\pi},
\end{array}
\right.$$$$B_{0}=2\pi ic_{0}^{2}\left( 4\pi i\tilde{h}-\eta_{1}(\tau_{0})\right)
-\sum_{j=1}^{3}n_{j}(n_{j}+1)e_{j}(\tau_{0}).$$
Theorem \[thm-II-9 copy(1)\] will be proven in Section 3. In Section 4, we will give another application of our isomonodromy theory (see Corollary \[corcor\]). More precisely, we will establish a one to one correspondence between the generalized Lamé equation and the Fuchsian equation on $\mathbb{CP}^{1}$. Furthermore, we will prove that if one of them is monodromy preserving then so is the other one. We remark that all the results above have important applications in our coming paper [@Chen-Kuo-Lin]. For example, Theorem \[thm-II-9 copy(1)\] can be used to study the converge of even solutions of the mean field equation (\[501\]) as $p\left( \tau \right)
\rightarrow0$ when $\tau \rightarrow \tau_{0}$.
We conclude this section by comparing our result Theorem \[theorem1-2\] with the paper [@Kawai] by Kawai. Define the Fuchsian equation in $E_{\tau}$ by$$y^{\prime \prime}\left( z\right) =q\left( z\right) y\left( z\right)
,\label{OK}$$ where$$\begin{aligned}
q\left( z\right) = & L+\sum_{i=0}^{m}\left[ H_{i}\zeta \left( z-t_{i}|\tau \right) +\frac{1}{4}\left( \theta_{i}^{2}-1\right) \wp \left(
z-t_{i}|\tau \right) \right] \label{OK-1}\\
& +\sum_{\alpha=0}^{m}\left[ -\mu_{\alpha}\zeta \left( z-b_{\alpha}|\tau \right) +\frac{3}{4}\wp \left( z-b_{\alpha}|\tau \right) \right]
\nonumber\end{aligned}$$ with$$\sum_{i=0}^{m}H_{i}-\sum_{\alpha=0}^{m}\mu_{\alpha}=0.\label{OK-2}$$ Here $L$, $H_{i}$, $t_{i}$, $\theta_{i}$, $\mu_{\alpha}$, $b_{\alpha}$ are complex parameters with $t_{0}=0$. The isomonodromic deformation of equation (\[OK\]) was first treated by Okamoto [@Okamoto] without varying the underlying elliptic curves and then generalized by Iwasaki [@IW] to the case of higher genus. Let $\mathcal{R}$ be the space of conjugacy classes of the monodromy representation of $\pi_{1}((E_{\tau}\backslash S),q_{0})$, where $S$ denotes the set of singular points. Then it was known that $\mathcal{R}$ is a complex manifold. It was proved by Iwasaki [@IW] that there exists a natural symplectic structure $\Omega$ on the space $\mathcal{R}$. In [@Kawai], Kawai considered the same Fuchsian equation (\[OK\]) but allowed the underlying elliptic curves to vary as well. By using the pull-back principle to the symplectic 2-form $\Omega$, Kawai studied isomonodromic deformations for equation (\[OK\]) which are described as a completely integrable Hamiltonian system: for $1\leq i\leq m,$$$\frac{\partial b_{\alpha}}{\partial t_{i}}=\sum_{i=1}^{m}\frac{\partial H_{i}}{\partial \mu_{\alpha}}\text{, }\frac{\partial b_{\alpha}}{\partial \tau}=\frac{\partial \mathcal{H}}{\partial \mu_{\alpha}}\text{, }\frac{\partial
\mu_{\alpha}}{\partial t_{i}}=-\sum_{i=1}^{m}\frac{\partial H_{i}}{\partial
b_{\alpha}}\text{, }\frac{\partial \mu_{\alpha}}{\partial \tau}=-\frac
{\partial \mathcal{H}}{\partial b_{\alpha}},\label{Ham}$$ where$$\mathcal{H}=\frac{1}{2\pi i}\left[ L+\eta_{1}(\tau)\left( \sum_{\alpha
=0}^{m}b_{\alpha}\mu_{\alpha}-\sum_{i=1}^{m}t_{i}H_{i}\right) \right] .$$ Now considering the simplest case $m=0$ and by using (\[OK-2\]) and $t_{0}=0$, the potential $q(z)$ takes the simple form (the subscript $0$ is dropped for simplicity)$$q\left( z\right) =L+\mu \zeta \left( z|\tau \right) +\frac{1}{4}\left(
\theta^{2}-1\right) \wp \left( z|\tau \right) -\mu \zeta \left( z-b|\tau
\right) +\frac{3}{4}\wp \left( z-b|\tau \right) .\label{potential}$$ Consequently, the Hamiltonian system (\[Ham\]) is reduced to$$\left \{
\begin{array}
[c]{l}\frac{db}{d\tau}=\frac{-i}{2\pi}\left[ 2\mu-\zeta(b|\tau)+b\eta_{1}\right]
,\\
\\
\frac{d\mu}{d\tau}=\frac{i}{2\pi}\left[ \mu \wp(b|\tau)+\mu \eta_{1}-\frac
{1}{4}(\theta^{2}-1)\wp^{\prime}(b|\tau)\right] .
\end{array}
\right. \label{Ham-1}$$ Furthermore, the Hamiltonian system (\[Ham-1\]) is equivalent to$$\frac{d^{2}}{d\tau^{2}}\left( \frac{b}{2}\right) =-\frac{1}{4\pi^{2}}\sum_{k=0}^{3}\frac{\theta^{2}}{32}\wp^{\prime}\left( \frac{b}{2}+\frac{\omega_{k}}{2}|\tau \right) ,$$ which implies that $\frac{b}{2}$ satisfies the elliptic form (\[124\]) with $\left( \alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3}\right) =(\frac
{\theta^{2}}{32},\frac{\theta^{2}}{32},\frac{\theta^{2}}{32},\frac{\theta^{2}}{32})$; see [@Kawai Theorem 3]. It is clear that our potential $I\left(
z\right) $ is *different from* (\[potential\]) except for $n_{k}=0,k=0,1,2,3$ in $I\left( z\right) $ and $\theta=\pm2$ in (\[potential\]). Notice that the linear ODE (\[OK\]) with (\[potential\]) only has the apparent singularity at $b$. Thus it seems that the monodromy representation for (\[OK\]) could not be reduced to $\pi_{1}\left( E_{\tau
}\right) $ when $\theta \not =\pm2$. However, when $n_{k}\in \mathbb{N}\cup \left \{ 0\right \} $ for all $k$, the monodromy representation for (\[89-0\]) could be simplified. We remark it is an advantage when we study the elliptic form (\[124\]) with $\alpha_{k}=\frac{1}{2}\left( n_{k}+\frac{1}{2}\right) ^{2},\ k=0,1,2,3$. From Kawai’s result in [@Kawai] and ours, it can be seen that the elliptic form (\[124\]) governs isomonodromic deformations of different linear ODEs (e.g. (\[OK\]) with (\[potential\]) and (\[89-0\])). Therefore, it is important to choose a suitable linear ODE when generic parameters ** are considered.
Painlevé VI and Hamiltonian system on the moduli space
======================================================
In this section, we want to develop an isomonodromy theory on the moduli space of elliptic curves. For this purpose, there are two fundamental issues needed to be discussed: (i) to derive the Hamiltonian system for the isomonodromic deformation of the generalized Lamé equation (\[89-0\]) with $n_{i}\not \in \frac{1}{2}+\mathbb{Z}$, $i=0,1,2,3$; (ii) to prove the equivalence between the Hamiltonian system and the elliptic form (\[124\]). We remark that the (ii) part holds true without any condition. Recall the generalized Lamé equation defined by $$y^{\prime \prime}=I\left( z;\tau \right) y, \label{152}$$ where$$\begin{aligned}
I\left( z;\tau \right) = & \sum_{i=0}^{3}n_{i}\left( n_{i}+1\right)
\wp \left( z+\frac{\omega_{i}}{2}|\tau \right) +\frac{3}{4}\left( \wp \left(
z+p|\tau \right) +\wp \left( z-p|\tau \right) \right) \nonumber \\
& +A\left( \zeta \left( z+p|\tau \right) -\zeta \left( z-p|\tau \right)
\right) +B, \label{154}$$ and $p\left( \tau \right) \not \in E_{\tau}\left[ 2\right] $. By replacing $n_{i}$ by $-n_{i}-1$ if necessary, we always assume $n_{i}\geq-\frac{1}{2}$ for all $i$. Remark that, since we assume $n_{i}\not \in \frac{1}{2}+\mathbb{Z}$, the exponent difference of (\[152\]) at $\frac{\omega_{i}}{2}$ is $2n_{i}+1\not \in 2\mathbb{Z}$, implying that (\[152\]) has no logarithmic singularity at $\frac{\omega_{i}}{2}$, $0\leq i\leq3$.
For equation (\[152\]), the necessary and sufficient condition for apparent singularity at $\pm p$ is given by
\[lem-apparent\]$\pm p$ are apparent singularities of (\[152\]) iff $A$ and $B$ satisfy$$B=A^{2}-\zeta \left( 2p\right) A-\frac{3}{4}\wp \left( 2p\right) -\sum
_{i=0}^{3}n_{i}\left( n_{i}+1\right) \wp \left( p+\frac{\omega_{i}}{2}\right) . \label{101}$$
It suffices to prove this lemma for the point $p$. Let $y_{i}$, $i=1,2$, be two linearly independent solutions to (\[152\]). Define $f\doteqdot
\frac{y_{1}}{y_{2}}$ as a ratio of two independent solutions and $v\doteqdot \log f^{\prime}$. Then$$\{f;z\}=v^{\prime \prime}-\frac{1}{2}\left( v^{\prime}\right) ^{2}=-2I(z).
\label{78}$$ It is obvious that (\[152\]) has no solutions with logarithmic singularity at $p$ iff $f\left( z\right) $ has no logarithmic singularity at $p$. First we prove the necessary part. Without loss of generality, we may assume $f\left( z\right) $ is holomorphic at $p$. The local expansion of $f$ at $p$ is:$$f\left( z\right) =c_{0}+c_{2}\left( z-p\right) ^{2}+\cdot \cdot \cdot,$$$$v\left( z\right) =\log f^{\prime}\left( z\right) =\log2c_{2}+\log \left(
z-p\right) +\sum_{j\geq1}d_{j}\left( z-p\right) ^{j}, \label{78-0}$$$$v^{\prime}\left( z\right) =\frac{1}{z-p}+\sum_{j\geq0}\tilde{e}_{j}\left(
z-p\right) ^{j}, \label{78-1}$$$$v^{\prime \prime}\left( z\right) =\frac{-1}{\left( z-p\right) ^{2}}+\sum_{j\geq0}\left( j+1\right) \tilde{e}_{j+1}\left( z-p\right) ^{j},$$ where $\tilde{e}_{j}=\left( j+1\right) d_{j+1}$. Thus,[$$\begin{aligned}
v^{\prime \prime}-\frac{1}{2}\left( v^{\prime}\right) ^{2}= & \frac
{-1}{\left( z-p\right) ^{2}}+\sum_{j\geq0}\left( j+1\right) \tilde
{e}_{j+1}\left( z-p\right) ^{j}\label{78-2}\\
& -\frac{1}{2}\Big[\frac{1}{z-p}+\sum_{j\geq0}\tilde{e}_{j}\left(
z-p\right) ^{j}\Big]^{2}.\nonumber\end{aligned}$$ ]{}Recalling $I(z)$ in (\[154\]), we compare both sides of (\[78\]). The $\left( z-p\right) ^{-2}$ terms match automatically. For the $\left(
z-p\right) ^{-1}$ term, we get$$-\tilde{e}_{0}=2A. \label{79}$$ For the $\left( z-p\right) ^{0}$, i.e. the constant term, we have[$$\begin{aligned}
& \tilde{e}_{1}-\frac{2}{2}\tilde{e}_{1}-\frac{1}{2}\tilde{e}_{0}^{2}\label{80}\\
= & -2\sum_{i=0}^{3}n_{i}\left( n_{i}+1\right) \wp \left( p+\frac
{\omega_{i}}{2}\right) -\frac{3}{2}\wp \left( 2p\right) -2A\zeta \left(
2p\right) -2B.\nonumber\end{aligned}$$ ]{}Then (\[101\]) follows from (\[79\]) and (\[80\]) immediately.
For the sufficient part, if (\[101\]) holds, then $\tilde{e}_{0}$ is given by (\[79\]). By any choice of $\tilde{e}_{1}$ and comparing (\[78\]) and (\[78-2\]), $\tilde{e}_{j}$ is determined for all $j\geq2$. Then it follows from (\[78-0\])-(\[78-1\]) that $f\left( z\right) $ is holomorphic at $p$. Since its Schwarzian derivative satisfies (\[78\]), $f$ is a ratio of two linearly independent solutions of (\[152\]). This implies that (\[152\]) has no solutions with logarithmic singularity at $p$, namely $p$ is an apparent singularity.
Isomonodromic equation and Hamiltonian system
---------------------------------------------
The 2nd order generalized Lamé equation (\[152\]) can be written into a 1st order linear system$$\frac{d}{dz}Y=Q\left( z;\tau \right) Y\text{ \ in }E_{\tau},\label{153}$$ where$$Q\left( z;\tau \right) =\left(
\begin{matrix}
0 & 1\\
I\left( z;\tau \right) & 0
\end{matrix}
\right) .\label{176}$$ The isomonodromic deformation of the generalized Lamé equation (\[152\]) is equivalent to the isomonodromic deformation of the linear system (\[153\]). Let $y_{1}\left( z;\tau \right) $ and $y_{2}\left(
z;\tau \right) $ be two linearly independent solutions of (\[152\]), then $Y\left( z;\tau \right) =\left(
\begin{matrix}
y_{1}\left( z;\tau \right) & y_{2}\left( z;\tau \right) \\
y_{1}^{\prime}\left( z;\tau \right) & y_{2}^{\prime}\left( z;\tau \right)
\end{matrix}
\right) $ is a fundamental system of solutions to (\[153\]). In general, $Y\left( z;\tau \right) $ is multi-valued with respect to $z$ and for each $\tau \in \mathbb{H}$, $Y\left( z;\tau \right) $ might have branch points at $S\doteqdot \left \{ \pm p,\frac{\omega_{k}}{2}|\text{ }k=0,1,2,3\right \} $. The fundamental solution $Y\left( z;\tau \right) $ is called $M$-invariant ($M$ stands for monodromy) if there is some $q_{0}\in E_{\tau}\backslash S$ and for any loop $\ell \in \pi_{1}(E_{\tau}\backslash S$, $q_{0})$, there exists $\rho \left( \ell \right) \in SL\left( 2,\mathbb{C}\right) $ *independent of* $\tau$ such that$$\ell^{\ast}Y\left( z;\tau \right) =Y\left( z;\tau \right) \rho \left(
\ell \right)$$ holds for $z$ near the base point $q_{0}$. Here $\ell^{\ast}Y\left(
z;\tau \right) $ denotes the analytic continuation of $Y\left( z;\tau \right)
$ along $\ell$. Let $\gamma_{k}\in \pi_{1}(E_{\tau}\backslash S,q_{0}),$ $k=0,1,2,3,\pm$, be simple loops which encircle the singularties $\frac
{\omega_{k}}{2},$ $k=0,1,2,3$ and $\pm p$ once respectively, and $\ell_{j}\in \pi_{1}(E_{\tau}\backslash S,q_{0})$, $j=1,2$, be two fundamental cycles of $E_{\tau}$ such that its lifting in $\mathbb{C}$ is a straight line connecting $q_{0}$ and $q_{0}+\omega_{j}$. We also require these lines do not pass any singularties. Of course, all the pathes do not intersect with each other except at $q_{0}$. We note that when $\tau$ varies in a neighborhood of some $\tau_{0}$, $\gamma_{k}$ and $\ell_{1}$ can be choosen independent of $\tau$.
Clearly the monodromy group with respect to $Y\left( z;\tau \right) $ is generated by $\left \{ \rho \left( \ell_{j}\right) ,\rho \left( \gamma
_{k}\right) |\text{ }j=1,2\text{ and }k=0,1,2,3,\pm \right \} $. Thus, $Y\left( z;\tau \right) $ is $M$-invariant if and only if the matrices $\rho \left( \ell_{j}\right) ,\rho \left( \gamma_{k}\right) $ are independent of $\tau$. Notice that $I\left( \cdot;\tau \right) $ is an elliptic function, so we can also treat (\[153\]) as a equation defined in $\mathbb{C}$, i.e.,$$\frac{d}{dz}Y=Q\left( z;\tau \right) Y\text{ \ in }\mathbb{C}. \label{153-1}$$ Furthermore, we can identify solutions of (\[153\]) and (\[153-1\]) in an obvious way. For example, after analytic continuation, any solution $Y(\cdot;\tau)$ of (\[153\]) can be extended to be a solution of (\[153-1\]) as a multi-valued matrix function defined in $\mathbb{C}$ (still denote it by $Y(\cdot;\tau)$). In the sequel, we always identify solutions of (\[153\]) and (\[153-1\]). Then we have the following theorem:
\[thm M\]System (\[153\]) is monodromy preserving as $\tau$ deforms if and only if there exists a single-valued matrix function $\Omega \left(
z;\tau \right) $ defined in $\mathbb{C}\times \mathbb{H}$ satisfying$$\left \{
\begin{array}
[c]{l}\Omega \left( z+1;\tau \right) =\Omega \left( z;\tau \right) \\
\Omega \left( z+\tau;\tau \right) =\Omega \left( z;\tau \right) -Q\left(
z;\tau \right) ,
\end{array}
\right. \label{204}$$ such that the following Pfaffian ** system$$\left \{
\begin{array}
[c]{l}\frac{\partial}{\partial z}Y(z;\tau)=Q\left( z;\tau \right) Y(z;\tau
)\smallskip \\
\frac{\partial}{\partial \tau}Y(z;\tau)=\Omega \left( z;\tau \right) Y(z;\tau)
\end{array}
\right. \text{ \ in \ }\mathbb{C}\times \mathbb{H} \label{184}$$ is completely integrable.
\[remk\]The classical isomonodromy theory in $\mathbb{C}$ (see e.g. [@GP Proposition 3.1.5]) says that system (\[153-1\]) is monodromy preserving if and only if there exists a single-valued matrix function $\Omega \left( z;\tau \right) $ defined in $\mathbb{C}\times \mathbb{H}$ such that (\[184\]) is completely integrable. Theorem \[thm M\] is the counterpart of this classical theory in the torus $E_{\tau}$. The property (\[204\]) comes from the preserving of monodromy matrices $\rho \left(
\ell_{j}\right) $, $j=1,2$ during the deformation (see from the proof of Theorem \[thm M\] below). Notice that $\rho \left( \ell_{j}\right) $ can be considered as connection matrices along the straight line $\ell_{j}$ connecting $q_{0}$ and $q_{0}+\omega_{j}$ for system (\[153-1\]).
Notice that system (\[184\]) is completely integrable if and only if $$\frac{\partial}{\partial \tau}Q\left( z;\tau \right) =\frac{\partial}{\partial
z}\Omega \left( z;\tau \right) +\left[ \Omega \left( z;\tau \right) ,Q\left(
z;\tau \right) \right] ,\text{ and } \label{185}$$$$d(\Omega \left( z;\tau \right) d\tau)=\left[ \Omega \left( z;\tau \right)
d\tau \right] \wedge \left[ \Omega \left( z;\tau \right) d\tau \right] ,
\label{185-1}$$ where $d$ denotes the exterior differentiation with respect to $\tau$ in (\[185-1\]). See Lemma 3.14 in [@GP] for the proof. Clearly (\[185-1\]) holds automatically since there is only one deformation parameter. We need the following lemma to prove Theorem \[thm M\].
\[lemma\]Let $Y\left( z;\tau \right) $ be an $M$-invariant fundamental solution of system (\[153\]) and define a $2\times2$ matrix-valued function $\Omega \left( z;\tau \right) $ in $E_{\tau}$ by$$\Omega \left( z;\tau \right) =\frac{\partial}{\partial \tau}Y\cdot Y^{-1}.
\label{w}$$ Then $\Omega \left( z;\tau \right) $ can be extended to be a globally defined matrix-valued function in $\mathbb{C\times H}$ by analytic continuation (still denote it by $\Omega \left( z;\tau \right) $). In particular, (\[w\]) holds in $\mathbb{C\times H}$ by considering $Y\left( z;\tau \right) $ as a solution of system (\[153-1\]).
The proof is the same as that in the classical isomonodromy theory in $\mathbb{C}$. Indeed, since $Y\left( z;\tau \right) $ is $M$-invariant, we have[$$\begin{aligned}
\gamma_{k}^{\ast}\Omega \left( z;\tau \right) & =\gamma_{k}^{\ast}\left(
\frac{\partial}{\partial \tau}Y\cdot Y^{-1}\right) =\frac{\partial}{\partial \tau}\gamma_{k}^{\ast}Y\cdot \gamma_{k}^{\ast}Y^{-1}\label{qqq6}\\
& =\frac{\partial}{\partial \tau}\left( Y\rho \left( \gamma_{k}\right)
\right) \cdot \rho \left( \gamma_{k}\right) ^{-1}Y^{-1}\nonumber \\
& =\frac{\partial}{\partial \tau}Y\cdot Y^{-1}=\Omega \left( z;\tau \right)
\nonumber\end{aligned}$$ ]{}for $k=0,1,2,3,\pm$, namely $\Omega \left( \cdot;\tau \right) $ is invariant under the analytic continuation along $\gamma_{k}$. Thus, $\Omega \left(
\cdot;\tau \right) $ is single-valued in any fundamental domain of $E_{\tau}$ for each $\tau$. Then for each $\tau \in \mathbb{H}$, we could extend $\Omega \left( z;\tau \right) $ to be a globally defined matrix-valued function in $\mathbb{C}$ by analytic continuation.
From now on, we consider equation (\[153\]) defined in $\mathbb{C}$, i.e., (\[153-1\]). The analytic continuation along any curve in $\mathbb{C}$ always keep the relation (\[w\]) between $Y\left( z;\tau \right) $ and $\Omega \left( z;\tau \right) $.
\[Proof of Theorem \[thm M\]\]First we prove the necessary part. Let $Y\left(
z;\tau \right) $ be an $M$-invariant fundamental solution of system (\[153\]) and define $\Omega \left( z;\tau \right) $ by $Y\left(
z;\tau \right) $. By Lemma \[lemma\], $\Omega \left( z;\tau \right) $ is a single-valued matrix function in $\mathbb{C}\times \mathbb{H}$ and $Y(z;\tau)$ is a solution of (\[184\]), which implies (\[185\]). Hence the Pfaffian ** system (\[184\]) is completely integrable.
It suffices to prove that $\Omega(z;\tau)$ satisfies (\[204\]). Note that $\Omega(z;\tau)$ is single-valued in $\mathbb{C\times H}$. Therefore, to prove (\[204\]), we only need to prove its validity in a small neighborhood $U_{q_{0}}\times V_{\tau_{0}}$ of some $(q_{0},\tau_{0})$, where $q_{0}$ is the base point. By considering $Y\left( z;\tau \right) $ as a solution of system (\[153-1\]), we see from Remark \[remk\] and Lemma \[lemma\] that, for $(z,\tau)\in U_{q_{0}}\times V_{\tau_{0}}$,$$Y(z+\omega_{i};\tau)=Y(z;\tau)\rho \left( \ell_{i}\right) , \label{1001}$$$$\Omega \left( z;\tau \right) =\frac{\partial}{\partial \tau}Y(z;\tau)\cdot
Y(z;\tau)^{-1}, \label{1002}$$$$\Omega \left( z+\omega_{i};\tau \right) =\frac{\partial}{\partial \tau
}Y(z+\omega_{i};\tau)\cdot Y(z+\omega_{i};\tau)^{-1}. \label{1003}$$ Therefore, (\[1001\]) and (\[1003\]) give[$$\begin{aligned}
& \Omega \left( z+\omega_{i};\tau \right) \\
& =\left[ \frac{d}{d\tau}Y\left( z+\omega_{i};\tau \right) -\frac{\partial
}{\partial z}Y\left( z+\omega_{i};\tau \right) \frac{d}{d\tau}\omega
_{i}\right] \cdot Y\left( z+\omega_{i};\tau \right) ^{-1}\\
& =\left[ \frac{d}{d\tau}\left( Y\left( z;\tau \right) \rho \left(
\ell_{i}\right) \right) -\frac{\partial}{\partial z}\left( Y\left(
z;\tau \right) \rho \left( \ell_{i}\right) \right) \frac{d}{d\tau}\omega
_{i}\right] \cdot \left( Y\left( z;\tau \right) \rho \left( \ell_{i}\right)
\right) ^{-1}.\end{aligned}$$ ]{}Since $\rho \left( \ell_{i}\right) ,i=1,2$, are independent of $\tau,$ we have$$\Omega \left( z+1;\tau \right) =\frac{d}{d\tau}Y\left( z;\tau \right) \cdot
Y\left( z;\tau \right) ^{-1}=\text{$\Omega \left( z;\tau \right) ,$}$$ and$$\begin{aligned}
\Omega \left( z+\tau;\tau \right) & =\frac{d}{d\tau}Y\left( z;\tau \right)
\cdot Y\left( z;\tau \right) ^{-1}-\frac{\partial}{\partial z}Y\left(
z;\tau \right) \cdot Y\left( z;\tau \right) ^{-1}\\
& =\text{$\Omega \left( z;\tau \right) $}-Q\left( z;\tau \right) .\end{aligned}$$ This proves (\[204\]).
Conversely, suppose there exists a single-valued matrix function $\Omega \left( z;\tau \right) $ in $\mathbb{C}\times \mathbb{H}$ satisfying (\[204\]) such that (\[184\]) is completely integrable. Let $Y\left(
z;\tau \right) $ be a solution of the Pfaffian ** system (\[184\]). Then (\[1001\])-(\[1003\]) hold and $Y\left( z;\tau \right) $ satisfies system (\[153\]) in $E_{\tau}$. Hence$$\frac{\partial}{\partial \tau}Y\left( z+\omega_{i};\tau \right) =\frac
{d}{d\tau}Y\left( z+\omega_{i};\tau \right) -\frac{\partial}{\partial
z}Y\left( z+\omega_{i};\tau \right) \frac{d}{d\tau}\omega_{i},$$ which implies[$$\begin{aligned}
\frac{\partial}{\partial \tau}Y\left( z+1;\tau \right) & =\frac{d}{d\tau
}\left( Y\left( z;\tau \right) \rho \left( \ell_{1}\right) \right)
\label{qqq2}\\
& =\frac{\partial}{\partial \tau}Y\left( z;\tau \right) \cdot \rho \left(
\ell_{1}\right) +Y\left( z;\tau \right) \frac{d}{d\tau}\rho \left( \ell
_{1}\right) \nonumber \\
& =\text{$\Omega \left( z;\tau \right) $}Y\left( z;\tau \right) \rho \left(
\ell_{1}\right) +Y\left( z;\tau \right) \frac{d}{d\tau}\rho \left( \ell
_{1}\right) \nonumber\end{aligned}$$ ]{}and [$$\begin{aligned}
& \frac{\partial}{\partial \tau}Y\left( z+\tau;\tau \right) \label{qqq3}\\
& =\frac{d}{d\tau}\left( Y\left( z;\tau \right) \rho \left( \ell
_{2}\right) \right) -\frac{\partial}{\partial z}\left( Y\left(
z;\tau \right) \rho \left( \ell_{2}\right) \right) \nonumber \\
& =\frac{\partial}{\partial \tau}Y\left( z;\tau \right) \cdot \rho \left(
\ell_{2}\right) +Y\left( z;\tau \right) \frac{d}{d\tau}\rho \left( \ell
_{2}\right) -\frac{\partial}{\partial z}Y\left( z;\tau \right) \cdot
\rho \left( \ell_{2}\right) \nonumber \\
& =\left[ \text{$\Omega \left( z;\tau \right) -Q\left( z;\tau \right) $}\right] Y\left( z;\tau \right) \rho \left( \ell_{2}\right) +Y\left(
z;\tau \right) \frac{d}{d\tau}\rho \left( \ell_{2}\right) .\nonumber\end{aligned}$$ ]{}On the other hand, by (\[1001\]) and (\[1003\]), we also have$$\frac{\partial}{\partial \tau}Y\left( z+\omega_{i};\tau \right) =\text{$\Omega
\left( z+\omega_{i};\tau \right) $}Y\left( z;\tau \right) \rho \left(
\ell_{i}\right) . \label{qqq4}$$ Then by (\[qqq2\]), (\[qqq3\]), (\[qqq4\]) and (\[204\]), we have$$Y\left( z;\tau \right) \frac{d}{d\tau}\rho \left( \ell_{1}\right) =Y\left(
z;\tau \right) \frac{d}{d\tau}\rho \left( \ell_{2}\right) =0.$$ Also, by the same argument as (\[qqq6\]), we could prove$$Y\left( z;\tau \right) \frac{d}{d\tau}\rho \left( \gamma_{k}\right)
=0\text{, }k=0,1,2,3,\pm.$$ Because of $\det Y\not =0$, we conclude that$$\frac{d}{d\tau}\rho \left( \ell_{j}\right) =\frac{d}{d\tau}\rho \left(
\gamma_{k}\right) =0.$$ Thus, $Y$ is an $M$-invariant solution of (\[153\]). That is, system (\[153\]) is monodromy preserving. This completes the proof.
Write $\Omega \left( z;\tau \right) =\left(
\begin{matrix}
\Omega_{11} & \Omega_{12}\\
\Omega_{21} & \Omega_{22}\end{matrix}
\right) $. Since $Q\left( z;\tau \right) $ has the special form (\[176\]), by a straightforward computation, the integrability condition (\[185\]) is equivalent to$$\Omega_{12}^{\prime \prime \prime}-4I\Omega_{12}^{\prime}-2I^{\prime}\Omega
_{12}+2\frac{\partial}{\partial \tau}I=0\text{ \ in }\mathbb{C\times
H},\label{186}$$ where we denote $^{\prime}=\frac{\partial}{\partial z}$ to be the partial derivative with respect to the variable $z$. This computation is the same as the case in $\mathbb{C}$ (see e.g. [@GP Proposition 3.5.1]), so we omit the details. Then we have the following fundamental theorem for isomonodromic deformations of (\[153\]) in the moduli space of elliptic curves:
\[thm M1\]System (\[153\]) is monodromy preserving as $\tau$ deforms if and only if there exists a single-valued solution $\Omega_{12}\left(
z;\tau \right) $ to (\[186\]) satisfying $$\begin{aligned}
\Omega_{12}\left( z+1;\tau \right) & =\Omega_{12}\left( z;\tau \right)
,\label{qqq}\\
\Omega_{12}\left( z+\tau;\tau \right) & =\Omega_{12}\left( z;\tau \right)
-1.\nonumber\end{aligned}$$
By Theorem \[thm M\], it suffices to prove the sufficient part. Suppose there exists a single-valued solution $\Omega_{12}\left( z;\tau \right) $ to (\[186\]) satisfying (\[qqq\]). Then we define $\Omega \left(
z;\tau \right) =\left(
\begin{matrix}
\Omega_{11} & \Omega_{12}\\
\Omega_{21} & \Omega_{22}\end{matrix}
\right) $ by setting[$$\begin{aligned}
\Omega_{11}\left( z;\tau \right) & =-\frac{1}{2}\Omega_{12}^{\prime}\left(
z;\tau \right) ,\label{qq}\\
\Omega_{21}\left( z;\tau \right) & =\Omega_{11}^{\prime}\left(
z;\tau \right) +\Omega_{12}\left( z;\tau \right) I\left( z;\tau \right)
,\nonumber \\
\Omega_{22}\left( z;\tau \right) & =\Omega_{12}^{\prime}\left(
z;\tau \right) +\Omega_{11}\left( z;\tau \right) .\nonumber\end{aligned}$$ ]{}By (\[186\]), it is easy to see that $\Omega \left( z;\tau \right) $ satisfies the integrability condition (\[185\]) (see e.g. [@GP Proposition 3.5.1]), namely (\[184\]) is completely integrable. Finally, (\[204\]) follows from (\[qqq\]). This completes the proof.
The first main result of this section is as follows:
\[thm4-2\]Let $n_{k}\not \in \frac{1}{2}+\mathbb{Z}$, $k=0,1,2,3$ and $p\left( \tau \right) $ is an apparent singular point of the generalized Lamé equation (\[152\]) with (\[154\]). Then (\[152\]) with $\left(
p,A\right) =\left( p\left( \tau \right) ,A\left( \tau \right) \right) $ is an isomonodromic deformation with respect to $\tau$ if and only if $\left(
p\left( \tau \right) ,A\left( \tau \right) \right) $ satisfies the Hamiltonian system:$$\frac{dp\left( \tau \right) }{d\tau}=\frac{\partial K\left( p,A,\tau \right)
}{\partial A},\text{ \ }\frac{dA\left( \tau \right) }{d\tau}=-\frac{\partial
K\left( p,A,\tau \right) }{\partial p}, \label{142}$$ where$$K\left( p,A,\tau \right) =\frac{-i}{4\pi}\left(
\begin{array}
[c]{l}A^{2}+\left( -\zeta \left( 2p|\tau \right) +2p\eta_{1}\left( \tau \right)
\right) A-\frac{3}{4}\wp \left( 2p|\tau \right) \\
-\sum_{k=0}^{3}n_{k}\left( n_{k}+1\right) \wp \left( p+\frac{\omega_{k}}{2}|\tau \right)
\end{array}
\right) . \label{143}$$
To prove Theorem \[thm4-2\], we need the following formulae for theta functions and functions in Weierstrass elliptic function theory.
\[lem4-1\] The following formulae hold:
(i)$$\frac{\partial}{\partial \tau}\ln \sigma \left( z|\tau \right) =\frac{i}{4\pi
}\left[ \wp \left( z|\tau \right) -\zeta^{2}\left( z|\tau \right) +2\eta
_{1}\left( z\zeta \left( z|\tau \right) -1\right) -\frac{1}{12}g_{2}z^{2}\right] ,$$ (ii)$$\frac{\partial}{\partial \tau}\zeta \left( z|\tau \right) =\frac{i}{4\pi
}\left[
\begin{array}
[c]{l}\wp^{\prime}\left( z|\tau \right) +2\left( \zeta \left( z|\tau \right)
-z\eta_{1}\left( \tau \right) \right) \wp \left( z|\tau \right) \\
+2\eta_{1}\zeta \left( z|\tau \right) -\frac{1}{6}zg_{2}\left( \tau \right)
\end{array}
\right] ,$$ (iii)$$\frac{\partial}{\partial \tau}\wp \left( z|\tau \right) =\frac{-i}{4\pi}\left[
\begin{array}
[c]{l}2\left( \zeta \left( z|\tau \right) -z\eta_{1}\left( \tau \right) \right)
\wp^{\prime}\left( z|\tau \right) \\
+4\left( \wp \left( z|\tau \right) -\eta_{1}\right) \wp \left(
z|\tau \right) -\frac{2}{3}g_{2}\left( \tau \right)
\end{array}
\right] ,$$ (iv)$$\frac{\partial}{\partial \tau}\wp^{\prime}\left( z|\tau \right) =\frac
{-i}{4\pi}\left[
\begin{array}
[c]{l}6\left( \wp \left( z|\tau \right) -\eta_{1}\right) \wp^{\prime}\left(
z|\tau \right) \\
+\left( \zeta \left( z|\tau \right) -z\eta_{1}\left( \tau \right) \right)
\left( 12\wp^{2}\left( z|\tau \right) -g_{2}\left( \tau \right) \right)
\end{array}
\right] ,$$ (v)$$\frac{d}{d\tau}\eta_{1}\left( \tau \right) =\frac{i}{4\pi}\left[ 2\eta
_{1}^{2}-\frac{1}{6}g_{2}\left( \tau \right) \right] ,$$ (vi)$$\frac{d}{d\tau}\ln \theta_{1}^{\prime}\left( \tau \right) =\frac{3i}{4\pi}\eta_{1},$$ where$$g_{2}\left( \tau \right) =-4\left( e_{1}\left( \tau \right) e_{2}\left(
\tau \right) +e_{1}\left( \tau \right) e_{3}\left( \tau \right)
+e_{2}\left( \tau \right) e_{3}\left( \tau \right) \right) ,$$$$\theta_{1}^{\prime}\left( \tau \right) \doteqdot \frac{d}{dz}\vartheta
_{1}\left( z;\tau \right) |_{z=0},\text{ \ }\frac{d}{dz}\ln \sigma \left(
z|\tau \right) \doteqdot \zeta \left( z|\tau \right) ,$$$$\vartheta_{1}\left( z;\tau \right) \doteqdot-i\sum_{n=-\infty}^{\infty
}(-1)^{n}e^{(n+\frac{1}{2})^{2}\pi i\tau}e^{(2n+1)\pi iz}.\label{ccc}$$
Those formulae in Lemma \[lem4-1\] are known in the literature; see e.g. [@YB] and references therein for the proofs.
To give a motivation for our proof of Theorem \[thm4-2\], we first consider the simplest case $n_{k}=0,\forall k$: Let $a_{1}=r+s\tau$ where $\left(
r,s\right) \in \mathbb{C}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$ is a fixed pair and $\pm p\left( \tau \right) $, $A\left( \tau \right) $, $B\left(
\tau \right) $ be defined by$$\zeta \left( a_{1}\left( \tau \right) +p\left( \tau \right) \right)
+\zeta \left( a_{1}\left( \tau \right) -p\left( \tau \right) \right)
-2\left( r\eta_{1}(\tau)+s\eta_{2}(\tau)\right) =0, \label{132}$$$$A=\frac{1}{2}\left[ \zeta \left( p+a_{1}\right) +\zeta \left( p-a_{1}\right) -\zeta \left( 2p\right) \right] , \label{44}$$$$B=A^{2}-\zeta \left( 2p\right) A-\frac{3}{4}\wp \left( 2p\right) ,
\label{45}$$ respectively. In [@Chen-Kuo-Lin] we could prove that under (\[132\])-(\[45\]), the two functions$$y_{\pm a_{1}}\left( z;\tau \right) =e^{\pm \frac{z}{2}\left( \zeta \left(
a_{1}+p\right) +\zeta \left( a_{1}-p\right) \right) }\frac{\sigma \left(
z\mp a_{1}\right) }{\left[ \sigma \left( z+p\right) \sigma \left(
z-p\right) \right] ^{\frac{1}{2}}}$$ are two linearly independent solutions to the generalized Lamé equation (\[152\]) with $n_{k}=0$, $k=0,1,2,3$, i.e., $$y^{\prime \prime}=\left[ \frac{3}{4}\left( \wp \left( z+p\right) +\wp \left(
z-p\right) \right) +A\left( \zeta \left( z+p\right) -\zeta \left(
z-p\right) \right) +B\right] y. \label{187}$$ Observe that (\[187\]) has singularties only at $\pm p$. Thus, the monodromy representation of (\[187\]) is a group homomorphism $\rho:\pi_{1}\left(
E_{\tau}\backslash \left \{ \pm p\right \} ,q_{0}\right) \rightarrow SL\left(
2,\mathbb{C}\right) $. Then we could also compute the monodromy group of (\[187\]) with respect to $\left( y_{a_{1}}\left( z;\tau \right)
,y_{-a_{1}}\left( z;\tau \right) \right) ^{t}$ as following [@Chen-Kuo-Lin]:$$\rho(\gamma_{\pm})\left(
\begin{matrix}
y_{a_{1}}\left( z;\tau \right) \\
y_{-a_{1}}\left( z;\tau \right)
\end{matrix}
\right) =\left(
\begin{matrix}
-1 & 0\\
0 & -1
\end{matrix}
\right) \left(
\begin{matrix}
y_{a_{1}}\left( z;\tau \right) \\
y_{-a_{1}}\left( z;\tau \right)
\end{matrix}
\right) , \label{155-1}$$$$\rho(\ell_{1})\left(
\begin{matrix}
y_{a_{1}}\left( z;\tau \right) \\
y_{-a_{1}}\left( z;\tau \right)
\end{matrix}
\right) =\left(
\begin{matrix}
e^{-2\pi is} & 0\\
0 & e^{2\pi is}\end{matrix}
\right) \left(
\begin{matrix}
y_{a_{1}}\left( z;\tau \right) \\
y_{-a_{1}}\left( z;\tau \right)
\end{matrix}
\right) , \label{155}$$$$\rho(\ell_{2})\left(
\begin{matrix}
y_{a_{1}}\left( z;\tau \right) \\
y_{-a_{1}}\left( z;\tau \right)
\end{matrix}
\right) =\left(
\begin{matrix}
e^{2\pi ir} & 0\\
0 & e^{-2\pi ir}\end{matrix}
\right) \left(
\begin{matrix}
y_{a_{1}}\left( z;\tau \right) \\
y_{-a_{1}}\left( z;\tau \right)
\end{matrix}
\right) . \label{156}$$ By (\[45\]) and Lemma \[lem-apparent\], $\pm p\left( \tau \right) $ are apparent singularities. Since the pair $\left( r,s\right) $ is fixed, we see from (\[155-1\])-(\[156\]) that the generalized Lamé equation (\[187\]) is monodromy preserving. Thus $Y=\left(
\begin{matrix}
y_{a_{1}}\left( z;\tau \right) & y_{-a_{1}}\left( z;\tau \right) \\
y_{a_{1}}^{\prime}\left( z;\tau \right) & y_{-a_{1}}^{\prime}\left(
z;\tau \right)
\end{matrix}
\right) $ is an $M$-invariant fundamental solution for the system (\[153\]). Then by Theorem \[thm M\], the single-valued matrix $\Omega \left( z;\tau \right) $ could be defined by$$\begin{aligned}
\Omega \left( z;\tau \right) & =\frac{\partial}{\partial \tau}Y\cdot Y^{-1}\\
& =\frac{1}{\det Y}\left(
\begin{matrix}
\frac{\partial}{\partial \tau}y_{a_{1}} & \frac{\partial}{\partial \tau
}y_{-a_{1}}\\
\frac{\partial}{\partial \tau}y_{a_{1}}^{\prime} & \frac{\partial}{\partial
\tau}y_{-a_{1}}^{\prime}\end{matrix}
\right) \left(
\begin{matrix}
y_{-a_{1}}^{\prime} & -y_{-a_{1}}\\
-y_{a_{1}}^{\prime} & y_{a_{1}}\end{matrix}
\right) ,\end{aligned}$$ which gives us [$$\Omega_{12}=\frac{y_{a_{1}}\frac{\partial}{\partial \tau}y_{-a_{1}}-y_{-a_{1}}\frac{\partial}{\partial \tau}y_{a_{1}}}{y_{a_{1}}y_{-a_{1}}^{\prime
}-y_{-a_{1}}y_{a_{1}}^{\prime}}=\frac{\frac{\partial}{\partial \tau}\ln
\frac{y_{a_{1}}}{y_{-a_{1}}}}{\frac{\partial}{\partial z}\ln \frac{y_{a_{1}}}{y_{-a_{1}}}}=\frac{\frac{\partial}{\partial \tau}\ln f\left( z;\tau \right)
}{\frac{\partial}{\partial z}\ln f\left( z;\tau \right) },$$ ]{}where $f\doteqdot \frac{y_{a_{1}}}{y_{-a_{1}}}$ is given by$$f\left( z;\tau \right) =e^{z\left( \zeta \left( a_{1}+p\right)
+\zeta \left( a_{1}-p\right) \right) }\frac{\sigma \left( z-a_{1}\right)
}{\sigma \left( z+a_{1}\right) }.$$ Using (\[132\]) and Legendre relation $\tau \eta_{1}-\eta_{2}=2\pi i$, we have$$f\left( z;\tau \right) =e^{2za_{1}\eta_{1}-4\pi isz}\frac{\sigma \left(
z-a_{1}\right) }{\sigma \left( z+a_{1}\right) }. \label{172}$$ In order to compute $\Omega_{12}$, we compute $\frac{\partial}{\partial \tau
}\ln f\left( z;\tau \right) $ and $\frac{\partial}{\partial z}\ln f\left(
z;\tau \right) $, respectively. By Lemma \[lem4-1\] and (\[172\]), we have[$$\begin{aligned}
& \frac{\partial}{\partial \tau}\ln f\left( z;\tau \right) \label{173}\\
= & 2zs\eta_{1}+2za_{1}\frac{d\eta_{1}}{d\tau}-\left( \zeta \left(
z-a_{1}|\tau \right) +\zeta \left( z+a_{1}|\tau \right) \right) s\nonumber \\
& +\frac{\partial}{\partial \tau}\ln \sigma \left( z-a_{1}|\tau \right)
-\frac{\partial}{\partial \tau}\ln \sigma \left( z+a_{1}|\tau \right) \nonumber \\
= & \frac{i}{4\pi}\left[ \zeta \left( z+a_{1}\right) -\zeta \left(
z-a_{1}\right) -2\eta_{1}a_{1}+4\pi is\right] \nonumber \\
& \times \left[ \zeta \left( z-a_{1}\right) +\zeta \left( z+a_{1}\right)
-2z\eta_{1}\right] +\frac{i}{4\pi}\left[ \wp \left( z-a_{1}\right)
-\wp \left( z+a_{1}\right) \right] ,\nonumber\end{aligned}$$ ]{}and $$\frac{\partial}{\partial z}\ln f\left( z;\tau \right) =2a_{1}\eta_{1}-4\pi
is+\zeta \left( z-a_{1}\right) -\zeta \left( z+a_{1}\right) . \label{174}$$ Thus from (\[173\]), (\[174\]) and (\[132\]), we have[$$\begin{aligned}
\Omega_{12}\left( z;\tau \right) = & -\frac{i}{4\pi}\left[ \zeta \left(
z-a_{1}\right) +\zeta \left( z+a_{1}\right) -2z\eta_{1}\right] \nonumber \\
& +\frac{i}{4\pi}\frac{\wp \left( z-a_{1}\right) -\wp \left( z+a_{1}\right)
}{2a_{1}\eta_{1}-4\pi is+\zeta \left( z-a_{1}\right) -\zeta \left(
z+a_{1}\right) }\nonumber \\
= & -\frac{i}{4\pi}\left[ \zeta \left( z-a_{1}\right) +\zeta \left(
z+a_{1}\right) -2z\eta_{1}\right] \label{175}\\
& +\frac{i}{4\pi}\frac{\wp \left( z-a_{1}\right) -\wp \left( z+a_{1}\right)
}{\zeta \left( a_{1}+p\right) +\zeta \left( a_{1}-p\right) +\zeta \left(
z-a_{1}\right) -\zeta \left( z+a_{1}\right) }.\nonumber\end{aligned}$$ ]{}From (\[175\]), we see that $\pm a_{1}$ are not poles of $\Omega
_{12}\left( z;\tau \right) $. In fact, $\pm p$ are the only simple poles and $0$ is a zero of $\Omega_{12}\left( z;\tau \right) $. Furthermore, we have$$\underset{z=\pm p}{\text{Res}}\Omega_{12}\left( z;\tau \right) =\frac
{-i}{4\pi}. \label{157}$$ By (\[175\]), it is easy to see that$$\Omega_{12}\left( -z;\tau \right) =-\Omega_{12}\left( z;\tau \right) .
\label{188}$$ By (\[157\]), (\[188\]) and (\[qqq\]), $\Omega_{12}\left(
z;\tau \right) $ has a simpler expression as follows:$$\Omega_{12}\left( z;\tau \right) =-\frac{i}{4\pi}\left( \zeta \left(
z-p\right) +\zeta \left( z+p\right) -2z\eta_{1}\right) .$$
For the general case, we do not have the explicit expression of the two linearly independent solutions. But the discussion above motivates us to find the explicit form of $\Omega_{12}$. For example, we might ask whether there exists $\Omega_{12}$ satisfying the property (\[188\]) or not. Thus, we need to study it via a different way. More precisely, we prove the following theorem:
\[thmA\]Under the assumption of Theorem \[thm4-2\], suppose the generalized Lamé equation (\[152\]) with $\left( p,A\right) =\left(
p\left( \tau \right) ,A\left( \tau \right) \right) $ is an isomonodromic deformation with respect to $\tau$. Then there exists an $M$-invariant fundamental solution $Y\left( z;\tau \right) $ of system (\[153\]) such that $\Omega_{12}\left( z;\tau \right) $ is of the form:$$\Omega_{12}\left( z;\tau \right) =-\frac{i}{4\pi}\left( \zeta \left(
z-p\left( \tau \right) \right) +\zeta \left( z+p\left( \tau \right)
\right) -2z\eta_{1}\right) , \label{c}$$ where $\Omega_{12}\left( z;\tau \right) $ is the (1,2) component of $\Omega \left( z;\tau \right) $ which is defined by $Y\left( z;\tau \right) $.
We remark that Theorem \[thmA\] is a result locally in $\tau$. In the following, we always assume that $V_{0}$ is a small neighborhood of $\tau_{0}$ such that $p\left( \tau \right) \not \in E_{\tau}\left[ 2\right] $ and $A\left( \tau \right) $, $B\left( \tau \right) $ are finite for $\tau \in
V_{0}$. First, we study the singularities of $\Omega_{12}\left(
z;\tau \right) $:
\[lem4-5\]Under the assumption and notations of Theorem \[thm4-2\], suppose $Y\left( z;\tau \right) $ is an $M$-invariant fundamental solution of (\[153\]) with $\left( p,A\right) =\left( p\left( \tau \right) ,A\left(
\tau \right) \right) $ and $\Omega \left( z;\tau \right) $ is defined by $Y\left( z;\tau \right) $. Then
- $\Omega_{12}\left( \cdot;\tau \right) $ is meromorphic in $\mathbb{C}$ and holomorphic for all $z\not \in \{ \pm p(\tau),\frac
{\omega_{i}}{2}$, $i=0,1,2,3\}+\Lambda_{\tau}$.
- If there exist $i\in \{0,1,2,3\}$ and $\left( b_{1},b_{2}\right)
\in \mathbb{Z}^{2}$ such that $\frac{\omega_{i}}{2}+b_{1}+b_{2}\tau$ is a pole of $\Omega_{12}\left( \cdot;\tau \right) $ with order $m_{i}$, then $m_{i}=2n_{i}$, and any point in $\frac{\omega_{i}}{2}+\Lambda_{\tau}$ is also a pole of $\Omega_{12}\left( \cdot;\tau \right) $ with the same order $m_{i}$. Consequently, if $\Omega_{12}\left( \cdot;\tau \right) $ has a pole at $\frac{\omega_{i}}{2}+\Lambda_{\tau}$, then $n_{i}\in \mathbb{N}$.
- $\Omega_{12}\left( \cdot;\tau \right) $ has poles at $\left \{
\pm p\right \} +\Lambda_{\tau}$ of order at most one.
\(i) Since equation (\[186\]) has singularities only at $\{ \pm p\left(
\tau \right) ,\frac{\omega_{i}}{2},i=0,1,2,3\}+\Lambda_{\tau}$, $\Omega
_{12}\left( \cdot;\tau \right) $ is holomorphic for all $z\not \in \{ \pm
p(\tau),\frac{\omega_{i}}{2},i=0,1,2,3\}+\Lambda_{\tau}$. On the other hand, if $z_{0}\in \{ \pm p\left( \tau \right) ,\frac{\omega_{i}}{2},i=0,1,2,3\}+\Lambda_{\tau}$ is a singularity of $\Omega_{12}\left(
\cdot;\tau \right) $, then by using (\[w\]) and the local behavior of $Y\left( \cdot;\tau \right) $ at $z_{0}$, it is easy to prove$$\Omega_{12}\left( z;\tau \right) =\frac{c(\tau)}{(z-z_{0})^{m}}(1+\text{higher order term})\text{ near }z_{0}$$ for some $c(\tau)\not =0$ and $m\in \mathbb{C}$. Since $\Omega_{12}\left(
\cdot;\tau \right) $ is single-valued, we conclude that $m\in \mathbb{N}$, namely $z_{0}$ must be a pole of $\Omega_{12}\left( \cdot;\tau \right) $. This proves (i).
The proof of (ii) and (iii) are similar, so we only prove (ii) for $i=0$. Without loss of generality, we may assume $b_{1}=b_{2}=0$. Suppose $0$ is a pole of $\Omega_{12}\left( z;\tau \right) $ with order $m_{0}\in \mathbb{N}$. By (\[qqq\]), it is obvious that for any $\left( b_{1},b_{2}\right)
\in \mathbb{Z}^{2}$, $b_{1}+b_{2}\tau$ is also a pole with the same order $m_{0}$. Suppose$$\Omega_{12}\left( z;\tau \right) =z^{-m_{0}}\left( \sum_{k=0}^{\infty}c_{k}z^{k}\right) =\frac{c_{0}}{z^{m_{0}}}+O\left( \frac{1}{z^{m_{0}-1}}\right) \text{ near }0, \label{190}$$ where $c_{0}\neq0$. Then we have$$\Omega_{12}^{\prime}(z;\tau)=-m_{0}\frac{c_{0}}{z^{m_{0}+1}}+O\left( \frac
{1}{z^{m_{0}}}\right) , \label{192}$$$$\Omega_{12}^{\prime \prime \prime}\left( z;\tau \right) =-m_{0}\left(
m_{0}+1\right) \left( m_{0}+2\right) \frac{c_{0}}{z^{m_{0}+3}}+O\left(
\frac{1}{z^{m_{0}+2}}\right) , \label{191}$$ and[$$\begin{aligned}
I\left( z;\tau \right) & =n_{0}\left( n_{0}+1\right) \frac{1}{z^{2}}\nonumber \\
& +\left[ \sum_{i=1}^{3}n_{i}\left( n_{i}+1\right) \wp \left( \frac
{\omega_{i}}{2}\right) +\frac{3}{2}\wp \left( p\right) +2A\zeta \left(
p\right) +B\right] +O\left( z\right) \nonumber \\
& =n_{0}\left( n_{0}+1\right) \frac{1}{z^{2}}+D\left( \tau \right)
+O\left( z\right) , \label{193}$$ ]{}where $D\left( \tau \right) $ is a constant depending on $\tau$. Thus$$I^{\prime}(z;\tau)=-2n_{0}\left( n_{0}+1\right) \frac{1}{z^{3}}+O\left(
1\right) ,\text{ \ \ }\frac{\partial I}{\partial \tau}(z;\tau)=O\left(
1\right) . \label{194}$$ Substituting (\[190\])-(\[194\]) into (\[186\]), we easily obtain$$m_{0}\left( m_{0}+1\right) \left( m_{0}+2\right) c_{0}=4n_{0}\left(
n_{0}+1\right) \left( m_{0}+1\right) c_{0}.$$ Since $n_{0}\geq-\frac{1}{2}$, we have $m_{0}=2n_{0}\in \mathbb{N}$. Together with the assumption that $n_{0}\not \in \frac{1}{2}+\mathbb{Z}$, we have $n_{0}\in \mathbb{N}$. This completes the proof.
For the isomonodromic deformation of the 2nd order Fuchsian equation (\[90\]) on $\mathbb{CP}^{1}$, if the non-resonant condition $n_{i}\not \in \frac{1}{2}+\mathbb{Z}$ holds, then $\Omega_{12}$ is independent of the choice of $M$-invariant fundamental solutions. See [@GP]. However, the same conclusion is not true in our study of equations defined in tori; see Remark \[rmk\] below. The following lemma is to classify the structure of solutions of (\[186\]).
\[lemI\]Under the assumption and notations of Lemma \[lem4-5\]. Then
- If $\tilde{Y}\left( z;\tau \right) $ is another $M$-invariant fundamental solution of (\[153\]), then $\Omega_{12}\left( z;\tau \right)
-\tilde{\Omega}_{12}\left( z;\tau \right) $ is an elliptic function with periods $1$ and $\tau$, and satisfies the following second symmetric product equation of (\[152\]):$$\Phi^{\prime \prime \prime}-4I\Phi^{\prime}-2I^{\prime}\Phi=0. \label{189}$$
- Let $\Phi(z;\tau)$ be an elliptic solution of (\[189\]). For any $c\in \mathbb{C}$, define $\tilde{\Omega}_{12}\left( z;\tau \right) $ by$$\tilde{\Omega}_{12}\left( z;\tau \right) \doteqdot \Omega_{12}\left(
z;\tau \right) +c\Phi \left( z;\tau \right) .$$ Then there exists an $M$-invariant fundamental solution $\tilde{Y}\left(
z;\tau \right) $ of system (\[153\]) such that $\tilde{\Omega}_{12}\left(
z;\tau \right) $ is the (1,2) component of $\tilde{\Omega}\left(
z;\tau \right) $ which is defined by $\tilde{Y}\left( z;\tau \right) $.
\(i) This follows directly from that $\Omega_{12}\left( z;\tau \right) $ and $\tilde{\Omega}_{12}\left( z;\tau \right) $ are both single-valued and satisfy (\[186\]) and (\[qqq\]).
\(ii) It is trivial to see that $\tilde{\Omega}_{12}\left( z;\tau \right) $ satisfies (\[186\]) and (\[qqq\]). Moreover, since both $\Phi \left(
z;\tau \right) $ and $\Omega_{12}\left( z;\tau \right) $ are single-valued, $\tilde{\Omega}_{12}\left( z;\tau \right) $ is single-valued. By Theorem \[thm M1\], there exists an $M$-invariant fundamental solution $\tilde
{Y}\left( z;\tau \right) $ of system (\[153\]) such that $\tilde{\Omega
}\left( z;\tau \right) $ is defined by $\tilde{Y}\left( z;\tau \right) $.
\[lem4-4\]Under the assumption and notations of Lemma \[lem4-5\]. Then there exists an $M$-invariant fundamental solution $\tilde{Y}\left(
z;\tau \right) $ such that $$\tilde{\Omega}_{12}\left( z;\tau \right) =-\Omega_{12}\left( -z;\tau \right)
.$$
Recall that $Y\left( z;\tau \right) =\left(
\begin{matrix}
y_{1}\left( z;\tau \right) & y_{2}\left( z;\tau \right) \\
y_{1}^{\prime}\left( z;\tau \right) & y_{2}^{\prime}\left( z;\tau \right)
\end{matrix}
\right) $ is an $M$-invariant fundamental solution of (\[153\]) in a neighborhood $U_{q_{0}}$ of $q_{0}$. Then for $z\in-U_{q_{0}}$, a neighborhood of $-q_{0}$, we define $$\tilde{Y}\left( z;\tau \right) :=\left(
\begin{matrix}
y_{1}\left( -z;\tau \right) & y_{2}\left( -z;\tau \right) \\
-y_{1}^{\prime}\left( -z;\tau \right) & -y_{2}^{\prime}\left( -z;\tau
\right)
\end{matrix}
\right) .$$ It is easy to see that $\tilde{Y}\left( z;\tau \right) $ is a fundamental solution to (\[153\]) in $-U_{q_{0}}$. Define $\tilde{\Omega}\left(
z;\tau \right) $ by $\tilde{Y}\left( z;\tau \right) $, then we have$$\det \tilde{Y}\left( z;\tau \right) \cdot \tilde{\Omega}_{12}\left(
z;\tau \right) =y_{1}\left( -z;\tau \right) \frac{\partial}{\partial \tau
}y_{2}\left( -z;\tau \right) -y_{2}\left( -z;\tau \right) \frac{\partial
}{\partial \tau}y_{1}\left( -z;\tau \right) ,$$ and since $\det \tilde{Y}\left( z;\tau \right) =-\det Y\left( -z;\tau \right)
$, we obtain$$\tilde{\Omega}_{12}\left( z;\tau \right) =-\Omega_{12}\left( -z;\tau \right)
\label{181}$$ for $z\in-U_{q_{0}}$. Since $\Omega_{12}$ is globally defined and single-valued, by analytic continuation, (\[181\]) holds true globally. Thus, $\tilde{\Omega}_{12}$ is globally defined and single-valued. Moreover, $\tilde{\Omega}_{12}\left( z;\tau \right) $ satisfies (\[186\]) and (\[qqq\]) which implies that $\tilde{Y}\left( z;\tau \right) $ is $M$-invariant. This completes the proof.
\[Proof of Theorem \[thmA\]\]Since the generalized Lamé equation (\[152\]) with (\[154\]) is monodromy preserving as $\tau$ deforms, by Theorem \[thm M1\] and Lemma \[lem4-5\], there exists a single-valued meromorphic function $\hat{\Omega}_{12}\left( z;\tau \right) $ satisfying (\[186\]) and (\[qqq\]). Define $\Omega_{12}\left( z;\tau \right) $ by $$\Omega_{12}\left( z;\tau \right) \doteqdot \frac{1}{2}\left[ \hat{\Omega
}_{12}\left( z;\tau \right) -\hat{\Omega}_{12}\left( -z;\tau \right)
\right] . \label{206}$$ To prove Theorem \[thmA\], we divide it into three steps:
**Step 1.** We prove that there exists an $M$-invariant fundamental solution $Y\left( z;\tau \right) $ of system (\[153\]) such that ** $$\Omega \left( z;\tau \right) =\frac{\partial}{\partial \tau}Y\left(
z;\tau \right) \cdot Y^{-1}\left( z;\tau \right) \label{c3}$$ and $\Omega_{12}\left( z;\tau \right) $ is the (1,2) component of $\Omega \left( z;\tau \right) $.
Let $$\Phi(z;\tau)=-\frac{1}{2}\left[ \hat{\Omega}_{12}\left( z;\tau \right)
+\hat{\Omega}_{12}\left( -z;\tau \right) \right] .$$ By Lemmas \[lem4-4\] and \[lemI\], $\Phi$ is an elliptic solution of equation (\[189\]) and$$\Omega_{12}\left( z;\tau \right) =\hat{\Omega}_{12}\left( z;\tau \right)
+\Phi(z;\tau)\text{.}$$ By Lemma \[lemI\] (ii), there exists an $M$-invariant fundamental solution $Y\left( z;\tau \right) $ of system (\[153\]) such that (\[c3\]) holds.
**Step 2.** We prove that $\Omega_{12}\left( z;\tau \right) $ is an odd meromorphic function and only has poles at $\left \{ \pm p\right \}
+\Lambda_{\tau}$ of order at most one. Furthermore, $\Omega_{12}^{\prime
}\left( z;\tau \right) $ is an even elliptic function.
Clearly (\[206\]) and Lemma \[lem4-5\] imply that $\Omega_{12}\left(
z;\tau \right) $ is an odd meromorphic function. Now we claim that:$$\Omega_{12}\left( z;\tau \right) \text{ only has poles at}\left \{ \pm
p\right \} +\Lambda_{\tau}\text{ of order at most one.} \label{213}$$ By Lemma \[lem4-5\] (i), $\Omega_{12}\left( z;\tau \right) $ is holomorphic for all $z\not \in \{ \pm p,\frac{\omega_{i}}{2},i=0,1,2,3\}+\Lambda_{\tau}$. If $\Omega_{12}\left( z;\tau \right) $ has a pole at $\frac{\omega_{i}}{2}+\Lambda_{\tau}$, then the order of the pole is $2n_{i}\in2\mathbb{N}$ by Lemma \[lem4-5\] (ii), which yields a contradiction to the fact that $\Omega_{12}\left( z;\tau \right) $ is odd and satisfies (\[qqq\]).
**Step 3.** We prove that $\Omega_{12}\left( z;\tau \right) $ is of the form (\[c\]):$$\Omega_{12}\left( z;\tau \right) =-\frac{i}{4\pi}\left( \zeta \left(
z-p\right) +\zeta \left( z+p\right) -2z\eta_{1}\right) .$$
By Step 2 **** and (\[213\]), we know that $\Omega_{12}^{\prime}\left(
z;\tau \right) $ must be of the following form$$\Omega_{12}^{\prime}\left( z;\tau \right) =-C\left( \wp \left( z+p\right)
+\wp \left( z-p\right) \right) +D$$ for some constants $C,D\in \mathbb{C}$. Thus by integration, we get $$\Omega_{12}\left( z;\tau \right) =C\left( \zeta \left( z+p\right)
+\zeta \left( z-p\right) \right) +Dz+E$$ for some $E\in \mathbb{C}$. Since $\Omega_{12}\left( z;\tau \right) $ is odd, we have $E=0$. Furthermore,$$\Omega_{12}\left( z+1;\tau \right) =\Omega_{12}\left( z;\tau \right)
+2C\eta_{1}+D,$$ and$$\Omega_{12}\left( z+\tau;\tau \right) =\Omega_{12}\left( z;\tau \right)
+2C\eta_{2}+D\tau.$$ By (\[204\]), we have$$2C\eta_{1}+D=0,\text{ }2C\eta_{2}+D\tau=-1.$$ By Legendre relation $\tau \eta_{1}-\eta_{2}=2\pi i$, we have$$C=\frac{-i}{4\pi}\text{ \ and \ }D=\frac{i}{2\pi}\eta_{1},$$ which implies (\[c\]). This completes the proof.
Under the assumption and notations of Lemma \[lem4-5\] and assume $n_{i}\not \in \mathbb{Z}$ for some $i\in \{0,1,2,3\}$. Then $\Omega
_{12}\left( z;\tau \right) $ is unique, i.e., $\Omega_{12}\left(
z;\tau \right) $ is independent of the choice of $M$-invariant solution $Y\left( z;\tau \right) $ of system (\[153\]).
For any $M$-invariant solution $Y\left( z;\tau \right) $ of system (\[153\]), by Theorem \[thm M1\], there exists a single-valued function $\Omega_{12}\left( z;\tau \right) $ satisfying (\[186\]) and (\[204\]). Let$$\Phi \left( z;\tau \right) =\Omega_{12}\left( z;\tau \right) +\Omega
_{12}\left( -z;\tau \right) .$$ If $\Phi \left( z;\tau \right) \not \equiv 0$, then $\Phi \left(
z;\tau \right) $ is an even elliptic solution of (\[189\]). Without loss of generality, we may consider the case $n_{1}\not \in \mathbb{Z}$. Then $2n_{1}\not \in \mathbb{Z}$ since $n_{1}\not \in \frac{1}{2}+\mathbb{Z}$. Since the local exponents of (\[189\]) at $\frac{\omega_{1}}{2}$ are $-2n_{1},1,2n_{1}+2$ and $\Phi \left( z;\tau \right) $ is elliptic, the local exponent of $\Phi \left( z;\tau \right) $ at $z=\frac{\omega_{1}}{2}$ must be $1$, i.e., $\frac{\omega_{1}}{2}$ is a simple zero. But again by $\Phi \left(
z;\tau \right) $ is even elliptic, we have $\Phi^{\prime}\left( \frac
{\omega_{1}}{2};\tau \right) =0$, which leads to a contradiction. Thus, $\Phi \left( z;\tau \right) \equiv0$, i.e., $\Omega_{12}\left( z;\tau \right)
$ is odd. Then by Theorem \[thmA\], $\Omega_{12}\left( z;\tau \right) $ is of the form (\[c\]).
\[rmk\]When $n_{i}\in \mathbb{Z}$ for all $i=0,1,2,3$, $\Omega_{12}\left(
z;\tau \right) $ might not be unique. For example, when $n_{i}=0$ for all $i=0,1,2,3$, we define$$\Phi \left( z;\tau \right) \doteqdot \zeta \left( z+p|\tau \right)
-\zeta \left( z-p|\tau \right) -\zeta \left( 2p|\tau \right) -2A,$$ then $\Phi \left( z;\tau \right) $ is an even elliptic solution of (\[189\]). So for any $c\in \mathbb{C}$,$$\tilde{\Omega}_{12}\left( z;\tau \right) \doteqdot \frac{-i}{4\pi}\left(
\zeta \left( z-p\right) +\zeta \left( z+p\right) -2z\eta_{1}\right)
+c\Phi \left( z;\tau \right)$$ satisfies (\[186\]) and (\[qqq\]). By Lemma \[lemI\], there exists an $M$-invariant solution $\tilde{Y}\left( z;\tau \right) $ such that $\tilde{\Omega}\left( z;\tau \right) $ is defined by $\tilde{Y}\left(
z;\tau \right) $.
Define $U\left( z;\tau \right) $ by$$U\left( z;\tau \right) \doteqdot \Omega_{12}^{\prime \prime \prime}\left(
z;\tau \right) -4I\left( z;\tau \right) \Omega_{12}^{\prime}\left(
z;\tau \right) -2I^{\prime}\left( z;\tau \right) \Omega_{12}\left(
z;\tau \right) +2\frac{\partial}{\partial \tau}I\left( z;\tau \right) ,$$ where $\Omega_{12}\left( z;\tau \right) $ is given in Theorem \[thmA\] (\[c\]), i.e.,$$\Omega_{12}\left( z;\tau \right) =-\frac{i}{4\pi}\left( \zeta \left(
z-p\right) +\zeta \left( z+p\right) -2z\eta_{1}\right) .$$ In order to prove Theorem \[thm4-2\], we need the following local expansions for $\Omega_{12}\left( z;\tau \right) $ and $I\left( z;\tau \right) $ at $p$ and $\frac{\omega_{k}}{2}$, $k=0,1,2,3$, respectively.
\[lem-expand\]$\Omega_{12}\left( z;\tau \right) $ and $I\left(
z;\tau \right) $ have local expansions at $p$ and $\frac{\omega_{k}}{2}$, $k=0,1,2,3$ as follows:
- Near $p$, let $u=z-p$. Then we have$$\Omega_{12}\left( z;\tau \right) =\frac{-i}{4\pi}\left(
\begin{array}
[c]{l}u^{-1}+\left( \zeta \left( 2p\right) -2p\eta_{1}\right) -\left( \wp \left(
2p\right) +2\eta_{1}\right) u\\
-\frac{1}{2}\wp^{\prime}\left( 2p\right) u^{2}-\frac{1}{6}\left(
\frac{g_{2}}{10}+\wp^{\prime \prime}\left( 2p\right) \right) u^{3}+O\left(
u^{4}\right)
\end{array}
\right) , \label{390}$$ and$$I\left( z;\tau \right) =\frac{3}{4}u^{-2}-Au^{-1}+A^{2}+H_{1}\left(
\tau \right) u+H_{2}\left( \tau \right) u^{2}+O\left( u^{3}\right) ,
\label{391}$$ where$$H_{1}\left( \tau \right) =\sum_{k=0}^{3}n_{k}\left( n_{k}+1\right)
\wp^{\prime}\left( p+\frac{\omega_{k}}{2}\right) +\frac{3}{4}\wp^{\prime
}\left( 2p\right) -A\wp \left( 2p\right) , \label{404}$$ and$$H_{2}\left( \tau \right) =\frac{1}{2}\left[ \sum_{k=0}^{3}n_{k}\left(
n_{k}+1\right) \wp^{\prime \prime}\left( p+\frac{\omega_{k}}{2}\right)
+\frac{3}{4}\wp^{\prime \prime}\left( 2p\right) +\frac{3}{40}g_{2}-A\wp^{\prime}\left( 2p\right) \right] .$$
- Near $\frac{\omega_{k}}{2}$, $k\in \{0,1,2,3\}$, let $u_{k}=z-\frac{\omega_{k}}{2}$. Then we have$$\Omega_{12}\left( z;\tau \right) =\frac{i}{4\pi}\left[
\begin{array}
[c]{l}-\left( \zeta \left( \frac{\omega_{k}}{2}+p\right) +\zeta \left(
\frac{\omega_{k}}{2}-p\right) -\omega_{k}\eta_{1}\right) \\
+2\left( \wp \left( \frac{\omega_{k}}{2}+p\right) +\eta_{1}\right)
u_{k}+O\left( u_{k}^{3}\right)
\end{array}
\right] , \label{392}$$ and$$I\left( z;\tau \right) =n_{k}\left( n_{k}+1\right) u_{k}^{-2}+\Lambda
_{k}\left( \tau \right) +O\left( u_{k}^{2}\right) , \label{393}$$ where[$$\begin{aligned}
\Lambda_{k}\left( \tau \right) = & \sum_{j\not =k}^{3}n_{j}\left(
n_{j}+1\right) \wp \left( \frac{\omega_{k}+\omega_{j}}{2}\right) +\frac
{3}{2}\wp \left( \frac{\omega_{k}}{2}+p\right) \label{406}\\
& +A\left( \tau \right) \left( \zeta \left( \frac{\omega_{k}}{2}+p\right)
-\zeta \left( \frac{\omega_{k}}{2}-p\right) \right) +B\left( \tau \right)
.\nonumber\end{aligned}$$ ]{}
Recall the following expansions:$$\zeta \left( u\right) =\frac{1}{u}-\frac{g_{2}}{60}u^{3}-\frac{g_{3}}{140}u^{5}+O\left( u^{7}\right) , \label{e1}$$$$\wp \left( u\right) =\frac{1}{u^{2}}+\frac{g_{2}}{20}u^{2}+\frac{g_{3}}{28}u^{4}+O\left( u^{6}\right) . \label{e2}$$ The proof follows from a direct computation by using (\[e1\]) and (\[e2\]).
By using Lemma \[lem-expand\], we have
\[lem-U\]$U\left( \cdot;\tau \right) $ is an even elliptic function and has poles only at $\pm p$ of order at most $3$. More precisely, $U\left(
z;\tau \right) $ is expressed as follows:[$$\begin{aligned}
U\left( z;\tau \right) = & L\left( \tau \right) \left( \wp^{\prime
}\left( z-p\right) -\wp^{\prime}\left( z+p\right) \right) \label{384}\\
& +M\left( \tau \right) \left( \wp \left( z-p\right) +\wp \left(
z+p\right) \right) \nonumber \\
& +N\left( \tau \right) \left( \zeta \left( z-p\right) -\zeta \left(
z+p\right) \right) +C\left( \tau \right) ,\nonumber\end{aligned}$$ ]{}where the coefficients $L\left( \tau \right) $, $M\left( \tau \right) $, $N\left( \tau \right) $ and $C\left( \tau \right) $ are given by$$L\left( \tau \right) =-\frac{1}{2}\left( 3\frac{dp}{d\tau}+\frac{i}{4\pi
}\left[ 6A-3\left( \zeta \left( 2p\right) -2p\eta_{1}\right) \right]
\right) , \label{398}$$$$M\left( \tau \right) =-2A\frac{dp}{d\tau}+\frac{i}{4\pi}\left[
-4A^{2}+2A\left( \zeta \left( 2p\right) -2p\eta_{1}\right) \right] ,
\label{399}$$$$N\left( \tau \right) =-2\frac{dA}{d\tau}+\frac{i}{4\pi}\left[
\begin{array}
[c]{l}4A\left( \wp \left( 2p\right) +\eta_{1}\right) -3\wp^{\prime}\left(
2p\right) \\
-2\sum_{k=0}^{3}n_{k}\left( n_{k}+1\right) \wp^{\prime}\left(
p+\frac{\omega_{k}}{2}\right)
\end{array}
\right] , \label{401}$$ [$$\begin{aligned}
C\left( \tau \right) = & 4A\frac{dA}{d\tau}-2H_{1}\left( \tau \right)
\frac{dp}{d\tau}\label{402}\\
& +\frac{i}{4\pi}\left[
\begin{array}
[c]{l}-4A^{2}\left( \wp \left( 2p\right) +2\eta_{1}\right) +3A\wp^{\prime}\left(
2p\right) \\
+2H_{1}\left( \tau \right) \left( \zeta \left( 2p\right) -2p\eta
_{1}\right)
\end{array}
\right] .\nonumber\end{aligned}$$ ]{}Here $H_{1}\left( \tau \right) $ is given in (\[404\]).
Since $I\left( \cdot;\tau \right) $ is elliptic, we have $I\left(
z;\tau \right) =I\left( z+\tau;\tau \right) $. Thus$$\begin{aligned}
\frac{\partial}{\partial \tau}I\left( z;\tau \right) & =I^{\prime}\left(
z+\tau;\tau \right) +\frac{\partial}{\partial \tau}I\left( z+\tau;\tau \right)
\label{382}\\
& =I^{\prime}\left( z;\tau \right) +\frac{\partial}{\partial \tau}I\left(
z+\tau;\tau \right) .\nonumber\end{aligned}$$ By using (\[382\]) and the translation property (\[qqq\]) of $\Omega
_{12}\left( z;\tau \right) $, we have$$U\left( z+\omega_{k};\tau \right) =U\left( z;\tau \right) ,\text{ }k=1,2,$$ that is, $U\left( \cdot;\tau \right) $ is elliptic. Moreover, since $I\left(
\cdot;\tau \right) $ is even, we have$$\frac{\partial}{\partial \tau}I\left( z;\tau \right) =\frac{\partial}{\partial \tau}I\left( -z;\tau \right) . \label{383}$$ By using (\[383\]) and $\Omega_{12}\left( \cdot;\tau \right) $ is odd, we see that $U\left( \cdot;\tau \right) $ is even.
Next, we claim that: $$U\left( \cdot;\tau \right) \text{ is holomorphic at }\frac{\omega_{k}}{2},\text{ }k=0,1,2,3. \label{408}$$ Since the proof is similar, we only give the proof for $k=2$. In this case, by (\[392\]) and (\[393\]) in Lemma \[lem-expand\], near $\frac{\tau}{2}$, we have, $$\Omega_{12}\left( z;\tau \right) =\frac{i}{4\pi}\left[ 2\pi i+2\left(
\wp \left( \frac{\tau}{2}+p\right) +\eta_{1}\right) u_{2}+O\left( u_{2}^{3}\right) \right] , \label{385}$$ and$$I\left( z;\tau \right) =n_{2}\left( n_{2}+1\right) u_{2}^{-2}+\Lambda
_{2}\left( \tau \right) +O\left( u_{2}^{2}\right) , \label{386}$$ Then near $\frac{\tau}{2}$, we have$$\Omega_{12}^{\prime}\left( z;\tau \right) =\frac{i}{4\pi}\left[ 2\left(
\wp \left( \frac{\tau}{2}+p\right) +\eta_{1}\right) +O\left( u_{2}^{2}\right) \right] , \label{387}$$$$I^{\prime}\left( z;\tau \right) =-2n_{2}\left( n_{2}+1\right) u_{2}^{-3}+O\left( u_{2}\right) , \label{388}$$ and$$\frac{\partial}{\partial \tau}I\left( z;\tau \right) =n_{2}\left(
n_{2}+1\right) u_{2}^{-3}+\frac{\partial}{\partial \tau}\Lambda_{2}\left(
\tau \right) +O\left( u_{2}\right) . \label{389}$$ By (\[385\])-(\[389\]) and $\Omega_{12}\left( z;\tau \right) $ is holomorphic at $\frac{\omega_{k}}{2},$ $k=0,1,2,3$, we have[$$\begin{aligned}
& U\left( z;\tau \right) \label{407}\\
= & \Omega_{12}^{\prime \prime \prime}\left( z;\tau \right) -4I\left(
z;\tau \right) \Omega_{12}^{\prime}\left( z;\tau \right) -2I^{\prime}\left(
z;\tau \right) \Omega_{12}\left( z;\tau \right) +2\frac{\partial}{\partial \tau}I\left( z;\tau \right) \nonumber \\
= & \Omega_{12}^{\prime \prime \prime}\left( z;\tau \right) -4\left[
n_{2}\left( n_{2}+1\right) u_{2}^{-2}+\Lambda_{2}\left( \tau \right)
+O\left( u_{2}^{2}\right) \right] \nonumber \\
& \times \left( \frac{i}{4\pi}\right) \left[ 2\left( \wp \left( \frac
{\tau}{2}+p\right) +\eta_{1}\right) +O\left( u_{2}^{2}\right) \right]
\nonumber \\
& -2\left[ -2n_{2}\left( n_{2}+1\right) u_{2}^{-3}+O\left( u_{2}\right)
\right] \nonumber \\
& \times \left( \frac{i}{4\pi}\right) \left[ 2\pi i+2\left( \wp \left(
\frac{\tau}{2}+p\right) +\eta_{1}\right) u_{1}+O\left( u_{2}^{3}\right)
\right] \nonumber \\
& +2\left[ n_{2}\left( n_{2}+1\right) u_{2}^{-3}+\frac{\partial}{\partial \tau}\Lambda_{2}\left( \tau \right) +O\left( u_{2}^{2}\right)
\right] .\nonumber\end{aligned}$$ ]{}From (\[407\]), it is easy to see that the coefficients of $u_{2}^{-3}$, $u_{2}^{-2}$, $u_{2}^{-1}$ are all vanishing which implies that $U\left(
z;\tau \right) $ is holomorphic at $\frac{\tau}{2}$.
Now we prove $U\left( z;\tau \right) $ can be written as (\[384\]). To compute the coefficients $L\left( \tau \right) $, $M\left( \tau \right) $, $N\left( \tau \right) $ and $C\left( \tau \right) $, we only need to compute near $p$. By (\[390\]) and (\[391\]), near $p$, we have$$\Omega_{12}^{\prime}\left( z;\tau \right) =\frac{-i}{4\pi}\left(
\begin{array}
[c]{l}-u^{-2}-\left( \wp \left( 2p\right) +2\eta_{1}\right) -\wp^{\prime}\left(
2p\right) u\\
-\frac{1}{2}\left( \frac{g_{2}}{10}+\wp^{\prime \prime}\left( 2p\right)
\right) u^{2}+O\left( u^{3}\right)
\end{array}
\right) , \label{394}$$$$\Omega_{12}^{\prime \prime \prime}\left( z;\tau \right) =\frac{-i}{4\pi}\left(
-6u^{-4}-\left( \frac{g_{2}}{10}+\wp^{\prime \prime}\left( 2p\right)
\right) +O\left( u\right) \right) , \label{395}$$$$I^{\prime}\left( z;\tau \right) =-\frac{3}{2}u^{-3}+Au^{-2}+H_{1}\left(
\tau \right) +2H_{2}\left( \tau \right) u+O\left( u^{2}\right) ,
\label{396}$$$$\begin{aligned}
\frac{\partial}{\partial \tau}I\left( z;\tau \right) & =\frac{3}{2}\frac
{dp}{d\tau}u^{-3}-A\frac{dp}{d\tau}u^{-2}-\frac{dA}{d\tau}u^{-1}\label{397}\\
& +\left( 2A\frac{dA}{d\tau}-H_{1}\left( \tau \right) \frac{dp}{d\tau
}\right) +O\left( u\right) .\nonumber\end{aligned}$$ By (\[390\]), (\[391\]) and (\[394\])-(\[397\]), near $p$, after computation, we have [$$\begin{aligned}
& U\left( z;\tau \right) \label{403}\\
= & \left( 3\frac{dp}{d\tau}+\frac{i}{4\pi}\left[ 6A-3\left( \zeta \left(
2p\right) -2p\eta_{1}\right) \right] \right) u^{-3}\nonumber \\
& +\left( -2A\frac{dp}{d\tau}+\frac{i}{4\pi}\left[ -4A^{2}+2A\left(
\zeta \left( 2p\right) -2p\eta_{1}\right) \right] \right) u^{-2}\nonumber \\
& +\left( -2\frac{dA}{d\tau}+\frac{i}{4\pi}\left[
\begin{array}
[c]{l}4A\left( \wp \left( 2p\right) +\eta_{1}\right) -3\wp^{\prime}\left(
2p\right) \\
-2\sum_{k=0}^{3}n_{k}\left( n_{k}+1\right) \wp^{\prime}\left(
p+\frac{\omega_{k}}{2}\right)
\end{array}
\right] \right) u^{-1}\nonumber \\
& +\left(
\begin{array}
[c]{l}4A\frac{dA}{d\tau}-2H_{1}\left( \tau \right) \frac{dp}{d\tau}\\
+\frac{i}{4\pi}\left[
\begin{array}
[c]{l}-4A^{2}\left( \wp \left( 2p\right) +2\eta_{1}\right) +3A\wp^{\prime}\left(
2p\right) \\
+2H_{1}\left( \tau \right) \left( \zeta \left( 2p\right) -2p\eta
_{1}\right)
\end{array}
\right]
\end{array}
\right) +O\left( u\right) .\nonumber\end{aligned}$$ ]{}Obviously, (\[403\]) implies that $U\left( z;\tau \right) $ has pole at $p$ with order at most $3$. Since $U\left( z;\tau \right) $ is an even elliptic function, $U\left( z;\tau \right) $ also has pole at $-p$ with order at most $3$. From here and (\[408\]), we conclude that $U\left(
z;\tau \right) $ has poles only at $\pm p$ with order at most $3$. Moreover, from (\[403\]), it is easy to see that the coefficients $L\left(
\tau \right) $, $M\left( \tau \right) $, $N\left( \tau \right) $ and $C\left( \tau \right) $ are given by (\[398\])-(\[402\]).
\[Proof of Theorem \[thm4-2\]\]By Theorem \[thm M1\] and Lemma \[lem-U\], the generalized Lamé equation (\[152\]) with $\left( p,A\right)
=\left( p\left( \tau \right) ,A\left( \tau \right) \right) $ is monodromy preserving as $\tau$ deforms if and only if $$U\left( z;\tau \right) =0,$$ if and only if $$L\left( \tau \right) =M\left( \tau \right) =N\left( \tau \right) =C\left(
\tau \right) =0. \label{403-1}$$ By (\[398\])-(\[402\]), a straightforward computation shows that (\[403-1\]) is equivalent to that $\left( p,A\right) =\left( p\left(
\tau \right) ,A\left( \tau \right) \right) $ satisfies the Hamiltonian system (\[142\]) (see (\[144\]) below).
Hamiltonian system and Painlevé VI
----------------------------------
Next, we will study the Hamiltonian structure for the elliptic form (\[124\]) with $\alpha_{i}$ defined by (\[125\]). Our second main theorem is the following:
\[thm4-1\]The elliptic form (\[124\]) with $\alpha_{k}=\frac{1}{2}\left( n_{k}+\frac{1}{2}\right) ^{2}$, $k=0,1,2,3$ is equivalent to the Hamiltonian system defined by (\[142\]) and (\[143\]).
Suppose $\left( p\left( \tau \right) ,A\left( \tau \right) \right) $ satisfies the Hamiltonian system (\[142\]), i.e.,$$\left \{
\begin{array}
[c]{l}\frac{dp\left( \tau \right) }{d\tau}=\frac{\partial K\left( p,A,\tau \right)
}{\partial A}=\frac{-i}{4\pi}\left( 2A-\zeta \left( 2p|\tau \right)
+2p\eta_{1}\left( \tau \right) \right) \\
\frac{dA\left( \tau \right) }{d\tau}=-\frac{\partial K\left( p,A,\tau
\right) }{\partial p}=\frac{i}{4\pi}\left(
\begin{array}
[c]{l}\left( 2\wp \left( 2p|\tau \right) +2\eta_{1}\left( \tau \right) \right)
A-\frac{3}{2}\wp^{\prime}\left( 2p|\tau \right) \\
-\sum_{k=0}^{3}n_{k}\left( n_{k}+1\right) \wp^{\prime}\left( p+\frac
{\omega_{k}}{2}|\tau \right)
\end{array}
\right)
\end{array}
\right. . \label{144}$$ Then we compute the second derivative $\frac{d^{2}p\left( \tau \right)
}{d\tau^{2}}$ of $p\left( \tau \right) $ as follows:[$$\begin{aligned}
\frac{d^{2}p\left( \tau \right) }{d\tau^{2}}= & \frac{-i}{4\pi}\left[
2\frac{dA\left( \tau \right) }{d\tau}+2\wp \left( 2p|\tau \right)
\frac{dp\left( \tau \right) }{d\tau}-\frac{\partial}{\partial \tau}\zeta \left( 2p|\tau \right) \right. \label{144-1}\\
& \left. +2\eta_{1}\left( \tau \right) \frac{dp\left( \tau \right) }{d\tau}+2p\left( \tau \right) \frac{d\eta_{1}\left( \tau \right) }{d\tau
}\right] .\nonumber\end{aligned}$$ ]{}By Lemma \[lem4-1\], we have$$-\frac{\partial}{\partial \tau}\zeta \left( 2p|\tau \right) =\frac{-i}{4\pi
}\left[
\begin{array}
[c]{l}\wp^{\prime}\left( 2p|\tau \right) +2\left( \zeta \left( 2p|\tau \right)
-2p\eta_{1}\left( \tau \right) \right) \wp \left( 2p|\tau \right) \\
+2\eta_{1}\zeta \left( 2p|\tau \right) -\frac{1}{3}pg_{2}\left( \tau \right)
\end{array}
\right] , \label{148}$$$$2p\left( \tau \right) \frac{d\eta_{1}\left( \tau \right) }{d\tau}=\frac
{i}{2\pi}p\left( \tau \right) \left[ 2\eta_{1}^{2}-\frac{1}{6}g_{2}\left(
\tau \right) \right] . \label{150}$$ Substituting (\[144\]), (\[148\]) and (\[150\]) into (\[144-1\]), we have[$$\begin{aligned}
\frac{d^{2}p\left( \tau \right) }{d\tau^{2}} & =\frac{-1}{4\pi^{2}}\left(
\wp^{\prime}\left( 2p|\tau \right) +\frac{1}{2}\sum_{k=0}^{3}n_{k}\left(
n_{k}+1\right) \wp^{\prime}\left( p+\frac{\omega_{k}}{2}|\tau \right)
\right) \nonumber \\
& =\frac{-1}{4\pi^{2}}\left[ \frac{1}{8}\sum_{k=0}^{3}\wp^{\prime}\left(
p+\frac{\omega_{k}}{2}|\tau \right) +\frac{1}{2}\sum_{k=0}^{3}n_{k}\left(
n_{k}+1\right) \wp^{\prime}\left( p+\frac{\omega_{k}}{2}|\tau \right)
\right] \label{151}\\
& =\frac{-1}{4\pi^{2}}\sum_{k=0}^{3}\frac{1}{2}\left( n_{k}+\frac{1}{2}\right) ^{2}\wp^{\prime}\left( p+\frac{\omega_{k}}{2}|\tau \right)
,\nonumber\end{aligned}$$ ]{}implying that $p\left( \tau \right) $ is a solution of the elliptic form (\[124\]) with $\alpha_{k}=\frac{1}{2}\left( n_{k}+\frac{1}{2}\right)
^{2}$, $k=0,1,2,3$.
Conversely, suppose $p\left( \tau \right) $ is a solution of the elliptic form (\[124\]) with $\alpha_{k}=\frac{1}{2}\left( n_{k}+\frac{1}{2}\right)
^{2}$, $k=0,1,2,3$. We define $A\left( \tau \right) $ by the first equation of (\[144\]), i.e.,$$A\left( \tau \right) \doteqdot2\pi i\frac{dp\left( \tau \right) }{d\tau
}+\frac{1}{2}\left( \zeta \left( 2p|\tau \right) -2p\eta_{1}\left(
\tau \right) \right) . \label{145}$$ Then[$$\begin{aligned}
\frac{dA\left( \tau \right) }{d\tau}= & 2\pi i\frac{d^{2}p\left(
\tau \right) }{d\tau^{2}}+\frac{1}{2}\left( -2\wp \left( 2p|\tau \right)
\frac{dp\left( \tau \right) }{d\tau}+\frac{\partial}{\partial \tau}\zeta \left( 2p|\tau \right) \right) \\
& -\left( \eta_{1}\left( \tau \right) \frac{dp\left( \tau \right) }{d\tau
}+p\left( \tau \right) \frac{d\eta_{1}\left( \tau \right) }{d\tau}\right) ,\end{aligned}$$ ]{}and by using (\[144\]), (\[148\]) and (\[150\]), we have[$$\begin{aligned}
\frac{dA\left( \tau \right) }{d\tau}= & \frac{i}{4\pi}\left[ 2\left(
\wp \left( 2p|\tau \right) +\eta_{1}\left( \tau \right) \right) A-\frac
{3}{2}\wp^{\prime}\left( 2p|\tau \right) \right. \\
& \left. -\sum_{k=0}^{3}n_{k}\left( n_{k}+1\right) \wp^{\prime}\left(
p+\frac{\omega_{k}}{2}|\tau \right) \right] .\end{aligned}$$ ]{}Thus, $\left( p\left( \tau \right) ,A\left( \tau \right) \right) $ is a solution to the Hamiltonian system (\[144\]).
Moreover, from (\[144\]), we could obtain the integral formula for $A\left(
\tau \right) $ in Theorem \[theorem1-5\].
\[Proof of Theorem \[theorem1-5\]\]Let us consider $F\left( \tau \right)
=A+\frac{1}{2}\left( \zeta \left( 2p\right) -2\zeta \left( p\right)
\right) $ and compute $\frac{d}{d\tau}F\left( \tau \right) $. By (\[144\]) and Lemma \[lem4-1\], we have[$$\begin{aligned}
& \frac{d}{d\tau}F\left( \tau \right) \\
= & \frac{dA}{d\tau}+\frac{1}{2}\frac{d}{d\tau}\left( \zeta \left(
2p\right) -2\zeta \left( p\right) \right) \\
= & \frac{i}{4\pi}\left( 2\left( \wp \left( 2p\right) +\eta_{1}\left(
\tau \right) \right) A-\frac{3}{2}\wp^{\prime}\left( 2p\right) -\sum
_{k=0}^{3}n_{k}\left( n_{k}+1\right) \wp^{\prime}(p(\tau)+\frac{\omega_{k}}{2})\right) \\
& -\left( \wp \left( 2p\right) -\wp \left( p\right) \right) \frac
{dp}{d\tau}+\frac{1}{2}\left( \frac{\partial}{\partial \tau}\zeta \left(
2p\right) -2\frac{\partial}{\partial \tau}\zeta \left( p\right) \right) \\
= & \frac{i}{2\pi}\left( 2\wp \left( 2p\right) -\wp \left( p\right)
+\eta_{1}\right) F\left( \tau \right) -\frac{i}{4\pi}\sum_{k=0}^{3}n_{k}\left( n_{k}+1\right) \wp^{\prime}(p(\tau)+\frac{\omega_{k}}{2}).\end{aligned}$$ ]{}Thus, $$\begin{aligned}
& F\left( \tau \right) \label{381}\\
= & \exp \left \{ \frac{i}{2\pi}\int^{\tau}\left( 2\wp \left( 2p(\hat{\tau
})|\hat{\tau}\right) -\wp \left( p(\hat{\tau})|\hat{\tau}\right) +\eta
_{1}(\hat{\tau})\right) d\hat{\tau}\right \} \cdot J\left( \tau \right)
,\nonumber\end{aligned}$$ where $$J\left( \tau \right) =\int^{\tau}\frac{-\frac{i}{4\pi}\left( \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp^{\prime}(p(\hat{\tau})+\frac{\omega_{k}}{2}|\hat{\tau
})\right) }{\exp \left \{ \frac{i}{2\pi}\int^{\hat{\tau}}\left( 2\wp
(2p(\tau^{\prime})|\tau^{\prime})-\wp(p(\tau^{\prime})|\tau^{\prime})+\eta
_{1}\left( \tau^{\prime}\right) \right) d\tau^{\prime}\right \} }d\hat
{\tau}+c_{1}$$ for some constant $c_{1}\in \mathbb{C}$. By Lemma \[lem4-1\], we have $$\frac{3i}{4\pi}\int^{\tau}\eta_{1}(\hat{\tau})d\hat{\tau}=\ln \theta
_{1}^{\prime}(\tau). \label{theta}$$ Then (\[514\]) follows from (\[381\]) and (\[theta\]).
Collapse of two singular points
===============================
In this section, we study the phenomena of collapsing two singular points $\pm
p(\tau)$ to $0$ in the generalized Lamé equation (\[89-0\]) when $p(\tau)$ is a solution of the elliptic form (\[124\]). As an application of Theorems \[thm4-2\] and \[thm4-1\], it turns out that the classical Lamé equation$$y^{\prime \prime}(z)=\left( n(n+1)\wp(z)+B\right) y(z)\text{ \ in \ }E_{\tau}
\label{5033}$$ appears as a limiting equation if $n_{k}=0$ for $k=1,2,3$ (see Theorem \[thm-II-9\] below). First we recall the following classical result.
**Theorem A.[@GP Proposition 1.4.1]** *Assume* $\theta_{4}=n_{0}+\frac{1}{2}\not =0$. *Then for any* $t_{0}\in \mathbb{CP}^{1}\backslash \{0,1,\infty \}$*, there exist two* $1$-*parameter families of solutions* $\lambda(t)$ *of Painlevé VI (\[46\]) such that*$$\lambda(t)=\frac{\beta}{t-t_{0}}+h+O(t-t_{0})\text{ as }t\rightarrow t_{0},
\label{II-132}$$ *where* $h\in \mathbb{C}$ *can be taken arbitrary and*$$\beta=\beta(\theta,t_{0})\in \left \{ \pm \tfrac{t_{0}(t_{0}-1)}{\theta_{4}}\right \} . \label{II-133}$$ *Furthermore, these two* $1$-*parameter families of solutions give all solutions of Painlevé VI (\[46\]) which has a pole at* $t_{0}$*.*
In this paper, we always identify the solutions $p(\tau)$ and $-p(\tau)$ of the elliptic form (\[124\]). As a consequence of Theorem A and the transformation (\[tr\]), we have the following result.
\[lemII-4\] *Assume* $n_{0}+\frac{1}{2}\not =0$. *Then for any* $\tau_{0}\in \mathbb{H}$*, by the transformation (\[tr\]) solutions* $\lambda(t)$ *in Theorem A give two* $1$-*parameter families of solutions* $p(\tau)$ *of the elliptic form (\[124\]) such that*$$p(\tau)=c_{0}(\tau-\tau_{0})^{\frac{1}{2}}(1+\tilde{h}(\tau-\tau_{0})+O(\tau-\tau_{0})^{2})\text{ as }\tau \rightarrow \tau_{0}, \label{515-5}$$ where $\tilde{h}\in \mathbb{C}$ *can be taken arbitrary,*$$c_{0}^{2}=\left \{
\begin{array}
[c]{l}i\frac{n_{0}+\frac{1}{2}}{\pi}\text{ \ \ if \ \ }\beta=-\frac{t_{0}(t_{0}-1)}{\theta_{4}}\\
-i\frac{n_{0}+\frac{1}{2}}{\pi}\text{ \ \ if \ \ }\beta=\frac{t_{0}(t_{0}-1)}{\theta_{4}}\end{array}
\right. , \label{515-3}$$ and $t_{0}=t(\tau_{0})$. *Furthermore, these two* $1$-*parameter families of solutions give all solutions* $p(\tau)$ *of the elliptic form (\[124\]) such that* $p(\tau_{0})=0$*.*
It suffices to prove (\[515-3\]), which follows readily from $$t^{\prime}(\tau_{0})=-i\frac{t_{0}(t_{0}-1)}{\pi}\left( e_{2}(\tau_{0})-e_{1}(\tau_{0})\right) . \label{II-134}$$ Remark that $t_{0}\not \in \{0,1\}$, so (\[II-134\]) implies $t^{\prime
}(\tau_{0})\not =0$.
The formula (\[II-134\]) is known in the literature. Here we give a proof for the reader’s convenience. Recalling theta functions $\vartheta_{2}(\tau),\vartheta_{3}(\tau)$ and $\vartheta_{4}(\tau)$, it is well-known that (cf. see [@YB] for a reference)$$e_{3}(\tau)-e_{2}(\tau)=\pi^{2}\vartheta_{2}(\tau)^{4},\text{ \ }e_{1}(\tau)-e_{3}(\tau)=\pi^{2}\vartheta_{4}(\tau)^{4}, \label{II-134-2}$$$$e_{1}(\tau)-e_{2}(\tau)=\pi^{2}\vartheta_{3}(\tau)^{4},$$$$\frac{d}{d\tau}\ln \vartheta_{4}(\tau)=\frac{i}{12\pi}\left[ 3\eta_{1}(\tau)-\pi^{2}(2\vartheta_{2}(\tau)^{4}+\vartheta_{4}(\tau)^{4})\right] ,$$$$\frac{d}{d\tau}\ln \vartheta_{3}(\tau)=\frac{i}{12\pi}\left[ 3\eta_{1}(\tau)+\pi^{2}(\vartheta_{2}(\tau)^{4}-\vartheta_{4}(\tau)^{4})\right] .$$ Therefore, $t=\vartheta_{4}^{4}/\vartheta_{3}^{4}$ and then$$\begin{aligned}
t^{\prime}(\tau) & =4t\left( \frac{d}{d\tau}\ln \vartheta_{4}-\frac{d}{d\tau}\ln \vartheta_{3}\right) =-i\pi t\cdot \vartheta_{2}^{4}\label{II-134-1}\\
& =-i\pi \frac{\vartheta_{2}^{4}\vartheta_{4}^{4}}{\vartheta_{3}^{4}}=-i\frac{t(t-1)}{\pi}\left( e_{2}-e_{1}\right) .\nonumber\end{aligned}$$ This completes the proof.
\[thm-II-9\]Assume that $n_{k}\not \in \mathbb{Z}+\frac{1}{2},$ $k\in \{0,1,2,3\}$, and (\[101\]) hold. Let $(p(\tau),A(\tau))$ be a solution of the Hamiltonian system (\[aa\]) such that $p(\tau_{0})=0$ for some $\tau_{0}\in \mathbb{H}$. Then$$p(\tau)=c_{0}(\tau-\tau_{0})^{\frac{1}{2}}(1+\tilde{h}(\tau-\tau_{0})+O(\tau-\tau_{0})^{2})\text{ as }\tau \rightarrow \tau_{0}, \label{515-4}$$ for some $\tilde{h}\in \mathbb{C}$ and the generalized Lamé equation (\[89-0\]) converges to$$y^{\prime \prime}=\left[ \sum_{j=1}^{3}n_{j}\left( n_{j}+1\right) \wp \left(
z+\frac{\omega_{j}}{2}\right) +m(m+1)\wp(z)+B_{0}\right] y\text{ in }E_{\tau_{0}}, \label{503-2}$$ where $c_{0}$ is seen in (\[515-3\]),$$m=\left \{
\begin{array}
[c]{l}n_{0}+1\text{ \ if \ }c_{0}^{2}=i\frac{n_{0}+\frac{1}{2}}{\pi}\text{\ i.e.,\ }\beta=-\frac{t_{0}(t_{0}-1)}{n_{0}+\frac{1}{2}},\\
n_{0}-1\text{ \ if \ }c_{0}^{2}=-i\frac{n_{0}+\frac{1}{2}}{\pi}\text{\ i.e.,\ }\beta=\frac{t_{0}(t_{0}-1)}{n_{0}+\frac{1}{2}},
\end{array}
\right. \label{503-3}$$$$B_{0}=2\pi ic_{0}^{2}\left( 4\pi i\tilde{h}-\eta_{1}(\tau_{0})\right)
-\sum_{j=1}^{3}n_{j}(n_{j}+1)e_{j}(\tau_{0}). \label{503-4}$$
Clearly (\[515-4\]) follows from Lemma \[lemII-4\], by which we have (write $p=p(\tau)$)$$(\tau-\tau_{0})^{\frac{1}{2}}=\frac{1}{c_{0}}p\left( 1-\frac{1}{c_{0}^{2}}\tilde{h}p^{2}+O(p^{4})\right) \text{ as }\tau \rightarrow \tau_{0}.$$ Consequently,$$\begin{aligned}
p^{\prime}(\tau) & =\frac{1}{2}c_{0}(\tau-\tau_{0})^{-\frac{1}{2}}[1+3\tilde{h}(\tau-\tau_{0})+O((\tau-\tau_{0})^{2})]\\
& =\frac{c_{0}^{2}}{2p}\left[ 1+\frac{4}{c_{0}^{2}}\tilde{h}p^{2}+O(p^{4})\right] \text{\ as\ }\tau \rightarrow \tau_{0}.\end{aligned}$$ This, together with the first equation of the Hamiltonian system (\[142-0\])-(\[143-0\]), gives[$$\begin{aligned}
A(\tau) & =\frac{1}{2}\left[ 4\pi ip^{\prime}(\tau)+\zeta(2p(\tau
))-2p(\tau)\eta_{1}(\tau)\right] \label{II-123}\\
& =\frac{\pi ic_{0}^{2}}{p}\left[ 1+\frac{4}{c_{0}^{2}}\tilde{h}p^{2}+O(p^{4})\right] +\frac{1}{4p}-\eta_{1}(\tau_{0})p+O(p^{3})\nonumber \\
& =\frac{c}{p}+ep+O(p^{3})\text{ \ as \ }\tau \rightarrow \tau_{0},\nonumber\end{aligned}$$ ]{}where $e=4\pi i\tilde{h}-\eta_{1}(\tau_{0})$ and$$c=\pi ic_{0}^{2}+\frac{1}{4}=\left \{
\begin{array}
[c]{c}-n_{0}-\frac{1}{4}\text{ \ \ if \ \ }c_{0}^{2}=i\frac{n_{0}+\frac{1}{2}}{\pi
},\\
n_{0}+\frac{3}{4}\text{ \ \ if \ \ }c_{0}^{2}=-i\frac{n_{0}+\frac{1}{2}}{\pi}.
\end{array}
\right. \label{II-124}$$ Clearly $c$ satisfies $$c^{2}-\frac{c}{2}-\frac{3}{16}-n_{0}(n_{0}+1)=0.$$ Consequently, we have[$$\begin{aligned}
B(\tau)= & A^{2}-\zeta \left( 2p\right) A-\frac{3}{4}\wp \left( 2p\right)
-\sum_{j=0}^{3}n_{j}\left( n_{j}+1\right) \wp \left( p+\frac{\omega_{j}}{2}\right) \\
= & \left( \frac{c}{p}+ep+O(p^{3})\right) ^{2}-\left( \frac{1}{2p}+O(p^{3})\right) \left( \frac{c}{p}+ep+O(p^{3})\right) \\
& -\frac{\left( \frac{3}{16}+n_{0}(n_{0}+1)\right) }{p^{2}}-\sum_{j=1}^{3}n_{j}\left( n_{j}+1\right) e_{j}(\tau_{0})+O(p^{2})\\
= & \frac{4c-1}{2}e-\sum_{j=1}^{3}n_{j}\left( n_{j}+1\right) e_{j}(\tau_{0})+O(p^{2})=B_{0}+O(p^{2})\end{aligned}$$ ]{}as $p=p(\tau)\rightarrow0$ since $\tau \rightarrow \tau_{0}$, where $B_{0}$ is given by (\[503-4\]). Furthermore, (\[II-123\]) implies$$A(\zeta(z+p)-\zeta(z-p))=A(-2p\wp(z)+O(p^{2}))\rightarrow-2c\wp(z)$$ uniformly for $z$ bounded away from the lattice points. Therefore, the potential of the generalized Lamé equation (\[89-0\]) converges to[$$\sum_{j=1}^{3}n_{j}\left( n_{j}+1\right) \wp \left( z+\frac{\omega_{j}}{2}\right) +\left[ n_{0}(n_{0}+1)+\frac{3}{2}-2c\right] \wp(z)+B_{0}$$ ]{}uniformly for $z$ bounded away from the lattice points as $\tau
\rightarrow \tau_{0}$. Using (\[II-124\]) we easily obtain (\[503-2\])-(\[503-3\]).
Correspondence between generalized Lamé equation and Fuchsian equation
======================================================================
In this section, we want to establish a one to one correspondence between the generalized Lamé equation (\[89-0\]) and a type of Fuchsian equations on $\mathbb{CP}^{1}$. After the correspondence, naturally we ask the question: *Is the isomonodromic deformation for the generalized Lamé equation in* $E_{\tau}$ *equivalent to the isomonodromic deformation for the corresponding Fuchsian equation on* $\mathbb{CP}^{1}$*?* Notice that we establish the correspondence by using the transformation $x=\frac{\wp \left(
z\right) -e_{1}}{e_{2}-e_{1}}$ (see (\[123\]) below) which is a double cover from $E_{\tau}$ onto $\mathbb{CP}^{1}$. Hence, it is clear that the isomonodromic deformation for the Fuchsian equation could imply the the isomonodromic deformation for the generalized Lamé equation. However, the converse assertion is not easy at all, because the lifting of a closed loop in $\mathbb{CP}^{1}$ via $x=\frac{\wp \left( z\right) -e_{1}}{e_{2}-e_{1}}$ is not necessarily a closed loop in $E_{\tau}$. As an application of Theorems \[thm4-2\] and \[thm4-1\], we could give a positive answer.
First we review the Fuchs-Okamoto theory. Consider a second order Fuchsian equation defined on $\mathbb{CP}^{1}$ as follows:$$y^{\prime \prime}+p_{1}\left( x\right) y^{\prime}+p_{2}\left( x\right) y=0,
\label{90}$$ which has five regular singular points at $\left \{ t,0,1,\infty
,\lambda \right \} $ and $p_{j}\left( x\right) =p_{j}(x;t$, $\lambda$, $\mu
)$, $j=1,2$, are rational functions in $x$ such that the Riemann scheme of (\[90\]) is$$\left(
\begin{array}
[c]{ccccc}t & 0 & 1 & \infty & \lambda \\
0 & 0 & 0 & \hat{\alpha} & 0\\
\theta_{t} & \theta_{0} & \theta_{1} & \hat{\alpha}+\theta_{\infty} & 2
\end{array}
\right) , \label{91}$$ where $\hat{\alpha}$ is determined by the Fuchsian relation, that is, $$\hat{\alpha}=-\tfrac{1}{2}\left( \theta_{t}+\theta_{0}+\theta_{1}+\theta_{\infty}-1\right) .$$ Throughout this section we always assume that $$\text{ }\lambda \not \in \left \{ 0,1,t\right \} \text{ and }\lambda \text{
\textit{is an apparent singular point}.} \label{137}$$ Since one exponent at any one of $0,1,\lambda,t$ is $0$ (see (\[91\])), $p_{2}(x)$ has only simple poles at $0,1,\lambda,t$. The residue of $p_{1}\left( x\right) $ at $x=\lambda$ is $-1$ because another exponent at $x=\lambda$ is $2$. Define $\mu$ and $K$ as follows:$$\mu \doteqdot \underset{x=\lambda}{\text{ Res }}p_{2}\left( x\right) ,\text{
}K\doteqdot-\underset{x=t}{\text{Res}}\text{ }p_{2}\left( x\right) .
\label{92}$$ By (\[91\])-(\[92\]), we have $$p_{1}\left( x\right) =\frac{1-\theta_{t}}{x-t}+\frac{1-\theta_{0}}{x}+\frac{1-\theta_{1}}{x-1}-\frac{1}{x-\lambda}, \label{96}$$$$p_{2}\left( x\right) =\frac{\hat{\kappa}}{x\left( x-1\right) }-\frac{t\left( t-1\right) K}{x\left( x-1\right) \left( x-t\right)
}+\frac{\lambda \left( \lambda-1\right) \mu}{x\left( x-1\right) \left(
x-\lambda \right) }, \label{97}$$ where $$\hat{\kappa}=\hat{\alpha}\left( \hat{\alpha}+\theta_{\infty}\right)
=\frac{1}{4}\left \{ \left( \theta_{0}+\theta_{1}+\theta_{t}-1\right)
^{2}-\theta_{\infty}^{2}\right \} . \label{a}$$ By the condition (\[137\]), i.e., $\lambda$ is apparent, $K$ can be expressed explicitly by $$K\left( \lambda,\mu,t\right) =\frac{1}{t\left( t-1\right) }\left \{
\begin{array}
[c]{l}\lambda \left( \lambda-1\right) \left( \lambda-t\right) \mu^{2}+\hat
{\kappa}\left( \lambda-t\right) \\
-\left[
\begin{array}
[c]{l}\theta_{0}\left( \lambda-1\right) \left( \lambda-t\right) +\theta
_{1}\lambda \left( \lambda-t\right) \\
+\left( \theta_{t}-1\right) \lambda \left( \lambda-1\right)
\end{array}
\right] \mu
\end{array}
\right \} . \label{98}$$ For all details about (\[96\])-(\[98\]), we refer the reader to [@GP].
Now let $t$ be the deformation parameter, and assume that (\[90\]) with $\left( \lambda \left( t\right) ,\mu \left( t\right) \right) $ preserves the monodromy representation. In [@Fuchs; @Okamoto2], it was discovered that under the non-resonant condition, $\left( \lambda \left( t\right)
,\mu \left( t\right) \right) $ must satisfy the following Hamiltonian system:$$\frac{d\lambda \left( t\right) }{dt}=\frac{\partial K}{\partial \mu},\text{
\ }\frac{d\mu \left( t\right) }{dt}=-\frac{\partial K}{\partial \lambda}.
\label{aa}$$ Indeed, the following theorem was proved in [@Fuchs; @Okamoto2].
**Theorem B.[@Fuchs; @Okamoto2]** *Suppose that* $\theta_{t},\theta_{0},\theta_{1},\theta_{\infty}\notin \mathbb{Z}$ *(i.e. the non-resonant condition)* *and* $\lambda$* is an apparent singular point. Then the second order ODE (\[90\]) preserves the monodromy as* $t$* deforms if and only if* $\left( \lambda \left( t\right)
,\mu \left( t\right) \right) $* satisfies the Hamiltonian system (\[aa\]).*
It is well-known in the literature that a solution of Painlevé VI (\[46\]) can be obtained from the Hamiltonian system (\[aa\]) with the Hamiltonian $K\left( \lambda,\mu,t\right) $ defined in (\[98\]). Let $\left( \lambda \left( t\right) ,\mu \left( t\right) \right) $ be a solution to the Hamiltonian system (\[aa\]). Then $\lambda \left( t\right)
$ satisfies the Painlevé VI (\[46\]) with parameters$$\left( \alpha,\beta,\gamma,\delta \right) =\left( \tfrac{1}{2}\theta
_{\infty}^{2},\, -\tfrac{1}{2}\theta_{0}^{2},\, \tfrac{1}{2}\theta_{1}^{2},\,
\tfrac{1}{2}\left( 1-\theta_{t}^{2}\right) \right) . \label{67}$$
Conversely, if $\lambda \left( t\right) $ is a solution to Painlevé VI (\[46\]), then we define $\mu \left( t\right) $ by the first equation of (\[aa\]), where $(\theta_{0},\theta_{1},\theta_{t},\theta_{\infty})$ and $\hat{\kappa}$ are given by (\[67\]) and (\[a\]), respectively. Consequently, $\left( \lambda \left( t\right) ,\mu \left( t\right) \right)
$ is a solution to (\[aa\]). The above facts can be proved directly. For details, we refer the reader to [@GR; @GP]. Together with this fact and Theorem B, we have
**Theorem C.** *Assume the same hypotheses of Theorem B. Then the second order ODE (\[90\]) preserves the monodromy as* $t$ *deforms if and only if* $\lambda(t)$ *satisfies Painlevé VI (\[46\]) with parameters (\[67\]).*
Now let us consider the following generalized Lamé equation in $E_{\tau}$:$$y^{\prime \prime}=\left[
\begin{array}
[c]{l}\sum_{i=0}^{3}n_{i}\left( n_{i}+1\right) \wp \left( z+\frac{\omega_{i}}{2}\right) +\frac{3}{4}\left( \wp \left( z+p\right) +\wp \left( z-p\right)
\right) \\
+A\left( \zeta \left( z+p\right) -\zeta \left( z-p\right) \right) +B
\end{array}
\right] y, \label{89}$$ and suppose that $p$ is an apparent singularity of (\[89\]). Then we shall prove that the generalized Lamé equation (\[89\]) is 1-1 correspondence to the 2nd order Fuchsian equation (\[90\]) with $\lambda$ being an apparent singularity. To describe the 1-1 correspondence between (\[90\]) and (\[89\]), we set$$x=\frac{\wp \left( z\right) -e_{1}}{e_{2}-e_{1}}\text{ and\ }\mathfrak{p}\left( x\right) =4x\left( x-1\right) \left( x-t\right) . \label{123}$$ Then we have the following theorem:
\[thm1\]Given a generalized Lamé equation (\[89\]) defined in $E_{\tau}$. Suppose $p\not \in E_{\tau}\left[ 2\right] $ is an apparent singularity of (\[89\]). Then by using $x=\frac{\wp \left( z\right) -e_{1}}{e_{2}-e_{1}}$, there is a corresponding 2nd order Fuchsian equation (\[90\]) satisfying (\[91\]) and (\[137\]) whose coefficients $p_{1}\left( x\right) $ and $p_{2}\left( x\right) $ are expressed by (\[96\])-(\[98\]), where$$t=\frac{e_{3}-e_{1}}{e_{2}-e_{1}},\text{ \ }\lambda=\frac{\wp \left( p\right)
-e_{1}}{e_{2}-e_{1}}, \label{138}$$$$\left( \theta_{0},\theta_{1},\theta_{t},\theta_{\infty}\right) =\left(
n_{1}+\tfrac{1}{2},n_{2}+\tfrac{1}{2},n_{3}+\tfrac{1}{2},n_{0}+\tfrac{1}{2}\right) , \label{103}$$$$\hat{\alpha}=-\frac{1}{2}\left( 1+n_{0}+n_{1}+n_{2}+n_{3}\right) ,
\label{139}$$$$\mu=\frac{2n_{3}-1}{4\left( \lambda-t\right) }+\frac{2n_{2}-1}{4\left(
\lambda-1\right) }+\frac{2n_{1}-1}{4\lambda}+\frac{3}{8}\frac{\mathfrak{p}^{\prime}\left( \lambda \right) }{\mathfrak{p}\left( \lambda \right) }+\frac{A\wp^{\prime}\left( p\right) }{b^{2}\mathfrak{p}\left(
\lambda \right) }, \label{104}$$$$\begin{aligned}
K & =-\frac{2n_{2}n_{3}-n_{2}-n_{3}}{4\left( t-1\right) }-\frac
{2n_{1}n_{3}-n_{1}-n_{3}}{4t}-\frac{2n_{3}-1}{4\left( t-\lambda \right)
}\label{105}\\
& +\frac{1}{4t\left( t-1\right) }\left[
\begin{array}
[c]{l}\frac{3}{2}\frac{\lambda \left( \lambda-1\right) }{\left( \lambda-t\right)
}-\frac{3}{2}\frac{\wp \left( p\right) +e_{3}}{e_{2}-e_{1}}+\frac
{A\wp^{\prime}\left( p\right) }{\left( \lambda-t\right) (e_{2}-e_{1})^{2}}\\
+\frac{n_{0}\left( n_{0}+1\right) e_{3}}{e_{2}-e_{1}}+\frac{n_{1}\left(
n_{1}+1\right) e_{2}}{e_{2}-e_{1}}+\frac{n_{2}\left( n_{2}+1\right) e_{1}}{e_{2}-e_{1}}\\
-\frac{2n_{3}\left( n_{3}+1\right) e_{3}}{e_{2}-e_{1}}+\frac{2}{e_{2}-e_{1}}A\wp \left( p\right) +\frac{B}{e_{2}-e_{1}}\end{array}
\right] .\nonumber\end{aligned}$$ Conversely, given a 2nd order Fuchsian equation (\[90\]) satisfying (\[91\]) and (\[137\]), there is a corresponding generalized Lamé equation (\[89\]) defined in $E_{\tau}$ where $\tau,\pm p,n_{i}$ are defined by (\[138\])-(\[103\]), the constant $A$ is defined by solving (\[104\]), and the constant $B$ is defined by (\[101\]). In particular, $p$ is an apparent singularity of (\[89\]).
For the second part of Theorem \[thm1\], the condition $\lambda
\not \in \{0,1,t\}$ is equivalent to $p\not \in E_{\tau}[2]$, which implies $\wp^{\prime}(p)\not =0$. Thus, $A$ is well-defined via (\[104\]). The proof of Theorem \[thm1\] will be given after Corollary \[corcor\].
Let $\rho:\pi_{1}(E_{\tau}\backslash \left( E_{\tau}\left[ 2\right]
\cup \left \{ \pm p\right \} \right) ,q_{0})\rightarrow SL\left(
2,\mathbb{C}\right) $, $\tilde{\rho}:\pi_{1}(\mathbb{CP}^{1}\backslash
\left \{ 0,1,t,\infty \right \} $, $\lambda_{0})\rightarrow GL\left(
2,\mathbb{C}\right) $ where $\lambda_{0}=x\left( q_{0}\right) ,$ be the monodromy representations of the generalized Lamé equation (\[89\]) and the corresponding Fuchsian equation (\[90\]) respectively. Let $Y\left(
z\right) =\left( y_{1}\left( z\right) ,y_{2}\left( z\right) \right) $ be a fixed fundamental solution of (\[89\]). We denote $\mathcal{N}$ and $\mathcal{M}$ to be the monodromy groups of (\[89\]) and (\[90\]) with respect to $Y\left( z\right) $ and $\hat{Y}\left( x\right) $ respectively. Here $\hat{Y}\left( x\right) =\left( \hat{y}_{1}\left( x\right) ,\hat
{y}_{2}\left( x\right) \right) $ with $\hat{y}_{j}\left( x\right) $ defined by$$\begin{aligned}
{y_{j}\left( z\right) } & =\psi(x){}\hat{y}_{j}\left( x\right) \\
& \doteqdot \left( x-\lambda \right) ^{-\frac{1}{2}}x^{-\frac{n_{1}}{2}}(x-1)^{-\frac{n_{2}}{2}}(x-t)^{-\frac{n_{3}}{2}}\hat{y}_{j}\left( x\right)
\text{, }j=1,2,\end{aligned}$$ and $x=\frac{\wp \left( z\right) -e_{1}}{e_{2}-e_{1}}$ is a fundamental solution of equation (\[90\]); see the proof of Theorem \[thm1\] below. Let $\gamma_{1}\in \pi_{1}\left( E_{\tau}\backslash \left( E_{\tau}\left[
2\right] \cup \left \{ \pm p\right \} \right) ,q_{0}\right) $ be a loop which encircles the singularity $\frac{\omega_{1}}{2}$ once. Then $x\left(
\gamma_{1}\right) \in \pi_{1}\left( \mathbb{CP}^{1}\backslash \left \{
0,1,t,\infty \right \} ,\lambda_{0}\right) $. Since $x=\frac{\wp \left(
z\right) -e_{1}}{e_{2}-e_{1}}$ is a double cover, the loop $x\left(
\gamma_{1}\right) $ encircles the singularity $0$ twice. Thus, $x\left(
\gamma_{1}\right) =\beta^{2}$ for some $\beta \in \pi_{1}\left( \mathbb{CP}^{1}\backslash \left \{ 0,1,t,\infty \right \} ,\lambda_{0}\right) $. Let $\rho \left( \gamma_{1}\right) =N_{1}$ and $\tilde{\rho}\left( \beta \right)
=M_{0}$. Then$$\begin{aligned}
Y\left( z\right) N_{1} & =\gamma_{1}^{\ast}Y\left( z\right) =\left(
\beta^{2}\right) ^{\ast}\left( {\psi(x)}\hat{Y}\left( x\right) \right)
\label{1}\\
& =C\left( \beta^{2}\right) {\psi(x)}\hat{Y}\left( x\right) M_{0}^{2}\nonumber \\
& =Y\left( z\right) C\left( \beta^{2}\right) M_{0}^{2}\nonumber\end{aligned}$$ for some constant $C\left( \beta^{2}\right) \in \mathbb{C}$ which comes from the analytic continuation of ${\psi(x)}$ along $\beta^{2}$. From (\[1\]), we see that $N_{1}=C\left( \beta^{2}\right) M_{0}^{2}$. By the same argument, we know that any element $N\in \mathcal{N}$ could be written as $$N=CM_{1}M_{2}\label{2}$$ for some $M_{i}\in \mathcal{M}$, $i=1,2$ and some constant $C\in \mathbb{C}$ coming from the gauge transformation ${\psi(x)}$. In general, $\mathcal{N}$ is not a subgroup of $\mathcal{M}$ because of ${\psi(x)}$. By (\[2\]), the isomonodromic deformation of (\[90\]) implies the isomonodromic deformation of (\[89\]). However, it is not clear to see whether the converse assertion is true or not from (\[2\]). Here we can give a confirmative answer. In fact, by (\[125\]), (\[126\]) and (\[67\]), we have (\[103\]) holds. Since $n_{i}\not \in \frac{1}{2}+\mathbb{Z}$ for $i\in \{0,1,2,3\}$, we have $\theta_{0},\theta_{1},\theta_{t},\theta_{\infty}\notin \mathbb{Z}$, i.e., non-resonant. Then as a consequence of Theorems \[thm1\], \[theorem1-2\] and C, we have
\[corcor\]Suppose $n_{i}\not \in \frac{1}{2}+\mathbb{Z}$ for $i=0,1,2,3$. If the generalized Lamé equation (\[89\]) in $E_{\tau}$ preverses the monodromy, then so does the corresponding Fuchsian equation (\[90\]) on $\mathbb{CP}^{1},$ and vice versa.
\[Proof of Theorem \[thm1\]\][ Let us first consider the generalized Lamé equation (\[89\]). By applying $$x=\frac{\wp \left( z\right) -e_{1}}{e_{2}-e_{1}},\text{ \ }t=\frac
{e_{3}-e_{1}}{e_{2}-e_{1}},\text{ \ }\lambda=\frac{\wp \left( p\right)
-e_{1}}{e_{2}-e_{1}},$$ and the addition formula$$\wp \left( z+p\right) +\wp \left( z-p\right) =\frac{\wp^{\prime}\left(
z\right) ^{2}+\wp^{\prime}\left( p\right) ^{2}}{2\left( \wp \left(
z\right) -\wp \left( p\right) \right) ^{2}}-2\wp \left( z\right)
-2\wp \left( p\right) , \label{362}$$ the equation (\[89\]) becomes the following second order Fuchsian equation defined on $\mathbb{CP}^{1}$:$$y^{\prime \prime}(x)+\frac{1}{2}\frac{\mathfrak{p}^{\prime}\left( x\right)
}{\mathfrak{p}\left( x\right) }y^{\prime}(x)-\frac{q(x)}{\mathfrak{p}\left(
x\right) }y(x)=0, \label{106}$$ where $\mathfrak{p}(x)$ is defined in (\[123\]), $b\doteqdot e_{2}-e_{1}$ and]{}$${\small q(x)=\left[
\begin{array}
[c]{l}n_{0}\left( n_{0}+1\right) \left( x+\frac{e_{1}}{b}\right) +\frac
{n_{1}\left( n_{1}+1\right) }{2}\left( \frac{\mathfrak{p}\left( x\right)
}{2x^{2}}-2x+\frac{4e_{1}}{b}\right) +\frac{B}{b}\\
+\frac{n_{2}\left( n_{2}+1\right) }{2}\left( \frac{\mathfrak{p}\left(
x\right) }{2\left( x-1\right) ^{2}}-2x+\frac{2e_{3}}{b}\right)
+\frac{n_{3}\left( n_{3}+1\right) }{2}\left( \frac{\mathfrak{p}\left(
x\right) }{2\left( x-t\right) ^{2}}-2x+\frac{2e_{2}}{b}\right) \\
+\frac{3}{4}\left( \frac{\mathfrak{p}\left( x\right) +\mathfrak{p}\left(
\lambda \right) }{2\left( x-\lambda \right) ^{2}}-2x-\frac{2}{b}(\wp \left(
p\right) +e_{1})\right) +A\left( \frac{2}{b}\zeta \left( p\right)
-\frac{\wp^{\prime}\left( p\right) }{b^{2}\left( x-\lambda \right)
}\right)
\end{array}
\right] .}$$ [Since $p$ is an apparent singularity of (\[89\]), equation (\[106\]) has no logarithmic solutions at $\lambda$. The Riemann scheme of (\[106\]) is as follows$$\left(
\begin{array}
[c]{ccccc}0 & 1 & t & \infty & \lambda \\
-\frac{n_{1}}{2} & -\frac{n_{2}}{2} & -\frac{n_{3}}{2} & -\frac{n_{0}}{2} &
-\frac{1}{2}\\
\frac{n_{1}+1}{2} & \frac{n_{2}+1}{2} & \frac{n_{3}+1}{2} & \frac{n_{0}+1}{2}
& \frac{3}{2}\end{array}
\right) . \label{107}$$ Now consider a gauge transformation $y\left( x\right) =\left(
x-\lambda \right) ^{-\frac{1}{2}}x^{-\frac{n_{1}}{2}}(x-1)^{-\frac{n_{2}}{2}}(x-t)^{-\frac{n_{3}}{2}}\hat{y}\left( x\right) $. Then the Riemann scheme for $\hat{y}\left( x\right) $ is $$\left(
\begin{array}
[c]{ccccc}0 & 1 & t & \infty & \lambda \\
0 & 0 & 0 & \hat{\alpha} & 0\\
n_{1}+\frac{1}{2} & n_{2}+\frac{1}{2} & n_{3}+\frac{1}{2} & \hat{\alpha}+n_{0}+\frac{1}{2} & 2
\end{array}
\right) , \label{108}$$ where ]{}$\hat{\alpha}$[$=-\frac{1}{2}\left( 1+n_{0}+n_{1}+n_{2}+n_{3}\right)
$. Moreover, $\hat{y}\left( x\right) $ satisfies the second order Fuchsian equation$$\hat{y}^{\prime \prime}\left( x\right) +\hat{p}_{1}\left( x,t\right)
\hat{y}^{\prime}\left( x\right) +\hat{p}_{2}\left( x,t\right) \hat
{y}\left( x\right) =0, \label{109}$$ where$$\hat{p}_{1}\left( x,t\right) =\frac{\frac{1}{2}-n_{1}}{x}+\frac{\frac{1}{2}-n_{2}}{x-1}+\frac{\frac{1}{2}-n_{3}}{x-t}-\frac{1}{x-\lambda} \label{110}$$ and[$$\begin{aligned}
& \hat{p}_{2}\left( x,t\right) =\frac{3}{4\left( x-\lambda \right) ^{2}}\label{111}\\
& +\frac{2n_{1}n_{2}-n_{1}-n_{2}}{4x\left( x-1\right) }+\frac{2n_{2}n_{3}-n_{2}-n_{3}}{4\left( x-1\right) \left( x-t\right) }+\frac
{2n_{1}n_{3}-n_{1}-n_{3}}{4x\left( x-t\right) }\nonumber \\
& +\frac{2n_{1}-1}{4x\left( x-1\right) }+\frac{2n_{2}-1}{4\left(
x-1\right) \left( x-\lambda \right) }+\frac{2n_{3}-1}{4\left( x-t\right)
\left( x-\lambda \right) }\nonumber \\
& -\frac{1}{\mathfrak{p}\left( x\right) }\left[
\begin{array}
[c]{l}n_{0}\left( n_{0}+1\right) \left( x+\frac{e_{1}}{b}\right) -n_{1}\left(
n_{1}+1\right) \left( x+\frac{2e_{1}}{b}\right) \\
-n_{2}\left( n_{2}+1\right) \left( x-\frac{e_{3}}{b}\right) -n_{3}\left(
n_{3}+1\right) \left( x-\frac{e_{2}}{b}\right) \\
+\frac{3}{4}\left( \frac{\mathfrak{p}\left( x\right) +\mathfrak{p}\left(
\lambda \right) }{2\left( x-\lambda \right) ^{2}}-2x-2\frac{\wp \left(
p\right) +e_{1}}{b}\right) \\
+A\left( \frac{2}{b}\zeta \left( p\right) -\frac{\wp^{\prime}\left(
p\right) }{b^{2}\left( x-\lambda \right) }\right) +\frac{B}{b}\end{array}
\right] .\nonumber\end{aligned}$$ ]{}Since $p\not \in E_{\tau}\left[ 2\right] $ and equation (\[106\]) has no logarithmic solutions at $\lambda$, it follows that $\lambda \not \in \left \{
0,1,t,\infty \right \} $ and $\lambda$ is an apparent singularity of (\[109\]). Thus, $\hat{p}_{2}\left( x,t\right) $ can be written into the form of (\[97\]) with$$\hat{\kappa}=-\frac{1}{4}\left( n_{0}-n_{1}-n_{2}-n_{3}\right) \left(
1+n_{0}+n_{1}+n_{2}+n_{3}\right) , \label{114}$$ [$$\begin{aligned}
\mu & =\underset{x=\lambda}{\text{ Res }}\hat{p}_{2}\left( x,t\right)
\label{112}\\
& =\frac{2n_{3}-1}{4\left( \lambda-t\right) }+\frac{2n_{2}-1}{4\left(
\lambda-1\right) }+\frac{2n_{1}-1}{4\lambda}+\frac{3}{8}\frac{\mathfrak{p}^{\prime}\left( \lambda \right) }{\mathfrak{p}\left( \lambda \right) }+\frac{A\wp^{\prime}\left( p\right) }{b^{2}\mathfrak{p}\left(
\lambda \right) },\nonumber\end{aligned}$$ ]{}$$K=-\underset{x=t}{\text{Res}}\text{ }\hat{p}_{2}\left( x,t\right) =\tilde
{K},$$ where[$$\begin{aligned}
\tilde{K} & \doteqdot-\frac{2n_{2}n_{3}-n_{2}-n_{3}}{4\left( t-1\right)
}-\frac{2n_{1}n_{3}-n_{1}-n_{3}}{4t}-\frac{2n_{3}-1}{4\left( t-\lambda
\right) }\label{113}\\
& +\frac{1}{4t\left( t-1\right) }\left[
\begin{array}
[c]{l}\frac{3}{2}\frac{\lambda \left( \lambda-1\right) }{\left( \lambda-t\right)
}-\frac{3}{2}\frac{\wp \left( p\right) +e_{3}}{b}+\frac{A\wp^{\prime}\left(
p\right) }{\left( \lambda-t\right) b^{2}}\\
+\frac{n_{0}\left( n_{0}+1\right) e_{3}}{b}+\frac{n_{1}\left(
n_{1}+1\right) e_{2}}{b}+\frac{n_{2}\left( n_{2}+1\right) e_{1}}{b}\\
-\frac{2n_{3}\left( n_{3}+1\right) e_{3}}{b}+\frac{2}{b}A\wp \left(
p\right) +\frac{B}{b}\end{array}
\right] .\nonumber\end{aligned}$$ ]{}Since[ $\lambda$ is an apparent singularity,]{} we conclude from (\[98\]) that$$\tilde{K}=\frac{1}{t\left( t-1\right) }\left \{
\begin{array}
[c]{l}\lambda \left( \lambda-1\right) \left( \lambda-t\right) \mu^{2}+\hat
{\kappa}\left( \lambda-t\right) \\
-\left[
\begin{array}
[c]{l}\theta_{0}\left( \lambda-1\right) \left( \lambda-t\right) +\theta
_{1}\lambda \left( \lambda-t\right) \\
+\left( \theta_{t}-1\right) \lambda \left( \lambda-1\right)
\end{array}
\right] \mu
\end{array}
\right \} . \label{113-0}$$ ]{}
[ Conversely, for a given second order Fuchsian equation (\[90\]) satisfying (\[91\]) and (\[137\]), we know that $p_{1}(x)$, $p_{2}(x)$ and $K$ are given by (\[96\])-(\[98\]), where$$\hat{\kappa}=\hat{\alpha}\left( \hat{\alpha}+\theta_{\infty}\right) .
\label{116}$$ Define ]{}$\pm$[$p$, $n_{i}$ ($i=0,1,2,3$), $A$, and $B$ by $$\lambda=\frac{\wp \left( p\right) -e_{1}}{e_{2}-e_{1}}, \label{117}$$$$\left( \theta_{0},\theta_{1},\theta_{t},\theta_{\infty}\right) =\left(
n_{1}+\frac{1}{2},n_{2}+\frac{1}{2},n_{3}+\frac{1}{2},n_{0}+\frac{1}{2}\right) , \label{115}$$$$\mu=\frac{2n_{3}-1}{4\left( \lambda-t\right) }+\frac{2n_{2}-1}{4\left(
\lambda-1\right) }+\frac{2n_{1}-1}{4\lambda}+\frac{3}{8}\frac{\mathfrak{p}^{\prime}\left( \lambda \right) }{\mathfrak{p}\left( \lambda \right) }+\frac{A\wp^{\prime}\left( p\right) }{b^{2}\mathfrak{p}\left(
\lambda \right) }, \label{118}$$ and$$B=A^{2}-\zeta \left( 2p\right) A-\frac{3}{4}\wp \left( 2p\right) -\sum
_{i=0}^{3}n_{i}\left( n_{i}+1\right) \wp \left( p+\frac{\omega_{i}}{2}\right) . \label{119}$$ Since $\lambda \not \in \{0,1,t,\infty \}$, $p\not \in E_{\tau}\left[ 2\right]
$. Thus $\wp^{\prime}\left( p\right) \not =0$ and $A$ is well-defined by (\[118\]). In order to obtain the corresponding generalized Lamé equation (\[89\]), it suffices to prove that $p_{1}\left( x,t\right) $ and $p_{2}\left( x,t\right) $ can be expressed in the form of (\[110\]) and (\[111\]). By (\[96\]) and (\[115\]), [it is easy to see that]{} $p_{1}\left( x,t\right) $ is of the form (\[110\]). By (\[113\]) and (\[113-0\]), we see that $K$ can be written into (\[113\]), so $p_{2}\left( x,t\right) $ can also be expressed in the form of (\[111\]). Finally, the assertion that ]{}$p$ is an apparent singularity follows from the assumption that [$\lambda$ is an apparent singularity of (\[90\]) (or follows from (\[119\]) and Lemma \[lem-apparent\]). This completes the proof.]{}
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|
---
abstract: 'We consider, within the statistical hadronization model, the near central rapidity $y\simeq 0$ integrated hadron yields expected at LHC $\sqrt{s_{\rm NN}} = 2.76$ TeV ion reactions, for which the total charged hadron rapidity most central head-on collision yield is $dh/dy|_{y=0}\simeq 1800$. For the chemical equilibrium SHM, we discuss composition of $dh/dy$ as function of hadronization temperature. For chemical non-equilibrium SHM, we input the specific strangeness yield $s/S$, demand explosive disintegration and study the break up as a function of the critical hadronization pressure $P$. We develop observables distinguishing the hadronization models and conditions.'
author:
- Johann Rafelski$^1$
- 'Jean Letessier$^{1,2}$'
date: ' December 7, 2010'
title: 'Particle Production in $\sqrt{s_{\rm NN}} = 2.76$ TeV Heavy Ion Collisions'
---
[**Introduction:**]{} According to the theory of strong interactions, Quantum Chromo-Dynamics (QCD), quarks and gluons are confined inside hadrons. At sufficiently high temperature lattice computations demonstrate that the deconfined Quark–Gluon Plasma (QGP) prevails [@Borsanyi:2010cj]. We seek to prove this QCD paradigm in high energy heavy ion collisions where heavy nuclei are crushed on each other, forming a small drop of thermalized deconfined QGP matter. This drop hadronizes into a high multiplicity of particles and we seek to determine the physical properties of QGP considering this multiparticle final state.
At LHC energy $\sqrt{s_{\rm NN}} = 2.76$ TeV, for near head-on 5% most central Pb–Pb collisions, the pseudo-rapidity density of primary charged particles at mid-rapidity is $dh/d\eta={1584} \pm {4} \pm {76} $ [*syst*]{}, an increase of about a factor [2.2]{} to central Au–Au RHIC collisions at $\sqrt{s_{\rm NN}} = 0.2$ TeV [@Aamodt:2010pb]. The study of particle yields per unit of rapidity obtained after integration of transverse momentum particle spectra eliminates the need to model the distortion of spectra introduced by explosive dynamics (see, e.g., [@Bozek:2010wt]) of highly compressed matter created in high energetic collisions.
Following the reference [@Bozek:2010wt], where deformation of the spectra were studied in model cases, we interpret the measured $dh/d\eta$ to be equivalent to a central rapidity density $dh/dy\simeq1800$. This experimental result enables us to offer prediction for a variety of different particle multiplicities, dependent on adopted hadronization model and a first interpretation in terms of particle source bulk properties.
[**SHM — Statistical hadronization model:**]{} QGP hadronic particle production yields are generally considered within the statistical hadronization model (SHM) [@Torrieri:2004zz; @Letessier:2005qe]. SHM has been successful in describing hadron production in heavy ion collisions for different colliding systems and energies. Some view the SHM as a qualitative model and as such one is tempted to seek [*simplicity*]{} in an effort to obtain an estimate of the yields for all hadrons with just a small number of parameters [@Andronic:2005yp; @Andronic:2008gu; @Becattini:2010sk].
Improving experimental precision, along with physics motivation based in qualitative dynamics of the hadronization process has stimulated refinements involving a greater parameter set allowing to control the dynamically established yield of different quark flavors, generally referred to as chemical -equilibrium SHM [@Letessier:2005qe]. This is achieved by introducing statistical occupancy parameters $\gamma_i>1,\ i=q,\,s,\,c$, where $s$ is the strange and $c$ the charm quark flavor. It can be assumed that up and down quark yields $q=u,\,d$ are equally equilibrated. We will not discuss the charm flavor here.
Moreover, we are interested in precise description of the bulk properties of the particle source, such as size, energy and entropy content of the QGP fireball. This requires precise capability to extrapolate observed hadron yields to unobserved kinematic domains and unobserved particle types. This is the case for chemical -equilibrium approach as demonstrated by the smooth systematic behavior of physical observables as a function of collision conditions such as reaction energy [@Letessier:2005qe] or collision centrality [@Rafelski:2004dp].
With increasing collision energy, the baryon content at central rapidity decreases rapidly. It is expected that there remains a small excess of matter over antimatter at central rapidity at LHC [@Andronic:2009gj]. In SHM, this is governed by chemical parameters $\lambda_q,\ \lambda_s$ (equivalent to $\mu_B,\ \mu_S$, the baryon and strangeness chemical potentials of matter). Determination of values of $\mu_B,\ \mu_S$ must await additional experimental data. We present, for orientation of how the baryon–antibaryon yields vary, results testing different values of the expected potentials choosing a fixed input value $\lambda_q =1.0055$, which results for in $\mu_B\simeq 2.9\pm0.3$MeV, $\mu_S\simeq 0.7\pm0.3$MeV, and $\mu_B\simeq 2.0\pm0.2$MeV, $\mu_S\simeq 0.45\pm0.05$MeV in case of equilibrium and non-equilibrium models, respectively.
Note that since the number of strange and antistrange quarks in hadrons has to be equal, $\lambda_s$, and thus $\mu_S$ is determined in order to satisfy this constraint. Another constraint is also implemented, the total charge per baryon has to be $Q/B=0.4$, since the stopping of electrically charged matter (protons) within a rapidity interval is the same as that of all baryonic matter (protons and neutrons). This is achieved by a suitable tiny up-down quark asymmetry, a feature implemented in the SHARE suit of programs we are using [@Torrieri:2004zz].
In the usual procedure of statistical hadronization modeling, the particle yields are used to find best statistical parameters. The SHARE suit of programs was written in a more flexible way to allow also mixed a fit, that is a fit where a few particle yields can be combined with the given ‘measured’ statistical parameters to obtain best fit of other statistical parameters. When we were developing SHARE, this feature was created since a parameter such as temperature could be measured using spectral shape and thus should be not fitted again in the yield description but be used as an experimental input. To be general, this feature was extended to all statistical parameters of the SHARE program.
This feature allows us to perform a fit of the mix of statistical parameters and particle yields. Our procedure has been outlined before [@Rafelski:2008an] and is, here, applied for the first time including LHC experimental data. This allows us to predict within the chemical equilibrium and non-equilibrium SHM models the differing pattern of particle production such that $dh/dy$ is fixed.
[**Importance of understanding chemical (non)-equilibrium:**]{} When chemical -equilibrium is derived from the particle yields, generating in the fit to data $\gamma_i\ne 1$, this suggests a dynamical picture of an explosively expanding and potentially equilibrated QGP, decaying rapidly into free streaming hadrons. Without a significant re-equilibration, the (nearly) equilibrated QGP cannot produce chemically equilibrated hadron yield. The high intrinsic QGP entropy content explains why equilibrated QGP turns into chemically overpopulated (over-saturated) HG phase space. — The fast breakup of QGP means that the emerging hadrons do not have opportunity to re-establish chemical equilibrium in the HG phase.
The differentiation of chemical equilibrium and non-equilibrium SHM models will be one of the challenges we address in our present discussion facing the SHM model and interpretation of the hadron production. One could think that resolution of this matter requires a good fit of SHM parameters to the data. However, with the large errors on particle yields and lack of sensitivity to $\gamma_q$, this is not easy.
The light quark phase space occupancy parameter $\gamma_q$ can be only measurable by determining overall baryon to meson yield, and this cannot be done without prior measurement of hadronization temperature $T$. When particle yield data is not available to measure both $T$ and $\gamma_q$, one can fit only $\gamma_s/\gamma_q$ to the data, which is then reported as $\gamma_s$ accompanied by the tacit assumption $\gamma_q=1$. Since $\gamma_s$ (or $\gamma_s/\gamma_q$) controls the overall (relative) yield of strange quarks, one expects and finds in most environments $\gamma_s\ne 1$ (or $\gamma_s/\gamma_q\ne 1$) and a value which increases with system size, and often with energy.
We recognize considerable physics implication of understanding the value of $\gamma_q$, as this relates directly to the measurement of $T$, related to the phase transformation condition $T_{\rm tr}$ of QGP to hadrons, studied within lattice QCD. In the chemical equilibrium context, $T\simeq T_{\rm tr}$. On the other hand, the chemical non-equilibrium SHM implies rapid expansion and supercooled transformation and hence $T< T_{\rm tr}$, with estimated difference at 10–15 MeV [@Rafelski:2000by].
For lower heavy ion reaction energies as compared to LHC, one can also determine the baryochemical potential at hadronization. This then allows to compare lattice QCD transformation condition, $T_{\rm tr},\ \mu_{\rm B, tr}$ [@deForcrand:2010ys; @Endrodi:2009sd], with the data fit in the chemical equilibrium SHM [@Andronic:2009gj; @Cleymans:2010dw] and chemical non-equilibrium SHM [@Letessier:2005qe]. One finds that lattice results are much flatter compared to the equilibrium SHM, that is $T_{\rm tr}$ drops off much slower with $\mu_{\rm B, tr}$. On the other hand, the non-equilibrium SHM parallels the lattice data 15 MeV below the transformation boundary. This favors, on theoretical grounds, the chemical non-equilibrium approach.
[**LHC predictions assuming chemical equilibrium:**]{} Just one observable, the number of charged particles $dh/dy\simeq 1800$, fixes within statistical model a large number of particle yields. This is allowing to test SHM model. We explore chemical equilibrium and non-equilibrium conditions in turn. We will show that many of the particle yields vary little as function of hadronization condition for a prescribed $dh/dy$. We present an unusual number of digits, a precision which has nothing to do with experiment but is needed to facilitate reproducibility of our results. We further state the propagation error of the error $\Delta\, dh/dy=100$ which is mostly confined to the volume, but is in some cases also visible in $T$ and also further below in the chemical non-equilibrium parameters $\gamma_q,\ \gamma_s$,
Our procedure for the equilibrium SHM is as follows: in the chemical equilibrium model, see table \[eq\_fit\], we set $\gamma_q=1,\ \gamma_s=1$ and as noted also on other grounds $\lambda_q=1.0055$. $\lambda_s$ follows from the constraint of strangeness balance $s-\bar s =0$. All our results maintain a fixed $Q/B=0.4$. We then see that, at fixed hadronization temperature $T$ chosen in table \[eq\_fit\] to be 159, 169, 179, 189 MeV, the yield of charged hadrons $dh/dy$ is, correlated to the source volume $dV/dy$. Volume varies strongly with temperature, since particle yields scale, for $T\gg m$, as $VT^3$. Actually, since $T\gg m$ condition is not satisfied, the volume is changing more rapidly so that $T^kdV/dy={\rm Const.},\ k\simeq 7.2$.
Since charged particle number is fixed, we also expect that entropy of the bulk is fixed and that is true; up to a small variation due to variation with $T$ in relative yield of heavy hadrons, the entropy content is $dS/dy= 14800 \pm 400$. As temperature increases, the proportion of heavy mass charged particles increases, and thus the pion yield and even the kaon yield slightly decrease with increasing $T$. Baryon yields is most sensitive to $T$: $\Omega$ doubles in yield in the temperature interval considered. Our choice of $\lambda_q$ fixes for each hadronization temperature the per rapidity net baryon yield also shown in table \[eq\_fit\]. — We believe our choice is reasonable and has been made also so that we see that there is no need to distinguish particles from antiparticles.
Yield of strangeness is slightly increasing, this increase is in heavy mass strange baryons, e.g., $\Lambda$, and this depletes slightly the yield of kaons. Overall the specific strangeness per entropy yield grows very slowly from $s/S=0.0245$ to 0.027. Several ratios, such as $\phi/{\rm K}^{0*}(892)\simeq 0.45 $, where several effects compensate are nearly constant.
We also show the post-weak decay $\pi^0_{\rm WD}$ yield which is relatively large and independent of hadronization $T$. The decay $\pi^0\to \gamma\gamma$ generates a strong electromagnetic energy component.
---------------------------- -------------- ------------- ------------- -------------
$T^*$\[MeV\] $159$ $169$ $179$ $189$
$\gamma_{q}^*$ $ 1$ $1$ $ 1$ $ 1$
$\gamma_{s}^*$ $1$ $1$ $ 1$ $ 1$
$\lambda_{q}^*$ $ 1.0055$ $1.0055$ $ 1.0055$ $ 1.0055$
$10^3(\lambda_{s}-1)^{**}$ $2.06$ $1.45$ $ 0.89 $ $ 0.39 $
$(Q/B)^*$ $0.4000$ $0.4000$ $0.4000$ $0.4000$
$(s-\bar s)^*$ $0.0000$ $0.0000$ $0.0000$ $0.0000$
$(dh/dy)^*$ $1800$ $1800 $ $1800$ $1800$
$dV/dy$ \[fm$^3$\] 5285$\pm$147 3452$\pm$96 2286$\pm$64 1538$\pm$43
$dS/dy$ 15155 14940 14690 14420
$s/S$ $0.0245$ 0.0255 0.0263 0.0270
$P$ \[MeV/fm$^3$\] $64.1$ 100 153 231
$E/TS$ 0.859 0.86 0.87 0.87
$P/E$ 0.164 0.158 0.153 0.150
$E/V$ \[GeV/fm$^3$\] 0.392 0.632 0.997 1.54
$\pi^-$, $\pi^+$ 839 830 821 813
K$^-$ 141.3 140.8 139.0 136.8
K$^+$ 142.1 141.6 140.1 137.8
p 53.6 63.1 72.0 79.8
$\bar{\rm p}$ 51.9 61.2 69.7 77.3
$\Lambda$ 30.0 36.3 42.1 47.3
$\overline\Lambda$ 29.2 35.4 41.1 46.2
$\Xi^-$ 4.45 5.47 6.41 7.23
$\overline\Xi^+$ 4.36 5.38 6.31 7.14
$\Omega^-$ 0.772 1.038 1.314 1.586
$(B-\overline B)^{**}$ 4.81 5.60 6.29 6.88
$\rho$ 92.4 96.6 99.1 100.3
$\phi$ 19.0 20.5 21.4 21.9
K$^{0*}(892)$ 42.6 45.3 46.9 47.6
K$^{0*}(892)$/K$^-$ 0.301 0.322 0.337 0.348
$\phi$/K$^{0*}(892)$ 0.446 0.452 0.456 0.460
$\pi^0$ 942 933 925 916
$\eta$ 110 111 111 110
$\eta'$ 9.67 10.4 10.8 11.1
$\pi^0_{\rm WD}$ 1251 1251 1249 1245
---------------------------- -------------- ------------- ------------- -------------
: \[eq\_fit\] Chemical equilibrium particle yields at $\sqrt{s_{NN}}= 2.76$ TeV. Top section: input properties; middle section: properties of the fireball associated with central rapidity; bottom section: expected particle yields, and some select ratios. \* signals an input value, and \*\* a result directly following from input value (combined often with a constraint). All yields, but $\pi^0_{\rm WD}$, without week decay feed to particle yields. Error in $dV/dy$ corresponds to error in $dh/dy$ equal 100.
------------------------------ ---------------- ---------------- ---------------- ----------------
$P^*$ \[MeV/fm$^3$\] $60.3 $ $70.0 $ $82.2 $ $90.1 $
$(s/S)^*$ $0.0367 $ $0.0370 $ $0.0370$ $0.0373 $
$\lambda_{q}^*$ $ 1.0055$ $1.0055$ $ 1.0055$ $ 1.0055$
$10^3(\lambda_{s}-1)^{**}$ $2.69$ $2.45$ $ 2.19 $ $2.04 $
$(Q/B)^*$ $0.400$ $0.400$ $0.4000$ $0.4000$
$(s-\bar s)^*$ $0.0000$ $0.0000$ $0.0000$ $0.0000$
$(dh/dy)^*$ $1800$ $1800$ $1800$ $1800$
$T$ \[MeV\] $131.2$ $134.3\pm0.1 $ $137.7\pm0.1$ $139.6\pm0.1$
$\gamma_{q}$ $1.599 $ $1.600 $ $1.601 $ $1.599 $
\[-0.1cm\] $ \pm0.001$ $ \pm0.008$ $ \pm0.009$ $ \pm0.011$
$\gamma_{s}$ $2.913$ $2.842$ $2.745$ $2.721$
\[-0.1cm\] $\pm0.008$ $\pm0.030$ $\pm 0.030$ $\pm 0.016$
\[0.1cm\] $dV/dy$ \[fm$^3$\] $5469$$\pm542$ $4731$$\pm136$ $4043$$\pm119$ $3705$$\pm168$
$dS/dy$ 13924 13879 13794 13797
$E/TS$ 1.060 1.060 1.059 1.059
$P/E$ 0.170 0.168 0.165 0.164
$E/V$ \[GeV/fm$^3$\] 0.354 0.417 0.497 0.550
$\pi^-$, $\pi^+$ 858 854 850 848
K$^-$ 192.0 190.2 186.5 186.2
K$^+$ 192.9 191.2 187.5 187.3
p 32.9 36.3 40.3 42.5
$\bar{\rm p}$ 31.8 35.2 39.0 41.2
$\Lambda$ 28.9 31.8 34.8 36.9
$\overline\Lambda$ 28.1 31.0 33.9 35.9
$\Xi^-$ 6.92 7.56 8.12 8.60
$\overline\Xi^+$ 6.77 7.40 7.96 8.43
$\Omega^-$ 1.56 1.73 1.89 2.03
$(B-\overline B)^{**}$ 3.640 3.973 4.328 4.539
$\rho$ 56.1 58.8 61.7 63.2
$\phi$ 30.0 30.7 30.8 31.4
K$^{0*}(892)$ 39.9 41.4 42.6 43.6
K$^{0*}(892)$/K$^-$ 0.208 0.218 0.228 0.234
$\phi$/$K^{0*}(892)$ 0.751 0.741 0.722 0.721
$\pi^0$ 988 983 979 977
$\eta$ 134 132 128 128
$\eta'$ 10.4 10.7 10.8 11.0
$\pi^0_{\rm WD}$ 1398 1396 1389 1391
------------------------------ ---------------- ---------------- ---------------- ----------------
: \[neq\_fit\] Chemical -equilibrium particle yields, each column for different hadronization pressure. See caption of table \[eq\_fit\] for further details.
[**LHC prediction within non-equilibrium SHM:**]{} Within the non-equilibrium hadronization approach, we need to further anchor the two quark pair abundance parameters $\gamma_q$ and $\gamma_s$. In absence of experimental data, we introduce additional hadronization conditions, the relative strangeness yield $s/S$ and hadronization pressure $P$. We vary $P$, see table \[neq\_fit\], just as we varied $T$ the hadronization condition in the chemical equilibrium model. In the chemical non-equilibrium SHM, two conditions suffice to narrow considerably the values of three SHM parameters ($\gamma_q$, $\gamma_s$ and $T$), but only if we insist that a third condition $E/TS>1$ is qualitatively satisfied.
Strangeness yield is a natural hadronization condition of QGP. We consider the ratio of strangeness per entropy $s/S$ in which $T^3$ coefficients and other systematic dependencies cancel. Since the entropy contents is directly related to the particle multiplicity $dh/dy$, in our case $s/S$ implies $s$-yield and directly relates to strangeness pair yield. In QGP, $s\bar s$-pairs are produced predominantly in thermal gluon processes and their yield can be obtained within the QCD perturbative approach. In a study which was refined to agree with the strangeness yield observed at RHIC, we predicted the value $s/S\simeq 0.037$ for LHC [@Letessier:2006wn]. We use here this result, noting that higher $s/S$ values are possible, depending on LHC formed QGP dynamics. The QGP expected dynamic strangeness yield is considerably higher than the chemical equilibrium yield, table \[eq\_fit\], $0.0270\le s_{\rm eq}/S \le 0.0245$. The greater strangeness content in QGP is, in fact, the reason behind the interest in strangeness as signature of QGP.
The second condition arises from the observation that once the statistical parameters were fitted across diverse reaction conditions at RHIC, the one constant outcome was that the hadronization pressure $P= 82$ MeV/fm$^3$ [@Rafelski:2009jr]. Choice of pressure as a natural QGP hadronization constraint is further rooted in the observation that the vacuum confinement phenomenon can be described within the qualitative MIT-bag model of hadrons introducing vacuum pressure is $B_{\rm MIT}=58$ MeV/fm$^3$, while in a bag-motivated fit to hadron spectra which allows additional flexibility in parameters one finds $B_{\rm fit}=112$ MeV/fm$^3$ [@Aerts:1984vv]. Clearly, a range of values is possible theoretically, with the hadronization condition $P=82$ MeV/fm$^3$ right in the middle of this domain. We will use this ‘critical pressure’ hadronization condition as our constraint, but also vary it such that $60 \le P\le 90$ MeV/fm$^3$ so that we can be sure that our prediction is not critically dependent on the empirical value. The pressure seen in equilibrium model case, table \[eq\_fit\], has a range $64\le P\le 230$ MeV/fm$^3$.
The third constraint is not imposed in its precise value, but we require that hadronization occurs under the constraint that $E/TS>1$. In comparison, for the equilibrium case, table \[eq\_fit\], we have $0.86<E/TS<0.87$ a relatively small variation. The importance of this quantity $E/TS>1$ as a diagnostic tool for explosive QGP outflow and hadronization was discussed in [@Rafelski:2000by]. In fact, we find that a reasonable and stable hadronization arises in chemical non-equilibrium within a narrow interval $1.059<E/TS<1.060$.
In table \[neq\_fit\], the outcome of this procedure is presented. We state the actual values of parameters for which solution of all constraints was numerically obtained, thus in first column pressure is not 60 but 60.3 MeV/fm$^3$. With rising hadronization pressure, the hadronization temperature rises, but it remains well below the phase transformation temperature. As we have discussed, the low value of $T$ in the chemical non-equilibrium SHM is consistent with the dynamics of the expansion, the flow of matter reduces the phase balance $T$. This in turn is then requiring that the light quark abundance parameter $\gamma_q\simeq 1.6$. This is the key distinction of the chemical non-equilibrium. It further signals enhancement of production of baryons over mesons by just this factor. Note that $\gamma_s/\gamma_q\simeq 1.72$. This large ratio means that the yield of $\Lambda$ and $p$ do not differ much. This indicates strong enhancement of strangeness, a first day observable of QGP formation, along with $\phi$ enhancement [@Rafelski:1982ii].
[**Comparison of SHM results:**]{} The large bulk hadronization volume $dV/dy\simeq 4500$fm$^3$ is suggesting that there will be noticeable changes in the HBT observables allowing to produce such a great hadronization volume. The bulk energy content is found in both approaches to be $dE/dy=E/V\times dV/dy= 2.00\pm 0.05$ TeV per unit of rapidity at $y=0$. This is the thermal energy of QGP prior to hadronization measured in the local fluid element rest frame.
The entropy content in the bulk for non-equilibrium, $dS/dy=13860\pm64$, is 5% smaller compared to equilibrium case. This is due to the fact that non-equilibrium particle yields do not maximize entropy. We note that the yields of pions, kaons, and even single strange hyperons are remarkably independent of hadronization pressure, or, in the equilibrium case, temperature, even though the volume parameter changes greatly. This effect is counteracted by a balancing change in hadronization temperature since the yield of charged hadrons is fixed. The hadronization energy density is very close to $E/V\simeq 0.5$ GeV/fm$^3$, and it tracks the pressure since the ratio $P/E$ is found to be rather constant.
The yield of multistrange hadrons is much enhanced in chemical non-equilibrium model, compared to the equilibrium model, on account of 50% increased yield of strangeness, which is potentiated for multistrange particles as was predicted to be the signature of QGP [@Rafelski:1982ii]. The yield of $\Lambda$, for the most favored hadronization condition in both equilibrium and non-equilibrium, can in fact be lower in the non-equilibrium case than in the equilibrium, yet at LHC the yield of K is always 40% greater. Like in the equilibrium results, we observe that several yields are largely independent of the hadronization condition, meaning that ratios such as $\phi/h$ could be a distinctive signature of hadronization, differentiating the two primary models. This is illustrated in figure \[phih\] where the ratio is shown as function of resultant hadronization $T$.
![\[phih\] (color on-line) Specific yield $\phi/h$ for chemical equilibrium model to right bottom (blue) and chemical non-equilibrium model left up (red). ](PLPHITLHC1800.ps){width="2.9in"}
One can, however, argue that $\phi/h$, seen in Fig.\[phih\], could be brought about by strangeness enhancement, not requiring that $\gamma_q>1$, for a more complete discussion see [@Petran:2009dc]. To narrow the choices, we propose to study two more ratios, shown in Fig.\[KstarKpi\]. The top section shows ${\rm K}^*/{\rm K}^-$, as function of the yield ${\rm K}^-/h$. The lower right (red) non-equilibrium result shows strangeness enhancement at low hadronization $T$ since ${\rm K}^*/{\rm K}^-$ mainly depends on $T$. We have shown both particle and antiparticle ratios derived from our fixed input for net baryon yield to illustrate that differentiation of these results will not be easily possible.
![\[KstarKpi\] (color on-line) Top: ratio of resonance K$^*$ to kaon yield K, as function of specific kaon yield K$^-/h$, the equilibrium model (blue) is left up and chemical non-equilibrium (red) is bottom right. Bottom frame: the K$/\pi$ ratio as function of $p/K$ ratio. Bottom right (blue) is equilibrium model and upper left is non-equilibrium model. ](PLKSTKKHLHC1800.ps){width="2.8in"}
![\[KstarKpi\] (color on-line) Top: ratio of resonance K$^*$ to kaon yield K, as function of specific kaon yield K$^-/h$, the equilibrium model (blue) is left up and chemical non-equilibrium (red) is bottom right. Bottom frame: the K$/\pi$ ratio as function of $p/K$ ratio. Bottom right (blue) is equilibrium model and upper left is non-equilibrium model. ](PLKPIPKPLUSLHC1800.ps){width="2.8in"}
The bottom section, in Fig. \[KstarKpi\], shows the strangeness enhancement in format of ${\rm K}/\pi$ as function of the easiest to measure baryon to meson ratio which is a measure of absolute magnitude of $\gamma$, here specifically, $\gamma_q^3/\gamma_s\gamma_q$. We see the equilibrium model to lower right (blue) while the non-equilibrium model is upper left.
[**Summary and conclusions:**]{} We have obtained particle yields within the statistical hadronization model for the LHC-ion run at $\sqrt{s_{\rm NN}} = 2.76$ TeV. We have discussed both bulk properties of QGP at breakup and the resulting particle yields. These can vary significantly depending on the hadronization mechanism. Distinctive features associated with QGP based strangeness enhancement and final state chemical non-equilibrium were described and strategies leading to better understanding of chemical (non-)equilibrium were proposed.
We have shown how enhanced yield of strangeness with the phase space occupancy $\gamma_s\simeq 2.75$ when $\gamma_q\simeq 1.6$ modifies the yield of strange hadrons and detailed predictions for the observables such as $\phi/h$, ${\rm K}^*/{\rm K}$, ${\rm K}/\pi$, $p/{\rm K}$, $\Lambda/p$ were offered. Enhanced yields of (multi)strange particles are tabulated. We note that absolute yield of $\phi$ is enhanced by a factor 1.5 in the non-equilibrium compared to equilibrium hadronization. There is no significant dependence of the $\phi$ yield on hadronization condition making it an ideal first day differentiating chemical equilibrium from non-equilibrium.
The large bulk hadronization volume $dV/dy\simeq 4500$fm$^3$ related to HBT observables, the local rest frame thermal energy content $dE/dy|_0=2$ TeV constrains hydrodynamic models. A large yield of $\pi^0$, $\eta$ and thus of associated decay photons is noted, enhanced somewhat in the.chemical non-equilibrium case.
[**Acknowledgments**]{} LPTHE: Laboratoire de Physique Th[' e]{}orique et Hautes Energies, at University Paris 6 and 7 is supported by CNRS as Unit[' e]{} Mixte de Recherche, UMR7589. This work was supported by a grant from the U.S. Department of Energy, DE-FG02-04ER41318
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abstract: 'Various interpretations of the notion of a trend in the context of global warming are discussed, contrasting the difference between viewing a trend as the deterministic response to an external forcing and viewing it as a slow variation which can be separated from the background spectral continuum of long-range persistent climate noise. The emphasis in this paper is on the latter notion, and a general scheme is presented for testing a multi-parameter trend model against a null hypothesis which models the observed climate record as an autocorrelated noise. The scheme is employed to the instrumental global sea-surface temperature record and the global land-temperature record. A trend model comprising a linear plus an oscillatory trend with period of approximately 60 yr, and the statistical significance of the trends, are tested against three different null models: first-order autoregressive process, fractional Gaussian noise, and fractional Brownian motion. The linear trend is significant in all cases, but the oscillatory trend is insignificant for ocean data and barely significant for land data. By means of a Bayesian iteration, however, using the significance of the linear trend to formulate a sharper null hypothesis, the oscillatory trend in the land record appears to be statistically significant. The results suggest that the global land record may be better suited for detection of the global warming signal than the ocean record.'
title: Statistical significance of rising and oscillatory trends in global ocean and land temperature in the past 160 years
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Introduction
============
At the surface of things, the conceptually simplest approach to detection of anthropogenic global warming should be the estimation of trends in global surface temperature throughout the instrumental observation era starting in the mid-nineteenth century. These kinds of estimates, however, are subject to deep controversy and confusion originating from disagreement about how the notion of a trend should be understood. In this paper we adopt the view that there are several, equally valid, trend definitions. Which one that will prove most useful depends on the purpose of the analysis and the availability and quality of observation data.
At the core of the global change debate is how to distinguish anthropogenically forced warming from natural variability. A complicating factor here is that natural variability has forced as well as internal components. Power spectra of climatic time series also suggest to separate internal dynamics into quasi-coherent oscillatory modes and a continuous and essentially scale-free spectral background. Over a vast range of time scales this background takes the form of a persistent, fractional noise or motion [@Lovejoybook; @Markonis2013]. Hence, the issue is threefold: (i) to distinguish the climate response to anthropogenic forcing from the response to natural forcing, (ii) to distinguish internal dynamics from forced responses, and (iii) to distinguish quasi-coherent, oscillatory modes from the persistent-noise background. This conceptual structure is illustrated by the Venn diagram in Figure 1(a). Figure 1(b) illustrates three possible trend notions based on this picture. Fundamental for all is the separation of the observed climate record into a trend component (also termed the [*signal*]{}) and a [*climate noise*]{} component. The essential difference between these notions is how to make this separation.
![Venn diagrams illustrating the interplay between forced, internal, and natural variability and various definitions of trend. (a): Natural variability can be both forced and internal. Forced variability can be both anthropogenic and natural. Internal variability is natural, but can consist of quasiperiodic oscillatory modes as well as a continuum of persistent noise. (b): The three different trend notions discussed in the text.[]{data-label="fig:Venndiagram"}](Fig1){width=".49\textwidth"}
The widest definition of the trend is to associate it with all forced variability and oscillatory modes as illustrated by the upper row in Figure 1(b). With this notion the methodological challenge will be to develop a systematic approach to extract the persistent noise component from the observed record, and then to subtract this component to establish the trend. The physical relevance of this separation will depend on to what extent we can justify to interpret the extracted trend as a forced response with internally generated oscillatory modes superposed. If detailed information on the time evolution of the climate forcing is not used or is unavailable such a justification is quite difficult. In this case we will first construct a parametrized model for the trend based on the appearance of the climate record at hand and our physical insight about the forcing and the nature of the dynamics. The next step will be to estimate the parameters of the trend model by conventional regression analysis utilizing the observed climate record. The justification of interpreting this trend as something forced and/or coherent different from background noise will be done through a test of the null hypothesis which states that the climate record can be modeled as a long-range memory (LRM) stochastic process. Examples of such processes are persistent fractional Gaussian noises (fGns) or fractional Brownian motions (fBms). For comparison we will also test the null hypothesis against a conventional short-memory notion of climate noise, the first-order autoregressive process (AR(1)). Rejection of this null hypothesis will be taken as an acceptance of the hypothesis that the estimated trend is significant, and will strengthen our confidence that these trends represent identifiable dynamical features of the climate system.
It could be argued that the value of this kind of analysis of statistical significance is of little interest since the result depends on the choice of null model for the climate noise. One can, however, test the null models against the observation data, and here our analysis seems to favor the fGn/fBm models over short-memory models.
If forcing data are available over the time span of the observed temperature record we can utilize this information in a parametrized, linear, dynamic-stochastic model for the climate response. The trend then corresponds to the deterministic solution to this model, i.e., the solution with the known (deterministic) component of the forcing. In this model the persistent-noise component of the temperature record is the response to a white-noise stochastic forcing. The method is described in [@RR2013], where only exponential and scale-free long-range persistent responses are modeled, without allowing for quasi-coherent oscillations. The approach in that paper adopts the trend definition described in the second row of Figure 1(b). Here the trend is the forced variability, while all unforced variability is relegated to the realm of climate noise. It is possible, however, to incorporate forced and natural oscillatory dynamics into such a response model. The simplest way could be to add the response of a forced, damped harmonic oscillator to the scale-free response. These extra degrees of freedom would add an oscillatory response to the deterministic forcing (this would be a forced, oscillatory response), but also an oscillatory response to the stochastic forcing which would be interpreted as an internal oscillatory mode. According to the approach described in [@RR2013] we have to classify all deterministic forced responses as trends, implying that a trend defined this way is not necessarily slow. For instance, the irregular sequence of volcanic eruptions provides a shot-noise like forcing signal. After having estimated the parameters of the forced response model using the full forcing data and the observed temperature record, the residual can be analyzed to assess the validity of different noise models. The responses to fast components in the forcing (like volcanic spikes) will be shifted to the forced response, rather than being incorrectly represented as parts of the internal noise. The test of different noise models via analysis of the residual will therefore give more correct results in the forced-response model than the trend-fit approach employed in the present paper.
The lower row in Figure 1(b) depicts the trend notion of foremost societal relevance; the forced response to anthropogenic forcing. Once we have estimated the parameters of the forced-response model, we can also compute the deterministic response to the anthropogenic forcing separately. One of the greatest advantages of the forced-response methodology is that it allows estimation of this anthropogenic trend/response and prediction of future trends under given forcing scenarios, subject to rigorous estimates of uncertainty. This will be a topic for a forthcoming paper.
Trend Detection Without Forcing Data {#seclot}
====================================
The noise modeling in this paper makes use of the concept of long-range memory (LRM), or (equivalently) long-term persistence (LTP) [@beran1994]. In global temperature records this has been studied in e.g., @pelletier1999 [@lennartz2009; @rybski2006; @Rypdal2010; @efstathiou2011; @rypdalJGR2013; @RR2013]. Emanating from these studies is the recognition that ocean temperature is more persistent than land temperature and that the 20’th century rising trend is stronger for land than for ocean. LRM is characterized by a time-asymptotic ($t\rightarrow \infty)$ autocorrelation function (ACF) of power-law form $C(t)\sim t^{\beta-1}$ for which the integral $\int_0^\infty C(t) \mathrm{d}t$ diverges. Here $\beta$ is a power-law exponent indicating the degree of persistence. The corresponding asymptotic ($f\rightarrow 0$) power spectral density (PSD) has the form $S(f)\sim f^{-\beta}$, hence $\beta$ is also called the spectral index of the LRM process. For $0<\beta<1$ the process is stationary and is termed a persistent fGn. For $1<\beta<3$ the process is non-stationary and termed an fBm. As a short-memory alternative we shall also consider the AR(1) process which has an exponentially decaying ACF and is completely characterized by the one-time-lag autocorrelation $\phi$ [@Storch].
Significance of linear trends under various null models, some exhibiting LRM, was studied by [@CohnandLins] in the context of northern-hemisphere temperature data. One of their main points was that trends classified as statistically significant under a short-memory null hypothesis might end up as insignificant under an LRM hypothesis. Here we will consider the instrumental data record HadSST3 for global ocean temperature [@kennedy2011] and the land temperature record HadCRUT3 [@jones2012]. These records are sea-surface and land-air temperature anomalies relative to the period 1961-90, with monthly resolution from 1850 to date. The analysis is made using a trend model which contains a linear plus a sinusoidal trend, although the methodology developed works for any parametrized trend model. We test this model against the null model that the full temperature record is a realization of an AR(1) process, an fGn, or an fBm (the fBm model is of interest only for the strongly persistent ocean data).
The significance tests are based on generation of an ensemble of synthetic realizations of the null models; AR(1) processes ($\phi<1$), fGns ($0<\beta<1$), and fBms ($1<\beta<3$). Each realization is fully characterized by a pair of parameters; $\theta\equiv (\sigma,\phi)$ for AR(1) and $\theta\equiv (\sigma,\beta)$ for fGn and fBm, where $\sigma$ is the standard deviation of the stationary AR(1) and fGn processes and the standard deviation of the differenced fBm. For an LRM null model the estimated value of $\hat{\beta}$ depends on which null model (fGn or fBm) one adopts. As we will show below, for ocean data, it is not so clear whether an fGn or an fBm is the most proper model [@lennartz2009; @rypdalJGR2013], so we will test the significance of the trends under both hypotheses.
Technically, we make use of the R package by @mcleod2007 to generate synthetic fGns and to perform a maximum-likelihood estimation of $\beta$. Synthetic fBms with memory exponent $1<\beta<3$ are produced by generating an fGn with exponent $\beta-2$ and then forming the cumulative sum of that process. This is justified because the one-step differenced fBm with $1<\beta<3$ is an fGn with memory exponent $\beta-2$ [@beran1994]. Maximum-likelihood estimation of $\beta$ for synthetic fBms and observed data records modeled as an fBm is done by forming the one-time-step increment (differentiation) process, estimate the memory exponent $\beta_\text{incr}$ for that process and find $\beta=\beta_\text{incr}+2$. There are some problems with this method when $\beta \approx 1$. Suppose we have a data record (like the global ocean record) and we don’t know whether $\beta <1$ or $\beta >1$. For all estimation methods there are large errors and biases for short data records of fGns/fBms for $\beta\approx 1$ [@rypdalJGR2013]. This means that there is an ambiguity as to whether a record is a realization of an fGn or an fBm when we obtain estimates of $\beta$ in the vicinity of 1. For the MLE method this ambiguity becomes apparent from Figure 2. Here we have plotted the MLE estimate $\hat{\beta}$ with error bars for an ensemble of realizations of fGns (for $0<\beta<1$) and of fBms ($1<\beta <2$) with 2000 data points. The red symbols are obtained by adopting an fGn model when $\beta$ is estimated. Hence, for $\beta >1$ we find the estimate $\hat{\beta}$ from a realization of an fBm with a model that assumes that it is an fGn. It would be expected that the analysis would give $\hat{\beta} \approx 1$ for an fBm, but we observe that it gives $\hat{\beta}$ considerably less than 1 in the range $1<\beta<1.4$, so if we observe a $\hat{\beta}$ in the vicinity of 1 by this analysis we cannot know whether it is an fGn or an fBm. The ambiguity remains by estimating with a model that assumes that the record is an fBm, because this yields a corresponding positive bias as shown by the green symbols when the record is an fGn. This ambiguity seems difficult to resolve for ocean data as short as the monthly instrumental record.
![The red symbols and 95% confidence intervals represent the maximum-likelihood estimate $\hat{\beta}$ for realizations of fGns/fBms with memory parameter $\beta$ by adopting an fGn model. Hence, for $\beta >1$ we find the estimate $\hat{\beta}$ from a realization of an fBm with a model that assumes that it is an fGn. The green symbols represent the corresponding estimate by adopting an fBm model, i.e., for $\beta<1$ we we find the estimate $\hat{\beta}$ from a realization of an fGn with a model that assumes that it is an fBm. “Adopting an fBm model” means that the synthetic record is differentiated, then analyzed as an fGn by the methods of [@mcleod2007] to obtain $\hat{\beta}_{\text{incr}}$, and then finally $\beta=\hat{\beta}_{\text{incr}}+2$.[]{data-label="fig:errormle"}](Fig2){width=".45\textwidth"}
The standard method for establishing a trend in time-series data is to adopt a parametrized model $T(A;t)$ for the trend, e.g., a linear model $A_1+A_2t$ with parameters $A=(A_1,A_2)$, and estimate the model parameters by a least-square fit of the model to the data. Another method, which brings along additional meaning to the trend concept, is the MLE method. This method adopts a model for the stochastic process; $x(t)=T(A;t)+\sigma w(t)$, where $w(t)$ is a correlated or uncorrelated random process and establishes the set of model parameters $A$ for which the likelihood of the stochastic model to produce the observed data attains its maximum. The method applied to uncorrelated and Gaussian noise models is described in many standard statistics texts [@Storch], and its application to fGns is described in [@mcleod2007]. If $w(t)$ is a Gaussian, independent and identically distributed (i.i.d.) random process, the MLE is equivalent to the least square fit. If $w(t)$ is a strongly correlated (e.g., LRM) process, and the trend model provides a poor description of the large-scale structures in the data, MLE may assign more weight to the random process (greater $\sigma$) than the least-square method. On the other hand, if the trend model is chosen such that it can be fitted to yield a good description of the large-scale structure, the parameters estimated by the two methods are quite similar, even if $w(t)$ used in the MLE method is an LRM process. In this case we can use least-square fit to establish the trend parameters without worrying about whether the residual noise obtained after subtracting the estimated trend can be modeled as a Gaussian, i.i.d. random process.
In the following, we make some definitions and outline the methodology we adopt to assess the significance of the estimated trend. The method is based on standard hypothesis testing, where the trend hypothesis (termed the “alternative hypothesis”) is accepted (although not verified, which is stronger) by rejection of a “null hypothesis.” Failure of rejection of the null hypothesis implies failure of acceptance of the alternative hypothesis, and hence the trend will be characterized as insignificant under this null hypothesis. Hence, it is clear that the outcome of the significance test will depend on the choice of alternative hypothesis (trend model) as well as on the null hypothesis (noise model).
Let us take the pragmatic point of view that a trend is a simple and slowly varying (compared with a predefined time scale $\tau$) function $T(A;t)$ of $t$, parametrized by the trend coefficients $A=(A_1,\ldots,A_n)$. It is also required that for the optimal choice of parameters, $A=\hat{A}_{\text{obs}}$ the trend $T(\hat{A}_{\text{obs}};t)$ makes a good fit to the large-scale structure of the data record. In practice, this means that the trend should be close to a low-pass filtered version of the signal, for instance a moving average over time-scale $\tau$. The trend is significant with respect to a particular null model if the fitted $T(\hat{A}_{\text{obs}};t)$ is very unlikely to be realized in an ensemble of fits $T(\hat{A};t)$ to realizations of the null model.
[*Remark 1*]{}: There is an infinity of measures that one may use to reject the null model, given the data, which use no information about the trend model $T(A;t)$. For instance, the structure of the record on time scales $<\tau$ could be inconsistent with the null hypothesis, while the trend could be consistent with it. In that case we would judge the trend as insignificant, although the null model is rejected by the observed data record.
[**The alternative hypothesis**]{} can be formulated as follows: The observed record $x(t)$ is a realization of the stochastic process
$$T(A;t)+\sigma w(t), \label{altmodel}$$
where the trend $T(A;t)$ is a specified function of $t$ depending on the trend coefficients $A=(A_1,\ldots,A_n)$, and $w(t)$ is a Gaussian stationary random process of unit variance. These coefficients are estimated from a least-square fit to $x(t)$ and have the values $\hat{A}_{\text{obs}}$. We assume that the trend model can be fitted so well to the data that MLE-estimates of $A$ with different noise models (white noise vs. strongly persistent fGn) give approximately the same $\hat{A}_{\text{obs}}$. It is part of the alternative hypothesis that the true value of $A$ is close to the estimated value $\hat{A}_{\text{obs}}$.
[*Remark 2*]{}: Without specifying that $A\approx \hat{A}_{\text{obs}}$ the alternative hypothesis will not set a criterion that can be used to reject the null hypothesis.
[**The null hypothesis**]{} states that the record $x(t)$ is a realization of a stochastic process
$$\varepsilon(\theta;t), \label{nullmodel}$$
with certain properties to be specified (e.g., the process is AR(1), fGn, or fBm). Like for the alternative hypothesis, the parameters $\theta$ should be restricted to be close to the values $\hat{\theta}_{\text{obs}}$ found from estimating it from fitting the null model (\[nullmodel\]) to the data record by means of MLE. How close will be discussed in Remark 4.
[*Remark 3*]{}: The properties of the null model for time scales $<\tau$ are irrelevant (see Remark 1), so a test of the null model should ignore these scales.
[**The Monte Carlo null ensemble**]{} is the collection of realizations $x_i(\theta)\,, i=1,2,\ldots, $ of the null model process (\[nullmodel\]).
[*Remark 4*]{}: The best choice of null model would be to utilize all our possible knowledge about the true parameter set $\theta$. This implies considering $\theta$ as a random variable, and hence a Bayesian approach [@Gelman2004]. We generate the null ensemble by drawing $\theta$ from the conditional distribution $P(\theta|\hat{\theta}_{\text{obs}})$, i.e., the probability that the “real" parameters of the observed process are $\theta$ given that the estimated parameters from the observed data are $ \hat{\theta}_{\text{obs}}$. One way of establishing this distribution is to generate an ensemble of realizations of the noise process with $\theta$ varied in the relevant range $\theta\approx \hat{\theta}_{\text{obs}}$ and establish the conditional distribution $P(\hat{\theta}|\theta)$. From Bayes’ theorem one then has $P(\theta|\hat{\theta})=P(\hat{\theta}|\theta)P(\theta)/P(\hat{\theta})$. By setting $\hat{\theta}=\hat{\theta}_{\text{obs}}$, and assuming a flat prior distribution $P(\theta)$ in the relevant range in the vicinity of $\theta_{\text{obs}}$, we the find $P(\theta|\hat{\theta}_{\text{obs}})=P(\hat{\theta}_{\text{obs}}|\theta)$.
[*Remark 5:*]{} As an alternative to the Bayesian ideas described in remark 4 one could employ a frequentist approach. This means that we assume that the null model has a fixed true parameter value $\theta$. This parameter value is unknown, and the strategy is to create the Monte Carlo null ensemble $x_i(\hat{\theta}_{\text{obs}})\,, i=1,2,\ldots, $ using the $\theta$-values estimated from the observed data. We must then take the uncertainty in the $\theta$-estimates into account, since $\hat{\theta}_{\text{obs}}$ may deviate from the true $\theta$. This estimation error can be quantified using the bootstrap method, which assumes that the error in the parameter estimates in the null model with parameters $\theta$ can be well approximated by the corresponding errors for the null model with parameters $\hat{\theta}_{\text{obs}}$. When estimation errors are quantified one can easily adjust for these in the hypothesis tests.
[**Pseudotrend estimates $\hat{A}^{(i)}$**]{} are the coefficients obtained by least-square fit of the trend model $T(A;t)$ to the realizations $x_i(\theta;t)$ of the null ensemble.
[**Pseudotrend distribution**]{} is the $n$-dimensional PDF $P(\hat{A})$ over the null ensemble.
[**Null-hypothesis confidence region**]{} is the region $\Omega$ in $n$-dimensional $A$-space for which $P(A)>P_{\text{thr}}$, where $P_{\text{thr}}$ is chosen such that $\int_{\Omega}P(A)\, \mathrm{d}A=0.95$.
[**Significance of the trend model**]{} is established if the null hypothesis is rejected, e.g., the full $n$-dimensional trend is 95% significant if $\hat{A}_{\text{obs}}\notin \Omega$.
[*Remark 6*]{}: If the null hypothesis is rejected by this procedure, we are rejecting only those aspects of the null model that are relevant to the full trend model, i.e., the trend model (alternative hypothesis) produces trend coefficients $\hat{A}_{\text{obs}}$ that give a good fit to the large-scale structure of the data, while it is very improbable that the null model can produce $\hat{A}$ in the vicinity of $\hat{A}_{\text{obs}}$.
We will apply the method to global temperature record using the following trend model:
$$T(A;t)=\delta +A_1t+A_2\sin (2\pi f t+\varphi ).
\label{eq:trend}$$
------- --------------------------- ---------------------------- ----------------------------- ---------------------------- ----------------------------- -------------------------- -------------------------- ------------------------
AR(1)
$\hat{\tau}_{\text{obs}}$ $\hat{\beta}_{\text{obs}}$ $\hat{\sigma}_{\text{obs}}$ $\hat{\beta}_{\text{obs}}$ $\hat{\sigma}_{\text{obs}}$ $\hat{A}_{1,\text{obs}}$ $\hat{A}_{2,\text{obs}}$ $\hat{T}_{\text{obs}}$
Ocean 21.3 0.994 0.25 1.45 0.086 4.21 0.128 69.7
Land 3.43 0.654 0.49 6.34 0.186 73.4
------- --------------------------- ---------------------------- ----------------------------- ---------------------------- ----------------------------- -------------------------- -------------------------- ------------------------
: Estimated noise parameters $\hat{\theta}_{\text{obs}}$ from the null hypotheses in and trend parameters $\hat{A}_{\text{obs}}$ estimated from the trend model .
\[tab:trend\]
This is a simplified version of the models used in several works by N. Scafetta (e.g., in [@Scafetta2011; @Scafetta2012]) and the oscillation is supposed to model the 60-yr cycle observed in the instrumental record [@schlesinger1994]. The model contains five parameters to be estimated from the observed temperature records by least-square fit. The frequency $f$ estimated for land and ocean records are slightly different, but both correspond to a period close to 60 yr. When estimating pseudotrends it has little meaning to let $f$ be a free parameter, since the synthetic noise records contain no preferred frequencies. We therefore fix $f$ equal to the value estimated from the observed records. Of the estimated pseudotrend coefficients $(\hat{A}_1,\hat{A}_2, \hat{\delta}, \hat{\varphi})$ only $(\hat{A}_1,\hat{A}_2)$ quantify the strength of the trend, so the relevant pseudotrend distribution to establish is $P(\hat{A}_1,\hat{A}_2)$ irrespective of the values of irrelevant parameters $(\hat{\delta},\hat{\varphi})$. Table 1 shows the estimated $\hat{\theta}_{\text{obs}}$ according to the null model in () using AR(1), fGn and fBm as the stochastic process $\varepsilon(\theta;t)$. Also in this table are the estimated trend parameters $(\hat{A_1},\hat{A_2},\hat{T})_{\text{obs}}$ from applying the trend model in , where $\hat{T}_{\text{obs}}=1/\hat{f}_{\text{obs}}$ is the estimated period of the oscillatory trend.
![In panels (a-e) the red dots represent the estimated trend coefficients $(\hat{A}_1,\hat{A}_2)_{\text{obs}}$ and the dashed, closed curve the 95% confidence contour of the distribution $P(\hat{A}_1,\hat{A}_2)$. (a): ocean data and AR(1) null model. (b): land data and AR(1) null model. (c): ocean data and fGn null model. (d): land data and fGn null model. (e): ocean data and fBm null model. (f): Black curves: The global ocean and land temperature records. Red curves: the linear and sinusoidal trends.[]{data-label="fig:oceantrend"}](Fig3){width=".5\textwidth"}
The results of the analysis are shown in Figure 3. We observe that the trend parameters $(\hat{A}_1,\hat{A}_2)_{\text{obs}}$ are outside the null-hypothesis 95% confidence region for all three noise models and for ocean as well as land records. But we also observe that the significance is more evident for land than for ocean, and is reduced as more strongly persistent noise models are employed. For the fBm model applied to ocean data the trend is barely outside the 95% confidence region.
It is the full n-dimensional trend model that is accepted by this test, but something can also be said about the separate significance of the individual trends represented by the individual trend coefficients from the pseudotrend distribution $P(\hat{A}_1,\hat{A}_2)$. For the AR(1) and fGn null models it is apparent from Figure 3(a)-(d) that the linear trend is highly significant since $\hat{A}_{1,\text{obs}}$ is located far to the right of the confidence region. On the other hand, except for the AR(1) model applied to land data in Figure 3(b), $A_{2,\text{obs}}$ is not totally above the confidence region. This means that the linear pseudotrends observed in the null ensemble has negligible chance of getting near the observed trend, while there is some chance to find oscillatory trends in the null ensemble which are as large as $\hat{A}_{2,\text{obs}}$. The significance of those separate trends against these null models is determined by forming the separate one-dimensional PDFs, $P(\hat{A}_1)\equiv \int P(\hat{A}_1,\hat{A}_2)\mathrm{d}\hat{A}_2$ and $P(\hat{A}_2)\equiv \int P(\hat{A}_1,\hat{A}_2)\mathrm{d}\hat{A}_1$ and form the confidence intervals in the standard way. In Figure 4 we have formed the corresponding one-dimensional cumulative distribution functions (CDFs) from the two-dimensional PDFs for ocean data shown in Figure 3(a), (c), and (e). We observe that the linear trend is significant for the AR(1) and fGn null models, but barely significant for the fBm model. The oscillatory trend is insignificant for all models. The corresponding CDFs for land data are shown in Figure 5. The linear trend is even more significant than for ocean data, while the oscillatory trend is significant for the AR(1) model, but barely significant for the fGn model.
One important lesson to learn from this analysis is that the stronger persistence in the ocean temperature record makes it harder to detect significant trends as compared to the land record. This is contrary to the common belief that the higher noise levels on short time scales in land records will make trend detection more difficult in these records. Another is that the land record analysis establishes beyond doubt that there is a significant global linear trend throughout the last century, and that the reality of an oscillatory trend is probable, but not beyond the 95% confidence limit.
![Curved lines are CDFs for trend coeffecients $\hat{A}_1$ and $\hat{A}_2 $ established from the null model ensemble for land data. Vertical dashed line marks the upper 95% confidence limit. Vertical solid lines mark $\hat{A}_{1,2,\text{obs}}$. (a) and (b): AR(1) null model. (c) and (d): fGn null model.[]{data-label="fig:cdfland"}](Fig5){width=".5\textwidth"}
Sharpening and Evaluating the Null Hypothesis
=============================================
In a Bayesian spirit, it would be appropriate to investigate the oscillatory trend further by including the linear trend as an established fact and construct a sharper null model;
$$\hat{\delta}_{\text{obs}} +\hat{A}_{1,\text{obs}}t+\varepsilon(\theta;t).
\label{newnullmodel}$$
------ --------------------------- ---------------------------- -----------------------------
AR(1)
$\hat{\tau}_{\text{obs}}$ $\hat{\beta}_{\text{obs}}$ $\hat{\sigma}_{\text{obs}}$
Land 2.04 0.584 0.391
------ --------------------------- ---------------------------- -----------------------------
: Estimated noise parameters $\hat{\theta}_{\text{obs}}$ from the new null hypotheses in . The units are same as in Table 1.
\[tab:newnull\]
We now first estimate a new $\hat{\theta}_{\text{obs}}$ by fitting the new null model (\[newnullmodel\]) to the observed land record. The new estimated noise parameters are shown in Table 2. Then we produce a new null ensemble of records from the null model by drawing $\theta$ from the conditional distribution $P(\theta|\hat{\theta}_{\text{obs}})$ as explained in Remark 4. Finally we fit the trend model (\[eq:trend\]) to each realization in the ensemble and form $P(\hat{A}_1,\hat{A}_2)$. The result is shown for land data and $\varepsilon(\theta;t)$ modeled as an fGn in Figure 6(a). The inclusion of the linear trend in the null model will imply that we shall fit $\varepsilon(\theta;t)$ to the record $\tilde{x}(t)\equiv x(t)-(\hat{\delta}_{\text{obs}} +\hat{A}_{1,\text{obs}}t)$ rather than to $x(t)$. Since we already have established that $x(t)$ contains a significant linear trend the variability of $\tilde{x}(t)$ may be considerably less than the variability of $x(t)$ and hence the new estimated noise parameters $\hat{\theta}_{\text{obs}}$ may correspond to smaller $\hat{\sigma}_{\text{obs}}$ and $\hat{\beta}_{\text{obs}})$ than we obtained for the original null model. This reduction in noise parameters leads to narrowing of $P(\hat{A}_1,\hat{A}_2)$, and a narrower CDF for the oscillation trend parameter $\hat{A_2}$, as shown in Figure 6(b). The result is that this sharper test establishes that the oscillatory trend is also significant.
The long-range memory associated with fractional noises and motions allows larger fluctuations on long time scales that allows description of such variability as part of the noise background rather as trends. The implication is that variability which has to be described as significant trends under white-noise or short-memory noise hypotheses are insignificant trends under an LRM null hypothesis. But how do we decide on the proper null hypothesis? One way to deal with this question is to use some estimator that characterizes the correlation structure of the observed record and compare that to the same estimator for different noise models. We have used the Mexican-hat wavelet variance due to its conceptual simplicity and its ability to extract the noise component and eliminate obvious linear trends, and because it works without modification for both fGns and fBms [@rypdalJGR2013]. In Figure 7(a) and (b) we have plotted the wavelet variance versus time scale for the observed ocean and land record and for ensembles of synthetic realizations of short-memory AR(1) processes, fGns and (for ocean) fBms, with parameters estimated from the observed records by the MLE method. The first thing to notice is that the wavelet variance curve for the ocean seems to fit reasonably well within the confidence limits for the fBm null model, but not that well for the fGn model and the AR(1) model. For the land record the wavelet-variance curve is also way outside the limits for both AR(1) and fGn. The reason for this is that the MLE estimates of $\hat{\theta}_{\text{obs}}$ attempts to fit the noise process to a record that consists of a linear trend in addition to a noise, and hence overestimates the noise parameters. The wavelet variance will therefore be estimated from synthetic realizations of too strong and too persistent noise processes. The wavelet variance of the observed signal will be closer to that of the true noise component, because the method automatically removes the effect of a linear trend. This is why the wavelet-variance curve of the observed signal is below the confidence region for the null ensemble in Figure 7(b), and is an additional confirmation that the linear trend is real.
In Figure 7(c) and (d) we have plotted the wavelet variance curves of the linearly detrended observed records. They are identical to the wavelet variance for the full records, which demonstrates that the wavelet variance is insensitive to a linear trend. The colored curves and confidence regions are produced from the null ensembles produced from the new null model (\[newnullmodel\]). As discussed above, this null model has weaker and less persistent noise, and this reduces the wavelet variances and bring the curves more in line with those obtained from the observed records. Now it appears that the observed wavelet variance is within the confidence limits for the LRM processes fGn/fBm, but somewhat outside those limits for the short-range AR(1) process, suggesting a preference for the LRM null models. These features appear considerably clearer when residuals between the observed records and the deterministic records from the forced response models are analyzed by wavelet variance [@RR2013], the reason being primarily that the small-scale features of the forced response in that case is subtracted, and hence this residual will give a better representation of the internal climate-noise component.
The differences between Figure 7(a), (b) and Figure 7(c), (d) show that the residual after linear detrending is much better described as a noise process than the undetrended record, and hence gives a clear indication that the new null model (\[newnullmodel\]) is better than the original model (\[nullmodel\]).
Conclusions
===========
In this paper we have attempted to classify the various possible ways to understand the notion of a trend in the climate context, and then we have focused on the detection of a combination of a rising and oscillatory trend in global ocean and land instrumental data when no information about the climate forcing is used. It is well known that the statistical significance of the trends depends on the degree of autocorrelation (memory) assumed for the random noise component of the climate record [@CohnandLins; @rybski2006; @rybski2009]. It is also known that the linear trends are easier to detect and appear to be more significant in global than in local data [@lennartz2009], although local records exhibit weaker long-term persistence than global records. Despite this fact, much effort is spent on establishing trends and their significance in data from local stations (e.g., [@Franzke2012]) with variable results. The failure of detecting consistent trends in local data records reflects the tendency of internal spatiotemporal variability to mask the trend that signals global warming, and we believe therefore that investigation of such trends should be performed on globally averaged data. For global data records our study demonstrates very clearly that the long-range memory observed in sea-surface temperature data leads to lower significance of detected trends compared to land data. This does not mean, of course, that the global warming signal and internal oscillations are not present in the local records or in the global ocean record. It is just not possible to establish the statistical significance of these trends from these records alone, since the large short-range weather noise in local temperatures and the slower fluctuations in ocean temperature both reduce the possibilities of trend detection. Hence, one needs to search for the optimal climate record to analyze for detection of the global warming signal, and our results suggest that the global land temperature signal may be the best candidate for such trend studies.
![(a): The 95% confidence contour of the distribution $P(\hat{A}_1,\hat{A}_2)$ for land data obtained by the new null model (\[newnullmodel\]) with $\varepsilon(\theta;t)$ an fGn process. (b): The CDF derived from $P(\hat{A}_2)$ for this null model, with upper 95% confidence limit marked as dotted vertical line.[]{data-label="fig:cdfnewnull"}](Fig6){width=".5\textwidth"}
![Panels (a) and (b) show the wavelet variance versus time scale for the observed ocean (a) and land (b) records (black crosses) and for ensembles of synthetic realizations of AR(1) processes (blue), fGns (red) and (for ocean) fBms (green), with parameters estimated from the observed records by the MLE method. The shaded areas are 95% confidence regions for these estimates. Panels (c) and (d) show the wavelet-variance curves of the linearly detrended, observed records (black crosses) and for the synthetic realizations of the processes generated from the new null model (\[newnullmodel\]). []{data-label="fig:waveletvariogram"}](Fig7){width=".5\textwidth"}
While a linear trend is only marginally significant under the long-range memory null hypothesis in ocean data, it is clearly significant in land data. Hence, there should be no doubt about the significance of a global warming signal over the last 160 years even under null hypotheses presuming strong long-range persistence of the climate noise.
There is some more doubt about the significance of a 60 year oscillatory mode in the global signal, as shown in Figure 4(d) and 4(f) and Figure 5(d). By mean of a Bayesian iteration, however, utilizing the established significance of a linear trend to formulate a sharper null hypothesis, we are able to establish statistical significance of the oscillatory trend in the land data record. We believe this is an important result, because it means that we cannot dismiss this oscillation as a spontaneous random fluctuation in the climate noise background. By the analysis presented here we cannot decide whether this oscillation is an internal mode in the climate system or an oscillation forced by some external influence. Such insights can be obtained from a generalization of the response model of [@RR2013] by employing information about the climate forcing, and will be the subject of a forthcoming paper. There are various published hypotheses about the nature of this oscillation. The least controversial is that this is a global manifestation of the Atlantic Multidecadal Oscillation (AMO) which is essentially an internal climate mode [@schlesinger1994]. Some authors go further and suggest that this oscillation is synchronized and phase locked with some astronomical influence [@Scafetta2011; @Scafetta2012]. Although some of these suggestions seem very speculative, there are some quite well-documented connections between periodic tidal effects on the Sun from the motion of the giant planets and radioisotope paleorecord proxies for solar activity on century and millennium time scales [@Abreu]. So far there exists no solid evidence that these, and multidecadal, variations in solar activity have a strong influence on terrestrial climate, but the issue will probably be in the frontline of research on natural climate variability in the time to come.
The authors are grateful to Ola L[ø]{}vsletten for illuminating discussions and comments.
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abstract: 'In this paper, we study the asymptotic behavior of a supercritical $(\xi,\psi)$-superprocess $(X_t)_{t\geq 0}$ whose underlying spatial motion $\xi$ is an Ornstein-Uhlenbeck process on $\mathbb R^d$ with generator $L = \frac{1}{2}\sigma^2\Delta - b x \cdot \nabla$ where $\sigma, b >0$; and whose branching mechanism $\psi$ satisfies Grey’s condition and some perturbation condition which guarantees that, when $z\to 0$, $\psi(z)=-\alpha z + \eta z^{1+\beta} (1+o(1))$ with $\alpha > 0$, $\eta>0$ and $\beta\in (0, 1)$. Some law of large numbers and $(1+\beta)$-stable central limit theorems are established for $(X_t(f) )_{t\geq 0}$, where the function $f$ is assumed to be of polynomial growth. A phase transition arises for the central limit theorems in the sense that the forms of the central limit theorem are different in three different regimes corresponding the branching rate being relatively small, large or critical at a balanced value.'
address:
- |
Yan-Xia Ren\
LMAM School of Mathematical Sciences & Center for Statistical Science\
Peking University\
Beijing, P. R. China, 100871
- |
Renming Song\
Department of Mathematics\
University of Illinois at Urbana-Champaign\
Urbana, IL, USA, 61801
- |
Zhenyao Sun\
School of Mathematics and Statistics\
Wuhan University\
Hubei, P. R. China, 100871
- |
Jianjie Zhao\
School of Mathematical Sciences\
Peking University\
Beijing, P. R. China, 100871
author:
- 'Yan-Xia Ren, Renming Song, Zhenyao Sun and Jianjie Zhao'
title: 'Stable Central Limit Theorems for Super Ornstein-Uhlenbeck Processes'
---
[^1]
[^2]
[^3]
Introduction
============
Motivation {#subsec:M}
----------
Let $d \in \mathbb N:= \{1,2,\dots\}$ and $\mathbb R_+:= [0,\infty)$. Let $\xi=\{(\xi_t)_{t\geq 0}; (\Pi_x)_{x\in \mathbb R^d}\}$ be an $\mathbb R^d$-valued Ornstein-Uhlenbeck process (OU process) with generator $$\begin{aligned}
Lf(x)
= \frac{1}{2}\sigma^2\Delta f(x)-b x \cdot \nabla f(x)
, \quad x\in \mathbb R^d, f \in C^2(\mathbb R^d),\end{aligned}$$ where $\sigma > 0$ and $b > 0$ are constants. Let $\psi$ be a function on $\mathbb R_+$ of the form $$\begin{aligned}
\label{eq: honogeneou branching mechanism}
\psi(z)
=- \alpha z + \rho z^2 + \int_{(0,\infty)} (e^{-zy} - 1 + zy)~\pi(dy)
, \quad z \in \mathbb R_+,\end{aligned}$$ where $\alpha > 0 $, $\rho \geq0$ and $\pi$ is a measure on $(0,\infty)$ with $\int_{(0,\infty)}(y\wedge y^2)~\pi(dy)< \infty$. $\psi$ is referred to as a branching mechanism and $\pi$ is referred to as the Lévy measure of $\psi$. Denote by $\mathcal M(\mathbb R^d)$ the space of all finite Borel measures on $\mathbb R^d$. For $f,g\in \mathcal B(\mathbb R^d, \mathbb R)$ and $\mu \in \mathcal M(\mathbb R^d)$, write $\mu(f)= \int f(x)\mu(dx)$ and $\langle f, g\rangle = \int f(x)g(x) dx$ whenever the integrals make sense. We say a real-valued Borel function $f:(t,x)\mapsto f(t,x)$ on $\mathbb R_+\times \mathbb R^d$ is *locally bounded* if, for each $t\in \mathbb R_+$, we have $ \sup_{s\in [0,t],x\in \mathbb R^d} |f(s,x)|<\infty. $ We say that an $\mathcal M(\mathbb R^d)$-valued Hunt process $X = \{(X_t)_{t\geq 0}; (\mathbb{P}_{\mu})_{\mu \in \mathcal M(\mathbb R^d)}\}$ on $(\Omega, \mathscr{F})$ is a *super Ornstein-Uhlenbeck process (super-OU process)* with branching mechanism $\psi$, or a $(\xi, \psi)$-superprocess, if for each non-negative bounded Borel function $f$ on $\mathbb R^d$, we have $$\begin{aligned}
\label{eq: def of V_t}
\mathbb{P}_{\mu}[e^{-X_t(f)}]
= e^{-\mu(V_tf)}
, \quad t\geq 0, \mu \in \mathcal M(\mathbb R^d),\end{aligned}$$ where $(t,x) \mapsto V_tf(x)$ is the unique locally bounded non-negative solution to the equation $$\begin{aligned}
V_tf(x) + \Pi_x \Big[ \int_0^t\psi (V_{t-s}f(\xi_s) )~ds\Big]
= \Pi_x [f(\xi_t)]
, \quad x\in \mathbb R^d, t\geq 0.\end{aligned}$$ The existence of such super-OU process $X$ is well known, see [@Dynkin1993Superprocesses] for instance.
Recently, there have been quite a few papers on laws of large numbers for superdiffusions. In [@Englander2009Law; @EnglanderWinter2006Law; @EnglanderTuraev2002A-scaling], some weak laws of large numbers (convergence in law or in probability) were established. The strong law of large numbers for superprocesses was first studied in [@ChenRenWang2008An-almost], followed by [@ChenRenSongZhang2015Strong-law; @ChenRenYang2019Skeleton; @EckhoffKyprianouWinkel2015Spines; @KouritzinRen2014A-strong; @LiuRenSong2013Strong; @Wang2010An-almost] under different settings. For a good survey on recent developments in laws of large numbers for branching Markov processes and superprocesses, see [@EckhoffKyprianouWinkel2015Spines].
The strong law of large numbers for the super-OU process $X$ above can be stated as follows: Under some conditions on $\psi$ (these conditions are satisfied under our Assumptions 1 and 2 below), there exists an $\Omega_0$ of $\mathbb{P}_\mu$-full probability for every $\mu\in\mathcal M(\mathbb R^d)$ such that on $\Omega_0$, for every Lebesgue-a.e. continuous bounded non-negative function $f$ on $\mathbb R^d$, we have $\lim_{t\to\infty} e^{-\alpha t} X_t(f) =H_\infty\langle f, \varphi\rangle $, where $H_\infty$ is the limit of the martingale $e^{-\alpha t}X_t(1)$ and $\varphi$ is the invariant density of the OU process $\xi$ defined in below. See [@ChenRenYang2019Skeleton Theorem 2.13 & Example 8.1] and [@EckhoffKyprianouWinkel2015Spines Theorem 1.2 & Example 4.1].
In this paper, we will establish some spatial central limit theorems (CLTs) for the super-OU process $X$ above. Our key assumption is that $\psi$ satisfies Grey’s condition and some perturbation condition which guarantees that, when $z\to 0$, $\psi(z)=-\alpha z + \eta z^{1+\beta} (1+o(1))$ with $\alpha > 0$, $\eta>0$ and $\beta\in (0, 1)$. Our goal is to find $(F_t)_{t\geq 0}$ and $(G_t)_{t\geq 0}$ so that $ (X_t(f) -G_t)/F_t $ converges weakly to some non-degenerate random variable as $t\rightarrow\infty$, for a large class of functions $f$. Note that, in the setting of this paper, $X_t(f)$ typically has infinite second moment.
There are many papers on CLTs for branching processes, branching diffusions and superprocesses under the second moment condition. See [@Heyde1970A-rate; @HeydeBrown1871An-invariance; @HeydeLeslie1971Improved] for supercritical Galton-Watson processes (GW processes), [@KestenStigum1966Additional; @KestenStigum1966A-limit] for supercritical multi-type GW processes, [@Athreya1969Limit; @Athreya1969LimitB; @Athreya1971Some] for supercritical multi-type continuous time branching processes and [@AsmussenHering1983Branching] for general supercritical branching Markov processes under certain conditions. Some spatial CLTs for supercritical branching OU processes with binary branching mechanism were proved in [@AdamczakMilos2015CLT] and some spatial CLTs for supercritical super-OU processes with branching mechanisms satisfying a fourth moment condition were proved in [@Milos2012Spatial]. These two papers made connections between CLTs and branching rate regimes. Some spatial CLTs for supercritical super-OU processes with branching mechanisms satisfying only a second moment condition were established in [@RenSongZhang2014Central]. Moreover, compared with the results of [@AdamczakMilos2015CLT; @Milos2012Spatial], the limit distributions in [@RenSongZhang2014Central] are non-degenerate. Since then, a series of spatial CLTs for a large class of general supercritical branching Markov processes and superprocesses with spatially dependent branching mechanisms were proved in [@RenSongZhang2014CentralB; @RenSongZhang2015Central; @RenSongZhang2017Central]. The functional version of the CLTs were established in [@Janson2004Functional] for supercritical multitype branching processes, and in [@RenSongZhang2017Functional] for supercritical superprocesses.
There are also many limit theorems for supercritical branching processes and branching Markov processes with branching mechanisms of infinite second moment. Heyde [@Heyde1971Some] established some CLTs for supercritical GW processes when the offspring distribution belongs to the domain of attraction of a stable law of index $\alpha\in (1, 2]$, and proved that the limit laws are stable laws. Similar results for supercritical multi-type GW processes and supercritical continuous time branching processes, under some $p$-th ($p\in(1,2]$) moment condition on the offspring distribution, were given in Asmussen [@Asmussen76Convergence]. Recently, Marks and Miloś [@MarksMilos2018CLT] considered the limit behavior of supercritical branching OU processes with a special stable offspring distribution. They established some spatial CLTs in the small and critical branching rate regimes, but they did not prove any CLT type result in the large branching rate regime. We also mention here that very recently [@IksanovKoleskoMeiners2018Stable-like] considered stable fluctuations of Biggins’ martingales in the context of branching random walks and [@RenSongSun2018Limit] considered the asymptotic behavior of a class of critical superprocesses with spatially dependent stable branching mechanism.
As far as we know, this paper is the first to study spatial CLTs for supercritical superprocesses without the second moment condition.
Main results {#sec:I:R}
------------
We will always assume that the following assumption holds.
\[asp: Greys condition\] The branching mechanism $\psi$ satisfies Grey’s condition, i.e., there exists $z' > 0$ such that $\psi(z) > 0$ for all $z>z'$ and $\int_{z'}^\infty \psi(z)^{-1}dz < \infty$.
For $\mu \in \mathcal M(\mathbb R^d)$, write $\|\mu\| = \mu(1)$. It is known (see [@Kyprianou2014Fluctuations Theorems 12.5 & 12.7] for example) that, under Assumption \[asp: Greys condition\], the *extinction event* $D :=\{\exists t\geq 0,~\text{s.t.}~ \|X_t\| =0 \}$ has positive probability with respect to $\mathbb P_\mu$ for each $\mu \in \mathcal M(\mathbb R^d)$. In fact, $ \mathbb{P}_{\mu} (D) = e^{-\bar v \|\mu\|}$ where $ \bar v := \sup\{\lambda \geq 0: \psi(\lambda) = 0\} \in (0,\infty) $ is the largest root of $\psi$.
Denote by $\Gamma$ the gamma function. For any $\sigma$-finite signed measure $\mu$, we use $|\mu|$ to denote the total variation measure of $\mu$. In this paper, we will also assume the following:
\[asp: branching mechanism\] There exist constants $\eta > 0$ and $\beta \in (0,1)$ such that $$\begin{aligned}
\label{eq: asp of branching mechanism}
\int_{(1,\infty)}y^{1+\beta +\delta}~\Big|\pi(dy)-\frac{\eta~dy}{\Gamma(-1-\beta)y^{2+\beta}}\Big| <\infty
\end{aligned}$$ for some $\delta > 0$.
We will show in Subsection \[sec: branching mechanism\] that if Assumption \[asp: branching mechanism\] holds, then $\eta$ and $\beta$ are uniquely determined by the Lévy measure $\pi$. In the reminder of the paper, we will always use $\eta$ and $\beta$ to denote the constants in Assumption \[asp: branching mechanism\]. Note that $\delta$ is not uniquely determined by $\pi$. In fact, if $\delta>0$ is a constant such that holds, then replacing $\delta$ by any smaller positive number, still holds. Therefore, Assumption \[asp: branching mechanism\] is equivalent to the following statement: There exist constants $\eta > 0$ and $\beta \in (0,1)$ such that, for all small enough $\delta>0$, holds.
\[rem:SP\] Roughly speaking, Assumption \[asp: branching mechanism\] says that $\psi$ is “not too far away” from $\widetilde \psi(z) := - \alpha z + \eta z^{1+\beta}$ near $0$. In fact, if we consider their difference $$\begin{aligned}
\label{eq:PB}
& \psi_1(z)
:= \psi(z) - \widetilde \psi(z)
\\ &= \rho z^2+ \int_{(0,\infty)}(e^{-yz}-1+yz) \Big(\pi(dy) - \frac{\eta~dy}{\Gamma(-1-\beta) y^{2+\beta}}\Big),
\quad z\geq 0,\end{aligned}$$ then it can be verified that (see Lemma \[lem:CEP\] below) $\psi_1(z)/z^{1+\beta} \xrightarrow[z\to 0]{} 0$. Therefore, we can write $ \psi(z) = - \alpha z + z^{1+\beta}(\eta + o(1))$ as $z\to 0$. One can further write that $\psi(z) = - \alpha z + z^{1+\beta} l(z)$ where $l$ is a function on $[0,\infty)$ which is slowly varying at $0$.
It will be proved in Lemma \[lem: LlogL criterion\] that, under Assumption \[asp: branching mechanism\], $\psi$ satisfies the $L \log L$ condition, i.e., $ \int_{(1,\infty)} y\log y~\pi(dy) < \infty. $ This guarantees that $H_\infty$, the limit of the non-negative martingale $(e^{-\alpha t} \|X_t\|)_{t\geq 0}$, is non-degenerate.
Let us introduce some notation in order to give the precise formulation of our main result. Denote by $\mathcal B(\mathbb R^d, \mathbb R)$ the space of all $\mathbb R$-valued Borel functions on $\mathbb R^d$. Denote by $\mathcal B(\mathbb R^d, \mathbb R_+)$ the space of all $\mathbb R_+$-valued Borel functions on $\mathbb R^d$. We use $(P_t)_{t\geq 0}$ to denote the transition semigroup of $\xi$. Define $
P^{\alpha}_t f(x)
:= e^{\alpha t} P_t f(x)
= \Pi_x [e^{\alpha t}f(\xi_t)]
$ for each $x\in \mathbb R^d$, $t\geq 0$ and $f\in \mathcal B(\mathbb R^d, \mathbb R_+)$. It is known that, see [@Li2011Measure-valued Proposition 2.27] for example, $(P^\alpha_t)_{t\geq 0}$ is the *mean semigroup* of $X$ in the sense that $
\mathbb{P}_{\mu}[X_t (f)] = \mu( P^\alpha_t f)
$ for all $\mu\in \mathcal M(\mathbb R^d)$, $t\geq 0$ and $f\in \mathcal B(\mathbb R^d, \mathbb R_+)$.
The limit behavior of $X$ is closely related to the spectral property of the OU semigroup $(P_t)_{t\geq 0}$ which we now recall (See [@MetafunePallaraPriola2002Spectrum] for more details). It is known that the OU process $\xi$ has an invariant probability on $\mathbb R^d$ $$\begin{aligned}
\label{invariantdensity}
\varphi(x)dx
:=\Big (\frac{b}{\pi \sigma^2}\Big )^{d/2}\exp \Big(-\frac{b}{\sigma^2}|x|^2 \Big)dx\end{aligned}$$ which is a symmetric multivariate Gaussian distribution. Let $L^2(\varphi)$ be the Hilbert space with inner product $$\begin{aligned}
\langle f_1, f_2 \rangle_{\varphi}
:= \int_{\mathbb R^d}f_1(x)f_2(x)\varphi(x) dx, \quad f_1,f_2 \in L^2(\varphi).\end{aligned}$$ Let $\mathbb Z_+ := \mathbb N\cup\{0\}$. For each $p = (p_k)_{k = 1}^d \in \mathbb{Z}_+^{d}$, write $|p|:=\sum_{k=1}^d p_k$, $p!:= \prod_{k= 1}^d p_k!$ and $\partial_p:= \prod_{k = 1}^d(\partial^{p_k}/\partial x_k^{p_k})$. The *Hermite polynomials* are defined by $$\begin{aligned}
H_p(x)
:=(-1)^{|p|}\exp(|x|^2) \partial_p \exp(-|x|^2)
, \quad x\in \mathbb R^d, p \in \mathbb{Z}_+^{d}.\end{aligned}$$ It is known that $(P_t)_{t\geq 0}$ is a strongly continuous semigroup in $L^2(\varphi)$ and its generator $L$ has discrete spectrum $\sigma(L)= \{-bk: k \in \mathbb Z_+\}$. For $k \in \mathbb Z_+$, denote by $\mathcal{A}_k$ the eigenspace corresponding to the eigenvalue $-bk$, then $ \mathcal{A}_k = \operatorname{Span} \{\phi_p : p\in \mathbb Z_+^d, |p|=k\}$ where $$\begin{aligned}
\label{eigenfunction}
\phi_p(x)
:= \frac{1}{\sqrt{ p! 2^{|p|} }} H_p \Big(\frac{ \sqrt{b} }{\sigma}x \Big)
, \quad x\in \mathbb R^d, p\in \mathbb Z_+^d.\end{aligned}$$ In other words, $
P_t\phi_p(x)
= e^{-b|p|t}\phi_p(x)
$ for all $t\geq 0$, $x\in \mathbb R^d$ and $p\in \mathbb Z_+^d$. Moreover, $\{\phi_p: p \in \mathbb Z_+^d\}$ forms a complete orthonormal basis of $L^2(\varphi)$. Thus for each $f\in L^2(\varphi)$, we have $$\begin{aligned}
\label{semicomp1}
f
= \sum_{k=0}^{\infty}\sum_{p\in \mathbb Z_+^d:|p|=k}\langle f, \phi_p \rangle_{\varphi} \phi_p
, \quad \text{in~} L^2(\varphi).\end{aligned}$$ For each function $f\in L^2(\varphi)$, define the order of $f$ as $$\kappa_f
:= \inf \left \{k\geq 0: \exists ~ p\in \mathbb Z_+^d , {\rm ~s.t.~} |p|=k {\rm ~and~} \langle f, \phi_p \rangle_{\varphi}\neq 0\right \}$$ which is the lowest non-trivial frequency in the eigen-expansion . Note that $ \kappa_f\geq 0$ and that, if $f\in L^2(\varphi)$ is non-trivial, then $\kappa_f<\infty$. In particular, the order of any constant non-zero function is zero.
Denote by $\mathcal M_c(\mathbb R^d)$ the space of all finite Borel measures of compact support on $\mathbb R^d$. For $p\in \mathbb{Z}_+^d$, define $
H_t^p
:= e^{-(\alpha-|p|b)t}X_t(\phi_p)
$ for all $t\geq 0$. If $\alpha \tilde \beta>|p|b, \tilde \beta := \beta/(1+\beta)$, then for all $\gamma\in (0, \beta)$ and $\mu\in \mathcal M_c(\mathbb R^d)$, we will prove in Lemma \[lem:M:L:ML\] that $(H_t^p)_{t\geq 0}$ is a $\mathbb{P}_{\mu}$-martingale bounded in $L^{1+\gamma}(\mathbb{P}_{\mu})$. Thus the limit $H^p_{\infty}:=\lim_{t\rightarrow \infty}H_t^p$ exists $\mathbb{P}_{\mu}$-almost surely and in $L^{1+\gamma}(\mathbb{P}_{\mu})$.
We first present a law of large numbers for our model which extends the strong laws of large numbers of [@ChenRenYang2019Skeleton; @EckhoffKyprianouWinkel2015Spines] in which the first order asymptotic ($\kappa_f=0$) was identified. Denote by $\mathcal P$ the class of functions of polynomial growth on $\mathbb R^d$, i.e., $$\begin{aligned}
\label{eq: polynomial growth function}
\mathcal{P}
:= \{f\in \mathcal B(\mathbb R^d, \mathbb R):\exists C>0, n \in \mathbb Z_+ \text{~s.t.~} \forall x\in \mathbb R^d, |f(x)|\leq C(1+|x|)^n \}.\end{aligned}$$ It is clear that $\mathcal{P} \subset L^2(\varphi)$.
\[thm: law of large number\] If $f \in \mathcal{P}$ satisfies $\alpha\tilde \beta>\kappa_f b$, then for all $\gamma\in (0, \beta)$ and $\mu\in \mathcal M_c(\mathbb R^d)$, $$e^{-(\alpha-\kappa_fb)t}X_t(f)
\xrightarrow[t\to \infty]{}\sum_{p\in \mathbb Z_+^d:|p|=\kappa_f}\langle f, \phi_p\rangle_{\varphi} H_{\infty}^p
\quad in~ L^{1+\gamma}(\mathbb{P}_{\mu}).$$ Moreover, if $f$ is twice differentiable and all its second order partial derivatives are in $\mathcal{P}$, then we also have almost sure convergence.
If $f\in \mathcal B(\mathbb R^d, \mathbb R_+)$ is non-trivial and bounded, then $\kappa_f=0$. Hence, Theorem \[thm: law of large number\] says that for any $\gamma\in (0, \beta)$ and $\mu\in \mathcal M_c(\mathbb R^d)$, as $t\rightarrow \infty$, $
e^{-\alpha t}X_t(f)
\rightarrow \langle f, \varphi\rangle H_{\infty}
$ in $L^{1+\gamma}(\mathbb{P}_{\mu})$. Moreover, if $f$ is twice differentiable and all its second order partial derivatives are in $\mathcal{P}$, then we also have a.s. convergence. However, to get a.s. convergence for bounded non-negative Lebesgue-a.e. continuous functions $f$, we do not need $f$ to be twice differentiable. See [@ChenRenYang2019Skeleton Theorem 2.13 & Example 8.1] and [@EckhoffKyprianouWinkel2015Spines Theorem 1.2 & Example 4.1].
For the rest of this subsection, we focus on the CLTs of $X_t(f)$ for a large collection of $f\in \mathcal P\setminus \{0\}$. Write $\tilde u = \frac{u}{ 1+ u}$ for each $u \neq -1$. It turns out that there is a phase transition in the sense that the results are different in the following three cases:
1. the small branching rate case where $f$ satisfies $\alpha \tilde \beta < \kappa_f b$;
2. the critical branching rate case where $f$ satisfies $\alpha \tilde \beta = \kappa_f b$; and
3. the large branching rate case where $f$ satisfies $\alpha \tilde \beta > \kappa_f b$.
Here, the small (resp. large) branching rate case means that the branching rate $\alpha$ is small (resp. large) compared to $\kappa_f$; and the critical branching rate means that the branching rate $\alpha$ is at a critical balanced value compared to $\kappa_f$. To present our result, we define a family of operators $(T_t)_{t\geq 0}$ on $\mathcal P$ by $$\begin{aligned}
\label{eq:I:R:1}
T_t f
:= \sum_{p \in \mathbb Z_+^d} e^{-| |p|b - \alpha \tilde \beta |t} \langle f, \phi_p \rangle_{\varphi} \phi_p
,\quad t\geq 0, f\in \mathcal P,\end{aligned}$$ and a family of $\mathbb C$-valued functionals $(m_t)_{0 \leq t < \infty}$ on $\mathcal P$ by $$\begin{aligned}
\label{eq:I:R:2}
m_t[f]
:= \eta \int_0^t ~du \int_{\mathbb R^d} (-iT_u f(x))^{1+\beta} \varphi(x) ~dx
, \quad 0 \leq t< \infty, f\in \mathcal P.\end{aligned}$$ Define $ \mathcal C_s := \mathcal P \cap \overline{\operatorname{Span}} \{ \phi_p: \alpha \tilde \beta < |p| b \}$, $\mathcal C_c := \mathcal P \cap \operatorname{Span} \{ \phi_p : \alpha \tilde \beta = |p| b \} $ and $ \mathcal C_l := \mathcal P \cap \operatorname{Span} \{ \phi_p: \alpha \tilde \beta > |p| b \}$. Note that $\mathcal C_s$ is an infinite dimensional space, $ \mathcal C_l$ and $\mathcal C_c$ are finite dimensional spaces, and $\mathcal C_c$ might be empty. For $f\in \mathcal P\setminus \{0\}$, in Lemma \[lem:m\] and Proposition \[prop:PL:S\] below, we will show that $$\begin{aligned}
\label{eq:I:R:3}
m[f]
:= \begin{cases}
\lim_{t\to \infty} m_t[f], &
f \in \mathcal C_s \oplus \mathcal C_l, \\
\lim_{t\to \infty} \frac{1}{t} m_t[f], & f\in \mathcal P \setminus \mathcal C_s \oplus \mathcal C_l,
\end{cases}\end{aligned}$$ is well defined, and moreover, there exists a $(1+\beta)$-stable random variable $\zeta^f$ with characteristic function $\theta \mapsto e^{m[\theta f]}$. The main result of this paper is as follows.
\[thm:M\] If $\mu\in \mathcal M_c(\mathbb R^d)\setminus \{0\}$, then under $\mathbb{P}_{\mu}(\cdot|D^c)$, the following hold:
1. \[thm:M:1\] if $f\in \mathcal C_s\setminus\{0\}$, then $\|X_t\|^{- \frac{1}{1+\beta}} X_t(f) \xrightarrow[t\to \infty]{d} \zeta^f$;
2. \[thm:M:2\] if $f\in \mathcal C_c\setminus\{0\}$, then $ \|t X_t\|^{-\frac{1}{1+\beta}} X_t(f) \xrightarrow[t\to \infty]{d} \zeta^f$;
3. \[thm:M:3\] if $f\in \mathcal C_l\setminus\{0\}$, then $$\|X_t\|^{-\frac{1}{1+\beta}} \Big( X_t(f) - \sum_{p\in \mathbb Z^d_+:\alpha \tilde \beta>|p|b}\langle f,\phi_p\rangle_\varphi e^{(\alpha-|p|b)t}H^p_{\infty}\Big)
\xrightarrow[t\to \infty]{d}
\zeta^{-f}.$$
At this point, we should mention that the theorem above does not cover all $f\in \mathcal P$. Theorem \[thm:M\].(1) can be rephrased as if $f\in \mathcal P\setminus\{0\}$ satisfies $\alpha \tilde \beta < \kappa_f b$, then under $\mathbb{P}_{\mu}(\cdot|D^c)$, $\|X_t\|^{- \frac{1}{1+\beta}} X_t(f) \xrightarrow[t\to \infty]{d} \zeta^f$. Combining the first two parts of Theorem \[thm:M\], one can easily get that if $f\in \mathcal P$ satisfies $\alpha \tilde \beta = \kappa_f b$, then under $\mathbb{P}_{\mu}(\cdot|D^c)$, $ \|t X_t\|^{-\frac{1}{1+\beta}} X_t(f) \xrightarrow[t\to \infty]{d} \zeta^f$. A general $f \in \mathcal P$ can be decomposed as $f_s + f_c + f_l$ with $f_s \in \mathcal C_s$, $f_c \in \mathcal C_c$ and $f_l \in \mathcal C_l$. For $f\in \mathcal P$ satisfying $\alpha \tilde \beta > \kappa_f b$, $f_s$ and $f_c$ maybe non-trivial. In this case, we do not have a CLT yet. We conjecture that the limit random variables in Theorem \[thm:M\] for $ f\in \mathcal C_s$, $f\in \mathcal C_c$ and $ f\in \mathcal C_l$ are independent. If this is valid, we can get a CLT for $ X_t(f)$ for all $f\in \mathcal P$. This independence is valid under the second moment condition, see [@RenSongZhang2015Central]. We leave the question of independence of the limit stable random variables to a future project.
We now give some intuitive explanation of the branching rate regimes and the phase transition. Similar explanation has been given in the context of branching-OU processes, see [@MarksMilos2018CLT]. First we mention that a super-OU process arises as the “high density” limit of a sequence of branching-OU processes, see [@Li2011Measure-valued] for example. A superprocess can be thought of as a cloud of infinitesimal branching “particles” moving in space. The phase transition is due to an interplay of two competing effects in the system: coarsening and smoothing. The coarsening effect corresponds to the increase of the spatial inequality and is a consequence of the branching: simply an area with more particles will produce more offspring. The smoothing effect corresponds to the decrease of the spatial inequality and is a consequence of the mixing property of the OU processes: each OU “particle” will “forget” its initial position exponentially fast.
Let us consider $X_t(\phi_p)$ as an example and discuss how the parameters $\alpha, \beta, b$ and $|p|$ influence those two effects:
- The branching rate $\alpha$ captures the mean intensity of the branching in the system. Therefore, the lager the branching rate $\alpha$, the stronger the coarsening effect.
- The tail index $\beta$ describes the heaviness of the tail of the offspring distribution which belongs to the domain of attraction of some $(1+\beta)$-stable random variable. When $\beta$ is smaller i.e. the tail is heavier, then it is more likely that one particle can suddenly have a large amount of offspring. In other words, the larger the tail index $\beta$, the smaller the fluctuation of offspring number, and then the stronger the coarsening effect.
- The drift parameter $b$ is related to the level of the mixing property of the OU particles. The larger the drift parameter $b$, the faster the OU-particles forgetting their initial position, and therefore the stronger the smoothing effect.
- The order $|p|$ is related to the capability of $\phi_p$ capturing the mixing property of the OU particles. In particular, in the case that $|p| = 0$, no mixing property can be captured by $\phi_p \equiv 1$ since we are only considering the total mass $\|X_t\|$. In general, the higher the order $|p|$, the more mixing property can be captured by $\phi_p$, and therefore the stronger the smoothing effect.
Here we discuss the role of the other parameters $\rho, \eta$ and $\sigma$ in our model:
- The coefficient $\rho$ dose not influence the result since $\rho z^2$ in the branching mechanism $\psi$ is a part of the small perturbation $\psi_1$ (see Remark \[rem:SP\]).
- The coefficients $\eta$ and $\sigma$ are hidden in the definition of the functional $m[f]$, and therefore influence the actual distribution of the limiting $(1+\beta)$-stable random variable $\xi^f$. Their role in the coarsening and smoothing effects are negligible compared to the four parameters $\alpha, \beta, b$ and $|p|$ mentioned above.
An outline of the methodology
-----------------------------
Let us give some intuitive explanation of the methodology used in this paper. For any $\mu\in \mathcal M_c(\mathbb R^d)$ and any random variable $Y$ with finite mean, we define $
\mathcal I_s^t Y
:= \mathcal I_s^t [Y, \mu]
:= \mathbb P_\mu[Y|\mathscr F_t] - \mathbb P_\mu[Y|\mathscr F_s]
$ where $0 \leq s \leq t <\infty.$ We will use the shorter notation $\mathcal I_s^t Y$ when there is no danger of confusion. For $f\in \mathcal{P}$, consider the following decomposition over the time interval $[0,t]$: $$\begin{aligned}
X_t(f)
:= \sum_{k=0}^{\lfloor t \rfloor-1} \mathcal I_{t-k-1}^{t-k} X_t (f)+\mathcal I_0^{t-\lfloor t \rfloor} X_t(f) + X_0( P^\alpha_tf),
\quad t\geq 0.\end{aligned}$$ To find the fluctuation of $X_t(f)$, we will investigate the fluctuation of each term on the right hand side above. The second term and third term are negligible after the rescaling, and for the first term we will establish a multi-variate unit interval CLT which says that $$\Big( \|X_t\|^{-\frac{1}{1+\beta}}\mathcal I^{t-k}_{t-k-1} X_t(f) \Big)_{k=0}^n
\xrightarrow [t\to \infty]{d} (\zeta^f_k)_{k=0}^n,$$ where $(\zeta^f_k)_{k \in \mathbb N}$ are independent $(1+\beta)$-stable random variables. If $f \in \mathcal C_s\setminus\{0\}$, then it can be argued that $\sum_{k=0}^{\lfloor t \rfloor} \zeta^f_k \xrightarrow[t\to \infty]{d} \zeta^f$ and then intuitively we have $
\|X_t\|^{-\frac{1}{1+\beta}} X_t(f)
\xrightarrow[t\to \infty]{d} \zeta^f.
$ If $f \in \mathcal C_c \setminus \{0\}$, then it can be argued that $
t^{-\frac{1}{1+\beta}} \sum_{k=0}^{\lfloor t\rfloor} \zeta_k \xrightarrow[t\to \infty]{ d} \zeta^f
$ and then intuitively we have $
\|tX_t\|^{-\frac{1}{1+\beta}} X_t(f)
\xrightarrow[t\to \infty]{d} \zeta^f.
$ If $f\in \mathcal C_l$, the general idea is almost the same, except that we need to consider the decomposition over the time interval $[t,\infty)$.
This paper is our first attempt on stable CLTs for superprocesses. There are still many open questions. Ren, Song and Zhang have established some spatial CLTs in [@RenSongZhang2015Central] for a class of superprocesses with general spatial motions under the assumption that the branching mechanisms satisfy a second moment condition. We hope to prove spatial CLTs for superprocesses with general motions without the second moment assumption on the branching mechanism in a future project.
Recall that our Assumption \[asp: branching mechanism\] says that the branching mechanism $\psi$ is $-\alpha z +\eta z^{1+\beta}$ plus a small perturbation $\psi_1(z)$ which satisfies with some $\delta>0$. It would be interesting to consider more general branching mechanisms.
The following correspondence between (sub)critical branching mechanisms and Bernstein functions is well known, see, for instance, [@Bertoin Theorem VII.4(ii)] and [@BRY Proposition 7]. Suppose that $f, g:(0, \infty)\to [0, \infty)$ are related by $f(x)=xg(x)$. Then $f$ is a (sub)critical branching mechanism with $\lim_{x\to 0}f(x)=0$ iff $g$ is a Bernstein function with a decreasing Lévy density. We now use this correspondence to give some examples of branching mechanisms satisfying Assumptions \[asp: Greys condition\] and \[asp: branching mechanism\]. If $h$ is a complete Bernstein function which is regularly varying at 0 with index $\beta_1\in (\beta, 1)$, then $$\psi(z)
:= -\alpha z + \rho z^2+\eta z^{1+\beta}+zh(z)
, \qquad z>0,$$ satisfies Assumptions \[asp: Greys condition\] and \[asp: branching mechanism\]. If $\beta_1\in (\beta, 1)$, $c_1\in (0, \eta/\Gamma(-1-\beta))$ and $c_2\ge 1$, then $$\psi(z)
:=-\alpha z + \rho z^2+\eta z^{1+\beta}-\int^\infty_{c_2} (e^{-yz}-1+yz)\frac{c_1dy}{y^{1+\beta_1}}
, \qquad z\in \mathbb R_+,$$ satisfies Assumptions \[asp: Greys condition\] and \[asp: branching mechanism\].
The rest of the paper is organized as follows: In Subsection \[sec: branching mechanism\] we will give some preliminary results for the branching mechanism $\psi$. In Subsections \[sec: controller\] and \[sec: h-controller\] we will give some estimates for some operators related to the super-OU process $X$. In Subsection \[sec: stable distributions\] we will give the definitions of the $(1+\beta)$-stable random variables involved in this paper. In Subsection \[sc:refined\] we will give some refined estimate for the OU semigroup. In Subsection \[sec: Small value probability\] we will give some estimates for the small value probability of continuous state branching processes. In Subsection \[sec: Moments for super-OU processes\] we will give upper bounds for the $(1+\gamma)$-moments for our superprocesses. These estimates and upper bounds will be crucial in the proofs of our main results. In Subsection \[sec: large rate lln\], we will give the proof of Theorem \[thm: law of large number\]. In Subsections \[sec:critical\]–\[sec: large rate clt\], we will give the proof of Theorem \[thm:M\]. In the Appendix, we consider a general superprocess $(X_t)_{t\geq 0}$ and we prove that the characteristic exponent of $X_t(f)$ satisfies a complex-valued non-linear integral equation. This fact will be used at several places in this paper, and we think it is of independent interest.
Preliminaries
=============
Branching mechanism {#sec: branching mechanism}
-------------------
Let $\psi$ be the branching mechanism given in . Suppose that Assumptions \[asp: Greys condition\] and \[asp: branching mechanism\] hold. In this subsection, we give some preliminary results on $\psi$. Recall that $\eta$ and $\beta$ are the constants in Assumption \[asp: branching mechanism\]. Let $\mathbb C_+:= \{x+iy: x\in \mathbb R_+, y \in \mathbb R\}$ and $\mathbb C^0_+:= \{x+iy: x\in (0,\infty), y \in \mathbb R\}$.
\[lem:CEP\] The function $\psi_1$ given by can be uniquely extended as a complex-valued continuous function on $\mathbb C_+$ which is holomorphic on $\mathbb C^0_+$. Moreover, for all $\delta > 0$ small enough, there exists $C>0$ such that for all $z\in \mathbb C_+$, we have $|\psi_1(z)| \leq C |z|^{1+\beta+\delta} + C|z|^2.$
According to Lemma \[lem: extension lemma for branching mechanism\] below and the uniqueness of holomorphic extensions, we know that $\psi_1$ can be uniquely extended as a complex-valued continuous function on $\mathbb C_+$ which is holomorphic on $\mathbb C^0_+$. The extended $\psi_1$ has the following form: $$\psi_1(z)
= \rho z^2 + \int_{(0,\infty)}(e^{-yz}-1+yz) \Big(\pi(dy) - \frac {\eta~dy} {\Gamma(-1-\beta)y^{2+\beta}} \Big)
, \quad z\in \mathbb C_+.$$ Now, according to Assumption \[asp: branching mechanism\], for all small enough $\delta > 0$, we have $$\begin{aligned}
|\psi_1(z)|
& \leq \rho |z|^2 + \int_{(0,\infty)} (|yz|\wedge |yz|^2) \Big|\pi(dy) - \frac{\eta~dy}{\Gamma(-1-\beta)y^{2+\beta}}\Big| \\
& \leq |z|^2 \Big(\rho + \int_{(0,1)} y^2 \Big|\pi(dy) - \frac{\eta~dy}{\Gamma(-1-\beta)y^{2+\beta}}\Big|\Big) \\
& \quad + |z|^{1+\beta +\delta}\int_{(1,\infty)} y^{1+\beta + \delta} \Big|\pi(dy) - \frac{\eta~dy}{\Gamma(-1-\beta)y^{2+\beta}}\Big|,
\quad z \in \mathbb C_+,
\end{aligned}$$ as desired.
The following lemma says that the constants $\eta, \beta$ in Assumption \[asp: branching mechanism\] are uniquely determined by the Lévy measure $\pi$.
\[lem: unique of beta and eta\] Suppose Assumption \[asp: branching mechanism\] holds. Suppose that there are $\eta', \delta'>0$ and $\beta'\in (0,1)$ such that $$\int_{(1,\infty)} y^{ 1 + \beta' + \delta' }~ \Big| \pi(dy) - \frac {\eta' ~dy} {\Gamma (- 1 - \beta ) y^{2 + \beta'}} \Big|
< \infty.$$ Then $\eta'= \eta$ and $\beta ' = \beta$.
Without loss of generality, we assume that $\beta+\delta \leq \beta'+ \delta'$. Using the fact that $y^{1+\beta+ \delta} \leq y^{1+\beta'+\delta'}$ for $y \geq 1$, we get $$\int_{(1, \infty)} y^{1 + \beta + \delta} \Big| \pi(dy) - \frac {\eta' ~dy} {\Gamma( - 1 - \beta)y^{2 + \beta'}} \Big|
< \infty .$$ Comparing this with Assumption \[asp: branching mechanism\], we get $$\int_{(1,\infty)} y^{ 1 + \beta + \delta} \Big| \frac { \eta ~dy} {\Gamma (- 1 - \beta) y^{2 + \beta}} - \frac {\eta' ~dy} {\Gamma (- 1 - \beta) y^{2 + \beta'}} \Big| < \infty.$$ In other words, if we denote by $\widetilde \pi(dy)$ the measure $\eta' \Gamma(-1-\beta)^{-1} y^{-2-\beta'} dy$, then $\widetilde \pi$ is a Lévy measure which satisfies Assumption \[asp: branching mechanism\]. Applying Lemma \[lem:CEP\] to $\widetilde \pi$, we have that there exists $c>0$ such that $$| \eta z^{ 1 + \beta } - \eta' z^{ 1 + \beta' } |
\leq c z^{ 1 + \beta + \delta } + c z^2
, \quad z \in \mathbb R_+.$$ Dividing both sides by $z^{1+\beta}$ we have $
| \eta - \eta' z^{ \beta' - \beta } |
\leq cz^{\delta}+cz^{1-\beta}
, z \in \mathbb R_+.
$ This implies that $ \eta' z^{\beta' - \beta} \xrightarrow[\mathbb R^+\ni z\to 0]{} \eta >0. $ So we must have $\beta'= \beta$ and $\eta'= \eta$.
\[lem: LlogL criterion\] If $\psi$ satisfies Assumption \[asp: branching mechanism\], then $\psi$ satisfies the $L \log L$ condition, i.e., $
\int_{(1,\infty)} y \log y~\pi(dy)
< \infty.
$
Using Assumption \[asp: branching mechanism\] and the fact that $y\log y \leq y^{1+\beta+\delta}$ for $y$ large enough, we get $$\int_{(1,\infty)} y \log y ~\Big| \pi(dy) - \frac { \eta ~dy } { \Gamma ( - 1 - \beta ) y^{ 2 + \beta } } \Big|
< \infty.$$ Therefore we have $$\int_{ ( 1, \infty ) } y \log y ~\Big( \pi(dy) - \frac { \eta ~dy } { \Gamma ( - 1 - \beta ) y^{ 2 + \beta } } \Big)
< \infty.$$ Combining this with $
\int_{ ( 1, \infty ) } \frac { \eta \log y ~dy } { \Gamma ( - 1 - \beta ) y^{ 1 + \beta } }
< \infty,
$ we immediately get the desired result.
Definition of controller {#sec: controller}
------------------------
Denote by $\mathcal B(\mathbb R^d, \mathbb C)$ the space of all $\mathbb C$-valued Borel functions on $\mathbb R^d$. Recall that $\mathcal P$ is given in . Define $\mathcal P^+:= \mathcal P \cap \mathcal B(\mathbb R^d, \mathbb R_+)$ and $\mathcal P^*:= \{f\in \mathcal B(\mathbb R^d, \mathbb C): |f|\in \mathcal P^+\}$.
In this paper, we say $R$ is a *monotone* operator on $\mathcal P^+$ if $R:\mathcal P^+ \to \mathcal P^+$ satisfies that $Rf\leq Rg$ for all $f\leq g$ in $\mathcal P^+$. For a function $h: [0,\infty) \to [0,\infty)$, we say $R$ is an *$h$-controller* if $R$ is a monotone operator on $\mathcal P^+$ and that $R(\theta f)\leq h(\theta) Rf$ for all $f\in \mathcal P^+$ and $\theta \in [0,\infty)$. For subsets $\mathcal D, \mathcal I\subset \mathcal P^*$ and an operator $R$ on $\mathcal P^+$, we say an operator $A$ is *controlled by $R$ from $\mathcal D$ to $\mathcal I$* if $A:\mathcal D \to \mathcal I$ and that $|Af| \leq R|f|$ for all $f\in \mathcal D$; we say a family of operators $\mathscr O$ is *uniformly controlled by $R$ from $\mathcal D$ to $\mathcal I$* if each operator $A\in \mathscr O$ is controlled by $R$ from $\mathcal D$ to $\mathcal I$. For subsets $\mathcal D, \mathcal I\subset \mathcal P^*$ and a function $h:[0,\infty) \to [0,\infty)$, we say an operator $A$ (resp. a family of operators $\mathscr O$) is *$h$-controllable* (resp. *uniformly $h$-controllable*) from $\mathcal D$ to $\mathcal I$ if there exists an $h$-controller $R$ such that $A$ (resp. $\mathscr O$) is controlled (resp. uniformly controlled) by $R$ from $\mathcal D$ to $\mathcal I$.
For two operators $A: \mathcal D_A \subset \mathcal P^*\to \mathcal P^*$ and $B: \mathcal D_B \subset \mathcal P^*\to \mathcal P^*$, define $(A \times B)f (x):= Af(x) \times Bf(x)$ for all $f\in \mathcal D_A \cap \mathcal D_B$ and $x\in \mathbb{R}^d$. For any $a \in \mathbb R$ and any operator $A :\mathcal D_A \to \mathcal B(\mathbb R^d, \mathbb C\setminus (-\infty, 0])$, define $A^{\times a}f(x):= (Af(x))^a$ for all $f\in \mathcal D_A$ and $x\in \mathbb R^d$.
The following lemma is easy to verify.
\[lem: property of controllable operators\] For each $i \in \{0,1\}$, let $\mathscr O_i$ be a family of operators which is uniformly controlled by an $h_i$-controller $R_i$ from $\mathcal D_i \subset \mathcal P^*$ to $ \mathcal I_i \subset \mathcal P^*$. Then the followings hold:
1. If $\mathcal I_0 \subset \mathcal D_1$, then $\{A_1A_0: A_i \in \mathscr O_i, i = 0,1\}$ is uniformly controlled by the $(h_1 \circ h_0)$-controller $R_1R_0$ from $\mathcal D_0$ to $\mathcal I_1$.
2. $\{ A_1 \times A_0: A_i \in \mathscr O_i, i = 0,1\}$ is uniformly controlled by the $(h_1\times h_0)$-controller $R_1 \times R_0$ from $\mathcal D_0 \cap \mathcal D_1$ to $\mathcal P^*$.
3. $\{ A_1 + A_0: A_i \in \mathscr O_i, i = 0,1\}$ is uniformly controlled by the $(h_1 \vee h_0)$-controller $R_1 + R_0$ from $\mathcal D_0 \cap \mathcal D_1$ to $\mathcal P^*$.
4. If $\mathcal I_0 \subset \mathcal B(\mathbb R^d, \mathbb C \setminus (\infty, 0])$ and $a>0$, then $\{A^{\times a} : A \in \mathscr O_0\}$ is uniformly controlled by the $(h_0^a)$-controller $R_0^{\times a}$ from $\mathcal D_0$ to $\mathcal P^*$.
5. Suppose that $\mathscr O_0 = \{A_\theta: \theta \in \Theta \}$ where $\Theta$ is an index set. Further suppose that $(\Theta, \mathcal J )$ is a measurable space and that $(\theta,x) \mapsto A_\theta f(x)$ is $\mathcal J \otimes \mathcal B(\mathbb R^d)$-measurable for each $f\in \mathcal D$. Then the following space of operators $$\Big\{ f \mapsto \int_{\Theta} A_\theta f~\nu(d\theta) : \nu \text{ is a probability measure on } (\Theta, \mathcal J) \Big\}$$ is uniformly controlled by $R_0$ from $\mathcal D_0$ to $\mathcal P^*$.
Controllers for the super-OU processes {#sec: h-controller}
--------------------------------------
Let $X$ be our super-OU process with branching mechanism $\psi$ satisfying Assumptions \[asp: Greys condition\] and \[asp: branching mechanism\]. In this subsection, we will define several operators and study some of their properties that will be used in this paper.
Define $\psi_0(z) = \psi(z) + \alpha z$ for $z\in \mathbb{R}_+$. According to Lemma \[lem:CEP\], $\psi, \psi_1$ and $\psi_0$ can all be uniquely extended as complex-valued continuous functions on $\mathbb C_+$ which are also holomorphic on $\mathbb C^0_+$. For all $f\in \mathcal B(\mathbb R^d, \mathbb C_+)$ and $x\in \mathbb R^d$, define $\Psi f (x) = \psi\circ f(x)$, $\Psi_0 f(x)= \psi_0 \circ f(x)$ and $\Psi_1 f(x)= \psi_1 \circ f(x)$.
For all $t\in [0,\infty), x\in \mathbb R^d $ and $f \in \mathcal{P}$, let $ U_tf(x) := \operatorname{Log} \mathbb P_{\delta_x}[e^{i\theta X_t(f)}]|_{\theta = 1} $ be the value of the characteristic exponent of the infinitely divisible random variable $X_t(f)$ (See the paragraph after Lemma \[lem: Lip of power function\]). It follows from that $-U_tf(x)$ takes values in $\mathbb C_+$. Furthermore, we know from Proposition \[prop: complex FKPP-equation\] that $$\begin{aligned}
\label{eq:chareq2}
U_tf(x) - \int_0^t P^\alpha_{t-s} \Psi_0(-U_sf)(x)ds
= i P^{\alpha}_t f(x)
, \quad t\in [0,\infty), x\in \mathbb{R}^d, f\in \mathcal P.\end{aligned}$$
For all $t\geq 0$ and $f\in \mathcal P$, we define $$\begin{aligned}
\label{eq: def of Zf}
Z_t f
:= \int_0^t P^\alpha_{t-s}\big( \eta (-i P^\alpha_sf)^{1+\beta}\big)ds,
& \qquad Z'_t f
:= \int_0^t P^\alpha_{t-s}\big( \eta (-U_s f)^{1+\beta}\big)ds,
\\ Z''_t f
:= \int_0^t P^\alpha_{t-s}\Psi_1(-U_s f)ds,
& \qquad\ Z'''_t f
:= (Z'_t - Z_t+ Z''_t)f.\end{aligned}$$ Then we have that $$\begin{aligned}
\label{eq: key equality}
U_t - i P^\alpha_t
= Z'_t + Z''_t
= Z_t + Z'''_t
, \quad t\geq 0.\end{aligned}$$ For all $\kappa \in \mathbb Z_+$ and $f\in \mathcal P$, define $$\begin{aligned}
\label{eq:Q}
Q_\kappa f
:= \sup_{t\geq 0} e^{\kappa b t}|P_t f|,
\qquad Q f
:= Q_{\kappa_f}f.\end{aligned}$$ Then according to [@MarksMilos2018CLT Fact 1.2], $Q$ is an operator from $\mathcal P$ to $\mathcal P$.
\[lem: upper bound for usgx\] Under Assumptions \[asp: Greys condition\] and \[asp: branching mechanism\], the following statements are true:
1. $(-U_t)_{0\leq t\leq 1}$ is uniformly $\theta$-controllable from $\mathcal P$ to $\mathcal P^*\cap \mathcal B(\mathbb R^d, \mathbb C_+)$.
2. $(P^\alpha_t)_{0\leq t\leq 1}$ is uniformly $\theta$-controllable on $\mathcal P^*$.
3. $\Psi_0$ is $(\theta^2\vee \theta^{1+\beta})$-controllable from $\mathcal P^* \cap \mathcal B(\mathbb R^d, \mathbb C_+)$ to $\mathcal P^*$.
4. $(U_t- iP_t^{\alpha})_{0\leq t\leq 1}$ is uniformly $(\theta^2\vee \theta^{1+\beta})$-controllable from $\mathcal P$ to $\mathcal P^*$.
5. $(Z'_t-Z_t)_{0\leq t\leq 1}$ is uniformly $(\theta^{2+\beta}\vee \theta^{1+2\beta})$-controllable from $\mathcal P$ to $\mathcal P^*$.
6. For all $\delta > 0$ small enough, we have that $(Z''_t)_{0\leq t\leq 1}$ is uniformly $(\theta^2\vee \theta^{1+\beta+\delta})$-controllable from $\mathcal P$ to $\mathcal P^*$.
7. For all $\delta > 0$ small enough, we have that $(Z'''_t)_{0\leq t\leq 1}$ is uniformly $(\theta^{2+\beta}\vee \theta^{1+\beta+\delta})$-controllable from $\mathcal P$ to $\mathcal P^*$.
(1). According to , $-U_t$ is an operator from $\mathcal P$ to $\mathcal B(\mathbb R^d, \mathbb C_+)$. It follows from that for all $g\in \mathcal P$, $0\leq t\leq 1$ and $x\in \mathbb R^d$, we have $ |U_t g(x)| \leq \sup_{0\leq u\leq 1}P_u^\alpha |g| (x). $ We claim that $f\mapsto\sup_{0\leq u\leq 1}P^{\alpha}_u f$ is a map from $\mathcal P^+$ to $\mathcal P^+$. In fact, if $f\in \mathcal P^+$, there exists constant $c>0$ such that $$0
\leq \sup_{0\leq u\leq 1}P^{\alpha}_u f
\leq \sup_{0\leq u\leq 1} P_u (e^{\alpha u} e^{-\kappa_f u} e^{\kappa_f u} f )
\leq c \sup_{0\leq u\leq 1} (e^{\kappa_fu}P_u f) \leq c Qf \in \mathcal P.$$ It is clear that $f\mapsto\sup_{0\leq u\leq 1}P^{\alpha}_u f$ is a $\theta$-controller.
(2). Similar to the proof of (1).
(3). By Lemma \[lem:CEP\], there exist $C, \delta >0$ satisfying $\beta+\delta< 1$ such that for all $ f \in \mathcal P^* \cap \mathcal B( \mathbb R^d, \mathbb C_+ )$, it holds that $ |\Psi_0 f| \leq \eta |f|^{1+\beta} + |\Psi_1 f| \leq \eta |f|^{1+\beta} + C|f|^2+ C |f|^{1+\beta + \delta}.$ Note that the operator $$f \mapsto \eta f^{1+\beta} + Cf^2+ Cf^{1+\beta + \delta}
, \quad f\in \mathcal P^+,$$ is a $(\theta^2 \vee \theta^{1+\beta})$-controller.
(4). From (1)–(3) above and Lemma \[lem: property of controllable operators\].(1), we know that the operators $$f \mapsto P^\alpha_{t-s}\Psi_0(-U_sf)
, \quad 0\leq s\leq t\leq 1,$$ are uniformly $(\theta^2\vee \theta^{1+\beta})$-controllable. Combining this with and Lemma \[lem: property of controllable operators\].(5), we get the desired result.
(5). Notice that from Lemma \[lem: Lip of power function\], $$|(-U_t f)^{1+\beta} - (-iP^\alpha_t f)^{1+\beta} |
\leq (1+\beta) |U_t f-iP^\alpha_t f|(|U_t f|^{\beta}+|i P^\alpha_t f|^{\beta}).$$ Now using (1), (2) and (4) above, and Lemma \[lem: property of controllable operators\], we get that the operators $$f \mapsto (-U_t f)^{1+\beta} - (-iP^\alpha_t f)^{1+\beta},\quad 0\leq t\leq 1,$$ are uniformly $(\theta^{2+\beta}\vee \theta^{1+2\beta})$-controllable. Combining with Lemma \[lem: property of controllable operators\], and $$(Z'_t - Z_t)f
= \int_0^t P^\alpha_{t-s}\Big( \eta ((-U_s f)^{1+\beta} - (-iT_s^\alpha f)^{1+\beta} )\Big)ds
, \quad 0\leq t\leq 1, f\in \mathcal P,$$ we get the desired result.
(6). By Lemma \[lem:CEP\], for all $\delta > 0$ small enough, there exists $C>0$ such that $$|\Psi_1(f)|
\le C(|f|^2+|f|^{1+\beta+ \delta})
, \quad f\in \mathcal P^*\cap\mathcal B(\mathbb R^d, \mathbb C_+).$$ Note that, for all $\delta, C>0$, $$f \mapsto C(f^2+f^{1+\beta+\delta})
, \quad f\in \mathcal P^+$$ is a $(\theta^2 \vee \theta^{1+\beta+\delta})$-controller. Therefore, for all $\delta > 0$ small enough, we have that $\Psi_1$ is a $(\theta^2 \vee \theta^{1+\beta+\delta})$-controllable operator from $\mathcal P^*\cap\mathcal B(\mathbb R^d, \mathbb C_+)$ to $\mathcal P^*$. Combining this with (1)–(2) above, and Lemma \[lem: property of controllable operators\], we get that, for all $\delta > 0$ small enough, the operators $$f
\mapsto Z_t'' f
= \int_0^t P_{t-s}^\alpha \Psi_1(-U_sf)ds
, \quad 0\leq t\leq 1,$$ are uniformly $(\theta^2 \vee \theta^{1+\beta+\delta})$-controllable from $\mathcal P$ to $\mathcal P^*$.
(7). Since $Z'''_t = (Z'_t-Z_t)+Z''_t$, the desired result follows from (5)–(6) above and Lemma \[lem: property of controllable operators\].(3).
Stable distributions {#sec: stable distributions}
--------------------
Recall that the operators $(T_t)_{t\geq 0}$ are defined by , and the functionals $(m_{t})_{0\leq t< \infty}$ and $m$ are given by and respectively.
\[lem:m\] $(T_t)_{t\geq 0}$, $(m_{t})_{0\leq t< \infty}$ and $m$ are well defined.
*Step 1.* We will show that for each $f \in \mathcal P$, there exists $h \in \mathcal P$ such that $ |T_tf| \leq e^{- \delta t} h$ for each $t\geq 0$, where $$\begin{aligned}
\label{eq:m:1}
\delta
:= \inf \big\{ |\tilde \beta \alpha - |p|b| : p \in \mathbb Z_+^d, \langle f, \phi_p\rangle_\varphi \neq 0 \big\}
\geq 0.
\end{aligned}$$ From this upper bound, it can be verified that $(T_t)_{t\geq 0}$ and $(m_{t})_{0 \leq t < \infty}$ are well defined. In fact, we can write $f = f_0 + f_1$ with $f_0\in \mathcal C_s \oplus \mathcal C_c$ and $f_1 \in \mathcal C_l$. According to [@MarksMilos2018CLT Lemma 2.7], there exists $h_0 \in \mathcal P$ such that for each $t\geq 0$, $$|T_t f_0|
= \Big| \sum_{p \in \mathbb Z_+^d: \tilde \beta \alpha \leq |p|b } e^{- ( |p| b - \tilde \beta \alpha ) t} \langle f, \phi_p \rangle_\varphi \phi_p \Big|
= e^{\tilde \beta \alpha t} | P_t f_0 |
\leq e^{- ( \kappa_{(f_0)} b - \tilde \beta \alpha) t} h_0
\leq e^{- \delta t} h_0.$$ On the other hand $$|T_t f_1|
\leq e^{- \delta t}\sum_{p \in \mathbb Z_+^d : \tilde \beta \alpha > |p|b} |\langle f, \phi_p \rangle_\varphi \phi_p|
=: e^{- \delta t} h_1,
\quad t\geq 0.$$ So the desired result in this step follows with $h := h_0 + h_1$.
*Step 2.* We will show that if $f \in \mathcal C_s \oplus \mathcal C_l$, then $m[f]$ is well defined. In fact, let $\delta$ be given by , then in this case $\delta > 0$. Now, according to Step 1 there exists $h \in \mathcal P$ such that $|T_tf| \leq e^{- \delta t} h$ for each $t\geq 0$. This exponential decay implies the desired result in this step.
*Step 3.* We will show that if $f\in \mathcal P \setminus (\mathcal C_s \oplus \mathcal C_l)$, then $m[f]$ is also well defined. In fact, $f$ can be decomposed as $f = f_c + f_{sl}$ where $f \in \mathcal C_c\setminus \{0\}$ and $f_{sl}\in \mathcal C_s \oplus \mathcal C_l$. Note that $T_t f_c = f_c$ for each $t\geq 0$. Also note that in Step 2, we already have shown that there exist $\delta > 0$ and $h \in \mathcal P^+$ such that for each $t\geq 0$, we have $|T_t f_{sl}| \leq e^{- \delta t}h$. Therefore, using Lemma \[lem: Lip of power function\] we have $$\begin{aligned}
&|(-iT_t f)^{1+\beta} - (-i f_c)^{1+\beta}|
\leq (1+\beta) ( |T_tf|^\beta + |f_c|^\beta) |T_tf_{sl}|
\\&\leq (1+\beta) ( |f_c + T_t f_{sl}|^\beta + |f_c|^\beta) e^{- \delta t} h
\leq (1+\beta) ( (|f_c| + |h|)^\beta + |f_c|^\beta) e^{- \delta t} h
=: e^{- \delta t} g,
\end{aligned}$$ where $g\in \mathcal P^+$. Therefore $$\begin{aligned}
\label{eq:P:S:1}
&\Big| \frac{1}{t} m_t[f] - \langle (-if_c)^{1+\beta}, \varphi\rangle\Big|
= \Big| \frac{1}{t} \cdot t \int_0^1 \big\langle (-iT_{rt}f)^{1+\beta} - (-if_c)^{1+\beta}, \varphi\big \rangle ~dr \Big|\\
&\leq \int_0^1 \big\langle |(-iT_{rt}f)^{1+\beta} - (-if_c)^{1+\beta}|, \varphi\big \rangle ~dr
\leq \langle g,\varphi\rangle\int_0^1 e^{-\delta rt} ~dr
\xrightarrow[t\to \infty]{} 0.
\qedhere
\end{aligned}$$
\[prop:PL:S\] For each $f \in \mathcal P\setminus \{0\}$, there exists a non-degenerate $(1+\beta)$-stable random variable $\zeta^f$ such that $ E[e^{i\theta\zeta^f}] = e^{m[\theta f]}$ for all $\theta \in \mathbb R$.
The proof of the above proposition relies on the following lemma:
\[lem: charactreisticfunction\] Let $q$ be a measure on $\mathbb R^d\setminus\{0\}$ with $\int_{\mathbb R^d\setminus\{0\}} |x|^{1+\beta} q(dx) \in (0,\infty)$. Then $$\theta
\mapsto \exp\Big\{\int_{\mathbb R^d\setminus\{0\}} (i\theta \cdot x)^{1+\beta} q(dx)\Big\},
\quad \theta \in \mathbb R^d,$$ is the characteristic function of an $\mathbb R^d$-valued $(1+\beta)$-stable random variable.
It follows from disintegration that there exist a measure $\lambda$ on $S:= \{\xi\in \mathbb R^d:|\xi| = 1\}$ and a kernel $k(\xi,dt)$ from $S$ to $\mathbb R_+$ such that $$\int_{\mathbb R^d\setminus \{0\}} f(x)q(dx)
= \int_S \lambda(d\xi) \int_{\mathbb R_+} f(t \xi)k(\xi,dt)
, \quad f\in \mathcal B(\mathbb R^d\setminus \{0\}, \mathbb R_+).$$ We define another measure $\lambda_0$ on $S$ by $$\lambda_0(d\xi)
:= \frac1{\Gamma(-1-\beta)}\int_0^\infty t^{1+\beta}k(\xi,dt) \lambda (d\xi),$$ where $\Gamma$ is the Gamma function. Then $\lambda_0$ is a non-zero finite measure, since $$\begin{aligned}
\lambda_0(S)
= &\frac{1}{\Gamma(-1-\beta)} \int_S \lambda (d\xi) \int_0^\infty |t\xi|^{1+\beta}k(\xi,dt) \\
= & \frac{1}{\Gamma(-1-\beta)} \int_{\mathbb R^d\setminus\{0\}} |x|^{1+\beta} q(dx) \in (0,\infty).
\end{aligned}$$ Define a measure $\nu$ on $\mathbb R^d\setminus\{0\}$ by $$\int_{\mathbb R^d\setminus\{0\}}f(x)\nu(dx)
= \int_{S} \lambda_0(d\xi) \int_0^\infty f(r\xi) \frac{dr}{r^{2+\beta}}
,\quad f\in \mathcal B(\mathbb R^d\setminus \{0\}, \mathbb R_+).$$ Then, according to [@Sato2013Levy Remark 14.4], $\nu$ is the Lévy measure of a $(1+\beta)$-stable distribution on $\mathbb R^d$, say $\mu$, whose characteristic function is $$\hat \mu(\theta)
= \exp \Big \{ \int_{\mathbb R^d\setminus\{0\}} (e^{-i\theta \cdot y}-1+i\theta \cdot y) \nu(dy) \Big \}
, \quad \theta \in \mathbb R.$$ Finally, according to , we have $$\begin{aligned}
& \int_{\mathbb R^d\setminus\{0\}} (e^{-i\theta \cdot y}-1+i\theta \cdot y) \nu(dy)
= \int_S \lambda_0(d\xi) \int_0^\infty (e^{-ir\theta \cdot \xi}-1+ir\theta \cdot \xi) \frac{dr}{r^{2+\beta}} \\
& = \int_S \lambda (d\xi) \int_0^\infty (e^{-ir\theta \cdot \xi}-1+ir\theta \cdot \xi) \frac{dr}{\Gamma(-1-\beta)r^{2+\beta}}\int_0^\infty t^{1+\beta} k(\xi,dt) \\
& = \int_S \lambda (d\xi) \int_0^\infty (i\theta\cdot \xi)^{1+\beta} t^{1+\beta} k(\xi,dt)
= \int_S \lambda(d\xi) \int_0^\infty (i\theta \cdot t\xi)^{1+\beta} k(\xi,dt) \\
& = \int_{\mathbb R^d} (i\theta \cdot x)^{1+\beta} q(dx).
\qedhere
\end{aligned}$$
Suppose that $f\in \mathcal C_s \oplus \mathcal C_l$. Note that $m[\theta f]$ can be written as $$\begin{aligned}
\label{eq:PL:S:1}
m[\theta f]
= \eta \int_0^{\infty}~ds\int_{\mathbb R^d} (-i\theta T_s f(x))^{1+\beta} \varphi(x)~dx,
\quad \theta \in \mathbb R.
\end{aligned}$$ Therefore, according to Lemma \[lem: charactreisticfunction\], in order to show that $\zeta^f$ is a $(1+\beta)$-stable random variable we only need to show that $$\label{e:new}
\int_0^{\infty}~ds\int_{\mathbb R^d} | T_{s} f(x)|^{1+\beta} \varphi(x)~dx
\in (0, \infty).$$ According to the Step 1 in the proof of Lemma \[lem:m\], we know that there exist $\delta> 0$ and $h \in \mathcal P$ such that $|T_sf| \leq e^{- \delta s} h$ for each $s\geq 0$. The claim then follows.
If $f \in \mathcal P \setminus (\mathcal C_s \oplus \mathcal C_l)$, then $f$ can be written by $f = f_c +(f - f_c)$ where $f_c \in \mathcal C_c\setminus\{0\}$ and $f - f_c \in \mathcal C_s \oplus \mathcal C_l$. In this case, according to , $m[\theta f]$ has an integral representation: $$\begin{aligned}
\label{eq:PL:S:2}
m[\theta f]
= \int_{\mathbb R^d} (-i\theta f_c(x))^{1+\beta} \varphi(x) ~dx,
\quad \theta \in \mathbb R.
\end{aligned}$$ Finally, according to Lemma \[lem: charactreisticfunction\] and the fact that $
\int_{\mathbb R^d} | f_c(x)|^{1+\beta} \varphi(x)~dx
\in (0, \infty),
$ We have that $\zeta^f$ is a non-degenerate $(1+\beta)$-stable random variable.
A refined estimate for the OU semigroup {#sc:refined}
---------------------------------------
It turns out that our proof of the CLT relies on the following refined estimate for the OU semigroup.
\[lem:P:R\] Suppose that $g \in \mathcal P$, then there exists $h \in \mathcal P^+$ such that for all $ f \in \mathcal P_g := \{\theta T_n g: n \in \mathbb Z_+, \theta \in [-1,1]\} $ and $t\geq 0$, we have $ | P_t (Z_1 f - \langle Z_1 f, \varphi \rangle )| \leq e^{-bt} h$.
Fix $g \in \mathcal P$. We write $g = g_0 + g_1$ with $g_0 \in \mathcal C_s \oplus \mathcal C_c$ and $g_1 \in \mathcal C_l$, and $q_f:=Z_1f - \langle Z_1f, \varphi \rangle\in \mathcal P^*$ for each $f\in \mathcal P$. We need to prove that there exists $h \in \mathcal P^+$ such that for each $f\in \mathcal P_g$, $|P_tq_f| \leq e^{-bt} h$.
*Step 1.* We claim that we only need to prove the result for all $f \in \widetilde{\mathcal P}_g:= \{T_{n+1} g : n \in \mathbb Z_+\}$. In fact, both $\operatorname{Re} q_g$ and $\operatorname{Im} q_g$ are functions in $\mathcal P$ of order $\geq 1$. The result is valid for $f = T_0 g = g$ according to [@MarksMilos2018CLT Fact 1.2]. Also, note that if the result is valid for some $f \in \mathcal P$, it is also valid for any $\theta f$ with $\theta \in [-1,1]$.
*Step 2.* We show that $\{T_s g: s> 0\} \subset C_\infty (\mathbb R^d) \cap \mathcal P$. In fact, for each $s > 0$, $$T_s g
= T_s (g_0 + g_1)
= e^{\alpha \tilde \beta s}P_s g_0 + \sum_{p \in \mathbb Z_+^d: \alpha \tilde \beta > |p|b}
\langle g_1, \phi_p \rangle_\varphi e^{-(\alpha \tilde \beta - |p|b)s} \phi_p.$$ Notice that the second term is in $C_\infty(\mathbb R^d)\cap \mathcal P$ since it is a finite sum of polynomials, and the first term is also in $C_\infty (\mathbb R^d) \cap \mathcal P$ according to [@MarksMilos2018CLT Fact 1.1].
*Step 3.* We show that there exists $h_3 \in \mathcal P^+$ such that for all $j \in \{1,\dots, d\}$ and $f \in \widetilde {\mathcal P}_g$, it holds that $|\partial_j f| \leq h_3$. In fact, it is known that (see [@MetafunePallaraPriola2002Spectrum] for example) $$\begin{aligned}
\label{eq:P:R:3:-1}
P_t f(x)
= \int_{\mathbb R^d} f\big(x e^{-bt} + y \sqrt{1-e^{-2bt}}\big) \varphi(y)~dy,
\quad t\geq 0, x\in \mathbb R^d, f\in \mathcal P.
\end{aligned}$$ For $f \in C_\infty(\mathbb R^d)\cap \mathcal P$ it can be verified from above that $$\begin{aligned}
\label{eq:P:R:3:1}
\partial_j P_t f
= e^{-bt} P_t \partial_j f,
\quad t \geq 0, j \in \{1,\dots, d\}.
\end{aligned}$$ Thanks to Step 2, $T_1 g_0 \in C_\infty(\mathbb R^d)\cap \mathcal P$. According to [@MarksMilos2018CLT Fact 1.3] and the fact that $\alpha \tilde \beta \leq \kappa _{g_0} b$, we have for each $j \in \{1,\dots, d\}$, $$\kappa_{(\partial_j T_1 g_0)}
\geq \kappa_{(T_1 g_0)} - 1
= \kappa_{g_0} - 1
\geq \frac{\alpha \tilde \beta}{b} - 1.$$ Therefore, there exists $h'_3\in \mathcal P^+$ such that for all $n \in \mathbb Z_+$ and $j\in \{1,\dots,d\}$, $$\begin{aligned}
& | \partial_j T_{n+1}g_0 |
= | \partial_j e^{\alpha \tilde \beta n}P_n T_1g_0 |
= e^{\alpha \tilde \beta n-bn} |P_n \partial_j T_1 g_0| \\
& \leq e^{\alpha \tilde \beta n-bn} \exp\{-\kappa_{(\partial_j T_1 g_0)}bn\}Q \partial_j T_1g_0
\leq Q\partial_j T_1g_0
\leq h'_3.
\end{aligned}$$ On the other hand, there exists $h_3''\in \mathcal P^+$ such that for all $n \in \mathbb Z_+$ and $j\in \{1,\dots,d\}$, $$\begin{aligned}
& |\partial_j T_{n+1}g_1 |
= \Big| \sum_{p\in \mathbb Z_+^d: \alpha \tilde \beta > |p|b} e^{- (\alpha \tilde \beta - |p|b)(n+1)} \langle g_1, \phi_p \rangle_\varphi \partial_j \phi_p \Big| \\
& \leq \sum_{p\in \mathbb Z_+^d: \alpha \tilde \beta > |p|b} |\langle g_1, \phi_p \rangle_\varphi \partial_j \phi_p |
\leq h_3''.
\end{aligned}$$ Then the desired result in this step follows.
*Step 4.* We show that there exists $h_{4} \in \mathcal P^+$ such that for all $j \in \{1,\dots, d\}, u \in [0, 1]$ and $f \in \widetilde {\mathcal P}_g$, it holds that $ | \partial_j P_{1-u}^\alpha (- i P_u^\alpha f)^{1+\beta} | \leq h_4$. In fact, thanks to Step 2 and , for all $j \in \{1,\dots, d\}, u \in [0, 1]$ and $f \in \widetilde{\mathcal P}_g$, we have $$\begin{aligned}
& \partial_j P_{1-u}^\alpha (- i P_u^\alpha f)^{1+\beta}
= e^{-(1-u)b} P_{1-u}^\alpha \partial_j (- i P_u^\alpha f)^{1+\beta}
\\ & = (1+\beta) e^{-(1-u)b} P_{1-u}^\alpha [ (- i P_u^\alpha f)^\beta \partial_j (- i P_u^\alpha f) ]
\\ & = -i(1+\beta) e^{-(1-u)b} P_{1-u}^\alpha[ (- i P_u^\alpha f)^\beta e^{-ub} P_u^\alpha \partial_j f]
\\ & = -i(1+\beta) e^{-b} e^{(1-u)\alpha} e^{u\alpha (1+\beta)} P_{1-u} [ (- i P_u f)^\beta P_u \partial_j f ].
\end{aligned}$$ Recall from Step 1 in the proof of Lemma \[lem:m\] there exists $h'_4\in \mathcal P^+$ such that for each $f \in \{T_sg:s\geq 0\}$ it holds that $|f| \leq h'_4$. Therefore, using Step 3, we have for all $j \in \{1,\dots, d\}, u \in [0, 1]$ and $f \in \widetilde {\mathcal P}_g$, $$\begin{aligned}
& |\partial_j P_{1-u}^\alpha (- i P_u^\alpha f)^{1+\beta}|
\leq (1+\beta) e^{\alpha (1+\beta)} P_{1-u} [ (P_u |f|)^\beta P_u |\partial_j f| ]
\\ & \leq (1+\beta) e^{\alpha (1+\beta)} P_{1-u} [ (P_u h'_4)^\beta P_u h_3 ]
\leq (1+\beta) e^{\alpha (1+\beta)} Q_0 [ (Q_0 h_4')^\beta Q_0 h_3 ],
\end{aligned}$$ where $Q_0$ is defined by . This implies the desired result in this step.
*Step 5.* We show that there exists $h_5 \in \mathcal P^+$ such that for each $f \in \widetilde {\mathcal P}_g$, we have $ |\nabla (Z_1f)| \leq h_5$. In fact, according to Step 4, for all $j \in \{1,\dots, d\}$, $f \in \widetilde{\mathcal P}_g$ and compact $A \subset \mathbb R^d$, we have $$\int_0^1 \sup_{x\in A} | (\partial_j P_{1-u}^\alpha (-i P_u^\alpha f)^{1+\beta}) (x) |~du
\leq \sup_{x\in A} h_4(x) < \infty.$$ Using this and [@Durrett2010Probability Theorem A.5.2], for all $j \in \{1,\dots, d\}$, $f\in \widetilde {\mathcal P}_g$ and $x\in \mathbb R^d$, it holds that $$| \partial_j Z_1 f(x)|
= \Big| \int_0^1 ( \partial_jP_{1-u}^\alpha (-iP_u^\alpha f)^{1+\beta} ) (x) ~du \Big|
\leq h_4(x).$$ Now, the desired result for this step is valid.
*Step 6.* Let $h_5$ be the function in Step 5. There are $c_0, n_0> 0$ such that for all $x\in \mathbb R^d$, $h_5(x) \leq c_0(1+|x|)^{n_0}$. Note that for all $x, y \in \mathbb R^d$, $1+|x|+|y|\leq (1+|x|) (1+|y|)$; and that for all $\theta \in [0,1]$, $|\sqrt {1 - \theta} - 1| \leq \theta$. Write $D_{x,y} = \{ax+by: a, b \in [0,1]\}$ fo $x, y \in \mathbb R^d$. Using and Step 5, there exists $h_6 \in \mathcal P^+$ such that for all $t \geq 0$, $f \in \widetilde {\mathcal P}_g$ and $x \in \mathbb R^d$, $$\begin{aligned}
& |P_t q_f(x)|
= \Big| \int_{\mathbb R^d} ( (Z_1f)(x e^{-bt}+ y \sqrt{1 - e^{-2bt}}) - Z_1 f (y) ) \varphi(y) ~dy \Big| \\
& \leq \int_{\mathbb R^d} \Big(\sup_{z\in D_{x,y}} |\nabla (Z_1f) (z) |\Big) | x e^{-bt} + y \sqrt{1 - e^{-2bt}} - y | \varphi(y) ~dy \\
& \leq e^{-bt} \int_{\mathbb R^d} c_0(1+|x|+|y|)^{n_0} (|x|+|y|) \varphi(y) ~dy \\
& \leq c_0 e^{-bt}(1+|x|)^{n_0}\Big(|x|\int_{\mathbb R^d} (1+|y|)^{n_0}\varphi(y) ~dy + \int_{\mathbb R^d} (1+ |y|)^{n_0} |y| \varphi(y) ~dy \Big) \\
& \leq e^{-bt} h_6(x).
\qedhere
\end{aligned}$$
Small value probability {#sec: Small value probability}
-----------------------
In this subsection, we digress briefly from our super-OU process and consider a (supercritical) *continuous-state branching process (CSBP)* $\{(Y_t)_{t\geq 0}; \mathbf P_x\}$ with branching mechanism $\psi$ given by . Such a process $\{(Y_t)_{t\geq 0}; \mathbf P_x\}$ is defined as an $\mathbb R^+$-valued Hunt process satisfying $$\mathbf P_x[e^{-\lambda Y_t}] = e^{- x v_t(\lambda)},
\quad x\in \mathbb R^+, t\geq 0, \lambda \in \mathbb R^+,$$ where for each $\lambda\geq 0$, $t\mapsto v_t(\lambda)$ is the unique positive solution to the equation $$\begin{aligned}
\label{eq: fkpp equation for CSBP}
v_t(\lambda) - \int_0^t \psi(v_s(\lambda))~ds = \lambda,
\quad t\geq 0.\end{aligned}$$ It can be verified that for each $\mu \in \mathcal M(\mathbb R^d)$ with $x = \| \mu \|$, we have $ \{(\|X_t\|)_{t\geq 0}; \mathbb P_\mu\} \overset{\text{law}}{=} \{(Y_t)_{t\geq 0}; \mathbf P_x\}$.
Our goal in this subsection is to determine how fast the probability $\mathbf P_x(0<e^{-\alpha t}Y_t \leq k_t)$ converges to $0$ when $t\mapsto k_t$ is a strictly positive function on $[0,\infty)$ such that $k_t \to 0$ and $k_t e^{\alpha t} \to \infty$ as $t\to \infty$. Suppose that Grey’s condition is satisfied i.e., there exists $z' > 0$ such that $\psi(z) > 0$ for all $z>z'$, and that $\int_{z'}^\infty \psi(z)^{-1}dz < \infty$. Also suppose that the $L \log L$ condition is satisfied i.e., $
\int_1^\infty y \log y~\pi(dr)
< \infty.
$ We write $W_t = e^{-\alpha t}Y_t$ for each $t\geq 0$.
\[lem: control of XT\] Suppose that $t\mapsto k_t$ is a strictly positive function on $[0,\infty)$ such that $k_t \to 0$ and $k_t e^{\alpha t} \to \infty$ as $t\to \infty$. Then, for each $x\geq 0$, there exist $C,\delta>0$ such that $$\mathbf P_x(0<W_t\leq k_t)
\leq C(k_t^\delta + e^{-\delta t}), \quad t\geq 0.$$
*Step 1.* We recall some known facts about the CSBP $(Y_t)$. For each $\lambda \geq 0$, we denote by $t\mapsto v_t(\lambda)$ the unique positive solution of . Letting $\lambda \to \infty$ in , we have by monotonicity that $\bar v_t:= \lim_{\lambda \to \infty}v_t(\lambda)$ exists in $(0,\infty]$ for all $t\geq 0$, and that $$\begin{aligned}
\label{eq: svp1}
\mathbf P_x(Y_t = 0)=e^{-x\bar v_t}, \quad t\geq 0, x\ge 0.
\end{aligned}$$ It is known, see [@Li2011Measure-valued Theorems 3.5–3.8] for example, that under Grey’s condition $\bar v:= \lim_{t\to \infty} \bar v_t \in [0,\infty)$ exists and is the largest root of $\psi$ on $[0,\infty)$. Letting $t \to \infty$ in , we have by monotonicity that $$\mathbf P_x(\exists t \geq 0, Y_t = 0)
= e^{-x\bar v}, \quad x\geq 0.$$ Note the derivative of $\psi$, i.e., $$\psi'(z)
= -\alpha + 2\rho z + \int_{(0,\infty)}(1-e^{-zy})y\pi(dy),\quad z\geq 0,$$ is non-decreasing. This says that $\psi$ is a convex function. Also notice that $\psi'(0+)=-\alpha <0$ and that there exists $z>0$ such that $\psi(z)>0$. Therefore we have (i) $\bar v > 0$; (ii) $\psi(z) < 0$ on $z\in (0,\bar v)$; and (iii) $\psi(z) > 0 $ on $z\in (\bar v, \infty)$. It is also known, see [@Li2011Measure-valued Proposition 3.3] for example, that (i) if $\lambda \in (0,\bar v)$, then $0<\lambda \leq v_t(\lambda)<\bar v $; (ii) if $\lambda \in (\bar v, \infty)$, then $\bar v < v_t(\lambda)\leq \lambda< \infty$; and (iii) for each $\lambda \in (0,\infty)\setminus \{\bar v\}$ and $t\geq 0$, we always have $
\int_{v_t(\lambda)}^\lambda \psi(z)^{-1}dz = t.
$ Taking $\lambda \to \infty$ and using the monotone convergence theorem, we have that $$\begin{aligned}
\label{eq:svp2}
\int_{\bar v_t}^\infty \frac{dz}{\psi(z)} = t, \quad t\geq 0.
\end{aligned}$$
*Step 2.* We will show that, for each $x \geq 0$ there exists a constant $c_1>0$ such that $$\mathbf P_{x}(0< W_t\leq k_t)
\leq c_1\big(|\bar v- v_t(k_t^{-1}e^{-\alpha t})|+|\bar v_t - \bar v|\big),
\quad t\geq 0.$$ In fact, for all $x\geq 0$ and $t\geq 0$, we have $$\begin{aligned}
& \mathbf P_{x}(0<W_t \leq k_t)
= \mathbf P_{x}( e^{-k_t^{-1}W_t}\geq e^{-1},W_t > 0) \\
& \leq e \mathbf P_{x}[e^{-k_t^{-1} W_t};W_t > 0]
= e\big(\mathbf P_x[e^{-k_t^{-1} W_t}]-\mathbf P_x(W_t = 0)\big) \\
& = e\big(e^{-xv_t(k_t^{-1} e^{-\alpha t})}-e^{-x\bar v_t}\big)
\leq ex \big(|\bar v-v_t(k_t^{-1} e^{-\alpha t})|+ |\bar v_t- \bar v|\big),
\end{aligned}$$ as desired in this step.
*Step 3.* We will show that there exist $c_2, \delta_1, t_0 > 0$ such that $$|\bar v_t-\bar v|
\leq c_2e^{-\delta_1 t}
, \quad t\geq t_0.$$ In fact, since $\psi$ is a convex function, we must have $\tau:=\psi'(\bar v)>0$ and that $\psi(z) \geq (z-\bar v)\tau$ for each $z\geq \bar v$. According to Grey’s condition, we can find $z_0 >\bar v $ such that $t_0 := \int^\infty_{z_0}\psi(z)^{-1}dz<\infty$. For each $t > t_0$, according to , we have $$\begin{aligned}
& t - t_0 =
\int^\infty_{\bar v_t} \frac{dz}{\psi(z)} - \int_{z_0}^\infty \frac{dz}{\psi(z)}
= \int_{\bar v_t}^{z_0} \frac{dz}{\psi(z)} \\
& \leq \int_{\bar v_t}^{z_0} \frac{dz}{(z-\bar v)\tau}
= \frac{1}{\tau} \big(\log (z_0-\bar v) - \log(\bar v_t-\bar v)\big).
\end{aligned}$$ Rearranging, we get $ \bar v_t - \bar v \leq (z_0 - \bar v)e^{-\tau(t-t_0)}, $ for all $t\geq t_0$. This implies the desired result in this step.
*Step 4.* We will show that there exist $c_3, \delta_2, t_1>0$ such that $$|\bar v - v_t(k_t^{-1} e^{-\alpha t})|\leq
c_3k_t^{\delta_2}, \quad t\geq t_1.$$ Define $\rho_t := 1+(\log k_t)/(t\alpha)$ for all $t\geq 0$. By the fact that $k_t^{-1}e^{-\alpha t} = e^{-\alpha \rho_t t}$ for all $t\geq 0$ and the condition that $k_t e^{\alpha t} \xrightarrow[t\to \infty]{} \infty$, we have $\rho_t t \xrightarrow[t\to \infty]{} \infty $. Since the $L\log L$ condition is satisfied, we have (see [@LiuRenSong2009Llog] for example), $W_t \xrightarrow[t\to \infty]{a.s.} W_\infty$, where the martingale limit $W_\infty$ is a non-degenerate positive random variable. This implies that $$v_t(e^{-\alpha t})
= -\log \mathbf P_1[e^{-W_t}]\xrightarrow[t\to \infty]{} - \log \mathbf P_{1}[e^{-W_\infty}]
=: z^* \in (0,\infty).$$ The $L \log L$ condition also guarantees that (see again [@LiuRenSong2009Llog] for example) $\{W_\infty = 0\} = \{\exists t \geq 0, X_t= 0\}$ a.s. in $\mathbf P_1$. This and the non-degeneracy of $W_\infty$ imply that $$z^*
= -\log \mathbf P_1[e^{-W_\infty}]
< -\log \mathbf P_1(W_\infty = 0) = \bar v.$$
Fix an arbitrary $\epsilon \in (0,\tau)$. According to the fact that $\tau=\psi'(\bar v)>0$, there exists $z_0 \in (0,\bar v)$ such that for all $z\in (z_0, \bar v)$, we have $-\psi(z)\geq (\bar v - z)(\tau- \epsilon)$. Fix this $z_0$. For $t$ large enough, we have $0<k_t^{-1}e^{-\alpha t} < v_t(k_t^{-1}e^{-\alpha t})< \bar v$. Then we have for $t>0$ large enough, $$\begin{aligned}
t
& =\int^{v_t(k_t^{-1} e^{-\alpha t})}_{k_t^{-1} e^{-\alpha t}}\frac{dz}{-\psi(z)}
= \Big(\int^{v_{\rho_t t}(e^{-\alpha \rho_t t})}_{e^{-\alpha \rho_t t}} + \int^{z_0}_{v_{\rho_t t}(e^{-\alpha \rho_t t})} +\int^{v_t(k_t^{-1}e^{-\alpha t})}_{z_0}\Big)\frac{dz}{-\psi(z)} \\
& = \rho_t t + O(1) +\int^{v_t(k_t^{-1}e^{-\alpha t})}_{z_0} \frac{dz}{-\psi(z)},
\end{aligned}$$ where we used the fact that $$\int_{v_{\rho_t t}(e^{-\alpha \rho_tt})}^{z_0} \frac{dz}{-\psi(z)}
\xrightarrow[t\to \infty] {} \int_{z^*}^{z_0} \frac{dz}{-\psi(z)}.$$ Now we have, for $t$ large enough, $$\begin{aligned}
& t
\leq \rho_t t + O(1) + \int_{z_0}^{v_t(k_t^{-1}e^{-\alpha t})} \frac{dz}{(\bar v-z)(\tau - \epsilon)} \\
& = \rho_t t +O(1)- \frac{1}{\tau-\epsilon}\Big( \log \big(\bar v-v_t(e^{-\alpha \rho_t t})\big) - \log(\bar v-z_0)\Big).
\end{aligned}$$ Rearranging, we get, for $t$ large enough, $$e^{-t(\tau - \epsilon)}
\geq e^{-\rho_t t(\tau - \epsilon)+O(1)}(\bar v - v_t(e^{-\alpha \rho_t t})).$$ Therefore, there exist $c_3>0$ and $t_1>0$ such that for all $t\geq t_1$, $$\bar v - v_t(k_t^{-1} e^{-\alpha t})
\leq e^{-t(\tau -\epsilon)+ (1+\frac{\log k_t}{t\alpha})t(\tau - \epsilon)+O(1)}
\leq c_3k_t^{\frac{\tau - \epsilon}{\alpha}}.$$ This implies the desired result in this step.
Finally, by Steps 2-4, we have for each $x\geq 0$, there exist $c_4, \delta_3, t_2 > 0$ such that $$\mathbf P_{x}(0< W_t\leq k_t)
\leq c_4(k_t^{\delta_3}+e^{-\delta_3 t})
, \quad t\geq t_2.$$ Note that the left hand side is always bounded from above by $1$, so we can take $t_2 =0$ in the above statement.
Moments for super-OU processes {#sec: Moments for super-OU processes}
------------------------------
In this subsection, we want to find some upper bound for the $(1+\gamma)$-th moment of $X_t(g)$, where $\gamma \in (0,\beta)$ and $g\in \mathcal P$.
\[lem: control pair for P(M>lambda)\] There is a $(\theta^2\vee\theta^{1+\beta})$-controller $R$ such that for all $0\leq t\leq 1$, $g\in \mathcal P$, $\lambda >0$ and $\mu\in \mathcal M_c(\mathbb R^d)$, we have $$\mathbb P_\mu ( |\mathcal{I}_0^tX_t(g)| > \lambda)
\leq \frac{\lambda}{2}\int_{-2/\lambda}^{2/\lambda}\mu(R|\theta g|) d\theta.$$
It is elementary calculus (see the proof of [@Durrett2010Probability Theorem 3.3.6] for example) that for $u>0$ and $x\neq0$, $$\frac{1}{u}\int_{-u}^u (1- e^{i\theta x})~d\theta = 2 - \frac{2\sin ux}{ux} \geq \mathbf 1_{ux>2}.$$ Denote by $R$ the $(\theta^2\vee\theta^{1+\beta})$-controller in Lemma \[lem: upper bound for usgx\].(4). Then, using Lemma \[lem: estimate of exponential remaining\] we get $$\begin{aligned}
& |\mathbb P_\mu (|\mathcal{I}_0^tX_t(g)| > \lambda)|
\leq \Big|\frac{\lambda}{2}\int_{-2/\lambda}^{2/\lambda}(1 - \mathbb P_\mu[e^{i\theta \mathcal{I}_0^tX_t(g)}])d\theta\Big| \\
& \leq \frac{\lambda}{2}\int_{-2/\lambda}^{2/\lambda}|1-e^{\mu(U_t(\theta g)-iP^\alpha_t (\theta g))}|d\theta
\leq \frac{\lambda}{2}\int_{-2/\lambda}^{2/\lambda}\mu(|U_t(\theta g) - iP^\alpha_t(\theta g)|) d\theta \\
& \leq \frac{\lambda}{2}\int_{-2/\lambda}^{2/\lambda}\mu(R|\theta g|) d\theta.
\qedhere
\end{aligned}$$
\[lem: temp\] For all $h \in \mathcal P^+$ and $\mu \in \mathcal M_c(\mathbb R^d)$, there exists $C > 0$ such that for all $\kappa \in \mathbb Z_+ $, $\lambda > 0$ and $0\leq r\leq s\leq t<\infty$ with $s-r \leq 1$, we have $$\sup_{g \in \mathcal P: Q_\kappa g\leq h}\mathbb P_{\mu}(|\mathcal I_r^sX_t(g)|>\lambda)
\leq C e^{\alpha r} \Big(\Big( \frac{e^{(t-s)(\alpha - \kappa b)}}{\lambda}\Big)^{1+\beta} + \Big( \frac{e^{(t-s)(\alpha - \kappa b)}}{\lambda}\Big)^{2} \Big).$$
Denote by $R$ the $(\theta^2\vee\theta^{1+\beta})$-controller in Lemma \[lem: control pair for P(M>lambda)\]. Fix $h \in \mathcal P^+$, $\mu \in \mathcal M_c(\mathbb R^d)$ $\kappa \in \mathbb Z_+ $ and $0\leq r\leq s\leq t < \infty$ with $s-r \leq 1$. Suppose that $g\in \mathcal P$ satisfies $Q_\kappa g \leq h$. Using the Markov property of $X$, we get $$\begin{aligned}
& \mathbb P_{\mu}(|\mathcal I_r^sX_t(g)|>\lambda)
= \mathbb P_\mu \Big[\mathbb P_\mu [| X_{s}(P_{t-s}^\alpha g) - X_{r}(P_{t-r}^\alpha g)|> \lambda | \mathscr F_r ]\Big] \\
& = \mathbb P_\mu \big[\mathbb P_{X_r}(| X_{s-r}(P_{t-s}^\alpha g) - X_{0}(P_{t-r}^\alpha g)|> \lambda)\big] \\
& = \mathbb P_\mu \big[\mathbb P_{X_r}(|\mathcal I_0^{s-r} X_{s-r}( P_{t-s}^\alpha g) |> \lambda)\big]
\leq \mathbb P_\mu \Big[ \frac{\lambda}{2}\int_{-2/\lambda}^{2/\lambda}X_r(R|\theta P^\alpha_{t-s}g|) d\theta \Big] \\
& \leq \mathbb P_\mu \Big[ \frac{\lambda}{2}\int_{-2/\lambda}^{2/\lambda}X_r(R|\theta e^{(t-s)(\alpha- \kappa b)}h|) d\theta \Big] \\
& \leq \mathbb P_\mu [X_r(Rh) ]
\frac{\lambda}{2}\int_{-2/\lambda}^{2/\lambda}(|\theta e^{(t-s)(\alpha- \kappa b)}|^{1+\beta} + |\theta e^{(t-s)(\alpha- \kappa b)}|^{2})d\theta
\\ & = \mu(P_r^\alpha Rh)
\Big( \frac{2^{2+\beta}}{2+\beta}\Big(\frac{e^{(t-s)(\alpha- \kappa b)}}{\lambda}\Big)^{1+\beta} + \frac{2^{3}}{3}\Big(\frac{e^{(t-s)(\alpha- \kappa b)}}{\lambda}\Big)^2\Big)
\\ & \leq C e^{\alpha r} \Big(\Big( \frac{e^{(t-s)(\alpha - \kappa b)}}{\lambda}\Big)^{1+\beta} + \Big( \frac{e^{(t-s)(\alpha - \kappa b)}}{\lambda}\Big)^{2} \Big),
\end{aligned}$$ where $C := \Big(\frac{2^{2+\beta}}{2+\beta} + \frac{2^{3}}{3} \Big) \mu(Q_0Rh)>0$.
For each random variable $\{Y; \mathbb P\}$ and $p \in [1,\infty)$, we write $ \|Y\|_{\mathbb P;p} := \mathbb P[|Y|^p]^{1/p}$. Recall that we write $\tilde u = \frac{u}{1+u}$ for each $u\neq -1$.
\[lem: control of mgtrs\] For all $h \in \mathcal P$, $\mu \in \mathcal M_c(\mathbb R^d)$ and $\gamma\in (0, \beta)$, there exists $C > 0$ such that for all $\kappa \in \mathbb Z_+$ and $0\leq r \leq s\leq t<\infty$ with $s-r \leq 1$, we have $$\sup_{g \in \mathcal P: Q_\kappa g \leq h} \|\mathcal I_r^s X_t(g) \|_{\mathbb P_\mu;1+\gamma}
\leq C e^{t\alpha (1- \tilde \gamma)+(t-s) (\alpha \tilde \gamma - \kappa b)}.$$
Fix $h \in \mathcal P$ and $\mu \in \mathcal M_c(\mathbb R^d)$. Let $C_0$ be the constant in the Lemma \[lem: temp\]. For all $\kappa \in \mathbb Z_+$, $0\leq r\leq s\leq t$ with $s-r \leq 1$, $g\in \mathcal P$ with $Q_{\kappa} g \leq h$, and $c>0$, we have $$\begin{aligned}
& \mathbb P_\mu[|\mathcal I_r^sX_t(g)|^{1+\gamma}]
= (1+\gamma)\int_0^\infty \lambda^{\gamma} \mathbb P_{\mu}(|\mathcal I_r^sX_t(g)|>\lambda) d\lambda \\
& \leq (1+\gamma)\int_0^c \lambda^{\gamma} d\lambda +(1+\gamma)\int_c^\infty \lambda^{\gamma}\mathbb P_\mu(|\mathcal I_r^sX_t(g)|> \lambda) d\lambda \\
& \leq c^{1+\gamma} + C_0 e^{\alpha r}(1+\gamma)\int_c^\infty \bigg(\Big(\frac{e^{(t-s)(\alpha - \kappa b)}}{\lambda}\Big)^{1+\beta}+\Big(\frac{e^{(t-s)(\alpha - \kappa b)}}{\lambda}\Big)^{2}\bigg)\lambda^{\gamma}d\lambda \\
& \leq c^{1+\gamma} e^{\alpha r} + C_0e^{\alpha r}(1+\gamma)\Big( \frac{e^{(1+\beta)(t-s)(\alpha- \kappa b)}}{(\beta - \gamma)c^{\beta - \gamma}} + \frac{e^{2(t-s)(\alpha- \kappa b)}}{(1 - \gamma)c^{1 - \gamma}} \Big).
\end{aligned}$$ Taking $c = e^{(t-s)(\alpha- \kappa b)}$, we get $$\begin{aligned}
& \mathbb P_\mu\big[|\mathcal I_r^s X_t(g)|^{1+\gamma}\big]
\leq e^{(1+\gamma)(t-s)(\alpha- \kappa b)} e^{\alpha r}\Big(1+ C_0 \frac{1+\gamma}{\beta - \gamma}+ C_0 \frac{1+\gamma}{1 - \gamma}\Big).
\end{aligned}$$ Note that $$\begin{aligned}
& (1+\gamma) (t-s) (\alpha- \kappa b) + \alpha r
= (t-s)\alpha+(t-s) (\gamma \alpha- (1+\gamma )\kappa b) \\
& \leq t\alpha+(t-s) (\gamma \alpha- (1+\gamma)\kappa b).
\end{aligned}$$ So the desired result is true.
\[lem:P:M:uc\] For all $h \in \mathcal P$, $\mu \in \mathcal M_c(\mathbb R^d)$, $\gamma\in (0, \beta)$ and $\kappa \in \mathbb Z_+$, there exists a constant $C > 0$ such that for all $t\geq 0$, we have
1. \[item:P:M:uc:1\] $\sup_{g\in \mathcal P: Q_\kappa g \leq h}\|X_t(g)\|_{\mathbb{P}_{\mu};1+\gamma}\leq C e^{(\alpha-\kappa b)t}$ provided $\alpha \tilde \gamma > \kappa b$;
2. \[item:P:M:uc:2\] $\sup_{g\in \mathcal P: Q_\kappa g \leq h}\|X_t(g)\|_{\mathbb{P}_{\mu};1+\gamma}\leq C te^{\frac{\alpha}{1+\gamma}t}$ provided $\alpha \tilde \gamma = \kappa b$;
3. \[item:P:M:uc:3\] $\sup_{g\in \mathcal P: Q_\kappa g \leq h} \|X_t(g)\|_{\mathbb{P}_{\mu};1+\gamma}\leq C e^{\frac{\alpha}{1+\gamma}t}$ provided $\alpha \tilde \gamma < \kappa b$.
Fix $\gamma \in (0,\beta)$ and $\mu \in \mathcal M_c(\mathbb R^d)$. Let $C$ be the constant in Lemma \[lem: control of mgtrs\]. Using the triangle inequality, for all $\kappa\in \mathbb Z_+$, $g \in \mathcal P$ with $Q_\kappa g \leq h$ and $t\geq 0$, we have $$\begin{aligned}
& \|X_t(g)\|_{\mathbb P_\mu;1+\gamma}
\leq \sum_{l=0}^{\lfloor t\rfloor - 1}\big\| \mathcal{I}_{t-l-1}^{t-l}X_t(g) \big\|_{\mathbb P_\mu;1+\gamma}+\big\| \mathcal{I}_{0}^{t-\lfloor t \rfloor}X_t(g) \big\|_{\mathbb P_\mu;1+\gamma} + |\mu(P^\alpha_t g)| \\
& \leq C^{\frac{1}{1+\gamma}} e^{\frac{\alpha}{1+\gamma}t} \sum_{l=0}^{\lfloor t\rfloor} e^{\frac{\gamma\alpha-\kappa (1+\gamma)b}{1+\gamma} l} + e^{(\alpha - \kappa b)t} \mu(h).
\end{aligned}$$ By calculating the sum on the right, we get the desired result.
Proofs of main results {#proofs of main results}
======================
In this section, we will prove the main results of this paper. For simplicity, we will write $\mathbb{\widetilde{P}}_{\mu}=\mathbb{P}_{\mu}(\cdot|D^c)$ in this section.
Law of large numbers {#sec: large rate lln}
--------------------
In this subsection, we prove Theorem \[thm: law of large number\]. For this purpose, we first prove the almost sure and $L^{1+\gamma}(\mathbb{P}_{\mu})$ convergence of a family of martingales for $\gamma\in (0, \beta)$. Recall that $L$ is the infinitesimal generator of the OU-process. For $f\in \mathcal{P}\cap C^2(\mathbb R^d)$ and $a\in \mathbb R$, we define $$\begin{aligned}
\label{defmartingale}
M_t^{f,a}
:=e^{-(\alpha-ab)t}X_t(f)-\int_0^t e^{-(\alpha-ab)s} X_s((L+ab)f)~ ds.\end{aligned}$$ Let $(\mathscr{F}_t)_{t\geq 0}$ be the natural filtration of $X$. The following lemma says that $\{M_t^{f,a}: t\geq 0\}$ is a martingale with respect to $(\mathscr{F}_t)_{t\geq 0}$.
\[lemma25\] For all $f\in \mathcal{P}\cap C^2(\mathbb R^d)$, $a\in \mathbb R$ and $\mu\in \mathcal M_c(\mathbb R^d)$, the process $(M_t^{f,a})_{t\geq 0}$ is a $\mathbb P_\mu$-martingale with respect to $(\mathscr F_t)_{t\geq 0}$.
Put$\bar{f} :=(L+ab)f$. It follows easily from Ito’s formula that $$\begin{aligned}
\label{Theorem55}
P_t^{ab}f(x)
= f(x)+\int_0^t P_s^{ab}\bar{f}(x)~ds,\quad t\geq 0,x\in \mathbb R^d,
\end{aligned}$$ where $P_t^{ab} := e^{abt}P_t$. For $0\leq s\leq t$, we have $$\begin{aligned}
\label{martingale1}
& \quad\mathbb{P}_{\mu}[M_t^{f,a}|\mathscr{F}_s]
=e^{-(\alpha-ab)t}\mathbb{P}_{\mu}\left[X_t(f)|\mathscr{F}_s\right]-\mathbb{P}_{\mu}\Big[\int_0^t e^{-(\alpha-ab)u}X_u(\bar{f})~ du\Big|\mathscr{F}_s\big] \\
& =e^{-(\alpha-ab)t} X_s(P_{t-s}^{\alpha}f)-\int_0^s e^{-(\alpha-ab)u} X_u(\bar{f})~ du - \int_s^t e^{-(\alpha-ab)u}X_s(P_{u-s}^{\alpha} \bar{f})~ du.
\end{aligned}$$ Using and Fubini’s theorem, we have $$\begin{aligned}
& \int_s^t e^{-(\alpha-ab)u}X_s(P_{u-s}^{\alpha} \bar{f})~ du=e^{-(\alpha-ab)s}\int_s^tX_s(P_{u-s}^{ab}\bar{f})~du\\
& = e^{ - ( \alpha - ab ) s } X_s\left( \int_0^{t-s} P_{u}^{ab} \bar{f}~ du\right)
= e^{-(\alpha-ab)s}\left(X_s(P_{t-s}^{ab}f) - X_s(f) \right) \\
& = e^{-(\alpha-ab)t} X_s( P_{t-s}^{\alpha}f) - e^{ - ( \alpha - ab ) s} X_s(f).
\end{aligned}$$ Using this and , we get the desired result.
Recall that, for $p\in \mathbb Z_+^d$, $\phi_p$ is an eigenfunction of $L$ corresponding to the eigenvalue $-|p|b$ and $ H_t^p =e^{-(\alpha-|p|b)t}X_t(\phi_p)$ for each $t\geq 0$.
\[lem:M:L:ML\] For all $\mu\in \mathcal M_c(\mathbb R^d)$ and $p \in \mathbb Z_+^d$, $(H^p_t)_{t\geq 0}$ is a $\mathbb P_{\mu}$-martingale with respect to $(\mathscr F_t)_{t\geq 0}$. Moreover if $\alpha\tilde \beta>|p|b$, the martingale is bounded in $L^{1+\gamma}(\mathbb P_\mu)$ for each $\gamma\in (0, \beta)$. Thus the limit $ H_{\infty}^p := \lim_{t\rightarrow \infty}H_t^p $ exists $\mathbb{P}_{\mu}$-a.s. and in $L^{1+\gamma}(\mathbb{P}_{\mu})$ for each $\gamma \in (0,\beta)$.
Fix a $\mu \in \mathcal M_c(\mathbb R^d)$ and a $p \in \mathbb Z_+^d$. It follows from Lemma \[lemma25\] that $(H_t^p)_{t\geq 0}$ is a $\mathbb P_\mu$-martingale. Further suppose that $\alpha \tilde \beta > |p| b$. Then there exists a $\gamma_0 \in (0,\beta)$ which is close enough to $\beta$ so that $\alpha\tilde \gamma>|p|b$ for each $\gamma\in [\gamma_0, \beta)$. Using Lemma \[lem:P:M:uc\] and the fact $\kappa_{\phi_p}=|p|$, we get that, for each $\gamma\in [\gamma_0, \beta)$, there exists a constant $C>0$ such that $$\|H_t^p\|_{\mathbb P_\mu;1+\gamma}
\leq C e^{-(\alpha-|p|b)t}e^{(\alpha-|p|b)t}
= C
, \quad t\geq 0.$$ For each $\gamma\in (0, \gamma_0)$ there exists a constant $C'>0$ such that $$\| H_t^p \|_{\mathbb P_\mu;1+\gamma}
\leq \| H_t^p \|_{\mathbb P_\mu;1+\gamma_0}
\leq C',
\quad t\geq 0.$$ Therefore, for each $\gamma \in (0,\beta)$, the martingale $(H_t^p)_{t\geq 0}$ is bounded in $L^{1+\gamma}(\mathbb{P}_{\mu})$.
\[lem: control of wt\] Suppose that $\mu\in \mathcal M_c(\mathbb R^d)$ and that $p \in \mathbb Z_+^d$ satisfies $\alpha \tilde \beta > |p|b$. Then for each $\gamma \in (0,\beta)$ satisfying $\alpha \tilde \gamma > |p|b$, there exists a constant $C> 0$ such that, $$\|H^p_t-H^p_s\|_{\mathbb{P}_{\mu};1+\gamma}
\leq C e^{-(\alpha \tilde \gamma-|p|b)s},
\quad 0 \leq s < t \leq \infty.$$
Thanks to Lemma \[lem:M:L:ML\], we only need to prove the inequality when $0\leq s < t<\infty$. Suppose $p\in \mathbb{Z}_+^d$, $\mu\in \mathcal M_c(\mathbb R^d)$ and $\gamma \in (0,\beta)$ with $\alpha \tilde \gamma > |p|b$ are fixed. Using Lemma \[lem: control of mgtrs\] with $g=\phi_p$ and $k=|p|$, we know that there exists a constant $C_1>0$ such that for all $0\leq r\leq s $ with $s-r\leq1$, $$\begin{aligned}
\|H^p_s-H^p_r\|_{\mathbb P_\mu; 1+\gamma}
\leq C_1 e^{-(\alpha\tilde \gamma-|p|b)s}.
\end{aligned}$$ Thus there exists $C_2>0$ such that for all $0\leq s<t$, $$\begin{aligned}
& \|H^p_t-H^p_s\|_{\mathbb{P}_{\mu};1+\gamma} \\
& \leq \|H^p_{\lfloor s \rfloor+1}-H^p_s\|_{\mathbb{P}_{\mu};1+\gamma}+\sum_{k=\lfloor s \rfloor+1}^{\lfloor t \rfloor}\|H^p_{k+1}-H^p_{k}\|_{\mathbb{P}_{\mu};1+\gamma}+\|H^p_t-H^p_{\lfloor t \rfloor+1}\|_{\mathbb{P}_{\mu};1+\gamma} \\
& \leq C_1 \Big(e^{-(\alpha \tilde \gamma- |p|b) s}+\sum_{k=\lfloor s \rfloor+1}^{\lfloor t \rfloor} e^{-(\alpha \tilde \gamma- |p|b) k} + e^{-(\alpha \tilde \gamma-|p|b) t}\Big)
\leq C_2e^{-(\alpha \tilde \gamma-|p|b)s}.
\qedhere
\end{aligned}$$
Fix $f \in \mathcal P$ such that $\alpha \beta > \kappa_f b (1+\beta)$ and $\mu \in \mathcal M_c(\mathbb R^d)$. Write $$f
= \sum_{p\in \mathbb Z_+^d:|p|\geq \kappa_f}\langle f,\phi_p\rangle_\varphi \phi_p
=: \sum_{p\in \mathbb Z_+^d:|p|= \kappa_f}\langle f,\phi_p\rangle_\varphi \phi_p+\widetilde{f}.$$ Then $$\begin{aligned}
& e^{-(\alpha-\kappa_fb)t}X_t(f)=
\sum_{p\in \mathbb Z_+^d:|p|= \kappa_f}\langle f,\phi_p\rangle_\varphi H_t^p+e^{-(\alpha-\kappa_fb)t} X_t(\widetilde{f}),
\quad t\geq 0.
\end{aligned}$$ According to Lemma \[lem:M:L:ML\], we have $$\begin{aligned}
\label{as convergence}
\sum_{p\in \mathbb{Z}_+^d:|p|= \kappa_f}\langle f,\phi_p\rangle_\varphi H_t^p
\xrightarrow[t\to \infty]{} \sum_{p\in \mathbb{Z}_+^d:|p|=\kappa_f}\langle f, \phi_p\rangle_{\varphi} H_{\infty}^p,
\end{aligned}$$ $\mathbb{P}_{\mu}$-a.s. and in $L^{1+\gamma}(\mathbb{P}_{\mu})$ for each $\gamma\in(0,\beta)$. Therefore, it suffices to show that $$\begin{aligned}
J_t
:=e^{-(\alpha-\kappa_fb)t}X_t( \widetilde{f}),
\quad t\geq 0,
\end{aligned}$$ converges to $0$ in $L^{1+\gamma}(\mathbb{P}_{\mu})$ for all $\gamma\in(0,\beta)$, and converges almost surely provided $f$ is twice differentiable and all its second order partial derivatives are in $\mathcal{P}$.
*Step 1.* Let $g\in \mathcal P$. Let $\kappa > 0$ be such that $\kappa < \kappa_g$ and $\kappa b < \alpha \tilde \beta$. We will show that for each $\gamma \in (0,\beta)$ there exist $C_1,\delta_1 > 0$ such that $$\|e^{-(\alpha - \kappa b)t} X_t(g)\|_{\mathbb P_\mu;1+\gamma}
\leq C_1 e^{-\delta_1 t},
\quad t\geq 0.$$ In order to do this, we choose a $\gamma_0 \in (0,\beta)$ close enough to $\beta$ such that $\kappa b< \alpha \tilde \gamma$ for each $\gamma \in [\gamma_0, \beta)$. According to Lemma \[lem:P:M:uc\], we have for each $\gamma \in (0,\beta)$,
1. if $\gamma \in [\gamma_0, \beta)$ and $\alpha\tilde \gamma> \kappa_g b$, then there exists $C_2>0$ such that $$\|e^{-(\alpha - \kappa b)t} X_t(g)\|_{\mathbb P_\mu;1+\gamma}
\leq C_2 e^{-(\alpha-\kappa b)t}e^{(\alpha-\kappa_g b)t}
\leq C_2 e^{-(\kappa_g - \kappa )bt},
\quad t\geq 0;$$
2. if $\gamma \in [\gamma_0, \beta)$ and $\alpha\tilde \gamma=\kappa_g b$, then there exists $C_3>0$ such that $$\|e^{-(\alpha - \kappa b)t} X_t(g)\|_{\mathbb P_\mu;1+\gamma}
\leq C_3 t e^{-(\alpha - \kappa b)t}e^{\frac{\alpha}{1+\gamma}t}
= C_3 t e^{-(\alpha \tilde \gamma - \kappa b)t},
\quad t\geq 0;$$
3. if $\gamma \in [\gamma_0, \beta)$ and $\alpha\tilde \gamma < \kappa_g b$, then there exists $C_4>0$ such that $$\|e^{-(\alpha - \kappa b)t} X_t(g)\|_{\mathbb{P}_{\mu};1+\gamma}
\leq C_4 e^{-(\alpha - \kappa b)t}e^{\frac{\alpha}{1+\gamma}t}
= C_4 e^{-(\alpha \tilde \gamma - \kappa b)t},
\quad t\geq 0;$$
4. if $\gamma \in (0,\gamma_0)$, then thanks to (1)–(3) above and the fact that $$\|e^{-(\alpha - \kappa b)t} X_t(g)\|_{\mathbb{P}_{\mu};1+\gamma}
\leq \|e^{-(\alpha - \kappa b)t} X_t(g)\|_{\mathbb{P}_{\mu};1+\gamma_0},$$ there exist $C_5, \delta_2 >0$ such that $$\|e^{-(\alpha - \kappa b)t} X_t(g)\|_{\mathbb{P}_{\mu};1+\gamma}
\leq C_5e^{-\delta_2 t},
\quad t\geq 0.$$
Thus, the desired conclusion in this step is valid. In particular, by taking $g = \widetilde f$ and $\kappa = \kappa_f$, we get that $J_t$ converges to $0$ in $L^{1+\gamma}(\mathbb{P}_{\mu})$ for any $\gamma\in(0,\beta)$.
*Step 2.* We further assume that $f\in C^2(\mathbb R^d)$ and $D^2f \in \mathcal{P}$. We will show that $J_t$ converges to $0$ almost surely. For $a \geq 0$, $ t\geq 0$, and $g\in \mathcal{P}\cap C^2(\mathbb{R}^d)$ satisfying $D^2g\in \mathcal{P}$, we define $$\begin{aligned}
L_t^{g,a}
& :=\int_0^t e^{-(\alpha-ab)s}X_s((L+ab)g) ds,
\quad
Y_t^{g,a}
:=\int_0^t e^{-(\alpha-ab)s}|X_s((L+ab)g)|ds.
\end{aligned}$$ Now choose $a_0 \in (\kappa_{f}, \kappa_f + 1)$ close enough to $\kappa_f$ so that $a_0 b < \alpha \tilde \beta$. According to , $$\begin{aligned}
J_t
= e^{-(a_0-\kappa_f)bt} (M_t^{\widetilde{f}, a_0}+L_t^{\widetilde{f}, a_0}),
\quad t\geq 0.
\end{aligned}$$ So we only need to show that $$\begin{aligned}
e^{-(a_0-\kappa_f)b t}M_t^{\widetilde{f},a_0}
\xrightarrow[t\to \infty]{} 0,
\quad e^{-(a_0-\kappa_f)b t}L_t^{\widetilde{f},a_0}
\xrightarrow[t\to \infty]{} 0,
\quad \mathbb{P}_{\mu}\text{-a.s.}
\end{aligned}$$ Notice that $\kappa_{(L+a_0 b)\widetilde{f}}\geq \kappa_{\widetilde{f}}\geq \kappa_f+1 > a_0$. By Step 1, for any fixed $\gamma\in (0,\beta)$, there exist $C_6, \delta_3>0$ such that for each $t\geq 0$, $$\| e^{-(\alpha-a_0 b)t}X_t(\widetilde{f}))\|_{\mathbb{P}_{\mu};1+\gamma}
\leq C_6 e^{-\delta_3 t},
\quad \|e^{-(\alpha-a_0 b)t}X_t(L\widetilde{f}+a_0 b\widetilde{f})\|_{\mathbb{P}_{\mu};1+\gamma}
\leq C_6 e^{-\delta_3 t}.$$ Now, by the triangle inequality, for each $t\geq 0$, $$\begin{aligned}
& \|L_t^{\widetilde{f},a_0}\|_{\mathbb{P}_{\mu};1+\gamma}
\leq\|Y_t^{\widetilde{f},a_0}\|_{\mathbb{P}_{\mu};1+\gamma} \\
& \leq \int_0^t \|e^{-(\alpha-a_0 b)s}X_s( L\widetilde{f}+a_0 b\widetilde{f})\|_{\mathbb{P}_{\mu};1+\gamma}ds\leq C_6 \int_0^t e^{-\delta_3 s}ds\leq\frac{C_6}{\delta_3}.
\end{aligned}$$ Since $Y_t^{\widetilde{f},a_0}$ is increasing in $t$, it converges to some finite random variable $Y_{\infty}^{\widetilde{f},a_0}$ almost surely and in $L^{1+\gamma}(\mathbb{P}_{\mu})$. Consequently, we have $$\begin{aligned}
\lim_{t\rightarrow \infty}e^{-(a_0 - \kappa_f)bt}|L_t^{\widetilde{f},a_0}|
\leq \lim_{t\rightarrow \infty}e^{-(a_0 - \kappa_f)bt}|Y_t^{\widetilde{f},a_0}|=0,
\quad \mathbb P_\mu\text{-a.s.}
\end{aligned}$$ On the other hand, the martingale $M_t^{\widetilde{f},a_0}$ satisfies $$\begin{aligned}
\|M_t^{\widetilde{f},a_0}\|_{\mathbb{P}_{\mu};1+\gamma}
\leq \|e^{-(\alpha-a_0 b)t}X_t(\widetilde{f})\|_{\mathbb{P}_{\mu};1+\gamma}+\|L_t^{\widetilde{f},a_0}\|_{\mathbb{P}_{\mu};1+\gamma}
\leq C_6(e^{-\delta_3 t}+\frac{1}{\delta_3}),
\quad t\geq 0.
\end{aligned}$$ This implies that the martingale converges almost surely. Consequently, $$\lim_{t\rightarrow\infty} e^{-(a_0-\kappa_f)bt}M_t^{\widetilde{f},a_0}
= 0,
\quad \mathbb P_\mu\text{-a.s.}.
\qedhere$$
Central limit theorems for unit time intervals {#sec:critical}
----------------------------------------------
In this subsection, we will establish the following CLT.
\[lem:PR:LC\] If $\mu \in \mathcal M_c(\mathbb R^d)$ and $f\in \mathcal{P}\setminus \{0\}$, then under $\mathbb{P}_{\mu}(\cdot | D ^c)$, we have $$\begin{aligned}
\label{eq:PR:LC:1}
\Upsilon^f_t
:= \frac{X_{t+1} (f) - X_t(P_1^\alpha f)}{\| X_t\|^{1-\tilde \beta}}
\xrightarrow[t\to \infty]{d}\zeta^f_0,
\end{aligned}$$ where $\zeta^f_0$ is a $(1+\beta)$-stable random variable with characteristic function $\theta\mapsto e^{\langle Z_1(\theta f), \varphi\rangle}$.
In fact, we prove a stronger result:
\[thm:Key\] For all $\mu \in \mathcal M_c(\mathbb R^d)$ and $g \in \mathcal P \setminus \{0\}$, there exist $C,\delta>0$ such that for all $t\geq 1$ and $f \in \mathcal P_g:= \{\theta T_ng:n \in \mathbb Z_+, \theta \in [-1,1]\}$, we have $$\mathbb P_\mu
\Big[ |\mathbb P_\mu [e^{i\Upsilon^f_t} - e^{\langle Z_1f, \varphi\rangle}; D^c | \mathscr F_t ] |\Big]
\leq C e^{- \delta t}.$$
Fix $\mu \in \mathcal M_c(\mathbb R^d)$ and $g \in \mathcal P\setminus \{0\}$.
*Step 1.* Write $ A_t(\epsilon) :=\{ \|X_t\| \geq e^{(\alpha - \epsilon)t} \} $ for $t\geq 0$ and $\epsilon > 0$. We will show that for all $f\in \mathcal P \setminus \{0\}$, $\epsilon > 0$ and $t\geq 0$, it holds that $$\mathbb P_\mu \Big[ | \mathbb P_\mu [e^{i\Upsilon^f_t} - e^{\langle Z_1(\theta f), \varphi\rangle}; D^c | \mathscr F_t ]| \Big]
\leq J^f_1(t,\epsilon)+J^f_2(t,\epsilon)+J^f_3(t,\epsilon),$$ where $$\begin{aligned}
\label{eq: Def of Ji}
&J^f_1(t,\epsilon):= \mathbb{P}_{\mu} [ | X_t(Z'''_1(\theta_t f)) |; A_t(\epsilon) ],
\quad
J^f_2(t,\epsilon):= \mathbb{P}_{\mu}[|X_t( Z_1(\theta_t f))-\langle Z_1f, \varphi\rangle |; A_t(\epsilon)],
\\ & J_3(t,\epsilon):=2\mathbb{P}_{\mu}(A_t (\epsilon)\Delta D^c),
\quad
\theta_t := \|X_t\|^{-(1 - \tilde \beta)}.\end{aligned}$$ In fact, it follows from , the definitions of $U_1$, $Z'''_1$ and $Z_1$, that for all $t\geq 0$, $$\begin{aligned}
\label{eq: need1}
& \mathbb{P}_{\mu}[e^{i\Upsilon^f_t}|\mathscr{F}_t]
= \mathbb{P}_{\mu}[\exp\{i\theta_t X_{t+1} (f) - i \theta_t X_t(P_1^\alpha f)\} |\mathscr{F}_{t}] \\
& = \exp\{X_t((U_1 - iP^\alpha_1 ) (\theta_t f))\}
= \exp\{X_t((Z_1 + Z'''_1) (\theta_t f))\}.\end{aligned}$$ From Lemma \[lem: charactreisticfunction\], we get that $\theta\mapsto \langle Z_1(\theta f),\varphi\rangle$ is the characteristic function of some $(1+\beta)$-stable random variable, and then $\operatorname{Re} \langle Z_1f, \varphi\rangle \leq 0$. Using this, , and the fact $|e^{-x} - e^{-y}| \leq |x-y|$ for all $x,y \in \mathbb C_+$, we get for each $t\geq 0$ and $\epsilon> 0$, $$\begin{aligned}
\label{eq: inequality that will used later}
& \mathbb{P}_\mu \Big[ | \mathbb{P}_\mu [ e^{i\Upsilon^f_t} - e^{\langle Z_1f,\varphi \rangle} ; D^c | \mathscr F_{t}] |\Big] \\
& \leq \mathbb{P}_\mu \Big[ | \mathbb{P}_\mu [ e^{i \Upsilon^f_t }-e^{\langle Z_1f, \varphi\rangle}; A_{t}(\epsilon) | \mathscr F_{t}] | + 2\mathbb P_\mu ( A_{t}(\epsilon) \Delta D^c | \mathscr F_{t}) \Big] \\
& = \mathbb{P}_{\mu}\Big[ |\mathbb{P}_\mu [e^{i\Upsilon^f_t}| \mathscr F_{t}]-e^{\langle Z_1f, \varphi\rangle}| ; A_{t}(\epsilon) \Big] + J_3(t,\epsilon) \\
& \leq \mathbb{P}_\mu \Big[ |e^{X_{t}((Z_1+Z'''_1) (\theta_t f))}-e^{\langle Z_1f, \varphi\rangle} | ; A_{t}(\epsilon) \Big]+ J_3(t,\epsilon) \\
& \leq \mathbb{P}_\mu \Big[ | X_{t} ( (Z_1+Z'''_1)(\theta_t f)) - \langle Z_1f, \varphi\rangle | ;A_{t}(\epsilon)\Big]+ J_3(t,\epsilon) \\
& \leq J^f_1(t,\epsilon) + J^f_2(t,\epsilon)+ J_3(t,\epsilon).\end{aligned}$$
*Step 2.* We will show that for $\epsilon>0$ small enough, there exist $C_2, \delta_2>0$ such that for all $t\geq 1$ and $f \in \mathcal P_g$, we have $ J^f_1(t,\epsilon) \leq C_2e^{-\delta_2 t}$.
In fact, let $\delta_0 >0$ be the constant in Lemma \[lem: upper bound for usgx\].(7) and let $R$ be the corresponding $(\theta^{2+\beta}\vee \theta^{1+\beta+\delta_0})$-controller. Acording to Step 1 in the proof of Lemma \[lem:m\], there exists $h_{2} \in \mathcal P^+$ such that for each $f \in \mathcal P_g$ it holds that $|f| \leq h_{2}$. Then, we have for all $t\geq 0$, $\epsilon> 0$ and $f\in \mathcal P_g$, $$\begin{aligned}
& |Z'''_1(\theta_t f)|\mathbf{1}_{A_{t}(\epsilon)}
\leq R(|\theta_{t} f|)\mathbf{1}_{A_{t}(\epsilon)}
\leq R \Big(\frac{h_{2}}{e^{(\alpha-\epsilon)t(1-\tilde \beta)}}\Big)
\leq \sum_{\rho \in \{\delta_0,1\}}e^{-\frac{1+\beta+\rho}{1+\beta}(\alpha-\epsilon)t}Rh_{2}.\end{aligned}$$ Thus for all $t\geq 0$, $\epsilon> 0$ and $f\in \mathcal P_g$, $$\begin{aligned}
\label{eq: estimate of J1}
J^f_1(t,\epsilon)
& \leq \sum_{\rho \in \{\delta_0,1\}}e^{-\frac{1+\beta+\rho}{1+\beta}(\alpha-\epsilon)t}\mathbb{P}_{\mu}[X_{t}(Rh_2)]
\leq \sum_{\rho \in \{\delta_0,1\}} \mu(Q_0 R h_{2}) e^{-(\alpha\frac{\rho}{1+\beta}-\epsilon\frac{1+\beta+\rho}{1+\beta})t},\end{aligned}$$ where $Q_0$ is defined by . By taking $\epsilon>0$ small enough, we get the desired result in this step.
*Step 3.* We will show that for $\epsilon>0$ small enough there exist $C_3, \delta_3 > 0$ such that for all $t \geq 0$ and $f\in \mathcal P_g$, we have $ J^f_2(t,\epsilon) \leq C_3 e^{-\delta_3 t}$. In fact, for all $t\geq 0$, and $f\in \mathcal P_g$, $$\begin{aligned}
& X_{t}(Z_1(\theta_t f))- \langle Z_1f, \varphi\rangle
= \theta_t^{1+\beta} X_t(Z_1 f) - \langle Z_1 f,\varphi \rangle
= \frac{1}{\|X_{t}\|}X_t(Z_1f - \langle Z_1 f ,\varphi \rangle),\end{aligned}$$ and therefore, $$\begin{aligned}
\label{eq: prevJ2}
J^f_2(t,\epsilon)
& = \mathbb P_\mu\Big[\Big| \frac{1}{\|X_{t}\|}X_t(Z_1f - \langle Z_1 f ,\varphi \rangle) \Big|;A_t(\epsilon)\Big]
\leq e^{-(\alpha-\epsilon)t} \mathbb{P}_{\mu}[|X_t (q_f) |].\end{aligned}$$ where $ q_f = Z_1 f-\langle Z_1 f,\varphi\rangle \in \mathcal P^*$. It follows from Lemma \[lem:P:R\] that there exists $h_{3}\in \mathcal{P}$ such that for each $ f\in \mathcal P_g$, we have $Q_1 (\operatorname{Re} q_f) \leq h_{3} \text{ and } Q_1 (\operatorname{Im} q_f)\leq h_3$, where $Q_1$ is given by with $\kappa=1$. In the rest of this step, we fix a $\gamma\in(0,\beta)$ small enough such that $\alpha \gamma < b < (1+\gamma)b$. According to Lemma \[lem:P:M:uc\].(3) (with $\kappa=1$), there exists $C_{3}>0$ such that for all $t\geq 0$ and $f\in \mathcal P_g$, $$\begin{aligned}
& \mathbb{P}_{\mu}\left[\left|X_{t}(q_f)\right|\right]
\leq \| X_{t}( \operatorname{Re} q_f)\|_{\mathbb{P}_{\mu,1+\gamma}} + \| X_{t}(\operatorname{Im} q_f)\|_{\mathbb{P}_{\mu,1+\gamma}} \\
& \leq 2\sup_{q\in \mathcal P: Q_1 q\leq h_{3}} \|X_t(q)\|_{\mathbb P_\mu; 1+\gamma} \leq C_{3} e^{\frac{\alpha t}{1+\gamma}}.\end{aligned}$$ Therefore, for all $t\geq 0, \epsilon > 0$ and $f \in \mathcal P_g$, we have $$\begin{aligned}
\label{eq: right bound for J2}
J^f_2(t, \epsilon)
\leq C_3 e^{-(\alpha-\epsilon)t}e^{\frac{\alpha t}{1+\gamma}}
\leq C_{3} e^{-(\alpha\tilde \gamma -\epsilon)t}.\end{aligned}$$ By taking $\epsilon >0$ small enough, we get the required result in this step.
*Step 4.* We will show that, for each $\epsilon\in (0, \alpha)$, there exist $C_4,\delta_4>0$ such that for all $t\geq 1$, $J_3(t,\epsilon)\leq C_4e^{-\delta_4 t}.$ In fact, we have for all $t\geq 0, \epsilon >0$, $$\mathbb P_{\mu}(A_{t}(\epsilon), D)
= \mathbb P_{\mu}[\mathbb P_{\mu}(D|\mathscr F_t);A_t(\epsilon)]
= \mathbb P_\mu[e^{-\bar v\|X_t\|};A_t(\epsilon)]
\leq \exp({-\bar v \|\mu\|e^{(\alpha - \epsilon)t}}).$$ On the other hand, by Proposition \[lem: control of XT\], for each $\epsilon \in (0, \alpha)$, there exists $C_{4}, \delta_{4}>0$ such that for all $t\geq 0$, $$\begin{aligned}
\mathbb P_\mu(A_t(\epsilon)^c,D^c)
\leq \mathbb P_\mu(0 < e^{-\alpha t}\|X_t\|
\leq e^{ - \epsilon t}) \leq C_{4} (e^{-\epsilon \delta_{4} t}+e^{-\delta_{4} t}).\end{aligned}$$ Combining these results, we get the desired result in this step.
*Step 5.* Combining the results in Steps 1–4, we immediately get the desired result.
The following corollary will be used later in the proof of Theorem \[thm:M\].
\[cor:MI\] If $g\in \mathcal{P}\setminus\{0\}$ and $\mu\in \mathcal M_c(\mathbb R^d)$, then there exist $C,\delta>0$ such that for all $l\leq n$ in $\mathbb Z_+$ and $(f_j)_{j=l}^n\subset \mathcal P_g$, $$\begin{aligned}
\label{32corollary}
\Big|\mathbb{\widetilde{P}}_{\mu}\Big[\prod_{k=l}^ne^{i \Upsilon^{f_k}_{k} }-\prod_{k=l}^n e^{\langle Z_1f_k, \varphi\rangle}\Big]\Big|\leq C e^{-\delta l}.\end{aligned}$$
For $l\leq n$ in $\mathbb Z_+$, $k \in \{l,\dots,n\}$ and $(f_j)_{j=l}^n\subset \mathcal P_g$, define $$a_k
:= \mathbb{\widetilde{P}}_{\mu}\Big[\prod_{j=l}^{k} e^{i\Upsilon_j^{f_j}}\Big] \times \Big(\prod_{j=k+1}^{n} e^{ \langle Z_1f_j,\varphi \rangle} \Big).$$ Then for all $l\leq n$ in $\mathbb Z_+$, $k \in \{l,\dots,n\}$ and $(f_j)_{j=l}^n\subset \mathcal P_g$, we have $$\begin{aligned}
& a_{k} - a_{k-1}
=\mathbb{P}_{\mu}(D^c)^{-1} \mathbb{P}_{\mu}\Big[(e^{i\Upsilon^{f_k}_k}-e^{\langle Z_1f_k, \varphi\rangle})\prod_{j=l}^{k-1} e^{i\Upsilon_j^{f_j}};D^c\Big] \Big(\prod_{j=k+1}^n e^{\langle Z_1f_j, \varphi\rangle}\Big)\\
& =\mathbb{P}_{\mu}(D^c)^{-1} \mathbb{P}_{\mu}\Big[\mathbb P_\mu[e^{i\Upsilon_k^{f_k}}-e^{\langle Z_1f_k, \varphi \rangle}; D^c|\mathscr F_k] \prod_{j=l}^{k-1} e^{i\Upsilon_j^{f_j}}\Big] \Big(\prod_{j=k+1}^{n}e^{\langle Z_1f_j, \varphi\rangle}\Big).
\end{aligned}$$ By Lemma \[thm:Key\], there exist $C_0,\delta_0 >0$ such that for all $l\leq n$ in $\mathbb Z_+$, $k \in \{l,\dots , n\}$, and $(f_j)_{j=l}^n\subset \mathcal P_g$, we have $$\begin{aligned}
| a_{k} - a_{k-1}|
& \leq \mathbb{P}_{\mu}(D^c)^{-1}\mathbb{P}_{\mu}\Big[\big|\mathbb P_\mu[e^{i\Upsilon_k^{f_k}}-e^{\langle Z_1f_k, \varphi \rangle}; D^c | \mathscr{F}_k]\big|\Big]
\leq C_0 e^{-\delta_0 k}.
\end{aligned}$$ Therefore, there exist $C,\delta >0$ such that for all $l\leq n$ in $\mathbb Z_+$ and each $(f_j)_{j=l}^n\subset \mathcal P_g$, we have $$\begin{aligned}
\text{LHS of \eqref{32corollary}}
& = \left|a_n-a_{l-1}\right|
\leq \sum_{k=l}^n\left|a_{k}-a_{k-1}\right|
\leq \sum_{k=l}^n C_0 e^{-\delta_0 k}
\leq C e^{- \delta l}.
\qedhere
\end{aligned}$$
Central limit theorem for $f\in \mathcal C_s$ {#sec: small rate}
---------------------------------------------
Fix $\mu\in \mathcal M_c(\mathbb R^d)$, $f\in \mathcal C_s$ and $t_0 > 1$ large enough so that $ \lceil t - \ln t\rceil \leq \lfloor t \rfloor - 1$ for all $t\geq t_0$. For each $t\geq t_0$, in this proof we write $\theta_t = \|X_t\|^{\tilde \beta - 1}$, $$\begin{gathered}
\label{eq:PM:CLTS:1}
\theta_t X_t(f)
= I^f_1(t) + I^f_2(t) + I^f_3(t)
:= \Big(\sum_{k=0}^{\lfloor t-\ln t \rfloor} \theta_t \mathcal I_{t-k-1}^{t-k} X_t(f) \Big)\\
+ \Big( \theta_t \mathcal I_0^{t-\lfloor t \rfloor} X_t(f) + \sum_{k=\lceil t-\ln t \rceil}^{\lfloor t \rfloor-1} \theta_t \mathcal I_{t-k-1}^{t-k} X_t(f) \Big) + (\theta_t X_0(P_t^\alpha f) ),
\end{gathered}$$ and $ I^f_0(t) := \sum_{k=0}^{\lfloor t-\ln t \rfloor} \Upsilon_{t-k-1}^{T_k \tilde f},$ where $\tilde f:= e^{\alpha(\tilde \beta - 1)} f$.
*Step 1.* We show that $I^f_0(t) \xrightarrow[t\to \infty]{d} \zeta^f$. In fact, for each $k \in \mathbb Z_+$, we have $T_{k} \tilde f \in \mathcal P_{\tilde f}:=\{\theta T_n \tilde f: n \in \mathbb Z_+, \theta \in [-1,1]\}$. Therefore from Corollary \[cor:MI\] we get that there exist $C_1,\delta_1 > 0$ such that $$\begin{aligned}
\Big|\mathbb{\widetilde{P}}_{\mu} [e^{i I^f_0(t)} ]-\exp\Big(\sum_{k=0}^{\lfloor t-\ln t \rfloor} \langle Z_1T_{k}\tilde f, \varphi\rangle \Big)\Big|
\leq C_1 e^{-\delta_1(t - \lfloor t - \ln t\rfloor)},
\quad t\geq t_0.
\end{aligned}$$ On the other hand, using and the fact that $\varphi(x)dx$ is the invariant probability of the semigroup $(P_t)_{t\geq 0}$, we have $$\begin{aligned}
\label{eq:PM:CLTS:2}
& \sum_{k=0}^\infty \langle Z_1 T_{k} \tilde f, \varphi \rangle
= \sum_{k=0}^\infty \int_0^1 \langle P_u^\alpha ((-iP_{1 - u}^\alpha T_k \tilde f)^{1+\beta}), \varphi\rangle ~du
\\& = \sum_{k=0}^\infty \int_0^1 e^{\alpha u} \langle (-iP_{1 - u}^\alpha T_{k}\tilde f)^{1+\beta}, \varphi \rangle ~du
\\& = \sum_{k=0}^\infty \int_0^1 \langle (-iT_{k+1 - u} f)^{1+\beta}, \varphi\rangle~du
= \int_0^\infty \langle (-iT_{u} f)^{1+\beta}, \varphi\rangle~du = m[f].
\end{aligned}$$ Therefore, we have $\mathbb{\widetilde{P}}_{\mu} [e^{i I^f_0(t)} ] \xrightarrow[t\to \infty]{} e^{m[f]}$. Since $I_0^f(t)$ is linear in $f$, we can replace $f$ with $\theta f$, $\theta \in \mathbb R$, and then the desired result in this step follows.
*Step 2.* We show that $I^f_1(t) - I^f_0(t) \xrightarrow[t\to \infty]{d} 0$. In fact, by [@Durrett2010Probability Lemma 3.4.3] we have that for each $t\geq t_0$, $$\begin{aligned}
\label{eq:PM:S:1}
|\mathbb{\widetilde{P}}_{\mu}[e^{i (I^f_1(t) - I^f_0(t) ) }] - 1|
\leq \sum_{k=0}^{\lfloor t-\ln t \rfloor}\mathbb{\widetilde{P}}_{\mu}\big[|Y_{t,k}|\big],
\end{aligned}$$ where $ Y_{t,k} := \exp(i \Upsilon_{t-k-1}^{T_{k} \tilde f} - i\theta_t \mathcal I_{t-k-1}^{t-k} X_t(f)) - 1. $ We claim that there exist $C_2, \delta_2>0$ such that $\widetilde {\mathbb P}_\mu [|Y_{t,k}|] \leq C_2 e^{-\delta_2 (t-k-1)}$ for all $k\in \mathbb Z_+$ and $t\geq k+1$. Then there exists $C_2'>0$ such that for each $t \geq t_0$, $|\mathbb{\widetilde{P}}_{\mu}[e^{i (I^f_1(t)- I^f_0(t))}]-1| \leq C_2't^{-\delta_1}$ which, combined with the fact that $I^f_1(t) - I^f_0(t)$ is linear in $f$, completes this step.
We will show the claim above in the following substeps 2.1 and 2.2. First we choose $\gamma \in (0,\beta)$ close enough to $\beta$ so that there exist $\eta,\eta'>0$ with $ \alpha \tilde \gamma > \eta > \eta - 3\eta' > \alpha \tilde \beta - \alpha \tilde \gamma > 0;$ and define for $k \in \mathbb Z_+$ and $t\geq k+1$, $$\mathcal{D}_{t,k}
:=\{|H_t-H_{t-k-1}|\leq e^{-\eta (t-k-1)}, H_{t-k-1}> 2e^{-\eta' (t-k-1)}\},$$ where $H_t := e^{-\alpha t}\|X_t\|$.
*Substep 2.1.* We show that there exist $C_{2.1},\delta_{2.1} >0$ such that for all $k \in \mathbb Z_+$ and $t\geq k+1$, $ \mathbb{\widetilde{P}}_{\mu} \big[ |Y_{t,k}| ;\mathcal{D}^c_{t,k} \big] \leq C_{2.1} e^{-\delta_{2.1} (t-k)}.$ In fact, it follows from Proposition \[lem: control of XT\], Lemma \[lem: control of wt\] with $|p|=0$ and Chebyshev’s inequality that there exist $C_{2.1}', \delta_{2.1}'>0$ such that for all $k \geq 0$ and $t\geq k+1$, $$\begin{aligned}
\label{eq: prob of Dtkc}
& \mathbb{\widetilde{P}}_{\mu}(\mathcal{D}_{t,k}^c)
\leq \mathbb{\widetilde{P}}_{\mu}(|H_t-H_{t-k-1}| > e^{-\eta (t-k-1)})+\mathbb{\widetilde{P}}_{\mu}(H_{t-k-1}\leq 2e^{-\eta'(t-k-1)}) \\
& \leq \mathbb{P}_{\mu}(D^c)^{-1}e^{\eta(t-k-1)}\mathbb{P}_{\mu}[|H_t-H_{t-k-1}|] + \mathbb{P}_{\mu}(D^c)^{-1} \mathbb P_\mu(H_{t-k-1}\leq 2e^{-\eta'(t-k-1)}; D^c) \\
& \leq \mathbb{P}_{\mu}(D^c)^{-1} e^{\eta(t-k-1)}\|H_t - H_{t-k-1}\|_{\mathbb P_\mu; 1+\gamma} + \mathbb{P}_{\mu}(D^c)^{-1} \mathbb P_\mu(0<H_{t-k-1}\leq 2e^{-\eta'(t-k-1)}) \\
& \leq C'_{2.1} e^{-(\alpha \tilde \gamma - \eta)(t-k-1)}+C'_{2.1} e^{-\delta'_{2.1}(t-k-1)}.
\end{aligned}$$ This implies the desired result in this substep, since $|Y_{t,k}| \leq 2$ a.s..
*Substep 2.2.* We will show that there exist $C_{2.2},\delta_{2.2} > 0$ such that for all $k\in \mathbb Z_+$ and $t\geq k+1$, it holds that $ \mathbb{\widetilde{P}}_{\mu} [|Y_{t,k}|;\mathcal{D}_{t,k}] \leq C_{2.2} e^{-\delta_{2.2} (t-k)}$. In fact, noticing that for $f\in \mathcal C_s$ and $k\in \mathbb Z_+$, we have $T_kf = e^{\alpha (\tilde \beta - 1 )k}P_k^\alpha f $; and therefore for all $k\in \mathbb Z_+$ and $t \geq k + 1$, $$\begin{aligned}
\label{eq:gammafunction11}
\Upsilon_{t-k-1}^{T_{k} \tilde f}
= \frac{X_{t-k}(T_{k} \tilde f) - X_{t -k-1}(P_1^\alpha T_{k} \tilde f)}{\|X_{t-k-1}\|^{1-\tilde \beta}}
= \frac{\mathcal I_{t - k - 1}^{t - k} X_t(f)}{\|e^{\alpha (k+1)}X_{t-k-1} \|^{1 -\tilde \beta}}.
\end{aligned}$$ Since $|e^{ix}-e^{iy}|\leq|x-y|$ for all $x,y\in \mathbb R$, we have for all $k \in \mathbb Z_+$ and $t\geq k+1$, $$\begin{aligned}
\label{eq: control of Ykt}
\mathbb{\widetilde{P}}_{\mu}[|Y_{t,k}|;\mathcal{D}_{t,k}]
& \leq \mathbb{\widetilde{P}}_{\mu}\Big[|\mathcal I_{t-k-1}^{t-k} X_t(f) | \cdot \Big| \| e^{\alpha(k+1)}X_{t-k-1}\| ^{ \tilde \beta - 1} - \|X_t\|^{ \tilde \beta - 1}\Big|; \mathcal D_{t,k}\Big] \\
& \leq e^{\alpha(\tilde \beta - 1)t}\mathbb{\widetilde{P}}_{\mu}\big[|\mathcal I_{t-k-1}^{t-k}X_t(f)|\cdot K_{t,k}\big],
\end{aligned}$$ where $$K_{t,k}
:= \Big| \frac {H_t^{1- \tilde \beta} - H_{t-k-1}^{1 - \tilde \beta}} {H_t^{1 - \tilde \beta} H_{t-k-1}^{ 1- \tilde \beta }} \Big| \mathbf{1}_{\mathcal{D}_{t,k}}.$$ Note that, since $\eta' < \eta$, we have almost surely on $\mathcal D_{t,k}$, $$\begin{aligned}
H_t
& \geq H_{t-k-1}- e^{-\eta (t-k-1)}
\geq 2e^{-\eta'(t-k-1)}-e^{-\eta(t-k-1)}
\geq e^{-\eta'(t-k-1)}.
\end{aligned}$$ Therefore, for all $k \in \mathbb Z_+$ and $t\geq k+1$, almost surely on $\mathcal D_{t,k}$, $$\begin{aligned}
& \Big|H_t^{1- \tilde \beta}-H_{t-k-1}^{1- \tilde \beta}\Big|
\leq (1- \tilde \beta) \max \{ H_t^{-\tilde \beta }, H_{t-k-1}^{ -\tilde \beta} \} | H_t - H_{t-k-1} | \\
& \leq (1- \tilde \beta ) \max\{e^{\eta' (t-k-1)}, \frac{1}{2}e^{\eta'(t-k-1)}\}^{\tilde \beta} e^{-\eta(t-k-1)} \leq (1- \tilde \beta) e^{-(\eta - \eta') (t-k-1)}
\end{aligned}$$ and $ |H_t^{1 - \tilde \beta} H_{t-k-1}^{ 1 - \tilde \beta}| \geq 2^{\frac{1}{1+\beta}} e^{-2\eta'(t-k-1)}$. Thus, there exists $C_{2.2}'> 0$ such that for all $k \geq 0, t\geq k+1$, almost surely $$\begin{aligned}
K_{t,k}
\leq C_{2.2}' e^{-(\eta - 3\eta')(t-k-1)}.
\end{aligned}$$ Now, by Lemma \[lem: control of mgtrs\], there exists $C''_{2.2}>0$ such that for all $k\geq 0$ and $t\geq k+1$, $$\begin{aligned}
\label{eq: Y in D}
& \mathbb{\widetilde{P}}_{\mu} [|Y_{t,k}| ; \mathcal{D}_{t,k} ]
\leq C_{2.2}' e^{\alpha (\tilde \beta - 1)t} \mathbb{\widetilde{P}}_{\mu} [ | \mathcal{I}_{t-k-1}^{t-k}X_t(f)| ] e^{-(\eta - 3\eta')(t-k-1)} \\
& \leq \frac{C_{2.2}' } {\mathbb{P}_{\mu}(D^c)} e^{ \alpha (\tilde \beta - 1)t} \|\mathcal{I}_{t-k-1}^{t-k} X_t(f)\|_{\mathbb P_\mu; 1+\gamma} e^{-(\eta - 3\eta')(t-k - 1)} \\
& \leq C_{2.2}'' e^{\alpha(\tilde \beta - \tilde \gamma)t} e^{ (\alpha \tilde \gamma - \kappa_f b)k} e^{-(\eta - 3\eta')(t-k)}
\leq C_{2.2}'' e^{\alpha(\tilde \beta - \tilde \gamma)(t-k)} e^{-(\eta - 3\eta')(t-k)},
\end{aligned}$$ as desired in this step. In the last inequality, we used the fact that $f\in \mathcal C_s$ and therefore $\alpha \tilde \beta < \kappa_f b$.
*Step 3.* We show that $I^f_2(t)\xrightarrow[t\to \infty]{d} 0$. First fix a $\gamma \in (0,\beta)$ in this step. From the fact that $\kappa_f b -\alpha \tilde \gamma > \alpha (\tilde \beta - \tilde \gamma)$, we can choose $\epsilon >0$ small enough so that $q:=\kappa_fb- \alpha \tilde \gamma > \alpha (\tilde \beta - \tilde \gamma) + 2\epsilon (1 - \tilde \beta)$. Now writing $\mathcal{E}_t:=\{\|X_t\|>e^{(\alpha-\epsilon) t}\}$, according to Proposition \[lem: control of XT\], there exist $C_3, \delta_3>0$ such that $$\begin{aligned}
\mathbb{\widetilde{P}}_{\mu}(\mathcal{E}^c_t)
\leq \frac{1}{\mathbb{P}_{\mu}(D^c)}\mathbb{P}_{\mu}(0<e^{-\alpha t}\|X_t\|\leq e^{-\epsilon t})\leq C_3e^{-\delta_3 t}
, \quad t\geq0.
\end{aligned}$$ Therefore, $$\begin{aligned}
\label{Theorem123}
|\mathbb{\widetilde{P}}_{\mu}[e^{i I^f_2(t)}-1;\mathcal{E}^c_t]|
\leq 2\mathbb{\widetilde{P}}_{\mu}(\mathcal{E}^c_t)
\leq 2C_3e^{-\delta_3 t},
\quad t\geq t_0.
\end{aligned}$$ According to Lemma \[lem: control of mgtrs\], there exist $C_3',C_3'',C_3'''>0$ such that for each $t\geq t_0 >1$, $$\begin{aligned}
& |\mathbb{\widetilde{P}}_{\mu} [ (e^{i I^f_2(t)}-1);\mathcal{E}_t]|
\leq \mathbb{\widetilde{P}}_{\mu} [ |I^f_2(t)|;\mathcal{E}_t] \\
& \leq ( e^{(\alpha-\epsilon) t} )^{\tilde \beta - 1}\Big(\sum_{k=\lceil t-\ln t \rceil}^{\lfloor t \rfloor - 1}\mathbb{\widetilde{P}}_{\mu} [| \mathcal{I}_{t-k-1}^{t-k} X_t(f) |] + \mathbb{\widetilde{P}}_{\mu}[| \mathcal{I}_{0}^{t-\lfloor t\rfloor} X_t(f)|]\Big) \\
& \leq ( e^{(\alpha-\epsilon) t} )^{\tilde \beta - 1}\Big(\sum_{k=\lceil t-\ln t \rceil}^{\lfloor t \rfloor - 1}\|\mathcal{I}_{t-k-1}^{t-k} X_t(f) \|_{\mathbb P_\mu; 1+\gamma} + \|\mathcal I_0^{t-\lfloor t \rfloor} X_t(f)\|_{\mathbb P_\mu;1+\gamma}\Big) \\
& \leq C_3' e^{\alpha (\tilde \beta - \tilde \gamma)t} e ^{\epsilon (1-\tilde \beta) t}\sum_{k=\lceil t-\ln t \rceil}^{\lfloor t \rfloor} e^{(\alpha\tilde \gamma-\kappa_f b)k}
\leq C_3' e^{q t}e^{-\epsilon ( 1 - \tilde \beta)t}\sum_{k=\lceil t-\ln t \rceil}^{\lfloor t \rfloor} e^{-q k}
\\ & \leq C_3'' e^{q(t - \lceil t - \ln t\rceil)}e^{-\epsilon(1 - \tilde \beta) t}
\leq C_3'' t^q e^{- \epsilon(1 - \tilde \beta) t}.
\end{aligned}$$ From this and , we get that $\widetilde {\mathbb P}_\mu[e^{i I^f_2(t)}] \xrightarrow[t\to \infty]{} 1$. Note that $I^f_2(t)$ is linear in $f$ so we can replace $f$ with $\theta f$ for $\theta \in \mathbb R$ and get the desired result in this step.
*Step 4.* We will show that $I^f_3(t) \xrightarrow[t\to \infty]{\widetilde {\mathbb P}_\mu \text{-} a.s.} 0$. In fact, we have $$\begin{aligned}
& |I^f_3(t)|
\leq \frac{X_0(|P^\alpha_tf|)}{\|X_t\|^{1 - \tilde \beta }}
\leq \frac{e^{\alpha t - \kappa_f b t}X_0(Qf)}{(e^{\alpha t} H_t)^{1 - \tilde \beta}}
= e^{(\alpha \tilde \beta - k_fb)t} H_t^{\tilde \beta - 1} X_0(Qf)
\xrightarrow[t\to \infty]{\widetilde {\mathbb P}_\mu \text{-} a.s.} 0.
\end{aligned}$$
*Step 5.* Combining Steps 1–4, we complete the proof of Theorem \[thm:M\]..
Central limit theorem for $f \in \mathcal C_c$
----------------------------------------------
Fix $\mu\in \mathcal M_c(\mathbb R^d)$, $f\in \mathcal C_c$ and $t_0 > 1$ large enough so that $ \lceil t - \ln t\rceil \leq \lfloor t \rfloor - 1$ for each $t\geq t_0$. For each $t\geq t_0$, in this proof we write $\theta_t = \|t X_t\|^{\tilde \beta - 1}$, define $I_i^f(t)$ using for $i = 1,2,3$, and set $ I^f_0(t) := t^{\tilde \beta - 1}\sum_{k=0}^{\lfloor t-\ln t \rfloor} \Upsilon_{t-k-1}^{T_{k} \tilde f}$, where $\tilde f = e^{\alpha(\tilde \beta - 1)} f$.
*Step 1.* We show that $I^f_0(t) \xrightarrow[t\to \infty]{d} \zeta^f$. In fact, for each $t \geq t_0( > 1)$ we have $t^{\tilde \beta - 1} < 1$; and therefore, for each $k \in \mathbb Z_+$, we have $t^{\tilde \beta - 1} T_{k+1} f \in \mathcal P_f:=\{\theta T_n f: n \in \mathbb Z_+, \theta \in [-1,1]\}$. Therefore from Proposition \[cor:MI\] and that $\tilde \beta - 1 = -\frac{1}{1+\beta}$ we get that there exist $C_1,\delta_1 > 0$ such that $$\begin{aligned}
\Big|\mathbb{\widetilde{P}}_{\mu} [e^{i I^f_0(t)} ]-\exp\Big(\frac{1}{t}\sum_{k=0}^{\lfloor t-\ln t \rfloor} \langle Z_1T_{k}\tilde f, \varphi\rangle \Big)\Big|
\leq C_1 e^{-\delta_1(t - \lfloor t - \ln t\rfloor)}
\leq \frac{C_1}{t^{\delta_1}},
\quad t\geq t_0.
\end{aligned}$$ Since $f \in \mathcal C_c\setminus \{0\}$, we have $T_k \tilde f = \tilde f$ for each $k \in \mathbb Z_+$. Similar to the argument in we have $$\begin{aligned}
\label{CLT:C:eq:m}
\lim_{t\to \infty} \frac{1}{t}\sum_{k=0}^{\lfloor t-\ln t \rfloor} \langle Z_1 T_{k}\tilde f, \varphi\rangle
= \langle Z_1 \tilde f,\varphi \rangle
= \langle (-if)^{1+\beta}, \varphi \rangle
= m[f].
\end{aligned}$$ Therefore $\mathbb {\widetilde P}_\mu[e^{i I^f_0(t)}] \xrightarrow[t\to \infty]{} e^{m[f]}$. The desired result in this step follows.
*Step 2.* We show that $ I^f_1(t) - I^f_0 (t) \xrightarrow[t\to \infty]{d} 0$. In fact, similar to Step 2 in the proof of Theorem \[thm:M\].(\[thm:M:1\]), we have is valid with $ Y_{t,k} := \exp(i t^{\tilde \beta - 1} \Upsilon_{t-k-1}^{T_{k}\tilde f} - i\theta_t \mathcal I_{t-k-1}^{t-k} X_t(f)) - 1$. Similarly, we claim that there exist $C_2, \delta_2>0$ such that $\widetilde {\mathbb P}_\mu [|Y_{t,k}|] \leq C_2 e^{-\delta_2 (t-k-1)}$ for all $k\in \mathbb N$ and $t\geq k+1$, and then the desired result in this step follows.
We will show the claim above in the following substeps 2.1 and 2.2. First we choose $\gamma \in (0,\beta)$ close enough to $\beta$ so that there exist $\eta,\eta'>0$ with $ \alpha \tilde \gamma > \eta > \eta - 3\eta' > \alpha \tilde \beta - \alpha \tilde \gamma > 0$; and define, for $k \in \mathbb N$ and $t\geq k+1$, $ \mathcal{D}_{t,k} := \{|H_t-H_{t-k-1}| \leq e^{-\eta (t-k-1)}, H_{t-k-1}> 2e^{-\eta' (t-k-1)}\}$.
*Substep 2.1.* Similar to Substep 2.1 in the proof of Theorem \[thm:M\].(\[thm:M:1\]), there exist $C_{2.1},\delta_{2.1} >0$ such that for all $k \in \mathbb N$ and $t\geq k+1$, $\mathbb{\widetilde{P}}_{\mu}[|Y_{t,k}|;\mathcal{D}^c_{t,k}] \leq C_{2.1} e^{-\delta_{2.1} (t-k)}$. We omit the details.
*Substep 2.2.* We will show that there exist $C_{2.2},\delta_{2.2} > 0$ such that for all $k\in \mathbb N$ and $t\geq k+1$, $ \mathbb{\widetilde{P}}_{\mu}[|Y_{t,k}|;\mathcal{D}_{t,k}] \leq C_{2.2} e^{-\delta_{2.2} (t-k)}.$ In fact, noticing that for $f\in \mathcal C_c$ and $k\in \mathbb Z_+$, we have $T_kf = e^{\alpha (\tilde \beta - 1 )k}P_k^\alpha $; and therefore for all $k\in \mathbb Z_+$ and $t \geq k + 1$, $$t^{\tilde \beta - 1} \Upsilon_{t-k-1}^{T_{k} \tilde f}
= \frac{X_{t-k}(T_{k} \tilde f) - X_{t -k-1}(P_1^\alpha T_{k} \tilde f)}{\|t X_{t-k-1}\|^{1-\tilde \beta}}
= \frac{\mathcal I_{t - k - 1}^{t - k} X_t(f)}{\|te^{\alpha (k+1)}X_{t-k-1} \|^{1 -\tilde \beta}}.$$ The rest is similar to Substep 2.2 in the proof of Theorem \[thm:M\].(\[thm:M:2\]). We omit the details.
*Step 3.* We show that $ I^f_2(t)\xrightarrow[t\to \infty]{d} 0$. In fact, writing $\mathcal{E}_t:=\{\|X_t\|>t^{-1/2}e^{\alpha t}\}$, according to Proposition \[lem: control of XT\], there exist $C_3, \delta_3>0$ such that $$\mathbb{\widetilde{P}}_{\mu}(\mathcal{E}^c_t)
\leq \frac{1}{\mathbb{P}_{\mu}(D^c)}\mathbb{P}_{\mu}(0<e^{-\alpha t}\|X_t\|\leq t^{-1/2})\leq C_3( t^{-\delta_3}+e^{-\delta_3 t})
, \quad t\geq0.$$ Therefore, $$\begin{aligned}
\label{Theorem123a}
|\mathbb{\widetilde{P}}_{\mu}[e^{i I^f_2(t)}-1;\mathcal{E}^c_t]|
\leq 2\mathbb{\widetilde{P}}_{\mu}(\mathcal{E}^c_t)
\leq C_3(t^{-\delta_3}+e^{-\delta_3 t}),
\quad t\geq t_0.
\end{aligned}$$ Choose a $\gamma\in (0,\beta)$ close enough to $\beta$ so that $\alpha(\tilde \beta - \tilde \gamma) \leq \frac{1}{2}(1- \tilde \beta)$. According to Lemma \[lem: control of mgtrs\], there exist $C_3',C_3'',C_3'''>0$ such that for each $t\geq t_0 (>1)$, $$\begin{aligned}
& |\mathbb{\widetilde{P}}_{\mu} [ (e^{i I^f_2(t)}-1)\mathbf{1}_{\mathcal{E}_t}]|
\leq \mathbb{\widetilde{P}}_{\mu} [ |I^f_2(t)|\mathbf{1}_{\mathcal{E}_t}] \\
& \leq (t^{\frac{1}{2}} e^{\alpha t} )^{\tilde \beta - 1}\Big(\sum_{k=\lceil t-\ln t \rceil}^{\lfloor t \rfloor - 1}\mathbb{\widetilde{P}}_{\mu} [| \mathcal{I}_{t-k-1}^{t-k} X_t(f) |] + \mathbb{\widetilde{P}}_{\mu}[| \mathcal{I}_{0}^{t-\lfloor t\rfloor} X_t(f)|]\Big) \\
& \leq C_3' t^{\frac{1}{2}(\tilde \beta - 1)} e^{\alpha(\tilde \beta - 1)t}\Big(\sum_{k=\lceil t-\ln t \rceil}^{\lfloor t \rfloor - 1}\|\mathcal{I}_{t-k-1}^{t-k} X_t(f) \|_{\mathbb P_\mu; 1+\gamma} + \|\mathcal I_0^{t-\lfloor t \rfloor} X_t(f)\|_{\mathbb P_\mu;1+\gamma}\Big) \\
& \leq C_3' t^{\frac{1}{2}(\tilde \beta - 1)} e^{\alpha (\tilde \beta - \tilde \gamma)t}\sum_{k=\lceil t-\ln t \rceil}^{\lfloor t \rfloor} e^{(\alpha\tilde \gamma-\kappa_f b)k}
= C_3' t^{\frac{1}{2}(\tilde \beta - 1)} e^{\alpha(\tilde \beta - \tilde \gamma) t}\sum_{k=\lceil t-\ln t \rceil}^{\lfloor t \rfloor} e^{-\alpha (\tilde \beta -\tilde \gamma) k}
\\ & \leq C_3'' t^{\frac{1}{2}(\tilde \beta - 1)} e^{\alpha (\tilde \beta - \tilde \gamma)(t - \lceil t - \ln t\rceil)}
\leq C_3'' t^{\frac{1}{2}(\tilde \beta - 1)} t^{\alpha (\tilde \beta - \tilde \gamma)}.
\end{aligned}$$ From this and , we get the desired result in this step.
*Step 4.* Similar to Step 4 in the proof of Theorem \[thm:M\].(\[thm:M:1\]), we can verify that $I_3(t) \xrightarrow[t\to \infty]{\widetilde {\mathbb P}_\mu \text{-} a.s.} 0$. We omit the details.
*Step 5.* Combining Steps 1–4, we complete the proof of Theorem \[thm:M\]..
Central limit theorem for $f\in \mathcal C_l$ {#sec: large rate clt}
---------------------------------------------
Fix $\mu \in \mathcal M_c(\mathbb R^d)$ and $f \in \mathcal C_l$. Define $\mathcal N:= \{p\in \mathbb Z_+^d: \alpha \tilde \beta > |p|b\}$. In this proof we write for each $t\geq 0$, $$\begin{aligned}
& \frac{X_t(f) - \sum_{p\in \mathbb Z_+^d: \alpha \tilde \beta \geq |p|b} \langle f,\phi_p\rangle_\varphi e^{(\alpha - |p|b)t}H_\infty^p}{\|X_t\|^{1- \tilde \beta}}
= \sum_{p\in \mathcal N}\frac{ \langle f,\phi_p\rangle_\varphi [X_t(\phi_p) - e^{(\alpha - |p|b)t}H_\infty^p]}{\|X_t\|^{1- \tilde \beta}}
\\& = \sum_{p \in \mathcal N} \frac{\langle f,\phi_p\rangle_\varphi e^{(\alpha - |p|b)t}(H_t^p - H_\infty^p)}{\|X_t\|^{1- \tilde \beta}}
= \sum_{k=0}^\infty \sum_{p \in \mathcal N} \langle f,\phi_p\rangle_\varphi e^{(\alpha - |p|b)t}\frac{ H_{t+k}^p - H_{t+k+1}^p}{\|X_t\|^{1- \tilde \beta}}
\\ & =: \sum_{k=0}^\infty \widetilde \Upsilon_{t,k}
= \Big(\sum_{k = 0}^{\lfloor t^2 \rfloor} \widetilde \Upsilon_{t,k} \Big) + \Big(\sum_{k = \lceil t^2 \rceil}^\infty \widetilde \Upsilon_{t,k}\Big)
= : I^f_1(t) + I^f_2(t),
\end{aligned}$$ and $I^f_0(t):= \sum_{k = 0}^{\lfloor t^2 \rfloor} \Upsilon_{t+k}^{- T_k \tilde f}$ where $\tilde f := \sum_{p\in \mathcal N} e^{-(\alpha - |p|b)}\langle f, \phi_p \rangle_\varphi \phi_p$.
*Step 1.* We show that $I^f_0(t) \xrightarrow [t\to \infty]{d} \zeta^{-f}$. In fact, since $T_k\tilde f \in \mathcal P_{\tilde f}$ for each $k\in \mathbb Z_+$, from Corollary \[cor:MI\] we have $\widetilde{ \mathbb P}_\mu[e^{i I_0^f(t)}]\xrightarrow[t\to \infty]{}\exp\{\sum_{k=0}^\infty \langle Z_1T_k(-\tilde f),\varphi\rangle\}$. Using and the fact that $\varphi(x)dx$ is the invariant probability of the semigroup $(P_t)_{t\geq 0}$ we have $$\begin{aligned}
\label{eq:PM:CLTS:2a}
& \sum_{k=0}^\infty \langle Z_1 T_{k} (-\tilde f), \varphi \rangle
= \sum_{k=0}^\infty \int_0^1 \langle P_u^\alpha ((iP_{1 - u}^\alpha T_k \tilde f)^{1+\beta}), \varphi\rangle ~du
\\& = \sum_{k=0}^\infty \int_0^1 e^{\alpha u} \langle (iP_{1 - u}^\alpha T_{k}\tilde f)^{1+\beta}, \varphi \rangle ~du
\\& = \sum_{k=0}^\infty \int_0^1 \langle (iT_{k+ u} f)^{1+\beta}, \varphi\rangle~du
= \int_0^\infty \langle (iT_{u} f)^{1+\beta}, \varphi\rangle~du = m[-f].
\end{aligned}$$ The result in this step follows.
*Step 2.* We show that $I^f_1(t) - I^f_0(t) \xrightarrow[t\to \infty]{d} 0$. In fact, by [@Durrett2010Probability Lemma 3.4.3] we have, for each $t\geq 0$, that $|\widetilde {\mathbb P}_{\mu}[e^{i(I_{1}^{f}(t) - I_0^f(t))} - 1]| \leq \sum_{k=0}^{\lfloor t^2 \rfloor} \widetilde {\mathbb {P}}_\mu[|Y_{t,k}|]$ where $Y_{t,k} := e^{i(\widetilde {\Upsilon}_{t,k} - \Upsilon_{t+k}^{-T_{k}\widetilde {f}})} - 1. $ We claim that there exist $C_2, \delta_2>0$ such that $\widetilde {\mathbb {P}}_\mu[|Y_{t,k}|] \leq C_2e^{-\delta_2 t}$ for all $t\geq 0$ and $k \in \mathbb Z_+$. Then $|\widetilde {\mathbb P}_{\mu}[e^{i(I_{1}^{f}(t) - I_0^f(t))} - 1]| \leq (t^2+1)C_2e^{-\delta_2 t}$ which completes this step.
We will show the claim above in the following substeps 2.1 and 2.2. First we choose $\gamma\in(0, \beta)$ close enough to $\beta$ so that $\alpha \tilde \gamma > |p|b$ for each $p\in \mathcal N$; and even closer so that there exist $\eta,\eta'>0$ satisfying $\alpha \tilde \gamma > \eta>\eta - 3\eta'> \alpha (\tilde \beta - \tilde \gamma)>0$. We also define $\mathcal{D}_{t,k} :=\{|H_t-H_{t+k}|\leq e^{-\eta t}, H_{t}> 2e^{-\eta' t}\}$.
*Substep 2.1.* Similar to Substep 2.1 in the proof of Theorem \[thm:M\].(\[thm:M:1\]), we have that there exist $C_{2.1},\delta_{2.1} >0$ such that for all $k \in \mathbb Z_+$ and $t\geq 0$, $ \mathbb{\widetilde{P}}_{\mu}[|Y_{t,k}|;\mathcal{D}^c_{t,k}] \leq C_{2.1} e^{-\delta_{2.1} t}$. We omit the details.
*Substep 2.2.* We show that there exist $C_{2.2}, \delta_{2.2}>0$ such that for all $k \in \mathbb Z_+$ and $t\geq 0$, we have $\widetilde {\mathbb {P}}_\mu[|Y_{t,k}|; \mathcal D_{t,k}]\leq C_{2.2}e^{- \delta_{2,2} t}$. In fact, it can be verified that for all $k \in \mathbb Z_+$ and $t\geq 0$, $$\begin{aligned}
& \Upsilon_{t+k}^{-T_k\tilde f}
= \frac{X_{t+k}(P^\alpha_1T_k\tilde f) - X_{t+k+1}(T_k \tilde f)}{\|X_{t+k}\|^{1 - \tilde \beta}}
\\& = \sum_{p\in \mathcal N}
\langle\tilde f,\phi_p\rangle_\varphi e^{-(\alpha \tilde \beta - |pb|)k}\frac{ X_{t+k}(P_1^\alpha \phi_p) - X_{t+k+1}(\phi_p)}{\|X_{t+k}\|^{1 - \tilde \beta}}
\\& = \sum_{p\in \mathcal N}
\langle f,\phi_p\rangle_\varphi e^{(\alpha -|p|b)t}\frac{H_{t+k}^p-H_{t+k+1}^p }{\|e^{-\alpha k}X_{t+k}\|^{1 - \tilde \beta}}.
\end{aligned}$$ Therefore for all $k\in \mathbb Z_+$ and $t\geq 0$, $$\begin{aligned}
&|Y_{t,k}| \mathbf 1_{\mathcal D_{t,k}}
\leq \Big( \sum_{p\in \mathcal N}|\langle f,\phi_p\rangle_\varphi| e^{(\alpha -|p|b)t} | H_{t+k}^p-H_{t+k+1}^p |\Big) \Big( \frac{1}{\|X_t\|^{1 - \tilde \beta}} - \frac{1}{\|e^{-\alpha k}X_{t+k}\|^{1 - \tilde \beta}} \Big)\mathbf 1_{\mathcal D_{t,k}}.
\\ &= \Big( \sum_{p\in \mathcal N}|\langle f,\phi_p\rangle_\varphi| e^{(\alpha -|p|b)t} | H_{t+k}^p-H_{t+k+1}^p |\Big)e^{\alpha (\tilde \beta - 1)t} K_{t,k}
\\ &= \Big( \sum_{p\in \mathcal N}|\langle f,\phi_p\rangle_\varphi| e^{(\alpha \tilde \beta -|p|b)t} | H_{t+k}^p-H_{t+k+1}^p |\Big) K_{t,k},
\end{aligned}$$ where $$K_{t,k}
:= \Big| \frac {H_t^{1- \tilde \beta} - H_{t+k}^{1 - \tilde \beta}} {H_t^{1 - \tilde \beta} H_{t+k}^{ 1- \tilde \beta }} \Big| \mathbf{1}_{\mathcal{D}_{t,k}}.$$ Similar to Substep 2.2 in the proof of Theorem \[thm:M\].(\[thm:M:1\]), we can verify that for all $k\in \mathbb Z_+$ and $t\geq 0$, almost surely $K_{t,k} \leq C_{2.2}'' e^{- (\eta - 3\eta')t}$. From this and Lemma \[lem: control of wt\] we know that there exists $C'''_{2.2}$ such that for all $k\in \mathbb Z_+$ and $t\geq 0$, $$\begin{aligned}
& \widetilde{\mathbb P}_\mu[|Y_{t,k}|; \mathcal D_{t,k}]
\leq \mathbb P_\mu(D)^{-1}\mathbb P_\mu[ |Y_{t,k}| ;\mathcal D_{t,k} ]
\\ & \leq \mathbb P_{\mu}(D)^{-1} C_{2.2}'' e^{- (\eta - 3\eta') t}\sum_{p\in \mathcal {N}} |\langle f,\phi_p\rangle_\varphi| e^{(\alpha \tilde \beta -|p|b)t} \mathbb P_\mu[| H_{t+k}^p-H_{t+k+1}^p |]
\\ & \leq \mathbb P_{\mu}(D)^{-1} C_{2.2}'' e^{- (\eta - 3\eta') t}\sum_{p\in \mathcal {N}} |\langle f,\phi_p\rangle_\varphi| e^{(\alpha \tilde \beta -|p|b)t} \| H_{t+k}^p-H_{t+k+1}^p \|_{\mathbb P_\mu; 1+\gamma}
\\&\leq \mathbb P_{\mu}(D)^{-1} C_{2.2}'' e^{- (\eta - 3\eta') t}\sum_{p\in \mathcal N} |\langle f,\phi_p\rangle_\varphi| e^{(\alpha \tilde \beta -|p|b)t} e^{-(\alpha \tilde \gamma - |p|b)(t+k)} \\
& \leq C_{2.2}''' e^{- (\eta - 3\eta') t} e^{(\alpha \tilde \beta - \alpha \tilde \gamma)t},
\end{aligned}$$ as desired in this substep.
*Step 3.* We show that $I^f_2(t) \xrightarrow[t\to \infty]{d} 0$. In order to do this, choose an $\epsilon \in (0,\alpha)$ and a $\gamma \in (0,\beta)$ close enough to $\beta$ so that for each $p\in \mathcal N$, it holds that $\alpha \tilde \gamma > |p|b$. Define $\mathcal E_t:= \{\|X_t\| > e^{(\alpha - \epsilon)t}\}$. According to Proposition \[lem: control of XT\], there exist $C_3, \delta_3 > 0$ such that for each $t\geq 0$, $|\widetilde {\mathbb {P}}_\mu[e^{i I_2^f(t)} - 1; \mathcal E_t^c]|\leq 2\widetilde {\mathbb {P}}_\mu(\mathcal E_t^c) \leq C_3 e^{- \delta_3 t}$. On the other hand, according to Lemma \[lem: control of wt\], we know that there exist $C_3',C_3''>0$ and $\delta_3'>0$ such that $$\begin{aligned}
& |\widetilde {\mathbb {P}}_\mu[e^{i I_{2}^{f}(t)} - 1; \mathcal {E}_t]|
\leq \widetilde {\mathbb {P}}_\mu[ | I_{2}^{f}(t)|; \mathcal {E}_t]
\leq \sum_{k = \lceil t^2\rceil}^\infty \widetilde {\mathbb {P}}_\mu[ |\widetilde {\Upsilon}_{t,k}|; \mathcal {E}_t]
\\ & \leq \mathbb P_\mu(D^c)^{-1} \sum_{k = \lceil t^2\rceil}^\infty \sum_{p \in \mathcal N} |\langle f,\phi_p\rangle_\varphi| e^{(\alpha - |p|b)t}\mathbb {P}_\mu\Big[\frac{ |H_{t+k}^p - H_{t+k+1}^p|}{\|X_t\|^{1- \tilde \beta}}; \mathcal E_t\Big]
\\ & \leq \mathbb P_\mu(D^c)^{-1} e^{(\alpha - \epsilon) (\tilde \beta - 1) t} \sum_{k = \lceil t^2\rceil}^\infty \sum_{p \in \mathcal N} |\langle f,\phi_p\rangle_\varphi| e^{(\alpha - |p|b)t}\|H_{t+k}^p - H_{t+k+1}^p\|_{\mathbb P_\mu; 1+\gamma}
\\ & \leq C_3' e^{(\alpha - \epsilon) (\tilde \beta - 1) t} \sum_{k = \lceil t^2\rceil}^\infty \sum_{p \in \mathcal N} |\langle f,\phi_p\rangle_\varphi| e^{(\alpha - |p|b)t} e^{- (\alpha \tilde \gamma - |p|b)(t+k)}
\\ & = C_3'' e^{ \alpha (\tilde \beta - \tilde \gamma) t } e^{ \epsilon (1 - \tilde \beta) t}e^{- \delta'_3 t^2}.
\end{aligned}$$ To sum up we have that $\widetilde {\mathbb P}_\mu[e^{iI_2^f(t)}] \xrightarrow[t\to \infty]{} 1$, which completes this step.
*Step 4.* Combining Steps 1–3, we complete the proof of Theorem \[thm:M\]..
=
Analytic facts
--------------
In this subsection, we collect some useful analytic facts.
\[lem: estimate of exponential remaining\] For $z\in \mathbb C_+$, we have $$\begin{aligned}
\label{eq: estimate of exponential remaining}
\Big|e^{-z} - \sum_{k=0}^n \frac{(-z)^k}{k!} \Big|
\leq \frac{|z|^{n+1}}{(n+1)!} \wedge \frac{2|z|^{n}}{n!}, \quad n\in \mathbb Z_+.
\end{aligned}$$
Notice that $|e^{-z}| = e^{- \operatorname{Re} z} \leq 1$. Therefore, $ |e^{-z} - 1| = \Big| \int_0^1 e^{-\theta z} z d\theta\Big| \leq |z|. $ Also, notice that $|e^{-z} - 1| \leq |e^{-z}|+1 \leq 2$. Thus is true when $n = 0$. Now, suppose that is true when $n = m$ for some $m \in \mathbb Z_+$. Then $$\begin{aligned}
&\Big|e^{-z} - \sum_{k=0}^{m+1} \frac{(-z)^k}{k!}\Big|
= \Big| \int_0^1\Big(e^{-\theta z} - \sum_{k=0}^m \frac{(-\theta z)^k}{k!} \Big) z d\theta \Big| \\
& \quad \leq \Big(\int_0^1 \frac{|\theta z|^{m+1}}{(m+1)!} |z| d\theta\Big) \wedge \Big(\int_0^1 \frac{2|\theta z|^{m}}{m!} |z| d\theta\Big)
= \frac{|z|^{m+2}}{(m+2)!} \wedge \frac{2|z|^{m+1}}{(m+1)!},
\end{aligned}$$ which says that is true for $n = m + 1$.
\[lem: extension lemma for branching mechanism\] Suppose that $\pi$ is a measure on $(0,\infty)$ with $\int_{(0,\infty)} (y \wedge y^2) \pi(dy)< \infty$. Then the functions $$\begin{aligned}
& h (z)
= \int_{(0,\infty)} (e^{-zy} - 1 + zy) \pi(dy), \quad z \in \mathbb C_+, \\
& h'(z)
= \int_{(0,\infty)}(1- e^{-zy})y \pi(dy), \quad z \in \mathbb C_+
\end{aligned}$$ are well defined, continuous on $\mathbb C_+$ and holomorphic on $\mathbb C_+^0$. Moreover, $$\frac{h(z)-h(z_0)}{z-z_0}
\xrightarrow[\mathbb C_+\ni z \to z_0]{} h'(z_0),\quad z_0 \in \mathbb C_+.$$
It follows from Lemma \[lem: estimate of exponential remaining\] that $h$ and $h'$ are well defined on $\mathbb C_+$. According to [@SchillingSongVondravcek2010Bernstein Theorems 3.2. & Proposition 3.6], $h'$ is continuous on $\mathbb C_+$ and holomorphic on $\mathbb C_+^0$.
It follows from Lemma \[lem: estimate of exponential remaining\] that, for each $z_0 \in \mathbb C_+$, there exists $C>0$ such that for $z \in \mathbb C_+$ close enough to $z_0$ and any $y>0$, $$\begin{aligned}
& \Big| \frac{e^{-zy} - e^{-z_0 y}+(z-z_0) y}{z-z_0} \Big|
= \frac{1}{|z-z_0|}\Big| \int_0^1 (-y e^{-(\theta z+(1-\theta)z_0)y}+y)(z-z_0)d\theta\Big| \\
& \leq y\int_0^1 |1-e^{-(\theta z +(1-\theta)z_0)y}| d\theta
\leq (2y) \wedge\Big( y^2\int_0^1|\theta z+(1-\theta)z_0|d\theta\Big)
\leq C(y\wedge y^2).
\end{aligned}$$ Using this and the dominated convergence theorem, we have $$\begin{aligned}
& \frac{h(z)-h(z_0)}{z-z_0} = \int_{(0,\infty)} \frac{e^{-zy}+zy -(e^{-z_0 y}+z_0 y)}{z-z_0} \pi(dy) \\
& \xrightarrow[\mathbb C_+\ni z\to z_0]{} \int_{(0,\infty)}(1 - e^{-z_0 y} )y\pi(dy) = h'(z_0),
\end{aligned}$$ which says that $h$ is continuous on $\mathbb C_+$ and holomorphic on $\mathbb C_+^0$.
For each $z\in \mathbb C\setminus (-\infty,0]$, we define $ \log z := \log |z| + i \arg z$ where $\arg z \in (-\pi,\pi)$ is uniquely determined by $ z = |z|e^{i \arg z}$. For all $z\in \mathbb C\setminus (-\infty,0]$ and $\gamma \in \mathbb C$, we define $ z^\gamma := e^{\gamma \log z}. $ Then it is known, see [@SteinShakarchi2003Complex Theorem 6.1] for example, that $z\mapsto \log z$ is holomorphic in $\mathbb C\setminus (-\infty,0]$. Therefore, for each $\gamma \in \mathbb C$, $z\mapsto z^\gamma$ is holomorphic in $\mathbb C\setminus (-\infty,0]$. (We use the convention that $0^\gamma := \mathbf 1_{\gamma = 0}$.) Using the definition above we can easily show that $(z_1z_0)^\gamma = z_1^\gamma z_0^\gamma$ provided $\arg (z_1z_0)=\arg (z_1) + \arg(z_0)$.
It is known, see, for instance, [@SteinShakarchi2003Complex Theorem 6.1.3] and the remark following it, that the Gamma function $\Gamma$ has an unique analytic extension in $\mathbb C\setminus\{0, -1,-2,\dots\}$ and that $$\Gamma(z+1)
= z \Gamma(z),\quad z\in \mathbb C\setminus\{0, -1,-2,\dots\}.$$ Using this recursively, one gets that $$\begin{aligned}
\label{eq: definition of Gamma function}
\Gamma(x)
:= \int_0^\infty t^{x-1} \Big(e^{-t} - \sum_{k=0}^{n-1} \frac{(-t)^k}{k!}\Big) dt,
\quad -n< x< -n+1, n\in \mathbb N.\end{aligned}$$
Fix a $\beta \in (0,1)$. Using the uniqueness of holomorphic extension and Lemma \[lem: extension lemma for branching mechanism\], we get that $$\begin{aligned}
z^{\beta}
= \int_0^\infty (e^{-zy}-1) \frac{dy}{\Gamma(-\beta)y^{1+\beta}},
\quad z\in \mathbb C_+,\end{aligned}$$ and similarly that $$\begin{aligned}
\label{eq: stable branching on C+}
z^{1+\beta}
= \int_0^\infty (e^{-zy}-1+zy)\frac{dy}{\Gamma(-1-\beta)y^{2+\beta}},
\quad z\in \mathbb C_+.\end{aligned}$$ Lemma \[lem: extension lemma for branching mechanism\] also says that the derivative of $z^{1+\beta}$ is $(1+\beta)z^{\beta}$ on $\mathbb C^0_+$.
\[lem: Lip of power function\] For all $z_0,z_1 \in \mathbb C_+$, we have $$\begin{aligned}
\label{eq: Lip of power function}
|z_0^{1+\beta} - z_1^{1+\beta}|
\leq (1+\beta)(|z_0|^{\beta}+|z_1|^{\beta})|z_0 - z_1|.\end{aligned}$$
Since $z^{1+\beta}$ is continuous on $\mathbb C_+$, we only need to prove the lemma assuming $z_0,z_1 \in \mathbb C^0_+$. Notice that $$\begin{aligned}
\label{eq: upper bound for beta power of z}
|z^\beta|
= |e^{\beta \log |z| +i\beta \operatorname {arg}z}| = e^{\beta \log |z|} = |z|^\beta,
\quad z \in \mathbb C\setminus (-\infty, 0].
\end{aligned}$$ Define a path $\gamma: [0,1] \to \mathbb C^0_+$ such that $$\gamma(\theta)
= z_0 (1-\theta) + \theta z_1,
\quad \theta \in [0,1].$$ Then, we have $$\begin{aligned}
|z_0^{1+\beta} - z_1^{1+\beta}|
& \leq (1+\beta) \int_0^1 |\gamma(\theta)^{\beta}|\cdot |\gamma'(\theta)|d\theta
\leq (1+\beta) \sup_{\theta \in [0,1]} |\gamma(\theta)|^{\beta} \cdot |z_1-z_0| \\
& \leq (1+\beta) ( |z_1|^{\beta}+|z_0|^{\beta} ) |z_1-z_0|.
\qedhere
\end{aligned}$$
Suppose that $\varphi(\theta)$ is a continuous function from $\mathbb R$ into $\mathbb C$ such that $\varphi(0) = 1$ and $\varphi(\theta) \neq 0$ for all $\theta \in \mathbb R$. Then according to [@Sato2013Levy Lemma 7.6], there is a unique continuous function $f(\theta)$ from $\mathbb R$ into $\mathbb C$ such that $f(0) = 0$ and $e^{f(\theta)} = \varphi(\theta)$. Such a function $f$ is called the distinguished logarithm of the function $\varphi$ and is denoted as $\operatorname{Log} \varphi(\theta)$. In particular, when $\varphi$ is the characteristic function of an infinitely divisible random variable $Y$, $\operatorname{Log} \varphi(\theta)$ is called the Lévy exponent of $Y$. This distinguished logarithm should not be confused with the $\log$ function defined on $\mathbb C\setminus (-\infty, 0]$. See the paragraph immediately after [@Sato2013Levy Lemma 7.6].
Feynman-Kac formula with complex valued functions {#seq: complex Feynman-Kac transform}
-------------------------------------------------
In this subsection we give a version of the Feynman-Kac formula with complex valued functions. Suppose that $\{(\xi_t)_{t \in [r,\infty)}; (\Pi_{r,x})_{r\in [0,\infty), x\in E}\}$ is a (possibly non-homogeneous) Hunt process in a locally compact separable metric space $E$. We write $$\begin{aligned}
H^{(h)}_{(s,t)}
:= \exp\Big\{\int_s^t h(u,\xi_u) du\Big\},
\quad 0 \leq s \leq t, h \in \mathcal B_b([0,t] \times E,\mathbb C).\end{aligned}$$
\[eq: complex FK\] Let $t \geq 0$. Suppose that $\rho_1, \rho_2\in \mathcal B_b([0,t] \times E, \mathbb C)$ and $f\in \mathcal B_b(E, \mathbb C)$. Then $$\begin{aligned}
\label{eq: expresion of g}
g(r,x)
:= \Pi_{r,x}[ H_{(r,t)}^{(\rho_1+\rho_2)} f(\xi_t)],\quad r \in [0,t], x\in E,
\end{aligned}$$ is the unique locally bounded solution to the equation $$g(r,x)
= \Pi_{r,x} [ H_{(r,t)}^{(\rho_1)} f(\xi_t)]+\Pi_{r,x} \Big[ \int_r^tH_{(r,s)}^{(\rho_1)}\rho_2(s,\xi_s) g(s,\xi_s)~ds \Big],\quad r \in [0,t], x\in E.$$
The proof is similar to that of [@Dynkin1993Superprocesses Lemma A.1.5]. We include it here for the sake of completeness. We first verify that is a solution. Notice that $$\begin{aligned}
\Pi_{r,x} \Big[ \int_r^t | H_{(r,t)}^{(\rho_1)}\rho_2(s,\xi_s) H_{(s,t)}^{(\rho_2)} f(\xi_t)| ~ds \Big]
\leq \int_r^t e^{(t-r)\|\rho_1\|_\infty}e^{(t-s)\|\rho_2\|_\infty}\|\rho_2\|_\infty\|f\|_\infty ~ds
< \infty.
\end{aligned}$$ Also notice that $$\begin{aligned}
\label{eq: crucial for Feynman-Kac}
\frac{\partial}{\partial s} H^{(\rho_2)}_{(s,t)}= -H^{(\rho_2)}_{(s,t)}\rho_2(s,\xi_s),
\quad s\in (0,t).
\end{aligned}$$ Therefore, from the Markov property of $\xi$ and Fubini’s theorem we get that $$\begin{aligned}
& \Pi_{r,x} \Big[ \int_r^tH_{(r,s)}^{(\rho_1)}~(\rho_2 g)(s,\xi_s)~ds \Big]
=\Pi_{r,x} \Big[ \int_r^t H_{(r,s)}^{(\rho_1)}\rho_2(s,\xi_s) \Pi_{s,\xi_s}[ H_{(s,t)}^{(\rho_1+\rho_2)} f(\xi_t)]~ds \Big] \\
& = \Pi_{r,x} \Big[ \int_r^t H_{(r,t)}^{(\rho_1)}\rho_2(s,\xi_s) H_{(s,t)}^{(\rho_2)} f(\xi_t) ~ds \Big]
= \Pi_{r,x} [ H_{(r,t)}^{(\rho_1)}f(\xi_t)(H_{(r,t)}^{(\rho_2)} - 1)] \\
& = g(r,x) - \Pi_{r,x} [ H_{(r,t)}^{(\rho_2)} f(\xi_t)].
\end{aligned}$$ For uniqueness, suppose $\widetilde g$ is another solution. Put $h(r) = \sup_{x\in E}|g(r,x) - \widetilde g(r,x)|$. Then $$h(r)
\leq e^{t\|\rho_1\|_\infty}\|\rho_2\|_\infty \int_r^t h(s)ds,
\quad r\le t.$$ Applying Gronwall’s inequality, we get that $h(r) = 0$ for $r\in [0,t]$.
Superprocesses {#sec: definition of superprocess}
--------------
In this subsection, we will give the definition of a general superprocess. Let $E$ be locally compact separable metric space. Denote by $\mathcal M(E)$ the collection of all the finite measures on $E$ equipped with the topology of weak convergence. For each function $F(x,z)$ on $E\times \mathbb R_+$ and each $\mathbb R_+$-valued function $f$ on $E$, we use the convention: $
F(x,f)
:= F(x,f(x)),
x\in E.
$ A process $X=\{(X_t)_{t\geq 0}; (\mathbf P_\mu)_{\mu \in \mathcal M(E)}\}$ is said to be a $(\xi,\psi)$-superprocess if
- the spatial motion $\xi=\{(\xi_t)_{t\geq 0};(\Pi_x)_{x\in E}\}$ is an $E$-valued Hunt process with its lifetime denoted by $\zeta$;
- the branching mechanism $\psi: E\times[0,\infty) \to \mathbb R$ is given by $$\begin{aligned}
\label{eq: branching mechanism}
\psi(x,z)=
-\rho_1(x) z + \rho_2 (x) z^2 + \int_{(0,\infty)} (e^{-zy} - 1 + zy) \pi(x,dy).\end{aligned}$$ where $\rho_1 \in \mathcal B_b(E)$, $\rho_2 \in \mathcal B_b(E, \mathbb R_+)$ and $\pi(x,dy)$ is a kernel from $E$ to $(0,\infty)$ such that $\sup_{x\in E} \int_{(0,\infty)} (y\wedge y^2) \pi(x,dy) < \infty$;
- $X=\{(X_t)_{t\geq 0}; (\mathbf P_\mu)_{\mu \in \mathcal M(E)}\}$ is an $\mathcal M(E)$-valued Hunt process with transition probability determined by $$\begin{aligned}
\mathbf P_\mu [e^{-X_t(f)}] = e^{-\mu(V_tf)},
\quad t\geq 0, \mu \in \mathcal M(E), f\in \mathcal B^+_b(E),
\end{aligned}$$ where for each $f\in \mathcal B_b(E)$, the function $(t,x)\mapsto V_tf(x)$ on $[0,\infty) \times E$ is the unique locally bounded non-negative solution to the equation $$\begin{aligned}
\label{eq:FKPP_in_definition}
V_tf(x) + \Pi_x \Big[ \int_0^{t\wedge \zeta} \psi(\xi_s,V_{t-s}f)ds \Big]
= \Pi_x [ f(\xi_t)\mathbf 1_{t<\zeta} ],
\quad t \geq 0, x \in E.
\end{aligned}$$
We refer our readers to [@Li2011Measure-valued] for more discussions about the definition and the existence of superprocesses. To avoid triviality, we assume that $\psi(x,z)$ is not identically equal to $-\rho_1(x)z$.
Notice that the branching mechanism $\psi$ can be extended into a map from $E \times \mathbb C_+$ to $\mathbb C$ using . Define $$\begin{aligned}
\psi'(x,z)
:= - \rho_1(x) + 2\rho_2(x) z + \int_{(0,\infty)} (1-e^{-zy})y\pi(x,dy),
\quad x\in E, z\in \mathbb C_+.\end{aligned}$$ Then according to Lemma \[lem: extension lemma for branching mechanism\], for each $x \in E$, $z \mapsto \psi(x,z)$ is a holomorphic function on $\mathbb C_+^0$ with derivative $z \mapsto \psi'(x,z)$. Define $\psi_0(x,z) := \psi(x,z)+ \rho_1(x)z $ and $\psi'_0(x,z) := \psi'(x,z) + \rho_1(x)$.
Denote by $\mathbb W$ the space of $\mathcal M(E)$-valued càdlàg paths with its canonical path denoted by $(W_t)_{t\geq 0}$. We say $X$ is *non-persistent* if $\mathbf P_{\delta_x}(\|X_t\|= 0) > 0$ for all $x\in E$ and $t> 0$. Suppose that $(X_t)_{t\geq 0}$ is non-persistent, then according to [@Li2011Measure-valued Section 8.4], there is a unique family of measures $(\mathbb N_x)_{x\in E}$ on $\mathbb W$ such that (i) $\mathbb N_x (\forall t > 0, \|W_t\|=0) =0$; (ii) $\mathbb N_x(\|W_0 \|\neq 0) = 0$; and (iii) if $\mathcal N$ is a Poisson random measure defined on some probability space with intensity $\mathbb N_\mu(\cdot):= \int_E \mathbb N_x(\cdot )\mu(dx)$, then the superprocess $\{X;\mathbf P_\mu\}$ can be realized by $\widetilde X_0 := \mu$ and $\widetilde X_t(\cdot) := \mathcal N[W_t(\cdot)]$ for each $t>0$. We refer to $(\mathbb N_x)_{x\in E}$ as the *Kuznetsov measures* of $X$.
Semigroups for superprocesses {#sec: definition of vf}
-----------------------------
Let $X$ be a non-persistent superprocess with its Kuznetsov measure denoted by $(\mathbb N_x)_{x\in E}$. We define the mean semigroup $$\begin{aligned}
P_t^{\rho_1} f(x)
:= \Pi_{x}[e^{\int_0^t \rho_1(\xi_s)ds}f(\xi_t) \mathbf 1_{t< \zeta}],
\quad t\geq 0, x\in E, f\in \mathcal B_b(E,\mathbb R_+).\end{aligned}$$ It is known from [@Li2011Measure-valued Proposition 2.27] and [@Kyprianou2014Fluctuations Theorem 2.7] that for all $t > 0$, $\mu \in \mathcal M(E)$ and $f\in \mathcal B_b(E,\mathbb R_+)$, $$\begin{aligned}
\label{eq: mean formula for superprocesses}
\mathbb N_{\mu}[W_t(f)]
=\mathbf P_{\mu}[X_t(f)]
=\mu(P^{\rho_1}_t f).\end{aligned}$$
Define $$\begin{aligned}
L_1(\xi)
&:= \{f\in \mathcal B(E): \forall x\in E, t\geq 0, \quad \Pi_x[|f(\xi_t)|]< \infty\}, \\
L_2(\xi)
&:= \{f \in \mathcal B(E): |f|^2 \in L_1(\xi)\}.\end{aligned}$$ Using monotonicity and linearity, we get from that $$\begin{aligned}
\mathbb N_x[W_t(f)]
= \mathbf P_{\delta_x}[X_t(f)]
= P^{\rho_1}_t f(x) \in \mathbb R,
\quad f\in L_1(\xi), t > 0,x\in E.\end{aligned}$$ This says that the random variable $\langle X_t, f\rangle$ is well defined under probability $\mathbf P_{\delta_x}$ provided $f\in L_1(\xi)$. By the branching property of the superprocess, $X_t(f)$ is an infinitely divisible random variable. Therefore, we can write $$U_t(\theta f)(x)
:= \operatorname{Log} \mathbf P_{\delta_x}[e^{i \theta X_t( f)}],
\quad t\geq 0, f\in L_1(\xi), \theta \in \mathbb R, x\in E,$$ as its characteristic exponent. According to Campbell’s formula, see [@Kyprianou2014Fluctuations Theorem 2.7] for example, we have $$\mathbf P_{\delta_x} [e^{i\theta X_t(f)}]
= \exp(\mathbb N_x[ e^{i\theta W_t(f)} - 1]),
\quad t>0, f\in L_1(\xi), \theta \in \mathbb R, x\in E.$$ Noticing that $\theta \mapsto \mathbb N_x[e^{i\theta W_t(f)} - 1]$ is a continuous function on $\mathbb R$ and that $\mathbb N_x[e^{i\theta W_t(f)} - 1] = 0$ if $\theta = 0$, according to [@Sato2013Levy Lemma 7.6], we have $$\begin{aligned}
\label{eq: N and characteristic exponent}
U_t(\theta f)(x)
= \mathbb N_x[e^{i W_t(\theta f)} - 1],
\quad t>0, f\in L_1(\xi), \theta \in \mathbb R, x\in E.\end{aligned}$$
There exists a constant $C\geq 0$ such that for all $f \in L_1(\xi),x\in E$ and $t\geq 0$, we have $$\begin{aligned}
\label{eq: upper bound of psi(v)}
|\psi (x,-U_tf)|
\leq C P^{\rho_1}_t |f|(x) + C (P^{\rho_1}_t |f| (x))^2.
\end{aligned}$$
Noticing that $
e^{\operatorname{Re} U_tf(x)}
= |e^{U_tf(x)}|
= |\mathbf P_{\delta_x}[e^{i X_t(f)}]|
\leq 1,
$ we have $$\begin{aligned}
\label{eq: -v has positive real part}
\operatorname{Re} U_tf(x)
\leq 0.
\end{aligned}$$ Therefore, we can speak of $\psi(x,-U_tf)$ since $z\mapsto \psi(x,z)$ is well defined on $\mathbb C_+$. According to Lemma \[lem: estimate of exponential remaining\], we have that $$\begin{aligned}
\label{eq: upper bound for vf}
|U_tf(x)|
\leq \mathbb N_x[|e^{i W_t(f)} - 1|]
\leq \mathbb N_x[|i W_t(f)|]
\leq (P^{\rho_1}_t |f|)(x).
\end{aligned}$$ Notice that, for any compact $K \subset \mathbb R$, $$\begin{aligned}
\label{eq: estimate of deriavetive of v(theta)}
\mathbb N_x \Big[\sup_{\theta \in K} \Big|\frac{\partial}{\partial \theta} (e^{i\theta W_t(f)} - 1) \Big|\Big]
\leq \mathbb N_x[|W_t(f)|] \sup_{\theta \in K}|\theta|
\leq (P^{\rho_1}_t |f|)(x) \sup_{\theta \in K}|\theta| < \infty.
\end{aligned}$$ Therefore, according to [@Durrett2010Probability Theorem A.5.2] and , $ U_t( \theta f)( x )$ is differentiable in $\theta \in \mathbb R$ with $$\frac{\partial}{\partial \theta} U_t(\theta f)(x)
= i\mathbb N_x[W_t(f) e^{i\theta W_t(f)}],
\quad \theta \in \mathbb R.$$ Moreover, from the above, it is clear that $$\begin{aligned}
\label{eq: upper bounded for derivative of v(theta)}
\sup_{\theta \in \mathbb R}\Big| \frac{\partial}{\partial \theta}U_t(\theta f)(x)\Big|
\leq ( P^{\rho_1}_t |f|)(x).
\end{aligned}$$ It follows from the dominated convergence theorem that $(\partial/\partial \theta)U_t(\theta f)(x)$ is continuous in $\theta$. In other words, $\theta \mapsto -U_t(\theta f)(x)$ is a $C^1$ map from $\mathbb R$ to $\mathbb C_+$. Thus, $$\begin{aligned}
\label{eq: path integration representation of psi(v)}
\psi(x,-U_tf)
= -\int_0^1 \psi' (x,-U_t(\theta f) ) \frac{\partial}{\partial \theta} U_t(\theta f)(x)~d\theta.
\end{aligned}$$ Notice that $$\begin{aligned}
& |\psi'(x, -U_tf)| \\
& = \Big| -\rho_1(x)- 2\rho_2(x) U_tf(x)+ \int_{(0,\infty)} y (1- e^{y U_tf(x)} ) \pi(x,dy)\Big| \\
& = \Big| - \rho_1(x)- 2\rho_2(x)\mathbb N_x[e^{i W_t(f)} - 1] + \int_{(0,\infty)} y \mathbf P_{y \delta_x}[1-e^{i X_t(f)}] \pi(x,dy) \Big| \\
& \leq \|\rho_1\|_\infty + 2\rho_2(x)\mathbb N_x[W_t(|f|)]+ \int_{(0,\infty)} y\mathbf P_{y\delta_x}[2\wedge X_t(|f|)] \pi(x,dy) \\
& \leq \|\rho_1\|_\infty + 2\|\rho_2\|_\infty P^{\rho_1}_t |f|(x) + \Big(\sup_{x\in E}\int_{(0,1]}y^2 \pi(x,dy)\Big)~P^{\rho_1}_t |f|(x) + 2\sup_{x\in E}\int_{(1,\infty)} y \pi(x,dy) \\
& =: C_1 + C_2(P^{\rho_1}_t |f|)(x), \label{eq: upper bound of psi'(v)}
\end{aligned}$$ where $C_1, C_2$ are constants independent of $f,x$ and $t$. Now, combining the display above with and we get the desired result.
This lemma also says that if $f\in L^2(\xi)$, then $
\Pi_x\Big[\int_0^t \psi(\xi_s,- U_{t-s}f)ds\Big]
\in \mathbb C,
x\in E, t\geq 0,$ is well defined. In fact, using Jensen’s inequality and the Markov property, we have $$\begin{aligned}
\label{eq: domination of psi(v)}
& \Pi_x\Big[\int_0^t |\psi (\xi_s,-U_{t-s}f )|ds\Big]
\leq \Pi_x\Big[\int_0^t (C_1 P_{t-s}^{\rho_1}|f|(\xi_s)+C_2 P_{t-s}^{\rho_1}|f|(\xi_s)^2 )ds\Big] \\
& \leq \int_0^t (C_1 e^{t\|\rho_1\|}\Pi_x [ \Pi_{\xi_s}[|f(\xi_{t-s})|] ]+C_2 e^{2t\|\rho_1\|}\Pi_x [ \Pi_{\xi_s}[|f (\xi_{t-s})|]^2 ] )~ds \\
& \leq \int_0^t (C_1 e^{t\|\rho_1\|}\Pi_x [ |f(\xi_{t})|]+C_2e^{2t\|\rho_1\|}\Pi_x [ |f (\xi_{t})|^2 ])~ds < \infty.\end{aligned}$$
A complex-valued non-linear integral equation
---------------------------------------------
Let $X$ be a non-persistent superprocess. In this subsection, we will prove the following:
\[prop: complex FKPP-equation\] If $f\in L_2(\xi)$, then for all $t\geq 0$ and $x\in E$, $$\begin{aligned}
\label{eq: complex FKPP-equation}
U_tf(x) - \Pi_x \Big[\int_0^t \psi (\xi_s, - U_{t-s}f ) ds \Big]
= i \Pi_x [f(\xi_t)].\end{aligned}$$ $$\begin{aligned}
\label{eq: complex FKPP-equation with FK-transform}
U_tf(x) - \int_0^t P_{t-s}^{\rho_1} \psi_0(\cdot,-U_sf) (x)~ds
= iP_t^{\rho_1} f(x).\end{aligned}$$
To prove this, we will need the generalized spine decomposition theorem from [@RenSongSun2017Spine]. Let $f\in \mathcal B_b(E,\mathbb R_+)$, $T >0$ and $x\in E$. Suppose that $\mathbf P_{\delta_x}[X_T(f)] = \mathbb N_x[ W_T(f)] = P^{\rho_1}_T f(x) \in (0,\infty)$, then we can define the following probability transforms: $$\begin{aligned}
d\mathbf P_{\delta_x}^{ X_T(f)}
:= \frac{X_T(f)}{P_T^{\rho_1} f(x)} d\mathbf P_{\delta_x};
\quad d\mathbb N_x^{W_T(f)}
:= \frac{W_T(f)}{P_T^{\rho_1} f(x)} d\mathbb N_x.\end{aligned}$$ Following the definition in [@RenSongSun2017Spine], we say that $\{\xi, \mathbf n;\mathbf Q_{x}^{(f,T)}\}$ is a spine representation of $\mathbb N_x^{\langle W_T, f\rangle}$ if
- the spine process $\{(\xi_t)_{0\leq t\leq T}; \mathbf Q^{(f,T)}_x\}$ is a copy of $\{(\xi_t)_{0\leq t\leq T}; \Pi^{(f,T)}_{x}\}$, where $$\begin{aligned}
d\Pi_x^{(f,T)}
:= \frac{f(\xi_T)e^{\int_0^T \rho_1(\xi_s)ds}}{P^{\rho_1}_T f(x)} d \Pi_x;
\end{aligned}$$
- given $\{(\xi_t)_{0\leq t\leq T}; \mathbf Q^{(f,T)}_x\}$, the immigration measure $
\{\mathbf n(\xi,ds,dw); \mathbf Q^{(f,T)}_x[\cdot |(\xi_t)_{0\leq t\leq T}]\}
$ is a Poisson random measure on $[0,T] \times \mathbb W$ with intensity $$\begin{aligned}
\label{eq: conditional intensity}
\mathbf m(\xi,ds,dw)
:= 2 \rho_2(\xi_s) ds \cdot \mathbb N_{\xi_s}(dw) + ds \cdot \int_{y\in (0,\infty)} y \mathbf P_{y\delta_{\xi_s}}(X\in dw) \pi(\xi_s,dy);\end{aligned}$$
- $\{(Y_t)_{0\leq t\leq T}; \mathbf Q^{(f,T)}_x\}$ is an $\mathcal M(E)$-valued process defined by $$\begin{aligned}
Y_t
:= \int_{(0,t] \times \mathbb W} w_{t-s} \mathbf n(\xi,ds,dw),
\quad 0 \leq t\leq T.
\end{aligned}$$
According to the spine decomposition theorem in [@RenSongSun2017Spine], we have that $$\begin{aligned}
\label{eq: Spine decomposition 1}
\{(X_s)_{s \geq 0};\mathbf P_{\delta_x}^{X_T(f)}\}
\overset{f.d.d.}{=} \{(X_s + W_s)_{s \geq 0};\mathbf P_{\delta_x} \otimes \mathbb N_x^{W_T(f)} \},\end{aligned}$$ $$\begin{aligned}
\label{eq: Spine decomposition 2}
\{(W_s)_{0\leq s\leq T};\mathbb N_x^{W_T(f)}\}
\overset{f.d.d.}{=} \{(Y_s)_{s \geq 0};\mathbf Q_x^{(f,T)}\}.\end{aligned}$$
Assume that $f\in \mathcal B_b(E)$. Fix $t>0, r\in [0,t), x\in E$ and a strictly positive $g\in \mathcal B_b(E)$. Denote by $\{\xi, \mathbf n; \mathbf Q_x^{(g,t)}\}$ the spine representation of $\mathbb N_x^{W_t(g)}$. Conditioned on $\{\xi; \mathbf Q_x^{(g,t)}\}$, denote by $\mathbf m(\xi, ds,dw)$ the conditional intensity of $\mathbf n$ in . Denote by $\Pi_{r,x}$ the probability of Hunt process $\{\xi; \Pi\}$ initiated at time $r$ and position $x$. From Lemma \[lem: estimate of exponential remaining\], we have $\mathbf Q^{(g,t)}_{x}$-almost surely $$\begin{aligned}
& \int_{[0,t]\times \mathbb W}|e^{i w_{t-s}(f)} - 1| \mathbf m(\xi, ds,dw)
\leq \int_{[0,t]\times \mathbb W} (| w_{t-s}(f)| \wedge 2 ) \mathbf m(\xi, ds,dw) \\
& \leq \int_0^t \Big(2\rho_2(\xi_s)\mathbb N_{\xi_s} ( W_{t-s}(|f|) ) + \int_{(0,1]} y \mathbf P_{y \delta_{\xi_s}}[X_{t-s}(|f|)] \pi(\xi_s,dy)
+ 2\int_{(1,\infty)}y\pi(\xi_s,dy)\Big) ds
\\ & \leq \int_0^t (P_{t-s}^{\rho_1} |f|)(\xi_s)\Big(2\rho_2(\xi_s) + \int_{(0,1]} y^2 \pi(\xi_s,dy)\Big) ds + 2t \sup_{x\in E}\int_{(1,\infty)}y\pi(x,dy)
\\ & \leq \Big(2\|\rho_2\|_\infty +\sup_{x\in E}\int_{(0,1]} y^2 \pi(x,dy)\Big) t e^{t\|\rho_1\|_\infty}\|f\|_\infty + 2t \sup_{x\in E}\int_{(1,\infty)}y\pi(x,dy)
< \infty.
\end{aligned}$$ Using this, Fubini’s theorem, and we have $\mathbf Q^{(g,t)}_{x}$-almost surely, $$\begin{aligned}
& \int_{[0,t]\times \mathbb N}(e^{i w_{t-s}(f)} - 1) \mathbf m(\xi, ds,dw)
\\ & =\int_0^t \Big(2\rho_2(\xi_s)\mathbb N_{\xi_s}(e^{i W_{t-s}(f)} - 1) + \int_{(0,\infty)} y \mathbf P_{y \delta_{\xi_s}}[e^{i X_{t-s}(f)} - 1] \pi(\xi_s,dy)\Big) ds
\\ & =\int_0^t \Big( 2\rho_2(\xi_s) U_{t-s} f(\xi_s) + \int_{(0,\infty)} y (e^{y U_{t-s}f(\xi_s)} - 1) \pi(\xi_s,dy) \Big) ds
\\ & = -\int_0^t \psi'_0 (\xi_s, -U_{t-s}f )ds.
\end{aligned}$$ Therefore, according to , Campbell’s formula and above, we have that $$\begin{aligned}
\label{eq: N to Pi}
& \mathbb N_x^{ W_t( g)}[e^{i W_t(f)}]
= \mathbf Q_x^{(g,t)} \Big[\exp\Big\{\int_{[0,t]\times \mathbb N}(e^{i w_{t-s}(f)} - 1) \mathbf m(\xi, ds,dw)\Big\}\Big]
\\ & = \Pi_x^{(g,t)} [e^{-\int_0^t \psi'_0(\xi_s, -U_{t-s}f)ds}]
= \frac{1}{P_t^{\rho_1} g (x)} \Pi_x[ g(\xi_t) e^{-\int_0^t \psi'(\xi_s, -U_{t-s}f)ds} ].
\end{aligned}$$ Let $\epsilon >0$. Define $f^+ = (f \vee 0) + \epsilon$ and $f^- = (-f) \vee 0 + \epsilon$, then $f^\pm$ are strictly positive and $f = f^+ - f^-$. According to , we have that $$\begin{aligned}
\frac{\mathbf P_{\delta_x}[X_t(f^{\pm}) e^{i X_t(f)}]}{\mathbf P_{\delta_x}[X_t(f^{\pm}) ]}
= \mathbf P_{\delta_x}[e^{i X_t(f)}] \mathbb N_x^{ W_t(f^{\pm})}[e^{i X_t(f)}].
\end{aligned}$$ Using and the above, we have $$\begin{aligned}
\frac{\mathbf P_{\delta_x}[X_t(f) e^{i X_t(f)}] }{\mathbf P_{\delta_x}[e^{i X_t(f)}]}
& = \mathbf P_{\delta_x}[X_t(f^+)] \mathbb N_x^{W_t(f^+)} [e^{i X_t(f)}] - \mathbf P_{\delta_x}[X_t(f^-)]\mathbb N_x^{W_t(f^-)}[e^{i X_t(f)}]
\\ & = \Pi_x[ f(\xi_t) e^{- \int_0^t \psi'(\xi_s, -U_{t-s}f) ds} ].
\end{aligned}$$ Therefore, we have $$\begin{aligned}
\frac{\partial}{\partial \theta} {U_t(\theta f)(x)}
= \frac{\mathbf P_{\delta_x}[i X_t(f) e^{i X_t(f)}] }{\mathbf P_{\delta_x}[e^{i X_t(f)}]}
= \Pi_x[ if(\xi_t) e^{ - \int_0^t \psi'(\xi_s, -U_{t-s}(\theta f)) ds} ].
\end{aligned}$$ Since $\{(\xi_{r+t})_{t \geq 0}; \Pi_{r,x}\} \overset{d}{=} \{(\xi_{t})_{t\geq 0}; \Pi_{x}\} $, we have $$\begin{aligned}
& \frac{\partial}{\partial \theta} U_{t-r}(\theta f)( x)
= \Pi_x[ i f(\xi_{t-r}) e^{-\int_0^{t-r} \psi'(\xi_s, -U_{t-r-s}(\theta f)) ds} ] \\
& = \Pi_{r,x}[i f(\xi_t)e^{-\int_0^{t-r} \psi'(\xi_{r+s}, -U_{t-r-s}(\theta f)) ds} ]
= \Pi_{r,x}[if(\xi_t)e^{-\int_r^t \psi'(\xi_{s}, -U_{t-s}(\theta f)) ds} ].
\end{aligned}$$
From , we know that for each $\theta\in \mathbb R$, $(t,x) \mapsto |\psi'(x,-U_tf(x))|$ is locally bounded (i.e. bounded on $[0,T]\times E$ for each $T \geq 0$). Therefore, we can apply Lemma \[eq: complex FK\] and get that $$\frac{\partial}{\partial \theta} U_{t-r}(\theta f)(x) + \Pi_{r,x} \Big[\int_r^t \psi' (\xi_s,- U_{t-s}(\theta f) )\frac{\partial}{\partial \theta} U_{t-s}(\theta f)(\xi_s)~ds\Big]
= \Pi_{r,x} [i f(\xi_t)]$$ and $$\begin{aligned}
& \frac{\partial}{\partial \theta} U_{t-r}(\theta f)(x) + \Pi_{r,x} \Big[\int_r^t e^{\int_r^s \rho_1(\xi_u)du}\psi_0' (\xi_s,- U_{t-s}(\theta f) )\frac{\partial}{\partial \theta} U_{t-s}(\theta f)(\xi_s)~ds\Big]\\
& = \Pi_{r,x} [i e^{\int_r^t \rho_1(\xi_s)ds}f(\xi_t)].
\end{aligned}$$ Integrating the two displays above with respect to $\theta$ on \[0,1\], using Fubini’s theorem, , and , we get $$\begin{aligned}
U_{t-r}f(x) - \Pi_{r,x} \Big[\int_r^t \psi (\xi_s,-U_{t-s}f ) ~ds\Big]
= i \theta \Pi_{r,x} [f(\xi_t)]
\end{aligned}$$ and $$\begin{aligned}
U_{t-r}f(x) - \Pi_{r,x} \Big[\int_r^t e^{\int_r^s \rho_1(\xi_u)du} \psi_0 (\xi_s,- U_{t-s}f ) ~ds\Big]
= i \Pi_{r,x} [e^{\int_r^t\rho_1(\xi_u)du}f(\xi_t)].
\end{aligned}$$ Taking $r = 0$, we get that and are true if $f\in \mathcal B_b(E)$.
The rest of the proof is to evaluate and for all $f\in L_2(\xi)$. We only do this for since the argument for is similar. Let $n \in \mathbb N$. Writing $f_n := (f^+ \wedge n) - (f^- \wedge n)$, then $f_n \xrightarrow[n\to \infty]{} f$ pointwise. From what we have proved, we have $$\begin{aligned}
\label{eq: complex FKPP-equation for fn}
U_tf_n(x) - \Pi_{x} \Big[\int_0^t \psi (\xi_s, - U_{t-s}f_n ) ~ds\Big]
= i \Pi_{x} [f_n(\xi_t)].
\end{aligned}$$ Note that (i) $\Pi_{x}[f_n(\xi_t)] \xrightarrow[n\to \infty]{} \Pi_{x}[f(\xi_t)]$; (ii) by , the dominated convergence theorem and the fact that $$|e^{i W_t(f_n)} - 1| \leq W_t(|f|);
\quad \mathbb N_x[W_t(|f|)] = (P_t^{\rho_1} |f|)(x) < \infty,$$ we have $U_tf_n(x) \xrightarrow[n\to \infty]{} U_tf(x)$, and (iii) by the dominated convergence theorem, and the fact (see ) that $$|\psi(\xi_s,- U_{t-s}f_n) |
\leq C_1 P_{t-s}^{\rho_1}|f|(\xi_s)+C_2 P_{t-s}^{\rho_1}|f|(\xi_s)^2,$$ we get that $\Pi_{x} [\int_0^t \psi(\xi_s,- U_{t-s}f_n)ds] \xrightarrow[n\to \infty]{} \Pi_{x} [\int_0^t \psi(\xi_s,- U_{t-s}f)ds]$. Using these, letting $n \to \infty$ in , we get the desired result.
Acknowledgment {#acknowledgment .unnumbered}
--------------
We thank Zenghu Li and Rui Zhang for helpful conversations. We also thank the referee for very helpful comments.
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[^1]: The research of Yan-Xia Ren is supported in part by NSFC (Grant Nos. 11671017 and 11731009) and LMEQF.
[^2]: The Research of Renming Song is support in part by a grant from the Simons Foundation (\#429343, Renming Song)
[^3]: Jianjie Zhao is the corresponding author
|
---
abstract: 'We propose a reinforcement learning solution to the *soccer dribbling task*, a scenario in which a soccer agent has to go from the beginning to the end of a region keeping possession of the ball, as an adversary attempts to gain possession. While the adversary uses a stationary policy, the dribbler learns the best action to take at each decision point. After defining meaningful variables to represent the state space, and high-level macro-actions to incorporate domain knowledge, we describe our application of the reinforcement learning algorithm *Sarsa* with CMAC for function approximation. Our experiments show that, after the training period, the dribbler is able to accomplish its task against a strong adversary around $58\%$ of the time.'
author:
- Arthur Carvalho and Renato Oliveira
bibliography:
- 'SamplePaper.bib'
title: |
\
**Reinforcement Learning for the Soccer Dribbling Task [^1] [^2]**
---
Introduction
============
Soccer dribbling consists of the ability of a soccer agent to go from the beginning to the end of a region keeping possession of the ball, while an adversary attempts to gain possession. In this work, we focus on the dribbler’s learning process, *i.e.*, the learning of an effective policy that determinesa good action for the dribbler to take at each decision point.
We study the soccer dribbling task using the RoboCup soccer simulator [@Noda:soccerserver]. Specific details of this simulator increase the complexity of the learning process. For example, besides the adversarial and real-time environment, agents’ perceptions and actions are noisy and asynchronous.
We model the soccer dribbling task as a *reinforcement learning* problem. Our solution to this problem combines the Sarsa algorithm with CMAC for function approximation. Despite the fact that the resulting learning algorithm is not guaranteed to converge to the optimal policy in all cases, many lines of evidence suggest that it converges to near-optimal policies (for example, see [@Gordon:Reinforcement_Learning; @Sutton:generalizationin; @Tsitsiklis:Function_Approximation; @Perkins:Convergence]).
Besides this introductory section, the rest of this paper is organized as follows. In the next section, we describe the soccer dribbling task. In Section 3, we show how to map this task onto an episodic reinforcement learning framework. In Section 4 and 5, we present, respectively, the reinforcement learning algorithm and its results against a strong adversary. In Section 6, we review the literature related to our work. In Section 7, we conclude and present future research directions.
Soccer Dribbling
================
Soccer dribbling is a crucial skill for an agent to become a successful soccer player. It consists of the ability of a soccer agent, henceforth called the *dribbler*, to go from the beginning to the end of a region keeping possession of the ball, while an adversary attempts to gain possession. We can see soccer dribbling as a subproblem of the complete soccer domain. The main simplification is that the players involved are only focused on specific goals, without worrying about team strategies or unrelated individual skills (*e.g.*, passing and shooting). Nevertheless, a successful policy learned by the dribbler can be used in the complete soccer domain whenever a soccer agent faces a dribbling situation.
Since our focus is on the dribbler’s learning process, an omniscient coach agent is used to manage the play. At the beginning of each trial (*episode*), the coach resets the location of the ball and of the players within a *training field*. The dribbler is placed in the center-left region together with the ball. The adversary is placed in a random position with the constraint that it does not start with possession of the ball. An example of a starting configuration is shown in Figure 1.
{width="0.75\columnwidth"}
Whenever the adversary gains possession for a set period of time or when the ball goes out of the training field by crossing either the left line or the top line or the bottom line, the coach declares the adversary as the winner of the episode. If the ball goes out of the training field by crossing the right line, then the winner is the first player to intercept the ball. After declaring the winner of an episode, the coach resets the location of the players and of the ball within the training field and starts a new episode. Thus, the dribbler’s goal is to reach the right line that delimits the training field with the ball. We call this task the *soccer dribbling task*.
We argue that the soccer dribbling task is an excellent benchmark for comparing different machine learning techniques since it involves a complex problem, and it has a well-defined objective, which is to maximize the number of episodes won by the dribbler. We study the soccer dribbling task using the RoboCup soccer simulator [@Noda:soccerserver].
The RoboCup soccer simulator operates in discrete time steps, each representing 100 milliseconds of simulated time. Specific details of this simulator increase the complexity of the learning process. For example, random noise is injected into all perceptions and actions. Further, agents must sense and act asynchronously. Each soccer agent receives visual information about other objects every 150 milliseconds, *e.g.*, its distance from other players in its current field of view. Each agent has also a body sensor, which detects its current “physical status" every 100 milliseconds, *e.g.*, that agent’s stamina and speed. Agents may execute a parameterized primitive action every 100 milliseconds, *e.g.*, *turn*(angle), *dash*(power), and *kick*(power, angle). Full details of the RoboCup soccer simulator are presented by Chen *et al.* [@Manual_RoboCup].
Since possession is not well-defined in the RoboCup soccer simulator, we consider that an agent has possession of the ball whenever the ball is close enough to be kicked, *i.e.*, it is in a distance less than $1.085$ meters from the agent.
The Soccer Dribbling Task as a Reinforcement Learning Problem
=============================================================
In the soccer dribbling task, an episode begins when the dribbler may take the first action. When an episode ends (*e.g.*, when the adversary gains possession for a set period of time), the coach starts a new one, thereby giving rise to a series of episodes. Thus, the interaction between the dribbler and the environment naturally breaks down into a sequence of distinct episodes. This point, together with the fact that the RoboCup soccer simulator operates in discrete time steps, allows the soccer dribbling task to be mapped onto a discrete-time, episodic reinforcement-learning framework.
Roughly speaking, reinforcement learning is concerned with how an agent must take actions in an environment so as to maximize the expected long-term reward [@Sutton:Reinforcement_learning_book]. Like in a trial-and-error search, the learner must discover which action is the most rewarding one in a given state of the world. Thus, solving a reinforcement learning problem means finding a function (*policy*) that maps *states* to *actions* so that it maximizes a *reward* over the long run. As a way of incorporating domain knowledge, the actions available to the dribbler are the following high-level *macro-actions*, which are built on the simulator’s *primitive actions*[^3]:
- *HoldBall*(): The dribbler holds the ball close to its body, keeping it in a position that is difficult for the adversary to gain possession;
- *Dribble*($\Theta, k$): The dribbler turns its body towards the global angle $\Theta$, kicks the ball $k$ meters ahead of it, and moves to intercept the ball.
The global angle $\Theta$ is in the range $[0, 360]$. In detail, the center of the training field has been chosen as the origin of the system, where the zero-angle points towards the middle of the right line that delimits the training field, and it increases in the clockwise direction. Those macro-actions are based on high-level skills used by the UvA Trilearn 2003 team [@uva]. The first one maps directly onto the primitive action *kick*. Consequently, it usually takes a single time step to be performed. The second one, however, requires an extended sequence of the primitive actions *turn*, *kick*, and *dash*. To handle this situation, we treat the soccer dribbling task as a *semi-Markov decision process* (SMDP) [@Puterman:MDP].
Formally, an SMDP is a 5-tuple $<S, A, P, r, F>$, where $S$ is a countable set of states, $A$ is a countable set of actions, $P(s^{\prime}|s, a)$, for $s^{\prime}, s \in S$, and $a \in A$, is a probability distribution providing the transition model between states, $r(s, a) \in \Re$ is a reward associated with the transition $(s, a)$, and $F( \tau | s, a)$ is a probability distribution indicating the sojourn time in a given state $s \in S$, *i.e.*, the time before transition provided that action $a$ was taken in state $s$.
Let $a_i \in A$ be the $i$th macro-action selected by the dribbler. Thus, several simulator’s time steps may elapse between $a_i$ and $a_{i+1}$. Let $s_{i+1} \in S$ and $r_{i+1} \in \Re$ be, respectively, the state and the reward following the macro-action $a_{i}$. From the dribbler’s point of view, an episode consists of a sequence of SMDP steps, *i.e.*, a sequence of states, macro-actions, and rewards: $s_0, a_0, r_1, s_1, \dots, s_i, a_i, r_{i+1}, s_{i+1}, \dots, a_{n-1}, r_n, s_n$, where $a_i$ is chosen based exclusively on the state $s_i$, and $s_n$ is a terminal state in which either the adversary or the dribbler is declared the winner of the episode by the coach. In the formercase, the dribbler receives the reward $r_n = -1$, while in the latter case its reward is $r_n = 1$. The intermediate rewards arealways equal to zero, *i.e.*, $r_1 = r_2 = \dots = r_{n-1} = 0$. Thus, our objective is to find a policy that maximizes the dribbler’s reward, *i.e.*, the number of episodes in which it is the winner.
Dribbler
--------
The dribbler must take a decision at each SMDP step by selecting an available macro-action. Besides the macro-action *HoldBall*, the set of actions available to the dribbler contains four instances of the macro-action *Dribble*: *Dribble*($30^\circ, 5$), *Dribble*($330^\circ, 5$), *Dribble*($0^\circ, 5$), and *Dribble*($0^\circ, 10$). Thus, besides hiding the ball from the adversary, the dribbler can kick the ball forward (strongly and weakly), diagonally upward, and diagonally downward. If at some time step the dribbler has not possession of the ball and the current state is not a terminal state, then it usually means that the dribbler chose an instance of the macro-action *Dribble* before and it is currently moving to intercept the ball.
We turn now to the state representation used by the dribbler. It consists of a set of state variables which are based on information related to the ball, the adversary, and the dribbler itself. Let $ang(x)$ be the global angle of the object $x$, and $ang(x, y)$ and $dist(x, y)$ be, respectively, the relative angle and the distance between the objects $x$ and $y$. Further, let $w$ and $h$ be, respectively, the width and the height of the training field. Finally, let $posY(x)$ be a function indicating whether the object $x$ is close to (less than 1 meter away from) the top line or the bottom line that delimits the training field. In the former case, $posY(x) = 1$, whereas in the latter case $posY(x) = -1$, and otherwise $posY(x) = 0$. Table 1 shows the state variables together with their ranges.
State Variable Range
------------------------------------------ -------------------------
$posY(\mbox{dribbler})$ $\{-1, 0 ,1\}$
$ang(\mbox{dribbler})$ $[0, 360]$
$ang(\mbox{dribbler}, \mbox{adversary})$ $[0, 360]$
$ang(\mbox{ball}, \mbox{adversary})$ $[0, 360]$
$dist(\mbox{ball}, \mbox{adversary})$ $[0, \sqrt{w^2 + h^2}]$
: Description of the state representation.
The first three variables help the dribbler to locate itself and the adversary inside the training field. Together, the last two variables can be seen as a point describing the position of the adversary in a polar coordinate system, where the ball is the pole. Thus, these variables are used by the dribbler to locate the adversary with respect to the ball. It is interesting to note that a more informative state representation can be used by adding more state variables, *e.g.*, the current speed of the ball and the dribbler’s stamina. However, large domains can be impractical due to the “curse of dimensionality", *i.e.*, the general tendency of the state space to grow exponentially in the number of state variables [@cursedimensionality]. Consequently, we focus on a state representation that is as concise as possible.
Adversary
---------
The adversary uses a fixed, pre-specified policy. Thus, we can see it as part of the environment in which the dribbler is interacting with. When the adversary has possession of theball, it tries to maintain possession for another time step byinvoking the macro-action *HoldBall*. If it maintains possession for two consecutive time steps, then it is the winner of the episode. When the adversary does not have the ball, it uses an iterative scheme to compute a near-optimal interception point based on the ball’s position and velocity. Thereafter, the adversary moves to that point as fast as possible. This procedure is the same used by the dribbler when it is moving to intercept the ball after invoking the macro-action *Dribble*. More details about this iterative scheme can be found in the description of the UvA Trilearn 2003 team [@uva].
The Reinforcement Learning Algorithm
====================================
Our solution to the soccer dribbling task combines the reinforcement learning algorithm Sarsa with CMAC for function approximation. In what follows, we briefly introduce both of them before presenting the final learning algorithm.
Sarsa
-----
The Sarsa algorithm works by estimating the action-value function $Q^\pi(s, a)$, for the current policy $\pi$ and for all state-action pairs $(s, a)$ [@Sutton:Reinforcement_learning_book]. The $Q$-function assigns to each state-action pair the expected return from it. Given a quintuple of events, $(s_t, a_t, r_{t+1}, s_{t+1}, a_{t+1})$, that make up the transition from the state-action pair $(s_t, a_t)$ to the next one, $(s_{t+1}, a_{t+1})$, the $Q$-value of the first state-action pair is updated according to the following equation:
$$Q(s_t,a_t) \leftarrow Q(s_t,a_t) + \alpha\delta_t,$$
where $\delta_t$ is the traditional *temporal-difference error*,
$$\delta_t = r_{t+1} + \lambda Q(s_{t+1},a_{t+1}) - Q(s_t,a_t),$$
$\alpha$ is the learning rate parameter, and $\lambda$ is a discount rate governing the weight placed on future, as opposed to immediate, rewards. Sarsa is an on-policy learning method, meaning that it continually estimates $Q^\pi$, for the current policy $\pi$, and at the same time changes $\pi$ towards greediness with respect to $Q^\pi$. A typical policy derived from the $Q$-function is an *$\epsilon$-greedy* policy. Given the state $s_t$, this policy selects a random action with probability $\epsilon$ and, otherwise, it selects the action with the highest estimated value, *i.e.*, $a = {\operatornamewithlimits{argmax}}_a Q(s_t, a)$.
CMAC
----
In tasks with a small number of state-action pairs, we can represent the action-value function $Q^\pi$ as a table with one entry for each state-action pair. However, this is not the case of the soccer dribbling task. For illustration’s sake, suppose that all variables in Table 1 are discrete. If we consider the $5$ actions available to the dribbler and a 20m x 20m training field, we end up with more than $1.9 \times 10^{10}$ state-action pairs. This would not only require an unreasonable amount of memory, but also an enormous amount of data to fill up the table accurately. Thus, we need to *generalize* from previously experienced states to ones that have never been seen. For dealing with this task, we use a technique commonly known as *function approximation*.
By using a function approximation, the action-value function $Q^\pi$ is now represented as a parameterized functional form [@Sutton:Reinforcement_learning_book]. Now, whenever we make a change in one parameter value, we also change the estimated value of many state-action pairs, thus obtaining generalization. In this work, we use the Cerebellar Model Arithmetic Computer (CMAC) for function approximation [@Albus:Tile_Coding_1; @Albus:Tile_Coding_2].
CMAC works by partitioning the state space into multi-dimensional *receptive fields*, each of which is associated with a *weight*. In this work, receptive fields are hyper-rectangles in the state space. Nearby states share receptive fields. Thus, generalization occurs between them. Multiple partitions of the state space (*layers*) are usually used, which implies that any input vector falls within the range of multiple *excited receptive fields*, one from each layer.
Layers are identical in organization, but each one is offset relative to the others so that each layer cuts the state space in a different way. By overlapping multiple layers, it is possible to achieve quick generalization while maintaining the ability to learn fine distinctions. Figure 2 shows an example of two grid-like layers overlaid over a two-dimensional space.
The receptive fields excited by a given state $s$ make up the feature set $\mathbb{F}_s$, with each action $a$ indexing their weights in a different way. In other words, each macro-action is associated with a particular CMAC. Clearly, the number of receptive fields inside each feature set is equal to the number of layers. The CMAC’s response to a feature set $\mathbb{F}_s$ is equal to the sum of the weights of the receptive fields in $\mathbb{F}_s$. Formally, let $\theta_a(i)$ be the weight of the receptive field $i$ indexed by the action $a$. Thus, the CMAC’s response to $\mathbb{F}_s$ is equal to $\sum_{i \in \mathbb{F}_s} \theta_a(i)$, which represents the $Q$-value $Q(s,a)$.
CMAC is trained by using the traditional *delta rule* (also known as the least mean square). In detail, after selecting an action $a$, the weight of an excited receptive field $i$ indexed by $a$, $\theta_a(i)$, is updated according to the following equation:
$$\theta_{a}(i) \leftarrow \theta_a(i) +\alpha\delta,$$
where $\delta$ is the temporal-difference error. A major issue when using CMAC is that the total number of receptive fields required to span the entire state space can be very large. Consequently, an unreasonable amount of memory may be needed. A technique commonly used to address this issue is called *pseudo-random hashing* [@Sutton:Reinforcement_learning_book]. It produces receptive fields consisting of noncontiguous, disjoint regions randomly spread throughout the state space, so that only information about receptive fields that have been excited during previous training is actually stored.
Linear, Gradient-Descent Sarsa
------------------------------
Our solution to the soccer dribbling task combines the Sarsa algorithm with CMAC for function approximation. We use an $\epsilon$-greedy policy for action selection. Sutton and Barto [@Sutton:Reinforcement_learning_book] provide a complete description of this algorithm under the name of *linear, gradient-descent Sarsa*. Our implementation follows the solution proposed by Stone *et al.* [@Stone:Keepaway]. It consists of three routines: *RLstartEpisode*, to be run by the dribbler at the beginning of each episode; *RLstep*, run on each SMDP step; and *RLendEpisode*, to be run when an episode ends. In what follows, we present each routine in detail.
### RLstartEpisode
Given an initial state $s_0$, this routine starts by iterating over all available actions. In line 2, it finds the receptive fields excited by $s_0$, which compose the feature set $\mathbb{F}_{s_0}$. Next, in line 3, the estimated value of each macro-action $a$ in $s_0$ is calculated as the sum of the weights of the excited receptive fields. In line 5, this routine selects a macro-action by following an $\epsilon$-greedy policy and sends it to the RoboCup soccer simulator. Finally, the chosen action and the initial state $s_0$ are stored, respectively, in the variables $LastAction$ and $LastState$.
$\mathbb{F}_{s_0} \leftarrow $ receptive fields excited by $s_0$ $Q_a \leftarrow \sum_{i \in \mathbb{F}_{s_0}} \theta_a(i)$ $
LastAction \leftarrow \left\{ \begin{array}{ll}
{\operatornamewithlimits{argmax}}_a Q_a & \mbox{w/ prob. } 1 - \epsilon \\
\mbox{random action} & \mbox{w/ prob. } \epsilon \\
\end{array} \right.
$ $LastState \leftarrow s_0 $
### RLstep
This routine is run on each SMDP step, whenever the dribbler has to choose a macro-action. Given the current state $s$, it starts by calculating part of the temporal-difference error (Equation 2), namely the difference between the intermediate reward $r$ and the expected return of the previous SMDP step, $Q_{LastAction}$. In lines 2 to 5, this routine finds the receptive fields excited by $s$ and uses their weights to compute the estimated value of each action $a$ in $s$. In line 6, the next action to be taken by the dribbler is selected according to an $\epsilon$-greedy policy. In line 7, this routine finishes to compute the temporal-difference error by adding the discount rate $\lambda$ times the expected return of the current SMDP step, $Q_{CurrentAction}$. Next, in lines 8 to 10, this routine adjusts the weights of the receptive fields excited in the previous SMDP step by the learning factor $\alpha$ times the temporal-difference error $\delta$ (see Equation 3). Since the weights have changed, we must recalculate the expected return of the current SMDP step, $Q_{CurrentAction}$ (line 11). Finally, the chosen action and the current state are stored, respectively, in the variables $LastAction$ and $LastState$.
$\delta \leftarrow r - Q_{LastAction}$ $\mathbb{F}_{s} \leftarrow $ receptive fields excited by $s$ $Q_a \leftarrow \sum_{i \in \mathbb{F}_{s}} \theta_a(i)$ $
CurrentAction \leftarrow \left\{ \begin{array}{ll}
{\operatornamewithlimits{argmax}}_a Q_a & \mbox{w/ prob. } 1 - \epsilon \\
\mbox{random action} & \mbox{w/ prob. } \epsilon \\
\end{array} \right.$ $\delta \leftarrow \delta + \lambda Q_{CurrentAction}$ $\theta_{LastAction}(i) \leftarrow \theta_{LastAction}(i) + \alpha\delta $ $Q_{CurrentAction} \leftarrow \sum_{i \in \mathbb{F}_{s}} \theta_{CurrentAction}(i)$ $LastAction \leftarrow CurrentAction$ $LastState \leftarrow s$
### RLendEpisode
This routine is run when an episode ends. Initially, it calculates the appropriate reward based on who won the episode. Next, it calculates the temporal-difference error in the action-value estimates (line 6). There is no need to add the expected return of the current SMDP step ($Q_{CurrentAction}$) since this value is defined to be $0$ for terminal states. Lastly, this routine adjusts the weights of the receptive fields excited in the previous SMDP step.
$r \leftarrow 1$ $r \leftarrow -1$ $\delta \leftarrow r - Q_{LastAction}$ $\theta_{LastAction}(i) \leftarrow \theta_{LastAction}(i) + \alpha\delta $
Empirical Results
=================
In this section, we report our experimental results with the soccer dribbler task. In all experiments, we used the standard RoboCup soccer simulator (version 14.0.3, protocol 9.3) and a 20m x 20m training region. In that simulator, agents typically have limited and noisy visual sensors. For example, each player can see objects within a $90^\circ$ view cone, and the precision of an object’s sensed location degrades with distance. To simplify the learning process, we removed those restrictions. Both the dribbler and the adversary were given $360^\circ$ of noiseless vision to ensure that they would always have complete and accurate knowledge of the environment.
Related to parameters of the reinforcement learning algorithm[^4], we set $\epsilon = 0.01$, $\alpha = 0.125$, and $\lambda = 1$. By no means do we argue that these values are optimal. They were set based on results of brief, informal experiments.
The weights of first-time excited receptive fields were set to $0$. The bounds of the receptive fields were set according to the generalization that we desired: angles were given widths of about 20 degrees, and distances were given widths of approximately 3 meters. We used 32 layers. Each dimension of every layer was offset from the others by $1/32$ of the desired width in that dimension. We used the CMAC implementation proposed by Miller and Glanz [@cmac_code], which uses pseudo-random hashing. To retain previously trained information in the presence of subsequent novel data, we did not allow hash collisions.
To create episodes as realistic as possible, agents were not allowed to recover their staminas by themselves. This task was done by the coach after five consecutive episodes. This enabled agents to start episodes with different stamina values. We ran this experiment 5 independent times, each one lasting 50,000 episodes, and taking, on average, approximately 74 hours. Figure 3 shows the histogram of the average number of episodes won by the dribbler during the training process. Bins of 500 episodes were used.
Throughout the training process, the dribbler won, on average, $23,607$ episodes ($\approx 47\%$). From Figure 3, we can see that it greatly improves its average performance as the number of episodes increases. At the end of the training process, it is winning slightly less than $53\%$ of the time.
Qualitatively, the dribbler seems to learn two major rules. In the first one, when the adversary is at a considerable distance, the dribbler keeps kicking the ball to the opposite side in which the adversary is located until the angle between them is in the range $[90, 270]$, *i.e.*, when the adversary is behind the dribbler. After that, the dribbler starts to kick the ballforward. An illustration of this rule can be seen in Figure 4.
The second rule seems to occur when the adversary is relatively close to and in front of the dribbler. Since there is no way for the dribbler to move forward or diagonally without putting the possession at risk, it then holds the ball until the angle between it and the adversary is in the range $[90, 270]$. Thereafter, it starts to advance by kicking the ball forward. An illustration of this rule can be seen in Figure 5.
After the training process, we randomly generated 10,000 initial configurations to test our solution. This time, the dribbler always selected the macro-action with the highest estimated value, *i.e.*, we set $\epsilon = 0$. Further, the weights of the receptive fields were not updated, *i.e.*, we set $\alpha = 0$. We used the receptive fields’ weights resulting from the simulation where the dribbler obtained the highest success rate. The result of this experiment was even better. The dribbler won 5,795 episodes, thus obtaining a success rate of approximately $58\%$.
![Example of the first major rule learned by the dribbler. (Top Left) The adversary is at a considerable distance from the dribbler. (Top Right) The dribbler starts to kick the ball to the opposite side in which the adversary is located. (Bottom Left) The angle between the adversary and the dribbler is in the range $[90, 270]$. Consequently, the dribbler starts to kick the ball forward. (Bottom Right) The dribbler keeps kicking the ball forward.](policy1.eps)
![Example of the second major rule learned by the dribbler. (Top Left) The adversary is close to and in front of the dribbler. (Top Right) The dribbler holds the ball so as not to lose possession. (Bottom Left) The dribbler keeps holding the ball. (Bottom Right) The angle between the adversary and the dribbler is the range $[90, 270]$. Consequently, the dribbler starts to advance by kicking the ball forward.](policy2.eps)
One-Dimensional CMACs
---------------------
For comparison’s sake, we repeated the above experiment using the original solution proposed by Stone *et al.* [@Stone:Keepaway]. It consists of the same learning algorithm presented in Section 3, but using one-dimensional CMACs. In detail, each layer is an interval along a state variable. In this way, the feature set $\mathbb{F}_s$ is now composed by $32 \times 5 = 160$ excited receptive fields, *i.e.*, $32$ excited receptive fields for each state variable.
One of the main advantages of using one-dimensional CMACs is that it is possible to circumvent the curse of dimensionality. In detail, the state space does not grow exponentially in the number of state variables because dependence between variables is not taken into account.
Figure 6 shows the histogram of the average number of episodes won by the dribbler during the training process. Each simulation took, on average, approximately 43 hours. Throughout the training process, the dribbler won, on average, $16,278$ episodes ($\approx 33\%$). From Figure 6, we can see that the learning algorithm converges much faster when using one-dimensional CMACs. However, its average performance is considerably worse. At the end of the training process, the dribbler is winning, on average, less than $30\%$ of the time.
After the training process, we tested this solution using the same 10,000 initial configurations previously generated. Again, we set $\epsilon = \alpha = 0$, and used the receptive fields’ weights resulting from the simulation where the dribbler obtained the highest success rate. The result of this experiment was slightly better. The dribbler won 3,701 episodes, thus obtaining a success rate of approximately $37\%$.
Qualitatively, the dribbler seems to learn a rule similar to the one shown in Figure 4. The major difference is that it always kicks the ball to the opposite side in which the adversary is located, it does not matter its distance from the adversary’s location. Consequently, it is highly unlikely that the dribbler succeeds when the adversary is close to it.
We conjecture that one of the main reasons for such a poor performance of the reinforcement learning algorithm when using one-dimensional CMACs is that it does not take into account dependence between variables, *i.e.*, they are treated individually. Hence, such approach may throw away valuable information. For example, the variables $ang(\mbox{ball}, \mbox{adversary})$ and $dist(\mbox{ball}, \mbox{adversary})$ together describe the position of adversary with respect to the ball. However, they do not make as much sense when considered individually.
Related Work
============
Reinforcement learning has long been applied to the robot soccer domain. For example, Andou [@Andou:Reinforcement_Learning_Robocup] uses “observational reinforcement learning" to refine a function that is used bythe soccer agents for deciding their positions on the field. Riedmiller *et al.* [@Riedmiller:Reinforcement_Learning_Robocup] use reinforcement learning to learn low-level soccer skills, such as kicking and ball-interception. Nakashima *et al.* [@fuzzyQlearning] propose a reinforcement learning meth-od called “fuzzy Q-learning", where an agent determines itsaction based on the inference result of a fuzzy rule-based system. The authors apply the proposed method to the sce-nario where a soccer agent learns to intercept a passed ball.
Arguably, the most successful application is due to Stone *et al.* [@Stone:Keepaway]. They propose the “keepaway task", which consists of two teams, the keepers and the takers, where the former tries to keep control of the ball for as long as possible, while the latter tries to gain possession. Our solution to the soccer dribbling task follows closely the solution proposed by thoseauthors to learn the keepers’ behavior. Iscen and Erogul [@Iscen:takes] use similar solution to learn a policy for the takers.
Gabel *et al.* [@NeuroHassle] propose a task which is the opposite of the soccer dribbling task, where a defensive player must interfere and disturb the opponent that has possession of the ball. Their solution to that task uses a reinforcement learning algorithm with a multilayer neural network for function approximation.
Kalyanakrishnan *et al.* [@Stone:half_field] present the “half-field offense task", a scenario in which an offense team attempts to outplay a defense team in order to shoot goals. Those authors pose that task as a reinforcement learning problem, and propose a new learning algorithm for dealing with it.
More closely related to our work are reinforcement learn-ing-based solutions to the task of conducting the ball (*e.g.*, [@riedmiller2008learning]), which can be seen as a simplification of the dribbling task since it usually does not include adversaries.
Conclusion
==========
We proposed a reinforcement learning solution to the soccer dribbling task, a scenario in which an agent has to go from the beginning to the end of a region keeping possession of the ball, while an adversary attempts to gain possession. Our solution combined the Sarsa algorithm with CMAC for function approximation. Empirical results showed that, after the training period, the dribbler was able to accomplish its task against a strong adversary around $58\%$ of the time.
Although we restricted ourselves to the soccer domain, dribbling, as defined in this paper, is also common in other sports, *e.g.*, hockey, basketball, and football. Thus, the proposed solution can be of value to dribbling tasks of other sports games. Furthermore, we believe that the soccer dribbling task is an excellent benchmark for comparing different machine learning techniques because it involves a complex problem, and it has a well-defined objective.
There are several exciting directions for extending this work. From a practical perspective, we intend to analyze the scalability of our solution, *i.e.*, to study how it performs with training fields of distinct sizes and against different adversaries. Further, we are considering schemes to extend our solution to the original partially observable environment, where the available information is incomplete and noisy.
As stated before, a more informative state representation could be obtained by using more state variables. The major problem of adding extra variables to our solution is that CMAC’s complexity increases exponentially with its dimensionality. Due to this fact, we are considering other solutions which use function approximations whose complexity is unaffected by dimensionality *per se*, *e.g.*, the Kanerva coding (for example, see Kostiadis and Hu’s work [@kostiadis2001kabage]).
Finally, we note that when modeling the soccer dribbling task as a reinforcement learning problem, we do not directly use intermediate rewards (they are all set to zero). However, they may make the learning process more efficient (for example, see [@Ng:policy_invariance]). Thus, we intend to investigate the influence of intermediate rewards on the final solution in future work.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank W. Thomas Miller, Filson H. Glanz, and others from the Department of Electrical and Computer Engineering at the University of New Hampshire for making their CMAC code available.
[^1]: Arthur Carvalho is with the David Cheriton School of Computer Science at the University of Waterloo, Waterloo, Ontario, Canada (email: [[email protected]]{}).
[^2]: Renato Oliveira is with the Center of Informatics at the Federal University of Pernambuco, Recife, Pernambuco, Brazil (email: [[email protected]]{}).
[^3]: Henceforth, we use the terms *action* and *macro-action* interchangeably, while always distinguishing *primitive actions*.
[^4]: The implementation of the learning algorithm can be found at:<http://sites.google.com/site/soccerdribbling/>
|
---
abstract: 'An analysis of adiabatic perturbations of a perfect fluid is performed to first-order about a general FLRW background using the $1+3$ covariant and gauge-invariant formalism. The analog of the Mukhanov-Sasaki variable and the canonical variables needed to quantise respectively the scalar and tensor perturbations in a general FLRW background space-time are identified. The dynamics of the vector perturbations is also discussed.'
author:
- 'Sylvain D. Brechet'
- 'Michael P. Hobson'
- 'Anthony N. Lasenby'
bibliography:
- 'references.bib'
title: 'First-order adiabatic perturbations of a perfect fluid about a general FLRW background using the $\mathbf{1+3}$ covariant and gauge-invariant formalism'
---
Introduction
============
Cosmological perturbation theory about a given background metric was initiated by Lifshitz [@Lifshitz:1946] and extended notably by Bardeen [@Bardeen:1980], Mukhanov, Feldmann & Brandenberger [@Mukhanov:1992], and Kodama & Sasaki [@Sasaki:1984]. This is the standard approach to performing a perturbation analysis of general relativity in order to describe the ‘realistic’ dynamics of cosmological models. It has the disadvantage that truly physical results can be obtained only after completely specifying the correspondence between the ‘real’ perturbed space-time and a ‘fictitious’ unperturbed background space-time, which is chosen to be highly symmetric. Such a correspondence is not uniquely defined and changes under a gauge transformation. The degrees of freedom in the definition of the correspondence, or the gauge freedom, leave unphysical gauge modes in the dynamical equations describing the evolution of the perturbations [@Bruni:1992]. This gauge problem [@Lifshitz:1963; @Sachs:1967] is inherent to such a ‘background-based’ perturbation approach. Indeed, the metric, and consequently the Einstein equations, have $10$ degrees of freedom whereas the dynamics is determined by $6$ parameters only. Therefore, the $4$ remaining degrees of freedom are directly related to the gauge.
To solve this problem, Bardeen [@Bardeen:1980] determined a set of gauge-invariant quantities to describe the perturbations and derived their dynamical equations. These quantities are mathematically well defined but do not have a transparent geometrical meaning since they are defined with respect to a particular coordinate system [@Stewart:1990], and their physical meaning is obscure [@Ellis:1989b].
An alternative way to circumvent the gauge problem is to follow the $1+3$ covariant approach, which was developed by Hawking [@Hawking:1966] and extended by Olsen [@Olson:1976] and Ellis & Bruni [@Ellis:1989]. The aim of this approach is to study the dynamics of real cosmological fluid models in a physically transparent manner. This formalism relies on covariantly defined variables, which are gauge-invariant by construction [@Bruni:1992], thus simplifying the methodology and clarifying the physical interpretation of the models. It also allows the metric to be arbitrary. This approach admits a covariant and gauge-invariant linearisation that allows a perturbation analysis to be performed in a direct manner [@Challinor:2000].
The basic philosophy of the perturbation theory based on the $1+3$ covariant formalism is different from the ‘background-based’ perturbation theory [@Ellis:1990]. Instead of starting with a background space-time and then perturbing it, the approach begins with an inhomogeneous and anisotropic (real) space-time, which reduces to the background space-time on large scales. In our case, we will take the background to be a homogeneous and isotropic FLRW space-time, although this formalism allows in principle for more complicated backgrounds. Therefore, the ‘real’ space-time has been appropriately called an almost FLRW space-time [@Ellis:1989b]. In this perturbation theory, the approximation takes place by neglecting higher-order terms in the exact equations when the values of the kinematic and dynamic variables are close to those they would take in the background FLRW space-time. The analysis is performed in the real space-time and the dynamical equations are subsequently linearised. The background solution is simply the zero-order approximation of the exact solution of the dynamical equations.
Although in the $1+3$ covariant formalism, the scalar, vector and tensor perturbations are handled in a unified way [@Challinor:2000], they decouple to first-order and can be studied independently [@Mukhanov:1992]. For a perfect fluid in a spatially-curved case, the Bardeen variable for the scalar perturbations in the $1+3$ covariant approach was first identified by Woszczyna & Kulak [@Woszczyna:1989] and the curvature perturbation by Bruni, Dunsby & Ellis [@Bruni:1992]. The first attempt to establish an explicit relation between the perturbation variables in the $1+3$ covariant formalism and the corresponding variables in the ‘background-based’ approach was made by Goode [@Goode:1989].
In the $1+3$ formalism, the time derivative of a physical quantity is defined usually as the projection of the covariant derivative of the quantity on the worldline as outlined in Sec. \[Kinematics in the 1+3 covariant formalism\], but this is not the only way to define covariantly a time derivative as Thiffeault showed in [@Thiffeault:2001]. As Langlois & Vernizzi suggested [@Langlois:2005], the Lie derivative along the worldline of a fluid element is another possible definition. However, the Lie derivative of a scalar field along the worldline is identical to time derivative of the scalar field along the worldline. Since, in this publication, we aim to identify scalar quantities, which are respectively scalar perturbations and the scalar amplitude of vector and tensor perturbations, we will define the time derivative of physical quantities to be the projection of the covariant derivative of the quantity on the worldline.
It is worth mentioning that, recently, Pitrou & Uzan [@Pitrou:2007], working in a spatially-flat case, used Lie derivatives to recast the dynamical equations for the scalar and tensor perturbations in the $1+3$ covariant formalism in order to identify perturbation variables which are similar to the Sasaki-Mukhanov and canonical variables. In particular, they sought to identify the scalar and tensor variables that map to the Mukhanov-Sasaki and to the tensor canonical variables when considering a spatially-flat almost FLRW universe. In the current work, we seek the extension to general FLRW universes. However, there are some further differences with the work of Pitrou & Uzan. Firstly, in the scalar case, the perturbation variable $v_a$ that Pitrou & Uzan obtained is not a scalar but a covector. We will seek a scalar here. Secondly, for tensor perturbations in the spatially-flat case, Pitrou & Uzan obtained a wave equation in terms of Lie derivatives of tensor fields. In the current work, we will extend their analysis by seeking to identify canonical variables, which correspond to the scalar amplitudes of the tensor perturbations. These scalar amplitudes are the canonical variables defined by Grishchuk [@Grishchuk:1974]. We will be using the name ‘canonical’ to refer to these variables henceforth in this publication. Finally, Pitrou & Uzan did not consider a perfect fluid analysis as we carry out here, but instead restricted their analysis to a single scalar field.
Thus the aim of this publication is to identify, in the $1+3$ covariant approach, for adiabatic perturbations of a perfect fluid in spatially-curved FLRW models, the analog of the Mukhanov-Sasaki variable [@Mukhanov:1992; @Sasaki:1984] and the canonical variables [@Grishchuk:1974] needed to quantise respectively the scalar and tensor perturbations. The quantisation of fields in a spatially-curved space-time lies outside the scope of this paper. However, this topic has been addressed in detail by Birrell & Davis [@Birrell:1982] and Fulling [@Fulling:1989]. Our approach broadens the scalar perturbations analysis performed by Woszczyna & Kulak [@Woszczyna:1989], Bruni, Dunsby & Ellis [@Bruni:1992] and Lyth & Woszczyna [@Lyth:1995] in a spatially-curved case.
The structure of this publication is as follows. In Sec. \[Kinematics in the 1+3 covariant formalism\], we give a concise description of the kinematics in the $1+3$ covariant approach. In Sec. \[First-order adiabatic dynamics of a perfect fluid about an FRW background\], we establish the first-order dynamical equations for adiabatic perturbations of a perfect fluid in the 1+3 covariant approach. In Sec. \[Scalar perturbations\], we identify the analog of the Mukhanov-Sasaki variable for the scalar perturbations. In Sec. \[Vector perturbations\], we determine the dynamics of the scalar amplitude of the vector perturbations. In Sec. \[Tensor perturbations\], we finally identify the canonical variables for the tensor perturbations.
For convenience, we follow Hawking [@Hawking:1966] and Ellis [@Ellis:1989] by adopting the $(-,+,+,+)$-signature for the metric. This choice of signature is particularly appropriate for the $1+3$ covariant formalism. Indeed, the dynamical equations are projected on the local spatial hypersurfaces, which are positively defined for our signature convention. The correspondence for dynamical quantities expressed in terms of the opposite signature can be found in [@Brechet:2007].
Kinematics in the 1+3 covariant formalism {#Kinematics in the 1+3 covariant formalism}
=========================================
We will briefly outline the basics of the $1+3$ covariant formalism describing the fluid kinematics. To introduce this formalism, we follow Ellis & van Elst’s approach [@Ellis:1999]. The approach is based on the $1+3$ decomposition of geometric quantities with respect to a fundamental $4$-velocity $u^{a}$ that uniquely determines the frame and the worldline of every infinitesimal volume element of fluid, $$u^{a}=\frac{dx^{a}}{d\tau}\ ,\ \ \ \ \ \ \ u_{a}u^{a}=-1\ ,$$ where $x^{a}$ are arbitrary cosmic coordinates, and $\tau$ is the proper time measured along the worldlines. In the context of a general cosmological model, we require that the $4$-velocity be chosen in a physical manner such that in the FLRW limit the dipole of the cosmic microwave background radiation vanishes. This condition is necessary to ensure the gauge-invariance of the approach.
The $4$-velocity $u^{a}$ defines locally and in a unique fashion two projection tensors, $$\begin{split}
&U_{ab} = -u_{a}u_{b}\ ,\\
&h_{ab} = g_{ab}+u_{a}u_{b}\ ,
\end{split}$$ which satisfy the following properties, $$\begin{split}
&{U^{a}}_{c}{U^{c}}_{b} = {U^{a}}_{b} \ , \ \ {U^{a}}_{a} = 1 \ , \ \ U_{ab}u^{b} = u_{a}\ ,\\
&{h^{a}}_{c}{h^{c}}_{b} = {h^{a}}_{b} \ , \ \ \ \ {h^{a}}_{a} = 3 \ , \ \, \ h_{ab}u^{b} = 0\ .
\end{split}$$
The first projects parallel to the 4-velocity vector $u^{a}$, and the second determines the metric properties of the instantaneous rest frame of the fluid. It is useful to introduce the totally antisymmetric Levi-Civita tensor $\varepsilon_{abc}$, which is defined in the rest frame of the fluid and thus satisfies, $$\varepsilon_{abc}u^{c} = 0\ .$$
Moreover, we define two projected covariant derivatives which are the time projected covariant derivative along the worldline (denoted $\mathbf{\dot{}}\ $) and the spatially projected covariant derivative (denoted $D_{a}$). For any quantity ${Q^{a\dots}}_{b\dots}$, these are respectively defined as $$\begin{split}
&{{\dot Q}^{a\dots}}_{\ \ \ \, b\dots}\equiv u^{c}\nabla_{c}{Q^{a\dots}}_{b\dots}\ ,\label{time covariant}\\
&D_{c}{Q^{a\dots}}_{b\dots}\equiv {h^{f}}_{c}{h^{a}}_{d}\dots {h^{e}}_{b}\dots\nabla_{f}{Q^{d\dots}}_{e\dots}\ .
\end{split}$$ Furthermore, the kinematics and the dynamics are determined by projected tensors that are orthogonal to $u^{a}$ on every index. The angle brackets are used to denote respectively orthogonal projections of vectors $V^{a}$ and the orthogonally projected symmetric trace-free part $(\mathrm{PSTF})$ of rank-$2$ tensors $T^{ab}$ according to, $$\begin{split}
&V^{\langle a\rangle} ={h^{a}}_{b}V^{b}\ ,\\
&T^{\langle ab\rangle} =\left({h^{(a\vphantom)}}_{c}{h^{\vphantom(b)}}_{d}-{\textstyle\frac{1}{3}}h^{ab}h_{cd}\right)T^{cd}\ .
\end{split}$$ For convenience, the angle brackets are also used to denote the orthogonal projections of covariant time derivatives of vectors and tensors along the worldline $u^{a}$ as follows, $$\begin{split}
&\dot{V}^{\langle a\rangle} ={h^{a}}_{b}\dot{V}^{b}\ ,\\
&\dot{T}^{\langle ab\rangle} =\left({h^{(a\vphantom)}}_{c}{h^{\vphantom(b)}}_{d}-{\textstyle\frac{1}{3}}h^{ab}h_{cd}\right)\dot{T}^{cd}\ .
\end{split}$$ Note that, in general, the time derivative of vectors and tensors does not commute with the projection of these quantities on the spatial hypersurfaces according to, $$\begin{split}
&\dot{V}^{\langle a\rangle} \neq({V^{\langle a\rangle}})^{\cdot}\ ,\\
&\dot{T}^{\langle ab\rangle} \neq({T^{\langle ab\rangle}})^{\cdot}\ .
\end{split}$$ The projection of the covariant time derivative of a quantity ${Q^{a\dots}}_{b\dots}$ on the spatial hypersurfaces is defined as, $$^{(3)}\left({Q^{a\dots}}_{b\dots}\right)^{\cdot}\equiv{h^{a}}_{c}\dots {h^{d}}_{b}\dots u^{e}\nabla_{e}{Q^{c\dots}}_{d\dots}\ .$$ It is also useful to define the projected covariant curl as, $${\,\mbox{curl}\,}Q_{a\dots b}\equiv\varepsilon_{cd\langle a}D^{c}{Q^{d}}_{\dots b\rangle}\ .\label{curl}$$
Information relating to the kinematics is contained in the covariant derivative of $u^{a}$ which can be split into irreducible parts, defined by their symmetry properties, $$\begin{aligned}
\begin{split}
\nabla_{a}u_{b}&=-u_{a}a_{b}+D_{a}u_{b}\\
&=-u_{a}a_{b}+{\textstyle\frac{1}{3}}\Theta h_{ab}+\sigma_{ab}+\omega_{ab}\ ,\label{kinematics}
\end{split}\end{aligned}$$ where
- $a^{a}\equiv u^{b}\nabla_{b}u^{a}$ is the relativistic acceleration vector, representing the degree to which matter moves under forces other than gravity.
- $\Theta\equiv D_{a}u^{a}$ is the scalar describing the volume rate of expansion of the fluid (with $H={\textstyle\frac{1}{3}}\Theta$ the Hubble parameter).
- $\sigma_{ab}\equiv D_{\langle a}u_{b\rangle}$ is the trace-free rate-of-shear tensor describing the rate of distortion of the fluid flow.
- $\omega_{ab}\equiv D_{[a\vphantom]}u_{\vphantom[b]}$ is the antisymmetric vorticity tensor describing the rotation of the fluid relative to a non-rotating frame.
These kinematical quantities have the following properties, $$\begin{aligned}
&a_{a}u^{a}=0\ ,\nonumber\\
&\sigma_{ab}u^{b}=0\ ,\ \ \sigma_{ab}=\sigma_{(ab)}\ ,\ \ {\sigma^{a}}_{a}=0\ ,\\
&\omega_{ab}u^{b}=0\ ,\ \ \omega_{ab}=\omega_{[ab]}\ ,\ \ {\omega^{a}}_{a}=0\ .\nonumber\end{aligned}$$
Note that in presence of vorticity the spatially projected covariant derivatives do not commute, which implies that any scalar field $S$ has to satisfy the non-commutation relation, $$D_{[a\vphantom]}D_{\vphantom[b]}S=\omega_{ab}\dot{S}\ .\label{kin non com}$$
It is useful to introduce a vorticity pseudovector $\omega^{a}$, which is the dual of the vorticity tensor $\omega_{bc}$ and is defined as, $$\omega^{a}\equiv{\textstyle\frac{1}{2}}\varepsilon^{abc}\omega_{bc}\ ,$$ and a vorticity scalar given by, $$\omega=\left(\omega_{a}\omega^{a}\right)^{1/2}\ .\label{Vorticity scal}$$
To build a covariant linear perturbation theory, we will linearise the quantities of the $1+3$ covariant formalism about an FLRW background space-time which is, by definition, homogeneous and isotropic. Gauge invariance of the perturbed quantities is guaranteed by the Gauge Invariance Lemma, which states that if a quantity vanishes in the background space-time, then it is gauge invariant at first-order [@Stewart:1974]. The homogeneity of the background space-time implies that the acceleration $a^{a}$ is a first-order variable since it vanishes in the background space-time (i.e. at zeroth order). Similarly, the isotropy of the background space-time implies that rate of shear $\sigma_{ab}$ is also a first-order variable. Finally, the existence of hypersurfaces orthogonal to the worldline in the background space-time implies that vorticity tensor $\omega_{ab}$ and the vorticity covector $\omega_{a}$ are first-order variables. Therefore, the only zero-order kinematic quantity is the rate of expansion $\Theta$.
In the background space-time, which means to zero-order in the dynamical variables, the covariant derivative of the worldline thus becomes, $$\nabla_{a}u_{b}={\textstyle\frac{1}{3}}\Theta h_{ab}\ .\label{zero kinematics}$$
It is also useful to define (up to some constant factor) a zero-order scale factor $R$ such that, $$\Theta=3H\equiv\frac{\dot{R}}{R}\ ,\label{scale factor}$$ where $H$ is the cosmic Hubble scale factor.
Finally, it is also convenient to introduce a conformal time variable $\hat{\tau}$ satisfying the differential relation, $$d\hat{\tau}\equiv\frac{d\tau}{R}\ ,\label{conformal time}$$ which implies that the derivatives with respect to cosmic time $\tau$ and conformal time $\hat{\tau}$ of a quantity ${Q^{a\cdots}}_{b\cdots}$ are related by, $${Q^{\prime\,a\cdots}}_{b\cdots}=R{\dot{Q}^{a\cdots}}_{\phantom{a\cdots}b\cdots}\ ,\label{derivatives relation}$$ where a prime denotes a derivative with respect to conformal time $\hat{\tau}$. The conformal Hubble scale factor $\mathcal{H}$ is related to the cosmic scale factor $H$ by $\mathcal{H}=RH$.
First-order adiabatic dynamics of a perfect fluid about an FLRW background {#First-order adiabatic dynamics of a perfect fluid about an FRW background}
==========================================================================
We will now use the $1+3$ covariant formalism, outlined in Sec. \[Kinematics in the 1+3 covariant formalism\], to describe the adiabatic dynamics of a perfect fluid to first-order in the dynamical variables, which means neglecting second- and higher-order products of first-order dynamical variables. We then perform a perturbation analysis of such a fluid about an FLRW background, which will be used in Sec. \[Scalar perturbations\]-\[Tensor perturbations\] to describe scalar, vector and tensor perturbations respectively. Thus, we require the cosmological fluid to be highly symmetric on large scales but allow for generic inhomogeneities on small scales.
The dynamics of a perfect fluid is described by the Einstein field equations, which read, $$\begin{aligned}
R_{ab}-{\textstyle\frac{1}{2}}g_{ab}\mathcal{R}=\kappa T_{ab}\ ,\label{Einstein eq} \end{aligned}$$ where $R_{ab}$ and $\mathcal{R}$ are respectively the Ricci tensor and scalar. The dynamical model is fully determined by the matter content and the curvature. The matter content is described by the stress-energy momentum tensor $T_{ab}$. For a perfect fluid, using the $1+3$ formalism, it can be recast as, $$\begin{aligned}
T_{ab}=\rho u_{a}u_{b}+ph_{ab}\ ,\label{stress energy mom 1+3}\end{aligned}$$ where $\rho$ is the energy density and $p$ the pressure of the fluid. We assume the fluid to be a specific linear barotropic fluid so that it satisfies the equation-of-state, $$\begin{aligned}
p=w\rho\ ,\label{eq of state}\end{aligned}$$ where $w$ is the equation-of-state parameter. The energy density $\rho$ and the pressure $p$ are zero-order variables that do not vanish on the background.
For an adiabatic flow, the speed of sound is defined as $$c_s^2\equiv\frac{dp}{d\rho}\ ,\label{speed of sound}$$ and the time derivative of the equation-of-state parameter satisfies, $$\dot{w}=-\Theta(c_s^2-w)(1+w)\ ,\label{deriv w}$$ and is derived using the energy conservation equation . Note that from , it follows that $c_s^2=w$ if $\dot{w}=0$.
All the information related to the curvature is encoded in the Riemann tensor which can be decomposed as [@Hawking:1966], $${R^{ab}}_{cd}={C^{ab}}_{cd}+2{\delta^{[a\vphantom]}}_{[c\vphantom]}{R^{\vphantom[b]}}_{\vphantom[d]}- {\textstyle\frac{1}{3}}\mathcal{R}{\delta^{a}}_{[c\vphantom]}{\delta^{b}}_{\vphantom[d]}\ ,\label{Riemann tens}$$ where ${C^{ab}}_{cd}$ is the Weyl tensor constructed to be the trace-free part of the Riemann tensor.
By analogy to classical electrodynamics, the Weyl tensor itself can be split, relative to the worldline $u^{a}$, into an ‘electric’ and a ‘magnetic’ part [@Hawking:1966] according to, $$\begin{aligned}
&E_{ab} = C_{acbd}u^{c}u^{d}\ ,\label{Elec}\\
&H_{ab} = \,^{\ast}C_{acbd}u^{c}u^{d} =
{\textstyle\frac{1}{2}}\varepsilon_{ade}{C^{de}}_{bc}u^{c}\ ,\label{Magn}\end{aligned}$$ where $\vphantom{}^{\ast}C_{abcd}$ is the dual of the Weyl tensor. Their properties follow directly from the symmetries of the Weyl tensor, $$\begin{split}
&E_{ab}u^{b}=0\ ,\ \ E_{ab}=E_{(ab)}\ ,\ \ \, {E^{a}}_{a}= 0\ ,\\
&H_{ab}u^{b}=0\ ,\ \ H_{ab}=H_{(ab)}\ ,\ \ {H^{a}}_{a}= 0\ .\label{Electric magnetic prop}
\end{split}$$ These electric and magnetic parts of the Weyl tensor represent the ‘free gravitational field’, enabling gravitational action at a distance and describing tidal forces and gravitational waves. The Weyl tensor vanishes on a conformally flat background space-time such as FLRW models. Thus, the electric $E_{ab}$ and magnetic part $H_{ab}$ of the Weyl tensor are first-order variables.
The Ricci tensor $R_{ab}$ is simply obtained by substituting the expression for the stress energy momentum tensor into the Einstein field equations , $$\begin{aligned}
R_{ab}={\textstyle\frac{\kappa}{2}}\left(\rho+3p\right)u_{a}u_{b}
+{\textstyle\frac{\kappa}{2}}\left(\rho-p\right)h_{ab}\ .\label{Ef Ricci tensor}\end{aligned}$$
The Riemann tensor $R_{abcd}$ can now be recast in terms of the Ricci tensor , the electric and magnetic parts of the Weyl tensor according to the decomposition in the following way, $$\begin{split}
{R^{ab}}_{cd}\ = &{\textstyle\frac{2}{3}}\kappa\left(\rho + 3p\right)u^{[a\vphantom]}u_{[c\vphantom]}{h^{\vphantom[b]}}_{\vphantom[d]}
+ {\textstyle\frac{2}{3}}\kappa\rho {h^{a}}_{[c\vphantom]}{h^{b}}_{\vphantom[d]}\\
\phantom{{R^{ab}}_{cd}\ = }&+4u^{[a\vphantom]}u_{[c\vphantom]}{E^{\vphantom[b]}}_{\vphantom[d]} + 4{h^{[a\vphantom]}}_{[c\vphantom]}{E^{\vphantom[b]}}_{\vphantom[d]}\\
\phantom{{R^{ab}}_{cd}\ = }&+2\varepsilon^{abe}u_{[c\vphantom]}H_{\vphantom[d]e} + 2\varepsilon_{cde}u^{[a\vphantom]}H^{\vphantom[b]e}\ .\label{Riemann tensor}
\end{split}$$ The Riemann tensor restricted to the spatial hypersurface, ${^{(3)}R^{ab}}_{cd}$, is related to the Riemann tensor defined on the whole space-time, ${R^{ab}}_{cd}$, by $$\begin{aligned}
^{(3)}{R^{ab}}_{cd}={h^{a}}_{e}{h^{b}}_{f}{h^{g}}_{c}{h^{h}}_{d}{R^{ef}}_{gh}-2{v^{a}}_{[c\vphantom]}{v^{b}}_{\vphantom[d]}\ .\label{Riemann tensor 3} \end{aligned}$$ where the tensor $v_{ab}$ is defined as, $$v_{ab}\equiv D_{a}u_{b}\ .\label{Expansion tensor}$$ For a perfect fluid, using the decomposition , the spatial Riemann tensor becomes, $$^{(3)}{R^{ab}}_{cd}={\textstyle\frac{2}{3}}\kappa\rho{h^{a}}_{[c\vphantom]}{h^{b}}_{\vphantom[d]}+4{h^{a}}_{[c\vphantom]}{E^{b}}_{\vphantom[d]}-2{v^{a}}_{[c\vphantom]}{v^{b}}_{\vphantom[d]}\ .\label{3 Riemann tensor}$$ To zero-order, the decomposition of the spatially projected Riemann tensor reduces to, $$^{(3)}{R^{ab}}_{cd}=\frac{2K}{R^2}{h^{a}}_{[c\vphantom]}{h^{b}}_{\vphantom[d]}\ .\label{Riemann 3 zero}$$ where $K$ is the curvature parameter, which is related to the Gaussian curvature $\mathcal{K}$ by, $${^{(3)}R}=\mathcal{K}=\frac{6K}{R^2}\ ,\label{Gaussian curvature}$$ where $^{(3)}\mathcal{R}$ is the spatial curvature scalar, which is obtained by twice contracting the spatially projected Riemann tensor .
In general, there are three sets of dynamical equations for a perfect fluid. These sets are derived, respectively, from the Ricci identities, the Bianchi identities, once- and twice-contracted. We present now each set in turn and expand the dynamical equations to first-order about an FLRW background space-time.
Ricci identities
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The first set of dynamical equations arises from the Ricci identities. These identities can firstly be applied to the whole space-time and secondly to the spatial hypersurfaces according to, $$\begin{aligned}
&\nabla_{[a\vphantom]}\nabla_{\vphantom[b]}u_{c}={\textstyle\frac{1}{2}}R_{abcd}u^{d}\ ,\label{Ricci identities}\\
&D_{[a\vphantom]}D_{\vphantom[b]}v_{c}={\textstyle\frac{1}{2}}{\vphantom{a}^{(3)}}R_{abcd}v^{d}\ ,\label{Projected Ricci}\end{aligned}$$ where the spatial vectors $v^{a}$ are orthogonal to the worldline, i.e. $v^{a}u_{a}=0$.
The information contained in the Ricci identities - can be extracted by projecting them on different hypersurfaces using the decomposition of the corresponding Riemann tensors - and following the same procedure as in [@Brechet:2007].
To first-order, the Ricci identities applied to the whole space-time yield three propagation equations, which are respectively the Raychaudhuri equation, the rate of shear propagation equation and the vorticity propagation equation, $$\begin{aligned}
&\dot{\Theta}=-{\textstyle\frac{1}{3}}\Theta^2-{\textstyle\frac{\kappa}{2}}\left(\rho+3p\right)+D^ba_b\ ,\label{Raychaudhuri eq}\\
&\dot{\omega}_{\langle a\rangle}=-{\textstyle\frac{2}{3}}\Theta\,\omega_{a}+{\textstyle\frac{1}{2}}{\,\mbox{curl}\,}a_{a}\ ,\label{Vorticity prop eq}\\
&\dot{\sigma}_{\langle ab\rangle}=-{\textstyle\frac{2}{3}}\Theta\,\sigma_{ab}-E_{ab}+D_{\langle a}a_{b\rangle}\ ,\label{Rate shear prop eq}\end{aligned}$$ and three constraint equations, $$\begin{aligned}
&D^{a}\omega_{a}=0\ ,\label{Constr eq 0}\\
&D^{b}\sigma_{ab}={\textstyle\frac{2}{3}}D_{a}\Theta-{\,\mbox{curl}\,}\omega_{a}\ ,\label{Constr eq 1}\\
&H_{ab}={\,\mbox{curl}\,}\sigma_{ab}-D_{\langle a}\omega_{b\rangle}\ .\label{Constr eq 2}\end{aligned}$$
The Ricci identities applied to the spatial hypersurfaces express the spatial curvature. Their contractions yield the spatial Ricci tensor $^{(3)}R_{ab}$ and scalar $^{(3)}\mathcal{R}$ respectively, which to first-order are given by, $$\begin{aligned}
&^{(3)}R_{ab}={\textstyle\frac{1}{3}}^{(3)}\mathcal{R}{h_{ab}}-{\textstyle\frac{1}{3}}\Theta\left(\sigma_{ab}-\omega_{ab}\right)+E_{ab}\ ,\label{Ricci tensor eq}\\
&\mathcal{K}=-{\textstyle\frac{2}{3}}\Theta^2+2\kappa\rho\ .\label{Gauss Codacci}\end{aligned}$$ To zero-order, using the expression for the Gaussian curvature , the contractions of the spatial Ricci identities and reduce respectively to, $$\begin{aligned}
&^{(3)}R_{ab}={\textstyle\frac{1}{3}}^{(3)}\mathcal{R}{h_{ab}}\ ,\label{Ricci tens first}\\
&\frac{1}{9}\Theta^2=\frac{\kappa}{3}\rho-\frac{K}{R^2}\ .\label{Friedmann}\end{aligned}$$ Note that the above expression is the Friedmann equation. It is useful to recast the Friedmann and Raychaudhuri equations in terms of conformal time. To zero-order, these equations are respectively given by, $$\begin{aligned}
&\mathcal{H}^2={\textstyle\frac{\kappa}{3}}\rho R^2-K\ ,\label{Friedmann 0}\\
&\mathcal{H}^{\prime}=-{\textstyle\frac{\kappa}{6}}\rho R^2(1+3w)\ .\label{Raychaudhuri 0}\end{aligned}$$ Finally, it is convenient to recast them as, $$\begin{aligned}
&\mathcal{H}^2-\mathcal{H}^{\prime}+K={\textstyle\frac{\kappa}{2}}\rho R^2\left(1+w\right)\ ,\label{Conformal 1}\\
&\mathcal{H}^{\prime}=-{\textstyle\frac{\kappa}{2}}\left(1+3w\right)(\mathcal{H}^2+K)\ .\label{Conformal 2}\end{aligned}$$
Once-contracted Bianchi identities {#Once-contracted Bianchi identities}
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The second and third set of dynamical equations are contained in the Bianchi identities. The Riemann tensor satisfies the Bianchi identities as follows, $$\nabla^{[e\vphantom]}{R^{\vphantom[ab]}}_{cd}=0 \
.\label{Riemann Bianchi}$$ By substituting the expression for the Riemann tensor decomposition and the effective Einstein field equations into the Bianchi identities and contracting two indices ($d=e$), the once-contracted Bianchi identities are found to be, $$\nabla^{d}{C^{ab}}_{cd}+\nabla^{[a\vphantom]}{R^{\vphantom[b]}}_{c}+{\textstyle\frac{1}{6}}{\delta_{c}}^{[a\vphantom]}\nabla^{\vphantom[b]}\mathcal{R}=0\ .\label{Simple Bianchi}$$ In a similar manner to the Ricci identities, the information stored in the once-contracted Bianchi identities has to be projected along the worldlines $u^{a}$ and on the spatial hypersurfaces ${h^{a}}_{b}$.
To first order, the once-contracted Bianchi identities yield two propagation equations, which are respectively the electric and magnetic propagation equations, $$\begin{aligned}
&\dot{E}_{\langle ab\rangle}=-\Theta E_{ab}+{\,\mbox{curl}\,}H_{ab}-{\textstyle\frac{\kappa}{2}}\left(\rho+p\right)\sigma_{ab}\ ,\label{Electric prop eq}\\
&\dot{H}_{\langle ab\rangle}=-\Theta H_{ab}-{\,\mbox{curl}\,}E_{ab}\ ,\label{Magnetic prop eq}\end{aligned}$$ and two constraint equations, $$\begin{aligned}
&D^{b}E_{ab}={\textstyle\frac{\kappa}{3}}D_{a}\rho\ ,\label{Constr eq 3}\\
&D^{b}H_{ab}=-\kappa\left(\rho+p\right)\omega_{a}\ .\label{Constr eq 4}\end{aligned}$$
Twice-contracted Bianchi identities {#Twice-contracted Bianchi identities}
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The third set of equations is given by the twice-contracted Bianchi identities which represent the conservation of the effective stress energy momentum tensor. They are obtained by performing a second contraction ($b=c$) on the once-contracted Bianchi identities , $$\nabla^{b}\left(R_{ab}+{\textstyle\frac{1}{2}}g_{ab}\mathcal{R}\right)=\kappa\nabla^{b}T_{ab}=0\ .\label{Twice Bianchi}$$ To first order, the twice-contracted Bianchi identities yield one propagation equations, which is the energy conservation equation, $$\dot{\rho} = -\Theta\,\left(\rho+p\right)\ ,\label{En cons
eq}$$ and one constraint equation, which is the momentum conservation equation, $$D_{a}p=-a_{a}\left(\rho+p\right)\ .\label{Mom cons eq}$$
Scalar perturbations {#Scalar perturbations}
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Bardeen equation
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Physically, scalar perturbations represent spatial variations of zero-order scalar quantities. Thus, spatial Laplacians of zero-order scalars are natural candidates to describe such perturbations. Since we are interested in determining the time evolution of scalar perturbations in a comoving frame, we choose comoving spatial Laplacians of zero-order scalars as scalar perturbations variables. For the linearised dynamics of a perfect fluid about a homogeneous and isotropic background, there are only four zero-order scalars, the energy density $\rho$, the pressure $p$, the expansion rate $\Theta$ and the Gaussian curvature $\mathcal{K}$. For an adiabatic flow, the pressure is a function of the energy density . Thus, to describe the dynamics of adiabatic scalar perturbations, we define three scalar perturbation variables, which are respectively the comoving spatial Laplacian of the energy density $\Phi$, the comoving spatial Laplacian of the expansion rate $\Psi$ and the comoving spatial Laplacian of the Gaussian curvature $\chi$, $$\begin{aligned}
&\Phi\equiv R\kappa\Delta\rho\ ,\label{phi}\\
&\Psi\equiv R\Delta\Theta\ ,\label{psi}\\
&\chi\equiv{\textstyle\frac{1}{2}}R\Delta\mathcal{K}\ ,\label{chi 0}\end{aligned}$$ where the comoving spatial Laplacian $\Delta$ is related to the cosmic spatial Laplacian $D^2$ by, $$\Delta=R^2D^2=R^2D^{a}D_{a}\ ,\label{spatial com Laplacian}$$ and the factor of a half in is included to be consistent with the usual definition of the corresponding variable in the ‘background-based’ approach [@Mukhanov:1992]. Similar definitions for the comoving spatial Laplacian of the energy density and the comoving spatial Laplacian of the Gaussian curvature were used respectively by Woszczyna & Kulak [@Woszczyna:1989] and Bruni, Dunsby & Ellis [@Bruni:1992]. For a vanishing background curvature, it is worth mentioning that the curvature perturbation does not vanish, since it is a first-order variable. The dynamics of the scalar perturbations is obtained by taking the comoving spatial Laplacian of the scalar propagation equations , , and the spatial gradient of the constraint in order to express the constraint in terms of a comoving spacial Laplacian of a zero-order scalar. The dynamical equations of the scalar perturbations are respectively the comoving spatial Laplacian of the energy conservation equation , the comoving spatial Laplacian of the Raychaudhuri equation , the comoving spatial Laplacian of the Gauss-Codacci equation and the spatial gradient of the momentum conservation equation, which to first-order reduce to, $$\begin{aligned}
&\Delta\dot{\rho}+\Theta\Delta(\rho+p)+(\rho+p)\Delta\Theta=0\ ,\label{D rho}\\
&\Delta\dot{\Theta}+{\textstyle\frac{2}{3}}\Theta\Delta\Theta+{\textstyle\frac{\kappa}{2}}\Delta(\rho+3p)-\Delta\left(D^{b}a_{b}\right)=0\ ,\label{D Theta}\\
&\Delta\mathcal{K}+{\textstyle\frac{4}{3}}\Theta\Delta\Theta-2\kappa\Delta\rho=0\ ,\label{D K}\\
&\Delta p+R^2\left(\rho+p\right)D^{b}a_{b}=0\ .\label{D p}\end{aligned}$$ In order to reverse the order of the comoving spatial Laplacian and the time derivative of a scalar field $S$ in the propagation equations and , it is useful to introduce the first-order scalar identity, $$\Delta\dot{S}=\left(\Delta S\right)^{\cdot}-{\textstyle\frac{1}{3}}\Theta\Delta S-R^2\dot{S}D^{b}a_{b}\ .\label{f identity}$$ The dynamical equations , , and are recast in terms of the comoving spatial Laplacian , and using the the scalar identity , the gradient of the momentum conservation equation , the Friedmann equation , the Raychaudhuri equation and the energy conservation equation . To first-order, in a comoving frame, the dynamical equations reduce to, $$\begin{aligned}
&\dot{\Phi}+{\textstyle\frac{1}{3}}\Theta\Phi+\kappa\rho(1+w)\Psi=0\ ,\label{t phi}\\
&\dot{\Psi}+\left(\frac{1}{2}+\frac{3Kc_s^2}{R^2\kappa\rho(1+w)}\right)\Phi+\frac{c_s^2}{R^2\kappa\rho(1+w)}\Delta\Phi=0\ ,\label{t Psi}\\
&\chi+{\textstyle\frac{2}{3}}\Theta\Psi-\Phi=0\ .\label{t chi}\end{aligned}$$ In order to determine the dynamics of the scalar perturbations, it is useful to express the dynamical equations , and in terms of conformal time according to, $$\begin{aligned}
&\Phi^{\prime}+\mathcal{H}\Phi+R\kappa\rho(1+w)\Psi=0\ ,\label{conf phi}\\
&\Psi^{\prime}+\left(\frac{R}{2}+\frac{3Kc_s^2}{R\kappa\rho(1+w)}\right)\Phi+\frac{c_s^2}{R\kappa\rho(1+w)}\Delta\Phi=0\ ,\label{conf Psi}\\
&\chi+\frac{2\mathcal{H}}{R}\Psi-\Phi=0\ .\label{conf chi}\end{aligned}$$
In order to determine the conformal time evolution of the density perturbation variable $\Phi$, the dynamics has to be recast in terms of a second-order differential equation for $\Phi$. By differentiating the $\Phi$-propagation equation with respect to conformal time, using the $\Psi$-propagation equation to substitute $\Phi$ for $\Psi$ and the zero-order relations and , a second-order differential equation for $\Phi$ is obtained according to, $$\begin{split}
\Phi^{\prime\prime}&+3\mathcal{H}\left(1+c_s^2\right)\Phi^{\prime}\\
&+\left[(1+3c_s^2)(\mathcal{H}^2-K)+2\mathcal{H}^{\prime}\right]\Phi-c_s^2\Delta\Phi=0\ ,\label{Bardeen eq}
\end{split}$$ which is the Bardeen equation, denoted \[4.9\] in [@Bardeen:1980] and identified by Woszczyna & Kulak [@Woszczyna:1989] in the $1+3$ covariant formalism.
We now formally relate the energy density perturbation variable $\Phi$ used in the $1+3$ covariant approach to the Bardeen variables $\Phi_A$ and $\Phi_H$ used in the ‘background-based’ approach. By taking the comoving spatial Laplacian of the divergence of the electric part of the Weyl tensor $E_{ab}$ , the energy density perturbation $\Phi$ is found to be related to $E_{ab}$ as, $$\Phi=3R^3D^aD^bE_{ab}\ .\label{Bardeen Weyl}$$ The expression of the electric part of the Weyl tensor $E_{ab}$ in terms of the Bardeen variables $\Phi_A$ and $\Phi_H$ was first established by Bruni, Dunsby & Ellis [@Bruni:1992] in equations \[113-114\]. For a perfect fluid (i.e. in absence of anisotropic stress), the Bardeen variables have the same norm but opposite signs (i.e. $\Phi_H=-\Phi_A$). Substituting the relations \[113-114\] derived by Bruni, Dunsby & Ellis [@Bruni:1992] into , the energy density perturbation $\Phi$ is found to be the fourth-order derivative of the Bardeen variable $\Phi_A$ according to, $$\Phi=\Box\Phi_A\ ,\label{Bardeen Weyl 2}$$ where, $$\Box\equiv 3D^aD^bD_{\langle a}D_{b\rangle}\ .$$ Note that by taking spatial derivatives of the electric part of the Weyl tensor, the vector and tensor perturbation terms contained in equations \[113-114\] vanish. To first-order, the conformal time derivative and the comoving spatial Laplacian commute with the spatial differential operator $\Box$, $$\begin{aligned}
&\Phi^{\prime}=\Box\Phi_A^{\prime}\ ,\\
&\Delta\Phi=\Box\Delta\Phi_A\ .\end{aligned}$$ Thus, to first-order, the dynamics of the energy density perturbation $\Phi$ is identical to the dynamics of the Bardeen variable $\Phi_A$ since the Bardeen equation is entirely determined by conformal time derivatives and spatial Laplacians of $\Phi$. Therefore, in the $1+3$ covariant formalism, $\Phi$ is the analog of $\Phi_A$.
Mukhanov-Sasaki equation
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The Bardeen equation describes the time evolution of the density perturbation variable $\Phi$. Similarly, the Mukhanov-Sasaki equation describes the time evolution of the curvature perturbation variable, which in the spatially-curved case we will denote by $\zeta$, and which reduces to $\chi$ defined in in the spatially-flat case ($K=0$), as we will now show.
To make contact with the standard definition of the curvature perturbation in the ‘background-based’ approach, it is useful to express $\chi$ in terms of $\Phi$ only. By substituting into , using the Friedmann equation , the expression for $\chi$ to first-order becomes, $$\chi=\frac{2}{3(1+w)}\left(1+\frac{K}{\mathcal{H}^2}\right)^{-1}\left(\Phi+\frac{\Phi^{\prime}}{\mathcal{H}}\right)+\Phi\ ,\label{curvature guess}$$ which, for a spatially-flat background space-time ($K=0$), reduces to, $$\chi=\frac{2}{3(1+w)}\left(\Phi+\frac{\Phi^{\prime}}{\mathcal{H}}\right)+\Phi\ ,\label{curvature perturb}$$ which is the standard definition of the curvature perturbation (see Mukhanov [@Mukhanov:2005], Durrer [@Durrer:2008]). In the spatially-flat case, the conformal time derivative of the curvature perturbation $\chi$ is given by, $$\chi^{\prime}=\frac{2c_s^2}{3\mathcal{H}(1+w)}\Delta\Phi\ ,\label{curvature pert deriv flat}$$ which shows that the curvature perturbation $\chi$ is a conserved quantity on large scales. To show this explicitly a Fourier transform has to be performed (see Durrer [@Durrer:2008]). However, we note that on large scales the comoving spatial Laplacian of the analog of the Bardeen variable is negligible compared to the comoving scale $\mathcal{H}$, thus satisfying $\Delta\Phi\ll\mathcal{H}$. Nonetheless, the curvature perturbation is not conserved on super-Hubble scales for a spatially-curved background space-time [@Bruni:1992].
In the spatially-curved case, Bruni, Dunsby & Ellis [@Bruni:1992] mention that there is a generalised curvature perturbation $\tilde{C}$, which is conserved on large scales and defined up to a constant amplitude. Thus, we define the generalised curvature variable $\zeta$ as $$\zeta\equiv\frac{R}{2}\left(\Delta\mathcal{K}-\frac{4K}{R^2(1+w)}\frac{\Delta\rho}{\rho}\right)\ ,\label{zeta}$$ where $\zeta\equiv{\textstyle\frac{1}{2}}\tilde{C}$ in order for $\zeta$ to reduce to $\chi$ in the spatially-flat case ($K=0$). The generalised curvature perturbation variable $\zeta$ in the spatially-curved case is related to the curvature perturbation $\chi$ in the spatially-flat case by, $$\zeta=\chi-\frac{2K}{3\mathcal{H}^2\left(1+w\right)}\left(1+\frac{K}{\mathcal{H}^2}\right)^{-1}\Phi\ .\label{zeta chi}$$
We now briefly show that $\zeta$ is conserved on large scales. By substituting into , the generalised curvature perturbation $\zeta$ is recast in terms of the analog of the Bardeen variable $\Phi$ only according to, $$\zeta=\frac{2}{3(1+w)}\left(1+\frac{K}{\mathcal{H}^2}\right)^{-1}\left[\left(1-\frac{K}{\mathcal{H}^2}\right)\Phi+\frac{\Phi^{\prime}}{\mathcal{H}}\right]+\Phi\ .\label{curvature pert gen}$$ For a spatially-curved case, by differentiating and substituting the Bardeen equation , the conformal time derivative of the generalised curvature perturbation $\zeta^{\prime}$ is found to be, $$\zeta^{\prime}=\frac{2c_s^2}{3\mathcal{H}(1+w)}\left(1+\frac{K}{\mathcal{H}^{2}}\right)^{-1}\Delta\Phi\ ,\label{curvature pert deriv curved}$$ which means that the generalised curvature perturbation $\zeta$ is a conserved quantity on super-Hubble scales. To show this explicitly a harmonic decomposition has to be performed. However, similarly to the spatially-flat case, we note that on large scale the comoving spacial Laplacian of the analog of the Bardeen variable is negligible compared to the comoving scale $\mathcal{H}$, thus satisfying $\Delta\Phi\ll\mathcal{H}$.
In order to determine the conformal time evolution of the curvature perturbation variable $\zeta$, the dynamics has to be recast in terms of a second-order differential equation for $\zeta$. By differentiating the $\zeta$-propagation equation with respect to time and using the comoving Laplacian of the generalised curvature perturbation $\zeta$ to substitute $\zeta$ for $\Phi$, a second order differential equation for $\zeta$ is obtained according to, $$\zeta^{\prime\prime}+2\left(\frac{\mathcal{H}}{2}\left(1-3c_s^2\right)-\frac{\mathcal{H}^{\prime}}{\mathcal{H}}\right)\zeta^{\prime}
-c_s^2\Delta\zeta=0\ .\label{zeta second order}$$
It is convenient to introduce a new variable $z$, $$z\equiv\frac{R^2}{c_s\mathcal{H}}\left(\frac{\kappa}{3}\rho(1+w)\right)^{1/2}\ ,\label{z}$$ which allows the $\zeta$-propagation equation to be recast in a more convenient form as, $$\zeta^{\prime\prime}+2\frac{z^{\prime}}{z}\zeta^{\prime}
-c_s^2\Delta\zeta=0\ .\label{zeta second order 2}$$ Note that using the dynamical equation , $z$ is rewritten as, $$z=\frac{R^2}{c_s\mathcal{H}}\left(\mathcal{H}^2-\mathcal{H}^{\prime}+K\right)^{1/2}\ ,\label{z bis}$$ which corresponds to the variable defined by Mukhanov et al. [@Mukhanov:1992] in equation \[$10.43$b\].
It is also useful to define another variable, $$v\equiv z\zeta\ ,\label{v}$$ which is the analog of the variable $v$ defined by Mukhanov et al. [@Mukhanov:1992] in equation \[$10.61$\]. Using $v$, the dynamics of the second-order dynamical equation is explicitly recast in terms of a wave equation given by, $$v^{\prime\prime}-\left(c_s^2\Delta+\frac{z^{\prime\prime}}{z}\right)v=0\ .\label{v second order}$$ This wave equation is the Mukhanov-Sasaki equation in a spatially-curved FLRW background space-time denoted \[$11.7$\] in [@Mukhanov:1992].
Mukhanov-Sasaki variable
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The scalar variable $v=z\zeta$ is the analog of the Mukhanov-Sasaki variable associated to the scalar perturbations [@Mukhanov:1992; @Sasaki:1984] (note that in a spatially-flat case, the analog of the Mukhanov-Sasaki variable reduces to $v=z\chi$). In the general case, the homogeneity and isotropy of the spatial hypersurfaces enable us to perform a harmonic decomposition of the scalar perturbation variable $v$, where $v$ is decomposed into components that transform irreducibly under translations and rotations, and evolve independently, as explained in [@Durrer:2008]. The harmonic analysis of the scalar perturbation $v$ consists of a decomposition into eigenfunctions of the comoving spatial Laplacian of $v$ according to, $$\Delta v_k=-k^2v_k\ ,\label{harmonic scalar}$$ where $k$ is the eigenvalue of the associated harmonic mode and the $k$-index denotes the eigenvector of the mode. The comoving wavenumber $\nu$ of the scalar mode is defined as, $$\nu^2=k^2+K\ ,\label{comoving wavenumber scal}$$ where $K=\{-1,0,1\}$ is normalised. The comoving wavenumber $\nu$ takes continuous values when $K=\{-1,0\}$ and discrete ones for $K=1$. In particular, the regular normalisable eigenmodes have $\nu\geq0$ for flat and hyperbolic spatial hypersurfaces, and an integer satisfying $\nu\geq1$ for spheric hypersurfaces as explained in [@Lyth:1995; @Tsagas:2008].
To first order, the dynamics of the scalar perturbations can be rewritten as a serie of decoupled harmonic oscillators. Using the harmonic decomposition in terms of $k$, the Mukhanov-Sasaki wave equation in the $k$-mode is given by, $${v_k}^{\prime\prime}+\left(c_s^2k^2-\frac{z^{\prime\prime}}{z}\right)v_k=0\ .\label{scal pert eq}$$ Note that corresponds to a simple harmonic oscillator in the $k$-mode, $${v_k}^{\prime\prime}+\omega_k^2v_k=0\ ,\label{scal pert eq II}$$ where the conformal time dependent frequency $\omega_k(\hat{\tau})$ is given by, $$\omega_k=\left(c_s^2k^2-\frac{z^{\prime\prime}}{z}\right)^{1/2}\ .\label{time frequency scal}$$
Comparison with scalar perturbations of a scalar field in the ‘background-based’ approach
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The dynamics of the early universe is believed to undergo an inflation phase described by a scalar field [@Guth:1981; @Linde:1982]. In a ‘background-based’ approach, the scalar field $\phi$ in the ‘real’ space-time is decomposed into a background component $\phi_0$ and a gauge-dependent perturbation $\delta\phi$ according to, $$\phi(t,\mathbf{x})={\phi}_0(t)+\delta\phi(t,\mathbf{x})\ .\label{scal field}$$ According to Weinberg [@Weinberg:2008], the energy-momentum tensor of an unperturbed scalar field takes the perfect fluid form with an energy density and pressure respectively given by, $$\begin{split}
&\rho_0={\textstyle\frac{1}{2}}\dot{\phi}_0^2+V(\phi_0)\ ,\label{rho scal}\\
&p_0={\textstyle\frac{1}{2}}\dot{\phi}_0^2-V(\phi_0)\ ,
\end{split}$$ where $V(\phi_0)$ is an arbitrary real potential. The initial conditions for inflation needed to perform the quantisation are set as the model emerges from the Big Bang, where the scalar field dynamics is dominated by the kinetic term and satisfies $\dot{\phi}_0\gg V(\phi_0)$. Therefore, at very early times, the scalar field behaves like stiff matter and the equation of state parameter $w$ is related to the speed of sound by $w=c_s^2=1$ [@Durrer:2008]. The dynamics of scalar perturbations of a scalar field in a $1+3$ covariant approach was first investigated by Bruni, Ellis & Dunsby [@Bruni:1992b]. Langlois& Vernizzi [@Langlois:2007] later generalised the $1+3$ covariant approach to multi-scalar fields.
To first order, the dynamical equation for the Bardeen variable is given by, $$\dot{\Phi}_A+H\Phi_A={\textstyle\frac{\kappa}{2}}\dot{\phi}_0\delta\phi\ ,\label{scal field eq 1}$$ which corresponds to equation \[$10.1.12$\] presented by Weinberg [@Weinberg:2008]. By comparing this dynamical relation with the corresponding result found in the $1+3$ covariant approach using the zero-order expression for the energy density and pressure of a scalar field and by using the correspondence relation , the $1+3$ covariant perturbation variable $\Psi$ is found to be related to the scalar field perturbation $\delta\phi$ by, $$\Psi=-\Box\left(\frac{\delta\phi}{2\dot{\phi}_0}\right)\ .\label{expansion pert 1}$$ The curvature perturbation variables in the flat case $\chi$ and the curved case $\zeta$ can be recast in terms of the Bardeen variable $\Phi_A$ and the scalar field perturbation variable $\delta\phi$ according to, $$\begin{split}
&\chi=\Box\left(\Phi_A+\mathcal{H}\frac{\delta\phi}{\phi_0^{\prime}}\right)\ ,\\
&\zeta=\Box\left(\left[1-\frac{K}{3\mathcal{H}^2}\left(1+\frac{K}{\mathcal{H}^2}\right)^{-1}\right]\Phi_A+\mathcal{H}\frac{\delta\phi}{\phi_0^{\prime}}\right)\ .\label{curv pert scal}
\end{split}$$ The quantisation variable $z$ can also be expressed in terms of the scalar field perturbation according to, $$z=\left(\frac{\kappa}{3}\right)^{1/2}\frac{R\phi_0^{\prime}}{\mathcal{H}}\ .\label{z scal field}$$ Finally, for a scalar field behaving like a stiff fluid (i.e. $c_s=1$), the Mukhanov-Sasaki wave equation in the $k$-mode reduces to, $${v_k}^{\prime\prime}+\left(k^2-\frac{z^{\prime\prime}}{z}\right)v_k=0\ .\label{scal pert eq scal field}$$ Thereby, with the identities presented above, we formally related the first-order scalar perturbation variables in the $1+3$ covariant formalism to the corresponding variables in the ‘background-based’ approach.
Vector perturbations {#Vector perturbations}
====================
Vector perturbations are described by spatially projected and divergence-free vectors [@Tsagas:2008]. The only dynamical variable which satisfies these constraints to first-order is the vorticity pseudo-vector $\omega^{a}$. The vanishing divergence of $\omega^{a}$ to first-order can be deduced from .
The dynamics of the vorticity covector $\omega_{a}$ is determined by the vorticity propagation equation . To obtain an explicit evolution equation in terms of the vorticity covector only, the term involving the acceleration has to be recast in terms of the vorticity. Using the momentum conservation equation and the kinetic non-commutation identity , we find to first-order, $${\,\mbox{curl}\,}a_{a}=2c_s^2\Theta\omega_{a}\ .\label{vorticity acc}$$ Thus, to first-order, the vorticity propagation equation can be recast as, $$\dot{\omega}_{\langle a\rangle}+{\textstyle\frac{2}{3}}\Theta\left(1-{\textstyle\frac{3}{2}}c_s^2\right)\omega_{a}=0\ ,\label{vorticity cosmic}$$ as shown by Hawking [@Hawking:1966]. It is convenient to express this dynamical equation in a comoving frame according to, $$\omega^{\prime}_{\langle a\rangle}+2\mathcal{H}\left(1-{\textstyle\frac{3}{2}}c_s^2\right)\omega_{a}=0\ .\label{vorticity conformal}$$ Using the vorticity contraction identity, $${w^2}^{\prime}=2\omega_{\langle a\rangle}^{\prime}\omega^{a}\ ,\label{vorticity contr}$$ a first-order propagation equation for the vorticity scalar $w$ is obtained, $$\omega^{\prime}=-2\mathcal{H}\left(1-{\textstyle\frac{3}{2}}c_s^2\right)\omega\ ,\label{vorticity scal eq}$$ and implies that the vorticity scalar scales as [@Hawking:1966; @Tsagas:2008], $$\omega\propto R^{-2+3c_s^2}\ .\label{vorticity scaling}$$
The scaling relation implies that during the expansion phase ($R^{\prime}>0$), the scalar amplitude of the vorticity $\omega$ decays if $c_s^2<{\textstyle\frac{2}{3}}$. The inflationary scenario is the simplest known generating mechanism for the initial density fluctuations. For an inflaton field in slow-roll ($w=-1$), the vorticity scalar scales as $\omega\propto R^{-5}$. Hence, the value of the vorticity scalar at the end of slow-roll inflation $\omega_f$ is related to the value of the vorticity scalar at the onset of the slow-roll inflation $\omega_i$ by, $$\frac{\omega_f}{\omega_i}=\mathrm{exp}\left(-5N\right)\ ,\label{vorticity inflation}$$ where $N$ is the number of e-fold during the slow-roll phase. Hence, if the vector perturbations were initially significant, they have decayed by a factor $\mathrm{exp}(5N)$ during slow-roll inflation and can safely be neglected in a subsequent quantitative perturbation analysis.
Tensor perturbations {#Tensor perturbations}
====================
Tensor perturbations are described by spatially projected, symmetric, trace-free and transverse second rank tensors, as explained in [@Tsagas:2008]. To find a suitable tensor to describe such perturbations, it is useful to split the magnetic part of the Weyl tensor $H_{ab}$ into a transverse part denoted $H_{ab}^{(T)}$ and a non-transverse part denoted $H_{ab}^{(V)}$ according to, $$H_{ab}=H_{ab}^{(T)}+H_{ab}^{(V)}\ .$$ where the $^{(T)}$ and $^{(V)}$ indices refer respectively to tensorial and vectorial degrees of freedom. By construction, $H_{ab}^{(T)}$ is divergence-free and satisfies the requirements for a tensor perturbation. For convenience, in this section, we will not use explicitly the $^{(T)}$ index to refer to the transverse part of the magnetic part of the Weyl tensor, since we are only considering tensorial degrees of freedom. Note that in the irrotational case, the magnetic part of the Weyl tensor is divergence-free to first-order $\--$ this can be deduced from the constraint . Thus, in that case, the analysis is restricted to the tensorial degrees of freedom only and $H_{ab}=H_{ab}^{(T)}$. It is also worth mentioning that the electric part of the Weyl tensor $E_{ab}$ cannot qualify as a tensor perturbation, since it is not divergence-free in presence of matter $\--$ this can be inferred from the constraint .
Grishchuk equation
------------------
In order to obtain a second-order differential equation in terms of $H_{ab}$, it is useful to introduce linearised identities found in [@Challinor:2000]. Using the Ricci identities -, the expression for the Riemann tensor to zero-order and the definition of a ${\,\mbox{curl}\,}$ , a symmetric and spatially projected tensor $T_{\langle ab\rangle}$ to first order has to satisfy the geometric linearised identity, $$^{(3)}({\,\mbox{curl}\,}T_{ab})^{\displaystyle\cdot}={\,\mbox{curl}\,}\dot{T}_{\langle ab\rangle}-{\textstyle\frac{1}{3}}\Theta{\,\mbox{curl}\,}T_{ab}\ ,\label{identities GW I}\\$$ and if $T_{ab}$ is transverse to first-order, $$D^bT_{ab}=0\ ,\label{T trans}$$ then it also satisfies the identity, $${\,\mbox{curl}\,}({\,\mbox{curl}\,}T_{ab})=-D^2T_{ab}+\frac{3K}{R^2}T_{ab}\ .\label{identities GW II}$$ By differentiating the magnetic propagation equation , substituting the electric propagation equation and using the linearised identities and , the dynamical equation for $H_{ab}$ to first order reduces to, $$\ddot{H}_{\langle ab\rangle}+\frac{7}{3}\Theta\dot{H}_{\langle ab\rangle}+2\left((1-w)\kappa\rho-\frac{3K}{R^2}\right)H_{ab}-D^2H_{ab}=0\ ,\label{H eq cosmic}$$ which has been established in [@Challinor:2000]. Using the zero-order dynamical relations and , it is convenient to recast the dynamics of $H_{ab}$ in terms of conformal time $\hat{\tau}$ according to, $$H^{\prime\prime}_{\langle ab\rangle}+6\mathcal{H}H^{\prime}_{\langle ab\rangle}-\left(\Delta-2K-8\mathcal{H}^2-4\mathcal{H}^{\prime}\right)H_{ab}=0\ .\label{H eq conformal}$$ In order to eliminate the second term on the LHS of and find a suitable wave equation for the gravitational waves, it is useful to introduce the rescaled magnetic part of the Weyl tensor $\tilde{H}_{ab}$ defined as $$\tilde{H}_{ab}\equiv R^3H_{ab}\ .\label{H rescaled}$$ The propagation equation of the tensor perturbations can be now elegantly reformulated to first-order in terms of $\tilde{H}_{ab}$ and yields, $$\tilde{H}_{\langle ab\rangle}^{\prime\prime}-\left(\Delta-2K+\frac{R^{\prime\prime}}{R}\right)\tilde{H}_{ab}=0\ ,\label{GW eq}$$ which is our version of the Grishchuk equation [@Grishchuk:1974] describing the dynamics of primordial gravitational waves in a spatially-curved case.
Tensor decomposition of $\tilde{H}_{ab}$
----------------------------------------
In order to determine the scalar canonical variables associated to the tensor perturbations, which are the scalar amplitudes of the tensor perturbations and will be referred to as canonical variables [@Grishchuk:1974] in the current work, a tensor decomposition of $\tilde{H}_{ab}$ has to be performed. From the properties of the magnetic part of the Weyl tensor $H_{ab}$, we deduce that the key tensor $\tilde{H}_{ab}$ is symmetric, trace-free and transverse to first-order. The transversality of $\tilde{H}_{ab}$ can be expressed as, $$D^{b}\tilde{H}_{ab}=k^{b}\tilde{H}_{ab}=0\ ,\label{transversality}$$ where $k^{b}$ is the spatial wavevector of the transverse gravitational waves satisfying $u^b k_b=0$. For convenience, we now define two vectors $e^{(1)a}$ and $e^{(2)a}$ that provide an orthonormal basis for the two-dimensional spatial hypersurface orthogonal to the propagation direction $k^{a}$ of the gravitational waves, and thus satisfy the following constraints, $$\begin{split}
&u^{a}e^{(1)}_{a}=u^{a}e^{(2)}_{a}=0\ ,\\
&k^{a}e^{(1)}_{a}=k^{a}e^{(2)}_{a}=0\ ,\\
&e^{(1)a}e^{(1)}_{a}=e^{(2)a}e^{(2)}_{a}=1\ ,\\
&e^{(1)a}e^{(2)}_{a}=0\ .\label{trans vec}
\end{split}$$ Note that these two vectors are not uniquely defined, which does not hinder our perturbation analysis since any orthonormal vector basis of the two dimensional hypersurface can be rotated to recover our vector basis $\{e^{(1)a}, e^{(2)a}\}$.
For an irrotational fluid, the Fermi-Walker transport of the basis vectors $e^{(1)}_{a}$ and $e^{(2)}_{a}$ vanishes [@Weinberg:1972], $$\begin{split}
&u^b\nabla_b e^{(1)}_a - u_a a^be^{(1)}_{b}=0\ ,\\
&u^b\nabla_b e^{(2)}_a - u_a a^be^{(2)}_{b}=0\ .\label{Fermi Transport}
\end{split}$$ Projecting the Fermi-Walker transported basis vectors on the spatial hypersurface yields, $$e^{(1)\prime}_{\langle a\rangle}=e^{(2)\prime}_{\langle a\rangle}=0\ .\label{time deriv basis}$$
Taking the comoving spatial Laplacian of the constraints , we deduce the following identities, $$\begin{split}
&e^{(1)a}\Delta e^{(1)}_{a}=e^{(2)a}\Delta e^{(2)}_{a}=0\ ,\\
&e^{(1)a}\Delta e^{(2)}_{a}=-e^{(2)a}\Delta e^{(1)}_{a}\ .\label{Laplacian vec id}
\end{split}$$
In order to decompose the tensor perturbation tensor $\tilde{H}_{ab}$ into two polarisation modes, we define two covariant, trace-free and linearly independent polarisation tensors, $$\begin{split}
&e^{+}_{ab}={\textstyle\frac{1}{2}}\left(e^{(1)}_{a}e^{(1)}_{b}-e^{(2)}_{a}e^{(2)}_{b}\right)\ ,\\
&e^{\times}_{ab}={\textstyle\frac{1}{2}}\left(e^{(1)}_{a}e^{(2)}_{b}+e^{(2)}_{a}e^{(1)}_{b}\right)\ ,\label{poltens}
\end{split}$$ which satisfy the orthonormality conditions, $$\begin{split}
&e^{+}_{ab}e^{+ab}=e^{\times}_{ab}e^{\times ab}=1\ ,\\
&e^{+}_{ab}e^{\times ab}=0\ ,\label{ortho cond}
\end{split}$$ and thus form an orthonormal tensor basis for the polarisation modes [@Durrer:2008]. From the identities , we deduce that the spatially projected time derivatives of the polarisation tensors vanish, $$\begin{aligned}
e^{+\prime}_{\langle ab\rangle}=\,e^{\times\prime}_{\langle ab\rangle}=0\ .\label{polar tens const}\end{aligned}$$ Taking the comoving Laplacian of the orthonormality conditions and using the identities , we deduce the following constraints, $$\begin{split}
&e^{+}_{ab}\Delta e^{+ab}=e^{\times}_{ab}\Delta e^{\times ab}=0\ ,\\
&e^{+}_{ab}\Delta e^{\times ab}=e^{\times}_{ab}\Delta e^{+ab}=0\ .\label{ortho constr}
\end{split}$$
The tensor perturbation variable $\tilde{H}_{ab}$ is spatially projected, transverse, traceless, and therefore can be decomposed into two polarisation modes $\{+, \times\}$ according to, $$\tilde{H}_{ab}=h^{+}e^{+}_{ab}+h^{\times}e^{\times}_{ab}\ ,\label{decom pol}$$ where $$\begin{split}
h^{+}=\tilde{H}^{ab}e^{+}_{ab}\ ,\\
h^{\times}=\tilde{H}^{ab}e^{\times}_{ab}\ ,
\end{split}$$ are the scalar amplitudes of the tensor perturbations. The corresponding decomposition in the ‘background-based’ approach is mentioned by Durrer in [@Durrer:2008]. The linear independence of the polarisation tensors in the first-order perturbation analysis allows us to study separately the dynamics of the two decoupled polarisation modes. To keep the notation compact, we introduce the polarisation mode superscript $\lambda=\{+, \times\}$.
Canonical variables
-------------------
Contracting the Grishchuk equation with the polarisation basis tensors $e^{\lambda}_{ab}$ and using the identities and , we obtain the Grishchuk equation for the scalar amplitude of the tensor perturbations associated to the polarisation mode $\lambda$, $$h^{\lambda\prime\prime}-\left(\Delta-2K+\frac{R^{\prime\prime}}{R}\right)h^{\lambda}=0\ .\label{tens pert eq full}$$ where $h^{\lambda}$ are the canonical variables, which describe the scalar amplitude of the tensor perturbations in the polarisation modes $\lambda$. There are two canonical variables $\{ h^{+}, h^{\times}\}$ associated to the polarisation modes of the tensor perturbations.
In a similar manner than for the scalar perturbations, we perform a harmonic decomposition of the tensors perturbations. The harmonics analysis of the tensor perturbations consists in a decomposition into eigenfunctions of the comoving spatial Laplacian of the scalar amplitude of the tensor perturbation variable $h^{\lambda}$ according to, $$\Delta h^{\lambda}_k=-k^2 h^{\lambda}_k\ ,\label{harmonic tensor}$$ where $k$ is the eigenvalue of the associated harmonic mode and the suffix $k$ denotes the eigenvector of the mode.
Finally, to first-order, the dynamics of the tensor perturbations for each polarisation mode $\lambda$ can be rewritten as a serie of decoupled harmonic oscillators. Using the harmonic decomposition of the scalar amplitudes in terms of $k$ , the evolution equation for the scalar amplitude of the tensor perturbations in the $k$-mode satisfies, $${h^{\lambda}_{k}}^{\prime\prime}+\left(k^2+2K-\frac{R^{\prime\prime}}{R}\right)h^{\lambda}_{k}=0\ ,\label{tens pert eq}$$ for each polarisation $\lambda=\{+, \times\}$. The tensor perturbation amplitudes variables $h^{\lambda}$ are identified as the two scalar canonical variables associated to the tensor perturbations [@Grishchuk:1974]. Note that corresponds to a simple harmonic oscillator with a polarisation $\lambda$ in the $k$-mode, $${h^{\lambda}_{k}}^{\prime\prime}+{\omega^{\lambda}_k}^{\,2}h^{\lambda}_{k}=0\ ,\label{tens pert eq II}$$ where the conformal time dependent frequency $\omega^{\lambda}_k(\hat{\tau})$ is given by, $$\omega^{\lambda}_k=\left(k^2+2K-\frac{R^{\prime\prime}}{R}\right)^{1/2}\ .\label{time frequency tens}$$
Conclusion
==========
We performed a perturbation analysis of an adiabatic perfect fluid to first order using the $1+3$ covariant and gauge-invariant formalism and identified the analog of the Mukhanov-Sasaki variable and the canonical variables needed to quantise the scalar and tensor perturbations respectively about a spatially-curved FLRW background space-time. We also determined the dynamics of the vector perturbations, which does not lead to a second order wave-equation, unlike the scalar and tensor perturbation, but to a first order equation involving the vorticity. In an expanding universe, the vorticity decays provided the sound speed squared satisfies $c_s^2<{\textstyle\frac{2}{3}}$ and the vector perturbations can therefore be neglected in the analysis.
In cosmological perturbation theory, there are in general six degrees of freedom: two scalars, two vectors and two tensors as mentioned by Bertschinger [@Schaeffer:1996]. However, in the perfect fluid case, the two Bardeen potential are related by $\Phi_A=-\Phi_H$, and thus there is only one remaining scalar degree of freedom. To sum up, to first-order, there is one degree of freedom associated with the scalar analog of the Mukhanov-Sasaki variable $v$, two degrees of freedom associated with the divergence-free vorticity covector $\omega_{a}$ and two degrees of freedom related to the two canonical variables $h^{+}$ and $h^{\times}$ representing the scalar amplitudes of the tensor perturbations. Hence, to first-order, the dynamics of the adiabatic perturbations of a perfect fluid are described by five parameters, as expected.
S. D. B. thanks the Isaac Newton Studentship and the Sunburst Fund for their support. The authors also thank Marco Bruni, Stephen Gull, Anthony Challinor and Pierre Dechant for useful discussions.
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